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abstract: 'Arcminute-resolution radio continuum images at 408 and 1420 MHz from the Canadian Galactic Plane Survey (CGPS) have been used to reexamine radio sources listed in the @kal80 catalogue. This catalogue is of particular interest to Galactic studies as it lists both extended and compact radio sources found in the second Galactic quadrant. We have determined the nature (extended vs. compact, Galactic vs. extragalactic) of all of these bright radio sources. A number of large regions with no optical counterparts are highlighted along with a sample of large radio galaxies. Many sources previously thought to be extended Galactic objects are shown to be point sources. A sample of point sources with flat or rising spectra between 408 and 1420 MHz has been compiled, and within this sample likely Gigahertz Peaked Spectrum sources have been identified.'
author:
- |
C. R. Kerton,$^1$[^1]\
$^1$Iowa State University, Department of Physics and Astronomy, Ames, IA, 50011, USA
title: 'A sharper view of the outer Galaxy at 1420 and 408 MHz from the Canadian Galactic Plane Survey I: Revisiting the KR catalogue and new Gigahertz Peaked Spectrum sources'
---
surveys – catalogues – Galaxy: disc – radio continuum: general.
Introduction {#sec:intro}
============
Radio continuum observations of the second quadrant of our Galaxy ($90\degr < l < 180\degr$) provide an unmatched opportunity for studying the structure and content of a spiral arm in detail. The Perseus Arm dominates Galactic structure in this quadrant and is viewed almost perpendicular to its long axis over the entire longitude range. The more distant Outer Arm is also well placed for study in this quadrant and in both cases confusion from Local Arm sources is minimal (cf. the view of the Galaxy around $l\sim
75\degr$ looking along the Local Arm). The previously best view of this region in the radio continuum (at 1420 MHz) was a series of surveys done by the Effelsberg 100-m telescope at 9-arcmin resolution. The surveys were summarized in the @kal80 and @rrf97 catalogues (KR and RRF respectively). RRF provides a listing of small diameter sources ($<16$ arcmin in extent) with an 80 mJy flux density limit (for point sources). The KR catalogue has a higher flux density limit (0.3 Jy) but is of particular interest to Galactic studies as it lists both compact and extended objects.
The new Canadian Galactic Plane Survey (CGPS; @tay03) data provide an unprecedented view of the continuum radiation at both 1420 and 408 MHz from the outer Galaxy. The data have arcminute-scale resolution and have full spatial frequency sensitivity crucial for the detection of extended structures.
In this paper we first revisit the sources found in the KR catalogue. @fic86 obtained high resolution VLA images of the sources originally classified as point sources in KR. For these sources we are primarily interested in observing the few of them that had poor VLA observations and to look for inverted spectrum sources. @tru90 obtained one-dimensional scans at 7.6 and 31.3 cm of most of the extended KR sources using the RATAN-600 telescope and found that many of the apparently extended KR objects were compact sources ($\leq$ 1-arcmin scale). @tru90 also suggested that a number of the KR objects were previously unknown compact Galactic supernova remnants (SNRs). We have reexamined all of these sources using the higher resolution and regular beamshape of the CGPS data and have been able to better determine the nature of all of the extended KR objects.
In the course of this study a new sample of extragalactic Gigahertz Peaked Spectrum (GPS) sources has been compiled. CGPS data have also revealed numerous new extended emission features in the second quadrant including both low-surface brightness extended emission and narrow filamentary features – both of which tend to be missed in the lower resolution surveys. The second paper in this series will present a complete catalogue of all extended emission features seen in the CGPS radio continuum data thus providing an updated version of the comprehensive catalogue compiled by @fic86.
In the next section we review the properties of the CGPS 1420 and 408 MHz data. In Sections \[sec:kr\] and \[sec:kr-p\] the CGPS view of the KR sources is presented. Flat and inverted spectrum sources are discussed in Section \[sec:fiss\] and conclusions are presented in Section \[sec:conc\].
Observations {#sec:observe}
============
The goal of the CGPS is to enhance the study of our Galaxy by obtaining arcminute-resolution images of all of the major components of the interstellar medium (ISM) in our Galaxy. Radio continuum observations made as part of this project were obtained using the seven-element interferometer at the Dominion Radio Astrophysical Observatory (DRAO) in Penticton, Canada [@lan00]. Details of the CGPS radio continuum observations, data reduction and data distribution are discussed at length in @tay03. CGPS observations currently cover $65\degr < l < 175\degr$ between $-3\fdg5
< b < +5\fdg5$ encompassing almost the entire second quadrant. The 1420 MHz observations have a nominal 1-arcmin resolution and both the 1420 and 408 MHz survey images were constructed with full spatial frequency coverage by combining the interferometer data with data from surveys using the Effelsberg single-dish and the Stockert single-dish telescopes. This provides sensitivity to extended structure which is very important for Galactic studies.
The simultaneous 408 MHz images, with nominal 3-arcmin resolution, provide invaluable data on the shape of the radio continuum spectrum as parameterized by the spectral index ($\alpha_{408}^{1420}$) between 408 and 1420 MHz (where flux density F$_\nu \propto \nu^\alpha$). In this paper we refer to inverted-spectrum sources as those with $\alpha_{408}^{1420} \geq +0.25$ and flat-spectrum sources as those with $|\alpha_{408}^{1420}| < 0.25$.
We also make use of the Mid-infrared Galaxy Atlas (MIGA; @ker00) and Infrared Galaxy Atlas (IGA; @cao97) arcminute resolution infrared images which make up part of the larger CGPS data collection. These infrared images are very useful in the identification of Galactic regions in cases where there is no associated optical emission or available radio recombination line observations.
Flux density measurements were made using software contained in the DRAO Export Software Package. Point source flux densities were obtained using the “fluxfit” program which fits Gaussians to the image and makes use of the beam shape information available in the CGPS data. Extended sources were measured using the “imview” program which allows the user to interactively derive background levels to use in determining the flux densities.
Extended sources in the KR catalogue {#sec:kr}
====================================
The KR catalogue is based on 1420 MHz radio continuum observations made at 9-arcmin resolution with the Effelsburg 100-m telescope. @kal80 identified 236 radio sources with flux density $F_\nu > 0.3$ Jy including point sources and extended objects up to 30-arcmin in diameter. The catalogue covered $l=93\degr$ to $l=162\degr$ and $|b| < 4\degr$. Extended sources were subdivided into three categories depending upon their apparent size: EP (partially extended), E (extended) and VE (very extended). EP sources had a greatest extent of $<$ 9-arcmin, E sources had greatest extents between 11-arcmin and 20-arcmin, while VE sources had greatest extents between 20-arcmin and 30-arcmin.
![1420 MHz images of KR 1, an enormous region in the Perseus Arm. The top panel shows the full extent of the region including extensive filamentary structure seen between $l=92\degr$ and $l=92\fdg5$. KR 4 is located in the lower left corner of this panel around $l=93\fdg75$. The lower panel shows the central region and reveals an intricate combination of filaments and bubble-like structures.[]{data-label="fig:kr1"}](fig1.eps){width="84mm"}
{width="140mm"}
Very-extended (VE) sources {#sec:kr-ve}
--------------------------
Data on the twelve very-extended (VE) objects identified by @kal80 are listed in Table \[tab:ve\]. The first column gives the KR catalogue number. Letters following the KR number are used in cases where the object is actually a multiple source at arcminute resolution and are not part of the original classification (e.g., KR206A). Columns 2 through 5 give the flux density measurements and 1$\sigma$ error estimates at 1420 and 408 MHz from the CGPS data. The spectral index between 408 and 1420 MHz ($\alpha_{408}^{1420}$) is given in column 6 followed by the angular scale of the source as seen in the 1420 MHz images in column 7. The final column provides extra information about the source, such as an association with well-known optically visible region or SNR. For extended (at 1-arcmin resolution) sources the RRF catalogue number is given if applicable, and for all of the arcminute-scale point sources the NRAO VLA Sky Survey (NVSS; @con98) catalogue designation is provided.
------ ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
1 $3.26\times10^4$ $9.9\times10^2$ $3.35\times10^4$ $7.9\times10^2$ $-0.02$ 120 RRF 861; Region
3 $4.48\times10^3$ $1.0\times10^2$ $4.29\times10^3$ $2.5\times10^2$ $+0.03$ 18 RRF 863; Region
6 $7.92\times10^2$ $5.0\times10^1$ $5.01\times10^2$ $2.9\times10^1$ $+0.4$ 12 Region
20 $1.01\times10^3$ $5.7\times10^1$ $9.68\times10^2$ $1.1\times10^2$ $+0.03$ 15 Region
47 $2.99\times10^3$ $1.3\times10^1$ $2.08\times10^3$ $7.5\times10^1$ $+0.3$ 20 Sh 2-135
65 $1.10\times10^3$ $5.4\times10^1$ $9.68\times10^2$ $2.6\times10^2$ $+0.1$ 12 Sh 2-151
122 $6.43\times10^2$ $3.9\times10^1$ $4.36\times10^2$ $1.8\times10^1$ $+0.3$ 24 Region
166A $7.35\times10^3$ $2.2\times10^2$ $1.52\times10^4$ $4.6\times10^2$ $-0.6$ 1 NVSS J032719+552029
166B $1.23\times10^3$ $3.9\times10^1$ $2.77\times10^3$ $8.4\times10^1$ $-0.7$ 1 NVSS J032744+552226
175A $2.31\times10^3$ $7.0\times10^1$ $4.86\times10^3$ $1.5\times10^2$ $-0.6$ 1 NVSS J032952+533236
175B $7.45\times10^1$ $5.3\times10^0$ $1.51\times10^2$ $4.5\times10^0$ $-0.6$ 1 NVSS J033003+532944
180 $4.5\times10^ 2$ $1.4\times10^1$ $1.03\times10^3$ $3.1\times10^1$ $-0.7$ 1 NVSS J035927+571706
206A $3.37\times10^2$ $1.0\times10^1$ $4.96\times10^2$ $1.5\times10^1$ $-0.3$ 1 NVSS J043523+511422
206B $2.28\times10^2$ $6.8\times10^0$ $1.08\times10^2$ $3.2\times10^0$ $+0.6$ 1 NVSS J043621+511253
210A $1.84\times10^2$ $5.6\times10^0$ $5.22\times10^2$ $1.6\times10^1$ $-0.8$ 1 NVSS J043342+502428
210B $7.89\times10^1$ $2.7\times10^0$ $1.52\times10^2$ $6.3\times10^0$ $-0.5$ 1 NVSS J043357+502420
------ ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Seven of these sources are Galactic regions. These sources all have flat or inverted spectral indices and have extensive infrared emission visible in the *IRAS* images. Five of the regions have no optical counterparts. KR 1 is an enormous region stretching up to 2in size (see Figure \[fig:kr1\]). Radio recombination line emission has been detected from the region at V$_\mathrm{LSR} \sim -60$ km s$^{-1}$ [@fic86] yielding a kinematic distance (accounting for known streaming motions) of $\sim 4.5$ kpc, which implies that the region is also physically large ($\sim 200$ pc). Note that the RRF 861 source associated with the region refers only to a compact source making up only a small portion of this extensive region.
KR 3, often incorrectly classified as a SNR, is a Galactic region with a blister morphology which was extensively studied by @fos01. In addition to the flat radio spectrum and extensive associated infrared emission, radio recombination line emission from the region has also been detected [@fos01] solidifying its classification as an region. RRF 863 is centered on the bright radio emission associated with the region/molecular cloud interface while the entire region extends up to 03 in size.
KR 6, KR 20 and KR 122 are all classified as extended Galactic regions on the basis of their radio spectrum and associated infrared emission. None of these regions have known optical counterparts. Finally there are two radio sources associated with optically visible regions. KR 47 is radio emission, about 20-arcmin in extent, associated with the Sh 2-135 region, while KR 65 is diffuse radio emission, about 12-arcmin in extent, that is apparently associated with Sh 2-151.
The remaining five VE sources turn out to be point sources at arcmin-scale resolution. KR 180 appears to have been misclassified because of nearby diffuse radio emission associated with Sh 2-214. This object was also listed by @tru90 as being extended and being a possible SNR but the CGPS data show this is not the case. The other sources tend to be pairs of point sources with separations $<$9-arcmin. All but one of the point sources have a non-thermal spectral index and no detectable infrared emission, consistent with them being distant extragalactic objects. The exception is the compact massive star-forming region KR 206B (NVSS J043621+511254) which has an inverted spectrum ($\alpha =
+0.6$) and is associated with the bright infrared source IRAS 04324+5106 (RAFGL 5124).
Extended (E) sources {#sec:kr-e}
--------------------
@kal80 listed 48 of these sources. Table \[tab:e\] summarizes the CGPS view of this sample using the same notation as in Table \[tab:ve\]. Note that KR 86 was not observed in the CGPS and KR 35 is apparently a spurious source; no bright point source or region of diffuse emission was detected near its catalogued position.
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
4 $1.06\times10^3$ $3.7\times10^1$ $9.01\times10^2$ $1.7\times10^1$ $+0.1$ 12 RRF 865; Region
7 $2.69\times10^3$ $8.1\times10^1$ $2.45\times10^3$ $7.3\times10^2$ $+0.07$ 12 RRF 874; Region
19A $1.70\times10^2$ $7.3\times10^0$ $7.85\times10^2$ $3.6\times10^0$ $+0.1$ 5 RRF 903; Region
19B $1.47\times10^2$ $3.4\times10^0$ $1.29\times10^2$ $3.9\times10^0$ $+0.1$ 4 RRF 903; Region
21A $3.78\times10^2$ $1.1\times10^1$ $9.09\times10^2$ $2.7\times10^1$ $-0.7$ 1 NVSS J214343+523958
21B $3.64\times10^2$ $1.1\times10^1$ $1.05\times10^2$ $3.1\times10^1$ $-0.8$ 1 NVSS J214418+524501
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Table \[tab:e\] is presented in its entirety in the electronic edition of the journal.
One source, KR 196, is a very large ($\sim$25-arcmin diameter) region of bright radio emission associated with the optical region Sh 2-206. Seven other sources match the original classification (diameters between 11-arcmin and 20-arcmin). Three of these (KR 55, 91 and 98) are associated with radio emission from known optical regions, while three others (KR 4, 7, and 80) are regions with no optical counterparts. All of these objects have flat or inverted radio spectra and have associated infrared emission. Finally KR 101 is the well-studied SNR 3C 10 (Tycho’s SNR).
Five other regions (KR 19, 46, 48, 171 and 198) are smaller extended regions. KR 19 consists of two compact regions with the western (19A) region being associated with IRAS 21336+5333 and the eastern one (19B) being associated with two infrared sources IRAS 21340+5339 and IRAS 21340+5337 (see Figure \[fig:small\_ex\]). KR 46 is a compact region that shows hints of a blister morphology at 1-arcmin resolution. The radio spectrum is thermal and there is bright infrared emission associated with the region. @tru90 suggested that KR 48 and KR 171 were possible Galactic supernova remnants. However the CGPS data show the regions have inverted (KR 48) and flat (KR 171) radio spectra and are associated with bright diffuse infrared emission and IRAS point sources. Thus it is more likely that they are both Galactic regions. Finally KR 198 is associated with the optical region Sh 2-207.
KR 168 consists of two slightly elongated sources separated by $\sim
4.5$ arcmin. It is likely that these sources are extragalactic jets that are just barely resolved at 1-arcmin resolution. It is not clear that the two sources are physically associated. KR 188 also consists of two elongated sources with a similar point source plus faint jet structure with the point sources being separated by $\sim$4 arcmin. In this case the two objects do share common diffuse emission and the jet structures both point back to a common point suggesting that they are physically related. In Table \[tab:e\] the NVSS designations for the point-like portions of these objects are given.
The remaining “extended” KR sources are all actually point sources at 1-arcmin resolution. The majority of these sources are extragalactic as they have strongly non-thermal spectral indices, are unresolved at 1-arcmin resolution, and have no associated infrared emission. Three of the sources have flat spectra (KR 63, 189 and 192A) and two have inverted spectra (KR 53 and 60A). None of the flat spectrum sources have associated infrared emission and, given that they all have $\alpha_{408}^{1420} = -0.2$, they are also most likely extragalactic objects. KR 53 is associated with the optical region Sh 2-138. Finally, KR 60A is apparently a flat-spectrum radio galaxy. There is no associated infrared emission and, combining the CGPS flux density measurements with data obtained using SPECFIND [@vol05], we find a very flat spectral index of $+0.09\pm0.05$ over the range from 325 to 4800 MHz as illustrated in Figure \[fig:kr60a\].
![KR 60A, a flat-spectrum radio galaxy. CGPS data are at 408 and 1420 MHz. Other data points were obtained from @vol05.[]{data-label="fig:kr60a"}](fig3.eps){width="84mm"}
Partially-extended (EP) sources {#sec:kr-ep}
-------------------------------
The KR catalogue lists 41 of these sources. Table \[tab:ep\] summarizes the CGPS view of this sample using the same notation as in the previous tables. One source (KR 145) appears to have been a spurious object as there are no strong point sources or regions of extended emission near the catalogued coordinates.
Three of the sources have diameters greater than 11-arcmin. KR 200 is a large ($\sim 30$ arcmin) region of radio emission a portion of which is directly associated with the optical region Sh 2-209. KR 140 is a 12-arcmin scale region and KR 130 is the well-studied SNR 3C 58.
There are 13 sources which are not point sources but have diameters $<9$ arcmin. Nine of these objects are radio sources associated with known small-diameter optical regions and one is associated with the nearby galaxy Maffei 2.
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
13 $1.13\times10^3$ $2.6\times10^2$ $7.32\times10^2$ $9.4\times10^1$ $+0.3$ 6 RRF 888; BFS 6
15 $3.10\times10^2$ $9.3\times10^0$ $5.55\times10^2$ $1.7\times10^1$ $-0.5$ 1 NVSS J212305+550027
17 $6.45\times10^2$ $1.9\times10^1$ $5.46\times10^2$ $1.6\times10^1$ $+0.1$ 2 RRF 899; Sh 2-187
18 $6.52\times10^2$ $8.7\times10^0$ $4.30\times10^2$ $1.5\times10^1$ $+0.3$ 6 RRF 929; BFS 8
28A $2.56\times10^2$ $7.9\times10^0$ $7.27\times10^2$ $2.3\times10^1$ $-0.8$ 1 NVSS J213932+554030
28B $1.64\times10^2$ $5.3\times10^0$ $5.03\times10^2$ $1.7\times10^1$ $-0.9$ 1 NVSS J213934+554445
28C $5.32\times10^1$ $2.5\times10^0$ $\cdots$ $\cdots$ $\cdots$ 1 NVSS J213943+554340
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Table \[tab:ep\] is presented in its entirety in the electronic edition of the journal.
KR 45 (RRF 981) is a combination of extended and point source emission (see Figure \[fig:kr45\]). The extended radio emission is associated with the distant region IRAS 22181+5716. Molecular line (CO) observations towards this source detect emission at V$_\mathrm{LSR} = -63$ km s$^{-1}$ placing the region at a heliocentric distance of $\sim 7$ kpc. There is also a close double point source (denoted 45A and 45B) which is unresolved in the lower resolution 408 MHz images. These non-thermal point sources have no infrared counterparts and are apparently just background extragalactic sources. The remaining two extended objects (KR 144 and 172) both appear to be radio galaxies with a distinct core/lobe morphology (see Figure \[fig:rgals\]). The objects shown in Figure \[fig:rgals\] appear to be similar to the giant radio source WN 1626+5153 discovered in the Westerbork Northern Sky Survey [@rot96].
![KR 45 at 1420 MHz. The original single source is actually a Galactic region and a pair of bright extragalactic sources. Contours are at 7, 8, 9, 10, 20, and 30 K. The cross indicates the position of the infrared source IRAS 22181+5716.[]{data-label="fig:kr45"}](fig4.eps){width="84mm"}
Finally the remaining EP sources are all point sources at 1-arcmin resolution. All but one (KR 58) are likely extragalactic sources having a non-thermal spectral index and no detectable infrared emission. KR 58 has an inverted spectrum and is the planetary nebula NGC 7354 (IRAS 22384+6101).
{width="140mm"}
The nature of the point sources in the KR Catalogue {#sec:kr-p}
===================================================
All of the KR point sources (135 in total) except one (KR 195) were observed by the CGPS. Table \[tab:p\] summarizes the CGPS view of this sample using the same notation as in the previous tables.
The vast majority of these sources are point sources at 1-arcmin resolution. As first demonstrated by @fic86 most of these are extragalactic sources as indicated in this study by their strongly negative spectral index between 408 and 1420 MHz and lack of associated infrared emission.
There are a few small extended sources in this subsample. KR 77, 212 and 228 are all regions of extended thermal emission associated with the optical regions Sh 2-159, Sh 2-212 and Sh 2-217 respectively. Perhaps more interesting are the extended extragalactic sources KR 2 and KR 226. Both of these objects are clearly radio galaxies (see Figure \[fig:rgals\]) and were noted by @fic86 as being overresolved in his VLA images. KR 2 extends for about 10-arcmin in its longest direction. Optical spectroscopy of this source places it at a redshift of z=0.02 [@mas04]. KR 226 extends for about 5-arcmin and no studies of this object beyond cataloging have been made.
---- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
2 $2.87\times10^3$ $8.6\times10^1$ $6.16\times10^3$ $1.8\times10^2$ $-0.6$ 6 RRF 862
5 $4.37\times10^2$ $1.3\times10^1$ $1.39\times10^3$ $4.2\times10^1$ $-0.9$ 1 NVSS J213646+495318
8 $1.77\times10^3$ $5.3\times10^1$ $1.07\times10^3$ $3.3\times10^1$ $+0.4$ 1 NVSS J213701+510136
9 $3.22\times10^2$ $9.9\times10^0$ $7.56\times10^2$ $2.4\times10^1$ $-0.7$ 1 NVSS J213158+521415
10 $6.69\times10^2$ $2.0\times10^1$ $1.24\times10^3$ $3.7\times10^1$ $-0.5$ 1 NVSS J213340+521951
11 $7.72\times10^2$ $2.3\times10^1$ $1.49\times10^3$ $4.5\times10^1$ $-0.5$ 1 NVSS J213833+513550
---- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Table \[tab:p\] is presented in its entirety in the electronic edition of the journal.
There are 14 flat spectrum sources of which three (KR 23, 208, and 212) are associated with optical regions (Sh 2-148, Sh 2-211 and Sh 2-212 respectively). The remaining 11 sources have no associated infrared emission and thus inferred to be extragalactic sources. We examined the four flat spectrum sources with positive spectral indices in more detail. CGPS data were combined with data from @vol05 and @fic86 to obtain the spectra shown in Figure \[fig:ps-flat-pos\].
The radio spectrum of KR 24 is very flat over a wide frequency range, and certainly flatter than expected just from 408 and 1420 MHz data. A least absolute deviation fit to the data gives an overall spectral index of $\alpha = -0.06$. KR 178 is another very flat spectrum source with least absolute deviation spectral index of $\alpha = +0.04$ over the entire range of observations. KR 30 shows a slightly rising spectrum with $\alpha = +0.2$. The highest frequency point suggests that the spectrum may be flattening above 10 GHz. Finally the KR 234 radio spectrum has a shallow negative slope spectrum of $\alpha = -0.2$. The low frequency data points for KR 234 are in good agreement but there is increased scatter at the higher frequencies. The large scatter observed in the spectra of KR 24, 178 and 234 at particular wavelengths suggests that these sources are variable. This is the likely reason that the overall spectral index for these three sources is shallower than the spectral index determined by the simultaneous CGPS observations.
{width="140mm"}
There are also eight inverted spectrum point sources. Three of the sources (KR 61, 67 and 72) are associated with optical regions (Sh 2-146, Sh 2-152 and Sh 2-156 respectively) and KR 138 is the compact region IRAS 02044+6031. Molecular line emission at V$_\mathrm{LSR} \sim -55$ km s$^{-1}$ has been detected towards this *IRAS* source placing it at a kinematic distance of $\sim 5.5$ kpc. Unfortunately the velocity field model of @bra93 is quite uncertain around this longitude ($l\sim
130\degr$) for this velocity making corrections for streaming motions problematic. Given its small angular size it it quite possible that KR 138 lies beyond the Perseus Arm. The remaining four sources have no infrared counterpart and are most likely extragalactic.
Such extragalactic radio sources with inverted spectra are interesting because of the possibility that they are Gigahertz Peaked Spectrum (GPS) sources. Astronomically these objects are of interest because they may represent an early stage in the evolution of radio galaxies [@ort06; @ode98]. Observationally these objects are defined as having a convex radio spectrum that peaks between 500 MHz and 10 GHz. The shape of the spectrum is most likely due to synchrotron self-absorption [@ort06]. Below the peak frequency the average spectral index is $0.51\pm0.03$ and above the peak it is $-0.73\pm0.06$ [@dev97].
For each of the extragalactic inverted spectrum sources we combined flux density measurements at other wavelengths from @vol05 and @fic86 with the CGPS measurements. The spectra are shown in Figure \[fig:kr-invert\]. Following @mar99 we fit a second order polynomial of the form $ \log F_\nu = a + b \log \nu - c(\log \nu
)^2$. This curve is not physically motivated, rather it simply allows us to easily identify sources with sufficiently high spectral curvature. Sources with $c > 1.0$ have sufficient spectral curvature to be considered GPS sources.
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{width="140mm"}
KR 8 does appear to have a convex spectra but the data above the peak has a large amount of scatter and the curvature is not as high as one would expect for a true GPS source ($c = 0.6$). KR 125 has a very low curvature spectrum ($c= 0.25$) with the curvature arising almost entirely from the highest frequency data point. Except for this point the spectrum is consistent with a rising spectrum with $\alpha = +0.3$ from 300 to 4800 MHz. KR 135 has a very steep low frequency spectral index and the cuvature of the spectrum is quite high ($c=0.96$). Unfortunately the data above the apparent peak in the spectrum are quite scattered and its status as a GPS source is very uncertain. Finally, KR 182 shows a rising spectrum with $\alpha = +0.3$ with no signs of any spectral curvature. There is a large amount of scatter in the spectrum at both low and high frequency.
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Flat and Inverted-spectrum point sources {#sec:fiss}
========================================
The presence of extragalactic sources with both flat and inverted spectra within the KR sample led us to examine all of the CGPS second quadrant data for similar sources. To rapidly search for other point sources with flat or inverted spectra the 1420 MHz images were first convolved to the 408 MHz resolution. The brightness of the convolved 1420 MHz images were then scaled to the expected brightness at 408 MHz assuming an optically thin thermal spectrum between 408 and 1420 MHz. The true 408 images were then subtracted from the scaled images resulting in a series of difference images. Point sources with steep negative spectral indices show up as distinct negative-valued sources on the difference images thus allowing the rapid identification of flat and inverted-spectrum sources. After candidate sources were identified in this manner, flux densities were measured at 1420 and 408 MHz. Sources in the final sample had both measurable 408 flux densities (complete to $\sim 50$ mJy at 408 MHz) and no visible infrared emission in the ancillary CGPS infrared images.
Table \[tab:if\] shows the resulting sample of flat-spectrum and inverted-spectrum sources. Column 1 gives the NVSS catalogue designation, columns 2-5 give the flux density and error estimates at 1420 and 408 MHz, and column 6 gives the spectral index.
---------------- ------------------ ----------------- ------------------ ----------------- -----------------------
NVSS F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$
(mJy) (mJy) (mJy) (mJy)
J054044+391612 $1.53\times10^2$ $4.7\times10^0$ $4.74\times10^1$ $4.2\times10^0$ $+0.9$
J054052+372847 $1.74\times10^2$ $5.3\times10^0$ $1.30\times10^2$ $8.2\times10^0$ $+0.2$
J050905+352817 $3.85\times10^2$ $1.2\times10^1$ $1.49\times10^2$ $3.8\times10^1$ $+0.8$
J050920+385046 $9.40\times10^1$ $2.9\times10^0$ $8.22\times10^1$ $7.5\times10^0$ $+0.1$
J051346+400618 $3.55\times10^2$ $1.1\times10^1$ $3.35\times10^2$ $1.1\times10^1$ $+0.0$
J050948+395154 $7.83\times10^1$ $2.4\times10^0$ $3.52\times10^1$ $1.5\times10^0$ $+0.6$
---------------- ------------------ ----------------- ------------------ ----------------- -----------------------
Table \[tab:if\] is presented in its entirety in the electronic edition of the journal.
In order to identify potential GPS sources we examined in more detail 43 of the sources which had $\alpha_{408}^{1420} \geq +0.4$. As before, radio data from the compilation of @vol05 were used to construct spectra over as wide a range of frequencies as possible. Of these objects eight of them were found to have a curvature of $c >
+1$. The radio spectra of these objects are shown in Figure \[fig:curve\].
We also found four other objects in the sample that had rising spectra ($\alpha \geq +0.3$ over the entire spectral range) combined with little scatter (see Figure \[fig:rise\]). These sources may be examples of, relatively rare, GPS sources with a peak above 5 GHz similar to the point source 71P 52 (NVSS 213551+471022) examined by @hig01.
Conclusions {#sec:conc}
===========
The KR catalogue is very useful for Galactic studies as it contains information on both compact and extended radio sources in the outer Galaxy. Unfortunately the relatively low resolution of the survey means that it overestimates the number of extended sources in the outer Galaxy. This paper updates this catalogue based primarily on new higher resolution images of the outer Galaxy at 1420 MHz obtained as part of the CGPS. We have clearly identified sources that were misclassified as extended objects and have determined which sources remain unresolved at 1-arcmin scale resolution. The simultaneous 408 MHz CGPS observations, combined with ancillary infrared data, also have allowed the nature of all of the observed KR sources to be determined with some confidence. Attention has been drawn particularly to a large number of unstudied Perseus Arm regions (including the extremely large KR 1 complex), objects previously considered to be SNR candidates (e.g., KR 171), and a sample of large radio galaxies (e.g., KR 144).
In addition, through the examination of the 408 and 1420 MHz CGPS images, this study has identified a sample of flat-spectrum and inverted-spectrum extragalactic radio sources based upon their 408 and 1420 MHz flux densities. A subset of these objects was examined in more detail and a new sample of GPS sources has been compiled.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank ISU undergraduate students Jason Murphy and Jon Patterson for their assistance on this project. The Dominion Radio Astrophysical Observatory is operated by the National Research Council of Canada. The Canadian Galactic Plane Survey is supported by a grant from Natural Science and Engineering Research Council of Canada.
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[^1]: E-mail: kerton@iastate.edu
|
---
abstract: 'CdS nanotubes with wall thickness comparable to excitonic diameter of the bulk material are synthesized by a chemical route. A change in experimental conditions result in formation of nanowires, and well-separated nanoparticles. The diameter and wall thickness of nanotubes measured to be 14.4 $\pm$ 6.1 and 4.7 $\pm$ 2.2 nm, respectively. A large number of CdS nanocrystallites having wurzite structure constitute these nanotubes. These nanotubes show high energy shifting of optical absorption and photoluminescence peak positions, compared to its bulk value, due to quantum confinement effect. It is proposed that nucleation and growth of bubbles and particles in the chemical reaction, and their kinetics and interactions are responsible for the formation of nanotubes.'
author:
- 'A. K. Mahapatra[^1]'
title: 'Synthesis of quantum-confined CdS nanotubes'
---
Introduction {#introduction .unnumbered}
============
The discovery of carbon nanotubes (Iijima 1991) has generated considerable research interest to synthesize such type of tubular nanostructures of other materials and study its properties. Carbon nanotubes are conceptualized as the wrapping of graphite layers into a seamless cylinder. Synthesis of single crystalline nanotubes of other similar type of layered materials like BN (Chopra et al. 1995), $MoS_2$ (Feldman et al. 1995) and $WS_2$(Tenne et al. 1992) are reported. In these materials, there exists a strong force within the layer plane, but a weak van der waals force between the inter-layer planes. It helps such materials to self assemble in the form of nanotubes. However, comparatively more isotropic materials like CdS (Zhan et al. 2000), CdSe (Duan and Lieber 2000), GaAs (Duan et al. 2000), and Si (Yu et al. 1998) tend to form nanowires instead of nanotubes. Therefore, nanotubes of these isotropic materials are generally synthesized by using nanowires as templates. CdS nanotubes are also synthesized by using Sn nanowires as the template (Hu et al. 2005). However, those synthesized CdS nanotubes have a wall thickness much larger than the excitonic diameter of the bulk CdS (6 nm). Hence, no quantum confinement effects could be observed. Aspect ratio and wall thickness of nanotubes play a major role in its mesoscopic properties (Masale et al. 1992). The quantum confinement effect can be observed if the wall thickness of nanotube is comparable to the excitonic diameter of bulk material. It should be noted that quantum confinement effects are least studied in one-dimensional systems, particularly with tubular structure, as compared to other low-dimensional systems.
The present work reports a single-step chemical process to synthesize micron length CdS nanotubes with wall thickness comparable to the excitonic diameter of the bulk material. A change in experimental conditions result in formation of nanowires, and well separated nanoparticles. A mechanism of formation of the nanotube is described. It is proposed that bubbles are playing the key role in formation of nanotubes. Optical absorption and photoluminescence measurements are carried out in order to study the quantum confinement effect.
Experimental {#experimental .unnumbered}
============
The precursors used for the synthesis of CdS nanotubes are thiourea (NH$_2$CSNH$_2$), Cadmium sulfate (CdSO$_4$), ammonia (NH$_3$), and poly vinyl alcohol (PVA). 0.01 M aqueous solution of CdSO$_4$ and thiourea are prepared separately. About 5 mL of 3 M aqueous ammonia solution is added through a buret into 20 mL of CdSO$_4$ solution in a slow stirring condition. During addition of ammonia it turns to a turbid white solution and then becomes completely transparent. A total of 15 mL of 4$\%$ aquous solution of PVA (degree of polymerization: 1,700 - 1,800) is added to it. About 20 mL of thiourea solution is then put in the same stirring condition, and left in the ambient condition without any further stirring. After around 10 min the solution becomes pale yellow color, which suggests formation of CdS. After 90 min, carbon coated Cu-grids are dipped into this yellow color solution and held aloft to dry. These dried grids are used for transmission electron microscopy (TEM) analysis. For other experiments nanotubes are collected by centrifugation.
TEM is performed using JEOL-2010 operated at 200 KeV electron beam energy. X-ray diffraction (XRD) was conducted on a philips PW1877 diffractometer using Cu $K_\alpha$ radiation. Optical absorption measurements are carried out by using a dual beam Shimadzu UV-3101 PC spectrophotometer. The photoluminescence (PL) measurements are carried out using 369 nm line from a Oriel Hg/Xe lamp as the source of excitation. Luminescence is detected by a PMT detector attached with Jovin Yvon TRIAX-180 spectrometer. Experiments are carried out in ambient temperature.
Results and discussions {#results-and-discussions .unnumbered}
=======================
![TEM micro-graphs of CdS nanotubes (a) with lower magnification (b) with higher magnification. (c) Histogram of outer diameter of nanotubes (d) histogram of thickness of nanotubes.[]{data-label="tem1"}](8k-22.eps "fig:"){width="7.0cm"} ![TEM micro-graphs of CdS nanotubes (a) with lower magnification (b) with higher magnification. (c) Histogram of outer diameter of nanotubes (d) histogram of thickness of nanotubes.[]{data-label="tem1"}](30k-2a.eps "fig:"){width="7.0cm"} ![TEM micro-graphs of CdS nanotubes (a) with lower magnification (b) with higher magnification. (c) Histogram of outer diameter of nanotubes (d) histogram of thickness of nanotubes.[]{data-label="tem1"}](width.eps "fig:"){width="4.3cm"}![TEM micro-graphs of CdS nanotubes (a) with lower magnification (b) with higher magnification. (c) Histogram of outer diameter of nanotubes (d) histogram of thickness of nanotubes.[]{data-label="tem1"}](thickness.eps "fig:"){width="4.3cm"}
The formation of CdS nanotubes is confirmed by transmission electron microscopy (TEM). The typical TEM micrographs at a lower and at a higher magnification are shown in Fig.\[tem1\]a, b, respectively. From Fig.\[tem1\]a, it is not possible to conclude whether these are nanotubes or nanowires. However, it is conformed that these nanostructures are of micron lengths and have high aspect ratio. With higher magnification (Fig.\[tem1\]b), a contrast between the solid side wall (darker contrast) and hollow middle part (lighter contrast) of these one-dimensional structure is observed. It gives the signature of formation of nanotubes. Outer diameter and wall thickness of several nanotubes are measured. A histogram of outer diameter of CdS nanotubes along with a fitted Gaussian , is shown in Fig. \[tem1\]c. The mean diameter of the nanotube is 14.4 nm with a standard deviation of 6.1 nm. The histogram of wall thickness, along with a fitted Gaussian, is shown in Fig. \[tem1\]d. The mean wall thickness of the nanotubes is 4.7 nm with a standard deviation of 2.2 nm. TEM image of a nanotube with a smaller diameter and a nanotube with a larger diameter are shown in Fig. \[tem2\] a,b respectively. The fluctuation observed in the thickness of wall is comparatively less than the fluctuation in diameter of the nanotubes.
![TEM micro-graphs of (a) a nanotube with smaller diameter (b) a nanotube with bigger diameter.[]{data-label="tem2"}](40k-2a.eps "fig:"){width="1.8cm"} ![TEM micro-graphs of (a) a nanotube with smaller diameter (b) a nanotube with bigger diameter.[]{data-label="tem2"}](120k-1b.eps "fig:"){width="6.4cm"}
The electron diffraction (ED) pattern is shown in Fig. \[tem3\]a. The rings in ED pattern suggests polycrystalline nature of the nanotubes. These nanotubes are not single crystalline, but consists of numerous nanocrystallites as it can be seen in high resolution electron micrograph (Fig.\[tem3\]b). It seems large number of nanocrystallites bind together and form nanotube. X-ray diffraction (Fig. \[xrd\]) peaks obtained are also broad due to small dimensions of these constituent crystallites. The XRD pattern shows characteristic peaks of wurzite structure of CdS (JCPDS card No. 41-1049). The interplanar distance of 0.24 nm, as shown in Fig.\[tem3\]b, matches for (102) planes and rings in ED pattern corresponds to (110) and (302) planes of wurzite structure of CdS. The formula used to calculate $d$-value in the ED pattern is $d_{hkl}$ =$\lambda L$/${R}$. where L is the Camera length, R is the radius of the ring and $\lambda$ is the wavelength of the electron beam.
=8.8 true cm =8.8 true cm
=8.8true cm
The general chemical equation for the formation of CdS nanotube can be written as follows (Pavaskar et al. 1977; Hariskos et al. 2001) :\
$CdSO_4+4(NH_3) \rightarrow [Cd(NH_3)_4]SO_4 \\
C(NH_2)_2S+ OH^- \rightarrow CH_2N_2+H_2O+HS^{-} \\
CH_2N_2+H_2O \rightarrow (NH_2)_2CO\\
(NH_2)_2CO + 2 OH^- \rightarrow CO_3^{2-} + 2NH_3(g)\\
HS^{-}+OH^- \rightarrow S^{--}+H_2O \\
$~$ [Cd(NH_3)_4]^{++} + S^{--} \rightarrow CdS (s)+4NH_3 (g) \\$
The chemical reaction mentioned above, leads to supersaturation of CdS concentrations. According to classical nucleation theory (Markov 1995), the energy required to form a critical particle homogeneously is given by: $$W_p=\frac{1}{3}\sum_n\sigma_n A_n$$ where $A_n$ is surface area and $\sigma_n$ is specific surface free energy of the $n$-th surface of the particle. Since crystal faces with different crystallographic orientation have different specific surface energy, the crystallite form a shape in which the total surface energy will be minimum.
It should be noted that the chemical route adopted to synthesize CdS nanotube is similar to the chemical bath deposition technique to make bulk CdS thin films (Pavaskar et al. 1977). The optimization parameters like concentration of the reactant and working temperature are varied. PVA is also used during these chemical synthesis process. PVA is a surface active and water soluble polymer. Although, it does not take part in the chemical reaction, its addition increases viscosity (Briscoe et al. 2000) and decreases surface tension (Bhattacharya and Ray 2004) of the solution.
According to the Henry’s law, the amount of a gas dissolved in a given volume of liquid,$C_{eq}$, is directly proportional to the partial pressure of that gas,$P$, above the liquid. It can be written as: $$C_{eq}=kP$$
where $k$ is the Henry’s law constant.
As viscosity of the solution is increased due to addition of PVA, the diffusional escape of the ammonia gas decreased and higher concentration of gas generated inside the solution than its equilibrium concentration. It results in supersaturation and nucleation of ammonia gas bubble inside the solution.
The energy required to form a critical bubble homogeneously (Landau and Lifshitz 1999; Bowers et al. 1995) is given by $$W_b= \frac{1}{3}\sigma A= \frac{16\pi{\sigma}^3 k^2}{3(C_s-C_{eq})^2}$$ here $\sigma$ is the surface tension and $C_s$ is the supersaturation concentration.
The probability of formation of critical bubble is proportional to exp$(-W_b/T)$. Due to presence of $\sigma^3$ term in the numerator of $W_b$ , the probability of formation of gas bubbles is increased significantly by the reduction of surface tension.
Although we considered the case of a homogeneous nucleation, the colloids ( both bubbles and particles ) most likely nucleate heterogeneously, as later process is energetically favorable than the former.
These colloids experience the gravitational forces due to its mass, the buoyant force due to the displaced fluid and the frictional force due to viscosity of the medium. This leads to a terminal velocity at which these colloids will move and is given by the equation (Lamb 1945) $$V=\frac{2}{3}\cdot \frac{gr^2(\rho'-\rho)}{\eta}\cdot\frac{\eta+\eta'}{2\eta+3\eta'}$$ Here $\rho'$ and $\eta'$ are the density and dynamic viscosity of the colloids, respectively. Similarly $\rho$ and $\eta$ are the density and dynamic viscosity of the medium, respectively. $g$ is the acceleration due to gravity and $r$ is the radius of the colloids. For particles, we can approximate $\eta' \gg \eta$; and for bubbles, $\eta'=0$ and $\rho'=0$. Hence, according to the above equation, the terminal velocity of colloids with size in nano range have a negligibly small terminal velocity. These colloids also experience a random force that originates from fast collisions with molecules of the medium. This gives rise to Brownian motion of the colloids. The average squared displacement of the colloid, $\langle x^2 \rangle $, that follows Brownian motion, after time $t$ from its initial position is given by (Reif 1965) $$\langle x^2 \rangle=( \frac {k_B T}{3\pi \eta r}) t
\label{diffusion}$$ The average squared displacement is inversely proportional to the radius of the colloid and to the viscosity of the medium. Hence, diffusive motion dominates the motion of the particles and bubbles. These particles and bubbles encounter during their random motion inside the solution, and during this process, the particles get attached with the bubbles. It should be noted that particle-bubble attachment can occur when particle-bubble contact time is longer than the induction time (Dai et al. 1999). Hence, reduction in induction time enhance the attachment efficiency. As bubbles and particles are very small in size, the induction time is very small and attachment efficiency is very high. The induction time is defined as the time for the liquid film between the particle and the bubble to thin, rupture and form a equilibrium three-phase contact.
It is observed that CdTe nanoparticles spontaneously aggregate into a pearl-necklace like structure upon controlled removal of the protective shell of organic stabilizer, and subsequently recrystallize into nanowires (Tang et al. 2002). These CdTe nanoparticles have a large dipole moment and the dipole-dipole interaction between them is responsible for their unidirectional self-organization (Sinyagin et al. 2005). CdS nanoparticles with wurzite crystal structure also have a large dipole moment (Sinyagin et al. 2005; Blanton et al. 1997; Shanbhag and Kotov 2006). Hence, particle-attached-bubble as a whole may has a strong net dipole moment; and dipole-dipole interaction between these particle-attached-bubbles is primarily responsible for their unidirectional aggregation. The adjacent nanoparticles on the bubble surface probably get attached at a planar interface, reduce total surface energy and transform into a stable nanotube. In the aggregation-based crystal growth ( Banfield et al. 2000; Penn and Banfield 1998), random force imparted on the nanoparticles by the medium molecules helps them in rotation and attachment at a planar interface so that they can share a common crystallographic orientation. However, even a small misorientation can lead to dislocation at the interfaces. As the medium is viscous and CdS nanoparticles are attached on a curved surface with equilibrium three phase contact, particles could not perfectly orient and attach in a atomically flat interfaces to give a dislocation free single crystalline nanotube. However, annealing after synthesis can help in improving the crystallinity.
It should be noted that colloids after nucleation goes through subsequent growth dynamics along with the above mention steps like bubble-particle attachment and unidirectional aggregation of particle-attached-bubbles. The colloids smaller than its critical size dissolve as surface energy is large. Colloids bigger than the critical size only grow. The attachment of colloids also reduces the total surface energy and effective size of the colloid become more than its critical radius. Hence, formation of nanotubes with a high aspect ratio are favorable and stable. In the TEM measurement, it is observed that inner wall of nanotubes are comparatively less smooth than the outside wall. A TEM micrograph of a nanotube in a formative stage, in which bubbles are coalescing, is shown in figure (Fig.\[bulge\]a). Other possible nanostructures like nanoparticles, nanowires, nanowires with spherical cavity(Fig.\[bulge\]b) are also observed in TEM measurements, although they are few in number. These observations lead to believe that bubbles are responsible for hollowness inside the nanotube.
=8.8true cm =8.8true cm
In order to confirm the role of PVA, experiments are carried out in similar experimental condition by not using PVA, and by using much higher amount of PVA (25 mL of 20$\%$ aquous solution). Nanowires are seen in TEM measurements instead of nanotubes, when PVA is not used in the experiment. The typical TEM micrographs at a lower and at a higher magnification are shown in Fig.\[wire\]a,b respectively. In this case probably bubbles did not nucleate. Nanoparticles get aligned unidirectionally due to their dipole moment and nanowires are formed. Formation of nanowires even in the absence of PVA exclude the possibility that linear chain structure of polymer is someway acting as a template for unidirectional aggregation of the nanoparticles, and forming nanotubes.
=8.8true cm =8.8true cm
When excess amount of PVA is used, the solution becomes very viscous. Only after $\approx$4 h ( in contrast to $\approx$10 min), the solution becomes pale yellow color. The solution then kept for another 90 min, and then spin-coated on a Cu-grid to carry out TEM measurements. Even though TEM is carried out by taking the grid on a liquid nitrogen cooled sample stage, the PVA matrix forms voids due to the heating effect of electron beams. However, it can be seen that nanocrystals are well separated. With too much increase in viscosity, the decrease in diffusive length of nanocrystals is significant (see Eq. \[diffusion\]). So nanocrystals could not aggregate to form nanowires or nanotubes. A typical TEM micrograph is shown in Fig. \[nanocrystal\]. It should be noted that well separated HgS nanocrystals are also synthesized by a similar chemical procedure using PVA (Mahapatra and Dash 2006).
=8.8true cm
Most of the interesting properties exhibited by semiconducting nanomaterials are attributed to quantum confinement effect. The electronic energy levels are strongly dependent on the size and also on the shape of the nanostructure (Kayanuma 1991). In the one-dimensional systems charge carriers are confined in two dimensions and free in one dimension. The spatial confinement of carriers leads to band gap widening and most directly realized by a high energy shift in optical absorption and photoluminescence peak. Quantum confinement effect for zero-dimensional system is studied both theoretically (Kayanuma 1988) and experimentally (Vossmeyer et al. 1994) in detail. However, it is not well understood for one-dimensional systems, particularly with tubular structure.
The optical absorption spectrum and photoluminescence (PL) spectrum of CdS nanotubes are shown in Figs. \[optical\], \[pl\], respectively. An excitonic peak appears at 464 nm in the optical absorption spectrum. The peak position is blue shifted by 48nm from its bulk band gap value (512 nm). The experimental PL data is fitted as the sum of two Gaussian functions, among which one is due to incident line. The PL peak is best fitted with the Gaussian of 492 nm mean and 26 nm standard deviation. It should be noted that, the PL peak position is also blue shifted from its bulk band gap value. High energy peak shifting in the optical absorption and PL spectra are expected due to quantum confinement effect as nanotube wall thickness is comparable to excitonic diameter of bulk CdS.
=8.8true cm
=8.8true cm
Luminescence that are observed in semiconductor nanomaterials are generally excitonic and trapped emissions. Excitonic emission is sharp and can be observed near the absorption edge if the material is pure (Pankove 1975). However, if the material is impure or off-stoichiometric then a broad and intense emission occur at higher wavelength due to recombination of charge carriers at trapped states. Hence the band edge luminescence that appears at 492 nm is due to excitonic transition. Band edge luminescence at 470 nm is also observed for CdS nanotubes synthesized by sacrificial template method (Li et al. 2006). However, CdS nanotubes synthesized by sacrificial template method also show a very intense and broad peak centered around 560 nm due to presence of trapped states. No such peak is observed in our synthesized CdS nanotubes. This suggests purity and stoichiometric nature of our synthesized CdS nanotubes.
Excitonic transition in a semiconductor can be observed well only at low temperature. However, it can be observed even at room temperature due to enhancement of oscillator strength in the low-dimensional systems (Kayanuma 1991). It should be noted that excitonic transitions is not observed in optical absorption and PL measurements for bulk CdS which was prepared by the similar chemical route (Pavaskar et al. 1977). However, a clear excitonic feature arises in both absorption and PL measurements when it forms nanotubes. The PL peak is red shifted by 150 meV with respect to absorption peak. Such a large red shift, known as stokes shift, of 147 meV is also reported for CdS nanoparticles (Tamborra et al. 2004). The difference in the absorption and emission states avoid sample self-absorption and could be very useful in making LEDs (Sze 1981).
Conclusion {#conclusion .unnumbered}
==========
CdS nanotubes with wall thickness comparable to excitonic diameter of the bulk material are synthesized by a chemical synthesis process. These synthesized nanotubes show band gap widening and enhanced oscillator strength due to quantum confinement effects. A large stokes shift of 150 meV is also observed. Bubbles are responsible for the hollowness of nanotubes; and bubbles of dissolved gases can be utilized to make nanostructures with hollow interior.
Acknowledgment {#acknowledgment .unnumbered}
==============
The help and encouragement received from Dr S. N. Sahu is gratefully acknowledged. Mr A. K. Dash, Mr U. M. Bhatta, and Dr P. V. Satyam are acknowledged for their help in TEM measurements.
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[^1]: e-mail: amulya@iopb.res.in
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abstract: |
This is a sequel to [@li-qva1] and [@li-qva2] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian $DY_{\hbar}(sl_{2})$, denoted by $DY_{q}(sl_{2})$ and $DY_{q}^{\infty}(sl_{2})$ with $q$ a nonzero complex number. For each nonzero complex number $q$, we construct a quantum vertex algebra $V_{q}$ and prove that every $DY_{q}(sl_{2})$-module is naturally a $V_{q}$-module. We also show that $DY_{q}^{\infty}(sl_{2})$-modules are what we call $V_{q}$-modules-at-infinity. To achieve this goal, we study what we call $\S$-local subsets and quasi-local subsets of $\Hom
(W,W((x^{-1})))$ for any vector space $W$, and we prove that any $\S$-local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with $W$ as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.
---
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ł Ł **
\[section\] \[thm\][Proposition]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Example]{} \[thm\][Lemma]{} \[thm\][Remark]{} \[thm\][Definition]{} \[thm\][Hypothesis]{} addtoreset[equation]{}[section]{}
[**Modules-at-infinity for quantum vertex algebras**]{}
[Haisheng Li[^1]\
Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102]{}
Introduction
============
This is a sequel to [@li-qva1] and [@li-qva2] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In [@li-qva1] and [@li-qva2], partially motivated by Etingof-Kazhdan’s notion of quantum vertex operator algebra over $\C[[\hbar]]$ (see [@ek]), we formulated and studied a notion of quantum vertex algebra over $\C$ and we established general constructions of (weak) quantum vertex algebras and modules. The general constructions were illustrated by examples in which quantum vertex algebras were constructed from certain Zamolodchikov-Faddeev-type algebras. The main goal of this paper is to establish a natural connection of (centerless) double Yangians with quantum vertex algebras over $\C$.
For each finite-dimensional simple Lie algebra $\g$, Drinfeld introduced a Hopf algebra $Y(\g)$, called Yangian (see [@dr1]), as a deformation of the universal enveloping algebra of the Lie algebra $\g\otimes \C[t]$. The Yangian double $DY_{\hbar}(\g)$ is a deformation of the algebra $U(\g\otimes \C[t,t^{-1}])$. In the simplest case with $\g=sl_{2}$ (cf. [@sm], [@kh]), the following is one of the defining relations in terms of generating functions $$\begin{aligned}
\label{e-introd-ee}
e(x_{1})e(x_{2})=\frac{x_{1}-x_{2}+\hbar}{x_{1}-x_{2}-\hbar}e(x_{2})e(x_{1}).\end{aligned}$$ In this paper we study two versions of $DY_{\hbar}(sl_{2})$ with the formal parameter $\hbar$ being evaluated at a nonzero complex number $q$. With a direct substitution, the defining relation (\[e-introd-ee\]) becomes $$\begin{aligned}
\label{e-introd-qee}
e(x_{1})e(x_{2})=\frac{x_{1}-x_{2}+q}{x_{1}-x_{2}-q}e(x_{2})e(x_{1}),\end{aligned}$$ where $$\begin{aligned}
\frac{x_{1}-x_{2}+q}{x_{1}-x_{2}-q}
=(x_{1}-x_{2}+q)\sum_{i\ge 0} q^{i}(x_{1}-x_{2})^{-i-1}
\in \C(((x_{1}-x_{2})^{-1}))\subset \C[x_{2}]((x_{1}^{-1})).\end{aligned}$$ In this way we get a version of $DY_{\hbar}(sl_{2})$, which we denote by $DY_{q}^{\infty}(sl_{2})$. Notice that for a module $W$ of highest weight type where the generating functions are elements of $\Hom (W,W((x)))$, the expression $$\frac{x_{1}-x_{2}+q}{x_{1}-x_{2}-q}e(x_{2})e(x_{1})$$ does not exist in general. Because of this, $DY_{q}^{\infty}(sl_{2})$ admits only modules $W$ of lowest weight type where the generating functions such as $e(x)$ are elements of $\Hom (W,W((x^{-1})))$.
Note that (quantum) vertex algebras and their modules are modules of highest weight type in nature and so far we have used only modules of highest weight type for various algebras to construct (quantum) vertex algebras and their modules. Motivated by this, we then consider another version of $DY_{\hbar}(sl_{2})$, which we denote by $DY_{q}(sl_{2})$, by expanding the same rational function $\frac{x_{1}-x_{2}+q}{x_{1}-x_{2}-q}$ as follows: $$\frac{x_{1}-x_{2}+q}{-q+x_{1}-x_{2}}
=-(x_{1}-x_{2}+q)\sum_{i\ge 0} q^{-i-1}(x_{1}-x_{2})^{i}
\in \C[[(x_{1}-x_{2})]]\subset \C[[x_{1},x_{2}]].$$ Contrary to the situation with $DY_{q}^{\infty}(sl_{2})$, the algebra $DY_{q}(sl_{2})$ admits only modules of highest weight type, including vacuum modules. For every nonzero complex number $q$, we construct a universal vacuum $DY_{q}(sl_{2})$-module $V_{q}$ and by applying the general construction theorems of [@li-qva1] and [@li-qva2] we show that there exists a canonical quantum vertex algebra structure on $V_{q}$ and that on any $DY_{q}(sl_{2})$-module there exists a canonical $V_{q}$-module structure.
As it was mentioned before, the algebra $DY_{q}^{\infty}(sl_{2})$ admits only lowest-weight-type modules. This is also the case for the Lie algebra of pseudo-differential operators. Nevertheless, we hope to associate such algebras with (quantum) vertex algebras in some natural way. Having this in mind, we systematically study how to construct quantum vertex algebras from suitable subsets of the space $$\E^{o}(W)=\Hom (W,W((x^{-1})))$$ for a general vector space $W$, developing a general theory analogous to that of [@li-qva1]. For a vector space $W$, a subset $T$ of $\E^{o}(W)$ is said to be $\S$-local if for any $a(x),b(x)\in T$, there exist $$a_{i}(x), b_{i}(x)\in T,\; f_{i}(x)\in \C(x),\; i=1,\dots,r$$ and a nonnegative integer $k$ such that $$(x_{1}-x_{2})^{k}a(x_{1})b(x_{2})
=(x_{1}-x_{2})^{k}\sum_{i=1}^{r}\iota_{x,\infty}(f_{i})(x_{1}-x_{2})
b_{i}(x_{2})a_{i}(x_{1}),$$ where $\iota_{x,\infty}f_{i}(x)$ denotes the formal Laurent series expansion of $f_{i}(x)$ at infinity. A subset $T$ of $\E^{o}(W)$ is said to be quasi-local if for any $a(x),b(x)\in T$, there exists a nonzero polynomial $p(x_{1},x_{2})$ such that $$p(x_{1},x_{2})a(x_{1})b(x_{2})=p(x_{1},x_{2})b(x_{2})a(x_{1}).$$ We prove that any $\S$-local subset generates a weak quantum vertex algebra and that any quasi-local subset of $\E^{o}(W)$ generates a vertex algebra in a certain natural way. To describe the structure on $W$ we formulate a notion of (left) quasi module-at-infinity for a vertex algebra and for a weak quantum vertex algebra. For a vertex algebra $V$, a (left) quasi $V$-module-at-infinity is a vector space $W$ equipped with a linear map $Y_{W}$ from $V$ to $\Hom
(W,W((x^{-1})))$, satisfying the condition that $Y_{W}({\bf
1},x)=1_{W}$ (the identity operator on $W$) and that for $u,v\in V$, there exists a nonzero polynomial $p(x_{1},x_{2})$ such that $$\begin{aligned}
& &p(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})
=p(x_{1},x_{2})Y_{W}(v,x_{2})Y_{W}(u,x_{1}),\\
& &p(x_{2}+x_{0},x_{2})Y_{W}(Y(u,x_{0})v,x_{2})
=\left(p(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})\right)|_{x_{1}=x_{2}+x_{0}}.\end{aligned}$$ For a module-at-infinity, the following opposite Jacobi identity holds for $u,v\in V$: $$\begin{aligned}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y_{W}(v,x_{2})Y_{W}(u,x_{1})
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{W}(u,x_{1})Y_{W}(v,x_{2})\\
& &\ \ \ \ =x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y_{W}(Y(u,x_{0})v,x_{2}).\end{aligned}$$ This notion of a left module-at-infinity for a vertex algebra $V$ coincides with the notion of a right module, which was suggested in [@hl].
As an application, we show that $DY_{q}(sl_{2})$-modules are canonical modules-at-infinity for the quantum vertex algebra $V_{q}$. We also show that lowest-weight type modules for the Lie algebras of pseudo-differential operators on the circle are quasi modules-at-infinity for some vertex algebras associated with the affine Lie algebra of a certain infinite-dimensional Lie algebra. In a sequel, we shall study general double Yangians $DY_{\hbar}(\g)$ and their central extensions $\widehat{DY_{\hbar}(\g)}$ (see [@kh]) in terms of quantum vertex algebras.
This paper is organized as follows: In Section 2, we introduce a version of the double Yangian $DY_{\hbar}(sl_{2})$ and we associate it with quantum vertex algebras and modules. In Section 3, we study quasi-compatible subsets and prove that any quasi-compatible subset canonically generates a nonlocal vertex algebra. In Section 4, we study $\S$-local subsets and modules-at-infinity for quantum vertex algebras. In Section 5, we study quasi local subsets and quasi modules-at-infinity for vertex algebras.
Associative algebra $DY_{q}(sl_{2})$ and quantum vertex algebras
================================================================
In this section, we first recall the notions of weak quantum vertex algebra and quantum vertex algebra and we then define an associative algebra $DY_{q}(sl_{2})$ over $\C$ with $q$ an arbitrary nonzero complex number, which is a version of the (centerless) double Yangian $DY_{\hbar}(sl_{2})$, and we associate a quantum vertex algebra to the algebra $DY_{q}(sl_{2})$.
We begin with the notion of nonlocal vertex algebra ([@kacbook2], [@bk], [@li-g1]). A [*nonlocal vertex algebra*]{} is a vector space $V$, equipped with a linear map $$\begin{aligned}
Y: V &\rightarrow & \Hom (V,V((x)))\subset (\End V)[[x,x^{-1}]]\nonumber\\
v&\mapsto& Y(v,x)=\sum_{n\in \Z}v_{n}x^{-n-1}\;\;\; (v_{n}\in \End V)\end{aligned}$$ and equipped with a distinguished vector ${\bf 1}$, such that for $v\in V$ $$\begin{aligned}
& &Y({\bf 1},x)v=v,\\
& &Y(v,x){\bf 1}\in V[[x]]\;\;\mbox{ and }\;\; \lim_{x\rightarrow
0}Y(v,x){\bf 1}=v\end{aligned}$$ and such that for $u,v,w\in V$, there exists a nonnegative integer $l$ such that $$\begin{aligned}
(x_{0}+x_{2})^{l}Y(u,x_{0}+x_{2})Y(v,x_{2})w=
(x_{0}+x_{2})^{l}Y(Y(u,x_{0})v,x_{2})w.\end{aligned}$$ For a nonlocal vertex algebra $V$, we have $$\begin{aligned}
\label{edproperty}
[{\cal{D}},Y(v,x)]=Y({\cal{D}}v,x)={d\over dx}Y(v,x) \;\;\;\mbox{
for }v\in V,\end{aligned}$$ where $\D$ is the linear operator on $V$, defined by $$\begin{aligned}
{\cal{D}}(v)=\left({d\over dx}Y(v,x){\bf 1}\right)|_{x=0}
\left(=v_{-2}{\bf 1}\right)
\;\;\;\mbox{ for }v\in V.\end{aligned}$$ Furthermore, for $v\in V$, $$\begin{aligned}
& &e^{x{\cal{D}}}Y(v,x_{1})e^{-x{\cal{D}}}=Y(e^{xD}v,x_{1})=Y(v,x_{1}+x),
\label{econjugationformula1}\\
& &Y(v,x){\bf 1}=e^{x{\cal{D}}}v.\label{ecreationwithd}\end{aligned}$$
A [*weak quantum vertex algebra*]{} (see [@li-qva1], [@li-qva2]) is a vector space $V$ (over $\C$) equipped with a distinguished vector ${\bf 1}$ and a linear map $$\begin{aligned}
Y:& &V\rightarrow \Hom (V,V((x)))\subset (\End V)[[x,x^{-1}]],\\
& &v\mapsto Y(v,x)=\sum_{n\in \Z}v_{n}x^{-n-1}\;\;
(\mbox{where }v_{n}\in \End V)\end{aligned}$$ satisfying the condition that $$\begin{aligned}
& &Y({\bf 1},x)v=v,\\
& &Y(v,x){\bf 1}\in V[[x]]\;\;\mbox{ and }\;\;\lim_{x\rightarrow
0}Y(v,x){\bf 1}=v\ \ \ \mbox{ for }v\in V,\end{aligned}$$ and that for any $u,v\in V$, there exist $u^{(i)},v^{(i)}\in V,\; f_{i}(x)\in \C((x)),\; i=1,\dots,r$, such that $$\begin{aligned}
\label{eS-jacobi}
&&x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)Y(u,x_{1})Y(v,x_{2})
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
\sum_{i=1}^{r}f_{i}(-x_{0})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})\nonumber\\
& &\hspace{2cm} = x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y(Y(u,x_{0})v,x_{2}).\end{aligned}$$
In terms of the notion of nonlocal vertex algebra, a weak quantum vertex algebra is simply a nonlocal vertex algebra that satisfies the $\S$-locality (cf. [@ek]) in the sense that for any $u,v\in V$, there exist $u^{(i)},v^{(i)}\in V,\; f_{i}(x)\in \C((x)),\; i=1,\dots,r$, such that $$\begin{aligned}
(x_{1}-x_{2})^{k}Y(u,x_{1})Y(v,x_{2})
=(x_{1}-x_{2})^{k}\sum_{i=1}^{r}f_{i}(x_{2}-x_{1})
Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})\end{aligned}$$ for some nonnegative integer $k$ depending on $u$ and $v$.
The notion of quantum vertex algebra involves the notion of unitary rational quantum Yang-Baxter operator, which here we recall: A [*unitary rational quantum Yang-Baxter operator*]{} on a vector space $H$ is a linear map $\S(x): H\otimes H\rightarrow H\otimes H\otimes \C((x))$ such that $$\begin{aligned}
& &\S_{21}(-x)\S(x)=1,\\
& &\S_{12}(x)\S_{13}(x+z)\S_{23}(z)=\S_{23}(z)\S_{13}(x+z)\S_{12}(x).\end{aligned}$$ In this definition, $\S_{21}(x)=\sigma_{12}\S(x)\sigma_{12}$, where $\sigma_{12}$ is the flip map on $H\otimes H$ $(u\otimes v\mapsto
v\otimes u)$, $$\S_{12}(x)=\S(x)\otimes 1: H\otimes H\otimes H
\rightarrow H\otimes H\otimes H\otimes \C((x)),$$ and $\S_{13}(x),\S_{23}(x)$ are defined accordingly.
A [*quantum vertex algebra*]{} ([@li-qva1], cf. [@ek]) is a weak quantum vertex algebra $V$ equipped with a unitary rational quantum Yang-Baxter operator $\S(x): V\otimes V\rightarrow V\otimes
V\otimes \C((x))$ such that for $u,v\in V$, (\[eS-jacobi\]) holds with $\S(x)(v\otimes u)=\sum_{i=1}^{r}v^{(i)}\otimes u^{(i)}\otimes
f_{i}(x)$.
Let $V$ be a nonlocal vertex algebra. A [*$V$-module*]{} is a vector space $W$ equipped with a linear map $Y_{W}: V\rightarrow \Hom
(W,W((x)))$ satisfying the condition that $Y_{W}({\bf 1},x)=1$ (the identity operator on $W$) and for any $u,v\in V,\; w\in W$, there exists a nonnegative integer $l$ such that $$\begin{aligned}
(x_{0}+x_{2})^{l}Y_{W}(u,x_{0}+x_{2})Y_{W}(v,x_{2})w
=(x_{0}+x_{2})^{l}Y_{W}(Y(u,x_{0})v,x_{2})w.\end{aligned}$$ Now, assume that $V$ is a weak quantum vertex algebra and let $(W,Y_{W})$ be a module for $V$ viewed as a nonlocal vertex algebra. It was proved ([@li-qva1], Lemma 5.7) that for any $u,v\in V$, $$\begin{aligned}
&&x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y_{W}(u,x_{1})Y_{W}(v,x_{2})\nonumber\\
& &\hspace{1cm}
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
\sum_{i=1}^{r}f_{i}(-x_{0})Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1})\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y_{W}(Y(u,x_{0})v,x_{2}),\end{aligned}$$ where $u^{(i)},v^{(i)}\in V,\; f_{i}(x)\in \C((x))$ are the same as those in (\[eS-jacobi\]).
Next, we introduce a version of the double Yangian $DY_{\hbar}(sl_{2})$ (associated to the three-dimensional simple Lie algebra $sl_{2}$). Let $T(sl_{2}\otimes \C[t,t^{-1}])$ denote the tensor algebra over the vector space $sl_{2}\otimes \C[t,t^{-1}]$. [*From now on we shall simply use $T$ for this algebra.*]{} We equip $T$ with the $\Z$-grading which is uniquely defined by $$\deg (u\otimes t^{n})=n\ \ \
\mbox{ for }u\in sl_{2},\; n\in \Z,$$ making $T$ a $\Z$-graded algebra $T=\coprod_{n\in \Z}T_{n}$. For $n\in \Z$, set $$I[n]=\coprod_{m\ge n} T_{m}\subset T.$$ This defines a decreasing filtration of $T=\cup_{n\in \Z}I[n]$ with $\cap_{n\in \Z} I[n]=0$. Denote by $\overline{T}$ the completion of $T$ associated with this filtration.
For $u\in sl_{2}$, set $$u(x)=\sum_{n\in \Z}u(n)x^{-n-1},$$ where $u(n)=u\otimes t^{n}$. We also write $sl_{2}(n)=sl_{2}\otimes t^{n}$ for $n\in \Z$. Let $e,f,h$ be the standard Chevalley generators of $sl_{2}$.
It is straightforward to see that $DY_{q}(sl_{2})$ admits a (unique) derivation $d$ such that $$\begin{aligned}
[d,u(x)]=\frac{d}{dx}u(x)\ \ \ \mbox{ for }u\in sl_{2}.\end{aligned}$$ That is, $$\begin{aligned}
[d,u(n)]=-nu(n-1)\ \ \ \mbox{ for }u\in sl_{2},\; n\in \Z.\end{aligned}$$
The following are some basic properties of a general vacuum $DY_{q}(sl_{2})$-module:
Let $W$ be a vacuum $DY_{q}(sl_{2})$-module with a vacuum vector $w_{0}$ as a generator. Set $F_{0}=\C w_{0}$ and $F_{k}=0$ for $k<0$. For any positive integer $k$, we define $F_{k}$ to be the linear span of the vectors $$a_{1}(-m_{1})\cdots a_{r}(-m_{r})w_{0}$$ for $r\ge 1,\; a_{1},\dots, a_{r}\in sl_{2},\;
m_{1},\dots,m_{r}\ge 1$ with $m_{1}+\cdots +m_{r}\le k$. Then the subspaces $F_{k}$ for $k\in \Z$ form an increasing filtration of $W$ and for any $a\in sl_{2},\; m,k\in \Z$, $$\begin{aligned}
\label{eproperty}
a(m)F_{k}\subset F_{k-m}.\end{aligned}$$ Furthermore, $T_{m}w_{0}=0$ for $m\ge 1$.
We first prove (\[eproperty\]). It is true for $k<0$ as $F_{k}=0$ by definition. With $F_{0}=\C w_{0}$, we see that (\[eproperty\]) holds for $k=0$. Assume $k\ge 1$. From definition, (\[eproperty\]) always holds for $m<0$. Let $a,b\in \{ e,f,h\}$. From the defining relations of $DY_{q}(sl_{2})$ we have $$\begin{aligned}
\label{erelation-dy}
a(m)b(n)=\pm b(n)a(m)+\sum_{i,j\ge 0, \;i+j\ge 1}\lambda_{ij}
b(n+i)a(m+j)+\alpha h(m+n)\end{aligned}$$ for all $m,n\in \Z$, where $\lambda_{ij},\alpha\in \C$, depending on $a,b$. Using this fact and induction on $k$ we obtain (\[eproperty\]), noticing that $a(m)F_{0}=0$ for $m\ge 0$. From (\[eproperty\]) we get $$T_{m}w_{0}= T_{m}F_{0}\subset F_{-m}=0\ \ \ \mbox{ for }m\ge 1.$$ It also follows from (\[eproperty\]) that $\cup_{k\ge 0}F_{k}$ is a submodule of $W$. Since $w_{0}$ generates $W$, we must have $W=\cup_{k\ge 0}F_{k}$. This proves that the subspaces $F_{k}$ for $k\in \Z$ form an increasing filtration of $W$.
Let $W$ be a vacuum $DY_{q}(sl_{2})$-module with a vacuum vector $w_{0}$ as a generator. For $n\in \N$, define $E_{n}$ to be the linear span of the vectors $$u^{(1)}(m_{1})\cdots u^{(r)}(m_{r})w_{0}$$ for $0\le r\le n,\; u^{(i)}\in \{ e,f,h\},\; m_{i}\in \Z$. Then the subspaces $E_{n}$ for $n\in \N$ form an increasing filtration of $W$ and for each $n\in \N$, $E_{n}$ is linearly spanned by the vectors $$\begin{aligned}
e(-m_{1})\cdots e(-m_{r})f(-n_{1})\cdots f(-n_{s})h(-k_{1})\cdots
h(-k_{l})w_{0},\end{aligned}$$ where $r,s,t\ge 0$ and $m_{i},n_{j},k_{t}$ are positive integers such that $$m_{1}>\cdots >m_{r},\ \ n_{1}>\cdots >n_{s},\ \ \
k_{1}\ge \cdots \ge k_{l}, \ \ \ r+s+l\le n.$$
As $w_{0}$ generates $W$, the subspaces $E_{n}$ for $n\in \N$ form an increasing filtration for $W$. It remains to prove the spanning property. For any nonnegative integer $n$, let $E'_{n}$ be the span of the vectors $$a_{1}(-m_{1})\cdots a_{r}(-m_{r})w_{0}$$ for $0\le r\le n,\; a_{1},\dots, a_{r}\in sl_{2},\;
m_{1},\dots,m_{r}\ge 1$. By definition, $E_{n}'\subset E_{n}$ for $n\ge 0$. Using induction (on $k$) and (\[erelation-dy\]) we get $$\begin{aligned}
& &a(m)E'_{k}\subset E'_{k+1}\;\;\;\mbox{ if }m< 0,\\
& &a(m)E'_{k}\subset E'_{k}\;\;\;\mbox{ if }m\ge 0\end{aligned}$$ for any $a\in sl_{2},\; m\in \Z,\; k\in \N$. Using this and induction we have $E_{n}\subset E_{n}'$ for $n\ge 0$. Thus $E_{n}=E_{n}'$ for all $n\ge 0$. For every nonnegative integer $n$, from Lemma \[lvacuum-property\], the subspaces $E_{n}\cap F_{m}$ for $m\in \N$ form an increasing filtration of $E_{n}$. The spanning property of $E_{n}$ follows from this filtration and (\[erelation-dy\]).
The following is a tautological construction of a vacuum module. Let $d$ be the derivation of $T$ such that $$d(a\otimes t^{n})=-n(a\otimes t^{n-1})\ \ \ \mbox{ for }a\in
sl_{2},\; n\in \Z.$$ Set $$T_{+}=\sum_{n\ge 1}T_{n}\;\;\;\mbox{ and }\ J=T\C[d]T_{+}.$$ With $J$ a left ideal of $T$, $T/J$ is a (left) $T$-module and for any $v\in T$, $sl_{2}(n)(v+J)=0$ for $n$ sufficiently large.
From the construction, $(V_{q},{\bf 1})$ is a vacuum $DY_{q}(sl_{2})$-module. As $dJ\subset J$, $V_{q}$ admits an action of $d$ such that $$\begin{aligned}
\label{ed-operator}
d{\bf 1}=0,\ \ \ \ [d,u(x)]=\frac{d}{dx}u(x)\ \ \ \mbox{ for }u\in sl_{2}.\end{aligned}$$ It is clear that for any vacuum $DY_{q}(sl_{2})$-module $(W,w_{0})$ on which $d$ acts such that $$dw_{0}=0\;\;\mbox{ and }\;\; [d,u(x)]=\frac{d}{dx}u(x)
\;\;\;\mbox{ for }u\in sl_{2},$$ there exists a unique $DY_{q}(sl_{2})$-module homomorphism from $V_{q}$ to $W$, sending ${\bf 1}$ to $w_{0}$.
We are going to show that $V_{q}$ has a certain normal basis and there is a canonical quantum vertex algebra structure on $V_{q}$. To show that $V_{q}$ has a certain normal basis, we shall construct a vacuum $DY_{q}(sl_{2})$-module with this property.
We are going to define a vacuum $DY_{q}(sl_{2})$-module structure on the vertex superalgebra $V=V_{L(\g)}$.
There exists a unique element $\Phi(t)\in \Hom (V,
V\otimes \C[t])$ such that $$\begin{aligned}
& &\Phi(t){\bf 1}={\bf 1},\ \ \ \Phi(t)\bar{e}=\bar{e}\otimes t,\ \ \ \
\Phi(t)\bar{f}=\bar{f}\otimes t, \ \ \ \
\Phi(t)\bar{h}= \bar{h}\otimes t^{2},\\
& &\ \ \ \ \ \Phi(t)Y(v,x)=Y(\Phi(t-x)v,x)\Phi(t)\ \ \ \mbox{ for }v\in V.\end{aligned}$$ Furthermore, we have $$\begin{aligned}
& &\Phi(t)\bar{e}(x)=(t-x)\bar{e}(x)\Phi(t), \ \ \
\Phi(t)\bar{f}(x)=(t-x)\bar{f}(x)\Phi(t), \\
& &\Phi(t)\bar{h}(x)=(t-x)^{2}\bar{h}(x)\Phi(t),\end{aligned}$$ and $[\D,\Phi(t)]=\frac{d}{dt}\Phi(t)$, $\Phi(x)\Phi(t)=\Phi(t)\Phi(x)$.
Let us equip $\C[t]$ with the vertex algebra structure for which $1$ is the vacuum vector and $$Y(p(t),x)q(t)=\left(e^{-x(d/dt)}p(t)\right) q(t)
=p(t-x)q(t)$$ for $p(t),q(t)\in \C[t]$. Then equip $V\otimes \C[t]$ with the tensor product vertex superalgebra structure where we denote the vertex operator map by $Y_{ten}$. Thus $$Y_{ten}(u\otimes t^{n},x)=Y(u,x)\otimes (t-x)^{n}
\ \ \ \mbox{ for }u\in V,\; n\in \Z.$$ We have $$\begin{aligned}
& &[Y_{ten}(\bar{e}\otimes t,x_{1}),Y_{ten}(\bar{e}\otimes t,x_{2})]_{+}
=[Y(\bar{e},x_{1}), Y(\bar{e},x_{2})]_{+}\otimes (t-x_{1})(t-x_{2})=0,\\
& &[Y_{ten}(\bar{f}\otimes t,x_{1}),Y_{ten}(\bar{f}\otimes t,x_{2})]_{+}
=[Y(\bar{f},x_{1}), Y(\bar{f},x_{2})]_{+}\otimes (t-x_{1})(t-x_{2})=0,\\
& &[Y_{ten}(\bar{h}\otimes t^{2},x_{1}),Y_{ten}(\bar{e}\otimes t,x_{2})]
=[Y(\bar{h},x_{1}), Y(\bar{e},x_{2})]\otimes (t-x_{1})^{2}(t-x_{2})=0,\\
& &[Y_{ten}(\bar{h}\otimes t^{2},x_{1}),Y_{ten}(\bar{f}\otimes t,x_{2})]
=[Y(\bar{h},x_{1}), Y(\bar{f},x_{2})]\otimes (t-x_{1})^{2}(t-x_{2})=0,\\
& &[Y_{ten}(\bar{e}\otimes t,x_{1}),Y_{ten}(\bar{f}\otimes t,x_{2})]_{+}
=[Y(\bar{e},x_{1}),Y(\bar{f},x_{2})]_{+}\otimes (t-x_{1})(t-x_{2})\\
& &\ \
=x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)
Y(\bar{h},x_{2})\otimes (t-x_{1})(t-x_{2})
=x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)Y(\bar{h},x_{2})\otimes
(t-x_{2})^{2},\\
& &[Y_{ten}(\bar{h}\otimes t^{2},x_{1}),Y_{ten}(\bar{h}\otimes t^{2},x_{2})]
=[Y(\bar{h},x_{1}),Y(\bar{h},x_{2})]\otimes (t-x_{1})^{2}(t-x_{2})^{2}=0.\end{aligned}$$ It follows that there exists a (unique) vertex-superalgebra homomorphism $\theta$ from $V$ to $V\otimes \C[t]$ such that $$\begin{aligned}
\theta (\bar{e})=\bar{e}\otimes t,\ \ \ \
\theta (\bar{f})=\bar{f}\otimes t, \ \ \ \
\theta (\bar{h})= \bar{h}\otimes t^{2}.\end{aligned}$$ Let us alternatively denote by $\Phi(t)$ the vertex superalgebra homomorphism $\theta$ (from $V$ to $V\otimes \C[t]$). Then $\Phi(t){\bf 1}={\bf 1}$, $\Phi(t)(\bar{e})=\bar{e}\otimes t$, $\Phi(t)(\bar{f})=\bar{f}\otimes t$, $\Phi(t)(\bar{h})=\bar{h}\otimes t^{2}$. Furthermore, for $u,v\in
V$, we have $$\Phi(t)Y(u,x)v=\theta(Y(u,x)v)=Y_{ten}(\theta(u),x)\theta(v)
=Y(\Phi(t-x)u,x)\Phi(t)v,$$ where $Y$ is viewed as a $\C[t]$-map. The rest follows immediately.
Let $q$ be any nonzero complex number and let $V=V_{L(\g)}$ be the vertex superalgebra as in Lemma \[lprepare\]. The assignment $$\begin{aligned}
e(x)=\bar{e}(x)\Phi(q+x),\ \ \
f(x)=\bar{f}(x)\Phi(-q+x),\ \ \
h(x)=q \bar{h}(x)\Phi(q+x)\Phi(-q+x)\end{aligned}$$ uniquely defines a vacuum $DY_{q}(sl_{2})$-module structure on $V$ with ${\bf 1}$ as the generating vacuum vector and $$\begin{aligned}
[\D,e(x)]=\frac{d}{dx}e(x),\ \ \ [\D,f(x)]=\frac{d}{dx}f(x),\ \ \
[\D,h(x)]=\frac{d}{dx}h(x).\end{aligned}$$ Furthermore, for $n\in \N$, define $E_{n}$ to be the linear span of the vectors $$u^{(1)}(m_{1})\cdots u^{(r)}(m_{r}){\bf 1}$$ for $0\le r\le n,\; u^{(i)}\in \{ e,f,h\},\; m_{i}\in \Z$. Then $E_{n}$ has a basis consisting of the vectors $$\begin{aligned}
e(-m_{1})\cdots e(-m_{r})f(-n_{1})\cdots f(-n_{s})h(-k_{1})\cdots
h(-k_{l}){\bf 1},\end{aligned}$$ where $r,s,t\ge 0$ and $m_{i},n_{j},k_{t}$ are positive integers such that $$m_{1}>\cdots >m_{r},\ \ n_{1}>\cdots >n_{s},\ \ \
k_{1}\ge \cdots \ge k_{l},\ \ \ r+s+l\le n.$$
Using Lemma \[lprepare\] we have $$\begin{aligned}
e(x_{1})e(x_{2})
&=&\bar{e}(x_{1})\Phi(q+x_{1})\bar{e}(x_{2})\Phi(q+x_{2})\\
&=&\bar{e}(x_{1})\bar{e}(x_{2})\Phi(q+x_{1})\Phi(q+x_{2})(q+x_{1}-x_{2})\\
&=&-\bar{e}(x_{2})\bar{e}(x_{1})\Phi(q+x_{2})\Phi(q+x_{1})(q+x_{1}-x_{2})\\
&=&-\bar{e}(x_{2})\Phi(q+x_{2})\bar{e}(x_{1})
\Phi(q+x_{1})(q+x_{1}-x_{2})(q+x_{2}-x_{1})^{-1}\\
&=&\frac{q+x_{1}-x_{2}}{-q-x_{2}+x_{1}}
e(x_{2})e(x_{1}),\\
f(x_{1})f(x_{2})
&=&\bar{f}(x_{1})\Phi(-q+x_{1})\bar{f}(x_{2})\Phi(-q+x_{2})\\
&=&\bar{f}(x_{1})\bar{f}(x_{2})\Phi(-q+x_{1})\Phi(-q+x_{2})(-q+x_{1}-x_{2})\\
&=&-\bar{f}(x_{2})\bar{f}(x_{1})\Phi(-q+x_{2})\Phi(-q+x_{1})(-q+x_{1}-x_{2})\\
&=&-\bar{f}(x_{2})\Phi(-q+x_{2})\bar{f}(x_{1})
\Phi(-q+x_{1})(-q+x_{1}-x_{2})(-q+x_{2}-x_{1})^{-1}\\
&=&\frac{-q+x_{1}-x_{2}}{q-x_{2}+x_{1}}
f(x_{2})f(x_{1}),\end{aligned}$$ $$\begin{aligned}
& &[e(x_{1}),f(x_{2})]\\
&=&\bar{e}(x_{1})\Phi(q+x_{1})\bar{f}(x_{2})\Phi(-q+x_{2})
-\bar{f}(x_{2})\Phi(-q+x_{2})\bar{e}(x_{1})\Phi(q+x_{1})\\
&=&(q+x_{1}-x_{2})\bar{e}(x_{1})\bar{f}(x_{2})\Phi(q+x_{1})\Phi(-q+x_{2})\\
& &\ \ \ \
-(-q+x_{2}-x_{1})\bar{f}(x_{2})\bar{e}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{2})\\
&=&(q+x_{1}-x_{2})(\bar{e}(x_{1})\bar{f}(x_{2})+\bar{f}(x_{2})\bar{e}(x_{1}))
\Phi(q+x_{1})\Phi(-q+x_{2})\\
&=&(q+x_{1}-x_{2})x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)
\bar{h}(x_{2})
\Phi(q+x_{1})\Phi(-q+x_{2})\\
&=&q x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)\bar{h}(x_{2})
\Phi(q+x_{2})\Phi(-q+x_{2})\\
&=&x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)h(x_{2}),\end{aligned}$$ $$\begin{aligned}
& &[h(x_{1}),h(x_{2})]\\
&=&q^{2}\bar{h}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{1})
\bar{h}(x_{2})\Phi(q+x_{2})\Phi(-q+x_{2})\\
& &\ \ -q^{2}\bar{h}(x_{2})\Phi(q+x_{2})\Phi(-q+x_{2})
\bar{h}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{1})\\
&=&q^{2} (q+x_{1}-x_{2})(-q+x_{1}-x_{2})
\bar{h}(x_{1})\bar{h}(x_{2})\Phi(q+x_{1})\Phi(-q+x_{1})
\Phi(q+x_{2})\Phi(-q+x_{2})\\
& &\ \ -q^{2}(q+x_{2}-x_{1})(-q+x_{2}-x_{1})
\bar{h}(x_{2})\bar{h}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{1})
\Phi(q+x_{2})\Phi(-q+x_{2})\\
&=&q^{2}(xq+_{1}-x_{2})(-q+x_{1}-x_{2}) [\bar{h}(x_{1}),\bar{h}(x_{2})]
\Phi(q+x_{1})\Phi(-q+x_{1})
\Phi(q+x_{2})\Phi(-q+x_{2})\\
&=&0,\end{aligned}$$ $$\begin{aligned}
& &h(x_{1})e(x_{2})\\
&=&q\bar{h}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{1})\bar{e}(x_{2})\Phi(q+x_{2})\\
&=&q \bar{h}(x_{1})\bar{e}(x_{2})
\Phi(q+x_{1})\Phi(-q+x_{1})\Phi(q+x_{2})
(q+x_{1}-x_{2})(-q+x_{1}-x_{2})\\
&=&q \bar{e}(x_{2})\bar{h}(x_{1})\Phi(q+x_{2})\Phi(q+x_{1})\Phi(-q+x_{1})
(q+x_{1}-x_{2})(-q+x_{1}-x_{2})\\
&=&\frac{q (q+x_{1}-x_{2})(-q+x_{1}-x_{2})}{(q+x_{2}-x_{1})^{2}}
\bar{e}(x_{2})\Phi(q+x_{2})\bar{h}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{1})\\
&=&\frac{q+x_{1}-x_{2}}{-q-x_{2}+x_{1}}
e(x_{2})h(x_{1}),\\
& &h(x_{1})f(x_{2})\\
&=&q \bar{h}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{1})\bar{f}(x_{2})\Phi(-q+x_{2})\\
&=&q \bar{h}(x_{1})\bar{f}(x_{2})
\Phi(q+x_{1})\Phi(-q+x_{1})\Phi(-q+x_{2})
(q+x_{1}-x_{2})(-q+x_{1}-x_{2})\\
&=&q \bar{f}(x_{2})\bar{h}(x_{1})\Phi(-q+x_{2})\Phi(q+x_{1})\Phi(-q+x_{1})
(q+x_{1}-x_{2})(-q+x_{1}-x_{2})\\
&=&\frac{q(q+x_{1}-x_{2})(-q+x_{1}-x_{2})}{(-q+x_{2}-x_{1})^{2}}
\bar{f}(x_{2})\Phi(-q+x_{2})\bar{h}(x_{1})\Phi(q+x_{1})\Phi(-q+x_{1})\\
&=&\frac{-q+x_{1}-x_{2}}{q-x_{2}+x_{1}}
f(x_{2})h(x_{1}).\end{aligned}$$ This proves that $V$ becomes a $DY_{q}(sl_{2})$-module. As $\Phi(x){\bf 1}={\bf 1}$, it is clear that ${\bf 1}$ is a vacuum vector for $DY_{q}(sl_{2})$. Now it remains to prove that ${\bf 1}$ generates $V$ as an $DY_{q}(sl_{2})$-module. Let $W$ be the $DY_{q}(sl_{2})$-submodule of $V$ generated by ${\bf 1}$. Using Lemma \[lprepare\] we have $$\Phi(x_{1})e(x)=\Phi(x_{1})(\bar{e}(x)\Phi(q+x))
=\bar{e}(x)\Phi(q+x)(x_{1}-x)\Phi(x_{1})=(x_{1}-x)e(x)\Phi(x_{1}).$$ Similar relations also hold for $\bar{f}(x)$ and $\bar{h}(x)$. As $\Phi(x){\bf 1}={\bf 1}$, by induction we have $\Phi(x)W\subset W((x))$. Then it follows that $W$ is stable under the actions of $\bar{e}(n),\bar{f}(n),\bar{h}(n)$ for $n\in \Z$. Thus $W=V$. This proves that ${\bf 1}$ generates $V$ as a $DY_{q}(sl_{2})$-module and then proves that $V$ is a vacuum $DY_{q}(sl_{2})$-module.
Now we prove the last assertion. With the spanning property having been established in Lemma \[cspanning-property2\] we only need to prove the independence. Recall that $V=\coprod_{n\in \N}V_{(n)}$ is $\N$-graded with $V_{(0)}=\C {\bf 1}$. For $n\in \N$, set $$\bar{F}_{n}=V_{(0)}\oplus V_{(1)}\oplus \cdots \oplus V_{(n)}\subset V.$$ We know that $\Bar{F}_{n}$ has a basis consisting of the vectors $$\begin{aligned}
\bar{e}(-m_{1})\cdots \bar{e}(-m_{r})\bar{f}(-n_{1})\cdots
\bar{f}(-n_{s})\bar{h}(-k_{1})\cdots
\bar{h}(-k_{l}){\bf 1},\end{aligned}$$ where $r,s,t\ge 0$ and $m_{i},n_{j},k_{t}$ are positive integers such that $$m_{1}>\cdots >m_{r},\ \ n_{1}>\cdots >n_{s},\ \ \
k_{1}\ge \cdots \ge k_{l},\ \ \ \sum m_{i}+\sum n_{j}+\sum k_{t}\le n.$$ From the commutation relations in Lemma \[lprepare\], we have $$\begin{aligned}
& &\Phi(t)\bar{e}(m)=(t\bar{e}(m)-\bar{e}(m+1))\Phi(t),\ \ \ \
\Phi(t)\bar{f}(m)=(t\bar{f}(m)-\bar{f}(m+1))\Phi(t),\\
& &\Phi(t)\bar{h}(m)=(t^{2}\bar{h}(m)-2t\bar{h}(m+1)+\bar{h}(m+2))\Phi(t)\end{aligned}$$ for $m\in \Z$. With $\Phi(t){\bf 1}={\bf 1}$, using induction we get $$\Phi(t)w\equiv t^{m} w\ \ \ \mod \; \bar{F}_{n-1}[t]
\ \ \ \mbox{ for }w\in \bar{F}_{n},\; n\ge 0,$$ where $m$ is a nonnegative integer depending on $w$. As $$e(x)=\bar{e}(x)\Phi(q+x)=\sum_{j\ge 0}\frac{1}{j!}x^{j}\bar{e}(x)\Phi^{(j)}(q),$$ for any $m\in \Z$ we have $$e(m)=\sum_{i\ge 0}\frac{1}{i!}\bar{e}(m+i)\Phi^{(i)}(q).$$ For $u\in \{ e,f,h\}$ and for $m\in \Z$, $w\in \bar{F}_{k}$, we have $$u(m)w\equiv \alpha \bar{u}(m)w \;\;\; \mod\; \bar{F}_{k-m-1}$$ for some nonzero complex number $\alpha$. It follows immediately that $E_{n}$ has a basis as claimed.
With $V_{q}$ being universal, from Proposition \[pyangian-double-q\] we immediately have:
For $n\in \N$, let $E_{n}$ be the subspace of $V_{q}$, linearly spanned by the vectors $$u^{(1)}(m_{1})\cdots u^{(r)}(m_{r}){\bf 1}$$ for $0\le r\le n,\; u^{(i)}\in \{ e,f,h\},\; m_{i}\in \Z$. Then the subspaces $E_{n}$ for $n\ge 0$ form an increasing filtration of $V_{q}$ and for each $n\ge 0$, $E_{n}$ has a basis consisting of the vectors $$\begin{aligned}
e(-m_{1})\cdots e(-m_{r})f(-n_{1})\cdots f(-n_{s})h(-k_{1})\cdots
h(-k_{l}){\bf 1},\end{aligned}$$ where $r,s,t\ge 0$ and $m_{i},n_{j},k_{t}$ are positive integers such that $$m_{1}>\cdots >m_{r},\ \ n_{1}>\cdots >n_{s},\ \ \
k_{1}\ge \cdots \ge k_{l},\ \ \ r+s+l\le n.$$
In view of Corollary \[cnormal-basis\], we can and we should consider $sl_{2}$ as a subspace of $V_{q}$ through the map $u\mapsto
u(-1){\bf 1}$ for $u\in sl_{2}$. The following is our main result:
Let $q$ be any nonzero complex number and let $(V_{q},{\bf 1})$ be the universal vacuum $DY_{q}(sl_{2})$-module. There exists one and only one weak quantum vertex algebra structure on $V_{q}$ with ${\bf 1}$ as the vacuum vector such that $$Y(e,x)=e(x),\ \ Y(f,x)=f(x), \ \ \ Y(h,x)=h(x),$$ and the weak quantum vertex algebra $V_{q}$ is nondegenerate. Furthermore, for any $DY_{q}(sl_{2})$-module $W$, there exists one and only one $V_{q}$-module structure $Y_{W}$ on $W$ such that $$Y_{W}(e,x)=e(x),\ \ Y_{W}(f,x)=f(x),
\ \ Y_{W}(h,x)=h(x).$$
We shall follow the procedure outlined in [@li-qva1] and [@li-qva2]. Let $W$ be any $DY_{q}(sl_{2})$-module and let $\overline{W}=V_{q}\oplus W$ be the direct sum module. Set $U=\{ e(x),f(x),h(x)\}\subset \E(\overline{W})$. From the defining relations, $U$ is an $\S$-local subset. By ([@li-qva1], Theorem 5.8), $U$ generates a weak quantum vertex algebra $V_{\overline{W}}$ where the identity operator $1_{\overline{W}}$ is the vacuum vector and $Y_{\E}$ denotes the vertex operator map. Furthermore, the vector space $\overline{W}$ is a faithful $V_{\overline{W}}$-module with $Y_{\overline{W}}(a(x),x_{0})=a(x_{0})$ for $a(x)\in V_{\overline{W}}$. It follows from ([@li-qva1], Proposition 6.7) and the defining relations of $DY_{q}(sl_{2})$ that $V_{\overline{W}}$ is a vacuum $DY_{q}(sl_{2})$-module with $e(x_{0}), f(x_{0}),h(x_{0})$ acting as $Y_{\E}(e(x),x_{0}), Y_{\E}(f(x),x_{0}), Y_{\E}(h(x),x_{0})$. As $V_{q}$ is universal, there exists a $DY_{q}(sl_{2})$-module homomorphism $\psi$ from $V_{q}$ to $V_{\overline{W}}$, sending ${\bf 1}$ to $1_{\overline{W}}$. Since $V_{q}$ is a $DY_{q}(sl_{2})$-submodule of $\overline{W}$, it follows that $\psi$ maps $V_{q}$ into $V_{q}\subset \overline{W}$. Notice that $V_{q}$ as a $DY_{q}(sl_{2})$-module is generated by ${\bf 1}$ and that we have the operator $d$ on $V_{q}$ with the property (\[ed-operator\]). Now we can apply Theorem 6.3 of [@li-qva1], asserting that there exists one and only one weak quantum vertex algebra structure on $V_{q}$ with the required properties. It follows from Theorem 6.5 of [@li-qva1] that $\overline{W}$ is a $V_{q}$-module with $W$ as a submodule.
Now it remains to prove that $V_{q}$ is nondegenerate. For $n\in \N$, define $E_{n}$ to be the linear span of the vectors $$u^{(1)}(m_{1})\cdots u^{(r)}(m_{r}){\bf 1}$$ for $0\le r\le n,\; u^{(i)}\in \{ e,f,h\},\; m_{i}\in \Z$. By Proposition 3.15 of [@li-qva2], the subspaces $E_{n}$ $(n\in
\N)$ form an increasing filtration of $V_{q}$ with $E_{0}=\C {\bf
1}$ such that $a_{k}E_{n}\subset E_{m+n}$ for $a\in E_{m},\; m,n\in
\N,\; k\in \Z$. Denote by $Gr_{E}(V_{q})$ the associated nonlocal vertex algebra. Notice that $e,f,h\in E_{1}$. Let $\hat{e},\hat{f},\hat{h}$ denote the images of $e,f,h$ in $E_{1}/E_{0}\subset Gr_{E}(V_{q})$. Then $\{\hat{e},\hat{f},\hat{h}\}$ is a generating subset of $Gr_{E}(V_{q})$ and we have $$\begin{aligned}
& &\hat{e}(x_{1})\hat{e}(x_{2})
=\frac{q+x_{1}-x_{2}}{-q+x_{1}-x_{2}}\hat{e}(x_{2})\hat{e}(x_{1}),\\
& &\hat{f}(x_{1})\hat{f}(x_{2})
=\frac{-q+x_{1}-x_{2}}{q+x_{1}-x_{2}}\hat{f}(x_{2})\hat{f}(x_{1}),\\
& &\hat{e}(x_{1})\hat{f}(x_{2})=\hat{f}(x_{2})\hat{e}(x_{1}),\\
& &\hat{h}(x_{1})\hat{h}(x_{2})=\hat{h}(x_{2})\hat{h}(x_{1}),\\
& &\hat{h}(x_{1})\hat{e}(x_{2})
=\frac{q+x_{1}-x_{2}}{-q+x_{1}-x_{2}}\hat{e}(x_{2})\hat{h}(x_{1}),\\
& &\hat{h}(x_{1})\hat{f}(x_{2})
=\frac{-q+x_{1}-x_{2}}{q+x_{1}-x_{2}}\hat{f}(x_{2})\hat{h}(x_{1}).\end{aligned}$$ From Corollary \[cnormal-basis\], for each $n\ge 0$, $E_{n+1}/E_{n}$ has a basis consisting of the vectors $$\begin{aligned}
\hat{e}(-m_{1})\cdots \hat{e}(-m_{r})\hat{f}(-n_{1})\cdots
\hat{f}(-n_{s})\hat{h}(-k_{1})\cdots
\hat{h}(-k_{l}){\bf 1},\end{aligned}$$ where $r,s,t\ge 0$ and $m_{i},n_{j},k_{t}$ are positive integers such that $$m_{1}>\cdots >m_{r},\ \ n_{1}>\cdots >n_{s},\ \ \
k_{1}\ge \cdots \ge k_{l},\ \ \ r+s+l=n+1.$$ It was proved in \[KL\] that $Gr_{E}(V_{q})$ is nondegenerate. Then by ([@li-qva2], Proposition 3.14), $V_{q}$ is nondegenerate.
Quasi-compatibility and quasi-modules-at-infinity for nonlocal vertex algebras
==============================================================================
In this section we study quasi-compatible subsets of $\Hom(W,W((x^{-1})))$ for a general vector space $W$ and we show that from any quasi-compatible subset, one can construct a canonical nonlocal vertex algebra. We formulate a notion of quasi module-at-infinity for a nonlocal vertex algebra and we show that the starting vector space $W$ is naturally a quasi module-at-infinity for the nonlocal vertex algebra generated by a quasi-compatible subset. The theory and the results of this section are analogous to those in [@li-qva1].
Let $W$ be any vector space over $\C$, which is fixed throughout this section. Set $$\begin{aligned}
\E^{o}(W)=\Hom (W,W((x^{-1})))\subset (\End W)[[x,x^{-1}]].\end{aligned}$$ Denote by $1_{W}$ the identity operator on $W$, a distinguished element of $\E^{o}(W)$. Note that $\E^{o}(W)$ is naturally a vector space over the field $\C((x^{-1}))$. Let $G$ denote the group of linear transformations on $\C$: $$\begin{aligned}
G=\{g(z)=c_{0}z+c_{1}\;|\; c_{0}\in \C^{\times},\; c_{1}\in \C\}.\end{aligned}$$ Group $G$ acts on $\E^{o}(W)$ with $g\in G$ acting as $R_{g}$ defined by $$\begin{aligned}
R_{g}a(x)=a(g(x))\ \ \ \mbox{ for }a(x)\in \E^{o}(W),\end{aligned}$$ where as a convention $$a(g(x))=\sum_{n\in \Z}a_{n}(c_{0}x+c_{1})^{-n-1}
=\sum_{n\in \Z}\sum_{i\in \N}\binom{-n-1}{i}
c_{0}^{-n-1-i}c_{1}^{i}a_{n}x^{-n-1-i}$$ for $a(x)=\sum_{n\in \Z}a_{n}x^{-n-1}$.
Let $\C(x_{1},x_{2})$ denote the field of rational functions. We have fields $\C((x_{1}^{-1}))((x_{2}^{-1}))$ and $\C((x_{1}^{-1}))((x_{2}))$ of formal series. As these fields contain $\C[x_{1},x_{2}]$ as a subring, there exist unique field-embeddings $$\begin{aligned}
\iota_{x_{1},\infty;x_{2},\infty}:& & \C(x_{1},x_{2})\rightarrow
\C((x_{1}^{-1}))((x_{2}^{-1})),\\
\iota_{x_{1},\infty;x_{2},0}:& & \C(x_{1},x_{2})\rightarrow
\C((x_{1}^{-1}))((x_{2})).\end{aligned}$$ Let $\C(x)$ denote the field of rational functions. Define field-embeddings $$\iota_{x,0}: \C(x)\rightarrow \C((x)),$$ sending $f(x)$ to the formal Laurent series expansion of $f(x)$ at $x=0$, and $$\iota_{x,\infty}: \C(x)\rightarrow \C((x^{-1})),$$ sending $f(x)$ to the formal Laurent series expansion of $f(x)$ at $x=\infty$.
Note that for a quasi-compatible pair $(a(x),b(x))$ in $\E^{o}(W)$, by definition there exists a nonzero polynomial $p(x_{1},x_{2})$ such that $$\begin{aligned}
\label{ecompatible-condition}
p(x_{1},x_{2})a(x_{1})b(x_{2})\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1}))).\end{aligned}$$ As $$\begin{aligned}
& &\iota_{x,\infty;x_{0},0}\left(1/p(x_{0}+x,x)\right)\in
\C((x^{-1}))((x_{0})),\\
& &\left( p(x_{1},x)a(x_{1})b(x)\right)|_{x_{1}=x+x_{0}}
\in (\Hom(W,W((x^{-1}))))[[x_{0}]],\end{aligned}$$ we have $$\iota_{x,\infty;x_{0},0}\left(1/p(x_{0}+x,x)\right)
\left( p(x_{1},x)a(x_{1})b(x)\right)|_{x_{1}=x+x_{0}}
\in (\Hom (W,W((x^{-1}))))((x_{0})).$$
It is easy to show that $Y_{\E^{o}}(a(x),x_{0})b(x)$ is well defined, i.e., the expression on the right hand side does not depend on the choice of polynomial $p(x_{1},x_{2})$.
Write $$\begin{aligned}
Y_{\E^{o}}(a(x),x_{0})b(x)=\sum_{n\in \Z}a(x)_{n}b(x) x_{0}^{-n-1}.\end{aligned}$$
The following is an immediate consequence:
Let $(a(x),b(x))$ be a quasi-compatible pair in $\E^{o}(W)$. Then $a(x)_{n}b(x)\in \E^{o}(W)$ for $n\in \Z$. Furthermore, let $p(x_{1},x_{2})$ be a nonzero polynomial such that $$p(x_{1},x_{2})a(x_{1})b(x_{2})\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1})))$$ and let $k$ be an integer such that $$x_{0}^{k}\iota_{x,\infty;x_{0},0}\left(1/p(x_{0}+x,x)\right)
\in \C((x^{-1}))[[x_{0}]].$$ Then $$\begin{aligned}
a(x)_{n}b(x)=0\;\;\;\mbox{ for }n\ge k.\end{aligned}$$
We shall need the following result:
Let $(a_{i}(x),b_{i}(x))$ $(i=1,\dots,n)$ be quasi-compatible ordered pairs in $\E^{o}(W)$. Suppose that $$\begin{aligned}
\label{esum-lemma-case1}
\sum_{i=1}^{n}g_{i}(z,x)a_{i}(z)b_{i}(x)\in \Hom (W,W((z^{-1},x^{-1})))\end{aligned}$$ for some polynomials $g_{1}(z,x),\dots,g_{n}(z,x)$. Then $$\begin{aligned}
\label{eassoc-sum}
\sum_{i=1}^{n}g_{i}(x+x_{0},x)Y_{\E^{o}}(a_{i}(x),x_{0})b_{i}(x)
=\left(\sum_{i=1}^{n}g_{i}(z,x)a_{i}(z)b_{i}(x)\right)|_{z=x+x_{0}}.\end{aligned}$$
Let $g(z,x)$ be a nonzero polynomial such that $$g(z,x)a_{i}(z)b_{i}(x)\in \Hom (W,W((z^{-1},x^{-1})))
\;\;\;\mbox{ for }i=1,\dots,n.$$ From Definition \[doperation-same-order\], we have $$g(x+x_{0},x)Y_{\E^{o}}(a_{i}(x),x_{0})b_{i}(x)
=\left(g(z,x)a_{i}(z)b_{i}(x)\right)\mid_{z=x+x_{0}}
\;\;\;\mbox{ for }i=1,\dots,n.$$ Then using (\[esum-lemma-case1\]) we have $$\begin{aligned}
& &g(x+x_{0},x)\sum_{i=1}^{n}g_{i}(x+x_{0},x)Y_{\E^{o}}(a_{i}(x),x_{0})b_{i}(x)
\nonumber\\
&=&\sum_{i=1}^{n}g_{i}(x+x_{0},x)\left(g(z,x)a_{i}(z)b_{i}(x)\right)|_{z=x+x_{0}}
\nonumber\\
&=&\left(g(z,x)\sum_{i=1}^{n}g_{i}(z,x)a_{i}(z)b_{i}(x)\right)|_{z=x+x_{0}}
\nonumber\\
&=&g(x+x_{0},x)\left(\sum_{i=1}^{n}g_{i}(z,x)a_{i}(z)b_{i}(x)\right)|_{z=x+x_{0}}.\end{aligned}$$ As both $\sum_{i=1}^{n}g_{i}(x+x_{0},x)
Y_{\E^{o}}(a_{i}(x),x_{0})b_{i}(x)$ and $$\left(\sum_{i=1}^{n}g_{i}(z,x)a_{i}(z)b_{i}(x)\right)|_{z=x+x_{0}}$$ lie in $(\Hom
(W,W((x^{-1}))))((x_{0}))$, from Remark \[rcancellation\], (\[eassoc-sum\]) follows immediately.
A quasi-compatible subspace $U$ of $\E^{o}(W)$ is said to be [*closed*]{} if $$\begin{aligned}
a(x)_{n}b(x)\in U\;\;\;\mbox{ for }a(x),b(x)\in U,\; n\in \Z.\end{aligned}$$ We are going to prove that any closed quasi-compatible subspace containing $1_{W}$ of $\E^{o}(W)$ is a nonlocal vertex algebra. First we prove the following result:
Let $V$ be a closed quasi-compatible subspace of $\E^{o}(W)$. Let $\psi(x),\phi(x),\theta(x)\in V$ and let $f(x,y)$ be a nonzero polynomial such that $$\begin{aligned}
& &f(x,y)\phi(x)\theta(y)\in \Hom (W,W((x^{-1},y^{-1}))),
\label{e4.31}\\
& &f(x,y)f(x,z)f(y,z)
\psi(x)\phi(y)\theta(z)\in \Hom (W,W((x^{-1},y^{-1},z^{-1}))).\label{e4.32}\end{aligned}$$ Then $$\begin{aligned}
& &f(x+x_{1},x)f(x+x_{2},x)f(x+x_{1},x+x_{2})
Y_{\E^{o}}(\psi(x),x_{1})Y_{\E^{o}}(\phi(x),x_{2})\theta(x)\nonumber\\
&=&\left(f(y,x)f(z,x)f(y,z)
\psi(y)\phi (z)\theta(x)\right)|_{y=x+x_{1},z=x+x_{2}}.\end{aligned}$$
With (\[e4.31\]), from Definition \[doperation-same-order\] we have $$\begin{aligned}
\label{ehphi-theta}
f(x+x_{2},x)Y_{\E^{o}}(\phi (x),x_{2})\theta (x)
=\left(f(z,x)\phi(z)\theta(x)\right)|_{z=x+x_{2}},\end{aligned}$$ which gives $$\begin{aligned}
& &f(y,x)f(y,x+x_{2})f(x+x_{2},x)\psi(y)
Y_{\E^{o}}(\phi (x),x_{2})\theta(x)\nonumber\\
&=&\left(f(y,x)f(y,z) f(z,x)
\psi(y)\phi(z)\theta(x)\right)|_{z=x+x_{2}}.\end{aligned}$$ From (\[e4.32\]) the expression on the right-hand side lies in $(\Hom (W,W((y^{-1}, x^{-1}))))[[x_{2}]]$, so does the expression on the left-hand side. That is, $$\begin{aligned}
f(y,x)f(y,x+x_{2})f(x+x_{2},x)\psi(y)
Y_{\E^{o}}(\phi(x),x_{2})\theta(x)
\in (\Hom (W,W((y^{-1},x^{-1}))))[[x_{2}]].\end{aligned}$$ Multiplying by $\iota_{x,\infty;x_{2},0}(f(x+x_{2},x)^{-1})$, which lies in $\C ((x^{-1}))((x_{2}))$, we have $$\begin{aligned}
f(y,x)f(y,x+x_{2})
\psi(y)Y_{\E^{o}}(\phi(x),x_{2})\theta(x)
\in (\Hom (W,W((y^{-1},x^{-1}))))((x_{2})).\end{aligned}$$ In view of Lemma \[lproof-need\], by considering the coefficient of each power of $x_{2}$, we have $$\begin{aligned}
& &f(x+x_{1},x)f(x+x_{1},x+x_{2})
Y_{\E^{o}}(\psi(x),x_{1})Y_{\E^{o}}(\phi(x),x_{2})\theta(x)
\nonumber\\
&=&\left(f(y,x)f(y,x+x_{2})
\psi(y)(Y_{\E^{o}}(\phi(x),x_{2})\theta(x)\right)|_{y=x+x_{1}}.\end{aligned}$$ Using this and (\[ehphi-theta\]) we have $$\begin{aligned}
& &f(x+x_{1},x)f(x+x_{2},x)f(x+x_{1},x+x_{2})
Y_{\E^{o}}(\psi(x),x_{1})Y_{\E^{o}}(\phi(x),x_{2})\theta(x)\nonumber\\
&=&\left(f(y,x)f(x+x_{2},x)f(y,x+x_{2})
\psi(y)Y_{\E^{o}}(\phi (x),x_{2})\theta(x)\right)|_{y=x+x_{1}}\nonumber\\
&=&\left(f(y,x)f(z,x)f(y,z)
\psi(y)\phi (z)\theta(x)\right)|_{y=x+x_{1},z=x+x_{2}},\end{aligned}$$ as desired.
To state our first result we shall need a new notion.
Here we make a new notation for convenience. Let $a(x)=\sum_{n\in \Z}a_{n}x^{-n-1}$ be any formal series (with coefficients $a_{n}$ in some vector space). For any $m\in \Z$, we set $$\begin{aligned}
a(x)_{\ge m}=\sum_{n\ge m}a_{n}x^{-n-1}.\end{aligned}$$ Then for any polynomial $q(x)$ we have $$\begin{aligned}
\label{esimplefactresidule}
\Res_{x}x^{m}q(x)a(x)=\Res_{x}x^{m}q(x)a(x)_{\ge m}.\end{aligned}$$ Now we are in a position to prove our first key result:
Let $V$ be a closed quasi-compatible subspace of $\E^{o}(W)$, containing $1_{W}$. Then $(V,Y_{\E^{o}},1_{W})$ carries the structure of a nonlocal vertex algebra with $W$ as a faithful (left) quasi module-at-infinity where the vertex operator map $Y_{W}$ is given by $Y_{W}(\alpha(x),x_{0})=\alpha(x_{0})$. Furthermore, if $V$ is compatible, $W$ is a $V$-module-at-infinity.
For any $a(x)\in \E^{o}(W)$, as $1_{W}a(x_{2})=a(x_{2})\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1})))$, by definition we have $$\begin{aligned}
& &Y_{\E^{o}}(1_{W},x_{0})a(x)=1_{W}a(x)=a(x),\\
& &Y_{\E^{o}}(a(x),x_{0})1_{W}=(a(x_{1})1_{W})|_{x_{1}=x+x_{0}}=a(x+x_{0})
=e^{x_{0}\frac{d}{dx}}a(x).\end{aligned}$$ For the assertion on the nonlocal vertex algebra structure, it remains to prove the weak associativity, i.e., for $\psi,\phi,\theta\in V$, there exists a nonnegative integer $k$ such that $$\begin{aligned}
\label{eweakassocmainthem}
(x_{0}+x_{2})^{k}Y_{\E^{o}}(\psi,x_{0}+x_{2})Y_{\E^{o}}(\phi,x_{2})\theta
=(x_{0}+x_{2})^{k}Y_{\E^{o}}(Y_{\E^{o}}(\psi,x_{0})\phi,x_{2})\theta.\end{aligned}$$ Let $f(x,y)$ be a nonzero polynomial such that $$\begin{aligned}
& &f(x,y)\psi(x)\phi(y)\in \Hom (W,W((x^{-1},y^{-1}))),\\
& &f(x,y)\phi(x)\theta(y)\in \Hom (W,W((x^{-1},y^{-1}))),\\
& &f(x,y)f(x,z)f(y,z)
\psi(x)\phi(y)\theta(z)\in \Hom (W,W((x^{-1},y^{-1},z^{-1}))).\end{aligned}$$ By Lemma \[lclosed\], we have $$\begin{aligned}
\label{e5.76}
& &f(x+x_{2},x)f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
Y_{\E^{o}}(\psi(x),x_{0}+x_{2})Y_{\E^{o}}(\phi(x),x_{2})\theta(x)
\nonumber\\
& &=\left(f(z,x)f(y,x)f(y,z)
\psi(y)\phi(z)\theta(x)\right)|_{y=x+x_{0}+x_{2},z=x+x_{2}}.\end{aligned}$$
On the other hand, let $n\in \Z$ be [*arbitrarily fixed*]{}. Since $\psi(x)_{m}\phi(x)=0$ for $m$ sufficiently large, there exists a nonzero polynomial $p(x,y)$, depending on $n$, such that $$\begin{aligned}
\label{eges}
p(x+x_{2},x)
(Y_{\E^{o}}(\psi(x)_{m}\phi(x), x_{2})\theta(x)
=\left(p(z,x)(\psi(z)_{m}\phi(z))\theta(x)\right)|_{z=x+x_{2}}\end{aligned}$$ for [*all*]{} $m\ge n$. With $f(x,y)\psi(x)\phi(y)\in \Hom (W,W((x^{-1},y^{-1})))$, from Definition \[doperation-same-order\] we have $$\begin{aligned}
\label{ethis}
f(x_{2}+x_{0},x_{2})(Y_{\E^{o}}(\psi(x_{2}),x_{0})\phi(x_{2}))\theta(x)
=\left(f(y,x_{2})\psi(y)\phi(x_{2})\theta(x)\right)|_{y=x_{2}+x_{0}}.\end{aligned}$$ Using (\[esimplefactresidule\]), (\[eges\]) and (\[ethis\]) we get $$\begin{aligned}
\label{e5.78}
& &\Res_{x_{0}}x_{0}^{n}f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
p(x+x_{2},x)\nonumber\\
& &\ \ \ \ \cdot Y_{\E^{o}}(Y_{\E^{o}}(\psi(x),x_{0})\phi(x), x_{2})\theta(x)\nonumber\\
&=&\Res_{x_{0}}x_{0}^{n}f(x+x_{0}+x_{2},x)
f(x+x_{0}+x_{2},x+x_{2})p(x+x_{2},x)\nonumber\\
& &\ \ \ \ \cdot
Y_{\cal{E}}(Y_{\E^{o}}(\psi(x),x_{0})_{\ge n}\phi(x), x_{2})\theta(x)\nonumber\\
&=&\Res_{x_{0}}x_{0}^{n}
f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})\nonumber\\
& &\ \ \ \ \cdot \left(p(z,x)
Y_{\E^{o}}(\psi(z),x_{0})_{\ge n}\phi(z))\theta(x)\right)|_{z=x+x_{2}}\nonumber\\
&=&\Res_{x_{0}}x_{0}^{n}
f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
\nonumber\\
& &\ \ \ \ \cdot \left(p(z,x)Y_{\E^{o}}(\psi(z),x_{0})\phi(z))\theta(x)\right)|_{z=x+x_{2}}\nonumber\\
&=&\Res_{x_{0}}x_{0}^{n}\left(f(z+x_{0},x)f(z+x_{0},z)
p(z,x)(Y_{\E^{o}}(\psi(z),x_{0})\phi(z))\theta(x)\right)|_{z=x+x_{2}}\nonumber\\
&=&\Res_{x_{0}}x_{0}^{n}\left(f(y,x)f(y,z)
p(z,x)\psi(y)\phi(z)\theta(x)\right)|_{y=z+x_{0},z=x+x_{2}}.\end{aligned}$$ Combining (\[e5.78\]) with (\[e5.76\]) we get $$\begin{aligned}
\label{e5.79}
& &\Res_{x_{0}}x_{0}^{n}
f(x+x_{2},x)f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
\nonumber\\
& &\ \ \ \ \cdot
p(x+x_{2},x)Y_{\E^{o}}(\psi(x),x_{0}+x_{2})
Y_{\E^{o}}(\phi(x),x_{2})\theta (x)\nonumber\\
&=&\Res_{x_{0}}x_{0}^{n}f(x_{2}+x,x)f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
\nonumber\\
& &\ \ \ \ \cdot p(x+x_{2},x)
Y_{\E^{o}}(Y_{\E^{o}}(\psi(x),x_{0})\phi(x), x_{2})\theta(x).\end{aligned}$$ Notice that both sides of (\[e5.79\]) involve only finitely many negative powers of $x_{2}$. In view of Remark \[rcancellation\] we can multiply both sides by $\iota_{x,\infty;x_{2},0}(p(x+x_{2},x)^{-1}f(x+x_{2},x)^{-1})$ (in $\C((x^{-1}))((x_{2}))$ to get $$\begin{aligned}
& &\Res_{x_{0}}x_{0}^{n}f(x+x_{0}+x_{2},x)
f(x+x_{0}+x_{2},x+x_{2})
Y_{\E^{o}}(\psi(x),x_{0}+x_{2})Y_{\E^{o}}(\phi(x),x_{2})\theta(x)\nonumber\\
& &=\Res_{x_{0}}x_{0}^{n}
f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
Y_{\E^{o}}(Y_{\E^{o}}(\psi(x),x_{0})\phi(x), x_{2})\theta(x).\ \ \ \\end{aligned}$$ Since [*$f(x,y)$ does not depend on $n$ and since $n$ is arbitrary*]{}, we have $$\begin{aligned}
\label{enearfinal-new}
& &f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
Y_{\E^{o}}(\psi(x),x_{0}+x_{2})Y_{\E^{o}}(\phi(x),x_{2})\theta(x)\nonumber\\
&=&f(x+x_{0}+x_{2},x)f(x+x_{0}+x_{2},x+x_{2})
Y_{\E^{o}}(Y_{\E^{o}}(\psi(x),x_{0})\phi(x), x_{2})\theta(x).\end{aligned}$$ Write $f(x,y)=(x-y)^{k}g(x,y)$ for some $k\in \N,\; g(x,y)\in \C[x,y]$ with $g(x,x)\ne 0$. Then $$\begin{aligned}
& &f(x+x_{0}+x_{2},x)=(x_{0}+x_{2})^{k}g(x+x_{0}+x_{2},x),\\
& &f(x+x_{0}+x_{2},x+x_{2})=x_{0}^{k}g(x+x_{0}+x_{2},x+x_{2}).\end{aligned}$$ Since $g(x,x)\ne 0$, we have $$\iota_{x,\infty;z,0}g(x+z,x)^{-1}\in \C((x^{-1}))[[z]],$$ so that $$\iota_{x,\infty;z,0}g(x+z,x)^{-1}|_{z=x_{0}+x_{2}},
\ \ \ \iota_{z,\infty;x_{0},0}g(z+x_{0},z)^{-1}|_{z=x+x_{2}}
\in \C((x^{-1}))[[x_{0},x_{2}]].$$ By cancellation, from (\[enearfinal-new\]) we obtain $$\begin{aligned}
& &(x_{0}+x_{2})^{k}Y_{\E^{o}}(\psi(x),x_{0}+x_{2})Y_{\E^{o}}(\phi(x),x_{2})
\theta(x)\\
&=&(x_{0}+x_{2})^{k}
Y_{\E^{o}}(Y_{\E^{o}}(\psi(x),x_{0})\phi(x),x_{2})\theta(x),\end{aligned}$$ as desired. This proves that $(V,Y_{\E^{o}},1_{W})$ carries the structure of a nonlocal vertex algebra.
Next, we prove that $W$ is a quasi module-at-infinity. For $a(x),b(x)\in V$, there exists a nonzero polynomial $h(x,y)$ such that $$h(x,y)a(x)b(y)\in \Hom (W,W((x^{-1},y^{-1}))).$$ Then $$h(x_{1},x_{2})Y_{W}(a(x),x_{1})Y_{W}(b(x),x_{2})
=h(x_{1},x_{2})a(x_{1})b(x_{2})
\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1})))$$ and $$\begin{aligned}
& &h(x_{0}+x_{2},x_{2})Y_{W}(Y_{\E^{o}}(a(x),x_{0})b(x),x_{2})\nonumber\\
&=&h(x_{0}+x_{2},x_{2})(Y_{\E^{o}}(a(x),x_{0})b(x))|_{x=x_{2}}\nonumber\\
&=&\left(h(x_{1},x_{2})
a(x_{1})b(x_{2})\right)|_{x_{1}=x_{2}+x_{0}}\nonumber\\
&=&\left(h(x_{1},x_{2})
Y_{W}(a(x),x_{1})Y_{W}(b(x),x_{2})\right)|_{x_{1}=x_{2}+x_{0}}.\end{aligned}$$ Therefore $W$ is a (left) quasi $V$-module-at-infinity with $Y_{W}(\alpha(x),x_{0})=\alpha(x_{0})$ for $\alpha(x)\in V$. Finally, if $V$ is compatible, the polynomial $h(x,y)$ is of the form $(x-y)^{k}$ with $k\in \N$. Then $W$ is a $V$-module-at-infinity, instead of a quasi $V$-module-at-infinity.
In practice, we are often given an unnecessarily closed quasi-compatible subspace. Next, we are going to show that every quasi-compatible subset is contained in some closed quasi-compatible subspace.
The following is an analogue of a result in [@li-g1] and [@li-qva1]:
Let $\psi_{1}(x),\dots,\psi_{r}(x),
a(x),b(x),\phi_{1}(x), \dots,\phi_{s}(x) \in \E^{o}(W)$. Assume that the ordered sequences $(a(x), b(x))$ and $(\psi_{1}(x),\dots,\psi_{r}(x), a(x),b(x),\phi_{1}(x),
\dots,\phi_{s}(x))$ are quasi-compatible (compatible). Then for any $n\in \Z$, the ordered sequence $$(\psi_{1}(x),\dots,\psi_{r}(x),a(x)_{n}b(x),\phi_{1}(x),\dots,\phi_{s}(x))$$ is quasi-compatible (compatible).
Let $f(x,y)$ be a nonzero polynomial such that $$f(x,y)a(x)b(y)\in \Hom (W,W((x^{-1},y^{-1})))$$ and $$\begin{aligned}
\label{elong-exp}
& &\left(\prod_{1\le i<j\le r}f(y_{i},y_{j})\right)
\left(\prod_{1\le i\le r, 1\le j\le s}f(y_{i},z_{j})\right)
\left(\prod_{1\le i<j\le s}f(z_{i},z_{j})\right)\nonumber\\
& &\;\;\cdot f(x_{1},x_{2})
\left(\prod_{i=1}^{r}f(x_{1},y_{i})f(x_{2},y_{i})\right)
\left(\prod_{i=1}^{s}f(x_{1},z_{i})f(x_{2},z_{i})\right)\nonumber\\
& &\;\;\cdot \psi_{1}(y_{1})\cdots \psi_{r}(y_{r})
a(x_{1})b(x_{2})\phi_{1}(z_{1})\cdots \phi_{s}(z_{s})\nonumber\\
& &\in \Hom (W,W((y_{1}^{-1},\dots, y_{r}^{-1},x_{1}^{-1},x_{2}^{-1},z_{1}^{-1},\dots,z_{s}^{-1}))).\end{aligned}$$ Set $$P=\prod_{1\le i<j\le r}f(y_{i},y_{j}),\;\;\;\;
Q=\prod_{1\le i<j\le s}f(z_{i},z_{j}),\;\;\;\;
R=\prod_{1\le i\le r,\; 1\le j\le s}f(y_{i},z_{j}).$$ Let $n\in \Z$ be [*arbitrarily fixed*]{}. There exists a nonnegative integer $k$ such that $$\begin{aligned}
\label{etruncationpsiphi}
x_{0}^{k+n}\iota_{x_{2},\infty;x_{0},0}\left(f(x_{0}+x_{2},x_{2})^{-1}\right)
\in \C ((x_{2}^{-1}))[[x_{0}]].\end{aligned}$$ Using (\[etruncationpsiphi\]) and Definition \[doperation-same-order\] we obtain $$\begin{aligned}
\label{ecompatibilitythreeproof}
& &\prod_{i=1}^{r}f(x_{2},y_{i})^{k}
\prod_{j=1}^{s}f(x_{2},z_{j})^{k}\nonumber\\
& &\ \ \ \ \cdot \psi_{1}(y_{1})\cdots\psi_{r}(y_{r})
(a(x_{2})_{n}b(x_{2}))\phi_{1}(z_{1})\cdots\phi_{s}(z_{s})\nonumber\\
&=&\Res_{x_{0}}x_{0}^{n}\prod_{i=1}^{r}f(x_{2},y_{i})^{k}
\prod_{j=1}^{s}f(x_{2},z_{j})^{k}\nonumber\\
& &\ \ \ \ \cdot \psi_{1}(y_{1})\cdots \psi_{r}(y_{r})
(Y_{\cal{E}}(a,x_{0})b)(x_{2})\phi_{1}(z_{1})\cdots \phi_{s}(z_{s})
\nonumber\\
&=&\Res_{x_{1}}\Res_{x_{0}}x_{0}^{n}\prod_{i=1}^{r}f(x_{2},y_{i})^{k}
\prod_{j=1}^{s}f(x_{2},z_{j})^{k}\nonumber\\
& &\ \ \ \ \cdot \iota_{x_{2},\infty;x_{0},0}(f(x_{2}+x_{0},x_{2})^{-1})
x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)\nonumber\\
& &\ \ \ \ \cdot \left(f(x_{1},x_{2})\psi_{1}(y_{1})\cdots \psi_{r}(y_{r})
a(x_{1})b(x_{2})\phi_{1}(z_{1})\cdots
\phi_{s}(z_{s})\right)\nonumber\\
&=&\Res_{x_{1}}\Res_{x_{0}}x_{0}^{n}\prod_{i=1}^{r}f(x_{1}-x_{0},y_{i})^{k}
\prod_{j=1}^{s}f(x_{1}-x_{0},z_{j})^{k}\nonumber\\
& &\ \ \ \ \cdot \iota_{x_{2},\infty;x_{0},0}(f(x_{2}+x_{0},x_{2})^{-1})
x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)\nonumber\\
& &\ \ \ \ \cdot \left(f(x_{1},x_{2})\psi_{1}(y_{1})\cdots \psi_{r}(y_{r})
a(x_{1})b(x_{2})\phi_{1}(z_{1})\cdots
\phi_{s}(z_{s})\right)\nonumber\\
&=&\Res_{x_{1}}\Res_{x_{0}}x_{0}^{n}
e^{-x_{0}\frac{\partial}{\partial x_{1}}}
\left(\prod_{i=1}^{r}f(x_{1},y_{i})
\prod_{j=1}^{s}f(x_{1},z_{j})\right)^{k}\nonumber\\
& &\ \ \ \ \cdot \iota_{x_{2},\infty;x_{0},0}(f(x_{2}+x_{0},x_{2})^{-1})
x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)\nonumber\\
& &\ \ \ \ \cdot \left(f(x_{1},x_{2})\psi_{1}(y_{1})\cdots \psi_{r}(y_{r})
a(x_{1})b(x_{2})\phi_{1}(z_{1})\cdots
\phi_{s}(z_{s})\right)\nonumber\\
&=&\Res_{x_{1}}\Res_{x_{0}}\sum_{t=0}^{k-1}\frac{(-1)^{t}}{t!}x_{0}^{n+t}
\left(\frac{\partial}{\partial x_{1}}\right)^{t}
\left(\prod_{i=1}^{r}f(x_{1},y_{i})
\prod_{j=1}^{s}f(x_{1},z_{j})\right)^{k}\nonumber\\
& &\ \ \ \ \cdot \iota_{x_{2},\infty;x_{0},0}(f(x_{2}+x_{0},x_{2})^{-1})
x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)\nonumber\\
& &\ \ \ \ \cdot \left(f(x_{1},x_{2})\psi_{1}(y_{1})\cdots \psi_{r}(y_{r})
a(x_{1})b(x_{2})\phi_{1}(z_{1})\cdots
\phi_{s}(z_{s})\right).\end{aligned}$$ Notice that for any polynomial $B$ and for $0\le t\le k-1$, $\left(\frac{\partial}{\partial x_{1}}\right)^{t}B^{k}$ is a multiple of $B$. Using (\[elong-exp\]) we have $$\begin{aligned}
& &P Q R \prod_{i=1}^{r}f(x_{2},y_{i})
\prod_{j=1}^{s}f(x_{2},z_{j})
\sum_{t=0}^{k-1}\frac{(-1)^{t}}{t!}x_{0}^{n+t}
\left(\frac{\partial}{\partial x_{1}}\right)^{t}
\left(\prod_{i=1}^{r}f(x_{1},y_{i})
\prod_{j=1}^{s}f(x_{1},z_{j})\right)^{k}\nonumber\\
& &\ \ \ \ \cdot \iota_{x_{2},\infty;x_{0},0}(f(x_{2}+x_{0},x_{2})^{-1})
x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)\nonumber\\
& &\ \ \ \ \cdot \left(f(x_{1},x_{2})\psi_{1}(y_{1})\cdots \psi_{r}(y_{r})
a(x_{1})b(x_{2})\phi_{1}(z_{1})\cdots
\phi_{s}(z_{s})\right)\nonumber\\
&\in& \left(\Hom (W,W((y_{1}^{-1},\dots,
y_{r}^{-1},x_{2}^{-1},z_{1}^{-1},\dots,z_{s}^{-1})))\right)
((x_{0}))[[x_{1}^{\pm 1}]].\end{aligned}$$ Then $$\begin{aligned}
& &P Q R\prod_{i=1}^{r}f(x_{2},y_{i})^{k+1}
\prod_{j=1}^{s}f(x_{2},z_{j})^{k+1}
\psi_{1}(y_{1})\cdots\psi_{r}(y_{r})
(a(x)_{n}b(x))(x_{2})\phi_{1}(z_{1})\cdots\phi_{s}(z_{s})\nonumber\\
&\in& \Hom (W,W((y_{1}^{-1},\dots,
y_{r}^{-1},x_{2}^{-1},z_{1}^{-1},\dots,z_{s}^{-1}))).\end{aligned}$$ This proves that $(\psi_{1}(x),\dots,\psi_{r}(x),a(x)_{n}b(x),
\phi_{1}(x), \dots,\phi_{s}(x))$ is quasi-compatible. It is clear that the assertion with compatibility holds.
The following is our second key result:
Every maximal quasi-compatible subspace of $\E^{o}(W)$ is closed and contains $1_{W}$. Furthermore, for any quasi-compatible subset $S$, there exists a (unique) smallest closed quasi-compatible subspace $\<S\>$ containing $S$ and $1_{W}$, and $(\<S\>,Y_{\E^{o}},1_{W})$ carries the structure of a nonlocal vertex algebra with $W$ as a faithful (left) quasi module-at-infinity where the vertex operator map $Y_{W}$ is given by $Y_{W}(\psi(x),x_{0})=\psi(x_{0})$. If $S$ is compatible, then $W$ is a module-at-infinity for $\<S\>$.
Let $K$ be any maximal quasi-compatible subspace of $\E^{o}(W)$. Clearly, $K+\C 1_{W}$ is quasi-compatible. With $K$ maximal we must have $1_{W}\in K$. Let $a(x),b(x)\in K,\;n\in \Z$. It follows from Proposition \[pgeneratingcomplicatedone\] and an induction that any finite sequence in $K\cap \{ a(x)_{n}b(x)\}$ is quasi-compatible. Again, with $K$ maximal we must have $a(x)_{n}b(x)\in K$. This proves that $K$ is closed. The rest assertions follow immediately from Theorem \[tclosed\].
Recall that $\C(x)$ denotes the field of rational functions and $\iota_{x,0}$ and $\iota_{x,\infty}$ are the field embeddings of $\C(x)$ into $\C((x))$ and $\C((x^{-1}))$, respectively.
Let $V$ be a nonlocal vertex algebra generated by a quasi-compatible subset of $\E^{o}(W)$. Suppose that the following relation holds $$\begin{aligned}
& &(x_{1}-x_{2})^{k}p(x_{1},x_{2})a(x_{1})b(x_{2})\nonumber\\
&=&(x_{1}-x_{2})^{k}p(x_{1},x_{2})
\sum_{i=1}^{r}\iota_{x,\infty}(q_{i})(x_{1}-x_{2})
u_{i}(x_{2})v_{i}(x_{1}),\end{aligned}$$ where $a(x),b(x),u_{i}(x),v_{i}(x)\in V$ and $p(x,y)\in \C[x,y],\; q_{i}(x)\in \C(x),\; k\in \N$ with $p(x,x)\ne 0$. Then there exists a nonnegative integer $k'$ such that $$\begin{aligned}
\label{epqsum}
& &(x_{1}-x_{2})^{k'}Y_{\E^{o}}(a(x),x_{1})Y_{\E^{o}}(b(x),x_{2})
\nonumber\\
&=&(x_{1}-x_{2})^{k'}\sum_{i=1}^{r}\iota_{x,0}(q_{i})(-x_{2}+x_{1})
Y_{\E^{o}}(u_{i}(x),x_{2})Y_{\E^{o}}(v_{i}(x),x_{1}).\end{aligned}$$
Let $\theta(x)\in V$. By Lemma \[lclosed\], there exists a nonzero polynomial $f(x,y)$ such that $$\begin{aligned}
& &f(x+x_{1},x)f(x+x_{1},x+x_{2})f(x+x_{2},x)
Y_{\E^{o}}(a(x),x_{1})Y_{\E^{o}}(b(x),x_{2})\theta (x)\nonumber\\
&=&\left(f(y,x)f(y,z)f(z,x)a(y)b(z)\theta (x)\right)|_{y=x+x_{1},z=x+x_{2}}\end{aligned}$$ and such that $$\begin{aligned}
& &f(x+x_{1},x)f(x+x_{1},x+x_{2})f(x+x_{2},x)
Y_{\E^{o}}(u_{i}(x),x_{1})Y_{\E^{o}}(v_{i}(x),x_{2})\theta (x)\nonumber\\
&=&\left(f(y,x)f(y,z) f(z,x)
u_{i}(y)v_{i}(z)\theta (x)\right)|_{y=x+x_{1},z=x+x_{2}}\end{aligned}$$ for $i=1,\dots,r$. Let $0\ne g(x)\in \C[x]$ such that $g(x)q_{i}(x)\in \C[x]$ for $i=1,\dots,r$. Then $$\begin{aligned}
& &f(x+x_{1},x)f(x+x_{1},x+x_{2})f(x+x_{2},x)
(x_{1}-x_{2})^{k}p(x+x_{1},x+x_{2})\nonumber\\
& &\ \ \ \ \cdot g(x_{1}-x_{2})
Y_{\E^{o}}(a(x),x_{1})Y_{\E^{o}}(b(x),x_{2})\theta (x)\nonumber\\
&=&(x_{1}-x_{2})^{k}
\left(f(y,x)f(y,z)f(z,x)g(y-z)p(y,z)a(y)b(z)\theta
(x)\right)|_{y=x+x_{1},z=x+x_{2}}\nonumber\\
&=&(x_{1}-x_{2})^{k}
\left(f(y,x)f(y,z)f(z,x)p(y,z)\sum_{i=1}^{r}(gq_{i})(y-z)
u_{i}(y)v_{i}(z)\theta (x)\right)|_{y=x+x_{1},z=x+x_{2}}
\nonumber\\
&=&f(x+x_{1},x)f(x+x_{1},x+x_{2})f(x+x_{2},x)
(x_{1}-x_{2})^{k}p(x+x_{1},x+x_{2})\nonumber\\
& &\ \ \ \ \cdot\sum_{i=1}^{r}(gq_{i})(x_{1}-x_{2})
Y_{\E^{o}}(u_{i}(x),x_{2})Y_{\E^{o}}(v_{i}(x),x_{1})\theta
(x)\nonumber\\
&=&f(x+x_{1},x)f(x+x_{1},x+x_{2})f(x+x_{2},x)
(x_{1}-x_{2})^{k}p(x+x_{1},x+x_{2})\nonumber\\
& &\ \ \ \ g(x_{1}-x_{2})
\cdot\sum_{i=1}^{r}\iota_{x,0}(q_{i})(-x_{2}+x_{1}))
Y_{\E^{o}}(u_{i}(x),x_{2})Y_{\E^{o}}(v_{i}(x),x_{1})\theta.\end{aligned}$$ Notice that we can multiply both sides by $\iota_{x,\infty;x_{1},0}f(x+x_{1},x)^{-1}
\iota_{x,\infty;x_{2},0}f(x+x_{2},x)^{-1}$ to cancel the factors $f(x+x_{1},x)$ and $f(x+x_{2},x)$ (recall Remark \[rcancellation\]). Since $p(x,x)\ne 0$, we can also cancel the factor $p(x+x_{1},x+x_{2})$. By cancelation we get $$\begin{aligned}
& &(x_{1}-x_{2})^{k}f(x+x_{1},x+x_{2})g(x_{1}-x_{2})
Y_{\E^{o}}(a(x),x_{1})Y_{\E^{o}}(b(x),x_{2})\theta (x)\nonumber\\
&=&(x_{1}-x_{2})^{k}f(x+x_{1},x+x_{2})g(x_{1}-x_{2})\\
& &\ \ \cdot \sum_{i=1}^{r}\iota_{x,0}(q_{i})(-x_{2}+x_{1}))
Y_{\E^{o}}(u_{i}(x),x_{2})Y_{\E^{o}}(v_{i}(x),x_{1})\theta (x).\ \ \
\\end{aligned}$$ Write $g(x)=x^{l}\bar{g}(x)$, where $l\ge 0,\; \bar{g}(x)\in \C[x]$ with $\bar{g}(0)\ne 0$. Similarly, write $f(x,z)=(x-z)^{t}\bar{f}(x,z)$, where $t\ge 0,\; \bar{f}(x,z)\in
\C[x,z]$ with $\bar{f}(x,x)\ne 0$. By a further cancelation we get (\[epqsum\]) with $k'=k+l+t$.
Recall that $G$ denotes the group of linear transformations on $\C$.
Let $\Gamma$ be a group of linear transformations and let $V$ be a vertex algebra generated by a quasi-compatible subset of $\E^{o}(W)$. Assume that $$R_{g}a(x)\;(=a(g(x)))\in V\;\;\;\mbox{ for } g\in \Gamma,\; a(x)\in V.$$ Then $$\begin{aligned}
\label{einvariance}
Y_{\E^{o}}(R_{g}a(x),x_{0})R_{g}b(x)=R_{g}Y_{\E^{o}}(a(x),g_{0}x_{0})b(x)\end{aligned}$$ for $g\in \Gamma,\; a(x),b(x)\in V$, where $g(x)=g_{0}x+g_{1}$.
Let $g\in \Gamma,\;a(x),b(x)\in V$. There exists a nonzero polynomial $p(x_{1},x_{2})$ such that $$p(x_{1},x_{2})a(x_{1})b(x_{2})\in
\Hom(W,W((x_{1}^{-1},x_{2}^{-1}))).$$ Then $$\begin{aligned}
p(x+g_{0}x_{0},x)Y_{\E^{o}}(a(x),g_{0}x_{0})b(x)
=\left(p(x_{1},x)a(x_{1})b(x)\right)|_{x_{1}=x+g_{0}x_{0}}.\end{aligned}$$ Substituting $x$ with $g(x)$ we get $$\begin{aligned}
&&p(g(x)+g_{0}x_{0},g(x))R_{g(x)}
\left(Y_{\E^{o}}(a(x),g_{0}x_{0})b(x)\right)\\
& =&
\left(p(x_{1},g(x))a(x_{1})b(g(x))
\right)|_{x_{1}=g(x)+g_{0}x_{0}=g(x+x_{0})}\\
&=&\left(p(g(x_{1}),g(x))a(g(x_{1})b(x)\right)|_{x_{1}=x+x_{0}}.\end{aligned}$$ We also have $$p(g(x_{1}),g(x_{2}))a(g(x_{1}))b(g(x_{2}))\in
\Hom(W,W((x_{1}^{-1},x_{2}^{-1}))),$$ so that $$p(g(x)+g_{0}x_{0},g(x))Y_{\E^{o}}(a(g(x)),x_{0})b(g(x))
=\left(p(g(x_{1}),g(x))a(g(x_{1}))b(g(x))\right)|_{x_{1}=x+x_{0}}.$$ Consequently, $$p(g(x)+g_{0}x_{0},g(x))Y_{\E^{o}}(a(g(x)),x_{0})b(g(x))
=p(g(x)+g_{0}x_{0},g(x))R_{g(x)}Y_{\E}^{o}(a(x),g_{0}x_{0})b(x).$$ By cancelation, we obtain (\[einvariance\]).
The following is an analogue of ([@li-gamma], Proposition 4.3):
Let $a(x),b(x),c(x)\in \E^{o}(W)$. Assume that $$\begin{aligned}
&&f(x_{1},x_{2})a(x_{1})b(x_{2})=f(x_{1},x_{2})b(x_{2})a(x_{1}),
\label{efab}\\
&&g(x_{1},x_{2})a(x_{1})c(x_{2})=\tilde{g}(x_{1},x_{2})c(x_{2})a(x_{1}),
\label{egca}\\
&&h(x_{1},x_{2})b(x_{1})c(x_{2})=\tilde{h}(x_{1},x_{2})c(x_{2})b(x_{1}),
\label{ehcb}\end{aligned}$$ where $f(x,y), g(x,y),\tilde{g}(x,y), h(x,y), \tilde{h}(x,y)$ are nonzero polynomials. Then for any $n\in \Z$, there exists $k\in \N$, depending on $n$, such that $$\begin{aligned}
\label{einduction-locality}
f(x_{3},x)^{k}g(x_{3},x)a(x_{3})(b(x)_{n}c(x))
=f(x_{3},x)^{k}\tilde{g}(x_{3},x)(b(x)_{n}c(x))a(x_{3}).\end{aligned}$$
Let $n\in \Z$ be arbitrarily fixed. Let $k$ be a nonnegative integer such that $$x_{0}^{k+n}\iota_{x,\infty;x_{0},0}(h(x+x_{0},x)^{-1})
\in \C((x^{-1}))[[x_{0}]].$$ In the proof of Proposition 4.3 of [@li-gamma], take $\alpha=1$ and replace $\iota_{x,x_{0}}$ with $\iota_{x,\infty'x_{0},0}$. Then the same arguments prove (\[einduction-locality\]).
Associative algebra $DY_{q}^{\infty}(sl_{2})$ and modules-at-infinity for quantum vertex algebras
=================================================================================================
In this section we continue to study $\S$-local subsets of $\E^{o}(W)$ for a vector space $W$ and we prove that any $\S$-local subset generates a weak quantum vertex algebra with $W$ as a canonical module-at-infinity. We introduce another version $DY_{q}^{\infty}(sl_{2})$ of the double Yangian and we prove that every $DY_{q}^{\infty}(sl_{2})$-module $W$ is naturally a module-at-infinity for the quantum vertex algebra $V_{q}$ which was constructed in Section 2.
First we prove a simple result that we shall need later:
Let $V$ be a nonlocal vertex algebra and let $(W,Y_{W})$ be a (left) quasi $V$-module-at-infinity. Then $$\begin{aligned}
\label{e3.1}
Y_{W}(\D v,x)=\frac{d}{dx}Y_{W}(v,x)\ \ \ \mbox{ for }v\in V.\end{aligned}$$
For any $v\in V$, by definition there exists a nonzero polynomial $p(x_{1},x_{2})$ such that $$\begin{aligned}
& &p(x_{1},x_{2})Y_{W}(v,x_{1})Y_{W}({\bf 1},x_{2})
\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1}))),\\
& & p(x_{2}+x_{0},x_{2})Y_{W}(Y(v,x_{0}){\bf 1},x_{2})
=\left( p(x_{1},x_{2})Y_{W}(v,x_{1})Y_{W}({\bf
1},x_{2})\right)|_{x_{1}=x_{2}+x_{0}}.\end{aligned}$$ With $Y(v,x_{0}){\bf 1}=e^{x_{0}\D}v$ and $Y_{W}({\bf
1},x_{2})=1_{W}$, we get $$\begin{aligned}
p(x_{2}+x_{0},x_{2})Y_{W}(e^{x_{0}\D}v,x_{2})
&=&\left( p(x_{1},x_{2})Y_{W}(v,x_{1})\right)|_{x_{1}=x_{2}+x_{0}}\\
&=&p(x_{2}+x_{0},x_{2})Y_{W}(v,x_{2}+x_{0}).\end{aligned}$$ As both $Y_{W}(e^{x_{0}\D}v,x_{2})$ and $Y_{W}(v,x_{2}+x_{0})$ lie in $(\Hom (W,W((x_{2}^{-1}))))[[x_{0}]]$, in view of Remark \[rcancellation\] we have $$\begin{aligned}
Y_{W}(e^{x_{0}\D}v,x_{2})=Y_{W}(v,x_{2}+x_{0})
=e^{x_{0}\frac{d}{dx_{2}}}Y_{W}(v,x_{2}),\end{aligned}$$ which implies (\[e3.1\]).
The following is straightforward to prove:
Let $W$ be any vector space and let $$\begin{aligned}
& &A(x_{1},x_{2})\in \Hom (W,W((x_{2}^{-1}))((x_{1}^{-1}))),\ \
B(x_{1},x_{2})\in \Hom (W,W((x_{1}^{-1}))((x_{2}^{-1}))),\\
& & \ \ \ \ \ \ \ C(x_{2},x_{0})\in (\Hom
(W,W((x_{2}^{-1})))((x_{0})).\end{aligned}$$ Then $$\begin{aligned}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
A(x_{1},x_{2})-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
B(x_{1},x_{2})\nonumber\\
& &\ \ \ =x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
C(x_{2},x_{0})\end{aligned}$$ if and only if there exist a nonnegative integer $k$ and $$F(x_{1},x_{2})\in \Hom(W,W((x_{1}^{-1},x_{2}^{-1})))$$ such that $$\begin{aligned}
& &(x_{1}-x_{2})^{k}A(x_{1},x_{2})=F(x_{1},x_{2})=(x_{1}-x_{2})^{k}B(x_{1},x_{2}),\\
& &x_{0}^{k}C(x_{2},x_{0})=F(x_{2}+x_{0},x_{2}).\end{aligned}$$
Using Lemma \[lsimple-factx\] we immediately have:
Let $V$ be a nonlocal vertex algebra, let $(W,Y_{W})$ be a $V$-module-at-infinity, and let $$u,v, u^{(i)},v^{(i)}\in V,\; f_{i}(x)\in \C(x)
\ \ (i=1,\dots,r).$$ Then $$\begin{aligned}
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{W}(u,x_{1})Y_{W}(v,x_{2})\nonumber\\
& &\hspace{1cm}
+x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
\sum_{i=1}^{r}\iota_{x,\infty}(f_{i})(-x_{0})
Y_{W}(v_{(i)},x_{2})Y_{W}(u_{(i)},x_{1})\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y_{W}(Y(u,x_{0})v,x_{2})\end{aligned}$$ if and only if there exists a nonnegative integer $k$ such that $$\begin{aligned}
& &(x_{1}-x_{2})^{k}Y_{W}(u,x_{1})Y_{W}(v,x_{2})\\
&=&(x_{1}-x_{2})^{k}
\sum_{i=1}^{r}\iota_{x,\infty}(f_{i})(-x_{1}+x_{2})
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1}).\end{aligned}$$
The following is an analogue of ([@li-qva1], Proposition 6.7):
Let $V$ be a nonlocal vertex algebra, let $(W,Y_{W})$ be a $V$-module-at-infinity, and let $$n\in \Z,\; u,v,
u^{(i)},v^{(i)}\in V,\; f_{i}(x)\in \C(x) \ \ (i=1,\dots,r), \;
c^{(0)},\dots,c^{(s)}\in V.$$ If $$\begin{aligned}
\label{ecross-va}
& &(x_{1}-x_{2})^{n}Y(u,x_{1})Y(v,x_{2})-(-x_{2}+x_{1})^{n}
\sum_{i=1}^{r}\iota_{x,0}(f_{i})(x_{2}-x_{1})
Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})\nonumber\\
&=&\sum_{j=0}^{s} Y(c^{(j)},x_{2})
\frac{1}{j!}\left(\frac{\partial}{\partial x_{2}}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)\end{aligned}$$ on $V$, then $$\begin{aligned}
\label{ecross-module}
& &(x_{1}-x_{2})^{n}
\sum_{i=1}^{r}\iota_{x,\infty}(f_{i})(-x_{1}+x_{2})
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1})\nonumber\\
& &\ \ \ \ \ \ \ -(-x_{2}+x_{1})^{n}Y_{W}(u,x_{1})Y_{W}(v,x_{2})\nonumber\\
&=&\sum_{j=0}^{s} Y_{W}(c^{(j)},x_{2})
\frac{1}{j!}\left(\frac{\partial}{\partial x_{2}}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)\end{aligned}$$ on $W$. If $(W,Y_{W})$ is faithful, the converse is also true.
Let $k$ be a nonnegative integer such that $k>s$ and $n+k\ge 0$. From (\[ecross-va\]) we get $$(x_{1}-x_{2})^{k+n}Y(u,x_{1})Y(v,x_{2})=\sum_{i=1}^{r}(x_{1}-x_{2})^{k+n}
\iota_{x,0}(f_{i})(x_{2}-x_{1})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1}).$$ By Corollary 5.3 in [@li-qva2] (cf. [@ek]) we have the $\S$-skew symmetry $$\begin{aligned}
Y(u,x)v=\sum_{i=1}^{r}\iota_{x,0}(f_{i}(-x))e^{x\D}Y(v^{(i)},-x)u^{(i)}.\end{aligned}$$ We also have the following $\S$-Jacobi identity $$\begin{aligned}
\label{e4.10proof}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y(u,x_{1})Y(v,x_{2})\nonumber\\
& &\hspace{1cm}
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
\sum_{i=1}^{r}\iota_{x_{0},0}(f_{i}(-x_{0}))
Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y(Y(u,x_{0})v,x_{2}).\end{aligned}$$ By taking $\Res_{x_{0}}x_{0}^{n}$ we get $$\begin{aligned}
& &(x_{1}-x_{2})^{n}Y(u,x_{1})Y(v,x_{2})-(-x_{2}+x_{1})^{n}
\sum_{i=1}^{r}\iota_{x,0}(f_{i})(x_{2}-x_{1})
Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})\ \ \ \nonumber\\
& &\ \ \ \ =\sum_{j\ge 0}Y(u_{n+j}v,x_{2})
\frac{1}{j!}\left(\frac{\partial}{\partial x_{2}}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right).\end{aligned}$$ Combining this with (\[ecross-va\]) we obtain $$\begin{aligned}
\label{euv=c}
u_{n+j}v=c^{(j)}\;\;\;\mbox{ for }j=0,\dots,s,\;\;\mbox{ and }\;
u_{n+j}v=0\;\;\;\mbox{ for }j>s.\end{aligned}$$
Let $l$ be a sufficiently large nonnegative integer such that $$\begin{aligned}
& &(x_{1}-x_{2})^{l}Y_{W}(u,x_{1})Y_{W}(v,x_{2})=F(x_{1},x_{2})
\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1}))),\\
& &x_{0}^{l}Y_{W}(Y(u,x_{0})v,x_{2})
=F(x_{2}+x_{0},x_{2})\end{aligned}$$ and $$\begin{aligned}
& &(x_{1}-x_{2})^{l}Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1})=G_{i}(x_{1},x_{2})
\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1}))),\\
& &x_{0}^{l}Y_{W}(Y(v^{(i)},-x_{0})u^{(i)},x_{1})
=G_{i}(x_{1},x_{1}-x_{0})\end{aligned}$$ for all $i=1,\dots,r$.
Let $p(x)$ be a nonzero polynomial such that $$p(x)f_{i}(x)\in \C[x]\ \ \ \mbox{ for all }i=1,\dots,r.$$
Note that by Lemma \[lD-infinity\], we have $Y_{W}(\D
v,x)=(d/dx)Y_{W}(v,x)$ for $v\in V$. Using all of these and the $\S$-skew symmetry we get $$\begin{aligned}
p(-x_{0})F(x_{2}+x_{0},x_{2})
&=&p(-x_{0})x_{0}^{l}Y_{W}(Y(u,x_{0})v,x_{2})\\
&=&\sum_{i=1}^{r}p(-x_{0})f_{i}(-x_{0})x_{0}^{l}
Y_{W}(e^{x_{0}\D}Y(v^{(i)},-x_{0})u^{(i)},x_{2})\\
&=&\sum_{i=1}^{r}p(-x_{0})f_{i}(-x_{0})x_{0}^{l}
Y_{W}(Y(v^{(i)},-x_{0})u^{(i)},x_{2}+x_{0})\\
&=&\sum_{i=1}^{r}p(-x_{0})f_{i}(-x_{0})
\left( G_{i}(x_{1},x_{1}-x_{0})\right)|_{x_{1}=x_{2}+x_{0}}\\
&=&\sum_{i=1}^{r}p(-x_{0})f_{i}(-x_{0})
G_{i}(x_{2}+x_{0},x_{2}),\end{aligned}$$ which implies $$\begin{aligned}
p(x_{2}-x_{1})F(x_{1},x_{2})=\sum_{i=1}^{r}(pf_{i})(x_{2}-x_{1})
G_{i}(x_{1},x_{2}).\end{aligned}$$ Then $$\begin{aligned}
& &p(x_{2}-x_{1})(x_{1}-x_{2})^{l}Y_{W}(u,x_{1})Y_{W}(v,x_{2})\\
&=&p(x_{2}-x_{1})
\sum_{i=1}^{r}(x_{1}-x_{2})^{l}\iota_{x,\infty}(f_{i})(-x_{1}+x_{2})
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1}).\end{aligned}$$ As both $(x_{1}-x_{2})^{l}Y_{W}(u,x_{1})Y_{W}(v,x_{2})$ and $$\sum_{i=1}^{r}(x_{1}-x_{2})^{l}\iota_{x,\infty}(f_{i})(-x_{1}+x_{2})
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1})$$ lie in $\Hom(W,W((x_{2}^{-1}))((x_{1}^{-1})))$, by Remark \[rcancellation\] we get $$\begin{aligned}
& &(x_{1}-x_{2})^{l}Y_{W}(u,x_{1})Y_{W}(v,x_{2})\\
&=&(x_{1}-x_{2})^{l}\sum_{i=1}^{r}\iota_{x,\infty}(f_{i})(-x_{1}+x_{2})
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1}).\end{aligned}$$ Now by Lemma \[lsimple-factx\] we have $$\begin{aligned}
\label{eS-jacobi-module}
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{W}(u,x_{1})Y_{W}(v,x_{2})\nonumber\\
& &\hspace{1cm}
+x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
\sum_{i=1}^{r}\iota_{x_{0},\infty}(f_{i}(-x_{0}))
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1})\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y_{W}(Y(u,x_{0})v,x_{2}).\end{aligned}$$ Using this and (\[euv=c\]) we obtain (\[ecross-module\]).
For the converse, we trace back, assuming that $W$ is faithful and (\[ecross-module\]) holds. Let $k$ be a nonnegative integer such that $k+n\ge 0$ and $k>s$. Then $$\begin{aligned}
\label{e4.14proof}
& &(x_{1}-x_{2})^{k+n}Y_{W}(u,x_{1})Y_{W}(v,x_{2})\\
&=&(x_{1}-x_{2})^{k+n}\sum_{i=1}^{r}\iota_{x,\infty}(f_{i})(-x_{1}+x_{2})
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1}).\end{aligned}$$ Using Lemma \[lsimple-factx\] we get (\[eS-jacobi-module\]). Combining (\[eS-jacobi-module\]) with (\[ecross-module\]) we obtain (\[euv=c\]), using the assumption that $Y_{W}$ is injective. Using (\[e4.14proof\]) we also have $$\begin{aligned}
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{W}(u,x_{1})Y_{W}(v,x_{2})\nonumber\\
& &\hspace{1cm}
+x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
\sum_{i=1}^{r}\iota_{x_{0},\infty}(f_{i}(-x_{0}))
Y_{W}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1})\nonumber\\
&=&x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)
\sum_{i=1}^{r}\iota_{x_{0},\infty}(f_{i}(-x_{0}))
Y_{W}(Y(v^{(i)},-x_{0})u^{(i)},x_{1})\\
&=&x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)
\sum_{i=1}^{r}\iota_{x_{0},\infty}(f_{i}(-x_{0}))
Y_{W}(Y(v^{(i)},-x_{0})u^{(i)},x_{2}+x_{0})\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
\sum_{i=1}^{r}\iota_{x_{0},\infty}(f_{i}(-x_{0}))
Y_{W}(e^{x_{0}\D}Y(v^{(i)},-x_{0})u^{(i)},x_{2}).\end{aligned}$$ Combining this with (\[eS-jacobi-module\]) we get the $\S$-skew symmetry, with which we obtain the $\S$-Jacobi identity (\[e4.10proof\]). Then using (\[euv=c\]) we obtain (\[ecross-va\]).
Let $W$ be a vector space and let $U$ be an $\S$-local subset of $\E^{o}(W)$. Then $U$ is compatible and the nonlocal vertex algebra $\<U\>$ generated by $U$ is a weak quantum vertex algebra with $W$ as a module-at-infinity.
Notice that the relation (\[eslocality-relation\]) implies that $$(x_{1}-x_{2})^{k}a(x_{1})b(x_{2})\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1}))).$$ Thus any ordered pair in $U$ is compatible. As in the proof of Lemma 3.2 of [@li-qva1], using induction we see that any finite sequence in $U$ is compatible. That is, $U$ is compatible. By Theorem \[tmaximal\], $U$ generates a nonlocal vertex algebra $\<U\>$ inside $\E^{o}(W)$ with $W$ as a module-at-infinity. As $U$ is $\S$-local, from Proposition \[pconvert\], the vertex operators $Y_{\E^{o}}(a(x),x_{0})$ for $a(x)\in U$ form an $\S$-local subset of $\<U\>$. Because $U$ generates $\<U\>$, by Lemma 2.7 of [@li-qva2], $\<U\>$ is a weak quantum vertex algebra.
Let $a(x),b(x)\in \E^{o}(W)$. Suppose that there exist $$u^{(i)}(x),v^{(i)}(x)\in \E^{o}(W),\ \ f_{i}(x)\in \C(x) \ (i=1,\dots,r)$$ such that $$\begin{aligned}
(x_{1}-x_{2})^{k}a(x_{1})b(x_{2})= (x_{1}-x_{2})^{k} \sum_{i=1}^{r}
\iota_{x,\infty}(f_{i})(x_{1}-x_{2}) u^{(i)}(x_{2})v^{(i)}(x_{1})\end{aligned}$$ for some nonnegative integer $k$. Then $(a(x),b(x))$ is compatible and $$\begin{aligned}
& &Y_{\E^{o}}(a(x),x_{0})b(x)\nonumber\\
&=& \Res_{x_{1}}x_{0}^{-1}\delta\left(\frac{x-x_{1}}{x_{0}}\right)
a(x_{1})b(x) -x_{0}^{-1}\delta\left(\frac{x_{1}-x}{-x_{0}}\right)
\sum_{i=1}^{r}\iota_{x,\infty}(f_{i})(-x_{0})
u^{(i)}(x)v^{(i)}(x_{1}).\ \ \ \\end{aligned}$$
We have $$(x_{1}-x_{2})^{k}a(x_{1})b(x_{2})\in
\Hom(W,W((x_{1}^{-1},x_{2}^{-1}))),$$ so that $$x_{0}^{k}Y_{\E^{o}(W)}(a(x),x_{0})b(x)
=\left((x_{1}-x_{2})^{k}a(x_{1})b(x_{2})\right)|_{x_{1}=x_{2}+x_{0}}.$$ Then we have the Jacobi identity, then the iterate formula.
Now we come back to double Yangians. Recall that $T$ is the tensor algebra over the space $sl_{2}\otimes \C[t,t^{-1}]$ and $T=\coprod_{n\in \Z}T_{n}$ is $\Z$-graded with $\deg (sl_{2}\otimes
t^{n})=n$ for $n\in \Z$. For $n\in \Z$, set $J[n]=\coprod_{m\le
-n}T_{m}$. This gives a decreasing filtration. Denote by $\tilde{T}$ the completion of $T$ associated with this filtration.
We define a [*$DY_{q}^{\infty}(sl_{2})$-module*]{} to be a $T(sl_{2}\otimes \C[t,t^{-1}])$-module $W$ such that for every $w\in W$, $$\begin{aligned}
sl(n)w=0\ \ \ \mbox{ for $n$ sufficiently small}\end{aligned}$$ and such that all the defining relations for $DY_{q}^{\infty}(sl_{2})$ hold. Then for any $DY_{q}^{\infty}(sl_{2})$-module $W$, the generating functions $e(x),f(x), h(x)$ are elements of $\E^{o}(W)$. Recall from Section 2 the quantum vertex algebra $V_{q}$. Then we have:
Let $q$ be any nonzero complex number and let $W$ be any $DY_{q}^{\infty}(sl_{2})$-module. There exists one and only one structure of a $V_{q}$-module-at-infinity on $W$ with $$Y_{W}(e,x)=e(x),\ \ Y_{W}(f,x)=f(x),\ \ Y_{W}(h,x)=h(x).$$
The uniqueness is clear as $e,f,h$ generate $V_{q}$. The proof for the existence is similar to the proof of Theorem \[tmain-dy-0\]. Set $U=\{ e(x),f(x),h(x)\}\subset \E^{o}(W)$. From the defining relations of $DY_{q}^{\infty}(sl_{2})$, $U$ is an $\S$-local subset. Then, by Theorem \[tqva-main\], $U$ generates a weak quantum vertex algebra $V_{W}$ with $W$ as a faithful module-at-infinity where $Y_{W}(a(x),x_{0})=a(x_{0})$ for $a(x)\in V_{W}$. Using Proposition \[pqva-module-infty\], we see that $V_{W}$ is a $DY_{q}(sl_{2})$-module with $e(x_{0}), f(x_{0}),h(x_{0})$ acting as $Y_{\E^{o}}(e(x),x_{0}), Y_{\E^{o}}(f(x),x_{0}),
Y_{\E^{o}}(h(x),x_{0})$. Clearly, $1_{W}$ is a vacuum vector of $V_{W}$ viewed as a $DY_{q}(sl_{2})$-module. Then $(V_{W},1_{W})$ is a vacuum $DY_{q}(sl_{2})$-module with an operator $\D$ such that $\D (1_{W})=0$ and $$[\D,u(x)]=\frac{d}{dx} u(x)\ \ \ \mbox{ for }u\in sl_{2}.$$ By the universal property of $V_{q}$, there exists a $DY_{q}(sl_{2})$-module homomorphism $\theta$ from $V_{q}$ to $V_{W}$, sending ${\bf 1}$ to $1_{W}$. Since $sl_{2}$ generates $V_{q}$ as a nonlocal vertex algebra, it follows that $\theta$ is a homomorphism of nonlocal vertex algebras. Using $\theta$ we obtain a structure of a $V_{q}$-module-at-infinity on $W$ with the desired property.
Quasi modules-at-infinity for vertex algebras
=============================================
In this section we study quasi-local subsets of $\E^{o}(W)$ for a vector space $W$ and we prove that every quasi-local subset generates a vertex algebra with $W$ as a quasi module-at-infinity. We give a family of examples related to infinite-dimensional Lie algebras of a certain type, including the Lie algebra of pseudo-differential operators on the circle.
First we prove:
Let $V$ be a vertex algebra and let $(W,Y_{W})$ be a quasi module-at-infinity for $V$ viewed as a nonlocal vertex algebra. Then for $u,v\in V$, there exists a nonzero polynomial $p(x_{1},x_{2})$ such that $$\begin{aligned}
\label{epcomm}
p(x_{1},x_{2})Y_{W}(v,x_{2})Y_{W}(u,x_{1})=
p(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2}).\end{aligned}$$ Furthermore, if $(W,Y_{W})$ is a module-at-infinity, then for $u,v\in V$, $$\begin{aligned}
\label{eqjacobi-lemma}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y_{W}(v,x_{2})Y_{W}(u,x_{1})
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{W}(u,x_{1})Y_{W}(v,x_{2})
\nonumber\\
& &\ \ \ \ \ \ =x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y_{W}(Y(u,x_{0})v,x_{2}).\end{aligned}$$
It basically follows from the arguments of [@ll] (Theorem 3.6.3). Let $u,v\in V$. From definition, there exist $$F(x_{1},x_{2}), G(x_{1},x_{2})\in \Hom(W,W((x_{1}^{-1},x_{2}^{-1}))),\ \
0\ne p(x_{1},x_{2})\in \C[x_{1},x_{2}]$$ such that $$p(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})=F(x_{1},x_{2}),\ \
p(x_{1},x_{2})Y_{W}(v,x_{2})Y_{W}(u,x_{1})=G(x_{1},x_{2})$$ and $$\begin{aligned}
& &p(x_{2}+x_{0},x_{2})Y_{W}(Y(u,x_{0})v,x_{2})
=F(x_{2}+x_{0},x_{2})\label{e3.2}\\
& &p(x_{1},x_{1}-x_{0})Y_{W}(Y(v,-x_{0})u,x_{1})
=G(x_{1},x_{1}-x_{0}).\ \ \ \ \label{e3.3}\end{aligned}$$ Using the skew symmetry of the vertex algebra $V$ and Lemma \[lD-infinity\], we have $$Y_{W}(Y(v,-x_{0})u,x_{1})=Y_{W}(e^{-x_{0}\D}Y(u,x_{0})v,x_{1})
=Y_{W}(Y(u,x_{0})v,x_{1}-x_{0}).$$ Now (\[e3.3\]) is rewritten as $$p(x_{1},x_{1}-x_{0})Y_{W}(Y(u,x_{0})v,x_{1}-x_{0})
=G(x_{1},x_{1}-x_{0}),$$ which gives $$p(x_{2}+x_{0},x_{2})Y_{W}(Y(u,x_{0})v,x_{2})=G(x_{2}+x_{0},x_{2}).$$ Combining this with (\[e3.2\]) we get $F(x_{1},x_{2})=G(x_{1},x_{2})$, proving (\[epcomm\]). For the second assertion, the polynomial $p(x_{1},x_{2})$ in the above argument is of the form $(x_{1}-x_{2})^{k}$ for $k\in \N$. Then it follows from Lemma \[lsimple-factx\].
The following is a counterpart of the notion of quasi-local subset in [@li-gamma]:
Specializing Theorem \[tmaximal\] we have:
Let $W$ be a vector space. Every quasi-local subset $U$ of $\E^{o}(W)$ is quasi-compatible and the nonlocal vertex algebra $\<U\>$ generated by $U$ inside $\E^{o}(W)$ is a vertex algebra with $W$ as a (left) quasi module-at-infinity, where $Y_{W}(a(x),x_{0})=a(x_{0})$ for $a(x)\in \<U\>$. If $U$ is local, $W$ is a module-at-infinity.
It is clear that any quasi-local subset $U$ is quasi-compatible. By Theorem \[tmaximal\], $U$ generates a nonlocal vertex algebra $\<U\>$ with $W$ as a quasi module-at-infinity. From Proposition \[pconvert\], $U$ is a local subspace of the nonlocal vertex algebra $\<U\>$ in the sense that the adjoint vertex operators associated to the vectors of $U$ are mutually local. As $U$ generates $\<U\>$ as a nonlocal vertex algebra, from [@li-g1] (Proposition 2.17) $\<U\>$ is a vertex algebra. As locality implies compatibility, the last assertion follows from Theorem \[tmaximal\].
The following notion was due to [@gkk]:
Recall the following notion from [@li-gamodule] (cf. [@li-gamma]):
Note that the projection $\frac{d}{dx}: G\rightarrow \C^{\times}$ is a group homomorphism. Then any group homomorphism $\Phi:
\Gamma\rightarrow G$ gives rise to a group homomorphism $\Phi_{0}=\frac{d}{dx}\circ \Phi:\Gamma\rightarrow \C^{\times}$.
We shall need the following technical result:
Let $V$ be a $\Gamma$-vertex algebra, let $\Phi: \Gamma\rightarrow G$ be a group homomorphism with $\Phi_{0}=\phi$, and let $(W,Y_{W})$ be a quasi module-at-infinity for $V$ viewed as a nonlocal vertex algebra. Assume that $\{
Y_{W}(u,x)\;|\; u\in U\}$ is $\Phi(\Gamma)$-local and $$Y_{W}(R_{g}u,x)=Y_{W}(u,\Phi(g)(x))\ \ \ \mbox{ for }g\in \Gamma,\; u\in U,$$ where $U$ is a $\Gamma$-submodule and a generating subspace of $V$. Then $(W,Y_{W})$ is a quasi $V$-module-at-infinity.
First we prove that for $u,v\in V$, if $Y_{W}(u,x)$ and $Y_{W}(v,x)$ are quasi compatible, then $$\begin{aligned}
\label{eyqumv}
Y_{W}(u_{n}v,x)=Y_{W}(u,x)_{n}Y_{W}(v,x)\ \ \ \mbox{ for }n\in \Z.\end{aligned}$$ Let $p(x_{1},x_{2})$ be a nonzero polynomial such that $$p(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})
\in \Hom (W,W((x_{1}^{-1},x_{2}^{-1}))).$$ We have $$p(x_{0}+x,x)Y_{\E^{o}}(Y_{W}(u,x),x_{0})Y_{W}(v,x)
=\left( p(x_{1},x)Y_{W}(u,x_{1})Y_{W}(v,x)\right)|_{x_{1}=x+x_{0}}.$$ On the other hand, there exists a nonzero polynomial $q(x_{1},x_{2})$ such that $$q(x_{0}+x,x)Y_{W}(Y(u,x_{0})v,x)
=\left( q(x_{1},x)Y_{W}(u,x_{1})Y_{W}(v,x)\right)|_{x_{1}=x+x_{0}}.$$ Then $$p(x_{0}+x)q(x_{0}+x,x)Y_{W}(Y(u,x_{0})v,x)
=p(x_{0}+x)q(x_{0}+x,x)Y_{\E^{o}}(Y_{W}(u,x),x_{0})Y_{W}(v,x).$$ Consequently, we get $$Y_{W}(Y(u,x_{0})v,x)
=Y_{\E^{o}}(Y_{W}(u,x),x_{0})Y_{W}(v,x),$$ proving (\[eyqumv\]). It follows from Proposition \[pgamma-locality-key\] and induction that $\{ Y_{W}(v,x)\;|\; v\in V\}$ is $\Phi(\Gamma)$-local.
Suppose that $Y_{W}(R_{g}u,x)=Y_{W}(u,\Phi(g)(x))$ and $Y_{W}(R_{g}v,x)=Y_{W}(v,\Phi(g)(x))$ for some $g\in \Gamma,\;
u,v\in V$. By suitably choosing a nonzero polynomial $p(x_{1},x_{2})$ we have $$\begin{aligned}
& &p(\phi(g)^{-1}x_{0}+x,x)Y_{W}(R_{g}Y(u,x_{0})v,x)\\
&=&p(\phi(g)^{-1}x_{0}+x,x)Y_{W}(Y(R_{g}u,\phi(g)^{-1}x_{0})R_{g}v,x)\\
&=&\left(p(x_{1},x)Y_{W}(R_{g}u,x_{1})Y_{W}(R_{g}v,x)\right)|_{x_{1}=
x+\phi(g)^{-1}x_{0}}\\
&=&\left(p(x_{1},x)
Y_{W}(u,\Phi(g)(x_{1}))Y_{W}(v,\Phi(g)(x))\right)|_{x_{1}=
x+\phi(g)^{-1}x_{0}}\\
&=&p(\phi(g)^{-1}x_{0}+x,x)Y_{W}(Y(u,x_{0})v,\Phi(g)(x)),\end{aligned}$$ which implies $$Y_{W}(R_{g}Y(u,x_{0})v,x)=Y_{W}(Y(u,x_{0})v,\Phi(g)(x)),$$ noticing that $\Phi(g)(x+\phi(g)^{-1}x_{0})=\Phi(x)+x_{0}$. Then it follows from induction that $$Y_{W}(R_{g}v,x)=Y_{W}(v,\Phi(g)(x))\ \ \ \mbox{ for all }g\in \Gamma,\; v\in
V.$$ Thus $W$ is a quasi $V$-module-at-infinity.
Now we have:
Let $W$ be a vector space, let $\Gamma$ be a subgroup of $G$ (the group of linear transformations on $\C$), and let $S$ be any $\Gamma$-local subset of $\E^{o}(W)$. Set $$\Gamma\cdot S={\rm span}\{ R_{g}a(x)=a(g(x))\;|\;g\in \Gamma,\; a(x)\in
S\}.$$ Then $\Gamma \cdot S$ is $\Gamma$-local and $\<\Gamma\cdot
S\>$ is a $\Gamma$-vertex algebra with group homomorphisms $$R: \Gamma\rightarrow GL(\<\Gamma\cdot S\>) \ \ \mbox{ and }\ \
\phi: \Gamma\rightarrow \C^{\times}$$ defined by $$\begin{aligned}
& &R_{g}(\alpha(x))=\alpha (g(x))\;\;\;\mbox{ for }g\in \Gamma,\;
\alpha(x)\in \<\Gamma\cdot S\>,\\
& &\phi(g(x))=g_{0}\;\;\;\mbox{ for }g(x)=g_{0}x+g_{1}\in \Gamma.\end{aligned}$$ Furthermore, $W$ is a quasi $\<\Gamma\cdot S\>$-module-at-infinity with $\Phi$ being the identity map.
For $g(x),h(x)\in \Gamma$, we have $$g(x_{1})-h(x_{2})
=g_{0}\left(x_{1}-(g_{0}^{-1}h_{0}x_{2}+g_{0}^{-1}(h_{1}-g_{1}))\right)
=g_{0}(x_{1}-(g^{-1}h)(x_{2})).$$ With this, it is clear that $\Gamma \cdot S$ is $\Gamma$-local. By Theorem \[tquasi-local\], $\Gamma\cdot S$ generates a vertex algebra $\<\Gamma\cdot S\>$ inside $\E^{o}(W)$ and $W$ is a quasi module-at-infinity for $\<\Gamma\cdot S\>$. It follows from Lemma \[lconnection\] and induction that $\<\Gamma \cdot S\>$ is $\Gamma$-stable. In view of Lemma \[lconnection\], $\<\Gamma \cdot S\>$ equipped with the action of $\Gamma$ and with the group homomorphism $\phi$ is a $\Gamma$-vertex algebra. Furthermore, for $\alpha(x),\beta(x)\in
\Gamma\cdot S$, as $Y_{W}(\alpha(x),x_{1})=\alpha(x_{1})$ and $Y_{W}(\beta(x),x_{2})=\beta(x_{2})$, $Y_{W}(\alpha(x),x_{1})$ and $Y_{W}(\beta(x),x_{2})$ are $\Gamma$-local. For $g\in \Gamma,\;
\alpha(x)\in \<\Gamma\cdot S\>$, we have $$Y_{W}(R_{g}\alpha(x),x_{0})=Y_{W}(\alpha(g(x)),x_{0})=\alpha(g(x_{0}))
=Y_{W}(\alpha(x),g(x_{0})).$$ It follows from Lemma \[lva-gamma-quasi-module\] that $W$ is a quasi module-at-infinity for $\<\Gamma \cdot S\>$ viewed as a $\Gamma$-vertex algebra.
We shall need the following result:
Let $W$ be a vector space and let $a(x),b(x)\in \E^{o}(W)$. Suppose that $$\begin{aligned}
\label{eab-bracket}
-[a(x_{1}),b(x_{2})]=\sum_{i=1}^{k}\sum_{j=0}^{r}\Psi_{i,j}(x_{2})
\frac{1}{j!}\left(\frac{\partial}{\partial x_{2}}\right)^{j}
x_{1}^{-1}\delta\left(\frac{\beta_{i}x_{2}}{x_{1}}\right),\end{aligned}$$ where $\beta_{1},\dots,\beta_{k}$ are distinct nonzero complex numbers with $\beta_{1}=1$ and $\Psi_{i,j}(x)\in \E^{o}(W)$. Then $(a(x),b(x))$ is quasi local and $a(x)_{n}b(x)=0$ for $n>r$ and $a(x)_{n}b(x)=\Psi_{1,n}(x)$ for $0\le n\le r$.
Set $p(x,z)=(x-\beta_{1}z)^{r+1}\cdots
(x-\beta_{k}z)^{r+1}$ and $q(x,z)=(x-\beta_{2})^{r+1}\cdots (x-\beta_{k}z)^{r+1}$. From (\[eab-bracket\]) we have $p(x_{1},x_{2})[a(x_{1}),b(x_{2})]=0$. Thus $(a(x),b(x))$ is quasi local. Furthermore, we have $$\begin{aligned}
p(x+x_{0},x)Y_{\E^{o}}(a(x),x_{0})b(x)
=\left(p(x_{1},x)a(x_{1})b(x)\right)|_{x_{1}=x+x_{0}}.\end{aligned}$$ In view of Lemma \[lsimple-factx\] we have $$\begin{aligned}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x}{x_{0}}\right)
p(x_{1},x)b(x)a(x_{1})-
x_{0}^{-1}\delta\left(\frac{x-x_{1}}{-x_{0}}\right)
p(x_{1},x)a(x_{1})b(x)\\
& &\ \ \ \ =x_{1}^{-1}\delta\left(\frac{x+x_{0}}{x_{1}}\right)
p(x_{1},x)Y_{\E^{o}}(a(x),x_{0})b(x).\end{aligned}$$ Multiplying both sides by $x_{0}^{-r-1}$ and using delta-function substitution we get $$\begin{aligned}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x}{x_{0}}\right)
q(x_{1},x)b(x)a(x_{1})-
x_{0}^{-1}\delta\left(\frac{x-x_{1}}{-x_{0}}\right)
q(x_{1},x)a(x_{1})b(x)\\
& &\ \ \ \ =x_{1}^{-1}\delta\left(\frac{x+x_{0}}{x_{1}}\right)
q(x_{1},x)Y_{\E^{o}}(a(x),x_{0})b(x).\end{aligned}$$ Taking $\Res_{x_{0}}$ we get $$\begin{aligned}
\label{efirst}
-q(x_{1},x)[a(x_{1}),b(x)]&=&
\Res_{x_{0}}x_{1}^{-1}\delta\left(\frac{x+x_{0}}{x_{1}}\right)
q(x_{1},x)Y_{\E^{o}}(a(x),x_{0})b(x)\nonumber\\
&=&\sum_{j\ge 0}q(x_{1},x)a(x)_{j}b(x)
\frac{1}{j!}\left(\frac{\partial}{\partial x}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x}{x_{1}}\right).\end{aligned}$$ On the other hand, from (\[eab-bracket\]) we have $$\begin{aligned}
\label{esecond}
-q(x_{1},x)[a(x_{1}),b(x)]=\sum_{j=0}^{r}q(x_{1},x)\Psi_{1,j}(x)
\frac{1}{j!}\left(\frac{\partial}{\partial x}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x}{x_{1}}\right).\end{aligned}$$ Assume $$\sum_{j=0}^{s}q(x_{1},x)A_{j}(x)
\frac{1}{j!}\left(\frac{\partial}{\partial x}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x}{x_{1}}\right)=0,$$ where $A_{j}(x)\in
\E^{o}(W)$ for $0\le j\le s$. As $q(x,1)$ and $(x-1)$ are relatively prime, we have $1=q(x,1)f(x)+(x-1)^{s+1}g(x)$ for some $f(x),g(x)\in
\C[x]$. Then $$\begin{aligned}
& &\sum_{j=0}^{s}A_{j}(x) \frac{1}{j!}\left(\frac{\partial}{\partial
x}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x}{x_{1}}\right)\\
&=&\sum_{j=0}^{s}A_{j}(x)\left(q(x_{1}/x,1)f(x_{1}/x)+(x_{1}/x-1)^{s+1}g(x_{1}/x)\right)
\frac{1}{j!}\left(\frac{\partial}{\partial x}\right)^{j}
x_{1}^{-1}\delta\left(\frac{x}{x_{1}}\right)
\\ &=&0,\end{aligned}$$ which implies $A_{j}(x)=0$ for $0\le j\le s$. Using this fact, combining (\[efirst\]) with (\[esecond\]) we obtain the desired relations.
Next, we study certain infinite-dimensional Lie algebras including the Lie algebra of pseudo-differential operators on the circle. Let $\g$ be a (possibly infinite-dimensional) Lie algebra equipped with a nondegenerate symmetric invariant bilinear form $\<\cdot,\cdot\>$. Associated with the pair $(\g,\<\cdot,\cdot\>)$, one has an (untwisted) affine Lie algebra $$\hat{\g}=\g\otimes \C[t,t^{-1}]\oplus \C {\bf k},$$ where ${\bf k}$ is central and $$[a\otimes t^{m},b\otimes
t^{n}]=[a,b]\otimes t^{m+n}+m\delta_{m+n,0}\<a,b\>{\bf k}$$ for $a,b\in \g,\; m,n\in \Z$. Defining $\deg (\g\otimes t^{m})=-m$ for $m\in \Z$ and $\deg {\bf k}=0$ makes $\hat{\g}$ a $\Z$-graded Lie algebra. For $a\in \g$, form the generating function $$a(x)=\sum_{n\in \Z}(a\otimes t^{n})x^{-n-1}.$$
We say that a $\hat{\g}$-module $W$ is of [*level*]{} $\ell\in \C$ if ${\bf k}$ acts on $W$ as scalar $\ell$. A [*vacuum vector*]{} in a $\hat{\g}$-module is a nonzero vector $v$ such that $(\g\otimes
\C[t])v=0$ and a [*vacuum $\hat{\g}$-module*]{} is a $\hat{\g}$-module $W$ equipped with a vacuum vector which generates $W$. Let $\ell$ be any complex number. Denote by $\C_{\ell}$ the $1$-dimensional $(\g\otimes \C[t]\oplus \C {\bf k})$-module with $\g\otimes \C[t]$ acting trivially and with ${\bf k}$ acting as scalar $\ell$. Form the induced $\hat{\g}$-module $$V_{\hat{\g}}(\ell,0)=U(\hat{\g})\otimes_{U(\g\otimes \C[t]
\oplus \C {\bf k})} \C_{\ell}.$$ Set ${\bf 1}=1\otimes 1$, which is a vacuum vector. From definition, $V_{\hat{\g}}(\ell,0)$ is a universal vacuum $\hat{\g}$-module of level $\ell$. Identify $\g$ as a subspace of $V_{\hat{\g}}(\ell,0)$ through the linear map $a\rightarrow
a(-1){\bf 1}$. It is now well known (cf. [@fz]) that there exists one and only one vertex-algebra structure on $V_{\hat{\g}}(\ell,0)$ with ${\bf 1}$ as the vacuum vector and with $Y(a,x)=a(x)$ for $a\in
\g$. Defining $\deg {\bf 1}=0$ makes $V_{\hat{\g}}(\ell,0)$ a $\Z$-graded $\hat{\g}$-module and the vertex algebra $V_{\hat{\g}}(\ell,0)$ equipped with this $\Z$-grading is a $\Z$-graded vertex algebra.
Let $\Gamma$ be a subgroup of $\Aut(\g)$, preserving the bilinear form $\<\cdot,\cdot\>$. Each $g\in \Gamma$ canonically lifts to an automorphism of the $\Z$-graded Lie algebra $\hat{\g}$. Then $\Gamma$ acts on the vertex algebra $V_{\hat{\g}}(\ell,0)$ by automorphisms preserving the $\Z$-grading. Let $\phi: \Gamma\rightarrow \C^{\times}$ be a group homomorphism. For $g\in \Gamma$, set $$R_{g}=\phi(g)^{-L(0)}g\in GL(V_{\hat{\g}}(\ell,0)),$$ where $L(0)$ denotes the $\Z$-grading operator. This defines a $\Gamma$-vertex-algebra structure on $V_{\hat{\g}}(\ell,0)$.
Consider the following completion of the $\Z$-graded affine Lie algebra $\hat{\g}$: $$\begin{aligned}
\hat{\g}(\infty)=\g\otimes \C((t^{-1}))\oplus \C {\bf k},\end{aligned}$$ where $$\begin{aligned}
[a\otimes p(t),b\otimes q(t)]=[a,b]\otimes
p(t)q(t)+\Res_{t}p'(t)q(t)\<a,b\>{\bf k}\end{aligned}$$ for $a,b\in \g,\; p(t),q(t)\in \C((t^{-1}))$.
Let $\g$ be a Lie algebra equipped with a nondegenerate symmetric invariant bilinear form $\<\cdot,\cdot\>$ and let $\Gamma$ be a group acting on $\g$ by automorphisms preserving the bilinear form $\<\cdot,\cdot\>$ and satisfying the condition that for any $u,v\in \g$, $$[gu,v]=0\;\;\mbox{ and }\;\; \<gu,v\>=0\;\;\;\mbox{ for all but
finitely many }g\in \Gamma.$$ Let $\Phi: \Gamma\rightarrow G$ be a group homomorphism. Define a new bilinear multiplicative operation $[\cdot,\cdot]_{\Gamma}$ on the vector space $\hat{\g}(\infty)=\g\otimes \C((t^{-1}))\oplus \C {\bf k}$ by $$\begin{aligned}
& &[a\otimes p(t),{\bf k}]_{\Gamma}=0=[{\bf k},a\otimes p(t)]_{\Gamma}, \\
& &[a\otimes p(t),b\otimes q(t)]_{\Gamma}=\sum_{g\in
\Gamma}[ga,b]\otimes p(g(t))q(t)+
\Res_{t}p'(g(t))g'(t)q(t)\<ga,b\>{\bf k}\ \ \ \ \ \\end{aligned}$$ for $a,b\in \g,\; p(t),q(t)\in \C((t^{-1}))$. Then the subspace, linearly spanned by the elements $$ga\otimes p(g(t))-a\otimes p(t)$$ for $g\in \Gamma,\;a\in \g,\;p(t)\in\C((t^{-1}))$, is a two-sided ideal of the nonassociative algebra and the quotient algebra, which we denote by $\hat{\g}(\infty)[\Gamma]$, is a Lie algebra.
Let $\Gamma$ act on the Lie algebra $\hat{\g}(\infty)$ by $$g(a\otimes p(t)+\lambda {\bf k})
=ga\otimes p(g(t))+\lambda {\bf k}\;\;\;\mbox{ for }g\in \Gamma,\;
a\in \g,\; p(t)\in \C((t^{-1})),\; \lambda\in \C.$$ It is straightforward to see that $\Gamma$ acts on $\hat{\g}(\infty)$ by automorphisms. We have $$\begin{aligned}
\sum_{g\in \Gamma}[g(a\otimes p(t)),b\otimes q(t)]
&=&\sum_{g\in \Gamma}[ga\otimes p(g(t)),b\otimes q(t)]\\
&=&\sum_{g\in \Gamma}[ga,b]\otimes p(g(t))q(t)
+\Res_{t}q(t)p'(g(t))g'(t)\<ga,b\>{\bf k}\end{aligned}$$ for $a,b\in \g,\; p(t),q(t)\in \C((t^{-1}))$ and $$g(a\otimes p(t)+\lambda {\bf k})-(a\otimes p(t)+\lambda {\bf
k})=ga\otimes p(g(t))-a\otimes p(t).$$ Now the assertions follow immediately from ([@li-gamodule], Lemma 4.1).
Let $$\pi:\hat{\g}(\infty)\rightarrow \hat{\g}(\infty)[\Gamma]$$ be the natural map. For $a\in \g$, set $$a_{\Gamma}(x)=\sum_{n\in \Z}\pi (a\otimes t^{n}) x^{-n-1}
\in \left(\hat{\g}(\infty)[\Gamma]\right)[[x,x^{-1}]].$$
For $g\in \Gamma,\; a\in \g$, we have $$\begin{aligned}
\label{e5.15}
(ga)_{\Gamma}(x)=\phi(g)a_{\Gamma}(g(x)).\end{aligned}$$ For $a,b\in \g$, we have $$\begin{aligned}
\label{ecomm-liealgebra}
[a_{\Gamma}(x_{1}),b_{\Gamma}(x_{2})]=\sum_{g\in
\Gamma}[ga,b]_{\Gamma}(x_{2})
x_{1}^{-1}\delta\left(\frac{g(x_{2})}{x_{1}}\right) +\<ga,b\>{\bf
k}\frac{\partial}{\partial
x_{2}}x_{1}^{-1}\delta\left(\frac{g(x_{2})}{x_{1}}\right),\end{aligned}$$ where $g(x)=\Phi(g)(x)\in G$.
Let $g(x)=g_{0} x+g_{1}\in G$, where $g_{0}\in \C^{\times},\;
g_{1}\in \C$. Then $g^{-1}(x)=g_{0}^{-1}(x-g_{1})$ and $$\begin{aligned}
x^{-1}\delta\left(\frac{g^{-1}(t)}{x}\right)
=x^{-1}\delta\left(\frac{t-g_{1}}{g_{0}x}\right)
=g_{0} t^{-1}\delta\left(\frac{g_{0} x+g_{1}}{t}\right)
=g_{0} t^{-1}\delta\left(\frac{g(x)}{t}\right).\end{aligned}$$ From definition we have $\pi(ga\otimes t^{n})=\pi(a\otimes (g^{-1}(t))^{n})$ for $g\in \Gamma,\;a\in \g,\; n\in \Z$. Then we get $$\begin{aligned}
(ga)_{\Gamma}(x)&=&\sum_{n\in\Z}\pi(ga\otimes t^{n})x^{-n-1}
= \sum_{n\in\Z}\pi(a\otimes
(g^{-1}(t))^{n})x^{-n-1}\\
&=&\pi\left(a\otimes x^{-1}\delta\left(\frac{g^{-1}(t)}{x}\right)\right)\\
&=&g_{0} \pi\left(a\otimes
t^{-1}\delta\left(\frac{g(x)}{t}\right)\right)\\
&=&g_{0}a_{\Gamma}(g(x)).\end{aligned}$$ proving (\[e5.15\]). As for (\[ecomm-liealgebra\]), notice that $$\sum_{m,n\in \Z}g(t)^{m}t^{n}x_{1}^{-m-1}x_{2}^{-n-1}
=x_{1}^{-1}\delta\left(\frac{g(t)}{x_{1}}\right)
x_{2}^{-1}\delta\left(\frac{t}{x_{2}}\right)
=x_{1}^{-1}\delta\left(\frac{g(x_{2})}{x_{1}}\right)
x_{2}^{-1}\delta\left(\frac{t}{x_{2}}\right)$$ and $$\begin{aligned}
\Res_{t}\sum_{m,n\in \Z}mg(t)^{m-1}g'(t)t^{n}x_{1}^{-m-1}x_{2}^{-n-1}
&=&-\Res_{t}\sum_{m,n\in \Z}ng(t)^{m}t^{n-1}x_{1}^{-m-1}x_{2}^{-n-1}\\
&=&\Res_{t}\frac{\partial}{\partial x_{2}}
x_{1}^{-1}\delta\left(\frac{g(t)}{x_{1}}\right)
t^{-1}\delta\left(\frac{x_{2}}{t}\right)\\
&=&\frac{\partial}{\partial x_{2}}
x_{1}^{-1}\delta\left(\frac{g(x_{2})}{x_{1}}\right).\end{aligned}$$ Now (\[ecomm-liealgebra\]) follows.
Now we are in a position to present our main result of this section, which is an analogue of a theorem of [@li-gamodule]:
Let $\g,\<\cdot,\cdot\>, \Gamma, \Phi$ be given as in Proposition \[ptwisted-affine-comp\] and let $W$ be any $\hat{\g}(\infty)[\Gamma]$-module of level $\ell\in \C$ such that $a_{\Gamma}(x)\in \E^{o}(W)$ for $a\in \g$. Then on $W$ there exists one and only one structure of a quasi module-at-infinity for $V_{\hat{\g^{o}}}(-\ell,0)$ viewed as a $\Gamma$-vertex algebra with $Y_{W}(a,x)=a_{\Gamma}(x)$ for $a\in \g$, where $\g^{o}$ denotes the opposite Lie algebra of $\g$.
Set $$U=\{ a_{\Gamma}(x)\;|\; a\in \g\}\subset \E^{o}(W).$$ For $a,b\in \g$, let $g_{1},\dots,g_{r}\in \Gamma$ such that $[ga,b]=0$ and $\<ga,b\>=0$ for $g\notin \{g_{1},\dots,g_{r}\}$. It follows from (\[ecomm-liealgebra\]) that $$(x_{1}-g_{1}(x_{2}))^{2}\cdots
(x_{1}-g_{r}(x_{2}))^{2}[a_{\Gamma}(x_{1}),b_{\Gamma}(x_{2})]=0.$$ Thus $U$ is a $\Gamma$-local subspace of $\E^{o}(W)$. From (\[e5.15\]), $\Gamma\cdot U=U$. By Theorem \[tmain-2\], $U$ generates a $\Gamma$-vertex algebra $\<U\>$ with $W$ as a quasi module-at-infinity with $Y_{W}(\alpha(x),x_{0})=\alpha(x_{0})$. Combining (\[ecomm-liealgebra\]) with Lemma \[ldecomposition\] we get $$\begin{aligned}
a_{\Gamma}(x)_{0}b_{\Gamma}(x)=-[a,b]_{\Gamma}(x),\;\;
a_{\Gamma}(x)_{1}b_{\Gamma}(x)=-\ell \<a,b\>1_{W}, \;\mbox{ and
}\;a_{\Gamma}(x)_{n}b_{\Gamma}(x)=0\end{aligned}$$ for $n\ge 2$. In view of the universal property (cf. [@pr], [@li-gamodule]) of $V_{\hat{\g^{o}}}(-\ell,0)$, there exists a (unique) vertex-algebra homomorphism from $V_{\hat{\g^{o}}}(-\ell,0)$ to $\<U\>$, sending $a$ to $a_{\Gamma}(x)$ for $a\in \g$. Consequently, $W$ is a quasi module-at-infinity for $V_{\hat{\g^{o}}}(-\ell,0)$ viewed as a vertex algebra. Furthermore, for $g\in \Gamma,\; a\in \g$ we have $$Y_{W}(R_{g}a,x)=\phi(g)^{-1}(ga)_{\Gamma}(x)
=a_{\Gamma}(g(x))=Y_{W}(a,g(x)).$$ As $\g$ generates $V_{\widehat{\g^{o}}}(-\ell,0)$ as a vertex algebra, it follows from Lemma \[lva-gamma-quasi-module\] that $W$ is a quasi module-at-infinity for $V_{\hat{\g^{o}}}(-\ell,0)$ viewed as a $\Gamma$-vertex algebra.
Now, let $\Gamma$ be any abstract group. As in [@li-gamodule], we define an associative algebra $gl_{\Gamma}$ with a $\C$-basis consisting of symbols $E_{\alpha,\beta}$ for $\alpha,\beta\in
\Gamma$ and with $$E_{\alpha,\beta}E_{\mu,\nu}=\delta_{\beta,\mu}E_{\alpha,\nu}
\ \ \ \mbox{ for }\alpha,\beta,\mu,\nu\in \Gamma,$$ and we equip $gl_{\Gamma}$ with a nondegenerate symmetric associative bilinear form defined by $$\<E_{\alpha,\beta},E_{\mu,\nu}\>=\delta_{\alpha,\nu}\delta_{\beta,\mu}
\;\;\;\mbox{ for }\alpha,\beta,\mu,\nu\in \Gamma.$$ Defining $T_{\alpha}\in
GL(gl_{\Gamma})$ for $\alpha\in \Gamma$ by $$T_{\alpha}E_{\mu,\nu}=E_{\alpha \mu,\alpha\nu}
\;\;\;\mbox{ for }\alpha,\beta,\mu,\nu\in \Gamma$$ (cf. [@gkk]) we have a group action of $\Gamma$ on $gl_{\Gamma}$ by automorphisms preserving the bilinear form. We can also view $gl_{\Gamma}$ as a Lie algebra with $\<\cdot,\cdot\>$ an invariant bilinear form. Furthermore, for any $\alpha,\beta,\mu,\nu\in
\Gamma$, we have $$[T_{g}E_{\alpha,\beta},E_{\mu,\nu}]=0\;\;\;\mbox{ and }
\;\;\; \< T_{g}E_{\alpha,\beta},E_{\mu,\nu}\>=0$$ for all but finitely many $g\in \Gamma$. Associated with the pair $(gl_{\Gamma},\<\cdot,\cdot\>)$, we have an (untwisted) affine Lie algebra $\widehat{gl_{\Gamma}}$ and its completion $\widehat{gl_{\Gamma}}(\infty)$.
Let $\Phi: \Gamma\rightarrow G$ be a group homomorphism. For $\alpha\in \Gamma$, we set $$\alpha (x)=\Phi(\alpha)=\alpha_{0}x+\alpha_{1}\in G,$$ where $\alpha_{0},\alpha_{1}\in \C$ with $\alpha_{0}\ne 0$. From Proposition \[ptwisted-affine-comp\] we have a Lie algebra $\widehat{gl_{\Gamma}}(\infty)[\Gamma]$.
For $\alpha,\beta\in \Gamma$, we have $$\begin{aligned}
\label{egeneral-comm}
& &[E_{\alpha,e}(x_{1}),E_{\beta,e}(x_{2})]_{\Gamma}\nonumber\\
&=&\sum_{g\in
\Gamma}[E_{g\alpha,g},E_{\beta,e}](x_{2})
x_{1}^{-1}\delta\left(\frac{g(x_{2})}{x_{1}}\right)
+\<E_{g\alpha,g},E_{\beta,e}\>{\bf k}\frac{\partial}{\partial
x_{2}}x_{1}^{-1}\delta\left(\frac{g(x_{2})}{x_{1}}\right)\nonumber\\
&=&E_{\beta\alpha,e}(x_{2})
x_{1}^{-1}\delta\left(\frac{\beta(x_{2})}{x_{1}}\right)
-E_{\beta,\alpha^{-1}}(x_{2})
x_{1}^{-1}\delta\left(\frac{\alpha^{-1}(x_{2})}{x_{1}}\right)
\nonumber\\
& &\ \ \ \ +\delta_{\alpha\beta,e}{\bf k} \frac{\partial}{\partial
x_{2}}x_{1}^{-1}\delta\left(\frac{\beta(x_{2})}{x_{1}}\right).\end{aligned}$$ For $\alpha\in \Gamma$, denote by $A_{\alpha}(x)$ the image of $E_{\alpha,e}(x)$ in $\left(\widehat{gl_{\Gamma}}(\infty)[\Gamma]\right)[[x,x^{-1}]]$. Note that $$E_{\beta,\alpha^{-1}}(x)=(T_{\alpha^{-1}}E_{\alpha\beta,e})(x)
=\alpha_{0}E_{\alpha\beta,e}(\alpha^{-1}(x)).$$ By (\[egeneral-comm\]) we have $$\begin{aligned}
[A_{\alpha}(x_{1}),A_{\beta}(x_{2})]
&=&A_{\beta\alpha}(x_{2})
x_{1}^{-1}\delta\left(\frac{\beta(x_{2})}{x_{1}}\right)
-\alpha_{0}A_{\alpha\beta}(\alpha^{-1}(x_{2}))
x_{1}^{-1}\delta\left(\frac{\alpha^{-1}(x_{2})}{x_{1}}\right)
\nonumber\\
& &\ \ \ \ \ \ +\delta_{\alpha\beta,e}{\bf k} \frac{\partial}{\partial
x_{2}}x_{1}^{-1}\delta\left(\frac{\beta(x_{2})}{x_{1}}\right).\end{aligned}$$ Let $W$ be any $\widehat{gl_{\Gamma}}(\infty)[\Gamma]$-module of level $\ell\in \C$ such that $E_{\alpha,\beta}(x)\in \E^{o}(W)$ for all $\alpha,\beta\in \Gamma$. In view of Theorem \[tquasi-module-lie\], there exists one and only one quasi module-at-infinity structure on $W$ for the $\Gamma$-vertex algebra $V_{\widehat{gl_{\Gamma}^{o}}}(-\ell,0)$.
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[^1]: Partially supported by NSA grant H98230-05-1-0018 and NSF grant DMS-0600189
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abstract: 'We study the modifications on the metric of an isolated self-gravitating bosonic superconducting cosmic string in a scalar-tensor gravity in the weak-field approximation. These modifications are induced by an arbitrary coupling of a massless scalar field to the usual tensorial field in the gravitational Lagrangian. The metric is derived by means of a matching at the string radius with a most general static and cylindrically symmetric solution of the Einstein-Maxwell-scalar field equations. We show that this metric depends on five parameters which are related to the string’s internal structure and to the solution of the scalar field. We compare our results with those obtained in the framework of General Relativity.'
author:
- |
C. N. Ferreira$^1$[^1] , M. E. X. Guimarães$^2$[^2] and J. A. Helayël-Neto$^1$[^3]\
\
\
\
title: 'Current-Carrying Cosmic Strings in Scalar-Tensor Gravities'
---
Introduction
============
The assumption that gravity may be intermediated by a scalar field (or, more generally, by many scalar fields) in addition to the usual symmmetric rank-2 tensor has considerably revived in the recent years. From the theoretical point of view, they seem to be the most natural alternative to General Relativity. Indeed, most attempts to unify gravity with the other interactions predict the existence of one (or many) scalar(s) field(s) with gravitational-strength couplings. If gravity is essentially scalar-tensorial, there will be direct implications for cosmology and experimental tests of the gravitational interaction (we refer the reader to Damour’s recent account on “Experimental Tests of GR" [@dam]). In particular, any gravitational phenomena will be affected by the variation of the gravitational “constant" $\tilde{G}_{0}$. At sufficiently high energy scales where gravity becomes scalar-tensor in nature [@green], it seems worthwhile to analyse the behaviour of matter in the presence of a scalar-tensorial gravitatinal field, specially those which originated in the early universe, such as cosmic strings. In this context, some a uthors have studied solutions for cosmic strings and domain walls in Brans-Dicke [@rom], in dilaton theory [@greg] and in more general scalar-tensor couplings [@mexg].
On the other hand, topological defects are expected to be formed during phase transitions in the early universe. Among them, cosmic strings have been widely studied in cosmology in connection with structure formation. In 1985, Witten showed that in many field theories cosmic strings behave as superconducting tubes and they may generate enormous currents of order $10^{20} A$ or more [@wit]. This fact has raised interest to current-carrying strings and their eventual explanations to many astrophysical phenomena, such as origin of the primordial magnetic fields [@tan], charged vaccum condensates [@nas] and sources of ultrahigh-energy cosmic rays [@hill], among others.
In ref. [@sen], Sen have considered solutions of a superconducting string in the Brans-Dicke theory. The aim of this paper is to study the implications of a class of more general scalar-tensor gravities for a superconducting, bosonic cosmic string. In particular, we will be interested on the modifications induced on the string metric and their possible observable consequences on the current carried by the string. These modifications come from an arbitrary coupling of a massless scalar field to the tensor field in the gravitational Lagrangian. The action which describes these theories (in the Jordan-Fierz frame) is $${\cal S} = \frac{1}{16\pi} \int d^4x \sqrt{-\tilde{g}}
\left[\tilde{\Phi}\tilde{R} - \frac{\omega(\tilde{\Phi})}{\tilde{\Phi}}
\tilde{g}^{\mu\nu}
\partial_{\mu}\tilde{\Phi}\partial_{\nu}\tilde{\Phi} \right]
+ {\cal S}_{m}[\Psi_m, \tilde{g}_{\mu\nu}] ,$$ where $\tilde{g}_{\mu\nu}$ is the physical metric in this frame, $\tilde{R}$ is the curvature scalar associated to it and ${\cal S}_m$ denotes the action describing the general matter fields $\Psi_m$. These theories are metric, e.g., matter couples minimally and universally to $\tilde{g}_{\mu\nu}$ and not to $\tilde{\Phi}$.
The main purpose of this paper is to study the influence of a scalar-tensorial coupling on the gravitational field of a current-carrying cosmic string described by Witten’s model [@wit]. For this purpose, we need to solve the modified Einstein’s equations having a current-carrying vortex as source of the spacetime. In General Relativity, the gravitational field of superconducting strings has been studied by many authors [@moss; @ams; @babul; @lin; @helli; @pet2]. In particular, the following technics have b een employed to derive the spacetime surrounding superconducting vortex: analytic integration of the Einstein’s equations over the string’s energy-momentum tensor [@moss]; linearization of the Einstein’s equations using distribution’s functions [@lin; @pet2]; numerical integrations of the fields equations (Einstein plus material fields) [@ams], among others.
In this paper, we will make an adaptation of Linet’s method [@lin] to our model. That is, we will solve the linearised (modified) Einstein’s equations using distribution’s functions while taking into account the scalar-tensor feature of gravity. This work is outlined as follows. In section 2, we describe the configuration of a superconducting string in scalar-tensor gravities. In section 3, we start by solving the equations for the exterior region. In the subsection 3.2, we solve the linearised equations by applying Linet’s method, introduced in ref. [@lin]. Then, we match the exterior solution with the internal parameters. In 3.3, we derive the deficit angle associated to the metric found previously. We also compare our results with previous results obtained in the framework of General Relativity. Finally, in section 4, we end with some conclusions and discussions.
Superconducting String Configuration in Scalar-Tensor Gravities
===============================================================
In what follows, we will search for a regular solution of a self-gravitating superconducting vortex in the framework of a scalar-tensor gravity. Hence, the simplest bosonic vortex arises from the action of the Abelian-Higgs $U(1) \times U'(1)$ model containing two pairs of complex scalar and gauge fields $$\begin{aligned}
{\cal S}_m & = & \int d^4x \sqrt{-\tilde{g}} \{
- \frac{1}{2}\tilde{g}^{\mu\nu}D_{\mu}\varphi D_{\nu}\varphi^* - \frac{1}{2}
\tilde{g}^{\mu\nu}D_{\mu}\sigma D_{\nu}\sigma^* \nonumber \\
&& - \frac{1}{16\pi}\tilde{g}^{\mu\nu}\tilde{g}^{\alpha\beta}H_{\mu\alpha}H_{\nu\beta}
- \frac{1}{16\pi} \tilde{g}^{\mu\nu}\tilde{g}^{\alpha\beta}F_{\mu\alpha}
F_{\nu\beta} - V(\mid\varphi\mid , \mid\sigma\mid) \}\end{aligned}$$ with $D_{\mu}\varphi \equiv (\partial_{\mu} + iqC_{\mu})\varphi$, $D_{\mu}\sigma \equiv (\partial_{\mu} + ieA_{\mu})\sigma$ and $F_{\mu\nu}$ and $H_{\mu\nu}$ are the field-strengths associated to the electromagnetic $A_{\mu}$ and gauge $C_{\mu}$ fields, respectively. The potential is “Higgs inspired" and contains appropriate $\varphi-\sigma$ interactions so that there occurs a spontaneous symmetry breaking $$V(\mid\varphi\mid , \mid\sigma\mid) = \frac{\lambda_{\varphi}}{4}
(\mid\varphi\mid^2 - \eta^2)^2 + f\mid\varphi\mid^2\mid\sigma\mid^2 + \frac{\lambda_{\sigma}}{4}
\mid\sigma\mid^4 - \frac{m^2}{2}\mid\sigma\mid^2 ,$$ with positive $\eta , f , \lambda_{\sigma}, \lambda_{\varphi}$ parameters. A vortex configuration arises when the $U(1)$ symmetry associated to the $(\varphi, C_{\mu})$ pair is spontaneously broken. The superconducting feature of this vortex is produced when the pair $(\sigma, A_{\mu})$, associated to the other $U'(1)$ symmetry of this model, is spontaneously broken in the core of the vortex.
We restrict ourselves to contemplate configurations of an isolated and static vortex in the $z$-axis. In a cylindrical coordinate system $(t,r,\theta,z)$, such that $r \geq 0$ and $0 \leq \theta <2\pi$, we make the choice
$$\varphi = R(r)e^{i\theta} \;\; \mbox{and} \;\;
C_{\mu} = \frac{1}{q}[P(r) - 1]
\delta^{\theta}_{\mu} ,$$
in much the same way as we proceed with ordinary (non-conducting) cosmic strings. The functions $R, P$ are functions of $r$ only. We also require that these functions be regular everywhere and that they satisfy the usual boundary conditions for a vortex configuration [@niel]
$$R(0) = 0 \;\; \mbox{and} \;\; P(0) =1$$ $$\lim_{r\rightarrow\infty} R(r)=\eta \;\; \mbox{and}
\lim_{r \rightarrow \infty} P(r) = 0 .$$
The $\sigma$-field is responsible for the bosonic current along the string, and the $A_{\mu}$ is the gauge field which produces an external magnetic field; their configuration are taken in the form
$$\sigma = \sigma(r)e^{i\psi(z)} \;\; \mbox{and} \;\;
A_{\mu} = \frac{1}{e}[A(r)- \frac{\partial{\psi}}{\partial z}]
\delta^{z}_{\mu}$$
The pair $(\sigma,A_{\mu})$ is subjected to the following boundary conditions
$$\frac{d\sigma(0)}{dr}=0 \;\; \mbox{and} \;\; A(0)=\frac{dA(0)}{dr}=0$$ $$\lim_{r\rightarrow \infty}\sigma (r) =0 \;\; \mbox{and} \;\;
\lim_{r \rightarrow \infty}A(r) \neq 0.$$ With this choice, we can see that $\sigma$ breaks electromagnetism inside the string and can form a charged scalar condensate in the string core. Outside the string, the $A_{\mu}$ field has a non-vanishing component along the $z$-axis which indicates that there will be a non-vanishing energy-momentum tensor in the region exterior to the string.
Although action (1) shows explicitly this gravity’s scalar-tensorial character, for technical reasons, we choose to work in the conformal (Einstein) frame in which the kinematic terms of the scalar and the tensor fields do not mix
$${\cal S} = \frac{1}{16\pi G} \int d^4x \sqrt{-g} \left[ R -
2g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi \right]
+ {\cal S}_{m}[\Psi_m,\Omega^2(\phi)g_{\mu\nu}] ,$$
where $g_{\mu\nu}$ is a pure rank-2 tensor in the Einstein frame, $R$ is the curvature scalar associated to it and $\Omega(\phi)$ is an arbitrary function of the scalar field.
Action (8) is obtained from (1) by a conformal transformation $$\tilde{g}_{\mu\nu} = \Omega^2(\phi)g_{\mu\nu} ,$$ and by a redefinition of the quantity $$G\Omega^2(\phi) = \tilde{\Phi}^{-1}$$ which makes evident the feature that any gravitational phenomena will be affected by the variation of the gravitation “constant" $G$ in the scalar-tensorial gravity, and by introducing a new parameter $$\alpha^2 \equiv \left( \frac{\partial \ln \Omega(\phi)}{\partial
\phi} \right)^2 = [2\omega(\tilde{\Phi}) + 3]^{-1} ,$$ which can be interpreted as the (field-dependent) coupling strength between matter and the scalar field. In order to make our calculations as broad as possible, we choose not to specify the factors $\Omega(\phi)$ and $\alpha (\phi)$ (the field-dependent coupling strength between matter and the scalar field), leaving them as arbitrary functions of the scalar field.
In the conformal frame, the Einstein equations are modified. A straightforward calculus shows that the “Einstein" equations are $$\begin{aligned}
G_{\mu\nu} & = & 2\partial_{\mu}\phi\partial_{\nu}\phi -
g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}
\phi + 8\pi G T_{\mu\nu} \nonumber \\
\Box_{g}\phi & = & -4\pi G \alpha(\phi) T .\end{aligned}$$ We note that the last equation brings a new information and shows that the matter distribution behaves as a source for $\phi$ and $g_{\mu\nu}$ as well. The energy-momentum tensor is defined as usual $$T_{\mu\nu} \equiv \frac{2}{\sqrt{-g}}\frac{\delta {\cal S}_m}
{\delta g_{\mu\nu}} ,$$ but in the conformal frame it is no longer conserved $\nabla_{\mu}T^{\mu}_{\nu} = \alpha(\phi)T\nabla_{\nu}\phi$. It is clear from transformation (9) that we can relate quantities from both frames in such a way that $\tilde{T}^{\mu\nu} =
\Omega^{-6}(\phi)T^{\mu\nu}$ and $\tilde{T}^{\mu}_{\nu} =
\Omega^{-4}(\phi)T^{\mu}_{\nu}$.
Guided by the symmetry of the source, we impose that the metric is static and cylindrically symmetric. We choose to work with a general cylindrically symmetric metric written in the form $$ds^2 = e^{2(\gamma -\Psi)}(-dt^2 + dr^2) + \beta^2 e^{-2\Psi}
d\theta^2 + e^{2\Psi}dz^2 ,$$ where the metric functions $\gamma,\Psi,$ and $\beta$ are functions of $r$ only. In addition, the metric functions satisfy the regularity conditions at the axis of symmetry $r=0$ $$\gamma = 0, \;\; \Psi =0, \;\; \frac{d\gamma}{dr}=0, \;\;
\frac{d\Psi}{dr} =0, \;\; \mbox{and} \;\; \frac{d\beta}{dr} =0 .$$
With metric given by expression (12) we are in a position to write the full equations of motion for the self-gravitating superconducting vortex in scalar-tensorial gravity. In the conformal frame, these equations are $$\begin{aligned}
\beta^{''} & = & 8\pi G\beta e^{2(\gamma - \Psi)} [T^{t}_{t} + T^{r}_{r}] \nonumber \\
(\beta\Psi^{'})^{'} & = & 4\pi G\beta e^{2(\gamma - \Psi)} [T^{t}_{t} + T^{r}_{r}
+T^{\theta}_{\theta} - T^{z}_{z}] \nonumber \\
\beta^{'}\gamma^{'} & = & \beta(\Psi^{'})^{2} - \beta(\phi^{'})^{2} +
8\pi G e^{2(\gamma - \Psi)}T^{r}_{r} \nonumber \\
(\beta\phi^{'})^{'} & = & - 4\pi G \alpha(\phi)\beta e^{2(\gamma - \Psi)} T ,\end{aligned}$$ where $(')$ denotes “derivative with respect to r". The non-vanishing components of the energy-momentum tensor (computed using equation (11)) are $$\begin{aligned}
T^{t}_{t} & = & - \frac{1}{2}\Omega^{2}(\phi) \{
e^{2(\Psi - \gamma)}(R'^{2} + \sigma'^{2}) + \frac{e^{2\Psi}}
{\beta^2}R^2P^2 + e^{-2\Psi}\sigma^2A^2 \nonumber \\
& & + \Omega^{-2}(\phi)e^{-2\gamma}(\frac{A'^2}{4\pi e^2}) +
\Omega^{-2}(\phi)\frac{e^{2(2\Psi - \gamma)}}{\beta^2}(\frac{P'^2}
{4\pi q^2}) + 2\Omega^2(\phi)V(R,\sigma) \} \nonumber \\
T^{r}_{r} & = & \frac{1}{2}\Omega^2(\phi) \{ e^{2(\Psi - \gamma)}
(R'^2 + \sigma'^2) - \frac{e^{2\Psi}}{\beta^2} R^2P^2 - e^{-2\Psi}\sigma^2A^2 \nonumber \\
& & + \Omega^{-2}(\phi)e^{-2\gamma}(\frac{A'^2}{4\pi e^2}) + \Omega^{-2}(\phi)
\frac{e^{2(2\Psi - \gamma)}}{\beta^2}(\frac{P'^2}{4\pi q^2}) - 2\Omega^2(\phi)
V(R,\sigma) \} \nonumber \\
T^{\theta}_{\theta} & = & - \frac{1}{2}\Omega^2(\phi) \{ e^{2(\Psi - \gamma)}
(R'^2 + \sigma'^2) - \frac{e^{2\Psi}}{\beta^2} R^2P^2 +
e^{-2\Psi}\sigma^2A^2 \\
& & + \Omega^{-2}(\phi)e^{-2\gamma}(\frac{A'^2}{4\pi e^2}) - \Omega^{-2}(\phi)
\frac{e^{2(2\Psi - \gamma)}}{\beta^2}(\frac{P'^2}{4\pi q^2}) + 2\Omega^2(\phi)
V(R,\sigma) \} \nonumber \\
T^{z}_{z} & = & - \frac{1}{2}\Omega^2(\phi) \{ e^{2(\Psi - \gamma)}
(R'^2 + \sigma^2) + \frac{e^2\Psi}{\beta^2}R^2P^2 - e^{-2\Psi}\sigma^2
A^2 \nonumber \\
& & - \Omega^{-2}(\Phi)e^{-2\gamma}(\frac{A'^2}{4\pi e^2}) + \Omega^{-2}(\phi)
\frac{e^{2(2\Psi -\gamma)}}{\beta^2}(\frac{P'^2}{4\pi q^2}) + 2\Omega^2(\phi)
V(R,\sigma) \nonumber \}\end{aligned}$$
As we said before, the energy-momentum tensor is not conserved in the conformal frame. Instead, the equation $$\nabla_{\mu}T^{\mu}_{\nu} = \alpha(\phi) T \nabla_{\nu}\phi ,$$ where $T$ is the trace of the energy-momentum tensor, gives an additional relation between the scalar field $\phi$ and the source.
In the next section, we will attempt to solve the field equations (14). For the purpose of these calculations, we can divide the space into two regions: an exterior region $r > r_0$, where all the fields drop away rapidly and the only survivor is the magnetic field; and an interior region $r \leq r_0$, where all the string’s field contribute to the energy-momentum tensor. Conveniently, $r_0$ has the same order of magnitude of the string radius. Then, we match the exterior and the interior solutions (to first order in $\tilde{G}_{0} = G\Omega^2(\phi_0)$, where $\phi_0$ is a constant) providing a relationship between the internal parameters of the string and the spacetime geometry.
Superconducting String Solution in Scalar-Tensor Gravities
==========================================================
The Exterior Solution and the Modified Rainich Algebra:
-------------------------------------------------------
In this region, $r > r_0$, the electromagnetic field is the only field which contributes to the energy-momentum tensor. Therefore, the energy-momentum tensor has the form[^4]
$$T^{\mu\nu} = \frac{1}{4\pi} \left[ F^{\mu\alpha}F^{\nu}_{\alpha} -
\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta} \right]$$
with the following algebraic properties
$$T^{\mu}_{\mu} = 0 \;\;\; \mbox{and} \;\;\;
T^{\alpha}_{\nu}T^{\mu}_{\alpha}=\frac{1}{4}\delta^{\mu}_{\nu}(T_{\alpha\beta}T^{\alpha\beta}).$$ which leads expression (15) to take a simple form
$$T^t_t = - T^r_r= T^{\theta}_{\theta}= - T^z_z =
- \frac{1}{2}e^{-2\gamma} \left( \frac{A'^2}{4\pi e^2} \right) .$$
Thus, our problem is reduced to solve the modified Einstein’s equations with source given by (16). That is,
$$\begin{aligned}
\beta'' & = & 0 \nonumber \\
(\beta\Psi')' & = & 4\pi G\beta e^{2(\gamma - \Psi)} [T^t_t - T^z_z] \nonumber \\
\beta'\gamma' & = & 8\pi G\beta e^{2(\gamma - \Psi)} T^r_r +
\beta(\Psi')^2 - \beta (\phi')^2 \nonumber \\
(\beta\phi')' & = & 0 \; . \end{aligned}$$
In General Relativity (i.e, in the absence of the dilaton field), these equations have been previously investigated by many authors [@louis]. A source of the form (16) leads to some algebraic conditions on the curvature scalar and the Ricci tensor, known as the Rainich conditions: $$R \equiv R^t_t + R^r_r + R^{\theta}_{\theta} + R^z_z = 0 ,$$ and $$(R^t_t)^2 = (R^r_r)^2 = (R^{\theta}_{\theta})^2 = (R^z_z)^2 .$$ The two equations above admit three sets of solutions: the magnetic case, the electric case and a third case which can correspond to either a static electric or a static magnetic field aligned along the $z$-axis [@louis]. The superconducting string defined in Witten’s model (2) correponds to the magnetic case: $$R^t_t= R^{\theta}_{\theta} \;\;\; R^{\theta}_{\theta} = R^z_z \;\;
\mbox{and} \;\; R^t_t = - R^r_r .$$
In a scalar-tensor gravity, we notice however that the Rainich conditions are no longer valid because of the very nature of the modified Einstein’s equations (actually an Einstein-Maxwell-dilaton system). Instead of the algebraic conditions stated above, we have now: $$R \equiv R^t_t + R^r_r + R^{\theta}_{\theta} + R^z_z =
2(\phi')^2 e^{2(\Psi - \gamma)} ,$$ and the analogous to the magnetic case in the scalar-tensor gravity is a solution of the form: $$R^t_t = R^{\theta}_{\theta} \;\;\; R^{\theta}_{\theta} = - R^z_z
\;\; \mbox{and} \;\; R^t_t = - R^r_r - 2(\phi')^2 e^{2(\Psi - \gamma)}.$$
We are now in a position to solve the modified Einstein’s equations. The first and last equations in (18) can be solved straightforwardly: $$\begin{aligned}
\beta(r) & = & B r \nonumber \\
\phi(r) & = & l\ln(r/r_0) .\end{aligned}$$ The second and third equations in (18) are solved with the help of the algebraic conditions (20) and (21): $$\gamma'' +\frac{1}{r} \gamma' = 0 ,$$ $$\Psi'' +\frac{1}{r}\Psi' - \Psi'^2 = -\frac{n^2}{r^2} .$$ We, thus, find the remaining metric functions: $$\begin{aligned}
\gamma(r) & = & m^2 \ln(r/r_0) \nonumber \\
\Psi(r) & = & n\ln(r/r_0) - \ln \left[ \frac{(r/r_0)^{2n} + \kappa}{(1+\kappa)} \right] ,\end{aligned}$$ where the constant $n$ is related to $l$ and $m$ through the expression $n^2 = l^2 + m^2$.
Therefore, the exterior metric is given by: $$ds^2 = \left( \frac{r}{r_0} \right)^{-2n} W^2(r) \left[
\left( \frac{r}{r_0}\right)^{2m^2} (-dt^2 +dr^2) + B^2r^2d\theta^2 \right] +
\left( \frac{r}{r_0} \right)^{2n}
\frac{1}{W^2(r)} dz^2 ,$$ where $$W(r) \equiv \frac{(r/r_0)^{2n} + \kappa}{(1+\kappa)} .$$ Besides, the solution for the scalar field $\phi(r)$ in the exterior region is given by equation (22). The integration constants $B,l,n,m$ will be fully determined after the introduction of the matter fields. In the particular case of Brans-Dicke, metric (24) belongs to a class of metrics corresponding to the [*case 1*]{} in Sen’s paper [@sen], with an appropriate adjustment in the parameters.
The Internal Solution and Matching:
-----------------------------------
We start by considering the full modified Einstein’s equations (14) with source (15) in the internal region defined by $ r \leq r_0$. In this region, all fields contribute to the energy-momentum tensor. In what follows, we will consider the solution for t he superconducting string to linear order in $\tilde{G}_{0}$. Therefore, we assume that the metric $g_{\mu\nu}$ and the scalar field $\phi$ can be written as: $$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} ,$$ $$\phi = \phi_0 + \phi_{(1)} ,$$ where $\eta_{\mu\nu} = diag(-,+,+,+)$ is the Minskowski metric tensor and $\phi_0$ is a constant. Thus, our problem reduces to solve the linearised Einstein’s equations[^5] $$\nabla^2 h_{\mu\nu}= - 16\pi G\Omega^2(\phi_0)(T_{\mu\nu}^{(0)} -
\frac{1}{2}\eta_{\mu\nu}T^{(0)}) ,$$ in a harmonic coordinate system such that $(h^{\mu}_{\nu}-\frac{1}{2}\delta^{\mu}_{\nu}h)_{,\nu}=0$. $T^{(0)}_{\mu\nu}$ is the string’s energy-momentum tensor to zeroth-order in $\tilde{G}_{0}=G\Omega^2(\phi_0)$ (evaluated in flat space) and $T^{(0)}$ its trace. Besides, we also need to solve the linearised equation for the scalar field $$\nabla^2 \phi_{(1)}= - 4\pi G\Omega^2(\phi_0)\alpha(\phi_0)T^{(0)}.$$ Then, we proceed with the junction between the internal and external solutions at $r= r_0$, with both solutions evaluated to linear order in $\tilde{G}_{0}$.
While doing these calculations, we will briefly recall the method of linearization using distribution functions (presented in Linet’s paper [@lin] and applied later by Peter and Puy in [@pet2], in the framework of General Relativity).
### The Linearised Field Equations:
First of all, let us evaluate the superconducting string’s energy-momentum tensor to zeroth-order in $\tilde{G}_{0}$ in cartesian coordinates $(t,x,y,z)$. The non-vanishing components of the energy-momentum tensor can now be re-written under the form: $$\begin{aligned}
T^{(0)t}_{t} & = & -\frac{1}{2} \left[ R'^2 + \sigma'^2 +\frac{R^2P^2}{r^2} + \sigma^2A^2
+ (\frac{A'^2}{4\pi e^2}) + (\frac{P'^2}{4\pi q^2}) + 2V \right] \nonumber \\
T^{(0)x}_{x} & = & (\cos^2\theta -\frac{1}{2}) \left[ R'^2 + \sigma'^2 -
\frac{R^2P^2}{r^2} + (\frac{A'^2}{4\pi e^2}) \right]
- \frac{1}{2} \left[ \sigma^2A^2 - (\frac{P'^2}{4\pi q^2}) + 2V \right] \nonumber \\
T^{(0)y}_{y} & = & (\sin^2\theta -\frac{1}{2}) \left[ R'^2 + \sigma'^2 -
\frac{R^2P^2}{r^2} + (\frac{A'^2}{4\pi e^2}) \right]
- \frac{1}{2} \left[ \sigma^2A^2 - (\frac{P'^2}{4\pi q^2}) + 2V \right] \nonumber \\
T^{(0)z}_{z} & = & -\frac{1}{2} \left[ R'^2 + \sigma'^2 +\frac{R^2P^2}{r^2} - \sigma^2A^2
- (\frac{A'^2}{4\pi e^2}) + (\frac{P'^2}{4\pi q^2})^2 + 2V \right] \end{aligned}$$
With the help of the source tensor defined by Thorne [@kip], we can stablish some linear densities which will be very useful in our further analysis. Let $$M^{\mu}_{\nu}(r) \equiv -2\pi \int_0^r T^{\mu}_{\nu}(r') r' dr'$$ be the source tensor. Let us define its components (to zeroth-order in $\tilde{G}_{0}$) as follows. The energy per unit length $U$: $$U \equiv M^t_t = -2 \pi \int_0^{r_0} T^t_t r dr ;$$ the tension per unit length $\tau$: $$\tau \equiv M^z_z = - 2\pi \int_0^{r_0} T^z_z r dr ;$$ and the remaining transversal components as: $$\begin{aligned}
X & \equiv & M^r_r = -2\pi \int_0^{r_0} T^r_r r dr \\
Y & \equiv & M^{\theta}_{\theta} = -2\pi \int_0^{r_0} T^{\theta}_{\theta} r dr .\end{aligned}$$ Now, in terms of the cartesian components of the energy-momentum tensor we can define a quantity $Z$ such that $$Z = -\int r dr d\theta T^x_x = - \int r dr d\theta T^y_y .$$ If we assume that the string is (idealistically) infinetely thin, then its energy-momentum tensor may be described in terms of distribution functions. Namely, $$T^{\mu\nu} = diag (U, -Z, -Z, -\tau) \delta(x)\delta(y) .$$ Equation (27) represents the string’s energy-momentum tensor with all quantities integrated in the internal region $r \leq r_0$, in the cartesian coordinate system. Let us now evaluate the electromagnetic energy-momentum tensor (16) to zeroth-order in $\tilde{G}_{0}$ in cartesian coordinates. We can find easily that: $$\begin{aligned}
T^{tt}_{em} & = & T^{zz}_{em} = \frac{I^2}{2\pi r^2} \nonumber \\
T^{ij}_{em} & = & \frac{I^2}{2\pi r^4} (2x^i x^j - r^2 \delta_{ij})\end{aligned}$$ where $i,j = x,y$. Though the energy-momentum tensor (29) expresses the string’s energy in the exterior region, one can still write it in terms of distributions, taking into account the relations $$\nabla^2 \left( \ln \frac{r}{r_0} \right)^2 = \frac{2}{r^2} \;\; \mbox{and} \;\;
\partial_i\partial_j \ln \left( \frac{r}{r_0} \right) = \frac{(r^2\delta^{ij} - 2x^i x^j )}{r^4} .$$ Therefore, (29) becomes $$\begin{aligned}
T^{tt}_{em} & = & T^{zz}_{em} = \frac{I^2}{4\pi} \nabla^2 \left(
\ln \frac{r}{r_0} \right)^2 \nonumber \\
T^{ij}_{em} & = & - \frac{I^2}{2\pi} \partial_i\partial_j \ln \frac{r}{r_0} .\end{aligned}$$ We are now in a position to calculate the linearised Einstein’s equations (25) with source identified by: $$\begin{aligned}
T^{tt}_{(0)} & = & U\delta(x)\delta(y) + \frac{I^2}{4\pi}\nabla^2 \left (
\ln \frac{r}{r_0} \right)^2 , \nonumber \\
T^{zz}_{(0)} & = & -\tau \delta(x)\delta(y) + \frac{I^2}{4\pi}\nabla^2 \left(
\ln \frac{r}{r_0} \right)^2 ,
\nonumber \\
T^{ij}_{(0)} & = & I^2 \left[ \delta^{ij} \delta(x)\delta(y) -
\frac{\partial_i\partial_j \ln \frac{r}{r_0}}{2\pi} \right] , \end{aligned}$$ and trace given by: $$T_{(0)} = - (U +\tau - I^2 )\delta(x)\delta(y) .$$ A straightforward calculus lead to the following solution of eq. (25): $$\begin{aligned}
h_{00} & = & - 4\tilde{G}_{0} \left[ I^2 \ln^2 \frac{r}{r_0} +
(U -\tau + I^2) \ln \frac{r}{r_0} \right] , \nonumber \\
h_{zz} & = & - 4\tilde{G}_{0} \left[ I^2 \ln^2 \frac{r}{r_0} +
(U-\tau -I^2) \ln \frac{r}{r_0} \right] , \nonumber \\
h_{ij} & = & -4 \tilde{G}_{0} \left[ \frac{I^2}{2} r^2 \partial_i\partial_j
\ln \frac{r}{r_0} + (U +\tau + I^2) \delta_{ij} \ln \frac{r}{r_0} \right] .\end{aligned}$$ One can easily verify that the harmonic conditions $(h^{\mu}_{\nu} - \frac{1}{2}\delta^{\mu}_{\nu} h)_{,\nu} = 0$, with $h_{\mu\nu}$ given by (33), are identically satisfied. Using expression (32) for the trace of the energy-momentum tensor, we can solve eq. (26) straightforwardly: $$\phi_{(1)} = 2\tilde{G}_{0} \alpha(\phi_0) (U+\tau -I^2)\ln \frac{r}{r_0} .$$ As expected, since the linearised (modified) Eisntein’s equations are the same as in General Relativity, we re-obtained here the same solutions (34) as in refs. [@lin; @pet2]. However, the scalar-tensor feature still brings a new information coming from solution (34).
We return now to the original cylindrical coordinates system and obtain: $$\begin{aligned}
g_{tt} & = & - \left\{ 1 + 4\tilde{G}_{0} \left[ I^2 \ln^2 \frac{r}{r_0} +
(U-\tau +I^2)\ln \frac{r}{r_0} \right]\right\} , \nonumber \\
g_{zz} & = & 1 - 4\tilde{G}_{0} \left[ I^2 \ln^2 \frac{r}{r_0}
+ (U-\tau - I^2)\ln \frac{r}{r_0} \right] , \nonumber \\
g_{rr} & = & 1 + 2\tilde{G}_{0}I^2 - 4\tilde{G}_{0}(U +\tau + I^2) \ln \frac{r}{r_0} , \nonumber \\
g_{\theta\theta} & = & r^2 \left[ 1 - 2\tilde{G}_{0}I^2 -
4\tilde{G}_{0} (U+\tau + I^2) \ln \frac{r}{r_0} \right] .\end{aligned}$$
In order to preserve our previous assumption that $g_{tt}=-g_{\rho\rho}$ (corresponding to the particular case of a magnetic solution of the Einstein-Maxwell-dilaton eqs.), we make a change of variable $r \rightarrow \rho$, such that $$\rho = r \left[ 1 + \tilde{G}_{0} (4U + I^2) - 4\tilde{G}_{0} U \ln \frac{r}{r_0} -
2\tilde{G}_{0}I^2 \ln^2 \frac{r}{r_0}\right] ,$$ and, thus, we have $$\begin{aligned}
ds^2 & = & \left\{ 1 + 4\tilde{G}_0 \left[ I^2\ln^2 \frac{\rho}{r_0} + (U-\tau +I^2)\ln
\frac{\rho}{r_0}\right] \right\} (-dt^2 + d\rho^2) \nonumber \\
& & + \left\{ 1 - 4\tilde{G}_0 \left[ I^2 \ln^2\frac{\rho}{r_0} +
(U-\tau -I^2) \ln \frac{\rho}{r_0} \right]\right\} dz^2 \\
& & + \rho^2 \left[ 1 - 8\tilde{G}_0 (U+ \frac{I^2}{2}) +
4\tilde{G}_0 (U-\tau -I^2)\ln \frac{\rho}{r_0} + 4\tilde{G}_0 I^2
\ln^2\frac{\rho}{r_0} \right] d\theta^2 \nonumber . \end{aligned}$$ Expressions (34) and (36) represent, respectively, the solutions of the scalar field and an isolated current-carrying string in the conformal frame, as long as the weak-field approximation is valid. Comparison with the external solutions (22) and (24) requires a linearision of these ones since they are exact solutions. Expanding them in power series of the paramenters $m$ and $n$, we find $$\begin{aligned}
g_{\rho\rho} & = & - g{tt} = 1 + 2m^2 \ln \frac{\rho}{r_0} + h(\rho) \\
g_{zz} & = & \frac{1}{1+h(\rho)} \\
g_{\theta\theta} & = & B^2 \rho^2 [ 1 + h(\rho) ] ,\end{aligned}$$ with $$h(\rho) = 2n \frac{1 - \kappa}{1+\kappa} \ln \frac{\rho}{r_0} +
2n^2 \frac{1 + \kappa^2}{(1+\kappa)^2} \ln^2 \frac{\rho}{r_0} .$$ Making the identification of the coefficients of both linearised metrics, we finally obtain $$\begin{aligned}
m^2 & = & 4\tilde{G}_0 I^2 \nonumber \\
B^2 & = & 1 - 8\tilde{G}_0 (U+\frac{I^2}{2}) \nonumber \\
l & = & 2\tilde{G}_0 \alpha(\phi_0) (U+\tau-I^2) \nonumber \\
\kappa & = & 1 + \tilde{G}^{1/2}_0 (U - \tau -I^2) .\end{aligned}$$ Calculating now the deficit angle for metric (36) $$\Delta\theta = 2\pi \left[ 1 - \frac{1}{\sqrt{g_{\rho\rho}}}\frac{d}{d\rho}
\sqrt{g_{\theta\theta}} \right] ,$$ we finally obtain $$\Delta\theta = 4\pi \tilde{G}_0 (U +\tau + I^2) .$$
Bending of Light Rays:
----------------------
A light ray coming from infinity in the transverse plane has its trajectory deflected, for an observer at infinity, by an angle given by: $$\Delta\theta = 2 \int_{\rho_{min}}^{\infty} d\rho
[-\frac{g^2_{\theta\theta}p^{-2}}{g_{\rho\rho}g_{tt}} -
\frac{g_{\theta\theta}}{g_{\rho\rho}}]^{-1/2} \,\, - \pi$$ where $\rho_{min}$ is the distance of closest approach, given by $\frac{d\rho}{d\theta} =0$: $$\frac{g_{\theta\theta}(\rho_{min})}{g_{tt}(\rho_{min})} = -p^2$$ which gives in turn: $$\frac{\rho_{min}}{r_0} = (\frac{p}{Br_0})^{1/(1-m^2)}.$$
We can now evaluate the deficit angle to first order in $\tilde{G}_0$. Performing an expansion to linear order in this factor, in much the same way as Peter and Puy [@pet2], we find: $$\Delta\theta = \frac{2}{B(1-m^2)}[\frac{\pi}{2}(1+m^2\ln\frac{p}{Br_0})-m^2\nu]
-\pi ,$$ where we have defined the quantity $\nu$ as $$\nu \equiv - \int_0^1\frac{\ln s}{\sqrt{1-s^2}} ds = \frac{\pi}{2}\ln 2 ,$$ with $s \equiv \frac{p}{Br_0}(\frac{\rho}{r_0})^{m^2 -1}$. Using expressions (37), we have $$\Delta\theta = 4\pi \tilde{G}_0 \left[ U + I^2 \left(\frac{3}{2} +
\ln \frac{\rho}{r_0}\right)\right] + 8\nu \tilde{G}_0 I^2 .$$
Conclusion
==========
In this work we studied the modifications induced by a scalar-tensor gravity on the metric of a current-carrying string described by model with action given by eq. (2). For this purpose, we made an adaptation of Linet’s method which consists in linearising the Einstein’s and dilaton’s equations using distribution’s functions while taking into account the scalar-tensor feature of gravity. We found that the metric depends on five parameters which are related to the string’s internal structure and to the scalar field (dilaton) solution. Concerning the deflection of light, if we compare our results with those obtained in General Relativity, we see that expression (39) does not change substantially, albeit the metric structure is indeed modified with respect to the one in General Relativity.
Now, an interesting investigation that opens up, and we have already initiated to pursue, is the analysis of the properties of the cosmic string generated by the action (2) in the supersymmetrized version, where it is implicit that the scalar-tensor degrees of freedom of the gravity sector are accomodated in a suitable supergravity multiplet. The study of such a model raises the question of understanding the rôle played by the fermionic partners of the bosonic matter and by the gravitino in the configuration of a string. Also, it might be of relevance to analyse the possibility of gaugino and gravitino condensation in this scenario.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to Brandon Carter, Bernard Linet and Patrick Peter for many discussions, suggestions and a critical reading of this manuscript. One of the authors (MEXG) thanks to the Centro Brasileiro de Pesquisas Físicas (in particular, the Departamento de Campos e Partículas) and to the Abdus Salam ICTP-Trieste for hospitality during the preparation of part of this work. CNF thanks to CNPq for a PhD grant.
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[^1]: crisnfer@cat.cbpf.br
[^2]: emilia@mat.unb.br
[^3]: Also at Grupo de Física Teórica, Universidade Católica de Petrópolis. helayel@cat.cbpf.br
[^4]: Just as a reminder, throughout this paper we will work in the conformal frame for the sake of simplicity. Also, for convenience, we work in units such that $\hbar = c=1 $ and keep Newton’s “constant" $G$.
[^5]: To linear order in $\tilde{G}_{0}$, the modified Einstein’s equations (10) reduce to the usual linearised Einstein’s equations [@dam1], the electromagnetic field being as in Minkowski spacetime.
|
---
abstract: 'The 5D Cosmological General Relativity theory developed by Carmeli reproduces all of the results that have been successfully tested for Einstein’s 4D theory. However the Carmeli theory because of its fifth dimension, the velocity of the expanding universe, predicts something different for the propagation of gravity waves on cosmological distance scales. This analysis indicates that gravitational radiation may not propagate as an unattenuated wave where effects of the Hubble expansion are felt. In such cases the energy does not travel over such large length scales but is evanescent and dissipated into the surrounding space as heat.'
author:
- |
**John G. Hartnett**\
School of Physics, the University of Western Australia,\
35 Stirling Hwy, Crawley 6009 WA Australia\
*john@physics.uwa.edu.au*\
\
**Michael E. Tobar**\
School of Physics, the University of Western Australia,\
35 Stirling Hwy, Crawley 6009 WA Australia\
*mike@physics.uwa.edu.au*
date:
title: '**Properties of gravitational waves in Cosmological General Relativity** '
---
Key words: cosmology, Carmeli, gravitational waves, 5 dimensions, expanding universe
\[sec:Intro\]Introduction
=========================
In recent decades, the search for gravity waves has intensified with large high powered laser-based interferometic detectors coming on line. See LIGO [@LIGO] and TAMA [@TAMA] for example. These detectors have already reached sensitivities that should enable them to “see” well beyond the local galactic Group. On the other hand, the Hulse-Taylor binary [@Taylor1979] ring-down energy budget is a precise test of general relativity and a clear indication of the existence of gravitational radiation, and it seems that the first direct detection is just a matter of time.
In standard General Relativity the expanding universe has no impact on the properties of gravitational waves, except the the well known effect of redshift. However, in Carmeli cosmology the expansion of the universe (or redshift of the gravitational wave) manifests as a fifth dimension [@Carmeli2002c] and in this paper we calculate the effect and how this might impact on a possible direct detection.
\[sec:CGR\]Cosmological General Relativity
==========================================
In the late 1990s Moshe Carmeli proposed a new cosmology, Cosmological General Relativity (CGR). [@Behar2000; @Carmeli1998; @Carmeli2002a; @Carmeli2002b; @Carmeli2002c] It is a generally covariant theory and extends the number of dimensions of the universe by the addition of a new dimension – the radial velocity of the galaxies in the Hubble flow. The Hubble law is assumed as a fundamental axiom for the universe and the galaxies are distributed accordingly.
As a result we have a 5D *spacetimevelocity* universe with two timelike and three spacelike coordinates in the metric. The signature is then $(+\,-\,-\,-\,+)$. The universe is represented by a 5-dimensional Riemannian manifold with a metric $g_{ \mu \nu}$ and a line element $ds^{2}=g_{ \mu \nu}dx^{\mu}dx^{\nu}$. This differs from general relativity in that here the $x^{4} = \tau v$ coordinate is more correctly *velocitylike* instead of *timelike* as is the case of $x^{0} = ct$, where $c$ is the speed of light, a universal constant and $t$ is the time coordinate. In this theory $x^{4} = \tau v$, where $\tau$ is also a universal constant, the Hubble-Carmeli time constant. The other three coordinates $x^{k}, k = 1,2,3$, are spatial and *spacelike*, as in general relativity.
It has been shown that all the results predicted by general relativity and experimentally verified are also predicted by CGR [@Carmeli2002b]. However in [@Carmeli2002c] Carmeli discussed the one consequence that was not exactly reproduced, and that was gravity waves in 5 dimensions. The new metric resulted in a redshift dependence with a more general wave equation incorporating 5 dimensions $(ct, x^{1}, x^{2}, x^{3}, \tau v)$.
\[sec:linearCGR\]Linearized general relativity
----------------------------------------------
As is the usual practice in a weak gravitational field we write the metric $$\label{eqn:gmetric}
g_{\mu \nu}=\eta_{\mu \nu} + h_{\mu \nu}$$ where the metric $\eta_{\mu \nu}$ is Minkowskian but extended here to 5D from the usual 4D Minkowski metric but with signature $(+\,-\,-\,-\,+)$. Here $\eta_{\mu \nu}$ is perturbed due to gravitating sources with $h_{\mu \nu} \ll 1$.
A useful tool is to define the trace-reversed $h_{\mu \nu}$ as $$\label{eqn:hbar}
\bar{h}_{\mu \nu}=h_{\mu \nu}-\frac{1}{2} \eta_{\mu \nu} h$$ where $h = \eta^{\alpha \beta} h_{\alpha \beta}$ is the trace of $h_{\mu \nu}$. Consequently $$\label{eqn:h}
h_{\mu \nu} = \bar{h}_{\mu \nu}-\frac{1}{2} \eta_{\mu \nu} \bar{h}$$ where $\bar{h} = \eta^{\alpha \beta} \bar{h}_{\alpha \beta}$ and $\bar{h} = -h$.
Then the linearized Einstein field equations to first order in $\bar{h}_{\mu \nu}$ yield $$\label{eqn:linearized}
\bigcirc \bar{h}_{\mu \nu} = -2 \kappa T_{\mu \nu} \qquad \textrm{plus} \; \eta^{\alpha \beta}\bar{h}_{\mu \alpha, \beta} = 0$$ where $\bigcirc$ is the D’Alembertian operator in 5D and may be expressed as $$\label{eqn:DAlamb}
\bigcirc = \left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} + \frac{1}{\tau^{2}}\frac{\partial^{2}}{\partial v^{2}}\right).$$ For conservation of energy and momentum, excluding gravity, it follows from (\[eqn:linearized\]) that $$\label{eqn:conserved}
\eta^{\alpha \beta}T_{\mu \alpha, \beta} = 0.$$
From (\[eqn:linearized\]) and (\[eqn:DAlamb\]) it is clear that (\[eqn:linearized\]) is a generalized wave equation that reduces to $$\label{eqn:homo}
\bigcirc \bar{h}_{\mu \nu}=0$$ in vacuum.
So the gravitational waves depend not only on space and time but also on the expansion velocity of the source in the Hubble flow. Here $v$ is the fifth co-ordinate, which represents the velocity of the expansion of the space through which the wave passes.
The solution of (\[eqn:linearized\]) is the sum of the solution of the homogeneous equation (\[eqn:homo\]), and a particular solution. The following is the special time independent retarded solution in the absence of source-less radiation. The contravariant form is [@Ohanian1976] $$\label{eqn:hetro}
\bar{h}^{\mu \nu}= -2\kappa \int \frac{T^{\mu \nu}d^{3}x'}{|\bf{x}-\bf{x'}|},$$ where the source mass is located at $\bf{x'}$ and the potential measured at $\bf{x}$. To evaluate the integral in (\[eqn:hetro\]), provided the measurement point determined by the vector $\bf{x}$ is far away from the source, a Taylor expansion of $1/|\bf{x}-\bf{x'}|$ about $\bf{x'}$ $= 0$ is taken retaining only the first two terms, $$\label{eqn:Taylor}
\frac{1}{|\bf{x}-\bf{x'}|} \approx \frac{1}{r} + \frac{x^{k}x'^{k}}{r^{3}}$$ where $r^2=x^{k}x^{k}$.
The integral in (\[eqn:hetro\]) is then written as $$\label{eqn:hetro2}
\bar{h}^{\mu \nu}= -\frac{GM}{r}-\frac{G}{2}\epsilon^{kln} S^{n}\frac{x^{k}}{r^{3}},$$ where the dipole term has been eliminated by choosing the origin of the coordinates to coincide with the center of the source mass. The first term in (\[eqn:hetro2\]) involves the integral $\int T^{00}(\bf{x'})$$d^{3}x' = M$ identified with the source mass and the second term $\int x'^{k}T^{l0}(\bf{x'})$$d^{3}x' = 1/2\epsilon^{k l n} S^{n}$ where $S^{n}$ is the spin angular momentum of the system. Here $k,l,n = 1, 2, 3$ for the spatial coordinates.
\[sec:waveeqn\]Wave equation in curved *spacevelocity*
------------------------------------------------------
Now considering the time dependence again we can write (\[eqn:homo\]) as $$\label{eqn:homo2}
\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2}\right) \barhup = -\frac{1}{c^2 \tau^{2}}\frac{\partial^{2}}{\partial z^{2}}\barhup,$$ where the substitution $v/c \rightarrow z$ has been made. Here $z$ is the redshift of the wave, and the substitution is valid where $z =< 0.1$, which is approximately 400 $Mpc$. We assume it is approximately valid beyond that. Now the solution to (\[eqn:homo2\]) is the sum of the solution to the homogeneous equation (\[eqn:homo2\]), which is the usual gravity wave solution in general relativity, and a particular solution of (\[eqn:homo2\]), which has a redshift dependent source term.
In fact, in CGR, because the Hubble law is assumed *a priori*, the expansion velocity $v$ (or gravitational wave redshift $z$) is not independent of $r$ and depends on the matter density of the universe. In fact, in the case of CGR, (\[eqn:homo2\]) can be written as $$\label{eqn:DAlambcurtved}
\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} + \frac{1}{c^{2}\tau^{2}}\frac{\partial^{2}}{\partial z^{2}}\right) \barhup = 0,$$ with $$\label{eqn:chain}
\frac{\partial^{2}}{\partial z^{2}} = \left\{\left(\frac{\partial r}{\partial z}\right)^{2}\frac{\partial^{2}}{\partial r^{2}}+
\frac{\partial^2 r}{\partial z^2}\frac{\partial}{\partial r}\right\}$$ when the chain rule is applied.
Let us look for a plane wave solution of the form $$\label{eqn:planewave}
\barhup = \varepsilon^{\mu \nu} \cos k_{\alpha}x^{\alpha},$$ where the 3-space co-ordinates are ($x^1$, $x^2$, $x^3$). Here $x^1$ and $x^2$ are orthogonal to the direction of propagation $x^3$ from source to detector. Here $\varepsilon^{\mu \nu}$ is a constant tensor and $k_{\alpha}$ is a constant vector. Therefore we look for a wave propagating in the $r$ direction which would have $k^{\alpha} = (\omega/c,0,0,k_{r})$.
This means we can retain only the $r$ derivative in $\nabla^2$ and effectively re-write (\[eqn:DAlambcurtved\]) in spherical co-ordinates as $$\label{eqn:DAlambcurtved2}
\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}} - \frac{\partial^{2}}{\partial r^{2}} - \frac{2}{r} \frac{ \partial}{\partial r} + \frac{1}{c^{2}\tau^{2}}\frac{\partial^{2}}{\partial z^{2}}\right) \barhup = 0.$$
Eqs (\[eqn:DAlambcurtved\]) and (\[eqn:DAlambcurtved2\]) are only valid where the Hubble law applies. When it doesn’t apply, that is, where $\partial r/\partial z =0$ in (\[eqn:chain\]), (\[eqn:DAlambcurtved\]) becomes the normal wave equation for gravity waves in source free regions.
However where the Hubble law is applicable, in flat (i.e. $\Omega = 1$) *spacevelocity* $\partial r/\partial z = c \tau$, which is the Hubble law in the zero distance/zero gravity limit. In the general curved *spacevelocity*, the form of the derivative is given by [@Hartnett2005] $$\label{eqn:deriv}
\frac{1}{c^{2} \tau^{2}}\left(\frac{\partial r}{\partial z}\right)^{2} = 1+ (1-\Omega) \frac{r^{2}}{c^{2} \tau^{2}},$$ where $\Omega = \rho/\rho_{c}$ is the mass/energy density at some epoch expressed as a fraction of the ‘critical’ density, $\rho_{c} = 3/8\pi G \tau^{2}$.
Substituting (\[eqn:deriv\]) into (\[eqn:DAlambcurtved2\]) with (\[eqn:chain\]) we get $$\label{eqn:DAlambcurtved3}
\left(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}- \frac{2}{r} \frac{ \partial}{\partial r} + \frac{1-\Omega}{c^{2}\tau^{2}}\left\{r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}\right\} \right) \barhup = 0,$$ which only has dependence on $r$ and $t$. This is a new equation depends on the surrounding matter density $\Omega$.
When the gravity wave is very distance from the source and when $r \gg c\tau/\sqrt{1-\Omega}$ the second term of (\[eqn:DAlambcurtved3\]) is much smaller than the term in curly brackets, we assume the second term negligible. Therefore (\[eqn:DAlambcurtved3\]) can be approximated for large $r$ as $$\label{eqn:DAlambcurtved4}
\left(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} + \frac{1-\Omega}{c^{2}\tau^{2}}\left\{r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}\right\} \right) \barhup = 0.$$
The solution of (\[eqn:DAlambcurtved4\]) can be obtained by separation of variables assuming a solution of the form $\barhup \propto R(r)e^{i (\omega t+k_z)}$. Substituting the latter into (\[eqn:DAlambcurtved4\]) yields $$\label{eqn:solnwavenumber}
\frac{\omega^{2}}{c^{2}} +\frac{\Omega - 1}{c^{2}\tau^{2}} = 0,$$ with $k_z \approx 0$ and $R(r) = a_{1}r^{-1}$ for $r \gg c\tau/\sqrt{1-\Omega}$. More generally $R(r) = \sum a_{n}r^{-n}$ a polynomial expression with an index $n > 0$. Furthermore by taking a hint from the solution of the usual heterogeneous equation shown in (\[eqn:hetro2\]) $R(r)$ is determined as $$\label{eqn:solnR}
R(r)= -\frac{GM}{r}- \mathcal{O}(\frac{1}{r})^{3}.$$ Equation (\[eqn:solnwavenumber\]) is a resonance condition. For this solution, which spans the whole extent of the Universe, the Universe acts like a resonant mode with a characteristic scale radius [@Oliveira2005a; @Oliveira2005b] of $$\label{eqn:scaleR}
R_{\Omega}= \sqrt{|R_{\Omega}^2|}=\sqrt{|\frac{c^2}{\omega^2}|}=\frac{c\tau}{\sqrt{|1-\Omega|}},$$ and resonance frequency $$\label{eqn:freq}
\omega = \frac{\sqrt{1-\Omega}}{\tau}.$$ For values of $\tau = 4.2 \times 10^{17}\;s$ and $\Omega = 0.02$ the scale radius is $R_{\Omega} \approx 4.13 \,Gpc$ and the characteristic frequency is $\omega/2\pi \approx 3.66 \times 10^{-19}$ Hz.
When $r \approx < c\tau/\sqrt{1-\Omega}$ the second and fourth terms of (\[eqn:DAlambcurtved3\]) are much smaller than the third, and hence can be neglected. Therefore (\[eqn:DAlambcurtved3\]) can be approximated as $$\label{eqn:DAlambcurtved5}
\left(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} + \frac{1-\Omega}{c^{2}\tau^{2}}r^{2}\frac{\partial^{2}}{\partial r^{2}} \right) \barhup = 0.$$
Now for a spherically symmetric expanding universe Carmeli [@Carmeli2002a; @Hartnett2005] obtains the relation between redshift and distance of the emitting source, $$\label{eqn:rz}
\frac{r}{c \tau} = \frac{\sinh(\varsigma \sqrt{1-\Omega})}{\sqrt{1-\Omega}},$$ where $\varsigma = ((1+z)^2-1)/((1+z)^2+1)$. Assuming this relation also holds for gravity waves (\[eqn:DAlambcurtved4\]) becomes $$\label{eqn:DAlambcurtved6}
\left(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} + \sinh^2(\varsigma \sqrt{1-\Omega})\frac{\partial^{2}}{\partial r^{2}} \right) \barhup = 0,$$
This is now a wave equation with a solution of the form $\barhup \propto e^{i (k_{r}r+\omega t)}$, which results in the following dispersion relation $$\label{eqn:dispersion}
k_{r}^{2} \approx -\frac{\omega^{2}/c^{2}}{\sinh^2(\varsigma \sqrt{1-\Omega})},$$ which can be approximated for $z \ll 1$ as $$\label{eqn:dispersion2}
k_{r}^{2} \approx \frac{\omega^{2}/c^{2}}{(\Omega-1)z^{2}}.$$ When $\Omega > 1$ the wave number is real and approximately $k_{r}=\omega/(c z \sqrt{\Omega - 1})$ and hence gravity waves propagate yet are dependent on redshift. When $\Omega < 1$ the wave number is imaginary and the amplitude is attenuated with a decay constant $\kappa = ik_{r}= \omega/(c z \sqrt{1-\Omega})$. In Fig. \[fig:fig1\] we plot the decay constant $\kappa$ normalized by $\omega/c$, using $k_{r}$ from (\[eqn:dispersion\]). From the figure it is apparent that the approximation of (\[eqn:dispersion2\]) is good for $z < 0.1$.
It follows from (\[eqn:dispersion\]) the phase and group velocities are determined from $$\label{eqn:velocity}
\frac{\partial \omega}{\partial k_{r}} = \frac{\omega}{k_{r}}= c\sqrt{-\sinh^2(\varsigma \sqrt{1-\Omega})}.$$ Using the identity $i\sin \theta = \sinh (i\theta)$ (\[eqn:velocity\]) becomes $$\label{eqn:velocity2}
\frac{\partial \omega}{\partial k_{r}} = \frac{\omega}{k_{r}}= c \sin(\varsigma \sqrt{\Omega-1}),$$ where $\Omega \geq 1$. The latter can be approximated for $z \ll 1$ as $c z \sqrt{\Omega-1}$. In the general cosmos where $\Omega > 1$ gravity wave propagate with the velocity $c \sin(\varsigma \sqrt{\Omega-1}) \rightarrow c$ where $v \rightarrow c$. In the cosmos where $\Omega <1$ we have evanescent decay.
The dependence in (\[eqn:deriv\]) is not the same within a bound galaxy of stars and gas as it is for the large scale structure of the expanding universe, which considers only the center of mass motion of galaxies within it. This is because within a galaxy (or cluster) the full effect of the Hubble expansion is not felt with respect to the center of mass, of which the gravitational radiation must travel with respect to, and here we show it results in the same solution as standard General Relativity.
For spherically symmetric distribution of matter in a galaxy with the region of interest far from the central potential of a fixed mass, in the disk region, it may be derived (Eqs B.63a and B.67 of Carmeli [@Carmeli2002a]) that $$\label{eqn:derivGalaxy}
\frac{1}{c^{2} \tau^{2}}\left(\frac{\partial r}{\partial z}\right)^{2} = (\Omega-1) \frac{r^{2}}{c^{2} \tau^{2}},$$ where $z = v/c$ and $\Omega$ is the mass density, with the origin of coordinates coinciding with the origin of the spherically symmetric gravitational potential.
Using (\[eqn:derivGalaxy\]) in (\[eqn:DAlambcurtved2\]) with (\[eqn:chain\]) results in a wave equation $$\label{eqn:DAlambcurtved7}
\left(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}- \frac{ \partial^2}{\partial r^2}-\frac{2}{r} \frac{ \partial}{\partial r} + \frac{\Omega-1}{c^{2}\tau^{2}}\left\{r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}\right\} \right) \barhup = 0,$$ where we can neglect the terms in the curly brackets because they are insignificant on the scale of a galaxy. Also after neglecting the third term for distant sources, we get the normal gravity wave equation of GR, that is, $$\label{eqn:DAlambcurtved8}
\left(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}- \frac{ \partial^2}{\partial r^2} \right) \barhup = 0.$$
\[sec:scales\]Density scales in the universe
--------------------------------------------
On the local scale the Hubble law $v = r/ \tau$ does not apply or is so insignificant as to be negligible. On that scale therefore effects of *spacevelocity* are negligible, or in other words, $dv \rightarrow 0$. That is the realm where GGR reduces to the usual special and general relativity theory. Gravity waves propagate as is usually expected according to (\[eqn:DAlambcurtved8\]).
On the cosmological scale we expect the Hubble law to be very significant and hence that is the realm where *spacevelocity* is operative. On that scale we can now also consider (\[eqn:DAlambcurtved5\]) to represent a genuine modification to the usual 4D *spacetime* equation found in general relativity textbooks. The effect of *spacevelocity* is contained in the modified D’Alembertian operator. This means close to the source gravitational energy is emitted in the usual fashion as described by (\[eqn:hetro2\]). Over cosmological length scales, however, CGR predicts the gravitational waves from distant galaxies will be fully attenuated by the time the reach the Earth. Thus, projects such as the Large-scale Cryogenic Gravitational-wave Telescope (LCGT) in Japan could be very important to test CGR theory [@Kuroda]. For example this project will detect gravity-waves from coalescing neutron-star binary systems 200 $Mpc$ away at a SNR of 10. In contrast the TAMA field of view is 1 $Mpc$ and LIGO is 20 $Mpc$. Thus, if the LGCT field of view includes a much larger volume of the Universe than LIGO or TAMA, and a smaller event rate than predicted by GR, this could be evidence for CGR.
The best estimates of the local baryonic matter density puts it around $\Omega_{m} \approx 0.02$ [@Fukugita1998] for the present epoch. As has been shown [@Hartnett2004a; @Hartnett2005], using the Carmeli theory it is not necessary to assume any dark matter in the cosmos, therefore the matter density between galaxies should be $\Omega < 1$ even out to a redshift $z = 2$ [@Hartnett2005]. Equation (\[eqn:deriv\]) is valid for large $z$ therefore the analysis applies.
Accordingly the averaged matter density of the universe has been determined at the current epoch $\Omega_m = 0.021 \pm 0.042$ [@Hartnett2005] and can be approximately related by $\Omega=\Omega_m (1+z)^3$ as a function of redshift for $z \leq 1$. Using this relation and the form of (\[eqn:dispersion\]) the decay constant ($\kappa$) is shown as $c \kappa /\omega$ in Fig. \[fig:fig1\] as a function of redshift, $z$. The value of $c \kappa /\omega$ in Fig. \[fig:fig1\] is limited by the unknown form of $\Omega(z)$ for $z \gg 1$. However, over the epochs shown, CGR predicts that gravitational waves do not propagate at scales beyond galaxies (and clusters).
Gravity waves that are generated within a galaxy quickly decay in the void between. Gravitational radiation therefore leaks into the surrounding space according to (\[eqn:DAlambcurtved5\]) with attenuated amplitudes when $\Omega$ drops below unity. According to CGR, gravity waves will not propagate far in an expanding universe. Therefore we would expect to see no stochastic gravity wave background spectrum. Instead we conjecture that the energy is deposited into space as heat. As a result they may contribute to the CMB blackbody temperature.
The binary pulsar PSR B1913+16, discovered in 1974 by Russell Hulse and Joseph Taylor [@Hulse1975], for which they won the 1993 Nobel prize, consists of two neutron stars closely orbiting their common center of mass. One of them is a pulsar with a rotational period of 59 ms and extremely stable compared to other pulsars. The two neutron stars slowly spiral toward their common center of mass radiating energy. The orbit period however is declining by about $7.5 \times 10^{-5}$ seconds per year on an orbit period of $7.75$ hours [@psr1913]. This change, believed to result from the system emitting energy in the form of gravitational waves, has been a very precise and successful test of general relativity [@Taylor1979].
On the scale of the Galaxy it expected that there are still some small effects of the Hubble law [@Hartnett2004b; @Hartnett2005b]. These effects modify the dynamics of the motion of tracer gases in the outlying regions of the Galaxy but in regard to gravity waves in galaxies they propagate unhindered, as in normal GR. Therefore it is expected that the energy dissipated by the Hulse-Taylor binary does travel as gravity waves within the Galaxy, but will be attenuated outside the region of the Galaxy where the mass density $\Omega$ drops below unity.
\[sec:Conclusion\]Conclusion
============================
This paper derives the wave propagation equation for gravitational radiation in an expanding universe where an additional constraint has been placed on the nature of space itself. This is the introduction into the metric the fundamental assumption of the expansion of space according to the Hubble law. It is then found that no unattenuated gravity wave propagation may be possible in regimes where the Hubble expansion has effect. Then, depending on the density of matter, the propagation constant of any gravitational wave is either real or imaginary. If imaginary it represents an evanescent wave, which we conjecture means the energy is dissipated into the surrounding space as heat. When gravity waves are eventually detected, a test of this theory would be the detection of gravity waves from within the Galaxy but not from extra-galactic sources.
Acknowledgment
==============
The authors wish to thank Dr Paul Abbott and Prof. Bahram Mashhoon for their helpful advice and suggestions.
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//astro201/psr1913.htm 14 December 2005 http://tamago.mtk.nao.ac.jp/
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abstract: 'Intratube quantum dots showing particle-in-a-box-like states with level spacings up to 200 meV are realized in metallic single-walled carbon nanotubes by means of low dose medium energy Ar$^+$ irradiation. Fourier transform scanning tunneling spectroscopy compared to results of a Fabry-Pérot electron resonator model yields clear signatures for inter- and intra-valley scattering of electrons confined between consecutive irradiation-induced defects (inter-defects distance $\le 10$ nm). Effects arising from lifting the degeneracy of the Dirac cones within the first Brillouin zone are also observed.'
author:
- 'G. Buchs'
- 'D. Bercioux'
- 'P. Ruffieux'
- 'P. Gröning'
- 'H. Grabert'
- 'O. Gröning'
title: 'Electron scattering in intra-nanotube quantum dots'
---
The experimental realization of quantum dots (QDs) [@QD_Leo], sometimes called “artificial atoms”, has led to a variety of new concepts in nanotechnology underlying advanced QD-based devices for applications in promising fields like nanoelectronics, nanophotonics and quantum information/computation [@Review_Hanson; @Review_Kern; @Chuang_book]. Frequently, for these applications a QD needs to be contacted by source, drain, and gate electrodes. In the field of semiconductor heterostructures the excitation energies of contacted QDs are usually so small that the devices can only be operated at cryogenic temperatures. A promising candidate for room temperature active dots are intra-nanotube QDs formed within a single-walled carbon nanotube (SWNT) by means of two local defects [@grifoni:2001]. For defect separations of order 10 nm the dot excitation energies are well above 100 meV and thus large compared with $k_\text{B}T$ at room temperature. Furthermore, the remaining sections of the SWNT to either side of the confining defects provide natural source and drain electrodes. So far, SWNT-based QD prototypes have been realized by tunneling barriers at metal-nanotube interfaces and/or by gate electrodes [@Review_Christian]. Several authors have analyzed defect-induced standing waves by means of scanning tunneling microscopy (STM) [@Lemay_nat01; @lieber_PRL; @lee:2004]. However, a detailed description of the scattering dynamics of electrons in and out of the QD is absent. Elaborate studies have only been reported for epitaxial graphene with defects, where an analysis of standing waves in Fourier space has permitted to distinguish between contributions to the wave modulation due to inter- and intra-valley scattering [@rutter:2007].
![(a) 3D topography image [@WSXM] of a $\sim$ 50 nm long portion of an armchair SWNT exposed to 200 eV Ar$^{+}$ ions, recorded in the constant current mode with a sample-tip bias voltage ($V_\text{s}$) of 1 V (sample grounded) and a tunneling current ($I_\text{s}$) of 0.1 nA, $T = 5.3$ K. (b) Corresponding $dI/dV$-scan with background subtraction, recorded along the horizontal dashed line in (a), $\Delta x =$ 0.34 nm. The $dI/dV$ spectra are recorded through a lock-in detection of a 12 mV rms ($\sim$ 600 Hz) a.c. tunneling current signal added to the d.c. sample bias under open-loop conditions ($V_\text{s}=$ 0.9 V, $I_\text{s}=$ 0.3 nA). (c) Energy spacings between discrete states visible in (b) in the negative bias range between defect sites d3-d4 ($\sim$ 7.9 nm) and d5-d6 ($\sim$ 9.9 nm).[]{data-label="QBS_1"}](Figure1){width="0.75\columnwidth"}
In this Letter we investigate electron standing waves in intra-tube QDs created in SWNTs irradiated with medium energy Ar$^+$ ions. This promising alternative to build intra-tube QDs has been suggested by observations of electronic confinement in metallic SWNTs due to intrinsic defects [@Bockrath01]. We first show that by virtue of this technique it is indeed possible to realize QDs with a level spacing considerably larger than the thermal broadening at room temperature. Then, by means of Fourier-transform scanning tunneling spectroscopy (FTSTS) combined with a Fabry-Pérot electron resonator model we are able to describe the dominant scattering mechanisms and to identify contributions from inter- and intra-valley scattering.
Our measurements were performed in a commercial (Omicron), ultrahigh vacuum LT-STM setup at $\sim$ 5 K. Extremely pure HipCo SWNTs [@Smalley01] with an intrinsic defect density $<$ 0.005 nm$^{-1}$ were deposited onto Au(111) surfaces from a 1,2-dichloroethane suspension [@Buchs_NJP_07]. *In situ* irradiation with medium energy Ar$^{+}$ ions was performed in a way to achieve a defect density of about 0.1 nm$^{-1}$ [@Buchs_Ar]. Figure \[QBS\_1\]a shows a 3D STM image of a $\sim50$ nm long portion of an armchair SWNT irradiated with $\sim200$ eV ions. Defects induced by medium energy Ar$^{+}$ ions appear typically as hillocks with an apparent height ranging from 0.5 [Å]{} to 4 [Å]{} and a lateral extension between 5 [Å]{} and 30 [Å]{}. We recorded consecutive and equidistant $dI/dV$ spectra (proportional to the local density of states (LDOS) [@Tersoff85]) along the tube axis. Typical $dI/dV(x,V)$ data sets, called $dI/dV$-scans in the following, consist of 150 $dI/dV$ spectra recorded on topography line scans of 300 pts. Figure \[QBS\_1\]b shows a $dI/dV$-scan with a spatial resolution $\Delta x =$ 0.34 nm recorded along the horizontal dashed line drawn in (a), running over seven defect sites (d1-d7). A third order polynomial fit has been subtracted from each $dI/dV$ spectrum to get a better contrast. Defect-induced modifications in the LDOS are revealed as one or more new electronic states at different energy values, spatially localized on the defect sites. First-principle calculations show that medium energy Ar$^{+}$ ions essentially give rise to single vacancies (SV), double vacancies (DV) and also C adatoms (CAd) on SWNTs [@Antti_07]. Based on these results we can confidently assume that the created defects in the present work are mainly of vacancy-type.
![(a) 3D STM topography image [@WSXM] of a metallic SWNT treated with 200 eV Ar$^{+}$ ions, showing four defects d1-d4. (b) Line-by-line flattened topography image of the tube in (a) including defects d2-d4, with the corresponding $dI/dV$-scan recorded along the horizontal dashed line. $V_\text{s} = 0.8$ V, $I_\text{s} = 0.32$ nA, $T = 5.21$ K, $\Delta x =$ 0.1 nm. (c) $dI/dV$ line profiles of the first two modes in the negative bias range, recorded between the red arrows drawn in (b). (d) $\left|dI/dV(k,V)\right|^{2}$ map calculated from the $dI/dV$-scan in (b) between the red arrows. (e) Differential conductance calculated within the Fabry-Pérot electron resonator model for a $(7,4)$ SWNT with a defect distance of 9.5 nm, intra- and inter-valley scattering parameters equal to $0.35$ for both impurities, with the corresponding $\left| dI/dV (k,V) \right|^{2}$ map in (f).[]{data-label="QBS_Fourier"}](Figure2){width="0.85\columnwidth"}
Several broad discrete states characterized by a modulation of the $dI/dV$ signal in the spatial direction are observed in the negative bias range between d3-d4 and d5-d6, and in the positive bias range between d2-d3. These states show a discrete number of equidistant maxima following a regular sequence $i,\, i+1,\, i+2...$ for increasing $\left|V_\text{bias}\right|$, similar to the textbook 1D particle-in-a-box model. Within this model it is possible to estimate the level spacing around the charge neutrality point (CNP) for discrete states observed for example in short SWNTs [@Lemay_nat01; @Rubio_prl99]. Assuming a linear dispersion $E=\hbar v_\text{F} k$ around the two inequivalent Fermi points $\mathbf{K}$ and $\mathbf{K'}$ for a SWNT with finite length $L$, the energy spacing is then given by: $$\Delta E = \hbar v_\text{F} \frac{\pi}{L} = \frac{h v_\text{F}}{2L} \simeq \frac{1.76}{L}\, \text{eV} \cdot \text{nm}
\label{deltaE}$$ with $L$ in nm and the Fermi velocity $v_\text{F}= 8.5 \cdot 10^{5}$ m$\cdot$s$^{-1}$ [@Lemay_nat01]. The energy spacings $\Delta E_\text{a1}$-$\Delta E_\text{a4}$ in the negative bias range between d3-d4 and $\Delta E_\text{b1}$-$\Delta E_\text{b4}$ between d5-d6 are reported in Fig. \[QBS\_1\]c. Using the sequence of maxima we can determine the level spacing closest to the CNP: $\Delta E_\text{a1} = 0.22$ eV \[$\Delta E_\text{b1} = 0.18$ eV\] between d3-d4 \[d5-d6\]. This corresponds to a defect distance of $L = 8$ nm \[$L = 9.78$ nm\] for the defect separation d3-d4 \[d5-d6\], in good agreement with the measured value at the center of the defect sites $L \simeq 7.9$ nm \[$L \simeq 9.9$ nm\] [@note:one]. These results show artificial defect-induced electron confinement regions in metallic SWNTs, *i.e.* intratube QDs. Importantly, spatially close defects can be generated with our method, allowing level spacings which are much larger than the thermal broadening at room temperature of $k_\text{B} T \simeq 25$ meV.
Figure \[QBS\_Fourier\]a shows a $\sim$16 nm long section of a metallic SWNT exposed to 200 eV Ar$^{+}$ ions with four defect sites (d1-d4). A line-by-line flattened topography image of the same tube between defects d2-d4 is displayed in panel (b) with the corresponding $dI/dV$-scan recorded along the horizontal dashed line. Two discrete states are clearly visible in the negative bias range between d3 and d4, at energies $E = -0.22$ eV and $E = -0.39$ eV. The measured energy spacing of about 170 meV fits well with the value of 177 meV obtained from Eq. (\[deltaE\]) for a defect separation of about 9.5 nm. Line profiles of $dI/dV$ signals recorded between the drawn red arrows in (b) and displayed in (c) show a clear oscillatory behavior characterized by a rapid oscillation with an average wavelength of 0.7 nm modulated by a slower variation of the amplitude. This slow modulation, which shows a decreasing wavelength for increasing $\left| V_\text{bias} \right|$, has been fitted with the function $\left| \psi \left( x \right) \right|^{2}=A+B \sin \left( 2 k x + \phi \right)$, where $\phi$ is an arbitrary phase and the factor 2 originates from the fact that $\left| \psi \left( x \right) \right|^{2}$ is probed.
More details on the observed oscillatory behavior are obtained by means of FTSTS, where line-by-line Fourier transforms are performed on the $dI/dV$-scan in Fig. \[QBS\_Fourier\]b, between the positions indicated by the red arrows. From the resulting $\left| dI/dV(k,V) \right|^{2}$ map in (d), we observe that the Fourier spectrum of each discrete state is composed of several components [@note:two]. Whereas the individual low frequency peaks between $k=0$ and $k=4$ nm$^{-1}$ with a high intensity for each discrete state correspond to the slow modulation discussed above, the rapid oscillation in $dI/dV$ is produced by several components around $k=11$ nm$^{-1}$ and $k=17$ nm$^{-1}$. These Fourier components are aligned along sloped lines, indicating the energy dispersive nature of these features. Around $k=17$ nm$^{-1}$, a unique positively sloped line is clearly visible, with $dE/dk \simeq$ 0.32 eVnm, whereas two lines with positive and negative slopes can be distinguished around $k=11$ nm$^{-1}$, with a measured slope of about 0.3 eVnm.
![(a) Outline of the experimental set-up and of the Fabry-Pérot resonator model. (b) Sketch of the graphene valleys at a fixed energy for a $(7,4)$ metallic SWNT. The parallel horizontal $\textbf{k}$-lines are reminiscent from the quantization in the circumferential direction. (c) Sketch of the 1D-scattering processes between two nonequivalent valleys: intra-valley (green arrows) and inter-valley scattering (red, blue and black arrows). The processes indicated by black arrows are only relevant if a commensurability condition is fulfilled. $k_{\text{A}_{i}}$ and $k_{\text{A}_{j}}$ are the CNP axial momenta components of valleys $i$ and $j$, respectively. \[fig3\]](Figure3){width="0.85\columnwidth"}
![(a) STM current error image [@WSXM] of a metallic SWNT with two defect sites d1 and d2 produced by an exposition to 1.5 keV Ar$^+$ ions, with the corresponding $dI/dV$-scan recorded along the horizontal dashed line. Interference pattern visibility is improved via a background subtraction. (b) Corresponding $\left| dI /dV (k, V ) \right|^{2}$ map limited to low frequencies contributions. $V_\text{s} =$ 0.8 V, $I_\text{s} =$ 0.3 nA, $T = 5.3$ K, $\Delta x =$ 0.22 nm. (c) Calculated $dI/dV$-scan for a $(10,7)$ SWNT with a length of $18.5$ nm corresponding to the average distance between the defects d2 and d3. The intra- and inter-valley impurity strengths are equal to $0.025$ and $0.05$ for the left and the right impurity, respectively. (d) Corresponding $\left| dI /dV (k, V ) \right|^{2}$ map limited to low frequencies contributions. \[fig4\]](Figure4){width="0.65\columnwidth"}
In order to fully explain the experimental features, we use a Fabry-Pérot electron resonator model (see Fig. \[fig3\]a) considering interference at fixed energy of electron states scattered by impurities. These states can be easily identified considering an unrolled SWNT, *i.e.* a graphene sheet showing periodic strings of defects along the circumferential direction. The impurities break the translational invariance along the SWNT axis, allowing low energy electron scattering among the six valleys or *Dirac cones* of the first Brillouin zone. The momenta exchanged in these processes can be decomposed in axial $k_\text{A}$ and circumferential $k_\text{C}$ components with respect to the tube axis. The former give rise to the interference pattern resulting in the standing waves, whereas the latter modulate the intensity of the standing waves in a non-linear way, *i.e.* a larger $k_\text{C}$ component leads to a lower intensity. In the calculated $\left| dI/dV(k,V) \right|^{2}$ maps, these standing waves give rise to intensities at $k$-values corresponding to the axial component $k_\text{A}$. Therefore, the $\left| dI/dV(k,V) \right|^{2}$ maps show a weighted projection of the 2D space of possible scattering vectors along the axial direction. The situation is depicted in Fig. \[fig3\]b. Two distinct scattering mechanisms take place: *intra*- and *inter*-valley scattering. For the first process within the same valley the momentum exchange is small, even zero at the CNP (green process in (c)); the second process connects different valleys (blue, red, and black processes in (b) and (c)). Both scattering mechanisms are related to the presence of SVs and DVs [@ando05].
This analysis reduces the scattering processes to a series of weighted 1D scattering events among electrons with a linear energy dispersion and axial momenta $k_{\text{A}_i},k_{\text{A}_j}$ (see Fig. \[fig3\]c). We model the impurities as delta-like potentials placed at a distance $L$, and the STM tip is included by allowing electron tunneling to an external electrode [@bercioux:2009; @note:coulomb] (see Fig. \[fig3\]a). Figure \[QBS\_Fourier\] shows a comparison between the measured (b) and the calculated (e) LDOS for the case of a SWNT with two identical impurities. The measured SWNT has a chiral angle $\theta \approx 21^\circ$ and shows three dispersion lines at $k=6.1,\, 10.7$ and $16.8$ nm$^{-1}$, compatible with a $(7,4)$ metallic SWNT. The numerically evaluated $\left| dI/dV(k,V) \right|^{2}$ map shown in (f) unveils richer structure than the experimental one. These differences can be attributed to the finite resolution of the tip. The components centered around $k=0$ nm$^{-1}$ are more intense than the others because they are associated with intra-valley scattering occurring at all six valleys. In the measured $\left| dI/dV(k,V) \right|^{2}$ the dispersion lines around $k=10.7$ and $16.8$ nm$^{-1}$ show a more intense signal for the positive slope branch than for the negative one. For the component centered at 16.8 nm$^{-1}$, the negative slope branch is almost missing. Similar behavior has been observed in all samples investigated. For armchair SWNTs, this effect has been related to an interplay between symmetry properties of defects and electronic bands resulting in a suppression of $\pi\to\pi$ scattering [@lieber_PRL]. However, in our case of chiral SWNTs the relation between the parity of the $\pi$- and $\pi^*$-band is more complex and there is no obvious explanation of the observed branch asymmetry. Since our observations are of pivotal importance to the electric transport properties of real SWNT devices, this issue certainly deserve further in depth experimental characterization and theoretical explanation.
If the commensurability condition $L= m \pi/\Delta k_\text{A}$ with $m$ integer and $L$ the defect separation is not fulfilled, inter-valley scattering takes place only between electrons with opposite direction of motion and implies an asymmetry of the spots in the $\left| dI/dV(k,V) \right|^{2}$ along the positive and negative slop branches. Contrarily, intra-valley scattering always fulfils this condition with $m=0$, therefore showing symmetric spots around $\Delta k_\text{A}=0$ (see Fig. \[fig3\](c)).
Figure \[fig4\]a shows a current error image of a metallic SWNT which has been exposed to 1.5 keV Ar$^{+}$ ions, and the corresponding $dI/dV(x,V)$-scan recorded along the tube axis through two defect sites labeled d1-d2. Here, instead of clear “textbook-like” modes as shown in Fig. \[QBS\_Fourier\], we observe curved stripes in the interference pattern between defects d1-d2. However, the corresponding $\left| dI/dV(k,V) \right|^{2}$ on the right hand side clearly shows well-defined small momentum spots with an average energy separation of about 90 meV in good agreement with Eq. (\[deltaE\]) giving $\Delta E \simeq 95$ meV for a measured defect separation $L$ of about 18.5 nm. These features are also captured by the Fabry-Pérot electron resonator model if the two impurities have different scattering strengths $\lambda$, as shown for the simulated $dI/dV$-scan with $\lambda_\text{L}=0.05$ and $\lambda_\text{R}=0.15$ in Fig. \[fig4\]b. There is now a clear asymmetry characterized by stripes showing an increasing curvature when moving from the weaker left to the stronger right impurity. There can also be energy dependent differences in the scattering strengths of the defects leading to energy dependent asymmetries in the standing wave pattern as seen in Fig. \[QBS\_1\]. The choice of ion energy can significantly change the type of defects produced and could therefore potentially be used as a parameter to control to some extend the scattering configuration [@Buchs_Ar].
In summary, studying intratube QDs in SWNT by FTSTS in combination with simulation, we provided an analysis of the dominant electron scattering processes. Clear signatures for inter- and inta-valley scattering were observed, and scattering effects arising from lifting the degeneracy of the Dirac cones were identified. The here applied strategy will be useful to investigate the scattering properties of other local modifications of SWNTs like e.g. chemical functionalities.
We thank M. Grifoni, O. Johnsen, S. G. Lemay, T. Nakanishi, Y. Nazarov, D. Passerone, C. Pignedoli, and in particular Ch. Schönenberger for fruitful discussions. This work was supported by the Swiss National Center of Competence in Research MANEP, and the Deutsche Forschungsgemeinschaft (DFG).
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abstract: 'Recent researches show that millimeter wave (mmWave) communications can offer orders of magnitude increases in the cellular capacity. However, the *secrecy* performance of a mmWave cellular network has not been investigated so far. Leveraging the new path-loss and blockage models for mmWave channels, which are significantly different from the conventional microwave channel, this paper comprehensively studies the network-wide physical layer security performance of the downlink transmission in a mmWave cellular network under a stochastic geometry framework. We first study the secure connectivity probability and the average number of perfect communication links per unit area in a noise-limited mmWave network for both non-colluding and colluding eavesdroppers scenarios, respectively. Then, we evaluate the effect of the artificial noise (AN) on the secrecy performance, and derive the analysis result of average number of perfect communication links per unit area in an interference-limited mmWave network. Numerical results are demonstrated to show the network-wide secrecy performance, and provide interesting insights into how the secrecy performance is influenced by various network parameters: antenna array pattern, base station (BS) intensity, and AN power allocation, etc.'
author:
- '[^1]'
title: Physical Layer Security in Millimeter Wave Cellular Networks
---
4.2ex
Millimeter wave network, physical layer security, stochastic geometry, Poisson point process, artificial noise
Introduction
============
Within the next 20 years, wireless data traffic can be anticipated to skyrocket 10,000 folds, spurred by the popularity of various intelligent devices. Conventional communication means are difficult to meet such incredible increase in the wireless data traffic. Millimeter wave cellular communication has received an increasing attention due to the large available bandwidth at millimeter wave frequencies [@5GMillimeter]. Recent field measurements have shown the huge advantages of mmWave networks, compared with the conventional microwave network in band below 6 GHz [@5GMillimeter; @outdoormmWave; @Millimeterpotentials]. Due to the small wavelength, the mmWave cellular network is different from the conventional microwave network in the following ways: large number of antennas, sensitivity to blockages, and variable propagation laws, etc [@CoverageMillimeter]. Recently, based on the real-world measurements in [@Millimeterpotentials], spatial statistical models of the mmWave channel have been built in [@MillimeterChannelModeling], which reveal the different path loss characteristics of the line-of-sight (LOS) and non-line-of-sight (NLOS) links. Under the new channel model, the network-wide performance of a mmWave cellular has attracted increasing attentions, and many works have investigated the SINR distribution, coverage, and average ergodic rate of the network under a stochastic geometry framework [@AnalysisBlockage]-[@MultiTierMillimiterWave]. They show that the mmWave network has a great potential to provide tremendous data traffic increase.
All the above works have focused on the *rate/reliability* performance of the mmWave network, however, its *secrecy* performance has not been investigated so far. Given the ubiquitousness of wireless connections, an enormous amount of sensitive and confidential information, e.g. financial data, electronic cryptography, and private video, have been transmitted via wireless channels. Thus, providing a secure service is one of the top priorities in the design and implementation of mmWave networks [@Safeguarding5G]. In this paper, we investigate the physical layer security performance of mmWave networks by adopting the stochastic geometry framework.
Background
----------
Physical layer security has been identified as a promising strategy that exploits randomness of wireless medium to protect the confidential information from wiretapping [@Wyner], [@Mukherjee2014Principles]. Recently, multiple-antenna technology becomes a powerful tool for enhancing the physical layer security in random networks [@PHYsignalProcessing]. With the degrees of freedom provided by multiple antennas, the transmitter can adjust its antenna steering orientation to exploit the maximum directivity gain while reducing the signal leakage to eavesdroppers [@Secureconnectivity]-[@Geraci2014Physical], or radiate the artificial nosie (AN) for jamming potential eavesdroppers [@Goel:TWC08]-[@Enhancing]. The secure connectivity, secrecy rate and secrecy outage with multi-antenna transmissions in wireless random networks have been studied in [@Secureconnectivity]-[@Geraci2014Physical], respectively. The impact of AN on the security of random networks has been studied in [@Enhancing].
However, all of the above works focus on conventional microwave networks, and the obtained results can not be applied to mmWave networks directly, due to the distinctive features of mmWave channel characteristics. For example, mmWave signals are more sensitive to blockage effects, and the fading statistical characteristics of the LOS link and NLOS link are totally different [@Millimeterpotentials]. For characterizing the blockage effects of mmWave signals, different mmWave channel models have been proposed in [@CoverageMillimeter]-[@MultiTierMillimiterWave]. In [@AnalysisBlockage], an exponential blockage model has been proposed, and such model has been approximated as a LOS ball based blockage model for the coverage analysis in [@CoverageMillimeter; @Coverageandcapacity]. In [@MillimeterAdhoc], the authors adopted the exponential blockage model to perform the coverage and capacity analysis for mmWave ad hoc networks. In [@TractableModelMillimiterWave], the authors proposed a ball based blockage model which is validated by using field measurements in New York and Chicago. Taking the outage state emerging in the mmWave communication into consideration, in [@MultiTierMillimiterWave], the authors have proposed a two-ball approximate blockage model for the analysis of the coverage and average rate of the multi-tier mmWave cellular network.
Under these new characteristics of mmWave channels, the secrecy performance of a mmWave cellular network will be significantly different from the conventional microwave network, which should be re-evaluated. The efficiency of traditional physical layer security techniques should be re-checked as well. Recently, the secrecy performance of a point-to-point mmWave communication has been studied in [@SecureMillimeter], which has shown that mmWave systems can enable significant secrecy improvement compared with conventional microwave systems. However, the *network-wide* secrecy performance of the mmWave cellular communication is still unknown, which motivates our work.
Contribution
------------
In this paper, using the stochastic geometry framework and the blockage model proposed in [@TractableModelMillimiterWave], we proposed a systematic secrecy performance analysis approach for the mmWave cellular communication, by modeling the random locations of the BSs and eavesdroppers as two independent homogeneous Possion point processes (PPPs). Our contributions can be summarized as follows:
1. **Secrecy performance analysis of noise-limited mmWave cellular networks.** We characterize the secrecy performance of a noise-limited mmWave cellular network that is applicable to medium/sparse network deployments, where each BS only adopts the directional beamforming to transmit the confidential information. Considering two cases: the non-colluding eavesdropper case and the colluding eavesdroppers case, we derive the analysis result of the secure connectivity probability and the cumulative distribution function (CDF) of the received SNR at the typical receiver and eavesdropper, respectively. The secure connectivity probability facilitates the evaluation of the probability of the existence of secure connections from a typical transmitter to its intended receiver. With the CDF of the received SNR at the typical receiver and eavesdropper, we can characterize the average number of perfect communication links per unit area statistically in the random network. We show that the high gain narrow beam antenna is very important for enhancing the secrecy performance of mmWave networks.
2. **Secrecy performance analysis of AN assisted mmWave cellular networks.** When AN is transmitted concurrently with the confidential information for interfering potential eavesdroppers, the AN radiation would increase the network interference. Thus, taking the network interference into consideration, we characterize the CDF of received SINRs at the intended receiver and non-colluding eavesdroppers. The secrecy probability and average number of perfect communication links per unit area for the AN assisted transmission have also been derived. The optimal power allocation between the AN and confidential signal is shown to depend on the array pattern and the intensity of eavesdroppers.
Paper Organization and Notations
--------------------------------
In Section II, the system model and mmWave channel characteristics are introduced. In Section III, considering the noise-limited mmWave cellular communication, we characterize the secure connectivity probability and average number of perfect communication links per unit area. In Section IV, taking the inter-cell interference into consideration, we characterize the average number of perfect communication links per unit area of the AN-assisted mmWave communication. Numerical results are provided in Section V and the paper is concluded in Section VI. *Notation:* $x\sim\textrm{gamma}(k,m)$ denotes the gamma-distributed random variable with shape $k$ and scale $m$, $\gamma(x,y)$ is the lower incomplete gamma function [@Table 8.350.1], $\Gamma(x)$ is the gamma function [@Table eq. (8.310)], and $\Gamma(a,x)$ is the upper incomplete function [@Table 8.350.2]. $b(o,D)$ denotes the ball whose center is origin and radius is $D$. The factorial of a non-negative integer $n$ is denoted by $n!$, $\mathbf{x} \sim \mathcal{CN}\left(\mathbf{\Lambda}, \mathbf{\Delta}\right)$ denotes the circular symmetric complex Gaussian vector with mean vector $\mathbf{\Lambda}$ and covariance matrix $\mathbf{\Delta}$, $\binom{n}{k}=\frac{n!}{\left(n-k\right)!k!}$. $\mathcal{L}_X(s)$ denotes the Laplace transform of $X$, i.e., $\mathbb{E}\left(e^{-sX}\right)$. $_2F_1(\alpha,\beta;\gamma,z)$ is the Gauss hypergeometric function [@Table eq. (9.100)].
System Model and Problem Formulation
====================================
We consider the downlink secure communication in the mmWave cellular network, where multiple spatially distributed BSs transmit the confidential information to authorized users in the presence of multiple malicious eavesdroppers. In the following subsections, we first introduce the system model and channel characteristics adopted in this paper, which have been validated in [@CoverageMillimeter; @MillimeterAdhoc; @TractableModelMillimiterWave]. With such models, we give some important results on probability theory which will be used in the performance analysis. The secrecy performance metrics adopted are given in Section II-G.
BS and eavesdropper layout
--------------------------
The locations of the BSs are modeled by a homogeneous PPP $\Phi_B$ of intensity $\lambda_B$. Using PPP for modeling the irregular BSs locations has been shown to be an accurate and tractable approach for characterizing the downlink performance of the cellular network [@wirelessnetworks]. Just as [@Enhancing]-[@D2Dsecure], the locations of multiple eavesdroppers are modeled as an independent homogeneous PPP, $\Phi_E$, of intensity $\lambda_E$. Such random PPP model is well motivated by the random and unpredictable eavesdroppers’ locations. Furthermore, just as [@Enhancing], [@worstcase1]-[@worstcase3], we consider the **worst-case** scenario by facilitating the eavesdroppers’ multi-user decodability, i.e., eavesdroppers can perform successive interference cancellation [@FundamentalsWirelessCommunication] to eliminate the interference due to the information signals from other interfering BSs. The total transmit power of each BS is $P_t$.
Directional beamforming
-----------------------
For compensating the significant path-loss at mmWave frequencies, highly directional beamforming antenna arrays are deployed at BSs to perform the directional beamforming. For mathematical tractability and similar to [@CoverageMillimeter; @MillimeterAdhoc; @TractableModelMillimiterWave; @MultiTierMillimiterWave], the antenna pattern is approximated by a sectored antenna model in [@directionalAntenna]. In particular, $$\begin{aligned}
G_b(\theta)=\left\{
\begin{array}{ll} M_s,&\mathrm{if}\quad |\theta|\leq\theta_b
\\ m_s,
&\textrm{Otherwise},
\end{array}
\right.\label{SectorAntennaModel}\end{aligned}$$ where $\theta_b$ is the beam width of the main lobe, $M_s$ and $m_s$ are the array gains of main and sidelobes, respectively. In this paper, we assume that each BS can get the perfect CSI estimation, including angles of arrivals and fading, and then, they can adjust their antenna steering orientation array for adjusting the boresight direction of antennas to their intended receivers and maximizing the directivity gains. In the following, we denote the boresight direction of the antennas as $0^o$. Therefore, the directivity gain for the intended link is $M_s$. For each interfering link, the angle $\theta$ is independently and uniformly distributed in $\left[-\pi,\pi\right]$, which results in a random directivity gain $G_b(\theta)$. For simplifying the performance analysis, just as [@TractableModelMillimiterWave; @SecureMillimeter], the authorized users and malicious eavesdroppers are both assumed to be equipped with a single omnidirectional antenna in this paper. [^2]
Small-scale fading
------------------
Just as [@CoverageMillimeter; @MillimeterAdhoc], we assume that the small-scale fading of each link follows independent Nakagami fading, and the Nakagami fading parameter of the LOS (NLOS) link is $N_L$ ($N_N$). For simplicity, $N_L$ and $N_N$ are both assumed to be positive integers. In the following, the small-scale channel gain from the BS at $x\in\mathbb{R}^2$ to the authorized user (eavesdropper) at $y\in\mathbb{R}^2$ is expressed as $h_{xy}$ ($g_{xy}$).
Blockage Model
--------------
The blockage model proposed in [@TractableModelMillimiterWave] is adopted, which can be regarded as an approximation of the statistical blockage model in [@MillimeterChannelModeling eq. (8)], [@MultiTierMillimiterWave], and incorporates the LOS ball model proposed in [@CoverageMillimeter; @Coverageandcapacity] as a special case. As shown by [@RateTrendsMillimiterWave; @TractableModelMillimiterWave], the blockage model proposed in [@TractableModelMillimiterWave] is simple yet flexible enough to capture blockage statistics, coverage and rate trends in mmWave cellular networks. In particular, defining $q_L(r)$ as the probability that a link of length $r$ is LOS, $$\begin{aligned}
q_L(r)=\left\{
\begin{array}{ll} C,&\mathrm{if}\quad r\leq D,
\\ 0,
&\textrm{Otherwise},
\end{array}
\right.\label{BlockageModel}\end{aligned}$$ for some $0\leq C\leq 1$. The parameter $C$ can be interpreted as the average LOS area in the spherical region around a typical user. The empirical $(C,D)$ for Chicago and Manhattan are $(0.081,250)$ and $(0.117,200)$, respectively [@TractableModelMillimiterWave], which would be adopted in the simulation results. With such blockage model, the BS process in $b(o,D)$ can be divided into two independent PPPs: the LOS BS process $\Phi_{L}$ with intensity $C\lambda_B$ and NLOS BS process with intensity $(1-C)\lambda_B$ [@wirelessnetworks Proposition 1.3.5]. Outside $b(o,D)$, only the NLOS BS process exists with intensity $\lambda_B$. We denote the whole NLOS BS process as $\Phi_{N}$.
Path loss model
---------------
Just as [@CoverageMillimeter; @TractableModelMillimiterWave], different path loss laws are applied to LOS and NLOS links. In particular, given a link from $x\in\mathbb{R}^2$ to $y\in\mathbb{R}^2$, its path loss $L(x,y)$ can be calculated by $$\begin{aligned}
L(x,y)=\left\{
\begin{array}{ll} C_L||x-y||^{-\alpha_L},&\textrm{ if link } x\rightarrow y\textrm{ is LOS link},
\\
C_N||x-y||^{-\alpha_N},
&\textrm{ if link } x\rightarrow y\textrm{ is NLOS link},
\end{array}
\right.\label{BlockageModel}\end{aligned}$$ where $\alpha_L$ and $\alpha_N$ are the LOS and NLOS path loss exponents, and $C_L\triangleq10^{-\frac{\beta_L}{10}}$ and $C_N\triangleq10^{-\frac{\beta_N}{10}}$ can be regarded as the path-loss intercepts of LOS and NLOS links at the reference distance. Typical $\alpha_j$ and $\beta_j$ for $j\in\{L,N\}$ are defined in [@MillimeterChannelModeling Table I]. For exmple, for 28 GHz bands, $\beta_L=61.4$, $\alpha_L=2,$ and $\beta_N=72$, $\alpha_N=2.92$. From the measured values of $C_j$ and $\alpha_j,$ $j\in\{L,N\}$ in [@MillimeterChannelModeling Table I], we know that it satisfies $C_L>C_N$ and $\alpha_L<\alpha_N$.
User association
----------------
For maximizing the receiving quality of authorized users [@CoverageMillimeter; @TractableModelMillimiterWave], one authorized user is assumed to be associated with the BS offering the lowest path loss to him, since the network considered is homogeneous. Thus, for the typical authorized user at origin, its serving BS is located at $x^*\triangleq\arg\max_{x\in\Phi_B}L(x,o)$. Denoting the distance from the typical authorized user to the nearest BS in $\Phi_j$ as $d^*_{j}$ for $j=\{L,N\}$, the following Lemma 1 provides their probability distribution functions (pdf), and the obtained statistics hold for a generic authorized user, due to Slivnyak’s theorem [@wirelessnetworks].
Given the typical authorized user observes at least one LOS BS, the pdf of $d^*_{L}$ is $$\begin{aligned}
f_{d^*_{L}}(r)=\frac{2\pi C\lambda_B r\textrm{exp}(-\pi C\lambda_B r^2)}{1-\textrm{exp}(-\pi C\lambda_B D^2)},\textrm{ for } r\in\left[0,D\right].\label{pdfdl}\end{aligned}$$ On the other hand, the pdf of $d^*_{N}$ is given by $$\begin{aligned}
f_{d^*_N}(r)=2\pi (1-C)\lambda_Bre^{-\pi(1-C)\lambda_Br^2}\mathbb{I}\left(r\leq D\right)+2\lambda_B\pi re^{-\lambda_B\pi\left(r^2-D^2\right)}e^{-\pi(1-C)\lambda_BD^2}\mathbb{I}\left(r> D\right),
\label{distancedistribution}\end{aligned}$$ where $\mathbb{I}\left(.\right)$ is the indicator function.
The proof is given in Appendix A.
Then, the following lemma gives the probability that the typical authorized user is associated with a LOS or NLOS BS.
The probability that the authorized user is associated with a NLOS BS, $A_N$, is given by $$\begin{aligned}
A_N=&\int^{\mu}_0{\left(\textrm{e}^{-\pi C\lambda_B\left(\frac{C_L}{C_N}\right)^{\frac{2}{\alpha_L}}x^{\frac{2\alpha_N}{\alpha_L}}}-\textrm{e}^{-\pi C\lambda_B D^2}\right)2\pi (1-C)\lambda_Bx}
\textrm{e}^{-\pi (1-C)\lambda_Bx^2}dx+\textrm{e}^{-\pi C\lambda_BD^2},\label{AN}\end{aligned}$$ where $\mu\triangleq\left(\frac{C_L}{C_N}\right)^{-\frac{1}{\alpha_N}}D^{\frac{\alpha_L}{\alpha_N}}$. The probability that the typical authorized user is associated with a LOS BS is given by $A_L=1-A_N$.
The proof is given in Appendix B.
With the smallest path loss association rule, the typical authorized user would be associated with the nearest LOS BS in $\Phi_L$ or the nearest NLOS BS in $\Phi_N$. The following lemma gives the pdf of the distance between the typical authorized user and its serving BS in $\Phi_j$, i.e., $r_j$, $\forall j\in\{L,N\}$.
On the condition that the serving BS is in $\Phi_L$, the pdf of the distance from the typical authorized user to its serving BS in $\Phi_L$ is $$\begin{aligned}
f_{r_L}(r)=\frac{\textrm{exp}\left(-(1-C)\lambda_B\pi\left(\frac{C_N}{C_L}\right)^{\frac{2}{\alpha_N}}r^{\frac{2\alpha_L}{\alpha_N}}\right)2\pi C\lambda_Br\textrm{exp}\left(-C\lambda_B\pi r^2\right)}{A_L}\textrm{, } r\in\left[0,D\right].\label{frL}\end{aligned}$$ On the condition that the serving BS is in $\Phi_N$, the pdf of the distance from the typical authorized user to its serving BS in $\Phi_N$ is $$\begin{aligned}
&f_{r_N}(r)=\frac{2\pi\lambda_Br\textrm{exp}\left(-\pi\lambda_Br^2\right)\left(\left(1-C\right)\textrm{exp}\left(\pi C\lambda_B\left(r^2-D^2\right)\right)\mathbb{I}\left(r\leq D\right)+{\mathbb{I}\left(r\geq D\right)}\right)}{A_N}
\nonumber\\
&+\frac{2\pi (1-C)\lambda_Br e^{-(1-C)\lambda_B\pi r^2}\left(e^{-C\lambda_B\pi\left(\frac{C_L}{C_N}\right)^{\frac{2}{\alpha_L}}r^{\frac{2\alpha_N}{\alpha_L}}}-e^{-C\lambda_B\pi D^2}\right)}{A_N}\mathbb{I}\left(r\leq\left(\frac{C_N}{C_L}\right)^{\frac{1}{\alpha_N}}D^{\frac{\alpha_L}{\alpha_N}}\right).\label{frN}\end{aligned}$$
The proof is given in Appendix C.
Secrecy Performance Metric
--------------------------
In this paper, we assume that the channels are all quasi-static fading channels. The legitimate receivers and eavesdroppers can obtain their own CSI, but mmWave BSs do not know the instantaneous CSI of eavesdroppers. For protecting the confidential information from wiretapping, each BS encodes the confidential data by the Wyner code [@Wyner]. Then two code rates namely, the rate of the transmitted codewords $R_b$, and the rate of the confidential information $R_s$ should be determined before the data transmission, and $R_b-R_s$ is the cost for securing the confidential information. The details of the code construction can be found in [@Wyner; @NewPhysicalLayerSecurity]. In this paper, just as [@Enhancing; @D2Dsecure; @NewPhysicalLayerSecurity], we adopt the fixed rate transmission, where $R_b$ and $R_s$ are fixed during the information transmission. For the secrecy transmission over quasi-static fading channels, the perfect secrecy can not always be guaranteed. Therefore, as indicated in [@Secureconnectivity; @D2Dsecure; @NewPhysicalLayerSecurity], an outage-based secrecy performance metric is more suitable. Therefore, we analyze the secrecy performance of the mmWave communication by considering both the secure connectivity probability and average number of perfect communication links per unit area.
1. [*Secure connectivity probability [@Secureconnectivity]*]{}. Secure connectivity probability introduced in [@Secureconnectivity], is defined as the probability that the secrecy rate is nonnegative. Using the secure connectivity probability, we aim to statistically characterize the existence of secure connection between any randomly chosen BS and its intended authorized user in the presence of multiple eavesdroppers.
2. *Average number of perfect communication links per unit area [@D2Dsecure]*.When $R_b$ and $R_s$ are given, we define the links that have perfect connection and secrecy as perfect communication links. Then, the mathematical definition of the average number of perfect communication links per unit area is given as follows.
- **Connection Probability.** When $R_b$ is below the capacity of legitimate links, authorized users can decode signals with an arbitrary small error, and thus perfect connection can be assured. Otherwise, connection outage would occur. The connection probability is denoted as $p_{con}$.
- **[Secrecy Probability.]{}** When the wiretapping capacity of eavesdroppers is below the rate redundancy $R_e\triangleq R_b-R_s$, there will be no information leakage to potential eavesdroppers, and thus perfect secrecy of the link can be assured [@Wyner]. Otherwise, secrecy outage would occur. The secrecy probability is denoted as $p_{sec}$.
Following [@D2Dsecure eq. (29)], the average number of perfect communication links per unit area is $$\begin{aligned}
N_p = \lambda_Bp_{con}p_{sec}. \label{PerfectCommunicationLink}\end{aligned}$$
With the given $R_b$ and $R_s$, the average achievable secrecy throughput per unit area $\omega$ can be calculated by $
\omega=N_pR_s.
$
Secrecy Performance of the Noise-Limited Millimeter Wave Communication
======================================================================
In this section, we evaluate the secrecy performance of the direct transmission for the noise-limited mmWave communication. As pointed out by [@Millimeterpotentials; @MillimeterChannelModeling; @TractableModelMillimiterWave; @MultiTierMillimiterWave], highly directional transmissions used in mmWave systems combined with short cell radius results in links that are noise-dominated, especially for densely blocked settings (e.g., urban settings) and medium/sparse network deployments [@TractableModelMillimiterWave; @MultiTierMillimiterWave]. This distinguishes from current dense cellular deployments where links are overwhelmingly interference-dominated. Therefore, just as [@TractableModelMillimiterWave; @MultiTierMillimiterWave], we first study the secrecy performance of the noise-limited mmWave communication without considering the effect of inter-cell interference. The received SNR by the typical authorized user at origin and the eavesdropper at $z$ with respect to the serving BS can be expressed as $\textrm{SNR}_U=\frac{P_tM_sL(x^*,o)h_{x^*o}}{N_0}$ and $\textrm{SNR}_{E_z}=\frac{P_tG_b(\theta)L(x^*,z)g_{x^*z}}{N_0}$. $N_0$ is the noise power in the form of $N_0=10^{\frac{N_0(dB)}{10}}$, where $N_0(dB)=-174+10\textrm{log}_{10}(\textrm{BW})+\mathcal{F}_{dB}$, BW is the transmission bandwidth and $\mathcal{F}_{dB}$ is the noise figure [@MultiTierMillimiterWave]. With the array pattern in (\[SectorAntennaModel\]), $G_b(\theta)$ seen by the eavesdropper is a Bernoulli random variable whose probability mass function (PMF) is given by $$\begin{aligned}
G_b(\theta)=\left\{\begin{matrix}
M_s,& \textrm{Pr}_{G_b}(M_s)\triangleq \textrm{Pr}\left(G_b(\theta)=M_s\right)= \frac{\theta_b}{180},
\\
m_s,&\ \ \textrm{Pr}_{G_b}(m_s) \triangleq \textrm{Pr}\left(G_b(\theta)=m_s\right)=\frac{180-\theta_b}{180}.
\end{matrix}
\right.
$$
Non-colluding Eavesdroppers
---------------------------
In this subsection, assuming that the random distributed eavesdroppers are **non-colluding**, we evaluate the secrecy performance of the mmWave cellular network.
### Secure Connectivity Probability
We first study the secure connectivity probability, $\tau_n$, of the mmWave communication in the presence of multiple [non-colluding eavesdroppers]{}. A secure connection is possible if the condition $\frac{M_sL(x^*,o)h_{x^*o}}{\max_{z\in\Phi_E}G_b(\theta)L(x^*,z)g_{x^*z}}\geq 1$ holds [@Secureconnectivity], and the secure connectivity probability can be calculated by $\tau_n=\textrm{Pr}\left(\frac{M_sL(x^*,o)h_{x^*o}}{\max_{z\in\Phi_E}G_b(\theta)L(x^*,z)g_{x^*z}}\geq 1\right)$. We can see that the wiretapping capability of multiple eavesdroppers is determined by the path loss process $G_b(\theta)L(x^*,z)g_{x^*z}$. Thus, for facilitating the performance evaluation, the following process is introduced.
The path loss process with fading (PLPF), denoted as $\mathcal{N}_E$, is the point process on $\mathbb{R}^+$ mapped from $\Phi_E$, where $\mathcal{N}_E\triangleq\left\{\varsigma_z=\frac{1}{G_b(\theta)g_{xz}L(x,z)},z\in\Phi_E\right\}$ and $x$ denotes the location of the wiretapped BS. We sort the elements of $\mathcal{N}_E$ in ascending order and denote the sorted elements of $\mathcal{N}_E$ as $\left\{\xi_i,i=1,\ldots\right\}$. The index is introduced such that $\xi_i\leq \xi_j$ for $\forall i<j$.
Note that $\mathcal{N}_E$ involves both the impact of small fading and spatial distribution of eavesdroppers, which is an ordered process. Consequently, $\mathcal{N}_E$ determines the wiretapping capability of eavesdropper. We then have the following lemma.
The PLPF $\mathcal{N}_E$ is an one-dimensional nonhomogeneous PPP with the intensity measure $$\begin{aligned}
\Lambda_E\left(0,t\right)=&2\pi\lambda_E\left(\sum_{j\in\left\{L,N\right\}}q_j\left(\Omega_{j,\textrm{in}}\left(M_s,t\right)+\Omega_{j,\textrm{in}}\left(m_s,t\right)\right)+\Omega_{N,\textrm{out}}\left(M_s,,t\right)
+\Omega_{N,\textrm{out}}\left(m_s,t\right)\right),\label{LambdaE}\end{aligned}$$ where $q_L\triangleq C$, $q_N\triangleq 1-C$, and $\Omega_{j,\textrm{in}}(V,t) \triangleq \textrm{Pr}_{G_b}(V)\frac{\left(VC_jt\right)^{\frac{2}{\alpha_j}}}{\alpha_j}\sum^{N_j-1}_{m=0}\frac{\gamma\left(m+\frac{2}{\alpha_j},\frac{D^{\alpha_j}}{VC_jt}\right)}{m!}
$, $\Omega_{j,\textrm{out}}(V,t) \triangleq \textrm{Pr}_{G_b}(V)\frac{\left(VC_jt\right)^{\frac{2}{\alpha_j}}}{\alpha_j}\sum^{N_j-1}_{m=0}\frac{\Gamma\left(m+\frac{2}{\alpha_j},\frac{D^{\alpha_j}}{VC_jt}\right)}{m!}
$, with $V\in\left\{M_s,m_s\right\}$.
The proof is given in Appendix D.
The following theorem gives the analysis result of the secure connectivity probability in the presence of non-colluding eavesdroppers.
In the case of non-colluding eavesdroppers, the secure connectivity probability is $$\begin{aligned}
\tau_n=\sum_{j\in\{L,N\}}
A_j\int^D_0f_{r_j}(r)dr\int^{+\infty}_0\frac{\textrm{e}^{-\Lambda_E\left(0,\frac{r^{\alpha_j}}{M_swC_j}\right)}w^{N_j-1}e^{-w}}{\Gamma(N_j)}dw.
\label{SecureConnectivityProbability}\end{aligned}$$
We have the following derivations: $$\begin{aligned}
\textrm{Pr}&\left(\frac{M_sL(x^*,o)h_{x^*o}}{\max_{z\in\Phi_E}G_b(\theta)L(x^*,z)g_{x^*z}}\geq 1\right)=
\textrm{Pr}\left(\min_{z\in\Phi_E}\frac{1}{G_b(\theta)L(x^*,z)g_{x^*z}}\geq \frac{1}{M_sL(x^*,o)h_{x^*o}}\right)
\nonumber\\
&\overset{(e)}{=}\textrm{Pr}\left(\xi_1\geq \frac{1}{M_sL(x^*,o)h_{x^*o}}\right)\overset{(f)}{=}\mathbb{E}_{L(x^*,o),h_{x^*o}}\left(\textrm{exp}\left(-\Lambda_E\left(0,\frac{1}{M_sL(x^*,o)h_{x^*o}}\right)\right)\right)
\nonumber\\
&\overset{(g)}{=}A_L\mathbb{E}_{r_L,h_{x^*o}}\left(\textrm{exp}\left(-\Lambda_E\left(0,\frac{r_L^{\alpha_L}}{M_sC_Lh_{x^*o}}\right)|\textrm{Serving BS is a LOS BS}\right)\right)+
\nonumber\\
&\qquad A_N\mathbb{E}_{r_N,h_{x^*o}}\left(\textrm{exp}\left(-\Lambda_E\left(0,\frac{r_N^{\alpha_N}}{M_sC_Nh_{x^*o}}\right)|\textrm{Serving BS is a NLOS BS}\right)\right),\label{derivationofsecureconnectivity}\end{aligned}$$ where step $(e)$ is due to Definition 1, step $(f)$ follows the PPP’s void probability [@wirelessnetworks], and step ($g$) is due to the law of total probability. When the serving BS is a LOS BS, $h_{x^*o}\sim\textrm{gamma}\left(N_L,1\right)$ and the pdf of $r_L$ is given by (\[frL\]), and when the serving BS is a NLOS BS, $h_{x^*o}\sim\textrm{gamma}\left(N_N,1\right)$ and the pdf of $r_N$ is given by (\[frN\]). Finally, substituting the pdf of $h_{x^*o}$, $r_L$, and $r_N$ into (\[derivationofsecureconnectivity\]), $\tau_n$ can be obtained.
![Secure connectivity probability of the mmWave communication in the presence of multiple non-colluding eavesdroppers vs $\lambda_E$ with BW=2GHz, $P_t=30$ dB, $\mathcal{F}_{dB}=10$, $\theta_b=9^o$, $M_s=15$ dB, and $m_s=-3$ dB.[]{data-label="noncolludingEvessecureconnectivity"}](noncolludingEve2MorePoints){width="2.5in"}
Theoretical results in Theorem 1 are validated in Fig. \[noncolludingEvessecureconnectivity\], where we plot the secure connectivity probability $\tau_{n}$ versus $\lambda_E$. For all the simulations in this paper, 100000 trials are used. From Fig. \[noncolludingEvessecureconnectivity\], we can find that theoretical curves coincide with the simulation ones well, which validates the theoretical result in Theorem 1.
### Average number of perfect communication links per unit area
In the following, we study the average number of perfect communication links per unit area, $N_p$, of the mmWave communication in the presence of **non-colluding** eavesdroppers. Firstly, we should derive the analytical result of the connection probability and secrecy probability of a mmWave communication link, given by $$\begin{aligned}
p_{con}\triangleq\textrm{Pr}\left(\textrm{SNR}_U\geq T_c\right)\textrm{ and }p_{sec,n}\triangleq\textrm{Pr}\left(\max_{z\in\Phi_E}\textrm{SNR}_{E_z}\leq T_e\right),\end{aligned}$$ respectively, where $T_c\triangleq 2^{R_c-1}$ and $T_e\triangleq 2^{R_e-1}$. We have the following theorem.
For the non-colluding eavesdroppers case, the analytical result of $p_{con}$ is given by $$\begin{aligned}
p_{con} = \int^{D}_0\frac{\Gamma\left(N_L,\frac{N_0T_cr^{\alpha_L}}{P_tM_SC_L}\right)}{\Gamma(N_L)}f_{r_L}(r)dr A_L+
\int^{+\infty}_0\frac{\Gamma\left(N_N,\frac{N_0T_cr^{\alpha_N}}{P_tM_SC_N}\right)}{\Gamma(N_N)}f_{r_N}(r)dr A_N,
\label{connectionnon}\end{aligned}$$ and the analytical result of $p_{sec,n}$ is given by $$\begin{aligned}
p_{sec,n}=\textrm{exp}\left(-\Lambda_E\left(0,\frac{1}{TeN_0}\right)\right).\label{Secrecynon}\end{aligned}$$
$p_{con}$ can be derived as follows $$\begin{aligned}
p_{con}=&\textrm{Pr}\left(h_{x^*o}\geq \frac{N _0T_c}{P_tM_SL(x^*,o)}|\textrm{Serving BS is a LOS BS}\right)A_L
+\nonumber\\
&\textrm{Pr}\left(h_{x^*o}\geq \frac{N _0T_c}{P_tM_SL(x^*,o)}|\textrm{Serving BS is a NLOS BS}\right)A_N
\nonumber\\
=&\int^{D}_0\frac{\Gamma\left(N_L,\frac{N_0T_cr^{\alpha_L}}{P_tM_SC_L}\right)}{\Gamma(N_L)}f_{r_L}(r)drA_L+
\int^{+\infty}_0\frac{\Gamma\left(N_N,\frac{N_0T_cr^{\alpha_N}}{P_tM_SC_N}\right)}{\Gamma(N_N)}f_{r_N}(r)drA_N.\end{aligned}$$ $p_{sec,n}$ can be derived as follows $$\begin{aligned}
p_{sec,n}&=\textrm{Pr}\left\{\frac{\max_{z\phi_{E_z}}G_b(\theta)L(x^*,z)g_{x^*,z}}{N_0}\leq T_e\right\}\overset{(g)}{=}\textrm{Pr}\left\{\frac{1}{\xi_1N_0}\leq T_e\right\}\overset{(h)}{=}\textrm{exp}\left(-\Lambda_E\left(0,\frac{1}{T_e N_0}\right)\right),\end{aligned}$$ where step $(g)$ is due to Definition 1, and step $(h)$ is due to the PPP’s void probability [@wirelessnetworks].
Colluding Eavesdroppers
-----------------------
In this subsection, we study the secrecy performance of the mmWave communication by considering the worst case: **colluding eavesdroppers**, where distributed eavesdroppers adopt the maximal-ratio combining to process the wiretapped confidential information.
### Secure Connectivity Probability
The secure connectivity probability $\tau_c$ in the presence of multiple colluding eavesdroppers can be calculated by $$\begin{aligned}
\tau_c=\textrm{Pr}\left(\frac{M_sL(x^*,o)h_{x^*o}}{I_E}\geq 1\right),\end{aligned}$$ where $I_E\triangleq {\sum_{z\in\Phi_E}G_b(\theta)L(x^*,z)g_{x^*z}}$. We have the following theorem.
In the case of colluding eavesdroppers, $\tau_{c}$ can be calculated by $$\begin{aligned}
&\tau_{c} =\mathbb{E}_{r_j}\left[\sum_{j\in\{L,N\}}\sum^{N_{j}-1}_{m=0}\left(\frac{r_j^{\alpha_j}}{M_SC_j}\right)^m\frac{A_j}{\Gamma(m+1)}(-1)^m\mathcal{L}^{(m)}_{I_E}\left(\frac{r_j^{\alpha_j}}{M_SC_j}\right)\right],\end{aligned}$$ where $\mathcal{L}_{I_E}(s)\triangleq\textrm{exp}\left(\Xi(s)\right)$ and $$\begin{aligned}
\Xi(s)\triangleq&-s\left(2\pi\lambda_E\sum_{j\in\{L,N\}}q_j\left(\sum_{V\in\{M_s,m_s\}}\textrm{Pr}_{G_b}(V)\frac{\left(VC_j\right)^{\frac{2}{\alpha_j}}}{\alpha_j}\sum^{N_j-1}_{m=0}\frac{\left(D^{\alpha_j}/(VC_j)\right)^{m+\frac{2}{\alpha_j}}}{\left(m+\frac{2}{\alpha_j}\right)\left(s+{D^{\alpha_j}/(VC_j)}\right)^{m+1}}
\right.\right.
\nonumber\\
&\left.
{_2}F_1\left(1,m+1;m+\frac{2}{\alpha_j}+1;\frac{{D^{\alpha_j}/(VC_j)}}{{D^{\alpha_j}/(VC_j)}+s}\right)\Bigg)+
2\pi\lambda_E\sum_{V\in\{M_s,m_s\}}\textrm{Pr}_{G_b}(V)\frac{\left(VC_N\right)^{\frac{2}{\alpha_N}}}{\alpha_N}
\right.\nonumber\\
&\left.\sum^{N_N-1}_{m=0}
\frac{\left(D^{\alpha_N}/(VC_N)\right)^{m+\frac{2}{\alpha_N}}}{\left(1-\frac{2}{\alpha_N}\right)\left(s+D^{\alpha_N}/(VC_N)\right)^{m+1}}
{_2}F_1\left(1,m+1;2-\frac{2}{\alpha_N};\frac{s}{s+D^{\alpha_N}/(VC_N)}\right)
\right).\end{aligned}$$
The proof is given in Appendix E.
Although the analytical result given in Theorem 3 is general and exact, it is rather unwieldy, motivating the interest in acquiring a more compact expression. Exploring the tight lower bound of the CDF of the gamma random variable in [@Inequality], a tight upper bound of $\tau_{c}$ can be calculated as follows.
$\tau_c$ can be tightly upper bounded by $$\begin{aligned}
&\tau_c\lessapprox
\sum_{j\in\{L,N\}}\sum^{N_j}_{n=1}\binom{N_j}{n}(-1)^{n+1}\int^{+\infty}_0f_{r_j}(r)\mathcal{L}_{I_E}\left(\frac{a_jnr^{\alpha_j}}{M_SC_j}\right)dr,\end{aligned}$$ where $a_L\triangleq(N_L)^{-\frac{1}{N_L}}$ and $a_N\triangleq(N_N)^{-\frac{1}{N_N}}$.
We leverage the tight lower bound of the CDF of a normalized gamma random variable, $g$ with $N$ degrees of freedom as $\textrm{Pr}\left(x\leq y\right)\gtrapprox \left(1-e^{-\kappa y}\right)^N$ [@Inequality], where $\kappa=\left(N!\right)^{-\frac{1}{N}}$. Since $h_{x^*,o}$ is a normalized gamma random variable, we have $$\begin{aligned}
\tau_c\lessapprox 1-\sum_{j\in\{L,N\}}\mathbb{E}_{I_E,r_j}\left(\left(1-\textrm{exp}\left(-\frac{na_jI_Er_j^{\alpha_j}}{M_SC_j}\right)\right)^{N_j}\right).\label{Bound}\end{aligned}$$ Using the binomial expansion, we can obtain $$\begin{aligned}
\tau_c&\lessapprox \sum_{j\in\{L,N\}}\sum^{N_j}_{n=1}\binom{N_j}{n}(-1)^{n+1}\mathbb{E}_{r_j}\left(\mathbb{E}_{I_E}\left(\textrm{exp}\left(-\frac{a_jnr_j^{\alpha_j}I_E}{M_SC_j}\right)\right)\right)
\nonumber\\
&=\sum_{j\in\{L,N\}}\sum^{N_j}_{n=1}\binom{N_j}{n}(-1)^{n+1}\int^{+\infty}_0f_{r_j}(r)\mathcal{L}_{I_E}\left(\frac{a_jnr^{\alpha_j}}{M_SC_j}\right).\end{aligned}$$
![Secure connectivity probability of mmWave communication in the presence of multiple colluding eavesdroppers vs $\lambda_E$. The system parameters are $P_t=30$ dB, $\beta_L=61.4$ dB, $\alpha_L = 2,$ $\beta_N=72$ dB, $\alpha_N=2.92$, BW = 2 GHz, $\mathcal{F}_{dB}=10$, $\lambda_B=0.0005,0.0002,0.00006,C=0.12$, $D=200$ m, $\theta_b=9^o$, $M_s=15$ dB, and $m_s=-3$ dB.[]{data-label="colludingEvessecureconnectivity"}](colludingEavesdropper){width="2.5in"}
The bounds in Theorem 4 are validated in Fig. \[colludingEvessecureconnectivity\], where we plot the secure connectivity probability $\tau_{c}$ versus $\lambda_E$. From Fig. \[colludingEvessecureconnectivity\], we can find that theoretical curves coincide with simulation ones well, which show that the upper bound given in Theorem 4 is tight.
### Average number of perfect communication links per unit area
In the case of colluding eavesdroppers, the connection probability $p_{con}$ of the typical authorized user can be still calculated by (\[connectionnon\]) in Theorem 2, and the achievable secrecy probability $p_{sec}$ can be calculated by $$\begin{aligned}
p_{sec,c}=\textrm{Pr}\left\{\frac{P_tI_E}{N_0}\leq T_e\right\}.\label{colludingEavesdropperpsec2}\end{aligned}$$ For getting the analysis result of $p_{sec,c}$ in (\[colludingEavesdropperpsec2\]), the CDF of $I_E$ should be available. Although the CDF of $I_E$ can be obtained from its Laplace transform $\mathcal{L}_{I_E}(s)$ by using the inverse Laplace transform calculation [@InverseLaplaceTransform], it could get computationally intensive in certain cases and may render the analysis intractable. As an alternative, we resort to an approximation method widely adopted in [@CoverageMillimeter; @MillimeterAdhoc; @TractableModelMillimiterWave] for getting an approximation of $p_{sec,c}$ which is given in the following theorem.
In the case of multiple colluding eavesdroppers, the approximation of $p_{sec,c}$ is given by $$\begin{aligned}
p_{sec,c}\lessapprox\sum^{N}_{n=1}\left(-1\right)^{n+1}\mathcal{L}_{I_E}\left(-\frac{an}{n_0T_e}\right),\label{SecrecyProbabilityBound}\end{aligned}$$ where $\mathcal{L}_{I_E}(s)$ is given in Theorem 3, $a\triangleq(N!)^{\frac{1}{N}}$ and $N$ is the number of terms used in approximation.
$$\begin{aligned}
p_{sec,c}=\textrm{Pr}\left\{\frac{P_tI_E}{N_0T_e}\leq 1\right\}\overset{(i)}{\approx}\textrm{Pr}\left\{\frac{P_tI_E}{N_0T_e}\leq w\right\}.\end{aligned}$$
In $(i)$, $w$ is a normalized gamma random variable with a shape parameter, $N$, and the approximation in $(i)$ is due to the fact that a normalized gamma random variable converges to identity when its shape parameter goes to infinity [@CoverageMillimeter; @MillimeterAdhoc].
Then, using the tight lower of the CDF of a normalized gamma random variable in [@Inequality], $p_{sec}$ can be tightly upper bounded by $$\begin{aligned}
p_{sec,c}\lessapprox 1-\left(1-\textrm{exp}\left(-\frac{a P_tI_E}{N_0T_e}\right)\right)^N.\label{boundDerivation}\end{aligned}$$ Finally, using the binomial expansion, (\[boundDerivation\]) can be further rewritten as (\[SecrecyProbabilityBound\]).
The approximate analysis result in Theorem 5 is validated in Fig. \[colludingEvessecureconnectivity2\]. From Fig. \[colludingEvessecureconnectivity2\], we can find that when $N=5,$ (\[SecrecyProbabilityBound\]) can give an accurate approximation. Then, in the following simulations, we set $N=5$ to calculate $p_{sec,c}$, approximately.
![Secrecy Probability of the mmWave communication in the presence of multiple colluding eavesdroppers vs $T_e$. The system parameters are $P_t=30$ dB, $\theta_b=9^o$, $M_s=15$ dB, $m_a=-3$ dB, $\beta_L=61.4$ dB, $\alpha_L=2,$ $\beta_N=72$ dB, $\alpha_N=2.92$, BW = 2 GHz, $\mathcal{F}_{dB}=10$, $\lambda_B=0.0005,C=0.081$, and $D=250$ m.[]{data-label="colludingEvessecureconnectivity2"}](SecureProbabilityApproximation){width="2.5in"}
Secrecy Performance of the Interference-Limited mmWave Network with AN
======================================================================
AN has been proved to be an efficient secure transmission strategy for the conventional cellular network [@Goel:TWC08; @Enhancing]. But for the mmWave network, the utility of the AN should be re-evaluated due to the distinguishing features of the mmWave communication. In this section, we will analyze the secrecy performance of the AN-assisted mmWave communication. Since the additional AN would increase the network interference, different from the previous section, we analyze the secrecy performance of the AN assisted mmWave communication by taking the inter-cell interference into consideration[^3]. For obtaining a tractable problem, we only study the second secrecy performance metric: the average number of perfect communication links per unit area for the **non-colluding** eavesdroppers case.
By introducing different phase shifts in each directional antenna, each BS can concentrate the transmit power of the confidential information signals into the direction of its intended receiver, while radiating AN uniformly in all other directions. For tractability of the analysis, the actual array pattern of each BS is approximated by the model of sectoring with artificial noise proposed in [@Enhancing Section II-A]. In particular, for the confidential information signals, it has main lobe of gain $M_s$ and angle of spread $\theta_b$, and just as [@Enhancing; @MicrowaveCommunication], the sidelobes of the confidential information signals are suppressed sufficiently, which can be omitted in the following[^4]. Accordingly, for the AN, it has main lobe of gain $M_a$ and angle of spread $360-\theta_b$, and the sidelobes of the AN are suppressed sufficiently, which can also be omitted. The sectors of the confidential signals and AN are non-overlapping.
Assuming that $\phi P_t$ is allocated to transmit the confidential information in the intended sector, and $(1-\phi) P_t$ is allocated to transmit AN concurrently out of intended sectors, where $0\leq \phi\leq 1$. The transmit power $x(\theta)$ of each BS is $$\begin{aligned}
x(\theta){=}
\left\{
\begin{array}{lll}
M_s\phi P_t,&\mathrm{if}\quad |\theta|\leq \theta_b, &\textrm{Pr}_{x}(M_s)\triangleq \textrm{Pr}\left(x(\theta)=M_s\phi P_t\right) =\frac{\theta}{180},
\\ M_a(1-\phi)P_t,
&\textrm{Otherwise},&\textrm{Pr}_{x}(M_a)\triangleq\textrm{Pr}\left(x(\theta)=M_a(1-\phi)P_t\right) =\frac{180-\theta}{180}.
\end{array}
\right.
\label{SectorAntennaModelArtificialNoise}\end{aligned}$$
Then, according to the mapping theorem, for any receiver (authorized user or eavesdropper), the interfering BSs can be divided into two independent PPPs: 1) the one transmitting the confidential signals to the receiver, which is denoted by $\Phi_I$ of intensity ${\lambda_B}\textrm{Pr}_{x}(M_s)$; 2) the one transmitting the AN to the receiver, which is denoted by $\Phi_A$ of intensity ${\lambda_B}\textrm{Pr}_{x}(M_a)$.
Connection Probability
----------------------
Considering the typical authorized user at the origin, its received SINR$_U$ can be calculated by $$\begin{aligned}
\textrm{SINR}_U=\frac{\phi P_tM_sh_{x^*o}L\left(x^*,o\right)}{I_B+N_0},\end{aligned}$$ where the interference from multiple interfering BSs: $I_B\triangleq\sum_{y\in\Phi_I/{x^*}}M_s\phi P_th_{yx^*}L(y,x^*)+\sum_{y\in\Phi_A}M_a(1-\phi) P_th_{yx^*}L(y,x^*)$. We have the following theorem.
The connection probability of the typical communication link can be tightly upper bounded by $$\begin{aligned}
p_{con}\triangleq\textrm{Pr}\left(\textrm{SINR}_U\geq {T}_c\right)\lessapprox
A_L\int^{D}_{0}\Xi_Lf_{r_L}(r)dr+A_N\int^{+\infty}_{0}\Xi_Nf_{r_N}(r)dr,\end{aligned}$$ where $$\begin{aligned}
&\Xi_L\triangleq\sum^{N_L}_{n=1}(-1)^{n+1}\binom{N_L}{n}\textrm{exp}\left(-\Theta_{L}(n)r^{\alpha_L}N_0\right)
\prod_{k=1}^3\varpi_{k}(M_s,\phi,n)\prod_{k=1}^3\varpi_{k}(M_a,1-\phi,n),
\\
&\Xi_N\triangleq\sum^{N_L}_{n=1}(-1)^{n+1}\binom{N_L}{n}\textrm{exp}\left(-\Theta_{N}(n)r^{\alpha_N}N_0\right)
\prod_{k=1}^3\varphi_{k}(M_s,\phi,n)\prod_{k=1}^3\varphi_{k}(M_a,1-\phi,n),
\\
&\varpi_1(a,b,n)\triangleq\Delta(C,r,D,\Theta_{L}(n),r^{\alpha_L},a,b),
\nonumber\\
&\varpi_2(a,b,n)\triangleq\Delta\left(1-C,{\min\left(\left(\frac{C_N}{C_L}\right)^{\frac{1}{\alpha_N}}r^{\frac{\alpha_L}{\alpha_N}},D\right)},D,\Theta_{L}(n),r^{\alpha_L},a,b\right)
,\nonumber\\
&\varpi_3(a,b,n)\triangleq\Delta\left(1,{\max\left(\left(\frac{C_N}{C_L}\right)^{\frac{1}{\alpha_N}}r^{\frac{\alpha_L}{\alpha_N}},D\right)},+\infty,\Theta_{L}(n),r^{\alpha_L},a,b\right)
,\nonumber\\
&\varphi_1(a,b,n)\triangleq\Delta\left(C,{\min\left(\left(\frac{C_L}{C_N}\right)^{\frac{1}{\alpha_L}}r^{\frac{\alpha_N}{\alpha_L}},D\right)},D,\Theta_{N}(n),r^{\alpha_N},a,b\right),
\nonumber\\
&\varphi_2(a,b,n)\triangleq\Delta\left(1-C,{\min\left(r,D\right)},D,\Theta_{N}(n),r^{\alpha_N},a,b\right),
\nonumber\\
&\varphi_3(a,b,n)\triangleq\Delta\left(1,\min(r,D),+\infty,\Theta_{N}(n),r^{\alpha_N},a,b\right),
$$ $a_L$ and $a_N$ have been defined in Theorem 4, the analysis results of $f_{r_L}(r)$ and $f_{r_N}(r)$ have been given in Lemma 3, $\Theta_{L}(n)\triangleq\frac{{T}_c n a_L}{\phi P_tM_s C_L}$, $\Theta_{N}(n)\triangleq\frac{{T}_c n a_N}{\phi P_tM_s C_N}$, $F(c,d,a,b) \triangleq1-\frac{1}{\left(1+cdab P_tC_Ly^{-\alpha_L}\right)^{N_L}}
$, $\Delta(z,v,y,c,d,a,b)\triangleq\textrm{exp}\left(-{2\pi z\lambda_B\textrm{Pr}_{x}(a)}\int^{y}_vF\left(c,d,a,b\right)ydy\right)$,
The proof is given in Appendix F.
![Connection probability of mmWave communication with AN vs $T_c$. The system parameters are $P_t=30$ dB, $\phi=0.5,\theta_b=9, M_s=15$dB, $M_a=3$dB, $\beta_L=61.4$ dB, $\alpha_L=2,$ $\beta_N=72$ dB, $\alpha_N=2.92$, BW = 2 GHz, $\mathcal{F}_{dB}=10$, $C=0.12$, and $D=200$ m.[]{data-label="ConnectionprobabilityArtificialNoise"}](ConnectionOutage){width="2.5in"}
In Fig. \[ConnectionprobabilityArtificialNoise\], we plot the connection probability $p_{con}$ versus $T_c$. From Fig. \[ConnectionprobabilityArtificialNoise\], we can find that approximate results coincide with simulation ones well, which show that the approximate analysis result given in Theorem 6 is tight. In addition, from simulation results, we can find an interesting phenomenon that for some $T_c$, a larger $\lambda_B$ may result in a smaller $p_{con}$. Therefore, we can conclude that $p_{con}$ is not a monotonically increasing function of $\lambda_B$ for the whole range of $T_c$. This can be explained by the fact that although the distance from the authorized user to its serving BS decreases with the increasing $\lambda_B$, the network interference also increases. Therefore, increasing $\lambda_B$ may not always improve $p_{con}$. This further shows that the mmWave network with AN is interference-limited.
Secrecy Probability
-------------------
In this subsection, we characterize the secrecy probability of the AN assisted mmWave communication. With the approximation (\[SectorAntennaModelArtificialNoise\]), only eavesdroppers inside the intended sector of the serving BS would wiretap the confidential information. Those eavesdroppers form a fan-shaped PPP and by the mapping theorem [@wirelessnetworks], they can be mapped as a homogeneous PPP on the whole plane, denoted by $\Phi_Z$ with density ${\lambda_E}\textrm{Pr}_{x}(M_s)$. Since we consider the worst-case where each eavesdropper can eliminate the interference due to the information signals from other interfering BSs, only the AN would deteriorate the receiving performance of eavesdroppers. Then, the received SINR by the eavesdropper at $z$ can be calculated as $$\begin{aligned}
\textrm{SINR}_z=\frac{\phi P_tM_SL(x^*,z)g_{x^*z}}{I_{A_z}+N_0},\label{SINREZ}\end{aligned}$$ where $I_{A_z}=\sum_{y\in\Phi_{A}}\left(1-\phi\right)P_tM_ag_{y,z}L(y,z)$.
In the case of non-colluding eavesdroppers, the secrecy probability of the mmWave network with AN can be calculated as $p_{sec}=\mathbb{E}_{\Phi_Z,\Phi_{A}}\left(\prod_{z\in\Phi_Z}\textrm{Pr}\left(\textrm{SINR}_z\leq T_e\right)\right)$, which is characterized by the following theorem.
The secrecy probability can be tightly lower bounded by $$\begin{aligned}
p_{sec}\gtrapprox&\textrm{exp}\left(-2\pi C\lambda_E\textrm{Pr}_{x}(M_s)\int^{D}_0\Omega_L(r)rdr-2\pi (1-C)\lambda_E\textrm{Pr}_{x}(M_s)\int^{D}_0\Omega_N(r)rdr\right)
\nonumber\\
&\textrm{exp}\left(-2\pi \lambda_E\textrm{Pr}_{x}(M_s)\int^{+\infty}_D\Omega_N(r)rdr\right),\end{aligned}$$ where $a_j\triangleq\left(N_j!\right)^{-\frac{1}{N_j}}$, $
\Omega_j(r)\triangleq\sum^{N_j}_{n=1}(-1)^{n+1}\binom{N_j}{n}\textrm{exp}\left(-\frac{na_jT_er^{\alpha_j}N_0}{\phi P_tM_sC_j}\right)
\textrm{exp}\left(\Psi\left(\frac{(1-\phi)na_jT_er^{\alpha_j}}{\phi M_sC_j}\right)\right),\quad j=\{L,N\},
$ and $$\begin{aligned}
\Psi(s)\triangleq&-s\left(2\pi\lambda_B\sum_{j\in\{L,N\}}q_j\left(\textrm{Pr}_{x}(M_a)\frac{\left(M_aC_j\right)^{\frac{2}{\alpha_j}}}{\alpha_j}\sum^{N_j-1}_{m=0}\frac{\left(D^{\alpha_j}/(yC_j)\right)^{m+\frac{2}{\alpha_j}}}{\left(m+\frac{2}{\alpha_j}\right)\left(s+{D^{\alpha_j}/(yC_j)}\right)^{m+1}}
\right.\right.
\nonumber\\
&\left.\left.
{_2}F_1\left(1,m+1;m+\frac{2}{\alpha_j}+1;\frac{{D^{\alpha_j}/(M_aC_j)}}{{D^{\alpha_j}/(M_aC_j)}+s}\right)\right)+
2\pi\lambda_E\textrm{Pr}_{x}(M_a)\frac{\left(M_aC_N\right)^{\frac{2}{\alpha_N}}}{\alpha_N}
\right.\nonumber\\
&\left.\sum^{N_N-1}_{m=0}
\frac{\left(D^{\alpha_N}/(M_aC_N)\right)^{m+\frac{2}{\alpha_N}}}{\left(1-\frac{2}{\alpha_N}\right)\left(s+D^{\alpha_N}/(M_aC_N)\right)^{m+1}}
{_2}F_1\left(1,m+1;2-\frac{2}{\alpha_N};\frac{s}{s+D^{\alpha_N}/(M_aC_N)}\right)
\right).\end{aligned}$$
The proof is given in Appendix G.
![Secrecy probability of mmWave communication with AN versus $T_e$. The system parameters are $P_t=30$ dB, $\theta_b=9^o$, $M_s=15$dB, $M_a=3$dB, BW = 2 GHz, $\beta_L=61.4$ dB, $\alpha_L=2,$ $\beta_N=72$ dB, $\alpha_N=2.92$, $\mathcal{F}_{dB}=10$, $\phi=0.5$, $P_t=30$dB, $\lambda_B=0.00005,C=0.12$, and $D=200$ m.[]{data-label="SecrecyProbabilityValidation"}](SecrecyProbability){width="2.5in"}
Theoretical results in Theorem 7 are validated in Fig. \[SecrecyProbabilityValidation\]. In Fig. \[SecrecyProbabilityValidation\], we plot the secrecy probability $p_{sec}$ versus $T_e$. From Fig. \[SecrecyProbabilityValidation\], we can find that the approximate results coincide with the simulation ones well, which show that the lower bound given in Theorem 7 is tight.
Simulation Result
=================
In this section, more representative simulation results are provided to characterize the secrecy performance of mmWave networks and the effect of different network parameters. Considering mmWave networks operating at a carrier frequency $F_c=28$ GHz, the path-loss model are taken from [@MillimeterChannelModeling Tables I]. Specially, the transmission bandwidth BW = 2 GHz, the noise figure $\mathcal{F}_{dB}=10$, the BS’s transmit power $P_t=30$ dB, the Nakagami fading parameters of the LOS (NLOS) link are $N_L=3$ ($N_N=2$), and the path-loss model: $\beta_L=61.4$ dB, $\alpha_L=2,$ $\beta_N=72$ dB, $\alpha_N=2.92$. Since the theoretical analysis results obtained in this paper have been validated by the simulation results in Fig. \[noncolludingEvessecureconnectivity\]-Fig. \[SecrecyProbabilityValidation\], all of the simulation results in this section are theoretical analysis results.
Secrecy performance evaluation of noise-limited mmWave cellular networks
------------------------------------------------------------------------
In this subsection, employing analysis results in Section III, we illustrate the secrecy performance of noise-limited mmWave networks in the presence of **non-colluding** and **colluding** eavesdroppers.
![Secrecy connectivity probability of the noise-limited mmWave communication in the presence of multiple eavesdroppers versus $\lambda_E$. The system parameters are $\lambda_B=0.00005,C=0.081$, and $D=250$ m.[]{data-label="SecrecyconnectivityVersusLambdaE"}](SectorGainSecureConnection){width="2.5in"}
Fig. \[SecrecyconnectivityVersusLambdaE\] plots the secrecy connectivity probability of the mmWave communication in the presence of multiple non-colluding and colluding eavesdroppers versus $\lambda_E$. Obviously, the wiretapping capability of the colluding eavesdroppers is larger than the non-colluding case. Therefore, compared with the non-colluding eavesdroppers, the secrecy connectivity probability for the colluding case deteriorates. With the increasing $\lambda_E$, the wiretapping capability of eavesdroppers increases and the secrecy connectivity probability decreases. Furthermore, the secrecy performance would be improved with the improving directionality of the beamforming of each BSs. This can be explained by the fact that the high gain narrow beam antenna decreases the information leakage, improves the receive performance of the authorized user, and increases the secure connectivity probability.
Fig. \[NumberNPversusLambdaE\] plots $N_p$ versus $\lambda_E$. Compared with the non-colluding eavesdroppers, the performance deterioration of the colluding case increases with the increasing $\lambda_E$, especially for the BS equipped with highly directional antenna arrays. Furthermore, the simulation results show that the directional beamforming is very important for the secrecy communication. For example, for the non-colluding eavesdroppers case, when $\theta_b=9^o,M_s=15$dB, $M_a=3$dB, $\lambda_E=4\times 10^{-4}$, $N_p\approx 1.1\times 10^{-4}$ and more than half of communication links is perfect, on average. However, for other two cases of array patterns, $N_p$ reduces greatly due to the increasing beam width of the main lobe and the decreasing array gains of the intended sector.
![Average number of perfect communication links per unit, $N_p$ of the noise-limited mmWave communication in the presence of multiple eavesdroppers versus $\lambda_E$. The system parameters are $P_t=30$ dB, $T_c$=10dB, $T_e$=0 dB, $\lambda_B=0.0002,C=0.12$, and $D=200$ m.[]{data-label="NumberNPversusLambdaE"}](NumberNP){width="2.5in"}
The simulation results above show that the directional beamforming of BSs is very important for the secrecy performance of noise-limited mmWave networks. Therefore, in practice, for improving the security of noise-limited mmWave networks, BSs should perform the highly directional beamforming.
Secrecy performance evaluation of interference-limited mmWave cellular networks with AN
---------------------------------------------------------------------------------------
In this subsection, employing analysis results in Section IV, we illustrate the impact of the AN on the secrecy performance of interference-limited mmWave networks in the presence of **non-colluding** eavesdroppers.
![Average number of perfect communication links per unit, $N_p$ of the interference-limited mmWave communication in the presence of multiple non-colluding eavesdroppers versus the power allocation coefficient $\phi$ and $\lambda_E$. The system parameters are $\theta_b=30^o$, $M_s=10$ dB, $M_a=3$ dB, $T_c$ = 10dB, $T_e$ = 0 dB, $\lambda_B=0.001,C=0.12$, and $D=200$ m.[]{data-label="TwoDimensionalNp"}](NpANversusphi){width="2.5in"}
![Average number of perfect communication links per unit, $N_p$ of the interference-limited mmWave communication in the presence of multiple non-colluding eavesdroppers versus the power allocation coefficient $\phi$ and $\lambda_E$. The system parameters are $\theta_b=30^o$, $M_s=10$ dB, $M_a=3$ dB, $T_c$ = 10dB, $T_e$ = 0 dB, $\lambda_B=0.001,C=0.12$, and $D=200$ m.[]{data-label="TwoDimensionalNp"}](TwoDimensionalNp){width="2.5in"}
Fig. \[NumberNPArtificialNoiseVersusPhi\] plots $N_p$ achieved by the mmWave communication versus the power allocation coefficient $\phi$ for different antenna patterns and $\lambda_E$. From the simulation results in Fig. \[NumberNPArtificialNoiseVersusPhi\], we find that the optimal fraction of the power allocated to the AN decreases with the decreasing $\lambda_E$ and improving directivity of the antenna array equipped at each BS. This can be explained by the fact that with the decreasing $\lambda_E$, the wiretapping capability of eavesdroppers decreases, and the optimal fraction of the power allocated to the AN can be reduced. Accordingly, with the highly directional beamforming, the information leakage decreases, and the receiving performance of eavesdroppers decreases. Therefore, the power allocated to the AN can be reduced. For showing the effect of $\lambda_E$ and $\theta_b$ on the optimal $\phi$ further, we plot $N_p$ versus $\phi$ and $\lambda_E$ in Fig. \[TwoDimensionalNp\], and $N_p$ versus $\phi$ and $\theta_b$ in Fig. \[NumberNPversusThetab\]. From the simulation results, it is clear that the optimal $\phi$ for maximizing $N_P$ increases with the decreasing $\lambda_E$ and decreasing $\theta_b$, which validates the conclusions draw above.
![Average number of perfect communication links per unit, $N_p$ of the interference-limited mmWave communication in the presence of multiple non-colluding eavesdroppers versus the power allocation coefficient $\phi$ and $\theta_b$. The system parameters are $P_t=30$ dB, $M_s=15$ dB, $M_a=3$ dB, $T_c$=10dB, $T_e$=0 dB, $\lambda_B=0.0008$, $\lambda_E=0.001$, $C=0.12$, and $D=200$ m.[]{data-label="NumberNPversusThetab"}](Twodimisinalthetabandphi){width="2.5in"}
Simulation results show that the optimal power allocated to AN depends on $\lambda_E$ and antenna pattern. The highly directional antenna array and small $\lambda_E$ both would decrease the power allocated to AN.
Secrecy performance comparison between the AN assisted microwave network and the AN assisted mmWave network
-----------------------------------------------------------------------------------------------------------
![Secrecy performance comparison between the microwave network and mmWave network. The system parameters of the mmWave network are $P_t=30$ dB, $T_c$=0dB, $T_e$=-30 dB, $\theta_b=9^o$, $M_s=15$ dB, $M_a=3$ dB, $\lambda_B=0.0008,C=0.12$, and $D=200$ m. The system parameters of the microwave network are $P_t=30$ dB, $T_c$=0dB, $T_e$=-30 dB, the beam width of the confidential signal is $60^o$, the beam width of AN is $300^o$, the number of antennas equipped at each microwave BS is 6, $\lambda_B=0.0008$, the small scale channel fading follows the normalized Rayleigh fading. .[]{data-label="ComparingwithMicrowaveNetwork"}](MicroWaveNetwork){width="2.5in"}
For validating the secrecy performance of the mmWave communication, we perform the secrecy performance comparison between the microwave network and mmWave network in Fig. \[ComparingwithMicrowaveNetwork\], where the intensities of microwave BSs and mmWave BSs are both set to be 0.0008. The carrier frequency of the microwave communication is $F_c=2.5$ GHz. Just as [@Enhancing; @MicrowaveCommunication], the antenna pattern of the microwave BS is approximated by (\[SectorAntennaModelArtificialNoise\]), where the beam width of the confidential signal is set to be $60^o$, and the beam width of AN is set to be $300^o$, the small scale channel fading of the microwave communication is the normalized Rayleigh fading. The large-scale path loss of the urban area cellular radio communication is used to model the path-loss of the microwave communication [@3GPP], where the path-loss exponent is 2.7. Since the microwave network is interference-limited [@Enhancing], the received noise power is ignored in the simulation. Therefore, only the ratio between the antenna gain of the confidential signals and the antenna gain of AN would determine the SINR received at the typical authorized user and eavesdropper in the microwave network [@Enhancing]. Just as [@Enhancing], the ratio between the antenna gain of the confidential signals and the antenna gain of AN in the microwave BS is set to be $M-1$, where $M$ is the number of antennas equipped at the microwave BS. In the simulation results of Fig. \[ComparingwithMicrowaveNetwork\], we set $M=6$. From the simulation results in Fig. \[ComparingwithMicrowaveNetwork\], we can find that the mmWave network can achieve better secrecy performance than the microwave network. This is because the unique characteristic of the mmWave communication: blockage effects and highly directional beamforming antenna arrays. Due to blockage effects, the wiretapping capability of eavesdroppers would decreases, since the blockage effects would deteriorate the reception quality of a large portion of eavesdroppers. Due to the highly directional beamforming antenna arrays equipped at each mmWave BS, the reception quality of the intended receiver would be improved, and the confidential information leakage would be decreased.
Conclusions
===========
In this paper, considering distinguishing features of the mmWave cellular network, we characterize the secrecy performance of the noise-limited mmWave network and the AN-assisted mmWave network. For the noise-limited case, we analyze the secure connectivity probability and average number of perfect communication links per unit area for colluding and non-colluding eavesdroppers. For the AN-assisted mmWave network which is interference-limited, by taking the network interference into consideration, we characterize the distributions of the received SINRs at the intended receiver and eavesdroppers, and average number of perfect communication links per unit area for non-colluding eavesdroppers. Simulation results show that the array pattern and intensity of eavesdroppers are very important system parameters for improving the secrecy performance of the mmWave communication. In particular, for the AN-assisted mmWave networks, the power allocated to AN depends on the array pattern and the intensity of eavesdroppers. It decreases with the decreasing beam width of the main lobe and decreasing intensity of eavesdroppers.
Proof of Lemma 1
================
We first show the derivation of $f_{d_ L^*}(r)$. Given the typical authorized user observes at least one LOS BS, the complementary cumulative distribution function (CCDF) of $d_ L^*$ can be derived as $$\begin{aligned}
\textrm{Pr}\left(d_ L^*\geq r\right)&\triangleq\textrm{Pr}\left(\Phi_{B_L}\left(B(o,r)\right)=0|\Phi_{B_L}\left(B(o,D)\right)\neq 0\right)
=
\frac{\textrm{e}^{-C\lambda_B\pi r^2}\left(1-\textrm{e}^{-C\lambda_B\pi \left(D^2-r^2\right)}\right)}{1-\textrm{e}^{-C\lambda_B\pi D^2}},\label{CDFDL}\end{aligned}$$ Then, with (\[CDFDL\]), the pdf $f_{d_ L^*}(r)=-\frac{d \textrm{Pr}\left(d_ L^*\geq r\right)}{dr}$ that can be derived as (\[pdfdl\]).
Secondly, invoking the PPP’s void probability [@wirelessnetworks], the CCDF $\textrm{Pr}\left(d^*_N\geq r\right)$ can be derived as $$\begin{aligned}
&\textrm{Pr}\left(d^*_N\geq r\right)=
\nonumber\\
&\textrm{Pr}\left(\Phi_{B_N}\left(B(o,r)\right)=0\right)\mathbb{I}(r\leq D)+\textrm{Pr}\left(\Phi_{B_N}\left(B(o,D)\right)=0,\Phi_{B_N}\left(B(o,r)/B(o,D)\right)=0\right)
\mathbb{I}(r> D)
\nonumber\\
&=\textrm{exp}\left(\left(1-C\right)\lambda_B\pi r^2\right)\mathbb{I}(r\leq D)
+\textrm{exp}\left(\left(1-C\right)\lambda_B\pi D^2\right)
\textrm{exp}\left(-\lambda_B\pi \left(r^2-D^2\right)\right)
\mathbb{I}(r> D).\end{aligned}$$ Finally, calculating $-\frac{d \textrm{Pr}\left(d^*_N\geq r\right)}{dr}$, the pdf $f_{d_ N^*}(r)$ can be derived as (\[distancedistribution\]).
Proof of Lemma 2
================
We do the following derivations. $$\begin{aligned}
A_N&=\textrm{Pr}\left(C_L(d^*_L)^{-\alpha_L}\leq C_N(d^*_N)^{-\alpha_N}\right)\textrm{Pr}\left(\Phi_{B_L}(B(o,D))\neq 0\right)+\textrm{Pr}\left(\Phi_{B_L}(B(o,D))= 0\right)
\nonumber\\
&={\textrm{Pr}\left(\left(\frac{C_L}{C_N}\right)^{\frac{1}{\alpha_L}}(d^*_N)^{\frac{\alpha_N}{\alpha_L}}\leq d^*_L\right)\left(1-\textrm{e}^{-\pi C\lambda_BD^2}\right)}+\textrm{e}^{-\pi C\lambda_BD^2}
\nonumber\\
&=
\mathbb{E}_{d^*_N\leq \left(\frac{C_L}{C_N}\right)^{-\frac{1}{\alpha_N}}D^{\frac{\alpha_L}{\alpha_N}}}\left({ \textrm{e}^{-\pi C\lambda_B\left(\frac{C_L}{C_N}\right)^{\frac{2}{\alpha_L}}(d_N^*)^{\frac{2\alpha_N}{\alpha_L}}}-\textrm{e}^{-\pi C\lambda_B D^2} }\right)
+{\textrm{e}^{-\pi C\lambda_B D^2}}.\label{ALproof}\end{aligned}$$ Finally, (\[AN\]) can be obtained by employing the pdf $f_{d^*_N}(r)$ in (\[distancedistribution\]). Accordingly, $A_L$ can be derived as $$\begin{aligned}
&A_L=\textrm{Pr}\left(C_L(d^*_L)^{-\alpha_L}\geq C_N(d^*_N)^{-\alpha_N}\right)\textrm{Pr}\left(\Phi_{B_L}(B(o,D))\neq 0\right)
\nonumber\\
&=\left(\textrm{Pr}\left(D\geq\left(\frac{C_L}{C_N}\right)^{\frac{1}{\alpha_L}}\left(d^*_N\right)^{\frac{\alpha_N}{\alpha_L}}\geq d_L^*\right)+\textrm{Pr}\left(D\leq\left(\frac{C_L}{C_N}\right)^{\frac{1}{\alpha_L}}\left(d^*_N\right)^{\frac{\alpha_N}{\alpha_L}}\right)\right)\textrm{Pr}\left(\Phi_{B_L}(B(o,D))\neq 0\right)
\nonumber\\
&{=}\left(\int^\mu_0\left(\textrm{Pr}\left(\left(\frac{C_L}{C_N}\right)^{\frac{1}{\alpha_L}}r^{\frac{\alpha_N}{\alpha_L}}\geq d^*_L\right)\right)f_{d_N^*}(r)dr+\textrm{Pr}\left(d^*_N\geq\mu\right)\right)\textrm{Pr}\left(\Phi_{B_L}(B(o,D))\neq 0\right).\label{additonalproof}\end{aligned}$$ Since $\textrm{Pr}\left(d_ L^*\leq r\right)=1-\textrm{Pr}\left(d_ L^*\geq r\right)$ and $\textrm{Pr}\left(d_ L^*\geq r\right)$ has been defined in (\[CDFDL\]), substituting the analytical expression of $\textrm{Pr}\left(d_ L^*\leq r\right)$ and $f_{d_N^*}(r)$ in (\[distancedistribution\]) into (\[additonalproof\]), we obtain $A_L=1-A_N$.
Proof of Lemma 3
================
We first show the derivation of $f_{r_L}(r)$. The CCDF $\textrm{Pr}\left(r_L\geq r\right)$ can be derived as $$\begin{aligned}
&\textrm{Pr}\left(r_L\geq r\right)\triangleq\textrm{Pr}\left(\Phi_{B_L}(B(o,r))=0\left|C_L(d^*_L)^{-\alpha_L}\geq C_N(d^*_N)^{-\alpha_N} \right.\right)
\nonumber\\
&=
\frac{\textrm{Pr}\left(\Phi_{B_L}(B(o,D))\neq 0\right)\textrm{Pr}\left(r\leq d^*_L,C_L(d^*_L)^{-\alpha_L}\geq C_N(d^*_N)^{-\alpha_N}\right)}{\textrm{Pr}\left(C_L(d^*_L)^{-\alpha_L}\geq C_N(d^*_N)^{-\alpha_N}\right)}
\nonumber\\
&\overset{(a)}{=}\frac{1}{A_L}\int^{D}_r\textrm{Pr}\left(\Phi_N\cap b\left(o,\left(\frac{C_N}{C_L}\right)^{\frac{1}{\alpha_N}}y^{\frac{\alpha_L}{\alpha_N}}\right)=\emptyset\right)2\pi C\lambda_By\textrm{exp}\left(-\pi C\lambda_By^2\right)dy.\label{distancefrlproof}\end{aligned}$$ Step $(a)$ can be derived by the PPP’s void probability and $f_{d^*_L}(r)$. Then calculating $f_{r_L}(r)=-\frac{d \textrm{Pr}\left(r_L\geq r\right)}{dr}$, the pdf $f_{r_L}(r)$ can be derived as (\[frL\]).
We show the derivation of $f_{r_N}(r)$. The CCDF $\textrm{Pr}\left(r_N\geq r\right)$ is equivalent to the conditional CCDF $$\begin{aligned}
&\textrm{Pr}\left(r_N\geq r\right)\triangleq\textrm{Pr}\left(d^*_N\geq r\left| C_L(d^*_L)^{-\alpha_L}\leq C_N(d^*_N)^{-\alpha_N}\right.\right)=
\nonumber\\
&\frac{\textrm{Pr}\left(r\leq d^*_N,C_L(d^*_L)^{-\alpha_L}\leq C_N(d^*_N)^{-\alpha_N}\right)\textrm{Pr}\left(\Phi_{B_L}\left(B(o,D)\right)\neq 0\right)}{A_N}+\frac{\textrm{Pr}\left(r\leq d^*_N,\Phi_{B_L}\left(B(o,D)\right)= 0\right)}{A_N}.
\label{CCDFdN}\end{aligned}$$ We first derive the first term in (\[CCDFdN\]) as $$\begin{aligned}
&\frac{\mathbb{E}_{d^*_N\geq r}\left(\textrm{Pr}\left(\left(\frac{C_L}{C_N}\right)^{\frac{1}{\alpha_L}}(d_N^*)^{\frac{\alpha_N}{\alpha_L}}\leq d^*_L\right)\right)\textrm{Pr}\left(\Phi_{B_L}\left(B(o,D)\right)\neq 0\right)}{A_N}
\label{Derivation1}\\
&\overset{(b)}{=}\frac{\mathbb{E}_{d^*_N\geq r}\left(\textrm{exp}\left(-C\lambda_B\pi\left(\frac{C_L}{C_N}\right)^{\frac{2}{\alpha_L}}(d_N^*)^{\frac{2\alpha_N}{\alpha_L}}\right)
-\textrm{exp}\left(-C\lambda_B\pi D^2\right)
\right)}{A_N}\mathbb{I}\left(r\leq \left(\frac{C_N}{C_L}\right)^{\frac{1}{\alpha_N}}D^{\frac{\alpha_L}{\alpha_N}}\right)
\label{DerivationdN}\end{aligned}$$ Step $(b)$ is obtained according to the PPP’s void probability. Finally, we derive the second term in (\[CCDFdN\]). Since the LOS BS process and NLOS BS process are two independent PPPs, we have $$\begin{aligned}
\frac{\textrm{Pr}\left(r\leq d^*_N,\Phi_{B_L}\left(B(o,D)\right)= 0\right)}{A_N}&=\frac{\textrm{exp}\left({-(1-C)\lambda_B\pi r^2-C\lambda_B\pi D^2}\right)}{{A_N}}\mathbb{I}\left(r\leq D\right)
+\frac{\textrm{e}^{-\lambda_B\pi r^2}}{A_N}\mathbb{I}\left(r\geq D\right).
\label{secondTerm}\end{aligned}$$ Substituting (\[secondTerm\]) and (\[DerivationdN\]) into (\[CCDFdN\]), $f_{r_N}(r)=-\frac{d\textrm{Pr}\left(r_N\geq r\right)}{d r}$, which can be derived as (\[frN\]).
Proof of Lemma 4
================
The point process $\mathcal{N}_E$ can be regarded as a transformation of the point process $\Phi_E$ by the probability kernel $
p\left(z,A\right)=\textrm{Pr}\left(\frac{1}{G_b(\theta)g_{xz}L(x,z)}\in A\right),z\in\mathbb{R}^2,A\in\mathcal{B}(\mathbb{R}^+).
$ According to the displacement theorem [@wirelessnetworks], $\mathcal{N}_E$ is a PPP on $\mathbb{R}^+$ with the intensity measure $\Lambda_E\left(0,t\right)$ given by $$\begin{aligned}
\Lambda_E\left(0,t\right)=\lambda_E\int_{\mathbb{R}^2}\textrm{Pr}\left(\frac{1}{G_b(\theta)g_{xz}L(x,z)}\in\left[0,t\right]\right)dz.
\label{lambdaE}\end{aligned}$$
From the blockage model in Section II-B, we know that $\Phi_E$ is divided into two independent point processes, i.e., the LOS and NLOS eavesdropper processes. Furthermore, the directivity gains received at the eavesdroppers in the main and sidelobes are different. Therefore, considering these and changing to a polar coordinate system, $\Lambda_E\left(0,t\right)$ in (\[lambdaE\]) can be further derived as $$\begin{aligned}
\Lambda_E\left(0,t\right)=&2\pi\lambda_E\sum_{V\in\{M_s,m_s\}}\textrm{Pr}_{G_b}(V)\sum_{j\in\{L,N\}}q_j\int^D_0\textrm{Pr}\left(\frac{r^{\alpha_j}}{G_b(\theta)g_rC_j}\leq t|G_b(\theta)=V\right)rdr
\nonumber\\
&+2\pi\lambda_E\sum_{V\in\{M_s,m_s\}}\textrm{Pr}_{G_b}(V)\int^{+\infty}_D\textrm{Pr}\left(\frac{r^{\alpha_N}}{G_b(\theta)g_rC_N}\leq t|G_b(\theta)=V\right)rdr,\label{DerivationofLambdaE}\end{aligned}$$ where $g_r$ denotes the small-scale fading of eavesdropper which is $r$ distant from the target BS at $x$. $g_r\sim\textrm{gamma}\left(N_L,1\right)$ if the link between the eavesdropper and the target BS is LOS, otherwise, $g_r\sim\textrm{gamma}\left(N_N,1\right)$.
For getting the analysis result $\Lambda_E\left(0,t\right)$, the analysis results of integral formulaes in (\[DerivationofLambdaE\]) should be derived. Firstly, the integral $\int^D_0\textrm{Pr}\left(\frac{r^{\alpha_j}}{G_b(\theta)g_rC_j}\leq t|G_b(\theta)=V\right)rdr$ can be derived with the procedures in (\[analysisIntegral1\]). $$\begin{aligned}
&\int^D_0\textrm{Pr}\left(g_r\geq \frac{r^{\alpha_j}}{VC_jt}\right)rdr\overset{(a)}{=}
\int^D_0\left(1-\frac{\gamma\left(N_j,\frac{r^{\alpha_j}}{VC_jt}\right)}{\Gamma(N_j)}\right)rdr
\nonumber\\
&\overset{(b)}{=}\int^D_0\frac{\Gamma\left(N_j,\frac{r^{\alpha_j}}{VC_jt}\right)}{\Gamma(N_j)}rdr\overset{(c)}{=}\int^D_0e^{-\frac{r^{\alpha_j}}{VC_jt}}\sum^{N_j-1}_{m=0}\left(\frac{r^{\alpha_j}}{VC_jt}\right)^m\frac{1}{m!}rdr
\overset{(d)}{=}\frac{\left(VC_jt\right)^{\frac{2}{\alpha_j}}}{\alpha_j}\sum^{N_j-1}_{m=0}\frac{\gamma\left(m+\frac{2}{\alpha_j},\frac{D^{\alpha_j}}{VC_jt}\right)}{m!}.
\label{analysisIntegral1}\end{aligned}$$ Step (a) is due to $g_r\sim\textrm{gamma}\left(N_j,1\right)$, step (b) is due to [@Table eq.(8.356.3)], step (c) is due to [@Table eq.(8.352.2)], and step (d) is due to [@Table eq.(3.381.1)].
With a similar procedure, the integral $\int^{+\infty}_D\textrm{Pr}\left(\frac{r^{\alpha_j}}{G_b(\theta)g_rC_j}\leq t|G_b(\theta)=V\right)rdr$ can be derived as $$\begin{aligned}
\int^{+\infty}_D\textrm{Pr}\left(\frac{r^{\alpha_j}}{G_b(\theta)g_rC_j}\leq t|G_b(\theta)=V\right)rdr=\frac{\left(VC_jt\right)^{\frac{2}{\alpha_j}}}{\alpha_j}\sum^{N_j-1}_{m=0}\frac{\Gamma\left(m+\frac{2}{\alpha_j},\frac{D^{\alpha_j}}{VC_jt}\right)}{m!}.
\label{analysisIntegral2}\end{aligned}$$ Finally, substituting (\[analysisIntegral1\]) and (\[analysisIntegral2\]) into (\[DerivationofLambdaE\]), the proof can be completed.
Proof of Theorem 3
==================
The achievable secure connectivity probability, $\tau_{c}$ can be calculated as $$\begin{aligned}
&\tau_{c}\overset{(g)}{=}
\mathbb{E}_{r_j}\left[\mathbb{E}_{I_E}\left[\sum_{j\in\{L,N\}}e^{-\frac{I_Er_j^{\alpha_j}}{M_sC_j}}\sum^{N_{j}-1}_{m=0}\left(\frac{I_Er_j^{\alpha_j}}{M_sC_j}\right)^m\frac{A_j}{\Gamma(m+1)}\right]\right]
\nonumber\\
&\overset{(h)}{=}\mathbb{E}_{r_j}\left[\sum_{j\in\{L,N\}}\sum^{N_{j}-1}_{m=0}\left(\frac{r_j^{\alpha_j}}{M_sC_j}\right)^m\frac{A_j}{\Gamma(m+1)}(-1)^m\mathcal{L}^{(m)}_{I_E}\left(\frac{r_j^{\alpha_j}}{M_sC_j}\right)\right].\end{aligned}$$ step $(g)$ holds, since the serving BS at $x^*$ can be a LOS or NLOS BS, and step $(h)$ is due to the Laplace transform property $t^nf(t)\overset{\mathcal{L}}{\leftrightarrow} (-1)^n\frac{d^n}{d s^n}\mathcal{L}_{f(t)}(s)$.
In the following, we derive the analysis result of $\mathcal{L}_{I_E}\left(s\right)$. $$\begin{aligned}
&\mathcal{L}_{I_E}\left(s\right)=\mathbb{E}\left(\textrm{e}^{-s{\sum_{z\in\Phi_E}G_b(\theta)L(x^*,z)g_{x^*z}}}\right)\overset{(i)}{=}\textrm{exp}\left(\int^{+\infty}_0\left(\textrm{e}^{-\frac{s}{x}}-1\right)\Lambda_E(0,dx)\right)
\nonumber\\
&\overset{(k)}{=}\textrm{exp}\left(-\int^{+\infty}_0\Lambda_E(0,x)\frac{s}{x^2}\textrm{e}^{-\frac{s}{x}}dx\right)\overset{(v)}{=}
\textrm{exp}\left(\underset{T}{\underbrace{-\int^{+\infty}_0\Lambda_E\left(0,\frac{1}{z}\right)s\textrm{e}^{-sz}dz}}\right),
\label{LaplaceTrasformIE}\end{aligned}$$ step $(i)$ is obtained by using the probability generating functional (PGFL), step $(k)$ is obtained by using integration by parts, and step $(v)$ is obtained by the variable replacing $z=\frac{1}{x}$. Then, we concentrate on deriving the analysis result of $T$ in (\[LaplaceTrasformIE\]). Substituting $\Lambda_E(0,\frac{1}{z})$ in (\[LambdaE\]) into $T$, $T$ can be rewritten as $$\begin{aligned}
T=&-s\left(2\pi\lambda_E\sum_{j\in\{L,N\}}q_j\left(\sum_{V\in\{M_s,m_s\}}\textrm{Pr}_{G_b}(V)\frac{\left(VC_j\right)^{\frac{2}{\alpha_j}}}{\alpha_j}
\sum_{m=0}^{N_j-1}\frac{1}{m!}\underset{H_1}{\underbrace{\int^{\infty}_0\frac{\gamma\left(m+\frac{2}{\alpha_j},\frac{D^{\alpha_j}z}{VC_j}\right)}{z^{\frac{2}{\alpha_j}}}
\textrm{e}^{-sz}dz}}
\right)\right.
\nonumber\\
&\left.
\sum_{V\in\{M_s,m_s\}}\textrm{Pr}_{G_b}(V)\frac{\left(VC_N\right)^{\frac{2}{\alpha_N}}}{\alpha_N}
\sum_{m=0}^{N_N-1}\frac{1}{m!}\underset{H_2}{\underbrace{\int^{\infty}_0\frac{\Gamma\left(m+\frac{2}{\alpha_N},\frac{D^{\alpha_N}z}{VC_N}\right)}{z^{\frac{2}{\alpha_N}}}
\textrm{e}^{-sz}dz}}\label{T}
\right).\end{aligned}$$
Finally, using [@Table eq. (6.455.1)] and [@Table eq. (6.455.2)], the integral terms $H_1$ and $H_2$ can be calculated as $$\begin{aligned}
H_1=\frac{\left(D^{\alpha_j}/(VC_j)\right)^{m+\frac{2}{\alpha_j}}\Gamma(m+1)}{\left(m+\frac{2}{\alpha_j}\right)\left(s+D^{\alpha_j}/(VC_j)\right)^{m+1}}
{_2}F_1\left(1,m+1;m+\frac{2}{\alpha_j}+1;\frac{D^{\alpha_j}/(VC_j)}{s+D^{\alpha_j}/(VC_j)}\right),
\label{H1}\\
H_2=\frac{\left(D^{\alpha_N}/(VC_N)\right)^{m+\frac{2}{\alpha_N}}\Gamma(m+1)}{\left(1-\frac{2}{\alpha_N}\right)\left(s+D^{\alpha_N}/(VC_j)\right)^{m+1}}
{_2}F_1\left(1,m+1;2-\frac{2}{\alpha_N}+1;\frac{S}{s+D^{\alpha_j}/(VC_j)}\right).\label{H2}\end{aligned}$$ Finally, substituting (\[H1\]) and (\[H2\]) into (\[T\]), the closed-form result of $\mathcal{L}_{I_E}\left(s\right)$ can be obtained.
Proof of Theorem 6
==================
Using total probability theorem, we have $$\begin{aligned}
\textrm{Pr}\left(\textrm{SINR}_U\geq T_c\right)&= A_L\underset{Q_1}{\underbrace{\textrm{Pr}\left(\textrm{SINR}_U\geq T_c|\textrm{Serving BS is a LOS BS}\right)}}
\nonumber\\
&+A_N\underset{Q_2}{\underbrace{\textrm{Pr}\left(\textrm{SINR}_U\geq T_c|\textrm{Serving BS is a NLOS BS}\right)}}.\end{aligned}$$
In the following, we detail the calculation of the conditional probability $Q_1$. The conditional probability $Q_2$ can be calculated with a similar procedure which is omitted for brevity. In the following, we denote the LOS terms in $\Phi_i$ as $\Phi_{i,L}$ and the NLOS terms in $\Phi_i$ as $\Phi_{i,N}$, for $i=I,A$. $$\begin{aligned}
&Q_1=
\textrm{Pr}\left(h_{x^*o}\geq \frac{T_cr_L^{\alpha_L}\left({\sum_{y\in\Phi_I}M_s\phi P_th_{yx^*}L(y,x^*)+\sum_{y\in\Phi_A}M_a(1-\phi) P_th_{xo}L(y,x^*)}+N_0\right)}{\phi P_t M_sC_L}\right)\overset{(z)}\lessapprox
\nonumber\\
&1-\mathbb{E}\left(1-\textrm{exp}\left(-{\frac{a_LT_cr_L^{\alpha_L}}{\phi P_t M_sC_L}\left({\sum_{y\in\Phi_I}M_s\phi P_th_{yx^*}L(y,x^*)+\sum_{y\in\Phi_A}M_a(1-\phi) P_th_{yx^*}L(y,x^*)+N_0}\right)}\right)\right)^{N_L}\nonumber\\
&\overset{(x)}{=}\mathbb{E}_{r_L}\left(\sum_{n=1}^{N_L}(-1)^{n+1}\binom{N_L}{n}\textrm{exp}\left(-\Theta_L(n)r_L^{\alpha_L}N_0\right)\prod_{j\in\{L,N\}}\mathbb{E}_{\Phi_{I,j}}\left(\prod_{y\in\Phi_{I,j}}\left(1+\Theta_L(n)r_L^{\alpha_L}M_s\phi P_tL(y,x^*)\right)^{-N_j}\right)\right.
\nonumber\\
&\left.\prod_{j\in\{L,N\}}\mathbb{E}_{\Phi_{A,j}}\left(\prod_{y\in\Phi_{A,j}}\left(1+\Theta_L(n)r_L^{\alpha_L}M_a(1-\phi) P_tL(y,x^*)\right)^{-N_j}\right)\right).\label{prooftheorem6LOS}\end{aligned}$$ Step $(z)$ is due to the tight lower bound of the CDF of the gamma random variable given in [@Inequality]; step $(x)$ is due to the multinomial expansion and Laplace transform of the gamma random variable.
On the condition that the serving BS is a LOS BS and the distance from the serving BS to the typical user is $r_L$, from the user association policy in Section II-F, we know that the nearest distance from the interfering BS in $\Phi_{I,L}$ and $\Phi_{A,L}$ to the typical authorized user should be larger than $r_L$, and the nearest distance from the interfering BS in $\Phi_{I,N}$ and $\Phi_{A,N}$ to the typical authorized user should be larger than $\left(\frac{C_N}{C_L}\right)^{\frac{1}{\alpha_N}}r_L^{\frac{\alpha_L}{\alpha_N}}$. Then, using PGFL, we have $$\begin{aligned}
&\mathbb{E}_{\Phi_{I,L}}\left(\prod_{y\in\Phi_{I,L}}\left(1+\Theta_L(n)r_L^{\alpha_L}M_s\phi P_tL(y,x^*)\right)^{-N_L}\right)=\varpi_1(M_s,\phi,n)
\nonumber\\
&\mathbb{E}_{\Phi_{I,N}}\left(\prod_{y\in\Phi_{I,N}}\left(1+\Theta_L(n)r_L^{\alpha_L}M_s\phi P_tL(y,x^*)\right)^{-N_N}\right)=\varpi_2(M_s,\phi,n)\varpi_3(M_s,\phi,n)\end{aligned}$$ Similarly, the analysis resut of $\mathbb{E}_{\Phi_{A,j}}\left(\prod_{y\in\Phi_{A,j}}\left(1+\Theta_L(n)r_L^{\alpha_L}M_a(1-\phi) P_tL(y,x^*)\right)^{-N_j}\right)$, $j\in\{L,N\}$ can be obtained. Finally, substituting the pdf of $r_L$ into (\[prooftheorem6LOS\]), the proof can be completed.
Proof of Theorem 7
==================
In the following, for the convenience of expression, we denote the set of the LOS eavesdroppers in $\Phi_Z$ as $\Phi_{Z,L}$, and denote the set of the NLOS eavesdroppers as $\Phi_{Z,N}$.
Since the AN signals received at multiple eavesdroppers are not independent, we resort to the technique in [@Enhancing] to derive a lower bound of $p_{sec}$. We first derive the conditional secrecy probability conditioned on $\Phi_A$. Then, with the Jensen’s inequality, we derive a lower bound of $p_{sec}$. Specially $$\begin{aligned}
p_{sec}
=&\mathbb{E}_{\Phi_{A}}\left(\mathbb{E}_{\Phi_{Z,L},\Phi_{Z,N}}\left(\left.\prod_{z\in\Phi_{Z,L}}\mathrm{Pr}\left(\textrm{SINR}_z\leq \mathrm{T}_e|\Phi_{A}\right)\prod_{z\in\Phi_{Z,N}}\mathrm{Pr}\left(\textrm{SINR}_z\leq \mathrm{T}_e|\Phi_{A}\right)\right.\right)
\right)
\nonumber\\
=&\mathbb{E}_{\Phi_{A}}\left(\textrm{exp}\left(-C\lambda_E\textrm{Pr}_{x}(M_s)\int_{\mathbb{R}^2\cap B(o,D)}\mathrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,L},\Phi_{A}\right)dz\right.\right.
\nonumber\\
&\qquad\qquad\quad-(1-C)\lambda_E\textrm{Pr}_{x}(M_s)\int_{\mathbb{R}^2\cap B(o,D)}\mathrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,N},\Phi_{A}\right)dz
\nonumber
\\
&\left.\left.\qquad\qquad\quad-\lambda_E\textrm{Pr}_{x}(M_s)\int_{\mathbb{R}^2/ B(o,D)}\mathrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,N},\Phi_{A}\right)dz\right)\right)\nonumber
\\
\overset{(n)}{\gtrapprox} & \textrm{exp}\left(-2\pi C\lambda_E\textrm{Pr}_{x}(M_s)\int^D_{0}\mathrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,L}\right)rdr\right)
\nonumber\\
&\textrm{exp}\left(-2\pi(1-C)\lambda_E\textrm{Pr}_{x}(M_s)\int^D_{0}\mathrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,N}\right)rdr\right)
\nonumber\\
&\textrm{exp}\left(-2\pi\lambda_E\textrm{Pr}_{x}(M_s)\int^{+\infty}_{D}\mathrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,N}\right)rdr\right),\label{ProofTheorem7processpsec}\end{aligned}$$ where $r$ denotes the link length between the eavesdropper and the serving BS. We should point out that the conditional probabilities in the first line of the equation (\[ProofTheorem7processpsec\]) denote the probabilities that the SINR received by eavesdroppers at different positions is not larger than T$_e$, conditioned on a common $\Phi_{A}$. Changing to a polar coordinate system and using Jensen’s inequality, we can obtain step $(n)$.
In the following, we first show the derivation of $\textrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,L}\right)$. Then, similar procedures can be adopted to derive $\textrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,N}\right)$, which are omitted, for brevity. $$\begin{aligned}
&\textrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,L}\right)=
\textrm{Pr}\left(g_{x^*z}\geq \frac{r^{\alpha_L}\textrm{T}_e\left(I_{A_z}+N_0\right)}{\phi P_tM_sC_L}|z\in\Phi_{Z,L}\right)\overset{(v)}{\lessapprox}
\nonumber\\
& 1-\mathbb{E}_{\Phi_{A}}\left(1-\textrm{e}^{-\frac{\textrm{T}_er^{\alpha_L}a_L\left(I_{A_z}+N_0\right)}{\phi P_tM_sC_L}}\right)^{N_L}
=\sum^{N_i}_{n=1}(-1)^{n+1}\binom{N_L}{n}\textrm{e}^{-\frac{na_L\textrm{T}_eN_0r^{\alpha_L}}{\phi P_tM_sC_L}}
\mathcal{L}_{I_{A_z}}\left({\frac{na_L\textrm{T}_er^{\alpha_L}}{\phi P_tM_sC_L}}\right),\label{ProofTheorem7process}
$$ Since $g_{x^*z}\sim\textrm{gamma}\left(N_L,1\right)$, with the inequality in [@Inequality], step $(v)$ can be obtained.
For facilitating the derivations, we first introduce the following definition.
The path loss process with fading (PLPF) $\mathcal{N}_z$ is the point process on $\mathbb{R}^+$ mapped from $\Phi_{A}$, where $\mathcal{N}_z\triangleq\left\{\zeta_y=\frac{1}{M_ag_{yz}L(y,z)},y\in\Phi_A\right\}$ and $z\in\mathbb{R}^2$. We sort the elements of $\mathcal{N}_z$ in ascending order and introduce the index such that $\zeta_i\leq \zeta_j$ for $\forall i<j$.
Then, following the proof of Lemma 4, the intensity measure of $\mathcal{N}_z$ can be calculated as $$\begin{aligned}
\Lambda_z\left(0,t\right)=&2\pi\lambda_B\sum_{j\in\left\{L,N\right\}}q_j\textrm{Pr}_{x}(M_a)\frac{\left(M_aC_jt\right)^{\frac{2}{\alpha_j}}}{\alpha_j}\sum^{N_j-1}_{m=0}\frac{\gamma\left(m+\frac{2}{\alpha_j},\frac{D^{\alpha_j}}{M_aC_jt}\right)}{m!}
\nonumber\\
&+2\pi\lambda_B\textrm{Pr}_{x}(M_a)\frac{\left(M_aC_Nt\right)^{\frac{2}{\alpha_N}}}{\alpha_N}\sum^{N_N-1}_{m=0}\frac{\Gamma\left(m+\frac{2}{\alpha_N},\frac{D^{\alpha_N}}{M_aC_Nt}\right)}{m!},
\nonumber\end{aligned}$$ where $q_L=C$ and $q_N=1-C$.
The the Laplace transform $\mathcal{L}_{I_{A_z}}\left(s\right)$ can be calculated as $$\begin{aligned}
&\mathcal{L}_{I_{A_z}}\left(s\right){=}\textrm{exp}\left(\int^{+\infty}_0\left(\textrm{e}^{-\frac{s\left(1-\phi\right)P_t}{x}}-1\right)\Lambda_z(0,dx)\right)
{=}\textrm{exp}\left(-\int^{+\infty}_0\Lambda_z(0,x)\frac{s\left(1-\phi\right)P_t}{x^2}\textrm{e}^{-\frac{s\left(1-\phi\right)P_t}{x}}dx\right)\nonumber\\
&\overset{(k)}{=}
\textrm{exp}\left({{-\int^{+\infty}_0\Lambda_z\left(0,\frac{1}{z}\right)s\left(1-\phi\right)P_t\textrm{e}^{-s\left(1-\phi\right)P_tz}dz}}\right)\end{aligned}$$ Step $(k)$ is obtained by the variable replacing $z=\frac{1}{x}$. Following the derivation of the analysis result of (\[LaplaceTrasformIE\]) in the proof of Theorem 3, we can derive the analysis result of $\mathcal{L}_{I_{A_z}}\left(s\right)$. Substituting the analysis result of $\mathcal{L}_{I_{A_z}}\left(s\right)$ into (\[ProofTheorem7process\]), the analysis result of $\textrm{Pr}\left(\textrm{SINR}_z\geq \textrm{T}_e|z\in\Phi_{Z,L}\right)$ can be obtained.
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[^1]: C. Wang and H.-M. Wang are with the School of Electronic and Information Engineering, and also with the MOE Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China. Email: [wangchaoxuzhou@stu.xjtu.edu.cn]{} and [xjbswhm@gmail.com]{}. The contact author is Hui-Ming Wang.
[^2]: This assumption is just for simplifying the performance analysis. However, the obtained analysis methods can be extended to the multiple antennas case directly by modeling the array pattern at authorized users and malicious eavesdroppers in a similar way as (\[SectorAntennaModel\]).
[^3]: In the mmWave network with AN, the transmitted AN simultaneously from each BS has made the transmissions of mmWave signals no longer highly directional. Therefore, different from mmWave network without AN, the out-cell interference should be taken into consideration in the mmWave network with AN.
[^4]: We should point out that the analysis results obtained in this section can be generalized to incorporate the sidelobe leakage signals, by considering the eavesdroppers in the intended sector and outside the intended sector separately, just as Section III. However, the analysis results in such case would become more complicated, whilst few design insights can be brought. Furthermore, as we know, massive antenna array would be deployed at the mmWave BS for improving the mmWave signal transmission performance [@5GMillimeter; @MillimeterChannelModeling]. Therefore, the sidelobes of the antenna pattern at the mmWave BS can be suppressed sufficiently, and it is reasonable to omit the sidelobe in the theoretical analysis.
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abstract: 'In this note we assess the validity and uncertainties in the predictions of the eikonalised mini-jet model for $\sigma^{inel}_{\gamma \gamma}$. We are able to find a choice of parameters where the predictions are compatible with the current data. Even for this restricted range of parameters the predictions at the high c.m. energies, which can be reached at the TeV energy $e^+e^-$ colliders, differ by about $\pm 25\%$. LEP 2 data can help pinpoint these parameters and hence reduce the uncertainties in the predictions.'
---
hep-ph/mmnnyyy\
IISc-CTS-9/96\
LNF-96/018(P)
Eikonalized mini-jet cross-sections in $\gamma \gamma$ collisions[^1]\
A. Corsetti\
INFN, Physics Department, University of Rome La Sapienza, Rome, Italy\
R. M. Godbole[^2]\
Center for Theoretical Studies, Indian Institute of Sciences\
Bangalore 560 012, India\
and\
G. Pancheri\
INFN - Laboratori Nazionali di Frascati , I00044 Frascati\
**Eikonalized mini-jet cross-sections** {#eikonalized-mini-jet-cross-sections .unnumbered}
=======================================
A. Corsetti$^1$, R.M. Godbole$^2$, G. Pancheri$^3$
------------------------------------------------- --
[*$^1$ INFN, Univ. La Sapienza, Roma, Italy*]{}
[*$^3$ INFN, Frascati, Italy*]{}
------------------------------------------------- --
In this note we wish to assess the validity and uncertainties of the eikonalized mini-jet model in predicting $\sigma_{\gamma \gamma}^{inel} $ and further to ascertain whether measurements at LEP-200 and HERA can constrain various parameters of the model. In its simplest formulation, the eikonalized mini–jet cross-section is given by $$\label{eikonal}
\sigma^{inel}_{ab} = P^{had}_{ab}\int d^2\vec{b}[1-e^{-n(b,s)}]$$ where the average number of collisions at a given impact parameter $\vec{b}$ is obtained from $$\label{av_n}
n(b,s)=A_{ab} (b) (\sigma^{soft}_{ab} + {{1}\over{P^{had}_{ab}}}
\sigma^{jet}_{ab})$$ with $A_{ab} (b)$ the normalized transverse overlap of the partons in the two projectiles and $P^{had}_{ab}$ to give the probability that both colliding particles $a,b$ be in a hadronic state. $\sigma^{soft}_{ab}$ is the non-perturbative part of the cross-section from which the factor of $P_{ab}^{had} $ has already been factored out and $\sigma^{jet}_{ab} $ is the hard part of the cross–section. The rise in $\sigma^{jet}_{ab} $ drives the rise of $\sigma_{ab}^{inel}$ with energy [@cline]. We have also assumed the factorization property $$P_{\gamma p}^{had} = P_{\gamma}^{had} ;
\ \ \ P_{\gamma \gamma}^{had} = (P_{\gamma}^{had})^2.$$ The predictions of the eikonalised mini-jet model [@minijets] for photon induced processes [@ladinsky] depend on 1) the assumption of one or more eikonals, 2) the hard jet cross-section $\sigma^{jet}_{ab}=\int_{p_{tmin}} {{d^2\hat{\sigma}}\over{dp_t^2}} dp_t^2$ which in turn depends on the minimum $p_t$ above which one can expect perturbative QCD to hold, viz. $ p_{tmin}$, and the parton densities in the colliding particles $a$ and $b$, 3) the soft cross–section $\sigma^{soft}_{ab}$, 4) the overlap function $ A_{ab}(b) $, defined as $$\label{aob}
A_{ab}(b)={{1}\over{(2\pi)^2}}\int d^2\vec{q}{\cal F}_a(q) {\cal F}_b(q)
e^{i\vec{q}\cdot \vec{b}}$$ where ${\cal F}$ is the Fourier transform of the b-distribution of partons in the colliding particles and 5) last but not the least $P_{ab}^{had}$.
In this note we shall restrict ourselves to a single eikonal. The hard jet cross-sections have been evaluated in LO perturbative QCD. The dependence of $\sigma_{ab}^{jet}$ on $ p_{tmin}$ is strongly correlated with the parton densities used. Here we show the results using GRV densities [@GRVo] (see ref. [@fatlinac] for the results using the DG densities [@DG]). For the purposes of this note, we determine $\sigma_{\gamma \gamma}^{soft}$ from $\sigma_{\gamma p}^{soft}$ which is obtained by a fit to the photoproduction data. We use the Quark Parton Model suggestion $\sigma_{\gamma \gamma}^{soft} = {{2}\over{3}}
\sigma_{\gamma p}^{soft}$.
In the original use of the eikonal model, the overlap function $A_{ab} (b)$ of eq. (\[aob\]) is obtained using for ${\cal F}$ the electromagnetic form factors and thus, for photons, a number of authors [@SARC; @FLETCHER] have assumed for ${\cal F }$ the pole expression used for the pion electromagnetic form factor, on the basis of Vector Meson Dominance (VMD). We shall investigate here another possibility, i.e. that the b-space distribution of partons in the photon is the Fourier transform of their intrinsic transverse momentum distributions. This will correspond to use the functional expression expected for the perturbative part [@pertint] $$\label{intrinsicphot}
{{d N_{\gamma}}\over{dk_t^2}}={{1}\over{k_t^2+k_o^2}}$$ Recently this expression was confirmed by the ZEUS [@ZEUS] Collaboration, with $k_o=0.66 \pm 0.22$ GeV. For $\gamma \gamma$ collisions, the overlap function is now simply given by $$\label{aobgg}
A(b)={{1}\over{4 \pi}} k_o^3 b K_1(b k_o)$$ with $K_1$ the Bessel function of the third kind. It is interesting to notice that for photon-photon collisions the overlap function will have the same analytic expression for both our ansätze: the VMD inspired pion form factor or the intrinsic transverse momentum; the only difference being that the former corresponds to a fixed value of $k_0 = 0.735$ GeV whereas the latter allows us to vary the value of the parameter $k_0$. Thus both possibilities can be easily studied by simply changing $k_0$ appropriately. Notice that the region most important to this calculation is for large values of the parameter b, where the overlap function changes trend, and is larger for smaller $k_o$ values.
As for $P^{had}_\gamma$, this is clearly expected to be ${\cal O} (\alpha_{em})$ and from VMD one would expect $1/250$. From phenomenological considerations [@FLETCHER]and fits to HERA data, one finds a value $1/200$, which indicates at these energies a non-VMD component of $\approx 20\%$. It should be noticed that the eikonalised minijet cross–sections do not depend on $A_{\gamma \gamma}$ and $P^{had}_{\gamma \gamma}$ separately, but depend only on the ratio of the two [@drees1; @dgo10]. Having thus established the range of variability of the quantities involved in the calculation of total photonic cross sections, we now proceed to calculate and compare with existing data the eikonalized minijet cross-section for $\gamma \gamma$ collisions. We use GRV (LO) densities and values of $p_{tmin}$ deduced from a best fit to photoproduction. As discussed in [@ourpaper], it is possible to include the high energy points in photoproduction using GRV densities and $p_{tmin}=2$ GeV, but the low energy region would be better described by a smaller $p_{tmin}$. This is the region where the rise, according to some authors, notably within the framework of the Dual Parton Model, is attributed to the so-called [*soft Pomeron*]{}. For our studies here we use $p_{tmin}=2.$ GeV. We also use $P_{\gamma}^{had} =1/204$ and A(b) from eq.(\[aobgg\]) with different values of $k_0$. One choice for $k_0$ is the pole parameter value in the photon b-distribution expression, which includes both the intrinsic transverse momentum option $0.66 \pm 0.22$ GeV as well as the pion form factor value, 0.735 GeV. The other value, 1 GeV, is a possible choice which appears to fit the present data better than everything else.
Our predictions are shown in Fig.(\[gamgam\]). A comparison with existing $\gamma \gamma$ data shows that all of our choices are compatible with the data within the present experimental errors. At high energies, however, like the ones reachable with the proposed linear photon colliders, these predictions vary by about $\pm 25 \%$. Reducing the error in the LEP1 region and adding new data points in the c.m. region attainable at LEP2, can help pinpoint and restrict the choices. Were the LEP1 and LEP2 data to confirm the present values, we believe that the best representation of the present data is obtained with the higher $k_0$ value.
0.5cm 0.1cm
R.M.G. wishes to acknowledge support from C.S.I.R. (India) under grant no.\
03(0745)/94/EMR-II. This research is supported in part by the EEC program “Human Capital and Mobility", contract CT92-0026 (DG 12 COMA).
[99]{} \[messy\]
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M. Drees, Univ. Wisconsin report MADPH-95-867, [*Proceedings of the 4th workshop on TRISTAN physics at High Luminosities*]{}, KEK, Tsukuba, Japan, Nov. 1994. \[drees1\]
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C. Corsetti, R.M. Godbole and G. Pancheri, in preparation.
[^1]: To appear in the proceedings of the workshop on [*$e^+e^-$ 2000 GeV Linear Colliders*]{} Annecey, Gran Sasso, Hamburg (1995).
[^2]: On leave of absence from Dept. of Physics, Uni. of Bombay, Bombay, India.
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abstract: 'We consider asynchronous CDMA systems in no-fading environments with a particular focus on a certain user. This certain user is called a desired user in this paper. In such a situation, an optimal sequence, maximum Signal-to-Interference plus Noise Ratio (SINR) and the maximum capacity for a desired user are derived with other spreading sequences being given and fixed. In addition, the maximum SINR and the optimal sequence for a desired user are written in terms of the minimum eigenvalue and the corresponding eigenvector of a matrix, respectively. Since it is not straightforward to obtain an explicit form of the maximum SINR, we evaluate SINR and obtain the lower and upper bounds of the maximum SINR. From these bounds, the maximum SINR may get larger as the quantities written in terms of quadratic forms of other spreading sequences decrease. Further, we propose a method to obtain spreading sequences for all the users which achieve large SINRs. The performance of our proposed method is numerically verified.'
author:
- 'Hirofumi Tsuda, [^1] [^2]'
title: Optimal Sequence and Performance for Desired User in Asynchronous CDMA System
---
Asynchronous CDMA systems, Spreading sequence, Signal-to-Interference Noise Ratio, Capacity, Rayleigh quotient
Introduction
============
evaluate channel capacity is a significant task since channel capacity is the maximum achievable rate [@thomas_cover]. If the rate is smaller than given capacity, then there is a code whose maximum error converges to zero as the length of code words goes to infinity [@csiszar] [@gallager]. Thus, if large channel capacity is achieved, then information can be sent at a high rate. From such a reason, large capacity has been demanded. These results have been proven in [@shannon]. Furthermore, in a general channel, capacity has been obtained in [@general_channel] [@output_stat]. With a practical scheme, channel capacity has been evaluated in [@multilevel]. Further investigations of capacity are expected to contribute to improvement in communication systems.
In Orthogonal Frequency Division Multiplexing (OFDM) systems, channel capacity with a non-linear amplifier has been obtained in [@clip]. In Multiple Input-Multiple Output (MIMO) systems, capacity has been investigated [@mimo]. By contrast, in some situations, capacity with MIMO systems has not been evaluated.
In Code Division Multiple Access (CDMA) systems, capacity has also been evaluated. One of representative characteristics of CDMA systems is to use spreading sequences to communicate each other. Therefore, capacity may depend on spreading sequences. Further, it is known that capacity increases as Signal-to-Interference plus Noise Ratio (SINR) increases in practical schemes [@multilevel]. There are many works to obtain spreading sequences which achieve large SINR.
CDMA systems are divided into three kinds of systems. In synchronized CDMA systems, it is known that the Welch bound equality (WBE) sequences achieve the maximal capacity [@sync]. In chip synchronized CDMA systems with given and fixed delays, an algorithm to obtain sequences which achieves nearly maximum SINR has been suggested [@chipsync]. However, maximum channel capacity in asynchronous CDMA systems have not been evaluated. In asynchronous CDMA systems, many kinds of spreading sequences have been suggested to obtain large SINR. For more details, we refer the reader to [@mazzini]-[@gold] and asynchronous CDMA systems have been investigated in [@fundamental] [@criteria] [@pursley]. Since it is known that correlations play important roles in CDMA systems, correlations of sequences have been investigated and bounds of correlations have been obtained [@welch] [@sarwate]. Further, sequences which achieve the equalities of such bounds have been obtained [@zadoff] [@chu] [@meet].
In this paper, we show a optimal sequence for a desired user in a sense of SINR with no fading environments. Further, we show that the maximum SINR and an optimal sequence for a desired user are written in terms of the minimum eigenvalue and the corresponding eigenvector of a matrix, respectively. Since we derive the maximum SINR, the maximum capacity is derived under an approximation. Although we show an expression for the maximum SINR, it does not seem to be straightforward to obtain its closed form. To overcome this obstacle, we evaluate the maximum SINR and derive lower and upper bounds of SINR. From these bounds, it turns out that maximum SINR gets larger as the quantities written in terms of quadratic forms of other spreading sequences decrease. It is numerically verified that the maximum SINR for a desired user depends on the spreading sequences for the other users. From the derivation of the optimal sequence for a desired user, we propose a method to obtain spreading sequences for all the users which achieve large SINRs. In numerical results, we verify the performance of our method.
System Description
==================
In this section, we show a model of asynchronous CDMA systems. This model has been investigated in [@pursley] [@borth] [@mypaper]. We make the following assumptions.
1. a modulation scheme is Binary Shift Phase Keying (BPSK)
2. there is no fading effect.
3. the spreading sequences for the other users are given and fixed. Only a spreading sequence for a certain user is regarded as a variable.
4. channel noise follows Gaussian.
5. interference noise follows Gaussian.
6. interference noise is independent of Gaussian channel noise.
7. the phase of a transmitted signal, the time delay, and the transmitted symbols are random variables and uniformly distributed on their domains.
The assumptions 1, 2 and 3 are often made and CDMA systems in no-fading effect have been investigated [@pursley] [@chipsync]. The assumptions 4 and 6 are usually made to analyze communication systems [@pursley] [@ofdmcdma]. The assumption 5 has been made in [@pursley]. Further, in analysis of Signal-to-Noise Ratio (SNR), this assumption is often made since Gaussian noise is the worst kind of additive noise in the view of capacity [@clip] [@gallager]. Thus, we consider the worst case in the view of capacity in asynchronous CDMA systems. The assumption 7 is often made to analyze asynchronous CDMA systems [@pursley] [@borth].
Let $N$ be the length of spreading sequences and $N$ is common for all the users. From the assumption 1, a data signal of the user $k$, $b_k(t)$, is written as $$b_k(t) = \sum_{n=-\infty}^{\infty} b_{k,n} p_{T}(t - nT),$$ where $b_{k,n} \in \{-1,1\}$ is the $n$-th component of the transmitted symbols which the user $k$ sends, $T$ is the duration of one symbol and $p_{T}(t)$ is the rectangular pulse written as $$p_T(t) = \left\{ \begin{array}{c c}
1 & 0 \leq t < T\\
0 & \mbox{otherwise}
\end{array} \right. .$$ Then, the code waveform of the user $k$, $s_k(t)$, is written as $$s_k(t) = \sum_{n=-\infty}^{\infty} s_{k,n} p_{T_c}(t - nT_c),$$ where $s_{k,n}$ is the $n$-th component of the spreading sequence of the user $k$ and $T_c$ is the width of each chip such that $NT_c = T$. Here, we assume that the sequence $(s_{k,n})$ is periodic, that is, $s_{k,n}=s_{k,n+N}$. Moreover, we assume the power normalization condition $$\sum_{n=1}^N \left|s_{k,n}\right|^2 = N.
\label{eq:power_cons}$$ This condition is often used [@sarwate] [@welch]. With the above signals, the transmitted signal of the user $k$, $\zeta_k$, is written as $$\zeta_k(t) = \sqrt{2P} \operatorname{Re}[s_k(t)b_k(t)\exp(j \omega_c t + j\theta_k)],
\label{eq:carrer}$$ where $P$ is the common signal power to all the users, $\operatorname{Re}[z]$ is the real part of $z$, $j$ is the unit imaginary number, $\omega_c$ is the common carrier frequency to all the users and $\theta_k$ is the phase of the user $k$. Note that the signal $\zeta_k(t)$ is called a Radio Frequency (RF) signal.
We assume that there are $K$ users and that all the users are not synchronized. Then, the received signal $r(t)$ is written as $$r(t) = \sum_{k=1}^K \zeta_k(t -\tau_k) + n(t),$$ where $\tau_k$ is the delay time of the user $k$ and $n(t)$ is additive white Gaussian noise (AWGN). Note that the quantities $b_{k,n}$, $\theta_k$, and $\tau_k$ are random variables.
To analyze SINR for a certain user, we focus on the user $i$ and the user $i$ is called desired user in this paper. If the user $i$ is a desired user and the received signal $r(t)$ is the input to a correlation receiver matched to $\zeta_i(t)$, then the corresponding output $Z_i$ is written as $$Z_i = \int_{0}^T r(t) \operatorname{Re}[s_i(t-\tau_i)\exp(j \omega_c t + j \psi_i)]dt.
\label{eq:output}$$ Without loss of generality, we assume $\tau_i = 0$ and $\theta_i = 0$. With a low-pass filter, we can ignore double frequency terms, and then rewrite Eq. (\[eq:output\]) as $$\begin{split}
Z_i &= \frac{1}{2} \sum_{k=1}^K \int_{0}^T \sqrt{2P} \operatorname{Re}[s_k(t)b_k(t)\overline{s_i(t)}\exp(j \psi_k)]dt\\
&+ \int_0^T n(t)\operatorname{Re}[s_i(t)\exp(j \omega_c t)]dt,
\label{eq:output2}
\end{split}$$ where $\overline{z}$ is the complex conjugate of $z$, $$\overline{s_i(t)} = \sum_{n=-\infty}^{\infty} \overline{s_{i,n}} p_{T_c}(t - nT_c),$$ and $\psi_i = \theta_i - \omega_c\tau_i$.
In Eq. (\[eq:output2\]), there are three kinds of random variables, the phases $\psi_k$, time delays $\tau_k$ and symbols $b_{k,n}$. From the assumption 7, these random variables, $\psi_k$, $\tau_k$ and $b_{k,n}$ are uniformly distributed on $[0, 2\pi)$, $[0,T)$ and $\{-1 ,1\}$, respectively. Without loss of generality, we assume that $b_{i,0} = +1$. To evaluate SINR, we define $$\mu_{i,k}(\tau; t) = b_k(t - \tau)s_k(t - \tau)\overline{s_i(t)}.$$ Then, the output value $Z_i$ is divided into three signals, the desired signal $D_i$, the interference signal $I_i$ and the AWGN signal $N_i$. They are written as $$\begin{split}
D_i &= \sqrt{\frac{P}{2}} \int_0^T b_i(t)dt\\
I_i &= \sqrt{\frac{P}{2}} \sum_{\substack{k=1 \\ k \neq i}}\operatorname{Re}[\tilde{I}_{i,k}]\\
N_i &= \int_0^Tn(t)\operatorname{Re}[s_i(t)\exp(j\omega_ct)]
\end{split}$$ where $$\tilde{I}_{i,k} = \int_0^T \mu_{i,k}(\tau_k; t) \exp(j \psi_k) dt.$$ Thus, the output $Z_i$ is rewritten as $$Z_i = D_i + I_i + N_i.$$ Note that the quantities $I_i$, and $N_i$ are random variables. Since $\operatorname{E}\{I_i\} = \operatorname{E}\{N_i\} = 0$ and $\displaystyle\operatorname{E}\{D_i\} = T\sqrt{P/2}$, we have $\displaystyle \operatorname{E}\{Z_i\} = T\sqrt{P/2}$, where $\operatorname{E}\{X\}$ is the mean of $X$. Then, SINR of the user $i$ is defined as $$\operatorname{SINR}_i =\sqrt{\frac{PT^2/2}{\operatorname{Var}\{I_i\} + \operatorname{Var}\{N_i\} }},
\label{eq:SINR_def}$$ where $\operatorname{Var}\{X\}$ is the variance of $X$. From [@pursley] [@borth], the variance of $N_i$ is written as $$\operatorname{Var}\{N_i\} = \frac{1}{4}N_0T$$ if $n(t)$ has a two-sided spectral density denoted as $\frac{1}{2}N_0$.
In [@mypaper], the formula of SINR has been proposed as $$\operatorname{SINR}(\mathbf{s}_i)_i = \left\{ \frac{1}{6N^2}\sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} + \frac{N_0}{2PT} \right\}^{-1/2},
\label{eq:SINR}$$ where $$S_m^{i,k} = \left( \mathbf{s}_i^* Q_m \mathbf{s}_i\right)\left( \mathbf{s}_k^* Q_m \mathbf{s}_k\right) + \left( \mathbf{s}_i^* \hat{Q}_m \mathbf{s}_i\right)\left(\mathbf{s}_k^* \hat{Q}_m \mathbf{s}_k \right).
\label{eq:quad_S}$$ In this paper, attention is drawn to this formula. The symbols in Eq. (\[eq:SINR\_def\]) are explained as follows. First, $\mathbf{s}_k$ is the vector written as $$\mathbf{s}_k = (s_{k,1},s_{k,2},\ldots,s_{k,N})^\top,$$ the matrices $Q_m$ and $\hat{Q}_m$ are given by $$Q_m = V^* C_m V, \hspace{3mm}\hat{Q}_m = \hat{V}^* \hat{C}_m \hat{V},
\label{eq:def_Q}$$ where $V$ and $\hat{V}$ are unitary matrices whose $(m,n)$-th component is written as $$\begin{split}
V_{m,n} &= \frac{1}{\sqrt{N}}\exp\left(-2 \pi j \frac{mn}{N}\right),\\
\hat{V}_{m,n} &= \frac{1}{\sqrt{N}}\exp\left(-2 \pi j n\left(\frac{m}{N} + \frac{1}{2N}\right)\right),
\end{split}$$ and $C_m$ and $\hat{C}_m$ are diagonal matrices whose $(m,m)$-th elements are given by $$\begin{split}
\left(C_m\right)_{m,m} &= \sqrt{1+\frac{1}{2}\cos\left(2 \pi \frac{m}{N}\right)},\\
\left(\hat{C}_m\right)_{m,m} &= \sqrt{1+\frac{1}{2}\cos\left(2 \pi \left(\frac{m}{N} + \frac{1}{2N}\right)\right)},
\end{split}$$ and the other elements are zero. In the above equations, $\mathbf{x}^\top$ and $\mathbf{z}^*$ denote the transpose of $\mathbf{x}$ and the conjugate transpose of $\mathbf{z}$, respectively. Note that the matrices $Q_m$ and $\hat{Q}_m$ are positive semidefinite matrices since $Q_m$ and $\hat{Q}_m$ are Gram matrices. It is obvious that Eq. (\[eq:SINR\]) depends on the vector $\mathbf{s}_i$.
Optimal Sequence and SINR for Desired User in No Fading
=======================================================
In this section, we derive an optimal spreading sequence in no fading situation for the user $i$. Since the optimal spreading sequence is derived, the maximum SINR and the maximum capacity for the user $i$ are obtained.
In the previous section, we have made seven assumptions. These are also assumed in this section. In this case where all the spreading sequence $\mathbf{s}_k$ are given, by assumption 1, 2, 3, and 6, SINR for the user $i$ is written as Eq. (\[eq:SINR\]). From assumption 3, Eq. (\[eq:SINR\]) depends on only $\mathbf{s}_i$ since the other spreading sequences $\mathbf{s}_k$ are fixed for $k \neq i$. Therefore, to maximize SINR, we consider the following optimization problem $$\begin{split}
(P_i)& \hspace{3mm} \min \hspace{2mm} \sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} \\
& \mbox{subject to} \hspace{3mm} \|\mathbf{s}_i\|^2 = N,
\end{split}$$ where $\|\mathbf{z}\|$ is the Euclidean norm of $\mathbf{z}$. Note that the constraint is obtained from Eq. (\[eq:power\_cons\]). It is clear that maximum SINR is obtained from the above optimization problem. In what follows, the problem $(P_i)$ is rewritten in another form.
To analyze the optimization problem, we define the following matrix $\Sigma_i$ $$\Sigma_i = \sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N \left( \mathbf{s}_k^* Q_m \mathbf{s}_k\right)Q_m + \left( \mathbf{s}_k^* \hat{Q}_m \mathbf{s}_k\right)\hat{Q}_m.
\label{eq:sigma}$$ The matrix $\Sigma_i$ is constant since $\mathbf{s}_k$ is given and fixed for $k \neq i$ under assumption 3. Further, the matrix $\Sigma_i$ is positive semidefinite since the quantities $\left( \mathbf{s}_k^* Q_m \mathbf{s}_k\right)$ and $\left( \mathbf{s}_k^* \hat{Q}_m \mathbf{s}_k\right)$ are non-negative, and the matrices $Q_m$ and $\hat{Q}_m$ are positive semidefinite.
With the matrix $\Sigma_i$, the optimization problem $(P_i)$ is rewritten as $$\begin{split}
(P_i)& \hspace{3mm} \min \hspace{2mm} \mathbf{s}_i^* \Sigma_i \mathbf{s}_i\\
& \mbox{subject to} \hspace{3mm} \|\mathbf{s}_i\|^2 = N.
\end{split}$$ Further, the above problem is equivalent to the following one $$\begin{split}
(P_i)& \hspace{3mm} \min \hspace{2mm} \frac{\mathbf{s}_i^* \Sigma_i \mathbf{s}_i}{\|\mathbf{s}_i\|^2/N}\\
& \mbox{subject to} \hspace{3mm} \|\mathbf{s}_i\|^2 = N.
\end{split}$$ Let the vector $\mathbf{u}_i$ be $\mathbf{u}_i = \frac{1}{\sqrt{N}}\mathbf{s}_i$. With $\mathbf{u}_i$, the problem $(P_i)$ is rewritten as $$\begin{split}
(P_i)& \hspace{3mm} \min \hspace{2mm} \frac{N \cdot \mathbf{u}_i^* \Sigma_i \mathbf{u}_i}{\|\mathbf{u}_i\|^2}\\
& \mbox{subject to} \hspace{3mm} \|\mathbf{u}_i\|^2 = 1.
\end{split}$$ It is obvious that the value of the objective function is invariant under the action $\mathbf{u}_i \mapsto c \mathbf{u}_i$, where $c \in \mathbb{C}$ is a non-zero scalar. This observation yields that if we obtain a non-zero solution $\tilde{\mathbf{u}}'$ which minimizes the objective function of $(P_i)$, then we can obtain the feasible optimal solution $\tilde{\mathbf{u}}$ as $\tilde{\mathbf{u}} = \tilde{\mathbf{u}}'/\|\tilde{\mathbf{u}}'\|$. Thus, we consider the following problem $$\begin{split}
(P'_i)& \hspace{3mm} \min_{\mathbf{u}_i \neq \mathbf{0}} \hspace{2mm} \frac{N \cdot \mathbf{u}_i^* \Sigma_i \mathbf{u}_i}{\|\mathbf{u}_i\|^2}.
\end{split}$$ This is the Rayleigh quotient of $N\Sigma_i$ [@manifold]. It is known that the optimal value coincides with the product of $N$ and the minimum eigenvalue of $\Sigma_i$, $\lambda^{(i)}_{\min} \geq 0$, and that the global minimizer of the problem $(P_i)$ is the eigenvector corresponding to $\lambda^{(i)}_{\min}$. Let $\mathbf{u}$ be such a minimizer. When the minimizer $\mathbf{u}_i$ is normalized as $\|\mathbf{u}_i\|=1$, the optimal spreading sequence for the user $i$, $\mathbf{s}_i^\star$, is written as $$\mathbf{s}_i^\star = \sqrt{N}\mathbf{u}_i.$$ Then, the maximum SINR is written as $$\operatorname{SINR}_i^\star = \operatorname{SINR}(\mathbf{s}_i^\star)_i = \left\{\frac{\lambda^{(i)}_{\min}}{6N} +\frac{N_0}{2PT}\right\}^{-1/2}.
\label{eq:optimal_SINR_i}$$ Further, it is known that the channel capacity is written in terms of Signal-to-Noise Ratio (SNR) if an input is continuous and channel noise is Gaussian [@thomas_cover] [@el_gamal]. The sum of interference noise and channel noise follow Gaussian since the sum of the two independent Gaussian variables follow Gaussian under assumptions 5 and 6 [@prob]. Even in a case where noise follows Gaussian, the channel capacity with a practical scheme is complicated [@gallager] [@multilevel]. In [@multilevel], the channel capacity with BPSK scheme is close to one with a continuous channel in low SNR. Taking into account these reasons, we approximate the maximum channel capacity of the user $i$ by one with a continuous channel. Under this approximation, from Eq. (\[eq:optimal\_SINR\_i\]), the maximum channel capacity for the user $i$, $C^\star_i$, is evaluated as $$C^\star_i \approx \frac{1}{2}\log\left[1 + \left\{\frac{\lambda^{(i)}_{\min}}{6N} +\frac{N_0}{2PT}\right\}^{-1}\right].
\label{eq:opt_capacity}$$ As seen in the above discussions, the maximum SINR and the maximum channel capacity depend on the minimum eigenvalue of the matrix $\Sigma_i$, and these maximums are achieved with the eigenvector corresponding to the minimum eigenvalue.
Estimating Maximum SINR
=======================
We have evaluated the maximum SINR in asynchronous CDMA systems for a desired user. Since the matrix $\Sigma_i$ depends on $\mathbf{s}_k$ $(k \neq i)$, the minimum eigenvalue $\lambda^{(i)}_{\min}$ may depend on other spreading sequences $\mathbf{s}_k$ $(k \neq i)$. Thus, to analyze the maximum SINR, it is necessary to obtain the explicit form of $\lambda^{(i)}_{\min}$. However, it is not straightforward to obtain the explicit form of $\lambda^{(i)}_{\min}$. Instead, we derive the lower and upper bounds of the maximum SINR in this section. From these bounds, we can estimate the maximum SINR and the know what the dominant factor related to SINR is.
As seen in Eq. (\[eq:sigma\]), the matrix $\Sigma_i$ consists of two kinds of the matrices, $Q_m$ and $\hat{Q}_m$. From Eq. (\[eq:def\_Q\]), the eigenvalues of the matrices $Q_m$ and $\hat{Q}_m$ are represented as the matrices $C_m$ and $\hat{C}_m$, respectively. Further, the matrices $C_m$ and $\hat{C}_m$ have one non-zero component at the $(m,m)$-th entry. Therefore, the matrix $\Sigma_i$ is written as $$\Sigma_i = V^* \Lambda_i V + \hat{V}^* \hat{\Lambda}_i \hat{V},
\label{eq:decomp_sigma}$$ where $\Lambda_i$ and $\hat{\Lambda}_i$ are diagonal matrices whose $m$-th diagonal components, $\lambda^{(i)}_m$ and $\hat{\lambda}^{(i)}_m$, are written as $$\begin{split}
\lambda^{(i)}_m &= \sqrt{1+\frac{1}{2}\cos\left(2 \pi \frac{m}{N}\right)} \sum_{\substack{k=1 \\ k \neq i}}^K \left( \mathbf{s}_k^* Q_m \mathbf{s}_k\right)\\
\hat{\lambda}^{(i)}_m &= \sqrt{1+\frac{1}{2}\cos\left(2 \pi \left(\frac{m}{N} + \frac{1}{2N}\right)\right)} \sum_{\substack{k=1 \\ k \neq i}}^K \left( \mathbf{s}_k^* \hat{Q}_m \mathbf{s}_k\right).
\end{split}$$ Since the matrices $V$ and $\hat{V}$ are unitary, the quantities $\lambda^{(i)}_m$ and $\hat{\lambda}^{(i)}_m$ are the eigenvalues of the matrices $V^*\Lambda_i V$ and $\hat{V}^*\hat{\Lambda}_i\hat{V}$, respectively. Note that the quantities $\lambda^{(i)}_m$ and $\hat{\lambda}^{(i)}_m$ depend on the spreading sequences $\mathbf{s}_k$ for $k \neq i$.
With the above eigenvalues, the bounds of the maximum SINR for the user $i$ are derived. First, we derive the upper bound. As seen in Eq. (\[eq:optimal\_SINR\_i\]), the maximum SINR is written with the minimum eigenvalue of $\Sigma_i$, $\lambda^{(i)}_{\min}$. Since $\lambda^{(i)}_{\min}$ is the optimal value of the Rayleigh quotient of $\Sigma_i$, the following relations are obtained $$\begin{split}
\lambda^{(i)}_{\min} =& \min_{\mathbf{u} \neq \mathbf{0}} \frac{\mathbf{u}^*\Sigma_i \mathbf{u}}{\|\mathbf{u}\|^2}\\
=& \min_{\mathbf{u} \neq \mathbf{0}} \frac{\mathbf{u}^* \left(V^* \Lambda_i V + \hat{V}^* \hat{\Lambda}_i^* \hat{V}\right) \mathbf{u}}{\|\mathbf{u}\|^2}\\
=& \min_{\substack{\mathbf{u}_1 \neq \mathbf{0}, \mathbf{u}_2 \neq \mathbf{0} \\ \mathbf{u}_1 = \mathbf{u}_2}}\left[ \frac{\mathbf{u}^*_1 V^* \Lambda_i V \mathbf{u}_1}{\|\mathbf{u}_1\|^2} + \frac{\mathbf{u}_2^* \hat{V}^* \hat{\Lambda}^*_i \hat{V} \mathbf{u}_2}{\|\mathbf{u}_2\|^2}\right]\\
\geq& \min_{\mathbf{u}_1 \neq \mathbf{0}} \frac{\mathbf{u}^*_1 V^* \Lambda_i V \mathbf{u}_1}{\|\mathbf{u}_1\|^2} + \min_{\mathbf{u}_2 \neq \mathbf{0}} \frac{\mathbf{u}_2^* \hat{V}^* \hat{\Lambda}^*_i \hat{V} \mathbf{u}_2}{\|\mathbf{u}_2\|^2}\\
=& \min_m \lambda^{(i)}_m + \min_m \hat{\lambda}^{(i)}_m,
\end{split}
\label{eq:lower_lambda}$$ where we have used Eq. (\[eq:decomp\_sigma\]) and the inequality in Eq. (\[eq:lower\_lambda\]) is established since the feasible region gets larger. Then, the upper bound of the maximum SINR is written as $$\left\{\frac{1}{6N}(\min_m \lambda^{(i)}_m + \min_m \hat{\lambda}^{(i)}_m) + \frac{N_0}{2PT}\right\}^{-1/2} \geq \operatorname{SINR}^\star_i.$$ On the other hand, to derive the lower bound of the maximum SINR, we use the following theorem [@Weyl_eigen].
Let $A$ and $B$ be the $n \times n$ Hermitian matrices whose eigenvalues are written as $\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_n$ and $\beta_1 \geq \beta_2 \geq \cdots \geq \beta_n$, respectively. Further, we define the Hermitian matrix $C=A+B$ whose eigenvalues are written as $\gamma_1 \geq \gamma_2 \geq \cdots \geq \gamma_n$. Then, the following relation holds for $k + l -1 \leq n$ $$\gamma_{k+l-1} \leq \alpha_k + \beta_l.$$
The eigenvalues of sums of Hermitian matrices have been investigated in [@wielandt] [@fulton] [@horn]. From the above theorem, it follows that $$\lambda^{(i)}_{\min} \leq \min \left\{\min_m \lambda^{(i)}_m+ \max_m \hat{\lambda}^{(i)}_m, \max_m \lambda^{(i)}_m + \min_m \hat{\lambda}^{(i)}_m\right\}.
\label{eq:upper_lambda}$$ Equation (\[eq:upper\_lambda\]) is obtained when we set $(k,l)=(n,1)$ and $(k,l)=(1,n)$ in the theorem. With Eq. (\[eq:upper\_lambda\]), a lower bound of the maximum SINR is written as $$\left\{\frac{1}{6N} \gamma + \frac{N_0}{2PT}\right\}^{-1/2} \leq \operatorname{SINR}^\star_i,$$ where $$\gamma = \min \left\{\min_m \lambda^{(i)}_m+ \max_m \hat{\lambda}^{(i)}_m, \max_m \lambda^{(i)}_m + \min_m \hat{\lambda}^{(i)}_m\right\}.$$ From Eqs. (\[eq:lower\_lambda\]) and (\[eq:upper\_lambda\]), we observe that the maximum SINR is related to the quantities $\lambda_m$ and $\hat{\lambda}_m$, that is, the maximum SINR for a desired user may depend on the sequences for the other users. This relation is numerically verified in Section VI. These observations yield that the maximum SINR is improved if the quantities $\lambda^{(i)}_m$ and $\hat{\lambda}^{(i)}_m$ are reduced. Therefore, if the spreading sequences $\mathbf{s}_k$ for $k \neq i$ are designed to achieve lower $\lambda^{(i)}_m$ and $\hat{\lambda}^{(i)}_m$ for $m=1,2,\ldots,N$, then larger SINR is obtained with the optimal sequence for the user $i$, $\mathbf{s}^\star_i$.
Algorithm to Obtain Large SINRs
===============================
In the previous sections, we have discussed the SINR and capacity for a certain user with the optimal sequence. In this section, we discuss the way to obtain sequences for all the users which achieve large SINRs. To take into account sequences for all the users, we consider the following sum of the squared SINRs $$\frac{1}{K}\sum_{i=1}^K \left( \operatorname{SINR}(\mathbf{s}_i)_i \right)^2 = \frac{1}{K}\sum_{i=1}^K \left\{ \frac{1}{6N^2}\sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} + \frac{N_0}{2PT} \right\}^{-1}.
\label{eq:sum_squared_SINR}$$ Note that the above quantity is the average of the squared SINRs, and is expected to yield SINR for all the users. Then, our goal is to obtain the sequences which make the quantity shown in Eq. (\[eq:sum\_squared\_SINR\]) large.
However, it is not straightforward to analyze Eq. (\[eq:sum\_squared\_SINR\]) since there is the sum of inverse numbers. To overcome this obstacle, we consider the harmonic mean of squared SINRs which is written as $$\begin{split}
&K \left\{ \sum_{i=1}^K \left( \operatorname{SINR}(\mathbf{s}_i)_i \right)^{-2} \right\}^{-1} \\
=& K \left\{ \frac{1}{6N^2} \sum_{i=1}^K \sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} + \frac{KN_0}{2PT} \right\}^{-1}.
\label{eq:harm_ave_SINR}
\end{split}$$ From the relation between the arithmetic mean and the harmonic mean, the following relation is established $$\begin{split}
& K \left\{ \frac{1}{6N^2} \sum_{i=1}^K \sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} + \frac{KN_0}{2PT} \right\}^{-1}\\
\leq & \frac{1}{K}\sum_{i=1}^K \left\{ \frac{1}{6N^2}\sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} + \frac{N_0}{2PT} \right\}^{-1}.
\label{eq:harm_ave_SINR}
\end{split}$$
From the above inequality, it is expected that the average of the SINRs increases as the harmonic mean increases. Thus, instead of the average of SINRs, we consider the harmonic mean of SINRs. Then, we consider the following problem $$\begin{split}
(P)& \hspace{3mm} \min \hspace{2mm} \sum_{i=1}^K \sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} \\
& \mbox{subject to} \hspace{3mm} \|\mathbf{s}_i\|^2 = N\hspace{3mm}(i=1,\ldots,K).
\end{split}$$ Similar to the discussion in Section III, we consider only the user $i$. Here, we assume that only the sequence for the user $i$, $\mathbf{s}_i$, is a variable and that the other sequences $\mathbf{s}_k$ are given and fixed for $k \neq i$. This idea is seen as an alternating direction method of multipliers (ADMM) technique [@admm]. Under this assumption, we solve the following problem $$\begin{split}
(P_i)& \hspace{3mm} \min \hspace{2mm} \sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} \\
& \mbox{subject to} \hspace{3mm} \|\mathbf{s}_i\|^2 = N.
\end{split}$$ We emphasize that the other sequences $\mathbf{s}_k$ for $k \neq i$ are given and fixed (see assumption 3 in Section II). As seen in Section III, the optimal value and minimizer are written in terms of the minimum eigenvalue and the corresponding eigenvector of the matrix $\Sigma_i$, respectively. Our algorithm is written in Algorithm \[algo:ours\].
Set the initial sequences $\mathbf{s}_k$ for $k=1,\ldots,K$ and $l=0$. Set $L \geq 1$.\
For $k=1,\ldots,K$, solve the problem $(P_i)$, obtain the optimal solution $\mathbf{s}^\star_k$, and set $\mathbf{s}_k \leftarrow \mathbf{s}^\star_k$.\
$l \leftarrow l+1$.\
If $\{ \mathbf{s}_k \}_{k=1,\ldots,K}$ converge or $l = L$, then go to Step 5. Otherwise, go to step 2.\
Output $\mathbf{s}_k$.
Note that when the problem $(P_i)$ is solved, the sequences $\mathbf{s}_k$ ($k=1,\ldots,i-1$) have already been updated.
Here we give an explanation about why large SINRs will be achieved with Algorithm \[algo:ours\]. As seen in the problem $(P)$, our aim is to achieve the large harmonic mean of squared SINRs. For $i$, the objective function of the problem $(P)$ is evaluated as $$\begin{split}
\sum_{i=1}^K \sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} &= 2\sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k} + \sum_{\substack{k_1=1 \\ k_1 \neq i}}^K \sum_{\substack{k_2=1 \\ k_2 \neq i,k_1}}^K \sum_{m=1}^N S_m^{k_1,k_2}\\
& = 2 \mathbf{s}^*_i \Sigma_i \mathbf{s}_i + \sum_{\substack{k_1=1 \\ k_1 \neq i}}^K \sum_{\substack{k_2=1 \\ k_2 \neq i,k_1}}^K \sum_{m=1}^N S_m^{k_1,k_2}\\
& \geq 2 N \lambda^{(i)}_{\min} + \sum_{\substack{k_1=1 \\ k_1 \neq i}}^K \sum_{\substack{k_2=1 \\ k_2 \neq i,k_1}}^K \sum_{m=1}^N S_m^{k_1,k_2},
\end{split}
\label{eq:decomp_harm}$$ where we have used the fact that $S_m^{i,k} = S_m^{k,i}$ for all $m$ and the results obtained in Section III. In the right hand side of the first line in Eq. (\[eq:decomp\_harm\]), the first term depends on $\mathbf{s}_i$ and the last term is independent of $\mathbf{s}_i$. Further, the first term is the objective function of the problem $(P_i)$. Thus, the first term can be minimized with the sequence $\mathbf{s}^\star_i$ and its value equals to $2N\lambda^{(i)}_{\min}$. This observation yields that solving the problem $(P_i)$ is equivalent to minimizing the terms relating the sequence $\mathbf{s}_i$ in the harmonic mean of squared SINRs.
From the above discussions, it has been shown that solving the problem $(P_i)$ leads to reducing the harmonic mean of squared SINRs.
Numerical Results
=================
We obtain the sequences $\mathbf{s}_k$ for $k=1,\ldots,K$ with Algorithm \[algo:ours\]. We set the number of users $K = 7$ and the length of sequences $N=31$. As the initial sequences (Step 1 in Algorithm \[algo:ours\]), the Gold codes [@gold] and random sequences are used. We calculate Bit Error Rate (BER) as $$\mbox{BER} = \frac{1}{K}\frac{1}{U} \sum_{k=1}^K \sum_{u=1}^U \mbox{BER}_{k,u},$$ where $\mbox{BER}_{k,u}$ is the BER of the user $k$ at the $u$-th iteration and $U$ is the number of iterations. Here, we set $U=1.0 \times 10^4$. As seen in Section II, the modulation scheme is BPSK. There is no fading effect.
Figure \[fig:ber\_gold\] shows the BER in the case where gold codes are used as the initial sequences. Here, $E_b$ denotes the average power per bit. Further, in the legend, “iteration” means $L$ in Algorithm \[algo:ours\]. As seen in Fig. \[fig:ber\_gold\], BER gets reduced when the number of iterations gets large. This observation yields that reducing the harmonic mean of squared SINRs leads to enlarging SINR for each user. In particular, the BER with one iteration is larger than ones with the other iterations. This observation yields that the maximum SINR for a desired user depends on the sequences for the other users. The reason is as follows. As seen in Eqs. (\[eq:lower\_lambda\]) and (\[eq:upper\_lambda\]), the maximum SINR for a desired user is written in terms of the sequences for the other users. In step 2 in our algorithm, the optimal sequence for the user $k$ is obtained. Then, SINR for the user $k$ is maximized. If the maximum SINR for a desired user is independent of the sequences for the other users, then the maximum SINR is constant for every iteration number $L$. However, as seen in Fig. \[fig:ber\_gold\], it is observed that the BER is varied for every iteration number $L$. This yields that the maximum SINR is varied for every $L$. From the above discussions, it is numerically verified that the maximum SINR for a desired user depends on the sequences for the other users.
Figure \[fig:ber\_random\] show the BER in the case where random sequences are used as the initial sequences. The aim is to verify whether the performance depends on initial sequences or not. As seen in Fig. \[fig:ber\_random\], BER gets smaller as the number of iterations increases. Since the initial sequences are generated randomly, the initial sequences have large BER. However, the BER with one iteration reduces significantly. The BER with 50 iterations is the smallest in this figure and its value is nearly equivalent to one with 50 iterations in Fig. \[fig:ber\_gold\]. From this observation, it is expected that our algorithm can always achieve low BER when the number of iterations is sufficiently large. Note that we have not proven the convergence of the objective function of the problem $(P)$.
Figures \[fig:sir\_0\] and \[fig:sir\_ave\] show the Signal-to-Interference noise Ratio (SIR) obtained with our algorithm. Here, SIR of the user $i$ is defined as $$\operatorname{SIR}_i= \operatorname{SIR}(\mathbf{s}_i)_i = \left\{ \frac{1}{6N^2}\sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k}\right\}^{-1/2}.
\label{eq:SIR_i}$$ By definition, SIR is equivalent to SINR with $N_0 = 0$. To depict these two figures, the Gold codes are used as the initial sequences. Figure \[fig:sir\_0\] shows the SIR raised to the power -2 for the user 1 at each iteration in our algorithm, that is, the vertical axis shows $\operatorname{SIR}^{-2}_1$. Since this quantity is in the objective function of the problem $(P_1)$, this figure also shows the value of the objective function of the problem $(P_1)$. As seen in this figure, the inverse of the squared SIR at 1 iteration is larger than the others except for the original one (SIR with original sequences). From this observation, SIR of the optimal sequence for a desired user depends on the sequences for the other users. This result explains why the BER at 1 iteration is larger than the other ones except for the Gold codes (see Fig. \[fig:ber\_gold\]). Further, as the number of the iteration gets larger, the quantity $\operatorname{SIR}^{-2}_1$ gets closer to 0.
Figure \[fig:sir\_ave\] shows the average of the inverses of squared SIRs at each iteration, that is, the vertical axis in Fig. \[fig:sir\_ave\] shows $$\begin{split}
&\mbox{(vertical axis in Fig. \ref{fig:sir_ave})}\\
=&\frac{1}{K}\sum_{i=1}^K\operatorname{SIR}^{-2}_i = \frac{1}{K}\sum_{i=1}^K \frac{1}{6N^2}\sum_{\substack{k=1 \\ k \neq i}}^K \sum_{m=1}^N S_m^{i,k}.
\end{split}
\label{eq:vert_fig2}$$ This quantity is in the objective function of the problem $(P)$. As seen in Fig. \[fig:sir\_ave\], the value of Eq. (\[eq:vert\_fig2\]) gets smaller and closer to 0 as the number of iterations gets larger. In Fig. \[fig:sir\_0\], we have seen that $\operatorname{SIR}^{-2}_1$ gets closer to 0 as the number of iterations gets larger. In Section V, we have considered the problem $(P)$ to take into account the SINRs for all the users. Further, from the relation between the arithmetic mean and the harmonic mean (see Eq. (\[eq:harm\_ave\_SINR\])), when the quantity shown in Fig. \[fig:sir\_ave\] and Eq. (\[eq:vert\_fig2\]) gets reduced, the arithmetic mean of squared SIRs gets large. Thus, Fig. \[fig:sir\_ave\] numerically verifies that our algorithm can achieve large SIR for each user $i$ and large SINRs for all the users are achieved with our algorithm.
![Bit Error Rate with sequences of each iteration: Initial sequences are the Gold codes.[]{data-label="fig:ber_gold"}](db_ber.eps){width="2.8in"}
![Bit Error Rate with sequences of each iteration: Initial sequences are generated randomly.[]{data-label="fig:ber_random"}](db_random_ber.eps){width="2.8in"}
![SIR for the user 1 at each iteration: Initial sequences are the Gold codes.[]{data-label="fig:sir_0"}](sir_0.eps){width="2.8in"}
![Average of inverses of squared SIRs: Initial sequences are the Gold codes.[]{data-label="fig:sir_ave"}](sir_ave.eps){width="2.8in"}
Conclusion
==========
In this paper, we have derived the optimal spreading sequence for the user $i$, which achieves maximum SINR and maximum capacity under an approximation. It has turned out that the maximum SINR is written in terms of the minimum eigenvalue of the matrix $\Sigma_i$ and that the optimal spreading sequence is obtained as a corresponding eigenvector. Further, we have derived the lower and upper bounds of maximum SINR. From these bounds, the maximum SINR will get larger as the quantities $\lambda^{(i)}_m$ and $\hat{\lambda}^{(i)}_m$ get smaller. From the derivation of the optimal sequence for a desired user, we have proposed the algorithm to obtain the sequences which achieve large SINRs for all the users. In numerical results, the performance of our algorithm has been verified. These results have also shown that the performance of the optimal sequence for a desired user depends on the sequences for the other users.
To consider the practical situations, we have to take into account fading effects. One issue is to derive optimal sequences in a sense of SINR under fading effects. This should be considered somewhere as a remaining issue.
Acknowledgment {#acknowledgment .unnumbered}
==============
The author would like to thank Dr. Shin-itiro Goto for his advise.
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[^1]: This work was partially supported by JSPS (KAKENHI) Grant Number 18J12903. The material in this paper was presented in part at the EuCNC 2019, Valencia, Spain, June 2019.
[^2]: The author is with the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan (e-mail: tsuda.hirofumi.38u@st.kyoto-u.ac.jp).
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---
abstract: 'Observations of the frequency dependence of the global brightness temperature of the redshifted 21 cm line of neutral hydrogen may be possible with single dipole experiments. In this paper, we develop a Fisher matrix formalism for calculating the sensitivity of such instruments to the 21 cm signal from reionization and the dark ages. We show that rapid reionization histories with duration $\Delta z\lesssim 2$ can be constrained, provided that local foregrounds can be well modelled by low order polynomials. It is then shown that observations in the range $\nu=50-100{\,\rm MHz}$ can feasibly constrain the [Ly$\alpha$ ]{}and X-ray emissivity of the first stars forming at $z\sim15-25$, provided that systematic temperature residuals can be controlled to less than 1 mK. Finally, we demonstrate the difficulty of detecting the 21 cm signal from the dark ages before star formation.'
author:
- 'Jonathan R. Pritchard'
- Abraham Loeb
title: Constraining the unexplored period between reionization and the dark ages with observations of the global 21 cm signal
---
[^1]
Introduction {#sec:intro}
============
The transition of the Universe from the dark ages following hydrogen recombination through to the epoch of reionization remains one of the least constrained frontiers of modern cosmology. Observing the sources responsible for heating and ionizing the intergalactic medium (IGM) at redshifts $z\gtrsim6$ pushes current observational techniques to the limit. Plans are underway to construct low-frequency radio telescopes, such as LOFAR[^2], MWA[^3], PAPER[^4], and SKA[^5], to observe the red-shifted 21 cm line of neutral hydrogen. These experiments aim to map the state of the intergalactic medium via tomographic observations of 3D fluctuations in the 21 cm brightness temperature. A simpler and significantly lower cost alternative to this would be measurements of the global 21 cm signal integrated over the sky [@mmr1997; @shaver1999; @sethi2005], which can be achieved by single dipole experiments like EDGES [@bowman2007edges] or CoRE [@chippendale2005]. Although such experiments are today in their infancy, their potential is large. In this paper, we explore the potential for these global sky experiments to measure the 21 cm signal and constrain the high redshift Universe.
We may draw a historical analogy with the Cosmic Background Explorer (COBE), whose FIRAS instrument measured the blackbody spectrum of the cosmic microwave background (CMB) [@mather1994] while the DMR instrument measured the level of temperature fluctuations [@smoot1992]. The precise measurement of a $T_{\rm CMB}=2.726{\,\rm K}$ blackbody spectrum placed tight constraints on early energy injection, since no Compton-$y$ or $\mu$-distortion were seen, and provided important evidence confirming the big bang paradigm. The detection of angular fluctuations paved the way for more sensitive experiments such as BOOMERANG [@lange2001] and WMAP [@spergel2003], which provided precision measurements of the CMB acoustic peaks. While, at the moment, attention is focussed on experiments designed to measure 21 cm fluctuations, it is important not to neglect the possibility of measuring the global signal.
The evolution of the 21 cm signal is driven primarily by the amount of neutral hydrogen and the coupling between the 21 cm spin temperature and the gas temperature. It is able to act as a sensitive thermometer when the IGM gas temperature is less than the CMB temperature placing constraints on energy injection that leads to heating. For example, the first black holes to form generate X-rays, which heat the gas. More exotic processes such as annihilating dark matter might have also been important. Additionally, energy injection in the form of [Ly$\alpha$ ]{}production modifies the strength of the coupling. This provides a way of tracking star formation, which will be the dominant source of [Ly$\alpha$ ]{}photons. As we show, the spectral structure of the 21 cm signal is much richer than that of a blackbody so that many things can be learnt about the early Universe. Given the uncertainties, we develop a model approach based upon those physical features most likely to be present.
The single most important factor determining the sensitivity of dipoles to astrophysics will be their ability to remove galactic foregrounds [e.g. @oh2003; @dimatteo2004]. Exploitation of spectral smoothness to remove foregrounds by fitting low order polynomials is key to avoiding throwing the signal away with the foreground. To quantify this, we develop a simple Fisher matrix formalism and validate it against more detailed numerical parameter fitting. This provides us with a way of quantitatively addressing the ability of global 21 cm experiments to constrain reionization and the astrophysics of the first galaxies [@loeb2010book]. Similar work on the subject [@sethi2005] ignored the influence of foregrounds limiting its utility considerably.
Much of the power of this technique stems from the limitations of other observational probes. While next generation telescopes such as JWST[^6], GMT[^7], EELT[^8] or TMT[^9] may provide a glimpse of the Universe at $z\gtrsim12$ they peer through a narrow field of view and are unlikely to touch upon redshifts $z\gtrsim20$. As we will show, 21 cm global experiments could potentially provide crude constraints on even higher redshifts at a much lower cost.
The structure of this paper is as follows. In §\[sec:physics\], we begin by describing the basic physics that drives the evolution of the 21 cm global signature and drawing attention to the key observable features. We follow this in §\[sec:foreground\] with a discussion of the foregrounds, which leads into our presenting a Fisher matrix formalism for predicting observational constraints in §\[sec:fisher\]. In §\[sec:reion\] and §\[sec:astro\] we apply this formalism to the signal from reionization and the first stars, respectively. After a brief discussion in §\[sec:darkages\] of the prospects for detecting the signal from the dark ages before star formation, we conclude in §\[sec:conclude\].
Throughout this paper where cosmological parameters are required we use the standard set of values $\Omega_m=0.3$, $\Omega_\Lambda=0.7$, $\Omega_b=0.046$, $H=100h\,\rm{km\,s^{-1}\,Mpc^{-1}}$ (with $h=0.7$), $n_S=0.95$, and $\sigma_8=0.8$, consistent with the latest measurements [@komatsu2009].
Physics of the 21 cm global signal {#sec:physics}
==================================
The physics of the cosmological 21 cm signal has been described in detail by a number of authors [@fob; @pritchard2008] and we focus here on those features relevant for the global signal. It is important before we start to emphasise our uncertainty in the sources of radiation in the early Universe, so that we must of necessity extrapolate far beyond what we know to make predictions for what we may find. Nonetheless the basic atomic physics is well understood and a plausible understanding of the likely history is possible.
The 21 cm line frequency $\nu_{\rm21\,cm}=1420{\,\rm MHz}$ redshifts for $z=6-27$ into the range 200-50 MHz. The signal strength may be expressed as a differential brightness temperature relative to the CMB $$\begin{gathered}
\label{tb}
T_b=27 x_{\rm{HI}}\left(\frac{T_S-T_\gamma}{T_S}\right)\left(\frac{1+z}{10}\right)^{1/2}\\ \times(1+\delta_b)\left[\frac{\partial_r v_r}{(1+z)H(z)}\right]^{-1}\,\rm{mK},
\end{gathered}$$ where $x_{\rm HI}$ is the hydrogen neutral fraction, $\delta_b$ is the overdensity in baryons, $T_S$ is the 21 cm spin temperature, $T_\gamma$ is the CMB temperature, $H(z)$ is the Hubble parameter, and the last term describes the effect of peculiar velocities with $\partial_r v_r$ the derivative of the velocities along the line of sight. Throughout this paper, we will neglect fluctuations in the signal so that neither of the terms $\delta_b$ nor the peculiar velocities will be relevant. Fluctuations in $x_H$ and $\delta_b$ will be relevant for the details of the signal, but are not required to get the broad features of the signal, on which we focus here.
![Evolution of the 21 cm global signal for different scenarios. [*Solid blue curve:* ]{} no stars; [*solid red curve:* ]{}$T_S\gg T_\gamma$; [*black dotted curve:* ]{}no heating; [*black dashed curve:* ]{}no ionization; [*black solid curve:* ]{}full calculation.[]{data-label="fig:pedplot"}](pedplot_nu.eps)
The evolution of $T_b$ is thus driven by the evolution of $x_H$ and $T_S$ and is illustrated for redshifts $z<100$ in Figure \[fig:pedplot\]. Early on, collisions drive $T_S$ to the gas temperature $T_K$, which after thermal decoupling (at $z\approx1000$) has been cooling faster than the CMB leading to a 21 cm absorption feature ($[T_S-T_\gamma]<0$). Collisions start to become ineffective at redshifts $z\sim80$ and scattering of CMB photons begins to drive $T_S\rightarrow T_\gamma$ causing the signal to disappear. In the absence of star formation, this would be the whole story [@loeb_zald2004].
Star formation leads to the production of [Ly$\alpha$ ]{}photons, which resonantly scatter off hydrogen coupling $T_S$ to $T_K$ via the Wouthysen-Field effect [@wouth1952; @field1958]. This produces a sharp absorption feature beginning at $z\sim30$. If star formation also generates X-rays they will heat the gas, first causing a decrease in $T_b$ as the gas temperature is heated towards $T_\gamma$ and then leading to an emission signal, as the gas is heated to temperatures $T_K>T_\gamma$. For $T_S\gg T_\gamma$ all dependence on the spin temperature drops out of equation and the signal becomes saturated. This represents a hard upper limit on the signal. Finally reionization will occur as UV photons produce bubbles of ionized hydrogen that percolate, removing the 21 cm signal.
We may thus identify five main events in the history of the 21 cm signal: (i) collisional coupling becoming ineffective (ii) [Ly$\alpha$ ]{}coupling becoming effective (iii) heating occurring (iv) reionization beginning (v) reionization ending. In the scenario described above the first four of these events generates a turning point (${\mbox{d}}T_b/{\mbox{d}}z=0$) and the final event marks the end of the signal. We reiterate that the astrophysics of the sources driving these events is very uncertain, so that when or even if these events occur as described is currently unknown. Figure \[fig:param\_comp\] shows a set of histories for different values of the X-ray and [Ly$\alpha$ ]{}emissivity, parametrized about our fiducial model by $f_X$ and $f_\alpha$ representing the product of the emissivity and the star formation efficiency following Ref. [@pritchard2008]. Clearly the positions of these features may move around both in the amplitude of $T_b$ and the frequency at which they occur.
![Dependence of 21 cm signal on the X-ray (top panel) and [Ly$\alpha$ ]{}(bottom panel) emissivity. In each case, we consider examples with the emissivity reduced or increased by a factor of up to 100. Note that in our model $f_X$ and $f\alpha$ are really the product of the emissivity and the star formation efficiency.[]{data-label="fig:param_comp"}](param_comp.eps)
We view this to be the most likely sequence of events for plausible astrophysical models. We are reassured in this sequencing since, in the absence of [Ly$\alpha$ ]{}photons escaping from galaxies [@higgins2009], X-rays will also produce [Ly$\alpha$ ]{}photons [@chen2006; @chuzhoy2006] and so couple $T_S$ to $T_K$ and, in the absence of X-rays, scattering of [Ly$\alpha$ ]{}photons heats the gas [@ciardi2003]. In each case the relative sequence of events is likely to be maintained. We will return to how different models may be distinguished later and now turn to the presence of foregrounds between us and the signal.
Foregrounds {#sec:foreground}
===========
At the frequencies of interest (10-250 MHz), the sky is dominated by synchrotron emission from the galaxy. A useful model of the sky has been put together by Ref. [@angelica2008] using all existing observations. The sky at 100 MHz is shown in Figure \[fig:skymaps\], where the form of the galaxy is clearly visible. In this paper, we will be focusing upon observations by single dipole experiments. These have beam shapes with a typical field-of-view of tens of degrees. The lower panel of Figure \[fig:skymaps\] shows the beam of dipole (approximated here as a single $\cos^2\theta$ lobe) sitting at the MWA site in Australia (approximate latitude 26$^\circ$59’S), observing at zenith, and integrated over a full day. Although the dipole does not see the whole sky at once it does average over large patches. We will therefore neglect spatial variations (although we will return to this point in our conclusions).
![[*Top panel:* ]{}Radio map of the sky at 100 MHz generated from Ref. [@angelica2008]. [*Bottom panel:* ]{} Ideal dipole response averaged over 24 hours.[]{data-label="fig:skymaps"}](plot_example_execute.eps "fig:") ![[*Top panel:* ]{}Radio map of the sky at 100 MHz generated from Ref. [@angelica2008]. [*Bottom panel:* ]{} Ideal dipole response averaged over 24 hours.[]{data-label="fig:skymaps"}](plot24.eps "fig:")
Averaging the foregrounds over the dipole’s angular response gives the spectrum shown in the top panel of Figure \[fig:fitresidual\_nice\]. First note that the amplitude of the foregrounds is large $\sim100{\rm\,K}$ compared to the 10 mK signal. Nonetheless, given the smooth frequency dependence of the foregrounds we are motivated to try fitting the foreground out using a low order polynomial in the hope that this leaves the signal behind. This has been shown by many authors [e.g. @mcquinn2005; @wang2006] to be a reasonable procedure in the case of 21 cm tomography. There the inhomogeneities fluctuate rapidly with frequency, so that only the largest Fourier modes of the signal are removed. In the case of the global 21 cm signal our signal is relatively smooth in frequency, especially if the bandwidth of the instrument is small. Throwing the signal out with the foregrounds is therefore a definite concern.
![Foreground (top panel) and residuals (bottom panel) left over after fitting a N-th order polynomial in $\log\nu$ to the foreground.[]{data-label="fig:fitresidual_nice"}](fitresidual_nice.eps)
Throughout this paper, we will fit the foregrounds using a polynomial of the form $$\log T_{\rm fit}=\sum_{i=0}^{N_{\rm poly}}a_i \log(\nu/\nu_0)^i.$$ Here $\nu_0$ is a pivot scale and we will generally recast $a_0\rightarrow \log T_0$ to emphasise that the zeroth order coefficient more naturally has units of temperature. The lower panel of Figure \[fig:fitresidual\_nice\] shows the residuals left over after fitting and subtracting polynomials of different order to the foregrounds. It is apparent that a polynomial of at least $N_{\rm poly}=3$ is necessary to remove the foreground. Unfortunately, our current knowledge of the low frequency sky is not sufficient for us to conclusively say that we will not need a higher order polynomial or to accurately quantify the minimum level of residuals that will be left on fitting the signal. The residuals visible in Figure \[fig:fitresidual\_nice\] for $N_{\rm poly}=3$ are dominated by numerical limitations of the sky model being used and have $\sqrt{\langle (T_{\rm sky}-T_{\rm fit})^2\rangle}\lesssim1{\,\rm mK}$ averaged over the band.
![Dependence of the best fit values for the first six parameters from the foreground fitting process on the order of the polynomial, $N_{\rm poly}$.[]{data-label="fig:fitvalues"}](fitvalues.eps)
Figure \[fig:fitvalues\] shows the evolution of the best fit values as we change the order of the fit. The first four values are non-zero and therefore important to the fit. The next two hover around zero (although as the order increases they move away from zero). This supports the inference that only the first four parameters are necessary and after that we are beginning to over fit. We therefore take as our fiducial model for the foreground the form $$\begin{gathered}
\log T_{\rm sky}= \log T_0\\+a_1\log(\nu/\nu_0) + a_2[\log(\nu/\nu_0)]^2 +a_3 [\log(\nu/\nu_0)]^3, \end{gathered}$$ with parameter values $\nu_0=150{\rm\,MHz}$, $T_0=320{\rm\,K}$, $a_1=-2.54$, $a_2=-0.074$, $a_3=0.013$, chosen from fitting to the band $\nu=100-200$ MHz. These values are roughly consistent with those found by the observations reported in Ref. [@rogers2008], which found $T_0=237\pm10{\,\rm K}$ and $a_1=-2.5\pm0.1$ over the same band. Where necessary we include additional terms as $a_i=0$ for $i\geq4$. Fitting to a different bandwidth and pivot frequency will modify these values. For example, fitting to $\nu=50-150$ MHz with $\nu_0=100$ MHz yields, $T_0=875{\rm\,K}$, $a_1=-2.47$, $a_2=-0.089$, $a_3=0.013$. Aside from the overall normalisation, there is little qualitative change in the shape.
Fisher calculation {#sec:fisher}
==================
The main objective of this paper is to develop a formalism for quantifying the ability of global 21 cm experiments to constrain astrophysical parameters. A straightforward, but brute force approach, is to model the signal, add a foreground, and then use Monte-Carlo (MC) fitting techniques to see how well model parameters may be constrained. When faced with the large space of model parameters to be explored this is inadequate. We therefore explore the use of the Fisher matrix approach, applicable if the model likelihood is well approximated by a multivariate Gaussian. We will later show that this is a good approximation by testing it directly against the results of direct MC fitting.
The Fisher matrix takes the form [@EHT99] $$F_{ij}=\frac{1}{2}{\rm Tr}\left[C^{-1}C_{,i}C^{-1}C_{,j}+C^{-1}(\mu_{,i}\mu^T_{,j}+\mu_{,j}\mu^T_{,i})\right].$$ where $C\equiv\langle x x^T\rangle$ is the covariance matrix and $\mu=\langle x\rangle$. For the 21 cm global signature, our observable is the antennae temperature $T_{\rm sky}(\nu)=T_{\rm fg}(\nu)+T_{b}(\nu)$, where we assume the dipole sees the full sky so that spatial variation can be ignored. We divide the signal into $N_{\rm channel}$ frequency bins {$\nu_n$} of bandwidth $B$ running between \[$\nu_{\rm min}$, $\nu_{\rm max}$\]. The covariance matrix is taken to be diagonal, since errors in different frequency bins are expected to be uncorrelated, so that it is given by $$C_{ij}=\delta_{ij}\sigma_i^2,$$ with the thermal noise given by the radiometer equation $$\sigma_i^2=\frac{T_{\rm sky}^2(\nu_i)}{B t_{\rm int}},$$ assuming an integration time $t_{\rm int}$. In this paper, we will consider single dipole experiments, but the noise could be further reduced by a factor $N_{\rm dipole}$ through the incoherent summing of the signal from multiple dipoles. Finally, we can allow for a limiting floor in the noise due to foreground fitting residuals or instrumental noise by setting $\sigma_i^2\rightarrow\sigma_i^2+\sigma_{i,{\rm res}}^2$.
Under these assumptions the Fisher matrix takes the form $$F_{ij}=\sum_{n=1}^{N_{\rm channel}}(2+B t_{\rm int})\frac{{\mbox{d}}\log T_{\rm sky}(\nu_n)}{{\mbox{d}}p_i}\frac{{\mbox{d}}\log T_{\rm sky}(\nu_n)}{{\mbox{d}}p_j},$$ where the parameter set $\{p_i\}$ includes both foreground and signal model parameters. Here the first term is the information contained in the amplitude of the noise and is subdominant for reasonable experiments ([cf. @sethi2005]). Given this Fisher matrix, the best parameter constraints achievable on parameter $p_i$ are given by the Cramer-Rao inequality $\sigma_i\geq\sqrt{F^{-1}_{ii}}$. This Fisher matrix offers a fast and, as we will show in the next section, reliable means of calculating the expected constraints for 21 cm global experiments.
The assumption of a full sky observation is not strictly valid, since the dipole sees the sky with a beam tens of degrees across. Both foreground and signal will show spatial variation. Fluctuations in the 21 cm signal can be large in amplitude, but span a characteristic scale of order a few arcminutes corresponding to the size of the ionized bubbles. As such our beam will average over many of these, so that we do not expect significant spatial fluctuations to survive. The foregrounds are another matter and spatial variation may be a mixed blessing. In practice, each foreground parameters should be fitted independently in each pixel. Since the signal is common to all pixels, exploiting the spatial variation of the foregrounds could be used to remove them more efficiently.
So far, we have assumed that the instrument’s frequency response can be calibrated out perfectly. At present one of the limiting factors of the EDGES experiment is that the dipole’s frequency response is uncalibrated. This has the effect of convolving both foregrounds and signal with some unknown function of frequency. Provided that this function is smooth the main complication so introduced is that the convolved foregrounds are no longer easily described by a low order polynomial. In Ref. [@bowman2007edges], a 12th order polynomial in $\nu$ was used for the foreground fitting, primarily in order to fit out the instrumental response. Since this is very much a prototype experiment, we will optimistically assume that this instrumental problem can be dealt with in more advanced designs.
Reionization {#sec:reion}
============
Next, we will consider the possibility of constraining the evolution of the hydrogen neutral fraction from the global 21 cm signal. Predicting the reionization history has attracted a great deal of attention in recent years [@loeb2010book]. Constraints arise from the [Ly$\alpha$ ]{}forest, the optical depth to the CMB, and numerous other locations. Although these may be combined to constrain the reionization history [e.g. @pritchard2009], the quality of current constraints is poor. In general though, reionization is expected to be a relatively extended process.
Given the uncertainty associated with making detailed predictions for the evolution of $x_H$, we adopt as a toy model for reionization a [*tanh*]{} step (as used by the WMAP7 analysis [@larson2010]) with parameters describing the two main features of reionization: its mid point $z_r$ and duration $\Delta z$. We will further assume that the 21 cm spin temperature is saturated at the relevant redshifts (a reasonable although not guaranteed simplifying assumption [@ciardi2003; @pritchard2007xray]). Under these assumptions, the 21 cm brightness temperature is given by $$\label{tanh}
T_{b}(z)=\frac{T_{21}}{2}\left(\frac{1+z}{10}\right)^{1/2}\left[\tanh\left(\frac{z-z_r}{\Delta z}\right)+1\right].$$ In principle, the amplitude of the signal $T_{21}$ is calculable from first principles ($T_{21}=27{\,\rm mK}$ for our fiducial cosmology), but we leave it as a free parameter. This helps us gauge how well the experiment is really detecting the 21 cm signal. Figure \[fig:nuhistory\_s\] shows a few different histories for this model.
![Evolution of the neutral fraction $x_H$ and brightness temperature $T_b$ for a [*tanh*]{} model of reionization (see Eq.\[tanh\]).[]{data-label="fig:nuhistory_s"}](nuhistory_s.eps)
Before exploring the detection space for 21 cm experiments, we validate our Fisher matrix against a more numerically intensive Monte-Carlo. We consider an experiment covering the frequency range $100-250{\,\rm MHz}$ in 50 bins and integrating for 500 hours (these parameters mimic EDGES with an order of magnitude longer integration time). Taking fiducial values of $z_r=8$, $\Delta z=1$, and $N_{\rm poly}=3$, we fit the model and foreground for $10^6$ realisations of the thermal noise. This yields an estimate of the parameter uncertainty that can be expected from observations and can be used to test our Fisher matrix calculation. The resulting parameter contours are shown in Figure \[fig:mc\_comp\] along with the Fisher matrix constraints. That they are in good agreement validates our underlying formalism.
![Comparison of 68 and 95% confidence regions between our MC likelihood (green and red coloured regions) and Fisher matrix (solid ellipses) calculations for a [*tanh*]{} model of reionization with $z_r=8$ and $\Delta z=1$ and fitting four foreground parameters.[]{data-label="fig:mc_comp"}](mc_comp.eps)
The error ellipses show that there is a strong degeneracy between $T_{21}$ and $\Delta z$. This is a consequence of the way in which foreground fitting removes power from more extended histories making it difficult to distinguish a larger amplitude extended scenario from a lower amplitude sharper scenario.
Despite the good agreement, this formalism breaks down when the Fisher matrix errors become large enough that reionization parameters are not well constrained. Although this is not a major hurdle here, caution should be used when errors are much larger than the parameters being constrained.
![95% detection region for global experiments assuming $N_{\rm poly}=3$ (solid curve), 6 (dashed curve), 9 (dotted curve), and 12 (dot-dashed curve). Also plotted are the 68 and 95% contours for WMAP5 with a prior that $x_i(z=6.5)>0.95$ (green and red coloured regions).[]{data-label="fig:zr_dz"}](plane_zr_dz.eps)
The resulting potential detection region for the above experiment is shown in Figure \[fig:zr\_dz\], where we consider several different orders of polynomial fit. The detection region shows a number of wiggles associated with points in the frequency range where the shape of the 21 cm signal becomes more or less degenerate with the polynomial fitting. We also show the 1- and $2-\sigma$ constraint regions from WMAP’s optical depth measurement. These constrain the redshift of reionization, but say little about how long it takes. Adding in a prior based upon [Ly$\alpha$ ]{}forest observations that the Universe is fully ionized by $z=6.5$ (specified here as $x_i(z=6.5)>0.95$) removes the region of parameter space with large $\Delta z$ and low $z_r$.
Global experiments can take a good sized bite out of the remaining parameter space. They are sensitive to the full range of redshifts, but primarily to the sharpest reionization histories. Only if $N_{\rm poly}\le6$ can histories with $\Delta z>1$ be constrained and histories with $\Delta z\gtrsim2.5$ appear too extended for high significance detections.
This is unfortunate, since @pritchard2009 found that most reionization histories compatible with the existing data have $\Delta z\gtrsim2$, suggesting it will be difficult for global experiments to probe the most likely models. An important caveat to these conclusions is that the [*tanh*]{} model that we have used here is a toy model of reionization. More realistic models may have more detectable features since they often end rapidly, but have a long tail to high redshifts.
First sources {#sec:astro}
=============
We now turn from reionization to the signal produced by the first galaxies, which generate an early background of [Ly$\alpha$ ]{}and X-ray photons. This region is essentially unconstrained by existing observations and global 21 cm experiments represent one of the only upcoming ways of probing this epoch.
Although models for the signal during this epoch exist [@furlanetto2006; @pritchard2008], it will be useful to focus on physical features of the signal that are both observable and model independent. With this in mind, we parametrize the signal in terms of the turning points of the 21 cm signal. Figure \[fig:spline\_comp\] shows the evolution of $T_b$ and its frequency derivative. As discussed in §\[sec:physics\], there are four turning points associated with: (0) a minimum during the dark ages where collisional coupling begins to become ineffective, (1) a maximum at the transition from the dark ages to the [Ly$\alpha$ ]{}pumping regime as [Ly$\alpha$ ]{}pumping begins to be effective, (2) an absorption minimum as X-ray heating begins to raise the signal towards emission, (3) an emission maximum as the signal becomes saturated and starts to decrease with the cosmic expansion. Finally reionization completes providing a fifth point. Asymptotically the signal goes to zero at very low and high frequencies.
![Evolution of the 21 cm global signal and its derivative. Vertical dashed lines indicate the locations of the turning points. In the top panel, we also show a cubic spline fit to the turning points (blue dotted curve) as described in the text.[]{data-label="fig:spline_comp"}](spline_comp.eps)
In order to have a simple model for the evolution of the signal, we adopt parameters $(\nu_0,T_{b0})$, $(\nu_1,T_{b1})$, $(\nu_2,T_{b2})$, $(\nu_3,T_{b3})$, and $\nu_4$ for the frequency and amplitude of the turning points and the frequency at the end of reionization. For clarity of notation we will label these points as $\mathbf{x}_i=(\nu_i,T_{bi})$ (with $\mathbf{x}_4=(\nu_4,0{\,\rm mK})$). We then model the signal with a simple cubic spline between these points with the additional condition that the derivative should be zero at the turning points (enforced by doubling the data points at the turning points and offsetting them by $\Delta\nu=\pm1{\,\rm MHz}$).
For our fiducial model, we adopt the fiducial parameter set of Ref. [@pritchard2008], assuming a star forming efficiency $f_*=0.1$, a [Ly$\alpha$ ]{}emissivity expected for Population II stars $f_\alpha=1$, and X-ray emissivity appropriate for extrapolating the locally observed X-ray-FIR correlation, $f_X=1$. This gives turning points $\mathbf{x}_0$=(16.1 MHz, -42 mK), $\mathbf{x}_1$=(46.2 MHz, -5 mK), $\mathbf{x}_2$=(65.3 MHz, -107 mK), $\mathbf{x}_3$=(99.4 MHz, 27 mK), and $\mathbf{x}_4$=(180 MHz, 0 mK). The resulting spline fit is shown in the top panel of Figure \[fig:spline\_comp\]. The model does a good job of capturing the general features of the 21 cm signal, although there are clear differences in the detailed shape. Since global experiments are unlikely to constrain more than the sharpest features, this approach should be adequate for our purposes.
There is considerable uncertainty in the parameters of this model, and so to gauge the likely model dependence of the turning points, we make use of the model of Ref. [@pritchard2008]. Varying the [Ly$\alpha$ ]{}, X-ray, and UV emissivity by two orders of magnitude on either side of their fiducial values we find the position and amplitude of the turning points to give the parameter space shown in Figure \[fig:turning\_map\]. This provides a useful guide to targeting observations in frequency space. We have found that a global experiment has very little sensitivity to features lying outside of the observed frequency band.
![Parameter space for the frequency and brightness temperature of the four turning points of the 21 cm signal calculated by varying parameters over the range $f_X=[0.01,100]$ and $f_\alpha=[0.01,100]$ for fixed cosmology and star formation rate $f_*=0.1$. Green region indicates $f_\alpha>1$, red region indicates $f_X>1$, blue regions indicates both $f_\alpha>1$ and $f_X>1$, while the black region has $f_\alpha<1$ and $f_X<1$.[]{data-label="fig:turning_map"}](turning_map.eps)
Since we fix the cosmology, $\mathbf{x}_0$ appears as a single point. The locations of $\mathbf{x}_1$ and $\mathbf{x}_3$ are controlled by the [Ly$\alpha$ ]{}and X-ray emissivity respectively. Only $\mathbf{x}_2$ shows significant dependence on both [Ly$\alpha$ ]{}and X-ray emissivity leading to a large uncertainty in its position. This is good news observationally, since even a poor measurement of the position of $\mathbf{x}_2$ is likely to rule out a wide region of parameter space. Since $\mathbf{x}_2$ is the feature with both the largest amplitude and sharpest shape, we expect that this is the best target for observation and makes experiments covering $\nu=50-100{\,\rm MHz}$ of great interest.
Since our model is approximate, it is important to check whether it leads to significantly biased constraints on the features of interest. One could imagine that fitting the splined shape might lead to biased estimates of the position of the turning points, for example. We have checked this through Monte-Carlo simulation by fitting the turning-point model to the full calculation signal for $10^6$ realisations of the thermal noise. As seen in Figure \[fig:mc\_comp\_high\_full\] for an experiment covering $\nu=$45-145 MHz in 50 bins and integrating for 500 hours, the MC calculation shows no sign of significant biasing and is in good agreement with the Fisher matrix calculation using the turning-point model.
![Comparison of the 68 and 95% confidence regions for our MC likelihood (green and red coloured regions) and Fisher matrix (solid contours). The MC calculation fits the turning point model to the full signal while the Fisher matrix calculation is for the turning point model only.[]{data-label="fig:mc_comp_high_full"}](mc_comp_high.eps)
The final panel of Figure \[fig:mc\_comp\_high\_full\] shows a degeneracy between $T_{b2}$ and $T_{b3}$. This might be expected for an experiment whose sensitivity is primarily to the derivative of the signal, which is left unchanged by shifting both of these points up or down.
As we examine lower frequencies where the foregrounds are brighter, we must increasingly worry about foreground removal leaving behind systematic residuals that limit the sensitivity of the experiment. In Figure \[fig:fgerror\_tsys\_nu3\], we plot the sensitivity of the same experiment to $\mathbf{x}_3$ as a function of this residual floor $T_{\rm res}$ for different values of $N_{\rm poly}$. Polynomials with $N_{\rm poly}<9$ are required to have any chance of detecting the signal. Sensitivity to the signal begins to degrade once $T_{\rm res}$ becomes greater than 0.1 mK corresponding roughly to the thermal noise for this experiment. A detection of $\mathbf{x}_3$ is still possible until $T_{\rm sys}\sim1{\,\rm mK}$.
![Dependence of ($\nu_3,T_{b3}$) and ($\nu_2,T_{b2}$) errors with level of systematic residuals for $N_{\rm poly}=3$ (black solid curve), 6 (red dotted curve), and 9 (blue dashed curve). The dashed vertical lines indicates the fiducial values $T_{b3}=27{\,\rm mK}$ and $|T_{b3}|=107{\,\rm mK}$.[]{data-label="fig:fgerror_tsys_nu3"}](fgerror_tsys_full.eps)
We finish this section by comparing the Fisher matrix constraints from Figure \[fig:mc\_comp\_high\_full\] on top of the region spanned by the turning points in Figure \[fig:turning\_map\]. This is shown in Figure \[fig:turning\_contour\] and gives a sense of the large space of astrophysical models that may be ruled out with a single global experiment. While the experiment has trouble constraining $\mathbf{x}_1$ and $\mathbf{x}_3$ with any significance, it places relatively good constraints on $\mathbf{x}_2$.
![Experimental constraints overlaid on the allowed region for the turning points. Shaded regions (dashed curves) illustrate contours of $f_X$ and $f_\alpha$ by an order of magnitude (red to yellow).[]{data-label="fig:turning_contour"}](turning_contour.eps)
Throughout this section we have chosen to model the 21 cm global signal by a simple cubic spline based upon the turning points of the signal. While this model is simple, one can imagine alternative approaches. Since the experiments are primarily sensitive to the derivative of the 21 cm signal, we might imagine taking the positions of the extrema of the derivative ${\mbox{d}}T_b/{\mbox{d}}\nu$ as our parameters and seek to constrain those. We leave the exploration of alternatives such as this to future work.
Dark Ages {#sec:darkages}
=========
The physics of the period before star formation at $z\sim30$ is determined by well known atomic processes and so has much in common with the CMB. However, many models have been put forward that would modify this simple picture with exotic energy deposition via annihilating or decaying dark matter [@furlanetto2006dm] or evaporating black holes [@mack2008], for example. During the dark ages, the 21 cm signal acts as a sensitive thermometer, potentially capable of constraining these exotic processes. Here we will focus on the standard history and leave consideration of the possibility of detecting other scenarios to future work.
The signal during the dark ages reaches a maximum at $\mathbf{x}_0=(16{\,\rm MHz}$, $-42{\,\rm mK})$, somewhat larger in amplitude than the reionization emission signal. However, at these low frequencies the foregrounds are extremely large, $T_{\rm fg}\approx10^4{\,\rm K}$ at $\nu=30{\,\rm MHz}$, making detection very difficult. Its is worth noting however that global experiments have an advantage over tomographic measurements here, since at these early times structures have had little chance to grow, making the fluctuations much smaller than during reionization. Further, it is easier to imagine launching a single dipole experiment beyond the Earth’s ionosphere rather than the many km$^2$ of collecting area needed for interferometers to probe this epoch [@gordon2009; @jester2009].
Given the large foregrounds, long integration times or many dipoles are required to reach the desired sensitivity level. Taking $T_{\rm fg}=10^4{\,\rm K}$ at $\nu=30{\,\rm MHz}$ a single dipole would need to integrate for $t_{\rm int}=1000{\,\rm hours}$ to reach 4 mK sensitivity. Removing the foregrounds over this dynamic range without leaving considerable residuals will clearly require very precise instrumental calibration. Given the challenges, we look at the most optimistic case as a limit of what could be accomplished.
Taking an experiment covering $\nu=5-60{\,\rm MHz}$ in 50 channels and integrating for 8000 hours, we assume a minimal $N_{\rm poly}=3$ polynomial fit leaving no residuals. The resulting constraint on the position and amplitude of the dark ages feature are shown in Figure \[fig:mc\_comp\_dark\]. Such an experiment is capable of detecting the signal, but only barely. For comparison, we have plotted the uncertainty arising from cosmological measurements of $\Omega_mh^2$ and $\Omega_bh^2$, the two main parameters determining the 21 cm signal. This uncertainty is much less than the experimental uncertainty.
![68 and 95% error ellipses on the amplitude and frequency of the dark ages minima for a single dipole experiment (solid curves, see text for details). For comparison, we show the spread in these quantities from the WMAP5 1- and 2-$\sigma$ uncertainties in $\Omega_m h^2$ and $\Omega_b h^2$ (green and red coloured region).[]{data-label="fig:mc_comp_dark"}](mc_comp_dark.eps)
Although we have shown that detecting the dark ages feature from the standard history would be extremely challenging, modified histories arising from exotic energy injection may lead to larger features more easily detected. Since there is no other probe of physics at $30<z<150$ global 21 cm experiments offer a unique if extremely challenging probe of this period.
Conclusions {#sec:conclude}
===========
Observations of the redshifted 21 cm line potentially provide a new window into the high redshift Universe. Detecting this signal in the presence of large foregrounds is challenging and it is important to explore all avenues for exploiting the signal. In this paper, we have focussed upon the possibility of using single dipole experiments to observe the all-sky 21 cm signal, in contrast to the 21 cm fluctuations targeted by MWA, LOFAR, PAPER, and SKA. Experiments targeting this global signal are in their infancy. We emphasise that instruments built from a few dipoles targeting the global 21 cm signal can be several orders of magnitude cheaper to build than interferometers targeting the fluctuations. Their scientific return will be similarly less, but at this stage where we know so little about the first sources, even that little is extremely valuable.
As we have outlined in this paper, the 21 cm signal generated by astrophysical processes has a well defined form, although the input parameters are only poorly understood. We have demonstrated that, at the level of our current knowledge, describing the Galactic foregrounds requires at least a 3rd order polynomial. At this level, we are able to remove the foregrounds to the sub-mK level, although in practice this procedure may be more complicated. In order to characterise the sensitivity of these experiments to the signal, we developed a Fisher matrix formalism and validated it against more numerical fitting of the model parameters. This Fisher matrix approach allows rapid calculations of the experimental sensitivity and appears to reproduce more detailed calculations very well.
Having developed this formalism we applied it to the signal from reionization and the epoch of the first stars. Using a toy model of reionization, we demonstrated that EDGES-like experiments should be capable of constraining rapid reionization histories with $\Delta z\lesssim2$. More promisingly, these experiments can rule out a wide variety of astrophysical models for the signal from the first stars where the evolution of the spin temperature is important. We used a straightforward fitting form for the signal based upon the positions of the turning points and showed that these features could be constrained, with the deepest absorption trough providing the best observational target.
Finally, we briefly explored the possibility of detecting the absorption feature present before star formation began. The increased foreground brightness at low frequencies make it very difficult to constrain this feature and will require long integration times and more sophisticated methods of foreground removal.
This paper represents a first serious look at the prospects for using global measurements of the 21 cm signal to constrain astrophysics. As a result, there are a number of places where future work might improve upon our calculations. These include investigating the effects of finite sky coverage, incorporating an arbitrary instrumental frequency response, and allowing for the removal of frequency channels corrupted by terrestrial radio interference.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Judd Bowman and Angelica de Oliveira-Costa for useful conversations. Figure \[fig:skymaps\] was generated using HEALpix [@gorski2005] and the global sky model software of Ref. [@angelica2008]. JRP is supported by NASA through Hubble Fellowship grant HST-HF-01211.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. AL acknowledges funding from NSF grant AST-0907890 and NASA grants NNA09DB30A and NNX08AL43G.
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[^1]: Hubble Fellow
[^2]: http://www.lofar.org/
[^3]: http://www.MWAtelescope.org/
[^4]: @parsons2009
[^5]: http://www.skatelescope.org/
[^6]: http://www.jwst.nasa.gov/
[^7]: http://www.gmto.org/
[^8]: http://www.eso.org/sci/facilities/eelt/
[^9]: http://www.tmt.org/
|
---
abstract: 'We investigate the classical and semiclassical features of generic 2D, matter-coupled, dilaton gravity theories. In particular, we show that the mass, the temperature and the flux of Hawking radiation associated with 2D black holes are invariant under dilaton-dependent Weyl rescalings of the metric. The relationship between quantum anomalies and Hawking radiation is discussed.'
address: |
Dipartimento di Scienze Fisiche, Universitá di Cagliari,\
Via Ospedale 72, I-09100 Cagliari, Italy\
and Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari, I-09100 Cagliari, Italy
author:
- 'Mariano Cadoni [^1]'
title: Trace anomaly and Hawking effect in 2D dilaton gravity theories
---
=10000 =10000 ł
INTRODUCTION
============
These notes summarize the content of two of our papers dealing with two-dimensional (2D) dilaton gravity theories [@CA1; @CA2]. Our presentation in Santa Margherita was essentially based on the first paper [@CA1], whereas the content of the second one [@CA2] was briefly sketched as work in progress. For sake of completeness it seems to us advisable to give a full account of the recent results of Ref. [@CA2] in these notes.
2D dilaton gravity theories have become very popular in recent years. They represent simple toy models for studying 4D black hole physics and its related challenging issues such us the ultimate fate of black holes or the loss of quantum coherence in the evaporation process. Moreover, the relation with non-critical string theory and the renormalizability of gravity in two space-time dimensions give to these theories an intrinsic interest, which goes beyond black hole physics. Even though 2D dilaton gravity contains, as particular cases, models with different features (e.g. the Callan-Giddings-Harvey-Strominger (CGHS) model [@CGHS], the Jackiw-Teitelboim (JT) model [@JT]), a simple, unified description of the general theory still exists. In this paper we will discuss three crucial issues of 2D dilaton gravity, which, as we shall see, are deeply interconnected. These issues are the equivalence of 2D dilaton gravity models under Weyl rescalings of the metric, the existence of black hole solutions and the relationship between quantum anomalies and Hawking radiation.
This paper is structured as follows: In Sect. 2 we consider the generic, matter-coupled, classical 2D dilaton gravity theory. In particular, we study the behaviour of physical observables under Weyl transformations and the black hole solutions of the theory. In Sect. 3 we investigate the semiclassical theory. In Sect. 4 we discuss some particular cases of the generic model. In Sect. 5 we present our conclusions.
CLASSICAL 2D DILATON GRAVITY
============================
The most general action of 2D dilaton gravity, conformally coupled to a set on $N$ matter scalar fields has the form [@MA] S\[g,,f\] =[12]{}, where $D,H,V$ are arbitrary functions of the dilaton $\f$ and $\l$ is a constant. An important issue in the context of 2D dilaton gravity is the equivalence of models connected by conformal transformations of the metric. From a purely field theoretical point of view, performing dilaton-dependent Weyl rescalings of the metric in the 2D dilaton gravity action we should get equivalent models, since these transformations are nothing but reparametrizations of the field space. The space-time interpretation of this equivalence presents, however, some problems because geometrical objects such us the scalar curvature of the space-time or the equation for the geodesics are not invariant under Weyl transformations. Let us consider the following Weyl transformation of the metric = e\^[P()]{} , where the function $P$ is constrained by requiring the transformation (\[e2\]) to be non-singular and invertible. Whereas the matter part of the action (\[e1\]) is invariant under the transformation (\[e2\]), the gravitational part is not, but it maintains its form, with the functions $D,H,V$ transforming as ($'=d/d\f$)
=D,=H +D’P’,=e\^[P]{}V.
The transformation laws (\[e2\]) and (\[e3\]) enable us to find out how the physical parameters characterizing the solutions of the theory transform under the Weyl transformation (\[e2\]). Following Mann [@MA], one can define the conserved quantity M=[12ł]{} \^dD V(-d) - [12ł]{} (D)\^[2]{} (-d). $M$ is constant whenever the equations of motion are satisfied and, in this case, it can be interpreted as the mass of the solution. Using Eqs. (\[e2\]), (\[e3\]), one can easily demonstrate that the mass $M$ given by the expression (\[e4\]) is invariant under Weyl transformations of the metric.
Choosing $P=-\int^{\f}d\tau \left[H(\tau)/ D'(\tau)\right]$, we can always achieve ${\h H}=0$. The generic static solutions in this conformal frame have already been found in Ref. [@MK], \^[2]{}=(J-[2Mł]{})dt\^[2]{}+ (J-[2Mł]{})\^[-1]{}dr\^[2]{}, D()=łr, where $d\h J/d{\h D}={\h V}$ and $M$ is the mass of the solution given by Eq. (\[e4\]). If the equation $\h J=2M/\l$ has at least one solution $\f=\f_{0}$, with $\h J$ monotonic, one is lead to interpret the solution as a black hole. However, a rigorous proof of the existence of black holes involves a detailed analysis of the global structure of the space-time. The dilaton $\f$ gives a coordinate-independent notion of location and it can therefore be used to define the asymptotic region, the singularities and the event horizon of our 2D space-time. Moreover, the natural coupling constant of the theory is $D^{-1/2}$ so that we have a natural division of our space-time in a strong-coupling region ($D=0$) and a weak-coupling region ($D=\infty$). These considerations enable us to identify the weak-coupling region $D=\infty$ as the asymptotic region of our space-time. This notion of location is Weyl-invariant because $D$ behaves as a scalar under the Weyl transformation (\[e2\]). One can now write down a set of conditions that, if satisfied, makes the interpretation of the solution (\[e5\]) as a black hole meaningful. One weakness of this kind of approach is that we need to use the scalar curvature $R$ to define the singularities and the asymptotic behaviour of the space-time. $R$ is not Weyl rescaling invariant and cannot be taken as a good quantity for a conformal invariant characterization of black holes. One has to perform the analysis in a particular conformal frame. In the conformal frame defined by Eq. (\[e5\]) black holes exist provided the function $\h V$ behaves asymptotically as [@CA1] V\~D\^, -1<1. A broader class of models whose static solutions can be interpreted as black holes can be obtained choosing a conformal frame in which the metric is asymptotically Minkowskian [@CA2].
The Hawking temperature of the generic black hole solution of the action (\[e1\]) is given by T=[ł4]{} K(\_[0]{}), where K()=V()(-\^ d). The temperature is invariant under Weyl transformations. This can be easily checked using Eqs. (\[e3\]) in Eq. (\[e8\]) and taking into account that the transformations (\[e2\]) do not change the position $\f_{0}$ of the event horizon. It is interesting to note that the mass (\[e4\]) and the temperature (\[e7\]) are invariant not only under Weyl transformations but also under reparametrizations of the dilaton field.
QUANTUM ANOMALIES AND HAWKING RADIATION
=======================================
It is well-known that in quantizing the scalar matter fields $f$ in a fixed background geometry the Weyl rescaling and/or part of the diffeomorphism invariance of the classical action for the matter fields has to be explicitly broken. If one decides to preserve diffeomorphism invariance the semiclassical action is given by the usual Liouville-Polyakov action but one has still the freedom of adding local, covariant, dilaton-dependent counterterms to the semiclassical action [@ST; @RST]. In the path integral formulation this ambiguity is related to the choice of the metric to be used in the measure. The semiclassical action has the general form S=S\_[cl]{}-[N96]{}. The second term above is the usual non-local Liouville-Polyakov action, $S_{cl}$ is the classical action (\[e1\]) and $N(\phi),G(\phi)$ are two arbitrary functions. The form of the functions $N$and $G$ depends on the conformal frame we choose, but, as we shall see later on in this paper, the final result for the Hawking radiation rate is invariant under Weyl transformations. In the conformal frame where the metric is asymptotically Minkowskian, $N$ and $G$ can be fixed requiring the expectation value of the stress-energy tensor to vanish when evaluated for the $M=0$ ground state solution (Minkowsky space) [@CA2].
The black hole radiation can now be studied working in the conformal gauge $ds^{2}=-e^{2\rho}dx^{+}dx^{-}$ and considering a black hole formed by collapse of a $f$-shock-wave, travelling in the $x^{+}$ direction and described by a classical stress-energy tensor $T_{++}=M\delta(x^{+}-x^{+}_{0})$. The classical solution describing the collapse of the shock-wave, for $\xp\le\xp _0$, is given by e\^[2]{}=K(-\^ d),
\^=[ł2]{}(-), and, for $\xp\ge\xp_0$, it is given by e\^[2]{}=(-\^ d) (K-)F’(), \^=[ł2]{} , F’()=[dFd]{}=([K]{})\_[=\_0]{}, where $K$ is given as in Eq. (\[e8\]). The next step in our semiclassical calculation is to use the effective action (\[e9\]) to derive the expression for the quantum contributions of the matter to the stress-energy tensor. The flux of Hawking radiation across spatial infinity is given by $<T_{--}>$ evaluated on the asymptotic $D=\infty$ region. For the class of models in Eq. (\[e6\]) and for the shock-wave solution described previously a straightforward calculation leads to [@CA2] < T\_[–]{}>\_[as]{}=[N24]{}[1(F’)\^2]{}{F,}, where $\{F,\xm\}$ denotes the Schwarzian derivative of the function $F(\xm)$. This is a Weyl rescaling and dilaton reparametrization invariant result for the Hawking flux. In fact the function $F(\xm)$ is defined entirely in terms of the function $K(\f)$ (see Eq. (\[f2\])), which in turn is invariant under both transformations (see Eq. (\[e8\])). Though the trace anomaly is Weyl rescaling dependent, the Hawking radiation seen by an asymptotic observer is independent of the particular conformal frame chosen. When the horizon $\f_{0}$ is approached, the Hawking flux reaches the thermal value < T\_[–]{}>\_[as]{}\^[h]{}=[N12]{}[ł\^[2]{}16]{}\[K(\_[0]{})\]\^[2]{}, which is the result found in Ref. [@CA1], written in a manifest Weyl rescaling and dilaton reparametrization invariant form.
PARTICULAR CASES
================
The general model (\[e1\]) contains, as particular cases, models that have already been investigated in the literature. In this section we will show how previous results for the CGHS model and the JT theory can be obtained as particular cases of our formulation.
String inspired dilaton gravity
-------------------------------
This is the most popular 2D dilaton gravity model. In its original derivation [@CGHS] the action has the form (\[e1\]) with D=V=e\^[-2]{},H=4e\^[-2]{}. The model admits asymptotically flat black hole solutions. Using Eq. (\[e7\]), (\[f4\]) we find for the temperature and magnitude of the Hawking effect T=[ł4]{}, \_[as]{}\^h=[N12]{}[ł\^216]{}. This result coincides, after the redefinition $\l\to 2\l$ with the CGHS result [@CGHS]. All the 2D dilaton gravity models obtained from the CGHS model using Weyl transformation are characterized by the same values of mass, temperature and Hawking radiation rate. In particular, this is true for the model investigated in Ref. [@CM]. This model is characterized by $H=0$ and its black hole solutions are described by a Rindler space-time. Also the models of Ref. [@FR] can be obtained from the CGHS model through a Weyl transformation of the metric [@CA2]. The black hole solutions of these models have, therefore, the same values of mass, temperature and Hawking radiation rate as the CGHS black holes.
The Jackiw-Teitelboim theory
-----------------------------
The JT theory is obtained from the action (\[e1\]) by taking $H=0$ and $2D=V=2\exp(-2\f)$. The model admits black hole solutions with asymptotic anti-de Sitter behaviour. More precisely, as shown in Ref. [@CM1], the black hole space-time is obtained from a particular parametrization of 2D anti-de Sitter space-time endowed with a boundary. Eqs. (\[e7\]) and (\[f4\]) give now T=[12]{}, \_[as]{}\^h= [N24]{}Mł. The same result for the Hawking radiation rate has been obtained in Ref. [@CM1] performing the canonical quantization of the scalar fields $f$ in the anti- de Sitter background geometry.
CONCLUSIONS
===========
In this paper we have discussed classical and semiclassical features of generic 2D dilaton gravity theories. We have shown that the usual relationship between conformal anomalies and Hawking radiation can be extended to a broad class of 2D dilaton gravity models. In particular, we have found a simple, general formula for the magnitude of the Hawking effect. We have also shown that physical observables associated with 2D black holes such as the mass, the temperature and the hawking radiation rate are invariant both under Weyl transformations and dilaton reparametrizations. This implies a conformal equivalence between 2D dilaton gravity models. An important issue we have not discussed in this paper is the physical meaning of this equivalence. For a detailed discussion of this issue see Ref. [@CA2].
[9]{} M. Cadoni, (1996) 4413. M. Cadoni, hep-th/9610201. C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, (1992) 1005. C. Teitelboim, in [*Quantum Theory of gravity*]{}, S.M. Christensen, ed. (Adam Hilger, Bristol, 1984); R. Jackiw, [*ibidem*]{}. R. B. Mann, (1993) 4438. D. Louis-Martinez and G. Kunstatter, (1994) 5227.
A. Strominger, , 4396 (1992).
J.C. Russo, L. Susskind, L. Thorlacius, (1992) 3444.
M. Cadoni, S. Mignemi (1995) 217.
A. Fabbri, J.G. Russo, hep-th/9510109.
M. Cadoni and S. Mignemi, (1995) 4319.
[^1]: Talk given at the Second Conference on Constrained Dynamics and Quantum Gravity, Santa Margherita Ligure, Italy, September 1996, to appear in the Proceedings
|
---
abstract: |
Massive clusters in our Galaxy are an ideal testbed to investigate the properties and evolution of high mass stars. They provide statistically significant samples of massive stars of uniform ages. To accurately determine the intrinsic physical properties of these stars we need to establish the distances, ages and reddening of the clusters. One avenue to achieve this is the identification and characterisation of the main sequence members of red supergiant rich clusters.
Here we utilise publicly available data from the UKIDSS galactic plane survey. We show that point spread function photometry in conjunction with standard photometric decontamination techniques allows us to identify the most likely main sequence members in the 10-20Myr old clusters RSGC1, 2, and 3. We confirm the previous detection of the main sequence in RSGC2 and provide the first main sequence detection in RSGC1 and RSGC3. There are in excess of 100 stars with more than 8$M_\odot$ identified in each cluster. These main sequence members are concentrated towards the spectroscopically confirmed red supergiant stars. We utilise the $J$$-$$K$ colours of the bright main sequence stars to determine the $K$-band extinction towards the clusters. The differential reddening is three times as large in the youngest cluster RSGC1 compared to the two older clusters RSGC2 and RSGC3. Spectroscopic follow up of the cluster main sequence stars should lead to more precise distance and age estimates for these clusters as well as the determination of the stellar mass function in these high mass environments.
author:
- |
Dirk Froebrich$^{1}$[^1], Alexander Scholz$^{2,3}$\
$^{1}$Centre for Astrophysics and Planetary Science, University of Kent, Canterbury, CT2 7NH, United Kingdom\
$^{2}$Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland\
$^{3}$School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, KY16 9SS, United Kingdom
bibliography:
- 'references.bib'
date: 'Received today / Accepted tomorrow'
title: The Main Sequence of three Red Supergiant Clusters
---
\[firstpage\]
open clusters and associations: general; galaxies: star clusters: general
Introduction {#intro}
============
Massive stars are most commonly formed in clusters and associations [@2012MNRAS.424.3037G], even if there are potential exceptions (e.g. @2013ApJ...768...66O). These building blocks are the sole observational characteristic of star formation that is observable in more distant galaxies. It is thus of importance to investigate the details of such massive clusters locally in our own Galaxy.
The identification of local examples of clusters and associations of massive stars is complicated by a number of facts. Such objects are generally rare in the Milky Way and are thus typically at large distances. The Sun’s position near the Galactic Plane, hence makes it difficult to identify and investigate these objects due to the large amounts of extinction along the line of sight. Furthermore, most massive star formation is projected towards the general direction of the Galactic Centre, which causes additional difficulties due to crowding.
Thus, the identification and characterisation of massive star clusters in our own Galaxy has only recently gained considerable momentum due to advances in infrared astronomy that allow us to probe these distant and obscured objects. Many of them have been known for a considerable time, but have not been recognised as massive star clusters. For example Westerlund1 was discovered by @1961PASP...73...51W but only recognised as young, very massive cluster more than 30yrs later . Similarly, the cluster Stephenson2 [@1990AJ.....99.1867S] had been known for many years until its true nature as Red Supergiant Cluster2 (RSGC2) was uncovered . Further, more systematic searches to uncover the population of the most massive clustes in the Galaxy are ongoing, e.g. @2013ApJ...766..135R.
Clusters at the top end of the mass distribution in our Galaxy (above $10^4$M$\odot$) have, depending on their age, a sizable number of evolved massive stars. These are either Wolf-Rayet stars, or blue/yellow/red supergiants. There are now a number of galactic clusters with a large population of Red Supergiant (RSG) stars known, many of which are in a region confined to where the Scutum spiral arm meets the near side of the Galactic Bar. These are e.g. RSGC1 to RSGC5 . All these clusters are about 6kpc from the Sun, highly extincted, between 7Myr and 20Myr old and have masses in excess of 10.000$M_\odot$. These clusters are thus an ideal testbed to study the formation and evolution of massive stars as well as their influence on the environment in great detail. However, it is important in the context of understanding the formation and evolution of massive clusters to investigate the distribution and properties of the lower mass stars as well as the bright supergiants and Wolf Rayet stars. In particular since the intrinsic properties of lower mass stars are much better understood, this should enable us to determine the distances, ages and reddening of these clusters more accurately.
The main sequence (MS) stars of these clusters have so far, however, not been investigated (though the slightly less massive and less reddened cluster NGC7419 has recently been studied by ). Given that the RSG cluster members have an apparent brightness of about $K$=6mag, one would expect to detect the MS in deep near-infrared (NIR) images, in particular the Galactic Plane Survey (GPS, @2008MNRAS.391..136L) data from the UK Deep Sky Survey (UKIDSS, @2007MNRAS.379.1599L) seems to be ideal to detect the fainter members of the RSG clusters mentioned above. But even data from the 2–Micron All Sky Survey (2MASS, @2006AJ....131.1163S) should be deep enough to detect the brightest main sequence stars in these clusters.
Only for the cluster RSGC2 has the main sequence been detected so far [@2013IJAA...03..161F]. The author uses 2MASS and GPS data to identify the top of the MS at colours of about $J$$-$$K$=1.5mag and at a brightness of slightly fainter than $K$=10mag. The author also investigates the data for RSGC1 and RSGC3 with the same methods but is unable to detect the main sequence for these clusters. It is speculated that this is caused by problems with accurate aperture photometry in the vicinity of the bright RSG stars in those objects. The main sequence in RSGC2 is the easiest to detect due to the spatial extent of the cluster, which is the largest amongst the RSG clusters. Here we hence try to identify the main sequences of other known and candidate RSG clusters by means of point spread function (PSF) photometry in the JHK images of the UKIDSS GPS data. This will include some of the new RSG cluster candidates (F3 and F4) identified by [@2013IJAA...03..161F] based on colour selected star density maps from 2MASS.
This paper is structured as follows. In Sect.\[data\] we briefly outline the data and analysis methods employed. We then discuss our results in Sect.\[results\] with focus on the properties of the main sequences detected in RSGC1, 2, 3.
------- --------- ----------- -------- ----------- ----------- ------------ ----------- ----------- ------------ ----------- ----------- ------------ -------
Field RA DEC $J_{min}$ $J_{max}$ $\sigma_J$ $H_{min}$ $H_{max}$ $\sigma_H$ $K_{min}$ $K_{max}$ $\sigma_K$
Input Output \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] \[mag\]
RSGC1 279.488 $-$6.880 31562 11976 14.0 17.0 0.024 13.0 16.0 0.034 12.0 15.0 0.045
RSGC2 279.838 $-$6.029 42313 17872 13.0 16.0 0.021 12.0 15.0 0.024 11.0 14.0 0.028
RSGC3 281.350 $-$3.387 43557 20960 13.0 16.0 0.021 12.0 15.0 0.026 12.0 15.0 0.046
F3 274.910 $-$14.340 34837 15312 13.0 16.0 0.022 12.5 15.0 0.027 12.0 15.0 0.045
F4 276.030 $-$13.330 43445 19664 13.0 16.0 0.036 12.0 15.0 0.029 11.0 14.0 0.035
------- --------- ----------- -------- ----------- ----------- ------------ ----------- ----------- ------------ ----------- ----------- ------------ -------
Data and Analysis Methods {#data}
=========================
UKIDSS data
-----------
In @2013IJAA...03..161F the main sequence of RSGC2 has been detected in aperture photometry of UKIDSS GPS data between $K$=11mag and $K$=14.5mag. All stars brighter than this are saturated. Tentatively, this main sequence was also visible in 2MASS data starting from about $K$=10mag. The red supergiants in this cluster are between 5$^{th}$ and 6$^{th}$ magnitude in the $K$-band [@2007ApJ...671..781D]. Thus, the brightest main sequence members in this cluster are five or six magnitudes fainter than the red supergiants. Given that several of the other known RSG clusters are in a similar position on the sky and potentially have similar ages, distances and reddening (@2006ApJ...643.1166F, @2007ApJ...671..781D, @2008ApJ...676.1016D, , @2009AJ....137.4824A) one can assume that their main sequences should have analogue properties, i.e. should have similarly bright members. However, for none of the other investigated RSG clusters or candidates in @2013IJAA...03..161F has the main sequence been detected in the GPS aperture photomety data. This was attributed to crowding in the clusters, and thus low quality aperture photometry of potential main sequence stars in the vicinity of the bright RSG cluster members. In other words, the GPS aperture photometry catalogue is highly incomplete near the bright RSG cluster stars. We hence selected RSGC1–5 and the cluster candidates F3 and F4 from @2013IJAA...03..161F for further investigations, i.e. PSF-photometry, to reveal their main sequences.
We downloaded 10$\times$10 cut-outs of the JHK images from the UKIDSS GPS around the nominal position of each of these clusters via the Widefield Camera Science Archive[^2] webpage. We further obtained a complete list of all source detections (independent of the photometric quality) in each field. Typically there are about 30–45,000 detections in each of the fields. We then manually removed all detections which were obvious image artefacts (detector cross talk, persistence, etc.) and added all visible real stars missed by the source detection software to these catalogues. In particular near the bright red supergiant cluster members a number of real stars is missing. Several hundred detections have been removed/added for each field by visually inspecting the $K$-band images and manually deleting all obvious false detections and appending every object that has the appearance of a star but was missing in the photometric catalogue. In Table\[calibration\] we list the number of objects in these input catalogues for each cluster field.
PSF Photometry
--------------
We performed PSF fitting photometry in each image at all positions from the above generated input catalogue. For this purpose, we used the standard routines from DAOPHOT, as implemented in IRAF [@1987PASP...99..191S]. To define the PSF for a given frame, about 10–15 isolated reference stars distributed across the images were selected. Since the crowding was significant near the field center, most of these reference stars are located in the outskirts of the clusters. We did not notice any sign of a spatially varying PSF. The average PSF of these reference stars is modelled in DAOPHOT with a combination of an analytical function and an empirical image, which better represents the wings of the PSF. This model PSF was then fit to all objects in the catalogue, to create the final list of magnitudes. As measured by the $\chi^2$ and the PSF subtracted images, the PSF fitting performs well in the UKIDSS GPS images. Only in one case, the $K$-band image of the cluster candidate F3, is the PSF undersampled and larger residuals are visible in the PSF subtracted images. However, the quality of the photometry is not significantly degraded in this image (see Table\[calibration\]).
We merged the photometric catalogues for each of the JHK filters and used the GPS aperture photometry of sources in each field to calibrate the instrumental magnitudes. Only objects with [pstar]{}$\ge$0.99965 (see @2008MNRAS.391..136L for details on how this is defined) are used in the calibration and a nominal shift of the magnitudes and no colour terms are considered. The root mean square ([*rms*]{}) scatter of the calibrated magnitudes is listed in Table\[calibration\] together with the magnitude range in which the calibration was performed. There are some images where we detect a clear non-linearity for bright sources. This is expected, since the PSF fitting should be reliable over a wider range of the nonlinear regime. Thus, the colours and magnitudes for the bright stars could be systematically off. Note that this only influences the very top of the potential cluster main sequences and will have no influence on our analysis.
All point sources which did lack a detection by the PSF fitting routine in at least one of the three NIR bands were removed from the final catalogues. We also removed all stars which were fainter than the completeness limit (determined as the peak in the magnitude distribution) in at least one filter. Finally all objects that had a photometric uncertainty which differed by more than 2$\sigma$ from the average of stars with the same apparent magnitude were removed. The number of stars in this output catalogue for each cluster is listed in Table\[calibration\].
Photometric Decontamination
---------------------------
Using the calibrated JHK PSF-photometry we performed a photometric decontamination of the stars in the cluster field to establish which stars are the most likely cluster members. The method is based on the technique described in @2007MNRAS.377.1301B and references therein. It uses the $J$-band magnitudes and $J$$-$$H$ and $J$$-$$K$ colours to distinguish field stars from cluster members based on their apparent magnitudes and colours. This particular choice of colours provides the maximum variance among stellar cluster’s colour-magnitude sequences for open clusters of various ages . We use a slight adaptation of this method, outlined in detail in @2010MNRAS.409.1281F. For each cluster we define as [*cluster area*]{} everything closer than the radius ($r$, as specified in Table\[properties\]) around the nominal cluster centre. Note that the radii for the clusters are based either on literature estimates or are chosen by us to include most of the confirmed or suspected RSG stars in each object. The actual value of the radius will not influence our conclusions. As [*control field*]{} we use all objects within the entire 10$\times$10 field but further away than two cluster radii from the centre of the cluster. See Fig.B1 in the Appendix for a $K$-band and Glimpse 8$\mu$m image of RSGC1, 2, 3 with circles indicating the cluster and control fields. We also show the $J$$-$$K$ vs $K$ colour magnitude diagrams of the cluster and control fields for these clusters in Fig.C1 in the Appendix. For each star ($i$) in the cluster field we then determine the colour-colour-magnitude distance ($r_{ccm}$) to every other star ($j$) in the cluster field as:
$$\label{eq_rccm}
r_{ccm}=\sqrt{\frac{1}{2} \left( J_i - J_j \right)^2 + \left( JK_i - JK_j
\right)^2 + \left( JH_i - JH_j \right)^2}$$
Where $JK = J-K$ and $JH = J-H$ are the above mentioned near infrared colours. The $r_{ccm}$ distance in which there are 20 stars in the cluster area is denoted as $r^{20}_{ccm}$. It essentially defines the local density of stars in the near infrared colour-colour-magnitude space. Note that the specific choice of 20 stars does not influence any of our results and is a compromise between the accuracy of the membership probabilities (see below) and the ’resolution’ at which we can determine the position of the main sequence. We then determine the number of stars ($N^{con}_{ccm}$) at the same position and within the same radius of the colour-colour-magnitude space but for the stars in the control field. With this number, as well as the respective surface area of the control field ($A_{con}$) and cluster area ($A_{cl}$), we can determine the membership-likelihood index or cluster membership probability ($P^i_{cl}$) of the star $i$ as:
$$\label{eq_pcl}
P^i_{cl}=1.0-\frac{N^{con}_{ccm}}{20}\frac{A_{cl}}{A_{con}}.$$
Note that these cluster membership probabilities are strictly speaking not real probabilities (Buckner & Froebrich 2013, subm.), since fluctuations of the field star density in principle allow negative values for $P^i_{cl}$. If this occurs in our analysis the $P^i_{cl}$ value is set to zero. However, the sum of all $P^i_{cl}$ values equals the excess number of stars in the cluster area compared to the control field. High values of $P^i_{cl}$ identify the stars in the cluster field which are the most likely members and thus allow us to establish the overall population of cluster stars statistically. Typically this method identifies a few hundred stars in the fields of the clusters RSGC1–3 which have a membership probability above 50% (see later).
\
\
\
------- ------- --------- ---------- ------------- ------------ -------------------- -------------- --------------- ------------------------ -----------
Name r d age $A^{Lit}_K$ $A^{MS}_K$ $\Delta$$A^{MS}_K$ $K^{3}_{MS}$ $K^{10}_{MS}$ $N_{MS}^{>8\,M_\odot}$ $N_{RSG}$
\[ \] \[kpc\] \[Myrs\] \[mag\] \[mag\] \[mag\] \[mag\] \[mag\]
RSGC1 1.5 6 10 2.74 2.3 0.75 10.5 (-5.9) 12.5 (-3.7) 210 14
RSGC2 1.8 6 17 1.47 1.0 0.25 10.0 (-5.0) 11.0 (-4.0) 115 26
RSGC3 1.8 6 20 1.50 1.4 0.25 11.5 (-3.6) 13.0 (-2.1) 115 16
------- ------- --------- ---------- ------------- ------------ -------------------- -------------- --------------- ------------------------ -----------
Results
=======
Here we discuss the results obtained for all the investigated clusters. Some further details on the individual objects can be found in AppendixA.
General
-------
Theoretically one would expect the main sequence of these massive clusters to appear as a vertical accumulation of high probability cluster members in a $K$ vs. $J$$-$$K$ colour-magnitude diagram (CMD), since most stars visible should be of high mass and thus have similar intrinsic near infrared colours. Some scatter in colour is expected due to differential reddening along the line of sight. Furthermore, these high mass main sequence stars should be situated at the bottom of the reddening band in a near infrared $H$$-$$K$ vs. $J$$-$$H$ colour-colour diagram (CCD) and not in the middle/top, which is usually occupied by giant stars.
In the left column of panels in Fig.\[rsgcfig\] we show as examples the decontaminated $K$ vs. $J$$-$$K$ CMD of RSGC1, 2, 3. All stars fainter than 8$^{th}$ magnitude in $K$ are from our PSF photometry. All bright, potential RSGs (large black squares) are taken from 2MASS since they are saturated in the UKIDSS images. All small black dot symbols are stars in the cluster area with less than 50% cluster membership probability $P^i_{cl}$. Green plus signs indicate stars with $50\,\% \leq P^i_{cl} < 60\,\% $, blue triangles stars with $60\,\% \leq P^i_{cl} < 70\,\% $ and red squares $P^i_{cl} > 70\,\%$.
We determined a running weighted average of the $J$$-$$K$ colours of the most likely cluster members along the detected main sequence. This is indicated by the solid black line in the CMDs in Fig.\[rsgcfig\]. As weighting factor for each star we used the square of the membership probability $P^i_{cl}$. To compare the cluster data with model isochrones, we utilise the Geneva isochrones by which are overplotted in each panel as a blue solid line, using the parameters specified in Table\[properties\]. We also overplot as a dashed line the isochrones for low and intermediate mass stars from .
In the right column of panels in Fig.\[rsgcfig\] we show the corresponding $H$$-$$K$ vs. $J$$-$$H$ CCDs for the same three clusters. Symbols and colours are identical in their meaning to the CMDs. However, we plot all stars in the field as small black dots and only high probability cluster members ($P^i_{cl} >
50\,\%$) from the potential main sequences (as indicated by the dotted boxes in the CMDs) are shown in large coloured symbols. These boxes exclude bright, potentially saturated stars as well as faint, low signal to noise objects and obvious background giants. The indicated reddening band for each cluster indicated, is based on the reddening law by @2005ApJ...619..931I.
From all the clusters investigated, we can detect a main sequence only in the known objects RSGC1, 2 and 3 (see Fig.\[rsgcfig\]). We also investigated the fields around RSGC4 and RSGC5 but there are no apparent overdensities of stars, in particular none that would indicate a main sequence (see Fig.D1 in the Appendix). There are several possibilities that could explain this. i) These clusters have less mass, i.e. fewer members, than the other objects and thus they do not manifest themselves as overdensities in colour magnitude space. ii) The clusters are much more extended spatially than our search area for potential main sequence stars; they are more association like in appearance than cluster like. Both points seem to contribute, since both clusters have fewer confirmed members than the brighter RSGCs and they seem to be embedded in more extended regions of massive young stars .
For the new cluster candidates F3 and F4 from @2013IJAA...03..161F only features that look like a tentative MS in the CMDs are found (see Fig.D2 in the Appendix). For F3, a clump of stars at $K$=13mag and $J$$-$$K$=2.4mag can be identified, while for F4 a more MS like feature can be seen at $J$$-$$K$=2.6mag. Both of these features contain a few hundred high probability members. However, when utilising the CCDs for both cluster candidates, one can identify that these features are not caused by main sequence stars. The high probability members clearly are not situated near the bottom of the reddening band, indicating they are giants. Only in the case of F3, there are some potential main sequence stars. Thus, both candidates are most likely holes in the general extinction and not real clusters.
\
Main Sequence Properties
------------------------
Here we will concentrate on determining the principle properties of the detected main sequences for RSGC1, 2 and 3. The isochrones overplotted to the CMDs and CCDs use age and distance estimates from the literature (see Table\[properties\] for these parameters). We adopt a distance of 6kpc for all clusters, which seems an appropriate average of the published values . The ages are taken for each individual cluster from the same references. We only vary the extinction in the $K$-band to shift the isochrone onto the detected main sequence. Since the upper end of the MS is almost vertical in the $K$ vs. $J$$-$$K$ CMDs, the actual choise of age and distance will not influence the required extinction value.
All main sequences are ’vertical’ in the CMDs for the top 2–4mag in the $K$-band. Hence, these are clearly massive MS stars and we can try to estimate the colour excess towards the cluster by shifting an isochrone until it fits the MS. This will determine the extinction towards potential cluster members, i.e. the column density of material along the line of sight that is not associated with the cluster itself, and is independent of the actual age chosen for the isochrone. Utilising an extinction law (we use @2005ApJ...619..931I), we can convert this to a foreground extinction value. These are the values we used to overplot the isochrones in Fig.\[rsgcfig\] and which are listed as $A_K^{MS}$ in Table\[properties\]. In the colour-colour diagrams in the right hand side panels of Fig.\[rsgcfig\] one can see that the reddening direction of the background giants confirms the validity of this reddening law. In essence the reddening law in these fields is in agreement with @2005ApJ...619..931I or @2009MNRAS.400..731S, but using a less steep dependence of extinction on wavelength such as in or @1985ApJ...288..618R can be ruled out from the CCDs. Thus, please note that the use of an extinction law in agreement with the CCDs will change the infered $K$-band extinction by only about 0.1mag. However, larger differences are expected when a less steep extinction law is applied. The cluster RSGC1 has the largest extinction of 2.4mag in the $K$-band, while the other two clusters have about half this value of reddening, i.e. $A_K$$\approx$1.2mag. Compared to the literature values (listed as $A_K^{Lit}$ in Table\[properties\]), our isochrone fit to the main sequence systematically finds smaller extinction values. This could be caused by the fact that: i) the literature extinction values are estimated using a different extinction law (such as by @2007ApJ...671..781D for RSGC2); ii) the observations are taken in different filters, e.g. 2MASS $K_S$ vs. UKIDSS $K$; iii) the extinction values are determined from the spectral types of the red supergiants in the cluster and not by the better understood main sequence stars. Recently @2013ApJ...767....3D have shown that there can indeed by issues with the RSG temperature scale.
We further estimate the amount of material associated with the cluster itself. This can be done by measuring the width of the main sequence in $J$$-$$K$ and convert this to a value of differential reddening. We list these values in the $\Delta A_K$ column in Table\[properties\]. As for the general interstellar extinction, RSGC1 shows the highest amount of differential reddening with about 0.75mag in the $K$-band. This is about three times as high as for the other two clusters, but comparable to, or even smaller than for other young embedded clusters (e.g. the Orion Nebula Cluster ). There are several possible explanations for the differences: i) This cluster is younger and thus still more deeply embedded in its parental molecular cloud – however this is unlikely given the age of the cluster. ii) The differential reddening is not caused by intrinsic dust, but by the variations in extinction of the foreground material. The larger value for $A^{MS}_K$ for this cluster could support this.
In order to investigate the number of potential main sequence stars in each cluster we defined a colour range in $J$$-$$K$ for each cluster (as indicated in the CMDs in Fig.\[rsgcfig\]) that encloses all the potential MS stars. We select all stars within this colour range and determine the $K$-band luminosity function along the main sequence. These are shown as dotted blue lines in Fig.E1 in the Appendix. If we only count the membership probabilities $P^i_{cl}$ for each star, then we obtain a more realistic luminosity function which is shown as solid red line in Fig.E1. The smallest difference between the two luminosity functions is evident for RSGC2. Thus, this is the cluster where the MS stands out most significantly from the field stars. This is in agreement with the fact that this is the only cluster where the main sequence has been detected previously [@2013IJAA...03..161F]. The typical membership probability for the MS cluster members is about 80% for RSGC2, while it is of the order of 60% for the other two clusters.
We investigate the brightest MS stars in each cluster and define $K^{3}_{MS}$ as the apparent $K$-band magnitude where there are at least three stars per 0.5mag bin along the MS. In other words we treat this as the top of the main sequence. Similarly we define $K^{10}_{MS}$ and list both values in Table\[properties\]. The most populated cluster main sequence at bright $K$-band magnitudes occurs in RSGC2. There, $K^{10}_{MS}$=11mag, while $K^{3}_{MS}$ is one magnitude brighter. RSGC3 has by far the fewest bright cluster main sequence stars, or the MS starts only at fainter magnitudes. The absolute magnitudes for $K^{3}_{MS}$ and $K^{10}_{MS}$ are determined from our adopted distance and de-reddened with $A_K^{MS}$. They are listed in brackets in Table\[properties\] and show that RSGC1 has the brightest ($M_K$=-5.9mag) end of the MS while RSGC3 has intrinsically the faintest end of the MS ($M_K$=-3.7mag). This is in good agreement with the ages for the clusters determined in the literature, which range from 8–12Myrs for RSGC1 [@2006ApJ...643.1166F; @2008ApJ...676.1016D] to about 20Myrs for RSGC3 [@2009AJ....137.4824A]. Note that according to the isochrones used , stars with a mass above 20M$_\odot$ (or O-type stars) have $M_K$=-2.9mag or brighter on the MS. Thus, in particular RSGC1 and RSGC2, could contain a significant number of massive MS or post-MS objects, that can easily be verified spectroscopically. Note that the total number of these cluster members is likely to be larger by about 50%, since we have only analysed the stars within one cluster radius. There are spectroscopically confirmed cluster members in the region between one a two cluster radii; typically only about 2/3$^{\rm rd}$ of the known members are within one cluster radius.
The completeness limit determined as the peak of the $K$-band luminosity function, for all clusters is between $K$=15mag and 16mag. In all cases, this is fainter than stars of about 8M$_\odot$ which have $M_K$=-0.9mag or an apparent magnitude of $K$=13.0mag at our adopted distance and without considering extinction. We thus can compare the total number of stars along the main sequence, brighter than these stars. The numbers are weighted by the membership probabilities and are listed as $N_{MS}^{>8\,M_\odot}$ in Table\[properties\]. We find that RSGC1 has about twice as many of these OB-type stars than RSGC2 and 3. Please note that we expect increased crowding in the cluster centres and thus the estimated OB-type cluster member numbers should be treated as lower limits. Since RSGC1 is the most compact of the three clusters, its numbers should be most affected. The number of confirmed RSGs (see column $N_{RSG}$ in Table\[properties\]) in RSGC2 is much higher than in RSGC1, which might be due to the lower age of the latter. Hence, out of the three clusters, RSGC1 seems to be the most massive object as estimated in @2006ApJ...643.1166F and @2008ApJ...676.1016D. Based on the number of stars along the main sequence and the age, RSGC2 and 3 might be less massive, partly (for RSGC3) in agreement with the predictions from @2009AJ....137.4824A and at the lower end of the mass range suggested in .
In Fig.\[position\_plot\] we show the spatial distribution of all stars along the MS for the three clusters. Only stars within the cluster radius are shown, since we have not determined membership probabilities outside this area. In all three cases the most likely MS stars are concentrated towards the nominal centre of the clusters. However, the distribution seems not be centrally condensed, but rather filamentary, especially for RSCG2. If this is a real effect, or caused by crowding in the cluster centre and ’missing’ objects near the bright RSGs is unclear. The spatial density of the main sequence cluster members (each star is weighted by its membership probability) is between two and three times higher in the cluster centre compared to the outer regions as defined by the cluster radius.
Conclusions
===========
We have used PSF photometry on deep NIR JHK imaging data from the UKIDSS GPS to investigate the fields of known and candidate red supergiant clusters. We confirm the detection by @2013IJAA...03..161F of the upper main sequence of the cluster RSGC2 and for the first time detect the main sequence for RSGC1 and RSGC3.
We use the age and distance estimates from the literature to overplot isochrones on the NIR colour-magnitude diagrams for all clusters in order to establish the reddening of the main sequences by utilising the reddening law from @2005ApJ...619..931I. In all cases the infered $K$-band extinction values for the main sequences are smaller than the quoted values in the literature, which are determined for the red supergiant stars. We also infer the differential reddening towards each cluster based on the width of the detected main sequence. The youngest of the clusters (RSGC1) has the highest extinction and differential reddening in accordance with its evolutionary status. It also contains the most number of stars (about 200) with masses above 8$M_\odot$.
The spatial distribution of the candidate main sequence stars in all clusters shows a concentration towards the nominal cluster centre. However, there is no indication of a centrally condensed distribution, which could either be real or caused by increased crowding and blending effects from the bright red supergiant cluster members.
We also investigated fields near the clusters RSGC4 and RSGC5, as well as the candidate RSG clusters F3 and F4 from @2013IJAA...03..161F. In all cases no main sequence could be detected. In the case of the already known clusters this could be caused by them having less members or being spatially more extended, i.e. more association like. Our results indicate that the new candidates can most likely be interpreted as holes in the background extinction.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank I.Negueruela and C.Gonzalez for fruitful discussions during the earlier stages of the project. We further acknowledge the constructive comments by the referee B. Davies which helped to improve the paper. Part of this work was funded by the Science Foundation Ireland through grant no. 10/RFP/AST2780.
\[lastpage\]
[^1]: E-mail: df@star.kent.ac.uk
[^2]: http://surveys.roe.ac.uk/wsa/index.html
|
---
abstract: 'We consider the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on ${{\mathbb R}}.$ We prove that if a solution $u$ of this equation is bounded and its initial value $u(x,0)$ has distinct limits at $x=\pm\infty,$ then the solution is quasiconvergent, that is, all its limit profiles as $t\to\infty$ are steady states.'
author:
- |
Antoine Pauthier and Peter Poláčik[^1]\
[School of Mathematics, University of Minnesota]{}\
[Minneapolis, MN 55455]{}
title: 'Large-time behavior of solutions of parabolic equations on the real line with convergent initial data'
---
[*Key words*: Parabolic equations on the real line, convergent initial data, quasiconvergence, convergence]{}
Introduction
============
Consider the Cauchy problem $$\begin{aligned}
u_t=u_{xx}+f(u), & \qquad x\in{{\mathbb R}},\ t>0, \label{eq1}\\
u(x,0)=u_0(x), & \qquad x\in{{\mathbb R}}, \label{ic1}
\end{aligned}$$ where $f$ is a locally Lipschitz function on ${{\mathbb R}}$ and $u_0\in C_b({{\mathbb R}}):=C({{\mathbb R}})\cap L^\infty({{\mathbb R}})$. We denote by $u(\cdot,t,u_0)$ the unique classical solution of (\[eq1\])-(\[ic1\]) and by $T(u_0)\in(0,+\infty]$ its maximal existence time. If $u$ is bounded on ${{\mathbb R}}\times[0,T(u_0))$, then necessarily $T(u_0)=+\infty,$ that is, the solution is global. In this paper, we are concerned with the behavior of bounded solutions as $t\to\infty$. A basic question we specifically want to address is whether, or to what extent, the large-time behavior of bounded solutions is governed by steady states of .
This question has long been settled for equation considered on a bounded interval, instead of ${{\mathbb R}}$, and complemented by one of common boundary conditions, say Dirichlet, Neumann, Robin, or periodic. Namely, in that case each bounded solution converges, uniformly on the spatial interval, to a steady state [@Chen-M:JDE; @Matano:conv; @Zelenyak]. In contrast, the large-time behavior of equation on ${{\mathbb R}}$ is not generally so simple and is much less understood.
To talk about the behavior in more specific terms, recall that, by standard parabolic regularity estimates, any bounded solution of has relatively compact orbit in $L^\infty_{loc}({{\mathbb R}})$. In other words, any sequence $t_n\to\infty$ has a subsequence $\{t_{n_k}\}$ such that $u(\cdot,t_{n_k})\to \varphi$ in $L^\infty_{loc}({{\mathbb R}})$ for some continuous function $\varphi$. It is therefore natural to use the topology of $L^\infty_{loc}({{\mathbb R}})$ when considering the convergence of solutions and related issues. Thus, we say that a bounded solution $u$ is *convergent* if for some $\varphi$ one has $u(\cdot,t)\to\varphi$ locally uniformly on ${{\mathbb R}}$. Of course, the convergence may take place in stronger topologies, but we take the convergence in $L_{loc}^\infty({{\mathbb R}})$, the topology in which the orbit is compact, as a natural minimal requirement.
While the convergence of the solution of , has been proved under various conditions on $u_0$ and $f$ [@Chen-L-Z-G; @Du-M; @p-Du; @Fasangova; @Fasangova-F; @Feireisl:long-time; @p-Fe; @p-Ma:1d; @Muratov-Z; @P:unbal; @Zlatos:sharp], it is not the general behavior of bounded solutions even when $f\equiv 0$, that is, when is the linear heat equation. As observed in [@Collet-E], if $u_0$ takes values $0$ and $1$ on suitably spaced long intervals with sharp transitions between them, then, as $t\to\infty$, $u({\cdot},t)$ approaches $0$ along a sequence of times $t_n\to
\infty$ and $1$ along another such sequence (the convergence is in $L_{loc}^\infty({{\mathbb R}})$ in both cases).
As we explain shortly, for the linear equation the large-time behavior of any bounded solution is still governed by steady states in the sense that every limit profile of any such solution is a steady state. Here a *limit profile* of a bounded solution $u$ of refers to any element of the $\omega$-limit set of $u$: $$\label{defomega}
\omega(u):=\left\{ \vp\in C_b({{\mathbb R}}):\ u(\cdot,t_n)\to\vp \textrm{
for some sequence}t_n\to\infty\right\},$$ where the convergence is in $L^\infty_{loc}({{\mathbb R}})$. If the solution $u$ corresponds to a given initial datum $u_0$, we also write ${\omega}(u_0)$ for ${\omega}(u)$. We say that a bounded solution $u$ of is *quasiconvergent* if ${\omega}(u)$ consists entirely of steady states. Thus, a quasiconvergent solution approaches a set of steady states, from which it follows that $u_t(\cdot,t)\to 0$, locally uniformly on ${{\mathbb R}}$, as $t\to\infty$. This makes quasiconvergent solutions hard to distinguish—numerically, for example—from convergent solutions; they move very slowly at large times.
In the case of the linear heat equation, the quasiconvergence of each bounded solution follows from the invariance property of the ${\omega}$-limit set: ${\omega}(u)$ consists of *entire* solutions of $\eqref{eq1}$, by which we mean solutions defined for all $t\in {{\mathbb R}}$. If $u$ is bounded, then the entire solutions in ${\omega}(u)$ are bounded as well and, by the Liouville theorem for the linear heat equation, all such solutions are constant.
In nonlinear equations, a common way to prove the quasiconvergence of a solution is by means of a Lyapunov functional. For equation , the following energy functional is used frequently: $$\label{eq:1}
E(v):=\int_{-\infty}^\infty \big(\,\frac{v^2_x(x)
}2-F(v(x))\big)\,dx,\qquad F(v):=\int_0^vf(s)\,ds.$$ Of course, for this functional to be defined along a solution, one needs assumptions on $f$ and $u$; but when such assumptions are made, it can be shown that $t\mapsto E(u({\cdot},t))$ is nonincreasing and consequently $u$ is quasiconvergent (see, for example, [@Feireisl:long-time] for results of this form).
For solutions which are not assumed to be bounded in an integral norm, the energy $E$ is usually not very useful.[^2] In fact, bounded solutions of nonlinear equations are not quasiconvergent in general. The existence of non-quasiconvergent solutions for some equations of the form was strongly indicated by results of [@Eckmann-R]. It was later demonstrated by various examples in [@P:examples; @P:unbal]. Moreover, the results of [@P:examples; @P:unbal] show that non-quasiconvergent bounded solutions occur quite “frequently” in . They exist whenever there is an interval $[a,b]$ on which $f$ is bistable: $ f(a)=f(b)=0$, $ f'(a),f'(b)<0$, and there is ${\gamma}\in (a,b)$ such that $f<0$ in $(a,{\gamma})$ and $f>0$ in $({\gamma},b)$. This is clearly a robust class of nonlinearities.
On the other hand, several classes of initial data $u_0$ have been identified for which the solutions are quasiconvergent, if bounded. These include nonnegative localized data, or, nonnegative elements of $C_0({{\mathbb R}})$, in the case $f(0)=0$ [@p-Ma:1d]; as well as front-like initial data, by which we mean functions $u_0\in C({{\mathbb R}})$ satisfying $a\le u_0\le b$, $u_0(-\infty)=b$, $u_0(\infty)=a$ for some zeros $a<b$ of $f$ [@P:prop-terr] (see [@P:quasiconv-overview] for a more detailed overview of quasiconvergence and related results). Here and below, $C_0({{\mathbb R}})$ stands for the space of all continuous functions on ${{\mathbb R}}$ converging to $0$ at $x=\pm\infty$. Note that the sign restriction in the case of localized initial data is essential; examples of non-quasiconvergent solutions [@P:examples; @P:unbal] do include some with sign-changing initial data in $C_0({{\mathbb R}})$.
In this paper, we consider a class of initial data with includes in particular all front-like initial data, but without any sign restrictions like $a\le u_0\le b$. Namely, we consider initial data $u_0$ in the space $$\label{spacelimit}
{{\mathcal V}}:=\left\{ v\in C_b({{\mathbb R}}):\textrm{ the limits }
v(-\infty),\,v(+\infty)\in {{\mathbb R}}\textrm{ exist}\right\}.$$ Note that the property of having finite limits at $\pm\infty$ is preserved by the solutions of , : if $u_0(\pm \infty)$ exist, then $u(\pm\infty,t,u_0)$ exist for all $t\in (0,T(u_0))$ (these limits vary with $t$ in general, see Lemma \[valueinfty\] below for a more precise statement). This means that the space ${{\mathcal V}}$ is an invariant space for , just like the space $C_b({{\mathbb R}})$, or the space $C_0({{\mathbb R}})$ in the case $f(0)=0$. Since ${{\mathcal V}}$ is a closed subspace of $C_b({{\mathbb R}})$, it is a Banach space when equipped with the supremum norm.
Of course, ${{\mathcal V}}$ contains functions $u_0$ with $u_0(-\infty)=u_0(\infty)$—in particular, it contains $C_0({{\mathbb R}})$—so we do not have quasiconvergence of all bounded solutions with initial data $u_0\in {{\mathcal V}}$. As it turns out, however, $u_0(-\infty)=u_0(\infty)$ is the only case when the quasiconvergence may fail to hold. This is a part of our main theorem, which we state precisely after introducing some notation and terminology.
Consider the ordinary differential equation for the steady states of : $$\label{steadyeq}
u_{xx}+f(u)=0,\qquad x\in{{\mathbb R}}.$$ The corresponding first-order system, $$\label{sys}
u_x=v,\qquad v_x=-f(u),$$ is a Hamiltonian system, which has only four types of bounded orbits: equilibria, nonstationary periodic orbits (or, closed orbits), homoclinic orbits, and heteroclinic orbits. We adopt the following common terminology concerning steady states $\varphi$ of . We say $\varphi$ is a *ground state* of if the orbit of $(\varphi,\varphi')$ is a homoclinic orbit of ; and $\varphi$ is a *standing wave* of if the orbit of $(\varphi,\varphi')$ is a homoclinic orbit of .
Under our standing hypothesis that $f$ is locally Lipschitz on ${{\mathbb R}}$, we have the following result.
\[mainthm\] Assume that $u_0\in{{\mathcal V}}$ and $u_0(-\infty)\ne u_0(\infty)$. If the solution $u(\cdot,\cdot,u_0)$ of *(\[eq1\]), (\[ic1\])* is bounded, then it is quasiconvergent: $\omega(u_0)$ consists entirely of steady states of *(\[eq1\]).* More specifically, if $\varphi\in \omega(u_0)$, then it is a constant steady, or a ground state of *,* or a standing wave of *(\[eq1\])*.
\[rmtothm\]
- An even more precise description of $\omega(u_0)$ will come out of the proof of the theorem. Namely, consider the following possibility:
- $u$ is eventually monotone in space: given any $k\in {{\mathbb R}}$, one has $u_x(x,t)\ne 0$ for all $x\in (-k,k)$ if $t$ is sufficiently large.
If (M) holds, we will prove that $u$ is quasiconvergent and each element of ${\omega}(u_0)$ is a constant steady state or a standing wave of (\[eq1\]). If (M) does not hold, we show that $u$ is even convergent, with ${\omega}(u_0)=\{\varphi\}$, where $\vp$ is a constant steady or a ground state of .
- Theorem \[mainthm\] in particular shows that nonconstant periodic steady states are never elements of ${\omega}(u_0)$ for $u_0\in
{{\mathcal V}}$.
- Clearly, the set of all functions $u_0\in {{\mathcal V}}$ satisfying $u_0(-\infty)\ne u_0(+\infty)$ is open and dense in ${{\mathcal V}}$ (with the supremum norm). Thus, Theorem \[mainthm\] has an interesting additional feature in that it shows that quasiconvergence is generic in ${{\mathcal V}}$: the solution of is quasiconvergent, if bounded, for an open and dense set of initial data in ${{\mathcal V}}$. In contrast, using the constructions from [@P:examples], one can show that this genericity statement is not valid if one replaces ${{\mathcal V}}$ with $C_b({{\mathbb R}})$ or $C_0({{\mathbb R}})$.
Theorem \[mainthm\] is proved in Section \[proof\]. Several preliminary results concerning steady states, zero number, and ${\omega}$-limit sets that are needed for the proof are recalled in Section \[prelims\].
As Theorem \[mainthm\] concerns bounded solutions only, modifying $f$ outside the range of the solution, we may assume without loss of generality that $f$ satisfies the following condition: $$\label{coercivity}
\text{there exists $\kappa>0$ such that for all $|u|>\kappa$
one has $f(u)=\frac{u}{2}$.}$$ This will be convenient in the next section.
Preliminaries {#prelims}
=============
Steady states and their trajectories in the phase plane {#stst}
-------------------------------------------------------
In this subsection, we recall several technical results concerning the steady states of (\[eq1\]), or, solutions of (\[steadyeq\]). The first-order system corresponding to (\[steadyeq\]) is Hamiltonian with respect to the energy $$\label{energy}
H(u,v)=\frac{v^2}{2}+F(u),$$ where $\displaystyle F(u)=\int_0^u f(s)ds.$ Thus, each orbit of (\[sys\]) is contained in a level set of $H.$ The level sets are symmetric with respect to the $v-$axis, and our extra hypothesis (\[coercivity\]) implies that they are all bounded. Therefore, all orbits of (\[sys\]) are bounded and, as already mentioned in the introduction, there are only four types of them: equilibria (all of which are on the $u-$axis), non-stationary periodic orbits, homoclinic orbits (corresponding to ground states of (\[steadyeq\])), and heteroclinic orbits (corresponding to standing waves of (\[eq1\])).
Each non-stationary periodic orbit $\MO$ is symmetric about the $u-$axis and for some $p<q$ one has $$\begin{aligned}
\MO\cap\{ (u,0):u\in{{\mathbb R}}\} & = \left\{(p,0),(q,0)\right\} \nonumber \\
\MO\cap \left\{(u,v):v>0\right\} & = \left\{\lp u,\sqrt{2(F(p)-F(u))}\rp:u\in(p,q)\right\}. \label{periodicorbits}\end{aligned}$$
The following result of [@p-Ma:1d] gives a description of the phase plane portraits of (\[sys\]) with all the periodic orbits removed. Let $$\begin{aligned}
\ME & := \{ (a,0):f(a)=0\} \textrm{ (the set of all equilibria of (\ref{sys}))}, \nonumber \\
\MP_0 & :=\{(a,b)\in{{\mathbb R}}^2: (a,b)\textrm{ lies on a non-stationary periodic orbit of (\ref{sys})}\} \nonumber \\
\MP & := \MP_0\cup\ME. \nonumber\end{aligned}$$
\[MatPolLemma\] *[@p-Ma:1d Lemma 3.1]* The following two statements are valid.
1. Let $\Sigma$ be a connected component of ${{\mathbb R}}^2\setminus\MP_0.$ Then $\Sigma$ is a compact set contained in a level set of the Hamiltonian $H$ and one has $$\Sigma = \left\{(u,v)\in{{\mathbb R}}^2:u\in J,\ v=\pm\sqrt{2(c-F(u))}\right\}$$ where $c$ is the value of $H$ on $\Sigma$ and $J=[p,q]$ for some $p,q\in{{\mathbb R}}$ with $p\leq q.$ Moreover, if $(u,0)\in\Sigma$ and $p<u<q,$ then $(u,0)$ is an equilibrium. The point $(p,0)$ is an equilibrium or it lies on a homoclinic orbit; the same is true for the point $(q,0).$
2. Each connected component of the set ${{\mathbb R}}^2\setminus\MP$ consists of a single orbit of *(\[sys\])*, either a homoclinic orbit or a heteroclinic orbit.
The following lemma is a simple consequence of the continuity of the solutions of (\[sys\]) with respect to the initial conditions.
\[lemperiods\] Let $\MO_n$, $n=1,2,\dots$ be a sequence of non-stationary periodic orbits of *(\[sys\])* with the minimal periods $\rho_n,$ $n=1,2,\dots$, respectively. Suppose that for some bounded set $K\subset {{\mathbb R}}^2\setminus\MP$ one has $\displaystyle\textrm{dist}\lp\MO_n,K\rp\underset{n\to\infty}{\longrightarrow}0.$ Then $\displaystyle \rho_n\underset{n\to\infty}{\longrightarrow}\infty.$
If $\vp$ is a $C^1$ bounded function on ${{\mathbb R}},$ we let $$\tau(\vp):=\left\{ \lp \vp(x),\vp_x(x)\rp:x\in{{\mathbb R}}\right\}$$ and refer to this set as the *spatial trajectory (or orbit)* of $\vp.$ If $\vp$ is a solution of (\[steadyeq\]), then $\tau(\vp)$ is the usual orbit of the solution $(\varphi,\varphi_x)$ of (\[sys\]).
Invariance of the ${\omega}$-limit set {#invariance}
--------------------------------------
Recall that the $\omega-$limit set of a bounded solution $u$ of , denoted by $\omega(u)$, or $\omega(u_0)$ if the initial value of $u$ is given, is defined as in , with the convergence in $L^\infty_{loc}({{\mathbb R}})$. By standard parabolic estimates the trajectory $\{ u(\cdot,t),\ t\geq1\}$ of $u$ is relatively compact in $L^\infty_{loc}({{\mathbb R}}).$ This implies that $\omega(u)$ is nonempty, compact, and connected in (the metric space) $L^\infty_{loc}({{\mathbb R}})$ and it attracts the solution in the following sense: $$\label{eq:11}
\textrm{dist}_{L^\infty_{loc}({{\mathbb R}})}\lp u(\cdot,t),\omega(u)\rp\underset{t\to\infty}{\longrightarrow}0.$$ It is also a standard observation that if $\vp\in\omega(u),$ there exists an entire solution $U(x,t)$ of (\[eq1\]) such that $$\label{entiresol}
U(\cdot,0)=\vp,\qquad U(\cdot,t)\in\omega(u)\quad (t\in{{\mathbb R}}).$$ Here, *an entire solution* of (\[eq1\]) refers to a solution defined for all $x\in{{\mathbb R}}$, $t\in{{\mathbb R}}.$ Let us briefly recall how such an entire solution $U$ is found. By parabolic regularity estimates, $u_t,u_x,u_{xx}$ are bounded on ${{\mathbb R}}\times[1,\infty)$ and are globally $\a-$Hölder for any $\a\in(0,1).$ If $u(\cdot,t_n)\underset{n\to\infty}{\longrightarrow}\vp$ in $L^\infty_{loc}({{\mathbb R}})$ for some $t_n\to\infty,$ we consider the sequence $u_n(x,t):=u(x,t+t_n)$, $n=1,2\dots$. Passing to a subsequence if necessary, we have $u_n\to U$ in $C^1_{loc}({{\mathbb R}}^2)$ for some function $U$; this function $U$ is then easily shown to be an entire solution of (\[eq1\]). By definition, $U$ satisfies . Note that the entire solution $U$ is determined uniquely by $\varphi$; this follows from the uniqueness and backward uniqueness for the Cauchy problem , .
Using similar compactness arguments, one shows easily that $\omega(u)$ is connected in $C_{loc}^1({{\mathbb R}}).$ Hence, the set $$\left\{ (\vp(x),\vp_x(x)):\vp\in\omega(u),x\in{{\mathbb R}}\right\} = \underset{\vp\in\omega(u)}{\cup}\tau(\vp)$$ is connected in ${{\mathbb R}}^2.$ Also, obviously, $\tau(\vp)$ is connected in ${{\mathbb R}}^2$ for all $\vp\in\omega(u).$
We will also use the following result (see [@p-Ma:1d Lemma 4.3] or [@P:prop-terr Lemma 6.10] for a proof).
\[translatedorbit\] Let $u$ be a bounded solution of . If $\vp\in\omega(u),$ $\psi$ is a solution of (\[steadyeq\]), and $\tau(\vp)\subset\tau(\psi),$ then $\vp$ is a shift of $\psi.$
Zero number for linear parabolic equations
------------------------------------------
In this subsection, we consider solutions of a linear parabolic equation $$\label{eqlin}
v_t=v_{xx}+c(x,t)v,\qquad x\in{{\mathbb R}},\ t\in\lp s,T\rp,$$ where $-\infty\leq s<T\leq \infty$ and $c$ is a bounded measurable function. For an interval $I=(a,b),$ with $-\infty\leq a < b\leq
\infty,$ we denote by $z_I(v(\cdot,t))$ the number, possibly infinite, of zeros $x\in I$ of the function $x\mapsto v(x,t).$ If $I={{\mathbb R}}$ we usually omit the subscript ${{\mathbb R}}$: $$z(v(\cdot,t)):=z_{{\mathbb R}}(v(\cdot,t)).$$ The following intersection-comparison principle holds [@Angenent:zero; @Chen:strong].
\[lemzero\] Let $v$ be a nontrivial solution of *(\[eqlin\])* and $I=(a,b),$ with $-\infty\leq a < b\leq \infty.$ Assume that the following conditions are satisfied:
- if $b<\infty,$ then $v(b,t)\neq0$ for all $t\in\lp s,T\rp,$
- if $a>-\infty,$ then $v(a,t)\neq0$ for all $t\in\lp s,T\rp.$
Then the following statements hold true.
1. For each $t\in\lp s,T\rp,$ all zeros of $v(\cdot,t)$ are isolated. In particular, if $I$ is bounded, then $z_I(v(\cdot,t))<\infty$ for all $t\in\lp s,T\rp.$
2. The function $t\mapsto z_I(v(\cdot,t))$ is monotone non-increasing on $(s,T)$ with values in ${{\mathbb N}}\cup\{0\}\cup\{\infty\}.$
3. If for some $t_0\in(s,T)$ the function $v(\cdot,t_0)$ has a multiple zero in $I$ and $z_I(v(\cdot,t_0))<\infty,$ then for any $t_1,t_2\in(s,T)$ with $t_1<t_0<t_2,$ one has $$\label{zerodrop}
z_I(v(\cdot,t_1))>z_I(v(\cdot,t_0))\ge z_I(v(\cdot,t_2)).$$
If (\[zerodrop\]) holds, we say that $z_I(v(\cdot,t))$ drops in the interval $(t_1,t_2).$
\[convzero\][ It is clear that if the assumptions of Lemma \[lemzero\] are satisfied and for some $t_0\in(s,T)$ one has $z_I(v(\cdot,t_0))<\infty,$ then $z_I(v(\cdot,t))$ can drop at most finitely many times in $(t_0,T)$; and if it is constant on $(t_0,T),$ then $v(\cdot,t)$ has only simple zeros in $I$ for all $t\in (t_0,T).$ In particular, if $T=\infty,$ there exists $t_1<\infty$ such that $t\mapsto z_I(v(\cdot,t))$ is constant on $(t_1,\infty)$ and all zeros are simple. ]{}
Using the previous remark and the implicit function theorem, we obtain the following corollary.
\[zeroIFT\] Assume that the assumptions of Lemma \[lemzero\] are satisfied and that the function $t\mapsto z_I(v(\cdot,t))$ is constant on $(s,T).$ If for some $(x_0,t_0)\in I\times(s,T)$ one has $v(x_0,t_0)=0,$ then there exists a $C^1$- function $t\mapsto\eta(t)$ defined for $t\in(s,T)$ such that $\eta(t_0)=x_0$ and $v(\eta(t),t)=0$ for all $t\in(s,T).$
We will also need the following robustness lemma (see [@Du-M Lemma 2.6]).
\[robustnesszero\] Let $w_n(x,t)$ be a sequence of functions converging to $w(x,t)$ in $\displaystyle C^1\lp I\times(s,T)\rp$ where $I$ is an open interval. Assume that $w(x,t)$ solves a linear equation (\[eqlin\]), $w\not\equiv0$, and $w(\cdot,t)$ has a multiple zero $x_0\in I$ for some $t_0\in(s,T)$. Then there exist sequences $x_n\to x_0$, $t_n\to t_0$ such that for all sufficiently large $n$ the function $w_n(\cdot,t_n)$ has a multiple zero at $x_n$.
In the next section we frequently use the following standard facts, often without notice. If $u$, $\bar u$ are bounded solutions of the nonlinear equation with a Lipschitz nonlinearity, then their difference $v=u-\bar u$ satisfies a linear equation with some bounded measurable function $c$. Similarly, $v=u_x$ and $v=u_t$ are solutions of such a linear equation.
Proof of the main result {#proof}
========================
Throughout this section we assume the hypotheses of Theorem \[mainthm\] to be satisfied: $u_0\in {{\mathcal V}}$ (cp. ) and $$\label{hyponu0}
{\alpha}:=u_0(-\infty)\ne {\beta}:= u_0(+\infty).$$ Further, we assume that the solution $u(x,t)$ of (\[eq1\])-(\[ic1\]) is bounded.
We denote $$\label{limits-t}
\theta_-(t):=\lim_{x\to-\infty}u(x,t),\qquad \theta_+(t):=\lim_{x\to\infty}u(x,t).$$ These limits exist according to the following lemma (the proof can be found in [@Volpert-V-V Theorem 5.5.2], for example).
\[valueinfty\] The limits $\theta_-(t),\theta_+(t)$ exist for all $t> 0$ and are solutions of the following initial-value problems: $$\label{valueinftyeq}
\dot{\theta}_\pm=f(\theta_\pm),\qquad \theta_-(0)=\a,\ \theta_+(0)=\b.$$
The reflection principle and stabilization of the critical points {#refl-sec}
-----------------------------------------------------------------
We employ the reflection-invariance of equation in a way quite common in studies of spatially homogeneous parabolic equations. For any ${\lambda}\in{{\mathbb R}},$ consider the function $V_{\lambda}u$ defined by $$\label{reflexion}
V_{\lambda}u(x,t)=u(2{\lambda}-x,t)-u(x,t),\quad x\in{{\mathbb R}},\,t\ge 0.$$ Being the difference of two solutions of , $V_{\lambda}u$ is a solution of the linear equation (\[eqlin\]) for some bounded function $c$.
We apply zero-number results to the functions $V_{\lambda}u$, ${\lambda}\in
{{\mathbb R}}$. First observe that for any ${\lambda}\in{{\mathbb R}}$, hypothesis (\[hyponu0\]) and Lemma \[valueinfty\] imply that for $t\ge 0$ the function $V_{\lambda}u(x,t)$ has the limits as $x\to\pm\infty$ given by $\pm(\theta^+(t)-\theta^-(t))$, and these limits are both nonzero for all sufficiently small $t>0$. Therefore, by Lemma \[lemzero\], $z(V_{\lambda}u(\cdot,t))$ is finite for all $t>0.$ By Remark \[convzero\], there is $T=T({\lambda})$ such that the function $t\mapsto z(V_{\lambda}u(\cdot,t))$ is constant on $\lp T({\lambda}),\infty\rp$ and, for all $t>T({\lambda}),$ all zeros of $V_{\lambda}u(\cdot,t)$ are simple. In particular, since $x={\lambda}$ is always a zero of $V_{\lambda}u$ by the definition of $V_{\lambda}u$, we have $$\label{uxneqzero}
-2u_x({\lambda},t)=\partial_x V_{\lambda}u(x,t)\vert_{x={\lambda}}\neq0\quad (t>T({\lambda})).$$ We use this to prove the following result (a similar theorem for solutions periodic in space can be found in [@Chen-M:JDE]).
\[loczero\] For any open bounded interval $I\subset{{\mathbb R}},$ there exist $T_1=T_1(I)>0$ and an integer $N\ge 0$ such that, for all $t>T_1,$ the function $u_x(\cdot,t)$ has exactly $N$ zeros in $I$, all of them simple. Moreover, if $N>0$ and $\eta_1(t)<\dots \eta_N(t)$ denote the zeros of $u_x(\cdot,t)$ in $I$ for $t>T_1$, then the functions $\eta_i(t),$ $i=1,\dots, N,$ are of class $C^1$, and for some $x_i^\infty\in\overline{I}$ one has $$\label{loczeroeq}
\eta_i(t)\underset{t\to\infty}{\longrightarrow}x_i^\infty\quad(i=1,\dots,N).$$
Let $I=(a,b)$ with $a,b\in{{\mathbb R}}$ be any open bounded interval. Applying (\[uxneqzero\]) to ${\lambda}\in \{a,b\}$ and setting $T_0:=\max\lp T(a),T(b)\rp$, we obtain $$u_x(a,t)\neq0,\ u_x(b,t)\neq0\quad (t>T_0).$$ The function $u_x(x,t)$ is a solution of a linear equation (\[eqlin\]). Hence, by Lemma \[lemzero\], there exist $T_1\geq T_0$ and $N\geq0$ such that for all $t>T_1$ the function $u_x(\cdot,t)$ has exactly $N$ zeros in $I$, all of them simple. If $N=0$, the proof of Proposition \[loczero\] is finished.
Assume that $N\geq1.$ Let $\eta_1(t)<\dots \eta_N(t)$ the zeros of $u_x(\cdot,t)$ in $I$. The simplicity of these zeros and the implicit function theorem imply that the functions $\eta_i(t),$ $i=1,\dots, N,$ are of class $C^1$. It remains to show that these functions are all convergent. Assume for a contradiction that for some $i\in \{1,\dots,N\}$ the function $\eta_i(t)$ is not convergent as $t\to\infty.$ Then it admits at least two accumulation points $x_i^\infty\neq\tilde{x}_i^\infty.$ Consequently, for ${\lambda}:=(x_i^\infty+\tilde{x}_i^\infty)/{2}$ there is a sequence $t_n\to\infty$ such that $\eta_i(t_n)={\lambda}$ for all $n,$ which contradicts (\[uxneqzero\]).
We complement the previous results with the following useful information.
\[le-conv0\] Under the notation of Proposition \[loczero\], if $N>0$, then for any ${\lambda}\in \{x_i^\infty: i=1,\dots,N\}$ one has $$\label{eq:3}
V_{{\lambda}}u({\cdot},t)\underset{t\to\infty}{\longrightarrow}0\ \text{
in $C^1_{loc}({{\mathbb R}})$.}$$
It is sufficient to prove the following statement. Given any sequence $t_n\to\infty$, one can pass to a subsequence such that holds with $t$ replaced by $t_n$.
This will be shown using an entire solution $U$, as constructed in Section \[invariance\]. Passing to a subsequence of $\{t_n\}$, we may assume that $u(\cdot,\cdot+t_n)\to U$ in $C^1_{loc}({{\mathbb R}}^2)$, where $U$ is an entire solution of . Then also $V_{\lambda}u(\cdot,\cdot+t_n)\to V_{\lambda}U$ in $C^1_{loc}({{\mathbb R}}^2)$, for any ${\lambda}$. If now ${\lambda}\in \{x_i^\infty: i=1,\dots,N\}$, then $$\partial_x V_{\lambda}u(x,t)=-2u_x({\lambda},t)\underset{t\to\infty}{\longrightarrow}0.$$ It follows that $$\partial_x V_{\lambda}U({\lambda},t)=0 \quad (t\in{{\mathbb R}}).$$ Since we also have $V_{\lambda}U({\lambda},t)=0$ (cp. ), $x={\lambda}$ is a multiple zero of $V_{\lambda}U(\cdot,t)$ for all $t\in{{\mathbb R}}$. Using Lemma \[lemzero\], one shows easily that this is only possible if $V_{\lambda}U\equiv 0$. This in particular yields the desired conclusion: $$V_{{\lambda}}u({\cdot},t_n){\to}0\ \text{
in $C^1_{loc}({{\mathbb R}})$.} \qedhere$$
Take now the intervals $I_k:=\lp-k,k\rp,$ $k=1,2,\dots$, and let $N_k$ be the number of zeros of $u_x(\cdot,t)$ in $I_k$ for $t>T_1(I_k)$, where $T_1$ is as in Proposition \[loczero\]. We distinguish the following mutually exclusive cases.
- There is $k_0$ such that $N_k=0$ for $k=k_0,k_0+1,\dots$.
- There is $k_0$ such that $N_k=1$ for $k=k_0,k_0+1,\dots$.
- There is $k_0$ such that $N_k\ge 2$ for $k=k_0,k_0+1,\dots$.
According to Proposition \[loczero\], (C1) means that each bounded interval is free of critical points of $u(\cdot,t)$ for $t$ large enough. In the case (C2), $u(\cdot,t)$ has exactly one critical point $\eta(t)$ such that $\eta(t)$ has a finite limit as $t\to\infty$; moreover, in any bounded interval, $u(\cdot,t)$ has no critical points different from $\eta_1(t)$ for $t$ large enough. In the case (C3), there are more than one critical points of $u(\cdot,t)$ with finite limit as $t\to\infty$.
We give the proof of Theorem \[mainthm\] in each of these cases separately.
Case (C1): no limit critical point
----------------------------------
We consider case (C1) here. Clearly, (C1) implies that $u_x(\cdot,t)$ is of one sign in $I_k$ for large $t$ and this sign is independent of $t$. Without loss of generality, replacing $u(x,t)$ by $u(-x,t)$ if necessary, we assume that for all $k$ one has $$\label{uxnegative}
u_x(x,t)<0\quad (x\in(-k,k),\ t>T(I_k)).$$ In this situation, we have the following result concerning the ${\omega}$-limit set ${\omega}(u)$:
\[nointersection\] Let $\psi$ be any nonconstant periodic solution of [(\[steadyeq\])]{}. Then $$\tau(\vp)\cap\tau(\psi)=\emptyset \quad (\varphi\in {\omega}(u)).$$
We go by contradiction. Assume that there is $\varphi\in {\omega}(u)$ such that $\tau(\vp)\cap\tau(\psi)\neq \emptyset.$ This means that, possibly after replacing $\psi$ by a translation, there exists $x_0\in{{\mathbb R}}$ such that $$\psi(x_0)=\vp(x_0),\qquad \psi'(x_0)=\vp'(x_0).$$ To simplify the notation, we will assume without loss of generality that $x_0=0$ (this can be achieved by a translation in the original equation, with no effect on the validity of the conclusion).
Let $U(x,t)$ be an entire solution as in (\[entiresol\]). There exists a sequence $t_n\to\infty$ such that $$u(\cdot,\cdot+t_n)\underset{n\to\infty}{\longrightarrow}U\textrm{ in }C^1_{loc}({{\mathbb R}}^2).$$ Then the sequence $\displaystyle w_n:=u(\cdot,\cdot +t_n)-\psi$ converges in $C^1_{loc}({{\mathbb R}}^2)$ to the function $w(x,t):=U(x,t)-\psi(x).$ The function $w$ solves a linear equation (\[eqlin\]) and $w(\cdot,0)$ admits a multiple zero at $x=0.$ Also, $w(\cdot,0)=\varphi -\psi\not\equiv 0$, for $\psi$ is nonconstant periodic, whereas for $\vp$ we have, as a direct consequence of , that $\vp'\leq0.$ Applying Lemma \[robustnesszero\], we obtain that there exist sequences $x_n\to0,$ $\d_n\to0,$ such that $w_n(\cdot,\d_n)$ has a multiple zero at $x=x_n.$ Consequently, with $\tilde{t}_n:=t_n+{\delta}_n$ we have $\tilde{t}_n \to\infty$, $x_n\to0$, and $$\label{doublezeronocritical}
u(\cdot,\tilde{t}_n)-\psi \textrm{ has a multiple zero at }x={x_n}.$$ We show that (\[doublezeronocritical\]) contradicts (\[uxnegative\]).
Since $\psi$ is periodic, there are $\rho_-<0<\rho_+$ such that $\psi(\rho_-)=\min\psi,$ $\psi(\rho_+)=\max\psi,$ and ${x_n}\in(\rho_-,\rho_+),$ for all $n.$ Consider the function $$z(t):=z_{(\rho_-,\rho_+)}\lp u(\cdot,t)-\psi\rp.$$ By (\[uxnegative\]), there is $T_0>0$ such that for all $t>T_0$ we have $u_x(\cdot,t)<0$ on $(\rho_--1,\rho_++1).$ Thus, if $z(t)>0,$ then $$\label{eq:2}
u(\rho_-,t)>\psi(\rho_-)\text{ and }u(\rho_+,t)<\psi(\rho_+).$$ Therefore, Lemma \[lemzero\] implies that $$\label{zdecreasing}
t\mapsto z(t) \textrm{ is nonincreasing on any subinterval of }\{t:t>T_0 \textrm{ and }z(t)>0\}.$$ Assertion (\[doublezeronocritical\]) implies that for some $T_1>T_0$ one has $z(T_1)>0.$ We consider two complementary cases:
*Case 1:* $z(t)>0,$ for all $t>T_1.$ In this case, shows that Lemma \[lemzero\] and Remark \[convzero\] apply to $v=u-\psi$. Therefore, for all large $t$, $u(\cdot,t)-\psi$ has only simple zeros in $(\rho_-,\rho_+)$. This is a contradiction to (\[doublezeronocritical\]).
*Case 2:* there exists $T_2>T_1$ such that $z(T_2)=0.$ Pick large enough $n_0$ such that $\tilde t_{n_0}>T_2$ and set $$\displaystyle T_3:=\sup\{t\in [T_2,\tilde t_{n_0}):
z(t)=0\}.$$ From (\[doublezeronocritical\]) we know that $T_3<\tilde t_{n_0}.$ The definition of $T_3$ and the monotonicity of $x\mapsto u(x,T_3)$ (cp. \[uxnegative\]) implies that either $u(\rho_-,T_3)=\psi(\rho_-)$ or $u(\rho_+,T_3)=\psi(\rho_+).$ We only consider the first possibility, the other being similar. In addition to the relation $u(\rho_-,T_3)=\psi(\rho_-)$, we have, by (\[uxnegative\]) and the definition of $\rho_-$, $u_x(\rho_-,T_3)<0=\psi_x(\rho_-)$. The implicit function theorem therefore implies that there exists a continuous function $t\mapsto\eta(t)$ defined on a neighborhood of $T_3$ such that on a neighborhood of the point of $(\rho_-,T_3)\in {{\mathbb R}}^2$ one has $u(x,t)=\psi(x)$ if and only if $x=\eta(t).$ Using this, (\[uxnegative\]) and the continuity of $u,$ we find $\e>0$ such that $u(x,t)=\psi(x)$ holds with $x\in(\rho_--\e,\rho_+]$, $t\in[T_3,T_3+\e)$ only if $x=\eta(t)$. As a result, for $t>T_3$ close enough to $T_3,$ we have $z(t)\le 1$. The definition of $T_3$ implies that $z(t)>0$ for all $t\in (T_3,t_{n_0}]$. From this obtain that, first, $z(t)= 1$ for all $t>T_3$, $t\approx
T_3$, and, second, (\[zdecreasing\]) applies to the interval $(T_3,t_{n_0}]$. Consequently, $z(t)= 1$ for all $t\in (T_3,t_{n_0}]$. Using this, , and (\[doublezeronocritical\]) with $n=n_0$, we now obtain a contradiction to Lemma \[lemzero\](iii) (take $t_0=t_{n_0}$ in ).
Assuming (\[uxnegative\]), we show that any $\varphi\in {\omega}(u)$ is either a constant steady state or a standing wave of .
By (\[uxnegative\]), $\vp_x\leq0$. Also, from Lemma \[nointersection\] we know that, in the notation of Lemma \[MatPolLemma\], $\tau(\vp)\subset {{\mathbb R}}^2\setminus\MP_0$.
If $\vp_x\equiv0$, then $\tau(\vp)$ consists of the single point $(\varphi(0),0)$. According to Lemma \[MatPolLemma\], this point is an equilibrium of or is contained in $\tau(\psi)$, where $\psi$ is a ground state solution of (\[steadyeq\]). In the later case, from Lemma \[translatedorbit\] we obtain $\vp$ is a shift of $\psi,$ which is impossible as $\vp_x\equiv 0$. Thus, in this case, $\vp$ is a constant steady state.
Assume now that $\vp_x\not\equiv 0.$ We first show that $\vp_x<0$ on ${{\mathbb R}}.$ Indeed, let $U(x,t)$ be the entire solution of (\[eq1\]) as in (\[entiresol\]): $U(\cdot,0)=\vp$ and $U(\cdot,t)\in {\omega}(u)$ for all $t\in{{\mathbb R}}$. The latter implies that $U_x (\cdot,t)\le 0$ for all $t$ and the former gives $U_x (\cdot,0)\not\equiv 0$. Therefore, applying the maximum principle to $U_x$, we obtain $\vp_x=U_x (\cdot,0)< 0$, as desired. In particular, $\tau(\vp)$ does not intersect the $u-$axis in the $u-v$ plane. This and Lemma \[MatPolLemma\](ii) imply that $\tau(\vp)\subset\tau(\psi)$, where $\tau(\psi)$ is either a heteroclinic orbit of or a homoclinic orbit of . The latter is impossible due to $\vp_x<0$. Thus $\tau(\psi)$ is a heteroclinic orbit of , meaning that $\psi$ is a standing wave of . By Lemma \[translatedorbit\], $\vp$ is a shift of $\psi$, hence it is a standing wave itself.
Case (C2): a unique limit critical point
----------------------------------------
In the case (C2), there exists a $C^1$ function $t\mapsto \eta(t)$ defined on an interval $(T_0,\infty)$ with with $\eta(t)\underset{t\to\infty}{\longrightarrow}\eta^\infty\in{{\mathbb R}}$ and with the following property. For each $k\in\{k_0,k_0+1,\dots\}$ there is $T(I_k)$ such that $$\label{uxuniquezero}
\{(x,t):u_x(x,t)=0, x\in(-k,k),\ t>T(I_k)\}=\{(\eta(t),t):t>T(I_k)\}.$$ Without loss of generality, using a shift if necessary, we will further assume that $\eta^\infty=0.$
By Proposition \[loczero\], $x=\eta(t)$ is a simple zero of $u_x(\cdot,t)$, so $u(\cdot,t)$ has a strict local minimum or a strict local maximum at $\eta(t)$. We only consider the latter, the former is analogous. Thus, we henceforth assume that $$\label{maxu}
u\lp \eta(t),t\rp = \underset{x\in(-k,k)}{\max}u(\cdot,t),\qquad t>T(I_k).$$
From , , and Lemma \[le-conv0\], we obtain that each $\vp\in\omega(u)$ has the following properties: $$\label{vpeven}
\max_{{{\mathbb R}}}\vp = \vp(0), \quad \vp_x(x)\le 0\quad(x>0),
\quad \vp(-x)=\vp(x)\quad(x\in{{\mathbb R}}).$$ Moreover, for each $\vp\in\omega(u)$ $$\label{monotonicity2}
\textrm{either }\vp_x\equiv0 \textrm{ or }\vp_x(x)<0 \textrm{ for all } x>0.$$ To prove this, let $U$ be the entire solution of (\[eq1\]) with $U(\cdot,0)=\vp$ and $U(\cdot,t)\subset\omega(u)$ for all $t\in {{\mathbb R}}$. Then, by , for all $t\in {{\mathbb R}}$ we have $U_x(0,t)=0$ and $U_x(\cdot,t)\leq0$ on $[0,\infty)$. Since the function $U_x$ satisfies a linear equation (\[eqlin\]), the maximum principle implies that $\vp_x=U_x(\cdot,0)$ is either identical to zero or strictly negative on $(0,\infty)$.
Our goal now is to prove the following result.
\[propcontinuum\] For some $\gamma\in{{\mathbb R}}$, one has $
\{\vp(0):\vp\in\omega(u)\}=\{\gamma\}.
$
Assuming that this true, we now complete the proof of Theorem \[mainthm\] in the case (C2); we show that $u$ is even convergent in this case. Then we give the proof of Proposition \[propcontinuum\].
We prove that $\omega(u)$ consists of a single element, either a ground state or a constant steady state.
Given any $\vp\in \omega(u).$ Let $U$ be the entire solution of (\[eq1\]) with $U(\cdot,0)=\vp,$ and $U(\cdot,t)\in\omega(u)$ for all $t\in{{\mathbb R}}.$ By Proposition \[propcontinuum\], $U(0,t)=\gamma$ for all $t\in{{\mathbb R}}$. Therefore, $U_t(0,\cdot)\equiv 0$. Moreover, by (\[vpeven\]), we have $U_x(0,t)=0$ for all $t$, thus $U_{xt}(0,t)=0$ for all $t$. This means that $U_t$ has a multiple zero at $x=0$ for all $t\in{{\mathbb R}}$. Since $U_t$ satisfies a linear equation (\[eqlin\]), Lemma \[lemzero\] implies that $U_t\equiv0.$ This shows that $\vp=U(\cdot,0)$ is a steady state.
We have thus proved that every function $\vp\in \omega(u)$ is a solution of the second order equation (\[steadyeq\]), satisfying $\vp(0)=\gamma$ and $\vp_x(0)=0.$ The uniqueness of this solution gives $\omega(u)=\{\vp\}$, for some $\vp$. Condition (\[vpeven\]) together with the description of the solutions of (\[steadyeq\]) given in Subsection \[stst\] imply that $\vp$ is either a ground state or a constant equilibrium.
For the proof of Proposition \[propcontinuum\], we need several preliminary results. Denote $$\label{eq:5}
J:= \{\vp(0):\vp\in\omega(u)\}.$$ By compactness and connectedness of $\omega(u)$ in $L^\infty_{loc}({{\mathbb R}})$, we have $$\label{Iinterval0}
J=[\gamma^-,\gamma^+] \text{ for some }\gamma^-\le \gamma^+,$$ that is, $J$ is a singleton or a compact interval. Proposition \[propcontinuum\] says that $\gamma^-= \gamma^+$, so this is what we want to show at the end. First, we establish some properties of $\gamma^-$, $\gamma^+$. In the formulations of the lemmas below, we consider the open interval $(\gamma^-,\gamma^+)$ with the understanding that it is empty if $\gamma^-=\gamma^+$.
\[lemmacase2\] The following assertions hold:
1. $f(s)>0$ for each $s\in(\gamma^-,\gamma^+)$;
2. if $\gamma^-<\gamma^+$, then $\gamma^+\in\omega(u)$ and $f(\gamma^+)=0$.
(It is perhaps needless to say that in $\gamma^+\in\omega(u)$, ${\gamma}^+$ refers to the constant function taking the value $\gamma^+$.)
We prove (i) by contradiction. Assume there exists $s\in (\gamma^-,\gamma^+)$ with $f(s)\le 0$ (in particular, $(\gamma^-,\gamma^+)\ne\emptyset$). Since $\eta(t)\to0$, the definition of $J$ implies that $$\label{eq:15}
\liminf_{t\to\infty}u(\eta(t),t)=\gamma^-<s,\qquad
\limsup_{t\to\infty}u(\eta(t),t)=\gamma^+>s.$$ Now, for all large $t$ the function $u({\cdot},t)$ has a local maximum at $x=\eta(t)$. Therefore, equation gives $$(u(\eta(t),t) )'=u_t(\eta(t),t) = u_{xx}(\eta(t),t) +
f(u(\eta(t),t)) \le f(u(\eta(t),t)).$$ This and the assumption $f(s)\le 0$ imply, via an elementary comparison argument for the equation $\dot \xi=f(\xi)$, that if the relation $u(\eta(t),t) <0$ is valid for some $t$, then it remains valid for all larger $t$. This contradiction to proves statement (i).
For the proof of statement (ii), we assume that ${\gamma}^-<{\gamma}^+$. For a contradiction, we assume also that $\gamma^+\not\in\omega(u).$ Then, by relations , , and compactness of $\omega(u)$, there exists $\e$ with $0<\e<\gamma^+-\gamma^-$ such that for all $\vp\in\omega(u)$ one has $\vp(\pm 1)<\gamma^+-\e.$ By statement (i), $f>0$ on $(\gamma^+-\e,\gamma^+)$. Therefore, we can choose $s\in (\gamma^+-\e,\gamma^+)$ such that the solution $\psi$ of (\[steadyeq\]) with $\psi(0)=s$, $\psi'(0)=0$ is a nonstationary periodic solution of (\[steadyeq\]). (The existence of such $s$ follows from Lemma \[MatPolLemma\], but one can also give more direct arguments, see for example [@p-Ma:1d Lemma 3.2]). Let $\rho>1$ be a period of $\psi.$ Then $-\rho$ is also a period of $\psi$ and we have $$\psi(\pm\rho) = s,\qquad \vp(\pm\rho)\le \vp(\pm1)<s \quad (\vp\in\omega(u)).$$ Hence, there is $T_1>0$ such that for all $t>T_1$ one has $u(\pm\rho,t)<s$. Therefore, by Lemma \[lemzero\] and Remark \[convzero\], $z_{(-\rho,\rho)}(u(\cdot,t)-\psi)$ is finite for $t> T_1$ and for all sufficiently large $t$ the function $u(\cdot,t)-\psi$ has only simple zeros in $(-\rho,\rho)$. On the other hand, the definition of ${\gamma}^\pm$ (cp. , ) yields $\vp\in\omega(u)$ with $\vp(0)=s=\psi(0)$. Since also $\vp'(0)=0$ (see ), $\vp-\psi$ has a multiple zero at $x=0.$ Applying Lemma \[robustnesszero\] as in the proof of Lemma \[nointersection\], we conclude that there exist sequences $t_n\to\infty,$ $x_n\to0$ such that $u(\cdot, t_n)-\psi$ has a multiple zero at $x=x_n,$ and we have a contradiction. This contradiction proves that $\gamma^+\in\omega(u)$.
It remains to show that $f({\gamma}^+)=0$. If this is not true, then, by statement (i), $f({\gamma}^+)>0$. Let $U$ be the entire solution of (\[eq1\]) with $U(\cdot,0)\equiv {\gamma}^+$ and $U(\cdot,t)\in\omega(u)$ for all $t\in{{\mathbb R}}.$ Since $U_x$ solves a linear equation , the identity $U_x(\cdot,0)\equiv 0$ implies that $U_x(\cdot,t)\equiv 0$ for all $t\in{{\mathbb R}}$. Thus $U=U(t)$ is a solution of $U_t=f(U)$ and $U'(0)=f({\gamma}^+)>0$. Thus $U(t)>{\gamma}^+$ for $t>0$, which contradicts the definition of ${\gamma}^+$. The proof is now complete.
\[le-consts\] If ${\lambda}\in({\gamma}^-,{\gamma}^+)$, then (the constant function) ${\lambda}$ is not contained in ${\omega}(u)$.
Assuming ${\lambda}\in {\omega}(u)$, let $U$ be the entire solution of (\[eq1\]) with $U(\cdot,0)\equiv {\lambda}$ and $U(\cdot,t)\in\omega(u)$ for all $t\in{{\mathbb R}}.$ Then, as at the end of the previous proof, $U=U(t)$ is a solution of $U_t=f(U),$ $U(0)={\lambda}$. By Lemma \[lemmacase2\](i), the range of the function $U$ is an interval on which $f>0$. We can choose $\tilde {\lambda}$ in this interval such that the solution $\psi$ of (\[steadyeq\]) with $\psi(0)=\tilde{\lambda},$ $\psi'(0)=0$ is a nonconstant periodic solution. This and symmetries of $\psi$ (cp. (\[periodicorbits\])) imply that if $\rho>0$ is the minimal period of $\psi$, then $$\psi\lp\frac{\rho}{2}\rp=\min\psi,\qquad\psi(\rho)=
\psi(2\rho)=\tilde {\lambda}=\max\psi.$$ We still have $\tilde {\lambda}\in {\omega}(u)$ as $\tilde {\lambda}=U\lp\tilde t\rp$ for some some $\tilde t$.
Now, as $\psi-U\lp\tilde t\rp=\psi-\tilde{\lambda}$ has a multiple zero at $x=\rho$ and $\psi-U\not\equiv 0$ ($\psi$ is nonconstant), Lemma \[robustnesszero\] yields sequences $t_n\to\infty,$ $x_n\to\rho$ such that $u(\cdot,t_n)-\psi$ has a multiple zero at $x=x_n$ for all $n.$ On the other hand, by (\[uxuniquezero\]), there exists $T_2>0$ such that for all $t>T_2$ one has $u_x(\cdot,t)<0$ on $\lp\frac{\rho}{2},2\rho\rp.$ One can now obtain a contradiction by considering $z_{\lp\frac{\rho}{2},2\rho\rp}(u(\cdot,t)-\psi)$ and using very similar arguments as in the proof of Lemma \[nointersection\]. We omit the details.
\[lemmacase22\] If ${\lambda}\in(\gamma^-,\gamma^+)$ and the solution $\psi$ of *(\[steadyeq\])* with $\psi(0)={\lambda}$, $\psi'(0)=0$ is periodic, then there is no $\vp\in\omega(u)$ such that $\vp\leq\psi$ on ${{\mathbb R}}$.
By Lemma \[lemmacase2\], the periodic solution $\psi$ is nonconstant. By Lemma \[lemmacase2\](ii) and (\[periodicorbits\]), $\psi$ satisfies $\max\psi={\lambda}$ and $\min\psi<\gamma^-.$ Assume for a contradiction that there exists $\vp\in\omega(u)$ such that $\vp\leq\psi.$ Consider the following set $$\label{eq:8}
K:=\{\xi\in{{\mathbb R}}:
\text{there exists $\vp\in\omega(u)$ such that $\vp\leq \psi(\cdot-\xi)$}\}.$$ By our assumption, $K$ contains $\xi=0$. By compactness of ${\omega}(u)$ in $L^\infty_{loc}({{\mathbb R}})$, $K$ is closed. We show that $K$ is also open, thereby proving that actually $K={{\mathbb R}}$.
Fix any $\xi\in K$ and take $\vp\in {\omega}(u)$ as in . Let $U$ be the entire solution of (\[eq1\]) with $U(\cdot,0)=\vp$ and $U(\cdot,t)\in\omega(u)$ for all $t.$ Since $ \psi(\cdot-\xi)$ is a steady state of (\[eq1\]), the strong comparison argument gives $$\label{eq:9}
\tilde \vp:=U(\cdot,1)<\psi(\cdot-\xi).$$ Since $\tilde \vp\in{\omega}(u)$, relations and the periodicity of $\psi$ imply that relation remains valid if $\xi$ is replaced by $\tilde \xi$ with $\tilde\xi \approx \xi$. This shows the openness of $K$, hence $K={{\mathbb R}}$.
Take now $\xi$ such that $\psi(-\xi)=\min \psi$. For some $\vp\in {\omega}(u)$ one has $\vp\le \psi({\cdot}-\xi)$. In particular, $$\vp(0)\le \psi(-\xi)=\min \psi<\gamma^-,$$ which is a contradiction (cp. , ). This contradiction completes the proof.
As a direct corollary of Lemma \[lemmacase22\](i), (\[monotonicity2\]), and the compactness of $\omega(u)$ in $C^1_{loc}({{\mathbb R}})$, we obtain the following result:
\[corcase2\] If ${\lambda}\in(\gamma^-,\gamma^+)$, then for all $\rho>0$ there exist $\kappa>0$ and $\e_1\in (0,1),$ depending on $\rho,$ such that for any $\vp\in\omega(u)$ with $\displaystyle \lb\vp(0)-{\lambda}\rb\leq\kappa$ one has $ \vp'<-\e_1$ on $\displaystyle\lp\frac{\rho}{2},\rho\rp.$
\[lemmacase23\] If ${\lambda}\in(\gamma^-,\gamma^+)$ and the solution $\psi$ of *(\[steadyeq\])* with $\psi(0)={\lambda}$, $\psi'(0)=0$ is periodic, then there exist $T>0,$ $\e>0$ with the following property. Denoting by $\rho>0$ is the minimal period of $\psi$, we have $$\label{eq:7}
z_{(-\rho,\rho)}(u(\cdot,t)-\psi)\le 2$$ whenever $t>T$ is such that $\displaystyle
u(\pm\rho,t)\in\lp\psi(0)-\e,\psi\rp$.
As in the proof of Lemma \[lemmacase22\], we have ${\lambda}=\psi(0)=\psi(\rho)=\max\psi$ and $\psi(\rho/2)=\min \psi$. Also recall that $u_x$ is uniformly bounded for $t>1.$
With $\kappa>0$ and $\e_1\in (0,1)$ as in Corollary \[corcase2\], we define the following positive quantities: $$ \e_2=\min\lp\frac{\kappa}{2},\frac{\rho\e_1}{8}\rp,\qquad
\d=\frac{\e_2}{\lV u_x\rV_{L^\infty({{\mathbb R}}\times(1,\infty))}+1}.$$ Note that ${\delta}<\rho/8$. In particular, $\psi({\delta})<\psi(0)$. We will show that the conclusion of Lemma \[lemmacase23\] is valid with $$\e:=\min\lp\psi(0)-\psi(\d),\e_1,\e_2\rp.$$
First, we claim that for any $\vp\in {\omega}(u)$ with $\varphi(\rho)\in (\psi(0)-\e,\psi(0))$ one has $\vp(0)>\psi(0)+\e_2$. Indeed, if not, then $ \vp(0)\le \psi(0)+\e_2<\psi(0)+\kappa$. Since also $\vp(\rho)>\psi(0)-\e>\psi(0)-\kappa$, using first Corollary \[corcase2\] with the mean value theorem, and then , we obtain $$\vp(\rho)\le \vp\lp\frac{\rho}2\rp-\e_1\frac{\rho}2\le
\vp(0)-\e_1\frac{\rho}2 \le \psi(0)+\e_2-\e_1\frac{\rho}2.$$ However, since $\e_2-\e_1\rho/2<-\e_2\le \e$, we have a contradiction to the assumption $\varphi(\rho)\in (\psi(0)-\e,\psi(0))$. Thus, our claim is true.
In view of compactness of ${\omega}(u)$ and , the above claim implies that there exists $T>1$ such that if $t>T$ and $u(\pm\rho,t)\in\lp\psi(0)-\e,\psi(0)\rp$, then $u(0,t)>\psi(0)+\e_2.$ Using our definition of $\d$ and the mean value theorem, we infer from $u(0,t)>\psi(0)+\e_2$ that $u(\cdot,t)>\psi(0)\ge\psi$ on $(-\d,\d).$ Next, we make $T$ larger, if necessary, so as to guarantee that if $t>T$ we have $u_x(\cdot,t)>0$ on $[-\rho,-\d]$ and $u_x(\cdot,t)<0$ on $[\d,\rho]$ (cp. , ). Thus, if $t>T$ and $u(\pm\rho,t)\in\lp\psi(0)-\e,\psi(0)\rp$, then for any $x\in [{\delta},{\rho}/{2}]$ we have $$u(x,t)>u(\rho,t)>\psi(0)-\e>\psi({\delta})\ge \psi(x),$$ where we have used the definition of $\e$ and the monotonicity of $\psi$ in $(0,\rho/2)$. Similarly one shows that $u(\cdot,t)>\psi$ on $[-{\rho}/{2},{\delta}]$. Combining these estimates with the previous one, we conclude that $u(\cdot,t)-\psi>0$ on $[-{\rho}/{2},{\rho}/{2}]$ whenever $t>T$ and $u(\pm\rho,t)\in\lp\psi(0)-\e,\psi(0)\rp$. Since the function $u(\cdot,t)-\psi$ is increasing on $\lp-\rho,-\frac{\rho}{2}\rp$ and decreasing on $\lp\frac{\rho}{2},\rho\rp$, it can have at most one zero in each of this intervals. This implies the conclusion of the lemma (in fact, the conclusion holds with the equality sign in , but this is of no significance to us).
We can now complete the proof of Proposition \[propcontinuum\]
The proof is by contradiction. Assume that $\gamma^-<\gamma^+$. As already noted in the proof of Lemma \[lemmacase2\], we can then choose ${\lambda}\in (\gamma^-,\gamma^+)$ such that the solution $\psi$ of (\[steadyeq\]) with $\psi(0)={\lambda}$, $\psi'(0)=0$ is nonconstant and periodic, and $\max \psi={\lambda}<{\gamma}^+$. Let $\rho$ be the minimal period of $\psi$ and let $\e$, $T$ be as in Lemma \[lemmacase23\]. Making $\e>0$ smaller, with no effect on the conclusion of Lemma \[lemmacase23\], we may assume that $ {\lambda}-\e>{\gamma}^-$. Also, in view of , making $T$ larger, if necessary, we may assume that $$\label{eq:12}
|u(-\rho,t)-u(\rho,t)|<\frac{\e}{2}\quad (t>T).$$
We next pick $s\in (\gamma^-,{\lambda}-\e)$. Then there is $\vp\in\omega(u_0)$ with $\vp(0)=s$. Lemma \[lemmacase2\] rules out the possibility that $\vp\le \psi$ in $(-\rho,\rho)$. Therefore, using the evenness and periodicity of $\psi$ in conjunction with , , one shows easily that $ \psi-\vp$ has at least 4 zeros in $(-\rho,\rho)$. Let now $U$ be the entire solution of (\[eq1\]) with $U(\cdot,0)=\vp$ and $U(\cdot,t)\in\omega(u)$ for all $t\in{{\mathbb R}}$. For $t\approx 0$, we have $$U(\pm \rho,t)\approx \vp(\pm\rho)<\varphi(0)=s<\psi(0)-\e=\psi(\pm\rho)-\e.$$ Therefore, an application of Lemma \[lemzero\] shows that arbitrarily close to 0 there is $t<0$ such that $z_{(-\rho,\rho)}(\psi -U(\cdot,t))\ge 4$ and all zeros of $\psi -U(\pm \rho,t)$ in $(-\rho,\rho)$ are simple. Replacing $\vp$ by $U(\cdot,t)$ for such $t$, we have thus found an element $\vp\in\omega(u_0)$ such that $$\vp (\pm\rho)<{\lambda}-\e=\psi(0)-\e$$ and $ \psi-\vp$ has at least 4 simple zeros in $(-\rho,\rho)$.
Since ${\gamma}^+$, $\varphi$ are elements of ${\omega}(u_0)$, we can approximate them arbitrarily closely in $C^1_{loc}({{\mathbb R}})$ by $u(\cdot,T_1)$, $u(\cdot,T_2)$ with $T_2>T_1>T$. In particular, we can choose $T_2>T_1>T$ such that $$\label{eq:13}
u({\cdot},T_1)>\psi\text{ on $[-\rho,\rho]$}$$ and $$\label{eq:14}
z_{(-\rho,\rho)}(\psi -u(\cdot,T_2))\ge 4,\quad u(\pm\rho,T_2)<\psi(0)-\e.$$ Denote $$\tau:=\inf\{s\in (T_1,T_2]: u(\pm\rho,t)<\psi(0)-\frac{\e}2\quad
(s\le t\le T_2)\}.$$ By , $\tau$ is a well defined element of $[T_1,T_2)$. By , $\tau>T_1$. Therefore, at least one of the values $u(\pm\rho,\tau)$ is equal to $\psi(0)-{\e}/2$ and the relations $\tau>T_1\ge T$ and consequently give $$u(\pm\rho,\tau) \in (\psi(0)-\e,\psi(0)).$$ It now follows from Lemma \[lemmacase23\] that $$\label{eq:71}
z_{(-\rho,\rho)}(u(\cdot,\tau)-\psi)\le 2.$$ Since $u(\pm\rho,t)<\psi(0)=\psi(\rho)$ on $[\tau,T_2]$ (see the the definition of $\tau$), the monotonicity of the zero number gives $$z_{(-\rho,\rho)}(u(\cdot,T_2)-\psi)\le 2,$$ in contradiction to . This contradiction shows that ${\gamma}^-<{\gamma}^+$ is impossible, which completes the proof.
Case (C3): two or more limit critical points
--------------------------------------------
In this last case, we assume that there exist two $C^1$ functions $\eta_1(t),$ $\eta_2(t)$ with $\eta_1(t)<\eta_2(t)$ and $\eta_i(t)\underset{t\to\infty}{\longrightarrow}\eta_i^\infty,$ $i=1,2,$ such that and for all $t$ large enough one has $$\label{uxtwozeros}
u_x(\eta_1(t),t)=u_x(\eta_2(t),t)=0.$$ In view of Proposition \[loczero\], $\eta_1(t)<\eta_2(t)$ can be selected such that they are two successive critical points of $u(\cdot,t)$, one of them a local minimum point, the other one a local maximum point.
\[propperiodic\] Set $\xi:=\eta_2^\infty-\eta_1^\infty$. If $\xi>0$, then each function $\vp\in\omega(u)$ is $2\xi-$periodic.
Lemma \[le-conv0\] implies that each function $\vp\in\omega(u)$ is even about each of the two distinct points $\eta_1^\infty$, $\eta_2^\infty$. Therefore it is also even about the points $2\eta_1^\infty-\eta_2^\infty$, and $2\eta_2^\infty-\eta_1^\infty$. Repeating such reflections arguments one obtains the $\xi$-periodicity easily.
\[cor-const\] Each $\vp\in {\omega}(u_0)$ is a constant function.
Suppose first that $\eta_1^\infty=\eta_2^\infty$. Since these are the limits of the critical points $\eta_1(t)<\eta_2(t)$, it follows that $$\lim_{t\to\infty} u_{xx}( \eta_1^\infty,t)= \lim_{t\to\infty} u_{x}(
\eta_1^\infty,t)=0$$ Consequently, for each $\vp\in {\omega}(u_0)$ we have $\vp_{xx}(
\eta_1^\infty)=\vp_{x}(\eta_1^\infty)=0$. Therefore, if $U$ is the entire solution of (\[eq1\]) with $U(\cdot,0)=\vp$ and $U(\cdot,t)\in\omega(u)$ for all $t\in{{\mathbb R}}$, we have $U_x(\eta_1^\infty,t)=U_{xx}(\eta_1^\infty,t)=0$ for all $t\in{{\mathbb R}}$. An application of Lemma \[lemzero\] on a suitable interval, one shows easily that these relations can hold only if $U_x\equiv 0$. In particular, $\vp$ is constant.
Let now $\eta_1^\infty<\eta_2^\infty$. Suppose $\vp\in {\omega}(u_0)$ and there is ${\lambda}\in {{\mathbb R}}$ with $\vp_x({\lambda})\ne 0$. Then $x={\lambda}$ is a simple zero of the function $V_{\lambda}(\vp)(x)=\vp(2{\lambda}-x)-\vp(x)$. Lemma \[propperiodic\] implies that $V_{\lambda}(\vp)$ is a periodic function, hence it has infinitely many simple zeros. Consequently, taking $t_n\to\infty$ such that $u({\cdot},t_n)\to \vp$ in $C^1_{loc}({{\mathbb R}})$, we have $z(V_{\lambda}u(\cdot,t_n))\to\infty$. However, as noted in Subsection \[refl-sec\], the condition $u_0(-\infty)\ne u_0(\infty)$ implies that $z(V_{\lambda}u(\cdot,t))$ is bounded from above as $t\to\infty$. This contradiction completes the proof.
We show that ${\omega}(u_0)=\{\vp\}$ for some constant $\vp$. By Lemma \[cor-const\]—and compactness and connectedness—${\omega}(u_0)$ is an interval $[a_1,b_1]$ of constants (which we identify here with the corresponding constant functions). Here $a_1\le b_1$ and we want to show that $a_1=b_1$.
We go by contradiction. Assume $a_1<b_1$. Then, clearly, there are $a<b$ such that $(a,b)\subset (a_1,b_1)$ and either $f \ge 0$ on $(a,b)$ or $f \le 0$ on $(a,b)$. Assume the former, the latter is analogous.
For large $t$, one of points $\eta_1(t)$, $\eta_2(t)$, further denoted by $\eta(t)$, is a local minimum point of $u(\cdot,t)$. From the fact that ${\omega}(u_0)$ consists of constants, we infer that given any $k>0$, $$\label{eq:10}
\sup_{x\in
(-k,k)}|u(\eta(t),t)-u(x,t)|\underset{t\to\infty}{\longrightarrow} 0.$$ As $a,b\in {\omega}(u_0)$, implies in particular that for some sequences $t_n\to\infty$, $t'_n\to\infty$ one has $u(\eta(t_n),t_n)\to b$, $u(\eta(t'_n),t'_n)\to a$. However, since $\eta(t)$ is a local minimum point and $f\ge 0$ on $(a,b)$, equation (\[eq1\]) gives $$(u(\eta(t),t) )'=u_t(\eta(t),t) = u_{xx}(\eta(t),t) + f(u(\eta(t),t)) \ge 0,$$ whenever $u(\eta(t),t) \in (a,b)$. This implies that if $n$ is so large that $\displaystyle u(\eta(t_n),t_n)>(a+b)/2,$ then $u(\eta(t),t)>(a+b)/2$ for all $t>t_n$ and we have a contradiction.
This contradiction shows that ${\omega}(u_0)$ consists of a single constant ${\lambda}$. The invariance of ${\omega}(u_0)$ shows that this constant is a steady state of . The proof of Theorem \[mainthm\] is now complete.
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[^1]: Supported in part by the NSF Grant DMS-1565388
[^2]: Note, however, that [@Gallay-S; @Gallay-S2] made a good use of with the integral taken over the intervals $(-R,R)$, $R\gg 1$, instead of $(-\infty,\infty)$. As proved in [@Gallay-S], the ${\omega}$-limit set of each bounded solution contains a steady state.
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abstract: 'We consider the numerical simulation of Hamiltonian systems of ordinary differential equations. Two features of Hamiltonian systems are that energy is conserved along trajectories and phase space volume is preserved by the flow. We want to determine if there are integration schemes that preserve these two properties for all Hamiltonian systems, or at least for all systems in a wide class. This paper provides provides a negative result in the case of two dimensional (one degree of freedom) Hamiltonian systems, for which phase space volume is identical to area. Our main theorem shows that there are no computationally reasonable numerical integrators for which all Hamiltonian systems of one degree of freedom can be integrated while conserving both area and energy. Before proving this result we define what we mean by a computationally reasonable integrator. We then consider what obstructions this result places on the existence of volume- and energy-conserving integrators for Hamiltonian systems with an arbitrary number of degrees of freedom.'
address: |
Department of Mathematics and Statistics, McGill University,\
Montréal QC, H3A 2K6 Canada.
author:
- 'P. F. Tupper'
title: |
A Non-Existence Result for\
Hamiltonian Integrators
---
ordinary differential equations,numerical integration ,Hamiltonian systems ,geometric integration,no-go theorems ,volume-conservation ,energy-conservation 65P10
Introduction
============
We consider a system of Hamiltonian differential equations on $\mathbb{R}^{2n}$ $$\label{eq:Ham0}
\frac{ dq}{dt} = \frac{\partial H}{\partial p}, \ \ \ \ \frac{dp}{dt} = -\frac{\partial H}{\partial q},$$ defined by the Hamiltonian function $H: \mathbb{R}^{2n} \rightarrow
\mathbb{R}$. We denote the time $t$ flow map of these equations by $S^t$. The flow of these differential equations has two important features. The first is that $H$ is conserved along trajectories. That is, $
H(S^t (q,p)) = H(q,p),
$ for all $t\in \mathbb{R}$ and all $(q,p) \in \mathbb{R}^{2n}$. The second is that phase space volume is conserved by the flow: if $A \subset \mathbb{R}^{2n}$ is a bounded open set, then $
\mathrm{vol} (S^t A) = \mathrm{vol}(A)
$ for all $t$. The latter property is a consequence of the symplecticity of the flow. (See, for example, [@hairer]).
For certain molecular dynamics applications, the ideal numerical integrator would retain these two properties of the flow [@tupper]. That is, if the integrator with time step $\Delta t$ defines a map $\Phi_{\Delta t}$ on $\mathbb{R}^{2n}$, we would like both $H(\Phi_{\Delta t}(q,p)) = H(q,p)$ for all $(q,p)$ (energy conservation) and $\mbox{vol}(\Phi_{\Delta t}( A) ) = \mbox{vol}(A)$ for all bounded open subsets $A$ of $\mathbb{R}^{2n}$(volume conservation).
Symplectic integrators such as the implicit midpoint rule conserve volume exactly for all Hamiltonian systems, with any number of degrees of freedom, but do not conserve energy [@hairer]. It has already been shown that in certain circumstances it is unreasonable to expect that symplectic integrators which also conserve energy exist [@zhong]. However, volume-conservation is a weaker property than symplecticity for Hamiltonian systems of more than one degree of freedom ($n>1$). Thus it seems plausible that there is a consistent integration scheme which for any Hamiltonian function $H$ and time step $\Delta t$ yields a map $\Phi_{\Delta t}$ which conserves both volume and energy.
In this article we will argue that there is no such integration scheme. We do this by showing that no numerical integrator is able to integrate *all* Hamiltonian systems in $\mathbb{R}^2$ while simultaneously conserving energy and phase space volume. For special Hamiltonian systems in $\mathbb{R}^2$ such an integrator is possible. (See [@quispel] for examples of such systems of any number of degrees of freedom.) However, our theorem states that for any energy-conserving numerical integrator from a very broad class, there will be at least one Hamiltonian system on $\mathbb{R}^2$ (and in fact very many) for which it does not conserve phase space volume.
Before stating and proving our main theorem, in Section \[sec:integrator\] we will define what we mean by an integrator. From the class of integrators we then define the *computationally reasonable* integrators. This will include any explicit or implicit formula for defining a new state $(q^{k+1},p^{k+1})$ as a function of a state $(q^k,p^k)$ and a timestep $\Delta t$ such that the only information used about $H$ is its value and the value of its derivatives at a finite number of points $(r^j,s^j) \in \mathbb{R}^{2n}$, $j=1,\ldots,m$. In Section \[sec:main\] we will state and prove our main result in terms of this definition. We imagine that a numerical analyst has devised a computationally reasonable numerical integrator that is energy-conserving for all Hamiltonian systems in $\mathbb{R}^2$. We apply this integrator to the system with Hamiltonian function $H(q,p)= p^2/2$ in $\mathbb{R}^2$. For two different inputs we observe at which points $(r^j,s^j), j=1,\ldots,m$ the integrator depends on the function $H$ and its derivatives. Using this information, we construct another Hamiltonian function $\widetilde{H}(q,p)=p^2/2+V(q)$ which is arbitrarily close to $H$ and has the following property: the numerical integrator cannot be volume-conserving for both $H$ and $\widetilde{H}$.
The main result demonstrates that there is no integration scheme that conserves volume and energy for arbitrary Hamiltonian systems of any number of degrees of freedom. However, we cannot conclude that there is no scheme that is energy- and volume-conserving for Hamiltonian systems in a particular number of dimensions $2n$, $n>1$. This question is open. To partially address this issue, in Section \[sec:multi\] we will show that under some reasonable— but not essential— conditions on an integrator, the problem reduces to that of the $n=1$ case. Thus, there is no energy- and volume-conserving integrator for fixed $n$ that satisfies these additional assumptions.
We now explain the relation between the results here and those of the well-known paper of Zhong and Marsden [@zhong]. In that paper the authors consider a Hamiltonian system for which there are no invariants except energy. They show that any integrator that is both symplectic and energy-conserving actually computes exact trajectories of the system up to a time reparametrization. From this result they conclude that energy-conserving symplectic integration is not possible in general, since presumably the set of Hamiltonians for which one could compute trajectories exactly up to a time reparametrization are very small. Though this argument is very plausible, it leaves open two questions:
1. How do we make precise the idea of something not being possible for any numerical integrator?
2. Is it possible to perform energy-conserving and symplectic integration for general Hamiltonians when we restrict ourselves to the $\mathbb{R}^2$ case?
The importance of the latter question is that if it were possible, then volume- and energy-conserving integration would be possible for general Hamiltonian systems via $J$-splitting [@quispel; @faou].
We provide an answer to the first question in Section \[sec:integrator\] with the definition of a computationally reasonable integrator. As for the second question, since volume-conservation and symplecticity are identical in $\mathbb{R}^2$, Zhong and Marsden’s result shows that in $\mathbb{R}^2$ volume- and energy-conserving integration is equivalent to solving the original system exactly up to a time-reparametrization. (This result is stated and proved for this case as Lemma \[lem:reparam\] in Section \[sec:main\].) The main result of our paper answers the second question in the negative by showing that it is not possible for general Hamiltonian systems in $\mathbb{R}^2$ using computationally reasonable integrators.
Before we begin we discuss some interesting related work. Even though energy and volume conserving integrators may not exist, the paper [@faou] does the next best thing. There the authors show how to approximate any Hamiltonian function arbitrarily well by a special piece-wise smooth function of a form described by [@quispel] whose trajectories can be integrated while conserving volume and energy. The original Hamiltonian function is not conserved. However, unlike for standard symplectic methods, a modified Hamiltonian function close to the original is conserved exactly [@faou] for all time.
What is a numerical integrator? {#sec:integrator}
===============================
An integrator for a Hamiltonian system of ordinary differential equations takes a Hamiltonian $H$, a step length $\Delta t$, and an initial value $(q^k,p^k)\in \mathbb{R}^{2n}$, and produces a value $(q^{k+1},p^{k+1}) \in \mathbb{R}^{2n}$. Typically, $(q^{k+1},p^{k+1})$ depends on $H$ through components of $\nabla H$ and perhaps $H$ itself. If $(q^k,p^k)$ is an approximation to $(q(k \Delta t),p(k \Delta t))$, we take $(q^{k+1},p^{k+1})$ to be an approximation to $(q((k+1)\Delta t),p((k+1)\Delta t))$.
\[defn:integrator\] An integrator $\Phi$ is a function that takes arguments $H : \mathbb{R}^{2n} \rightarrow
\mathbb{R}$, $\Delta t \in [0,\infty)$, $(q^k,p^k) \in \mathbb{R}^{2n}$ and either returns $(q^{k+1},p^{k+1}) \in \mathbb{R}^{2n}$ or is not defined. We write $$(q^{k+1},p^{k+1}) = \Phi( H, (q^k,p^k),\Delta t).$$
We have allowed the integrator to not be defined for certain input values. This is often the case for implicit integrators when the vector field is insufficiently smooth or the time step is too large.
In order to get meaningful constraints on what is computationally reasonable, we cannot let arbitrary maps be included in the class of algorithms we study. After all, the exact flow map has all the qualitative features one could want, but it is not feasible to compute it (even to machine precision) for most applications. We would like our definition to be broad enough to include most existing numerical integrators.
Informally, we say an integrator $\Phi$ is *computationally reasonable* if for each $(q^k,p^k)$ and $\Delta t$, $(q^{k+1},p^{k+1})=\Phi(H,(q^k,p^k),
\Delta t)$ depends on $H$ only through its value and the value of its derivatives at a finite number of points $(r^j,s^j) \in \mathbb{R}^{2n}$, $j=1,\ldots,n$. In the following formal definition we use multi-index notation to define higher-order derivatives: for $H : \mathbb{R}^{2n} \rightarrow \mathbb{R}$ and $\alpha \in \mathbb{N}^{2n}_0$ we let $$\partial_\alpha H :=
\frac{\partial^{\alpha_1}}{\partial q_1^{\alpha_1}}
\cdots
\frac{\partial^{\alpha_n}}{\partial q_n^{\alpha_n}}
\frac{\partial^{\alpha_{n+1}}}{\partial p_1^{\alpha_{n+1}}}
\cdots
\frac{\partial^{\alpha_{2n}}}{\partial p_n^{\alpha_{2n}}} H$$
\[defn:reasonable\] An integrator $\Phi$ is *computationally reasonable* if for each $H: \mathbb{R}^{2n} \rightarrow \mathbb{R}$, $(q^k,p^k) \in \mathbb{R}^{2n}$ and $\Delta t \geq
0$ there exists
1. $m \in \mathbb{N}_0$,
2. $(r^j,s^j) \in \mathbb{R}^{2n}, j=1, \ldots, m$,
3. $\alpha^j \in \mathbb{N}_0^{2n}, j=1, \ldots, m$,
such that for any function $\widetilde{H}: \mathbb{R}^{2n} \rightarrow \mathbb{R}$ that satisfies $$\partial_{\alpha^j} \widetilde{H} (r^j) = \partial_{\alpha^j} H(r^j), \ \ \ j=1,\ldots,m$$ either $\Phi(H,(q^k,p^k),\Delta t) =
\Phi(\widetilde{H},(q^k,p^k),\Delta t)$ or both are not defined.
In the remainder of this section we discuss examples of the class of computationally reasonable integrators. First note that all explicit methods fit into this class. We informally define an integrator to be explicit if it can be implemented by an algorithm that terminates in a finite number of steps using function evaluations of $H$ or its derivatives, arithmetic operations, and logical operations. This includes all of the explicit Runge-Kutta and partitioned Runge-Kutta methods, for example. It also includes integrators that collect information adaptively to perform a step, such as the Bulirsch-Stoer method [@bs]. Taylor series methods are also included.
The class of computationally reasonable integrators also includes implicit methods, such as the implicit Runge-Kutta methods. Here we define an implicit method to be one where $(q^{k+1},p^{k+1})$ is specified by requiring it to be the solution to a nonlinear system of equations in $H$ and its derivatives. Implicit algorithms cannot in general be implemented exactly in a finite number of steps, but they still fit into the framework of Definition \[defn:reasonable\]. To see this, note that even though solving a system of nonlinear equations exactly typically requires looking at $H$ and its derivatives at an infinite number of points (while performing the Newton Iteration, for example), determining if we have a solution to a nonlinear system of equations only requires examining a finite number of points. So if $(q^{k+1},p^{k+1})$ solves the equations for $H$, it will still solve the equation for any $\widetilde{H}$ which is identical to $H$ at the points. Similarly, step-and-project methods are included in this class. These are methods that consist of one step of a simpler method followed by a projection onto a manifold [@hairer IV.4]. What integrators do not satisfy Definition \[defn:reasonable\]? Integrators that require the exact computation of integrals of $H$ or its derivatives do not. Computing the integral of a general function requires knowing its value at an infinite set of points on the domain of integration. Unlike in the case of solving nonlinear equations, if we are given a value for the integral, there is no finite set of points at which we can examine the function to verify that the value is correct. Of course, it is possible to define an numerical integrator that uses integrals of functions, but the actual computation of these integrals for general Hamiltonian systems would require numerical quadrature. This in turn would require sampling the function at a finite number of points, and introducing truncation error. The new method with this additional truncation error does form a computationally reasonable integrator, while the original method with the exact integral does not.
Finally, we note that multistep methods are not even integrators according to Definition \[defn:integrator\]. We believe our framework could be extended to multistep method but we do not do so here.
Main Theorem {#sec:main}
============
To prove our main result Theorem \[thm:main\] we use the following lemma. It shows that for a Hamiltonian system in $\mathbb{R}^{2}$, the map defined by an energy and area preserving integrator is just a time-reparametrization of the flow map. As discussed in the introduction, this is essentially Zhong and Marsden’s result [@zhong] in the two dimensional case. The only addition is that we show that the time-reparametrization is just a constant rescaling of time where locally the constant does not depend on energy.
\[lem:reparam\] Let $H:\mathbb{R}^2 \rightarrow \mathbb{R}$ be a smooth function and $S^t$ its induced Hamiltonian flow map. Let $E \in \mathbb{R}$ be a particular energy. Let $U$ be an open set whose intersection with $\{ (q,p) | H(q,p) = E \}$ is a simple curve $\Gamma$. Suppose that $\nabla H \neq 0$ on $\Gamma$. Let $\Phi_{\Delta t}$, $\Delta t>0$ be a continuous area-conserving map defined on $U$ that conserves $H$. Then there is a constant $c$ such that $\Phi_{\Delta t}(q,p) = S^{c \Delta t}(q,p)$ for all $(q,p) \in \Gamma$.
[**Proof:** ]{} On $U$ we can define canonical action-angle coordinates $(\theta, \phi)$ in which the Hamiltonian function is $H(\theta,\phi) =
H(\theta)$. The flow map is then $
S^t(\theta,\phi) = (\theta,\phi + t H'(\theta) ),
$ where $H'(\theta) \neq 0$, since we still have $[H'(\theta), 0]^T = \nabla H \neq 0$ in the new coordinates. To study the Jacobian of $\Phi_{\Delta t}$ in the coordinates $(\theta,\phi)$ we let $
(\theta_{1}, \phi_{1}) = \Phi_{\Delta t}( \theta_0, \phi_0)
$ for $(\theta_0,\phi_0) \in U$. Then the Jacobian of $\Phi_{\Delta t}$ is $$\left[ \begin{array}{cc}
1 & 0 \\
\frac{\partial \phi_1}{\partial \theta_0} & \frac{\partial \phi_1}{\partial \phi_0}
\end{array}
\right].$$ Area conservation implies that the determinant is $1$ and so $
\partial \phi_1/\partial \phi_0 = 1
$ for all $\theta_0, \phi_0$. This yields $\phi_1 = \phi_0 + \tau$ for all $\phi_0$ for some $\tau$ which may depend on $\theta_0$. Hence, if we choose $c$ so that $c \Delta t/H'(\theta_0) = \tau$ we have $$\Phi_{\Delta t}(\theta_0,\phi_0) = (\theta_0, \phi_0 + \tau) = S^{c \Delta t}(\theta_0,\phi_0),$$ for all $(\theta_0,\phi_0) \in \Gamma$, as required. $\Diamond$
For the main result we will use the following very weak definition of consistency.
\[defn:consistent\] Let $S^t$ be the flow map for the Hamiltonian $H: \mathbb{R}^{2n} \rightarrow \mathbb{R}$. An integrator $\Phi$ is consistent at $(q,p) \in \mathbb{R}^{2n}$ if $$\lim_{\Delta t \rightarrow 0} \frac{ \|S^{\Delta t}(q,p) - \Phi(H,\Delta t,(q,p)) \|}{\Delta t} = 0.$$
\[thm:main\] Let $\Phi$ be an integrator for a Hamiltonian system in $\mathbb{R}^2$. Let $H(q,p)=p^2/2$. Let $U$ be an open neighbourhood of $\{(q,1)| q \in \mathbb{R} \}$. Suppose:
1. $\Phi$ is consistent at $(0,1)$ (Definition \[defn:consistent\]).
2. $\Phi$ is computationally reasonable (Definition \[defn:reasonable\]).
3. $\Phi$ conserves energy for any $H$, $(q,p)$, and $\Delta t$ for which it is defined.
4. For $\Delta t \leq \Delta t_0$, $\Phi(H,(q,p),\Delta t)$ is defined, depends continuously on $(q,p)$, and conserves volume on $U$.
Then for all sufficiently small $\Delta t$ there is a $C_0^\infty$ function $V: \mathbb{R} \rightarrow \mathbb{R}$ such that if $\widetilde{H}(q,p)=p^2/2 + V(q)$ then $\Phi(\widetilde{H},\cdot,\Delta t)$ is not simultaneously defined, continuous, and volume-conserving on $U$. For each such $\Delta t$, the $V$ constructed can be replaced by $\lambda V$ for $\lambda \in (0,1)$ and the same result holds.
[**Proof:** ]{} For any $\Delta t \in (0, \Delta t_0]$, since $\Phi(H,\cdot,\Delta t)$ conserves energy and is continuous on $U$, the set $\{(q,1) | q\in \mathbb{R} \}$ is mapped onto itself. As $\Phi(H,\cdot,\Delta t)$ conserves volume, Lemma \[lem:reparam\] shows that it is identical to the flow of the original Hamiltonian system on $\{(q,1)| q\in \mathbb{R}\}$ with a rescaling of time: $$\Phi(H,(q,1),\Delta t) = (q+c \Delta t, 1),$$ for all $q \in \mathbb{R}$, where $c$ does not depend $q$. The consistency condition at $(0,1)$ implies that for small enough $\Delta t$ we have that $c>0$. From now on, we assume $\Delta t$ is small enough so that $c>0$.
Consider the integrator applied to $H$ at the point $(q,p)=(0,1)$. Since $\Phi$ is computationally reasonable (Definition \[defn:reasonable\]), there are a finite number of points $(r^j,s^j) \in \mathbb{R}^2$, $j=1,\ldots,m$, such that $\Phi$ only depends on $H$ at these points. Choose a $q_0 \in \mathbb{R}$ big enough so that the interval $[q_0,q_0+ c
\Delta t ]$ contains none of the points $r^j, j=1,\ldots,m$ and is disjoint from the interval $[0,c \Delta t]$.
Consider the integrator applied to $H$ at the point $(q_0,1)$. There are points $(\bar{r}^j,\bar{s}^j), j=1,\ldots,\bar{m}$ such that $\Phi(H,(q_0,1),\Delta t)$ only depends on $H$ at these points. Let $V$ be a $C^\infty_0$ function such that
1. $V(q)=0$ for $q$ not in $[q_0, q_0+c \Delta t]$, and $V(\bar{r}^j)=0, j=1,\ldots,\bar{m}$,
2. $0 \leq V(q) < 1/2$ for all $q$,
3. for some $\epsilon>0$, $V(q)>0$ for $q \in (q_0,q_0+\epsilon)$.
Note that multiplying $V$ by any factor in $(0,1)$ gives a function satisfying the same conditions. Let $\widetilde{H}(q,p) = p^2/2 + V(q)$ and let $\widetilde{S}^t$ denote the flow map of the system with this Hamiltonian. Now if $\Phi(\widetilde{H},\cdot,\Delta t)$ were defined, continuous and volume-conserving on $U$, then by Lemma \[lem:reparam\] $$\Phi(\widetilde{H},(q,1),\Delta t) = \widetilde{S}^{d \Delta t} (q,1)$$ for all $q\in \mathbb{R}$ for some constant $d$. We will show that this is impossible.
First note that the integrator gives the same result for both Hamiltonians at $(0,1)$. This is because here the result of the integrator only depends on $H$ only at the points $(r^j,s^j)$, at which $H$ and $\widetilde{H}$ agree. Since $$(d \Delta t,1) = \Phi(\widetilde{H},(0,1),\Delta t) = \Phi(H,(0,1),\Delta t) = (c \Delta t,1)$$ we must have $c=d$.
On the other hand, the integrators also give the same results for both Hamiltonians at $(q_0,1)$. This implies $c \neq d$ in the following way. We have that $$\begin{aligned}
(q_0+c \Delta t,1) & = & \Phi (H,(q_0,1),\Delta t) \\
& = & \Phi(\widetilde{H},(q_0,1),\Delta t)\\
& = & \widetilde{S}^{d \Delta t} (q_0,1) \\
& \neq & S^{d \Delta t}(q_0,1)\\
& = & (q_0+d \Delta t,1).\end{aligned}$$ The inequality follows from Lemma \[lem:differ\], since the Hamiltonians differ on the interval $[q_0,q_0 + \epsilon]$. So $c \neq d$. This is a contradiction. Therefore, $\Phi(\widetilde{H},\cdot,\Delta t)$ cannot be simultaneously defined, continuous, and volume-conserving on $U$. $\Diamond$
The following lemma asserts the intuitively clear fact that if $V(q)$ is positive for $q \in [q_0, q_0+\epsilon]$ then the flows starting from $(q_0,1)$ corresponding to $H$ and $\widetilde{H}$ are different for positive times.
\[lem:differ\] Let $V: \mathbb{R}^{2n} \rightarrow \mathbb{R}$ be $C^\infty_0$ with $V(q_0)=0$, $1/2>V(q) \geq 0$ for all $q\in
\mathbb{R}$ and $V(q)>0$ for all $q \in [q_0, q_0+\epsilon]$. Let $H(q,p) = p^2/2$ and $\widetilde{H}(q,p) = p^2/2 + V(q)$. Let $S^t$ and $\widetilde{S}^t$ be the respective Hamiltonian flow maps of $H$ and $\widetilde{H}$. Then $S^t(q_0,1) \neq \widetilde{S}^t(q_0,1)$ for $t> 0$.
[**Proof.**]{} The trajectory for the Hamiltonian $H$ as a function of time is $(q(t),p(t))=(t,1)$ for all $t$. Letting $\tilde{q}(t)$ describe the position for the Hamiltonian $\widetilde{H}$, the usual solution technique gives $$t = \int_{q_0}^{\tilde{q}} \frac{1}{\sqrt{1-2V(x)}} dx=: \widetilde{F}(\tilde{q})$$ The function $\widetilde{F}$ is strictly increasing and so has a well defined inverse. We can write $
\tilde{q}(t) = \widetilde{F}^{-1}(t).
$ Now $\widetilde{F}'(\tilde{q}) > 1$ for $\tilde{q} >0$, so $\tilde{q}'(t) < 1$ for $t>0$. Hence $\tilde{q}(t) < t$ for $t>0$ and the two flow maps cannot be equal for $t>0$. $\Diamond$
Multiple Degrees of Freedom {#sec:multi}
===========================
In the previous section we showed that there can be no general energy- and volume-conserving integration schemes because there are no integrators that conserve energy and volume for all Hamiltonians in $\mathbb{R}^2$. However, suppose we ask if such integrators exists for Hamiltonian systems of dimension $2n$, $n>1$. We conjecture that a result like Theorem \[thm:main\] still holds in this case. However, the method of proof for the $n=1$ case does not extend to this case.
Instead we will state two conditions on an integrator, either one of which prevents it from being volume- and energy-conserving for general Hamiltonian systems in $\mathbb{R}^{2n}$. Both of these conditions are desirable for an integrator to have, but unlike computational reasonibility, it is not difficult to imagine a practical integrator that did not satisfy them. The proof of the theorems in this section will work by showing that, if a computational reasonable energy-conserving integrator with either condition exists, a special Hamiltonian in $\mathbb{R}^{2n}$ can be constructed for which it does not conserve volume.
In the following, fix $n \geq 2$. Let $q,p \in \mathbb{R}^{n}$ have components $q_i, p_i \in \mathbb{R}$. Define the projections $\pi_i$, $i=1,\ldots,n$ by $\pi_i(q,p)=(q_i,p_i)$.
The first condition asserts that if variables $(q_i,p_i)$ do not occur in the Hamiltonian function then the integrator does not change their value.
\[cond:untouch\] If $H(q,p)$ is independent of the variables $(q_i,p_i)$ then $$\pi_i \Phi( H, (q,p), \Delta t) = (q_i,p_i)$$ for all $(q,p)$ and $\Delta t$ for which $\Phi$ is defined.
Let $\Phi$ be an integrator for Hamiltonian systems in $\mathbb{R}^{2n}$. Let $H(q,p)=p_1^2/2$. Let $U$ be an open neighbourhood of the set $$\left\{ \left( (q_1,0, \ldots,0), (1,0,\ldots,0) \right) | q_1 \in \mathbb{R} \right\}.$$ Suppose
1. $\Phi$ is consistent (Definition 3).
2. $\Phi$ is computationally reasonable (Definition 2).
3. $\Phi$ conserves energy for any $H$, $(q,p)$ and $\Delta t$ for which it is defined.
4. $\Phi$ satisfies Condition \[cond:untouch\]
5. For sufficiently small $\Delta t$, $\Phi(H,(q,p),\Delta t)$ is defined, continuous, and conserves volume for $(q,p) \in U$.
Then for sufficiently small $\Delta t$, there is a $C_0^\infty$ function $V: \mathbb{R} \rightarrow \mathbb{R}$ such that if $\widetilde{H}(q,p) = p_1^2 + V(q_1)$, then $\Phi(\widetilde{H},\cdot, \Delta t)$ is not simultaneously defined, continuous, and volume-conserving on $U$. For each such $\Delta t$, the $V$ constructed can be replaced by $\lambda V$ for $\lambda \in (0,1)$ and the same result holds.
[**Proof:** ]{} We will use the hypothesized integrator on $\mathbb{R}^{2n}$ to construct an integrator $\phi$ on $\mathbb{R}^2$ satisfying the conditions of Theorem \[thm:main\]. Let $H:\mathbb{R}^2 \rightarrow \mathbb{R}$ be a given Hamiltonian function. Define $H^*: \mathbb{R}^{2n} \rightarrow \mathbb{R}$ by $H^*(q,p)= H(q_1,p_1)$. We define the integrator $\phi$ by $$\phi(H,(q_1,p_1),\Delta t) = \pi_1 \Phi(H^*,((q_1, 0 ,\ldots,0),(p_1,0,\ldots,0)),\Delta t).$$ Let $\bar{U} \subset \mathbb{R}^2$ be given by $$\bar{U} := \left\{ (q_1,p_1) | ((q_1,0,\ldots,0),(p_1,0,\ldots,0)) \in U \right\}.$$ It is straightforward to check that $\phi$ satisfies the conditions of Theorem \[thm:main\] on $\bar{U}$. Thus, by the theorem, we have an arbitrarily small $C^\infty_0$ function $V$ such that $\phi$ is not simultaneously defined, continuous, and volume-conserving on $\bar{U}$ for $\widetilde{H}(q_1,p_1)=p_1^2+V(q_1)$.
Let $\widetilde{H}^*(q,p) = p_1^2/2 + V(q_1)$ for $(q,p)\in \mathbb{R}^{2n}$. Now suppose that $\Phi(\widetilde{H}^*,\cdot,\Delta t)$ is defined, continuous, and volume preserving on $U$. We will derive a contradiction by showing this implies that $\phi(\widetilde{H},\cdot,\Delta t)$ is, in fact, defined, continuous, and volume-conserving on $\bar{U}$.
First note that $\Phi(\widetilde{H}^*,\cdot,\Delta t)$ being defined and continuous on $U$ implies that $\phi(\widetilde{H},\cdot,\Delta t)$ is defined and continuous on $\bar{U}$. To check volume conservation, note that the Jacobian of the map $\Phi(\widetilde{H}^*,\cdot,\Delta t)$ has structure $$\left[
\begin{array}{cc}
J_{11} & J_{12} \\
0 & I
\end{array}
\right]$$ where we have put the variables in order $(q_1,p_1,\ldots,q_n,p_n)$ and $J_{11}$ is a 2-by-2 matrix. Since the determinant of this matrix is 1 by volume-conservation, the determinant of $J_{11}$ must be 1. But $J_{11}$ is the Jacobian of $\phi(\widetilde{H},\cdot,\Delta t)$, so this latter map must be area preserving. This contradicts our earlier assumption. $\diamond$
The second condition states that if the Hamiltonian system consists of $n$ identical uncoupled one-degree-of-freedom systems, then the integrator itself should consist of $n$ identical uncoupled maps on the state-space of each subsystem.
\[cond:prod\] If $H(q,p)= \sum_{i=1}^n h(q_i,p_i)$ then there is an integrator $\phi$ on $\mathbb{R}^2$ such that $$\pi_i \Phi(H,(q,p),\Delta t) = \phi(h,(q_i,p_i),\Delta t)$$ for $i=1,\ldots,n$, for any $(q,p)$ and $\Delta t$ for which the integrator is defined.
Though this is certainly a nice property for the integrator to have (since the flow map has the same property) there are many integrators for which it does not hold. For example, step-and-project methods may not satisfy this condition, even if the underlying one-step method does.
Let $\Phi$ be an integrator for Hamiltonian systems in $\mathbb{R}^{2n}$. Let $H(q,p)=\sum_i p_i^2/2$. Let $U \subset \mathbb{R}^{2n}$ be an open neighbourhood of the set $$\left\{ \left((q_1, \ldots,q_1), (1,\ldots,1)\right) | q_1 \in \mathbb{R} \right\}.$$ Suppose
1. $\Phi$ is consistent (Definition 3).
2. $\Phi$ is computationally reasonable (Definition 2).
3. $\Phi$ conserves energy for any $H$, $(q,p)$ and $\Delta t$ for which it is defined.
4. $\Phi$ satisfies Condition \[cond:prod\]
5. For sufficiently small $\Delta t$, $\Phi(H,(q,p),\Delta t)$ is defined, continuous, and conserves volume for $(q,p) \in U$.
Then for sufficiently small $\Delta t$, there is a $C_0^\infty$ function $V: \mathbb{R} \rightarrow \mathbb{R}$ such that if $\widetilde{H}(q,p) = \sum_{i=1}^n (p_i^2 + V(q_i))$ then $\Phi(\widetilde{H},\cdot, \Delta t)$ is not simultaneously defined, continuous, and volume-conserving on $U$. For each such $\Delta t$, the $V$ constructed can be replaced by $\lambda V$ for $\lambda \in (0,1)$ and the same result holds.
[**Proof:** ]{} This theorem is proven analogously to the previous theorem. For any $H: \mathbb{R}^{2} \rightarrow \mathbb{R}$ we define $H^* :\mathbb{R}^{2n} \rightarrow \mathbb{R}$ by $H^*(q,p)= \sum_i H(q_i,p_i)$. We define the integrator $\phi$ by $$\phi(H,(q_1,p_1),\Delta t) = \pi_1 \Phi(H^*,((q_1,\ldots,q_1),(p_1,\ldots,p_1)),\Delta t).$$ We define $\bar{U} \in \mathbb{R}^2$ by $$\bar{U} := \left\{ (q_1,p_1) | \left( (q_1,\ldots,q_1), (p_1,\ldots,p_1)\right) \in U \right\}.$$ As in the proof of the previous theorem, $\phi$ and $\bar{U}$ satisfy the conditions of Theorem \[thm:main\]. Thus, by the theorem, we have an arbitrarily small $C^\infty_0$ function $V$ such that $\phi$ is not defined continuous and volume-conserving on $\bar{U}$ for $\widetilde{H}=p_1^2+V(q_1)$.
Let $\widetilde{H}^*(q,p) = \sum_i (p_i^2/2 + V(q_i))$ for $(q,p)\in \mathbb{R}^{2n}$. Now suppose that $\Phi(\widetilde{H}^*,\cdot,\Delta t)$ is defined, continuous, and volume preserving on $U$. We will derive a contradiction.
Now $\phi(\widetilde{H},\cdot, \Delta t)$ is defined and continuous on $\bar{U}$. To check volume conservation, note that the Jacobian of the map $\Phi(\widetilde{H}^*,\cdot,\Delta t)$ in this case has structure $$\left[
\begin{array}{cccc}
J_{11} & 0 & \ldots & 0 \\
0 & J_{22} & \ldots & 0 \\
\vdots & \vdots & \ddots & 0 \\
0 & \ldots & 0 & J_{nn}
\end{array}
\right]$$ where we have put the variables in order $(q_1,p_1,\ldots,q_n,p_n)$ and each $J_{ii}$ is 2-by-2. Since the determinant of this matrix must be 1 by volume-conservation and the determinants of the $J_{ii}$ are identical, the determinant of $J_{11}$ must be $\pm 1$. As in the proof of the previous theorem this implies $\phi(\widetilde{H},\cdot,\Delta t)$ is area conserving on $\bar{U}$ which is a contradiction. $\diamond$
[**Acknowledgments.**]{} The author thanks Nilima Nigam for her comments on this work. The author was supported by an NSERC Discovery Grant.
[00]{}
P. Chartier and E. Faou, Volume-energy preserving integrators for piece-wise smooth approximations of Hamiltonian systems. (2006).
E. Hairer, C. Lubich and G. Wanner, [*Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations*]{}, Springer, Berlin, 2002.
R. I. McLachlan and G. R. W. Quispel, Geometric integration of conservative polynomial ODEs, [*Applied Numerical Mathematics*]{}, [**45**]{} (2003) 411–418.
J. Stoer and R. Bulirsch, [*Introduction to Numerical Analysis*]{}, Springer-Verlag, New York, 1980.
P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems, [*SIAM J. Appl. Dyn. Sys.*]{} [**4**]{} (2005) 563–587.
G. Zhong, J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, [*Phys. Lett. A*]{}, (1988) 133:134-139.
|
---
author:
- 'G. Magkos'
- 'and V. Pavlidou'
title: 'Deflections of ultra-high energy cosmic rays by the Milky Way magnetic field: how well can they be corrected?'
---
Introduction {#sec:intro}
============
Ultra high energy cosmic rays (UHECRs) are the most energetic particles ever detected, with energies in some cases exceeding $10^{20} \ {\rm eV}$. Their sources, mechanism of acceleration, and composition are currently unknown and debated [@Aloisio]. One reason for the difficulty to reach safe conclusions in any of the above areas is the extremely small flux of the incoming UHECRs, which leads to poor statistics. For example, for energies of about $10^{20} \ {\rm eV}$, the flux is of the order of one particle per ${\rm km}^2$ per century [@TA]. Another reason is the fact that UHECRs, consisting of protons or heavier nuclei, therefore charged particles, are deflected by the Galactic magnetic field (GMF) as well as the intergalactic magnetic field. In the present work, we focus on the role of the GMF.
The main limitation in studies of the GMF is that currently available observables that probe the GMF are integrated along the line of sight (LOS). Such observables include Faraday rotation measures, synchrotron intensity and polarization of dust emission (see, e.g., [@Han] for a recent review on magnetic field observations). In the absence of 3D tomographic information, the approach that has been used for the mapping of the GMF is parameter fitting of different GMF components, including a large random component (e.g., [@JF12_ord; @JF12_rand; @Sun8; @Sun10; @Planck]).
However, the advent of the Gaia mission [@Gaia], which is expected to provide parallaxes for a, for the first time, significant fraction of the Galactic stellar population, is redefining our ability to conduct tomographic studies of the Galaxy including its magnetic field. Gaia stellar distances, combined with optopolarimetric measurements of starlight, will provide a unique and previously unavailable handle on the GMF structure. As dust grains in interstellar clouds tend to align themselves perpendicular to the direction of the magnetic field, optical light passing through a cloud will be absorbed preferentially in the direction of the grain long axis and will be polarized in the plane-of-the-sky (POS) direction of the magnetic field there [@Dust]. Upcoming optopolarimetric surveys such as PASIPHAE [@PASIPHAE; @PASIPHAE_paper] are expected to provide a large number ($>10^6$) of high-quality stellar polarization measurements that can yield 3D information for the POS direction (measured directly) as well as the POS magnitude of the GMF (inferred using e.g. the Davis-Chandrasekhar-Fermi method [@Davis; @ChF; @Hildebrand; @Panopoulou; @Soam_Pattle; @Liu; @Beuther; @Kwon; @Mao; @Soam_Lee; @Clemens]).
Any such 3D measurements of the GMF could be used to reconstruct the trajectories of individual cosmic rays through the Galaxy, in order to obtain the original arrival direction of a cosmic ray before it was deflected by the GMF. The effectiveness of such a correction would depend on the accuracy of any such future measurement. If an effective correction were possible in this way, it would constitute an important development in cosmic-ray astronomy, as it could result in increased clustering of cosmic-ray arrival directions, and improved estimates for the location of sources.
In the present work, we study the limits of this method of trajectory reconstruction for individual UHECRs, independently of any specific dataset, experiment, or technique for the determination of the magnetic field direction and strength. We simulate this process by using hypothetical 3D measurements of the GMF. We examine how the precision of the measurements can affect the quality of the reconstruction and how this quality is affected by the rigidity (i.e. the energy of the particle divided by its charge) and arrival direction of the cosmic ray at the Earth. This is accomplished as follows. First, given the rigidity and arrival direction, we find the “true” path of the cosmic ray by using a specific GMF model to provide the “true” values of the GMF. We then obtain the original arrival direction of the cosmic ray before it entered the Galaxy. Next, we introduce random errors to the “true” GMF values along the trajectory in order to simulate the uncertainty of some hypothetical GMF measurements. Again, we find the estimated cosmic ray trajectory using the GMF values with errors and obtain the estimated original arrival direction. Finally, we quantify the effectiveness of our correction by calculating how close the estimated original arrival direction is to the “true” original arrival direction.
Methods {#sec:Methods}
=======
Propagation of an UHECR in a constant and uniform magnetic field
----------------------------------------------------------------
An UHECR, thought to be a proton or a nucleus, has charge $q=Z e$, where $Z$ is the number of protons in the nucleus and $e$ the charge of a proton. As it is an ultra-relativistic particle of $\gamma \sim 10^{9}$–$10^{10}$ for $E \sim 10^{19} \ {\rm eV}$, we can safely neglect its rest mass and consider its speed to be equal to the speed of light $c$.
The velocity $\bm{v}$ of a relativistic charge $q$ with energy $E$ in a constant and uniform magnetic field $\bm{B}$ follows:
$$\dfrac{d\bm{v}}{dt}=\dfrac{c^2 q}{E} (\bm{v} \times \bm{B} ).
\label{eq:velocity}$$
The quantity that defines how much the cosmic ray is deflected in a given magnetic field is the rigidity $R=E/q=E/Z e$. Using eq. \[eq:velocity\] for small time intervals $\delta t$ we can numerically calculate the evolution of the velocity and position of the particle. The time interval that we use is such that $c \, \delta t = 1 \, {\rm pc}$, which is much smaller than the typically assumed coherence length $\sim 100 \, {\rm pc}$ of the GMF (e.g. [@Golup]).
Reconstruction of the “true” cosmic ray trajectory
--------------------------------------------------
To reconstruct the cosmic ray trajectory, we construct, within the Galaxy, a cubic grid with side $L=100 \ {\rm pc}$. We assume that the magnetic field is uniform inside each cube. The length $L$ has been chosen to be similar to the coherence length of the GMF.
Before the trajectory reconstruction begins, the magnetic field for each grid cube is calculated using a GMF model and our result is assumed to be the “true” value within that cube. We use two different GMF models, so as to have an indication of the influence that the choice of GMF model has on our results. The models used are the Jansson & Farrar model (hereafter JF12) [@JF12_ord; @JF12_rand] and the Sun & Reich model (hereafter Sun10) [@Sun8; @Sun10], as updated recently by the Planck collaboration [@Planck]. Details on the models and parameter values are given in appendix \[sec:GMF\].
Both models, besides a regular component, include a random component, which is not explicitly specified. The only information given is its RMS value $B_{RMS} (\bm{r})$ as a function of the position. It is considered isotropic, i.e. it has equal probability to point in any direction. We model the random component using three random gaussian variables with a mean of 0 and a standard deviation of $B_{RMS}/\sqrt{3}$, representing the x, y and z components of the magnetic field. Note that this method does not enforce the requirement $\nabla \cdot \bm{B} = 0$ for the random field. A more detailed way to simulate the random field would be to use a Kolmogorov random field, as explained in [@Keivani_diss] and implemented in an updated version of the CRT numerical propagation code [@CRT]. However, this choice in implementation is expected to affect our results negligibly compared to the much more significant uncertainties in the overall GMF geometry itself, as we will see in section \[sec:Results\].
After calculating the magnetic field, we let time run backwards in steps of $c \, \delta t$ and, using eq. (\[eq:velocity\]), we calculate for each step the position and velocity of the cosmic ray, as it propagates in the cubic grid. The only information needed to reconstruct the cosmic ray trajectory, given the magnetic field, is the rigidity and arrival direction of the cosmic ray.
Eventually, as time runs backwards, the cosmic ray reaches a point where the strength of the GMF is negligible, and so there is no more Galaxy-induced deflection. The choice of a specific point to place such a cutoff is somewhat arbitrary. We have chosen to place our cutoff at a height of 10 kpc north or south of the Galactic plane and at a radius (in cylindrical coordinates) of 20 kpc from the Galactic center.
When the cosmic ray reaches this cutoff, we obtain the direction of its velocity vector, from which we infer the “true” original arrival direction of the particle before it entered the Galaxy. If the particle has not suffered significant deflections during its propagation in the intergalactic medium, this direction should be close to the direction of its source.
Reconstruction of the cosmic ray trajectory with uncertainties {#subsec:errors}
--------------------------------------------------------------
Having found the “true” trajectory and original arrival direction of the cosmic ray, we then simulate attempts to reconstruct the trajectory given hypothetical measurements of the GMF with some uncertainty. We perform this by repeating the above process of reconstruction, but this time introducing random errors to the “true” magnetic field values found using a GMF model. When the cosmic ray reaches a new grid cube during its backward propagation, the “measured” value of the magnetic field for that cube is calculated.
Errors are introduced separately for the POS magnitude, the LOS magnitude and the POS direction of the magnetic field. For the POS and LOS magnitudes, the “measured” values of the respective magnetic field component are considered to follow a lognormal distribution. The mean of this distribution is placed at the “true” value $B$ of the respective component received from the GMF model, and the standard deviation $\sigma_B$ is proportional to $B$ so that the ratio $\sigma_B/B$ stays the same for all grid cubes. This is implemented by multiplying the “true” value $B$ by a random lognormal variable $r$ with a mean of 1 and standard deviation $\sigma_r = \sigma_B/B$. For the POS direction, the “true” POS direction is rotated by a random angular variable that follows a von Mises distribution with a mean of zero and different values of the parameter $1/\sqrt{\kappa}$ (see appendix \[sec:vonMises\]), which, for large $\kappa$, approaches the standard deviation of a normal distribution.
Again, we obtain the direction of the velocity of the cosmic ray when it reaches the cutoff, from which we can infer the estimated original arrival direction. We then repeat the above process for a large number of iterations with different random errors.
Quality of the reconstruction
-----------------------------
To quantify how well the cosmic ray trajectory has been reconstructed given the uncertainty of our hypothetical GMF measurements, we calculate the post-correction residual angle $\phi_{res}$ between the “true” and the estimated original arrival direction. We also calculate the total deflection angle $\phi_{defl}$ of the cosmic ray, as the angle between the “true” original arrival direction and the arrival direction at the Earth. We then compare the two angles, introducing the “effectiveness coefficient”
$$a=\frac{\phi_{res}}{\phi_{defl}} .$$
This coefficient measures the effectiveness of our correction. Smaller values of $a$ mean a better correction. A value of $a=1$ means that the residual angle is equal to the deflection, and thus the correction is not helpful. A value of $a>1$ means that the residual angle is even larger than the deflection angle, which means that the correction has actually made matters worse.
During each iteration with different random errors, we collect the values of $\phi_{res}$ and $a$ to find their distribution. We do this for different choices of uncertainties and different initial conditions (arrival direction and rigidity).
Validation
----------
We have tested our code for the simple case of a uniform magnetic field. Further, we have successfully reproduced the deflection map for the regular field of the original JF12 model (Figure 11 of [@JF12_ord]), using, only in this occasion, the original parameters without the update from the Planck collaboration.
Additionally, we have performed a test of how the discretization of the magnetic field in cubes of uniform field with side 100 pc can affect the simulated propagation of the cosmic ray. Using the regular field of the (updated) JF12 model, we have simulated the propagation of a cosmic ray using three different discretization scenarios for the magnetic field. The first scenario is the same as before, using cubes of side 100 pc containing a uniform field. For the second scenario, we have doubled the cube side to 200 pc, expecting that this would make possible effects of the discretization more apparent. Finally, for the third scenario we did not use a cubic grid, but used the value for the magnetic field at the exact location of the cosmic ray during each 1-pc step of the numerical propagation, essentially using an almost continuous field.
We performed this test using 60 EeV particles (either protons or iron nuclei) for 10 arrival directions. To compare between the results of the three scenarios, we calculated the difference between the total deflections experienced by the cosmic ray in each scenario. The resulting differences between the first and third scenarios were usually less than 1% and the largest difference found was $\sim 3\%$. The differences between the second and third scenarios were in general larger, but still smaller than $\sim 6\%$, except for a single value that was at $\sim 21\%$. [^1] These results suggest that using cubes of side 100 pc containing a uniform magnetic field does not have a significant effect on the propagation, while using larger cubes could have a more pronounced effect.
Results {#sec:Results}
=======
Effectiveness Coefficient
-------------------------
First, we examine the dependence of the effectiveness coefficient $a$ on the uncertainty of the POS magnitude (figure \[fig:a\_POSmagn\]), the LOS magnitude (figure \[fig:a\_LOSmagn\]) and the POS direction (figure \[fig:a\_POSdir\]) of the magnetic field. Each of the three sources of uncertainty is examined independently. For example, when varying the uncertainty of the POS magnitude, the LOS magnitude and the POS direction are assumed to be known exactly. Details on what the uncertainty values on the horizontal axis of each figure represent can be found in §\[subsec:errors\].[^2]
![Median and 68% confidence interval of the effectiveness coefficient $a$ as a function of the uncertainty $\sigma_B/B$ of the POS magnitude of the magnetic field for different GMF models, nuclei and Galactic longitudes $l$. Particles of energy 60 EeV and Galactic latitude $b=40\degree$ are assumed. Lower values of $a$ indicate a better correction. []{data-label="fig:a_POSmagn"}](a_POSmagn){width="\textwidth"}
![As in figure \[fig:a\_POSmagn\], but for the uncertainty $\sigma_B/B$ of the LOS magnitude.[]{data-label="fig:a_LOSmagn"}](a_LOSmagn){width="\textwidth"}
![As in figure \[fig:a\_POSmagn\], but for the uncertainty $1/\sqrt{\kappa}$ of the POS direction.[]{data-label="fig:a_POSdir"}](a_POSdir){width="\textwidth"}
The values of $a$ were collected by using one specific realization of the GMF with 10000 iterations with different random errors. The points show the *median* of the values of $a$ received for each value of uncertainty and the error bars show the 68% confidence interval (i.e. the 68% range of values around the median). This choice was made, instead of using the typically used mean and standard deviation, because the values of $\phi_{res}$ and $a$ were found to follow a quite skewed distribution, as can be seen from the asymmetry of the error bars.
Our results are shown for two different values of the Galactic longitude and rigidity, and using two different GMF models. We assume particles with energy $E=60 \ {\rm EeV}$, charge $Z e$ and arrival direction of Galactic longitude $l$ and Galactic latitude $b=40 \degree$.
Accuracies of $\sigma_{B}/B \leq 2$ are shown in figure \[fig:a\_POSmagn\] because, for larger uncertainties, blue points (arrival directions away from the Galactic center) generally exhibit prohibitively high effectiveness coefficients for any meaningful arrival direction corrections. The same range of accuracies is shown in figure \[fig:a\_LOSmagn\], as the behavior of the effectiveness coefficient $a$ demonstrated for this range does not change abruptly for higher values of $\sigma_{B}/B$.
As expected, for small deflections (as is the case for 60 EeV protons), the sensitivity to the LOS magnetic field is very small, as the particle velocity stays approximately parallel to the LOS throughout its trajectory. In contrast, for larger deflections (as happens for 60 EeV iron), a significant POS velocity component is present away from the Earth, making the correction sensitive to the LOS uncertainty.
The figures show that a quite good correction is often feasible in the range of uncertainties that we have investigated. There are much better prospects of correction if the cosmic rays are protons, compared to iron nuclei of the same energy. There is also significant difference between arrival directions pointing towards the Galactic center ($l=0\degree$) and away from the Galactic center ($l=180\degree$). Cosmic rays with arrival directions closer to the Galactic center, where the deflection is larger, are good candidates for a better correction. Differences between the two GMF models can be very pronounced. For example, in the case of iron with $l=0\degree$, the JF12 model suggests that a good correction can be made, while the Sun10 model suggests that it is very difficult to make a correction.
It is also apparent that some plots for iron have a more “irregular” form than for protons. This seems to happen in general for large deflections $(\gtrsim 90\degree)$ and especially when $a$ is also large. That is because $a$ and $\phi_{res}$ do not capture the actual details during the process of the trajectory reconstruction. For example, for some uncertainty value, the cosmic ray might statistically tend to go through a region of large magnetic field during the reconstruction, which will change its trajectory.
It should be noted that, while, as expected, the effectiveness coefficient (and so the residual angle) does decrease to zero as the uncertainty approaches zero, in the case of 60 EeV iron this decrease starts to happen at very low uncertainties. For example, using the JF12 model, an arrival direction with $l=180 \degree$ and $b=40 \degree$, and POS magnitude uncertainty of $\sigma_B/B \approx 10^{-3}$, the median of the effectiveness coefficient remains $a \approx 0.2$, while for $\sigma_B/B \approx 10^{-4}$, the median is $a \approx 0.01$.
Some of the figures show that it is in some cases possible for $a$ to exceed the value of 1. This means that the residual angle becomes larger than the deflection. In that case, instead of being corrected, the deflection has been amplified. A useful question, then, is to ask what the probability of getting $a<1$ and thus managing to shrink the deflection is. This question is approximately answered in figure \[fig:p100\], where the percentage of the values of $a$ which satisfy $a<1$ is given for the same cases as before.
![Percentage of cases with effectiveness coefficient $a < 1$, for the same parameters and choices of uncertainties as those in figures \[fig:a\_POSmagn\], \[fig:a\_LOSmagn\] and \[fig:a\_POSdir\]. []{data-label="fig:p100"}](Percentage100){width="93.00000%"}
The results are quite sensitive to the realization of the random magnetic field. Thus, figures \[fig:a\_POSmagn\]-\[fig:p100\] are only meant to showcase some general features, while the actual values of $a$ shown can vary significantly between different realizations.
To give an example of how the random field can affect our results, we have repeated the above correction process for 100 realizations of the random field, each with 500 iterations with different random errors. In figure \[fig:a\_histogram\] we present histograms showing the distribution of the median of the effectiveness coefficient $a$ for two different cases. Both cases assume $60 \ \rm{EeV}$ protons arriving from a Galactic latitude $b=40\degree$ and an uncertainty of $\sigma_{B}/B=1$ only for the POS magnitude of the GMF. The JF12 model is used. In the first case (left panel) the arrival direction has a Galactic latitude $l=0\degree$ and in the second case (right panel) $l=180\degree$. These results can be compared with the top left panel of figure \[fig:a\_POSmagn\] for $\sigma_B/B=1$. These distributions can change depending on the rigidity, arrival direction, uncertainty of the magnetic field values and on how dominant the random field is throughout the cosmic ray trajectory compared to the regular field.
Sky maps
--------
To showcase how the effectiveness of the correction depends on the arrival direction of the cosmic ray, we have constructed sky maps in Mollweide projection of the median of the deflection, the residual angle and the effectiveness coefficient. These are shown in figure \[fig:sky\_map\]. The results are for 60 EeV protons and uncertainties of $\sigma_B/B=1$ for the POS and LOS magnitudes and $1/\sqrt{\kappa}=40 \degree$ for the POS direction, using both the JF12 and Sun10 models of the GMF. [^3] To create these maps, we performed simulations for 2520 arrival directions in total, covering the whole sky, with the Galactic longitude ranging from $-180 \degree$ to $175 \degree$ with a step of $5\degree$, and the Galactic latitude from $-85 \degree$ to $+85 \degree$, again with a $5\degree$ step.
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In this case, the correction process was repeated for 100 realizations of the magnetic field and 50 iterations with different random errors for each realization. Values for the deflection, the residual angle and the effectiveness coefficient were collected during each iteration and the median of all collected values was calculated. Thus, here the median is over both the different random field realizations and the iterations with different random errors (of course, the deflection changes only with the different realizations). This way, we can get a sense of the “general” behaviour of each GMF model, independent of the specific realization of the random field.
Several features can be observed from these sky maps. First, it is apparent that there are significant differences between the two models, as seen most dramatically in the deflection sky maps. The Sun10 model predicts in general larger deflections than the JF12 model, while the location of regions of large deflection also differs. Next, the residual angle tends to be larger for arrival directions near the Galactic center, which is expected, as the deflection is also larger. Also, looking at the sky maps of the effectiveness coefficient $a$, we can see that, to a large extent, their form can be predicted from the respective deflection maps. Arrival directions with larger deflection have lower $a$.
Discussion
==========
We have found that there are important differences between the two GMF models used and also that the random component can influence significantly the effectiveness of the correction. This showcases the fact that our knowledge of the GMF is still quite limited, at least for the purposes of reconstructing the trajectory of UHECRs, and highlights the importance of making actual 3D measurements of the GMF. Our most significant differences between different models arise in the southern Galactic hemisphere, in agreement with the results of [@uncertainty], who have recently studied the uncertainties involved in the parametrization of the GMF. After an experiment has been carried out to make such measurements, a similar analysis to that presented in this paper can be performed using the specific measurements and error distribution of that experiment, in order to estimate more accurately the uncertainty of any backtracing.
It should be noted that, as reported in [@Deflections], which provides a detailed study of the magnetic deflections of UHECRs using the JF12 model, the deflections are found to be sensitive to the coherence length of the random component of the GMF. Thus, the coherence length that we have assumed in our simulations might have influenced our results for the effectiveness of the correction as well. It is also possible that the coherence length presents large variations throughout the Galaxy, while we have assumed it to be the same everywhere in the Galaxy.
It is important to stress that our work has assumed random errors within each grid cube, which are only realizable in a bona-fide tomographic experiment rather than the currently-available parameter-fitting modeling. As such, we strongly caution the reader that our results cannot be regarded as predictive of the effectiveness of deflection corrections using current GMF models.
For completeness, we have assumed that tomographic measurements of the GMF are available everywhere in the sky. However, any specific experiment might yield measurements for limited parts of the sky only. In addition, we have used a spatial resolution of 100 pc as a reasonable choice for the coherence length of the Galactic magnetic field, as our purpose here is to examine the effect of measurement uncertainties alone, assuming that the sampling is dense enough to provide an adequate/complete set of measurements. Additional uncertainties in the de-propagation may, thus, be introduced by incomplete sampling along the cosmic ray trajectory.
Finally, we found that the effectiveness of the correction is likely to be high for light cosmic ray composition. Therefore, given relatively accurate measurements of the GMF, the process of correction can be used as a composition probe. If no increased clustering is found assuming proton or light nuclei composition, where a source and good correction can be reasonably expected, then this result can be interpreted as evidence for a heavy composition.
Summary and Conclusions
=======================
In our present work, we have focused on the possibility of correcting the magnetic deflection of UHECRs using 3D measurements of the GMF. We have attempted to examine how the effectiveness of such a correction depends on the uncertainty of our measurements and how it is influenced by the rigidity and arrival direction of the cosmic ray, assuming that these are perfectly known. To that end, we have constructed a numerical code, which simulates attempts to make such corrections using hypothetical measurements of the GMF, based on two recently updated GMF models, with varying uncertainty for the POS magnitude, the LOS magnitude and the POS direction of the magnetic field.
Our results highlight the conditions under which an effective correction is achievable. A better correction can be achieved in general if the UHECRs are of high rigidity (ideally if they are high-energy protons) and if they have arrival directions that imply a larger deflection, which can be more easily corrected. In the case of ultra-high-energy protons, knowledge of the magnitude of the LOS component of the magnetic field is unnecessary, while in the case of iron nuclei, it becomes of similar importance to the POS component.
The significant differences in predictions using different GMF models underlines the sensitivity of our results to the actual GMF geometry. This however will be much better constrained should a 3D tomographic GMF mapping become available.
We wish to thank Kostas Tassis and Aris Tritsis for valuable discussions that helped to improve the present work.
Galactic Magnetic Field Models {#sec:GMF}
==============================
Here we briefly present the GMF models that we have used in our code. These are the Jansson & Farrar model (JF12) [@JF12_ord; @JF12_rand] and the Sun & Reich model (Sun10) [@Sun8; @Sun10]. Both of these models have recently been updated by the Planck Collaboration [@Planck], and these updates, which will also be mentioned, have been implemented into our code.
The parameters in the formulas below that are not explained are free parameters of the models, and the values that we have used for each of them (based on the original papers and the Planck update) can be found in the respective tables.
JF12
----
The JF12 magnetic field consists of a large-scale regular component, a random component and a “striated” component, often mentioned in other models as the “ordered random” component. Our choice of parameters is based on the updated model called “Jansson12b” in the Planck update.
### Regular Component
The regular field for this model consists of a disk component, a toroidal halo component and an out-of-plane component, also referred to as the “X-field” component. The parameter values used in our code for the regular component can be found in table \[tab:JF12 regular\]. This table also shows which of these parameter values have been updated by the Planck Collaboration and which have not.
The coordinates used are Cartesian $(x,y,z)$, as well as cylindrical $(r,\phi ,z)$, where the Galactic center is placed at the origin, the Galactic plane lies on the $x-y$ plane, and the Sun is placed at $x=-8.5 \ {\rm kpc}$ and $y=0$. The Galactic North is towards the positive z-direction.
The disk field is defined for $3 \ {\rm kpc} \leq r \leq 20 \ {\rm kpc}$. For $3 \ {\rm kpc} \leq r \leq 5 \ {\rm kpc}$, there is a “molecular ring”, which has a purely azimuthal field $\bm{b} = b_{ring} \hat{\bm{\phi}}$. For larger $r$, eight logarithmic spiral regions are defined, with boundaries obeying the equation
$$r=r_{-x}\exp \left[(\phi-\pi) \tan i \right] ,$$
where $i=11.5 \deg$ is the opening angle of the spirals and $r_{-x}$ is the radius where each spiral crosses the negative $x$-axis.[^4] We take $r_{-x}= \lbrace 5.1, 6.3, 7.1, 8.3, 9.8, 11.4, 12.7, 15.5 \rbrace {\rm kpc} $. The disk field is then given by the formula
$$b_{disk}= (1-L(z,h_{disk},w_{disk})) \cdot b_i \cdot 5 \ {\rm kpc} / r ,$$
where $b_i$ is the field strength of the i-th spiral region at $r=5\ {\rm kpc}$, and
$$L(z,h,w)=(1+\exp \left( -2(\vert z \vert - h)/w\right))^{-1}$$
is a logistic function used to describe the transition from the disk field to the halo field. The field direction is $\hat{\bm{b}}_{disk}= \sin i \ \ru + \cos i \ \phiu $.
The toroidal halo field, as its name implies, has a purely azimuthal component. For $z>0$ (north half of the halo), the field is
$$B_{\phi}^{tor} = \exp \left(-\vert z \vert /z_0 \right) L(z,h_{disk},w_{disk}) B_n (1-L(r,r_n,w_h)) ,$$
while for $z<0$ the parameters $B_n$ and $r_n$ are substituted by the different parameters $B_s$ and $r_s$ respectively.[^5] Thus, the field strength and radial extent of the halo field are different for the north and south halves of the halo.
Finally, the “X-field” component is axisymmetric and purely poloidal (i.e. it does not have an azimuthal component). Thus the field changes only in the $(r,z)$ plane, remaining the same for any angle $\phi$. The field lines are straight lines, which run from the southern direction towards the z-axis with a certain “elevation angle” $\Theta_X$ with the $z=0$ plane, and, when they cross the $z=0$ plane, they run away from the $z$-axis and towards the northern direction with the same angle $\Theta_X$. The radius at which a field line crosses the $z=0$ plane is defined as $r_p$. To every point $(r,z)$ corresponds one field line, and therefore one radius $r_p$. If $r_p$ is larger than a certain radius $r_X^c$, the field lines are taken to have a constant elevation angle $\Theta_X^0$. If $r_p < r_X^c$, then the elevation angle increases as $r_p$ approaches zero, and reaches the value $\Theta_X = 90 \degree $ at $r_p=0$.
The strength of the magnetic field at the $z=0$ plane is defined as
$$b_X (r_p)=B_X \exp (-r_p/r_X) ,$$
where $B_X$ and $r_X$ are free parameters of the model. With the above definitions and geometry, the requirement $\nabla \cdot \bm{B} = 0$ gives the following formulas for the magnetic field.
When
$$r_p \geq r_X^c \Leftrightarrow r \geq r_X^c + \frac{z}{\tan{\Theta_X^0}},$$
the magnetic field strength is
$$b_X(r)= b_X(r_p) \cdot r_p/r ,$$
with
$$r_p=r-\frac{\mid z \mid}{\tan \Theta_X^0} .$$
When $r_p < r_X^c$, the magnetic field strength is
$$b_X(r)=b_X(r_p) \left(r_p/r\right) ^2,$$
with
$$r_p=\frac{r \ r_X^c}{r_X^c + \mid z \mid / \tan \Theta_X^0 } ,$$
and the elevation angle changes as
$$\Theta_X = \tan ^{-1} \left(\frac{\mid z \mid }{r -r_p} \right) .$$
\[h!\]
Parameter Value Updated?
-------------- -------------- ----------
$b_1$ 0.1 $\mu$G No
$b_2$ 3.0 $\mu$G No
$b_3$ -0.9 $\mu$G No
$b_4$ -0.8 $\mu$G No
$b_5$ -2.0 $\mu$G No
$b_6$ -3.5 $\mu$G Yes
$b_7$ 0.0 $\mu$G No
$b_8$ 2.7 $\mu$G No
$b_{ring}$ 0.1 $\mu$G No
$h_{disk}$ 0.4 kpc No
$w_{disk}$ 0.27 kpc No
$B_n$ 1.4 $\mu$G No
$B_s$ -1.1 $\mu$G No
$r_n$ 9.22 kpc No
$r_s$ $>$ 16.7 kpc No
$w_h$ 0.2 kpc No
$z_0$ 5.3 kpc No
$B_X$ 1.8 $\mu$G Yes
$\Theta_X^0$ 49$\degree$ No
$r_X^c$ 4.8 kpc No
$r_X$ 2.9 kpc No
: Parameter values used for the JF12 regular field.[]{data-label="tab:JF12 regular"}
### Random Component
The random component is taken to be isotropic and its root mean square (RMS) value has different contributions from the disk and the halo. Parameter values can be found in table \[tab:JF12 random\].
The disk random field is given by $$B_{disk}^{rand} = f(r) \cdot e^{-z^2/2 (z_0^{disk})^2}$$ where
$$f(r) = \left\{\begin{array}{lr}
b_{int}, & \text{for }r < 5 \ {\rm kpc} \\
b_i \cdot \frac{5 \ {\rm kpc}}{r}, & \text{for } r \geq 5 \ {\rm kpc}
\end{array}\right\}$$
The value of the parameter $b_i$ is different in each spiral region defined as in the regular field (note that this parameter is different from the regular component parameter of the same name; we understand that this choice of notation might be somewhat confusing, but we have decided to retain the notation of the original papers).
The halo field is
$$B_{halo}^{rand}= B_0 e^{-r/r_0} e^{-z^2/2 z_0^2}.$$
The total random field strength is then taken to have the RMS value
$$B_{rand}= \sqrt{(B_{disk}^{rand})^2 + (B_{halo}^{rand})^2} .$$
To implement this field in our code, we use three random gaussian variables with a mean of 0 and a standard deviation of $B_{rand}/ \sqrt{3}$ , which represent the x, y and z components of the magnetic field. This choice satisfies the requirements that the field is isotropic and that the RMS value for the magnetic field strength is $B_{rand}$.
\[h!\]
Parameter Value Updated?
------------------------------ ----------------------------------- ----------
$\langle B_{iso}^2 \rangle $ 5 $\mu$G Yes
$b_{even}$ 0.8 $\langle B_{iso}^2 \rangle $ Yes
$b_{odd}$ 0.4 $\langle B_{iso}^2 \rangle $ Yes
$b_{int}$ 0.5 $\langle B_{iso}^2 \rangle $ Yes
$z_0^{disk}$ 0.61 kpc No
$B_0$ 0.94 $\langle B_{iso}^2 \rangle $ Yes
$r_0$ 10.97 kpc No
$z_0$ 2.84 kpc No
: Parameter values used for the JF12 random field.[]{data-label="tab:JF12 random"}
### Striated Component
The JF12 model also includes a “striated” component, often called in other models the “ordered random” component. A striated field is thought to be produced when an isotropic random or perhaps a coherent field experiences stress or shear. It is assumed to have a preferred large-scale direction parallel to the regular field, but experiences sign reversals on a small scale, thus its average value is zero.
We have neglected this component in our code. A discussion of its effects on UHECR propagation can be found in [@Deflections_Centaurus] and a specific way to implement it is mentioned in [@Keivani_diss].
Sun10
-----
The paper presenting the Sun10 model actually contains three distinct regular field models, as well as a simple treatment for the random component. We focus on the model that has been updated by the Planck Collaboration, which is the ASS+RING model. ASS stands for axi-symmetric spiral and RING refers to its structure, as it uses concentric rings in which the sign of the field is reversed. Throughout this paper, the updated ASS+RING model is referred to simply as Sun10. In the Planck update, a more detailed random component, as well as an ordered random field (the equivalent of the striated field in the JF12 model) have been added to this model. As in the JF12 model, we have neglected the ordered random component.
### Regular Component
The regular component of the Sun10 model is significantly simpler than that of the JF12 model. It includes a disk component $\bm{B}^D$ and a halo component $\bm{B}^H$. Parameter values can be found in table \[tab:Sun10 regular\].
The disk component is written in cylindrical coordinates $(R,\phi,z)$ with the Galactic centre at the origin as:
$$\begin{aligned}
B_R^D &= D_1(R,\phi,z) D_2(R,\phi,z) \sin p \\
B_{\phi}^D &=-D_1(R,\phi,z) D_2(R,\phi,z) \cos p \\
B_z^D &=0\end{aligned}$$
where
$$D_1(R,z) = \left\{\begin{array}{lr}
B_0 \exp \left(-\frac{R-R_{\bigodot}}{R_0} - \frac{\mid z \mid}{z_0} \right), & \text{for } R>R_c\\
B_c, & \text{for } R \leq R_c
\end{array}\right\}$$
with $R_{\bigodot}=8.5 \ {\rm kpc}$ being the Galactic radius of the Sun, and
$$D_2(R) = \left\{\begin{array}{lr}
+1, & \text{for } R>7.5 \ {\rm kpc} \\
-1, & \text{for } 6 \ {\rm kpc} < R \leq 7.5 \ {\rm kpc}\\
+1, & \text{for } 5 \ {\rm kpc} < R \leq 6 \ {\rm kpc} \\
-1, & \text{for } R \leq 5 \ {\rm kpc}
\end{array}\right\}.$$
The halo field is purely azimuthal and is written as:
$$B_{\phi}^H (R,z) = \text{sign} (z) \ B_0^H \frac{1}{1+ \left( \frac{\mid z \mid - z_0^H}{z_1^H} \right) ^2} \frac{R}{R_0^H} \exp \left( - \frac{R-R_0^H}{R_0^H} \right).$$
\[h!\]
Parameter Value Updated?
----------- -------------------------- ----------
$R_0$ 10 kpc No
$z_0$ 1 kpc No
$R_c$ 5 kpc No
$B_0$ 2 $\mu$G No
$B_c$ 0.5 $\mu$G Yes
$z_0^H$ 1.5 kpc No
$z_1^H$ 0.2 kpc, for $|z|<z_0^H$ No
0.4 kpc, otherwise No
$B_0^H$ 10 $\mu$G No
$R_0^H$ 4 kpc No
: Parameter values used for the Sun10 regular field.[]{data-label="tab:Sun10 regular"}
### Random Component
The random component in the updated Sun10 model has an RMS strength that can be written as:
$$B_{RMS} (R,z) = \langle B_{iso}^2 \rangle ^{1/2} f(R) g(z)$$
where
$$f(R)= \exp \left(- \frac{R-R_{\bigodot}}{r_0^{ran}} \right) ,$$
and
$$g(z)= (1-f_{disk}^{ran}) {\rm sech} ^2 \left( \frac{z}{h_{halo}^{ran}} \right)
+ f_{disk}^{ran} {\rm sech} ^2 \left( \frac{z}{h_{disk}^{ran}} \right) .$$
Again, to model this isotropic random field in our code, we use three random gaussian variables with a mean of 0 and a standard deviation of $B_{RMS}/\sqrt{3}$. Parameter values can be found in table \[tab:Sun10 random\].
\[h!\]
Parameter Value Updated?
----------------------------- ------------ ----------
$\langle B^2_{iso} \rangle$ 4.8 $\mu$G Yes
$r_0^{ran}$ 30 kpc Yes
$f_{disk}^{ran}$ 0.5 Yes
$h_{halo}^{ran}$ 3 kpc Yes
$h_{disk}^{ran}$ 1 kpc Yes
: Parameter values used for the Sun10 random field.[]{data-label="tab:Sun10 random"}
The von Mises distribution {#sec:vonMises}
==========================
To simulate a random angular variable, one needs to use a circular distribution. The generalization of the normal distribution to circular variables is called the wrapped normal distribution. In our simulation, we have chosen to use the von Mises distribution, which involves simpler computations and closely approximates the wrapped normal distribution.
A random angular variable $\Theta$ obeying the von Mises distribution $VM(\mu,\kappa)$, has a probability density function of the form
$$f(\theta) = \frac{1}{2 \pi I_0(\kappa)} e^{\kappa \cos(\theta - \mu)} ,$$
where $I_0$ is the modified Bessel function of the first kind and order 0, $\mu$ is the mean of the distribution, and $\kappa$ is the “concentration parameter” [@Dir_stat]. For large $\kappa$ the von Mises distribution approaches a normal distribution with standard deviation $1/\sqrt{\kappa}$.
In our code, the von Mises distribution is simulated using the method of [@von_Mises].
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[^1]: The 21% difference was found for an arrival direction with Galactic longitude $l=90\degree$ and Galactic latitude $b=-40\degree$, assuming iron composition, which in general means larger deflections than protons, and thus larger differences between scenarios are to be expected.
[^2]: We should stress that the quantity $1/\sqrt{\kappa}$ used in figure \[fig:a\_POSdir\] is only a useful measure of the uncertainty and does not in general represent the actual standard deviation of the distribution. Only for small uncertainties, where the von Mises distribution approaches a gaussian, does this quantity approximate the standard deviation.
[^3]: For the JF12 model, the parameter $r_s$, which describes the toroidal halo field for the southern Galactic hemisphere (see appendix \[sec:GMF\]), remains unspecified as only a lower limit is given to be 16.7 kpc. In our simulation, we have considered its value to be equal to that lower limit. While the previous results are not affected by the choice of $r_s$, because they concern the northern Galactic hemisphere, these results are somewhat affected. After constructing the same sky maps using the highest possible value of 20 kpc, it was found that the residual angle and deflection angle were slightly affected in the southern hemisphere, but the effectiveness coefficient remained relatively unchanged, at least for the choice of uncertainties and rigidity tested.
[^4]: Note typo in the original paper. Also, we have changed the origin of the angle $\phi$ so that $\phi = \pi$ at the negative x-axis.
[^5]: As can be seen in table \[tab:JF12 regular\], the parameter $r_s$ is not specified, but only a lower limit is given. In our code we have chosen $r_s$ to be equal to this lower limit.
|
---
abstract: 'We study the black hole shape using the C-metric solution for a matter trapped near the IR-brane in the $\mbox{AdS}_4$ space. In the AdS/CFT duality, the IR-brane is introduced by embedding a 2-brane at the $\mbox{AdS}_4$ radius $z=l$, while the UV-brane is defined by the $\mbox{AdS}_4$ boundary. We find the C-metric solution generates a negative tension IR-brane and a negative thermal energy gas of colliding particles. We analyze the momentum energy tensor at the $\mbox{AdS}_4$ boundary. We find that the negative energy black hole solution is entirely unstable even for a small perturbation in the Poincare coordinate space. However, such a black hole decays very rapidly due to the imaginary part emerges in the ADM mass. This imaginary part appears because of the orbifold constraints of the C-metric solution in the unusual coordinates. Moreover, this decay rate diverges at the UV-brane. This implies that the black hole evaporates instantaneously otherwise the boundary itself will collapse.'
author:
- 'I. Zakout'
title: 'The C-metric black hole near the IR-brane in the $\mbox{AdS}_4$ space'
---
Introduction
============
The production of black holes in the higher dimensions in the heavy ions collisions physics has received much attention recently[@Giddings2; @Dimopolous1]. It might be the end of short-range physics. In the Randall-Sundrum scenario[@Randall1; @Randall2], the 4-dimensional world is a brane embedded in a higher dimensional space. The black hole can be produced in the brane and the bulk as well. However, the black hole solution for a matter trapped in the 3-brane embedded in AdS$_{5}$ is very important to shed some information about the black hole production in the QCD[@Polchinski1]. It has been argued the supergravity solution with warped metric that approximately AdS in large region corresponds to the high energy scattering in a large-N ${\cal N}$ supersymmetric gauge theory with broken conformal symmetry and partial broken supersymmetry[@Maldacena1].
Unfortunately, the nonlinear gravitational solution of the black hole in $\mbox{AdS}_{5}$ is not available. Giddings[@Giddings1] has studied a linearized solution of the black hole near the IR-brane, where the brane is the end of space. He found the black hole shape saturates the Froissart bound cross section. When the energy increases, the black hole summit reaches smaller values of $z$ further toward the AdS boundary. They will, however, be completely smeared out over the compact manifold $X$. At this energy, the approximation of these black holes in a background flat space breaks down. Therefore, the nonlinear solution becomes essential to describe the shape of such black hole. The analysis of a lower dimensional world might give some hints what could be happened in the higher dimensional world. Emparan, Horowitz and Myers [@Emparan1] used the C-metric solution of two accelerated black holes to analyze the exact solution of black hole near the 2-brane embedded in $\mbox{AdS}_4$ in the Randall-Sundrum scenario[@Randall1; @Randall2]. In their treatment, they produced the 2-brane by removing the space between the brane and the $\mbox{AdS}_4$ boundary. The full space is completed by gluing the space between the 2-brane and the $\mbox{AdS}_4$ horizon in both sides of the 2-brane.
To study the black hole shape near the IR-brane, unlike to Ref.[@Emparan1], the infrared 2-brane is embedded by removing the space between the brane and $\mbox{AdS}_4$ horizon and then gluing the space from the IR-brane to the $\mbox{AdS}_4$ boundary in both sides of the 2-brane to complete the space. The 2-brane and $\mbox{AdS}_{4}$ boundary become the IR-brane and UV-brane, respectively, in the dual AdS/CFT. Although, the present problem has been mentioned shortly in Ref.[@Emparan1], we revisit it more thoroughly in the present work to study the stability of such configuration.
The outline is that. In sect. II, we present the analysis of the black hole solution near the IR-brane. At first, we present the orbifold constraints. Then we show that the produced thermal gas has a negative thermal energy that diverges at $\mu_{\mbox{max}}$. Finally, we calculate the asymptotic black hole ADM mass at the boundary. The conclusion is presented in sect. III.
The C-metric Black hole near $\mbox{AdS}_4$ IR-brane
====================================================
The $\mbox{AdS}_{n+1}$ metric in a large region reads $$\begin{aligned}
ds^2\approx \frac{l^2}{z^2}\left[ dz^2+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right]
+l^2ds^2_{X},\end{aligned}$$ where $l$ is the $\mbox{AdS}_{n+1}$ radius and $X$ is some appropriate compact manifold. At long distance, the smooth geometry given above is truncated in the infrared and an infrared ($n$-1)-brane is embedded at this end of space at $z=l$. The UV-limit of the geometry is the $\mbox{AdS}_{n+1}$ boundary at $z=0$. The action in the ($n$+1)-dimensional gravity reads $$\begin{aligned}
I=M^{n-1}_P\int d^n x dz \sqrt{-\hat{g}}\left(R-\Lambda/M^{n-1}_P\right)
-\int d^n x \sqrt{-g_{\mbox{brane}}} T_{n-1}
+\int d^n x dz{\cal L}_{\mbox{matter}},\end{aligned}$$ where $M^{n-1}_P=\frac{1}{16\pi G_{n+1}}$. The variation of this action gives Einstein’s equations, $$\begin{aligned}
\sqrt{-\hat{g}}(R_{MN}-\frac{1}{2}\hat{g}_{MN}R)=&-&\frac{1}{2M^{n-1}_P}\left[\sqrt{-\hat{g}}\hat{g}_{MN}\Lambda
-\sqrt{-g_{\mbox{brane}}}
\hat{g}_{\mu\nu}\delta^{\mu}_M\delta^{\nu}_N \delta(z-l)T_{n-1}\right. \nonumber\\
&-&\left.\sqrt{-\hat{g}}T^{\mbox{matter}}_{MN}\delta^{n-1}(x)\delta(z-l)
\right],\end{aligned}$$ where $T^{\mbox{matter}}_{MN}$ the momentum energy tensor for the source of matter trapped at the ($n$-1)-brane. The Einstein’s equations have a delta function $\delta^{n-1}(x)\delta(z-l)$ because of the source of matter localized on the brane and another delta function $\delta(z-l)$ because of the IR-brane. To deal with these two delta functions, the exact solution needs a very sophisticated Green function tools. Nonetheless, it is thought that the problem can be simplified drastically by finding a solution for the black hole in the $\mbox{AdS}_{n+1}$ with a source of matter localized at $\delta^{(n-1)}(x)\delta(z-l)$ and then embedding the ($n-1$)-brane by somehow. Unfortunately, even in this simplified version without any IR-brane, it is hard to find a stationary solution in the Poincare coordinates.
There is a solution called the AdS C-metric solution[@Plebanski1], which satisfies the Einstein’s equation with a negative cosmological constant in 3+1 space. The C-metric solution reads, $$\begin{aligned}
ds^{2}=
\frac{l^2}{(y-x)^2}
\left[-U(y)dt^2+\frac{dy^2}{U(y)}+\frac{dx^2}{G(x)}+G(x)d\phi^2\right],\end{aligned}$$ where $$\begin{aligned}
U(y)=y^2(1-2\mu y),\end{aligned}$$ and $$\begin{aligned}
G(x)=1-x^2(1-2\mu x). \end{aligned}$$ It is invariant under translations of $t$ and $\phi$. The metric solution is written in the unusual coordinates system $x$ and $y$ and satisfies the Einstein’s equation with a negative cosmological constant $R_{AB}=-(3/l^2)g_{AB}$.
However, to understand this solution, we have to analyze the orbifold constraints of these coordinates. It is seemed that $y$ acts as a radial variable, while $x$ is analogous to a polar coordinate. The range limits for the unusual variables $x$ and $y$ are crucial to characterize the shape of black hole. The factor $(y-x)^{-2}$ in front of the metric implies that $y=x$ is infinitely far away from points with $y\ne x$ and corresponds to the UV-limit in the dual AdS/CFT. Furthermore, there is another horizon at $y=0$ which is degenerate and has a zero Hawking temperature. This adds an additional constraint given by $0\le y\le \infty$. The black hole horizon is found at $y_{\mbox{BH}}=\frac{1}{2\mu}$. The range, $y_{\mbox{BH}}\le y \le\infty$, is the domain of the gravitational collapse. Hence, the allowed interval outside the black hole horizon is given by $0\le y\le y_{\mbox{BH}}$.
The function $G(x)$ plays an important rule to determine the interval for the $x$-polar orbifold and the periodicity for the $\phi$-azimuthal angle. It should be a positive definite in order the metric has a Lorentz signature. For $\mu=0$ case, there is only two roots $x_0=-1$ and $x_1=1$. The interval for $x$-polar orbifold is defined by $x_0<x<x_1$. When the 2-brane is embedded at $x=0$, it acts as the IR-boundary limit of the geometry. Hence the space will be defined only for $x\ge 0$. Since the space for $x\le 0$ will be discarded, we shall focus the discussion for the positive root $x_1$. However, to avoid the canonical singularity at $x=x_1$, the metric can be written as $$\begin{aligned}
ds^2=\left[\frac{1}{G(x)}dx^2+G(x)d\phi^2\right]_{x=x_1}
\rightarrow d\lambda^2+\lambda^2 d\left(\frac{\phi^2}{a^2_{\mu}}\right),\end{aligned}$$ where $$\begin{aligned}
a_{\mu}=\frac{2}{|G'(x_1)|}.\end{aligned}$$ Hence periodicity for $\phi$-azimuthal angle becomes $$\begin{aligned}
\Delta \phi=2\pi a_{\mu} \label{deficit1}.\end{aligned}$$ This periodicity is $2\pi$ for $\mu=0$. When $\mu$ becomes finite but small the function $G(x)$ has three roots $x_0$, $x_1$ and $x_2$. To keep the Lorentz identity signature, the orbifold interval for the $x$-polar is determined by the roots $x_0$ and $x_1$. The root $x_2$ is discarded from the discussion since it is not needed any more. As far $\mu$ increases, the root $x_1$ increases as $1\le x_1\le \sqrt{3}$ for $\mu\le \mu_{\mbox{max}}$ where $\mu_{\mbox{max}}=\frac{1}{3\sqrt{3}}$. However, when $\mu$ exceeds $\mu_{\mbox{max}}$, the root $x_1$ disappears and the function $G(x)$ becomes a positive definite for all positive $x$-orbifold $x\ge 0$. The periodicity $\Delta \phi$ increases and exceeds $2\pi$ as $\mu$ increases and finally diverges at $\mu_{\mbox{max}}$. This means that $\mu_{\mbox{max}}$ is the upper limit for the parameter $\mu$. Furthermore, the point $x=y$ is always located outside the black hole horizon for $\mu\le\mu_{\mbox{max}}$ because of $y_{\mbox{BH}}>x_1$.
To embed the infrared 2-brane at $x=0$, we follow the procedure given in Ref[@Emparan1] but instead we cut off the space between IR-boundary and the $\mbox{AdS}_4$ horizon and then complete the space by gluing onto the surface at $x=0$ a mirror copy of the space extended from the IR-boundary to the $\mbox{AdS}_4$ boundary. The procedure is illustrated by pseudo-Penrose diagram displayed in Fig.\[fig2\]. To be more precise, we take a time-like three-surface $\Sigma$ whose extrinsic curvature is proportional to its intrinsic metric to be the surface at $x=0$ (or $z=l$) and the resulting space-time will be free of conical singularities for the region $x\ge 0$. Then we take two copies of this side of the space-time and glue them together along $\Sigma$. The C-metric becomes continuous but not differentiable around $x$. The extrinsic curvature at $x=0$ is given by[@Emparan1], $$\begin{aligned}
K_{\mu\nu}=\frac{1}{2} n^{\sigma}\partial_{\sigma} \hat{g}_{\mu\nu}|_{x=0}&=&
-\frac{(y-x)\sqrt{G(x)}}{2l} \frac{\partial \hat{g}_{\mu\nu}}{\partial x}|_{x=0}
\nonumber\\
&=&-\frac{1}{l}\hat{g}_{\mu\nu}|_{x=0}.\end{aligned}$$ The discontinuous in the extrinsic curvature is interpreted as a delta-function $\delta(x)$ (or $\delta(z-l)$) source of stress-energy. The Poincare coordinate $z$ will be given below. It is interpreted as a thin relativistic $(n-1)$brane embedded at $x=0$ with negative tension $T_{n-1}=-\frac{n-1}{4\pi G_{n+1} l}$ where $G_{n+1}$ is the Newton’s constant in $n+1$ dimensions. This means the brane is expanding out and this is opposed to the physical case where the brane is supposed to has a positive tension. The orbifold constraints for the unusual coordinates $x$, $y$ and $\phi$ read $$\begin{aligned}
0\le &x&\le x_1, \nonumber \\
0\le &y&\le y_{\mbox{BH}}=\frac{1}{2\mu}, \nonumber \\
0\le &\phi& \le 2\pi a_\mu.\end{aligned}$$ Note that constraint $x\ge 0$ appears because of the 2-brane is embedded at $x=0$. There is an additional constraint for a finite matter trapped near the infrared 2-brane $$\begin{aligned}
\mu\le \frac{1}{3\sqrt{3}}. \end{aligned}$$
![\[fig2\] Pseudo-Penrose diagram shows the cut and glue of the space. The shaded area is removed from the space. The space is completed by gluing a mirror copy of the uncut space extended from the brane. The dashed line represents the thin 2-brane. The thin solid line represents the black hole horizon. (a) It covers the space between the 2-brane and the $\mbox{AdS}_4$ horizon. (b) It covers the space between the 2-brane and the $\mbox{AdS}_4$ boundary.](adshorizon)
The black hole horizon never reaches the boundary $y=x$ since $x_1\le x\le y_{\mbox{BH}}$ is forbidden by the orbifold cone constraint. The black hole horizon saturates at least $(y_{\mbox{BH}}-x_1)$ away from the $\mbox{AdS}_4$ boundary in the C-metric coordinates.
The Poincare transformation is essential to understand the black hole solution in the $\mbox{AdS}_{4}$. If there is no matter trapped on the IR-brane ($\mu=0$), the function $G(x)$ has only two roots $x_0=-1$ and $x_1=1$. Since the region for $x\le 0$ is removed from the space, the allowed interval for the polar coordinate becomes $0\le x\le 1$. The azimuthal periodicity at $x_1$ is $\Delta \phi=2\pi$. In this case, the solution in the Poincare coordinates reduces to the $\mbox{AdS}_{4}$ metric $$\begin{aligned}
ds^2=\frac{{\it l}^2}{z^2}\left[-d{\hat{t}}^2+dr^2+r^2d\hat{\phi}^2+dz^2\right],\end{aligned}$$ by using the transformation $$\begin{aligned}
z&=&{\it l}\frac{(y-x)}{y}, \nonumber \\
r&=&{\it l}\frac{\sqrt{1-x^2}}{y}, \nonumber \\
\hat{\phi}&=&\phi, \nonumber \\
\hat{t}&=&{\it l}t.\end{aligned}$$ The condition $x=y$ corresponds to the $\mbox{AdS}_{4}$ boundary at $z=0$. However, The condition $y=0^{+}$ with $-1\le x <0$ corresponds to the usual $\mbox{AdS}_{4}$ horizon at $z=\infty$. Indeed, we have only the space bounded between the IR-brane at $z=l$ and the UV-brane at $z=0$ since the space between the IR-brane and the $\mbox{AdS}_{4}$ horizon is removed. The curvature singularity at $y=\infty$ is located at the 2-brane $z=l$ and $r=0$ in the Poincare coordinates. When a finite amount of matter trapped near the IR-brane (i.e.: $\mu\neq 0$), the solution in the Poincare coordinates becomes $$\begin{aligned}
ds^2=\frac{l^2}{z^2}\left[-(1-h_{00})d\hat{t}^2 +
ds_{rr}^2+ds^2_{\hat{\phi}\hat{\phi}}+ds_{zz}^2\right],\end{aligned}$$ where $h_{00}=2\mu y(z,r)$. The coordinates transformation are modified as $$\begin{aligned}
z&=&{\it l}\frac{(y-x)}{y}, \nonumber \\
r&=&\frac{l f(x,y)}{y}+2\mu l, \nonumber \\
\hat{\phi}&=&\phi, \nonumber \\
\hat{t}&=&{\it l}t.\end{aligned}$$ The black hole solution with the C-metric coordinate set transformed to the Poincare coordinates set is illustrated in Fig.\[fig3\]. Note that the azimuthal angle $\phi$ in the C-metric coordinate set is the same for that $\hat{\phi}$ in the Poincare coordinate set and their periodicity increase and diverges at $\mu_{\mbox{max}}$. Furthermore the black hole horizon looks like a black cigar solution where its summit is located at $z=l(1-2\mu x_1)$ and reaches it maximum value $z_{max}=l/3$ when $\mu$ reaches its critical value. Furthermore, the solution doesn’t approach the asymptotic AdS space at the boundary. This means that the deformation of the space because of the negative energy source persists to exist even at the boundary.
The black hole thermodynamics is important to shed some information about the center of mass energy. The temperature is calculated as $$\begin{aligned}
T=\frac{1}{8\pi\mu l}.\end{aligned}$$ Furthermore, the entropy reads, $$\begin{aligned}
S=\frac{\cal A}{4G_4},\end{aligned}$$ where the area is calculated by $$\begin{aligned}
{\cal A}&=&2l^2\Delta \phi \int_0^{x_1} dx \frac{1}{(y_{\mbox{BH}}-x)^2}.\end{aligned}$$ The center of mass energy is calculated from the first law of black hole thermodynamics $$\begin{aligned}
d{\cal E} = T dS.\end{aligned}$$ It is evaluated as $$\begin{aligned}
{\cal E}_{\mbox{c.m.}}=G_4M_4=\frac{l}{2}\left(1-\frac{\sqrt{1-2\mu x_1}}{1-3\mu x_1}\right).\end{aligned}$$ The thermal energy is found negative. It diverges at $\mu=\mu_{\mbox{max}}$ where $3\mu x_1=1$. Moreover, we have $\sqrt{1-2\mu x_1}>1$ for $\mu\le \mu_{\mbox{max}}$. The resultant thermodynamical quantities are corresponding to negative energy particles. However, these results are already found in Ref.[@Emparan1] and are expected because of the negative IR-brane tension.
![\[fig3\] The black hole solution through the space between the 2-brane and the $\mbox{AdS}_4$ boundary.](bhbrane2)
When a finite but small amount of matter is trapped near the IR-boundary the solution in the Poincare coordinate is perturbed as follows $$\begin{aligned}
ds^2&=&\frac{l^2}{z^2}\left[-d\hat{t}^2+dr^2+a_{\mu}^2r^2d\tilde{\phi}^2+dz^2
\right],
\nonumber \\
&=&\frac{l^2}{z^2}\left[h_{ab}dx^adx^b+dz^2\right]=\gamma_{ab}dx^adx^b+\frac{l^2}{z^2}dz^2,\end{aligned}$$ where $\hat{\phi}=a_\mu\tilde{\phi}$ and $\Delta \tilde\phi=2\pi$. The stress energy tensor is defined by[@Myers2; @Balasubramanian1] $$\begin{aligned}
\tau^{a b}=
\frac{1}{8\pi G_n}\left(\Theta^{a b}-\gamma^{a b }\Theta^c\!_c\right),\end{aligned}$$ where $a,b,c$ denotes directions parallel to the boundary. The extrinsic curvature reads $$\begin{aligned}
\Theta_{ab}=-\gamma_\mu\!^\rho D_{\rho} n_\mu.\end{aligned}$$ Denote the space time metric as $g_{\mu\nu}$ and $n^{\mu}$ is the outward pointing normal to the boundary normalized with $n^{\mu}n_{\mu}=1$, the induced metric on the boundary $\gamma_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$ acts a projection tensor onto the boundary. The background subtraction procedure yield a finite surface stress tensor[@Myers2; @Balasubramanian1] $$\begin{aligned}
\hat{\tau}^{a b}=\tau^{a b}-(\tau^0)^{a b}.\end{aligned}$$
The conformal transformation can be accounted for by writing the stress tensor expectation values in the field theory as follows $$\begin{aligned}
\sqrt{-h}h^{ab}<\hat{T}_{bc}>=\lim_{z\rightarrow 0}
\sqrt{-\gamma}\gamma^{ab}\hat{\tau}_{bc},\end{aligned}$$ where $h_{ab}$ is the background metric of the field theory. The background metric for the field theory is defined by stripping off the divergent conformal factor from the boundary $$\begin{aligned}
h_{ab}=\lim_{z\rightarrow 0} \frac{z^2}{l^2} \gamma_{ab}.\end{aligned}$$ The total energy in the field theory becomes $$\begin{aligned}
E_{\mbox{ADM}}=\oint d^{n-2}x \sqrt{-h}<T_{tt}>-\oint d^{n-2}x \sqrt{-h}<T^0_{tt}>.\end{aligned}$$ When a small amount of matter is introduced, the metric is slightly perturbed by modifying the periodicity of the azimuthal angle. The ADM mass at UV-brane becomes $$\begin{aligned}
E_{\mbox{ADM}}=\lim_{z\rightarrow 0} -\frac{2}{l}\frac{l^3}{z^3}(a_{\mu}-1)\int r dr d\tilde{\phi},\end{aligned}$$ and $$\begin{aligned}
E_{\mbox{ADM}}/\mbox{Area}=
\lim_{z\rightarrow 0} -\frac{2}{l}\frac{l^3}{z^3}(a_{\mu}-1).\end{aligned}$$ The surface energy density diverges even for a very small perturbation of matter. This means that present solution is unstable and the space should collapse with any fluctuation. In C-metric solution, the ADM mass at the boundary $y=x$ is calculated as follows $$\begin{aligned}
E_{\mbox{ADM}}=\lim_{\zeta\rightarrow 0}
-\frac{2}{l}\frac{l^3}{(\sqrt{2}\zeta)^3}\int d\tilde{\phi}d\xi \sqrt{\frac{2U_0(y)G_0(x)}{U_0(y)+G_0(x)}}
\left[a_{\mu} U(y)/U_0(y) -1\right],\end{aligned}$$ where $$\begin{aligned}
y=\frac{\xi+\zeta}{\sqrt{2}}, \nonumber \\
x=\frac{\xi-\zeta}{\sqrt{2}}. \end{aligned}$$ If we assume $U(y)/U_0(y)=1$ and $x_1=1$ at the boundary, then the energy becomes $$\begin{aligned}
E_{\mbox{ADM}}=
\lim_{\zeta\rightarrow 0} -\frac{2}{l}\frac{l^3}{(\sqrt{2}\zeta)^3}
(2/3)2\pi(a_{\mu}-1).\end{aligned}$$ This quantity diverges at $\zeta=0$. In the real case, we have $x_1\ge 1$. Therefore, the ADM mass has an imaginary part $$\begin{aligned}
\mbox{Mass}\propto (-\epsilon+i\Gamma_{\epsilon})/\zeta^3.\end{aligned}$$ The decay rate factor $\Gamma_{\epsilon}$ increases as $\mu$ increases. Nonetheless, the decay rate $\Gamma=\frac{1}{\zeta^3}\Gamma_{\epsilon}$ diverges at the boundary $z=0$.
Conclusion
==========
We have revisited the C-metric solution for a matter trapped near the IR-brane in the $\mbox{AdS}_4$ space bounded by the IR-brane and UV-brane in the AdS/CFT duality. As noted in Ref[@Emparan1], the C-metric solution generates a negative 2-brane tension and a negative thermal energy gas. It is claimed that this negative energy gas corresponds to the anti-gravity scenario due to the negative IR-brane tension. We have demonstrated that the black hole summit never reaches the $\mbox{AdS}_4$ boundary even if its thermal energy diverges at $\mu_{\mbox{max}}$. The black hole height tends to saturate at $z_{\mbox{max}}=l/3$.
We have calculated the asymptotic ADM mass at $\mbox{AdS}_4$ boundary. We have shown that the C-metric solution implies to entirely unstable configuration in the Poincare coordinates even for a small perturbation. The ADM mass diverges at the boundary for the C-metric solution. Furthermore, the C-metric solution in the unusual coordinates space produces a decay rate factor because of the imaginary part in the ADM mass. This imaginary part appears because of the orbifold constraints. This decay rate diverges at the $\mbox{AdS}_4$ boundary. This means that either the black hole is unstable and evaporates instantaneously or the space configuration is unstable and subsequently the $\mbox{AdS}_4$ boundary collapses.
I am grateful to S. Giddings, S. Shenker and L. Susskind for helpful and stimulating discussions. I also thanks D. Bak, K. Dasgupta, C. Herdeiro, S. Hirano, B. Kol, M. Sheikh-Jabbari and V. Hubeny for valuable conversations. This work is supported by Fulbright FY2002 grant.
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R. C. Myers,“Stress tensors and Casimir energies in the AdS-CFT correspondence”, Phys. Rev. D 60, 046002 (1999). V. Balasubramanian and P. Kraus,“A Stress Tensor For Anti-de Sitter Gravity”, Commun Math Phys [**208**]{} 413 (1999) \[arXiv:hep-th/9902121\].
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abstract: 'This is supplementary material for the main *Geodesics* article by the authors. In Appendix \[app\_gaussian\], we present some general results on the construction of Gaussian random fields. In Appendix \[app\_shape\], we restate our Shape Theorem from [@lagatta2009shape], specialized to the setting of this article. In Appendix \[app\_geomgeod\], we state some straightforward consequences on the geometry of geodesics for a random metric. In Appendix \[geombg\], we provide a rapid introduction to Riemannian geometry for the unfamiliar reader. In Appendix \[analytictools\], we present some analytic estimates which we use in the article. In Appendix \[proof\_mo\_lem\], we present the construction of the conditional mean operator for Gaussian measures. In Appendix \[fermiproof\], we describe Fermi normal coordinates, which we use in our construction of the bump metric.'
address:
- |
Courant Institute of Mathematical Sciences\
New York University\
251 Mercer St.\
New York, New York 10012
- |
Department of Mathematics\
The University of Arizona\
617 N. Santa Rita Ave.\
P.O. Box 210089\
Tucson, AZ 85721
author:
- Tom LaGatta
- Jan Wehr
title: |
Geodesics of Random Riemannian Metrics:\
Supplementary Material
---
[How04]{}
V.I. Arnold. . The MIT Press, 1998.
S.N. Armstrong and P.E. Souganidis. Stochastic homogenization of $l^\infty$ variational problems. , 2011.
S.N. Armstrong and P.E. Souganidis. Stochastic homogenization of level-set convex hamilton-jacobi equations. , 2012.
N.D. Blair-Stahn. . , 2010.
R. Durrett. . Duxbury Press Belmont, CA, 4th edition, 1996.
T. Gneiting. Compactly supported correlation functions. , 83(2):493–508, 2002.
C.D. Howard. Models of first-passage percolation. , pages 125–173, 2004.
H. Kesten. Aspects of first passage percolation. , 1180:125–264, 1984.
H. Kesten. Percolation theory and first-passage percolation. , 15(4):1231–1271, 1987.
J.F.C. Kingman. The ergodic theory of subadditive stochastic processes. , 30(3):499–510, 1968.
J.M. Lee. . Springer, 1997.
T. LaGatta and J. Wehr. . , 51(5), 2010.
E. Poisson. . Cambridge Univ Pr, 2004.
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abstract: 'We consider the Hartree-Fock approximation of Quantum Electrodynamics, with the exchange term neglected. We prove that the probability of static electron-positron pair creation for the Dirac vacuum polarized by an external field of strength $Z$ behaves as $1-\exp(-\kappa Z^{2/3})$ for $Z$ large enough. Our method involves two steps. First we estimate the vacuum expectation of general quasi-free states in terms of their total number of particles, which can be of general interest. Then we study the asymptotics of the Hartree-Fock energy when $Z\to+{{\ensuremath {\infty}}}$ which gives the expected bounds.'
address: 'Département de Mathématiques, CNRS UMR 8088, Université de Cergy-Pontoise, 95000 Cergy-Pontoise, France'
author:
- Julien Sabin
title: |
Static Electron-Positron Pair Creation in\
Strong Fields for a Nonlinear Dirac model
---
Introduction {#introduction .unnumbered}
============
In 1930, Dirac [@Dirac-30] suggested the idea of identifying the vacuum with a sea of virtual electrons with negative kinetic energy. His theory implies that when a sufficiently strong source of energy is provided to the vacuum, some virtual electrons are excited into real electrons, leaving “holes” in the Dirac sea. These holes can be interpreted as positrons, the anti-particles of the electrons, which were experimentally observed in 1933 by Anderson [@Anderson-33]. The extraction of an electron from the Dirac sea is usually called *electron-positron pair creation*. Sauter [@Sauter-31], and Heisenberg-Euler [@HeiEul-36] considered the possibility that an external electromagnetic field could excite the Dirac sea to create those pairs. Schwinger [@Schwinger-51a] then computed the probability of dynamical pair creation by a constant, uniform, external electric field in the framework of Quantum Electrodynamics (QED). The specific phenomenon of pair production triggered by an external, non-quantized field is thus labeled the *Schwinger effect*. It is remarkable that this effect is different from the absorption of photons by the vacuum, which is another possible source for pair creation. The Schwinger effect is based on the fact that the vacuum acts as a polarizable medium which can decay into electron-positron pairs when excited by a sufficiently strong electric field. Although the modern formulation of QED no longer describes the vacuum as a sea of virtual particles, Dirac’s theory is still valid in the mean-field approximation [@HaiLewSerSol-07].
Experimentally, pair creation in electric fields has not been observed yet because it is only non-negligible in a very strong field. However, recent progress in laser physics have permitted to create very strong fields, making the observation of the Schwinger effect possible in the near future [@Dunne-09; @Tajima-09; @Bulanov-10].
One has to distinguish between *dynamical* and *static* pair creation. Dynamical pair creation consists in studying the time evolution of the vacuum state when an external field is progressively turned on, so that a pair consisting of a scattering electron and a corresponding hole in the Dirac sea is created. The external field is then progressively switched off, and one has to check if the pair still exists when the field is completely turned off. Static pair creation, on the other hand, consists in the study of the absolute ground state (the polarized vacuum) of the Hamiltonian in an external field. Therefore, it is a time-independent process. In this context, the vacuum with an additional particle is energetically more favorable than the vacuum without particle. Static pair creation is easier to study than dynamical pair creation, but it is also a bit less relevant from the physical point of view.
When the interactions between particles are neglected (the so-called *linear* case), *static* pair creation was mathematically studied by Klaus and Scharf [@KlaSch-77a]. They proved that the probability of pair creation becomes 1 when the strength of the positive external field sufficiently increases such that an eigenvalue of the Hamiltonian of the system crosses zero. In the linear case, *dynamical* pair creation is a very involved phenomenon, whose properties were mathematically understood very recently. Nenciu [@Nenciu-80; @Nenciu-87] proved that there is a discontinuity in the probability to create pairs as the strength of a specific external field increases, in the adiabatic limit. Later on, Pickl and Dürr [@PicDur-08; @PicklPhD] proved that the probability of pair creation tends to 1 in the adiabatic limit, for general over-critical external fields, by carefully studying the resonances created by the eigenvalues diving into the essential spectrum of the Hamiltonian of the system.
This article is devoted to the mathematical study of *static* pair creation in a *nonlinear* model describing the polarized vacuum, taking into account the interactions between particles. This model was first proposed by Chaix and Iracane [@ChaIra-89] in 1989 and it has recently been given a solid mathematical ground in a series of papers by Gravejat, Hainzl, Lewin, Séré, and Solovej [@HaiLewSer-05a; @HaiLewSer-05b; @HaiLewSol-07; @HaiLewSer-08; @GraLewSer-09]. As in those papers, the main difficulty of our work is the nonlinearity of the model. The more involved study of dynamical pair creation for the same model will be the subject of future work.
In the considered model, the polarized vacuum in a potential generated by a density of charge $Z\nu$ is described by an operator $P_Z$ on $L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)$ (a density matrix). This operator is a solution to the nonlinear equation $$\left\{\begin{array}{ccc}
P_Z & = & {{\ensuremath{\chi_{(-{{\ensuremath {\infty}}},0]}(D_Z)}}} +\delta \\
D_Z & = & D^0 +\alpha\left(\rho_{P_Z-\frac{1}{2}} -Z\nu\right)\star|\cdot|^{-1}
\end{array}\right. ,$$ where $D^0:=-i{\boldsymbol{\alpha}}\cdot\nabla +\beta$ is the (free) Dirac operator and $\delta$ is any self-adjoint operator such that $0{\leqslant}\delta{\leqslant}1$ with ${\rm rank}(\delta)\subset\ker(D_Z)$. Discarding the operator $\delta$, we see that $P_Z$ is the ground state in the grand canonical ensemble of a dressed Dirac operator with density of charge $Z\nu$ perturbed by the density $\rho_{P_Z-\frac{1}{2}}$ of the vacuum. The polarized vacuum therefore interacts with itself.
We shall consider the operator $P_Z$ in the limit $Z\to+{{\ensuremath {\infty}}}$. Of our particular interest is the probability that pairs are generated, which is a nonlinear function of $P_Z$ (see Section \[sec:def-proba\] below). The usual picture [@KlaSch-77a; @SchSei-82; @Hainzl-04] is that if the first eigenvalue $\lambda_1(Z)$ of $D_Z$ is negative, as showed in Figure \[fig:spectre\], then the vacuum becomes charged and the probability of creating at least one pair is 1. In the linear case, Hainzl [@Hainzl-04] showed that the charge of the vacuum in the external density $Z\nu$ is exactly the number of eigenvalues (counted with multiplicity) of the operator $D^0-tZ\nu\star|\cdot|^{-1}$ crossing 0 when we increase $t$ from $0$ to $1$. However, because of the nonlinearity of the model we study, detecting for which values of $Z$ the first eigenvalue will cross 0 is very difficult. However, the probability of pair creation can be very close to 1 without any crossing, as we will explain in Section \[sec:def-proba\].
More precisely, we prove that the probability of static pair creation behaves as $1-\exp(-\kappa Z^{2/3})$ (see Theorem \[th:main\]), where $Z$ is the charge of a nucleus put in the vacuum, and $\kappa$ is a constant depending on different parameters of the model such as the cut-off or the shape of the nucleus. The proof relies on the large-$Z$ asymptotics of the polarized vacuum energy, which is obtained by using an appropriate trial state. This implies that the average number of particles of the polarized vacuum is of order $Z^{2/3}$. We then use general estimates showing that the probability to create pairs for a quasi-free quantum state is bigger than $1-\exp(-\kappa' N)$, where $N$ is the average number of particles of the quantum state and $\kappa'$ is a universal constant. Since for the polarized vacuum $N\simeq Z^{2/3}$, the result follows.
The paper is organized as follows. In Section 1 we introduce the Bogoliubov-Dirac-Fock model and we state our main result. In Section 2, we prove the general estimates on quasi-free states on Fock space, which are of independent interest. In the end of Section 2 we come back to our particular setting. In Section 3, we study the large-$Z$ asymptotics of the polarized vacuum energy. Finally, in Section 4 we prove Theorem \[th:main\] using the tools developed in Section 2 and 3. In Appendix A, we recall some properties of product states, which are used in Section 2.
**Acknowledgments.** I sincerely thank Mathieu Lewin for his precious guidance and constant help. I also acknowledge support from the ERC MNIQS-258023 and from the ANR “NoNAP” (ANR-10-BLAN 0101) of the French ministry of research.
Estimate on the probability to create pairs
===========================================
Probability to create a pair {#sec:def-proba}
----------------------------
The first quantity to define is the probability to create a pair. Let ${\mathfrak{H}}_+,{\mathfrak{H}}_-$ be (separable) Hilbert spaces, representing the one particle (resp. anti-particle) space. The natural space to describe a system with an arbitrary number of particles/anti-particles is the Fock space $${\mathcal{F}}_0:={\mathcal{F}}({\mathfrak{H}}_+)\otimes{\mathcal{F}}({\mathfrak{H}}_-),$$ with the usual notation ${\mathcal{F}}({\mathfrak{H}}):=\oplus_{N{\geqslant}0}\wedge_1^N{\mathfrak{H}}$ for any Hilbert space ${\mathfrak{H}}$ and with the convention $\wedge_1^0{\mathfrak{H}}:={{\ensuremath {\mathbb C} }}$. We also define the *vacuum state* $\Omega:=\Omega_+\otimes\Omega_-\in{\mathcal{F}}_0$ where $\Omega_\pm:=1\oplus0\oplus0\cdots\in{\mathcal{F}}({\mathfrak{H}}_\pm)$. Recall that a state over ${\mathcal{F}}_0$ can be defined [^1] as a positive linear functional $\omega:{\mathcal{B}}({\mathcal{F}}_0)\to{{\ensuremath {\mathbb C} }}$ with $\omega(\rm{Id}_{{\mathcal{F}}_0})=1$, where ${\mathcal{B}}({\mathcal{F}}_0)$ is the set of all bounded linear operators on ${\mathcal{F}}_0$. Notice that any normalized $\psi\in{\mathcal{F}}_0$ defines a state $\omega_\psi$ (called *pure* state) by the formula $\omega_\psi(A)=\langle\psi,A\psi\rangle_{{\mathcal{F}}_0}$, where $\langle\cdot,\cdot\rangle_{{\mathcal{F}}_0}$ is the usual inner product on ${\mathcal{F}}_0$. Following [@PicklPhD Corollary 4.1], [@Thaller Eq. (10.154)], and [@Nenciu-87 Section 2], we define the probability $p(\omega)$ for a state $\omega$ to create a particle/anti-particle pair by $$\boxed{
p(\omega):=1-\omega\left({{\ensuremath{| \Omega\rangle\langle \Omega|}}}\right),
}$$ where ${{\ensuremath{| \Omega\rangle\langle \Omega|}}}\in{\mathcal{B}}({\mathcal{F}}_0)$ is the orthogonal projection on ${{\ensuremath {\mathbb C} }}\Omega$. For a pure state $\psi=\psi_{0,0}\oplus\psi_{0,1}\oplus\psi_{1,0}\oplus\cdots\in{\mathcal{F}}_0$, we have $p(\omega_\psi)=1-\left|\psi_{0,0}\right|^2$.Therefore, $p(\omega_\psi)=0$ if and only if $\psi=\Omega$ (the vacuum has probability zero to create pairs), while $p(\omega_\psi)=1$ if and only if $\psi_{0,0}=0$. In the latter case, notice that $\psi$ does not litterally contain pairs, in the sense that its number of particles may not be equal to its number of anti-particles. This definition merely measures the probability that a state contains real particles/anti-particles.
Typically, $\Omega$ represents the free (or bare) vacuum and we want to measure the probability of a perturbation $\Omega'$ of $\Omega$, representing the polarized (or dressed) vacuum in the presence of an external electric field, to have pairs. Assuming that $\Omega'$ is a pure quasi-free state, we have the well-known formula (see e.g. [@Thaller Theorem 10.6], [@BacLieSol-94 Theorem 2.2], or [@HaiLewSer-08 Theorem 5]) $$\label{formula-vacuum}
\Omega'=\prod_{i{\geqslant}1}\frac{1}{\sqrt{1+\lambda_i^2}}\prod_{n=1}^N a_0^*(f_n)\prod_{m=1}^M b_0^*(g_m)\prod_{i{\geqslant}1}\left(1+\lambda_ia_0^*(v_i)b^*_0(u_i)\right)\Omega,$$ where $a_0^*$ (resp. $b_0^*$) is the free particle (resp. anti-particle) creation operator, $(f_n)_n\cup(v_i)_i$ (resp. $(g_m)_m\cup(u_i)_i$) is an orthonormal set for ${\mathfrak{H}}_+$ (resp. ${\mathfrak{H}}_-$), and $(\lambda_i)_i\in\ell^2({{\ensuremath {\mathbb R} }}_+)$.
From the formula , we see that $p(\omega_{\Omega'})=1$ as soon as $N>0$ or $M>0$. Moreover, in this case real particles in the states $(f_n)_n$ and real anti-particles in the states $(g_m)_m$ have been created. In the linear case, Klaus and Scharf [@KlaSch-77a] proved that $N,M\neq0$ if the external field is strong enough. However, there can be a high probability to create pairs even when $N=M=0$. Indeed, since in this case we have $$|\langle\Omega',\Omega\rangle_{{\mathcal{F}}_0}|^2=\prod_{i{\geqslant}1}\frac{1}{1+\lambda_i^2},$$ one sees that $p(\omega_{\Omega'})$ is close to 1 when the $\lambda_i$ are large enough. One simple condition is that $\sum_i\lambda_i^2(1+\lambda_i^2)^{-1}$ is large enough, by the inequality $$\prod_{i{\geqslant}1}\frac{1}{1+\lambda_i^2}{\leqslant}\exp\left[-\sum_i\frac{\lambda_i^2}{1+\lambda_i^2}\right].$$ Note that this is indeed (half) the average total number of particle of the state $\Omega'$ (number of particle + number of anti-particle), $$\omega_{\Omega'}({\mathcal{N}})=2\sum_i\frac{\lambda_i^2}{1+\lambda_i^2},$$ where ${\mathcal{N}}$ is the usual number operator on ${\mathcal{F}}_0$ (see formula ). Hence $p(\omega_{\Omega'})$ is close to 1 when $\omega_{\Omega'}({\mathcal{N}})$ is large enough. While the non-vanishing $N,M$ case can be interpreted as the creation of real particles, this second explanation for an increasing $p(\omega_{\Omega'})$ can be interpreted as a “virtual pair creation”. In this article, we study an analog of “virtual pair creation” for more general states than those given by formula .
Static pair creation in the reduced BDF approximation {#sec:BDF-notations}
-----------------------------------------------------
For noninteracting electrons in an external field $V$, the polarized vacuum $\Omega'$ is the unique Hartree-Fock state whose density matrix is [@KlaSch-77a; @Hainzl-04] $$P={{\ensuremath{\chi_{(-{{\ensuremath {\infty}}},0]}(D^0+V)}}}.$$ In this article, we will rather use the reduced Bogoliubov-Dirac-Fock approximation, a non-linear model enabling to describe an interacting vacuum in which $V$ is a function of $P$ itself. It was introduced by Hainzl, Lewin, Séré and Solovej in a series of articles [@HaiLewSer-05a; @HaiLewSer-05b; @HaiLewSol-07; @HaiLewSer-08] after the pioneering work of Chaix, Iracane, and Lions [@ChaIra-89; @ChaIraLio-89]. We will now briefly recall the model and the results needed for our study.
In units where $m=c=\hbar=1$, the reduced Bogoliubov-Dirac-Fock (rBDF) energy functional is the (formal) difference between the energy of the state $P$ and that of the free vacuum $P^0_-={{\ensuremath{\chi_{(-{{\ensuremath {\infty}}},0]}(D^0)}}}$, with the exchange term dropped. It depends only on the variable $Q=P-P^0_-$, $$\label{rBDF-energy}
{{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{\nu}(Q)}}}:={{\ensuremath{\operatorname{Tr}_0(D^0Q)}}}-\alpha D(\rho_Q,\nu)+\frac{\alpha}{2}D(\rho_Q,\rho_Q).$$ Here, $\alpha>0$ is the coupling constant and $\nu:{{\ensuremath {\mathbb R} }}^3\to{{\ensuremath {\mathbb R} }}$ is the external charge density belonging to the Coulomb space $${\mathcal{C}}:=\left\{f\in\mathscr{S}'({{\ensuremath {\mathbb R} }}^3):\:\int_{{{\ensuremath {\mathbb R} }}^3}\frac{|\widehat{f}(k)|^2}{|k|^2}{{\ensuremath{\,\text{d}k}}}<+{{\ensuremath {\infty}}}\right\},$$ endowed with the inner product $D(\rho_1,\rho_2)=\int|k|^{-2}\widehat{\rho_1}(k)\overline{\widehat{\rho_2}(k)}$ (the hat denotes the Fourier transform[^2]). We also use the notation $\|\rho\|_{\mathcal{C}}=D(\rho,\rho)^{1/2}$ for any $\rho\in{\mathcal{C}}$. In order to define the domain of the rBDF energy functional, let us fix a cut-off $\Lambda>0$ and define the one-particle Hilbert space $${\mathfrak{H}}_\Lambda:=\left\{f\in L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4),\quad\operatorname{supp}\widehat{f}\subset B(0,\Lambda)\right\}.$$ The operator $D^0=-i{\boldsymbol{\alpha}}\cdot\nabla+\beta$ is the usual Dirac operator on $L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)$, where $\alpha_1,\alpha_2,\alpha_3,\beta$ are the Dirac matrices acting on ${{\ensuremath {\mathbb C} }}^4$, $$\alpha_i=\left(\begin{array}{cc}
0 & \sigma_i\\
\sigma_i & 0
\end{array}\right),\quad i=1,2,3, \qquad
\beta=\left(\begin{array}{cc}
\text{Id}_{{{\ensuremath {\mathbb C} }}^2} & 0\\
0 & -\text{Id}_{{{\ensuremath {\mathbb C} }}^2}
\end{array}\right),$$ and $(\sigma_i)_{i=1,2,3}$ are the Pauli matrices, $$\sigma_1=\left(\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right),\qquad
\sigma_2=\left(\begin{array}{cc}
0 & -i\\
i & 0
\end{array}\right),\qquad
\sigma_3=\left(\begin{array}{cc}
1 & 0\\
0 & -1
\end{array}\right).$$
The operator $D^0$ stabilizes ${\mathfrak{H}}_\Lambda$, and its restriction to ${\mathfrak{H}}_\Lambda$ defines a bounded operator on ${\mathfrak{H}}_\Lambda$, which we still denote by $D^0$. For convenience we introduce $P^0_+:=1-P^0_-$. We denote by ${\mathfrak{S}}_p({\mathfrak{H}})$ the Schatten class of all bounded operators $A$ on the Hilbert space ${\mathfrak{H}}$ such that $\operatorname{Tr}(|A|^p)<{{\ensuremath {\infty}}}$. For any operator $Q$ on ${\mathfrak{H}}_\Lambda$ and for any ${\varepsilon},{\varepsilon}'\in\{+,-\}$, we let $Q_{{\varepsilon}{\varepsilon}'}:=P^0_{\varepsilon}QP^0_{{\varepsilon}'}$ and we define $${\mathfrak{S}}_{1,{P^0_-}}({\mathfrak{H}}_\Lambda):=\left\{ Q\in{\mathfrak{S}}_2({\mathfrak{H}}_\Lambda),\quad Q_{++},Q_{--}\in{\mathfrak{S}}_1({\mathfrak{H}}_\Lambda)\right\}.$$ It is a Banach space endowed with the norm $$\|Q\|_{1,P^0_-}:=\|Q_{++}\|_{{\mathfrak{S}}_1}+\|Q_{--}\|_{{\mathfrak{S}}_1}+\|Q_{+-}\|_{{\mathfrak{S}}_2}+\|Q_{-+}\|_{{\mathfrak{S}}_2}.$$
For any $Q\in{\mathfrak{S}}_{1,{P^0_-}}({\mathfrak{H}}_\Lambda)$, we define its generalized trace by $${{\ensuremath{\operatorname{Tr}_0(Q)}}}:=\operatorname{Tr}\left(Q_{++}+Q_{--}\right),$$ and its density $\rho_Q$ by $\rho_Q(x):=\operatorname{Tr}_{{{\ensuremath {\mathbb C} }}^4}(Q(x,x))$ for all $x\in{{\ensuremath {\mathbb R} }}^3$, where $Q(x,y)$ denotes the $4\times4$ matrix kernel of $Q$. This density $\rho_Q$ is well defined since $\operatorname{supp}{\widehat}{Q}(\cdot,\cdot)\subset B(0,\Lambda)\times B(0,\Lambda)$ implies that $Q(\cdot,\cdot)$ is smooth. Furthermore, it is proved in [@HaiLewSer-08 Lemma 1] that $\rho_Q\in L^2({{\ensuremath {\mathbb R} }}^3)\cap{\mathcal{C}}$ for any $Q\in{\mathfrak{S}}_{1,{P^0_-}}({\mathfrak{H}}_\Lambda)$. We conclude that the rBDF energy functional is well-defined on the convex set $$\label{eq:cK}
{\mathcal{K}}:=\left\{Q\in{\mathfrak{S}}_{1,{P^0_-}}({\mathfrak{H}}_\Lambda),\quad Q=Q^*,\quad -P^0_-{\leqslant}Q{\leqslant}1-P^0_-\right\}.$$ Notice that the kinetic part of the rBDF energy is well defined since $${{\ensuremath{\operatorname{Tr}_0(D^0Q)}}}=\operatorname{Tr}(|D^0|(Q_{++}-Q_{--})).$$ The variational set ${\mathcal{K}}$ is the convex hull of $\{P-P^0_-,\,P=P^2=P^*,\,P-P^0_-\in{\mathfrak{S}}_2({\mathfrak{H}}_\Lambda)\}$, where $P$ is the density matrix of a pure Hartree-Fock state, which is a Hilbert-Schmidt perturbation of the free vacuum $P^0_-$. The rigorous derivation of the rBDF energy functional and the motivation for this functional setting can be found in [@HaiLewSol-07; @HaiLewSer-05a].
For any $Z>0$ and $\nu\in{\mathcal{C}}$, the rBDF energy functional ${\mathcal{E}}_{\text{rBDF}}^{Z\nu}$ admits global minimizers $Q_Z$ on ${\mathcal{K}}$. Minimizers are not necessarily unique, but they always share the same density $\rho_Z:=\rho_{Q_Z}$. Any minimizer $Q_Z$ satisfies the self-consistent equation $$\left\{\begin{array}{ccc}
Q_Z & = & {{\ensuremath{\chi_{(-{{\ensuremath {\infty}}},0]}(D_Z)}}}-P^0_- +\delta \\
D_Z & = & D^0 +\alpha\left(\rho_Z-Z\nu\right)\star|\cdot|^{-1}
\end{array}\right. ,$$ where $\delta$ is a self-adjoint operator such that $0{\leqslant}\delta{\leqslant}1$ and ${\rm rank}(\delta)\subset\ker(D_Z)$. Hence, uniqueness holds if and only if $\ker(D_Z)=\{0\}$. Notice that since the density $\rho_Z$ is unique, the operator $D_Z$ is itself unique. Any minimizer of ${\mathcal{E}}_{\text{rBDF}}^{Z\nu}$ on ${\mathcal{K}}$ is interpreted as a generalized one-particle density matrix of a BDF state $\omega_{\text{vac}}^{Z\nu}$ (see Section \[sec:BDFstates\]) representing the *polarized vacuum* in the potential $Z\nu\star|\cdot|^{-1}$. When there is a unique minimizer $Q_Z$, it is a difference of two projectors and hence it is the generalized one-particle density matrix of a pure state. We emphasize that there is no charge constraint in this minimization problem: In fact, the polarized vacuum could (and should) be charged when $Z$ is very large. In this case, one may think that an electron-positron pair is created, with the positron sent to infinity.
We want to estimate $p(\omega_{\text{vac}}^{Z\nu})$ in terms of $Z$, and confirm the picture that the stronger the field, the more pairs are created. As a consequence, we will fix a non-zero density $\nu$ (interpreted as the shape of the external charge density) and study $p_Z:=p(\omega_{\text{vac}}^{Z\nu})$ for $Z>0$ large. Our main result is the following.
\[th:main\] Let $\alpha>0$ and $\Lambda>0$. Let $\nu\in{\mathcal{C}}$ such that $\int_{{{\ensuremath {\mathbb R} }}^3}(1+|x|)|\nu(x)|{{\ensuremath{\,\text{d}x}}}<{{\ensuremath {\infty}}}$ and $q:=\int_{{{\ensuremath {\mathbb R} }}^3}\nu\neq0$. Then, there exists a constant $Z_1>0$ and a constant $\kappa>0$ such that for all $Z> Z_1$ we have $$\label{est:main}
\boxed{
p_Z{\geqslant}1-e^{-\kappa Z^{2/3}}.
}$$ The constant $Z_1$ is equal to ${\widetilde}{Z_1}/\int_{{{\ensuremath {\mathbb R} }}^3}|\nu|$ where ${\widetilde}{Z_1}$ is defined in Equation and ${\widetilde}{Z_1}$ depends only on $\Lambda$, $\alpha$, $|q|$, and $\int_{{{\ensuremath {\mathbb R} }}^3}|x||\nu|$. The constant $\kappa$ equals $0.0941248\ldots\times C(\int_{{{\ensuremath {\mathbb R} }}^3}|\nu|)^{2/3}$, where $C$ is defined in Equation below.
The assumption $|x|\nu\in L^1({{\ensuremath {\mathbb R} }}^3)$ allows us to have an explicit estimate. If we remove this assumption, we can still prove the weaker result that $p_Z\to1$ as $Z\to{{\ensuremath {\infty}}}$. In the sequel, by rescaling $Z$ if needed, we will assume that $$\boxed{
\int_{{{\ensuremath {\mathbb R} }}^3}|\nu|=1.
}$$ If $\int\nu=0$, we expect an asymptotics lower than $Z^{2/3}$, but we are unable to prove it.
We will see that $Z_1\sim\text{const.}\times\alpha^{-3/2}$ as $\alpha\to0$.
Theorem \[th:main\] says that in a very strong field, $Z\gg1$, the probability to create at least one electron-positron pair is very close to 1. It is reasonable to think that for some sufficiently large $Z$, the first eigenvalue of $D_Z$ crosses 0 in which case $p_Z=1$. However, determining the behaviour of the eigenvalue of $D_Z$ as $Z$ increases is difficult because of the nonlinearity of the model and because we are in a regime far from being perturbative. For all these reasons, the estimate on $p_Z$ is the best we have so far. For very large $Z$, one expects that many electron-positron pairs will be generated. We conjecture that we have indeed $\omega_{\text{vac}}^{Z\nu}({\mathcal{P}}_k)\to0$ as $Z\to+{{\ensuremath {\infty}}}$ for all $k\in{{\ensuremath {\mathbb N} }}$, where ${\mathcal{P}}_k$ is the orthogonal projector on the $k$-particle space in Fock space (see Section \[sec:quasi-free\]). This would mean that for large $Z$, the probability to create at least $k$-pairs is very close to 1. Our method of proof only gives this result for $k=0$. However, if we assume that $\omega_{\text{vac}}^{Z\nu}$ is a *pure* quasi-free state for all $Z$ large enough (which is the case if $0\notin\sigma(D_Z)$), then the conjecture follows from Proposition \[prop:HFBpure\] below.
Strategy of the proof
---------------------
The proof is separated into two parts. The first one consists in estimating the energy of the polarized vacuum, $$E(Z):={{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q_{\text{vac}}^{Z\nu})}}}=\inf\left\{{{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q)}}},\quad Q\in{\mathcal{K}}\right\},$$ from above by $- cZ^{5/3}$. We will also give a lower bound $E(Z)\gtrsim -Z^{5/3}$ to show that the power $5/3$ is optimal, although we only need the upper bound for the proof of Theorem \[th:main\]. From this estimate, we then infer that the average number of particles (counted relatively to that of the free vacuum, see Section \[sec:BDFstates\]) in the polarized vacuum satisfies $$\operatorname{Tr}\left((Q_{\text{vac}}^{Z\nu})_{++}-(Q_{\text{vac}}^{Z\nu})_{--}\right)\gtrsim Z^{2/3}.$$ The precise statements of these results and their proofs can be found in Section \[sec:asymptotics\].
In a second part, we prove an estimate on the vacuum expectation $\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}})$ for a quasi-free state $\omega$, in terms of its average number of particle $\omega({\mathcal{N}})$. These estimates are of independent interest and therefore we also provide several other estimates for the distribution of quasi-free states in the $k$-particle spaces. These results are contained in Section \[sec:quasi-free\]. Finally, in Section \[sec:proof\], we combine the two parts and prove Theorem \[th:main\].
On the distribution of quasi-free states in the $k$-particle spaces {#sec:quasi-free}
===================================================================
In this section, we consider general quasi-free states. Only in Section \[sec:BDFstates\] we come back to our particular situation of pair creation. We start by introducing the notation used throughout this section.
Notation {#sec:notation}
--------
Let $({\mathfrak{H}},\langle\cdot,\cdot\rangle)$ be a complex, separable Hilbert space whose inner product $\langle\cdot,\cdot\rangle$ is linear in the second argument. We also need an anti-linear operator ${J:{\mathfrak{H}}\to{\mathfrak{K}}}$ such that $J^*J=\text{Id}_{\mathfrak{H}}$, where $({\mathfrak{K}},\langle\cdot,\cdot\rangle_{\mathfrak{K}})$ is another complex Hilbert space[^3]. Let ${\mathcal{F}}:=\oplus_{N{\geqslant}0}{\mathfrak{H}}^N$ be the associated Fock space with ${\mathfrak{H}}^N:=\wedge_1^N{\mathfrak{H}}$. We still denote by $\Omega=1\oplus0\oplus\cdots\in{\mathcal{F}}$ the vacuum vector. For $k\in{{\ensuremath {\mathbb N} }}$, we denote by ${\mathcal{P}}_k\in{\mathcal{B}}({\mathcal{F}})$ the orthogonal projection on ${\mathfrak{H}}^k\subset{\mathcal{F}}$. We recall from Section \[sec:def-proba\] that ${\mathcal{B}}({\mathcal{F}})$ is the space of all linear bounded operators on ${\mathcal{F}}$. Let $\operatorname{\mathfrak{A}}[{\mathfrak{H}}]$ be the CAR unital $C^*$-subalgebra of ${\mathcal{B}}({\mathcal{F}})$ generated by the usual creation (resp. annihilation operators) $a^*(f)$ (resp. $a(f)$), for $f\in{\mathfrak{H}}$. We denote by ${\mathcal{N}}$ the particle number operator on ${\mathcal{F}}$, $${\mathcal{N}}:=\bigoplus_{k{\geqslant}0}k\,\text{Id}_{{\mathfrak{H}}^k}=\sum_{i{\geqslant}0}a^*(f_i)a(f_i)$$ for any orthonormal basis $(f_i)_{i{\geqslant}0}$ in ${\mathfrak{H}}$. Then ${\mathcal{P}}_k={\boldsymbol{1}}_{\{{\mathcal{N}}=k\}}$ for all $k\in{{\ensuremath {\mathbb N} }}$. A state on $\operatorname{\mathfrak{A}}[{\mathfrak{H}}]$ is a non-negative linear functional $\omega:\operatorname{\mathfrak{A}}[{\mathfrak{H}}]\to{{\ensuremath {\mathbb C} }}$ which is normalized: $\omega(\text{Id}_{\mathcal{F}})=1$. A state $\omega$ is called *normal* if there exists a non-negative operator $G$ on ${\mathcal{F}}$ (sometimes called the density matrix of $\omega$) such that $\operatorname{Tr}_{\mathcal{F}}(G)=1$ and $\omega(A)=\operatorname{Tr}_{\mathcal{F}}(GA)$ for all $A\in\operatorname{\mathfrak{A}}[{\mathfrak{H}}]$. Of particular interest are the *pure* states which are normal states with $G={{\ensuremath{| \psi\rangle\langle \psi|}}}$ for $\psi\in{\mathcal{F}}$ with $\|\psi\|_{\mathcal{F}}=1$. We define the average particle number of $\omega$ as $$\omega({\mathcal{N}}):=\sum_{i{\geqslant}0}\omega(a^*(f_i)a(f_i))\in[0,+{{\ensuremath {\infty}}}].$$ The *one-particle density matrix* (1-pdm) $\gamma$ of $\omega$ is the operator defined by $$\langle g,\gamma f\rangle:=\omega(a^*(f)a(g)),$$ for all $f,g\in{\mathfrak{H}}$. It is a self-adjoint operator on ${\mathfrak{H}}$, satisfying $0{\leqslant}\gamma{\leqslant}1$. In the same fashion, we define its *pairing matrix* $\alpha:{\mathfrak{K}}\to{\mathfrak{H}}$ which is a linear operator on ${\mathfrak{K}}$ by $$\langle \alpha J f,g\rangle:=\omega(a^*(f)a^*(g)),$$ for all $f,g\in{\mathfrak{H}}$. It satisfies $(\alpha J)^*=-\alpha J$. Moreover, if we define the operator $\Gamma(\gamma,\alpha)$ on ${\mathfrak{H}}\oplus{\mathfrak{K}}$ by block $$\label{Gamma}
\Gamma(\gamma,\alpha):=\left(
\begin{array}{cc}
\gamma & \alpha \\
\alpha^* & 1-J\gamma J^*
\end{array}
\right),$$ then $0{\leqslant}\Gamma(\gamma,\alpha){\leqslant}1$, see [@BacLieSol-94 Lemma 2.1]. This last relation implies that $$\label{ineq:gammalpha}
\gamma^2+\alpha\alpha^*{\leqslant}\gamma,$$ in the sense of quadratic forms on ${\mathfrak{H}}$. Notice also that $\omega({\mathcal{N}})=\operatorname{Tr}(\gamma)$. A state $\omega$ is called *quasi-free* if for any operators $e_1,\ldots,e_{2p}$ which are either a $a^*(f)$ or a $a(g)$ for any $f,g\in{\mathfrak{H}}$, then $\omega(e_1 e_2\ldots e_{2p-1})=0$ for any $p{\geqslant}1$ and $$\label{wick}
\omega(e_1 e_2\ldots e_{2p})=\sum_{\pi\in\widetilde{{\mathcal{S}}_{2p}}} (-1)^{{\varepsilon}(\pi)}\omega(e_{\pi(1)}e_{\pi(2)})\ldots\omega(e_{\pi(2p-1)}e_{\pi(2p)}),$$ where $\widetilde{{\mathcal{S}}_{2p}}$ is the set of permutations of $\{1,\ldots,2p\}$ which verify $\pi(1)<\pi(3)<\cdots<\pi(2p-1)$ and $\pi(2j-1)<\pi(2j)$ for all $1\leqslant j \leqslant p$, and ${\varepsilon}(\pi)$ is the parity of the permutation $\pi$. The relation (\[wick\]) is called the Wick formula. From this definition, we see that a quasi-free state is completely determined by its density matrices $(\gamma,\alpha)$. We recall [@BacLieSol-94 Theorem 2.3]
\[BacLieSol\] For any $(\gamma,\alpha)$ such that $0{\leqslant}\Gamma(\gamma,\alpha){\leqslant}1$ with additionally $\operatorname{Tr}(\gamma)<+{{\ensuremath {\infty}}}$, there exists a unique quasi-free state $\omega$ on $\operatorname{\mathfrak{A}}[{\mathfrak{H}}]$ with finite number of particle such that $\gamma$ is its 1-particle density matrix and $\alpha$ its pairing matrix. Furthermore, $\omega$ is normal: there exists $G:{\mathcal{F}}\to{\mathcal{F}}$ with $0{\leqslant}G{\leqslant}1$ and $\operatorname{Tr}_{\mathcal{F}}(G)=1$ such that $\omega(A)=\operatorname{Tr}_{\mathcal{F}}(GA)$ for all $A\in\operatorname{\mathfrak{A}}[{\mathfrak{H}}]$.
Now, we need some terminology, which is not universal in the literature. We call *Hartree-Fock* (HF) states the quasi-free states with $\operatorname{Tr}(\gamma)<+{{\ensuremath {\infty}}}$ and $\alpha=0$, because when such states are pure, they are usual Slater determinants. Quasi-free states with $\operatorname{Tr}(\gamma)<+{{\ensuremath {\infty}}}$ and $\alpha\neq0$, are called *Hartree-Fock-Bogoliubov* (HFB) states. Pure HFB states are particularly simple since they are Bogoliubov rotations of the vacuum $\Omega$. The aim of this section is to study the distribution of quasi-free states in the particle subspaces ${\mathfrak{H}}^k$, in terms of $\operatorname{Tr}(\gamma)$. Our results are different for HF or HFB, pure or mixed states.
Motivation
----------
Quasi-free states are also called *Gaussian* states, in particular because they can be written as (limits of) Gibbs states of quadratic Hamiltonians (i.e. normal states with density matrices $e^{-\beta\mathbb{H}}/\operatorname{Tr}(e^{-\beta\mathbb{H}})$, where $\mathbb{H}$ is a quadratic Hamiltonian). The Gaussian character of quasi-free states is however deeper. In this section we will show that the distribution of a quasi-free state $\omega$ over the different ${\mathfrak{H}}^k$, that is $(\omega({\mathcal{P}}_k))_{k{\geqslant}0}$, also has some Gaussian characteristics. More precisely, we will provide estimates of the form $$\omega({\mathcal{P}}_k){\leqslant}c_k e^{-c'\omega({\mathcal{N}})}.$$ This estimate means that a quasi-free state which has a large average number of particles $\omega({\mathcal{N}})\gg1$ necessarily has an exponentially small vacuum expectation $$\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}}){\leqslant}c_0 e^{-c'\omega({\mathcal{N}})}.$$
Let us explain the picture in a commutative setting. Let $f(x)=\pi^{-1/2}e^{-|x-a|^2}$ for $a\in{{\ensuremath {\mathbb R} }}_+$ be a Gaussian function such that $\int_{{\ensuremath {\mathbb R} }}f=1$. Then $a=\int_{{\ensuremath {\mathbb R} }}xf(x){{\ensuremath{\,\text{d}x}}}$ is the average position of $f$, as desbribed in Figure \[fig:gaussian\]. Now if $a$ goes to $+{{\ensuremath {\infty}}}$, the whole function moves to infinity and in particular $f(0)$ becomes smaller and smaller: $f(0)=\pi^{-1/2}e^{-a^2}$. In other words, as the average position of $f$ goes to infinity, $f(0)$ goes to zero (and this is true for any $f(x_0)$ with $x_0$ fixed). We will prove a similar fact for quasi-free states. Indeed, for any quasi-free state $\omega$ we have $1=\omega(\text{Id}_{\mathcal{F}})=\sum_{k{\geqslant}0}\omega({\mathcal{P}}_k)$, which is the analog of $\int_{{\ensuremath {\mathbb R} }}f(x){{\ensuremath{\,\text{d}x}}}=1$. We also know that $\omega({\mathcal{N}})=\sum_{k{\geqslant}0}k\omega({\mathcal{P}}_k)$ is the average number of particle of $\omega$; it is the analog of $\int_{{\ensuremath {\mathbb R} }}xf(x){{\ensuremath{\,\text{d}x}}}$. We want to prove that when $\omega({\mathcal{N}})$ is large, then the main part of $\omega$ lives in the high-$k$ particle spaces, that is $\omega$ “follows” its average number of particles, as shown in Figure \[fig:gaussian\]. The analog of $f(0)$ in this case is $\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}})$ and we thus want to prove that $\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}})$ goes to zero as $\omega({\mathcal{N}})$ goes to $+{{\ensuremath {\infty}}}$, for any quasi-free state $\omega$. A natural extension of this result would be that $\omega({\mathcal{P}}_{k_0})$ also goes to zero for any fixed $k_0$.
We will provide explicit estimates depending on the properties of the quasi-free state (pure, mixed, HF or HFB). In the most general case of mixed HFB states, we only derive a bound on the vacuum expectation. The following table tells us where each case is treated.
Pure Mixed
--------------------- ------------------------- --------------------------
HF ($\alpha=0$)
HFB ($\alpha\neq0$) Section \[sec:HFBpure\] Section \[sec:HFBmixed\]
In spite of their usefulness, we have not found the following estimates in the literature. One main reason is probably that $e^{-\beta{\mathcal{N}}}$ does *not* belong to the CAR algebra, hence $\omega(e^{-\beta{\mathcal{N}}})$ only makes sense for normal states.
A useful tool for the proofs of the following results is the notion of *product state*. A product state $\otimes_i\omega_i$ is a state on ${\mathcal{F}}(\oplus_i{\mathfrak{H}}_i)\simeq\otimes_i{\mathcal{F}}({\mathfrak{H}}_i)$ when each $\omega_i$ is a state on ${\mathcal{F}}({\mathfrak{H}}_i)$. While this notion is intuitive, we recall how it is precisely defined in Appendix \[prodstates\].
Hartree-Fock case {#sec:HF}
-----------------
\[prop:HF\] Let $\omega$ be a quasi-free state with $\operatorname{Tr}(\gamma)<+{{\ensuremath {\infty}}}$ and $\alpha=0$. Then for any $\beta{\geqslant}0$ we have $$\label{eq:HFequality}
\boxed{
\omega\left(e^{-\beta{\mathcal{N}}}\right)=\operatorname{Det}_{\mathfrak{H}}\left(1+(e^{-\beta}-1)\gamma\right).
}$$ We also have the following estimate $$\label{eq:HFestimate}
\boxed{
\omega({\mathcal{P}}_k){\leqslant}\frac{(e\operatorname{Tr}(\gamma))^k}{k!}e^{-\operatorname{Tr}(\gamma)},
}$$ for all $k{\geqslant}k_0$ while $\omega({\mathcal{P}}_k)=0$ if $k<k_0$, where $k_0:=\dim\ker(\gamma-1)$.
\[rk:1\] This estimate implies that for any fixed $k\in{{\ensuremath {\mathbb N} }}$, $\omega({\mathcal{P}}_k)\to0$ as $\operatorname{Tr}(\gamma)\to+{{\ensuremath {\infty}}}$, which is the expected behaviour.
\[rk:2\] A more theoretical corollary of is that for any fixed $f:{{\ensuremath {\mathbb N} }}\to{{\ensuremath {\mathbb C} }}$ vanishing at infinity, $\omega(f({\mathcal{N}}))$ goes to zero as $\operatorname{Tr}(\gamma)$ goes to $+{{\ensuremath {\infty}}}$. This uses the fact that the algebra of these $f$s is generated by the $(e^{-\beta\cdot})_{\beta{\geqslant}0}$. In the same fashion, one can also prove that $\omega(K)\to0$ as $\operatorname{Tr}(\gamma)\to+{{\ensuremath {\infty}}}$, for any fixed compact operator $K$.
Since $\gamma$ is trace-class it can be diagonalized in an orthonormal basis $(f_i)_{i\in{{\ensuremath {\mathbb N} }}}$, $\gamma=\sum_{i{\geqslant}0}\lambda_i{{\ensuremath{| f_i\rangle\langle f_i|}}}$. For all $i$, let $\omega_i$ be the unique quasi-free state on ${\mathcal{F}}({{\ensuremath {\mathbb C} }}f_i)$ having $\lambda_i\text{Id}_{{{\ensuremath {\mathbb C} }}f_i}$ as 1-pdm and $0_{{{\ensuremath {\mathbb C} }}f_i}$ as its pairing matrix. Then by Proposition \[propprodstates\] in Appendix \[prodstates\], one has $\omega=\otimes_i\omega_i$. Moreover, since $$Te^{-\beta{\mathcal{N}}}T^*=\bigotimes_{i\in{{\ensuremath {\mathbb N} }}}\left( 1+(e^{-\beta}-1)a^*(f_i)a(f_i)\right),$$ where $T$ is the isometry between ${\mathcal{F}}(\oplus_i{\mathfrak{H}}_i)$ and $\otimes_i{\mathcal{F}}({\mathfrak{H}}_i)$ defined in Appendix \[prodstates\], we have $$\begin{aligned}
\omega(e^{-\beta{\mathcal{N}}}) & = & \prod_{i\in{{\ensuremath {\mathbb N} }}}\omega_i\left( 1+(e^{-\beta}-1)a^*(f_i)a(f_i)\right)=\prod_{i\in{{\ensuremath {\mathbb N} }}}\left(1+(e^{-\beta}-1)\lambda_i\right)\\
& = & \operatorname{Det}_{\mathfrak{H}}\left(1+(e^{-\beta}-1)\gamma\right).
\end{aligned}$$ To prove , we notice that for all $\beta{\geqslant}0$, $\omega(e^{-\beta{\mathcal{N}}})=\sum_{k{\geqslant}0}e^{-\beta k}\omega({\mathcal{P}}_k)$, and we identify the coefficients of $e^{-\beta k}$ in $\prod_{i\in{{\ensuremath {\mathbb N} }}}\left(1+(e^{-\beta}-1)\lambda_i\right)$. This yields $$\begin{aligned}
\omega({\mathcal{P}}_k) & = & \sum_{\substack{I\subset{{\ensuremath {\mathbb N} }}\\ \#I=k}}\prod_{i\in I}\lambda_i\prod_{j\notin I}(1-\lambda_j)\\
& {\leqslant}& \sum_{\substack{I\subset{{\ensuremath {\mathbb N} }}\\ \#I=k}}\left(\prod_{i\in I}\lambda_i\right)e^{-\sum_{j\notin I}\lambda_j} \\
& {\leqslant}& e^{-\operatorname{Tr}(\gamma)}\sum_{\substack{I\subset{{\ensuremath {\mathbb N} }}\\ \#I=k}}\prod_{i\in I}\lambda_i e^{\lambda_i} \\
& {\leqslant}& \frac{e^k}{k!}e^{-\operatorname{Tr}(\gamma)}\sum_{i_1,\ldots,i_k\in{{\ensuremath {\mathbb N} }}}\lambda_{i_1}\cdots\lambda_{i_k}=\frac{(e\operatorname{Tr}(\gamma))^k}{k!}e^{-\operatorname{Tr}(\gamma)},
\end{aligned}$$ where we used that $0{\leqslant}\lambda_i{\leqslant}1$ for all $i$. Notice from the first equality that $\omega({\mathcal{P}}_k)=0$ if $k<\dim\ker(\gamma-1)$.
Pure Hartree-Fock-Bogoliubov case {#sec:HFBpure}
---------------------------------
\[prop:HFBpure\] Let $\omega$ a quasi-free *pure* state with $\operatorname{Tr}(\gamma)<+{{\ensuremath {\infty}}}$. Then for any $\beta{\geqslant}0$ we have $$\label{eq:HFBpureequality}
\boxed{
\omega(e^{-\beta{\mathcal{N}}})=\operatorname{Det}_{\mathfrak{H}}\sqrt{1+(e^{-2\beta}-1)\gamma}.
}$$ We also have the following estimate for all $k=k_0+2\ell$ with $k_0:=\dim\ker(\gamma-1)$ and $\ell{\geqslant}0$ $$\boxed{
\omega({\mathcal{P}}_k){\leqslant}\frac{e^{k/2}}{\ell!}\left(\frac{\operatorname{Tr}(\gamma)}{2}\right)^\ell e^{-\frac{\operatorname{Tr}(\gamma)}{2}},
}$$ while $\omega({\mathcal{P}}_k)=0$ if $k<k_0$ or $k=k_0+2\ell+1$.
It is well-known [@BacLieSol-94 Theorem 2.6] that $\omega$ is pure if and only if $\Gamma(\gamma,\alpha)^2=\Gamma(\gamma,\alpha)$, which is equivalent to $\gamma^2+\alpha\alpha^*=\gamma$ and $[\gamma,\alpha J]=0$. The operator $\gamma$ is trace-class and $\alpha J$ is anti-hermitian and Hilbert-Schmidt, hence both $\gamma$ and $\alpha J$ are diagonalizable. Since they commute, they are simultaneously diagonalizable. Remember that any anti-hermitian can be diagonalized in $1\times1$ blocks corresponding to its kernel and $2\times2$ blocks. Hence there exists a decomposition ${\mathfrak{H}}=\oplus_{i{\geqslant}0}{\mathfrak{H}}_i$, with $\dim({\mathfrak{H}}_i){\leqslant}2$ such that
- For all $i$, $\gamma$ and $\alpha J$ stabilize ${\mathfrak{H}}_i$;
- If $\dim({\mathfrak{H}}_i)=1$ then $\gamma_{|{\mathfrak{H}}_i}=\lambda_i\text{Id}_{{\mathfrak{H}}_i},\alpha J_{|{\mathfrak{H}}_i}=0$;
- If $\dim({\mathfrak{H}}_i)=2$ then $\gamma_{|{\mathfrak{H}}_i}=\left(
\begin{array}{cc}
\lambda_i & 0\\
0 & \lambda_i
\end{array}\right),
\alpha J_{|{\mathfrak{H}}_i}=\left(
\begin{array}{cc}
0 & \alpha_i\\
-\alpha_i & 0
\end{array}\right)$ with ${\alpha_i\in{{\ensuremath {\mathbb R} }}}$ and $\alpha_i^2=\lambda_i-\lambda_i^2$.
In particular, $\omega=\otimes_{i{\geqslant}0}\omega_i$ where $\omega_i$ is the quasi-free state on ${\mathcal{F}}({\mathfrak{H}}_i)$ with 1-pdm $\gamma_{|{\mathfrak{H}}_i}=:\gamma_i$ and pairing matrix $\alpha J_{|{\mathfrak{H}}_i}$. Let us now prove that for all $i$ $$\label{eq:HFB-pure-ineq_i}
\omega_i( e^{-\beta{\mathcal{N}}_i})=\operatorname{Det}_{{\mathfrak{H}}_i}\sqrt{1+(e^{-2\beta}-1)\gamma_i},$$ where ${\mathcal{N}}_i$ is the number operator on ${\mathcal{F}}({\mathfrak{H}}_i)$. First we consider the case $\dim({\mathfrak{H}}_i)=1$, and let $f_i\in{\mathfrak{H}}_i$ be a normalized vector. Then ${\mathcal{F}}({\mathfrak{H}}_i)={{\ensuremath {\mathbb C} }}\oplus{{\ensuremath {\mathbb C} }}f_i$ and ${\mathcal{N}}_i=a^*(f_i)a(f_i)$ so that $e^{-\beta{\mathcal{N}}_i}=1+(e^{-\beta}-1)a^*(f_i)a(f_i)$. Therefore $$\omega_i(e^{-\beta{\mathcal{N}}_i})=1+(e^{-\beta}-1)\lambda_i.$$ Since $\gamma^2+\alpha\alpha^*=\gamma$, we have $\alpha_i=0$ if $\dim({\mathfrak{H}}_i)=1$ hence $\lambda_i=0$ or $\lambda_i=1$. In both cases we have $$\omega_i( e^{-\beta{\mathcal{N}}_i})=\operatorname{Det}_{{\mathfrak{H}}_i}\sqrt{1+(e^{-2\beta}-1)\gamma_i}.$$ Now suppose $\dim({\mathfrak{H}}_i)=2$ and let $(f_i,g_i)$ be an orthonormal basis of ${\mathfrak{H}}_i$ such that in this basis $\gamma_{|{\mathfrak{H}}_i}$ and $\alpha J_{|{\mathfrak{H}}_i}$ have the form given above. Then ${\mathcal{F}}({\mathfrak{H}}_i)={{\ensuremath {\mathbb C} }}\oplus{{\ensuremath {\mathbb C} }}f_i\oplus{{\ensuremath {\mathbb C} }}g_i\oplus{{\ensuremath {\mathbb C} }}f_i\wedge g_i$ and ${\mathcal{N}}_i=a^*(f_i)a(f_i)+a^*(g_i)a(g_i)$, so that $$e^{-\beta{\mathcal{N}}_i}=(1+(e^{-\beta}-1)a^*(f_i)a(f_i))(1+(e^{-\beta}-1)a^*(g_i)a(g_i)).$$ We deduce that $$\begin{aligned}
\omega_i(e^{-\beta{\mathcal{N}}_i}) & = & 1+(e^{-\beta}-1)\omega_i({\mathcal{N}}_i)+(e^{-\beta}-1)^2\omega_i(a^*(f_i)a(f_i)a^*(g_i)a(g_i)) \\
& = & 1+(e^{-\beta}-1)\omega_i({\mathcal{N}}_i)+ (e^{-\beta}-1)^2[\omega_i(a^*(f_i)a(f_i))\omega_i(a^*(g_i)a(g_i))\\
& & -\omega_i(a^*(f_i)a^*(g_i))\omega_i(a(f_i)a(g_i))+ \omega_i(a^*(f_i)a(g_i))\omega_i(a(f_i)a^*(g_i))] \\
& = & 1+2(e^{-\beta}-1)\lambda_i+(e^{-\beta}-1)^2(\lambda_i^2+\alpha_i^2) \\
& = & 1+2(e^{-\beta}-1)\lambda_i+(e^{-\beta}-1)^2\lambda_i \\
& = & 1+(e^{-2\beta}-1)\lambda_i=\operatorname{Det}_{{\mathfrak{H}}_i}\sqrt{1+(e^{-2\beta}-1)\gamma_i},
\end{aligned}$$ where in the second equality we used Wick’s relation for $\omega_i$. The equality then follows by taking the product of the relations . Putting aside the indices $i$ such that $\lambda_i=1$, we obtain $$\omega(e^{-\beta{\mathcal{N}}})=e^{-\beta k_0}\prod_{i\in{{\ensuremath {\mathbb N} }}}(1+(e^{-2\beta}-1)\lambda_i),$$ where $k_0:=\dim\ker(\gamma-1)$. Identifying the coefficient of $e^{-\beta k}$ in both sides, as in the proof of Proposition \[prop:HF\], we find that $\omega({\mathcal{P}}_k)=0$ for all $k<k_0$, and that $$\begin{aligned}
\omega({\mathcal{P}}_k) & = & \sum_{\substack{I\subset{{\ensuremath {\mathbb N} }}\\ \#I=\ell}}\prod_{i\in I}\lambda_i\prod_{j\notin I}(1-\lambda_j)\\
& {\leqslant}& \frac{e^\ell}{\ell !}\left(\frac{\operatorname{Tr}(\gamma)-k_0}{2}\right)^\ell e^{-\frac{\operatorname{Tr}(\gamma)-k_0}{2}} {\leqslant}\frac{e^{k/2}}{\ell!}\left(\frac{\operatorname{Tr}(\gamma)}{2}\right)^\ell e^{-\frac{\operatorname{Tr}(\gamma)}{2}},\end{aligned}$$ for $k=k_0+2\ell$ with $\ell{\geqslant}0$. This concludes the proof of Proposition \[prop:HFBpure\].
Mixed Hartree-Fock-Bogoliubov case {#sec:HFBmixed}
----------------------------------
In the most general case of a mixed HFB state, we cannot apply the same strategy as in the previous cases, i.e. identify $\omega$ as a product of states living on smaller dimensional spaces. Indeed, $\gamma$ and $\alpha J$ can have no common stable finite-dimensional subspaces. However, we can still prove an estimate on the vacuum expectation.
\[prop:HFBmixed\] Let $\omega$ a quasi-free state with $\operatorname{Tr}(\gamma)<+{{\ensuremath {\infty}}}$. Then we have the following estimate $$\boxed{
\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}}){\leqslant}e^{-a\operatorname{Tr}(\gamma)}
}$$ where $a=\max_{\beta{\geqslant}0}\frac{3(e^{\beta}-1)}{4e^{3\beta}+7e^{\beta}-8}=0.0941248\ldots$.
We believe that for any sequence of quasi-free state $(\omega_n)_n$ with finite number of particles, we have $\omega_n(e^{-\beta{\mathcal{N}}})\to0$ as $\omega_n({\mathcal{N}})\to+{{\ensuremath {\infty}}}$ for all $\beta>0$, as it was the case in the previous sections. However, our method does not provide this result.
Let $\omega$ be a quasi-free state with $\operatorname{Tr}(\gamma)<+{{\ensuremath {\infty}}}$. According to [@BacLieSol-94 Eq. (2b.26)], there exists a Bogoliubov map ${\mathcal{V}}:{\mathfrak{H}}\oplus{\mathfrak{K}}\to{\mathfrak{H}}\oplus{\mathfrak{K}}$, i.e. a unitary operator of the form $$\label{bogo}
{\mathcal{V}}=\left(
\begin{array}{cc}
U & J^* V J^*\\
V & JUJ^*
\end{array}\right).$$ with $\operatorname{Tr}(V^*V)<+{{\ensuremath {\infty}}}$, such that $$\label{eq:GammaU}
{\mathcal{V}}\Gamma(\gamma,\alpha){\mathcal{V}}^*=\Gamma(D,0)$$ with $D=\text{diag}(\lambda_i)_{i{\geqslant}0}$, $\sum_i\lambda_i<+{{\ensuremath {\infty}}}$, and $0{\leqslant}\lambda_i{\leqslant}1/2$ for all $i$. Let $\omega'$ be the unique HF state associated with $\Gamma(D,0)$ given by Proposition \[BacLieSol\]. Let us also denote by ${{\ensuremath {\mathbb U} }}$ the unitary operator on ${\mathcal{F}}$ lifting ${\mathcal{V}}$ [@Solovej-07 Theorem 9.5]. Then $$\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}})=\omega'({{\ensuremath {\mathbb U} }}|\Omega\rangle\langle\Omega|{{\ensuremath {\mathbb U} }}^*)=\omega'(|\Omega_{{\ensuremath {\mathbb U} }}\rangle\langle\Omega_{{\ensuremath {\mathbb U} }}|),$$ with $|\Omega_{{\ensuremath {\mathbb U} }}\rangle={{\ensuremath {\mathbb U} }}|\Omega\rangle$. We now estimate $|\Omega_{{\ensuremath {\mathbb U} }}\rangle\langle\Omega_{{\ensuremath {\mathbb U} }}|$ by $e^{-\beta{\mathcal{N}}}$ for any $\beta{\geqslant}0$. Thus, let us fix $\beta{\geqslant}0$. At the end, we will optimize over $\beta$. By [@Solovej-07 Eq. (67)] we can write $$|\Omega_{{\ensuremath {\mathbb U} }}\rangle=\prod_{i=-K}^{-1}a^*(\eta_i)\prod_{i{\geqslant}0}(\alpha_i-\beta_i a^*(\eta_{2i})a^*(\eta_{2i+1}))|\Omega\rangle,$$ where $(\eta_i)_{i\in{{\ensuremath {\mathbb Z} }}}$ is an orthonormal basis in ${\mathfrak{H}}$, $(\eta_i)_{-K{\leqslant}i{\leqslant}-1}$ are eigenvectors of $V^*V$ for the eigenvalue 1, $\alpha_i^2+\beta_i^2=1$ for all $i{\geqslant}0$, and $(\beta_i^2)_{i{\geqslant}0}$ are the eigenvalues of $V^*V$ strictly between 0 and 1, which are all of multiplicity 2. We interpret this equality by saying that $$|\Omega_{{\ensuremath {\mathbb U} }}\rangle=T^*(\otimes_i|\psi_i\rangle)\in T^*(\otimes_i{\mathcal{F}}({\mathfrak{H}}_i))={\mathcal{F}},$$ where ${\mathfrak{H}}_i={{\ensuremath {\mathbb C} }}\eta_i$, $|\psi_i\rangle=\eta_i$ for $-K{\leqslant}i{\leqslant}-1 $, ${\mathfrak{H}}_i={{\ensuremath {\mathbb C} }}\eta_{2i+1}\oplus{{\ensuremath {\mathbb C} }}\eta_{2i}$, $|\psi_i\rangle=\alpha_i-\beta_i\eta_{2i}\wedge\eta_{2i+1}$ for $i{\geqslant}0$, and ${\mathfrak{H}}_i={{\ensuremath {\mathbb C} }}\eta_i$, $|\psi_i\rangle=\Omega_i$ (the vacuum in ${\mathcal{F}}({\mathfrak{H}}_i)$) if $i< -K$. Recall that the operator $T$ is the unitary transformation between ${\mathcal{F}}(\oplus_i{\mathfrak{H}}_i)$ and $\otimes_i{\mathcal{F}}({\mathfrak{H}}_i)$ defined in Appendix \[prodstates\] . If $-K{\leqslant}i{\leqslant}-1$, then ${\mathcal{F}}({\mathfrak{H}}_i)={{\ensuremath {\mathbb C} }}\oplus{{\ensuremath {\mathbb C} }}\eta_i$ and the matrix of $|\psi_i\rangle\langle\psi_i|$ in the basis $(1,\eta_i)$ can be dominated by $$|\psi_i\rangle\langle\psi_i|=\left(
\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right)
{\leqslant}e^\beta\left(
\begin{array}{cc}
1 & 0\\
0 & e^{-\beta}
\end{array}\right)
=e^{\beta-\beta{\mathcal{N}}_i}{\leqslant}e^{\frac{e^{2\beta}-1}{2}-\beta{\mathcal{N}}_i},$$ where as usual ${\mathcal{N}}_i$ is the number operator on ${\mathcal{F}}({\mathfrak{H}}_i)$. If $i<-K$ then ${\mathcal{F}}({\mathfrak{H}}_i)={{\ensuremath {\mathbb C} }}\oplus{{\ensuremath {\mathbb C} }}\eta_i$ and we have $$|\psi_i\rangle\langle\psi_i|=\left(
\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right)
{\leqslant}\left(
\begin{array}{cc}
1 & 0\\
0 & e^{-\beta}
\end{array}\right)
=e^{-\beta{\mathcal{N}}_i}.$$ Now if $i{\geqslant}0$ then ${\mathcal{F}}({\mathfrak{H}}_i)={{\ensuremath {\mathbb C} }}\oplus{{\ensuremath {\mathbb C} }}\eta_{2i}\oplus{{\ensuremath {\mathbb C} }}\eta_{2i+1}\oplus{{\ensuremath {\mathbb C} }}\eta_{2i}\wedge\eta_{2i+1}$. In the basis $(1,\eta_{2i},\eta_{2i+1},\eta_{2i}\wedge\eta_{2i+1})$ the matrix of $|\psi_i\rangle\langle\psi_i|$ can be dominated by $$\begin{aligned}
|\psi_i\rangle\langle\psi_i| = \left(
\begin{array}{cccc}
\alpha_i^2 & 0 & 0 & -\alpha_i\beta_i\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
-\alpha_i\beta_i & 0 & 0 & \beta_i^2
\end{array}\right) & {\leqslant}& e^{(e^{2\beta}-1)\beta_i^2} \left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & e^{-\beta} & 0 & 0\\
0 & 0 & e^{-\beta} & 0\\
0 & 0 & 0 & e^{-2\beta}
\end{array}\right)\\
& = & e^{(e^{2\beta}-1)\beta_i^2-\beta{\mathcal{N}}_i},
\end{aligned}$$ since with $$A=\left( \begin{array}{cc}
\alpha_i^2 & -\alpha_i\beta_i\\
-\alpha_i\beta_i & \beta_i^2
\end{array}
\right), \qquad
B=\left(\begin{array}{cc}
1 & 0\\
0 & e^{-2\beta}
\end{array}
\right)$$ we have $B^{-1/2}AB^{-1/2}{\leqslant}(1+(e^{2\beta}-1)\beta_i^2)\text{Id}_{{{\ensuremath {\mathbb C} }}^2}{\leqslant}e^{(e^{2\beta}-1)\beta_i^2}\text{Id}_{{{\ensuremath {\mathbb C} }}^2}$. We know that $$\operatorname{Tr}(V^*V)=\sum_{i=-K}^{-1}1+\sum_{i{\geqslant}0}2\beta_i^2,$$ so that $$\begin{aligned}
|\Omega_{{\ensuremath {\mathbb U} }}\rangle\langle\Omega_{{\ensuremath {\mathbb U} }}| & = & T^*\left[\bigotimes_{i\in{{\ensuremath {\mathbb Z} }}}|\psi_i\rangle\langle\psi_i|\right]T\\
& {\leqslant}& T^*\left[\bigotimes_{i=-K}^{-1}e^{\frac{e^{2\beta}-1}{2}-\beta{\mathcal{N}}_i}\otimes\bigotimes_{i{\geqslant}0}e^{\frac{e^{2\beta}-1}{2}2\beta_i^2-\beta{\mathcal{N}}_i}\otimes\bigotimes_{i<-K}e^{-\beta{\mathcal{N}}_i}\right]T\\
& = & \exp\left(\frac{e^{2\beta}-1}{2}\left(\sum_{i=-K}^{-1}1+\sum_{i{\geqslant}0}2\beta_i^2\right)-\beta\sum_i{\mathcal{N}}_i\right)\\
& = & e^{\frac{e^{2\beta}-1}{2}\operatorname{Tr}(V^*V)}e^{-\beta{\mathcal{N}}}.
\end{aligned}$$ Here we have used that for any operators $A,B,C,D,$ such that $0{\leqslant}A{\leqslant}B$ and $0{\leqslant}C{\leqslant}D$ it holds $0{\leqslant}A\otimes C{\leqslant}B\otimes D$. We obtain the estimate $$\label{firstestimate}
\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}}){\leqslant}e^{\frac{e^{2\beta}-1}{2}\operatorname{Tr}(V^*V)}\omega'(e^{-\beta{\mathcal{N}}}){\leqslant}e^{\frac{e^{2\beta}-1}{2}\operatorname{Tr}(V^*V)+(e^{-\beta}-1)\operatorname{Tr}(D)},$$ where we have used that $\omega'(e^{-\beta{\mathcal{N}}}){\leqslant}e^{(e^{-\beta}-1)\omega'({\mathcal{N}})}$, which is a consequence of the equality applied to the HF state $\omega'$. Unfortunately, the estimate (\[firstestimate\]) is not good enough because the constant $(e^{2\beta}-1)/2$ in front of $\operatorname{Tr}(V^*V)$ is positive, while $\operatorname{Tr}(V^*V)$ represents the number of particles of $|\Omega_{{\ensuremath {\mathbb U} }}\rangle$. We will thus get another estimate by exchanging the roles of $\omega'$ and $|\Omega_{{\ensuremath {\mathbb U} }}\rangle$. The idea is to see $|\Omega_{{\ensuremath {\mathbb U} }}\rangle$ as the state and $\omega'$ as the observable. Since $\ker(D-1)=\{0\}$, it is well-known that $\omega'$ is a normal state with density matrix $G=Z^{-1}\Upsilon(M)$ where $M=\frac{D}{1-D}$, $Z=\operatorname{Tr}_{\mathcal{F}}(\Upsilon(M))$, and $$\label{Upsilon}
\Upsilon(M):=\oplus_{N{\geqslant}0}M^{\otimes N}:{\mathcal{F}}\to{\mathcal{F}}$$ (see for instance [@JakOgaPauPil-11 Proposition 6.6 (1)]). Hence we can write $$\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}})=\omega'(|\Omega_{{\ensuremath {\mathbb U} }}\rangle\langle\Omega_{{\ensuremath {\mathbb U} }}|)=\operatorname{Tr}(Z^{-1}\Upsilon(M)|\Omega_{{\ensuremath {\mathbb U} }}\rangle\langle\Omega_{{\ensuremath {\mathbb U} }}|)=\omega_{{\ensuremath {\mathbb U} }}(Z^{-1}\Upsilon(M)),$$ where $\omega_{{\ensuremath {\mathbb U} }}$ is the pure state associated with the vector $|\Omega_{{\ensuremath {\mathbb U} }}\rangle$. We know that $\omega_{{\ensuremath {\mathbb U} }}({\mathcal{N}})=\operatorname{Tr}(V^*V)$. Hence if we can dominate $Z^{-1}\Upsilon(M)$ by $e^{-\beta'{\mathcal{N}}}$ for a certain $\beta'{\geqslant}0$, we will get another estimate on $\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}})$ by applying equality to the pure state $\omega_{{\ensuremath {\mathbb U} }}$. Let $(\mu_i)_i$ the eigenvalues of $M$, $\mu_i=\frac{\lambda_i}{1-\lambda_i}$. The spectrum of $\Upsilon(M)$ on ${\mathfrak{H}}^k$ is $$\sigma(\Upsilon(M)_{|{\mathfrak{H}}^k})=\left\{\prod_{i\in I}\mu_i,\quad I\subset{{\ensuremath {\mathbb N} }},\quad\# I=k\right\},$$ so that for $I\subset{{\ensuremath {\mathbb N} }}$ of cardinal $k$ we have $$Z^{-1}\prod_{i\in I}\mu_i=\prod_{i{\geqslant}0}\frac{1}{1+\mu_i}\prod_{i\in I}\mu_i=\prod_{i{\geqslant}0}(1-\lambda_i)\prod_{i\in I}\frac{\lambda_i}{1-\lambda_i}=\prod_{i\notin I}(1-\lambda_i)\prod_{i\in I}\lambda_i.$$ Using $0{\leqslant}\lambda_i{\leqslant}1/2$ for all $i$, we finally get $Z^{-1}\Upsilon(M)_{|{\mathfrak{H}}^k}{\leqslant}1/2^k$ so that $$Z^{-1}\Upsilon(M){\leqslant}e^{-(\ln 2){\mathcal{N}}}.$$ Equality now implies that $\omega_{{\ensuremath {\mathbb U} }}(e^{-\beta'{\mathcal{N}}}){\leqslant}e^{\frac{e^{-2\beta'}-1}{2}\omega_{{\ensuremath {\mathbb U} }}({\mathcal{N}})}$ for all $\beta'{\geqslant}0$. Choosing $\beta'=\ln 2$, one gets $$\label{secondestimate}
\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}}){\leqslant}\omega_{{\ensuremath {\mathbb U} }}(e^{-(\ln 2){\mathcal{N}}}){\leqslant}e^{-\frac{3}{8}\operatorname{Tr}(V^*V)}.$$ Interpolating the inequalities (\[firstestimate\]) and (\[secondestimate\]) we get $$\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}}){\leqslant}e^{\theta(e^{-\beta}-1)\operatorname{Tr}(D)+\left[\theta\frac{e^{2\beta}-1}{2}-(1-\theta)\frac{3}{8}\right]\operatorname{Tr}(V^*V)},$$ for all $\beta{\geqslant}0$ and $0{\leqslant}\theta{\leqslant}1$. We choose $\theta$ such that the coefficients before $\operatorname{Tr}(D)$ and $\operatorname{Tr}(V^*V)$ are equal since we have $\operatorname{Tr}(V^*V)+\operatorname{Tr}(D){\geqslant}\operatorname{Tr}(U^*DU)+\operatorname{Tr}(V^*(1-D)V)=\operatorname{Tr}(\gamma)$ by , using $UU^*{\leqslant}1$ and $1-D{\leqslant}1$. We thus choose $$\theta=\frac{3}{7-8e^{-\beta}+4e^{2\beta}},$$ and we obtain $$\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}}){\leqslant}e^{\frac{3(e^{\beta}-1)}{8-7e^{\beta}-4e^{3\beta}}\operatorname{Tr}(\gamma)},$$ for all $\beta{\geqslant}0$. Optimizing the coefficient before $\operatorname{Tr}(\gamma)$, we get the desired estimate with $\beta\simeq0.36443$ and $\theta\simeq0.308194$.
Bogoliubov-Dirac-Fock case {#sec:BDFstates}
--------------------------
In this section we introduce the correct setup for studying electron-positron pair creation. Let ${\mathfrak{H}}$ be a Hilbert space and $\Pi$ be an orthogonal projection on ${\mathfrak{H}}$. We also need an anti-unitary operator $J$ as in Section \[sec:notation\], with the additional assumptions that ${\mathfrak{K}}={\mathfrak{H}}$ (i.e. $J$ maps ${\mathfrak{H}}$ to ${\mathfrak{H}}$) and that $J\Pi J^*=\Pi$ or $J\Pi J^*=1-\Pi$. The particle/anti-particle spaces are given by ${\mathfrak{H}}_+=(1-\Pi){\mathfrak{H}}$ and ${\mathfrak{H}}_-=J \Pi{\mathfrak{H}}$. Notice that ${\mathfrak{H}}_-=\Pi{\mathfrak{H}}$ or ${\mathfrak{H}}_-=(1-\Pi){\mathfrak{H}}$. In the context given by Section \[sec:BDF-notations\], we have ${\mathfrak{H}}={\mathfrak{H}}_\Lambda$, $\Pi=P^0_-$ and $J=i\beta\alpha_2{\mathscr{C}}$ the charge conjugation operator on ${\mathfrak{H}}_\Lambda$ (i.e. such that $J(D^0+V)J^*=-(D^0-V)$ for any scalar potential $V$), with ${\mathscr{C}}$ the complex conjugation on ${\mathfrak{H}}_\Lambda$. With this choice, vectors of ${\mathfrak{H}}_-$ are interpreted as states with a positive energy relatively to the Hamiltonian with an opposite charge. Hence, they represent positronic states. Notice that this specific $J$ verifies $JP^0_-J^*=P^0_+=1-P^0_-$. In the sequel, we will keep a triplet $({\mathfrak{H}},\Pi,J)$ satisfying the assumptions given above.
The mathematical description of Bogoliubov-Dirac-Fock states is a special case of the well known Araki-Wyss representation [@Ara-64] (see [@JakOgaPauPil-11 Section 6.4] for a review). Let ${\mathcal{F}}_0={\mathcal{F}}({\mathfrak{H}}_+)\otimes{\mathcal{F}}({\mathfrak{H}}_-)$. For $f\in{\mathfrak{H}}_+$ and $g\in{\mathfrak{H}}_-$ we denote by $a_+^*(f)$ and $a_-^*(g)$ the usual creation operators on ${\mathcal{F}}({\mathfrak{H}}_+)$ and ${\mathcal{F}}({\mathfrak{H}}_-)$, respectively. We now define the “creation operator” on ${\mathcal{F}}_0$ for all $f\in{\mathfrak{H}}$ by $$\psi^*(f):=a_+^*((1-\Pi)f)\otimes \text{Id}_{{\mathcal{F}}({\mathfrak{H}}_-)}+\Upsilon\left(-\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_+)}\right)\otimes a_-(J \Pi f),$$ where $\Upsilon$ is the operation defined by Equation . The operators $(\psi^*(f))_f$ are not exactly the usual creation operators in the full Fock space since they create a particle in the state $(1-\Pi)f$, and at the same time they annihilate a anti-particle in the state $J \Pi f$, according to the “particle-hole” picture of Dirac’s theory. However, they still satisfy the CAR thanks to the “twist” $\Upsilon\left(-\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_+)}\right)$ on the second term. A *Bogoliubov-Dirac-Fock* (BDF) state is, by definition, a quasi-free state $\omega$ on the $C^*$-algebra ${\mathfrak{A}}_0\subset{\mathcal{B}}({\mathcal{F}}_0)$ generated by the $(\psi(f))_{f\in{\mathfrak{H}}}$. We define the normal ordering $:\hspace{-0.1cm}\psi^*(f)\psi(g)\hspace{-0.1cm}:$ of the operator $\psi^*(f)\psi(g)$ by $$\begin{gathered}
:\hspace{-0.1cm}\psi^*(f)\psi(g)\hspace{-0.1cm}:\: =a^*_+(f_+)a_+(g_+)\otimes\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_-)}+a^*_+(f_+)\Upsilon\left(-\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_+)}\right)\otimes a^*_-(J g_-)\\
+\Upsilon\left(-\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_+)}\right)a_+(g_+)\otimes a_-(J f_-)-\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_+)}\otimes a^*_-(J g_-)a_-(J f_-),\end{gathered}$$ where $h_+=(1-\Pi)h$ and $h_-=\Pi h$ for all $h$. It corresponds to moving all the creation operators $a^*$ to the left of annihilation operators $a$. For any BDF state $\omega$, we define its renormalized one-particle density matrix $Q:{\mathfrak{H}}\to{\mathfrak{H}}$ by $$\langle g,Q f\rangle=\omega(:\hspace{-0.1cm}\psi^*(f)\psi(g)\hspace{-0.1cm}:).$$ and its pairing matrix $p:{\mathfrak{H}}\to{\mathfrak{H}}$ by the usual formula $$\langle p J f, g\rangle = \omega(\psi^*(f)\psi^*(g)).$$ Recall that we have already defined the 1-pdm $\gamma$ by $\langle g,\gamma f\rangle=\omega(\psi^*(f)\psi(g))$. Therefore, we have the relation $\gamma=\Pi+Q$. If ${\mathcal{N}}$ is the number operator on ${\mathcal{F}}_0$, $$\label{eq:numberop}
{\mathcal{N}}=\sum_i a^*_+({\varphi}_{i,+})a_+({\varphi}_{i,+})\otimes\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_-)}+\text{Id}_{{\mathcal{F}}({\mathfrak{H}}_+)}\otimes a^*_-({\varphi}_{i,-})a_-({\varphi}_{i,-}),$$ where $({\varphi}_{i,+})_i$, $({\varphi}_{i,-})_i$ are orthonormal basis for, respectively, ${\mathfrak{H}}_+$ and ${\mathfrak{H}}_-$, then $$\omega({\mathcal{N}})=\operatorname{Tr}(Q_{++}-Q_{--}),$$ where $Q_{++}:=(1-\Pi)Q(1-\Pi)$ and $Q_{--}=\Pi Q\Pi$. Hence, $\omega({\mathcal{N}})<+{{\ensuremath {\infty}}}$ is equivalent to having $Q\in{\mathfrak{S}}_{1,\Pi}({\mathfrak{H}}):=\{Q\in{\mathfrak{S}}_2({\mathfrak{H}}),\,Q_{++},Q_{--}\in{\mathfrak{S}}_1({\mathfrak{H}})\}$. Notice also that as in the HFB case we have $pp^*{\leqslant}\gamma-\gamma^2=Q_{++}-Q_{--}-Q^2$, thus $p\in{\mathfrak{S}}_2({\mathfrak{H}})$ as soon as $\omega({\mathcal{N}})<+{{\ensuremath {\infty}}}$.
The other natural number operator $\sum_i \psi^*(f_i)\psi(f_i)$ gives the total number of particles in the system, that is also those of the vacuum $\Pi$. If $\dim({\mathfrak{H}}_-)=+{{\ensuremath {\infty}}}$, this number is just $+{{\ensuremath {\infty}}}$. However, we only want to count the number of particles relative to the vacuum $\Pi$. That is why the operator ${\mathcal{N}}$ is chosen here: It counts the number of “real” electrons ${\varphi}_{i,+}$ and the number of “holes” ${\varphi}_{i,-}$ in the vacuum.
The following proposition in an easy adaptation of the arguments given in [@BacBarHelSie-99 pp. 449–450]. We give the complete proof here to clarify the link between BDF states and HFB states.
\[prop:BDF-existence\] Let $(Q,p)\in{\mathfrak{S}}_{1,\Pi}({\mathfrak{H}})\times{\mathfrak{S}}_2({\mathfrak{H}})$ such that $$0{\leqslant}\Gamma(\Pi+Q,p)=\left(
\begin{array}{cc}
\Pi+Q & p\\
p^* & 1- J(\Pi+Q)J^*
\end{array}
\right){\leqslant}1$$ as an operator on ${\mathfrak{H}}\oplus{\mathfrak{H}}$. Then there exists a unique, normal, BDF state on ${\mathcal{F}}_0$ having $Q$ as renormalized 1-pdm and $p$ as pairing matrix.
We are going to construct an HFB state on $\operatorname{\mathfrak{A}}[{\mathfrak{H}}_+\oplus{\mathfrak{H}}_-]$ using Proposition \[BacLieSol\], and then transform it into a BDF state having the desired property via the unitary transformation $$T:{\mathcal{F}}({\mathfrak{H}}_+\oplus{\mathfrak{H}}_-)\to{\mathcal{F}}_0={\mathcal{F}}({\mathfrak{H}}_+)\otimes{\mathcal{F}}({\mathfrak{H}}_-),$$ defined linearly by its action on each $N$-particle space $$\begin{gathered}
T{\varphi}_{i_1,+}\wedge\cdots\wedge{\varphi}_{i_k,+}\wedge{\varphi}_{j_1,-}\wedge\cdots\wedge{\varphi}_{j_{N-k},-}=\\
\left({\varphi}_{i_1,+}\wedge\cdots\wedge{\varphi}_{i_k,+}\right)\otimes\left({\varphi}_{j_1,-}\wedge\cdots\wedge{\varphi}_{j_{N-k},-}\right),\end{gathered}$$ with the convention $T\Omega=\Omega_+\otimes\Omega_-$. Since $T$ leaves the scalar product invariant and maps an orthonormal basis for ${\mathcal{F}}({\mathfrak{H}}_+\oplus{\mathfrak{H}}_-)$ onto an orthonormal basis for ${\mathcal{F}}_0$, it naturally extends to a unique unitary operator on ${\mathcal{F}}({\mathfrak{H}}_+\oplus{\mathfrak{H}}_-)$. It induces the following transformation of the CAR: $$\label{AW-CAR}
T a({\varphi}_+\oplus{\varphi}_-)T^*=\psi({\varphi}_+\oplus{\varphi}_-).$$ Define $p_{++}=(1-\Pi) p (1-\Pi)$, $p_{+-}=(1-\Pi)p\Pi$, etc. Suppose also that we are in the case where $J\Pi J^*=1-\Pi$, and write ${\mathfrak{H}}^0_+=(1-\Pi){\mathfrak{H}}={\mathfrak{H}}_+$, ${\mathfrak{H}}^0_-=\Pi{\mathfrak{H}}$. We now introduce $$\gamma_0:=\left(
\begin{array}{cc}
Q_{++} & p_{++}\\
p_{++}^* & -J Q_{--}J^*
\end{array}
\right),\quad \alpha_0:=\left(
\begin{array}{cc}
p_{+-} & Q_{+-}\\
-J Q_{-+}J^* & p_{-+}^*
\end{array}
\right),$$ where $\gamma_0:{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_+^0\to{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_+^0$ and $\alpha_0:{\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_-^0\to{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_+^0$. Let us define ${\mathcal{J}}:{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_+^0\to{\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_-^0$ by $${\mathcal{J}}=\left(
\begin{array}{cc}
J & 0\\
0 & J^*
\end{array}
\right),$$ and the operator $\Gamma_0$ on $({\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_+^0)\oplus({\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_-^0)$ by $$\Gamma_0=\Gamma(\gamma_0,\alpha_0)=\left(
\begin{array}{cc}
\gamma_0 & \alpha_0\\
\alpha_0^* & 1-{\mathcal{J}}^*\gamma_0{\mathcal{J}}\end{array}
\right).$$ We now show that there exists a HFB state $\omega_0$ on $\operatorname{\mathfrak{A}}[{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_+^0]$ having $\Gamma_0$ as density matrix. Hence we prove that $0{\leqslant}\Gamma_0{\leqslant}1$. Let us first write the block decomposition of $\Gamma(\Pi+Q,p)$ as an operator on ${\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0={\mathfrak{H}}\oplus{\mathfrak{H}}$ $$\Gamma(\Pi+Q,p)=\left(
\begin{array}{cccc}
Q_{++} & Q_{+-} & p_{++} & p_{+-}\\
Q_{-+} & 1+Q_{--} & p_{-+} & p_{--}\\
p^*_{++} & p^*_{-+} & -JQ_{--}J^* & -JQ_{-+}J^*\\
p^*{+-} & p^*_{--} & -JQ_{+-}J^* & 1-JQ_{++}J^*
\end{array}\right),$$ where we have used that $J{\mathfrak{H}}^0_\pm={\mathfrak{H}}^0_\mp$ to write the lower right block. Let us consider the unitary operator $W:{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_-^0\to{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0$ whose matrix is $$W=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
0 & 1 & 0 & 0\\
\end{array}\right).$$ Then, as it was noticed in [@BacBarHelSie-99], we have the relation $$\Gamma_0=W^*\Gamma(\Pi+Q,p) W.$$ By assumption, we have $0{\leqslant}\Gamma(\Pi+Q,p){\leqslant}1$, so that $0{\leqslant}\Gamma_0{\leqslant}1$ as well. Now we have $\operatorname{Tr}(\gamma_0)=\operatorname{Tr}(Q_{++}-Q_{--})<{{\ensuremath {\infty}}}$ hence by Proposition \[BacLieSol\] with ${\mathfrak{H}}={\mathfrak{H}}^0_+\oplus{\mathfrak{H}}^0_+={\mathfrak{H}}_+\oplus{\mathfrak{H}}_-$ and ${\mathfrak{K}}={\mathfrak{H}}^0_-\oplus{\mathfrak{H}}^0_-$, there exists a unique, normal, HFB state $\omega_0$ on ${\mathcal{F}}({\mathfrak{H}}_+\oplus{\mathfrak{H}}_-)$ with finite number of particles having $\Gamma_0$ as density matrix. We define a state on ${\mathcal{F}}_0$ via the unitary operator $T$ by $$\omega(A)=\omega_0(T^*AT),\quad\forall A\in{\mathcal{B}}({\mathcal{F}}_0).$$ By (\[AW-CAR\]), $\omega$ is the quasi-free state with renormalized 1-pdm $Q$ and pairing matrix $p$. Furthermore, $\omega$ is obviously normal since $\omega_0$ is normal. This concludes the proof of Proposition \[prop:BDF-existence\], in the case where $J\Pi J^*=1-\Pi$. If $J\Pi J^*=\Pi$, the proof is the same with $W:{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0\to{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0\oplus{\mathfrak{H}}_+^0\oplus{\mathfrak{H}}_-^0$ whose matrix is $$W=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
\end{array}\right),$$ which is the same as in [@BacBarHelSie-99] where $J$ was the complex conjugation. With this choice of $W$, $W^*\Gamma(\Pi+Q,p) W$ is the density matrix of a HFB state with ${\mathfrak{H}}={\mathfrak{H}}^0_+\oplus{\mathfrak{H}}^0_-={\mathfrak{K}}$.
\[coro:estimate-BDF\] Let $\omega$ be a BDF state with renormalized one-particle density matrix $Q$ such that $\operatorname{Tr}(Q_{++}-Q_{--})<+{{\ensuremath {\infty}}}$. Then we have the estimate $$\label{eq:BDF-ineq}
\boxed{
\omega({{\ensuremath{| \Omega_0\rangle\langle \Omega_0|}}}){\leqslant}e^{-a\operatorname{Tr}(Q_{++}-Q_{--})},
}$$ where $a$ is the same constant as in Proposition \[prop:HFBmixed\] and $\Omega_0$ is the vacuum in ${\mathcal{F}}_0$.
Use Proposition \[prop:HFBmixed\] to the HFB mixed state $\omega_0$ constructed in the proof of Proposition \[prop:BDF-existence\].
In [@HaiLewSol-07], the rBDF energy is derived by evaluating the QED Hamiltonian on BDF states. As usual in mean-field theories, this energy only depends on $(Q,p)$. However, since the interaction between the particules is repulsive, any minimizer of the rBDF energy has $p=0$. This explains why we can take $p=0$ for the polarized vacuum in the proof of Theorem \[th:main\].
Asymptotics of the Polarized Vacuum Energy in strong external fields {#sec:asymptotics}
====================================================================
Main Results
------------
In this section, we study the asymptotics of the reduced BDF ground state energy $E(Z)$ for large $Z$, where we recall that $E(Z) = \inf \{ {{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q)}}}, Q\in{\mathcal{K}}\}$, with ${\mathcal{K}}$ defined by Equation .
\[prop:limsup\] Let $\alpha>0$ and $\Lambda>0$. Let $\nu\in{\mathcal{C}}\cap L^1({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb R} }})$ be such that $\int|\nu|=1$ and $q=\int\nu\neq0$. Then $$\boxed{
\limsup_{Z\to+{{\ensuremath {\infty}}}}\frac{E(Z)}{Z^{5/3}}{\leqslant}-c_1\alpha\Lambda |q|^{5/3},
}$$ where $c_1=2^{-23/3}3^{1/3}\pi^{-4/3}=0.001543\ldots$.
To estimate the convergence rate of $E(Z)Z^{-5/3}$, we need a further assumption on the decay of $\nu$ at infinity.
\[prop:speed\] Let $\alpha>0$ and $\Lambda>0$. Let $\nu\in{\mathcal{C}}\cap L^1({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb R} }})$ be such that $\int|x||\nu(x)|{{\ensuremath{\,\text{d}x}}}<+{{\ensuremath {\infty}}}$, $\int|\nu|=1$ and $q=\int\nu\neq0$. Then there exists a constant ${\widetilde}{Z_1}={\widetilde}{Z_1}(\Lambda,\alpha,|q|,\||x|\nu\|_1)>0$ such that $$\label{eq:speed}
\forall Z> {\widetilde}{Z_1},\qquad E(Z){\leqslant}-\frac{c_1}{2}\alpha\Lambda |q|^{5/3}Z^{5/3},$$ where $c_1$ is defined in Proposition \[prop:limsup\].
The constant ${\widetilde}{Z_1}$ behaves as $\Lambda^3$ when $\Lambda\to{{\ensuremath {\infty}}}$, as $\alpha^{-3/2}$ when $\alpha\to0$ and as $|q|^{-4}$ when $q\to0$. It is probably not optimal.
We also give a lower bound for $E(Z)$, proving that the power $Z^{5/3}$ is optimal.
\[prop:liminf\] Let $\alpha>0$, $\Lambda>0$, and $\nu\in{\mathcal{C}}\cap L^1({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb R} }})$ with $\int|\nu|=1$. Then for all $Z>0$ we have $$\boxed{
E(Z){\geqslant}-c_2\alpha\Lambda Z^{5/3},
}$$ where $c_2:=2^{-4/3}3^{2/3}\pi^{-1/3}= 0.563626\ldots$.
From the asymptotics of $E(Z)$ in Proposition \[prop:speed\] we can now derive a lower bound on the total number of particles and anti-particles in the polarized vacuum.
\[coro:number-estimate\] Let $\alpha>0$, $\Lambda>0$. Let $\nu\in{\mathcal{C}}\cap L^1({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb R} }})$ with $\int|\nu|=1$, $\int|x||\nu(x)|{{\ensuremath{\,\text{d}x}}}<+{{\ensuremath {\infty}}}$, and $q=\int\nu\neq0$. Then for any minimizer $Q$ for $E(Z)$ and for all $Z>{\widetilde}{Z_1}$, we have $$\operatorname{Tr}(Q_{++}-Q_{--}){\geqslant}CZ^{2/3},$$ where ${\widetilde}{Z_1}$ is the same as in Proposition \[prop:speed\] and $C$ is a constant independent of $Z$, given in Equation below.
Proof of the lower bound
------------------------
We first give the proof of the lower bound in Proposition \[prop:liminf\], which is easier than the upper bound.
\[key-lemma\] For any $Q\in{\mathcal{K}}$, $\rho_Q\in L^{{{\ensuremath {\infty}}}}({{\ensuremath {\mathbb R} }}^3)$ and $\|\rho_Q\|_{L^{{\ensuremath {\infty}}}}{\leqslant}\frac{\Lambda^3}{6\pi^2}$.
The proof will only use the fact that any $Q\in{\mathcal{K}}$ is a bounded operator on ${\mathfrak{H}}_\Lambda$. Hence, let $Q\in{\mathcal{K}}$ and $V\in L^1({{\ensuremath {\mathbb R} }}^3)\cap L^2({{\ensuremath {\mathbb R} }}^3)$. Since $\rho_Q\in L^2({{\ensuremath {\mathbb R} }}^3)$ we know that $$\int\rho_Q V=\operatorname{Tr}_0(QV),$$ where in the trace $V$ is seen as a multiplication operator on $L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)$. Let us denote by $\Pi_\Lambda$ the multiplication operator by ${\boldsymbol{1}}_{B(0,\Lambda)}$ in Fourier space. Since $Q$ is an operator on ${\mathfrak{H}}_\Lambda$, we have $\Pi_\Lambda Q\Pi_\Lambda=Q$. Now assume that $V{\geqslant}0$. Then $$\begin{aligned}
\left|\int\rho_Q V\right| & = & \left|\operatorname{Tr}_0\left(Q\Pi_\Lambda V\Pi_\Lambda\right)\right|\\
& {\leqslant}& \|\Pi_\Lambda V\Pi_\Lambda\|_{{\mathfrak{S}}_1}=\|\sqrt{V}\Pi_\Lambda\|_{{\mathfrak{S}}_2}^2{\leqslant}(2\pi)^{-3}\int V\times\frac{4}{3}\pi\Lambda^3,
\end{aligned}$$ where we used $\|Q\|{\leqslant}1$ and the Kato-Seiler-Simon inequality: $$\forall p{\geqslant}2,\qquad\|f(x)g(p)\|_{{\mathfrak{S}}_p}{\leqslant}(2\pi)^{-3/p}\|f\|_{L^p}\|g\|_{L^p}.$$ Now if $V$ is not necessarily non-negative, we split $V=V_+-V_-$ with $V_+=\max(V,0)$ and $V_-=\max(-V,0)$. Then we apply the previous bound twice to obtain $$\left|\int\rho_Q V\right|{\leqslant}\left|\int\rho_Q V_+\right|+\left|\int\rho_Q V_-\right|{\leqslant}\frac{\Lambda^3}{6\pi^2}\left(\int V_+ +\int V_-\right)=\frac{\Lambda^3}{6\pi^2}\|V\|_{L^1}.$$ By the density of $L^1\cap L^2$ in $L^1$ and by the fact that $(L^1)'\simeq L^{{\ensuremath {\infty}}}$, we get the result.
Lemma \[key-lemma\] is crucial to understand the $Z^{5/3}$ behaviour of $E(Z)$. Indeed, an easy lower bound to ${\mathcal{E}}_{\text{rBDF}}^{Z\nu}$ is ${\mathcal{E}}_{\text{rBDF}}^{Z\nu}(Q){\geqslant}-\alpha Z^2 D(\nu,\nu)/2$ for all $Q$, using the positivity of the kinetic energy and completing the square in the other terms. One may think that this lower bound would be attained by a $Q$ such that $\rho_Q\simeq Z\nu$, i.e. by a state which density of charge compensates the external field. However, Lemma \[key-lemma\] implies that such a state cannot exist in ${\mathcal{K}}$, precisely because of the cut-off $\Lambda$. In other words, the vacuum cannot “follow” the external field when the field is too strong.
. Let $Q\in{\mathcal{K}}$, then $${{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q)}}}{\geqslant}\widetilde{{\mathcal{E}}}^{Z\nu}(\rho_Q):=-\alpha D(\rho_Q,Z\nu)+\frac{\alpha}{2}D(\rho_Q,\rho_Q).$$ We also know that for all $Q\in{\mathcal{K}}$, one has $\|\rho_Q\|_{L^{{\ensuremath {\infty}}}}{\leqslant}\frac{\Lambda^3}{6\pi^2}=:\delta$, so that $$E(Z){\geqslant}\widetilde{E}(Z,\delta):=\inf\left\{\widetilde{{\mathcal{E}}}^{Z\nu}(\rho),\:\rho\in{\mathcal{C}}\cap L^{{\ensuremath {\infty}}}({{\ensuremath {\mathbb R} }}^3), \|\rho\|_{L^{{\ensuremath {\infty}}}}{\leqslant}\delta\right\}.$$ The variational problem $\widetilde{E}(Z,\delta)$ has the scaling property $$\widetilde{E}(Z,\delta)=Z^2\widetilde{E}(1,\delta/Z).$$ Define ${\varepsilon}=\delta/Z$. We will now show that $$\widetilde{E}(1,{\varepsilon}){\geqslant}-\frac{3}{4}(8\pi{\varepsilon})^{1/3},$$ which then implies Proposition \[prop:liminf\]. Let $\rho$ be a trial state for ${\widetilde}{E}(1,{\varepsilon})$. Let $R>0$ and write $|\cdot|^{-1}=V_1+V_{{\ensuremath {\infty}}}$ with $V_1:={\boldsymbol{1}}_{|x|{\leqslant}R}|\cdot|^{-1}\in L^1({{\ensuremath {\mathbb R} }}^3)$ and $V_{{\ensuremath {\infty}}}:={\boldsymbol{1}}_{|x|{\geqslant}R}|\cdot|^{-1}\in L^{{\ensuremath {\infty}}}({{\ensuremath {\mathbb R} }}^3)$. On the one hand $$\left|\int_{{{\ensuremath {\mathbb R} }}^3}\rho(\nu\star V_1)\right|{\leqslant}{\varepsilon}\|\nu\star V_1\|_{L^1}{\leqslant}{\varepsilon}\|\nu\|_{L^1}\|V_1\|_{L^1}=2\pi{\varepsilon}R^2,$$ where in the last inequality we used Young’s inequality and $\int|\nu|=1$. On the other hand, $$\int_{{{\ensuremath {\mathbb R} }}^3}\rho(\nu\star V_{{\ensuremath {\infty}}})=\int_{{{\ensuremath {\mathbb R} }}^3}\rho(x)\left(\int_{|y|{\geqslant}R}\frac{\nu(x-y)}{|y|}\text{d}y\right)\text{d}x=\int_{|y|{\geqslant}R}\frac{\widetilde{\rho}(y)}{|y|}\text{d}y,$$ where $\widetilde{\rho}:={\widetilde}{\nu}\star\rho$ and ${\widetilde}{\nu}(x)=\nu(-x)$ for all $x\in{{\ensuremath {\mathbb R} }}^3$. Since $\int_{|y|{\geqslant}R}\frac{{\widetilde}{\rho}(y)}{|y|}=D({\widetilde}{\rho},f)$ with $f=(4\pi)^{-1}\delta_{|x|=R}$, as in [@LieSim-77b Proof of Theorem II.3], we have $$-\int_{|y|{\geqslant}R}\frac{\widetilde{\rho}(y)}{|y|}\text{d}y+\frac{1}{2}D(\widetilde{\rho},\widetilde{\rho}){\geqslant}-\frac{1}{2}D(f,f)=-\frac{1}{2R}.$$ Notice that $$D(\widetilde{\rho},\widetilde{\rho})=4\pi(2\pi)^3\int_{{{\ensuremath {\mathbb R} }}^3}\frac{|\widehat{\rho}(k)|^2|\widehat{\nu}(k)|^2}{|k|^2}\text{d}k{\leqslant}D(\rho,\rho),$$ since $|\widehat{\nu}(k)|{\leqslant}(2\pi)^{-3/2}\|\nu\|_{L^1}=(2\pi)^{-3/2}$ for all $k$. Hence, $$\widetilde{E}(1,{\varepsilon}){\geqslant}-2\pi\alpha{\varepsilon}R^2-\frac{\alpha}{2R}.$$ Optimizing over $R$, one gets the result. This finishes the proof of Proposition \[prop:liminf\].
Proof of the upper bound
------------------------
Both Propositions \[prop:limsup\] and \[prop:speed\] are proved via appropriate trial states. Before turning to the estimates, we start by explaining our choice of the trial states. In the sequel, we will assume that $\int\nu>0$. The case $\int\nu<0$ is treated in the same fashion, except for the choice of the trial state (see Remark \[rk:qneg\]). We define the operator $Q$ on ${\mathfrak{H}}_\Lambda$ by its kernel in Fourier space. For all $p,q\in{{\ensuremath {\mathbb R} }}^3$, let $${\widehat}{Q}(p,q)=\widehat{\gamma}(p,q)X(p)X(q)^*,$$ with $$\widehat{\gamma}(p,q)=(2\pi)^{-3/2}\int_{B(0,\Lambda/2)}\text{d}\ell\, g_r(p-\ell)\widehat{F_r}(p-q)g_r(q-\ell)$$ and $$X(p)=U(p)\left[\left(1\;0\;0\;0\right)^t\right]$$ with $$U(p)=a_+(p)-a_-(p)\beta\frac{{\boldsymbol{\alpha}}\cdot p}{|p|},\qquad a_\pm(p)=\frac{1}{\sqrt{2}}\sqrt{1\pm\frac{1}{1+|p|^2}}.$$ The operator $U(p)$ is unitary on ${{\ensuremath {\mathbb C} }}^4$ for all $p$ and it diagonalizes $D^0(p):={\boldsymbol{\alpha}}\cdot p+\beta$ as $U(p)D^0(p)U(p)^*=\sqrt{1+|p|^2}\beta$. In the definition of ${\widehat}{\gamma}$, we choose $0<r<\Lambda/2$ (a small number which will eventually tend to zero), $g_r=r^{-3/2}g(\cdot/r)$ with $g\in L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb R} }})$, $\int g^2=1$ and $\text{supp}(g)\subset B(0,1)$. We also choose $F_r=F(r\cdot)$ with $F\in L^1({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb R} }})$ such that $0{\leqslant}F(x){\leqslant}1$ for all $x$.
The operator $Q$ belongs to the variational set ${\mathcal{K}}$.
First, notice that $Q$ defines a Hilbert-Schmidt operator on $L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)$ since for instance $$\begin{gathered}
\|Q(\cdot,\cdot)\|_{L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)}^2{\leqslant}(2\pi)^{-3}\|g_r\|_{L^2}^2\int_{B(0,\Lambda/2)}\text{d}\ell\int_{{{\ensuremath {\mathbb R} }}^3}\text{d}p\left( |\widehat{F_r}|^2\star |\tau_\ell g_r|^2\right)(p)\\
{\leqslant}(2\pi)^{-3} \|g_r\|_{L^2}^2\int_{B(0,\Lambda/2)}\text{d}\ell \left\| |\widehat{F_r}|^2\star |\tau_\ell g_r|^2 \right\|_{L^1}\\
{\leqslant}\frac{\text{vol}(B(0,\Lambda/2))}{(2\pi)^3}\|g_r\|_{L^2}^4\|F_r\|_{L^1}^2 <{{\ensuremath {\infty}}}\end{gathered}$$ by Young’s inequality. It is self-adjoint because $\widehat{Q}(p,q)=\overline{\widehat{Q}(q,p)}$ for all $p,q$, and we have $\operatorname{supp}\widehat{Q}(\cdot,\cdot)\subset (B(0,\Lambda/2)+\operatorname{supp}g_r )^2\subset B(0,\Lambda)^2$ hence $Q$ is an operator with range in ${\mathfrak{H}}_\Lambda$. Since for all ${\varphi}\in{\mathfrak{H}}_\Lambda$ we have $${\widehat}{Q{\varphi}}(p)=\left(\int_{{{\ensuremath {\mathbb R} }}^3}{\widehat}{\gamma}(p,q)\left\langle X(q),{\widehat}{{\varphi}}(q)\right\rangle_{{{\ensuremath {\mathbb C} }}^4}{{\ensuremath{\,\text{d}q}}}\right)X(p),$$ we conclude $Q_{--}=Q_{+-}=Q_{-+}=0$ so that $Q=Q_{++}$. Let ${\varphi}\in{\mathfrak{H}}_+^0$. Then $$\begin{aligned}
\langle Q{\varphi},{\varphi}\rangle_{L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)} & = & (2\pi)^{-3/2} \int_{B(0,\Lambda/2)}\text{d}\ell\,\left\langle (\tau_\ell g)X\widehat{{\varphi}},\widehat{F}\star((\tau_\ell g)X\widehat{{\varphi}})\right\rangle_{L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)}\\
& = & \int_{B(0,\Lambda/2)}\text{d}\ell\int_{{{\ensuremath {\mathbb R} }}^3}\text{d}x\, F(x) \left\|{\mathcal{F}}^{-1}\left((\tau_\ell g)X\widehat{{\varphi}}\right)(x)\right\|_{{{\ensuremath {\mathbb C} }}^4}^2\\
& {\leqslant}& \int_{B(0,\Lambda/2)}\text{d}\ell\int_{{{\ensuremath {\mathbb R} }}^3}\text{d}p\,\left\|g(p-\ell)X(p)\widehat{{\varphi}}(p)\right\|_{{{\ensuremath {\mathbb C} }}^4}^2\\
& {\leqslant}& \|{\varphi}\|_{L^2({{\ensuremath {\mathbb R} }}^3,{{\ensuremath {\mathbb C} }}^4)}^2,\end{aligned}$$ where we have denoted by ${\mathcal{F}}^{-1}$ the inverse Fourier transform. Hence $-P^0_-{\leqslant}Q{\leqslant}1-P^0_-$. Finally, $\operatorname{Tr}(Q)=\int_{{{\ensuremath {\mathbb R} }}^3}{\widehat}{\gamma}(p,p)\|X(p)\|_{{{\ensuremath {\mathbb C} }}^4}^2{{\ensuremath{\,\text{d}p}}}{\leqslant}(2\pi)^{-3}\int F<+{{\ensuremath {\infty}}}$, so that $Q\in{\mathfrak{S}}_1({\mathfrak{H}}_\Lambda)\subset{\mathfrak{S}}_{1,P^0_-}({\mathfrak{H}}_\Lambda)$.
Since for any $R\in{\mathcal{K}}$ we have the formula $${\widehat}{\rho_R}(k)=\frac{1}{(2\pi)^{3/2}}\int_{{{\ensuremath {\mathbb R} }}^3}\operatorname{Tr}_{{{\ensuremath {\mathbb C} }}^4}({\widehat}{R}(p+k,p)){{\ensuremath{\,\text{d}p}}},\qquad \forall k\in{{\ensuremath {\mathbb R} }}^3,$$ the density of $Q$ can be written as $\rho_Q=\rho_1+\rho_2$ where for all $k$, $${\widehat}{\rho_1}(k)=(2\pi)^{-3}V_\Lambda{\widehat}{F_r}(k)g_r\star{\widetilde}{g_r}(-k)$$ and $${\widehat}{\rho_2}(k)=(2\pi)^{-3}V_\Lambda{\widehat}{F_r}(k)\int_{{{\ensuremath {\mathbb R} }}^3}g_r(p)g_r(p+k)\left\langle X(p),(X(p+k)-X(p))\right\rangle_{{{\ensuremath {\mathbb C} }}^4}{{\ensuremath{\,\text{d}p}}}$$ with $V_\Lambda:=\text{vol}(B(0,\Lambda/2))$ and ${\widetilde}{g}:=g(-\cdot)$.
We start by estimating the terms giving the $Z^{5/3}$ behaviour. We have $$D(\rho_1,Z\nu)=4\pi Z\frac{V_\Lambda}{(2\pi)^3} r^{-2}\int_{B(0,2\Lambda)}\widehat{F}(k)\frac{g\star\widetilde{g}(-k)\overline{\widehat{\nu}(rk)}}{|k|^2}\,\text{d}k$$ and $$D(\rho_1,\rho_1)=4\pi\left(\frac{V_\Lambda}{(2\pi)^3}\right)^2 r^{-5}\int_{B(0,2\Lambda)}|\widehat{F}(k)|^2\frac{|g\star\widetilde{g}(-k)|^2}{|k|^2}\,\text{d}k.$$ We choose $r$ such that $(2\pi)^{-3/2}qZ\frac{V_\Lambda}{(2\pi)^3} r^{-2}=\left(\frac{V_\Lambda}{(2\pi)^3}\right)^2 r^{-5}$, ie $r=\frac{1}{\sqrt{2\pi}}\left(\frac{qZ}{V_\Lambda}\right)^{-1/3}$. The constraint $r<\Lambda/2$ is equivalent to $qZ>\frac{1}{(2\pi)^{3/2}}\frac{4\pi}{3}$, which is automatically satisfied in the limit $Z\to+{{\ensuremath {\infty}}}$. We will come back to it in the proof of Proposition \[prop:speed\]. We thus get by the dominated convergence theorem $$\begin{gathered}
\label{eq:lim}
\lim_{Z\to+{{\ensuremath {\infty}}}}Z^{-5/3}\left(-\alpha D(\rho_1,Z\nu)+\frac{\alpha}{2}D(\rho_1,\rho_1)\right)=2^{-11/6}3^{-1/3}\pi^{-13/6}\alpha\Lambda q^{5/3}\times \\
\left(-\int_{B(0,2\Lambda)}\frac{{\widehat}{F}(k)g\star{\widetilde}{g}(-k)}{|k|^2}{{\ensuremath{\,\text{d}k}}}+\frac{1}{2}\int_{B(0,2\Lambda)}\frac{|{\widehat}{F}(k)g\star{\widetilde}{g}(-k)|^2}{|k|^2}{{\ensuremath{\,\text{d}k}}}\right).\end{gathered}$$ We now want to optimize the right side with respect to $g$ and $F$. We choose $F={\boldsymbol{1}}_{B(0,a)}$ and $g=(3(4\pi)^{-1}b^{-3})^{1/2}{\boldsymbol{1}}_{B(0,b)}$ with $a>0$ and $b\in(0,1)$, and we optimize over $a$ and $b$. Since $g\star{\widetilde}{g}(k)=(\text{vol}(B(0,b)))^{-1}\text{vol}(B(0,b)\cap B(-k,b))$ for all $k$ and since $B(-k/2,b-|k|/2)\subset B(0,b)\cap B(-k,b)$ for all $|k|{\leqslant}2b$, we have $$g\star\widetilde{g}(k){\geqslant}{\boldsymbol{1}}_{B(0,2b)}(k)\left(1-\frac{|k|}{2b}\right)^3.$$ For all $|k|{\leqslant}2b$, we also have $$\widehat{F}(k){\geqslant}\widehat{F}(0)-\frac{\int|x|F}{(2\pi)^{3/2}}2b=\frac{a^3}{\sqrt{2\pi}}\left(\frac{2}{3}-ab\right).$$ Therefore, $$\begin{aligned}
\int_{B(0,2\Lambda)}\frac{\widehat{F}(k)g\star\widetilde{g}(-k)}{|k|^2}\text{d}k & {\geqslant}& \frac{a^3}{\sqrt{2\pi}}\left(\frac{2}{3}-ab\right)\int_{B(0,2b)}\left(1-\frac{|k|}{2b}\right)^3\frac{\text{d}k}{|k|^2}\\
& = & \sqrt{2\pi}a^3 b\left(\frac{2}{3}-ab\right).\end{aligned}$$ Then, using $|\widehat{F}(k)|{\leqslant}4\pi a^3/(3(2\pi)^{3/2})$ and $|g\star\widetilde{g}(k)|{\leqslant}1$ for all $k$, we obtain $$\label{eq:maj1}
\int_{B(0,2\Lambda)}\frac{|\widehat{F}(k)g\star\widetilde{g}(-k)|^2}{|k|^2}{\leqslant}\frac{16}{9}a^6b.$$ Hence for the $Z^{5/3}$ term we find that $$\begin{gathered}
\lim_{Z\to+{{\ensuremath {\infty}}}}Z^{-5/3}\left(-\alpha D(\rho_1,Z\nu)+\frac{\alpha}{2}D(\rho_1,\rho_1)\right){\leqslant}2^{-11/6}3^{-1/3}\pi^{-13/6}\alpha\Lambda q^{5/3}\times\\ a^3b\left(\sqrt{2\pi}\left(ab-\frac{2}{3}\right)+\frac{8}{9}a^3\right).\end{gathered}$$ Optimizing the right side over all $a>0$ and $b\in(0,1)$ we get the result because $$\min_{\substack{a>0 \\ b\in(0,1)}}a^3b\left(\sqrt{2\pi}\left(ab-\frac{2}{3}\right)+\frac{8}{9}a^3\right)=-2^{-35/6}3^{2/3}\pi^{5/6}.$$
It now remains to prove that the other terms in ${{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q)}}}$ are of lower order than $Z^{5/3}$. We begin with the kinetic energy, $$\operatorname{Tr}_0(D^0Q){\leqslant}\sqrt{1+\Lambda^2}\operatorname{Tr}(Q)=(2\pi)^{-3}r^{-3}\int F=(2\pi)^{-3/2}\frac{\sqrt{1+\Lambda^2}}{\Lambda^3}a^3qZ.$$ For the terms involving $\rho_2$, we first use that for all $p,k$, $$\|X(p+k)-X(p)\|{\leqslant}\|U(p+k)-U(p)\|{\leqslant}\frac{7}{\sqrt{2}}|k|.$$ Consequently, for all $k$, $$|{\widehat}{\rho_2}(k)|{\leqslant}\frac{7V_\Lambda}{\sqrt{2}(2\pi)^3}r^{-3}|{\widehat}{F}(k/r)||g\star{\widetilde}{g}(-k/r)||k|.$$ Using this bound together with the estimates leading to , one finds that $$\left\{
\begin{array}{ccc}
|D(\rho_2,Z\nu)| & {\leqslant}& \beta_1\Lambda^2q^{1/3}Z^{4/3},\\
|D(\rho_1,\rho_2)| & {\leqslant}& \beta_2\Lambda^2q^{4/3}Z^{4/3},\\
|D(\rho_2,\rho_2)| & {\leqslant}& \beta_3\Lambda^3qZ,
\end{array}
\right.$$ with $$\left\{
\begin{array}{ccc}
\beta_1 & = & 2^{19/3}3^{-2/3}7\pi^{23/6}a^3b^2,\\
\beta_2 & = & 2^{11/6}3^{-8/3}7\pi^{-7/3}a^6b^2,\\
\beta_3 & = & 2^{-3/2}3^{-4}7^2\pi^{-5/2}a^6b^3.
\end{array}
\right.$$ We conclude that $$\limsup_{Z\to+{{\ensuremath {\infty}}}}\frac{E(Z)}{Z^{5/3}}{\leqslant}\lim_{Z\to+{{\ensuremath {\infty}}}}\frac{{{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q)}}}}{Z^{5/3}}=-c_1\alpha\Lambda q^{5/3}.$$
\[rk:qneg\] In the case $\int\nu<0$, the proof is the same except that we take $$X(p)=U(p)\left[\left(0\;0\;0\;1\right)^t\right],\qquad\forall p\in{{\ensuremath {\mathbb R} }}^3,$$ and $${\widehat}{Q}(p,q)=-\widehat{\gamma}(p,q)X(p)X(q)^*,\qquad\forall p,q\in{{\ensuremath {\mathbb R} }}^3.$$ The trial state $Q$ now verifies $Q=Q_{--}$, so that ${\widehat}{\rho_Q}$ is locally negative around $0$.
To estimate the convergence rate of $E(Z)$ towards $-c_1\alpha\Lambda |q|^{5/3}Z^{5/3}$, we will use the first moment $\int|x||\nu|$ to control the convergence of ${\widehat}{\nu}(k)$ to ${\widehat}{\nu}(0)$.
We assume $q>0$ (the case $q<0$ follows from obvious modifications). We split the term $D(\rho_1,Z\nu)$ into $$D(\rho_1,Z\nu)=\frac{4\pi |q|Z}{(2\pi)^{3/2}}\int_{B(0,2\Lambda)}\frac{{\widehat}{\rho_1}(k)}{|k|^2}{{\ensuremath{\,\text{d}k}}}+R_1,$$ with $$R_1=4\pi Z\int_{B(0,2\Lambda)}\frac{{\widehat}{\rho_1}(k)\overline{{\widehat}{\nu}(k)-{\widehat}{\nu}(0)}}{|k|^2}{{\ensuremath{\,\text{d}k}}}.$$ We use that $|{\widehat}{\nu}(k)-{\widehat}{\nu}(0)|{\leqslant}(2\pi)^{-3/2}\||x|\nu\|_1 |k|$ for all $k$ to estimate $$|R_1|{\leqslant}2^{5/3}3^{-5/3}\pi^{-11/6}a^3b^2\||x|\nu\|_1\Lambda^2|q|^{1/3}Z^{4/3}.$$ Hence we have for all $|q|Z>\frac{1}{(2\pi)^{3/2}}\frac{4\pi}{3}$, $$\label{eq:maj2}
E(Z){\leqslant}{{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q)}}}{\leqslant}-c_1\alpha\Lambda |q|^{5/3}Z^{5/3}(1-B_1Z^{-1/3}-B_2Z^{-2/3}),$$ with $$\begin{gathered}
B_1 = 2^{28/3}3^{-2}\pi^{-1/2}a^3b^2\||x|\nu\|_1\Lambda |q|^{-4/3}+2^{4/3}3^{-1}7\pi^{31/6}a^3b^2\Lambda |q|^{-4/3}\\
+2^{51/6}3^{-3}7\pi^{-1}a^6b^2\Lambda |q|^{-1/3}\end{gathered}$$ and $$B_2 = 2^{37/6}3^{-1/3}\pi^{4/3}a^3\frac{\sqrt{1+\Lambda^2}}{\Lambda^4}\alpha^{-1}|q|^{-2/3}+2^{31/6}3^{-11/3}7^2\pi^{-7/6}a^6b^3\Lambda^2|q|^{-2/3}.$$ We furthermore have $1-B_1X-B_2X^2{\geqslant}1/2$ for $0{\leqslant}X{\leqslant}X_0$ with $$X_0=\frac{B_1}{2B_2}\left( \sqrt{1+\frac{2B_2}{B_1^2}}-1\right),$$ therefore Proposition \[prop:speed\] holds with $$\label{eq:Z_1}
{\widetilde}{Z_1}:=\max\left(X_0^{-3},\frac{1}{(2\pi)^{3/2}}\frac{4\pi}{3|q|}\right).$$
For any $Q\in{\mathcal{K}}$, we have the estimate [@HaiLewSer-08 Lemma 1] $$D(\rho_Q,\rho_Q)^{1/2}{\leqslant}C_\Lambda\|Q\|_{1,P^0_-},$$ with $C_\Lambda<+{{\ensuremath {\infty}}}$. From the constraint $-P^0_-{\leqslant}Q{\leqslant}1-P^0_-$ we also get $Q_{++}-Q_{--}{\geqslant}Q^2$, hence defining $X:=\left[\operatorname{Tr}(Q_{++}-Q_{--})\right]^{1/2}$ we have $$2\|Q_{+-}\|_{{\mathfrak{S}}_2}^2=2\|Q_{-+}\|_{{\mathfrak{S}}_2}^2=\|Q_{+-}\|_{{\mathfrak{S}}_2}^2+\|Q_{-+}\|_{{\mathfrak{S}}_2}^2{\leqslant}\|Q\|_{{\mathfrak{S}}_2}^2{\leqslant}X^2,$$ hence $\|Q\|_{1,P^0_-}{\leqslant}X^2+\sqrt{2}X$. Using the Cauchy-Schwarz inequality for the Coulomb scalar product we also get that for all $Q\in{\mathcal{K}}$ $$D(\rho_Q,Z\nu){\leqslant}C_\Lambda Z \|\nu\|_{\mathcal{C}}(X^2+\sqrt{2}X).$$ From this estimate and the inequality , we can see that for all $Z>{\widetilde}{Z_1}$ and for all minimizer $Q^{\text{vac}}$ for $E(Z)$, $$-\alpha C_\Lambda Z \|\nu\|_{\mathcal{C}}(X^2+\sqrt{2}X){\leqslant}{{\ensuremath{{\mathcal{E}}_{\rm{rBDF}}^{Z\nu}(Q^{\text{vac}})}}}=E(Z){\leqslant}-\frac{c_1}{2}\alpha\Lambda |q|^{5/3}Z^{5/3}.$$ Hence, $$X{\geqslant}\frac{1}{\sqrt{2}}\left( \sqrt{1+2a}-1\right),$$ with $a=\frac{c_1\Lambda |q|^{5/3}}{2C_\Lambda\|\nu\|_{\mathcal{C}}}Z^{2/3}$, so that $$\operatorname{Tr}(Q_{++}^{\text{vac}}-Q_{--}^{\text{vac}})=X^2{\geqslant}\frac{1}{2}\left( \sqrt{1+2a}-1\right)^2{\geqslant}CZ^{2/3},$$ where $$\label{eq:cste}
C=\left(\sqrt{1+\frac{c_1\Lambda |q|^{5/3}}{C_\Lambda\|\nu\|_{\mathcal{C}}}{\widetilde}{Z_1}^{2/3}}-1\right)^2{\widetilde}{Z_1}^{-2/3},$$ using that $x\mapsto(\sqrt{1+x}-1)/\sqrt{x}$ is increasing on $(0,+{{\ensuremath {\infty}}})$.
Proof of Theorem \[th:main\] {#sec:proof}
============================
Let $Q$ be any minimizer for $E(Z)$, and let $\omega$ be the unique BDF state on ${\mathcal{F}}({\mathfrak{H}}_+)\oplus{\mathcal{F}}({\mathfrak{H}}_-)$ having $Q$ as its generalized 1-pdm and $p=0$ as its pairing matrix, defined by Proposition \[prop:BDF-existence\]. Then by Corollary \[coro:estimate-BDF\], we have $$p_Z=1-\omega({{\ensuremath{| \Omega\rangle\langle \Omega|}}}){\geqslant}1-e^{-a\operatorname{Tr}(Q_{++}-Q_{--})}.$$ Now by Corollary \[coro:number-estimate\], we know that for all $Z>{\widetilde}{Z_1}$, $$\operatorname{Tr}(Q_{++}-Q_{--}){\geqslant}CZ^{2/3}.$$ Thus $p_Z{\geqslant}1-e^{-\kappa Z^{2/3}}$ with $\kappa:=aC$.
Product States {#prodstates}
==============
Given a (at most) countable family of separable Hilbert spaces $({\mathfrak{H}}_i)_{i\in {{\ensuremath {\mathbb N} }}}$ and a family $(\omega_i)_{i\in{{\ensuremath {\mathbb N} }}}$ of quasi-free states such that $\omega_i$ is a state on ${\mathcal{F}}({\mathfrak{H}}_i)$ for all $i\in{{\ensuremath {\mathbb N} }}$, we want to give a meaning to the product state $\otimes_{i\in {{\ensuremath {\mathbb N} }}}\omega_i$ as a quasi-free state on ${\mathcal{F}}(\oplus_{i\in {{\ensuremath {\mathbb N} }}}{\mathfrak{H}}_i)$ . We first consider the unitary transformation $$\begin{array}{cccc}
T: & {\mathcal{F}}\left(\bigoplus_{i\in {{\ensuremath {\mathbb N} }}}{\mathfrak{H}}_i\right) & \longrightarrow & \bigotimes_{i\in {{\ensuremath {\mathbb N} }}}{\mathcal{F}}({\mathfrak{H}}_i) \\
& \bigwedge_{j\in J_{i_1}}{\varphi}_{j,i_1}\wedge\cdots\wedge\bigwedge_{j\in J_{i_k}}{\varphi}_{j,i_k} & \longmapsto & \bigwedge_{j\in J_{i_1}}{\varphi}_{j,i_1}\otimes\cdots\otimes\bigwedge_{j\in J_{i_k}}{\varphi}_{j,i_k},
\end{array}$$ where $i_1 < \cdots < i_k$ are elements of ${{\ensuremath {\mathbb N} }}$ and for each $1{\leqslant}\ell{\leqslant}k$, $({\varphi}_{j,i_\ell})_{j\in{{\ensuremath {\mathbb N} }}}$ is an orthonormal basis for ${\mathfrak{H}}_{i_\ell}$ and $J_{i_\ell}\subset{{\ensuremath {\mathbb N} }}$ is finite. We recall the definition of a product state on a tensor product of $C^*$-algebras [@Gui-66 Proposition 2.9].
Let $({\mathfrak{A}}_i)_{i\in{{\ensuremath {\mathbb N} }}}$ be a collection of (unital) $C^*$-algebras and let $\omega_i$ be a state on ${\mathfrak{A}}_i$ for all $i$. There exists a unique state on $\otimes_i {\mathfrak{A}}_i$, denoted by $\otimes_i\omega_i$ such that for any $(A_{i_1},\ldots,A_{i_\ell})\in{\mathfrak{A}}_{i_1}\times\cdots\times{\mathfrak{A}}_{i_\ell}$ with any indices $i_1<\cdots< i_\ell$, we have $\otimes_i\omega_i\left(A_{i_1}\otimes\cdots\otimes A_{i_\ell}\right)=\prod_{k=1}^\ell\omega_{i_k}(A_{i_k})$.
Recall that the tensor product $\otimes_i{\mathfrak{A}}_i$ is defined as the inductive limit of the $C^*$-algebras $\otimes_{i\in J}{\mathfrak{A}}_i$ with $J$ finite, or equivalently as the completion (for a certain $*$-norm) of $\otimes_i{\mathfrak{A}}_i$ seen as a tensor product of unital algebras [@Gui-66].
In the particular case where ${\mathfrak{A}}_i=\operatorname{\mathfrak{A}}[{\mathfrak{H}}_i]$ for all $i\in{{\ensuremath {\mathbb N} }}$, the tensor product $\otimes_i{\mathfrak{A}}_i$ is also a CAR algebra generated by the operators $$\widehat{a}(f_j):=\Upsilon\left(-\text{Id}_{{\mathfrak{H}}_0}\right)\otimes\cdots\otimes\Upsilon\left(-\text{Id}_{{\mathfrak{H}}_{j-1}}\right)\otimes a_j(f_j),$$ where $j\in{{\ensuremath {\mathbb N} }}$, $f_j\in{\mathfrak{H}}_j$ and $a_j$ is the annihilation operator on ${\mathcal{F}}({\mathfrak{H}}_j)$ for all $j$. Notice the “twisting” operator $\Upsilon\left(-\text{Id}\right)$ on the left used to ensure that the $(\widehat{a}(f_j))_{f_j,j}$ satisfy the CAR. Notice also that we have the relation $$\widehat{a}(f_j)=T a(f_j) T^*,$$ where $a(f_j)$ is the usual annihilation operator on ${\mathcal{F}}(\oplus_i{\mathfrak{H}}_i)$.
\[propprodstates\] Let $(\omega_i)_{i\in{{\ensuremath {\mathbb N} }}}$ be a collection of quasi-free states such that $\omega_i$ is defined on $\operatorname{\mathfrak{A}}[{\mathfrak{H}}_i]$ for all $i$. Let $(\gamma_i,\alpha_i)_i$ be the collection of their density matrices, and assume that $\sum_i\operatorname{Tr}_{{\mathfrak{H}}_i}(\gamma_i)<+{{\ensuremath {\infty}}}$. Let $\omega$ be the unique quasi-free state on $\operatorname{\mathfrak{A}}[\oplus_i{\mathfrak{H}}_i]$ having $\gamma:=\oplus_i\gamma_i$ as 1-pdm and $\alpha:=\oplus_i\alpha_i$ as pairing matrix. Then we have $$\forall A\in\operatorname{\mathfrak{A}}[\oplus_i{\mathfrak{H}}_i],\quad \omega(A)=\otimes_i\omega_i\left(TAT^*\right).$$ As a consequence, the product state $\otimes_i\omega_i$ can be considered as a state on $\operatorname{\mathfrak{A}}[\oplus_i{\mathfrak{H}}_i]$.
The state $\omega$ is well-defined by Proposition \[BacLieSol\], since $\operatorname{Tr}(\gamma)=\sum_i\operatorname{Tr}_{{\mathfrak{H}}_i}(\gamma_i)$ is finite.
We will show that the state $\otimes_i\omega_i$ is quasi-free on the CAR generated by the $(\widehat{a}(f_j))_{f_j,j}$. This proves the result because the state $\otimes_i\omega_i(T\cdot T^*)$ is then quasi-free and one easily shows that it has $\oplus_i\gamma_i$ as 1-pdm and $\oplus_i\alpha_i$ as pairing matrix. Since the density matrices determine uniquely the quasi-free state, we must have $\otimes_i\omega_i(T\cdot T^*)=\omega$. Therefore we only have to show the Wick relation for the state $\otimes_i\omega_i$. Let us first notice that it is enough to prove it for products of the form $$\widehat{a}^\sharp\left(f_1^{(i_1)}\right)\cdots\widehat{a}^\sharp\left(f_{2k_1}^{(i_1)}\right)\cdots\widehat{a}^\sharp\left(f_1^{(i_N)}\right)\cdots\widehat{a}^\sharp\left(f_{2k_N}^{(i_N)}\right),$$ with $i_1<\cdots<i_N$, $k_\ell\in{{\ensuremath {\mathbb N} }}$, $f_p^{(i_\ell)}\in{\mathfrak{H}}_{i_\ell}$ for all $1{\leqslant}p{\leqslant}2k_\ell$, $1{\leqslant}\ell{\leqslant}N$, and where $\sharp$ means star or no star. This means two things:
1. We can restrict to ordered products with respect to the decomposition $\oplus_{i\in{{\ensuremath {\mathbb N} }}}{\mathfrak{H}}_i$: The $2k_1$ first creation/annihilation operators $\widehat{a}^\sharp\left(f_p^{(i_1)}\right)$ with $1{\leqslant}p{\leqslant}2k_1$ all create/annihilate particles belonging to ${\mathfrak{H}}_{i_1}$, then the following $2k_2$ creation/annihilation $\widehat{a}^\sharp\left(f_p^{(i_2)}\right)$ with $1{\leqslant}p{\leqslant}2k_2$ create/annihilate particles belonging to ${\mathfrak{H}}_{i_2}$, etc. We can always order in this way any product of $a^\sharp$s because Wick’s formula does not depend on the choice of an ordering.\
2. We can restrict to products where there is an even number of particles created/annihilated in each space ${\mathfrak{H}}_{i_\ell}$. If not the case, both sides of Wick relation can easily be shown to vanish.
This being said, let us compute the left-hand side of Wick’s formula: $$\begin{aligned}
X & = & \otimes_i\omega_i\left[\widehat{a}^\sharp\left(f_1^{(i_1)}\right)\cdots\widehat{a}^\sharp\left(f_{2k_1}^{(i_1)}\right)\cdots\widehat{a}^\sharp\left(f_1^{(i_N)}\right)\cdots\widehat{a}^\sharp\left(f_{2k_N}^{(i_N)}\right)\right] \\
& = & \otimes_i\omega_i\left[ \Upsilon\left(-\text{Id}_{{\mathfrak{H}}_0}\right)^{\sum_{\ell=1}^N 2k_\ell}\otimes\cdots\otimes\Upsilon\left(-\text{Id}_{{\mathfrak{H}}_{i_1-1}}\right)^{\sum_{\ell=1}^N 2k_\ell}\otimes\right. \\
& & \left.\otimes a^\sharp\left(f_1^{(i_1)}\right)\cdots a^\sharp\left(f_{2k_1}^{(i_1)}\right)\Upsilon\left(-\text{Id}_{{\mathfrak{H}}_{i_1}}\right)^{\sum_{\ell=2}^N 2k_\ell}\otimes\cdots\right.\\
& & \left.\qquad\qquad\qquad\qquad\qquad\qquad\cdots\otimes a^\sharp\left(f_1^{(i_N)}\right)\cdots a^\sharp\left(f_{2k_N}^{(i_N)}\right)\right]\\
& = & \prod_{\ell=1}^N\omega_{i_\ell}\left[a^\sharp\left(f_1^{(i_\ell)}\right)\cdots a^\sharp\left(f_{2k_\ell}^{(i_\ell)}\right)\right]\\
& =& \prod_{\ell=1}^N\sum_{\pi_\ell\in\widetilde{{\mathcal{S}}_{2k_\ell}}}(-1)^{{\varepsilon}(\pi_\ell)}\omega_{i_\ell}\left[a^\sharp\left(f_{\pi_\ell(1)}^{(i_\ell)}\right)a^\sharp\left(f_{\pi_\ell(2)}^{(i_\ell)}\right)\right]\times\cdots \\
& & \qquad\qquad\qquad\qquad\qquad\qquad \cdots\times\omega_{i_\ell}\left[a^\sharp\left(f_{\pi_\ell(2k_\ell-1)}^{(i_\ell)}\right)a^\sharp\left(f_{\pi_\ell(2k_\ell)}^{(i_\ell)}\right)\right].\end{aligned}$$ Let us reindex the product $$\begin{gathered}
\widehat{a}^\sharp\left(f_1^{(i_1)}\right)\cdots\widehat{a}^\sharp\left(f_{2k_1}^{(i_1)}\right)\cdots\widehat{a}^\sharp\left(f_1^{(i_N)}\right)\cdots\widehat{a}^\sharp\left(f_{2k_N}^{(i_N)}\right)=\\
\widehat{a}^\sharp\left(f_1\right)\cdots\widehat{a}^\sharp\left(f_{2k_1}\right)\widehat{a}^\sharp\left(f_{2k_1+1}\right)\cdots\widehat{a}^\sharp\left(f_{2(k_1+\ldots+k_N)}\right),\end{gathered}$$ such that the sum of Wick’s formula (with $K:=k_1+\ldots+k_N$), $$Y=\sum_{\pi\in\widetilde{{\mathcal{S}}_{2K}}}(-1)^{{\varepsilon}(\pi)}(\otimes_i\omega_i)\left[\widehat{a}^\sharp\left(f_{\pi(1)}\right)\widehat{a}^\sharp\left(f_{\pi(2)}\right)\right]\cdots(\otimes_i\omega_i)\left[\widehat{a}^\sharp\left(f_{\pi(2K -1)}\right)\widehat{a}^\sharp\left(f_{\pi(2K)}\right)\right]$$ contains only the terms with a permutation $\pi$ leaving invariant every interval of the form $[2k_\ell +1,2k_{\ell+1}]$, corresponding to the Hilbert space ${\mathfrak{H}}_{i_\ell}$, $1{\leqslant}\ell{\leqslant}N$. This is proved by induction on $K$: We introduce $$W_\rho^K(e_1\ldots e_{2K}):=\sum_{\pi\in\widetilde{{\mathcal{S}}_{2K}}} (-1)^{{\varepsilon}(\pi)}\rho(e_{\pi(1)}e_{\pi(2)})\ldots\rho(e_{\pi(2K-1)}e_{\pi(2K)}).$$ For $\pi\in\widetilde{{\mathcal{S}}_{2K}}$, we have $\pi(1)=1$, hence we have the induction formula $$W_\rho^K(e_1\ldots e_{2K})=\sum_{i=2}^{2K}(-1)^i\rho(e_1e_i)W_\rho^{K-1}(e_2\ldots\widehat{e_i}\ldots e_{2K}).$$ In our case where $e_j=\widehat{a}^\sharp(f_j)$ and $\rho=\otimes_i\omega_i$, we see that $\rho(e_1e_i)\neq0$ if and only if $i{\leqslant}2k_1$. Hence, by induction on $K$, the only permutations $\pi\in\widetilde{{\mathcal{S}}_{2K}}$ giving rise to a non-zero term in $Y$ are those which leave invariant the intervals $[2k_\ell +1,2k_{\ell+1}]$ with $1{\leqslant}\ell{\leqslant}N$. We can thus write $$\begin{aligned}
Y & = &\prod_{\ell=1}^N \sum_{\pi_\ell\in\widetilde{{\mathcal{S}}_{2k_\ell}}}(-1)^{{\varepsilon}(\pi_\ell)}(\otimes_i\omega_i)\left[\widehat{a}^\sharp\left(f_{\pi_\ell(2k_\ell-1)}^{(i_\ell)}\right)\widehat{a}^\sharp\left(f_{\pi_\ell(2k_\ell)}^{(i_\ell)}\right)\right]\times\cdots\\
& & \qquad\qquad\qquad\qquad\qquad\qquad\cdots
\times(\otimes_i\omega_i)\left[\widehat{a}^\sharp\left(f_{\pi_\ell(1)}^{(i_\ell)}\right)\widehat{a}^\sharp\left(f_{\pi_\ell(2)}^{(i_\ell)}\right)\right]\\
& = & \prod_{\ell=1}^N \sum_{\pi_\ell\in\widetilde{{\mathcal{S}}_{2k_\ell}}} (-1)^{{\varepsilon}(\pi_\ell)}\omega_{i_\ell}\left[a^\sharp\left(f_{\pi_\ell(2k_\ell-1)}^{(i_\ell)}\right)a^\sharp\left(f_{\pi_\ell(2k_\ell)}^{(i_\ell)}\right)\right]\times\cdots\\
& & \qquad\qquad\qquad\qquad\qquad\qquad \cdots\times\omega_{i_\ell}\left[a^\sharp\left(f_{\pi_\ell(1)}^{(i_\ell)}\right)a^\sharp\left(f_{\pi_\ell(2)}^{(i_\ell)}\right)\right]\\
& = & X.\end{aligned}$$ This proves that the state $\otimes_i\omega_i$ is quasi-free, which concludes the proof.
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[^1]: For convenience, we will later define a state as a linear form on the tensor product of CAR algebras $\operatorname{\mathfrak{A}}[{\mathfrak{H}}_+]\otimes\operatorname{\mathfrak{A}}[{\mathfrak{H}}_-]\subset{\mathcal{B}}({\mathcal{F}}_0)$, which in our context does not change anything since all the states we consider are normal.
[^2]: Our convention is ${\widehat}{f}(k)=(2\pi)^{-3/2}\int_{{{\ensuremath {\mathbb R} }}^3}f(x)e^{-ik\cdot x}{{\ensuremath{\,\text{d}x}}}, \quad\forall k\in{{\ensuremath {\mathbb R} }}^3$.
[^3]: Recall that the adjoint $J^*$ of an anti-linear operator is defined as $\langle J^*f,g\rangle:=\langle J g,f\rangle_{\mathfrak{K}}$ for all $f\in{\mathfrak{K}}$ and $g\in{\mathfrak{H}}$. Typically, one chooses ${\mathfrak{K}}={\mathfrak{H}}$ and $J$ the complex conjugation, or ${\mathfrak{K}}={\mathfrak{H}}^*$ and $J(f)=\langle f,\cdot\rangle$ [@Solovej-07]. Here we keep ${\mathfrak{K}}$ abstract because this will be useful for the construction of BDF states in Section \[sec:BDFstates\].
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abstract: 'Large spatiotemporal demand datasets can prove intractable for location optimization problems, motivating the need to aggregate such data. However, demand aggregation introduces error which impacts the results of the location study. We introduce and apply a framework for comparing both deterministic and stochastic aggregation methods using distance-based and volume-based aggregation error metrics. In addition we introduce and apply weighted versions of these metrics to account for the reality that demand events are non-homogeneous. These metrics are applied to a large, highly variable, spatiotemporal demand dataset of search and rescue events in the Pacific ocean. Comparisons with these metrics between six quadrat aggregations of varying scales and two zonal distribution models using hierarchical clustering is conducted. We show that as quadrat fidelity increases the distance-based aggregation error decreases, while the two deliberate zonal approaches further reduce this error while utilizing fewer zones. However, the higher fidelity aggregations have a detrimental effect on volume error. In addition, by splitting the search and rescue dataset into a training and test set we show that stochastic aggregation of this highly variable spatiotemporal demand appears to be effective at simulating actual future demands.'
address: 'Air Force Institute of Technology, 2950 Hobson Way, Wright Patterson Air Force Base, Ohio, 45433'
author:
- 'Zachary T. Hornberger'
- 'Bruce A. Cox'
- 'Raymond R. Hill'
bibliography:
- 'mybibfile.bib'
title: Effects of Aggregation Methodology on Uncertain Spatiotemporal Data
---
Location; Aggregation; Modifiable Areal Unit Problem; Aggregation Error
Introduction
============
Location modeling is a branch of operations research with vast real-world applicability and thus has been studied for a number of decades. Location modeling typically considers the location and time of demand signals over a network and optimizes the corresponding location of a servicing asset, such as a factory or vehicle. The underlying spatiotemporal demand signal data points are thus instrumental to the quality of the resulting model.
When considering large spatiotemporal datasets, there is frequently a need to aggregate demand points to make the problem more tractable for the solver, clearer for the analyst, and comprehensible for the end-user. Aggregation, while of practical use, is not a lossless compression, and introduces aggregation error into the model. When location data is aggregated, the resulting grouping’s location is traditionally represented by an aggregated data point. The distances between the actual demand points and the aggregated data points depend on the size of the aggregated region and the manner of aggregation. Similarly, the magnitude of uncertainty in the aggregated demand volumes is influenced by the nature of the aggregation. Therefore, great consideration must be given to the aggregation technique used when solving location problems.
The impact of aggregation becomes more pronounced when the geographic region expands in size and there is high variability in demand density across the region; this struggle is actualized in studying the United State Coast Guard (USCG) District 14’s search and rescue (SAR) mission. The international community recognizes the need for global cooperation in responding to emerging crises around the world. Nations have entered into SAR agreements, dividing the globe into respective search and rescue regions (SSRs). Per the United States National Search and Rescue Supplement to the International Aeronautical and Maritime Search and Rescue Manual [@NatSAR], the USCG is the federal SAR coordinator for SAR missions within the United States’ maritime SSRs and the aeronautical SSRs that do not overlay the continental United States or Alaska.
USCG District 14 is headquartered in Honolulu, Hawaii and is responsible for USCG statutory missions across the Pacific region. In particular, the district’s SSR spans more than 12 million square nautical miles, though the preponderance of SAR emergencies occur in the vicinity of Guam and the Hawaiian Islands. Additionally, District 14 has among the fewest assets in the USCG fleet, increasing the necessity to optimally posture those assets across the Pacific. Given the time-sensitive nature of rescue operations, it is imperative the USCG be optimally postured to ensure rapid response. Over the past decade, researchers have partnered with Coast Guard units - USCG and international - to solve these variations of the traditional facility location problem. These studies typically use historic SAR event data as the foundation of either a deterministic or simulation-based location model.
This study quantifies the effects of the aggregation trade-off for spatiotemporal data over a large region, using District 14 SAR emergency data as a practical basis for consideration. Section 2 of this paper reviews previous works related to the aggregation of data for location models in general and coast guard SAR missions in particular. In section 3, we outline the methodology for implementing various aggregation techniques, both deterministic and stochastic, using a training data set. In section 4, we evaluate the effectiveness of these techniques by quantifying the aggregation errors between the modelled demand and actual demand over a two-year period. In section 5, we review our findings and provide recommendations for future research.
Related Works
=============
Researchers have long been cognizant of a relationship between the methods used to aggregate location data and the resulting solutions generated by location models using this data. [@Gehlke1934] were among the first to note this problem, observing that the smoothing of census data inherent in aggregation resulted in a loss of valuable information and impacted the corresponding correlation coefficients of their models. [@Hillsman1978] laid a foundation for aggregation theory when they classified three sources of error (type A, B, and C) associated with representing individual demand points using aggregated demand points for solving factory location problems. Source A refers to the difference in distance from the aggregated demand points to the placed factory and the sum of distances from individual demand points to the factory. Source B is similar to Source A, if the factory were required to be collocated with an aggregated demand point. Source C refers to the phenomena where individual demand points are erroneously assigned to inefficient factories due to the zone in which it is aggregated.
Several research teams have subsequently sought to quantify and minimize these aggregation errors. [@Papadimitriou1981] presents two heuristics for aggregating data points in a manner that reduces the worst-case aggregation error and [@Zemel1984] produced a theorem for the worst-case bounds on Papadimitriou’s honeycomb approach. [@Qi2010] note the underlying assumption to Zemel’s work of uniformly distributed demand points, and propose a multi-pattern tiling approach for considering arbitrarily distributed demand. Works by [@Current1987; @Current1990], outline methods for eliminating Source A and B error when solving P-Median, set covering, and maximal covering location problems. [@Francis2014] present a metric for measuring the error bounds for a P-Median problem, and [@Francis2004b] discuss formulations for minimizing the aggregation error using a penalty function approach.
In the fields of geography and ecology, aggregation error of spatial data points is dubbed the modifiable areal unit problem (MAUP) [@Openshaw; @Dark2007] or the zone definition problem [@Fotheringham1995; @Curtis1996]. Research into MAUP typically decomposes the problem into two main effects: the scale effect and the zone effect. The scale effect refers to the impact on the spatial analysis results that are caused by the fidelty of the aggregation; for example, the impact of aggregating demand in a city using 200m x 200m grids versus 1km x 1km grids. Conversely, the zone effect refers to the impact caused by the way in which aggregation zones are bounded; for example, the impact of aggregating demand in a state using county lines versus city limits versus a grid overlay [@Openshaw; @Dark2007]. created a seminal contrived demonstration of these effects, which we replicate for completeness in Figure \[JelWU97\].
![(a-c) show effect of scale effect. As scale of aggregation increases mean does not change but variance declines. (d-e) show effect of aggregation effect. Keeping scale equal but changing method of aggregation changes variance. (c,e,f) show that even when number of zones is constant (4) mean and variance can change. []{data-label="JelWU97"}](JelWu.PNG){width="85.00000%"}
Previous research on MAUP has cautioned against arbitrary aggregation of spatial data and stressed its threat on the reliability of the resulting location analysis. [@Openshaw] was foundational in the study of MAUP and called for developing better methods for aggregating spatial data due to MAUP’s impact on the reliability of geographic studies. [@Curtis1996] studied data for New York and concluded that researchers can bias the results of their analysis based on the means of aggregation, even if there appears to be a logical basis for the employed method of aggregation. [@Fotheringham1995] go so far as to question the accuracy of any location-based analysis conducted using aggregated data because of the effects of MAUP.
In studies of MAUP, and aggregation theory in general, trends have emerged. Increases in the number of aggregated zones are typically proportional to decreases in distance-based aggregation error; distance-based aggregation error disappears when each distinct demand point is assigned to a unique zone (i.e., the number of aggregation zones equals the number of demand points). As any grouping introduces an associated level of distance-based error, it follows that reducing the amount of aggregation would subsequently reduce this error. [@Francis2004] notes the law of diminishing returns applies in this context, however, suggesting that iterative reductions in the number of aggregate groups shows diminishing improvements to error reduction. [@Francis1992] discuss the *paradox of aggregation*, noting that solving formulations to minimize error can be more cumbersome than the original location problem being solved, which is counter-intuitive as aggregation is employed to simplify the resolution of these original location problems. [@Dark2007] consider the trends corresponding to both the scale effect and the zone effect. A known benefit of aggregation is tied to the scale effect; predictions of aggregated demand levels tend to be more accurate with fewer, larger aggregate zones. This is because when there are more demand points consolidated in each zone, the demand variance between zones decreases. The impact of zone effect is less understood and tends to differ from problem-to-problem.
The importance of careful aggregation has been thoroughly studied and is synthesized by [@Francis2008]. In their survey of previous literature regarding aggregation error associated with location problems, Francis et al. note that there is an inherent tradeoff when aggregating data points; although aggregation has a tendency to decrease computational requirements and statistical uncertainty within the grouped data, it increases the error within the model by introducing aggregation error. Thus there does not exist a singular “best” level of aggregation and the tradeoffs inherent in aggregation must be considered.
In addition to the theoretical work on this problem, there has been applied work specifically relating to Coast Guard SAR missions. Although some research into this area was conducted in the late 1970s [@Armstrong], the preponderance of studies relating to Coast Guard posturing has emerged in the past decade. Studies researching the allocation of SAR assets, or facilities, typically adopt a quadrat modeling technique for aggregating location data [@AkbariINFOR; @Akbari; @Karatas; @Afshartous]. This technique consists of decomposing the region in question into square cells using a grid overlay. Notably, the quadrat method is frequently adopted in crime data analyses, which typically seek to quantify spatial trends in criminal activity across a city or state [@Anselin; @Chainey].
[@Armstrong] constructed a goal-programming model for assigning SAR aircraft, incorporating probabilistic consideration for the time required by the aircraft to locate distress events in different areas of the corresponding region, using a grid overlay to create a collection of square zones. These zones were then assigned deterministic values, representing the average number of distress events per month. Similarly, [@Karatas] utilized a quadrat model for simulating the location and volume of distress calls for the Turkish Coast Guard in the Aegean Sea. They first determined the optimal resource allocation strategy using individual events as separate demand nodes, and then evaluated the effectiveness of this strategy using simulated demand.
The incorporation of kernel density estimation with the quadrat model, popular in crime data analysis [@Anselin], has been previously implemented in SAR location problems. The kernel density estimation method composes the region into grid cells and assigns a density function to each data point ($s_i$). Points that are within proximity to each other relative to a specified bandwidth ($\tau$), are grouped into a kernel ($k$) and their density functions are combined. The resulting image is a smooth heat map with greater densities illustrated over areas that have the most activity clustered closely together [@Anselin; @Chainey]. [@Erdemir] utilized kernel density estimation when considering the problem of locating aeromedical bases across the state of New Mexico. Similarly, [@Akbari] implemented a kernel density estimation approach to approximate the intensity of distress calls received by the Canadian Coast Guard. They varied the size of the grid overlay based upon the proximity to shoreline. This decision was based upon the assumption that since most distress events occurred closer to shore, the analysis would benefit from greater fidelity in aggregation along the coastline.
Though not specifically kernel density estimation, [@Afshartous] implemented an intensity function-based approach for solving the Coast Guard SAR location problem. They first constructed a non-parametric statistical simulation of distress calls within USCG District 7 (headquartered in Miami, Florida) and then utilized their simulation to model demand for a facility location problem. This simulation was constructed by overlaying the region with a relatively fine grid and estimating the intensity of distress calls for each cell.
While most work regarding SAR posturing has incorporated quadrat techniques, [@Azofra] introduced an intuitive method that has been applied to maritime research. Instead of defaulting to grids, Azofra’s zonal distribution model allows for flexibility in the definition of emergency zones, such as zones based upon subject matter expertise. Once the zones are determined, the centroids of distress calls, dubbed *superaccidents*, are computed for each zone. The zonal distribution model is a gravitational model, with the determination in optimal SAR operational response based upon the distance to the superaccidents and their associated weight. They demonstrate the implementation of this model using a notional example involving three superaccidents and three ports.
Since the introduction of the zonal distribution model, some researchers have opted to expand upon it by applying it to real-world problems. [@Ai] utilize this model for locating supply bases and positioning vessels for maritime emergencies for a portion of the coastline of China along the Yellow Sea. While not adhering to the strict grid cells of previous studies, their zones remained rectangular in shape and varied in size across the region. [@Razi] improved upon the zonal distribution model by utilizing a *k*-means clustering algorithm for defining the zones and implementing a weighted approach for locating the superaccidents. By adopting this approach, Razi et al. define the aggregated zones and corresponding representative demand nodes based upon historical trends in distress calls in the Aegean Sea rather than arbitrary cells. [@Hornberger] propose an extension to the work of Razi and Karatas, which they dub the stochastic zonal distribution model. Their model implements hierarchical *k*-means clustering algorithm to define the aggregation zones, fits probability distributions to model the SAR demand for each zone, and then uses empirically constructed discrete distributions to model the corresponding rescue response for each emergency.
A review of the existing literature regarding SAR asset posturing models finds a lack of explicit consideration regarding the impact of aggregation. Additionally, as SAR research expands to larger regions of consideration (e.g., oceans vs. seas or shorelines), it is necessary to more thoroughly consider the effects of various aggregation methods. Outside of SAR, and more generally emergency response asset modeling (e.g., [@araz2007fuzzy]), other transportation resource posturing problems which utilize massive demand data-sets assume or require demand aggregation (e.g., taxi service areas ([@li2019taxi], [@rajendran2019insights]), and should also be concerned with how such aggregation effects the associated location modeling. To provide such consideration, our study utilizes historic SAR data from across the Pacific Ocean to compare the effectiveness of a zonal aggregation technique compared to quadrats of varying fidelity. Additionally, we evaluate these tradeoffs in the aggregation as applied to deterministic and stochastic implementations.
Methodology
===========
In this section, we consider two key characteristics that define a zonal aggregation of demand signals: dividing the region into zones, and modeling the demand level. Using these two characteristics as the framework, we model and compare the following methodologies: deterministic quadrat approaches of various fidelities, the [@Razi] zonal distribution model, and the [@Hornberger] stochastic zonal distribution model.
These methodologies are compared using the District 14 SAR region, an interesting test case due to its large area and highly variable demand levels; Figure \[SAR\_Region\] depicts the Honolulu Maritime Search and Rescue Region [@SARPlan]. Historic search and rescue demand data was obtained from the Marine Information for Safety and Law Enforcement (MISLE) database to form both a training set and a test set. The training set is comprised of SAR events from a 5 year span (January 2011 - December 2015) and is utilized to construct the models of spatiotemporal SAR demand. The accuracy of the aggregated demand methodologies is then evaluated using historic SAR data for the same region from January 2016 - December 2017.
![Honolulu Maritime SAR Region[]{data-label="SAR_Region"}](SAR_Region.PNG){width="85.00000%"}
The training and test data is scoped to only consider events that occurred within the District 14 area of responsibility (AOR). Additionally, demand points missing GPS coordinates were removed as were data points classified as medical consultations since these consultations only require a discussion with a medical professional over the phone and resources are not dispatched. The final training set contains 2629 demand points and the test set contains 1080 demand points.
Modeling Spatiotemporal Demand
------------------------------
The quadrat aggregation approach was implemented with 6 different quadrat scales to test the impact of the scale effect. These six grid-based decompositions of the region are labelled Aggregations A - F. Aggregation A considered the region of study as a singular zone, consolidating all demand points; see Figure \[AggA\]. Aggregation B divided the region into two zones along the antimeridiean; see Figure \[AggB\]. Aggregations C, D, and E are iterative increases in fidelity, decomposing the region into eight, fifteen, and forty-three zones, respectively; see Figures \[AggC\], \[AggD\], and \[AggE\]. Aggregation F adopts the approach employed by [@Akbari] and allows for smaller grid cells in sections of higher demand. Specifically, the two zones from Aggregation E with the greatest proportion of Guam and Hawaiian Island workloads are further decomposed into $1^o$ x $1^o$ cells; Aggregation F results in 212 zones. Aggregation F is depicted in Figure \[AggF\].
![Aggregation A (1 Zone)[]{data-label="AggA"}](AggAZones.PNG){width="85.00000%"}
![Aggregation B (2 Zones)[]{data-label="AggB"}](AggBZones.PNG){width="85.00000%"}
![Aggregation C (8 Zones)[]{data-label="AggC"}](AggCZones.PNG){width="85.00000%"}
![Aggregation D (15 Zones)[]{data-label="AggD"}](AggDZones.PNG){width="85.00000%"}
![Aggregation E (43 Zones)[]{data-label="AggE"}](AggEZones.PNG){width="85.00000%"}
![Aggregation F (212 Zones)[]{data-label="AggF"}](AggFZones.PNG){width="85.00000%"}
Aggregation ZDM was constructed utilizing [@Razi] general implementation of the zonal distribution model and divided the AOR using a weighted *k*-means clustering algorithm; see Figure \[ZDM\]. Razi and Karatas defined the weight of each SAR event using an analytical hierarchy process based upon the level of fatality, material damage, response arduousness, and environmental impact. Their weighting scheme was not viable for this study based on the available information in MISLE, so this implementation of Razi and Karatas’s procedure utilizes *total activities* as a weighting. The metric of total activities represents the number of resources assigned to a rescue operations, in addition to the instances when a significant change occurred in the course of the rescue operation; this metric of total activities serves as a proxy for the complexity of a SAR event. Razi and Karatas determine the number of zones to cluster demand points into based upon a *rule of thumb method* proposed by [@Kodinariya]. This method suggests that the number of zones Z is based upon the total number of events K, such that $|Z| \approx \sqrt{|K|/2}$.
![Aggregation ZDM (36 Zones)[]{data-label="ZDM"}](AggZDM.PNG){width="85.00000%"}
Aggregation SZDM was developed by implementing the stochastic zonal distribution model approach proposed by [@Hornberger]; see Figure \[SZDM\]. Hornberger et al. utilized a hierarchical *k*-means clustering algorithm to aggregate demand points into zones. All demand points are sorted into mutually exclusive groups based upon the unit that coordinated the response and the types of assets utilized in the response. District 14 is divided into Sector Guam and Sector Honolulu, which split the coverage of the AOR around longitude $160^o$ E. Current policy dictates that the mission range for USCG boats is 50 nautical miles from the shoreline of an island on which there exists a USCG boat station; District 14 has boat stations located on the islands of Guam, O’ahu, Kaua’i, and Maui. Hornberger et al. note that a reasonable approximation of asset utilization would be a combination of boats and helicopter aircraft responding to SAR events within the 50 nautical mile boundary of these islands while a combination of cutters and aeroplane aircraft respond to SAR events beyond these boundaries. Therefore, all demand points where sorted into the following mutually exclusive groups: Guam Boat/Helicopter Events, Guam Cutter/Airplane Events, Hawaii Boat/Helicopter Events, and Hawaii Cutter/Airplane Events. These groups are further decomposed into clusters based upon the geographic proximity of the data points by employing a *k*-means clustering algorithm. The number of zones was determined by considering the relationship between the number of zones and the corresponding within-cluster variance. A plot of this relationship forms an *elbow curve*, whose name is tied to the phenomena that initial groupings account for a greater reduction in variance compared to subsequent groupings; the ‘elbow’ of the curve occurs at the suggested number of zones for the data set.
![Aggregation SZDM (15 Zones)[]{data-label="SZDM"}](AggSZDM.PNG){width="85.00000%"}
Methods of Comparative Analysis
-------------------------------
This study evaluates the effectiveness of various methods of aggregation when conducting spatiotemporal forecasting. Specifically, we seek to assess the merit of the [@Razi] deterministic zonal distribution mode, and the [@Hornberger] stochastic zonal distribution model, comparing their effectiveness against traditional quadrat methods of varying fidelity’s. To conduct these comparisons, two metrics are considered: distance-based aggregation error and volume-based aggregation error.
The distance-based aggregation error ($d_e$) represents the total distance between where events were modelled as occurring ($\hat{x_j}$) and the actual location of their occurrence ($x_{i, j}$), for each event ($i \in I$) in the zone ($j \in J$). The anticipated event locations for all zones are weighted centroids for the each zone. In the quadrat models, the centroids are computed as an average of the latitudes/longitudes, multiplied by the events’ corresponding total activities, for all events in the zone. In the zonal and stochastic zonal distribution models, the clustering algorithm yields a weighted centroid. The distance-based aggregation error metric is:
$$\label{DistMetric}
d_e = \sum_{i \in I} \sum_{j \in J} |x_{i, j} - x_j|$$
where the Haversine formula,
$$\label{haversine}
d = 2r \text{ arcsin} \left( \sqrt{\text{sin}^2\left( \frac{\phi_2 - \phi_1}{2} \right) + \text{cos}(\phi_1)\text{cos}(\phi_2)\text{sin}^2\left( \frac{\theta_2 - \theta_1}{2}\right)} \right)$$
which, given latitudes $\phi$, and longitudes $\theta$, calculates the great-circle distance between two points, is used to calculate each individual distance.
The weighted distance-based aggregation error ($d_{we}$) is the sum of the differences in distance between where individual assets are modelled as being deployed to ($\hat{x_j}$) and the actual location assets are dispatched to. The weighting ($w_i$) is the number of assets assigned to the rescue operation. The difference between $d_e$ and $d_{we}$ is that the former treats individual SAR events as being equal in magnitude, whereas the latter incorporates the number of deployed assets. As with $d_e$ the individual distances in $d_{we}$ are calculated using the Haversine formula.
$$\label{DistMetric}
d_{we} = \sum_{i \in I} \sum_{j \in J} w_i|x_{i, j} - x_j|$$
The distance-based aggregation error ($d_e$), and the weighted distance-based aggregation error ($d_{we}$) are both computed for all aggregations A-F, as well as for the ZDM and the SZDM.
The volume-based aggregation error ($v_e$) represents the total difference between the predicted level of monthly demand ($\hat{l}_{j, k}$) and the actual level of monthly demand ($l_j$), for each month in the considered time frame ($k \in K$). The metric is computed as:
$$\label{VolMetric}
v_e = \sum_{j \in J} \sum_{k \in K} |l_{j, k} - l_j|$$
Given that a primary difference between ZDM and SZDM is the integration of stochastic elements in the modeling of the demand, both deterministic and stochastic demand comparisons for volume-based aggregation error are conducted. For purposes of consistency, all frequency considerations are made on a *per month* basis.
Aggregations A-F are compared to the ZDM using a deterministic demand signal. This requires a singular, static value which represents the typical demand volume for each zone. Two methods are frequently used to identify these deterministic values: averages and medians. The average value is a common metric and is familiar to an end-user decision maker, but can be easily skewed by the presence of outliers. Median values tend to be more stable in the presence of outliers and thus more representative of the typical demand volume. As such, median values are implemented as the metric for deterministic demand volume in this study.
The stochastic modelling approach utilized in SZDM considers the inherent uncertainty present in SAR events by fitting probability distributions to demand volumes in each zone. As noted by [@Afshartous] and [@Akbari], SAR events can often be viewed as Poisson processes. In particular, [@Hornberger] found the emergence of SAR events in District 14’s AOR could be modelled using poisson and gamma-poisson distributions. This study implements stochastic demand modeling in SZDM, and compares this to aggregations C and D to compare the impact of aggregation method on the simulation of future SAR demand. (Aggregations A and B were deemed too trivial to be of real interest, and stochastic models of Aggregations E and F proved intractable on the authors’ hardware.)
A modification of the volume-based aggregation error, $v_e = \sum_{j \in J} \sum_{k \in K} ( l_{j, k} - l_j)$, is also considered providing a distinction between over- and under-forecasting events. Stochastic models are compared graphically, plotting the simulated output for each month of the 24-month test period against the actual demand volume observed.
Analysis
========
Distance-Based Aggregation Error
--------------------------------
The distances, in nautical miles, between the aggregated demand point and the subsequent demand nodes during 2016 - 2017 are shown in Table \[DeterDemandDist\]. The resulting distance-based aggregation error for the quadrat models reflect the law of diminishing returns, as described by [@Francis2004]. The first division of the region of study, from Aggregation A to Aggregation B, results in an 82.3% reduction to the locational aggregation error. This error was continuously diminished with additional divisions. These results support the trend of location error generally reducing with additional zones.
**Aggregation** **Number of Zones** $d_e$ $d_{we}$
----------------- --------------------- ------------- -------------
A 1 1,471,479 2,195,276
B 2 251,042.3 312,118.6
C 8 171,531.3 225,615
D 15 158,119.1 208,812.7
E 43 86,745.88 119,741.5
F 212 51,553.33 66,668.67
*ZDM* *36* *80,165.06* *92,669.37*
*SZDM* *15* *92,067.72* *97,425.77*
: Distance-Based Aggregation Error[]{data-label="DeterDemandDist"}
Aggregations ZDM and SZDM perform very well compared to the quadrat models. The zonal distribution model has a lower associated location error than Aggregation E, despite only having 36 zones compared to Aggregation E’s 43 zones. This runs counter to the general claim that more zones always improves the accuracy of the location model, suggesting instead that deliberate steps can be implemented to aggregate spatial demand points in fewer clusters while still achieve competitively low levels of location error. The stochastic zonal distribution model’s results support this observation, achieving a 41.7% reduction in distance-based aggregation error compared to Aggregation D despite using the same number of zones.
Similar trends are observed when the attention is shifted from the error in SAR event distances to the error in resource dispatch distances. There is a steady improvement in accuracy as the number of zones is increased, with the exception of Aggregations ZDM and SZDM. Additionally, the differences between $d_e$ and $d_{we}$ are notably larger for the quadrat models compared to Aggregations ZDM and SZDM; the stochastic zonal distribution model had the smallest increase in location error when weighting by the number of resources dispatched. These observations suggest that deliberate zoning of demand point can enhance the robustness of aggregate zones to weighted events, particularly when the zones are developed with consideration to both geographic proximity and the operational characteristics that are tied to the event weights.
Deterministic Volume-Based Aggregation Error
--------------------------------------------
The total error in volume based upon the median monthly demand for each zone compared to the actual demand volumes as depicted in Table \[DeterDemandVol\]. The phenomena described by [@Francis2008] and [@Dark2007] is observed; there is a general increase in total volume-based aggregation error as the number of zones increases.
**Aggregation** **Number of Zones** $v_e$
----------------- --------------------- -------
A 1 139
B 2 189
C 8 288
D 15 306
E 44 372
F 212 584
*ZDM* *36* *458*
: Volume-Based Aggregation Error for Deterministic Demand Modeling[]{data-label="DeterDemandVol"}
Interestingly, implementing the zonal distribution model corresponds to a large volume-based aggregation area, second only to Aggregation F; see Figure \[Total\_Vol\]. This suggests deliberate clustering based on geographic proximity does not correspond to improvements in deterministic demand volume modeling.
![Comparison of the Total Volume-Based Aggregation Error for Deterministic Demand Modeling[]{data-label="Total_Vol"}](Deter_Total_Vol2.png){width="100.00000%"}
Additional analysis compared the tendency for different aggregation models to overpredict versus underpredict demand volume. A plot of this analysis is shown in Figure \[OverUnder\], colorcoding the region of overprediction as red and underprediction as blue. For each month, Aggregation A and B perform equally well; the lines overlap in the plot. With the exception of Aggregation F, all methods adhere to similar trends in spikes and drops throughout the test timeframe. The general trend is for models to underpredict more consistently as they incorporate more aggregated zones. The exception to this trend is the zonal distribution model, which continues to have greater volume-based aggregation error compared to Aggregation E.
![Comparison of Over- and Under-predictions fo Deterministic Demand Modeling[]{data-label="OverUnder"}](Deter_Vol_Over_Under2.png){width="100.00000%"}
Stochastic Volume-Based Aggregation Error
-----------------------------------------
A comparison of stochastic demand models was used probability distributions fit to each zone in Aggregations C, D, and SZDM. The results from these simulations are compared to the actual observed demand levels for the two-year test period; see Figure \[Stoch\_Vol\]. Note that since the demand distributions were observed to be relatively stationary at large, each month’s simulated volume from each model is determined by random draws from static probability distributions assigned to each zone (i.e., poisson and gamma-poisson distributions).
![Comparison of Stochastic Demand Models and Observed Demand Levels[]{data-label="Stoch_Vol"}](Stoch_Vol.PNG){width="100.00000%"}
Since the results from Figure \[Stoch\_Vol\] are randomly generated, the emphasis is less on the specific results from month-to-month and more on whether overall trend appears similar to the observed trend. This analysis shows similar trends for the three stochastic demand models, suggesting that they all could be used to effectively simulate the stochastic demand of the AOR. Aggregation C does make a notable spike in simulated SAR activity at the end of the test period, caused by the coincidence of multiple zones within the model simulating larger-than-normal demand volume. This phenomena was investigated further.
While the observed demand volume fluctuates from month-to-month, it stays within the bounds of 30 and 60 events per month. Using these levels as thresholds, a monte carlo simulation of 10,000 2-year models was constructed. For each of the 240,000 simulated months, Table \[ExtMonth\] shows the number that were beyond the thresholds of 30 and 60 events per month. All models appear relatively stable compared to these bounds; Aggregation C, with the greatest number of ‘extreme months’, only had approximately 4.6% of the 240,000 months classified as ‘extreme’. The stochastic zonal distribution model appeared to be the most stable of the three considered models, having the fewest months classified as ‘extreme’ on either side of the bound. These findings suggests that while extreme months are not likely to be a significant occurrence in a simulation of SAR demand, the stochastic zonal distribution model minimizes the likelihood this will occur.
**Aggregation** **Below 30 Events** **Above 60 Events**
----------------- --------------------- ---------------------
C 6175 5056
D 5402 4660
*SZDM* 4727 3854
: Comparison of Extreme Months over 10,000 2-Year Simulations[]{data-label="ExtMonth"}
Conclusion
==========
The method used to aggregate spatiotemporal demands affects the outcome of location models built using the aggregated data, thus an understanding of the impacts of aggregation methods is fundamental. We have presented a framework for comparison of both static and stochastic spatiotemporal aggregation models, utilizing both a distance based aggregation error metric, an event magnitude weighted distance based aggregation error metric, and a volume based aggregation error metric. We further applied this framework to test six quadrat aggregation models of varying fidelity’s, and two zonal based models, using historical search and rescue data from a massive scale region possessing highly variable demands. As expected aggregations with greater fidelity tend to reduce the distance-based aggregation error. In addition implementation of a deliberate zoning approach (e.g., ZDM and SDZM) further reduce this error while utilizing fewer zones. However, higher fidelity aggregations with increased number of zones has a detrimental effect on the modelling of demand volumes. Finally, stochastic representations of SAR demand appears to be effective at simulating actual SAR demand.
Based on the results of our aggregation analysis we propose the following as potential exploratory efforts. Zonal techniques based on hierarchies and clustering techniques seem very promising, additional research on the impacts of clustering techniques could be fruitful. Additionally combining these zonal techniques, with their associated reduced location errors, with a lower fidelity aggregation model to project region level demands may be useful. Finally, a study examining possible nonlinear dynamic effects on the resulting output of location models as a result of changes in aggregation method may be informative.
|
---
abstract: 'Recent work has proposed various adversarial losses for training generative adversarial networks. Yet, it remains unclear what certain types of functions are valid adversarial loss functions, and how these loss functions perform against one another. In this paper, we aim to gain a deeper understanding of adversarial losses by decoupling the effects of their component functions and regularization terms. We first derive some necessary and sufficient conditions of the component functions such that the adversarial loss is a divergence-like measure between the data and the model distributions. In order to systematically compare different adversarial losses, we then propose DANTest—a new, simple framework based on discriminative adversarial networks. With this framework, we evaluate an extensive set of adversarial losses by combining different component functions and regularization approaches. This study leads to some new insights into the adversarial losses. For reproducibility, all source code is available at <https://github.com/salu133445/dan>.'
bibliography:
- 'ref.bib'
---
Introduction {#sec:intro}
============
Generative adversarial networks (GANs) [@goodfellow2014] are a class of unsupervised machine learning algorithms. In essence, a GAN learn a generative model with the guidance of another discriminative model which is trained jointly. However, the idea of adversarial losses is not limited to unsupervised learning. Adversarial losses can also be applied to supervised and semi-supervised scenarios (e.g., [@isola2017; @dossantos2017]). Over the past few years, adversarial losses have advanced the state of the art in many fields [@goodfellow2016].
Despite the success, there are several open questions that need to be addressed. On one hand, although plenty adversarial losses have been proposed, we have little theoretical understanding of what makes a loss function a valid one.
On the other hand, we note that any two adversarial losses can differ in terms of not only the *component functions* (e.g., minimax or hinge; see [Section \[sec:background\]]{}) used in the main loss function that sets up the two-player adversarial game, but also the *regularization approaches* (e.g., gradient penalties [@gulrajani2017]) used to regularize the models. However, it remains unclear how they respectively contribute to the performance of an adversarial loss. In other words, when empirically compare two adversarial losses, we need to decouple the effects of the component functions and the regularization terms, otherwise we cannot tell which one of them makes an adversarial loss better than the other.
Among existing comparative analysis of adversarial losses, to the best of our knowledge, only @lucic2018 and @kurach2018 attempted to decouple the effects of the component functions and regularization approaches. But, only few combinations of component functions and regularization approaches were tested in these two prior works, only seven and nine respectively. We attribute this to the high computational cost that may involve to conduct the experiments, and, more importantly, the lack of a framework to systematically evaluate adversarial losses.
$f$ $g$ $h$ $y^*$
--------------------------------- --------------------- ------------------------- ------------------------- ---------------
minimax [@goodfellow2014] $-\log(1 + e^{-y})$ $-y - \log(1 + e^{-y})$ $-y - \log(1 + e^{-y})$ $0$
nonsaturating [@goodfellow2014] $-\log(1 + e^{-y})$ $-y - \log(1 + e^{-y})$ $\log(1 + e^{-y})$ $0$
Wasserstein [@arjovsky2017wgan] $y$ $-y$ $-y$ $0$
least squares [@mao2017] $-(y - 1)^2$ $-y^2$ $(y - 1)^2$ $\frac{1}{2}$
hinge [@lim2017; @tran2017] $\min(0, y - 1)$ $\min(0, -y - 1)$ $-y$ $0$
These two research questions can be summarized as follows:
1. What certain types of component functions are theoretically valid adversarial loss functions?
2. How different combinations of the component functions and the regularization approaches perform empirically against one another?
We aim to tackle these two RQs in this paper to advance our understanding of the adversarial losses. Specifically, our contribution to RQ1 is based on the intuition that a favorable adversarial loss should be a divergence-like measure between the distribution of the real data and the distribution of the model output, since in this way we can use the adversarial loss as the training criterion to learn the model parameters. We derive necessary and sufficient conditions such that an adversarial loss has such a favorable property (Sections \[sec:necessary\_conditions\] and \[sec:sufficient\_conditions\]). Interestingly, our theoretical analysis leads to a new perspective to understand the underlying game dynamics of adversarial losses ([Section \[sec:psi\_function\_analysis\]]{}).
For RQ2, we need an efficient way to compare different adversarial losses. Hence, we adopt the discriminative adversarial networks (DANs) [@mirza2014], which are essentially conditional GANs with both the generator and the discriminator being discriminative models. Based on DANs, we propose *DANTest*—a new, simple framework for comparing adversarial losses ([Section \[sec:dantest\]]{}). The main idea is to first train a number of DANs for a supervised learning task (e.g., classification) using different adversarial losses, and then compare their performance using standard evaluation metrics for supervised learning (e.g., classification accuracy). With the DANTest, we systematically evaluate 168 adversarial losses featuring the combination of ten existing component functions, two new component functions we originally propose in this paper in light of our theoretical analysis, and 14 existing regularization approaches ([Section \[sec:experiments\]]{}). Moreover, we use the DANTest to empirically study the effect of the Lipschitz constant [@arjovsky2017wgan], penalty weights [@mescheder2018], momentum terms [@kingma2014], and others. We discuss the new insights that are gained, and their implications to the design of adversarial losses in future research.
Background {#sec:background}
==========
Generative Adversarial Networks {#sec:gan}
-------------------------------
A generative adversarial network [@goodfellow2014] is a generative latent variable model that aims to learn a mapping from a latent space $\mathcal{Z}$ to the data space $\mathcal{X}$, i.e., a generative model $G$, which we will refer to as the *generator*. A discriminative model $D$ (i.e., the *discriminator*) defined on $\mathcal{X}$ is trained alongside the $G$ to provide guidance for it. Let $p_d$ denote the *data distribution* and $p_g$ be the *model distribution* implicitly defined by $G({\mathbf{z}})$ when ${\mathbf{z}}\sim p_{\mathbf{z}}$. In general, most GAN loss functions proposed in the literature can be formulated as: $$\begin{aligned}
\label{eq:discriminator}
\max_{D}\;&{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,,\\
\label{eq:generator}
\min_{G}\;&{\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[h(D({\tilde{{\mathbf{x}}}}))]\,,\end{aligned}$$ where $f$, $g$ and $h$ are real functions defined on the data space (i.e., ${\mathcal{X}}\to {\mathbb{R}}$) and we will refer to them as the *compoenent functions*. We summarize in [ \[tab:loss\_functions\]]{} the component functions $f$, $g$ and $h$ used in some existing adversarial losses.
$p_{{\hat{{\mathbf{x}}}}}$ $R(x)$
--------------------------------------------- ------------------------------ -----------------------------
coupled gradient penalties [@gulrajani2017] $p_d + U[0, 1]\,(p_g - p_d)$ $(x - k)^2$ or $\max(x, k)$
local gradient penalties [@kodali2017] $p_d + c\,N(0, I)$ $(x - k)^2$ or $\max(x, k)$
R~1~ gradient penalties [@mescheder2018] $p_d$ $x$
R~2~ gradient penalties [@mescheder2018] $p_g$ $x$
-------------------------------------------------- ------------------------------------------------ --------------------------------------------- ---------------------------------------------
{width=".135\linewidth"} {width=".135\linewidth"} {width=".135\linewidth"} {width=".135\linewidth"}
\(a) coupled gradient penalties \(b) local gradient penalties \(c) R~1~ gradient penalties \(d) R~2~ gradient penalties
-------------------------------------------------- ------------------------------------------------ --------------------------------------------- ---------------------------------------------
Some prior work has also investigated the so-called IPM-based GANs, where the discriminator is trained to estimate an integral probability metric (IPM) between $p_d$ and $p_g$: $$\begin{aligned}
\label{eq:ipm_distance}
d(p_d, p_g) = -\sup_{D\in\mathcal{D}}\;{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[D({\mathbf{x}})] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[D({\tilde{{\mathbf{x}}}})]\,,\end{aligned}$$ where $\mathcal{D}$ is a set of functions from ${\mathcal{X}}$ to ${\mathbb{R}}$. For example, the Wasserstein GANs [@arjovsky2017wgan] consider $\mathcal{D}$ to be the set of all 1-Lipschitz functions. Other examples include McGAN [@mroueh2017mcgan], MMD GAN [@li2017] and Fisher GAN [@mroueh2017fishergan]. Please note that the main difference between and is that in the latter we constrain $D$ to be in some set of functions $\mathcal{D}$.
Gradient Penalties {#sec:gradient_penalties}
------------------
As the discriminator is often found to be too strong to provide reliable gradients to the generator, one regularization approach is to use some gradient penalties to constrain the modeling capability of the discriminator. Most gradient penalties proposed in the literature take the following form: $$\label{eq:gradient_penalties}
\lambda\,{\mathbb{E}}_{{\hat{{\mathbf{x}}}}\sim p_{{\hat{{\mathbf{x}}}}}}[R(||\nabla_{{\hat{{\mathbf{x}}}}} D({\hat{{\mathbf{x}}}})||)]\,,$$ where the *penality weight* $\lambda \in {\mathbb{R}}$ is a pre-defined constant, and $R(\cdot)$ is a real function. The distribution $p_{{\hat{{\mathbf{x}}}}}$ defines where the gradient penalties are enforced. [ \[tab:gradient\_penalties\]]{} shows the distribution $p_{{\hat{{\mathbf{x}}}}}$ and function $R$ used in some common gradient penalties. And, [ \[fig:gradient\_penalties\]]{} illustrates $p_{{\hat{{\mathbf{x}}}}}$.
When gradient penalties are enforced, the loss function for training the discriminator contains not only the component functions $f$ and $g$ in but also the *regularization term* .
Spectral Normalization {#sec:spectral_normalization}
----------------------
Another regularization approach we consider is the spectral normalization proposed by @miyato2018. It normalizes the spectral norm of each layer in a neural network to enforce the Lipschitz constraints. While the gradient penalties introduced in [Section \[sec:gradient\_penalties\]]{} impose local regularizations, the spectral normalization imposes a global regularization on the discriminator. Therefore, it is possible to combine the spectral normalization with the gradient penalties. We will examine this in [Section \[sec:exp\_adversarial\_losses\]]{}.
Theoretical Results {#sec:theory}
===================
In the following analysis, we follow the notations in and . Proofs can be found in [Appendix \[app:sec:proofs\]]{}.
Favorable properties for adversarial losses
-------------------------------------------
Let us first consider the minimax formulation: $$\begin{aligned}
\label{eq:minimax}
\min_{G}\;\max_{D}\;&{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,.\end{aligned}$$ We can see that if the discriminator is able to reach optimality, the training criterion for the generator is $$\begin{aligned}
\label{eq:g_loss}
L_G &= \max_{D}\;{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,.\end{aligned}$$
In general, for a valid adversarial loss, the discriminator is responsible for providing a measure of the discrepancy between the data distribution $p_d$ and the model distribution $p_g$. In principle, this will then serve as the training criterion for the generator to push $p_g$ towards $p_d$. Hence, we would like such an adversarial loss to be a divergence-like measure between $p_g$ and $p_d$. From this view, we can now define the following two favorable properties of adversarial losses.
[(Weak favorable property)]{.nodecor} For any fixed $p_d$, $L_G$ has a global minimum at $p_g = p_d$. \[prop:weak\]
[(Strong favorable property)]{.nodecor} For any fixed $p_d$, $L_G$ has a unique global minimum at $p_g = p_d$. \[prop:strong\]
We can see that [Property \[prop:strong\]]{} makes $L_G - L^*_G$ a divergence of $p_d$ and $p_g$ for any fixed $p_d$, where $L^*_G = L_G\,\big\rvert_{\,p_g = p_d}$, and [Property \[prop:weak\]]{} provides a weaker version when the identity of indiscernibles is not necessary. Note that $L_G$ is not a divergence since $L_G \geq 0$ does not always hold.
$\Psi$ and $\psi$ functions
---------------------------
In order to derive some necessary and sufficient conditions for Properties \[prop:weak\] and \[prop:strong\], we first observe from that $$\begin{aligned}
&L_G = \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_g({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
\begin{split}
&= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\quad\;\;\, \max_{D} \left(\frac{p_d({\mathbf{x}})\,f(D({\mathbf{x}}))}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} + \frac{p_g({\mathbf{x}})\,g(D({\mathbf{x}}))}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\,.
\end{split}\end{aligned}$$ Now, if we let $\tilde{\gamma} = \frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}$ and $\tilde{y} = D({\mathbf{x}})$, we get $$\label{eq:g_loss_expanded}
\begin{split}
L_G &= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\quad\qquad\; \max_{\tilde{y}}\;\tilde{\gamma}\,f(\tilde{y}) + (1 - \tilde{\gamma})\,g(\tilde{y})\,d{\mathbf{x}}\,.
\end{split}$$ Please note that $\tilde{\gamma}({\mathbf{x}}) = \frac{1}{2}$ if and only if $p_d({\mathbf{x}}) = p_g({\mathbf{x}})$. Let us now consider the terms inside the integral and define the following two functions: $$\begin{aligned}
\label{eq:big_psi_function}
&\Psi(\gamma, y) = \gamma\,f(y) + (1 - \gamma)\,g(y)\,,\\
\label{eq:small_psi_function}
&\psi(\gamma) = \max_y\;\Psi(\gamma, y)\,,\end{aligned}$$ where $\gamma \in [0, 1]$ and $y \in {\mathbb{R}}$ are two variables independent of ${\mathbf{x}}$. We visualize in Figures \[fig:psi\_functions\](a)–(d) the $\Psi$ and $\psi$ functions for different common adversarial losses (see [Appendix \[app:sec:psi\_function\_graphs\]]{} for the graphs of the $\psi$ functions alone). These two functions actually reflect some important characteristics of the adversarial losses (see [Section \[sec:psi\_function\_analysis\]]{}) and will be used intensively in our theoretical analysis.
Necessary conditions for the favorable properties {#sec:necessary_conditions}
-------------------------------------------------
For the necessary conditions of Properties \[prop:weak\] and \[prop:strong\], we have the following two theorems.
If [Property \[prop:weak\]]{} holds, then for any $\gamma \in [0, 1]$, $\psi(\gamma) + \psi(1 - \gamma) \geq 2\,\psi(\frac{1}{2})$. \[theo:weak\_necessary\_condition\]
If [Property \[prop:strong\]]{} holds, then for any $\gamma \in [0, 1] \setminus \{\frac{1}{2}\}$, $\psi(\gamma) + \psi(1 - \gamma) > 2\,\psi(\frac{1}{2})$. \[theo:strong\_necessary\_condition\]
With Theorems \[theo:weak\_necessary\_condition\] and \[theo:strong\_necessary\_condition\], we can easily check if a pair of component functions $f$ and $g$ form a valid adversarial loss.
Sufficient conditions for the favorable properties {#sec:sufficient_conditions}
--------------------------------------------------
For sufficient conditions, we have two theorems as follows.
If $\psi(\gamma)$ has a global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:weak\]]{} holds. \[theo:weak\_sufficient\_condition\]
If $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:strong\]]{} holds. \[theo:strong\_sufficient\_condition\]
We also have the following theorem for a more specific guideline for choosing the component functions $f$ and $g$.
If $f'' + g'' \leq 0$ and there exists some $y^*$ such that $f(y^*) = g(y^*)$ and $f'(y^*) = -g'(y^*) \neq 0$, then $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$. \[theo:strong\_sufficient\_condition2\]
By Theorems \[theo:strong\_sufficient\_condition\] and \[theo:strong\_sufficient\_condition2\], we now see that any component function pair $f$ and $g$ that satisfies the prerequisites in [Theorem \[theo:strong\_sufficient\_condition2\]]{} makes $L_G - L^*_G$ a divergence between $p_d$ and $p_g$ for any fixed $p_d$. Interestingly, while such a theoretical analysis has not been done before, it happens that all the adversarial loss functions listed in [ \[tab:loss\_functions\]]{} have such favorable properties. We intend to examine in [Section \[sec:exp\_properties\]]{} empirically the cases when the prerequisites of [Theorem \[theo:strong\_sufficient\_condition2\]]{} do not hold.
In practice, the discriminator often cannot reach optimality at each iteration. Therefore, as discussed by @nowozin2016 [@fedus2018], the objective of the generator is similar to variational divergence minimization (i.e., to minimize a lower bound of some divergence between $p_d$ and $p_g$), where the divergence is estimated by the discriminator.
Loss functions for the generator {#sec:g_loss}
--------------------------------
Intuitively, the generator should minimize the divergence-like measure estimated by the discriminator. We have accordingly $h = g$. However, some prior works have investigated setting $h$ different from $g$. In general, most of these alternative generator losses do not change the solutions of the game and are proposed base on some heuristics. While our theoretical analysis concerns with only $f$ and $g$, we intend to empirically examine the effects of the generator loss function $h$ in [Section \[sec:exp\_g\_loss\]]{}.
Analyzing the adversarial game by the $\Psi$ functions {#sec:psi_function_analysis}
------------------------------------------------------
[ \[fig:psi\_functions\]]{} gives us some new insights regarding the adversarial behaviors of the discriminator and the generator. On one hand, if we follow and consider $\tilde{y} = D({\mathbf{x}})$ and $\tilde{\gamma}({\mathbf{x}}) = \frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}$, then the discriminator can be viewed as maximizing $\Psi$ along the $\tilde{y}$-axis. On the other hand, since the generator is trained to push $p_g$ towards $p_d$, it can be viewed as minimizing $\Psi$ along the $\tilde{\gamma}$-axis. In this way, we can see why all these $\Psi$ functions are saddle-shaped and have saddle points at $\gamma = \frac{1}{2}$ (i.e., when $p_d({\mathbf{x}}) = p_g({\mathbf{x}})$).
Ideally, if the discriminator can be trained till optimality, then we will be on the green line, the domain of the $\psi$ function. In this case, the generator can be viewed as minimizing $\Psi$ along the green line (i.e., minizing $\psi$). Note that as $L_G$ is an integral over all possible ${\mathbf{x}}$, such adversarial game is actually being played in a (usually) high dimensional space.
By designing the landscape of $\Psi$, we propose and consider two new losses in our empirical study in [Section \[sec:exp\_adversarial\_losses\]]{}:
- The *absolute* loss, with $f(y) = -h(y) = -|1 - y|$, $g(y) = -|y|$. Its $\Psi$-landscape is similar to those of the least squares and the hinge losses (see [ \[fig:psi\_functions\]]{}(e)).
- The *asymmetric* loss, with $f(y) = -|y|$, $g(y) = h(y) = -y$. Its $\Psi$-landscape is similar to that of the Wasserstein loss, but the positive part of $y$ is blocked (see [ \[fig:psi\_functions\]]{}(f)).
DANTest {#sec:dantest}
=======
Discriminative adversarial networks (DANs) [@dossantos2017] are essentially conditional GANs [@mirza2014] where both the generator and the discriminator are discriminative models, as shown in [ \[fig:dan\]]{}. Based on DANs, we propose a new, simple framework, dubbed *DANTest*, for systematically comparing different adversarial losses. Specifically, the DANTest goes as follows:
1. Build several DANs. For each of them, the generator $G$ takes as input a real sample and outputs a fake label. The discriminator takes as input a real sample with either its true label, or a fake label made by $G$, and outputs a scalar indicating if the “sample–label” pair is real.
2. Train the DANs with different component loss functions, regularization approaches or hyperparameters.
3. Predict the labels of test data by the trained models.
4. Compare the performance of different models with standard evaluation metrics used in supervised learning.
Note that the generator is no longer a generative model in this framework, while the discriminator is still trained by the same loss function to measure the discrepancy between $p_d$ and $p_g$. This way, we can still gain insight into the performance and stability for different adversarial losses. Moreover, although we take a classification task as an example here, the proposed framework is generic and can be applied to other supervised learning tasks as well, as long as the evaluation metrics for that task are well defined.
An extension of the proposed framework is the *imbalanced dataset test*, where we examine the ability of different adversarial losses on datasts that feature class imbalance. This can serve as a measure of the *mode collapse* phenomenon [@che2017mdgan], which is a commonly-encountered failure case in GAN training. By testing on datasets with different levels of imbalance, we can examine how different adversarial losses suffer from the mode collapse problem.
Experiments and Results {#sec:experiments}
=======================
Datasets and Implementation Details {#sec:dataset}
-----------------------------------
All the experiments reported here are done based on the DANTest. If not otherwise specified, we use the MNIST handwritten digits database [@lecun1998], which we refer to as the **standard** dataset. As it is class-balanced, we create two imbalanced versions of it. The first one, referred to as the **imbalanced** dataset, is created by augmenting the training samples for digit ‘0’ by shifting them each by one pixel to the top, bottom, left and right, so that it contains *five* times more training samples of ‘0’ than the standard dataset. Moreover, we create the **very imbalanced** dataset, where we have *seven* times more training samples for digit ‘0’ than the standard dataset. For other digits, we randomly sample from the standard dataset and intentionally make the sizes of the resulting datasets identical to that of the standard dataset. We use the same test set for all the experiments.
We implement $G$ and $D$ as convolutional neural networks (see [Appendix \[app:sec:net\_architectures\]]{} for the network architectures). We use the batch normalization [@ioffe2015] in $G$. If the spectral normalization is used, we only apply it to $D$, otherwise we use the layer normalization [@ba2017] in $D$. We concatenate the label vector to each layer of $D$. For the gradient penalties, we use Euclidean norms and set $\lambda$ to $10.0$ (see ), $k$ to $1.0$ and $c$ to $0.01$ (see [ \[tab:gradient\_penalties\]]{}). We use the Adam optimizers [@kingma2014] with $\alpha = 0.001$, $\beta_1 = 0.0$ and $\beta_2 = 0.9$. We alternatively update $G$ and $D$ once in each iteration and train the model for 100,000 generator steps. The batch size is $64$. We implement the model in Python and TensorFlow [@abadi2016]. We run each experiment for ten runs and report the mean and the standard deviation of the error rates.
nonsaturating Wasserstein hinge
------------------ ------------------- --------------------- ---------------------
$\epsilon = 0.5$ 8.47$\pm$0.36 73.16$\pm$6.36 15.20$\pm$2.46
$\epsilon = 0.9$ 8.96$\pm$0.63 57.66$\pm$5.13 8.94$\pm$0.87
$\epsilon = 1.0$ **8.25$\pm$0.35** **5.89$\pm$0.26** **6.59$\pm$0.31**
$\epsilon = 1.1$ 8.62$\pm$0.45 60.30$\pm$7.61 8.02$\pm$0.35
$\epsilon = 2.0$ 9.18$\pm$0.94 69.54$\pm$5.37 11.87$\pm$0.85
: Error rates (%) for the $\epsilon$-weighted versions of the nonsaturating, the Wasserstein and the hinge losses (see ) on the standard dataset. Here, $\epsilon = 1.0$ corresponds to the original losses.[]{data-label="tab:exp_properties"}
**unregularized** **TCGP** **TLGP** **R~1~ GP** **R~2~ GP** **SN** **SN + TCGP** **SN + TLGP** **SN + R~1~ GP** **SN + R~2~ GP**
------------------------------------------------------ ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
**classic (M)** ([-@goodfellow2014]) 9.11$\pm$0.63 5.65$\pm$0.27 5.42$\pm$0.17 19.01$\pm$3.73 12.91$\pm$1.13 7.37$\pm$0.52 5.55$\pm$0.37 5.57$\pm$0.28 11.16$\pm$2.66 14.00$\pm$2.49
**classic (N)** ([-@goodfellow2014]) 26.83$\pm$7.17 5.64$\pm$0.23 5.56$\pm$0.31 14.67$\pm$4.86 13.80$\pm$3.20 8.25$\pm$0.35 5.52$\pm$0.16 5.61$\pm$0.50 12.98$\pm$2.71 13.50$\pm$3.78
**classic (L)** 17.38$\pm$5.16 5.66$\pm$0.36 5.55$\pm$0.16 18.49$\pm$5.51 14.92$\pm$5.20 7.98$\pm$0.36 5.70$\pm$0.36 5.48$\pm$0.29 15.45$\pm$6.54 17.61$\pm$7.60
**hinge (M)** **4.83$\pm$0.34** **4.88$\pm$0.25** 9.49$\pm$5.30 **6.22$\pm$0.23** **5.06$\pm$0.33** 10.62$\pm$2.10 12.91$\pm$4.29
**hinge (N)** 37.55$\pm$20.22 **5.00$\pm$0.24** 4.97$\pm$0.24 **7.34$\pm$1.83** 7.54$\pm$1.31 6.90$\pm$0.33 5.05$\pm$0.22 **5.06$\pm$0.39** 11.91$\pm$4.02 12.10$\pm$4.74
**hinge (L)** ([-@lim2017; -@tran2017]) 11.50$\pm$5.32 5.01$\pm$0.26 **4.89$\pm$0.18** 8.96$\pm$3.55 7.71$\pm$1.82 6.59$\pm$0.31 **4.97$\pm$0.19** 5.18$\pm$0.27 13.63$\pm$4.13 11.35$\pm$3.40
**Wasserstein** ([-@arjovsky2017wgan]) 7.69$\pm$0.33 5.04$\pm$0.19 4.92$\pm$0.23 13.89$\pm$20.64 **7.25$\pm$1.19** 5.50$\pm$0.18 5.76$\pm$0.70 13.74$\pm$5.47 13.82$\pm$4.93
**least squares** ([-@mao2017]) **7.15$\pm$0.47** 7.27$\pm$0.44 6.70$\pm$0.44 30.12$\pm$28.43 32.44$\pm$21.05 7.88$\pm$0.45 6.69$\pm$0.25 7.11$\pm$0.37 **9.91$\pm$1.55** 11.56$\pm$4.09
**relativistic** ([-@jolicoeur-martineau2018]) 90.20$\pm$0.00 5.25$\pm$0.25 5.01$\pm$0.31 **8.00$\pm$1.63** 8.75$\pm$5.83 7.14$\pm$0.39 5.35$\pm$0.29 5.25$\pm$0.26 **9.31$\pm$2.01**
**relativistic hinge** ([-@jolicoeur-martineau2018]) 52.01$\pm$9.38 8.28$\pm$10.26 8.39$\pm$1.92 7.67$\pm$1.82 6.44$\pm$0.16 **5.02$\pm$0.31** 12.56$\pm$4.42 12.40$\pm$4.55
**absolute** **6.69$\pm$0.24** 5.23$\pm$0.29 5.20$\pm$0.26 8.01$\pm$1.96 6.79$\pm$0.45 5.23$\pm$0.13 5.18$\pm$0.35 10.42$\pm$3.07 **9.93$\pm$2.28**
**asymmetric** 7.81$\pm$0.27 4.94$\pm$0.14 8.79$\pm$3.18 **7.33$\pm$1.01** **5.98$\pm$0.40** 5.60$\pm$0.29 5.82$\pm$0.44 **8.80$\pm$1.18**
Examining the necessary conditions for favorable adversarial loss functions {#sec:exp_properties}
---------------------------------------------------------------------------
As discussed in [Section \[sec:sufficient\_conditions\]]{}, we examine here the cases when the prerequisites in [Theorem \[theo:strong\_sufficient\_condition2\]]{} do not hold. We consider the classic nonsaturating, the Wasserstein and the hinge losses and change the training objective for the discriminator into $$\label{eq:imbalanced_loss}
\max_{D}\;\epsilon\,{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,,$$ where $\epsilon \in {\mathbb{R}}$ is a constant. The prerequisites in [Theorem \[theo:strong\_sufficient\_condition2\]]{} do not hold when $\epsilon \neq 1$. We illustrate the $\Psi$ functions of these *$\epsilon$-weighted losses* in [Appendix \[app:sec:psi\_function\_graphs\]]{}.
[ \[tab:exp\_properties\]]{} shows the results for $\epsilon = 0.5$, $0.9$, $1.0$, $1.1$, $2.0$, using the spectral normalization for regularization. We can see that all the original losses (i.e., $\epsilon = 1$) result in the lowest error rates. In general, the error rates increase as $\epsilon$ goes away from $1.0$. Notably, the Wasserstein loss turn out failing with error rates over 50% when $\epsilon \neq 1$.
On different discriminator loss functions {#sec:exp_adversarial_losses}
-----------------------------------------
In this experiment, we aim to compare different discriminator loss functions. Specifically, we evaluate an comprehensive set (in total 168) of different combinations of component functions and regularization approaches.
For the component functions, we consider the classic minimax and the classic nonsaturating losses [@goodfellow2014], the Wasserstein loss [@arjovsky2017wgan], the least squares loss [@mao2017], the hinge loss [@lim2017; @tran2017], the relativistic average and the relativistic average hinge losses [@jolicoeur-martineau2018], as well as the absolute and the asymmetric losses we propose and describe in [Section \[sec:psi\_function\_analysis\]]{}.
For the regularization approaches, we consider the coupled, the local, the R~1~ and the R~2~ gradient penalties (**GP**) and the spectral normalization (**SN**). For the coupled and the local gradient penalties, we examine both the two-side and the one-side versions (see [ \[tab:gradient\_penalties\]]{}). We will use in the captions **OCGP** and **TCGP** as the shorthands for the one-side and the two-side coupled gradient penalties, respectively, and **OLCP** and **TLCP** for the one-side and the two-side local gradient penalties, respectively. We also consider the combinations of the SN with different gradient penalties.
We report in [ \[tab:exp\_adversarial\_losses\]]{} the results for all the combinations and present in [ \[fig:training\_progress\]]{} the training progress for the nonsaturating and the hinge losses. We can see that *there is no single winning component functions and regularization approach across all different settings*. Some observations are:
With respect to the **component functions**—
- The classic minimax and nonsaturating losses never get the lowest three error rates for all different settings.
- The hinge, the asymmetric and the two relativistic losses are robust to different regularization approaches and tend to achieve lower error rates.
- The relativistic average loss outperforms both the classic minimax and nonsaturating losses across all regularization approaches. But, the relativistic average hinge loss does not always outperform the standard hinge loss.
With respect to the **regularization approaches**—
- The coupled and the local GPs outperform the R~1~ and the R~2~ GPs across nearly all different component functions, no matter whether the SN is used or not.
- The coupled and the local GPs stabilize the training (see [ \[fig:training\_progress\]]{}) and tend to have lower error rates.
- The R~2~ gradient penalties achieve lower error rates than the R~1~ gradient penalties. In some cases, they can be too strong and even stop the training early (see [ \[fig:training\_progress\]]{} (a)).[^1]
- Combining either the coupled or the local GP with the SN usually leads to higher error rates than using the coupled or the local GP only.
- Similarly, combining either the R~1~ or the R~2~ GP with the SN degrades the result. Moreover, it leads to unstable training (see Figures \[fig:training\_progress\](b) and (d)). This result implies that R~1~ and R~2~ GPs do not work well with the SN.
- Using the one-side GPs instead of their two-side counterparts increase the error rates by 0.1–9.5%. (We report the results for the one-side GPs in [Appendix \[app:sec:results\]]{} due to page limit.)
We also note that some combinations result in remarkably high error rates, e.g., “least squares loss + R~1~ GP”, “least squares loss + R~2~ GP” and “classic minimax loss + R~1~ GP”.
In sum, according to the overall performance and the robustness to different settings, for the component functions, *we recommend the hinge, the asymmetric and the two relativistic losses*. We note that these functions also feature lower computation costs as all their components functions are piecewise linear (see [ \[tab:loss\_functions\]]{} and [Section \[sec:psi\_function\_analysis\]]{}). For the regularization approaches, *we recommend the two-side coupled and the two-side local gradient penalties*.
We also conduct the imbalanced dataset test (see [Section \[sec:dantest\]]{}) on the two imbalanced datasets described in [Section \[sec:dataset\]]{} to compare the regularization approaches. We use the classic nonsaturating loss. As shown in [ \[tab:exp\_imbalanced\_dataset\]]{}, the error rates increase as the level of imbalance increases. The two-side local GP achieve the lowest error rates across all three datasets. The error rates for the R~1~ and the R~2~ GPs increase significantly when the dataset goes imbalanced.
### Effects of the Lipschitz constants {#sec:exp_lipschitz}
In this experiment, we examine the effects of the Lipschitz constant ($k$) used in the coupled and the local GPs (see [ \[tab:gradient\_penalties\]]{}). We use the classic nonsaturating loss here. We report in [ \[fig:exp\_lipschitz\]]{} the results for $k = 0.01$, $0.1$, $1$, $10$, $100$. We can see that the error rate increases as $k$ goes away from $1.0$, suggesting that $k = 1$ is indeed a good default value. Moreover, the two-side GPs are more sensitive to $k$ than their one-side counterparts.
We note that @petzka2018 suggested that the one-side coupled GP are preferable to the two-side version and showed empirically that the former has more stable behaviors. However, we observe in our experiments that the two-side penalties usually lead to faster convergence to lower error rates compared to the one-side penalties.[^2]
standard imbalanced very imbalanced
--------- --------------------- --------------------- ---------------------
TCGP 5.64$\pm$0.23 7.09$\pm$0.64 8.12$\pm$0.31
OCGP 7.20$\pm$0.39 8.86$\pm$0.65 10.23$\pm$0.75
TLGP **5.51$\pm$0.27** **6.94$\pm$0.28** **8.10$\pm$0.55**
OLGP 6.92$\pm$0.21 8.63$\pm$0.75 10.21$\pm$0.52
R~1~ GP 14.67$\pm$4.86 18.66$\pm$5.60 27.90$\pm$9.59
R~2~ GP 13.80$\pm$3.20 15.70$\pm$2.07 29.97$\pm$12.4
: Error rates (%) for different gradient penalties (using the nonsaturating loss) on datasets with different levels of imbalance.[]{data-label="tab:exp_imbalanced_dataset"}
### Effects of the penalty weights {#sec:exp_penalty_weights}
We then examine the effects of the penalty weights ($\lambda$) for the R~1~ and the R~2~ GPs (see ). We consider the classic nonsaturating, the Wasserstein and the hinge losses. We present in [ \[fig:exp\_r1r2\_penalty\_weight\]]{} the results for $\lambda = 0.01$, $0.1$, $1$, $10$, $100$. We can see that the R~1~ GP tends to outperform the R~2~ GP, while they are both sensitive to the value of $\lambda$. Hence, future research should run hyperparmeter search for $\lambda$ to find out its optimal value. When the spectral normalization is not used, the hinge loss is less sensitive to $\lambda$ than the other two losses. However, when spectral normalization is used, the error rate increases as $\lambda$ increases, which again implies that the R~1~/R~2~ GPs and the SN do not work well together.
\
(a) without the spectral normalization\
\
(b) with the spectral normalization
On different generator loss functions {#sec:exp_g_loss}
-------------------------------------
As discussed in [Section \[sec:g\_loss\]]{}, we also aim to examine the effects of the generator loss function $h(\cdot)$. We consider the classic and the hinge losses for the discriminator and the following three generator loss functions: minimax (**M**)—$h(x) = g(x)$, nonsaturating (**N**)—$h(x) = \log(1 + e^{-x})$, and linear (**L**)—$h(x) = -x$. We report the results in the first six rows of [ \[tab:exp\_adversarial\_losses\]]{}. For the classic discriminator loss, we see no single winner among the three generator loss functions across all the regularization approaches, which implies that the heuristics behind these alternative losses might not be true. For the hinge discriminator loss, the minimax generator loss is robust to different regularization approaches and achieves three lowest and four lowest-three scores. Hence, *we recommend to use hinge loss for the discriminator and minimax loss for the generator* as the overall best choice according to our experimental results.
Effects of the momentum terms of the optimizers {#sec:exp_momentum}
-----------------------------------------------
We observe a trend towards using smaller momentum [@radford2016] or even no momentum [@arjovsky2017wgan; @gulrajani2017; @miyato2018; @brock2018] in GAN training. Hence, we would also like to examine the effects of momentum terms in the optimizers with the proposed framework. As suggested by @gidel2018, we also include a negative momentum value of $-0.5$. We use the classic nonsaturating loss and the SN along with the coupled GPs for regularization. [ \[fig:exp\_momentum\]]{} shows the results for all combinations of $\beta_1 = -0.5$, $0.0$, $0.5$, $0.9$ for $G$ and $D$. We can see that for the two-side coupled GP, using larger momenta in both $G$ and $D$ leads to lower error rates, while there is no specific trend for the one-side coupled GP.
Discussions and Conclusions {#sec:discussions_and_conclusions}
===========================
In this paper, we have shown in theory what certain types of component functions form a valid adversarial loss. We have also introduced a new framework called DANTest for comparing adversarial losses. With DANTest, we systematically compared combinations of different component functions and regularization approaches to decouple their effects. Our empirical results show that there is no single winning component functions or regularization approach across all different settings. Our theoretical and empirical results can together serve as a reference for choosing or designing adversarial training objectives in future research.
As compared to the commonly used metrics for evaluating generative models, such as the Inception Score [@salimans2016] and Fréchet Inception Distance [@heusel2017] adopted in @lucic2018 and @kurach2018, the DANTest is simpler and is easier to control and extend. This allows us to easily evaluate new adversarial losses. However, we note that while the discriminator in a DAN is trained to optimize the same objectives as in a conditional GAN, the generators in the two models actually work in opposite ways ($\mathcal{X} \to \mathcal{Z}$ in a DAN versus $\mathcal{Z} \to \mathcal{X}$ in a GAN). Hence, it is unclear whether the empirical results can be generalized to conditional and unconditional GANs. Nonetheless, recent work has also adapted adversarial losses to plenty discriminative models (e.g., image-to-image translation [@isola2017] and image super-resolution [@ledig17cvpr]). Therefore, it is worth investigating the behaviors of adversarial losses in different scenarios.
In addition, our theoretical analysis provides a new perspective on adversarial losses and reveals a large class of component functions valid for adversarial losses. We note that @nowozin2016 has also shown a certain class of component functions can result in theoretically valid adversarial losses. However, in their formulations, the component functions $f$ and $g$ are not independent of each other as they considered only the *f*-divergences. A future direction is to investigate the necessary and sufficient conditions for the existence and the uniqueness of a Nash equilibrium.
Proofs of the Theorems {#app:sec:proofs}
======================
If [Property \[prop:weak\]]{} holds, then for any $\gamma \in [0, 1]$, $\psi(\gamma) + \psi(1 - \gamma) \geq 2\,\psi(\frac{1}{2})$. \[app:theo:weak\_necessary\_condition\]
Since [Property \[prop:weak\]]{} holds, we have for any fixed $p_d$, $$\label{app:eq:theo1_prerequisite}
L_G \geq L_G\,\big\rvert_{\,p_g = p_d}\,.$$
Let us consider $$\begin{aligned}
p_d({\mathbf{x}}) = \gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t})\,,\\
p_g({\mathbf{x}}) = (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{s}) + \gamma\,\delta({\mathbf{x}}- \mathbf{t})\,.
\end{aligned}$$ for some $\gamma \in [0, 1]$ and $\mathbf{s}, \mathbf{t} \in {\mathcal{X}}, \mathbf{s} \neq \mathbf{t}$. Then, we have $$\begin{aligned}
&L_G\,\big\rvert_{\,p_g = p_d}\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_d({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,(f(D({\mathbf{x}})) + g(D({\mathbf{x}})))\,d{\mathbf{x}}\\
\begin{split}
&= \max_{D}\;\int_{\mathbf{x}}\big(\,(\gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t}))\\
&\qquad\qquad\qquad\qquad\qquad (f(D({\mathbf{x}})) + g(D({\mathbf{x}})))\,\big)\,d{\mathbf{x}}\end{split}\\
\begin{split}
\label{app:eq:theo1_note1}
&= \max_{D}\;\big(\,\gamma (f(D(\mathbf{s})) + g(D(\mathbf{s})))\\
&\qquad\qquad\qquad + (1 - \gamma)\,(f(D(\mathbf{t})) + g(D(\mathbf{t})))\,\big)
\end{split}\\
\begin{split}
\label{app:eq:theo1_note2}
&= \max_{y_1, y_2}\;\big(\,\gamma (f(y_1) + g(y_1))\\
&\qquad\qquad\qquad + (1 - \gamma) (f(y_2) + g(y_2))\,\big)
\end{split}\\
\begin{split}
&= \max_{y_1}\;\gamma\,(f(y_1) + g(y_1))\\
&\qquad\qquad\qquad + \max_{y_2}\;(1 - \gamma)\,(f(y_2) + g(y_2))
\end{split}\\
&= \max_{y}\;f(y) + g(y)\\
\label{app:eq:theo1_note3}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$
Moreover, we have $$\begin{aligned}
&L_G\nonumber\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_g({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
\begin{split}
&= \max_{D}\;\int_{\mathbf{x}}\big(\,(\gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t}))\,f(D({\mathbf{x}}))\\
&\;\qquad + ((1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{s}) + \gamma\,\delta({\mathbf{x}}- \mathbf{t}))\,g(D({\mathbf{x}}))\,\big)\,d{\mathbf{x}}\end{split}\\
\begin{split}
\label{app:eq:theo1_note4}
&= \max_{D}\;\big(\,\gamma\,f(D(\mathbf{s})) + (1 - \gamma)\,f(D(\mathbf{t}))\\
&\qquad\qquad\qquad + (1 - \gamma)\,g(D(\mathbf{s})) + \gamma\,g(D(\mathbf{t}))\,\big)
\end{split}\\
\begin{split}
\label{app:eq:theo1_note5}
&= \max_{y_1, y_2}\;\big(\,\gamma\,f(y_1) + (1 - \gamma)\,g(y_1))\\
&\qquad\qquad\qquad + (1 - \gamma)\,f(y_2) + \gamma\,g(y_2)\,\big)
\end{split}\\
\begin{split}
&= \max_{y_1}\;\gamma\,f(y_1) + (1 - \gamma)\,g(y_1))\\
&\qquad\qquad\qquad + \max_{y_2} (1 - \gamma)\,f(y_2) + \gamma\,g(y_2)
\end{split}\\
\label{app:eq:theo1_note6}
&= \psi(\gamma) + \psi(1 - \gamma)\,.
\end{aligned}$$ (Note that we can obtain from and from because $D$ can be any function and thus $D(\mathbf{s})$ is independent of $D(\mathbf{t})$.)
As holds for any fixed $p_d$, by substituting and into , we get $$\psi(\gamma) + \psi(1 - \gamma) \geq 2\,\psi(\tfrac{1}{2})$$ for any $\gamma \in [0, 1]$, which concludes the proof.
If [Property \[prop:strong\]]{} holds, then for any $\gamma \in [0, 1] \setminus \{\frac{1}{2}\}$, $\psi(\gamma) + \psi(1 - \gamma) > 2\,\psi(\frac{1}{2})$. \[app:theo:strong\_necessary\_condition\]
Since [Property \[prop:strong\]]{} holds, we have for any fixed $p_d$, $$\label{app:eq:theo2_prerequisite}
L_G\big\rvert_{\,p_g \neq p_d} > L_G\,\big\rvert_{\,p_g = p_d}\,.$$
Following the proof of [Theorem \[app:theo:weak\_necessary\_condition\]]{}, consider $$\begin{aligned}
\label{app:eq:theo2_note1}
&p_d({\mathbf{x}}) = \gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t})\,,\\[1ex]
\label{app:eq:theo2_note2}
&p_g({\mathbf{x}}) = (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{s}) + \gamma\,\delta({\mathbf{x}}- \mathbf{t})\,,
\end{aligned}$$ for some $\gamma \in [0, 1]$ and some $\mathbf{s}, \mathbf{t} \in {\mathcal{X}}, \mathbf{s} \neq \mathbf{t}$. It can be easily shown that $p_g = p_d$ if and only if $\gamma = \tfrac{1}{2}$.
As holds for any fixed $p_d$, by substituting and into , we get $$\psi(\gamma) + \psi(1 - \gamma) > 2\,\psi(\tfrac{1}{2})\,,$$ for any $\gamma \in [0, 1] \setminus \{\tfrac{1}{2}\}$, concluding the proof.
If $\psi(\gamma)$ has a global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:weak\]]{} holds. \[app:theo:weak\_sufficient\_condition\]
First, we see that $$\begin{aligned}
&L_G\,\big\rvert_{\,p_g = p_d}\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_d({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
&= \max_{y}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(y) + p_d({\mathbf{x}})\,g(y)\,d{\mathbf{x}}\\
&= \max_{y}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,(f(y) + g(y))\,d{\mathbf{x}}\\
&= \max_{y}\;(f(y) + g(y)) \int_{\mathbf{x}}p_d({\mathbf{x}})\,d{\mathbf{x}}\\
&= \max_{y}\;f(y) + g(y)\\
\label{app:eq:theo3_note1}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$ On the other had, we have $$\begin{aligned}
&L_G\nonumber\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_g({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
&= \max_{y}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(y) + p_g({\mathbf{x}})\,g(y)\,d{\mathbf{x}}\\
\begin{split}
&= \max_{y}\;\int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\qquad\qquad \left(\frac{p_d({\mathbf{x}})\,f(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} + \frac{p_g({\mathbf{x}})\,g(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\end{split}\\
\begin{split}
&= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\qquad\qquad \max_{y} \left(\frac{p_d({\mathbf{x}})\,f(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} + \frac{p_g({\mathbf{x}})\,g(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\,.
\end{split}
\end{aligned}$$ Since $\frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} \in [0, 1]$, we have $$\label{app:eq:theo3_note2}
L_G = \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi\left(\frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\,.$$ As $\psi(\gamma)$ has a global minimum at $\gamma = \frac{1}{2}$, now we have $$\begin{aligned}
L_G &\geq \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi(\tfrac{1}{2})\,d{\mathbf{x}}\\
&= \psi(\tfrac{1}{2}) \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,d{\mathbf{x}}\\
\label{app:eq:theo3_note3}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$ Finally, combining and yields $$L_G \geq L_G\,\big\rvert_{\,p_g = p_d}\,,$$ which holds for any $p_d$, thus concluding the proof.
If $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:strong\]]{} holds. \[app:theo:strong\_sufficient\_condition\]
Since $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$, we have for any $\gamma \in [0, 1] \setminus \frac{1}{2}$, $$\psi(\gamma) > \psi(\tfrac{1}{2})\,.$$ When $p_g \neq p_d$, there must be some ${\mathbf{x}}_0 \in {\mathcal{X}}$ such that $p_g({\mathbf{x}}_0) \neq p_d({\mathbf{x}}_0)$. Thus, $\frac{p_d({\mathbf{x}}_0)}{p_d({\mathbf{x}}_0) + p_g({\mathbf{x}}_0)} \neq \frac{1}{2}$, and thereby $\psi\left(\frac{p_d({\mathbf{x}}_0)}{p_d({\mathbf{x}}_0) + p_g({\mathbf{x}}_0)}\right) > \psi(\frac{1}{2})$. Now, by we have $$\begin{aligned}
&L_G\,\big\rvert_{\,p_g \neq p_d}\\
&= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi\left(\frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\\
&> \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi(\tfrac{1}{2})\,d{\mathbf{x}}\\
&= \psi(\tfrac{1}{2}) \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,d{\mathbf{x}}\\
\label{app:eq:theo4_note1}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$ Finally, combining and yields $$L_G\,\big\rvert_{\,p_g \neq p_d} > L_G\,\big\rvert_{\,p_g = p_d}\,,$$ which holds for any $p_d$, thus concluding the proof.
If $f'' + g'' \leq 0$ and there exists some $y^*$ such that $f(y^*) = g(y^*)$ and $f'(y^*) = -g'(y^*) \neq 0$, then $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$. \[app:theo:strong\_sufficient\_condition2\]
First, we have by definition $$\Psi(\gamma, y) = \gamma\,f(y) + (1 - \gamma)\,g(y)\,.$$ By taking the partial derivatives, we get $$\begin{aligned}
\label{app:eq:theo5_partial1}
&\frac{\partial\Psi}{\partial\gamma} = f(y) - g(y)\,,\\[1ex]
\label{app:eq:theo5_partial2}
&\frac{\partial\Psi}{\partial y} = \gamma\,f'(y) + (1 - \gamma)\,g'(y)\,,\\[1ex]
\label{app:eq:theo5_partial3}
&\frac{\partial^2\Psi}{\partial y^2} = \gamma\,f''(y) + (1 - \gamma)\,g''(y)\,.
\end{aligned}$$
We know that there exists some $y^*$ such that $$\begin{aligned}
\label{app:eq:theo5_prerequisite1}
&f(y^*) = g(y^*)\,,\\[1ex]
\label{app:eq:theo5_prerequisite2}
&f'(y^*) = -g'(y^*) \neq 0\,.
\end{aligned}$$
1. By and , we see that $$\begin{aligned}
\label{app:eq:theo5_note1}
&\frac{\partial\Psi}{\partial\gamma}\,\Big\rvert_{\,y = y^*} = 0\,,\\[1ex]
\label{app:eq:theo5_note2}
&\frac{\partial\Psi}{\partial y}\,\Big\rvert_{\,(\gamma, y) = (\frac{1}{2}, y^*)} = 0\,.
\end{aligned}$$ Now, by we know that $\Psi$ is constant when $y = y^*$. That is, for any $\gamma \in [0, 1]$, $$\Psi(\gamma, y^*) = \Psi(\tfrac{1}{2}, y^*)\,.$$
2. Because $f'' + g'' \leq 0$, by we have $$\begin{aligned}
\label{app:eq:theo5_note3}
\frac{\partial^2\Psi}{\partial y^2}\,\Big\rvert_{\,\gamma = \tfrac{1}{2}} &= \tfrac{1}{2}\,f''(y) + \tfrac{1}{2}\,g''(y)\\
&\leq 0\,.
\end{aligned}$$ By and , we see that $y^*$ is a global minimum point of $\Psi\big\rvert_{\gamma = \tfrac{1}{2}}$. Thus, we now have $$\begin{aligned}
\Psi(\tfrac{1}{2}, y^*) &= \max_y\;\Psi(\tfrac{1}{2}, y)\\
\label{app:eq:theo5_note4}
&= \psi(\tfrac{1}{2})\,.
\end{aligned}$$
3. By , we see that $$\begin{aligned}
\frac{\partial\Psi}{\partial y}\,\Big\rvert_{\,y = y^*}
&= \gamma\,f'(y^*) + (1 - \gamma)\,g'(y^*)\\
&= \gamma\,f'(y^*) + (1 - \gamma)\,(-f'(y^*))\\
&= (2 \gamma - 1)\,f'(y^*)\,.
\end{aligned}$$ Since $f'(y^*) \neq 0$, we have $$\frac{\partial\Psi}{\partial y}\,\Big\rvert_{\,y = y^*} \neq 0\quad\forall\,\gamma \in [0, 1] \setminus \tfrac{1}{2}\,.$$ This shows that for any $\gamma \in [0, 1] \setminus \tfrac{1}{2}$, there must exists some $y^\circ$ such that $$\label{app:eq:theo5_note5}
\Psi(\gamma, y^\circ) > \Psi(\gamma, y^*)\,.$$ And by definition we have $$\begin{aligned}
\label{app:eq:theo5_note6}
\Psi(\gamma, y^\circ) &< \max_y\;\Psi(\gamma, y)\\
&= \psi(\gamma)\,.
\end{aligned}$$ Hence, by and we get $$\label{app:eq:theo5_note7}
\psi(\gamma) > \Psi(\gamma, y^*)\,.$$
Finally, combining , and yields $$\psi(\gamma) > \psi(\tfrac{1}{2})\quad\forall\,\gamma \in [0, 1] \setminus \tfrac{1}{2}\,,$$ which concludes the proof.
More Graphs of the $\Psi$ and $\psi$ Functions {#app:sec:psi_function_graphs}
==============================================
We show in [ \[app:fig:small\_psi\_functions\]]{} the graphs of the $\psi$ functions for different adversarial losses. Note that for the Wasserstein loss, the $\psi$ function is only defined at $\gamma = 0.5$, where it takes the value of zero, and for the asymmetric loss, the $\psi$ function is only defined when $\gamma > 0.5$, where it takes the value of zero. Hence, we do not include them in [ \[app:fig:small\_psi\_functions\]]{}.
We also present in [ \[app:fig:psi\_functions\_imbalanced\]]{} the graphs of the $\Psi$ functions for the $\epsilon$-weighted versions of the classic, the Wasserstein and the hinge losses. Moreover, Figures \[app:fig:small\_psi\_functions\](b) and (c) show the graphs of the $\psi$ functions for the $\epsilon$-weighted versions of the classic and the hinge losses, respectively.
Network Architectures {#app:sec:net_architectures}
=====================
We present in [ \[app:tab:network\_architectures\]]{} the network architectures for the generator and the discriminator used for all the experiments.
More Results {#app:sec:results}
============
We report in [ \[app:tab:exp\_loss\_functions\]]{} the results for the one-side coupled and local gradient penalties.
We also present in [ \[app:fig:exp\_momentum\]]{} the results for the experiment on the momentum terms using the hinge loss.
\
(a) common adversarial losses\
------------------------------------------------------- -----------------------------------------------------
\(b) $\epsilon$-weighted versions of the classic loss \(c) $\epsilon$-weighted versions of the hinge loss
------------------------------------------------------- -----------------------------------------------------
[lccc]{}\
*conv* &32 &3$\times$3 &3$\times$3\
*conv* &64 &3$\times$3 &3$\times$3\
*maxpool* &- &2$\times$2 &2$\times$2\
*dense* &128\
*dense* &10\
[lccc]{}\
*conv* &32 &3$\times$3 &3$\times$3\
*conv* &64 &3$\times$3 &3$\times$3\
*maxpool* &- &2$\times$2 &2$\times$2\
*dense* &128\
*dense* &1\
OCGP OLGP SN + OCGP SN + OLGP
----------------------------------------------- ------------------- ------------------- ------------------- -------------------
classic (M) [@goodfellow2014] 7.15$\pm$0.77 **6.95$\pm$0.51** 7.16$\pm$0.31 6.86$\pm$0.29
classic (N) [@goodfellow2014] 7.20$\pm$0.39 6.98$\pm$0.22 7.47$\pm$0.62 7.15$\pm$0.36
classic (L) 7.12$\pm$0.61 7.00$\pm$1.00 7.29$\pm$0.35 7.18$\pm$0.54
hinge (M) **5.82$\pm$0.31** 7.33$\pm$1.35 **5.80$\pm$0.24** 5.83$\pm$0.20
hinge (N) 7.88$\pm$1.33 5.92$\pm$0.36 **5.74$\pm$0.27**
hinge (L) [@lim2017; @tran2017] **5.77$\pm$0.29** **6.22$\pm$1.04** **5.77$\pm$0.30** **5.82$\pm$0.20**
Wasserstein [@arjovsky2017wgan] 7.60$\pm$3.02 13.34$\pm$1.49 6.35$\pm$0.43 6.06$\pm$0.45
least squares [@mao2017] 7.99$\pm$0.35 8.06$\pm$0.49 8.43$\pm$0.50 8.31$\pm$0.52
relativistic [@jolicoeur-martineau2018] 8.03$\pm$3.32 9.41$\pm$2.90 6.18$\pm$0.29 6.03$\pm$0.24
relativistic hinge [@jolicoeur-martineau2018] 10.70$\pm$2.51 14.17$\pm$1.79
absolute 5.95$\pm$0.19 6.22$\pm$0.25 6.08$\pm$0.32
asymmetric 5.85$\pm$0.35 7.57$\pm$0.98 6.21$\pm$0.34 5.92$\pm$0.37
[^1]: This is possibly because the R~1~ and the R~2~ gradient penalties encourage $D$ to have small gradients, and thus the gradients for both $D$ and $G$ might vanish when $p_g$ and $p_d$ are close enough.
[^2]: A possible reason is that as $p_g$ move towards $p_d$, the gradients for $G$ become smaller (and eventually zero when $p_d = p_g$), which can slow down the training. The two-side penalties can alleviate this by encouraging the norm of the gradients to be a fixed value.
|
---
author:
- 'Maria Chudnovsky[^1]'
- 'Shenwei Huang[^2]'
- 'Pawe[ł]{} Rzżewski[^3]'
- 'Sophie Spirkl[^4]'
- 'Mingxian Zhong[^5]'
bibliography:
- 'main.bib'
title: 'Complexity of $C_k$-coloring in hereditary classes of graphs[^6]'
---
Introduction
============
Preliminaries {#sec:pre}
=============
Polynomial algorithm for $P_9$-free graphs {#sec:poly}
==========================================
Hardness results {#sec:NPc}
================
Conclusion {#sec:conclusion}
==========
[^1]: Princeton University, Princeton, NJ 08544, USA. Supported by NSF grant DMS-1763817. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under grant number W911NF-16-1-0404.
[^2]: College of Computer Science, Nankai University, Tianjin 300350, China. This research is partially supported by the National Natural Science Foundation of China (11801284).
[^3]: Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland. Supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.
[^4]: Rutgers University, Piscataway, NJ 08854, USA. This material is based upon work supported by the National Science Foundation under Award No. DMS1802201.
[^5]: Lehman College, CUNY, Bronx, NY 10468, USA
[^6]: The extended abstract of the paper was presented on ESA 2019.
|
---
abstract: 'We report on the detection of the alignment between galaxies and large-scale structure at $z\sim0.6$ based on the CMASS galaxy sample from the Baryon Oscillation Spectroscopy Survey data release 9. We use two statistics to quantify the alignment signal: 1) the alignment two-point correlation function which probes the dependence of galaxy clustering at a given separation in redshift space on the projected angle ($\theta_p$) between the orientation of galaxies and the line connecting to other galaxies, and 2) the $\cos(2\theta)-$statistic which estimates the average of $\cos(2\theta_p)$ for all correlated pairs at given separation $s$. We find significant alignment signal out to about 70 $h^{-1}$Mpc in both statistics. Applications of the same statistics to dark matter halos of mass above $10^{12} h^{-1}M_\odot$ in a large cosmological simulation show similar scale-dependent alignment signals to the observation, but with higher amplitudes at all scales probed. We show that this discrepancy may be partially explained by a misalignment angle between central galaxies and their host halos, though detailed modeling is needed in order to better understand the link between the orientations of galaxies and host halos. In addition, we find systematic trends of the alignment statistics with the stellar mass of the CMASS galaxies, in the sense that more massive galaxies are more strongly aligned with the large-scale structure.'
author:
- 'Cheng Li, Y. P. Jing, A. Faltenbacher, and Jie Wang'
title: 'Detection of the large scale alignment of massive galaxies at $z\sim0.6$'
---
Introduction {#sec:introduction}
============
Galaxies are not oriented at random, but show various forms of spatial alignment [@Carter-Metcalfe-80; @Binggeli-82; @Dekel-85; @West-89b; @Struble-90; @Plionis-94; @Plionis-03; @Hashimoto-Henry-Boehringer-08]. In particular, recent studies of galaxies in the Sloan Digital Sky Survey [SDSS; @York-00] have revealed that satellite galaxies are preferentially distributed along the major axis of the central galaxies [@Brainerd-05; @Yang-06a; @Azzaro-07; @Faltenbacher-07; @Faltenbacher-09], and tend to be preferentially oriented toward the central galaxy [@Pereira-Kuhn-05; @Agustsson-Brainerd-06a; @Faltenbacher-07]. These studies were mostly limited to the local universe and intermediate-to-small scales (less than a few tens of Mpc). Similar alignment signals have been detected for galaxies at intermediate redshifts [$0.2<z<0.5$; @Donoso-O'Mill-Lambas-06; @Okumura-Jing-Li-09]. In a recent study @Smargon-12 reported on the detection of intrinsic alignment between clusters of galaxies at $0.08<z<0.44$ out to 100 $h^{-1}$Mpc in the SDSS cluster catalogs. There have also been many observational studies which include galaxy ellipticity and measure both the galaxy orientation-density correlation and the intrinsic shear-density correlation [e.g. @Mandelbaum-06a; @Hirata-07; @Blazek-McQuinn-Seljak-11; @Joachimi-11], in order to better understand the potential contamination of galaxy alignment to cosmic shear surveys.
The various forms of the alignment of galaxies and clusters are generally expected in the $\Lambda$ cold dark matter cosmological paradigm in which galaxies are hosted by dark matter halos and are embedded in a cosmic web containing a varity of structures. The shapes of halos (and galaxies) are predicted to be aligned with each other due to the large-scale tidal field and the preferred accretion of matter along filaments [@Pen-Lee-Seljak-00; @Croft-Metzler-00; @Heavens-Refregier-Heymans-00; @Catelan-Kamionkowski-Blandford-01; @Crittenden-01; @Jing-02; @Porciani-Dekel-Hoffman-02b]. Therefore, measuring the alignment of galaxies/clusters, as function of redshift, spatial scale and galaxy properties, is expected to provide useful constraints on both galaxy formation and structure formation models.
In this work we extend the effort of detecting galaxy alignment to higher redshifts ($0.4<z<0.7$) and larger scales ($<200 h^{-1}$Mpc). For this we use the recently-released CMASS galaxy sample from the ninth data release [DR9; @Ahn-12] of the Baryon Oscillation Spectroscopic Survey [BOSS; @Schlegel-White-Eisenstein-09; @Dawson-13], which is a part of the SDSS-III [@Eisenstein-11]. We apply two different statistics suitable for quantifying the spatial alignment of galaxies to the CMASS sample, and show that the alignment between the orientation of the CMASS galaxies and the large-scale galaxy distribution extends out to $120 h^{-1}$Mpc. Applying the same statistics to dark matter halos in a large cosmological simulation, we detect similar alignment signals for the halos. This indicates that the observed large-scale alignment of galaxies can be explained by the anisotropy in the large-scale matter distribution, as we have recently found from a theoretical analysis of dark matter halos [@Faltenbacher-Li-Wang-12].
Methodology
===========
We use two different statistics to quantify the alignment between the orientation of galaxies and their large-scale spatial distribution: the alignment correlation function (ACF) and the $\cos(2\theta)-$statistic, which were originally introduced in @Faltenbacher-09. Here we briefly describe the statistics and refer the reader to that paper for details.
Alignment correlation function
------------------------------
The ACF extends the conventional two-point correlation function (2PCF) by including the angle between the major axis of a galaxy and the line connecting to another galaxy ($\theta_p$, projected on the sky for a survey sample) as an additional property of galaxy pairs. For a pair of galaxies with one member in the sample in question (called Sample Q hereafter) and another member in the reference sample (called Sample G hereafter), we consider $\theta_p$ as a secondary property of the pair, in addition to the separation of the paired galaxies. The estimator for the conventional 2PCF is then easily modified to give a measure of the ACF: $$\label{eqn:estimator}
\xi(\theta_p,s) = \frac{N_{R}}{N_{G}}
\frac{QG(\theta_p,s)}{QR(\theta_p,s)} -1,$$ where $s$ is the redshift-space pair separation, $N_G$ and $N_R$ are the number of galaxies in the reference and random samples. $QG(\theta_p,s)$ and $QR(\theta_p,s)$ are the counts of cross pairs between the given samples for given $\theta_p$ and $s$. The value of $\theta_p$ ranges from zero (parallel to the major axis of the main galaxy) to 90 degrees (perpenticular). Thus, higher amplitudes of $\xi(\theta_p,s)$ at small (large) $\theta_p$ indicate the galaxies in G are preferentially aligned along the major (minor) axis of the galaxies in Q. Sample Q is either the same as, or a subset of Sample G. In the former case the ACF is actually the alignment [*auto*]{}-correlation function, thus probing the alignment between galaxies within the same sample.
The $\cos(2\theta)-$statistic
-----------------------------
The $\cos(2\theta)-$statistic measures the average value of $\cos(2\theta)$ over all [*correlated*]{} pairs for a given spatial separation. This statistic is related to the ACF by $$\langle\cos(2\theta_p)_{\mbox{cor}}\rangle (s) =
\frac{\int_0^{\pi/2}\cos(2\theta_p)\xi(\theta_p,s)d\theta_p}
{\int_0^{\pi/2}\xi(\theta_p,s)d\theta_p},$$ and estimated by $$\langle \cos(2\theta_p)_{\mbox{cor}} \rangle(s) =
\frac{QG_{\theta_p}(s)}{QG(s)-(N_G/N_R)\cdot QR(s)},$$ where $QG_{\theta_p}(s)$ is the sum of $\cos(2\theta_p)$ for all the cross pairs between samples $Q$ and $G$ at separation $s$: $$QG_{\theta_p}(s) = \sum_{(i,j)\in QG(s)}\cos(2\theta_{p}^{i,j}).$$ The statistic so-defined ranges between -1 and 1, with positive and negative values indicating a preference for small ($<45^\circ$) and large ($>45^\circ$) angles. Values of zero means isotropy.
Data
====
The BOSS/CMASS galaxy sample
----------------------------
By selection the CMASS is a roughly volume-limited sample of massive galaxies in the redshift range of $0.4<z<0.7$ [@Eisenstein-11; @Anderson-12]. Clustering measurements and halo occupation distribution modeling [@White-11] revealed that the CMASS galaxies are hosted by dark matter halos with mass above $10^{12}h^{-1}M_\odot$, with the majority ($90\%$) being central galaxies in halos of mass $\sim10^{13}h^{-1}M_\odot$. The CMASS galaxy sample from BOSS/DR9 and the corresponding random sample suitable for large-scale structure analyses are generated by @Anderson-12, and are publicly available at the SDSS-III website[^1]. For this work we restrict ourselves to the survey area in the northern Galactic cap, including a total number of 207,246 galaxies. We exclude the sourthern Galactic cap from our analysis in order to avoid possible effects of the systematic differences between the northern and southern parts as found in recent BOSS-based studies [e.g. @Sanchez-12a]. The orientation of the galaxies is given by the position angle (PA) of the major axis of their $r$-band images, determined from the de Vaucouleurs model fit by the SDSS photometric pipeline [Photo]{}[@Lupton-01; @Stoughton-02].
The MultiDark Run 1 Simulation
------------------------------
In addition to analyzing the CMASS galaxy sample, we also apply our alignment statistics to dark matter halos in the MultiDark Run 1 simulation [MDR1; @Prada-12] [^2]. Assuming the WMAP7 concordant $\Lambda$CDM cosmology, the simulation uses $2048^3$ particles to follow the dark matter distribution in a cubic region with 1 $h^{-1}$Gpc on a side, which corresponds to a particle mass of $8.72\times10^9h^{-1}M_\odot$. Dark matter halos are identified by means of a friends-of-friends [@Davis-85] algorithm with a linking length of 0.17 times the mean particle separation. For the comparison with the CMASS galaxy sample we use snapshot 60 which corresponds to a redshift of $z\sim0.6$.
Following @Joachimi-13 we determine the projected orientations of the dark matter halos based on the 3D mass ellipsoids which are provided in the Multidark database. For this approach the line of sight is assumed to be parallel to the $z$-axis. We limit our analysis to dark matter halos with masses above $10^{12}h^{-1}M_\odot$, i.e., the aforementioned lower limit of the host halo mass for CMASS galaxies as found by @White-11. Halos of this mass are identified with a number of 115 particles. In this case an uncertainty of 10% is expected for the halo orientation determination [@Bett-07; @Joachimi-13], which we expect not to introduce significant bias into our result in the next section.
Results
=======
We have obtained the alignment correlation function $\xi(\theta_p,s)$ from the CMASS galaxy sample for three successive angular intervals: $0^\circ\leq\theta_p<30^\circ$, $30^\circ\leq\theta_p<60^\circ$, and $60^\circ\leq\theta_p<90^\circ$, as well as the conventional two-point correlation function, $\xi(s)$, which is a function of only the redshift-space separation and can be regarded as an average of the alignment correlation function over the full range of $\theta_p$.
In Figure \[fig:ratio\_xis\_all\] (left panel) we plot the difference in the alignment correlation function at small/large angles with respect to the conventional correlation function $\xi(s)$. The error bars plotted in the figure and in what follows are estimated using the bootstrap resampling technique [@Barrow-Bhavsar-Sonoda-84]. We have constructed 100 bootstrap samples based on the real sample, and we estimate the difference between $\xi(\theta,s)$ and $\xi(s)$ for each sample. The error at given scale is then estimated from the $1\sigma$ variance between the bootstrap samples.
As can be seen, $\xi(\theta_p,s)$ differ from $\xi(s)$ at both small and large angles, with stronger clustering at smaller angles and weaker clustering at larger angles, consistent with the picture that the major axis of the galaxies is preferentially aligned with their spatial distribution.
It is essential to perform systematics tests on any clustering measurements [e.g. @Mandelbaum-05; @Sanchez-12b]. As one of such tests, we have repeated the same analysis as above for a set of 100 random samples in which the position angles are shuffled at random among the main galaxies (Sample Q). The hatched regions plotted in red/green/blue in Figure \[fig:ratio\_xis\_all\] show the $1\sigma$ variance of the alignment correlation function between the random samples, measured for the three angle bins separately. It is interesting that the alignment signal detected in the real sample is significantly seen for a wide range of scales, from the smallest scales probed ($\sim 5 h^{-1}$Mpc) out to $\sim70 h^{-1}$Mpc according to both the bootstrap errors of the measurements and the $1\sigma$ regions of the random samples.
The right-hand panel of Figure \[fig:ratio\_xis\_all\] shows the $\cos(2\theta)-$statistic, plotted in solid circles for the CMASS sample and in green hatched region for the 100 randomly shuffled samples. The statistic for the real sample shows positive values on all scales probed, while its difference from the random samples is significantly seen only for scales below $\sim70 h^{-1}$Mpc, consistent with what the left-hand panel reveals. As mentioned above, a positive value in the $\cos(2\theta)-$statistic indicates a preference for angles smaller than $45^\circ$, thus implying that the major axis of the galaxies tends to be aligned with the large-scale distribution of galaxies. The $\cos(2\theta)-$statistic of the random samples shows a systematic positive bias at scales above $\sim40 h^{-1}$Mpc, implying that the position angle of the CMASS galaxies is not randomly distributed on the sky, a cosmic variance effect due to the limited survey area and probably also the quite irregular shape of the survey geometry. This can be tested in future with mock catalogs or later data releases of the BOSS survey.
For comparison the same statistics obtained for dark matter halos of mass $M_h>10^{12}h^{-1}M_\odot$ in the MDR1 simulation are shown in Figure \[fig:ratio\_xis\_all\] as solid lines. Alignment signal is seen in both statistics and on all the scales up to 120 $h^{-1}$Mpc. Both statistics show strong dependence on the spatial scale, which is very similar to what is seen for the CMASS galaxies. At fixed scale, however, the alignment of the halos is systematically stronger than that of the galaxies. This discrepancy might be partially (if not totally) due to the misalignment between the orientation of central galaxies and that of their host halos. A previous study done by @Okumura-Jing-Li-09 on the alignment of luminous red galaxies (LRGs) at $0.16<z<0.47$ in the SDSS/DR6 suggested that the misalignment angle between a central LRG and its host halo follows a Gaussian distribution with a zero mean and a typical width $\sigma_\theta=35.4$ deg [see also @Faltenbacher-09]. Such misalignment is expected to smooth out the alignment to some extent, leading the alignment of galaxies to be weaker than that of their host halos. Those authors found no evidence for the redshift evolution of their results. Thus, it is likely that a similar misalignment occurs also at the redshift of the CMASS sample.
To test the effect of such misalignment on our statistics, we artificially assign misalignment to the orientation of the dark matter halo before the alignment statistics are measured, assuming a Gaussian distribution with $\sigma_\theta=35^\circ$ for the misalignment angle. The results are plotted as dashed lines in Figure \[fig:ratio\_xis\_all\]. As expected, the amplitude of both statistics decreases considerably, becoming comparable with the data on scales below $\sim70 h^{-1}$Mpc. This simple experiment implies that the observed large-scale alignment signal for the massive galaxies at $z\sim0.6$ is real and can be explained by the alignment between dark matter halos and the large-scale matter distribution after the misalignment between the galaxies and their halos has been considered properly.
In order to test whether the uncertainties in the position angle measurements of the CMASS galaxies can introduce any systematic errors, we have repeated the analysis for a subset of $\sim73,000$ galaxies with the weight of the de Vaucouleurs model component [fracDev]{}$>0.8$, the de Vaucouleurs model scale radius $R_{deV}>1^{\prime\prime}$, and ellipticity $1-b/a>0.2$. The results are shown in Figure \[fig:sys\_tests\], with the grey/black symbols for the real sample and the hatched regions for the randomly shuffled samples. In addition, we have constructed 51 jacknife samples by dividing the CMASS/North area into 51 non-overlapping subregions and dropping one of the subregions from each of the 51 samples. The results of these samples are plotted in the red/blue lines in the same figure. These tests demonstrate that the alignment signal is reliably detected in the CMASS sample, at least to $\sim70 h^{-1}$Mpc, and is robust to the position angle measurements and the statistical error estimation.
Finally, we focus on the CMASS galaxies and examine the dependence of the alignment statistics on the stellar mass of the galaxies. For this we take the stellar mass esimates from the Wisconsin group [^3] derived by @Chen-12 from a BOSS spectrum principal component analysis (PCA) using the stellar population models of @Bruzual-Charlot-03. We divide the CMASS galaxies into two subsamples, with stellar mass either below or above $10^{11.6}M_\odot$. We take each of the subsamples as Sample Q, and we measure the alignment cross-correlation function with respect to the full CMASS galaxy sample (Sample G) in the same way as above. The results are shown in Figure \[fig:mass\_dependence\], with the two panels for the two subsamples separately. The result of the full sample is repeated in both panels as red/blue solid lines, for reference. Both subsamples show systematic differences from the full sample, in the sense that the high-mass subsample shows stronger-than-average alignment signals and the low-mass subsample shows weaker-than-average signals.
Summary
=======
We have applied two statistics, that are defined to be suitable for quantifying the spatial alignment of galaxies, to the CMASS galaxy sample from the SDSS-III/BOSS DR9, which consist of about $2\times10^{5}$ massive galaxies with mass above $\sim10^{11}M_\odot$ and redshift in the range $0.4<z<0.7$. Both statistics have revealed significant alignment, out to $\sim70 h^{-1}$Mpc, between the major axis of the CMASS galaxies and the large-scale distribution of the galaxies in the same sample. In addition, we have also detected a systematic trend of the alignment with the stellar mass of the galaxies, in the sense that more massive galaxies are more strongly aligned with the large-scale structure.
We have applied the same statistics to dark matter halos with mass above $10^{12}h^{-1}M_\odot$ in the MultiDark Run 1 (MDR1) simulation, and obtained very similar alignment sginals to what we have seen for the CMASS galaxies. This is consistent with previous studies of halo occupation distribution models on the CMASS sample which indicated that the majority of the CMASS galaxies are central galaxies in halos of mass $M_h>10^{12}h^{-1}M_\odot$. Furthermore, to test whether and how the possible misalignment between galaxies and host halos may affect our results for dark matter halos, we have performed a simple experiment in which we artifically asign a misalignment to the orientation of the halos, assuming a Gaussian distribution function for the misalignment angle with a width of 35 degrees. With such misalignment being included, the alignment statistics for the halos become substantially weaker, thus agreeing better with the observational results from the CMASS. This suggests that the large-scale alignment detected in the BOSS data is physically real and can be explained by the large-scale alignment of dark matter halos with respect to the matter distribution, as recently found by @Faltenbacher-Li-Wang-12 from cosmological simulations (also see a follow-up work by @Papai-Sheth-13 who developed a theoretical model to explain this finding). Detailed modeling of the observed alignment statistics should be able to provide powerful constraints on many aspects in both galaxy formation and structure formation theories, and would need to include a number of effects that are not considered in this work, including the contamination of satellite galaxies in the CMASS sample and the too simple mass cut in the dark matter halo sample. We will come back to this point in next studies.
We are grateful to the anonymous referee whose comments have helped us to significantly improve our paper. CL acknowledges the support of the 100 Talents Program of Chinese Academy of Sciences (CAS), Shanghai Pujiang Programme (no. 11PJ1411600) and the exchange program between Max Planck Society and CAS. This work is sponsored by NSFC (11173045, 11233005, 10878001, 11033006, 11121062) and the CAS/SAFEA International Partnership Program for Creative Research Teams (KJCX2-YW-T23). This work has made use of the public data from the SDSS-III. The MultiDark Database used in this paper and the web application providing online access to it were constructed as part of the activities of the German Astrophysical Virtual Observatory as result of a collaboration between the Leibniz-Institute for Astrophysics Potsdam (AIP) and the Spanish MultiDark Consolider Project CSD2009-00064. The Bolshoi and MultiDark simulations were run on the NASA’s Pleiades supercomputer at the NASA Ames Research Center.
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\[lastpage\]
[^1]: http://data.sdss3.org/datamodel/fiels/BOSS\_LSS\_REDUX/
[^2]: http://www.multidark.org/MultiDark/
[^3]: http://www.sdss3.org/dr9/algorithms/galaxy\_wisconsin.php
|
---
abstract: |
The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the $k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, *an $S$-Steiner tree* or *a Steiner tree connecting $S$* (or simply, *an $S$-tree*) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a *vertex-rainbow $S$-tree* if the vertices of $V(T)\setminus S$ have distinct colors. For a fixed integer $k$ with $2\leq k\leq n$, the vertex-coloring $c$ of $G$ is called a *$k$-vertex-rainbow coloring* if for every $k$-subset $S$ of $V(G)$ there exists a vertex-rainbow $S$-tree. In this case, $G$ is called *vertex-rainbow $k$-tree-connected*. The minimum number of colors that are needed in a $k$-vertex-rainbow coloring of $G$ is called the *$k$-vertex-rainbow index* of $G$, denoted by $rvx_k(G)$. When $k=2$, $rvx_2(G)$ is nothing new but the vertex-rainbow connection number $rvc(G)$ of $G$. In this paper, sharp upper and lower bounds of $srvx_k(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvx_k(G)\leq
n-2$. We obtain the Nordhaus-Guddum results for $3$-vertex-rainbow index, and show that $rvx_3(G)+rvx_3(\overline{G})=4$ for $n=4$ and $2\leq rvx_3(G)+rvx_3(\overline{G})\leq n-1$ for $n\geq 5$. Let $t(n,k,\ell)$ denote the minimal size of a connected graph $G$ of order $n$ with $rvx_k(G)\leq \ell$, where $2\leq \ell\leq n-2$ and $2\leq k\leq n$. The upper and lower bounds for $t(n,k,\ell)$ are also obtained.\
[**Keywords:**]{} vertex-coloring; connectivity; vertex-rainbow $S$-tree; vertex-rainbow index; Nordhaus-Guddum type.\
[**AMS subject classification 2010:**]{} 05C05, 05C15, 05C40, 05C76.
author:
- |
Yaping Mao[^1]\
Department of Mathematics, Qinghai Normal\
University, Xining, Qinghai 810008, China\
title: '**The vertex-rainbow index of a graph** [^2]'
---
Introduction
============
The rainbow connections of a graph which are applied to measure the safety of a network are introduced by Chartrand, Johns, McKeon and Zhang [@Chartrand]. Readers can see [@Chartrand; @Chartrand4; @Chartrand3] for details. Consider an edge-coloring (not necessarily proper) of a graph $G=(V,E)$. We say that a path of $G$ is *rainbow*, if no two edges on the path have the same color. An edge-colored graph $G$ is *rainbow connected* if every two vertices are connected by a rainbow path. The minimum number of colors required to rainbow color a graph $G$ is called *the rainbow connection number*, denoted by $rc(G)$. In [@M.Krivelevich], Krivelevich and Yuster proposed a similar concept, the concept of vertex-rainbow connection. A vertex-colored graph $G$ is *vertex-rainbow connected* if every two vertices are connected by a path whose internal vertices have distinct colors, and such a path is called a *vertex-rainbow path*. The *vertex-rainbow connection number* of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ vertex-rainbow connected. For more results on the rainbow connection and vertex-rainbow connection, we refer to the survey paper [@LiSun] of Li, Shi and Sun and a new book [@LiSun1] of Li and Sun. All graphs considered in this paper are finite, undirected and simple. We follow the notation and terminology of Bondy and Murty [@Bondy], unless otherwise stated.
For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, *an $S$-Steiner tree* or *a Steiner tree connecting $S$* (or simply, *an $S$-tree*) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. A tree $T$ in $G$ is a *rainbow tree* if no two edges of $T$ are colored the same. For $S\subseteq V(G)$, a *rainbow $S$-Steiner tree* (or simply, *rainbow $S$-tree*) is a rainbow tree connecting $S$. For a fixed integer $k$ with $2\leq k\leq n$, the edge-coloring $c$ of $G$ is called a *$k$-rainbow coloring* if for every $k$-subset $S$ of $V(G)$ there exists a rainbow $S$-tree. In this case, $G$ is called *rainbow $k$-tree-connected*. The minimum number of colors that are needed in a $k$-rainbow coloring of $G$ is called the *$k$-rainbow index* of $G$, denoted by $rx_k(G)$. When $k=2$, $rx_2(G)$ is the rainbow connection number $rc(G)$ of $G$. For more details on $k$-rainbow index, we refer to [@CLS; @CLS2; @Chartrand2; @CLYZ; @LSYZ; @LSYZ2].
Chartrand, Okamoto and Zhang [@Chartrand3] obtained the following result.
[[@Chartrand2]]{}\[th1-1\] For every integer $n\geq 6$, $rx_3(K_n)=3$.
As a natural counterpart of the $k$-rainbow index, we introduce the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this paper. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a *vertex-rainbow $S$-tree* or *vertex-rainbow tree connecting $S$* if the vertices of $V(T)\setminus S$ have distinct colors. For a fixed integer $k$ with $2\leq k\leq n$, the vertex-coloring $c$ of $G$ is called a *$k$-vertex-rainbow coloring* if for every $k$-subset $S$ of $V(G)$ there exists a vertex-rainbow $S$-tree. In this case, $G$ is called *vertex-rainbow $k$-tree-connected*. The minimum number of colors that are needed in a $k$-vertex-rainbow coloring of $G$ is called the *$k$-vertex-rainbow index* of $G$, denoted by $rvx_k(G)$. When $k=2$, $rvx_2(G)$ is nothing new but the vertex-rainbow connection number $rvc(G)$ of $G$. It follows, for every nontrivial connected graph $G$ of order $n$, that $$rvx_2(G)\leq rvx_3(G)\leq \cdots \leq rvx_n(G).$$
Let $G$ be the graph of Figure 1 $(a)$. We give a vertex-coloring $c$ of the graph $G$ shown in Figure 1 $(b)$. If $S=\{v_1,v_2,v_3\}$ (see Figure 1 $(c)$), then the tree $T$ induced by the edges in $\{v_1u_1,v_2u_1,u_1u_4,u_4v_3\}$ is a vertex-rainbow $S$-tree. If $S=\{u_1,u_2,v_3\}$, then the tree $T$ induced by the edges in $\{u_1u_2,u_2u_4,u_4v_3\}$ is a vertex-rainbow $S$-tree. One can easily check that there is a vertex-rainbow $S$-tree for any $S\subseteq V(G)$ and $|S|=3$. Therefore, the vertex-coloring $c$ of $G$ is a $3$-vertex-rainbow coloring. Thus $G$ is vertex-rainbow $3$-tree-connected.
\
Figure 1: Graphs for the basic definitions.
In some cases $rvx_k(G)$ may be much smaller than $rx_k(G)$. For example, $rvx_k(K_{1,n-1})=1$ while $rx_k(K_{1,n-1})=n-1$ where $2\leq k\leq n$. On the other hand, in some other cases, $rx_k(G)$ may be much smaller than $rvx_k(G)$. For $k=3$, we take $n$ vertex-disjoint cliques of order $4$ and, by designating a vertex from each of them, add a complete graph on the designated vertices. This graph $G$ has $n$ cut-vertices and hence $rvx_3(G)\geq n$. In fact, $rvx_3(G)=n$ by coloring only the cut-vertices with distinct colors. On the other hand, from Theorem \[th1-1\], it is not difficult to see that $rx_3(G)\leq 9$. Just color the edges of the $K_n$ with, say, color $1,2,3$ and color the edges of each clique with the colors $4,5,\cdots,9$.
Steiner tree is used in computer communication networks (see [@Du]) and optical wireless communication networks (see [@Cheng]). As a natural combinatorial concept, the rainbow index and the vertex-rainbow index can also find applications in networking. Suppose we want to route messages in a cellular network in such a way that each link on the route between more than two vertices is assigned with a distinct channel. The minimum number of channels that we have to use is exactly the rainbow index and vertex-rainbow index of the underlying graph.
The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou [@Chartrand2] in 1989, is a natural generalization of the concept of classical graph distance. Let $G$ be a connected graph of order at least $2$ and let $S$ be a nonempty set of vertices of $G$. Then the *Steiner distance* $d(S)$ among the vertices of $S$ (or simply the distance of $S$) is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. The *Steiner $k$-eccentricity $e_k(v)$* of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d(S)\,|\,S\subseteq V(G), |S|=k,~and~v\in
S \}$. The *Steiner $k$-diameter* of $G$ is $sdiam_k(G)=\max
\{e_k(v)\,|\,v\in V(G)\}$. Clearly, $sdiam_k(G)\geq k-1$.
Then, it is easy to see the following results.
\[pro1\] Let $G$ be a nontrivial connected graph of order $n$. Then $rvx_k(G)=0$ if and only if $sdiam_k(G)=k-1$.
\[pro2\] Let $G$ be a nontrivial connected graph of order $n \ (n\geq 5)$, and let $k$ be an integer with $2\leq k\leq n$. Then $$0\leq rvx_k(G)\leq n-2.$$
We only need to show $rvx_k(G)\leq n-2$. Since $G$ is connected, there exists a spanning tree of $G$, say $T$. We give the internal vertices of the tree $T$ different colors. Since $T$ has at most two leaves, we must use at most $n-2$ colors to color all the internal vertices of the tree $T$. Color the leaves of the tree $T$ with the used colors arbitrarily. Note that such a vertex-coloring makes $T$ vertex-rainbow $k$-tree-connected. Then $rvx_k(T)\leq n-2$ and hence $rvx_k(G)\leq rvx_k(T)\leq n-2$, as desired.
------------------------------------------------------------------------
\[obs1\] Let $K_{s,t}$, $K_{n_1,n_2,\ldots,n_k}$, $W_{n}$ and $P_n$ denote the complete bipartite graph, complete multipartite graph, wheel and path, respectively. Then
$(1)$ For integers $s$ and $t$ with $s\geq 2,t \geq 1$, $rvc(K_{s,t})=1$.
$(2)$ For $k\geq 3$, $rvx_k(K_{n_1,n_2,\ldots,n_k})=1$.
$(3)$ For $n\geq 4$, $rvx_k(W_{n})=1$.
$(4)$ For $n\geq 3$, $rvx_k(P_n)=n-2$.
Let $\mathcal {G}(n)$ denote the class of simple graphs of order $n$ and $\mathcal {G}(n,m)$ the subclass of $\mathcal {G}(n)$ having graphs with $n$ vertices and $m$ edges. Give a graph parameter $f(G)$ and a positive integer $n$, the *Nordhaus-Gaddum (**N-G**) Problem* is to determine sharp bounds for: $(1)$ $f(G)+f(\overline{G})$ and $(2)$ $f(G)\cdot f(\overline{G})$, as $G$ ranges over the class $\mathcal {G}(n)$, and characterize the extremal graphs. The Nordhaus-Gaddum type relations have received wide attention; see a recent survey paper [@Aouchiche] by Aouchiche and Hansen.
Chen, Li and Lian [@CLLian] gave sharp lower and upper bounds of $rx_k(G)+rx_k(\overline{G})$ for $k=2$. In [@CLLiu], Chen, Li and Liu obtained sharp lower and upper bounds of $rvx_k(G)+rvx_k(\overline{G})$ for $k=2$. In Section $2$, we investigate the case $k=3$ and give lower and upper bounds of $rvx_3(G)+rvx_3(\overline{G})$.
\[th2\] Let $G$ and $\overline{G}$ be a nontrivial connected graph of order $n$. If $n=4$, then $rvx_3(G)+rvx_3(\overline{G})=4$. If $n\geq 5$, then we have $$2\leq rvx_3(G)+rvx_3(\overline{G})\leq n-1.$$ Moreover, the bounds are sharp.
Let $s(n,k,\ell)$ denote the minimal size of a connected graph $G$ of order $n$ with $rx_k(G)\leq \ell$, where $2\leq \ell\leq n-1$ and $2\leq k\leq n$. Schiermeyer [@Schiermeyer] focused on the case $k=2$ and gave exact values and upper bounds for $s(n,2,\ell)$. Later, Li, Li, Sun and Zhao [@LLSZ] improved Schiermeyer’s lower bound of $s(n,2,2)$ and get a lower bound of $s(n,2,\ell)$ for $3\leq \ell \leq \lceil\frac{n}{2}\rceil$.
In Section $3$, we study the vertex case. Let $t(n,k,\ell)$ denote the minimal size of a connected graph $G$ of order $n$ with $rvx_k(G)\leq \ell$, where $2\leq \ell\leq n-2$ and $2\leq k\leq n$. We obtain the following result in Section $3$.
\[th3\] Let $k,n,\ell$ be three integers with $2\leq \ell\leq n-3$ and $2\leq k\leq n$. If $k$ and $\ell$ has the different parity, then $$n-1\leq t(n,k,\ell)\leq n-1+\frac{n-\ell-1}{2}.$$ If $k$ and $\ell$ has the same parity, then $$n-1\leq t(n,k,\ell)\leq n-1+\frac{n-\ell}{2}.$$
Nordhaus-Guddum results
=======================
To begin with, we have the following result.
\[pro3\] Let $G$ be a connected graph of order $n$. Then the following are equivalent.
$(1)$ $rvx_3(G)=0$;
$(2)$ $sdiam_3(G)=2$;
$(3)$ $n-2\leq \delta(G)\leq n-1$.
For Proposition \[pro1\], $rvx_3(G)=0$ if and only if $sdiam_3(G)=2$. So we only need to show the equivalence of $(1)$ and $(3)$. Suppose $n-2\leq \delta(G)\leq n-1$. Clearly, $G$ is a graph obtained from the complete graph of order $n$ by deleting some independent edges. For any $S=\{u,v,w\}\subseteq V(G)$, at least two elements in $\{uv,vw,uw\}$ belong to $E(G)$. Without loss of generality, let $uv,vw\in E(G)$. Then the tree $T$ induced by the edges in $\{uv,vw\}$ is an $S$-Steiner tree and hence $d_G(S)\leq
2$. From the arbitrariness of $S$, we have $sdiam_3(G)\leq 2$ and hence $sdiam_3(G)=2$. Therefore, $rvx_3(G)=0$.
Conversely, we assume $rvx_3(G)=0$. If $\delta(G)\leq n-3$, then there exists a vertex $u\in V(G)$ such that $d_{G}(u)\leq n-3$. Furthermore, there are two vertices, say $v,w$, such that $uv,uw\notin E(G)$. Choose $S=\{u,v,w\}$. Clearly, any rainbow $S$-tree must occupy at least a vertex in $V(G)\setminus S$, which implies that $rvx_3(G)\geq 1$, a contradiction. So $n-2\leq
\delta(G)\leq n-1$.
------------------------------------------------------------------------
After the above preparation, we can derive a lower bound of $rvx_3(G)+rvx_3(\overline{G})$.
\[lem1\] Let $G$ and $\overline{G}$ be a nontrivial connected graph of order $n$. For $n\geq 5$, we have $rvx_3(G)+rvx_3(\overline{G})\geq 2$. Moreover, the bound is sharp.
From Proposition \[pro2\], we have $rvx_3(G)\geq 0$ and $rvx_3(\overline{G})\geq 0$. If $rvx_3(G)=0$, then we have $n-2\leq
\delta(G)\leq n-1$ by Proposition \[pro3\] and hence $\overline{G}$ is disconnected, a contradiction. Similarly, we can get another contradiction for $rvx_3(\overline{G})=0$. Therefore, $rvx_3(G)\geq 1$ and $rvx_3(\overline{G})\geq 1$. So $rvx_3(G)+rvx_3(\overline{G})\geq 2$.
------------------------------------------------------------------------
To show the sharpness of the above lower bound, we consider the following example.
**Example 1:** Let $H$ be a graph of order $n-4$, and let $P=a,b,c,d$ be a path. Let $G$ be the graph obtained from $H$ and the path by adding edges between the vertex $a$ and all vertices of $H$ and adding edges between the vertex $d$ and all vertices of $H$; see Figure 2 $(a)$. We now show that $rvx_3(G)=rvx_3(\overline{G})=1$. Choose $S=\{a,b,d\}$. Then any $S$-Steiner tree must occupy at least one vertex in $V(G)\setminus
S$. Note that the vertices of $V(G)\setminus S$ in the tree must receive different colors. Therefore, $rvx_3(G)\geq 1$. We give each vertex in $G$ with one color and need to show that $rvx_3(G)\leq 1$. It suffices to prove that there exists a vertex-rainbow $S$-tree for any $S\subseteq V(G)$ with $|S|=3$. Suppose $|S\cap V(H)|=3$. Without loss of generality, let $S=\{x,y,z\}$. Then the tree $T$ induced by the edges in $\{xa, ya,za\}$ is a vertex-rainbow $S$-tree. Suppose $|S\cap V(H)|=2$. Without loss of generality, let $x,y\in S\cap V(H)$. If $a\in S$, then the tree $T$ induced by the edges in $\{xa,ya\}$ is a vertex-rainbow $S$-tree. If $b\in S$, then the tree $T$ induced by the edges in $\{xa,ya,ab\}$ is a vertex-rainbow $S$-tree.
\
Figure 2: Graphs for Example $1$.
Suppose $|S\cap V(H)|=1$. Without loss of generality, let $x\in
S\cap V(H)$. If $a,b\in S$, then the tree $T$ induced by the edges in $\{xa,ab\}$ is a vertex-rainbow $S$-tree. If $b,c\in S$, then the tree $T$ induced by the edges in $\{xd,cd,bc\}$ is a vertex-rainbow $S$-tree. If $a,c\in S$, then the tree $T$ induced by the edges in $\{xa,ab,bc\}$ is a vertex-rainbow $S$-tree. Suppose $|S\cap
V(G')|=0$. If $a,b,c\in S$, then the tree $T$ induced by the edges in $\{ab,bc\}$ is a vertex-rainbow $S$-tree. If $a,b,d\in S$, then the tree $T$ induced by the edges in $\{ab,bc,cd\}$ is a vertex-rainbow $S$-tree. From the arbitrariness of $S$, we conclude that $rvx_3(G)\leq 1$. Similarly, one can also check that $rvx_3(\overline{G})=1$. So $rvx_3(G)+rvx_3(\overline{G})=2$.
------------------------------------------------------------------------
We are now in a position to give an upper bound of $rvx_3(G)+rvx_3(\overline{G})$. For $n=4$, we have $G=\overline{G}=P_4$ since we only consider connected graphs. Observe that $rvx_3(G)=rvx_3(\overline{G})=rvx_3(P_4)=2$.
\[obs2\] Let $G,\overline{G}$ be connected graphs of order $n \ (n=4)$. Then $rvx_3(G)+rvx_3(\overline{G})=n$.
For $n\geq 5$, we have the following upper bound of $rvx_3(G)+rvx_3(\overline{G})$.
\[lem2\] Let $G,\overline{G}$ be connected graphs of order $n \ (n=5)$. Then $rvx_3(G)+rvx_3(\overline{G})\leq n-1$.
If $G$ is a path of order $5$, then $rvx_3(G)=3$ by Observation \[obs1\]. Observe that $sdiam_3(\overline{G})=3$. Then $rvx_3(\overline{G})\leq 1$ and hence $rvx_3(G)+rvx_3(\overline{G})\leq 4$, as desired.
\
Figure 3: Graphs for Lemma \[lem2\].
If $G$ is a tree but not a path, then we have $G=H_1$ since $\overline{G}$ is connected (see Figure 3 $(a)$). Clearly, $rvx_3(G)\leq 2$. Furthermore, $\overline{G}$ consists of a $K_2$ and a $K_3$ and two edges between them (see Figure 3 $(a)$). So we assign color $1$ to the vertices of $K_2$ and color $2$ to the vertices of $K_3$, and this vertex-coloring makes the graph $G$ vertex-rainbow $3$-tree-connected, that is, $rvx_3(\overline{G})\leq
2$. Therefore, $rvx_3(G)+rvx_3(\overline{G})\leq 4$, as desired.
Suppose that both $G$ and $\overline{G}$ are not trees. Then $e(G)\geq 5$ and $e(\overline{G})\geq 5$. Since $e(G)+e(\overline{G})=e(K_5)=10$, it follows that $e(G)=e(\overline{G})=5$. If $G$ contains a cycle of length $5$, then $G=\overline{G}=C_5$ and hence $rvx_3(G)=rvx_3(\overline{G})=2$. If $G$ contains a cycle of length $4$, then $G=H_2$ (see Figure 3 $(b)$). Clearly, $rvx_3(G)=rvx_3(\overline{G})=2$. If $G$ contains a cycle of length $3$, then $G=\overline{G}=H_3$ (see Figure 3 $(c)$). One can check that $rvx_3(G)=rvx_3(\overline{G})=2$. Therefore, $rvx_3(G)+rvx_3(\overline{G})=4$, as desired.
------------------------------------------------------------------------
\[lem3\] Let $G$ be a nontrivial connected graph of order $n$, and $rvx_3(G)=\ell$. Let $G'$ be a graph obtained from $G$ by adding a new vertex $v$ to $G$ and making $v$ be adjacent to $q$ vertices of $G$. If $q \geq n-\ell$, then $rvx_3(G')\leq \ell$.
Let $c: V(G)\rightarrow \{1,2,\cdots,\ell\}$ be a vertex-coloring of $G$ such that $G$ is vertex-rainbow $3$-tree-connected. Let $X=\{x_1, x_2,\cdots,x_q\}$ be the vertex set such that $vx_i\in
E(G')$. Set $V(G)\setminus X=\{y_1,y_2,\cdots, y_{n-q}\}$. We can assume that there exist two vertices $y_{j_1},y_{j_2}$ such that there is no vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$; otherwise, the result holds obviously.
\
Figure 4: Four type of the Steiner tree $T_i$.
We define a minimal $S$-Steiner tree $T$ as a tree connecting $S$ whose subtree obtained by deleting any edge of $T$ does not connect $S$. Because $G$ is vertex-rainbow $3$-tree-connected, there is a minimal vertex-rainbow tree $T_i$ connecting $\{x_i,y_{j_1},y_{j_2}\}$ for each $x_i \ (i\in \{1,2,\cdots,q\})$. Then the tree $T_i$ has four types; see Figure $4$. For the type shown in $(c)$, the Steiner tree $T_i$ connecting $\{x_i,y_{j_1},y_{j_2}\}$ is a path induced by the edges in $E(P_1)\cup E(P_2)$ and hence the internal vertices of the path $T_i$ must receive different colors. Therefore, the tree induced by the edges in $E(P_1)\cup E(P_2)\cup \{vx_i\}$ is a vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$, a contradiction. So we only need to consider the other three cases shown in Figure 4 $(a),(b),(d)$. Obviously, $T_i\cap T_j$ may not be empty. Then we have the following claim.
**Claim 1:** No other vertex in $\{x_1,x_2,\cdots,x_q\}$ different from $x_i$ belong to $T_i$ for each $1\leq i\leq q$.
[*Proof of Claim $1$*]{}: Assume, to the contrary, that there exists a vertex $x_i'\in \{x_1,x_2,\cdots,x_q\}$ such that $x_i'\neq x_i$ and $x_i'\in V(T_i)$. For the type shown in Figure 4 $(a)$, the Steiner tree $T_i$ connecting $\{x_i,y_{j_1},y_{j_2}\}$ is a path induced by the edges in $E(P_1)\cup E(P_2)$ and hence the internal vertices of the path $T_i$ receive different colors. If $x_i'\in V(P_1)$, then the tree induced by the edges in $E(P_1')\cup
E(P_2)\cup \{vx_i\}$ is a vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$ where $P_1'$ is the path between the vertex $x_i'$ and the vertex $y_{j_1}$ in $P_1$, a contradiction. If $x_i'\in V(P_2)$, then the tree induced by the edges in $E(P_2)\cup
\{vx_i\}$ is a vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$, a contradiction. The same is true for the type shown in Figure 4 $(b)$. For the type shown in Figure 4 $(c)$, the Steiner tree $T_i$ connecting $\{x_i,y_{j_1},y_{j_2}\}$ is a tree induced by the edges in $E(P_1)\cup E(P_2)\cup E(P_3)$ and hence the internal vertices of the tree $T_i$ receive different colors. Without loss of generality, let $x_i'\in V(P_1)$. Then the tree induced by the edges in $E(P_1')\cup E(P_2)\cup E(P_3)$ is a vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$ where $P_1'$ is the path between the vertex $x_i'$ and the vertex $v$ in $P_1$, a contradiction.
------------------------------------------------------------------------
0.5em
From Claim $1$, since there is no vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$, it follows that there exists a vertex $y_{k_i}$ such that $c(x_i)=c(y_{k_i})$ for each tree $T_i$, which implies that the colors that are assigned to $X$ are among the colors that are assigned to $V(G)\setminus X$. So $rvx_3(G)=\ell\leq
n-q$. Combining this with the hypothesis $q\geq n-\ell$, we have $rvx_3(G)=n-q$, that is, all vertices in $V(G)\setminus X$ have distinct colors. Now we construct a new graph $G'$, which is induced by the edges in $E(T_1)\cup E(T_2)\cup \cdots \cup E(T_q)$.0.5em
**Claim 2**: For every $y_t$ not in $G'$, there exists a vertex $y_s\in G'$ such that $y_ty_s\in E(G)$.0.5em
[*Proof of Claim $2$*]{}: Assume, to the contrary, that $N(y_t)\subseteq \{x_1,x_2,\cdots,x_q\}$. Since $G$ is vertex-rainbow $3$-tree-connected, there is a vertex-rainbow tree $T$ connecting $\{y_t,y_{j_1},y_{j_2}\}$. Let $x_r$ be the vertex in the tree $T$ such that $x_r\in N_G(y_t)$. Then tree induced by the edges in $(E(T)\setminus \{y_tx_r\})\cup \{vx_r\}$ is a vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$, a contradiction.
------------------------------------------------------------------------
0.5em
From Claim $2$, $G[y_1,y_2,\cdots, y_{n-q}]$ is connected. Clearly, $G[y_1,y_2,\cdots, y_{n-q}]$ has a spanning tree $T$. Because the tree $T$ has at least two pendant vertices, there must exist a pendant vertex whose color is different from $x_1$, and we assign the color to $x_1$. One can easily check that $G$ is still vertex-rainbow $3$-tree-connected, and there is a vertex-rainbow tree connecting $\{v,y_{j_1},y_{j_2}\}$. If there still exist two vertices $y_{j_3},y_{j_4}$ such that there is no vertex-rainbow tree connecting $\{v,y_{j_3},y_{j_4}\}$, then we do the same operation until there is a vertex-rainbow tree connecting $\{v,y_{j_r},y_{j_s}\}$ for each pair $y_{j_r},y_{j_s}\in
\{1,2,\cdots, n-q\}$. Thus $G'$ is vertex-rainbow $3$-tree-connected. So $rvc(G')\leq \ell$.
------------------------------------------------------------------------
**Proof of Theorem \[th2\]:** We prove this theorem by induction on $n$. By Lemma \[lem2\], the result is evident for $n=5$. We assume that $rvx_3(G)+rvx_3(\overline{G})\leq n-1$ holds for complementary graphs on $n$ vertices. Observe that the union of a connected graph $G$ and its complement $\overline{G}$ is a complete graph of order $n$, that is, $G\cup \overline{G}=K_n$. We add a new vertex $v$ to $G$ and add $q$ edges between $v$ and $V(G)$. Denoted by $G'$ the resulting graph. Clearly, $\overline{G'}$ is a graph of order $n+1$ obtained from $\overline{G}$ by adding a new vertex $v$ to $\overline{G}$ and adding $n-q$ edges between $v$ and $V(\overline{G})$.0.5em
**Claim 3:** $rvx_3(G')\leq rvx_3(G)+1$ and $rvx_3(\overline{G'})\leq rvx_3(\overline{G})+1$.0.5em
[*Proof of Claim $3$*]{}: Let $c$ be a $rvx_3(G)$-vertex-coloring of $G$ such that $G$ is vertex-rainbow $3$-tree-connected. Pick up a vertex $u\in N_G(v)$ and give it a new color. It suffices to show that for any $S\subseteq V(G')$ with $|S|=3$, there exists a vertex-rainbow $S$-tree. If $S\subseteq
V(G)$, then there exists a vertex-rainbow $S$-tree since $G$ is vertex-rainbow $3$-tree-connected. Suppose $S\nsubseteq V(G)$. Then $v\in S$. Without loss of generality, let $S=\{v,x,y\}$. Since $G$ is vertex-rainbow $3$-tree-connected, there exists a vertex-rainbow tree $T'$ connecting $\{u,x,y\}$. Then the tree $T$ induced by the edges in $E(T')\cup \{uv\}$ is a vertex-rainbow $S$-tree. Therefore, $rvx_3(G')\leq rvx_3(G)+1$. Similarly, $rvx_3(\overline{G'})\leq
rvx_3(\overline{G})+1$.
------------------------------------------------------------------------
0.5em
From Claim $3$, we have $rvx_3(G')+rvx_3(\overline{G'})\leq
rvx_3(G)+1+rvx_3(\overline{G})+1\leq n+1$. Clearly, $rvx_3(G')+rvx_3(\overline{G'})\leq n$ except possibly when $rvx_3(G')=rvx_3(G)+1$ and $rvx_3(\overline{G'})=
rvx_3(\overline{G})+1$. In this case, by Lemma \[lem3\], we have $q\leq n-rvx_3(G)-1$ and $n-q\leq n-rvx_3(\overline{G})-1$. Thus, $rvx_3(G)+rvx_3(\overline{G})\leq (n-1-q)+(q-1)=n-2$ and hence $rvx_3(G')+rvx_3(\overline{G'})\leq n$, as desired. This completes the induction.
------------------------------------------------------------------------
To show the sharpness of the above bound, we consider the following example.
**Example $2$:** Let $G$ be a path of order $n$. Then $rvx_3(G)=n-2$. Observe that $sdiam_3(\overline{G})=3$. Then $rvx_3(\overline{G})=1$, and so we have $rvx_3(G)+rvx_3(\overline{G})=(n-2)+1=n-1$.
The minimal size of graphs with given vertex-rainbow index
==========================================================
Recall that $t(n,k,\ell)$ is the minimal size of a connected graph $G$ of order $n$ with $rvx_k(G)\leq \ell$, where $2\leq \ell\leq
n-2$ and $2\leq k\leq n$. Let $G$ be a path of order $n$. Then $rvx_k(G)\leq n-2$ and hence $t(n,k,n-2)\leq n-1$. Since we only consider connected graphs, it follows that $t(n,k,n-2)\geq n-1$. Therefore, the following result is immediate.
\[th3-1\] Let $k$ be an integer with $2\leq k\leq n$. Then $$t(n,k,n-2)=n-1.$$
A *rose graph $R_{p}$ with $p$ petals* (or *$p$-rose graph*) is a graph obtained by taking $p$ cycles with just a vertex in common. The common vertex is called the *center* of $R_{p}$. If the length of each cycle is exactly $q$, then this rose graph with $p$ petals is called a *$(p,q)$-rose graph*, denoted by $R_{p,q}$. Then we have the following result.
**Proof of Theorem \[th3\]:** Suppose that $k$ and $\ell$ has the different parity. Then $n-\ell-1$ is even. Let $G$ be a graph obtained from a $(\frac{n-\ell-1}{2},3)$-rose graph $R_{\frac{n-\ell-1}{2},3}$ and a path $P_{\ell+1}$ by identifying the center of the rose graph and one endpoint of the path. Let $w_0$ be the center of $R_{\frac{n-\ell-1}{2},3}$, and let $C_i=w_0v_iu_iw_0 \ (1\leq i\leq \frac{n-\ell-1}{2})$ be the cycle of $R_{\frac{n-\ell-1}{2},3}$. Let $P_{\ell+1}=w_0w_1\cdots
w_{\ell}$ be the path of order $\ell+1$. To show the $rvx_k(G)\leq
\ell$, we define a vertex-coloring $c: V(G)\rightarrow
\{0,1,2,\cdots,\ell-1\}$ of $G$ by $$c(v)=\left\{
\begin{array}{ll}
i, &if~v=w_i \ (0\leq i\leq \ell-1);\\
1,&if~v=u_i~or~v=v_i \ (1\leq i\leq \frac{n-\ell-1}{2})\\
1,&if~v=w_{\ell}.
\end{array}
\right.$$ One can easily see that there exists a vertex-rainbow $S$-tree for any $S\subseteq V(G)$ and $|S|=3$. Therefore, $rvx_k(G)\leq \ell$ and $t(n,k,\ell)\leq n-1+\frac{n-\ell-1}{2}$.
Suppose that $k$ and $\ell$ has the same parity. Then $n-\ell$ is even. Let $G$ be a graph obtained from a $(\frac{n-\ell}{2},3)$-rose graph $R_{\frac{n-\ell}{2},3}$ and a path $P_{\ell}$ by identifying the center of the rose graph and one endpoint of the path. Let $w_0$ be the center of $R_{\frac{n-\ell}{2},3}$, and let $C_i=w_0v_iu_iw_0
\ (1\leq i\leq \frac{n-\ell}{2})$ be the cycle of $R_{\frac{n-\ell}{2},3}$. Let $P_{\ell}=w_0w_1\cdots w_{\ell-1}$ be the path of order $\ell$. To show the $rvx_k(G)\leq \ell$, we define a vertex-coloring $c: V(G)\rightarrow \{0,1,2,\cdots,\ell-1\}$ of $G$ by $$c(v)=\left\{
\begin{array}{ll}
i, &if~v=w_i \ (0\leq i\leq \ell-1);\\
1,&if~v=u_i~or~v=v_i \ (1\leq i\leq \frac{n-\ell}{2})\\
\end{array}
\right.$$ One can easily see that there exists a vertex-rainbow $S$-tree for any $S\subseteq V(G)$ and $|S|=3$. Therefore, $rvx_k(G)\leq \ell$ and $t(n,k,\ell)\leq n-1+\frac{n-\ell}{2}$.
------------------------------------------------------------------------
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[^1]: E-mail: maoyaping@ymail.com
[^2]: Supported by the National Science Foundation of China (No. 11161037) and the Science Found of Qinghai Province (No. 2014-ZJ-907).
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abstract: 'The quasi-biennial oscillation (QBO) of equatorial winds on Earth is the clearest example of the spontaneous emergence of a periodic phenomenon in geophysical fluids.In recent years, observations have revealed intriguing disruptions of this regular behaviour, and different QBO-like regimes have been reported in a variety of systems. Here we show that part of the variability in mean flow reversals can be attributed to the intrinsic dynamics of wave-mean flow interactions in stratified fluids. Using a constant-in-time monochromatic wave forcing, bifurcation diagrams are mapped for a hierarchy of simplified models of the QBO, ranging from a quasilinear model to fully nonlinear simulations. [The existence of new bifurcations associated with faster and shallower flow reversals, as well as a quasiperiodic route to chaos are reported in these models.]{} The possibility for periodicity disruptions is investigated by probing the resilience of regular wind reversals to external perturbations.'
author:
- Antoine Renaud$^1$
- 'Louis-Philippe Nadeau$^2$'
- Antoine Venaille$^1$
title: 'Periodicity disruption of a model quasi-biennial oscillation'
---
Earth’s equatorial stratospheric winds [oscillate between westerly and easterly mean flow]{} every 28 months. These low-frequency reversals known as quasi-biennial oscillations are driven by high-frequency waves emitted in the lower part of the atmosphere, and supported by the presence of stable density stratification [@baldwin2001quasi]. It is an iconic example of the spontaneous emergence of a periodic phenomenon in a turbulent geophysical flow [@vallis2017atmospheric], with analogues in other planetary stratospheres [@dowling2008planetary], in laboratory experiments [@plumb1978instability; @semin2018nonlinear], as well as in idealized numerical simulations [@wedi2006direct; @couston]. In recent years, increasing attention has been given to the robustness of these regular reversals to external wave forcing and perturbations. Disruptions of this type of oscillations have been observed both in the Earth’s atmosphere [@osprey2016unexpected; @newman2016anomalous] and in Saturn’s atmosphere [@fletcher2017disruption]. In addition, a variety of oscillatory regimes, including non-periodic ones, have been reported in direct numerical simulations of a stratified fluid forced by an oscillating boundary [@wedi2006direct] or driven by an explicitly resolved turbulent convective layer [@couston]. Non-periodic oscillations have also been reported in global circulation model simulations of the solar interior and Giant planets [@rogers2006angular; @showman2018atmospheric].
Until now, the non-periodic nature of the reversals were interpreted as the system’s response to transient external variations. For example, the non-periodic disruption of the Earth’s QBO and Saturn’s QBO-like oscillation have been attributed to the response of equatorial stratospheric dynamics to extratropical perturbations [@osprey2016unexpected; @newman2016anomalous; @fletcher2017disruption]. Also, the existence of nonperiodic regimes in direct numerical simulations of stratified flows has been related to the time variability of the underlying turbulent convective layer [@couston; @showman2018atmospheric]. Here, we show that the non-periodic nature of the reversals is a fundamental characteristic of stratified fluids by revealing the existence of a vast diversity of oscillatory regimes obtained using a simple steady monochromatic forcing. We further demonstrate that this rich intrinsic variability effectively controls part of the system’s response to a transient external variation. Periodicity disruptions are more easily triggered and are increasingly lengthened when the system approaches a bifurcation point.\
**Model.** The simplest configuration capturing the dynamics of the quasi-biennial oscillation (QBO, Fig. 1a) is given by a vertical 2D section of a stably stratified Boussinesq fluid, periodic in the zonal (longitudinal) direction, and forced by upward propagating internal gravity waves. This wave forcing is typically generated by an oscillating bottom boundary meant to represent the effect of tropopause height variations on the stratosphere. The evolution of the horizontally averaged zonal velocity, $\overline{u}$, is governed by a simplified version of the momentum equation $$\partial_{t}\overline{u}-\nu\partial_{zz}\overline{u}=-\partial_{z}\overline{u^{\prime}w^{\prime}},\label{eq:MeanEvol}$$ where $\nu$ is the kinematic viscosity, $z$ is the upward direction, and $\overline{u^{\prime}w^{\prime}}$ is the Reynolds stress due to velocity fluctuations around the zonal average. In weakly nonlinear regimes, this stress is carried by internal gravity waves, and any process damping the wave amplitude leads to a transfer of momentum from the waves to the mean-flow through the Reynolds stress divergence. Wave properties are also affected by the mean-flow and this interplay results in a complex coupled system.
To close the dynamical system, one needs to compute the Reynolds stress in (\[eq:MeanEvol\]). In this study, the wave field is simulated either by taking into account all nonlinear interactions between waves and mean-flow (hereafter “nonlinear 2D model") or by considering a simplified closure that neglects wave-wave interactions, together with a WKB approach [@plumb1977interaction]. The latter approach (hereafter “quasilinear 1D model") has proven to be successful in explaining the spontaneous emergence of low-frequency periodic flow reversals [@plumb1977interaction; @vallis2017atmospheric] and synchronisation with an external forcing [@rajendran2016synchronisation]. By assuming horizontally averaged dynamics, this quasilinear model is much simpler than the original flow equations but has nevertheless a large number of degrees of freedom since an infinite number of vertical oscillatory modes are possible. The 1D model is thus a natural starting point to investigate how periodic reversals are destabilized when the forcing strength is increased.
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We consider a standing wave pattern with wavenumber $k$ and frequency $\omega$, forcing a stratified fluid with buoyancy frequency $N$, for which the background stratification is maintained by Newtonian cooling with damping rate $\gamma$. Together, the Newtonian cooling $\gamma$, and the viscosity $\nu$, damp the wave amplitude over a characteristic e-folding length $\Lambda=\alpha k c^{4} /(\nu N^3)$, where $c=\omega/k$ is the zonal phase speed, and where $\alpha=\nu N^2/(\nu N^2 +\gamma c^2)$ is the ratio of viscosity to Newtonian cooling in wave damping. Another essential parameter of the problem is the effective Reynolds number $\mathrm{Re}=\mathcal{F}_{0} \Lambda/\left(c \nu \right)$, where $\mathcal{F}_{0}={\left(u^{\prime}_0w^{\prime}_0\right)}_{\mathrm{r.m.s}.}$ is the wave forcing strength at the bottom boundary. This wave forcing strength further sets a characteristic time scale of low-frequency flow reversals $\mathcal{T}=c\Lambda/\mathcal{F}_{0}$ [@vallis2017atmospheric]. Supplemental Material provides details on the simulations as well as estimates of the key parameters of the the quasilinear and nonlinear models, as well as for the Earth’s stratosphere.
The parameter range used in the simulations is close to that used in the pioneering work on the subject [@holton1972updated; @plumb1977interaction], and presented in standard textbooks on geophysical fluid dynamics [@vallis2017atmospheric]. Notice that the effective Reynolds number is based on an eddy viscosity, meant to represent the turbulent eddy motion at scales smaller than the internal gravity waves and used as a subgrid-scale parameterization for turbulence in coarse-grained climate models. The actual Reynolds number of the atmosphere based on the kinematic viscosity of the air is higher by many orders of magnitude than the effective Reynolds number. A self-consistent theory for the QBO would require to infer the eddy viscosity from the knowledge of the actual Reynolds number and other problem parameters, but this conundrum has up to now be out of reach. Here we follow a common practice in geophysical fluid dynamics that amounts to: (i) use an eddy viscosity to describe bifurcations occurring under an increase in forcing amplitude, and (ii) test the robustness of these bifurcations in more complex members of the hierarchy of geophysical flow models [@dijkstra2013nonlinear].\
**Bifurcation diagrams.** To map the bifurcation diagram of the quasilinear 1D model, we performed a large number of simulations spanning effective Reynolds numbers between $\mathrm{Re}=2$ and $330$, covering roughly the relevant range for the Earth’s stratosphere (Table 1). For sufficiently low values of $\mathrm{Re}$, the system has only one attractor: a stable point at $\overline{u}=0$. A first bifurcation occurs above the critical value $\mathrm{Re}_{c1}\approx 4.25/(1+\alpha)$ [@yoden1988new], for which the zonally averaged velocities are attracted towards a limit cycle [@plumb1977interaction; @yoden1988new] corresponding to horizontal mean-flow reversals and downward phase propagation (Fig. 1b). This period-1 cycle arguably reproduces the salient features of the observed QBO before the disruption event of 2016 (Fig. 1a). Figure 1d shows a bifurcation diagram plotted for increasing Reynolds numbers. A second bifurcation from periodic to quasi-periodic regimes occurs above the critical value $\mathrm{Re}_{c2}$. Additional bifurcations occur at higher Reynolds numbers, with transitions to frequency-locked regimes, and chaotic regimes. The term ’frequency-locking’ is often used where a nonlinear oscillator forced at some frequency exhibits, as a dominant response, an oscillation at the forcing frequency. By extension, we use this term here to describe synchronisation between oscillating modes of the dynamical system. As $\mathrm{Re}$ increases, new oscillating modes appear in the vertical structure of the mean flow. For example, a unique frequency is observed at all heights for the period-1 limit cycle shown in Fig. 1b, while faster reversals are observed in the lower levels for the frequency-locked regime shown in Fig. 1c. Ultimately, in chaotic regimes, the superposition of these modes yields a fractal-like structure of nested flow reversals (Fig. S1 in Supplemental Material). Such regimes with faster reversals in the lower layers have also been reported in direct numerical simulations driven by a convective boundary layer [@couston; @showman2018atmospheric].
The quasiperiodic regime occurring at $\mathrm{Re}>\mathrm{Re}_{c2}$ is embedded with a complicated set of frequency-locked regimes (Fig. 1d). The global structure of the bifurcation diagrams is better appreciated by considering, in Fig. 1e, the two-dimensional parameter space spanned by the Reynolds number $Re$ and the parameter $\alpha$. This figure shows a range of parameters where frequency locked regions are organized into a sequence of staircases, qualitatively similar to Arnold’s tongues [@arnold1961small].
Transition to chaos in a similar 1D quasilinear model was reported in Ref. [@kim2001gravity], which focused only on the purely viscous case, $\alpha=1$, with other boundary conditions relevant for the solar tachocline. In fact, this behavior occurs generically in nonlinear systems, with numerous examples in hydrodynamics [@swinney1983observations].
In the case of internal gravity wave streaming, frequency locked states organized into Arnold’s tongues were found when the 1D quasilinear model is coupled to an external low frequency forcing mimicking seasonal forcing [@read2015], which is reminiscent of synchronisations phenomena [in models]{} of El Nino Southern Oscillations [@tziperman1994nino; @jin1994nino]. Here, by considering a simple monochromatic forcing, and by covering the full parameter space $Re-\alpha$, we bring to light an unforeseen intrinsic dynamical structure of the underlying quasilinear model.
![\[fig:BifDiag\_GCM\]**Bifurcations in Navier-Stokes simulations. a.** As in Fig 1d, but showing the bifurcation diagram obtained with the nonlinear simulations, using $\alpha=0.6$. Selected values of $\mathrm{Re}$ (marked in orange, red and purple) correspond to panels b, c, and d. The dashed blue line corresponds to the second bifurcation (from period-1 to quasiperiodic) occuring at $\mathrm{Re}_{c2}$. **b.** Phase space trajectory for $\mathrm{Re}^{-1}=0.009$, projected on a 3D space defined by velocities at three different heights: $(\overline{u}_{1},\overline{u}_{2},\overline{u}_{3})=(\overline{u}(z=0.5\Lambda,t),\overline{u}(z=1.5\Lambda,t),(z=3\Lambda,t))$. **c.** Same as b, but using $\mathrm{Re}^{-1}=0.0068$. **d.** Same as b, but using $\mathrm{Re}^{-1}=0.0058$.](Fount_GCM.png){width="\columnwidth"}
The 1D quasilinear model is a highly truncated version of the original flow equations. It is thus crucial to see whether the aforementioned bifurcations occur in Navier-Stokes simulations of the fully nonlinear dynamics, including both wave-mean and wave-wave interactions. In Fig. 2a, we performed more than 200 two-dimensional numerical simulations to build a diagram similar to the one obtained with the 1D quasilinear model. These simulations show that the route to chaos is robust to the presence of nonlinear interactions between waves and mean-flow, with transitions from periodic solutions (Fig. 2b) to quasiperiodicity (Fig. 2c), to frequency locking (Fig. 2d) and eventually to chaos. However, significant differences from the quasilinear case are observed in the nonlinear simulations, where bifurcations occur at different effective Reynolds numbers, and where new dynamical regimes emerge. For instance, the large region of period-3 frequency locking obtained in the 1D quasilinear model (Fig. 1c) is replaced by a thin region of period-2 frequency locking for which the symmetry $U \rightarrow -U$ is broken (Figs. 2a and 2d).\
**Response to external perturbations.** By considering a fixed monochromatic wave forcing, we show above that quasi-periodicity arises naturally at steady state in the stratified fluid. This fixed forcing contrasts however with the actual QBO signal, which is driven by time-varying wave forcing and extra-tropical perturbations. In the following, we investigate how the presence of a bifurcation point influences the resilience of a given period-1 QBO-like oscillation to external variability by considering the effect of a time-dependent perturbation superimposed on its reference monochromatic wave forcing.
We first consider the effect of a time-dependent pulse in wave forcing strength, $\mathcal{F}_{0}$, mimicking the reported sudden increase in wave activity at the equator in the winter preceding the observed periodicity disruption of 2016 (see Supplemental Material for details on the perturbation). From a dynamical point of view, this perturbation suddenly drives the system out of its limit cycle, until it eventually relaxes back to its original period-1 oscillation over a characteristic time $\tau$. Figures 3a and 3b show examples of transient recovery periods for two values of the effective Reynolds number using the nonlinear model. In each case, the time evolution of the mean-flow displays short eastward-flow structures sandwiched between broader westward wind patterns. [These higher vertical modes of oscillations, frequently excited in transient disrupted regimes, share qualitative similarities with the periodicity disruption observed in 2016. ]{}
Figure 3d shows that the characteristic timescale for recovery diverges as the system approaches the bifurcation point $\mathrm{Re}_{c2}$. We found similar responses to a pulse in zonal mean momentum, and for the spin-up of the system from a state of rest. The recovery time’s divergence is observed both in the quasilinear model and the nonlinear model. The swift increase in recovery timescale observed as the system approaches a bifurcation point is [a generic feature of dynamical systems]{} often referred to as “critical slowing down" [@scheffer2009early; @kuehn2011mathematical]. In the climate system context, critical slowing down has proven to be useful in detecting early warnings of a bifurcation point [@lenton2011early].\
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**Conclusions and perspectives.** Our study demonstrates that erratic mean flow reversals are recovered with a simple monochromatic wave-forcing, provided that the forcing strength is sufficiently large. This suggests that similar states previously observed with more complex forcing [@couston; @showman2018atmospheric] can partly be attributed to the intrinsic dynamics of stably stratified fluids, rather than to fluctuations of the forcing itself. The quasiperiodic route to chaos found both in our quasilinear and fully nonlinear simulations reveal that increasing the forcing strength leads to the excitation of fast and shallow bottom-trapped modes nested in deeper and slower vertical modes. These fast bottom-trapped reversals are also excited during the transient response to an external perturbation. Most important, our results have crucial implications for the interpretation of the variability of a QBO-like oscillation: (i) the existence of a second bifurcation is robust in a hierarchy of models (suggesting that it may exist for the atmosphere), (ii) the proximity to this second bifurcation has a strong effect on the response of the oscillation to external perturbations, and consequently (iii) the intrinsic variability of a given oscillation is key to interpret its response to external perturbations.
Several aspects of actual planetary flow such as seasonal forcing, rotation, meridional circulation and two-way coupling between stratospheric and tropospheric dynamics are omitted in the simplified flow models considered in this letter. The interplay between intrinsic modes of variability and these additional features will need to be addressed in future work, but we expect that the the existence of a second bifurcation as well as the critical slowing down approaching this bifurcation will be robust through the whole hierarchy of geophysical flow models with stable stratification. Exploratory 3D simulations with rotation - presented in supplementary materials - comfort our 2D results.
[11]{} M.P. Baldwin [*et. al.*]{}, *The quasi-biennial oscillation*, Reviews of Geophysics (2001) G.K. Vallis, *Atmospheric and oceanic fluid dynamics* second edition, Cambridge University Press (2017) T.E. Dowling, *Planetary science: Music of the stratospheres*, Nature (2008) R.A. Plumb & A.D. McEwan, *The instability of a forced standing wave in a viscous stratified fluid: A laboratory analogue of. the quasi-biennial oscillation*, Journal of the Atmospheric Sciences (1978) B. Sémin [*et. al.*]{}, *Nonlinear saturation of the large scale flow in a laboratory model of the quasibiennial oscillation*, Physical review letters (2018) N.P. Wedi & P.K. Smolarkiewicz, *Direct numerical simulation of the [P]{}lumb–[M]{}c[E]{}wan laboratory analog of the [QBO]{}*, Journal of the Atmospheric Sciences (2006) L.A. Couston [*et. al.*]{}, *Order Out of Chaos: Slowly Reversing Mean Flows Emerge from Turbulently Generated Internal Waves*, Physical Review Letters (2018) S.M. Osprey [*et. al.*]{}, *An unexpected disruption of the atmospheric quasi-biennial oscillation*, Science (2016) P.A. Newman [*et. al.*]{}, *The anomalous change in the QBO in 2015–2016*, Geophysical Research Letters (2016) L.N. Fletcher [*et. al.*]{}, *Disruption of Saturn[’]{}s quasi-periodic equatorial oscillation by the great northern storm*, Nature Astronomy (2017) T.M. Rogers & G.A. Glatzmaier, *Angular momentum transport by gravity waves in the solar interior*, The Astrophysical Journal (2006) A.P. Showman [*et. al.*]{}, *Atmospheric Circulation of Brown Dwarfs and Jupiter and Saturn-like Planets: Zonal Jets, Long-term Variability, and QBO-type Oscillations*, arXiv preprint arXiv:1807.08433 (2018) R.A. Plumb, *The interaction of two internal waves with the mean flow: Implications for the theory of the quasi-biennial oscillation*, Journal of the Atmospheric Sciences (1977) K. Rajendran [*et. al.*]{}, *Synchronisation of the equatorial [QBO]{} by the annual cycle in tropical upwelling in a warming climate*, Quarterly Journal of the Royal Meteorological Society (2016) J.R. Holton & R.S. Lindzen, *An updated theory for the quasi-biennial cycle of the tropical stratosphere*, Journal of the Atmospheric Sciences (1972) H.A. Dijkstra, *Nonlinear climate dynamics*, Cambridge University Press (2013) S. Yoden & J.R. Holton, *A new look at equatorial quasi-biennial oscillation models*, Journal of the atmospheric sciences (1988) V.I. Arnold, *Small denominators. I. Mapping the circle onto itself*,Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya (1961) H.L. Swinney, *Observations of order and chaos in nonlinear systems*, Physica D: Nonlinear Phenomena (1983) K. Rajendran [*et. al.*]{}, *Synchronisation of the equatorial QBO by the annual cycle in tropical upwelling in a warming climate*, Quarterly Journal of the Royal Meteorological Society (2015) E. Tziperman [*et. al.*]{}, *El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator*, Science (1994) F.F. Jin [*et. al.*]{}, *El [N]{}i[ñ]{}o on the devil’s staircase: annual subharmonic steps to chaos*, Science (1994) M. Scheffer [*et. al.*]{}, *Early-warning signals for critical transitions*, Nature (2009) C. Kuehn, *A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics*, Physica D: Nonlinear Phenomena (2011) T.M. Lenton, *Early warning of climate tipping points*, Nature Climate Change (2011) E.J. Kim & K.B. MacGregor, *Gravity wave-driven flows in the solar tachocline*, The Astrophysical Journal Letters (2001) *The Quasi-Biennial-Oscillation (QBO) Data Series*, [www.geo.fu-berlin.de/en/met/ag/strat/produkte/qbo/index.html](www.geo.fu-berlin.de/en/met/ag/strat/produkte/qbo/index.html) (Accessed: 2018-03)
**Supplemental Material for ‘Periodicity disruption of a model quasi-biennial oscillation’**\
Antoine Renaud$^1$, Louis-Philippe Nadeau$^2$, Antoine Venaille$^{1}$\
*$^1$ Univ Lyon, Ens de Lyon, Univ Claude Bernard,\
CNRS, Laboratoire de Physique, F-69342 Lyon, France\
$^2$ Institut des Sciences de la Mer de Rimouski, Université du Québec à Rimouski. Rimouski, Québec, Canada*\
(Dated: )
Methods and details on numerical simulations {#methods-and-details-on-numerical-simulations .unnumbered}
============================================
**1D quasilinear simulations.** Using a static Wentzel-Kramers-Brillouin (WKB) approximation to compute the wave field for a given mean-flow [@Splumb1978instability], the wave-induced Reynolds stress $\overline{u^{\prime}w^{\prime}}$ is parametrised by the formula given in Eq. (\[eq:Parametrisation\]).
$$\overline{u^{\prime}w^{\prime}}\left(z\right)=\sum_{i=1}^{2} (-1)^i\mathcal{F}_{0}\exp\left\{-\frac{1}{\Lambda}\int_{0}^{z}\mathrm{d}z^{\prime}\left(\frac{\alpha}{\left(1-(-1)^i\overline{u}\left(z^{\prime}\right)/c\right)^{4}}+\frac{1-\alpha}{\left(1-(-1)^i\overline{u}\left(z^{\prime}\right)/c\right)^{2}}\right)\right\}\label{eq:Parametrisation}.$$
This formula is derived under the hydrostatic balance assumption (valid in the limit $k|c\pm\overline{u}|/N\rightarrow 0$) and the weak damping assumption (valid in the limit $\gamma/(k|c\pm \overline{u}|)\ll1$ and $\nu N^{2}/(k|c\pm\overline{u}|^{3})\ll 1$). Assuming that the characteristic vertical length for $\overline{u}$ is $\Lambda$, then the small parameter needed in the WKB approach is the Froude number $Fr = c/(\Lambda N)\rightarrow 0$. In practice, the different assumptions are most certainly violated. However, this set of equations has long been recognized as a useful model to probe the salient features of QBO reversals.
We solve numerically using a centered second-order finite difference method with grid size $\delta_{z}=H/60$, and a second-order Adams-Bashforth scheme with time-step $\delta_{t}=0.005\mathcal{T}$; $\mathcal{T}=c\Lambda/\mathcal{F}_{0}$. A no-slip condition is used at the bottom boundary, $z=0$, and a free-slip condition is used at the upper boundary, $z=H$. Singularities in Eq. (\[eq:Parametrisation\]) appear when $\overline{u}=c$ (critical layers). These singularities are treated as follows: at a given height $z=z_c$, if the absolute value of $\overline{u}$ reaches locally a value higher than $c$, then the corresponding exponential is set to zero for all $z\geq z_{c}$. The definition and value of each of the model’s dimensionless numbers are given in table \[tab:Param\].\
**2D nonlinear simulations.** The fully nonlinear simulations are conducted using the MIT general circulation model [@SMarshall1997] solving the 2D Navier-Stokes equations under the Boussinesq and hydrostatic approximations $$\begin{cases}
\partial_{t}u+\mathbf{u}\cdot\nabla u & =-\partial_{x} \phi+\nu\nabla^{2}u-\gamma_u \left(u-u_0\right) \\
0 & = -\partial_{z}\phi+b\\
\partial_{t}b+\mathbf{u}\cdot\nabla b&= \kappa\nabla^{2} b-\gamma\left(b-b_0\right)\\
\nabla\cdot\mathbf{u}&=0
\end{cases},\label{eq:NumModel}$$ where $\mathbf{u} = u {\hat{\textbf{\i}}}+ w {\hat{\textbf{k}}} $ is the velocity field, $\mathbf{u}_{\rm h}$ is its projection on the horizontal plane $(x,y)$; $b=g\left(\rho_{0}-\rho\right)/\rho_{0}$ is the buoyancy; $\rho$ is the density and $\rho_{0}$ is a reference density; $g$ is the gravitational acceleration; $\phi=P/\rho_{0}+gz$; $P$ is the pressure; $\nu$ is the viscosity coefficient; $\kappa$ is the buoyancy diffusion coefficient; $\gamma_u$ and $\gamma$ are the rates at which the momentum and buoyancy are linearly restored to the reference profiles $u_0$ and $b_0$, respectively.
The domain is a Cartesian grid, periodic in the zonal direction, with zonal length $L_{x}=2\pi/k$ and height $H$ . The horizontal and vertical resolutions are respectively $\delta_x=L/26$ and $\delta_{z}=H/200$. A free-slip condition is used at the bottom boundary, while a free-surface condition is used at the top.
The zonal momentum equation is forced at the bottom boundary using a linear velocity relaxation $\gamma_u=\delta_b/\tau_{u}$, where $\tau_{u}$ is a relaxation timescale, and $\delta_b$ is a delta function equal to $1$ for the bottom grid-point and $0$ for all other vertical levels. In this last grid-point, velocity is relaxed to a zonally periodic standing wave pattern $$u_0=\sqrt{\frac{4N\mathcal{F}_0}{\omega}}\cos\left(kx\right)\cos\left(\omega t\right), \label{eq:bottom_dns}$$ where $\mathcal{F}_0$ controls the wave momentum flux amplitude at the bottom. This forcing is thought to generate a standing internal gravity wave field while enforcing an effective no-slip condition for the mean flow $\overline{u}$. Buoyancy is relaxed to the linear profile $b_0=N^{2}z$. To avoid any wave reflection at the upper free-surface, the vertical grid spacing and the Newtonian cooling are both increased in the $20$ upper grid layers.\
**Poincaré sections.** For each combination of parameters $\left(\mathrm{Re},\alpha\right)$, experiments are first spun-up over a time $t_{\rm e}=1500\mathcal{T}$ where $\mathcal{T}=c\Lambda/\mathcal{F}_{0}$. This time is sufficient for the system to reach its attractor. To combine the information of more than $10^{6}$ simulations into a single bifurcation diagram, we first select two vertical levels: $z_1$ near the surface and $z_2$ aloft. Resuming the simulation at statistical equilibrium ($t>t_{\rm e}$), we store the values of $\overline{u}(z_2)$ that intersects $\overline{u}(z_1)=0$ in the set $$\mathbb{O}_{\mathrm{Re},\alpha}=\left\{\;\overline{u}\left(z_{2},t\right)\left|\;\overline{u}\left(z_{1},t\right)=0\right.\right\}.\label{eq:Mat_PS}$$ The simulations are stopped once $200$ values are stored (i.e. after $200$ reversals of the lower-level mean-flow $\overline{u}\left(z_{1}\right)$). For each simulation associated with couples of parameters $(\mathrm{Re},\alpha)$, we build an histogram of the values stored in (\[eq:Mat\_PS\]), using $1000$ bins in the range $\left[-c,c\right]$. Histograms corresponding to all values of $\mathrm{Re}$ for a fixed $\alpha=0.6$ are drawn horizontally in figure 1c using a binary colour-map.
1D quasilinear simulations 2D Nonlinear simulations Stratosphere
--------------- -------------------------------------------------- -------------------------- -------------- -------------------
$\mathrm{Re}$ $=F_{0}\Lambda/\left(c\nu\right)$ $15-350$ $50-600$ $2-400$
$\alpha$ $=\nu N^{2}/\left(\nu N^{2}+\gamma c^{2}\right)$ $0-1$ $0.25$ $0-0.3$
$Pr$ $=\nu/\kappa$ $\infty$ $740$ N.A.
$\mathrm{Fr}$ $=c / \left(\Lambda N\right)$ $ \mathrm{Fr} \to 0 $ $ 0.06$ $0.1$
${\color{white} =}\omega/N$ $ \omega/N \to 0$ $0.1$ $10^{-4}-10^{-3}$
${\color{white} =}\omega\tau_{u}$ $0$ $0.1$ N.A.
${\color{white} =}H/\Lambda$ $3.5$ $4.1$ $1.5$
To collapse the information of the Poincaré sections into a 2D bifurcation diagram $(\alpha, \mathrm{Re}^{-1})$, we compute the ratio of populated bins to the total number of reversals for each histogram in the set (\[eq:Mat\_PS\]). This ratio with values in $\left[0,1\right]$ provides an empirical estimate of each histogram’s distribution and allows for an extensive classification of the different dynamical regimes (see Fig. 1d in the letter)\
**Recovery from a perturbation.** We consider perturbations to a given period-1 QBO-like oscillation. Three types of external perturbations are considered at $t=t_p$: (i) a pulse in wave amplitude (representing a sudden increase of the underlying tropical convection) (ii) a body force acting directly on the mean flow (representing a reorganization of the mean flow due to extratropical perturbations) (iii) a reboot of the oscillations from a state of rest (the recovery time is then equivalent to the spin-up time).
Perturbation (i) is modeled using a time-dependent momentum flux amplitude $\mathcal{F}_0$ in Eq. (\[eq:Parametrisation\]) for the quasilinear model and in Eq. (\[eq:bottom\_dns\]) for the nonlinear simulations: $$\mathcal{F}_0\left(t\right)=\mathcal{F}_{0,p}\left(1+9e^{-\frac{1}{2}\left(10\frac{t-t_{p}}{T_{\rm qbo}}\right)^{10}}\right),$$ where $\mathcal{F}_{0,p}$ is a constant forcing amplitude corresponding to a periodic regime with period $T_{\rm qbo}$. Perturbation (ii) is represented in the quasilinear model by an additional body forcing term $\mathcal{F}_{\mathrm{bulk}}$ in the r.h.s. of the mean flow equation (1): $$\mathcal{F}_{\mathrm{bulk}}\left(z,t\right)=
\frac{10}{T_{\rm qbo}} e^{-\frac{1}{2}\left(10\frac{t-t_{p}}{T_{\rm qbo}}\right)^{10}} e^{-800\left(\frac{z-z_{p}}{z_{\rm max}}\right)^{2}},$$ where $z_{p}=0.2z_{\rm max}$ sets the height. Results are insensitive to the specific choices of $z_{p}$.
For all three types of external perturbations, the system is driven away from its steady state period-1 limit cycle and then freely recovers back to the cycle. To estimate the recovery timescale, we first introduce the running mean-square $$\left\langle\overline{u}^{2}\right\rangle\left(t\right)=\frac{1}{T_{\rm qbo}H}\int_{t-T_{\rm qbo}/2}^{t+T_{\rm qbo}/2}\,\int_{0}^{H}\,\overline{u}^{2}\left(z,t^{\prime}\right)\mathrm{d}z\mathrm{d}t^{\prime},$$ where $T_{\rm qbo}$ is the period of the limit cycle. At steady state equilibrium, this running mean-square has a constant value $\langle\overline{u}^{2}\rangle_{\infty}$. Assuming a pulse shorter than the period of the limit cycle (see Fig. 3c in the letter), occurring at time $t_{p}$, the recovery timescale is then defined by $$\label{recovT}
\tau=\min_{\Delta t\geq 0} \left\{ \frac{\langle\overline{u}^{2}\rangle\left(t_{p}+\Delta t\right)-\langle\overline{u}^{2}\rangle_{\infty}}{\langle\overline{u}^{2}\rangle_{\infty}}\leq 0.2 \right\}.$$
We reproduced figure 3d of the letter in logarithmic scale in order to exhibit the power-law like scaling of the recovery timescale as the system approaches $\mathrm{Re}_{c2}$. It proved very difficult to deduce a precise value for the critical exponent as the uncertainty on the value of $\mathrm{Re}_{c2}$ echoes on it. However, the critical exponent remains close to $-1$.
![\[fig:BifDiag\_Plumb\_S6\]**Spin-up and recovery times in log-scale.** Characteristic recovery and spin-up timescales as a function of the relative distance to the second bifurcation point, $1-\mathrm{Re}/\mathrm{Re}_{c2}$, represented in log-scale with revert $x$-axis. The orange markers correspond to the spin-up time for the 1D quasi-linear model. The blue and purple markers are obtained with the quasilinear model, and correspond to the recovery from a pulse in the wave amplitude and from a direct body force, respectively. The red markers correspond to the spin-up time with the 2D nonlinear model. We used $\mathrm{Re}_{c2}=24.3$ for the 1D quasilinear model and $\mathrm{Re}_{c2}=135$ for the 2D nonlinear model. A dashed line representing a power law with exponent $-1$ is provided to guide the reader.](SupplMat_6.png){width="0.4\linewidth"}
Vertical flow structure in different regimes and effect of resolution {#vertical-flow-structure-in-different-regimes-and-effect-of-resolution .unnumbered}
=====================================================================
![\[fig:BifDiag\_Plumb\_S1\]**Bifurcations in the 1D quasilinear model. a.** A Poincaré section is shown for varying values of $\mathrm{Re}^{-1}$ and $\alpha=0.6$ (see Methods). **b.** Projection of phase-space trajectory in a 3D space $(\overline{u}_{1},\overline{u}_{2},\overline{u}_{3})=(\overline{u}(z=0.1\Lambda),\overline{u}(z=1.5\Lambda,t),\overline{u}(z=3\Lambda,t))$ for $\mathrm{Re}^{-1}=0.059$. **c.** Same for $\mathrm{Re}^{-1}=0.045$. **d.** Same for $\mathrm{Re}^{-1}=0.025$. **e.** Same for $Re^{-1}=250$. **f.** Hovmöller diagram of the mean-flow $\overline{u}\left(z,t\right)$ for $\mathrm{Re}^{-1}=0.059$. Time is rescaled by $\mathcal{T}=c\Lambda/\mathcal{F}_{0}$. The velocity $\overline{u}$ ranges from to $-c$ (blue) to $+c$ (red). The horizontal dotted lines highlight the height $z=0.1\Lambda$, $z=1.5\Lambda$ and $z=3\Lambda$, associated with the 3D projections plotted in panels b to e. **g.** Same for $\mathrm{Re}^{-1}=0.045$. **h.** Same for $\mathrm{Re}^{-1}=0.025$. **i.** Same for $\mathrm{Re}^{-1}=0.0004$.](SupplMat_1.png){width="0.8\linewidth"}
**Bifurcations in the quasilinear model.** In order to develop intuition for the underlying dynamics of the bifurcation diagrams of Fig. 1d, we show in Fig. \[fig:BifDiag\_Plumb\_S1\] phase space trajectories (panels b-e) and hovmöller diagrams of the mean-flow (panels f-i) for four selected values of the Reynolds number. Shown are examples for a period-1 limit cycle (panels b and f), a quasiperiodic oscillation (panels c and g), a frequency locked oscillation with frequency ratio $1/3$ (panels d and h), and a chaotic oscillation (panels e and i).
![\[fig:BifDiag\_Plumb\_S2\]**Additional bifurcation diagrams of the 1D quasilinear model. a.** Poincaré sections for each value of $\mathrm{Re}^{-1}$ using $\alpha=0$ (see Methods). **b.** Same for $\alpha=1/3$. **c.** Same for $\alpha=2/3$. **d.** Same for $\alpha =1$.](SupplMat_2.png){width="0.75\linewidth"}
Additional bifurcation diagrams obtained for different values of $\alpha$ are shown in Fig. \[fig:BifDiag\_Plumb\_S2\]. Although sharing a common qualitative structure, each bifurcation diagrams show distinct interesting features. For example, Fig. \[fig:BifDiag\_Plumb\_S2\]b shows that the upper quasiperiodic region vanishes almost entirely when $\alpha$ approaches $1/3$. Fig. \[fig:BifDiag\_Plumb\_S2\]d ($\alpha=1$) shows a spontaneous breaking of the symmetry $U\leftrightarrow -U$ occurring in one of the frequency locked states ($\mathrm{Re}^{-1}\sim 0.045$), while all the frequency-locked regimes preserve this symmetry at lower values of $\alpha$ in panels a, b and c.\
**Effect of the resolution in the quasilinear model.** In order to test the robustness of the quasilinear model results to resolution, we show in Fig. \[fig:BifDiag\_Plumb\_S4\] five bifurcation diagrams for which the vertical resolution has been successively doubled from $\delta_z=3.5/15$ to $3.5/240$. Panel c corresponds to the reference resolution used in Fig. 1d. Results show a strong dependence on vertical resolution, in particular for the structure of the embedded frequency locked regimes. However, the essential feature relevant to the periodicity disruption is the second bifurcation point $Re_{c2}$, marking the transition from periodic to quasiperiodic oscillations. Results of Fig. \[fig:BifDiag\_Plumb\_S4\] show that the value of $Re_{c2}$ is converging for a resolution $\delta_z=3.5/60$, corresponding to the reference resolution used in this work.
![\[fig:BifDiag\_Plumb\_S4\]**Resolution dependence in the 1D quasilinear model** **a** Poincaré sections for each value of $\mathrm{Re}^{-1}$ with $\alpha=0.6$. The spatial resolution used is $\delta_z=3.5/15$. **b.** Same for $\delta_z=3.5/30$. **c.** Same for $\delta_z=3.5/60$. **d.** Same for $\delta_z=3.5/120$. **e.** Same for $\delta_z=3.5/240$.](SupplMat_4.png){width="1\linewidth"}
As far as critical slowing down is concerned, the response of the system approaching the bifurcation from periodic to quasi-periodic state will be robust to higher resolutions, as the threshold $Re_{c2}$ and the nature of the bifurcation remains the same. However, the details of the response, including the vertical structure of transient oscillations and the prefactor of the recovery time power law may be affected by a change in resolution.\
3D nonlinear simulations with rotation {#d-nonlinear-simulations-with-rotation .unnumbered}
======================================
In this section, we solve the 3D Navier-Stokes equations with rotation approximated by an equatorial beta-plane. The horizontal momentum equation in (\[eq:NumModel\]) now writes $$\begin{cases}
\partial_{t}u+\mathbf{u}.\nabla u -\beta y v & = -\partial_{x}\phi+\nu\delta^2 u - \gamma_u \left(u-u_0\right)\\
\partial_{t}v+\mathbf{u}.\nabla v + \beta y u &= -\partial_{y}\phi+\nu\delta^2 v
\end{cases},\label{eq:NumModel3D}$$ where the velocity vector field is now 3D with $\mathbf{u}=u\hat{\mathbf{i}}+v\hat{\mathbf{j}}+w\hat{\mathbf{k}}$. $\beta$ is the Rossby parameter. Let us denote $L_{y}$ the length of the added meridional dimension and $\delta_y$ the associated resolution. We consider an horizontal aspect ratio $L_y /L_x=1$ and resolution ratio $\delta_y/\delta_x=1$, with free-slip lateral boundary condition at $y=\pm L_y /2$. We explore a weak rotation case, for which the equatorial Radius of deformation $L_d=\sqrt{N\Lambda/\beta}$ is much larger than the meridional extension of domain: $L_d /L_y =96$. All other parameters are identical to the 2D nonlinear simulations, including the forcing, constant along the $y$ direction. The chosen initial condition breaks the meridional invariance.
Fig. \[fig:Bif3D\_S3\] shows that bifurcation from a periodic regime (panel a) to a quasiperiodic regime (panel b) occurs when the Reynolds number is increased from $\mathrm{Re}=125$ to $\mathrm{Re}=250$. This demonstrates that the intrinsic variability observed in the 1D quasilinear model is robust to the presence of 3D wave-wave interactions.
In the Earth’s stratosphere, the parameter $L_d/L_y$ is smaller than one, and equatorial waves are trapped along the equator over a typical scale of the order of the deformation radius. Future work is needed to explore the robustness of the bifurcation point $Re_{c2}$ in the presence of strong rotation.
![\[fig:Bif3D\_S3\]**Bifurcations in 3D nonlinear simulation with weak rotation. a.** Hovmöller diagrams of the mean-flow $\overline{u}(z,t)$ for $\mathrm{Re}^{-1}=0.08$. **b** Same for $\mathrm{Re}^{-1}=0.04$.](SupplMat_3.png){width="0.5\linewidth"}
[11]{} J. Marshall [*et. al.*]{}, *A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers*, Journal of Geophysical Research (1997). P. Haynes, *Stratospheric Dynamics*, Annual Review of Fluid Mechanics (2005). G.K. Vallis, *Atmospheric and oceanic fluid dynamics* second edition, Cambridge University Press (2017) R.A. Plumb & A.D. McEwan, *The instability of a forced standing wave in a viscous stratified fluid: A laboratory analogue of. the quasi-biennial oscillation*, Journal of the Atmospheric Sciences (1978) J.R. Holton & R.S. Lindzen, *An updated theory for the quasi-biennial cycle of the tropical stratosphere*, Journal of the Atmospheric Sciences (1972)
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abstract: 'The thermal conductivity $\kappa$ of the iron-arsenide superconductor [Ba$_{1-x}$K$_x$Fe$_2$As$_2$]{} was measured for heat currents parallel and perpendicular to the tetragonal down to 50 mK in magnetic fields up to 15 T. Measurements were performed samples with compositions optimal doping down to into the region superconductivity . residual linear term in $\kappa(T)$ as $T\to 0$ for in-plane inter-plane transport. superconducting gap. However, the superconducting energy gap We propose that gap structure the Fermi surface by the antiferromagnetic order.'
author:
- 'J.-Ph. Reid'
- 'M. A. Tanatar'
- 'X. G. Luo'
- 'H. Shakeripour'
- 'S. René de Cotret'
- 'A. Juneau-Fecteau'
- 'J. Chang'
- 'B. Shen'
- 'H.-H. Wen'
- 'H. Kim'
- 'R. Prozorov'
- 'N. Doiron-Leyraud'
- Louis Taillefer
title: 'Doping evolution of the superconducting gap structure in the underdoped iron arsenide [Ba$_{1-x}$K$_x$Fe$_2$As$_2$]{} revealed by thermal conductivity'
---
Introduction
============
Soon after the discovery of superconductivity in iron-based materials, [@Hosono] it was recognized that a conventional phonon-mediated pairing the high critical temperature [$T_{\rm c}$]{}.[@phonon] of superconductivity in proximity to a magnetic quantum critical point [@Louisreview] magnetically-mediated pairing,[@magneticpairing] a scenario discussed for cuprate and heavy-fermion materials.[@Normanscience] [@Scalapino]
the superconducting gap structure oxygen-free materials with BaFe$_2$As$_2$ (Ba122) as a parent compound.[@Rotter] High-quality single crystals types of dopants to induce superconductivity in the parent hole doping with potassium in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ (K-Ba122),[@Wencrystals] electron doping with cobalt in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ (Co-Ba122),[@Athena; @CB] and iso-electron substitution of arsenic with phosphorus in BaFe$_2$(As$_{1-x}$P$_x$)$_2$ (P-Ba122).[@Kasahara]
Early on, an ARPES study of optimally-doped K-Ba122 found a full superconducting gap on all sheets of the Fermi surface.[@Ding] This was explained $s_\pm$ scenario.[@MazinNature] However, studies of the superconducting gap structure in Ba122 diversity. In , the gap is nodal for all .[@HashimotoScience; @ShiyanRu] In , the gap is isotropic at optimal doping but it develops in both under- and over-doped compositions.[@GordonPRB; @Martin3D; @TanatarPRL; @Reid3D; @Goffryk] In , isotropic at optimal doping,[@Ding; @ReidSUST] but ,[@XGLuo; @MartinK] and ,[@Fukazawa; @Hashimoto; @ShiyanK; @ReidPRL; @Watanabe; @Okazaki]
This diversity in the gap structure competition intra-band and inter-band pairing interactions.[@Chubukovreview; @Hirschfeld-ROPP] we report a study of the superconducting gap structure in using transport measurements,
\[t\] {width="18cm"}
Methods
=======
Single crystals of [Ba$_{1-x}$K$_x$Fe$_2$As$_2$]{} were grown using a self-flux technique.[@Wencrystals] Details of the sample preparation, screening, compositional analysis and resistivity measurements The superconducting [$T_{\rm c}$]{} of samples changes monotonically with $x$. formula [$T_{\rm c}$]{} $=38.5-54~(0.345-x)-690~(0.345-x)^2$. The thermal conductivity was measured in a standard one-heater two-thermometer technique ,[@Reid3D] for two directions of the heat flow: parallel ($Q \parallel c$; $\kappa_c$) and perpendicular ($Q \parallel a$; $\kappa_a$) to the \[001\] tetragonal $c$ axis. The magnetic field $H$ was applied along the $c$ axis. Measurements were done on warming after cooling from above [$T_{\rm c}$]{} in a constant field, to ensure a homogeneous field distribution in the sample. At least two samples were measured for all compositions to reproducibility. Resistivity measurements the upper critical field were performed in [*Quantum Design*]{} PPMS down to 1.8 K.
Results
=======
Electrical resistivity
----------------------
right panels Fig. \[KTv2\], for both $J \parallel a$ and $J \parallel c$. $\rho$(300 K), do not $\mu \Omega$ cm, respectively.[@caxis; @YLiudoping] $\kappa_{\rm N}/T$, via the Wiedemann-Franz law, $\kappa_{\rm N}/T = L_0 / \rho_0$, where $L_0 \equiv (\pi^2/3)(k_{\rm B}/e)^2$.
Thermal conductivity
--------------------
The thermal conductivity samples, , is also displayed in Fig. \[KTv2\]. The data in the top row are for a heat current along , $\kappa_a$, the data in the bottom panels are for the inter-plane heat current, $\kappa_c$. The fits show that the data below 0.3 K are well described by the power-law function $\kappa/T = a + b T^\alpha$. The first term, $a \equiv \kappa_0 /T$, is the residual linear term, entirely due to electronic excitations.[@Shakeripour2009a] The second term is due to phonons, which at low temperature are scattered by the sample boundaries, with $1 < \alpha < 2$.[@Sutherland2003; @Li2008] with [$T_{\rm c}$]{} $=26$ K.[@XGLuo]
how the residual linear term $\kappa_0/T$ evolves as a function of magnetic field $H$, for both in-plane (top panel) and inter-plane (bottom panel) heat current directions. conductivity, $\kappa_{\rm N}/T$, and the magnetic field is upper critical field $H_{\rm c2}$ . the $\left(\kappa_{0}/T\right) / \left(\kappa_{\rm N}/T \right)$, $\kappa_{0} / \kappa_{\rm N}$ for , is plotted as a function of K concentration $x$, $H = 0$ and $H = 0.15$ [$H_{\rm c2}$]{}.
\[t\] ![ Residual linear term $\kappa_0/T$, normalized by the normal-state conductivity, $\kappa_{\rm N}/T$, as a function of The data for in-plane ($J \parallel a$, open symbols) and inter-plane ($J \parallel c$, closed symbols) transport are shown for five K concentrations, indicated by their [$T_{\rm c}$]{} values. For comparison, we reproduce corresponding data for the isotropic $s$-wave superconductor Nb,[@Shakeripour2009a], the multi-band $s$-wave superconductor NbSe$_2$,[@Boaknin2003] and the nodal $d$-wave superconductor Tl-2201.[@Proust2002] []{data-label="KHKn"}](Fig2_Final "fig:"){width="8.15cm"}
\[t\] ![ Left panel: [$H_{\rm c2}$]{} is determined from electrical resistivity measurements.[@YLiudoping] The black square shows the [$H_{\rm c2}$]{} value determined from thermal conductivity measurements on the [$T_{\rm c}$]{} $=7$ K sample. Right panel: [@YLiudoping] []{data-label="Hc2evolution"}](Fig3_Final "fig:"){width="9cm"}
Upper critical field
--------------------
In the left panel of Fig. \[Hc2evolution\], we plot the , for four , as determined from resistivity measurements for $H \parallel c$. For the is sufficient to reach the normal state. For the other dopings, we obtain [$H_{\rm c2}$]{}(0), the value of [$H_{\rm c2}$]{}$(T)$ at $T \to 0$, by linear extrapolation. slope of the [$H_{\rm c2}$]{}$(T)$ curves increases with increasing [$T_{\rm c}$]{}, as expected for superconductors in the clean limit, [@YLiudoping] which holds for K-Ba122 at all dopings. data from a sample with a slightly higher concentration.[@YLiudoping]
Discussion
==========
In the absence of nodes, quasiparticle conduction proceeds by tunnelling between states localized in the cores of adjacent vortices, which grows exponentially as the inter-vortex separation decreases with increasing field,[@Boaknin2003]
borocarbide superconductors.[@borocarbide] NbSe$_2$ (ref. ). See Fig. \[KHKn\] for the data on NbSe$_2$.
![ Top panel: Doping phase diagram of K-Ba122, showing the onset of the superconducting phase (SC) below the critical temperature [$T_{\rm c}$]{} as a function of the K concentration (doping) $x$. Open (closed) red circles give the [$T_{\rm c}$]{} values of the $a$-axis ($c$-axis) samples used in this study. For compositions to the left of the dashed blue line at $x \simeq 0.25$ ([$T_{\rm c}$]{} $\simeq 26$ K),[@caxis; @Avci] superconductivity coexists with antiferromagnetism (AFM). Bottom: Residual linear term in the thermal conductivity $\kappa$ as $T \to 0$, $\kappa_0/T$, plotted as a fraction of the normal-state conductivity, $\kappa_{\rm N}/T$, for both $\kappa_a$ (open symbols) and $\kappa_c$ (closed symbols), for magnetic fields $H=0$ (red) and $H=0.15~H_{c2}$ (black). Error bars reflect the combined uncertainties of extrapolating $\kappa/T$ and $\rho$ to $T=0$, to get $\kappa_0/T$ and $\rho_0$. The red vertical dashed line at marks the end of the superconducting phase. []{data-label="PDKKn"}](Fig4_Final){width="8.5cm"}
\[t\] ![ Sketch of the evolution of the gap structure in K-Ba122 with doping $x$, The gap is isotropic at optimal doping (c). of the gap starts upon entering the where antiferromagnetism (AFM) appears, and coexists with superconductivity (panel b). []{data-label="Sketch"}](Fig5_Final "fig:"){height="3.5cm"}
At high $x$, the gap is isotropic (panel c), meaning that there is no indication of any modulation of the gap with angle. Upon lowering $x$, the acquires a , with a minimum gap $\Delta_{\min}$ along some direction , This explains
In a number of calculations applied to pnictides, the so-called $s_\pm$ state is the most stable. This is a state with $s$-wave symmetry but with a gap that changes sign in going from the hole-like Fermi surface centred at $\Gamma$ ($\Delta_h > 0 $) to the electron-like Fermi surfaces centred at $X$ and $M$ ($\Delta_e < 0 $).[@Wang2009; @Graser2009; @Chubukovanisotropy] Although fundamentally nodeless, the associated gap function strong modulations, depending on details of the Fermi surface and the interactions, possibly leading to accidental nodes. [@Hirschfeld-ROPP] The gap modulation comes from a strongly anisotropic pairing interaction, which is also band-dependent, involving interplay of intra-band and inter-band interactions. It is typically the gap on the electron Fermi surface centred at the $M$ point of the Brillouin zone shows a strong angular dependence within the basal plane. [@Graser2009; @Wang2009] Therefore, the evolution of the gap structure detected here in K-Ba122, from isotropic to modulated , is compatible with the general findings of such calculations.
This is based on the fact that the modulation of the gap and the magnetic order appear at the same concentration, as seen in Fig. \[PDKKn\].
Neutron scattering studies show that antiferromagnetic order in K-Ba122 coexists with superconductivity over a broad range of doping, up to $x\simeq 0.25$ ([$T_{\rm c}$]{} $\simeq 26$ K), and both magnetism and superconductivity are bulk and occupy at least 95% of the sample volume.[@Avci] (The fact that $\kappa_0/T = 0$ for $H=0$ in all our samples This bulk coexistence is deemed [@Parker; @Fernandesspm]
Maiti [*et al.*]{} [@Chubukovreconstruction] showed theoretically that such a reconstruction triggers a strong modulation of the superconducting gap, which develops strong minima, and possibly even (accidental) nodes, at the crossing points. It therefore seems natural to attribute the appearance of gap minima in underdoped K-Ba122 to the onset of magnetic order.
\[t\] ![ (a) Sketch of the evolution of the superconducting gap structure (dashed line) in K-Ba122 as the Fermi surface (solid line) is reconstructed by antiferromagnetic order with a wave-vector $\bf Q$ as drawn.[@Chubukovreconstruction] (b) When the hole (red) and electron (blue) pockets overlap as a result of Fermi-surface reconstruction, an energy gap opens at the crossing points, and this leads to the formation of small crescent-like pieces. Calculations show that this can lead to the development of minima in the superconducting gap, or even nodes.[@Chubukovreconstruction] []{data-label="Sketchfolding"}](Fig6_Final "fig:"){height="3.5cm"}
\[t\] ![ Comparison of the superconducting gap anisotropy thermal conductivity studies in electron-doped Co-Ba122 (left column)[@TanatarPRL; @Reid3D] and hole-doped K-Ba122 (right column, this work). The top panels show the phase diagrams of both materials. and bottom panels show $\kappa_0/\kappa_{\rm{N}}$ vs $x$ for $H=0.15$ [$H_{\rm c2}$]{} and $H=0$, respectively. Open (closed) symbols correspond to transport along $J \parallel a$ ($J \parallel c$). []{data-label="Co-K"}](Fig7_Final "fig:"){width="8.3cm"}
In Fig. \[Co-K\], we compare the evolution of the gap as found in thermal conductivity measurements on the two sides of the phase diagram of BaFe$_2$As$_2$: the electron-doped side (Co-Ba122) and the hole-doped side (K-Ba122). In both cases, the gap is isotropic close to optimal doping, and it develops a strong modulation with underdoping,
Summary
=======
In summary, the thermal conductivity of K-Ba122 in the $T=0$ limit reveals three main facts. First, the superconducting gap at optimal doping, where [$T_{\rm c}$]{} is maximal, is isotropic, with no sign of significant modulation anywhere on the Fermi surface. This reinforces the statement made earlier on the basis of thermal conductivity data in Co-Ba122, [@Reid3D] that superconductivity in pnictides is strongest when isotropic, pointing fundamentally to a state with $s$-wave symmetry, at least in the high-[$T_{\rm c}$]{} members of the pnictide family. Secondly, Third,
Acknowledgements
================
We thank A. V. Chubukov, R. M. Fernandes, P. J. Hirschfeld, D.-H. Lee and I. I. Mazin for fruitful discussions and J. Corbin for his assistance with the experiments.
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abstract: 'I describe the long-standing search for a “smoking-gun" signal for the manifestation of (scale-)chiral symmetry in nuclear interactions. It is prompted by Gerry Brown’s last unpublished note, reproduced verbatim below, on the preeminent role of pions and vector ($\rho$,$\omega$) mesons in providing a simple and elegant description of strongly correlated nuclear interactions. In this note written in tribute to Gerry Brown, I first describe a case of an unambiguous signal in axial-charge transitions in nuclei and then combine his ideas with the more recent development on the role of hidden symmetries in nuclear physics. What transpires is the surprising conclusion that the Landau-Migdal fixed point interaction $G_0^\prime$, the nuclear tensor forces and Brown-Rho scaling, all encoded in scale-invariant hidden local symmetry, as Gerry put, “run the show and make all forces equal."'
address: |
Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France\
mannque.rho@cea.fr
author:
- Mannque Rho
---
In Search of a Pristine Signal\
for (Scale-)Chiral Symmetry in Nuclei {#ra_ch1}
=====================================
Introduction
------------
The currently active theoretical nuclear physics research is to calculate, “ab initio," nuclear properties in an effective field theory starting from chiral symmetry of QCD associated with the light-mass quarks relevant in nuclear interactions. This approach consists of calculating $m$-nucleon potentials for $m\geq 2$ with “irreducible diagrams" to high orders $\kappa \gg 1$ in N$^\kappa$LO in the standard chiral counting and computing many-body nuclear correlations summing “reducible diagrams" in a variety of sophisticated many-body techniques. Here the term “ab initio" refers then to the putative contact with QCD via effective field theory in the spirit of Weinberg’s “folk theorem" [@weinberg-folk-theorem]. In consistency with the folk theorem, the higher one can go up in $\kappa$, while preserving the required conditions such as symmetries etc., the better the calculation will fare in confronting Nature. Of course, given the nature of effective field theory, one is currently limited in scope by rapidly increasing number of parameters as $\kappa$ is increased, but unless the effective field theory in question breaks down – which could happen under certain extreme conditions – with more refined experimental information and increasing computer power, it is reasonable to expect that our understanding of what goes on in nuclear systems will be greatly improved in the years to come. One could say this is a nuclear physics proof of the “folk theorem."
In this note, I would like to describe what Gerry and I undertook, initially in 1970’s, then more intensively in 1980’s and 1990’s, to uncover what we considered as “preeminent features" of chiral symmetry, combined with a presumed scale symmetry, in QCD. Looked at from the present state of art in high-order chiral effective field theory, most, if not all, of those features could very well be captured in high $\kappa$ calculations. Thus one might say, “what’s the big deal?"
Our philosophy has been this: Whatever beautiful phenomena there may be in the processes accessed – and however well they are described – by high-order and consistent chiral perturbation theory, they are likely buried and difficult to single out in full-fledged high-$\kappa$ computations. By identifying the preeminent features by relying on simplicity and intuition, one can reveal in what elegant way Nature works and make certain predictions that are hidden in potentially accurate “ab initio" approaches.
Low-Energy Theorems
-------------------
What one might identify as the first “signal" for chiral symmetry was “seen" in the 1972 calculation by Riska and Brown for the thermal $np$ capture $n+p\rightarrow d+\gamma$ [@riska-brown] where the dominant source for $\sim 10 \%$ meson-exchange effects in the cross section was identified. This calculation was prompted by the observation that the soft-pion theorems given by the current algebra relations, fairly well established then, could give an important contribution to the M1 matrix element figuring in low-energy processes of the $np$-capture type [@chemtob-rho]. Soon afterwards, it was realized that the soft-pion theorems must play an even more important role in weak processes in nuclei. Indeed it was predicted that the exchange of a nearly zero-mass pion could give a lot bigger contribution to the exchange axial-charge matrix elements in nuclear beta decay than in the M1 process [@kubodera-delorme-rho]. This prediction was confirmed, convincingly, by several experiments of first-forbidden beta decay transitions as I will describe below (see for an early discussion, Ref. [@BR-comments]). The corollary to these two (confirmed) observations was that both the EM charge operators and the weak Gamow-Teller operators could not receive contributions from soft-pion exchanges and hence must be subject to higher-order corrections. These observations were made before Weinberg’s 1979 paper [@weinberg-chpt] established that these current algebra terms are the first term in the chiral perturbation expansion, which is the source of the subsequent developments in EFT for QCD and the current lively activity in the nuclear physics community. In the modern language, therefore, the soft-pion terms operative in the M1 and axial-charge operators are the leading exchange-current operators with next-order corrections suppressed by one or two (in the latter case) chiral orders. This was already understood in 1981 when we started [@BR-comments; @MR-Erice] to formulate the chiral counting rule in nuclear interactions motivated by Weinberg’s 1979 paper. It is perhaps fair to say that our work foresaw the arrival of nuclear EFT largely triggered by Weinberg’s influential paper on nuclear chiral effective field theory [@weinberg-nuclearEFT].[^1]
As I shall stress below, the enhanced soft-pion exchange currents offer a [*clear signal*]{} of how chiral symmetry (more precisely scale-chiral symmetry, specified below) manifests in nuclei. This point, reinforced later in the context of what’s known as “BR scaling" as mentioned below, was underlined in our 1981 note [@BR-comments] where it was stated “Meson exchange currents, therefore, probe the structure of the ‘vacuum’ inside the nucleus." I will come back to this matter to argue for a “pristine" signal for scale-chiral symmetry.
Scale-Chiral Symmetry
---------------------
### Vector mesons and scalar meson
When Gerry Brown received the first draft of Weinberg’s 1991 article with a request for comments, his first reaction was that he preferred that the vector-meson degrees of freedom, in particular, that of the $\rho$ meson, be explicit in the Lagrangian, instead of being generated at higher chiral orders as in the chiral Lagrangian with pions only, adopted by Weinberg and in almost all of the current applications. The reason for this was Gerry’s conviction that certain properties of nuclear forces could be most economically and efficiently captured if the vector mesons were treated explicitly. He was persuaded on this by a variety of nuclear observables connected to the nuclear tensor force [@BR-tensorforces], which is one of the most important component of nuclear forces, in particular, spin-isospin response functions, the vector dominance and most significantly the effect of vacuum change in dense nuclear medium.
Although Gerry relied mostly on intuitive reasoning at the early stage, the most rigorous way presently available to address the problems involved is now recognized to be to resort to flavor gauge symmetry for the vector mesons $V=(\rho,\omega)$ supplemented by scale symmetry for a scalar degree of freedom, with the vector and scalar degrees of freedom making up the crucial ingredients of the argument.
It is now pretty convincingly established (as is summarized for example in the review [@HRW]) that up to nuclear matter density, most, if not all, of nuclear properties are well described by chiral EFT[^2]. This means that relevant fluctuations with the quantum numbers of vector and scalar mesons, if important, could be properly captured in higher-loop terms in chiral EFT. However such an EFT must break down when the energy scale probed becomes comparable to the mass of heavier mesons, possibly at some density above normal. This could happen if the “effective" vector meson mass in medium went down as suggested in the structure of the tensor forces [@BR-tensorforces], and more seriously at high density if the mass went to near zero according to the “vector manifestation" (VM) [@HY:PR]. One way the dropping vector meson mass can be handled is to treat the vector meson as a local gauge boson. In fact, it is only in gauge symmetry that one can take the $\rho$ meson as light and gets, at the leading order, the KSRF relations and vector dominance that agree well with experiments. In addition, one can set up a chiral perturbative scheme, if the vector meson mass is formally considered as light as the pion mass (although it is $\sim 6$ times the pion mass in the vacuum), with a systematic chiral expansion [@HY:PR] that works fairly well in the vacuum. In medium it would work even better, the lighter the $\rho$ mass dropped as predicted in HLS at high density. There are only two cases known in gauge theory where the notion of “light" (strong-interaction) vector meson makes sense; one is the case of hidden gauge symmetry we are dealing with and the other is a supersymmetric QCD in some special parameter space [@komargodski].
An equally important degree of freedom in nuclear physics is a scalar meson of mass $\sim 600$ MeV that [*effectively*]{} provides the attraction that binds nuclei. In the particle data booklet, there is a scalar of comparable mass, namely, $f_0(500)$, with a large width. In the view Gerry and I have advocated since 1991, the scalar is a dilaton resulting from spontaneously broken scale symmetry. We consider it as “light" in the same sense as the $\rho$ mass ($m_\rho\approx 770$ MeV) is “light." This is an assumption that of course needs still to be confirmed by higher-order calculations (in the scheme mentioned below). In fact it is a long-standing controversy, with no clear consensus, whether such a scalar – that we will denote as $\phi$ – can be associated with scale invariance. On the one hand, lattice calculations indicate there can be an infrared (IR) fixed point within the conformal window but at large number of flavors $N_f\sim 8$. This is the case for which the Higgs may be identified as a dilaton [@yamawaki-higgs]. An active work on this issue is in progress in anticipation of further discoveries at LHCb [@yamawaki-higgs; @yamawaki-technirho]. But so far there is no firm nonperturbative evidence, lattice or otherwise, for an infrared (IR) fixed point for $N_f\leq 3$ that we are concerned with in QCD. There is therefore a school – call it “no-go school" – that dismisses the notion of a dilaton scalar for $f_0(500)$. On the other hand, there is a conjecture that $f_0(500)$ could be interpreted as a dilaton with an IR fixed point with the $\beta$ function for the QCD gauge coupling $\alpha_s(=g_{QCD}^2/4\pi)$ vanishing and hence the trace of the energy-momentum tensor $\theta_\mu^\mu$ vanishing in the chiral limit [@CT]. In this scheme, scale symmetry and chiral symmetry merge into what is called “scale-chiral symmetry" [@CT] with their scales locked to each other, $4\pi f_\chi\approx 4\pi f_\pi$ where $f_\chi$ is the dilaton decay constant. At present, there is no rigorous no-go theorem against scale-chiral symmetry either, so that possibility cannot be ruled out on theoretical ground. Furthermore this scheme has a great advantage not only for particle physics (such as, among others, giving a simple explanation for the famous $\Delta I=1/2$ rule[^3]) but also for nuclear physics where the dilaton scalar $\phi$ can provide a systematic scale-chiral expansion including a scalar meson, generalizing the standard chiral expansion. It can provide justification to the long-standing use – with success – of a local scalar field for nuclear potentials (e.g., Bonn potential), Walecka-type mean-field models etc. In addition, it offers an additional procedure to calculate its mass, width etc. at low loop-orders both in and out of medium. Perhaps more importantly, it would provide a more efficient method to do calculations where strange hadrons, such as hyperons and kaons, relevant for compact-star matter – as suggested in the counting rule in [@CT] – are involved. We will see below that this scalar as a dilaton plays a key role in what Gerry and I have been doing all along.
The question remains, however, as to whether the failure for the lattice calculations to “see" the putative IR fixed point will not invalidate what I will be discussing below. I have no clear answer to this. In my opinion, one way to address this issue is to view the scale symmetry we are exploiting is an emergent symmetry in a way analogous to hidden local symmetry in baryonic medium. The $U_A (1)$ anomaly offers another analogy.
The “no-go school" argument against a possible IR fixed point in QCD is anchored on the trace anomaly which cannot be turned off in the vacuum. The trace anomaly is due to the regularization required for the quantum theory, or put differently, the dimensional transmutation, and is renormalization-group invariant[^4]. Similarly the $U_A(1)$ anomaly cannot be turned off. It can be tuned to zero if the number of colors $N_c$ is tuned to $\infty$. However, in Nature, $N_c\ll \infty$, so the axial anomaly is there to stay. Nonetheless it has been argued that the $U_A(1)$ symmetry could be restored at high temperature [@pisarski-wilczek].[^5] In a similar vein, it is possible that the trace anomaly could be turned off, in the chiral limit, by density with the symmetry exposed at some high density. This possibility is being explored [@PKLR]. Since I am not concerned here with densities much higher than that of normal nuclear matter, I will not go into it.
### Hidden symmetries
It should be recognized that [*both*]{} the local symmetry for the $\rho$ and the scale symmetry for the $\phi$ are “hidden" symmetries: They are not visible or may even be absent in QCD proper. The hidden local symmetry for the $\rho$ becomes manifest only when the $\rho$ mass is driven toward – but not exactly onto – zero [@HY:PR][^6]. Since this flavor local symmetry is not present in QCD proper, the $m_\rho=0$ limit may not be accessible in QCD. I will however suggest that it can emerge via strong nuclear correlations in dense medium. As for scale symmetry, it can be shown that the familiar linear sigma model has the scale symmetry [*hidden*]{} in it. It has been shown [@yamawaki-technirho] that by dialing one parameter $\lambda$ in a potential term in the standard linear sigma model (which is equivalent to the standard Higgs model) from $\infty$ to 0, the Lagrangian can go from the non-linear sigma model with no conformal (scale) symmetry to a conformal-invariant nonlinear sigma model. With hidden local gauge fields suitably incorporated, the latter turns into scale-invariant hidden local symmetry (sHLS for short). Below I will use sHLS with the scale symmetry spontaneously broken by a potential $V(\chi)$ where $\chi$ is what is referred to as “conformal compensator field" connected to the dilaton $\phi$ (defined below). What the baryon density does is to drive the parameter between $\lambda=\infty$ and $\lambda=0$ and expose the hidden symmetries somewhere along the way.
Scale-Invariant Hidden Local Symmetric Nuclear EFT
--------------------------------------------------
Let me start with the mesonic Lagrangian denoted as sHLS that combines scale symmetry and hidden local symmetry that can be written in a schematic form: \_[sHLS]{}=[L]{}\_0 (U, , V\_) +[L]{}\_[SB]{} (U,)\[sHLS\] where the conformal compensator field $\chi$ is related to the dilaton field $\phi$ as =f\_e\^[/f\_]{} and the chiral field is given by the familiar form $U=e^{i\frac{2\pi}{f_\pi}}$. In this article I will deal with flavor $SU_f(2)$ and assume $U(2)$ symmetry for the vector mesons $V=(\rho,\omega)$.[^7] The mass dimension-one $\chi$ transforms linearly under scale transformation while $\phi$ transforms nonlinearly as the pion field $\pi$ does under chiral transformation, that is, as a Nambu-Goldstone. The first term (\[sHLS\]) is of scale dimension 4, so gives scale-invariant action and also HLS (chiral) invariant. The second term contains the pseudo-scalar meson mass term, hence breaking explicitly the chiral symmetry, and a potential that breaks scale symmetry both explicitly (due to the trace anomaly) and spontaneously. Although most of my discussions in application to nuclei can be done in the chiral limit, the chiral symmetry breaking pion term will be necessary to fix the property of the pion decay constant which sets the density-scaling behavior in nuclear matter.
Expanded in fields, the Lagrangian (\[sHLS\]) in unitary gauge in hidden gauge symmetry is, to ${\cal O}(p^2)$ chiral order, of the form \_[\_s]{} &=& \^2 (\_U\^\^U) + \^3 v\^3 (U+U\^)\
& & - a\^2 \^2\
& & - \_ \^ - \_ \^ +12 \_\^+ V() \[shls\] where $a$ is related to the ratio $f_\pi/f_\sigma$ (where $f_\pi$ is the pion decay constant and $f_\sigma$ is the decay constant of the would-be Goldstone boson Higgsed to give the vector meson mass), $\kappa=\chi/f_{\chi}$ with $f_{\chi}=\la 0|\chi|0\ra$, $l_\mu=\del_\mu\xi\xi^\dagger$, $r_\mu=\del_\mu\xi^\dagger\xi$ with $\xi=\sqrt{U}$ and $v$ is a constant of mass dimension 1.
### Intrinsic density dependence
In order to apply (\[sHLS\]) to baryonic matter, baryon degrees of freedom are needed. There are two ways to bring in baryon fields. One way, perhaps most consistent with QCD, is to generate baryons as skyrmions of the mesonic Lagrangian. There is a work along this line with some progress. At present, however, it is not developed well enough to quantitatively describe nuclear processes. The alternative is to put baryon fields explicitly in consistency with the folk theorem, staying as faithful as possible to the symmetries involved. I will follow this approach below. Let me call the baryon-implemented effective Lagrangian bsHLS.
Since the ultimate aim is to probe the density regime where the vector mesons, i.e., $\rho$ and $\omega$, and the scalar are relevant, perhaps overlapping the regime where the explicit QCD degrees of freedom may intervene, the strategy to take is to have the EFT matched to QCD at the scale where the cutoff for the EFT is set. In [@HY:PR], the matching was done by means of the vector and axial-vector correlators, tree order in HLS (i.e., “bare" HLS) and OPE in QCD, at $\Lambda_M=\Lambda_\chi\sim 4\pi f_\pi$. With sHLS, the energy-momentum tensor needs also to be matched. What the matching does is then to endow the “bare" parameters of the EFT Lagrangian with dependence on the condensates, i.e., the quark condensate $\la\bar{q}q\ra$, the gluon condensate $\la G^2\ra$ and mixed forms etc. Since those condensates depend on the vacuum, if the vacuum is modified by density, then they will necessarily depend on density. This dependence will then render the parameters of the Lagrangian [*intrinsically*]{} density-dependent. This density dependence, of QCD origin, is called “intrinsic density dependence" (IDD for short).
Given the Lagrangian so matched to QCD, then one has an effective Lagrangian, the “bare" parameters of which are density-dependent, with which one can do quantum theory. The IDD so defined is related – but not identical – to what is known in the literature as “Brown-Rho scaling" (BR scaling for short)[^8].
The most efficient and flexible approach, presently available, to treat many-body nuclear dynamics with bsHLS is the renormalization-group approach employing $V_{lowk}$. It involves “double decimations" [@BR:dd]. For nuclear processes, one should be able to do the decimation from a cutoff $\tilde{\Lambda}$ somewhat lower than $\rho$ mass. From the vector meson mass scale, the “bare" parameters of the EFT Lagrangian – except for $f_\pi$[^9]\[6\] – do not flow to the scale picked, $\tilde{\Lambda}$, from which the decimation is to be done, so it should be justified to lower the cutoff to $\tilde{\Lambda}$ without modifying the “bare" Lagrangian. In practice, the first decimation is made from $\sim (2-3)$ fm$^{-1}$ to obtain the $V_{lowk}$ and then the 2nd decimation consists of doing Fermi-liquid calculations with this $V_{lowk}$ [@holt-fermiliquid] .
In HLS taken to ${\cal{O}}(p^2)$ in the chiral (derivative) counting, there are only three parameters $g$, $f_\pi$ and $a$. In the skyrmion description, nucleon properties including couplings to the vector mesons involved do not require additional parameters. With the scalar field included, there is of course an additional parameter, namely, $f_\chi$. However the locking of scale symmetry and chiral symmetry makes $f_\chi$ equal to $f_\pi$, so it does not require additional IDD.
Now the question is how these parameters vary as a function of density and how their dependence affects hadron masses and coupling constants of the “bare" Lagrangian?
The answer to this question requires knowing whether there is any phase change in the matter structure as density increases. It is obvious that the parameters will not necessarily continue moving smoothly in density. For example, at some density, QCD degrees of freedom could enter. In the skyrmion description of baryonic matter, there is a robust topological transition from a skyrmion matter to a half-skyrmion matter at a density around $n_{1/2} \sim (2-3)n_0$. In fact this transition is in a sense equivalent to what is called “quarkyonic" in which quark degrees of freedom figure at about the same density [@fukushima; @PKLR]. In terms of the “bare" Lagrangian, such a transition would imply changes in the density dependence of the bare parameters. I won’t go into what happens after such transition which matters for compact star structure – which has been studied [@PKLR], so let me focus on the density regime $n<n_{1/2}$.
First consider the $\rho$ mass. The “bare" mass at the matching scale $\Lambda_M\sim \Lambda_\chi$ is given by m\_\^2=af\_\^2 g\^2.\[ksrf\] What is remarkable about this relation, known as KSRF formula, is that it holds to all orders of loop corrections with the HLS Lagrangian taken to ${\cal O}(p^2)$ – and believed to be valid at higher chiral (derivative) orders – with corrections of ${\cal O}(m_\rho^2/\Lambda_\chi^2)$ [@ksrf-allorders; @HY:PR] . This expression therefore becomes more accurate, the lighter the vector mass becomes as is predicted in dense medium [@HY:PR]. This means that the “bare" $\rho$ mass in the Lagrangian will always be of the form of (\[ksrf\]), regardless of the cutoff for decimation, with the IDD reflecting [*entirely*]{} the effect of density. The explicit calculation of the EFT-QCD matching formulas shows that both $g$ and $a$ depend quite weakly on the quark and gluon condensates [@HY:PR] and hence the density dependence will be mainly in the pion decay constant, hence in the dilaton condensate since $f_\pi\approx f_\chi$. Therefore the only scaling factor in the density regime $n\leq n_{1/2}$ is f\_\^/f\_f\_\^/f\_(n). Thus via (\[ksrf\]) m\_\^/m\_. It follows from the bare Lagrangian (\[shls\]) with the expansion $\chi=\la 0^\ast|\chi|0^\ast\ra +\chi^\prime$ (where $0^*$ is the in-medium vacuum) that m\_H\^/m\_H(n)\[BR\] for $H=N,\rho,\omega,\phi$ assuming $U(2)$ symmetry for $(\rho,\omega)$ for $n<n_{1/2}$. The pion mass, with broken chiral symmetry – and hence broken scale symmetry, scales differently, m\_\^/m\_.\[BRpi\] Equations (\[BR\]) and (\[BRpi\]) are the same as the expressions derived in 1991 using the skyrmion model. Note that they follow not directly from chiral symmetry but from scale symmetry locked to chiral symmetry. In other words, [*it is the dilaton condensate that “runs" the show*]{}. This means that the symmetry involved is the “scale-chiral symmetry" as defined precisely in [@CT].
### Soft-pion signal for scale-chiral symmetry
The first “pristine" signal for scale-chiral symmetry in nuclei is in the axial-charge beta decay process in nuclei. It is in the first-forbidden beta transition of the form I(0\^-)F(0\^+) +e\^- +|\_e T=1. This beta decay process from the initial nucleus $I$ to the final nucleus $F$ goes via the axial change operator $A_0^a$.
As first recognized in 1978 [@kubodera-delorme-rho] from current algebras and later confirmed in chiral perturbation theory [@MR-chpt], the exchange axial-charge two-body operator receives a large contribution from a soft-pion exchange term, with the next contribution suppressed by two chiral order. Furthermore the leading one-body operator, being first-forbidden, is kinematically suppressed. Therefore the two-body “correction" term is expected to contribute to the decay at an order comparable to or bigger than the “leading" single-particle operator. Written in effective one-body operator, the corresponding Feynman diagram is of the form Fig. \[axial-charge\].
[![Effective single-particle soft-pion-exchange axial charge operator. The solid line is the nucleon and the wiggly line the external weak field. The right vertex $A_\mu^a \pi$NN is large for the time component $A_0^a$ and is suppressed for the space component $\mathbf{A}^a$.[]{data-label="axial-charge"}](exchange-charge.pdf "fig:"){width="6cm"}]{} -0.5cm
The sum of the one-body and two-body axial charge operators with the IDD incorporated into the constants of the EFT Lagrangian has the extremely simple form [@kk-br] A\_0\^a=g\_A (1+) where $\mathbf{p}$ is the nucleon momentum, $R$ is the ratio of the matrix element of the two-body operator ($M_2$) to that of one-body operator ($M_1$). The factor $\Phi$ corresponds to the IDD dependence in the EFT Lagrangian. It may be that the numerical values of $M_1$ and $M_2$ depend on how nuclear wave functions are calculated. However the ratio $R$ is highly insensitive to it. Thus one can take either Fermi-liquid model or Fermi-gas model in place of more sophisticated wave functions. One gets essentially the same value. What is significant is that the $R$ is big $R\sim {\cal O}(1)$, and varies slowly in density, reflecting the robust nature of the soft-pion exchange.
One can readily make a simple estimate of what comes out. Let’s look at the quantity defined and measured experimentally by Warburton [@warburton] \_[MEC]{}= (1+).\[epsilon\] This represents the enhancement factor due to both the exchange-current contribution in the transition matrix element relative to the single-particle operator contribution [*and*]{} the IDD. Take the lead $A=205-212$ nuclei for which data are available which have densities comparable to nuclear matter density. Calculating $R$ in Fermi-gas model, it is found at nuclear matter density $n_0$ that $R(n_0) \approx 0.5$. Now from pionic atom data, one has $\Phi (n_0)\approx 0.8$ [@yamazaki]. Thus $(\ref{epsilon})$ gives $\epsilon_{MEC}\approx 2.0$. This agrees very well with the measured enhancement factor $\epsilon^{exp}_{MEC}=2.01\pm 0.05$. Taking into account the density dependence of $\Phi$ and $R$, one can also reproduce the observed enhancements in $A=12$ and $A=16$ systems, $\epsilon^{exp}_{MEC} (A=12)=1.64\pm 0.05$ [@minamisono] and $\epsilon^{exp}_{MEC} (A=16)\approx 1.7$ [@garvey]. Although the theoretical estimate as well as the experimental values for $\epsilon_{MEC}$ are rough, this is a clear evidence for both the soft-pion and IDD effects: The density dependence of $\epsilon_{MEC}$ is found to be consistent with what’s predicted of $\Phi$ and $R$.
Two remarks are in order here.
One is the crucial role of soft pions. It zeroes in on the Nambu-Golddstone-boson nature of the pion with a mass nearly vanishing (on the strong interaction scale). For very low-energy processes, $E\ll m_\pi\approx 140$ MeV, according to the standard lore of EFT, one may be justified to integrate out the pions leaving only the nucleons as relevant degrees of freedom. One then gets what is called “pionless effective field theory" ($\not{\pi}$EFT), which is generally thought to be consistent with the “folk theorem." With the pions absent, the resulting Lagrangian is blind to chiral symmetry but it does not mean chiral symmetry is violated. So does it always work? The question is: Is $\not{\pi}$EFT applicable to the axial-charge transition which receives big contributions from soft pions? If the pion mass were strictly zero for which the soft-pion theorems hold, clearly the pion could not be integrated out from the chiral Lagrangian. Thus it seems inevitable that the $\not{\pi}$EFT Lagrangian with the pions gotten rid of would miss the soft-pion effect (at the $\kappa=1$ order) and hence fail even if it were applicable to Gamow-Teller transitions.
The other remark is on the possibility to do precision calculations and test the combined enhancement by soft pions and IDD. The operators are well defined to the leading chiral order with higher-order terms strongly suppressed, so given accurate wave functions, one could then do a precision test of the scaling parameter $\Phi$. Recent developments on “ab initio" approaches with sophisticated many-body techniques could be exploited to calculate $\epsilon_{MEC}$ with accurate error estimates for both theory and experiment.
What runs the show in nuclear interactions?
-------------------------------------------
Let me now come to the startling, if not puzzling, observation – the main thrust of Gerry’s note – that the $\pi$, $\rho$, $\omega$ and $\phi$, the principal degrees of freedom of bsHLS at mean-field, play the dominant and even clear-cut role in nuclear dynamics. For this part, reading Gerry’s note (added below) will be helpful.
Given the bsHLS Lagrangian, one could perhaps perform a (covariant) density functional analysis for the nuclear ground-state properties along the line set up by the Hohenberg-Kohn theorem for atomic/molecular physics and chemical physics. The currently popular covariant energy-density functional approaches employed in nuclear theory typically have six or more free parameters. In contrast, up to at least nuclear matter density, the bsHLS Lagrangian was found to have basically only one parameter [^10] governing the d-scaling factor $\Phi$ associated with the dilaton condensate. It would be extremely interesting to see how an “ab initio" covariant density functional given by bsHLS compares with the standard approach with many more parameters. However one is ultimately interested in the equation of state relevant to compact stars. For this the density functional approach does not seem appropriate.
Instead the strategy followed below is the double-decimation renormalization-group (RG) procedure in terms of $V_{lowk}$ [@BR:dd], which is to start with the first RG decimation to go from the effective cutoff $\tilde{\Lambda}$ down to the scale at which the $V_{lowk}$ is gotten. For this, in principle, the “irreducible graphs" are to be summed to high orders in scale-chiral expansion to give the potential with which to do the decimation. In doing this, the IDDs enter. In practice, BonnS-type potentials are used and the cutoff is put at (2-3) fm$^{-1}$, somewhat lower than $\tilde{\Lambda}$. Performing the second decimation corresponds to doing Landau Fermi-liquid theory with $V_{lowk}$ as formulated in [@vlowk; @holt-fermiliquid]. The result is then the set of Fermi-liquid fixed point parameters, i.e., the effective masses and interactions. I will focus on the Fermi-liquid parameters and the fixed point quasiparticle interactions in which the pion and the $\rho$ play the main role, in particular, the Landau parameter $G_0^\prime$ and the tensor forces.
### The EELL effect or $G_0^\prime$
In his note, Gerry Brown presents strong intuitive arguments, drawing from the previous works [@vlowk; @holt-fermiliquid], to show that the Landau parameter $G_0^\prime$, coming from the pion and $\rho$ exchanges, is by far the largest among the Fermi-liquid interactions and dominates the Kuo-Brown effective interactions at mean-field level, with higher order terms suppressed [@holtetal]. How and to what extent the suppression takes place more generally is yet to be worked out. However in the Wilsonian RG approach, the beta function for the quasiparticle interactions should tend to zero in the large $N$ limit where $N$ is related to the Fermi momentum $k_F$, as beautifully explained in [@shankar]. Gerry relies on the double-decimation $V_{lowk}$ approach with BR scaling implemented to argue that the Kuo-Brown interaction, with just one bubble, which is classical in nature, has the most of the physics in it, giving an extremely simple interpretation of why and how the high-order core polarization contributions are suppressed as found in [@holtetal]. Both the scalar $\phi$ and the vector $\omega$ figure in bringing the interaction $\sim G_0^\prime (\tau_1\cdot\tau_2)(\sigma_1\cdot\sigma_2)$ to the fixed point, the former “holding the ball of pions together" and the latter providing the short-range repulsion giving rise to the Ericson-Ericson-Lorentz-Lorenz effect.
### Nuclear tensor forces
The exchange of $\pi$ and $\rho$ with the parameters endowed with IDD of the EFT Lagrangian gives the effective in-medium nuclear tensor forces. While the pion tensor is more or less unaffected by density, an effect which could be attributed to protection by chiral symmetry, the $\rho$ tensor, with the dropping mass, increases in magnitude with a sign opposite to that of the pion tensor as density increases. Because of the cancellation between the two, the net tensor force strength gets weaker due to the BR scaling at increasing density. The attraction in the tensor channel goes nearly to zero when density reaches 2$n_0$. This has been well known. In fact precisely this effect has been exploited with an impressive success to explain the long C14 life-time [@holt-c14]. What’s involved in this process is a delicate density-dependent cancelation in the Gamow-Teller matrix element, which nearly vanishes in the density regime involved. This is a spectacular signal of working of the BR scaling, although it is not as clear-cut an evidence as in the case of the first-forbidden beta decay process described above. As noted by Gerry, in the context of double decimation, the effect of BR scaling here is equivalent to the effect of contact three-body forces in the sense described in footnote 8.
I will now propose that there is a possibility of “seeing" the IDD scaling – not just BR scaling – in nuclei via the tensor forces. I would like to describe this using the $V_{lowk}$ formalism.
In a series of beautiful papers [@otsuka], Takaharu Otsuka showed that the tensor forces played a remarkable role in the “monopole" matrix element of the two-body interaction between two single-particle states labeled $j$ and $j^\prime$ and total two-particle isospin $T$ V\_[j,j\^]{}\^T= . What is special with this matrix element is that it affects the evolution of single-particle energy; \_p (j)=12(V\_[jj\^]{}\^[T=0]{} +V\_[jj\^]{}\^[T=1]{})n\_n(j\^) where $\Delta\epsilon_p (j)$ represents the change of the single-particle energy of protons in the state $j$ when $n_n(j^\prime)$ neutrons occupy the state $j^\prime$. It turns out that the matrix elements $V_{jj^\prime}$ and $V_{j^\prime j}$ have opposite sign for the tensor forces if $j$ and $j^\prime$ are spin-orbit partners.
Let me summarize the salient features of the shell evolution connected to the tensor force found by Otsuka:
Otsuka works with the phenomenological potential Av18’ and the one given by ChPT at N$^3$LO, both treated à la $V_{\rm lowk}$. Other realistic potentials are found to give the same results. He varies the cutoff $\tilde{\Lambda}$ and finds $\tilde{\Lambda}$ independence around 2.1 ${\rm fm}^{-1}$.
Otsuka calculated the shell evolution in the pf and sd regions by including high-order correlations using the Q-box formalism to 3rd order. While he finds the central part of the potential strongly renormalized by high-order terms, the tensor forces are left unrenormalized, leaving the “bare" tensors more or less intact. Shown in Fig. \[pfshell\] is Otsuka’s result (copied from his paper) in the pf shell region.
![Tensor forces in AV8’ interaction, in low-momentum interactions in the pf shell obtained from AV8’, and in the 3rd-order Q$_{box}$ interaction for (a) T=0 and (b) T=1. []{data-label="pfshell"}](Fig4.pdf){width="8cm"}
The result shows that the sum of the short-range correlation and medium effects as taken into account by the 3rd order Q-box leaves the bare tensor force unchanged. Otsuka looks at a variety of other realistic potentials, both phenomenological as well as ChPT at N$^3$LO, and finds, remarkably, that they all give the same result, showing that the effect is robust. It implies V\_[low k]{}\^[tensor]{}=(\[V\_[low k]{}\^[tensor]{}\], ) 0.\[beta\] Some experimental data are available, e.g., Jahn-Teller effect [@otsuka], that verify the tensor implemented calculations to be in agreement with experimental data quite well. Forthcoming experiments in RIB accelerators promise to reveal more surprising results.
The result (\[beta\]) says that the beta function is zero [*both*]{} in the first decimation and in the second. The latter could perhaps be understood as the net tensor force at a given density being at the Landau fixed point with all correlation effects suppressed. However the former is surprising since it implies that the tensor force does not RG-flow in the vacuum. Why the net tensor force is free of all strong interaction effects, in and out of medium, is mysterious. Learning of Otsuka’s results, Tom Kuo kindly performed a $V_{lowk}$ analysis in the vacuum. Shown in Fig. \[tensor-free\] are his results of the tensor potentials in momentum space $V^{tensor} (k_1,k_2)$ for $k_1=k_2$ and $k_1\neq k_2$. The $V^{tensor}_{lowk}$ potential is identical to the “bare" BonnS tensor potential, independent of $\tilde{\Lambda}$ .
[ ![“Bare" BonnS and $V_{low k}$ tensor forces in matter-free space. Courtesy of Tom Kuo. []{data-label="tensor-free"}](Kuo1.pdf "fig:"){width="5.5cm"} ![“Bare" BonnS and $V_{low k}$ tensor forces in matter-free space. Courtesy of Tom Kuo. []{data-label="tensor-free"}](Kuo2.pdf "fig:"){width="5.5cm"}]{}
Seeing “pristine" signals
-------------------------
Let me close this note with what “pristine" signal could mean for chiral symmetry (or more precisely scale-chiral symmetry). At the early stage of dilepton production in relativistic heavy-ion collisions, Gerry imagined that it would mean the $\rho$ mass going to zero at the critical temperature $T_c$ (in the chiral limit). Gerry’s idea was that in medium the $\rho$ mass, rather than the quark condensate, is the relevant order parameter to measure for chiral restoration, and in HLS which is a natural framework for addressing the issue, this meant going toward the VM fixed point as $T_c$ is approached. However the VM fixed point with an enhanced symmetry turns out not to be in QCD in the vacuum [@georgi; @HY:PR]. It thus could make sense – if at all – only as an emergent phenomenon. Furthermore HLS predicts that in approaching the VM fixed point, while the $\rho$ mass could go to zero with a vanishing width, the photon tends to largely decouple from the dropping-mass $\rho$. This is quite unlike seeing the $\rho$ in the vacuum by measuring the dileptons of the given invariant mass of $\rho$ with a detector in the vacuum. In our view, dileptons in heavy-ion processes are not at all a suitable probe for scale-chiral symmetry: the $\rho$ meson produced in the process is so strongly distorted by background nuclear correlations in the medium so that the signal for the $\rho$ meson carrying the order parameter subject to the VM, when measured with a detector outside of the medium, will be like a “needle in the haystack" as Gerry and I, with our collaborators, have argued.
So how does one go about “seeing" the signal?
I have argued in this note that “seeing" (scale-)chiral symmetry in action in nuclei is much like “seeing" the meson-exchange currents in nuclei. Three such signals for scale-chiral symmetry in nuclei are described.
The first is the combined effect of soft pions and IDD, giving a whopping factor of $\sim 4$ effect in the decay rate. It reveals [*both*]{} the presence of meson-exchange currents [*and*]{} the influence of scale-chiral symmetry manifested by the excitation of pions. It further shows that pions cannot be integrated out for certain processes – such as first-forbidden transitions - that single out soft-pion dominated effects in near zero-energy processes.
The second is the Landau parameter $G_0^\prime$ dominated by the $\pi$ and $\rho$-meson exchanges in strong correlations with the scalar and $\omega$ mesons (binding and short-range) which in Gerry’s words, runs the “show" in nuclear dynamics.
The third is the non-renormalization of the net tensor force in and out of medium, offering the possibility of a pristine probing for IDD by zeroing in on those processes that are controlled by the (net) tensor force and by dialing the density of the system. Of course how to dial the density is a big open issue.
Finally all these are simple and elegant aspects of nuclear interactions which could be sharpened in “ab initio" precision calculations – in progress and to come.
### Acknowledgments {#acknowledgments .unnumbered}
I am grateful for fruitful discussions and collaborations with Masa Harada, Tom Kuo, Hyun Kyu Lee, Yong-Liang Ma and Won-Gi Paeng and would like to acknowledge extensive comments and tutorials from Rod Crewther, Lewis Tunstall and Koichi Yamawaki on scale symmetry in gauge theory. I would particularly like to thank Tom Kuo for his help on the nuclear tensor force at its putative fixed point.
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**Note Added: Unpublished Paper by G.E. Brown**
**“The Dominant Role of the EELL Interaction in Nuclear Structure"**
**Forewords**
0.5cm [*This is Gerry’s last manuscript, hand-written and as he used to do, faxed to me on April 4th, 2007, the cover letter and the first page of which are scanned and put in Fig. \[fax\]. Below, the article is transcribed verbatim, totally unedited and unrevised. It contained no abstract, but a short statement on the cover letter conveyed the essence of the idea. Referring to the Ericson-Ericson-Lorentz-Lorenz (EELL or E$^2$L$^2$ for short) interaction, which captures the RG fixed point interaction $g_0^\prime$ in Landau-Migdal Fermi liquid theory, Gerry stated “I believe this is as universal as Brown-Rho scaling. It’s magical how E$^2$L$^2$ runs the show and makes, together with B-R scaling, all forces equal." In the contribution to this volume given above, I describe how Gerry’s ideas can be synthesized into the more recently developed notion of scale-invariant hidden local symmetry in nuclear interactions.*]{}
![Cover letter (left) and title page (right) of the faxed paper.[]{data-label="fax"}](titlepage.pdf){width="6.5cm"}
### Introduction {#introduction-1 .unnumbered}
We develop the renormalization group description of Schwenk et al. [@schwenk] which introduced the effective interaction $V_{lowK}$, The effective pion energies in terms of low energy scale Fermi-liquid interactions G\_0\^=1.00.2, G\_1\^=00.2 2/3 of which arise in the nucleonic sector and 1/3 in the $\Delta$-hole sector. These appear as the Ericson-Ericson Lorentiz-Lorenz interaction, in the sense that they occur in the pionic channel due to the short-range repulsion between nucleons; i.e., this inteaction has the quantum numbers given by pion exchange.
These same interactions arise from Brown-Rho scaled Fermi-liquid theory [@holt] as repulsive interactions between nucleons, replacing the attractive piece of the zero range pion-nucleon interaction which is removed by the short-range repulsion between nucleons. The effective $\vec{\sigma}_1\cdot\vec{\sigma}_2 \mathbf{\tau}_1\cdot\mathbf{\tau}_2$ component of the interaction found by Holt et al. [@holt] is not only the strongest component of the effective interaction, but is exactly that found by Schwenk et al. [@schwenk]. The long-range scalar exchange through the $\sigma$-meson and coupled two-pion exchange tends to hold the ball of pions together.
As far as we can see, were we to include the $\Delta$ with coupling constant $g_{N\Delta}^\prime=0.3$ and the double $\Delta$ with $g_{\Delta\Delta}^\prime=0.3$ then we would have a description in the double-decimation decimation of Brown and Rho [@BR:DD] dual to that of Schwenk, Brown and Friman [@schwenk] of the screening by the $\Delta$-hole channel of the screening of the interaction in the pionic interaction. We return to this later.
### Nuclear Matter with Brown-Rho Scaled Fermi-liquid Interactons {#nuclear-matter-with-brown-rho-scaled-fermi-liquid-interactons .unnumbered}
We believe that it is useful to review the work of Holt et al. [@holt] quantitatively in comparison with that of Schwenk et al. [@schwenk] because the double decimation is essentially the same as the latter with introduction of three-body forces. In any case, the substantial improvement in going over to the double decimation from the usual Fermi-liquid approach in the long wave-length limit should serve a stimulus for pursuing this direction. We reproduce Table 4 of Ref.[@holt] as Tabel 1 below.
\[table1\]
In Table \[table1\] we show the effective mass, compression modulus, symmetry energy, and anomalous orbital gyromagnetic ratio for the Nijmegen I ($V_{NI}$) and II ($V_{NII}$) and CD-Bonn ($V_{CDB}$) potentials with the in-medium modification à la Brown-Rho. We also show for comparison the results from the Nijmegen 93 ($V_{N93}$) one-boson exchange potential, which has only 15 parameters and is not fine-tuned separately in each partial wave. The iterative solution is in better agreement with all nuclear variables. The anomalously large compression modulus in the CD-Bonn potential results almost completely from the presence of an $\omega$ coupling $g_{\omega NN}^2/4\pi=20$ as discussed in [@holt]. Otherwise we do not see much difference between the generally good fit to observables.
Since the double decimation, although crudely done, generally gives observables close to the empirical ones we shall not try to distinguish between them and following Ref.[@holt] we take the average of them and quote a deviation, reproducing Table 5 of [@holt] as Table 2 below.
$l$ $F_l$ $G_l$ $F_l^\prime$ $G_l^\prime$
----- ------------------ ---------------- ---------------- ----------------
0 $-0.20\pm 0.39$ $0.04\pm 0.11$ $0.24\pm 0.16$ $0.53\pm 0.09$
1 $-0.86\pm 0.10$ $0.19\pm 0.06$ $0.18\pm 0.05$ $0.17\pm 0.01$
2 $-0.21\pm 0.01$ $0.12\pm 0.01$ $0.10\pm 0.02$ $0.01\pm 0.02$
3 $ -0.09\pm 0.01$ $0.05\pm 0.01$ $0.05\pm 0.01$ $0.01\pm 0.01$
\[table2\]
Aside from the very large compression modulus $K=495$ MeV in the CDB the variuous observables are not significantly different from each other.
The results of Holt et al. differ from those of Schwenk et al. in that they do not include the $\Delta$ isobar. The latter renormalize $G_0^\prime$ to take effects from the $\Delta$ into account.. Without their effects, Kawahigashi et al. [@kawa] find $g_{NN}^\prime=0.6$ to compare with $G_0^\prime=0.53\pm 0.09$ from our Table 2. Generalization by Schwenk et al. to a model corrected for the screening due to $\Delta$-hole excitations to all orders with $NN\rightarrow N\Delta$ and $N\Delta\rightarrow N\Delta$ interaction strengths of $g_{N\Delta}^\prime=0.3$ and $g_{\Delta\Delta}^\prime=0.3$ by Körfgen et al. [@kor1] gives $G_0^\prime=1.0$. Thus, we believe that adding the $\Delta$ to our description will have the same consequene and that we are starting with a $g_{NN}^\prime$ close enough so that we have essentially the same screening by a small change in our $\Delta$ coupling.
### $G_0^\prime$ as the Main Interaction Strength {#g_0prime-as-the-main-interaction-strength .unnumbered}
Were there no two-body correlation function keeping the two nucleons apart, there would be no pionic interaction with them in the long wavelength limit since it is derivative in nature H= | (r) and the momentum $\vec{p}$ = goes to zero as the volume goes to $\infty$. The total interaction by way of pion exchange is V(r)= 13 \_1\_2 \_1\_2(-(r))\[pipot\] plus a tensor interaction which averages to zero over angles. The integration of the above $V(r)$ d\^3r V(r)=0 does go to zero. The Ericsons [@ericsons] included short-range correlations by multiplying $V(r)$ by a short-range correlation function $g(r_{12})$ which had the property that g(r\_[12]{})=0, r\_[12]{}=0 but otherwise was of sufficiently short range that it did not affect anything.
If we leave the $\delta(r)$ out we have a minimalist description of the effect of short-range correlations. We are not finished because it is well-known that the exchange of $\rho$-mesons with tensor coupling between two nucleons contributes to the Lorentz-Lorenz effect. Brown [@brown] found that inclusion of this contribution increases the pionic one by a factor of 1.8, increasing the 1/3 in Eq. \[pipot\] to 0.6. We note that this is close to the $G_0^\prime$, the average $\sigma\tau$ Fermi liquid interaction in the $l=0$ state of the interactions shown in Table 2. Furthemore, the 0.6 is precisely what Kawahigashi et al. [@kawa] find for the contribution to the E$^2$L$^2$ interaction from the nucleon channel alone.
We see that the $G_0^\prime=1$, $G_1^\prime=0$ is the largest interaction by far. (It will be clear below why we group them.) In Table 2 the next largest ones are $F_l=-0.20$ and $F_l^\prime=0.24$. We divide $F_l$ by 3 because of its angle dependence.
We see that $G_0^\prime=1$, $G_1^\prime=0$ implies a $\delta$-function interaction, the potential V(r)= \_1\_2 \_1\_2 (r) representing the potential energy necessary to pull the nucleon and antinucleon apart from each other to overcome the pionic attraction. Note that this potential is always attractive, because the two fermions are at the same point and since they are antisymmetrical, if they are spin triplet they must be isospin singlet and vice versa.
Note that $V_{lowk}$ gives quite a good description of the $^{18}$O and $^{18}$F spectra [@holt2]. These calculations do not have the Brown-Rho-scaled masses in them, however, and would be expected to change as have results of Table 1 and Table 2.
### Conclusion {#conclusion .unnumbered}
We believe that we can offer a qualitative understanding of one of the main points of nuclear structure physics without detailed calculations; namely, why the one-bubble correction to the mean-field (shell-model) spectrum is very important as is well known in the Kuo-Brown interactions, whereas higher order corrections, as found in Holt et al. [@holt2] do not change the pattern qualitatively. This is because the E$^2$L$^2$ interaction is by far the strongest one and it is largely spent in the single bubble. In preliminary estimates we find that the direct and exchange effects largely cancel each other in two-bubble corrections and suggest that the one bubble which is classical in nature has most of the physics in it. The last paragraph is highly speculative.
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[^1]: The formulation that we initiated in early 1980’s was interrupted by the rediscovery in the context of QCD of the Skyrme soliton model for nucleons in 1983 which took us away from our activity on chiral EFT, and was made more complete after Weinberg’s paper. See [@MR-chpt].
[^2]: By chiral EFT, I mean EFT based on chiral Lagrangian with pions only (with or without baryon fields).
[^3]: This is somewhat like the status of the KSRF relations before the notion of hidden local symmetry was introduced.
[^4]: I am grateful to Koichi Yamawaki for his point of view on this matter.
[^5]: Up to date lattice calculations fail to see the phase transition up to $\sim 1.5$ times the chiral transition temperature for $N_f=2$ [@karsch].
[^6]: Unless otherwise noted, I will be working with the chiral limit.
[^7]: There is a strong indication that this symmetry is badly broken in sHLS at a density denoted $n_{1/2}\sim (2-3)n_0$ where $n_0$ stands for normal nuclear matter density. I will not deal with this high density which is relevant for compact stars [@PKLR].
[^8]: The BR scaling as applied to certain nuclear processes may contain other density dependence than IDD. For example, when short-range contact 3-body forces are integrated out, the resulting two-body force can inherit the three-body force effect in the form of BR scaling. One can understand this by that the zero-range three-body forces figuring in chiral EFT, involving $\omega$ and heavier meson exchanges of bsHLS, are of the same or higher scale than the cutoff scale $\tilde{\Lambda}$, hence their effects get captured in BR scaling when integrated out to arrive at chiral EFT. This will be the case with the C14 dating problem mentioned below. This means that the BR scaling used there has a contribution from the three-body force effect in addition to IDD. Another example: The axial current coupling $g_A$ is different for Gamow-Teller transitions (space component of the current) from axial charge transitions, also discussed below. The former is BR and the latter is IDD. Note that IDD in EFT Lagrangian is a Lorentz-invariant object while BR in physical observables may contain Lorentz-breaking contributions. A most prominent example where BR scaling in practice can contain more than IDD is the anomalous orbital gyromagnetic ratio $\delta g_l$ which can be well described by a BR scaling expressed in terms of certain Fermi-liquid parameters. It is in Fermi-liquid parameters that the IDD is lodged [@friman-rho]. Here the link between BR scaling and IDD is indirect and complicated.
[^9]: The pion decay constant does, however, flow by pion loops below $\tilde{\Lambda}$.
[^10]: Or at most three if fine-tuning is needed to obtain a precision fit to data. Since the only parameter of the theory, $\Phi$, can be fixed by experiments at least up to nuclear matter density, there is no real free parameter in the theory.
|
---
abstract: 'The clockwork mechanism for gravity introduces a tower of massive graviton modes, “[*clockwork gravitons*]{},” with a very compressed mass spectrum, whose interaction strengths are much stronger than that of massless gravitons. In this work, we compute the lowest order contributions of the clockwork gravitons to the anomalous magnetic moment, $g-2$, of muon in the context of extra dimensional model with a five dimensional Planck mass, $M_5$. We find that the total contributions are rather insensitive to the detailed model parameters, and determined mostly by the value of $M_5$. In order to account for the current muon $g-2$ anomaly, $M_5$ should be around $0.2~{\rm TeV}$, and the size of the extra dimension has to be quite large, $l_5 \gtrsim 10^{-7}\,$m. For $M_5\gtrsim1~{\rm TeV}$, the clockwork graviton contributions are too small to explain the current muon $g-2$ anomaly. We also compare the clockwork graviton contributions with other extra dimension models such as Randall-Sundrum models or large extra dimension models. We find that the leading contributions in the small curvature limit are universal, but the cutoff-independent subleading contributions vary for different background geometries and the clockwork geometry gives the smallest subleading contributions.'
author:
- Deog Ki Hong
- Du Hwan Kim
- Chang Sub Shin
title: 'Clockwork graviton contributions to muon $g-2$'
---
Introduction
------------
After the Higgs boson was discovered to complete the standard model (SM) of particle physics, there has been an intense search for new particles at the large hadron collider (LHC) that probes ${\rm TeV}$ energy scales. At LHC Run 2 the mass limit of new particles has been pushed up above $1~{\rm TeV}$ [@Khachatryan:2016yec; @ATLAS:2016eeo; @Aaboud:2017efa], putting most models for physics beyond SM (BSM) in great tension with their naturalness criterion, advocated by ’t Hooft [@'tHooft:1979bh].
Recently an interesting mechanism, called clockwork (CW), is proposed to generate naturally an exponential hierarchy for a given theory with multicomponents of fields [@Choi:2014rja; @Choi:2015fiu; @Kaplan:2015fuy]. Giudice and McCullough then proposed a clockwork solution to the electroweak hierarchy problem [@Giudice:2016yja], which exhibits rather rich structure and phenomenology [@Giudice:2017fmj]. The clockwork scenario addresses, similarly to other extra dimensional scenarios, the hierarchy problem by assuming that the fundamental scale of the theory is not much higher than the electroweak scale. The simplest clockwork model can be constructed with a set of 4D theories at $ N+1$ sites in a theory space with asymmetric couplings of link fields between nearby sites so that the zero mode of link fields is highly localized at a single site, while all SM particles reside at a site which has the least overlap with the zero mode to suppress its coupling to SM particles. If one identifies this zero mode as the 4D massless graviton[^1], the clockwork setup solves the naturalness problem associated with the weak scale and becomes in the large $N$ limit the linear dilaton model [@Antoniadis:2011qw; @Baryakhtar:2012wj] of 5D little string theory [@Aharony:1998ub; @Aharony:2004xn]. The clockwork theory then predicts an infinite tower of massive Kaluza-Klein (KK) gravitons, with unique spectrum, that couple to SM particles in a specific way. Especially the low-lying states of clockwork KK gravitons exhibit rather interesting signatures at colliders, compared to other models of extra dimensions, as studied in detail [@Giudice:2017fmj]. In this paper we study the contributions of the clockwork gravitons to the anomalous magnetic moments of muon and constrain the parameters of the clockwork gravitons, which will be complementary to collider searches. We find that the intrinsic scale, $M_5$, of the clockwork graviton has to be around $0.2~{\rm TeV}$ or higher to be compatible with the current muon $g-2$ anomaly.
It is well known that the standard model estimation of the anomalous magnetic moment of muon has quite a significant deviation from the experiments, which thus provides interesting constraints for new physics, if it were to explain the deviation [^2]. The current deviation between the experimental value, obtained at the Brookhaven National Laboratory (BNL) [@Bennett:2006fi], and the SM estimate, based on $e^{+}e^{-}$ hadronic cross sections [@Jegerlehner:2009ry], is found to be $$\Delta a_{\mu}=a_{\mu}^{\rm exp}- a_{\mu}^{\rm th}=(290\pm90)\times10^{-11}\,,$$ where $a_{\mu}\equiv \left(g-2\right)/2$ and $g$ is the gyromagnetic ratio of the magnetic moment of muon. The current deviation corresponds to about $3.2~\sigma$. An improved muon $(g-2)$ experiment at the Fermilab is about to take data, aiming to achieve a precision of $0.14$ ppm [@Miller:2007kk; @Gray:2015qna], which will then move the current deviation, if persistent, to more than 5 $\sigma$.
Massive graviton constributions to muon $g-2$
---------------------------------------------
While a massless spin 2 particle, that couples to the energy-momentum tensor, necessarily leads at low energy to Einstein’s general relativity, a consistent description of massive gravitons, respecting the general coordinate invariance, has been found only recently [@deRham:2014zqa]. To describe a (massive) graviton in a flat spacetime, we write the metric as $$g_{\mu\nu}=\eta_{\mu\nu}+2\kappa\,h_{\mu\nu}\,$$ where $\eta_{\mu\nu}$ is the Minkowski metric for the flat spacetime and $h_{\mu\nu}$ is the graviton field with coupling $\kappa\equiv\sqrt{8\pi G}$ for Einstein gravity with Newton’s gravitational constant $G$. Under a general coordinate transformation, $x^{\mu}\mapsto x^{\mu}+\xi^{\mu}(x)$, the graviton field transforms as $$h_{\mu\nu}\mapsto h_{\mu\nu}-\frac{1}{2\kappa}\left(\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}\right)\,.$$ Fixing the above gauge degrees of freedom, the massive graviton propagator of mass $M$ in $D$ dimensional spacetime becomes $$\int_xe^{ip\cdot x}\left<0\right|T\left\{h_{\mu\nu}(x)h_{\alpha\beta}(0)\right\}\left|0\right>=\frac{i}{2}\frac{{\tilde \eta}^{\mu\alpha}
{\tilde \eta}^{\nu\beta}+{\tilde \eta}^{\mu\beta}{\tilde \eta}^{\nu\alpha}-\beta{\tilde \eta}^{\mu\nu}{\tilde \eta}^{\alpha\beta}}{p^2-M^2+i\epsilon}\,,$$ where ${\tilde \eta}^{\mu\nu}=\eta_{\mu\nu}-p^{\mu}p^{\nu}/M^2$ and $\beta=\frac{2}{D-1}$. (For massless gravitons, ${\tilde \eta}^{\mu\nu}=\eta^{\mu\nu}-p^{\mu}p^{\nu}/p^2$ and $\beta=\frac{2}{D-2}$. Here we consider $D=4$ only.) The gravitational interactions are given at the linear level as $${\cal L}_{\rm int}=-{\kappa}\, h_{\mu\nu}T^{\mu\nu}\,,$$ where $T^{\mu\nu}\equiv\frac{-2}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu\nu}}$ with $S$ being the SM action is a symmetric and conserved energy-momentum tensor of SM that sources (massive) gravitons.
The one-loop contributions of massless graviton to muon anomalous magnetic moment was calculated by Berendes and Gastmans [@Berends:1974tr] and that of massive graviton was calculated in [@Graesser:1999yg]. Both are found to be finite. We briefly discuss the single graviton contributions first. There are 5 diagrams that contribute at one-loop to muon $g-2$, shown in Fig. \[fig1\].
0.1in
![ Graviton contributions at one-loop to the anomalous magnetic moment of muon. The dashed lines denote gravitons and the curly lines are photons. []{data-label="fig1"}](amm1a.pdf){width="1\linewidth" height="0.12\textheight"}
![ Graviton contributions at one-loop to the anomalous magnetic moment of muon. The dashed lines denote gravitons and the curly lines are photons. []{data-label="fig1"}](amm2aa.pdf){width="1\linewidth" height="0.12\textheight"}
![ Graviton contributions at one-loop to the anomalous magnetic moment of muon. The dashed lines denote gravitons and the curly lines are photons. []{data-label="fig1"}](amm2pp.pdf){width="1\linewidth" height="0.12\textheight"}
![ Graviton contributions at one-loop to the anomalous magnetic moment of muon. The dashed lines denote gravitons and the curly lines are photons. []{data-label="fig1"}](amm3a){width="1\linewidth" height="0.12\textheight"}
![ Graviton contributions at one-loop to the anomalous magnetic moment of muon. The dashed lines denote gravitons and the curly lines are photons. []{data-label="fig1"}](amm3p.pdf){width="1\linewidth" height="0.12\textheight"}
All the five diagrams are ultraviolet (UV) divergent and hence need to be regularized. But, the sum of all five diagrams for muon $g-2$ turns out to be finite. For massless graviton, one finds $$a_{\mu}^{\rm gr}=\frac{7}{32\pi^2}\kappa^2m^2\,,$$ where $m$ is the mass of muon [@Berends:1974tr]. For the one-loop contribution of massive gravitons of mass $M$ to muon $g-2$, we find $$a_{\mu}^{\rm massive}=\frac{5}{16\pi^2}\kappa^2m^2f\left(\frac{m}{M}\right)\,,$$ where $f(x)$ is a monotonically decreasing function from $1$ to $2/3$ as $x$ increases from 0 to $\infty$ (See Fig. \[fig2\]), which agrees with the previous result in the integral form in [@Graesser:1999yg] [^3].
![The massive graviton contributions in the unit of $a_{\mu}^{\rm massive}(M\to\infty)$.[]{data-label="fig2"}](asymptotic.pdf){width="50.00000%"}
We note that the massive graviton does not decouple in the loop corrections to muon $g-2$, as its mass goes to infinity, $M/m\to\infty$, since the gravity is non-renormalizable [^4]. In the large mass limit, $M\gg m$, $$a_{\mu}^{\rm massive}=\frac{5}{16\pi^2}\kappa^2m^2\left[1+\left\{\frac{1}{3}\ln\left(\frac{m}{M}\right)+\frac{11}{72}\right\}\frac{m^2}{M^2}+\cdots\right]\,,
\label{massive}$$ where the ellipsis denotes the higher order terms in $m/M$. The massless limit of massive graviton is known to be discontinuous due to the non-decoupling of the longitudinal mode [@vanDam:1970vg]. For the muon anomalous magnetic moment one finds a following discontinuity [@Graesser:1999yg]: $$a_{\mu}^{\rm massive}(M\to0)=a_{\mu}^{\rm gr}+\frac{\kappa^2m^2}{48\pi^2}\left(1-\frac{D-1}{D-2}\right)\,,$$ where $D=4$ is the dimension of the spacetime in which massive gravitons propagate.
Clockwork gravitons and other extra dimension models
----------------------------------------------------
If gravitons propagate in extra dimensions as well as in 4D spacetime where SM particles reside, either in a continuous extra dimension such as the large extra dimension (LED) model [@ArkaniHamed:1998rs], and the Randall-Sundrum (RS) model [@Randall:1999ee], or in a discrete extra dimension as the deconstructed gravity [@ArkaniHamed:2002sp] and the clockwork (CW) gravity [@Giudice:2016yja], there will be a (infinite) tower of massive gravitons, $h^{(n)}_{\mu\nu}$, that interact with 4D SM particles with specific couplings, $${\cal L}_{\rm int}=-\sum_{n=1}\frac{1}{\Lambda_n}h_{\mu\nu}^{(n)}T^{\mu\nu}.
\label{massive_int}$$ The mass of the $n$-th graviton $M_{(n)}$ and the coupling $1/\Lambda_n$ are the function of the model parameters and its intrinsic scale $M_5$ that describes the extra dimension, discussed below. In this paper, we are mostly interested in the clockwork case. However it turns out that the leading contribution to the muon $g-2$ is quite independent of detailed model parameters, and is only the function of $m$ and $M_5$. Let us start from the CW case first, and we will discuss later the cases of RS and LED backgrounds.
In the clockwork theory the massive gravitons, $h_{\mu\nu}^{(n)}$, have mass, given as $$M_{(n)}^2=k^2+\frac{n^2}{R^2}+{\cal O}\left(\frac{1}{N}\right),\quad n=1,\cdots,N\,,
\label{mass}$$ where $k$, $R$ correspond respectively to the warped factor or the clockwork spring and the radius of the extra dimension, orbifolded by $Z_2$. In the continuum limit ($N\to\infty$) the clockwork geometry becomes, having the SM particles localized at the $y=0$ brane, $$ds^2=e^{\frac{4k\left|y\right|}{3}}\left(dx_{\mu}dx^{\mu}-dy^2\right)\,,$$ which is nothing but the linear dilaton model studied in [@Antoniadis:2011qw]. Being a 5D theory, the continuum clockwork theory has an intrinsic 5D Planck scale $M_5$, which is an additional parameter of the clockwork theory, in addition to the radius $R$ and the warped factor $k$ or the 5D cosmological constant $-2k^2$. Upon the Kaluza-Klein (KK) reduction of the 5D clockwork gravity, one finds the clockwork gravitons’ couplings to the SM particles at the linear level [@Giudice:2016yja], given as the inverse of $$\Lambda_n=\sqrt{M_5^3\pi R\left(1+\frac{k^2R^2}{n^2}\right)}\,$$ and the effective 4D Planck mass $M_P$ or the effective coupling of the massless graviton, $\kappa=1/M_P$, is defined as $$M_P^2=\frac{M_5^3}{k}\left(e^{2k\pi R}-1\right)\,.
\label{planck}$$
The $n$-th clockwork graviton contribution to the muon $g-2$ at one-loop is from Eqs. (\[massive\]) and (\[massive\_int\]) $$a_{\mu}^{(n)}=\frac{5}{16\pi^2}\frac{m^2}{\Lambda_n^2}\left[1+{\cal O}\left(\frac{m^2}{M_n^2}\right)\right]\,.$$ Since massive gravitons do not decouple to the muon anomalous magnetic moment in the large mass limit ($M_{(n)}\gg m$), one might assume all towers of gravitons do contribute to the muon $g-2$. However, since the effective description of massive gravitons in 4D will breakdown at very short distances, smaller than the UV cutoff, $\Lambda_{\rm cut}$, of the clockwork theory, only finite number of gravitons are relevant in the 4D effective theory. The relevant gravitons should have therefore mass $M_{(n)}\lesssim \Lambda_{\rm cut}$ or the highest level $n_c$ that massive gravitons can be excited in the 4D effective theory should be from Eq. (\[mass\]) $$n_c=\sqrt{(\Lambda_{\rm cut}R)^2-(kR)^2}\,.$$
Since the choice of the cutoff $\Lambda_{\rm cut}$ depends on the regularization scheme of the 5D effective theory [@Contino:2001nj], we parametrize the regularization scheme dependence by a constant $\alpha$ as $$\Lambda_{\rm cut}=\alpha M_5\,,$$ where $\alpha$ should be positive and expected to be ${\cal O}(1)$ by the naive dimensional analysis but not predictable in the 5D effective theory. We will keep this parameter in our discussion in order to understand its physical meaning, and then take $\alpha=1$ just to estimate some numerical values. The 5D Planck mass $M_5$ of the clockwork theory also sets the upper bound of the curvature of 5D clockwork geometry. For the clockwork to work the warped factor should be smaller than the intrinsic scale, $k\ll M_5$, and we have $n_c\gg 1$ for a given $kR$. Furthermore, since we are considering only the linear terms for the graviton interactions, our approximation will break down when massive gravitons couple strongly, where the effects of UV completed quantum gravity are important. Therefore, there should be an upper bound for the highest level, $n_c\le n_{*}$, the massive gravitons of the 4D effective theory can reach. Namely at the upper bound $n=n_{*}$, the graviton mass is of the order of its effective Planck scale, $\Lambda_{n_*}\approx M_{(n_{*})}$. Approximately we then have $$M_5^3\pi R\left(1+\frac{k^2R^2}{n_*^2}\right)=k^2+\frac{n_*^2}{R^2}\,.$$ The maximum upper bound for graviton mass is therefore given by $n_*=M_5R\sqrt{M_5\pi R}$, as three parameters of the clockwork theory $k,R$ and $M_5$ are related to each other by Eq. (\[planck\]). The number of allowed KK levels, $n_c$, or the hierarchies between $M_{(1)}\simeq k$, $1/R$ and $M_5$ are very sensitive to the value of $kR$, as shown in Tables \[tab1\] and \[tab2\], which leads to very different collider phenomenology. However the muon $g-2$ is rather insensitive to such model parameters for $k\ll M_5$. Since the more the allowed KK levels are, the weaker their couplings, two effects cancel with each other.
----------------- ------------------- --------------------- ------------------- ----------------------------------
$M_5~(\rm TeV)$ $M_5R$ $n_c$ $n_*$ $a_{\mu}^{\rm CW}\times 10^{10}$
\[0.5ex\] 0.5 $1.2\times 10^5$ $1.2\times 10^5$ $7.4\times 10^7$ 4.5
1 $3.1\times 10^4$ $3.0\times 10^4$ $9\times 10^6$ 1.1
5 $1.2\times 10^3$ $1.2\times 10^3$ $7.3\times 10^4$ 0.045
10 306 305 $9.5\times 10^3$ 0.011
50 12 7 74 $4.0\times 10^{-5}$
\[1ex\]
----------------- ------------------- --------------------- ------------------- ----------------------------------
: The maximum Kaluza-Klein graviton level $n_c$ in the clockwork geometry with $k=10/R$, $n_*$ in the 4D effective theory, and the contribution to the muon $g-2$ for different $M_5$ with $\alpha=1$.
\[tab1\]
----------------- ------- -------- ------------------ ----------------------------------
$M_5~(\rm TeV)$ $kR$ $n_c$ $n_*$ $a_{\mu}^{\rm CW}\times 10^{10}$
\[0.5ex\] 0.5 11.1 111 $2.1\times 10^3$ 3.9
1 10.9 109 $2.1\times 10^3$ 1.0
5 10.4 102 $1.9\times 10^3$ 0.038
10 10.2 99 $1.8\times 10^3$ 0.010
50 9.7 93 $1.7\times 10^3$ $3.9\times 10^{-4}$
\[1ex\]
----------------- ------- -------- ------------------ ----------------------------------
: Same as Table \[tab1\], but fixing $k$ as $k = 0.1 M_5$.
\[tab2\]
Summing up the contributions of the clockwork gravitons up to the $n_c$-th level, we get $$a_{\mu}^{\rm CW}\simeq\sum_{n=1}^{n_c}a_{\mu}^{(n)}=\frac{5}{16\pi^3}\left(\frac{m}{M_5}\right)^2\left[\alpha
\sqrt{1-\left(\frac{k}{\alpha M_5}\right)^2}-\frac{k}{M_5}\sum_{n=1}^{n_c}\frac{kR}{n^2+(kR)^2}\right]\,,
%\frac{m}{M_5}\right)^2\left[\frac{2n_c+1}{2\pi M_5R}-\frac{k}{2M_5}\coth\left(\pi kR\right)\right]\,.$$ which becomes for $n_c\gg1$ or $M_5\gg k$ $$a_{\mu}^{\rm CW}\simeq \frac{5}{16\pi^3}\left(\frac{m}{M_5}\right)^2\left[\alpha-\frac{k}{M_5}\left\{-\frac{1}{2kR}+\frac{\pi}{2}\coth(\pi kR)\right\}+{\cal O}\left(\frac{k^2}{M_5^2}\right)\right]\,.$$
We note that the contribution of clockwork gravitons could be divided by the regularization scheme-dependent part, $$\Delta_{(1)} a_\mu^{\rm CW} \simeq\frac{5 \alpha }{16\pi^3}\left(\frac{m}{M_5}\right)^2 ,$$ which is independent of detailed graviton spectrum, and the scheme-independent part $$\Delta_{(2)} a_\mu^{\rm CW} \simeq - \frac{5}{32\pi^2}\frac{k}{M_5}\left(\frac{m}{M_5}\right)^2 ,$$ which depends on the 5D geometry. The fact that $\Delta_{(1)}a_\mu^{\rm CW}$ is independent of background geometry is easy to understand. In a 5D theory, this term represents a linear divergence that is dominated by large graviton momentum, $p_5$, along the fifth dimension. Such a graviton of large momentum would not see the background geometry. Also this term could be absorbed by a local counter term for the graviton-loop contribution such as $${\cal L}_{\rm c.t.} \sim \frac{\lambda}{M_5^2}( \bar \psi \gamma^{\mu\nu} \gamma^\rho D_\rho \psi) F_{\mu\nu},$$ which is consistent with symmetries of theory and independent of the background geometry. On the other hand, the contribution $\Delta_{(2)}a_\mu^{\rm CW}$ is UV finite and scheme-independent. Therefore it can be considered as the contribution that represents a genuine feature of CW geometry. In order to see that $\Delta_{(2)} a_\mu^{\rm CW}$ is scheme-independent, we may regularize the KK sum by different methods. For example, using $$\begin{aligned}
\sum_{n=1}^{\infty} \frac{n^2}{n^2 + (kR)^2}
&= \left[\frac{\partial^2}{\partial \epsilon^2} \sum_{n=1}^{\infty} \frac{ e^{- \epsilon n}}{n^2 + (kR)^2}\right]_{\epsilon \to 0^+}\nonumber\\
&= \frac{1}{\epsilon} +\frac{1}{2} +\frac{i kR}{2}\left[\frac{\Gamma'( i k R)}{\Gamma( i k R)}- {\rm h.c.}\right]\nonumber\\
&\approx \frac{1}{\epsilon} - \frac{\pi k R}{2} \quad{\rm for}\ kR\gg 1, \end{aligned}$$ where ${\rm h.c.}$ denotes the Hermitian conjugate and $\Gamma(x)$, $\Gamma^{\prime}(x)$ the gamma function and its derivative, respectively, we get the same finite contribution for $\Delta_{(2)}a_\mu^{\rm CW}$.
In the absence of a computation in a UV complete theory, however, it is still useful to calculate whole clockwork graviton contributions, taking $\alpha=1$, although a UV complete theory is needed to determine $\alpha$. Taking $kR\simeq10$, we get respectively for $M_5=0.5,1,10,50~{\rm TeV}$ as $$a_{\mu}^{\rm CW}\times 10^{10}\simeq 4.5,\,1.1,\,0.045,\,0.01,\,4.5\times 10^{-4}\,.$$ For $M_5=0.2~{\rm TeV}$, we find that the CW theory does explain the muon $g-2$ anomaly for any value of $k$ up to its maximum value $k_{\rm max}\sim M_5$, which corresponds to $k_{\rm max}R\sim11$ or $n_c\sim {\cal O}(1)$ (see Fig. \[fig3\]).
![The clockwork graviton contributions, shown in solid line, to the muon $g-2$ for $M_5=0.2~{\rm TeV}$ and $kR\in[7.5,11.3]$. The $1\sigma$ band of the muon $g-2$ anomaly lies in the shaded region between two dashed lines. []{data-label="fig3"}](amu02.pdf){width="60.00000%"}
![The contribution of the KK gravitons to the muon $g-2$ for $\Delta a_\mu = 10^{-8},\cdots, 10^{-12}$ in the $M_5-kR$ (left) and $M_5-k$ (right) plots. The contributions from the clockwork gravitons are shown by the solid lines. Here we also show the contributions with the Randall-Sundrum background, Eq. (\[muong-2RS\]) by dashed lines. The gray region is the $1\sigma$ band of the current muon $g-2$ anomaly [@Bennett:2006fi]. In the left panel, the size of extra dimension, $l_5$ (Eq. (\[extrasize\])) is shown by black lines for $l_5=10^{-6}\,$m and $10^{-9}\,$m.[]{data-label="fig4"}](muong-2M5vskR.pdf "fig:"){width="45.00000%"} ![The contribution of the KK gravitons to the muon $g-2$ for $\Delta a_\mu = 10^{-8},\cdots, 10^{-12}$ in the $M_5-kR$ (left) and $M_5-k$ (right) plots. The contributions from the clockwork gravitons are shown by the solid lines. Here we also show the contributions with the Randall-Sundrum background, Eq. (\[muong-2RS\]) by dashed lines. The gray region is the $1\sigma$ band of the current muon $g-2$ anomaly [@Bennett:2006fi]. In the left panel, the size of extra dimension, $l_5$ (Eq. (\[extrasize\])) is shown by black lines for $l_5=10^{-6}\,$m and $10^{-9}\,$m.[]{data-label="fig4"}](muong-2M5vsk.pdf "fig:"){width="46.00000%"}
The KK graviton contributions for generic values of $k$ and $M_5$ are plotted in Fig. \[fig4\] [^5]. In Fig. 4, we also show that there is a lower bound for the size of extra dimension $l_5$, defined as $$l_5 \equiv\int_0^{\pi R} dy \sqrt{ -g_{55}} = \frac{3(e^{\frac{2}{3} k\pi R} - 1)}{2k},
\label{extrasize}$$ to explain the muon $g-2$ anomaly. If $l_5$ is smaller than $10^{-7}$m, the CW gravitons contribution to the muon $g-2$ is always smaller than the current muon $g-2$ anomaly (See the left panel of Fig. \[fig4\]).
In the large $n_c$ limit or in the limit of small curvature of the extra dimension, $k\ll M_5$, the leading contribution, $\Delta_{(1)}a_{\mu}^{\rm CW}$ to the muon $g-2$ is independent of the background geometry of extra dimension. Such a universal form of the leading contribution to the muon $g-2$ in the large $n_c$ limit encourages us to calculate the value in other types of background geometry. Being a 5D theory in a warped geometry, the continuum clockwork theory is quite similar to the Randall-Sundrum model [@Randall:1999ee]. But, the differences lie in the mass spectrum and the graviton couplings, which lead to quite different features generically in low-energy physics such as the muon $g-2$ as well as in collider physics. The metric of RS models takes for $0\le y\le \pi R$ $$ds^2=e^{2ky}dx_{\mu}dx^{\mu}-dy^2$$ but the 4D effective Planck mass is same as that of CW theory, Eq. (\[planck\]). The RS graviton has mass given by the zeros of the Bessel function of first kind $J_1(j_n)=0$ as $$M_{(n)}=kj_n\approx \left(n+\frac14\right)\pi k, \quad n=1,2,3,\cdots\,.$$ and its 4D effective gravitational inverse-coupling $$\Lambda_n\simeq\sqrt{\frac{M_5^3}{k}}\,.$$ Similarly to the CW gravitons, the highest level that RS gravitons can be excited to is given by $$M_{(n_c)}\approx \alpha M_5 \quad {\rm or}\quad {n}_c\approx\frac{\alpha M_5}{\pi k}-\frac14\,,$$ which is always smaller than the maximum value of the highest level, $n_c<{n_*}=n_c^{3/2}\sqrt{\pi}$, at which $M_{(n_*)}\sim\Lambda_{n_*}$. The one-loop contribution of RS gravitons to the muon $g-2$ anomaly is summed up to its allowed highest level $n_c$ to get [^6] $$a_{\mu}^{\rm RS}=\sum_{n=1}^{n_c}a_{\mu}^{(n)}\approx\frac{5}{16\pi^3}\left(\frac{m}{M_5}\right)^2\left[\alpha-\frac{\pi k}{4M_5}\right]\,.\label{muong-2RS}$$ We see that for large $n_c\gg1$ or $M_5\gg k$, both the clockwork and Randall-Sundrum models give at the leading order the same contribution of gravitons to the muon $g-2$, namely $\Delta_{(1)}a_{\mu}^{\rm RS}=\Delta_{(1)}a_{\mu}^{\rm CW}$. On the other hand the subleading but scheme-independent contribution of RS background is bigger than that of CW background. The CW contribution in the small curvature limit ($M_5\gg k$) is therefore smaller than the RS contribution.
Finally we brief consider the massive graviton contributions to the muon $g-2$ in the large extra dimensions (LED), which has a flat 5D metric with the effective 4D Planck mass, given in terms of 5D intrinsic scale $M_5$ and the radius, $R$, of the 5th direction, $$M_P^2=M_5^3\pi R\,.$$ Since the metric is flat and by the boundary condition of the graviton wave-function, $\psi(y+\pi R)=\pm\psi(y)$, the Kaluza-Klein mass spectrum of LED graviton takes with $n=1,2,3,\cdots$ $$M_{(n)}=\frac{n}{R}\,.$$ Taking the highest level of LED gravitons $n_c=\alpha M_5R$, we find the total contribution of LED gravitons, $$a_{\mu}^{\rm LED}\approx \frac{5\alpha}{16\pi^3}\left(\frac{m}{M_5}\right)^2=4.5\times10^{-10}\alpha\left(\frac{0.5~{\rm TeV}}{M_5}\right)^2\,,$$ which is nothing but the lowest upper bound of all warped extra dimensions as $\alpha$ is independent of the background geometry. For the next order contributions, if $k\sim M_5$, only a few massive gravitons are allowed, $n_c\sim {\cal O}(10)$, and in this case RS gravitons always give bigger contributions to the muon $g-2$ than CW gravitons but smaller contributions, compared to LED gravitons, $a_{\mu}^{\rm CW}\lesssim a_{\mu}^{\rm RS}\lesssim a_{\mu}^{\rm LED}$, though all the extra dimensional models give similar contributions for the same intrinsic scale, $M_5$.
Results and Discussion
----------------------
We have calculated in the clockwork theory the contributions of massive gravitons to the anomalous magnetic moment of muon. We find that the clockwork gravitons do explain the current muon $g-2$ anomaly or $\Delta a_{\mu}\approx 28.8\times 10^{-10}$, if the intrinsic scale of the extra dimension $M_5\sim 0.2~{\rm TeV}$ with $k\lesssim 0.1~{\rm TeV}$, which corresponds to a quite large extra dimension, $l_5\gtrsim 10^{-7}\,{\rm m}$. For generic clockwork models, however, with $M_5\sim 1 - 100~{\rm TeV}$ and $M_5> k$, the contributions of clockwork gravitons are too small to account for the current muon $g-2$ anomaly. We also find that the lowest upper bound of the graviton contributions to the muon $g-2$ is given by the large extra dimension models among all the extra dimension models with the same 5D intrinsic scale $M_5$, if the UV sensitive contribution is same, which might be useful in top-down model building. This property could be confirmed for extra dimension models with more generic 5D metric background and KK graviton spectra that solve the gauge hierarchy problem [@Choi:2017ncj].
Acknowledgements
----------------
We thank A. Strumia for a useful comment. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B06033701) (DKH) and also by IBS under the project code, IBS-R018-D1 (DHK,CSS). This work was initiated at the CERN-CKC workshop “What’s going on at the weak scale?", Jeju Island, June 2017, and completed while one of the author (DKH) was enjoying the hospitality of the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. CSS is supported in part by the Ministry of Science, ICT&Future Planning and by Gyeongsangbuk-Do and Pohang City.
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[^1]: The clockwork symmetry between two neighboring sites is shown to be respected only for the abelian case [@Craig:2017cda]. But, here we take the effective field theory approach, as emphasized in [@Giudice:2017suc], for the clockwork gravity that is nothing but the discretized linear-dilaton model [@Antoniadis:2011qw; @Baryakhtar:2012wj].
[^2]: The muon $g-2$ would also provide interesting constraints on new physics even if there were not any deviation.
[^3]: There is a minor typo in Eq. (16) of [@Graesser:1999yg] that the function in the integrand should be $R$ instead of $R/L$.
[^4]: The graviton coupling to muon grows in energy as $\kappa^2E^2$ and the massive graviton contribution to the muon $g-2$ is thus not suppressed. The unitarity is also expected to be violated at energy $E\sim\kappa^{-1}\equiv M_{P}\,(=2.4\times 10^{18}~{\rm GeV})$. But, because the gravity is nonlinear, the theory necessarily has a UV cutoff below the Planck scale, $M_P$ [@ArkaniHamed:2002sp].
[^5]: Note that in this plot we did not impose the constraints on the low values of $M_5$ from other experiments just to see the parameter dependence of $a_\mu$ more clearly. However, the muon $g-2$ anomaly alone already excludes the clockwork theory with $M_5<0.2~{\rm TeV}$, unless there are negative contributions from other new particles.
[^6]: The muon $g-2$ in the Randall-Sundrum model has been calculated with different cutoff, $n_c$ in [@Park:2001uc; @Kim:2001rc]. The subleading part $\Delta_{(2)}a_{\mu}^{\rm RS}$ also could be shown scheme-independent by other regularization methods as well.
|
---
abstract: 'High-precision spectroscopy of large stellar samples plays a crucial role for several topical issues in astrophysics. Examples include studying the chemical structure and evolution of the Milky Way galaxy, tracing the origin of chemical elements, and characterizing planetary host stars. Data are accumulating from instruments that obtain high-quality spectra of stars in the ultraviolet, optical and infrared wavelength regions on a routine basis. These instruments are located at ground-based 2- to 10-m class telescopes around the world, in addition to the spectrographs with unique capabilities available at the Hubble Space Telescope. The interpretation of these spectra requires high-quality transition data for numerous species, in particular neutral and singly ionized atoms, and di- or triatomic molecules. We rely heavily on the continuous efforts of laboratory astrophysics groups that produce and improve the relevant experimental and theoretical atomic and molecular data. The compilation of the best available data is facilitated by databases and electronic infrastructures such as the NIST Atomic Spectra Database, the VALD database, or the Virtual Atomic and Molecular Data Centre (VAMDC). We illustrate the current status of atomic data for optical stellar spectra with the example of the Gaia-ESO Public Spectroscopic Survey. Data sources for 35 chemical elements were reviewed in an effort to construct a line list for a homogeneous abundance analysis of up to $10^5$ stars.'
address:
- '$^1$Institutionen för fysik och astronomi, Uppsala universitet, Box 516, 751 20 Uppsala, Sweden'
- '$^2$Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek, ACT 2611, Australia'
- '$^3$Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany'
- '$^4$INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Florence, Italy'
- '$^5$Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom'
- '$^6$Laboratoire Lagrange (UMR7293), Université de Nice Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 04, France'
- '$^7$Institute of Theoretical Physics and Astronomy, Vilnius University, A. Goštauto 12, 01108 Vilnius, Lithuania'
- '$^8$Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom'
author:
- 'U. Heiter$^1$, K. Lind$^1$, M. Asplund$^2$, P. S. Barklem$^1$, M. Bergemann$^3$, L. Magrini$^4$, T. Masseron$^5$, Š. Mikolaitis$^{6,7}$, J. C. Pickering$^8$, M. P. Ruffoni$^8$'
bibliography:
- 'heiter.bib'
title: Atomic and Molecular Data for Optical Stellar Spectroscopy
---
=4
Received 19 Oct 2014 / Accepted 29 Dec 2014 by Phys. Scr.
[*Keywords*]{}: atomic data, cool stars, surveys
Stellar spectroscopy – topical issues and data needs
====================================================
High-precision analysis of spectra of large numbers of stars is currently routinely used for
- studying the chemo-dynamical structure and evolution of the Milky Way galaxy,
- studying the evolution of stars,
- tracing the origin of chemical elements,
- or characterizing planetary host stars.
High-quality spectra are accumulating in the archives of various observatories from dedicated surveys and individual programs. The interpretation of these spectra requires high-quality data for numerous atomic and molecular transitions.
Large amounts of data are needed from the ultraviolet through the optical and into the infrared wavelength regions. Stellar spectroscopy in the ultraviolet region is mainly based on high-resolution spectra obtained with current instruments on the Hubble Space Telescope. The optical and infrared regions are surveyed with ground-based 2- to 10-m telescopes around the world. A few examples are the optical NARVAL spectrograph at the 2m telescope on Pic du Midi (France); the APOGEE (Apache Point Observatory Galactic Evolution Experiment) spectrograph at the 2.5m telescope in New Mexico (USA); the ESPaDOnS (Echelle SpectroPolarimetric Device for the Observation of Stars) spectrograph at the 3.6m telescope on Hawaii; the HARPS (High Accuracy Radial velocity Planet Searcher) spectrograph at the 3.6m telescope at ESO (Chile); and the UVES (Ultraviolet and Visual Echelle Spectrograph) and CRIRES (CRyogenic high-resolution InfraRed Echelle Spectrograph) instruments at the 8m VLT at ESO (Chile).
Cool stars with surface temperatures between about 3500 K and 6500 K are the most suitable objects to study the astrophysical topics mentioned above. Their spectra are dominated by absorption lines of mainly neutral and singly ionized atoms, as well as diatomic and triatomic molecules.
The types of data which are most important for spectrum analysis (apart from transition wavelengths, and assuming local thermodynamic equilibrium) are on the one hand transition probabilities (oscillator strengths, $gf$-values), and on the other hand data describing the effects of elastic collisions. Oscillator strengths can be either measured by laboratory astrophysics groups or calculated by atomic physics groups. Parameters for line-broadening by collisions with neutral or charged particles are obtained from calculations by atomic physics groups, e.g. or . Experimental data for collisional line broadening are only available for very few lines and have rather large uncertainties .
Extraction of data from publications in regular scientific journals can be a tedious task. However, databases and electronic infrastructures are available to facilitate this part of the work of stellar spectroscopists. Examples for relevant databases are the NIST Atomic Spectra Database [@NIST], the VALD database [@Pisk:95; @Kupk:99; @Ryab:99; @2008JPhCS.130a2011H][^1], or the STARK-B database [@starkb]. These databases can be queried either through their proprietory web interfaces, or through the VAMDC electronic infrastructure [@2010JQSRT.111.2151D][^2], which currently provides access to about 30 databases simultaneously.
As an example, we describe a simple query for data for ionized calcium using the VAMDC portal. The web portal shows a list of all databases and dynamically highlights those which are available to answer the constructed query. It also provides a graphical interface, which allows one to select a species – e.g. an atom with symbol Ca and ion charge 1, and the type of process, e.g. radiative transitions, and to specify further constraints. Setting the lower and upper wavelength limits to 864 and 867 nm and executing the query will result in a list of databases with an overview of the query results for each database (number of different species, states, and processes satisfying the constraints), as well as a datafile with the results for each database in a standard format. The web portal can be used to display the results in tabular form. In the simple example, two transitions of Ca$^+$ are found in two databases (Chianti and VALD), including wavelengths, oscillator strengths, and state descriptions (see Fig. \[fig:VAMDC\]). In addition, the VALD database node returns detailed references to the data sources.
Gaia-ESO Public Spectroscopic Survey
====================================
We illustrate the current status of atomic data for stellar spectrum modelling in the optical region with the example of the Gaia-ESO Public Spectroscopic Survey[^3] [@2012Msngr.147...25G]. The goal of the Gaia-ESO survey is to provide a homogeneous overview of the distributions of motions and chemical abundances in the Milky Way. The survey is carried out as an ESO programme with more than 300 co-investigators, using the multi-object instrument FLAMES at the 8-m Very Large Telescope in Chile. About $10^5$ stars within the Milky Way, in the field or in stellar clusters, will be observed over five years (having started in 2012). Spectra are obtained in several wavelength regions, mostly covering 480 to 680 nm and 850 to 900 nm at different resolutions ($R=\lambda/\Delta\lambda=$47000 and $<$20000). These spectra are analysed to determine velocities, surface temperatures, surface gravities, and chemical abundances.
All data are processed by several groups using different analysis codes. Early in the project it was decided that the spectrum analysis codes should use standardized input data as far as possible, such as model atmospheres, and atomic and molecular data. This approach allows one to rule out input data as sources of possible discrepancies between the results of different codes. Thus, we needed to define a standard line list, and to compile the best atomic data to be used by all groups. This task was assigned to a group of initially eight people within the Gaia-ESO collaboration. The first step was to identify spectral lines which allow accurate determination of stellar parameters and abundances of many chemical elements for F-, G-, and K-type stars. Most of the Gaia-ESO targets belong to these stellar spectral types. Figure \[fig:benchmarkstars\] illustrates the parameter ranges of typical Gaia-ESO targets by showing the parameters of the Gaia FGK benchmark stars . The insets show the increasing number of absorption lines appearing in the optical spectra for decreasing surface temperatures. The Gaia-ESO spectra cover the green-to-red parts of the images, and a section in the near infrared.
A line list for Gaia-ESO
========================
The approach for defining the Gaia-ESO line list was as follows. First, a list of 1341 preselected transitions for 35 elements (44 species comprising neutral and singly ionized atoms) was created[^4]. Next, the best atomic data were selected for these lines from the literature. We preferred the most precise laboratory measurements of $gf$-values (oscillator strengths), where available. This work led to a collaboration with the Laboratory Astrophysics group at Imperial College London, resulting in new data for neutral Fe lines [@2014MNRAS.441.3127R]. These data were supplemented by less precise laboratory $gf$-values and calculated data. We did not include or derive any astrophysical $gf$-values, as these would be dependent on the reference object and models used, and would not be applicable to all targets and analysis groups. Finally, simple quality flags for recommended use were assigned to each line (*Yes*/*Undecided*/*No*), based on the quality of the selected data and evaluated with spectral syntheses for the Sun and Arcturus.
As an example, we describe the data selection for neutral Fe (the details for all species will be presented in a forthcoming publication). The preselected line list includes 545 Fe lines. For 42% of these lines, precise laboratory measurements [@BKK; @BK; @BIPS; @GESB79b; @GESB82c; @GESB82d; @GESB86; @BWL; @2014MNRAS.441.3127R; @2014ApJS..215...23D], were available, and they were assigned the usage flag *Yes*. Older, less precise laboratory data were used for 33% of the lines, which were assigned the usage flag *Undecided*. Finally, the semi-empirical calculations by were assigned to the remaining 25% of the lines together with the usage flag *No*.
Figure \[fig:Fe1\] shows example spectra for three Fe lines with different flags for recommended usage for the Sun and Arcturus. Observed spectra are from and , respectively and were convolved to $R=$47000. For the calculated spectra MARCS atmospheric models [@Gust:08] and solar abundances by were used. In the case of Arcturus, the model parameters are $T_{\rm eff}$=4247 K and $\log g$(cm s$^{-2}$)=1.59, and the abundances were reduced by $-$0.54 dex, with 0.2 dex enhancement for $\alpha$-process elements. Observed and synthetic spectra agree for both stars in the case of the line with the *Yes* flag, while they disagree to different degrees for the lines with *Undecided* and *No* flags. The astrophysical performance of lines with different flags was investigated through abundance determinations for each individual Fe line using the observed spectrum of the Sun. The lines with a *Yes* flag showed a somewhat smaller dispersion around the mean than the lines with an *Undecided* flag. Abundances for lines with a *No* flag deviated from the mean by up to 1.5 dex.
Even if the analysis focuses on wavelength regions around the recommended list of preselected lines, these data are not sufficient for a thorough analysis. We need complete information (as far as possible) on all transitions visible in the observed wavelength ranges, in order to identify blends of the preselected lines, as “background” for synthetic spectrum calculations, and for an evaluation of the quality of spectrum processing (e.g. continuum normalization). Therefore, the preselected lines were complemented with data for $\sim$72000 atomic lines extracted from the VALD database with default configuration and stellar parameters corresponding to those of the target stars, as well as data for 27 molecular species. Priority was given to molecules which contribute significantly to the absorption in the spectra of G or K-type stars. For TiO, ZrO, SiH, CaH, VO, and FeH (and their isotopologues), the best line lists available in the literature were used. For CH, C$_2$, NH, OH, and MgH, improved line lists were computed using recent laboratory measurements. A detailed explanation of the procedure for the case of CH can be found in . Figure \[fig:master\] shows observed [@2000vnia.book.....H] and calculated spectra for Arcturus, convolved to $R=$47000 in an interval of 10 nm near the Na doublet lines at 589 nm, using the full line list. Most of the observed features are reproduced by the calculations, but in several places the calculated lines are too weak or completely missing, indicating a lack of atomic data.
Summary and Outlook
===================
Accurate laboratory data in the optical and IR wavelength regions are needed for the analysis of stellar spectra from ongoing or planned large-scale surveys. The Gaia-ESO collaboration provides a list of recommended lines for the analysis of FGK stars. The Gaia-ESO line list is regularly updated, resulting in a new version about once a year. The tests of the performance of the preselected lines should be extended to all of the Gaia FGK benchmark stars. Work in this direction has started within the Gaia-ESO collaboration. It is worth noting that numerous lines in the spectra of FGK stars are still unidentified. This problem can be remedied either by analysis of laboratory spectra, or analysis of carefully selected stellar spectra . We acknowledge the contributions of H. Jönsson, P. de Laverny, E. Maiorca, N. Ryde, and J. Sobeck to the Gaia-ESO line list work. UH acknowledges support from the Swedish National Space Board (Rymdstyrelsen). JCP and MR thank STFC of the UK for support of the laboratory astrophysics programme at Imperial College.
References {#references .unnumbered}
==========
[^1]: <http://vald.astro.uu.se>
[^2]: Virtual Atomic and Molecular Data Centre, <http://www.vamdc.eu>
[^3]: <http://www.gaia-eso.eu/>
[^4]: The species are H, Li, C, O, Na, Mg, Al, Si, Si$^+$, S, Ca, Ca$^+$, Sc, Sc$^+$, Ti, Ti$^+$, V, V$^+$, Cr, Cr$^+$, Mn, Fe, Fe$^+$, Co, Ni, Cu, Zn, Sr, Y, Y$^+$, Zr, Zr$^+$, Nb, Mo, Ru, Ba$^+$, La$^+$, Ce$^+$, Pr$^+$, Nd$^+$, Sm$^+$, Eu$^+$, Gd$^+$, Dy$^+$.
|
---
abstract: |
BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.
**AMS Subject Classification:** 65N55, 65M55, 65Y05
**Key words:** Iterative substructuring, additive Schwarz method, balancing domain decomposition, BDD, BDDC, Multispace BDDC, Multilevel BDDC
author:
- Jan Mandel
- Bedřich Sousedík
- 'Clark R. Dohrmann'
bibliography:
- '../../bibliography/bddc.bib'
title: Multispace and Multilevel BDDC
---
[Structural Dynamics Research Department, Sandia National Laboratories, Albuquerque NM 87185-0847, USA. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94-AL85000.]{}
Introduction {#sec:introduction}
============
The BDDC (Balancing Domain Decomposition by Constraints) method by Dohrmann [@Dohrmann-2003-PSC] is the most advanced method from the BDD family introduced by Mandel [@Mandel-1993-BDD]. It is a Neumann-Neumann iterative substructuring method of Schwarz type [@Dryja-1995-SMN] that iterates on the system of primal variables reduced to the interfaces between the substructures. The BDDC method is closely related to the FETI-DP method (Finite Element Tearing and Interconnecting - Dual, Primal) by Farhat et al. [@Farhat-2001-FDP; @Farhat-2000-SDP]. FETI-DP is a dual method that iterates on a system for Lagrange multipliers that enforce continuity on the interfaces, with some coarse variables treated as primal, and it is a further development of the FETI method by Farhat and Roux [@Farhat-1991-MFE]. Polylogarithmic condition number estimates for BDDC were obtained in [@Mandel-2003-CBD; @Mandel-2005-ATP] and a proof that the eigenvalues of BDDC and FETI-DP are actually the same except for eigenvalue equal to one was given in Mandel et al. [@Mandel-2005-ATP]. Simpler proofs of the equality of eigenvalues were obtained by Li and Widlund [@Li-2006-FBB], and also by Brenner and Sung [@Brenner-2007-BFW], who also gave an example when BDDC has an eigenvalue equal to one but FETI-DP does not. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. However, since the coarse problem in BDDC has the same form as the original problem, the BDDC method can be applied recursively to solve the coarse problem only approximately. This leads to a multilevel form of BDDC in a straightforward manner, see Dohrmann [@Dohrmann-2003-PSC]. Polylogarithmic condition number bounds for three-level BDDC (BDDC with two coarse levels) were proved in two and three spatial dimensions by Tu [@Tu-2007-TBT; @Tu-2007-TBT3D].
In this paper, we present a new abstract Multispace BDDC method. The method extends a simple variational setting of BDDC from Mandel and Sousedík [@Mandel-2007-ASF], which could be understood as an abstract version of BDDC by partial subassembly in Li and Widlund [@Li-2007-UIS]. However, we do not adopt their change of variables, which does not seem to be suitable in our abstract setting. We provide a condition number estimate for the abstract Multispace BDDC method, which generalizes the estimate for a single space from [@Mandel-2007-ASF]. The proof is based on the abstract additive Schwarz theory by Dryja and Widlund [@Dryja-1995-SMN]. Many BDDC formulations (with an explicit treatment of substructure interiors, after reduction to substructure interfaces, with two levels, and multilevel) are then viewed as abstract Multispace BDDC with a suitable choice of spaces and operators, and abstract condition number estimates for those BDDC methods follow. This result in turn gives a polylogarithmic condition number bound for Multilevel BDDC applied to a second-order scalar elliptic model problems, with an arbitrary number of levels. A brief presentation of the main results of the paper, without proofs and with a simplified formulation of the Multispace BDDC estimate, is contained in the conference paper [@Mandel-2007-OMB].
The paper is organized as follows. In Sec. \[sec:problem-setting\] we introduce the abstract problem setting. In Sec. \[sec:abstract-multispace\] we formulate an abstract Multispace BDDC as an additive Schwarz preconditioner. In Sec. \[sec:FE-setting\] we introduce the settings of a model problem using finite element discretization. In Sec. \[sec:one-level\] we recall the algorithm of the original (two-level) BDDC method and formulate it as a Multispace BDDC. In Sec. \[sec:multilevel-bddc\] we generalize the algorithm to obtain Multilevel BDDC and we also give an abstract condition number bound. In Sec. \[sec:multilevel-condition\] we derive the condition number bound for the model problem. Finally, in Sec. \[sec:numerical-examples\], we report on numerical results.
Abstract Problem Setting {#sec:problem-setting}
========================
We wish to solve an abstract linear problem $$u\in X:a(u,v)=\left\langle f_{X},v\right\rangle ,\quad\forall v\in X,
\label{eq:problem}$$ where $X$ is a finite dimensional linear space, $a\left( \cdot,\cdot\right)
$ is a symmetric positive definite bilinear form defined on $X$, $f_{X}\in
X^{\prime}$ is the right-hand side with $X^{\prime}$ denoting the dual space of $X$, and $\left\langle \cdot,\cdot\right\rangle $ is the duality pairing. The form $a\left( \cdot,\cdot\right) $ is also called the energy inner product, the value of the quadratic form $a\left( u,u\right) $ is called the energy of $u$, and the norm $\left\Vert u\right\Vert _{a}=a\left( u,u\right)
^{1/2}$ is called the energy norm. The operator $A_{X}:X\mapsto X^{\prime}$ associated with $a$ is defined by $$a(u,v)=\left\langle A_{X}u,v\right\rangle ,\quad\forall u,v\in X.$$
A preconditioner is a mapping $B:X^{\prime}\rightarrow X$ and we will look for preconditioners such that $\left\langle r,Br\right\rangle $ is also symmetric and positive definite on $X^{\prime}$. It is well known that then $BA_{X}:X\rightarrow X$ has only real positive eigenvalues, and convergence of the preconditioned conjugate gradients method is bounded using the condition number$$\kappa=\frac{\lambda_{\max}(BA_{X})}{\lambda_{\min}(BA_{X})},$$ which we wish to bound above.
All abstract spaces in this paper are finite dimensional linear spaces and we make no distinction between a linear operator and its matrix.
Abstract Multispace BDDC {#sec:abstract-multispace}
========================
To introduce abstract Multispace BDDC preconditioner, suppose that the bilinear form $a$ is defined and symmetric positive semidefinite on some larger space $W\supset X$. The preconditioner is derived from the abstract additive Schwarz theory, however we decompose some space between $X$ and $W$ rather than $X$ as it would be done in the additive Schwarz method: In the design of the preconditioner, we choose spaces $V_{k}$, $k=1,\ldots,M$, such that $$X\subset\sum_{k=1}^{M}V_{k}\subset W. \label{eq:multispace-BDDC-spaces}$$
\[assum:pos-def\]The form $a\left( \cdot,\cdot\right) $ is positive definite on each $V_{k}$ separately.
\[Abstract Multispace BDDC\]\[alg:multispace-bddc\] Given spaces $V_{k}$ and linear operators $Q_{k}$, $k=1,\ldots,M$ such that $a\left( \cdot
,\cdot\right) $ is positive definite on each space $V_{k}$, and$$X\subset\sum_{k=1}^{M}V_{k},\quad Q_{k}:V_{k}\rightarrow X,$$ define the preconditioner $B:r\in X^{\prime}\longmapsto u\in X$ by$$B:r\mapsto u=\sum_{k=1}^{M}Q_{k}v_{k},\quad v_{k}\in V_{k}:\quad a\left(
v_{k},z_{k}\right) =\left\langle r,Q_{k}z_{k}\right\rangle ,\quad\forall
z_{k}\in V_{k}. \label{def:abs-mult-BDDC}$$
We formulate the condition number bound first in the full strength allowed by the proof. The bound used in the rest of this paper will be a corollary.
\[thm:BDDC-M\]Define for all $k=1,\ldots,M$ the spaces $V_{k}^{\mathcal{M}}$ by$$V_{k}^{\mathcal{M}}=\left\{ v_{k}\in V_{k}:\forall z_{k}\in V_{k}:Q_{k}v_{k}=Q_{k}z_{k}\Longrightarrow\left\Vert v_{k}\right\Vert _{a}^{2}\leq\left\Vert z_{k}\right\Vert _{a}^{2}\right\} .$$ If there exist constants $C_{0},$ $\omega,$ and a symmetric matrix $\mathcal{E}=(e_{ij})_{i,j=1}^{M}$, such that$$\begin{aligned}
& \forall u\in X\quad\exists v_{k}\in V_{k},\text{ }k=1,\ldots,M:u=\sum
_{k=1}^{M}Q_{k}v_{k},\text{ }\sum_{k=1}^{M}\left\Vert v_{k}\right\Vert
_{a}^{2}\leq C_{0}\left\Vert u\right\Vert _{a}^{2}\label{eq:upper-Wi}\\
& \forall k=1,\ldots,M\quad\forall v_{k}\in V_{k}^{\mathcal{M}}:\left\Vert
Q_{k}v_{k}\right\Vert _{a}^{2}\leq\omega\left\Vert v_{k}\right\Vert _{a}^{2}\label{eq:lower-Wi}\\
& \forall z_{k}\in Q_{k}V_{k},\text{ }k=1,\ldots,M:a\left( z_{i},z_{j}\right) \leq e_{ij}\left\Vert z_{i}\right\Vert _{a}\left\Vert
z_{j}\right\Vert _{a},\quad\label{eq:cauchy-W}$$ then the preconditioner from Algorithm \[alg:multispace-bddc\] satisfies $$\kappa=\frac{\lambda_{\max}(BA_{X})}{\lambda_{\min}(BA_{X})}\leq C_{0}\omega\rho(\mathcal{E}).$$
We interpret the Multispace BDDC preconditioner as an abstract additive Schwarz method. An abstract additive Schwarz method is specified by a decomposition of the space $X$ into subspaces,$$X=X_{1}+...+X_{M}, \label{eq:decomposition-schwarz}$$ and by symmetric positive definite bilinear forms $b_{i}$ on $X_{i}$. The preconditioner is a linear operator $$B:X^{\prime}\rightarrow X,\qquad B:r\mapsto u,$$ defined by solving the following variational problems on the subspaces and adding the results,$$B:r\mapsto u=\sum\limits_{k=1}^{M}u_{k},\quad u_{k}\in X_{k}:\quad b_{k}(u_{k},y_{k})=\left\langle r,y_{k}\right\rangle ,\quad\forall y_{k}\in X_{k}.
\label{eq:def-uM}$$
Dryja and Widlund [@Dryja-1995-SMN] proved that if there exist constants $C_{0},$ $\omega,$ and a symmetric matrix $\mathcal{E}=(e_{ij})_{i,j=1}^{M}$, such that$$\begin{aligned}
\forall u & \in X\text{ }\exists u_{k}\in X_{k},\text{ }k=1,\ldots
,M:u=\sum_{k=1}^{M}u_{k},\text{ }\sum_{k=1}^{M}\left\Vert u_{k}\right\Vert
_{b_{k}}^{2}\leq C_{0}\left\Vert u\right\Vert _{a}^{2}\label{eq:upper-bi}\\
\forall k & =1,\ldots,M\text{ }\forall u_{k}\in X_{k}:\left\Vert
u_{k}\right\Vert _{a}^{2}\leq\omega\left\Vert u_{k}\right\Vert _{b_{k}}^{2}\label{eq:lower-bi}\\
\forall u_{k} & \in X_{k},\text{ }k=1,\ldots,M:a(u_{i},u_{j})\leq
e_{ij}\left\Vert u_{i}\right\Vert _{a}\left\Vert u_{j}\right\Vert _{a}
\label{eq:str-cauchy}$$ then $$\kappa=\frac{\lambda_{\max}(BA_{X})}{\lambda_{\min}(BA_{X})}\leq C_{0}\omega\rho(\mathcal{E}),$$ where $\rho$ is the spectral radius.
Now the idea of the proof is essentially to map the assumptions of the abstract additive Schwarz estimate from the decomposition (\[eq:decomposition-schwarz\]) of the space $X$ to the decomposition (\[eq:multispace-BDDC-spaces\]). Define the spaces $$X_{k}=Q_{k}V_{k}.$$ We will show that the preconditioner (\[def:abs-mult-BDDC\]) satisfies (\[eq:def-uM\]), where $b_{k}$ is defined by$$b_{k}(u_{k},y_{k})=a\left( G_{k}x,G_{k}z\right) ,\quad x,z\in X,\quad
u_{k}=Q_{k}G_{k}x,\quad y_{k}=Q_{k}G_{k}z. \label{eq:def-bk}$$ with the operators $G_{k}:X\rightarrow V_{k}^{\mathcal{M}}$ defined by$$G_{k}:u\mapsto v_{k},\quad\frac{1}{2}a\left( v_{k},v_{k}\right)
\rightarrow\min,\text{ s.t. }v_{k}\in V_{k}^{\mathcal{M}},\text{ }u=\sum
_{k=1}^{M}Q_{k}v_{k}, \label{eq:op-G-minim}$$ First, from the definition of operators $G_{k}$, spaces $X_{k},$ and because $a$ is positive definite on $V_{k}$ by Assumption (\[assum:pos-def\]), it follows that $G_{k}x$ and $G_{k}z$ in (\[eq:def-bk\]) exist and are unique, so $b_{k}$ is defined correctly. To prove (\[eq:def-uM\]), let $v_{k}$ be as in (\[def:abs-mult-BDDC\]) and note that $v_{k}$ is the solution of $$\frac{1}{2}a\left( v_{k},v_{k}\right) -\left\langle r,Q_{k}v_{k}\right\rangle \rightarrow\min,\quad v_{k}\in V_{k}.$$ Consequently, the preconditioner (\[def:abs-mult-BDDC\]) is an abstract additive Schwarz method and we only need to verify the inequalities (\[eq:upper-bi\])–(\[eq:str-cauchy\]). To prove (\[eq:upper-bi\]), let $u\in X$. Then, with $v_{k}$ from the assumption (\[eq:upper-Wi\]) and with $u_{k}=Q_{k}G_{k}v_{k}$ as in (\[eq:def-bk\]), it follows that$$u=\sum_{k=1}^{M}u_{k},\quad\sum_{k=1}^{M}\left\Vert u_{k}\right\Vert _{b_{k}}^{2}=\sum_{k=1}^{M}\left\Vert v_{k}\right\Vert _{a}^{2}\leq C_{0}\left\Vert
u\right\Vert _{a}^{2}.$$ Next, let $u_{k}\in X_{k}$. From the definitions of $X_{k}$ and $V_{k}^{\mathcal{M}}$, it follows that there exist unique $v_{k}\in V_{k}^{\mathcal{M}}$ such that $u_{k}=Q_{k}v_{k}$. Using the assumption (\[eq:lower-Wi\]) and the definition of $b_{k}$ in (\[eq:def-bk\]), we get $$\left\Vert u_{k}\right\Vert _{a}^{2}=\left\Vert Q_{k}v_{k}\right\Vert _{a}^{2}\leq\omega\left\Vert v_{k}\right\Vert _{a}^{2}=\omega\left\Vert
u_{k}\right\Vert _{b_{k}}^{2},$$ which gives (\[eq:lower-bi\]). Finally, (\[eq:cauchy-W\]) is the same as (\[eq:str-cauchy\]).
The next Corollary was given without proof in [@Mandel-2007-OMB Lemma 1]. This is the special case of Theorem \[thm:BDDC-M\] that will be actually used. In the case when $M=1$, this result was proved in [@Mandel-2007-ASF].
\[cor:multispace-bddc\]Assume that the subspaces $V_{k}$ are energy orthogonal, the operators $Q_{k}$ are projections, $a\left( \cdot
,\cdot\right) $ is positive definite on each space $V_{k}$, and$$\forall u\in X:\left[ u=\sum_{k=1}^{M}v_{k},\ v_{k}\in V_{k}\right]
\Longrightarrow u=\sum_{k=1}^{M}Q_{k}v_{k}\text{.} \label{eq:dec-unity}$$ Then the abstract Multispace BDDC preconditioner from Algorithm \[alg:multispace-bddc\] satisfies$${\kappa=\frac{\lambda_{\max}(BA_{X})}{\lambda_{\min}(BA_{X})}\leq\omega
=\max_{k}\sup_{v_{k}\in V_{k}}\frac{\left\Vert Q_{k}v_{k}\right\Vert _{a}^{2}}{\left\Vert v_{k}\right\Vert _{a}^{2}}}\text{ }{.}
\label{eq:multispace-bddc-bound}$$
We only need to verify the assumptions of Theorem \[thm:BDDC-M\]. Let $u\in
X$ and choose $v_{k}$ as the energy orthogonal projections of $u$ on $V_{k}$. First, since the spaces $V_{k}$ are energy orthogonal, $u={\textstyle\sum}
v_{k}$, $Q_{k}$ are projections, and from (\[eq:dec-unity\]) $u={\textstyle\sum}
Q_{k}v_{k}$, we get that $\left\Vert u\right\Vert _{a}^{2}={\textstyle\sum}
\left\Vert v_{k}\right\Vert _{a}^{2}$ which proves (\[eq:upper-Wi\]) with $C_{0}=1$. Next, the assumption (\[eq:lower-Wi\]) becomes the definition of ${\omega} $ in (\[eq:multispace-bddc-bound\]). Finally, (\[eq:cauchy-W\]) with $\mathcal{E}=I$ follows from the orthogonality of subspaces $V_{k}$.
The assumption (\[eq:dec-unity\]) can be written as$$\left. \sum_{k=1}^{M}Q_{k}P_{k}\right\vert _{X}=I,$$ where $P_{k}$ is the $a$-orthogonal projection from $\bigoplus\nolimits_{j=1}^{M}V_{j}$ onto $V_{k}$. Hence, the property (\[eq:dec-unity\]) is a type of decomposition of unity.
In the case when $M=1$, (\[eq:dec-unity\]) means that the projection $Q_{1}
$ is onto $X$.
Finite Element Problem Setting {#sec:FE-setting}
==============================
Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$, $d=2$ or $3$, decomposed into $N$ nonoverlapping subdomains $\Omega^{s}$, $s=1,...,N$, which form a conforming triangulation of the domain $\Omega$. Subdomains will be also called substructures. Each substructure is a union of Lagrangian $P1$ or $Q1$ finite elements with characteristic mesh size $h$, and the nodes of the finite elements between substructures coincide. The nodes contained in the intersection of at least two substructures are called boundary nodes. The union of all boundary nodes is called the interface $\Gamma$. The interface$~\Gamma$ is a union of three different types of open sets: *faces*, *edges*, and *vertices*. The substructure vertices will be also called corners. For the case of regular substructures, such as cubes or tetrahedrons, we can use standard geometric definition of faces, edges, and vertices; cf., e.g., [@Klawonn-2006-DPF] for a more general definition.
In this paper, we find it more convenient to use the notation of abstract linear spaces and linear operators between them instead of the space $\mathbb{R}^{n}$ and matrices. The results can be easily converted to the matrix language by choosing a finite element basis. The space of the finite element functions on $\Omega$ will be denoted as $U$. Let $W^{s}$ be the space of finite element functions on substructure $\Omega^{s}$, such that all of their degrees of freedom on $\partial\Omega^{s}\cap\partial\Omega$ are zero. Let$$W=W^{1}\times\cdots\times W^{N},$$ and consider a bilinear form arising from the second-order scalar elliptic problem as$$a\left( u,v\right) =\sum_{s=1}^{N}\int_{\Omega^{S}}\nabla u\nabla
v\,dx,\quad u,v\in W. \label{eq:scalar-bilinear-form}$$
Now $U\subset W$ is the subspace of all functions from $W$ that are continuous across the substructure interfaces. We are interested in the solution of the problem (\[eq:problem\]) with $X=U$,$$u\in U:a(u,v)=\left\langle f,v\right\rangle ,\quad\forall v\in U,
\label{eq:problem-full-space}$$ where the bilinear form $a$ is associated on the space $U$ with the system operator$~A$, defined by $$A:U\mapsto U^{\prime},\quad a(u,v)=\left\langle Au,v\right\rangle \text{ for
all }u,v\in U, \label{eq:def-A}$$ and $f\in U^{\prime}$ is the right-hand side. Hence, (\[eq:problem-full-space\]) is equivalent to$$Au=f. \label{eq:problem-full-space-algebraic}$$
Define $U_{I}\subset U$ as the subspace of functions that are zero on the interface $\Gamma$, i.e., the interior functions. Denote by $P$ the energy orthogonal projection from $W$ onto $U_{I}$,$$P:w\in W\longmapsto v_{I}\in U_{I}:a\left( v_{I},z_{I}\right) =a\left(
w,z_{I}\right) ,\quad\forall z_{I}\in U_{I}.$$ Functions from $\left( I-P\right) W$, i.e., from the nullspace of $P,$ are called discrete harmonic; these functions are $a$-orthogonal to $U_{I}$ and energy minimal with respect to increments in $U_{I}$. Next, let $\widehat{W}$ be the space of all discrete harmonic functions that are continuous across substructure boundaries, that is $$\widehat{W}=\left( I-P\right) U. \label{eq:discrete-harm}$$ In particular, $$U=U_{I}\oplus\widehat{W},\quad U_{I}\perp_{a}\widehat{W}.
\label{eq:int-harm-dec}$$
A common approach in substructuring is to reduce the problem to the interface. The problem (\[eq:problem-full-space\]) is equivalent to two independent problems on the energy orthogonal subspaces $U_{I}$ and $\widehat{W}$, and the solution $u$ satisfies $u=u_{I}+\widehat{u}$, where $$\begin{aligned}
u & \in U_{I}:a(u_{I},v_{I})=\left\langle f,v_{I}\right\rangle ,\quad\forall
v_{I}\in U_{I},\label{eq:problem-int-1}\\
u & \in\widehat{W}:a(\widehat{u},\widehat{v})=\left\langle f,\widehat
{v}\right\rangle ,\quad\forall\widehat{v}\in\widehat{W}.
\label{eq:problem-reduced}$$ The solution of the interior problem (\[eq:problem-int-1\]) decomposes into independent problems, one per each substructure. The reduced problem (\[eq:problem-reduced\]) is then solved by preconditioned conjugate gradients. The reduced problem (\[eq:problem-reduced\]) is usually written equivalently as$$u\in\widehat{W}:s(\widehat{u},\widehat{v})=\left\langle g,\widehat
{v}\right\rangle ,\quad\forall\widehat{v}\in\widehat{W},$$ where $s$ is the form $a$ restricted on the subspace $\widehat{W}$, and $g$ is the reduced right hand side, i.e., the functional $f$ restricted to the space $\widehat{W}$. The reduced right-hand side $g$ is usually written as$$\left\langle g,\widehat{v}\right\rangle =\left\langle f,\widehat
{v}\right\rangle -a(u_{I},\widehat{v}),\quad\forall\widehat{v}\in\widehat{W},
\label{eq:reduced-rhs}$$ because $a(u_{I},\widehat{v})=0$ by (\[eq:int-harm-dec\]). In the implementation, the process of passing to the reduced problem becomes the elimination of the internal degrees of freedom of the substructures, also known as static condensation. The matrix of the reduced bilinear form $s$ in the basis defined by interface degrees of freedom becomes the Schur complement, and (\[eq:reduced-rhs\]) becomes the reduced right-hand side. For details on the matrix formulation, see, e.g., [Smith-1996-DD]{} or [@Toselli-2005-DDM Sec. 4.3].
The BDDC method is a two-level preconditioner characterized by the selection of certain *coarse degrees of freedom*, such as values at the corners and averages over edges or faces of substructures. Define $\widetilde{W}\subset W$ as the subspace of all functions such that the values of any coarse degrees of freedom have a common value for all relevant substructures and vanish on $\partial\Omega,$ and $\widetilde{W}_{\Delta}\subset W$ as the subspace of all function such that their coarse degrees of freedom vanish. Next, define $\widetilde{W}_{\Pi}$ as the subspace of all functions such that their coarse degrees of freedom between adjacent substructures coincide, and such that their energy is minimal. Clearly, functions in $\widetilde{W}_{\Pi}$ are uniquely determined by the values of their coarse degrees of freedom, and $$\widetilde{W}_{\Delta}\perp_{a}\widetilde{W}_{\Pi},\text{\quad and\quad
}\widetilde{W}=\widetilde{W}_{\Delta}\oplus\widetilde{W}_{\Pi}.
\label{eq:tilde-dec}$$
We assume that$$a\text{ is positive definite on }\widetilde{W}. \label{eq:pos-def}$$ That is the case when $a$ is positive definite on the space $U$, where the problem (\[eq:problem\]) is posed, and there are sufficiently many coarse degrees of freedom. We further assume that the coarse degrees of freedom are zero on all functions from $U_{I}$, that is,$$U_{I}\subset\widetilde{W}_{\Delta}. \label{eq:coarse-int}$$ In other words, the coarse degrees of freedom depend on the values on substructure boundaries only. From (\[eq:tilde-dec\]) and (\[eq:coarse-int\]), it follows that the functions in $\widetilde{W}_{\Pi}$ are discrete harmonic, that is,$$\widetilde{W}_{\Pi}=\left( I-P\right) \widetilde{W}_{\Pi}.
\label{eq:coarse-is-discrete-harmonic}$$
Next, let $E$ be a projection from $\widetilde{W}$ onto $U$, defined by taking some weighted average on substructure interfaces. That is, we assume that$$E:\widetilde{W}\rightarrow U,\quad EU=U,\quad E^{2}=E. \label{eq:E-onto-U}$$ Since a projection is the identity on its range, it follows that $E$ does not change the interior degrees of freedom, $$EU_{I}=U_{I}, \label{eq:int-unchanged}$$ since $U_{I}\subset U$. Finally, we show that the operator $\left(
I-P\right) E$ is a projection. From (\[eq:int-unchanged\]) it follows that $E$ does not change interior degrees of freedom, so $EP=P$. Then, using the fact that $I-P$ and $E$ are projections, we get$$\begin{aligned}
\left[ \left( I-P\right) E\right] ^{2} & =\left( I-P\right) E\left(
I-P\right) E\nonumber\\
& =\left( I-P\right) \left( E-P\right) E\label{eq:(I-P)E}\\
& =\left( I-P\right) \left( I-P\right) E=\left( I-P\right) E.\nonumber\end{aligned}$$
In [@Mandel-2005-ATP; @Mandel-2007-ASF], the whole analysis was done in spaces of discrete harmonic functions after eliminating $U_{I}$, and the space $\widehat{W}$ was the solution space. In particular, $\widetilde{W}$ consisted of discrete harmonic functions only, while the same space here would be $(I-P)\widetilde{W}$. The decomposition of this space used in [@Mandel-2005-ATP; @Mandel-2007-ASF] would be in our context written as $$(I-P)\widetilde{W}=(I-P)\widetilde{W}_{\Delta}\oplus\widetilde{W}_{\Pi},\quad(I-P)\widetilde{W}_{\Delta}\perp_{a}\widetilde{W}_{\Pi}.
\label{eq:dharm-dec}$$
In the next section, the space $X$ will be either $U$ or $\widehat{W}$.
Two-level BDDC as Multispace BDDC {#sec:one-level}
=================================
We show several different ways the original, two-level, BDDC algorithm can be interpreted as multispace BDDC. We consider first BDDC applied to the reduced problem (\[eq:problem-reduced\]), that is, (\[eq:problem\]) with $X=\widehat{W}$. This was the formulation considered in [@Mandel-2005-ATP]. Define the space of discrete harmonic functions with coarse degrees of freedom continuous across the interface$$\widetilde{W}_{\Gamma}=\left( I-P\right) \widetilde{W}.$$ Because we work in the space of discrete harmonic functions and the output of the averaging operator $E$ is not discrete harmonic, denote$$E_{\Gamma}=\left( I-P\right) E. \label{eq:E-gamma}$$ In an implementation, discrete harmonic functions are represented by the values of their degrees of freedom on substructure interfaces, cf., e.g. [@Toselli-2005-DDM]; hence, the definition (\[eq:E-gamma\]) serves formal purposes only, so that everything can be written in terms of discrete harmonic functions without passing to the matrix formulation.
\[[@Mandel-2007-ASF], BDDC on the reduced problem\]\[alg:bddc-elim-int\]Define the preconditioner $r\in\widehat{W}^{\prime}\longmapsto u\in\widehat
{W}$ by$$u=E_{\Gamma}w_{\Gamma},\quad w_{\Gamma}\in\widetilde{W}_{\Gamma}:a\left(
w_{\Gamma},z_{\Gamma}\right) =\left\langle r,E_{\Gamma}z_{\Gamma
}\right\rangle ,\quad\forall z_{\Gamma}\in\widetilde{W}_{\Gamma}.
\label{eq:bddc-elim-int-alg-1}$$
\[[@Mandel-2007-ASF]\]The BDDC preconditioner on the reduced problem in Algorithm \[alg:bddc-elim-int\] is the abstract Multispace BDDC from Algorithm \[alg:multispace-bddc\] with $M=1$ and the space and operator given by $$X=\widehat{W},\quad V_{1}=\widetilde{W}_{\Gamma},\quad Q_{1}=E_{\Gamma}.
\label{eq:bddc-elim-int-1}$$ Also, the assumptions of Corollary \[cor:multispace-bddc\] are satisfied.
We only need to note that the bilinear form $a(\cdot,\cdot)$ is positive definite on $\widetilde{W}_{\Gamma}\subset\widetilde{W}$ by (\[eq:pos-def\]), and the operator $E_{\Gamma}$ defined by (\[eq:E-gamma\]) is a projection by (\[eq:(I-P)E\]). The projection $E_{\Gamma}$ is onto $\widehat{W}$ because $E$ is onto $U$ by (\[eq:E-onto-U\]), and $I-P$ maps $U$ onto $\widehat{W}$ by the definition of $\widehat{W}$ in (\[eq:discrete-harm\]).
Using the decomposition (\[eq:dharm-dec\]), we can split the solution in the space $\widetilde{W}_{\Gamma}$ into the independent solution of two subproblems: mutually independent problems on substructures as the solution in the space $\widetilde{W}_{\Gamma\Delta}=(I-P)\widetilde{W}_{\Delta}$, and a solution of global coarse problem in the space $\widetilde{W}_{\Pi}$. The space $\widetilde{W}_{\Gamma}$ has a decomposition $$\widetilde{W}_{\Gamma}=\widetilde{W}_{\Gamma\Delta}\oplus\widetilde{W}_{\Pi
},\text{\quad and\quad}\widetilde{W}_{\Gamma\Delta}\perp_{a}\widetilde{W}_{\Pi}, \label{eq:dharm-dec-interface}$$ the same as the decomposition (\[eq:dharm-dec\]), and Algorithm \[alg:bddc-elim-int\] can be rewritten as follows.
\[[@Mandel-2005-ATP], BDDC on the reduced problem\]\[alg:bddc-elim-int-2\]Define the preconditioner $r\in\widehat{W}^{\prime
}\longmapsto u\in\widehat{W}$ by $u=E_{\Gamma}\left( w_{\Gamma\Delta}+w_{\Pi
}\right) $, where$$\begin{aligned}
w_{\Gamma\Delta} & \in\widetilde{W}_{\Gamma\Delta}:a\left( w_{\Gamma\Delta
},z_{\Gamma\Delta}\right) =\left\langle r,E_{\Gamma}z_{\Gamma\Delta
}\right\rangle ,\quad\forall z_{\Gamma\Delta}\in\widetilde{W}_{\Gamma\Delta
},\label{eq:discr-harm-subs-corr}\\
w_{\Pi} & \in\widetilde{W}_{\Pi}:a\left( w_{\Pi},z_{\Gamma\Pi}\right)
=\left\langle r,E_{\Gamma}z_{\Gamma\Pi}\right\rangle ,\quad\forall
z_{\Gamma\Pi}\in\widetilde{W}_{\Pi}. \label{eq:discr-harm-coarse-corr}$$
The BDDC preconditioner on the reduced problem in Algorithm \[alg:bddc-elim-int-2\] is the abstract Multispace BDDC from Algorithm \[alg:multispace-bddc\] with $M=2$ and the spaces and operators given by $$X=\widehat{W},\quad V_{1}=\widetilde{W}_{\Gamma\Delta},\quad V_{2}=\widetilde{W}_{\Pi},\quad Q_{1}=Q_{2}=E_{\Gamma}. \label{eq:bddc-elim-int-2}$$ Also, the assumptions of Corollary \[cor:multispace-bddc\] are satisfied.
Let $r\in\widehat{W}^{\prime}$. Define the vectors $v_{i}$, $i=1,2$ in Multispace BDDC by (\[def:abs-mult-BDDC\]) with $V_{i}$ and $Q_{i}$ given by (\[eq:bddc-elim-int-2\]). Let $u,$ $w_{\Gamma\Delta},$ $w_{\Pi}$ be the quantities in Algorithm \[alg:bddc-elim-int-2\], defined by (\[eq:discr-harm-subs-corr\])-(\[eq:discr-harm-coarse-corr\]). Using the decomposition (\[eq:dharm-dec-interface\]), any $w_{\Gamma}\in\widetilde
{W}_{\Gamma}$ can be written uniquely as $w_{\Gamma}=w_{\Gamma\Delta}$+$w_{\Pi}$ for some $w_{\Gamma\Delta}$ and $w_{\Pi}$ corresponding to (\[def:abs-mult-BDDC\]) as $v_{1}=w_{\Gamma\Delta}$ and $v_{2}=w_{\Pi}$, and $u=E_{\Gamma}\left( w_{\Gamma\Delta}+w_{\Pi}\right) $.
To verify the assumptions of Corollary \[cor:multispace-bddc\], note that the decomposition (\[eq:dharm-dec-interface\]) is $a-$orthogonal, $a(\cdot,\cdot)$ is positive definite on both $\widetilde{W}_{\Gamma\Delta}$ and $\widetilde{W}_{\Pi}$ as subspaces of $\widetilde{W}_{\Gamma}$ by (\[eq:pos-def\]), and $E_{\Gamma}$ is a projection by (\[eq:(I-P)E\]).
Next, we present a BDDC formulation on the space $U$ with explicit treatment of interior functions in the space $U_{I}$ as in [@Dohrmann-2003-PSC; @Mandel-2003-CBD], i.e., in the way the BDDC algorithm was originally formulated.
\[[@Dohrmann-2003-PSC; @Mandel-2003-CBD], original BDDC\]\[alg:original-bddc\] Define the preconditioner $r\in U^{\prime
}\longmapsto u\in U$ as follows. Compute the interior pre-correction$$u_{I}\in U_{I}:a\left( u_{I},z_{I}\right) =\left\langle r,z_{I}\right\rangle
,\quad\forall z_{I}\in U_{I}. \label{eq:int-corr}$$ Set up the updated residual$$r_{B}\in U^{\prime},\quad\left\langle r_{B},v\right\rangle =\left\langle
r,v\right\rangle -a\left( u_{I},v\right) ,\quad\forall v\in U.
\label{eq:def-rb}$$ Compute the substructure correction $$u_{\Delta}=Ew_{\Delta},\quad w_{\Delta}\in\widetilde{W}_{\Delta}:a\left(
w_{\Delta},z_{\Delta}\right) =\left\langle r_{B},Ez_{\Delta}\right\rangle
,\quad\forall z_{\Delta}\in\widetilde{W}_{\Delta}. \label{eq:subs-corr}$$ Compute the coarse correction$$u_{\Pi}=Ew_{\Pi},\quad w_{\Pi}\in\widetilde{W}_{\Pi}:a\left( w_{\Pi},z_{\Pi
}\right) =\left\langle r_{B},Ez_{\Pi}\right\rangle ,\quad\forall z_{\Pi}\in\widetilde{W}_{\Pi}. \label{eq:coarse}$$ Add the corrections$$u_{B}=u_{\Delta}+u_{\Pi}.$$ Compute the interior post-correction$$v_{I}\in U_{I}:a\left( v_{I},z_{I}\right) =a\left( u_{B},z_{I}\right)
,\quad\forall z_{I}\in U_{I}. \label{eq:int-post-corr}$$ Apply the combined corrections$$u=u_{B}-v_{I}+u_{I}. \label{eq:sol}$$
The interior corrections (\[eq:int-corr\]) and (\[eq:int-post-corr\]) decompose into independent Dirichlet problems, one for each substructure. The substructure correction (\[eq:subs-corr\]) decomposes into independent constrained Neumann problems, one for each substructure. Thus, the evaluation of the preconditioner requires three problems to be solved in each substructure, plus solution of the coarse problem (\[eq:coarse\]). In addition, the substructure corrections can be solved in parallel with the coarse problem.
As it is well known [@Dohrmann-2003-PSC], the first interior correction (\[eq:int-corr\]) can be omitted in the implementation by starting the iterations from an initial solution such that the residual in the interior of the substructures is zero, $$a\left( u,z_{I}\right) -\left\langle f_{X},z_{I}\right\rangle =0,\quad
\forall z_{I}\in U_{I},$$ i.e., such that the error is discrete harmonic. Then the output of the preconditioner is discrete harmonic and thus the errors in all the CG iterations (which are linear combinations of the original error and outputs from the preconditioner) are also discrete harmonic by induction.
The following proposition will be the starting point for the multilevel case.
\[prop:original-bddc-as-multispace\]The original BDDC preconditioner in Algorithm \[alg:original-bddc\] is the abstract Multispace BDDC from Algorithm \[alg:multispace-bddc\] with $M=3$ and the spaces and operators given by$$\begin{aligned}
X & =U,\quad V_{1}=U_{I},\quad V_{2}=(I-P)\widetilde{W}_{\Delta},\quad
V_{3}=\widetilde{W}_{\Pi},\label{eq:3-space}\\
Q_{1} & =I,\quad Q_{2}=Q_{3}=\left( I-P\right) E, \label{eq:3-proj}$$ and the assumptions of Corollary \[cor:multispace-bddc\] are satisfied.
Let $r\in U^{\prime}$. Define the vectors $v_{i}$, $i=1,2,3$, in Multispace BDDC by (\[def:abs-mult-BDDC\]) with the spaces $V_{i}$ given by (\[eq:3-space\]) and with the operators $Q_{i}$ given by (\[eq:3-proj\]). Let $u_{I}$, $r_{B}$, $w_{\Delta}$, $w_{\Pi}$, $u_{B}$, $v_{I}$, and $u$ be the quantities in Algorithm \[alg:original-bddc\], defined by (\[eq:int-corr\])-(\[eq:sol\]).
First, with $V_{1}=U_{I}$, the definition of $v_{1}$ in (\[def:abs-mult-BDDC\]) with $k=1$ is identical to the definition of $u_{I}$ in (\[eq:int-corr\]), so $u_{I}=v_{1}$.
Next, consider $w_{\Delta}\in\widetilde{W}_{\Delta}$ defined in (\[eq:subs-corr\]). We show that $w_{\Delta}$ satisfies (\[def:abs-mult-BDDC\]) with $k=2$, i.e., $v_{2}=w_{\Delta}$. So, let $z_{\Delta}\in\widetilde{W}_{\Delta}$ be arbitrary. From (\[eq:subs-corr\]) and (\[eq:def-rb\]), $$a\left( w_{\Delta},z_{\Delta}\right) =\left\langle r_{B},Ez_{\Delta
}\right\rangle =\left\langle r,Ez_{\Delta}\right\rangle -a\left(
u_{I},Ez_{\Delta}\right) . \label{eq:w-delta}$$ Now from the definition of $u_{I}$ by (\[eq:int-corr\]) and the fact that $PEz_{\Delta}\in U_{I}$, we get $$\left\langle r,PEz_{\Delta}\right\rangle -a\left( u_{I},PEz_{\Delta}\right)
=0, \label{eq:pez}$$ and subtracting (\[eq:pez\]) from (\[eq:w-delta\]) gives$$\begin{aligned}
a\left( w_{\Delta},z_{\Delta}\right) & =\left\langle r,\left( I-P\right)
Ez_{\Delta}\right\rangle -a\left( u_{I},\left( I-P\right) Ez_{\Delta
}\right) \\
& =\left\langle r,\left( I-P\right) Ez_{\Delta}\right\rangle ,\end{aligned}$$ because $a\left( u_{I},\left( I-P\right) Ez_{\Delta}\right) =0$ by orthogonality. To verify (\[def:abs-mult-BDDC\]), it is enough to show that $Pw_{\Delta}=0;$ then $w_{\Delta}\in(I-P)\widetilde{W}_{\Delta}=V_{2}$. Since $P$ is an $a$-orthogonal projection, it holds that$$a\left( Pw_{\Delta},Pw_{\Delta}\right) =a\left( w_{\Delta},Pw_{\Delta
}\right) =\left\langle r_{B},EPw_{\Delta}\right\rangle =0, \label{eq:APP}$$ where we have used $EU_{I}\subset U_{I}$ following the assumption (\[eq:int-unchanged\]) and the equality$$\left\langle r_{B},z_{I}\right\rangle =\left\langle r,z_{I}\right\rangle
-a\left( u_{I},z_{I}\right) =0$$ for any $z_{I}\in U_{I}$, which follows from (\[eq:def-rb\]) and (\[eq:int-corr\]). Since $a$ is positive definite on $\widetilde{W}\supset
U_{I} $ by assumption (\[eq:pos-def\]), it follows from (\[eq:APP\]) that $Pw_{\Delta}=0$.
In exactly the same way, from (\[eq:coarse\]) – (\[eq:sol\]), we get that if $w_{\Pi}\in$ $\widetilde{W}_{\Pi}$ is defined by (\[eq:coarse\]), then $v_{3}=w_{\Pi}$ satisfies (\[def:abs-mult-BDDC\]) with $k=3$. (The proof that $Pw_{\Pi}=0$ can be simplified but there is nothing wrong with proceeding exactly as for $w_{\Delta}$.)
Finally, from (\[eq:int-post-corr\]), $v_{I}=P\left( Ew_{\Delta}+Ew_{\Pi
}\right) $, so$$\begin{aligned}
u & =u_{I}+\left( u_{B}-v_{I}\right) \\
& =u_{I}+\left( I-P\right) Ew_{\Delta}+\left( I-P\right) Ew_{\Pi}\\
& =Q_{1}v_{1}+Q_{2}v_{2}+Q_{3}v_{3}.\end{aligned}$$
It remains to verify the assumptions of Corollary \[cor:multispace-bddc\].
First, the spaces $\widetilde{W}_{\Pi}$ and $\widetilde{W}_{\Delta}$ are $a
$-orthogonal by (\[eq:tilde-dec\]) and, from (\[eq:coarse-int\]), $$\left( I-P\right) \widetilde{W}_{\Delta}\subset\widetilde{W}_{\Delta},$$ thus $\left( I-P\right) \widetilde{W}_{\Delta}\perp_{a}\widetilde{W}_{\Pi}
$. Clearly, $\left( I-P\right) \widetilde{W}_{\Delta}\perp_{a}U_{I}$. Since $\widetilde{W}_{\Pi}$ consists of discrete harmonic functions from (\[eq:coarse-is-discrete-harmonic\]), so $\widetilde{W}_{\Pi}\perp_{a}U_{I}$, it follows that the spaces $V_{i}$, $i=1,2,3$, given by (\[eq:3-space\]), are $a$-orthogonal.
Next, $\left( I-P\right) E$ is by (\[eq:(I-P)E\]) a projection, and so are the operators $Q_{i}$ from (\[eq:3-proj\]).
It remains to prove the decomposition of unity (\[eq:dec-unity\]). Let$$u^{\prime}=u_{I}+w_{\Delta}+w_{\Pi}\in U,\quad u_{I}\in U_{I},\quad w_{\Delta
}\in\left( I-P\right) \widetilde{W}_{\Delta},\quad w_{\Pi}\in\widetilde
{W}_{\Pi}, \label{eq:dec-u}$$ and let$$v=u_{I}+\left( I-P\right) Ew_{\Delta}+\left( I-P\right) Ew_{\Pi}.$$ From (\[eq:dec-u\]), $w_{\Delta}+w_{\Pi}\in U$ since $u^{\prime}\in U$ and $u_{I}\in U_{I}\subset U$. Then $E\left( w_{\Delta}+w_{\Pi}\right)
=w_{\Delta}+w_{\Pi}$ by (\[eq:E-onto-U\]), so$$\begin{aligned}
v & =u_{I}+\left( I-P\right) E\left( w_{\Delta}+w_{\Pi}\right) \\
& =u_{I}+\left( I-P\right) \left( w_{\Delta}+w_{\Pi}\right) \\
& =u_{I}+w_{\Delta}+w_{\Pi}=u^{\prime},\end{aligned}$$ because both $w_{\Delta}$ and $w_{\Pi}$ are discrete harmonic.
The next Theorem shows an equivalence of the three Algorithms introduced above.
\[thm:equiv\]The eigenvalues of the preconditioned operators from Algorithm \[alg:bddc-elim-int\], and Algorithm \[alg:bddc-elim-int-2\] are exactly the same. They are also the same as the eigenvalues from Algorithm \[alg:original-bddc\], except possibly for multiplicity of eigenvalue equal to one.
From the decomposition (\[eq:dharm-dec-interface\]), we can write any $w\in\widetilde{W}_{\Gamma}$ uniquely as $w=w_{\Delta}+w_{\Pi}$ for some $w_{\Delta}\in\widetilde{W}_{\Gamma\Delta}$ and $w_{\Pi}\in\widetilde{W}_{\Pi
}$, so the preconditioned operators from Algorithms \[alg:bddc-elim-int\] and \[alg:bddc-elim-int-2\] are spectrally equivalent and we need only to show their spectral equivalence to the preconditioned operator from Algorithm \[alg:original-bddc\]. First, we note that the operator $A:U\mapsto U^{\prime}$ defined by (\[eq:def-A\]), and given in the block form as$$A=\left[
\begin{array}
[c]{cc}\mathcal{A}_{II} & \mathcal{A}_{I\Gamma}\\
\mathcal{A}_{\Gamma I} & \mathcal{A}_{\Gamma\Gamma}\end{array}
\right] ,$$ with blocks $$\begin{aligned}
\mathcal{A}_{II} & :U_{I}\rightarrow U_{I}^{\prime},\quad\mathcal{A}_{I\Gamma}:U_{I}\rightarrow\widehat{W}^{\prime},\\
\mathcal{A}_{\Gamma I} & :\widehat{W}\rightarrow U_{I}^{\prime},\quad\mathcal{A}_{\Gamma\Gamma}:\widehat{W}\rightarrow\widehat{W}^{\prime},\end{aligned}$$ is block diagonal and $\mathcal{A}_{\Gamma I}=\mathcal{A}_{I\Gamma}=0$ for any $u\in U$, written as $u=u_{I}+\widehat{w}$, because $U_{I}\perp_{a}\widehat
{W}$. Next, we note that the block $\mathcal{A}_{\Gamma\Gamma}:\widehat
{W}^{\prime}\rightarrow\widehat{W}$ is the Schur complement operator corresponding to the form $s$. Finally, since the block $\mathcal{A}_{II}$ is used only in the preprocessing step, the preconditioned operator from Algorithms \[alg:bddc-elim-int\] and \[alg:bddc-elim-int-2\] is simply $M_{\Gamma\Gamma}\mathcal{A}_{\Gamma\Gamma}:r\in\widehat{W}^{\prime
}\rightarrow$ $u\in\widehat{W}.$
Let us now turn to Algorithm \[alg:original-bddc\]. Let the residual $r\in
U$ be written as $r=r_{I}+r_{\Gamma}$, where $r_{I}\in U_{I}^{\prime}$ and $r_{\Gamma}\in\widehat{W}^{\prime}$. Taking $r_{\Gamma}=0$, we get $r=r_{I}$, and it follows that $r_{B}=u_{B}=v_{I}=0$, so $u=u_{I}$. On the other hand, taking $r=r_{\Gamma}$ gives $u_{I}=0$, $r_{B}=r_{\Gamma}$, $v_{I}=Pu_{B}$ and finally $u=\left( I-P\right) E(w_{\Delta}+w_{\Pi})$, so $u\in\widehat{W}$. This shows that the off-diagonal blocks of the preconditioner $M$ are zero, and therefore it is block diagonal $$M=\left[
\begin{array}
[c]{cc}M_{II} & 0\\
0 & M_{\Gamma\Gamma}\end{array}
\right] .$$ Next, let us take $u=u_{I},$ and consider $r_{\Gamma}=0$. The algorithm returns $r_{B}=u_{B}=v_{I}=0$, and finally $u=u_{I}$. This means that $M_{II}\mathcal{A}_{II}u_{I}=u_{I},$ so $M_{II}=\mathcal{A}_{II}^{-1}$. The operator $A:U\rightarrow U^{\prime}$, and the block preconditioned operator $MA:r\in U^{\prime}\rightarrow u\in U$ from Algorithm \[alg:original-bddc\] can be written, respectively, as$$A=\left[
\begin{array}
[c]{cc}\mathcal{A}_{II} & 0\\
0 & \mathcal{A}_{\Gamma\Gamma}\end{array}
\right] ,\quad MA=\left[
\begin{array}
[c]{cc}I & 0\\
0 & M_{\Gamma\Gamma}\mathcal{A}_{\Gamma\Gamma}\end{array}
\right] ,$$ where the right lower block $M_{\Gamma\Gamma}\mathcal{A}_{\Gamma\Gamma}:r\in\widehat{W}^{\prime}\rightarrow$ $u\in\widehat{W}$ is exactly the same as the preconditioned operator from Algorithms \[alg:bddc-elim-int\] and \[alg:bddc-elim-int-2\].
The BDDC condition number estimate is well known from [@Mandel-2003-CBD]. Following Theorem \[thm:equiv\] and Corollary \[cor:multispace-bddc\], we only need to estimate $\left\Vert \left( I-P\right) Ew\right\Vert _{a}$ on $\widetilde{W}$.
\[[@Mandel-2003-CBD]\]\[thm:element-bound\]The condition number of the original BDDC algorithm satisfies ${\kappa\leq\omega}$, where$$\omega=\max\left\{ {\sup_{w\in\widetilde{W}}\frac{\left\Vert \left(
I-P\right) Ew\right\Vert _{a}^{2}}{\left\Vert w\right\Vert _{a}^{2}},1}\right\} \text{ }\leq C\left( 1+\log\frac{H}{h}\right) ^{2}{.}
\label{eq:element-bound}$$
In [@Mandel-2003-CBD], the theorem was formulated by taking the supremum over the space of discrete harmonic functions $(I-P)\widetilde{W}$. However, the supremum remains the same by taking the larger space $\widetilde{W}\supset(I-P)\widetilde{W}$, since$${\frac{\left\Vert \left( I-P\right) Ew\right\Vert _{a}^{2}}{\left\Vert
w\right\Vert _{a}^{2}}\leq\frac{\left\Vert \left( I-P\right) E\left(
I-P\right) w\right\Vert _{a}^{2}}{\left\Vert \left( I-P\right) w\right\Vert
_{a}^{2}}}$$ from $E\left( I-P\right) =E$, which follows from (\[eq:int-unchanged\]), and from $\left\Vert w\right\Vert _{a}\geq\left\Vert \left( I-P\right)
w\right\Vert _{a}$, which follows from the $a$-orthogonality of the projection $P$.
Before proceeding into the Multilevel BDDC section, let us write concisely the spaces and operators involved in the two-level preconditioner as$$U_{I}{\textstyle\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{P}{\leftarrow}}{\subset}}U{\textstyle\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{E}{\leftarrow}}{\subset}}\widetilde{W}_{\Delta}\oplus\widetilde{W}_{\Pi}=\widetilde{W}\subset W.$$ We are now ready to extend this decomposition into the multilevel case.
Multilevel BDDC and an Abstract Bound {#sec:multilevel-bddc}
=====================================
In this section, we generalize the two-level BDDC preconditioner to multiple levels, using the abstract Multispace BDDC framework from Algorithm \[alg:multispace-bddc\]. The substructuring components from Section \[sec:one-level\] will be denoted by an additional subscript $_{1},$ as $\Omega_{1}^{s},$ $s=1,\ldots N_{1}$, etc., and called level $1$. The level $1$ coarse problem (\[eq:coarse\]) will be called the level $2$ problem. It has the same finite element structure as the original problem (\[eq:problem\]) on level $1$, so we put $U_{2}=\widetilde{W}_{\Pi1}$. Level $1 $ substructures are level $2$ elements and level $1$ coarse degrees of freedom are level $2$ degrees of freedom. Repeating this process recursively, level $i-1$ substructures become level $i$ elements, and the level $i$ substructures are agglomerates of level $i$ elements. Level $i$ substructures are denoted by $\Omega_{i}^{s},$ $s=1,\ldots,N_{i},$ and they are assumed to form a conforming triangulation with a characteristic substructure size $H_{i}$. For convenience, we denote by $\Omega_{0}^{s}$ the original finite elements and put $H_{0}=h$. The interface$~\Gamma_{i}$ on level$~i$ is defined as the union of all level$~i$ boundary nodes, i.e., nodes shared by at least two level$~i$ substructures, and we note that $\Gamma
_{i}\subset\Gamma_{i-1}$. Level $i-1$ coarse degrees of freedom become level $i$ degrees of freedom. The shape functions on level $i$ are determined by minimization of energy with respect to level $i-1$ shape functions, subject to the value of exactly one level $i$ degree of freedom being one and others level $i$ degrees of freedom being zero. The minimization is done on each level $i$ element (level $i-1$ substructure) separately, so the values of level $i-1$ degrees of freedom are in general discontinuous between level $i-1$ substructures, and only the values of level $i$ degrees of freedom between neighboring level $i$ elements coincide.
The development of the spaces on level $i$ now parallels the finite element setting in Section \[sec:FE-setting\]. Denote $U_{i}=\widetilde{W}_{\Pi
i-1}$. Let $W_{i}^{s}$ be the space of functions on the substructure $\Omega_{i}^{s}$, such that all of their degrees of freedom on $\partial
\Omega_{i}^{s}\cap\partial\Omega$ are zero, and let$$W_{i}=W_{i}^{1}\times\cdots\times W_{i}^{N_{i}}.$$ Then $U_{i}\subset W_{i}$ is the subspace of all functions from $W$ that are continuous across the interfaces $\Gamma_{i}$. Define $U_{Ii}\subset U_{i}$ as the subspace of functions that are zero on$~\Gamma_{i}$, i.e., the functions interior to the level$~i$ substructures. Denote by $P_{i}$ the energy orthogonal projection from $W_{i} $ onto $U_{Ii}$,$$P_{i}:w_{i}\in W_{i}\longmapsto v_{Ii}\in U_{Ii}:a\left( v_{Ii},z_{Ii}\right) =a\left( w_{i},z_{Ii}\right) ,\quad\forall z_{Ii}\in
U_{Ii}.$$ Functions from $\left( I-P_{i}\right) W_{i}$, i.e., from the nullspace of $P_{i},$ are called discrete harmonic on level $i$; these functions are $a$-orthogonal to $U_{Ii}$ and energy minimal with respect to increments in $U_{Ii}$. Denote by $\widehat{W}_{i}\subset U_{i}$ the subspace of discrete harmonic functions on level$~i$, that is $$\widehat{W}_{i}=\left( I-P_{i}\right) U_{i}. \label{eq:discrete-harm-ML}$$ In particular, $U_{Ii}\perp_{a}\widehat{W}_{i}$. Define $\widetilde{W}_{i}\subset W_{i}$ as the subspace of all functions such that the values of any coarse degrees of freedom on level$~i$ have a common value for all relevant level$~i$ substructures and vanish on $\partial\Omega_{i}^{s}\cap\partial\Omega,$ and $\widetilde{W}_{\Delta i}\subset W_{i}$ as the subspace of all functions such that their level $i$ coarse degrees of freedom vanish. Define $\widetilde{W}_{\Pi i}$ as the subspace of all functions such that their level $i$ coarse degrees of freedom between adjacent substructures coincide, and such that their energy is minimal. Clearly, functions in $\widetilde{W}_{\Pi i}$ are uniquely determined by the values of their level $i$ coarse degrees of freedom, and $$\widetilde{W}_{\Delta i}\perp_{a}\widetilde{W}_{\Pi i},\text{\quad}\widetilde{W}_{i}=\widetilde{W}_{\Delta i}\oplus\widetilde{W}_{\Pi i}.
\label{eq:tilde-dec-ML}$$ We assume that the level$~i$ coarse degrees of freedom are zero on all functions from $U_{Ii}$, that is,$$U_{Ii}\subset\widetilde{W}_{\Delta i}. \label{eq:coarse-int-ML}$$ In other words, level $i$ coarse degrees of freedom depend on the values on level$~i$ substructure boundaries only. From (\[eq:tilde-dec-ML\]) and (\[eq:coarse-int-ML\]), it follows that the functions in $\widetilde{W}_{\Pi
i}$ are discrete harmonic on level$~i$, that is$$\widetilde{W}_{\Pi i}=\left( I-P_{i}\right) \widetilde{W}_{\Pi i}.
\label{eq:coarse-is-discrete-harmonic-ML}$$ Let $E$ be a projection from $\widetilde{W}_{i}$ onto $U_{i}$, defined by taking some weighted average on $\Gamma_{i}$$$E_{i}:\widetilde{W}_{i}\rightarrow U_{i},\quad E_{i}U_{Ii}=U_{Ii},\quad
E_{i}^{2}=E_{i}.$$ Since projection is the identity on its range, $E_{i}$ does not change the level$~i$ interior degrees of freedom, in particular$$E_{i}U_{Ii}=U_{Ii}. \label{eq:int-unchanged-ML}$$
Finally, we introduce an interpolation $I_{i}:U_{i}\rightarrow\widetilde
{U}_{i}$ from level $i$ degrees of freedom to functions in some classical finite element space $\widetilde{U}_{i}$ with the same degrees of freedom as $U_{i}$. The space $\widetilde{U}_{i}$ will be used for comparison purposes, to invoke known inequalities for finite elements. A more detailed description of the properties of $I_{i}$ and the spaces $\widetilde{U}_{i}$ is postponed to the next section.
The hierarchy of spaces and operators is shown concisely in Figure \[fig:multi-bddc\]. The Multilevel BDDC method is defined recursively [@Dohrmann-2003-PSC; @Mandel-2007-OMB] by solving the coarse problem on level $i $ only approximately, by one application of the preconditioner on level $i-1$. Eventually, at level, $L-1$, the coarse problem, which is the level $L$ problem, is solved exactly. We need a more formal description of the method here, which is provided by the following algorithm.
$$\fbox{$\begin{array}
[c]{ccccccccccccccc}
& & U & = & \widetilde{W}_{\Pi0} & & & & & & & & & & \\
& & \shortparallel & & & & & & & & & & & & \\
U_{I1} & {\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{P_{1}}{\leftarrow}}{\subset
}} & U_{1} & {\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{E_{1}}{\leftarrow
}}{\subset}} & \widetilde{W}_{\Pi1} & \oplus & \widetilde{W}_{\Delta1} & = &
\widetilde{W}_{1} & \subset & W_{1} & & & & \\
& & & & \shortparallel & & & & & & & & & & \\
& & U_{I2} & {\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{P_{2}}{\leftarrow
}}{\subset}} & U_{2} & {\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{E_{2}}{\leftarrow}}{\subset}} & \widetilde{W}_{\Pi2} & \oplus & \widetilde
{W}_{\Delta2} & = & \widetilde{W}_{2} & \subset & W_{2} & & \\
& & & & {\scriptstyle\ \downarrow I_{2}} & & \shortparallel & & & & &
& & & \\
& & & & \widetilde{U}_{2} & & \vdots & & & & & & & & \\
& & & & & & \shortparallel & & & & & & & & \\
& & & & U_{I,L-1} & {\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{P_{L-1}}{\leftarrow}}{\subset}} & U_{L-1} &
{\genfrac{}{}{0pt}{}{\genfrac{}{}{0pt}{}{E_{L-1}}{\leftarrow}}{\subset}} &
\widetilde{W}_{\Pi L-1} & \oplus & \widetilde{W}_{\Delta L-1} & = &
\widetilde{W}_{L-1} & \subset & W_{L-1}\\
& & & & & & {\scriptstyle\ \downarrow I_{L-1}} & & \shortparallel & &
& & & & \\
& & & & & & \widetilde{U}_{L-1} & & U_{L} & & & & & & \\
& & & & & & & & {\scriptstyle\ \downarrow I_{L}} & & & & & & \\
& & & & & & & & \widetilde{U}_{L} & & & & & &
\end{array}
$}$$
\[Multilevel BDDC\]\[alg:multilevel-bddc\]Define the preconditioner $r_{1}\in U_{1}^{\prime}\longmapsto u_{1}\in U_{1}$ as follows:
**for** $i=1,\ldots,L-1$**,**
Compute interior pre-correction on level $i$,$$u_{Ii}\in U_{Ii}:a\left( u_{Ii},z_{Ii}\right) =\left\langle r_{i},z_{Ii}\right\rangle ,\quad\forall z_{Ii}\in U_{Ii}. \label{eq:ML-uIi}$$
Get updated residual on level $i$,$$r_{Bi}\in U_{i},\quad\left\langle r_{Bi},v_{i}\right\rangle =\left\langle
r_{i},v_{i}\right\rangle -a\left( u_{Ii},v_{i}\right) ,\quad\forall v_{i}\in
U_{i}. \label{eq:ML-rBi}$$
Find the substructure correction on level $i$: $$w_{\Delta i}\in W_{\Delta i}:a\left( w_{\Delta i},z_{\Delta i}\right)
=\left\langle r_{Bi},E_{i}z_{\Delta i}\right\rangle ,\quad\forall z_{\Delta
i}\in W_{\Delta i}. \label{eq:ML-wDi}$$
Formulate the coarse problem on level $i$, $$w_{\Pi i}\in W_{\Pi i}:a\left( w_{\Pi i},z_{\Pi i}\right) =\left\langle
r_{Bi},E_{i}z_{\Pi i}\right\rangle ,\quad\forall z_{\Pi i}\in W_{\Pi i},
\label{eq:ML-coarse}$$
If $\ i=L-1$, solve the coarse problem directly and set $u_{L}=w_{\Pi
L-1}$, otherwise set up the right-hand side for level $i+1$,$$r_{i+1}\in\widetilde{W}_{\Pi i}^{\prime},\quad\left\langle r_{i+1},z_{i+1}\right\rangle =\left\langle r_{Bi},E_{i}z_{i+1}\right\rangle
,\quad\forall z_{i+1}\in\widetilde{W}_{\Pi i}=U_{i+1}, \label{eq:ML-ri+1}$$
**end.**
**for** $i=L-1,\ldots,1\mathbf{,}$** **
Average the approximate corrections on substructure interfaces on level $i$,$$u_{Bi}=E_{i}\left( w_{\Delta i}+u_{i+1}\right) . \label{eq:ML-uBi-1}$$
Compute the interior post-correction on level $i$,$$v_{Ii}\in U_{Ii}:a\left( v_{Ii},z_{Ii}\right) =a\left( u_{Bi},z_{Ii}\right) ,\quad\forall z_{Ii}\in U_{Ii}. \label{eq:ML-vIi}$$
Apply the combined corrections, $$u_{i}=u_{Ii}+u_{Bi}-v_{Ii}. \label{eq:ML-ui}$$
**end.**
We can now show that the Multilevel BDDC can be cast as the Multispace BDDC on energy orthogonal spaces, using the hierarchy of spaces from Figure \[fig:multi-bddc\].
\[lem:bddc-ML-spaces\]The Multilevel BDDC preconditioner in Algorithm \[alg:multilevel-bddc\] is the abstract Multispace BDDC preconditioner from Algorithm \[alg:multispace-bddc\] with $M=2L-1,$ and the spaces and operators$$\begin{aligned}
X & =U_{1},\quad V_{1}=U_{I1},\quad V_{2}=(I-P_{1})\widetilde{W}_{\Delta
1},\quad V_{3}=U_{I2},\nonumber\\
V_{4} & =(I-P_{2})\widetilde{W}_{\Delta2},\quad V_{5}=U_{I3},\quad
\ldots\label{eq:ML-spaces}\\
V_{2L-4} & =(I-P_{L-2})\widetilde{W}_{\Delta L-2},\quad V_{2L-3}=U_{IL-1},\nonumber\\
V_{2L-2} & =(I-P_{L-1})\widetilde{W}_{\Delta L-1},\quad V_{2L-1}=\widetilde{W}_{\Pi L-1},\nonumber\end{aligned}$$$$\begin{aligned}
Q_{1} & =I,\quad Q_{2}=Q_{3}=\left( I-P_{1}\right) E_{1},\quad\nonumber\\
Q_{4} & =Q_{5}=\left( I-P_{1}\right) E_{1}\left( I-P_{2}\right)
E_{2},\quad\ldots\label{eq:ML-operators}\\
Q_{2L-4} & =Q_{2L-3}=\left( I-P_{1}\right) E_{1}\,\cdots\,\left(
I-P_{L-2}\right) E_{L-2},\nonumber\\
Q_{2L-2} & =Q_{2L-1}=\left( I-P_{1}\right) E_{1}\,\cdots\,\left(
I-P_{L-1}\right) E_{L-1},\nonumber\end{aligned}$$ and the assumptions of Corollary \[cor:multispace-bddc\] are satisfied.
Let $r_{1}\in U_{1}^{\prime}$. Define the vectors $v_{k},$ $k=1,\ldots
,2L-1$ by (\[def:abs-mult-BDDC\]) with the spaces and operators given by (\[eq:ML-spaces\])-(\[eq:ML-operators\]), and let $u_{Ii}$, $r_{Bi}$, $w_{\Delta i}$, $w_{\Pi i}$, $r_{i+1}$, $u_{Bi}$, $v_{Ii}$, and $u_{i}$ be the quantities in Algorithm \[alg:multilevel-bddc\], defined by (\[eq:ML-uIi\])-(\[eq:ML-ui\]).
First, with $V_{1}=U_{I1}$, the definition of $v_{1}$ in (\[def:abs-mult-BDDC\]) is (\[eq:ML-uIi\]) with $i=1$ and $u_{I1}=v_{1}$. We show that in general, for level $i=1,\ldots,L-1$, and space $k=2i-1,$ we get (\[def:abs-mult-BDDC\]) with $V_{k}=U_{Ii}$, so that $v_{k}=u_{Ii}$ and in particular $v_{2L-3}=u_{IL-1}$. So, let $z_{Ii}\in U_{Ii}$, $i=2,\ldots,L-1,$ be arbitrary. From (\[eq:ML-uIi\]) using (\[eq:ML-ri+1\]) and (\[eq:ML-rBi\]), $$\begin{aligned}
a(u_{Ii},z_{Ii}) & =\left\langle r_{i},z_{Ii}\right\rangle =\left\langle
r_{Bi-1},E_{i-1}z_{Ii}\right\rangle =\label{eq:PF-uIi}\\
& =\left\langle r_{i-1},E_{i-1}z_{Ii}\right\rangle -a\left( u_{Ii-1},E_{i-1}z_{Ii}\right) .\nonumber\end{aligned}$$ Since from (\[eq:ML-uIi\]) using the fact that $P_{i-1}E_{i-1}z_{Ii}\in
U_{Ii-1}$ it follows that$$\left\langle r_{i-1},P_{i-1}E_{i-1}z_{Ii}\right\rangle -a\left(
u_{Ii-1},P_{i-1}E_{i-1}z_{Ii}\right) =0,$$ we get from (\[eq:PF-uIi\]),$$a(u_{Ii},z_{Ii})=\left\langle r_{i-1},\left( I-P_{i-1}\right) E_{i-1}z_{Ii}\right\rangle -a\left( u_{Ii-1},(I-P_{i-1})E_{i-1}z_{Ii}\right) ,$$ and because $a\left( u_{Ii-1},(I-P_{i-1})E_{i-1}z_{Ii}\right) =0$ by orthogonality, we get$$a(u_{Ii},z_{Ii})=\left\langle r_{i-1},\left( I-P_{i-1}\right) E_{i-1}z_{Ii}\right\rangle .$$ Repeating this process recursively using (\[eq:PF-uIi\]), we finally get$$\begin{aligned}
a(u_{Ii},z_{Ii}) & =\left\langle r_{i-1},\left( I-P_{i-1}\right)
E_{i-1}z_{Ii}\right\rangle =\quad...\\
& =\left\langle r_{1},\left( I-P_{1}\right) E_{1}\,\cdots\,\left(
I-P_{i-1}\right) E_{i-1}z_{Ii}\right\rangle .\end{aligned}$$
Next, consider $w_{\Delta i}\in\widetilde{W}_{\Delta i}$ defined by (\[eq:ML-wDi\]). We show that for $i=1,\ldots,L-1$, and $k=2i$, we get (\[def:abs-mult-BDDC\]) with $V_{k}=\widetilde{W}_{\Delta i}$, so that $v_{k}=w_{\Delta i}$ and in particular $v_{2L-2}=w_{\Delta L-1}$. So, let $z_{\Delta i}\in\widetilde{W}_{\Delta i}$ be arbitrary. From (\[eq:ML-wDi\]) using (\[eq:ML-rBi\]), $$a\left( w_{\Delta i},z_{\Delta i}\right) =\left\langle r_{Bi},E_{i}z_{\Delta
i}\right\rangle =\left\langle r_{i},E_{i}z_{\Delta i}\right\rangle -a\left(
u_{Ii},E_{i}z_{\Delta i}\right) . \label{eq:PF-wDelta-i}$$ From the definition of $u_{Ii}$ by (\[eq:ML-uIi\]) and since $P_{i}E_{i}z_{\Delta i}\in U_{Ii}$ it follows that $$\left\langle r_{i},P_{i}E_{i}z_{\Delta i}\right\rangle -a\left( u_{Ii},P_{i}E_{i}z_{\Delta i}\right) =0,$$ so (\[eq:PF-wDelta-i\]) gives$$a\left( w_{\Delta i},z_{\Delta i}\right) =\left\langle r_{i},\left(
I-P_{i}\right) E_{i}z_{\Delta i}\right\rangle -a\left( u_{Ii},\left(
I-P_{i}\right) E_{i}z_{\Delta i}\right) .$$ Next, because $a\left( u_{Ii},\left( I-P_{i}\right) E_{i}z_{\Delta
i}\right) =0$ by orthogonality, and using (\[eq:ML-ri+1\]), $$a\left( w_{\Delta i},z_{\Delta i}\right) =\left\langle r_{i},\left(
I-P_{i}\right) E_{i}z_{\Delta i}\right\rangle =\left\langle r_{Bi-1},E_{i-1}\left( I-P_{i}\right) E_{i}z_{\Delta i}\right\rangle .$$ Repeating this process recursively, we finally get$$\begin{aligned}
a\left( w_{\Delta i},z_{\Delta i}\right) & =\left\langle r_{i},\left(
I-P_{i}\right) E_{i}z_{\Delta i}\right\rangle =\quad\ldots\nonumber\\
& =\left\langle r_{1},\left( I-P_{1}\right) E_{1}\,\cdots\,\left(
I-P_{i}\right) E_{i}z_{\Delta i}\right\rangle .\nonumber\end{aligned}$$ To verify (\[def:abs-mult-BDDC\]), it remains to show that $P_{i}w_{\Delta
i}=0;$ then $w_{\Delta i}\in(I-P_{i})\widetilde{W}_{\Delta i}=V_{k}$. Since $P_{i}$ is an $a$-orthogonal projection, it holds that$$a\left( P_{i}w_{\Delta i},P_{i}w_{\Delta i}\right) =a\left( w_{\Delta
i},P_{i}w_{\Delta i}\right) =\left\langle r_{Bi},E_{i}P_{i}w_{\Delta
i}\right\rangle =0,$$ where we have used $E_{i}U_{Ii}\subset U_{Ii}$ following the assumption (\[eq:int-unchanged-ML\]) and the equality$$\left\langle r_{Bi},z_{Ii}\right\rangle =\left\langle r_{i},z_{Ii}\right\rangle -a\left( u_{Ii},z_{Ii}\right) =0$$ for any $z_{Ii}\in U_{Ii}$, which follows from (\[eq:ML-uIi\]) and (\[eq:ML-rBi\]).
In exactly the same way, we get that if $w_{\Pi L-1}\in\widetilde{W}_{\Pi
L-1}$ is defined by (\[eq:ML-coarse\]), then $v_{2L-1}=w_{\Pi L-1}$ satisfies (\[def:abs-mult-BDDC\]) with $k=2L-1$.
Finally, from (\[eq:ML-uBi-1\])-(\[eq:ML-ui\]) for any $i=L-2,\ldots,1$, we get $$\begin{aligned}
u_{i} & =u_{Ii}+u_{Bi}-v_{Ii}\\
& =u_{Ii}+\left( I-P_{i}\right) E_{i}\left( w_{\Delta i}+u_{i+1}\right) \\
& =u_{Ii}+\left( I-P_{i}\right) E_{i}\left[ w_{\Delta i}+u_{Ii+1}+\left(
I-P_{i+1}\right) E_{i+1}\left( w_{\Delta i+1}+u_{i+2}\right) \right] \\
& =u_{Ii}+\\
& +\left( I-P_{i}\right) E_{i}\left[ w_{\Delta i}+\ldots+\left(
I-P_{L-1}\right) E_{L-1}\left( w_{\Delta L-1}+u_{\Pi L-1}\right) \right] ,\end{aligned}$$
and, in particular for $u_{1}$, $$\begin{aligned}
u_{1} & =u_{I1}+\\
& +\left( I-P_{1}\right) E_{1}\left[ w_{\Delta1}+\ldots+\left(
I-P_{L-1}\right) E_{L-1}\left( w_{\Delta L-1}+u_{\Pi L-1}\right) \right] \\
& =Q_{1}v_{1}+Q_{2}v_{2}+\ldots+Q_{2L-2}v_{2L-2}+Q_{2L-1}v_{2L-1}.\end{aligned}$$
It remains to verify the assumptions of Corollary \[cor:multispace-bddc\].
The spaces $\widetilde{W}_{\Pi i}$ and $\widetilde{W}_{\Delta i}$, for all $i=1,\ldots,L-1$, are $a$-orthogonal by (\[eq:tilde-dec-ML\]) and from (\[eq:coarse-int-ML\]), $$\left( I-P_{i}\right) \widetilde{W}_{\Delta i}\subset\widetilde{W}_{\Delta
i},$$ thus $\left( I-P_{i}\right) \widetilde{W}_{\Delta i}$ is $a$-orthogonal to $\widetilde{W}_{\Pi i}$. Since$\ \widetilde{W}_{\Pi i}=U_{i+1}$ consists of discrete harmonic functions on level$~i$ from (\[eq:coarse-is-discrete-harmonic-ML\]), and $U_{Ii+1}\subset U_{i+1}$, it follows by induction that the spaces $V_{k}$, given by (\[eq:ML-spaces\]), are $a$-orthogonal.
We now show that the operators $Q_{k}$ defined by (\[eq:ML-operators\]) are projections. From our definitions, coarse degrees of freedom on substructuring level $i$ (from which we construct the level$i+1$ problem) depend only on the values of degrees of freedom on the interface$~\Gamma_{i}$ and $\Gamma_{j}\subset\Gamma_{i}$ for $j\geq i$. Then,$$(I-P_{j})E_{j}(I-P_{i})E_{i}(I-P_{j})E_{j}=(I-P_{i})E_{i}(I-P_{j})E_{j}.
\label{eq:DH-unchanged-ML}$$
Using (\[eq:DH-unchanged-ML\]) and since $\left( I-P_{1}\right) E_{1}$ is a projection by (\[eq:(I-P)E\]), we get$$\begin{aligned}
\left[ \left( I-P_{1}\right) E_{1}\cdots\left( I-P_{i}\right)
E_{i}\right] ^{2} & =\left( I-P_{1}\right) E_{1}\left( I-P_{1}\right)
E_{1}\cdots\left( I-P_{i}\right) E_{i}\\
& =\left( I-P_{1}\right) E_{1}\cdots\left( I-P_{i}\right) E_{i},\end{aligned}$$ so the operators $Q_{k}$ from (\[eq:ML-operators\]) are projections.
It remains to prove the decomposition of unity (\[eq:dec-unity\]). Let $u_{i}\in U_{i}$, such that$$\begin{aligned}
u_{i}^{\prime} & =u_{Ii}+w_{\Delta i}+u_{i+1},\qquad\label{eq:pf-dec-u}\\
u_{Ii} & \in U_{Ii},\quad w_{\Delta i}\in\left( I-P_{i}\right)
\widetilde{W}_{\Delta i},\quad u_{i+1}\in U_{i+1}$$ and$$v_{i}=u_{Ii}+\left( I-P_{i}\right) E_{i}w_{\Delta i}+\left( I-P_{i}\right)
E_{i}u_{i+1}. \label{eq:pf-dec-v}$$ From (\[eq:pf-dec-u\]), $w_{\Delta i}+u_{i+1}\in U_{i}$ since $u_{i}\in
U_{i}$ and $u_{Ii}\in U_{Ii}\subset U_{i}$. Then $E_{i}\left[ w_{\Delta
i}+u_{i+1}\right] =w_{\Delta i}+u_{i+1}$ by (\[eq:int-unchanged-ML\]), so$$\begin{aligned}
v_{i} & =u_{Ii}+(I-P_{i})E_{i}\left[ w_{\Delta i}+u_{i+1}\right]
=u_{Ii}+(I-P_{i})\left[ w_{\Delta i}+u_{i+1}\right] =\\
& =u_{Ii}+w_{\Delta i}+u_{i+1}=u_{Ii}+w_{\Delta i}+u_{i+1}=u_{i}^{\prime},\end{aligned}$$ because $w_{\Delta i}$ and $u_{i+1}$ are discrete harmonic on level $i$. The fact that $u_{i+1}$ in (\[eq:pf-dec-u\]) and (\[eq:pf-dec-v\]) are the same on arbitrary level $i$ can be proved in exactly the same way using induction and putting $u_{i+1}$ in (\[eq:pf-dec-u\]) as$$\begin{aligned}
u_{i+1} & =u_{Ii+1}+\ldots+w_{\Delta L-1}+w_{\Pi L-1},\\
u_{Ii+1} & \in U_{Ii+1},\quad w_{\Delta L-1}\in\left( I-P_{L-1}\right)
\widetilde{W}_{\Delta L-1},\quad w_{\Pi L-1}\in\widetilde{W}_{\Pi L-1},\end{aligned}$$ and in (\[eq:pf-dec-v\]) as $$u_{i+1}=u_{Ii+1}+\ldots+\left( I-P_{i+1}\right) E_{i+1}\cdots\left(
I-P_{L-1}\right) E_{L-1}\left( w_{\Delta L-1}+w_{\Pi L-1}\right) .\;$$
The following bound follows from writing of the Multilevel BDDC as Multispace BDDC in Lemma \[lem:bddc-ML-spaces\] and the estimate for Multispace BDDC in Corollary \[cor:multispace-bddc\].
\[lem:bddc-ML-estimate-MLspaces\]If for some $\omega\geq1$,$$\begin{aligned}
\left\Vert (I-P_{1})E_{1}w_{\Delta1}\right\Vert _{a}^{2} & \leq
\omega\left\Vert w_{\Delta1}\right\Vert _{a}^{2}\quad\forall w_{\Delta1}\in\left( I-P_{1}\right) \widetilde{W}_{\Delta1},\nonumber\\
\left\Vert (I-P_{1})E_{1}u_{I2}\right\Vert _{a}^{2} & \leq\omega\left\Vert
u_{I2}\right\Vert _{a}^{2}\quad\forall u_{I2}\in U_{I2},\nonumber\\
& \ldots\label{eq:est-omega-MLspaces}\\
\left\Vert (I-P_{1})E_{1}\,\cdots\,(I-P_{L-1})E_{L-1}w_{\Pi L-1}\right\Vert
_{a}^{2} & \leq\omega\left\Vert w_{\Pi L-1}\right\Vert _{a}^{2}\quad\forall
w_{\Pi L-1}\in\widetilde{W}_{\Pi L-1},\nonumber\end{aligned}$$ then the Multilevel BDDC preconditioner (Algorithm \[alg:multilevel-bddc\]) satisfies $\kappa\leq\omega.$
Choose the spaces and operators as in (\[eq:ML-spaces\])-(\[eq:ML-operators\]) so that $u_{I1}=v_{1}\in V_{1}=U_{I1}$, $w_{\Delta
1}=v_{2}\in V_{2}=\left( I-P_{1}\right) \widetilde{W}_{\Delta1}$, $\ldots$, $w_{\Pi L-1}=v_{2L-1}\in V_{2L-1}=\widetilde{W}_{\Pi L-1}$. The bound now follows from Corollary \[cor:multispace-bddc\].
\[lem:bddc-ML-estimate\]If for some $\omega_{i}\geq1$, $$\left\Vert (I-P_{i})E_{i}w_{i}\right\Vert _{a}^{2}\leq\omega_{i}\left\Vert
w_{i}\right\Vert _{a}^{2},\quad\forall w_{i}\in\widetilde{W}_{i},\quad
i=1,\ldots,L-1, \label{eq:est-omega-k}$$ then the Multilevel BDDC preconditioner (Algorithm \[alg:multilevel-bddc\]) satisfies $\kappa\leq{\textstyle\prod_{i=1}^{L-1}}
\omega_{i}.$
Note from Lemma \[lem:bddc-ML-estimate-MLspaces\] that $\left(
I-P_{1}\right) \widetilde{W}_{\Delta1}\subset\widetilde{W}_{\Delta1}\subset\widetilde{W}_{1}$, $U_{I2}\subset\widetilde{W}_{\Pi1}\subset
\widetilde{W}_{1}$, and generally $\left( I-P_{i}\right) \widetilde
{W}_{\Delta i}\subset\widetilde{W}_{\Delta i}\subset\widetilde{W}_{i}$, $U_{Ii+1}\subset\widetilde{W}_{\Pi i}\subset\widetilde{W}_{i}.$
Condition Number Bound for the Model Problem {#sec:multilevel-condition}
============================================
Let $\left\vert w\right\vert _{a(\Omega_{i}^{s})}$ be the energy norm of a function $w\in\widetilde{W}_{\Pi i},$ $i=1,\ldots,L-1,$ restricted to subdomain $\Omega_{i}^{s},$ $s=1,\ldots N_{i}$, i.e., $\left\vert w\right\vert
_{a(\Omega_{i}^{s})}^{2}=\int_{\Omega_{i}^{s}}\nabla w\nabla w\,dx,$ and let $\left\Vert w\right\Vert _{a}$ be the norm obtained by piecewise integration over each $\Omega_{i}^{s}$. To apply Lemma \[lem:bddc-ML-estimate\] to the model problem presented in Section \[sec:one-level\], we need to generalize the estimate from Theorem \[thm:element-bound\] to coarse levels. To this end, let $I_{i+1}:\widetilde{W}_{\Pi i}\rightarrow\widetilde{U}_{i+1}$ be an interpolation from the level$~i$ coarse degrees of freedom (i.e., level $i+1$ degrees of freedom) to functions in another space $\widetilde{U}_{i+1}$ and assume that, for all $i=1,\ldots,L-1,$ and $s=1,\ldots,N_{i},$ the interpolation satisfies for all $w\in\widetilde{W}_{\Pi i}$ and for all $\Omega_{i+1}^{s}$ the equivalence $$c_{i,1}\left\vert I_{i+1}w\right\vert _{a(\Omega_{i+1}^{s})}^{2}\leq\left\vert
I_{i}w\right\vert _{a(\Omega_{i+1}^{s})}^{2}\leq c_{i,2}\left\vert
I_{i+1}w\right\vert _{a(\Omega_{i+1}^{s})}^{2}, \label{eq:Ii-equiv}$$ which implies by Lemma \[lem:equivalence\] also the equivalence$$c_{i,1}\left\vert I_{i+1}w\right\vert _{H^{1/2}(\partial\Omega_{i+1}^{s})}^{2}\leq\left\vert I_{i}w\right\vert _{H^{1/2}(\partial\Omega_{i+1}^{s})}^{2}\leq c_{i,2}\left\vert I_{i+1}w\right\vert _{H^{1/2}(\partial\Omega
_{i+1}^{s})}^{2}, \label{eq:Ii-equiv-seminorm}$$ with $c_{i,2}/c_{i,1}\leq\operatorname*{const}$ bounded independently of $H_{0},\ldots,H_{i+1}$.
Since $I_{1}=I$, the two norms are the same on $\widetilde{W}_{\Pi
0}=\widetilde{U}_{1}=U_{1}.$
For the three-level BDDC in two dimensions, the result of Tu [@Tu-2007-TBT Lemma 4.2], which is based on the lower bound estimates by Brenner and Sung [@Brenner-2000-LBN], can be written in our settings for all $w\in\widetilde{W}_{\Pi1}$ and for all $\Omega_{2}^{s}$ as$$c_{1,1}\left\vert I_{2}w\right\vert _{a(\Omega_{2}^{s})}^{2}\leq\left\vert
w\right\vert _{a(\Omega_{2}^{s})}^{2}\leq c_{1,2}\left\vert I_{2}w\right\vert
_{a(\Omega_{2}^{s})}^{2}, \label{eq:one-equiv-2D}$$ where $I_{2}$ is a piecewise (bi)linear interpolation given by values at corners of level $1$ substructures, and $c_{1,2}/c_{1,1}\leq
\operatorname*{const}$ independently of $H/h$.
For the three-level BDDC in three dimensions, the result of Tu [@Tu-2007-TBT3D Lemma 4.5], which is based on the lower bound estimates by Brenner and He [@Brenner-2003-LBT], can be written in our settings for all $w\in\widetilde{W}_{\Pi1}$ and for all $\Omega_{2}^{s}$ as $$c_{1,1}\left\vert I_{2}w\right\vert _{H^{1/2}(\partial\Omega_{2}^{s})}^{2}\leq\left\vert w\right\vert _{H^{1/2}(\partial\Omega_{2}^{s})}^{2}\leq
c_{1,2}\left\vert I_{2}w\right\vert _{H^{1/2}(\partial\Omega_{2}^{s})}^{2},
\label{eq:one-equiv-3D}$$ where $I_{2}$ is an interpolation from the coarse degrees of freedom given by the averages over substructure edges, and $c_{1,2}/c_{1,1}\leq
\operatorname*{const}$ independently of $H/h$.
We note that the level $2$ substructures are called subregions in [@Tu-2007-TBT; @Tu-2007-TBT3D]. Since $I_{1}=I$, with $i=1$ the equivalence (\[eq:one-equiv-2D\]) corresponds to (\[eq:Ii-equiv\]), and (\[eq:one-equiv-3D\]) to (\[eq:Ii-equiv-seminorm\]).
The next Lemma establishes the equivalence of seminorms on a factor space from the equivalence of norms on the original space. Let $V\subset U$ be finite dimensional spaces and $\left\Vert \cdot\right\Vert _{A}\ $a norm on $U$ and define $$\left\vert u\right\vert _{U/V,A}=\min\limits_{v\in V}\left\Vert u-v\right\Vert
_{A}. \label{eq:factor-norm}$$
We will be using (\[eq:factor-norm\]) for the norm on the space of discrete harmonic functions $(I-P_{i})\widetilde{W}_{i}$ with $V$ as the space of interior functions $U_{Ii}$, and also with $V$ as the space $\widetilde
{W}_{\Delta i}$. In particular, since $\widetilde{W}_{\Pi i}\subset
(I-P_{i})\widetilde{W}_{i}$, we have$$w\in\widetilde{W}_{\Pi i},\quad\left\Vert w\right\Vert _{a}=\min_{w_{\Delta
}\in\widetilde{W}_{\Delta i}}\left\Vert w-w_{\Delta}\right\Vert _{a}
\label{eq:i-factor}$$
\[lem:equivalence\]Let $\left\Vert \cdot\right\Vert _{A}$, $\left\Vert
\cdot\right\Vert _{B}$ be norms on $U$, and $$c_{1}\left\Vert u\right\Vert _{B}^{2}\leq\left\Vert u\right\Vert _{A}^{2}\leq
c_{2}\left\Vert u\right\Vert _{B}^{2},\quad\forall u\in U.
\label{eq:equiv-norms}$$ Then for any subspace $V\subset U$,$$c_{1}\left\vert u\right\vert _{U/V,B}^{2}\leq\left\vert u\right\vert
_{U/V,A}^{2}\leq c_{2}\left\vert u\right\vert _{U/V,B}^{2},$$ resp., $$c_{1}\min_{v\in V}\left\Vert u-v\right\Vert _{B}^{2}\leq\min_{v\in
V}\left\Vert u-v\right\Vert _{A}^{2}\leq c_{2}\min_{v\in V}\left\Vert
u-v\right\Vert _{B}^{2}. \label{eq:equiv-factor}$$
From the definition (\[eq:factor-norm\]) of the norm on a factor space, we get $$\left\vert u\right\vert _{U/V,A}=\min\limits_{v\in V}\left\Vert u-v\right\Vert
_{A}=\left\Vert u-v_{A}\right\Vert _{A}.$$ for some $v_{A}$. Let $v_{B}$ be defined similarly. Then $$\begin{aligned}
\left\vert u\right\vert _{U/V,A}^{2} & =\min\limits_{v\in V}\left\Vert
u-v\right\Vert _{A}^{2}=\left\Vert u-v_{A}\right\Vert _{A}^{2}\leq\left\Vert
u-v_{B}\right\Vert _{A}^{2}\leq\\
& \leq c_{2}\left\Vert u-v_{B}\right\Vert _{B}^{2}=c_{2}\min\limits_{v\in
V}\left\Vert u-v\right\Vert _{B}^{2}=c_{2}\left\vert u\right\vert _{U/V,B}^{2},\end{aligned}$$ which is the right hand side inequality in (\[eq:equiv-factor\]). The left hand side inequality follows by switching the notation for $\left\Vert
\cdot\right\Vert _{A}$ and $\left\Vert \cdot\right\Vert _{B}$.
\[lem:W\_i norm equiv\]For all $i=0,\ldots,L-1$, and $s=1,\ldots,N_{i}$, $$c_{i,1}\left\vert I_{i+1}w\right\vert _{a(\Omega_{i+1}^{s})}^{2}\leq\left\vert
w\right\vert _{a(\Omega_{i+1}^{s})}^{2}\leq c_{k,2}\left\vert I_{i+1}w\right\vert _{a(\Omega_{i+1}^{s})}^{2},\quad\forall w\in\widetilde{W}_{\Pi
i},\;\forall\,\Omega_{i+1}^{s}, \label{eq:ML-equiv}$$ with $c_{i,2}/c_{i,1}\leq C_{i}$, independently of $H_{0}$,…, $H_{i+1}$.
The proof follows by induction. For $i=0$, (\[eq:ML-equiv\]) holds because $I_{1}=I$. Suppose that (\[eq:ML-equiv\]) holds for some $i<L-2$ and let $w\in\widetilde{W}_{\Pi i+1}$. From the definition of $\widetilde{W}_{\Pi
i+1}$ by energy minimization,$$\left\vert w\right\vert _{a(\Omega_{i+1}^{s})}=\min_{w_{\Delta}\in
\widetilde{W}_{\Delta i+1}}\left\vert w-w_{\Delta}\right\vert _{a(\Omega
_{i+1}^{s})}. \label{eq:def-wa}$$ From (\[eq:def-wa\]), the induction assumption, and Lemma \[lem:equivalence\] eq. (\[eq:equiv-factor\]), it follows that$$\begin{aligned}
\lefteqn{c_{i,1}\min_{w_{\Delta}\in\widetilde{W}_{\Delta i+1}}\left\vert
I_{i+1}w-I_{i+1}w_{\Delta}\right\vert _{a(\Omega_{i+2}^{s})}^{2}}\label{eq:Ii1w}\\
& \qquad\leq\min_{w_{\Delta}\in\widetilde{W}_{\Delta i+1}}\left\vert
w-w_{\Delta}\right\vert _{a(\Omega_{i+2}^{s})}^{2}\leq c_{i,2}\min_{w_{\Delta
}\in\widetilde{W}_{\Delta i+1}}\left\vert I_{i+1}w-I_{i+1}w_{\Delta
}\right\vert _{a(\Omega_{i+2}^{s})}^{2}\nonumber\end{aligned}$$ From the assumption (\[eq:Ii-equiv\]), applied to the functions of the form $I_{i+1}w$ on $\Omega_{i+2}^{s}$, $$c_{1}\left\vert I_{i+2}w\right\vert _{a(\Omega_{i+2}^{s})}^{2}\leq
\min_{w_{\Delta}\in\widetilde{W}_{\Delta i+1}}\left\vert I_{i+1}w-I_{i+1}w_{\Delta}\right\vert _{a(\Omega_{i+2}^{s})}^{2}\leq c_{2}\left\vert
I_{i+2}w\right\vert _{a(\Omega_{i+2}^{s})}^{2} \label{eq:Tuk}$$ with $c_{2}/c_{1}$, bounded independently of $H_{0},\ldots,H_{i+1}$. Then (\[eq:def-wa\]), (\[eq:Ii1w\]) and (\[eq:Tuk\]) imply (\[eq:ML-equiv\]) with $C_{i}=C_{i-1}c_{2}/c_{1}$.
Next, we generalize the estimate from Theorem \[thm:element-bound\] to coarse levels.
\[lem:W\_i operator equiv\]For all substructuring levels $i=1,\ldots,L-1$, $$\left\Vert (I-P_{i})E_{i}w_{i}\right\Vert _{a}^{2}\leq C_{i}\left(
1+\log\frac{H_{i}}{H_{i-1}}\right) ^{2}\left\Vert w_{i}\right\Vert _{a}^{2},\quad\forall w_{i}\in U_{i}. \label{eq:ML-bound}$$
From (\[eq:ML-equiv\]), summation over substructures on level$~i$ gives$$c_{i,1}\left\Vert I_{i}w\right\Vert _{a}^{2}\leq\left\Vert w\right\Vert
_{a}^{2}\leq c_{i,2}\left\Vert I_{i}w\right\Vert _{a}^{2},\quad\forall w\in
U_{i}. \label{eq:equiv-i1}$$ Next, in our context, using the definition of $P_{i}$ and (\[eq:equiv-factor\]), we get $$\left\Vert I_{i}(I-P_{i})E_{i}w_{i}\right\Vert _{a}^{2}=\min_{u_{Ii}\in
U_{Ii}}\left\Vert I_{i}E_{i}w_{i}-I_{i}u_{Ii}\right\Vert _{a}^{2},$$ so from (\[eq:element-bound\]) for some $\overline{C}_{i}$ and all $i=1,\ldots,L-1$, $$\min_{u_{Ii}\in U_{Ii}}\left\Vert I_{i}E_{i}w_{i}-I_{i}u_{Ii}\right\Vert
_{a}^{2}\leq\overline{C}_{i}\left( 1+\log\frac{H_{i}}{H_{i-1}}\right)
^{2}\left\Vert I_{i}w_{i}\right\Vert _{a}^{2},\quad\forall w_{i}\in U_{i}.
\label{eq:log-linear}$$ Similarly, from (\[eq:equiv-i1\]) and (\[eq:log-linear\]) it follows that$$\begin{aligned}
\left\Vert (I-P_{i})E_{i}w_{i}\right\Vert _{a}^{2} & =\min_{u_{Ii}\in
U_{Ii}}\left\Vert E_{i}w_{i}-u_{Ii}\right\Vert _{a}^{2}\\
& \leq c_{i,2}\min_{u_{Ii}\in U_{Ii}}\left\Vert I_{i}E_{i}w_{i}-I_{i}u_{Ii}\right\Vert _{a}^{2}\\
& \leq c_{i,2}\overline{C}_{i}\left( 1+\log\frac{H_{i}}{H_{i-1}}\right)
^{2}\left\Vert I_{i}w_{i}\right\Vert _{a}^{2}\\
& \leq\frac{c_{i,2}\overline{C}_{i}}{c_{i,1}}\left( 1+\log\frac{H_{i}}{H_{i-1}}\right) ^{2}\left\Vert w_{i}\right\Vert _{a}^{2},\end{aligned}$$ which is (\[eq:ML-bound\]) with$$C_{i}=\frac{c_{i,2}\overline{C}_{i}}{c_{i,1}},$$ and $c_{i,2}/c_{i,1}$ from Lemma \[lem:W\_i norm equiv\].
\[thm:ML-bound\]The Multilevel BDDC for the model problem and corner coarse function in 2D and edge coarse functions in 3D satisfies the condition number estimate$$\kappa\leq{\textstyle\prod_{i=1}^{L-1}}
C_{i}\left( 1+\log\frac{H_{i}}{H_{i-1}}\right) ^{2}.$$
The proof follows from Lemmas \[lem:bddc-ML-estimate\] and \[lem:W\_i operator equiv\], with $\omega_{i}=C_{i}\left( 1+\log\frac{H_{i}}{H_{i-1}}\right) ^{2}$.
For $L=3$ in two and three dimensions we recover the estimates by Tu [@Tu-2007-TBT; @Tu-2007-TBT3D], respectively.
\[rem:nonmonotone\]While for standard (two-level) BDDC it is immediate that increasing the coarse space and thus decreasing the space $\widetilde{W}$ cannot increase the condition number bound, this is an open problem for the multilevel method. In fact, the 3D numerical results in the next section suggest that this may not be the case.
In the case of uniform coarsening, i.e. with $H_{i}/H_{i-1}=H/h$ and the same geometry of decomposition on all levels $i=1,\ldots L-1,$ we get $$\kappa\leq C^{L-1}\left( 1+\log H/h\right) ^{2\left( L-1\right) }.
\label{eq:all-same}$$
Numerical Examples {#sec:numerical-examples}
==================
\[c\][|c|c|c|c|c|c|c|]{}$L$ & & & $n$ & $n_{\Gamma}$\
& iter & cond & iter & cond & &\
\
2 & 8 & 1.92 & 5 & 1.08 & 144 & 80\
3 & 13 & 3.10 & 7 & 1.34 & 1296 & 720\
4 & 17 & 5.31 & 9 & 1.60 & 11,664 & 6480\
5 & 23 & 9.22 & 10 & 1.85 & 104,976 & 58,320\
6 & 31 & 16.07 & 11 & 2.12 & 944,748 & 524,880\
7 & 42 & 28.02 & 13 & 2.45 & 8,503,056 & 4,723,920\
\
2 & 9 & 2.20 & 6 & 1.14 & 256 & 112\
3 & 15 & 4.02 & 8 & 1.51 & 4096 & 1792\
4 & 21 & 7.77 & 10 & 1.88 & 65,536 & 28,672\
5 & 30 & 15.2 & 12 & 2.24 & 1,048,576 & 458,752\
6 & 42 & 29.7 & 13 & 2.64 & 16,777,216 & 7,340,032\
\
2 & 10 & 2.99 & 7 & 1.33 & 1024 & 240\
3 & 19 & 7.30 & 11 & 2.03 & 65,536 & 15,360\
4 & 31 & 18.6 & 13 & 2.72 & 4,194,304 & 983,040\
5 & 50 & 47.38 & 15 & 3.40 & 268,435,456 & 62,914,560\
\
2 & 11 & 3.52 & 8 & 1.46 & 2304 & 368\
3 & 21 & 10.12 & 12 & 2.39 & 331,776 & 52,992\
4 & 39 & 29.93 & 15 & 3.32 & 47,775,744 & 7,630,848\
\
2 & 11 & 3.94 & 8 & 1.56 & 4096 & 496\
3 & 23 & 12.62 & 13 & 2.67 & 1,048,576 & 126,976\
4 & 43 & 41.43 & 16 & 3.78 & 268,435,456 & 32,505,856\
\[tab1\]
\[c\][|c|c|c|c|c|c|c|c|c|]{}$L$ & & & & $n$ & $n_{\Gamma}$\
& iter & cond & iter & cond & iter & cond & &\
&\
2 & 10 & 1.85 & 8 & 1.47 & 5 & 1.08 & 1728 & 1216\
3 & 14 & 3.02 & 12 & 2.34 & 8 & 1.50 & 46,656 & 32,832\
4 & 18 & 4.74 & 18 & 5.21 & 11 & 2.20 & 1,259,712 & 886,464\
5 & 23 & 7.40 & 26 & 14.0 & 16 & 3.98 & 34,012,224 & 23,934,528\
&\
2 & 10 & 1.94 & 9 & 1.66 & 6 & 1.16 & 4096 & 2368\
3 & 15 & 3.51 & 14 & 3.24 & 10 & 1.93 & 262,144 & 151,552\
4 & 20 & 6.09 & 22 & 9.95 & 14 & 3.05 & 16,777,216 & 9,699,328\
&\
2 & 12 & 2.37 & 11 & 2.24 & 8 & 1.50 & 32,768 & 10,816\
3 & 19 & 5.48 & 20 & 7.59 & 14 & 3.32 & 16,777,216 & 5,537,792\
&\
2 & 12 & 2.56 & 12 & 2.47 & 9 & 1.69 & 64,000 & 17,344\
3 & 20 & 6.39 & 22 & 10.1 & 16 & 3.85 & 64,000,000 & 17,344,000\
Numerical examples are presented in this section for the Poisson equation in two and three dimensions. The problem domain in 2D (3D) is the unit square (cube), and standard bilinear (trilinear) finite elements are used for the discretization. The substructures at each level are squares or cubes, and periodic essential boundary conditions are applied to the boundary of the domain. This choice of boundary conditions allows us to solve very large problems on a single processor since all substructure matrices are identical for a given level.
The preconditioned conjugate gradient algorithm is used to solve the associated linear systems to a relative residual tolerance of $10^{-8}$ for random right-hand-sides with zero mean value. The zero mean condition is required since, for periodic boundary conditions, the null space of the coefficient matrix is the unit vector. The coarse problem always has $4^{2}$ ($4^{3}$) subdomains at the coarsest level for 2D (3D) problems.
The number of levels ($L$), the number of iterations (iter), and condition number estimates (cond) obtained from the conjugate gradient iterations are reported in Tables \[tab1\] and \[tab2\]. The letters C, E, and F designate the use of corners, edges, or faces in the coarse space. For example, C+E means that both corners and edges are used in the coarse space. For 2D and 3D problems, the theory is applicable to coarse spaces C and E, respectively. Also shown in the tables are the total number of unknowns ($n$) and the number of unknowns ($n_{\Gamma}$) on subdomain boundaries at the finest level.
The results in Tables \[tab1\] and \[tab2\] are displayed in Figure \[ne:fig1\] for a fixed value of $H_{i}/H_{i-1}=3$. In two dimensions we observe very different behavior depending on the particular form of the coarse space. If only corners are used in 2D, then there is very rapid growth of the condition number with increasing numbers of levels as predicted by the theory. In contrast, if both corners and edges are used in the 2D coarse space, then the condition number appears to vary linearly with $L$ for the the number of levels considered. Our explanation is that a bound similar to Theorem \[thm:ML-bound\] still applies to the favorable 2D case, though possibly with (much) smaller constants, so the exponential growth of the condition number is no longer apparent. The results in Tables \[tab1\] and \[tab2\] are also displayed in Figure \[ne:fig2\] for fixed numbers of levels. The observed growth of condition numbers for the case of uniform coarsening is consistent with the estimate in (\[eq:all-same\]).
Similar trends are present in 3D, but the beneficial effects of using more enriched coarse spaces are much less pronounced. Interestingly, when comparing the use of edges only (E) with corners and edges (C+E) in the coarse space, the latter does not always lead to smaller numbers of iterations or condition numbers for more than two levels. The fully enriched coarse space (C+E+F), however, does give the best results in terms of iterations and condition numbers. It should be noted that the present 3D theory in Theorem \[thm:ML-bound\] covers only the use of the edges only, and the present theory does not guarantee that the condition number (or even its bound) decrease with increasing the coarse space (Remark \[rem:nonmonotone\]).
In summary, the numerical examples suggest that better performance, especially in 2D, can be obtained when using a fully enriched coarse space. Doing so does not incur a large computational expense since there is never the need to solve a large coarse problem exactly with the multilevel approach. Finally, we note that a large number of levels is not required to solve very large problems. For example, the number of unknowns in 3D for a 4-level method with a coarsening ratio of $H_{i}/H_{i-1}=10$ at all levels is $(10^{4})^{3} =
10^{12}$.
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abstract: 'The article considers the generalized $k$-Bessel functions and represents it as Wright functions. Then we study the monotonicity properties of the ratio of two different orders $k$- Bessel functions, and the ratio of the $k$-Bessel and the $m$-Bessel functions. The log-convexity with respect to the order of the $k$-Bessel also given. An investigation regarding the monotonicity of the ratio of the $k$-Bessel and $k$-confluent hypergeometric functions are discussed.'
address:
- |
Department of Mathematics\
King Faisal University, Al Ahsa 31982, Saudi Arabia
- |
Department of Mathematics\
Prince Sattam bin Abdulaziz University, Saudi Arabia
author:
- 'Saiful. R. Mondal'
- 'Kottakkaran S. Nisar'
title: 'Inequalities for the modified $k$- Bessel function'
---
Introduction {#Intro}
============
One of the generalization of the classical gamma function $\Gamma$ studied in [@Diaz] is defined by the limit formula $$\begin{aligned}
\label{eqn-1}
\Gamma_k(x) := \lim_{n \to \infty} \frac{n! \; k^n (n^k)^{\tfrac{x}{k}-1}}{(x)_{n, k}}, \quad k>0,\end{aligned}$$ where $(x)_{n, k}:=x(x+k) (x+2k)\ldots (x+(n-1)k)$ is called $k$-Pochhammer symbol. The above $k-$gamma function also have an integral representation as $$\begin{aligned}
\label{eqn-2}
\Gamma_k(x)= \int_0^\infty t^{x-1} e^{-\frac{t^{k}}{k}} dt, \quad \operatorname{Re}(x)>0.\end{aligned}$$ Properties of the $k$-gamma functions have been studies by many researchers [@CGK; @CGK2; @VK; @MM; @Mubeen13]. Follwoing properties are required in sequel:
- $\Gamma _{k}\left( x+k\right) =x\Gamma _{k}\left( x\right)$
- $\Gamma _{k}\left( x\right) =k^{\frac{x}{k}-1}\Gamma \left( \frac{x}{k}\right)$
- $\Gamma _{k}\left( k\right)=1$
- $\Gamma _{k}\left( x+nk\right)=\Gamma _{k}(x) (x)_{n, k}$
Motivated with the above generalization of the $k$-gamma functions, Romero et. al.[@Romero-Cerutti] introduced the $k-$Bessel function of the first kind defined by the series $$\begin{aligned}
\label{k1}
J_{k,\nu }^{\gamma ,\lambda }\left( x\right) :=\sum_{n=0}^{\infty }\frac{\left( \gamma \right) _{n,\;k}}{\Gamma _{k}\left( \lambda n+\upsilon +1\right)
}\frac{\left( -1\right) ^{n}\left( x/2\right) ^{n}}{\left( n!\right) ^{2}},\end{aligned}$$where $k\in \mathbb{R^{+}}$; $\alpha,\lambda,\gamma,\upsilon \in C$; $\operatorname{Re}(\lambda)>0$ and $\operatorname{Re}(\upsilon) >0$. They also established two recurrence relations for $J_{k,\nu }^{\gamma ,\lambda }$.
In this article, we are considering the following function: $$\begin{aligned}
\label{eqn-modfb}
I_{k,\nu }^{\gamma ,\lambda }\left( x\right) :=\sum_{n=0}^{\infty }\frac{\left( \gamma \right) _{n,\;k}}{\Gamma _{k}\left( \lambda n+\upsilon +1\right)
}\frac{\left( x/2\right) ^{n}}{\left( n!\right) ^{2}},\end{aligned}$$ Since $$\lim_{k, \lambda, \gamma \to 1 }I_{k,\nu }^{\gamma ,\lambda }\left( x\right)=
\sum_{n=0}^{\infty }\frac{1}{\Gamma\left( n+\upsilon +1\right)
}\frac{\left( x/2\right) ^{n}}{n!}=\left(\frac{2}{x}\right)^{\frac{\nu}{2}} I_\nu(\sqrt{2x}),$$ the classical modified Bessel functions of first kind. In this sense, we can call $I_{k,\nu }^{\gamma ,\lambda }$ as the modified $k$-Bessel functions of first kind. In fact, we can express both $J_{k,\nu }^{\gamma ,\lambda }$ and $I_{k,\nu }^{\gamma ,\lambda }$ together in $$\label{eqn-genb}
\mathtt{W}_{k,\nu, c }^{\gamma,\lambda }(x) :=\sum_{n=0}^{\infty}\frac{
( \gamma )_{n,\;k}}{\Gamma_{k}(\lambda n+ \nu +1)
}\frac{(-c)^n( x/2) ^{n}}{\left( n!\right) ^{2}}, \quad c \in \mathbb{R}.$$ We can termed $\mathtt{W}_{k,\nu }^{\gamma ,\lambda }$ as the generalized $k$-Bessel function.
First we study the representation formulas for $\mathtt{W}_{k,\nu }^{\gamma ,\lambda }$ in term of the classical Wright functions. Then we will study about the monotonicity and log-convexity properties of $I_{k,\nu }^{\gamma ,\lambda }$.
Representation formula for the generalized $k$-Bessel function {#sec1}
==============================================================
The generalized hypergeometric function ${}_pF_q(a_1,\ldots, a_p;c_1,\ldots,c_q;x)$, is given by the power series $$\label{eqn:gen-hyp-func}
{}_pF_q(a_1,\ldots, a_p;c_1,\ldots,c_q;z) = \sum_{k=0}^{\infty}\dfrac{(a_1)_{k}
\cdots (a_p)_{k}}{(c_1)_{k}\cdots(c_q)_{k}(1)_{k}}z^k, \quad \quad |z|<1,$$ where the $c_{i}$ can not be zero or a negative integer. Here $p$ or $q$ or both are allowed to be zero. The series $(\ref{eqn:gen-hyp-func})$ is absolutely convergent for all finite $z$ if $p\leq q$ and for $|z|<1$ if $p=q+1$. When $p>q+1$, then the series diverge for $z \not=0$ and the series does not terminate.
The generalized Wright hypergeometric function ${}_p\psi_q(z)$ is given by the series $$\label{eqn-9-bessel}
{}_p\psi_q(z)={}_p\psi_q\left[\begin{array}{c}
(a_i,\alpha_i)_{1,p} \\
(b_j,\beta_j)_{1,q}
\end{array}\bigg|z\right]=\displaystyle\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{p}\Gamma(a_{i}+\alpha_{i}k)}
{\prod_{j=1}^{q}\Gamma(b_{j}+\beta_{j}k)}
\frac{z^{k}}{k!},$$ where $a_i, b_j\in \mathbb{C}$, and real $\alpha_i, \beta_j\in \mathbb{R}$ ($i=1,2,\ldots,p; j =1,2,\ldots,q$). The asymptotic behavior of this function for large values of argument of $z\in \mathbb{C}$ were studied in [@CFox; @Kilbas] and under the condition $$\label{eqn-10-bessel}
\displaystyle\sum_{j=1}^{q}\beta_{j}-\displaystyle\sum_{i=1}^{p}\alpha_{i}>-1$$ in literature [@Wright-2; @Wright-3]. The more properties of the Wright function are investigated in [@Kilbas; @Kilbas-itsf; @KST].
Now we will give the representation of the generalized $k$-Bessel functions in terms of the Wright and generalized hypergeometric functions.
Let, $k \in \mathbb{R}$ and $\lambda ,\gamma ,\nu \in \mathbb{C}$ such that $\operatorname{Re}(
\lambda) >0,\operatorname{Re}( \nu ) >0.$ Then $$\mathtt{W}_{k,\nu,c }^{\gamma ,\lambda }(x) =\frac{1}{k^{\frac{\nu+k+1}{k}}\Gamma\left(\frac{\gamma}{k}\right)}
{}_1\psi_2\left[\begin{array}{ccc}
\left(\frac{\gamma}{k}, 1\right)& \\
\left(\frac{\nu+1}{k}, \frac{\gamma}{k}\right) & (1, 1)
\end{array}\bigg|-\frac{c x}{2 k^{\frac{\lambda}{k}-1}}\right]$$
Using the relations $\Gamma _{k}\left( x\right) =k^{\frac{x}{k}-1}\Gamma \left( \frac{x}{k}\right)$ and $\Gamma _{k}\left( x+nk\right)=\Gamma _{k}(x) (x)_{n, k}$, the generalized $k$-Bessel functions defined in $\eqref{eqn-genb}$ can be rewrite as $$\begin{aligned}
\mathtt{W}_{k,\nu,c }^{\gamma ,\lambda }(x)
&=\sum_{n=0}^\infty \frac{\Gamma_k(\gamma+n k)}{\Gamma_k(\lambda n+\nu+1) \Gamma_k(\gamma)} \frac{(-c)^n}{(n!)^2} \left(\frac{x}{2}\right)^n\\
&=\frac{1}{k^{\frac{\nu+k+1}{k}} \Gamma\left(\frac{\gamma}{k}\right)}\sum_{n=0}^\infty \frac{\Gamma\left(\frac{\gamma}{k}+n \right)}{\Gamma\left(\frac{\lambda}{k} n+\frac{\nu+1}{k}\right) \Gamma\left(\frac{\gamma}{k}\right)} \frac{(-c)^n}{\Gamma(n+1)\Gamma(n+1)} \left(\frac{x}{2 k^{\frac{\lambda}{k}-1}}\right)^n\\
&=\frac{1}{k^{\frac{\nu+k+1}{k}}\Gamma\left(\frac{\gamma}{k}\right)}
{}_1\psi_2\left[\begin{array}{ccc}
\left(\frac{\gamma}{k}, 1\right)& \\
\left(\frac{\nu+1}{k}, \frac{\gamma}{k}\right) & (1, 1)
\end{array}\bigg|-\frac{c x}{2 k^{\frac{\lambda}{k}-1}}\right]\end{aligned}$$ Hence the result follows.
Monotonicty and log-convexity properties {#sec2}
========================================
This section discuss the monotonicity and log-convexity properties for the modified $k$-Bessel functions $\mathtt{W}_{k,\nu,-1 }^{\gamma ,\lambda }(x)=\mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)$.
Following lemma due to Biernacki and Krzyż [@Biernacki-Krzy] will be required.
\[lemma:1\][@Biernacki-Krzy] Consider the power series $f(x)=\sum_{k=0}^\infty a_k x^k$ and $g(x)=\sum_{k=0}^\infty b_k x^k$, where $a_k \in \mathbb{R}$ and $b_k > 0$ for all $k$. Further suppose that both series converge on $|x|<r$. If the sequence $\{a_k/b_k\}_{k\geq 0}$ is increasing (or decreasing), then the function $x \mapsto f(x)/g(x)$ is also increasing (or decreasing) on $(0,r)$.
The above lemma still holds when both $f$ and $g$ are even, or both are odd functions.
The following results holds true for the modified $k$-Bessel functions.
1. For $\mu \geq \nu>-1$, the function $x \mapsto \mathtt{I}_{k,\mu}^{\gamma ,\lambda }(x)/\mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)$ is increasing on $(0, \infty)$ for some fixed $k >0$.
2. If $k\geq \lambda \geq m>0$, the function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\mathtt{I}_{m,\nu}^{\gamma ,\lambda }(x)$ is increasing on $(0, \infty)$ for some fixed $\nu >-1$ and $\gamma \geq \nu+1$.
3. The function $\nu \mapsto \mathcal{I}_{k,\nu}^{\gamma ,\lambda }(x)$ is log-convex on $(0, \infty)$ for some fixed $k, \gamma>0$ and $x>0$. Here, $\mathcal{I}_{k,\nu}^{\gamma ,\lambda }(x):=\Gamma_k(\nu+1)\mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)$.
4. Suppose that $\lambda \geq k>0$ and $\nu>-1$. Then
1. The function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\Phi _{k}\left( a,c;x\right)$ is decreasing on $(0, \infty)$ for $a \geq c >0$ and $0<\gamma \leq \nu+1$. Here, $\Phi _{k}\left( a; c; x\right)$ is the $k$-confluent hypergeometric functions.
2. The function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\Phi _{k}\left( \gamma; \lambda; x/2\right)$ is decreasing on $(0, 1)$ for $\gamma >0$ and $0< k \leq \lambda \leq \nu+1$.
3. The function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\Phi _{k}\left( \gamma; \lambda; x/2\right)$ is decreasing on $[1, \infty)$ for $\gamma >0$ and $0< k \leq \min\{\lambda,\nu+1\}$.
${\bf (1)}$ Form it follows that $$\mathtt{I}_{ k, \nu}^{\gamma ,\lambda }(x)= \sum_{n=0}^\infty a_n(\nu) x^n\quad
\text{and} \quad \mathtt{I}_{ k, \nu}^{\gamma ,\lambda }(x)= \sum_{n=0}^\infty a_n(\mu) x^n,$$ where $$a_n(\nu)= \frac{(\gamma)_{n,k}}{\Gamma_k(\lambda n+\nu+1) (n!)^2 2^n}
\quad \text{and} \quad
a_n(\mu)= \frac{(\gamma)_{n,k}}{\Gamma_k(\lambda n+\mu+1) (n!)^2 2^n}$$ Consider the function $$f(t):= \frac{\Gamma_k(\lambda t+\mu+1)}{\Gamma_k(\lambda t+\nu+1)}.$$ Then the logarithmic differentiation yields $$\begin{aligned}
\frac{f'(t)}{f(t)}= \lambda( \Psi_k(\lambda t+\mu+1)-\Psi_k(\lambda t+\nu+1)).\end{aligned}$$ Here, $\Psi_k=\Gamma_k'/\Gamma_k $ is the $k$-digamma functions studied in [@Kwara14] and defined by $$\begin{aligned}
\label{def-digamma}
\Psi_k(t)=\frac{\log(k)-\gamma_1}{k}-\frac{1}{t}+\sum_{n=1}^\infty \frac{t}{nk(nk+t)}\end{aligned}$$ where $\gamma_1$ is the Euler-Mascheroni’s constant.
A calculation yields $$\begin{aligned}
\label{def-digamma-2} \Psi_k'(t)=\sum_{n=0}^\infty \frac{1}{(nk+t)^2}, \quad k>0 \quad \text{and} \quad t>0.\end{aligned}$$ Clearly, $\Psi_k$ is increasing on $(0, \infty)$ and hence $f'(t)>0$ for all $t\geq0$ if $\mu \geq \nu>-1$. This, in particular, implies that the sequence $\{d_n\}_{n \geq 0}=\{a_n(\nu)/a_n(\mu)\}_{n \geq 0}$ is increasing and hence the conclusion follows from Lemma $\ref{lemma:1}$.
[**(2)**]{}. This result also follows from Lemma $\ref{lemma:1}$ if the sequence $\{d_n\}_{n \geq 0}=\{a_n^k(\nu)/a_n^m(\mu)\}_{n \geq 0}$ is increasing for $k \geq m >0$. Here, $$a_{n}^{k}\left( \nu \right) =\frac{\left( \gamma \right) _{n,k}}{\Gamma _{k}\left( \lambda n+\nu+1\right) \left( n!\right) ^{2}} \quad \text{and} \quad
a_{n}^{m}\left( \nu \right) =\frac{\left( \gamma \right) _{n,m}}{\Gamma _{m}\left( \lambda n+\nu+1\right) \left( n!\right) ^{2}},$$ which together with the identity $\Gamma _{k}\left( x+nk\right)=\Gamma _{k}(x) (x)_{n, k}$ gives $$\begin{aligned}
d_n&=\frac{\left( \gamma \right) _{n,k}}{\left( \gamma \right) _{n,m}} \frac{
\Gamma _{m}\left( \lambda n+\nu+1\right) }{\Gamma _{k}\left( \lambda n+\nu+1\right)}\\
&= \frac{
\Gamma _{k}\left( \gamma +nk\right)\Gamma _{m}\left( \lambda n+\nu+1\right) }{\Gamma _{k}\left( \gamma +nm\right)\Gamma _{k}\left( \lambda n+\nu+1\right)}.\end{aligned}$$ Now to show that $\{d_n\}$ is increase, consider the function $$f(y):=\frac{
\Gamma _{k}\left( \gamma +yk\right)\Gamma _{m}\left( \lambda y+\nu+1\right) }{\Gamma _{k}\left( \gamma +ym\right)\Gamma _{k}\left( \lambda y+\nu+1\right)}$$ The logarithmic differentiation of $f$ yields $$\begin{aligned}
\label{3}
\frac{f'(y)}{f(y)}= k \Psi_k(\gamma +yk)+ \lambda \Psi_m\left( \lambda y+\nu+1\right)-m \Psi_m(\gamma +ym )-\lambda \Psi_k\left( \lambda y+\nu+1\right)\end{aligned}$$ If $\gamma \geq \nu+1$ and $k \geq \lambda \geq m $, then can be rewrite as $$\begin{aligned}
\label{44}
\frac{f'(y)}{f(y)}\geq \lambda \big(\Psi_k(\nu+1 +yk)- \Psi_k\left( \lambda y+\nu+1\right)\big)+ m\big( \Psi_m\left( \lambda y+\nu+1\right)- \Psi_m(\nu+1 +ym )\big) \geq 0.\end{aligned}$$ This conclude that $f$, and consequently the sequence $\{d_n\}_{n\geq 0}$, is increasing. Finally the result follows from the Lemma \[lemma:1\].
[**(3).**]{} It is known that sum of the log-convex functions is log-convex. Thus, to prove the result it is enough to show that $$\nu \mapsto a_{n}^{k}\left( \nu \right) :=\frac{\left( \gamma \right) _{n,k}\Gamma _{k}\left( \nu+1\right)}{\Gamma _{k}\left( \lambda n+\nu+1\right) \left( n!\right) ^{2}}$$ is log-convex.
A logarithmic differentiation of $a_n(\nu)$ with respect to $\nu$ yields $$\begin{aligned}
\frac{\partial}{\partial \nu} \log\left(a_{n}^{k}\left( \nu \right)\right)=\Psi_k\left(\nu+1\right) - \Psi_k\left( \lambda n+\nu+1\right).\end{aligned}$$ This along with gives $$\begin{aligned}
\frac{\partial^2}{\partial\nu^2}\log\left(a_{n}^{k}\left( \nu \right)\right)
&=\Psi'_k\left(\nu+1\right) - \Psi'_k\left( \lambda n+\nu+1\right)\\
&=\sum_{r=0}^\infty \frac{1}{(rk+\nu+1)^2} - \sum_{r=0}^\infty \frac{1}{(rk+\lambda n+\nu+1)^2}\\
&=\sum_{r=0}^\infty \frac{\lambda n(2 rk+\lambda n+2\nu+2)}{(rk+\nu+1)^2(rk+\lambda n+\nu+1)^2} >0,\end{aligned}$$ for all $n \geq 0$, $k >0$ and $\nu>-1$. Thus, $\nu \mapsto a_{n}^{k}\left( \nu \right)$ is log-convex and hence the conclusion.\
[**(4).**]{} Denote $\Phi _{k}\left( a,c;x\right)=\sum_{n=0}^\infty c_{n, k}( a, c) x^{n}$ and $\mathtt{I}_{ k, \nu}^{\gamma ,\lambda }(x)= \sum_{n=0}^\infty a_n(\nu) x^n,$ where $$a_n(\nu)= \frac{(\gamma)_{n,k}}{\Gamma_k(\lambda n+\nu+1) (n!)^2 2^n}\quad \text{and} \quad
d_{n,k}\left( a,c\right) =\frac{\left( a\right) _{n,k}}{\left( c\right)
_{n,k}n!}$$with $v>-1$ and $a,c, \lambda, \gamma, k>0.$ To apply Lemma \[lemma:1\], consider the sequence $\left\{ w_{n}\right\} _{n\geq 0}$ defined by $$\begin{aligned}
w_{n} =\frac{a_{n}\left( \nu \right) }{d_{n, k}\left( a,c\right) }&=&\frac{\Gamma _{k}\left( \gamma +nk\right) }{2^{n}\Gamma _{k}\left( \gamma \right)
\Gamma _{k}\left( \lambda n+\alpha +1\right) \left( n!\right) ^{2}}.\frac{\Gamma _{k}\left( a\right) \Gamma _{k}\left( c+nk\right) n!}{\Gamma
_{k}\left( a+nk\right) \Gamma _{k}\left( c\right) } \\
&=&\frac{\Gamma _{k}\left( a\right) }{\Gamma _{k}\left( \gamma \right)
\Gamma _{k}\left( c\right) }\rho _{k}\left( n\right)\end{aligned}$$where $$\rho _{k}\left( x\right) =\frac{\Gamma _{k}\left( \gamma +xk\right) \Gamma
_{k}\left( c+xk\right) }{\Gamma _{k}\left( \lambda x+\nu +1\right) \Gamma
_{k}\left( a+xk\right) 2^x \Gamma(x+1) }.$$In view of the increasing properties of $\Psi_k$ on $(0, \infty)$, and $$\frac{\rho ^{\prime }\left( x\right) }{\rho \left( x\right) }= k \psi
_{k}\left( \gamma +xk\right) +k\psi _{k}\left( c+xk\right) -\lambda \psi _{k}\left(
\lambda x+\alpha +1\right) -k \psi _{k}\left( a+xk\right),$$ it follows that for $a\geq c>0$, $\lambda \geq k$ and $\nu+1\geq \gamma$, the function $\rho$ is decreasing on $
\left( 0,\infty \right) $ and thus the sequence $\left\{ w_{n}\right\} _{n\geq
0} $ also decreasing. Finally the conclusion for $(a)$ follows from the Lemma \[lemma:1\].
In the case $(b)$ and $(c)$, the sequence $\{w_n\}$ reduces to $$\begin{aligned}
w_{n} =\frac{a_{n}\left( \nu \right) }{d_{n, k}\left(\gamma, \lambda\right) }&=&\frac{\rho _{k}\left( n\right)}{\Gamma _{k}\left( \lambda \right)
}\end{aligned}$$where $$\rho _{k}\left( x\right) =\frac{\Gamma _{k}\left( \lambda +xk\right) }{\Gamma _{k}(\nu +1+\lambda x) \Gamma(x+1) }.$$Now as in the proof of part (a) $$\frac{\rho _{k}'\left( x\right)}{\rho _{k}\left( x\right)} =k \Psi_k(\lambda +xk)-\lambda \Psi_k(\nu+1 +xk)-\Psi(x+1)>0,$$ if $\nu+1+ \lambda x \geq \lambda + xk$. Now for $x \in (0,1)$, this inequality holds if $0< k \leq \lambda \leq \nu+1$, while for $x \geq 1$, it is required that $k \leq \min\{\lambda, \nu+1\}.$
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abstract: 'We report on the mesoscale fabrication and characterization of polymeric templates for isotropic photonic materials derived from hyperuniform point patterns using direct laser writing in a polymer photoresist. We study experimentally the microscopic structure by electron microscopy and small angle light scattering. Reducing the refractive index mismatch by liquid infiltration we find good agreement between the scattering data and numerical calculations based on a discrete dipole approximation. Our work demonstrates the feasibility of fabricating such random designer materials on technologically relevant length scales.'
address:
- '$^1$Physics Department and Fribourg Center for Nanomaterials, University of Fribourg, Chemin de Musée 3, 1700 Fribourg, Switzerland'
- '$^2$Currently with the Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland '
author:
- 'Jakub Haberko$^{1,2}$ and Frank Scheffold$^{1}$'
title: 'Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials'
---
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Introduction
============
Coherent scattering can strongly influence the macroscopic transport of electrons or photons. In the limit of weak scattering from periodic lattices the classical concept of Bloch allows to distinguish propagating and non-propagating wave vectors $\bf{k}$. For a periodic one-dimensional structure with a lattice constant $d$ forbidden wave vectors appear around the edge of the Brillouin zone $k =\pi/a$, the latter being a signature of back-reflection from Bragg planes with scattering vectors $q=2k=2\pi/a$ [@ref1]. Despite the beauty of BlochÕs theory it is well known that the model covers only a small part of actual electron transport phenomena in metals or semi-conductors. A prime example is the fact that many alloys and even liquids display metallic or semiconducting properties [@Wa04]. The broader concept of electronic scattering from ordered or disordered structures can be readily transferred to the case of electromagnetic waves as shown in the pioneering work of Yablonovitch and John [@ref9; @ref10]. Following their discovery there have been enormous research efforts aimed at manufacturing photonic semi-conducting materials. Again these efforts have largely concentrated on crystalline structures, owing to the simple deterministic design rules and the fact that band-structure calculations are well established for crystals [@ref1; @ref11; @ref12]. The structural anisotropy of periodic structures, both in real space and reciprocal-space, however, imposes physical limitations for three-dimensional photonic materials. A direct consequence of anisotropy is the appearance of stop bands with varying strengths depending on the direction of wave propagation. In order to achieve complete band-gaps in three dimensions the gaps in all directions have to overlap, a condition that is difficult to achieve or that requires a very large dielectric contrast. As emphasized previously [@ref7; @ref8], the availability of isotropic photonic materials would thus be highly desirable. A step in this direction has been made with the discovery that quasi-crystalline structures equally possess photonic properties while having a much higher degree of rotational symmetry [@Flor09QC; @ref13]. Quasicrystalline structures are aperiodic but nevertheless exhibit sharp Bragg peaks [@ref14]. Since they are ordered and can be projected from higher-dimensional crystalline structures they can be considered a case in between crystalline and amorphous. Disordered dielectric heterostructures have also been considered candidates for photonic band gap materials [@ref2; @Rec11]. However, until recently, the design rules needed to derive such structures have remained obscure. A number of studies have looked into materials with short-range order such as photonic liquids and glasses derived from self-assembled colloidal [@ref3; @ref4; @ref5] or biomimetic materials [@ref6]. Although Bragg-like scattering and pseudo gaps have been observed no conceptual proof for the existence of a complete band gap could be derived even at arbitrarily high dielectric contrast. Moreover, the interplay between short range-order and Anderson localization of light in three dimension, highlighted in the early work of Sajeev John [@ref10], remains an unsolved question.
Recently Florescu, Torquato and Steinhardt have taken a different route to design amorphous structures with a full photonic bandgap (PBG)[@ref7]. The authors make the general claim that three structural conditions have to be satisfied to obtain a PBG: hyperuniformity, uniformlocal topology, and short-range geometric order. Structures that fulfill these conditions display a vanishing structure factor $S(q)$ at small but finite wavenumbers $\left| \bf{q} \right| \le \left| {\bf{q_c}} \right|$, a structural property coined ÔstealthyÕ hyperuniformity. Above a threshold dielectric contrast such structures are predicted to display a full PBG. A trivial finding is that all crystalline and quasicrystalline materials fall into this class. Their main point however is that they could identify peculiar structures that are disordered and thus completely isotropic but still fulfill the above mentioned conditions [@ref7]. It has been shown that for exampled randomly jammed packings of spheres posses such hyperuniform long-range correlations [@Zac11]. In addition to the underlying geometric order, the local topology plays an important role [@ref8]. Mapping hyperuniform point patterns with short-range geometric order into tessellations allows the design of interconnected networks that gives rise to enhanced photonic properties. Numerical calculations for such two- and three-dimensional disordered designer materials [@ref7; @ref8] indicate the presence of a robust full band-gap in the limit of sufficiently high dielectric contrast, typically $n>3$ in air. Preliminary experimental data obtained from 2D hyperuniform structures in the microwave regime seem to confirm these predictions [@Man10].
Here we report on the first experimental realization of micron scale three-dimensional polymeric templates, n $\sim 1.52$, based on the design rules derived by Florescu and coworkers [@ref7]. Our present aim is to demonstrate the feasibility of fabricating such random designer materials on technologically relevant length scales, a first major step towards the fabrication of three dimensional amorphous full PBG materials.
![Design of amorphous photonic structures based upon a tetrahedral network of elliptical rods. (a) Enlarged view of part of the network obtained by tessellation of a hyperuniform seed pattern, rod cross section $840 \times280$nm$^2$. (b) close-up showing also the grid points lying inside or at the surface of the rods used for the numerical analysis of the scattering pattern.[]{data-label="fig1"}](fig1){width="11cm"}
Results
=======
Nanofabrication of designer disordered materials in a polymer photoresist.
--------------------------------------------------------------------------
We fabricate mesoscale polymer structures by direct laser writing (DLW) (Photonic Professional, Nanoscribe, Germany). The method allows to replicate a 3D structure voxel by voxel into a polymer photoresist with submicron resolution using a two-photon polymerization process [@ref12]. We first implement the protocol suggested by Florescu and coworkers to map a hyperuniform point pattern into tessellations for photonic materials design [@ref7]. As a seed structure we use the centroid positions from a maximally randomly jammed assembly of spheres of diameter $a = 3.31\mu$m with a volume filling fraction of $\phi \simeq 0.64$ [@ref15]. As shown by Torquato and coworkers [@Zac11] such structures indeed possess the required hyperuniform long-range correlations. Locally the seed pattern displays pronounced short-range correlations due to close packing of the original assembly of spheres. Therefore the real space pair correlation function $g(r)$ is sharply peaked at contact $\left| \bf{r} \right|= d$ and thus the structure factor $S(q)$ is peaked at $\left| \bf{q} \right|_{max}=2\pi/d$ [@Don05]. The latter is closely related to the well-known diffraction ring observed in experiments on dense molecular and complex liquids or glasses [@Bar03]. Next we perform a 3D Delaunay tessellation of the spheres center positions. In this scheme tetrahedrons are formed in such a way that no sphere center is contained in the circumsphere of any tetrahedron in the tessellation. The center-of-mass of neighboring tetrahedrons are then connected resulting in a 3D random tetrahedral network with the desired hyperuniform properties. The only parameters not yet fixed are related to the shape and cross section of the dielectric rods replacing the fictitious connection lines (Fig. \[fig1\] (a)). We find optimal conditions in the writing process when the rods are written into IPG-780 negative tone photoresist (Nanoscribe, Germany). The laser writing pen has an elliptical cross section of about $840 \times 280$ nm$^2$. When writing thinner rods the structure becomes mechanically unstable while thicker rods lead to overfilling of the structure. The aspect ratio of ca. $3.0$ of the rods is dictated by the point spread function of the illuminating microscope objective during the writing process which is set by the refractive index of about $1.52$ of the material and the numerical aperture of $1.4$. For the parameters chosen we estimate the volume-filling fraction of the rods to be roughly 8 $\%$. In comparison to the well-established case of periodic rod assemblies [@ref12] the fabrication of these designer disordered structures is very demanding and required several months of optimization in order to obtain the results reported here. Particular care must be taken how to set up the writing protocol and moreover all writing parameters have to be optimized in order to create mechanically stable structures. We note that precision direct laser writing is almost always operated by sequential writing of lines in 3D (with $840 \times 280$ nm$^2$ cross section in our case) and not by a layer-by-layer process, commonly employed in lower-resolution commercial 3D printing techniques used for rapid prototyping or manufacturing. Therefore, in our fabrication process particular care must be taken how to set up the writing protocol since there exist no obvious rules how to write a random free-standing network structure at optimal resolution. Moreover it must be ensured that the structure remains mechanically stable in a soft gel photoresist throughout the writing process of about 1 h. This task is further complicated by intrinsic mechanical stresses created upon exposure of the photoresist. The latter leads to substantial deformations if the written lines are not attached within seconds to a mechanically rigid superstructure. To overcome these challenging problems, encountered in our initial fabrication attempts, we have developed a optimized writing protocol. We first divide the entire volume in cubic sub-volumes of side length roughly $1.5 D$ (where $D$ is the average distance between nearest points in the underlying point pattern). We sort the rods in such a way that i) once all rods belonging to a certain cube are written, we proceed to a neighboring cube, ii) first all cubes closest to the substrate are filled with rods, then the ones lying higher above and so on until the whole network has been written. Fig. \[fig2\] displays a representative set of electron micrographs of fabricated structures. We succeeded to write structures with either a square ($65 \times 65 \mu$m$^2$) or circular (diameter $65 \mu$m) footprint with heights varying between $h=4-12 \mu$m. Structures higher than $4\mu$m were surrounded by a massive wall (Fig. \[fig2\]) for enhanced mechanical stability [@ref12].
![Electron micrographs of fabricated hyperuniform three-dimensional disordered structures. (a) Normal view of a structure with height $h = 8\mu$m and inner diameter $d =65 \mu m$ (b) close-up view (c) focused ion beam cut of the same structure.[]{data-label="fig2"}](fig2){width="10cm"}
Structural analysis by light scattering
---------------------------------------
We characterize the properties of our samples by measurements of the scattering patterns using visible light. For the optical characterization of our sub-mm sized samples we have built a small angle scattering instrument consisting of a helium neon laser ($\lambda = 632.8$ nm), a focusing lens (focal length $f=50$ mm), two diaphragms to suppress stray light and a white screen, positioned at a distance of $z= 125$mm from the sample, with a central absorbing beam block. The scattered light pattern is photographed off the white screen using a digital camera. The setup has been calibrated using a small pinhole. Histogram normalization has been applied to all images displayed in Fig. \[fig3\]. This procedure represents a linear transformation of an image where the value $P_{in}$ of each pixel is scaled according to [@ref16]: $P_{out}=255 (P_{in}-c)/(d-c)$. Here $c$ and $d$ are the $x$-th and $(100-x)$th percentile in the histogram of pixel intensity values. All values $P_{out}$ smaller than zero are set to $0$, while those larger than $255$ are set to $255$. In our case $x=0.1$. Before calculating radial averages of experimental diffraction patterns, several data processing steps were performed. Despite the use of diaphragms we could still observe some stray light contributing to the image. This contribution has to be subtracted from the raw data. To this end we have acquired an image from a bare glass substrate (empty cell) inserted into the laser beam. This image is subsequently subtracted from the raw data. The modulus of the scattering vector $\bf{q}$ is calculated from the radial distance $u$ from the center via the relation $\tan(\theta)=u/z$ and $q=(4¹/\lambda)\sin(\theta/2)$. These relations apply also for the (toluene or toluene/chlorobenzene) infiltrated structures since for small angles the reduction of the wavelength $\lambda/n$ within the sample is to a good approximation offset by refraction at the flat sample-air interface. Another small correction results from the actual detector acceptance angle being different for each scattering angle $\theta$. This gives rise to a correction [@ref19] of the measured data by a factor $\cos(\theta)^{-3}$. Examples for light scattering data recorded for $h = 4\mu$m structures are shown in Fig. \[fig3\]. The data clearly reveals a concentric ring profile without Bragg peaks. The pronounced maximum indicates short-range order while the ring-shape is a signature of structural isotropy without any long-range order. Another feature related to disorder is the speckled appearance of the scattering pattern, reminiscent of the laser speckle observed for an arbitrary random structure. Although our first observations confirm the overall picture, the ring pattern obtained in air, Fig. \[fig3\](a), appears blurred with a weaker than expected maximum and substantial low angle scattering. Interestingly similar observations have been made for quasicristalline structures [@ref17]. In our case we can clearly attribute the blurring to multiple scattering. From the attenuation of the direct laser beam intensity we can estimate a scattering length of $l_s \simeq 3\mu$m, short even compared to the lowest sample studied with $h = 4\mu$m. In order to reduce scattering we infiltrate the sample first with isopropanol ($n=1.377$) and then with toluol ($n=1.496$), which leads to a gradual reduction of multiple scattering and a sharpening of the diffraction ring, Fig. \[fig3\](b),(c), while at the same time the ring position remains unchanged. For the latter case the refractive index of the polymeric structure is almost matched and the direct beam is attenuated only by a few percent signaling the absence of multiple scattering. Similar results are obtained for the $h = 8\mu$m structures when using a 1:2 mixture of toluene/chlorobenzene ($n=1.517$) as an index matching fluid (Fig. \[fig4\] (b). In both cases a clear and pronounced ring appears in the scattering pattern at $q\sim 2 \pi/d$ and for smaller $q$-values scattering is strongly suppressed. Spatial speckle fluctuations can be largely (although not entirely) suppressed by taking radial averages as shown in Fig. \[fig4\] .
![Small angle light scattering pattern using a $\lambda=632.8$nm laser. (a),(b),(c), scattering pattern for a structure of thickness $h = 4\mu$m in air, infiltrated with isopropanol ($n=1.377$) and with toluene ($n=1.496$). (d) calculated scattering pattern in the single scattering limit for the same structure based on a discrete dipole approximation (DDA).[]{data-label="fig3"}](fig3){width="10cm"}
Comparison with numerical calculations
--------------------------------------
For a quantitative analysis of the experimental results we numerically solve the scattering problem using a discrete dipole approximation (DDA) in the single scattering limit [@ref18]. Given the weak scattering contrast in the toluene- (or toluene/chlorobenzene) infiltrated structures we expect this approximation to hold very well. First, rods coordinates generated in the manner described above are used to create a 3D binary representation of the network (Fig. \[fig1\] (a)). Namely a sufficiently dense grid is defined and then each grid point is set to 1 if the point belongs to a rod and 0 otherwise (Fig. \[fig1\] (b)). Using this procedure the lithographic voxel size and ellipsoidal shape (as obtained in our direct laser writing system) can be fully taken into account. Consequently, the binary network represents a good approximation of the true structure as manufactured by laser nanolithography. Following that, a 3D Fast Fourier Transform of this data is calculated. The squared modulus of FFT is proportional to the intensity $I(\bf{q})$ scattered by the structure for a scattering wave vector $\bf{q}=\bf{k}-\bf{k_0}$, where $\bf{k}$ denotes the scattered wave vector and $\bf{k_0}$ the incident wave vector. Lines plotted in Fig. \[fig4\] have been obtained by averaging over several realizations of a structure of a given height $h$. As shown in Fig. \[fig3\] (d) the numerical results reproduce both the ring diffraction as well as the superimposed random specular structure. A comparison of the radially averaged numerical data with experiment, Fig. \[fig4\] , reveals a very good match with no adjustable parameters except for the absolute scale of intensities.
![Radially averaged scattering intensity. Symbols: experimental data $I(q)$ for $h = 4\mu$m, immersed in toluene ($n=1.496$), and $h = 8\mu$m, immersed in toluene/chlorobenzene 1:2 ($n=1.517$). Solid lines: theoretical calculations for the same heights.[]{data-label="fig4"}](fig4){width="9cm"}
Discussion
==========
Our optical characterization confirms the high quality of our fabricated samples. Moreover the results readily show that due to the finite size the optical properties of the material are not yet fully developed. As can be seen in Fig. \[fig4\] the peak height increases from $h = 4\mu$m to $h = 8\mu$m. Preliminary numerical results (data not shown) indicate that the peak height saturates for heights larger than 20$\mu m$m. Moreover both the experimental and numerical results show that the extrapolated values for $S(q\to0)$ are finite. Due to experimental difficulties accessing wavenumbers close to the primary beam we are unable to clearly distinguish finite-size effects from residual contaminations and experimental artifacts. Similarly the accuracy of our numerical calculations for small wavenumbers is limited due to finite size effects. Nevertheless, since the seed structure is hyperuniform and since the quality of our polymeric templates is very high, we do believe that the polymeric template possesses the contemplated structural properties and thus should likely give rise to a full PBG when transferred into high-index dielectric. We now turn our attention to the spectral properties of the interconnected network structures. Numerical results predict a broad isotropic bandgap in three dimensions for a refractive index of $n=3.6$ and a volume filling fraction of about $20\%$ [@ref8]. In the following we discuss the additional processing steps required in order to obtain such strong photonic properties. Building upon the successful fabrication reported here the following parameters have to be optimized: a) the material refractive index needs to be increased b) the volume filling fraction must be increased to about 20$\%$ and finally the structural features ideally should be further scaled down. To realize the first point one can rely on well-established procedures reported in the literature. It has been shown that polymeric templates very similar to ours can be transferred into materials such as silicon ($n = 3.6$ for infrared wavelengths) by double inversion retaining the original topology [@ref17; @Stau10]. Optimal filling fractions have been predicted to be around 20$\%$ while currently, in our case, the polymer content is only around $8\%$. This means that for best results one either has to increase the polymer volume fraction and perform a double inversion or alternatively it should be possible to coat directly the polymeric template with a high index material. Finally, reducing the structural length scales will require further incremental optimisation of the delicate fabrication process. Such experiments are currently underway in our laboratory. For the polymeric structures reported here one would expect a mid gap wavelength in air at around $\lambda \sim 4\pi/q_{max} \sim 6\mu$m, about four times larger than typical telecommunication wavelengths of $\lambda \sim 1.5\mu$m [@ref8]. Based on preliminary results we expect to be able to reduce the typical length scales by at least a factor of two within the next months. We note that if the feature sizes are scaled down this also favourably affects the filling fraction as long as the size of the laser-writing pen is kept constant.
Summary and conclusion
======================
We have successfully demonstrated the fabrication of high quality three-dimensional polymeric templates for disordered photonic materials. Although a number of further processing steps are still required to obtain strong photonic properties, our results already demonstrate the feasibility of creating such complex materials on length scales comparable to optical wavelengths. As shown previously such polymeric templates can be reliably replicated into materials such as silicon. We thus envision a successful transfer of our polymer templates into a high index material within the near future [@Stau10]. Moreover, in the present study we have shown that the design parameters of polymeric templates can be set precisely using direct laser writing lithography. This in turn will allow rigorous experimental testing of the existing theoretical concepts when applied to the high index replica.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Georg Maret and Hui Cao for illuminating discussions and Matteo Molteni and Fabio Ferri for verifying some of our numerical calculations using an independent algorithm. We are grateful to the MPI for polymer research (Mainz) for giving us access to their focused-ion-beam (FIB) instrument and we thank Michael Kappl for help with the experiments. JH acknowledges funding from a Sciex Swiss Research Fellowship No. 10.030. We thank the Swiss National Science Foundation (projects 132736 and 128729) and the Adolphe Merkle Foundation for financial support.
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abstract: 'In bayesian UQ most relevant cases of forward maps (FM, or regressor function) are defined in terms of a system of (O, P)DE’s with intractable solutions. These necessarily involve a numerical method to find approximate versions of such solutions which lead to a numerical/approximate posterior distribution. In the past decade, several results have been published on the regularity conditions required to ensure converge of the numerical to the theoretical posterior. However, more practical guidelines are needed to ensure a suitable working numerical posterior. [@Capistran2016] prove for ODEs that the Bayes Factor (BF) of the approximate vs the theoretical model tends to 1 in the same order as the numerical method approximation order. In this work we generalize the latter paper in that we consider 1) the use of expected BFs, 2) also PDEs, 3) correlated observations, which results in, 4) more practical and workable guidelines in a more realistic multidimensional setting. The main result is a bound on the absolute global errors to be tolerated by the FM numerical solver, which we illustrate with some examples. Since the BF is kept near 1 we expect that the resulting numerical posterior is basically indistinguishable from the theoretical posterior, even though we are using an approximate numerical FM. The method is illustrated with an ODE and a PDE example, using synthetic data.'
author:
- |
J. Andrés Christen$^{1,2}$, Marcos A. Capistrán$^{1}$\
and Miguel Ángel Moreles$^{1}$
bibliography:
- 'PostErrControl\_and\_EABF.bib'
date: 29 AUG 2017
title: |
Numerical posterior distribution error control\
and expected Bayes Factors in the Bayesian\
Uncertainty Quantification of inverse problems
---
KEYWORDS: Inverse Problems, Bayesian Inference, Bayes factors, ODE solvers, PDE solvers.
Introduction {#sec:intro}
============
Bayesian Uncertainty Quantification (UQ) has attracted substantial attention in recent years, covering a wide range of applications both in well established fields as well as in emerging areas. Some recent examples may be found in . For reviews on the subject see .
The usual parametric (finite dimensional) Bayesian formulation of Inverse Problems, in broad terms, is that given a Forward Map (FM) $F_\theta$, a noise model is assumed for the observations $y_j \sim G_{\sigma}(F^j_\theta)$, for some noise level $\sigma$ and typically additive gaussian errors with known standard deviation $\sigma$ are assumed. This observation model creates a probability density of all data ${\mathbf{Y}}$ given all parameters $\Phi$ namely $P_{ {\mathbf{Y}}| \Phi} ({\mathbf{y}}| \theta,\sigma)$. For fixed data ${\mathbf{y}}$ this forms the likelihood, regarding the latter as a function of $\theta,\sigma$, and is the basis of the statistical analysis of Inverse Problems. Using Bayesian inference one establishes a prior distribution $P_{\Phi}(\theta,\sigma)$ and defines the posterior distribution $$\label{eqn.exact_post}
P_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) = \frac{P_{ {\mathbf{Y}}| \Phi }( {\mathbf{y}}| \theta,\sigma) P_{\Phi}(\theta,\sigma)}{P_{{\mathbf{Y}}} ({\mathbf{y}})} .$$ This probability distribution on the unknowns $\theta$ and $\sigma$ quantifies the uncertainty on the possible values for these parameters coherent with the data ${\mathbf{y}}$. However, the common denominator in this particular Bayesian inference problem is that we do not have an analytical or computationally simple and precise implementation of the FM.
Instead, a numerical approach is required to create a solver and find a numerical approximation of the FM $F^{\alpha}_\theta$, for some discretization parameter $\alpha$ (eg. step size, grid norm, terms in a series, etc. concrete examples will be given in section \[sec.setting\]). Since we can only use the numeric approximation, this in turn leads to a numeric likelihood $P^\alpha_{ {\mathbf{Y}}| \Phi} ({\mathbf{y}}| \theta,\sigma)$ and therefore a numeric posterior $$\label{eqn.num_post}
P^\alpha_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) = \frac{P^\alpha_{ {\mathbf{Y}}| \Phi }( {\mathbf{y}}| \theta,\sigma) P_{\Phi}(\theta,\sigma)}{P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})} .$$ $P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}}) = \int P^\alpha_{ {\mathbf{Y}}| \Phi }( {\mathbf{y}}| \theta,\sigma) P_{\Phi}(\theta,\sigma) d\theta d\sigma$ and $P_{{\mathbf{Y}}} ({\mathbf{y}}) = \int P_{ {\mathbf{Y}}| \Phi }( {\mathbf{y}}| \theta,\sigma) P_{\Phi}(\theta,\sigma) d\theta d\sigma$ are the normalization constants of the two models, also called the *marginal likelihoods* of data ${\mathbf{y}}$.
Numerical methods are designed so as, if the discretization tends to zero $|\alpha| \rightarrow 0$, for some norm or functional $|\cdot|$, then the numeric FM tends to the theoretical FM at some order $O(|\alpha|^p)$. This is the global error control for the numerical method or solver, which we dicuss in detail in section \[sec.global\_err\]. However, it is of great interest to prove that the same happens with the theoretical vs the numeric posteriors in (\[eqn.exact\_post\]) and (\[eqn.num\_post\]) respectively, in order to make sense of our Bayesian approach.
Recently a number of papers have dealt with this problem in a theoretical sense by establishing regularity conditions so as $$\lim_{|\alpha| \rightarrow 0}
|| P^\alpha_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) - P_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) || = 0 ,$$ for some (eg. Hellinger) measure metric $|| \cdot ||$; see [@COTTER2010] for a review. This forms a sound theoretical basis for the Bayesian analysis of inverse problems. However, in applications we have to choose a discretization $\alpha$. More practical guidelines are needed to choose the numerical solver precision and how this controls the level of approximation between $P^\alpha_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) $ and $P_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) $.
On the other hand [@Capistran2016] present an approach to address the above problem using Bayes factors (BF; the odds in favor) of the numerical model vs the theoretical model (further details will be given in section \[sec.setting\]). With equal prior probability for both models, this BF is $\frac{P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})}$. In an ODE framework, these odds are proved in [@Capistran2016] to converge to 1 (that is, both models would be equal) in the same order as the numerical solver used. For high order solvers [@Capistran2016] illustrates, by reducing the step size in the numerical solver, that there should exist a point at which the BF is basically 1, but for fixed discretization $\alpha$ (step size) greater than zero. This is the main point made by [@Capistran2016]: it could be possible to calculate, for solver orders of 2 or more, a threshold for the tolerance such that the *numerical* posterior is basically equal to the theoretical posterior so, although we are using an approximate FM, the resulting posterior is error free. [@Capistran2016] illustrate, with some examples, that such optimal solver discretization leads to basically no differences in the numerical and the theoretical posterior (since the BF is basically 1). Moreover, since for most solvers its computational complexity goes to infinity as $|\alpha| \rightarrow 0$, using the optimal $\alpha$ led in their examples to a 90% save in CPU time. However, [@Capistran2016] still has a number of shortcomings. First, it depends crucially on estimating the normalizing constants $P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})$ from Monte Carlo samples of the unnormalized posterior, for a range of discretizations $|\alpha|$. This is a very complex estimation problem and is the subject of current research and is in fact very difficult to reliably estimate these normalizing constants in mid to high dimension problems. Second, [@Capistran2016] approach is as yet incomplete since one would need to decrease $|\alpha|$ systematically, calculating $P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})$ to eventually estimate $P_{{\mathbf{Y}}} ({\mathbf{y}})$, which in turn will pin point a discretization at which both models are indistinguishable. Being this a second complex estimation problem, the main difficulty here is that one has already calculated the posterior for small $|\alpha|$ and therefore it renders useless the selection of the optimal step size.
To improve on [@Capistran2016], the idea of this paper is to consider the *expected* value of the BFs, before data is observed. We will try to bound this expected BF to find general guidelines to establish error bounds on the numerical solver, depending on the specific problem at hand and the sample design used, but not on particular data. These guidelines will be solely regarding the forward map and, although conservative, represent useful bounds to be used in practice.
We do not discuss in this paper the infinite dimension counterpart of this approach, of interest when inference is needed over function spaces as it is the case in some general PDE inverse problems, see for example [@COTTERetAl2009; @Dunlop2015] and references therein. As mentioned above, we restrict ourselves to the finite dimensional parametric case where we establish our results.
The paper is organized as follows. Our formal setting will be discussed in section \[sec.setting\]. In section \[sec:main\] we present our main result, including several comments of some implications and practical guidelines for its use. In sections \[sec:exaODE\] and \[sec:exaPDE\] we present prove of concept examples, considering an ODE and a PDE, respectively. In both cases, using error estimated on the numeric forward maps we were able to very sustantially reduce CPU time while obtaining basically the same posterior. Finally, a discussion of the paper is presented in section \[sec:discussion\].
Setting {#sec.setting}
=======
Assume that we observe a process ${\mathbf{y}}= (y_1,\dots,y_n)$ at *locations* $x_1, \dots, x_n \in D \subset {\mathbb{R}}^m$. This is a general setting, to include ODEs and PDEs and other inverse problems, in which the domain may include, for example, space and time: $x_i = [ (z_{ix}, z_{iy}), t_i ]$. That is, $x_i$ is an observation at coordinates $(z_{ix}, z_{iy})$ and at time $t_i$, etc.
We assume that the Forward Map $F_{\theta} : {\mathbb{R}}^m \rightarrow {\mathbb{R}}^q$ is well defined for all parameters $\theta$ where $\theta \in A \subset {\mathbb{R}}^d$. Typically, as mentioned above, $F_{\theta}(x)$, for all $x \in D$, is the solution of a system of ODE’s or PDE’s. This means that $F_{\theta}(x)$ are the $q$ state variables representing the solution of the ODE or PDE system, with parameters $\theta$, at location $x$. In many cases, specailly dealing with PDEs, the actual unknown is a function in which the inference problem at hand is infinite dimensional. As mentioned in the introduction, in this paper we confine ourselves to the finite dimensional parametric problem, that is, the unknown is $\theta$ of dimension $d$. The initial or boundary conditions are taken as known, although these may be turned to be part of the unknown parameters, using common techniques.
Let $f: {\mathbb{R}}^q \rightarrow {\mathbb{R}}$ be the observational functional, in the sense that $y_i$ is an observation of $f(F_{\theta}(x_i))$. For example, $f(F_{\theta}(x_i))$ is one particular state variable, for which we have observations. We only consider univariate observations at each location $x_i$.
We assume gaussian errors on the observations, however, we consider the possibility of correlated observations, namely, let ${\mathbf{f}}_{\theta} = ( f(F_{\theta}(x_1)), \ldots , f(F_{\theta}(x_n)) )'$ then $${\mathbf{y}}\mid \theta, \sigma^2, {\mathbf{A}}\sim N_n ( {\mathbf{f}}_{\theta}, \sigma^{-2} {\mathbf{A}}),$$ where $\sigma^{-2} {\mathbf{A}}$ is the *precision* matrix (inverse of the variance-covariance matrix) of the $n$-dimensional Gaussian distribution with mean ${\mathbf{f}}_{\theta}$. ${\mathbf{A}}^{-1}$ is a correlation matrix with some correlation structure on $D$. ${\mathbf{A}}^{-1}$ will be taken as known, but $\sigma^{2}$ will be considered unknown in general. This is a particular covariance structure, very common in time series or spatial statistics. Indeed, if ${\mathbf{A}}= \mathbf{I}$ we go back to the common uncorrelated error structure. A specific example is to use a correlation function $\rho$ an a metric $d(x_i,x_j)$ in $D$ to give ${\mathbf{A}}^{-1} = (\rho(d(x_i,x_j)))$, and therefore correlation decreases as the location points separate. This is an isotropic correlation structure. Many other structures may be considered and this has been extensively studied in the statistics literature [@christakos:1992]. Note that the marginal distribution for the observations is $$\label{eqn.obsmodel}
y_i = f(F_{\theta}(x_i)) + \varepsilon_i ,~~ \varepsilon_i \sim \mathcal{N}(0, \sigma b_i) ,$$ where $b_i^2 = [A^{-1}]_{ii}$, that is, $\sigma b_i$ is the standard error of the noise.
Let also ${\mathbf{f}}_{\theta}^{\alpha} = ( f(F_{\theta}^{\alpha}(x_1)), \ldots , f(F_{\theta}^{\alpha}(x_n)) )'$ and ${\mathbf{y}}\mid \theta, \sigma^2, {\mathbf{A}}\sim N_n ( {\mathbf{f}}_{\theta}^{\alpha}, \sigma^{-2} {\mathbf{A}})$ for the numerical model. From this the likelihood of the approximated model is $$\label{eqn.ll}
P^\alpha_{ {\mathbf{Y}}| \Phi} ({\mathbf{y}}| \theta,\sigma) =
(2 \pi \sigma^2)^{-\frac{n}{2}} \left| {\mathbf{A}}\right|^{ \frac{1}{2} }
\exp\left\{-\frac{1}{2 \sigma^2} ({\mathbf{y}}- {\mathbf{f}}_{\theta}^{\alpha})' {\mathbf{A}}({\mathbf{y}}- {\mathbf{f}}_{\theta}^{\alpha})\right\} ,$$ with the corresponding expression for the exact model.
Regarding the prior distribution for the parameters $P_{\Phi}(\theta,\sigma)$, we assume that $P_{\Phi}(\theta,\sigma) = g(\sigma) P_\Theta (\theta)$. A common and reasonable assumption in which the prior on the observational noise is independent of the prior regarding the actual model parameters.
Global Error Control {#sec.global_err}
--------------------
We assume we use a numerical method to obtain an approximation of the Forward Map, that is $F^{\alpha}_{\theta}(x_i)$, for some discretization $\alpha$. As a general setting for approximation strategies, the numerical method or solver discretization $\alpha$ may now be multidimensional. We assume that the global error control of the solver states that $$\label{eqn:global_error}
|| F^{\alpha}_{\theta}(x_i) - F_{\theta}(x_i) || \leq K_{x_i , \theta} |\alpha|^p ,$$ for some functional $| \cdot |$ for the discretization. That is, the solver is of order $p$. For example $\alpha = ( \Delta x, \Delta t )$ in finite element PDE solvers, etc. and, perhaps $|\alpha| = sup |\alpha_i|$. In section \[sec:exaODE\] we present an ODE example in which $\alpha = \Delta t$, ie. the time step size, $| \cdot |$ is the identity and the solver has order $p=4$. Moreover, in section \[sec:exaPDE\] we present a PDE example where $\alpha = ( \Delta x, \Delta t )$, $|\alpha| = \Delta x$ and the solver has order $p=2$.
Using this asymptotic behavior of the global error and assuming that $f$ is differentiable, we can write $$\label{eqn:Dh}
f(F^{\alpha}_{\theta}(x)) - f(F_{\theta}(x)) =
\nabla f \left( F_{\theta}(x) \right)(F^{\alpha}_{\theta}(x) -F_{\theta}(x)) + O(|\alpha|^{2p}) = O(|\alpha|^p) .$$ for all $x \in D$ and $\theta \in A$. We further assume that $\nabla f$ is bounded and that $K_{x_i , \theta} < K'$ for all locations $x_i$ and all $\theta \in A$. From this we conclude that the global error is $$\label{eqn.global_error}
| f(F^{\alpha}_{\theta}(x_i)) - f(F_{\theta}(x_i)) | \leq K |\alpha|^p ,$$ for some global $K$. This approximation is also of order $p$.
Bayes Factors {#sec.BF}
-------------
As mentioned in the introduction the Bayes Factor (BF) of the exact vs the approximated model, for some fixed data ${\mathbf{y}}$ is $\frac{P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})}$. Assuming an equal prior probability for both models, the BF is the posterior odds of one model against the other (ie. $\frac{p}{1-p}$ with $p=\frac{P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}}) + P_{{\mathbf{Y}}} ({\mathbf{y}})}$ the posterior probability of the numerical model with discretization $\alpha$). In terms of model equivalence an alternative expression conveying the same odds is $$\frac{1}{2} \left| 1 - \frac{P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})} \right| .$$ We will call the above absolute deviance of the BF from 1 the “ABF”. In terms of the Jeffreys scale if the ABF is less than 1 (ie. 1 $\leq$ BF $\leq$ 3) the difference regarding both models is “not worth more than a bare mention” [@KASS1995; @Jeffreys61]. An even more stringent requirement would be an ABF less than $\frac{1}{20} = 0.05$, for example, to practically ensure no difference in both the numerical and the approximated posteriors. In the next section we see how to bound the *expected* ABF in terms of the absolute maximum global error for the numeric FM.
Main result {#sec:main}
===========
We now present our main result.
\[theo.main\] For the Expected ABF (EABF) we have that $$\label{eqn.main}
|| P_{{\mathbf{Y}}} (\cdot) - P^{\alpha}_{{\mathbf{Y}}} (\cdot) ||_{TV} =
\int \frac{1}{2} \left| 1 - \frac{P^{\alpha}_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})} \right| P_{{\mathbf{Y}}} ({\mathbf{y}}) d{\mathbf{y}}\leq \sqrt{\frac{1}{2 \pi}} \frac{n}{\sigma^*} K |\alpha|^p \frac{b_i}{n} \sum_{i=1}^n \sum_{j=1}^n | a_{ij} | .$$ where $\sigma^* = \left( \int \frac{1}{\sigma} g(\sigma) d\sigma \right)^{-1}$.
As in [@Capistran2016] define the likelihood ratio $$R_{\alpha}(\theta) = \frac{P^\alpha_{ {\mathbf{Y}}| \Phi} ({\mathbf{y}}| \theta,\sigma)}
{P_{ {\mathbf{Y}}| \Phi} ({\mathbf{y}}| \theta,\sigma)} .$$ Using (\[eqn.ll\]) and after a simple manipulation we see that $$R_{\alpha}(\theta) =
\exp\left[ -\frac{1}{2\sigma^2} \left\{
-2 ({\mathbf{f}}_{\theta}^{\alpha} - {\mathbf{f}}_{\theta})'{\mathbf{A}}({\mathbf{y}}- {\mathbf{f}}_{\theta}) +
({\mathbf{f}}_{\theta}^{\alpha} - {\mathbf{f}}_{\theta})'{\mathbf{A}}({\mathbf{f}}_{\theta}^{\alpha} - {\mathbf{f}}_{\theta}) \right\}\right] .$$ Let ${\mathbf{D}}_{\theta}^{\alpha} = ({\mathbf{f}}_{\theta}^{\alpha} - {\mathbf{f}}_{\theta})'$. Using (\[eqn.global\_error\]) we have $$\label{eqn.quad_bound}
|{\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}({\mathbf{D}}_{\theta}^{\alpha})'| <
||{\mathbf{D}}_{\theta}^{\alpha}||_2^2 ||{\mathbf{A}}||_2 = O(|\alpha|^{2p})$$ where $||{\mathbf{A}}||_2$ refers to the induced $L_2$ matrix norm. Therefore for $|\alpha|$ small $$R_{\alpha}(\theta) -1 = \frac{1}{\sigma^2} {\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}({\mathbf{y}}- {\mathbf{f}}_{\theta}) + O(|\alpha|^{2p})$$ since $e^{-x} = 1 - x + O(x^2)$ for $|x|$ small. Note now that $$P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}}) = P_{{\mathbf{Y}}} ({\mathbf{y}}) +
\int P_{ {\mathbf{Y}}| \Phi }( {\mathbf{y}}| \theta,\sigma) (R_{\alpha}(\theta) -1) P_{\Phi}(\theta,\sigma) d\theta d\sigma .$$ Dividing by $P_{{\mathbf{Y}}} ({\mathbf{y}})$ we have $$\begin{aligned}
\left| 1 - \frac{P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})} \right| &=&
\left| \int \left(\frac{1}{\sigma^2} {\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}({\mathbf{y}}- {\mathbf{f}}_{\theta}) + O(|\alpha|^{2p})\right)
P_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) d\theta d\sigma \right| \nonumber \\
&\leq& \frac{1}{\sigma^2} \int \left| {\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}({\mathbf{y}}- {\mathbf{f}}_{\theta})\right|
P_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) d\theta d\sigma + |O(|\alpha|^{2p})| .\label{eqn.ABF}\end{aligned}$$ Now for the EABF we have $$\int \left| 1 - \frac{P^{\alpha}_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})} \right| P_{{\mathbf{Y}}} ({\mathbf{y}}) d{\mathbf{y}}\leq
\frac{1}{\sigma^2} \int \int \left| {\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}({\mathbf{y}}- {\mathbf{f}}_{\theta})\right|
P_{ {\mathbf{Y}}| \Phi }( {\mathbf{y}}| \theta,\sigma) d{\mathbf{y}}P_{\Phi}(\theta,\sigma) d\theta d\sigma ,$$ ignoring the higher order term. Let ${\mathbf{u}}= {\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}$ with ${\mathbf{u}}= (u_i)$, then $$\label{eqn.bound1}
\sigma^{-2} \left| {\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}({\mathbf{y}}- {\mathbf{f}}_{\theta})\right| =
\sigma^{-2} \left| \sum_{i=1}^n u_i (y_i - f(F_{\theta}(x_i))) \right|
\leq \frac{1}{\sigma} \sum_{i=1}^n |u_i| b_i \left| \frac{y_i - f(F_{\theta}(x_i))}{\sigma b_i} \right| .$$ Since $\int |x| \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} dx = \sqrt{\frac{2}{\pi}}$ we have $$\int \left| 1 - \frac{P^{\alpha}_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})} \right| P_{{\mathbf{Y}}} ({\mathbf{y}}) d{\mathbf{y}}\leq
\sqrt{\frac{2}{\pi}} \int \frac{b_i}{\sigma} \sum_{i=1}^n |u_i| P_{\Phi}(\theta,\sigma) d\theta d\sigma ,$$ and therefore $$\label{eqn.main1}
\int \frac{1}{2} \left| 1 - \frac{P^{\alpha}_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})} \right| P_{{\mathbf{Y}}} ({\mathbf{y}}) d{\mathbf{y}}\leq
\sqrt{\frac{1}{2 \pi}} \frac{b_i}{\sigma^*} \int \sum_{i=1}^n |u_i| P_\Theta (\theta) d\theta =
\sqrt{\frac{1}{2 \pi}} \frac{b_i}{\sigma^*} \int ||{\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}||_1 P_\Theta (\theta) d\theta .$$ From (\[eqn.global\_error\]) note that $$\label{eqn.bound2}
|u_i| \leq K |\alpha|^p \sum_{j=1}^n | a_{ij} |$$ and from this we obtain the result.
The EABF tends to zero in the same order as the numerical solver, that is $O(|\alpha|^p)$; the same happens to the ABF assuming $$\int \left| \frac{y_i - f(F_{\theta}(x_i))}{\sigma b_i} \right|
P_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) d\theta d\sigma < \infty.$$ for all $x_i$.
The result is immediate for the EABF since the bound in (\[eqn.main\]) is $O(|\alpha|^p)$. For the ABF use (\[eqn.ABF\]) and (\[eqn.bound1\]) to obtain the result.
Remarks on Theorem \[theo.main\]
--------------------------------
- Note that if, as in many inverse problems formulations, the gaussian observation noise variance $\sigma^2$ assumed known, then we could think that its prior is a Dirac delta and indeed $\sigma^* = \left( \int \frac{1}{s} g(s) ds \right)^{-1} = \sigma$. In fact, we assume in the examples in sections \[sec:exaODE\] and \[sec:exaPDE\] that $\sigma$ is known.
- The bound in Theorem \[theo.main\] is a result of the uniform bound for the error in the numerical solver, as stated in (\[eqn.global\_error\]). However, a more precise and at least theoretically interesting bound would be using the integral in the rhs of (\[eqn.main1\]) $$\int ||{\mathbf{D}}_{\theta}^{\alpha} {\mathbf{A}}||_1 P_\Theta (\theta) d\theta < C .$$ With this one may obtain an *expected* bound for the EABF. Since it is an expected and not an uniform bound, potentially we may allow more solver error in regions with less a priori mass, etc.
However, our present approach considers using a simpler global bound uniform for all $\theta$, perhaps less precise, but importantly not adding more computational burden on problems already computationally very demanding, as explained in the next remark.
- If we set the absolute uniform global error as $| f(F^{\alpha}_{\theta}(x_i)) - f(F_{\theta}(x_i)) | \leq K_0 = K |\alpha|^p$ and, as explained in section \[sec.BF\], if we let the $EABF \leq \frac{1}{20} = 0.05$ we expect nearly no difference in the numerical and the theoretical posterior. For uncorrelated data we have $a_{ij} = \delta_{i,j}$ and therefore $\frac{b_i}{n} \sum_{i=1}^n \sum_{i=1}^n | a_{ij} | = 1$. Then, setting $\sqrt{\frac{1}{2\pi}} \frac{n K_0 }{\sigma^*} \leq \frac{1}{20} $ we need $$\label{eqn.main_bound}
K_0 \leq \frac{k}{n} \sigma^*$$ where $k = \frac{1}{20} \sqrt{2 \pi} \approx 0.12$. That is, we may tolerate an absolute uniform error of up to 12% of the expected standard deviation observation noise for $n=1$, 0.12% for $n=10$, 0.012% for $n=100$ etc. Reasonably, in the light of this analysis of the Bayesian inverse problems, the tolerated error in the numerical solver is put in terms of the observational noise expected and the sample size. When more noise is expected more error may be tolerated and as the sample size increases the numerical solver needs to be more precise.
This is the main application of our theorem, the uniform global error to be accepted should be bounded in terms of the observational noise and sample size considered and not solely as a property of the forward map at hand. Our bound could be conservative, but it is intended to be an useful and applicable tool serving as a reference for the numerical solver precision.
- Note that the bound in (\[eqn.global\_error\]) may be realized during computation. That is, the solver may be applied and its error estimated at the design points $x_i$, when needing the FM for some specific $\theta$. If the required bound is exceeded then the discretization $\alpha$ could be altered to comply with the error bound. That is, in practice there is no need to establish (\[eqn.global\_error\]) theoretically, but rather by a careful strategy for actual global error estimation (in the numerical analysis literature these error estimates are derived from the so called output or *a posteriori* error estimates, but here we use a different name for obvious reasons). The posterior distribution in most cases is sampled using MCMC, which requires the approximated likelihood at each of many iterations; an automatic process of global error estimation and control will be required in order to comply with (\[eqn.global\_error\]).
- The result in the corollary regarding the ABF means that, for any fixed data ${\mathbf{y}}$ the BF $\frac{P^\alpha_{{\mathbf{Y}}} ({\mathbf{y}})}{P_{{\mathbf{Y}}} ({\mathbf{y}})}$ tends to one in the same order as the solver order, that is $O(|\alpha|^p)$. This is a more general version of the main result of [@Capistran2016] using the assumption $$\int \left| \frac{y_i - f(F_{\theta}(x_i))}{\sigma b_i} \right|
P_{ \Phi | {\mathbf{Y}}}( \theta,\sigma | {\mathbf{y}}) d\theta d\sigma < \infty$$ (i.e. the posterior expected standardrized absolute residuals are bounded). In [@Capistran2016] they assumed instead that the parameter space $A$ is compact, which implies the above. The setting in [@Capistran2016] is only for ODE’s while ours considers more general forward maps, including ODEs and PDEs.
Example using an ODE: Logistic growth {#sec:exaODE}
=====================================
![\[fig.LogGr\_DataAndFit\] Logistic growth example data, true model (blue) and best (MAP) fit (red). True parameters are $r=1$, $K=1000$ and $\sigma=30$. Shaded areas represent the uncertainty in the model fit, as draws from the posterior distribution.](LogGr_DataAndFit_30.pdf){height="5.5cm" width="8cm"}
As a prove of concept, we base our first numerical study on the logistic growth model which is a common model of population growth in ecology, medicine, among many other applications [@Forys2003]. Let $X(t)$ be, for example, the size of a tumor to time $t$. The logistic growth dynamics are governed by the following differential equation $$\label{eq:logistic}
\frac{dX}{dt} = r X(t) (1-X(t)/K), \quad X(0)=X_0$$ with $r$ being the growth rate and $K$ the carrying capacity e.g. $\lim_{t\rightarrow \infty} X(t) = K$. The ODE in (\[eq:logistic\]) has an explicit solution equal to $$X(t) = \frac{KX_0}{X_0 + (K-X_0)e^{- r t}}.$$ We simulate a synthetic data set with the error model $ y_i =X(t_i) + \varepsilon_i$, where $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$, and the following parameters $X(0)=100, \quad r= 1, \quad K=1000,$ $\sigma = 30$. The data are plotted in Figure \[fig.LogGr\_DataAndFit\]. We consider $26$ observations at times $t_i$ regularly spaced between $0$ and $10$. [@Capistran2016] also studied this example.
Since we have an analytic solution, if we run a numerical solver on the system we may calculate the maximum absolute error of the solver, $K_0$ in (\[eqn.main\_bound\]), exactly by comparing with the analytic solution. Moreover, in Appendix \[sec.appendErrODE\] we explain how the global error may be estimated using Runge-Kutta solver methods. We run a Cash–Karp RK method, of order 5 [@Cash1990], which enables us to produce and estimate $\hat{K}_0$ of $K_0$.
The error bound for the FM as stated in (\[eqn.main\_bound\]) is $\frac{k}{26} 30 \approx 0.13$. The ODE with the initial condition defines our forward map $F_{\theta}$ ($q=p=1$) and we take $f(x) = x$ as the observational functional. To sample from the posterior distribution we use a generic MCMC algorithm called the t-walk [@Christen2010].
Regarding the numerical solver we start with a large step size of $0.1$, that would maintain numerical stability, and calculate $\hat{K}_0$ and $K_0$. If the solution does not comply with the bound as calculated by the estimate, that is $\hat{K}_0 > 0.13$, a new solution is attempted by reducing the step size by half, until the Runge-Kutta estimated global absolute errors is within the bound, $\hat{K}_0 \leq 0.13$. The results are shown in figure \[fig.LogGr\_PostComp\]. Moreover, a fine step size Runge-Kutta solver was also ran with step size of 0.005, for comparisons. No difference was observed in both posterior distributions, and the posterior mean and MAP estimators were basically identical. However, a 93% save was obtain by reducing CPU time from approximately 100 to 7 min, with 40,000 MCMC iterations.
Since in this case an analytic solution is available we also calculate the exact maximum absolute error $K_0$. The average estimated error was $7.8\cdot10^{-3}$ while the average exact error was $3.8\cdot10^{-5}$, for the adaptive step size method as describe above. For the fixed time step method we had estimated and average errors $2.1\cdot10^{-8}$ and $6.2\cdot10^{-12}$, respectively. In both cases our error estimates where some orders of magnitude higher than the true errors. Certainly the fine (fixed) step solver is quite more precise reaching more than 5 orders of magnitude larger precision. However, in the light of theorem \[theo.main\] this increase precision is not worth the great increase in CPU time since indeed results are basically the same using a coarser solver. And a 93% CPU time reduction was obtained nevertheless our error estimates where two orders of magnitude higher. Numerical errors are now viewed, not in a context free situation, but in the context of this inverse problem, relative to sample size and expected observational errors, by using the bound in (\[eqn.main\_bound\]).
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![\[fig.LogGr\_PostComp\] Logistic growth example marginal posterior distributions of $r$ (a) and $K$ (b), $\sigma=30$. For a fine step size of 0.005 (blue histograms) we obtain basically the same results (green histograms) as starting from a step size of 0.1 and restricting the estimated maximum absolute error to the proposed bound, that is $\hat{K}_0 \leq \frac{k}{26} \sigma = 0.13$. A 92% CPU time reduction was obtained, from approximately 103 to 7 min, running 40,000 iterations of the MCMC.](LogGr_Postr_30.pdf "fig:"){height="4.5cm" width="6.5cm"} ![\[fig.LogGr\_PostComp\] Logistic growth example marginal posterior distributions of $r$ (a) and $K$ (b), $\sigma=30$. For a fine step size of 0.005 (blue histograms) we obtain basically the same results (green histograms) as starting from a step size of 0.1 and restricting the estimated maximum absolute error to the proposed bound, that is $\hat{K}_0 \leq \frac{k}{26} \sigma = 0.13$. A 92% CPU time reduction was obtained, from approximately 103 to 7 min, running 40,000 iterations of the MCMC.](LogGr_PostK_30.pdf "fig:"){height="4.5cm" width="6.5cm"}
(a) (b)
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Example using a PDE: Burgers’ equation {#sec:exaPDE}
======================================
Burgers’ is a fundamental PDE with many applications in fluid mechanics, acoustics, traffic flow modeling, among others [@leveque2002finite].
Let us consider the Riemann problem for the viscous Burgers’ equation $$\label{eq:burgers}
\begin{split}
u_{t} + uu_{z}&= \epsilon u_{zz}\\
u(z,0)&=
\begin{cases}
u_{L} & z < z_{0}\\
u_{R} & z_{0} < z,
\end{cases}
\end{split}$$ where $u_{L}$, $u_{R}$, $z_{0}$ and $\epsilon$ are real numbers. The direct problem is to compute $u = u( z, t)$ solving the initial value problem (\[eq:burgers\]). We assume $u_{L}>u_{R}$ which would correspond to a shock wave in the inviscid limit $\epsilon=0$.
Using the Cole-Hopf transformation we can obtain the solution of problem (\[eq:burgers\]) in closed form to obtain $$\label{eq:soln_burgers}
u( z, t) = u_{L}-\frac{u_{L}-u_{R}}{1 + q(z-z_{0},t) \exp(-\frac{u_{L}-u_{R}}{2\epsilon}(z -z_{0} - ct))}$$ where $q( z, t) = \text{erfc}(\frac{ z - u_{R} t}{\sqrt{4\epsilon t}})/\text{erfc}(\frac{ z - u_{L}t}{\sqrt{ 4\epsilon t}})$ and $c=(u_{L}+u_{R})/2$; see Chapter 4 of Whitham [@Whitham1999], for details.
The inference problem is to estimate $\theta=(u_{L}-u_{R}, z_{0})$ given observations $u_{j} = u( z_{1}, t_{j} )$ at a fixed point in space $z_{1}$ for $j=1,..,k$. That is, the locations are $x_j = ( z_{1}, t_{j} )$. As in the previous example, the observation operator $f(\cdot)$ is the identity.
To numerically solve the Riemann problem in (\[eq:burgers\]), we use a classical second-order accurate finite-volume implementation of the viscous Burgers’ equation with piecewise linear slope reconstruction with outflow boundary conditions, see Leveque [@leveque2002finite]. In this case, we build a grid with both space and time steps, namely $\alpha = (\Delta z, \Delta t )$. The standard procedure is to fix $\Delta z$ and the temporal grid is determined by a Courant-Friedricks-Levy condition $$\label{eq:cfl}
\Delta t_n = c \frac{\Delta z}{\max | u(\cdot, t_{n-1}) |} .$$
In the numerical example below we use $c=0.1$. In this case $|\alpha| = \Delta z$ and $p=2$, that is, the numerical method is order 2, $O(|\alpha|^2)$. Moreover, error estimates may be obtained from this solution, as explained in Appendix \[sec.appenErrPDE\].
Using the analytic solution in (\[eq:soln\_burgers\]) and the error estimates, we can establish the exact error $K_0$ and its estimate $\hat{K}_{0}$ as in the previous example. From $\hat{K}_{0}$ we check if the adaptive method is within the bounds established in (\[eqn.main\_bound\]). Note that, in the current case $\hat{K}_{0}\approx K_{0}$, provided $K_{0}$ is estimated through interpolation, as shown in Appendix \[sec.appenErrPDE\].
Again we use synthetic data using the observation point $z_1=2.0$, $n=6$, $t=\{0.0,0.1,0.2,\allowbreak 0.3,0.4,0.5\}$ and $\sigma = 0.0115$ (we choose this standard error to obtain a signal-to-noise ratio of $100$). The true parameter values are $u_{R}=2$, $u_{L}=1$, $z_{0}=1$.
We estimate the numerical posterior distribution with a high resolution spatial grid of $512$ points for $z \in [0,4]$, and an adaptive grid, starting with a grid of 128. If the bound is not met then the grid was doubled to 256 and thereafter to 512. We used 20,000 iterations of the twalk MCMC algorithm [@Christen2010] to simulate from the posterior distributions using the high-resolution and adaptive grid solvers. The execution time for the high resolution and adaptive grids are respectively $10.95$ h and $4.27$ h respectively. That is, in this case we obtained a 60% decrease in CPU, again leading to basically the same posterior distribution.
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![\[fig.Burgers\_PostComp\] Burgers equation example, marginal posterior distribution for (a) $u_{L}-u_{R}$ and (b) $z_{0}$. Blue histograms are the result of the adaptive grid using our bound, transparent green histogram denotes the posterior distribution from the high resolution grid (see text). Both histograms are basically the same, with a 60% decrease in CPU time.](burgers_uL-uR.png "fig:"){height="5.cm" width="6cm"} ![\[fig.Burgers\_PostComp\] Burgers equation example, marginal posterior distribution for (a) $u_{L}-u_{R}$ and (b) $z_{0}$. Blue histograms are the result of the adaptive grid using our bound, transparent green histogram denotes the posterior distribution from the high resolution grid (see text). Both histograms are basically the same, with a 60% decrease in CPU time.](burgers_x0.png "fig:"){height="5.cm" width="6cm"}
(a) (b)
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Discussion {#sec:discussion}
==========
The Bayesian UQ analysis of inverse problems continues to be a very challenging research topic. In this paper we tried to contribute to the development of this discipline by analyzing the relationship between error in the numerical solver of the differential system under study and the corresponding induced error in the numerical posterior distribution. Once it is established, under regularity conditions, that the numerical posterior distribution tends to the theoretical posterior as the solver error tends to zero, we still need to decide to which precision run the solver. Elaborating on previous work [@Capistran2016], theorem \[theo.main\] suggest that by carefully choosing a threshold for the global error in the solver we may obtain a posterior distribution that is basically error free. An intuitive perspective on our result is the following: since there is observational error, we may tolerate certain small amount of error in the solver, which will end up blurred by the observational error and therefore not noticeable in the posterior distribution.
Indeed, “small” is relative to the data error standard deviation $\sigma$ (and to the sample size), as expressed in (\[eqn.main\_bound\]). This also suggest that, in Bayesian UQ, solver error could be viewed in the perspective of the inference problem at hand, potentially allowing for less precise and less computational demanding solvers that, nevertheless induce basically no error in the resulting posterior distribution and therefore in the uncertainty quantification of the problem at hand.
On the other hand, it is well known that the numerical solver discretization $\alpha$ must be carefully tuned and $|\alpha|$ cannot be increased to arbitrary values. In complex cases, beyond some tight limits for the discretization the numerical solver becomes unstable and results as in (\[eqn:global\_error\]) cease to be valid. Indeed, solver order and global error control are properties valid for “small” $|\alpha|$. Therefore, in a large set of case studies, little room will be available to decide on an optimal solver discretization. Moreover, in very complex PDE solvers, just changing the discretization is a very demanding enterprise, since there is as yet no automatic and reliable way to define the solver grid, as in for example some spacio-temporal 3D PDE case studies [@Cui2011].
Our examples were proof of concept only, but show promising results. More importantly, we believe, since the bound in (\[eqn.main\_bound\]) has a simple form, the posterior adaptive control described in both the ODE and the PDE examples could be applied in other case studies. However, this hinges on the availability of reliable, adaptive-like, solver error estimates which, unfortunately, are not always readily available.
A natural continuation of this work is to consider the performance of embedded methods such as Cash-Karp RK45, as well as other feasible alternatives, on *stiff* ODE problems in this Bayesian context. For PDEs our current interest is on conservation laws approximating the Forward Map by the Discontinuous Galerkin method. For this method, error estimates rest on a solid foundation making our approach promising, see [@hesthaven2007nodal; @di2011mathematical].
The study of our results in more complex UQ problems is also left for future research. Note also that in many PDE problems the actual unknown is a function, eg. an unknown boundary condition on a scattering problem, needed to be recovered from data. In such a case, the theoretical posterior distribution is infinite dimensional and, as we explained from the onset, we did not consider such case. The proof of our results in such arbitrary space setting is also left for future research.
Global error estimation in Runge-Kutta methods {#sec.appendErrODE}
==============================================
For the reader’s convenience, here we briefly describe how we may estimate the global error in solving the following ODE initial value problem $$\frac{X(t)}{dt} = G( t, X, \theta), X(0) = X_0,$$ when using a Runge-Kutta (RK) type numerical method, see [@Quarteroni2006] chap. 11 for details. We need the parameters $\theta$ of the ODE system to be fixed and therefore we write $G( t, X) = G( t, X, \theta)$ to ease notation.
A RK method can be written as follows. Let $\alpha = h > 0$ be a (time) step size and define a uniform grid such that $t_{n+1} = t_n + h$. In this case $|\alpha| = h$. Let $
u_{n+1} = u_n + h \sum_{i=1}^s b_i K_i;
$ $u_{n+1}$ is the approximation for $X(t_n)$ and $
K_i = G(t_n+c_ih,u_n+h\sum_{j=1}^s a_{ij}K_j),\quad i=1,2,\ldots,s .
$ $s$ denotes the number of *stages* of the method.
The components of the vector ${\mathbf{c}}' = (c_1,c_2,\dots,c_s)$ need to satisfy $
c_i=\sum_{j=1}^s a_{ij},\quad i=1,2,\ldots,s
$. Let ${\mathbf{A}}= (a_{ij})$ and ${\mathbf{b}}' = (b_1,b_2,\dots,b_s)$, any RK method is defined by the matrix ${\mathbf{A}}$ and the vectors ${\mathbf{b}}$ and ${\mathbf{c}}$. To have an explicit method we require $a_{ij}=0$ for $j \geq i$, with $i=1,2,\ldots,s$. We used an explicit method in our implementation.
The local truncation error $\tau_{n+1}$ at node $t_{n+1}$ of the RK method is defined as the error made in step $n+1$ of the solver if starting at the exact value $X(t_n)$, that is $
\tau_{n+1} = X(t_{n+1}) - X(t_n) - h \sum_{i=1}^s b_i K_i .
$. The RK method is *consistent*, if $\tau = max_n\vert \tau_n \vert \to 0$ as $h\to 0$. This happens if and only if $\sum_{i=1}^sb_i=1$. The RK method is of order $p$ if $\tau = O(h^{p+1})$ as $h\to 0$ and it is known that $s \geq p$. The global (truncation) error at knot $t_n$ is defined as the error made by the solver, that is $
e_n = X(t_n) - u_n .
$ It is clear that $e_n = \sum_{i=1}^{n} \tau_n$. Under regularity conditions for a RK method of order $p$ we have $K_0 = max_n \vert e_n \vert = O(h^p)$ as $h\to 0$. That is, the maximum absolute global error $K_0$ is of order $p$ as $h\to 0$.
The strategy described here to estimate $e_n$ is to consider two *embedded* RK methods to solve the system, one with order $p$ and one with order $p-1$, both with the same number of stages $s$ and the same matrix ${\mathbf{A}}$ and vector ${\mathbf{c}}$, only with different vectors ${\mathbf{b}}$ and $\hat{{\mathbf{b}}}$, respectively.
Let $u_{n+1}$ be the $n+1$ estimation of $X(t_{n+1})$ of the $p$ order method and let $y_{n+1}$ be obtained by the $p-1$ order method by starting at $u_n$, namely $$u_{n+1} = u_n + h\sum_{i=1}^s b_i K_i ~~\text{and}~~ y_{n+1}=u_n+h\sum_{i=1}^{s} \hat{b}_i K_i .$$ An estimation of the local truncation error at $t_{n+1}$ is $\hat{\tau}_n = u_{n+1} - y_{n+1} = h\sum_{i=1}^{s} (b_i - \hat{b}_i) K_i$ which is basically a byproduct of the $p$ order solver.
The estimate of the global truncation error at knot $t_n$ is then $\hat{e}_n = \sum_{i=1}^{n} \hat{\tau}_n$.
Global error estimation in the Burgers PDE solver {#sec.appenErrPDE}
=================================================
In order to solve numerically the initial condition problem for the viscous Burgers in (\[eq:burgers\]) we used a second order explicit finite-volume method to handle the advective flux. On the other hand, second order time-stepping is accomplished through Crank-Nicolson updating implicitly in the viscosity term. We discretize the solution using homogeneous Neumann conditions on a space interval $I=[0,4]$ adding two ghost cells at each endpoint. In order to march in time we enforce the Courant-Friedrichs-Levy condition through the time-stepping rule (\[eq:cfl\]) where $c=0.1$ and $u_{h}$ is the numerical solution at time $t$. Discretization is implemented setting a number of space points, e.g. $N=2^{9}$. Of note, the time step is adapted to obtain the solution at prescribed observation times $t=\{0.0,0.1,0.2,0.3,0.4,0.5\}$.
If we denote the numerical solution by $u_{h}$, then the residue is $$\label{eq:residue}
R_{h}(u)=
\frac{\partial u_{h}}{\partial t} + u \frac{\partial u_{h}}{\partial z} - \epsilon \frac{\partial^{2}u_{h}}{\partial z^{2}} .$$
We apply our main result using the following after the fact error estimate for continuous approximations to nonlinear viscous hyperbolic conservation laws; see theorem 5.2 of Cockburn [@cockburn1999simple]. Let $v$ be the entropy solution (of problem (\[eq:burgers\])) and let $u$ be a continuos approximation. Then $$\label{eq:traffic_bound}
||u(T)-v(T)||_{L^{1}(\mathbb{R})}\leq\Phi(v_{0},u,T)$$ where $$\Phi(v_{0},u,T)=||u(0)-v_{0}||_{L^{1}(\mathbb{R})}+||R_{h}(u)||_{L^{1}(0,T)\times\mathbb{R}}
+C(u)\sqrt{\epsilon}$$ and $$C^{2}(u) = 8|u|_{L^{\infty}(0,T;TV(R))}|u|_{L^{1}(0,T;TV(R))} .$$
Note that the finite volume method that we have used lets the residual go to zero quadratically. Hence, we have the following application of Cockburn theorem. Let $v(t)$ denote the analytic solution (\[eq:soln\_burgers\]) of problem (\[eq:burgers\]), and let $u_{h}$ denote the finite volume solution at times $t^{n}$, $n=1,...,N$. Following Cockburn analysis, let us denote by $\mathcal{U}$ the set of all the interpolates $u$ such that $u(t^{n})=u_{h}(t^{n})$ for $n=1,...,N$. Then application of Theorem 5.2 of Cockburn gives $$||u_{h}(t^{n})-v(t^{n})||_{L^{1}(\mathbb{R})}\leq\inf_{u\in\mathcal{U}}\Phi(v_{0},u;t^{n}),\;n=1,..,N$$
Let us define $$\label{eq:ratio}
r(u_{h},t^{n})=\frac{\Phi(v_{0},u,T)}{||u_{h}(t^{n})-v(t^{n})||_{L^{1}(\mathbb{R})}}.$$ In order to estimate $\hat{K}_{0}\approx K_{0}$ we take $2^{N}+1$ grid points on the spatial domain $z\in[0,4]$ for $N=6,7,8,9,19$ and interpolate $r(u_{h},1)=1+K_{0}h^{2}$, where $h=1/\Delta z$.
|
---
abstract: |
The Newman-Penrose formalism is used to deal with the quasinormal modes(QNM’s) of Rarita-Schwinger perturbations outside a Reissner-Nordström black hole. We obtain four kinds of possible expressions of effective potentials, which are proved to be of the same spectra of quasinormal mode frequencies. The quasinormal mode frequencies evaluated by the WKB potential approximation show that, similar to those for Dirac perturbations, the real parts of the frequencies increase with the charge $Q$ and decrease with the mode number $n$, while the dampings almost keep unchanged as the charge increases.\
PACS: number(s): 04.70.Dy, 04.70.Bw, 97.60.Lf\
Keywords: Black hole, QNM’s, Rarita-Schwinger field, Reissner-Nordström, WKB approximation.
address:
- 'Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, People’s Republic of China'
- 'National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, People’s Republic of China'
- 'Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China'
- 'Graduate School of Chinese Academy of Sciences, Beijing 100039, People’s Republic of China.'
author:
- 'Fu-Wen Shu'
- 'You-Gen Shen'
title: 'Quasinormal modes of Rarita-Schwinger field in Reissner-Nordström black hole spacetimes'
---
For several decades, the QNM’s of black holes have been of great interest both to gravitation theorists and to gravitational-wave experimentalists [@ZH; @HW; @VC] due to the remarkable fact that QNM’s allow us not only to test the stability of the event horizon against small perturbations, but also to probe the parameters of black hole, such as its mass, electric charge, and angular momentum. QNM’s are induced by the external perturbations. For instance, if an unfortunate astronaut fall into a black hole, the surrounding geometry will undergo damped oscillations. They can be accurately described in terms of a set of discrete spectrum of complex frequencies, whose real parts determine the oscillation frequency, and whose imaginary parts determine the damped rate. Mathematically, they are defined as solutions of the perturbation equations belonging to certain complex characteristic frequencies which satisfy the boundary conditions appropriate for purely ingoing waves at the event horizon and purely outgoing waves at infinity[@CH].\
Recent observational results suggest that our Universe in large scale is described by Einstein equations with a cosmology constant. Motivated by the recent anti-de Sitter (AdS) conformal field theory (CFT) correspondence conjecture[@JM], much attention has been paid to the QNM’s in AdS spacetimes[@HW; @VC]. The latest studies on quantum gravity show that QNM’s also play an important role in this realm due to their close relations to the Barbero-Immirzi parameter, a factor introduced by hand in order that loop quantum gravity reproduces correctly entropy of the black hole[@SH; @OD; @AC; @JB].\
A well-known fact is that quasinormal mode (QN) frequencies are closely related to the spin of the exterior perturbation fields[@SS]. Previous works on QNM’s in black holes has concentrated on the perturbation for scalar, neutrino, electromagnetic and gravitational fields[@VC]. However, few has been done for the case of Rarita-Schwinger fields, which closely relate to supergravity. According to the supergravity theory, the Rarita-Schwinger field acts as a source of torsion and curvature, the supergravity field equations reduce to Einstein vacuum field equations when Rarita-Schwinger field vanishes. This can be seen from the action of supergravity, namely[@DZF], $$\nonumber I=\int d^{4}x (\textit{L}_2+\textit{L}_{3/2}),$$ where $\textit{L}_2$ and $\textit{L}_{3/2}$ represent the Lagrangian of gravitational and Rarita-Schwinger fields, respectively. We hence expect to obtain some interesting and new physics by investigating ONM’s of Rarita-Schwinger field.\
As to QNM’s, the first step is to obtain the one-dimensional radial wave-equation. We usually start with linearized perturbation equations. Two often used ways are available for obtaining the linearized perturbation equations. One is a straightforward but usually complicated way that linearize the Rarita-Schwinger equation directly and deduce a set of partial differential equations (one can see Ref.[@CVV] for details); The other is provided by the Newman-Penrose (N-P) formalism, which end up with partial differential equations in $r$ and $\theta$ instead of ordinary differential equations in $r$. Torres[@GFT] have deduced the linearized Rarita-Schwinger equation in N-P formalism for a type D vacuum background. We start with the Rarita-Schwinger equation in a curved background space-time $$\nabla_{A\dot{D}}\psi^{A}_{B\dot{C}}=\nabla_{B\dot{C}}\psi^{A}_{A\dot{D}}.$$ or, in the Newman-Penrose notation, namely, the Teukolsky’s master equations[@SAT; @SAT1] $$\begin{aligned}
\nonumber\{[D-2\epsilon+\epsilon^{*}-3\rho-\rho^{*}](\bar{\Delta}-3\gamma+\mu)\\
-[\delta-2\beta-\alpha^{*}-3\tau+\pi^{*}](\bar{\delta}-3\alpha+\pi)
-\Psi_{2}\}\Phi_{3/2}=0,\label{sp1}
\end{aligned}$$ and $$\begin{aligned}
\nonumber \{[\bar{\Delta}+2\gamma-\gamma^{*}+3\mu+\mu^{*}](D+3\epsilon-\rho)\\
-[\bar{\delta}+2\alpha+\beta^{*}+3\pi-\tau^{*}](\delta+3\beta-\tau)
-\Psi_{2}\}\Phi_{-3/2}=0.\label{sp2}
\end{aligned}$$ Here we have introduced a null tetrad $(l^\mu,n^\mu,m^\mu,\bar{m}^\mu)$ which satisfies the orthogonality relations $l_{\mu}n^{\mu}=-m_{\mu}\bar{m}^{\mu}=1$ and $l_{\mu}m^{\mu}=l_{\mu}\bar{m}^{\mu}=n_{\mu}m^{\mu}=n_{\mu}\bar{m}^{\mu}=0$, and the metric conditions $g_{\mu\nu}=l_{\mu}n_{\nu}+n_{\mu}l_{\nu}-
m_{\mu}\bar{m}_{\nu}-\bar{m}_{\mu}m_{\nu}$. According to these conditions, we can take the null tetrad as $ l^{\mu}=(e^{-2U(r)},1,0,0)$, $n^{\mu}=\frac{1}{2} (1,-e^{2U(r)},0,0)$, $m^{\mu}=\frac{1}{\sqrt{2}r}\left(0,0,1,\frac{i}{\sin\theta}\right)$, $\bar{m}^{\mu}=\frac{1}{\sqrt{2}r}\left(0,0,1,-\frac{i}{\sin\theta}\right).
$ The corresponding covariant null tetrad is $l_{\mu}=
(1,-e^{-2U(r)},0,0)$, $n_{\mu}=\frac{1}{2} (e^{2U(r)},1,0,0)$ , $m_{\mu}=-\frac{r}{\sqrt{2}}\left(0,0,1,i\sin\theta\right)$, $\bar{m}_{\mu}=-\frac{r}{\sqrt{2}}\left(0,0,1,-i\sin\theta\right)$. The non-vanishing spin coefficients read $$\begin{aligned}
\rho=-\frac{1}{r},\quad
\alpha=-\beta=-\frac{\cot\theta}{2\sqrt{2}r},\quad
\mu=-\frac{e^{2U(r)}}{2r},\quad
\gamma=\frac{1}{4}(e^{2U(r)})^{\prime},\end{aligned}$$ and only one of Weyl tensors is not zero, i.e., $\Psi_{2}=-\frac{(e^{2U(r)})^{\prime}}{2r}$, where a prime denotes the partial differential with respect to $r$.\
In standard coordinates, the line element for the Reissner-Nordström spacetime can be expressed as $$ds^{2} =-e^{2U(r)}dt^{2}+e^{-2U(r)}dr^{2}+r^{2}\left( {d\theta
^{2} + sin^{2}\theta d\varphi ^{2}} \right),$$ with $$e^{2U(r)}
=1-\frac{{2M}}{{r}}+\frac{Q^{2}}{r^{2}},$$ where $M$ and $Q$ are the mass and charge of the black hole, respectively.\
The directional derivatives given in Eqs.(\[sp1\]) and (\[sp2\]), when applied as derivatives to the functions with a $t$- and a $\varphi$-dependence specified in the form $e^{i(\omega t+m\varphi)}$, become the derivative operators $$\begin{aligned}
D=\mathscr{D}_{0},\quad
\bar{\Delta}=-\frac{\Delta}{2r^{2}}\mathscr{D}^{\dag}_{0},\quad
\delta=\frac{1}{\sqrt{2}r}\mathscr{L}^{\dag}_{0},\quad
\bar{\delta}=\frac{1}{\sqrt{2}r}\mathscr{L}_{0},\end{aligned}$$ where $$\begin{aligned}
\mathscr{D}_{n}&=&\partial_{r}+\frac{i\omega
r^{2}}{\Delta}+n\cdot\frac{\Delta^{\prime}}{\Delta},\quad
\mathscr{D}^{\dag}_{n}=\partial_{r}-\frac{i\omega
r^{2}}{\Delta}+n\cdot\frac{\Delta^{\prime}}{\Delta},
\nonumber \\
\mathscr{L}_{n}&=&\partial_{\theta}+\frac{m}{\sin\theta}+n
\cot\theta,\quad
\mathscr{L}^{\dag}_{n}=\partial_{\theta}-\frac{m}{\sin\theta}+n
\cot\theta,\end{aligned}$$ and $$\Delta=r^{2}-2Mr+Q^{2}.$$ It is obvious that $\mathscr{D}_{n}$ and $\mathscr{D}^{\dag}_{n}$ are purely radial operators, while $\mathscr{L}_{n}$ and $\mathscr{L}^{\dag}_{n}$ are purely angular operators. After some transformations are made, Eqs.(\[sp1\]) and (\[sp2\]) can be decoupled as the two pairs of equations[@SS], $$\begin{aligned}
[\Delta\mathscr{D}_{-1/2}\mathscr{D}^{\dag}_{0}-4i\omega
r]P_{+3/2}&=&\lambda P_{+3/2},\label{sp5}\\
\mathscr{L}^{\dag}_{-1/2}\mathscr{L}_{3/2}A_{+3/2}&=&-\lambda
A_{+3/2},\label{sp6}\end{aligned}$$ and $$\begin{aligned}
[\Delta\mathscr{D}^{\dag}_{-1/2}\mathscr{D}_{0}+4i\omega
r]P_{-3/2}&=&\lambda P_{-3/2},\label{sp7}\\
\mathscr{L}_{-1/2}\mathscr{L}^{\dag}_{3/2}A_{-3/2}&=&-\lambda
A_{-3/2},\label{sp8}\end{aligned}$$ where $\lambda$ is a separation constant. The reason we have not distinguished the separation constants in Eqs.(\[sp6\])- (\[sp8\]) is that $\lambda$ is a parameter that is to be determined by the fact that $A_{+3/2}$ should be regular at $\theta=0$ and $\theta=\pi$, and thus the operator acting on $A_{-3/2}$ on the left-hand side of Eq.(\[sp8\]) is the same as the one on $A_{+3/2}$ in Eq.(\[sp6\]) if we replace $\theta$ by $\pi- \theta$.\
In Reissner-Nordström black hole, the separation constant can be determined analytically[@SAT1; @NP1; @NP] $$\lambda=\begin{cases}(l+3)(l+1)\quad\quad\quad & \text{for} \quad j=l+s,\\
l(l-2)\quad\quad\quad & \text{for}\quad j=l-s,
\end{cases}$$ where $l=2,3,4,\cdots$. Note we only consider the case for $j=l+s$ in our following discussions, the case for $j=l-s$ can be easily obtained in the same way. Since $P_{+3/2}$ and $P_{-3/2}$ satisfy complex-conjugate equations (\[sp5\]) and (\[sp7\]), it will suffice to consider the equation (\[sp5\]) only.\
By introducing a tortoise coordinate transformation $dr_{*}=\frac{r^{2}}{\Delta}dr$, and defining $\Lambda_{\pm}=\frac{d}{dr_{*}}\pm i\omega$, $Y=r^{-2}P_{+3/2}$, one can rewrite Eq.(\[sp5\]) in a simplified form $$\Lambda^{2}Y+\tilde{P}\Lambda_{-}Y-\tilde{Q}Y=0,\label{sp10}$$ where $$\tilde{P}=\frac{d}{dr_{*}}\ln\frac{r^{6}}{\Delta^{3/2}},\quad
\tilde{Q}=\frac{\Delta}{r^{4}}\left[\lambda-(\frac{2\Delta}{r^{2}}
-\frac{\Delta^{\prime}}{r})\right].$$ Transformation theory [@CH] shows that one can transform Eq.(\[sp10\]) to a one-dimensional wave-equation of the form $\Lambda^{2}Z=VZ$ by introducing some parameters (certain functions of $r_*$ to be determined) $\xi(r_*)$, $\chi(r_*)$, $\beta_1(r_*)$, $T_1(r_*)$, and several constants (to be specified) $\beta_2$, $T_2$, $\kappa$, $\kappa_1$. If we assume that $Y$ is related to $Z$ in the manner $Y=\xi VZ+T\Lambda_{+}Z$, and the relations $T=T_{1}(r_{*})+2i\omega,
\beta=\beta_{1}(r_{*})+2i\omega\beta_{2}$, one can obtain a equation governing $\beta_2$, $\kappa$, $\kappa_1$ derived from eq.(\[sp10\])[@SS] $$\frac{\Delta^{3/2}}{r^{6}}(F+\beta_{2})^{2}-\frac{(F+\beta_{2})
F_{,r_*,r_*}}{F-\beta_{2}}+\frac{({F_{,r_*}}^{2}-\kappa_{1}^{2})F}
{(F-\beta_{2})^{2}}=\kappa,\label{sp23}$$ where we have defined $F=\frac{r^{6}\tilde{Q}}{\Delta^{3/2}}$, and ‘$,r_*$’ denotes the differential with respect to $r_{*}$. A key step to obtain the expression of potential $V$ is to seek available $\beta_{2}$, $\kappa$, and $\kappa_{1}$ whose values satisfy Eq.(\[sp23\]). Further study shows that $F$ as defined in the previous text does satisfy Eq.(\[sp23\]) with the choice $$\beta_{2}=\pm2Q,\quad \kappa=0, \quad
\kappa_{1}=\pm\lambda\sqrt{1+\lambda}.$$ The potential $V$ can then be expressed as $$V=-\frac{\Delta^{3/2}}{r^{6}}\beta_2-\frac{(F_{,r_*}-\kappa_{1})(\kappa_1
F-\beta_2F_{,r_*})}{(F-\beta_2)(F^2-\beta_2^2)},\label{potential}$$ Note that the sign of $\beta_2$ and $\kappa_1$ can be assigned independently, so there exists four sets of expressions of $V$. Here we denote them by $V^j (j=1,2,3,4)$. Chandrasekhar pointed out that there has a closed relation between the solutions belonging to the different potentials showed in equation(\[potential\]) (see Ref.[@CH] §97(d) for details) $$K^i Z^i=\left\{
\frac{F+\beta_2^i}{F+\beta_2^j}K^j+\left[i\omega\frac{F-\beta_2^j}
{F+\beta_2^j}-\frac{\kappa_1^jF-\beta_2^jF^{\prime}}{F^2-\beta_2^2}\right]
D^{ij}\right\}Z^j+D^{ij}\frac{dZ^j}{dr_*},\label{zre}$$ where $$\begin{aligned}
K&=&-4\omega^2\beta_2+2i\omega \kappa_1,\\
D^{ij}&=&2i\omega(\beta_2^i-\beta_2^j)+(\kappa_1^i-\kappa_1^j)
\frac{F^2-\beta_2^2}{(F-\beta_2^i)(F-\beta_2^j)}\end{aligned}$$ One can easily prove that the potentials vanish when we let $r\rightarrow\pm\infty$ $$\begin{aligned}
{2}
V^j&\to e^{\frac{r_*}{2M}}, &\quad\quad& \text{as\quad $ r_* \to - \infty$,}\\
V^j&\to r^{-2}, && \text{as\quad $ r_* \to + \infty$.}
\end{aligned}$$ A direct consequence of this property is that the wave-function has an asymptotically flat behavior for $r\rightarrow\pm\infty$, i.e., $Z\rightarrow e^{\mp i\omega
r_*}$ (this is just the boundary conditions of QNM’s). It has shown that in asymptotically flat spacetimes, solutions related in the way showed in Eq.(\[zre\]) yield the same reflexion and transmission coefficients, and hence possess the same spectra of QN frequencies[@CH]. Moreover, we can easily obtain the potentials for negative charge by rearranging the order of $V^j$ for positive ones since $\beta_2$ equals to $\pm2Q$. Therefore, we shall concentrate just on potential with $\beta_2=2Q (Q>0, say)$ and $\kappa_1=\lambda\sqrt{1+\lambda}$ in our following works.\
We hence know that the radial equation (\[sp5\]) can be simplified to a one-dimensional wave-equation of the form $$\frac{d^{2}Z}{dr_{*}^{2}}+\omega^{2}Z=VZ,$$ where $$V=-\frac{2Q\Delta^{3/2}}{r^{6}}-\frac{(F_{,r_*}-\lambda\sqrt{1+\lambda})(\lambda\sqrt{1+\lambda}
F-2QF_{,r_*})}{(F-2Q)(F^2-4Q^2)}.$$ Note that we have written $V^j$ as $V$ because we only work with one case of the potentials.\
The effective potential $V(r,Q,l)$, which depends only on the value of $r$ for fixed $Q$ and $l$, has a maximum over $r\in(r_+,+\infty)$. The location $r_{0}$ of the maximum has to be evaluated numerically. An interesting phenomenon is that the position of the potential peak approaches a critical value when $l\rightarrow\infty$, i.e., $$r_{0}(l\rightarrow\infty)\rightarrow3M.$$ Obviously, the effective potential relates to the electric charge of black hole. Figure 1 demonstrates the variation of the effective potential $V(r,Q,l)$ with respect to charge $Q$ for fixed $l=2$. From this we can see that the peak value of the effective potential $V$ increases with $Q$, but the location of the peak decreases with charge. This is quite consistent with the case for Dirac perturbation in Reissner-Nordström black hole spacetimes.\
{width="2.9in" height="2.5in"}
We now evaluate their frequencies by using third-order WKB potential approximation[@SS], a numerical method devised by Schutz and Will[@SW], and was extended to higher orders in[@SI1; @RA]. Due to its considerable accuracy for lower-lying modes[@SI2], this analytic method has been used widely in evaluating QN frequencies of black holes. Noting that during our evaluating procedures, we have let the mass $M$ of the black hole as a unit of mass so as to simplify the calculation. The values are listed in Table 1, where we only list the values for $l=5$ as an example. Values for other mode numbers can easily obtained in the same way. As a reference, we have also evaluated the values (listed in Table 2) for $l=5$ by using the first-order WKB potential approximation[@SW]. Obviously, compared to the first-order approximation, great improvement, especially for larger $n$, has been made for third-order approximation.\
Figure 2 demonstrates the variation of real and imaginary part of the QN frequencies with different $Q$ and $n$ for $l=5$. It shows that the real part of the quasinormal mode frequencies increases with the charge $Q$, while decreases with $n$. But things are totally different for the imaginary part as showed in the figure, whose values almost keep unchanged as the charge increasing, whereas them increase very quickly with the mode number. Furthermore, there is also an interesting phenomena that the larger the charge is, the smaller effect of $n$ on the real part of QN frequencies may have.
$Q$ $n=0$ $n=1$ $n=2$ $n=3$ $n=4$
----- ---------------- ---------------- ---------------- ---------------- ---------------- --
0 1.3273+0.0958i 1.3196+0.2881i 1.3048+0.4824i 1.2839+0.6795i 1.2582+0.8794i
0.1 1.3295+0.0959i 1.3218+0.2883i 1.3070+0.4827i 1.2863+0.6798i 1.2606+0.8799i
0.2 1.3364+0.0960i 1.3287+0.2888i 1.3140+0.4835i 1.2933+0.6809i 1.2678+0.8813i
0.3 1.3481+0.0963i 1.3405+0.2895i 1.3260+0.4847i 1.3055+0.6826i 1.2802+0.8834i
0.4 1.3653+0.0966i 1.3579+0.2906i 1.3436+0.4864i 1.3234+0.6849i 1.2985+0.8863i
0.5 1.3890+0.0970i 1.3817+0.2918i 1.3677+0.4884i 1.3481+0.6876i 1.3238+0.8896i
0.6 1.4206+0.0974i 1.4136+0.2930i 1.4001+0.4903i 1.3811+0.6902i 1.3576+0.8928i
0.7 1.4627+0.0977i 1.4560+0.2938i 1.4432+0.4916i 1.4251+0.6917i 1.4027+0.8945i
0.8 1.5197+0.0976i 1.5135+0.2932i 1.5016+0.4904i 1.4848+0.6898i 1.4639+0.8916i
: QN frequencies of Rarita-Schwinger field in RN black hole for $l=5$ (third-order WKB approximation)
$Q$ $n=0$ $n=1$ $n=2$ $n=3$ $n=4$
----- ---------------- ---------------- ---------------- ---------------- ---------------- --
0 1.3358+0.0957i 1.3618+0.2817i 1.4077+0.4542i 1.4657+0.6108i 1.5300+0.7522i
0.1 1.3380+0.0958i 1.3640+0.2819i 1.4099+0.4545i 1.4679+0.6112i 1.5323+0.7528i
0.2 1.3448+0.0959i 1.3708+0.2824i 1.4166+0.4554i 1.4747+0.6125i 1.5391+0.7546i
0.3 1.3566+0.0962i 1.3825+0.2832i 1.4282+0.4569i 1.4862+0.6147i 1.5507+0.7575i
0.4 1.3737+0.0966i 1.3995+0.2843i 1.4451+0.4589i 1.5031+0.6177i 1.5677+0.7615i
0.5 1.3973+0.0970i 1.4229+0.2857i 1.4683+0.4614i 1.5262+0.6215i 1.5908+0.7666i
0.6 1.4288+0.0974i 1.4541+0.2871i 1.4991+0.4641i 1.5567+0.6257i 1.6213+0.7724i
0.7 1.4707+0.0977i 1.4954+0.2882i 1.5398+0.4664i 1.5968+0.6297i 1.6610+0.7783i
0.8 1.5273+0.0975i 1.5512+0.2880i 1.5942+0.4671i 1.6500+0.6318i 1.7133+0.7823i
: QN frequencies of Rarita-Schwinger field in RN black hole for $l=5$ (first-order WKB approximation)
{width="5in" height="3in"}
**Acknowledgements**
One of the authors(Fu-wen Shu) wishes to thank Doctor Xian-Hui Ge for his valuable discussion. The work was supported by the National Natural Science Foundation of China under Grant No. 10273017.
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---
abstract: 'In capillary electrophoresis, sample ions migrate along a micro-capillary filled with a background electrolyte under the influence of an applied electric field. If the sample concentration is sufficiently high, the electrical conductivity in the sample zone could differ significantly from the background. Under such conditions, the local migration velocity of sample ions becomes concentration dependent resulting in a nonlinear wave that exhibits shock like features. If the nonlinearity is weak, the sample concentration profile, under certain simplifying assumptions, can be shown to obey Burgers’ equation (S. Ghosal and Z. Chen [*Bull. Math. Biol.*]{} 2010 [**72**]{}(8), pg. 2047) which has an exact analytical solution for arbitrary initial condition. In this paper, we use a numerical method to study the problem in the more general case where the sample concentration is not small in comparison to the concentration of background ions. In the case of low concentrations, the numerical results agree with the weakly nonlinear theory presented earlier, but at high concentrations, the wave evolves in a way that is qualitatively different.'
author:
- Zhen Chen
- Sandip Ghosal
title: Strongly nonlinear waves in capillary electrophoresis
---
Introduction {#sec:Intro}
============
In capillary electrophoresis (CE), separation of charged molecular species is accomplished by exploiting the differential migration of ions in a narrow channel (10–100 $\mu$m) in which a strong electric field ($\sim 100$ V/m) is applied in the axial direction [@czebook1; @czebook2]. The sample ions exist in solution in an electrolytic buffer which is referred to as the background electrolyte (BGE). Separation is accompanied by the competing process of diffusive spreading in the axial direction which causes peak dispersion. Dispersion reduces resolution of the separation and may lower the peak concentration to below the detection threshold. It is therefore detrimental. Any effect that tends to increase axial spreading over the minimum imposed purely by molecular diffusion in the axial direction is referred to as “anomalous dispersion” [@ghosal_annrev06]. The transport problem of ions in the capillary is of considerable interest as it determines the amount of dispersion of the sample peak.
In this paper we are concerned with an effect known as “electromigration dispersion” (EMD) that causes significant anomalous dispersion when the ratio of sample to background ion concentration becomes large enough. For this reason it is also known as the “sample overloading effect”. In CE, it is desirable to have the sample concentration at the inlet as high as possible (to ensure that even trace components are within detectable limits) and buffer conductivity as low as possible (to minimize Joule heat), so that the limitation imposed by EMD quickly becomes significant [@jorgenson_lukacs_ac_81].
The physical mechanism of EMD may be explained roughly in the following way: when the concentration of sample ions is sufficiently high in comparison to that of the background electrolyte, the local electrical conductivity of the solution is altered in the region around the sample peak. However, charge conservation requires the electric current to be the same at all points along the axis of the capillary. If diffusion currents due to concentration inhomogeneities are ignored for the moment, it follows, that the electric field must change axially. This is because Ohm’s law, taken together with current conservation, implies that the product of the conductivity and electric field must remain constant along the capillary. The axially varying electric field then alters the effective migration speed of the sample ions, which in turn alters its concentration distribution. Thus, in the continuum limit, the concentration of sample ions is described by a nonlinear transport equation. As expected, the CE signal exhibits features reminiscent of nonlinear waves familiar from other physical contexts [@ghosal_chen10; @whitam_book].
A one dimensional nonlinear hyperbolic equation for the sample ion concentration may be derived using simplifications that arise from assuming local electroneutrality and from neglecting the diffusivity of ions [@mi_ev_ve_79a]. The restriction to zero ionic diffusivities was recently removed by Ghosal and Chen [@ghosal_chen10]. They considered the minimal model of a three ion system – the sample ion, a co-ion and counter-ion. The diffusivities of the three ionic species were assumed equal, though not necessarily zero. The sample ion concentration was then shown to obey a one dimensional nonlinear advection-diffusion equation which reduced to Burgers’ equation if the sample concentration was not too high relative to that of the background ions.
In this paper, we focus on the minimal three ion system considered by Ghosal and Chen [@ghosal_chen10] but we do not assume that the concentration of sample ions is small. Local electro-neutrality is however an excellent approximation in CE systems, since characteristic length scales are much larger than the Debye length which is on the order of nanometers. We therefore exploit it to reduce the numerical stiffness of the coupled ion transport equations. We identify a small number of parameters that primarily determine the system evolution and study the dynamics for a representative range of these parameters. We show that at low concentrations the peak evolves in accordance with the weakly nonlinear theory [@ghosal_chen10], but at high enough concentrations, the dynamics of peak evolution is qualitatively different as the system is dominated by the nonlinearity. Surprisingly, in the strongly nonlinear regime, the peak breaks up into two zones marked by a critical concentration ($\phi = \phi_c$) and separated by a diffusive boundary. The high concentration zone ($\phi > \phi_c$) remains quasi-stationary whereas the low concentration zone propagates forward forming a “surge front” superficially resembling nonlinear wave phenomena familiar in the context of water waves, such as a river bore [@stoker_formation_1948]. The critical concentration ($\phi_c$) can be predicted by a simple argument based on flux conservation. At late times, dispersion ensures that concentrations throughout the domain get smaller and the peak once again may be described by Burgers’ equation. The complex nonlinear behavior is a consequence of the nonlinearity inherent in the Nerst-Planck equations of ion transport, just as the behavior of large amplitude water waves arise from the nonlinear nature of the Navier-Stoke’s equations of hydrodynamics.
Numerical Simulations
=====================
We set up and solve numerically an idealized problem in which a sample peak migrates in a background electrolyte. The channel walls are assumed charge neutral, so that electro-osmotic flow is absent [^1]. Further, local electro-neutrality is invoked which enables us to express the electric field in terms of the instantaneous concentration distributions rather than solve the Poisson’s equation for the electric potential. This considerably simplifies the numerical work as the Poisson’s equation is stiff on account of the smallness of the Debye length. Thus, the problem is reduced to solving a set of one dimensional coupled partial differential equations for the ion concentration fields.
Model System
------------
We will consider a three ion system consisting of sample ions, co-ions and counter-ions. Results will be expressed in terms of dimensionless variables: all lengths are in units of a characteristic length $w_0$ determined by the initial peak width, time is in units of $w_0/v$, where $v$ is the migration velocity of an isolated sample ion in the applied field ($E^{\infty}$) and the electric potential is in units of $E^{\infty} w_0$. All concentrations are in units of $c_n^{\infty}$, where $c_n^{\infty}$ is the concentration of negative ions in the background electrolyte. In order to define a minimal problem with the fewest possible parameters, we assume that the mobility ($u$) is the same for all the species, and therefore, so is the diffusivity ($D$), in accordance with the Einstein relation ($D_{i}/u_{i} = D/u = k_{B} T$ where $k_{B}$ is Boltzmann’s constant and $T$ is the absolute temperature). Note however, since the valence $z_i$ are different, the electrophoretic mobilities of the species $\mu_{i} = z_{i} e u$ are not identical. Then the only parameters in the problem are $Pe = vw_0/D$, which may be regarded as a “Péclet number” based on the electromigration velocity $v$, and the two valence ratios $z_n/z,z_p/z$, where $z_p$, $z_n$ and $z$ are respectively the valence of cations, anions and sample. We present results for two values of the Péclet number: $Pe=100$ and $200$ and fix the valence ratio at $z:z_p:z_n=1:2:-1$. For other values of these parameters the results are qualitatively similar. The parameter of greatest interest is the degree of sample loading or the amplitude of the initial peak. The shape of the wave is insensitive to initial conditions, so for convenience we take the initial peak shape to have a rectangular [^2] profile of height $c_{m}$ and width $2 w_0$ centered at $x=10w_{0}$. This is also the most common initial shape encountered in practice where the sample is introduced by electrokinetic injection. The degree of sample loading is conveniently characterized [@ghosal_chen10] in terms of the quantity $$\Gamma = \int_{-\infty}^{+\infty} \phi(x,t) \; dx = \int_{-\infty}^{+\infty} \frac{c_n}{c_n^{\infty}} \; dx,$$ which has units of length. The length scale $\Gamma$ may be used to define a second Péclet number $P = v \Gamma /D$ which may be treated as a dimensionless measure of sample loading. A series of simulations are conducted with peak heights in the range $\phi_{m} = c_{m}/c^{\infty}_n = 0.01$ (low sample loading) to $0.8$ (high sample loading). The initial co-ion concentration $c_p$ is assumed constant throughout the domain. Then the counter-ion concentration is determined by the local electro-neutrality constraint, Eq. (\[eq:LEN\]). An infinite domain is approximated by a finite computational box of length much greater than $w_{0}$. The values of the concentrations are held fixed at the domain boundaries and $\partial \phi_e / \partial x$ is set to the constant value $-E^{\infty}$. The domain is chosen to be sufficiently large that the perturbations of the concentrations and fields are always negligible near the domain boundaries.
Numerical Method
----------------
We will solve the governing equations for ion transport in solution, which are $$\frac{\partial c_{i}}{\partial t} + \frac{\partial}{\partial x}\left[ -\mu_{i} c_{i} \frac{\partial \phi_e}{\partial x} - D_{i} \frac{\partial c_{i}}{\partial x} \right]= 0
\label{eq:transport}$$ where $c_{i}$ is the concentration of species $i$ ($i=1,2,\ldots,N$) with electrophoretic mobility $\mu_{i}$ and diffusivity $D_{i}$. Electro-osmotic flow is neglected so that the problem is one dimensional and may be described using the co-ordinate $x$ along the capillary and time since injection, $t$. On account of the requirement of local electroneutrality [@rubinstein_book] $$\sum_{i=1}^{N} z_{i} c_{i} = 0
\label{eq:LEN}$$ ($z_i$ is the valence of the $i$th species). The electric potential $\phi_e$ may be found from the equation of current conservation: $$\frac{\partial}{\partial x}\left[ -\sum^N_{i = 1} z_i \mu_{i} c_{i} \frac{\partial \phi_e}{\partial x} - \sum^N_{i = 1} z_iD_{i} \frac{\partial c_{i}}{\partial x} \right]= 0.
\label{eq:current}$$ Eq. (\[eq:current\]) may be readily integrated to yield the local electric field, $E = - \partial_{x} \phi_{e}$: $$E(x,t) = \frac{E^{\infty} \sum_{i} z_i \mu_{i} c_{i}^{\infty} +
\sum_{i} z_i D_{i} \partial_{x} c_{i}}{\sum_{i} z_i \mu_{i} c_{i}},
\label{eq:E(x)}$$ where the superscript $\infty$ indicates the value of the respective variable far away from the peak and the summation is over all species.
A finite volume method is used to discretize equations (\[eq:transport\]) and (\[eq:current\]) in space using an adaptive grid refinement algorithm that is enabled by applying the Matlab library “MatMOL” [@vande_wouwer_simulation_2004]. The spatially discretized system of equations is then integrated in time using the Matlab solver “ode45” [@shampine_matlab_1997] which is based on an explicit Runge-Kutta (4,5) formula. Equations (\[eq:transport\]) and (\[eq:current\]) automatically ensure that the electro-neutrality condition, Eq. (\[eq:LEN\]), is satisfied and this is verified at each time step.
Results
-------
Figure 1(a) shows the profiles of the normalized sample concentration $\phi(x,t) = c_n/c_n^{\infty}$ at fixed times $vt/w_{0}=0,0.5,2.0,4.0$ and $8.0$ for the case of low sample loading. Fig. 1(b) and (c) show respectively the profiles of the corresponding electric field $E(x,t)$ and the Kohlrausch regulating function $K(x,t) = (c_p + c_n + c)/u$. The Kohlrausch regulating function is a useful quantity for describing electrokinetic transport. If all ionic species have the same diffusivity, $K(x,t)$ evolves as a passive scalar [@ghosal_chen10]. If ionic diffusivities are treated as zero, then $K(x,t)$ is a conserved quantity [@kohlrausch]. It is seen that $K(x,t)$ remains localized near the injection zone and spreads only slowly by molecular diffusion. The sample peak on the other hand moves to the right and after a short time, the sample peak essentially lies in a zone where $K=K_{\infty}$. This illustrates the behavior postulated earlier that makes possible a simplified description in terms of the one dimensional nonlinear equation [@ghosal_chen10]: $$\frac{\partial \phi}{\partial t} + \frac{\partial}{\partial x} \left( \frac{v \phi}{1 - \alpha \phi} \right)
= D \frac{\partial^{2} \phi }{\partial x^{2}}.
\label{1Dnlwave}$$ If $\phi$ is small, Eq.(\[1Dnlwave\]) reduces to the Burgers’ equation on Taylor expansion of $(1 - \alpha \phi)^{-1}$. In the vicinity of the sample peak, the electric field is functionally related to the normalized sample concentration; $E = E^{\infty}/(1 - \alpha \phi)$. Here $\alpha$ is the “velocity slope parameter” introduced in [@ghosal_chen10].
It may be shown [@ghosal_chen10] that the requirement of positivity of co- and counter-ion concentrations implies that only sample profiles satisfying the condition $\phi < \phi_c$, where $\phi_c$ is a positive number, may be described by the theory. We will call such profiles “realizable”. The critical concentration, $\phi_c$, is given by $\phi_{c} = (z_p - z_n)/(z_p - z)$ when $z < 0$ and $\phi_{c} = - [ z_n (z_p - z_n) / [ z_p (z - z_n) ]$ when $z > 0$. When the parameter $\alpha >0$, it may be shown with some simple algebra that $\phi_c < \phi_c^{\prime} \equiv \alpha^{-1}$ (see Appendix), so that the singularity implicit in Eq.(\[1Dnlwave\]) when $\phi = \phi_c^{\prime}$ is never reached for realizable solutions. Fig. 2 shows the behavior of the system for initial conditions that are not realizable. In this situation, a stationary “barrier” develops at a fixed spatial location corresponding to a certain value $\phi = \phi^{inter} < \phi_{c}$. The sample ions move more or less freely on crossing the barrier but are effectively immobilized on the left of the barrier. This is due to the greatly reduced strength of the electric field in the injection zone where the electrical conductivity is high. This is clearly seen in Fig. 2(b) which shows a sharp reduction in the electric field in the injection zone. Only sample ions near the edges of the zone are able to “leak out” and are carried to the right as an advancing wave. Since part of the sample profile remains quasi stationary, the assumption of the constancy of the Kohlrausch function, $K = K_{\infty}$ can no longer be made for non-realizable concentrations. Thus, Eq.(\[1Dnlwave\]), which would have led to unphysical negative concentrations for such non-realizable profiles, is not applicable until after a sufficient time has evolved so that $\phi$ is reduced to a value below $\phi^{inter}$ throughout the domain.
Fig. 3 shows the variation in time of the quantity $(2D)^{-1} d \sigma^{2} / dt$ for a series of different values of the diffusivity and sample loading characterized by the pair of Péclet numbers $(\text{Pe},P)$. If the profile spread purely by molecular diffusivity, this quantity should approach one asymptotically. However, it is seen that the long time asymptotic value is not one but rather $D_{\mbox{eff}}$ which depends solely on $P$. The dashed line shows the theoretical value of $D_{\mbox{eff}}$ predicted by the weakly nonlinear theory based on solutions of the Burgers’ equation [@ghosal_chen10]. Thus, once the system has evolved long enough, and dispersion has caused the amplitude to drop sufficiently, Burgers’ equation provides a valid description of the peak evolution. However, a real separation happens in a finite capillary and the long time limit may not necessarily apply. A quantity of interest is the timescale characterized by $t_{*}$: the time needed for the quantity $(2D)^{-1} d \sigma^{2} / dt$ to relax to $0.95$ of its asymptotic value $D_{\mbox{eff}}$. If the separation is conducted in a capillary of length $L$, the question of interest is whether $t_{*}$ is small or large compared to the total separation time $T = L/v$. In Fig. 4 we show the normalized time $v t_{*} /w_{0}$ from a series of simulations with different values of $(Pe,P)$. Clearly, $v t_{*}/w_{0}$ is a monotonically decreasing function of $P$. This can be anticipated from the theory of nonlinear waves [@whitam_book]: the higher the amplitude, the quicker a shock or shock like structure is formed. In contrast to the effective diffusivity shown in Fig. 3 which depends on $P$ but not on $Pe$, the time to reach the asymptotic state does depend on $Pe$. In fact, as Fig. 4 shows, the curve $v t_{*} /w_{0}$ as a function of $P$ is shifted upwards as $Pe$ is increased. Indeed, larger $Pe$ corresponds to lower diffusivity and therefore a longer time for the peak to spread and its amplitude to fall sufficiently for the weakly nonlinear description to be valid. Typical values of the physical parameters in a microchip based system may be $w_{0} \sim 100 \mu$m, $L \sim 5$ cm, so that $v T/w_{0} \sim 500$. Thus, Fig. 4 suggests that the Burgers’ solution does describe the peak dynamics for most of the separation time except for possibly a relatively short initial transient.
Analysis
--------
An approximate theoretical determination of the concentration $\phi^{inter}$ may be provided using the conservation equations. The method of doing this is in fact entirely analogous to the “Moving Boundary Equations” (MBE) [@dole_theory_1945] for describing advancing fronts (e.g. in isotachophoresis), except, in this case, the front happens to be quasi stationary. The conceptual framework is illustrated in Fig. 3. The domain is decomposed into three parts: the “Initial Zone” where the sample is injected, the “Background Zone” ahead of the advancing wave where all concentrations equal their initial values and an “Interzone” between them. All variables are assumed constant within each zone but undergo a discontinuous change across zone boundaries. The values of the variables in each zone are indicated in Fig. 3. The boundary between the Initial Zone and the Interzone is stationary whereas the boundary between the Interzone and the Background Zone moves to the right. The arrows indicate fluxes of ions across the stationary zone boundary. Conservation of these ionic fluxes require $$\begin{aligned}
E^{ini}\phi^{ini} &=& E^{inter}\phi^{inter}\\
E^{ini}\phi^{ini}_n &=& E^{inter}\phi^{inter}_n\end{aligned}$$ where $E$ represents the electric field and $\phi$ represents the concentration (normalized by $c_{n}^{\infty}$). The superscript (“ini” for the Initial Zone, “inter” for the Interzone and “$\infty$” for the Background Zone) indicates the zone in which the variable is evaluated and the subscript ($p$ for cation, $n$ for anion and no subscript for the sample) identifies the species. Therefore, $$\phi^{inter}_n = ( \phi^{ini}_n / \phi^{ini} ) \phi^{inter}
\label{eq:phi_n}$$ For the inter zone, $$\begin{aligned}
K^{inter} &=& c_{n}^\infty \left( \phi^{inter} + \phi^{inter}_p + \phi^{inter}_n \right) / u \nonumber\\
&=& K_{\infty} = c^{\infty}_n \left( \phi^{\infty}_p + 1 \right) / u \nonumber\\
&=& c^{\infty}_n \left( 1 - z_n / z_p \right) /u.
\label{eq:Kinter}\end{aligned}$$ The electro-neutrality condition (valid in all zones) is: $$z_p \phi_p + z_n \phi_n + z \phi = 0.$$ By combining Eq. (\[eq:phi\_n\]) and (\[eq:Kinter\]) and using the electro-neutrality condition we get an equation for determining $\phi^{inter}$ $$(1 - z/z_p) \phi^{inter} + r (1 - z_n/z_p) \phi^{inter} = 1 - z_n/z_p,$$ where the ratio $\phi^{ini}_n / \phi^{ini} = r$ is a constant determined by the ionic composition of the injected zone. Solving the above linear equation for $\phi^{inter}$ we have $$\phi^{inter} = \left[ r + (z_p - z)/(z_p - z_n) \right]^{-1}.
\label{eq: phi inter}$$ In our numerical experiment the cation concentration was chosen to be uniform, so that $\phi^{ini}_p = \phi^{\infty}_p = -z_n/z_p$. Since $z:z_p:z_n=1:2:-1$, $r = 1/\phi^{ini} -z/z_n = 1.5$, and, $\phi^{inter} = 0.55$. This value is indicated by the dashed line in Fig. 2(a). Clearly, it correctly describes the concentration of sample in the Interzone.
Thus, the theoretical description developed in [@ghosal_chen10] may be used in the Interzone ($\phi < \phi^{inter}$) but not in the Initial Zone. In order that all ion concentrations be non-negative in the Interzone we must have $\phi^{inter} < \phi_{c} < \phi_{c}^{\prime}$. This inequality is indeed true as can be shown by some simple algebra (see Appendix).
Conclusions
===========
The development of nonlinear waves in capillary electrophoresis in the limit of low as well as high concentration of sample ions was studied by numerical integration of the governing equations. An idealized minimal model was considered consisting of a three ion (sample, co-ion and counter-ion) system of strong electrolytes [^3]. This study complements an earlier paper by the authors. There it was shown that, in the weakly nonlinear limit, the evolution of the sample concentration may be reduced to Burgers’ equation, which admits an exact analytical solution.
Numerical simulation revealed that the evolution of the peak proceeds in a way that is qualitatively different when the sample concentration is high. As a consequence of the sharp reduction of the electric field in the region of sample injection, the ion migration velocity in this zone is very small. Ahead of this zone the ions form a surge front with a step-like profile propagating to the right. This state of affairs continues until the dimensionless ion concentration ($\phi$) in the injection zone drops sufficiently so that $\phi < \phi^{inter}$. The subsequent dynamics then proceeds in accordance with the weakly nonlinear theory [@ghosal_chen10]. The value of $\phi^{inter}$ may be approximately calculated by using a simple model based on conservation of ionic fluxes.
This qualitative change in the dynamics of peak evolution explains the breakdown of the weakly nonlinear theory when the concentration $\phi$ exceeds the critical value $\phi_c$. When $\phi$ exceeds a certain value $\phi^{inter} < \phi_{c}$ part of the propagating wave is effectively immobilized in the injection zone. It is then no longer correct to assume [@ghosal_chen10] that the sample pulse would quickly move out to a region where the Kohlrausch function is constant.
The model studied here is clearly oversimplified. In particular, real electrophoresis buffers contain many more than three ions including one or more weak acids or bases to maintain a stable pH. Further, complex effects due to inhomogeneities in the electroosmotic flow may be relevant [@chen_effect_2008]. In this paper we ignore these complexities and attempt to produce a detailed understanding of a “minimal” model problem. One may question whether the strongly nonlinear regime considered here is of relevance to actual laboratory practice. The answer depends on the numerical values of the critical concentrations $\phi^{inter} < \phi_{c} < \phi_{c}^{\prime}$. If the sample and carrier ions have similar valences then all of these critical concentrations are of order unity. Thus, to exceed these critical values the sample ions in the injected plug will need to be present at concentrations approaching that of the carrier electrolyte. Such high concentrations are rarely employed in laboratory practice. However, if the sample is a macro-ion the critical values may actually be quite small. For example, at pH 2.0 Bovine serum albumin has a valence, $z \sim 55$ [@ford_measurement_1982]. Then, in a univalent carrier electrolyte we have $\phi_{c} \sim 0.04$, so that the strongly nonlinear regime studied here may be easily reached.\
This work was supported by the National Institute of Health under grant R01EB007596.
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Appendix: Proof of the inequality $\phi^{inter} < \phi_{c} < \phi_{c}^{\prime}$ {#appendix-proof-of-the-inequality-phiinter-phi_c-phi_cprime .unnumbered}
===============================================================================
The critical concentration $\phi_c$ is defined as [@ghosal_chen10] $$\phi_c = \left\{
\begin{array}{ll}
\frac{z_p - z_n}{z_p - z} & \mbox{if $z<0$}\\
- \frac{z_n}{z_p} \frac{z_p-z_n}{z - z_n} & \mbox{if $z>0$}
\end{array}
\right.$$ whereas $$\phi_{c}^{\prime} = \frac{1}{\alpha} = \frac{z_n (z_p - z_n )}{(z-z_n)(z - z_p)}.$$ We need to show that $\phi_c < \phi_c^{\prime}$ when $\alpha > 0$, that is, when $z_p > z > z_n$. To do this, evaluate the ratio $\phi_{c}/\phi_{c}^{\prime}$ when $z_p > z > z_n$: $$\frac{\phi_c}{\phi_{c}^{\prime}} = \left\{
\begin{array}{ll}
\frac{z_p - z_n}{z_p - z} \frac{(z-z_n)(z - z_p)}{z_n (z_p - z_n )} = \frac{-z_n + z}{-z_n} < 1& \mbox{if $z<0$}\\
- \frac{z_n}{z_p} \cdot \frac{z_p-z_n}{z - z_n} \frac{(z-z_n)(z - z_p)}{z_n (z_p - z_n )} = \frac{z_p - z}{z_p} < 1& \mbox{if $z>0$}
\end{array}
\right.$$ which completes the proof.
To prove the remaining inequality, $\phi^{inter} < \phi_{c}$, we first show that $r > -z /z_n$ when $z<0$. To do this, we use the electro-neutrality condition to express $\phi_{p}^{ini}$ in terms of the other variables $$\phi_{p}^{ini} = - \frac{z}{z_p} \phi^{ini} - \frac{z_n}{z_p} \phi_{n}^{ini} = \frac{\phi^{ini}}{z_p} (- z - r z_n ).$$\
Now we must have $\phi_{p}^{ini} > 0$. This is always true if $z<0$, but if $z>0$ then we require that $r > - z/z_n$.
First suppose that $z <0$. Then $$\begin{aligned}
\phi^{inter} &=& \frac{1}{ r + (z_p - z)/(z_p - z_n) } \nonumber \\
&<& \frac{1}{ (z_p - z)/(z_p - z_n) } \nonumber \\
&=& \frac{z_p - z_n}{z_p - z} = \phi_{c}\end{aligned}$$
Now suppose that $z >0$. Then $$\begin{aligned}
\phi^{inter} &=& \frac{1}{ r + (z_p - z)/(z_p - z_n) } \nonumber \\
&<& \frac{1}{ -(z/z_n) + (z_p - z)/(z_p - z_n) } \nonumber \\
&=& - \frac{z_n}{z_p} \frac{z_p - z_n}{z - z_n} = \phi_{c}\end{aligned}$$ Thus, in all cases, $\phi^{inter} < \phi_{c}$ which completes the proof.
[^1]: The effect of a wall zeta potential has recently been investigated [@ghosal_chen_emd_eof].
[^2]: to reduce numerical errors, the corners of the rectangle were slightly “rounded” by using a tan hyperbolic function.
[^3]: the situation of a weak electrolytic buffer was recently investigated by the authors [@EMD1].
|
---
abstract: 'This paper deals with the infinite-horizon optimal control problem for Boolean control networks (BCNs) with a discounted-cost criterion. This problem has been investigated in existing studies with algorithms characterized by high computational complexity. We thus attempt to develop more efficient approaches for this problem from a deterministic Markov decision process (DMDP) perspective. First, we show the eligibility of a DMDP to model the control process of a BCN and the existence of an optimal solution. Next, two approaches are developed to handle the optimal control problem in a DMDP. One approach adopts the well-known value iteration algorithm, and the other resorts to the Madani’s algorithm specifically designed for DMDPs. The latter approach can find an exact optimal solution and outperform existing methods in terms of time efficiency, while the former value iteration based approach usually obtains a near-optimal solution much faster than all others. The 9-state-4-input *ara* operon network of the bacteria *E. coli* is used to verify the effectiveness and performance of our approaches. Results show that both approaches can reduce the running time dramatically by several orders of magnitude compared with existing work.'
author:
- 'Shuhua Gao, Cheng Xiang, and Tong Heng Lee [^1]'
bibliography:
- 'boolnet.bib'
title: ' Optimal Control of Boolean Control Networks with Discounted Cost: An Efficient Approach based on Deterministic Markov Decision Process '
---
Introduction
============
An effective and widely used model of gene regulatory networks [@barabasi2004network] is the Boolean network (BN) model, first proposed by Kauffman in 1969 [@kauffman1969metabolic], that describes gene expression state with binary values. Since then, BNs have drawn a lot of research interest and been applied to various fields beyond biomolecular networks, such as information mining in consumer community networks [@meng2018properties] and analysis of social consensus impacted by peer interactions [@Emergence2007]. We can further incorporate binary control inputs into a BN to manipulate its states and get a control system commonly referred to as a *Boolean control network* (BCN) [@zhao2010input].
A considerable number of studies on BCNs emerged in the last decade thanks to the development of a novel mathematical tool called the semi-tensor product (STP) [@zhao2010input; @cheng2010linear]. An equivalent algebraic state-space representation (ASSR) can be built using STP, which makes it possible to adapt established techniques in traditional control theory for similar investigations of BCNs. Based on the STP and the ASSR of BCNs, quite a few control-theoretical problems have been tackled in the recent literature, for example, controllability and observability [@cheng2009controllability; @laschov2013observability; @zhao2010input], stabilization [@cheng2011stability], pinning control [@lu2019pinning], and output tracking [@zhang2019output], to name a few. Following this mainstream, we also initiate our study on infinite-horizon optimal control of BCNs with the ASSR here.
Optimal control is a classic topic that deals with the design of an *optimal* control law according to a given performance index. Specifically, optimal control of BCNs can be used to develop medical intervention strategies for an underlying GRN to treat diseases like cancers while minimizing expenses or maximizing the therapeutic effect [@faryabi2008optimal]. A variety of optimal control problems regarding BCNs have been studied in recent years, which are divided into two broad categories depending on the optimization horizon length. In the first class, the horizon length is finite, and the performance criterion is the summation of stage costs at a countable number of time steps as well as one terminal cost. An early study was conducted in [@laschov2010maximum] towards the Mayer-type optimal control (i.e., only considering the terminal cost) of single-input BCNs by a maximum principle. Two common objectives in optimal control, minimum energy, and minimum time, have been attempted in [@li2013minimum] and [@laschov2013minimum], respectively. E. Fornasini *et al.* investigate more general cases of such finite-horizon problems in [@fornasini2013optimal] and present recursive algorithms that are analogous to the discrete-time Riccati equation. The second class of problems, i.e., infinite-horizon optimal control, are generally more challenging, of which the objective function takes either an average-cost form or a discounted-cost form to ensure the convergence of the total cost [@faryabi2008optimal]. The first attempt for infinite-horizon optimal control with an average-cost criterion was presented in [@zhao2010optimal] by enumerating all cycles in the input-state space with prohibitively high time complexity. Several improvements were proposed later, including a Floyd-like algorithm [@zhao2011floyd], a value iteration algorithm [@fornasini2013optimal], and a policy iteration approach [@wu2019optimal]. By contrast, the discounted-cost counterpart has got less attention, which was first addressed in [@cheng2014optimal] using a Floyd-like algorithm similar to that in [@zhao2011floyd]. The algorithm [@zhao2011floyd] has been modified in a recent study [@zhu2018optimal] to operate in the state space instead of the input-state space of a BCN for further speedup.
A major issue of the STP-based algebraic methods discussed above is their prohibitively high computational cost once the size of the BCN is large. It has been proved in [@AKUTSU2007670] that, in general, control problems on BCNs are NP-hard. Consequently, it is hopeless to seek polynomial-time algorithms since P $ \ne $ NP is a widely believed conjecture. This is indeed an intuitive fact because all algorithms above run in a polynomial time of $ N $, where $ N {\coloneqq}2^n$ and $ n $ is the number of state variables in a BCN. Nevertheless, even faced with the NP-hardness, we can still pursue shorter running time in practice by designing algorithms whose time complexity is a lower-order polynomial in $ N $. For example, by resorting to the Warshall algorithm, Liang *et al.* [@liang2017improved] proposed an improved controllability criterion for BCNs with time complexity reduced from $ O(N^4) $ to $ O(N^3) $. Our latest work [@gao2019infinite] (preprint) investigates infinite-horizon optimal control of BCNs with average cost using Karp’s minimum mean cycle (MMC) algorithm and achieves the lowest time complexity so far. Notably, regarding the discounted-cost optimal control problem considered in this paper, the existing two studies [@cheng2014optimal] and [@zhu2018optimal] both attempt to locate the overall optimal cycle by examining individual optimal cycles of length ranging from 1 to $ N $ iteratively, which consequently leaves ample space for further efficiency improvement.
The primary goal of this study is to develop more efficient algorithms for discounted-cost infinite-horizon optimal control of BCNs. As a natural choice, the Markov decision process (MDP) theory has been extensively used in optimal control of probabilistic and stochastic Boolean networks, e.g., see [@faryabi2008optimal] and [@wu2017finite]. Though a deterministic BCN considered here can undoubtedly be treated as a special stochastic BCN, more complexity will be introduced that causes unnecessary deterioration of computational efficiency. To the best of our knowledge, there is currently no work on optimal control of BCNs that views the control process as a deterministic Markov decision process (DMDP). The interesting point is that, by adopting the equivalent DMDP description, we can resort to established algorithms, like Madani’s algorithm [@madani2010discounted], to solve the discounted-cost optimal control problem for BCNs with reduced time complexity. The development of such efficient, DMDP-based algorithms forms the main contribution of this paper.
The rest of this paper is organized as follows. First, in Section \[sec: preliminary\], we introduce the algebraic representation of BCNs. We then formulate the optimal control problem in Section \[sec: problem\]. The main results of our study are presented in Section \[sec: results\], which detail the development of two efficient approaches. We compare the performance of the proposed approaches and existing ones on a biological network in Section \[sec: example\]. Finally, Section \[sec: conclusion\] concludes this study. The Python implementation of all algorithms in this paper is available at <https://github.com/ShuhuaGao/bcn_opt_dc>.
Preliminaries {#sec: preliminary}
=============
Notations
---------
- $ {\mathbb R}$, $ {\mathbb N}$, and $ {\mathbb N}^+ $ denote the sets of real numbers, nonnegative integers, and positive integers, respectively. Given $ k , n \in {\mathbb N}$ with $ k \le n $, $ [k, n] {\coloneqq}\{k, k + 1, \cdots, n \}$.
- $\mathcal{A}_{p\times q}$ denotes the set of all $p\times q$ matrices. Given $ A \in {\mathcal A}$, $ A_{ij} $ is its $ (i, j) $-th entry, and $ \textrm{Row}_i(A) $, $\textrm{Col}_j(A) $ denote its $ i $-th row and $ j $-th column respectively.
- $ \delta_n^i {\coloneqq}\textrm{Col}_i(I_n) $, where $ I_n $ is the $ n $-dimensional identity matrix. $\Delta_n {\coloneqq}\{ \delta_n^i | i = 1, 2, \cdots, n \}$, and $ \Delta {\coloneqq}\Delta_2 $. The shorthand of $ \{\delta_n^{i_1}, \delta_n^{i_2}, \cdots, \delta_n^{i_k}\} $ is $ \delta_n\{i_1, i_2, \cdots, i_k\} $.
- A matrix $ L \in {\mathcal A}_{n \times q} $ with $ \text{Col}_i(L) \in \Delta_{n}, \forall i \in [1, q], $ is called a *logical matrix*. Let $\mathcal{L}_{n\times q} $ denote the set of all $ n\times q $ logical matrices.
- $ {\mathcal D}{\coloneqq}\{0, 1\}. $ Logical operators [@cheng2010linear]: $\land$, conjunction; $\lor$, disjunction; $ \lnot $, negation; and $ \oplus $, exclusive or.
Algebraic Representation of BCNs
--------------------------------
[@zhao2010optimal] The semi-tensor product (STP) of two matrices $ A \in \mathcal{M}_{m\times n}$ and $ B \in \mathcal{M}_{p\times q}$ is defined by $$A \ltimes B = (A \otimes I_{\frac{s}{n}})(B \otimes I_{\frac{s}{p}}),$$ where $\otimes$ denotes the Kronecker product, and $ s $ is the least common multiple of $ n $ and $ p $. $ \ltimes_{i=1}^n A_i {\coloneqq}A_1\ltimes A_2\ltimes \cdots\ltimes A_n $.
The STP generalizes the traditional matrix product while preseving most fundamental properties [@cheng2010linear]. For notational simplicity, the symbol $\ltimes$ is omitted hereafter.
Identify Boolean values in $ {\mathcal D}$ by $ 0 \sim \delta_2^1 $ and $ 1 \sim \delta_2^2 $.
[@cheng2010linear] \[lemma: structure matrix\] Any Boolean function $ f(x_1, x_2, \cdots, x_n): \Delta^n \rightarrow \Delta $ can be expressed uniquely in a multi-linear form as $$f(x_1, x_2, \cdots, x_n) = M_f x_1 x_2 \cdots x_n,$$ where $ M_f \in \mathcal{L}_{2\times 2^n}$ is the unique *structure matrix* of $ f $.
Consider a BCN with $ n $ nodes and $m $ control inputs: $$\label{eq: bcn}
\begin{cases}
x_1(t+1) = f_1(x_1(t), \cdots, x_n(t), u_1(t), \cdots, u_m(t))\\
\vdots \\
x_n(t+1) = f_n(x_1(t), \cdots, x_n(t), u_1(t), \cdots, u_m(t)),
\end{cases}$$ where $ x_i(t) \in \Delta, u_j(t) \in \Delta, $ denote states and control inputs respectively, and $ f_i : \Delta^{m+n} \rightarrow \Delta $ is the Boolean function associated with the state variable $ x_i $, $ i \in [1, n], j \in [1, m] $.
Using the STP, the ASSR of the BCN is $$\label{eq: ASSR}
x(t+1) = Lu(t)x(t),$$ where $ x(t) {\coloneqq}x_1(t) \ltimes \cdots \ltimes x_n(t) \in \Delta_{2^n} $ and $ u(t) {\coloneqq}u_1(t) \ltimes \cdots \ltimes u_m(t) \in \Delta_{2^m} $ are canonical vectors. Let $ N {\coloneqq}2^n $ and $ M {\coloneqq}2^m $. We have the logical matrix $ L \in {\mathcal L}_{N \times MN} $. Ref. [@cheng2010linear] details the computation of . Note that the two notations $ N $ and $ M $ defined here are used throughout this paper.
Problem Formulation {#sec: problem}
===================
Given the BCN , let the cost of applying control $ u \in {\Delta_M}$ at state $ x \in {\Delta_N}$ be $ g(x, u) $. The bounded function $ g: {\Delta_N}\times {\Delta_M}\rightarrow {\mathbb R}$ is called the *stage cost* function. We seek a control sequence that minimizes the discounted cost for BCN accumulated in an infinite horizon. Note that we consider a more general and challenging scenario here beyond that in [@cheng2014optimal] and [@zhu2018optimal], which involves various constraints on both states and inputs. The problem is formalized as follows.
\[prob: 1\] Consider BCN . Solve the following constrained optimization problem for optimal control: $$\begin{aligned}
\label{eq: problem}
\min_{{\bm u}} J({\bm u}) = \lim_{T\rightarrow\infty} \sum_{t=0}^{T-1} \lambda^tg(x(t), u(t)), \nonumber\\
\textrm{s.t.} \begin{cases}
x(t+1) = Lu(t)x(t) \\
x(t) \in C_{\textnormal{\textrm{x}}} \\
u(t) \in C_{\textnormal{\textrm{u}}}(x(t)) \\
x(0) = x_0
\end{cases},
\end{aligned}$$ where $ {\bm u}= \big(u(t) \in \Delta_M\big)_{t=0}^{T-1} $ denotes a control sequence; $ \lambda \in (0, 1) $ is the discount factor; $ C_{\textnormal{\textrm{x}}} \subseteq {\Delta_N}$ and $ C_{\textnormal{\textrm{u}}}(x(t)) \subseteq {\Delta_M}$ denote the state constraints and the state-dependent control input constraints respectively; and $ x_0 \in C_{\textnormal{\textrm{x}}}$ is the initial state of the BCN.
\[rmk: problem\] No constraints are considered in [@cheng2014optimal], and only the avoidance of undesirable states is handled in [@zhu2018optimal]. By contrast, the above problem formulation emerges as the most generic one, which can incorporate state constraints, control constraints, and transition constraints [@zhang2017finite]. We assume that Problem \[prob: 1\] is feasible, that is, at least one control sequence exists that allows the indefinite evolution of the BCN.
Main Results {#sec: results}
============
In this section, we first show that the control of a BCN can be handled elegantly in an MDP framework. Then, we propose two methods to solve Problem \[prob: 1\]: a general value iteration approach commonly used in MDP optimization and a more efficient approach specialized for a DMDP.
Deterministic Markov Decision Process (DMDP)
--------------------------------------------
An MDP is a widely used mathematical model in sequential decision making under uncertaintis, that is, choosing differente actions in different situations [@RL2]. Specificially, in our application with BCNs, the *action* at time point $ t $ refers to the control input $ u(t) $, and the *situation* is represented by the network state $ x(t) $. In the MDP framework, each decision is associated with a *reward*. The essential property of an MDP is that the next state and the reward depend only on the current state and the current action, known as the *Markov property* [@RL2]. Obviously, we see from that the control process of a BCN is indeed an MDP, because $ x(t+1) $ is completed determined by $ x(t) $ and $ u(t) $.
In an MDP, the goal of the controller is to maximize the cumulative reward from any initial state in the long run [@RL2; @busoniu2017reinforcement]. A *policy* is a decision rule that specifies which action should be chosen for each state. In our BCN application, the *reward* is replaced by the *cost* in Problem \[prob: 1\]. Accordingly, we aim to find a policy that minimizes the aggregated discounted cost over the infinite horizon for optimal control of BCNs.
Unlike general MDPs considered in reinforcement learning, a useful property of the BCN control process is that its state transition and rewarding are both deterministic. That is, given the current state $ x \in {\Delta_N}$ and the control action $ u \in {\Delta_M}$, the next state is definitely $ Lux $ by , and the cost is fixed to $ g(x, u) $ in Problem \[prob: 1\]. Formally, the control process of a BCN is called a deterministic Markov decision process (DMDP). As we will show later, such determinism allows the development of time-bounded optimization algorithms compared with those for general MDPs.
Existence of Optimal Solutions
------------------------------
In control of BCNs, a policy $ \pi $ refers to a mapping from states to control inputs, i.e., $ \pi: {\Delta_N}\rightarrow {\Delta_M}$. A feasible policy must respect the constraints of Problem \[prob: 1\]: for any $ x \in C_{\textrm{x}} $, it must satisfy $$\pi(x) \in C_{\textrm{u}}(x), \ L\pi(x)x \in C_{\textrm{x}}.$$
Now we can restate Problem \[prob: 1\] using the MDP terminology as follows: find an optimal policy $\pi_* $, which conforms to all constraints, such that the performance index function $ J $ is minimized. The first question coming to our mind is whether an optimal policy exists for Problem \[prob: 1\]. In the following illustration, we mainly borrow the notations and terminology from the monograph [@RL2]. Note that we are dealing with a DMDP, and all probabilistic expectations in the general MDP framework can thereby be omitted.
The quality of a policy can be evaluated by a value function [@busoniu2017reinforcement]. Given a policy $ \pi $, the *value function* of a state $ x $, termed $ v_{\pi}(x) $, is the performance index obtained with the initial state $ x $ and the control sequence $ {\bm u}$ generated by $ \pi $: $$\label{eq: value function}
v_{\pi}(x) = \sum_{t=0}^{\infty} \lambda^tg(x(t), \pi(x(t))) \bigg|_{x(0) = x}, \ x \in C_{\textrm{x}}.$$ For simplicity, we set $ v_{\pi}(x) = \infty $ for $ x \notin C_{\textrm{x}} $. Let the next state be $ x' = L\pi(x)x $. From , the recursion below holds $$\label{eq: vpi}
v_{\pi}(x) = g(x, \pi(x)) + \lambda v_{\pi}(x'), \ x \in C_{\textrm{x}}.$$
Since we aim to minimize the cost, we say a policy $ \pi $ is better than another policy $ \pi' $ if and only if $ v_{\pi}(x) \le v_{\pi'}(x), \forall x \in C_{\textrm{x}} $. The optimal value function $ v_* $ and the optimal policy $ \pi_* $ are specified by $$\begin{aligned}
v_*(x) &= \min_{\pi}v_{\pi}(x), \label{eq: v*}\\
\pi_*(x) &= \operatorname*{arg\,min}_{u \in C_{\textrm{u}}(x)} g(x, u) + \lambda v_*(Lux). \label{eq: pi*}\end{aligned}$$ Further, there holds obviously $ v_{\pi_*}(x) = v_*(x) , \forall x \in C_{\textrm{x}},$ by the Bellman optimality equation [@RL2; @busoniu2017reinforcement], given below $$\label{eq: Bellman}
v_{*}(x) = \min_{u \in C_{\textrm{u}}(x)} g(x, u) + \lambda v_*(Lux), \ x \in C_{\textrm{x}},$$
A fundamental result in the MDP theory is that the infinite sum in has a finite value as long as the reward sequence is bounded [@RL2]. As aforementioned in Section \[sec: problem\], it is natural and common to set up a bounded stage cost function $ g $ [@cheng2014optimal; @zhu2018optimal; @zhao2010optimal; @wu2019optimal], which implies a finite value function for each state. Additionally, recall that the number of states and the number of control inputs are both finite in BCN , i.e., $ N $ and $ M $, respectively. Consequently, the number of possible policies in our case is also finite, which is at most $ M^N $ after constraint-violating ones are eliminated. Note that we assume Problem \[prob: 1\] is feasible, i.e., at least one policy exists that violates no constraints (see Remark \[rmk: problem\]). By the policy improvement theorem [@RL2], an optimal policy always exists that minimizes the value function for all states, from which we can construct the optimal control sequence for Problem \[prob: 1\] (see Section \[sec: vi\]). The correctness of the following proposition is obvious.
Consider Problem \[prob: 1\]. There exists an optimal control sequence if the stage cost function $ g $ is bounded.
The existence of solutions to infinite-horizon optimal control of BCNs with discounted cost (no constraints involved) has been shown in [@cheng2014optimal] and [@zhu2018optimal] from other aspects instead of the DMDP here. Note that the optimal control strategies for Problem \[prob: 1\] may not be unique.
Value Iteration based Approach {#sec: vi}
------------------------------
A widely used method in searching optimal policies for finite MDPs is *value iteration*, a dynamic programming based algorithm, which attempts to estimate the optimal value function of each state via iterative update [@RL2; @busoniu2017reinforcement]. It is intuitive to derive the update rule in value iteration from the Bellman optimality equation. Recall that the BCN control process is essentially a DMDP, and its optimality equation has been presented in .
Given an initial guess of the value function, termed $ V(\cdot) $, value iteration works by updating the value function following a rule similar to the optimality equation : $$\label{eq: Bellman update}
V(x) = \min_{u \in C_{\textrm{u}}(x)} g(x, u) + \lambda V(Lux), \ x \in C_{\textrm{x}}.$$ Such update is repeated iteratively until the value function converges for all states, i.e., the change between two iterations gets small enough below a threshold $ \theta \ge 0$. After the update loop is terminated, we can determine an (approximate) optimal policy from the value function by $$\label{eq: optimal policy}
\pi_*(x) = \operatorname*{arg\,min}_{u \in C_{\textrm{u}}(x)} g(x, u) + \lambda V(Lux), \ x \in C_{\textrm{x}}.$$
Next, a state feedback control law for optimal control can be directly constructed from the optimal policy with the following proposition.
\[prop: feedback matrix\] Consider Problem \[prob: 1\]. If $ \pi_* $ is an optimal policy for the associated discounted-cost DMDP, then infinite-horizon optimal control can be achieved by stationary state feedback $ u=Kx $, where nontrivial columns of the matrix $ K \in {\mathcal L}_{M \times N} $ are specified by $$\label{eq: state feedback}
\textnormal{\textrm{Col}}_i(K) = \pi_*({\delta_N}^i), \ \textrm{if } {\delta_N}^i \in C_{\textrm{x}},$$ with the other columns arbitrarily set.
Note that states and control inputs of BCN are both logical vectors filled with all zeros except a single entry of value 1. We thus have $ K{\delta_N}^i = \textrm{Col}_i(K) = \pi_*({\delta_N}^i)$ for any $ {\delta_N}^i \in C_{\textrm{x}} $. That is, we are exactly taking the optimal policy by applying the state feedback law . By the definitions in and , the optimal policy minimizes the value function for each state $ x \in C_{\textrm{x}} $, and $ v_*(x_0) $ is therefore the minimum of the performance index $ J(\cdot) $.
The value iteration routine for Problem \[prob: 1\] is listed in Algorithm \[alg: value iteration\]. In practice, a small positive threshold $ \theta > 0 $ is used to acquire a sub-optimal solution with an affordable computational cost, since this algorithm generally cannot converge to the exact optimimum in a finite number of iterations [@RL2; @busoniu2017reinforcement]. Supposing there are $ P $ iterations required for a specific $ \theta $, the computational cost of the loop (Line \[line: repeat\] - \[line: until\] ) is $ O(PMN) $. The computation of and runs in $ O(MN) $ and $ O(N) $ respectively. In summary, the time complexity of Algorithm \[alg: value iteration\] is $ O(PMN) $. Finally, we note that the state feedback controller is independent of the initial state $ x_0 $. Given an initial state $ x_0 $, the optimal control sequence can be computed readily from by evolving the BCN from state $ x_0 $ with the control law .
Problem \[prob: 1\]: $ L, C_{\textrm{u}}(\cdot), C_{\textrm{x}}, \lambda $. Threshold $ \theta \ge 0 $. Optimal state feedback matrix $ K $ Initialize the value function $ V(x) $ arbitrarily for $ x \in C_{\textrm{x}} $ \[line: repeat\] $ \psi \gets 0 $ $ v \leftarrow V(x) $ Update $ V(x) $ by $ \psi \gets \max(\psi, | v - V(x)|) $ \[line: until\] Resolve the optimal policy $ \pi^* $ by Construct the matrix $ K $ by Proposition \[prop: feedback matrix\]
Madani’s Algorithm based Approach
---------------------------------
The primary drawback of the basic value iteration approach in Algorithm \[alg: value iteration\] is that the number of iterations to get the exact optimal control strategy is not bounded [@RL2; @busoniu2017reinforcement]. Consequently, only a sub-optimal solution can be acquired in practice. On the other hand, recall that value iteration is a general algorithm for MDPs, especially stochastic ones, while our BCN control is more precisely a DMDP. In [@madani2010discounted], exploiting the determinism of a DMDP, Madani *et al.* develops a specialized and more efficient algorithm for solving discounted-cost DMDP problems. A more desirable advantage of this algorithm is its guarantee that exact solutions can be obtained in finite steps. In this section, we develop a more efficient and effective method to solve Problem \[prob: 1\] by resorting to Madani’s algorithm [@madani2010discounted].
Madani’s algorithm handles discounted-cost DMDPs from a graphical perspective and can be viewed as an adaptation of Karp’s algorithm for average-cost DMDPs [@gao2019infinite]. In the context of optimal BCN control, the DMDP is described by the state transition graph (STG) of the BCN, termed $ G = (V, E) $, where each vertex represents a state, i.e., $ V {\coloneqq}C_{\textrm{x}}$, and each edge denotes a state transition, i.e., $$E = \{(x, x') \in C_{\textrm{x}} \times C_{\textrm{x}} | \exists u \in C_{\textrm{u}}(x), x' = Lux \}.$$
The weight of each edge is the minimal cost of the corresponding state transition, since a transition may be attained by more than one control input at different costs. Consider two connected states (vertices) in $ G $, say $ (x, x') \in E $. The set of admissible control inputs for this transition (edge) is $$U_{xx'} = \{ u \in C_{\textrm{u}}(x) | x' = Lux \},$$ and the weight of this edge is $$\label{eq: w}
w(x, x') = \min_{u \in U_{xx'}} g(x, u),$$ along with the best control input enabling this transition $$\label{eq: u*}
u^*(x, x') = \operatorname*{arg\,min}_{u \in U_{xx'}} g(x, u).$$ Note that the best control input in may not be unique, and we can choose an arbitrary one in that case. Besides, the technique by and can also be adapted to the above value iteration approach to first filter out unlikely actions for specific states to improve computational efficiency.
Given BCN with constraints in Problem \[prob: 1\], it is easy to construct the STG $ G $ following a breadth-first search (BFS) routine, whose details can be found in our previous work [@gao2019infinite]. After the STG is available, Madani’s algorithm works in three stages, like follows.
1. Compute the minimal discounted cost of a $ k $-edge path starting from each vertex $ x \in C_{\textrm{x}} $, termed $ d_k(x) $, for each $ k \in [1, |C_{\textrm{x}}| ]$ with $ d_0(x) = 0 $.
2. Compute the quantity below for each vertex $ x \in C_{\textrm{x}}$: $$\label{eq: y0}
y_0(x) = \max_{0 \le k < |C_{\textrm{x}}| } \frac{d_{|C_{\textrm{x}}|}(x) - \lambda^{|C_{\textrm{x}}| - k}d_k(x)}{1 - \lambda^{|C_{\textrm{x}}| - k}}.$$
3. Recompute the the minimal discounted cost of a $ k $-edge path from each vertex $ x \in C_{\textrm{x}} $, termed $ y_k(x) $, but with the initial value $ y_0(x) $ in , for $ 1 \le k < |C_{\textrm{x}}| $.
4. The optimal value function of each state (vertex) $ x \in C_{\textrm{x}} $ is obtained by $$\label{eq: v*2}
v_*(x) = \min_{0 \le k < |C_{\textrm{x}}| } y_k(x).$$
Interested readers can refer to [@madani2010discounted] for detailed proof of the correctness of this algorithm. In practical implementation, the above tasks 1) and 3) can be done efficiently via dynamic programming in a form like Bellman optimality equation . The corresponding pseudocode is presented in Algorithm \[alg: Madani\]. Once the optimal value function $ v_* $ is obtained, we can again, just like Algorithm \[alg: value iteration\], get the optimal policy by and the optimal state feedback law by Proposition \[prop: feedback matrix\].
Problem \[prob: 1\]: $ L, C_{\textrm{u}}(\cdot), C_{\textrm{x}}, \lambda $. Optimal state feedback matrix $ K $ Build the STG $ G = (V, E) $ (see [@gao2019infinite] for details) $ d_0(x) \gets 0 $ for each $ x \in V$ $ d_k(x) \gets \min_{(x, x') \in E} w(x, x') + \lambda d_{k-1}(x')$ Compute $ y_0(x) $ by $ y_k(x) \gets \min_{(x, x') \in E} w(x, x') + \lambda y_{k-1}(x')$ Compute $ v^*(x) $ by Get the optimal policy $ \pi^* $ by Construct the matrix $ K $ by Proposition \[prop: feedback matrix\]
As we have analyzed in [@gao2019infinite], the time complexity to build the STG $ G = (V, E) $ subject to constraints in Problem \[prob: 1\] is $ O(MN) $. The running time of Madani’s algorithm in the graph $ G $ is $ O(|V||E|) $ [@madani2010discounted]. Note that there are at most $ N $ vertices in the STG, i.e., $ |V| \le N $, and each vertex has at most $ M $ outgoing edges, which means $ |E| \le M|V| \le MN $. Therefore, the running time of Algorithm \[alg: Madani\] is dominated by the Madani’s part, which is consequently $ O(MN^2) $.
A Biological Example: *Ara* Operon Network {#sec: example}
===========================================
In this section, we apply the two approaches proposed above to the *ara* operon network in the bacteria *E.coli* and compare its performance with that of existing methods. The ara operon network is a well studied GRN that plays a key role in metablism of the sugar *L-arabinose* in the absence of glucose. The GRN’s BCN model has 9 state variables (nodes), listed in Table \[tbl:ara\], and 4 control inputs, $ A_e $, $ A_{em} $, $ A_{ra\_} $, and $ G_e $. The Boolean functions associated with each node are also listed in Table \[tbl:ara\]. More biological knowledge of this network is available in [@jenkins2017bistability]. Its ASSR has a structure matrix $ L \in {\mathcal L}_{512 \times 8192} $ with $ M = 16 $ and $ N = 512 $, which is presented in the online material.
Node Function Node Function
--------------- ---------------------------------- --------- ---------------------------------------------
$ A $ $ A_e \land T $ $ D $ $\lnot A _{ra_+} $ $ \land $ $A _{ra_-} $
$ A_m $ $ (A_{em} \land T) \lor A_e$ $ M_S $ $ A _{ra_+} \land C \land \lnot D $
$ A _{ra_+} $ $ (A_m \lor A) \land A _{ra_-} $ $ M_T $ $ A _{ra_+}\land C $
$C $ $ \lnot G_e $B $ T $ $ M_T $
$ E $ $ M_S $
: BCN model of the *ara* operon network[]{data-label="tbl:ara"}
Wu *et al.* have investigated the infinite-horizon optimal control of the *ara* operon network with average cost in [@wu2019optimal]. We reuse their stage cost function in this study as follows: $$g(x, u) = AX + BU$$ with the column vectors $ X= [x_1, x_2, \cdots, x_9]^{\top}, U = [u_1, u_2, u_3, u_4]^{\top}$ and the two weight vectors as $$A = [-28, -12, 12, 16, 0, 0, 0, 20, 16], \ B = [-8, 40, 20, 40].$$ We assume an initial state $ x_0 = \delta_{512}^{10} $ and a discount factor $ \lambda = 0.5 $. No constraints are applied here for comparison purpose, since existing methods are not designed to handle constraints. In the value iteraton approach, the $ \epsilon $-suboptimal solutions are obtained. We implement all algorithms in Python 3.7 and measure their running time for Problem \[prob: 1\] on a laptop PC with a 1.8 GHz Core i7-8550U CPU, 8 GB RAM, and 64-bit Windows 10. All methods obtain the same optimal value, $ J^* = 5.232 $, except that the value iteration approach gets an approximate one.
We gather the theoretical time complexity and the measured running time of each method in Table \[tbl: comparison\]. As we see, the huge difference in running time between different methods accords well with previous time complexity analysis. Clearly, the two DMDP based approaches proposed in this paper can significantly reduce the running time. Note that, though Algorithm \[alg: value iteration\] has no upper bound on the number of iterations to get an exact optimum, it usually converges very fast in practice if only a suboptimal solution is desired. For example, only 9, 13, and 18 iterations are needed in this case for the three thresholds in Table \[tbl: comparison\]. Overall, the take-home message is that one can first try Algorithm \[alg: value iteration\] based on value iteration and then resorts to Algorithm \[alg: Madani\] that depends on Madani’s algorithm if the former cannot work properly.
[width=1]{}
Method [@cheng2014optimal] [@zhu2018optimal] Algorithm \[alg: value iteration\] Algorithm \[alg: Madani\]
----------------- --------------------- ------------------- ------------------------------------ ---------------------------
Time complexity $O(N^4)$ $ O(N^4) $ $ O(MN^2) $
0.21 ($ \theta = 0.1 $)
0.28 ($ \theta = 0.01 $)
0.36 ($ \theta = 0.001 $)
: Comparion of time complexity and measured running time in optimal control of the *Ara* operon network[]{data-label="tbl: comparison"}
$ P $ refers to the number of iterations and is not bounded for an exact optimium.
The time complexity is stated to be $ O(MN + N^4) $ in [@zhu2018optimal]. We note that, in general, there exists $ M < N $ or even $ M \ll N $ in practice, i.e., fewer control inputs than state variables, especially for large networks [@lu2019pinning]. Besides, we can always assume $ M \le N $, since a state can transit to at most $ N $ succeeding states regardless of the number of control inputs, and it is useless to have more inputs than state variables. Thus, the time complexity of Algorithm \[alg: Madani\] is equivalently $ O(N^3) $. Though the running time listed in Table \[tbl: comparison\] may partly depend on implementation details, the difference in orders of magnitude demonstrates obviously the superiority of our approaches in terms of time efficiency.
Conclusions {#sec: conclusion}
===========
We tackled the infinite-horizon optimal control of BCNs with discounted cost in this paper. Unlike the existing methods, we solved this problem from the perspective of a deterministic Makov decision process (DMDP). We first showed that the control of a BCN could be well described by a DMDP and then proposed two approaches for the optimization of this DMDP, one based on value iteration and the other based on Madani’s algorithm, while the latter can obtain the exact optimum with lower time complexity than existing work. Besides, the value iteration based approach can potentially get a near-optimal solution with much less running time than all other methods. A benchmark example using the *ara* operon network has demonstrated the superior time efficiency of both proposed approaches. The DMDP view of BCN control may be promising for other problems as well and deserves more investigations.
[^1]: All authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, 117583. Email: [elexc@nus.edu.sg]{}
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author:
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M.A.Braun, A.Tarasov\
S.Peterburg State University, Russia
title: '**Forward pomeron propagator in the external field of the nucleus**'
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=-0.5in =-0.0in =8.75in =6.5in
epsf
[**Abstract**]{}
It is shown by numerical calculations that the convoluted forward pomeron propagator in the external field created by a solution of the Balitski-Kovchegov equation in the nuclear matter vanishes at high rapidities. This may open a possibility to apply the perturbative approach for the calculation of pomeron loops.
Introduction
============
In the QCD, in the limit of large number of colours, strong interaction at high energies is mediated by the exchange of BFKL pomerons, which interact via their splitting and fusion. In the quasi-classical approximation for photon (hadron)-nucleus scattering the relevant tree (fan) diagrams are summed by the well-known Balitski-Kovchegov (BK) evolution equation [@bal; @kov; @bra1]. For nucleus-nucleus scattering appropriate quasi-classical equations were derived in [@bra2; @bra3]. In both cases pomeron loops were neglected. This approximation can be justified if the parameter $\gamma=\lambda\exp{\Delta y}$ is small, with $y$ the rapidity and $\Delta$ and $\lambda$ the pomeron intercept and triple pomeron coupling. Then for a large nuclear target, such that $A^{1/3}\gamma\sim 1$, the tree diagrams indeed give the dominant contribution and loops can be dropped. However with the growth of $y$ the loop contribution becomes not small and this approximation breaks down.
Direct calculation of the loop contribution seems to be a formidable task for the non-local BFKL pomeron. Simplest loops have been studied in several papers for purely hadronic scattering [@peschansky; @bartels; @bra4]. In particular in [@bra4] it has been found that pomeron loops become essential already at rapidities of the order 10$\div 15$. They shift the position of the pomeron pole to the complex plane and thus lead to oscillations in cross-sections. However with the growth of energy loop contributions begin to dominate and one needs to sum all of them. There have been many attempts to do this in the framework of the so-called reaction-diffusion formulation of the QCD dynmaics and the following correspondence with the statistical approach [@iancu; @AHM; @levlub; @lev; @marquet; @levmil] (see also a review [@soyez] and references therein). Unfortunately concrete results could be obtained only with very crude approximations for the basic BFKL interaction and the stochastical noise in the statistical formulation. The conclusions of different groups are incomplete and contradictory. So in [@levmil] it was found that the geometric scaling following from the BK equation was preserved with loops taken into account, although going to the black disc limit was much slower. On the contrary in papers based on the analogy with statistical phyiscs (see [@marquet; @soyez]) it was argued that the BK scaling was changed to the so called diffusive scaling (with an extra $\sqrt{y}$ in the denominator of the argument) but the speed of achieving the black disk limit was essentially unchanged.
In our previous study of pomeron loops [@BT] we considered a much simpler model with the local supercritical pomeron in the Regge-Gribov formalism. Instead of trying to solve the model for the purely hadronic scattering we considered the hadron-nucleus scattering and propagation of the pomeron inside the heavy nucleus target. Moreover to avoid using numerical solution of the tree diagrams contribution with diffusion in the impact parameter, we concentrated on the case of a constant nuclear density which allowed to start with the known analytical solutions. We have found that the nuclear surrounding transforms the pomeron from the supercritical one with intercept $\epsilon>0$ to a subcritical one with the intercept $-\epsilon$. Then Regge cuts, corresponding to loop diagrams, start at branch points located to the left of the pomeron pole and their contribution is subdominant at high energies. As a result the theory aquired the properties similar to the Regge-Gribov with a subcritical pomeron and allows for application of the perturbation theory. In [@BT] we expressed our hopes that a similar phenomenon might occur in the QCD with BFKL pomerons.
In this note we demonstrate that such hopes are possibly founded. We consider the pomeron propagator in the external pomeron field created inside the nucleus and give arguments that, similar to the local Regge-Gribov case, it vanishes at large rapidity distances. We stress that at present we are unable to give the full proof for this behaviour. Our study is based on numerical calculations. This makes us to choose a relatively small subset of initial conditions out of the complete set necessary for the study of the pomeron propagator. Moreover, due to technical difficulties, in this note we restricy ourselves to the much simpler pomeron propagator in the forward direction. Our numerical results show that, with the chosen set of initial conditions, this forward propagator vanishes at large rapidity distances. This result is insufficient for the study of loops, where non-forward propagator are involved. However it can be applied for double inclusive cross-sections in the nucleus-nucleus scattering, in which only forward propagators are important.
Main equation
=============
We are going to study the behaviour of the BFKL pomeron propagator in the external field, generated by a solution of the BK equation. We consider the simplified case of the nuclear matter, when the dependence on the impact parameter $b$ is absent. This propagator corresponds to a sum of diagrams shown in Fig. \[fig1\].
As mentioned, in this note we restrict ourself to the forward propagator, $P(y,x,x')$ when the transferred momentum is zero. It depends on rapidity $y$ and two 2-dimensional coordinate vectors $x$ and $x'$ corresponding to the initial and final distances between the reggeized gluons in the pomeron. At $y=0$ we have $P(y,x,x')=\nabla^{-4}\delta^2(x-x')$. Since the study is only possible numerically, to avoid using this singular initial condition, we shall consider a convolution of $P(y,x,x')$ with an arbitrary initial function $\nabla^4\psi(x)$ P(y,x)=d\^2x’P(y,x,x’)\^4(x’). This convolution satisfies the same equation as the propagator itself but at $y=0$ we have P(y=0,x)=(x). \[ini\] Obviously properties of the propagator can be studied taking a full set of functions $\psi(x)$.
The equation for $P(y,x)$ can be conveniently obtained from the BK equation for the sum of fan diagrams. In the forward direction this sum $\Phi(y,x)$ satisfies a non-linear eqiation = d\^2x\_1 ((y,x\_1)+(y,x\_2)-(y,x)-(y,x\_1)(y,x\_2)), \[bk\] where standardly $$\bar{\alpha}=\frac{\alpha_sN_c}{\pi}.$$ The equation for the convoluted propagator $P(y,x)$ in the presence of nuclear medium is obtained when one of the $\Phi$ in the non-linear term in (\[bk\]) is substituted by a particular solution of (\[bk\]) with a given boundary condition. The equation thus obtained is = d\^2x\_1 (P(y,x\_1)+P(y,x\_2)-P(y,x)-2(y,x\_1)P(y,x\_2)). \[eq1\] Note that the initial condition for $\Phi(y=0,x)=\Phi_0(x)$ is fixed by the properties of the nuclear medium, whereas the initial condition (\[ini\]) for $P(y,x)$ is arbitrary, since we we are interested in the propagator in a given nuclear surrounding.
Equation (\[eq1\]) is a linear equation for $P(y,x)$ in contrast to the BK equation. At $y\to\infty$ $\Phi(y,x)\to 1$ independent of the chosen initial condition. One may think that at $y\to\infty$ the behaviour of $P(y,x)$ can be derived from the asymptotic equation $$\frac{\partial P(y,x)}{\partial y}\Big|_{y\to\infty}=
\frac{\bar{\alpha}}{2\pi}\int d^2x_1\frac{x^2}{x_1^2x_2^2}
\Big(P(y,x_1)+P(y,x_2)-P(y,x)-2P(y,x_2)\Big)$$= -P(y,x)d\^2x\_1. \[eq11\] However the integral on the right-hand side has become divergent (although it converges at finite $y$). This means that the limit $y\to\infty$ is more delicate and cannot be taken under the sign of integral over $x_1$. And indeed we shall see by numerical calculation that the behaviour of the solution at $y\to\infty$ is not solely determined by the limiting value of $\Phi(y,x)$ but depends on its behaviour at finite $y$.
For numerical studies both the BK equation and linear equation (\[eq1\]) in the momentum space are more convenient. Introducing (y,x)=, p(y,x)= and then passing to the momentum space we obtain the following equations for $\phi(y,k)$ and $p(y,k)$ = -|(H\_[BFKL]{}(y,k)+\^2(y,k)) \[bkmom\] and = -|(H\_[BFKL]{}+2(y,k))p(y,k), \[eq1mom\] where H\_[BFKL]{}=k\^2+x\^2-2((1)+2). To study the behaviour of the propagator in the external field $\phi$ one has to solve this pair of equations with the initial conditions (y,k)\_[y=0]{}=\_0(k), p(y,k)\_[y=0]{}=p\_0(k), \[inimom\] with some fixed $\phi_0$ and for a complete set of function $p_0(k)$.
Numerical studies
=================
We have set up a program which simultaneously solves the pair of equations (\[bkmom\]) and (\[eq1mom\]) for a given pair of initial conditions (\[inimom\]). For the BK evolution we have fixed the initial condition as \_0(k)=-(1/2)[Ei]{}(-k\^2/0.3657) used in our previous calculations. The behaviour of $\phi(k)$ with $k^2$ at different values of the scaled rapidity $Y=\bar{\alpha}y=2,4,6,8$ and 10 is shown in Fig. \[fig2\]. (Note that the maximal value of the scaled rapidity $Y=10$ corresponds to the natural rapidity of order 50).
For the BFKL evolution in the external field $\phi$, in the first run (A), we have taken the same form of the initial condition but with a variable slope p\_0(k)=-(1/2)[Ei]{}(-k\^2/a). \[ini1\] We have performed calculations for $a=0.2,\ 0.6,\ 1.0,\ 1.4$ and 1.8. In the second run (B) the initial condition was taken with extra powers of $k^2$. p\_0(k)=-(1/2)k\^[2n]{}[Ei]{}(-k\^2/0.3657) \[ini2\] with $n=0,1,2,3$ and 4.
In all cases the behaviour of the solution $p(y,k)$ was found to be universal. At large enough $y$ the solution becomes independent of $k^2$ up to a certain maximal $k^2_{max}(y)$, starting from which it goes to zero. Roughly p(y,k)\~A(y)(k\^2\_[max]{}-k\^2). \[pyk\] As $y$ grows $A(y)$ goes to zero and $k^2_{max}(y)$ goes to infinity. . So on the whole the solution vanishes as $y\to\infty$, its $x$ dependence tending to $\delta^2(x)$.
We illustrate this behaviour in Figs. \[fig3\] and \[fig4\], in which we show the solution $p(y,k)$ for run A with $a=1.0$ and run B with $n=2$ as a function of $k^2$. One observes that although the values of $p(y,k)$ for the two cases are different, their behavior with $y$ is the same: they vanish as $y\to\infty$.
This unversality is especially obvious if one calculates the slope $\Delta(y,k)$ of the $y$-dependence of $p(y,k)$ at fixed $k$ presenting p(y,k)e\^[Y(y,k)]{}. It turns out that at $Y>1$ the slope $\Delta(y,k)$ is independent of $k$ and identical for all considered cases (run A with all studied $a$ and run B with all studied $n$). Its smooth behaviour with $Y$ is shown in Fig. \[fig5\]. One observes that starting from $Y=5$ the slope becomes negative indicating that the solution goes to zero at $Y>>1$.
One has to take into account that in the external field $\phi(y,k)$ depending on rapidity the pomeron prpagator ceases to depend only on the rapidity difference. Rather the initial and final rapidities become two independent variables. To see what influence it has on the behaviour of the propagator at large rapidities we varied the initial rapidity $y=y_0$ for the evolution of $p(y,k)$, leaving unchanged the initial rapidity $y=0$ for the evolution of $\phi(y,k)$, which is the rapidity of the nucleus. One finds that although at initial stages of evolution the behavior of $p(y,k)$ strongly depends on the value of $y_0$, at higher rapidities this behaviour is essentially the same for any $y_0$, namely the convoluted propagator goes down with rapidity with the slope independent of $y_0$. This is illustrated in Figs. \[fig6\] and \[fig7\] which show results for $y_0=3/\bar{\alpha}\ (Y_0=3)$ In Fig. \[fig6\] we show the solution $p(y,k)$ for run A with $a=1$. One observes, that although absolute values of $p(y,k)$ are quite different from the case $y_0=0$ shown in Fig \[fig3\], the behaviour with the growth of rapidity is the same. It is especially clear from the values for the slope $\Delta$ shown in Fig. \[fig7\] together with those for the case $y=0$ (Fig. \[fig5\]). Again at the initial stage of the evolution the behavior with $Y_0=3$ is quite different from that with $Y_0=0$. However at higher rapidities the values for the slope become the same.
It is remarkable that this behaviour takes place only with $\phi$ given by the exact solution of the BK equation. Taking an approximate form (y,x)1-e\^[-Q\^2(y)x\^2]{}, \[apprphi\] where the “saturation momentum” $Q^2(y)\sim e^{2.05\bar{\alpha}y}$ we obtain an equation for $p(y,k)$ in the momentum space =-|\[evoleq\] Taking for simplicity the initial condition $p_0(k)=\phi_0(k)$ at $y=0$ we get the solution shown in Fig. \[fig8\].
One observes that at large $y$ the solution acquires the same form (\[pyk\]) where however $A(y)$ grows with $y$: A(y)\~e\^[1.4 Y]{} This implies that with the approximate form (\[apprphi\]) of $\Phi$ the final solution $P(y,x)$ in the coordinate space behaves in a singular manner at $y\to\infty$. Effectively P(y,x)\_[y]{}e\^[1.4 y]{}x\^2\^2(x) and it is impossible to say that it vanishes in this limit.
Conclusions
===========
We have studied numerically the BFKL pomeron forward propagator in the external field created by the solution of the BK equation in the nuclear matter. We have found that for more or less arbitrary set of initial conditions the convoluted propagator vanishes at large rapidities, its coordinate dependence tending to the $\delta$-function. This gives reasons to believe that the forward propagator itsef vanishes at large rapidities in the nuclear background. This result follows only with the field being the exact solution of the BK equation.
Our results are obviously insufficient for the calculation of pomeron loops, which requires the non-forward pomeron propagator. However they can be directly applied to the study of double inclusive cross-section for gluon jet production in nucleus-nucleus collisions. This highly complicated problem is left for future investigation.
Acknowledgments
===============
This work has been supported by grants RFFI 09-012-01327-a and RFFI-CERN 08-02-91004.
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abstract: 'Let $M$ be a geometrically finite $m$-dimensional hyperbolic manifold, where $m \geq 2$, and let $G$ be a finite group of its isometries. We provide a formula for the number of connected components of the locus of fixed points of any element of $G$ in terms of its action. This formula extends Macbetah’s one for the case of compact Riemann surfaces of genus at least two, which was initially used to obtain the character of the representation associated to the induced action of $G$ on the first homology group, and later turned out to be extremely useful in many other contexts.'
address:
- 'Faculty of Mathematics, Physics and Informatics, Gdańsk University, Gdańsk, Poland'
- 'Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile'
author:
- 'Grzegorz Gromadzki and Ruben A. Hidalgo'
title: On the number of connected components of the set of fixed points of isometries of geometrically finite hyperbolic manifolds
---
[^1]
Introduction
============
Let $M$ be a geometrically finite hyperbolic manifold of dimension $m \geq 2$ and let $G$ be a finite group of its hyperbolic isometries. The locus of fixed points of $g \in G$ consists of a finite collection of pairwise disjoint totally geodesic subspaces (including isolated points). In this paper we provides a formula that permits to obtain the number of the connected components of the fixed points of each of the isometries in $G$ (Theorem \[Macbeath-Kleinian\]). In the last section we provide a couple of examples for $m=3$.
In 1973, for $m=2$ and $M$ is compact and orientable (i.e., a compact Riemann surface), Macbeath [@Mcb] found a formula for the number of fixed points of each $g \in G$ in terms of the topological type of the action of $G$ (we recall it in Section 4). Generalization of this formula to count the number of connected components of fixed points has been found later for anti-conformal automorphisms of compact Riemann surfaces and also for dianalytic automorphisms of bordered and unbordered compact Klein surfaces (both in orientable and non-orientable cases) in [@GG; @G1; @G2; @IS]. In this dimension, in contrast with the dimension $m >2$, there are two extra features. Namely due to the Nielsen realization theorem, given a finite group $G$ of self-homeomorphism of a closed orientation surface $X$ there is a Riemann surface structure on it so that $G$ becomes its group of conformal automorphisms. So actually Macbeath’s formulas can be applied for arbitrary periodic self-homeomorphisms. Furthermore, due to well known description of discrete cocompact groups of isometries of the hyperbolic plane, the formulas in two dimensional case have more explicit character.
Let us observe that, for $m=3$, Marden’s (or the tame ends) conjecture, proved by Agol [@A] and Calegari-Gabai [@CG] (see also the preliminary results of Cannary and Misnky [@CM]), states that if $M$ has finitely generated fundamental group, then it is homeomorphic to the interior of a compact $3$-manifold. Also, in the papers [@B-B; @N-S; @Ohshika] it has been proved that the locus of geometrically finite Kleinian groups is dense on the space of finitely generated Kleinian groups. This, in particular, makes us to believe that our formula could have some relevance.
Our formula in Theorem \[Macbeath-Kleinian\] still valid for manifolds obtained as quotient of the Teichmüller space ${\mathcal T}_{g}$, of genus $g \geq 1$ Riemann surfaces, or the Siegel space ${\mathfrak H}_{g}$ that parametriizes principally polarized abelian varieties (see Remark \[siegel\]).
Preliminaries
=============
Here we shall recall some concepts and facts concerning isometries of hyperbolic spaces, Kleinian groups and associated manifolds which shall be used. A good references on these are, for instance, the classical books [@M; @MT]. For $n \geq 1$, we shall use as a model of the $(n+1)$-dimensional hyperbolic space $\mathcal{H}^{n+1}$ the $(n+1)$-dimensional upper-half space $\{x=(x_{1},\ldots ,x_{n+1}) \in {\mathbb R}^{n+1}: x_{n+1}>0\}$ equipped with the Riemannian metric $ds=\|dx\|/x_{n+1}$. Its conformal boundary is the $n$-dimensional sphere $\mathcal{S}^{n}={\mathbb R}^{n} \cup \{\infty\}$. Each $(n-1)$-dimensional sphere $\Sigma \subset \mathcal{S}^{n}$ (for $n=1$, $\Sigma$ is understood as two different points) has the associated reflection $\sigma= \sigma_{\Sigma}$ having $\Sigma$ as its locus of fixed points. By the Poincaré extension theorem, the reflection $\sigma$ extends naturally to an order two orientation reversing isometry of $\mathcal{H}^{n+1}$; this being the reflection on the half-$n$-dimensional sphere inside $\mathcal{H}^{n+1}$ induced by $\Sigma$. A Möbius (respectively, extended Möbius) transformation of $\mathcal{S}^{n}$ is the composition of an even (respectively odd) number of reflections. By the classical complex analysis, for $n=2$, and the Liouville Theorem, for $n\geq 3$, the group $\widehat{\mathcal M}^{n}$, composed by all Möbius and extended Möbius transformations, is the full group of conformal automorphisms of $\mathcal{S}^{n}$. We shall denote denote by ${\mathcal M}^{n}$ its canonical subgroup of index two consisting of all Möbius transformations. Again, by the Poincaré extension theorem, every Möbius (respectively, extended Möbius) transformation extends to an orientation-preserving (respectively, orientation-reversing) isometry of $\mathcal{H}^{n+1}$ and all isometries of $\mathcal{H}^{n+1}$ are obtained in this way. This allows us to identify the group $\widehat{\mathcal M}^{n}$ with the group ${\rm Isom}(\mathcal{H}^{n+1})$ of all isometries of $\mathcal{H}^{n+1}$ and ${\mathcal M}^{n}$ with the index two subgroup ${\rm Isom}^+(\mathcal{H}^{n+1})$ of all its orientation-preserving isometries.
An element of $\widehat{\mathcal M}^{n}$, viewed as an isometry of $\mathcal{H}^{n+1}$, may or may not have fixed points and if the former is the case, then it is called [*elliptic*]{} if it preserves orientation and [*pseudo-elliptic*]{} otherwise. The locus of fixed points of an elliptic or pseudo-elliptic transformation is known to be either a point or a totally geodesic subspace of $\mathcal{H}^{n+1}$.
A [*Kleinian group*]{} is a discrete subgroup of ${\mathcal M}^{n}$ and an [*extended Kleinian group*]{} is a discrete subgroup of $\widehat{\mathcal M}^{n}$ not contained in ${\mathcal M}^{n}$. Elliptic or pseudo-elliptic transformations of a (extended) Kleinian group have necessarily finite orders. A subgroup $\mathcal{K}$ of $\widehat{\mathcal M}^{n}$ is an extended Kleinian group if and only if $\mathcal{K}^{+}=\mathcal{K} \cap {\mathcal M}^{n}$ is a Kleinian group. To each Kleinian group ${\mathcal F}<{\mathcal M}^{n}$ there is associated a $(n+1)$-dimensional orientable hyperbolic orbifold $M_{\mathcal F}=\mathcal{H}^{n+1}/{\mathcal F}$.
If ${\mathcal F}$ is torsion free, then $M_{\mathcal F}$ is a $(n+1)$-dimensional hyperbolic manifold, which means that it carries a natural complete Riemannian metric of constant negative curvature inherited from the one of $\mathcal{H}^{n+1}$. In this case, a [*conformal automorphism*]{} (respectively, [*anti-conformal automorphism*]{}) of $M_{{\mathcal F}}$ is an orientation-preserving (respectively, orientation-reversing) self-isometry. We denote by ${\rm Aut }(M_{{\mathcal F}})$ the group of all automorphisms of $M_{{\mathcal F}}$ and by ${\rm Aut}^{+}(M_{{\mathcal F}})$ its subgroup of conformal automorphisms.
The Kleinian group $\mathcal{F}$ is called [*geometrically finite*]{} if it has a finite-sided fundamental polyhedron in ${\mathcal H}^{n+1}$, in particular, it is finitely generated and the hyperbolic volume of ${\rm Hull}_{\epsilon}(\Lambda(\mathcal{F}))/\mathcal{F}$ is finite, where $\Lambda(\mathcal{F}) \subset \mathcal{S}^{n}$ stands for the limit set of $\mathcal{F}$ and ${\rm Hull}_{\epsilon}(\Lambda(\mathcal{F}))$ is the $\epsilon$-neighborhood of the convex hull of $\Lambda(\mathcal{F})$ in $\mathcal{H}^{n+1}$. An extended Kleinian group is geometrically finite if its index two orientation-preserving half Kleinian group is so. Finite index extensions of geometrically finite groups are still geometrically finite. Another properties of geometrically finite groups can be found, for instance, in [@Apanasov; @M].
Let us consider a finitely generated (extended) Kleinian group ${\mathcal K}<{\mathcal M}^{n}$. If $n \in \{1,2\}$, then i${\mathcal K}$ contains a finite number of conjugacy classes of elements of finite order [@FM], but for $n \geq 3$, this property may be false in general[@KP], but it holds true if ${\mathcal K}$ is known to be geometrically finite.
So, if ${\mathcal K}<{\mathcal M}^{n}$ is geometrically finite, then are able to find a collection of finite order elements $\{\kappa_{1},\ldots, \kappa_{r}\} \subset \mathcal{K} $ so that the following holds:
1. each $\kappa_{i}$ generates a maximal cyclic subgroup of $\mathcal{K}$;
2. the cyclic subgroups generated by $\kappa_{1} ,\ldots, \kappa_{r} $ are pairwise non-conjugate in $\mathcal{K}$;
3. the collection is maximal with respect to the above two properties.
In the above, following the terminology used by Maclachlan in [@Mcl] for Fuchsian groups, we will say that the collection $\{\kappa_{1},\ldots, \kappa_{r}\}$ is an [*elliptic complete system* ]{} ([*e.c.s.*]{} in short) and we will say that any of its elements is a [*canonical generating symmetry of ${\mathcal K}$*]{}.
Quantitative aspects of the set of fixed points
===============================================
In this section, ${\mathcal F}<{\mathcal M}^{n}$ will be a fixed torsion free geometrically finite Kleinian group. As previously mentioned, the hyperbolic manifold $M_{\mathcal F}$ has finitely generated fundamental group (as it is isomorphic to ${\mathcal F}$). Lets us denote by $\pi: \mathcal{H}^{n+1} \to M_{{\mathcal F}}$ a universal covering with ${\mathcal F}$ as its group of deck transformations.
Let $G$ be a finite subgroup of ${\rm Aut }(M_{{\mathcal F}})$ and let $\mathcal{K}$ be the subgroup of ${\rm Isom}(\mathcal{H}^{n+1})$ defined by all the lifts under $\pi$ of the elements of $G$. In this way, as $\mathcal{K}$ is a (normal) finite extension of ${\mathcal F}$, it is also geometrically finite, so a finitely generated Kleinian or extended Kleinian group of isometries of $\mathcal{H}^{n+1}$. There is a natural short sequence $$1 \to {\mathcal F} \to {\mathcal K} \stackrel{\theta}{\to} G \to 1$$ so that, if $g = \theta (\kappa)$ and $x= \pi(y)$, then $g(x) = \pi (\kappa y)$, for arbitrary $\kappa \in \mathcal{K}$. We shall keep all these notations throughout the rest of this paper
Each element $\kappa \in \mathcal{K} $ of finite order defines an automorphism $\theta(\kappa) \in G$ acting with fixed points. The converse is clear as $\pi$ is a local homeomorphism.
\[equal o disjoint\] The sets of fixed points of two distinct non-trivial elements of $\mathcal{K}$ of finite orders inducing the same automorphisms of $M_{\mathcal F}$ are disjoint.
Let $\kappa, \kappa' $ be elements of finite order of $\mathcal{K}$ and let $\theta(\kappa)=\theta(\kappa')$. Let us assume they have non-disjoint connected components, say $C$ and $C'$, of their loci of fixed points. If $y \in C\cap C'$, then $y$ is a fixed point of $\kappa^{-1}\kappa' \in \ker \theta = {\mathcal F}$. As ${\mathcal F}$ is torsion free and $\kappa^{-1}\kappa'$ has a fixed point, we must have that $\kappa^{-1}\kappa'=1$.
Since $\mathcal{K}$ is geometrically finite, we may find an e.c.s. $\{\kappa_1, \ldots , \kappa_r\}$ for ${\mathcal K}$, which we assume, from now on, to be fixed.
If $g \in G$ is a non-trivial element with fixed points, then property $(ecs3)$ ensures that $g= \theta (\kappa)$ for some elliptic element $\kappa$ of $\mathcal{K}$ which is conjugated to a power of some canonical generating symmetry $\kappa_{j}$. Let $J(g)$ be the set of such $j \in \{1,\ldots,r\}$ for which $g=\theta(\omega \kappa_{j}^{n_{j}} \omega^{-1})$ for some $\omega \in \mathcal{K}$. Let us observe, as $\ker \theta={\mathcal F}$ is torsion free, that the equality $$\label{difference}
\theta(\omega_{j_1} \kappa_{j_1}^{n_{j_1}} \omega_{j_1}^{-1}) = \theta(\omega_{j_2} \kappa_{j_2}^{n_{j_2}} \omega_{j_2}^{-1}).$$ for $j_1, j_2 \in J(g)$ shows in particular that the isometries $\omega_{j_1} \kappa_{j_1}^{n_{j_1}} \omega_{j_1}^{-1}$ and $\omega_{j_2} \kappa_{j_2}^{n_{j_2}} \omega_{j_2}^{-1}$ have the same finite order.
For $n=1$, $\mathcal{K}$ is either a Fuchsian or an NEC group, so the set of fixed points of an elliptic element $\kappa \in {\mathcal K}$ consist in a single point and, as the only finite order orientation reversing isometries of the hyperbolic plane are reflections, the locus of fixed points in this case is a geodesic line. In particular, different elliptic elements of the same order have different sets of fixed points. Unfortunately, this is no longer true for symmetries of higher dimensional spaces and this is a one of the essential differences between Macbeath’s formula for Riemann surfaces and our formula for hyperbolic manifolds of higher dimensions. For instance, Let $n\geq 4$ and take $A, B \in {\rm O}_{n}(\mathbb R)$ generating a non-cyclic finite group ${\mathcal U}$. Assume that none of them has eigenvalue equal to $1$ and so that they are non-conjugate in ${\mathcal U}$ (this can be done for $n \geq 4$). Let us consider the isometries of the hyperbolic $(n+1)$-space, in this example modeled by the unit ball in ${\mathbb R}^{n+1}$, given by $$T_{A}=\left[ \begin{array}{cc}
A & 0\\
0 & 1
\end{array}
\right], \; T_{B}=\left[ \begin{array}{cc}
B & 0\\
0 & 1
\end{array}
\right].$$ Then, $\mathcal{K}=\langle T_{A}, T_{B}\rangle \cong {\mathcal U}$ is a finite extended Kleinian group, where $T_{A}$ and $T_{B}$ both share the same geodesic set of fixed points, but they are non-conjugated in $\mathcal{K}$.
As note in the above remark, although different canonical generating symmetries give rise to different connected components of their sets of fixed points, it is no longer true for their powers of the same order. In order to neutralize that deficiency while we count the number of connected components of ${\rm Fix}(g)$, for $g \in G-\{I\}$, we must define the following equivalence relation on $J(g)$: $$j_1, j_2 \in J(g): \;
j_1 \sim j_2 \Leftrightarrow
{\rm Fix}(\kappa_{j_1}^{n_{1}}) \cap {\rm Fix}(\kappa_{j_2}^{n_{2}}) \neq \emptyset .$$
The above relation is equivalent for the sets of fixed points of elliptic elements to be projected on the same subset of ${M}_{\mathcal{F}}$. Let $I(g)$ be a set of representatives for the quotient set $J(g)/\! \! \sim$.
Finally, as the set ${\mathcal C}$ of fixed point of an isometry of finite order $\kappa \in \mathcal{K}$ is a totally geodesic subspace of $\mathcal{H}^{n+1}$, its image $\ell=\pi({\mathcal C})$ is a connected component of the set of fixed points of $\theta(\kappa)$; this being again a totally geodesic submanifold of $M_{\mathcal F}$. Let $G_{\ell}$ be the subgroup of $G$ leaving $\ell$ set-wise invariant (the elements of $G_{\ell}$ permutes the points on $\ell$ and may or may not have fixed points on it).
We have now introduced all notations and facts needed to state and prove our main result concerning the number of connected components of the locus of fixed points of isometries in $G$.
\[Macbeath-Kleinian\] Let $\mathcal{K}$ be a geometrically finite Kleinian group, $G$ a finite abstract group, $\theta:\mathcal{K} \to G$ a surjective homomorphism whose kernel is a torsion free geometrically finite Kleinian group ${\mathcal F}$. Fix an elliptic complete system $\kappa_{1},\ldots, \kappa_{r}$ for $\mathcal{K}$ and suppose that $\kappa_i$ has order $m_i$. Then the number of connected components of the set of fixed points of a non-trivial element $g \in G$ of order $n$ in the hyperbolic manifold $M_{\mathcal F}$ is $$|{\mathcal N}_G\langle g \rangle | \sum_{i \in I(g)} {1}/{n_{i}},$$ where ${\mathcal N}_{G}\langle g \rangle$ stands for the normalizer in $G$ of the cyclic subgroup $\langle g \rangle$ and $n_i$ is the order of the $\theta$-image of the $\mathcal{K}$-normalizer of $\langle \kappa^{m_i/n}_i\rangle$.
Denote $m_i/n$ by $s_i$ and let, as before, $\pi$ be the universal covering $\mathcal{H}^{n+1} \to M_{\mathcal F}$ induced by ${\mathcal F}$. Then $x=\pi (h)$ is a fixed point of $g= \theta (\kappa)$ if and only if $\pi (h) = \pi ( \kappa h)$ and so if and only if $\gamma h = \kappa h$ for some $\gamma \in \mathcal{F}$. This means that $\gamma ^{-1} \kappa \in \mathcal{K}$ has a fixed point and hence it is conjugate to a power of some element $\kappa_i$ of [*e.c.s.*]{} and therefore $g=\theta(\omega \kappa_{i}^{\alpha s_{i}} \omega^{-1})$ for some $\alpha$ coprime with $n$ and $\omega \in \mathcal{K}$. Clearly neither $i$, nor $\alpha$ and nor $\omega$ must be unique here. So given $i \in J(g)$ consider $$N_{i}(g)= \{ \omega \in \mathcal{K} :
g = \theta (\omega \kappa_i^{t_i} \omega^{-1}) \; {\rm for \; some} \;t_i \}$$
If now we let ${\mathcal C}_{i}$ to be the totally geodesic subspace of the set of fixed points of $\kappa_{i}^{t_{i}}$ and if we denote by ${\mathcal K}_{i}$ its stabilizer in ${\mathcal K}$ which is the normalizer of $\langle \kappa^{m_i/n}_i \rangle$, then $x \in \pi( \omega (\mathcal{C}_i))$ and conversely given $\omega \in N_i(g)$, $\pi( \omega (\mathcal{C}_i)) \subseteq {\rm Fix}(g)$ and as a result we obtain $${\rm Fix}(g) = \bigcup_{i \in I(g)}\bigcup_{\, \omega \in N_i(g)} \pi(\omega ({\mathcal C}_i)),$$ Now fix $\omega_i$ in $N_{i}(g)$. Then for any orther $\omega \in N_{i}(g)$, $\omega \mathcal{K}_i = \omega_i \mathcal{K}_i$ and so $N_{i}(g)$ is the left coset $\omega_i \mathcal{K}_i$. Furthermore $\omega \in N_i(g)$, gives $\theta(\omega_{i}^{-1} \omega ) \in {\mathcal N}_G\langle g
\rangle $ and so $\omega_{i}^{-1} \omega \in \theta^{-1}( {\mathcal
N}_G\langle g \rangle )$ which in turn means that $N_i(g)$ is also left coset $\omega_i\,\theta^{-1}( {\mathcal N}_G\langle g \rangle)
$
Now, $\ell_{i} = \pi (\omega({\mathcal C}_{i}))$ is a one of the connected components of the set of fixed points of $g$ and notice that $\theta(\omega {\mathcal K}_{i} \omega^{-1})= G_{\ell_i}$.
Now, given $\nu, \nu' \in \theta^{-1}({\mathcal N}_G\langle g \rangle )$, we have the following chain of equivalences $$
--------------------------------------------------------------------------------------- ------------------- ----------------------------------------------------------------------------------------------------------------
$\pi( \omega_{i}\nu ({\mathcal C}_i))\cap \pi( \omega_{i} \nu' ({\mathcal C}_i))\neq $\Leftrightarrow$ $\pi(\omega_{i}\nu ({\mathcal C}_i))= \pi(\omega_{i} \nu' ({\mathcal C}_i))$
\emptyset$
$\Leftrightarrow$ $\gamma \omega_{i} \nu ({\mathcal C}_{i}))= \omega_{i} \nu' ({\mathcal C}_{i})$, for $\gamma \in {\mathcal F}$
$\Leftrightarrow$ $\nu'^{-1} \nu
({\mathcal C}_{i}) = {\mathcal C}_{i}$
$\Leftrightarrow$ $\theta (\nu'^{-1}\nu) \in G_{\ell_i} $
$\Leftrightarrow$ $\nu'^{-1}\nu \in \theta^{-1}(G_{\ell_i}) $
$\Leftrightarrow$ $ \theta(\nu'^{-1})\theta (\nu ) \in \theta (\omega \mathcal{K} \omega^{-1})$.
--------------------------------------------------------------------------------------- ------------------- ----------------------------------------------------------------------------------------------------------------
$$
The first equivalence follows from Lemma \[equal o disjoint\], the third is a consequence of the normality of ${\mathcal F}$ in $\mathcal{K}$; the remainder are rather clear. Thus, each $i
\in I(g)$ produces $$[\theta^{-1}({\mathcal N}_G\langle g \rangle ):\theta^{-1}(G_{\ell_i})]
=\displaystyle{\frac{|{\mathcal N}_G\langle g \rangle |}{|G_i|} =
\frac{|{\mathcal N}_G\langle g \rangle |}{n_i}
}$$ connected components of the locus of fixed points of $g$.
Finally, in order to get the desired formula, we need to prove that $\pi( \omega_{i}({\mathcal C}_{i})) \cap \pi ( \omega_{j}({\mathcal C}_{j})) = \emptyset$, if $i, j \in I(g)$ with $i \neq j$. In fact, otherwise (by Lemma \[equal o disjoint\]) if they intersect, then necessarily $\pi(\omega_{i} ( {\mathcal C}_{i}) ) = \pi (\omega_{j}({\mathcal C}_{j}))$; so for arbitrary $c_i \in {\mathcal C}_i$ we have $\gamma \omega_{i} (c_i)= \omega_{j} (c_j)$ for some $c_j \in {\mathcal C}_j$ and some $\gamma \in {\mathcal F}$. Therefore $\omega_{j}^{-1} \gamma \omega_{i} ({\mathcal C}_{i})={\mathcal C}_{j}$. In other words, there is an element $\eta \in \mathcal{K}$ so that $\eta \kappa_{i}^{t_{i}} \eta^{-1}$ and $\kappa_{j}^{t_{j}}$ have the same set of fixed points, contradicting the definition of the set $I(g)$.
It follows from the proof of the above Theorem the following upper bound.
\[Macbeath-Kleinian inequality\] Let $\mathcal{K}, {\mathcal F}, G, \theta, \pi$, $\kappa_{i}$ and $m_{i}$ be as in Theorem $\ref{Macbeath-Kleinian}$. Then the number of connected components of the set of fixed points of $g \in G$ does not exceed $$|{\mathcal N}_G\langle g \rangle | \sum_{j \in J(g)} {1}/{m_{j}} $$
Indeed $|I(g)|\leq |J(g)|$ and $m_i \leq n_i$.
\[siegel\] Note, from the proof, that Theorem \[Macbeath-Kleinian\] still valid for $G$ being a finite group of isometries of a Riemannian manifold $M$ under the assumption that the universal Riemannian cover $\widetilde M$ of $M$ has the property that its finite order isometries have non-empty and connected set of fixed points. Examples of these situations are when $\widetilde M$ is the Teichmüller space ${\mathcal T}_{g}$ of genus $g \geq 1$ Riemann surfaces or the Siegel space ${\mathfrak H}_{g}$ parametrizing principally polarized abelian varieties.
Comments concerning dimension two
=================================
As already mentioned in the introduction, due to Nielsen’s realization theorem, if $G$ is a finite group of self-homeomorphism of a closed orientation surface $X_g$, then there is a Riemann surface structure on $X_{g}$ so that $G$ becomes it group of conformal automorphisms and so actually formulas we recall here for conformal automorphisms holds for periodic self-homeomorphisms. Furthermore, due to a better understanding and description of discrete cocompact groups of isometries of the hyperbolic plane, the formulas in two dimensional case have a more explicit character. As the locus ${\rm Fix}(\kappa_i)$ of any canonical elliptic generator $x_{i}$ of a Fuchsian group, is a single point $p_i$ and $G_{\{p_i\}}=\langle x_i \rangle$, Theorem \[Macbeath-Kleinian\] reduces to Macbeath’s counting formula in [@Mcb].
\[Macbeath-Riemann\] Let ${\mathcal K}$ a finitely generated discrete group of isometries of the hyperbolic plane $\mathcal{H}^{2}$ containing a Fuchsian group ${\mathcal F}$ as a finite index normal subgroup . Let $G={\mathcal K}/{\mathcal F}$ be the group of orientation preserving automorphisms of the Riemann surface $X=\mathcal{H}^{2}/{\mathcal F}$. Let $x_1, \ldots, x_r$ be the set of canonical elliptic generators of ${\mathcal K}$ of orders $m_1,\ldots,m_r$ respectively. Denote by $\theta:{\mathcal K} \to G$ the canonical projection. Then the number of points of $X$ fixed by $g\in G$ is given by the formula $$|{\mathcal N}_G(\langle g \rangle)|\sum 1/m_i,$$ where ${\mathcal N}$ stands for the normalizer and the sum is taken over those $i$ for which $g$ is conjugate to a power of the image $\theta(x_i)$. In particular the number of fixed points of $g$ is finite.
An anti-holomorphic automorphism of a compact Riemann surface of genus $g$, with fixed points, must be an involution; its locus of fixed points consist of $s \in \{1,\ldots,g+1\}$ disjoint sets, each of which is homeomorphic to a circle (ovals) by the well known result of Harnack. A canonical elliptic generator inducing an anti-holomorphic automorphisms (with fixed points) is a reflection, which is determined by its axis. In this way, we see that $G_{\ell}= \theta( {C} (\Lambda, c_i)))$ and therefore Theorem \[Macbeath-Kleinian\] reduces to the main result from [@G1].
\[centralizer\] Let ${\mathcal K}$ be an NEC-group and let $\theta:{\mathcal K} \to G$ be an epimorphism defing an action of $G$ on a compact Riemann surface $X$ as a group of conformal and anticonformal automorphisms. Then a symmetry $\sigma$ with fixed points is conjugate to $\theta (c)$ for some canonical reflection $c$ of ${\mathcal K}$ and it has $$\sum \; [\, {C} (G, \theta (c_i) ) :
\theta ({C} ({\mathcal K}, c_i)) \,]$$ ovals, where $C$ stands for thre centralizer, $c_i$ run over nonconjugate canonical reflections of ${\mathcal K}$, whose images under $\theta$ belongs to the orbit of $\sigma$ in $G$.
An algebraic structure of the centralizers of reflections in an NEC-group was found by Singerman in his thesis nearly fifty years ago and published in [@S]. There is a simple method, based on the geometry of the hyperbolic plane, to find explicit formulas for them as described in [@G3]. Similarly, effective formulas are also known for periodic self-homeomorphisms of non-orientable or bordered compact surfaces [@GG; @G2].
A couple of examples in hyperbolic $3$-dimensional case
=======================================================
We shall give two examples, for $n=2$, i.e. $3$-dimensional hyperbolic world, to see how our formula works in practice.
Let $m,k \geq 3$ be integers. A [*generalized Fermat manifold of type $(m,k)$*]{} is a compact hyperbolic $3$-manifold $N$ admitting a group $H \cong {\mathbb Z}_{m}^{k}$ of isometries so that the hyperbolic orbifold $M/G$ is homeomorphic (as orbifolds) to the orbifold ${\mathcal O}$ whose underlying space is the unit $3$-dimensional sphere $\mathcal{S}^{3}$ and the conical locus is given by $k$ disjoint loops(each one of index $m$) as shown in Figure \[ref:figure2\] (for the case $k=10$). In this case, the group $H$ is called a [*generalized Fermat group of type $(m,k)$*]{} and the pair $(N,H)$ a [*generalized Fermat pair of type $(m,k)$*]{}. By Mostow’s rigidity theorem, up to isometry, there is only one generalized Fermat pair of type $(m,k)$. As the $3$-orbifold ${\mathcal O}$ is closed, Haken and homotopically atoroidal, it has a hyperbolic structure [@BMP; @HKM], that is, there is Kleinian group $\mathcal{K}$ for which ${\mathcal O}=\mathcal{H}^{3}/\mathcal{K}$. We have that $\mathcal{K}$ is generated by $x_{1},\ldots ,x_{k}$ subject to the relations: $$x_{1}^{m}=\ldots =x_{k}^{m}=1,
x_{i}^{}x_{i+1}^{-1}x_{i}^{-1}x^{}_{i+1}= x_{i+1}^{}x_{i+2}^{-1}x_{i+1}^{-1}x^{}_{i+2},$$ where $i$ are taken modulo $k$ (see Figure \[ref:figure4\]). The collection $x_{1},\ldots ,x_{k}$ is a elliptic complete system of $\mathcal{K}$ and the derived subgroup $\mathcal{K}'$ of $\mathcal{K}$ is torsion free [@HM]. So $M=\mathcal{H}^{3}/\mathcal{K}'$ is a closed hyperbolic $3$-manifold with abelian group $G=\mathcal{K}/\mathcal{K}' \cong {\mathbb Z}_{m}^{k}$ of automorphisms.
Let us now consider the canonical projection $\theta:\mathcal{K} \to G$ and set $a_{i}=\theta(x_{i})$. By Theorem \[Macbeath-Kleinian\], the number of connected components of fixed points of each $a_{i}$ is exactly $m^{k-1}$. In fact, in this example we have that ${\mathcal N}_{G}\langle a_{i} \rangle
=G$, so $|{\mathcal N}_{G}\langle a_{i} \rangle |=m^{k}$, and ${\mathcal N}_{\mathcal{K}}\langle x_{i} \rangle =\langle x_{i} \rangle$.
It is expected that (in the generic situation) the generalized Fermat group is unique. Two-dimensional generalized Fermat manifolds (called generalized Fermat curves) have been considered in [@GenFerCur; @AutCurFerGen].
![$k=10$[]{data-label="ref:figure2"}](figure2.eps){width="4.7cm"}
![[]{data-label="ref:figure4"}](figure4.eps){width="4.7cm"}
An [*extended Schottky group of rank $g$*]{} is an extended Kleinian group whose canonical subgroup of orientation preserving isometries is a Schottky group of rank $g$. We proceed to construct one of these groups (of rank $g=5$) below. Choose three pairwise disjoint circles on the complex plane, all of them bounding a common $3$-connected region. For each of these circles, we take either a reflection or an imaginary reflection that permutes both discs bounded by such a circle. Let us denote these transformations by ${\kappa}_{1}$, ${\kappa}_{2}$ and ${\kappa}_{3}$ and let $\mathcal{K}$ be the extended Kleinian group generated by them. We have that $\mathcal{K}$ is isomorphic to the free product of three copies of ${\mathbb Z}_{2}$, ${\mathcal N}_{\mathcal{K}}\langle {\kappa}_{i}\rangle =\langle {\kappa}_{i} \rangle$ and $\{{\kappa}_{1}, {\kappa}_{2}, {\kappa}_{3}\}$ is a elliptic complete system of it. Consider the surjective homomorphism $\theta:\mathcal{K} \to G= {\mathbb Z}_{2}^{3} = \langle a_{1},a_{2},a_{3}\rangle $ defined by $\theta({\kappa}_{i})=a_{i}$, for $i=1,2,3$. Then, the Kleinian group $${\mathcal F}=\ker\theta=\langle \! \langle ({\kappa}_{2}{\kappa}_{1})^{2},
({\kappa}_{2}{\kappa}_{3})^{2}, ({\kappa}_{1}{\kappa}_{3})^{2}\rangle \! \rangle,$$ were the last stands for the normal closure, is a Schottky group of rank $5$. So $M=\mathcal{H}^{3}/{\mathcal F}$ is homeomorphic to the interior of a handlebody of genus $5$ admitting three symmetries $a_{1}$, $a_{2}$ and $a_{3}$, each one of order two. Since $G$ is abelian, we have that $|{\mathcal N}_{G}\langle a_{i} \rangle |=8$ and $|\theta({\mathcal N}_{\mathcal{K}}\langle {\kappa}_{i}
\rangle |=|\langle a_{i} \rangle|=2$. It follows from the counting formula of Theorem \[Macbeath-Kleinian\] that each $a_{i}$ has exactly $4$ connected components of its set of fixed points; all of them being either isolated points or discs.
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[^1]: G. Gromadzki was supported by Polish National Sciences Center by the grant NCN 2015/17/B/ST1/03235, R. A. Hidalgo was supported by Projects FONDECYT 1150003 and Anillo ACT 1415 PIA-CONICYT
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---
abstract: 'The unoccupied states of complex materials are difficult to measure, yet play a key role in determining their properties. We propose a technique that can measure the unoccupied states, called time-resolved Compton scattering, which measures the time-dependent momentum distribution (TDMD). Using a non-equilibrium Keldysh formalism, we study the TDMD for electrons coupled to a lattice in a pump-probe setup. We find a direct relation between temporal oscillations in the TDMD and the dispersion of the underlying unoccupied states, suggesting that both can be measured by time-resolved Compton scattering. We demonstrate the experimental feasibility by applying the method to a model of MgB$_2$ with realistic material parameters.'
author:
- 'A. F. Kemper'
- 'M. Sentef'
- 'B. Moritz'
- 'C. C. Kao'
- 'Z. X. Shen'
- 'J. K. Freericks'
- 'T. P. Devereaux'
bibliography:
- 'wigner.bib'
- 'timedomain.bib'
title: Mapping of the unoccupied states and relevant bosonic modes via the time dependent momentum distribution
---
introduction
============
In understanding the emergent properties of complex materials, it is insufficient to limit ones’ study solely to the occupied electronic states. The unoccupied states play an important role in determining, for example, the nature of the gap in charge- and spin-density waves, the absorption properties of semiconductors, or the magnetic properties of Mott systems. For example, without knowledge of the unoccupied states, it is difficult to measure both the size as well as the ordering wave vector $\vec q$ of a gap in the electronic spectrum. Unfortunately, the most direct experimental measurements of momentum-resolved states in quantum materials, angle-resolved photoemission spectroscopy (ARPES), only determines the occupied states. This leaves the alternative methods of inverse photoemission spectroscopy, which lacks the signal strength to measure with sufficient resolution, or indirectly inferring the unoccupied dispersion from two-particle quantities such as optical spectroscopy. Pump-probe spectroscopy, and in particular time-resolved (tr-) ARPES, can measure part of the unoccupied states, but is limited by the inherent competition between temporal and energy resolution.
Here we consider a different quantity, where one sacrifices the energy information in favor of high time resolution, while maintaining high momentum resolution. We propose to measure the (gauge-invariant) time-dependent momentum distribution (TDMD) of quasiparticles, defined by $\langle c_{\mathbf{k}}^\dagger(t) c_{\mathbf{k}}(t)\rangle \equiv n_{\mathbf{k}}(t)$, which is similar to the so-called Wigner distribution.[@g_mahan] It contains a rich amount of information about the behavior of pumped quasiparticles while avoiding some of the resolution-related complications of other measurements.
The TDMD is commonly measured via time-of-flight absorption images in cold atomic gases, where it was recently used to map out the Fermi surface of a gas of $^{40}$K atoms.[@t_drake_12] In the context of condensed matter systems, we propose two new types of measurements to access the TDMD. First, we propose to extend the technique of Compton scattering for solids into the time domain[@[Time-resolved; @Compton; @scattering; @in; @1D; @was; @previously; @studied; @by; @]r_wagner_10]. Compton scattering has a long history as a probe of the electron momentum distribution for solids in equilibrium [@m_cooper_99; @a_bansil_01; @y_sakurai_11]. With the advent of x-ray free-electron laser sources, the photon energy of the ultrashort x-ray probe pulses has been extended into the hard x-ray regime, where the fractional Compton cross-section becomes appreciable. With the advent of a hard x-ray free electron laser at the Linac Coherent Light Source, the photon energy will extend the energy of the x-ray free electron laser to beyond 20 keV. There is also significant progress in crystal optics at these energies, such that time-resolved high resolution Compton scattering with momentum resolution of a few percent of the Brillouin zone will be feasible. Additionally, recent developments in photonics suggest the future availability of ultrashort pulses ranging from x-rays to gamma rays[@k_taphuoc_12]. Time-resolved Compton scattering has the advantage of using high-momentum photons, which can readily access the Brillouin zone edges in all directions. Secondly, one can access the TDMD with tr-ARPES by using similar time delays and probe pulse widths and integrating the tr-ARPES signal over energy (assuming the tr-ARPES signal comes from a single band near the Fermi level).
{width="\textwidth"}
We illustrate the use of the TDMD by demonstrating how it can be used to map out the unoccupied dispersion in a pump-probe setup on a model system with electron-lattice interactions appropriate to a large class of complex materials. The TDMD exhibits two ubiquitous phenomena both due to electron-phonon interactions. First, the phonons dissipate the energy delivered to the electronic system by the pump. Second, after pumping, oscillations related to the phonon frequencies (and independent of the initial pump parameters) are commonly observed in the resulting spectra measured by the probe. [@f_schmitt_08; @l_perfetti_08; @l_rettig_10; @k_kim_12] We will show that both of these features are clearly seen in the TDMD, and that the oscillation frequencies can be analyzed to extract the underlying dispersion, in particular for the unoccupied states. Finally, we will demonstrate the feasibility of this technique by studying a model band structure for MgB$_2$.
The paper shall proceed as follows. In Sec. \[sec:method\], we outline our method for calculations of two-time Green’s functions on the Keldysh contour. In Sec. \[sec:results\], we present our results for the TDMD of a model system, and in Sec. \[sec:mgb2\], we apply the technique to the case of MgB$_2$ to demonstrate realistic capabilities of our method to map unoccupied states and their coupling to phonon spectra. A summary is presented in Sec. \[sec:summary\].
Method {#sec:method}
======
We study a two-dimensional system of electrons coupled to a bath of non-dispersive phonons with frequency $\Omega$ via a constant coupling $g$ (known as the Holstein model), [@t_holstein_59a; @*t_holstein_59b]. $$\begin{aligned}
\mathcal H = \sum_{{\mathbf{k}}\sigma} (\epsilon({\mathbf{k}}) - \mu) c^\dagger_{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma}+ \sum_{\mathbf{q}}\Omega b^\dagger_{\mathbf{q}}b_{\mathbf{q}}\nonumber\\
+ g \sum_{{\mathbf{k}}{\mathbf{q}}\sigma} c^\dagger_{{\mathbf{k}}+{\mathbf{q}},\sigma} c_{{\mathbf{k}},\sigma} \left( b_{\mathbf{q}}+ b^\dagger_{-{\mathbf{q}}} \right)\end{aligned}$$ where $c_{\mathbf{k}}$ ($b_{\mathbf{q}}$) annihilates an electron (phonon) of momentum ${\mathbf{k}}$ (${\mathbf{q}}$).
While the TDMD is useful independent of the underlying model, here we use a band structure motivated by the transition-metal oxides: a 2D tight-binding $\epsilon({\mathbf{k}})$ with (next) nearest-neighbor hoppings $V_{nn}$ and $V_{nnn} = 0.3 V_{nn}$, and $\mu=-1.02V_{nn}$. We use the conventions $\hbar = e = c = 1$. Below, we will report timescales in units of a characteristic phonon timescale $\tau_p = 1/\Omega$. For example, a phonon with an energy of $\Omega=10$ meV has $\tau_p\approx 0.4$ ps. Similarly, there is a characteristic electron timescale $\tau_e = 1/V_{nn} \approx 16$ fs for $V_{nn}=250$ meV. Although in this work we will explicitly examine higher energy phonons for numerical stability reasons, the results below occur due to the relevant energy scales in the problem, and will simply be rescaled to lower phonon energies.
To describe the non-equilibrium pump-probe process, we propagate the system on the Kadanoff-Baym-Keldysh (Keldysh) contour[@kadanoff_baym; @*l_keldysh_64], which has been described in detail elsewhere [@g_mahan; @a_jauho_84; @*j_davies_88; @v_turkowski_05; @*j_freericks_08]. The non-equilibrium Keldysh formalism has been successfully used to describe the time evolution of correlated electrons within dynamical mean-field theory [@m_eckstein_08; @*m_eckstein_09; @*m_eckstein_09b; @b_moritz_10; @m_eckstein_10; @b_moritz_11; @*b_moritz_12]. The electric field is included through the standard Peierls’ substitution ${\mathbf{k}}\rightarrow {\mathbf{k}}- {\mathbf{A}}(t)$, where the vector potential ${\mathbf{A}}$ is related to the applied electric field $\mathbf{F}$ via $-\partial{\mathbf{A}}(t)/\partial t = \mathbf{F}$ and we work in the Hamiltonian gauge.
Within the Migdal limit, we consider the renormalization of the electron Green’s function by a single phonon emission/absorption and ignore the phonon self-energy (as is done in conventional Migdal-Eliashberg theory). In this limit, we solve the Dyson equation $$\begin{aligned}
G_{\mathbf{k}}(t,t') = G^0_{\mathbf{k}}(t,t') + \int_\mathcal{C} dt_1 dt_2 G^0_{\mathbf{k}}(t,t_1) \Sigma(t_1,t_2) G_{\mathbf{k}}(t_2,t')\end{aligned}$$ with the self-energy $\Sigma(t,t') = i g^2 \sum_{\mathbf{k}}D^0(t,t') G^0_{{\mathbf{k}}}(t,t')$, where $G^0_{\mathbf{k}}(t,t')$ and $D^0(t,t')$ are the non-interacting electron and phonon Green’s functions, respectively [@[For; @a; @complete; @description; @of; @the; @non-interacting; @Green's; @functions; @see; @e.g.; @]g_mahan]. The Dyson equation can be solved by recasting it as a matrix equation[@j_freericks_08] or by decomposing into Volterra-type equations through the Langreth rules[@r_van_leeuwen_05], which is the approach used here.
Results {#sec:results}
=======
Figure \[fig:wigner2d\] shows the gauge-invariant TDMD $n_{\mathbf{k}}(t)=iG_{{\mathbf{k}}+\vec A(t)}^<(t,t'\rightarrow t)$ for $g = \sqrt{0.02}V_{nn}$, $\Omega = 0.4V_{nn}$ and at an initial temperature $T=0.04 V_{nn}$. Fig. \[fig:wigner2d\] (a) shows $n_{\mathbf{k}}(t)$ at times far before the pump, where the system is fully described by the equilibrium problem. To clearly show the changes for the small pump fluences considered ($F_{max} \le 4V_{nn}$), Fig. \[fig:wigner2d\] (b) shows the change in the TDMD from equilibrium: $\delta n_{\mathbf{k}}(t) \equiv n_{\mathbf{k}}(t) - n_{\mathbf{k}}(t\rightarrow -\infty)$.
We apply a pulse centered at time $t=0$ in the $(11)$ (diagonal) direction of the form $A(t) = \left(F_\mathrm{max}/\omega_A\right) \exp(-t^2/(2\sigma^2)) \sin(\omega_A t)$, with a maximum field strength of $F_\mathrm{max}=4V_{nn}$, a frequency $\omega_A=2V_{nn}$ and $\sigma \approx 0.4 \tau_p$. At the time shown in Fig. \[fig:wigner2d\] (b), the pump has passed, and the system is relaxing towards equilibrium. The $\mathcal C_4$ symmetry of $n_{\mathbf{k}}(t)$ in equilibrium is broken by the field (which points in the $(11)$ direction as shown by the black arrow), and this is reflected in the transient change. In fact, this can be used to emphasize particular regions of interest in the Brillouin zone. By rotating the field, the regions exhibiting the largest change in $n_{\mathbf{k}}(t)$ will shift with the field direction, providing access to new regions by emphasizing the experimental signal near different momenta.
![a) (Color online) Change in the TDMD $n_{\mathbf{k}}(t)$ from the equilibrium value (at $t=0$) for the points in the Brillouin zone indicated in b). Symbols are calculated data points (for clarity, not all are shown), solid lines are fits to an exponential decay using a fixed relaxation rate $1/\tau_{\mathbf{k}}= -2\textrm{Im}\ \Sigma(\omega=\epsilon_{\mathbf{k}})$. c) Real and imaginary parts of the equilibrium self-energy $\Sigma(\omega)$ for the parameters in a) ($g=\sqrt{0.02}V_{nn}$, $T=0.04V_{nn}$, $\Omega=0.4V_{nn}$).[]{data-label="fig:decayfits"}](fig2.pdf){width="\columnwidth"}
First, we consider the anisotropic redistribution of quasiparticles as a function of time. In Figure \[fig:decayfits\] (a), we show $\delta n_{\mathbf{k}}(t)$ at the momenta indicated in the inset; the blue (black) momenta have the same energy $\epsilon_{\mathbf{k}}$ but due to the $\mathcal C_4$ symmetry breaking have different time traces (Fig. \[fig:decayfits\] (b)). From the figure, one can clearly see that there is a difference in the overall magnitude of the change at the equivalent momenta (with the same color). The solid lines are fits to the amplitude of a simple decaying exponential, where we have taken the decay constant at a momentum ${\mathbf{k}}$ from the on-shell *equilibrium* self-energy $1/\tau_{\mathbf{k}}(\omega)=-2\textrm{Im}\ \Sigma(\omega=\epsilon_{\mathbf{k}})$ shown in Fig. \[fig:decayfits\] (c), as suggested by a recent study[@m_sentef_12]. The remarkable agreement with the calculated data points confirms that it is indeed the equilibrium self-energy that controls the decay rate. Since the equilibrium self-energy is $\mathcal C_4$ symmetric, the anisotropic redistribution in $\delta n_{\mathbf{k}}(t)$ is caused by the $\mathcal C_4$ symmetry breaking of the applied pump field.
![a) (Color online) Oscillatory part of $\delta n_{\mathbf{k}}(t)$ for ${\mathbf{k}}$ along (11) at the energies $\epsilon_{\mathbf{k}}$ indicated (offset for clarity). b) Fourier transform of a). Vertical lines indicate the frequencies corresponding to the strong oscillations on the unoccupied side, $\omega = \epsilon_{\mathbf{k}}+ \Omega$ and $\omega = \epsilon_{\mathbf{k}}-W_- + \Omega$. In both a) and b) the oscillations on the unoccupied side close to the Fermi level (blue curves) have been scaled down by 4 for visibility. (see text for details).[]{data-label="fig:oscillations"}](fig3.pdf){width="\columnwidth"}
On top of the decay are oscillations dominated by a few characteristic frequencies, as often observed in pump-probe experiments. [@f_schmitt_08; @l_perfetti_08; @l_rettig_10; @k_kim_12] These are clearly observed as rings in $\delta n_{\mathbf{k}}(t)$ shown in Fig. \[fig:wigner2d\], as well as oscillations in the time traces in Fig. \[fig:decayfits\] (a). We further emphasize these in Fig. \[fig:oscillations\], where we have subtracted the decaying exponential from several points along the Brillouin zone diagonal in Fig. \[fig:wigner2d\]. From Fig. \[fig:oscillations\] (a), one can observe two main features. First, the oscillation frequencies in the time traces depend strongly on momentum. Second, although there is only a single phonon frequency in our model, several oscillation frequencies can be observed. Figure \[fig:oscillations\] (b) shows the Fourier transform power spectrum of the oscillations in Fig. \[fig:oscillations\] (a). Each curve has readily visible maxima, in addition to some smaller structure across the frequency spectrum. For each energy (momentum) $\epsilon_{\mathbf{k}}$, the vertical lines indicate the energies $\omega = \epsilon_{\mathbf{k}}+ \Omega$ and $\omega=\epsilon_{\mathbf{k}}- W_- + \Omega$, where $W_-$ is the energy at the bottom of the band (at ${\mathbf{k}}= \Gamma$).
![(Color online) Fourier power spectrum of the oscillation frequencies along the $\Gamma$-M and M-X directions in the unoccupied Brillouin zone. The two strong peaks in the Fourier spectra follow the lines corresponding to phonon emission processes; a small tertiary line of peaks can be seen at lower frequencies corresponding to a phonon absorption process. The Fermi surface is shown in green on the base plane.[]{data-label="fig:ft_powerspec"}](fig4.pdf){width="0.9\columnwidth"}
The temporal dynamics of the Holstein model leads to this rich physics. The decay and the oscillations observed are momentum (energy) dependent, and their amplitudes depend on the electron-phonon coupling, as well as the applied field. Describing the full temporal dynamics of $n_{\mathbf{k}}(t)$ is complicated, yet we can gain some insight by focusing on the times where the pump is off, while the system is still displaced from its equilibrium configuration. The subsequent relaxation occurs from the new configuration through the equations of motion. In equilibrium, the TDMD is balanced by equal rates of in- and out-scattering; once the pump breaks this symmetry, the system relaxes back to the equilibrium according to the dynamics contained within the self-energy and Green’s functions. In addition to the relaxation, the TDMD oscillates at the band energy shifted by the characteristic energies in the problem (the phonon frequency and bandwidth) at which the self-energy is largest. This is shown analytically at $T=0$ in the appendix; the full calculation requires the numerical methods used here.
At low temperatures, the real part of the self-energy $\Sigma^\prime$ is large and peaked near four frequencies, $\omega = \pm \Omega$ and $\omega = W_\pm \pm \Omega$, where $W_\pm$ is the upper (lower) band edge (see Fig. \[fig:oscillations\](b)). The resulting $n_{\mathbf{k}}(t)$ oscillates at the band energy $\epsilon_{\mathbf{k}}$ shifted by those frequencies. In principle, all four frequencies should be observable in the power spectrum of Fig. \[fig:oscillations\]; however, two frequencies (with the positive sign above) correspond to phonon absorption, which is small at low temperatures, and thus the corresponding peaks in the power spectrum are reduced significantly. The two strong peaks expected for the unoccupied side are indicated in Fig. \[fig:oscillations\] by vertical lines, which agree well with the observed frequencies, up to a small shift in energy due to $\Sigma^\prime(\omega=0)$ which is normally absorbed into the chemical potential $\mu$. An interesting consequence of the expected oscillation frequencies is that if one considers a measurement near the Fermi level, the oscillations will appear to be strongest at just the phonon frequency (as in recent tr-ARPES experiments[@f_schmitt_08; @l_perfetti_08; @l_rettig_10; @k_kim_12]).
The simple dependence of the oscillation frequencies on the underlying band structure suggests a novel method to measure the electron dispersion by looking at the oscillations in the TDMD. The oscillations are most clearly visible in the unoccupied region of the Brillouin zone, which is exactly the region that traditional methods for measuring the dispersion have difficulty accessing. By tracking the oscillation frequencies as a function of momentum one can directly map out the dispersion. In Fig. \[fig:ft\_powerspec\], we plot the Fourier transform power spectrum in the unoccupied portion of the Brillouin zone along the zone diagonal and zone face. This shows most clearly the two strong peaks in the Fourier transform associated with phonon emission, although a weaker line corresponding to phonon absorption is also visible below the two main lines. All three lines, up to the constant shifts from $\Sigma^\prime(0)$,$W_\pm$ and $\Omega$, follow the unoccupied dispersion. Thus, from the maxima in the power spectrum the unoccupied dispersion can be directly measured.
The high temporal resolution afforded by the neglect of any frequency information (in contrast with tr-ARPES) allows these oscillations to be clearly resolved. Furthermore, the oscillations are strongest in the direction of the applied field. This gives the method an additional degree of freedom, where the field direction can be used to select the momentum cuts of interest. Measuring the TDMD thus provides complementary information to that gained from other time-resolved experiments. Additionally, we have shown that there is a direct connection between the oscillations in the TDMD and $\Sigma^\prime(\omega)$. This is the complement to the recent work by Sentef et al.[@m_sentef_12], where it was shown that the *imaginary* part of the self-energy $\Sigma^{\prime\prime}(\omega)$ can be inferred from decay rates in tr-ARPES.
A case study: MB$_2$ {#sec:mgb2}
====================
{width="38.00000%"} {width="61.00000%"}
The above analysis, applied to theoretical results, can be applied to experimental results equally well. Here, the experimental capabilities must be considered. The salient points to consider are as follows. First, the experimental time resolution will place an upper bound on the oscillation frequencies that can be resolved. This will limit the measurement capability to regions near the Fermi level; fortunately, this is generally where one wishes to measure, and thus the experimental resolution only limits the upper bound of resolvability. Second, materials have more than a single Holstein phonon mode, which should be taken into account. In this section, we repeat the calculations done previously, but for a band structure appropriate to MgB$_2$.[@k_szalowski_06] Furthermore, we couple the electronic system to a distribution of phonons through a model $\alpha^2F(\omega)$, which we have constructed based on experimentally broadened phonon frequencies calculated from first-principles.[@k_bohnen_01]
Fig. \[fig:ft\_powerspec\_mgb2\] shows the model $\alpha^2F(\omega)$ used in the calculations. For calculations of the TDMD, we focus on the $\pi$ bonding band of MgB$_2$, along the line $k_y=0, k_z=0$ of the band structure $$\begin{aligned}
\epsilon_\pi({\mathbf{k}}) =& e_\pi + 2t_\perp \cos k_z \nonumber \\
&- t^\prime_{||}\sqrt{1+4\cos \frac{k_y}{2}\left(\cos \frac{k_y}{2}+\cos\frac{k_x\sqrt{3}}{2}\right)},\end{aligned}$$ where $e_\pi=0.04$ eV, $t_\perp=0.92$ eV, and $t^\prime_{||}=1.60$ eV, and the Fermi level is set to 0.[@k_szalowski_06]
In principle, the redistribution of quasiparticles during the pump in a multi-band system is complex; however, here we focus entirely on the decay of the excitations after the pump, which are determined by the equilibrium, intra-band self-energy. Furthermore, our intent here is to include realistic material properties to show that this signal can, in fact, be measured. As such, we perform the calculations for the $\pi$ bonding band only. Since the self-energy is a priori unknown (unlike in the model calculation), we have fitted an exponential to the time traces. The fitted exponential is subtracted from the time traces, which are subsequently Fourier transformed. The Fourier transform power spectrum is shown as a false-color intensity map for a cut along the zone boundary in Fig. \[fig:ft\_powerspec\_mgb2\]. The red lines on the plot indicate the expected oscillation frequencies for a single mode at $\Omega=65$ meV, namely (the larger) $\omega_1 = |\epsilon({\mathbf{k}}) + \Omega|$ and (near the Fermi level) $\omega_2 = |\epsilon({\mathbf{k}}) + \Omega+W|$, where $W$ is the top of the band. Although a full phonon distribution is included through $\alpha^2F(\omega)$, the presence of a strongly coupled mode allows for the resolution of a single dispersive mode, although broadened. The agreement with the predicted frequencies and the observed ones indicates that our previous conclusions hold, even in the presence of multiple phonon modes. The observation that the power spectrum is dominated by the single broadened mode for a realistic phonon spectrum justifies the use of a single Holstein mode previously, as it captures the essential features seen here. The final, orange dashed line on the plot indicates the experimental resolution bound for a probe resolution of 10 fs.
The further experimental considerations involve scattering matrix elements, and the distinction between real and crystal momentum. These issues have been discussed in some detail (in equilibrium) previously, and the arguments presented there hold in non-equilibrium as well.[@w_schulke_96; @m_cooper_99].\
Summary {#sec:summary}
=======
We have shown the use of the time-dependent momentum distribution as a novel concept in the understanding of time-resolved spectroscopy, and non-equilibrium phenomena in general. The decay rates and the oscillations in the time traces of the TDMD were shown to be directly related to the underlying equilibrium self-energy. This result is of importance in time-resolved spectroscopy, where it has long been assumed that the equilibrium properties can be studied in a time-resolved experiment. We further have shown that the TDMD can be directly measured by an experiment that only has momentum and time resolution, such as time-resolved Compton scattering and energy-integrated time-resolved ARPES. Experimentally, the width of the probe pulse limits the temporal resolution of the TDMD. However, in many systems of interest, the features one wishes to investigate lie near the Fermi level, both above and below. For example, in the high-Tc cuprates, several features are predicted to lie within this energy range.[@b_moritz_09] In other gapped systems, without the knowledge of the spectra above the Fermi level it is impossible to correctly assign the gap magnitude. Similar effects due to spin fluctuations, phonons, and other interesting collective and bosonic features lie at low energy, and are thus accessible by this technique.
Analytic calculation of oscillations
====================================
We can explain the oscillations in the TDMD by examining the equations of motion for the mixed and lesser Green’s functions at $T=0$ (where the state is fully described without a thermal average). Following the Langreth rules, we obtain the following Volterra-type equations[@r_van_leeuwen_05],
$$\begin{aligned}
\left[i\partial_t - h(t)\right]G_{\mathbf{p}}^{ri}(t,0) =& \int_0^t d\bar t\ \Sigma^R(t,\bar t) G_{\mathbf{p}}^{ri}(\bar t,\tau)
\label{eq:Gri}\\
G_{\mathbf{p}}^{ri}(0,0) =&\ i n_{\mathbf{p}}\nonumber \\
\left[i\partial_t - h(t)\right] G_{\mathbf{p}}^<(t,t') =& \int_0^t d\bar t\ \Sigma^R(t,\bar t) G_{\mathbf{p}}^<(\bar t,t')
+\int_0^{t'} d\bar t\ \Sigma^<(t,\bar t) G_{\mathbf{p}}^A(\bar t,t')
\label{eq:glesser}\\
G_{\mathbf{p}}^<(0,t') =& -G_{\mathbf{p}}^{ri}(t',0)^* \nonumber\end{aligned}$$
where $h(t) = \epsilon({\mathbf{k}}-{\mathbf{A}}(t))$, $\epsilon({\mathbf{k}})$ is the dispersion, and $n_{\mathbf{p}}$ is the Fermi function at energy $\epsilon_{\mathbf{p}}$. The superscripts $<$ and $ri$ denote the “lesser” and mixed components of the Green’s function and self-energy, respectively.
In the absence of a driving field or interactions, we obtain the known results for the mixed and lesser Green’s functions $$\begin{aligned}
G_{0{\mathbf{p}}}^{ri}(t,0) =& i n_{\mathbf{p}}e^{-i \epsilon_{\mathbf{p}}t}\\
G_{0{\mathbf{p}}}^<(t,t') =& i n_{\mathbf{p}}e^{-i\epsilon_{\mathbf{p}}(t-t')}\end{aligned}$$ We are interested in the oscillations of the TDMD after the pump is off, when the system is out of its equilibrium state. Since we are interested in the dynamics, in lieu of driving the system explicitly with a field, we turn on the interactions at an infinitesimal time after zero ($t=0^+$). By construction, this problem has the same dynamics as those of the relaxation after driving by a pump. This, coupled with the approximation of weak interactions (where we replace the full Green’s function on the right hand side with the bare one), allows us to obtain analytic formulae which show the oscillation dynamics. We shall consider the change of the equilibrium Green’s functions $\Delta G = G - G_0$. First, we solve equation of motion for $\Delta G^{ri}(t)$, which we shall need as an initial condition for the TDMD. $\Delta G^{ri}(t)$ has no initial condition beyond that for the bare Green’s function $G_0^{ri}(t)$. $$\begin{aligned}
\Delta G_{\mathbf{p}}^{ri}(t,0) =& i g^2 n_{\mathbf{p}}\sum_{\mathbf{k}}\bigg[
\frac{n_{\mathbf{k}}}{(\epsilon_{\mathbf{k}}- \Omega - \epsilon_{\mathbf{p}})^2} F(-\Omega) \nonumber \\
&+ \frac{1-n_{\mathbf{k}}}{(\epsilon_{\mathbf{k}}+ \Omega - \epsilon_{\mathbf{p}})^2} F(\Omega) \bigg] \\
\nonumber \\
F(\Omega) = e^{-i(\epsilon_{\mathbf{k}}+ \Omega)t} &+ e^{-i \epsilon_{\mathbf{p}}t} \big[i(\epsilon_{\mathbf{k}}+\Omega - \epsilon_{\mathbf{p}})t-1 \big] \nonumber\end{aligned}$$ We proceed by similarly solving the equation for the lesser component. The full expressions are complex, but can be significantly simplified by considering the case where $t' \rightarrow t$,
$$\begin{aligned}
\Delta G^<_{\mathbf{p}}(t,t') =& -G^{ri}(t)^* e^{-i\epsilon_{\mathbf{p}}t} + i g^2 n_{\mathbf{p}}\sum_{\mathbf{k}}\bigg[
\frac{n_{\mathbf{k}}}{(\epsilon_{\mathbf{k}}- \Omega - \epsilon_{\mathbf{p}})^2} F(-\Omega)
+ \frac{1-n_{\mathbf{k}}}{(\epsilon_{\mathbf{k}}+ \Omega - \epsilon_{\mathbf{p}})^2} F(\Omega)
- \frac{ 2\cos\left[ (\epsilon_{\mathbf{k}}- \Omega - \epsilon_{\mathbf{p}}) t\right] - 2}{(\epsilon_{\mathbf{k}}- \Omega + \epsilon_{\mathbf{p}})^2}
\bigg] \nonumber \\
=& 2 i g^2 n_{\mathbf{p}}\sum_{\mathbf{k}}(1-n_{\mathbf{k}}) \bigg[
\frac{ \cos\left[ (\epsilon_{\mathbf{k}}+ \Omega - \epsilon_{\mathbf{p}})t\right]-1}{(\epsilon_{\mathbf{k}}+ \Omega - \epsilon_{\mathbf{p}})^2}
-\frac{ \cos\left[ (\epsilon_{\mathbf{k}}- \Omega - \epsilon_{\mathbf{p}})t\right]-1}{(\epsilon_{\mathbf{k}}- \Omega - \epsilon_{\mathbf{p}})^2}
\bigg]\end{aligned}$$
Now, only the integral over the ${\mathbf{k}}$ states remains. We shall assume that the density of states is flat over the region we integrate over (and equal to $N_0$), and has band edges at $\pm W$. This results in integrals of the form $$\begin{aligned}
\int_0^W& dx\ \frac{ \cos\left[(x\pm\Omega - \epsilon_{\mathbf{p}})t\right]-1}{(x\pm\Omega-\epsilon_{\mathbf{p}})^2}=\nonumber\\
&\frac{\cos\left[(x\pm\Omega - \epsilon_{\mathbf{p}})t\right]-1}{x\pm\Omega - \epsilon_{\mathbf{p}}}
- t \mathcal{S}\left[(x\pm\Omega - \epsilon_{\mathbf{p}})t\right] \bigg|_0^W\end{aligned}$$ Here, $\mathcal{S}$ denotes the sine integral. Thus, the two terms each show two distinct oscillation frequencies, $\omega = \epsilon_{\mathbf{p}}\pm \Omega$ and $\omega = \epsilon_{\mathbf{p}}\pm (\Omega +W)$.
![Change in the TDMD after turning on interactions for an infinite band.[]{data-label="fig:deltank"}](figs1.pdf){width="\columnwidth"}
Figure \[fig:deltank\] shows the change in the TDMD as the interactions are turned on in an infinite band. Here, the oscillations occur at $\epsilon_{\mathbf{p}}\pm \Omega$, although those where $|\epsilon_{\mathbf{p}}- \Omega|$ is smallest dominate the signal.
In the course of this calculation, a number of approximations were made. Nevertheless, the oscillation frequencies agree with those observed from the full numerical simulations. The inclusion of the correct density of states, as well as the replacement of the bare Green’s functions will cause a small shift of the frequencies observed, and a distribution of frequencies centered around the bare one shown here.
|
---
abstract: 'Optimal sensor placement is an important yet unsolved problem in control theory. In biological organisms, genetic activity is often highly nonlinear, making it difficult to design libraries of promoters to act as reporters of the cell state. We make use of the Koopman observability gramian to develop an algorithm for optimal sensor (or reporter) placement for discrete time nonlinear dynamical systems to ease the difficulty of design of the promoter library. This ease is enabled due to the fact that the Koopman operator represents the evolution of a nonlinear system linearly by lifting the states to an infinite-dimensional space of observables. The Koopman framework ideally demands high temporal resolution, but data in biology are often sampled sparsely in time. Therefore we compute what we call the temporally *fine-grained* Koopman operator from the temporally *coarse-grained* Koopman operator, the latter of which is identified from the sparse data. The optimal placement of sensors then corresponds to maximizing the observability of the fine-grained system. We demonstrate the algorithm on a simulation example of a circadian oscillator.'
author:
- 'Aqib Hasnain, Nibodh Boddupalli, and Enoch Yeung[^1]'
bibliography:
- 'main.bib'
title: Optimal reporter placement in sparsely measured genetic networks using the Koopman operator
---
Introduction
============
Spectral methods have been increasingly popular in data-driven analysis of nonlinear dynamical systems. Recently, researchers working in Koopman operator theory have shown that it is possible to identify and learn the fundamental modes for a nonlinear dynamical system from data [@rowleySpectral2009; @proctor_brunton_kutz_2016]. This operator, originally defined nearly 100 years ago by Koopman [@koopman], is a linear infinite dimensional operator that fully describes the underlying nonlinear dynamical system. Identifying Koopman operators from data has become computationally tractable, largely due to advances integrating machine learning and deep learning to generate novel, efficient representations of observable subspaces for the Koopman operator [@yeung2017learning; @lusch2018deep].
In many high-dimensional nonlinear systems, typically it is not physically or economically feasible to measure every state with the resolution specified by a fine-grained temporal model. For example, the bacteria *E. coli* have approximately 4400 genes, making both spatially and temporally fine data collection nearly impossible. On one hand, high-coverage omics measurements provide a system-level view of all gene activity, but prohibitive costs and the laborious and destructive nature of sampling make it difficult to resolve dynamics at a high temporal resolution. On the other hand, fluorescently tagged genes can be measured at the second to minutes timescale, to profile bursty RNA dynamics and protein expression. Knowing which genes to tag with fluorescent markers is critical, since not every gene can be simultaneously tagged. This challenge motivates the need for algorithmic data-driven approaches which allow the user (e.g. biologists) to know *a priori* which genes should be sampled. Finally, is it possible to design a nonlinear observer that rather than measuring a single gene or a single node in the network, fuses the state of a select set of biomarker genes to report out an aggregate cellular state of the system? The fundamental question is how to use metrics for nonlinear observability to design observers or optimize sensor placement. Sinha et al. presented a systematic framework based on linear transfer operators for the optimal placement of sensors and actuators for control of nonequilibrium dynamics [@sinha2016operator].
Koopman operators have been used to characterize observability of a nonlinear system [@vaidya_2007; @surana_banaszuk_2016]. Yeung et al. formulated the Koopman gramian and showed they can be used to quantify controllability and observability and lend insight for the underlying nonlinear dynamical system [@yeung_liu_hodas_2018]. This recent development of the Koopman gramians can advance the imporant and unsolved problem of optimal sensor placement in control theory. The Koopman framework embeds nonlinear dynamics in a linear framework for optimal nonlinear estimation and control [@korda_mezic_2018; @abraham2017model; @arbabi2018data]. For sensor placement search spaces that are reasonable in size, there are model-based solutions using optimal experiment design [@boyd2004convex; @joshi2009sensor], information theoretic and Bayesian criteria [@caselton1984optimal; @krause2008near; @lindley1956measure; @sebastiani2000maximum; @paninski2005asymptotic]. There is a need to develop purely data-driven methods for determining optimal sensor placement. Manohar et al. explored optimized sparse sensor placement for signal reconstruction based on a tailored library of features extracted from training data [@manohar2017data]. In [@sharma2018transfer], Sharma et al. extended the transfer operator based approach for optimal sensor placement, providing a probabilistic metric to gauge coverage under uncertain conditions. Fontanini et al. presented a data driven sensor placement algorithm based on a dynamical systems approach, utilizing the Perron-Frobenius operator [@fontanini2016methodology]. Our framework provides a method to determine optimal sensor placement, even in the presence of noisy and temporally sparse data using Koopman operator theory.
In this paper, we develop an algorithm for optimizing sensor placement from sparsely sampled time-series data. We use the Koopman observability gramian, developed by Yeung et al. [@yeung_liu_hodas_2018], to maximize the observability of the underlying discrete time nonlinear dynamical system. Section \[sec:Koop\] introduces the Koopman operator formulation and Section \[sec:ObsGram\] introduces the notion of a Koopman observability gramian [@yeung_liu_hodas_2018]. In Section \[sec:ObsPlaceSparse\], we show how to compute the temporally fine-grained Koopman operator from the temporally coarse-grained Koopman operator, which is learned from data that are temporally sparse. In the case of noisy data, a closed form expression for the error in computing the temporally coarse-grained Koopman operator is derived. In Section \[sec:OptObs\], we present a novel algorithm for optimal sensor design and placement. Finally, the algorithm is illustrated with a simulation example.
Koopman Operator Formulation {#sec:Koop}
============================
We briefly introduce Koopman operator theory, as we will use it extensively for the sensor placement problem. Consider a discrete time open-loop nonlinear system of the form $$\begin{split}
x_{t+1} &=f(x_t)\\
y_t &=h(x_t)
\end{split}
\label{eq:sys}$$ with $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is analytic and $h$ $\in$ $\mathbb{R}^p$. The Koopman operator of (\[eq:sys\]), $\mathcal{K}$ : $\mathcal{F}$ $\rightarrow$ $\mathcal{F}$, is a linear operator that acts on observable functions $\psi (x_k)$ and propagates them forward in time as $$\psi (x_{t+1})=\mathcal{K}\psi (x_t).
\label{eq:KoopEq}$$ Here $\mathcal{F}$ is the space of observable functions that is invariant under the action of $\mathcal{K}$.
Given system (\[eq:sys\]), we suppose that $y_k=h(x_k)$ $\in$ $\mathcal{F}$ and that $h$ $\in$ span$\left\{\psi_1,\psi_2,...\right\}$.
Then the output $y_t$ can be expressed as $$y_t = h(x_t) = W_h\psi(x_t)
\label{eq:out}$$ where the output matrix $W_h$ $\in$ $\mathbb{R}^{p\times n_L}$, $n_L \leq \infty$. We make this strong assumption since the structure of $W_h$ will be manipulated to achieve optimal sensor placement.
Throughout the paper, we take observable functions which are state-inclusive, i.e. $$\psi (x) = (x,\varphi(x)) \label{eq:stateIncObs}$$ where $\varphi$ $\in$ $\mathbb{R}^{n_L-n}$ are continuous functions in $\mathcal{F}$.
Koopman Observability Gramian {#sec:ObsGram}
=============================
The observability matrix of the transformed system may be obtained by showing how the Koopman operator maps initial conditions $x_0$ to $y$ [@yeung_liu_hodas_2018]. Using equations (\[eq:KoopEq\]) and (\[eq:out\]), we have $$y_t = W_h \mathcal{K}^t \psi (x_0)$$ Therefore, $W_h \mathcal{K}^t :$ $\mathbb{R}^{n_L}$ $\rightarrow$ $\mathbb{R}^p$ is the transformation that maps $\psi (x_0)$ to $y_t$. Given an initial condition $\psi (x_0)$ $\in$ $\mathbb{R}^{n_L}$, the energy of the output $y_t$ is given by $$\begin{split}
\Vert y \Vert ^2 & = \sum_{n} <y_t,y_t> \\
& = \sum_{n} \psi (x_0)^\top (\mathcal{K}^t)^\top W_h^\top W_h \mathcal{K}^t \psi (x_0) \\
& = \sum_{n} \psi (x_0)^\top X_o^{\psi} \psi (x_0)
\label{eq:normysquared}
\end{split}$$ where $<\cdotp,\cdotp>$ represents the inner product and as can be seen in the last equality of (\[eq:normysquared\]), the Koopman observability gramian is defined as $$X_o^{\psi} = \sum_{t=0}^{\infty} (\mathcal{K}^t)^\top W_h^\top W_h \mathcal{K}^t$$ and is an $n_L \times n_L$ matrix. The observability gramian can be obtained as a solution of following matrix Lyapunov equation $${\cal K}^\top X_o^{\psi} {\cal K} - X_o^{\psi} = -W_h^\top W_h.$$ The Koopman observability gramian quantifies the observability of the function $\psi (x)$. More importantly, when $\psi (x)$ includes observable functions related to the local observability of the underlying nonlinear system (\[eq:sys\]), the Koopman observability gramian retains that information [@yeung_liu_hodas_2018].
Sensor Placement from Temporally Sparse Data {#sec:ObsPlaceSparse}
============================================
*Fine-grained* models from *coarse-grained* models
---------------------------------------------------
We consider the scenario where high-resolution measurements of all genes in a single cell are infrequently sampled. This is a common scenario when tracking the state of biological, cyber-physical, and social networks. Exhaustive measurement of every state in the system is expensive (and often manual) and thus can only be performed infrequently.
Consider the case where the system in (\[eq:sys\]) is a biomolecular reaction network evolving with unknown governing equations. The precise functional form and parameters of $f$ are typically considered unknown. In some settings, [*a priori*]{} knowledge of the biomolecular reaction network can be utilized to bootstrap the modeling problem [@yeung2014modeling; @yeung2017biophysical] . We consider a data-driven operator theoretic approach, using the method of Koopman briefly introduced in Section \[sec:Koop\].
The discrete time Koopman representation for the system (\[eq:sys\]) is $$\psi(x_{t+1}) = K\psi(x_t)$$ where the matrix $K \in \mathbb{R}^{n_L\times n_L}$ is a finite dimensional approximation of the exact Koopman operator [$\cal K$]{} and $\psi (x_k) \in \mathbb{R}^{n_L}$. We suppose that full-state measurements are made available for $x_{t}, x_{t+N}$, with enough biological replicates that the [*temporally coarse-grained*]{} (approximate) Koopman operator is identifiable via the optimization problem $$\min_{K_N} || \Psi(X_f) - K_N\Psi(X_p)||$$ where $$\begin{aligned}
\Psi(X_f) &\equiv \begin{bmatrix} \psi(x_{t+N}(\omega_1)) & \hdots & \psi(x_{t+N}(\omega_R)) \end{bmatrix}, \\ \Psi(X_p) & \equiv \begin{bmatrix} \psi(x_{t}(\omega_1)) & \hdots & \psi(x_{t}(\omega_R))\end{bmatrix}.
\end{aligned}$$ and $\omega_R$ represents the number of replicates. In the presence of sparse and noisy data, [@sinha_yeung_2019] showed that the Koopman learning problem can be formulated as a robust optimization problem, which is equivalent to a specific regularized learning problem in which the LASSO penalty parameter corresponds to the upper bound on the noise i.e. the maximum Frobenius norm of the noise. We will suppose, for simplicity of exposition of the technique, that an exact Koopman operator for the coarse-time step mapping $t$ to $t+N$ is either known or obtained directly from data satisfying $$\label{eq:coarseKoop}
\psi(x_{t+N}) = K_{N} \psi(x_t).$$ Because of linearity of the Koopman operator, we know that the [*temporally fine-grained*]{} Koopman operator $K$ satisfies $$\psi (x_{t+1}) = K \psi (x_t)$$ and most importantly, $$\label{eq:coarsetofine}
K = K_N^{1/N}.$$ When the Koopman observable function includes the state as an element, this relationship allows the recovery of the fine-grained governing equations for $f$ directly from a temporally coarse-grained Koopman operator (and the corresponding data). To see this, take the state-inclusive observable functions (\[eq:stateIncObs\]) and partition the Koopman equation accordingly as $$\begin{bmatrix}
x_{t+1} \\ \varphi(x_{t+1})
\end{bmatrix}
= \begin{bmatrix} K_{xx} & K_{x\varphi} \\ K_{\varphi x} & K_{\varphi \varphi} \end{bmatrix}\begin{bmatrix}
x_{t} \\ \varphi(x_{t})
\end{bmatrix}.$$ Since the Koopman operator satisfies $$K \psi(x_t) = \psi(f(x_t))$$ for each row, then in particular, the upper half of the Koopman equation satisfies $$x_{t+1} = K_{xx} x_t + K_{x\varphi}\varphi(x_t) = f(x_t).$$ This provides a powerful scheme for estimating the governing equations of a fine-grained time-evolving biological process from sparse or coarse-grained temporal measurements, so long as the coarse-grained time measurement is a product of regularly spaced intervals of time in the fine-grained representation. Again, since RNAseq and proteomic measurements often provide full-state measurements of a network, this in theory can provide sufficient information to recover the Koopman operator, even in the presence of noise [@sinha_yeung_2019]. The key insight and property leveraged is the linearity of the lifted Koopman representation. One would not be able to obtain the fine-grained dynamics of the governing equations from a coarse grained representation of the governing equations as it is generally not feasible to compute the $n^{\text{th}}$ root of a $n$-layered function composition. Specifically, note that the $N$-step map for the underlying governing dynamics of system (\[eq:coarseKoop\]) is given as $$x_{t+N} = f^{(n)}(x_t) = f \circ f \circ \hdots f (x_t) \equiv f_N(x_t).$$ Given an arbitrary nonlinear function $f_N(x_t)$ that is the $N^{\text{th}}$ composition of $f(x_t)$, there is no general way to obtain the underlying $f(x_t)$. However, by using the Koopman operator lifting framework, we can express the governing equations with linear coordinates, which allows us to consider computing the $N^{\text{th}}$ root to obtain the single-step map from the $N$-step map.
Although, in general, the matrix root always exists, we note that we may not always obtain the desired fine-grained $K$ from $K_N$ due to there being multiple solutions to matrix roots. Yue et al [@yue2016systems] showed that similarly the matrix logarithm raises a concept of system aliasing. They describe the scenario where there might be multiple fine-grained systems which give the same coarse-grained system. In the case that multiple fine-grained Koopman operators exist, our method can be applied to each operator. We can distinguish which operator is the “correct” operator by collecting a few data points at a fine-grained temporal resolution and evaluating the predictive accuracy of the fine-grained Koopman operator models.
In the presence of noise, we approximate the fine-grained discrete time Koopman operator $K$ from the coarse-grained discrete time Koopman operator $K_N$ as $$\begin{aligned}\label{eq:ctofwitherror}
\hat{K} = \hat{K}_N^{(1/N)}& = (K_N + \epsilon(x))^{1/N}\\
& = \sum_{k=0}^\infty \binom{1/N}{k} K_N^{(1/N -k)}\epsilon(x)^k \\
&= \hat{K}_N^{1/N}+\frac{1}{N}K_N^{(1/N - 1)} \epsilon (x) \\ & \qquad
+ \frac{\frac{1}{N}(\frac{1}{N}-1)}{2!}K_N^{(1/N-2)} \epsilon (x)^2 + ...
\end{aligned}$$ where the last equality follows from Newton’s generalization of the binomial theorem [@liu2010essence]. Here we assume that $\epsilon(x)$ is bounded as in [@johnson_yeung_2018] for all $x \in {\cal M}$ $\subseteq$ $\mathbb{R}^n$. A closed form expression of the error term $\epsilon (x)$ is found by noting that $$\epsilon (x) = \hat{K}_N - K_N.$$ Then we have $$\begin{aligned}
\epsilon (x) \Psi(X_p) & = (\hat{K}_N - K_N) \Psi(X_p)
& = \hat{\Psi}(X_f) - \Psi(X_f)
\end{aligned}$$ $$\epsilon (x) \Psi(X_p) \Psi(X_p)^\dagger = (\hat{\Psi}(X_f) - \Psi(X_f)) \Psi(X_p)^\dagger.$$ giving the closed form expression of the error as $$\epsilon (x) = (\hat{\Psi}(X_f) - \Psi(X_f)) \Psi(X_p)^\dagger.$$
Once we obtain the one-step Koopman operator, notice that the Koopman invariant subspace of observable functions is the same as the $N$-step operator. We suppose, mirroring the scenario presented with transcriptomic and proteomic measurements, that the state is measured completely, in this setting. The precise coverage of the entire transcriptome and proteome is often a subject of debate, but relative to the spatial sparsity of fluorescence based readout approaches, we shall assume for our purposes that the full state of the network is measured sparsely.
The state-output equations of the coarse-grained system can then be written as $$\begin{aligned}
x_{t+N} &= f(x_t) \\
y_t & = x_t
\end{aligned}$$ and thus the corresponding Koopman equation can be written as $$\label{eq:coarseKoopSys}
\begin{aligned}
\psi(x_{t+N})& = K_N \psi(x_t)\\
y_t &= P_x \psi(x_t)
\end{aligned}$$ where $$P_x = \begin{bmatrix} I_{n} & 0 \\ 0 & 0 \end{bmatrix}$$ is the projection matrix that extracts the state observable from the vector observable $\psi(x_t).$
Fine-Grained Sensor Placement via Optimal Observability {#sec:OptObs}
-------------------------------------------------------
Often times, it is not physically or economically feasible to measure every state with the resolution specified by a fine-grained temporal model. We seek to develop an algorithm for identifying the design and placement of reporters that maximizes the observability of the underlying nonlinear system, as well as the corresponding Koopman representation. For this task, we find it convenient to pose this problem using the Koopman gramian as defined in Section \[sec:ObsGram\]. Specifically, we seek to construct an output observer for the fine-grained dynamical system (\[eq:sys\]) given full-state sparse temporal measurements at $t$, $t+N$, $t + jN$ in sufficient frequency to recover the temporally coarse-grained Koopman operator $K_N,$ so that it is possible to compute the fine-grained Koopman operator $K = K_N^{(1/N)}$. We suppose that the corresponding Koopman representation with output equation is thus written as $$\label{eq:fineKoopSys}
\begin{aligned}
\psi(x_{t+1}) &= K \psi(x_t) \\
y_t &= W_h \psi(x_t).
\end{aligned}$$ We seek to maximize the output energy $||y_t||^2$ for an initial condition $x_0$ at a time instant $t$ i.e. solve the nonlinear optimization problem $$\label{eq:NPoptProb}
\max_{h(x) \in {\cal L}^2\left({\cal M}\right)} ||y(t_j)||^2$$ for all initial conditions $x_0$ with $||x_0|| \leq 1$. This is an optimization problem of a nonlinear function space (i.e. an uncountably infinite dimensional space) and is generally intractable. However, if we were to find a basis for $h(x)$, we could express the problem in terms of a linear combination of the basis functions, which would yield a convex formulation of the problem. This is precisely what we can do using the spectral properties of the Koopman operator representation.
Following the formulation given in Section \[sec:ObsGram\], the system in (\[eq:fineKoopSys\]) has Koopman observability gramian $$\label{eq:ObsGramFine}
X_{o,f}^\psi = \sum_{j=0}^{t_N} (K^{j})^\top W_h^\top W_h (K^{j})$$ where the subscript $f$ is used to distinguish the fine-grained system from coarse-grained.
We want to identify the optimal sensor placement that informs the design of optimal observers. Utilizing the Koopman observability gramian, $X_{o,f}$, as defined in (\[eq:ObsGramFine\]), the output energy of system (\[eq:fineKoopSys\]) is written as $$\label{eq:outputNorm}
||y_{t_N}||^2 = \sum_{j=0}^{t_N} \psi (x_0)^\top (K^{j})^\top W_h^\top W_h (K^{j}) \psi (x_0).$$ Our goal is to now maximize the output energy (\[eq:outputNorm\]) of the lifted system up at time $t$ with the output matrix $W_h$ as the decision variable. If the output energy of the lifted system is maximized, then by proxy the output energy of the original nonlinear system is maximized.
For the purposes of this paper, we will suppose that we construct an observable function basis that results in a diagonalizable Koopman operator. The subsequent presentation can be generalized for Koopman operators that only admit a Jordan decomposition, but for simplicity of exposition, we consider the case of the diagonalizable Koopman operator.
We suppose that $\psi(x)$ and $K$ are provided or trained during the learning process to admit a diagonalizable $K$.
Thus, an eigendecomposition of $K$ gives $$KV = V\Lambda$$ where $V$ is an $n_L \times n_L$ matrix of eigenvectors. The $n_L \times n_L$ matrix $\Lambda$ is a diagonal matrix whose components are the eigenvalues $\lambda$ of the Koopman operator, $K$. The eigenfunctions of $K$ are then written as $$\phi (x_0) = V^{-1}\psi(x_0).$$ where $\phi \in \mathbb{R}^{n_L}$. Since (\[eq:outputNorm\]) has a symmetric form, let us deal with the right half of this equation. We have that $$\begin{aligned}
W_hK^j\psi(x_0) &= W_h V\Lambda^j V^{-1}\psi(x_0) \\
&= W_h V\Lambda^j V^{-1} V \phi (x_0) \\
&= W_h V\Lambda^j \phi(x_0).
\end{aligned}$$ The output energy can now be written in terms of the Koopman eigenfunctions as $$||y_{t_N}||^2 = \sum_{j=0}^{t_N} \bigg[\phi(x_0)^\top\Lambda^jV^\top W_h^\top W_hV\Lambda^j\phi(x_0) \bigg]$$ The optimization problem (\[eq:NPoptProb\]) can now be formulated as $$\label{eq:optProb}
\mathcal{J} = \max_{W_h} \sum_{j=0}^{t_N} \bigg[\phi(x_0)^\top\Lambda^jV^\top W_h^\top W_hV\Lambda^j\phi(x_0) \bigg]$$ with $|| W_h^\top W_h ||_2 \leq C$. The upper bound $C$ would vary between biological experiments and should be identified directly from data.
By picking out the $p$ $(\leq n_L)$ most observable modes of the system such that we can ensure the collection of measurements which correspond to maximal energy. If we define $W_h$ as $$W_h \triangleq \begin{bmatrix} I_{p\times p} & 0 \end{bmatrix} V^{-1}$$ the argument of (\[eq:optProb\]) becomes $$\begin{aligned}
& \sum_{j=0}^{t_N} \left( \phi(x_0)^\top\Lambda^jV^\top(V^{-1})^\top \begin{bmatrix} I_{p\times p} \\ 0 \end{bmatrix} \begin{bmatrix} I_{p\times p} & 0 \end{bmatrix} V^{-1}V\Lambda^j\phi(x_0) \right) \\
& \qquad = \sum_{j=0}^{t_N} \left(\phi(x_0)^\top diag(\lambda_1^{2j},\lambda_2^{2j},
...,\lambda_p^{2j},0,...,0) \phi(x_0) \right)
\end{aligned}$$ where $\lambda_1$ through $\lambda_p$ are the $p$ maximum eigenvalues of $K$. The maximum output energy comes from a choice of $W_h$ that depends on the eigenvectors of the Koopman operator.
### Example (Circadian oscillator)
To illustrate our sensor placement algorithm, we consider a model of a circadian oscillator, see Vilar et al. [@vilar2002mechanisms], that involves an activator $A$ and a repressor $R$. Both $A$ and $R$ are transcribed into $mRNA$ and subsequently translated into protein. Since $A$ can bind to both $A$ and $R$ promoters, it increases their transcription rates. $R$ acts as a negative element by hindering $A$. The deterministic dynamics are given by the following reaction rate equations
$$\label{eq:circadianModel}
\begin{aligned}
\dot{D}_A &= \theta_AD_A^{'} - \gamma_AD_AA \\
\dot{D}_R &= \theta_RD_R^{'} - \gamma_RD_RA \\
\dot{D}_A^{'} &= \gamma_AD_AA - \theta_AD_A^{'} \\
\dot{D}_R^{'} &= \gamma_RD_RA - \theta_RD_R^{'} \\
\dot{M}_A &= \alpha_A^{'}D_A^{'} + \alpha_AD_A - \delta_{MA}M_A \\
\dot{A} &= \beta_AM_A + \theta_AD_A^{'} + \theta_RD_R^{'} \\ & - A(\gamma_AD_AA + \gamma_RD_R + \gamma_CR + \delta_R) \\
\dot{M}_R &= \alpha_R^{'}D_R^{'} + \alpha_RD_R - \delta_{MR}M_R \\
\dot{R} &= \beta_RM_R - \gamma_CAR + \delta_AC - \delta_RR \\
\dot{C} &= \gamma_CAR - \delta_AC.
\end{aligned}$$
Extended dynamic mode decomposition (EDMD) [@williams_kevrekidis_rowley_2015] is used to compute the finite-dimensional approximation of the Koopman operator, $K_N$, for a coarse time step. A dictionary of state-inclusive observable functions, $\Psi$, is constructed using up to second-order polynomials. Often in biological systems, Hill function type nonlinearities appear in the dynamics. Even in these cases, the dictionary of polynomial functions should capture the dynamics well, according to the Weierstrass Approximation theorem, which states that any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials. [@stone]. Therefore, as long as the non-polynomial linearity is continuous, we expect that this dictionary of polynomials will result in accurate predictions, although the representation may not be as low dimensional as a representation drawn from a more efficient encoding [@lusch2018deep; @yeung2017learning].
In this example, initial conditions were chosen such that the trajectories converge to a limit cycle. From the coarse-grained Koopman operator obtained from simulation data, the fine-grained Koopman operator, $K$, is computed using the scheme outlined in Section \[sec:ObsPlaceSparse\]. Solving the optimization problem (\[eq:optProb\]), we can identify the optimal sensor placement. Choosing $p$, the number of rows in $W_h$, to be $p=20$, we get the output matrix structure as seen in figure \[fig:WhStructure2\]. A total of 55 observable functions were used which correspondingly sets the number of columns in the output matrix, $W_h$. The output matrix has a sparse structure with most elements of the matrix nearly zero. Using the criteria that the 1-norm of the columns of $W_h$ determine the most *active* states of the observable coordinates, we can determine optimal sensor placement. Using this criteria, the most active dynamics are $M_AC, M_RC, AC, R^2, RC, \text{and }C^2$ for a single initial condition where the trajectories converge to limit cycles. Figure \[fig:NetArch\] shows the entire network architecture of the circadian oscillator. The states highlighted in red are the active states and correspondingly are where the algorithm would dictate sensors should be placed. Figure \[fig:actDyn\] shows how frequently a state appears as an active state in the observable coordinates over 20 different initial conditions.
![Sparse structure of the output matrix $W_h$ with $p=20$ for the circadian oscillator simulation.[]{data-label="fig:WhStructure2"}](figs/W_hStructure_p20_Circadian_diffx0.pdf){width="0.85\columnwidth"}
![Network architecture of the circadian oscillator model in (\[eq:circadianModel\]). Arrows indicate activation, while bars indicate repression or degradation. Highlighted in red are the states which have the most active dynamics in the observable coordinates. Note that these active states were taken from the single initial condition used to produce figure \[fig:WhStructure2\].[]{data-label="fig:NetArch"}](figs/NetworkArchitecture_CircadianClock.pdf){width="0.61\columnwidth"}
![Histogram showing the frequency of a state of (\[eq:circadianModel\]) being in the 10 most active states of the observables over 20 different initial conditions.[]{data-label="fig:actDyn"}](figs/Freq_obs_coords.pdf){width="0.85\columnwidth"}
From this analysis, a nonlinear observable can be designed. For example, the state $C^2$ is highly active in the observable basis, therefore a nonlinear observer can be designed where a $C$ molecule binds with another $C$ molecule and integrated to obtain the output. We can then use this observer (and other observers) to act as a reporter for the cell state. This can enable rapid experimentation in synthetic biology since there would no longer be a need to collect expensive full state proteomics and transcriptomics data at a low temporal resolution. We can collect partial state measurements from states of interest at a high temporal resolution. The Koopman method thus can identify critical genes that serve as cell state biomarkers. These biomarkers provide a link between internal dynamics and observed phenotypes.
Conclusion
==========
In this work, we developed an algorithm for optimal sensor placement from sparsely sampled time-series data of discrete time nonlinear systems. The optimal sensor placement algorithm was formulated as maximizing the observability of a dynamical system or genetic network in the Koopman lifted space. We compute the temporally *fine-grained* Koopman operator from the temporally *coarse-grained* Koopman operator, the latter of which is identified directly from sparse biological data. In the case of noisy data, a closed form expression for the error in the coarse-grained Koopman operator is derived. Finally, we have illustrated the optimal sensor placement method on a simulation example of a circadian oscillator. This method can be utilized in the context of developing bacterial sensors where the design of a library of promoters is now informed by the sensor placement algorithm.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Professor Igor Mezic for insightful discussions. This material is based on work supported by DARPA and AFRL under contract number DEAC0576RL01830. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Defense Advanced Research Project Agency, the Department of Defense, or the United States government.
[^1]: A. Hasnain and N. Boddupalli are with the Department of Mechanical Engineering, University of California, Santa Barbara [aqib@ucsb.edu]{}, [nibodh@ucsb.edu]{} E. Yeung is with the Department of Mechanical Engineering, Center for Control, Dynamical Systems, and Computation, and Biomolecular Science and Engineering, University of California, Santa Barbara [eyeung@ucsb.edu]{}
|
---
abstract: 'We present a new derivation of the Casimir force between two parallel plane mirrors at zero temperature. The two mirrors and the cavity they enclose are treated as quantum optical networks. They are in general lossy and characterized by frequency dependent reflection amplitudes. The additional fluctuations accompanying losses are deduced from expressions of the optical theorem. A general proof is given for the theorem relating the spectral density inside the cavity to the reflection amplitudes seen by the inner fields. This density determines the vacuum radiation pressure and, therefore, the Casimir force. The force is obtained as an integral over the real frequencies, including the contribution of evanescent waves besides that of ordinary waves, and, then, as an integral over imaginary frequencies. The demonstration relies only on general properties obeyed by real mirrors which also enforce general constraints for the variation of the Casimir force.'
author:
- Cyriaque Genet
- Astrid Lambrecht
- Serge Reynaud
date: 'February 10, 2003'
title: The Casimir force and the quantum theory of lossy optical cavities
---
Introduction
============
An important prediction of quantum theory is the existence of irreducible fluctuations of electromagnetic fields in vacuum. Besides their numerous observable consequences in microscopic physics, vacuum fluctuations also have observable effects in macroscopic physics, for example the Casimir force they exert on mirrors [@Casimir48].
Casimir calculated this force in a geometrical configuration where two plane mirrors are placed a distance $L$ apart and parallel to each other, the area $A$ of the mirrors being much larger than the squared distance $A \gg L^2$. He considered the ideal case of perfectly reflecting mirrors and obtained an expression which, remarkably, depends only on the geometrical quantities $A$ and $L$ and on the fundamental constants $\hbar$ and $c$ $$F_\mathrm{Cas} = \frac{\hbar c \pi^2 A}{240 L^4}
\label{eqCasimir}$$
This attractive force has been observed in a number of ‘historical’ experiments [@Deriagin57; @Spaarnay58; @Tabor68; @Black68; @Sabisky73] which confirmed its existence and main properties [@Sparnaay89; @Milonni94; @Mostepanenko97]. Several recent experiments reached an accuracy in the % range by measuring the force between a plane and a sphere [@Lamoreaux97; @Mohideen98; @Roy99; @Harris00] or two cylinders [@Ederth00]. Similar experiments were also performed with MEMS [@Chan01s; @Chan01p] (see also [@Buks01]). An experiment studied the plane-plane configuration considered by Casimir [@Bressi02] but, as a consequence of the difficulties associated with this geometry, reached only a 15% accuracy (see reviews of recent experiments in [@Bordag01; @Lambrecht02]).
The Casimir force is the most accessible experimental consequence of vacuum fluctuations in the macroscopic world while vacuum energy is known to raise a serious problem with respect to gravity and cosmology (see references in [@Reynaud01; @Genet02]). This is a reason for testing the predictions of Quantum Field Theory concerning the Casimir effect with the greatest care and accuracy. The theory of the Casimir force is also a key point for the experiments searching for the new weak forces predicted by theoretical unification models to arise at distances between nanometer and millimeter [@Carugno97; @Fischbach98; @Bordag99; @Fischbach99; @Long99; @Hoyle01; @Adelberger02; @Long02]. The Casimir force is indeed the dominant effect between two neutral objects at $\mu$m or sub$\mu$m distances so that an accurate knowledge of its theoretical expectation is as crucial as the precision of measurements in such experiments [@Lambrecht00].
In this context, it is essential to account for the differences between the ideal case considered by Casimir and the real experimental situation. Recent experiments use metallic mirrors which show perfect reflection only at frequencies below their plasma frequency. They are performed at room temperature, with the effect of thermal fluctuations superimposed to that of vacuum fluctuations. In the most accurate experiments, the force is measured between a plane and a sphere, and not between two parallel planes. The surface state of the plates, in particular their roughness, should also affect the force. A large number of works have been devoted to the study of these effects and we refer the reader to [@Bordag01; @Lambrecht02] for a bibliography.
The evaluation of the Casimir force between imperfect lossy mirrors at non zero temperature has given rise to a burst of controversial results [@Bostrom00; @Svetovoy00; @Bordag00; @Lamoreaux01c; @Sernelius01r; @Sernelius01c; @Bordag01r; @Klimchitskaya01; @Bezerra02; @Lamoreaux02] which constitutes a part of the motivations for the present work. For the sake of comparing experimental measurements and theoretical expectations, it is necessary to have at one’s disposal a reliable expression of the Casimir force in the experimental situation. In the present paper, we focus our attention on the effect of imperfect reflection of the mirrors. Other effects, in particular the effect of temperature, will be addressed in follow-on papers.
We consider the original Casimir geometry with two perfectly plane and parallel mirrors. Except for these assumptions, we consider arbitrary frequency dependences for the mirrors which, in particular, may be lossy. We evaluate the Casimir force as the effect of vacuum radiation pressure on the Fabry-Perot cavity formed by the two mirrors. The net force results from the balance between the repulsive and attractive contributions associated respectively with resonant or antiresonant frequencies. It is obtained as an integral over the axis of real frequencies, including the contribution of evanescent waves besides that of ordinary waves. It is then transformed into an integral over imaginary frequencies by using physical properties fulfilled by all real mirrors.
The formula obtained here for the Casimir force turns out to be identical to the expression already published in [@Jaekel91] but the new derivation has a wider scope of validity than the previous one since it remains valid for lossy mirrors. The fact that the formula keeps the same form despite the widening of the assumptions is intimately related to a theorem which relates the spectral density of the fields inside the cavity to the reflection amplitudes seen by the same fields. This theorem was demonstrated in [@Jaekel91] and [@Barnett98] in specific cases and we prove it in the present paper without any restriction. To this aim, we introduce a systematic treatment of lossy mirrors and cavities as dissipative networks [@Meixner]. We define scattering and transfer matrices for elementary networks like the interface between two media or the propagation over a given length in a medium. We then deduce the matrices associated with composed networks, like the optical slab or the multilayer mirror.
The results obtained in this manner are therefore applicable to a large variety of mirrors, still with the assumption of perfect plane geometry. In the particular case of a slab with a large width, the Lifshitz expression [@Lifshitz56; @LLcasimir] is recovered. At the limit of perfectly reflectors, the ideal Casimir formula (\[eqCasimir\]) is obtained. More generally, the expression gives the Casimir force as an integral written in terms of the reflection amplitudes characterizing the two mirrors. This integral is finite as soon as the amplitudes obey the general properties of scattering theory already alluded to. In other words, the difficulties usually associated with the infiniteness of vacuum energy are solved by using the properties of real mirrors themselves rather than through an additional formal regularization technique.
We finally show that the same physical properties constrain the variation of the Casimir force. In particular, they invalidate proposals which has been done for ‘tayloring’ the force at will by using mirrors with specially designed scattering amplitudes [@Iacopini93; @Ford93]. In these proposals, the balance between attractive and repulsive contributions to the force is change, leading to the hope that the Casimir force could reach large or have its sign changed from an attractive force to a repulsive one [@Iacopini93]. Using the simple model of a one-dimensional space, it has already been shown [@Lambrecht97] that these hopes cannot be met for arbitrary mirrors built up with dielectric layers. Here, the argument is generalized to the Casimir geometry in three-dimensional space with the following conclusions : the Casimir force cannot exceed the value obtained for perfect mirrors, it remains attractive for any cavity length and its value is a decreasing function of the cavity length. This is true for any mirror obtained by piling up layers of media described by dielectric functions. This definition of multilayer dielectric mirrors includes the case of metallic layers, provided that magnetic effects play a negligible role in the optical response.
Vacuum field modes
==================
As explained in the Introduction, we consider in this paper the original Casimir geometry with perfectly plane and parallel mirrors aligned along the directions $x$ and $y$. This configuration obeys a symmetry with respect to time translation as well as transverse space translations along these directions. We use bold letters for two-dimensional vectors along these directions and denote $\mathbf{r} \equiv \left( x,y\right)$ the transverse position. As a consequence of this symmetry, the frequency $\omega $, the transverse vector $\mathbf{k} \equiv \left( k_{x},k_{y}\right)$ and the polarization $p=\mathrm{TE},\mathrm{TM}$ are preserved throughout the scattering processes on a mirror or a cavity. The scattering couples only the free vacuum modes which have the same values for the preserved quantum numbers and differ by the sign of the longitudinal component $k_{z}$ of the wavevector.
In the present section, we introduce notations for the vacuum field modes, first in empty space and then in a dielectric medium. These notations are chosen to be well-adapted to the symmetry of the problem.
Vacuum modes in empty space
---------------------------
In empty space, the components of the wavevector are given for each field mode by the frequency $\omega$, the incidence angle $\theta$ and the azimuthal angle $\varphi $ $$\begin{aligned}
k_{x}= |\mathbf{k}| \cos \varphi &&\qquad
|\mathbf{k}| = \frac\omega{c} \sin\theta \nonumber \\
k_{y}= |\mathbf{k}| \sin \varphi &&\qquad
k_{z}= \frac{\omega }{c} \cos \theta
\label{dispersionrelation}\end{aligned}$$ $|\mathbf{k}|$ is the modulus of the transverse wavevector and the longitudinal component $k_{z}$ may be expressed in terms of the preserved quantities $\omega$ and $\mathbf{k}$ $$k_{z}=\phi \sqrt{\frac{\omega ^{2}}{c^{2}}-\mathbf{k}^{2}}\qquad \phi =\pm 1$$ $\phi$ is defined as the sign of $\cos\theta$ and represents the direction of propagation with $+1$ and $-1$ corresponding respectively to rightward and leftward propagation.
The two polarizations $p=\mathrm{TE},\mathrm{TM}$ are defined by the transversality with the incidence plane of electric and magnetic fields respectively. They are given by the unit electric vectors $\widehat{\epsilon}$ $$\begin{aligned}
\widehat{\epsilon}_{x}^\mathrm{TM} = \cos \theta \cos \varphi &&\qquad
\widehat{\epsilon}_{x}^\mathrm{TE} = -\sin \varphi \nonumber \\
\widehat{\epsilon}_{y}^\mathrm{TM} = \cos \theta \sin \varphi &&\qquad
\widehat{\epsilon}_{y}^\mathrm{TE} = \cos \varphi \nonumber \\
\widehat{\epsilon}_{z}^\mathrm{TM} = -\sin \theta &&\qquad
\widehat{\epsilon}_{z}^\mathrm{TE} = 0
\label{Epolar}\end{aligned}$$ or, equivalently, the unit magnetic vectors $\widehat{\beta}^\mathrm{TM} =\widehat{\epsilon}^\mathrm{TE}$ and $\widehat{\beta}^\mathrm{TE}=-\widehat{\epsilon}^\mathrm{TM}$. For each mode, the wavevector and polarization vectors form an orthogonal spatial basis. We have chosen linear polarizations described by real components; hence the unit vectors $\widehat{\epsilon }$ and $\widehat{\beta}$ are not affected by the complex conjugation appearing below in the relation between positive and negative frequencies.
The two modes corresponding to the same values of $\omega$, $\mathbf{k}$ and $p$ but opposite values of $\phi$ are coupled by scattering on a mirror. For this reason, we introduce a label $m\equiv \left( \omega ,\mathbf{k},p\right)$ gathering the values of $\omega $, $\mathbf{k}$ and $p$. A mode freely propagating in vacuum is thus labeled by $m$ and $\phi $ and the summation over modes is described by the symbols $$\begin{aligned}
\sum_{m\phi } &\equiv& \sum_{p}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi ^{2}}
\int_{-\infty }^{\infty }\frac{\mathrm{d}k_{z}}{2\pi } \nonumber \\
&\equiv& \sum_{\phi}\sum_{p}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi ^{2}}
\int_{0}^{\infty }\frac{\omega }{ck_{z}}\frac{\mathrm{d}\omega }{2\pi c}
\label{mphi}\end{aligned}$$ Note that $\phi $ appears implicitly as the sign of $k_{z}$ in the first form whereas it appears explicitly in the second one.
The free vacuum fields are then written as linear superpositions of modes $$\begin{aligned}
E\left( \mathbf{r},z,t\right) &=&\sqrt{cZ_\mathrm{vac}}\sum_{m\phi }\
\sqrt{\frac{\hbar \omega }{2}}\widehat{\epsilon }_{m}^{\phi }\left( e_{m}^{\phi }\
e^{-i\left( \omega t-\mathbf{k.r}-k_{z}z\right) }+\left( e_{m}^{\phi }\right)
^\dagger \ e^{i\left( \omega t-\mathbf{k.r}-k_{z}z\right) }\right) \nonumber \\
B\left( \mathbf{r},z,t\right) &=&\sqrt{\frac{Z_\mathrm{vac}}{c}}\sum_{m\phi }\
\sqrt{\frac{\hbar \omega }{2}}\widehat{\beta }_{m}^{\phi }\left( e_{m}^{\phi
}\ e^{-i\left( \omega t-\mathbf{k.r}-k_{z}z\right) }+\left( e_{m}^{\phi
}\right) ^\dagger \ e^{i\left( \omega t-\mathbf{k.r}-k_{z}z\right) }\right)
\label{EBfields}\end{aligned}$$ The vacuum impedance $Z_\mathrm{vac}= \mu _{0}c \simeq 377\Omega$ describes the electromagnetic constants in vacuum. In the following, the symbol $\varepsilon $ will be reserved to the relative permittivity with the value 1 in vacuum.
The quantum field amplitudes $e_{m}^{\phi}$ and $\left( e_{m}^{\phi}\right)^\dagger $ correspond to positive and negative frequency components. They fit the definition of annihilation and creation operators of quantum field theory and obey the canonical commutation relations [@CCT87] $$\begin{aligned}
\left[ e_{m^\prime }^{\phi ^\prime },e_{m}^{\phi}{}^\dagger \right] &=&
\left( 2\pi \right) ^{3}\delta ^{(2)}\left( \mathbf{k-k}^{\prime
}\right) \delta \left( k_{z}-k_{z}^\prime \right) \delta _{pp^{\prime
}}\delta _{\phi \phi ^\prime } \nonumber \\
&\equiv& \delta _{mm^\prime }\delta _{\phi \phi ^{\prime}} \nonumber \\
\left[ e_{m^\prime }^{\phi ^\prime },e_{m}^{\phi}\right] &=&
\left[ e_{m^\prime }^{\phi ^\prime }{}^{\dagger},e_{m}^{\phi}{}^\dagger \right] =0
\label{commut}\end{aligned}$$ In the vacuum state, the anticommutators of quantum amplitudes are derived from the corresponding commutators $$\begin{aligned}
\left\langle e_{m^\prime }^{\phi ^\prime } \cdot e_{m}^{\phi}{}^\dagger \right\rangle
_{\mathrm{vac}} &=&
\frac{1}{2} \left[ e_{m^\prime }^{\phi ^\prime }, e_{m}^{\phi}{}^\dagger \right]
= \frac{1}{2} \delta _{mm^\prime }\delta _{\phi \phi ^{\prime}} \nonumber \\
\left\langle e_{m^\prime }^{\phi ^\prime } \cdot e_{m}^{\phi} \right\rangle
_{\mathrm{vac}} &=&
\frac{1}{2} \left[ e_{m^\prime }^{\phi ^\prime }, e_{m}^{\phi}{} \right] = 0
\label{anticommVacuum}\end{aligned}$$ The dot symbol represents a symmetrized product.
Stress tensor in empty space
----------------------------
The energy density per unit volume $T_{00}$ is a quadratic form of the fields $E$ and $B$ $$T_{00} \left( \mathbf{r},z,t\right) = \frac{1}{2cZ_\mathrm{vac}}
\left( E^{2} + c^{2} B^{2} \right)$$ When subsituting the expression of free fields, $T_{00}$ is obtained as a bilinear form of the field amplitudes. Here, we study the averaged radiation pressure in the vacuum state which leads to a contraction $m^\prime=m$ in the sums over modes. Using the vacuum property (\[anticommVacuum\]), we find the averaged energy density in vacuum equal to the sum over the modes of $\frac{\hbar \omega }{2}$ $$\left\langle T_{00} \left( \mathbf{r},z,t\right) \right\rangle _\mathrm{vac}=
\sum_{m\phi }\ \frac{\hbar\omega }{2}
\label{T00Vac}$$ As it is well-known, this energy density is infinite.
The radiation pressure on plane mirrors oriented along $xy$ directions is determined by the component $T_{zz}$ of the Maxwell stress tensor $$T_{zz}\left( \mathbf{r},z, t\right) = \frac{1}{2Z_\mathrm{vac}}
\left( E \cdot \overline{E} + c^{2} B \cdot \overline{B} \right)$$ Here, the dot symbol represents a symmetrized product of the quantum amplitudes and, simultaneously, a scalar product of the vectors; the overline symbol describes the mathematical reflexion of a vector with respect to the plane $xy$ $$\overline{E}_{x}= E_x \qquad \overline{E}_{y}= E_y \qquad
\overline{E}_{z}= - E_z$$
As for $T_{00}$, averaging $T_{zz}$ in vacuum state leads to a contraction over the modes with the result $$\begin{aligned}
\left\langle T_{zz} \left( \mathbf{r},z,t\right) \right\rangle _\mathrm{vac}
&=& \sum_{m\phi}\frac{\hbar\omega }{4} \left(
\widehat{\epsilon}_m^\phi . \overline{\widehat{\epsilon}_m^\phi}
+ \widehat{\beta}_m^\phi . \overline{\widehat{\beta}_m^\phi}
\right) \nonumber \\
&=&\sum_{m\phi }\ \frac{\hbar \omega }{2} \cos ^{2}\theta
\label{TzzVac}\end{aligned}$$ This expression is similar to the expression (\[T00Vac\]) of the energy density with an extra factor $\cos ^{2}\theta$ well-known in studies of radiation pressure. The sum over modes is still infinite but this infiniteness problem will be solved in the forthcoming calculation of the Casimir force.
Fields in dielectric media
--------------------------
In the following, we consider mirrors built up as dielectric multilayers. Each dielectric medium is characterized by a relative permittivity $\varepsilon \left[ \omega \right]$ or, equivalently, an index of refraction $n\left[ \omega \right] = \sqrt{\varepsilon \left[ \omega \right]}$ depending on frequency. The magnetic permeability is kept equal to its vacuum value since this corresponds to all experimental situations studied so far. We stress again that this definition of dielectric mirrors includes the case of metals as long as the magnetic response plays a negligible role. We consider layers thick enough so that the dielectric response is local, i.e. described by a wavevector-independent permittivity $\varepsilon \left[ \omega \right]$.
We will sometimes take the plasma model as a first description of metallic optical response $$\varepsilon\left[ \omega \right] = 1 - \frac{\omega_\mathrm{P}^2}{\omega^2}
\qquad \omega_\mathrm{P} = \frac{2\pi c}{\lambda_\mathrm{P}}
\label{PlasmaModel}$$ where $\omega_\mathrm{P}$ and $\lambda_\mathrm{P}$ represent respectively the plasma frequency and the plasma wavelength. This simple model is not sufficient for an accurate evaluation of the Casimir force between real mirrors [@Lambrecht00]. To this aim, it is necessary to describe the optical response of metals with a dissipative part associated with electronic relaxation processes. As a consequence of causality, the real and imaginary parts of $n$ are related to each other through the Kramers-Kronig dispersion relations [@LLcausality].
For any function of frequency more generally, causality is unambiguously characterized in terms of analyticity properties : $n\left[ \omega \right]$ or $\varepsilon \left[ \omega \right] $ are analytical functions of $\omega $ in the ‘physical domain’ of the complex frequency plane, that is the domain of frequencies $\omega$ with a positive imaginary part $\Im \omega >0$. This property is obeyed by other response functions to be encountered below and it will play an important role in the derivation of the Casimir force. We will introduce an equivalent notation $\xi$ for complex frequencies with the physical domain now defined by a positive real part for $\xi$ $$\omega \equiv i\xi \qquad \Re \xi >0$$
The dispersion relation (\[dispersionrelation\]) is changed inside a refractive medium to $$\begin{aligned}
k_{x}= |\mathbf{k}| \cos \varphi &&\qquad
|\mathbf{k}| = n \left[ \omega \right] \frac\omega{c} \sin\theta \nonumber \\
k_{y}= |\mathbf{k}| \sin \varphi &&\qquad
k_{z}= n \left[ \omega \right] \frac{\omega }{c} \cos \theta
\label{dispersionrelationIndex}\end{aligned}$$ The preservation of $\omega$ and $\mathbf{k}$ at the traversal of an interface is equivalent to the Snell-Descartes law of refraction.
The sign has to be carefully chosen when extracting the square root to express $k_{z}$ in terms of the conserved quantities $\omega $ and $\mathbf{k}$. As soon as the refractive index contains an imaginary part, this is also the case for $k_{z}$ and the dephasing $\exp\left( ik_{z} z\right)$ associated with propagation includes an extinction factor. In order to ensure that this factor is effectively a decreasing exponential, we have to choose a specific root defined differently for the two propagation directions $\phi=\pm 1$ of the field $$\begin{aligned}
&&k_{z}\equiv i \phi \kappa \nonumber \\
\kappa &=&\sqrt{\varepsilon \left[ i\xi \right] \frac{\xi ^{2}}{c^{2}}+\mathbf{k}^{2}}
\qquad \Re\kappa >0
\label{dispersionKappa}\end{aligned}$$ The argument has been presented for freely propagating modes but it holds as well for evanescent waves confined to the vicinity of an interface between two media. In this case, the sign of $k_{z}$ is also chosen so that it corresponds to an extinction when the distance to the interface increases and this choice is still described by equation (\[dispersionKappa\]). In the following, we will use systematically the notations $\xi$ and $\kappa$, keeping in mind that the causality relations have to be written for each value of the conserved quantity $\mathbf{k}$.
Besides the dispersion relation (\[dispersionrelationIndex\]), the dielectric medium also changes the impedance, that is the ratio between magnetic and electric field amplitudes. Precisely, the impedance is changed from the value $Z_\mathrm{vac}$ in empty space to the value $\frac{Z_\mathrm{vac}}n$ in a dielectric medium of index $n$, resulting in reflection at the interface.
Mirrors as optical networks
===========================
We now introduce the description of mirrors as optical networks. We present the scattering and transfer representations and the relations between them. The transfer approach is well adapted to the composition of networks which are piled up. We first consider elementary networks such as an interface or propagation inside a refractive medium. We then use the composition law to study composed networks such as the slab and multilayer. In the present section, we only consider classical fields or, equivalently, mean quantum fields. The next section will be devoted to the full quantum treatment including the addition of noise associated with the losses inside the mirror.
Scattering and transfer representations
---------------------------------------
We first introduce the scattering and transfer representations for an arbitrary network represented with two ports and four fields. These fields are identified as lefthand/righthand (symbols ‘L’ and ‘R’), rightward/leftward (arrows $\rightarrow$ and $\leftarrow$) or input/output fields (labels ‘in’ and ‘out’), as shown on Figure \[FigNetwork\].
Let us emphasize that the arrows are a symbolic representation of the two modes coupled by the network which correspond to the same label $m$ and to the two opposite signs $\phi = \pm 1$. The geometrical directions of propagation are given by the wavevectors of equation (\[dispersionrelationIndex\]). The coupling between the fields is described by reflection and transmission amplitudes represented below by scattering or transfer matrices.
In the scattering point of view, we gather the input and output fields in twofold columns related by a $S-$matrix $$\begin{aligned}
&&\left| \mathcal{E}^{\mathrm{in}}\right\rangle =\left(
\begin{tabular}{l}
$\mathcal{E}_{\mathrm{L}}^{\mathrm{in}}$ \\
$\mathcal{E}_{\mathrm{R}}^{\mathrm{in}}$\end{tabular}
\right) \qquad
\left| \mathcal{E}^{\mathrm{out}}\right\rangle =\left(
\begin{tabular}{l}
$\mathcal{E}_{\mathrm{L}}^{\mathrm{out}}$ \\
$\mathcal{E}_{\mathrm{R}}^{\mathrm{out}}$\end{tabular}
\right)
\nonumber \\
&&\left| \mathcal{E}^{\mathrm{out}}\right\rangle =S\left|
\mathcal{E}^{\mathrm{in}}\right\rangle \qquad
S=\left(
\begin{tabular}{ll}
$r$ & $\overline{t}$ \\
$t$ & $\overline{r}$\end{tabular}
\right)
\label{Smatrix}\end{aligned}$$ $r$ and $\overline{r}$ are the reflection amplitudes while $t$ and $\overline{t}$ are the transmission amplitudes. We will also use an equivalent convention where the output ket is defined with the upper and lower components exchanged $$\begin{aligned}
&&\widetilde{\left| \mathcal{E}^{\mathrm{out}}\right\rangle }
=\left(
\begin{tabular}{l}
$\mathcal{E}^{\mathrm{out}}_\mathrm{R}$ \\
$\mathcal{E}^{\mathrm{out}}_\mathrm{L}$\end{tabular}
\right) = \eta \left| \mathcal{E}^{\mathrm{out}}\right\rangle \qquad
\eta =\left(
\begin{tabular}{ll}
$0$ & $1$ \\
$1$ & $0$\end{tabular}
\right)
\nonumber \\
&&\widetilde{\left| \mathcal{E}^{\mathrm{out}}\right\rangle }=
\widetilde{S}\left| \mathcal{E}^{\mathrm{in}}\right\rangle \qquad
\widetilde{S}=\eta S=\left(
\begin{tabular}{ll}
$t$ & $\overline{r}$ \\
$r$ & $\overline{t}$\end{tabular}
\right)
\label{SmatrixTilde}\end{aligned}$$ This convention simplifies some algebraic manipulations while being completely equivalent to the former convention. For comparison with previous works, note that the former notation (\[Smatrix\]) was used in [@Jaekel91] whereas the latter one (\[SmatrixTilde\]) was used in [@Lambrecht97].
In the transfer point of view, the network is described by lefthand and righthand columns related by a $T-$matrix $$\begin{aligned}
&&\left| \mathcal{E}_{\mathrm{L}}\right\rangle =\left(
\begin{tabular}{l}
$\mathcal{E}_{\mathrm{L}}^{\rightarrow }$ \\
$\mathcal{E}_{\mathrm{L}}^{\leftarrow }$\end{tabular}
\right)
\qquad
\left| \mathcal{E}_{\mathrm{R}}\right\rangle =\left(
\begin{tabular}{l}
$\mathcal{E}_{\mathrm{R}}^{\rightarrow }$ \\
$\mathcal{E}_{\mathrm{R}}^{\leftarrow }$\end{tabular}
\right)
\nonumber \\
&&\left| \mathcal{E}_{\mathrm{L}}\right\rangle =T\left|
\mathcal{E}_{\mathrm{R}}\right\rangle \qquad T=\left(
\begin{tabular}{ll}
$a$ & $b$ \\
$c$ & $d$\end{tabular}
\right) \label{Tmatrix}\end{aligned}$$
The matrix $\eta$ introduced in (\[SmatrixTilde\]) exchanges the two directions of propagation. We also use in the following the matrices $\pi_\pm$ which project onto each direction $$\pi _{+} =\left(
\begin{tabular}{ll}
$1$ & $0$ \\
$0$ & $0$\end{tabular}
\right) \qquad
\pi _{-} =\left(
\begin{tabular}{ll}
$0$ & $0$ \\
$0$ & $1$\end{tabular}
\right)$$ These matrices obey simple rules which define an algebraic calculus in the space $\mathcal{M}_{2}\left( \mathbb{C}\right) $ of $2 \times 2$ matrices with complex coefficients $$\begin{aligned}
\pi _{+}^{2} = \pi _{+} &&\qquad \pi _{-}^{2}=\pi _{-}\qquad
\pi _{+}\pi _{-}=\pi _{-}\pi _{+}=0 \nonumber \\
\eta ^{2} = I &&\qquad \eta \pi _{+}=\pi _{-}\eta \qquad
\eta \pi _{-}=\pi _{+}\eta \end{aligned}$$
The identification of Figure (\[FigNetwork\]) is written as $$\begin{aligned}
\pi _{+}\left| \mathcal{E}_{\mathrm{R}}\right\rangle &=&\pi _{+}\widetilde{\left|
\mathcal{E}^{\mathrm{out}}\right\rangle }\qquad \pi _{-}\left|
\mathcal{E}_{\mathrm{R}}\right\rangle
=\pi _{-}\left| \mathcal{E}^{\mathrm{in}}\right\rangle \nonumber \\
\pi _{+}\left| \mathcal{E}_{\mathrm{L}}\right\rangle &=&\pi _{+}\left|
\mathcal{E}^{\mathrm{in}} \right\rangle \qquad \pi _{-}\left|
\mathcal{E}_{\mathrm{L}}\right\rangle =\pi_{-}\widetilde{\left|
\mathcal{E}^{\mathrm{out}}\right\rangle } \label{Identify}\end{aligned}$$ It relates the transfer and scattering amplitudes. We decompose the scattering equations (\[Smatrix\]) on the two components and use (\[Identify\]) to rewrite them as $$\begin{aligned}
\pi _{+}\left| \mathcal{E}_{\mathrm{R}}\right\rangle &=&\pi _{+}\widetilde{S}
\left( \pi _{+} \left| \mathcal{E}_{\mathrm{L}}\right\rangle +\pi _{-}\left|
\mathcal{E}_{\mathrm{R}}\right\rangle \right) \nonumber \\
\pi _{-}\left| \mathcal{E}_{\mathrm{L}}\right\rangle &=&\pi _{-}\widetilde{S}
\left( \pi _{+}\left| \mathcal{E}_{\mathrm{L}}\right\rangle +\pi _{-}\left|
\mathcal{E}_{\mathrm{R}}\right\rangle \right)\end{aligned}$$ This linear system may be put under a matrix form $$\left( \pi _{-}-\widetilde{S}\pi _{+}\right) \left| \mathcal{E}_{\mathrm{L}}
\right\rangle =-\left( \pi _{+}-\widetilde{S}\pi _{-}\right) \left|
\mathcal{E}_{\mathrm{R}}\right\rangle$$ It is equivalent to the transfer equation (\[Tmatrix\]) with the $T-$matrix obtained as $$T=-\left( \pi _{-}-\widetilde{S} \pi_{+}\right) ^{-1}
\left( \pi _{+}-\widetilde{S}\pi _{-}\right)
\label{StoT}$$
The converse transformation is obtained by performing the same manipulations in the reverse order. Starting from the transfer equation (\[Tmatrix\]) and using (\[Identify\]), one obtains a linear system which is equivalent to the scattering equation (\[SmatrixTilde\]) with $$\widetilde{S}
=-\left( \pi _{-}-T\pi _{+}\right) ^{-1}\left( \pi _{+}-T\pi _{-}\right)
\label{TtoS}$$ The relations (\[StoT\]) and (\[TtoS\]) have the same form. They represent an idempotent homographic transformation in the space $\mathcal{M}_{2}\left( \mathbb{C}\right)$, care being taken for the non-commutativity of multiplications in this space. When inverting algebraically the homographic relations (\[StoT\]) and (\[TtoS\]), one obtains equivalent expressions $$\begin{aligned}
&&\widetilde{S}=\left( \pi _{+}+\pi _{-}T\right)
\left( \pi _{-}+\pi _{+}T\right) ^{-1} \nonumber \\
&&T=\left( \pi _{+}+\pi _{-}\widetilde{S}\right)
\left( \pi _{-}+\pi _{+}\widetilde{S}\right) ^{-1}\end{aligned}$$ Other equivalent expressions are obtained from the equalities $$\begin{aligned}
\left( \pi _{-}-\widetilde{S}\pi _{+}\right)
\left( \pi _{-}-T\pi_{+}\right) &=& I \nonumber \\
\left( \pi _{-}+\pi _{+}\widetilde{S}\right)
\left( \pi _{-}+\pi_{+}T\right) &=& I
\label{SandTsym}\end{aligned}$$
All these expressions may be written in terms of the scattering and transfer amplitudes $$\begin{aligned}
a=\frac{1}{t} &&\qquad b=-\frac{\overline{r}}{t} \nonumber \\
c=\frac{r}{t} &&\qquad d=\frac{t\overline{t}-r\overline{r}}{t} \nonumber \\
r=\frac{c}{a} &&\qquad \overline{t}=\frac{ad-bc}{a} \nonumber \\
t=\frac{1}{a} &&\qquad \overline{r}=-\frac{b}{a}
\label{SandTamplitudes}\end{aligned}$$ The more formal homographic transformations written above are nevertheless useful, as it will become clear in forthcoming calculations.
Composition of optical networks
-------------------------------
The $T-$matrices are perfectly adapted to the composition of optical networks corresponding to a piling up process (see Figure \[FigComposition\]).
On each network, the transfer equations are written as $$\begin{aligned}
\left| \mathcal{E}_\mathrm{L} {\{\mathrm{A}\}} \right\rangle &=& T {\{\mathrm{A}\}} \left| \mathcal{E}
_{\mathrm{R}} {\{\mathrm{A}\}} \right\rangle \nonumber\\
\left| \mathcal{E}_{\mathrm{L}}
{\{\mathrm{B}\}} \right\rangle &=& T {\{\mathrm{B}\}} \left| \mathcal{E}_{\mathrm{R}} {\{\mathrm{B}\}}
\right\rangle\end{aligned}$$ The brackets ${\{\mathrm{\ }\}}$ specify the network for which the $T-$matrix or field column is written. Identifying the fields according to Figure (\[FigComposition\]) $$\begin{aligned}
&&\left| \mathcal{E}_{\mathrm{L}} {\{\mathrm{AB}\}} \right\rangle \equiv
\left| \mathcal{E}_{\mathrm{L}} {\{\mathrm{A}\}} \right\rangle \qquad
\left| \mathcal{E}_{\mathrm{R}} {\{\mathrm{A}\}} \right\rangle \equiv
\left| \mathcal{E}_{\mathrm{L}} {\{\mathrm{B}\}} \right\rangle \nonumber \\
&&\left| \mathcal{E}_{\mathrm{R}} {\{\mathrm{AB}\}} \right\rangle \equiv
\left| \mathcal{E}_{\mathrm{R}} {\{\mathrm{B}\}} \right\rangle
\label{identifyCompo}\end{aligned}$$ we deduce that the piling up process is equivalent to the product of $T$-matrices $$\begin{aligned}
\left| \mathcal{E}_{\mathrm{L}} {\{\mathrm{AB}\}} \right\rangle &=&
T {\{\mathrm{AB}\}} \left| \mathcal{E}_{\mathrm{R}} {\{\mathrm{AB}\}} \right\rangle
\nonumber \\
T {\{\mathrm{AB}\}} &=& T {\{\mathrm{A}\}} T {\{\mathrm{B}\}}
\label{compoT}\end{aligned}$$ We have assumed the two networks to be in the immediate vicinity of each other but without any electronic exchange between them, which again corresponds to the assumption of thick enough layers.
Elementary networks
-------------------
We now study two elementary networks, that is the traversal of an interface and the propagation over a given length inside a dielectric medium.
For the scattering at the plane interface between two media with indices $n_{0}$ and $n_{1}$, we write the reflection and transmission amplitudes as the Fresnel scattering amplitudes [@LLfresnel]. Reflection amplitudes $r^p {\{\mathrm{Int}\}}$ are obtained from characteristic impedances $z^p$ defined for plane waves with polarization $p$ in each medium and from the continuity equations at the interface $$\begin{aligned}
r^p {\{\mathrm{Int}\}} &&= - \overline{r}^p {\{\mathrm{Int}\}} = \frac{1-z^p}{1+z^p} \nonumber \\
z^\mathrm{TE} &&= \frac{n_1 \cos\theta _1} {n_0 \cos \theta_0}
= \frac{\kappa_1} {\kappa_0} \nonumber \\
z^\mathrm{TM} &&= \frac{n_1 \cos\theta _0} {n_0 \cos \theta_1}
= \frac{\varepsilon _{1}\kappa_{0}}{\varepsilon_0 \kappa _{1}}
\label{r01}\end{aligned}$$ Then the transmission amplitudes are obtained as $$\sqrt{\frac{\kappa_1}{\kappa_0}} t^p {\{\mathrm{Int}\}}
= \sqrt{\frac{\kappa_0}{\kappa_1}} \overline{t}^p {\{\mathrm{Int}\}}
= \sqrt{1- \left( r^p {\{\mathrm{Int}\}} \right) ^{2} }$$ We deduce the expression of the transfer matrix $$\begin{aligned}
&&T^p {\{\mathrm{Int}\}} = \sqrt{\frac{\kappa_1}{\kappa_0}}
\frac{1}{\sqrt{2 \sinh \beta^p }}
\left(
\begin{tabular}{ll}
$e^{\frac{\beta^p}{2}}\quad $ & $-e^{-\frac{\beta^p }{2}}$ \\
$-e^{-\frac{\beta^p}{2}}$ & $\quad e^{\frac{\beta^p }{2}}$\end{tabular}
\right) \nonumber \\
&&\beta ^p = \ln \frac{z^p + 1}{z^p - 1}
\label{Tint}\end{aligned}$$
We now consider the process of field propagation over a propagation length $\ell $ inside a dielectric medium characterized by a permittivity $\varepsilon$. For this elementary network, the $T-$matrix has the simple form $$\begin{aligned}
&&T {\{\mathrm{Prop}\}} =
\left(
\begin{tabular}{ll}
$e^{\alpha}$ & $0$ \\
$0$ & $e^{-\alpha}$\end{tabular}
\right) \nonumber \\
&&\alpha = \kappa \ell = \sqrt{\varepsilon
\frac{\xi ^2}{c^2}+\mathbf{k}^2} \ \ell
\label{Tprop}\end{aligned}$$ The optical depth $\alpha$ does not depend on the polarization.
Note that the composition is commutative within the class of interfaces or that of propagations : it corresponds to the multiplication of the $z-$parameters for interfaces and to the addition of $\alpha -$parameters for propagations. But the composition is no longer commutative when interfaces and propagations are piled up.
Reciprocity theorem
-------------------
We now prove a reciprocity theorem obeyed by arbitrary dielectric multilayers, [*i.e.*]{} networks obtained by piling up interfaces and propagations.
To this aim, we first remark that the ratio of the two transmission amplitudes is related to the determinant of the $T-$matrix $$\frac{\overline{t}}{t}=ad-bc=\det T
\label{determinant0}$$ This follows from the relations (\[SandTamplitudes\]) between $S-$ and $T-$amplitudes for an arbitrary network. Then, it is clear from (\[compoT\]) that the determinant of $T$ is simply multiplied under composition $$\det T {\{\mathrm{AB}\}} = \det T {\{\mathrm{A}\}} \ \det T {\{\mathrm{B}\}}$$ For the two kinds of elementary networks studied previously (see eqs \[Tint\]-\[Tprop\]), the determinant of $T$ is the ratio of the values of $\kappa$ at the right and left sides of the network $$\det T = \frac{\kappa _\mathrm{R}}{\kappa _\mathrm{L}}
\label{RecipTheorem}$$ It follows that this relation is valid for any optical network composed by piling up interfaces and propagations.
In the particular case where the network has its two ports corresponding to vacuum, which is the case for a mirror, the values of $\kappa$ are equal on its two sides and the $T-$matrix has a unit determinant $$\det T = 1 \qquad \overline{t}=t
\label{RecipTheorem1}$$ Note that reciprocity corresponds to a symmetrical $S-$matrix and has to be distinguished from the spatial symmetry of the network with respect to its mediane plane which entails $\overline{r}=r$.
This theorem is the specific form, when the symmetry of plane mirrors is assumed, of the general reciprocity theorem demonstrated by Casimir [@Casimir45] as an extension to electromagnetism of Onsager’s microreversibility theorem [@Onsager31]. We have disregarded any static magnetic field which could affect these reciprocity relations.
Slabs and multilayers
---------------------
We now consider the dielectric slabs and multilayers as composed networks and we deduce their transfer and scattering amplitudes from the preceding results.
The slab is obtained by piling up a vacuum/matter interface with indices $n_0=1$ and $n_1$ at its left and righthand sides, propagation over a length $\ell$ inside matter, and a matter/vacuum interface with now $n_1$ and $n_0=1$ at its left and righthand sides. We denote $T{\{\mathrm{Int}\}}$ the $T-$matrix associated with the first interface and obtain the $T-$matrix associated with the second interface as the inverse of $T{\{\mathrm{Int}\}}$. As a consequence of the composition law (\[compoT\]), the $T-$matrix associated with the slab is obtained as $$T {\{\mathrm{Slab}\}} = T {\{\mathrm{Int}\}} T {\{\mathrm{Prop}\}} T {\{\mathrm{Int}\}} ^{-1}$$
Using the expressions (\[Tint\],\[Tprop\]) of $T {\{\mathrm{Int}\}}$ and $T {\{\mathrm{Prop}\}}$, we evaluate $T {\{\mathrm{Slab}\}}$ as $$T{\{\mathrm{Slab}\}} = \frac{1}{\sinh \beta}
\left(
\begin{tabular}{cc}
$\sinh \left( \beta +\alpha \right)$ & $\sinh \alpha $ \\
$-\sinh \alpha $ & $\sinh \left( \beta -\alpha\right) $\end{tabular}
\right)
\label{Tslab}$$ We deduce the form of the $S-$matrix which is simultaneously reciprocal ($\overline{t}=t$) and symmetrical in the exchange of its two ports ($\overline{r}=r$) $$S{\{\mathrm{Slab}\}} = \frac{1}{\sinh \left( \beta +\alpha \right)}
\left(
\begin{tabular}{cc}
$-\sinh \alpha $ & $\sinh \beta $ \\
$\sinh \beta $ & $-\sinh \alpha $\end{tabular}
\right)
\label{slab}$$ In the limiting case of a small thickness $\alpha \rightarrow 0$, we find $t {\{\mathrm{Slab}\}} \rightarrow 1$ and $r {\{\mathrm{Slab}\}} \rightarrow 0$, which means that the slab tends to become transparent. In this case indeed, the propagation can be forgotten and the two inverse interfaces have their effects cancelled by each other.
The opposite limiting case of a large thickness is often considered since it fits the usual experimental situations. More precisely, experiments are performed with metallic mirrors having a thickness much larger than the plasma wavelength. This is why the limit of a total extinction of the field through the medium is assumed in most calculations. This corresponds to the so-called ‘bulk limit’ with $e^{-\alpha }\rightarrow 0$ and $r {\{\mathrm{Slab}\}} \rightarrow -e^{-\beta }= r {\{\mathrm{Int}\}}$ in eq.(\[slab\]) : the reflection amplitude is determined entirely by the first interface. Let us emphasize however that the bulk limit raises several delicate problems. First, the transmission amplitude $t{\{\mathrm{Slab}\}}$ vanishes in this limit so that the $T-$matrix is not defined, with the drawback of invalidating the general method used in the present paper. Then, the bulk limit cannot be met in the case of non absorbing media where $e^{-\alpha }$ remains a complex number with unit modulus for any value of $\ell$. Even in the presence of absorption, a large value of the width $\ell$ does not necessarily imply a large value of the optical thickness $\alpha$ since $\kappa$ may go to zero at normal incidence and zero frequency, leading to a transparent slab in contrast with the results of the bulk limit. Therefore a reliable calculation must consider the experimental situation of mirrors with a large but finite thickness. In the present paper, we consider the general case of arbitrary mirrors and test the reliability of the bulk limit in the end of the calculations.
We can deal with the case of dielectric multilayers similarly. If we consider as an example the multilayer obtained by piling up a vacuum/matter interface with indices $n_0=1$ and $n_1$ at its left and righthand sides, propagation over a length $\ell_1$ inside the medium 1, an interface between media 1 and 2, propagation over a length $\ell_2$ inside the medium 2, and an interface between medium 2 and vacuum, its $T-$matrix is obtained as the product $$\begin{aligned}
T {\{\mathrm{Multilayer}\}} &=& T {\{\mathrm{Int01}\}} T {\{\mathrm{Prop1}\}} T {\{\mathrm{Int12}\}} \nonumber \\
&& \times T {\{\mathrm{Prop2}\}} T {\{\mathrm{Int20}\}} \end{aligned}$$ Alternatively, the same multilayer may be obtained by piling up two slabs each corresponding to one of the layers $$\begin{aligned}
T {\{\mathrm{Multilayer}\}} &=& T {\{\mathrm{Slab010}\}} T {\{\mathrm{Slab020}\}} \end{aligned}$$ In the last two equations, the indices specify the different interfaces, propagations or slabs using an obvious convention.
Since any multilayer mirror is obtained by piling up slabs connecting two vacuum ports and thus obeying the reciprocity relation $\overline{t}=t$, we can use a simple form of the composition law written in terms of scattering amplitudes [@Lambrecht97] $$\begin{aligned}
r{_\mathrm{AB}} &=&r{_\mathrm{A}}+\frac{t{_\mathrm{A}}^{2} r{_\mathrm{B}}}
{1-\overline{r}{_\mathrm{A}} r{_\mathrm{B}}} \qquad
\overline{r}{_\mathrm{AB}} =\overline{r}{_\mathrm{B}}
+\frac{\overline{r}{_\mathrm{A}} t{_\mathrm{B}}^{2}}
{1-\overline{r}{_\mathrm{A}} r{_\mathrm{B}}} \nonumber \\
t{_\mathrm{AB}} &=&\frac{t{_\mathrm{A}} t{_\mathrm{B}}}
{1-\overline{r}{_\mathrm{A}} r{_\mathrm{B}}}
\label{compoSlab}\end{aligned}$$ For readibility, we have specified the networks by using subscripts rather than brackets. We will proceed similarly in forthcoming specific computations. Iterating this composition law, we can compute the scattering amplitudes for any dielectric multilayer. This systematic technique is quite similar to the classical computation techniques used for studying multilayers [@Abeles55]. It is generalized to the full quantum treatment in the next section. It also leads in the following to general results constraining the variation of the Casimir force for arbitrary dielectric mirrors. It reproduces the known results for the multilayer systems which have already been studied [@Zhou95; @Bordag01].
Quantum treatment of lossy mirrors
==================================
Up to now, we have performed a classical analysis which is not sufficient for the purpose of describing the scattering of vacuum fluctuations. Real mirrors consist of absorbing media which scatter incident fields to spontaneous emission modes and reciprocally scatter fluctuations from noise modes to the modes of interest. The $S-$matrix calculated previously cannot be unitary for a lossy mirror but it should be the restriction to the modes of interest of a larger $S-$matrix which includes the noise modes and obeys unitarity. In the present section, we characterize the additional fluctuations for a lossy mirror by using the corresponding ‘optical theorem’, that is also the unitarity of the larger $S-$matrix (see [@Barnett96; @Courty00] and references therein).
We assume that the scattering restricted to the modes of interest still fulfills the symmetry of plane mirrors considered in the previous classical calculations. This amounts to neglect multiple scattering processes which could couple different modes through their coupling with noise modes. Except for this assumption, we consider arbitrary dissipative media and discuss the optical theorem in the scattering and transfer points of view. We use the latter one to deal with composition of additional fluctuations when lossy mirrors are piled up.
Noise in the scattering approach
--------------------------------
Should we use the previous classical equations for the quantum amplitudes, we would find that the output fields cannot obey the canonical commutators, except in the particular case of lossless mirrors. This implies that the input/output transformation for quantum field must include additional fluctuations superimposed to the classical equations $$\left| e^\mathrm{out} \right\rangle = S \left| e^\mathrm{in}\right\rangle
+ \left| F\right\rangle
\label{SmatrixNoise}$$ $\left| e^\mathrm{out} \right\rangle$ and $\left| e^\mathrm{in} \right\rangle$ are defined as in (\[Smatrix\]) with the quantum amplitudes $e$ in place of the classical fields $\mathcal{E}$, $S$ is the same matrix as previously and $\left| F\right\rangle$ is a twofold column matrix describing the additional fluctuations. All these quantities depend on the quantum number $m$ which is common to all fields coupled in the scattering process.
The additional fluctuations are linear superpositions of all modes coupled to the main modes $e_{m}^{\phi }$ by the microscopic couplings which cause absorption. As an example, the atoms constituting a dielectric medium couple the main modes to all electromagnetic modes through spontaneous emission processes, represented symbolically by the wavy arrows on Figure \[FigAddFluct\]. The stationarity assumption implies that only modes having the same frequencies are coupled. In particular, it forbids parametric couplings which could couple modes with different frequencies and ‘squeeze’ the vacuum fluctuations [@Reynaud90]. The whole scattering matrix which takes into account all coupled field modes is unitary and this basic property makes the canonical commutation relations compatible for input and output fields. In contrast, the reduced scattering matrix containing only the classical scattering amplitudes coupling the main modes $e_{m}^{\phi }$ is not unitary, except in the particular case of lossless mirrors.
In order to write the unitarity property of the whole scattering matrix, it is convenient to represent the additional fluctuations $\left| F\right\rangle$ by introducing auxiliary noise modes $\left| f \right\rangle$ and auxiliary noise amplitudes gathered in a noise matrix $S^\prime$ $$\left| F\right\rangle =S^\prime \left| f\right\rangle \qquad
S^\prime =\left(
\begin{tabular}{ll}
$r^\prime $ & $t^\prime $ \\
$t^\prime $ & $\overline{r}^\prime $\end{tabular}
\right)$$ The components of the twofold column $\left| f\right\rangle$ are defined to have the same canonical commutators as the input fields in the main modes. In fact, they are linear superpositions of the input vacuum modes responsible for the fluctuation process. They are defined up to an ambiguity : any canonical transformation of the noise modes leads to an equivalent representation of the additional fluctuations, which corresponds to a different form for the noise amplitudes while leading to the same physical results at the end of the computations.
For any of these equivalent representations, the norm matrix $S^{\prime
}S^{\prime \ \dagger }$ has the same expression determined by the optical theorem, that is the unitarity condition for the whole scattering process, $$SS^\dagger +S^\prime S^{\prime \ \dagger }=I \label{unitarity}$$ where $I$ is the $2 \times 2$ unity matrix. This is easily proven by a direct inspection of the explicit expressions of the commutators of the output fields. The same inspection shows that noise modes corresponding to different values of $m$ are not correlated to each other. Condition (\[unitarity\]) is made more explicit when $SS^\dagger$ and $S^\prime S^{\prime \ \dagger}$ are developed in terms of scattering amplitudes $$\begin{aligned}
rr^{\ast }+tt^{\ast }+r^\prime r^{\prime \ast }+t^\prime t^{\prime \ast
} &=&tt^{\ast }+\overline{r}\overline{r}^{\ast }+t^\prime t^{\prime \ast }+
\overline{r}^\prime \overline{r}^{\prime \ast } \nonumber \\
&=&1 \nonumber \\
rt^{\ast }+t\overline{r}^{\ast }+r^\prime t^{\prime \ast }+t^\prime
\overline{r}^{\prime \ast } &=&tr^{\ast }+\overline{r}t^{\ast }+t^{\prime
}r^{\prime \ast }+\overline{r}^\prime t^{\prime \ast } \nonumber \\
&=&0\end{aligned}$$ More detailed discussions are presented for the case of the slab in appendix \[AppSlab\].
The description of noise may as well be represented with the alternative representation (\[SmatrixTilde\]) of the scattering process $$\begin{aligned}
&&\widetilde{\left| e^{\mathrm{out}} \right\rangle } =
\widetilde{S}\left| e^{\mathrm{in}}\right\rangle
+\widetilde{\left| F\right\rangle } \nonumber \\
&&\widetilde{\left| e^{\mathrm{out}} \right\rangle } =
\eta \left| e^{\mathrm{out}} \right\rangle \qquad
\widetilde{\left| F\right\rangle } = \eta \left| F\right\rangle
\label{SmatrixTildeNoise}\end{aligned}$$ The additional fluctuations are then represented in terms of the same noise modes and of a modified noise matrix $$\begin{aligned}
\widetilde{\left| F\right\rangle } &=&
\widetilde{S^\prime} \left| f \right\rangle \qquad
\widetilde{S^\prime} = \eta S^\prime \nonumber \\
&&\widetilde{S^\prime} \widetilde{S^\prime} ^\dagger =
I - \widetilde{S}\widetilde{S}^\dagger
\label{unitarityTilde}\end{aligned}$$
Noise in the transfer approach
------------------------------
We now present the description of additional fluctuations in the transfer approach. Performing the same manipulations as in the previous section, we transform equation (\[SmatrixTildeNoise\]) into $$\left( \pi _{-}-\widetilde{S}\pi _{+} \right) \left| e_{\mathrm{L}}
\right\rangle = -\left( \pi _{+}-\widetilde{S}\pi _{-}\right) \left|
e_{\mathrm{R}}\right\rangle +\widetilde{\left| F\right\rangle }$$ We thus get transfer equations with additional fluctuations described by a twofold column $\left| G\right\rangle$ $$\begin{aligned}
&&\left| e_{\mathrm{L}}\right\rangle = T \left| e_{\mathrm{R}}\right\rangle
+ \left| G\right\rangle \nonumber \\
&&\left| G\right\rangle = \left(\pi _{-}-\widetilde{S}\pi _{+}\right) ^{-1}
\widetilde{\left| F\right\rangle }
\label{FtoG}\end{aligned}$$ The $T-$ matrix has the same expression (\[StoT\]) as previously and the additional fluctuations $\left| G\right\rangle $ are a linear expression of the fluctuations $\left| F\right\rangle $ defined in the scattering approach. This linear relation may be written under alternative forms by using the relations (\[SandTsym\]) $$\begin{aligned}
\left| F\right\rangle &&= \left(\pi _{-}-\widetilde{S}\pi _{+}\right)
\widetilde{\left| G\right\rangle }
= \left( \pi _{-}-T\pi_{+}\right) ^{-1}
\widetilde{\left| G\right\rangle } \nonumber \\
\left| G\right\rangle &&= \left( \pi _{-}-T\pi_{+}\right)
\widetilde{\left| F\right\rangle } \end{aligned}$$
In the scattering approach, the norm of additional fluctuations is described by matrices $S^\prime S^{\prime \ \dagger}$ and $\widetilde{S^\prime} \widetilde{S^\prime} ^\dagger$ which are themselves determined by the optical theorem (\[unitarity\]) or (\[unitarityTilde\]). In order to translate these properties to the transfer approach, we rewrite (\[FtoG\]) in terms of the canonical noise modes $\left| f\right\rangle$ and of noise amplitudes gathered in a matrix $T^\prime $ $$\begin{aligned}
&&\left| G\right\rangle =T^\prime \left| f\right\rangle \nonumber \\
T^\prime &&=\left( \pi _{-} - \widetilde{S}\pi _{+}\right) ^{-1}
\widetilde{S^\prime }=\left( \pi _{-}-T\pi _{+}\right) \widetilde{S^\prime }\end{aligned}$$ The associated norm matrix is $$\begin{aligned}
T^\prime T^{\prime \ \dagger}&&=\left( \pi _{-}-T\pi _{+}\right)
\widetilde{S^\prime } \widetilde{S^\prime } ^\dagger
\left( \pi _{-}-T\pi _{+}\right) ^\dagger \end{aligned}$$ Using equations (\[unitarityTilde\]) and (\[TtoS\]), we rewrite it as $$T^\prime T^{\prime \ \dagger }=T\Phi T^\dagger - \Phi \qquad
\Phi =\pi _{+}-\pi _{-}
\label{unitaryT}$$ $\Phi $ is a diagonal matrix with two eigenvalues representing the directions of propagation $\phi=\pm 1$ of the field.
Composition of dissipative networks
-----------------------------------
Using these tools, we now write composition laws for the fluctuations and their norms.
We start from transfer equations written for each network A and B $$\begin{aligned}
\left| e_{\mathrm{L}} {\{\mathrm{A}\}} \right\rangle &=& T {\{\mathrm{A}\}} \left| e_{\mathrm{R}}
{\{\mathrm{A}\}} \right\rangle +\left| G {\{\mathrm{A}\}} \right\rangle \nonumber \\
\left| e_{\mathrm{L}} {\{\mathrm{B}\}} \right\rangle &=& T {\{\mathrm{B}\}} \left| e_{\mathrm{R}}
{\{\mathrm{B}\}} \right\rangle +\left| G {\{\mathrm{B}\}} \right\rangle\end{aligned}$$ Using the identifications (\[identifyCompo\]) associated with the composition law, we deduce for the composed network $$\begin{aligned}
\left| e_{\mathrm{L}} {\{\mathrm{AB}\}} \right\rangle &=&T {\{\mathrm{AB}\}} \left| e_{\mathrm{R}}
{\{\mathrm{AB}\}} \right\rangle +\left| G {\{\mathrm{AB}\}} \right\rangle \nonumber \\
\left| G {\{\mathrm{AB}\}} \right\rangle &=&
\left| G {\{\mathrm{A}\}} \right\rangle +T {\{\mathrm{A}\}} \left| G {\{\mathrm{B}\}}\right\rangle
\label{compoG}\end{aligned}$$ The fluctuations $\left| G {\{\mathrm{AB}\}} \right\rangle$ are a linear superposition of fluctuations $\left| G {\{\mathrm{A}\}} \right\rangle$ and $\left| G {\{\mathrm{B}\}} \right\rangle$ added in A and B.
In order to obtain the composition law for the norm matrices, we develop the additional fluctuations $\left| G {\{\mathrm{AB}\}} \right\rangle$ on the canonical noise modes associated with the two elements $$\left| G {\{\mathrm{AB}\}} \right\rangle = T^\prime {\{\mathrm{A}\}} \left| f {\{\mathrm{A}\}}
\right\rangle + T {\{\mathrm{A}\}} T^\prime {\{\mathrm{B}\}} \left| f{\{\mathrm{B}\}} \right\rangle$$ Since the noise modes associated with different elements are uncorrelated, $\left| G {\{\mathrm{AB}\}} \right\rangle$ may be rewritten in terms of new canonical noise modes and new noise amplitudes such that $$\begin{aligned}
\left| G {\{\mathrm{AB}\}} \right\rangle &=& T^\prime{\{\mathrm{AB}\}}
\left| f {\{\mathrm{AB}\}} \right\rangle \nonumber \\
T^\prime{\{\mathrm{AB}\}} T^\prime {\{\mathrm{AB}\}} ^\dagger &=&T^\prime{\{\mathrm{A}\}}
T^\prime {\{\mathrm{A}\}} ^\dagger \nonumber \\
&+&T{\{\mathrm{A}\}} T^\prime{\{\mathrm{B}\}}
T^\prime {\{\mathrm{B}\}} ^\dagger T{\{\mathrm{A}\}} ^\dagger
\label{compoNoiseT}\end{aligned}$$ Using expression (\[unitaryT\]) of the optical theorem for both networks A and B, we deduce that the composed network AB obeys the same relation $$\begin{aligned}
T^\prime{\{\mathrm{AB}\}} T^\prime {\{\mathrm{AB}\}} ^\dagger &&=
T {\{\mathrm{A}\}} \Phi T{\{\mathrm{A}\}} ^\dagger - \Phi \nonumber \\
&&+ T {\{\mathrm{A}\}} \left( T {\{\mathrm{B}\}} \Phi T{\{\mathrm{B}\}} ^\dagger - \Phi \right)
T{\{\mathrm{A}\}} ^\dagger \nonumber \\
&&= T {\{\mathrm{AB}\}} \Phi T {\{\mathrm{AB}\}} ^\dagger - \Phi\end{aligned}$$ Equivalently, the $S-$matrix of the composed network AB obeys the optical theorem (\[unitarity\]) as soon as the two networks A and B do.
Resonance for cavity fields
---------------------------
We have studied the scattering or, equivalently, the lefthand/righthand transfer of fields by a composed network AB. We want now to characterize the properties of the fields inside the cavity formed between A and B. This problem will play a key role in the evaluation of the Casimir force (see next section).
The situation is illustrated by Figure \[FigCavity\] which, in contrast to Figure \[FigComposition\], keeps the trace of the intracavity fields. In algebraic terms, the cavity fields are defined by rewriting the identifications (\[identifyCompo\]) as $$\begin{aligned}
&&\left| e_{\mathrm{L}} {\{\mathrm{AB}\}} \right\rangle \equiv
\left| e_{\mathrm{L}} {\{\mathrm{A}\}} \right\rangle \qquad
\left| e_{\mathrm{R}} {\{\mathrm{AB}\}} \right\rangle \equiv
\left| e_{\mathrm{R}} {\{\mathrm{B}\}} \right\rangle \nonumber \\
&&\left| e_{\mathrm{C}} {\{\mathrm{AB}\}} \right\rangle \equiv
\left| e_{\mathrm{R}} {\{\mathrm{A}\}} \right\rangle =
\left| e_{\mathrm{L}} {\{\mathrm{B}\}} \right\rangle \end{aligned}$$ From now on, we drop the label ${\{\mathrm{AB}\}}$ for the composed network and use subscripts for the networks A and B.
In order to express the cavity fields in terms of the input modes and additional fluctuations, we first write the cavity fields $\left| e_\mathrm{C} \right \rangle$ in terms of the righthand ones $\left| e_\mathrm{R} \right \rangle$ $$\left| e_{\mathrm{C}} \right \rangle =T {_\mathrm{B}} \left| e_{\mathrm{R}} \right\rangle
+\left| G{_\mathrm{B}} \right\rangle$$ We then identify the two components of $\left| e_\mathrm{R} \right \rangle$ as $$\begin{aligned}
\pi _{+}\left| e_{\mathrm{R}}\right\rangle &=& \pi _{+}\widetilde{\left|
e^{\mathrm{out}}\right\rangle }=\pi _{+}\left( \widetilde{S}\left| e^{\mathrm{in}}
\right\rangle +\widetilde{ \left| F\right\rangle }\right) \nonumber \\
\pi_{-}\left| e_{\mathrm{R}}\right\rangle &=& \pi _{-}\left| e^{\mathrm{in}}\right\rangle\end{aligned}$$ Using the expression of $\widetilde{ \left| F\right\rangle}$ in terms of $\left| G\right\rangle$ and the composition law (\[compoG\]) for $\left| G\right\rangle$, we deduce $$\begin{aligned}
&&\left| e_{\mathrm{C}}\right\rangle =
R \left| e^{\mathrm{in}}\right\rangle + R^\prime {_\mathrm{A}} \left| f{_\mathrm{A}}
\right\rangle + R^\prime {_\mathrm{B}} \left| f {_\mathrm{B}} \right\rangle \nonumber \\
&&R = T{_\mathrm{B}} N \qquad
N= \left( \pi _{+}\widetilde{S}+\pi _{-}\right) =
\left( \pi _{-}+\pi _{+}T\right) ^{-1} \nonumber \\
&&R^\prime {_\mathrm{A}} = T{_\mathrm{B}} P T^\prime {_\mathrm{A}} \qquad
P= - N \pi _{+} \nonumber \\
&&R^\prime {_\mathrm{B}} = \left( I + T{_\mathrm{B}} P T{_\mathrm{A}} \right)
T^\prime {_\mathrm{B}} \end{aligned}$$
As already explained, the unitarity of scattering entails that the output fields have the same commutators as the input ones. But this is not the case for the cavity fields which have their commutators determined by the matrix $$\mathcal{G} = RR^\dagger +R^\prime {_\mathrm{A}} R^{\prime\ \dagger} {_\mathrm{A}}
+R^\prime {_\mathrm{B}} R^{\prime\ \dagger} {_\mathrm{B}}$$ Expanding this quadratic form and using the composition law (\[compoNoiseT\]), we rewrite $\mathcal{G}$ as $$\begin{aligned}
\mathcal{G} &=& T{_\mathrm{B}} NN^\dagger T{_\mathrm{B}}^\dagger +T{_\mathrm{B}}
PT^\prime T^{\prime\ \dagger } P^\dagger T{_\mathrm{B}}^\dagger \nonumber \\
&&+ T{_\mathrm{B}} P T{_\mathrm{A}} T^\prime {_\mathrm{B}} T ^{\prime\ \dagger}
{_\mathrm{B}} \nonumber \\
&&+ T^\prime {_\mathrm{B}} T^{\prime\ \dagger} {_\mathrm{B}} T{_\mathrm{A}}^\dagger
P^\dagger T{_\mathrm{B}}^\dagger \nonumber \\
&&+ T^\prime {_\mathrm{B}} T^{\prime\ \dagger} {_\mathrm{B}} \end{aligned}$$ Using relation (\[unitaryT\]) for the three networks A, B and AB, we obtain a simpler expression after a few rearrangements $$\mathcal{G} = - \Phi - T{_\mathrm{B}} P T{_\mathrm{A}} \Phi
-\Phi T{_\mathrm{A}}^\dagger P^\dagger T{_\mathrm{B}}^\dagger$$
We now proceed to explicit calculations of these matrices. We note that $P = - t \pi_+$ where $t$ is the transmission amplitude of the network AB and deduce $$-T{_\mathrm{B}} P T{_\mathrm{A}}\Phi =
t \left(
\begin{tabular}{ll}
$a{_\mathrm{B}} a{_\mathrm{A}}$ & $-a{_\mathrm{B}} b{_\mathrm{A}}$ \\
$c{_\mathrm{B}} a{_\mathrm{A}}$ & $-c{_\mathrm{B}} b{_\mathrm{A}}$\end{tabular}
\right)$$ $t$ is simply the inverse of the transfer amplitude $a$ associated with the network AB (see eq.\[SandTamplitudes\]) and the latter is deduced from the composition law (\[compoT\]) $$t=\frac 1{a} \qquad a =a{_\mathrm{A}} a{_\mathrm{B}}
+b{_\mathrm{A}} c{_\mathrm{B}}$$ Then, the transfer amplitudes of the networks A and B may be substituted by the associated scattering amplitudes, leading to $$-T{_\mathrm{B}} P T{_\mathrm{A}}\Phi =\frac{1}{1-\overline{r}{_\mathrm{A}} r{_\mathrm{B}}}
\left(
\begin{tabular}{cc}
$1$ & $\overline{r}{_\mathrm{A}}$ \\
$r{_\mathrm{B}}$ & $\overline{r}{_\mathrm{A}} r{_\mathrm{B}}$\end{tabular}
\right)$$ Collecting these results and proceeding to slight rearrangements, we finally get $$\begin{aligned}
\mathcal{G} &=&I+\frac{1}{1-\overline{r}{_\mathrm{A}} r{_\mathrm{B}}} \left(
\begin{tabular}{cc}
$\overline{r}{_\mathrm{A}} r{_\mathrm{B}}$ & $\overline{r}{_\mathrm{A}}$ \\
$r{_\mathrm{B}}$ & $\overline{r}{_\mathrm{A}} r{_\mathrm{B}}$\end{tabular}
\right) \nonumber \\
&&+\frac{1}{\left( 1-\overline{r}{_\mathrm{A}} r{_\mathrm{B}}\right) ^\ast }
\left(
\begin{tabular}{cc}
$\overline{r}{_\mathrm{A}} r{_\mathrm{B}}$ & $\overline{r}{_\mathrm{A}}$ \\
$r{_\mathrm{B}}$ & $\overline{r}{_\mathrm{A}} r{_\mathrm{B}}$\end{tabular}
\right) ^\dagger \end{aligned}$$ In the following we will use the diagonal terms of the matrix $\mathcal{G}$ to evaluate the Casimir force.
Scattering on a Fabry-Perot cavity
----------------------------------
In order to prepare the evaluation of the Casimir force, we generalize the preceding expression to the case of the Fabry-Perot cavity containing a zone of field propagation between the two mirrors M1 and M2 (see Figure \[FigFabryPerot\]).
The distance between the two mirrors is denoted $L$ and the cavity fields are defined at an arbitrary position inside the cavity, say at distances $L_1$ from M1 and $L_2$ from M2 with $L_1 + L_2 = L$. In these conditions, the study of the Fabry-Perot cavity is reduced to the problem studied in the preceding subsection through the following identifications : the network A contains the mirror M1 and the propagation L1 with $T {_\mathrm{A}} = T{_\mathrm{M1}} T{_\mathrm{L1}}$ while the network B contains the propagation L2 and the mirror M2 with $T {_\mathrm{B}} = T{_\mathrm{L2}} T{_\mathrm{M2}}$. The transfer amplitudes for the networks A and B are derived from those corresponding to M1 and M2 and from phase factors corresponding to the propagations L1 and L2 $$\begin{aligned}
t{_\mathrm{A}} &=& \overline{t}{_\mathrm{A}} = t_1 e^{-\alpha _1} \qquad
\alpha_1 = \kappa_0 L_1 \nonumber \\
\overline{r}{_\mathrm{A}}&=&\overline{r}_1 e^{-2\alpha _1} \qquad
r{_\mathrm{A}}=r_1 \nonumber \\
t{_\mathrm{B}} &=& \overline{t}{_\mathrm{B}} = e^{-\alpha _2} t_2 \qquad
\alpha_2 = \kappa_0 L_2 \nonumber \\
\overline{r}{_\mathrm{B}} &=& \overline{r}_2 \qquad r{_\mathrm{B}} = r_2 e^{-2\alpha _2}\end{aligned}$$ We have labeled the amplitudes for the mirrors M1 and M2 with mere indices 1 and 2; $\kappa_0$ is defined in vacuum. These results entail that the reflection amplitudes $\overline{r}{_\mathrm{A}}$ and $r{_\mathrm{B}}$ are seen from a point inside the cavity as the product of phase factors by the reflection amplitudes $\overline{r}_1$ and $r_2$ seen from a point in the immediate vicinity of M1 and M2.
We then deduce the scattering amplitudes for the whole cavity $$\begin{aligned}
r &=& r_1 + \frac{t_1^2 r_2 e^{-2\alpha}}{D} \qquad
\overline{r} = \overline{r}_2 + \frac{\overline{r}_1 t_2^2 e^{-2\alpha}}{D} \nonumber \\
t &=& \overline{t} = \frac{t_1 t_2 e^{-\alpha}}{D} \nonumber \\
D &=& 1 - \overline{r}_1 r_2 e^{-2\alpha} \qquad \alpha =\alpha _1+\alpha _2 \end{aligned}$$ and the expression of $\mathcal{G}$ $$\begin{aligned}
\mathcal{G} &=&I+\frac{1}{D}\left(
\begin{tabular}{cc}
$\overline{r}_1 r_2 e^{-2\alpha}$ & $\overline{r}_1 e^{-2\alpha_1}$ \\
$r_2 e^{-2\alpha_2}$ & $\overline{r}_1 r_2 e^{-2\alpha}$ \end{tabular}
\right) \nonumber \\
&&+\frac{1}{D^\ast}\left(
\begin{tabular}{cc}
$\overline{r}_1 r_2 e^{-2\alpha}$ & $\overline{r}_1 e^{-2\alpha_1}$ \\
$r_2 e^{-2\alpha_2}$ & $\overline{r}_1 r_2 e^{-2\alpha}$ \end{tabular}
\right) ^\dagger
\label{Gmatrix}\end{aligned}$$ The diagonal terms in the matrix $\mathcal{G}$ coincide with the Airy function $$\begin{aligned}
&&g=1+f+f^{\ast }=\frac{1-\left| \overline{r}_{1}r_{2}e^{-2\alpha }\right| ^{2}}
{\left| 1-\overline{r}_{1}r_{2}e^{-2\alpha }\right| ^{2}} \nonumber \\
&&f=\frac{\overline{r}_{1}r_{2}e^{-2\alpha }}{1-\overline{r}_{1}r_{2}e^{-2\alpha}}
\label{gfunction}\end{aligned}$$ This result will play the central role in the derivation of the Casimir force in the next section. It means that the commutators of the intracavity fields are not the same as those of the input or output fields. They correspond to a spectral density modified through a multiplication by the Airy function $g$. This is the basic property used in Cavity Quantum ElectroDynamics [@Haroche84].
It is clear from the present derivation that this result has a quite general status : it is obtained for any inner field in any composed network, assuming the symmetry of plane mirrors. This property was already known for non absorbing mirrors [@Jaekel91] and for lossy mirrors symmetrical with respect to their mediane plane [@Barnett98]. The present derivation proves that it is also valid for arbitrary dielectric multilayers with dissipation. The final result only depends on the reflection amplitudes $\overline{r}_{1}$ and $r_{2}$ of the mirrors as they are seen from the inner side of the cavity. The reflection amplitudes seen from the outer side and the transmission amplitudes do not appear in expressions (\[Gmatrix\],\[gfunction\]). This can be interpreted as resulting from the unitarity of the whole scattering processes.
Casimir force between real mirrors
==================================
We may now deal with the radiation pressure of vacuum fields on the mirrors of a Fabry-Perot cavity. We show that the resulting Casimir force is a regular integral which can be written over real or imaginary frequencies. We then derive general constraints obeyed by the Casimir force for arbitrary dielectric mirrors.
Vacuum radiation pressure
-------------------------
If we first consider a mirror isolated in vacuum, the radiation pressure is obtained by adding the contributions of the 4 fields coupled in the scattering process $$\begin{aligned}
\left\langle P\right\rangle _{\mathrm{vac}} &=&\sum_{m}\ \hbar \omega _{m}\
\cos ^{2}\theta _{m}\ \left\langle e_{m\ \mathrm{L}}^{\rightarrow }\cdot e_{m\
\mathrm{L}}^{\rightarrow \ \dagger }+e_{m\ \mathrm{L}}^{\leftarrow }\cdot e_{m\
\mathrm{L}}^{\leftarrow \ \dagger }-e_{m\ \mathrm{R}}^{\rightarrow }\cdot e_{m\
\mathrm{R}}^{\rightarrow \ \dagger }-e_{m\ \mathrm{R}}^{\leftarrow }\cdot e_{m\
\mathrm{R}}^{\leftarrow \ \dagger }\right\rangle _{\mathrm{vac}}\end{aligned}$$ The identification of these fields is given by Figure (\[FigNetwork\]). We have developed the sum over $\phi $ and kept the symbol $m$ to represent the quantum numbers $\left( \omega ,\mathbf{k},p\right)$. We assume that the whole system is in vacuum, that is at zero temperature, so that the anticommutators of input fields are given by relation (\[anticommVacuum\]). Since the commutators are the same for the output and input fields, the vacuum radiation pressure vanishes in the case of an isolated mirror. In other words, the two sides of the mirror play equivalent roles so that no mean force can appear.
When we consider two mirrors forming a Fabry-Perot cavity, the two sides of a given mirror are no longer equivalent since one is an inner side and the other an outer side. It follows that the compensation observed for an isolated mirror does no longer hold, resulting in the appearance of the Casimir force. In order to evaluate the force, we write the mean radiation pressures $\left\langle
P_{1}\right\rangle _{\mathrm{vac}}$ and $\left\langle P_{2}\right\rangle
_{\mathrm{vac}}$ on mirrors M1 and M2 (see Figure \[FigFabryPerot\]) $$\begin{aligned}
\left\langle P_{1}\right\rangle _{\mathrm{vac}} &=&\sum_{m}\ \hbar \omega _{m}\
\cos ^{2}\theta _{m}\ \left\langle e_{m\ \mathrm{L}}^{\rightarrow }\cdot e_{m\
\mathrm{L}}^{\rightarrow \ \dagger }+e_{m\ \mathrm{L}}^{\leftarrow }\cdot e_{m\
\mathrm{L}}^{\leftarrow \ \dagger }-e_{m\ \mathrm{C}}^{\rightarrow }\cdot e_{m\
\mathrm{C}}^{\rightarrow \ \dagger }-e_{m\ \mathrm{C}}^{\leftarrow }\cdot e_{m\
\mathrm{C}}^{\leftarrow \ \dagger }\right\rangle _{\mathrm{vac}} \nonumber \\
\left\langle P_{2}\right\rangle _{\mathrm{vac}} &=&\sum_{m}\ \hbar \omega _{m}\
\cos ^{2}\theta _{m}\ \left\langle e_{m\ \mathrm{C}}^{\rightarrow }\cdot e_{m\
\mathrm{C}}^{\rightarrow \ \dagger }+e_{m\ \mathrm{C}}^{\leftarrow }\cdot e_{m\
\mathrm{C}}^{\leftarrow \ \dagger }-e_{m\ \mathrm{R}}^{\rightarrow }\cdot e_{m\
\mathrm{R}}^{\rightarrow \ \dagger }-e_{m\ \mathrm{R}}^{\leftarrow }\cdot e_{m\
\mathrm{R}}^{\leftarrow \ \dagger }\right\rangle _{\mathrm{vac}} \end{aligned}$$
For the same reasons as previously, the field anticommutators are given by (\[anticommVacuum\]) for input and output fields. For intracavity fields, they are multiplied by the Airy function (\[gfunction\]) like the commutators $$\begin{aligned}
\left\langle e_{m ^\prime \ \mathrm{C}}^{\phi ^\prime }\cdot e_{m\ \mathrm{C}}^{\phi
\ \dagger }\right\rangle _{\mathrm{vac}}
&=&\frac{1}{2}\left[ e_{m ^\prime \ \mathrm{C}}^{\phi ^\prime } , e_{m\
\mathrm{C}}^{\phi \ \dagger }\right] \nonumber \\
&=&\frac{1}{2}g_{m} \delta _{mm^\prime }\delta _{\phi \phi ^\prime}
\label{anticommCav}\end{aligned}$$ As shown in the previous section, these expressions do not depend on the position inside the cavity where the cavity fields are defined. We finally deduce the mean radiation pressures on mirrors M1 and M2 $$\begin{aligned}
\left\langle P_{1}\right\rangle _{\mathrm{vac}}&=&
- \left\langle P_{2}\right\rangle _{\mathrm{vac}} \nonumber \\
&=& \sum_{m}\ \hbar \omega _{m} \cos ^{2}\theta _{m} \left( 1-g_{m}\right) \end{aligned}$$ At this point, it is worth emphasizing that we have assumed equilibrium at zero temperature for the whole system : not only the input fields but also any fluctuations associated with loss mechanisms inside the mirrors correspond to zero-point fluctuations, whatever their microscopic origin may be. Otherwise, the expression of the force discussed in the following would be affected.
The pressures have opposite values on the two mirrors M1 and M2. This entails that the global force exerted by vacuum upon the cavity vanishes, in consistency with the translational invariance of vacuum. In the following, we denote $F$ the Casimir force calculated for M1 when considering the limit of a large area $A\gg L^{2}$ $$F=A\left\langle P_{1}\right\rangle _{\mathrm{vac}}=A\sum_{m}\ \hbar \omega
_{m}\ \cos ^{2}\theta _{m}\ \left( 1-g_{m}\right)$$ The sign conventions used here are such that the positive value obtained below for $F$ corresponds to an attraction of the two mirrors to each other.
The force as an integral over real frequencies
----------------------------------------------
We now perform a change of variable to rewrite the summation symbol as specified in (\[mphi\]) $$\begin{aligned}
F&=&A\left\langle P_{1}\right\rangle _{\mathrm{vac}} \nonumber \\
&=&A\sum_{p}\int \frac{\mathrm{d}^{2} \mathbf{k}}{4\pi ^{2}}
\int \frac{\mathrm{d}\omega }{2\pi } \hbar k_{z}\
\left( 1-g_{\mathbf{k}}^{p}\left[ \omega \right] \right)
\label{CasimirAiry}\end{aligned}$$ We will now specify the domain of integration for $\omega $.
Up to now, we have discussed the scattering for ordinary waves which freely propagate in vacuum and correspond to frequencies $\omega$ larger than the bound $c\left| \mathbf{k}\right|$ fixed by the norm of the transverse wavevector. But we must also take into account the contribution of evanescent waves which correspond to frequencies $\omega$ smaller than $c\left| \mathbf{k}\right|$. These waves are fed by the additional fluctuations coming from the noise lines into the dielectric medium and propagating with an incidence angle larger than the limit angle. They are thus transformed at the interface into evanescent waves decreasing exponentially when the distance from the interface increases. As is well known [@BWevanescent], the properties of these evanescent waves are conveniently described through an analytical continuation of those of ordinary waves. This analytical continuation can only be dealt with in terms of functions having a well defined analyticity behaviour. This is not the case for the Airy function $g_{\mathbf{k}}^{p}\left[ \omega \right]$ but we know that this function is the sum (\[gfunction\]) of parts having well defined analyticity properties $$\begin{aligned}
g_{\mathbf{k}}^{p}\left[ \omega \right] &=&1+f_{\mathbf{k}}^{p}\left[ \omega \right]
+f_{\mathbf{k}}^{p}\left[ \omega \right] ^{\ast }
=\frac{1-\left| \rho _{\mathbf{k}}^{p}\left[ \omega \right] \right| ^{2}}
{\left| 1-\rho _{\mathbf{k}}^{p}\left[\omega \right] \right| ^{2}} \nonumber \\
f_{\mathbf{k}}^{p}\left[ \omega \right] &=&\frac{\rho _{\mathbf{k}}^{p}\left[ \omega \right] }
{1-\rho _{\mathbf{k}}^{p}\left[ \omega \right] } \nonumber \\
\rho _{\mathbf{k}}^{p}\left[ \omega \right] &=&r_{\mathbf{k},1}^{p}\left[ \omega \right]
r_{\mathbf{k},2}^{p}\left[ \omega \right] e^{-2\kappa _{0}L}
\label{loopfunctions}\end{aligned}$$ $\rho _{\mathbf{k}}^{p}\left[ \omega \right]$ is the ‘open loop function’ corresponding to one round trip of the field inside the cavity and defined as the product of the reflection amplitudes $r_{\mathbf{k},1}^{p}\left[\omega \right] $ and $r_{\mathbf{k},2}^{p}\left[ \omega \right] $ of the two mirrors and of the propagation phaseshift $e^{-2\kappa _{0}L}$; it is an analytical function in the physical domain of complex frequencies $\Re\xi > 0$ with the branch of the square root chosen so that $\Re\kappa >0$. Since the transverse wavevector is spectator throughout the whole scattering process, analyticity is defined with $\mathbf{k}$ fixed.
Then, $f_{\mathbf{k}}^{p}\left[ \omega \right] $ is the ‘closed loop function’ built up on the open loop function $\rho _{\mathbf{k}}^{p}\left[ \omega \right]$. It is also an analytical function, thanks to analyticity of the open loop and to a stability property which has a natural interpretation : the system formed by the Fabry-Perot cavity and the vacuum fluctuations is stable because neither the mirrors nor the vacuum would have the ability to sustain an oscillation. In some cases, the stability can be derived from a more stringent passivity property [@Lambrecht97] which may essentially be written $\left| \rho _{\mathbf{k}}^{p}\left[ \omega \right] \right| <1$. However, the passivity property is sometimes too stringent to be obeyed by real mirrors (see more detailed discussions in appendix \[AppEvan\]). In any case, the stability property, [*i.e.*]{} the absence of self sustained oscillations, is sufficient for the present derivation of the Casimir force.
We are now able to give more precise specifications of the domain of integration in (\[CasimirAiry\]). Using the decomposition (\[loopfunctions\]), we write the contribution of ordinary waves to this integral as the sum of two conjugated expressions $$\begin{aligned}
F_{\mathrm{ord}}&=&\mathcal{F}_{\mathrm{ord}}+\mathcal{F}_{\mathrm{ord}}^{\ast } \nonumber \\
\mathcal{F}_{\mathrm{ord}} &=&-A\sum_{p}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi^{2}}
\int_{c\left| \mathbf{k}\right| }^{\infty }\frac{\mathrm{d}\omega }{2\pi }
\hbar k_{z}\ f_{\mathbf{k}}^{p}\left[ \omega \right] \end{aligned}$$ The integral $\mathcal{F}_{\mathrm{ord}}$ is built on the retarded function $f_{\mathbf{k}}^{p}\left[ \omega \right] $ which may be extended through an analytical continuation from the sector of ordinary waves to that of evanescent waves. The contribution of evanescent waves to the force is thus obtained as $$\begin{aligned}
F_{\mathrm{eva}}&=&\mathcal{F}_{\mathrm{eva}}+\mathcal{F}_{\mathrm{eva}}^{\ast } \nonumber \\
\mathcal{F}_{\mathrm{eva}}&=&-A\sum_{p}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi^{2}}
\int_{0}^{c\left| \mathbf{k}\right| }\frac{\mathrm{d}\omega }{2\pi }
\hbar k_{z}\ f_{\mathbf{k}}^{p}\left[ \omega \right] \end{aligned}$$ The final expression of the Casimir force is the sum of the contributions of ordinary and evanescent waves that is also the integral over the whole axis of real frequencies $$\begin{aligned}
F&=&F_{\mathrm{ord}}+F_{\mathrm{eva}}=\mathcal{F}+\mathcal{F}^{\ast } \nonumber \\
\mathcal{F}&=&-A\sum_{p}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi ^{2}}
\int_{0}^{\infty }\frac{\mathrm{d}\omega }{2\pi }
\hbar i\kappa_0 \ f_{\mathbf{k}}^{p}\left[ \omega \right]
\label{ForceReal}\end{aligned}$$ As far as ordinary waves are concerned, this corresponds to the intuitive picture where the Casimir force results from the radiation pressure of vacuum fluctuations filtered by the cavity [@Jaekel91]. The contribution of evanescent waves is but the extension of the domain of integration to the whole real axis with the cavity response function $f_{\mathbf{k}}^{p}\left[ \omega \right]$ extended through an analytical continuation. In the evanescent sector, the cavity function $f_{\mathbf{k}}^{p}\left[ \omega \right]$ is written in terms of reflection amplitudes calculated for evanescent waves and exponential factors corresponding to evanescent propagation through the cavity. This means that it describes the ‘frustration’ of total reflection on one mirror due to the presence of the other. This explains why the radiation pressure of evanescent waves is not identical on the two sides of a given mirror and, therefore, how evanescent waves have a non null contribution to the Casimir force.
The force as an integral over imaginary frequencies
---------------------------------------------------
Using the Cauchy theorem, we now rewrite the Casimir force (\[ForceReal\]) as an integral over the axis of imaginary frequencies.
Since $\kappa_0 f_{\mathbf{k}}^{p}\left[ i\xi \right] $ is analytical in the domain $\Re\xi >0$, its integral over a closed contour lying in this domain has to vanish. We choose the contour drawn on Figure \[FigContour\] which consists of the positive part of the real axis including ordinary (C$_\mathrm{o}$) and evanescent (C$_\mathrm{e}$) waves, a quarter of circle C$_{\infty}$ with a very large radius and, finally, the imaginary axis C$_\mathrm{i}$ run from infinity to zero. Now the function $\kappa_0 f_{\mathbf{k}}^{p}\left[ i\xi \right] $ goes to zero for large values of the frequency, as a consequence of transparency at high frequency, a property certainly valid for any realistic model of optical mirror. Thanks to this property, the contribution to the integral of C$_{\infty}$ vanishes. We then deduce that the integrals over the real axis $\left[0,+\infty \right[$ and over the imaginary axis $\left[ 0,+i\infty \right[ $ are equal.
We thus get a new expression of the force $F$ as an integral over imaginary frequencies $\omega$, that is also as an integral over real values of $\xi$, $$\begin{aligned}
F&=& \mathcal{F} +\mathcal{F}^{\ast } = 2\mathcal{F} \nonumber \\
\mathcal{F}&=&A\sum_{p}\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi ^{2}}
\int_{0}^{\infty }\frac{\mathrm{d}\xi }{2\pi }
\hbar\kappa_0 \ f_{\mathbf{k}}^{p}\left[ i\xi \right] \nonumber \\
&&\kappa _{0}=\sqrt{\mathbf{k}^{2}+ \frac{\xi ^{2}}{c^2}}
\label{Force}\end{aligned}$$ We have used the fact that $\mathcal{F}$ is real, so that $\mathcal{F}^{\ast }$ is simply equal to $\mathcal{F}$. This property is less obvious, but also true, with $\mathcal{F}$ written as an integral over real frequencies. We wish to emphasize more generally that expression (\[Force\]) is mathematically equivalent to (\[ForceReal\]). The former expression is closer to the physical intuition whereas the latter is better adapted to explicit computations of the force.
Expression (\[Force\]) gives the Casimir force between real mirrors described by arbitrary frequency dependent reflection amplitudes. It is a regular integral as soon as these amplitudes obey the physical assumptions used in the derivation : causality, unitarity and high frequency transparency for each mirror, stability of the system formed by the two mirrors and the scattered vacuum fields. The demonstration holds for dissipative mirrors and not only for lossless ones.
The limit of perfect mirrors is obtained in expression (\[Force\]) by letting the reflection amplitudes go to unity, which leads to the Casimir formula (\[eqCasimir\]). This can be considered as an alternative demonstration of the Casimir formula without any reference to a renormalization or regularization technique. Basically, the properties of real mirrors, in particular their high frequency transparency, are sufficient to provide a regular expression of the force, as it was guessed a long time ago by Casimir [@Casimir48].
As a simple model of the mirrors used in the experiments, let us consider a metallic slab with a large width, that is a width $\ell $ larger than a few plasma wavelengthes. We use expression (\[Force\]) of the force written as an integral over imaginary values of the frequencies ($\omega=i\xi$, $\xi$ real). Hence, the phase factor corresponding to one round trip inside the slab is a decreasing exponential with a real exponent $e^{-2\kappa_1\ell}$. For the plasma model (\[PlasmaModel\]), $\kappa_1$ is given by $\sqrt{ \frac{\xi^2}{c^2}+\frac{\omega_\mathrm{P}^2}{c^2}+\mathbf{k}^2}$ and it is larger than $\frac{2\pi}{\lambda_\mathrm{P}}$ for all values of $\xi$ and $\mathbf{k}$. When relaxation is taken into account, this is still the case except in a very narrow domain with values of $\xi$ and $\mathbf{k}$ both close to zero. This domain has a negligible contribution to the integral (\[Force\]) and it follows that the reflection amplitude of the slab may be replaced by the limiting expression obtained for the bulk. One thus recovers the Lifshitz expression for the Casimir force [@Lifshitz56] which is widely used for comparing experimental results with theoretical expectations [@Bordag01].
Constraints on the force
------------------------
We now deduce general constraints which invalidate proposals made for tayloring the Casimir force at will by using specially designed mirrors [@Iacopini93; @Ford93]. This generalizes to 3D space the results obtained for 1D space in [@Lambrecht97] to which the reader is referred for further discussions.
Expression (\[Force\]) is an integral over the axis of imaginary frequencies essentially determined by the reflection amplitudes $r_1\left[ i\xi \right]$ and $r_2\left[ i\xi \right]$ for $\xi$ real. These amplitudes always have a modulus smaller than unity, for arbitrary dielectric multilayers (see appendix \[AppPassiv\]). They are negative for arbitrary dielectric slabs (see appendix \[AppSlab\]) and we deduce from the composition law (\[compoSlab\]) that this is still the case for arbitrary dielectric multilayers. It follows that the product of the reflection amplitudes of the two mirrors is always positive with a modulus smaller than unity $$0<r_{1}\left[ i\xi \right] r_{2}\left[ i\xi \right] <1$$
From this, we deduce first that the Casimir force has an absolute value smaller than the value (\[eqCasimir\]) reached for perfect mirrors and that it remains attractive $$0\leq F\leq F_{\mathrm{Cas}}$$ We also derive that the Casimir force decreases as a function of the length $$\frac{\mathrm{d}F}{\mathrm{d}L}\leq 0$$ This means that the properties obeyed by real mirrors strongly constrain the possible variation of the Casimir force, contrarily to what might have been expected at first sight [@Iacopini93; @Ford93].
Note that we have considered mirrors used in the experiments which have electric permittivity but no magnetic permeability. Different results would be obtained with magnetic mirrors, precisely with one of the mirrors dominated by electric response and the other one by magnetic response. The product of the two reflection amplitudes would indeed be negative in this case and the Casimir force repulsive [@Boyer74; @Kupiszewska93; @Alves00; @Kenneth02].
Conclusion
==========
We have presented a derivation of the Casimir force between lossy mirrors characterized by arbitrary frequency dependent reflection amplitudes, in the Casimir geometry where the cavity is made with two parallel plane mirrors.
We have shown how mirrors and cavities may be dealt with by using a quantum theory of optical networks. We have deduced the additional fluctuations accompanying dissipation from expressions of the optical theorem adapted to quantum network theory. The optical theorem is equivalent to the unitarity of the whole scattering process which couples the modes of interest and the noise modes and it ensures that the quantum commutators of the output fields are the same as those of the input fields. The situation is different for the cavity fields which do not freely propagate. We have given a general proof of a theorem previously demonstrated in particular cases [@Jaekel91; @Barnett98] which states that the modification of the commutators is determined by the usual Airy function, that is the spectral density associated with the Fabry-Perot cavity. For arbitrary lossy mirrors, the spectral density is determined by the reflection amplitudes as they are seen by the intracavity fields. It determines the radiation pressure exerted by vacuum fluctuations upon the mirrors with repulsive and attractive contributions associated respectively with resonant or antiresonant frequencies. The Casimir force is then obtained as an integral over the whole axis of real frequencies, including the contribution of evanescent waves besides that of ordinary waves. It is equivalently expressed as an integral over imaginary frequencies. The derivation only uses a few general assumptions certainly valid for real optical mirrors, namely causality, unitarity, high-frequency transparency for each mirror and stability of the compound cavity-vacuum system. It leads to a finite result without any further reference to a regularization technique [@Jaekel91].
The formula obtained in the present paper for the Casimir force was already known [@Jaekel91] but its scope of validity is widened by the present demonstration. It has been used to discuss the effect of imperfect reflection for the metallic mirrors used in the experiments. Different descriptions of the optical response of metals have been used, from the crude application of the plasma model (\[PlasmaModel\]) to a more complete characterization of the dielectric constant derived from tabulated optical data and dispersion relations. This kind of calculations, discussed in great detail for the mirrors corresponding to the recent experiments (see for example [@Lambrecht00]), has not been reproduced here. Instead, we have presented general results valid for any real mirrors obeying the physical properties already evoked and shown that they strongly constrain the variation of the Casimir force.
In the present paper, we have restricted our attention on the limit of zero temperature although our work was partly motivated by a recent polemical discussion of the effect of temperature on the Casimir force between real mirrors [@Bostrom00; @Svetovoy00; @Bordag00; @Lamoreaux01c; @Sernelius01r; @Sernelius01c; @Bordag01r; @Klimchitskaya01; @Bezerra02; @Lamoreaux02]. Since contradictory results may have raised doubts about the validity and consistency of various derivations of the Casimir force, we have considered it was important to come back to the first principles in this derivation. This has been done in the present paper for the case of zero temperature. A follow-on publication will show how to include the effect of thermal fluctuations in the treatment in order to obtain an expression free from ambiguities for the Casimir force between arbitrary lossy mirrors at non zero temperatures.
The dielectric slab {#AppSlab}
===================
In this appendix, we discuss in more detail the specific case of the dielectric slab. We consider lossy as well as lossless slabs.
For a lossless dielectric medium, the permittivity $\varepsilon $ is real at real frequencies. For ordinary waves, $\kappa _{0}$ and $\kappa _{1}$ are purely imaginary, so that the impedance ratios are real for both polarizations. Hence $\beta$ is real ($\beta=\beta_r$) and $\alpha$ purely imaginary ($\alpha=i\alpha_i$) so that the scattering amplitudes (\[slab\]) are read as $$\begin{aligned}
t &=&\frac{\sinh \beta _{r}}{\sinh \left( \beta _{r}+i\alpha _{i}\right) } \nonumber \\
&=&\frac{\sinh \beta _{r}}{\sinh \beta _{r}\cos \alpha _{i}+i\cosh \beta
_{r}\sin \alpha _{i}} \nonumber \\
r &=&-\frac{\sinh \left( i\alpha _{i}\right) }{\sinh \left( \beta
_{r}+i\alpha _{i}\right) } \nonumber \\
&=&-\frac{i\sin \alpha _{i}}{\sinh \beta _{r}\cos
\alpha _{i}+i\cosh \beta _{r}\sin \alpha _{i}}\end{aligned}$$ The sum of the squared amplitudes is unity $\left| t\right| ^{2}+\left| r\right| ^{2}=1 $ while the reflexion and transmission amplitudes are in quadrature to each other $tr^{\ast }+rt^{\ast }=0$, which means that $S$ is a unitary $2\times 2$ matrix, as it was expected for a lossless mirror. This implies that the reflection amplitude has a modulus smaller than unity $\left| r\right| <1$. This property also holds for lossy mirrors thanks to positivity of dissipation (see appendix \[AppPassiv\]).
Unitarity is defined without ambiguity only in the case of ordinary waves. For a lossless slab and evanescent waves, $\kappa_0$ is real - it is just the inverse of the penetration length of evanescent wave in vacuum - whereas $\kappa _1$ remains purely imaginary. Hence $\beta$ as well as $\alpha$ are purely imaginary and it is no longer possible to obtain general bounds for the scattering amplitudes $$\begin{aligned}
t&=&\frac{\sinh \left( i\beta _{i}\right) }{\sinh \left( i\beta _{i}+i\alpha
_{i}\right) }=\frac{\sin \beta _{i}}{\sin \left( \beta _{i}+\alpha
_{i}\right) }\nonumber \\
r&=&-\frac{\sinh \left( i\alpha _{i}\right) }
{\sinh \left( i\beta _{i}+i\alpha _{i}\right) }=-\frac{\sin \alpha _{i}}
{\sin\left( \beta _{i}+\alpha _{i}\right) }\end{aligned}$$ In particular, $\left| r\right|$ does not remain always smaller than 1 (see more explicit discussions in appendix \[AppEvan\] with different results for the TE and TM polarizations).
For imaginary frequencies finally, $\beta $ and $\alpha $ are positive real numbers, for lossy as well as lossless slabs. In this case, general bounds are easily obtained for the amplitudes $$\begin{aligned}
&&0<t=\frac{\sinh \left( \beta _{r}\right) }{\sinh \left( \beta _{r}+\alpha
_{r}\right) }<1 \nonumber \\
&&0<-r=\frac{\sinh \left( \alpha _{r}\right) }
{\sinh \left( \beta _{r}+\alpha _{r}\right) }<1
\label{boundRTimaginary}\end{aligned}$$ The fact that $r$ is negative with a modulus smaller than unity plays an important role in the derivation of constraints on the Casimir force.
Interesting results are also obtained for the eigenvalues of the $S-$matrix, which have a simple form $s_\pm = r \pm t$ since the slab is symmetrical in the exchange of its two ports. In the sector of ordinary waves, unitarity (\[unitarity\]) has a simple form in terms of $s_\pm = r \pm t$ and of the similar quantities $s_\pm^\prime = r^\prime \pm t^\prime $ defined on the noise matrix $S^\prime $ $$\left| s_\pm \right| ^{2}+\left| s_\pm ^\prime \right| ^{2}=1
\label{unitaritySSprime}$$ For the lossless slab, $s_\pm$ have a unit modulus and $s_\pm ^\prime$ vanish. For a lossy slab, we have $$\left| s_\pm \right| ^{2} \leq 1
\label{boundS}$$ This can be considered as a consequence of (\[unitaritySSprime\]) with $\left| s_\pm^\prime \right| ^{2} \geq 0$. Equivalently, it can be considered that unitarity (\[unitaritySSprime\]) fixes the modulus of $s_\pm ^\prime$ when the modulus of $s_\pm$ is known.
Condition (\[boundS\]) will be found in appendix \[AppPassiv\] to express a passivity property for the slab, here for ordinary waves. This property still holds in the sector of imaginary frequencies, as a consequence of (\[boundRTimaginary\]) and of the following inequalities obeyed for all positive real numbers $\alpha$ and $\beta$ $$\begin{aligned}
\left| \sinh \beta \mp \sinh \alpha \right| \leq {\sinh \left( \alpha +\beta
\right) } \end{aligned}$$ Using the terms of appendix \[AppPassiv\], this means that the domain of passivity always includes the sectors of ordinary waves and imaginary frequencies, in the case of a dielectric slab. However, it does not necessarily include the sector of evanescent waves (see appendix \[AppEvan\]).
The sector of evanescent waves {#AppEvan}
==============================
Ordinary waves correspond to frequencies $\omega \geq c\left| \mathbf{k}\right|$ and real wavevectors $k_{z}$ whereas evanescent waves correspond to frequencies $\omega \leq c\left| \mathbf{k}\right|$ and imaginary values of $k_{z}$. Causal scattering amplitudes can be extended from ordinary to evanescent waves, by an analytical continuation through the physical domain of complex frequencies $\omega=i\xi$ with $\Re\xi > 0$ and $\Re\kappa >0$. The ‘energy conditions’ which bear on quadratic forms are not necessarily preserved in this process.
In order to illustrate the idea, let us consider the reflection amplitude (\[r01\]) at the interface between vacuum ($\varepsilon _0=1$) and a lossless dielectric medium ($\varepsilon _1$ real for $\omega$ real). In the sector of evanescent waves, $\kappa _1$ is imaginary and $\kappa _0$ real, so that $r$ and $\overline{r}$ are complex numbers with a unit modulus, that is also pure dephasings corresponding to the phenomenon of total reflection. Meanwhile, the transmission amplitudes differ from zero, which describes how evanescent waves in vacuum are fed by the fields coming from the dielectric medium with an incidence angle larger than the limit angle. In these conditions, it is clear that the condition $\left| r\right| ^{2}+\left| t\right| ^{2}\leq 1$ fails.
For the TE polarization, it turns out that $$\begin{aligned}
\left| r^\mathrm{TE}\right| \leq 1
\label{rTEeva}\end{aligned}$$ in the evanescent sector at the interface between vacuum and any dielectric medium. This property is always true in the sectors of ordinary waves and imaginary frequencies for an arbitrary mirror (see the appendices \[AppSlab\] and \[AppPassiv\]). Using high frequency transparency, it follows from the Phragmén-Lindelöf theorem [@Phragmen] that inequality (\[rTEeva\]) holds in the whole physical domain in the complex plane. This ensures that the closed loop function $f ^\mathrm{TE}$ is analytic and, in particular, has no pole in the domain $\Re\xi >0$. In other words, since the open loop gain is smaller than unity, the closed loop cannot reach the oscillation threshold, leading to the stability property used in the derivation of the Casimir force.
Although it seems quite natural, this argument is not valid in the general case. For metallic mirrors for example, the condition $\left| r \right| \leq 1$ is violated in the evanescent sector for TM modes. The reflection amplitude is even known to reach large resonant values at the plasmon resonances [@Barton79]. Of course, this does not prevent the stability property to be fulfilled : the Fabry-Perot cavity is in this case a stable closed loop built on an open loop exceeding the unit modulus but with a phase such that the oscillation threshold is not reached.
We stress again that the stability property is necessary in the derivation of the Casimir force since it entails that the closed loop function is properly defined in the evanescent sector. When the more stringent property $\left| r \right| \leq 1$ is also obeyed, it follows from expression (\[loopfunctions\]) that the Airy function, which has been defined with the significance of a positive spectral density on ordinary waves, remains positive in the evanescent sector. When the property $\left| r \right| \leq 1$ fails, the Airy function can no longer be thought of as a spectral density in the whole physical domain, but this does not invalidate the derivation of the Casimir force.
The domain of passivity {#AppPassiv}
=======================
In this appendix, we discuss the related but not identical properties corresponding to positivity of dissipation and passivity.
We consider an arbitrary mirror, that is a reciprocal network connecting two vacuum ports. For ordinary waves, we define the power dissipated by the mirror $$\begin{aligned}
\pi &=&\left( e_{\mathrm{L}}^{\mathrm{in}} {}^\dagger e_{\mathrm{L}}^{\mathrm{in}}
-e_{\mathrm{L}}^{\mathrm{out}} {}^\dagger e_{\mathrm{L}}^{\mathrm{out}} \right)
+\left( e_{\mathrm{R}}^{\mathrm{in}} {}^\dagger e_{\mathrm{R}}^{\mathrm{in}}
-e_{\mathrm{R}}^{\mathrm{out}} {}^\dagger e_{\mathrm{R}}^{\mathrm{out}}\right)
\nonumber \\
&=&\left\langle e^{\mathrm{in}}\right| \left| e^{\mathrm{in}}\right\rangle
-\left\langle e^{\mathrm{out}}\right| \left| e^{\mathrm{out}}\right\rangle\end{aligned}$$ where we have introduced row vectors conjugated to the column vectors $$\left\langle e^{\mathrm{out}}\right| =\left| e^{\mathrm{out}}\right\rangle
^\dagger \qquad \qquad \left\langle e^{\mathrm{in}}\right| =\left|
e^{\mathrm{in}}\right\rangle ^\dagger$$ This power is positive as a consequence of unitarity $$\begin{aligned}
\pi &=& \left\langle e^{\mathrm{in}}\right| I-S^\dagger S\left|
e^{\mathrm{in}}\right\rangle \nonumber \\
&=& \left\langle e^{\mathrm{in}}\right|
S^{\prime \ \dagger}S^\prime \left| e^{\mathrm{in}}\right\rangle \geq 0\end{aligned}$$ which corresponds to the positivity of the matrix $I- S^\dagger S$ $$\forall \left| e\right\rangle \qquad \qquad \left\langle e\right|
I-S^\dagger S\left| e\right\rangle \geq 0
\label{defPassiv}$$ where $\left| e\right\rangle $ represents arbitrary input fields. Positivity can also be expressed in terms of the eigenvalues $\ell $ of $S^\dagger S$ $$\det \left( S^\dagger S-\ell I\right) =0\qquad \qquad \ell \geq 0$$ These eigenvalues are always real and positivity of dissipation is equivalent to the fact that they are smaller than unity $$\ell \leq 1
\label{def2Passiv}$$
Passivity is a property directly related to positivity of dissipation but defined more generally for complex frequencies in the physical domain. In order to discuss it, we extend the matrix $S$ from the sector of ordinary waves through the analytical continuation already discussed. We extend $S^\dagger $ similarly, with the complex conjugation cautiously defined since it involves complex frequencies : conjugation corresponds to $\xi \rightarrow \xi ^\ast$ and $\kappa \rightarrow \kappa ^\ast$ and it preserves the physical domain $\Re\xi >0\ ,\ \Re\kappa >0$; the derivations performed for an amplitude in the domain $\Re\xi >0\ ,\ \Im\xi <0$ are thus translated to similar derivations for the conjugated amplitude in the quarter plane $\Re\xi >0\ ,\ \Im\xi >0$.
Then, the domain of passivity of $S$ is defined by the domain of $\xi$ for which $I-S^\dagger S$ is a positive matrix (eq.\[defPassiv\]) that is also for which the eigenvalues $\ell$ of $S^\dagger S$ are smaller than unity (eq.\[def2Passiv\]). An important feature of this property is that it is stable under composition : when two networks A and B are piled up as in Figure \[FigComposition\], the quadratic forms appearing in (\[defPassiv\]) simply add up so that passivity of the network AB follows from passivity of the two networks A and B. This is a special case of a general theorem [@Meixner] which states that networks built up with passive elements are passive.
Passivity means that the eigenvalues $1-\ell$ of the matrix $I-S^\dagger S$ are both positive, which is equivalent to the following inequalities $$\mathrm{Tr}\left( I-S^\dagger S\right) \geq 0 \qquad
\det \left(I-S^\dagger S\right) \geq 0$$ It may be written in terms of the scattering amplitudes $$\begin{aligned}
&&\left| r\right| ^{2}+\left| \overline{r}\right| ^{2}+2\left| t\right|
^{2}\leq 2\nonumber \\ &&\left| r\overline{r}-t^{2}\right| ^{2}\geq \left|
r\right| ^{2}+\left| \overline{r}\right| ^{2}+2\left| t\right| ^{2}-1\end{aligned}$$ Passivity implies that the scattering amplitudes have a modulus smaller than unity $$\left| r\right| \leq 1 \qquad \left| \overline{r}\right| \leq 1 \qquad
\left| t\right| \leq 1$$ Conversely, the latter conditions are necessary but not sufficient for passivity.
For a mirror symmetrical in the exchange of its two ports, a slab for example, the passivity conditions take the simple form $\left| r \pm t \right| ^{2} \leq 1$. The results of appendix \[AppSlab\] thus entail that the domain of passivity always includes the sectors of ordinary waves and imaginary frequencies, for arbitrary slabs (eq.\[boundS\]). Using the stability of passivity under composition, we deduce that this is also the case for arbitrary multilayers. It follows that the reflection amplitudes always have a modulus smaller than unity for imaginary frequencies ($\xi$ real) $$\left| r\left[ i\xi \right] \right| \leq 1
\label{passivImaginary}$$
Thanks are due to Gabriel Barton and Marc-Thierry Jaekel for their helpful comments.
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|
---
author:
- 'Daniel R. Hunter'
bibliography:
- 'ppsd.bib'
title: 'Derivation of the anisotropy profile, constraints on the local velocity dispersion, and implications for direct detection'
---
Introduction
============
The shape of the velocity distribution is an important influence on predictions for detection of galactic dark matter (DM) [@Bertone:2004pz], both indirect [@Robertson:2009bh; @Campbell:2010xc; @Ferrer:2013cla] and direct [@Gondolo:2002np; @Strigari:2009zb; @Catena:2011kv; @Frandsen:2011gi]. N-body simulations tell us the mass distribution of dark matter in galactic halos, but velocity data is more subject to numerical noise and is thus more difficult to measure. Making assumptions about the phase-space density, one can derive the velocity distribution from a mass density profile [@binney]. This usually involves constraining the anisotropy profile to a certain functional form [@Osipkov:1979; @Cuddeford:1991; @Gerhard:1991; @Baes:2002tw] (alternatively, see [@VanHese:2010qy]). Here we *derive* the anisotropy profile using only information from models of N-body simulations and the Jeans equation (see also [@Zait:2007es]). We impose the physical condition that the anisotropy profile does not rise above one inside the halo, and we discover that this constrains the velocity dispersion profile. In particular we discuss the *maximum value* implied for the local velocity dispersion. We assume halos are spherically symmetric (see [@Sparre:2012zk; @Wojtak:2013eia] for studies of velocity anisotropy in aspherical halos) and in equilibrium within their virial radius. It should be kept in mind that while any function that satisfies the collisionless Boltzmann equation also satisfies the Jeans equation, the converse is not necessarily true.
For some years now, it has been apparent that measurements of the pseudo-phase-space density (PPSD) of simulated halos follow a power-law over many decades of radius [@Taylor:2001bq; @VanHese:2008ce; @Ma:2009ek]. Some early work extrapolated central isotropy everywhere and derived the halo mass distribution from this power-law [@Hansen:2004gs; @Dehnen:2005cu; @Austin:2005ks]. Work has also been done to explain the dynamical origins of such a power-law. We will study how assuming a density profile and a PPSD power-law completely specifies the dispersion profile and anisotropy profile of a halo. The sensitivity to the precise slope of the PPSD power-law will be considered. This may be important since evidence for a PPSD power-law so far comes from DM-only simulations. It is becoming viable, however, to include complicated baryonic effects in simulations, which may result in a PPSD slope so far unmeasured or erase the power-law trend completely. Specifically we will use PPSD slopes of $2$, which corresponds to the isothermal profile, $35/18 = 1.9\overline{4}$, the critical value discussed by Dehnen & McLaughlin [@Dehnen:2005cu], and $15/8 = 1.875$, the value first found by Taylor & Navarro [@Taylor:2001bq]. We focus on a Milky Way-sized DM halo, which is specified by the halo parameters: the virial mass ${M_\mathrm{vir}}$, the scale radius ${r_\mathrm{s}}$, and the concentration $c \equiv {r_\mathrm{vir}}/{r_\mathrm{s}}$, and we use the profile by Navarro, Frenk, and White.
In Section \[sec:ani\_deriv\] we outline the derivation of the anisotropy profile from the Jeans equation, the halo profile, and the PPSD profile. Section \[sec:constraints\] shows how the anisotropy profile places upper limits on the local velocity dispersion. The anisotropy profile itself is presented in Section \[sec:ani\_profile\], accounting for uncertainty in the input parameters. Section \[sec:ani\_dist\] introduces an anisotropic model for the local velocity distribution, which is used to calculate basic predictions for a generic direct detection experiment. We conclude in Section \[sec:conclusions\], which is followed by appendices containing some details.
Deriving the Anisotropy Profile {#sec:ani_deriv}
===============================
We use the Navarro, Frenk, and White (NFW) profile [@Navarro:1996gj]. General expressions and some details specific to NFW are deferred to Appendix \[sec:details\]. The mass distribution is $$\rho(x) = \frac{{\rho_\mathrm{s}}}{x\left( 1+x \right)^2},
\label{eq:density}$$ where ${\rho_\mathrm{s}}$ is the scale density and $x \equiv r/{r_\mathrm{s}}$ is the dimensionless radius. Solving for the contained mass $M(x)$ gives us the scale density ${\rho_\mathrm{s}}$ in terms of the virial mass ${M_\mathrm{vir}}\equiv M(c)$ and concentration $c$.
Following Taylor and Navarro [@Taylor:2001bq] (also see [@Dehnen:2005cu]), we take the PPSD to be a power-law with negative slope ${\alpha}$: $$\frac{\rho}{{\sigma_\mathrm{r}}^3} = \frac{\rho_s}{{\sigma_{\mathrm{r,s}}}^3} x^{-{\alpha}}.
\label{eq:ppsd}$$ The radial velocity dispersion is now known ([eq. \[eq:sigr\]]{}), and its value at the scale radius ${\sigma_{\mathrm{r,s}}}$ may be set by assuming a local radial velocity dispersion ${\sigma_{\mathrm{r},\odot}}$.
The anisotropy parameter ${\beta}$ is defined as $${\beta}\equiv 1 - \frac{\sigma_\mathrm{t}^2}{2{\sigma_\mathrm{r}}^2},$$ where ${\sigma_\mathrm{r}}$ is the radial velocity dispersion and $\sigma_\mathrm{t}$ is the tangential velocity dispersion[^1]. From the integral Jeans equation [@binney], we can solve for the anisotropy parameter (compare with [@Zait:2007es; @Schmidt:2009kz]), $${\beta}(x) = \frac{5}{6}{\gamma}\left(x\right) - \frac{{\alpha}}{3} - \frac{G M(x)}{2x {r_\mathrm{s}}{\sigma_\mathrm{r}}^2(x)},
\label{eq:ani}$$ where we have defined the negative log-log slope of the density ${\gamma}(x) \equiv -\mathrm{d}\log(\rho)/\mathrm{d}\log(x)$. We know the contained mass $M(x)$, and we know the radial velocity dispersion ${\sigma_\mathrm{r}}$ from the mass density and PPSD, so we have $$\label{eq:ani_f}
{\beta}(x) = \frac{5}{6}{\gamma}\left(x\right) - \frac{{\alpha}}{3} - \Sigma^{-2} f(x;{\alpha}),$$ where $f(x;{\alpha})$ is a somewhat complicated function of $x$, with ${\alpha}$ its sole parameter (*i.e.* it does not depend on the halo parameters or ${\sigma_{\mathrm{r,s}}}$, see [eq. \[eq:nfw\_ani\]]{} for the full expression). The quantity $\Sigma$ is a dimensionless measure of the radial velocity dispersion at the scale radius, defined as $$\Sigma^2 \equiv \frac{{\sigma_{\mathrm{r,s}}}^2}{4\pi G {r_\mathrm{s}}^2 {\rho_\mathrm{s}}/3} = \frac{{\sigma_{\mathrm{r,s}}}^2}{V_{c,s}^2},
\label{eq:dimensionless_sigmars}$$ where $V_{c,s}$ is the circular velocity at the edge of a spherical mass of radius ${r_\mathrm{s}}$ and constant density ${\rho_\mathrm{s}}$.
To summarize the necessary ingredients that go into [eq. \[eq:ani\_f\]]{}, we need the PPSD slope ${\alpha}$, the halo parameters ${M_\mathrm{vir}}$, ${r_\mathrm{s}}$, and $c$ (one of which may be determined by the local halo density ${\rho_\odot}$), the local radial velocity dispersion ${\sigma_{\mathrm{r},\odot}}$, and the local (solar) radius ${r_\odot}$.
Constraints from the anisotropy parameter {#sec:constraints}
=========================================
It is shown in Appendix \[sec:ani\_limits\] that the anisotropy parameter for a NFW profile with PPSD slope ${\alpha}\approx 2$ has asymptotic limits $${\beta}(x) \rightarrow \begin{cases}
(5-2{\alpha})/6 & \text{for}\hspace{6pt}x\rightarrow 0\\
(15-2{\alpha})/6 & \text{for}\hspace{6pt}x\rightarrow \infty
\end{cases}$$ This is acceptable in the small-$x$ limit, where ${\beta}\rightarrow 1/6$ for, as an example, ${\alpha}= 2$. However, in the large-$x$ limit, with the same value of ${\alpha}$, $\beta \rightarrow 11/6$, which is greater than one, implying an imaginary velocity dispersion. Requiring that ${\beta}\le 1$ as $x\rightarrow\infty$ would imply ${\alpha}\ge 9/2$, which is a far steeper slope than seen in simulations. This unphysical behavior in ${\beta}$ may naively suggest that we cannot have a physical model that simultaneously exhibits an NFW density profile and power-law PPSD, but really this requirement for physical-ness is too restrictive. We do not expect the models or assumption of equilibrium (via the Jeans equation) to hold beyond around the virial radius. Requiring that these models are consistent and physical only up to just before they are expected to break down is, however, reasonable and still has consequences elsewhere in a halo. Thus, let us just require that the anisotropy parameter is no greater than one everywhere *within the virial radius*.
Mathematically, we require $$\forall x\le c : {\beta}(x) \le 1.$$ We can effectively satisfy this for our purposes by requiring that ${\beta}(c) \le 1$. This gives a maximum value for $\Sigma$ ([eq. \[eq:Sigma2\_upper\_limit\]]{}) that depends only on the concentration $c$ (by way of the virial mass) and PPSD log-slope ${\alpha}$. For reasonable values of $c$ and ${\alpha}$, this upper limit is of order one. From the definition of $\Sigma$ in [eq. \[eq:dimensionless\_sigmars\]]{}, this immediately gives an upper bound on ${\sigma_{\mathrm{r,s}}}$ ([eq. \[eq:sigrs\_max\]]{}) and thus also on ${\sigma_{\mathrm{r},\odot}}$ ([eq. \[eq:sigrlocal\_max\]]{}) in terms of the halo parameters and ${r_\odot}$.
![Maximum value of the local total velocity dispersion ${\sigma_{\mathrm{tot},\odot}}$. The gray band reflects the uncertainty in the halo parameters: the spread is over the 68% confidence intervals in Table \[tab:catena\_results\]. The solid red line marks the mean value for ${\sigma_{\mathrm{tot},\odot}}$ found by Catena and Ullio, while the dashed and dotted red lines mark their 68% and 95% confidence intervals [@Catena:2011kv].[]{data-label="fig:sigtlocal_max"}](sigtlocal_max_plot.eps){width="\textwidth"}
Once ${\sigma_{\mathrm{r},\odot}}$ is set, the anisotropy profile ${\beta}(x)$ is totally specified, including the local anisotropy parameter ${{\beta}_\odot}= {\beta}({x_\odot})$. The *total* velocity dispersion profile ${\sigma_\mathrm{tot}}$ is then also given, using the relation ${\sigma_\mathrm{tot}}^2 = (3 - 2{\beta}) {\sigma_\mathrm{r}}^2$. We find that ${\sigma_{\mathrm{tot},\odot}}$ depends monotonically on the choice of ${\sigma_{\mathrm{r},\odot}}$, so we finally have an upper bound on ${\sigma_{\mathrm{tot},\odot}}$ ([eq. \[eq:sigtlocal\_max\]]{}).
Uncertainty in the Maximum Local Dispersion and Anisotropy Profile {#sec:ani_profile}
==================================================================
We have derived an upper limit on the local total velocity dispersion ${\sigma_{\mathrm{tot},\odot}}$, subject to the constraint that the anisotropy profile ${\beta}(x)$ is no more than one up to the virial radius. This upper limit ([eq. \[eq:sigtlocal\_max\]]{}) depends on the PPSD slope, the halo parameters, and the solar radius. There is significant uncertainty in these quantities. To get an idea of the uncertainty in the upper limit of ${\sigma_{\mathrm{tot},\odot}}$, we will use the results of Catena and Ullio [@Catena:2009mf], summarized in Table \[tab:catena\_results\].
lower 95% lower 68% mean upper 68% upper 95%
---------------------------------------------------- ----------- ----------- --------- ----------- -----------
${M_\mathrm{vir}}\;[10^{12}\,{\mathcal{M}_\odot}]$ $1.23$ $1.33$ $1.49$ $1.64$ $1.86$
$c$ $13.93$ $16.59$ $19.70$ $22.90$ $24.6$
${\rho_\odot}\;[\mathrm{GeV/cm^3}]$ $0.338$ $0.365$ $0.389$ $0.414$ $0.435$
${r_\odot}\;[\mathrm{kpc}]$ $7.67$ $8.00$ $8.28$ $8.55$ $8.81$
${\sigma_{\mathrm{tot},\odot}}\;[\mathrm{km/s}]$ $276.7$ $281.7$ $287.0$ $292.2$ $297.2$
${v_\mathrm{esc}}\;[\mathrm{km/s}]$ $528.5$ $539.7$ $550.7$ $561.7$ $573.3$
: Assumed ranges for the halo parameters, solar radius, local total velocity dispersion, and local escape speed. Taken from Table 3 of [@Catena:2009mf] and Table 1 of [@Catena:2011kv].[]{data-label="tab:catena_results"}
With these ranges of parameters, we plot the upper limit of ${\sigma_{\mathrm{tot},\odot}}$ versus the PPSD slope ${\alpha}$ in Figure \[fig:sigtlocal\_max\]. The dark, solid line uses the mean values in Table \[tab:catena\_results\], while the upper and lower dashed lines take the extreme values of ${\sigma_{\mathrm{tot},\odot}}$ allowed by the 68% confidence intervals in Table \[tab:catena\_results\]. In other words, the band in Figure \[fig:sigtlocal\_max\] includes all combinations of parameters within the 68% confidence intervals. This is one of our main results. Also shown is the mean value and 68% and 95% confidence intervals for ${\sigma_{\mathrm{r},\odot}}$ in [@Catena:2011kv].
Actually choosing a value for ${\sigma_{\mathrm{r},\odot}}$ (or ${\sigma_{\mathrm{tot},\odot}}$) determines the anisotropy profile, but this quantity is also uncertain. We use the results for ${\sigma_{\mathrm{tot},\odot}}$ from [@Catena:2011kv] and then take ${\sigma_{\mathrm{r},\odot}}^2 = {\sigma_{\mathrm{tot},\odot}}^2/3$, which is used to find the anisotropy profile in [eq. \[eq:ani\]]{}. Note that the factor $1/3$ corresponds to the isotropic case. As we will see, we find only radial bias at the solar radius. Given the same value of ${\sigma_{\mathrm{tot},\odot}}$, radial bias implies a larger value of ${\sigma_{\mathrm{r},\odot}}$, which in turn gives a greater local radial bias[^2]. So as far as predicting departure from isotropy, this is a conservative approximation.
![The anisotropy profile for ${\alpha}= 2$, corresponding to the isothermal case, as a function of $x\equiv r/{r_\mathrm{s}}$. The gray band reflects the uncertainty in the halo parameters: the spread is over the 68% confidence intervals in Table \[tab:catena\_results\]. The two pairs of vertical lines represent the 68% confidence interval for the local radius ${r_\odot}$ and halo scale radius ${r_\mathrm{s}}$.[]{data-label="fig:ani_profile2"}](ani_plot_alpha2.eps){width="\textwidth"}
![The anisotropy profile for ${\alpha}= 35/18$, the critical value discussed in [@Dehnen:2005cu], as a function of $x\equiv r/{r_\mathrm{s}}$. The gray band reflects the uncertainty in the halo parameters: the spread is over the 68% confidence intervals in Table \[tab:catena\_results\]. The two pairs of vertical lines represent the 68% confidence interval for the local radius ${r_\odot}$ and halo scale radius ${r_\mathrm{s}}$.[]{data-label="fig:ani_profile35_18"}](ani_plot_alpha35_18.eps){width="\textwidth"}
![The anisotropy profile for ${\alpha}= 15/8$, the value found in [@Taylor:2001bq], as a function of $x\equiv r/{r_\mathrm{s}}$. The gray band reflects the uncertainty in the halo parameters: the spread is over the 68% confidence intervals in Table \[tab:catena\_results\]. The two pairs of vertical lines represent the 68% confidence interval for the local radius ${r_\odot}$ and halo scale radius ${r_\mathrm{s}}$.[]{data-label="fig:ani_profile15_8"}](ani_plot_alpha15_8.eps){width="\textwidth"}
We plot the anisotropy profile for fiducial values ${\alpha}= 2,35/18, 15/8$ in Figures \[fig:ani\_profile2\], \[fig:ani\_profile35\_18\], and \[fig:ani\_profile15\_8\]. The solid curve takes the mean values in Table \[tab:catena\_results\] while the dashed curves are the extreme cases, with all parameters within the 68% confidence interval in Table \[tab:catena\_results\]. The vertical lines mark the 68% lower and upper limits of ${x_\odot}= {r_\odot}/{r_\mathrm{s}}$ and $c$. For example, if we assume a PPSD slope of $35/18$ (Figure \[fig:ani\_profile35\_18\]), we might expect a local anisotropy parameter of at least approximately $0.2$ and no more than about $0.4$.
Generally, the profile is slightly radially biased near the center, reaches a minimum at around a tenth the scale radius, and rises to a (local) maximum of around $0.4$ to $0.6$ before the virial radius. We see in all cases that for $x\rightarrow 0$ the anisotropy parameter rises slowly to the value in [eq. \[eq:Aasym\]]{}, which is independent of the halo parameters. See [@Hansen:2004qs; @An:2009nc] for discussion of central anisotropy. Here we do not presume that either assumed model, of the mass distribution or PPSD, necessarily stays valid at very small or very large radii. See [@Ludlow:2011cs] for an investigation of the break-down of the PPSD power law.
Anisotropic velocity distributions and predictions {#sec:ani_dist}
==================================================
Recently, close attention has been paid to the form of the velocity distribution used to calculate predictions for indirect and direct DM detection. In some cases the functional form can make a significant difference. Especially, the assumed velocity distribution influences the interpretation of results from direct detection experiments [@Ullio:2000bf; @Vogelsberger:2008qb; @Catena:2011kv]. Here on we focus on the local distribution and suppress the subscript $\odot$. We introduce a new, anisotropic generalization of the model proposed by Mao, et al. [@Mao:2012hf]: $$f(\mathbf{v}) \propto \mathrm{exp}\left\{-\sqrt{\frac{{v_\mathrm{r}}^2}{{v_{\mathrm{r},0}}^2} + \frac{{v_\mathrm{t}}^2}{{v_{\mathrm{t},0}}^2}} \right\} \left({v_\mathrm{esc}}^2 - v^2 \right)^p,
\label{eq:animao}$$ where ${v_\mathrm{r}}= v\cos\left(\eta\right)$ and ${v_\mathrm{t}}= v\sin\left(\eta\right)$ are the radial and tangential velocity components and $\eta$ is the angle from the radial direction. The parameters ${v_{\mathrm{r},0}}$ and ${v_{\mathrm{t},0}}$ are not dispersions but just velocity scales. The exponent $p$ characterizes the high-velocity tail. The function is normalized so that $\int \mathrm{d}\mathbf{v} f\left(\mathbf{v}\right) = 1$. We choose this distribution because of its recent success in modeling the Eris simulation (see Fig. 3 in [@Kuhlen:2013tra]). For consistency with that study we take $p=1.5$, which was used to model the ErisDark results[^3]. The escape speed ${v_\mathrm{esc}}$ is given by the combined gravitational potential of both the DM halo and any other matter, and we use the mean value in Table \[tab:catena\_results\]. The total dispersion ${\sigma_\mathrm{tot}}$ and the anisotropy parameter ${\beta}$ are then determined by the parameters ${v_{\mathrm{r},0}}$ and ${v_{\mathrm{t},0}}$. We require that the total dispersion equals the mean value in Table \[tab:catena\_results\] and solve for ${v_{\mathrm{r},0}}$ and ${v_{\mathrm{t},0}}$ such that the desired anisotropy parameter is generated. Of course, the original, isotropic distribution is recovered when ${v_{\mathrm{r},0}}={v_{\mathrm{t},0}}$. See Appendix \[sec:vd\_details\] for details on the selection of values for ${v_{\mathrm{r},0}}$ and ${v_{\mathrm{t},0}}$.
We have checked that the uncertainties in the values of ${\sigma_\mathrm{tot}}$ and ${v_\mathrm{esc}}$ have a small impact on the following calculations. More importantly, the uncertainties affect both the isotropic and anisotropic cases equally once ${\beta}$ has been chosen. So for the purposes of investigating the importance of modeling deviation from isotropy, we show only results using the mean values in Table \[tab:catena\_results\].
We use the function in [eq. \[eq:animao\]]{} to model the local velocity distribution with the intention of understanding the impact that anisotropy can have on direct detection. For the purposes of this work, we assume a conservative value of $0.2$ for the anisotropy parameter ${\beta}$. It is straight-forward to calculate the function $$g(v,t) = \rho_\odot \int_0^\pi \mathrm{d}\eta\sin\left(\eta\right) \int_0^{2\pi}\mathrm{d}\psi\,v f\left(\mathbf{v}_\mathrm{halo}\right),
\label{eq:g}$$ where $\mathbf{v}_\mathrm{halo}$ is the velocity vector boosted from the detector frame to the halo frame [@Catena:2011kv]. The boost depends on the time of the year $t$. In Figure \[fig:g\] we plot this function for June and December; for the isotropic case and the anisotropic case.
![The function $g\left(v,t\right)$ defined in [eq. \[eq:g\]]{}. Solid lines are calculated in June; dashed lines are calculated in December.[]{data-label="fig:g"}](g_plot.eps){width="\textwidth"}
Using the function $g\left(v,t\right)$, the differential detection rate is found by specifying a velocity threshold ${v_\mathrm{th}}$ for DM particles in the detector frame: $${\frac{\mathrm{d}R}{\mathrm{d}Q}} \propto G\left({v_\mathrm{th}}, t\right) \equiv \int_{v \ge {v_\mathrm{th}}} \mathrm{d}v\,g\left(v,t\right).
\label{eq:rate}$$
[0.48]{} ![The time-averaged function $\langle G\left({v_\mathrm{th}},t\right)\rangle_t$ as a function of the velocity threshold (see [eq. \[eq:rate\]]{}) and the fractional difference between the isotropic to anisotropic cases.[]{data-label="fig:rate"}](rate_plot.eps "fig:"){width="\textwidth"}
[0.48]{} ![The time-averaged function $\langle G\left({v_\mathrm{th}},t\right)\rangle_t$ as a function of the velocity threshold (see [eq. \[eq:rate\]]{}) and the fractional difference between the isotropic to anisotropic cases.[]{data-label="fig:rate"}](rate_frac_change_plot.eps "fig:"){width="\textwidth"}
[0.48]{} ![Signal modulation amplitude as a function of the velocity threshold and the fractional difference between the isotropic to anisotropic cases.[]{data-label="fig:mod"}](mod_plot.eps "fig:"){width="\textwidth"}
[0.48]{} ![Signal modulation amplitude as a function of the velocity threshold and the fractional difference between the isotropic to anisotropic cases.[]{data-label="fig:mod"}](mod_frac_change_plot.eps "fig:"){width="\textwidth"}
The velocity threshold is determined by the specifics of any particular experiment and the DM particle mass, and we leave it free. Figure \[fig:rate\] plots the function $G$, averaged between June and December, for the isotropic and anisotropic cases, with the fractional difference $$\Delta G = \left( G_\mathrm{Ani} - G_\mathrm{Iso}\right)/G_\mathrm{Iso}.
\label{eq:G_frac_diff}$$
We also consider the modulation amplitude of the signal, defined here as half the difference between the rate in June and the rate in December: $$A\left({v_\mathrm{th}}\right) = \left| G\left({v_\mathrm{th}},t_\mathrm{June}\right) - G\left({v_\mathrm{th}},t_\mathrm{Dec}\right)\right|/2.$$ This is plotted in Figure \[fig:mod\] for the isotropic and anisotropic cases, with the fractional difference, analogous to [eq. \[eq:G\_frac\_diff\]]{}.
Conclusions {#sec:conclusions}
===========
Combining models of the mass distribution and pseudo-phase-space density, the Jeans equation gives us a particular anisotropy profile. We have plotted this profile for a few representative values of the PPSD slope and for a spread of parameters that describe the galactic halo. These profile shapes are consistent with those shown in Figure 1 of [@Zait:2007es], although those results exhibit less anisotropy overall. The anisotropy profiles found in [@Bozorgnia:2013pua] are also similar but were derived from models of the phase-space distribution. The difference in methods strengthens both their results and these.
We have used an anisotropic modification to the model proposed by Mao et al. [@Mao:2012hf], which was also used to model the Eris simulation. We find that assuming a local anisotropy of approximately $0.2$ is reasonable and conservative. In the Eris simulation, a comparable amount of radially biased anisotropy was found at the location corresponding to the solar radius (this is roughly seen by measuring the half-maximum width of the radial and azimuthal distributions in Figure 2 of [@Kuhlen:2013tra]). On the other hand, the results of [@Catena:2009mf] favor a local *tangential* bias, though the small local radial bias found in this work and others already mentioned is approximately within their 95% confidence interval.
Different direct detection collaborations have found contradictory results (*e.g.* see [@Gondolo:2012rs]). Part of the general goal in studying the local velocity distribution is to alleviate these discrepancies. Since different experiments can have different threshold velocities, Figure \[fig:rate\] suggests that the difference between observed signals can vary by several percent due to the effect of local anisotropy. This may seem small, but it is comparable to the uncertainty introduced by considering different density profiles [@Catena:2011kv]. The modification to the modulation amplitude can be even more significant and is sensitive to the value of the velocity threshold, but the signal itself is smallest where the modification is greatest.
In principle, a detector that can give information about the direction of a detected WIMP’s velocity would allow us to *measure* the local anisotropy. This is difficult, as it would require an individual WIMP to interact multiple times inside the detector or require a low detector density so the recoiled particle can be tracked. Once a discovery is confirmed, however, it may be viable to consider such an experiment. Future work will consider this possibility (also, see [@Gondolo:2002np]).
We note that the anisotropy at radii beyond about the scale radius is sensitive to the shape of the PPSD profile and to the other parameters, and it can also be quite large. However, it seems unlikely that this grants a viable observational effect, since the density is so low there and substructure would dominate any emission. The most novel result of this work is the constraint on the velocity dispersion profile. Requiring the anisotropy parameter to be physical (no greater than one) inside the virial radius implies a maximum value for the local total velocity dispersion of about $300\,\mathrm{km/s}$ or so. Typical assumed values for the local velocity dispersion (such as in the Standard Halo Model, $220\,\mathrm{km/s}$) do not seem to be in great danger, but this consistency check should be remembered in future model-building.
The author is grateful to Francesc Ferrer and Stanley D. Hunter for advice and discussions. The author also greatly appreciates the anonymous referee’s time and very helpful comments. This work was supported by the U.S. DOE at Washington University in St. Louis.
Detailed expressions {#sec:details}
====================
Here we discuss the derivation of the anisotropy parameter and related quantities in detail. We consider two-power-law density profiles, with inner slope $\gamma_0$ and outer slope $\gamma_\infty$, $$\label{eq:general_density}
\rho_{\gamma_0\gamma_\infty}\left( x\right) = \rho_s x^{-\gamma_0} \left( 1+x\right)^{-\gamma_\infty+\gamma_0}.$$ The contained mass is, omitting the constant factor $4\pi \rho_s r_s^3$, $$M_{\gamma_0\gamma_\infty}\left( x\right) = (-1)^{\gamma_0-1} B_{-x}\left( 3-\gamma_0, 1+\gamma_0-\gamma_\infty \right),$$ where $B_z(a,b)$ is the incomplete beta function. Note the following particular cases: $$\begin{aligned}
M_{1\gamma_\infty}(x) & = \frac{1 - \left[1 - x\left(2-\gamma_\infty\right)\right](1+x)^{2-\gamma_\infty} }{(3-\gamma_\infty)(2-\gamma_\infty)}, \\
M_{12}(x) & = x - \log(1+x),\\
M_\mathrm{NFW}(x) = M_{13}(x) & = -\frac{x}{1+x} + \log(1+x),\\
M_{\gamma_04}(x) & = \frac{1}{3-\gamma_0}\left(\frac{x}{1+x}\right)^{3-\gamma_0}.\end{aligned}$$ From the PPSD power-law in [eq. \[eq:ppsd\]]{} and the general density profile in [eq. \[eq:general\_density\]]{}, we have the radial velocity dispersion $${\sigma_\mathrm{r}}^2(x) = {\sigma_{\mathrm{r,s}}}^2 \left[ x^{-\gamma_0+{\alpha}} \left(\frac{2}{1+x}\right)^{\gamma_\infty-\gamma_0} \right]^{2/3}.$$ For the NFW profile this is $${\sigma_\mathrm{r}}^2(x) = {\sigma_{\mathrm{r,s}}}^2\left(\frac{4 x^{-1+{\alpha}}}{(1+x)^2}\right)^{2/3}.
\label{eq:sigr}$$ The expression for the anisotropy parameter ${\beta}\left(x\right)$ in [eq. \[eq:ani\]]{} is general. Specific to the case of the NFW profile, it is $$\label{eq:nfw_ani}
{\beta}(x) = \frac{5+15x}{6+6x} - \frac{{\alpha}}{3} - \Sigma^{-2} \cdot 3 x^{-(2{\alpha}+1)/3} \left(\frac{1+x}{2}\right)^{1/3} \left[-x + (1+x)\log( 1+x )\right].$$ with $\Sigma^2 \equiv {\sigma_{\mathrm{r,s}}}^2/(4\pi G {r_\mathrm{s}}^2 {\rho_\mathrm{s}}/3)$. The upper limit on $\Sigma$ in the case of a NFW profile is $$\Sigma^2 \le \Sigma^2_\mathrm{max} \equiv \frac{3^2 \cdot 2^{2/3}(1+c)^{4/3}\left[-c+(1+c)\log( 1+c)\right]}{c^{(1+2{\alpha})/3} \left[9c - 2{\alpha}(1+c) - 1\right]}.
\label{eq:Sigma2_upper_limit}$$ This translates to the upper limits on ${\sigma_{\mathrm{r,s}}}$ and ${\sigma_{\mathrm{r},\odot}}$: $$\begin{aligned}
{\sigma_{\mathrm{r,s}}}^2 \le \sigma^2_\mathrm{r,s,max} &\equiv& (4\pi G {r_\mathrm{s}}^2{\rho_\mathrm{s}}/3) \Sigma^2_\mathrm{max}\left({\alpha},c\right),\label{eq:sigrs_max}\\
{\sigma_{\mathrm{r},\odot}}^2 \le \sigma^2_{\mathrm{r},\odot,\mathrm{max}} &\equiv& \left(\frac{4\,{x_\odot}^{-1+{\alpha}}}{\left(1+{x_\odot}\right)^2}\right)^{2/3} \sigma^2_\mathrm{r,s,max}\left({\alpha},{M_\mathrm{vir}},{r_\mathrm{s}},c\right).\label{eq:sigrlocal_max}\end{aligned}$$ Finally, because ${\sigma_{\mathrm{tot},\odot}}$ increases monotonically with ${\sigma_{\mathrm{r},\odot}}$, its upper limit is $$\label{eq:sigtlocal_max}
{\sigma_{\mathrm{tot},\odot}}^2 \le (3 - 2{{\beta}_\odot}) \sigma^2_{\mathrm{r},\odot,\mathrm{max}},$$ which depends on ${\alpha}$, ${M_\mathrm{vir}}$, ${r_\mathrm{s}}$, $c$, and ${r_\odot}$.
Asymptotic behavior {#sec:ani_limits}
===================
We split the function for the NFW anisotropy parameter in [eq. \[eq:nfw\_ani\]]{} into two parts, so ${\beta}(x) = A(x) + B(x)$, with $$\begin{aligned}
A(x) & = & \frac{5+15x}{6+6x} - \frac{{\alpha}}{3}, \\
B(x) & = & -\Sigma^{-2}\cdot 3x^{-(2{\alpha}+1)/3} \left(\frac{1+x}{2}\right)^{1/3} \left[-x + (1+x)\log( 1+x )\right].\end{aligned}$$ The first part has simple asymptotic limits $$A(x) \rightarrow \begin{cases}
(5-2{\alpha})/6 & \text{for}\hspace{6pt}x\rightarrow 0\\
(15-2{\alpha})/6 & \text{for}\hspace{6pt}x\rightarrow \infty
\end{cases}
\label{eq:Aasym}$$ while the second is more complicated. In the limit $x \rightarrow 0$, we have $$B(x) \rightarrow \begin{cases}
-\infty & \text{if}\hspace{6pt}{\alpha}> 5/2\\
- 2^{-4/3}\cdot 3 \times \Sigma^{-2} & \text{if}\hspace{6pt}{\alpha}=5/2\\
0 & \text{if}\hspace{6pt}{\alpha}<5/2
\end{cases}
\label{eq:Basym0}$$ and in the limit $x \rightarrow \infty$, we have $$B(x) \rightarrow \begin{cases}
0 & \text{if}\hspace{6pt}{\alpha}> 3/2\\
-\infty & \text{if}\hspace{6pt}{\alpha}\le 3/2
\end{cases}
\label{eq:BasymInf}$$ As long as $3/2 < {\alpha}< 5/2$, the extreme values of ${\beta}(x)$ are determined solely by ${\alpha}$.
Details of anisotropic velocity distributions {#sec:vd_details}
=============================================
![Contours of the parameters ${v_{\mathrm{r},0}}$ and ${v_{\mathrm{t},0}}$ that give the specified values of the anisotropy parameter or total velocity dispersion.[]{data-label="fig:mao_contours"}](mao_beta_plot.eps){width="\textwidth"}
We use the velocity distribution in [eq. \[eq:animao\]]{} to model the local velocity distribution, with $p=1.5$ from [@Kuhlen:2013tra] and with ${v_\mathrm{esc}}= 550.7\,\mathrm{km/s}$ from [@Catena:2011kv]. The choice of parameters ${v_{\mathrm{r},0}}$ and ${v_{\mathrm{t},0}}$ determine the velocity dispersion and anisotropy parameter. Figure \[fig:mao\_contours\] plots contours that give the specified value of ${\beta}$ or ${\sigma_\mathrm{tot}}$. In this work we choose ${\sigma_\mathrm{tot}}= 287\,\mathrm{km/s}$ as the mean value [@Catena:2011kv]. For the isotropic case, this implies ${v_{\mathrm{r},0}}= {v_{\mathrm{t},0}}= 209.8\,\mathrm{km/s}$; for the anisotropic case, with ${\beta}= 0.2$, this implies ${v_{\mathrm{r},0}}= 270.4\,\mathrm{km/s}$ and ${v_{\mathrm{t},0}}= 187.1\,\mathrm{km/s}$.
[^1]: We define the tangential velocity dispersion such that $\sigma^2_\mathrm{t} = \sigma^2_\theta + \sigma^2_\phi = 2\sigma^2_\theta$.
[^2]: Successive adapting of the relation between ${\sigma_{\mathrm{r},\odot}}$ and ${\sigma_{\mathrm{tot},\odot}}$ would, of course, converge to the correct “trial value” for ${{\beta}_\odot}$.
[^3]: We do not take the Eris parameter $p=2.7$ for two reasons: we have not considered baryonic effects on the PPSD profile, and because such a steep cut-off makes it difficult to achieve anisotropy greater than ${\beta}\approx 1.0$ with the model in [eq. \[eq:animao\]]{}.
|
---
abstract: 'We produce a complete descrption of the lattice of gauge-invariant ideals in $C^*(\L)$ for a finitely aligned $k$-graph $\L$. We provide a condition on $\L$ under which every ideal is gauge-invariant. We give conditions on $\L$ under which $C^*(\L)$ satisfies the hypotheses of the Kirchberg-Phillips classification theorem.'
address: |
School of Mathematical and Physical Sciences\
University of Newcastle\
Callaghan\
NSW 2308\
AUSTRALIA
author:
- Aidan Sims
date: 'June 29, 2004'
title: 'Gauge-invariant ideals in the $C^*$-algebras of finitely aligned higher-rank graphs'
---
(\#1)[Ł\^[\#1]{}]{}
(\#1,\#2)[[Ł\^(\#1,\#2) ]{}]{} Ł
\[section\] \[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Conjecture]{} \[theorem\][Claim]{} \[theorem\][Remark]{} \[theorem\][Remarks]{} \[theorem\][Comment]{} \[theorem\][Example]{} \[theorem\][Examples]{} \[theorem\][Definition]{} \[theorem\][Definitions]{} \[theorem\][Notation]{}
[^1]
Introduction {#sec:intro}
============
Among the main reasons for the sustained interest in the $C^*$-algebras of directed graphs and their analogues in recent years are the elementary graph-theoretic conditions under which the associated $C^*$-algebra is simple and purely infinite, and the relationship between the gauge-invariant ideals in a graph $C^*$-algebra and the connectivity properties of the underlying graph.
A complete description of the lattice of gauge-invariant ideals of the $C^*$-algebra $C^*(E)$ of a directed graph $E$ was given in [@BHRS], and conditions on $E$ were described under which $C^*(E)$ is simple and purely infinite. Building upon these results, Hong and Szymański achieved a description of the primitive ideal space of $C^*(E)$ in [@HS]. The results of [@BHRS] were obtained by a process which builds from a graph $E$ and a gauge-invariant ideal $I$ in $C^*(E)$, a new graph $F = F(E,I)$ in such a way that the graph $C^*$-algebra $C^*(F)$ is canonically isomorphic to the quotient algebra $C^*(E)/I$. However, recent work of Muhly and Tomforde shows that the quotient algebra $C^*(E)$ can also be regarded as a *relative graph algebra* associated to a subgraph of $E$.
In this note, we turn our attention to the classification of the gauge-invariant ideals in the $C^*$-algebra of a finitely aligned higher-rank graph $\L$, and to the formulation of conditions under which these algebras are simple and purely infinite. Because of the combinatorial peculiarities of higher-rank graphs, constructive methods such as those employed in [@BHRS] are not readily available to us in this setting. However, the author has studied a class of *relative Cuntz-Krieger algebras* associated to a higher-rank graph $\L$ in [@Si1], and we use these results to analyse the gauge-invariant ideal structure of $C^*(\L)$. We use the results of [@Si1] to give conditions on $\L$ under which $C^*(\L)$ is simple and purely infinite; we also show that relative graph algebras $C^*(\L;\Ee)$, and in particular graph algebras $C^*(\L)$ always belong to the bootstrap class $\mathcal{N}$ of [@RSc], and hence are nuclear and satisfy the UCT.
We begin in Section \[sec:k-graphs\] by defining higher-rank graphs, and supplying the definitions and notation we will need for the remainder of the paper. In Section \[sec:hereditary subsets\], we introduce the appropriate analogue in the setting of higher-rank graphs of a *saturated hereditary* set of the vertices of $\L$, and show that such sets $H$ give rise to gauge-invariant ideals $I_H$ in $C^*(\L)$. In Section \[sec:ideals and quotients\], we use the gauge-invariant uniqueness theorem of [@Si1] to show that the quotient $C^*(\L)/I_H$ of $C^*(\L)$ by the gauge-invariant ideal associated to a saturated hereditary set $H$ is canonically isomorphic to a relative Cuntz-Krieger algebra $C^*(\L\setminus\L H; \Ee_H)$ associated to a subgraph of $\L$. Using this result, we show in Section \[sec:ideal listing\] that the gauge-invariant ideals of $C^*(\L)$ are in bijective correspondence with pairs $(H, B)$ where $H$ is saturated and hereditary, and $B \cup \Ee_H$ is *satiated* as in [@Si1 Definition 4.1]. In Section \[sec:lattice\], we describe the lattice order $\preceq$ on pairs $(H,B)$ which corresponds to the lattice order $\subset$ on gauge-invariant ideals of $C^*(\L)$. In Section \[sec:CK ideals\], we prove that for a certain class of higher-rank graphs $\L$, all the ideals of $C^*(\L)$ are gauge-invariant; however, whilst this result does generalise similar results of [@BPRS; @RSY1], the condition (D) which we need to impose on $\L$ to guarantee that all ideals are gauge-invariant is, in most instances, more or less uncheckable — the situation is not particulary satisfactory in this regard. In Section \[ch:classifiable\] we show that $C^*(\L)$ always falls into the bootstrap class $\mathcal{N}$ of [@RSc], and provide graph-theoretic conditions under which $C^*(\L)$ is simple and purely infinite.
Warning: for consistency with [@KP], the author has continued to use terminology such as “hereditary” and “cofinal” in this paper. Readers familiar with graph algebras should be wary as to the meaning of these terms because of the change of edge-direction conventions involved in going from directed graphs to $k$-graphs.
[**Acknowledgements.**]{} This article is based on part of the author’s PhD dissertation, which was written at the University of Newcastle, Australia, under the supervision of Iain Raeburn. The author would like to thank Iain for his insight and guidance. The author would also like to thank D. Gwion Evans for directing his attention to the results of [@PQR] on AF algebras associated to non-row-finite skew-product $k$-graphs.
Higher-rank graphs and their representations {#sec:k-graphs}
============================================
The definitions in this section are taken more or less wholesale from [@Si1].
We regard $\NN^k$ as an additive semigroup with identity 0. For $m,n \in \NN^k$, we write $m \vee n$ for their coordinate-wise maximum and $m \wedge n$ for their coordinate-wise minimum. We write $n_i$ for the $i^{\rm th}$ coordinate of $n \in \NN^k$, and $e_i$ for the $i^{\rm th}$ generator of $\NN^k$; so $n = \sum^k_{i=1} n_i \cdot e_i$.
\[dfn:k-graph\] Let $k \in \NN \setminus \{0\}$. A $k$-graph is a pair $(\L,d)$ where $\L$ is a countable category and $d$ is a functor from $\L$ to $\NN^k$ which satisfies the *factorisation property*: [*For all $\lambda \in \Mor(\L)$ and all $m,n \in \NN^k$ such that $d(\lambda) = m+n$, there exist unique morphisms $\mu$ and $\nu$ in $\Mor(\L)$ such that $d(\mu) = m$, $d(\nu) = n$ and $\lambda = \mu\nu$.*]{}
Since we are regarding $k$-graphs as generalised graphs, we refer to elements of $\Mor(\L)$ as *paths* and we write $r$ and $s$ for the codomain and domain maps.
The factorisation property implies that $d(\lambda) = 0$ if and only if $\lambda = \id_v$ for some $v \in \Obj(\L)$. Hence we identify $\Obj(\L)$ with $\{\lambda \in \Mor(\L) : d(\lambda) = 0\}$, and write $\lambda \in \L$ in place of $\lambda \in \Mor(\L)$.
Given $\lambda \in \L$ and $E \subset\L$, we define $\lambda E :=\{\lambda\mu : \mu \in E, r(\mu) = s(\lambda)\}$ and $E\lambda := \{\mu\lambda : \mu \in E, s(\mu) = r(\lambda)\}$. In particular if $d(v) = 0$, then $vE = \{\lambda \in E : r(\lambda) = v\}$. In analogy with the path-space notation for $1$-graphs, we denote by $\L^n$ the collecton $\{\lambda \in \L : d(\lambda) = n\}$ of paths of degree $n$ in $\L$.
The factorisation property ensures that if $l \le m \le n \in \NN^k$ and if $d(\lambda) = n$, then there exist unique elements, denoted $\lambda(0, l)$, $\lambda(l, m)$ and $\lambda(m,n)$, of $\L$ such that $d(\lambda(0,l)) = l$, $d(\lambda(l,m)) = m-l$, and $d(\lambda(m,n)) = n-m$ and such that $\lambda = \lambda(0,l)\lambda(l,m)\lambda(m,n)$.
\[dfn:common extensions\] Let $(\L,d)$ be a $k$-graph. For $\mu,\nu \in \L$ we denote the collection $\{\lambda \in \L : d(\lambda) = d(\mu) \vee d(\nu), \lambda(0, d(\mu)) = \mu, \lambda(0, d(\nu)) = \nu\}$ of *minimal common extensions* of $\mu$ and $\nu$ by $\MCE(\mu,\nu)$. We write $\Lmin(\mu,\nu)$ for the collection $$\Lmin(\mu,\nu) := \{(\alpha,\beta) \in \L \times \L : \mu\alpha = \nu\beta \in \MCE(\mu,\nu)\}.$$ If $E \subset \L$ and $\mu \in \L$, then we write $\Ext_\L(\mu;E)$ for the set $$\begin{split}
\Ext_\L(\mu;E) := \{\beta \in s(\mu)\L : \text{there }&\text{exists } \nu \in E \text{ such that } \mu\beta \in \MCE(\mu,\nu)\};
\end{split}$$ when the ambient $k$-graph $\L$ is clear from context, we write $\Ext(\mu;E)$ in place of $\Ext_\L(\mu; E)$. We say that $\L$ is finitely aligned if $|\MCE(\mu,\nu)| < \infty$ for all $\mu,\nu \in \L$.
Let $v \in \L^0$ and $E \subset v\L$. We say $E$ is *exhaustive* if $\Ext(\lambda;E) \not= \emptyset$ for all $\lambda \in v\L$.
Let $(\L,d)$ be a finitely aligned $k$-graph. Define $$\FE(\L) := \textstyle\bigcup_{v \in \L^0} \{E \subset v\L\setminus\{v\} :\text{ $E$ is finite and exhaustive}\}.$$ For $E \in \FE(\L)$ we write $r(E)$ for the vertex $v \in \L^0$ such that $E \subset v\L$.
Notice that whilst any finite subset of $v\L$ which contains $v$ is automatically finite exhaustive, we don not include such sets in $\FE(\L)$. Note also that since $v\L$ is never empty (in particular, it always contains $v$), finite exhausitve sets, and in particular elements of $\FE(\L)$, are always nonempty.
\[dfn:relCK family\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $\Ee$ be a subset of $\FE(\L)$. A *relative Cuntz-Krieger $(\L;\Ee)$-family* is a collection $\{t_\lambda : \lambda \in \L\}$ of partial isometries in a $C^*$-algebra satisfying
- $\{t_v : v \in \L^0\}$ is a collection of mutually orthogonal projections;
- $t_\lambda t_\mu = \delta_{s(\lambda), r(\mu)} t_{\lambda\mu}$ for all $\lambda, \mu \in \L$;
- $t^*_\lambda t_\mu = \sum_{(\alpha,\beta) \in \Lmin(\lambda,\mu)} t_\alpha t^*_\beta$ for all $\lambda,\mu \in \L$; and
- $\prod_{\lambda \in E} (t_{r(E)} - t_\lambda t^*_\lambda) = 0$ for all $E \in \Ee$.
When $\Ee = \FE(\L)$, we call $\{t_\lambda : \lambda \in \L\}$ a *Cuntz-Krieger $\L$-family*.
For each pair $(\L,\Ee)$ there exists a universal $C^*$-algebra $C^*(\L;\Ee)$, generated by a universal relative Cuntz-Krieger $(\L;\Ee)$-family $\{s_\Ee(\lambda) : \lambda \in \L\}$ which admits a *gauge-action* $\gamma$ of $\TT^k$ satisfying $\gamma_z(s_\Ee(\lambda)) = z^{d(\lambda)} s_\Ee(\lambda)$. We write $C^*(\L)$ for $C^*(\L;\FE(\L))$, and call it the *Cuntz-Krieger* algebra, and we denote the universal Cuntz-Krieger family by $\{s_\lambda : \lambda \in \L\}$; this agrees with the definitions given in [@RSY2].
There is also a Toeplitz algebra $\Tt C^*(\L)$ associated to each $k$-graph $\L$. By definition, this is the universal $C^*$-algebra generated by a family $\{s_{\Tt}(\lambda) : \lambda \in \L\}$ which satisfy (TCK1)–(TCK3), and hence is canonically isomorphic to $C^*(\L;\emptyset)$. Indeed, each $C^*(\L;\Ee)$ is a quotient of $\Tt C^*(\L)$:
\[lem:univ-quotients\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $\Ee \subset \FE(\L)$. Let $J_{\Ee}$ denote the ideal of $\Tt C^*(\L)$ generated by the projections $$\textstyle
\Big\{\prod_{\lambda \in E} \big(s_\Tt(r(E)) - s_\Tt(\lambda) s_\Tt(\lambda)^*\big) : E \in \Ee\Big\}.$$ Then $C^*(\L;\Ee)$ is canonically isomorphic to $\Tt C^*(\L) / J_\Ee$.
The universal property of $\Tt C^*(\L)$ gives a homomorphism $\pi : \Tt C^*(\L) \to C^*(\L;\Ee)$ satisfying $\pi(s_\Tt(\lambda)) = s_\Ee(\lambda)$ for all $\lambda$. Since $\{s_\Ee(\lambda) : \lambda \in \L\}$ satisfy (CK), we have $J_\Ee \subset \ker\pi$ and hence $\pi$ descends to a homomorphism $\tilde\pi : \Tt C^*(\L) / J_\Ee \to C^*(\L;\Ee)$ such that $\tilde\pi(s_\Tt(\lambda) + J_\Ee) = s_\Ee(\lambda)$ for all $\lambda$.
On the other hand, the family $\{s_\Tt(\lambda) + J_\Ee : \lambda \in \L\} \subset \Tt C^*(\L) / J_{\Ee}$ satisfy (CK) by definition of $J_\Ee$, so the universal property of $C^*(\L;\Ee)$ gives a homomorphism $\phi : C^*(\L;\Ee) \to \Tt C^*(\L) / J_{\Ee}$ such that $\phi(s_\Ee(\lambda)) = s_\Tt(\lambda) + J_\Ee$ for all $\lambda$. We have that $\tilde\pi$ and $\phi$ are mutually inverse, and the result follows.
Hereditary subsets and associated ideals {#sec:hereditary subsets}
========================================
\[dfn:sat,hered\] Let $(\L,d)$ be a finitely aligned $k$-graph. Define a relation $\le$ on $\L^0$ by $v \le w$ if and only if $v \L w
\not= \emptyset$.
- We say that a subset $H$ of $\L^0$ is *hereditary* if $v \in H$ and $v \le w$ imply $w \in H$.
- We say that $H \subset \L^0$ is *saturated* if, whenever $v \in \L^0$ and there exists a finite exhaustive subset $F \subset v \L$ with $s(F) \subset H$, we also have $v \in H$.
For $H \subset \L^0$ we call the smallest saturated set containing $H$ the *saturation* of $H$.
\[lem:saturation\] Let $(\L,d)$ be a finitely aligned $k$-graph and let $G \subset \L^0$. Let $\Sigma G := \{v \in \L^0 : \text{ there exists a finite exhaustive set } F \subset v\L G\}$. Then
- $\Sigma G$ is equal to the saturation of $G$; and
- if $G$ is hereditary, then $\Sigma G$ is hereditary.
First note that if $v \in G$ then $\{v\} \subset v\L G$ is finite and exhaustive so that $G \subset \Sigma G$. Note also that $\Sigma G$ is a subset of the saturation of $G$ by definition. To see that $\Sigma G$ is saturated, let $v \in \L^0$ and suppose $F \in v\L(\Sigma G)$ is finite and exhaustive. If $v \in F$, then $v \in \Sigma G $ by definition, so suppose that $v \not\in F$. Let $E := \{\lambda \in F : s(\lambda) \not \in G\}$. By definition of $\Sigma G $, for each $\lambda \in E$, there exists $E_\lambda \in s(\lambda)\FE(\L)$ with $s(E_\lambda) \subset G$. Then [@Si1 Lemma 5.3] shows that $F' := (F \setminus E) \cup (\bigcup_{\lambda \in E} \lambda E_\lambda)$ belongs to $\FE(\L)$. Since $F' \subset v\L G$, it follows that $v \in \Sigma G $ by definition. This establishes (1).
To prove claim (2), suppose $G$ is hereditary, and suppose $v,w \in \L^0$ satisfy $v \in \Sigma G$ and $v \le w$; say $\lambda \in \L$ with $r(\L) = v$, $s(\L) = w$. If $v \in G$ then $w \in G$ because $G$ is hereditary, so suppose that $v \in \Sigma G \setminus G$. By definition of $\Sigma$ there exists $F \in v \FE(\L)$ such that $s(F) \subset G$. By [@Si1 Lemma 2.3], $\Ext(\lambda;F)$ is a finite exhaustive subset of $w \L$. Since $s(F) \subset G$, and since, for $\alpha \in \Ext(\lambda;F)$, we have $s(\alpha) \le s(\mu)$ for some $\mu \in F$, we have $s(\Ext(\lambda;F)) \subset G$. It follows that $w \in \Sigma G $, completing the proof.
\[lem:H\_I sat,hered\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $I$ be an ideal of $C^*(\L)$. Then $H_I := \{v \in \L^0 : s_v \in I\}$ is saturated and hereditary.
To prove Lemma \[lem:H\_I sat,hered\], we first need to recall some notation from [@RS1].
\[ntn:vee E\] Let $(\L,d)$ be a finitely aligned $k$-graph and let $E$ be a finite subset of $\L$. As in [@RS1], we denote by $\vee E$ the smallest subset of $\L$ such that $E \subset \vee E$ and such that if $\lambda,\mu \in \vee E$, then $\MCE(\lambda,\mu) \subset \vee E$. We have that $\vee E$ is finite and that $\lambda \in \vee E$ implies $\lambda = \mu\mu'$ for some $\mu \in E$ by [@RS1 Lemma 8.4].
Suppose $v \in H_I$ and $w \in \L^0$ with $v \le w$. So there exists $\lambda \in v\L w$. Since $s_v \in I$, we have $s_w = s_\lambda^* s_v s_\lambda \in I$, and then $w \in H_I$; consequently $H_I$ is hereditary. Now suppose that $v \in \L^0$ and there is a finite exhaustive set $F \subset v \L$ with $s(F) \subset H_I$. By [@RSY2 Lemma 3.1], we have $s_v \in \lsp\{s_\lambda s^*_\lambda : \lambda \in \vee F\}$. Since $\lambda \in \vee F$ implies $\lambda = \alpha\alpha'$ for some $\alpha \in F$, and since $H_I$ is hereditary, we have $s(\vee F) \subset H_I$. Consequently, for $\lambda \in \vee F$, we have $s_\lambda s^*_\lambda = s_\lambda s_{s(\lambda)} s^*_\lambda \in I$, so $s_v \in I$, giving $v \in H_I$.
\[ntn:I\_H\] For $H \subset \L^0$, let $I_H$ be the ideal in $C^*(\L)$ generated by $\{s_v : v \in H\}$. Let $H\L$ denote the subcategory $\{\lambda \in \L : r(\lambda) \in H\}$ of $\L$.
\[lem:Lambda(H) Vs I\_H\] Let $(\L,d)$ be a finitely aligned $k$-graph, and suppose that $H \subset \L^0$ is saturated and hereditary. Then $(H\L, d|_{H\L})$ is also a finitely aligned $k$-graph, and $C^*(H\L) \cong C^*(\{s_\lambda : r(\lambda) \in H\}) \subset C^*(\L)$. Moreover this subalgebra is a full corner in $I_H$.
One checks that $(H\L, d|_{H\L})$ is a $k$-graph just as in [@RSY1 Theorem 5.2], and it is finitely aligned because $(H\L)^{\min}(\lambda,\mu) \subset \Lmin(\lambda,\mu)$.
The universal property of $C^*(H\L)$ ensures that there exists a homomorphism $\pi : C^*(H\L) \to C^*(\{s_\lambda : r(\lambda) \in H\})$. Write $\gamma_H$ for gauge action on $C^*(H\L)$ and $\gamma|$ for the restriction of the gauge action on $C^*(\L)$ to $C^*(\{s_\lambda : r(\lambda) \in H\})$. Then $\pi \circ (\gamma_H)_z = (\gamma|)_z \circ \pi$ for all $z \in \TT^k$, and [@RSY2 Theorem 4.2] shows that $\pi$ is injective.
For the final statement, just use the argument of [@BPRS Theorem 4.1(c)] to see that $C^*(\{s_\lambda : r(\lambda) \in H\})$ is the corner of $I_H$ determined by the projection $P_H := \sum_{v \in H} s_v \in \Mm(I_H)$, and that this projection is full.
Quotients of $C^*(\L)$ by $I_H$ {#sec:ideals and quotients}
===============================
We now want to show that the quotients of Cuntz-Krieger algebras by the ideals $I_H$ of section \[sec:hereditary subsets\] are relative Cuntz-Krieger algebras associated to $\L \setminus \L H$.
Let $(\L,d)$ be a $k$-graph, and let $H \subset \L^0$ be a saturated hereditary set. Consider the subcategory $\L\setminus\L H = \{\lambda \in \L : s(\lambda) \not \in H\}$.
\[lem:quotient graph\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $H \subset \L^0$ be saturated and hereditary. Then $(\L\setminus\L H, d|_{\L\setminus\L H})$ is also a finitely aligned $k$-graph.
We first check the factorisation property for $(\L\setminus\L H, d|_{\L\setminus\L H})$, and then that $(\L\setminus\L H, d|_{\L\setminus\L H})$ is finitely aligned. For the factorisation property, let $\lambda \in \L\setminus\L H$, and let $m,n \in \NN^k$, $m+n = d(\lambda)$. By the factorisation property for $\L$, there exist unique $\mu,\nu \in \L$ such that $d(\mu) = m$, $d(\nu) = n$ and $\lambda = \mu\nu$. Since $s(\nu) = s(\lambda) \not \in H$, we have $\nu \in \L\setminus\L H$. Since, by definition of $\le$, we have $r(\nu) \le s(\nu)$ it follows that $r(\nu) \not\in H$ because $H$ is hereditary. But $r(\nu) = s(\mu)$ so it follows that $\mu \in \L\setminus\L H$. Finite alignedness of the $k$-graph $\L\setminus\L H$ is trivial since $(\L\setminus\L H)^{\min}(\lambda,\mu) \subset \Lmin(\lambda,\mu)$ for all $\lambda,\mu \in \L\setminus\L H$.
\[dfn:EesubH\] Let $(\L,d)$ be a finitely aligned $k$-graph and let $H$ be a saturated hereditary subset of $\L^0$. Define $\Ee_H := \{E \setminus EH : E \in \FE(\L)\}$.
\[lem:EeH f.e.\] Let $(\L,d)$ be a finitely aligned $k$-graph, and suppose that $H \subset \L^0$ is saturated and hereditary. Then $\Ee_H \subset \FE(\L\setminus \L H)$.
Suppose that $E \in \Ee_H$ and that $\mu \in r(E)(\L \setminus \L H)$. Suppose for contradiction that $(\L \setminus \L H)^{\min}(\lambda,\mu) = \emptyset$ for all $\lambda \in E$. Since $E \in \Ee_H$, there exists $F \in \FE(\L)$ such that $F \setminus F H = E$. We have $$\label{eqn:Exts}
\Ext_\L(\mu; F) = \Ext_\L(\mu;E) \cup \Ext_\L(\mu; F \setminus E) = \Ext_\L(\mu;E) \cup \Ext_\L(\mu; F H).$$ Now $F H \subset \L H$ by definition, and then $\Ext(\mu; F H) \in \L H$ because $H$ is hereditary. Since $(\L \setminus \L H)^{\min}(\lambda,\mu) = \emptyset$ for all $\lambda \in E$, we must have $\Lmin(\lambda,\mu) \subset \L H \times \L H$ for all $\lambda \in E$, and hence we also have $\Ext_\L(\mu;E) \subset \L H$. Hence shows that $\Ext_\L(\mu;F) \subset \L H$. But $F$ is exhaustive in $\L$, so $\Ext(\mu;F)$ is also exhaustive by [@Si1 Lemma 2.3], and then since $H$ is saturated, it follows that $s(\mu) \in H$, contradicting our choice of $\mu$.
\[thm:quotient conjecture\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $H \subset \L^0$ be saturated and hereditary. Then $C^*(\L) / I_H$ is canonically isomorphic to $C^*((\L\setminus\L H) ; \Ee_H)$.
To prove Theorem \[thm:quotient conjecture\], we need to collect some additional results. Recall from [@Si1 Definition 4.1] that a subset $\Ee$ of $\FE(\L)$ is said to be *satiated* if it satisfies
- if $G \in \Ee$ and $E \in \FE(\L)$ with $G \subset E$, then $E \in \Ee$;
- if $G \in \Ee$ with $r(G) = v$ and $\mu \in v\L \setminus G\L$, then $\Ext(\mu;G) \in \Ee$;
- if $G \in \Ee$ and $0 < n_\lambda \le d(\lambda)$ for $\lambda \in G$, then $\{\lambda(0, n_\lambda) : \lambda \in G\} \in \Ee$; and
- if $G \in \Ee$, $G' \subset G$ and for each $\lambda \in G'$, $G'_\lambda$ is an element of $\Ee$ such that $r(G_\lambda') = s(\lambda)$, then $\textstyle\big((G\setminus G') \cup \big(\bigcup_{\lambda \in G'} \lambda
G'_\lambda\big)\big) \in \Ee$.
\[lem:EsubH its own bar\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $H \subset \L^0$ be saturated and hereditary. Then $\Ee_H$ is satiated.
For (S1), suppose that $E \in \Ee_H$ and $F \subset \L \setminus \L H$ is finite with $E \subset F$. By definition of $\Ee_H$, there exists $E' \in \FE(\L)$ such that $E' \setminus E' H = E$. But then $F' := F \cup E' H \in \FE(\L)$ by [@Si1 Lemma 5.3]. Since $F = F' \setminus F' H$, it follows that $F \in \Ee_H$.
For (S2), suppose that $E \in \Ee_H$, that $\mu \in r(E)(\L \setminus \L H)$ and that $\mu \not\in E\L$. Since $E \in \Ee_H$, there exists $E' \in \FE(\L)$ such that $E' \setminus E' H = E$. Since $\mu \in \L \setminus \L H$, we have $\mu \not\in E' H$, and hence $\Ext_\L(\mu;E') \in \FE(\L)$ by [@Si1 Lemma 2.3]. We also have $$\begin{split}
\Ext_\L(\mu;E')
&= \Ext_\L(\mu; E) \cup \Ext_\L(\mu; E' H) \\
&= \Ext_{\L \setminus \L H}(\mu; E) \cup \Ext_\L(\mu;E) H \cup \Ext_\L(\mu;E' H).
\end{split}$$ Since both $\Ext\L(\mu;E) H$ and $\Ext_\L(\mu; E'H)$ are subsets of $\L H$, it follows that $$\Ext_{\L \setminus \L H}(\mu;E) = \Ext_\L(\mu;E') \setminus \Ext_\L(\mu;E') H,$$ and hence belongs to $\Ee_H$.
For (S3), suppose that $E \in \Ee_H$, say $E' \in \FE(\L)$ and $E = E' \setminus E' H$. For each $\lambda \in E$, let $n_\lambda \in \NN^k$ with $0 < n_\lambda \le d(\lambda)$. For $\mu \in E' H$, let $n_\mu := d(\mu)$. Since $E'$ is exhaustive in $\L$, we have that $\{\mu(0,n_\mu) : \mu \in E'\}$ is also a finite exhaustive subset of $\L$ by [@Si1 Lemma 5.3], and since $$\{\lambda(0,n_\lambda) : \lambda \in E\} = \{\mu(0, n_\mu) : \mu \in E'\} \setminus \{\mu(0, n_\mu) : \mu \in E' H\},$$ it follows that $\{\lambda(0,n_\lambda) : \lambda \in E\} \in \Ee_H$.
Finally, for (S4), suppose that $E \in \Ee_H$, say $E' \in \FE(\L)$ and $E = E' \setminus E'H$. Let $F \subset E$, and for each $\lambda \in F$, suppose that $F_\lambda \in \Ee_H$ with $r(F_\lambda) = s(\lambda)$. We must show that $G := (E \setminus F) \cup \big(\bigcup_{\lambda \in F} \lambda F_\lambda\big) \in \Ee_H$. Since each $F_\lambda \in \Ee_H$, for each $\lambda \in F$, there exists a set $F'_\lambda \in \FE(\L)$ with $F_\lambda = F'_\lambda \setminus F'_\lambda H$. Let $G' := (E' \setminus F) \cup \big(\bigcup_{\lambda \in F} \lambda F'_\lambda\big)$. We will show that $G = G' \setminus G'H$, and that $G'$ is finite and exhaustive in $\L$; it follows from the definition of $\Ee_H$ that $G \in \Ee_H$, proving the result.
We have $G' \in \FE(\L)$ by [@Si1 Lemma 5.3], so it remains only to show that $G = G' \setminus G'H$. But since $H$ is hereditary, we have $$\begin{split}
G'H &=\textstyle \Big((E' \setminus F) \cup \big(\bigcup_{\lambda \in F} \lambda F'_\lambda\big)\Big) H \\
&=\textstyle (E' \setminus F) H \cup \big(\bigcup_{\lambda \in F} \lambda(F'_\lambda H)\big)
= E'H \cup \big(\bigcup_{\lambda \in F} \lambda F'_\lambda\big) H
\end{split}$$ because $F \subset E \subset \L\setminus\L H$. Consequently $$\textstyle
G' \setminus G'H = \big((E' \setminus F) \cup \big(\bigcup_{\lambda \in F} \lambda F'_\lambda\big)\big)
\setminus
\big(E'H \cup \big(\bigcup_{\lambda \in F} \lambda F'_\lambda H\big)\big) = G$$ as required.
\[lem:CK fams descend\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $H \subset \L^0$ be saturated and hereditary. Let $\{t_\lambda :\lambda \in \L\}$ be a Cuntz-Krieger $\L$-family, and let $I^t_H$ be the ideal in $C^*(\{t_\lambda : \lambda \in \L\})$ generated by $\{t_v : v \in H\}$. Then $\{t_\lambda + I^t_H : \lambda \in \L\setminus\L H\}$ is a relative Cuntz-Krieger $(\L \setminus\L H; \Ee_H)$-family in $C^*(\{t_\lambda : \lambda \in \L\}) / I^t_H$.
Relations (TCK1) and (TCK2) hold automatically since they also hold for the Cuntz-Krieger $\L$-family $\{t_\lambda : \lambda \in \L\}$. For (TCK3), let $\lambda,\mu \in \L \setminus\L H$ and notice that since $\{t_\lambda : \lambda \in \L\}$ is a Cuntz-Krieger $\L$-family, we have $$(t^*_\lambda + I^t_H)(t_\mu + I^t_H) = \sum_{(\alpha,\beta) \in \Lmin(\lambda,\mu)} t_\alpha t^*_\beta + I^t_H.$$ To show that this is equal to $\sum_{(\alpha,\beta) \in (\L\setminus\L H)^{\min}(\lambda,\mu)} t_\alpha t^*_\beta + I^t_H$, we need to show that $$(\alpha,\beta) \in \Lmin(\lambda,\mu) \setminus (\L \setminus \L H)^{\min}(\lambda, \mu) \text{ implies } t_\alpha t^*_\beta \in I^t_H.$$ So fix $(\alpha,\beta) \in \Lmin(\lambda,\mu) \setminus (\L \setminus \L H)^{\min}(\lambda, \mu)$. Then $s(\alpha) = s(\beta) \in H$, and hence $s_\alpha s^*_\beta = s_\alpha s_{s(\alpha)} s^*_\beta \in I^t_H$.
It remains to check (CK). Let $E \in \Ee_H$, say $E' \in \FE(\L)$ and $E = E' \setminus E' H$, and let $v := r(E)$. We must show that $\prod_{\lambda \in E} (t_v - t_\lambda t^*_\lambda)$ belongs to $I^t_H$. We know that $\prod_{\lambda \in E'} (t_v - t_\lambda t^*_\lambda) = 0$, and it follows that $$\label{eq:orth to FH}
\textstyle
\prod_{\lambda \in E}(t_v - t_\lambda t^*_\lambda) \Big(\prod_{\mu \in E' H} (t_v - t_\mu t^*_\mu)\Big) = 0.$$ Since $H$ is hereditary, Notation \[ntn:vee E\] gives $\vee (E'H) \subset \L H$, and $\prod_{\mu \in \vee (E' H)} (t_v - t_\mu t^*_\mu) \le \prod_{\mu \in E' H} (t_v - t_\mu t^*_\mu)$. Furthermore by [@RSY2 Proposition 3.5] we have $$\textstyle
t_v = \prod_{\mu \in \vee (E' H)} (t_v - t_\mu t^*_\mu) + \sum_{\mu \in \vee (E' H)} Q(t)^{\vee (E' H)}_\mu$$ where $Q(t)^{\vee (E' H)}_\mu := \prod_{\mu\mu' \in \vee(E' H) \setminus \{\mu\}} (t_\mu t^*_\mu - t_{\mu\mu'} t^*_{\mu\mu'})$.
Hence we can calculate $$\begin{aligned}
\prod_{\lambda \in E}(t_v - t_\lambda t^*_\lambda)
&= \Big(\prod_{\lambda \in E}(t_v - t_\lambda t^*_\lambda)\Big) t_v \\
&= \Big(\prod_{\lambda \in E}(t_v - t_\lambda t^*_\lambda)\Big) \Big(\prod_{\mu \in \vee (E' H)} (t_v - t_\mu t^*_\mu) + \sum_{\mu \in \vee (E' H)} Q(t)^{\vee (E' H)}_\mu \Big).\end{aligned}$$ Hence gives $\prod_{\lambda \in E}(t_v - t_\lambda t^*_\lambda) = \big(\prod_{\lambda \in E}(t_v - t_\lambda t^*_\lambda)\big) \big(\sum_{\mu \in \vee (E' H)} Q(t)^{\vee (E' H)}_\mu\big)$, and hence belongs to $I_H$ because $\vee (E' H) \subset \L H$, so each $Q(t)^{\vee (E' H)}_\mu \in I_H$.
Finally, before proving Theorem \[thm:quotient conjecture\], we need to recall some notation and definitions from [@RSY2] and [@Si1].
Let $(\L,d)$ be a finitely aligned $k$-graph, and let $G \subset \L$. As in [@RSY2 Definition 3.3], $\Pi G$ denotes the smallest subset of $\L$ which contains $G$ and has the property that if $\lambda,\mu$ and $\sigma$ belong to $G$ with $d(\lambda) = d(\mu)$ and $s(\lambda) = s(\mu)$ and if $(\alpha,\beta) \in \Lmin(\mu,\sigma)$, then $\lambda\alpha \in G$. If follows from [@RSY2 Lemma 3.2] that $\Pi G$ is finite when $G$ is. We denote by $\Pi G \times_{d,s} \Pi G$ the set of pairs $\{(\lambda,\mu) \in \Pi G \times \Pi G : d(\lambda) = d(\mu), s(\lambda) = s(\mu)\}$.
Let $\{t_\lambda : \lambda \in \L\}$ satisfy (TCK1)–(TCK3). As in [@RSY2 Proposition 3.5], for a finite set $G \subset \L$ and a path $\lambda \in \Pi G$, we write $Q(t)^{\Pi G}_\lambda$ for the projection $$\label{eq:Qdef}
Q(t)^{\Pi G}_\lambda :=
\prod_{\lambda\lambda' \in (\Pi G)\setminus \{\lambda\}} (t_\lambda t^*_\lambda - t_{\lambda\lambda'} t^*_{\lambda\lambda'}),$$ and for $(\lambda,\mu) \in \Pi G \times_{d,s} \Pi G$, we define $$\Theta(t)^{\Pi G}_{\lambda,\mu} := t_\lambda \Big(\prod_{\lambda\lambda' \in (\Pi G)\setminus \{\lambda\}} (t_{s(\lambda)} - t_{\lambda'} t^*_{\lambda'})\Big) t^*_\mu.$$ By [@RSY2 Lemma 3.10], we have $$Q(t)^{\Pi G}_\lambda t_\lambda t^*_\mu = \Theta(t)^{\Pi G}_{\lambda,\mu} = t_\lambda t^*_\mu Q(t)^{\Pi G}_\mu.$$
Finally, recall from [@Si1 Definition 4.4] that a graph morphism $x: \Omega_{k,m} \to \L$ is a *boundary path of $\L$* if, whenever $n \le m$ and $E \in x(n)\FE(\L)$, we have $x(n, n+d(\lambda)) = \lambda$ for some $\lambda \in E$. We write $r(x)$ for $x(0)$ and $d(x)$ for $m$. The collection $\partial\L := \{x : x\text{ is a boundary path of $\L$}\}$ is called the boundary-path space of $\L$. For $\lambda \in \L$ and $x \in \partial\L $ with $r(x) = s(\lambda)$, there is a unique boundary path $\lambda x$ such that $(\lambda x)(0, d(\lambda)) = \lambda$ and $(\lambda x)(d(\lambda), d(\lambda)+n) = x(0,n)$ for all $n \in \NN^k$. Likewise, given $x \in \partial\L $ and $n \le d(x)$, there is a unique boundary path $x|^{d(x)}_n$ such that $(x|^{d(x)}_n)(0, m) = x(n, n+m)$ for all $m \in \NN^k$. As in [@Si1 Definition 4.6], we define partial isometries $\{S_\lambda : \lambda \in \L\} \subset \Bb(\ell^2(\partial\L ))$ by $$S_\lambda e_x := \delta_{s(\lambda), r(x)} e_{\lambda }.$$ Lemma 4.7 of [@Si1] shows that $\{S_\lambda : \lambda \in \L\}$ is a Cuntz-Krieger $\L$-family called the *boundary-path representation* and that $$\label{eq:S_lambda*}
S^*_\lambda e_x =
\begin{cases}
e_{x|^{d(x)}_{d(\lambda)}} &\text{ if $x(0, d(\lambda)) = \lambda$} \\
0 &\text{ otherwise.}
\end{cases}$$
Fix $v \in \L^0 \setminus \L H$ and fix $E \in \FE(\L\setminus\L H)\setminus\Ee_H$.
\[clm:projection not in ideal\] Claim 1: For all $a \in \lsp\{s_\lambda s^*_\mu : \lambda,\mu \in \L H\}$, we have
- $\|s_v - a\| \ge 1$; and
- $\big\|\big(\prod_{\lambda \in E} (s_{r(E)} - s_\lambda s^*_\lambda)\big) - a\big\| \ge 1$ .
Express $a = \sum_{\lambda \in F} a_{\lambda,\mu} s_\lambda s^*_\mu$ where $F$ is a finite subset of $\L H$, and $\{a_{\lambda,\mu} : \lambda,\mu \in F\} \subset \CC$. Let $\pi_S$ be the boundary-path representation of $C^*(\L)$ and let $A := \pi_S(a) = \sum_{\lambda,\mu \in F} a_{\lambda,\mu} S_\lambda S^*_\mu$.
To check (1), note that since $v \not\in H$ and since $H$ is saturated, we have that $v F \cap \L^0 = \emptyset$ and that $v F \not\in \FE(\L)$. Hence there exists $\tau \in v\L$ such that $\Lmin(\tau,\lambda) = \emptyset$ for all $\lambda \in F$. By [@Si1 Lemma 4.7(1)], there exists a boundary path $x$ in $s(\tau)\partial\L$. By choice of $\tau$, we have that $\tau x \in v\partial\L \setminus F\partial\L$. But now $$\label{eq:norm is 1}\textstyle
\|S_v - A\|
\ge \|(S_v - A) e_{\tau x}\|
= \|S_v e_{\tau x} - \sum_{\lambda,\mu \in F} (a_{\lambda,\mu} S_\lambda S^*_\mu e_{\tau x})\|.$$ Since $\tau x \not\in F\partial\L$ by choice, gives $S^*_\mu e_{\tau x} = 0$ for all $\mu \in F$, and hence gives $\|S_v - A\| \ge \|S_v e_{\tau x}\| = \|e_{\tau x}\| = 1$. Since $\pi_S$ is a $C^*$-homomorphism, and hence norm-decreasing, this establishes (1).
For (2), note that $E \not\in \Ee_H$, and $F \subset \L H$ is finite, so we know that $E \cup F \not\in \FE(\L)$. Hence there exists $\tau \in \L$ such that $\Lmin(\sigma,\tau) = \emptyset$ for all $\sigma \in E \cup F$. By [@Si1 Lemma 4.7(1)], there exists $x \in \partial\L$ such that $r(x) = s(\tau)$. Set $y := \tau x \in \partial\L$. By choice of $\tau$, we have that $y(0, d(\sigma)) \not= \sigma$ for all $\sigma \in E \cup F$. Hence $S^*_\sigma e_y = 0$ for all $\sigma \in E \cup G$ by . In particular, $\sigma \in F$ implies $S_\sigma^* e_y = 0$, so $A e_y = 0$, and $\lambda \in E$ implies $S_\lambda^* e_y = 0$. It follows that $\big(\prod_{\lambda \in E}(S_{r(E)} - S_\lambda S^*_\lambda)\big) e_y = S_{r(E)} e_y = e_y$. Hence $$\textstyle
\big\|\big(\prod_{\lambda \in E}(S_{r(E)} - S_\lambda S^*_\lambda) - A\big)\big\| \ge
\big\|\big(\prod_{\lambda \in E}(S_{r(E)} - S_\lambda S^*_\lambda) - A\big) e_y\big\| = \|e_y\| = 1.$$ It follows that $\big\|\prod_{\lambda \in E}(S_{r(E)} - S_\lambda S^*_\lambda) - A\big\| \ge 1$. Again since $\pi_S$ is norm-decreasing, this establishes (2). 100
Since $I_H \subset C^*(\L)$ is fixed under the gauge action, $\gamma$ descends to a strongly continuous action $\theta$ of $\TT^k$ on $C^*(\L) / I_H$ such that $\theta_z \circ \pi^{\Ee_H}_{s + I_H} = \pi^{\Ee_H}_{s + I_H} \circ \gamma_z$ fo all $z \in \TT^k$.
It is easy to check using (TCK3) that $\lsp\{s_\lambda s^*_\mu : \lambda,\mu \in \L H\}$ is a dense subset of $I_H$. Hence Claim \[clm:projection not in ideal\] shows that neither $s_v$ nor $\prod_{\lambda \in E}(s_{r(E)} - s_\lambda s^*_\lambda)$ belongs to $I_H$. Since $v \in \L^0 \setminus H$ and $E \in \FE(\L\setminus\L H)\setminus \Ee_H$ were arbitrary, and since Lemma \[lem:EsubH its own bar\] shows that $\Ee_H$ is satiated, the gauge-invariant uniqueness theorem [@Si1 Theorem 6.1] shows that $\pi^{\Ee_H}_{s + I_H}$ is injective.
Gauge-invariant ideals in $C^*(\L)$ {#sec:ideal listing}
===================================
Theorem \[thm:quotient conjecture\] and [@Si1 Theorem 6.1] combine to show that every nontrivial gauge-invariant ideal in $C^*(\L\setminus\L H;\Ee_H)$ which contains no vertex projection $s_{\Ee_H}(v)$ must contain some collection of projections $$\textstyle\big\{\prod_{\lambda \in E} \big(s_{\Ee_H}(r(E)) - s_{\Ee_H}(\lambda) s_{\Ee_H}(\lambda)^*\big) : E \in B\big\}$$ where $B$ is a subset of $\FE(\L\setminus \L H) \setminus \Ee_H$.
Since $C^*(\L\setminus\L H; \Ee_H)$ itself is the quotient of $C^*(\L)$ by $I_H$, it follows that the ideals $I$ of $C^*(\L)$ such that the set $H_I$ defined in Lemma \[lem:H\_I sat,hered\] is equal to $H$ should be indexed by some collection of subsets of $\FE(\L\setminus\L H) \setminus \Ee_H$.
In this section, we show that the gauge-invariant ideals of $C^*(\L)$ are indexed by pairs $(H,B)$ where $H$ is a saturated hereditary subset of $\L^0$ and $B$ is a subset of $\FE(\L\setminus \L H) \setminus \Ee_H$ such that $B \cup \Ee_H$ is satiated.
Let $(\L,d)$ be a finitely aligned $k$-graph and let $H \subset \L^0$ be saturated and hereditary. Let $B$ be a subset of $\FE(\L\setminus \L H)$. We define $J_{H,B}$ to be the ideal of $C^*(\L)$ generated by $$\textstyle
\big\{s_v : v \in H\big\} \cup \big\{ \prod_{\lambda \in E} (s_{r(E)} - s_\lambda s^*_\lambda) : E \in B\big\}.$$ We define $I(\L\setminus\L H)_B$ to be the ideal of $C^*(\L \setminus \L H; \Ee_H)$ generated by $$\textstyle
\big\{\prod_{\lambda \in E} (s_{\Ee_H}(r(E)) - s_{\Ee_H}(\lambda) s_{\Ee_H}(\lambda)^*) : E \in B\big\}.$$
If $H \subset \L^0$ is saturated and hereditary, and if $B$ is a subset of $\FE(\L \setminus\L H)\setminus\Ee_H$ such that $\Ee_H \cup B$ is satiated, then $q(J_{H,B}) \cong I(\L\setminus\L H)_B$ where $q$ is the quotient map from $C^*(\L)$ to $C^*(\L) / I_H \cong C^*(\L\setminus\L H; \Ee_H)$.
We now investigate the structure of $C^*(\L)/J_{H,B}$.
\[lem:quotients equal\] Let $(\L,d)$ be a finitely aligned $k$-graph and let $H \subset \L^0$ be saturated and hereditary. Let $B$ be a subset of $\FE(\L\setminus \L H) \setminus \Ee_H$ such that $\Ee_H \cup B$ is satiated. Then $$C^*(\L\setminus \L H; \Ee_H) / I(\L\setminus\L H)_B = C^*(\L\setminus \L H; (\Ee_H \cup B)).$$
By Lemma \[lem:univ-quotients\], we have that $C^*(\L\setminus \L H; \Ee_H) \cong \Tt C^*(\L \setminus\L H) / J_{\Ee_H}$ and $C^*(\L\setminus \L H; (\Ee_H \cup B)) \cong \Tt C^*(\L \setminus\L H) / J_{\Ee_H \cup B}$. Hence we just need to show that $a \in \Tt C^*(\L \setminus\L H)$ belongs to $J_{\Ee_H \cup B}$ if and only if $q(a) \in I(\L\setminus\L H)_B$ where $q : \Tt C^*(\L \setminus\L H) \to C^*(\L\setminus \L H; \Ee_H)$ is the quotient map.
By definition of $I(\L\setminus\L H)_B$, the inverse image $q^{-1}(I(\L\setminus\L H)_B)$ under the quotient map is precisely the ideal in $\Tt C^*(\L \setminus \L H)$ generated by $$\begin{split}\textstyle
\{\prod_{\lambda \in E} &(s_{\Tt}(r(E)) - s_\Tt(\lambda) s_\Tt(\lambda)^*) : E \in B\} \\
&\textstyle\cup\ \{\prod_{\lambda \in E} (s_\Tt(r(E)) - s_\Tt(\lambda) s_\Tt(\lambda)^*) : E \in \Ee_H\};
\end{split}$$ that is, $q^{-1}(I(\L\setminus\L H)_B) = J_{\Ee_H \cup B}$ as required.
\[cor:2nd quotient a relCK\] Let $(\L,d)$ be a finitely aligned $k$-graph, let $H \subset \L^0$ be saturated and hereditary, and let $B \subset \FE(\L\setminus \L H) \setminus \Ee_H$. Then $$C^*(\L)/J_{H,B} \cong C^*(\L\setminus \L H; (\Ee_H \cup B)).$$
We will show that $C^*(\L)/ J_{H,B} = (C^*(\L)/I_H)/I(\L\setminus\L H)_B$; the result then follows from Lemma \[lem:quotients equal\]. Let $$\begin{aligned}
q_{H,B} &: C^*(\L) \to C^*(\L)/ J_{H,B}, \\
q_H &: C^*(\L) \to C^*(\L) / I_H, \\
q_B &: C^*(\L) / I_H \to (C^*(\L) / I_H)/ I(\L\setminus\L H)_B \end{aligned}$$ be the quotient maps. It is clear that the kernel of $q_{H,B}$ is contained in that of $q_B \circ q_H$, giving a canonical homomorphism $\pi_1$ of $C^*(\L)/ J_{H,B}$ onto $(C^*(\L)/I_H)/I(\L\setminus\L H)_B$. On the other hand, since $I_H \subset J_{H,B}$, there is a canonical homomorphism $\pi_2$ of $C^*(\L)/I_H$ onto $C^*(\L)/ J_{H,B}$ whose kernel contains $I(\L\setminus\L H)_B$ by definition. It follows that $\pi_2$ descends to a canonical homomorphism $\tilde\pi_2$ of $(C^*(\L)/I_H)/I(\L\setminus\L H)_B$ onto $C^*(\L)/ J_{H,B}$ which is inverse to $\pi_1$.
Let $(\L,d)$ be a finitely aligned $k$-graph. For each gauge-invariant ideal $I$ in $C^*(\L)$, recall that $H_I$ denotes $\{v \in \L^0 : s_v \in I\}$, and define $$\textstyle B_I := \big\{E \in \FE(\L \setminus \L H_I) \setminus \Ee_{H_I} :
\prod_{\lambda \in E} (s_{\Ee_{H_I}}(r(E)) - s_{\Ee_{H_I}}(\lambda) s_{\Ee_{H_I}}(\lambda)^*) \in q_{H_I}(I)\big\},$$ where $q_{H_I}$ is the quotient map from $C^*(\L)$ to $C^*(\L) / I_{H_I}$.
\[thm:every g-i ideal is a JH,B\] Let $(\L,d)$ be a finitely aligned $k$-graph.
- Let $I$ be a gauge-invariant ideal of $C^*(\L)$. Then $H_I \subset \L^0$ is nonempty saturated and hereditary, $\Ee_{H_I} \cup B_I$ is a satiated subset of $\FE(\L \setminus \L {H_I})$, and $I = J_{H_I, B_I}$.
- Let $H \subset \L^0$ be nonempty, saturated and hereditary, and let $B$ be a subset of $\FE(\L\setminus \L H) \setminus \Ee_H$ such that $\Ee_H \cup B$ is satiated in $\L \setminus \L H$. Then $H_{J_{H,B}} = H$ and $B_{J_{H,B}} = B$.
Theorem 6.1 of [@Si1] shows that $H_I$ is nonempty, and Lemma \[lem:H\_I sat,hered\] shows that it is saturated and hereditary. That $\Ee_H \cup B_I$ is satiated follows from [@Si1 Corollary 4.10].
Let $I$ be a gauge-invariant ideal of $C^*(\L)$. We have $J_{H_I, B_I} \subset I$ by definition, so there is a canonical homomorphism $\pi$ of $C^*(\L) / J_{H_I, B_I}$ onto $C^*(\L)/I$. By Corollary \[cor:2nd quotient a relCK\], this gives us a homomorphism, also denoted $\pi$ of $C^*(\L\setminus\L H_I; \Ee_{H_I} \cup B_I)$ onto $C^*(\L)/I$. Since $I$ is gauge-invariant, the gauge action on $C^*(\L)$ descends to an action $\theta$ of $\TT^k$ on $C^*(\L)/I$ such that $\theta_z \circ \pi = \pi \circ \gamma_z$ where $\gamma$ is the gauge action on $C^*(\L\setminus\L H_I; \Ee_{H_I} \cup B_I)$.
Suppose that $\pi(s_{\Ee_{H_I} \cup B_I}(v))$ is equal to $0$ in $C^*(\L)/I$. Then $s_v \in I$ by definition, so $v \in H_I$. Hence $\pi(s_{\Ee_{H_I} \cup B_I}(v)) \not= 0$ for all $v \in (\L \setminus \L H_I)^0$.
Now suppose that $E \in \FE(\L \setminus \L H_I)$ satisfies $$\textstyle
\pi\Big(\prod_{\lambda \in E} (s_{\Ee_{H_I} \cup B_I}(r(E)) - s_{\Ee_{H_I} \cup B_I}(\lambda)
s_{\Ee_{H_I} \cup B_I}(\lambda)^*)\Big) = 0_{C^*(\L)/I}.$$ Then either $E \in \Ee_{H_I}$, or else $E \in B_I$ by the definition of $B_I$. But then $\prod_{\lambda \in E} (s_{r(E)} - s_\lambda s^*_\lambda) \in J_{H_I, B_I}$, so that $$\prod_{\lambda \in E} (s_{\Ee_{H_I} \cup B_I}(r(E)) - s_{\Ee_{H_I} \cup B_I}(\lambda) s_{\Ee_{H_I} \cup B_I}(\lambda)^*) = 0_{C^*(\L\setminus \L H_I; \Ee_{H_I} \cup B_I)}.$$ Hence $\pi\Big(\prod_{\lambda \in E} (s_{\Ee_{H_I} \cup B_I}(r(E)) - s_{\Ee_{H_I} \cup B_I}(\lambda) s_{\Ee_{H_I} \cup B_I}(\lambda)^*)\Big) \not= 0$ for all $E \in \FE(\L) \setminus (\Ee_H \cup B)$.
By the previous three paragraphs we can apply [@Si1 Theorem 6.1] to see that $\pi$ is faithful, and hence that $I = J_{H_I, B_I}$ as required.
Now let $H \subset \L^0$ be saturated and hereditary, and let $B$ be a subset of $\FE(\L \setminus\L H)\setminus \Ee_H$ such that $\Ee_H \cup B$ is satiated.
We have $H \subset H_{J_{H,B}}$ and $B \subset B_{J_{H,B}}$ by definition. If $v \in H_{J_{H,B}}$, then $s_v \in J_{H,B}$ and hence its image in $C^*(\L \setminus \L H; \Ee_H \cup B)$ is trivial. It follows that either $v \in H$ or $s_{\Ee_H \cup B}(v) = 0$. But $s_{\Ee_H \cup B}(v) \not= 0$ for all $v \in (\L \setminus \L H)^0$ by [@Si1 Theorem 4.3], giving $v \in H$.
If $E \in B_{J_{H,B}}$, then we have $$\prod_{\lambda \in E} (s_{\Ee_H}(v) - s_{\Ee_H}(\lambda) s_{\Ee_H}(\lambda)^*) \in I(\L\setminus\L H)_B \subset C^*(\L\setminus\L H; \Ee_H).$$ Hence $\prod_{\lambda \in E} (s_{\Ee_H \cup B}(v) - s_{\Ee_H \cup B}(\lambda) s_{\Ee_H \cup B}(\lambda)^*)$ is equal to the zero element of $C^*(\L\setminus\L H; \Ee_H) / I(\L\setminus\L H)_B = C^*(\L\setminus\L H; \Ee_H \cup B)$. Since $\Ee_H \cup B$ is satiated, it follows that either $E \in \Ee_H$ or $E \in B$ by [@Si1 Theorem 4.3]. But $B_{J_{H,B}} \cap \Ee_H = \emptyset$ by definition, and it follows that $E \in B$ as required.
- Given a saturated hereditary $H \subset \L^0$, the ideal $I_H$ (see Notation \[ntn:I\_H\]) is listed by Theorem \[thm:every g-i ideal is a JH,B\] as $J_{H,\emptyset}$.
- It seems difficult to establish an analogue of Lemma \[lem:Lambda(H) Vs I\_H\] for arbitrary $J_{H,B}$. A good strategy would be to aim to describe $I(\L\setminus\L H)_B = J_{H,B} / I_H$ as (Morita equivalent to) a $k$-graph algebra. But this seems difficult even when $B$ is “singly generated:” i.e. when $\Ee_H \cup B$ is the satiation (see [@Si1 Definition 5.1]) of $\Ee_H \cup \{E\}$ where $E \in \FE(\L\setminus \L H) \setminus \Ee_H$.
The lattice order {#sec:lattice}
=================
In this section we describe the lattice ordering of the gauge-invariant ideals of $C^*(\L)$ in terms of a lattice order on the pairs $(H,B)$ where $H \subset \L^0$ is saturated and hereditary, and $B$ is a subset of $\FE(\L\setminus \L H) \setminus \Ee_H$ such that $\Ee_H \cup B$ is satiated.
\[dfn:(H,B) order\] Let $(\L,d)$ be a finitely aligned $k$-graph. Define $$\begin{split}
\Ipairs :=
\big\{(H,B) : {}&\emptyset\not= H \subset \L^0, H\text{ is saturated and hereditary, }\\
&B \subset \FE(\L\setminus\L H)\setminus\Ee_H\text{ and }\Ee_H \cup B\text{ is satiated}\big\}.
\end{split}$$ Define a relation $\preceq$ on $\Ipairs$ by $(H_1, B_1) \preceq (H_2, B_2)$ if and only if
- $H_1 \subset H_2$; and
- if $E \in B_1$ and $r(E) \not\in H_2$, then $E \setminus EH_2$ belongs to $\Ee_{H_2} \cup B_2$.
\[thm:gauge-invariant ideals\] Let $(\L,d)$ be a finitely aligned $k$-graph. The map $(H,B) \mapsto J_{H,B}$ is a lattice isomorphism between $(\Ipairs, \preceq)$ and $(I^\gamma(\L), \subset)$ where $I^\gamma(\L)$ denotes the collection of gauge-invariant ideals of $C^*(\L)$.
Theorem \[thm:every g-i ideal is a JH,B\] implies that $(H,B) \mapsto J_{H,B}$ is a bijection between $\Ipairs$ and $I^\gamma(C^*(\L))$. Hence, we need only establish that for $(H_1, B_1), (H_2, B_2) \in \Ipairs$, $$\label{eq:subset vs preceq}
\text{$J_{H_1, B_1} \subset J_{H_2, B_2}$ if and only if $(H_1, B_1) \preceq (H_2, B_2)$.}$$
First suppose that $J_{H_1, B_1} \subset J_{H_2, B_2}$. Theorem \[thm:every g-i ideal is a JH,B\] shows immediately that $H_1 \subset H_2$, so if we can show that $F \in B_1$ with $r(F) \not\in H_2$ implies $F\setminus FH_2 \in \Ee_{H_2} \cup B_2$, it will follow that $(H_1, B_1) \preceq (H_2, B_2)$.
Suppose that $E = F\setminus FH_2$ for some $F \in B_1$ with $r(F) \not\in H_2$. Suppose further for contradiction that $E \not\in \Ee_{H_2} \cup B_2$. Let $q_i : C^*(\L) \to C^*(\L)/ J_{H_i, B_i}$ where $i = 1,2$ denote the quotient maps; by Corollary \[cor:2nd quotient a relCK\], we can regard $q_i$ as a homomorphism of $C^*(\L)$ onto $C^*(\L\setminus\L H_i; \Ee_{H_i} \cup B_i)$ for $i=1,2$. Since $J_{H_1, B_1} \subset J_{H_2, B_2}$, there is a homomorphism $\pi : C^*(\L\setminus \L H_1; \Ee_{H_1}\cup B_1) \to C^*(\L\setminus \L H_2; \Ee_{H_2}\cup B_2)$ such that $\pi\circ q_1 = q_2$. Since $F \in B_1$, we have $q_1\big(\prod_{\lambda \in F} (s_{r(F)} - s_\lambda s^*_\lambda)\big) = 0$, and hence $$\label{eq:q2 image 0}\textstyle
q_2\big(\prod_{\lambda \in F} (s_{r(F)} - s_\lambda s^*_\lambda)\big) = \pi\Big(q_1\big(\prod_{\lambda \in F} (s_{r(F)} - s_\lambda s^*_\lambda)\big)\Big) = 0.$$ Since $s(\lambda) \in H_2$ implies $q_2(s_\lambda s^*_\lambda) = 0$ by definition, we have that $$\label{eq:q2 image}\textstyle
q_2\Big(\prod_{\lambda \in F} (s_{r(F)} - s_\lambda s^*_\lambda)\Big)
= \prod_{\lambda \in E}\big(s_{\Ee_{H_2} \cup B_2}(r(E)) - s_{\Ee_{H_2} \cup B_2}(\lambda)s_{\Ee_{H_2} \cup
B_2}(\lambda)^*\big),$$
We consider two cases:
Case 1: $E$ belongs to $\FE(\L\setminus\L H_2)$. Then since $E \not\in \Ee_{H_2} \cup B_2$, [@Si1 Corollary 4.10] ensures that $\prod_{\lambda \in E}\big(s_{\Ee_{H_2} \cup B_2}(r(E)) - s_{\Ee_{H_2} \cup B_2}(\lambda) s_{\Ee_{H_2} \cup B_2}(\lambda)^*\big)$ is nonzero.
Case 2: $E \not\in \FE(\L\setminus\L H_2)$. Then there exists $\mu \in r(E)\L \setminus \L H_2$ with $\Ext(\mu;E) = \emptyset$; we then have $$\begin{split}\textstyle
\prod_{\lambda \in E}\big(s_{\Ee_{H_2} \cup B_2}(r(E)) - s_{\Ee_{H_2} \cup B_2}(\lambda)s_{\Ee_{H_2} \cup
B_2}(\lambda)^*\big) &s_{\Ee_{H_2} \cup B_2}(\mu) s_{\Ee_{H_2} \cup B_2}(\mu)^* \\
&= s_{\Ee_{H_2} \cup B_2}(\mu) s_{\Ee_{H_2} \cup B_2}(\mu)^*
\end{split}$$ by (TCK3). Since $s_{\Ee_{H_2} \cup B_2}(\mu) s_{\Ee_{H_2} \cup B_2}(\mu)^* \not= 0$ by [@Si1 Corollary 4.10], it follows that $$\prod_{\lambda \in E}\big(s_{\Ee_{H_2} \cup B_2}(r(E)) - s_{\Ee_{H_2} \cup B_2}(\lambda)s_{\Ee_{H_2} \cup
B_2}(\lambda)^*\big) s_{\Ee_{H_2} \cup B_2}(\mu) s_{\Ee_{H_2} \cup B_2}(\mu)^* \not= 0.$$
In either case, shows that $q_2\big(\prod_{\lambda \in F} (s_{r(F)} - s_\lambda s^*_\lambda)\big)$ is nonzero, contradicting . This establishes the “only if” assertion of .
Now suppose that $(H_1, B_1) \preceq (H_2, B_2) \in \Ipairs$. Let $v \in H_1$. Since $(H_1, B_1) \preceq (H_2, B_2)$, we have that $H_1 \subset H_2$, and hence $v \in H_2$ giving $s_v \in J_{H_2, B_2}$ by definition. Now let $E \in B_1$. If $r(E) \in H_2$, then $s_{r(E)} \in J_{H_2, B_2}$ by definition, and hence $\prod_{\lambda \in E} (s_{r(E)} - s_\lambda s^*_\lambda) = \big(\prod_{\lambda \in E} (s_{r(E)} - s_\lambda s^*_\lambda)\big) s_{r(E)} \in J_{H_2, B_2}$. If $r(E) \not\in H_2$, then since $(H_1, B_1) \preceq (H_2, B_2)$, we have that $E \setminus E H_2 \in
\Ee_{H_2} \cup B_2$. For $\lambda \in \L H_2$, we have $s_\lambda s^*_\lambda = s_\lambda s_{s(\lambda)} s^*_\lambda \in J_{H_2, B_2}$ and hence $q_2(s_\lambda s^*_\lambda) = 0$, so $$\label{eq:quotient image}
q_2\Big(\prod_{\lambda \in E} (s_{r(E)} - s_\lambda s^*_\lambda)\Big) = \prod_{\lambda \in E\setminus E H_2}
(s_{\Ee_{H_2}\cup B_2}(r(E)) - s_{\Ee_{H_2}\cup B_2}(\lambda) s_{\Ee_{H_2}\cup B_2}(\lambda)^*).$$ Since $E \setminus E H_2 \in \Ee_{H_2} \cup B_2$, and since $\{s_{\Ee_{H_2} \cup B_2}(\lambda) : \lambda \in \L \setminus \L H_2\}$ is a relative Cuntz-Krieger $(\L\setminus\L H_2; E_{H_2} \cup B_2)$-family, relation (CK) gives $$\textstyle
\prod_{\lambda \in E \setminus E H_2} (s_{\Ee_{H_2} \cup B_2}(r(E)) - s_{\Ee_{H_2} \cup B_2}(\lambda) s_{\Ee_{H_2} \cup B_2}(\lambda)^*) = 0.$$ Hence $\prod_{\lambda \in E} (s_{r(E)} - s_\lambda s^*_\lambda) \in \ker q_2 = J_{H_2, B_2}$ by and Corollary \[cor:2nd quotient a relCK\].
Since all the generating projections of $J_{H_1, B_1}$ belong to $J_{H_2, B_2}$, it follows that $J_{H_1, B_1} \subset J_{H_2, B_2}$, establishing the “if” assertion of .
$k$-graphs in which all ideals are gauge-invariant {#sec:CK ideals}
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In this section we use the Cuntz-Krieger uniqueness theorem of [@Si1] to show that for a certain class of $k$-graphs, the ideals $J_{H,B}$ identified in Section \[sec:ideal listing\] are all the ideals in $C^*(\L)$; that is, every ideal in $C^*(\L)$ is gauge-invariant.
Recall from [@Si1 Definition 6.2] that if $x : \Omega_{k, d(x)} \to \L$ and $y : \Omega_{k, d(y)} \to \L$ are graph morphisms, then $\MCE(x,y)$ is the collection of all graph morphisms $z : \Omega_{k, d(z)} \to \L$ such that $d(z)_i = \max{d(x)_i, d(y)_i}$ for $1 \le i \le k$, and such that $z|_{\Omega_{k, d(x)}} = x$ and $z|_{\Omega_{k, d(y)}} = y$.
Recall also from [@Si1 Theorem 6.3] that if $(\L,d)$ is a finitely aligned $k$-graph and $\Ee$ is a subset of $\FE(\L)$, then $(\L,\Ee)$ is said to satisfy *condition [(C)]{}* if
- For all $v \in \L^0$ there exists $x \in v\partial(\L;\Ee)$ such that for distinct $\lambda,\mu$ in $\L r(x)$, we have $\MCE(\lambda x, \mu x) = \emptyset$; and
- for each $F \in v\FE(\L)\setminus\overline\Ee$, there is a path $x$ as in (1) such that $x \in v\partial(\L;\Ee) \setminus F\partial(\L;\Ee)$.
Let $(\L,d)$ be a finitely aligned $k$-graph. We say that $\L$ satisfies *condition * if $$\label{eq:new condition K}
\text{$(\L \setminus \L H, \Ee_H)$ satisfies condition~(C) for each saturated, hereditary $H \subset \L^0$.}\tag{D}$$
\[thm:cond(D)–> every ideal a JH,B\] Let $(\L,d)$ be a finitely aligned $k$-graph which satisfies condition .
- Let $I$ be an ideal of $C^*(\L)$. Then $H_I$ is nonempty, saturated and hereditary, $B_I \cup \Ee_{H_I}$ is satiated in $\L \setminus \L {H_I}$, and $I = J_{H_I, B_I}$.
- Let $H \subset \L^0$ be nonempty, saturated and hereditary, and let $B \subset \FE(\L\setminus \L H) \setminus \Ee_H$ be such that $B \cup \Ee_H$ is satiated in $\L \setminus \L H$. Then $H_{J_{H,B}} = H$ and $B_{J_{H,B}} = B$.
The proof of (1) is the same as the proof of of Theorem \[thm:every g-i ideal is a JH,B\](1) except that, since we do not know *a priori* that $I$ is gauge-invariant, we do not automatically have an action $\pi$ on $C^*(\L)/I$ such that $\theta_z \circ \pi = \pi \circ \gamma_z$. Consequently, we cannot apply [@Si1 Theorem 6.1] to deduce that $\pi$ is faithful; instead, we use our assumption that $(\L\setminus\L H, \Ee_H)$ satisfies condition (C) to apply [@Si1 Theorem 6.3].
The proof of (2) is identical the to proof of part (2) of Theorem \[thm:every g-i ideal is a JH,B\].
Classifiability {#ch:classifiable}
===============
In this section we investigate when $C^*(\L)$ is a Kirchberg-Phillips algebra. We show that all relative $k$-graph algebras $C^*(\L;\Ee)$ fall into the bootstrap class $\mathcal{N}$ of [@RSc]. We show that if $\L$ satisfies condition (C), then $C^*(\L)$ is simple if and only if $\L$ is *cofinal*. Finally, we show that if in addition every vertex of $\L$ can be reached from a *loop with an entrance*, then $C^*(\L)$ is purely infinite.
The main results in this section are generalisations to arbitrary finitely aligned $k$-graphs of the corresponding results of Kumjian and Pask for row-finite $k$-graphs with no sources in [@KP].
The author would like to thank D. Gwion Evans for drawing his attention to the results of [@PQR] which provide the necessary technical machinery for the proof of Proposition \[prp:rel algs nuclear\].
\[prp:rel algs nuclear\] Let $(\L,d)$ be a finitely aligned $k$-graph and let $\Ee$ be a subset of $\FE(\L)$. Then $C^*(\L;\Ee)$ is stably isomorphic to a crossed product of an AF algebra by $\ZZ^k$, and hence falls into the bootstrap class $\mathcal{N}$ of [@RSc]; in particular, $C^*(\L;\Ee)$ is nuclear and satisfies the Universal Coefficient Theorem.
This proposition generalises [@KP Theorem 5.5], and the overall strategy of the proof is the same, but the technical details are more complicated, and draw on [@RSY2] and [@PQR]. We first need to establish some preliminary lemmas, the first of which generalises [@KP Lemma 5.4].
\[lem:skew-product AF\] Let $(\L,d)$ be a finitely aligned $k$-graph and let $\Ee \subset \FE(\L)$. Suppose there is a function $b : \L^0 \to \ZZ^k$ such that $d(\lambda) = b(s(\lambda)) - b(r(\lambda))$ for all $\lambda \in \L$. Then $C^*(\L;\Ee)$ is AF.
The proof is based heavily on that of [@RSY2 Lemma 3.2].
It suffices to show that for $E \subset \L$ finite, we have that $C^*(\{s_\Ee(\lambda) : \lambda \in E\})$ is finite dimensional. Recalling the definition of $\vee E$ from Notation \[ntn:vee E\], define a map $M$ on finite subsets of $\L$ by $$\label{eq:Pi tilde}
\begin{split}
M(E) := \{(\lambda_1(0, d(\lambda_1))\lambda_2(n_2, d(\lambda_2)) \dots &\lambda_l(n_l, d(\lambda_l)) : \\
&l \in \NN\setminus\{0\}, \lambda_i \in \vee E, n_i \le d(\lambda_i)\}.
\end{split}$$ We claim that
- $M(E)$ is finite;
- $E \subset \vee E \subset M(E)$;
- $\bigvee_{\lambda \in M(E)} b(s(\lambda)) = \bigvee_{\mu \in E} b(s(\mu))$;
- $\lambda,\mu,\sigma,\tau \in E$ implies $s_\Ee(\lambda) s_\Ee(\mu)^* s_\Ee(\sigma) s_\Ee(\tau)^* \in \lsp\{s_\Ee(\eta) s_\Ee(\zeta)^* : \eta,\zeta \in M(E)\}$; and
- if $M^2(E) \not= M(E)$, then $\min\{\sum^k_{i=1}b(s(\lambda))_i : \lambda \in M^2(E) \setminus M(E)\}$ is strictly greater than $\min\{\sum^k_{i=1}b(s(\mu))_i : \mu \in M(E) \setminus E\}$.
For (a), note that each path in $M(E)$ can be factorised as $\alpha_1\dots\alpha_{|d(\lambda)|}$ where each $\alpha_i = \mu(n, n+e_l)$ for some $n \in \NN^k$, $1 \le l \le k$, and $\mu \in \vee E$. Moreover, $i < j \implies b(s(\alpha_i)) < \big(b(s(\alpha_i)) + d(\alpha_j)\big) \le b(s(\alpha_j)) \implies \alpha_i \not= \alpha_j$. Since $\vee E$ is finite, the number of possible values for $\alpha_i$ is finite, and it follows that $M(E)$ is finite.
We have $E \subset \vee E$ by definition, and $\vee E \subset M(E)$ by taking $l = 1$ in , establishing (b).
For (c), first note that $\lambda \in M(E) \implies s(\lambda) = s(\mu)$ for some $\mu \in \vee E$, so $$\label{eq:first le}
\textstyle \bigvee_{\lambda \in M(E)} b(s(\lambda)) \le \bigvee_{\mu \in \vee E} b(s(\mu)).$$ Next recall from [@RS1 Definition 8.3] that for finite $F \subset \L$ $$\MCE(F) := \{\lambda \in \L : d(\lambda) = \bigvee_{\mu \in F} d(\mu), \lambda(0, d(\mu)) = \mu\text{ for all } \mu \in F\},$$ and that $\vee E = \bigcup\{\MCE(F) : F \subset E\}$. So $\lambda \in \vee E \implies \lambda \in \MCE(F)$ for some subset $F$ of $E$. In particular, $\MCE(F)$ is nonempty, so we must have $F \subset v\L$ for some $v \in \L^0$. Write $n$ for $b(v)$, and calculate: $$\textstyle
b(s(\lambda)) = n + \bigvee_{\mu \in F} d(\mu) = n + \bigvee_{\mu \in F} (b(s(\mu)) - n) = \bigvee_{\mu \in F} b(s(\mu)).$$ Hence $\bigvee_{\lambda \in \vee E} b(s(\lambda)) \le \bigvee_{\mu \in E} b(s(\mu))$, so $\bigvee_{\lambda \in M(E)} b(s(\lambda)) \le \bigvee_{\mu \in E} b(s(\mu))$ by . The reverse inequality follows from (b), establishing (c).
Claim (d) follows from and (TCK3). Finally, (e) follows from an argument identical to the proof of (e) in [@RSY2 Lemma 3.2] but with $d(\lambda)$ replaced with $b(\lambda)$ throughout. This establishes the claim.
It now follows as in [@RSY2 Lemma 3.2] that $M^\infty(E) := \bigcup^\infty_{i=1} M^i(E)$ is finite and that $\lsp\{s_\Ee(\lambda) s_\Ee(\mu)^* : \lambda,\mu \in M^\infty(E)\}$ is a finite-dimensional subalgebra of $C^*(\L;\Ee)$ containing $C^*(\{s_\Ee(\lambda) : \lambda \in E\})$.
Let $\L \times_d \ZZ^k$ be the skew-product $k$-graph which is equal, as a set, to $\L \times \ZZ^k$ and has range, source and degree maps given by $r(\lambda,n) := (r(\lambda), n - d(\lambda))$, $s(\lambda,n) := (s(\lambda), n)$, and $d(\lambda,n) := d(\lambda)$ (see [@KP Definition 5.1]). For $E \in \Ee$ and $n \in \ZZ^k$, let $E \times_d \{n\} := \{(\lambda, n + d(\lambda)) : \lambda \in E\}$, and let $\Ee \times_d \ZZ^k := \{E \times_d \{n\} : E \in \Ee, n \in \ZZ^k\}$.
Recall that a *coaction* $\delta$ of a group $G$ on a $C^*$-algebra $A$ is an injective unital homomorphism $\delta : A \to A \otimes C^*(G)$ satisfying the *cocycle identity* $(\id \otimes \delta_G)\circ\delta = (\delta \otimes \id)\circ\delta$. The *fixed point algebra* is the subspace $A^\delta := \{a \in A : \delta(a) = a \otimes e\}$. There is a universal crossed product algebra $A \times_\delta G$ associated to the triple $(A,G,\delta)$, and this algebra admits a *dual action* $\hat\delta$ of $G$. Crossed product duality says that $A \times_\delta G \times_{\hat\delta} G \cong A \otimes \ell^2(G)$.
The following lemma generalises [@PQR Theorem 7.1] to relative $k$-graph algebras.
\[lem:coactions\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $\Ee$ be a subset of $\FE(\L)$. Then
- $\Ee \times_d \ZZ^k$ is a subset of $\FE(\L \times_d \ZZ^k)$;
- $C^*(\L \times_d \ZZ^k; \Ee \times_d \ZZ^k)$ is AF;
- there is a unique coaction $\delta$ of $\ZZ^k$ on $C^*(\L;\Ee)$ such that $\delta(s_\Ee(\lambda)) := s_\Ee(\lambda) \otimes d(\lambda)$ for all $\lambda \in \L$; and
- the crossed product algebra $C^*(\L;\Ee) \times_\delta \ZZ^k$ is isomorphic to $C^*(\L \times_d \ZZ^k; \Ee \times_d \ZZ^k)$.
For part (1), fix $E \times_d \{n\} \in \Ee \times_d \ZZ^k$, and suppose that $r(\lambda,m) = r(E \times_d \{n\})$. Then $m = n + d(\lambda)$ and $r(\lambda) = r(E)$. Since $E \in \FE(\L)$, there exists $\alpha \in \Ext(\lambda;E)$. It is straightforward to check that $(\alpha, m + d(\alpha)) \in \Ext((\lambda,m); E \times_d \{n\})$. Since $(\lambda,m)$ was arbitrary, it follows that $E \times_d \{n\} \in \FE(\L \times_d \ZZ^k)$, and since $E \times_d \{n\}$ was itself arbitrary in $\Ee \times_d \ZZ^k$, this establishes (1).
For (2), define $b : (\L \times_d \ZZ^k)^0 \to \ZZ^k$ by $b(\lambda,n) := n$. Then the pair $(\L \times_d \ZZ^k, b)$ satisfies the hypotheses of Lemma \[lem:skew-product AF\], so $C^*(\L \times_d \ZZ^k; \Ee \times_d \ZZ^k)$ is AF.
Parts (3) and (4) now follow exactly as (i) and (ii) of [@PQR Theorem 7.1].
We have that $C^*(\L;\Ee) \times_\delta \ZZ^k \cong C^*(\L \times_d \ZZ^k; \Ee \times_d \ZZ^k)$ is AF. But crossed product duality gives $C^*(\L;\Ee) \otimes \ell^2(\ZZ^k) \cong C^*(\L;\Ee) \times_\delta \ZZ^k \times_{\hat\delta} \ZZ^k$, so $C^*(\L;\Ee)$ is stably isomorphic to a crossed product of an AF algebra by $\ZZ^k$.
Our simplicity result is a direct generalisation of [@KP Proposition 4.8], though our proof is based on that of [@BPRS Proposition 5.1].
Let $(\L,d)$ be a finitely aligned $k$-graph. We say that $\L$ is *cofinal* if for all $v \in \L^0$ and $x \in \partial\L $, there exists $n \le d(x)$ such that $v \L x(n) \not= \emptyset$.
\[prp:simple graph alg\] Let $(\L,d)$ be a finitely aligned $k$-graph, and suppose that $\L$ satisfies condition [(C)]{}. Then $C^*(\L)$ is simple if and only if $\L$ is cofinal.
First suppose that $\L$ is cofinal, and suppose that $I$ is an ideal in $C^*(\L)$. If $s_v \in I$ for all $v \in \L^0$, then $I = C^*(\L)$ by (TCK2). Suppose that $v \in \L^0$ with $s_v \not \in I$. We must show that $H_I$ is empty, for if so then [@Si1 Theorem 6.3] shows that $I$ is trivial. Since $H_I$ is saturated, we have that $$\label{eq:saturation consequence}
\text{if $v' \not\in H_I$ and $E \in v\FE(\L)$, then there exists $\lambda \in E$ such that $s(\lambda) \not \in H_I$.}$$ To prove the proposition, we first establish the following claim:
\[clm:boundary path\] There exists a path $x \in \partial\L$ such that $x(n) \not\in H_I$ for all $n \le d(x)$.
The proof of the claim is very similar to the proof of [@Si1 Lemma 4.7(1)], but with minor technical changes needed to establish that we can obtain $x(n) \not\in H_I$ for all $n$. Consequently, we give a proof sketch with frequent references to the proof in [@Si1].
As in the proof of [@Si1 Lemma 4.7(1)], let $P : \NN^2 \to \NN$ be the position function associated to the diagonal listing of $\NN^2$: $$P(0,0) = 0,\quad P(0,1) = 1,\quad P(1,0) = 2,\quad P(0,2) = 3,\quad P(1,1) = 4,\quad \dots$$ For $l \in \NN$, let $(i_l, j_l)$ be the unique element of $\NN^2$ such that $P(i_l, j_l) = l$.
We will show by induction that there exists a sequence $\{\lambda_l : l \ge 0\} \subset v\L$ and enumerations $\{E_{l,j} : j \ge 0\}$ of $s(\lambda_l)\FE(\L)$ for all $l \ge 0$ such that
- $s(\lambda_l) \not\in H_I$ for all $l$;
- $\lambda_{l+1}(0, d(\lambda_l)) = \lambda_l$ for all $l \ge 1$; and
- $\lambda_{l+1}(d(\lambda_{i_l}), d(\lambda_{l+1})) \in E_{i_l, j_l}\L$ for all $l \ge 0$.
As in the proof of [@Si1 Lemma 4.7(1)], we proceed by induction on $l$; for $l = 0$ we take $\lambda_0 := v$ and fix $\{E_{0,j} : j \ge 0\}$ to be any enumeration of $\{E \in \FE(\L) : r(E) = v\}$. These satisfy (i) by definition of $H_I$, and trivially satisfy (ii) and (iii).
Now as an inductive hypothesis, suppose that $l \ge 0$ and that $\lambda_1, \dots, \lambda_l$ and $\{E_{1,j} : j \ge 1\}, \dots, \{E_{l,j} : j \ge 1\}$ have been chosen and satisfy (i)–(iii). Just as in the proof of [@Si1 Lemma 4.7(1)], we have that $l \ge i_l$ so that $E_{i_l, j_l}$ has already been defined. If $\lambda_l(d(\lambda_{i_{l+1}}, d(\lambda_l))) \in E_{i_{l+1}, j_{l+1}}$ already, then $l > 0$ because $E \in \FE(\L)$ implies $E \cap \L^0 = \emptyset$, so $\lambda_{l+1} := \lambda_l$ and $E_{l+1, j} := E_{l,j}$ for all $j$ satisfy (i)–(iii) by the inductive hypothesis. On the other hand, if $\lambda_l(d(\lambda_{i_{l+1}}, d(\lambda_l))) \not\in E_{i_{l+1}, j_{l+1}}$, then $E := \Ext\big(\lambda_l(d(\lambda_{i_{l+1}}, d(\lambda_l))); E_{i_{l+1}, j_{l+1}}\big) \in \FE(\L)$ by [@RSY2 Lemma C.5]. By , there exists $\nu_{l+1} \in E$ such that $s(\nu) \not\in H_i$. But now $\lambda_{l+1} := \lambda_l\nu_{l+1}$ satisfies (i) by choice of $\nu_{l+1}$, and taking $\{E_{l+1, j} : j \ge 1\}$ to be any enumeration of $\{E \in \FE(\L) : r(E) = s(\nu_{l+1})\}$ we have (ii) and (iii) satisfied just as in the proof of [@Si1 Lemma 4.7(1)].
The remainder of the proof of [@Si1 Lemma 4.7(1)] shows that $x(0, d(\lambda_l)) := \lambda_l$ for all $l$ defines an element of $v\partial\L$, and since $H_I$ is hereditary, condition (i) shows that $x(n) \not\in H_I$ for all $n \le d(x)$. 100
Now fix $w \in \L^0$. Let $x \in v\partial\L$ with $x(n) \not\in H_I$ for all $n$ as in Claim \[clm:boundary path\]. Since $\L$ is cofinal, there exists $n \le d(x)$ such that $w \L x(n) \not= \emptyset$. Since $x(n) \not\in H_I$ by construction of $x$, and since $H_I$ is hereditary, it follows that $w \not\in H_I$. Consequently $H_I = \emptyset$ as required.
Now suppose that $C^*(\L)$ is simple. Let $x \in \partial\L $, and let $$H_x := \{w \in \L^0 : w \L x(n) = \emptyset\text{ for all $n$}\}.$$ It is clear that $H_x$ is hereditary. We claim that $H_x$ is saturated: suppose that $E \in v\FE(\L)$ with $s(E) \in H_x$, and suppose for contradiction that $\lambda \in v \L x(n)$. If $\lambda = \mu\mu'$ for $\mu \in E$, then $\mu' \in s(\mu)\L x(n)$, contradicting $s(\mu) \in H_x$. On the other hand, if $\lambda \not\in E\L$, then $\Ext(\lambda;E)$ is exhaustive by [@Si1 Lemma 2.3]. Since $x \in \partial(\L;\Ee)$, it follows that $x(n, n + d(\alpha)) = \alpha$ for some $\alpha \in \Ext(\lambda;E)$; say $(\alpha,\beta) \in \Lmin(\lambda,\mu)$ where $\mu \in E$. Then $\beta \in s(\mu)\L x(n+d(\alpha))$, again contradicting $s(\mu) \in H_x$. This proves our claim.
Now $H_x \not = \L^0$ because, in particular, $r(x) \not \in H_x$. It follows that if $H_x$ is nonempty then it corresponds to a nontrivial ideal $I_{H_x}$ which is impossible since $C^*(\L)$ is simple by assumption. Hence $\L$ is cofinal as required.
\[dfn:loop with entrance\] Let $(\L,d)$ be a finitely aligned $k$-graph. We say that a path $\mu \in \L$ is a *loop with an entrance* if $s(\mu) = r(\mu)$ and there exists $\alpha \in s(\mu)\L$ such that $d(\mu) \ge d(\alpha)$ and $\mu(0, d(\alpha)) \not= \alpha$. We say that a vertex $v \in \L^0$ can be *reached from a loop with an entrance* if there exists a loop with an entrance $\mu \in \L$ such that $v \L s(\mu) \not= \emptyset$.
The following proposition rectifies a slight error in [@KP Proposition 4.9], specifically in the argument that $\mathcal{G}_\Lambda$ is locally contracting. Our condition that every vertex can be reached from a loop with an entrance is slightly than that in [@KP] that every vertex can be reached from a nontrivial loop, and this stronger condition is needed to make both our argument and that of [@KP] run.
\[prp:purely infinite\] Let $(\L,d)$ be a finitely aligned $k$-graph, and suppose that $\L$ satisfies condition [(C)]{}. Suppose also that every $v \in \L^0$ can be reached from a loop with an entrance. Then every nontrivial hereditary subalgebra of $C^*(\L)$ contains an infinite projection. In particular, if $\L$ is also cofinal, then $C^*(\L)$ is purely infinite.
The proof of Proposition \[prp:purely infinite\] is based heavily on the proof of [@BPRS Proposition 5.3]. First we need to recall some definitions and establish some technical results and notation. Definitions \[dfn:Ts and nus\] and \[dfn:Psub n,v\] and the proof of Lemma \[lem:norm equality\] are based almost entirely on the definitions and techniques used in [@RSY2] from [@RSY2 Notation 3.12] to the proof of [@RSY2 Proposition 3.13]. We present them seperately here because the conclusion of Lemma \[lem:norm equality\] is not stated explicitly in [@RSY2].
\[dfn:Ts and nus\] Let $(\L,d)$ be a finitely aligned $k$-graph, and let $E \subset \L$ be finite. As in [@RSY2 Notation 3.12], for all $n$ and $v$ such that $(\Pi E) v \cap \L^n$ is nonempty, we write $T^{\Pi E}(n,v)$ for the set $\{\nu \in v\L \setminus \{v\} : \lambda\nu \in \Pi E\text{ for some }\lambda \in (\Pi E) v \cap \L^n\}$. By the properties of $\Pi E$, the set $T(\lambda) := \{\nu \in s(\lambda)\L\setminus\{s(\lambda)\} : \lambda\nu \in \Pi E\}$ is equal to $T^{\Pi E}(n,v)$ for all $\lambda \in (\Pi E)v \cap \L^n$ [@RSY2 Remark 3.4]. If, in addition to $(\Pi E)v \cap \L^n \not= \emptyset$, we have $T^{\Pi E}(n,v) \not\in \FE(\L)$, we fix, once and for all, an element $\xi^{\Pi E}(n,v)$ of $v \L$ such that $\Ext(\xi^{\Pi E}(n,v); T^{\Pi E}(n,v)) = \emptyset$, and for $\lambda \in (\Pi E)v \cap \L^n$, we define $\xi_\lambda := \xi^{\Pi E}(n,v)$.
Notice that if $\lambda,\mu \in \Pi E$ satisfy $s(\lambda) = s(\mu)$ and $d(\lambda) = d(\mu)$, then we also have $T(\lambda) = T(\mu)$ and $\xi_\lambda = \xi_\mu$.
\[dfn:Psub n,v\] Let $(\L,d)$ be a finitely aligned $k$-graph, let $E \subset \L$ be finite, and let $\{t_\lambda : \lambda \in \L\}$ be a Cuntz-Krieger $\L$-family. For each $n, v$ such that $(\Pi E)v \cap \L^n$ is nonempty and $T^{\Pi E}(n,v)$ is not exhaustive, we define $$P_{n,v} := \sum_{\lambda\in (\Pi E)v \cap \L^n} s_{\lambda{\xi_\lambda}} s^*_{\lambda{\xi_\lambda}} \in C^*(\L).$$
Let $(\L,d)$ be a finitely aligned $k$-graph. We write $\Phi$ for the linear map from $C^*(\L)$ to $C^*(\L)^\gamma$ determined by $\Phi(a) := \int_\TT \gamma_z(a)\,dz$. We have that $\Phi$ is positive and is faithful on positive elements.
\[lem:norm equality\] Let $(\L,d)$ be a finitely aligned $k$-graph, let $E \subset \L$ be finite, and let $a = \sum_{\lambda,\mu \in \Pi E} a_{\lambda,\mu} s_\lambda s^*_\mu$ with $a \not= 0$. For $n \in \NN^k$ and $v \in \L^0$ such that $(\Pi E)v \cap \L^n$ is nonempty and $T^{\Pi E}(n,v)$ is not exhaustive, let $$\Ff_{\Pi E}(n,v) := \clsp\{s_{\lambda{\xi_\lambda}} s^*_{\mu{\xi_\lambda}} : \lambda,\mu \in (\Pi E) v \cap \L^n\}.$$ Then for all $n,v$ such that $(\Pi E) v \cap \L^n$ is nonempty and $T^{\Pi E}(n,v)$ is not exhaustive, we have that $P_{n,v} \Phi(a) \in \Ff_{\Pi E}(n,v)$. Furthermore, there exist $n_0, v_0$ such that $(\Pi E) v_0 \cap \L^{n_0}$ is nonempty and $T^{\Pi E}(n_0, v_0)$ is not exhaustive, and such that $\|P_{n_0, v_0} \Phi(a)\| = \|\Phi(a)\|$.
By [@RSY2 Lemma 3.15], we have that each $s_{\lambda\xi_\lambda} s^*_{\lambda\xi_\lambda} \le Q(s)^{\Pi E}_\lambda$ where $Q(s)^{\Pi E}_\lambda$ is defined by . Since the $Q(s)^{\Pi E}_\lambda$ are mutually orthogonal projections, it follows that $s_{\lambda\xi_\lambda} s^*_{\lambda\xi_\lambda} Q(s)^{\Pi E}_\mu = \delta_{\lambda,\mu} s_{\lambda\xi_\lambda} s^*_{\lambda\xi_\lambda}$. Hence, for $(\lambda,\mu) \in \Pi E \times_{d,s} \Pi E$, we have $$\label{eq:left mpctn}
P_{n,v} \Theta(s)^{\Pi E}_{\lambda,\mu}
= P_{n,v} Q(s)^{\Pi E}_\lambda s_\lambda s^*_\mu
= s_{\lambda\xi_\lambda} s^*_{\lambda\xi_\lambda} s_\lambda s^*_\mu
= s_{\lambda\xi_\lambda} s^*_{\mu\xi_\lambda},$$ and hence $P_{n,v} \Phi(a) \in \Ff_{\Pi E}(n,v)$. Moreover, taking adjoints in , shows that each $P_{n,v}$ commutes with each $\Theta(s)^{\Pi E}_{\lambda,\mu}$.
By definition of the $\Theta(s)^{\Pi E}_{\lambda,\mu}$, and by [@Si1 Corollary 4.10], we have that $\Theta(s)^{\Pi E}_{\lambda,\mu}$ is nonzero if and only if $T(\lambda)$ is not exhaustive. Moreover, since the $Q(s)^{\Pi E}_\lambda$ are mutually orthogonal and dominate the $s_{\lambda\xi_\lambda} s^*_{\lambda\xi_\lambda}$, we have that the latter are also mutually orthogonal. It follows from this and from that $$b \mapsto \sum_{\substack{(\Pi E)v \cap \L^n \not= \emptyset \\ T^{\Pi E}(n,v) \not\in \FE(\L)}} P_{n,v} b$$ is an injective homomorphism of $\clsp\{\Theta(s)^{\Pi E}_{\lambda,\mu} : \lambda,\mu \in \Pi E \times_{d,s} \Pi E\}$. Since injective $C^*$-homomorphisms are isometric, it follows that $\big\|\sum P_{n,v} \Phi(a)\big\| = \|\Phi(a)\|$.
Since the $P_{n,v}$ are mutually orthogonal and commute with $\Phi(a)$, there therefore exists a vertex $v_0$ and a degree $n_0$ such that $\|\Phi(a)\| = \|P_{n_0, v_0} \Phi(a)\|$. Clearly for this $n_0,v_0$ we must have $(\Pi E)v_0 \cap \L^{n_0}$ nonempty and $T(\lambda)$ non-exhaustive for $\lambda \in (\Pi E)v_0 \cap \L^{n_0}$, for otherwise we have $P_{n_0, v_0} = 0$ contradicting $a \not= 0$.
\[lem:infinite vert projs\] Let $(\L,d)$ be a finitely aligned $k$-graph, and suppose that every $v \in \L^0$ can be reached from a loop with an entrance. Then for each $v \in \L^0$, the projection $s_v$ is infinite, and hence for each $\lambda \in \L$, the range projection $s_\lambda s^*_\lambda$ is also infinite.
Fix $v \in \L^0$, and let $\mu$ be a loop with an entrance such that $v\L s(\mu)$ is nonempty. Fix $\lambda \in v\L s(\mu)$, and fix $\alpha \in s(\mu)\L$ such that $d(\alpha) \le d(\mu)$ and $\mu(0, d(\alpha)) \not= \alpha$. We have $s_v \ge s_\lambda s^*_\lambda \sim s^*_\lambda s_\lambda = s_{s(\mu)}$, so it suffices to show that $s_{s(\mu)}$ is infinite. But (TCK3) ensures that $s_\mu s^*_\mu s_\alpha s^*_\alpha = 0$, and it follows that $s_{s(\mu)} = s^*_\mu s_\mu \sim s_\mu s^*_\mu \le s_{s(\mu)} - s_\alpha s^*_\alpha < s_{s(\mu)}$.
For the last statement, notice that $s_{s(\lambda)}$ is infinite by the previous paragraph, and $s_\lambda s^*_\lambda \sim s^*_\lambda s_\lambda = s_{s(\lambda)}$.
\[lem:BPRS lem\] Let $E \subset \L^n$, let $w \in s(E)$, and let $t$ be a positive element of $\Ff_E(w) := \lsp\{s_\lambda s^*_\mu : \lambda,\mu \in Ew\}$. Then there is a projection $r$ in $C^*(t) \subset \Ff_E(w)$ such that $r t r = \|t\| r$.
The proof is formally identical to that of [@BPRS Lemma 5.4]
Our proof follows that of [@BPRS Proposition 5.3] very closely.
Fix a nontrivial hereditary subalgebra $A$ of $C^*(\L)$, and a positive element $a \in A$ such that $\Phi(a) \in C^*(\L)^\gamma$ satisfies $\|\Phi(a)\| = 1$. Let $b = \sum_{\lambda,\mu \in E} b_{\lambda,\mu} s_\lambda s^*_\mu$ be a finite linear combination such that $b > 0$ and $\|a - b\| \le \frac{1}{4}$; this is always possible because $\lsp\{s_\lambda s^*_\mu : \lambda,\mu \in \L\}$ is a dense $^*$-subalgebra of $C^*(\L)$. Let $b_0 := \Phi(b)$. Since $\Phi$ is norm-decreasing and linear, we have $$1 - \|b_0\| = \big|\|\Phi(a)\| - \|\Phi(b)\|\big| \le \|\Phi(a - b)\| \le \|a-b\| \le \frac{1}{4},$$ and hence $\|b_0\| \ge \frac{3}{4}$. Furthermore, $b_0 \ge 0$ because $\Phi$ is positive. Applying Lemma \[lem:norm equality\], we obtain a projection $P_{n_0, v_0}$ such that $b_1 := P_{n_0, v_0} b_0$ satisfies $b_1 \in \Ff_{\Pi E}(n_0,v_0)$ and $\|b_1\| = \|b_0\|$, where $(\Pi E)v_0 \cap \L^{n_0}$ is nonempty and $T^{\Pi E}(n_0, v_0)$ is not exhaustive. Notice that $b_1 \ge 0$. By Lemma \[lem:BPRS lem\] there exists a projection $r \in C^*(b_1)$ with $r b_1 r = \|b_1\| r$; note that $r$ is clearly nonzero. Let $v_1 := s(\xi^{\Pi E}(n_0, v_0))$, and let $S := \{\lambda{\xi_\lambda} : \lambda \in (\Pi E) v_0 \cap \L^{n_0}\}$.
Since $b_1 \in \lsp\{s_{\lambda} s^*_{\mu} : \lambda,\mu \in S\}$, which is a matrix algebra indexed by $S$, we can express $r$ as a finite sum $r = \sum_{\lambda,\mu \in S} r_{\lambda,\mu} s_\lambda s^*_\mu$, and the $S \times S$ matrix $(r_{\lambda,\mu})$ is a projection.
Since $(\L,d)$ satisfies condition (C), there exists $x \in v_1 \partial\L$ such that for $\lambda,\mu \in \L r(x)$ with $\lambda \not= \mu$, we have $\MCE(\lambda x, \mu x) = \emptyset$. By [@Si1 Lemma 6.4], for distinct $\lambda,\mu \in S$, there exists $n^x_{\lambda,\mu}$ such that $\Lmin(\lambda x(0, n^x_{\lambda,\mu}), \mu x(0, n^x_{\lambda,\mu})) = \emptyset$. Let $$\textstyle
M := \bigvee\{n^x_{\lambda,\mu} : \lambda,\mu \in S, \lambda\not=\mu\},$$ and let $x_M := x(0,M)$. Let $q := \sum_{\lambda,\mu \in S} r_{\lambda,\mu} s_{\lambda x_M} s^*_{\mu x_M}$. Since the matrix $(r_{\lambda,\mu})$ is a nonzero projection in $M_S(\CC)$, we know that $q$ is a nonzero projection in $\Ff_{N_E + d(x_M)}$, and since $s_{x_M} s^*_{x_M}$ is a subprojection of $s_{v_1}$, we have $q \le r$. Using the defining property of $x_M$ as in the proof of [@Si1 Lemma 6.7], we have that $q P_{n_0, v_0} b q = q P_{n_0, v_0} b_0 q = q b_1 q$. Now $q \le P_{n_0, v_0}$ by definition so our choice of $r$ gives $$q b q = q b_1 q = q r b_1 r q = \|b_1\|rq = \|b_0\| q \ge \frac{3}{4} q.$$ Since $\|a - b\| \le \frac{1}{4}$, we have $qaq \ge qbq - \frac{1}{4}q \ge \frac{3}{4} q - \frac{1}{4}q = \frac{1}{2} q$, and it follows that $q a q$ is invertible in $q C^*(\L) q$. Write $c$ for the inverse of $q a q$ in $q C^*(\L) q$, and let $$t := c^{1/2} q a^{1/2}.$$ Then $t^* t = a^{1/2} q c q a^{1/2} \le \|c\| a$, so $t^* t \in A$ because $A$ is hereditary.
We now need only show that $t^* t$ is an infinite projection. But $$t^* t \sim t t^* = c^{1/2} q a q c^{1/2} = 1_{q C^*(\L) q} = q,$$ so it suffices to show that $q$ is infinite. By choice of $n_0, v_0$, there exists $\sigma \in S$. By Lemma \[lem:infinite vert projs\], $s_{\sigma x_M} s^*_{\sigma x_M}$ is infinite. But $s_{\sigma x_M} s^*_{\sigma x_M}$ is a minimal projection in the finite-dimensional $C^*$-algebra $\lsp\{s_{\sigma x_M} s^*_{\tau x_M} : \sigma,\tau \in S\}$, which contains $q$. Since $q \not= 0$, $s_{\sigma x_M} s^*_{\sigma x_M}$ is equivalent to a subprojection of $q$, so $q$ is infinite.
Let $(\L,d)$ be a finitely aligned $k$-graph. Suppose that $\L$ satisfies condition [(C)]{} and is cofinal, and that every $v \in \L^0$ can be reached from a loop with an entrance. Then $C^*(\L)$ is determined up to isomorphism by its $K$-theory.
We have that $C^*(\L)$ is nuclear and satisfies UCT by Proposition \[prp:rel algs nuclear\], is simple by Proposition \[prp:simple graph alg\], and is purely infinite by Proposition \[prp:purely infinite\]. The result then follows from the Kirchberg-Phillips classification theorem [@P Theorem 4.2.4].
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[^1]: This research is part of the author’s Ph.D. thesis, supervised by Professor Iain Raeburn, and was supported by an Australian Postgraduate Award and by the Australian Research Council.
|
---
author:
- 'O. Sipilä'
- '[P. Caselli]{}'
bibliography:
- 'hydro.bib'
date: 'Received / Accepted'
title: 'Hydrodynamics with gas-grain chemistry and radiative transfer: comparing dynamical and static models'
---
Introduction
============
Models of interstellar chemistry are invoked to understand the chemical origin of line emission and absorption toward a variety of objects. Let us assume that an emission line is observed toward an object and one wants to derive the abundance distribution of the emitting molecules. The simplest approach is to assume that the molecule has a constant abundance along the line of sight, and to find the abundance that best fits the observation [e.g., @Jorgensen02; @Tafalla04a]. Alternatively, one can assume an exponentially decaying or power-law abundance profile [e.g., @Tafalla02; @Crapsi07]. A more elaborate approach is to use a chemical model to derive radially and temporally varying chemical abundance profiles, which is possible if the physical structure of the object is known [e.g. @Coutens14; @Harju17b]. In this case it is usually assumed that the physical structure of the object remains static as the chemistry evolves. Natural questions to ask are whether this assumption is valid and whether the simulated chemical abundances would turn out different if one took the dynamics into account. We explore these issues in the present paper. In what follows, we refer to models that keep the physical structure constant as the chemistry evolves simply as “static models”.
Static models have trouble reproducing observations of nitrogen-bearing species, in particular ammonia. The binding energy of ammonia onto water ice is very high (canonically $\sim$5500K; @Collings04), and so ammonia is expected to deplete strongly onto grain surfaces at high density and low temperature, corresponding to the centers of starless and prestellar cores. Indeed, several gas-grain chemical models show such strong ammonia depletion [@Semenov10; @LeGal14; @Sipila15b]. Observations on the other hand show that ammonia is not depleted toward the centers of starless and prestellar cores [e.g. @Tafalla02; @Crapsi07]. Because of the high binding energy, thermal or cosmic ray induced desorption are not strong enough to desorb ammonia from the grain surfaces. Another non-thermal alternative for desorption is related to the energy released in exothermic grain-surface reactions, the so-called chemical desorption process (@Garrod07; more recently @Minissale16a [@Vasyunin17]), but the inclusion of this process in a static model can produce too much gas-phase ammonia in lower-density gas away from the core center [@Caselli17]. The discrepancy between modeling and observations is still a mystery.
In the present work we discuss chemical abundance profiles calculated in the framework of a spherically symmetric one-dimensional hydrodynamical model and compare the results against those obtained from static models. One-dimensional hydrodynamical simulations of the collapse of starless cores in a similar context have been previously carried out by several groups, with varying degrees of complexity in terms of chemistry, for example: no chemistry [@Masunaga00]; limited gas-phase chemistry [@Keto08]; full gas-grain chemistry including deuterium chemistry [@Aikawa12]. Here we adopt an extensive chemical reaction scheme in the gas and on the grains that includes deuterium and spin-state chemistry, with an explicit treatment of the ortho and para states of ammonia and water [@Sipila15a; @Sipila15b], which is important for understanding observations (ammonia) and for an accurate treatment of line cooling (water). We combine the hydrodynamical and chemical calculations with radiative transfer methods so that the effect of chemistry on line cooling can be determined self-consistently. Our specific aim is to quantify the differences in chemical abundance profiles arising from the use of a dynamical instead of a static core model.
The paper is organized as follows. In Sect.\[s:model\] we describe the details of our hydrodynamical, chemical and radiative transfer models. In Sect.\[s:results\] we present the results of our modeling and discuss them in Sect.\[s:discussion\]. Conclusions are drawn in Sect.\[s:conclusions\]. In Appendix \[appendixa\] we present simple gas-phase and grain-surface chemical networks.
Model description {#s:model}
=================
Hydrodynamics
-------------
We developed a code that solves partial differential equations of hydrodynamics in one spatial dimension in the Lagrangian formalism, assuming spherical symmetry. The problem is described by the following equations: $$\label{eq_u}
\frac{\partial u}{\partial t} = -\frac{1}{\rho_0} \left(\frac{R}{r}\right)^2 \frac{\partial (p+q)}{\partial r} - \frac{GM}{r^2}$$ $$\label{eq_r}
\frac{\partial R}{\partial t} = u$$ $$\label{eq_rho}
\frac{1}{\rho} = \frac{1}{\rho_0} \left(\frac{R}{r}\right)^2 \frac{\partial R}{\partial r}$$ $$\label{eq_mass}
M(R) = 4\pi\int_0^{R} \rho R^{\prime 2}\,dR^{\prime}$$ $$\label{eq_E}
\frac{\partial E}{\partial t} = -(p + q) \frac{\partial}{\partial t}\left(\frac{1}{\rho}\right) + \frac{\Lambda}{\rho}$$ $$\label{eq_intene}
E = \frac{5}{2}R_{\rm sp}T$$ $$\label{eq_P}
p = \rho R_{\rm sp} T$$ $$\label{eq_q}
q =
\begin{cases}
l^2 \, \rho \, (\partial u / \partial R)^2 &\partial u / \partial R < 0 \\
0 &\partial u / \partial R \geq 0 \, .
\end{cases}$$ These equations represent: (\[eq\_u\]) the time evolution of the velocity field in the core ($G$ is the gravitational constant); (\[eq\_r\]) the location of a given grid cell as a function of time (in the equations, $r$ represents the original location of the cell while $R(t)$ represents its location in the moving frame); (\[eq\_rho\]) the scaling of the density profile as the core evolves ($\rho_0(r)$ represents the original density profile); (\[eq\_mass\]) the core mass as a function of radius; (\[eq\_E\]) the time evolution of the internal energy of the gas ($\Lambda$ parametrizes heating/cooling; see below); (\[eq\_intene\]) the internal energy, assuming diatomic gas ($R_{\rm sp} = k/m$ is the specific gas constant, where $k$ and $m$ are the Boltzmann constant and the average molecular mass of the gas); (\[eq\_P\]) the thermal pressure; (\[eq\_q\]) the pseudo-viscosity of the medium ($l = a\Delta R$, where $a$ is a dimensionless constant; we take $a = \sqrt3$). The above equations are essentially the same as those given in @Richtmyer67, although we have added the mass and cooling terms (see @Keto05 for a similar treatment of the problem). Equation (\[eq\_rho\]) can be substituted into (\[eq\_u\]) to consider the spatial derivative of $R$ instead of $r$.
The workflow of our code consists of solving Eqs.(\[eq\_u\]) to (\[eq\_q\]) in sequence. The maximum length of the time step is set by the Courant condition for this problem: $$\Delta t = C \, \frac{\Delta R}{c_s} \, ,$$ where C is a dimensionless constant and $c_s = \sqrt{kT/m}$ is the sound speed. In practice the value of C is limited to $0 < C \lesssim 1$; for larger values the solution of the hydrodynamics equations will diverge [@Richtmyer67]. In this paper we set $C = 0.6$. Another factor limiting the length of the time step is the spacing $\Delta R$ of the grid cells. The choice of the initial time step, i.e., the adopted amount of grid cells, is discussed in Sect.\[ss:iso\]. As the core starts to contract, the time step becomes shorter.
We use a simple first-order finite differencing method to solve the partial differential equations. The practical application of this method is detailed in @Richtmyer67, and is not repeated here.
The origin is not included in the grid considered in the calculations. We assume that the infall velocity is zero at the origin, which is reasonable given the symmetry of the problem and our first-order approach to solving the hydrodynamics equations. This assumption results in an infall profile that is consistent with previous models of the collapse of a spherical cloud [e.g., @Foster93; @Ogino99; @Aikawa05; @Keto05] which show that the infall velocity is close to zero near the origin before the formation of a central object. Here we do not attempt to follow the dynamical evolution all the way to the formation of a protostar; the calculational loop is designed to break when the flow becomes supersonic. If the supersonic condition is not reached until $t = 5 \times 10^6\,\rm yr$, the code terminates.
Integration of chemistry and radiative transfer {#ss:int_chem_rt}
-----------------------------------------------
In this work we incorporate a self-consistent treatment of gas-grain chemistry into the framework of hydrodynamics. Chemical abundances evolve in a time-dependent fashion in tandem with the physical evolution of the core. In particular, chemistry influences the dynamic evolution of the core through its effect on the line cooling. The chemical evolution is calculated in a subset of the full grid (25 cells), which are distributed in such a way that the grid resolution is higher near the origin than at the the outer, less dense parts of the core where chemical evolution is slow. Running the chemical calculations in a subset of the full grid is also imperative to keep the total run time of the model in a manageable scope (see Sect.\[ss:codeeff\]).
We use the chemical model described in detail in @Sipila15a and @Sipila15b. In short, the code considers gas-grain chemistry assuming that the ice on the grain surfaces consists of a single reactive layer, and that the grains themselves are spherical with a radius of 0.1$\mu$m. Our chemical model has been recently extended with the option of a multilayer approach to ice chemistry [@Sipila16b], but this issue is not considered here. Included desorption mechanisms are thermal desorption, cosmic-ray-induced desorption [@HH93], and reactive desorption for exothermic surface reactions assuming $\sim$1% efficiency [@Garrod07]. Quantum tunneling on grain surfaces is allowed for reactions with activation barriers, but diffusion is assumed to occur only thermally. Also, we do not consider grain coagulation, the inclusion of which would affect for example the charge balance in the gas phase [@Flower05]. A significant effect on the results presented here is not expected even if coagulation was included in the modeling, although a quantitative study would of course be needed to confirm this statement.
Both the gas-phase and grain-surface reaction networks used in this paper contain deuterated species with up to seven deuterium atoms, and explicit spin-state chemistry for $\rm H_2$, $\rm H_2^+$, $\rm H_3^+$, water and ammonia, and all of their deuterated isotopologs. Unlike in @Sipila15a [@Sipila15b], we use in the present paper the KIDA gas-phase network [@Wakelam15] as the base upon which the deuterium and spin-state chemistry is added with the method described in full in @Sipila15b. The grain-surface network is essentially the same as in @Sipila15b, although it has been modifed slightly so that the (surface counterparts of) species not included in the KIDA network have been removed. The full gas-phase network consists of $\sim 77000$ reactions while the grain-surface network includes $\sim 2200$ reactions. However, in the present work we only consider subsets of the full network (see Sect.\[ss:models\]). In the chemical calculations we assume that the gas is initially atomic with the exception of $\rm H_2$ and HD which are molecular. The adopted initial abundances are given in Table \[tab1\].
Species Initial abundance
---------------------------- ----------------------
$\rm H_2$ 0.5
$\rm He$ $9.00\times10^{-2}$
$\rm HD$ $1.60\times10^{-5}$
$\rm O$ $2.56\times10^{-4}$
$\rm C^+$ $1.20\times10^{-4}$
$\rm N$ $7.60\times10^{-5}$
$\rm S^+$ $8.00\times10^{-8}$
$\rm Si^+$ $8.00\times10^{-9}$
$\rm Na^+$ $2.00\times10^{-9}$
$\rm Mg^+$ $7.00\times10^{-9}$
$\rm Fe^+$ $3.00\times10^{-9}$
$\rm P^+$ $2.00\times10^{-10}$
$\rm Cl^+$ $1.00\times10^{-9}$
$\rm H_2\,(o/p)_{\rm ini}$ $1.00\times10^{-1}$
: Initial chemical abundances with respect to $n_{\rm H}$, and the adopted initial $\rm H_2$ o/p ratio.
\[tab1\]
We follow the chemical evolution as a function of time and use the abundance gradients of selected cooling species as input to a Monte Carlo non-LTE radiative transfer program (Juvela, in prep.; see also @Juvela97) to determine the total line cooling power ($\Lambda_{\rm line}$) at each time step. The included cooling species are presented in Sect.\[ss:models\]. The gas is heated mainly by cosmic rays: $$\Gamma_{\rm CR} = \Delta Q_{\rm CR} \, \zeta \, n({\rm H_2}) \, ,$$ where $\Delta Q_{\rm CR} = 20 \, \rm eV$, $\zeta = 1.3 \times 10^{-17} \, \rm s^{-1}$ is the cosmic-ray ionization rate of $\rm H_2$ molecules, and $n({\rm H_2})$ is the $\rm H_2$ density [@Goldsmith78][^1]. Gas-dust collisional coupling may also heat or cool the gas at number densities above a few $\times~10^4 \, \rm cm^{-3}$ [@Goldsmith01]: $$\begin{aligned}
\Lambda_{\rm gd} &= 2.0 \times 10^{33} \, \left( n({\rm H_2}) / {\rm cm^{-3}} \right)^2 \, \left[(T_{\rm gas} - T_{\rm dust}) / {\rm K} \right] \nonumber \\
&\times (T_{\rm gas} / 10 \, \rm K)^{0.5} \, erg \, cm^{-3} \, s^{-1} .\end{aligned}$$ In addition, we include heating by the photoelectric effect ($\Gamma_{\rm peh}$; see @Juvela03b [@Juvela11]). The photoelectric heating is calculated by another radiative transfer program [@Juvela05]. The unattenuated ISRF spectrum is taken from @Black94. We assume that the visual extinction at the edge of the model core is $A_{\rm V} = 2 \, \rm mag$. However, analogously to @Sipila17, we also added an extra layer outside the core corresponding to $A_{\rm V} = 1 \, \rm mag$ that is only used in the line cooling calculations. The abundances of the various species in this layer are assumed equal to the abundances at the core edge. If this extra layer is not included, the photon escape probability at the core edge is unphysically high because the radiative transfer program thinks there is nothing outside the model core to absorb the emitted photons, leading to low gas temperatures at the core edge. The extra layer is ignored in all other calculations besides line cooling.
Combining all of the above, the net cooling term in Eq.(\[eq\_E\]) becomes $$\label{heatcool}
\Lambda = \Gamma_{\rm CR} + \Gamma_{\rm peh} - \Lambda_{\rm gd} - \Lambda_{\rm line} \, .$$ Finally we note that the dust temperature is here calculated with the radiative transfer program of @Juvela05, using dust opacity data from @OH94 ([-@OH94]; thin ice mantles). The dust temperature is determined before the chemical calculations, i.e., between steps (\[eq\_mass\]) and (\[eq\_E\]) in the calculation workflow.
Chemical reaction sets {#ss:models}
----------------------
Network Description Cooling species included
--------- --------------------------------------------------------------------------------- --------------------------------------------------------------------------------------
A1 Full gas-phase and grain-surface reaction networks described $\rm ^{12}CO$, $\rm ^{13}CO$, $\rm C$, $\rm C^+$, $\rm O$, $\rm oH_2O$, p$\rm H_2O$,
in Sect.\[ss:int\_chem\_rt\], but considering only species with up to five $\rm NO$, $\rm HCN$, $\rm HNC$
atoms, $\sim$43500 (42000+1510) reactions in total
A1alt As A1, but with some coolants removed $\rm ^{12}CO$, $\rm ^{13}CO$, $\rm C$, $\rm C^+$, $\rm O$, $\rm oH_2O$, p$\rm H_2O$
B1 Simple chemical network based on the one used by $\rm ^{12}CO$, $\rm ^{13}CO$, $\rm C$, $\rm C^+$, $\rm O$, $\rm H_2O^{(a)}$
@Nelson99 ([-@Nelson99]; see Appendix \[appendixa\]), no
surface chemistry except for the formation of $\rm H_2$
B2 As B1, but with more surface chemistry (see Appendix \[appendixa\]) $\rm ^{12}CO$, $\rm ^{13}CO$, $\rm C$, $\rm C^+$, $\rm O$, $\rm H_2O^{(a)}$
B2alt As B2, but including photodesorption$^{(b)}$ of grain-surface CO and $\rm H_2O$ $\rm ^{12}CO$, $\rm ^{13}CO$, $\rm C$, $\rm C^+$, $\rm O$, $\rm H_2O^{(a)}$
\[tab2\]
In this paper we present the results of models that incorporate different chemical reaction sets and varying sets of cooling molecules. The models are tabulated in Table \[tab2\] along with a description of the network used in each case. The effect of the choice of cooling species is discussed in Sect.\[ss:dynamics\].
The main aim of the present paper is to study whether the chemical abundance gradients evolving dynamically with the core are significantly different from those deduced from static core models. To this end, we take three snapshots of the evolution of a collapsing core (see Sect.\[ss:collapsemodel\]), and run a pseudo-time-dependent chemical model using the density and temperature structures corresponding to these snapshots, starting from the same initial chemical abundances as the dynamical model (Table \[tab1\]). In the static model, the physical quantities of the model do not evolve as the chemistry runs from $t = 0 \, \rm yr$ up to the snapshot time. The results of this analysis are presented in Sect.\[ss:staticmodels\].
Efficiency of the code {#ss:codeeff}
----------------------
The majority of the total running time of the code is spent on the chemical calculations. It takes $\sim$4 minutes to calculate the chemical evolution in one grid cell when using the gas-phase and grain-surface network A1, depending also on the physical parameters (density, temperature etc.). The code is parallelized so that the chemical evolution can be computed simultaneously in multiple grid cells. The radiative transfer calculations, also parallelized, are completed in a timescale of two minutes per time step, so it is the chemistry that takes up the majority of the total computational time.
It takes about 14 hours of cpu time to calculate $1 \times 10^5 \, \rm yr$ of dynamical evolution using a standard desktop computer with four processor cores, using the A1 reaction set with (initial) time resolution of $\sim 10^3$yr (see Sect.\[ss:iso\]). Therefore, models such as those presented in this paper can be run with a personal computer in reasonable timescale of a few days per model.
For this paper we did not perform calculations using the entire chemical reaction sets described in Sect.\[ss:int\_chem\_rt\]. The advantage of using the full reaction scheme is that we could model the abundances of more complex species such as methanol, but at a great computational cost. Furthermore, the reaction set A1 used here captures all of the essential chemistry related to the cooling species (mainly CO, C, and water) and relatively simple species such as ammonia, and so our results, presented below, would remain unaffected even if we did switch to the full networks.
Results {#s:results}
=======
![The density ([*left panel*]{}) and velocity profiles ([*right panel*]{}) of a stable BES at different times, indicated in the figure. The green line shows the initial density profile. The velocity is initially zero across the core. []{data-label="fig:iso_stable"}](iso_stable.eps){width="1.0\columnwidth"}
![The density ([*left panel*]{}) and velocity profiles ([*right panel*]{}) of an unstable BES at different times, indicated in the figure. The green line shows the initial density profile. The velocity is initially zero across the core. []{data-label="fig:iso_unstable"}](iso_unstable.eps){width="1.0\columnwidth"}
Verification of the code: isothermal core {#ss:iso}
-----------------------------------------
A basic requirement of the hydrodynamics code is that it should produce expected behavior for (isothermal) Bonnor-Ebert spheres: unstable cores should ultimately collapse while stable ones should not. To verify that the code works as intended, we carried out test calculations using two Bonnor-Ebert spheres with central density $n(\rm H_2) = 2\times10^4\,cm^{-3}$ and temperature $T = 10\,\rm K$. We picked two different values of the non-dimensional radius $\xi$ which represents the stability of the core: 4 (stable configuration) and 15 (unstable configuration); for more details on the stability of the Bonnor-Ebert sphere, see @Bonnor56. For the stable core, the $\rm H_2$ density at the edge is $n^{\rm edge}_{\rm H_2} \sim 4.2 \times 10^3 \, \rm cm^{-3}$, outer radius is $R_{\rm out} \sim19900 \, \rm AU$, and the mass is $M \sim 1.6 \, M_{\odot}$. For the unstable core, the corresponding parameters are $n^{\rm edge}_{\rm H_2} \sim 1.8 \times 10^2 \, \rm cm^{-3}$, $R_{\rm out} \sim74300 \, \rm AU$, and $M \sim 7.15 \, M_{\odot}$.
In Fig.\[fig:iso\_stable\] we show the evolution of the density and velocity profiles of the stable BES at different times during the dynamical evolution. As expected, the core does not collapse; instead, it oscillates. The velocity is initially zero across the core. At the very start of the dynamical evolution, a shallow infall profile (maximum infall speed of a few $\times$ $10\,\rm cm\,s^{-1}$) develops because of the gravitational potential (the core is not homogeneous), but this turns to an expansion in a timescale of $\sim 8 \times 10^5$yr. Later, periods of contraction and expansion alternate. The infall/expansion velocity stays below $\lesssim 1$ ms$^{-1}$ at all times. The plot also shows that the velocity profile fluctuates particularly at the inner boundary, but these fluctuations do not drive the evolution of the large-scale behavior. In this case the code runs until the termination time, $t = 5 \times 10^6\,\rm yr$, is reached.
Figure \[fig:iso\_unstable\] shows three snapshots of the evolution of the unstable BES. Because the density contrast of this core is clearly higher than that of the stable BES (note the difference in density scale between Figs. \[fig:iso\_stable\] and \[fig:iso\_unstable\]), a steeper infall profile develops early on and the core begins to collapse gradually. In this case the collapse is never halted because the core is gravitationally supercritical. The last time step shown in the figure corresponds to a time very close to when the code breaks the calculation loop (the flow becomes supersonic). Similar velocity fluctuations as those evident in Fig.\[fig:iso\_stable\] are present here as well, but they cannot be seen in the velocity scale of this figure. Overall, from the tests discussed above we conclude that the hydrodynamics part of the code works as intended.
The tests were performed adopting 1000 grid points for the spatial resolution, corresponding to an initial time step of $t \sim 1.1 \times 10^3$yr in the unstable core model. The choice of the time resolution affects the dynamics. If the initial time step is long, the core develops a steep infall profile quickly and collapses rapidly. The better the time resolution (i.e., the more grid points in the model), the more accurate the representation of the dynamics. However, using a small time step ($\gg$1000 grid points) increases the total calculational time tremendously when the chemistry is included. We studied the effect of the time resolution on the dynamical timescale by calculating the unstable BES model with different amounts of grid points. Table \[tab3\] summarizes the results of the test. The total running time of the model, defined by the time when the infalling flow becomes supersonic, increases with the number of grid points. The test results are close to each other (within a few tens of percent for the collapse timescale) when the number of grid points is of the order of 1000. Therefore we employ the value of 1000 also in the models presented below, as a compromise between accuracy of the dynamics and the required computational time. We stress that the collapse timescale itself is not the issue studied in this paper; we concentrate on the differences in chemical abundances between static and dynamical models, and this comparison is unaffected by how long it takes for the core to collapse.
Grid points $t_{\rm ini}$ $t_{\rm f}$
------------- ---------------------------- ----------------------------
100 $1.01\times10^4 \, \rm yr$ $1.25\times10^6 \, \rm yr$
500 $2.19\times10^3 \, \rm yr$ $2.19\times10^6 \, \rm yr$
1000 $1.01\times10^3 \, \rm yr$ $2.61\times10^6 \, \rm yr$
3000 $3.02\times10^2 \, \rm yr$ $3.27\times10^6 \, \rm yr$
5000 $2.01\times10^2 \, \rm yr$ $3.54\times10^6 \, \rm yr$
: Summary of test runs exploring the effect of the spatial resolution on the collapse timescale.
\[tab3\]
{width="2.0\columnwidth"}
{width="1.8\columnwidth"}
Collapse of a nonisothermal core {#ss:collapsemodel}
--------------------------------
We present now the results of calculations from a model including chemistry and radiative transfer, i.e., in this case the core is no longer isothermal. We use the A1 model introduced in Sect.\[ss:models\]. The initial core is the same as the unstable ($\xi = 15$) BES described in Sect.\[ss:iso\].
Figure \[fig:noniso\] shows three snapshots of the evolution in the innermost 50000AU of the model core, and the breakdowns of the net cooling power and line cooling power for one of these snapshot times. As was the case with the unstable BES discussed in Sect.\[ss:iso\], a minor infall motion was introduced already at the beginning of the simulation and the core started to collapse gradually. The infall flow became supersonic at $t = 7.19 \times 10^5\,\rm yr$, stopping the calculation.
The dust temperature is strongly tied to the density profile of the core. As the core becomes more centrally concentrated, the dust temperature drops at the center. The gas temperature is affected by several processes depending on the distance from the center of the core. Figure \[fig:cooling\] demonstrates the situation at $t = 5 \times 10^5 \, \rm yr$ so that one obtains an overall idea of the relative strengths of the heating and cooling processes. Looking first at the innermost 10000AU, it can be seen that the heating of the gas is dominated by cosmic rays and compression caused by the infall motion, while line radiation is mainly responsible for the cooling (the gas-dust coupling is also important near the origin). Outwards from 10000AU, the energy input from cosmic rays and compressive heating decrease more rapidly than the total line cooling power and, as a result, the gas is cooler than near the core center. Towards the core edge the gas temperature begins to rise again as photoelectric heating becomes important. We note that the gas and dust temperatures are not yet equal even at the core center when the code terminates. In the @Goldsmith01 gas-dust coupling scheme, the two temperatures become coupled at a medium density of $\sim$$10^5 \, \rm cm^{-3}$ and finally equal to each other at a density of $\sim$$10^6 \, \rm cm^{-3}$ [@Goldsmith01; @Keto05]. The central density in the A1 model at the last time step ($\sim$$5\times10^5 \, \rm cm^{-3}$) is below the equalization threshold.
From the breakdown of the line cooling power at this time step ($t = 5 \times 10^5 \, \rm yr$, corresponding to the dashed lines in Fig.\[fig:noniso\]) one can see that the contributions of the cooling molecules to the total line cooling power are highly dependent on the distance from the core center. Cooling near the center of the core is dominated by HCN, CO, and NO, while atomic carbon is the most important coolant beyond $R \sim 40000$AU. ($\rm C^+$ is the most important coolant near the edge of the core, not shown.) Notably, NO and HCN are powerful coolants near the core center even though they are less abundant than CO by several orders of magnitude (Fig.\[fig:noniso\]). For $t \lesssim 2 \times 10^5\, \rm yr$, before CO and particularly the late-type molecule NO have formed efficiently, the most important coolant in the central areas is HCN. Our results highlight the need for including a variety of cooling molecules when one employs a complex chemical model. In fact, it turns out that truncating the list of coolant species may increase the collapse timescale or prevent the collapse altogether. The effect of the line cooling scheme in our models is discussed further in Sect.\[ss:dynamics\]. We note that the total line cooling power is lower than that of HCN alone at $R \sim 20000-25000$AU in Fig.\[fig:cooling\]. This is because atomic carbon is providing heating (i.e., a negative contribution to cooling) in the region $R \sim 15000-30000$AU, caused by optical thickness effects: the line radiation is unable to escape the cloud, unlike in the outer regions where the medium density is low.
Chemical abundances in the collapsing core
------------------------------------------
{width="2.0\columnwidth"}
![Visual extinction $A_{\rm V}$ of the collapsing model core as a function of radius at different times, indicated in the figure. []{data-label="fig:Av"}](Av.eps){width="0.9\columnwidth"}
Figure \[fig:abus\] shows the abundances of several species of interest in the collapsing core at different times. A significant amount of ammonia is present in the gas from the center of the core to above 20000AU. The same is true for $\rm N_2H^+$ and $\rm H_3^+$. The abundances of the deuterated isotopologs of these species show a marked increase in abundance toward the center of the core. Just before the termination of the calculation, all of the aforementioned species are depleted near the center of the core due to the high medium density, either because of direct adsorption onto dust grains (neutrals) or secondary effects (ions).
Notably, the $\rm H_2$ ortho/para (hereafter o/p) ratio is $\sim$0.1 through most of the core, dropping to lower values only in the innermost 10000 or 20000AU depending on the time. The reason for this is the low visual extinction due to the low medium density. The visual extinction $A_{\rm V}$ is shown in Fig.\[fig:Av\]. As the material flows inward, the density profile of the core becomes more centrally concentrated, and low values of $A_{\rm V}$ are found at increasingly smaller radii. At low $A_{\rm V}$, $\rm H_2$ is efficiently photodissociated and then reformed in the thermal o/p ratio of 3, while proton-exchange reactions of $\rm H_2$ with $\rm H^+$ and $\rm H_3^+$ drive the $\rm H_2$ o/p ratio towards the LTE value which is of the order of $10^{-3}$ at 20K. These competing processes lead to an $\rm H_2$ o/p ratio of $\sim 0.1$. Because of the close chemical relationship between the o/p ratio of $\rm H_2$ and that of $\rm H_2D^+$ [@Brunken14], the $\rm H_2D^+$ o/p ratio stays near its thermal ratio of 3 as long as the $\rm H_2$ o/p ratio is high, dropping to lower values near the core center as the $\rm H_2$ o/p ratio decreases. We note that our code does not include $\rm H_2$ self-shielding. If this effect was included in the modeling, the $\rm H_2$ o/p ratio would decrease in the outer core along with the decreased $\rm H_2$ photodissociation rate. We tested the influence of $\rm H_2$ photodissociation on our results by running the A1 model with $\rm H_2$ photodissociation completely removed. The removal does not infuence the dynamics in any appreciable way. However, in this case the $\rm H_2$ o/p ratio is low ($\lesssim 10^{-3}$) throughout the core which can boost the deuteration fractions by more than two orders of magnitude in the outer core, depending on the species. This underlines the requirement of an accurate treatment of photodissociation when comparing models with observed abundance profiles (which is beyond the scope of this paper).
The o/p ratios of $\rm NH_3$ and $\rm NH_2D$ are determined by many factors [@Sipila15b]. In the outer core, the o/p ratios of both $\rm NH_3$ and $\rm NH_2D$ are thermal at all times. In the core center, the model predicts similar deviations from the thermal value as was found by @Sipila15b using static models. We compare predictions from our dynamical model and those of static models in detail in the next section. Abundance profiles for a selection of species other than those plotted above are shown and discussed in Sect.\[ss:abus\].
Static models {#ss:staticmodels}
-------------
{width="2.0\columnwidth"}
{width="2.0\columnwidth"}
{width="2.0\columnwidth"}
Figures \[fig:static1\] to \[fig:static3\] show the abundances obtained from static models, calculated as described in Sect.\[ss:models\], compared to the abundances obtained from the dynamical model at different times. The results from the two types of model can be clearly different from each other, and the relative difference increases with time. At $t = 3.00 \times 10^5 \, \rm yr$ the results of the two models are nearly identical. However, at $t = 5.00 \times 10^5 \, \rm yr$ clear differences are already evident. The static model presents generally higher abundances and more extended profiles for most of the plotted species. We note that the high peak ammonia abundance (around $10^{-6}$) results from the adoption of the reactive desorption process, which leads to the desorption of the products of exothermic association reactions on grain surfaces with 1% efficiency. If this desorption mechanism was turned off, the ammonia abundance would peak at lower values, but would still likely peak in the same location.
At $t = 7.19 \times 10^5 \, \rm yr$, the static model overproduces the abundances of many of the species with respect to the dynamical model outside the core center, and underproduces them near the origin. For example, the abundance of singly deuterated ammonia is an order of magnitude higher in the static model at $R \sim 10000$AU than in the dynamical model. The reason for the differences evident in the models is in the density and temperature profiles. In the static model, the core spends the entirety of its lifetime in a strongly centrally concentrated configuration which facilitates fast depletion onto grain surfaces in the central regions. On the other hand, the temperature near the center is higher than 10K throughout the chemical evolution of the core, as opposed to the dynamical model core where the temperature in the central areas is <10K for times up to $\lesssim 5 \times 10^5 \, \rm yr$. These effects are the cause of the variations evident in the abundance profiles in the two types of model.
The spin-state abundance ratios are also different in the two types of model at late times. The dynamical model does not yield an $\rm H_2D^+$ o/p ratio below $\sim$unity at any of the time steps displayed, whereas in the static model the $\rm H_2$ o/p ratio drops to such low values in the center that the $\rm H_2D^+$ o/p ratio decreases to $\sim$0.1. The discrepancy between the two models at $t = 7.19 \times 10^5 \, \rm yr$ is significant. The $\rm NH_3$ and $\rm NH_2D$ o/p ratios are also underestimated by the static model at late times, as compared to the dynamical model. These trends are opposite to what we found for the total abundances which are generally overestimated by the static model. The results from the dynamical model are especially interesting given that static models have trouble reproducing observed o/p ratios in (deuterated) ammonia [@Harju17a], although we point out that we do not attempt to reproduce any particular observations in the present work. The differences in the $\rm H_2$ o/p ratio, and in the density and temperature profiles, lead to clear differences in the efficiency of deuteration in the two models. This point is further emphasized in Sect.\[ss:abus\].
In conclusion, our results show that the dynamical changes in temperature and density during the evolution of a molecular core are clearly reflected on the chemical abundance profiles, an effect which is completely missed by employing static models, and imply that the use of a static physical model is likely to lead to errors in the derivation of molecular abundances, for example. We discuss the observational impact of our modeling results in Sect.\[ss:obs\].
Discussion {#s:discussion}
==========
Impact of chemistry on the dynamics {#ss:dynamics}
-----------------------------------
![As the top row and lower left panel of Fig.\[fig:noniso\], but calculated using reaction set B1. []{data-label="fig:noniso_simple"}](noniso_simple.eps){width="1.0\columnwidth"}
Calculating chemical development with an extensive chemical network is computationally very expensive, and so it is desirable to use a limited chemical network in 1D/3D hydrodynamical models that include a large amount of grid cells to keep the total computational time in a manageable scope. It is however not evident if and how the dynamics is altered if one switches from a simple reaction network, describing for example only $\rm H_2$ and CO formation, to a full gas-grain network. We quantify this issue next. We note that similar studies already exist in the literature: for example @Hocuk14 and @Hocuk16 have recently studied the effect of chemistry on the results of hydrodynamical models. However, even the complete network used in the latter work is very limited compared to that adopted in the present paper, justifying additional investigation into this issue.
We ran the collapsing core model (Sect.\[ss:collapsemodel\]) using the simplified chemical networks B1, B2, and B2alt described in Sect.\[ss:models\] and Appendix \[appendixa\]. The results of the calculation using the B1 reaction set are presented in Fig.\[fig:noniso\_simple\]. The collapse timescale of the core – as determined by the termination condition of the code – using reaction set B1 is $\sim$2.1 times that of the A1 model (Fig.\[fig:noniso\]). The main reason for this difference is in the evolution of the C and CO abundances. In model A1, CO forms (at early times) efficiently through neutral-neutral reactions such as $\rm O + C_2 \longrightarrow C + CO$ and $\rm O + CH_2 \longrightarrow CO + H + H$. These reactions are not included in model B1, where CO forms through $\rm HCO^+ + e^-$, and the CO abundance only becomes similar to that of C at very late times near the center of the core, and stays at very low abundances in the outer parts of the core throughout the time evolution. The cooling power of C is low near the core center ($\sim 2 \times 10^{-24}\,\rm erg \, cm^{-3} \, s^{-1}$) even though it is abundant, and so the slow increase in CO abundance means that it takes a significant amount of time for the total line cooling power to rise to high levels and for the collapse to proceed efficiently. The gas temperature profile behaves similarly to that in the A1 model (Fig.\[fig:noniso\]), although the gas is overall warmer by about one K in the B1 model as long as the gas-dust coupling is not significant.
In model B2 where surface reactions are included, oxygen is locked in grain-surface water in a relatively short timescale, resulting in low gas-phase abundances for CO and water. This translates to low line cooling rates and the core oscillates instead of collapsing. In the B2alt model where surface CO and water are also capable of photodesorbing, the gas cooling rates are higher and the core ultimately collapses. However, in this case the code terminates at $2.68 \times 10^6$yr, i.e., it takes a factor of $\sim1.8$ longer for the core to collapse than when using the B1 model scheme – a factor of 3.8 higher than the collapse timescale of the A1 model.
We also ran the A1 model considering only the smaller set of coolants as in models B1 and B2 (marked as A1alt in Table \[tab2\]). In this case, the termination of the code is retarded to $2.93 \times 10^6$yr. A further test shows that if water is also removed from the set of coolants, the core never collapses. These results highlight the need to have a comprehensive description of cooling in order to obtain the most accurate estimate for the collapse timescale.
From the above it is clear that the dynamics of the core evolution can be strongly affected by the adopted chemical reaction network and by the set of cooling molecules used in the modeling. While disregarding depletion onto dust grains has an obvious impact on the modeling, it is also evident that limiting the gas-phase reaction network affects the results because of the differences in how the molecules are processed. A systematic study of the effect of considering chemical networks of varying complexity on the dynamics of core evolution, with the particular aim of obtaining information on the most essential processes, would be a worthwhile effort.
Observational difference in dynamical and static models {#ss:obs}
-------------------------------------------------------
{width="2.0\columnwidth"}
Figures \[fig:static1\] to \[fig:static3\] exemplify that the abundance profiles obtained from dynamical models can differ clearly from those obtained from static models. The differences in the central areas are especially relevant for species with high critical densities, such as ammonia. In the previous figures we plotted the abundances of some common tracer species expected a priori to be important near the core center. To quantify the differences in line profiles resulting from different radial abundance profiles depending on the physical model, we calculated simulated line emission profiles for ammonia and $\rm N_2H^+$ using the radiative transfer program of @Juvela97. We modeled the (1,1) inversion transition of $\rm pNH_3$ at 23.69GHz, the ($1-0$) rotational transition of $\rm oNH_3$ at 572.50GHz, and the ($1-0$) rotational transition of $\rm N_2H^+$ at 93.17GHz. For simplicity, all of the simulated spectra were convolved to a 10$\arcsec$ beam. We assumed that the distance to the model core is 100pc.
Figure \[fig:lines\] shows the simulated line profiles at $t = 7.19 \times 10^5 \, \rm yr$. The difference between the static and dynamical models is striking. Although the static model predicts a higher abundance for all of the plotted species off the center of the core (Fig.\[fig:static3\]), the emission is stronger in the dynamical model. This is because of the critical densities of the lines, which are $\sim$$2.0 \times 10^3 \, \rm cm^{-3}$, $\sim$$3.7 \times 10^7 \, \rm cm^{-3}$, and $\sim$$2.3 \times 10^5 \, \rm cm^{-3}$ (at 10K) for the modeled $\rm pNH_3$, $\rm oNH_3$, and $\rm N_2H^+$ transitions, respectively. For example, a density of $\sim 2.3 \times 10^5 \, \rm cm^{-3}$ corresponds to a radius of just a few thousand AU at this time step (Fig.\[fig:noniso\]), meaning that the large part of the core where the $\rm N_2H^+$ abundance is higher in the static model is actually not emitting, leading to stronger emission in the dynamical model because of the higher abundance near the very center of the core. We note that the line simulations pertaining to static physical models adopt the infall velocity profile from the dynamical model at the appropriate time step, resulting in infall asymmetry in the static case as well.
These tests show that there can be a significant difference in simulated lines depending on whether one uses a dynamical or static physical model. Given the tendency of the static model to underpredict the various abundances near the core center and to overpredict them away from the center, the emission lines from the various high-density-tracing species will tend to be weak in the static model, while simultaneously the lower-density-tracing lines are likely to be too strong (optical depth effects will of course influence the situation as well). The requirement for an accurate model of the (thermal) history of a core is apparent.
Finally we comment on the recent paper by @Caselli17, who found that the abundance of $\rm oNH_3$ and the associated emission in the (1,1) line are not well fitted by a chemical model coupled with a static physical model of the structure of L1544 [@Keto10; @Keto14]. In particular, the $\rm oNH_3$ abundance profile predicted by the model is at odds with the abundance profile deduced by @Crapsi07; the chemical model predicts strong ammonia depletion at the core center, and on the other hand too much ammonia toward the outer core. The results of the present paper imply that the situation could be remedied at least to some extent by employing the dynamical model, which predicts somewhat less depletion and also less ammonia in lower density gas. Quantitative conclusions on how the simulated lines would appear with the present model in the case of L1544 cannot be drawn without further modeling. We leave a detailed investigation of specific cores, such as L1544, for future work.
Further abundance profiles in dynamical and static models {#ss:abus}
---------------------------------------------------------
{width="2.0\columnwidth"}
In the preceding sections we showed the abundance profiles of selected species as given by the dynamical and static models at different time steps. For completeness, we show in Fig.\[fig:static\_more\] the abundance profiles of a variety of other species at $t = 7.19 \times 10^5 \, \rm yr$. The figure shows similar tendencies to Figs.\[fig:static1\] to \[fig:static3\]; the static model underpredicts abundances near the core center and overpredicts them in the outer core. This rule is not absolute, however, as evidenced by $\rm HCO^+$ which is more abundant in the central core in the static model than in the dynamical model, caused by a higher central gas-phase CO abundance in the static model (itself caused by the temperature which stays above 10K in the central areas throughout the chemical evolution in this particular static model, cf. the discussion in Sect.\[ss:staticmodels\]).
Fig.\[fig:static\_more\] further emphasizes the fact that deuteration is overestimated in the central regions of the core by employing a static physical model. The overestimate increases with the number of deuterium atoms, suggesting potentially major implications for the interpretation of observations of multiply-deuterated species. Of course, the strength of the effect depends on the excitation conditions as well. This issue will be explored in detail in a future work.
Conclusions {#s:conclusions}
===========
We presented a new one-dimensional spherically symmetric hydrodynamics code that integrates the solution of the basic hydrodynamics equations with radiative transfer calculations and a gas-grain chemical code that includes deuterium and spin-state chemistry. Time-dependent abundances from the chemical model are used as input to the radiative transfer calculations so that the cooling rates can be determined self-consistently. Our main aim in the present paper was to quantify whether the molecular abundances predicted by a model including dynamical evolution differ from those obtained with static physical models. The full gas-phase and grain-surface reaction sets considered here contain a combined total of $\sim$43500 reactions. The use of an extensive description of chemistry allows investigation of the effect of chemistry on the overall dynamics, when our fiducial model results are compared against those obtained with alternative chemical reaction sets with decreased degrees of complexity. We followed the dynamical evolution up to the point when the infall velocity becomes supersonic. The formation of a central source is not presently included in the code.
We found that the difference in predicted abundances between the dynamical and static models is generally significant, but this result is time-dependent. At early times, well before the core starts to collapse, the results from the two models are nearly identical. As the infall velocity increases and the core contracts, the modeling results begin to diverge. At late times the static models generally overpredict abundances in the outer parts of the core, and underpredict them near the core center, with respect to the dynamical model. The differences in the abundances can translate to striking differences in simulated emission lines. For the ammonia and $\rm N_2H^+$ lines simulated here, the dynamical model predicts clearly stronger emission than the static model because the lines originate near the center of the core where the dynamical model predicts higher abundances. Low-density-tracing lines are accordingly expected to appear clearly brighter in the static model.
Our results show that using static physical models on the one hand to simulate abundance gradients and on the other hand to reproduce observed line emission profiles may lead to large errors if the core has already started contracting. Also, an extensive chemical network is needed for the most accurate representation of the dynamics because of the effect of the chemistry on the total line cooling power; the collapse timescale depends on the number of grid points in the model. The downside of the present approach where the chemistry and line cooling are calculated self-consistently is that the total calculation time becomes very long if the time step is small. However, we found that the core collapse timescale reaches a plateau with increasing number of grid points, so that a fairly good approximation of the dynamics can be reached with a limited number of grid points, keeping the total calculation time in a manageable scope.
We also studied the effect of the complexity of the adopted chemical networks on the overall dynamical evolution. We found that using a very simple gas-phase chemical network limited to essential CO and water chemistry increases the collapse timescale of our model core by a factor of $\sim$2.1 with respect to the model with complete gas-grain chemistry. On other hand, adding a very simple set of grain-surface reactions to the gas-phase network leads to a solution where the core does not collapse, caused by the trapping of oxygen into grain-surface water, unless photodesorption of water (and CO) is included. In the latter case the collapse timescale is $\sim$3.7 times that of the model with full chemistry, indicating that the use of very simple chemical networks in hydrodynamical simulations may lead to overestimation of the collapse timescale.
Finally, we found that in a collapse model including complex chemistry, it is vital to consider a variety of cooling species. In the central regions, most of the molecular line cooling is in our model due to HCN, CO, and NO. If only a limited set of coolants is considered, the core collapse timescale increases or the collapse is prevented altogether because of insufficient cooling in the central regions of the core.
Our hydrodynamical scheme is based on thermal processes and does not at present include additional sources of pressure such as the magnetic field or turbulence. With the addition of these processes, left for future work, the versatility of our model would be improved as we would obtain a more realistic description of the physics of the collapse. This will be beneficial when real cores are simulated. However, the main conclusions of the present paper on the discrepancy between the dynamical and static models would not be affected. The code can also be extended to model the collapse up to the formation of a central source. This would yield interesting information on chemical abundances in the innermost areas of collapsing cores, on the sub-100AU scale.
We thank the anonymous referee for comments that helped to improve the paper. We also thank Mika Juvela for valuable discussions on the implementation of radiative transfer in the modeling. P.C. acknowledges financial support of the European Research Council (ERC; project PALs 320620).
Chemical networks B1 and B2 {#appendixa}
===========================
In this Appendix, we present the gas-phase and grain-surface networks that are used in the B1, B2, and B2alt models summarized in Table \[tab2\]. The gas-phase network given in Table \[tabb1\] is used in all of the B models. The difference between the B1 and B2 cases is in the surface chemistry; in model B1 only surface formation of $\rm H_2$ is considered, while models B2 and B2alt include all of the surface reactions given in Table \[tabb2\].
[ccccccccccc]{}
\
& & Chemical reaction & & & & $\alpha$ & $\beta$ & $\gamma$ & Type\
\
& & Chemical reaction & & & & $\alpha$ & $\beta$ & $\gamma$ & Type\
$\rm H $ & $\rm CRP $ & $\longrightarrow$ & $\rm H^+ $ & $\rm e^- $ & & 4.60e-01 & 0.00e+00 & 0.00e+00 & $1$\
$\rm He $ & $\rm CRP $ & $\longrightarrow$ & $\rm He^+ $ & $\rm e^- $ & & 5.00e-01 & 0.00e+00 & 0.00e+00 & $1$\
$\rm H_2 $ & $\rm CRP $ & $\longrightarrow$ & $\rm H $ & $\rm H $ & & 1.00e-01 & 0.00e+00 & 0.00e+00 & $1$\
$\rm H_2 $ & $\rm CRP $ & $\longrightarrow$ & $\rm H $ & $\rm H^+ $ & $\rm e^- $ & 2.20e-02 & 0.00e+00 & 0.00e+00 & $1$\
$\rm H_2 $ & $\rm CRP $ & $\longrightarrow$ & $\rm H_2^+ $ & $\rm e^- $ & & 9.30e-01 & 0.00e+00 & 0.00e+00 & $1$\
$\rm C $ & $\rm CRP $ & $\longrightarrow$ & $\rm C^+ $ & $\rm e^- $ & & 1.02e+03 & 0.00e+00 & 0.00e+00 & $1$\
$\rm OH $ & $\rm CRP $ & $\longrightarrow$ & $\rm H $ & $\rm O $ & & 5.10e+02 & 0.00e+00 & 0.00e+00 & $1$\
$\rm H_2O $ & $\rm CRP $ & $\longrightarrow$ & $\rm H $ & $\rm OH $ & & 9.70e+02 & 0.00e+00 & 0.00e+00 & $1$\
$\rm CO $ & $\rm CRP $ & $\longrightarrow$ & $\rm C $ & $\rm O $ & & 5.00e+00 & 0.00e+00 & 0.00e+00 & $1$\
$\rm CO $ & $\rm CRP $ & $\longrightarrow$ & $\rm CO^+ $ & $\rm e^- $ & & 3.00e+00 & 0.00e+00 & 0.00e+00 & $1$\
$\rm H_2 $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm H $ & $\rm H $ & & 3.40e-11 & 0.00e+00 & 2.50e+00 & $2$\
$\rm C $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm C^+ $ & $\rm e^- $ & & 3.10e-10 & 0.00e+00 & 3.33e+00 & $2$\
$\rm OH $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm H $ & $\rm O $ & & 3.90e-10 & 0.00e+00 & 2.24e+00 & $2$\
$\rm CO $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm C $ & $\rm O $ & & 2.60e-10 & 0.00e+00 & 3.53e+00 & $2$\
$\rm HCO^+ $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm H^+ $ & $\rm CO $ & & 1.50e-10 & 0.00e+00 & 2.50e+00 & $2$\
$\rm Fe $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm Fe^+ $ & $\rm e^- $ & & 2.80e-10 & 0.00e+00 & 2.20e+00 & $2$\
$\rm H_2O $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm H $ & $\rm OH $ & & 8.00e-10 & 0.00e+00 & 2.20e+00 & $2$\
$\rm H_2O $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm H_2O^+$ & $\rm e^- $ & & 3.10e-11 & 0.00e+00 & 3.90e+00 & $2$\
$\rm H^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm PHOTON$ & & 3.50e-12 & -7.00e-01 & 0.00e+00 & $3$\
$\rm H_2^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm H $ & & 2.53e-07 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_3^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm H $ & $\rm H $ & 4.36e-08 & -5.20e-01 & 0.00e+00 & $3$\
$\rm H_3^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm H_2 $ & & 2.34e-08 & -5.20e-01 & 0.00e+00 & $3$\
$\rm He^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm He $ & $\rm PHOTON$ & & 4.50e-12 & -6.70e-01 & 0.00e+00 & $3$\
$\rm HCO^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm CO $ & & 2.80e-07 & -6.90e-01 & 0.00e+00 & $3$\
$\rm C^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm C $ & $\rm PHOTON$ & & 4.40e-12 & -6.10e-01 & 0.00e+00 & $3$\
$\rm Fe^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm Fe $ & $\rm PHOTON$ & & 3.70e-12 & -6.50e-01 & 0.00e+00 & $3$\
$\rm CH^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm C $ & $\rm H $ & & 7.00e-08 & -5.00e-01 & 0.00e+00 & $3$\
$\rm OH^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm O $ & & 6.30e-09 & -4.80e-01 & 0.00e+00 & $3$\
$\rm CO^+ $ & $\rm e^- $ & $\longrightarrow$ & $\rm C $ & $\rm O $ & & 2.75e-07 & -5.50e-01 & 0.00e+00 & $3$\
$\rm H_2O^+$ & $\rm e^- $ & $\longrightarrow$ & $\rm O $ & $\rm H_2 $ & & 3.90e-08 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_2O^+$ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm OH $ & & 8.60e-08 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_2O^+$ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm H $ & $\rm O $ & 3.05e-07 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_3O^+$ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm H $ & $\rm OH $ & 2.60e-07 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_3O^+$ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm H_2O $ & & 1.10e-07 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_3O^+$ & $\rm e^- $ & $\longrightarrow$ & $\rm H_2 $ & $\rm OH $ & & 6.00e-08 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_3O^+$ & $\rm e^- $ & $\longrightarrow$ & $\rm H $ & $\rm O $ & $\rm H_2 $ & 5.60e-09 & -5.00e-01 & 0.00e+00 & $3$\
$\rm H_2 $ & $\rm H_2^+ $ & $\longrightarrow$ & $\rm H $ & $\rm H_3^+ $ & & 2.10e-09 & 0.00e+00 & 0.00e+00 & $3$\
$\rm H $ & $\rm H_2^+ $ & $\longrightarrow$ & $\rm H_2 $ & $\rm H^+ $ & & 6.40e-10 & 0.00e+00 & 0.00e+00 & $3$\
$\rm C $ & $\rm H_3^+ $ & $\longrightarrow$ & $\rm H_2 $ & $\rm CH^+ $ & & 2.00e-09 & 0.00e+00 & 0.00e+00 & $3$\
$\rm O $ & $\rm H_3^+ $ & $\longrightarrow$ & $\rm H_2 $ & $\rm OH^+ $ & & 7.98e-10 & -1.56e-01 & 1.41e+00 & $3$\
$\rm CO $ & $\rm H_3^+ $ & $\longrightarrow$ & $\rm H_2 $ & $\rm HCO^+ $ & & 9.45e-01 & 1.99e-09 & 2.51e-01 & $5$\
$\rm Fe $ & $\rm H_3^+ $ & $\longrightarrow$ & $\rm H $ & $\rm H_2 $ & $\rm Fe^+ $ & 4.90e-09 & 0.00e+00 & 0.00e+00 & $3$\
$\rm H_2 $ & $\rm He^+ $ & $\longrightarrow$ & $\rm H $ & $\rm He $ & $\rm H^+ $ & 3.30e-15 & 0.00e+00 & 0.00e+00 & $3$\
$\rm CO $ & $\rm He^+ $ & $\longrightarrow$ & $\rm He $ & $\rm O $ & $\rm C^+ $ & 1.00e+00 & 1.75e-09 & 2.51e-01 & $5$\
$\rm H_2 $ & $\rm He^+ $ & $\longrightarrow$ & $\rm He $ & $\rm H_2^+ $ & & 9.60e-15 & 0.00e+00 & 0.00e+00 & $3$\
$\rm H_2 $ & $\rm C^+ $ & $\longrightarrow$ & $\rm H $ & $\rm CH^+ $ & & 1.50e-10 & 0.00e+00 & 4.64e+03 & $3$\
$\rm OH $ & $\rm C^+ $ & $\longrightarrow$ & $\rm H $ & $\rm CO^+ $ & & 1.00e+00 & 9.15e-10 & 5.50e+00 & $4$\
$\rm H_2 $ & $\rm CO^+ $ & $\longrightarrow$ & $\rm H $ & $\rm HCO^+ $ & & 7.50e-10 & 0.00e+00 & 0.00e+00 & $3$\
$\rm H_2 $ & $\rm OH^+ $ & $\longrightarrow$ & $\rm H $ & $\rm H_2O^+$ & & 1.10e-09 & 0.00e+00 & 0.00e+00 & $3$\
$\rm H_2 $ & $\rm H_2O^+$ & $\longrightarrow$ & $\rm H $ & $\rm H_3O^+$ & & 6.10e-10 & 0.00e+00 & 0.00e+00 & $3$\
\
$\rm H_2O $ & $\rm H_3^+ $ & $\longrightarrow$ & $\rm H_2 $ & $\rm H_3O^+$ & & 1.00e+00 & 1.73e-09 & 5.41e+00 & $4$\
$\rm C $ & $\rm OH $ & $\longrightarrow$ & $\rm H $ & $\rm CO $ & & 1.15e-10 & -3.39e-01 & -1.08e-01 & $3$\
$\rm H $ & $\rm CO^+ $ & $\longrightarrow$ & $\rm CO $ & $\rm H^+ $ & & 4.00e-10 & 0.00e+00 & 0.00e+00 & $3$\
\[tabb1\]
[ccccccc]{}
\
& & Chemical reaction & & & $\alpha$ & $\beta$\
\
& & Chemical reaction & & & $\alpha$ & $\beta$\
$\rm CO* $ & $\rm CRP $ & $\longrightarrow$ & $\rm C* $ & $\rm O* $ & 5.00e+00 & 0.00e+00\
$\rm OH* $ & $\rm CRP $ & $\longrightarrow$ & $\rm O* $ & $\rm H* $ & 5.10e+02 & 0.00e+00\
$\rm H_2O* $ & $\rm CRP $ & $\longrightarrow$ & $\rm OH* $ & $\rm H* $ & 9.70e+02 & 0.00e+00\
$\rm CO* $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm C* $ & $\rm O* $ & 2.60e-10 & 3.53e+00\
$\rm OH* $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm O* $ & $\rm H* $ & 3.90e-10 & 2.24e+00\
$\rm H_2O* $ & $\rm PHOTON$ & $\longrightarrow$ & $\rm OH* $ & $\rm H* $ & 8.00e-10 & 2.20e+00\
$\rm C* $ & $\rm O* $ & $\longrightarrow$ & $\rm CO* $ & & 1.00e+00 & 0.00e+00\
$\rm H* $ & $\rm H* $ & $\longrightarrow$ & $\rm H_2* $ & & 1.00e+00 & 0.00e+00\
$\rm H* $ & $\rm O* $ & $\longrightarrow$ & $\rm OH* $ & & 1.00e+00 & 0.00e+00\
$\rm H* $ & $\rm OH* $ & $\longrightarrow$ & $\rm H_2O* $ & & 1.00e+00 & 0.00e+00\
$\rm C* $ & $\rm OH* $ & $\longrightarrow$ & $\rm CO* $ & $\rm H* $ & 1.00e+00 & 0.00e+00\
\[tabb2\]
[^1]: We note that @Glassgold12 have calculated smaller values of $\Delta Q \sim 10 - 13 \, \rm eV$ for the medium densities probed here.
|
---
abstract: 'Particle stabilized emulsions have received an enormous interest in the recent past, but our understanding of the dynamics of emulsion formation is still limited. For simple spherical particles, the time dependent growth of fluid domains is dominated by the formation of droplets, particle adsorption and coalescence of droplets (Ostwald ripening), which eventually can be almost fully blocked due to the presence of the particles. Ellipsoidal particles are known to be more efficient stabilizers of fluid interfaces than spherical particles and their anisotropic shape and the related additional rotational degrees of freedom have an impact on the dynamics of emulsion formation. In this paper, we investigate this point by means of simple model systems consisting of a single ellipsoidal particle or a particle ensemble at a flat interface as well as a particle ensemble at a spherical interface. By applying combined multicomponent lattice Boltzmann and molecular dynamics simulations we demonstrate that the anisotropic shape of ellipsoidal particles causes two additional timescales to be of relevance in the dynamics of emulsion formation: a relatively short timescale can be attributed to the adsorption of single particles and the involved rotation of particles towards the interface. As soon as the interface is jammed, however, capillary interactions between the particles cause a local reordering on very long timescales leading to a continuous change in the interface configuration and increase of interfacial area. This effect can be utilized to counteract the thermodynamic instability of particle stabilized emulsions and thus offers the possibility to produce emulsions with exceptional stability.'
author:
- Florian Günther
- Stefan Frijters
- Jens Harting
title: Timescales of emulsion formation caused by anisotropic particles
---
Introduction {#sec:introduction}
============
Particle stabilized emulsions play an important role in pharmaceutical, food, oil and cosmetic industries [@Dickinson2010a]. The particles are adsorbed at the interface between two immiscible fluids and as such stabilize the emulsion. The stability of the emulsions depends on several parameters like particle coverage at the interfaces and the wettability of the particles. It was found that the particle coverage at the interface is the most important parameter for stabilizing emulsions [@Fan2012a]. The colloidal particles act in a similar way as surfactants. In both cases the free energy of the interface is reduced. However, the fluid-fluid interfacial tension is not being modified by particles [@Frijters2012a].
Several types of particle stabilized emulsions are known including the bicontinuous interfacially jammed emulsion gel (bijel) and the more widely known Pickering emulsion. The Pickering emulsion was discovered in the beginning of the 20^th^ century independently by Pickering and Ramsden [@Pickering1907a; @Ramsden1903a]. It consists of discrete particle covered droplets of a fluid immersed in a second fluid. The bijel was predicted in 2005 by simulations and experimentally realized for the first time in 2007 [@Stratford2005a; @Herzig2007a]. It consists of two continuous phases. The choice of control parameters such as particle concentration, particle wettability and ratio between the two fluids determines if a bijel or a Pickering emulsion is obtained [@Guenther2012a; @Jansen2011a]. There are many kinds of particles/colloid types which can stabilize an emulsion. I.e., next to spheres [@He2007a; @Aveyard2003a], the colloidal particles can also be of more complex nature and include anisotropic shapes [@Kalashnikova2013a], magnetic interactions [@Kim2010a; @Melle2005a], or anisotropic Janus style properties [@Binks2001a].
The influence of the particle shape on the stabilization of Pickering emulsions was studied experimentally with prolate and oblate ellipsoids, e.g. in Ref. [@Madivala2009b]. As the degree of the particle anisotropy increases, the effective coverage area increases. In this way they are more efficient stabilizers for emulsions than spherical particles. Furthermore, the rheological properties of the emulsion vary with changing aspect ratio because the coverage of the fluid interfaces and the capillary interactions differ.
In Refs. [@Guenther2012a; @deGraaf2010a; @Dong2005a; @Bresme2007b; @Faraudo2003a] the adsorption of a single particle at a flat interface is studied in absence of external fields. The stable configuration for elongated ellipsoids is the orientation parallel to the interface [@Guenther2012a]. This state minimizes the free energy of the particle at the interface by reducing the interfacial area [@deGraaf2010a; @Bresme2007b; @Faraudo2003a]. If the particle shape is more complex like e.g. the super-ellipsoidal hematite particle [@Morgan2013a], several equilibrium orientations are possible.
Furthermore, if particles are adsorbed at an interface they generally deform the interface. This deformation can be caused for example by particle anisotropy [@Lehle2008a], external forces such as gravity or electromagnetic forces acting on the particles [@Bleibel2011b; @Bleibel2013], or non-constant interface curvature [@Zeng2012a]. This deformation leads to capillary interactions between the particles. In case of ellipsoids at a flat interface it is a quadrupolar potential [@Botto2012b], which leads to spatial ordering [@Madivala2009a].
In general, particle stabilized emulsions are thermodynamically unstable and just kinetically stable. The energetic penalty for creating the interface is much higher than the entropic increase. While thermodynamic stability for emulsions has been reported in some special cases, one can generally assume that this requires the interplay of several effects such as particle interactions due to charges, amphiphilic interactions (Janus particles) or additional degrees of freedom [@Sacanna2007b; @Kegel2009a; @Aveyard2012].
Due to the short timescales and limited optical accessibility, the dynamics of the formation of emulsions has only found limited attendance so far [@Dai2008]. The focus of the current article is to study the influence of the geometrical anisotropy and rotational degrees of freedom of ellipsoidal particles on the time development of fluid domain sizes in particle-stabilized emulsions. To obtain a deeper understanding of the individual contributions to the stabilization and formation process due to the particles we investigate model systems involving either a single particle or particle ensembles at a simple interface. We will demonstrate that the rotational degrees of freedom of ellipsoids can have an impact on the domain growth and might be a suitable way to generate particle stabilized emulsions with exceptional long-term stability.
This article is organized as follows: the simulation method is introduced in section II. Dynamic emulsion properties are studied in section III. Sections IV and V discuss a single particle and a particle ensemble at a flat interface, respectively. Section VI describes the behavior of a particle ensemble at a spherical interface. We finalize the paper with a conclusion.
Simulation method {#sec:sim-method}
=================
The lattice Boltzmann method {#ssec:lb}
----------------------------
For the simulation of the fluids the lattice Boltzmann method is used [@succi2001a]. The discrete form of the Boltzmann equation can be written as [@Frijters2012a] $$\label{eq:LBG}
{f_i}^c({{\mathbf{x}}}+ {{\mathbf{c}}_i}\Delta t , t + \Delta t)={f_i}^c({{\mathbf{x}}},t)+\Omega_i^c({{\mathbf{x}}},t)
\mbox{,}$$ where ${f_i}^c({{\mathbf{x}}},t)$ is the single-particle distribution function for fluid component $c$ with discrete lattice velocity ${{\mathbf{c}}_i}$ at time $t$ located at lattice position ${{\mathbf{x}}}$. The D3Q19 lattice with the lattice constant $\Delta x$ for three dimensions and with nineteen velocity directions is used. $\Delta t$ is the timestep and $$\label{eq:BGK_collision_operator}
\Omega_i^c({{\mathbf{x}}},t) = -\frac{{f_i}^c({{\mathbf{x}}},t)- {f_i}^\mathrm{eq}(\rho^c({{\mathbf{x}}},t), {\mathbf{u}}^c({{\mathbf{x}}},t))}{\left( \tau^c / \Delta t \right)}$$ is the Bhatnagar-Gross-Krook (BGK) collision operator [@bhatnagar1954a]. The density is defined as $\rho^c({{\mathbf{x}}},t)=\rho_0\sum_i{f_i}^c({{\mathbf{x}}},t)$ where $\rho_0$ is the proportionality factor of the density. $\tau^c$ is the relaxation time for the component $c$ and $$\begin{gathered}
\label{eq:equilibrium-distribution}
{f_i}^{\mathrm{eq}}(\rho^c,{\mathbf{u}}^c) = \zeta_i \rho^c \bigg[ 1 + \frac{{{\mathbf{c}}_i}\cdot {\mathbf{u}}^c}{c_s^2} + \frac{ \left( {{\mathbf{c}}_i}\cdot {\mathbf{u}}^c \right)^2}{2 c_s^4} \\ - \frac{ \left( {\mathbf{u}}^c \cdot {\mathbf{u}}^c \right) }{2 c_s^2} + \frac{ \left( {{\mathbf{c}}_i}\cdot {\mathbf{u}}^c \right)^3}{6 c_s^6} - \frac{ \left( {\mathbf{u}}^c \cdot {\mathbf{u}}^c \right) \left( {{\mathbf{c}}_i}\cdot {\mathbf{u}}^c \right )}{2 c_s^4} \bigg]
\mbox{}\end{gathered}$$ is the third order equilibrium distribution function. $$\label{eq:sos}
c_s = \frac{1}{\sqrt{3}} \frac{\Delta x}{\Delta t}
\mbox{}$$ is the speed of sound, ${\mathbf{u}}^c=\sum_i{f_i}^c({{\mathbf{x}}},t){{\mathbf{c}}_i}/\rho^c({{\mathbf{x}}},t)$ is the velocity and $\zeta_i$ is a coefficient depending on the direction: $\zeta_0=1/3$ for the zero velocity, $\zeta_{1,\dots,6}=1/18$ for the six nearest neighbors and $\zeta_{7,\dots,18}=1/36$ for the next nearest neighbors in diagonal direction. The kinematic viscosity can be calculated as $$\label{eq:kinvis}
\nu^c = c_s^2 \Delta t \left( \frac{\tau^c}{\Delta t} - \frac{1}{2} \right)
\mbox{.}$$ In the following we choose $\Delta x = \Delta t = \rho_0 = 1$ for simplicity. In all simulations the relaxation time is set to $\tau^c \equiv 1$.
Multicomponent lattice Boltzmann {#ssec:multicomponent-lb}
--------------------------------
There are different extensions for the lattice Boltzmann method to simulate multi-component and multiphase systems [@shan1993a; @Orlandini1995a; @Swift1996a; @Lishchuk2003a; @LeeFischer06]. An overview on different methods for multi-component fluid systems and the treatment of fluid-fluid interfaces is given in Ref. [@Krueger2012c]. In this paper, the method introduced by Shan and Chen is used [@shan1993a]. Every species has its own distribution function following [Eq. (\[eq:LBG\])]{}. To obtain an interaction between the different components a force $$\label{eq:sc}
{{\mathbf{F}}^c}({{\mathbf{x}}},t) = -{\Psi^c}({{\mathbf{x}}},t) \sum_{c'}g_{cc'} \sum_{{{\mathbf{x}}}'} \Psi^{c'}({{\mathbf{x}}}',t) ({{\mathbf{x}}}'-{{\mathbf{x}}})
\mbox{}$$ is calculated locally and is included in the equilibrium distribution function. it is summed up over the different fluid species $c'$ and ${{\mathbf{x}}}'$, the nearest neighbors of lattice positions ${{\mathbf{x}}}$. $g_{cc'}$ is the coupling constant between the species and ${\Psi^c}(\mathbf{x},t)$ is a monotonous weight function representing an effective mass. For the results presented here, the form $$\label{eq:psifunc}
{\Psi^c}({{\mathbf{x}}},t) \equiv \Psi(\rho^c({{\mathbf{x}}},t) ) = 1 - e^{-\rho^c({{\mathbf{x}}},t)}$$ is used. To incorporate ${{\mathbf{F}}^c}({{\mathbf{x}}},t)$ in ${f_i}^\mathrm{eq}$ we define $$\label{eq:delta-u}
\Delta {\mathbf{u}}^c({{\mathbf{x}}},t) = \frac{\tau^c {{\mathbf{F}}^c}({{\mathbf{x}}},t)}{\rho^c({{\mathbf{x}}},t)}
\mbox{.}$$ The macroscopic velocity included in ${f_i}^\mathrm{eq}$ is shifted by $\Delta {\mathbf{u}}^c$ as $$\label{eq:delta-u-bulk}
{\mathbf{u}}^c({{\mathbf{x}}},t) = \frac{\sum_i f^c_i({{\mathbf{x}}},t) {{\mathbf{c}}_i}}{\rho^c({{\mathbf{x}}},t)} - \Delta {\mathbf{u}}^c({{\mathbf{x}}},t)
\mbox{.}$$ As we are interested in immiscible fluids we choose a positive value for $g_{cc'}$ which leads to a repulsive interaction. This interaction has to be strong enough to obtain two separate phases but it should not be too high in order to keep the simulation stable. Here, we use the range of $0.08\le g_{cc'}\le 0.14$.
Nanoparticles {#ssec:nanoparticles}
-------------
Particles are simulated with molecular dynamics where Newton’s equations of motion $$\label{eq:MD}
{{\mathbf{F}}}={m}{\dot{{\mathbf{u}}}_{\rm par}}\mbox{ and }
{{\mathbf{D}}}={J}{\dot{{\mathbf{\omega}}}_{\rm par}}$$ are solved by a leap frog integrator. ${{\mathbf{F}}}$ and ${{\mathbf{D}}}$ are the force and torque acting on the particle with mass ${m}$ and moment of inertia ${J}$. ${{\mathbf{u}}_{\rm par}}$ and ${{\mathbf{\omega}}_{\rm par}}$ are the velocity and the rotation vector of the particle.\
The particles are also discretized on the lattice. They are coupled to both fluid species by a modified bounce-back boundary condition which was originally introduced by Ladd [@Jansen2011a; @aidun1998a; @ladd1994a; @ladd1994b; @ladd2001a]. This changes the lattice Boltzmann equation as follows: $$\label{eq:mbb}
{f_i}^c({{\mathbf{x}}}+{{\mathbf{c}}_i},t+1) = f^c_{\bar{i}}({{\mathbf{x}}}+{{\mathbf{c}}_i},t) + \Omega_{\bar{i}}^c({{\mathbf{x}}}+{{\mathbf{c}}_i},t) + {{\cal C}}\mbox{,}$$ where ${\mathbf{c}}_i$ is the velocity vector pointing to the next neighbor. ${{\cal C}}$ depends linearly on the local particle velocity, $\bar{i}$ is defined in a way that ${{\mathbf{c}}_i}= -{\mathbf{c}}_{\bar{i}}$ is fulfilled. A change of the fluid momentum due to a particle leads to a change of the particle momentum in order to keep the total momentum conserved: $$\label{eq:mbb2}
{{\mathbf{F}}}(t) = \big( 2f_{\bar{i}}^c({{\mathbf{x}}}+{{\mathbf{c}}_i},t) + {{\cal C}}\big) {\mathbf{c}}_{\bar{i}}
\mbox{.}$$ If the particle moves, some lattice nodes become free and others become occupied. The fluid on the newly occupied nodes is deleted and its momentum is transferred to the particle as $$\label{eq:mom-transfer-to-ptcl}
{\mathbf{F}}(t) = - \sum_c \rho^c({{\mathbf{x}}},t) {\mathbf{u}}^c({{\mathbf{x}}},t)
\mbox{.}$$ A newly freed node (located at ${{\mathbf{x}}}$) is filled with the average density of the $N_{\mathrm{FN}}$ neighboring fluid lattice nodes ${{\mathbf{x}}}_{i_\mathrm{FN}}$ for each component $c$, $$\label{eq:rho-surr}
\overline{\rho}^c({{\mathbf{x}}},t) \equiv \frac{1}{N_{\mathrm{FN}}} \sum_{i_\mathrm{FN}} \rho^c({{\mathbf{x}}}+{\mathbf{c}}_{i_{\mathrm{FN}}},t)
\mbox{.}$$ Hydrodynamics leads to a lubrication force between the particles. This force is reproduced automatically by the simulation for sufficiently large particle separations. If the distance between the particles is so small that no free lattice point exists between them this reproduction fails. If the smallest distance between two identical spheres with radius $R$ is smaller than a critical value $\Delta_c= \frac{2}{3}$ the correction term is given as [@ladd2001a]: $$\label{eq:lubrication}
\mathbf{F}_{ij}=\frac{3\pi\mu{R^2}}{2}{\mathbf{\hat{r}}_{ij}}({\mathbf{\hat{r}}_{ij}}(\mathbf{u}_i-\mathbf{u}_j))\left(\frac{1}{r_{ij}-2R}-\frac{1}{\Delta_c}\right)
\mbox{.}$$ $\mu$ is the dynamic viscosity, ${\mathbf{\hat{r}}_{ij}}$ a unit vector pointing from one particle center to the other one and $\mathbf{u}_i$ is the velocity of particle $i$. To use this potential for ellipsoidal particles [Eq. (\[eq:lubrication\])]{} is generalized in a way proposed by Berne and Pechukas [@berne1972a; @Janoschek2010b; @Guenther2012a]. We define $\sigma=2{R}$ and $\epsilon=\frac{3\pi\mu}{8}\sigma$. Both are extended to the anisotropic case as $$\begin{split}
\epsilon({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j)=\frac{\overline{\epsilon}}{\sqrt{1-{\Upsilon}^2({\mathbf{\hat{o}}}_i{\mathbf{\hat{o}}}_j)^2}}
\quad\mbox{and}\quad\quad\quad\quad\quad\quad\quad\quad\quad
\\
\label{eq:sigmaaniso}
\sigma({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j,{\mathbf{\hat{r}}_{ij}})
=\frac{\overline{\sigma}}{\sqrt{1-\frac{{\Upsilon}}{2}(\frac{({\mathbf{\hat{r}}_{ij}}{\mathbf{\hat{o}}}_i+{\mathbf{\hat{r}}_{ij}}{\mathbf{\hat{o}}}_j)^2}{1+{\Upsilon}{\mathbf{\hat{o}}}_i{\mathbf{\hat{o}}}_j}+\frac{({\mathbf{\hat{r}}_{ij}}{\mathbf{\hat{o}}}_i-{\mathbf{\hat{r}}_{ij}}{\mathbf{\hat{o}}}_j)^2}{1-{\Upsilon}{\mathbf{\hat{o}}}_i{\mathbf{\hat{o}}}_j})}}
\mbox{,}
\end{split}$$ with $\overline{\sigma}=2{R_{\perp}}$, $\overline{\epsilon}=\frac{3\pi\mu}{8}\overline{\sigma}$, ${\Upsilon}=\frac{{R_{\parallel}}^2-{R_{\perp}}^2}{{R_{\parallel}}^2+{R_{\perp}}^2}$ and ${\mathbf{\hat{o}}}_i$ the orientation unit vector of particle $i$. ${R_{\parallel}}$ and ${R_{\perp}}$ are the parallel and the orthogonal radius of the ellipsoid. Using [Eq. (\[eq:sigmaaniso\])]{} we can rewrite [Eq. (\[eq:lubrication\])]{} and obtain $$\label{eq:lubricationres}
\mathbf{F}_{ij}({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j,{\mathbf{r}_{ij}})=\epsilon({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j)\mathbf{\tilde{F}}_{ij}\left(\frac{r_{ij}}{\sigma({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j,{\mathbf{\hat{r}}_{ij}})}\right)
\mbox{.}$$ $\mathbf{\tilde{F}}$ is a dimensionless function taking the specific form of the force into account and in this example it is $\mathbf{\tilde{F}}(r)={\mathbf{\hat{r}}_{ij}}({\mathbf{\hat{r}}_{ij}}(\mathbf{u}_i-\mathbf{u}_j))(\frac{1}{r-1}-\frac{\sigma}{\Delta_c})$.\
The lubrication force (including the correction) already reduces the probability that the particles come closely together and overlap. For the few cases where the particles still would overlap we introduce the direct potential between the particles which is assumed to be a hard core potential. To approximate the hard core potential we use the Hertz potential [@Hertz1881a] which has the following shape for two identical spheres with radius $R$: $$\label{eq:Hpot}
\phi_H=K_H(2{R}-{r})^{5/2}\mbox{ for }{r}<2{R}\mbox{.}$$ ${r}$ is the distance between particle centers. For larger distances $\phi_H$ vanishes. $K_H$ is a force constant and is chosen to be $K_H=100$ for all simulations. To use this potential for ellipsoidal particles [Eq. (\[eq:Hpot\])]{} is generalized in a similar way as the lubrication force. Using [Eq. (\[eq:sigmaaniso\])]{}, $\sigma=2{R}$ and $\epsilon=K_H\sigma^{\frac{5}{2}}$ we can rewrite [Eq. (\[eq:Hpot\])]{} and obtain $$\phi_H({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j,\mathbf{r}_{ij})
=\epsilon({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j)\tilde{\phi}_H \left(\frac{r_{ij}}{\sigma({\mathbf{\hat{o}}}_i,{\mathbf{\hat{o}}}_j,{\mathbf{\hat{r}}_{ij}})}\right)
\mbox{.}$$ $\tilde{\phi}_H$ is a dimensionless function taking the specific form of the potential into account and in this example it is $\tilde{\phi}_H(x)=(1-x)^{5/2}$.\
The Shan-Chen forces also act between a node in the outer shell of a particle and its neighboring node outside of the particle. This would lead to an increase of the fluid density around the particle. Therefore, the nodes in the outer shell of the particle are filled with a virtual fluid corresponding to the average of the value in the neighboring free nodes for each fluid component: $\rho_{\mathrm{virt}}^c({{\mathbf{x}}},t) = \overline{\rho}^c({{\mathbf{x}}},t)$. This can be used to control the wettability properties of the particle surface for the special case of two fluid species which will be named red and blue. We define the parameter $\Delta\rho$ and call it particle color. For positive values of $\Delta\rho$ we add it to the red fluid component: $$\label{eq:red-colour}
\rho_{\mathrm{virt}}^r=\overline{\rho}^r+\Delta\rho
\mbox{.}$$ For negative values we add its absolute value to the blue component: $$\label{eq:blue-colour}
\rho_{\mathrm{virt}}^b=\overline{\rho}^b+|\Delta\rho|
\mbox{.}$$ In Ref. [@Guenther2012a] it is shown that there is a linear relation between $\Delta\rho$ and the three-phase contact angle ${\theta}$.
Emulsions
=========
![Snapshots of typical simulated Pickering emulsions (left) and bijels (right) after 10$^5$ timesteps. The emulsions are stabilized by prolate ellipsoids ($m=2$, top), spheres ($m=1$, center) and oblate ellipsoids ($m=1/2$, bottom). The parameter determining if one obtains a bijel or a Pickering emulsion is the fluid ratio which is chosen as 1:1 for the bijels and 5:2 for Pickering emulsions. []{data-label="PEBijelSnapshot"}](PE_bijel_6shots01b-crop.pdf){width="52.00000%"}
In this section the different types of particle stabilized emulsions and the effect of the particle shape on some of their properties are discussed. We find two different types of emulsions in our simulations, namely the Pickering emulsion (Fig. \[PEBijelSnapshot\], left) and the bijel (Fig. \[PEBijelSnapshot\], right). The choice of parameters (such as particle contact angle, particle concentration, fluid-fluid ratio, particle aspect ratio) determines the type of emulsions. Parameter studies for emulsions have been discussed in Refs. [@Jansen2011a] and [@Guenther2012a] for spherical and ellipsoidal particles, respectively. In the current publication we limit ourselves to anisotropy effects on the time dependence of the emulsion formation. We use the following particle shapes ($m={R_{\parallel}}/{R_{\perp}}$ is the particle aspect ratio. ${R_{\parallel}}$ and ${R_{\perp}}$ are the parallel and orthogonal radius of the particles, respectively): prolate ellipsoids ($m=2$; Fig. \[PEBijelSnapshot\], top), spheres ($m=1$; Fig. \[PEBijelSnapshot\], center) and oblate ellipsoids ($m=1/2$; Fig. \[PEBijelSnapshot\], bottom). For $m=1/2$ we choose ${R_{\parallel}}=5\Delta x$ and ${R_{\perp}}=10\Delta x$. For the other values of $m$ the radii ${R_{\parallel}}$ and ${R_{\perp}}$ are chosen as such that the particle volume is kept constant, resulting in ${R_{\parallel}}\approx12.6\Delta x$ and ${R_{\perp}}\approx6.3\Delta x$ for $m=2$ as well as ${R_{\parallel}}={R_{\perp}}\approx7.9\Delta x$ for spheres. The interaction parameter between the fluids (see [Eq. (\[eq:sc\])]{}) is chosen as $g_{br}=0.08$ which corresponds to a fluid-fluid interfacial tension of ${\sigma}=0.0138$. The particles are neutrally wetting (contact angle ${\theta}=90^\circ$) and the particle volume concentration is chosen as ${C}=0.24$. The simulated systems of volume ${V_S}={L_S}^3$ have periodic boundary conditions in all three directions and a side length of ${L_S}=256\Delta
x=32{R_{\parallel}}$. Initially, the particles are distributed randomly. At each lattice node a random value for each fluid component is chosen so that the designed fluid-fluid ratio is kept (1:1 for the bijels and 5:2 for the Pickering emulsions). When the simulation evolves in time, the fluids separate and droplets/domains with a majority of red or blue fluid form.
The average size of droplets/domains ${L(t)}$ can be determined by measuring $$\label{eq:L}
{L(t)}=\frac{1}{3}\sum_{i=x,y,z}{L(t)}_i
\mbox{.}$$ Here, $$\label{eq:Li}
{L(t)}_i=\frac{2\pi}{\sqrt{\langle{k}_i^2(t)\rangle}}$$ is the average domain size in direction $i$. $\langle{k}_i^2(t)\rangle=\sum_{{{\mathbf{k}}}}{k}_i^2(t){\varsigma}({{\mathbf{k}}},t)/\sum_{{{\mathbf{k}}}}{k}_i^2(t)$ is the second-order moment of the three-dimensional structure function ${\varsigma}({{\mathbf{k}}},t) = (1/{{\varsigma}_n})
|{{\varphi}'}_{{\mathbf{k}}}(t)|$. ${{\varphi}'}={\tilde{{\varphi}}}-\langle{\tilde{{\varphi}}}\rangle$ is the fluctuation of ${\tilde{{\varphi}}}$ which is the Fourier transform of the order parameter field ${\varphi}={\rho^r}-{\rho^b}$. In this publication, the time is given in simulation timesteps, which can be converted to physical units. We use [Eq. (\[eq:sos\])]{} and [Eq. (\[eq:kinvis\])]{} to relate the kinematic viscosity to $\Delta x$ and $\Delta t$. By assuming $\nu=10^{-6}m^2/s$, the kinematic viscosity for water, $R=125nm$ and $R=7.9\Delta x$ (this is the value used for the spherical particle, see above) we fix the chosen resolution of the simulation. Thus, we obtain $\Delta x=15.8nm$ and $\Delta t=4.2\times10^{-11}s$ and a total system size of $L_S\approx4\mu$. The interfacial tension is then $\sigma=3.14\times10^{-08}N/m$. Larger system sizes can be reached with the same computational effort by compromising on the resolution.
![Pickering emulsion and bijel: Time development of the average domain size ${L(t)}$ (see [Eq. (\[eq:L\])]{}) for $m=1$ and $m=2$. At first view, a steady state is reached after about $10^5$ timesteps. ${L(t)}$ is larger for bijels than for Pickering emulsions, which is due to the measurement being based on the Fourier transform of the order parameter. Ellipsoids are able to stabilize larger interface areas than spheres leading to smaller ${L(t)}$.[]{data-label="LtvglKugelPEuBijel"}](dsft_mr3_3u4c.pdf){width="40.00000%"}
The time development of ${L(t)}$ for the three different particle types (prolate, spherical and oblate ($m=2$, $1$ and $1/2$) and for Pickering emulsions and bijels is shown in Fig. \[LtvglKugelPEuBijel\]. We can identify three regimes: in the first few hundred timesteps the initial formation of the droplets/domains starts. Then, the growth of droplets/domains is being driven by Ostwald ripening. At even later times, droplets/domains grow due to coalescence. When two droplets unify, the area coverage fraction of the particles at the interface is increased because the surface area of the new droplet is smaller than that of the two smaller droplets before. At some point the area coverage fraction of the particles is sufficiently high to prevent further coalescence. The state which is reached at that time is (at least kinetically) stabilized and one obtains a stable emulsion. The values for ${L(t)}$ are larger for bijels than for Pickering emulsions. This can be explained by the way we calculate ${L(t)}$ (see [Eq. (\[eq:L\])]{} and related text) using a Fourier transformation of the order parameter field.
It can clearly be seen that anisotropic particles are more efficient in interface stabilization than spheres since they can cover larger interfacial areas leading to smaller fluid domains (note that the simulation volume is kept constant). However, the difference in ${L(t)}$ for ${m}=2$ and ${m}=1/2$ is small. This can be understood as follows: if a neutrally wetting prolate ellipsoid is adsorbed at a flat interface, it occupies an area $A_{P,F}({m}>1)={m}^{1/3}A_{p,s}$, where $A_{p,s}$ is the occupied interface area for a sphere with the same volume. This corresponds in the case of ${m}=2$ to the occupied interface being larger by a factor of $1.26$ as compared to spheres. For an oblate ellipsoid the occupied interface area is $A_{P,F}({m}<1)={m}^{-2/3}A_{p,s}$ which for ${m}=1/2$ is by a factor of $1.59$ larger than the area occupied by spheres. Since in emulsions the interfaces are generally not flat, these formulae can only provide a qualitative explanation of the behavior of ${L(t)}$: If the interface curvature is not neglectable anymore, we loose some of the efficiency of interface stabilization, which is more pronounced for ${m}<1$. This explains why the value of ${L(t)}$ for ${m}=1/2$ is only slightly smaller than for ${m}=2$.
It seems that ${L(t)}$ reaches a steady state after some $10^5$ timesteps for both types of emulsions and for all three values of ${m}$. However, if one zooms in one can observe that ${L(t)}$ develops for a longer time period if the particles have a non-spherical shape. As will be demonstrated below, the reason for this phenomenon is the additional rotational degrees of freedom due to the particle anisotropy. Furthermore, the time development of ${L(t)}$ for emulsions stabilized by prolate particles requires more time than that for the oblate ones. If a particle changes its orientation as compared to the interface or a neighboring particle this generally changes the interface shape. In this way the domain sizes are influenced, leading to changes of ${L(t)}$ – an effect which is not observed for $m=1$.
![Pickering emulsion: zoom of the time dependent average domain size ${L(t)}$ for $m=2$, $m=1$ and $m=1/2$. The slow but continuous decrease of ${L(t)}$ clearly shows the occurrence of additional timescales in the domain growth. The kink in the measurement for $m=2$ can be adhered to the coalescence of two droplets. []{data-label="dsft_zoom_PE"}](dsft_mr3_4zoom6.pdf){width="40.00000%"}
![Bijel: Zoom for $m=2$ and $m=1/2$: Time development of the average domain sizes depicting the impact of the additional timescales. The range of the variation of ${L(t)}$ is larger as compared to the Pickering emulsion due to the impact of a small deformation on the larger effective interface of the bijel.[]{data-label="dsft_zoom_bijel"}](dsft_mr3_3zoom6.pdf){width="40.00000%"}
Fig. \[dsft\_zoom\_PE\] and Fig. \[dsft\_zoom\_bijel\] depict a zoom-in of the time development of ${L(t)}$ for Pickering emulsions and bijels with $m=2$ and $m=1/2$, respectively. One observes that ${L(t)}$ decays in all four cases. The kink in Fig. \[dsft\_zoom\_PE\] after about 2.8 million timesteps is due to the coalescence of two droplets of the Pickering emulsion. A substantial difference is the range of the decay. It is larger for the bijel since it consists of a single large interface whereas the Pickering emulsion consists of many small interfaces. The large interface in the bijel is much more deformable. This explains the larger range of the decay of ${L(t)}$ for the bijel. The fluctuations are of the same order for Pickering emulsions and bijels. Furthermore, the range of the decay is larger for $m=2$ than for $m=1/2$. The time of reordering is much shorter for $m=1/2$ as compared to $m=2$. These effects can be explained by the presence of additional rotational degrees of freedom for the anisotropic particles. While oblate particles have only a single additional rotational degree of freedom as compared to spheres, prolate particles show an even more complex behavior due to their second additional rotational degree of freedom.
In this section we demonstrated that particle anisotropy causes additional timescales to influence the growth of domains in particle-stabilized emulsions. In the following sections we discuss model systems in order to obtain a deeper understanding of this effect. We will restrict ourselves to prolate particles with $m=2$. Furthermore, the high resolution of the particles in the current section was only chosen to be able to sufficiently resolve the oblate objects. In order to reduce the required computational resources, we use smaller particles in the model systems studied below (${R_{\parallel}}=8\Delta x$ and ${R_{\perp}}=4\Delta x$). It has been checked carefully that the reduced particle size does not have a qualitative impact on the results.
Single particle adsorption
==========================
In the previous section we demonstrated that there is an additional time development of the average domain size $L$ for emulsions stabilized by anisotropic particles. In the following sections we relate this behavior to the orientational degree of freedom of the particles at the interface. To obtain a more basic understanding of the additional timescales some simple model systems are discussed. The simplest possible example is the adsorption of a single particle at a flat fluid-fluid interface. To characterize the particle orientation we introduce the angles ${\vartheta}$ and ${\phi}$. ${\vartheta}$ is the angle between the particle main axis and the ${y}$-axis, where the ${y}$-axis is oriented perpendicular to the flat fluid-fluid interface. ${\phi}$ is the angle between the particle main axis and the ${x}$-axis, where the ${x}$-axis is orientated parallel to the interface. ${\xi}$ is the distance between the particle center and the undeformed interface in units of the long particle axis. In this section we consider the case of neutral wetting (${\theta}=90^\circ$) and restrict ourselves to an aspect ratio of $m={R_{\parallel}}/{R_{\perp}}=2$. The fluid-fluid interaction parameter is set to $g_{br}=0.1$ corresponding to an interfacial tension of ${\sigma}\approx0.041$. We use a cubic system with 64 lattice nodes in each direction. A wall is placed at the top and bottom in ${y}$-direction. Periodic boundary conditions are applied in the ${x}$- and ${z}$-direction. In order to obtain a flat interface the system is filled with two equally sized cuboid shaped lamellae with an interface orthogonal to the ${y}$-axis. The lamellae are mainly filled with red and blue fluid, respectively. The initial majority and minority species are set to ${\rho}^{\rm maj}=0.7$ and ${\rho}^{\rm
min}=0.04$.\
For this study a particle is placed so that it just touches the (undeformed) fluid-fluid interface. This is done for different initial orientations of the particle. The inset of Fig. \[adsorbmpua15h\] shows snapshots of a typical adsorption process.
![Outer plot: ${\vartheta}$-$\xi$-plot for neutral wetting (${\theta}=90^\circ$), ${m}=2$ and ${\sigma}\approx0.041$. A particle is placed as such that it just touches the undeformed interface. The dashed lines denote the adsorption trajectories, the solid lines show the points where the particle touches the undeformed interface. The circular points depict the stable and the metastable point. The square points are related to the snapshots describing the adsorption process in the inset. For initial particle orientations of ${\vartheta}(t=0)\neq0^\circ$ the particle ends in its stable configuration orientated parallel to the interface.[]{data-label="adsorbmpua15h"}](adsorbmp6x48ua01i_a15h01.pdf){width="8cm"}
In the beginning the particle is oriented almost orthogonally to the interface. In the first ca. 2000 timesteps the particle moves towards the interface without changing its orientation considerably. Then, the particle rotates and reaches its final orientation after 3600 timesteps. The outer plot of Fig. \[adsorbmpua15h\] shows a ${\vartheta}-{\xi}$ diagram of the adsorption. The points where the particle just touches a flat interface for the different orientations are marked with solid lines. The dotted lines indicate the adsorption trajectories. Each black square is related to one of the snapshots in the inset of Fig. \[adsorbmpua15h\]. Almost all dashed lines end in the upper circle which corresponds to the equilibrium point where the free energy function has a global minimum. Just the cases with an initial value of ${\vartheta}(t=0)=0^\circ$ end at the metastable point at ${\vartheta}=0^\circ$ as shown by the circle at the bottom of Fig. \[adsorbmpua15h\]. This metastable point might not be found in experiments: on the one hand fluctuations will cause a rotation of the particle towards the stable points and on the other hand, it is impossible to place the particle exactly at ${\vartheta}=0^\circ$.\
Fig. \[adsorbplotphit5\_1\] and \[ggerreicht0\_98\] depict the dynamics of the particle adsorption and the influence of the initial particle orientation ${\vartheta}$ with respect to the flat interface.
![Time development of the particle orientation ${\vartheta}(t)$ for different initial orientations. For ${\vartheta}(t=0)\neq0^\circ$ and ${\vartheta}(t=0)\neq90^\circ$ the particle rotates in the ‘wrong’ direction in the first timesteps. The time needed to be in the final orientation depends on the initial orientation.[]{data-label="adsorbplotphit5_1"}](phit7xp.pdf){width="8cm"}
![Outer plot: Time $t_e$ which the particle needs to reach the final orientation (${\vartheta}=90^\circ$) for different initial orientation angles ${\vartheta}_0={\vartheta}(t=0)$ from ${\vartheta}=0^\circ$ to ${\vartheta}=90^\circ$. $t_e$ diverges if ${\vartheta}_0$ approaches $0^\circ$. The reason for the divergence is the approach of ${\vartheta}_0$ to the orientation of the metastable point, as it is shown in the inset: If the starting angle (middle dashed line) approaches ${\vartheta}(t=0)=0$ (lower dashed line) the time required to reach the equilibrium point diverges.[]{data-label="ggerreicht0_98"}](ggerreicht0_98fxuadsorbvs7.pdf){width="8cm"}
Fig. \[adsorbplotphit5\_1\] shows the time development of ${\vartheta}$ for different values of ${\vartheta}(t=0)$. For ${\vartheta}(t=0)=0^\circ$ and ${\vartheta}(t=0)=90^\circ$ (upper and lower lines) the orientation remains unchanged and the adsorption at the interface causes only a translational particle movement. The lines for the three other simulation runs start at ${\vartheta}(t=0)=22.5^\circ$, ${\vartheta}(t=0)=45^\circ$ and ${\vartheta}(t=0)=67.5^\circ$. All of them go in the ‘wrong’ direction during the first few $10^2$ timesteps and end at ${\vartheta}=90^\circ$ corresponding to the stable point, but the time needed for reaching this value differs. Furthermore, in all cases during the first timesteps, ${\vartheta}$ decreases but then it increases up to this final value. The time $t_e$ the particle needs to reach the final orientation of ${\vartheta}=90^\circ$ depending on ${\vartheta}(t=0)$ is shown in the outer plot of Fig. \[ggerreicht0\_98\]. Due to the discretization of the particle on the lattice, its orientation shows small deviations from the theoretical final value. Therefore, we measure $t_e$ as the time when the angle reaches $98\%$ of the theoretical final angle. The particle oscillates arround this final value but these oscillations are very small and their magnitude falls below the threshold for the measurement of $t_e$. $t_e$ increases with decreasing ${\vartheta}$ and diverges for ${\vartheta}\rightarrow0$. This divergence can be understood using the inset of Fig. \[ggerreicht0\_98\]. If the starting angle ${\vartheta}(t=0)$ comes closer to ${\vartheta}=0^\circ$ (corresponding to the metastable case where the particle never flips) the capillary forces causing the particle rotation become smaller and vanish.\
We have seen that anisotropy of particles causes additional timescales in the development of the domain sizes in the emulsions, because of orientational ordering. This timescale is of the order of $10^{6}$ LB timesteps. In this section we have shown that the adsorption of a single particle at an interface and its orientational ordering takes of the order of $10^{3}$ timesteps and depending on the initial particle orientation towards the interface. We can identify one extra timescale where the particles rotate towards the interface. This timescale plays a role in the beginning of the emulsion formation (during droplet formation and droplet growth) when the particles come in contact with the interfaces. However, this timescale does not yet explain the full time development. We require additional model systems to obtain a full understanding of the additional timescales. Thus, we consider many particles at a flat interface as well as at a single droplet in the following sections.
Particle ensembles at a flat interface
======================================
After having studied the adsorption of a single particle we discuss the behavior of a many-particle ensemble at a flat interface. What is the influence of the hydrodynamic interaction between many particles on the timescales involved in emulsion formation? For the case of the single-particle adsorption the particle orientation towards the interface (${\vartheta}$) is an important parameter. For prolate particles, also the mutual orientation (${\phi}$) of the particles is important and one has an additional degree of freedom leading to particle orientational ordering. To characterize the ordering of the particles we use two order parameters and two correlation functions.
Measures for global ordering effects of the particles are the orientational order parameters ${S}$ and ${Q}$. We define the uniaxial order parameter ${S}$ [@Kralj1991a; @Collings2003a] as $$\label{eq:S}
{S}=\frac{1}{2}\left\langle3\cos^2{\vartheta}-1\right\rangle
\mbox{,}$$ where $\langle\rangle$ denotes the averaging over particles. Originally ${S}$ is an order parameter for studying liquid crystals which indicates the phase transition from the isotropic to the anisotropic/nematic phase. Here, the parameter ${S}$ is used as a measure for the orientation of the particle ensemble towards the interface. If all particles are oriented orthogonal to the interface we have ${S}={S}_{\perp{}}=1$ (see top right of Fig. \[ma2t\_initshot2\_gesamt\]). The orientation of all particles parallel to the interface leads to ${S}=S_{\parallel{}}=-0.5$ (see top left of Fig. \[ma2t\_initshot2\_gesamt\]).\
The biaxial order parameter ${Q}$ [@Collings2003a] is defined as $$\label{eq:Q}
{Q}=\frac{3}{2}\left\langle\sin^2{\vartheta}\cos(2{\phi})\right\rangle
\mbox{.}$$ The parameter ${Q}$ is a measure for the mutual orientation of the particles oriented parallel to the interface. If all particles lying parallel to the interface are oriented in the same direction it is ${Q}={Q}_{\rm
aniso}=1.5$. ${Q}={Q}_{\rm iso}=0$ means that the particles oriented parallel to the interface have a two-dimensional isotropic ordering.\
The local ordering effects are investigated by using two correlation functions. The discretized form of the pair correlation function $g({r})$ is defined as $$\label{eq:gd}
{g}({r})=\frac{1}{2\pi{{g}_n}{N}}\left\langle\sum_{i,j\neq i}\int_{{r}-\frac{1}{2}}^{{r}+\frac{1}{2}}\delta(\tilde{{r}}-{r}_{ij})d\tilde{{r}}\right\rangle
\mbox{,}$$ where ${N}$ is the number of particles, ${r}$ and ${r}_{ij}$ are the distance from a reference particle and the distance between the two particle centers of particle $i$ and $j$ in units of ${R_{\parallel}}$, respectively, and ${{g}_n}$ is a normalization factor chosen such that ${g}({r})\rightarrow1$ for $r\rightarrow\infty$. ${g}({r})$ gives a probability to find a particle at a distance ${r}$ from a reference particle. It is a measure for the ordering of the particle centers and ignores the orientation. As a measure for the local orientational ordering effects the angular correlation function [@Cuesta1990a] is defined as (in the discrete form) $$\label{eq:g2l}
{h}({r})=\int_{{r}-\frac{1}{2}}^{{r}+\frac{1}{2}}\left\langle\cos(2l({\vartheta}(0)-{\vartheta}({r}))\right\rangle
\mbox{,}$$ with $l=1$ in order to have the appropriate values of ${h}$ for a given value of ${\vartheta}$ discussed below. ${h}({r})$ gives a measure for the average orientation of particles at distance ${r}$ from a reference particle. If the particles at distance ${r}$ from the reference particle are all oriented parallel to the reference particle we have ${h}({r})=1$ (see right and left configuration
in the bottom of Fig. \[ma2t\_initshot2\_gesamt\]) and an orthogonal orientation leads to ${h}({r})=-1$ (see central configuration in the bottom of Fig. \[ma2t\_initshot2\_gesamt\]). In the following we use smoothed versions of ${g}$ and ${h}$, where we average over neighboring data points. The flat interface considered in this section is periodic in two dimensions parallel to the interface and each period has a size of ${A_I}={L_I}^2$, with ${L_I}=512=64{R_{\parallel}}$. The system is confined by walls $40$ lattice units distant from the interface in the third dimension. The particle coverage fraction for $N$ particles adsorbed at the interface is defined as ${\chi}({\xi},{\vartheta})=\frac{N{A_P}({\xi},{\vartheta})}{{A_I}}$. ${A_P}({\xi},{\vartheta})$ is the area which the particle would occupy on a hypothetical flat interface and depends on the distance between the particle center and the undeformed interface and the particle orientation relative to the flat interface and ${\xi}$ is the distance between particle center and undeformed interface. In the following we relate the coverage fraction to the case of ${\xi}=0$ and ${\vartheta}=90^{\circ}$ (${\chi}_I$) or ${\vartheta}=90^{\circ}$ (${\chi}_F$) corresponding to the initial state and the equilibrium state for ${\theta}=90^{\circ}$ (see previous section). This leads to ${\chi}_I=\frac{NA_{P,I}}{{A_I}}$ and ${\chi}_F=\frac{NA_{P,F}}{{A_I}}$ with $A_{P,I}=\pi{R_{\perp}}^2$ and $A_{P,F}=\pi{R_{\parallel}}{R_{\perp}}$.\
Initially, the particles are oriented almost orthogonally to the interface (see top right of Fig. \[ma2t\_initshot2\_gesamt\]). The initial value for the polar angle is chosen as ${\vartheta}\approx0.6^\circ$ for all particles, whereas ${\phi}$ and the particle positions are chosen randomly. Analogously to the case of the single-particle adsorption the particle flips to an orientation parallel to the interface (see Fig. \[ma2t\_shotafterfliptop\]). Fig. \[OP\_2\_5\_01\] shows the time development of ${S}$ for different values of ${\chi}_I$ (${\chi}_I\approx0.08$ (squares), ${\chi}_I\approx0.38$ (circles), ${\chi}_I\approx0.46$ (upward pointing triangles) and ${\chi}_I\approx0.52$ (downward pointing triangles)) and the time development of ${Q}$ for ${\chi}_I\approx0.38$ (diamonds).
[\[OP\_2\_5\_01\]![(a) Time development of the two order parameters ${S}(t)$ and ${Q}(t)$ (see [Eq. (\[eq:S\])]{} and [Eq. (\[eq:Q\])]{}) for $m=2$, ${\theta}=90^\circ$, $\sigma\approx0.041$. ${Q}(t)$ is shown for a single value of ${\chi}_I$ only since it stays at a value of approximately 0 for all ${\chi}_I$. ${S}(t)$ is shown for different values of ${\chi}_I$. In case of highly packed interfaces, i.e. for large values of ${\chi}_I$, not all particles are able to fully align with the interface. For larger values of ${\chi}_I$ ${S}$ needs a longer time to get into the equilibrium than shown here. (b) Outer plot: The final values of the order parameter ${S}$ are plotted for different particle densities ${\chi}_I$. As shown in \[OP\_2\_5\_01\] a transition from a fully ordered to a disordered state can be found at a critical value of ${\chi}_{I,C}\approx0.42$. Inset: the time the order parameter $S$ (defined in Eq. \[eq:S\]) requires to reach the final value (time which particles need to flip). For small values of ${\chi}_I$ $t_f$ is independent of ${\chi}_I$ but above a critical value of ${\chi}_I={\chi}_{I,C}$ $t_f$ increases with increasing ${\chi}$ by almost one order of magnitude. ](OPSQ09x.pdf "fig:"){width="8"}]{}
The parameter ${Q}$ starts at 0 and ends at a small value (${Q}_{\rm final}\approx0.05\ll {Q}_{\rm aniso}$) far away from the value of total ordering. A similar behavior is found for all values of ${\chi}_I$. Fig. \[ma2t\_shotafterfliptop\] shows that there are smaller domains where particles are oriented in the same direction. But every domain has a different preferred particle direction which might lead to small but still finite values of ${Q}$. Another reason for this effect is the finite system size and finite particle number which change the parameter as follows [@Cuesta1990a]: $$\label{eq:Qfinite}
{Q}={Q}_{\infty}+O\left(\frac{1}{\sqrt{N}}\right)
\mbox{.}$$ ${Q}_{\infty}$ is the value of the biaxial order parameter that the corresponding system with an infinite amount of particles would have.
The parameter ${S}$ starts for all values of ${\chi}_I$ with a value of ${S}_{\perp{}}=1$, corresponding to the initial configuration. For lower values of ${\chi}_I$ the parameter ${S}$ reaches ${S}_{\parallel{}}=-0.5$, corresponding to the case that the particles flip completely. For higher values of ${\chi}_I$ the final value of the parameter is ${S}_{\rm final}>{S}_{\parallel{}}=-0.5$. This corresponds to the case where some particles cannot flip completely to the equilibrium orientation because there is insufficient space. The final values of ${S}$ (obtained after $10^5$ timesteps) are shown in the outer plot of Fig. \[SD\_sigma\] as a function of ${\chi}_I$. We find a transition point at ${\chi}_{I,C}\approx0.42$ corresponding to ${\chi}_{F,C}\approx0.84$. If all particles are oriented parallel to the interface the system corresponds practically to a two dimensional system of ellipses. However, the value of ${\chi}_{I,C}$ found is below the value of the closest packing density for a two-dimensional system of ellipses with ${m}=2$, which is ${\chi}_{F,\rm max}\approx0.91$. Such a system was also studied in Ref. [@Cuesta1990a] with Monte Carlo simulations. For the case of an ellipse with an aspect ratio ${m}_{2d}=2$ a transition point of ${\chi}_{\rm 2dmc}\approx0.78$ from isotropy to a solid phase was found. The solid phase describes a state where the particle centers as well as the orientations are ordered. We do not reach the limit of the solid phase. This suggests that hydrodynamic interactions and absence confinement in the third dimension still play a dominant role. The biaxial order parameter in the MC system grows up to $Q\approx1$ (see Fig. 11 in Ref. [@Cuesta1990a]) corresponding to a global anisotropic state with a quite high degree of ordering for ${\chi}_F>{\chi}_{\rm 2dmc}$. This effect is not observed in our system. The reason for this difference is the method used to reach this state. A two-dimensional system of ellipses was studied in Ref. [@Cuesta1990a] wheres we simulated three-dimensional ellipsoids which form an effective two-dimensional system by flipping to the interface.\
We can see that in the many-particle system and for small and moderate ${\chi}_I$ about $10^3$ timesteps are required for the particles to flip which is the same order of magnitude as in the case of the single particle adsorption for small values of ${\chi}_I$. The inset in Fig. \[SD\_sigma\] shows the time the order parameter ${S}$ needs to reach its final value. This corresponds to the time required for the whole particle ensemble to be flipped completely (${\chi}_I<{\chi}_c$) or to reach the semi-flipped state for ${\chi}_I>{\chi}_c$. For ${\chi}_I<0.38$ $t_f$ stays almost constant at about 4500 timesteps. In this regime the distance between the particles is sufficient so that the influence of hydrodynamic interactions on the flipping behavior can be neglected. For higher values it increases very sharply and hydrodynamic interactions between the particles must not be neglected anymore. Furthermore, the time needed to flip completely for the very dense systems (jammed state) is about one order of magnitude larger. The biaxial order parameter does not show any global ordering but the snapshot in Fig. \[ma2t\_shotafterfliptop\] shows some local ordering effects. Hence, we need other ways to characterize the local ordering effects and utilize the two local correlation functions ${g}({r})$ and ${h}({r})$ defined above. The particles have a contact angle of $90^\circ$, so there are no capillary interactions between them in the final state when all of them have flipped completely and the system has reached an equilibrium. However, there are dipolar interface deformations and thus the interactions during the flipping process of the particles and for ${\chi}>{\chi}_c$ which causes capillary interactions at this time. After flipping there are still some capillary waves going through the system, leading to interactions between the particles. The pair correlation function
[\[Grm5\_01\]![(a) Pair correlation function ${g}({r})$ (defined in [Eq. (\[eq:gd\])]{}) (b) orientation correlation function $h({r})$ (defined in [Eq. (\[eq:g2l\])]{}). In both cases the ordering increases with increasing ${\chi}_I$.](g08x.pdf "fig:"){width="8"}]{}
${g}({r})$ is shown in Fig. \[Grm5\_01\] for three different values of ${\chi}_I$ (${\chi}_I\approx0.23$, ${\chi}_I\approx0.31$ and ${\chi}_I\approx0.38$) after $10^5$ timesteps. The first peak is pronounced in all three cases. The distance ${r}$ of this peak decreases for increasing ${\chi}_I$ as well as the degree of ordering. For the highest ${\chi}_I$ a depletion region leading to a minimum after the peak is pronounced. To obtain a measure of the local orientational ordering effects we investigate the orientational correlation function $h({r})$ as shown in Fig. \[G2thm2\_01\] for the same 3 values of ${\chi}_I$. The first two positive peaks and the first negative peak can be explained with the drawings in the bottom of Fig. \[ma2t\_initshot2\_gesamt\]. The first positive peak is due to a side-to-side alignment of two particles. Fig. \[ma2t\_shotafterfliptop\] shows several domains of side-to-side alignment. The first negative peak comes from an alignment where the particles are oriented perpendicular to each other and the second positive peak comes from a tip-to-tip alignment or second nearest neighbors of side-to-side orientation. The degree of translational and orientational ordering increases with increasing ${\chi}_I$.\
![Time development of ${g}({r})$ for ${\chi}_I\approx0.38$. The second peak is more pronounced at later timesteps. The particles reorder and the ordering increases. The reordering process is almost done after $4\cdot10^5$ timesteps.[]{data-label="Grm5_01tdev"}](g_a10x_tdev08.pdf){width="8cm"}
After having discussed the correlation functions we investigate the time development of $g(r)$ in order to understand the time development of the average domain size ${L(t)}$. Fig. \[Grm5\_01tdev\] shows ${g}({r})$ at different times between 10$^4$ and 10$^6$ timesteps. The first peak decreases but at later times the following peaks are more pronounced. Thus, the degree of ordering increases. After $4\cdot10^5$ timesteps this development has almost come to an end. The reason for this remaining development is the particle reordering. The particles form domains where they align parallel to each other. These domains become larger with time.
In this section we have shown shows that the presence of many particles at an interface leads to two additional timescales in the reordering. The first one is the rotation of the particle towards the interface. The particle rotates towards its final orientation parallel to the interface. For lower values of ${\chi}_I$ this process does not depend on ${\chi}_I$ and is not different from the single particle adsorption. For larger values of ${\chi}_I$ the time needed to come to its final orientation increases. Hydrodynamic as well as excluded volume effects become more important. Above a critical value not every particle reaches its ‘final’ orientation. The reordering of $h$ (corresponding to $g$ in Fig. \[Grm5\_01tdev\]) can also be observed. The first 2 peaks get more pronounced after several $10^5$ timesteps as compared to the state after 10$^4$ timesteps shown in Fig. \[G2thm2\_01\].
Particle ensembles at a spherical interface
===========================================
In the previous chapter the behavior of particle ensembles at a flat interface was discussed. However, in emulsions the interfaces are generally not flat. Pickering emulsions usually have (approximately) spherical droplets and a bijel has an even more complicated structure of curved interface. The simplest realization of a curved interface is a single droplet and as such is studied in this section.
The simulated system is periodic and each period has a size of $L_S=256$ lattice units. The droplet radius and the number of adsorbed particles are chosen to be $R_D=0.6L_S\approx76.8$ and 600, respectively. In the beginning of the simulation the particles are placed orthogonal to the local interface tangential plane. As we have seen already for the case of flat interfaces the particles flip to an orientation parallel to this tangential plane. This state is shown in Fig. \[snapshot\_droplet\] after $2\cdot10^5$ timesteps.
![Snapshot of a particle ensemble at a spherical interface after $2\cdot10^5$ timesteps.[]{data-label="snapshot_droplet"}](bright-rendered-lauf01_t00200000-1040493992_Droplet.png){width="8cm"}
A preliminary comparison between flat and spherical interfaces has already been given in our previous contribution [@Krueger2012c]. The time development of ${S}$ is shown in Fig. 11(a) in Ref. [@Krueger2012c]. It has been found that the influence of the interface curvature on the flipping process is larger than the influence of the particle coverage. The time needed for the particles to flip is about a factor two smaller in the case of the curved interface.
![Time development of the order parameter $g(r)$ for particles adsorbed at a spherical interface.[]{data-label="Grm5_droplet"}](g09_Droplet.pdf){width="8cm"}
Here, we investigate the particle correlation function (see [Eq. (\[eq:gd\])]{}) for the particle ensemble. Fig. \[Grm5\_droplet\] shows ${g}$ for ${\chi}_I\approx0.27$ at three different times. After 10$^4$ timesteps it is still close to the correlation function of the initial condition. After 10$^5$ timesteps some changes can be seen. The first peak is reduced but the second peak is more pronounced. There is no substantial change between $1\cdot 10^5$ and $2\cdot 10^5$ timesteps. Compared to the state at 10$^5$ timesteps the correlation function shows pronounced peaks at longer distances from the particle (about $6 R_p$). The particles mostly reorder during the first 10$^5$ timesteps since at later times only minor changes in the particle order can be observed. Similar to the case of flat interfaces that was discussed in the previous section, the particle ensemble forms domains where the particles are ordered in a nematic fashion. The peaks in the correlation function are more pronounced in the case of droplets than in the case of a flat interface. The reason is given by the capillary interactions between the particles which are much stronger in the case of curved interfaces. In particular, non-zero capillary interactions persist between spheroids even in the case of neutrally wetting particles.
The time development of ${g}$ at the droplet as discussed in this section differs from the behavior in the case of a flat interface. For the droplet, ${g}$ arrives at its final structure after about 10$^5$ timesteps whereas at the flat interface about four times more as many steps are required. In addition, for flat interfaces, ${g}$ only shows one or two peaks (depending on ${\chi}_I$), while for the particle covered droplet five peaks are found due to a larger range of ordering of the particles. This is a result of the stronger capillary interactions between the particles due to the interface curvature.
We can understand one of the additional timescales with the behavior of the ellipsoidal particles at a single droplet. The particles reorder and it can be shown that this leads to a small deviation of the shape of the droplet which is (almost) exactly spherical in the beginning [@Kim2008a]. A change of the interface shape caused by reordering of anisotropic particles leads to a change of $L(t)$. The reordering of particle ensembles at flat as well as spherical interfaces takes of the order of $10^5$ timesteps. This reordering takes place in idealized systems with constant interfaces which do not change their shape considerably. In real emulsions, however, the interface geometry changes substantially during their formation. For example, two droplets of a Pickering emulsion can coalesce. After this unification the particle ordering starts a new. This explains the fact that the additional timescale we find in our emulsions is of the order of several $10^6$ timesteps.
Conclusion
==========
In this article we have investigated the dynamics of the formation of Pickering emulsions and bijels stabilized by ellipsoidal particles. In contrast to emulsions stabilized by spherical particles, spheroids cause the average time dependent droplet or domain size to slowly [*decrease*]{} even after very long simulation times corresponding to several million simulation timesteps. The additional timescales related to this effect have been investigated by detailed studies of simple model systems. At first, the adsorption of single ellipsoidal particles was shown to happen on a comparably short timescale ($\approx 10^4$ timesteps). Second, many particle ensembles at flat interfaces, however, might require substantially more time in case of sufficiently densely packed interfaces. Here, local reordering effects induced by hydrodynamic interactions and interface rearrangements prevent the system from attaining a steady state and add a further timescale to the emulsion formation ($\approx 10^5$ timesteps). Third, this reordering is pronounced in the case of curved interfaces, where the movement of the particles leads to interface deformations and capillary interactions. During the formation of an emulsion, droplets might coalesce (Pickering emulsions) or domains might merge (bijels). After such an event the particles at the interface have to rearrange in order to adhere to the new interface structure. Due to this, the local reordering is practically being “restarted” leading to an overall increase of the interfacial area on a timescale of at least several $10^6$ timesteps. With the nanoscale resolution chosen above, this corresponds to physical times of the order of $10^{-5}s$.
Our findings provide relevant insight in the dynamics of emulsion formation which is generally difficult to investigate experimentally due to the required high temporal resolution of the measurement method and limited optical transparency of the experimental system. It is well known that in general particle-stabilized emulsions are not thermodynamically stable and therefore the involved fluids will always phase separate – even if this might take several months. Anisotropic particles, however, provide properties which might allow the generation of emulsions that are stable on substantially longer timescales. This is due to the continuous reordering of the particles at liquid interfaces which leads to an increase in interfacial area and as such counteracts the thermodynamically driven reduction of interface area.
Financial support is greatly acknowledged from NWO/STW (Vidi grant 10787 of J. Harting) and FOM/Shell IPP (09iPOG14 - “Detection and guidance of nanoparticles for enhanced oil recovery”). We thank the Jülich Supercomputing Centre, SARA Amsterdam, and HLRS Stuttgart for computing resources. J. de Graaf, M. Dijkstra, and R. van Roij are kindly acknowledged for fruitful discussions.
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abstract: 'We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study modules for Heisenberg algebras and we classify irreducible modules satisfying certain conditions and obtain a complete reducibility theorem.'
---
§
**
\[section\] \[thm\][Proposition]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Example]{} \[thm\][Lemma]{} \[thm\][Remark]{} \[thm\][Definition]{} \[thm\][Hypothesis]{} addtoreset[equation]{}[section]{}
[**On vertex algebras and their modules associated with even lattices**]{}
[Haisheng Li[^1] and Qing Wang[^2]\
Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102]{}
Introduction
============
Let $L$ be any nondegenerate even lattice in the sense that $L$ is a finite-rank free abelian group equipped with a symmetric $\Z$-bilinear form $\<\cdot,\cdot\>$ such that $\<\alpha,\alpha\>\in
2\Z$ for $\alpha\in L$. Associated to $L$ there exists a canonical vertex algebra $V_{L}$ ([@b86], [@flm]). These vertex algebras were originated from the explicit vertex-operator realizations of the basic modules for affine Kac-Moody Lie algebras and they form an important class in the theory of vertex algebras. For vertex algebras $V_{L}$, irreducible modules were classified in [@dong1] (cf. [@dlm-reg], [@ll]) where Dong proved that every irreducible $V_{L}$-module is isomorphic to one of those constructed in [@flm].
The vertex algebras $V_{L}$ were built up by using certain infinite-dimensional Heisenberg (Lie) algebras and their modules. For a nondegenerate even lattice $L$, set $\frak{h}=\C\otimes_{\Z}L$ and extend the $\Z$-bilinear form on $L$ to a nondegenerate symmetric $\C$-bilinear form $\<\cdot,\cdot\>$ on $\frak{h}$. Associated to the pair $(\frak{h},\<\cdot,\cdot\>)$, there is an affine Lie algebra $\hat{\frak{h}}=\frak{h}\otimes
\C[t,t^{-1}]\oplus \C {\bf k}$, whose subalgebra $\hat{\frak{h}}_{*}=\sum_{n\ne 0}\frak{h}(n)+\C {\bf k}$ is a Heisenberg algebra, where $\frak{h}(m)=\frak{h}\otimes t^{m}$ for $m\in \Z$. For each $\alpha\in \frak{h}=\frak{h}^{*}$, there is a canonical irreducible $\hat{\frak{h}}$-module $M(1,\alpha)$ on which ${\bf k}$ acts as identity and $h(0)$ acts as scalar $\<\alpha,h\>$ for $h\in \frak{h}$. The associated vertex algebra $V_{L}$ is built on the direct sum of the non-isomorphic irreducible $\hat{\frak{h}}$-modules $M(1,\alpha)$ for $\alpha\in L$, where $M(1,0)$ is a vertex subalgebra with $M(1,\alpha)$ as irreducible modules. Note that the vertex algebra $M(1,0)$ can also be constructed independently by starting with the finite-dimensional vector space $\frak{h}$ equipped with a nondegenerate symmetric bilinear form $\<\cdot,\cdot\>$, without referring to the lattice $L$. All of these motivated the work [@lx].
The main goal of [@lx] was to characterize the family of vertex algebras $V_{L}$. With vertex algebras $V_{L}$ as models, a class $\mathcal{A}$ of simple vertex algebras was formulated, where each $V$ in $\mathcal{A}$ contains $M(1,0)$ associated to some $\frak{h}$ as a vertex subalgebra and is a direct sum of some non-isomorphic irreducible $M(1,0)$-modules $M(1,\alpha)$ with $\alpha\in
\frak{h}$. It was proved therein that each $V$ in $\mathcal{A}$ is a direct sum of $M(1,\alpha)$ with $\alpha\in L$ for some additive subgroup $L$ of $\frak{h}$ such that $\<\alpha,\alpha\>\in 2\Z$ for $\alpha\in L$ and such that there exists a function (normalized $2$-cocycle) $\varepsilon: L\times L\rightarrow \C^{\times}$ satisfying the condition that $$\begin{aligned}
& &\varepsilon(\alpha,0)=\varepsilon(0,\alpha)=1,\\
& &\varepsilon(\alpha,\beta+\gamma)\varepsilon(\beta,\gamma)
=\varepsilon(\alpha+\beta,\gamma)\varepsilon(\alpha,\beta),\\
& &\varepsilon(\alpha,\beta)\varepsilon(\beta,\alpha)^{-1}
=(-1)^{\<\alpha,\beta\>} \ \ \ \ \mbox{ for }\alpha,\beta\in L.\end{aligned}$$ Such a vertex algebra $V$ was denoted therein by $V_{(\frak{h},L)}$. (However, the existence and uniqueness of the desired vertex algebra $V_{(\frak{h},L)}$ was neglected.) If the vertex algebra $V$ is finitely generated, one can readily see that $L$ is finitely generated, so that $L$ is free of finite rank (as $L$ is torsion-free). In this case, $L$ is a (finite rank) possibly degenerate even lattice.
In another work [@bdt], motivated by a certain connection between vertex algebras and toroidal Lie algebras, Berman, Dong and Tan studied modules for a class of vertex algebras which may be identified as $V_{(\frak{h},L)}$ with $\<\cdot,\cdot\>|_{L}=0$ and $\dim \frak{h}=2{\rm rank} (L)$ (finite). Among other results, they constructed and classified $\N$-graded irreducible modules.
In this current paper, we study vertex algebra $V_{(\frak{h},L)}$ and its modules associated with a general pair $(\frak{h},L)$ from the point of view of vertex algebra extensions. Specifically, we make use of the vertex algebra $M(1,0)$ and its modules to construct vertex algebra $V_{(\frak{h},L)}$ and its modules. This idea goes back to the so-called simple current extension of vertex operator algebras ([@li-phys], [@li-ext], [@dlm-simple], [@li-extaffine]). For the vertex algebra $V_{L}$ with $L$ a nondegenerate even lattice, irreducible modules have been constructed before in [@flm] and [@ll] with quite different methods. Our treatment here is further different from those of [@flm] and [@ll].
It is known (cf. [@ll]) that on every $\hat{\frak{h}}$-module $W$ of level $1$, which is [*restricted*]{} in the sense that for every $w\in W$, $\frak{h}(n)w=0$ for $n$ sufficiently large, there exists a unique module structure for vertex algebra $M(1,0)$, extending the action of $\hat{\frak{h}}$ in a certain canonical way and that every module for vertex algebra $M(1,0)$ is naturally a restricted $\hat{\frak{h}}$-module of level $1$. This leads us to study irreducible restricted modules for Heisenberg algebras, e.g., $\hat{\frak{h}}_{*}$. It is well known that Heisenberg algebras have canonical realizations on polynomial algebras by differential (annihilation) operators and left multiplication (creation) operators. For a Heisenberg algebra with a fixed nonzero level, there is a very nice module category in which there is only one irreducible module up to equivalence and each module is completely reducible; see [@lw2] ([@flm], Theorem 1.7.3), [@kac1]. In the case with the Heisenberg algebra $\hat{\frak{h}}_{*}$, the modules in this category are restricted and $\frak{h}(n)$ are locally nilpotent for $n\ge 1$. In this paper, we study a bigger category ${\mathcal{F}}$ of restricted modules on which $\frak{h}(n)$ are locally finite for $n\ge 1$. It turns out that ${\mathcal{F}}$ is also very nice in the sense that each module is completely reducible and each irreducible representation can be explicitly realized by using differential operators and left multiplication operators on a space of exponential functions and polynomials. In contrast to the old case, in the category ${\mathcal{F}}$ there are infinitely many unequivalent irreducible objects. We also study and construct certain irreducible restricted modules outside the category ${\mathcal{F}}$.
For a general pair $(\frak{h},L)$ with $\frak{h}\ne {\rm span}(L)$, the vertex algebra $V_{(\frak{h},L)}$ like $M(1,0)$ has irreducible modules other than those constructed from the canonical realization of Heisenberg algebras on polynomial algebras. We construct and classify irreducible $V_{(\frak{h},L)}$-modules in terms of irreducible restricted $\hat{\frak{h}}$-modules of a certain type. For the construction of irreducible modules, we use the idea from [@li-ext] and [@dlm-simple] and for the classification of irreducible modules, we use certain important ideas (and results) from [@dong1] and [@dlm-reg]. In the case that $\<\cdot,\cdot\>|_{L}=0$ we obtain certain irreducible $V_{(\frak{h},L)}$-modules other than those constructed by Berman, Dong and Tan in [@bdt]. Furthermore, we give another construction of the vertex algebra $V_{(\frak{h},L)}$ by using a certain affine Lie algebra.
This paper is organized in the following manner. In Section 2, we study certain categories of modules for Heisenberg algebras and we classify all the irreducible objects. In Section 3, we study the modules for vertex algebras associated with even lattices. In Section 4, we give a characterization of vertex algebras in terms of an affine Lie algebra.
Modules for Heisenberg Lie algebras
===================================
In this section we first associate a Heisenberg Lie algebra $\H_{I}$ to each nonempty set $I$ and we then construct and classify irreducible $\H_{I}$-modules of certain types. Especially, we define a category ${\mathcal{F}}$ and we establish a complete reducibility theorem, which generalizes a theorem of [@flm] (Section 1.7).
First, we start with introducing the Heisenberg Lie algebra $\H_{I}$. Let $I$ be any nonempty set, which is fixed throughout this section. Let $\H_{I}$ be the Heisenberg Lie algebra with a basis $\{{\bf p}_{i},{\bf q}_{i},{\bf k}\;|\; i\in I\}$ (over $\C$) and with Lie bracket relations $$[{\bf k},\H_{I}]=0,\ \
[{\bf p}_{i},{\bf p}_{j}]=0=[{\bf q}_{i},{\bf q}_{j}]\ \ \mbox{ and
}\ \ [{\bf p}_{i},{\bf q}_{j}]=\delta_{ij}{\bf k}\ \ \ \mbox{ for }
i,j\in I.$$ An $\H_{I}$-module $W$ is said to be of [*level*]{} $\ell\in \C$ if ${\bf k}$ acts as scalar $\ell$.
Let $W$ be any irreducible $\H_{I}$-module of level $1$. For $i\in I$, ${\bf p}_{i}$ has an eigenvector in $W$ if and only if ${\bf p}_{i}$ acts locally finitely. Furthermore, ${\bf
p}_{i}w=0$ for some nonzero $w\in W$ if and only if ${\bf p}_{i}$ acts locally nilpotently. The same assertions hold with ${\bf
p}_{i}$ replaced by ${\bf q}_{i}$.
It is clear that if ${\bf p}_{i}$ is locally finite, ${\bf p}_{i}$ has an eigenvector, as the scalar field is $\C$. Now, assume that ${\bf p}_{i}$ has an eigenvector $w_{0}$ of eigenvalue $\lambda$. Set $$A=\sum_{j\in I,\; j\ne i}(\C {\bf p}_{j}+\C {\bf q}_{j})+\C {\bf
p}_{i}+\C {\bf k},$$ a Lie subalgebra of $\H_{I}$. We have $\H_{I}=\C {\bf q}_{i}\oplus A$. Since $W$ is an irreducible $\H_{I}$-module, $W=U(\H_{I})w_{0}=U(A)\C[{\bf q}_{i}]w_{0}$. It follows from induction that for any nonnegative integer $k$, $\sum_{n=0}^{k}\C{\bf q}_{i}^{n}w_{0}$ is a finite-dimensional subspace which is closed under the action of ${\bf p}_{i}$. Furthermore, for any $a\in U(A)$, as $[{\bf p}_{i},A]=0$, $\sum_{n=0}^{k}a\C {\bf q}_{i}^{n}w_{0}$ is a finite-dimensional subspace which is closed under the action of ${\bf p}_{i}$. Now, it follows that ${\bf p}_{i}$ is locally finite. For the nilpotent case, it is also clear.
Let $W$ be an irreducible $\H_{I}$-module of level $1$ and let $i\in I$. If ${\bf p}_{i}$ is locally finite on $W$, then any nonzero ${\bf q}_{i}$-stable subspace is infinite-dimensional. In particular, ${\bf q}_{i}$ is not locally finite. The same assertion holds when ${\bf p}_{i}$ and ${\bf q}_{i}$ are exchanged.
Suppose that there exists a nonzero finite-dimensional subspace $U$ which is ${\bf q}_{i}$-stable. Then there exists $0\ne w\in U$ such that ${\bf q}_{i}w=\alpha w$ for some $\alpha\in \C$. Using induction we get $${\bf q}_{i}{\bf p}_{i}^{n}w=(\alpha {\bf p}_{i}^{n}-n{\bf p}_{i}^{n-1})w
\ \ \ \mbox{ for }n\ge 0.$$ It follows that ${\bf p}_{i}^{n}w\ne 0$ for $n\ge 0$. As ${\bf p}_{i}$ is locally finite, $\C[{\bf
p}_{i}]w$ is finite-dimensional. Let $n=\dim \C[{\bf p}_{i}]w\ge 1$. Then $w, {\bf p}_{i}w,\dots, {\bf p}_{i}^{n-1}w$ are linearly independent and $${\bf p}_{i}^{n}w=c_{n-1}{\bf
p}_{i}^{n-1}w+c_{n-2}{\bf p}_{i}^{n-2}w+\cdots + c_{0}w$$ for some $c_{0},\dots,c_{n-1}\in \C$. Applying ${\bf q}_{i}$ we get $$\alpha {\bf p}_{i}^{n}w-n{\bf p}_{i}^{n-1}w
=\alpha c_{n-1}{\bf p}_{i}^{n-1}w+u$$ for some $u\in
\sum_{r=0}^{n-2}\C{\bf p}_{i}^{r}w$. Then we obtain $$n{\bf p}_{i}^{n-1}w=\alpha (c_{n-2}{\bf p}_{i}^{n-2}w+\cdots + c_{0}w) -u.$$ This is a contradiction as $w, {\bf p}_{i}w,\dots, {\bf
p}_{i}^{n-1}w$ are linearly independent.
For our study in the next section on modules for vertex algebras associated to Heisenberg Lie algebras we are interested in $\H_{I}$-modules $W$ such that for every $w\in W$, ${\bf p}_{i}w=0$ for all but finitely many $i\in I$. For convenience, we call such an $\H_{I}$-module a [*restricted*]{} module. In view of Lemmas \[lnilp-finite\] and \[lpiqi\], if $W$ is an irreducible restricted $\H_{I}$-module, then $I_{W}^{\bf p}$ is a cofinite subset of $I$ and $I_{W}^{\bf p}\cap I_{W}^{\bf q}=\emptyset$.
Let $x_{i}$ $(i\in I)$ be mutually commuting independent formal variables. Denote by $F^{0}(I,\C)$ the set of functions ${\bf \mu}:
I\rightarrow \C$ such that $\mu_{i}=0$ for all but finitely many $i\in I$. For ${\bf \lambda}\in F^{0}(I,\C)$, set $${\bf \lambda} {\bf x}
=\sum_{i\in I}\lambda_{i}x_{i}\in \C[x_{i}\;|\; i\in I].$$ Furthermore, set $$M(1,{\bf \lambda)}=e^{\bf \lambda x}\C[x_{i}\;|\; i\in I],$$ a space of functions in $x_{i}$ $(i\in I)$. It is clear that $M(1,{\bf \lambda})$ is an $\H_{I}$-module with ${\bf p}_{i}$ acting as $\partial /\partial x_{i}$, ${\bf q}_{i}$ as (the left multiplication of) $x_{i}$, and ${\bf k}$ as identity.
The $\H_{I}$-module $M(1,{\bf \lambda})$ belongs to the category $\mathcal{F}$ and is irreducible. For ${\bf
\lambda},{\bf \mu}\in F^{0}(I,\C)$, $M(1,{\bf \lambda})\simeq
M(1,{\bf \mu})$ if and only if ${\bf \lambda}={\bf \mu}$.
Let $S$ be a finite subset of $I$ such that $\lambda_{i}=0$ for all $i\in I-S$. Then $\partial e^{\bf \lambda
x}/\partial x_{i}=0$ for all $i\in I-S$. On the other hand, for any polynomial $f({\bf x})$ in $x_{j}$ $(j\in I)$, $\partial f/\partial
x_{i}=0$ for all but finitely many $i\in I$. It follows that for any $w\in M(1,\lambda)$, ${\bf p}_{i}w=0$ for all but finitely many $i\in I$. This shows that $M(1,{\bf \lambda})$ is a restricted $\H_{I}$-module. Let $i\in I$ and let $f$ be any polynomial in $x_{j}$ with $j\ne i$. For any $r\in \N$, $\sum_{n=0}^{r}fe^{\bf
\lambda x}\C x_{i}^{n}$ is closed under the action of ${\bf p}_{i}$. It follows that ${\bf p}_{i}$ acts locally finitely on $M(1,{\bf
\lambda})$. This proves that $M(1,{\bf \lambda})$ belongs to the category $\mathcal{F}$. For any function ${\bf \mu}\in F^{0}(I,\C)$, set $$M(1,{\bf \lambda})_{\bf \mu}=\{ w\in M(1,{\bf \lambda})\;|\; {\bf
p}_{i}w=\mu_{i}w\ \ \ \mbox{ for }i\in I\}.$$ With ${\bf p}_{i}$ acting as $\partial/\partial x_{i}$ $(i\in I)$, for $f({\bf x})\in
\C[x_{i}\;|\; i\in I]$, $e^{\bf \lambda x}f({\bf x})\in M(1,{\bf
\lambda})_{\bf \mu}$ if and only if $$\partial f/\partial x_{i}=(\mu_{i}-\lambda_{i})f\ \ \ \mbox{ for all
}i\in I.$$ Furthermore, if $f({\bf x})\ne 0$, we have $f\in \C$ and $\lambda_{i}=\mu_{i}$ for all $i\in I$. Thus $M(1,{\bf
\lambda})_{\bf \mu}=0$ for ${\bf \mu}\ne {\bf \lambda}$ and $M(1,{\bf \lambda})_{\bf \lambda}=\C e^{\bf \lambda x}$. It now follows immediately that $M(1,{\bf \lambda})$ is an irreducible $\H_{I}$-module. From this proof, it is evident that for ${\bf
\lambda},{\bf \mu}\in F^{0}(I,\C)$, $M(1,{\bf \lambda})\simeq
M(1,{\bf \mu})$ if and only if ${\bf \lambda}={\bf \mu}$.
For every nonzero $\H_{I}$-module $W$ in the category ${\mathcal{F}}$, there exist a nonzero vector $w_{0}\in W$ and a function $\lambda\in F^{0}(I,\C)$ such that ${\bf
p}_{i}w_{0}=\lambda_{i}w_{0}$ for $i\in I$. Furthermore, the submodule generated by $w_{0}$ is isomorphic to $M(1,\lambda)$.
Let $0\ne w\in W$. From definition, ${\bf
p}_{i}w=0$ for $i\in I-S$, where $S$ is a finite subset of $I$. Set $\Omega'=\{ u\in W\;|\; {\bf p}_{i}u=0\ \ \mbox{ for }i\in I-S\}$. Then $w\in \Omega'\ne 0$. With $[{\bf p}_{r},{\bf p}_{s}]=0$ for $r,s\in I$, $\Omega'$ is closed under the actions of ${\bf p}_{j}$ for $j\in S$. With ${\bf p}_{j}$ $(j\in S)$ locally finite and mutually commuting, there exists $0\ne w_{0}\in \Omega'$ such that ${\bf p}_{j}w_{0}=\lambda_{j}w_{0}$ for $j\in S$ with $\lambda_{j}\in \C$. We also have ${\bf p}_{i}w_{0}=0$ for $i\in
I-S$. Defining $\lambda_{j}=0$ for $j\in I-S$ gives rise to a function ${\bf \lambda}\in F^{0}(I,\C)$. In view of the P-B-W theorem, we have $U(\H_{I})w_{0}=\C[{\bf q}_{i}\;|\; i\in I]w_{0}$. Define a linear map $$\psi: M(1,{\bf \lambda})\rightarrow U(\H_{I})w_{0},\ \ e^{\bf \lambda x}f({\bf
x})\mapsto f({\bf q})w_{0}.$$ It is straightforward to show that $\psi$ is an $\H_{I}$-module isomorphism.
The following is a generalization of a theorem of [@flm] (Theorem 1.7.3):
Every irreducible $\H_{I}$-module in the category $\mathcal{F}$ is isomorphic to $M(1,{\bf \lambda})$ for some ${\bf
\lambda}\in F^{0}(I,\C)$ and every $\H_{I}$-module in the category $\mathcal{F}$ is completely reducible.
The first assertion follows immediately from Lemma \[levery\]. For complete reducibility we first consider a special case. Let ${\mathcal{N}}$ be the subcategory of ${\mathcal{F}}$, consisting of restricted $\H_{I}$-modules of level $1$ on which ${\bf p}_{i}$ acts locally nilpotently for every $i\in I$. We see that for $\lambda\in
F^{0}(I,\C)$, $M(1,\lambda)$ is in ${\mathcal{N}}$ if and only if $\lambda=0$. Then $M(1,0)$ is the only irreducible module in ${\mathcal{N}}$ up to equivalence. With Lemma \[levery\], the same proof of Theorem 1.7.3 of [@flm] shows that every $\H_{I}$-module in ${\mathcal{N}}$ is completely reducible.
We now prove that every $\H_{I}$-module $W$ in $\mathcal{F}$ is completely reducible. For $\mu\in F^{0}(I,\C)$, set $$W_{\mu}=\{ w\in W\;|\; ({\bf p}_{i}-\mu_{i})^{r}w=0
\ \ \mbox{ for }i\in I \mbox{ and for some $r\ge 0$ depending on
}i\}.$$ Note that $$({\bf p}_{i}-\mu_{i})^{r}{\bf p}_{j}={\bf p}_{j}({\bf
p}_{i}-\mu_{i})^{r}, \ \ \ \ ({\bf p}_{i}-\mu_{i})^{r}{\bf
q}_{i}={\bf q}_{i}({\bf p}_{i}-\mu_{i})^{r}+r({\bf
p}_{i}-\mu_{i})^{r-1}{\bf k}$$ for $i,j\in I,\; r\ge 0$. It follows that $W_{\mu}$ is an $\H_{I}$-submodule of $W$. For any $w\in W$, since ${\bf p}_{i}w=0$ for all but finitely many $i\in I$ and since each ${\bf p}_{i}$ is locally finite, $\C[{\bf p}_{i}\;|\; i\in I]w$ is finite-dimensional. From this we have $W=\oplus_{\mu\in
F^{0}(I,\C)}W_{\mu}$. Now it suffices to prove that for each $\mu\in
F^{0}(I,\C)$, $W_{\mu}$ is completely reducible. Let $\mu\in
F^{0}(I,\C)$ be fixed. Then ${\bf p}_{i}-\mu_{i}$ is locally nilpotent on $W_{\mu}$ for $i\in I$. Define a linear endomorphism $\theta_{\mu}$ of $\H_{I}$ by $$\theta_{\mu}({\bf k})={\bf k},\ \ \theta_{\mu}({\bf p}_{i})={\bf
p}_{i}-\mu_{i} \ \mbox{ and }\ \theta_{\mu}({\bf q}_{i})={\bf q}_{i}
\ \ \ \mbox{ for }i\in I.$$ Clearly, $\theta_{\mu}$ is a Lie algebra automorphism of $\H_{I}$. Let $\rho: \H_{I}\rightarrow \End W_{\mu}$ denote the Lie algebra homomorphism for the $\H_{I}$-module $W_{\mu}$. Then $\rho \circ\theta_{\mu}$ is a representation of $\H_{I}$ on $W_{\mu}$ in the category ${\mathcal{N}}$, which is completely reducible. Consequently, $\rho$ is completely reducible.
Let $I_{1}$ be any cofinite subset of $I$ and let ${\bf \lambda}\in
F^{0}(I,\C)$. Define an action of $\H_{I}$ on the space $e^{\bf
\lambda x}\C[x_{i}\;|\; i\in I]$ by $$\begin{aligned}
{\bf p}_{i}\mapsto \begin{cases}\partial/\partial x_{i} &
\mbox{ for }i\in I_{1}\\
x_{i} &\mbox{ for }i\in I-I_{1},
\end{cases}
\ \ \ \
{\bf q}_{i}\mapsto \begin{cases}
x_{i} &\mbox{ for }i\in I_{1}\\
-\partial/\partial x_{i}&\mbox{ for }i\in I-I_{1}.
\end{cases}\end{aligned}$$ This makes $e^{\bf \lambda x}\C[x_{i}\;|\; i\in I]$ an $\H_{I}$-module of level $1$, which we denote by $M(1,I_{1},\lambda)$. In fact, this $\H_{I}$-module $M(1,I_{1},\lambda)$ is a twisting of the $\H_{I}$-module $M(1,{\bf
\lambda})$ by an automorphism of $\H_{I}$. Let $\theta_{I_{1}}$ be the linear endomorphism of $\H_{I}$ defined by $$\begin{aligned}
& &\theta_{I_{1}}({\bf p}_{i})={\bf p}_{i},\ \ \ \
\theta_{I_{1}}({\bf q}_{i})={\bf
q}_{i}\ \ \ \ \ \mbox{ for }i\in I_{1},\\
& &\theta_{I_{1}}({\bf p}_{i})={\bf q}_{i},\ \ \ \
\theta_{I_{1}}({\bf q}_{i})=-{\bf p}_{i}\ \ \ \mbox{ for }i\in
I-I_{1}.\end{aligned}$$ It is evident that $\theta_{I_{1}}$ is a Lie algebra automorphism and that $M(1,I_{1},{\bf \lambda})$ is isomorphic to the twisting of $M(1,{\bf \lambda})$ by the automorphism $\theta_{I_{1}}$. Consequently, $M(1,I_{1},{\bf \lambda})$ is an irreducible $\H_{I}$-module of level $1$. Furthermore, using the automorphism $\theta_{I_{1}}$ and Theorem \[theisenberg\] we immediately have:
Let $W$ be an irreducible restricted $\H_{I}$-module of level $1$ such that $I_{W}^{\bf p}\cup I_{W}^{\bf
q}=I$. Then $W$ is isomorphic to $M(1,I_{W}^{\bf p},{\bf \lambda})$ for some ${\bf \lambda}\in F^{0}(I,\C)$.
Next, we continue to investigate general irreducible restricted $\H_{I}$-modules of level $1$. Let $I=I_{0}\cup I_{1}$ be any disjoint decomposition of $I$ with $I_{0}\ne \emptyset$ and $I_{1}\ne \emptyset$. We view $\H_{I_{0}}$ and $\H_{I_{1}}$ as subalgebras of $\H_{I}$ in the obvious way. Note that the two subalgebras are commuting. Let $W_{0}$ and $W_{1}$ be irreducible modules of level $1$ for $\H_{I_{0}}$ and $\H_{I_{1}}$, respectively. Then $W_{0}\otimes W_{1}$ is naturally an $\H_{I}$-module of level $1$. Furthermore, if either $I_{0}$ or $I_{1}$ is countable, $W_{0}\otimes W_{1}$ is an irreducible $\H_{I}$-module. (Notice that either $\H_{I_{0}}$ or $\H_{I_{1}}$ is of countable dimension, which implies that either $\End_{\H_{I_{0}}}(W_{0})=\C$ or $\End_{\H_{I_{1}}}(W_{1})=\C$.)
Assume that $I$ is countable. Let $W$ be an irreducible restricted $\H_{I}$-module of level $1$ such that $I\ne
I(W)=I_{W}^{\bf p}\cup I_{W}^{\bf q}$. Set $I_{0}=I-I(W)$. Then $W\simeq M(1,I_{W}^{\bf p},{\bf \lambda})\otimes U$, where $M(1,I_{W}^{\bf p},{\bf \lambda})$ is an $\H_{I(W)}$-module for some ${\bf \lambda}\in F^{0}(I(W),\C)$ and $U$ is an irreducible $\H_{I_{0}}$-module such that $(I_{0})_{U}^{\bf p}=\emptyset
=(I_{0})_{U}^{\bf q}$.
We view $\H_{I(W)}$ and $\H_{I_{0}}$ as subalgebras of $\H_{I}$ in the obvious way. Note that the two subalgebras are commuting. From Theorem \[theisenberg\], $W$ as an $\H_{I(W)}$-module is completely reducible. Let $W_{1}$ be an irreducible $\H_{I(W)}$-submodule of $W$. Since $W$ is an irreducible $\H_{I}$-module, we have $$W=U(\H_{I})W_{1}=U(\H_{I_{0}})W_{1}.$$ As $[\H_{I(W)},\H_{I_{0}}]=0$, it follows that $W$ as an $\H_{I(W)}$-module is a sum of irreducible modules isomorphic to $W_{1}$. With $I$ countable, $W_{1}$ is of countable dimension, so that $\End_{\H_{I(W)}}(W_{1})=\C$. It follows that $W=W_{0}\otimes
W_{1}$, where $W_{0}=\Hom_{\H_{I(W)}}(W_{1},W)$ is naturally an $\H_{I_{0}}$-module. Furthermore, $W_{0} $ is an irreducible $\H_{I_{0}}$-module. In view of Lemma \[lnilp-finite\] we have $(I_{0})_{U}^{\bf p}=\emptyset =(I_{0})_{U}^{\bf q}$.
Having established Proposition \[pdecomposition\], we next study irreducible $\H_{I_{0}}$-modules $U$ of level $1$ with $(I_{0})_{U}^{\bf p}=(I_{0})_{U}^{\bf q}=\emptyset$ with $I_{0}$ a finite subset of $I$. For the rest of this section, let $I_{0}$ be a nonempty finite subset of $I$. For ${\bf \mu}\in F^{0}(I_{0},\C)$, set $${\bf x^{\mu}}=\prod_{i\in I_{0}}x_{i}^{\mu_{i}},$$ where as before $x_{i}$ $(i\in I_{0})$ are mutually commuting independent formal variables. Set $$\C_{*}\{x_{i}\;|\; i\in I_{0}\}=\coprod_{{\bf \mu}\in F^{0}(I_{0},\C)}\C {\bf x}^{\bf \mu},$$ a vector space. With ${\bf p}_{i}$ acting as $\partial/\partial
x_{i}$ (the formal partial differential operator), ${\bf q}_{i}$ as $x_{i}$, and ${\bf k}$ as identity, the space $\C_{*} \{ x_{i}\;|\;
i\in I_{0}\}$ becomes an $\H_{I_{0}}$-module of level $1$.
Denote by $F^{0}_{*}(I_{0},\C)$ the subset of $F^{0}(I_{0},\C)$, consisting functions ${\bf \mu}: I_{0}\rightarrow \C$ such that $\mu_{i}\notin \Z$ for all $i\in I_{0}$. For ${\bf \mu}\in
F^{0}_{*}(I_{0},\C)$, set $$M_{*}[{\bf \mu}]=\coprod_{\lambda\in F(I_{0},\Z)}\C {\bf x}^{\mu+\lambda}
={\bf x^{\mu}}\C[x_{i}^{\pm 1}\;|\; i\in I_{0}],$$ where $F(I_{0},\Z)$ denotes the set of integer-valued functions on $I_{0}$. It is clear that $M_{*}[\bf \mu]$ is an $\H_{I_{0}}$-submodule of $\C_{*}\{ x_{i}\;|\; i\in I_{0}\}$.
We say that an $\H_{I_{0}}$-module $W$ of level $1$ satisfies [*Condition $C_{I_{0}}$*]{} if for every $i\in I_{0}$, ${\bf q}_{i}{\bf
p}_{i}$ is semisimple and ${\bf p}_{i}w\ne 0$, ${\bf q}_{i}w\ne 0$ for any $0\ne w\in W$. In terms of this notion we have:
The $\H_{I_{0}}$-module $M_{*}[{\bf \mu}]$ is irreducible and satisfies Condition $C_{I_{0}}$. On the other hand, every irreducible $\H_{I_{0}}$-module $W$ satisfying Condition $C_{I_{0}}$ is isomorphic to $M_{*}[{\bf \mu}]$ for some ${\bf
\mu}\in F^{0}_{*}(I_{0},\C))$.
For any function ${\bf \beta}\in F^{0}(I_{0},\C)$, we have $$\left(x_{i}{\partial\over\partial x_{i}}\right){\bf
x^{\beta}}=\beta_{i} {\bf x^{\beta}}\ \ \ \mbox{ for }i\in I_{0}.$$ Then ${\bf q}_{i}{\bf p}_{i}$ for all $i\in I_{0}$ act semisimply on $M_{*}[\mu]$. We also have $$\frac{1}{r!}\left({\partial\over \partial x_{i}}\right)^{r}{\bf x}^{\bf \beta}
=\binom{\beta_{i}}{r}{\bf x}^{\bf \beta}x_{i}^{-r}\ \ \ \mbox{ for
}r\in \N,$$ where if $\beta_{i}\notin \Z$, $\binom{\beta_{i}}{r}\ne
0$ for $r\in \N$. It is then clear that $M_{*}[{\bf \mu}]$ is an irreducible $\H_{I_{0}}$-module. It is also clear that $${\bf p}_{i}w\ne 0,\ \ \ {\bf q}_{i}w\ne 0
\ \ \ \mbox{ for }i\in I_{0},\; 0\ne w\in M_{*}[\mu].$$ This proves that $M_{*}[\mu]$ satisfies Condition $C_{I_{0}}$.
Let $W$ be an irreducible $\H_{I_{0}}$-module of level $1$, satisfying Condition $C_{I_{0}}$. As ${\bf q}_{j}{\bf p}_{j}$ for $j\in I_{0}$ are mutually commuting and are semisimple on $W$ by assumption, there exists $0\ne w_{0}\in W$ such that $${\bf q}_{j}{\bf p}_{j}w_{0}=\beta_{j}w_{0}\ \ \ \mbox{ for }j\in
I_{0},$$ where $\beta_{j}\in \C$. This gives rise to a function ${\bf \beta}\in F^{0}(I_{0},\C)$. We claim that ${\bf \beta}\in F^{0}_{*}(I_{0},\C)$, i.e., $\beta_{j}\notin
\Z$ for all $j\in I_{0}$. For $j\in I_{0},\; n\in \N$, we have $$\begin{aligned}
& &({\bf q}_{j}{\bf p}_{j}){\bf p}_{j}^{n}w_{0}={\bf p}_{j}^{n}({\bf
q}_{j}{\bf p}_{j})w_{0}-n{\bf p}_{j}^{n}w_{0}=(\beta_{j}-n){\bf
p}_{j}^{n}w_{0},\\
& &({\bf q}_{j}{\bf p}_{j}){\bf q}_{j}^{n}w_{0}={\bf q}_{j}^{n}({\bf
q}_{j}{\bf p}_{j})w_{0}+n{\bf q}_{j}^{n}w_{0}=(\beta_{j}+n){\bf
q}_{j}^{n}w_{0}.\end{aligned}$$ Since ${\bf p}_{j}w\ne 0$ and ${\bf q}_{j}w\ne 0$ for any $0\ne w\in
W$, it follows that $\beta_{j}-n\ne 0$ and $\beta_{j}+n\ne 0$ for $n\ge 0$, proving $\beta_{j}\notin \Z$. Thus, $\beta\in
F_{*}^{0}(I_{0},\C)$.
Since $W$ is an irreducible $\H_{I_{0}}$-module, we have $W=U(\H_{I_{0}})w_{0}$. Define a linear map $\psi: M_{*}[{\bf
\beta}]\rightarrow W$ by $$\psi({\bf x}^{\bf \beta+m})={\bf q}^{\bf m}w_{0}$$ for ${\bf m}: I_{0}\rightarrow \Z$, where ${\bf q}^{\bf
m}=\prod_{i\in I_{0}}{\bf q}_{i}^{m_{i}}$ with ${\bf
q}_{i}^{m_{i}}={\bf q}_{i}^{m_{i}}$ for $m_{i}\ge 0$ and $${\bf
q}_{i}^{m_{i}}=\frac{1}{\binom{\beta_{i}}{-m_{i}}}\frac{1}{(-m_{i})!}{\bf
p}_{i}^{-m_{i}}$$ for $m_{i}<0$. One can show that $\psi$ is an $\H_{I_{0}}$-module isomorphism.
Vertex algebras $V_{(\frak{h},L)}$ and their modules
====================================================
In this section, we study vertex algebras associated with possibly degenerate even lattices. This slightly generalizes the vertex algebras associated with nondegenerate even lattices. We construct and classify irreducible modules satisfying certain conditions for the vertex algebras.
First, we start with vertex operator algebras associated with (infinite-dimensional) Heisenberg Lie algebras. Let $\frak{h}$ be a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form $\<\cdot,\cdot\>$, which is fixed throughout this section. Viewing $\frak{h}$ as an abelian Lie algebra equipped with $\<\cdot,\cdot\>$ as a nondegenerate symmetric invariant bilinear form, we have an affine Lie algebra $$\hat{\frak{h}}=\frak{h}\otimes \C[t,t^{-1}]\oplus \C {\bf k},$$ where ${\bf k}$ is central and for $u,v\in \frak{h},\; m,n\in \Z$, $$\begin{aligned}
[u(m),v(n)]=m\<u,v\>\delta_{m+n,0}{\bf k},\end{aligned}$$ where $h(n)=h\otimes t^{n}$ for $h\in \frak{h},\; n\in \Z$. For $n\in \Z$, set $$\frak{h}(n)=\{ h(n)\;|\; h\in \frak{h}\}\subset \hat{\frak{h}}$$ and we set $$\hat{\frak{h}}_{*}=\coprod_{n\ne 0}\frak{h}(n) +\C {\bf k}.$$ Note that $\frak{h}(0)$ is a central subalgebra, $\hat{\frak{h}}_{*}$ is a Heisenberg algebra, and $$\hat{\frak{h}}=\hat{\frak{h}}_{*}\oplus \frak{h}(0),$$ a Lie algebra product decomposition.
Let $W$ be any irreducible $\hat{\frak{h}}$-module. Then ${\bf k}$ and $h(0)$ for every $h\in \frak{h}$ act as scalars on $W$ and $W$ as an $\hat{\frak{h}}_{*}$-module is also irreducible.
Since $W$ is an irreducible $\hat{\frak{h}}$-module and $\frak{h}$ is finite-dimensional, $W$ is of countable dimension over $\C$. By a version of Schur lemma, we have $\End_{\hat{\frak{h}}}
(W)=\C$. With ${\bf k}$ and $h(0)$ being central in $\hat{\frak{h}}$, they must act as scalars. It is now clear that $W$ as an $\hat{\frak{h}}_{*}$-module is also irreducible.
An $\hat{\frak{h}}$-module $W$ is said to be of [*level*]{} $\ell$ in $\C$ if ${\bf k}$ acts on $W$ as scalar $\ell$, and an $\hat{\frak{h}}$-module $W$ is said to be [*restricted*]{} if for every $w\in W$, $\frak{h}(n)w=0$ for $n$ sufficiently large. For any $\ell\in \C$, denote by $\C_{\ell}$ the $1$-dimensional $\frak{h}\otimes \C[t]+\C {\bf k}$-module $\C$ with $\frak{h}\otimes
\C[t]$ acting trivially and with ${\bf k}$ acting as scalar $\ell$. Form the induced module $$V_{\hat{\frak{h}}}(\ell,0)
=U(\hat{\frak{h}})\otimes_{U(\frak{h}\otimes \C[t]+\C {\bf k})}
\C_{\ell}.$$ Set ${\bf 1}=1\otimes 1\in V_{\hat{\frak{h}}}(\ell,0)$ and identify $\frak{h}$ as a subspace of $V_{\hat{\frak{h}}}(\ell,0)$ through the linear map $h\mapsto
h(-1){\bf 1}$. It is well known now (cf. [@ll]) that there exists a unique vertex algebra structure on $V_{\hat{\frak{h}}}(\ell,0)$ with ${\bf 1}$ as the vacuum vector and with $Y(h,x)=h(x)=\sum_{n\in \Z}h(n)x^{-n-1}$ for $h\in \frak{h}$. Furthermore, for every nonzero $\ell$, $V_{\hat{\frak{h}}}(\ell,0)$ is a vertex operator algebra of central charge $d=\dim \frak{h}$ with conformal vector $$\omega=\frac{1}{2\ell}\sum_{r=1}^{d}h^{(r)}(-1)h^{(r)}(-1){\bf 1},$$ where $\{h^{(1)},\dots,h^{(d)}\}$ is any orthonormal basis of $\frak{h}$. It is also known (cf. [@ll]) that every module $(W,Y_{W})$ for vertex algebra $V_{\hat{\frak{h}}}(\ell,0)$ is a restricted $\hat{\frak{h}}$-module of level $\ell$ with $h(x)$ acting as $Y_{W}(h,x)$ for $h\in \frak{h}$ and the set of $V_{\hat{\frak{h}}}(\ell,0)$-submodules of $W$ coincides with the set of $\hat{\frak{h}}$-submodules of $W$. On the other hand, on every restricted $\hat{\frak{h}}$-module $W$ of level $\ell$, there is a unique module structure $Y_{W}$ for vertex algebra $V_{\hat{\frak{h}}}(\ell,0)$ with $Y_{W}(h,x)=h(x)$ for $h\in
\frak{h}$.
It is known (cf. [@ll]) that vertex algebras $V_{\hat{\frak{h}}}(\ell,0)$ for $\ell\ne 0$ are all isomorphic. In view of this, we restrict ourselves to the vertex operator algebra $V_{\hat{\frak{h}}}(1,0)$. In literature, the vertex operator algebra $V_{\hat{\frak{h}}}(1,0)$ is also often denoted by $M(1)$. In view of Lemma \[lnilp\], classifying irreducible modules for vertex algebra $M(1)$ amounts to classifying irreducible restricted $\hat{\frak{h}}_{*}$-modules of level $1$. One way to apply the results of Section 2 is to fix an orthonormal basis $\{h^{(1)},\dots,h^{(d)}\}$ of $\frak{h}$ and set $$I=\{(r,n)\;|\; 1\le r\le d,\; n\ge 1\}.$$ Then identify the Heisenberg Lie algebra $\hat{\frak{h}}_{*}$ with $\H_{I}$ by $${\bf p}_{(r,n)}=\frac{1}{n}h^{(r)}(n),\ \ {\bf
q}_{(r,n)}=h^{(r)}(-n)\ \ \ \mbox{ for }(r,n)\in I.$$
In view of Theorem \[theisenberg\] we immediately have:
Let $W$ be a restricted $\hat{\frak{h}}$-module of level $1$ such that $\frak{h}(0)$ is semisimple and $\frak{h}(n)$ is locally finite for $n\ge 1$. Then $W$ is completely reducible.
For $\alpha\in \frak{h}$, we set ([@lw], [@flm]) $$\begin{aligned}
E^{\pm}(\alpha,x) =\exp \left(\sum_{n=1}^{\infty}\frac{\alpha (\pm
n)}{\pm n}x^{\mp n}\right).\end{aligned}$$ The following are the fundamental properties: $$\begin{aligned}
& &E^{\pm}(0,x)=1,\\
& &E^{\pm}(\alpha,x_{1})E^{\pm }(\beta,x_{2})= E^{\pm
}(\beta,x_{2})E^{\pm}(\alpha,x_{1}),\\
& &E^{+}(\alpha,x_{1})E^{-}(\beta,x_{2})
=\left(1-\frac{x_{2}}{x_{1}}\right)^{\<\alpha,\beta\>}E^{-}(\beta,x_{2})E^{+}(\alpha,x_{1}),\\
& &E^{\pm}(\alpha,x)E^{\pm }(\beta,x)= E^{\pm }(\alpha+\beta,x).\end{aligned}$$
Set $$\begin{aligned}
\bar{\Delta}(\alpha,x)=(-x)^{\alpha(0)}\exp\left(\sum_{n=1}^{\infty}\frac{\alpha(n)}{-n}
(-x)^{-n}\right)=(-x)^{\alpha(0)}E^{+}(-\alpha,-x).\end{aligned}$$ This is a well defined element of $(\End W)[[x,x^{-1}]]$ for any module $W$ for vertex algebra $M(1)$, on which $\alpha(0)$ acts semisimply with only integer eigenvalues and $\alpha(n)$ for $n\ge
1$ act locally nilpotently. The following are immediate consequences: $$\begin{aligned}
& &\bar{\Delta}(0,x)=1,\\
& &\bar{\Delta}(\alpha,x_{1})\bar{\Delta}(\beta,x_{2})
=\bar{\Delta}(\beta,x_{2})\bar{\Delta}(\alpha,x_{1}),\\
& &\bar{\Delta}(\alpha,x)\bar{\Delta}(\beta,x)
=\bar{\Delta}(\alpha+\beta,x).\end{aligned}$$
For the rest of this section, we assume that $L$ is an additive subgroup of $\frak{h}$ such that $$\begin{aligned}
\label{eevenL}
\<\alpha,\alpha\>\in 2\Z \ \ \ \mbox{ for }\alpha \in L,\end{aligned}$$ equipped with a function $\varepsilon: L\times L\rightarrow
\C^{\times}$, satisfying the condition that $$\begin{aligned}
& &\varepsilon(\alpha,0)=\varepsilon(0,\alpha)=1,\label{enormalization}\\
& &\varepsilon(\alpha,\beta+\gamma)\varepsilon(\beta,\gamma)
=\varepsilon(\alpha+\beta,\gamma)\varepsilon(\alpha,\beta),\\
& &\varepsilon(\alpha,\beta)\varepsilon(\beta,\alpha)^{-1}
=(-1)^{\<\alpha,\beta\>}\label{e2cocycle}\end{aligned}$$ for $\alpha,\beta,\gamma\in L$.
Set $$\begin{aligned}
V_{(\frak{h},L)}=\C[L]\otimes M(1).\end{aligned}$$ For $\alpha,\beta\in L,\; u,v\in M(1)$, we define $$\begin{aligned}
\label{eexplicitformula1}
& &Y(e^{\alpha}\otimes u,x)(e^{\beta}\otimes v)\nonumber\\
&=&\varepsilon(\alpha,\beta)e^{\alpha+\beta}\otimes
x^{\<\alpha,\beta\>}E^{-}(-\alpha,x)Y(\bar{\Delta}(\beta,x)u,x)\bar{\Delta}(\alpha,-x)v.\end{aligned}$$ In particular, for $h\in \frak{h}$, $$\begin{aligned}
& &Y(e^{0}\otimes h,x)(e^{\beta}\otimes v)=e^{\beta}\otimes
(\<\beta,h\>x^{-1}+Y(h,x))v,
\label{eY-formula-special-h}\\
& & Y(e^{\alpha}\otimes {\bf
1},x)(e^{\beta}\otimes
v)=\varepsilon(\alpha,\beta)e^{\alpha+\beta}\otimes
x^{\<\alpha,\beta\>}E^{-}(-\alpha,x)E^{+}(-\alpha,x)v\ \ \ \ \
\label{eY-formula-special}\end{aligned}$$ as $\bar{\Delta}(\beta,x)h=h+\<\beta,h\>{\bf 1}x^{-1}$ and $\alpha(0)=0$ on $M(1)$. One can prove that the quadruple $(V_{(\frak{h},L)},Y, e^{0}\otimes {\bf 1})$ carries the structure of a vertex algebra, by using a theorem of [@fkrw] and [@mp]. Here, we give a uniform treatment for both the vertex algebras and their modules.
Let $U$ be an $\hat{\frak{h}}$-module of level $1$, satisfying Condition $C_{L}$. Set $$\begin{aligned}
V_{(\frak{h},L)}(U)=\C[L]\otimes U.\end{aligned}$$ For $\alpha,\beta\in L,\; v\in M(1),\; w\in U$, we define $$\begin{aligned}
\label{eformula-def}
& &Y_{W}(e^{\alpha}\otimes v,x)(e^{\beta}\otimes
w)\nonumber\\
&=&\varepsilon(\alpha,\beta)e^{\alpha+\beta}\otimes
x^{\<\alpha,\beta\>}E^{-}(-\alpha,x)Y_{U}(\bar{\Delta}(\beta,x)v,x)\bar{\Delta}(\alpha,-x)w.\end{aligned}$$ Then $(V_{(\frak{h},L)},Y, e^{0}\otimes {\bf 1})$ carries the structure of a vertex algebra with $M(1)$ as a vertex subalgebra and $(V_{(\frak{h},L)}(U),Y_{W})$ carries the structure of a $V_{(\frak{h},L)}$-module. Furthermore, if $U$ is irreducible, so is $V_{(\frak{h},L)}(U)$.
First, notice that the $\hat{\frak{h}}$-module $M(1)$ satisfies all the assumptions on $U$ and that $V_{(\frak{h},L)}=V_{(\frak{h},L)}(U)$ with $U=M(1)$ where $Y_{W}$ coincides with $Y$. Second, for any $\alpha,\beta\in L,\; v\in
M(1),\; w\in U$, we have $$Y_{W}(e^{\alpha}\otimes v,x)(e^{\beta}\otimes w)\in V_{(\frak{h},L)}(U)((x)).$$ Third, $Y_{W}(e^{0}\otimes {\bf 1},x)=1$ and when $U=M(1)$ we also have $$Y(e^{\alpha}\otimes u,x)(e^{0}\otimes {\bf 1})
=e^{\alpha}\otimes E^{-}(-\alpha,x)Y(u,x){\bf 1}\in
V_{(\frak{h},L)}[[x]]$$ with $$\lim_{x\rightarrow
0}Y(e^{\alpha}\otimes u,x)(e^{0}\otimes {\bf 1})=e^{\alpha}\otimes
u.$$ Next, we show that the Jacobi identity holds. Let $\alpha,\beta,\gamma\in L,\;
u,v\in M(1),\; w\in U$. Using definition and formulas we have $$\begin{aligned}
& &Y_{W}(e^{\alpha}\otimes u,x_{1})Y_{W}(e^{\beta}\otimes
v,x_{2})(e^{\gamma}\otimes w)\\
&=&Y_{W}(e^{\alpha}\otimes
u,x_{1})\left(\varepsilon(\beta,\gamma)e^{\beta+\gamma}\otimes
x_{2}^{\<\beta,\gamma\>}
E^{-}(-\beta,x_{2})Y_{U}(\bar{\Delta}(\gamma,x_{2})v,x_{2})\bar{\Delta}(\beta,-x_{2})w\right)\\
&=&\varepsilon(\alpha,\beta+\gamma)\varepsilon(\beta,\gamma)e^{\alpha+\beta+\gamma}\otimes
x_{1}^{\<\alpha,\beta+\gamma\>}x_{2}^{\<\beta,\gamma\>}
E^{-}(-\alpha,x_{1})\cdot\\
& &\cdot Y_{U}(\bar{\Delta}(\beta+\gamma,x_{1})u,x_{1})\bar{\Delta}(\alpha,-x_{1})
E^{-}(-\beta,x_{2})Y_{U}(\bar{\Delta}(\gamma,x_{2})v,x_{2})\bar{\Delta}(\beta,-x_{2})w\\
&=&\varepsilon(\alpha,\beta+\gamma)\varepsilon(\beta,\gamma)e^{\alpha+\beta+\gamma}\otimes
x_{1}^{\<\alpha,\beta+\gamma\>}x_{2}^{\<\beta,\gamma\>}
E^{-}(-\alpha,x_{1}) Y_{U}(\bar{\Delta}(\beta+\gamma,x_{1})u,x_{1})\cdot\\
& &\cdot (1-x_{2}/x_{1})^{\<\alpha,\beta\>}
E^{-}(-\beta,x_{2})Y_{U}(\bar{\Delta}(\alpha,-x_{1}+x_{2})\bar{\Delta}(\gamma,x_{2})v,x_{2})
\bar{\Delta}(\alpha,-x_{1})\bar{\Delta}(\beta,-x_{2})w\\\end{aligned}$$ By (\[e2.4\]) we have $$\begin{aligned}
& &Y_{U}(\bar{\Delta}(\beta+\gamma,x_{1})u,x_{1})E^{-}(-\beta,x_{2})\\
&=&E^{-}(-\beta,x_{2})
Y_{U}(\bar{\Delta}(\beta,x_{1}-x_{2})\bar{\Delta}(-\beta,x_{1})\bar{\Delta}(\beta+\gamma,x_{1})u,x_{1})\\
&=&E^{-}(-\beta,x_{2})
Y_{U}(\bar{\Delta}(\beta,x_{1}-x_{2})\bar{\Delta}(\gamma,x_{1})u,x_{1}).\end{aligned}$$ Then $$\begin{aligned}
& &Y_{W}(e^{\alpha}\otimes u,x_{1})Y_{W}(e^{\beta}\otimes
v,x_{2})(e^{\gamma}\otimes w)\\
&=&\varepsilon(\alpha,\beta+\gamma)\varepsilon(\beta,\gamma)e^{\alpha+\beta+\gamma}\otimes
x_{1}^{\<\alpha,\gamma\>}x_{2}^{\<\beta,\gamma\>}(x_{1}-x_{2})^{\<\alpha,\beta\>}
E^{-}(-\alpha,x_{1})E^{-}(-\beta,x_{2})\cdot\\
& &\cdot
Y_{U}(\bar{\Delta}(\beta,x_{1}-x_{2})\bar{\Delta}(\gamma,x_{1})u,x_{1})
Y_{U}(\bar{\Delta}(\alpha,-x_{1}+x_{2})\bar{\Delta}(\gamma,x_{2})v,x_{2})
\bar{\Delta}(\alpha,-x_{1})\bar{\Delta}(\beta,-x_{2})w.\end{aligned}$$ This also shows $$\begin{aligned}
& &Y_{W}(e^{\beta}\otimes v,x_{2})Y_{W}(e^{\alpha}\otimes u,x_{1})
(e^{\gamma}\otimes w)\\
&=&\varepsilon(\beta,\alpha+\gamma)\varepsilon(\alpha,\gamma)e^{\alpha+\beta+\gamma}\otimes
x_{1}^{\<\alpha,\gamma\>}x_{2}^{\<\beta,\gamma\>}(x_{2}-x_{1})^{\<\beta,\alpha\>}
E^{-}(-\alpha,x_{1})E^{-}(-\beta,x_{2})\cdot\\
& &\cdot
Y_{U}(\bar{\Delta}(\alpha,x_{2}-x_{1})\bar{\Delta}(\gamma,x_{2})v,x_{2})
Y_{U}(\bar{\Delta}(\beta,-x_{2}+x_{1})\bar{\Delta}(\gamma,x_{1})u,x_{1})
\bar{\Delta}(\alpha,-x_{1})\bar{\Delta}(\beta,-x_{2})w\\
&=&\varepsilon(\alpha,\beta)\varepsilon(\alpha+\beta,\gamma)e^{\alpha+\beta+\gamma}\otimes
x_{1}^{\<\alpha,\gamma\>}x_{2}^{\<\beta,\gamma\>}(-x_{2}+x_{1})^{\<\beta,\alpha\>}
E^{-}(-\alpha,x_{1})E^{-}(-\beta,x_{2})\cdot\\
& &\cdot
Y_{U}(\bar{\Delta}(\alpha,x_{2}-x_{1})\bar{\Delta}(\gamma,x_{2})v,x_{2})
Y_{U}(\bar{\Delta}(\beta,-x_{2}+x_{1})\bar{\Delta}(\gamma,x_{1})u,x_{1})
\bar{\Delta}(\alpha,-x_{1})\bar{\Delta}(\beta,-x_{2})w,\end{aligned}$$ where $\varepsilon(\beta,\alpha+\gamma)\varepsilon(\alpha,\gamma)
=(-1)^{\<\alpha,\beta\>}\varepsilon(\alpha,\beta)\varepsilon(\alpha+\beta,\gamma)$. On the other hand, we have $$\begin{aligned}
& &Y_{W}\left(Y(e^{\alpha}\otimes u,x_{0})(e^{\beta}\otimes
v),x_{2}\right)(e^{\gamma}\otimes w)\\
&=&Y_{W}(\varepsilon(\alpha,\beta)e^{\alpha+\beta}\otimes
x_{0}^{\<\alpha,\beta\>}
E^{-}(-\alpha,x_{0})Y(\bar{\Delta}(\beta,x_{0})u,x_{0})\bar{\Delta}(\alpha,-x_{0})v,x_{2})
(e^{\gamma}\otimes w)\\
&=&\varepsilon(\alpha+\beta,\gamma)\varepsilon(\alpha,\beta)e^{\alpha+\beta+\gamma}\otimes
x_{0}^{\<\alpha,\beta\>}x_{2}^{\<\alpha+\beta,\gamma\>}E^{-}(-\alpha-\beta,x_{2})\cdot\\
& &\cdot Y_{U}\left(\bar{\Delta}(\gamma,x_{2})E^{-}(-\alpha,x_{0})
Y(\bar{\Delta}(\beta,x_{0})u,x_{0})\bar{\Delta}(\alpha,-x_{0})v,x_{2}\right)
\bar{\Delta}(\alpha+\beta,-x_{2})w\\
&=&\varepsilon(\alpha+\beta,\gamma)\varepsilon(\alpha,\beta)e^{\alpha+\beta+\gamma}\otimes
x_{0}^{\<\alpha,\beta\>}x_{2}^{\<\alpha+\beta,\gamma\>}E^{-}(-\alpha-\beta,x_{2})
(1+x_{0}/x_{2})^{\<\alpha,\gamma\>}\cdot\\
& &\cdot Y_{U}\left(E^{-}(-\alpha,x_{0})\Delta(\gamma,x_{2})
Y(\bar{\Delta}(\beta,x_{0})u,x_{0})\bar{\Delta}(\alpha,-x_{0})v,x_{2}\right)
\bar{\Delta}(\alpha+\beta,-x_{2})w\\
&=&\varepsilon(\alpha+\beta,\gamma)\varepsilon(\alpha,\beta)e^{\alpha+\beta+\gamma}\otimes
x_{0}^{\<\alpha,\beta\>}x_{2}^{\<\beta,\gamma\>}E^{-}(-\alpha-\beta,x_{2})
(x_{2}+x_{0})^{\<\alpha,\gamma\>}\cdot\\
& &\cdot Y_{U}\left(E^{-}(-\alpha,x_{0})
Y(\bar{\Delta}(\gamma,x_{2}+x_{0})\bar{\Delta}(\beta,x_{0})u,x_{0})
\bar{\Delta}(\gamma,x_{2})\bar{\Delta}(\alpha,-x_{0})v,x_{2}\right)
\cdot\\
& &\ \ \ \ \cdot \bar{\Delta}(\alpha+\beta,-x_{2})w\\
&=&\varepsilon(\alpha+\beta,\gamma)\varepsilon(\alpha,\beta)e^{\alpha+\beta+\gamma}\otimes
x_{0}^{\<\alpha,\beta\>}x_{2}^{\<\beta,\gamma\>}E^{-}(-\alpha-\beta,x_{2})
(x_{2}+x_{0})^{\<\alpha,\gamma\>}
E^{-}(-\alpha,x_{0}+x_{2})\cdot\\
& &\cdot E^{-}(\alpha,x_{2})Y_{U}\left(
Y(\bar{\Delta}(\gamma,x_{2}+x_{0})\bar{\Delta}(\beta,x_{0})u,x_{0})
\bar{\Delta}(\gamma,x_{2})\bar{\Delta}(\alpha,-x_{0})v,x_{2}\right)
\cdot\\
& &\ \ \ \ \cdot
\bar{\Delta}(-\alpha,-x_{2})\bar{\Delta}(\alpha,-x_{2}-x_{0})
\bar{\Delta}(\alpha+\beta,-x_{2})w\\
&=&\varepsilon(\alpha+\beta,\gamma)\varepsilon(\alpha,\beta)e^{\alpha+\beta+\gamma}\otimes
x_{0}^{\<\alpha,\beta\>}x_{2}^{\<\beta,\gamma\>}(x_{2}+x_{0})^{\<\alpha,\gamma\>}E^{-}(-\beta,x_{2})
E^{-}(-\alpha,x_{0}+x_{2})\cdot\\
& &\cdot Y_{U}\left(
Y(\bar{\Delta}(\gamma,x_{2}+x_{0})\bar{\Delta}(\beta,x_{0})u,x_{0})
\bar{\Delta}(\gamma,x_{2})\bar{\Delta}(\alpha,-x_{0})v,x_{2}\right)
\cdot\\
& &\ \ \ \ \cdot \bar{\Delta}(\alpha,-x_{2}-x_{0})
\bar{\Delta}(\beta,-x_{2})w.\end{aligned}$$ Set $$\begin{aligned}
& &A=\bar{\Delta}(\gamma,x_{1})\bar{\Delta}(\beta,x_{0})u, \
\ B=\bar{\Delta}(\gamma,x_{2})\bar{\Delta}(\alpha,-x_{0})v\in
M(1)[x_{1}^{\pm
1},x_{2}^{\pm 1},x_{0}^{\pm 1}],\\
& &\ \ \ \ \ C=\bar{\Delta}(\alpha,-x_{1})
\bar{\Delta}(\beta,-x_{2})w\in U[x_{1}^{\pm 1},x_{2}^{\pm 1}].\end{aligned}$$ We have the following Jacobi identity $$\begin{aligned}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y_{U}(A,x_{1})Y_{U}(B,x_{2})C
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{U}(B,x_{2})Y_{U}(A,x_{1})C\ \ \\
&&\ \ \ =x_{1}^{-1}\delta\left(\frac{x_{2}+x_{0}}{x_{1}}\right)
Y_{U}(Y(A,x_{0})B,x_{2})C.\end{aligned}$$ Using this and delta-function substitutions we obtain the Jacobi identity as desired. This proves the assertions on vertex algebra structure and module structure. It follows from (\[eY-formula-special-h\]) that $M(1)$ is a vertex subalgebra with $e^{0}\otimes v$ identified with $v$ for $v\in M(1)$. Furthermore, by (\[eformula-def\]) (cf. (\[eY-formula-special-h\])) we have $$\begin{aligned}
& &h_{W}(n)(e^{\beta}\otimes w)=e^{\beta}\otimes h(n)w\ \ \ \mbox{ for }n\ne 0,\\
& &h_{W}(0)(e^{\beta}\otimes w)=e^{\beta}\otimes (\<\beta,h\>+h(0))w\end{aligned}$$ for $h\in \frak{h},\; \alpha,\beta\in L$, where $Y_{W}(h,x)=\sum_{n\in \Z}h_{W}(n)x^{-n-1}$.
Assume that $U$ is an irreducible $\hat{\frak{h}}$-module. In view of Lemma \[lnilp\], there exists $\lambda\in \frak{h}$ such that $h(0)$ acts as scalar $\<\lambda,h\>$ on $U$ for $h\in \frak{h}$. Furthermore, $h(0)$ acts on $\C e^{\alpha}\otimes U$ as scalar $\<\lambda+\alpha,h\>$ for $\alpha\in L$. Consequently, $\C
e^{\alpha}\otimes U$ for $\alpha\in L$ are non-isomorphic irreducible $\hat{\frak{h}}$-submodules of $V_{(\frak{h},L)}(U)$. Furthermore, for $\alpha,\beta\in L,\; w\in U$ we have $$Y_{W}(e^{\alpha},x)(e^{\beta}\otimes w)
=\varepsilon(\alpha,\beta)e^{\alpha+\beta}\otimes
x^{\<\alpha,\beta\>}E^{-}(-\alpha,x)E^{+}(-\alpha,x)x^{\alpha(0)}w,$$ which gives $$x^{-\alpha(0)}E^{+}(\alpha,x)\left(E^{-}(\alpha,x)Y_{W}(e^{\alpha},x)(e^{\beta}\otimes
w)\right) =\varepsilon(\alpha,\beta)e^{\alpha+\beta}\otimes
x^{\<\alpha,\beta\>}w.$$ Then for every $\beta\in L$, $0\ne w\in U$, $e^{\beta}\otimes w$ generates $V_{(\frak{h},L)}(U)$ as an $V_{(\frak{h},L)}$-module. It now follows that $V_{(\frak{h},L)}(U)$ is an irreducible $V_{(\frak{h},L)}$-module.
Let $W$ be any irreducible $V_{(\frak{h},L)}$-module. Then $W$ is an $\hat{\frak{h}}$-module with $h(x)=Y_{W}(h,x)$ for $h\in \frak{h}$, satisfying Condition $C_{L}$.
With $M(1)$ as a vertex subalgebra of $V_{(\frak{h},L)}$, $W$ is a module for vertex algebra $M(1)$, so that $W$ is a restricted $\hat{\frak{h}}$-module of level $1$. By Lemma 3.15 of [@dlm-reg], there exists a nonzero vector $w\in W$ such that $$\alpha(n)w=0\ \ \ \ \mbox{ for }\alpha\in L,\; n\ge 1.$$ For the rest of the proof we follow Dong’s arguments in [@dong1]. Let $\alpha$ be any nonzero element of $L$. As $V_{(\frak{h},L)}$ is simple, from \[DL\], we have $Y(e^{\alpha},x)w\ne 0$. Assume that $e^{\alpha}_{k}w\ne 0$ and $e^{\alpha}_{m}w=0$ for $m>k$. From the relation ${d\over dx}
e^{\alpha}(x)w=\alpha(x)e^{\alpha}(x)w$, extracting the coefficients of $x^{-k-2}$ we obtain $(-k-1)e^{\alpha}_{k}w=\alpha(0)e^{\alpha}_{k}w$. That is, $e^{\alpha}_{k}w$ is an eigenvector of $\alpha(0)$ with integer eigenvalue $-k-1$. We have $$[\alpha(m),h(n)]=m\<\alpha,h\>\delta_{m+n,0},\ \ \
[\alpha(m),e^{\beta}_{n}]=\<\alpha,\beta\>e^{\beta}_{m+n}$$ for $h\in \frak{h},\; \beta\in L,\; m,n\in \Z$. As $W$ is irreducible, $e^{\alpha}_{k}w$ generates $W$ by operators $h(m),\; e^{\beta}_{m}$ for $h\in \frak{h},\; \beta\in L,\; m\in \Z$. Then it follows that $\alpha(0)$ acts semisimply on $W$ with only integer eigenvalues and that $\alpha(n)$ for $n\ge 1$ act locally nilpotently.
Let $\{\alpha_{1},\dots,\alpha_{r}\}\subset L$ be such that $\{\alpha_{1},\dots,\alpha_{r}\}$ is a basis for the subspace $\C L$ of $\frak{h}$. Then extend $\{\alpha_{1},\dots,\alpha_{r}\}$ to a basis $\{ \alpha_{1},\dots,\alpha_{r},u_{1},\dots,u_{s}\}$ of $\frak{h}$. Let $\{\beta_{1},\dots,\beta_{r},v_{1},\dots,v_{s}\}$ be the dual basis. Set $$\begin{aligned}
L^{o}=\{\lambda\in \frak{h}\;|\; \<\alpha,\lambda\>\in \Z\ \ \
\mbox{ for all }\alpha\in L \}.\end{aligned}$$ Consider subalgebras of $\hat{\frak{h}}$: $$\begin{aligned}
{\mathcal{L}}_{0}&=&\sum_{i=1}^{r}\sum_{n\ge 1}(\C \alpha_{i}(n)+\C
\beta_{i}(-n))+\C {\bf k},\\
{\mathcal{L}}_{1}&=&\sum_{j=1}^{s}\sum_{n\ge 1}(\C u_{j}(n)+\C
v_{j}(-n))+\C {\bf k}.\end{aligned}$$ Then $\hat{\frak{h}}=({\mathcal{L}}_{0}+{\mathcal{L}}_{1})\oplus
\frak{h}(0)$ with ${\mathcal{L}}_{0}$, ${\mathcal{L}}_{1}$, and $\frak{h}(0)$ mutually commuting. If $\C L=\frak{h}$, we have ${\mathcal{L}}_{1}=\C {\bf k}$ and if $L=0$, we have ${\mathcal{L}}_{0}=\C {\bf k}$. Otherwise, ${\mathcal{L}}_{0}$ and ${\mathcal{L}}_{1}$ are infinite-dimensional Heisenberg algebras.
Let $M_{0}(1)$ $(=\C[\beta_{i}(-n)\;| \; 1\le i\le d,\; n\ge 1]$ as a vector space) denote the canonical irreducible ${\mathcal{L}}_{0}$-module of level $1$ on which $\alpha_{i}(n)$ acts locally nilpotent for $1\le i\le r,\; n\ge 1$. Let $U_{1}$ be an irreducible ${\mathcal{L}}_{1}$-module of level $1$ which is restricted in the sense that for every $w\in U_{1}$ and $1\le j\le
s$, $u_{j}(n)w=0$ for $n$ sufficiently large. Let $\lambda\in
L^{o}$ and denote by $\C e^{\lambda}$ (where $e^{\lambda}$ is just a symbol) the $1$-dimensional $\frak{h}(0)$-module with $h(0)$ acting as scalar $\<\lambda,h\>$ for $h\in \frak{h}$. Set $$M(1,\lambda,
U_{1})=\C e^{\lambda}\otimes U_{1}\otimes M_{0}(1).$$ Then $M(1,\lambda, U_{1})$ is an $\hat{\frak{h}}$-module of level $1$ with $\frak{h}(0)$ acting on $\C e^{\lambda}$, ${\mathcal{L}}_{1}$ acting on $U_{1}$, and ${\mathcal{L}}_{0}$ acting on $M_{0}(1)$.
The defined $\hat{\frak{h}}$-module $M(1,\lambda,U_{1})$ satisfies Condition $C_{L}$ and is irreducible. On the other hand, every irreducible $\hat{\frak{h}}$-module satisfying Condition $C_{L}$ is isomorphic to $M(1,\lambda, U_{1})$ for some $\lambda\in
L^{o}$ and some irreducible restricted ${\mathcal{L}}_{1}$-module $U_{1}$ of level $1$.
It is evident that $M(1,\lambda,U_{1})$ satisfies Condition $C_{L}$. Notice that the Schur lemma holds for the ${\mathcal{L}}_{0}$-module $M_{0}(1)$ as $M_{0}(1)$ is of countable dimension over $\C$. Then it follows that $M(1,\lambda,U_{1})$ is an irreducible $\hat{\frak{h}}$-module. Let $U$ be an irreducible $\hat{\frak{h}}$-module satisfying Condition $C_{L}$. By Theorem \[theisenberg\], $U$ viewed as an ${\mathcal{L}}_{0}$-module is completely reducible with each irreducible submodule isomorphic to $M_{0}(1)$. Consequently, $U=M_{0}(1)\otimes U^{1},$ where $U^{1}=\Hom_{\mathcal{L}_{0}}(M_{0}(1),U)$ is naturally an $({\mathcal{L}}_{1}+\frak{h}(0))$-module of level $1$. With $U$ an irreducible restricted $\hat{\frak{h}}$-module, $U^{1}$ irreducible and restricted. As $\frak{h}(0)$ is central in $\hat{\frak{h}}$ and commutes with ${\mathcal{L}}_{1}$, there exists $\lambda\in
\frak{h}$ such that $h(0)$ acts as scalar $\<\lambda,h\>$ on $U^{1}$ for $h\in \frak{h}$, and $U^{1}$ is an irreducible ${\mathcal{L}}_{1}$-module of level $1$. From Condition $C_{L}$, we have $\<\lambda,\alpha_{i}\>\in \Z$ for $1\le i\le r$. Thus $\lambda\in L^{o}.$ Taking $U_{1}=U^{1}$ viewed as an ${\mathcal{L}}_{1}$-module, we have $U\simeq M(1,\lambda,U_{1})$.
Let $\lambda\in L^{o}$ and $U_{1}$ an irreducible restricted ${\mathcal{L}}_{1}$-module of level $1$. In view of Proposition \[plast\] and Theorem \[tmain\], we have an irreducible $V_{(\frak{h},L)}$-module $V_{(\frak{h},L)}(U)$ with $U=M(1,\lambda,U_{1})$. We denote this module by $V_{(\frak{h},L)}(\lambda,U_{1})$, where $$V_{(\frak{h},L)}(\lambda,U_{1})=\C e^{\lambda}\otimes \C[L]\otimes
U_{1}\otimes M_{0}(1)$$ as a vector space. Set $${\frak{u}}(0)=\C u_{1}(0)\oplus \cdots \oplus \C u_{s}(0)\subset
\frak{h}(0)\subset \hat{\frak{h}}.$$
Let $W$ be an irreducible $V_{(\frak{h},L)}$-module on which $\frak{u}(0)$ acts semisimply. Then $W$ is isomorphic to $V_{(\frak{h},L)}(\lambda,U_{1})$ for some $\lambda\in L^{o}$ and for some irreducible restricted ${\mathcal{L}}_{1}$-module $U_{1}$ of level $1$.
For $\lambda\in \frak{h}$, set $$W_{\lambda}=\{ w\in W\;|\; h(0)w=\<\lambda,h\>w\ \ \mbox{ for
}h\in \frak{h}\}.$$ Because $\alpha_{1}(0),\dots,\alpha_{r}(0)$ are semisimple by Lemma \[lconverse\] and $\frak{u}(0)$ is assumed to be semisimple, we have $W=\coprod_{\lambda\in \frak{h}}W_{\lambda}$. As $[\frak{h}(0),\hat{\frak{h}}]=0$, $W_{\lambda}$ are $\hat{\frak{h}}$-submodules of $W$. For $\alpha\in L,\; h\in
\frak{h},\; v\in M(1)$, we have $$[h(0),Y_{W}(e^{\alpha}\otimes v,x)]=Y_{W}(h(0)(e^{\alpha}\otimes
v),x) =\<h,\alpha\>Y_{W}(e^{\alpha}\otimes v,x).$$ Consequently, $$\begin{aligned}
\label{especial-bracket}
Y_{W}(e^{\alpha}\otimes v,x)W_{\lambda}\subset
W_{\lambda+\alpha}[[x,x^{-1}]]\end{aligned}$$ for $\alpha\in L,\; v\in M(1),\; \lambda\in \frak{h}$. Suppose that $W_{\lambda_{0}}\ne 0$ and let $0\ne w\in W_{\lambda_{0}}$. By a result of [@dm] and [@li-thesis], the linear span of $\{
a_{m}w\;|\; a\in V_{(\frak{h},L)},\; m\in \Z\}$ is a $V_{(\frak{h},L)}$-submodule of $W$. Consequently, $$W={\rm span}\{ a_{m}w\;|\; a\in V_{(\frak{h},L)},\; m\in \Z\}.$$ Combining this with (\[especial-bracket\]) and the decomposition $W=\coprod_{\lambda\in \frak{h}}W_{\lambda}$, we get $$W_{\lambda_{0}}={\rm span}\{ v_{m}w\;|\; v\in M(1),\; m\in \Z\}
\ \ \mbox{ and }\ \ W=\coprod_{\alpha\in L}W_{\lambda_{0}+\alpha}.$$ It follows that $W_{\lambda_{0}}$ is an irreducible module for $M(1)$ viewed as a vertex algebra. As $\frak{h}$ generates $M(1)$ as a vertex algebra, $W_{\lambda_{0}}$ is an irreducible $\hat{\frak{h}}$-module. In view of Lemma \[lconverse\], $W_{\lambda_{0}}$ satisfies Condition $C_{L}$. We are going to prove that $W\simeq V_{(\frak{h},L)}(W_{\lambda_{0}})$.
Let $\alpha\in L$. We have $$\begin{aligned}
& &\frac{d}{dx}Y_{W}(e^{\alpha},x)=Y_{W}(L(-1)e^{\alpha},x)=
Y_{W}(\alpha(-1)e^{\alpha},x)\\
& &\ \ \ \ \
=\alpha(x)^{+}Y_{W}(e^{\alpha},x)+Y_{W}(e^{\alpha},x)\alpha(x)^{-},\end{aligned}$$ where $\alpha(x)^{+}=\sum_{n<0}\alpha(n)x^{-n-1}$ and $\alpha(x)^{-}=\sum_{n\ge 0}\alpha(n)x^{-n-1}$. We also have $\frac{d}{dx}E^{-}(\alpha,x)=-E^{-}(\alpha,x)\alpha(x)^{+}$, and $\frac{d}{dx}E^{+}(\alpha,x)x^{-\alpha(0)}=-\alpha(x)^{-}E^{+}(\alpha,x)x^{-\alpha(0)}$. Using these relations we obtain $$\frac{d}{dx}E^{-}(\alpha,x)Y_{W}(e^{\alpha},x)E^{+}(\alpha,x)x^{-\alpha(0)}=0.$$ Then we set $$E_{\alpha}=E^{-}(\alpha,x)Y_{W}(e^{\alpha},x)E^{+}(\alpha,x)x^{-\alpha(0)}\in \End _{\C}W.$$ Define a linear map $\psi:
V_{(\frak{h},L)}(W_{\lambda_{0}})\rightarrow W$ by $$\begin{aligned}
\psi( e^{\alpha}\otimes u)=E_{\alpha}u\ \ \ \mbox{ for }\alpha\in
L,\; u\in W_{\lambda_{0}}.\end{aligned}$$ We are going to prove that $\psi$ is a $V_{(\frak{h},L)}$-module isomorphism.
First, we establish some properties for $E_{\alpha}$ with $\alpha\in
L$. For $h\in \frak{h}$, we have $$\begin{aligned}
[h(0),E_{\alpha}]=E^{-}(\alpha,x)[h(0),Y_{W}(e^{\alpha},x)]E^{+}(\alpha,x)x^{-\alpha(0)}
=\<h,\alpha\>E_{\alpha}.\end{aligned}$$ As $h(i)e^{\alpha}=\delta_{i,0}\<h,\alpha\>e^{\alpha}$ for $i\ge 0$, we have $$[h(n),Y_{W}(e^{\alpha},x)]=\<h,\alpha\>x^{n}Y_{W}(e^{\alpha},x)
\ \ \ \ \mbox{ for }n\in \Z.$$ If $n>0$, from [@flm] (Proposition 4.1.1) we also have $$\begin{aligned}
& &[h(n),E^{-}(\alpha,x)]=-\<h,\alpha\>x^{n}E^{-}(\alpha,x),\\
& &[h(n),E^{+}(\alpha,x)]=0=[h(n),\alpha(0)].\end{aligned}$$ Then we get $[h(n),E_{\alpha}]=0$. Similarly, we have $[E_{\alpha},
h(n)]=0$ for $h\in \frak{h},\; n< 0$.
From the definition of $E_{\alpha}$ we have $$\begin{aligned}
\label{eYW}
Y_{W}(e^{\alpha},x)=E^{-}(-\alpha,x)E^{+}(-\alpha,x)E_{\alpha}x^{\alpha(0)}.\end{aligned}$$ For $\alpha,\beta\in L$, from (\[eY-formula-special\]) we have $$e^{\alpha}_{-\<\alpha,\beta\>-1}e^{\beta}=\varepsilon(\alpha,\beta)e^{\alpha+\beta}
\ \ \mbox{ and }\ \ e^{\alpha}_{m}e^{\beta}=0\ \ \mbox{ for }m\ge
-\<\alpha,\beta\>.$$ Let $0\ne w\in W_{\lambda_{0}}$ be such that $\gamma(n)w=0$ for $\gamma\in L,\; n\ge 1$. Using (\[eYW\]) we have $$e^{\gamma}_{-\<\gamma,\lambda_{0}\>-1}w=E_{\gamma}w \ \ \mbox{ and }\ \ e^{\gamma}_{m}w=0
\ \ \mbox{ for }\gamma\in L, \; m\ge -\<\gamma,\lambda_{0}\>.$$ In particular, this is true for $\gamma=\alpha$, or $\beta$. Combining this with Jacobi identity we get $$(x_{0}+x_{2})^{-\<\alpha,\lambda_{0}\>}Y_{W}(e^{\alpha},x_{0}+x_{2})Y_{W}(e^{\beta},x_{2})w
=(x_{2}+x_{0})^{-\<\alpha,\lambda_{0}\>}Y_{W}(Y(e^{\alpha},x_{0})e^{\beta},x_{2})w.$$ Then by applying $\Res_{x_{0}}\Res_{x_{2}}x_{0}^{-\<\alpha,\beta\>-1}x_{2}^{-\<\beta,\lambda_{0}\>-1}$ we obtain $$\begin{aligned}
E_{\alpha}E_{\beta}=\varepsilon(\alpha,\beta)E_{\alpha+\beta}.\end{aligned}$$
Finally, we are ready to finish the proof. We have $$\psi (h(0)(e^{\alpha}\otimes u))
=\<\alpha,h\>E_{\alpha}u+E_{\alpha}h(0)u=h(0)E_{\alpha}u
=h(0)\psi(e^{\alpha}\otimes u).$$ Then $\psi$ is an $\hat{\frak{h}}$-module homomorphism. Furthermore, we have $$\begin{aligned}
\psi\left( Y_{W}(e^{\alpha},x)(e^{\beta}\otimes u)\right)
&=&\psi\left(\varepsilon(\alpha,\beta)e^{\alpha+\beta}\otimes
x^{\<\alpha,\beta\>}E^{-}(-\alpha,x)E^{+}(-\alpha,x)x^{\alpha(0)}u\right)\\
&=&\varepsilon(\alpha,\beta)x^{\<\alpha,\beta\>}
E_{\alpha+\beta}E^{-}(-\alpha,x)E^{+}(-\alpha,x)x^{\alpha(0)}u\\
&=&x^{\<\alpha,\beta\>}E_{\alpha}E_{\beta}E^{-}(-\alpha,x)E^{+}(-\alpha,x)x^{\alpha(0)}u\\
&=&E^{-}(-\alpha,x)E^{+}(-\alpha,x)E_{\alpha}
x^{\alpha(0)}E_{\beta}u\\
&=&Y_{W}(e^{\alpha},x)\psi(e^{\beta}\otimes u).\end{aligned}$$ As $\frak{h}$ and $e^{\alpha}\ (\alpha\in L)$ generate $V_{(\frak{h},L)}$ as a vertex algebra, $\psi$ is a $V_{(\frak{h},L)}$-module isomorphism.
Let $\lambda_{1},\lambda_{2}\in L^{o}$ and let $U_{1}$ and $U_{2}$ be irreducible restricted ${\mathcal{L}}_{1}$-modules of level $1$. Then $V_{(\frak{h},L)}(\lambda_{1},U_{1})\simeq
V_{(\frak{h},L)}(\lambda_{2},U_{2})$ if and only if $\lambda_{1}+L=\lambda_{2}+L$ and $U_{1}\simeq U_{2}$.
Note that for each $\alpha\in L$, $\C e^{\alpha} \otimes \C e^{\lambda_{i}}\otimes U_{i}\otimes
M_{0}(1)$ is an $\hat{\frak{h}}$-submodule of $V_{(\frak{h},L)}(\lambda_{i},U_{i})$ and $$\C e^{\alpha} \otimes \C e^{\lambda_{i}}\otimes U_{i}\otimes
M_{0}(1)\simeq \C e^{\lambda_{i}+\alpha}\otimes U_{i}\otimes
M_{0}(1)=M(1,\lambda_{i}+\alpha,U_{i}).$$ We see that the set of $\frak{h}(0)$-weights of $V_{(\frak{h},L)}(\lambda_{i},U_{i})$ is $\lambda_{i}+ L$. If $V_{(\frak{h},L)}(\lambda_{1},U_{1})\simeq
V_{(\frak{h},L)}(\lambda_{2},U_{2})$, we must have $\lambda_{1}+L=\lambda_{2}+L$ and $U_{1}\otimes M_{0}(1)\simeq
U_{2}\otimes M_{0}(1)$ as $({\mathcal{L}}_{0}+{\mathcal{L}}_{1})$-modules, which implies that $U_{1}\simeq U_{2}$.
On the other hand, assume $\lambda_{1}+L=\lambda_{2}+L$ and $U_{1}=U_{2}$. Then $\lambda_{1}=\lambda_{2}+\gamma$ for some $\gamma\in L$. Define a linear isomorphism $\theta:
V_{(\frak{h},L)}(\lambda_{1},U_{1})\rightarrow
V_{(\frak{h},L)}(\lambda_{2},U_{2})$ by $$\theta (e^{\beta}\otimes e^{\lambda_{1}}\otimes w)
=\varepsilon(\beta,\gamma) (e^{\beta+\gamma}\otimes
e^{\lambda_{2}}\otimes w)$$ for $\beta\in L,\; w\in U_{1}\otimes
M_{0}(1)$. It is clear that $\theta$ is an $\hat{\frak{h}}$-module isomorphism. Furthermore, for $\alpha\in L$, we have $$\begin{aligned}
& &\theta( Y_{W}(e^{\alpha},x)(e^{\beta}\otimes
e^{\lambda_{1}}\otimes w))\\
&=&\theta\left( \varepsilon(\alpha,\beta)e^{\alpha+\beta} \otimes
x^{\<\alpha,\beta+\lambda_{1}\>}e^{\lambda_{1}}
\otimes E^{-}(-\alpha,x)E^{+}(-\alpha,x)w \right)\\
&=&\varepsilon(\alpha,\beta)\varepsilon(\alpha+\beta,\gamma)e^{\alpha+\beta+\gamma}
\otimes x^{\<\alpha,\beta+\lambda_{1}\>}e^{\lambda_{2}}
\otimes E^{-}(-\alpha,x)E^{+}(-\alpha,x)w\\
&=&\varepsilon(\alpha,\beta+\gamma)\varepsilon(\beta,\gamma)e^{\alpha+\beta+\gamma}
\otimes x^{\<\alpha,\beta+\gamma+\lambda_{2}\>}e^{\lambda_{2}}
\otimes E^{-}(-\alpha,x)E^{+}(-\alpha,x)w\\
&=&Y_{W}(e^{\alpha},x)\theta(e^{\beta}\otimes e^{\lambda_{1}}\otimes
w).\end{aligned}$$ Since $\frak{h}$ and $e^{\alpha}$ for $\alpha\in L$ generate $V_{(\frak{h},L)}$ as a vertex algebra, $\theta$ is a $V_{(\frak{h},L)}$-module homomorphism.
A characterization of vertex algebras $V_{(\frak{h},L)}$ with $\<\cdot,\cdot\>|_{L}=0$
======================================================================================
In this section we study vertex algebras $V_{(\frak{h},L)}$ with $\<\cdot,\cdot\>|_{L}=0$. In this case, we give a characterization of the vertex algebras in terms of a certain affine Lie algebra.
Let $\frak{h}$ be a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form $\<\cdot,\cdot\>$ and let $L\subset \frak{h}$ a free abelian group with $\<\alpha,\beta\>=0$ for $\alpha,\beta\in L$. View $\frak{h}$ and the group algebra $\C[L]$ as abelian Lie algebras. Let $\frak{h}$ act on $\C[L]$ by $$h\cdot e^{\alpha}=\<h,\alpha\>e^{\alpha}\ \ \ \mbox{ for }h\in
\frak{h},\; \alpha\in L.$$ Then $\frak{h}$ acts on $\C[L]$ (viewed as a Lie algebra) by derivations. Form the cross product Lie algebra $\frak{p}=\frak{h}\ltimes \C[L]$ and extend the bilinear form on $\frak{h}$ to $\frak{p}$ by $$\begin{aligned}
\<h+u,h'+v\>=\<h,h'\> \ \ \ \ \mbox{ for }h,h'\in \frak{h},\; u,v\in
\C[L].\end{aligned}$$ This form is symmetric and invariant. Then we have an affine Lie algebra $$\hat{\frak{p}}=\frak{p}\otimes \C[t,t^{-1}]\oplus \C {\bf k},$$ where ${\bf k}$ is central and $$\begin{aligned}
& &[h(m),h'(n)]=m\<h,h'\>\delta_{m+n,0}{\bf k},\\
& &[h(m),e^{\alpha}(n)]=\<h,\alpha\>e^{\alpha}(m+n),\\
& &[e^{\alpha}(m),e^{\beta}(n)]=0\end{aligned}$$ for $h,h'\in \frak{h},\; \alpha,\beta\in L,\; m,n\in \Z$. Let $\ell$ be a complex number. Denote by $\C_{\ell}$ the $1$-dimensional $(\frak{p}\otimes \C[t]+\C {\bf k})$-module $\C$ with $\frak{p}\otimes \C[t]$ acting trivially and with ${\bf k}$ acting as scalar $\ell$. Form the induced $\hat{\frak{p}}$-module $$V_{\hat{\frak{p}}}(\ell,0)
=U(\hat{\frak{p}})\otimes_{U(\frak{p}\otimes \C[t]+\C {\bf k})}
\C_{\ell}.$$ We identify $\frak{p}$ as a subspace of $V_{\hat{\frak{p}}}(\ell,0)$ through the linear map $a\mapsto
a(-1){\bf 1}$, where ${\bf 1}=1\otimes 1$. In particular, we identify $\alpha$ with $\alpha(-1){\bf 1}$ and $e^{\alpha}$ with $e^{\alpha}(-1){\bf 1}$ for $\alpha\in L$. Then there exists a vertex algebra structure on $V_{\hat{\frak{p}}}(\ell,0)$ with ${\bf
1}$ as the vacuum vector and with $Y(a,x)=a(x)$ for $a\in \frak{p}$ (cf. [@ll]). Recall that $\D$ is the linear operator on $V_{\hat{\frak{p}}}(\ell,0)$ defined by $\D v=v_{-2}{\bf 1}$. Note that the normalized $2$-cocycle $\varepsilon: L\times L\rightarrow
\C^{\times}$ defined in Remark \[rcocycle-exist\] is trivial in the sense that $\varepsilon(\alpha,\beta)=1$ for all $\alpha,\beta\in L$.
The vertex algebra $V_{(\frak{h},L)}$ is isomorphic to the quotient vertex algebra of $V_{\hat{\frak{p}}}(1,0)$ modulo the ideal $J$ generated by the elements $$e^{0}-{\bf 1},\ \ \ e^{\alpha}_{-1}e^{\beta}-e^{\alpha+\beta},
\ \ \ \D e^{\alpha}-\alpha(-1)e^{\alpha}\ \ \ \mbox{ for
}\alpha,\beta\in L.$$
First, we show that $V_{(\frak{h},L)}$ is a $\hat{\frak{p}}$-module of level $1$ with $h(x)=Y(h,x)$ and $e^{\alpha}(x)=Y(e^{\alpha},x)$ for $h\in \frak{h},\; \alpha \in L$. We know that $V_{(\frak{h},L)}$ is an $\hat{\frak{h}}$-module of level $1$ with $h(x)=Y(h,x)$. For $\alpha,\beta\in L$, since $\<\alpha,\beta\>=0$, we have $$Y(e^{\alpha},x)e^{\beta}
=E^{-}(-\alpha,x)E^{+}(-\alpha,x)e^{\alpha}x^{\alpha(0)}\cdot
e^{\beta} =E^{-}(-\alpha,x)e^{\alpha+\beta},$$ which contains only nonnegative powers of $x$, so that $$[Y(e^{\alpha},x_{1}),Y(e^{\beta},x_{2})]=0.$$ For $h\in \frak{h}$, since $h(n)e^{\alpha}=\delta_{n,0}\<h,\alpha\>e^{\alpha}$ for $n\ge 0$, we have $$[Y(h,x_{1}),Y(e^{\alpha},x_{2})]
=\<h,\alpha\>x_{2}^{-1}\delta\left(\frac{x_{1}}{x_{2}}\right)Y(e^{\alpha},x_{2}).$$ Then $V_{(\frak{h},L)}$ is a $\hat{\frak{p}}$-module of level $1$. Clearly, ${\bf 1}$ generates $V_{(\frak{h},L)}$ as a $\hat{\frak{p}}$-module with $a(n){\bf 1}=0$ for $a\in \frak{p},\;
n\ge 0$. It follows that there exists a unique $\hat{\frak{p}}$-module homomorphism $\psi$ from $V_{\hat{\frak{p}}}(1,0)$ onto $V_{(\frak{h},L)}$, sending the vacuum vector to the vacuum vector. That is, $\psi({\bf 1})={\bf 1}$ and $$\psi(a_{m}v)=a_{m}\psi(v)\ \ \ \mbox{ for }a\in \frak{p},
\ v\in V_{\hat{\frak{p}}}(1,0),\ m\in \Z.$$ As $\frak{p}$ generates $V_{\hat{\frak{p}}}(1,0)$ as a vertex algebra, it follows that $\psi$ is a vertex algebra homomorphism. It is easy to see that the following relations hold in $V_{(\frak{h},L)}$: $$e^{0}={\bf 1},\ \ e^{\alpha}_{-1}e^{\beta}=e^{\alpha+\beta},\ \
\D e^{\alpha}=e^{\alpha}_{-2}{\bf 1}=\alpha(-1)e^{\alpha}\ \ \
\mbox{ for }\alpha\in L,$$ so that the kernel of $\psi$ contains the ideal $J$. Set $V=V_{\hat{\frak{p}}}(1,0)/J$ (the quotient vertex algebra). Due to the linear map $\psi$, the linear map $a\in
\frak{p}\mapsto a(-1){\bf 1}\in V$ is also injective, so that $\frak{p}$ can be identified as a subspace of $V$. Set $$K=\sum_{\alpha\in L}U(\hat{\frak{h}})e^{\alpha}\subset V.$$ We shall show that $K=V$ by proving that $K$ is a vertex subalgebra of $V$, containing all the generators. As $[\D,h(m)]=-mh(m-1)$ for $h\in \frak{h},\; m\in \Z$ and $\D e^{\alpha}=\alpha(-1)e^{\alpha}$ for $\alpha\in L$, we see that $\D K\subset K$. For $\alpha,\beta\in
L$, we have $e^{\alpha}_{n}e^{\beta}=0$ for $n\ge 0$ and $e^{\alpha}_{-1}e^{\beta}=e^{\alpha+\beta}$. For $n\ge 1$, we have $$\begin{aligned}
& &ne^{\alpha}_{-n-1}e^{\beta}=[\D, e^{\alpha}_{-n}]e^{\beta} =\D
e^{\alpha}_{-n}e^{\beta}-e^{\alpha}_{-n}\D e^{\beta} =\D
e^{\alpha}_{-n}e^{\beta}-e^{\alpha}_{-n}\beta(-1)e^{\beta}\\
& &\ \ \ \ =(\D-\beta(-1))e^{\alpha}_{-n}e^{\beta},\end{aligned}$$ noting that $[\beta(-1),e^{\alpha}_{m}]=0$ for $m\in \Z$. It follows from induction that $e^{\alpha}_{-n-1}e^{\beta}\in K$ for $n\ge 0$. Thus $$e^{\alpha}_{m}e^{\beta}\in K\ \ \ \mbox{ for }\alpha,\beta,\; m\in
\Z.$$ For $h\in \frak{h},\;\alpha\in L,\; \;m, n\in \Z$, we have $$e^{\alpha}_{m}h(n)=h(n)e^{\alpha}_{m}+\<\alpha,h\>e^{\alpha}_{m+n}.$$ Again, it follows from induction that $e^{\alpha}_{m}K\subset K$ for $m\in \Z$. Thus $K$ is a $\hat{\frak{p}}$-submodule of $V$, containing the vacuum vector ${\bf 1}$. Consequently, $K=V$. For $\alpha\in L$, $U(\hat{\frak{h}})e^{\alpha}$ is an irreducible $\hat{\frak{h}}$-module of level $1$, which is isomorphic to $M(1)\otimes \C e^{\alpha}$. Then it follows that $\psi$ is an isomorphism.
For any $V_{(\frak{h},L)}$-module $(W,Y_{W})$, $W$ is a restricted $\hat{\frak{p}}$-module of level $1$ with $a(x)=Y_{W}(a,x)$ for $a\in \frak{p}$, satisfying the condition that $$\begin{aligned}
e^{0}(x)=1,\ \ \ e^{\alpha+\beta}(x)=e^{\alpha}(x)e^{\beta}(x),\ \ \
\frac{d}{dx}e^{\alpha}(x)=\alpha(x)e^{\alpha}(x)\end{aligned}$$ for $\alpha\in L$. On the other hand, if $W$ is a restricted $\hat{\frak{p}}$-module of level $1$, satisfying the above condition, then there exists a $V_{(\frak{h},L)}$-module structure $Y_{W}$ such that $Y_{W}(a,x)=a(x)$ for $a\in \frak{p}$.
Let $(W,Y_{W})$ be a $V_{(\frak{h},L)}$-module. Then $W$ is naturally a $V_{\hat{\frak{p}}}(1,0)$-module. We have $$e^{0}(x)=Y_{W}(e^{0},x)=Y_{W}({\bf 1},x)=1,$$ $$\begin{aligned}
& &\frac{d}{dx}e^{\alpha}(x)=\frac{d}{dx}Y_{W}(e^{\alpha},x)
=Y_{W}(\D e^{\alpha},x)=Y_{W}(\alpha(-1)e^{\alpha},x)
=Y_{W}(\alpha,x)Y_{W}(e^{\alpha},x)
\ \ \ \ \\
& &\ \ \ \ \ \ \ \ =\alpha(x)e^{\alpha}(x),\\
& &e^{\alpha}(x)e^{\beta}(x)=Y_{W}(e^{\alpha},x)Y_{W}(e^{\beta},x)
=Y_{W}(e^{\alpha}_{-1}e^{\beta},x)=Y_{W}(e^{\alpha+\beta},x)=e^{\alpha+\beta}(x).\end{aligned}$$
Conversely, let $W$ be a restricted $\hat{\frak{p}}$-module of level $1$, satisfying the conditions. Then $W$ is a $V_{\hat{\frak{p}}}(1,0)$-module with $Y_{W}(a,x)=a(x)$ for $a\in
\frak{p}$. We have $$\begin{aligned}
Y_{W}(e^{0}-{\bf 1},x)&=&e^{0}(x)-1=0,\\
Y_{W}(e^{\alpha}_{-1}e^{\beta}-e^{\alpha+\beta},x)
&=&Y_{W}(e^{\alpha},x)Y_{W}(e^{\beta},x)-Y_{W}(e^{\alpha+\beta},x)\\
&=&e^{\alpha}(x)e^{\beta}(x)-e^{\alpha+\beta}(x)=0,\\
Y_{W}(\D e^{\alpha}-\alpha(-1)e^{\alpha},x)
&=&\frac{d}{dx}Y_{W}(e^{\alpha},x)-Y_{W}(\alpha,x)Y_{W}(e^{\alpha},x)\\
&=&\frac{d}{dx}e^{\alpha}(x)-\alpha(x)e^{\alpha}(x)=0.\end{aligned}$$ Then $W$ is naturally a module for the quotient vertex algebra $V_{\hat{\frak{p}}}(1,0)/J$, where $J$ is the ideal generated by the vectors $$e^{0}-{\bf 1},\ \ e^{\alpha}_{-1}e^{\beta}-e^{\alpha+\beta},\;\;
\D e^{\alpha}-\alpha(-1)e^{\alpha}$$ for $\alpha,\beta\in L$. In view of Theorem \[tchara\], $W$ is a $V_{(\frak{h},L)}$-module.
[AAGBP]{}
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[^1]: Partially supported by NSF grant DMS-0600189
[^2]: Permanent address: Department of Mathematics, Xiamen University, Xiamen, China
|
---
abstract: 'Non-canonical scalar fields with the Lagrangian ${\cal L} = X^\alpha - V(\phi)$, possess the attractive property that the speed of sound, $c_s^{2} = (2\,\alpha - 1)^{-1}$, can be exceedingly small for large values of $\alpha$. This allows a non-canonical field to cluster and behave like warm/cold dark matter on small scales. We demonstrate that simple potentials including $V = V_0\coth^2{\phi}$ and a Starobinsky-type potential can unify dark matter and dark energy. [*Cascading dark energy*]{}, in which the potential cascades to lower values in a series of discrete steps, can also work as a unified model. In all of these models the kinetic term $X^\alpha$ plays the role of dark matter, while the potential term $V(\phi)$ plays the role of dark energy.'
author:
- 'Swagat S. Mishra and Varun Sahni'
title: 'Unifying Dark Matter and Dark Energy with non-Canonical Scalars'
---
Introduction {#sec:intro}
============
A key feature of our universe is that $96\%$ of its matter content is weakly interacting and non-baryonic. It is widely believed that this so-called dark sector consists of two distinct sub-components, the first of which, dark matter (DM), consists of a pressureless fluid which clusters, while the second, dark energy (DE), has large negative pressure and causes the universe to accelerate at late times.
Although numerous theoretical models have been advanced as to what may constitute dark matter, none so far has received unambiguous experimental support [@dark_matter]. The same may also be said of dark energy. The simplest model of DE, the cosmological constant $\Lambda$, fits most observational data sets quite well [@planck_2015]; see however [@bao_2014; @sss14]. Yet the fine tuning problem associated with $\Lambda$ and the cosmic coincidence issue, have motivated the development of dynamical dark energy (DDE) models in which the DE density and equation of state (EOS) evolve with time [@review_DE; @sahni04].
In view of the largely unknown nature of the dark sector several attempts have been made to describe it within a unified setting [@CG; @CG1; @tirth; @CG_pert; @sandvik04; @scherrer04; @tejedor_feinstein; @bertacca07; @fang07; @bertacca08; @bertacca11; @cervantes11; @asen14; @sahni_sen; @li-scherrer; @gurzadyan]. Perhaps the earliest prescription for a unified model of dark matter and dark energy was made in the context of the Chaplygin gas (CG) [@CG; @CG1; @CG_pert]. CG possesses an equation of state which is pressureless ($p \simeq 0$) at early times and $\Lambda$-like ($p \simeq -\rho$) at late times. This led to the hope that CG may be able to describe both dark matter and dark energy through a unique Lagrangian. Unfortunately an analysis of density perturbations dashed these early hopes for unification [@sandvik04]. In this paper we build on early attempts at unification and demonstrate that a compelling unified description of dark matter and dark energy can emerge from scalar fields with non-canonical kinetic terms.
Non-canonical scalar fields {#sec:NC}
===========================
The non-canonical scalar field Lagrangian [@Mukhanov-2006; @sanil-2008; @sanil13] (X,) = X()\^[-1]{} - V(), \[eqn: Lagrangian\] presents a simple generalization of the canonical scalar field Lagrangian (X,) = X - V(), X = \^2 \[eqn: Lagrangian0\] to which (\[eqn: Lagrangian\]) reduces when $\alpha=1$.
Two properties of non-canonical scalars make them attractive for the study of cosmology:
1. Their equation of motion + + ()()\^[- 1]{} = 0, \[eqn: EOM-model\] is of second order, as in the canonical case. Indeed, (\[eqn: EOM-model\]) reduces to the standard canonical form ${\ddot \phi}+ 3\, H {\dot \phi} + V'(\phi) = 0$ when $\alpha =1$.
2. The speed of sound [@Mukhanov-2006] c\_s\^[2]{} = \[eqn: sound speed model\] can become quite small for large values of $\alpha$ since $c_s \to 0$ when $\alpha \gg 1$.
This latter property ensures that non-canonical scalars can, in principle, play the role of dark matter. In this paper we shall show that, for a suitable choice of the potential $V(\phi)$, non-canonical scalars can also unify dark matter with dark energy.
This paper works in the context of a spatially flat Friedmann-Robertson-Walker (FRW) universe for which the energy-momentum tensor has the form T\^\_ = (\_[\_]{}, -p\_[\_]{}, - p\_[\_]{}, - p\_[\_]{}). In the context of non-canonical scalars, the energy density, $\rho_{_{\phi}}$, and pressure, $p_{_{\phi}}$, can be written as \_[\_]{} &=& ł(2-1)Xł()\^[-1]{} + V(),\
p\_[\_]{} &=& Xł()\^[-1]{} - V(). \[eqn:rho\] It is easy to see that (\[eqn:rho\]) reduces to the the canonical form $\rho_{_{\phi}} = X + V$, $p_{_{\phi}} = X - V$ when $\alpha = 1$.
The Friedmann equation which is solved in association with is H\^[2]{} = ł(\_[\_]{} + \_[\_r]{} +\_[\_b]{}) \[eqn:freid\] where $\rho_{_r}$ is the radiation term and $\rho_{_b}$ is the contribution from baryons. Note that we do not assume a separate contribution from dark matter or dark energy which are encoded in $\rho_{_{\phi}}$.
As shown in [@sahni_sen], for sufficiently flat potentials with $V' \simeq 0$ the third term in (\[eqn: EOM-model\]) can be neglected, leading to - a\^[-]{} . \[eq:phidot\] Substituting (\[eq:phidot\]) in \_[\_X]{} = ł(2-1)Xł()\^[-1]{}, X = \^2 \[eqn:rhoNC\] one finds \_[\_X]{} a\^[-]{} \[eq:kinetic\] which reduces to $\rho_{_X} \propto a^{-3}$ for $\alpha \gg 1$. Comparing with $\rho_{_X} \propto a^{-3(1+w_{_X})}$ one finds w\_[\_X]{} = , \[eq:state\] so that $w_{_X} \simeq 0$ for $\alpha \gg 1$. We therefore conclude that for flat potentials and large values of $\alpha$, the kinetic term, $\rho_{_X}$, plays the role of dark matter while the potential term, $V$, plays the role of dark energy in (\[eqn:rho\]).
The above argument is based on the requirement that the third term is much smaller than the first two terms in equation (\[eqn: EOM-model\]). In a recent paper Li and Scherrer [@li-scherrer] have made the interesting observation that the potential need not be flat in order that (\[eq:phidot\]) be satisfied. The analysis of Li and Scherrer suggests that the third term in (\[eqn: EOM-model\]) can be neglected under more general conditions than anticipated in [@sahni_sen] provided the potential is ‘sufficiently rapidly decaying’ [@li-scherrer]. Indeed it is easy to show that the requirement of the third term in (\[eqn: EOM-model\]) being much smaller than the second translates into the inequality V’ () , \[eq:inequality1\] which reduces to V’ [V]{} 3H\_[\_X]{}, \[eq:inequality2\] when $\alpha \gg 1$. Equation can be recast as . \[eq:inequality3\] Equations & inform us that the variation in $V$ can be quite large at early times when both $H$ and $\rho_{_X}$ are large. By contrast the same equations suggest that the potential should be quite flat at late times when $H$ and $\rho_{_X}$ are small. This latter property ensures that $V(\phi)$ can play the role of DE at late times and drive cosmic acceleration. The inequalities in , can be satisfied by a number of potentials some of which are discussed below.
We first consider the inverse-power-law (IPL) family of potentials [@ratra88a] V() = , p>0. \[eq:IPL\] Assuming that the background density falls off as \_[\_[B]{}]{} a\^[-m]{} \[eq:background\] (where $m=4, 3$ for radiative and matter dominated epochs respectively), one can show that (\[eq:phidot\]) is a late-time attractor provided the inequality p \[eq:inequality\] is satisfied [@li-scherrer]. For the large values $\alpha \gg 1$ which interest us, (\[eq:inequality\]) reduces to p for which the late-time attractor is [@li-scherrer] V ł(\_[\_[B]{}]{})\^ [and]{} \_[\_X]{} a\^[-3]{} . \[eq:PE\_att\] From one finds that for $p > 2$ the potential falls off [*faster*]{} than the background density $\rho_{_{\rm B}}$. The case $p=2$ is special since $V \propto \rho_{_{\rm B}}$, the potential scales exactly like the background density. This is illustrated in figure \[fig:rho\_ipl\]. In this case one finds (for $\alpha \gg 1$) ()\^2 \[eq:ratio\] where $M$ is a free parameter in the non-canonical Lagrangian .
![This figure illustrates the evolution of the density in radiation (red curve), baryons (dotted green curve) and a non-canonical scalar field with the potential $V \propto \phi^{-2}$. The solid green curve shows the evolution of the kinetic term $\rho_{_X} \propto X^\alpha$ which behaves like dark matter, $\rho_{_X} \propto a^{-3}$. The blue curve shows the evolution of the potential $V(\phi)$. Note that $V$ scales like the background fluid so that $V \propto a^{-4}$ during the radiative regime and $V \propto a^{-3}$ during matter domination. ($\alpha = 10^5$ has been assumed.)[]{data-label="fig:rho_ipl"}](figs/DE_nc_ipl_rho.eps){width="85.00000%"}
The above analysis suggests that the IPL potential $V \propto \phi^{-p}, ~ p\geq 2$ cannot give rise to cosmic acceleration at late times. Thus while being able to account for dark matter (through $\rho_{_X}$) this model is unable to provide a unified description of dark matter and dark energy.
However, as we show in the rest of this paper, a unified prescription for dark matter and dark energy is easily provided by any one of the following potentials[^1]: (i) $V(\phi) = V_0 \coth^2{\phi}$, (ii) the Starobinsky-type potential $V(\phi) = V_0 \left ( 1 - e^{-{\phi}}\right )^{2}$, (iii) the step-like potential $V(\phi) = A + B\tanh{\beta \phi}$. It is interesting that all of these potentials belong to the $\alpha$-attractor family [@linde1; @linde2] in the canonical case.
Unified models of Dark Matter and Dark Energy {#sec:unified}
=============================================
Dark Matter and Dark Energy from $V = V_0\coth^2{\phi}$ {#sec:coth}
-------------------------------------------------------
The potential [@bms17] V() &=& V\_0 \^p[()]{}, p>0\
\
&& V\_0 ł()\^p \[eq:coth\] can provide a compelling description of dark matter and dark energy on account of its two asymptotes: V() && , [for]{} m\_p\[eq:IPL1\]\
V() && V\_0 , [for]{} m\_p . \[eq:coth\_late\] As discussed in the previous section, the IPL asymptote (\[eq:IPL1\]) ensures that the kinetic term behaves like dark matter $\rho_{_X} \propto a^{-3}$, while $V(\phi)$ scales like the background density (for $p=2$) or faster (for $p > 2$); see eqn. (\[eq:PE\_att\]).
The late-time asymptote (\[eq:coth\_late\]) demonstrates that the potential flattens to a constant value at late times. This feature allows $V(\phi)$ to play the role of dark energy. Indeed, a detailed numerical analysis of the $\coth$ potential, summarized in figures \[fig:rho\_coth\] and \[fig:Omega\_coth\], demonstrates that (\[eq:coth\]) with $p = 2$ can provide a successful unified description of dark matter and dark energy in the non-canonical setting. (This is also true for $p > 2$. However for shallower potentials with $p < 2$ the third term in (\[eqn: EOM-model\]) cannot be neglected. This implies that for such potentials $\rho_{_X}$ does not scale as $a^{-3}$ and therefore cannot play the role of dark matter [@li-scherrer].)
$\begin{array}{@{\hspace{-0.5in}}c@{\hspace{-0.2in}}c}
\multicolumn{1}{l}{\mbox{}} &
\multicolumn{1}{l}{\mbox{}} \\ [-0.10in]
\epsfxsize=3.8in
\epsffile{figs/DE_nc_coth_p2_l1_rho.eps} &
\epsfxsize=3.8in
\epsffile{figs/DE_nc_coth_p2_l1_rho_MQ.eps} \\
\end{array}$
![The evolution of density parameters $(\Omega_i = \rho_i/\rho_{\rm cr})$ is shown for radiation (red), baryons (dotted green), kinetic term $\rho_{_X} \propto X^{\alpha}$ (solid green) and the potential $V(\phi)$ (solid blue) for $V(\phi) \propto \coth^{2}{{\phi}}$. This model gives rise to a matter dominated epoch (sourced by the kinetic term $\rho_{_X}$) when $z > 1$ and a DE dominated epoch, sourced by the potential $V(\phi)$, when $z < 1$. []{data-label="fig:Omega_coth"}](figs/DE_nc_coth_p2_l1_Omega.eps){width="70.00000%"}
For non-canonical models the equation of state can be determined from (\[eqn:rho\]), namely w\_= = -1 + ()() , \[eq:EOS\_nc\_1\] which simplifies to w\_= -1 + , [for]{} 1 . \[eq:EOS\_nc\_2\] For the potential (\[eq:coth\]) $\rho_{_X} \propto a^{-3}$ whereas $V(\phi)$ approaches a constant value at late times. Consequently one gets w\_(z=0)-1 + , \[eq:EOS\_nc\_0\] where $\Omega_{0V} = V_0/\rho_{cr,0}$. Assuming $\Omega_{0V}=0.7$, $\Omega_{0b}=0.04$, one finds $\Omega_{0X}=0.26$ so that w\_(z=0)-0.7292. \[eq:EOS\_nc\_att\] Our results for the deceleration parameter $q = -\frac{\ddot a}{aH^2}$ and $w_\phi$ are shown in figure \[fig:w\_q\_coth\] for the IPL model (\[eq:IPL\]) and the $\coth$ potential (\[eq:coth\]), both with $p=2$.
$\begin{array}{@{\hspace{-0.5in}}c@{\hspace{-0.2in}}c}
\multicolumn{1}{l}{\mbox{}} &
\multicolumn{1}{l}{\mbox{}} \\ [-0.10in]
\epsfxsize=3.8in
\epsffile{figs/DE_nc_coth_ipl_wf.eps} &
\epsfxsize=3.8in
\epsffile{figs/DE_nc_coth_ipl_q.eps} \\
\end{array}$
The effective equation of state of the kinetic and potential components can be determined from w\_[\_X]{} &=& - -1 ,\[eq:wX\]\
w\_[\_V]{} &=& --1 . \[eq:wV\] Substituting for $\rho_{_{X}}$ from one finds w\_[\_X]{} = -1- . which leads to w\_[\_X]{} = c\_[\_S]{}\^2 =(2- 1)\^[-1]{} after substitution for $3H{\dot\phi}$ from .
It is instructive to rewrite as 1+w\_[\_V]{} = - () . \[eq:wV1\] As noted in the inequality ${\dot V} \ll 3H\rho_{_X}$ should be satisfied in order for $\rho_{_{X}}$ to behave like dark matter. Since the densities in dark matter and dark energy are expected to be comparable at late times, one finds $\frac{\rho_{_{X}}}{V} \sim O(1)$ at $z \leq 1$. Substituting these results in one concludes that the EOS of DE is expected to approach $w_{_V} \simeq -1$ at late times in unified models of the dark sector. (One should also note that since $\frac{\rho_{_{X}}}{V}$ can be fairly large at early times, $w_{_V}$ is not restricted to being close to -1 at $z \gg 1$.) Figure \[fig:wXwV\_coth\_ipl\] compares the behaviour of $w_{_X}$ and $w_{_V}$ in the two potentials: $\coth^2{\phi}$ and $\phi^{-2}$. One notices that $w_{_V}\simeq 1/3$ at early times in both potentials, which is a reflection of the scaling behaviour $V \propto \rho_{_{\rm B}}$ noted in . At late times $w_{_V}$ in the coth potential drops to negative values causing the universe to accelerate. For $V \propto \phi^{-2}$ on the other hand, $w_{_V}$ always tracks the dominant background fluid which results in $w_{_V} = 0$ at late times. (Note that in this case the fluid which dominates at late times is the kinetic term, so that $V \propto \rho_{_X}$.)
$\begin{array}{@{\hspace{-0.5in}}c@{\hspace{-0.2in}}c}
\multicolumn{1}{l}{\mbox{}} &
\multicolumn{1}{l}{\mbox{}} \\ [-0.10in]
\epsfxsize=3.8in
\epsffile{figs/DE_nc_ipl_p2_w.eps} &
\epsfxsize=3.8in
\epsffile{figs/DE_nc_coth_p2_l1_w.eps} \\
\end{array}$
Dark Matter and Dark Energy from a Starobinsky-type potential {#sec:star}
-------------------------------------------------------------
A unified model of dark matter and dark energy can also arise from the potential [@linde1; @linde2; @sss17] V() = V\_0 ( 1 - e\^[-]{})\^[2]{} , >0 , \[eq:star1\] which reduces to the Starobinsky potential in the Einstein frame [@star] for $\lambda=\sqrt{\frac{2}{3}}$. The potential in is characterized by three asymptotic branches (see figure \[fig:staroughpot\]): V() && V\_0 e\^[-2/m\_p]{} , 3(1+w\_[\_[B]{}]{})$.
The situation radically changes for non-canonical scalars ($1$).
As shown in \cite{li-scherrer}, if the background density scales as $\_[\_[B]{}]{} a\^[-m]{}$, then for
$(2- 1) m > 6$
the late time attractor is
\beq
{\dot\phi}^{2\alpha - 1} \propto t^{-6/m}
~~~ \Rightarrow ~~ {\dot\phi} \propto a^{-\frac{3}{2\alpha - 1}}~.
\label{eq:attractor}
\eeq
Since $\_[\_X]{} \^[2]{}$ one finds
%X^\alpha$ one finds \_[\_X]{} a\^[-]{} which reduces to \_[\_X]{} a\^[-3]{} [for]{} 1. We therefore find that, as in the IPL case, for large values of the non-canonical parameter $\alpha$ the density of the kinetic term scales just like pressureless (dark) matter. From one also finds that ${\dot\phi} \sim$ [*constant*]{} when $\alpha \gg 1$. From this it is easy to show that for $\alpha \gg 1$ the amplitude of the scalar field grows as a\^[m/2]{} , \[eq:phi\_star\] so that $\phi \propto a^2$ during the radiative regime and $\phi \propto a^{3/2}$ during matter domination. This behaviour is illustrated in figure \[fig:phi\_att\_exp\]. Substituting in (\[eq:starpot1\]) one finds $V \propto \exp{[-2\lambda\phi]} \sim \exp{[-2\lambda a^{m/2}]}$, which implies an exponentially rapid decline in the value of the potential as the universe expands and $a(t)$ increases. One therefore concludes that like the IPL potential, an exponential potential too can never dominate the energy density of the universe and source cosmic acceleration.
![The evolution of $\phi$ is shown for the exponential potential $V(\phi)=V_0~ e^{-\frac{2\phi}{m_p}}$. Commencing at $z \sim 10^{15}$, different initial conditions (represented by dashed green and red curves) rapidly converge onto the attractor solution (\[eq:phi\_star\]) (solid blue curve) which corresponds to $\phi \propto a^2$ during radiation domination and $\phi \propto a^{3/2}$ during matter domination. The rapid growth in $\phi$ is accompanied by a steep decline in $V(\phi)$.[]{data-label="fig:phi_att_exp"}](figs/DE_nc_exp_phi.eps){width="80.00000%"}
### Accelerating Cosmology from a Starobinsky-type potential {#sec:star_acc}
In the context of the Starobinsky-type potential in , the rapid growth of $\phi$ in (\[eq:phi\_star\]) enables the scalar field to pass from the steep left wing to the flat right wing of $V(\phi)$. In other words the scalar field rolls from A to B in figure \[fig:staroughpot\]. Since $V \simeq V_0$ on the flat right wing, cosmological expansion in this model mimicks $\Lambda$CDM at late times. This is illustrated in figure \[fig:rho\_staro\]. (Note that $\rho_{_X} \propto a^{-3}$ on both wings of the potential.)
$\begin{array}{@{\hspace{-0.5in}}c@{\hspace{-0.2in}}c}
\multicolumn{1}{l}{\mbox{}} &
\multicolumn{1}{l}{\mbox{}} \\ [-0.10in]
\epsfxsize=3.8in
\epsffile{figs/DE_nc_staro_l1_rho.eps} &
\epsfxsize=3.8in
\epsffile{figs/DE_nc_staro_l1_w.eps} \\
\end{array}$
Figure \[fig:rho\_staro\] shows the behaviour of $w_{_X}$ and $w_{_V}$ as $\phi$ moves under the influence of the potential . A key feature to be noted is that $w_{_V}$ encounters a [*pole*]{} as $\phi$ rolls from the steep left wing to the flat right wing of $V(\phi)$ (from A to B in figure \[fig:staroughpot\]).
Cascading Dark Energy {#sec:cascade}
---------------------
A question occasionally directed towards dark energy is whether cosmic acceleration will continue forever (as in $\Lambda$CDM) or whether, like the earlier transient epochs (inflation, radiative/matter dominated) dark energy will also will be a fleeting phenomenon. In this section we investigate a transient model of DE in which the potential $V(\phi)$ is piece-wise flat and resembles a staircase; see figure \[fig:cascade\]. Such a potential might mimick a model in which an initially large vacuum energy cascades to lower values through a series of waterfalls – discrete steps [@watson06].
$\begin{array}{@{\hspace{-0.5in}}c@{\hspace{-0.2in}}c}
\multicolumn{1}{l}{\mbox{}} &
\multicolumn{1}{l}{\mbox{}} \\ [-0.10in]
\epsfxsize=3.8in
\epsffile{figs/pot_cascade.eps} &
\epsfxsize=3.8in
\epsffile{figs/DE_nc_cascade1_rho.eps} \\
\end{array}$
Locally the i-th step of this potential may be described by [@sahni_sen; @tower09] V() = A + B \[eq:step\] where $A+B = V_{\rm i}$ and $A-B = V_{\rm i+1}$. If $V_{\rm i+1} \simeq 10^{-47} {\rm GeV}^4$ then this potential could account for cosmic acceleration. (One might imagine yet another step at which $V_{\rm i+2} < 0$. In this case the universe would stop expanding and begin to contract at some point in the future.) Motion along the staircase potential leads to a cascading model of dark energy. Remarkably, the inequality in holds even as $\phi$ cascades from higher to lower values of $V$. This ensures that the kinetic term scales as $\rho_{_X} \propto a^{-3}$ and behaves like dark matter while $V(\phi)$ behaves like dark energy, as shown in fig. \[fig:cascade\].
Note that the cascading DE model runs into trouble in the canonical context since the kinetic energy of a canonical scalar field moving along a flat potential declines as $\frac{1}{2}{\dot\phi}^2 \propto a^{-6}$. This puts the brakes on $\phi(t)$ which soon approaches its asymptotic value $\phi_*$, resulting in inflation sourced by $V(\phi_*)$. By contrast ${\dot\phi} \sim$ [*constant*]{} in non-canonical models with $\alpha \gg 1$, see . This allows $\phi(t)$ to cross each successive step on the DE staircase in a finite amount of time, $\Delta t \simeq \frac{\Delta\phi}{\dot\phi}$, and drop to a lower value of $V(\phi)$; see left panel of fig. \[fig:cascade\].
Discussion
==========
In this paper we have demonstrated that a scalar field with a non-canonical kinetic term can play the dual role of dark matter and dark energy. The key criterion which must be satisfied by unified models of the dark sector is . This inequality ensures that the third term in the equation of motion is small and can be neglected, resulting in $\rho_{_X} \propto a^{-3}$ and $w_{_X} \simeq c_{_S} \simeq 0$. In other words if is satisfied the kinetic term behaves like dark matter with vanishing pressure and sound speed. Of equal importance is the fact that if holds then eqn. implies $w_{_V} \simeq -1$ at late times. This ensures that the potential $V(\phi)$ can dominate over $\rho_{_X}$ and source cosmic acceleration at late times.
The following unified models of the dark sector have been discussed in this paper: (i) Models with exactly flat potentials $V' = 0, ~V=V_0$. As shown in [@sahni_sen] the entire expansion history of this model resembles $\Lambda$CDM. (ii) Successful unification of the dark sector can also arise from potentials which are steep at early times and flatten out at late times. Both the $\coth$ potential and the Starobinsky-type potential provide us with examples of this category. (iii) The step-like potential also leads to unification [^2]. In this case the motion of $\phi$ resembles a series of waterfalls as $V(\phi)$ cascades to lower and lower values. It is interesting to note that for all of the above potentials [^3] the kinetic term scales as $\rho_{_X} \propto a^{-3}$ throughout the expansion history of the universe, even as the shape of the potential continuously changes. This property allows the kinetic term to play the role of dark matter while the potential term $V(\phi)$ plays the role of dark energy and leads to cosmic acceleration at late times.
As shown in [@sahni_sen] the small (but non-vanishing) speed of sound in non-canonical models suppresses gravitational clustering on small scales. Non-canonical models with $c_s \ll 1$ can therefore have a macroscopic Jeans length which might help in resolving the cusp–core and substructure problems which afflict the standard cold dark matter scenario. In this context the dark matter content of our model shares similarities with warm dark matter [@neutrino; @neutrino1] and fuzzy cold dark matter [@hu00; @sahni_wang; @Witten; @sss17] both of which are known to possess a large Jeans scale. Finally it is interesting to note that all of the potentials discussed in this paper belong to the $\alpha$-attractor family of potentials [@linde1; @linde2] and lead to interesting models of inflation, dark matter and dark energy [@sss17; @bms17] in the canonical case.
Acknowledgements {#acknowledgements .unnumbered}
================
S.S.M thanks the Council of Scientific and Industrial Research (CSIR), India, for financial support as senior research fellow. S.S.M also acknowledges the hospitality of the Yukawa Institute for Theoretical Physics (YITP) at Kyoto University, where a substantial part of this research work was conducted during the workshop “Gravity and Cosmology 2018”.
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[^1]: Perhaps the simplest potential for unification is $V(\phi) = V_0$ which results in a $\Lambda$CDM cosmology [@sahni_sen].
[^2]: Note that the width of each step is restricted by the fact that the universe does not appear to accelerate prior to $z \sim 1$ (with the exclusion of an early inflationary epoch).
[^3]: We note in passing that the asymptotically flat potential [@linde1] $V(\phi) = V_0 \tanh^2{\phi}$ also leads to a unified scenario of dark matter and dark energy although we do not discuss it in this paper.
|
---
abstract: 'This paper analyses the effect of preferential sampling in Geostatistics when the choice of new sampling locations is the main interest of the researcher. A Bayesian criterion based on maximizing utility functions is used. Simulated studies are presented and highlight the strong influence of preferential sampling in the decisions. The computational complexity is faced by treating the new local sampling locations as a model parameter and the optimal choice is then made by analysing its posterior distribution. Finally, an application is presented using rainfall data collected during spring in Rio de Janeiro. The results showed that the optimal design is substantially changed under preferential sampling effects. Furthermore, it was possible to identify other interesting aspects related to preferential sampling effects in estimation and prediction in Geostatistics.'
address:
- 'National School of Statistical Sciences, Brazilian Institute of Geography and Statistics, Rio de Janeiro, Brazil, '
- 'Department of Statistical Methods, Mathematical Institute, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, '
author:
-
-
title: Optimal Design in Geostatistics under Preferential Sampling
---
./style/arxiv-ba.cfg
Introduction
============
@diggle2010geostatistical presented a novel methodology to perform inference in the traditional Geostatistical model under preferential sampling. They assumed that the sample design could be described by a log-Gaussian Cox process [@moller1998log] and performed maximum likelihood estimation for the model parameters through simulation. In addition, they have made simulations to evaluate the effect of preferential sampling on parameter estimation in Geostatistics, concluding that this was not negligible. In all simulations, after obtaining unbiased estimates about the model parameters, the spatial prediction was made by the [*plug-in*]{} method, according to the classical approach to perform inference in Geostatistics [see @cressie1993statistics; @diggle1998model].
Based on this work, many others questions have emerged, and one of them was related to the influence of preferential sampling in the optimal design choice. This procedure is widespread in Geostatistics literature. There are several recent papers dealing with this, such as @zidek2000designing [@fernandez2005effect; @zhu2005spatial; @diggle2006bayesian; @gumprecht2009designs; @boukouvalas2009; @muller2010compound], among others. The advances made by @muller1999simulation [@muller2004optimal] and @muller2007simulation that propose methods based on maximization of utility functions are especially relevant. Since these procedures incorporate the Bayesian approach and intensive simulation methods in a natural way, they became quite appropriate to perform the optimal design choice in Geostatistics.
Here the uncertainties about model parameters will be considered allowing us to evaluate their impact on spatial prediction (or kriging) under preferential sampling. With this aim, spatial prediction about the underlying process are performed by analysing the predictive distribution conditional on the observed data and the design sample. Then, comparisons between these predictions and those obtained according to the classical approach are made.
However, the most important contribution of this paper is its analysis of preferential sampling effects in the process of obtaining the optimal design in Geostatistical models. Using an approach based on maximizing utility functions [@muller1999simulation] to obtain optimal design, the influence of preferential sampling is evaluated in situations where the researcher’s goal is to optimize some objective function, e.g. reduce predictive variances. It will be shown through simulations that the optimal decision about this choice is substantially modified under a preferential sampling effect.
This paper is organized as follows: Section 2 presents some background about spatial processes and a methodology for fully Bayesian inference, and spatial prediction in Geostatistics under preferential sampling using MCMC methods is described. Section 3 presents a method to obtain the optimal design based on maximization of utility functions. Section 4 combines this methodology to obtain the optimal design under preferential sampling. In Sections 5 and 6, we use the methodology to obtain the optimal design in some simulated examples, and we analyse a real dataset where the researcher is interested in monitoring the occurrence of extreme events. Finally, Section 7 discusses the results obtained.
Spatial Process & Optimal Design
================================
This section presents some background on Geostatistics and point processes, and presents a procedure for obtaining the optimal design through utility functions.
Geostatistics
-------------
Geostatistics deals with stochastic processes defined in a region $D$, $D \subset \Re^k$, where usually $k=$ 1 or 2. Following the approach in @diggle1998model, one can assume that the researcher is interested in studying the features of the stochastic process . Additionally, assume that $S(x)$ is a Gaussian stationary and isotropic process, with zero mean, constant variance $\sigma^2$ and autocorrelation function $\rho(S(x),S(x+h);\phi)=\break\rho(\parallel h \parallel;\phi)$, $\forall x
\in D$, which may depend on one or more parameters represented by $\phi$. Several families of autocorrelation function can be found in the literature [see @cressie1993statistics; @diggle2007model].
Assuming that $n$ observations $Y_i = Y( x_i )$, $i= 1,\dots,n$ are available and $$Y_i=\mu + S(x_i)+Z_i,$$ $$E[Z_i]=0, Var[Z_i]=\tau^2,\quad \forall i,\vadjust{\goodbreak}$$ one can consider ${\bf Y} = ( Y_1, \dots, Y_n )$ as a noisy version of the underlying process $S$. As usual, it will be assumed that ${\bf Z} = (Z_1, \dots, Z_n )$ has a Gaussian distribution independent of the process $S$.
Under the Bayesian paradigm, the posterior distribution of all model parameters must be obtained to make inference. Assuming that $\theta=(\tau^2,\sigma^2,\phi,\mu)$ represents the set of unknown parameters and that $p({\bf y} \mid \theta)$ is the likelihood function, we need to specify a prior distribution $p(\theta)$ to obtain the posterior distribution $p(\theta \mid {\bf y})$. It is usual to assign Gamma distributions to the parameters $\phi$, $\sigma^{-2}$ and $\tau^{-2}$ and a Gaussian distribution to $\mu$. Since the posterior distributions obtained have no closed form, we can approximate them using MCMC methods.
Usually there is an additional interest in obtaining predictions of the process $S$ at locations not observed. Computations required in inference are usually implemented via discretization of the region $D$ in $M$ subregions or cells. Thus, we can redefine the underlying Gaussian process over the centroids of these subregions, $S=\{S_1,\dots,S_M\}$. In addition, we define the partition $S=\{S_{\bf
y},S_N\}$ to distinguish the underlying process associated with the $n$ observed and the $N=M-n$ unobserved locations. The distribution of $S$ is a multivariate Gaussian distribution with dimension $M=n+N$, mean vector [**0**]{} and autocorrelation matrix given by $$R=\left(\begin{array}{cc} R_{n} & R_{n,N} \\ R_{N,n} & R_{N} \end{array}\right),$$ whose elements are defined by the autocorrelation function $\rho(\cdotp;\phi)$. Then, we have the model $$[{\bf y} \mid S_{\bf y},\mu,\tau^{-2}] \sim N({\bf 1}\mu + S_{\bf y},\tau^2 I_{n}), \label{obs}$$ $$[S \mid \sigma^{-2},\phi] \sim N({\bf 0}, \sigma^2R) \label{intensity}$$ that is completed with the priors $\mu \sim N(0,k)$, $\tau^{-2} \sim G(a_{\tau},b_{\tau})$ and $\sigma^{-2} \sim
G(a_{\sigma},b_{\sigma})$, where $k, a_{\tau},b_{\tau},a_{\sigma}$ and $b_{\sigma}$ are known hyperparameters.
Without loss of generality with respect to the objectives of this work, the exponential autocorrelation function $\rho(\parallel h
\parallel;\phi)=\exp (-\parallel h \parallel / \phi )$ will be used in the sequel.
The full conditional distributions for the parameters $\mu,\tau^{-2}$, $\sigma^{-2}$, and $S$ can be updated by Gibbs sampling steps, whereas the range parameter $\phi$ can be updated by Metropolis steps in the MCMC algorithm. The full conditional distribution for $S$ is of particular interest in Geostatistics and has Gaussian distribution with mean vector and covariance matrix given by $$\sigma^2 R_{N,n}(\tau^2 I_{n} + \sigma^2R_{n})^{-1}({\bf y}-{\bf 1}\mu)$$ and $$\sigma^2 R_{N}-\sigma^2 R_{N,n}(\tau^2 I_{n} + \sigma^2R_{n})^{-1}\sigma^2 R_{n,N}.$$
Expressions above are known as [*kriging predictor*]{} and [*kriging variance*]{}, respectively. There are extensions of the basic model of Geostatistics, most of them developed to deal with non-stationarity (@higdon1999non [@fuentes2001new; @fuentes2002spectral] and @bornn2012) or non-Gaussianity [@diggle1998model].
Spatial Point Process
---------------------
The use of point processes for modelling patterns of points in space has been intensified in the last decades, specially after the publication of the classic texts of @ripley2005spatial and @diggle1983. The study of point processes has also evolved, based in recent computational advances. @moller2007modern present an excellent review of the methods and models to spatial point processes and highlight several applications and computational aspects related to inference.
A spatial point process [**X**]{} can be understood as a random finite subset of event locations belonging to a certain limited region $D
\subset \Re ^k$, where usually $k=1,2$ or 3. A spatial point process, defined in a region $D \subset \Re^k$, governed by a non-negative random function $\Lambda=\{\Lambda(x): x \in \Re^k\}$, is a [*Cox process*]{} if the conditional distribution of $[{\bf X} \mid
\Lambda=\lambda]$ is a Poisson process with intensity function $\lambda(x)$. Additionally, if one can assume that $\Lambda(x)=\exp
\{Z(x)\},$ where $Z=\{Z(x) \in \Re^k\}$ is a stationary and isotropic Gaussian process, then it is said to be [*log-Gaussian Cox process*]{} [@moller1998log].
The likelihood function associated with this process is given by $$p({\bf x} \mid Z) \propto \exp \left(-\int_{D} \exp \{Z(x)\}dx\right) \prod_{i=1}^{n} \exp \{Z(x_i)\}.$$ Although this function is not analytically tractable, inference on a log-Gaussian Cox process can be performed in a reasonably simple way through Monte Carlo simulation methods [@moller1998log]. Again, it is usual to represent the domain $D$ of the point process as a grid and approximate the Gaussian process $Z(x)$ by a finite-dimensional normal distribution defined on the grid. @waagepetersen2004convergence showed that the discretized posterior for log-Gaussian Cox process converges to the exact posterior when the sizes of the grid cells tend to zero. According to @moller1998log, if the point process is reasonably aggregated and has moderate intensity, the choice of the grid does not need to be fine to produce good results in inference. The inference conducted under the Bayesian paradigm, using MCMC methods, makes it relatively easy to obtain approximations of $p({\bf x} \mid
Z)$. On the other hand, the computational cost may be high [@moller2007modern].
Geostatistics under Preferential Sampling
-----------------------------------------
In the Geostatistics literature, it is common to consider the sample points [**x**]{} as fixed or, if coming from a stochastic process, independent of the process $S(x)$. When the sample design is stochastic, we must specify the joint distribution of $[Y, S, X]$. We have a process under [*preferential sampling*]{} if $[S, X] \neq [S]
[X]$, i.e. the sampling design is dependent of the spatial process.
The class of models proposed by @diggle2010geostatistical to accommodate preferential sampling effect assumes that, conditional on $S$, $X$ is a [*log-Gaussian Cox process*]{} with intensity $\lambda(x)=\exp\{ \alpha + \beta S(x) \}$. In addition, conditional on $[S,X]$, we have that $Y_{i} \sim N[\mu+S(x_i),\tau^2]$, $i=1,\dots,n$. @diggle2010geostatistical presents a way to evaluate this distribution through a fine discretization of the region $D$. The region $D$ can be discretized into $M$ cells with centroids $x_i, i =
1, \dots, M$, where only one point is expected in each cell.
In a Bayesian approach, we need to obtain the posterior distributions to make inference about the model parameters. For this purpose we assign Gaussian priors to $\alpha$ and $\beta$. @pati2011bayesian proved that the use of improper priors for the parameter that controls the preferential sampling effects produces proper posteriors. Thus, the data provide enough information to perform inference for this model parameter, even under vague prior information.
Since the posterior distributions of these parameters have no closed form, we use an approximation making use of MCMC methods in a discretized version of the model given by $$p({\bf x} \mid S,\alpha,\beta) \propto \prod_{i=1}^{M} [\exp(\alpha + \beta S(x_i))]^{n_i}
\exp \left(-\sum_{i} \Delta_i \exp(\alpha+\beta S(x_i))\right),$$ where $n_i$ and $\Delta_i$ represent the counts and the volume of the $i$th subregion, $i =1, \dots, M$.
The full conditional distributions of the parameters $\mu, \phi,
\tau^{-2}$ and $\sigma^{-2}$ are given by the same expressions obtained in the case of non-preferential sampling. The full conditional distributions of $\beta$ and $\alpha$ are updated in a similar way. The full conditional distribution of $S$ is updated in MCMC by Metropolis steps (see details in Appendix A).
Optimal Design
==============
In general, finding an optimal design involves procedures for obtaining the maximum or minimum of an objective function. These objective functions usually quantify the gains and losses related to each possible decision. In this case, we need to decide which locations (in time or space) will be collected in order to better understand certain characteristics of the phenomenon. When the phenomenon of interest is studied assuming that it is governed by an underlying stochastic process, the methodology for obtaining the optimal design is usually performed via Decision Theory. For more details related to Decision Theory, see the classical textbook of @degroot2005optimal.
According to @muller1999simulation, the procedure for obtaining the optimal design can be performed by defining a utility function $u({\bf d},\theta,{\bf y_d})$, where ${\bf d}=(d_1,\dots,d_m)'$ represents the $m$ new sample locations, $d_i \in D$, and ${\bf
y_d}=(y_{d_1},\dots,y_{d_m})'$ is the vector of future observations arising from it, $i=1,\dots,m$. After a set of observations [**y**]{} is available, the optimal design is the vector ${\bf d}^{*}$ that maximizes the function $$U({\bf d})= \int u({\bf d},\theta,{\bf y_d})p_{{\bf d}}({\bf y_d} \mid \theta,
{\bf y})p(\theta \mid {\bf y})d\theta d{\bf y_d}=E_{\theta,{\bf y_d} \mid
{\bf y}}[u({\bf d},\theta,{\bf y_d})].$$
In other approaches, one can include additional information to obtain the optimal design. In order to achieve the optimal design in time, @stroud2001optimal used covariates to reduce the model parameter uncertainties, thereby obtaining the more appropriate point in time for the return of a patient undergoing treatment. On the other hand, @ding2008bayesian used a hierarchical model to relate the effects of different treatments in clinical studies to determine the optimal design associated with a specific treatment.
An interesting strategy to optimize the expected utility is known as [*an augmented model*]{} [@muller1999simulation]. In this case, the optimal design point is considered as a parameter and can be estimated through its posterior distribution. Thus, an artificial probability model $h({\bf d},\theta,{\bf y_d})$ is defined assuming that $D$ is bounded and $u({\bf d},\theta, {\bf y_d})$ is non-negative and limited. The distribution $h({\bf d},\theta,{\bf y_d})$ is given by $$h({\bf d},\theta,{\bf y_d}) \propto u({\bf d},\theta,
{\bf y_d})p_{{\bf d}}({\bf y_d} \mid \theta)p(\theta).$$
Under this assumptions, the marginal distribution of ${\bf d}$ is proportional to $$h({\bf d}) \propto \int u({\bf d},\theta,{\bf y_d})p_{{\bf d}}({\bf y_d} \mid \theta)
p(\theta)d\theta d{\bf y_d} = U({\bf d}).$$ Therefore, finding the mode of $h({\bf d})$ becomes equivalent to maximizing the expected utility of ${\bf d}$. Since the sampled values of ${\bf d}$ in the MCMC algorithm will be concentrated near regions of high expected utility, less time is consumed in the simulation procedure. In other words, one only needs to find the mode of the distribution, instead of dealing with an optimization problem.
It is important to remark that this methodology can be naturally performed in situations where one wants to choose $m>1$ new design points. The main difficulty in this case is to face the increased computational cost involved since the evaluation of $u(\bf d)$ is required at each iteration in MCMC. In addition, a large sample of ${\bf d}=(d_1,\dots,d_m)'$ would be needed to evaluate $h(\bf d)$ properly. Alternatively, it is also possible to obtain the $m$ new points sequentially. However, in this case, there is no guarantee that the resulting design will be optimal. The evaluation of the utility function also allows removing points instead of adding new points. In this case, the decision would be based on gains and losses in the utility after eliminating each site or group of sites.
Utility Functions
-----------------
Commonly in Geostatistics, the researcher focuses more on spatial prediction than on inference about its parameters. In these cases, the choice of a utility function that depends on the predicted values and their variances is a natural choice. In this sense, the prediction variance associated with the location $x \in D$, $V(S(x) \mid {\bf y})$ gives to the researcher the degree of uncertainty about his/her predictions after observing the data [**y**]{}. Thus, reducing this variance throughout the region $D$ constitutes a reasonable criterion for choosing a new sample location $d$, $d \in D$. Based on this principle, the following utility function is a natural choice for implementing the approach described in @muller1999simulation in the context of Geostatistics: $$\label{eq:utilidade1}
u({\bf d},\theta,{\bf y_d})=\int [V(S(x) \mid \theta,{\bf y})-V(S(x)
\mid \theta, {\bf y}, {\bf y_d})]dx,$$ which can be interpreted as the gain obtained in reduction of prediction uncertainty after observing ${\bf y_d}$. Thus we need to maximize its expectation given by $U({\bf d})$.
Finally, it is also important to note that the predictive variances used in the utility function $u({\bf d}, \theta,{\bf y_d})$ depend only on the location of ${\bf y_d}$ in $D$, instead of its value. This feature of the predictive variance avoids the necessity of evaluating all possible values of ${\bf y_d}$ while maximizing $U({\bf d})$, thus reducing the computational cost involved. More details about the evaluation of the utility function (\[eq:utilidade1\]) are presented in Appendix B.
Other utility functions could be chosen. In Section 6, we employed a utility function that is higher for locations where extreme values or exceedances are expected. In this case, we have $$\label{eq:utilidade2}
u(d,\theta,y_d)=P[|S(x_d)| > x_0 \mid \theta,{\bf y}]$$ where $x_d$ represents the location associated with $y_d$ and $x_0$ is an extreme threshold. In practical situations, this expression could be improved by considering costs and risks related to the occurrences.
Optimal Design under Preferential Sampling
==========================================
The problem of obtaining the optimal design ${\bf d}^*$ for spatial processes under preferential sampling can be performed based on optimization of $$U({\bf d})=E_{\theta,{\bf y_d} \mid {\bf x},{\bf y}}[u({\bf d},\theta,
{\bf y_d})] = \int u({\bf d},\theta,{\bf y_d})p({\bf y_d} \mid \theta,
{\bf x},{\bf y})p(\theta \mid {\bf x},{\bf y})d\theta d{\bf y_d}$$ where $p(\theta \mid {\bf x},{\bf y})$ is obtained in Section 2. Given the posterior distribution of $\theta$ and the utility function, we can achieve the optimal design by seeking the mode of the posterior pseudo-distribution of ${\bf d}$. The greatest difficulty in this step is evaluate the effects of preferential sampling in the utility function $u({\bf d},\theta,{\bf y_d})$. In many cases, like the situations presented in this paper, the evaluation of this function becomes challenging.
As will be noted in the next section, preferential sampling directly impacts on the estimation of the mean $\mu$ of the underlying Gaussian process. If the utility function $u({\bf d},\theta,{\bf y_d})$ depends on this parameter, the choice of optimal design will be greatly affected. On the other hand, it would be expected that preferential sampling will also affects utility functions defined to quantify reductions of uncertainty related to each choice. This actually happens since the spatial configuration of the sample points also provides information about the underlying process.
Using as an example the utility function defined in Section 3, one would need to include the information provided by the observed point process [**x**]{}, i.e. $$u({\bf d},\theta,{\bf y_d})=\int [V(S(x) \mid \theta,{\bf y},{\bf x})-V(S(x)
\mid \theta, {\bf y},{\bf x},{\bf y_d})]dx,$$ and one would still need to know the variance of the distribution of $[S \mid \theta,{\bf y},{\bf x}]$. Samples of this distribution are easily obtained during the implementation of MCMC, as described in Section 2, but we cannot directly obtain estimates of this variance at each iteration of the algorithm. To deal with this difficulty, one must resort to an approximation. A sampling-based approximation would require an additional MCMC sub-chain at each iteration, thus increasing substantially the already high computational cost. An analytic, cheaper alternative is to use a Gaussian approximation of this distribution in order to evaluate the variances required in the utility function (\[eq:utilidade1\]). Note that this approximation is only used to evaluate $u({\bf d},\theta,{\bf y_d})$. More detail about this approximation are presented in Appendix C. On the other hand, if the utility function (\[eq:utilidade2\]) is used, we can directly evaluate $U(d)$ from $[S \mid {\bf y,x}]$, since a sample of this distribution will be available after performing the MCMC.
Simulation Study
================
In this section, we simulate datasets according the model presented in Section 2.3 to illustrate the effects of preferential sampling on inference and optimal design choice. In all cases, we chose a set of parameters which do not produce very large samples, in order to enhance the effects of preferential sampling. We also consider situations where one needs to add $m=1$ and $m=2$ new locations to the sample design. In order to produce the observations $y_1,\dots,y_n$, we first generate a surface $S(x)$, according the Geostatistical model presented in Section 2.1, over the discretized region $D$. Conditioning on $S$, we then generate a point process from the log-Cox Gaussian model presented in Section 2.2 in order to obtain $n$ observations. We considered five simulated studies:
- Simulation considering a one-dimensional region $D=[0, 100]$, partitioned into $M=100$ sub-regions, where $(\alpha; \beta; \mu; \sigma^2; \phi; \tau^2; n)=(-3; 2; 12; 2;
20; 0.1; 18)$.
- Simulation considering a one-dimensional region $D=[0, 100]$, partitioned into $M\,{=}\,200$ sub-regions, where $(\alpha; \beta; \mu; \sigma^2; \phi; \tau^2; n)\,{=}\,(-3.5; 3; 12;
1; 20; 0.01; 123)$.
- Simulation considering a one-dimensional region $D=[0, 200]$, partitionedinto $M=200$ sub-regions, where $(\alpha; \beta; \mu; \sigma^2; \phi; \tau^2; n)=(-1.5; 0.5;
12; 1; 20;\break 0.01; 56)$.
- Simulation considering a two-dimensional region $D=[0, 100]$, partitioned into $M=225$ sub-regions, where $(\alpha; \beta; \mu; \sigma^2; \phi; \tau^2; n)=(-8; 2; 12; 2;
20; 0.1; 12)$.
- Exactly as Case IV but with $M=400$ sub-regions.
[*Case V*]{} was considered in order to evaluate the discretization effect. Figure 1 shows the Gaussian processes $S$ simulated for [*Cases I–IV*]{}. Since $\beta> 0$ in all simulations, the observations are concentrated near the sites where the process $S$ showed higher values.
Inference and Prediction
------------------------
Posterior inference about model parameters was performed. Inference was also performed without considering preferential sampling, i.e. using the Geostatistical model of Section 2.1 to allow comparison. Prior distributions $\mu \sim \alpha \sim \beta \sim N(0;10^3)$, $\tau^{-2}
\sim \sigma^{-2} \sim G(2; 0.5)$ and $\phi \sim G(2; 0.05)$ were used in all cases. Furthermore, in [*Cases I–III*]{}, 500,000 iterations were generated in the MCMC algorithm and only the last 100,000 were used to compose the posterior distribution samples of the model parameters. For the other cases, 400,000 iterations were generated and only the last 50,000 were considered. The convergence of the chains was assessed by visual inspection of several chains generated from different initial values.
Considering the effect of preferential sampling, the posterior distributions of each model parameters are concentrated around the true values of the model’s parameters, except for the $\sigma^2$ in [*Case I*]{}. However, this parameter was also underestimated considering a non-preferential model. Figure 2 shows that the variogram estimated assuming the preferential sampling effects is slightly closer to the true variogram for the [*Cases I*]{} and $V$. Figure 2 also shows that the variograms estimated by the preferential model are closer than those estimated by the non-preferential model in [*Cases II*]{} and $IV$ (Figure 2(a),(b),(e), and (f)). Finally, in [*Case III*]{}, since $\beta$ is small, the estimated variograms are very similar (also in Figure 2).
In particular, in [*Case IV*]{}, the posterior distributions are not concentrated around the true values for all parameters of the model. This difficulty is partly justified by the small sample size. However, the results obtained by the model with preferential sampling are more satisfactory. Besides a better estimated variogram, this also can be observed through the posterior distributions of $\mu$, shown in Figure 3.
Figure 4 shows the predicted values of $S$, represented by the median of a posterior $S$, and the respective 95% credibility intervals for each model in [*Cases I*]{} and $II$.
Analysing Figure 4(a)–(b), we can see that only the credible intervals which consider the effect of preferential sampling encompass the most extreme values of the simulated process $S$. Additionally, it can be seen that the point estimates of $S$ in the regions where the process was hardly observed are better with the model with preferential sampling. The differences are even more pronounced when we analyse the results for [*Case II*]{}, where the intervals are narrower for the preferential model (Figure 4(c)–(d)). In addition, the non-preferential model underestimates the process $S$.
Since the estimated variograms obtained by both models were similar in [*Cases I*]{} and $V$, it seems reasonable to conclude that the differences observed in prediction for these cases were caused mostly due to the differences between the predictive distributions $[S\mid
{\bf y}]$ and $[S\mid {\bf y,x}]$.
Finally, Figure 5 shows the predicted surfaces of $S+\mu$, represented by the posterior means, obtained for each model in [*Case IV*]{}. It can be observed that only the preferential model can identify regions where the underlying process presents low values. This feature avoid predictions concentrated around the mean, like those obtained by the traditional kriging methods. In addition, we can conclude that the small sample size intensifies the effects of preferential sampling in prediction.
![Posterior distributions of $\mu$ under preferential sampling (left) and without considering this effect (right) for [*Case IV*]{} (red points represent the true values of the parameters).](944f03)
![Simulated process (solid line), posterior median of the predictive distribution (red line) and the respective 95% credibility intervals (dashed lines) for $S$ obtained by considering (left) and by not considering (right) the effect of preferential sampling in [*Cases I*]{} and $II$.](944f04)
![Posterior mean of the predictive distribution of $[S+\mu]$ obtained considering (left) and without considering (right) the effect of preferential sampling for [*Case IV*]{}.](944f05)
All simulations presented in this section reflect situations where there are few observed sample points. This is not uncommon in practice, especially in monitoring studies of environmental or climatic phenomena which are rare or difficult to detect. However, even under these conditions, the use of models assuming the existence of preferential sampling effects has produced variogram and kriging surfaces generally closer to the true values when compared to the estimates produced without considering this effect. Another major advantage of using these models is the correction that is made during the inference about the underlying process mean. Finally, we showed the ability of this model to identify areas where the underlying process takes extreme values, even in situations where there are no samples nearby.
@diggle2010geostatistical evaluated the influence of preferential sampling influence in the predictions of Gaussian processes using $[S|{\bf y},\hat{\theta}]$ after correcting the bias caused by preferential sampling. However, these predictions did not take into account the information provided by [**x**]{}. In the context of environmental monitoring, @shaddick2014case also presented a methodology for correcting the bias caused by a selective reduction of sample sites. In contrast, the Bayesian approach provides samples directly of the distribution of interest $[S|{\bf y},{\bf x}]$. The comparison between the predictions assuming that $\theta$ is known reinforced the conclusion that methods based on corrections of the variogram’s bias are not sufficient to reproduce the true uncertainty associated with the predictive distribution of the underlying process $S$.
Performing different simulated scenarios of prediction under preferential sampling effects, @gelfand2012effect also concluded that they affect more significantly the spatial prediction than the estimation of the model parameters. In their paper, they discussed ways to evaluate the effects of preferential sampling by comparing two predicted surfaces. One of the forms of global comparison they mentioned is associated with the local and the global mean squared prediction error. The [*Local Prediction Error*]{} associated with $x_0$, denoted $LPE(x_0)$, is given by $$LPE(x_0)=E[\hat{S}(x_0)-S(x_0)]^2,$$ where $\hat{S}(x_0)$ is the predictor of $S$ in $x_0$. The [*Global Prediction Error*]{} is given by $$GPE=\frac{1}{|D|}\int_{D} LPE(x)dx.$$ Table 1 presents the $GPE$ values for each of the simulations. Based on this table, one can observe a significant reduction obtained by considering the effects of preferential sampling.
Simulation No preferential sampling Preferential sampling
---------------- -------------------------- -----------------------
[*Case I*]{} 0.9496 0.7301
[*Case II*]{} 0.5533 0.1783
[*Case III*]{} 0.5286 0.4512
[*Case IV*]{} 2.1336 1.6789
[*Case V*]{} 1.6214 1.3482
: Global Prediction Error (GPE) for each of the simulations.
The one-dimensional simulation showed that the $LPE$s are reduced when they refer to the locations where the underlying process $S$ has lower values. Similar conclusions were obtained in the second simulation, since errors remain smaller in regions where the magnitude of $S$ is lower when the preferential sampling effect is taken into account in the modelling. Further simulations without the effect of preferential sampling were also performed. In such cases, in general, the model assuming preferential sampling produced posterior distributions of $\beta$ centred at zero.
Optimal Design
--------------
Using the one-dimensional simulated data presented in [*Case I*]{}, 83 auxiliary points, required for evaluation of the utility function (\[eq:utilidade1\]) described in Section 3, were also used to form a grid and obtaining the predictive variance reductions. Using the samples of $\theta$, which were obtained from the posterior distribution in the MCMC algorithm, we can generate samples of $d$ as mentioned in Section 3 to obtain a new optimal sample point for the case where $m=1$.
Figure 6 shows the histograms of the posterior pseudo-distribution of $d$ with and without the preferential sampling assumption, respectively. It can be noted that the optimal design choice without preferential sampling leads the researcher to select locations where there are no nearby points, i.e. in the interval $[5, 35]$. On the other hand, under preferential sampling, the results lead the researcher to a different direction. In this case, except in the subregions where there are several observed samples, the other choices have similar expected utilities. In summary, under preferential sampling, the optimal design choice is notably changed when the researcher wants to reduce the predictive variance.
![Posterior pseudo-distributions of $d$ under the effect of preferential sampling (left) and without considering this effect (right) for [*Case I*]{}.](944f06)
To illustrate the effect of the preferential sampling where $m>1$, we obtained the pseudo-distribution of ${\bf d}=(d_1,d_2)'$ assuming that two new locations would need to be added in the sample design. The results are presented in Figure 7. Analysing the results for the non-preferential model, it can be seen that the optimal solution could be to add one point $d_1$ from interval $[0, 20]$ and to make no restrictions to the location of the other point $d_2$, since $h({\bf
d})$ does not vary much. On the other hand, the results for the preferential model indicate that the optimal design should include one point $d_1$ from interval $[0, 4]$ and another point $d_2$ from interval $[88, 92]$.
![Posterior pseudo-distributions of ${\bf d}=(d_1,d_2)'$ under the effect of preferential sampling (left) and without considering this effect (right) for [*Case I*]{}.](944f07)
In the two-dimensional [*Case IV*]{}, 900 auxiliary points were also used to form a grid and obtaining the predictive variance reductions, as in the previous section. Figure 8 shows the posterior pseudo-distributions of $d$ under preferential sampling and without considering this effect for the case where $m=1$.
![Posterior pseudo-distribution of $d$ under the effect of preferential sampling (left) and without considering this effect (right) for [*Case IV*]{}. The densities of these pseudo-distribution are multiplied by 100 for better visualization.](944f08)
It can be noted in Figure 8 that the areas with the highest expected utility are not the same for the two models. As expected, the results obtained without considering the effect of preferential sampling makes the researcher choose locations far from the observed points. In contrast, under preferential sampling, the largest utilities expected are more dispersed over region $D$. Again, under preferential sampling, the optimal design choice was changed, since the regions with few sample sites also provide useful information about $S$.
The two simulated studies have involved situations where the inference step has produced similar results (in the one-dimensional case) and different results (in the two-dimensional case). Surprisingly, even in the one-dimensional case, where the estimated variograms were similar, the process of choosing the optimal design led to quite different results.
However, the use of a different utility function can produce even more extreme results. Even though the utility function chosen in this paper was designed for our Geostatistics goals, other functions could be considered. According to the results obtained from the simulated studies, utility functions that depend on the underlying process $\mu$ may also be affected by the preferential sampling effect. This situation will be explored in next section. There are other effects that may affect the results, such as the choice of the auxiliary grid (used to evaluate the predictive variance reduction) and the discretization level of $D$. Current computational costs associated with this methodology are still a barrier for a more thorough evaluation of the degree of influence of each of these marginal effects.
### Effectiveness of the Optimal Decision
After obtaining the optimal design, one can evaluate if the results are better under preferential sampling. Thus, we performed an analysis of the $GPE$ after this decision to the two-dimensional simulated data [*Case IV*]{}. The coordinates of the optimal design was $x_d=(90.00;76.66)$, under preferential sampling, and $x_d=(30.00;50.00)$ without this effect. In addiction, it was assumed that $\tau^2=0$.
Finally, we proceed to the inference via classical Geostatistics methods but including the optimal data point $Y_d$ obtained under preferential sampling. The results provided a $GPE=1.8392$, which is lower than that obtained using the optimal data point $Y_d$ pointed out by the non-preferential model ($GPE=2.3348$). Thus, the optimal design obtaining under preferential sampling was more advantageous even when the inference is performed via classical Geostatistics methods.
Case Study: Rainfall Data in Rio de Janeiro
===========================================
The methodology is now applied to a real scenario in the context of monitoring networks. More specifically, we will analyse pluviometric precipitation data obtained from 32 monitoring stations located in the city of Rio de Janeiro, Brazil. The data refer to the period from 1 to 31 October 2005 and were obtained from the [*Pereira Passos*]{} Institute, an official agency associated with the local government. The rainfall during this month, which begins the rainy season in the Brazil’s Southeast, is of particular interest to meteorologists and government agencies [@alves2005]. Figure 9 shows the map of the Rio de Janeiro city with the respective precipitation levels observed.
Analysis of the spatial distribution of the precipitation seems to indicate that the stations are more concentrated near places where rainfall level is higher. Even though the geography and the spatial distribution of economic activities could be considered as possible causes of this sample design, the methodology for choosing a new design point under the effect of preferential sampling can be employed here.
For inference, we used the same priors of the previous simulations (by changing some hyperparameters) and the study area was partitioned into $M=332$ subregions. We monitored 100,000 iterations in the MCMC algorithm for both models and the first 10,000 were considered as burn-in. The convergence of the chains was assessed by visual inspection of several chains generated from different initial values. Table 2 presents summaries of the posterior distributions for all model parameters. An analysis of the results suggests that the effect of preferential sampling is significant, indicating a positive association between the sample design and the rainfall intensity.
Model parameters Preferential Non-Preferential
------------------ ----------------------------- ----------------------------
$\tau^2$ 1.25 (0.49; 2.90) 1.32 (0.51; 3.45)
$\sigma^2$ 4289.92 (2096.31; 10528.91) 4132.72 (2120.30; 8514.38)
$\mu$ 104.84 (97.73; 110.60) 119.88 (111.49; 130.32)
$\phi$ 10.69 (4.27; 26.75) 10.43 (4.51; 22.76)
$\alpha$ $-$3.84 ($-$4.24; $-$3.48) —
$\beta$ 0.008 (0.002; 0.014) —
: Posterior mean and 95% credibility interval (in parentheses) for model parameters.
As for the other model parameters, we observed differences between the estimates for the mean $\mu$, which can be explained by the presence of preferential sampling effects. The utility function (\[eq:utilidade2\]) was used to obtaining the optimal design, with $x_0 = 200$ mm. This utility function assigns greater utility for regions where it is most likely to observe a monthly rainfall above 200 mm. The expected utility $U(d)$ obtained for each model is shown in Figure 10.
The preferential model has concentrated high expected utility into a small region in map since the utility function favours regions with the highest probability of extremes values and due to the overestimation of $\mu$. On the other hand, the non-preferential model has produced expected utilities more spread out over the southern part of the city.=-1
Discussion
==========
The results obtained in Sections 5 and 6 help us to understand the effects of preferential sampling on inference and optimal design choice in the context of Geostatistics. The results of Section 5 showed that the bias corrections made during the estimation step are not enough to ensure a satisfactory prediction. On the other hand, the Bayesian approach presented here allows us to make predictions about any functional of $S$ directly from $[S\mid {\bf y,x}]$, which is the correct predictive distribution under the effect of preferential sampling. Furthermore, the inference about the parameters that define the effect of preferential sampling, i.e. $\alpha$ and $\beta$, was quite satisfactory in all simulations. Despite the high associated computational cost, the approach is computationally feasible and showed satisfactory results.
In practical situations, it may be unlikely to assume that the design is governed by a log-Gaussian Cox process. However, this model seems to be flexible to obtain understanding about the consequences if the researcher has no covariates or a better explanation for its true causes. Although not applicable in the strictly spatial context, an alternative approach to detect and deal with preferential sampling is spatio-temporal analysis, since the changes on site locations over time may be informative about the association between its configuration and the underlying process. @zidek2014reducing present a method that can learn about the preferential selection process over time and that allows the researcher to deal with its effects.
Traditional approaches to deal with preferential sampling also include the fit of trend surfaces to assess first order effects and, in the context of survey sampling methods, weighting schemes. However, both require some prior knowledge about the sample selection processes. The researcher must carefully evaluate the (theoretical or empirical) evidence of preferential sampling effects. The use of this approach can produce misleading results otherwise. If there is evidence of preferential sampling, the authors believe the use of models based on latent processes, such as log-Gaussian Cox process, should be used even when the dependence structure between $X$ and $S$ is not completely known.
As evidenced by simulations, the effects of preferential sampling on optimal design choice cannot be disregarded. The knowledge about the sampling pattern reduces the predictive variance of $S$ in areas poorly sampled, substantially changing the optimal decision. On the other hand, a utility function based on exceedances seems to overestimate $\mu$ under preferential sampling. As a result, the ideal region to receive a new sample becomes very small. However, obtaining the optimal design in situations where there is a need to monitor extreme events or exceedances may be not simple. @chang2007designing presented an approach to deal with some challenges arising in designing networks for monitoring fields of extremes, as the loss of spatial dependence and the limitations of conventional approaches.
Potential areas to apply this methodology include the study of phenomena scarcely observed and those in which, due to researcher’s interest or limited resources, can only be observed in locations considered critical. One can include the monitoring of mosquitoes or other disease spreading pests that only become detectable in places where its occurrence is very high among the phenomena that are scarcely observed and difficult to be detected. Another potential application is the monitoring of maximum and minimal temperatures in regions near airports or industrial plants. In both cases, it seems reasonable to infer that the way the phenomenon is observed can be related to the underlying process. In this situation, the optimal design choice can change significantly, since the spatial point pattern brings valuable information to the researcher.
The methodology employed can be adjusted in order to produce pseudo-distributions of $d$ more peaked around the mode. The strategy based on simulated annealing [@muller1999simulation] can be explored for this purpose. The approach of @muller1999simulation for optimal design choice has computational advantages in comparison with traditional procedures of optimization. The choice of a utility function based on predictive variance reduction is easily justified in Geostatistics. Under preferential sampling, the use of utility functions that directly depend on $\mu$ seems to be more affected than those based in variance reductions.=-1
This methodology also has an expensive computational cost and alternative procedures can be used to deal with this problem, e.g. @simpson2011going that use the [*Integrated Nested Laplace Aproximation*]{} – INLA methods [@rue2009approximate], the use of Predictive Process [@banerjee2008gaussian], methods to approximate likelihoods [see @stein2004approximating; @fuentes2007approximate] and the use of sparse covariance matrices [@furrer2006covariance].=-1
The authors recognize the limitations in the simulation studies presented here. The complexity and the variety of situations arising from designing under preferential sampling lead us to focus on specific features rather than obtaining more general conclusions. Finally, the authors also invite researchers interested in reproducing the methodology presented in this paper to contact the authors in order to obtain more details about the computational implementation.
Appendix A {#appendix-a .unnumbered}
==========
In the preferential model, the full conditional distributions of $S,\beta$ and $\alpha$ are proportional to $$\begin{aligned}
p(S \mid \mu,\tau^{-2},\sigma^{-2},\phi,\alpha,\beta,{\bf x,y})
& \propto\exp \left\{ -\frac{1}{2\tau^2} [S_{\bf y}'S_{\bf y} - 2S_{\bf y}'({\bf y}
- \mu {\bf 1}) ] + \beta S'{\bf n} - \frac{S'R_{M}^{-1}S}{2\sigma^2} \right\}\\
&\quad \times \exp \left\{ - \Delta e^{\alpha} \sum^{M} \exp(\beta S(x_i)) \right\},\end{aligned}$$ $$\begin{aligned}
p(\beta \mid S, \mu,\tau^{-2},\sigma^{-2},\phi,\alpha,{\bf x,y}) &\propto
p({\bf x} \mid S,\alpha,\beta)p(\beta)\\
&\propto \exp \left\{ \beta S'{\bf n} - \Delta e^{\alpha} \sum^{M}
\exp(\beta S(x_i)) - \frac{\beta^2}{2k} \right\} ,\end{aligned}$$ $$p(\alpha \mid S, \mu,\tau^{-2},\sigma^{-2},\phi,\beta,{\bf x,y}) \propto \exp
\left\{ n\alpha - \Delta e^{\alpha} \sum^{M} \exp(\beta S(x_i))
- \frac{\alpha^2}{2k} \right\}.$$ In the MCMC, these quantities can be updated in Metropolis steps assuming a Gaussian proposal distribution centred in the previous values sampled. Then, we accept the proposal with probability $$\begin{aligned}
p_S&=\exp \biggl\{ -\frac{1}{2\tau^2} [S_{\bf y}'^{prop}S_{\bf y}^{prop} -
S_{\bf y}'S_{\bf y} - 2(S_{\bf y}^{prop}-S_{\bf y})'({\bf y} - \mu {\bf 1}) ] +
\beta (S^{prop}-S)'{\bf n} \biggr\} \\
&\quad \times \exp \biggl\{ \Delta e^{\alpha} \sum^{M} [\exp(\beta S(x_i))
-\exp(\beta S(x_i)^{prop})] + \frac{(S'R_{M}^{-1} S
- S'^{prop} R_{M}^{-1} S^{prop})}{2\sigma^2} \biggr\},\end{aligned}$$ $$p_{\beta}=\exp \biggl\{ (\beta^{prop}\,{-}\,\beta) S'{\bf n} \,{+}\,
\Delta e^{\alpha} \sum^{M} (\exp(\beta S(x_i))\,{-}\,\exp(\beta^{prop} S(x_i))) \,{+}\,
\frac{(\beta^2-\beta^{2prop})}{2k} \biggr\},$$ $$p_{\alpha}=\exp \biggl\{ (\alpha^{prop}-\alpha) n + (e^{\alpha}-e^{\alpha^{prop}})
\Delta \sum^{M} \exp (\beta S(x_i)) + \frac{(\alpha^2-\alpha^{2prop})}{2k} \biggr\},$$ respectively. The vector ${\bf n}'=(n_1,n_2,\dots,n_M)$ represents the number of observations in each subregion, where $\sum_{i=1}^M n_i=n$.
In both models, the full conditional distribution of $\phi$ is proportional to $$p(\phi \mid S,\mu,\tau^{-2},\sigma^{-2},\alpha,\beta,{\bf
y}) \propto |R_M|^{-1/2} \phi^{a_{\phi}-1} \exp \left\{
-\frac{S'R_{M}^{-1}S}{2\sigma^2} -b_{\phi}\phi \right\}$$ and this parameter can be updated in Metropolis steps assuming the following proposal distribution $$q(\phi^{prop} \mid \phi) \sim Lognormal\left( \ln(\phi) - \delta/2; \delta \right),$$ where $\delta$ must be chosen in order to produce reasonable acceptance rates in MCMC. Then, we accept the proposal with probability $$\begin{aligned}
p_\phi&= \left( \frac{|R_M|^{prop}}{|R_M|} \right)^{-1/2}
\left( \frac{\phi^{prop}}{\phi}\right)^{a_{\phi}}
\times \exp \biggl\{ -\frac{(S'R_{M}^{prop-1}S-S'R_{M}^{-1}S)}{2\sigma^2}
+b_{\phi}(\phi-\phi^{prop})\\
&\quad - \frac{(\ln\phi-\ln\phi^{prop}+ \delta/2)^2-(\ln\phi^{prop}
-\ln\phi+ \delta/2)^2}{2\delta}\biggr\}.\end{aligned}$$
Appendix B {#appendix-b .unnumbered}
==========
To evaluate the integral in expression of $u({\bf d}, \theta, {\bf
y_d})$ a discretization of the region $D$ can be applied yielding $M$ subregions, as described in Section 2. Then, we have that $u({\bf d},
\theta, {\bf y_d})$ can be approximated by $$\tilde{u}({\bf d},\theta,{\bf y_d}) = \frac{1}{M} \sum_{i}
[V(S_i \mid \theta,{\bf y})-V(S_i \mid \theta, {\bf y}, {\bf y_d})],$$ where $[S_i \mid \theta,{\bf y}]$ and $[S_i \mid \theta, {\bf
y}, {\bf y_d}]$, $i =1,\dots,M$, are Gaussian with respective means $$\begin{aligned}
\sigma^2{\bf r}^{'}_{n}(\tau^2 I_{n} + \sigma^2R_{n})^{-1}({\bf
y}-{\bf 1}\mu) \quad \mbox{and} \quad \sigma^2{\bf r}^{'}_{n+m}(\tau^2
I_{n+m} + \sigma^2R_{n+m})^{-1}({\bf y}^{*}-{\bf 1}\mu)\end{aligned}$$ and variances $$\begin{aligned}
\sigma^2-\sigma^2{\bf r}^{'}_{n}(\tau^2 I_{n} +
\sigma^2R_{n})^{-1}\sigma^2{\bf r}_{n} \quad \mbox{and} \quad
\sigma^2-\sigma^2{\bf r}^{'}_{n+m}(\tau^2 I_{n+m} +
\sigma^2R_{n+m})^{-1}\sigma^2{\bf r}_{n+m},\end{aligned}$$ where ${\bf y}^{*}=({\bf y}, {\bf y_d})'$.
Appendix C {#appendix-c .unnumbered}
==========
Following the procedure proposed in Section 4, we can make an approximation in $p({\bf x} \mid S, \alpha, \beta)$ to obtain an analytical expression of $V[S \mid \theta,{\bf y},{\bf x}]$. In particular, expanding the exponential function in Taylor series around zero, up to the second order term, we obtain the following approximation $$\sum_i \exp(\alpha+\beta S(x_i)) \approx e^{\alpha}\left({\bf 1_{M}}
+ \beta{\bf 1_{M}}'S + \frac{\beta^2}{2}S'S \right).$$
By inserting this expression in $p({\bf x} \mid S, \alpha, \beta)$, it can be shown that the conditional distribution $[S \mid \theta,{\bf
y},{\bf x}]$ becomes Gaussian with mean vector $\Theta$ and covariance matrix $\Sigma$ given by $$\begin{aligned}
\Theta=\Sigma \times \left(\begin{array}{c} ({\bf y_n}-\mu {\bf
n})\tau^{-2} + \beta {\bf n} - \Delta \beta e^{\alpha}{\bf n} \\[3pt] -
\Delta \beta e^{\alpha}{\bf 1_N} \end{array} \right)\quad \mbox{and}\end{aligned}$$ $$\Sigma= \left(\begin{array}{cc}
(\tau^{-2} + \Delta \beta^2 e^{\alpha})I_{{\bf n}} + R_{n}^{-1}R_{n,N} A^{-1}
R_{N,n}R_{n}^{-1} + \sigma^{-2}R_{n}^{-1} & -R_{n}^{-1}R_{n,N} A^{-1} \\[3pt]
-A^{-1}R_{N,n}R_{n}^{-1} & \Delta \beta^{2} e^{\alpha} I_{N} + A^{-1} \end{array}\right)^{-1},$$ where $A = \sigma^2 R_N- \sigma^2 R_{N,n} R_{n}^{-1}
R_{n,N}$, and the vectors [**n**]{} and ${\bf y_n}$ represent the number of observations and the total observed in each subregion of $D$, respectively. If , this matrix becomes equal to the kriging variance matrix traditionally obtained in Geostatistics.
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This paper is based on the doctoral thesis of the first author under the supervision of the second author. The authors thank the referees for numerous comments and suggestions that led to substantial improvement of the presentation. The second author wishes to thank the support from CNPq-BRAZIL.
|
---
abstract: 'Training a good deep learning model often requires a lot of annotated data. As a large amount of labeled data is typically difficult to collect and even more difficult to annotate, data augmentation and data generation are widely used in the process of training deep neural networks. However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance. In this paper, we try to address such a question using vehicle license plate character recognition as an example application. We apply computer graphic scripts and Generative Adversarial Networks to generate and augment a large number of annotated, synthesized license plate images with realistic colors, fonts, and character composition from a small number of real, manually labeled license plate images. Generated and augmented data are mixed and used as training data for the license plate recognition network modified from DenseNet. The experimental results show that the model trained from the generated mixed training data has good generalization ability, and the proposed approach achieves a new state-of-the-art accuracy on Dataset-1 and AOLP, even with a very limited number of original real license plates. In addition, the accuracy improvement caused by data generation becomes more significant when the number of labeled images is reduced. Data augmentation also plays a more significant role when the number of labeled images is increased.'
author:
- Changhao Wu
- 'Shugong Xu[^1]'
- Guocong Song
- Shunqing Zhang
bibliography:
- 'myreference.bib'
title: 'How many labeled license plates are needed?'
---
Introduction
============
License plate recognition is one of the most important components of modern intelligent transportation systems. It has attracted the attention of many researchers. However, most existing algorithms[@Gou2016Vehicle; @li2016reading; @li2018toward; @wang2017adversarial] can only work normally under certain conditions. For example, some recognition systems require sophisticated hardware to shoot high-quality images, while other systems require the vehicle to slowly pass through a fixed access opening or even stop. Accurately detecting license plates and recognizing characters in an open environment is a challenging task. The main difficulties are different license plate fonts and colors, character distortion caused by the image capture process and non-uniform illumination, and low-quality images caused by occlusion or motion blur.
In this paper, we propose a license plate recognition system, in which we cope with challenge such as, low light, low resolution, motion blur, and other harsh conditions. Fig. \[fig:complex\] shows the license plates which can be correctly recognized by our proposed method. From top to bottom are the license plate images affected by the shooting angle, uneven illumination, low resolution, detection error and motion blur.
![The complex license plates images.[]{data-label="fig:complex"}](cplx.pdf){width="80.00000%"}
In general, supervised learning requires a large amount of labeled data in order to achieve good results. However, real data is not easy to obtain, the acquisition process is slow, and the data needs to be processed and annotated before it can be used for training. To achieve a higher accuracy of the annotation, manual inspection is also required.
However, the acquisition of a large amount of real data and manual annotations is very expensive. Therefore, data generation is very important for the training of license plate recognition network. We believe that the information contained in a small number of real license plates is sufficient to recognize most of the existing license plate images. However, there is no clear common understanding on how much labeled data is needed to get satisfactory performance. In this paper, we try to address such a question in vehicle license plate character recognition.
The main contributions of this paper can be summarized as the following three points:
1.We propose various methods of data generation and data augmentation. As long as we have a few labeled license plate images, a large amount of generated data can be created. We can achieve and even exceed the recognition accuracy and results of systems trained only on real images.
2.We compare the performance of various data generation and data augmentation methods to find that both data generation methods and data augmentation methods can significantly improve license plate recognition accuracy. Data augmentation plays a larger role in accuracy improvement when there are many labeled license plates but when the number of labelled license plates is small, data generation more significantly increases accuracy.
3.We apply a network that is modified from DenseNet to license plate recognition to reduce network parameters and inference time and improve accuracy.
The rest of paper is arranged as follows. In Section 2 we review the related works briefly. In Section 3 we describe the details of networks used in our approach. Experimental results are provided in Section 4, and conclusions are drawn in Section 5.
Related Work
============
The section introduces previous work on license plate recognition and GANs.
License Plate Recognition
-------------------------
Existing license plate recognition systems are either text segmentation-based [@Gou2016Vehicle; @Guo2008License], or non-segmentation-based [@li2016reading]. Methods that depend on segmentation first preprocess the license plate image and then segment individual characters through image processing. After this, each character is classified by a convolutional neural network. This method is very dependent on the accuracy of text segmentation, and the recognition speed is slower. A recognition method that does not require segmentation is proposed by Li et al. [@li2016reading]. It is composed of a deep convolutional network and a Long Short-Term Memory(LSTM), where the deep CNN is directly applied for feature extraction, and a bidirectional LSTM network is applied for sequence labeling. DenseNet[@huang2017densely] is a highly efficient convolutional neural network. Because of its low parameter number and fast inference time, DenseNet is widely used. Our method is also a segmentation-free approach based on the framework proposed by [@huang2017densely], where DenseNet is applied for feature extraction.
Data generation is used in license plate recognition to improve the accuracy of recognition. The labeled license plates generated by CycleGAN as a pre-training data set for the recognition network are used in [@wang2017adversarial], and the model is fine-tuned with the real license plate data set. This data generation method can significantly improve the recognition accuracy. License plate detection and recognition is combined in [@li2018toward], and it finally improves the recognition speed and recognition accuracy of the system.
Generative Adversarial Networks
-------------------------------
Generative adversarial networks(GANs) [@goodfellow2014generative; @radford2015unsupervised] train a generator and discriminator alternatively. The output of the discriminator acts as a generator’s loss function. Zhu et al. [@zhu2017unpaired] propose Cycle-Consistent Adversarial Networks(CycleGAN), which learns the mapping relationship from one domain to another and is mainly used for the style conversion of pictures. Wasserstein GANs(WGAN) [@arjovsky2017wasserstein] are proposed to improve the stability of GAN training. Applying Wasserstein loss to CycleGAN, creating CycleWGAN, also improves its training stability in [@wang2017adversarial]. Gradient penalties in WGAN(WGAN-GP) [@gulrajani2017improved] are proposed to solve the WGAN generator weight distribution problem.
Data Generation For Training
----------------------------
A large number of real labeled images are often difficult to obtain, so the role of data generation is very significant [@shrivastava2017learning]. The synthesized images are used to train scene text detection networks [@gupta2016synthetic] and recognition networks [@jaderberg2014synthetic]. The generated data is shown to improve the performance of person detection [@yu2010improving], font recognition [@wang2015deepfont], and semantic segmentation [@ros2016synthia]. However, when the difference between the generated data and the real data is very large, the performance is poor when applied to a real scene. Therefore, [@wang2017adversarial] applies CycleGAN to convert the style of license plate generated by the script into a real license plate, which can greatly reduce the gap between the generated image and the real image. We apply data generation and data augmentation methods at the same time, and use the data generated by different methods directly as the training set for recognition network. Therefore we need very little real data.
License plate recognition based on data generation and augmentation.
====================================================================
In this section, the pipeline of the proposed method is described. We train the GAN model using synthetic images and real images simultaneously. We then use the generated images to train a model modified from DenseNet.
CycleGAN
--------
CycleGAN [@zhu2017unpaired] learns to translate an image from a source domain X to a target domain Y in the absence of paired examples. Our goal is to train a mapping relationship G between the script license plate domain X and the real license plate domain Y. CycleGAN contains two mapping functions $G:X\rightarrow Y$ and $Y\rightarrow X$, and associated adversarial discriminators D$_Y$, D$_X$.
The techniques proposed in WGAN [@arjovsky2017wasserstein] are applied in CycleGAN, and CycleWGAN is proposed in [@wang2017adversarial]. WGAN points out why the traditional GAN is difficult to converge and improve during training, which greatly reduces the training difficulty and speeds up the convergence. There are two main improvements: the first one is to remove the log from the loss function, and the second is to perform weight clipping after each iteration to update the weight, and limit the weight to a range (eg, the limit range is \[-0.1, +0.1\]. Outside weights are trimmed to -0.1 or +0.1). CycleWGAN solves the problem of training instability and collapse mode, which makes the result more diverse.
We apply the techniques in WGAN-GP [@gulrajani2017improved] to CycleWGAN and propose the CycleWGAN-GP. WGAN-GP also proposes an improvement plan based on WGAN. WGAN reduces the training difficulty of GAN, but it is still difficult to converge in some conditions, and the generated pictures are worse than DCGAN. WGAN-GP applies gradient penalty, and solves the above problem along with the problems of vanishing gradient and exploding gradient during training. It also converges faster than CycleWGAN and produces higher quality pictures.
We apply a CycleGAN equipped with WGAN and WGAN-GP techniques to train the mapping relationship between the fake license plate and the real license plate. First of all, we apply OpenCV scripts to generate synthetic license plates as a source domain X, and then choose real license plates without labels as a target domain Y. Before the training of CycleWGAN-GP, these license plates are randomly cropped and randomly flipped horizontally or vertically.
Recognition network design
--------------------------
DenseNet is a densely connected convolutional neural network. In this network, there is a direct connection between any two layers. The input of each layer of the network is the union of the output of all previous layers, and the feature map learned by this layer is also directly transmitted to all subsequent layers. DenseNet allows the input of l$^{th}$ Layer to directly affect all subsequent layers. Its output is: $$x_l=H_l([x_0,x_1,...,x_{l-1}]) \label{equ:dense}$$ where $H_l(\cdot)$ refers to a composite function of three consecutive operations: batch normalization (BN) [@Ioffe2015Batch], followed by a rectified linear unit (ReLU), and a $3\times 3$ convolution (Conv). Additionally, since each layer contains the output information of all previous layers, it only needs a few feature maps, so the number of parameter of DenseNet is greatly reduced compared to other models.
Layers Output Size Recognition Network
---------------- ------------------------- ------------------------------------------
Input 136$\times$36$\times$1
Convolution 68$\times$18$\times$64 5$\times$5 conv, stride 2
Dense Block(1) 68$\times$18$\times$128 [\[]{} 3$\times$3 conv [\]]{} $\times$ 8
68$\times$18$\times$128 1$\times$1 conv
34$\times$9$\times$128 2$\times$2 average pool, stride 2
Dense Block(2) 34$\times$9$\times$192 [\[]{} 3$\times$3 conv [\]]{} $\times$ 8
34$\times$9$\times$128 1$\times$1 conv
17$\times$4$\times$128 2$\times$2 average pool, stride 2
Dense Block(3) 17$\times$4$\times$192 [\[]{} 3$\times$3 conv [\]]{} $\times$ 8
: Construction of recognition network. The output size represents w$\times$h$\times$c. Note that each ¡°conv¡± layer shown in the table corresponds the sequence BN-ReLU-Conv[]{data-label="tab:construction"}
Our network structure is shown in Table \[tab:construction\], which is different from the network structure of [@huang2017densely], because the input license plate image is smaller and is a gray scale image of 136$\times$36, so the network only has 3 dense blocks. The transition layers used in our network consist of a batch normalization layer and an 1$\times$1 convolutional layer followed by a 2$\times$2 average pooling layer. A 1$\times$1 convolution can be introduced as bottleneck layer before each 3$\times$3 convolution to reduce the number of input feature-maps. To improve model compactness, we reduce the number of feature-maps from 192 to 128 at transition layers 2.
The last DenseNet layer is followed by a fully-connected layer with 68 neurons for the 68 classes of label, including 31 Chinese characters, 26 letters, 10 digits and ¡°blank¡±. We train the networks with stochastic gradient descent (SGD). The labelling loss is derived using Connectionist Temporal Classification (CTC) [@graves2006connectionist]. The optimization algorithm Adam [@kingma2014adam] is then applied, as it converges quickly and does not require a complicated learning rate schedule. Another advantage of using the modified DenseNet network is that it does not require the Long Short-Term Memory(LSTM) networks. The use of LSTM complicates the solution and increases computational cost.
Experiment
==========
In this section, we conduct experiments to verify the effectiveness of the proposed methods. Our network is implemented capitalizing keras. The experiments are trained on a NVIDIA Tesla P40 with 24GB memory and are tested on a NVIDIA GTX745 GPU with 4GB memory.
Dataset
-------
The image in the Dataset-1 [@wang2017adversarial] are captured from a wide variety of real traffic monitoring scenes under various viewpoints, blurring and illumination. Dataset-1 contains a training set of 203,774 plates and a test set of 9,986 plates. The first character of Chinese license plates is a Chinese character which represents the province. While there are 31 abbreviations for all of the provinces, Dataset-1 contains 30 classes of them.
The second data set is the application-oriented license plate (AOLP) [@hsu2013application] benchmark database, which has 2049 images of Taiwan license plates. This database is categorized into three subsets: access control (AC) with 681 samples, traffic law enforcement (LE) with 757 samples, and road patrol (RP) with 611 samples.
Implementation Details
----------------------
### Network
The recognition network is shown in Table \[tab:construction\]. We implement it with Keras. The images are resized to 136$\times$36 and converted to gray scale and then fed to the recognition network. We change the last layer of fully connected layers to 68 neurons according to the 68 classes of characters-33 Chinese characters, 24 letters, 10 digits and “blank”. We train the networks with SGD and learning rate of 0.0001. The labelling loss is derived using CTC. We set the training batch size as 256 and predicting size as 1.
### Evaluation Criterion
In this work, we evaluate the model’s performance in terms of recognition accuracy and character recognition accuracy, which is similar to Wang et al.[@wang2017adversarial]. Recognition accuracy is defined as : $$RA = \frac{Number\ of\ correctly\ recognized\ license\ plates}{Number\ of\ all\ license\ plates}$$ Character recognition accuracy is defined as: $$CRA = \frac{Number\ of\ correctly\ recognized\ characters}{Number\ of\ all\ characters}$$
### GAN Training and Testing
Three data generation methods are shown in Fig. \[fig:tradeoff\]. To train CycleWGAN, first we use the OpenCV scripts to generate 1000 blue fake license plates as a source domain X, and then select 1000 real blue license plates from Dataset-1 as a target domain Y. We train the CycleWGAN model with these fake license plates and real license plates. The training real plates do not require character labels. All the images are resized to 143$\times$143, cropped to 128$\times$128 and randomly flipped for data augmentation. We use Adam with $L_{1}=0.9$, $L_{2}=0.999$ and learning rate of 0.0001. We stop training after 300,000 steps and save the model. When testing, first we use the OpenCV scripts to generate 40,000 blue fake license plates, and then we apply the last checkpoint to generate 40,000 license plates. The same goes for CycleWGAN-GP. Finally we get 80,000 blue license plates generated by CycleWGAN and CycleWGAN-GP.
### Data Augmentation
The six data augmentation methods are proposed in order to increase the training data of the recognition network. The data was augmented through affine transformation, motion blurring, uneven lighting, stretching, erosion and dilation, downsampling and the application of gaussian noise. Examples of these transformations are shown in Fig. \[fig:augm\]. A real license plate image randomly passes through the six data augmentation methods, allowing for the creation of much more training data. First, we select a small number of labeled real license plates from Dataset-1, such as 300. And then using data augmentation methods in Fig. \[fig:augm\], we generate 80,000 augmented license plates with these selected real license plates.
### Mixed Training Data
Our mixed training data consists of four parts, including 40,000 license plates generated by OpenCV scripts, 40,000 license plates generated by CycleWGAN, 40,000 license plates generated by CycleWGAN-GP, and 80,000 license plates augmented from a small number of labeled real license plates. All 200,000 training images are generated with license plate character labels. The license plates that need manual labeling are only selected from Dataset-1. After converting the training data to gray scale, 400,000 more training images are obtained by flipping pixels in order to simulate gray images of yellow and green license plates. Then, these images are fed to the recognition network modified from DenseNet.
Performance Evaluation on Dataset-1
-----------------------------------
[|p[3cm]{}<|p[2cm]{}<|p[2cm]{}<|p[2cm]{}<|]{} Method & Training Data & RA & CRA\
\*[baseline]{} & 9000 & 96.1 & 98.9\
& 50000 & 96.7 & 99.1\
& 200000 & 97.6 & 99.5\
\*[ours]{} &300 & 97.5 & 99.3\
& 700 & 98.2 & 99.5\
& 3333 & 98.6 & 99.8\
& 4750 & 99.0 & 99.9\
& 6000 & 99.0 & 99.9\
With the above methods, our mixed training data is generated from 300, 700, 3,333, 4,750 and 6,000 real license plates selected from Dataset-1 training set respectively. Our baseline is the [@wang2017adversarial] using the license plate images generated by the CycleWGAN pre-training recognition network, and then using 9,000, 50,000 and 200,000 real labeled license plate images in a fine-tuning model. From the results in Table \[tab:dataset1\], it is concluded that when data generation, data augmentation and DenseNet are used, we only need 300 real labeled license plates to achieve the effect of 200,000 real license plates. In the same way, when the number of real license plates reaches 4,750, the final recognition accuracy has reached 99.0%, an increase of 1.4%. When the number of real license plate images exceeds 4,750, license plate recognition accuracy and character recognition accuracy are not improving. We conjecture that 4,750 real images contain enough information to recognize most of the license plates. Thus, by increasing the number of real license plates, the total amount of information after data augmentation will not change, and the recognition accuracy will not increase any further.
Performance Evaluation on Data Generation
-----------------------------------------
In order to evaluate the effect of the data generated by different methods, we train the models using synthetic data generated by script, CycleGAN, CycleWGAN, and CycleWGAN-GP respectively. The results are shown in Table \[tab:augmentation\]. When we only use the data set generated by script for training, the recognition accuracy on the test set of Dataset-1 is 42.2%. As shown in Fig. \[fig:script\], our synthetic license plates generated by script also contain noise such as low light, low resolution, motion blur. The CycleGAN images achieve a recognition accuracy of 51.2%. Accuracy is not much improved because of the instability and lack of diversity in CycleGAN training. As shown in Fig. \[fig:WGAN\] and Fig. \[fig:WGAN-GP\], the CycleWGAN and CycleWGAN-GP images display more various styles and colors, and part of them can not really distinguish from real images. The CycleWGAN and CycleWGAN-GP images achieve a recognition accuracy of 62.5% and 64.5% respectively. We also compare the impact of data generation and data augmentation on accuracy. When the number of real license plates is 3333, the recognition accuracy of the augmented data on Dataset-1 is 97.9%, far exceeding the recognition accuracy of the generated data.
\[tab:augmentation\] \[my-label\]
[|p[2.4cm]{}<|p[1cm]{}<|p[1cm]{}<|p[1.1cm]{}<|p[1.1cm]{}<|p[1cm]{}<|p[1cm]{}<|p[1.1cm]{}<|p[1.35cm]{}<|]{} & &\
method & RA & CRA & CRA-C & & & CRA & CRA-C & CRA-NC\
Script & 42.2 & 80.3 & 43.8 & 90.8 & 4.4 & 30.0 & 20.0 & 31.7\
CycleGAN & 51.2 & 87.6 & 51.2 & 93.1 & 34.6 & 82.8 & 41.3 & 89.8\
CycleWGAN & 62.5 & 92.5 & 66.8 & 96.8 & 61.3 & 90.6 & 66.2 & 94.8\
CycleWGAN-gp & 64.5 & 93.7 & 65.2 & 98.4 & - & - & - & -\
Augmentation & 97.9 & 99.1 & 99.2 & 99.7 & - & - & - & -\
In order to understand how much number of real license plates improves recognition accuracy, we compare data augmentation results from 60 to 6000 real license plates. The result in Table \[tab:numdata\] shows that the greater the number of real license plates, the higher the recognition accuracy obtained. Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%. Even if the number of real license plates is increased from 4750, the result is no longer improved.
In order to understand the impact of GAN on recognition accuracy, we did some additional comparative experiments. It can also be seen in the Table \[tab:numdata\] that training data composed of data augmentation and data generation can get better results than training data composed of only data augmentation. The conclusion is that the recognition accuracy of augmented data can be improved with data generation. In addition, the fewer real license plates, the more recognition accuracy increases contributed from generated data.
\[tab:numdata\]
[|p[3cm]{}<|p[2cm]{}<|p[2cm]{}<|p[2cm]{}<|p[2cm]{}<|]{} Training Data & RA(A) & RA(A+G) & CRA(A) & CRA(A+G)\
60 & 47.5 & 79.3 & 88.3 & 94.4\
150 & 83.8 & 93.8 & 96.6 & 98.7\
200 & 92.4 & 96.7 & 98.4 & 98.7\
300 & 96.1 & 97.5 & 98.7 & 99.3\
700 & 97.1 & 98.2 & 98.9 & 99.5\
3333 & 97.9 & 98.6 & 99.2 & 99.8\
4750 & 98.8 & 99.0 & 99.8 & 99.9\
6000 & 98.9 & 99.0 & 99.8 & 99.9\
Performance Evaluation on AOLP
------------------------------
For the application-oriented license plate(AOLP) dataset, the experiments are carried out by using license plates from different sub-datasets for training and test. This data set is divided into three sub-datasets: access control (AC), traffic law enforcement (LE), and road patrol (RP). For example, in Table \[tab:alop\], we use the license plates from the LE and RP sub-datasets to train the DenseNet, and test its performance on the AC sub-dataset. Similarly, AC and RP are used for training and LE for test, and so on. Since there is no AOLP license plate font, only the data augmentation methods are used, without script and GAN generated license plates. In Table \[tab:alop\], through data augmentation and DenseNet, our method achieves the highest recognition accuracy on the AOLP dataset.
[|p[2cm]{}<|p[1.2cm]{}<|p[1.2cm]{}<|p[1.2cm]{}<|p[1.2cm]{}<|p[1.2cm]{}<|p[1.2cm]{}<|]{} & & &\
method & RA & CRA & RA & CRA & RA & CRA\
Hsu et al.[@hsu2013application] & - & 96 & - & 94 & - & 95\
Li et al.[@li2016reading] & 94.85 & - & 94.19 & - & 88.38 & -\
Li et al.[@li2018toward] & 95.71 & - & 97.21 & - & 84.60 & -\
ours & **96.61** & **99.08** & **97.80** & **99.65** & **91.00** & **97.22**\
Conclusion
==========
In this paper, we have investigated how many real labeled license plates are needed to train the license plate recognition system. We have proposed three data generation methods and six data augmentation methods in order to fully obtain all the information in a small number of images. The experimental results show that the proposed method only requires 300 real labeled license plates to achieve the effect achieved by 200,000 real license plates. The result shows that the greater the number of real license plates, the higher the recognition accuracy obtained. Up to 4750, the highest recognition accuracy of the Dataset-1 is 99.0%. Even if the number of real license plates is increased furthermore, the result is no longer improved. Additionally, training data composed of both augmented and generated data can achieve better results than training data composed of only augmented data. Furthermore, the fewer real license plates, the more recognition accuracy increases contributed from generated data.
[^1]: Corresponding author. Shanghai Institute for Advanced Communication and Data Science, Shanghai University, Shanghai, China(email: shugong@shu.edu.cn).
|
---
author:
- Caglar Gulcehre
- Ziyu Wang
- Alexander Novikov
- Tom Le Paine
- Sergio Gómez Colmenarejo
- Konrad Zolna
- Rishabh Agarwal
- Josh Merel
- Daniel Mankowitz
- Cosmin Paduraru
- 'Gabriel Dulac-Arnold'
- Jerry Li
- Mohammad Norouzi
- Matt Hoffman
- Ofir Nachum
- George Tucker
- Nicolas Heess
- Nando de Freitas
bibliography:
- 'refs.bib'
title: 'RL Unplugged: Benchmarks for Offline Reinforcement Learning'
---
Introduction
============
Reinforcement Learning (RL) has seen important breakthroughs, including learning directly from raw sensory streams [@mnih2015human], solving long-horizon reasoning problems such as Go [@silver2016mastering], StarCraft II [@vinyals2019grandmaster], DOTA [@berner2019dota], and learning motor control for high-dimensional simulated robots [@heess2017emergence; @akkaya2019solving]. However, many of these successes rely heavily on repeated online interactions of an agent with an environment. Despite its success in simulation, the uptake of RL for real-world applications has been limited. Power plants, robots, healthcare systems, or self-driving cars are expensive to run and inappropriate controls can have dangerous consequences. They are not easily compatible with the crucial idea of exploration in RL and the data requirements of online RL algorithms. Nevertheless, most real-world systems produce large amounts of data as part of their normal operation.
![**Task domains included in RL Unplugged.** We include several open-source environments that are familiar to the community, as well as recent releases that push the limits of current algorithms. The task domains span key environment properties such as action space, observation space, exploration difficulty, and dynamics.[]{data-label="fig:rl_unplugged"}](figures/tasks_current.pdf){width="0.85\linewidth"}
There is a resurgence of interest in offline methods for reinforcement learning,[^1] that can learn new policies from logged data, without any further interactions with the environment due to its potential real-world impact. Offline RL can help (1) pretrain an RL agent using existing datasets, (2) empirically evaluate RL algorithms based on their ability to exploit a fixed dataset of interactions, and (3) bridge the gap between academic interest in RL and real-world applications.
Offline RL methods [e.g @agarwal2019optimistic; @fujimoto2018off] have shown promising results on well-known benchmark domains. However, non-standardized evaluation protocols, differing datasets and lack of baselines make algorithmic comparisons difficult. Important properties of potential real-world application domains such as partial observability, high-dimensional sensory streams such as images, diverse action spaces, exploration problems, non-stationarity, and stochasticity are under-represented in the current offline RL literature. This makes it difficult to assess the practical applicability of offline RL algorithms.
The reproducibility crisis of RL [@henderson2018deep] is very evident in offline RL. Several works have highlighted these reproducibility challenges in their papers: [@peng2019advantage] discusses the difficulties of implementing the MPO algorithm, [@fujimoto2019benchmarking] mentions omitting results for SPIBB-DQN due to the complexity of implementation. On our part, we have had difficulty implementing SAC [@haarnoja2018soft]. We have also found it hard to scale BRAC [@wu2019behavior] and BCQ [@fujimoto2018off]. This does not indicate these algorithms do not work. Only that implementation details matter, comparing algorithms and ensuring their reproducibility is hard. The intention of this paper is to help in solving this problem by putting forward common benchmarks, datasets, evaluation protocols, and code.
The availability of large datasets with strong benchmarks has been the main factor for the success of machine learning in many domains. Examples of this include vision challenges, such as ImageNet [@deng2009imagenet] and COCO [@veit2016coco], and game challenges, where simulators produce hundreds of years of experience for online RL agents such as AlphaGo [@silver2016mastering] and the OpenAI Five [@berner2019dota]. In contrast, lack of datasets with clear benchmarks hinders the similar progress in RL for real-world applications. This paper aims to correct this such as to facilitate collaborative research and measurable progress in the field.
To this end, we introduce a novel collection of task domains and associated datasets together with a clear evaluation protocol. We include widely-used domains such as the DM Control Suite [@tassa2018controlsuite] and Atari 2600 games [@bellemare2013arcade], but also domains that are still challenging for strong online RL algorithms such as real-world RL (RWRL) suite tasks [@dulacarnold2020realworldrlempirical] and DM Locomotion tasks [@heess2017emergence; @merel2018hierarchical; @merel2018neural; @merel2020deep]. By standardizing the environments, datasets, and evaluation protocols, we hope to make research in offline RL more reproducible and accessible. We call our suite of benchmarks “RL Unplugged”[^2], because offline RL methods can use it without any actors interacting with the environment.
This paper offers four main contributions: (i) a unified API for datasets (ii) a varied set of environments (iii) clear evaluation protocols for offline RL research, and (iv) reference performance baselines. The datasets in RL Unplugged enable offline RL research on a variety of established online RL environments without having to deal with the exploration component of RL. In addition, we intend our evaluation protocols to make the benchmark more fair and robust to different hyperparameter choices compared to the traditional methods which rely on online policy selection. Moreover, releasing the datasets with a proper evaluation protocols and open-sourced code will also address the reproducibility issue in RL [@henderson2018deep]. We evaluate and analyze the results of several SOTA RL methods on each task domain in RL Unplugged. We also release our datasets in an easy-to-use unified API that makes the data access easy and efficient with popular machine learning frameworks.
RL Unplugged
============
The RL Unplugged suite is designed around the following considerations: to facilitate ease of use, we provide the datasets with a unified API which makes it easy for the practitioner to work with all data in the suite once a general pipeline has been established. We further provide a number of baselines including state-of-the art algorithms compatible with our API.
Properties of RL Unplugged {#sec:Unplugged:Properties}
--------------------------
Many real-world RL problems require algorithmic solutions that are general and can demonstrate robust performance on a diverse set of challenges. Our benchmark suite is designed to cover a range of properties to determine the difficulty of a learning problem and affect the solution strategy choice. In the initial release of RL Unplugged, we include a wide range of task domains, including Atari games and simulated robotics tasks. Despite the different nature of the environments used, we provide a unified API over the datasets. Each entry in any dataset consists of a tuple of state ($s_t$), action ($a_t$), reward ($r_t$), next state ($s_{t+1}$), and the next action ($a_{t+1}$). For sequence data, we also provide future states, actions, and rewards, which allows for training recurrent models for tasks requiring memory. We additionally store metadata such as episodic rewards and episode id. We chose the task domains to include tasks that vary along the following axes. In Figure \[fig:rl\_unplugged\], we give an overview of how each task domain maps to these axes.
**Action space** We include tasks with both discrete and continuous action spaces, and of varying action dimension with up to 56 dimensions in the initial release of RL Unplugged.
**Observation space** We include tasks that can be solved from the low-dimensional natural state space of the MDP (or hand-crafted features thereof), but also tasks where the observation space consists of high-dimensional images ([*e.g.,*]{} Atari 2600). We include tasks where the observation is recorded via an external camera (third-person view), as well as tasks in which the camera is controlled by the learning agent (e.g. robots with egocentric vision).
**Partial observability & need for memory** We include tasks in which the feature vector is a complete representation of the state of the MDP, as well as tasks that require the agent to estimate the state by integrating information over horizons of different lengths.
**Difficulty of exploration** We include tasks that vary in terms of exploration difficulty for reasons such as dimension of the action space, sparseness of the reward, or horizon of the learning problem.
**Real-world challenges** To better reflect the difficulties encountered in real systems, we also include tasks from the Real-World RL Challenges [@dulacarnold2020realworldrlempirical], which include aspects such as action delays, stochastic transition dynamics, or non-stationarities.
The characteristics of the data is also an essential consideration, including the behavior policy used, data diversity, [*i.e.,*]{} state and action coverage, and dataset size. RL Unplugged introduces datasets that cover those different axes. For example, on Atari 2600, we use large datasets generated across training of an off-policy agent, over multiple seeds. The resulting dataset has data from a large mixture of policies. In contrast, we use datasets from fixed sub-optimal policies for the RWRL suite.
Evaluation Protocols {#sec:Unplugged:Evaluation}
--------------------
In a strict offline setting, environment interactions are not allowed. This makes hyperparameter tuning, including determining when to stop a training procedure, difficult. This is because we cannot take policies obtained by different hyperparameters and run them in the environment to determine which ones receive higher reward (we call this procedure **online policy selection**[^3].) Ideally, offline RL would evaluate policies obtained by different hyperparameters using only logged data, for example using offline policy evaluation (OPE) methods [@voloshin2019empirical] (we call this procedure **offline policy selection**). However, it is unclear whether current OPE methods scale well to difficult problems. In RL Unplugged we would like to evaluate offline RL performance in both settings.
Evaluation by online policy selection (See Figure \[fig:evaluation\] (left)) is widespread in the RL literature, where researchers usually evaluate different hyperparameter configurations in an online manner by interacting with the environment, and then report results for the best hyperparameters. This enables us to evaluate offline RL methods in isolation, which is useful. It is indicative of performance given perfect offline policy selection, or in settings where we can validate via online interactions. This score is important, because as offline policy selection methods improve, performance will approach this limit. But it has downsides. As discussed before, it is infeasible in many real-world settings, and as a result it gives an overly optimistic view of how useful offline RL methods are today. Lastly, it favors methods with more hyperparameters over more robust ones.
![**Comparison of evaluation protocols.** (left) Evaluation using **online policy selection** allows us to isolate offline RL methods, but gives overly optimistic results because they allow perfect policy selection. (right) Evaluation using **offline policy selection** allows us to see how offline RL performs in situations where it is too costly to interact with the environment for validation purposes; a common scenario in the real-world. We intend our benchmark to be used for both. []{data-label="fig:evaluation"}](figures/policy-selection.pdf){width="\linewidth"}
Evaluation by offline policy selection (See Figure \[fig:evaluation\] (right)) has been less popular, but is important as it is indicative of robustness to imperfect policy selection, which more closely reflects the current state of offline RL for real-world problems. However it has downsides too, namely that there are many design choices including what data to use for offline policy selection, whether to use value functions trained via offline RL or OPE algorithms, which OPE algorithm to choose, and the meta question of how to tune OPE hyperparameters. Since this topic is still under-explored, we prefer not to specify any of these choices. Instead, we invite the community to innovate to find which offline policy selection method works best.
Importantly, our benchmark allows for evaluation in both online and offline policy selection settings. For each task, we clearly specify if it is intended for online vs offline policy selection. For offline policy selection tasks, we use a naive approach which we will described in Section \[sec:baselines\]. We expect future work on offline policy selection methods to improve over this naive baseline. If a combination of offline RL method and offline policy selection can achieve perfect performance across all tasks, we believe this will mark an important milestone for offline methods in real-world applications.
Tasks
=====
For each task domain we give a description of the tasks included, indicate which tasks are intended for online vs offline policy selection, and provide a description of the corresponding data.
DM Control Suite
----------------
DeepMind Control Suite [@tassa2018controlsuite] is a set of control tasks implemented in MuJoCo [@todorov2012mujoco]. We consider a subset of the tasks provided in the suite that cover a wide range of difficulties. For example, *Cartpole swingup* a simple task with a single degree of freedom is included. Difficult tasks are also included, such as *Humanoid run, Manipulator insert peg, Manipulator insert ball*. *Humanoid run* involves complex bodies with 21 degrees of freedom. And *Manipulator insert ball/peg* have not been shown to be solvable in any prior published work to the best of our knowledge. In all the considered tasks as observations we use the default feature representation of the system state, consisting of proprioceptive information such as joint positions and velocity, as well as additional sensors and target position where appropriate. The observation dimension ranges from 5 to 67.
**Data Description** Most of the datasets in this domain are generated using D4PG. For the environments *Manipulator insert ball* and *Manipulator insert peg* we use V-MPO [@song2019v] to generate the data as D4PG is unable to solve these tasks. We always use 3 independent runs to ensure data diversity when generating data. All methods are run until the task is considered solved. For each method, data from the entire training run is recorded. As offline methods tend to require significantly less data, we reduce the sizes of the datasets via sub-sampling. In addition, we further reduce the number of successful episodes in each dataset by $2/3$ so as to ensure the datasets do not contain too many successful trajectories. See Table \[tab:control\_suite\_data\] for the size of each dataset. Each episode in this dataset contains 1000 time steps.
[0.485]{}
------------------------- ------ ----
Cartpole swingup 40 1
Cheetah run 300 6
Humanoid run 3000 21
Manipulator insert ball 1000 5
Walker stand 200 6
Finger turn hard 500 2
Fish swim 200 5
Manipulator insert peg 1500 5
Walker walk 200 6
------------------------- ------ ----
[0.485]{}
-------------------- ------ ---- ----
Humanoid corridor 4000 2 56
Humanoid walls 4000 40 56
Rodent gaps 2000 2 38
Rodent two tap 2000 40 38
Humanoid gaps 4000 2 56
Rodent bowl escape 2000 40 38
Rodent mazes 2000 40 38
-------------------- ------ ---- ----
DM Locomotion {#sec:dm_locomotion}
-------------
These tasks are made up of the corridor locomotion tasks involving the CMU Humanoid, for which prior efforts have either used motion capture data [@merel2018hierarchical; @merel2018neural] or training from scratch [@song2019v]. In addition, the DM Locomotion repository contains a set of tasks adapted to be suited to a virtual rodent [see @merel2020deep]. We emphasize that the *DM Locomotion* tasks feature the combination of challenging high-DoF continuous control along with perception from rich egocentric observations.
**Data description** Note that for the purposes of data collection on the CMU humanoid tasks, we use expert policies trained according to [@merel2018neural], with only a single motor skill module from motion capture that is reused in each task. For the rodent task, we use the same training scheme as in [@merel2020deep]. For the CMU humanoid tasks, each dataset is generated by $3$ online methods whereas each dataset of the rodent tasks is generated by $5$ online methods. Similarly to the control suite, data from entire training runs is recorded to further diversify the datasets. Each dataset is then sub-sampled and the number of its successful episodes reduced by $2/3$. Since the sensing of the surroundings is done by egocentric cameras, all datasets in the locomotion domain include per-timestep egocentric camera observations of size $64\times64\times3$. The use of egocentric observation also renders some environments partially observable and therefore necessitates recurrent architectures. We therefore generate sequence datasets for tasks that require recurrent architectures. For dataset sizes and sequence lengths of see Table \[tab:locomotion\_data\].
Atari 2600
----------
The Arcade Learning environment (ALE) [@bellemare2013arcade] is a suite consisting of a diverse set of $57$ Atari 2600 games (Atari57). It is a popular benchmark to measure the progress of online RL methods, and Atari has recently also become a standard benchmark for offline RL methods [@agarwal2019optimistic; @fujimoto2019benchmarking] as well. In this paper, we are releasing a large and diverse dataset of gameplay following the protocol described by @agarwal2019optimistic, and use it to evaluate several discrete RL algorithms.
**Data Description** The dataset is generated by running an online DQN agent and recording transitions from its replay during training with sticky actions [@machado2018revisiting]. As stated in [@agarwal2019optimistic], for each game we use data from five runs with $50$ million transitions each. States in each transition include stacks of four frames to be able to do frame-stacking with our baselines.
In our release, we provide experiments on the $46$ of the Atari games that are available in OpenAI gym. OpenAI gym implements more than $46$ games, but we only include games where the online DQN’s performance that has generated the dataset was significantly better than the random policy. We provide further information about the games we excluded in Appendix \[sec:atari\_data\_selection\]. Among our $46$ Atari games, we chose nine to allow for online policy selection. Specifically, we ordered all games according to the their difficulty,[^4] and picked every fifth game as our offline policy section task to cover diverse set of games in terms of difficulty. In Table \[table:atari\_taxonomy\], we provide the full list of games that we decided to include in RL Unplugged.
----------------------------------------------------------- --------------------------------------------------------------- -------------------------------------------------------------- -------------------------------------------------------------- -------------------------------------------------------------
<span style="font-variant:small-caps;">BeamRider</span> <span style="font-variant:small-caps;">DoubleDunk</span> <span style="font-variant:small-caps;">Ms. Pacman</span> <span style="font-variant:small-caps;">Road Runner</span> <span style="font-variant:small-caps;">Zaxxon</span>
<span style="font-variant:small-caps;">DemonAttack</span> <span style="font-variant:small-caps;">Ice Hockey</span> <span style="font-variant:small-caps;">Pooyan</span> <span style="font-variant:small-caps;">Robotank</span>
<span style="font-variant:small-caps;">Alien</span> <span style="font-variant:small-caps;">Breakout</span> <span style="font-variant:small-caps;">Frostbite</span> <span style="font-variant:small-caps;">Name This Game</span> <span style="font-variant:small-caps;">Time Pilot</span>
<span style="font-variant:small-caps;">Amidar</span> <span style="font-variant:small-caps;">Carnival</span> <span style="font-variant:small-caps;">Gopher</span> <span style="font-variant:small-caps;">Phoenix</span> <span style="font-variant:small-caps;">Up And Down</span>
<span style="font-variant:small-caps;">Assault</span> <span style="font-variant:small-caps;">Centipede</span> <span style="font-variant:small-caps;">Gravitar</span> <span style="font-variant:small-caps;">Pong</span> <span style="font-variant:small-caps;">Video Pinball</span>
<span style="font-variant:small-caps;">Asterix</span> <span style="font-variant:small-caps;">Chopper Command</span> <span style="font-variant:small-caps;">Hero</span> <span style="font-variant:small-caps;">Q\*Bert</span> <span style="font-variant:small-caps;">Wizard of Wor</span>
<span style="font-variant:small-caps;">Atlantis</span> <span style="font-variant:small-caps;">Crazy Climber</span> <span style="font-variant:small-caps;">James Bond</span> <span style="font-variant:small-caps;">River Raid</span> <span style="font-variant:small-caps;">Yars Revenge</span>
<span style="font-variant:small-caps;">Bank Heist</span> <span style="font-variant:small-caps;">Enduro</span> <span style="font-variant:small-caps;">Kangaroo</span> <span style="font-variant:small-caps;">Seaquest</span>
<span style="font-variant:small-caps;">Battlezone</span> <span style="font-variant:small-caps;">Fishing Derby</span> <span style="font-variant:small-caps;">Krull</span> <span style="font-variant:small-caps;">Space Invaders</span>
<span style="font-variant:small-caps;">Boxing</span> <span style="font-variant:small-caps;">Freeway</span> <span style="font-variant:small-caps;">Kung Fu Master</span> <span style="font-variant:small-caps;">Star Gunner</span>
----------------------------------------------------------- --------------------------------------------------------------- -------------------------------------------------------------- -------------------------------------------------------------- -------------------------------------------------------------
: **Atari games.** We have 46 games in total in our Atari data release. We reserved 9 of the games for online policy selection (top) and the rest of the 37 games are reserved for the offline policy selection (bottom).[]{data-label="table:atari_taxonomy"}
Real-world Reinforcement Learning Suite
---------------------------------------
@dulacarnold2019challenges ([-@dulacarnold2019challenges]) and @dulacarnold2020realworldrlempirical ([-@dulacarnold2020realworldrlempirical]) identify and evaluate respectively a set of $9$ challenges that are bottlenecks to implementing RL algorithms, at scale, on applied systems. These include high-dimensional state and action spaces, large system delays, system constraints, multiple objectives, handling non-stationarity and partial observability. In addition, they have released a suite of tasks called [`realworldrl-suite`]{}[^5] which enables a practitioner to verify the capabilities of their algorithm on domains that include some or all of these challenges. The suite also defines a set of standardized challenges with varying levels of difficulty. As part of the “RL Unplugged” collection, we have generated datasets using the ‘easy‘ Combined Challenges on four tasks: Cartpole Swingup, Walker Walk, Quadruped Walk and Humanoid Walk.
**Data Description** The datasets were generated as described in Section 2.8 of @dulacarnold2020realworldrlempirical; note that this is the first data release based on those specifications. We used either the *no challenge* setting, which includes unperturbed versions of the tasks, or the *easy combined challenge* setting (see Section 2.9 of @dulacarnold2020realworldrlempirical), where data logs are generated from an environment that includes effects from combining all the challenges. Although the *no challenge* setting is identical to the control suite, the dataset generated for it is different as it is generated from fixed sub-optimal policies. These policies were obtained by training $3$ seeds of distributional MPO [@abbas2018mpo] until convergence with different random weight initializations, and then taking snapshots corresponding to roughly $75\%$ of the converged performance. For the *no challenge* setting, three datasets of different sizes were generated for each environment by combining the three snapshots, with the total dataset sizes (in numbers of episodes) provided in Table \[tab:batch\_rl\_data\]. The procedure was repeated for the *easy combined challenge* setting. Only the “large data” setting was used for the combined challenge to ensure the task is still solvable. We consider all RWRL tasks as online policy selection tasks.
Cartpole swingup Walker walk Quadruped walk Humanoid walk
---------------- ------------------ ------------- ---------------- ---------------
Small dataset 100 1000 100 4000
Medium dataset 200 2000 200 8000
Large dataset 500 5000 500 20000
: **real-world Reinforcement Learning Suite dataset sizes.** Size is measured in number of episodes, with each episode being 1000 steps long.[]{data-label="tab:batch_rl_data"}
Baselines {#sec:baselines}
=========
We provide baseline results for a number of published algorithms for both continuous (DM Control Suite, DM Locomotion), and discrete action (Atari 2600) domains. We will open-source implementations of our baselines for the camera-ready. We follow the evaluation protocol presented in Section \[sec:Unplugged:Evaluation\]. Our baseline algorithms include behavior cloning (BC [@pomerleau1989alvinn]); online reinforcement learning algorithms (DQN [@mnih2015human], D4PG [@barth2018distributed], IQN [@dabney2018implicit]); and recently proposed offline reinforcement learning algorithms (BCQ [@fujimoto2018off], BRAC [@wu2019behavior], RABM [@siegel2020keep], REM [@agarwal2019optimistic]). Some algorithms only work for discrete or continuous actions spaces, so we only evaluate algorithms in domains they are suited to. Detailed descriptions of the baselines and our implementations (including hyperparameters) are presented in Section \[sec:appendix:baselines\] in the supplementary material.
**Naive approach for offline policy selection** For the tasks we have marked for offline policy selection, we need a strategy that does not use online interaction to select hyperparameters. Our naive approach is to choose the set of hyperparameters that performs best overall on the online policy selection tasks from the same domain. We do this independently for each baseline. This approach is motivated by how hyperparameters are often chosen in practice, by using prior knowledge of what worked well in similar domains. If a baseline algorithm drops in performance between online and offline policy selection tasks, this indicates the algorithm is not robust to the choice of hyperparameters. This is also cheaper than tuning hyperparameters individually for all tasks, which is especially relevant for Atari. For a given domain, a baseline algorithm and a hyperparameter set, we compute the average[^6] score over all tasks allowing online policy selection. The best hyperparameters are then applied to all offline policy selection tasks for this domain. The details of the experimental protocol and the final hyperparameters are provided in the supplementary material.
DM Control Suite {#dm_control_suite}
----------------
![**Baselines on DM Control Suite.** (left) Performance using evaluation by online policy selection. (right) Performance using evaluation by offline policy selection. Horizontal lines for each task show 90th percentile of task reward in the dataset. Note that D4PG, BRAC, and RABM perform equally well on easier tasks e.g. Cartpole swingup. But BC, and RABM perform best on harder tasks e.g. Humanoid run.[]{data-label="fig:baselines_dm_control_suite"}](figures/dm_control_suite.pdf){width="\linewidth"}
In Figure \[fig:baselines\_dm\_control\_suite\], we compare baselines across the online policy selection tasks (left) and offline policy selection tasks (right). A table of results is included in Section \[sec:control\_suite\_results\] of the supplementary material. For the simplest tasks, such as Cartpole swingup, Walker stand, and Walker walk, where the performance of offline RL is close to that of online methods, D4PG, BRAC and RABM are all good choices. But the picture changes on the more difficult tasks, such as Humanoid run (which has high dimension action spaces), or Manipulator insert ball and manipulator insert peg (where exploration is hard). Strikingly, in these domains BC is actually among the best algorithms alongside RABM, although no algorithm reaches the performance of online methods. This highlights how including tasks with diverse difficulty conditions in a benchmark gives a more complete picture of offline RL algorithms.
DM Locomotion {#dm_locomotion}
-------------
![**Baselines on DM Locomotion.** (left) Performance using evaluation by online policy selection. (right) Performance using evaluation by offline policy selection. Horizontal lines for each task show 90th percentile of task reward in the dataset. The trend is similar to the harder tasks in DM Control Suite, i.e. BC and RABM perform well, while D4PG performs poorly.[]{data-label="fig:baselines_dm_locomotion"}](figures/dm_locomotion.pdf){width="\linewidth"}
In Figure \[fig:baselines\_dm\_locomotion\], we compare baselines across the online policy selection tasks (left) and offline policy selection tasks (right). A table of results is included in Section \[sec:locomotion\_results\] of the supplementary material. This task domain is made exclusively of tasks that are high action dimension, hard exploration, or both. As a result the stark trends seen above continue. BC, and RABM perform best, and D4PG performs quite poorly. We also could not make BCQ or BRAC perform well on these tasks, but we are not sure if this is because these algorithms perform poorly on these tasks, or if our implementations are missing a crucial detail. For this reason we do not include them. This highlights another key problem in online and offline RL. Papers do not include key baselines because the authors were not able to reproduce them, see eg [@peng2019advantage; @fujimoto2019benchmarking]. By releasing datasets, evaluation protocols and baselines, we are making it easier for researchers such as those working with BCQ to try their methods on these challenging benchmarks.
Atari 2600
----------
In Figure \[fig:atari\_aggregate\_improvement\], we present results for Atari using normalized scores. Due to the large number of tasks, we aggregate results using the median as done in [@agarwal2019optimistic; @hessel2018rainbow] (individual scores are presented in Appendix \[sec:atari\_results\]). These results indicate that DQN is not very robust to the choice of hyperparameters. Unlike REM or IQN, DQN’s performance dropped significantly on the offline policy selection tasks. BCQ, REM and IQN perform at least as well as the best policy in our training set according to our metrics. In contrast to other datasets (Section \[dm\_control\_suite\] and \[dm\_locomotion\]), BC performs poorly on this dataset. Surprisingly, the performance of off-the-shelf off-policy RL algorithms is competitive and even surpasses BCQ on offline policy selection tasks. Combining behavior regularization methods ([*e.g.*]{}, BCQ) with robust off-policy algorithms (REM, IQN) is a promising direction for future work.
[.5]{} ![**Baselines on Atari.** (left) Performance using evaluation by online policy selection. (right) Performance using evaluation by offline policy selection. The bars indicate the median normalized score, and the error bars show a bootstrapped estimate of the $\left[25,75\right]$ percentile interval for the median estimate computed across different games. The score normalization is done using the best performing policy among the mixture of policies that generated the offline Atari dataset (see Appendix \[atari:details\] for details). []{data-label="fig:atari_aggregate_improvement"}](figures/atari_tuning.pdf "fig:"){width="95.00000%"}
[.5]{} ![**Baselines on Atari.** (left) Performance using evaluation by online policy selection. (right) Performance using evaluation by offline policy selection. The bars indicate the median normalized score, and the error bars show a bootstrapped estimate of the $\left[25,75\right]$ percentile interval for the median estimate computed across different games. The score normalization is done using the best performing policy among the mixture of policies that generated the offline Atari dataset (see Appendix \[atari:details\] for details). []{data-label="fig:atari_aggregate_improvement"}](figures/atari_testing.pdf "fig:"){width="\textwidth"}
Related Work
============
There is a large body of work focused on developing novel offline reinforcement learning algorithms [@fujimoto2018off; @wu2019behavior; @agarwal2019optimistic; @siegel2020keep]. These works have often tested their methods on simple MDPs such as grid worlds [@laroche2017safe], or fully observed environments were the state of the world is given [@fujimoto2018off; @wu2019behavior; @fu2020d4rl]. There has also been extensive work applying offline reinforcement learning to difficult real-world domains such as robots [@cabi2019framework; @gu2017deep; @kalashnikov2018qt] or dialog [@henderson2008hybrid; @pietquin2011sample; @jaques2019way], but it is often difficult to do thorough evaluations in these domains for the same reason offline RL is useful in them, namely that interaction with the environment is costly. Additionally, without consistent environments and datasets, it is impossible to clearly compare these different algorithmic approaches. We instead focus on a range of challenging simulated environments, and establishing them as a benchmark for offline RL algorithms. There are two works similar in that regard. The first is [@agarwal2019optimistic] which release DQN Replay dataset for Atari 2600 games, a challenging and well known RL benchmark. We have reached out to the authors to include this dataset as part of our benchmark. The second is [@fu2020d4rl] which released datasets for a range of control tasks, including the Control Suite, and dexterous manipulation tasks. Unlike our benchmark which includes tasks that test memory and representation learning, their tasks are all from fully observable MDPs, where the physical state information is explicitly provided.
Conclusion
==========
We are releasing RL Unplugged, a suite of benchmarks covering a diverse set of environments, and datasets with an easy-to-use unified API. We present a clear evaluation protocol which we hope will encourage more research on offline policy selection. We empirically evaluate several state-of-art offline RL methods and analyze their results on our benchmark suite. The performance of the offline RL methods is already promising on some control suite tasks and Atari games. However, on partially-observable environments such as the locomotion suite the offline RL methods’ performance is lower. We intend to extend our benchmark suite with new environments and datasets from the community to close the gap between real-world applications and reinforcement learning research.
Broader Impact {#broader-impact .unnumbered}
==============
Online methods require exploration by having a learning agent interact with an environment. In contrast, offline methods learn from fixed dataset of previously logged environment interactions. This has three positive consequences: 1) Offline approaches are more straightforward in settings where allowing an agent to freely explore in the environment is not safe. 2) Reusing offline data is more environmentally friendly by reducing computational requirements, because in many settings exploration is the dominant computational cost and requires large-scale distributed RL algorithms. 3) Offline methods may be more accessible to the wider research community, insofar as researchers who do not have sufficient compute resources for online training from large quantities of simulated experience can reproduce results from research groups with more resources, and improve upon them.
But offline approaches also have potential drawbacks. Any algorithm that learns a policy from data to optimize a reward runs the risk of producing behaviors reflective of the training data or reward function. Offline RL is no exception. Current and future machine learning practitioners should be mindful of where and how they apply offline RL methods, with particular thought given to the scope of generalization they can expect of a policy trained on a fixed dataset.
Acknowledgements {#acknowledgements .unnumbered}
================
We want to thank Misha Denil for his valuable feedback and comments on our paper’s early draft. We appreciate all the help and support we received from Sarah Henderson and Claudia Pope throughout this project.
[^1]: Sometimes referred to as ‘Batch RL,’ but in this paper, we use ‘Offline RL’.
[^2]: Our benchmarks will be available under, <https://github.com/deepmind/deepmind-research/tree/master/rl_unplugged>.
[^3]: Sometimes referred to as online model selection, but we choose policy selection to avoid confusion with models of the environment as used in model based RL algorithms.
[^4]: The details of how we decide the difficulty of Atari games are provided in Appendix \[sec:atari\_game\_difficulty\].
[^5]: <https://github.com/google-research/realworldrl_suite>
[^6]: We always use the arithmetic mean with the exception of Atari where we use median following [@hessel2018rainbow].
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address: 'Author Affiliation(s)'
bibliography:
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title: AUTHOR GUIDELINES FOR ASRU 2017 PROCEEDINGS MANUSCRIPTS
---
One, two, three, four, five
Introduction {#sec:intro}
============
These guidelines include complete descriptions of the fonts, spacing, and related information for producing your proceedings manuscripts. Please follow them and if you have any questions, direct them to Conference Management Services, Inc.: Phone +1-979-846-6800 or email to\
`papers@asru2017.org`.
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=====================
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==============
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===================
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ILLUSTRATIONS, GRAPHS, AND PHOTOGRAPHS {#sec:illust}
======================================
Illustrations must appear within the designated margins. They may span the two columns. If possible, position illustrations at the top of columns, rather than in the middle or at the bottom. Caption and number every illustration. All halftone illustrations must be clear black and white prints. Colors may be used, but they should be selected so as to be readable when printed on a black-only printer.
Since there are many ways, often incompatible, of including images (e.g., with experimental results) in a LaTeX document, below is an example of how to do this [@Lamp86].
FOOTNOTES {#sec:foot}
=========
Use footnotes sparingly (or not at all!) and place them at the bottom of the column on the page on which they are referenced. Use Times 9-point type, single-spaced. To help your readers, avoid using footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence).
![Example of placing a figure with experimental results.[]{data-label="fig:res"}](image1){width="8.5cm"}
\(a) Result 1
![Example of placing a figure with experimental results.[]{data-label="fig:res"}](image3){width="4.0cm"}
\(b) Results 3
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\(c) Result 4
COPYRIGHT FORMS {#sec:copyright}
===============
You must include your fully completed, signed IEEE copyright release form when form when you submit your paper. We [**must**]{} have this form before your paper can be published in the proceedings.
REFERENCES {#sec:ref}
==========
List and number all bibliographical references at the end of the paper. The references can be numbered in alphabetic order or in order of appearance in the document. When referring to them in the text, type the corresponding reference number in square brackets as shown at the end of this sentence [@C2]. An additional final page (the fifth page, in most cases) is allowed, but must contain only references to the prior literature.
|
---
abstract: 'Pattern matching in time series data streams is considered to be an essential data mining problem that still stays challenging for many practical scenarios. Different factors such as noise, varying amplitude scale or shift, signal stretches or shrinks in time are all leading to performance degradation of many existing pattern matching algorithms. In this paper, we introduce a dynamic z-normalization mechanism allowing for proper signal scaling even under significant time and amplitude distortions. Based on that, we further propose a Dynamic Time Warping-based real-time pattern matching method to recover hidden patterns that can be distorted in both time and amplitude. We evaluate our proposed method on synthetic and real-world scenarios under realistic conditions demonstrating its high operational characteristics comparing to other state-of-the-art pattern matching methods.'
author:
- Renzhi Wu
- Sergey Sukhanov
- Christian Debes
bibliography:
- 'refs.bib'
title: Real Time Pattern Matching with Dynamic Normalization
---
<ccs2012> <concept> <concept\_id>10002951.10003227.10003351</concept\_id> <concept\_desc>Information systems Data mining</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Conclusion {#sec:conclusions}
==========
In this paper, we introduced a real-time pattern matching approach that is based on the dynamic z-normalization scheme and is robust to time and amplitude distortions of different degree. We proved that the introduced dynamic z-normalization provides similar results to the traditional z-normalization performed on the proper (but in practice unknown) window. We demonstrated that the proposed pattern matching method provides high operational performance on both synthetic and real-world scenarios outperforming the other state-of-the-art pattern matching methods.
|
---
abstract: 'Single-channel speech separation in time domain and frequency domain has been widely studied for voice-driven applications over the past few years. Most of previous works assume known number of speakers in advance, however, which is not easily accessible through monaural mixture in practice. In this paper, we propose a novel model of single-channel multi-speaker separation by jointly learning the time-frequency feature and the unknown number of speakers. Specifically, our model integrates the time-domain convolution encoded feature map and the frequency-domain spectrogram by attention mechanism, and the integrated features are projected into high-dimensional embedding vectors which are then clustered with deep attractor network to modify the encoded feature. Meanwhile, the number of speakers is counted by computing the Gerschgorin disks of the embedding vectors which are orthogonal for different speakers. Finally, the modified encoded feature is inverted to the sound waveform using a linear decoder. Experimental evaluation on the GRID dataset shows that the proposed method with a single model can accurately estimate the number of speakers with 96.7 % probability of success, while achieving the state-of-the-art separation results on multi-speaker mixtures in terms of scale-invariant signal-to-noise ratio improvement (SI-SNRi) and signal-to-distortion ratio improvement (SDRi).'
author:
- |
Yiming Xiao and Haijian Zhang\
Signal Processing Lab., School of Electronic Information, Wuhan University, China
bibliography:
- 'mybib.bib'
title: |
Improved Source Counting and Separation\
for Monaural Mixture
---
=1
Speech Separation, Unknown Number of Speakers, Joint Time-and-Frequency Feature, Attention Mechanism.
Through years of research, very impressive results have been achieved when the speakers are known in advance, but the task remains challenging when the speakers in the mixed voice are still unknown especially in single-channel scenario, referred as the single-channel speaker-independent source separation problem.
Introduction
============
speech separation with pretty performance is an important prerequisite for robust speech processing in real-world acoustic environments. For instance, automatic speech recognition (ASR) in multi-speaker conditions first requires the separation of individual speakers from their monaural mixture before identifying a target speaker or recognizing target speech [@QIAN20181; @KHADEMIAN20181]. The well-known cocktail party problem which is effortless for humans has been shown to be difficult for computer algorithms [@8369155; @8683850; @7979557; @Nie2015TwostageMJ; @fan2020deep]. Therefore, substantial efforts should be made to solve the cocktail-party problem based solely on a monaural mixture.
To tackle this problem, various methods including computational auditory scene analysis (CASA) [@1709891; @403c9e27c60d4acaa65c7720b40b949e; @7886000], non-negative matrix factorization (NMF) [@6854302; @6854305; @5495567] were proposed. Recently, the major development in deep learning techniques has led to a big step forward in solving the speech separation task. Most deep learning based techniques were studied in frequency domain [@8462471; @Wang2019; @Liu2019; @inproceedings; @7471631; @Isik+2016; @7952155; @8264702; @8462507], in which the deep clustering (DC) approach projected the mixture spectrogram to a high-dimensional embedding space which was more discriminative for speaker partitioning [@7471631; @Isik+2016]. Based on the DC, deep attractor network (DANet) [@7952155; @8264702] or loss functions [@8462507] were introduced to improve the separation performance. In addition, Luo *et al.* [@8462116; @8707065] introduced a new solution to speech separation in time domain, i.e., TasNet, which achieved an impressive performance compared against the frequency-domain solutions. In [@8707067], Yang *et al.* constructed a time-and-frequency (T-F) feature map by concatenating features for both time domain and frequency domain, and performed cross-domain joint embedding and clustering over this feature map, thus further improving the separation performance.
Despite considerable progress in recent years, the current literature has paid less attention to estimating the number of potential sources [@DBLP190403065; @Higuchi2017; @nachmani2020voice], i.e., most of the above methods assume the number of speakers is known. The deep clustering approach requires the number of sources to cluster embeddings and obtain time-frequency (TF) masks[@Isik+2016]. The TasNet needs the number of speakers to fix the dimension of the output embedding, which makes it inflexible to deal with varying number of sources [@8462116; @8707065]. Actually, the number of speakers in realistic scenarios is often uncertain or even time-varying, thus accurately estimating the potential number of sources would be critical to subsequent separation. One way is to use the orthogonality of the high-dimensional embeddings. In [@Higuchi2017], the number of speakers was estimated by computing the rank of the covariance matrix of high-dimensional embedding vectors. Although the existing works have made certain achievements in multi-channel scenarios [@ZHANG20171; @9004553; @gu2020multimodal], there is much room for improving source counting and separation performance in single-channel scenarios. Consequently, sophisticated monaural speech separation models with unknown number of speakers are strongly required.
{width="99.00000%"}
In this paper, we propose an attention-based T-F feature fusion model for speech separation with unknown number of speakers. To further exploit the T-F cross-domain feature, the attention mechanism is adopted in our encoding stage, which gives rise to better separation performance. Then, the encoded features are projected to high-dimensional vectors through an embedding part similar to TasNet. Theoretically, the orthogonal direction of above high-dimensional vectors is equivalent to the number of sources. Thus, the number of speakers could be estimated by computing the Gerschgorin disks of the embedding vectors. During the mask estimation stage, the ADANet [@8264702] is utilized to estimate the mask of each source from the mixture using the similarity between the embeddings and each attractor. This implies that our network can be extended to an arbitrary number of sources once the attractors are established.
Our contributions are threefold:
- We propose a speech separation method for separating a mixture of different numbers of speakers with a single model which can obtain the state-of-the-art results for both two- and three-speaker mixtures.
- We further show that our proposed method can accurately detect the number of speakers in a mixture.
- We adopt the attention model to further make use of the T-F cross-domain features, which brings better source separation performance.
Proposed Method
===============
The flowchart of the proposed method is depicted in Fig. 1, which consists of three processing modules: an attention-based encoder, a separator with unknown number of speakers, and a decoder. In the following subsections, each module of the proposed method is introduced in detail.
Attention-based Encoder
-------------------------
The monaural speech separation problem is formulated by estimating $C$ sources $ \{s_i(t)\}_{i=1,\cdots,C}$ given the monaural mixture $x(t) = \sum_{i=1} ^C s_i(t)$, where the number of speakers $C$ is unknown. In the encoder part, we jointly utilize both the 1-dimensional (1-D) time-domain mixture and the 2-dimensional (2-D) frequency-domain spectrogram obtained by short-time Fourier transform (STFT). As shown in Fig. 1, the attention-based encoder first encodes the mixture $x(t)$ into a hybrid-domain 2-D feature map $\mathbf{H}$, which consists of $\mathbf{H}_{conv} $ and $\mathbf{H}_{spec} $ with $F = F_{conv} + F_{spec}$ frequency channels and $T$ time frames. The previous $F_{conv}$ channels are generated through the 1-D convolution operation and the subsequent $F_{spec}$ channels are obtained by the 2-D spectrogram. To integrate these two extracted features from T-F domains, the same window length and overlapping size for both domains are used.
$$X(t,f) = \sum_{i=1} ^C S_i(t,f)$$
In order to effectively exploit the T-F across-domain features $\mathbf{H}$, the squeeze-and-excitation network (SENet) [@PMID31034408] is employed to selectively emphasize informative features and meanwhile suppress useless ones. As shown in Fig. 2, we squeeze the global time information into a channel/frequency descriptor. This is achieved by using global average pooling to generate channel/frequency-wise statistics. A statistic $\mathbf{z} \in \mathbb{R}^{F}$ is generated by shrinking $\mathbf{H}$ along its time dimension such that the $f$-th element of $\mathbf{z}$ is calculated by $$\mathbf{z}_{f}=\mathcal{F}_{s q}\left(\mathbf{H}_{f}\right)=\frac{1}{T} \sum_{t=1}^{T} \mathbf{H}_{f}(t), \quad f \in 1,2,...,F$$
The information aggregated in the squeeze operation is followed by a gating mechanism which To fulfill this objective, the function must meet two criteria: first, it must be flexible and second, it must learn a non-mutually-exclusive relationship since we would like to ensure that multiple channels are allowed to be emphasized (rather than enforcing a one-hot activation). consists of a bottleneck with two fully-connected layers around a non-linearity ReLU, i.e., a dimensionality-reduction layer with reduction ratio $r$, the ReLU and then a dimensionality-increasing layer returning to the channel dimension of the transformation output $$\mathbf{u}=\mathcal{F}_{e x}(\mathbf{z}, \mathbf{W})=\sigma \big(g(\mathbf{z}, \mathbf{W})\big)=\sigma \big(\mathbf{W}_{2} \delta\left(\mathbf{W}_{1} \mathbf{z}\right) \big),$$ where $\delta$ refers to the ReLU function [@Sanchez2013], $\sigma$ refers to the Sigmoid function, $\mathbf{W}_{1} \in \mathbb{R}^{\frac{F}{r} \times F}$ and $\mathbf{W}_{2} \in \mathbb{R}^{F \times \frac{F}{r}} .$ The selective T-F fusion features $\widetilde{\mathbf{H}}$ is obtained by rescaling $\mathbf{H}$ with the activations $\mathbf{u}=[u_1~ u_2~ \cdots ~ u_F]$ $$\widetilde{\mathbf{H}}_{f}=\mathcal{F}_{\text {scale}}\left(\mathbf{H}_{f}, {u}_{f}\right)={u}_{f} \mathbf{H}_{f}, \quad f \in 1,2,... ,F$$ which is adopted as the input of the separator module.
{width="49.90000%"}
Separator with Unknown Number of Speakers
------------------------------------------
The separator module contains three parts: an embedding network, an attractor network for mask estimation, and a source counting part. The overall proposed system allows us to separate the speech mixture without assuming the known number of speakers, which is automatically estimated by the source counting part.
### Embedding Network
In order to estimate the speaker assignment for each T-F index on the hybrid-domain feature map $\widetilde{\mathbf{H}}$, we project the elements in $\widetilde{\mathbf{H}}$ to $L$-dimensional embeddings $\mathbf{V} \in \mathbb{R}^{N \times L}$, where $N = T \times F_{conv}$, and $\mathbf{V}$ are in $C$ orthogonal directions through multiple layers of 1-D Conv blocks, as shown in Fig. 1, where the 1-D Conv block is actually a residual block consisting of a 1x1-conv, a dilated depth-wise convolution and a 1x1-conv module [@8707065]. Thus, the embeddings $\mathbf{V}$ based on $\widetilde{\mathbf{H}}$ are given as $$\mathbf{V} = Embed(\widetilde{\mathbf{H}}).$$
### Deep Attractor Network for Mask Estimation
To estimate masks of all the speakers in the mixture, we follow the ADANet in [@8264702] starting with $K$ initial centers $\{e_{k}\}_{k=1,\cdots,K}$. By arbitrarily choosing $C$ out of the $K $ initial centers ($C$ is known in training but unknown in test time), we acquire $C$ new centroids by performing *k*-means clustering with ${I}$ iterations on the embeddings $\mathbf{V}$. Considering there are all $\tbinom{K}{C}$ possible selections out of the $K$ initial centers, we can obtain a total of $\tbinom{K}{C}$ sets of centroids, among which we determine the set of centroids $\mathbf{A}$ with the largest in-set distance. The masks for each speaker $\mathbf{M} \in \mathbb{R}^{T \times F_{conv} \times C}$ are then estimated by the dot product of the chosen centroids in $\mathbf{A}$ and the embeddings $\mathbf{V}$. $
\mathbf{M}_{i_{t, f}}=\mathbf{V}_{t, f} \cdot \mathbf{a}_{i} \quad \text { for } \mathbf{a}_{i} \in \mathbf{A}
$ However, the above strategy accomplished in [@Higuchi2017] achieves a limited result, not only acquire a 67.3 % source counting accuracy but also need a experimental threshold.
### Source Counting
In the test stage, we estimate the number of sources based on $\mathbf{V}$. Theoretically, the high-dimensional embedding vectors $\mathbf{V}$ after training are in $C$ directions that are orthogonal to each other. This property allows us to estimate the number of sources by estimating the rank of the covariance matrix of $\mathbf{V}$ [@Higuchi2017]. However, the above strategy might be sensitive to noise intensity. To overcome this problem, we propose to utilize the Gerschgorin disk estimation (GDE) algorithm [@Dong2013A] to count the number of speakers. At first, we compute the covariance matrix of $\mathbf{V}=[\mathbf{v}_{1}~ \mathbf{v}_{2} ~\cdots ~\mathbf{v}_{N}]$ $$\begin{aligned}
\mathbf{B} \! =\!\frac{1}{N} \!\sum_{n=1}^N \!\mathbf{v}_{n} \!\mathbf{v}_{n}^{T} \! = \!\left(\begin{array}{cccc}
\!r_{11} & r_{12} & \cdots & r_{1 L}\! \\
\!r_{21} & r_{22} & \cdots & r_{2 L} \!\\
\!\vdots & \vdots & \ddots & \vdots \!\\
\!r_{L 1} & r_{L 2} & \cdots & r_{L L}\!
\end{array}\right) \! =\!\Bigg[\!\!\begin{array}{cc}
\!\mathbf{R}_{1}\! & \!\mathbf{r}\! \\
\!\mathbf{r}^{H} \!& \!r_{L L}\!
\end{array}\!\!\Bigg],
\end{aligned}\notag$$ where $T$ and $H$ denote transpose and conjugate transpose operators, respectively. $L$ is the dimension of $\mathbf{v}_n$, $\mathbf{R}_1$ is an $(L-1)\times (L-1)$ sub-matrix obtained by deleting the last row and column of $\mathbf{B}$, and $\mathbf{r}=\left[r_{1 L}, \ldots, r_{(L-1) L}\right]^{T}$. The eigenvectors $\mathbf{U}_{1}$ of $\mathbf{R}_1$ can be obtained via eigenvalue decomposition. The decomposition can be expressed as $$\begin{array}{c}
{\mathbf{R}_{1}=\mathbf{U}_{1} \mathbf{\Lambda}_{1} \mathbf{U}_{1}^{H}} \\
{\mathbf{U}_{1}=\left[\mathbf{u}_{1}, \mathbf{u}_{2}^{\prime}, \cdots, \mathbf{u}_{N-1}^{\prime}\right]}
\end{array}$$ Then we formulate a new matrix as $$\mathbf{U}_{2}=\left[\begin{array}{cc}
{\mathbf{U}_{1}} & {\mathbf{0}_{(L-1) \times 1}} \\
{\mathbf{0}_{1 \times(L-1)}} & {1}
\end{array}\right],$$ based on which we transform $\mathbf{B}$ as below $$\begin{aligned}
\mathbf{R}_{2} = \mathbf{U}_{2}^{H} \mathbf{B} \mathbf{U}_{2}=\left(\begin{array}{ccccc}
\lambda_{1} & 0 & \cdots & 0 & \rho_{1} \\
0 & \lambda_{2} & \cdots & 0 & \rho_{2} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & \lambda_{L-1} & {\rho}_{L-1} \\
{\rho}_{1}^{*} & {\rho}_{2}^{*} & \cdots & {\rho}_{L-1}^{*} & r_{L L}
\end{array}\right),
\end{aligned}$$ where $*$ denotes conjugate operator, $\rho_{l}, \lambda_{l}$ ($l=1,2,...,L-1$) are radii and centers of the Gerschgorin disks, respectively. It is believed that the smaller are the radii and centers of noise Gerschgorin disks, the larger are the radii and centers of remaining Gerschgorin disks which correspond to signal Gerschgorin disks. Therefore, we use Gerschgorin disk radii to estimate the number of sources through $$G D E(k)=\left|\rho_{k}\right|-\frac{F_{\text{GDE}}(N)}{L-1} \sum_{l=1}^{L-1}\left|\rho_{l}\right|, ~ k \in 1,2,...,L-1$$ where $F_{\text{GDE}}(N)$ is an adjustable factor and also a non-increasing function of the sample size $N$. By detecting the first non-positive value $G D E\left(k_{0}\right)$, we can estimate the number of speakers as $\hat{C}=k_{0}-1$, which is more robust to noise.
[p[3.5cm]{}<p[4.9cm]{}<p[1.5cm]{}<p[1.8cm]{}<p[1.5cm]{}<p[1.8cm]{}<]{} & &\
& & &SI-SNRi (dB) & SDRi (dB)&SI-SNRi (dB)\
& DPCL++ [@Isik+2016] & 10.3&10.1 &\
& ADANet [@8264702] & 10.1 &9.8 & &\
& Conv-TasNet-gLN [@8707065]& 14.4 &14.1 & &\
& Improved [@8707067]& 15.4 &15.1 & &\
&$\mathbf{Proposed}$& $\mathbf{16.2}$ & $\mathbf{16.0}$ & &\
& DPCL++ [@Isik+2016] & & 6.9 & 6.7\
&ADANet [@8264702]& & & 8.6 & 8.3\
&Conv-TasNet-gLN [@8707065]& & & 11.4 & 11.0\
&Improved [@8707067]& & & 12.3 & 12.0\
&$\mathbf{Proposed}$& & & $\mathbf{13.4}$ & $\mathbf{13.2}$\
& DPCL++ [@Isik+2016]& 10.7& 10.4 & 7.3 & 7.1\
& ADANet [@8264702]& 10.4 &10.2 & 8.5 & 8.2\
& $\mathbf{Proposed}$& $\mathbf{15.5}$ & $\mathbf{15.3}$ & $\mathbf{15.0}$ & $\mathbf{14.5}$\
Decoder
-------
After multiplying the feature map $\widetilde{\mathbf{H}}_{conv}$ by the estimated masks $\mathbf{M},$ we disassemble the masked encoded features into their original components. As shown in Fig. 1, the convolutional feature is dealt with through a deconvolution layer followed by overlap-add method to reconstruct the original signals $
\hat{s} =$ Decoder $ (\mathbf{M} \odot \widetilde{\mathbf{H}}_{conv})
$, where $\odot$ is element-wise multiplication. We choose the negative signal-to-distortion ratio as our training objective $$\begin{aligned}
\mathcal{L}_{o s s} =
-10 \log _{10} \frac{\langle s, \hat{s}\rangle^{2}}{\|s\|^{2}\|\hat{s}\|^{2}-\left\langle s, \hat{s} \rangle^{2}\right.}
\end{aligned},$$ where $\langle\cdot, \cdot\rangle$ is dot product, and $\|s\|^{2}$ denotes the signal power.
Experimental Evaluation
=======================
We evaluate the proposed method on 2-speaker and 3-speaker mixtures, which are derived from the GRID dataset [@12111]. The 30 hours of training set and 10 hours of validation set are generated by different speakers from GRID at various signal-to-noise ratio (SNR) from -2.5 dB to 2.5 dB. The 5 hours of test set is similarly generated with the validation set except that the speakers are different. All the speech waveforms are resampled to $8$ [kHz]{}. The window size of the STFT and the kernel size for the convolution layer in the encoder are both 2.5 ms, and the square root Hann window is used for STFT. 20-point DFT is performed to extract the 11-D log magnitude feature, combined with the 256-D feature extracted by the 1-D Conv, thus forming 267-D feature in $\mathbf{H}$. Then, the attention-based feature $\widetilde{\mathbf{H}}$ is gained by emphasizing different channels via SENet, where the reduction ratio $r$ is set to 16 [@PMID31034408]. For the separator, the feature $\widetilde{\mathbf{H}}$ first goes through a $1 \times 1$-conv with 256 filters, followed by 8 residual 1-D Conv blocks, with dilated rate of $1,2, \ldots, 128,$ repeated for 4 times. $L=20$ is chosen as the embedding dimension for better comparison [@7471631; @Isik+2016; @7952155; @8264702; @8462507]. We set $K=4$ initial centers and $I=1$ iteration for *k*-means [@8264702]. The networks are trained for 100 epochs using Adam algorithm with permutation invariant training [@7979557; @7952154]. The training is performed end-to-end so that all components are jointly learned.
We evaluate the approach by the signal-to-distortion ratio improvement SDRi[@1643671] and the scale-invariant signal-to-noise ratio improvement (SI-SNRi)[@8264702].
Source Separation Evaluation
----------------------------
The proposed method is compared with other state-of-the-art methods [@Isik+2016; @8264702; @8707065; @8707067] on the generated test set. These methods are categorized into three groups: the two-speaker model trained for two-speaker separation task, the three-speaker model trained for three-speaker separation task, and the two & three-speaker model which is trained so that it can be applied to both two-speaker and three-speaker separation tasks with a unified model.
The separation results of different methods in terms of signal-to-distortion ratio improvement (SDRi) and the scale-invariant SNR improvement (SI-SNRi) [@8264702; @8707065; @1643671] are shown in Table I. In the cases where the number of speakers in the mixture is different from that of the model target, we mark them as ’Not Applicable’. It is observed that the proposed method achieves the best performance on both SI-SNRi and SDRi through 2-speaker and 3-speaker model respectively. The proposed method solves the source separation problem in time domain similar to Conv-TasNet-gLN [@8707065], thus having much better separation performance compared with DPCL++ [@Isik+2016] and ADANet [@8264702] which formulate the separation problem in frequency domain. However, different from Conv-TasNet-gLN [@8707065], the proposed method incorporates the ADANet for generating mask which is proved to be more effective for speech separation [@8707067]. More importantly, the proposed network with a single model enables us to handle the speech separation problem with different number of speakers.
Compared to the improved method [@8707067], the attention module in our encoder part can emphasize more important information which contributes to better separation performance. In addition, it is seen from the results in 2 & 3-speaker model that our proposed method obtains a great improvement, i.e., more than 4 dB gains of SDRi and SI-SNRi on 2-speaker mixtures, and 6.5 dB gains of SDRi and SI-SNRi on 3-speaker mixtures compared with DPCL++ [@Isik+2016] and ADANet [@8264702]. As a result, our model can obtain state-of-the-art separation results on both 2-speaker and 3-speaker mixtures in a single model. It should be also noted that our 2 & 3-speaker model can obtain better performance on 3-speaker mixtures than that using our 3-speaker model. It suggests that parameters of the proposed model learned in 2-speaker mixtures contribute to the separation of 3-speaker mixtures, which can be used for studying speech separation with more speakers.
Source Counting Evaluation
--------------------------
The GDE algorithm is adopted based on the high-dimensional features to count the number of speakers. The method of estimating the rank of the covariance matrix $\mathbf{B}$ in [@Higuchi2017] is used as a comparison. For both methods, the high-dimensional features $\mathbf{V}$ obtained from the mixture are used as input features. The test set consists of both 2-speaker and 3-speaker mixtures which are randomly selected from 3000 samples. We tune the threshold to achieve the best source counting performance of the rank estimation method. As shown in Table II, the proposed method identifies the number of speakers more accurately compared with the rank estimation method, and moreover it does not need to tune threshold for different number of speakers. The experimental results in Table I and Table II confirm the suitability for estimating the necessary information on the number of sources, which is often assumed to be known in advance.
[p[2.5cm]{}<p[1.5cm]{}<p[1.5cm]{}<p[1.5cm]{}<]{} &\
& Two Speakers & Three Speakers & Avg.\
[Rank Esti. [@Higuchi2017]]{} & 84.9 & 75.2 & 80.1\
$\mathbf{Proposed}$ & $\mathbf{95.7}$& $\mathbf{97.6}$&$\mathbf{96.7}$\
Conclusion
==========
In this paper, we propose an attention-based network for source counting and separation in T-F fusion domain. The SENet is incorporated into the encoder part of our model to emphasize more useful separation information, and the source number is counted by computing the Gerschgorin disks based on the covariance matrix of embedding vectors. The ADANet is then followed for mask estimation and enables our network to handle the speech separation with different number of speakers in a single model. Experimental results show that our proposed method can not only separate a mixture of different numbers of speakers, but also can accurately detect the number of speakers. Most current single-channel speech separation methods only address the mixture data up to three speakers, and our method shows significant performance. However, for the separation of more speakers, the performance of our method might degrade. It has been concluded from Table I that the parameters of the proposed model learned in fewer speaker mixtures would be beneficial to handle the mixture separation with more speakers, which motivates us to follow this direction to further investigate multi-speaker speech separation.
Although our method can handle the separation of different number of mixtures with a single model and is proved to be effective in two- and three-speaker mixtures, it still has a long way to go in the case of processing more speaker mixtures.
Acknowledgements
================
This research was funded by Hubei Provincial Natural Science Foundation of China under grant 2019CFB512.
|
---
abstract: 'Solutions to flat space Friedmann-Robertson-Walker cosmologies in Brans-Dicke theory with a cosmological constant are investigated. The matter is modelled as a $\gamma$-law perfect fluid. The field equations are reduced from fourth order to second order through a change of variables, and the resulting two-dimensional system is analyzed using dynamical system theory. When the Brans-Dicke coupling constant is positive $(\omega > 0)$, all initially expanding models approach exponential expansion at late times, regardless of the type of matter present. If $\omega < 0$, then a wide variety of qualitatively distinct models are present, including nonsingular “bounce” universes, “vacillating” universes and, in the special case of $\omega = -1$, models which approach stable Minkowski spacetime with an exponentially increasing scalar field at late times. Since power-law solutions do not exist, none of the models appear to offer any advantage over the standard deSitter solution of general relativity in achieving a graceful exit from inflation.'
address: |
Department of Physics\
University of California\
Santa Barbara, CA 93106-9530
author:
- 'Shawn J. Kolitch [^1]\'
date: 'February , 1995'
---
[Qualitative Analysis of Brans-Dicke Universes with a Cosmological Constant]{}
\#1[Fig. [\#1]{}]{} \#1[Equation \#1)]{} \#1[Equations \#1)]{} \#1[Eq. \#1)]{} \#1[Eqs. \#1)]{} ==\#1)[()]{} \#1[[\#1]{}]{}
Introduction {#intro}
============
A recent renewal of interest in Brans-Dicke (BD) theory[@BD] can be traced to the discovery by La and Steinhardt that the use of BD theory in place of general relativity can ameliorate the exit problem of inflationary cosmology[@La]. This is possible because the interaction of the BD scalar field with the metric slows the expansion from exponential to power-law. Although the original “extended inflation” scenario appears to have been ruled out by observational constraints[@LL], models which survive include those based on more general scalar-tensor theories, such as “hyperextended inflation”[@S-A], and hybrid models which include both a first-order phase transition and a period of slow-roll[@Holman2]. The renewed interest in scalar-tensor gravitation has led to several recent investigations into the generation of exact solutions for cosmology in such theories[@Barrow93], as well as to some qualitative studies of the models which result[@Barrow90; @Burd; @Damour93; @Romero; @Wands; @Kolitch94]. It has also been pointed out recently that an inflationary era may result directly from the dynamics of the scalar field, without any potential or cosmological constant being necessary[@Levin].
In this paper we are concerned with the behavior of homogeneous and isotropic cosmological models in Brans-Dicke gravity, with the addition of a positive cosmological constant. This differs from standard extended inflationary scenarios in that the vacuum energy is decoupled from the scalar field. The goal is simply to analyze the cosmological models which such a theory gives rise to, with an emphasis on the question of whether a viable inflationary model might exist. Previously, similar analyses have been performed for the case $\Lambda \ne 0$ with no matter present[@Romero], and for the case $\Lambda = 0$ with additional matter present[@Wands; @Kolitch94]. The treatment will closely parallel that given in Ref. [@Kolitch94], and the interested reader is referred to that paper for more detail.
In Sec. II, it is shown that the field equations for this theory can be reduced to a two-dimensional dynamical system in the case of flat space. In Sec. III, the equilibrium points are found and the corresponding solutions are discussed. In Sec. IV we summarize the results.
The Field Equations {#BDcosmo}
===================
In this section the field equations are reduced to a planar dynamical system through a change of variables. For notation and conventions, the reader is referred to the parallel treatment given in[@Kolitch94]. Generalizing the action for Brans-Dicke theory to include a nonzero cosmological constant, it may be written as $$S_{BD} = \int d^4x \sqrt{-g} \left(-\phi [R - 2\Lambda]
+ \omega {{\phi^{,\mu}
\phi_{,\mu}}\over {\phi}} + 16\pi
{\cal \char'114}_m \right).
\label{BDaction}$$ Taking $\Lambda$ and $\omega$ constant, and varying this action with respect to the metric and the scalar field, one finds that the nontrivial components of the field equations in a homogeneous and isotropic (FRW) spacetime are $$\begin{aligned}
\left({\dot a \over a} + {\dot \phi \over {2\phi}} \right)^2
+ {k \over a^2} =&& \left({2\omega + 3} \over 12 \right)
\left({\dot \phi \over \phi} \right)^2 +
{8\pi \rho \over {3\phi}} + {\Lambda \over 3},
\label{BDFRW1} \\
-{1 \over {a^3}}{d \over dt}(\dot \phi a^3) =&&
\left({8\pi} \over {3 + 2\omega} \right)
\left(T^\mu{}_\mu - {{\Lambda \phi}\over{4\pi}}\right),
\label{BDFRW2}\end{aligned}$$ where $a(t)$ is the cosmic scale factor. Assuming a perfect fluid form for the stress-energy tensor, [*i.e., *]{} $T_{\mu \nu} = \hbox{diag}(\rho, p, p, p)$, the usual conservation equation is also satisfied (the zeroth component of $T^{\mu \nu}{}_{;\nu} = 0$): $$\dot \rho = -3 {\dot a \over a} (p + \rho).
\label{BDFRW3}$$ Assuming only that $\rho > 0$ and $\phi > 0$, inspection of Eq. \[BDFRW1\]) reveals that we must have ${\omega\ge-3/2}$ in order to satisfy the field equations for all values of $k$ and $\Lambda$. Furthermore, we see from the form of the action in Eq. (\[BDaction\]) that the integrity of the theory is lost if $\omega = 0$. We therefore take ${\omega\ge -3/2}$ and $\omega \ne 0$ in what follows.
Now we take $k=0$, and transform the fourth-order system specified by Eqs. \[BDFRW1\]-\[BDFRW3\]) into a pair of coupled second-order equations in which, however, only first derivatives of the new variables appear. It will be sufficient for our purposes to consider only the models with $k=0$, as any candidate for a viable inflationary model must at the very least solve the flatness problem. First, define the new variables $$\begin{aligned}
\Theta \equiv&& \left({\dot a \over a} + {\dot \phi \over {2\phi}}
\right),
\label{betadef} \\
\Sigma \equiv&& A {\dot \phi \over \phi},
\label{sigdef}\end{aligned}$$ where dots represent derivatives with respect to time, and $A \equiv
(2\omega+3)/ 12$. Next, parametrize the equation of state by writing $p = (\gamma - 1) \rho$, where, for example, $$\gamma = \cases{ 0, & false-vacuum energy; \cr
1, & pressureless dust; \cr
4/3, & radiation; \cr
2, & ``stiff'' matter. \cr}$$ Finally, take $k=0$ and rewrite Eqs. (\[BDFRW1\]-\[BDFRW3\]) in terms of these new variables. Then straightforward differentiation and resubstitution lead to the equivalent field equations $$\begin{aligned}
\dot \Sigma =&& \Theta^2(1-3\gamma /4)-{{\Sigma^2}\over 2A}
(1-3\gamma /2) - 3\Theta \Sigma - {\Lambda \over 6}
(1-3\gamma /2),
\label{DS1} \\
\dot \Theta =&& 3{{\Sigma^2}\over A}(\gamma /2 - 1) -
{{3\gamma\Theta^2}\over 2} + {{\Theta\Sigma}\over {2A}}
+ {{\Lambda\gamma}\over 2}.
\label{DS2}\end{aligned}$$ Eqs. \[DS1\]) and (\[DS2\]) constitute a planar dynamical system in the variables $\Theta$ and $\Sigma$, and are the desired results of this section.
The Equilibrium Points {#qualanal}
======================
The equilibrium points of the dynamical system are obtained by setting $\dot\Theta$ and $\dot\Sigma$ equal to zero in Eqs. \[DS1\]) and (\[DS2\]), and then solving the resulting equations for $\Theta$ and $\Sigma$. As each of these equations is a second order polynomial, we expect in general four equilibrium points. When this implies that $\dot a / a = \Theta - \Sigma/2A$ and $\dot\phi / \phi = \Sigma/A$ are both constants, so that $$\begin{aligned}
a(t) =&& a_0 \exp\left[\left(\Theta - {\Sigma \over {2A}}\right)t
\right], \\
\phi(t) = && \phi_0 \exp\left({\Sigma t}\over A \right).
\label{EQa}\end{aligned}$$ Therefore a fixed point in the $\Theta$–$\Sigma$ system represents deSitter spacetime, with the addition of an exponentially varying scalar field. In the special case $\Theta = \Sigma /2A$, the solution is not deSitter but rather Minkowski spacetime with a scalar field. Note also that only equilibrium points at finite values have been considered. The global picture, including the behavior of $\Theta$ and $\Sigma$ at infinity, may be obtained by various compactification methods[@DSref]. Such an analysis is not necessary for our purposes, however, as we are primarily interested in whether viable inflationary models exist in this theory. In particular, one sees by inspection of Eqs. \[DS1\]) and (\[DS2\]) that the origin of the $\Theta$–$\Sigma$ plane can never be an equilibrium point when $\Lambda\ne 0$. This immediately rules out the possiblity of a stable power-law solution, since such a solution would appear in that plane as line of constant slope, and would asymptotically approach equilibrium at the origin.
Although solution curves span the entire $\Theta$–$\Sigma$ plane, the requirement $\rho > 0$, where $\rho$ is the energy density of the perfect fluid matter, eliminates some regions on physical grounds. It follows from Eq. \[BDFRW1\]) with $k=0$ that (assuming $\phi > 0$) any point $(\Theta_0,\Sigma_0)$ satisfying ${\Theta_0}^2 > {{\Sigma_0^2} / A} + {\Lambda / 3}$ will lie in $\rho > 0$, whereas points satisfying $ {\Theta_0}^2 < {{\Sigma_0^2} / A} + {\Lambda / 3}$ lie in $\rho < 0$ and thus do not represent physical solutions. Now let us proceed with the analysis. Although we restrict ourselves to the consideration of models with $\Lambda > 0$, it is clear that the techniques can easily be extended to models with a negative cosmological constant.
One pair of equilibrium points are always present regardless of the value of $\gamma$; they satisfy the field equations for $\rho = 0$ and are thus vacuum solutions. These points are $$(\Theta_0,\Sigma_0)_{1,2} = \pm\left[{\Lambda (2\omega +3)}\over
{2(3\omega +4)}\right]^{1/2}(1,1/6)$$ and they represent the solutions $$\begin{aligned}
a(t) =&& a_0 \exp\left\{\pm(\omega +1)\left[{{2\Lambda}\over
{(2\omega +3)(3\omega +4)}}\right]^{1/2}t\right\},
\label{vac1} \\
\phi(t) =&& \phi_0\exp\left\{\pm\left[{{2\Lambda}\over
{(2\omega +3)(3\omega +4)}}\right]^{1/2}t\right\},
\label{vac2}\end{aligned}$$ These solutions have previously been noted in the literature[@Barrow90; @Romero], and are attractors for most, but not all, of the initially expanding models in this theory ([*cf.*]{} discussion below). Note that if $\omega = -1$, then there are solutions where the geometry is Minkowski and the scalar field either grows or shrinks exponentially. Under the field redefinition $\phi \to e^{-\Phi}$, these can be identified as the static “linear dilaton” solutions of string cosmology[@Myers].
The location and stability of the remaining equilibrium points depends upon the value of $\gamma$, as well as upon $\omega$ and $\Lambda$. Hence it is convenient to classify the models, and to discuss the overall character and stability of the solutions, according to the equation of state of the perfect fluid. The additional equilibrium points are as follows:
$$\begin{aligned}
(\gamma = 0): \quad (\Theta_0,\Sigma_0)_{3,4} =&&
\pm\left[\Lambda\over 6\right]^{1/2}(1,0) \\
(\gamma = 1): \quad (\Theta_0,\Sigma_0)_{3,4} =&&
\pm\left[\Lambda\over {6(3\omega +4)}\right]^{1/2}
\left(1,{{2\omega +3}\over 2}\right) \\
(\gamma = 4/3): \quad (\Theta_0,\Sigma_0)_{3,4} =&&
\pm\left[\Lambda\over {2(2\omega +3)}\right]^{1/2}
\left(1,{{2\omega +3}\over 3}\right) \\
(\gamma = 2): \quad (\Theta_0,\Sigma_0)_{3,4} =&&
\pm\left[{2\Lambda}\over 3\right]^{1/2}
\left(1,{{2\omega +3}\over 4}\right)\end{aligned}$$
The details of the stability analysis are given in Appendix A. Table 1 of that appendix lists the eigenvalues of the equilibrium points, and Table 2 explicity states the existence and stability of each equilibrium point as a function of $\omega$. Note that in some cases, the solutions represented by a given equilibrium point require negative energy density, and are therefore physically uninteresting. In such cases we have simply written “$\rho < 0$” in the tables.
The overall character of the solutions may be further examined by numerically integrating the solutions $\Theta(t)$ and $\Sigma(t)$ for a variety of initial conditions with each qualitatively distinct set of parameters $(\Lambda,\omega,\gamma)$. Figures 1–4 show the results of this procedure, where we have selected $\Lambda = 3$ arbitrarily. The shaded regions in each case require $\rho < 0$, and so are disallowed physically. Curves to the right of the line $da/dt = 0$ represent expanding universes, and those to its left represent contracting universes. Note that if $-1 < \omega < 0$, as for example in Fig. 1b, then there exist nonsingular “bounce” models which pass smoothly from contraction to expansion, and “vacillating” models which pass from expansion to contraction to reexpansion, or the time reversal of this behavior. There also exist extreme cases of “vacillation”; some models can vacillate several times before settling down to exponential contraction at late times, as shown in Fig. 4d.
Summary {#concl}
=======
We have shown that in Brans-Dicke theory with a positive cosmological constant, a wide range of flat-space models exist, including some with no analogues in general relativity. In general, these models are parametrized by the initial conditions of the scalar field, the value of the BD coupling constant, an initial expansion rate, an equation of state for the matter and the value of the cosmological constant. The first two parameters are, of course, absent in general relativity. Most, but not all, of the expanding models asymptotically approach vacuum deSitter spacetime at late times. Power-law expansion is not possible when $\Lambda$ is nonzero.
If $\omega > -1$, there exist two finite-valued equilibrium points of physical interest, representing the vacuum solutions (\[vac1\]) and (\[vac2\]). One of these is stable and corresponds to expanding deSitter spacetime, and all initially expanding models approach this solution asymptotically. As $\omega \to \infty$, the solution becomes identical to the deSitter universe where $a(t) \sim e^{\Lambda t / 3}$ and $\phi(t) = \hbox{constant}$, in accordance with the well-established correspondence between GR and BD theory in this limit. When $\omega > 0$, all contracting models contract to a singularity; however, if $-1 < \omega < 0$, then nonsingular “bounce” models are also possible, and these may or may not approach deSitter spacetime at late times, depending upon the initial conditions of the model. This behavior has been noted by other authors[@bounce]. Also there are “vacillating” models, which expand from a big bang, slow down, and recontract before continuing their expansion and approaching deSitter spacetime. The time-reversal of this behavior also exists.
If $\omega = -1$, there exist “static-exponential” solutions, where the geometry is Minkowski spacetime while the scalar field changes exponentially with time. The solution with increasing scalar is found to be stable, and all models which expand from a big bang approach it at late times, regardless of the type of matter present. The stability of these models may be explained by the fact that in the Newtonian limit, $\phi \sim G^{-1}$[@SW]. Hence the exponential increase in the scalar field corresponds to an exponential weakening of the gravitational interactions, ensuring that the universe does not recollapse regardless of its matter content.
If $-3/2 < \omega < -1$, then the behavior of the models depends upon the type of matter present, and we can distinguish two classes of behavior. [**(i)**]{} In the cases of false-vacuum energy and pressureless dust, only the equilibrium points representing the vacuum solutions (\[vac1\]) and (\[vac2\]) are present in the regime $\rho > 0$, and the contracting solution is stable. All models which start from a big bang eventually contract to a singularity; this collapse will become asymptotically exponential if $-4/3 < \omega < -1$, or superexponential if $-3/2 < \omega < -4/3$. There also exist models which start with a finite rate of expansion and expand perpetually. [**(ii)**]{} In the cases of radiation and “stiff” matter, the dynamical system undergoes a qualitative change at a particular value of $\omega$. This critical value is $\omega_c = -5/4$ for radiation, and $\omega_c = -7/6$ for “stiff” matter. If $\omega_c \le \omega < -1$, then only the equilibrium points representing (\[vac1\]) and (\[vac2\]) are present with $\rho \ge 0$, and the contracting solution is stable. If $-4/3 < \omega < \omega_c$, then there are four equilibrium points in the regime $\rho \ge 0$, representing both vacuum and non-vacuum deSitter solutions; however, only the contracting non-vacuum solution is stable. In these solutions, the exponential growth of the scalar field exactly balances that of the energy density, so that the ratio $\rho / \phi$ is constant. Thus the ordinary matter acts exactly like a cosmological constant, since it is this ratio which appears as the matter source term in the field equation (\[BDFRW1\]). All models will collapse to a singularity; the collapse will be asymptotically exponential for models starting from a big bang. If $-3/2 < \omega < -4/3$, the unstable vacuum equilibrium points no longer exist; otherwise the behavior of the models remains unchanged.
Although somewhat exotic, the models discussed here do not seem to have any hope of solving the graceful exit problem of inflationary cosmology. In models of extended inflation, the mediation of the scalar field slows the expansion from exponential to power-law, so that the Hubble parameter decreases with time and true-vacuum bubble nucleation may complete the inflation-ending phase transition. Here, however, the cosmological constant induces deSitter spacetime, and all of the problems of Guth’s “old” inflation recur. In addition, one is faced with the question of the origin of the cosmological constant in these models. Although there is no [*a priori*]{} reason to exclude such a term from the field equations, the usual explanation of a field with a nonzero potential as the source of the vacuum energy is not available in this case, since such a potential term would be coupled to the scalar field.
Acknowledgements
================
This research was supported in part by the National Science Foundation under Grants No. PHY89-04035 and PHY90-08502.
Defining $\xi^{(1)} \equiv \Theta - \Theta_0 \quad {\rm and} \quad \xi^{(2)}
\equiv \Sigma - \Sigma_0$, the dynamical system specified by (\[DS1\]) and (\[DS2\]) may be written in the form $${{d \roarrow{\xi}} \over {dt}} = {\bf {\rm J}} \roarrow{\xi}
\quad + \quad \dots \quad ,$$ where the Jacobian is $${\bf {\rm J}} =
\pmatrix{-3\gamma\Theta_0 + \Sigma_0 / 2A
&-6(\gamma /2 - 1)\Sigma_0 / A
+ \Theta_0 / A \cr
\noalign{\medskip}
2\Theta_0(1-3\gamma /4) - 3\Sigma_0
&-(1-3\gamma /2)\Sigma_0 / A - 3\Theta_0 \cr}.$$
In cases where the eigenvalues of the Jacobian all have nonvanishing real part, the fixed point is called hyperbolic and we can determine its stability from the signs of those real parts: if the real part of each of the eigenvalues is negative at a given equilibrium point, the solution is stable at that point; if the real part of each eigenvalue is positive, or if the real part of one eigenvalue is positive and that of the other is negative, then the solution is unstable at that point. Finally, if the real part of any of the eigenvalues is zero at a point, then the point is called nonhyperbolic and its stability in the neighborhood of that point cannot be determined by this method [@DSref].
Table 1 shows the eigenvalues of the Jacobian for each equilibrium point, and Table 2 explicitly states the existence and stability of the equilibrium points as a function of $\omega$. In cases where the solution represented by the equilibrium point requires negative energy density, we have simply written “$\rho < 0$”. The points are labelled in accordance with the conventions in the text, [*i.e., *]{} for each value of $\gamma$, points 1 and 2 represent vacuum solutions, and points 3 and 4 represent non-vacuum solutions.
-------------------- ----------------------------------------------------- ------------------------------------------------------------
Equilibrium Point $\lambda_1$ $\lambda_2$
$(\gamma = 0)_1$ $-[2\Lambda / (2\omega +3)(3\omega + 4)]^{1/2}$ $-[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$
$(\gamma = 0)_2$ $+[2\Lambda (3\omega +4) / (2\omega +3)]^{1/2}$ $+[2\Lambda / (2\omega +3)(3\omega + 4)]^{1/2}$
$(\gamma = 0)_3$ $\rho < 0$ $\rho < 0$
$(\gamma = 0)_4$ $\rho < 0$ $\rho < 0$
$(\gamma = 1)_1$ $-[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$ $-[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$
$(\gamma = 1)_2$ $+[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$ $+[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$
$(\gamma = 1)_3$ $\rho < 0$ $\rho < 0$
$(\gamma = 1)_4$ $\rho < 0$ $\rho < 0$
$(\gamma = 4/3)_1$ $-[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$ $-(4\omega +5)[2\Lambda / (2\omega +3)(3\omega +4)]^{1/2}$
$(\gamma = 4/3)_2$ $+(4\omega +5)[2\Lambda / (2\omega +3)(3\omega +4)] $+[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$
^{1/2}$
$(\gamma = 4/3)_3$ $C(-1 + \sqrt{64\omega + 81})$ $C(-1 - \sqrt{64\omega + 81})$
$(\gamma = 4/3)_4$ $C(+1 + \sqrt{64\omega + 81})$ $C(+1 - \sqrt{64\omega + 81})$
$(\gamma = 2)_1$ $-[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$ $-(6\omega +7)[2\Lambda / (2\omega +3)(3\omega +4)]^{1/2}$
$(\gamma = 2)_2$ $+(6\omega +7)[2\Lambda / (2\omega +3)(3\omega +4)] $+[2\Lambda(3\omega +4) / (2\omega +3)]^{1/2}$
^{1/2}$
$(\gamma = 2)_3$ $C(-\sqrt{3(2\omega +3)} + \sqrt{102\omega + 121}) $C(-\sqrt{3(2\omega +3)} - \sqrt{102\omega + 121})
$ $
$(\gamma = 2)_4$ $C(+\sqrt{3(2\omega +3)} + \sqrt{102\omega + 121}) $C(+\sqrt{3(2\omega +3)} - \sqrt{102\omega + 121})
$ $
-------------------- ----------------------------------------------------- ------------------------------------------------------------
: Eigenvalues of the Jacobian Matrix for BD Cosmology with $\Lambda > 0$
\[table1\]
------------------------ -------------------------- ---------------- -----------------------
Equilibrium Point $-3/2 < \omega \le -4/3$ $-4/3 < \omega $\omega \ge \omega_c$
< \omega_c$
$(\gamma = 0,\,1)_1$ nonexistent N/A stable
$(\gamma = 0,\,1)_2$ nonexistent N/A unstable
$(\gamma = 0,\,1)_3$ $\rho < 0$ N/A $\rho < 0$
$(\gamma = 0,\,1)_4$ $\rho < 0$ N/A $\rho < 0$
$(\gamma = 4/3,\,2)_1$ nonexistent unstable stable
$(\gamma = 4/3,\,2)_2$ nonexistent unstable unstable
$(\gamma = 4/3,\,2)_3$ stable stable $\rho < 0$
$(\gamma = 4/3,\,2)_4$ unstable unstable $\rho < 0$
------------------------ -------------------------- ---------------- -----------------------
: Existence and Stability of the Equilibrium Points
\[table3\]
\#1, \#2, \#3, 1\#4\#5\#6[ [*\#1 *]{}[**\#2**]{}, \#3 (1\#4\#5\#6)]{}
C. Brans and R.H. Dicke, 124, 925, 1961.
D.A. La and P.J. Steinhardt, 62, 376, 1989.
A.R. Liddle and D.H. Lyth, Phys. Lett., 291B, 391, 1992.
P.J. Steinhardt and F.S. Accetta, 64, 2740, 1990
R. Holman, E.W. Kolb, S. Vadas and Y. Wang, 43, 3833, 1991.
J.D. Barrow, 47, 5329, 1993; 48, 3592, 1993.
J.D. Barrow and K. Maeda, Nucl. Phys., B341, 294, 1990.
A. Burd and A. Coley, Phys. Lett., 267B, 330, 1991.
T. Damour and K. Nordvedt, 70, 2217, 1993; 48, 3436, 1993.
C. Romero and A. Barros, Gen. Rel. and Grav., 23, 491, 1993.
D. Wands, Unpublished Ph.D. Thesis, University of Sussex, 1993.
S. Kolitch and D. Eardley, to be published in [*Annals of Physics*]{}, gr-qc preprint 9405016.
J.A. Levin, lanl preprints gr-qc 9405061, hep-th 9407101.
For a pedagogical review of the stability analysis of dynamical systems, see, for example, M.W. Hirsch and S. Smale, “Differential Equations, Dynamical Systems and Linear Algebra”, Academic Press, New York, 1974.
R.C. Myers, 199B, 371, 1987; I. Antoniadis et. al., 211B, 393, 1988.
L.E. Gurevich, A.M. Finkelstein and V.A. Ruban, Ap. Space Sci., 22, 231, 1973.
S. Weinberg, “Gravitation and Cosmology”, p.245, Wiley and Sons, New York, 1972.
[^1]: E-Mail Address: kolitch@nsfitp.itp.ucsb.edu
|
---
abstract: 'The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixed distributions are an exciting new area for optimal quantization. In this paper, we have determined the optimal sets of $n$-means, the $n$th quantization error, and the quantization dimensions of different mixed distributions. Besides, we have discussed whether the quantization coefficients for the mixed distributions exist. The results in this paper will give a motivation and insight into more general problems in quantization for mixed distributions.'
address: |
School of Mathematical and Statistical Sciences\
University of Texas Rio Grande Valley\
1201 West University Drive\
Edinburg, TX 78539-2999, USA.
author:
- Mrinal Kanti Roychowdhury
title: An overview of the quantization for mixed distributions
---
[^1]
Introduction
============
The quantization problem for a probability distribution has a deep background in information theory such as signal processing and data compression (see [@GG; @GN; @Z]). Although the work of quantization in engineering science has a long history, rigorous mathematical treatment has given by Graf and Luschgy (see [@GL1]). Let us consider a Borel probability measure $P$ on ${\mathbb{R}}^d$ and a natural number $n \in {\mathbb{N}}$. Then, the $n$th [quantization error]{} for $P$ is defined by: $$V_n:=V_n(P)={\text}{inf}{\{\int \min_{a\in {\alpha}} \|x-a\|^2 dP(x): {\alpha}{\subset}{\mathbb{R}}^d, \, {\text}{card}({\alpha}) \leq n\}},$$ where $\|\cdot\|$ denotes the Euclidean norm on ${\mathbb{R}}^d$. A set ${\alpha}$ for which the infimum occurs and contains no more than $n$ points is called an [optimal set of $n$-means]{}, or [optimal set of $n$-quantizers]{}. Of course, this makes sense only if the mean squared error or the expected squared Euclidean distance $\int \| x\|^2 dP(x)$ is finite (see [@AW; @GKL; @GL; @GL1]). It is known that for a continuous probability measure an optimal set of $n$-means always has exactly $n$-elements (see [@GL1]). For a finite set ${\alpha}{\subset}{\mathbb{R}}^d$, the number $\int\min_{a\in {\alpha}}\|x-a\|^2 dP(x)$ is often refereed to as the [cost]{} or [distortion error]{} for ${\alpha}$ with respect to the probability distribution $P$. The numbers $${\underline}D(P):=\liminf_{n\to \infty} \frac{2\log n}{-\log V_n(P)}, {\text}{ and } {\overline}D(P):=\limsup_{n\to \infty} \frac{2\log n}{-\log V_n(P)},$$ are, respectively, called the [lower]{} and [upper quantization dimensions]{} of the probability measure $P$. If ${\underline}D(P)={\overline}D (P)$, the common value is called the [quantization dimension]{} of $P$ and is denoted by $D(P)$. For any $s\in (0, +\infty)$, the numbers $\liminf_n n^{\frac 2 s} V_n(P)$ and $\limsup_n n^{\frac 2s} V_n(P)$ are, respectively, called the $s$-dimensional [lower]{} and [upper quantization coefficients]{} for $P$. If the $s$-dimensional lower and upper quantization coefficients for $P$ are finite and positive, then $s$ coincides with the quantization dimension of $P$. Main concerns in quantization problem include $(i)$ the asymptotic properties of the quantization errors such as the quantization dimensions and the quantization coefficients; $(ii)$ the optimal sets in the quantization for a given measure. It is known that for any Borel probability measure $P$ on ${\mathbb{R}}^d$ with non-vanishing absolutely continuous part $\lim_n n^{\frac 2 d} V_n(P)$ is finite and strictly positive (see [@BW]); in other words, the quantization dimension of a Borel probability measure with non-vanishing absolutely continuous part equals the dimension $d$ of the underlying space. Although absolutely continuous probability measures have been well studied, there are not many results on the optimal sets for such a measure. In fact, to determine the optimal sets for a probability measure, singular or nonsingular, is much more difficult than to determine the quantization dimension of such a measure. For some work in the direction of optimal sets for a probability measure, one can see [@DR; @GL2; @R1; @R2]. For a finite set ${\alpha}{\subset}{\mathbb{R}}^d$, the [Voronoi region]{} generated by $a\in {\alpha}$, denoted by $M(a|{\alpha})$, is defined to be the set of all elements in ${\mathbb{R}}^d$ which are nearest to $a$. The set ${\{M(a|{\alpha}) : a \in {\alpha}\}}$ is called the [Voronoi diagram]{} or [Voronoi tessellation]{} of ${\mathbb{R}}^d$ with respect to ${\alpha}$. The point $a$ is called the centroid of its own Voronoi region if $a=E(X : X\in M(a|{\alpha}))$, where $X$ is a $P$-distributed random variable. Let us now state the following proposition (see [@GG; @GL1]).
\[prop0\] Let ${\alpha}$ be an optimal set of $n$-means, $a \in {\alpha}$, and $M (a|{\alpha})$ be the Voronoi region generated by $a\in {\alpha}$. Then, for every $a \in{\alpha}$, $(i)$ $P(M(a|{\alpha}))>0$, $(ii)$ $ P(\partial M(a|{\alpha}))=0$, $(iii)$ $a=E(X : X \in M(a|{\alpha}))$, and $(iv)$ $P$-almost surely the set ${\{M(a|{\alpha}) : a \in {\alpha}\}}$ forms a Voronoi partition of ${\mathbb{R}}^d$.
$(p_1, p_2, \cdots, p_N)$ is a probability vector, by that it is meant that $0<p_j<1$ for all $1\leq j\leq N$, and $\sum_{j=1}^N p_j=1$. We now give the following definition.
Let $P_1, P_2, \cdots, P_N$ be Borel probability measures on ${\mathbb{R}}^d$, and $(p_1, p_2, \cdots, p_N)$ be a probability vector. Then, a Borel probability measure $P$ on ${\mathbb{R}}^d$ is called a [mixed probability distribution]{}, or in short, [mixed distribution]{}, generated by $P_1, P_2, \cdots, P_N$ and the probability vector if for all Borel subsets $A$ of ${\mathbb{R}}^d$, $P(A)=p_1P_1(A)+p_2P_2(A)+\cdots+p_NP_N(A)$. Such a mixed distribution is denoted by $P:=p_1P_1+p_2P_2+\cdots+p_NP_N$, and $P_1, P_2, \cdots, P_N$ are called the [*components*]{} of the mixed distribution.
In this paper, in Section \[sec1\], we have considered a mixed distribution $P:=pP_1+(1-p)P_2$, where $p=\frac 12$, $P_1$ is a uniform distribution on the closed interval $C:=[0, \frac 12]$, and $P_2$ is a discrete distribution on $D:={\{\frac 23, \frac 56, 1\}}$. For this mixed distribution, in Subsection \[sec2\], we have determined the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. We further showed that the quantization dimension of $P$ exists, and equals the quantization dimension of $P_1$, which again equals one, which is the dimension of the underlying space. For such a mixed distribution quantization coefficient also exists. In Section \[sec3\], for a mixed distribution $P:=pP_1+(1-p)P_2$, where $P_1$ is an absolutely continuous probability measure supported by the closed interval $C:=[0, 1]$, and $P_2$ is discrete on $D:={\{0, 1\}}$, we mentioned a rule how to determine the optimal sets of $n$-means. In Proposition \[prop1111\], for a special case, we gave a closed formula to determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. In Remark \[rem4\], we proved a claim that the optimal sets for a mixed distribution may not be unique. In Section \[sec5\], we determined the optimal sets of $n$-means, and the $n$th quantization errors for all $n\geq 2$ for a mixed distribution $P:=\frac 12 P_1+\frac 12 P_2$, where $P_1$ is a Cantor distribution with support lying in the closed interval $[0, \frac 12]$, and $P_2$ is discrete on $D:={\{\frac 23, \frac 56, 1\}}$. We further showed that the quantization dimension of this mixed distribution exists, but the quantization coefficient does not exist. In Section \[sec61\], we mentioned some open problems to be investigated on mixed distributions. In Section \[sec7\], we considered a mixed distribution $P:=\frac 12 P_1+\frac 12 P_2$, where both $P_1$ and $P_2$ are Cantor distributions. For this mixed distribution, we determined the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. Further we showed that the quantization dimension of this $P$ exists, and satisfies $D(P)=\max {\{D(P_1), D(P_2)\}}$, but the quantization coefficient for $P$ does not exist. Finally, we would like to mention that mixed distributions are an exciting new area for optimal quantization, and the results in this paper will give a motivation and insight into more general problems.
Quantization with $P_1$ uniform and $P_2$ discrete {#sec1}
==================================================
Let $P_1$ be a uniform distribution on the closed interval $C:=[0, \frac 12]$, i.e., $P_1$ is a probability distribution on ${\mathbb{R}}$ with probability density function $g$ given by $$g(x)=\left\{\begin{array}{ccc}
2 & {\text}{ if }x \in C,\\
0 & {\text}{ otherwise}.
\end{array}\right.$$ Let $P_2$ be a discrete probability distribution on ${\mathbb{R}}$ with probability mass function $h$ given by $h(x)=\frac 13 $ for $x\in D$, and $h(x)=0$ for $x\in {\mathbb{R}}\setminus D$, where $D:={\{\frac 23, \frac 56, 1\}}$. Let $P$ be the mixed distribution on ${\mathbb{R}}$ such that $P=\frac 12 P_1+\frac 12 P_2$. Notice that the support of $P_1$ is $C$, and the support of $P_2$ is $D$ implying that the support of $P$ is $C{\cup}D$. Thus, for a Borel subset $A$ of ${\mathbb{R}}$, we can write $$P(A)=\frac 12 P_1(A{\cap}C)+\frac 12 P_2(A{\cap}D).$$ We now prove the following lemma.
\[lemma1\] Let $E(X)$ and $V:=V(X)$ represent the expected value and the variance of a random variable $X$ with distribution $P$. Then, $E(X)=\frac{13}{24}$ and $V=\frac{181}{1728}=0.104745$.
We have $$\begin{aligned}
&E(X)=\int x dP=\frac 12 \int x dP_1+\frac 12 \int x dP_2=\frac 12 \int_{[0, \frac{1}{2}]} x g(x)dx+\frac 12 \sum_{x\in D} x h(x)=\frac{13}{24}, {\text}{ and } \\ &E(X^2)=\int x^2 dP=\frac 12 \int x^2 dP_1+\frac 12 \int x^2 dP_2=\frac 12 \int_{[0, \frac 12]}x^2 g(x)dx+\frac 12 \sum_{x\in D} x^2 h(x)=\frac{43}{108},\end{aligned}$$ implying $V:=V(X)=E(X^2)-(E(X))^2 =\frac{43}{108}-\left(\frac{13}{24}\right)^2=\frac{181}{1728}$. Thus, the lemma is yielded.
Following the standard rule of probability, we see that $E\|X-a\|^2 =\int(x-a)^2 dP=V(X)+(a-E(X))^2=V+(a-\frac{13}{24})^2$, which yields the fact that the optimal set of one-mean consists of the expected value $\frac {13}{24}$, and the corresponding quantization error is the variance $V$ of the random variable $X$. By $P(\cdot|C)$, we denote the restriction of the probability measure $P$ on the interval $C$, i.e., $P(\cdot|C)=\frac {P(\cdot{\cap}C)}{P(C)}$, in other words, for any Borel subset $B$ of $C$ we have $P(B|C)=\frac{P(B{\cap}C)}{P(C)}$. Notice that $P(\cdot|C)$ is a uniform distribution with density function $f$ given by $$f(x)=\left\{\begin{array}{ccc}
2 & {\text}{ if }x \in C,\\
0 & {\text}{ otherwise},
\end{array}\right.$$ implying the fact that $P(\cdot|C)=P_1$. Similarly, $P(\cdot|D)=P_2$. In the sequel, for $n\in {\mathbb{N}}$ and $i=1, 2$, by ${\alpha}_n(P_i)$ and $V_n(P_i)$, it is meant the optimal sets of $n$-means and the $n$th quantization error with respect to the probability distributions $P_i$. If nothing is mentioned within a parenthesis, i.e., by ${\alpha}_n$ and $V_n$, it is meant an optimal set of $n$-means and the $n$th quantization error with respect to the mixed distribution $P$.
\[prop1111\] Let $P_1$ be the uniform distribution on the closed interval $[a, b]$ and $n\in {\mathbb{N}}$. Then, the set ${\{a + \frac{(2i-1)(b-a)}{2n} : 1\leq i\leq n\}}$ is a unique optimal set of $n$-means for $P_1$, and the corresponding quantization error is given by $V_n(P_1)=\frac{(a-b)^2}{12 n^2}$.
Notice that the probability density function $g$ of $P_1$ is given by $$g(x)=\left\{\begin{array}{ccc}
\frac 1{b-a} & {\text}{ if }x \in [a, b],\\
0 & {\text}{ otherwise}.
\end{array}\right.$$ Since $P_1$ is uniformly distributed on $[a, b]$, the boundaries of the Voronoi regions of an optimal set of $n$-means will divide the interval $[a, b]$ into $n$ equal subintervals, i.e., the boundaries of the Voronoi regions are given by $$\left\{a, \ a+\frac {(b-a)}{n}, \ a+ \frac {2(b-a)}{n},\ \cdots \ a+ \frac {(n-1)(b-a)}{n}, \ a+ \frac {n(b-a)} {n}\right\}.$$ This implies that an optimal set of $n$-means for $P_1$ is unique, and it consists of the midpoints of the boundaries of the Voronoi regions, i.e., the optimal set of $n$-means for $P_1$ is given by ${\alpha}_n(P_1):={\{a+\frac{(2i-1)(b-a)}{2n} : 1\leq i\leq n\}}$ for any $n\geq 1$. Then, the $n$th quantization error for $P_1$ due to the set ${\alpha}_n(P_1)$ is given by $$\begin{aligned}
&V_n(P_1)=n \int_{[a, a+\frac {b-a}{n}]} \Big(x-(a+\frac {b-a}{2n})\Big)^2 dP_1=n \int_{[0, \frac 1{2n}]} \frac 1{b-a} (x-\frac 1{4n})^2 dx=\frac{(a-b)^2}{12 n^2},\end{aligned}$$ which yields the proposition.
\[cor1\] Let $P_1$ be the uniform distribution on the closed interval $[0, \frac 12]$ and $n\in {\mathbb{N}}$. Then, the set ${\{\frac{2i-1}{4n} : 1\leq i\leq n\}}$ is a unique optimal set of $n$-means for $P_1$, and the corresponding quantization error is given by $V_n(P_1)=\frac 1{48n^2}$.
Notice that if ${\beta}{\subset}{\mathbb{R}}$, then $$\begin{aligned}
&\int \min_{b \in {\beta}}\|x-b\|^2 dP=\frac 12 \int_{[0, \frac 12]} \min_{b \in {\beta}}(x-b)^2 g(x) dx+\frac 12 \sum_{x\in D} \min_{b \in {\beta}}(x-b)^2h(x), {\text}{ and so, }\end{aligned}$$ $$\label{eq001} \int \min_{b \in {\beta}}\|x-b\|^2 dP=\int_{[0, \frac 12]} \min_{b \in {\beta}}(x-b)^2 dx+\frac 16 \sum_{x\in D} \min_{b \in {\beta}}(x-b)^2.$$
Optimal sets of $n$-means and the errors for all $n\geq 2$ {#sec2}
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In this subsection, we first determine the optimal sets of $n$-means and the $n$th quantization error for the mixed distribution $P$. Then, we show that the quantization dimension of $P$ exists and equals the quantization dimension of $P_1$, which again equals one, which is the dimension of the underlying space. To determine the distortion error in this subsection we will frequently use equation .
Let ${\alpha}$ be an optimal set of two-means. Then, ${\alpha}={\{\frac 14, \frac 56\}}$ with quantization error $V_2=\frac{17}{864}=0.0196759.$
Consider the set of two-points ${\beta}$ given by ${\beta}:={\{\frac 14, \frac 56\}}$. Then, the distortion error is $$\int \min_{b \in {\beta}}\|x-b\|^2 dP= \int_{[0, \frac 12]}(x-\frac 14)^2 dx+\frac 16 \sum_{x\in D} (x-\frac 56)^2=\frac{17}{864}=0.0196759.$$ Since $V_2$ is the quantization error for two-means we have $V_2\leq 0.0196759$. Let ${\alpha}:={\{a_1, a_2\}}$ be an optimal set of two-means with $a_1<a_2$. Since the optimal points are the centroids of their own Voronoi regions, we have $0<a_1<a_1<a_2\leq 1$. If $\frac{13}{32}\leq a_1$, then $$V_2\geq \int_{[0, \frac{13}{32}]} (x-\frac{13}{32})^2 dx=\frac{2197}{98304}=0.022349>V_2,$$ which is a contradiction. So, we can assume that $a_1\leq \frac{13}{32}$. We now show that the Voronoi region of $a_1$ does not contain any point from $D$. For the sake of contradiction, assume that the Voronoi region of $a_1$ contains points from $D$. Then, the following two case can arise:
Case 1. $\frac 23 \leq \frac 12(a_1+a_2)<\frac 56$.
Then, $a_1=E(X : X \in C {\cup}{\{\frac 23\}})=\frac{17}{48}$ and $a_2=E(X : X\in {\{\frac 56, 1\}})=\frac{11}{12}$, and so $\frac 12(a_1+a_2)=\frac{61}{96}<\frac 23$, which is a contradiction.
Case 2. $\frac 56 \leq \frac 12(a_1+a_2)<1$.
Then, $a_1=E(X : X \in C {\cup}{\{\frac 23, \frac 56\}})=\frac{9}{20}$ and $a_2=1$, and so $\frac 12(a_1+a_2)=\frac{29}{40}<\frac 56$, which is a contradiction.
By Case 1 and Case 2, we can assume that the Voronoi region of $a_1$ does not contain any point from $D$. We now show that the Voronoi region of $a_2$ does not contain any point from $C$. Suppose that the Voronoi region of $a_2$ contains points from $C$. Then, the distortion error is given by $$\begin{aligned}
&\int_{[0, \frac 12(a_1+a_2)]}(x-a_1)^2 dx+\int_{[\frac 12(a_1+a_2), \frac 12]}(x-a_2)^2 dx+\frac 16 \sum_{x\in D}(x-a_2)^2 \\
&=\frac{1}{108} \left(27 a_1^3+27 a_1^2 a_2-27 a_1 a_2^2-27 a_2^3+108 a_2^2-117 a_2+43\right),\end{aligned}$$ which is minimum when $a_1=\frac{5}{24}$ and $a_2=\frac{19}{24}$, and the minimum value is $\frac{37}{1728}=0.021412>V_2$, which leads to a contradiction. So, we can assume that the Voronoi region of $a_2$ does not contain any point from $C$. Thus, we have $a_1=\frac 14$ and $a_2=\frac 56$, and the corresponding quantization error is $V_2=\frac{17}{864}=0.0196759$. This, completes the proof of the lemma.
Let ${\alpha}$ be an optimal set of three-means. Then, ${\alpha}={\{0.191074, 0.573223, \frac {11}{12}\}}$ with quantization error $V_3=0.0106152.$
Let us consider the set of three-points ${\beta}:={\{0.191074, 0.573223, \frac {11}{12}\}}$. Since $0.382149=\frac{1}{2} (0.191074\, +0.573223)<\frac{1}{2}<\frac{2}{3}<\frac{1}{2} \left(0.573223\, +\frac{11}{12}\right)=0.744945<\frac{5}{6}$, the distortion error due to the set ${\beta}$ is given by $$\begin{aligned}
& \int \min_{b \in {\beta}}\|x-b\|^2 dP= \int_{[0, \, 0.382149]}(x-0.191074)^2 dx+\int_{[0.382149, \, \frac 12]}(x-0.573223)^2 dx\\
&+\frac 16 (\frac 23-0.573223)^2 +\frac 16(\frac 56-\frac {11}{12})^2+\frac 16(1-\frac {11}{12})^2=0.0106152.\end{aligned}$$ Since $V_3$ is the quantization error for three-means, we have $V_3\leq 0.0106152$. Let ${\alpha}:={\{a_1, a_2, a_3\}}$ be an optimal set of three-means with $a_1<a_2<a_3$. Since the optimal points are the centroids of their own Voronoi regions, we have $0<a_1<a_2<a_3\leq 1$. If $\frac{3}{8}\leq a_1$, then $$V_3\geq \int_{[0, \frac 3 8]}(x-\frac 38)^2 dx=\frac{9}{512}=0.0175781>V_3,$$ which leads to a contradiction. So, we can assume that $a_1<\frac 38$. If the Voronoi region of $a_2$ does not contain any point from $C$, then as the points of $D$ are equidistant from each other with equal probability, we will have either ($a_2=\frac 12(\frac 23+\frac 56)=\frac 34$ and $a_3=1$), or ($a_2=\frac 23$ and $a_3=\frac 12(\frac 56+1)=\frac {11}{12}$). In any case, the distortion error is $$\int_{[0, \frac 12]}(x-\frac 14)^2 dx+\frac 16((\frac 23-\frac 34)^2+(\frac 56-\frac 34)^2)=\frac{11}{864}=0.0127315>V_3,$$ which is a contradiction. So, we can assume that the Voronoi region of $a_2$ contains points from $C$. If the Voronoi region of $a_2$ does not contain any point from $D$, we must have $a_1=\frac 18$, $a_2=\frac 38$, and $a_3=\frac 56$. Then, the distortion error is $$\int_{[0, \frac 14]}(x-\frac 18)^2 dx+\int_{[\frac 14, \frac 12]}(x-\frac 38)^2 dx+\frac 16((\frac 23-\frac 56)^2+(1-\frac 56)^2)=\frac{41}{3456}=0.0118634>V_3,$$ which leads to a contradiction. Therefore, we can assume that the Voronoi region of $a_2$ contains points from $C$ as well as from $D$. We now show that the Vornoi region of $a_2$ contains only the point $\frac 23$ from $D$. On the contrary, assume that the Voronoi region of $a_2$ contains the points $\frac 23$ and $\frac 56$ from $D$. Then, we must have $a_3=1$, and so the distortion error is $$\begin{aligned}
&\int_{[0, \frac{a_1+a_2}{2}]} (x-a_1)^2 \, dx+\int_{[\frac{a_1+a_2}{2}, \frac{1}{2}]} (x-a_2)^2 \, dx+\frac{1}{6}\Big( (\frac{2}{3}-a_2)^2+(\frac{5}{6}-a_2)^2\Big)\\
&=\frac{1}{108} \Big(27 a_1^3+27 a_1^2 a_2-27 a_1 a_2^2-27 a_2^3+90 a_2^2-81 a_2+25\Big),\end{aligned}$$ which is minimum when $a_1=\frac 14$ and $a_2=\frac 34$, and the minimum value is $\frac{11}{864}=0.0127315>V_3$, which is a contradiction. Therefore, the Vornoi region of $a_2$ contains only the point $\frac 23$ from $D$. This implies $a_3=\frac 12(\frac 56+1)=\frac {11}{12}$, and then the distortion error is $$\begin{aligned}
& \int_{[0, \frac{a_1+a_2}{2}]} (x-a_1)^2 \, dx+\int_{[\frac{a_1+a_2}{2}, \frac{1}{2}]} (x-a_2)^2 \, dx+\frac{1}{6} (\frac{2}{3}-a_2)^2+\frac{1}{6} \left((1-\frac{11}{12})^2+(\frac{5}{6}-\frac{11}{12})^2\right)\\
&=\frac{1}{144} \left(36 a_1^3+36 a_1^2 a_2-36 a_1 a_2^2-36 a_2^3+96 a_2^2-68 a_2+17\right),\end{aligned}$$ which is minimum when $a_1=0.191074$ and $a_2=0.573223$, and the corresponding distortion error is $V_3=0.0106152$. Moreover, we have seen $a_3=\frac{11}{12}$. Thus, the proof of the lemma is complete.
Let ${\alpha}$ be an optimal set of four-means. Then, ${\alpha}={\{\frac 14, \frac 38, \frac 34, 1\}}$, or ${\alpha}={\{\frac 14, \frac 38, \frac 23, \frac {11}{12}\}}$, and the quantization error is $V_4=\frac{17}{3456}=0.00491898$.
Let us consider the set of four-points ${\beta}:={\{\frac 14, \frac 38, \frac 34, 1\}}$. Then, the distortion error due to the set ${\beta}$ is $$\begin{aligned}
& \int \min_{b \in {\beta}}\|x-b\|^2 dP=\int_{[0, \frac 14]}(x-\frac 18)^2 dx+\int_{[\frac 14, \frac 12]}(x-\frac 38)^2 dx+\frac 16\Big((\frac 23-\frac 34)^2+(\frac 56-\frac 34)^2\Big)=\frac{17}{3456}.\end{aligned}$$ Since $V_4$ is the quantization error for four-means, we have $V_4\leq\frac{17}{3456}=0.00491898$. Let ${\alpha}:={\{a_1<a_2<a_3<a_4\}}$ be an optimal set of four-means. Since the optimal points are the centroids of their own Voronoi regions, we have $0<a_1<\cdots<a_4\leq 1$. If the Voronoi region of $a_2$ does not contain points from $C$, then $$V_4\geq \int_{[0, \frac 12]}(x-\frac 14)^2 dx=\frac 1{96}=0.0104167>V_4,$$ which gives a contradiction, and so, we can assume that the Voronoi region of $a_2$ contains points from $C$. If the Voronoi region of $a_2$ contains points from $D$, then it can contain only the point $\frac 23$ from $D$, and in that case $a_3=\frac 56$ and $a_4=1$, which leads to the distortion error as $$\begin{aligned}
&\int_{[0, \frac{a_1+a_2}{2}]} (x-a_1)^2 \, dx+\int_{[\frac{a_1+a_2}{2}, \frac{1}{2}]} (x-a_2)^2 \, dx+\frac{1}{6} (\frac{2}{3}-a_2)^2\\
&=\frac{1}{216} \left(54 a_1^3+54 a_1^2 a_2-54 a_1 a_2^2-54 a_2^3+144 a_2^2-102 a_2+25\right),\end{aligned}$$ which is minimum when $a_1=0.191074$ and $a_2=0.573223$, and then, the minimum value is $0.00830043>V_4$, which is a contradiction. So, the Voronoi region of $a_2$ does not contain any point from $D$. If the Voronoi region of $a_3$ does not contain any point from $D$, then $a_4=\frac 56$ yielding $$V_4\geq \frac 16\Big((\frac 23-\frac 56)^2+(1-\frac 56)^2\Big)=\frac{1}{108}=0.00925926>V_4,$$ which leads to a contradiction. So, the Voronoi region of $a_3$ contains at least one point from $D$. Suppose that the Voronoi region of $a_3$ contains points from $C$ as well. Then, the following two cases can arise:
Case 1. $\frac 23 \in M(a_3|{\alpha})$.
Then, $a_4=\frac{11}{12}$, and the distortion error is $$\begin{aligned}
&\int_{[0, \frac{a_1+a_2}{2}]} (x-a_1)^2 \, dx+\int_{[\frac{a_1+a_2}{2}, \frac{a_2+a_3}{2}]} (x-a_2)^2 \, dx+\int_{[\frac{a_2+a_3}{2}, \frac{1}{2}]} (x-a_3)^2 \, dx+\frac{1}{6} (\frac{2}{3}-a_3)^2\\
&\qquad \qquad \qquad +\frac{1}{6} \Big((1-\frac{11}{12})^2+(\frac{5}{6}-\frac{11}{12})^2\Big)\\
&=\frac{1}{144} \Big(36 a_1^3+36 a_1^2 a_2-36 a_1 a_2^2+4 (9 a_2^2-17) a_3+(96-36 a_2) a_3^2-36 a_3^3+17\Big)\end{aligned}$$ which is minimum if $a_1= 0.118238$, $a_2= 0.354715$, and $a_3=0.645285$, and the minimum value is $0.00506623>V_4$, which is a contradiction.
Case 2. ${\{\frac 23, \frac 56\}} {\subset}M(a_3|{\alpha})$.
Then, $a_4=1$, and the corresponding distortion error is $$\begin{aligned}
&\int_{[0, \frac{a_1+a_2}{2}]} (x-a_1)^2 \, dx+\int_{[\frac{a_1+a_2}{2}, \frac{a_2+a_3}{2}]} (x-a_2)^2 \, dx+\int_{[\frac{a_2+a_3}{2}, \frac{1}{2}]} (x-a_3)^2 \, dx\\
&+\frac{1}{6} \Big((\frac{2}{3}-a_3)^2+(\frac{5}{6}-a_3)^2\Big)\\
&=\frac{1}{108} \Big(27 a_1^3+27 a_1^2 a_2-27 a_1 a_2^2+27 (a_2^2-3) a_3+(90-27 a_2) a_3^2-27 a_3^3+25\Big),\end{aligned}$$ which is minimum if $a_1=0.0990219, \, a_2=0.297066$, and $a_3=0.702934$, and the minimum value is $0.00680992>V_4$, which gives a contradiction.
By Case 1 and Case 2, we can assume that the Voronoi region of $a_3$ does not contain any point from $C$. Thus, we have $(a_1=\frac 14$, $a_2=\frac 38$, $a_3=\frac 34$, and $a_4=1$), or $(a_1=\frac 14$, $a_2=\frac 38$, $a_3=\frac 23$, and $a_4=\frac {11}{12})$, and the corresponding quantization error is $V_4=\frac{17}{3456}=0.00491898$.
\[lemma5111\] Let ${\alpha}$ be an optimal set of five-means. Then, ${\alpha}={\{\frac 18, \frac 38, \frac 23, \frac 56, 1\}}$, and the corresponding quantization error is $V_5=\frac{1}{384}=0.00260417$.
Consider the set of five points ${\beta}:={\{\frac 14, \frac 38, \frac 23, \frac 56, 1\}}$. The distortion error due to the set ${\beta}$ is given by $$\begin{aligned}
& \int \min_{b \in {\beta}}\|x-b\|^2 dP=\int_{[0, \frac 14]}(x-\frac 18)^2 dx+\int_{[\frac 14, \frac 12]}(x-\frac 38)^2 dx=\frac{1}{384}=0.00260417.\end{aligned}$$ Since $V_5$ is the quantization error for five-means, we have $V_5\leq 0.00260417$. Let ${\alpha}:={\{a_1<a_2<a_3<a_4<a_5\}}$ be an optimal set of five-means. Since the optimal points are the centroids of their own Voronoi regions, we have $0<a_1<\cdots<a_5\leq 1$. If the Voronoi region of $a_3$ does not contain any point from $D$, then we must have $(a_1=\frac{1}{12}$, $a_2=\frac 14$, $a_3=\frac 5{12}$, $a_4=\frac 34$, and $a_4=1)$, or $(a_1=\frac{1}{12}$, $a_2=\frac 14$, $a_3=\frac 5{12}$, $a_4=\frac 23$, and $a_4=\frac{11}{12})$ yielding the distortion error $$3 \int_{[0, \frac 16]}(x-\frac 1{12})^2 dx +\frac{1}{6} \Big((\frac{2}{3}-\frac{3}{4})^2+(\frac{5}{6}-\frac{3}{4})^2\Big)=\frac{1}{288}=0.00347222>V_5,$$ which is a contradiction. So, we can assume that the Voronoi region of $a_3$ contains a point from $D$. In that case, we must have $a_4=\frac 56$ and $a_5=1$. Suppose that the Voronoi region of $a_3$ contains points from $C$ as well. Then, the distortion error is $$\begin{aligned}
&\int_{[0, \frac{a_1+a_2}{2}]} (x-a_1)^2 \, dx+\int_{[\frac{a_1+a_2}{2}, \frac{a_2+a_3}{2}]} (x-a_2)^2 \, dx+\int_{[\frac{a_2+a_3}{2}, \frac{1}{2}]} (x-a_3)^2 \, dx+\frac{1}{6} \Big(\frac{2}{3}-a_3\Big)^2\\
&=\frac{1}{216} \left(54 a_1^3+54 a_1^2 a_2-54 a_1 a_2^2+6 \left(9 a_2^2-17\right) a_3-18 (3 a_2-8) a_3^2-54 a_3^3+25\right),\end{aligned}$$ which is minimum if $a_1=0.118238$, $a_2=0.354715$, and $a_3=0.645285$, and the minimum value is $0.00275142>V_5$, which is a contradiction. So, the Voronoi region of $a_3$ does not contain any point from $C$ yielding $a_1=\frac 18$, $a_2=\frac 38$, $a_3=\frac 23$, $a_4=\frac 56$ and $a_5=1$, and the corresponding quantization error is $V_5=\frac{1}{384}=0.00260417$. Thus, the proof of the lemma is complete.
\[th61\] Let $n\in {\mathbb{N}}$ and $n\geq 5$, and let ${\alpha}_n$ be an optimal set of $n$-means for $P$ and ${\alpha}_n(P_1)$ be the optimal set of $n$-means with respect to $P_1$. Then, $${\alpha}_n(P)={\alpha}_{n-3}(P_1){\cup}D, {\text}{ and } V_n(P)=\frac 12 V_{n-3}(P_1).$$
If $n=5$, by Lemma \[lemma5111\], we have ${\alpha}_5(P)={\{\frac 18, \frac 38, \frac 23, \frac 56, 1\}}$ and $V_5(P)=\frac 1{384}$, which by Corollary \[cor1\] yields that ${\alpha}_5(P)={\alpha}_2(P_1){\cup}D$ and $V_5(P)=\frac 12 V_2(P_1)$, i.e., the theorem is true for $n=5$. Proceeding in the similar way, as Lemma \[lemma5111\], we can show that the theorem is true for $n=6$ and $n=7$. We now show that the theorem is true for all $n\geq 8$. Consider the set of eight points ${\beta}:={\{\frac 1{20}, \frac{3}{20}, \frac 1 4, \frac 7{20}, \frac{9}{20}, \frac 23, \frac 56, 1\}}$. The distortion error due to set ${\beta}$ is given by $$\int \min_{b \in {\beta}}\|x-b\|^2 dP=5 \int_{[0, \frac 1{10}]}(x-\frac1 {20})^2 dx=\frac{1}{2400}=0.000416667.$$ Since $V_n$ is the $n$th quantization error for $n$-means for $n\geq 8$, we have $V_n\leq V_8\leq 0.000416667$. Let ${\alpha}_n:={\{a_1<a_2<\cdots <a_n\}}$ be an optimal set of $n$-means for $n\geq 8$, where $0<a_1<\cdots<a_n\leq 1$. To prove the first part of the theorem, it is enough to show that $M(a_{n-2}|{\alpha}_n)$ does not contain any point from $C$, and $M(a_{n-3}|{\alpha}_n)$ does not contain any point from $D$. If $M(a_{n-2}|{\alpha}_n)$ does not contain any point from $D$, then $$V_n\geq \frac 16\Big((\frac 23-\frac 34)^2+(\frac 56-\frac 34)^2\Big)=\frac{1}{432}=0.00231481>V_n,$$ which leads to a contradiction. So, $M(a_{n-2}|{\alpha}_n)$ contains a point, in fact the point $\frac 23$, from $D$. If $M(a_{n-2}|{\alpha}_n)$ does not contain points from $C$, then $a_{n-2}=\frac 23$. Suppose that $M(a_{n-2}|{\alpha}_n)$ contains points from $C$. Then, $\frac 23\leq \frac 12(a_{n-2}+a_{n-1})$ implies $a_{n-2}\geq \frac 43-a_{n-1}=\frac 43-\frac 56=\frac 12$. The following three cases can arise:
Case 1. $\frac 12\leq a_{n-2}\leq \frac 7{12}$.
Then, $V_n\geq \frac 16 (\frac 23-\frac 7{12})^2=\frac{1}{864}=0.00115741>V_n,$ which is a contradiction.
Case 2. $\frac 7{12}\leq a_{n-2}\leq \frac 58$.
Then, $\frac 12(a_{n-3}+a_{n-2})<\frac 12$ implying $a_{n-3}<1-a_{n-2}\leq 1-\frac 7{12}=\frac 5{12}$, and so $$V_n\geq \int_{[\frac{5}{12}, \frac{1}{2}]} \Big(x-\frac{5}{12}\Big)^2 \, dx+\frac{1}{6} \Big(\frac{2}{3}-\frac{5}{8}\Big)^2=\frac{5}{10368}=0.000482253>V_n,$$ which leads to a contradiction.
Case 3. $\frac 58 \leq a_{n-2}$.
Then, $\frac 12(a_{n-3}+a_{n-2})<\frac 12$ implying $a_{n-3}<1-a_{n-2}\leq 1-\frac 5{8}=\frac 38$, and so $$V_n\geq \int_{[\frac{3}{8}, \frac{1}{2}]} \Big(x-\frac{3}{8}\Big)^2 \, dx=\frac{1}{1536}=0.000651042>V_n,$$ which gives contradiction.
Thus, in each case we arrive at a contradiction yielding the fact that $M(a_{n-2}|{\alpha}_n)$ does not contain any point from $C$. If $M(a_{n-3}|{\alpha})$ contains any point from $D$, say $\frac 23$, then we will have $$M(a_{n-2}|{\alpha}){\cup}M(a_{n-1}|{\alpha}){\cup}M(a_n|{\alpha})={\{\frac 56, 1\}},$$ which by Proposition \[prop0\] implies that either $(a_{n-2}=a_{n-1}=\frac 56$, and $a_n=1$), or $(a_{n-2}=\frac 56$, and $a_{n-1}=a_n=1)$, which contradicts the fact that $0<a_1<\cdots<a_{n-2}<a_{n-1}<a_n\leq 1$. Thus, $M(a_{n-3}|{\alpha})$ does not contain any point from $D$. Hence, ${\alpha}_n(P)={\alpha}_{n-3}(P_1){\cup}D$, and so, $$V_n(P)=\int_{C}\min_{a\in{\alpha}_{n-3}(P_1)}(x-a)^2 dx+\frac 16 \sum_{x\in D}\min_{a\in D}(x-a)^2=\frac 12\int_{C}\min_{a\in{\alpha}_{n-3}(P_1)}(x-a)^2 2dx$$ implying $V_n(P)=\frac 12 V_{n-3}(P_1)$. Thus, the proof of the theorem is complete.
\[prop612\] Let $P$ be the mixed distribution as defined before. Then, $$\lim_{n\to \infty} n^2 V_n(P)=\frac 1{96}.$$
By Corollary \[cor1\] and Theorem \[th61\], we have $$\lim_{n\to \infty} n^2 V_n(P)=\frac 12 \lim_{n\to \infty} n^2 V_{n-3} (P_1)=\frac12 \lim_{n\to \infty} \frac {n^2} {48(n-3)^2}=\frac 1{96},$$ and thus, the proposition is yielded.
By Proposition \[prop612\], it follows that $\mathop{\lim}\limits_{n\to\infty} n^2 V_n(P)=\frac 1{96}$, i.e., one-dimensional quantization coefficient for the mixed distribution $P$ is finite and positive implying the fact that the quantization dimension of the mixed distribution $P$ exists, and equals one, which is the dimension of the underlying space. It is known that for a probability measure $P$ on ${\mathbb{R}}^d$ with non-vanishing absolutely continuous part $\mathop{\lim}\limits_{n\to\infty} n^{\frac 2d} V_n(P)$ is finite and strictly positive, i.e., the quantization dimension of $P$ exists, and equals the dimension $d$ of the underlying space (see [@BW]). Thus, for the mixed distribution $P$ considered in this section, we see that $D(P)=D(P_1)=1$.
A rule to determine optimal quantizers {#sec3}
======================================
Let $0<p<1$ be fixed. Let $P$ be a mixed distribution given by $P=pP_1+(1-p)P_2$ with the support of $P_1$ equals $C$ and the support of $P_2$ equals $D$, such that $P_1$ is continuous on $C$, and $P_2$ is discrete on $D$, and $D{\subset}C$. It is well-known that the optimal set of one-mean consists of the expected value and the corresponding quantization error is the variance $V$ of the $P$-distributed random variable $X$. Assume that $P_1$ is absolutely continuous on $C:=[0, 1]$, and $P_2$ is discrete on $D:={\{0, 1\}}$. Then, in the following note we give a rule how to obtain the optimal sets of $n$-means for the mixed distribution $P$ for any $n\geq 2$.
\[note2222\] Let ${\alpha}_n:={\{a_1, a_2, \cdots, a_n\}}$ be an optimal set of $n$-means for $P$ such that $0\leq a_1<a_2<\cdots<a_n\leq 1$. Write $$\begin{aligned}
\label{eq234}
M(a_i|{\alpha}_n):=\left\{\begin{array}{cc}
\left[0, \frac{a_1+a_2}{2}\right] & {\text}{ if } i=1, \\
\left[\frac{a_{i-1}+a_i}{2}, \frac{a_i+a_{i+1}}{2}\right] & {\text}{ if } 2 \leq i \leq n-1, \\
\left[\frac{a_{n-1}+a_n}{2}, 1\right] & {\text}{ if } i=n,
\end{array}
\right.\end{aligned}$$ where $M(a_i|{\alpha})$ represent the Voronoi regions of $a_i$ for all $1\leq i\leq n$ with respect to the set ${\alpha}_n$. Since the optimal points are the centroids of their own Voronoi regions, we have $a_i=E(X : X \in M(a_i|{\alpha}))$ for all $1\leq i\leq n$. Solving the $n$ equations one can obtain the optimal sets of $n$-means for the mixed distribution $P$. Once, an optimal set of $n$-means is known, the corresponding quantization error can easily be determined.
Let us now give the following proposition.
\[prop1111\] Let ${\alpha}_n$ be an optimal set of $n$-means and $V_n$ is the corresponding quantization error for $n\geq 2$ for the mixed distribution $P:=\frac 12 P_1+\frac 12 P_2$ such that $P_1$ is uniformly distributed on $C:=[0, 1]$ with probability density function $g$ given by $$g(x)=\left\{\begin{array}{ccc}
1 & {\text}{ if }x \in C,\\
0 & {\text}{ otherwise},
\end{array}\right.$$ and $P_2$ is discrete on $D:={\{1\}}$ with mass function $h$ given by $h(1)=1$. Then, for $n\geq 2$, $${\alpha}_n:=\left\{\frac{(2 i-1) \left(-\sqrt{n^2-n+1}+2 n-1\right)}{2 (n-1) n} : 1\leq i\leq n \right\}$$ and $V_n=\frac{4 n^2-4 \left(\sqrt{n^2-n+1}+1\right) n+2 \sqrt{n^2-n+1}+7}{12 \left(\sqrt{n^2-n+1}+2 n-1\right)^2}.$
As mentioned in Note \[note2222\], solving the $n$ equations $a_i=E(X : X \in M(a_i|{\alpha}))$, we obtain $$a_i=\frac{(2 i-1) \left(-\sqrt{n^2-n+1}+2 n-1\right)}{2 (n-1) n},$$ for all $1\leq i\leq n$, and hence, the corresponding quantization error is given by $$\begin{aligned}
V_n=\int_0^{\frac 12(a_1+a_2)}(x-a_1)^2 dx+ \sum_{i=2}^{n-1}\int_{\frac 12(a_{i-1}+a_{i})}^{\frac 12(a_{i}+a_{i+1})}(x-a_{i})^2 dx+\int_{\frac 12(a_{n-1}+a_{n})}^1(x-a_n)^2 dx+\frac 12(a_n-1)^2,\end{aligned}$$ which upon simplification yields $V_n=\frac{4 n^2-4 \left(\sqrt{n^2-n+1}+1\right) n+2 \sqrt{n^2-n+1}+7}{12 \left(\sqrt{n^2-n+1}+2 n-1\right)^2}$. Thus, the proof of the proposition is complete.
\[rem4\] Let $P_1$ be absolutely continuous on $C:=[0, 1]$ and $P_2$ be discrete on $D$ with $D{\subset}C$. Then, if $D:={\{0, 1\}}$, the system of equations in has a unique solution implying that there exists a unique optimal set of $n$-means for the mixed distribution $P:=pP_1+(1-p)P_2$ for each $n\in {\mathbb{N}}$. If $D{\cap}{\text}{Int}(C)$ is nonempty, where ${\text}{Int}(C)$ represents the interior of $C$, then the optimal sets of $n$-means for the mixed distribution $P$ for all $n\in {\mathbb{N}}$ is not necessarily unique, see Proposition \[prop1112\].
\[prop1112\] Let $P:=\frac 12 P_1+\frac 12 P_2$, where $P_1$ is uniformly distributed on $C:=[0, 1]$ and $P_2$ is discrete on $D:={\{\frac 12\}}$. Then, $P$ has two different optimal sets of two-means.
Let ${\alpha}:={\{a_1, a_2\}}$ be an optimal set of two means for $P$ with $0<a_1<a_2<1$. Then, $P$-almost surely, we have $C=M(a_1|{\alpha}){\cup}M(a_2|{\alpha})$ implying that either $\frac 12 \in M(a_1|{\alpha})$, or $\frac 12\in M(a_2|{\alpha})$. First, assume that $\frac 12 \in M(a_1|{\alpha})$, i.e., $0<a_1<\frac 12\leq \frac 12(a_1+a_2)$. Then, $$\begin{aligned}
& a_1=E(X : X \in [0, \frac 12(a_1+a_2)])=\frac{\int_0^{\frac{a+b}{2}} x \, dx+\frac{1}{2}}{\int_0^{\frac{a+b}{2}} 1 \, dx+1}=\frac{a^2+2 a b+b^2+4}{4 (a+b+2)}, {\text}{ and } \\
& a_2=E(X : X \in [\frac 12(a_1+a_2), 1])=\frac{\int_{\frac{a+b}{2}}^1 x \, dx}{\int_{\frac{a+b}{2}}^1 1 \, dx}=\frac{1}{4} (a+b+2).\end{aligned}$$ Solving the above two equations, we have $a_1=\frac{1}{4} (-5+3 \sqrt{5})$ and $a_2=\frac{1}{4} (1+\sqrt{5})$, and the corresponding quantization error is given by $$\begin{aligned}
& V_2(P)=\int\min_{a\in{\alpha}}\|x-a\|^2 dP=\frac 12 \int\min_{a\in{\alpha}}(x-a)^2 dP_1+\frac 12\int\min_{a\in{\alpha}}(x-a)^2 dP_2\\
&=\frac{1}{2} \int_0^{\frac{a_1+a_2}{2}} (x-a_1)^2 \, dx+\frac{1}{2} \int_{\frac{a_1+a_2}{2}}^1 (x-a_2)^2 \, dx+\frac{1}{2} \Big(\frac{1}{2}-a_1\Big)^2=0.0191242.
\end{aligned}$$ Next, assume that $\frac 12 \in M(a_2|{\alpha})$, i.e., $\frac 12(a_1+a_2)\leq \frac 12<a_2<1$. Then, $$\begin{aligned}
& a_1=E(X : X \in [0, \frac 12(a_1+a_2)])=\frac{\int_0^{\frac{a+b}{2}} x \, dx}{\int_0^{\frac{a+b}{2}} 1 \, dx}=\frac{a+b}{4}, {\text}{ and } \\
& a_2=E(X : X \in [\frac 12(a_1+a_2), 1])=\frac{\int_{\frac{a+b}{2}}^1 1 x \, dx+\frac{1}{2}}{\int_{\frac{a+b}{2}}^1 1 \, dx+1}=\frac{a^2+2 a b+b^2-8}{4 (a+b-4)}.\end{aligned}$$ Solving the above two equations, we have $a_1=\frac{1}{4} (3-\sqrt{5})$ and $a_2=\frac{3}{4}(3-\sqrt{5})$, and as before, the corresponding quantization error is give by $$\begin{aligned}
& V_2(P)=\frac{1}{2} \int_0^{\frac{a_1+a_2}{2}} (x-a_1)^2 \, dx+\frac{1}{2} \int_{\frac{a_1+a_2}{2}}^1 (x-a_2)^2 \, dx+\frac{1}{2} \Big(\frac{1}{2}-a_2\Big)^2=0.0191242.
\end{aligned}$$ Thus, we see that there are two different optimal sets of two-means with same quantization error, which is the proposition.
For each even positive integer $n$, for the mixed distribution $P:=\frac 12 P_1+\frac 12 P_2$ given by Proposition \[prop1112\], there are two different optimal sets of $n$-means, and between the two different optimal sets of $n$-means, one is the reflection of the other with respect to the point $\frac 12$.
Quantization with $P_1$ a Cantor distribution and $P_2$ discrete {#sec5}
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In this section, we consider a mixed distribution $P:=\frac 12 P_1+\frac 12 P_2$, where $P_1$ is a Cantor distribution given by $P_1=\frac 12 P_1\circ S_1^{-1}+\frac 12 P\circ S_2^{-1}$, where $S_1(x)=\frac 13 x$ and $S_2(x)=\frac 13 x+\frac 13$ for all $x\in {\mathbb{R}}$, and $P_2$ is a discrete distribution on $D:={\{\frac 23, \frac 56, 1\}}$ with density function $h$ given by $h(x)=\frac 13$ for all $x\in D$. By a [word]{}, or a [string]{} of length $k$ over the alphabet ${\{1, 2\}}$, it is meant ${\sigma}:={\sigma}_1{\sigma}_2\cdots {\sigma}_k$, where ${\sigma}_j\in {\{1, 2\}}$ for $1\leq j\leq k$. A word of length zero is called the empty word and is denoted by ${\emptyset}$. Length of a word ${\sigma}$ is denoted by $|{\sigma}|$. The set of all words over the alphabet ${\{1, 2\}}$ including the empty word ${\emptyset}$ is denoted by ${\{1, 2\}}^\ast$. For two words ${\sigma}:={\sigma}_1{\sigma}_2\cdots{\sigma}_{|{\sigma}|}$ and ${\tau}:={\tau}_1{\tau}_2\cdots {\tau}_{|{\tau}|}$, by ${\sigma}{\tau}$, it is meant the concatenation of the words ${\sigma}$ and ${\tau}$. If ${\sigma}={\sigma}_1{\sigma}_2\cdots{\sigma}_k$, we write $S_{\sigma}:=S_{{\sigma}_1}\circ S_{{\sigma}_2}\circ\cdots\circ S_{{\sigma}_k}$, and $J_{\sigma}=S_{\sigma}(J)$, where $J=J_{\emptyset}:=[0, \frac 12]$. $S_1$ and $S_2$ generate the Cantor set $C:=\bigcap_{k\in \mathbb N} \bigcup_{{\sigma}\in \{1, 2\}^k} J_{\sigma}$. $C$ is the support of the probability distribution $P_1$. Notice that the support of the Mixed distribution $P$ is $C{\cup}D$. For any ${\sigma}\in {\{1, 3\}}^k$, $k\geq 1$, the intervals $J_{{\sigma}1}$ and $J_{{\sigma}2}$ into which $J_{\sigma}$ is split up at the $(k+1)$th level are called the [children]{} of $J_{\sigma}$.
The following lemma is well-known and appears in many places, for example, see [@GL2; @R1].
\[lemma51\] Let $f : \mathbb R \to \mathbb R$ be Borel measurable and $k\in \mathbb N$. Then $$\int f dP=\sum_{\sigma \in \{1, 2\}^k} \frac 1 {2^k} \int f \circ S_\sigma dP.$$
\[lemma52\] Let $X_1$ be a $P_1$-distributed random variable. Then, its expectation and the variance are respectively give by $E(X_1)=\frac 14$ and $V(X_1)=\frac {1}{32},$ and for any $x_0 \in \mathbb R$, $\int (x-x_0)^2 dP_1(x) =V (X_1) +(x_0-\frac 14)^2.$
Using Lemma \[lemma51\], we have $E(X_1)=\int x \, dP_1=\frac 1 2\int \frac 1 3 x \, dP_1 +\frac 12 \int (\frac 13 x +\frac 13 ) \, dP_1= \frac 1{6} \,E(X_1) +\frac 1{6} \, E(X_1) +\frac 1{6}$ implying $E(X_1)=\frac 1 4.$ Again, $$\begin{aligned}
&E(X_1^2)=\int x^2 \, dP_1=\frac 1 2 \int \frac {1} {9}\, x^2 \, dP_1 +\frac 12 \int \Big(\frac 1 3 \, x +\frac 13 \Big)^2 \, dP_1= \frac {1}{9}\, E(X_1^2) +\frac {1}{9} \, E(X_1) +\frac {1} {18},\end{aligned}$$ which yields $E(X_1^2)=\frac {3}{32}$, and hence $V(X_1)=E(X_1-E(X_1))^2=E(X_1^2)-\left(E(X_1)\right)^2 =\frac {3}{32}-(\frac 14 )^2=\frac {1}{32}$. Then, following the standard theory of probability, we have $ \int(x-x_0)^2 \, dP_1 =V(X_1)+(x_0-E(X_1))^2,$ and thus the lemma is yielded.
\[defi5111\] For $n\in {\mathbb{N}}$ with $n\geq 2$, let $\ell(n)$ be the unique natural number with $2^{\ell(n)} \leq n<2^{\ell(n)+1}$. For $I{\subset}{\{1, 2\}}^{\ell(n)}$ with card$(I)=n-2^{\ell(n)}$ let ${\beta}_n(I)$ be the set consisting of all midpoints $a({\sigma})$ of intervals $J_{\sigma}$ with ${\sigma}\in {\{1,2\}}^{\ell(n)} \setminus I$ and all midpoints $a({\sigma}1)$, $a({\sigma}2)$ of the children of $J_{\sigma}$ with ${\sigma}\in I$, i.e., $${\beta}_n(I)={\{a({\sigma}) : {\sigma}\in {\{1,2\}}^{\ell(n)} \setminus I\}} {\cup}{\{a({\sigma}1) : {\sigma}\in I\}} {\cup}{\{a({\sigma}2) : {\sigma}\in I\}}.$$
The following proposition follows due to [@GL2 Definition 3.5 and Proposition 3.7].
\[prop5111\] Let ${\beta}_n(I)$ be the set for $n\geq 2$ given by Definition \[defi5111\]. Then, ${\beta}_n(I)$ forms an optimal set of $n$-means for $P_1$, and the corresponding quantization error is given by $$V_n(P_1)=\int\min_{a\in {\beta}_n(I)}\|x-a\|^2 \, dP_1=\frac 1{18^{\ell(n)}}\cdot\frac 1{32} \Big(2^{\ell(n)+1}-n+\frac 19\left(n-2^{\ell(n)}\right)\Big).$$
\[lemma53\] Let $E(X)$ and $V:=V(X)$ represent the expected value and the variance of a random variable $X$ with distribution $P$. Then, $E(X)=\frac{13}{24}$ and $V=\frac{95}{864}=0.109954$.
In this proof we use the results from Lemma \[lemma51\]. We have $$\begin{aligned}
&E(X)=\int x dP=\frac 12 \int x dP_1+\frac 12 \int x dP_2=\frac 12 \int x dP_1+\frac 12 \sum_{x\in D} x h(x)=\frac{13}{24}, {\text}{ and } \\ &E(X^2)=\int x^2 dP=\frac 12 \int x^2 dP_1+\frac 12 \sum_{x\in D} x^2 h(x)=\frac{697}{1728},\end{aligned}$$ implying $V:=V(X)=E(X^2)-(E(X))^2 =\frac{697}{1728}-\left(\frac{13}{24}\right)^2=\frac{95}{864}$. Thus, the lemma is yielded.
Since $E\|X-a\|^2 =\int(x-a)^2 dP=V(X)+(a-E(X))^2=V+(a-\frac{13}{24})^2$, it follows that the optimal set of one-mean for the mixed distribution $P$ consists of the expected value $\frac {13}{24}$, and the corresponding quantization error is the variance $V$ of the random variable $X$. For any ${\sigma}\in {\{1, 2\}}^\ast$, by $a({\sigma})$, it is meant $a({\sigma}):=E(X_1 : X_1\in J_{\sigma})$, where $X_1$ is a $P_1$ distributed random variable, i.e., $a({\sigma})=S_{\sigma}(\frac 14)$. Notice that for any ${\sigma}\in {\{1, 2\}}^\ast$, and for any $x_0 \in \mathbb R$, we have $$\label{eq234} \int_{J_{\sigma}} (x-x_0)^2 \, dP_1 =p_{\sigma}\Big(s_{\sigma}^2 V +(S_{\sigma}(\frac 14)-x_0)^2\Big).$$
Optimal sets of $n$-means and $n$th quantization error
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In this subsection, we determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$ for the mixed distribution $P$. To determine the distortion error, we will frequently use the equation .
Let ${\alpha}$ be an optimal set of two-means. Then, ${\alpha}={\{\frac 14, \frac 56\}}$ with quantization error $V_2=\frac{43}{1728}=0.0248843.$
Consider the set of two-points ${\beta}$ given by ${\beta}:={\{\frac 14, \frac 56\}}$. Then, the distortion error is $$\int \min_{b \in {\beta}}\|x-b\|^2 dP= \frac 12 \int_{C}(x-\frac 14)^2 dP_1+\frac 16 \sum_{x\in D} (x-\frac 56)^2=\frac{43}{1728}=0.0248843.$$ Since $V_2$ is the quantization error for two-means, we have $V_2\leq 0.0248843$. Let ${\alpha}:={\{a_1, a_2\}}$ be an optimal set of two-means with $a_1<a_2$. Since the optimal points are the centroids of their own Voronoi regions, we have $0<a_1<a_1<a_2\leq 1$. If $a_1\geq \frac {29}{72}>S_{21}(\frac 12)$, then $$V_2\geq \frac 12\int_{J_1{\cup}J_{21}}(x-\frac {29}{72})^2 dP_1=\frac{1105}{41472}=0.0266445>V_2,$$ which leads to a contradiction. We now show that the Voronoi region of $a_1$ does not contain any point from $D$. Notice that the Voronoi region of $a_1$ can not contain all the points from $D$ as by Proposition \[prop0\], $P(M(a_2|{\alpha}))>0$. First, assume that the Voronoi region of $a_1$ contains both $\frac 23$ and $\frac 56$. Then, $$a_1=E(X : X\in C{\cup}{\{\frac 23, \frac 56\}})=\frac{\frac 12 \frac 1 4+\frac 16 \frac 23+\frac 16 \frac 56}{\frac 12+\frac 16+\frac 16}=\frac{9}{20} {\text}{ and } a_2=1,$$ which yield $\frac 12(a_1+a_2)=\frac {29}{40}<\frac 56$, which is a contradiction, as we assumed ${\{\frac 23, \frac 56\}}{\subset}M(a_1|{\alpha})$. Next, assume that the Voronoi region of $a_1$ contains only the point $\frac 23$ from $D$. Then, $$a_1=E(X : X\in C{\cup}{\{\frac 23\}})=\frac{\frac 12 \frac 1 4+\frac 16 \frac 23}{\frac 12+\frac 16}=\frac{17}{48} {\text}{ and } a_2=\frac 12(\frac 56+1)=\frac {11}{12},$$ which yield $\frac 12(a_1+a_2)=\frac{61}{96}<\frac 23$, which is a contradiction, as the Voronoi region of $a_1$ contains $\frac 23$. Thus, we can assume that the Voronoi region of $a_1$ does not contain any point from $D$ implying that $a_1\leq \frac 14$. Notice that if the Voronoi region of $a_1$ does not contain any point from $D$ and the Voronoi region of $a_2$ does not contain any point from $C$, then $a_1=\frac 14$ and $a_2=\frac 56$. If $a_2< \frac{21}{32}$, then $$V_2\geq \frac 16\Big((\frac 23-\frac{21}{32})^2+(\frac 56-\frac{21}{32})^2+(1-\frac{21}{32})^2\Big)=\frac{1379}{55296}=0.0249385>V_2,$$ which gives a contradiction, and so $\frac{21}{32}\leq a_2\leq \frac 56$. Suppose that $\frac{21}{32}\leq a_2\leq \frac{17}{24}$. Since $a_1\leq \frac 14$, $E(X_1 : X_1\in J_{1}{\cup}J_{21})=\frac{19}{108}<\frac 14 $, and $S_{21} (\frac 12)< \frac 12(\frac{19}{108}+\frac {21}{32})<\frac{1}{2} (\frac{1}{4}+\frac{21}{32})<S_{2212}(0)$, we have $$\begin{aligned}
V_2&\geq \frac 12\Big(\int_{J_1{\cup}J_{21}}(x-\frac{19}{108})^2 dP_1+\int_{J_{2212}{\cup}J_{222}}(x-\frac {21}{32})^2 dP_1\Big)+\frac{1}{6} \Big((\frac 56-\frac{17}{24})^2+(1-\frac{17}{24})^2\Big)\\
&=\frac{1938409}{71663616}=0.0270487>V_2,\end{aligned}$$ which leads to a contradiction. So, we can assume that $\frac{17}{24}\leq a_2\leq \frac 56$. Suppose that $\frac{17}{24}\leq a_2\leq \frac 34$. Notice that $S_{221}(\frac 12)<\frac 12(\frac 14+\frac{17}{24})<S_{222}(0)$, and $E(X_1 : X_1\in J_{1}{\cup}J_{21}{\cup}J_{2211})=\frac{829}{4212}<\frac 14$, and so, we have $$\begin{aligned}
&V_2\geq \frac 12 \Big(\int_{J_1{\cup}J_{21}{\cup}J_{2211}}(x-\frac{829}{4212})^2 dP_1+\int_{J_{2212}}(x-\frac 14)^2 dP_1+\int_{J_{222}}(x-\frac {17}{24})^2 dP_1\Big)\\
&+\frac{1}{6} \Big((\frac{2}{3}-\frac{17}{24})^2+(\frac{5}{6}-\frac{3}{4})^2+(1-\frac{3}{4})^2\Big)=\frac{2242573}{87340032}=0.0256763>V_2,\end{aligned}$$ which is a contradiction. So, we can assume that $\frac 34\leq a_2\leq \frac 56$. Then, notice that $\frac 12(a_1+a_2)<\frac 12$ implying $a_1<1-a_2\leq \frac 14$, but $\frac 12(\frac 14+\frac 34)=\frac 12$, and thus, $P$-almost surely the Voronoi region of $a_2$ does not contain any point from $C$ yielding $a_1=\frac 14$, $a_2=\frac 56$, and the corresponding quantization error is $V_2=\frac{43}{1728}=0.0248843.$
Let ${\alpha}$ be an optimal set of three-means. Then, ${\alpha}={\{\frac{1}{12},\frac{31}{60},\frac{11}{12}\}}$ with quantization error $V_3=\frac{89}{8640}=0.0103009$.
Let us consider the set of three-points ${\beta}:=\left\{\frac{1}{12},\frac{31}{60},\frac{11}{12}\right\}$. The distortion error due to the set ${\beta}$ is given by $$\begin{aligned}
& \int \min_{b \in {\beta}}\|x-b\|^2 dP= \frac 12 \Big(\int_{J_1}(x-\frac 1{12})^2 dx+ \int_{J_2}(x-\frac {31}{60})^2 dx\Big)\\
&+\frac 16 \Big((\frac 23-\frac{31}{60})^2+(\frac 56-\frac {11}{12})^2 +(1-\frac {11}{12})^2\Big)=\frac{89}{8640}=0.0103009.\end{aligned}$$ Since $V_3$ is the quantization error for three-means, we have $V_3\leq 0.0103009$. Let ${\alpha}:={\{a_1, a_2, a_3\}}$ be an optimal set of three-means with $a_1<a_2<a_3$. Since the optimal points are the centroids of their own Voronoi regions, we have $0<a_1<a_2<a_3\leq 1$. If $a_3< \frac 34$, then $$V_3\geq \frac{1}{6} \Big((\frac 56-\frac{3}{4})^2+(1-\frac{3}{4})^2\Big)=\frac{5}{432}=0.0115741>V_3,$$ which is a contradiction. So, we can assume that $\frac 34\leq a_3$. We now show that the Voronoi region of $a_3$ does not contain any point from $J_2$. Suppose that the Voronoi region of $a_3$ contains points from $J_2$. Consider the following two cases:
Case 1. $\frac 34\leq a_3\leq \frac 56$.
Then, $\frac 12(a_2+a_3)<\frac 12$ implying $a_2<1-a_3\leq 1-\frac 34=\frac 14$, and so the distortion error is $$V_3\geq \frac 12 \int_{J_2}(x-\frac 14)^2 dP_1+\frac 16(1-\frac 56)^2=\frac{43}{3456}=0.0124421>V_3,$$ which leads to a contradiction.
Case 2. $\frac 56\leq a_3<1$.
Then, $\frac 12(a_2+a_3)<\frac 12$ implying $a_2<1-a_3\leq 1-\frac 56=\frac 16$, and so the distortion error is $$V_3\geq \frac 12 \int_{J_2}(x-\frac 16)^2 dP_1=\frac{19}{1152}=0.0164931>V_3,$$ which is a contradiction.
Thus, by Case 1 and Case 2, we can assume that the Voronoi region of $a_3$ does not contain any point from $J_2$, and so $\frac 56\leq a_3$. If the Voronoi region of $a_2$ does not contain any point from $D$, then we will have $a_1=a(1)$, $a_2=a(2)$, $a_3=\frac 56$ yielding the distortion error as $$\frac 12 \Big(\int_{J_1}(x-a(1))^2dP_1+ \int_{J_2}(x-a(2))^2dP_1\Big) +\frac 16 \sum_{x\in D}(x-\frac 56)^2 =\frac{19}{1728}=0.0109954>V_3,$$ which is a contradiction. So, we can assume that the Voronoi region of $a_2$ contains points from $D$. If the Voronoi region of $a_2$ does not contain any point from $C$, then $$V_3\geq \frac 12 \int_{C}(x-\frac 14)^2 dP_1=\frac 1{ 64}=0.015625>V_3,$$ which leads to a contradiction. So, the Voronoi region of $a_2$ contains points from both $C$ and $D$. Suppose that the Voronoi region of $a_2$ contains both $\frac 23$ and $\frac 56$ from $D$. Then, $a_3=1$, and $\frac 56\leq \frac 12 (a_2+a_3)<1$ implying $\frac 23\leq a_2<1$. Moreover, as the Voronoi region of $a_2$ contains points from $C$, $\frac 12 (a_1+a_2)<\frac 12 $ implying $a_1<1-a_2\leq 1-\frac 23=\frac 13$. Notice that $E(X_1 : X_1\in J_1{\cup}J_{21}{\cup}J_{221})=\frac{163}{756}<\frac 13$, and so, we have $$V_3\geq \frac 12\Big(\int_{J_1{\cup}J_{21}{\cup}J_{221}}(x-\frac{163}{756})^2 dP_1+\int_{J_{222}}(x-\frac 13)^2 dP_1\Big)=\frac{17027}{1306368}=0.0130338>V_3,$$ which yields a contradiction. Thus, we can assume that the Voronoi region of $a_2$ contains only the point $\frac 23$ from $D$. Then, $a_3=\frac 12(\frac 56+1)=\frac {11}{12}$, and $\frac 23\leq \frac 12(a_2+a_3)<\frac 56$ implying $\frac 5{12}\leq a_2<\frac 34$. If the Voronoi region of $a_2$ contains points from $J_1$, then $\frac 12(a_1+a_2)<\frac 16$ implying $a_1<\frac 13-a_2\leq \frac 13-\frac 5{12}=-\frac 1{12}$, which is a contradiction as $0<a_1$. Thus, the Voronoi region of $a_2$ does not contain any point from $J_1$ implying the fact that $a_1\geq a(1)=\frac 1{12}$, and $E(X_1 : X_1 \in J_2{\cup}{\{\frac 23\}})=\frac {31}{60}\leq a_2<\frac 34$. Suppose that $\frac{333}{640}\leq a_2<\frac 34$. Then, $\frac 12(a_1+a_2)<\frac 12$ implying $a_1<1-a_2\leq 1-\frac{333}{640}=\frac{307}{640}<S_{2222}(0)$. Moreover, $E(X_1 : X_1 \in J_1{\cup}J_{21}{\cup}J_{221}{\cup}J_{2221})=\frac{227}{972}<\frac{307}{640}$, and so, writing $A=J_1{\cup}J_{21}{\cup}J_{221}{\cup}J_{2221}$, we have $$V_3\geq \frac 12\Big(\int_A(x-\frac{227}{972})^2 dP_1+\int_{J_{2222}}(x-\frac{307}{640})^2dP_1+\frac{1}{6} \Big((\frac 56-\frac{11}{12})^2+(1-\frac{11}{12})^2\Big)=\frac{4106379547}{257989017600},$$ i.e., $V_3\geq\frac{4106379547}{257989017600}=0.0159169>V_3$, which is a contradiction. Thus, we can assume that $\frac {31}{60}\leq a_2\leq \frac{333}{640}$. If $a_1\geq \frac 5{24}$, then, $$V_3\geq \frac 12\int_{J_1}(x-\frac 5{24})^2dP_1+\frac{1}{6} \Big((\frac 23-\frac{333}{640})^2+(\frac 56-\frac{11}{12})^2+(1-\frac{11}{12})^2\Big)=\frac{78587}{7372800}=0.010659>V_3,$$ which gives a contradiction. So, we can assume that $a_1<\frac 5{24}$. Suppose that $\frac 16< a_1<\frac 5{24}$. Since, $S_{211}(\frac 12)<\frac 12(\frac 5{24}+\frac {31}{60})<S_{212}(0)$, we have $$\begin{aligned}
&V_3\geq \frac 12\Big(\int_{J_1}(x-\frac 16)^2 dP_1+\int_{J_{211}}(x-\frac 5{24})^2 dP_1+\int_{J_{212}{\cup}J_{22}}(x-\frac {31}{60})^2 dP_1\Big)\\
&+\frac{1}{6} \Big((\frac 23-\frac{333}{640})^2+(\frac 56-\frac{11}{12})^2+(1-\frac{11}{12})^2\Big)=\frac{735859}{66355200}=0.0110897>V_3,\end{aligned}$$ which leads to a contradiction. So, we can assume that $a_1\leq \frac 16$. Suppose that $\frac 18\leq a_1\leq \frac 16$. Then, $S_{2111}(\frac 12)<\frac 12(\frac 16+\frac{31}{60})<S_{2112}(0)$. Using equation , it can be proved that for $\frac 18\leq a_1\leq \frac 16$, the error $\int_{J_1}(x-a_1)^2$ is minimum if $a_1=\frac 18$. Thus, $$\begin{aligned}
&V_3\geq \frac 12\Big(\int_{J_1}(x-\frac 18)^2 dP_1+\int_{J_{2111}}(x-\frac 16)^2 dP_1+\int_{J_{2112}{\cup}J_{212}{\cup}J_{22}}(x-\frac{31}{60})^2dP_1\Big)\\
&+\frac 16\Big((\frac 23-\frac{333}{640})^2+(\frac 56-\frac {11}{12})^2+(1-\frac {11}{12})^2\Big)=\frac{138551}{13271040}=0.0104401>V_3,\end{aligned}$$ which leads to a contradiction. So, we can assume that $a_1\leq \frac 18$. Then, notice that $\frac 12(\frac 18+\frac {31}{60})<\frac 13=S_{2}(0)$, i.e., the Voronoi region of $a_1$ does not contain any point from $J_2$, implying $a_1=a(1)=\frac 1{12}$, $a_2=\frac{31}{60}$, and $a_3=\frac {11}{12}$, and the corresponding quantization error is given by $V_3=\frac{89}{8640}=0.0103009$. Thus, the lemma is yielded.
Let ${\alpha}$ be an optimal set of four-means. Then, ${\alpha}={\{\frac 1{12}, \frac 5{12}, \frac 34, 1\}}$, or ${\alpha}={\{\frac 1{12}, \frac 5{12}, \frac 23, \frac {11}{12}\}}$, and the quantization error is $V_4=\frac{7}{1728}=0.00405093$.
Let us consider the set of four points ${\beta}:={\{\frac 1{12}, \frac 5{12}, \frac 34, 1\}}$. Then, the distortion error due to the set ${\beta}$ is $$\begin{aligned}
& \int \min_{b \in {\beta}}\|x-b\|^2 dP\\
&=\frac 12 \Big(\int_{J_1}(x-\frac 1{12})^2 dP_1+\int_{J_2}(x-\frac 5{12})^2 dP_1\Big)+\frac 16\Big((\frac 23-\frac 34)^2+(\frac 56-\frac 34)^2\Big)=\frac{7}{1728}.\end{aligned}$$ Since $V_4$ is the quantization error for four-means, we have $V_4\leq\frac{7}{1728}=0.00405093$. Let ${\alpha}:={\{a_1<a_2<a_3<a_4\}}$ be an optimal set of four-means. Since the optimal points are the centroids of their own Voronoi regions, we have $0<a_1<a_2<a_3<a_4\leq 1$. If $a_1\geq \frac{19}{96}$, then $$V_4\geq \frac 12 \int_{J_1}(x-\frac{19}{96})^2dP_1=\frac{17}{4096}=0.00415039>V_4,$$ which is a contradiction. So, we can assume that $a_1<\frac{19}{96}$. Suppose that the Voronoi region of $a_2$ contains points from $D$. Then, it contains only the point $\frac 23$ from $D$, as it must be $a_3=\frac 56$, and $a_4=1$. Moreover, $\frac 23 \leq \frac 12(a_2+a_3)<\frac 56$ implying $\frac 12\leq a_2<\frac 56$. Then, $S_{21121}(\frac 12)<\frac 12(\frac {19}{96} + \frac 12)<S_{21122} (0)$ yielding $$V_4 \geq \frac 12\Big(\int_{J_1}(x-a(1))^2 dP_1+\int_{J_{2111}{\cup}J_{21121}}(x-\frac {19}{96})^2 dP_1+ \int_{J_{21122}{\cup}J_{212}{\cup}J_{22}}(x-\frac 12)^2 dP_1\Big)=\frac{153563}{23887872},$$ i.e., $V_4\geq \frac{153563}{23887872}=0.00642849>V_4$, which gives a contradiction. Therefore, we can assume that the Voronoi region of $a_2$ does not contain any point from $D$. If the Voronoi region of $a_3$ does not contain any point from $D$, then $$V_4\geq \frac 16\Big((\frac 23-\frac 56)^2+(1-\frac 56)^2\Big)=\frac{1}{108}=0.00925926>V_4,$$ which leads to a contradiction. So, the Voronoi region of $a_3$ contains points from $D$. Suppose that the Voronoi region of $a_3$ contains points from $C$ as well. Then, two cases can arise.
Case 1. ${\{\frac 23\}}{\subset}M(a_3|{\alpha})$ and ${\{\frac 56, 1\}}{\subset}M(a_4|{\alpha})$.
Then, $a_4=\frac 12(\frac 56+1)=\frac {11}{12}$, and $\frac 23 \leq \frac 12(a_3+a_4)<\frac 56 $ implying $\frac {5}{12}\leq a_3<\frac 34$. Assume that $a_3<\frac{9}{16}$. Then, $$V_4\geq \frac 16\Big((\frac 23-\frac{9}{16})^2+(\frac 56-\frac {11}{12})^2+(1-\frac {11}{12})^2\Big)=\frac{19}{4608}=0.00412326>V_4,$$ which gives a contradiction. So, we can assume that $\frac 9{16}\leq a_3<\frac 34$. Suppose that $\frac 9{16}\leq a_3<\frac 7{12}$. Then, if $a_2\leq \frac 13$, as $S_{22111}(\frac 12)<\frac 12 (\frac 13 + \frac 9{16})<S_{22112}(0)$, we have $$\begin{aligned}
& V_4\geq \frac 12\Big(\int_{J_{21}{\cup}J_{22111}}(x-\frac{1}{3})^2+\int_{J_{22112}{\cup}J_{2212}{\cup}J_{222}}(x-\frac{9}{16})^2\Big)\\
& +\frac{1}{6} \Big((\frac{2}{3}-\frac{7}{12})^2+(\frac 56-\frac{11}{12})^2+(1-\frac{11}{12})^2\Big)=\frac{55735}{11943936}=0.00466638>V_4,\end{aligned}$$ which leads to a contradiction. If $\frac 13<a_2$, then the Voronoi region of $a_2$ does not contain any point from $J_1$, and $\frac 12(a_2+a_3)<\frac 12$ implies that $a_2<1-a_3\leq 1-\frac {9}{16}=\frac 7{16}<S_{22}(0)$, and so, we have $$\begin{aligned}
& V_4\geq \frac 12\Big( \int_{J_1}(x-a(1))^2 dP_1+ \int_{J_{22}}(x-\frac 7{16})^2 dP_1\Big)\\
& +\frac 16\Big((\frac 23-\frac 7{12})^2+(\frac 56-\frac {11}{12})^2+(1-\frac {11}{12})^2\Big)=\frac{251}{55296}=0.00453921>V_4,\end{aligned}$$ which is a contradiction. So, we can assume that $\frac {7}{12}\leq a_3<\frac 34$. Suppose that $\frac 7{12}\leq a_3\leq \frac{29}{48}$. Then, $\frac 12(a_2+a_3)<\frac 12$ implying $a_2<1-a_3\leq \frac 5{12}$. First, assume that $\frac 13\leq a_2<\frac 5{12}$. Then, the Voronoi region of $a_2$ does not contain any point from $J_1$. Moreover, using equation , we see that for $\frac 13\leq a_2<\frac 5{12}$, the error $\int_{J_2}(x-a_2)^2dP_1$ is minimum if $a_2=\frac 5{12}$, and so, $$\begin{aligned}
& V_4\geq \frac 12 \Big(\int_{J_1}(x-a(1))^2 dP_1+ \int_{J_2}(x-\frac 5{12})^2 dP_1\Big)
+\frac 16\Big((\frac{2}{3}-\frac{29}{48})^2+(\frac 56-\frac {11}{12})^2+(1-\frac {11}{12})^2\Big)\\
&=\frac{65}{13824}=0.00470197>V_4,\end{aligned}$$ which gives a contradiction. Next, assume that $a_2<\frac 13$. Then, $S_{221212}(0)<\frac{1}{2}(\frac{1}{3}+\frac{7}{12})<S_{221212}(\frac{1}{2})$ implying $$\begin{aligned}
& V_4\geq \frac 12\Big(\int_{J_{21}{\cup}J_{2211}{\cup}J_{221211}}(x- \frac 13)^2 dP_1+ \int_{J_{22122}{\cup}J_{222}}(x-\frac 7{12})^2 dP_1\Big)\\
&+\frac 16\Big((\frac{2}{3}-\frac{29}{48})^2+(\frac 56-\frac {11}{12})^2+(1-\frac {11}{12})^2\Big)=\frac{3197515}{725594112}=0.00440675>V_4,\end{aligned}$$ which leads to a contradiction. So, we can assume that $\frac {29}{48}\leq a_3<\frac 34$. Then, $\frac 12(a_2+a_3)<\frac 12$ implies $a_2<1-a_3\leq \frac {19}{48}$. First, assume that $\frac 13\leq a_2<\frac {19}{48}$. Then, the Voronoi region of $a_2$ does not contain any point from $J_1$. Moreover, using equation , we see that for $\frac 13\leq a_2<\frac {19}{48}$, the error $\int_{J_2}(x-a_2)^2dP_1$ is minimum if $a_2=\frac {19}{48}$, and so, $$\begin{aligned}
& V_4\geq \frac 12 \Big(\int_{J_1}(x-a(1))^2 dP_1+\frac 12 \int_{J_2}(x-\frac {19}{48})^2 dP_1\Big)
+\frac 16\Big((\frac 56-\frac {11}{12})^2+(1-\frac {11}{12})^2\Big)\\
&=\frac{115}{27648}=0.00415943>V_4,\end{aligned}$$ which gives a contradiction. Next, assume that $a_2\leq \frac 13$, then $S_{221}(\frac 12)<\frac 12(\frac 13 + \frac {29}{48})<S_{222}(0)$ implying $$\begin{aligned}
& V_4\geq \frac 12\Big(\int_{J_{21}{\cup}J_{221}}(x- \frac 13)^2 dP_1+ \int_{J_{222}}(x-\frac{29}{48})^2 dP_1\Big)+\frac 16\Big((\frac 56-\frac {11}{12})^2+(1-\frac {11}{12})^2\Big)=\frac{1385}{331776},\end{aligned}$$ i.e., $V_4\geq \frac{1385}{331776}=0.0041745>V_4$, which yields a contradiction.
Case 2. ${\{\frac 23, \frac 56\}}{\subset}M(a_3|{\alpha})$ and $a_4=1$.
Then, $\frac 56\leq \frac 12(a_3+1)$ implying $a_3\geq \frac 53-1=\frac 23$. Since by the assumption, the Voronoi region of $a_3$ contains points from $C$, we have $\frac 12(a_2+a_3)<\frac 12$ implying $a_2<1-a_3\leq 1-\frac 23=\frac 13$. Then, $$V_4\geq \frac 12\int_{J_2}(x-\frac 13)^2 dP_1+\frac 16\Big((\frac 23-\frac 34)^2+(\frac 56-\frac 34)^2\Big)=\frac{17}{3456}=0.00491898>V_4,$$ which is a contradiction.
Thus, by Case 1 and Case 2, we can assume that the Voronoi region of $a_3$ does not contain any point from $C$. Again, we have proved that the Voronoi region of $a_2$ does not contain any point from $D$. Hence, ($a_1=a(1), a_2=a(2)$, $a_3=\frac 34$ and $a_4=1$), or ($a_1=a(1), a_2=a(2)$, $a_3=\frac 23$ and $a_4=\frac {11}{12}$), and the corresponding quantization error is $V_4=\frac{7}{1728}=0.00405093$, which is the lemma.
\[lemma551\] Let ${\alpha}$ be an optimal set of five-means. Then, ${\alpha}={\alpha}_2(P_1){\cup}D$, and the corresponding quantization error is $V_5=\frac{1}{576}=\frac 12 V_2(P_1)$.
Consider the set of five points ${\beta}:={\{\frac{1}{12},\frac{5}{12}, \frac 23, \frac 56, 1\}}$. The distortion error due to the set ${\beta}$ is given by $$\begin{aligned}
& \int \min_{b \in {\beta}}\|x-b\|^2 dP=\frac 12 \int_{J_1}(x-\frac 1{12})^2 dP_1+\frac 12 \int_{J_2}(x-\frac 5{12})^2 dP_1=\frac{1}{576}=0.00173611.\end{aligned}$$ Since $V_5$ is the quantization error for five-means, we have $V_5\leq 0.00173611$. Let ${\alpha}:={\{a_1<a_2<a_3<a_4<a_5\}}$ be an optimal set of five-means. Since the optimal points are the centroids of their own Vornoi regions, we have $0<a_1<a_2<a_3<a_4<a_5\leq 1$. Suppose that $\frac 16\leq a_1$. Then, $$V_5\geq \frac 12\int_{J_1}(x-\frac 16)^2 dP_1=\frac{1}{384}=0.00260417>V_5,$$ which is a contradiction. So, we can assume that $a_1<\frac 16$. If the Voronoi region of $a_1$ contains points from $J_2$, we must have $\frac 12(a_1+a_2)>\frac13$ implying $a_2>\frac 23-a_1>\frac 23-\frac 1 6=\frac 12$, and then the distortion error is $$V_5\geq \frac 12 \Big(\int_{J_1}(x-\frac 1{12})^2 dP_1+ \int_{J_2}(x-\frac 12)^2 dP_1\Big)=\frac{1}{288}=0.00347222>V_5,$$ which leads to a contradiction. So, the Voronoi region of $a_1$ does not contain any point from $J_2$ implying $a_1\leq \frac 1{12}$. Notice that the Voronoi region of $a_2$ can not contain any point from $D$, as ${\alpha}$ is an optimal set of five-means and $D$ contains only three points. Thus, we have $a_2\leq a(2)=\frac 5{12}$. If the Voronoi region of $a_3$ does not contain any point from $D$, then $$V_5\geq \frac{1}{6} \Big(\Big(\frac 23-\frac{5}{6}\Big)^2+\Big(1-\frac{5}{6}\Big)^2\Big)=\frac{1}{108}=0.00925926>V_5,$$ which is a contradiction. So, we can assume that the Voronoi region of $a_3$ contains a point from $D$. In that case, we must have $a_4=\frac 56$ and $a_5=1$. If the Voronoi region of $a_3$ does not contain any point from $C$, then $a_3=\frac 23$. Suppose that the Voronoi region of $a_3$ contains points from $C$. Then, $\frac 23\leq \frac 12(a_3+a_4)$ implying $a_3\geq \frac 43-a_4=\frac 43-\frac 56=\frac 12$, i.e., $\frac 12\leq a_3\leq \frac 23$. The following three cases can arise:
Case A. $\frac 12\leq a_3\leq \frac {7}{12}$.
If $a_2<\frac{7}{24}$, then $S_{21}(\frac 12)<\frac 12(\frac 7{24}+\frac 12)<S_{22}(0)$ yielding $$V_5\geq \frac 12\Big(\int_{J_{21}}(x-\frac 7{24})^2dP_1+\int_{J_{22}}(x-\frac 12)^2 dP_1\Big)+\frac 16(\frac 23-\frac 7{12})^2=\frac{1}{512}=0.00195313>V_5,$$ which leads to a contradiction. Assume that $\frac 7{24}\leq a_2\leq \frac 13$. Then, $\frac 12(a_1+a_2)<\frac 16$ implying $a_1<\frac 13-a_2\leq \frac 13-\frac 7{24}=\frac 1{24}$, and so, $$V_5\geq \frac 12\Big(\int_{J_{12}}(x-\frac 1{24})^2 dP_1 +\int_{J_{21}}(x-\frac 13)^2 dP_1+\int_{J_{22}}(x-\frac 12)^2 dP_1\Big)+\frac 16(\frac 23-\frac 7{12})^2=\frac{37}{13824},$$ i.e., $V_5\geq \frac{37}{13824}=0.0026765>V_5$, which is a contradiction. Next, assume that $\frac 13<a_2$, and then, the Voronoi region of $a_2$ does not contain any point from $J_1$. Recall that $a_2\leq \frac 5{12}$. Thus, we have $$V_5\geq \frac 12\Big(\int_{J_1}(x-a(1))^2dP_1+\int_{J_{21}}(x-a(21))^2 dP_1+\int_{J_{22}}(x-\frac 12)^2 dP_1\Big)+\frac 16(\frac 23-\frac 7{12})^2=\frac{23}{10368},$$ i.e., $V_5\geq\frac{23}{10368}=0.00221836>V_5$, which gives a contradiction. Case B. $\frac {7}{12}\leq a_3\leq \frac{5}{8}$.
As Case A, we can show that if $a_2<\frac{7}{24}$ a contradiction arises. Assume that $\frac 7{24}\leq a_1\leq \frac 13$, then $S_{221212}(0)<\frac 12 (\frac 13 + \frac 7{12})<S_{221212}(\frac 12)$, and so, $$\begin{aligned}
& V_5\geq \frac 12\Big(\int_{J_{11}}(x-a(11))^2 dP_1+\int_{J_{21}{\cup}J_{2211}{\cup}J_{221211}}(x-\frac 13)^2dP_1+\int_{J_{22122}{\cup}J_{222}}(x-\frac 7{12})^2dP_1\Big)\\
&+\frac 16(\frac 23-\frac 58)^2=\frac{1290451}{725594112}=0.00177848>V_5,\end{aligned}$$ which give a contradiction. Next, assume that $\frac 13<a_2$, and then the Voronoi region of $a_2$ does not contain any point from $J_1$. Again, $\frac 12(a_2+a_3)<\frac 12$ implies that $a_2<1-a_3\leq 1-\frac 7{12}=\frac 5{12}$. Moreover, for $\frac 13<a_2\leq \frac 5{12}$, the error $\int_{J_2}(x-a_2)^2dP_1$ is minimum if $a_2=\frac 5{12}$. Thus, $$V_3\geq \frac 12\Big(\int_{J_1}(x-a(1))^2 dP_1+\int_{J_2}(x-\frac 5{12})^2dP_1\Big)+\frac 16(\frac 23-\frac 58)^2=\frac{7}{3456}=0.00202546>V_5,$$ which leads to a contradiction.
By Case A and Case B, we can assume that $\frac 58\leq a_3\leq \frac 23$. We now show that the Voronoi region of $a_3$ does not contain any point from $C$. On the contrary, assume that $\frac12(a_2+a_3)<\frac 12$ implying $a_2<1-a_3\leq 1-\frac 58=\frac 38$. Then, if $a_2<\frac 13$, as $S_{221}(\frac 12)<\frac{1}{2} (\frac{1}{3}+\frac{5}{8})<S_{222}(0)$, we have $$V_5\geq \frac 12\Big(\int_{J_{21}{\cup}J_{221}}(x-\frac 13)^2 dP_1+\int_{J_{222}}(x-\frac 58)^2dP_1\Big)=\frac{181}{82944}=0.0021822>V_5$$ which gives a contradiction. Assume that $\frac 13<a_2$. Then, the Voronoi region of $a_2$ does not contain any point from $J_1$. Using equation , we can show that for $\frac 13\leq a_2\leq \frac 38$, the error $\int_{J_2}(x-a_2)^2 dP_1$ is minimum if $a_2=\frac 38$, and so $$V_5\geq \frac 12 \Big(\int_{J_1}(x-a(1))^2dP_1+\int_{J_2}(x-\frac 38)^2dP_1\Big)=\frac{5}{2304}=0.00217014,$$ which is a contradiction. So, we can assume that the Voronoi region of $a_3$ does not contain any point from $C$ yielding $a_1=a(1)$, $a_2=a(2)$, $a_3=\frac 23$, $a_4=\frac 56$, and $a_5=1$, and so, by Proposition \[prop5111\], we have ${\alpha}={\alpha}_2(P_1){\cup}D$, and the corresponding quantization error is $V_5=\frac{1}{576}=\frac 12 V_2(P_1)$. Thus, the proof of the lemma is complete.
\[th611\] Let $n\in {\mathbb{N}}$ and $n\geq 5$, and let ${\alpha}_n$ be an optimal set of $n$-means for $P$ and ${\alpha}_n(P_1)$ be the optimal set of $n$-means for $P_1$. Then, $${\alpha}_n(P)={\alpha}_{n-3}(P_1){\cup}D, {\text}{ and } V_n(P)=\frac 12 V_{n-3}(P_1).$$
If $n=5$, by Lemma \[lemma551\], we see that the theorem is true for $n=5$. Proceeding in the similar way, as Lemma \[lemma551\], we can show that the theorem is true for $n=6$ and $n=7$. We now show that the theorem is true for all $n\geq 8$. Consider the set of eight points ${\beta}:={\{a(11), a(12), a(21), a(221), a(222), \frac 23, \frac 56, 1\}}$. The distortion error due to set ${\beta}$ is given by $$\int \min_{b \in {\beta}}\|x-b\|^2 dP=\frac 12 V_5(P_1)=\frac{7}{46656}=0.000150034.$$ Since $V_n$ is the $n$th quantization error for $n$-means for $n\geq 8$, we have $V_n\leq V_8\leq0.000150034$. Let ${\alpha}_n:={\{a_1<a_2<\cdots <a_n\}}$ be an optimal set of $n$-means for $n\geq 8$, where $0<a_1<\cdots<a_n\leq 1$. To prove the first part of the theorem, it is enough to show that $M(a_{n-2}|{\alpha}_n)$ does not contain any point from $C$, and $M(a_{n-3}|{\alpha}_n)$ does not contain any point from $D$. If $M(a_{n-2}|{\alpha}_n)$ does not contain any point from $D$, then $$V_n\geq \frac 16\Big((\frac 23-\frac 34)^2+(\frac 56-\frac 34)^2\Big)=\frac{1}{432}=0.00231481>V_n,$$ which leads to a contradiction. So, $M(a_{n-2}|{\alpha}_n)$ contains a point, in fact the point $\frac 23$, from $D$. If $M(a_{n-2}|{\alpha}_n)$ does not contain points from $C$, then $a_{n-2}=\frac 23$. Suppose that $M(a_{n-2}|{\alpha}_n)$ contains points from $C$. Then, $\frac 23\leq \frac 12(a_{n-2}+a_{n-1})$ implies $a_{n-2}\geq \frac 43-a_{n-1}=\frac 43-\frac 56=\frac 12$. The following three cases can arise:
Case 1. $\frac 12\leq a_{n-2}\leq \frac 7{12}$.
Then, $V_n\geq \frac 16 (\frac 23-\frac 7{12})^2=\frac{1}{864}=0.00115741>V_n,$ which is a contradiction.
Case 2. $\frac 7{12}\leq a_{n-2}$.
Then, $\frac 12(a_{n-3}+a_{n-2})<\frac 12$ implying $a_{n-3}<1-a_{n-2}\leq 1-\frac 7{12}=\frac 5{12}$, and so $$V_n\geq \frac 12 \int_{J_{22}} \Big(x-\frac{5}{12}\Big)^2 dP_1=\frac{1}{2304}=0.000434028>V_n,$$ which leads to a contradiction.
By Case 1 and Case 2, we can assume that $M(a_{n-2}|{\alpha}_n)$ does not contain any point from $C$. If $M(a_{n-3}|{\alpha})$ contains any point from $D$, say $\frac 23$, then we will have $$M(a_{n-2}|{\alpha}){\cup}M(a_{n-1}|{\alpha}){\cup}M(a_n|{\alpha})={\{\frac 56, 1\}},$$ which by Proposition \[prop0\] implies that either ($a_{n-2}=a_{n-1}=\frac 56$ and $a_n=1$), or ($a_{n-2}=\frac 56$ and $a_{n-1}=a_n=1$), which contradicts the fact that $0<a_1<\cdots<a_{n-2}<a_{n-1}<a_n\leq 1$. Thus, $M(a_{n-3}|{\alpha})$ does not contain any point from $D$. Hence, ${\alpha}_n(P)={\alpha}_{n-3}(P_1){\cup}D$, and so, $$V_n(P)=\frac 12 \int_{C}\min_{a\in{\alpha}_{n-3}(P_1)}(x-a)^2 dP_1+\frac 16 \sum_{x\in D}\min_{a\in D}(x-a)^2=\frac 12\int_{C}\min_{a\in{\alpha}_{n-3}(P_1)}(x-a)^2 dP_1$$ implying $V_n(P)=\frac 12 V_{n-3}(P_1)$. Thus, the proof of the theorem is complete.
Let ${\beta}$ be the Hausdorff dimension of the Cantor set generated by the similarity mappings $S_1$ and $S_2$. Then, ${\beta}=\frac{\log 2}{\log 3}$. By [@GL2 Theorem 6.6], it is known that the quantization dimension of $P_1$ exists and equals ${\beta}$, i.e., $D(P_1)={\beta}$. Since $$D(P)=\lim_{n\to \infty} \frac{2\log n}{-\log 2-\log V_{n-m} (P_1)}=\lim_{n\to \infty} \frac{2\log (n-m)}{-\log V_{n-m}(P_1)}=D(P_1)={\beta},$$ we can say that the quantization dimension of the mixed distribution exists and equals the quantization dimension of the Cantor distribution $P_1$, i.e., $D(P)=D(P_1)={\beta}$. Again, by [@GL2 Theorem 6.3], it is known that the quantization coefficient for $P_1$ does not exits. By Theorem \[th611\], we have $\liminf_{n\to \infty} n^{\frac {2}{{\beta}}} V_n(P)=\frac 12 \liminf_{n\to \infty} n^{\frac {2}{{\beta}}} V_{n-3}(P_1)=\frac 12 \liminf_{n\to \infty} (n-3)^{\frac {2}{{\beta}}} V_{n-3}(P_1)$, and similarly, $\limsup_{n\to \infty} n^{\frac {2}{{\beta}}} V_n(P)=\frac 12 \limsup_{n\to \infty} (n-3)^{\frac {2}{{\beta}}} V_{n-3}(P_1)$. Hence, the quantization coefficient for the mixed distribution $P$ does not exist.
Some remarks {#sec61}
============
Theorem \[th61\] and Theorem \[th611\] motivate us to give the following remarks.
Let $0<p<1$ be fixed. Let $P$ be the mixed distribution given by $P=pP_1+(1-p)P_2$ with the support of $P_1=C$ and the support of $P_2=D$, such that $P_1$ is continuous on $C$ and $P_2$ is discrete on $D$. Let ${\text}{card}(D)=m$ for some positive integer $m$. Further assume that $C$ and $D$ are [strongly separated]{} : there exists a ${\delta}>0$ such that $d(C, D):=\inf{\{d(x, y) : x\in C {\text}{ and } y\in D\}}>{\delta}$. Then, there exists a positive integer $N$ such that for all $n\geq N$, we have ${\alpha}_n(P)={\alpha}_{n-m}(P_1){\cup}D$, and so $$\begin{aligned}
V_n(P)&=\int \min_{a\in\in {\alpha}_n(P)}(x-a)^2dP=p \int \min_{a \in {\alpha}_{n-m}(P_1)}(x-a)^2 dP_1 +\sum_{x\in D} \min_{a\in D}(x-a)^2 h(x),\end{aligned}$$ implying $$\begin{aligned}
V_n(P)=p \int \min_{a \in {\alpha}_{n-m}(P_1)}(x-a)^2 dP_1=pV_{n-m}(P_1).\end{aligned}$$ Thus, we have $$D(P)=\lim_{n\to \infty} \frac{2\log n}{-\log p-\log V_{n-m} (P_1)}=\lim_{n\to \infty} \frac{2\log (n-m)}{-\log V_{n-m}(P_1)}=D(P_1).$$
\[rem5\] Let $D$ be a finite discrete subset of $C:=[0, 1]$. If $P_1$ is continuous on $C$, singular or nonsingular, and $P_2$ is discrete on $D$, then for the mixed distribution $P:=pP_1+(1-p)P_2$, where $0<p<1$, the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$ and for all $D$ are not known yet. Some special cases to be investigated are as follows: Take $p=\frac 12$, $P_1$ as a uniform distribution on $C$, and $D={\{\frac 23, \frac 56, 1\}}$. The optimal sets of $n$-means and the $n$th quantization errors for such a mixed distribution for all $n\geq 2$ are not known yet. Such a problem can also be investigated by taking $P_1$ as a Cantor distribution, and $P_2$ discrete on $D$, for example, one can take $P_1$ the classical Cantor distribution, as considered in [@GL2], and $D={\{\frac 23, \frac 56, 1\}}$. Notice that $p$, $P_1$ and $D$ can be chosen in many different ways.
Quantization where $P_1$ and $P_2$ are Cantor distributions {#sec7}
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Let $P_1$ be the Cantor distribution given by $P_1=\frac 12 P_1\circ S_1^{-1}+\frac 12 P_2\circ S_2^{-1}$, where $S_1(x)=\frac 13 x$ and $S_2(x)=\frac 13 x+\frac 29$ for all $x\in {\mathbb{R}}$. Let $P_2$ be the Cantor distribution given by $P_2=\frac 12 P_2\circ T_1^{-1}+\frac 12 P_2\circ T_2^{-1}$, where $T_1(x)=\frac 14 x+\frac 12$ and $T_2(x)=\frac 14 x+\frac 34 $ for all $x\in {\mathbb{R}}$. Let $C$ be the Cantor set generated by $S_1$ and $S_2$, and $D$ be the Cantor set generated by $T_1$ and $T_2$. Let $P$ be the mixed distribution generated by $P_1$ and $P_2$ such that $P=\frac 12 P_1+\frac 12 P_2$. Let ${\{1, 2\}}^\ast$ be the set of all words over the alphabet ${\{1, 2\}}$ including the empty word ${\emptyset}$ as defined in Section \[sec5\]. Write $J:=[0, \frac 13]$ and $K:=[\frac 23, 1]$. Then, we have $C=\bigcap_{k\in \mathbb N} \bigcup_{{\sigma}\in \{1, 2\}^k} J_{\sigma}$ and $D=\bigcap_{k\in \mathbb N} \bigcup_{{\sigma}\in \{1, 2\}^k} K_{\sigma}$, where for ${\sigma}\in {\{1, 2\}}^\ast$, $J_{\sigma}=S_{\sigma}([0, \frac 13])$ and $K_{\sigma}=T_{\sigma}([\frac 23, 1])$. Thus, $C$ is the support of $P_1$, and $D$ is the support of $P_2$ implying the fact that $C{\cup}D$ is the support of the mixed distribution $P$. As before, if nothing is mentioned within a parenthesis, by ${\alpha}_n$ and $V_n$, we mean an optimal set of $n$-means and the corresponding quantization error for the mixed distribution $P$.
The following two lemmas are similar to Lemma \[lemma52\].
Let $E(P_1)$ and $V(P_1)$ denote the expected value and the variance of a $P_1$-distributed random variable. Then, $E(P_1)=\frac 16$ and $V(P_1)=\frac 1{72}$. Moreover, for any $x_0 \in {\mathbb{R}}$, $\int (x-x_0)^2 \, dP_1 =V (P_1) +(x_0-\frac 16)^2.$
Let $E(P_2)$ and $V(P_2)$ denote the expected value and the variance of a $P_2$-distributed random variable. Then, $E(P_2)=\frac 56$ and $V(P_2)=\frac{1}{60}$. Moreover, for any $x_0 \in {\mathbb{R}}$, $\int (x-x_0)^2 dP_2(x) =V (P_2) +(x_0-\frac 56)^2.$
We now prove the following lemma.
\[lemma71\] Let $E(P)$ and $V(P)$ denote the expected value and the variance of a $P$-distributed random variable, where $P$ is the mixed distribution given by $P=\frac 12 P_1+\frac 12 P_2$. Then, $E(P)=\frac 12$ and $V(P)=\frac{91}{720}$. Moreover, for any $x_0 \in {\mathbb{R}}$, $\int (x-x_0)^2 dP(x) =V (P) +(x_0-\frac 12)^2.$
Let $X$ be a $P$-distributed random variable. Then, $$E(X)=\int x dP(x)=\frac 12 \int x \, dP_1+\frac 12 \int x dP_2(x)=\frac 12\Big (\frac 16+\frac 56\Big)=\frac 12, {\text}{ and }$$ $$E(X^2)=\int x^2 dP(x)=\frac 12 \int x^2 \, dP_1+\frac 1 2 \int x^2 dP_2(x)=\frac 12 \Big(\frac{1}{24}+\frac{32}{45}\Big)=\frac{271}{720},$$ and so, $V(P)=E(X^2)-(E(X))^2=\frac{91}{720}$. Then, by the standard theory of probability, for any $x_0 \in {\mathbb{R}}$, $\int (x-x_0)^2 dP(x) =V (P) +(x_0-\frac 12)^2.$ Thus, the proof of the lemma is complete.
From Lemma \[lemma71\], it follows that the optimal set of one-mean for the mixed distribution $P$ is $\frac 12$ and the corresponding quantization error is $V(P)=\frac{91}{720}$. Again, notice that for any $x_0\in {\mathbb{R}}$, we have $$\int (x-x_0)^2 dP(x) =\frac 12 \Big (V (P_1)+V(P_2)+(x_0-\frac 16)^2+(x_0-\frac 56)^2\Big).$$
\[defi51\] For $n\in {\mathbb{N}}$ with $n\geq 2$, let $\ell(n)$ be the unique natural number with $2^{\ell(n)} \leq n<2^{\ell(n)+1}$. For ${\sigma}\in {\{1, 2\}}^\ast$, let $a({\sigma})$ and $b({\sigma})$, respectively, denote the midpoints of the basic intervals $J_{\sigma}$ and $K_{\sigma}$. Let $I{\subset}{\{1, 2\}}^{\ell(n)}$ with card$(I)=n-2^{\ell(n)}$. Define ${\beta}_n(P_1, I)$ and ${\beta}_n(P_2, I)$ as follows: $$\begin{aligned}
{\beta}_n(P_1, I)&={\{a({\sigma}) : {\sigma}\in {\{1,2\}}^{\ell(n)} \setminus I\}} {\cup}{\{a({\sigma}1) : {\sigma}\in I\}} {\cup}{\{a({\sigma}2) : {\sigma}\in I\}}, {\text}{ and } \\
{\beta}_n(P_2, I)&={\{b({\sigma}) : {\sigma}\in {\{1,2\}}^{\ell(n)} \setminus I\}} {\cup}{\{b({\sigma}1) : {\sigma}\in I\}} {\cup}{\{b({\sigma}2) : {\sigma}\in I\}}.\end{aligned}$$
The following proposition follows due to [@GL2 Definition 3.5 and Proposition 3.7].
\[prop51\] Let ${\beta}_n(P_1, I)$ and ${\beta}_n(P_2, I)$ be the sets for $n\geq 2$ given by Definition \[defi51\]. Then, ${\beta}_n(P_1, I)$ and ${\beta}_n(P_2, I)$ form optimal sets of $n$-means for $P_1$ and $P_2$, respectively, and the corresponding quantization errors are given by $$\begin{aligned}
V_n(P_1)&=\int\min_{a\in {\beta}_n(P_1, I)}\|x-a\|^2 \, dP_1=\frac 1{18^{\ell(n)}}\cdot\frac 1{72} \Big(2^{\ell(n)+1}-n+\frac 19\left(n-2^{\ell(n)}\right)\Big), {\text}{ and } \\
V_n(P_2)&=\int\min_{a\in {\beta}_n(P_2, I)}\|x-a\|^2 \, dP_2=\frac 1{32^{\ell(n)}}\cdot\frac 1{60} \Big(2^{\ell(n)+1}-n+\frac 1{16}\left(n-2^{\ell(n)}\right)\Big).\end{aligned}$$
\[prop10.0\] For $n\geq 2$, let ${\alpha}_n$ be an optimal set of $n$-means for $P$. Then, ${\alpha}_n{\cap}[0, \frac 13)\neq {\emptyset}$ and ${\alpha}_n{\cap}(\frac 23, 1]\neq {\emptyset}$.
Consider the set of two-points ${\beta}_2:={\{\frac 16, \frac 56\}}$. Then, $$\int \min_{a\in {\beta}_2}\|x-a\|^2dP=\frac 12 \Big(\int (x-\frac 16)^2 dP_1+\int(x-\frac 56)^2 dP_2\Big)=\frac{11}{720}=0.0152778.$$ Since $V_n$ is the quantization error for $n$-means for $n\geq 2$, we have $V_n\leq V_2\leq 0.0152778$. Let ${\alpha}_n={\{a_1, a_2, a_3, \cdots, a_n\}}$ be an optimal set of $n$-means such that $a_1<a_2<a_3<\cdots<a_n$. Since the optimal points are centroids of their own Voronoi regions, we have $0<a_1<\cdots<a_n<1$. Assume that $\frac 13\leq a_1$. Then, $$V_n\geq \int_{[0, \frac 13]}(x-\frac 13)^2 dP=\frac 12 \int_{[0, \frac 13]}(x-\frac 13)^2 dP_1=\frac{1}{48}=0.0208333>V_n,$$ which is a contradiction, and so we can assume that $a_1<\frac 13$. Next, assume that $a_n\leq \frac 23$. Then, $$V_n\geq \int_{[\frac 23, 1]}(x-\frac 23)^2 dP=\frac 12 \int_{[\frac 23, 1]}(x-\frac 23)^2 dP_2=\frac{1}{45}=0.0222222>V_n,$$ which leads to a contradiction, and so we can assume that $\frac 23<a_n$. Thus, we see that ${\alpha}_n{\cap}[0, \frac 13)\neq {\emptyset}$ and ${\alpha}_n{\cap}(\frac 23, 1]\neq {\emptyset}$, which proves the proposition.
\[prop10.00\] For $n\geq 2$, let ${\alpha}_n$ be an optimal set of $n$-means for $P$. Then, ${\alpha}_n$ does not contain any point from the open interval $(\frac 13, \frac 23)$. Moreover, the Voronoi region of any point from ${\alpha}_n{\cap}J$ does not contain any point from $K$, and the Voronoi region of any point from ${\alpha}_n{\cap}K$ does not contain any point from $J$.
By Proposition \[prop10.0\], the statement of the proposition is true for $n=2$. Now, we prove it for $n=3$. Consider the set of three points ${\beta}_3:={\{\frac 16, \frac{17}{24},\frac{23}{24}\}}$. Then, $$\int \min_{a\in {\beta}_3}\|x-a\|^2dP=\frac 12\Big( \int_J (x-\frac 1{6})^2 dP_1+\int_{K_1} (x-\frac{17}{24})^2 dP_2+\int_{K_2}(x-\frac{23}{24})^2 dP_2\Big)=\frac{43}{5760}.$$ Since $V_3$ is the quantization error for three-means, we have $V_3\leq\frac{43}{5760}=0.00746528$. Let ${\alpha}_3:={\{a_1, a_2, a_3\}}$ be an optimal set of three-means such that $0<a_1<a_2<a_3<1$. By Proposition \[prop10.0\], we have $a_1<\frac 13$ and $\frac 23<a_3$. Suppose that $a_2 \in (\frac 13, \frac 23)$. The following two cases can aries:
Case 1. $\frac 13<a_2\leq \frac 12$.
Then, $\frac 12(a_2+a_3)>\frac 23$ implying $a_3>\frac 43-a_2\geq \frac 43-\frac 12=\frac 56$. Using an equation similar to , we can show that for $\frac 56< a_3<1$, the error $\frac 12 \int_{K}(x-a_3)^2dP_2$ is minimum if $P$-almost surely, $a_3=\frac 56$, and the minimum value is $\frac{1}{120}$. Thus, $$V_3\geq \frac 12 \int_K(x-\frac 56)^2 dP_2=\frac{1}{120}=0.00833333>V_3,$$ which is a contradiction.
Case 2. $\frac 12\leq a_2<\frac 23$.
Then, $\frac 12(a_1+a_2)<\frac 13$ implying $a_1<\frac 23-a_2\leq \frac 23-\frac 12=\frac 16$. Similar in Case 1, for $0<a_1<\frac 16$, the error $\frac 12 \int_{J}(x-a_1)^2dP_1$ is minimum if $P$-almost surely, $a_1=\frac 16$, and the minimum value is $\frac{1}{144}$. Thus, $$V_3\geq \frac 1{144}+\frac 12\int_{K_1}(x-\frac 23)^2dP_2=\frac{11}{1440}=0.00763889>V_3,$$ which leads to a contradiction.
Thus, by Case 1 and Case 2, we see that ${\alpha}_3$ does not contain any point from $(\frac 13, \frac 23)$. We now prove the proposition for all $n\geq 4$. Consider the set of four points ${\beta}_4:={\{\frac 1{18}, \frac 5{18}, \frac{17}{24}, \frac{23}{24}\}}$. The distortion error due to the set ${\beta}_4$ is given by $$\int\min_{a\in {\beta}_4}\|x-a\|^2 dP=\frac 12(V_2(P_1)+V_2(P_2))=\frac{67}{51840}=0.00129244.$$ Since $V_n$ is the quantization error for $n$-means for all $n\geq 4$, we have $V_n\leq V_4\leq 0.00129244$. Let $j=\max{\{i : a_i<\frac 23 {\text}{ for all } 1\leq i\leq n\}}$. Then, $a_j<\frac 23$. We need to show that $a_j<\frac 13$. For the sake of contradiction, assume that $a_j\in (\frac 13, \frac 23)$. Then, two cases can arise:
Case A. $\frac 13<a_j\leq \frac 12$.
Then, $\frac 12 (a_j+a_{j+1})>\frac 23$ implying $a_{j+1}>\frac 43-a_j\geq \frac 43-\frac 12=\frac 56$, and so, $$V_n\geq \frac 12 \int_{K_1}(x-\frac 56)^2 dP_2=\frac{1}{240}=0.00416667>V_n,$$ which leads to a contradiction.
Case B. $\frac 12\leq a_j\leq \frac 23$.
Then, $\frac 12 (a_{j-1}+a_j)<\frac 13$ implying $a_{j-1}<\frac 23-a_j\leq \frac 23-\frac 12=\frac 16$, and so, $$V_n\geq \frac 12 \int_{J_2}(x-\frac 16)^2 dP_1=\frac{1}{288}=0.00347222>V_n,$$ which gives a contradiction.
Thus, by Case A and Case B, we can assume that $a_j\leq \frac 13$. If the Voronoi region of any point from ${\alpha}_n{\cap}J$ contains points from $K$, then we must have $\frac 12(a_j+a_{j+1})>\frac 23$ implying $a_{j+1}>\frac 43-a_j\geq \frac 43-\frac 13=1$, which is a contradiction since $a_{j+1}<1$. Similarly, the Voronoi region of any point from ${\alpha}_n{\cap}K$ does not contain any point from $J$. Thus, the proof of the proposition is complete.
From Proposition \[prop10.0\] and Proposition \[prop10.00\], it follows that for $n\geq 2$, if an optimal set ${\alpha}_n$ contains $n_1$ elements from $J$ and $n_2$ elements from $K$, then $n=n_1+n_2$. In that case, we write ${\alpha}_n:={\alpha}_{(n_1, n_2)}$ and $V_n:=V_{(n_1, n_2)}$. Thus, ${\alpha}_n={\alpha}_{(n_1, n_2)}={\alpha}_{n_1}(P_1){\cup}{\alpha}_{n_2}(P_2)$, and $V_n=V_{(n_1, n_2)}=\frac 12(V_{n_1}(P_1)+V_{n_2}(P_2))$.
\[lemma10.1\] Let ${\alpha}$ be an optimal set of two-means for $P$. Then, ${\alpha}={\alpha}_{(1,1)}$, and the corresponding quantization error is $V_2=\frac{5}{432}=0.0115741$.
Let ${\alpha}={\{a_1, a_2\}}$ be an optimal set of two-means such that $0<a_1<a_2<1$. By Proposition \[prop10.0\], we have $a_1<\frac 13$ and $\frac 23<a_2$ yielding $a_1=\frac 16$, $a_2=\frac 56$, i.e., ${\alpha}={\alpha}_1(P_1){\cup}{\alpha}_1(P_2)$, and $V_2=\frac{11}{720}=0.0152778$. Thus, the proof of the lemma is complete.
\[lemma10.2\] Let ${\alpha}$ be an optimal set of three-means. Then, ${\alpha}={\alpha}_{(1, 2)}$, and the corresponding quantization error is $V_3=\frac{43}{5760}=0.00746528$.
Let ${\alpha}$ be an optimal set of three-means. By Proposition \[prop10.0\] and Proposition \[prop10.00\], we can assume that either ${\alpha}={\alpha}_2(P_1) {\cup}{\alpha}_1(P_2)$, or ${\alpha}={\alpha}_1(P_1) {\cup}{\alpha}_2(P_2)$. Since $$\int\min_{a\in {\alpha}_1(P_1) {\cup}{\alpha}_2(P_2)}(x-a)^2 dP<\int\min_{a\in {\alpha}_2(P_1) {\cup}{\alpha}_1(P_2)}(x-a)^2 dP,$$ the set ${\alpha}={\alpha}_1(P_1) {\cup}{\alpha}_2(P_2)$ forms an optimal set of three-means, and the corresponding quantization error is $$V_3=\int\min_{a\in {\alpha}_1(P_1) {\cup}{\alpha}_2(P_2)}(x-a)^2 dP=\frac12(V_1(P_1)+V_2(P_2))=\frac{43}{5760}=0.00746528,$$ which yields the lemma.
\[lemma10.3\] Let ${\alpha}$ be an optimal set of four-means. Then, ${\alpha}={\alpha}_{(2,2)}$, and the corresponding quantization error is $V_4=\frac{67}{51840}=0.00129244$.
Let ${\alpha}$ be an optimal set of four-means. By Proposition \[prop10.0\] and Proposition \[prop10.00\], we can assume that either ${\alpha}={\alpha}_3(P_1) {\cup}{\alpha}_1(P_2)$, ${\alpha}={\alpha}_2(P_1) {\cup}{\alpha}_2(P_2)$, or ${\alpha}={\alpha}_1(P_1) {\cup}{\alpha}_3(P_2)$ . Among all these possible choices, we see that ${\alpha}={\alpha}_2(P_1) {\cup}{\alpha}_2(P_2)$ gives the minimum distortion error, and hence, ${\alpha}={\alpha}_2(P_1) {\cup}{\alpha}_2(P_2)$ is an optimal set of four-means, and the corresponding quantization error is $V_4=\frac 12(V_2(P_1)+V_2(P_2))=\frac{67}{51840}=0.00129244$, which is the lemma.
\[rem1001\] Proceeding in the similar way, as Lemma \[lemma10.3\], it can be proved that the optimal sets of $n$-means for $n=5,6,7,$ etc. are, respectively, ${\alpha}_{(3,2)}, \, {\alpha}_{(2^2, 2)} \, {\alpha}_{(2^2, 3)}$, etc.
We now prove the following lemma.
\[lemma11.0\] Let ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ be an optimal set of $2^{6n-4}+2^{5n-4}$-means for $P$ for some positive integer $n$. For $1\leq i\leq 5$ and $1\leq j\leq 6$, let $\ell_i, k_j \in {\mathbb{N}}$ be such that $1\leq \ell_i\leq 2^{5n-4+(i-1)}$ and $1\leq k_j\leq 2^{6n-4+(j-1)}$. Then, $(i)$ ${\alpha}_{(2^{6n-4}, 2^{5n-4}+\ell_1)}$ is an optimal set of $2^{6n-4}+2^{5n-4}+\ell_1$-means; $(ii)$ ${\alpha}_{(2^{6n-4}+k_1, 2^{5n-3})}$ is an optimal set of $2^{6n-4}+2^{5n-3}+k_1$-means; $(iii)$ ${\alpha}_{(2^{6n-3}, 2^{5n-3}+\ell_2)}$ is an optimal set of $2^{6n-3}+2^{5n-3}+\ell_2$-means; $(iv)$ ${\alpha}_{(2^{6n-3}+k_2, 2^{5n-2})}$ is an optimal set of $2^{6n-3}+2^{5n-2}+k_2$-means; $(v)$ ${\alpha}_{(2^{6n-2}, 2^{5n-2}+\ell_3)}$ is an optimal set of $2^{6n-2}+2^{5n-2}+\ell_3$-means; $(vi)$ ${\alpha}_{(2^{6n-2}+k_3, 2^{5n-1})}$ is an optimal set of $2^{6n-2}+2^{5n-1}+k_3$-means; $(vii)$ ${\alpha}_{(2^{6n-1}, 2^{5n-1}+\ell_4)}$ is an optimal set of $2^{6n-1}+2^{5n-1}+\ell_4$-means; $(viii)$ ${\alpha}_{(2^{6n-1}+k_4, 2^{5n})}$ is an optimal set of $2^{6n-1}+2^{5n}+k_4$-means; $(ix)$ ${\alpha}_{(2^{6n}, 2^{5n}+\ell_5)}$ is an optimal set of $2^{6n}+2^{5n}+\ell_5$-means; $(x)$ ${\alpha}_{(2^{6n}+k_5, 2^{5n+1})}$ is an optimal set of $2^{6n}+2^{5n+1}+k_5$-means; and $(xi)$ ${\alpha}_{(2^{6n+1}+k_6, 2^{5n+1})}$ is an optimal set of $2^{6n+1}+2^{5n+1}+k_6$-means.
By Remark \[rem1001\], it is known that ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ is an optimal set of $2^{6n-4}+2^{5n-4}$-means for $n=1$. So, we can assume that ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ is an optimal set of $2^{6n-4}+2^{5n-4}$-means for $P$ for some positive integer $n$. Recall that ${\alpha}_{(n_1, n_2)}$ is an optimal set of $n_1+n_2$-means, and contains $n_1$ elements from $C$ and $n_2$ elements from $D$, and so, an optimal set of $n_1+n_2+1$-means must contain at least $n_1$ elements from $C$, and at least $n_2$ elements from $D$. For all $n\geq 1$, since $$\frac 12(V_{2^{6n-4}}(P_1)+V_{2^{5n-4}+1}(P_2))<\frac 12(V_{2^{6n-4}+1}(P_1)+V_{2^{5n-4}}(P_2)),$$ we can assume that ${\alpha}_{(2^{6n-4}, 2^{5n-4}+\ell_1)}$ is an optimal set of $2^{6n-4}+2^{5n-4}+\ell_1$-means for $\ell_1=1$. Having known ${\alpha}_{(2^{6n-4}, 2^{5n-4}+1)}$ as an optimal set of $2^{6n-4}+2^{5n-4}+1$-means, we see that $$\frac 12(V_{2^{6n-4}}(P_1)+V_{2^{5n-4}+2}(P_2))<\frac 12(V_{2^{6n-4}+1}(P_1)+V_{2^{5n-4}+1}(P_2)),$$ and so, ${\alpha}_{(2^{6n-4}, 2^{5n-4}+\ell_1)}$ is an optimal set of $2^{6n-4}+2^{5n-4}+\ell_1$-means for $\ell_1=2$. Proceeding in this way, inductively, we can show that ${\alpha}_{(2^{6n-4}, 2^{5n-4}+\ell_1)}$ is an optimal set of $2^{6n-4}+2^{5n-4}+\ell_1$-means for $1\leq \ell_1\leq 2^{5n-4}$. Thus, $(i)$ is true. Now, by $(i)$, we see that ${\alpha}_{(2^{6n-4}, 2^{5n-3})}$ is an optimal set of $2^{6n-4}+2^{5n-3}$-means. Then, proceeding in the same way as $(i)$ we can show that $(ii)$ is true. Similarly, we can prove the statements from $(iii)$ to $(xi)$. Thus, the lemma is yielded.
\[prop12.0\] The sets ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$, ${\alpha}_{(2^{6n-4}, 2^{5n-3})}$, ${\alpha}_{(2^{6n-3}, 2^{5n-3})}$, ${\alpha}_{(2^{6n-3}, 2^{5n-2})}$, ${\alpha}_{(2^{6n-2}, 2^{5n-2})}$, ${\alpha}_{(2^{6n-2}, 2^{5n-1})}$, ${\alpha}_{(2^{6n-1}, 2^{5n-1})}$, ${\alpha}_{(2^{6n-1}, 2^{5n})}$, ${\alpha}_{(2^{6n}, 2^{5n})}$, ${\alpha}_{(2^{6n}, 2^{5n+1})}$, ${\alpha}_{(2^{6n+1}, 2^{5n+1})}$, and ${\alpha}_{(2^{6n+2}, 2^{5n+1})}$ are optimal sets for all $n\in {\mathbb{N}}$.
By Remark \[rem1001\], it is known that ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ is an optimal set of $2^{6n-4}+2^{5n-4}$-means for $n=1$. Then, by Lemma \[lemma11.0\], it follows that ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ is an optimal set of $2^{6n-4}+2^{5n-4}$-means for $n=2$, and so, applying Lemma \[lemma11.0\] again, we can say that ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ is an optimal set of $2^{6n-4}+2^{5n-4}$-means for $n=3$. Thus, by induction, ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ are optimal sets of $2^{6n-4}+2^{5n-4}$-means for all $n\geq 2$. Hence, by Lemma \[lemma11.0\], the statement of the proposition is true.
Because of Lemma \[lemma71\], Lemma \[lemma10.1\], Lemma \[lemma10.2\], Lemma \[lemma10.3\], and Remark \[rem1001\], the optimal sets of $n$-means are known for all $1\leq n\leq 6$. To determine the optimal sets of $n$-means for any $n\geq 6$, let $\ell(n)$ be the least positive integer such that $2^{6\ell(n)-4}+2^{5\ell(n)-4}\leq n<2^{6(\ell(n)+1)-4}+ 2^{5(\ell(n)+1)-4}$. Then, using Lemma \[lemma11.0\], we can determine $n_1$ and $n_2$ with $n=n_1+n_2$ so that ${\alpha}_n={\alpha}_{(n_1, n_2)}$ gives an optimal set of $n$-means. Once $n_1$ and $n_2$ are known, the corresponding quantization error is obtained by using the formula $V_n=\frac 12(V_{n_1}(P_1)+V_{n_2}(P_2))$.
Asymptotics for the $n$th quantization error $V_n(P)$
-----------------------------------------------------
In this subsection, we investigate the quantization dimension and the quantization coefficients for the mixed distribution $P$. Let ${\beta}_1$ be the Hausdorff dimension of the Cantor set $C$ generated by $S_1$ and $S_2$, and ${\beta}_2$ be the Hausdorff dimension of the Cantor set $D$ generated by $T_1$ and $T_2$. Then, $
{\beta}_1=\frac{\log 2}{\log 3}$ and ${\beta}_2=\frac 12$. If $D(P_i)$ are the quantization dimensions of $P_i$ for $i=1, 2$, then it is known that $D(P_1)={\beta}_1$ and $D(P_2)={\beta}_2$ (see [@GL2]).
\[Th2\] Let $D(P)$ be the quantization dimension of the mixed distribution $P:=\frac 12 P_1+\frac 12 P_2$. Then, $D(P)=\max {\{D(P_1), D(P_2)\}}$.
Define $F(n):=2^{5n-4}(2^n+1)=2^{6n-4}+2^{5n-4}$, where $n\in {\mathbb{N}}$. Notice that $F(n)\geq F(1)=6$. For $n\in {\mathbb{N}}$, $n\geq 6$, let $\ell(n)$ be the least positive integer such that $F(\ell(n))\leq n<F(\ell(n)+1)$. Then, $V_{F(\ell(n)+1)}<V_n\leq V_{F(\ell(n))}$. Thus, we have $$\begin{aligned}
\frac {2\log\left(F(\ell(n))\right)}{-\log\left(V_{F(\ell(n)+1)}\right)}< \frac {2\log n}{-\log V_n}< \frac {2\log\left(F(\ell(n)+1)\right)}{-\log\left(V_{F(\ell(n))}\right)}\end{aligned}$$ Notice that $$V_{F(n)}=V_{(2^{6n-4}, 2^{5n-4})}=\frac 12\Big(V_{2^{6n-4}}(P_1)+V_{2^{5n-4}}(P_2)\Big)=\frac{1}{240} \left(2^{17-20 n}+5\cdot 3^{7-12 n}\right).$$ Then, $$\begin{aligned}
&\lim_{\ell(n)\to \infty} \frac {2\log\left(F(\ell(n))\right)}{-\log\left(V_{F(\ell(n)+1)}\right)}=\lim_{\ell(n)\to \infty} \frac {2\log (2^{6\ell(n)-4}+2^{5\ell(n)-4})} {\log 240-\log \left(2^{-3-20 \ell(n)}+5\cdot 3^{-5-12 \ell(n)}\right)}\Big(\frac{\infty}{\infty} {\text}{ form}\Big)\\
& =\lim_{\ell(n)\to \infty} \frac {\frac{2^{6\ell(n)-4} 12 \log 2+2^{5\ell(n)-4} 10\log 2}{2^{6\ell(n)-4}+2^{5\ell(n)-4}}}{\frac{2^{-3-20 \ell(n)} 20\log 2+ 5\cdot 3^{-5-12 \ell(n)} 12 \log 3}{2^{-3-20 \ell(n)}+5\cdot 3^{-5-12 \ell(n)}}}=\frac{\log 2}{\log 3},\end{aligned}$$ and similarly, $$\lim_{\ell(n)\to \infty} \frac {2\log\left(F(\ell(n)+1)\right)}{-\log\left(V_{F(\ell(n))}\right)}=\frac{\log 2}{\log 3}.$$ Since $\ell(n)\to \infty$ whenever $n\to \infty$, we have $\frac{\log 2}{\log 3}\leq \liminf_n \frac{2\log n}{-\log V_n}\leq \limsup_n \frac{2\log n}{-\log V_n}\leq\frac{\log 2}{\log 3}$ implying the fact that the quantization dimension of the mixed distribution $P$ exists and equals ${\beta}_1$, i.e., $D(P)=D(P_1)$. Since $D(P_1)={\beta}_1>{\beta}_2=D(P_2)$, we have $D(P)=\max{\{D(P_1), D(P_2)\}}$. Thus, the proof of the theorem is complete.
Theorem \[Th2\] verifies the following well-known proposition in [@L] for $d=1$ and $r=1$.
(see [@L Theorem 2.1]) Let $0<r<+\infty$, and let $P_1$ and $P_2$ be any two Borel probability measures on ${\mathbb{R}}^d$ such that $D_r(P_1)$ and $D_r(P_2)$ both exist. If $P=pP_1+(1-p)P_2$, where $0<p<1$, then $D_r(P)=\max{\{D_r(P_1), D_r(P_2)\}}$.
\[Th3\] Quantization coefficient for the mixed distribution $P:=\frac 12 P_1 +\frac 12 P_2$ does not exist.
By Theorem \[Th2\], the quantization dimension of the mixed distribution exists and equals ${\beta}_1$, where ${\beta}_1=\frac {\log 2}{\log 3}$. To prove the theorem it is enough to show that the sequence $\Big(n^{\frac 2{{\beta}_1}}V_n(P)\Big)_{n\geq 1}$ has at least two different accumulation points. By Lemma \[lemma11.0\] (i), it is known that ${\alpha}_{(2^{6n-4}, 2^{5n-4})}$ is an optimal set of $2^{6n-4}+2^{5n-4}$-means. Again, by Lemma \[lemma11.0\] (ii), it is known that ${\alpha}_{(2^{6n-4}+2^{6n-5}, 2^{5n-3})}$ is an optimal set of $2^{6n-4}+2^{6n-5}+2^{5n-3}$-means. Write $F(n):=2^{6n-4}+2^{5n-4}$, and $G(n):=2^{6n-4}+2^{6n-5}+2^{5n-3}$ for $n\in {\mathbb{N}}$. Recall that $$\begin{aligned}
V_{F(n)}&=V_{(2^{6n-4}, 2^{5n-4})}=\frac 12\Big(V_{2^{6n-4}}(P_1)+V_{2^{5n-4}}(P_2)\Big)=\frac{1}{240} \left(2^{17-20 n}+5\cdot 3^{7-12 n}\right), \\
V_{G(n)}&=V_{(2^{6n-4}+2^{6n-5}, 2^{5n-3})}=\frac 12\Big(V_{2^{6n-4}+2^{6n-5}}(P_1)+V_{2^{5n-3}}(P_2)\Big)=\frac{1}{15} 2^{9-20 n}+\frac{5}{16} 81^{1-3 n}.\end{aligned}$$ Notice that $(2^{6n})^{\frac 2{{\beta}_1}}=2^{\frac {12n\log 3}{\log 2}}=3^{12n}$ and $\lim_{n\to \infty } \left(\frac{3^{12}}{2^{20}}\right)^n=0$, and so, we have $$\begin{aligned}
&\lim_{n\to \infty} F(n)^{\frac 2 {{\beta}1}}V_{F(n)}(P) =\lim_{n\to \infty} (2^{6n-4}+2^{5n-4})^{\frac 2{{\beta}_1}}\frac{1}{240} \left(2^{17-20 n}+5\cdot 3^{7-12 n}\right)\\
&=\lim_{n\to\infty} 3^{12n}\Big(\frac 1 {2^4}+\frac 1{2^4}\cdot \frac 1{2^n}\Big)^{\frac 2{{\beta}_1}}\frac{1}{240} \left(2^{17-20 n}+5\cdot 3^{7-12 n}\right)=2^{-\frac{8}{{\beta}_1}}\frac {5\cdot 3^7}{240}=\frac{1}{144}=0.00694444,\end{aligned}$$ and $$\begin{aligned}
&\lim_{n\to \infty} G(n)^{\frac 2 {{\beta}1}}V_{G(n)}(P) =\lim_{n\to \infty} (2^{6n-4}+2^{6n-5}+2^{5n-3})^{\frac 2{{\beta}_1}}(\frac{1}{15}\cdot 2^{9-20 n}+\frac{5}{16}\cdot 81^{1-3 n})\\
&=\lim_{n\to \infty}3^{12n}\Big(\frac 1{2^4}+\frac 1{2^5}+\frac 1{2^3}\frac 1{2^n}\Big)^{\frac 2{{\beta}_1}}(\frac{1}{15}\cdot 2^{9-20 n}+\frac{5}{16}\cdot 81 \cdot 3^{-12n})=\frac{5}{16}\cdot 3^{\frac{2 \log (3)}{\log (2)}-6}=0.0139496.\end{aligned}$$ Since $(F(n)^{\frac 2 {{\beta}1}}V_{F(n)}(P))_{n\geq 1}$ and $( G(n)^{\frac 2 {{\beta}1}}V_{G(n)}(P))_{n\geq 2}$ are two subsequences of $(n^{\frac 2{{\beta}_1}}V_n(P))_{n\in {\mathbb{N}}}$ having two different accumulation points, we can say that the sequence $(n^{\frac 2{{\beta}_1}}V_n(P))_{n\in {\mathbb{N}}}$ does not converge, in other words, the ${\beta}_1$-dimensional quantization coefficient for $P$ does not exist. This completes the proof of the theorem.
We now conclude the paper with the following remark.
Optimal quantization for a general probability measure, singular or nonsingular, is still open, which yields the fact that the optimal quantization for a mixed distribution taking any two probability measures is not yet known.
[9999]{}
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[^1]:
|
---
author:
- 'K. Lind'
- 'J. Melendez'
- 'M. Asplund'
- 'R. Collet'
- 'Z. Magic'
date: 'Received 5 March 2013; accepted 25 May 2013'
title: 'The lithium isotopic ratio in very metal-poor stars'
---
Introduction
============
To explain the complex abundance patterns of Li isotopes in the Galaxy, chemical evolution modelling must consider several poorly constrained production and destruction mechanisms. Stars, in particular, can act as both Li sources and sinks depending sensitively on the timescales and efficiencies of mixing processes in the stellar interior [for reviews see @Pinsonneault97; @Dantona10]. The most well-constrained production source is that of standard Big Bang nucleosynthesis based on the high-precision calculations of the baryon density in the early universe, as inferred from WMAP observations [@Komatsu11]. Using extended and improved nuclear networks, the most recent predictions by @Coc12 imply $A(\rm^7Li)=\log(N(\rm^7Li)/N(\rm H))+12=2.72$ and $A\rm(^6Li)=-1.91$. These primordial abundance ratios have proven highly nontrivial to reconcile with the those measured in Population II stars born shortly after the Big Bang. The well-documented, universal shortage of $^7\rm Li$ in these stars and the claimed detection of $^6\rm Li$ in at least some of them, form the basis for two separate – and opposite – cosmological Li problems [see e.g. @Asplund06; @Fields11].
A key aspect is that many proposed solutions to one of the problems aggravate, rather than alleviate, the other. The best example, which has slowly been gaining traction over the last decade, is the suggestion that low-mass stars deplete $^7$Li from their atmospheres by gravitational settling over the course of their main-sequence life times [e.g @Richard05b]. To constrain the physical processes that would regulate the depletion, many authors have endeavored to carefully map $^7$Li abundances in low-mass stars as function of age and/or mass and/or metallicity [@Korn07; @Lind09b; @Melendez10]. While the final word has not yet been said on the subject, it is considered a likely explanation that stellar depletion accounts for part or all of the missing $^7\rm Li$.
For the lighter and more fragile $^6\rm Li$ isotope, the surface drainage would have a similar or even greater impact and hence it is of fundamental importance to constrain its abundance levels in metal-poor stellar atmospheres. The standard Big Bang scenario produces insignificant amounts of $^6\rm Li$, not detectable through spectral analysis of stars. Specifically, the primordial isotopic number density ratio, here denoted by $\rm^6Li/^7Li$, amounts to no more than $2.35\times10^{-5}$ [@Coc12]. Cosmic ray spallation is a more efficient production mechanism that may be constrained by the evolution of Be as traced by un-evolved metal-poor stars. Still, at $\rm[Fe/H]<-2$, the predicted level of $\rm^6Li/^7Li\la1\%$ [@Prantzos12] fall below the detection limit in stars even with the highest quality observational data. Hence, $^6\rm Li$-detections in any stars at this low metallicity call for either a revision of one of the known production channels, e.g. non-standard physics in the Big Bang [@Jedamzik09], or for a third production channel, such as cosmic-ray production by PopIII stars [@Rollinde06] or during the shocks of large-scale structure formation [@Suzuki02].\
While upper limits to the $^6\rm Li$ abundances of metal-poor halo stars were determined already by @Maurice84, the first significant detection was claimed by @Smith93 for the turn-off star HD84937 (see also @Hobbs94 and @Cayrel99). When the isotopic analyses were extended to greater samples [@Smith98; @Asplund06], additional stars were found to have significant ($>2\sigma$) detections of $^6\rm Li$. Indeed, evidence for an upper envelope emerged, with several very metal-poor stars, two of them having $\rm[Fe/H]<-2.0$, clustered around $A\rm(^6Li)\approx1$. Correcting for the depletion factors of $^6\rm Li$ expected from standard stellar evolution theory before and during the main sequence, the highest initial abundances measured fall in the range $A\rm(^6Li)=1.1-1.3$. For very metal-poor stars, this is approximately one order of magnitude higher than expected from Galactic cosmic-ray production [@Prantzos12]. Such standard Li depletion is however not sufficient to explain the cosmological $^7$Li problem, which requires non-canonical processes to act below the convective envelope (e.g. gravitational settling of Li, rotation-induced shears, gravity waves). The predicted initial abundances of $^6\rm Li$ may therefore be even higher, further aggravating the problem.
A revised analysis of the Li isotopic abundances of metal-poor stars is prompted by the considerable improvement in line formation modelling that has recently become possible. Specifically, realistic 3D radiation-hydrodynamical models of stellar surface convection can be conveniently tailored to the stellar parameters of individual stars and the expected strong departures from local thermodynamic equilibrium (LTE) in the metal-poor atmospheres can be properly resolved. That the simplifying assumptions of 1D, hydrostatic equilibrium and LTE influence the line profiles and may bias the determination of the isotopic ratio has been shown e.g. by @Cayrel07. The novel approach of our study is to account for non-LTE (NLTE) effects in 3D atmospheric models for both the Li line and Na and Ca lines used for calibration of rotational line broadening, a parameter that is highly degenerate with the isotopic ratio. This is a substantial improvement with respect to previous analyses that either subjected the Li line and the calibration lines to LTE modelling in 1D or 3D [e.g. @Smith98; @Asplund06], or applied a 3D, NLTE method only to the Li line without any additional constraint on external line broadening [@Cayrel07; @Steffen10a; @Steffen10b]. Note that the results presented in this paper supersede the preliminary findings presented by @Lind12c.
Observations
============
Because of the high computational demand of 3D, NLTE analysis, we limit our first study on this subject to four metal-poor stars, spanning a range in well-constrained stellar parameters. HD19445 is a main-sequence star, HD84937 and G64-12 are turn-off stars, and HD140283 a subgiant.
For the above stars we have obtained spectra of superb quality using the HIRES spectrograph [@Vogt94] on the 10m KeckI telescope. We used the decker E4 that has a length of $7\arcsec$ (allowing a good sky subtraction) and a slit width of $0.4\arcsec$ (achieving a resolving power of about 10$^5$). The wavelength coverage is about $400-800$nm using a mosaic of three CCDs optimized for the blue, green and red spectral regions. The exposure time ranges from 10 min for the brightest star (HD140283) to six hours for G64-12, as detailed in Table \[tab:hires\].
We extracted the spectral orders by hand using the IRAF package[^1] and also using the data reduction package MAKEE[^2], which was developed by T. A. Barlow specifically for reduction of Keck HIRES data and is optimized for spectral extraction of single point sources. The manual reduction with IRAF followed the standard procedures, correcting for bias, flat-fielding, cosmic rays and scattered light, then extraction of the orders (and sky subtraction) and finally wavelength calibration using the ThAr frames. We verified that both IRAF and MAKEE gave similar results and adopted the orders extracted with the MAKEE package, which was specifically designed for HIRES. For each individual frame the barycentric correction was applied. Further data reduction (continuum normalization and combining the different frames) was performed with IRAF.
Before combining the spectra we corrected each individual exposure to the rest frame, thus cancelling shifts due to guiding errors at the slit, changes to the optics of the spectrograph (e.g., due to temperature variations), or even small intrinsic radial velocity shifts. We used robust, outlier-resistant statistics to find the trimean radial velocity of each frame, achieving an internal line-to-line scatter of about 0.4 km/s, which is about 7 times better than the spectral resolution of the spectrograph ($\sim$ 3 km/s), and a standard error of the mean of $\sigma_{m}$ = 0.06 km/s. Without discarding any potential outlier the achieved internal precision is $\sigma_{m}$ = 0.09 km/s, still significantly better than the resolution. Thus, precise zero-point corrections were applied to different exposures. Due to variations within the individual exposures, the resolving power of the combined spectrum is slightly lower than that measured using the ThAr frames (see below). We take this effect into account, so that our adopted resolving power is slightly lower than the measured value.
As the stars are very metal-poor and relatively warm (see Table\[tab:param\]) there is a large number of continuum points to normalise the spectra. We tried different approaches to normalise them and the best results were obtained when selecting small spectral regions (about $\pm$3 Å around the relevant lines) for continuum normalization, so that the lowest possible spline was used. The absolute continuum placement was further fine-tuned in the line-profile analysis (see Sect. 3.3).
We measured the resolving power in the different spectral orders using the ThAr exposures, so that we can precisely estimate the instrumental broadening for a given line. The achieved resolving power in the wavelength region around the Li6707Å line ranges from $R=90\,000-100\,000$, already taking into account the small corrections mentioned above. We verified that the ThAr lines can be reproduced very well with a Gaussian, thus our corrected resolving power was used to convolve the synthetic spectra using a Gaussian profile. The combined spectra have a signal-to-noise (S/N) per pixel of approximately $\rm S/N=800-1100$ around the Li6707Å line. Thus, our data have both the required spectral resolution and S/N for the analysis of the Li isotopic ratio.
[lccccc]{} Star & V & Observing date (UT) & Exp. time \[s\] & Total Exp. time \[s\] & S/N\
HD19445 & 8.06 & 2005/10/22 & 3 x 500 & 1500 & 740\
HD84937 & 8.32 & 2006/01/19 & 2 x 400 + 3 x 500 + 4 x 600 & 4700 & 1030\
HD140283 & 7.21 & 2005/06/16 & 2 x 300 & 600 & 990\
G64-12 & 11.45 & 2005/06/16-17 & 18 x 1200 & 21600 & 820\
To investigate the Ca ionisation balance we complement the optical spectra with near infra-red FOCES spectra, covering the wavelength regions of the CaII triplet lines at $\sim$8600Å. The data were acquired in several observing runs between 1995 and 1999[^3]. FOCES is a fiber-fed spectrograph mounted on the 2.2m telescope on Calar Alto observatory [@Pfeiffer98]. The data have a $R\approx30\,000$ and $S/N\approx400$, with excellent continuum definition necessary to retain the shapes of the broad Ca lines.\
Analysis
========
Model atmospheres and stellar parameters
----------------------------------------
For the spectral line formation calculations presented here, we adopted a set of time-dependent, 3D, hydrodynamical model stellar atmospheres of the halo stars in our sample [@Magic13]. The models are based on radiation-hydrodynamical stellar-surface convection simulations generated with a custom version of the <span style="font-variant:small-caps;">Stagger</span>-code originally developed by Nordlund and Galsgaard[^4]. The simulations are part of a set of simulations of standard stars computed by @Collet11a and two of them were used by @Bergemann12 for a NLTE study of Fe lines with average 3D model atmospheres. For a detailed account of the procedure used to generate such simulations, we refer to @Magic13. We used the <span style="font-variant:small-caps;">Stagger</span>-code to solve the discretized, time-dependent, 3D, hydrodynamical equations for the conservation of mass, momentum, and energy in a representative volume located at the stellar surface. The simulation domains comprise the photosphere and the upper portion of the convection zone; more specifically, they cover typically about twelve pressure scale heights vertically and about ten granules at the surface at any one time. The corresponding spatial scale on the surface ranges from $6^2$Mm for the dwarf to $35^2$Mm for the subgiant. The simulation domains are discretized using a Cartesian mesh with a numerical resolution of $240^{3}$. The total stellar time covered by the simulations is 3–4h for the turn-off and subgiant and 0.5h for the dwarf. Open boundary conditions are assumed in the vertical direction and periodic ones horizontally.
During the 3D atmosphere modelling, energy exchange between gas and radiation is accounted for by solving the radiative transfer equation along the vertical and eight ($2\,\mu~\times~4\,\phi$) inclined rays cast through the simulation domain using a Feautrier-like method [@Feautrier64]. The non-grey character of radiative transfer in stellar atmospheres is approximated using an opacity-binning method [@Nordlund82; @Skartlien00] with twelve opacity bins in the implementation described by @Collet11a. We use an updated version of the realistic equation of state by @Mihalas88 and state-of-the-art continuous and line opacities for preparing the opacity bins [see @Magic13 for the complete references].
We tailored the 3D models of our stars to reflect their expected parameters. Three stars have accurate parallax measurements that can be used to constrain the surface gravity to within $\pm0.05\rm\,dex$. For G64-12 we have instead considered the spectroscopic gravity estimate from Fe ionisation balance by @Bergemann12 and from Strömgren photometry by @Nissen07. We have targetted effective temperatures based on a comparison between Balmer line analysis and direct application of the infra-red flux method (IRFM), as summarised in Table \[tab:param\]. Only our model for HD84937 is somewhat cooler than the trusted indicators imply, which we account for in the error determination.
[lrrrr]{} & G64-12 & HD140283 & HD84937 & HD19445\
$T_{\rm eff}$ H$\alpha^a$ & ... & 5753 & ... & 5980\
$T_{\rm eff}$ H$\beta^b$ & 6435 & 5849 & 6357 & ...\
$T_{\rm eff}$ IRFM$^c$ & 6464 & 5777 & 6408 & 6135\
$T_{\rm eff}$ model & 6428 & 5780 & 6238 & 6061\
$\log(g)$ ast.$^{a,b}$ & (4.26) & 3.72 & 4.07 & 4.42\
$\log(g)$ spec.$^d$ & 4.34 & 3.63 & 4.28 & ...\
$\log(g)$ model & 4.20 & 3.70 & 4.00 & 4.50\
$\rm[Fe/H]$ spec. $^{a,b}$ & -3.24 & -2.38 & -2.11 & -2.02\
$\rm[Fe/H]$ spec. $^{d}$ & -3.16 & -2.41 & -2.04 & ...\
$\rm[Fe/H]$ model & -3.00 & -2.50 & -2.00 & -2.00\
\
\
\
\
\
LTE and NLTE line formation
---------------------------
![Term diagrams of the Ca model atom used in the NLTE analysis. The dashed horisontal line marks the ground state of CaII. The four states of CaI with highest excitation potential are super-levels, corresponding to all or some of the individual fine-structure levels for $n=5-8$. The transitions used for detailed spectral analysis are marked with thick lines, and their approximate wavelengths given in Å.[]{data-label="fig:termdiag"}](CaII_term){width="\textwidth"}
![Term diagrams of the Ca model atom used in the NLTE analysis. The dashed horisontal line marks the ground state of CaII. The four states of CaI with highest excitation potential are super-levels, corresponding to all or some of the individual fine-structure levels for $n=5-8$. The transitions used for detailed spectral analysis are marked with thick lines, and their approximate wavelengths given in Å.[]{data-label="fig:termdiag"}](CaI_term){width="\textwidth"}
To compute synthetic spectra of Li and Ca lines, we utilized two different 3D spectrum synthesis codes; <span style="font-variant:small-caps;">Scate</span> [@Hayek11] and <span style="font-variant:small-caps;">Multi3d</span> [@Botnen97; @Leenaarts10]. The technique was described in @Lind12c, but we reiterate the main points here.
The spatial and temporal average flux profile in LTE was computed with <span style="font-variant:small-caps;">Scate</span> solving the radiative transfer equation for $6\mu\times4\phi$-angles for 20 snapshots in time. For each line, the profile was resolved by 100 wavelength points equidistant by 0.5km/s in velocity space and the logarithmic oscillator strength was varied by $\pm0.3\,$dex to allow for accurate interpolation and $\chi^2$-analysis with respect to observed data. Four different isotopic ratios were assumed, spanning the range: $\rm^6Li/^7Li=0.0-0.06$. Negative ratios and intermediate ratios were obtained by inter/extrapolation of the line profiles. Negative values are unphysical, but required for accurate $\chi^2$-minimisation. The adopted line data for the main transitions are summarized in Table \[tab:abund\] and the fine- and hyper-fine subcomponents of the Li lines are detailed in @Lind09a.
NLTE calculations are significantly more time consuming than LTE calculations and a few simplifications are necessary to make these tractable. We chose to calculate the mean NLTE/LTE profile ratio from four snapshots using <span style="font-variant:small-caps;">Multi3d</span> and thereafter multiply the ratio with LTE profiles computed with <span style="font-variant:small-caps;">Scate</span> for the same abundances. The calculations were performed for a range of abundances for each individual line, except that $\rm^6Li/^7Li=0.0$ for the Li line, i.e. we assume that the NLTE/LTE profile ratio is independent of the isotopic ratio. In analogy with LTE calculations, $6\mu\times4\phi$-angles were used. We have confirmed that the agreement between the two codes is satisfactory in LTE.
The three model atoms include, respectively, 40 levels of CaI, five levels of CaII, and the CaIII ground state, 20 levels of LiI and the LiII ground state, and 20 levels of NaI and the NaII ground state. A more detailed description of the atomic data can be found in @Lind09a [@Lind11b; @Lind12c]. We note that the statistical equilibrium calculations for Li and Na are well constrained thanks to rigorous quantum mechanical calculations of radiative and collisional transition probabilities [see e.g. @Barklem10]. However, as is often the case in NLTE modelling [@Asplund05], the main uncertainty in the calculations for Ca arises from the unknown efficiency of inelastic collisions between Ca and HI atoms. We have here assumed a scaling factor $S_{\rm H}=0.1$ to the rates computed with the traditional @Drawin68 recipe, as suggested by @Mashonkina07, and investigate the consistency between NLTE abundances derived from different lines (see Sect. 4.2). Schematic term diagrams are illustrated for Ca in Fig.\[fig:termdiag\], with the important transitions marked with thick lines.
The NLTE effects on Li, Na, and Ca lines are qualitatively very similar in these atmospheres. The steep temperature stratification of the bright granules of up-flowing gas efficiently boosts the over-ionisation, by giving rise to a strongly super-thermal radiation field in the ultra-violet wavelength regions. This is particularly true at low metallicity where 3D hydrodynamical models have much cooler outer atmospheric layers than classical 1D hydrostatic models [see e.g. @Asplund99; @Asplund03]. Vice versa, the intergranular lanes of downflowing gas suffer from the opposite effect, which is over-recombination of the neutral atoms compared to LTE. As shown in Fig.\[fig:xy\], the Li line can be both strengthened and weakened by up to as much as a factor of three in NLTE with respect to LTE. Because the bright regions have the dominant influence over the spatially averaged line profile, the net effect is a considerable weakening of the lines, which must be compensated for by a significant abundance increase. Resonance lines are particularly sensitive to the effects of over-ionisation, because of their higher temperature sensitivity.
While the total line strength of the Li resonance line determines the $^7$Li-abundance, it is the shape of the line profile that determines the isotopic ratio due to the shift between $^6\rm Li$ and $^7\rm Li$ isotopic components. It is therefore critical to resolve the strongly differential NLTE effects on the granules and inter-granular lanes, because they have a preferential influence over the blue- and red-shifted part of the line profile, respectively. As seen in Fig.\[fig:xy\], the line is stronger in the granules in LTE, compared to the lanes, while the opposite is true in NLTE [@Asplund03]. At a given line strength, the net result is a stronger depression in the red wing of the line, as shown in Fig.\[fig:linediff\]. The contrast in line strength over the surface has also decreased significantly in NLTE, because the line formation is decoupled from the strong temperature and density inhomogeneities that depict the LTE line formation.
Because of the added absorption in the red wing, the spatially averaged line profiles are broadened in NLTE. Consequently, when constraining the unknown external line broadening that is caused by the rotation of the star, the resulting $v_{\rm rot}\sin{i}$-values decrease by $0.7-1.9\rm\,km/s$ (see Table\[tab:isotope\]). Less additional broadening is needed to reproduce the same observed line profile. Except for the difference in width that can be compensated for by external broadening, there is a small, but not completely negligible difference in the shape of the line profile when lifting the LTE assumption. This influences the isotopic ratios, as demonstrated in the following section.
Similar behavior is seen for all elements, but compared to Li the balance is shifted to a greater degree of over-recombination for Na and over-ionisation for Ca. The net result is smaller positive abundance corrections for Na and larger for Ca than for Li. In Fig.\[fig:hist\] we demonstrate with histograms how the line strength changes in NLTE for the different elements in a selected snapshot of the G64-12 model. Evidently, the important line-strengthening that occurs in the lanes due to over-recombination has an almost identical influence on Li and Na. For Ca, this effect is not as strong.
$\chi^2$-minimisation
---------------------
The Li isotopic abundances were determined by $\chi^2$-minimisation between observed and synthetic spectra in a region extending $\pm1.7$ Å from the line centre. A $\chi^2$-matrix was formed, with the number of dimensions equal to the number of free parameters, either four or five, as described below. The analysis was performed using the standard definition of $\chi^2$:
$$\chi^2 = \sum\limits_{N_{\rm data}}\frac{(F_{\rm obs}-F_{\rm mod})^2}{\sigma^2} \\$$
Here, $F$ denotes the normalised fluxes of model and observation, $\sigma=(S/N)^{-1}$ is the estimated error of the observed flux per pixel and $N_{\rm data}$ the number of pixels used to fit the full line profile. In order to make a fair comparison between solutions obtained using different number of free parameters, we define also the reduced $\chi^2$-statistic:
$$\chi_{\rm red}^2 = \frac{1}{N_{\rm d.o.f.}}\chi^2 \\$$
$N_{\rm d.o.f.} = N_{\rm data}-N_{\rm free}-1$ is the number of degrees of freedom. The number of free parameters is either $N_{\rm free}=5$ or $N_{\rm free}=4$, depending on the method. In the first case, the Li line itself was used to determine all five free parameters, i.e. continuum normalisation ($C_{\rm norm}$), relative radial velocity shift at central wavelength ($\Delta v_{\rm rad}$), $v_{\rm rot}\sin i$, $A\rm(^7Li)$, and $\rm^6Li/^7Li$, and in the latter case only four of these parameters, trusting $v_{\rm rot}\sin i$ determined using Na and Ca calibration lines. A $\chi_{\rm red}^2$-value close to unity is indicative of a good match between model and observations and a realistic estimate of the observational uncertainties.
The isotopic ratio and its associated error ($\sigma_{\rm obs}$ in Table\[tab:isotope\]) were found by analysing several 2D-surfaces in the $\chi^2$-space. These were formed by fixing the isotopic ratio and one free parameter at the time to a grid of values, while optimising all other free parameters. Examples of the resulting $\chi^2$-contours, which reveal the extent of the parameter degeneracies, are shown in Figs. \[fig:chisquare1\] and \[fig:chisquare2\]. The best-fit values and corresponding errors of the fitting parameters were found by parabolic fits to the $\chi^2$ data along the lines of maximum degeneracy. As seen in the figures, the full parameter space defined by the $1\sigma$-contours ($\chi^2=\chi^2_{\rm min}+1$) is adequately covered by the error bars. When $v_{\rm rot}\sin i$ was determined from calibration lines, the associated uncertainty was propagated and added to $\sigma_{\rm obs}$.
Finally, we have estimated the errors inherent in the synthetic profiles due to uncertainties in stellar parameters and to the limited sampling of the NLTE/LTE profile ratio. We refer to this error as $\sigma_{\rm model}$. We assumed that only the error in effective temperature plays a significant role for the line formation of neutral species, and hence in the determination of the isotopic ratio, and adopted 100K as a reasonable error bar (see Table \[tab:param\] ). As was pointed out by e.g. @Asplund06, errors of this magnitude do not contribute significantly to the error in the $\rm^6Li/^7Li$-ratio in a 1D analysis, but in 3D we must account for a non-negligible effect on the shape of the line profile. We have estimated this contribution by repeating the 3D, LTE analysis for a 140K hotter model of HD84937, and adopt 0.009 and 0.004 as reasonable estimates of $\sigma_{\rm model}$ when $N_{\rm free}=5$ and $N_{\rm free}=4$, respectively. The 3D, NLTE analysis was not repeated, however, since the convective motions of the higher temperature model are slightly too high, leading to negative $v_{\rm rot}\sin{i}$. This indicates that the star is indeed a very slow rotator, as expected for an old halo star, and that our 3D model is realistic in terms of predicting the intrinsic line broadening from convective motions. Instead, we adopted the same errors as for 3D, LTE and added to that an estimate of the influence of the limited number of snapshots used to sample the NLTE/LTE profile ratio. For all stars, and both methods, this error on $\rm^6Li/^7Li$ is equal to 0.002.
The isotopic ratios and associated errors due to random and systematic uncertainties should thus be read from Table \[tab:isotope\] as $\rm^6Li/^7Li\pm\sigma_{\rm obs}\pm\sigma_{\rm model}$.
![Example synthetic profiles in LTE (*dashed*) and NLTE (*solid*) of the Li resonance line, computed with the same abundance ($A\rm(^7Li)=2.0$, $\rm^6Li/^7Li=0.0$, $v_{\rm rot}\sin{i}=0.0$) for the model of HD140283. Also shown is an LTE line profile interpolated to meet the same equivalent width as the NLTE line (*dotted*). More absorption appears in the red wing relative to the blue in NLTE due to strongly differential effects in granules and inter-granular lanes. []{data-label="fig:linediff"}](linediff)
![Histograms of the ratio between the spatially resolved equivalent widths at disk center intensity found in NLTE and LTE for a snapshot of G64-12. []{data-label="fig:hist"}](Hist)
![$\chi^2$-surfaces (1$\sigma$, 2$\sigma$, and 3$\sigma$) obtained for G64-12 by varying two line parameters at the time, as indicated on the respective axes; *red dashed lines:* $N_{\rm free}=5$ and *black solid lines:* $N_{\rm free}=4$. The other free parameters have been optimised at each grid point. The best-fit value and associated error bars are indicated for $N_{\rm free}=5$ in the top panel and $N_{\rm free}=4$ in the two lower panels *(bullets)*.[]{data-label="fig:chisquare1"}](g64-12_LTE_chi2D1)
![$\chi^2$-surfaces (1$\sigma$, 2$\sigma$, and 3$\sigma$) obtained for G64-12 by varying two line parameters at the time, as indicated on the respective axes; *red dashed lines:* $N_{\rm free}=5$ and *black solid lines:* $N_{\rm free}=4$. The other free parameters have been optimised at each grid point. The best-fit value and associated error bars are indicated for $N_{\rm free}=5$ in the top panel and $N_{\rm free}=4$ in the two lower panels *(bullets)*.[]{data-label="fig:chisquare1"}](g64-12_NLTE_chi2D1)
![$\chi^2$-surfaces (1$\sigma$, 2$\sigma$, and 3$\sigma$) obtained for G64-12 by varying two line parameters at the time, as indicated on the respective axes; *red dashed lines:* $N_{\rm free}=5$ and *black solid lines:* $N_{\rm free}=4$. The other free parameters have been optimised at each grid point. The best-fit value and associated error bars are indicated for $N_{\rm free}=5$ in the top panel and $N_{\rm free}=4$ in the two lower panels *(bullets)*.[]{data-label="fig:chisquare1"}](g64-12_LTE_chi2Dfix1)
![$\chi^2$-surfaces (1$\sigma$, 2$\sigma$, and 3$\sigma$) obtained for G64-12 by varying two line parameters at the time, as indicated on the respective axes; *red dashed lines:* $N_{\rm free}=5$ and *black solid lines:* $N_{\rm free}=4$. The other free parameters have been optimised at each grid point. The best-fit value and associated error bars are indicated for $N_{\rm free}=5$ in the top panel and $N_{\rm free}=4$ in the two lower panels *(bullets)*.[]{data-label="fig:chisquare1"}](g64-12_NLTE_chi2Dfix1)
![$\chi^2$-surfaces (1$\sigma$, 2$\sigma$, and 3$\sigma$) obtained for G64-12 by varying two line parameters at the time, as indicated on the respective axes; *red dashed lines:* $N_{\rm free}=5$ and *black solid lines:* $N_{\rm free}=4$. The other free parameters have been optimised at each grid point. The best-fit value and associated error bars are indicated for $N_{\rm free}=5$ in the top panel and $N_{\rm free}=4$ in the two lower panels *(bullets)*.[]{data-label="fig:chisquare1"}](g64-12_LTE_chi2Dfix3)
![$\chi^2$-surfaces (1$\sigma$, 2$\sigma$, and 3$\sigma$) obtained for G64-12 by varying two line parameters at the time, as indicated on the respective axes; *red dashed lines:* $N_{\rm free}=5$ and *black solid lines:* $N_{\rm free}=4$. The other free parameters have been optimised at each grid point. The best-fit value and associated error bars are indicated for $N_{\rm free}=5$ in the top panel and $N_{\rm free}=4$ in the two lower panels *(bullets)*.[]{data-label="fig:chisquare1"}](g64-12_NLTE_chi2Dfix3)
![Same as Fig.\[fig:chisquare1\] but for HD84937.[]{data-label="fig:chisquare2"}](hd84937_LTE_chi2D1)
![Same as Fig.\[fig:chisquare1\] but for HD84937.[]{data-label="fig:chisquare2"}](hd84937_NLTE_chi2D1)
![Same as Fig.\[fig:chisquare1\] but for HD84937.[]{data-label="fig:chisquare2"}](hd84937_LTE_chi2Dfix1)
![Same as Fig.\[fig:chisquare1\] but for HD84937.[]{data-label="fig:chisquare2"}](hd84937_NLTE_chi2Dfix1)
![Same as Fig.\[fig:chisquare1\] but for HD84937.[]{data-label="fig:chisquare2"}](hd84937_LTE_chi2Dfix3)
![Same as Fig.\[fig:chisquare1\] but for HD84937.[]{data-label="fig:chisquare2"}](hd84937_NLTE_chi2Dfix3)
Calibration lines
-----------------
Following the same reasoning as detailed in previous studies [@Smith98; @Asplund06], simultaneous modelling of lines of other neutral species is important in order to constrain any intrinsic line broadening and thereby reduce the error bar on the isotopic ratio. The broadening due to non-thermal gas motions in the atmosphere is here assumed to be realistically captured by accounting for the velocity field of the hydrodynamical simulations. Hence any remaining line broadening is ascribed to the unknown rotation of the stars and therefore the projected rotational velocity, $v_{\rm rot}\sin{i}$, is treated as a free parameter, assuming an Unsöld profile [e.g. @Unsoeld38]. While the ability of hydrodynamical models to reproduce observed line profiles has been successfully demonstrated [e.g. @Asplund00], there is a possibility that the true $v_{\rm rot}\sin{i}$ becomes over- or underestimated with this assumption due to inadequate modelling of the convective motions, e.g. due to erroneous stellar parameters. Thereby, $v_{\rm rot}\sin{i}$ may here also compensate to some extent for missing non-rotational broadening. However, we have verified that the Li isotopic ratios obtained using a rotational or Gaussian velocity profile are the same. The uncertainty introduced in the shape of the convolving function used for the unknown line broadening is thus negligible.
The best-fit values for the free parameters (continuum normalisation, radial velocity, and Ca abundance) were determined from the Ca lines using the same $\chi^2$-minisation technique as described for Li. However, in the determination of the optimal $v_{\rm rot}\sin{i}$, we considered all the calibration lines simultaneously, i.e. we enforced a single value, rather than computed a mean value. Not all Ca lines that are detectable in the spectra were used as calibration lines, but only those with $W_\lambda<50$mÅ . The reason is that we do not wish the formation layers of the calibration lines to deviate too far from that of the Li line, having a total line strength of $23-46$mÅ. In addition, stronger lines become increasingly less sensitive to rotational broadening and are therefore less optimal as calibrators (see Appendix). To get a robust constraint on $v_{\rm rot}\sin{i}$ in the most metal-poor star in our sample, G64-12, we complemented the two detectable Ca lines with the NaD lines at $\sim$5890Å, which are of suitable strength ($20-30$mÅ); at higher \[Fe/H\] the lines are too strong to be useful. The lines used for calibration in each star are marked in Table \[tab:abund\].
In principle, also the central position of the Li line can be calibrated using other neutral lines, removing yet another free parameter from the determination of the isotopic ratio. However, when investigating this possibility we found that the central radial velocity shifts determined from different lines are not compatible with one another within the errors. For each individual line, the accuracy in $v_{\rm rad}$ implied by the $\chi^2$-analysis is typically $<0.05\,$km/s, while the line-to-line dispersion is closer to $0.1$km/s. This may be caused by errors in the adopted laboratory wavelengths or by imperfect modelling of the distribution of convective velocities at different heights of formation. We therefore chose to determine $v_{\rm rad}$ from the Li line itself. We note that the relative central wavelengths of $^6$Li and $^7$Li line components have been measured to a very high accuracy [@Sansonetti95].
The full set of Ca lines can be used to assess the realism of our stellar parameters and 3D modelling technique. By inclusion of the resonance line 4226Å, the neutral Ca lines span up to 2.7eV in lower level excitation potential, which is sufficient to inspect abundance trends with this parameter (see Fig.\[fig:calcium\]). In addition to this verification of the excitation balance, the near-infrared CaII triplet lines provide us with information about the ionisation balance.
[lrrrrllllllll]{} Species & $\lambda$ & $\epsilon_{\rm exc}$ & $\log(gf)$ & vdW$^a$ & & & &\
& & & & & LTE & NLTE & LTE & NLTE & LTE & NLTE & LTE & NLTE\
LiI & 6103.6$^b$ & 1.848 & 0.583 &837.274& 2.33(5)& 2.35(6) & 2.10(2) & 2.14(2) & 2.23(4) & 2.26(4) & 2.17(4) & 2.23(5)\
LiI & 6707.8$^b$ & 0.000 & 0.174 &346.236& 2.12& 2.23 & 1.82 & 2.12 & 1.94 & 2.15 & 1.99 & 2.25\
NaI & 5889.951& 0.000 & 0.117 & 407.273& 2.73$^*$ & 2.78$^*$ & ... & ... & ... & ... & ... & ...\
NaI & 5895.924& 0.000 & -0.184 & 407.273& 2.75$^*$ & 2.79$^*$ & ... & ... & ... & ... & ... & ...\
CaI & 4226.728& 0.000 & 0.243 & 372.238& 3.00 & 3.81 & 3.57 & 4.19 & 4.16 & 4.57 & 4.36 & 4.78\
CaI & 4544.879& 1.899 & 0.318 & 949.274& 3.46$^*$ & 3.63$^*$ & 3.84$^*$ & 4.18$^*$ & 4.32$^*$ & 4.62$^*$ & 4.39 & 4.70\
CaI & 5588.749& 2.526 & 0.358 & 400.282& ... & ... & 3.95$^*$ & 4.17$^*$ & 4.40$^*$ & 4.57$^*$ & 4.47$^*$ & 4.71$^*$\
CaI & 6102.723& 1.879 & -0.793& 876.233& ... & ... & 3.93$^*$ & 4.15$^*$ & 4.39$^*$ & 4.54$^*$ & 4.49$^*$& 4.70$^*$\
CaI & 6122.217& 1.886 & -0.316& 876.234& ... & ... & 3.93$^*$ & 4.18$^*$ & 4.37$^*$ & 4.56$^*$ & 4.47$^*$ & 4.73$^*$\
CaI & 6162.173& 1.899 & -0.090& 876.234& 3.54$^*$ & 3.68$^*$ & 3.89$^*$ & 4.19$^*$ & 4.35$^*$ & 4.60$^*$ & 4.44 & 4.76\
CaI & 6439.075& 2.526 & 0.390 & 366.242& ... & ... & 3.95$^*$ & 4.16$^*$ & 4.40$^*$ & 4.58$^*$ & 4.46$^*$ & 4.73$^*$\
CaI & 6717.681& 2.709 &-0.524& 992.255& ... & ... & ... & ... & 4.46$^*$ & 4.55$^*$ & 4.59$^*$ & 4.72$^*$\
CaII & 8498.023 & 1.692&-1.496& 291.275& 3.93 & 3.55& 4.25 & 4.05 & 4.79 & 4.71 & 4.76 & 4.70\
CaII & 8542.091 & 1.700&-0.514& 291.275 & 3.95 & 3.74 & 4.12 & 4.02 & 4.71 & 4.69 & 4.69& 4.67\
CaII & 8662.141 & 1.692&-0.770& 291.275& ... & ... & 4.16 & 4.04 & 4.77 & 4.73 & 4.69 & 4.67\
.\
\
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](g64-12_LTE_Ca)
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](g64-12_NLTE_Ca)
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](hd140283_LTE_Ca)
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](hd140283_NLTE_Ca)
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](hd84937_LTE_Ca)
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](hd84937_NLTE_Ca)
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](hd19445_LTE_Ca)
![Ca abundances plotted against lower level excitation potential, for CaI (*bullets*) and CaII lines (*squares*).[]{data-label="fig:calcium"}](hd19445_NLTE_Ca)
Results
=======
Ca abundances
-------------
Our NLTE modelling predicts very large positive abundance increase for neutral lines in all four stars compared to the LTE case. To investigate whether these effects are realistic we inspect the excitation and ionisation balance of Ca. The results are shown in Fig.\[fig:calcium\]. Evidently, 3D LTE modelling predicts large positive slopes of line abundance with excitation potential, which would require an increase of $\sim$400K to be removed. In NLTE the slopes are considerably flatter, but a minor over-estimation of the temperatures for G64-12 is implied. The ionisation balance is well established for this star and HD19945 in NLTE, while HD84937 and HD140283 show a small discrepancy ($\sim0.1\rm\,dex$) in opposite directions at these stellar parameters. The positive corrections to neutral lines and negative corrections to singly ionised lines lead to significant improvement of the ionisation balance in NLTE with respect to LTE.
To perfectly establish agreement between different Ca indicators would require fine-tuning of stellar parameters and the NLTE effects, which are controlled by the unknown efficiency of hydrogen-atom inelastic collisions. This is beyond the scope of this work and will be addressed in a separate publication. We note that the patterns described for the Ca excitation balance are qualitatively similar to those described in the <3D>, NLTE analysis of Fe lines in three of these stars in our sample by @Bergemann12.
Li isotopic abundances
----------------------
[lrrrrrrrr]{} & & & &\
& LTE & NLTE & LTE & NLTE & LTE & NLTE & LTE & NLTE\
$N_{\rm free}=5$\
$v_{\rm rot}\sin{i}$ \[km/s\] & 3.729 & 1.940 & 3.739 & 2.116 & 3.569 & 2.469 & 3.385 & 1.517\
$\sigma$ \[km/s\] & 0.440 & 0.900 & 0.121 & 0.229& 0.329 & 0.482 & 0.220 & 0.560\
$\rm^6Li/^7Li$ & 0.003 & 0.005 & 0.022 & 0.000 & 0.040 & 0.011 & 0.028 & 0.003\
$\sigma_{\rm obs}$ & 0.025 & 0.023 & 0.007 & 0.004 &0.016 & 0.010 & 0.013 & 0.006\
$\sigma_{\rm model}$ & 0.009 & 0.011 & 0.009 & 0.011 & 0.009 & 0.011 & 0.009 & 0.011\
$\chi^2_{\rm red}$ & 1.065 & 1.030 & 1.176 & 1.074 & 1.186 & 1.201 & 0.910 & 0.862\
$N_{\rm free}=4$\
$v_{\rm rot}\sin{i}$ \[km/s\] & 3.189 & 2.447 & 2.830 & 1.467 & 2.976 & 2.153 & 2.706 & 1.436\
$\sigma$ \[km/s\] & 0.067 & 0.093 & 0.025 & 0.048& 0.023 & 0.032 & 0.027 & 0.049\
$\rm^6Li/^7Li$ & 0.026 & -0.002 & 0.052 & 0.007 & 0.060 & 0.017 & 0.051 & 0.005\
$\sigma_{\rm obs}$ & 0.016 & 0.013 & 0.006 & 0.003 & 0.011 & 0.007 & 0.010 & 0.005\
$\sigma_{\rm model}$ & 0.004 & 0.006 & 0.004 & 0.006 & 0.004 & 0.006 & 0.004 & 0.006\
$\chi^2_{\rm red}$ & 1.069 & 1.022 & 1.632 & 1.148 & 1.204 & 1.198 & 0.988 & 0.855\
The Li isotopic abundances were derived from the 6707Å feature using either the line itself or other neutral lines to constrain the rotational line broadening, under the assumptions of LTE and NLTE. The four sets of results for $\rm^6Li/^7Li$, $v_{\rm rot}\sin{i}$, and associated $\chi_{\rm red}^2$ are summarised in Table \[tab:isotope\]. The best-fit 3D, NLTE Li synthetic profiles obtained when using the calibration lines are shown in Fig.\[fig:liprof\].
The shapes and sizes of the probability contours displayed in Fig.\[fig:chisquare1\] and \[fig:chisquare2\] determine the sizes of the error bars. It is evident that increasing Li isotopic ratio is partly degenerate with decreasing projected rotational velocity, increasing central radial-velocity shift, and decreasing $^7$Li-abundance. The degeneracy with continuum normalisation is very small in comparison and therefore not shown. A striking result is how much the definition of the minimum benefits from the use of calibration lines, which can be appreciated by comparing the two sets of contours in each panel in Fig.\[fig:chisquare1\] and \[fig:chisquare2\]. The random error component, $\sigma_{\rm obs}$, decreases by up to a factor of two. Indeed, for G64-12 in particular, the use of calibration lines is necessary to obtain a total random error in the isotopic ratio below 2.5 percentage points. This is partly a reflection of the weakness of the Li line in this hot star. However, the minimum $\chi_{\rm red}^2$-value is either decreased or increased when using calibration lines (see Table\[tab:isotope\]), indicating either a better or worse goodness-of-fit. The choice of which method is better is thus not straightforward, but if the modelling is sound the two should agree on the location of the minimum value within the error bars. Inspecting Table \[tab:isotope\], the agreement is substantially better in NLTE compared to LTE, with HD140283 being the most prominent example. For this star, the $v_{\rm rot}\sin{i}$-value derived from the Li line in LTE is almost $1$km/s greater than that from the calibration lines, which propagates into a significant difference in the Li isotopic ratio.
The great importance of accounting for NLTE effects in the line formation is evident from the Li isotopic ratios displayed in Table \[tab:isotope\]. The best-fit ratios decrease by up to five percentage points in NLTE compared to LTE, and it is clear that the latter assumption can lead to spurious detections. In NLTE, no star has a $2\sigma$-detection of $^6\rm Li$ and the reported values should be regarded as upper limits. For HD84937, the non-detection found using only the line itself ($0.011\pm0.010\pm0.011$) challenges the significance of the $1\sigma$-detection when calibration lines are used ($0.017\pm0.007\pm0.006$). Extending the analysis to include more calibration lines would be desirable to further decrease the error bar.
Finally, we emphasize that the small difference in profile shape in NLTE with respect to LTE acts to increase the error in $v_{\rm rot}\sin{i}$, but decrease the error in $\rm^6Li/^7Li$. This is not surprising since with the more realistic NLTE modelling, the convective broadening becomes relatively more influential, which forces a larger relative error on the rotational velocity.
Discussion
==========
It has only recently become numerically feasible to perform the complex 3D, NLTE calculations outlined in this study. Therefore, until a few years ago, all quantitative analyses of the lithium isotopic ratio were performed under the 1D, LTE assumptions. @Asplund06 were the first to investigate the influence of 3D, LTE modelling on the abundances and found minor differences with respect to 1D, with no large systematic influence. Due to the stronger departures from LTE expected in 3D, the authors put more trust in the 1D, LTE results. Including also the follow-up paper by @Asplund08, a total of 27 stars were analysed in the two studies, 11 of which were found to have a significant $2\sigma$-detection of $^6$Li. The studied sample contains all four of our stars and significant detections were then reported for HD84937 ($0.051\pm0.015$) and G64-12 ($0.059\pm0.021$), while the other two stars were found compatible with a vanishing $^6$Li-signature. In addition to the 1D, LTE modelling technique, their analysis differs from our new study in the choice of calibration lines; the studies by @Asplund06 and @Asplund08 used a larger set of 14–35 lines of NaI, MgI, CaI, ScII, TiII, CrI, FeI, and FeII, while we are here limited to 4–11 lines of NaI and CaI, however in our case analysed in 3D, NLTE for the first time.
We have confirmed that the conclusions of the two previous studies are verified using 1D, LTE calculations and the same set of calibration lines, but with our new $\chi^2$-analysis routines. However, trusting only the smaller set of calibration lines used in the present study, the detection vanishes for G64-12 and is thus in agreement with our 3D, LTE and 3D, NLTE results. The other stars are not much affected. We believe that an important reason why the results for G64-12 are less model dependent compared to the other stars in our analysis (see Table\[tab:isotope\]), is the strong similarity in the line formation of LiI and NaI resonance lines (see Fig.\[fig:hist\]), which here have similar strengths. The use of appropriate calibration lines thus effectively cancel out systematic uncertainties in the determination of the isotopic ratio. Of course, one should remember that the Na D lines may in general be less suitable for other reasons, e.g. interstellar absorption. From our results for the other stars, for which only CaI lines have been used, it seems that the cancellation of 3D and NLTE effects work less well between Li and Ca. 3D, NLTE modelling of both calibration lines and the Li line is required to not make false detections of $\rm^6Li$. We emphasize that it would be desirable to have 3D, NLTE modelling of Fe and other elements to increase the number of available calibration lines.
@Cayrel07 demonstrated with a mostly theoretical analysis how the neglected Li line asymmetry in 1D, LTE, with respect to 3D, NLTE, may lead to overestimation of the lithium isotopic ratio. Following this line of arguments, @Steffen10a [@Steffen10b] and @Steffen12 applied theoretical corrections to the 1D, LTE results of @Asplund06 and @Asplund08, based on modelling of only the Li line in 1D, LTE and 3D, NLTE. By repeating our isotopic analysis in 1D, LTE without use of calibration lines, we confirm that the typical sizes of such corrections would be of order $1-2\%$. However, as @Steffen12 also points out, this post-correction procedure is complicated by the fact that the results of @Asplund06 and @Asplund08 were based on simultaneous modelling of calibration lines. As is evident from the results presented in Table \[tab:isotope\], the lithium isotopic ratios are systematically amplified by the use of calibration lines in LTE, which is also in agreement with the findings of @Steffen10b. In 3D, NLTE, the calibration lines serve their intended purpose to decrease the observational error.
The main conclusions outlined in this paper agree with those presented in @Lind12c, where we first tested our new 3D, NLTE modelling technique. Quantitative differences in isotopic ratio lie within the error bars and are most notable for G64-12. They arise mainly because our previous study used observed spectra that had not yet been optimally processed on a line-by-line basis and a different set of calibration lines.
Further, we note that our 3D, NLTE result for the metal-poor turn-off star HD84937 ($0.011\pm0.010\pm0.011$ without the use of calibration lines) is barely in agreement with the corresponding result by @Steffen12, who find $0.051\pm0.023$, based on a direct comparison to observed spectra. However, the difference is critical because it leads to different conclusions whether the lighter isotope has been detected or not. We are confident that our upper limit is more realistic, based on our superior spectrum quality and important verification using calibration lines.
In this study, we have demonstrated that lithium isotopic abundances in metal-poor halo stars are prone to systematic uncertainties due to the common simplifying assumptions of 1D and LTE and that these uncertainties do not necessarily cancel out using calibration lines. A full understanding of 3D, NLTE line formation is necessary to make correct measurements of the level of $^6\rm Li$. We conclude from our study that only upper limits can be derived on the isotopic ratios in our studied stars; there is thus currently no empirical evidence for a high $^6\rm Li$ content in the early Galaxy that could signal a cosmological production, perhaps stemming from non-standard Big Bang nucleosynthesis. It will be of great value to extend our study to higher metallicity stars, in order to pinpoint when Galactic production of the lighter isotope starts to significantly influence spectral line profiles and thereby testing our understanding of the Galactic chemical evolution of the Li isotopic ratio.
We acknowledge W. Hayek for his help with 3D line formation calculations and M. Bergemann, T. Gehren. Klaus Fuhrmann, and Michael Pfeiffer for providing the FOCES spectra. The data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M.Keck Foundation.
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3D NLTE synthetic line profiles
===============================
[^1]: <http://iraf.noao.edu/>
[^2]: <http://www.astro.caltech.edu/~tb/ipac_staff/tab/makee/index.html>
[^3]: The FOCES spectra have been collected by the members of the group lead by Prof. Thomas Gehren (LMU, Munich) and were kindly provided to us by M. Bergemann.
[^4]: <http://www.astro.ku.dk/~kg/Papers/MHD_code.ps.gz>
|
---
abstract: 'Recently Féray, Goulden and Lascoux gave a proof of a new hook summation formula for unordered increasing trees by means of a generalization of the Prüfer code for labelled trees and posed the problem of finding a bijection between weighted increasing trees and Cayley trees. We give such a bijection, providing an answer to the problem posed by Féray, Goulden and Lascoux as well as showing a combinatorial connection to the theory of tree volumes defined by Kelmans. In addition we give two simple proofs of the hook summation formula. As an application we describe how the hook summation formula gives a combinatorial proof of a generalization of Abel and Hurwitz’ theorem, originally proven by Strehl.'
address: 'Department of Combinatorics & Optimization. University of Waterloo, Canada'
author:
- 'S.R. Carrell'
bibliography:
- 'document.bib'
title: 'Hook Weighted Increasing Trees, Cayley Trees and Abel-Hurwitz Identities'
---
Introduction
============
We begin by fixing some terminology. A tree $T$ is an acyclic connected graph and we denote by $V(T)$ the set of vertices of $T$ and $E(T)$ the set of edges. A tree is said to be rooted if one of its vertices is distinguished. This distinguished vertex is called the root. We only consider unordered trees, that is, trees in which the children of any vertex are unordered. Given a finite set $A$, we let ${\mathbf{m}}(A) = \min(A)$ and ${\mathbf{M}}(A) = \max(A)$.
Let $T$ be a labelled tree with vertex labels given by the finite set $A$ and rooted at the vertex labelled with ${\mathbf{m}}(A)$. We direct the edges of $T$ away from the root so that if $(i,j)$ is an edge in $T$ then $i$ is on the unique path from the root of $T$ to $j$. In this case we call $i$ the father of $j$ in $T$ and denote this by ${\mathfrak{f}}_T(j)$. A vertex is said to be increasing if ${\mathfrak{f}}_T(i) < i$ and is decreasing otherwise. Note that ${\mathfrak{f}}_T({\mathbf{m}}(A))$ is not defined and so the root of $T$ is neither increasing nor decreasing. We say that a tree $T$ is increasing if every non root vertex in $T$ is an increasing vertex.
Given a tree $T$ and a vertex $i \in V(T)$ we define the hook generated by $i$ in $T$, written ${\mathfrak{h}}_T(i)$, to be the set of vertices $j$ such that $i$ is in the unique path from the root to $j$ in $T$. In other words, ${\mathfrak{h}}_T(i)$ is the set of vertices in the subtree of $T$ rooted at $i$. Note that $i \in {\mathfrak{h}}_T(i)$.
Consider the family ${\mathfrak{T}}_A$ of increasing unordered labelled trees with vertex labels given by $A$. Féray, Goulden and Lascoux[@FGL2] studied a combinatorial sum involving a hook weight summed over increasing trees with a fixed number of vertices. Using a generalization of the Prüfer code, it was shown that these sums have an appealing multiplicative closed form. In particular, the following theorem is proven.
\[Thm.2\] For a tree $T \in {\mathfrak{T}}_A$ define a weight on $T$ as $$\widetilde{w}(T) = \prod_{i \in A \backslash {\mathbf{m}}(A)} x_{{\mathfrak{f}}_T(i)} \left( \sum_{ j \in {\mathfrak{h}}_T(i) } y_{i,j} \right).$$ Then the generating series is given by $$\begin{aligned}
{\Theta}_A &= \sum_{T \in {\mathfrak{T}}_A} \widetilde{w}(T) \\
&= x_{{\mathbf{m}}(A)} y_{{\mathbf{M}}(A),{\mathbf{M}}(A)} \prod_{i \in A \backslash \{{\mathbf{M}}(A),{\mathbf{m}}(A)\}} \left( y_{i,i}\sum_{\substack{j \in A \\ j \leq i}} x_j + x_i \sum_{\substack{j \in A \\ j > i}} y_{i,j} \right).
\end{aligned}$$
If one makes the specialization $x_i \to 1$ and $y_{i,j} \to 1$ for all $i$ and $j$ in Theorem \[Thm.2\] then the right hand side of the identity becomes $|A|^{|A|-2}$, which is the number of Cayley trees with vertices labelled by the set $A$ as shown by Cayley[@Cayley]. In other words, $|A|^{|A|-2}$ is the number of trees with labels given by $A$ and which are not rooted and not necessarily increasing (although for convenience we may assume that a Cayley tree is rooted at the vertex labelled by ${\mathbf{m}}(A)$). This observation prompted Féray, Goulden and Lascoux to ask for a combinatorial bijection between increasing trees and Cayley trees which could be used to prove Theorem \[Thm.2\]. Further evidence for the existence of such a bijection was provided by some results in Féray and Goulden’s earlier paper[@FG1] in which the authors study a specialization of Theorem \[Thm.2\] and are able to give a combinatorial bijection for the top degree of the polynomial identity (Section 2.2 in [@FG1]) which involves Cayley trees. In Section \[Section.Direct\] we give a bijective proof of Theorem \[Thm.2\] which involves Cayley trees, solving the problem posed by Féray, Goulden and Lascoux.
In addition to answering the question posed by Féray, Goulden and Lascoux, the contents of Section \[Section.Direct\] also indicates a connection between the hook sum formula in Theorem \[Thm.2\] and the theory of tree volume formulas defined by Kelmans[@K92] and further studied by Kelmans, Postnikov and Pitman[@P01; @P02; @KP08]. This connection comes from Theorem \[Bijection\] below which implies that the generating polynomial ${\Theta}_A$ in Theorem \[Thm.2\] is in fact a tree volume polynomial corresponding to the complete graph. More generally, this gives a connection between the hook sum formula in Theorem \[Thm.2\] and various generalizations of the binomial theorem, such as Abel and Hurwitz’ identities.
As an application of Theorem \[Thm.2\] we will take a moment to discuss more directly the connection to a multivariate generalization of the binomial theorem. In [@Strehl], Strehl proves the following multivariate generalization of the binomial theorem.
\[Thm.S\] Suppose $A$ is a finite set of positive integers and let $$w_A(z) = z \prod_{i \in A \backslash \{{\mathbf{M}}(A)\}} \left( z + \sum_{\substack{j \in A \\ j \leq i}} x_j + \sum_{\substack{j \in A \\ j > i}} y_{i,j} \right).$$ Then $$w_A(u+v) = \sum_{B \sqcup C = A} w_B(u)w_C(v),$$ where $B \sqcup C = A$ means that $B \cup C = A$ and $B \cap C = \emptyset$.
By specializing variables, Theorem \[Thm.S\] can be seen to be a generalization of the binomial theorem. In particular, if we let $y_{i,j} \to 0$ and $x_i \to 0$ for all $i$ and $j$ then it is easily seen that the identity in Theorem \[Thm.S\] is the binomial identity. If we let $y_{i,j} \to 1$ and $x_i \to 1$ for all $i$ and $j$ then Theorem \[Thm.S\] gives Abel’s generalization[@Abel; @Riordan] of the binomial theorem, $$(u+v)(u+v+n)^{n-1} = \sum_{k = 0}^n \binom{n}{k} u (u+k)^{k-1}v(v+(n-k))^{n-k-1},$$ where $n = |A|$. Lastly, if we let $y_{i,j} \to x_j$ for $i < j$ then Theorem \[Thm.S\] becomes Hurwitz’ generalization[@Hurwitz] of Abel’s identity, $$(u+v)\left(u+v+ \sum_{i \in A} x_i\right)^{|A|-1} = \sum_{B \sqcup C = A} u\left(u+\sum_{i \in B} x_i\right)^{|B|-1} v \left(v+\sum_{i \in C}x_i\right)^{|C|-1}.$$ We refer the reader to Strehl’s paper [@Strehl] for additional specializations of interest as well as a number of applications.
Theorem \[Thm.2\] can be used to give a new proof of Theorem \[Thm.S\].
\[Thm.Binom\] Let $x_i, i \geq 0$ and $y_{i,j}, 0 \leq i < j$ be indeterminates and for any finite set $A$ of integers let $${\Theta}_A = x_{{\mathbf{m}}(A)} y_{{\mathbf{M}}(A),{\mathbf{M}}(A)} \prod_{i \in A \backslash \{{\mathbf{M}}(A),{\mathbf{m}}(A)\}} \left( y_{i,i}\sum_{\substack{j \in A \\ j \leq i}} x_j + x_i \sum_{\substack{j \in A \\ j > i}} y_{i,j} \right).$$ Then for any finite set $A$ of positive integers, $$\left. {\Theta}_{A \cup \{0\}} \right|_{x_0 = u + v} = \sum_{B \sqcup C = A} \left. {\Theta}_{B \cup \{0\}} \right|_{x_0 = u} \left. {\Theta}_{C \cup \{0\}} \right|_{x_0 = v}.$$
Our Theorem \[Thm.Binom\] above follows directly from Theorem \[Thm.2\] since both sides of the equality in Theorem \[Thm.Binom\] count trees in which each root edge is coloured either red or blue and then each blue edge is marked with a $u$ and each red edge is marked with a $v$.
Note that if $A$ is a finite set of positive integers and we let $y_{i,i} \to 1$ for all $i$ and $y_{i,j} \to \frac{y_{i,j}}{x_i}$ for all $i < j$ then we recover Theorem \[Thm.S\] from Theorem \[Thm.Binom\] where $w_A(z) = \left. {\Theta}_{A \cup \{0\}} \right|_{x_0 = z}$. Similarly, if we let $z \to x_0 y_{0,0}$, $x_i \to x_i y_{i,i}$ for $i \in A$ and $y_{i,j} \to x_i y_{i,j}$ for $i < j \in A$ then we recover Theorem \[Thm.Binom\] from Theorem \[Thm.S\] where ${\Theta}_{A \cup \{0\}} = w_A$.
It should be noted that the method of proof for Theorem \[Thm.S\] and Theorem \[Thm.Binom\] is very similar, the main difference being the combinatorial description of the generating series involved. Strehl uses the description of the generating series given in Proposition \[MatrixTreeProp\] below as sums over Cayley trees. Instead, we use the description of the generating series as hook weighted sums over increasing trees as given in the statement of Theorem \[Thm.2\].
The remainder of this paper is organized as follows. In Section \[Section.Direct\] we give a combinatorial proof of Theorem \[Thm.2\] by describing an ‘unsorting’ operation which can be applied to increasing trees and, after repeated application, results in a Cayley tree. Following the combinatorial proof we also describe two simple proofs of Theorem \[Thm.2\]. In Section \[Section.Indirect\] we give an indirect combinatorial proof by showing that both expressions for the polynomials ${\Theta}_A$ given in Theorem \[Thm.2\] satisfy the same recursion and initial conditions and in Section \[Section.Algebraic\] we give a direct algebraic proof which uses the fact that increasing trees can be constructed inductively by adding leaves.
A Bijective Proof {#Section.Direct}
=================
Let ${\mathcal{L}}_{i,j}(A)$ be the set of pairs $(T, \phi)$ where $T$ is a labelled tree with vertex labels given by $A$, $\phi$ is a function from the set of increasing vertices in $T$ to $A$ and the pair $(T,\phi)$ satisfies the following conditions.
1. For any increasing vertex $v$ in $T$, $\phi(v) \in {\mathfrak{h}}_T(v)$ and $\phi(v) \geq v$.
2. If $v$ is an increasing vertex in $T$ and $\phi(v) \not = v$ then every vertex on the unique path from $v$ to the root (not including the root) is increasing.
3. If $v$ is a decreasing vertex in $T$ and $u$ is an increasing vertex with $\phi(u) \not = u$ then $u < v$.
4. $T$ has $i$ decreasing vertices.
5. $T$ has $j$ increasing vertices $v$ with $\phi(v) \not = v$.
Define a weight function on ${\mathcal{L}}_{i,j}(n)$ by $${\omega}(T, \phi) = \prod_{\substack{increasing \\ w \in V(T)}} x_{{\mathfrak{f}}_T(w)} y_{w,\phi(w)} \prod_{\substack{decreasing \\ w \in V(T)}} x_w y_{w,{\mathfrak{f}}_T(w)}.$$
The goal of the following theorem is to describe a method by which we can transform trees contained in the sets ${\mathcal{L}}_{0,j}(n)$ (increasing trees) into trees counted by the sets ${\mathcal{L}}_{i,0}(n)$. The reason for this is that the increasing trees contained in the ${\mathcal{L}}_{0,j}(n)$ sets are the objects of interest for the purposes of Theorem \[Thm.2\], however, the weight function depends on non-local information. In particular, for some vertex $v$ it may be the case that $\phi(v)$ is not adjacent to $v$ since the only condition is that $\phi(v) \in {\mathfrak{h}}_T(v)$. Fortunately, the following theorem says that we can repeatedly ‘unsort’ the increasing trees so that they become Cayley trees in which the weight function is entirely local. That is, in ${\mathcal{L}}_{i,0}(n)$ the weight of each tree depends only on vertices and their neighbors and so the generating series can be computed in a straightforward way.
\[Bijection\] There exists a weight preserving bijection between ${\mathcal{L}}_{i,j}(A)$ and ${\mathcal{L}}_{i+1,j-1}(A).$
Let $(T,\phi) \in {\mathcal{L}}_{i,j}(A)$ and let $v$ be the increasing vertex with greatest label such that $\phi(v) \not = v$. Let $b \in {\mathfrak{h}}_v(T)$ be the vertex adjacent to $v$ with $\phi(v) \in {\mathfrak{h}}_T(b)$ and $a$ be the vertex adjacent to $v$ on the unique path from $v$ to the root. In other words, $a = {\mathfrak{f}}_T(v)$ and $b$ is the child of $v$ whose hook contains $\phi(v)$. Note that condition 2 on $(T,\phi)$ implies that every vertex in the path from $v$ to the root is increasing.
Form a new tree $T'$ by removing edges $av$ and $vb$ and adding edges $ab$ and $\phi(v) v$. Since condition 3 implies that $b$ is an increasing vertex in $T$, it is still increasing in $T'$. Also, vertex $v$ is decreasing in $T'$ by condition 1. If we construct a function $\phi'$ from the set of increasing vertices in $T'$ to $\{1, 2, \cdots, n\}$ such that $\phi'(u) = \phi(u)$ for all increasing $u$ in $T'$ then it is easily checked that $(T', \phi')$ satisfies the conditions for ${\mathcal{L}}_{i+1,j-1}(A)$.
To see that this map is invertible we only need for $a, b$ and $v$ to be uniquely determined in $T'$ since $\phi$ must be equal to $\phi'$ for all increasing vertices in $T'$ and $\phi(v) = {\mathfrak{f}}_{T'}(v)$. However, $v$ is the unique decreasing vertex in $T'$ with the smallest label (this follows from condition 4 and the choice of $v$ in $T$). Once we know this, $a$ and $b$ must be the unique pair of adjacent vertices on the path from $v$ to the root in $T'$ such that $a < v < b$ and every vertex on the path from the root to $a$ in $T'$ is increasing (this follows from conditions 2 and 3).
It is then straightforward to check that the constructed bijection is weight preserving since the weight corresponding to vertices $v$ and $b$ in $(T,\phi)$ is equal to their weight in $(T', \phi')$ (although $v$ becomes decreasing in $T'$).
Now we need to determine the generating series for trees in the collection of sets of the form ${\mathcal{L}}_{i,0}(n)$. However, note that in this case the map $\phi$ is redundant since every vertex $v$ in such a tree must have $\phi(v) = v$. In other words, this amounts to determining the generating series for the set of Cayley trees.
\[MatrixTreeProp\] Let ${\mathfrak{D}}(A)$ be the set of Cayley trees labelled by $A$, rooted at ${\mathbf{m}}(A)$ and with edges directed toward the root. For $T \in {\mathfrak{D}}(A)$ let $$w(T) = \prod_{(i,j) \in E(T)} w_{i,j},$$ where $$w_{i,j} = \begin{cases}
x_i y_{i,j} & \mbox{ if } i < j, \\
x_j y_{i,i} & \mbox{ if } i > j.
\end{cases}$$ Then $$\sum_{T \in {\mathfrak{D}}(A)} w(T) = x_{{\mathbf{m}}(A)} y_{{\mathbf{M}}(A),{\mathbf{M}}(A)} \prod_{i \in A \backslash \{{\mathbf{M}}(A),{\mathbf{m}}(A)\}} \left( y_{i,i} \sum_{\substack{j \in A \\ j \leq i}} x_j + x_i \sum_{\substack{j \in A \\ j > i}} y_{i,j} \right).$$
This follows from a straightforward application of the matrix tree theorem and is essentially the same as the method used in part of the proof of Proposition 1 in Strehl[@Strehl]. Without loss of generality we may assume that $A = \{1, 2, \cdots, n\}$. $$\sum_{T \in {\mathfrak{D}}(\{1, 2, \cdots, n\})} w(T) = \det(K_{1,1}),$$ where $$k_{i,j} = \begin{cases}
-x_i y_{i,j} & \mbox{ if } 1 \leq i < j \leq n, \\
-x_j y_{i,i} & \mbox{ if } 1 \leq j < i \leq n, \\
y_{i,i} \sum_{m=1}^{i-1} x_m + x_i \sum_{m = i+1}^n y_{i,m} & \mbox{ if } i = j,
\end{cases}$$ and $K_{1,1}$ is the matrix $K$ with the first row and column removed. Adding each of the columns to the last column and then subtracting $\frac{y_{i-1,i-1}}{y_{i,i}}$ times row $i$ from row $i-1$ for each $i$ gives the matrix $L$ with $$\ell_{i,n} = 0 \mbox{ for } 1 \leq i < n-1, \qquad \ell_{i,j} = 0 \mbox{ for } 1 \leq j < i \leq n-1,$$ $$\ell_{i,i} = y_{i+1,i+1} \sum_{j=1}^{i+1} x_j + x_{i+1} \sum_{j=i+2}^n y_{i+1,j} \mbox{ for } 1 \leq i < n-1,$$ and $\ell_{n-1,n-1} = y_{n,n}x_1.$ Since $\det(K_{1,1}) = \det(L)$ is the product of the main diagonal of $L$, the result follows.
First note that by expanding it is easily seen that $$\sum_{T \in {\mathfrak{T}}_A} \prod_{v = 2}^n x_{{\mathfrak{f}}_T(v)} \left( \sum_{u \in {\mathfrak{h}}_T(v)} y_{v,u} \right) = \sum_{(T,\phi)} \prod_{v=2}^n x_{{\mathfrak{f}}_T(v)} y_{v,\phi(v)}$$ where the sum is over all pairs $(T, \phi)$ where $T$ is in ${\mathfrak{T}}_A$ and $\phi$ is a map from the set of increasing vertices in $T$ to $A$ with $\phi(v) \in {\mathfrak{h}}_T(v)$ for all increasing vertices $v$. However, this is equal to the sum $$\sum_{i \geq 0} \sum_{(T,\phi) \in {\mathcal{L}}_{0,i}(A)} {\omega}(T,\phi).$$ Applying Theorem \[Bijection\] then gives, letting ${\mathfrak{D}}(A)$ be the set of Cayley trees with vertex labels given by $A$ as in Proposition \[MatrixTreeProp\], $$\sum_{j \geq 0} \sum_{(T,\phi) \in {\mathcal{L}}_{j,0}(A)} {\omega}(T,\phi) = \sum_{T \in {\mathfrak{D}}(A)} \prod_{\substack{increasing \\ v \in V(T)}} x_{{\mathfrak{f}}_T(v)}y_{v,v} \prod_{\substack{decreasing \\ v \in V(T)}} x_v y_{v,{\mathfrak{f}}_T(v)}.$$ The result then follows by applying Proposition \[MatrixTreeProp\]
An Indirect Combinatorial Proof {#Section.Indirect}
===============================
We will now give an indirect combinatorial proof of Theorem \[Thm.2\] which relies on Proposition \[MatrixTreeProp\]. Given a set $A$ of positive integers, let $t_A = 1$ if $|A| = 1$ and for $|A| > 1$, $$t_A = x_{{\mathbf{m}}(A)} y_{{\mathbf{M}}(A),{\mathbf{M}}(A)} \prod_{i \in A \backslash \{{\mathbf{M}}(A),{\mathbf{m}}(A)\}} \left( y_{i,i} \sum_{\substack{j \in A \\ j \leq i}} x_j + x_i \sum_{\substack{j \in A \\ j > i}} y_{i,j} \right).$$ From Proposition \[MatrixTreeProp\] above we immediately get the following two results.
\[singleEdge\] Let ${\mathfrak{E}}(A)$ be the subset of trees in ${\mathfrak{D}}(A)$ which have a unique edge incident with ${\mathbf{m}}(A)$. Then if $$r_A = \sum_{T \in {\mathfrak{E}}(A)} w(T),$$ with $w(T)$ as defined in Proposition \[MatrixTreeProp\], then $$r_A = x_{{\mathbf{m}}(A)} y_{{\mathbf{M}}(A),{\mathbf{M}}(A)}\prod_{i \in A \backslash \{{\mathbf{M}}(A),{\mathbf{m}}(A)\}} \left( y_{i,i} \sum_{\substack{j \in A \backslash {\mathbf{m}}(A) \\ j \leq i}} x_j + x_i \sum_{\substack{j \in A \backslash {\mathbf{m}}(A) \\ j > i}} y_{i,j} \right).$$
This follows from the observation that $$r_A = x_{{\mathbf{m}}(A)} \left. \frac{d}{d x_{{\mathbf{m}}(A)}} t_A \right|_{x_{{\mathbf{m}}(A)} = 0}.$$
\[LittleRecursion\] With the polynomials $t_A$ and $r_A$ as defined above with $|A| > 1$ and for any $a \in A \backslash \{ {\mathbf{m}}(A) \}$, $$t_A = \sum_{\substack{B \sqcup C = A \\ {\mathbf{m}}(A) \in C \\ a \in B}} x_{{\mathbf{m}}(A)} \left( \sum_{j \in B} y_{{\mathbf{m}}(B), j} \right) t_B t_C.$$
By Proposition \[MatrixTreeProp\] we know that $t_A = \sum_{T \in {\mathfrak{D}}(A)} w(T)$. For any $T \in {\mathfrak{D}}(A)$ there is a unique child $v$ of ${\mathbf{m}}(A)$ for which the subtree rooted at $v$ contains $a$. Letting $B$ be the set of labels in the subtree it follows from Lemma \[singleEdge\] that $w(T) = w(T_1)w(T_2)$ where $T_1 \in {\mathfrak{E}}(B \cup {\mathbf{m}}(A))$ and $T_2 \in {\mathfrak{D}}(A \backslash B)$. Thus, $$t_A = \sum_{\substack{B \sqcup C \\ {\mathbf{m}}(A) \in C \\ a \in B}} r_{B \cup {\mathbf{m}}(A)} t_C.$$ We also see that $$r_{B \cup {\mathbf{m}}(A)} = x_{{\mathbf{m}}(A)} \left( \sum_{j \in B} y_{{\mathbf{m}}(B),j} \right) t_B,$$ from which the result follows.
Given a finite set $A$ of positive integers let $${\Gamma}_A = x_{{\mathbf{m}}(A)} \left. \frac{d}{d x_{{\mathbf{m}}(A)}} {\Theta}_A \right|_{x_{{\mathbf{m}}(A)} = 0} = \sum_{T \in {\mathfrak{R}}_A} \widetilde{w}(T)$$ where ${\mathfrak{R}}_A$ is the subset of ${\mathfrak{T}}_A$ in which there is a single edge incident with ${\mathbf{m}}(A)$.
\[BigRecursion\] With $|A| > 1$ and for any $a \in A \backslash \{{\mathbf{m}}(A)\}$, $${\Theta}_A = \sum_{\substack{B \sqcup C = A \\ {\mathbf{m}}(A) \in C \\ a \in B}} x_{{\mathbf{m}}(A)} \left( \sum_{j \in B} y_{{\mathbf{m}}(B), j} \right) {\Theta}_B {\Theta}_C.$$
As in the proof of Proposition \[LittleRecursion\] by considering the subtree of ${\mathbf{m}}(A)$ which contains $a$ we see that $${\Theta}_A = \sum_{\substack{B \sqcup C = A \\ {\mathbf{m}}(A) \in C \\ a \in B}} {\Gamma}_{B \cup {\mathbf{m}}(A)} {\Theta}_C.$$ Since for any tree $T \in {\mathfrak{R}}(B \cup {\mathbf{m}}(A))$ we have ${\mathfrak{h}}_T({\mathbf{m}}(B)) = B$ this gives $${\Gamma}_{B \cup {\mathbf{m}}(A)} = x_{{\mathbf{m}}(A)} \left( \sum_{j \in B} y_{{\mathbf{m}}(B),j} \right) {\Theta}_B$$ from which the result follows.
That $${\Theta}_A = x_{{\mathbf{m}}(A)} y_{{\mathbf{M}}(A),{\mathbf{M}}(A)} \prod_{i \in A \backslash \{{\mathbf{M}}(A),{\mathbf{m}}(A)\}} \left( y_{i,i}\sum_{\substack{j \in A \\ j \leq i}} x_j + x_i \sum_{\substack{j \in A \\ j > i}} y_{i,j} \right),$$ for $|A| > 1$ follows by induction after comparing Proposition \[LittleRecursion\] and Proposition \[BigRecursion\] and checking the base case $${\Theta}_A = 1 = t_A,$$ when $|A| = 1$.
The indirect combinatorial proof above uses the canonical decomposition of an unordered increasing tree by removing the edge with vertex labels $1$ and $2$. The same result can be obtained by using the decomposition in which the vertex labelled $1$ is removed. In either case the proof is essentially the same, the generating series for hook-weighted increasing trees and weighted labelled trees are shown to satisfy the same recursion.
An Algebraic Proof {#Section.Algebraic}
==================
Lastly we give an algebraic proof of Theorem \[Thm.2\] which proceeds by induction on the number of vertices. In fact, we prove a small variation of Theorem \[Thm.2\] as it will make the algebraic manipulations that follow a little easier.
\[Thm.3\] Let ${\mathfrak{T}}_n = {\mathfrak{T}}_{\{1, 2, \cdots, n\}}$ and let $${\Theta}_n = \sum_{T \in {\mathfrak{T}}_n} \left( \prod_{i = 2}^n x_{{\mathfrak{f}}_T(i)} \right)\left( \prod_{i = 1}^n \left( \sum_{j \in {\mathfrak{h}}_T(i)} y_{i,j} \right) \right).$$ Then for $n \geq 1$, $${\Theta}_n = y_{n,n} \prod_{i=1}^{n-1} \left( y_{i,i} \sum_{j=1}^i x_j + x_i \sum_{j=i+1}^n y_{i,j} \right).$$
Note that Theorem \[Thm.2\] follows very easily from Theorem \[Thm.3\].
Without loss of generality we may assume that the set $A$ in Theorem \[Thm.2\] is $A = \{1, 2, \cdots, n\}$. In this case, $${\Theta}_n = \left( \sum_{i = 1}^n y_{1,i} \right) {\Theta}_A,$$ and so the result follows.
First, notice that $${\Theta}_1 = y_{1,1}, \qquad \mbox{ and } \qquad {\Theta}_2 = x_1 (y_{1,1} + y_{1,2}) y_{2,2}$$ agree with the combinatorial definition. Now, suppose that ${\Theta}_n$ is as above and let, for $1 \leq i < j \leq n$, $\psi^i_j$ be the evaluation map which takes $y_{k,i}$ to $y_{k,i} + y_{k,j}$ for all $1 \leq k \leq i$. Then since every increasing tree on $n+1$ vertices is created by adding the vertex labelled $n+1$ to some other vertex, we see that $${\Theta}_{n+1} = y_{n+1,n+1} \sum_{i=1}^n x_i \psi^i_{n+1} {\Theta}_n.$$ Let ${\alpha}_i(n) = y_{i,i} \sum_{j=1}^i x_j + x_i(\sum_{j=i+1}^n y_{i,j})$ so that ${\Theta}_n = y_{n,n} \prod_{i=1}^{n-1} {\alpha}_i(n)$. For $1 \leq i \leq n-1$ we have $$\begin{aligned}
\psi^i_{n+1} {\Theta}_n &= y_{n,n} \prod_{k=1}^{n-1} \left( \psi^i_{n+1} y_{k,k} \sum_{j=1}^k x_j + x_k \sum_{j=k+1}^n \psi^i_{n+1} y_{k,j} \right) \\
&= y_{n,n} \left( \prod_{k=1}^i {\alpha}_k(n+1) \prod_{k=i+1}^{n-1} {\alpha}_k(n) + \sum_{j=1}^{i-1} x_j y_{i,n+1} \prod_{k=1}^{i-1} {\alpha}_k(n+1) \prod_{k=i+1}^{n-1} {\alpha}_k(n) \right).\end{aligned}$$ If we let $${\beta}_i(n) = \prod_{k=1}^{i-1} {\alpha}_k(n+1) \prod_{k=i+1}^{n-1} {\alpha}_k(n),$$ this shows that for $1 \leq i \leq n-1$, $$\psi^i_{n+1} {\Theta}_n = y_{n,n} \left( \prod_{k=1}^i {\alpha}_k(n+1) \prod_{k=i+1}^{n-1} {\alpha}_k(n) + \sum_{j=1}^{i-1} x_j y_{i,n+1} {\beta}_i(n) \right).$$ Also, $$\psi^n_{n+1} {\Theta}_n = (y_{n,n} + y_{n,n+1}) \prod_{k=1}^{n-1} {\alpha}_k(n+1).$$ Putting this together gives, after some algebraic manipulation, $$\begin{aligned}
\frac{{\Theta}_{n+1}}{y_{n+1,n+1}} &= \sum_{i=1}^n \psi^i_{n+1} {\Theta}_n \\
&= x_n(y_{n,n} + y_{n,n+1}) \prod_{k=1}^{n-1} {\alpha}_k(n+1) \\
&\qquad + \sum_{i=1}^{n-1} y_{n,n} x_i \left( \prod_{k=1}^i {\alpha}_k(n+1) \prod_{k=i+1}^{n-1} {\alpha}_k(n) + \sum_{j=i+1}^{n-1} x_j y_{j,n+1} {\beta}_j(n) \right).\end{aligned}$$ Now, since ${\alpha}_k(n+1) = {\alpha}_k(n) + x_k y_{k,n+1}$, by expanding from the largest index to the smallest, $$\prod_{k=1}^{n-1} {\alpha}_k(n+1) = \prod_{k=1}^i {\alpha}_k(n+1) \prod_{k=i+1}^{n-1} {\alpha}_k(n) + \sum_{j=i+1}^{n-1} x_j y_{j,n+1} {\beta}_j(n).$$ Thus, $$\begin{aligned}
\frac{{\Theta}_{n+1}}{y_{n+1,n+1}} &= x_n(y_{n,n} + y_{n,n+1}) \prod_{k=1}^{n-1} {\alpha}_k(n+1) + \sum_{i=1}^{n-1} y_{n,n} x_i \prod_{k=1}^{n-1} {\alpha}_k(n+1) \\
&= \prod_{k=1}^{n-1} {\alpha}_k(n+1) \left( y_{n,n} \sum_{i=1}^n x_i + x_n y_{n,n+1} \right) \\
&= \prod_{k=1}^n {\alpha}_k(n+1).\end{aligned}$$
|
---
abstract: 'We analyze a multi-type age dependent model for cell populations subject to unidirectional motion, in both a stochastic and deterministic framework. Cells are distributed into successive layers; they may divide and move irreversibly from one layer to the next. We adapt results on the large-time convergence of PDE systems and branching processes to our context, where the Perron-Frobenius or Krein-Rutman theorem can not be applied. We derive explicit analytical formulas for the asymptotic cell number moments, and the stable age distribution. We illustrate these results numerically and we apply them to the study of the morphodynamics of ovarian follicles. We prove the structural parameter identifiability of our model in the case of age independent division rates. Using a set of experimental biological data, we estimate the model parameters to fit the changes in the cell numbers in each layer during the early stages of follicle development.'
author:
- 'Frédérique Clément[^1], Frédérique Robin[^2], and Romain Yvinec [^3].'
bibliography:
- 'article\_2017\_ark.bib'
title: 'Analysis and calibration of a linear model for structured cell populations with unidirectional motion : application to the morphogenesis of ovarian follicles'
---
[^1]: Project team MYCENAE, Centre INRIA de Paris, France. (frederique.clement@inria.fr)
[^2]: Project team MYCENAE, Centre INRIA de Paris, France. (frederique.robin@inria.fr)
[^3]: PRC, INRA, CNRS, IFCE, Université de Tours, 37380 Nouzilly, France. (romain.yvinec@inra.fr)
|
---
abstract: 'A detailed empirical analysis of the productivity of non financial firms across several countries and years shows that productivity follows a non-Gaussian distribution with power law tails. We demonstrate that these empirical findings can be interpreted as consequence of a mechanism of exchanges in a social network where firms improve their productivity by direct innovation or/and by imitation of other firm’s technological and organizational solutions. The type of network-connectivity determines how fast and how efficiently information can diffuse and how quickly innovation will permeate or behaviors will be imitated. From a model for innovation flow through a complex network we obtain that the expectation values of the productivity level are proportional to the connectivity of the network of links between firms. The comparison with the empirical distributions reveals that such a network must be of a scale-free type with a power-law degree distribution in the large connectivity range.'
author:
- 'T. Di Matteo'
- 'T. Aste'
- 'M. Gallegati'
title: 'Innovation flow through social networks: Productivity distribution'
---
Introduction
============
Recently, the availability of huge sets of longitudinal firm-level data has generated a soars of productivity studies in the economic literature [@Ijiri; @Axtell; @Gaffeo; @Gibrat; @Sutton; @Barnes; @Kruger]. There are several measures of productivity [@Hulten2000], in this work we consider two basic measures: labour and capital productivity. The Labour productivity is defined as value added over the amount of employees (where value added, defined according to standard balance sheet reporting, is the difference between total revenue and cost of input excluding the cost of labour). Although elementary, this measure has the advantage of being accurately approximated given the available data. The other alternative measure is the capital productivity which is defined as the ratio between value added and fixed assets (i.e. capital). This second measure has some weakness since the firms’ assets change continuously in time (consider for instance the value associated with the stock price). Usually the literature recognizes that the productivity distribution is not normally distributed [@Kruger], and empirically ‘fat tails’ with power law behaviors are observed. But the mainstream proposed explanations cannot retrieve this power law tails yielding -at best- to log-normal distributions [@Hopenhayn; @Ericson]. According to the evolutionary perspective [@Nelson1982; @Nelson1995], firms improve their productivity implementing new technological and organizational solutions and, by this way, upgrading their routines. The search for more efficient technologies is carried out in two ways: (1) by [*innovation*]{} (direct search of more efficient routines); (2) by [*imitation*]{} of the most innovative firms [@Dosi; @Mazzuccato]. In practice, one can figure out that once new ideas or innovative solutions are conceived by a given firm then they will percolate outside the firm that originally generated them by imitation from other firms. In this way the innovation flows through the firms. Therefore, the network of contacts between firms which allows such a propagation must play a decisive role in the process.
In this paper we introduce a model for the production and flow of innovation in a complex network linking the firms. We show that the resulting productivity distribution is shaped by the connectivity distribution of this network and in particular we demonstrate that power law tails emerge when the contact-network is of a scale-free type. These theoretical finding are corroborated by a large empirical investigation based on the data set *Amadeus*, which records data of over 6 million European firms from 1990 to 2002 [@newpaper]. A statistical analysis of such a data reveals that: (i) the productivity is power law distributed in the tail region; (ii) this result is robust to different measures of productivity (added value-capital and capital-labor ratios); and (iii) it is persistent over time and countries [@newpaper]. A comparison with the theoretical prediction reveals that the empirical data are well interpreted by assuming that the contact network is of scale-free type with power law tailed degree distributions.
The paper is organized as follows: Section \[s.second\] recalls the concept of social network; Section \[S.m\] introduces the model supporting the technological distribution while Section \[s.EMTH\] describes the empirical findings. A conclusive section summarizes the main results.
Contact networks in social systems {#s.second}
==================================
Systems constituted of many elements can be naturally associated with networks linking interacting constituents. Examples in natural and artificial systems are: food webs, ecosystems, protein domains, Internet, power grids. In social systems, networks also emerge from the linkage of people or group of people with some pattern of contacts or interactions. Examples are: friendships between individuals, business relationships between companies, citations of scientific papers, intermarriages between families, sexual contacts. The relevance of the underlying connection-network arises when the collective dynamics of these systems is considered. Recently, the discovery that, above a certain degree of complexity, natural, artificial and social systems are typically characterized by networks with power-law distributions in the number of links per node (degree distribution), has attracted a great deal of scientific interest [@Barabasi; @Newman; @Amaral]. Such networks are commonly referred as scale-free networks and have degree distribution: $p_k \sim k^{-\alpha}$ (with $p_k$ the probability that a vertex in the network chosen uniformly at random has degree $k$). In scale-free networks most nodes have only a small number of links, but a significant number of nodes have a large number of links, and all frequencies of links in between these extremes are represented. The earliest published example of a scale-free network is probably the study of Price [@Price] for the network of citations between scientific papers. Price found that the exponent $\alpha$ has value $2.5$ (later he reported a more accurate figure of $\alpha=3.04$). More recently, power law degree distributions have been observed in several networks, including other citation networks, the World Wide Web, the Internet, metabolic networks, telephone calls and the networks of human sexual contacts [@Barabasi; @Newman; @Liljeros; @Mossa; @Gabor]. All theses systems have values of the exponents $\alpha$ in a range between 0.66 and 4, with most occurrences between $2$ and $3$ [@Bara04; @Aleberich02; @Bara4; @Watts98].
When analyzing the industrial dynamics, it is quite natural to consider the firms as interacting within a network of contacts and communications. In particular, when the productivity is concerned, such a network is the structure through which firms can imitate each-other. Our approach mimics such a dynamics by considering simple type of interactions but assuming that they take place through a complex network of contacts.
Innovation flow {#S.m}
===============
The innovation originally introduced in a given firm ‘$i$’ at a certain time $t$ can spread by imitation across the network of contacts between firms. In this way, interactions force agents to progressively adapt to an ever changing environment.
In this section we introduce a model for the flow of innovation through the system of firms. We start from the following equation describing the evolution in time of the productivity $x_l$ of a given firm ‘$l$’: $$\begin{aligned}
\label{W}
x_l(t+1)= x_l(t) + A_l(t)+ \sum_{j \in \mathcal{I}_l} Q_{j \to l}(t) [x_j(t) - x_j(t-1)] \\
-\sum_{\tau=l}^{t-1} q_{l}^{(\tau)}(t) [x_l(t-\tau) - x_l(t-\tau-1)] \nonumber.\end{aligned}$$ The term $A_l(t)$ is a stochastic additive quantity which accounts the progresses in productivity due to innovation. The terms $Q_{j \to l}$ are instead exchange factors which model the imitation between firms. These terms take into account the improvement of the productivity of the firm ’$l$’ in consequence of the imitation of the processes and innovations that had improved the productivity of the firm ’$j$’ at a previous time. Such coefficients are in general smaller than one because the firms tend to protect their innovation content and therefore the imitation is -in general- incomplete. In the following we will consider only the static cases where these quantity are independent on $t$. The term $q_l^{(\tau)}$ is: $$\begin{aligned}
\label{q}
q_l^{(1)} &=& \sum_{j \in \mathcal{I}_l} Q_{j \to l} Q_{l \to j} \;\; \mbox{for $\tau=1$} \\
q_l^{(\tau)} &=& \sum_{j \in \mathcal{I}_l} Q_{j \to l} \sum_{h_1 \ldots h_{\tau-1}} Q_{l \to h_1} Q_{h_1 \to h_2} \ldots Q_{h_{ {\tau-1} \to j}} \;\; \mbox{for $\tau \geq 2$}.\end{aligned}$$ This term excludes back-propagation: firm ‘$l$’ imitates only improvements of the productivity of firm ‘$j$’ which have not been originated by imitation of improvements occurred at the firm ‘$l$’ itself at some previous time. The system described by Equation \[W\] can be viewed as a system of self-avoiding random walkers with sources and traps.
The probability $P_{t+1}(y,l)dy$ that the firm $l$ at the time $t+1$ has a productivity between $y$ and $y+dy$ is related to the probabilities to have a set $\{Q_{j\to l} \}$ of interaction coefficients and a set of additive coefficients $\{A_l(t)\}$ such that a given distribution of productivity $\{x_j(t)\}$ at the time $t$ yields, through Equation \[W\], to the quantity $y$ for the agent $l$ at time $t+1$. This is: $$\begin{aligned}
\label{Pw-1}
P_{t+1}(y,l) &=&
\int_{-\infty}^\infty \ da \,
\Lambda_t(a,l) \prod_{\xi=0}^{t-1}
\int_{-\infty }^\infty dx_1^{(\xi)} P_{t-\xi}(x_1^{(\xi)},1)
\cdots \\ \nonumber
&&
\int_{-\infty }^\infty dx_N^{(\xi)} P_{t-\xi}(x_N^{(\xi)},N) \\ \nonumber
&&
\delta \big(y - a - x_l^{(0)} - \sum_{j \in \mathcal{I}_l} [x_{j}^{(0)}- x_{j}^{(1)}] Q_{j\to l} + \sum_{\tau=l}^{t-1} q_{l}^{(\tau)} [x_l^{(\tau)} - x_l^{(\tau+1)}] \big) ,\end{aligned}$$ where $\delta(y)$ is the Dirac delta function and $\Lambda_t(a,l)$ is the probability density to have at time $t$ on site $l$ an additive coefficient $A_l(t)=a$. Let us introduce the Fourier transformation of $ P_t(y,l)$ and its inverse $$\begin{aligned}
\label{FP}
\hat P_t(\varphi,l) &=& \int_{-\infty}^\infty dy
e^{+ i y \varphi} P_t(y,l)
\nonumber \\
P_t(y,l) &=& \frac{1}{2\pi} \int_{-\infty}^\infty d\varphi
e^{- i y \varphi}\hat P_t(\varphi,l) \;\;\;.\end{aligned}$$ In appendix \[A\], we show that Equation \[Pw-1\] can be re-written in term of these transformations, resulting in: $$\begin{aligned}
\label{Pw5}
\hat P_{t+1}(\varphi,l)
&=&
\hat \Lambda_t(\varphi,l)
\hat P_t(\varphi,l)
\prod_{\xi=2}^{t-1}\hat P_{t-\xi}((-q_l^{(\xi)} +q_l^{(\xi-1)}) \varphi,l) \nonumber \\
&& \hat P_0(q_l^{(t-1)} \varphi,l) \hat P_{t-1}(-q_l^{(1)} \varphi,l) \\
&& \prod_{j \in \mathcal{I}_l}
\hat P_t(Q_{j\to l} \varphi,j) \hat P_{t-1}(-Q_{j\to l} \varphi,j)\;\;, \nonumber\end{aligned}$$ with $\hat \Lambda_t(\varphi,l)$ being the Fourier transform of $\Lambda_t(a,l)$. From this equation we can construct a relation for the propagation of the cumulants of the productivity distribution. Indeed, by definition the cumulants of a probability distribution are given by the expression: $$\label{C-1}
k^{(\nu)}_l(t) = (-i)^{\nu}\frac{d^{\nu}}{d\varphi^{\nu}} \ln \hat P_t(\varphi,l) \Big|_{\varphi=0} \;\;,$$ where the first cumulant $k^{(1)}_l(t)$ is the expectation value of the stochastic variable $x_l$ at the time $t$ ($\left< x_l(t) \right> $) and the second cumulant $k^{(2)}_l(t)$ is its variance ($\sigma_l^2(t)$). By taking the logarithm of Equation \[Pw5\] and applying Equation \[C-1\] we get: $$\begin{aligned}
\label{C-2}
k^{(\nu)}_l(t+1) &=& c^{(\nu)} (t)
+ k^{(\nu)}_l(t) +\sum_{\xi=2}^{t-1} (q_l^{(\xi-1)} -q_l^{(\xi)})^{\nu} k^{(\nu)}_l(t-\xi) \nonumber \\
&& + (q_l^{(t-1)})^{\nu} k^{(\nu)}_l(0) +(-q_l^{(1)})^{\nu} k^{(\nu)}_l(t-1)
+ \\
&& \sum_{j \in \mathcal{I}_l} [\left(Q_{j\to l} \right)^\nu k^{(\nu)}_j(t) +\left(-Q_{j\to l} \right)^\nu k^{(\nu)}_j(t-1)] \;\;\;.\nonumber\end{aligned}$$
It has been established by Maddison that the average innovation rate of change in the OECD countries since $1870$ has been roughly constant [@Maddison]. In our formalism this implies $$\frac{\left< A_l(t+1) \right> - \left< A_l(t) \right> }{\left< A_l(t) \right>} \sim const.$$ Therefore, the mean of the additive term in Equation \[W\] ($\left< A_l(t) \right>$) must grow exponentially with time and consequently the first cumulant (the average indeed) reads: $c^{(1)}=c_0^{(1)} ( c_1^{(1)} )^t$. Equivalently we assume an exponential growth also for the other moments ($c^{(\nu)}=c_0^{(1)} ( c_1^{(\nu)} )^t$).
Equation \[C-2\] can now be solved by using a mean-field, self-consistent solution (neglecting correlations and fluctuations in the interacting firms) obtaining: $$\begin{aligned}
\label{S1}
k^{(1)}_l(t) &=& \frac{1}{A}\frac{c_0^{(1)} c_1^{(1)} }{(c_1^{(1)} -1)} \Big [1 + {\bar a } Q z_l \Big ] (c_1^{(1)} )^{t} \nonumber \;\; \;\; \;\; \;\; \mbox{for $\nu=1$}
\\
k^{(\nu)}_l(t) &=&
\frac{c_0^{(\nu)} }{B_{\nu}} \Big [1 + (1 + \frac{(-1)^{\nu}}{c_1^{(\nu)}}) {\bar b^{(\nu)} } Q^{\nu} z_l \Big ] (c_1^{(\nu)} )^t \;\; \;\; \;\;\;\;\mbox{for $\nu>1$}\end{aligned}$$ where $$\begin{aligned}
\label{abdf}
&& {\bar a} = \frac{1}{1 - \left<\frac{Q z_l}{A}\right>} \frac{1}{\left< A\right >} \\
&& {\bar b^{(\nu)} } = \frac{1}{1 + \left<\frac{(1 + (-1)^{\nu}/c_1^{(\nu)}) Q^{\nu} z_l}{B_{\nu}}\right>} \frac{1}{\left< B_{\nu}\right>}\end{aligned}$$ and $$\begin{aligned}
\label{A1B1AvBv}
A &=& c_1^{(1)} + z_l \sum_{\xi=1}^{t-1} \frac{Q^{\xi+1}} {(c_1^{(1)} )^{\xi}} \\
B_{\nu} &=& -1 + c_1 ^{(\nu)} - z_l^{\nu} \big[\frac{(-Q^2)^{\nu}}{c_1^{(\nu)}} \nonumber \\
&+& \sum_{\xi=2}^{t-1} \frac{(Q^{\xi} - Q^{\xi+1})^{\nu}}{(c_1^{(\nu)} )^{\xi}} + \frac{(Q^t)^{\nu}}{(c_1^{(\nu)} )^t} \big]\end{aligned}$$ with $Q$ being the average exchange factor. When this exchange term is small, Equation \[S1\] can be highly simplified by taking the first order in $Q$ only, leading to: $$\begin{aligned}
\label{TT1}
k^{(1)}_l(t) &\sim& \frac{c_0^{(1)}}{c_1^{(1)}-1}
\Big[1+ z_l \frac{Q}{c_1^{(1)}}\Big](c_1^{(1)})^{t} \nonumber \\
k^{(\nu)}_l(t) &\sim& \frac{c_0^{(\nu)}}{c_1^{(\nu)}-1} (c_1^{(\nu)})^{t}\end{aligned}$$ Equation \[S1\] (and its simplified form (Equation \[TT1\])) describes a mean productivity which grows at the same rate of the mean innovation growth (as a power of $c_1^{(1)}$) and is directly proportional to the number of connections that the firm has in the exchange network. From Equation \[S1\] we also have that all the cumulants increase with a corresponding power rate ($(c_1^{(\nu)})^t$). But, if we analyze the *normalized cumulants*: $\lambda^{(\nu)}(t) = k^{(\nu)}_l(t)/[k^{(2)}_l(t)]^{\nu/2}$ we immediately see that at large $t$ they all tend to zero excepted for the mean and the variance. Therefore the probability distributions tend to Gaussians at large times.
Summarizing, in this section we have shown that, at large $t$, the expectation value of the productivity level of a given firm is proportional to its connectivity in the network of interaction and the fluctuations around this expectation-value are normally distributed. Each firm has a different connectivity and therefore the probability distribution for the productivity of the ensemble of firms is given by a normalized sum of Gaussians with averages distributed according with the network connectivity. As discussed in the previous section, power-law-tailed degree distributions are very common in many social and artificial networks. It is therefore natural to hypotheses that also the social/information network through which firms can exchange and imitate productivity has a degree distribution characterized by a power law in the large connection-numbers region. If this is the case, then the whole productivity distribution will show a power-law tail characterized by the same exponent of the degree distribution [@footnote1].
Empirical analysis and comparison with theory {#s.EMTH}
=============================================
Figures \[f.P2\], \[f.P4\], \[f.P1\] and \[f.P3\] show the log-log plot of the frequency distributions (Left) and the complementary cumulative distributions (Right) of labour productivity and for capital productivity measured as quotas of total added value of the firms. In these figures the different data sets correspond to different years: $1996-2001$. For the sake of exposition, we illustrate the productivity distribution for France and Italy only, but similar results have been obtained for other Euroland countries of the AMADEUS dataset. The frequency distributions show a very clear non-Gaussian character: they are skewed with asymmetric tails and the labour productivity (Figures \[f.P2\] and \[f.P4\] (Left)) present a clear leptokurtic pick around the mode. The complementary cumulative distributions ($P_>(x)$, being the probability to find a firm with productivity larger than $x$) show a linear trend at large $x$ implying a non-Gaussian character with the probability for large productivities well mimicked by a power-law behavior.
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The model presented in this paper gives a simple explanation for the occurrence of such power law tails in the productivity distribution: they are a consequence of the social/information network which is of “scale-free” type (analogously with several other complex systems where such a connectivity-distribution can be measured [@DiMatteo; @Richmond2001; @Stanley1998; @Biham1998; @Bouchaud2000; @Solomon2002]). Indeed, we have shown that distribution for the productivity of the ensemble of firms is given by a normalized sum of Gaussians with averages distributed according with the network connectivity. As consequence, when the connection network is of scale-free type the productivity distribution must share with it the same exponent in the power-law-tail.
Comparisons between the theoretical predictions from Equation \[TT1\] associated with a scale-free network and the empirical findings are shown in the Figures \[f.P2\], \[f.P4\], \[f.P1\] and \[f.P3\] (Right). In particular, accordingly with Equation \[TT1\], we assume an average productivity given by $k^{(1)}_l = m + z_l n$, a variance equal to $\sigma$ and the degree distribution of the network given by $ p_k \propto k^{-\alpha} \exp(-\beta/k) $. The agreement with the empirical findings is quantitatively rather good. We note that, although there are several parameters, the behavior for large productivity is controlled only by the power-law exponent $-\alpha$. On the other hand, in the small and the middle range of the distribution the other parameters have a larger influence.
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From our analysis we observe that the theoretical curves fit well the empirical findings by assuming the power law exponent equal to $\alpha=2.7$ and $2.1$ for the labour productivity in Italy and France respectively. These exponents are in good agreement with the ones typical of the degree distribution in social networks. On the other hand the capital productivity presents much steeper decays which can be fitted with exponents $3.8$ and $4.6$ respectively. These very high values of the exponents might be consequence of the *irrational euphoria* of the late $90$es when the stock markets were hit by a speculative bubble ($1997$) and its subsequent crash ($2000$). The bubble increased the value of the firms’ asset thus lowering the value added-capital (i.e. capital productivity) ratio and soaring the power law coefficient of the power law distribution of the capital productivity distribution. However the very high capital productivity regions show a slowing down which could be fitted with lower exponents.
Conclusions
===========
In this paper we have shown that the productivity of non-financial firms is power law distributed. This result is robust to different measures of productivity, different industrial sectors, years and countries. We have also argued that the empirical evidence corroborates the prescription of the evolutionary approach to technical change and demonstrated that power law distributions in productivity can be interpreted as consequence of a simple mechanism of exchanges within a social network. In particular, we have shown that the expectation values of the productivity level are proportional to the connectivity of the network of links between firms. The comparison with the empirical data indicates that such a network is of a scale-free type with a power-law degree distribution. In the present formulation we have assumed an underlying network which is fixed in time. This allows obtaining equilibrium solutions. On the other hand, a more realistic analysis should consider a non-static underlying network and therefore non-equilibrium trajectories modulated by the fluctuation in the underlying network. This non-equilibrium dynamics can be studied numerically from Equation \[W\] by using fluctuating exchange coefficients $Q_{j \to l}(t) $ . This is left to future research. In this paper we had a narrower goal: to show that empirical evidence is very well fitted by the evolutionary view of technical change.
We thank Corrado Di Guilmi for excellent research assistance. T. Di Matteo benefited from discussions with the participants to the COST P10 ‘Physics of Risk’ meeting in Nyborg (DK), April 2004. TDM and TA acknowledge partially financial support from ARC Discovery project DP0344004 (2003).
Cumulant propagation {#A}
====================
By using the Fourier transformation (Equation \[FP\]), Equation \[Pw-1\] becomes: $$\begin{aligned}
\label{Pw1}
P_{t+1}(y,l)
&=&
\int_{-\infty}^\infty da \Big\{\Lambda_t(a,l)
\prod_{\xi=0}^{t-1} \Big[
\frac{1}{(2\pi)^N}
\int_{-\infty}^\infty dx_1^{(\xi)} \cdots \int_{-\infty}^\infty dx_N^{(\xi)}
\\
&&
\int_{-\infty}^\infty d\varphi_1^{(\xi)} e^{- i x_1^{(\xi)} \varphi_1^{(\xi)}}
\hat P_{t-\xi}(\varphi_1^{(\xi)},1)
\cdots \int_{-\infty}^\infty d\varphi_N^{(\xi)} e^{- i x_N^{(\xi)} \varphi_N^{(\xi)}}
\hat P_{t-\xi}(\varphi_N^{(\xi)},N) \Big]
\nonumber \\
&&
\frac{1}{2\pi}
\int_{-\infty}^\infty d\phi
e^{- i ( y - a - x_l^{(0)} - \sum_{j \in \mathcal{I}_l} [x_{j}^{(0)}- x_{j}^{(1)}] Q_{j\to l} + \sum_{\tau=l}^{t-1} q_{l}^{(\tau)} [x_l^{(\tau)} - x_l^{(\tau+1)}] )\phi} \Big\},\nonumber\end{aligned}$$ where the Dirac delta function has been written as $$\label{dir}
\delta(y-y_0) = \frac{1}{2\pi}
\int_{-\infty}^\infty d\phi e^{- i (y -y_0) \phi} \;\;\;.$$ Equation \[Pw1\] can be re-written as: $$\begin{aligned}
\label{Pw2}
&& P_{t+1}(y,l) =
\frac{1}{(2\pi)}
\int_{-\infty}^\infty da \Big\{ \Lambda_t(a,l)
\int_{-\infty}^\infty d\phi e^{- i (y-a) \phi} \\
&&
\prod_{\xi=0}^{t-1} \Big[
\frac{1}{(2\pi)^N}
\int_{-\infty}^\infty d\varphi_l^{(\xi)}
\Big(\hat P_{t-\xi}(\varphi_l^{(\xi)},l)
\int_{-\infty}^\infty dx_l^{(\xi)}
e^{- i ( \varphi_l^{(0)} - \phi) x_l^{(0)}} e^{- i \sum_{\tau=2}^{t-1} ( \varphi_l^{(\tau)} + q_l^{(\tau)} \phi - q_l^{(\tau-1)} \phi) x_l^{(\tau)}} \nonumber \\
&& e^{- i ( \varphi_l^{(t)} - q_l^{(t-1)} \phi) x_l^{(t)}} e^{- i ( \varphi_l^{(1)} - q_l^{(1)} \phi) x_l^{(1)}} \Big) \prod_{j \in \mathcal{I}_l}
\int_{-\infty}^\infty d\varphi_j^{(\xi)}
\Big(\hat P_{t-\xi}(\varphi_j^{(\xi)},j) \nonumber \\
&& \int_{-\infty}^\infty dx_j^{(\xi)}
e^{ - i [( \varphi_j^{(0)} - Q_{j\to l} \phi) x_j^{(0)} +( \varphi_j^{(1)} + Q_{j\to l} \phi) x_j^{(1)}]}
\Big) \Big]\Big\}.
\nonumber\end{aligned}$$ The integration over the $x$’s yields $$\begin{aligned}
\label{Pw3}
&& P_{t+1}(y,l) =
\frac{1}{2\pi}
\int_{-\infty}^\infty da \Big\{\Lambda_t(a,l)
\int_{-\infty}^\infty d\phi \Big[
e^{- i (y-a) \phi }
\hat P_t(\phi,l) \\
&&\prod_{\xi=2}^{t-1}\hat P_{t-\xi}((-q_l^{(\xi)} +q_l^{(\xi-1)}) \phi,l) \hat P_0(q_q^{(t-1)} \phi,l) \hat P_{t-1}(-q_l^{(1)}\phi,l) \nonumber \\
&& \prod_{j \in \mathcal{I}_l}
\hat P_t(Q_{j\to l} \phi,j) \hat P_{t-1}(-Q_{j\to l} \phi,j)
\Big]\Big\}.\nonumber\end{aligned}$$ Its Fourier transform is: $$\begin{aligned}
\label{Pw4}
\hat P_{t+1}(\varphi,l)
&=&
\frac{1}{2\pi}
\int_{-\infty}^\infty da \Big\{\Lambda_t(a,l)
\int_{-\infty}^\infty d\phi \Big[ e^{ i a \phi}
\int_{-\infty}^\infty d y e^{- i y (\phi - \varphi)}\\
&&
\hat P_t(\phi,l) \prod_{\xi=2}^{t-1}\hat P_{t-\xi}((-q_l^{(\xi)} +q_l^{(\xi-1)}) \phi,l) \hat P_0(q_q^{(t-1)} \phi,l) \hat P_{t-1}(-q_l^{(1)}\phi,l)\big] \nonumber \\
&& \prod_{j \in \mathcal{I}_l}
\hat P_t(Q_{j\to l} \phi,j) \hat P_{t-1}(-Q_{j\to l} \phi,j)
\Big\}.
\nonumber\end{aligned}$$ Equation \[Pw4\] can be integrated over $y$ giving the Fourier transform of Equation \[Pw-1\] which is Equation \[Pw5\] in Section \[S.m\].
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|
---
abstract: 'Understanding the thermal equilibrium (stability) curve may offer insights into the nature of the warm absorbers often found in active galactic nuclei. Its shape is determined by factors like the spectrum of the ionizing continuum and the chemical composition of the gas. We find that the stability curves obtained under the same set of the above mentioned physical factors, but using recently derived dielectronic recombination rates, give significantly different results, especially in the regions corresponding to warm absorbers, leading to different physical predictions. Using the current rates we find a larger probability of having thermally stable warm absorber at $10^5 \kel$ than previous predictions and also a greater possibility for its multiphase nature. the results obtained with the current dielectronic recombination rate coefficients are more reliable because the warm absorber models along the stability curve have computed coefficient values, whereas previous calculations relied on guessed averages for the same due to lack of available data.'
author:
- |
Susmita Chakravorty$^{1}$ [^1], Ajit K. Kembhavi$^{1*}$, Martin Elvis$^{2*}$, Gary Ferland$^{3*}$, N.R.Badnell$^{4*}$\
$^{1}$IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India;\
$^{2}$Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138;\
$^{3}$Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506;\
$^{4}$Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK.
title: Dielectronic recombination and stability of warm gas in AGN
---
\#1[\[eq:\#1\]]{} \#1[\[fig:\#1\]]{} \#1[\[sec:\#1\]]{} \#1[\[tab:\#1\]]{}
\#1[Equation \[eq:\#1\]]{} \#1[Figure \[fig:\#1\]]{} \#1[Section \[sec:\#1\]]{} \#1[Table \[tab:\#1\]]{}
\#1[[\#1]{}]{} \#1[10\^[\#1]{}]{}
quasars: absorption lines - galaxies : active - ISM - ISM: lines and bands - abundances - atoms
Introduction
============
Warm Absorbers are highly photoionised gas found along our line of sight to the continuum source of active galactic nuclei (AGN). Their signatures are a wealth of absorption lines and edges from highly ionized species, notably OVII, OVIII, FeXVII, NeX, CV and CVI in the soft X-ray (0.3-1.5 keV) spectra. The typical column density observed for the gas is $\nh \sim
10^{22\pm1}\cmsqi$ and the temperature $T$ is estimated to be a few times $10^5\kel$. For many objects the warm absorber exists as a multiphase absorbing medium with all the phases in near pressure equilibrium (Morales, Fabian & Reynolds, 2000; Collinge 2001; Kaastra 2002; Krongold 2003; Netzer 2003; Krongold 2005; Ashton 2006).
Any stable photoionised gas will lie on the thermal equilibrium curve or ‘stability’ curve, in the temperature - pressure phase space, where heating balances cooling; this curve is often used to study the multiphase nature of the photoionised gas. The equilibrium depends on the shape of the ionizing continuum and the chemical abundance of the gas (Krolik, McKee & Tarter, 1981; Krolik & Kriss, 2001; Reynolds & Fabian, 1995 and Komossa & Mathur, 2001). We are investigating this dependences in details and will report elsewhere (Chakravorty , to be submitted).
Over the past two decades the estimates of dielectronic recombination rate coefficients have improved. In this letter we show that these have affected the stability curves significantly which may lead to quite different physical models for the warm absorber gas. We conclude with a caution on the reliability of earlier results.
The Stability Curve
===================
As is customary, we consider an optically thin, plane parallel slab of gas being photoionised by the central source of the AGN. In ionization equilibrium, photoionisation is balanced by recombinations. Thermal equilibrium is achieved when photoionisation and Compton heating are balanced by collisional cooling, recombination, line-excitation, bremsstrahlung and Compton cooling. The conditions under which these equilibriums are achieved depend on the shape of the continuum, metallicity of the gas, the density, the column density, and the ratio of the ionizing photon flux to the gas density. Following the convention of Tarter, Tucker & Salpeter (1969) we specify this ratio through an [*ionization parameter*]{} $\xi=L/nR^2$, where $L$ is the luminosity of the source and $n$ the number density of gas at a distance $R$ from the center of the AGN.
We consider a sequence of models for a range of ionization parameters $\xi$, for optically thin gas having constant density of $\onlyten{9}\cmcui$, being irradiated by a power-law ionizing continuum with photon index $\Gamma=1.8$, so that $f(\nu) \sim \nu^{-(\Gamma-1)}$, and extending from 13.6 eV to 40 keV. The chemical composition of the gas (referred to as [*old Solar abundance*]{}) is is due to Grevesse and Anders (1989) with extensions by Grevesse and Noels (1993). These parameter values (hereafter called the [*standard set*]{} of parameters) have been considered because that allows us to compare our results easily with earlier work.
Using version C07.02.01[^2] (hereafter C07) of the photoionisation code CLOUDY (see Ferland 1998 for a description), we plot the thermal stability curve which is shown as the solid curve (labeled C07) in . $\xi/T$ is proportional to $p_{rad}/p$, where $p_{rad}$ is the radiation pressure and $p$ the gas pressure; so an isobaric perturbation of a system in equilibrium is represented by a vertical displacement from the curve, and only changes the temperature. If the system being perturbed is located on a part of the curve with positive slope, which covers most of the curve, then a perturbation corresponding to an increase in temperature leads to cooling, while a decrease in temperature leads to heating of the gas. Such a gas is therefore thermally stable. But if the system is located on one of the few parts of the curve with negative slope, then it is thermally unstable because isobaric perturbations will lead to runaway heating or cooling.
Comparison With Previous Work
=============================
Reynolds & Fabian (1995, hereafter RF95), have studied the stability curve for the warm absorber in MCG-6-30-15, using the standard set of parameters given above. We have reproduced their curve, as given in Figure 3 of RF95, using version C84.12a (hereafter C84) of CLOUDY, which was the stable version between 1993 and 1996; this is shown as the dashed curve in . The C84 and C07 curves match at high temperatures in the range $T>10^{7.2}\kel$, where Compton heating and cooling dominate. They also agree at low temperatures $T\lesssim10^{4.5}\kel$. However, the curves are significantly different in the intermediate temperature range $\onlyten{4.5}\le
T\le\onlyten{7.2}\kel$, which is the region of interest for the warm absorbers and where recombination and line excitation are dominant cooling mechanisms.
[c c c c c c]{} & & & &\
\[0cm\]\[0cm\][Version]{} & \[0cm\]\[0cm\][$\xi_5$]{} & \[0cm\]\[0cm\][$N_{\rm phases}$]{} & \[0cm\]\[0cm\][$\sim10^5\kel$]{} & \[0cm\]\[0cm\][$\sim10^6\kel$]{} & \[0cm\]\[0cm\][$\Delta_{\rm{M}}\log(\xi/T)$]{}\
C84 & 45 & 2 & 0.05 & 0.47 & 0.05\
\
C07 & 74 & 2 & 0.22 & 0.46 & 0.07\
A detailed comparison between the stable phases for warm absorbers predicted by the two different versions of CLOUDY is done in . The second column of the table shows that for an absorber at $T\sim10^5\kel$, C84 predicts $\xi_5\sim45$, as compared to $\xi_5\sim74$ obtained from the C07 stability curve, where $\xi_5$ is defined to be the ionization parameter corresponding to the middle of the $10^5\kel$ stable warm absorber phase. Both C84 and C07 predict two discrete phases of warm absorber at $\sim10^5$ and $\sim10^{5.7}\kel$ which are in pressure equilibrium with each other and have been highlighted in . The C07 curve continues to have stable thermal states at $\sim 10^6\kel$ which is not true in the C84 case. The range of $\log(\xi/T)$, over which the warm absorber exists are given in the fourth and fifth columns of respectively for the low and high temperature states. The extent of the $10^5\kel$ phase in $\log(\xi/T)$ is about four times larger in C07, predicting greater probability of finding $10^5\kel$ warm absorbers. In the sixth column we have compared the range $\Delta_{\rm{M}}\log(\xi/T)$ where the warm absorber exhibits multiple phases. C07 predicts a 40% larger range and hence greater possibility of a multiphase warm absorber.
In order to isolate the atomic physics underlying the change in the stability curve, we have plotted the fractional variation of temperature $\Delta T/T_{C07} = (T_{C84}-T_{C07})/T_{C07}$, from one version to another, against $\log \xi$ in the top panel of . We see that $\Delta T > 0$ for a major part of the range $1.0 < \log\xi < 4.5$, so that the gas is predicted to be cooler by C07. The cooling fractions ($\Delta C$) of the major cooling agents and the heating fractions ($\Delta H$) contributed by the principal heating agents using C07 are plotted against $\log \xi$ respectively in the middle and bottom panel of . The ions that contribute significantly where $\Delta T/T_{C07} \gtrsim 0.5$, are He$^{+1}$ and high-ionization species of silicon (+10 and +11) and iron (+21, +22 and +23). In the same $\log\xi$ range, the principal heating agents are highly ionized species of oxygen (+6 and +7) and iron (+17 to +25).
To identify the ions which are responsible for $\Delta T/T_{C07}
\gtrsim 50\%$, we compare their column densities predicted by C84 and C07. In the previous photoionisation calculations in this paper it was sufficient to use one zone models for optically thin gas. However, to calculate and compare the column densities of the ions over the range $1.0<\log\xi<4.5$, we chose to specify the gas to have total hydrogen column density $\nh=\onlyten{22}\cmsqi$, which is typical for warm absorbers. The column densities are plotted in with solid lines for C07 and dashed lines for C84. It is seen that the column densities of the major coolants changed significantly. The cooling agents are among the ions for which dielectronic recombination rate coefficients (hereafter DRRC) have been updated to the references below, as will be discussed in detail in . Thus the enhanced cooling in C07 due to the change in DRRC is the cause of the shift in the stability curves.
Changes in Dielectronic Recombination Rate Coefficients
========================================================
[c c c c c ]{} & & &\
\[0cm\]\[0cm\][Ion]{} & \[0cm\]\[0cm\][$\log T$]{} & \[0cm\]\[0cm\][$\log \xi$]{} & \[0cm\]\[0cm\][C07]{} & \[0cm\]\[0cm\][C84]{}\
\
He$^{+1}$ & 5.34 & 2.00 & 1.66 & -\
Si$^{+10}$ & “ & ” & 1.36 $\times 10^{-1}$ & 2.26 $\times 10^{-2}$\
Si$^{+11}$ & “ & ” & 8.85 $\times 10^{-2}$ & 2.50 $\times 10^{-2}$\
\
\
Fe$^{+21}$ & “ & ” & 9.06 $\times 10^{-2}$ & 2.20 $\times 10^{-2}$\
Fe$^{+22}$ & “ & ” & 8.36 $\times 10^{-2}$ & 2.37 $\times 10^{-2}$\
Fe$^{+23}$ & “ & ” & 5.77 $\times 10^{-2}$ & 2.28 $\times 10^{-2}$\
The evolution of the thermal phases from the nebular temperatures of $10^4\kel$ to the coronal temperatures of $10^6\kel$ depends sensitively on the detailed atomic physics of the various elements which contribute to photoelectric heating as well as to cooling due to recombination and collisionally excited lines. In , we have compared the total recombination rates (dielectronic + radiative) for the significant cooling agents as predicted by C07 and C84. The values for $\log T$ and $\log \xi$ at which the comparisons have been made are given in columns 2 and 3 respectively. For all the ions the total recombination rates are significantly higher in C07 than in C84 as shown by columns 4 and 5. Referring to , we see that the differences in total recombination rates are relatively larger for ions like Si$^{+10}$, Si$^{+11}$, Fe$^{+21}$ and Fe$^{+22}$ which are significant cooling agents for $\log \xi \sim 2.0$ and $\log \xi \sim 3.2$, corresponding to which we have maximum difference in predicted equilibrium temperatures $T_{C07}$ and $T_{C84}$.
In the warm absorber temperature range $10^5 \lesssim T \lesssim
10^7\kel$, dielectronic recombination dominates over radiative recombination for many ions (Osterbrock & Ferland, 2006). Unlike the radiative recombination rate coefficients, the DRRC have undergone significant changes over the last decade. C84 used DRRC from Nussbaumer & Storey (1983;1984; 1986; and 1987) and Arnaud & Raymond (1992). In C07 we have taken the DRRC for the isoelectronic sequences of lithium, beryllium, boron, carbon, nitrogen, oxygen and iron-like ions respectively from Colgan (2004), Colgan (2003), Altun (2004), Zatsarinny (2004a), Mitnik & Badnell (2004), Zatsarinny (2003), and Gu (2003). The DRRC for Ne to Na like ions and Na to Mg like ions are taken from Zatsarinny (2004b) and Gu (2004). The C07 DRRC for any given ion is usually substantially larger than the C84 DRRC when the temperature is much lower than the ionization potential of the ion. The significant cooling agents in C07 are among these ions. This indicates that the updated DRRC database in C07 is the cause of the changes in the stability curve.
It is seen from that the increase in recombination rates from C84 to C07 is larger for the lower ionization species. The reason for the large increase, in general, in the low-temperature DRRC (referenced above) is the explicit inclusion of the contribution from low-lying (in energy) level-resolved dielectronic recombination resonances which have been accurately positioned by reference to the observed core energies. Unlike high-temperature dielectronic recombination, where the full Rydberg series contributes, low-temperature dielectronic recombination is not amenable to simple scaling or empirical formulae or guesses. Zatsarinny et al (2003) illustrate the radical and erratic changes in the low-temperature DRRC on moving between adjacent ions of the same isoelectronic sequence and, even, on changing from a term-resolved picture to level-resolved one for the same ion.
The DRRC database is still not complete, specially for the lower ionization states. For ions which do not have computed DRRC values, C07 uses a solution, as suggested by Ali (1991): for any given kinetic temperature, ions that lack data are given DRRC values that are the averages of all ions with the same charge. The advantage of this method is that the assumed rates are within the range of existing published rates at the given kinetic temperature and hence cannot be drastically off. However we have checked whether the points in the C07 stability curves have computed DRRC values or guessed average values, concentrating on the parts of the curve which have multiphase solutions for the warm absorber ($10^5 \lesssim T \lesssim
10^6\kel$) and are different from the C84 stability curve. We find that all the ions which act as major cooling agents for each of these points in the stability curve have reliable computed DRRC values. Thus, the new data base provides a more robust measurement of the various physical parameters involved in studying the thermal and ionization equilibrium of photoionised gas.
Conclusion
==========
We have shown that stability curves for warm absorbers in AGN generated by two versions of the ionization code CLOUDY, C84 and C07, for the same physical conditions, are substantially different in shape, leading to different conclusions regarding the nature of the warm absorber. The differences in the results of the photoionisation calculations arise due to major changes in the dielectronic recombination rate coefficient (DRRC) data bases which have taken place over the last decade. The modern version C07 includes reliable computed DRRC values for many more ions than C84 for the entire part of the stability curve relevant for warm absorbers and does not rely on guessed average values. Thus, the physical nature of the warm absorber predicted by modern calculations are more reliable. We suggest that past calculations for photoionised plasma in thermal equilibrium should be reconsidered, taking into account these changes. Since, any other atomic physics code (eg. XSTAR), in popular use for warm absorber studies, is also likely to be affected by the changes in the DRRC data base, caution should be exercised while using or comparing results from them as well.
acknowledgements {#acknowledgements .unnumbered}
================
We thank Smita Mathur for reading an early version of the draft and providing helpful suggestions.
[99]{} Ali B., Blum R. D., Bumgardner T. E., Cranmer S. R., Ferland G. J., Haefner R. I., Tiede G. P., 1991, PASP, 103, 1182 Ali B., Blum R. D., Bumgardner T. E., Cranmer S. R., Ferland G. J., Haefner R. I., Tiede G. P., 1991, PASP, 103, 1182 Arnaud, M., Raymond, J., 1992, ApJ, 398, 394 Ashton C. E., Page M. J., Branduardi-Raymont G., Blustin A. J., 2006, MNRAS, 366, 521 Ashton C. E., Page M. J., Branduardi-Raymont G., Blustin A. J., 2006, MNRAS, 366, 521 Colgan, J., Pindzola, M.S. & Badnell, N.R. 2004, Astron. Astrophys. 417, 1183 Collinge, M. J., 2001, ApJ, 557,2 Ferland G. J., Korista K. T., Verner D. A., Ferguson J. W., Kingdon J. B., Verner E. M., 1998, PASP, 110, 761 Gehrels, N & Williams, E.D. 1993, ApJ 418, L25 Grevesse N., Anders E., 1989, in Waddington C. J., ed., AIP Conf. Proc. Vol. 183, Cosmic Abundances of Matter. Am. Inst. Phys., New York, p. 1 Grevesse N., Anders E., 1989, in Waddington C. J., ed., AIP Conf. Proc. Vol. 183, Cosmic Abundances of Matter. Am. Inst. Phys., New York, p. 1 Gu, M. F. 2003, ApJ, 590, 1131 Gu, M. F. 2004, ApJS, 153, 389 Kaastra J. S., Steenbrugge K. C., Raassen A. J. J., van der Meer R. L. J., Brinkman A. C., Liedahl D. A., Behar E., de Rosa A., 2002, A&A, 386, 427 Komossa & Mathur 2001, A&A, 374, 914 Krolik, J., & Kriss, G. A. ApJ 561:684, (2001) Krolik, J. H., McKee, C. F., & Tarter, C. B. 1981, ApJ, 249, 422 Krongold Y., Nicastro F., Brickhouse N. S., Elvis M., Liedahl D. A., Mathur S., 2003, ApJ, 597, 832 Krongold Y., Nicastro F., Brickhouse N. S., Elvis M., Mathur S., 2005, ApJ, 622, 842 Mitnik, D.M. & Badnell, N.R. 2004, Astron. Astrophys. 425, 1153 Morales, R., Fabian, A. C. & Reynolds, C. S. 2000, MNRAS, 315, 149 Netzer, H. 2003, ApJ, 599, 933 Nussbaumer, H., & Storey, P. J. 1983, A&A, 126, 75 Nussbaumer, H., & Storey, P. J. 1984, A&AS, 56, 293 Nussbaumer, H., & Storey, P. J. 1986, A&AS, 64, 545 Nussbaumer, H., & Storey, P. J. 1987, A&AS, 69, 123 Osterbrock, Ferland, 2006, in Osterbrock D. E., Ferland G. J., eds, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei 2nd edn. University Science Books, Sausalito CA p. 1 Reynolds, C.S. and Fabian, A.C. 1995, MNRAS 273:1167 Tarter, C.B., Tucker, W. & Salpeter, E.E. 1969, ApJ 156:943 Zatsarinny O., Gorczyca T. W., Korista K. T., Badnell N. R., Savin D. W., 2003, A&A, 412, 587 Zatsarinny O., Gorczyca T. W., Korista K. T., Badnell N. R., Savin D. W., 2004a, A&A, 417, 1173 Zatsarinny O., Gorczyca T. W., Korista K. T., Badnell N. R., Savin D. W., 2004b, A&A, 426, 699
[^1]: E-mail: susmita@iucaa.ernet.in (SC); akk@iucaa.ernet.in (AK); elvis@head.cfa.harvard.edu (MA); gary@pa.uky.edu (GF); badnell@phys.strath.ac.uk (NB)
[^2]: http://www.nublado.org/
|
---
abstract: 'In the last several years, remote sensing technology has opened up the possibility of performing large scale building detection from satellite imagery. Our work is some of the first to create population density maps from building detection on a large scale. The scale of our work on population density estimation via high resolution satellite images raises many issues, that we will address in this paper. The first was data acquisition. Labeling buildings from satellite images is a hard problem, one where we found our labelers to only be about 85% accurate at. There is a tradeoff of quantity vs. quality of labels, so we designed two separate policies for labels meant for training sets and those meant for test sets, since our requirements of the two set types are quite different. We also trained weakly supervised footprint detection models with the classification labels, and semi-supervised approaches with a small number of pixel-level labels, which are very expensive to procure.'
author:
- |
Amy Zhang[^1]\
Facebook\
`amyzhang@fb.com`\
Xianming Liu\
Facebook\
`xmliu@fb.com`\
Andreas Gros\
Facebook\
`andreasg@fb.com`\
Tobias Tiecke\
Facebook\
`ttiecke@fb.com`\
bibliography:
- 'egbib.bib'
title: Building Detection from Satellite Images on a Global Scale
---
Introduction
============
With the recent improvements in remote sensing technology, there has been a lot of work in building detection and classification from high resolution satellite imagery. However, we are the first to implement a system on a global scale. Other work uses handpicked features to define buildings [@DBLP:journals/corr/Cohen0KCD16] [@autorooftop] which would not scale well across countries with very different styles of buildings. The work closest to ours is done by Yuan [@DBLP:journals/corr/Yuan16], which also uses pixel level convolutional neural networks for building detection, but is only validated on a handful of cities in the US and likely would not transfer well to smaller settlements or other countries.
In order to speed up our pipeline we need a fast bounding box proposal algorithm to limit the number of images that need to be run through our convolutional neural network. To maintain high recall, however, we need to be careful to not filter out too many candidates. We used a naive bounding box proposal algorithm, by performing straight edge detection to extract smaller masks to run through our classification network. This reduced the amount of landmass to process by 50%. The distribution of buildings is still very negatively skewed, where only 2% of proposals are positive. This also means we need to sample a large number of masks in order to have confident precision and recall numbers by country. We also use a weak building classifier to filter masks with over 0.3 IoU (intersection over union) by choosing the mask with the highest probability of containing a building in the center, since these overlapping masks are likely to contain the same building.
Discovering systematic issues with our models is also a slow, manual problem that requires visualization of .kmz files, pinpointing large numbers of false positive or false negative areas, and debugging the causes. The problems encountered included noise, contrast issues, cloud cover, or just deficiencies in the model, and we set up a feedback loop to fix those problems.
We will be open sourcing our population density results as well as our labeled dataset as a benchmark for future efforts.
Dataset Collection Issues
=========================
We have two goals for data collection, obtaining labels for training, and accuracy numbers on a country level. Obtaining accuracy numbers of the entire pipeline for a single country requires randomly sampling from all possible 64x64 masks. That distribution is incredibly skewed, and randomly sampling enough masks to obtain a reasonable confidence interval on accuracy is expensive. Instead, we measure how well our neural network performs building classification by randomly sampling from the distribution of masks generated by our bounding box proposal algorithm. The assumption is that the bounding box proposal algorithm only eliminates clear negatives, so reduces skew on the underlying distribution without affecting recall of the overall pipeline. This drops the number of labels we need by a factor of 10, because our new distribution now is 2%-5% positive.
Collecting a training set went through several iterations because we want a more balanced dataset for training so the model can get enough samples of both the background and the building classes. We also employ simple active learning techniques by sampling from masks the network was “less sure” about, where the probability was closer to the threshold.
Generalizing a Global Model
===========================
Training a global building classification model has trade-offs. Buildings can look very different across different countries, but there is still a lot of information that can be transferred from country to country. We initially started with a model trained only on Tanzania, which when applied to a new country had a large drop in accuracy. However, we found that as we labeled data in more countries and re-trained our model with the new data, our new global model performed better on Tanzania than a Tanzania specific model. The generalizations learned from other countries made the model more robust.
Another argument for training a global model is that building a large training set takes time, and the amount of data required to train a model from scratch for each country was prohibitive.
The trade-off is that the global model doesn’t work equally well on all countries, and we found it necessary to perform some amount of model specialization. We fine-tuned the global model with the same samples it had seen from the initial training, but only from a handful of countries that we wanted it to improve upon. We saw gains of 20-40% in precision and recall on the validation set using the extra fine-tuning step, but noticed there were trade-offs. The training and validation sets gave no evidence of overfitting, but we saw an increase in systematic false positives when visualizing the results on a country level, in certain countries.
Building Classification Model
-----------------------------
The classification model we trained was a weakly supervised version of SegNet [@DBLP:journals/corr/BadrinarayananK15], which is a fast yet accurate pixel classification network that uses deconvolution layers. We trained with weak “pixel level” labels, and generate a mask level probability using global average pooling on the final pixel level probabilities over the 64x64 mask. We have 500TB of satellite imagery, and being able to run the model over all these countries (multiple times) is crucial for fast iteration. It was a non-trivial task to develop a model that was large enough to capture the complex idea of what defines a building, while also being small enough to run quickly during inference time. SegNet performed well on this by saving the indices from the max pooling layers to perform non-linear upsampling in the deconvolution layers.
Building Segmentation Model
---------------------------
![Semantic segmentation results using weakly-supervised model.[]{data-label="fig:segmentation"}](./footprints.png){width="50.00000%"}
Finely pixel-wise labeled data is extremely time consuming to acquire, and errors will accumulate especially for small foreground objects. Instead of utilizing fully supervised semantic segmentation method such as FCN [@long2015fully], we investigated weakly supervised segmentation models relying on feedback neural network [@cao2015look], which utilizes the large amount of “cheap” weakly-supervised training data. Notably, to increase the efficiency of semantic segmentation, the classification model is composed to filter out negative candidate regions. By combining results from both models, the segmentation model successfully suppress false positives and generate best results, with an example shown in Figure \[fig:segmentation\]
Dealing with Systematic Errors
==============================
Finding Systematic Errors
-------------------------
The precision and recall numbers we measure by randomly sampling from the mask candidates do not account for systematic errors arising from varying satellite image quality. To discover those systematic errors, we adopt both visually inspection and evaluation using external data.
Intuitively, we visualize our results by construction *KMZ* files and overlaying with Google Earth to manually pinpoint areas of concern. We also use this strategy to sample *ambiguous* training data to fine-tune our model to reduce the chance of further systematic errors. Moreover, we also quantitatively measure systematic errors at a coarser scale by comparing our results with external datasets on those areas with adequate data coverage. However, it is still an open question to discover systematic errors on large scale with less manual work.
Data Quality
------------
One of the reasons for systematic errors is also issues with data quality. The satellite images are taken at various times of day, and pre-processed across multiple layers for the highest quality image. However, areas with a lot of cloud cover tend to have much fewer clear images taken, and so quality suffers. This has an impact on our model, since most of the data is randomly or semi-randomly sampled, and so it does not get a lot of exposure to these poorer quality images during training. We use geographical meta-information to further detect the cloud occlusion during deploying stage.
Another key factor of low data quality comes from noise, which are introduced in either imaging or image enhancing phases. Traditional image denoising approach such as BM3D [@dabov2006image] is computationally expensive in handling large imagery files, and can only work for limited type of noises, such as white noise. To this end, we train a shallow neural network end-to-end by mimicking several kinds of noise existed in satellite images. The trained denoising model is appended as a transformer before imagery is fed to the classification network. Comparison of classification results of the same low data quality area before and after denoising is shown in Figure \[fig:denoising\].
Results
=======
Overall the SegNet model by itself achieves a precision and recall of $pr=0.9$, $re=0.89$ on a global dataset where the imbalance is such that $93\%$ of the randomly sampled testing data is not a building. Below we have some heat maps generated of building density in three countries: Mozambique, Madagascar, and India.
So far we have released datasets for 5 countries: Haiti, Malawi, Ghana, South Africa, and Sri Lanka. The rest are pending validation with third party groups. Below we show precision recall curves and best F-score with confidence intervals for each of the countries released.
The estimation of population density via settlement buildings as a proxy results in significant improvement compared with previous efforts. Figure \[fig:stateofart\] shows the comparison of previous highest resolution estimation from Galantis and our own results. This gives a totally new perspective to various social / economic research.
![Comparison of Galantis and our results[]{data-label="fig:stateofart"}](./stateofart.png){width="50.00000%"}
Conclusion
==========
We have built one of the first building detection systems that can be deployed at a global scale. Future work includes reducing the amount of iteration required to achieve a robust model as we roll out to more countries, the biggest problem of which is detecting systematic errors. Detecting and solving these systematic issues in classification is still a work in progress. We are still looking into ways to automate the data validation process and data collection methods further, which will also shorten the length of each iteration required to improve our dataset accuracy.
[^1]: Authors are of equal contribution
|
---
abstract: 'We treat the problem of characterizing the cyclic vectors in the weighted Dirichlet spaces, extending some of our earlier results in the classical Dirichlet space. The absence of a Carleson-type formula for weighted Dirichlet integrals necessitates the introduction of new techniques.'
address:
- |
Département de Mathématiques\
Université Mohamed V\
B.P. 1014 Rabat\
Morocco
- |
CMI\
LATP\
Université de Provence\
39 rue F. Joliot-Curie\
13453 Marseille\
Ê France
- |
Département de mathématiques et de statistique\
Université Laval\
Québec (QC)\
Canada G1V 0A6
author:
- 'O. El-Fallah$^1$'
- 'K. Kellay$^2$'
- 'T. Ransford$^3$'
title: Cantor sets and cyclicity in weighted Dirichlet spaces
---
[^1]
[^2]
[^3]
Introduction {#S:intro}
============
In this paper we study the weighted Dirichlet spaces ${{\mathcal{D}}}_\alpha~(0\le\alpha\le1)$, defined by $${{\mathcal{D}}}_\alpha:=
\Bigl\{f\in\text{hol}({\mathbb{D}}): {{\mathcal{D}}}_\alpha(f):=
\frac{1}{\pi} \int_{\mathbb{D}}|f'(z)|^2(1-|z|^2)^\alpha\,dA(z)<\infty\Bigr\}.$$ Here ${\mathbb{D}}$ denotes the open unit disk, and $dA$ is area measure on ${\mathbb{D}}$. Clearly ${{\mathcal{D}}}_\alpha$ is a Hilbert space with respect to the norm $\|\cdot\|_\alpha$ given by $$\|f\|_{\alpha}^2:= |f(0)|^2+{{\mathcal{D}}}_\alpha(f).$$ A classical calculation shows that, if $f(z)=\sum_{n\ge0}a_nz^n$, then $$\|f\|_\alpha^2 \asymp\sum_{n\ge0}(n+1)^{1-\alpha}|a_n|^2.$$ Note that ${{\mathcal{D}}}_1=H^2$ is the usual Hardy space, and ${{\mathcal{D}}}_0$ is the classical Dirichlet space (thus our labelling convention follows [@Al] rather than [@BS]).
An [*invariant subspace*]{} of ${{\mathcal{D}}}_\alpha$ is a closed subspace $M$ of ${{\mathcal{D}}}_\alpha$ such that $ zM\subset M$. Given $f\in {{\mathcal{D}}}_\alpha$, we denote by $[f]_{{{\mathcal{D}}}_\alpha}$ the smallest invariant subspace of ${{\mathcal{D}}}_\alpha$ containing $f$, namely the closure in ${{\mathcal{D}}}_\alpha$ of $\{pf: p \text{ a polynomial}\}$. We say that $f$ is [*cyclic*]{} for ${{\mathcal{D}}}_\alpha$ if $[f]_{{{\mathcal{D}}}_\alpha}={{\mathcal{D}}}_\alpha$. The survey article [@EKR2] gives a brief history of invariant subspaces and cyclic functions in the classical case $\alpha=0$.
Our goal is to characterize the cyclic functions of ${{\mathcal{D}}}_\alpha$. In order to state our results, we introduce the notion of $\alpha$-capacity. For $\alpha\in[0,1)$, we define the kernel function $k_{\alpha}:{\mathbb{R}}^+\to{\mathbb{R}}\cup\{\infty\}$ by $$k_{\alpha}(t):=
\begin{cases}
1/t^\alpha, & 0<\alpha<1,\\
\log(1/ t),& \alpha =0.
\end{cases}$$ The *$\alpha$-energy* of a (Borel) probability measure $\mu$ on ${\mathbb{T}}$ is defined by $$I_\alpha(\mu):=
\iint k_{\alpha}(|\zeta-\zeta'|)\,d\mu(\zeta)\,d\mu(\zeta').$$ A standard calculation gives $$I_\alpha(\mu) \asymp \sum _{n\ge 0}\frac{|{\widehat}{\mu}(n)|^2}{(1+n)^{1-\alpha}}.$$ The *$\alpha$-capacity* of a Borel subset $E$ of ${\mathbb{T}}$ is defined by $$C_{\alpha}(E):=1/\inf\{I_\alpha(\mu): \mathcal{P}(E)\},$$ where $\mathcal{P}(E)$ denotes the set of all probability measures supported on compact subsets of $E$. In particular, $C_\alpha(E)>0$ if and only if there exists a probability measure $\mu$ supported on a compact subset of $E$ and having finite $\alpha$-energy. If $\alpha=0$, then $C_0$ is the classical logarithmic capacity.
We recall a result due to Beurling and Salem–Zygmund [@Ca2 §V, Theorem 3] about radial limits of functions in the weighted Dirichlet spaces. If $f\in {{\mathcal{D}}}_\alpha$, then $f^*(\zeta):=\lim\limits_{r\to 1-}f(r\zeta) $ exists for all $\zeta\in{\mathbb{T}}$ outside a set of $\alpha$-capacity zero.
The following theorem gives two necessary conditions for cyclicity in ${{\mathcal{D}}}_\alpha$.
\[T:nec\] Let $\alpha\in[0,1)$. If $f$ is cyclic in ${{\mathcal{D}}}_\alpha$, then
- $f$ is an outer function,
- $\{\zeta\in{\mathbb{T}}:f^*(\zeta)=0\}$ is a set of $\alpha$-capacity zero.
The first part is [@BS Corollary 1]. For $\alpha=0$, the second part is [@BS Theorem 5], and for general $\alpha$ the proof is similar, the only difference being that the logarithmic kernel $k_0$ is replaced by $k_\alpha$. We omit the details.
Our main result is a partial converse to this theorem. To state it, we need to define the notion of a generalized Cantor set.
Let $(a_n)_{n\ge0}$ be a positive sequence such that $a_0\le2\pi$ and $$\sup_{n\ge0} \frac{a_{n+1}}{a_n}< \frac{1}{2}.$$ The *generalized Cantor set* $E$ associated to $(a_n)$ is constructed as follows. Start with a closed arc of length $a_0$ on the unit circle ${\mathbb{T}}$. Remove an open arc from the middle, to leave two closed arcs each of length $a_1$. Then remove two open arcs from their middles to leave four closed arcs each of length $a_2$. After $n$ steps, we obtain $E_n$, the union of $2^{n}$ closed arcs each of length $a_n$. Finally, the generalized Cantor set is $E:=\cap_n E_n$.
\[T:suff\] Let $\alpha\in[0,1)$ and let $f\in{{\mathcal{D}}}_\alpha$. Suppose that:
- $f$ is an outer function,
- $|f|$ extends continuously to $\overline{{\mathbb{D}}}$,
- $\{\zeta\in{\mathbb{T}}:|f(\zeta)|=0\}$ is contained in a generalized Cantor set $E$ of $\alpha$-capacity zero.
Then $f$ is cyclic for ${{\mathcal{D}}}_\alpha$.
Functions $f$ satisfying the hypotheses exist in abundance. Indeed, any generalized Cantor set is a so-called Carleson set, and is thus the zero set of some outer function $f$ such that $f$ and all its derivatives extend continuously to $\overline{{\mathbb{D}}}$. Moreover, it is very easy to determine which generalized Cantor sets have $\alpha$-capacity zero. More details will be given in §\[S:Cantor\].
To prove Theorem \[T:suff\], we adopt the following strategy. In §\[S:Korenblum\], using a technique due to Korenblum, we show that $[f]_{{{\mathcal{D}}}_\alpha}$ contains at least those functions $g\in{{\mathcal{D}}}_\alpha$ satisfying $|g(z)|\le \operatorname{dist}(z,E)^4$. The idea is then to take one simple such $g$, and gradually transform it into the constant function $1$ while staying inside $[f]_{{{\mathcal{D}}}_\alpha}$, thereby proving that $1\in[f]_{{{\mathcal{D}}}_\alpha}$. This requires three tools: a general estimate for weighted Dirichlet integrals of outer functions, some properties of generalized Cantor sets, and a regularization theorem. These tools are developed in §§\[S:Dint\],\[S:Cantor\],\[S:reg\] respectively, and all the pieces are finally assembled in §\[S:completion\], to complete the proof of Theorem \[T:suff\].
Theorem \[T:suff\] was established for the classical Dirichlet space, $\alpha = 0$, in [@EKR1 Corollary 1.2]. The proof there followed the same general strategy, but in several places key use was made of a formula of Carleson [@Ca1] expressing the Dirichlet integral of an outer function $f$ in terms of the values of $|f^*|$ on the unit circle. No analogue of Carleson’s formula is known in the case $0<\alpha<1$, and one of the main points of this note is to show how this difficulty may be overcome.
Throughout the paper, we use the notation $C(x_1,\dots,x_n)$ to denote a constant that depends only on $x_1,\dots,x_n$, where the $x_j$ may be numbers, functions or sets. The constant may change from one line to the next.
Korenblum’s method {#S:Korenblum}
==================
Our aim in this section is to prove the following theorem.
\[T:Korenblum\] Let $f\in {{{\mathcal{D}}}_\alpha}$ be an outer function such that $|f|$ extends continuously to $\overline{{\mathbb{D}}}$, and let $F:=\{\zeta\in {\mathbb{T}}: |f(\zeta)|=0\}$. If $g\in {{{\mathcal{D}}}_\alpha}$ and $$|g(z)|\le \operatorname{dist}(z,F)^4 \quad (z\in{\mathbb{D}}),$$ then $g\in [f]_{{{\mathcal{D}}}_\alpha}$.
This theorem is a ${{\mathcal{D}}}_\alpha$-analogue of [@EKR1 Theorem 3.1], which was proved using a technique of Korenblum. We shall use the same basic technique here. However, the proof in [@EKR1] proceeded via a so-called fusion lemma, which, being based on Carleson’s formula for the Dirichlet integral, is no longer available to us here. Its place is taken by Corollary \[C:Korenblum\] below.
We need to introduce some notation. Given an outer function $f$ and a Borel subset $\Gamma$ of ${\mathbb{T}}$, we define $$f_\Gamma (z):=\exp\Bigl( \frac{1}{2\pi}\int_{\Gamma}
\frac{\zeta+z}{\zeta-z}\log |f^*(\zeta)|\,|d\zeta|\Bigr) \quad(z\in{\mathbb{D}}).$$ We write $\partial\Gamma$ and $\Gamma^c$ for the boundary and complement of $\Gamma$ in ${\mathbb{T}}$ respectively.
\[L:Korenblum\] Let $f$ be a bounded outer function. For every Borel set $\Gamma\subset{\mathbb{T}}$, $$|f_\Gamma'(z)|\le C(f)(|f'(z)|+\operatorname{dist}(z,\partial\Gamma)^{-4}) \quad(z\in{\mathbb{D}}).$$
Without loss of generality, we may suppose that $\|f\|_\infty\le1$. Note that then $\|f_\Gamma\|_\infty\le1$ for all $\Gamma$. Also, obviously, $\log|f^*|\le0$ a.e. on ${\mathbb{T}}$, which will help simplify some of the calculations below.
We begin by observing that $$\frac{f_\Gamma'(z)}{f_\Gamma(z)}
=\frac{1}{2\pi}\int_\Gamma \frac{2\zeta}{(\zeta-z)^2}\log|f^*(\zeta)|\,|d\zeta|
\quad(z\in{\mathbb{D}}),$$ from which it follows easily that $$\label{E:Gamma}
|f_\Gamma'(z)|\le \frac{2\log(1/|f(0)|)}{\operatorname{dist}(z,\Gamma)^2} \quad(z\in{\mathbb{D}}).$$
Our aim now is to prove a similar inequality, but with $\partial\Gamma$ in place of $\Gamma$. Set $G:=\{z\in{\mathbb{D}}:\operatorname{dist}(z,\Gamma)\ge\operatorname{dist}(z,\Gamma^c)^2\}$. Clearly $\operatorname{dist}(z,\Gamma)\ge\operatorname{dist}(z,\partial\Gamma)^2$ for all $z\in G$, so implies $$\label{E:zinG}
|f_\Gamma'(z)|\le \frac{2\log(1/|f(0)|)}{\operatorname{dist}(z,\partial\Gamma)^4} \quad(z\in G).$$ Now suppose that $z\in{\mathbb{D}}\setminus G$. Then $\operatorname{dist}(z,\Gamma^c)^2>\operatorname{dist}(z,\Gamma)\ge (1-|z|^2)/2$, and hence $$|f_{\Gamma^c}(z)|
=\exp\Bigl(\frac{1}{2\pi}\int_{\Gamma^c}\frac{1-|z|^2}{|\zeta-z|^2}\log|f^*(\zeta)|\,|d\zeta|\Bigr)
\ge|f(0)|^2.$$ Since obviously $f_\Gamma=f/f_{\Gamma^c}$, it follows that, for all $z\in{\mathbb{D}}\setminus G$, $$|f_\Gamma'(z)|
\le \frac{|f'(z)|}{|f_{\Gamma^c}(z)|}+\frac{|f(z)|}{|f_{\Gamma^c}(z)|^2}|f_{\Gamma^c}'(z)|
\le \frac{|f'(z)|}{|f(0)|^2}+\frac{1}{|f(0)|^4}\frac{2\log(1/|f(0)|)}{\operatorname{dist}(z,\Gamma^c)^2},$$ where once again we have used , this time with $\Gamma$ replaced by $\Gamma^c$. Noting that $\operatorname{dist}(z,\Gamma^c)\ge \operatorname{dist}(z,\partial\Gamma)$ for all $z\in{\mathbb{D}}\setminus G$, we deduce that $$\label{E:znotinG}
|f_\Gamma'(z)|\le \frac{|f'(z)|}{|f(0)|^2}+\frac{1}{|f(0)|^4}\frac{2\log(1/|f(0)|)}{\operatorname{dist}(z,\partial\Gamma)^2} \quad(z\in{\mathbb{D}}\setminus G).$$ The inequalities and between them give the result.
\[C:Korenblum\] Let $\alpha\in[0,1)$ and $f\in{{\mathcal{D}}}_\alpha\cap H^\infty$ be an outer function. Then, for every Borel set $\Gamma\subset{\mathbb{T}}$ and every $g\in{{\mathcal{D}}}_\alpha$ satisfying $|g(z)|\le\operatorname{dist}(z,\partial\Gamma)^4$, we have $$\|f_\Gamma g\|_\alpha\le C(\alpha,f)(1+\|g\|_\alpha).$$
Using Lemma \[L:Korenblum\], we have $$|(f_\Gamma g)'|\le |f_\Gamma'||g|+|f_\Gamma||g'|\le C(f)(|f'|+1+|g'|).$$ The conclusion follows easily from this.
Let $I$ be a connected component of ${\mathbb{T}}\setminus F$, say $I= (e^{ia},e^{ib})$. Let $\rho>1$, and define $$\begin{aligned}
\psi_\rho(z)&:=(z-1)^4/(z-\rho)^4,\\
\phi_\rho(z)&:= \psi_\rho(e^{-ia}z)\psi_\rho(e^{-ib}z).\end{aligned}$$ The first step is to show that $\phi_\rho f_{{\mathbb{T}}\setminus I}\in[f]_{{{\mathcal{D}}}_\alpha}$.
Let $\epsilon >0$ and set $I_\epsilon:=(e^{i(a+\epsilon)},e^{i(b-\epsilon)})$ and $$\phi_{\rho,\epsilon}(z):=\psi_\rho(e^{-i(a+\epsilon)}z)\psi_\rho(e^{-i(b-\epsilon)}z).$$ By Corollary \[C:Korenblum\], $$\label{E:phieps}
\| \phi_{\rho,\epsilon} f_{{\mathbb{T}}\setminus I_\epsilon}\|_\alpha
\le C(f) \|\phi_{\rho,\epsilon}\|_\alpha \le C(f,\rho).$$ Note that $|f_{{\mathbb{T}}\setminus I_\epsilon}|= |f|$ in a neighborhood of ${\mathbb{T}}\setminus I$. Since $|f|$ does not vanish inside $I$, it follows that $|\phi_{\rho,\epsilon}f_{{\mathbb{T}}\setminus I_\epsilon}|/|f|$ is bounded on ${\mathbb{T}}$. Also $f$ is an outer function. Therefore, by a theorem of Aleman [@Al Lemma 3.1], $$\phi_{\rho ,\epsilon}f_{{\mathbb{T}}\setminus I_\epsilon} \in [f]_{{{\mathcal{D}}}_\alpha}.$$ Using , we see that $\phi_{\rho,\epsilon} f_{{\mathbb{T}}\setminus I_\epsilon}$ converges weakly in ${{\mathcal{D}}}_\alpha$ to $\phi_\rho f_{{\mathbb{T}}\setminus I}$ as $\epsilon\to 0$. Hence $\phi_\rho f_{{\mathbb{T}}\setminus I} \in [f]_{{{\mathcal{D}}}_\alpha}$, as claimed.
Next, we multiply by $g$. As $g\in{{\mathcal{D}}}_\alpha\cap H^\infty$, Aleman’s theorem immediately yields $\phi_\rho f_{{\mathbb{T}}\setminus I} g\in [f]_{{{\mathcal{D}}}_\alpha}$. Using the fact that $|g(z)|\le \operatorname{dist}(z,F)^4$, it is easy to check that $\|\phi_{\rho}g\| _{\alpha}$ remains bounded as $\rho\to1$. By Corollary \[C:Korenblum\] again, $\|\phi_\rho f_{{\mathbb{T}}\setminus I}g\|_{\alpha}$ is uniformly bounded, and $\phi_\rho f_{{\mathbb{T}}\setminus I}g$ converges weakly to $f_{{\mathbb{T}}\setminus I}g$. Hence $f_{{\mathbb{T}}\setminus I}g\in [f]_{{{\mathcal{D}}}_\alpha}$.
Now let $(I_j)_{j\ge1}$ be the complete set of components of ${\mathbb{T}}\setminus F$, and set $J_n:=\cup_1^nI_j$. An argument similar to that above gives $f_{{\mathbb{T}}\setminus J_n}g\in[f]_{{{\mathcal{D}}}_\alpha}$ for all $n$. Moreover, $\|f_{{\mathbb{T}}\setminus J_n}g\|_{\alpha}$ is uniformly bounded. Thus $f_{{\mathbb{T}}\setminus J_n}g$ converges weakly to $g$, and so finally $g\in[f]_{{{\mathcal{D}}}_\alpha}$.
Estimates for weighted Dirichlet integrals {#S:Dint}
==========================================
The following result will act as a partial substitute for Carleson’s formula.
\[T:Carleson\] Let $\alpha\in[0,1)$, and let $h:{\mathbb{T}}\to{\mathbb{R}}$ be a positive measurable function such that, for every arc $I\subset{\mathbb{T}}$, $$\label{E:hjensen}
\frac{1}{|I|}\int_I h(\zeta)\,|d\zeta|
\ge |I|^{\alpha}.$$ If $f$ is an outer function, then $${{\mathcal{D}}}_\alpha(f)\le
\frac{1}{\pi}\iint_{{\mathbb{T}}^2}
\frac{(|f^*(\zeta)|^2-|f^*(\zeta')|^2)(\log|f^*(\zeta)|-\log|f^*(\zeta')|)}
{|\zeta-\zeta'|^{2}} (h(\zeta)+ h(\zeta'))|d\zeta||d\zeta'|.$$
Given $z\in{\mathbb{D}}$, let $I_z$ be the arc of ${\mathbb{T}}$ with midpoint $z/|z|$ and arclength $|I_z|=2(1-|z|^2)$. For $\zeta\in I_z$, we have $|\zeta-z|\le 2(1-|z|^2)$, and so $$P(z,\zeta):=\frac{1-|z|^2}{|\zeta-z|^2}\ge \frac{1}{4(1-|z|^2)}=\frac{1}{2|I_z|}.$$ Hence, using , we have $$\int_{\mathbb{T}}P(z,\zeta) h(\zeta)\, |d\zeta|
\ge \frac{1}{2|I_z|}\int_{I_z} h (\zeta)\,|d\zeta|
\ge\frac{1}{2}{|I_z|^\alpha}
\ge\frac{1}{2}(1-|z|^2)^\alpha.$$ Therefore, by Fubini’s theorem, $${{\mathcal{D}}}_\alpha(f)
:=\frac{1}{\pi}\int_{\mathbb{D}}|f'(z)|^2(1-|z|^2)^\alpha\,dA(z)
\le 2\int_{\mathbb{T}}{{\mathcal{D}}}_\zeta(f)h(\zeta)\,|d\zeta|,$$ where $${{\mathcal{D}}}_\zeta(f):=\frac{1}{\pi}\int_{\mathbb{D}}|f'(z)|^2P(z,\zeta)\,dA(z) \quad(\zeta\in{\mathbb{T}}).$$ Now ${{\mathcal{D}}}_\zeta(f)$ is the so-called local Dirichlet of integral of $f$ at $\zeta$, which was studied in detail by Richter and Sundberg in [@RS]. In particular, they showed that, if $f$ is an outer function, then $${{\mathcal{D}}}_\zeta(f)
=\frac{1}{2\pi}\int_{{\mathbb{T}}}\frac{|f^*(\zeta)|^2-|f^*(\zeta')|^2 -2|f^*(\zeta')|^2\log|f^*(\zeta)/f^*(\zeta')|}{|\zeta-\zeta'|^{2}}\,|d\zeta'|.$$ Substituting this into the preceding estimate for ${{\mathcal{D}}}_\alpha(f)$, and noting the obvious fact that $h(\zeta)\le h(\zeta)+h(\zeta')$, we deduce that ${{\mathcal{D}}}_\alpha(f)$ is majorized by $$\frac{1}{\pi}\iint_{{\mathbb{T}}}\frac{|f^*(\zeta)|^2-|f^*(\zeta')|^2 -2|f^*(\zeta')|^2\log|f^*(\zeta)/f^*(\zeta')|}{|\zeta-\zeta'|^{2}}(h(\zeta)+h(\zeta'))\,|d\zeta'|\,|d\zeta|.$$ Exchanging the roles of $\zeta$ and $\zeta'$, we see that ${{\mathcal{D}}}_\alpha(f)$ is likewise majorized by $$\frac{1}{\pi}\iint_{{\mathbb{T}}}\frac{|f^*(\zeta')|^2-|f^*(\zeta)|^2 -2|f^*(\zeta)|^2\log|f^*(\zeta')/f^*(\zeta)|}{|\zeta'-\zeta|^{2}}(h(\zeta')+h(\zeta))\,|d\zeta|\,|d\zeta'|.$$ Taking the average of these last two estimates, we obtain the inequality in the statement of the theorem.
We are going to apply this result with $h(\zeta):=Cd(\zeta,E)^\alpha$, where $C$ is a constant, $d$ denotes arclength distance on ${\mathbb{T}}$, and $E$ is a closed subset of ${\mathbb{T}}$. Condition thus becomes $$\label{E:Kset}
\frac{1}{|I|}\int_I d(\zeta,E)^\alpha\,|d\zeta|\ge C^{-1}|I|^\alpha
\quad\text{for all arcs~}I\subset{\mathbb{T}}.$$ A set $E$ which satisfies this condition for some $\alpha,C$ is called a *K-set* (after Kotochigov). K-sets arise as the interpolation sets for certain function spaces, and have several other interesting properties. We refer to [@Br §1] and [@Dy §3] for more details. In particular, if $E$ satisfies , then it has measure zero and $\log d(\zeta,E)\in L^1({\mathbb{T}})$.
\[T:fw\] Let $\alpha\in(0,1)$, let $E$ be a closed subset of ${\mathbb{T}}$ satisfying , and let $w:[0,2\pi]\to {\mathbb{R}}^+ $ be an increasing function such that $t\mapsto \omega(t^\gamma)$ is concave for some $\gamma>2/(1-\alpha)$. Let $f_w$ be the outer function satisfying $$|f_w^*(\zeta)|=w(d (\zeta, E))\qquad \text{a.e on } {\mathbb{T}}.$$ Then $$\label{E:fw}
{{\mathcal{D}}}_\alpha(f_w)\le
C(\alpha, \gamma,E) \int_{{\mathbb{T}}}w'(d(\zeta,E))^2 d(\zeta,E)^{1+\alpha}\,|d\zeta|.$$ In particular $f_w\in {{\mathcal{D}}}_\alpha$ if the last integral is finite.
The proof is largely similar to that of [@EKR1 Theorem 4.1], so we give just a sketch, concentrating on those parts where the two proofs differ.
We begin by remarking that the concavity condition on $w$ easily implies that $|\log w(d(\zeta,E))|\le C(w)|\log d(\zeta,E)|$, so $\log w(d(\zeta,E))\in L^1({\mathbb{T}})$ and the definition of $f_w$ makes sense.
By Theorem \[T:Carleson\], we have $${{\mathcal{D}}}_\alpha(f_w)\le C(\alpha,E)\iint_{{\mathbb{T}}^2}
\frac{(w^2(\delta)-w^2(\delta'))(\log w(\delta)-\log w(\delta'))}{|\zeta-\zeta'|^2}
(\delta^\alpha+ \delta'^\alpha )\,|d\zeta|\,|d\zeta'|,$$ where we have written $\delta:=d(\zeta,E)$ and $\delta':=d(\zeta',E)$.
Let $(I_j)$ be the connected components of ${\mathbb{T}}\setminus E$, and set $$N_E(t):=2\sum_j 1_{\{|Ij|>2t\}} \quad(0<t\le\pi).$$ Then, for every measurable function $\Omega:[0,\pi]\to{\mathbb{R}}^+$, we have $$\int_{\mathbb{T}}\Omega(d(\zeta,E))\,|d\zeta|=\int_0^\pi \Omega(t)N_E(t)\,dt.$$ In particular, as in [@EKR1], it follows that $$\begin{aligned}
&\iint_{{\mathbb{T}}^2}\frac{(w^2(\delta)-w^2(\delta'))(\log w(\delta)-\log w(\delta'))}{|\zeta-\zeta'|^2}
(\delta^\alpha+ \delta'^\alpha )\,|d\zeta|\,|d\zeta'|\\
&\le C(\alpha)\int_0^\pi\int_0^\pi
\frac{(w^2(s+t)-w^2(t))(\log w(s+t)-\log w(t))}{s^2}
(s+t )^\alpha N_E(t)\,ds\,dt.\end{aligned}$$ The concavity assumption on $w$ implies that $t\to t^{1-1/\gamma}w'(t)$ is decreasing, and thus, as in [@EKR1], $$\begin{aligned}
w^2(t+s)-w^2(t)\le 2\gamma w(t+s)w'(t)t\bigl((1+s/t)^{1/\gamma}-1\bigr),\\
\log w(t+s)-\log w(t)\le tw'(t)\frac{(1+s/t)^{1/\gamma}}{w(t+s)}\log(1+s/t).\end{aligned}$$ Combining these estimates, we obtain $$\begin{aligned}
&\int_0^\pi \int_0^\pi\frac{(w^2(t+s)-w^2(t))(\log w(t+s)-\log w(t))}{s^2}(s+t)^\alpha \,ds N_E(t)\,dt\\
&\le \int_0^\pi \int_0^\pi 2\gamma w'(t)^2t^{2+\alpha}\bigl((1+s/t)^{1/\gamma}-1\bigr)(1+s/t)^{1/\gamma+\alpha}\log(1+s/t)\,\frac{ds}{s^2}N_E(t)\,dt\\
&= \int_0^\pi 2\gamma w'(t)^2t^{1+\alpha}\bigl( \int_0^{\pi/t} 2\gamma \bigl((1+x)^{1/\gamma}-1\bigr)(1+x)^{1/\gamma+\alpha}\log (1+x)\frac{dx}{x^2}\bigr)N_E(t)\,dt\\
&\le C(\alpha,\gamma) \int_0^\pi w'(t)^2t^{1+\alpha}N_E(t)\,dt.\end{aligned}$$ In the last inequality we used the fact that $\gamma>2/(1-\alpha)$.
Generalized Cantor sets {#S:Cantor}
=======================
The notion of the generalized Cantor set $E$ associated to a sequence $(a_n)$ was defined in §\[S:intro\]. In this section we briefly describe some pertinent properties of these sets. We shall write $$\lambda_E:=\sup_{n\ge0}\frac{a_{n+1}}{a_n}.$$ Recall that, by hypothesis, $\lambda_E<1/2$.
Our first result shows that generalized Cantor sets satisfy , and hence that Theorem \[T:fw\] is applicable to such sets.
Let $E$ be a generalized Cantor set and let $\alpha\in[0,1)$. Then, for each arc $I\subset{\mathbb{T}}$, $$\frac{1}{|I|}\int_I d(\zeta,E)^\alpha\,|d\zeta|\ge C(\alpha,\lambda_E)|I|^\alpha.$$
Let $I$ be an arc with $|I|\le 2a_0$, and choose $n$ so that $2a_n< |I|\le 2a_{n-1}$. Recall that the $n$-th approximation to $E$ consists of $2^n$ arcs, each of length $a_n$, and that the distance between these arcs is at least $a_{n-1}-2a_n$. If $I$ meets at least two of these arcs, then $I\setminus E$ contains an arc $J$ of length $a_{n-1}-2a_n$, and if $I$ meets at most one of these arcs, then $I\setminus E$ contains an arc $J$ of length $(|I|-a_n)/2$. Thus $I\setminus E$ always contains an arc $J$ such that $|J|/|I|\ge \min\{1/2-\lambda_E,\,1/4\}$. Consequently $$\frac{1}{|I|}\int_Id(\zeta,E)^\alpha\,|d\zeta|
\ge\frac{1}{|I|}\int_Jd(\zeta,E)^\alpha\,|d\zeta|
\ge\frac{1}{|I|}\frac{|J|^{\alpha+1}}{\alpha+1}
\ge C(\alpha,\lambda_E)|I|^\alpha.$$
In the next two results, we write $E_t:=\{\zeta\in{\mathbb{T}}:d(\zeta,E)\le t\}$.
\[P:|Et|\] If $E$ is a generalized Cantor set, then $|E_t|=O(t^\mu)$ as $t\to0$, where $\mu:=1-\log2/\log(1/\lambda_E)$.
Given $t\in(0,a_0]$, choose $n$ so that $a_n<t\le a_{n-1}$. Then clearly $|E_t|\le 2^n(a_n+2t)\le 3.2^n t$. Also $2^{n-1}=(1/\lambda_E^{n-1})^{\log2/\log(1/\lambda_E)}\le
(a_0/t)^{\log2/\log(1/\lambda_E)}$. Hence $|E_t|\le C(a_0,\lambda_E)t^{1-\log2/\log(1/\lambda_E)}$.
In particular, every generalized Cantor set $E$ is a *Carleson set*, that is, $\int_0^\pi( |E_t|/t)\,dt<\infty$. Taylor and Williams [@TW] showed that Carleson sets are zero sets of outer functions in $A^\infty({\mathbb{D}})$. This justifies a remark made in §\[S:intro\].
The final property that we need concerns the $\alpha$-capacity, $C_\alpha$, which was defined in §\[S:intro\].
\[T:capmeas\] Let $E$ be a generalized Cantor set and let $\alpha\in[0,1)$. Then $$C_\alpha(E)=0 \iff \int_0^\pi \frac{dt}{t^\alpha |E_t|}=\infty.$$
This follows easily from [@Ca2 §IV, Theorems 2 and 3].
Regularization {#S:reg}
==============
We shall need the following regularization result. The proof is the same, with minor modifications, as that of [@EKR1 Theorem 5.1].
\[T:reg\] Let $\alpha\in[0,1)$, let $\sigma\in(0,1)$ and let $a>0$. Let $\phi:(0,a]\to {\mathbb{R}}^+$ be a function such that
- $\phi(t)/t$ is decreasing,
- $0<\phi(t)\le t^\sigma$ for all $t\in (0,a]$,
- $\displaystyle \int_0^a \frac{dt}{t^\alpha\phi(t)}=\infty$.
Then, given $\rho\in(0,\sigma)$, there exists a function $\psi:(0,a]\to {\mathbb{R}}^+ $ such that
- $\psi(t)/t^\rho$ is increasing,
- $\phi(t)\le \psi(t)\le t^\sigma$ for all $t\in (0,a]$,
- $\displaystyle\int_{0}^{a} \frac{dt}{t^\alpha \psi(t)}=\infty$.
Completion of the proof of Theorem \[T:suff\] {#S:completion}
=============================================
For $\alpha=0$, this theorem was proved in [@EKR1 §6]. The proof for $\alpha\in(0,1)$ will follow the same general lines, and once again we shall concentrate mainly on the places where the proofs differ.
Let $f$ be the function in the theorem. Our goal is to show that $1\in[f]_{{{\mathcal{D}}}_\alpha}$.
Let $E$ be as in the theorem, and let $g$ be the outer function such that $$|g^*(\zeta)|=d(\zeta,E)^4 \quad \text{a.e.\ on~}{\mathbb{T}}.$$ Then $|g(z)|\le (\pi/2)^4\operatorname{dist}(z,E)^4~(z\in{\mathbb{D}})$: indeed, for every $\zeta_0\in E$, we have $$\log|g(z)|
\le \frac{1}{2\pi}\int_{\mathbb{T}}\frac{1-|z|^2}{|z-\zeta|^2}4\log\bigl((\pi/2)|\zeta-\zeta_0|\bigr)\,|d\zeta|
=4\log\bigl((\pi/2)|z-\zeta_0|\bigr).$$ By assumption, the zero set $F:=\{\zeta\in{\mathbb{T}}:|f(\zeta)|=0\}$ in contained in $E$, so $|g(z)|\le (\pi/2)^4\operatorname{dist}(z,F)^4~(z\in{\mathbb{D}})$. Theorem \[T:Korenblum\] therefore applies, and we can infer that $g\in[f]_{{{\mathcal{D}}}_\alpha}$. It thus suffices to prove that $1\in [g]_{{{\mathcal{D}}}_\alpha}$.
We shall construct a family of functions $w_\delta:[0,\pi]\to{\mathbb{R}}^+$ for $0<\delta<1$ such that the associated outer functions $f_{w_\delta}$ belong to $[g]_{{{\mathcal{D}}}_\alpha}$ and satisfy:
- $|f_{w_\delta}^*|\to 1$ a.e. on ${\mathbb{T}}$ as $\delta\to0$,
- $|f_{w_\delta}(0)|\to 1$ as $\delta\to0$,
- $\liminf_{\delta\to0}\|f_{w_\delta}\|_\alpha<\infty$.
If such a family exists, then a subsequence of the $f_{w_\delta}$ converges weakly to $1$ in ${{\mathcal{D}}}_\alpha$, and since they all belong to $[g]_{{{\mathcal{D}}}_\alpha}$, it follows that $1\in [g]_{{{\mathcal{D}}}_\alpha}$, as desired.
By Proposition \[P:|Et|\], there exists $\mu>0$ such that $|E_t|=O(t^\mu)$ as $t\to0$. Fix $\rho,\sigma$ satisfying $$\frac{1-\alpha}{2}<\rho<\sigma<\min\Bigl\{1-\alpha,~\frac{1-\alpha+\mu}{2}\Bigr\}.$$ Define $\phi:(0,\pi]\to{\mathbb{R}}^+$ by $$\phi(t):=\max\Bigl\{\min\{|E_t|,~t^\sigma\},~t^{1-\alpha}\Bigr\} \quad(t\in(0,\pi]).$$ Clearly $\phi(t)/t$ is increasing and $0\le \phi(t)\le t^\sigma$ for all $t$. We claim also that $$\label{E:claim}
\int_0^\pi \frac{dt}{t^\alpha \phi(t)}=\infty.$$ To see this, note that $$\int_t^\pi \frac{ds}{s^\alpha|E_s|}\ge
\frac{t}{|E_t|}\int_t^\pi\frac{ds}{s^{\alpha+1}}
=C(\alpha)\frac{t^{1-\alpha}}{|E_t|},$$ whence $$\begin{aligned}
\int_\epsilon^\pi \frac{dt}{t^\alpha\phi(t)}
&\ge \int_\epsilon^\pi \frac{dt}{\max\{t,t^\alpha|E_t|\}}\\
&\ge C(\alpha)\int_\epsilon^\pi \frac{ds}{t^\alpha|E_t|(\int_t^\pi ds/s^\alpha|E_s|)}\\
&\ge C(\alpha)\log \int_\epsilon^\pi \frac{dt}{t^\alpha|E_t|}.\end{aligned}$$ Since $E$ is a generalized Cantor set of $\alpha$-capacity zero, Theorem \[T:capmeas\] shows that $$\int_0^\pi \frac{dt}{t^\alpha|E_t|}=\infty.$$ Consequently holds, as claimed.
We have now shown that $\phi$ satisfies all the hypotheses of the regularization theorem, Theorem \[T:reg\]. Therefore there exists a function $\psi:(0,\pi]\to{\mathbb{R}}^+$ satisfying the conclusions of that theorem, namely: $\psi(t)/t^\rho$ is increasing, $t^{1-\alpha}\le\phi(t)\le \psi(t)\le t^\sigma$ for all $t$, and $\int_0^\pi dt/t^\alpha\psi(t)=\infty$.
For $0<\delta<1$, we define $w_\delta:[0,\pi]\to{\mathbb{R}}^+$ by $$w_\delta(t):=
\begin{cases}
\displaystyle \frac{\delta^\rho}{\psi(\delta)}t^{1-\alpha-\rho} &0\le t\le\delta\\
\displaystyle A_\delta-\log\int_t^\pi\frac{ds}{s^\alpha\psi(s)} &\delta<t\le\eta_\delta\\
\displaystyle1 &\eta_\delta<t\le\pi,
\end{cases}$$ where $A_\delta$ and $\eta_\delta$ are constants chosen to make $w_\delta$ continuous.
Let us show that $f_{w_\delta}\in[g]_{{{\mathcal{D}}}_\alpha}$. Note first that $w_\delta(t)/t^{1-\alpha-\rho}$ is a bounded function. Therefore $f_{w_\delta}/g^{(1-\alpha-\rho)/4}$ is bounded. Using Theorem \[T:fw\], we have $g^{(1-\alpha-\rho)/4}\in{{\mathcal{D}}}_\alpha$. Consequently, by a theorem of Aleman [@Al Lemma 3.1], $f_{w_\delta}\in[g^{(1-\alpha-\rho)/4}]_{{{\mathcal{D}}}_\alpha}$. Using another result of Aleman [@Al Theorem 2.1], we have $g^{(1-\alpha-\rho)/4}\in[g]_{{{\mathcal{D}}}_\alpha}$. Hence $f_{w_\delta}\in[g]_{{{\mathcal{D}}}_\alpha}$, as claimed.
It remains to check that the functions $f_{w_\delta}$ satisfy properties (i)–(iii) above. The verifications run along the same lines as those in [@EKR1 §6], using the properties of $\psi$ above, and Theorem \[T:fw\] in place of [@EKR1 Theorem 4.1].
[00]{}
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O. El-Fallah, K. Kellay, T. Ransford. On the Brown–Shields conjecture for cyclicity in the Dirichlet space, *Adv. Math.* 222 (2009), 2196–2214.
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[^1]: [1. Research partially supported by a grant from Egide Volubilis (MA09209)]{}
[^2]: [2. Research partially supported by grants from Egide Volubilis (MA09209) and ANR Dynop]{}
[^3]: [3. Research partially supported by grants from NSERC (Canada), FQRNT (Québec) and the Canada Research Chairs program]{}
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---
abstract: 'We describe the set of bounded harmonic functions for the Heckman–Opdam Laplacian, when the multiplicity function is larger than $1/2$. We prove that this set is a vector space of dimension the cardinality of the Weyl group. We give some consequences in terms of the associated hypergeometric functions.'
address: 'Département de Mathématiques, Bât. 425, Université Paris-Sud 11, F-91405 Orsay, cedex, France. '
author:
- Bruno Schapira
title: 'Bounded harmonic functions for the Heckman–Opdam Laplacian'
---
Introduction
============
In this paper we will consider the operator ${\mathcal{L}}$ (called here Heckman–Opdam Laplacian) on ${\mathbb{R}}^n$ defined, for $f$ a $C^2$ function, by $$\begin{aligned}
\label{explicitlaplacian} {\mathcal{L}}f(x) &=& \Delta f(x)+ \sum_{\alpha
\in {\mathcal{R}}^+}k_\alpha \coth \frac{{\langle}\alpha,x{\rangle}}{2}\partial_\alpha f(x) \\
&\quad& \nonumber - \sum_{\alpha \in {\mathcal{R}}^+}k_\alpha
\frac{|\alpha|^2}{4\sinh^2 \frac{{\langle}\alpha,x{\rangle}}{2}} \{f(x)-f(r_\alpha
x)\}.\end{aligned}$$ Here $\Delta$ is the usual Euclidean laplacian, ${\mathcal{R}}$ is a root system, ${\mathcal{R}}^+$ its positive part, the $r_\alpha $’s are the orthogonal reflexions associated to the roots and $k$ is a positive function invariant under the action of the $r_\alpha$’s (see the next section). We denote by $W$ the Weyl group, i.e. the finite group generated by the $r_\alpha$’s. We denote by $L$ the restriction of ${\mathcal{L}}$ to the set of $W$-invariant functions. A simpler formula for $L$ is given by $$\begin{aligned}
\label{radiallaplacian2} L f(x) = \Delta f(x)+ \sum_{\alpha
\in {\mathcal{R}}^+}k_\alpha \coth \frac{{\langle}\alpha,x{\rangle}}{2}\partial_\alpha f(x).\end{aligned}$$ Our main results are the two following:
\[theoradial\] Assume that $k\ge 1/2$. Then the set of bounded $W$-invariant harmonic functions for the Heckman–Opdam Laplacian is exactly the set of constant functions. In other words the Poisson boundary of $L$ is trivial.
\[theononradial\] Assume that $k\ge 1/2$. Then the set of bounded harmonic functions for the Heckman–Opdam Laplacian is a vector space of dimension $|W|$. In other words the Poisson boundary of ${\mathcal{L}}$ is $W$.
In the next section we will give a precise definition for the terminology “harmonic function”. We shall also discuss some consequences of our results in terms of the Heckman–Opdam hypergeometric functions, which are particular eigenfunctions of the operator ${\mathcal{L}}$.
The first result (Theorem \[theoradial\]) was already known for values of $k$ corresponding to the case of symmetric spaces of the noncompact type $G/K$. The second result (Theorem \[theononradial\]) is new even for these particular values of $k$, but should be also compared to the situation on symmetric spaces. There, according to the fundamental work of Furstenberg [@F] (see also [@GJT]), the Poisson boundary of the Laplace–Beltrami operator (but also of a large class of random walks) is $K/M$. But it was already observed that in the Heckman–Opdam (also called trigonometric Dunkl) theory the group $W$ often plays the same role than $K$ or $K/M$. First geometrically, since there is a kind of Cartan decomposition: any $x\in {\mathbb{R}}^d$ can be uniquely decomposed as $w\cdot x^W$, with $x^W$ the radial part of $x$ (lying in the positive Weyl chamber) and $w\in W$. In representation theory also [@O]: briefly if ${\mathcal{H}}$ is the graded Hecke algebra generated by $W$ and the Dunkl–Cherednik operators (see next section), then $({\mathcal{H}},W)$ shares some properties of the Gelfand pair $(G,K)$, like the fact that in any irreducible finite-dimensional ${\mathcal{H}}$-module the subspace of $W$-invariant vectors is at most $1$-dimensional. So in some sense Theorem \[theononradial\] is another manifestation (let say at an analytical or probabilistic level) of the strong analogy between $W$ and $K$.
We should add that the hypothesis $k>0$ is probably sufficient to get the results of Theorem \[theoradial\] and \[theononradial\]. Here we restrict us to the case $k\ge 1/2$, because then the stochastic process associated with $L$ (or ${\mathcal{L}}$) a.s. never hit the walls (the hyperplanes orthogonal to the roots, which correspond to the singularities of $L$), and we need it to be sure that the coupling we use is well defined.
The paper is organized as follows. In the next section we recall all necessary definitions. In section \[secradial\] we prove Theorem \[theoradial\], by using the probabilistic technique of mirror coupling. In section \[secnonradial\] we prove Theorem \[theononradial\], by extending the coupling to the non-radial process. Our main tool for this is the skew-product representation from Chybiryakov [@Chy], that we have to adapt to our setting.
*Acknowledgments: I warmly thank Marc Arnaudon for having explained to me the technique of mirror coupling, and Alano Ancona for enlightening discussions about the regularity of harmonic functions.*
Preliminaries
=============
Let ${\mathfrak{a}}$ be a Euclidean vector space of dimension $n$, equipped with an inner product ${\langle}\cdot ,\cdot {\rangle}$, and denote by ${\mathfrak{h}}:={\mathfrak{a}}+i{\mathfrak{a}}$ its complexification. We consider ${\mathcal{R}}\subset {\mathfrak{a}}$ an integral root system (see [@Bou]). We choose a subset of positive roots ${\mathcal{R}}^+$. Let $\alpha^\vee=2\alpha/|\alpha|^2$ be the coroot associated to a root $\alpha$ and let $$r_\alpha(x)=x-{\langle}\alpha^\vee,x{\rangle}\alpha,$$ be the corresponding orthogonal reflection. Remember that $W$ denotes the Weyl group associated to ${\mathcal{R}}$, i.e. the group generated by the $r_\alpha$’s. Let $k\ :\ {\mathcal{R}}\rightarrow [1/2,+\infty)$ be a multiplicity function, which by definition is $W$-invariant. We set $$\rho=\frac{1}{2}\sum_{\alpha \in {\mathcal{R}}^+}k_\alpha \alpha.$$ Let $${\mathfrak{a}}_+ = \{x \mid \forall \alpha \in {\mathcal{R}}^+,\ {\langle}\alpha,x{\rangle}>0\},$$ be the positive Weyl chamber. Let also $\overline{{\mathfrak{a}}_+}$ be its closure, $\partial {\mathfrak{a}}_+$ its boundary and ${\mathfrak{a}}_{\text{reg}}$ the subset of regular elements in ${\mathfrak{a}}$, i.e. those elements which belong to no hyperplane $\{\alpha=0\}$. As recalled in the introduction any $x\in {\mathfrak{a}}$ can be uniquely decomposed as $x=w x^W$, with $x^W\in \overline{{\mathfrak{a}}_+}$ and $w\in W$. We call $x^W$ the radial part of $x$ and $w$ its angular part.
For $\xi \in {\mathfrak{a}}$, let $T_\xi$ be the Dunkl–Cherednik operator [@C]. It is defined, for $f\in C^1({\mathfrak{a}})$, and $x\in
{\mathfrak{a}}_{\text{reg}}$, by $$T_\xi f(x)=\partial_\xi
f(x) + \sum_{\alpha \in {\mathcal{R}}^+}k_\alpha
\frac{{\langle}\alpha,\xi{\rangle}}{1-e^{-{\langle}\alpha,x{\rangle}}}\{f(x)-f(r_\alpha x)
\}-{\langle}\rho,\xi{\rangle}f(x).$$ The Dunkl-Cherednik operators form a commutative family of differential-difference operators (see [@C] or [@O]). The Heckman–Opdam Laplacian ${\mathcal{L}}$ is also given by the formula $${\mathcal{L}}+|\rho|^2=\sum_{i=1}^{n} T_{\xi_i}^2,$$ where $\{\xi_1,\dots,\xi_n\}$ is any orthonormal basis of ${\mathfrak{a}}$.
Let ${\lambda}\in {\mathfrak{h}}$. We denote by $F_{\lambda}$ the unique (see [@HO], [@O]) analytic $W$-invariant function on ${\mathfrak{a}}$, which satisfies the differential equations $$p(T_\xi)F_{\lambda}=p({\lambda})F_{\lambda}\text{ for all W-invariant polynomials }p$$ and which is normalized by $F_\lambda(0)=1$ (in particular ${\mathcal{L}}F_{\lambda}=({\langle}{\lambda},{\lambda}{\rangle}-|\rho|^2) F_{\lambda}$). We denote by $G_\lambda$ the unique analytic function on ${\mathfrak{a}}$, which satisfies the differential-difference equations (see [@O]) $$\begin{aligned}
\label{equations} T_\xi G_{\lambda}= {\langle}{\lambda},\xi{\rangle}G_{\lambda}\text{ for all }\xi
\in {\mathfrak{a}},\end{aligned}$$ and which is normalized by $G_\lambda(0)=1$. These functions are related by the formula: $$\begin{aligned}
\label{FG}
F_{\lambda}(x)=\frac{1}{|W|} \sum_{w\in W} G_{\lambda}(wx),\end{aligned}$$ for all $x\in {\mathfrak{a}}$ and all ${\lambda}\in {\mathfrak{h}}$.
It was shown in [@Sch2] that $\frac{1}{2}{\mathcal{L}}$ and $\frac{1}{2}L$ are generators of Feller semi-groups that we shall denote respectively by $(P_t,t\ge 0)$ and $(P^W_t,t\ge 0)$. We will use the following definition for harmonic functions:
\[defharmonic\] A bounded or nonnegative function $h:{\mathfrak{a}}\to {\mathbb{R}}$ is called harmonic if it is measurable and satisfies $P_th=h$ for all $t>0$.
*It is well known that if $h$ is a $C^2$ function such that ${\mathcal{L}}h=0$, then $h$ is harmonic in the sense of Definition \[defharmonic\]. Inversely Corollary \[coroG\] below shows, when $k\ge 1/2$, that any bounded harmonic function is regular, thus satisfies ${\mathcal{L}}h=0$. On the other hand, it is a general fact (which applies for any $k>0$), that bounded $W$-invariant harmonic functions are regular in ${\mathfrak{a}}_+$, but we will not use this fact here.*
Observe that by definition $F_\rho$ is a $W$-invariant harmonic function. Moreover it is known (see [@Sch2] Remark 3.1) that it is bounded. So Theorem \[theoradial\] shows that in fact $F_\rho$ is constant equal to $1$. Similarly the functions $G_{w\rho}$’s, for $w\in W$, are harmonic and also bounded. This last property follows from Formula , since the $G_{w\rho}$’s are real positive (see [@Sch2] Lemma 3.1). In fact one has the following
\[coroG\] If $k\ge 1/2$, then any bounded harmonic function is a linear combination of the $G_{w\rho}$’s, $w\in W$.
The only thing to prove is that the $G_{w\rho}$’s are linearly independent. This results from the fact that they are all eigenfunctions of the Dunkl–Cherednik operators but for different eigenvalues. More precisely, assume that for some real numbers $(c_w)_{w\in W}$, we have $$\sum_{w\in W} c_w G_{w\rho}=0.$$ By applying then the operators $p(T_\xi)$, with $p$ polynomial, we get $$\sum_{w\in W} c_w p(w\rho)G_{w\rho}=0 \quad \textrm{for all p}.$$ >From this, and the fact that $G_{w\rho}(0)=1$ for all $w$, it is easily seen that we must have $c_w=0$ for all $w$.
The $W$-invariant case: proof of Theorem \[theoradial\] {#secradial}
=======================================================
In this section we shall prove Theorem \[theoradial\]. For this we will use the stochastic process $(X^W_t,t\ge 0)$ associated with $L$, called radial HO-process, and the so-called mirror coupling technique.
First it is known [@Sch1] that $X^W$ is a strong solution of the SDE: $$X^W_t=x+B_t + V^1_t$$ where $(B_t,t\ge 0)$ is a Brownian motion on ${\mathfrak{a}}$ and $$V^1_t:=\sum_{\alpha\in {\mathcal{R}}^+} k_\alpha \alpha \int_0^t \coth {\langle}\alpha, X^W_s{\rangle}\ d s.$$ Moreover when $k\ge 1/2$, $X^W$ a.s. takes values in ${\mathfrak{a}}_+$, or in other words it never reaches $\partial {\mathfrak{a}}_+$ (see [@Sch1]). Now if $x,y\in {\mathfrak{a}}_+$, we define the couple $((X^W_t,Y^W_t),t\ge 0)$ as follows. Set $T=\inf\{s \mid X^W_s=Y^W_s\}$. Then by definition $X^W$ is as above, and $(X^W,Y^W)$ is the unique solution of the SDE: $$\begin{aligned}
\label{SDE}
(X^W_t,Y^W_t)=(x,y) + (B_t,B'_t) + (V^1_t,V^2_t), \quad \textrm{for } t<T,\end{aligned}$$ where $dB'_t=r_t dB_t$, with $r_t$ the orthogonal reflexion with respect to the hyperplane orthogonal to the vector $Y^W_t-X^W_t$ (in particular Levy criterion shows that $B'$ is a Brownian motion), and $$V^2_t:=\sum_{\alpha\in {\mathcal{R}}^+} k_\alpha \alpha \int_0^t \coth {\langle}\alpha, Y^W_s{\rangle}\ d s.$$ For $t\ge T$, we set $Y^W_t=X^W_t$. The existence of this coupling is guaranteed by the fact that the SDE has locally regular coefficients. We define also $Z^W$ by $$Z^W_t:=Y^W_t-X^W_t,$$ and set $z^W_t= |Z^W_t|$. It is known [@Sch1] that a.s. $X^W_t/t \to \rho$, and thus that ${\langle}\alpha,X^W_t{\rangle}\sim {\langle}\rho,\alpha{\rangle}t$, for all $\alpha \in {\mathcal{R}}^+$. From this we see that a.s. $\sup_{t\ge 0}|V^2_t-V^1_t|<+\infty$. Then Tanaka formula ([@RY] p.222) shows that $$z^W_t =\gamma_t + v_t, \quad \textrm{for } t<T,$$ with $\gamma$ a one-dimensional Brownian motion and a.s. $\sup_{t\ge 0}|v_t|<+\infty$. In particular $T$ is a.s. finite.
The end of the proof is routine now. Assume that $h$ is a bounded $W$-invariant harmonic function. Then it is well known, and not difficult to show, that $(h(X^W_t),t\ge 0)$ as well as $(h(Y^W_t),t\ge 0)$ are bounded martingales. Thus they are a.s. converging toward some limiting (random) values, respectively $l$ and $l'$. Since a.s. $X^W_t=Y^W_t$ for $t$ large enough, we have a.s. $l=l'$. Then usual properties of bounded martingales show that $$h(x) = {\mathbb{E}}[l] = {\mathbb{E}}[l']=h(y).$$ Since this holds for any $x,y \in {\mathfrak{a}}_+$, this proves well that $h$ is constant. $\square$
The non $W$-invariant case: proof of Theorem \[theononradial\] {#secnonradial}
==============================================================
In order to prove Theorem \[theononradial\], the first idea is to extend the previous coupling to the full process $(X_t,t\ge 0)$ with semi-group $(P_t,t\ge 0)$. For this our tool will be the skew-product representation founded by Chybiryakov [@Chy] (see [@GaY] and [@Chy2] for the one-dimensional case). Actually Chybiryakov dealt with Dunkl processes, so we shall first mention the changes needed to adapt his proof to the present setting, and then explain how to combine this representation with the coupling from the previous section.
Skew-product representation and extension of the coupling {#sprod}
---------------------------------------------------------
The skew-product representation gives a constructive way to define $X$ starting from $X^W$, by adding successively jumps in the direction of the roots. Let us sketch the main steps of the construction (for more details see [@Chy]). First one fixes arbitrarily an order for the positive roots: $\alpha_1,\dots,\alpha_{|{\mathcal{R}}^+|}$. Then for each $j \in [1,|{\mathcal{R}}^+|]$, set $${\mathcal{L}}^jf(f):= Lf(x) - \sum_{i\le j}c_{\alpha_i}(x) \{f(x)-f(r_{\alpha_i}
x)\},$$ where for any root $\alpha$, $$c_\alpha(x):= k_{\alpha}
\frac{|\alpha|^2}{4\sinh^2 \frac{{\langle}\alpha,x{\rangle}}{2}}.$$ Decide also that ${\mathcal{L}}^0=L$. Set $$\widetilde{{\mathcal{L}}}^j f(x) : = c_{\alpha_j}^{-1}(x) {\mathcal{L}}^j f(x),$$ and $${\mathcal{L}}^{j,j+1}f(x) := c_{\alpha_{j+1}}^{-1}(x) {\mathcal{L}}^j f(x).$$ The goal is to define inductively a sequence of processes $(X^j(t),t\ge 0)$, $j=0,\dots,|{\mathcal{R}}^+|$, associated to the operators ${\mathcal{L}}^j$’s. First $X^0$ is just the radial HO-process considered in the previous section. Next assume that ${\mathcal{L}}^j$ is the generator of a Markov process $(X^j(t),t\ge 0)$. Then set $$A_t^j=\int_0^t c_{\alpha_{j+1}}(X^j_s)\ ds,$$ and $$\tau_t^j= \inf\{s\ge 0\mid A_s^j>t\}.$$ Using the martingale problem characterization one can see that the radial part of $X^j$ is a radial HO-process. Thus for all $\alpha\in {\mathcal{R}}^+$, $|{\langle}\alpha,X^j_t{\rangle}|\ge c t$, for $t$ large enough and $c>0$ some constant. In particular the increasing process $A^j$ is bounded. Set $T^j=\lim_{t\to +\infty} A^j_t$. Then observe that $\tau_t^j=+\infty$, when $t\ge T^j$. This is essentially the only difference with the Dunkl case considered in [@Chy] (where $A^j$ was not bounded and $\tau_t^j$ finite for all $t$). But one can still see that if $$X^{j,j+1}(t):= X^j(\tau_t^j) \quad t< T^j,$$ then $X^{j,j+1}$, killed at time $T^j$, is solution of the martingale problem associated with ${\mathcal{L}}^{j,j+1}$ (see for instance [@EK] exercise 15 p.263 and section 6 p.306). The next step is to add jumps to $X^{j,j+1}$ in the direction of the root $\alpha_{j+1}$. Namely one define a new process $\widetilde{X}^j$, also denoted by $X^{j,j+1}*_{\alpha_{j+1}} N$ in [@Chy] section 2.5, which is solution of the martingale problem associated with $\widetilde{{\mathcal{L}}}^{j+1}$. Roughly $\widetilde{X}^j$ is constructed by gluing several paths, all with law $X^{j,j+1}$ or $r_{\alpha_{j+1}}X^{j,j+1}$, such that for any two consecutive path the starting point of the second is the image of the end point of the first path by the reflexion $r_{\alpha_{j+1}}$. The lengths of the paths are determined by independent exponentially distributed random variables. Here the only minor change is that $\widetilde{X}^j$ explodes at some time, let say $\widetilde{T}^j$. A change of variables shows that $$\lim_{t\to \widetilde{T}^j} \int_0^t c_{\alpha_{j+1}}^{-1}(\widetilde{X}^j(s))\ ds=+\infty.$$ So for any $t\ge 0$, one can define $\widetilde{A}^j(t)$ as solution of the equation $$t = \int_0^{\widetilde{A}^j(t)}c_{\alpha_{j+1}}^{-1}(\widetilde{X}^j(s))\ ds.$$ Differentiating this equation one get $$\frac{d}{dt} \widetilde{A}^j(t) = c_{\alpha_{j+1}}(\widetilde{X}^j(\widetilde{A}^j(t))).$$ Then set $X^{j+1}(t)=\widetilde{X}^j(\widetilde{A}^j(t))$, for all $t\ge 0$. The preceding equation gives $$\widetilde{A}^j(t)= \int_0^t c_{\alpha_{j+1}}(X^{j+1}(s))\ ds,$$ which in turn shows that $X^{j+1}$ is solution of the martingale problem associated with ${\mathcal{L}}^{j+1}$, as wanted.
The point now is to combine this construction of $X=X^{|{\mathbb{R}}^+|}$ with the coupling of the radial process from section \[secradial\]. We first take $(X^0,Y^0)$ with law given by this coupling. Then we define the sequence $((X^j(t),Y^j(t)),t\ge 0)$, $j= 1,\dots,|{\mathcal{R}}^+|$, simply by following the previous construction for the two coordinates. Actually this coupling is interesting only when $X=X^{|{\mathcal{R}}^+|}$ and $Y=Y^{|{\mathcal{R}}^+|}$ never jump, but this is precisely what we need. Indeed in this case we have $X_t=X^0(t)$ and $Y_t=Y^0(t)$, for all $t\ge 0$, so they coincide a.s. after some finite time.
End of the proof
----------------
For any $x\in {\mathfrak{a}}$, we denote by ${\mathbb{P}}_x$ the law of $(X_t,t\ge 0)$ starting from $x$. For $\epsilon \in (0,1)$, set $$A_\epsilon := \{z\in {\mathfrak{a}}\mid {\mathbb{P}}_z[X \textrm{ never jumps}] \ge 1-\epsilon\}.$$ We know that the process $(X_t,t\ge 0)$ can jump, so a priori $A_\epsilon \subsetneq {\mathfrak{a}}$. But we know also [@Sch1] that a.s. $X$ eventually stops to jump after some finite random time. This implies that $$\begin{aligned}
\label{eqjumps}
\lim_{t\to +\infty} {\mathbb{P}}_x[X \textrm{ never jumps after time } t]=1,\end{aligned}$$ for all $x\in {\mathfrak{a}}$. But by using the Markov property, we have for all $t>0$, $$\begin{aligned}
\label{eqjump2}
\nonumber {\mathbb{P}}_x[X \textrm{ never jumps after time } t] &=& {\mathbb{E}}_x\left[ {\mathbb{P}}_{X_t}[X \textrm{ never jumps}] \right]\\
&=& \int_{\mathfrak{a}}{\mathbb{P}}_z[X \textrm{ never jumps}]\ d\mu^x_t(z),\end{aligned}$$ where $\mu^x_t$ is the law of $X_t$ under ${\mathbb{P}}_x$. So and imply that for all $x\in {\mathfrak{a}}$, $\mu_t^x(A_\epsilon) \to 1$, when $t\to +\infty$. In particular $A_\epsilon$ is nonempty. Moreover, by invariance of ${\mathcal{L}}$ under $W$, we know that for any $w\in W$, the law of $(wX_t,t\ge 0)$ under ${\mathbb{P}}_x$ is ${\mathbb{P}}_{wx}$. In particular, for any $w\in W$ and any $\epsilon\in (0,1)$, we have $w(A_\epsilon\cap {\mathfrak{a}}_+)=A_\epsilon \cap w{\mathfrak{a}}_+$. Thus all these subsets of $A_\epsilon$ are nonempty as well.
Let now $h$ be some harmonic function. Fix $w\in W$, and take $x,y \in A_\epsilon \cap w{\mathfrak{a}}_+$. Consider the coupling $((X_t,Y_t),t\ge 0)$ as defined above. Since $(h(X_t),t\ge 0)$ and $(h(Y_t),t\ge 0)$ are bounded martingales, they converge a.s. toward some limits, respectively $l$ and $l'$. We already saw that $X^W$ and $Y^W$ a.s. coincide after some time. So if both processes $X$ and $Y$ never jump, they must also coincide after some time, and in this case we have $l=l'$. Since $x,y \in A_\epsilon$, this shows that $$|h(x)-h(y)|=|{\mathbb{E}}[l]-{\mathbb{E}}[l']| \le 2C\epsilon,$$ where $C=\sup h$. In particular, by completeness of ${\mathbb{R}}$, for any sequence $(x_\epsilon)_{\epsilon \in (0,1)}$, such that $x_\epsilon \in A_\epsilon\cap w{\mathfrak{a}}_+$ for all $\epsilon \in (0,1)$, the limit of $h(x_\epsilon)$ when $\epsilon $ tends to $0$ exists, and is independent of the chosen sequence. Call $l_w$ this limit.
For all $t\ge 0$, we denote by $w_t$ the angular part of $X_t$. Since $X$ eventually stops to jump, $(w_t,t\ge 0)$ a.s. converges, i.e. becomes stationary. Then for any $w\in W$, define the function $h_w$ on ${\mathfrak{a}}$ by $$h_w(x)={\mathbb{P}}_x\left[\lim_{t\to +\infty}w_t=w\right].$$ By standard properties of Markov processes, we know that these functions are measurable, and actually it is not difficult to see that they are harmonic. Moreover the above convergence result for harmonic functions shows that these functions $h_w$, $w\in W$, are linearly independent. Then set $$\tilde{h}(x):= \sum_{w\in W} l_w h_w(x),$$ for all $x\in {\mathfrak{a}}$. All that remains to do now is to prove that $\tilde{h}=h$. Indeed if this was true, this would prove that the vector space of bounded harmonic function has dimension $|W|$ as wanted. By using the martingale property, we have for any $t>0$ $$\begin{aligned}
\label{hhtilde}
|h(x)-\tilde{h}(x)|= |{\mathbb{E}}_x[h(X_t)-\tilde{h}(X_t)]| \le \int_{\mathfrak{a}}|h(z)-\tilde{h}(z)|\ d\mu_t^x(z).\end{aligned}$$ We have seen that for all $\epsilon \in (0,1)$, $$\begin{aligned}
\label{Aepsilon}
\mu_t^x(A_\epsilon) \to 1\end{aligned}$$ when $t\to +\infty$. But it is not difficult to see (by using the definition of the $l_w$’s), that for any $\epsilon'>0$, there exists $\epsilon>0$ such that $$|h(z)-\tilde{h}(z)|\le \epsilon' \quad \forall z\in A_\epsilon.$$ Since this holds for any $\epsilon'>0$, and show that $h=\tilde{h}$ as wanted. $\square$
*We have seen in the previous proof that the family $(h_w)_{w\in W}$ is a basis of the space of bounded harmonic functions. Since the family $(G_{w\rho})_{w\in W}$ is another basis, it would be interesting to know the coefficients relating these two basis.*
[99]{}
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---
abstract: 'By analysing the *K2* short-cadence photometry we detect starspot occultation events in the lightcurve of WASP-107, the host star of a warm-Saturn exoplanet. WASP-107 also shows a rotational modulation with a period of $17.5\pm1.4$d. Given that the rotational period is nearly three times the planet’s orbital period, one would expect in an aligned system to see starspot occultation events to recur every three transits. The absence of such occultation recurrences suggests a misaligned orbit unless the starspots’ lifetimes are shorter than the star’s rotational period. We also find stellar variability resembling $\gamma$Doradus pulsations in the lightcurve of WASP-118, which hosts an inflated hot Jupiter. The variability is multi-periodic with a variable semi-amplitude of $\sim$200ppm. In addition to these findings we use the *K2* data to refine the parameters of both systems, and report non-detections of transit-timing variations, secondary eclipses and any additional transiting planets. We used the upper limits on the secondary-eclipse depths to estimate upper limits on the planetary geometric albedos of 0.7 for WASP-107b and 0.2 for WASP-118b.'
author:
- |
T. Močnik,[^1] C. Hellier, D. R. Anderson, B. J. M. Clark and J. Southworth\
Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK
bibliography:
- 'bibliography.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Starspots on WASP-107 and pulsations of WASP-118'
---
\[firstpage\]
planetary systems – stars: individual: (WASP-107, WASP-118) – stars: oscillations – starspots.
INTRODUCTION
============
The *Kepler* [@Borucki10] and *K2* [@Howell14] missions have provided the community with high-precision photometric observations of 2449 confirmed transiting exoplanets to date among a total of 2716 confirmed transiting exoplanets, according to NASA Exoplanet Archive[^2]. In addition, *K2* is also observing exoplanets previously found by the ground-based transit surveys such as WASP [@Pollacco06]. WASP-107 and WASP-118 are among the brightest systems observed by *K2*, which allows for detailed characterisation owing to high-precision lightcurves combined with existing spectroscopic observations performed by @Anderson17 and @Hay16.
WASP-107b is a warm Saturn in a 5.7-d orbit around a *V*=11.6, K6 main-sequence star [@Anderson17]. The planet lies in the transition region between ice giants and gas giants, with a mass of 2.2$M_{\rm Nep}$ or 0.12$M_{\rm Jup}$, but an inflated radius of 0.94$R_{\rm Jup}$. The WASP discovery photometry revealed a possible stellar rotational modulation with a period of $\sim$17d and an amplitude of 0.4 per cent. This led @Anderson17 to propose that the host star is magnetically active.
WASP-118b is an inflated hot Jupiter with a mass of 0.51$M_{\rm Jup}$ and a radius of 1.4$R_{\rm Jup}$. It orbits a *V*=11.0 F6IV/V star every 4.0d [@Hay16].
If a transiting planet crosses a starspot it will produce a temporary brightening in the transit [@Silva03]. Starspot occultation events can provide an accurate measurement of the stellar rotational period and the obliquity, i.e. the angle between the stellar rotational axis and the planet’s orbital axis. Obliquities can tell us about a planet’s dynamical history and about planet migration mechanisms. If the obliquity is small, the same starspot can be occulted recurrently at different stellar longitudes, such as in the case of an aligned Qatar-2 system [@Mocnik16c; @Dai17]. Alternatively, if a system is misaligned, the transit chord will cross stellar active latitudes only at certain preferential phases, such as in the case of HAT-P-11b [@Sanchis11].
The presence of a massive close-in planet may induce tides that can lead to multi-periodic non-radial pulsations, and in special cases radial pulsations of the host star [@Schuh10; @Herrero11]. However, only a handful set of pulsating exoplanet hosts have been found so far. V391b, for example, has been found orbiting an sdB star through pulsation-timing variations [@Silvotti07]. The first main-sequence star where asteroseismology was applied to exoplanetary research is $\mu$Arae [@Bazot05]. High precision and long-term *Kepler* observations have lead to the discovery of several transiting exoplanet host stars which exhibited solar-like oscillations [@Davies16], whose typical semi-amplitudes are of the order of a few ppm for main-sequence stars [@Baudin11].
WASP-33 was the first transiting-exoplanet host star exhibiting $\delta$Scuti pulsations, with a semi-amplitude of about 900ppm [@CollierCameron10b; @Herrero11]. One of the harmonics has a frequency 26 times the orbital frequency, suggesting that WASP-33’s pulsations might be induced by the planet. A more confident claim for the planet-induced stellar pulsations was made recently for HAT-P-2 system, a possible low-amplitude $\delta$Scuti pulsator with a 87 minutes pulsation period and an amplitude of 40 ppm [@deWit17]. Its pulsation modes correspond to exact harmonics of the planet’s orbital frequency and are thought to be induced by the transient tidal interactions with its massive (8$M_{\rm Jup}$), short-period (5.6d) and highly eccentric (*e* = 0.5) planet [@deWit17]. Systems such as WASP-33 and HAT-P-2 provide a laboratory to study star–planet interactions. The observations provided by the *K2* mission have led to the discovery of another pulsating transiting exoplanet host star, namely HAT-P-56 which is likely a $\gamma$Doradus pulsator [@Huang15]. While the exact asteroseismic analysis approach can vary depending on the class of variability, one can investigate any potential star–planet interactions and derive precise stellar mass, radius and the depth-dependant chemical composition, given the appropriate modelling capabilities [@Schuh10].
In this paper, we present a detection of starspots in the *K2* lightcurve of WASP-107, and pulsations in the lightcurve of WASP-118. We also refine system parameters for both systems, search for transit-timing variations, phase-curve modulations and any additional transiting planets, and provide a measurement of the rotational period for WASP-107.
Simultaneously with our paper, @Dai17b have announced an analysis of the same *K2* short-cadence observations of WASP-107. Their results are in a good agreement with ours.
THE K2 OBSERVATIONS
===================
WASP-107 was observed by *K2* in the 1-min short-cadence observing mode during Campaign 10 between 2016 July 6 and 2016 September 20. The dominant systematics present in the *K2* lightcurves are the sawtooth-shaped artefacts caused by the drift of the spacecraft. We attempted to correct for the drift artefacts using the same self-flat-fielding (SFF) procedure as described in @Mocnik16a. However, since the drifts of the spacecraft were variable during this observing campaign, we obtained better results using the K2 Systematic Correction pipeline (K2SC; @Aigrain16), which uses break-points to isolate different drift behaviours to overcome the issue of inconsistent drifts. The K2SC procedure was designed for the *K2* long-cadence observing mode and had to be modified slightly to accept the short-cadence data. The main modification was to split the input short-cadence lightcurve into smaller overlapping sections, which were processed individually by K2SC and then merged back together using the overlapping sections. However, since the WASP-107 lightcurve exhibits pronounced modulations, the K2SC failed to correct the drift artefacts properly at three different 2-d-long sections centred at BJD 2457624.7, 2457631.8 and 2457638.3. We replaced these sections with the lightcurve we obtained using the above-mentioned SFF procedure. Using this approach we achieved a median 1-min photometric precision of 260ppm, compared to 870ppm before the artefact correction, and 300ppm using only the SFF procedure. The final drift-corrected lightcurve of WASP-107 is shown in Fig. 1. We used this lightcurve to study the rotational modulation of the host star in Section 6. For all other analyses we used the normalized version of the lightcurve, which we produced with PyKE tool [@Still12], which divides the measured flux with a low-order polynomial fit with window and step sizes of 3 and 0.3d, respectively, which effectively removed any low-frequency modulations.
{width="8.3cm"}
WASP-118 was observed during the *K2* Campaign 8 between 2016 January 4 and 2016 March 23, also in the short-cadence mode. Because the spacecraft’s drifts were more consistent during Campaign 8, we produced a slightly better lightcurve with our SFF procedure than with K2SC. The drift-corrected lightcurve of WASP-118 is shown in red in Fig. 2. In addition to the transits, the lightcurve exhibits variability with a $\sim$5-d time-scale. The variability is inconsistent and incoherent, and cannot be interpreted as a rotational modulation with confidence. We produced the normalized version of the lightcurve in the same way as for WASP-107. The normalized lightcurve revealed multi-periodic higher-frequency variability with a semi-amplitude of $\sim$200ppm (see Fig. 3). We suggest that this variability is produced by the weakly pulsating host star (see Section 7).
{width="8.3cm"}
{width="17.6cm"}
Because the presence of pulsations in the lightcurve of WASP-118 could affect the analysis, we used the lightcurve with embedded pulsations only for the pulsations analysis in Section 7. For every other aspect of analysis we produced another lightcurve in which we removed these higher-frequency, low-amplitude pulsations. This was done by modifying our SFF procedure. Instead of the usual 5-d time-steps, we split the lightcurve into sections of individual spacecraft drifts before applying the SFF correction. Since the typical 6-h time-scale at which the drifts occur is considerably shorter than the observed pulsation time-scales, our modified SFF procedure effectively removed the pulsations from the lightcurve, along with any other variabilities at longer time-scales. This produced the lightcurve with a final median 1-min photometric precision of 210ppm. We used this version of lightcurve for every aspect of the analysis, except for the analysis of pulsations in Section 7.
SYSTEM PARAMETERS
=================
To determine the system parameters we simultaneously analysed the *K2* transit photometry and radial-velocity measurements for both planets using a Markov Chain Monte Carlo (MCMC) code [@CollierCameron07; @Pollacco08; @Anderson15]. We used all the radial-velocity datasets that have been reported in the corresponding discovery papers (@Anderson17 and @Hay16), except the HARPS-N in-transit dataset for WASP-118. This dataset was excluded in order to prevent biasing the system parameters owing to a large scatter and underestimated error bars, as pointed out by the authors of the discovery paper. It may be that some of this scatter results from the pulsations that we now report. Limb darkening was accounted for using a four-parameter law, with coefficients calculated for the *Kepler* bandpass and interpolated from the tabulations of @Sing10.
For the MCMC analysis of WASP-107 we used the normalized lightcurve, from which we excluded any starspot occultation events (see Section 5), because failing to do so may lead to inaccurate system parameter determination, as discussed by @Oshagh13. Similarly, the presence of pulsations in the lightcurve of WASP-118 may have affected the precise transit analysis and so we instead used the lightcurve with pulsations removed.
We first imposed a circular orbit in the main MCMC analysis for both planets. We then estimated the upper limits on eccentricities in a separate MCMC run by allowing the eccentricities to be fitted as a free parameter. To improve the precision of the orbital ephemerides we also ran another MCMC analysis for both systems that included all the available ground-based photometric datasets used in the discovery papers which expanded the observational time-span and reduced the uncertainty on the orbital period by factors of 6.5 for WASP-107 and 2.3 for WASP-118. For these additional photometric datasets, we used limb-darkening coefficients from @Claret00 [@Claret04], as appropriate for different bandpasses.
The resulting system parameters are given in Table 1 and the corresponding transit models are shown in Figs. 4 and 5.
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{width="8.3cm"}
-------------------------------------- ---------------------------------------------------- ------------------------------------ ------------------------------------ ------------------ --
WASP-107 WASP-118
Parameter Symbol Value Value Unit
Transit epoch *t*$_{\rm 0}$ $2457584.329746\pm0.000011$ $2457423.044825\pm0.000020$ BJD
Orbital period *P* $5.72149242\pm0.00000046$ $4.0460407\pm0.0000026$ d
Area ratio $(R_{\rm p}/R_{\star})^{2}$ $0.020910\pm0.000058$ $0.006679\pm0.000010$ ...
Transit width *t*$_{\rm 14}$ $0.11411\pm0.000081$ $0.20464\pm0.00010$ d
Ingress and egress duration *t*$_{\rm 12}$, *t*$_{\rm 34}$ $0.01467\pm0.00012$ $0.01610\pm0.00012$ d
Impact parameter *b* $0.139\pm0.024$ $0.206\pm0.015$ ...
Orbital inclination *i* $89.560\pm0.078$ $88.24\pm0.14$ $^{\circ}$
Orbital eccentricity *e* 0 (adopted; $<$0.025 at $2\sigma$) 0 (adopted; $<$0.028 at $2\sigma$) ...
Orbital separation *a* $0.0553\pm0.0013$ $0.05450\pm0.00049$ au
Stellar mass *M*$_{\star}$ $0.691\pm0.050$ $1.319\pm0.035$ M$_\odot$
Stellar radius *R*$_{\star}$ $0.657\pm0.016$ $1.754\pm0.016$ R$_\odot$
Stellar density $\rho_{\star}$ $2.441\pm0.023$ $0.2445\pm0.0024$ $\rho_\odot$
Planet mass *M*$_{\rm p}$ $0.119\pm0.014$ $0.52\pm0.18$ $M_{\rm Jup}$
Planet radius *R*$_{\rm p}$ $0.924\pm0.022$ $1.394\pm0.013$ $R_{\rm Jup}$
Planet density $\rho_{\rm p}$ $0.152\pm0.017$ $0.193\pm0.066$ $\rho_{\rm Jup}$
Planet equilibrium temperature$^{a}$ *T*$_{\rm p}$ $736\pm17$ $1753\pm34$ K
Limb-darkening coefficients $a_{\rm 1}$, $a_{\rm 2}$, $a_{\rm 3}$, $a_{\rm 4}$ 0.710, –0.773, 1.520, –0.641 0.522, 0.313, –0.071, –0.045 ...
-------------------------------------- ---------------------------------------------------- ------------------------------------ ------------------------------------ ------------------ --
$^{a}$
: Planet equilibrium temperature is based on assumptions of zero Bond albedo and complete heat redistribution.
NO TTV OR TDV
=============
Inter-planet gravitational interactions can cause transit-timing variations (TTVs) and transit-duration variations (TDVs) [@Algol05]. The detection of these variations can therefore reveal additional planets in the system. Typical reported TTV amplitudes range from a few seconds and up to several hours with periods of the order of a few days [@Mazeh13]. TDVs are expected to be in phase with the TTVs but at a significantly lower amplitude [@Nesvorny13].
To search for TTVs and TDVs we ran another MCMC analysis on individual transits for both systems. We again removed the starspot occultation events from the WASP-107 lightcurve and used the pulsation-free lightcurve of WASP-118 since the lightcurve variability could affect the timing accuracy [@Oshagh13].
Against the hypothesis of equal transit timing spacings and constant transit durations the measured TTVs and TDVs for WASP-107 correspond to $\chi^{2}$ values of 11.1 and 6.6, respectively, for 10 degrees of freedom. Similarly, for WASP-118 the TTV and TDV $\chi^{2}$ values are 26.8 and 15.7, respectively, for 19 degrees of freedom. Thus there are no significant TTVs or TDVs. The upper limits for WASP-107 are 20 and 60s for TTVs and TDVs respectively, for periods shorter than 80d. For WASP-118 the upper limits are 40 and 100s. Given the absence of any statistically significant TTV or TDV variations, we can conclude that any additional close-in, massive planets are unlikely in either of the two systems.
STARSPOTS ON WASP-107
=====================
In Fig. 6 we show the lightcurve of WASP-107 centred at individual transits after subtracting the best-fitting MCMC transit model from Section 3. The residual lightcurve reveals several starspot occultation events. Each of us has individually examined the residual lightcurve by eye and marked the events as definite or possible occultations. Here we report occultation events that were marked by at least two colleagues. This gives 5 definite starspot occultation events (marked with dark-red ellipses in Fig. 6) and 4 possible events (marked with light-red ellipses) within the 10 observed transits. Table 2 lists all the marked definite and possible occultation events, measured orbital phases at which they occur and the corresponding stellar longitudes which we calculated using the system parameters from Table 1. There may well be additional, smaller-amplitude spots present in some of the transits, in addition to those listed.
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---- ----------- ---------- --------- -------
1 $-0.0033$ $0.0002$ $-22.4$ $1.6$
1 $0.0062$ $0.0002$ $45.0$ $2.2$
2 $-0.0064$ $0.0002$ $-47.6$ $2.4$
6 $-0.0018$ $0.0002$ $-12.1$ $1.4$
7 $0.0051$ $0.0002$ $36.2$ $1.9$
8 $-0.0028$ $0.0002$ $-18.5$ $1.5$
9 $-0.0059$ $0.0002$ $-42.9$ $2.1$
9 $0.0056$ $0.0002$ $40.3$ $2.0$
13 $0.0004$ $0.0002$ $2.4$ $1.3$
---- ----------- ---------- --------- -------
: Phase and longitude positions of every detected starspot occultation event.
$^{a}$
: Longitude runs from $-90^{\circ}$ (first planetary contact), through $0^{\circ}$ (central meridian) to $90^{\circ}$ (last contact).
Given the stellar rotational period of $17.5\pm1.4$d (see Section 6) and the planet’s orbital period of 5.72d (see Table 1), one would expect in an aligned system to see the same starspot being occulted again three transits later, with a longitude shift of $-7^{\circ +31}_{\,\,\, -26}$. Occultations one, two or four transits later would be hard to detect, since the phase shifts would be in multiples of $120^{\circ}$, and therefore the spots would either be close to the limb, where they are hard to detect owing to limb darkening, or not on the visible face at all.
There is only one pair of starspot occultations that might be a recurrence. If the occultation event in transit 6 and the first occultation in transit 9 (see Fig. 6) were caused by the same starspot, they would imply an orbital period of $18.8\pm0.2$d. Although this is compatible with the $17.5\pm1.4$-d rotational period derived from the rotational modulation (see Section 6) there are reasons to doubt that the pair was actually caused by the same starspot. Firstly, a starspot from transit 8 does not produce an occultation pair at a similar phase shift in transit 11. Secondly, the starspot lifetimes for main-sequence stars are of the order of days (@Bradshaw14 and citations therein), which means that in 17d, the time it takes for the planet to orbit its host star three times, a starspot could disappear.
Overall, we do not find compelling evidence for recurring starspots, which would suggest that the system might be misaligned. However, because of the 17-d time span between recurrences that could be readily observed, we cannot exclude the possibility that it is an aligned system with relatively short starspot lifetimes.
ROTATIONAL MODULATION OF WASP-107
=================================
The lightcurve of WASP-107 in Fig. 1 reveals a low-frequency modulation with a semi-amplitude of about 0.2 per cent. To measure the periodicity of this modulation we removed the planetary transits and calculated a Lomb-Scargle periodogram (see Fig. 7). The highest peak in the periodogram and its full width at half maximum correspond to a periodicity of $17.5\pm1.4$d.
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Knowing that the stellar surface harbours starspots (see Section 5), we believe that this modulation is caused by the stellar rotation. Using the stellar radius from Table 1 and assuming that the rotational axis is orthogonal to the line of sight, the measured rotational modulation period implies a stellar rotational velocity of $1.9\pm0.2$kms$^{-1}$. This value agrees with the spectroscopic projected rotational velocity of $2.5\pm0.8$kms$^{-1}$ [@Anderson17]. Our rotational modulation period and amplitude also agree well with the period of $17\pm1$d and amplitude of 0.4 per cent that were derived from the ground-based photometry and reported in the discovery paper by @Anderson17.
STELLAR PULSATIONS OF WASP-118
==============================
Fig. 3 provides a close-up view of the higher-frequency photometric variability in the lightcurve of WASP-118, after removing incoherent low-frequency modulations. This variability is not correlated with the spacecraft’s drifts, is preserved using four different data-reduction procedures, is not present in the *K2* lightcurves of other nearby stars, and is not correlated with the orbital phase.
The variability cannot be realistically considered as a rotational modulation because its short 2-d period would require the star to rotate three times faster than the spectroscopically measured projected rotational velocity. This would only be possible if the system were hugely misaligned, in contradiction with the Rossiter–McLaughlin measurements by @Hay16 who suggested that the system is aligned. The variability is therefore most likely to be weak pulsations of the host star.
The semi-amplitude of the pulsations is $\sim$200ppm. The Lomb-Scargle periodogram reveals that the variability is multi-periodic (see Fig. 8) with the highest peak at 1.9d. To check whether the normalization procedure affected the detected pulsations we ran signal injection tests and found that the applied normalization procedure preserves 80 per cent of the variability at the peak pulsation period near 1.9d.
{width="8.3cm"}
@Kaye99 introduced a $\gamma$Doradus pulsating type for main-sequence and subgiant stars of spectral types A7–F5. $\gamma$Doradus stars exhibit non-radial, high-order and low-degree multiperiodic pulsations with periods of 0.4–3d and amplitudes below 0.1mag. The pulsation amplitude of such stars may vary during an observing season by as much as a factor of 4.
The spectral type and variability characteristics of WASP-118 suggest that the star is probably a weak, late-type $\gamma$Doradus pulsator.
If the stellar pulsations were induced by the star–planet interactions, we would expect them to appear at exact harmonics of the planet’s orbital period, such as in the cases of WASP-33 [@Herrero11] and HAT-P-2 [@deWit17]. We found no match when comparing the main harmonics of the WASP-118b’s orbital period with the measured pulsations’ periodicities (see Fig. 8). Despite the lack of any commensurabilities with the main harmonics, we cannot yet reject any complex commensurabilities nor the possibility that the pulsations were induced by the orbiting planet. A detailed follow-up asteroseismic analysis is required to further investigate the star–planet interactions as a possible cause for the observed pulsations in the lightcurve of WASP-118.
NO PHASE-CURVE MODULATIONS
==========================
Phase-curves in exoplanet systems consist of three main components at optical wavelengths: 1) ellipsoidal modulation, 2) Doppler beaming, and 3) planetary reflection (e.g. @Esteves13). Additionally, a transiting planet may produce a secondary eclipse, an occultation of the planet by its host star, which blocks the reflected light during the occultation. Therefore, the depth of the secondary eclipse is twice the semi-amplitude of the reflectional modulation.
To produce a phase-curve of WASP-107 we had to remove the significant rotational modulation prior to phase-folding the lightcurve. However, because the ratio between the 5.7-d orbital period and the 17-d rotational modulation is not small enough, this would also remove any phase-curve modulations. For WASP-118 we again removed stellar variability which would again remove any phase-curve modulations. However, in both systems the procedures would not have removed any secondary eclipses present. A non-detection of secondary eclipses in both systems therefore allowed us to estimate conservative upper limits on the secondary-eclipse depths of 100ppm for WASP-107 and 50ppm for WASP-118.
Using the system parameters from Table 1, the theoretically expected semi-amplitudes of ellipsoidal, Doppler beaming and reflectional modulation for WASP-107 are 0.03, 0.2 and $75A_{\rm g}$ppm, respectively, where $A_{\rm g}$ is the planet’s geometrical albedo. For WASP-118, the expected semi-amplitudes are 1, 0.7 and $150A_{\rm g}$ppm. These amplitudes have been calculated using the relations from @Mazeh10. While the expected amplitudes for the ellipsoidal and Doppler beaming modulations are below the *K2* photometric precision, the inflated planetary radii could produce a significant reflectional modulation in both systems. Using the upper limits for secondary eclipse depths and theoretically expected amplitudes for reflectional modulations allows us to constrain the planetary geometric albedos to less than 0.7 for WASP-107b and less than 0.2 for WASP-118b.
NO ADDITIONAL TRANSITING PLANETS
================================
To search for signatures of any additional transiting planets in the normalized lightcurves of both systems, we first removed the transits of the known transiting planets and then calculated the box-least-square periodograms of any other periodic signals with the PyKE tool . The absence of any significant residual signals in the period range 0.5–30d results in transit-depth upper limits of 130ppm for any additional transits in the WASP-107 system and 140ppm in WASP-118.
AGES OF THE HOST STARS
======================
We estimated the ages of both host stars by comparing the measured stellar densities from Table 1 and the published spectroscopic effective temperatures to isochrones computed from the stellar evolution models. This was done with the Bayesian mass and age estimator [@Maxted15], which uses the code [@Weiss08] to compute the evolution models. The best-fitting stellar evolution tracks provided the age estimates of $8.3\pm4.3$Gyr for WASP-107 and $2.3\pm0.5$Gyr for WASP-118.
The rate at which a star rotates acts as another age estimator. Over time, stars lose angular momentum through magnetised stellar winds and gradually slow down [@Barnes03]. Knowing the rotational period of WASP-107 from the detected rotational modulation (see Section 6), we estimated the stellar age with the gyrochronological relation by @Barnes07 to obtain $0.6\pm0.2$Gyr.
Our isochronal age estimate for WASP-118 agrees well with the age provided by @Hay16 who used the same approach but using the system parameters derived only from the ground-based observations.
The age discrepancy between the isochronal and gyrochronological age estimate for WASP-107 is significant. Similar discrepancies have been observed for many other K-type stars hosting transiting exoplanets (e.g. @Maxted15b). It has been suggested that stars hosting massive short-period planets may have been spun-up by the tidal interaction with the planet and thus exhibit a lower gyrochronological age [@Maxted15b]. However, the warm-Saturn WASP-107b is not massive enough and does not orbit close enough to its host star to cause a significant tidal spin-up. A more probable reason for the observed age discrepancy in the case of WASP-107 is the radius anomaly, in which late-type stars exhibit larger radii than is predicted by the stellar models [@Popper97]. The radius anomaly is an active research topic driven by the advances in simulating convections in low-mass stars [@Ludwig08] and incorporating magnetic fields into stellar models [@Feiden13]. @Morales10 have demonstrated that the presence of starspots near the poles of low-mass stars could affect the stellar radii and cause the observed radius anomaly. The age discrepancy of magnetically active K-type WASP-107 may therefore be tentatively attributed to starspots.
CONCLUSIONS
===========
The two main results presented in this paper are the direct detection of magnetic activity in the short-cadence *K2* lightcurve of WASP-107 and the detection of stellar variability of WASP-118.
The magnetic activity of WASP-107 is manifest firstly as a rotational modulation which gives a stellar rotational period of $17.5\pm1.4$d. We also detect a total of 5 definite and 4 possible starspot occultation events. With the planet’s orbital period being nearly one-third of the rotational period of the star, we might expect to see the same starspot recurring every three transits. Since we found no evidence of recurring starspots, we suggest that the system is misaligned, unless the starspots’ lifetimes are shorter than the rotational period of the star.
The multi-periodic variability in the lightcurve of WASP-118 indicates that the star is likely a low-amplitude $\gamma$Doradus pulsator. WASP-118 is a good target for a follow-up asteroseismic analysis in order to obtain more precise stellar parameters and to investigate star–planet interactions as a possible cause for the observed stellar pulsations.
Our refinement of WASP-107 system parameters may also prove beneficial, since the planet lies in the transition region between ice and gas giants. Knowing precise system parameters of such planets is crucial for understanding why some ice giants do not become gas giants.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We thank the anonymous referee for their helpful comments. We also thank Prof. Suzanne Aigrain for her help modifying the K2SC data reduction procedure to accept the *K2* short-cadence data. We gratefully acknowledge the financial support from the Science and Technology Facilities Council (STFC), under grants ST/J001384/1, ST/M001040/1 and ST/M50354X/1. This paper includes data collected by the *K2* mission. Funding for the *K2* mission is provided by the NASA Science Mission directorate. This work made use of PyKE [@Still12], a software package for the reduction and analysis of *Kepler* data. This open source software project is developed and distributed by the NASA Kepler Guest Observer Office. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
\[lastpage\]
[^1]: E-mail: t.mocnik@keele.ac.uk
[^2]: http://exoplanetarchive.ipac.caltech.edu/
|
---
abstract: 'In neutrino experiments, neutrino energy reconstruction is crucial because neutrino oscillations and differential cross-sections are functions of neutrino energy. It is also challenging due to the complexity in the detector response and kinematics of final state particles. We propose a regression Convolutional Neural Network (CNN) based method to reconstruct electron neutrino energy and electron energy in the NOvA neutrino experiment. We demonstrate that with raw detector pixel inputs, a regression CNN can reconstruct event energy even with complicated final states involving lepton and hadrons. Compared with kinematics-based energy reconstruction, this method achieves a significantly better energy resolution. The reconstructed to true energy ratio shows comparable or less dependence on true energy, hadronic energy fractions, and interaction modes. The regression CNN also shows smaller systematic uncertainties from the simulation of neutrino interactions. The proposed energy estimator provides improvements of $16\%$ and $12\%$ in RMS for $\nu_e$ CC and electron, respectively. This method can also be extended to solve other regression problems in HEP, taking over kinematics-based reconstruction tasks.'
author:
- 'Pierre Baldi, Jianming Bian, Lars Hertel, Lingge Li'
title: Improved Energy Reconstruction in NOvA with Regression Convolutional Neural Networks
---
Introduction
============
Energy reconstruction plays a key role in High Energy Physics (HEP), as it converts detector unit readout into kinematics of interactions. In a HEP analysis, event and particle types are usually identified first and then reconstructed energies are assigned to individual final state particles and the overall event via the energy reconstruction process. Based on these energies, physical phenomena can be studied as functions of overall energy and internal kinematics of an event. In neutrino physics, energy reconstruction is essential and challenging for the neutrino oscillation studies. Traditionally, particle energies are reconstructed by adding up or fitting to the hits on detector readout units, and event energy is reconstructed as a function of particle energies in the event. In this paper, a deep-learning based method for energy reconstruction of neutrino oscillations will be discussed. The method directly uses detector hits as inputs without intermediate steps.
Neutrino oscillations are so far the only experimental observation beyond the standard model since its development about 30 years ago. Neutrinos are very elusive since they only interact via the weak nuclear force. They have three active flavor states $\nu_e$, $\nu_\mu$ and $\nu_\tau$. Each is a different superposition of three mass states $\nu_1$, $\nu_2$ and $\nu_3$. Neutrinos can oscillate between flavor states. The relationship between flavor and mass states, and the oscillation between flavors are commonly described by the Pontecorvo$-$Maki$-$Nakagawa$-$Sakata (PMNS) matrix [@ref:pmns].
Two fundamental questions of interest are remaining to be determined by studying neutrino oscillations. First, what is the CP phase $\delta$? The CP phase $\delta$ relates to the difference in oscillation behavior between neutrinos and anti-neutrinos. CP violation in the lepton sector holds implications for matter-antimatter asymmetry in the Universe through leptogenesis. Second, what is the mass ordering ($m_3>m_{1,2}$ or $m_{1,2}>m_3$) between neutrinos? The mass hierarchy provides key information for future searches of the neutrino-less double beta decay. Observing the neutrino-less double beta decay would imply that the neutrino is a Majorana particle meaning it is its own anti-particle. The mass hierarchy will also constrain the so far undetermined absolute neutrino masses.
Aiming to solve these two questions, current and future neutrino oscillation experiments focus on electron neutrino appearance ($\nu_\mu\to\nu_e$). Neutrino oscillation can be measured by sending a beam of neutrinos of one flavor through a detector. Oscillated neutrinos arriving in the detector will be of a different flavor than the one generated from the beam. Observed neutrino interactions need to be tagged by their flavors. Importantly, the energy of the incoming neutrino needs to be well reconstructed as the $\nu_\mu\to\nu_e$ oscillation probability changes as a function of neutrino energy.
One of the challenges is thus a good estimation of electron neutrino energy. The accuracy of neutrino energy reconstruction influences how precisely neutrino oscillation parameters can be estimated. An electron neutrino can only be identified in charged current (CC) interactions where the electron neutrino converts into an electron. The $\nu_e-CC$ events are characterized by an electron along with other potential activity produced by hadrons. Traditionally, the reconstructed neutrino energy is calculated as a function of electron and hadron visible energy deposits. However, the estimation of the energy with the kinematics based method is complicated by missing energy in dead material, non-linear detector energy responses, invisible energy and identities (mass) of hadrons, and overlaps between electron and hadron showers.
To address these issues, the neutrino energy can be predicted directly from images of interactions. These images provide additional information on interaction details such as trajectories and energy deposit patterns of electrons and hadrons. Deep learning and deep convolutional neural network methods are a natural choice for processing data produced by complex detectors in high energy physic [@Shimmin:2017mfk; @Sadowski:2017ilo; @Baldi:2016fql; @Baldi:2014kfa] and have demonstrated success in [**classification**]{} problems in collider and neutrino experiments. NOvA has pioneered this deep learning technique in flavor tagging problems and has used it to produce oscillation physics results [@Aurisano:2016jvx; @Adamson:2017gxd]. CNN based event identification and reconstruction have also been investigated in other neutrino experiments [@Racah:2016gnm; @Acciarri:2016ryt; @Renner:2016trj; @Perdue:2018ihs; @Delaquis:2018zqi]. In this work, we propose to develop a [**regression**]{} CNN based method to precisely reconstruct electron neutrino energy and electron energy in the NOvA neutrino experiment. Neutrino interactions at NOvA typically involve multiple final state particles with complicated kinematics. The convolutional filters in CNNs can extract a richer set of features of these events than the sum of energy deposits. For example, two neutrinos could deposit almost exactly the same amount of energy in the detector and convolutional neural network features learned from event topologies could be used to make a more refined prediction on the true neutrino energy than the kinematics based method.
Reconstruction of individual final state particle energy in an interaction is a basic task of energy reconstruction. Specifically, these particle energies are used to study the kinematics, such as the momentum and energy transfer, in neutrino interactions. To demonstrate that the regression CNN can also reconstruct single particle energy, a similar method to the electron neutrino energy estimator is used to estimate electron energy. This estimator can be used for cross-section measurements and shower reconstruction study.
The NOvA Experiment {#sec:novaexp}
===================
NOvA uses an intense neutrino beam and sends it through sensitive, fine-grained detectors for long periods of time. The NuMI (Neutrinos at the Main Injector) muon neutrino beam is produced at Fermilab, Illinois. Aimed at 3.3 degrees downward, the beam travels 810 km through the earth to the 14 kilotons far detector (FD) in Ash River, Minnesota [@Adamson:2015dkw]. The far detector measures electron neutrinos oscillated from muon neutrinos in the beam. The NOvA experiment has the longest beam-detector distance in the world which maximizes the matter effect and allows a measurement of the neutrino mass ordering. Additionally, a 330 ton functionally identical near detector at Fermilab measures unoscillated beam neutrinos and estimates backgrounds and signals at the far detector. Both detectors are located 14 milli-radians off the centerline of the neutrino beam. This allows the detectors to capture a narrow energy spectrum of neutrinos at approximately 2 GeV. This is the energy at which the oscillation probability from a muon neutrino to an electron neutrino is expected to be at its peak.
The NOvA detectors are constructed in layers of alternating vertical and horizontal PVC cells; activities in the cells are recorded in a top view and a side view. There are 344,064 cells in the far detector and 18,000 cells in the near detector. In the far detector, each cell is 3.9 cm wide, 6.0 cm deep and 15.6 m long. The cells are made of highly reflective plastic filled with liquid scintillator. The scintillation light produced by neutrino interactions in the detectors is collected by a wavelength shifting fiber connected to an avalanche photodiode installed on one end of each cell. The readouts from these photodiodes are converted to calorimetric energy for physics analyses.
Methods {#sec:method}
=======
Simulated Data Sample {#sec:sample}
---------------------
### Simulation
The standard NOvA simulation is used to generate training and validation samples for the regression CNN. The simulation of NuMI neutrino beam is described in Ref. [@NOvA:2018gge]. The beam is simulated by GEANT4 [@Agostinelli:2002hh] and corrected according to external thin-target hadroproduction data with the PPFX tool [@Aliaga:2016oaz]. The flux shape of the NuMI neutrino beam at NOvA is referred to as the regular flux in this paper. Since NOvA is an off-axis experiment, the neutrino spectrum at the NOvA far detector from the NuMI flux peaks at about $2$ GeV, close to the $\nu_\mu\to\nu_e$ oscillation maximum. Since there are few low energy neutrino ($<$ 1 GeV) events in the NOvA FD Monte Carlo sample with the regular NuMI flux, the regression CNN $\nu_e$ energy trained with it has a significant true energy dependence (see Section \[sec:result\]). To minimize the dependence of estimated neutrino energy on true neutrino energy in the $\nu_e$ energy training, a flat neutrino flux shape is used to generate the far detector $\nu_e$ CC Monte Carlo sample to train the regression CNN for $\nu_e$ energy reconstruction. In the case of electron shower energy estimation, electrons from the regular flux $\nu_e$ CC far detector Monte Carlo sample are used in the electron energy regression CNN training and its validation.
At NOvA, interactions of neutrinos on nuclei are simulated by GENIE [@Andreopoulos:2009rq], and detector responses are then simulated by GEANT4. The customized NOvA detector simulation chain is described in [@Aurisano:2015oxj].
To study the $\nu_e$ interactions in the far detector Monte Carlo, we generate $\nu_e$ interactions with energies taken from the $\nu_\mu$ flux distribution. After that, no neutrino oscillations are applied to the training samples. This is equivalent to assuming all muon neutrinos oscillate to electron neutrinos in the far detector. To study realistic energy reconstruction performances from different energy estimators, the real $\nu_e$ appearance signal in the far detector can be obtained by applying realistic oscillation weights to this sample.
The simulation produces image pairs of the entire detector. As explained in Section \[sec:novaexp\] the two images correspond to cells in the top view (X-view) planes and side view (Y-view) planes of the detector. The images have a size 896$\times$384 (horizontal $\times$ vertical), where each pixel corresponds to the energy deposited in the corresponding detector cell. The horizontal coordinate (0-895) of a pixel represents the plane index of the detector cell and the vertical coordinate (0-383) represents the cell index in that plane. Since X-view planes and Y-view planes are assembled alternatively, all pixels with odd (even) plane indices are set to zero for X-View (Y-View). The neutrino flavor and the type of interaction are tagged by the true neutrino interaction information in GENIE.
### Reconstruction
The overall reconstruction process at NOvA is described in [@ref:reco0]. First, different neutrino interactions captured in the same pair of detector views are separated [@ref:reco1]. Cell hits are clustered by space and time. This separates neutrino interactions caused by beam neutrinos from cosmic ray neutrinos in a time window. The procedure collects cell hits from a single neutrino interaction (slice). The slices then serve as the foundation for all later reconstruction stages. We will refer to one slice as neutrino interaction from here on.
For each neutrino interaction, the vertex is then identified. The vertex is where the neutrino interacts with the detector material. All particles created in the interaction originate at the vertex. In order to reconstruct the vertex position, a modified Hough transform is used to fit straight-lines to cell hits. Then the lines are tuned in an iterative procedure until they converge to the image’s reconstructed vertex [@ref:reco2; @ref:reco3; @ref:reco4; @ref:fuzzyk; @Niner:2015aya]. The cell closest to the reconstructed interaction vertex in each view is chosen as the reference cell in our pixel maps. We will refer to the reference cell as the reconstructed vertex from here on.
Since the size of the neutrino interaction is much smaller than the entire detector, the image can be cropped. An image can be cropped by considering the number of pixels that are occupied in all four directions from the reconstructed vertex. To determine a window size the distribution of electron neutrino interactions is inspected. The cropped image contains 30 pixels to the left and 120 pixels to the right of the vertex. In the vertical-direction, 70 pixels above and below the vertex are included. This produces images of $151 \times 141$ pixels in each view. On average, $99.5\%$ of the hits are contained in a cropped image.
![Electron neutrino image pair example.[]{data-label="fig:nue_image"}](nue_image.pdf){width="\linewidth"}
![Distributions of input cell energies (left) and true electron neutrino energies (right) from a subset of the regular flux training sample.[]{data-label="fig:nue_trainE_nonflat"}](traincellE_flat.pdf "fig:"){width="8cm"} ![Distributions of input cell energies (left) and true electron neutrino energies (right) from a subset of the regular flux training sample.[]{data-label="fig:nue_trainE_nonflat"}](trainnueE_flat.pdf "fig:"){width="8cm"}
![Distributions of input cell energies (left) and true electron neutrino energies (right) from a subset of the regular flux training sample.[]{data-label="fig:nue_trainE_nonflat"}](traincellE_nonflat.pdf "fig:"){width="8cm"} ![Distributions of input cell energies (left) and true electron neutrino energies (right) from a subset of the regular flux training sample.[]{data-label="fig:nue_trainE_nonflat"}](trainnueE_nonflat.pdf "fig:"){width="8cm"}
### The Electron Neutrino Dataset
True neutrino information is used to select $\nu_e$ CC events from all neutrino interactions. To to speed up the processing time, a loose pre-selection is applied to remove events with long prongs or too many hits. The pre-selection requires the number of occupied cells in the neutrino interaction to be less than 200. Additionally, the length of the longest prong is required to be less than 500 cm. Prongs are collections of cell hits with a start point and direction, which are reconstructed based on distances from hits to the lines associated with each of the particles that paths emanating from the reconstructed vertex [@ref:reco2; @ref:reco3; @ref:reco4; @ref:fuzzyk; @Niner:2015aya]. The pre-selection keeps most of the electron-neutrino appearance signal while rejecting a large fraction of background events with long muon tracks. No requirements are applied to calorimetric energy or reconstructed neutrino interaction identities.
We use 0.98 million simulated samples of electron neutrino interactions as the electron neutrino dataset. Each sample consists of a pair of images from the two detector views, the reconstructed vertex and the simulated truth of electron neutrino energy. We split the dataset into a training sample with 0.75 million events and a validation sample with 0.23 million events. One example pair of images from the training dataset is shown in Figure \[fig:nue\_image\]. In Figure \[fig:nue\_trainE\_flat\] (left) we show the spectrum of cell energy deposits in cell hits from a subset of the flat flux training sample. Figure \[fig:nue\_trainE\_flat\] (right) shows the spectrum of true $\nu_e$ energy from the subset of the flat flux training sample. One can find that there are enough events in the low $\nu_e$ energy region ($<$ 1 GeV) for training. As a comparison, in Figure \[fig:nue\_trainE\_nonflat\] the cell hit energy deposits and $\nu_e$energy from the regular flux $\nu_e$ FD Monte Carlo sample are shown. Since NOvA is an off-axis experiment, the $\nu_e$ energy in the FD is bell-shaped peaking around at $2$ GeV, and there are few events below 1 GeV for training.
![Electron shower image pair example.[]{data-label="fig:elec_image"}](electron_image.pdf){width="\linewidth"}
![Distributions of input cell energies (left) and true electron energies (right) from a subset of the regular flux training sample.[]{data-label="fig:elec_trainE_nonflat"}](trainshcellE_nonflat.pdf "fig:"){width="8cm"} ![Distributions of input cell energies (left) and true electron energies (right) from a subset of the regular flux training sample.[]{data-label="fig:elec_trainE_nonflat"}](trainshE_nonflat.pdf "fig:"){width="8cm"}
### The Electron Shower Dataset
Electron shower images are created from electron neutrino interactions by reconstructing the pixels corresponding to the electron shower and setting cell energy values for all other pixels to zero. We select electron showers from $\nu_e$ CC FD Monte Carlo events by matching the reconstructed shower direction to the true particle direction.
The creation of electron shower pixel maps starts with prongs. First, the shower core is defined based on the prong direction provided by the prong cluster. Then signal hits are collected in a column around this core. The electron deposits energy through ionization in the first few planes before it starts multiple scattering. In order to capture all deposits from the electron shower, the region corresponding to multiple scattering is enlarged. We require the radius to be twice the cell width for the first 8 planes from the start point of the shower. For the following planes, we require the radius to be $20$ times the cell width [@Bian:2015opa].
Electrons from the regular flux $\nu_e$ CC FD Monte Carlo sample are used for training. One example pair of images from the training dataset is shown in Figure \[fig:elec\_image\]. We use 660k simulated samples of electron showers as the electron shower dataset. Each sample consists of a pair of images, the reconstructed vertex and the true electron energy. We split the dataset into 610,000 training samples and 50,000 validation samples. In Figure \[fig:elec\_trainE\_nonflat\] (left) we show the spectrum of cell energy deposits from a subset of the regular flux training sample. Figure \[fig:elec\_trainE\_nonflat\] (right) shows the spectrum of true electron shower energies from the subset of training sample.
Neural Network Architecture {#sec:neural-network-architecture}
---------------------------
The electron neutrino energy model and electron shower energy model are equal in architecture but weights are not shared across the two predictors. The neural network input consists of two matrices of shape $(151, 141)$ which represent the pixel values of the images and the cell indices for each view given by the reconstructed vertex. The output is one positive real-valued number. Inputs and outputs for each model are illustrated in Figure \[fig:two\_predictors\].
{width=".8\linewidth"}
The neural network architecture is modified from the architecture used by NOvA’s CVN event classifier [@Aurisano:2016jvx]. This architecture is optimized from the convolutional neural network GoogLeNet [@ref:googlenet] developed for image recognition. In addition, NOvA’s CVN utilizes a siamese network structure [@baldi93finger]. The siamese network structure consists of two identical sub-networks whose outputs are merged to produce the final output. Each sub-network processes an image from one view. Weights are not shared between the sub-networks to provide independent information aggregation in each view. The sub-networks are constructed from convolutional layers and pooling layers.
Convolutional layers apply a weight matrix in a sliding window fashion to the input image. This allows computing the same feature at different locations in the input image producing an output referred to as a feature map. Using multiple weight matrices allows learning a variety of features from the input images. Stacking these convolutional layers makes it possible to learn higher level features with each additional layer. Pooling layers take a feature map and reduce its dimensionality. This is done by tiling the feature map and reducing each tile to its maximum or average. Pooling layers are normally used between convolution layers. The convolutional layers in our network are based on the Inception module introduced by Ref. [@ref:googlenet] as part of GoogLeNet.
GoogLeNet is the winner of the ImageNet Large Scale Visual Recognition Competition 2014 [@russakovsky2015imagenet] and state-of-the-art for convolutional neural networks with pixelmap inputs. Therefore, it is a good choice for our task. In particular, GoogLeNet uses the Inception module to efficiently extract features of different sizes from the input. This increases its modeling capacity without a significant increase in the computational cost.
Each sub-network here is constructed from a sequence of `Conv`-`MaxPool`-`Conv`-`Conv`-`MaxPool` layers followed by two Inception modules [@ref:googlenet]. The Inception module is a specific configuration of convolutional layers built to simultaneously extract features of different dimensions. The features are then concatenated and pooled. Each convolutional layer and Inception module has $32$ filters in the networks used. In the experiments, using additional Inception modules does not improve model performance. This is expected because the images here are sparse relative to natural images.
In the NOvA far detector, the scintillation light produced by neutrino interactions in each 15.5m-long detector cell is collected by the wavelength shifting fiber and read from the avalanche photodiode installed on one end of the cell. The attenuation of the scintillation light signal is a function of the distance from the interaction point to the readout photodiode, so the number of photoelectrons on the photodiode depends on the location of the interaction point. The readout threshold for each cell is a fixed number of photoelectrons, so the distribution of the cell energy deposit in each cell after the readout threshold cut is impacted by the position of the interaction with respect to the readout. This position dependence cannot be recovered by the cosmic attenuation calibration, which corrects the position dependence of the average number of photoelectrons for each cell using the minimum ionizing peak (MIP) position of cosmic muon hits. To consider this position effect in cell energies, we use the reconstructed vertex positions in the two views as neural network inputs. In our neural network architecture, after the inception modules and an average pooling layer, the output is flattened and concatenated with the reconstructed vertex position. A linear regression follows which produces the network output, the electron neutrino energy.
Neural Network Training {#sec:neural-network-training}
-----------------------
Neural networks for supervised learning are trained by defining a differentiable loss function $L$ between neural network outputs $f_{\mathbf{W}}(\mathbf{x}_i)$ and target values $y_i$. Here, $\mathbf{W}$ represents the weights of the neural network and $\mathbf{x}_i$ is the neural network input. The loss function represents the metric by which the neural network accuracy is assessed. During training the neural network weights $\mathbf{W}$ are iteratively updated to minimize $L$ using its gradient and a step size $\alpha$.
Typically these updates are computed over mini-batches of size $n$ which make up a partition of the total training dataset. Iteration over all mini-batches constitutes one epoch. Training parameters such as the step size $\alpha$ and the batch size $n$ are referred to as hyperparameters must be selected before training and possibly tuned using the loss function on the validation dataset. We utilize the hyperparameter optimization software SHERPA [@ref:sherpa2018] which implements a number of hyperparameter optimization strategies and visualization tools. By automating the task, this software significantly speeds up the computationally expensive of finding optimal hyperparameters.
Unlike classification problems such as image recognition and particle identification, the target value $y_i$ for the output (energy) of our regression neural network is a continuous variable varying over events. This requires the definition of an appropriate loss function for the task of the regression neural network. The goal is to minimize the standard deviation of a Gaussian fit to the peak of the histogram given by the energy resolution $\frac{E_{reco}-E_{true}}{E_{true}}$ on the test set. While this quantity cannot be directly optimized the absolute scaled error loss provides an appropriate surrogate for the task. The training loss function is then given by: $$L(\mathbf{W}, \{\mathbf{x}_i, y_i\}_{i=1}^n ) = \frac{1}{n} \sum_{i=1}^n |\frac{f_{\mathbf{W}}(\mathbf{x}_i)-y_i}{y_i}|.$$
Traditional loss functions for regression problems are the mean squared error $\frac{1}{n} \sum_{i=1}^n (f_{\mathbf{W}}(\mathbf{x}_i)-y_i)^2$ or the mean absolute error $\frac{1}{n} \sum_{i=1}^n |f_{\mathbf{W}}(\mathbf{x}_i)-y_i|$. The former is often used due to its relationship to the log-likelihood when the data distribution is assumed to be Normal, its strict convexity, the fact it can be decomposed into variance and bias, and many other desirable properties. In our case, however, the mean squared error is suboptimal, because its derivative with respect to $\mathbf{W}$ is $\frac{1}{n} \sum_{i=1}^n 2 (f_{\mathbf{W}}(\mathbf{x}_i)-y_i) \frac{\delta f_{\mathbf{W}}(\mathbf{x}_i)}{\delta \mathbf{W}}$. This increases proportionally with the distance of the predicted value from the truth. In other words, outliers will have increased impacts during gradient descent. In the training for neutrino energy, the events with large invisible energy due to dead material and hadronic interactions shouldn’t have much larger impacts than those whose visible energy is close to the true energy, so we choose the absolute error instead of the squared error in the loss function. Furthermore, the original CVN/GoogleNet are designed and trained for classification tasks, we optimized training hyperparameters for the regression task.
Input image pixels are typically normalized to increase numerical stability and gradient quality. Here most image pixels are zero and the non-zero ones tend to be small. We apply three normalization methods: mean zero unit variance standardization, log transformation, and constant scaling. The three methods produce similar results. Therefore, a constant scaling factor of $100$ is chosen after visual inspection of the input spectrum for $\nu_e$ (Figure \[fig:nue\_trainE\_nonflat\] (right)) and electron (Figure \[fig:elec\_trainE\_nonflat\] (right)).
The models are trained with stochastic gradient descent. The hyperparameter search yields as best hyperparameters: (1) a batch size of $n=32$; (2) an initial learning rate of $5\times10^{-4}$ with an exponential learning rate decay per batch of $1\times10^{-5}$; and (3) a momentum of $0.7$. Models are trained for 100 epochs, or until the validation loss does not increase by at least 0.001 for 5 epochs. The weights from the epoch with the best validation loss are kept.
Regularization techniques are also explored using random search implemented in SHERPA. Regularization refers to a set of methods that reduce modeling capacity to prevent fitting to noise in the training set (over-fitting). Here, the model training is optionally regularized with L2-penalty on all convolutional layer weights and on the fully connected layer weights. L2-penalty also referred to as weight-decay which adds the term $\lambda ||\mathbf{W}||_2^2$ to the loss function $L(\mathbf{W}, \{\mathbf{x}_i, y_i\}_{i=1}^n )$. The added term prevents weights from getting too large and thus reduces modeling capacity. Random search was applied to the L2-penalty multiplier $\lambda$ over a range of $1\times10^{-5}$ to $1\times10^{-7}$. To increase the robustness of learned features, dropout [@ref:srivastava2014dropout; @ref:baldidropout14] was also applied to the fully-connected layer. Dropout is a technique that randomly sets hidden layer units to zero with a given dropout-probability during the training. In the hyperparameter search, we let the dropout-probabilities range from $0$ to $0.4$. The best performing model found from random search had $\lambda=0$ and zero dropout-probability. While dropout and L2-penalty tend to be useful for classification there is an intuitive explanation as to why those methods decrease performance in our regression problem. In the case of classification, outputs do not directly depend on the magnitude of the neural network outputs since the outputs are normalized by the sum of all outputs. In regression, the output of the neural network has to exactly match the target value, which can take a wide range of values. If hidden layer units are randomly dropped as in dropout, this estimate may significantly change depending on what units are dropped. Similarly, L2-penalty may prevent weights from adopting the magnitude required by the scale of the targets of the prediction.
For the training and validation of the neural network, all models are implemented in Keras [@chollet2015keras] with Tensorflow backend.
Results {#sec:result}
=======
We use a simulated test data sample independent of the training and validation samples in Section \[sec:method\] to test the physics performance of the trained neural networks. The test $\nu_e$ CC sample is produced with the same simulation and reconstruction method as the training sample. Simulation and reconstruction at NOvA are described in Section \[sec:sample\]. The test sample has a regular flux. In order to mimic the real neutrino energy spectrum in the NOvA FD, we apply the neutrino oscillations to each FD MC sample by event weighting. To mimic overall energy resolution of the $\nu_e$ oscillation signal while keeping independent from specific CP and mass order choices, oscillation probabilities are calculated from the first-order terms in the full oscillation formula. The values of oscillation parameters are chosen to be $\sin^2\theta_{23} = 0.5$, $\sin^22\theta_{23} = 1$, $\Delta m^2_{32} = +2.35\times10^{-3}$eV$^{2}$ and $\sin^{2}2\theta_{13} = 0.1$.
$\nu_e$-CC neutrino energy
--------------------------
The proposed regression CNN energy estimator is compared with two methods used in previous NOvA $\nu_e$ analyses: calorimetric energy estimation and kinematics-based energy estimation. Used as the $\nu_e$ CC energy in NOvA’s first $\nu_e$ oscillation analysis in 2016 [@Adamson:2016tbq], the calorimetric energy estimator takes the sum of the calibrated calorimetric energy in each cell for an event and multiplies the sum by a scale factor. The scale factor corrects for the dead material in NOvA detectors and missing energy taken by undetected particles. It is estimated via simulated neutrino events. The kinematics-based energy estimator is based on the method used in NOvA’s $\nu_e$ analysis in 2017 [@ref:taenergy] (Kinematic Energy). This estimator is based on a quadratic function of the reconstructed electromagnetic and hadronic energy. The electromagnetic energy component is estimated by the sum of calorimetric energies from the electron and photons. The hadronic energy is estimated via the sum of calorimetric energies from hadrons such as pions, kaons, and protons. The electron, photons, and hadrons are identified by a deep-learning based particle identification algorithm called prong CVN [@Psihas:2018czu]. Parameters of the quadratic function are also determined using simulated data.
![Monte Carlo distributions of $\nu_e$ CC energy reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green), overlapping with true neutrino energy (dashed). The neutrino oscillations are applied.[]{data-label="fig:nueE"}](nueE.pdf){width="8cm"}
The true and reconstructed $\nu_e$ CC energy in the FD, weighted by the oscillation probabilities, are shown in Figure \[fig:nueE\]. The off-axis spectrum convoluted with the oscillation probability makes the $\nu_e$-CC energy spectrum peak at around 2 GeV.
![Monte Carlo distributions of ratios of differences between reconstructed and true $\nu_e$ CC energies to true $\nu_e$ CC energy in the calorimetric energy range of 0 to 5 GeV. Neutrino energy is reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:nueERes"}](nueERes.pdf){width="8cm"}
The overall performance is illustrated in Figure \[fig:nueERes\]. Shown are histograms of ($E_{reco}-E_{true})/E_{true}$, the ratio of the difference between reconstructed and true $\nu_e$ CC energy over true neutrino energy in the calorimetric energy range of 0 to 5 GeV. The neural network energy is the one with the best resolution. Gaussian fits to $(E_{reco}-E_{true})/E_{true}$ distributions provide relative resolutions of $8.9\%$ (CNN Energy), $10.1\%$ (Kinematic Energy) and $10.2\%$ (Calorimetric Energy), respectively. The relative resolution is defined as the ratio of the standard deviation to the peak value in a Gaussian fit. Relative RMSs (the ratio of RMS to Mean) are $11.1\%$ (CNN Energy), $13.2\%$ (Kinematic Energy) and $13.6\%$ (Calorimetric Energy).
Figure \[fig:nueEResvsTrueE\] shows means and RMSs of $(E_{reco}-E_{true})/E_{true}$ in each 1-GeV-wide true energy bin. Both Kinematic Energy and CNN energy estimators are determined from the NOvA FD $\nu_e$ CC signal sample with oscillated energy spectrum, peaking around 2 GeV. As shown in Figure \[fig:nueEResvsTrueE\] (left), the energy scale of the three estimators shows no significant biases with respect to the true neutrino energy, and the regression CNN has better energy resolutions.
The standard training sample of the described regression CNN $\nu_e$ energy estimator uses a flat flux. We also train the regression CNN using the regular flux with the peak around 2 GeV to understand the effect of the training energy spectrum on the linearity of the energy scale. The flat flux sample and the regular flux sample are defined in Section \[sec:neural-network-training\].
Energy scales for neutrino energy based on the flat flux training and regular flux training are shown in Figure \[fig:nueEResvsTrueE0\]. One can find that the energy scale from the flat flux training has less biases over true neutrino energy. The flat flux training, therefore, represents the preferred training mode to generate the regression CNN for the neutrino energy reconstruction.
![Means (left) and relative RMS (right, the ratio of RMS to the mean value of energy) of the Monte Carlo distributions of the ratios of differences between reconstructed and true $\nu_e$ CC energies to true $\nu_e$ CC energy for different true neutrino energy bins ranging from 0 to 5 GeV. Neutrino energy is reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:nueEResvsTrueE"}](FDMCMeanTrueE.pdf "fig:"){width="8cm"} ![Means (left) and relative RMS (right, the ratio of RMS to the mean value of energy) of the Monte Carlo distributions of the ratios of differences between reconstructed and true $\nu_e$ CC energies to true $\nu_e$ CC energy for different true neutrino energy bins ranging from 0 to 5 GeV. Neutrino energy is reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:nueEResvsTrueE"}](FDMCRMSTrueE.pdf "fig:"){width="8cm"}
![Means of the Monte Carlo distributions of ratios of differences between reconstructed and true $\nu_e$ CC energies to true $\nu_e$ CC energy for different true neutrino energy bins ranging from 0 to 5 GeV. Neutrino energy is reconstructed by CNN trained with flat flux (blue) and regular flux (red), the neutrino oscillations are applied.[]{data-label="fig:nueEResvsTrueE0"}](FDMCMeanTrueE0.pdf){width="8cm"}
Figure \[fig:nueEResMode\] shows estimator performance by interaction mode. $\nu_e$ CC interactions can be classified as quasi-elastic (QE), resonant (RES), and deep-inelastic scattering (DIS) modes. In a QE event, the nucleon ($p$ or $n$) recoils quasi-elastically from the scattering electron, and the electron, because of its small mass, takes the majority of the incident neutrino energy. Hadronic energy portions and hadron multiplicities vary in these three modes. For the RES mode, the nucleon is excited into baryonic resonances and decays to hadrons, so more neutrino energy is transferred into the hadronic system. In DIS events, the nucleon is smashed into several hadrons, requiring even larger neutrino energy transfer to the hadronic system. Figure \[fig:nueEResMode\] shows $(E_{reco}-E_{true})/E_{true}$ in these categories individually. The CNN Energy scale shows a better resolution and consistency among the interaction modes.
![Monte Carlo distributions of ratios of the difference between reconstructed and true $\nu_e$ CC energy to true neutrino energy for QE, RES and DIS modes. Neutrino energy is reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:nueEResMode"}](nueEResMode.pdf){width="16cm"}
![Monte Carlo distributions of ratios of differences between reconstructed and true $\nu_e$ CC energies to true $\nu_e$ CC energy in the calorimetric energy range of 0 to 5 GeV. Error bands represent systematic uncertainties evaluated by GENIE reweighting. Neutrino energy is reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:res_1d_syst"}](all_res_1d_syst.pdf){width="8cm"}
Systematic uncertainties in the energy reconstruction from the simulation of neutrino interactions are evaluated by using the reweighting knobs built into GENIE [@Andreopoulos:2015wxa]. Each reweighting knob computes a weighting factor that can be applied to MC events to vary normalization and/or shape of a specific type of interaction. In general, these GENIE reweighting knobs deal with systematic uncertainties from modeling of cross-sections, the hadronization, and final state interactions. The reweighting knobs used in this GENIE uncertainty study are similar to NOvA’s oscillation analysis Ref [@NOvA:2018gge]. We vary each reweighting knob by +1 $\sigma$ and -1 $\sigma$, where the size of the systematic variation $\sigma$ is the recommendation from the GENIE and NOvA authors, based on surveys of interaction models and existing experimental results. Both the background yield in the signal region before the background correction and the background correction factor determined by the Data-MC difference in the sideband are re-determined in the reweighted background MC.
The overall performance with GENIE systematic shifts is illustrated in Figure \[fig:res\_1d\_syst\]. Shown are histograms of $(E_{reco}-E_{true})/E_{true}$, the ratio of the difference between reconstructed and true $\nu_e$ CC energy over true neutrino energy in the calorimetric energy range of 0 to 5 GeV. The systematic errors of $(E_{reco}-E_{true})/E_{true}$ are $0.2\%$ (CNN Energy), $0.6\%$ (Kinematic Energy) and $0.9\%$ (Calorimetric Energy), respectively. The systematic errors of the relative RMSs are $0.3\%$ (CNN Energy), $0.4\%$ (Kinematic Energy) and $0.4\%$ (Calorimetric Energy). Systematic errors of mean and RMS in each energy bin are shown in \[fig:mean\_rms\_syst\]. The regression CNN shows smallest systematic uncertainties from the simulation of neutrino interactions.
![Means (left) and relative RMS (right) of the Monte Carlo distributions of the ratios of differences between reconstructed and true $\nu_e$ CC energies to true $\nu_e$ CC energy for different true neutrino energy bins ranging from 0 to 5 GeV. Error bars represent systematic uncertainties evaluated by GENIE reweighting. Neutrino energy is reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:mean_rms_syst"}](all_mean_syst.pdf "fig:"){width="8cm"} ![Means (left) and relative RMS (right) of the Monte Carlo distributions of the ratios of differences between reconstructed and true $\nu_e$ CC energies to true $\nu_e$ CC energy for different true neutrino energy bins ranging from 0 to 5 GeV. Error bars represent systematic uncertainties evaluated by GENIE reweighting. Neutrino energy is reconstructed by the regression CNN (CNN Energy, blue), particle kinematic information (Kinematic Energy, red), and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:mean_rms_syst"}](all_rms_syst.pdf "fig:"){width="8cm"}
Electron Shower Energy
----------------------
The reconstructed electron shower energy given by the regression CNN (CNN Energy) is compared to the sum of the calibrated calorimetric energies (Calorimetric Energy) in electron showers. In a simulated $\nu_e$ CC event, the most energetic shower matched to a true electron is chosen as the electron shower sample. The neutrino oscillation weights are applied to the $\nu_e$ CC events. True electron energy and reconstructed electron shower energy distributions in the FD are shown in Figure \[fig:elecE\]. The CNN electron shower energy is closer to the true electron energy than Calorimetric Energy. Overall $(E_{reco}-E_{true})/E_{true}$ for electron showers are shown in Figure \[fig:elecERes\], with relative Gaussian resolutions of $8.2\%$ (CNN Energy) and $9.6\%$ (Calorimetric Energy) and relative RMS of $13.4\%$ (CNN Energy) and $15.2\%$ (Calorimetric Energy) . Means and RMSs of $(E_{reco}-E_{true})/E_{true}$ in each 1-GeV-wide true electron energy bin, and histograms of $(E_{reco}-E_{true})/E_{true}$ in different interaction modes are shown in Figure \[fig:elecEResvsE\] and \[fig:elecEResMode\]. One can find that the CNN Energy has better resolutions. The bias in CNN electron energy at low energies is caused by the small proportion of low energy electrons in the regular flux sample used for training.
![Monte Carlo distributions of reconstructed electron shower energy and true electron energy (dashed) in $\nu_e$ CC events. The shower energy is reconstructed by CNN (CNN Energy, blue) and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:elecE"}](elecE.pdf){width="8cm"}
![Monte Carlo distributions of ratios of reconstructed electron shower energy to true electron energy in $\nu_e$ CC events in the shower calorimetric energy range of 0 to 5 GeV. Shower energy is reconstructed by CNN (CNN Energy, blue) and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:elecERes"}](elecERes.pdf){width="8cm"}
![Means (left) and RMSs (right) of the Monte Carlo distributions of the ratios of reconstructed electron shower energy to true electron energy in $\nu_e$ CC events for different true electron energy bins ranging from 0 to 5 GeV. Shower energy is reconstructed by CNN (CNN Energy, blue) and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:elecEResvsE"}](FDMCMeanElecTrueE.pdf "fig:"){width="8cm"} ![Means (left) and RMSs (right) of the Monte Carlo distributions of the ratios of reconstructed electron shower energy to true electron energy in $\nu_e$ CC events for different true electron energy bins ranging from 0 to 5 GeV. Shower energy is reconstructed by CNN (CNN Energy, blue) and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:elecEResvsE"}](FDMCRMSElecTrueE.pdf "fig:"){width="8cm"}
![Monte Carlo distributions of ratios of reconstructed electron shower energy to true electron energy in $\nu_e$ CC events for QE, RES and DIS modes. Shower energy is reconstructed by CNN (CNN Energy, blue) and summing the calibrated calorimetric energy in each cell (Calorimetric Energy, green). The neutrino oscillations are applied.[]{data-label="fig:elecEResMode"}](elecEResMode.pdf){width="16cm"}
Summary
=======
We developed regression CNNs with direct pixel-level inputs for electron neutrino energy and electron shower energy reconstruction. This is an early effort and proof-of-concept of using CNNs to solve regression problems such as energy reconstruction, vertex reconstruction, and track parameter determination in HEP. It was found that the absolute scaled error provides a useful neural network loss function when the goal is to optimize energy resolution histograms as commonly used in HEP. We also describe the best training parameters found from hyperparameter search for this regression task. This work demonstrates that energy reconstruction tasks can be simplified without elaborate energy scale calibration and model-dependent fits. The performance of the CNN energy reconstruction was verified for different energy bins, different hadronic energy fractions and different interaction modes. In all cases, the regression CNN energy achieves superior performance compared with kinematics based energy reconstruction methods. The regression CNN also shows smaller systematic uncertainties from the simulation of neutrino interactions.
Acknowledgments
===============
The authors thank the NOvA collaboration for use of its Monte Carlo simulation software and related tools. This work was supported by the US Department of Energy, the US National Science Foundation and University of California, Irvine. The DOE grant is “FY 2018 Research Opportunities in High Energy Physics" (Comparative Review) Funding Opportunity Announcement \[DE-FOA-0001781\]. NOvA receives additional support from the Department of Science and Technology, India; the European Research Council; the MSMT CR, Czech Republic; the RAS, RMES, and RFBR, Russia; CNPq and FAPEG, Brazil; and the State and University of Minnesota. We are grateful for the contributions of the staff at the Ash River Laboratory, Argonne National Laboratory, and Fermilab. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the US DOE.
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abstract: 'We study the asymptotic expansion for the Landau constants $G_n$, $$\pi G_{n}\sim \ln(16N)+\gamma+\sum^{\infty}_{k=1}\frac{\alpha_k}{N^k} ~~\mbox{as} ~ n\rightarrow\infty,$$ where $N=n+1$, and $\gamma$ is Euler’s constant. We show that the signs of the coefficients $\alpha_{k}$ demonstrate a periodic behavior such that $(-1)^{\frac {l(l+1)} 2} \alpha_{l+1}< 0$ for all $l$. We further prove a conjecture of Granath which states that $(-1)^{\frac {l(l+1)} 2} \varepsilon_l(N)<0$ for $l=0,1,2,\cdots$ and $n=0,1,2,\cdots$, $\varepsilon_l(N)$ being the error due to truncation at the $l$-th order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the form $$\ln(16N)+\gamma+\sum_{k=1}^{p}\frac{\alpha_{k}}{N^{k}}<\pi G_{n}<\ln(16N)+\gamma+\sum_{k=1}^{q}\frac{\alpha_{k}}{N^{k}}$$ for all $n=0,1,2\cdots$, all $p=4s+1,\; 4s+2$ and $q=4m,\; 4m+3$, with $s=0,1,2,\cdots$ and $m=0, 1, 2,\cdots$.'
author:
- 'Chun-Ru Zhao, Wen-Gao Long and Yu-Qiu Zhao[^1]'
date: ' [*[Department of Mathematics, Sun Yat-sen University, GuangZhou 510275, China]{}*]{} '
title: Proof of a conjecture of Granath on optimal bounds of the Landau constants
---
[*[MSC2010:]{}*]{} 39A60; 41A60; 41A17; 33C05
.3cm [*[Keywords: ]{}*]{} Landau constants; second-order linear difference equation; sharper bound; asymptotic expansion; hypergeometric function .3cm
Introduction and Statement of Results
=====================================
In 1913, Landau [@Landau] proved that if $f(z)$ is analytic in the unit disc, and $\left|f(z)\right| <1$ for $|z|<1$, with the Maclaurin expansion $$f(z)=a_0+a_1 z+a_2 z^2+\cdots+a_n z^n+\cdots,~~ \left|z\right|<1,$$ then there exist constants $G_n$ such that $$\left| a_0+a_1+\cdots+a_n \right|\leq G_n,~~n=0,1,2,\cdots,$$ and the bound is optimal for each $n$, where $G_0=1$, and $$\label{Landau-constants}
G_{n}=1+\left (\frac{1}{2}\right )^{2}+\left (\frac{1\cdot3}{2\cdot4}\right )^{2}+\cdots
+\left (\frac{1\cdot3\cdot\cdots(2n-1)}{2\cdot4\cdot\cdots(2n)}\right )^{2}~~\mbox{for}~~n=1,2,\cdots.$$
The constants $G_n$ are termed the Landau constants. The large-$n$ behavior is known from the very beginning. Landau [@Landau] derived that $$G_n\sim \frac{1}{\pi}\ln n ~~\mbox{as}~n\rightarrow\infty;$$see also Watson [@Watson]. It is worth mentioning that there exist generating functions for these constants; cf. [@Cvijovic-Srivastava], possible $q$-versions of the constants; cf. [@ismail-li-rahman2015], and an observation made by Ramannujan (cf. [@Cvijovic-Srivastava]) that relates the Landau constants to the generalized hypergeometric functions. Useful integral representations for $G_n$ have been obtained from such relations; cf., e.g., Watson [@Watson]; see also Cvijović and Srivastava [@Cvijovic-Srivastava].
The approximation of $G_n$ has gone in two related directions. One is to obtain large-$n$ asymptotic approximations for the constants, in a time period spanning from the early twentieth century [@Landau; @Watson] to very recently [@Cvijovic-Srivastava; @li-liu-xu-zhao1]. The other direction is to find sharper bounds of $G_n$ for all nonnegative integers $n$. Authors working on the sharper bounds includes Brutman [@Brutmam] and Falaleev [@Falaleev] (in terms of elementary functions), Alzer [@Alzer] and Cvijović and Klinowski [@Cvijovic-Klinowski] (using the digamma function), Zhao [@zhao], Mortici [@Mortici] and Granath [@Granath] (involving higher order terms), and Chen and Choi [@chen-choi2014] and Chen [@chen2014] (digamma function and higher order terms). The list is by no means complete. The reader is referred to [@Cvijovic-Srivastava; @li-liu-xu-zhao1; @li-liu-xu-zhao2] for a historic account.
Optimal bounds up to all orders
-------------------------------
Attempts have been made to seek bounds in a sense optimal, and up to arbitrary accuracy.
In 2012, Nemes [@nemes] derived full asymptotic expansions. For $0 <h < 3/2$, he shows that the Landau constants $G_n$ have the asymptotic expansion $$\label{nemes-expansion}
G_n\sim \frac 1 \pi \ln (n+h) +\frac 1 \pi (\gamma+4\ln 2 ) - \sum_{k\geq 1}\frac {g_k(h)}{(n+h)^k}~~\mbox{as}~~n\rightarrow +\infty,$$ where $\gamma=0.577215\cdots$ is Euler’s constant. Earlier in 2011, the special cases $h=\frac 12$ and $h=1$ were established by Nemes and Nemes [@nemes-nemes] using a formula in [@Cvijovic-Klinowski]. They also conjecture in [@nemes-nemes] a symmetry property of the computable constant coefficients such that $g_k(h)=(-1)^k g_k(3/2-h)$ for every $k\geq 1$. The conjecture has been proved by G. Nemes himself in [@nemes]. A natural consequence is that for $h=3/4$, all odd terms in the expansion vanish. In this important special case, Nemes [@nemes] has further proved that
(Nemes) The following asymptotic approximation holds: $$\label{beta-alternative}
\pi G_n\sim \ln (n+3/4) + \gamma+4\ln 2 + \sum_{s=1}^\infty \frac { \beta_{2s}}{ (n+3/4)^{2s}},~~n\rightarrow \infty,$$ where the coefficients $(-1)^{s+1}\beta_{2s}$ are positive rational numbers.
The derivation of Nemes [@nemes] is based on an integral representation of $G_n$ involving a Gauss hypergeometric function in the integrand. An entirely different difference equation approach is applied in Li [*et al.*]{} [@li-liu-xu-zhao1] to obtain full asymptotic expansions with coefficients iteratively given. What is more, in a follow-up paper [@li-liu-xu-zhao2], it is shown that the error due to truncation of is bounded in absolutely value by, and of the same sign as, the first neglected term for all $n=0,1,2,\cdots$. An immediate corollary is
(Li, Liu, Xu and Zhao) For $N=n+3/4$, it holds $$\label{beta-bounds}
\ln N+\gamma+4\ln 2+\sum_{s=1}^{2m}\frac{\beta_{2s}}{N^{2s}}< \pi G_n < \ln N+\gamma+4\ln 2+\sum_{s=1}^{2k-1}\frac{\beta_{2s}}{N^{2s}}$$ for all $n=0,1,2,\cdots$, $m=0,1,2,\cdots$, and $k=1,2,\cdots$.
In a sense, the formulas and in the above propositions seem to have ended a journey since one has thus obtained optimal bounds up to arbitrary orders. Yet there is an interesting observation worth mentioning, as presented in the 2012 paper [@Granath] of Granath; see also [@li-liu-xu-zhao2].
Granath derives an asymptotic expansion $$\label{Granath-expansion}
\pi G_{n}\sim \ln(16N)+\gamma +\sum_{k=1} ^\infty \frac{\alpha_k}{N^k}, ~~n\rightarrow\infty,$$ where $\alpha_k$ are effectively computable constants but not explicitly given, except for the first few. Here and hereafter we use the notation $N=n+1$.
Denoting the truncation of $$\label{Granath-truncated}
A_l(N)=\ln(16N)+\gamma +\sum_{k=1} ^l \frac{\alpha_k}{N^k},$$ then one of the main results in Zhao [@zhao] reads $A_2(N) < \pi G_n< A_3(N)$ for $n=0,1,2,\cdots$. Mortici [@Mortici] have actually proved that $
A_5(N)<\pi G_{n}<A_4(N)$ for all non-negative $n$.
In [@Granath], Granath proves that $A_5(N) <\pi G_{n} < A_7(N)$ and states that $A_9(N) <\pi G_{n} < A_{11}(N)$, for all non-negative $n$. Based on these formulas and numerical evidences, Granath proposes a conjecture.
(Granath) It holds $$\label{Granath-conjecture}
(-1)^{\frac {l(l+1)} 2} \left ( \pi G_{n}-A_l(N)\right ) <0$$for all $n=0, 1,2,\cdots$ and $l=0,1,2,\cdots$.
Statement of results
--------------------
We will show that the conjecture is true. To do so, we will make use of the second order difference equation for $G_n$ employed in [@li-liu-xu-zhao1], and some estimating techniques used in [@li-liu-xu-zhao2].
First we denote the error term $$\label{remainder}
\varepsilon_l(N)=\pi G_n -A_l(N)=\pi G_n -\left\{ \ln(16N)+\gamma +\sum_{k=1} ^l \frac{\alpha_k}{N^k} \right\};$$cf. , where $N=n+1$. It is readily seen that $\varepsilon_l(N)\sim {\alpha_{l+1}}/{N^{l+1}}$ as $N\to\infty$. Hence we may start by showing that holds for large $n$. To this aim, we have
\[coefficient\] The coefficients of the asymptotic expansion satisfy $$\label{the-coefficient-sgn}
(-1)^{\frac {l(l+1)} 2} \alpha_{l+1}< 0,~~l=0,1,2,\cdots.$$
Next, we will prove the conjecture for all non-negative $n$.
\[thm epsilon-N\] For $N=n+1$, it holds $$\label{the-remainder-sgn}
(-1)^{\frac {l(l+1)} 2} \varepsilon_l(N)<0$$ for $l=0, 1,2,\cdots$ and $n=0, 1,2,\cdots$.
As a straightforward application of Theorem \[thm epsilon-N\], we obtain the following sharp bounds up to arbitrary orders.
\[thm-inequalities\] For $N=n+1$, it holds $A_p(N) <\pi G_{n}<A_q(N)$, that is, $$\label{sharp-bounds}
\ln(16N)+\gamma+\sum_{k=1}^{p}\frac{\alpha_{k}}{N^{k}} <\pi G_{n}< \ln(16N)+\gamma+\sum_{k=1}^{q}\frac{\alpha_{k}}{N^{k}}$$ for all $n=0,1,2,\cdots$ and for all $p=4s+1,\;4s+2$ and $q=4m,\;4m+3$, with $s=0,1,2,\cdots$ and $m=0,1,2,\cdots$.
In view of Theorem \[coefficient\], we see that the bounds in are optimal as $n\to\infty$.
Theorem \[thm epsilon-N\] can actually be understood as an estimate of the error term, such that the error due to truncation is bounded in absolute value by, and of the same sign as, the first one or two neglected terms. Indeed, since $$\varepsilon_l(N)=\frac {\alpha_{l+1}}{N^{l+1}}+ \varepsilon_{l+1}(N)~~\mbox{and}~~\varepsilon_l(N)=\frac {\alpha_{l+1}} {N^{l+1}}+ \frac { \alpha_{l+2}}{N^{l+2}}+ \varepsilon_{l+2}(N),$$ taking into account the signs in Theorems \[coefficient\] and \[thm epsilon-N\], we have $$0< \varepsilon_{4k+1}(N)< \frac { \alpha_{4k+2}}{N^{4k+2}}+ \frac { \alpha_{4k+3}}{N^{4k+3}}~~\mbox{and}~~0< \varepsilon_{4k+2}(N)< \frac { \alpha_{4k+3}}{N^{4k+3}}$$for all non-negative integers $n$ and $k$, and $$\frac { \alpha_{4k+1}}{N^{4k+1}} < \varepsilon_{4k}(N)<0~~\mbox{and}~~ \frac { \alpha_{4k+4}}{N^{4k+4}} + \frac { \alpha_{4k+5}}{N^{4k+5}} < \varepsilon_{4k+3}(N)< 0$$for all non-negative integers $n$ and $k$.
As a by-product of the proof of Theorem \[thm epsilon-N\], we have approximations of the asymptotic coefficients, follows respectively from and :
\[cor-coefficient-approx\] Assume that $\alpha_k$ are the coefficients in the asymptotic expansion . Then we have $$\label{alpha-even-approx}
\alpha_{2k}=(-1)^{k+1} \frac { 2(2k-2)!} {(2\pi)^{2k}} \left (1+O\left (\frac 1 k\right )\right )$$and $$\label{alpha-odd-approx}
\alpha_{2k+1}=(-1)^{k+1} \frac {8 (2k)! \ln (2k+1)}{(2\pi)^{2k+2}} \left ( 1+ O \left (\frac 1 {\ln k}\right )\right )$$ as $k\to \infty$.
The asymptotic coefficients and the proof of Theorem \[coefficient\]
====================================================================
From the representation one obtains the recurrence relation $$G_{n+1}-G_n= \left (\frac {2n+1}{2n+2}\right )^2 \left (G_n-G_{n-1}\right ).$$ Set $N=n+1$, we may rewrite it as a standard second-order difference equation $$\label{difference-equation}
w(N+1)-\left (2-\frac 1 N+\frac 1{4N^2}\right ) w(N)+\left (1-\frac 1 {2N}\right )^2 w(N-1)=0,$$ where $w(N)=\pi G_n$. An interesting fact is that the formal solution to is an asymptotic solution; cf. Li and Wong [@Wong-Li1992a]; see also [@li-liu-xu-zhao1]. Hence the asymptotic series furnishes a formal solution of . Therefore, one way to determine the coefficients $\alpha_k$ is to substitute into and equalizing the coefficients of the same powers of $x=1/N$. We include some details as follows. $$\ln (1+x)+\sum^\infty_{k=1} \frac {\alpha_k x^k}{(1+x)^k}
-\left (2-x+\frac {x^2} 4\right ) \sum^\infty_{k=1} \alpha_k x^k
+\left (1-\frac x 2\right )^2 \left [ \ln (1-x)
+\sum^\infty_{k=1}\frac{\alpha_k x^k}{(1-x)^k}\right ]=0.$$ Using the Maclaurin series expansions, we have $$- \sum_{s=3}^\infty d_{0,s} x^s +\sum^\infty_{k=1} \alpha_k x^k \sum^\infty_{j=2} d_{k, j+k}x^j=
\sum^\infty_{s=3} \left ( \sum^{s-2}_{k=1} d_{k, s} \alpha_k - d_{0, s}\right ) x^s =0.$$ Accordingly, coefficients $\alpha_k$ are determined by $$\label{coefficient-alpha}
d_{s-2, s}\alpha_{s-2}+d_{s-3,s}\alpha_{s-3}+\cdots+d_{1, s}\alpha_1-d_{0,s}=0, ~~s=3,4, \cdots,$$ where the coefficients $d_{s-2, s}=(s-2)^2$ for $s=3,4,\cdots$, $$\begin{aligned}
& d_{0,s}=\frac {(-1)^{s} +1} s -\frac 1{s-1}+ \frac 1 {4(s-2)}~~\mbox{for}~s=3,4,\cdots,~~\mbox{and}\label{d-0-s} \\
& d_{k,s}=\frac {\left ( (-1)^{s-k} +1\right ) (k)_{s-k}} {(s-k)!} -\frac {(k)_{s-k-1}}{(s-k-1)!}+ \frac {(k)_{s-k-2}} {4(s-k-2)!} \label{d-k-s}
\end{aligned}$$ for $k=1,2,\cdots, s-3$ and $s=k+3, k+4,\cdots$.
Appealing to -, the first few coefficients $\alpha_k$ can be evaluated as $$\begin{array}{llll}
\alpha_1= -\frac 1 4 , &\alpha_2=\frac{5}{192} , &\alpha_3=\frac{3}{128} , &\alpha_4= -\frac{341}{122880}, \\[.1cm]
\alpha_5= -\frac{75}{8192}, &\alpha_6=\frac{7615}{8257536} , & \alpha_7=\frac{2079}{262144} , &\alpha_8= -\frac {679901}{1006632960} ,\\[.1cm]
\alpha_9=-\frac{409875}{33554432}, &\alpha_{10}=\frac{16210165}{17716740096} , & \alpha_{11}=\frac{31709469}{1073741824} , & \alpha_{12}=-\frac{568756771963}{281406257233920} .
\end{array}$$ One readily sees a periodic phenomenon of the signs of the coefficients, which agrees with Theorem \[coefficient\]. To give a full proof of the theorem, we may connect the coefficients with those in , and eventually with a certain hypergeometric function. Indeed, re-expanding the formula in descending powers of $N=n+1$ yields the expansion . Hence we have $$\label{alpha-beta}
\alpha_k=4^{-k} \left [ -\frac 1 k+\sum^k_{j=1} \frac {(k-1)! 4^j \beta_j} {(j-1)! (k-j)!}\right ],~~k=1,2,\cdots;$$ cf. [@li-liu-xu-zhao2 (4.4)], where $\beta_j$ vanish for odd integers $j$. We also note that the coefficients $\beta_{2k}$ possess a generating function, that is, $$\label{u-function}
u(x)= \sum^\infty_{k=0} \rho_k x^{2k} =\frac x {2\sin\frac x 2} F\left ( \frac 1 2, \frac 1 2; 1; \sin^2\frac x 4\right ):= \frac x {2\sin\frac x 2} F\left ( \sin^2\frac x 4\right ) ,$$ where $\rho_0=1$ and $\rho_s=\frac {(-1)^{s+1}\beta_{2s}}{(2s-1)!}$, $s=1,2,\cdots$ are the positive constants defined in [@li-liu-xu-zhao2 Sec. 3.1]. It is shown in [@li-liu-xu-zhao2] that the generating function $u$ solves a second-order differential equation, and consequently the hypergeometric function $F\left ( \frac 1 2, \frac 1 2; 1; t\right )$ is brought in. It is worth noting that the function also furnishes a generating relation for the Landau constants, namely $\frac {F(x)}{1-x}=\sum^\infty_{n=0} G_n x^n$ for small $x$; see [@nemes]. Here and hereafter we denote for short the hypergeometric function as $F(t)=F\left ( \frac 1 2, \frac 1 2; 1; t\right )$.
.4cm
[**[Proof of Theorem \[coefficient\]]{}**]{}. From we have $$\label{alpha-2k}
\frac {\alpha_{2k}}{(2k-1)!} =\sum^k_{s=0} \frac {(-1)^{s+1} \rho_s}{(2k-2s)!} \left (\frac 1 4\right )^{2k-2s}~~\mbox{for}~k=1,2,\cdots .$$Here use has been made of the fact that $\beta_{2s-1}=0$ for $s=1,2,\cdots$. From we further have $$\label{generating-function-2k1}
1+\sum^\infty_{k=1} \frac {(-1)^{k+1} \alpha_{2k}}{(2k-1)!}x^{2k} =\left\{\sum^\infty_{s=0} \rho_s x^{2s} \right\} \left\{
\sum^\infty_{s=0} \frac 1 {(2s)!} \left (-\frac {x^2} {16} \right )^{s}\right\} =u(x)\cos\frac x 4 .$$ Combining with , and applying a quadratic transformation formula, we have $$1+\sum^\infty_{k=1} \frac {(-1)^{k+1} \alpha_{2k}}{(2k-1)!}x^{2k} =\frac {\frac x 4}{\sin\frac x 4} F\left ( \sin^2\frac x 4\right )= \frac {\frac x 4}{\sin\frac x 4} \frac 1 {\cos^2\frac x 8} F\left ( \tan^4\frac x 8\right );$$see [@as (15.3.17)]. Each factor on the right-hand side possesses a Maclaurin expansion with positive coefficients; see Nemes [@nemes pp.842-843]. Hence we conclude that $$\label{alpha-even-sgn}
(-1)^{k+1} \alpha_{2k}>0~~\mbox{for}~k=1,2,\cdots.$$
Similarly, we may write $$\label{tlide a odd}
\sum^\infty_{k=0} \frac {(-1)^{k+1} \alpha_{2k+1}}{(2k)!}x^{2k}=\frac 1 4 \left\{\sum^\infty_{s=0} \rho_s x^{2s} \right\} \left\{
\sum^\infty_{s=0} \frac 1 {(2s+1)!} \left (-\frac {x^2} {16} \right )^{s}\right\} =u(x)\frac {\sin\frac x 4} x .$$Taking into account, we can write the right-hand side term as $$\frac 1 {4\cos\frac x 4} F\left ( \sin^2\frac x 4\right )= \frac 1 {4\cos\frac x 4} \frac 1 {\cos^2\frac x 8} F\left ( \tan^4\frac x 8\right ),$$which again has a Maclaurin expansion with all positive coefficients. Here we have used the formula $$\frac 1 {\cos t}=\sum^\infty_{k=0}\frac {(-1)^k E_{2k} }{(2k)!} t^{2k},$$where $E_{2k}$ are the Euler numbers such that $(-1)^k E_{2k}>0$ for $k=0,1,2,\cdots$; see [@nist (24.2.6)-(24.2.7)]. Accordingly we have $$\label{alpha-odd-sgn}
(-1)^{k+1} \alpha_{2k+1}>0~~\mbox{for}~k=0,1,2,\cdots.$$ A combination of and then gives . .5cm
Proof of Theorem \[thm epsilon-N\]
==================================
To give a rigorous proof of Theorem \[thm epsilon-N\], we introduce $$\label{R-l}
R_l(N)=\varepsilon_l(N+1)-\left (2-\frac 1 N +\frac 1 {4N^2}\right ) \varepsilon_l(N)+ \left (1-\frac 1 {2N}\right )^2 \varepsilon_l(N-1)$$for $l=0,1,2\cdots$ and $N=n+1=1,2,3,\cdots$, where $\varepsilon_l$ is the remainder term given in . Similar to the derivation of , substituting into , and again denoting $x=1/N$, we see that $R_l(N)$ is an analytic function of $x$ at the origin, with the Maclaurin expansion $$\label{R-l-expansion}
R_l(N)= \sum_{k=3}^\infty d_{0,k} x^k -\sum^l_{k=1} \alpha_k x^k \sum^\infty_{j=2} d_{k, j+k}x^j =\sum^\infty_{s=l+3} r_{l,s} x^s,$$ where, for $s=l+3, l+4,\cdots$, and $l=0,1,2,\cdots$, the coefficients in are $$\label{R-l-coefficient}
r_{l,s}=-\left(d_{l, s}\alpha_{l}+d_{l-1, s}\alpha_{l-1}+\cdots+d_{1, s}\alpha_1-d_{0, s}\right).$$
To justify Theorem \[thm epsilon-N\], we state a lemma as follows, leaving the proof of it to later sections.
[\[lemma-R(N)\]]{}For $N=n+1$, it holds $$\label{R-even-sgn}
\tilde R_{2l}(N):=(-1)^{l+1} R_{2l}(N)>0,~~n=1,2,3,\cdots,~~l=0,1,2,\cdots .$$
Now we prove the theorem, assuming that Lemma \[lemma-R(N)\] holds true..5cm
[**[Proof of Theorem \[thm epsilon-N\]]{}**]{}. For fixed $l$, $l=0,1,2\cdots$, first we show that $$\label{remainder-even-sgn}
\tilde\varepsilon_{2l}(N):=(-1)^{l+1} \varepsilon_{2l}(N)>0$$ for all $n=1,2,\cdots$, where $N=n+1$. To this aim, we note that $$\label{tilde-varepsilon-2l}
\tilde\varepsilon_{2l}(N) = (-1)^{l+1} \varepsilon_{2l}(N)= \frac {(-1)^{l+1}\alpha_{2l+1}}{N^{2l+1}}\left\{ 1+O\left ( \frac 1 N\right )\right \}
= \frac {\left |\alpha_{2l+1}\right |}{N^{2l+1}}\left\{ 1+O\left ( \frac 1 N\right )\right \}
>0$$ for $N$ large enough; cf. , and . Now assume that $\tilde \varepsilon_{2l}(N)>0$ is not true for some $N$. Then there exists a finite positive $M$ such that $$M=\max\{N=n+1: n\in \mathbb{N}~ \mbox{and}~ \tilde\varepsilon_{2l}(N)\leq 0\}.$$Thus for the positive integer $M$, we have $\tilde\varepsilon_{2l}(M)\leq 0$, while $\tilde\varepsilon_{2l}(M+1),~ \tilde\varepsilon_{2l}(M+2),~ \cdots > 0$.
Denoting $b(N)= \left (1-\frac 1 {2N}\right )^2$ for simplicity, from we have $$\tilde \varepsilon_{2l}(M+2)=(1+b(M+1)) \tilde \varepsilon_{2l}(M+1)+b(M+1) ( -\tilde \varepsilon_{2l}(M)) +\tilde R_{2l}(M+1).$$The later terms on the right-hand side are nonnegative (where $M+1\geq 2$), hence we have $$\tilde \varepsilon_{2l}(M+2)\geq (1+b(M+1)) \tilde \varepsilon_{2l}(M+1) >\tilde \varepsilon_{2l}(M+1).$$ Moreover, from we further have $$\tilde \varepsilon_{2l}(M+3)\geq (1+b(M+2)) \tilde \varepsilon_{2l}(M+2)+b(M+2) ( -\tilde \varepsilon_{2l}(M+1))
> \tilde \varepsilon_{2l}(M+2) .$$ Repeating the process gives $$\tilde \varepsilon_{2l}(M+k+1)
> \tilde \varepsilon_{2l}(M+k),~~k=1,2,\cdots .$$ By induction we conclude $$\label{inequality}
\tilde \varepsilon_{2l}(M+1) < \tilde \varepsilon_{2l}(M+k)$$for $k\geq 2$. Recalling that $\tilde \varepsilon_{2l}(N)=O\left ( N^{-2l-1}\right )$ for $N\to\infty$; cf. , letting $k\to\infty$ in gives $\tilde \varepsilon_{2l}(M+1)\leq 0$. This contradicts the fact that $\tilde\varepsilon_{2l}(M+1)>0$. Hence holds.
Now from , and , we have $$\label{remainder-odd-sgn}
(-1)^l \varepsilon_{2l-1}(N) = \frac { (-1)^{l} \alpha_{2l}} {N^{2l}} +(-1)^l \varepsilon_{2l}(N)<0~~\mbox{for}~~ l=1,2,3,\cdots,~~\mbox{and}~~ n=1,2,\cdots,$$ where $N=n+1$.
A combination of and gives . Thus completes the proof of the theorem.
Lemma \[lem-alpha-k-estimate\]: Estimating of the coefficients $\alpha_k$
=========================================================================
To prove Lemma \[lemma-R(N)\], first we estimate the coefficients $\alpha_k$, or, more precisely, the quantities $\tilde \alpha_{2k}= \frac {(-1)^{k+1} \alpha_{2k}} {(2k-1)!}$ and $\tilde \alpha_{2k+1}= \frac {(-1)^{k+1} \alpha_{2k+1}} {(2k)!}$ for $k=1,2,\cdots$. These are positive constants; cf. and . As a preparation, we give a brief account of the analytic continuation of the hypergeometric function. The reader is referred to [@li-liu-xu-zhao2 Sec. 3.2] for full details. We denote $$\label{varphi}
\varphi(z)=F\left(\sin^2 \frac{z}{4}\right )= F\left(\frac{1}{2},\frac{1}{2};1;\sin^2 \frac{z}{4}\right)~~\mbox{for}~~\Re z \in (-2\pi, 2\pi)\cup (2\pi, 6\pi).$$Then the piecewise-defined function $$\label{varphi-continuation}
v(z)=\left \{\begin{array}{ll}
\varphi(z), & 0\leq \Re z < 2\pi, \\
\varphi(z) \pm 2i \varphi(z-2\pi), & 2\pi< \Re z < 4\pi~\mbox{and}~\pm\Im z>0
\end{array}\right .$$furnishes an analytic continuation of $\varphi(z)$ in from the strip $\Re z \in [0, 2\pi)$ to the cut strip $0\leq \Re z<4\pi$ and $z\not\in [2\pi, +\infty)$. What is more, we have the connection formula (see [@li-liu-xu-zhao2 (3.17)]) $$\label{v-connection}
v(z)=v_A(z) -\frac 2 \pi \varphi (z-2\pi)\ln \left (2\pi -z \right )$$ for $0< \Re z <4\pi$, with $v_A(z)$ being analytic in the strip, and the branch of the logarithm being chosen as $\arg (2\pi -z)\in (-\pi, \pi)$. We proceed to show that
\[lem-alpha-k-estimate\] It holds $$\label{alpha-2k-estimate}
\frac{1.9621}{2k-1}\frac{1}{(2\pi)^{2k}}\leq \tilde \alpha_{2k}\leq\frac{2.2032}{2k-1}\frac{1}{(2\pi)^{2k}}
,~~ k=9, 10,11, \cdots$$ and $$\label{alpha-2k+1-estimate}
\frac{4\ln(2k+1)+0.6551}{\pi(2\pi)^{2k+1}}\leq \tilde \alpha_{2k+1}\leq\frac{4\ln(2k+1)+2.2048}{\pi(2\pi)^{2k+1}},~~k=9,10,11,\cdots .$$
![The deformed contour $\Gamma$: the oriented curve (see [@li-liu-xu-zhao2 Fig.2]).[]{data-label="contour-Gamma"}](figure2.eps){height="6cm"}
[**[Proof]{}**]{}. We understand as a generating relation for $\tilde \alpha_{2k}$. Using the Cauchy integral formula, and in view of , we have $$\tilde \alpha_{2k}=\frac{1}{2 \pi i}\oint\frac {u(z)\cos( {z}/{4})}{z^{2k+1}} dz=\frac{1}{8\pi i}\int_{\Gamma} \frac{v(z)}{\sin({z}/{4})}\frac {dz}{z^{2k}},$$ where initially the integration path $\Gamma$ is a loop encircling the origin anti-clockwise, and is then deformed to the oriented curve illustrated in Figure \[contour-Gamma\]; see also [@li-liu-xu-zhao2 Fig.2], and $v(z)$ is the function defined in . From , paying attention to the symmetric properties of $v(z)$ and $\Gamma$, we have $$\label{alpha-2k-tuta}
\tilde \alpha_{2k}=\frac{1}{4\pi i}\int_{\Gamma_v}\frac{v(z)}{\sin({z}/{4})}\frac { dz }{z^{2k}}
-\frac{1}{4\pi i}\int_{\Gamma_l}\frac{\frac{2}{\pi}\varphi(z-2\pi)\ln(2\pi-z)}{\sin({z}/{4})}\frac {dz}{ z^{2k}}:=I_v+I_l,$$where $\Gamma_v$ is the vertical part $\Re z=3\pi$, and $\Gamma_l$ is the remaining right-half part of $\Gamma$, consisting of a circular part around $z=2\pi$, and a pair of horizontal line segments, respectively along the upper and lower edges of $(2\pi, 3\pi)$, joining the circle with the vertical line; see Figure \[contour-Gamma\].
First, straightforward calculation gives $$\label{even-Iv}
\left|I_v\right|\leq \frac {M_v}{4\pi} \frac 1 {(3\pi)^{2k}} \int^\infty_{-\infty} \frac {dy}{\big | \sin \frac {3\pi+iy}4\big |}= \frac{M_v}{2\pi(3\pi)^{2k}}B\left (\frac{1}{4},\frac{1}{4}\right )\approx\frac{3.1153\cdots}{(3\pi)^{2k}},$$ where $|v(3\pi+iy)| \leq \sqrt 5 \max_{y\in \mathbb{R}} \left |\varphi(\pi+iy)\right |\leq M_v=2.6393\cdots$; cf. [@li-liu-xu-zhao2 p.297], and $B\left (\frac{1}{4},\frac{1}{4}\right )$ is the Beta function.
Now we turn to the dominant part $I_l$. It is readily seen that $$\label{even-Il}
I_l=\frac 1 \pi\int^{3\pi}_{2\pi} \frac {\varphi(x-2\pi)}{ \sin (x/4)}\frac {dx}{x^{2k}}= \frac{1}{\pi}\int_{2\pi}^{3\pi}\frac{dx}{x^{2k}} +\frac{1}{\pi}\int_{2\pi}^{3\pi}\left\{\frac{g(x)-1}{x-2\pi}\right\}\frac{(x-2\pi)}{x^{2k}}dx,$$ where $g(x)=\frac{\varphi(x-2\pi)}{\sin({x}/{4})}$ such that $g(2\pi)=1$. One can see that $\frac{g(x)-1}{x-2\pi}$ is positive and monotone increasing for $x\in (2\pi, 3\pi]$ since $$\frac{g(x)-1}{x-2\pi}=\left\{ \frac {\sin(t/4)}{t\cos(t/4)}\right \} \left\{ \frac {\varphi(t) -1}{\sin(t/4)}\right\} +\left\{ \frac 1 t \left (\frac 1 {\cos(t/4)}-1\right)\right\},~~t=x-2\pi,$$ and each right-hand side term in the curly braces is positive and monotone increasing for $t\in (0, \pi]$; see [@li-liu-xu-zhao2 p.299] for the monotonicity of $\frac {\varphi(t) -1}{\sin(t/4)}$. Therefore, we have for $x\in (2\pi, 3\pi]$, $$0\leq \frac{g(x)-1}{x-2\pi}\leq \frac{g(3\pi)-1}{\pi}:= M_g=\frac 1 \pi \left [\sqrt 2\; F\left (\frac{1}{2},\frac{1}{2};1;\frac{1}{2}\right )-1\right]= 0.2130\cdots.$$ Substituting it into , we have $$\label{even-I2}
I_l=\frac{2}{2k-1}\frac{1}{(2\pi)^{2k}}+\frac{\delta_{l,k}}{(2k-1)(2\pi)^{2k}}$$ with $-3\left (\frac{2}{3}\right )^{2k}<\delta_{l,k}\leq\frac{2\pi M_g}{k-1}$. Further substituting and into gives $$\label{alpha-2k-tuta-approx}
\tilde \alpha_{2k}=\frac{2}{2k-1}\frac{1}{(2\pi)^{2k}}+\frac{\delta_k}{(2k-1)(2\pi)^{2k}}$$ with $$ -\left \{ 3.1153\;(2k-1)+3\right \} \left (\frac 2 3\right )^{2k} <\delta_{k} < 3.1153\;(2k-1) \left (\frac 2 3\right )^{2k} +
\frac{2\pi M_g}{(k-1)}.$$Hence for $k\geq 9$, we obtain the inequalities in ..3cm
Now we turn to the inequality for the odd terms. From and we have $$\tilde \alpha_{2k+1}=\frac{1}{2\pi i}\oint \frac {u(z)\sin({z}/{4}) dz }{z^{2k+2}}=\frac{1}{8\pi i}\int_{\Gamma}\frac{v(z)dz}{\cos({z}/{4}) \; z^{2k+1}},$$where $\Gamma$ is the same path illustrated in Figure \[contour-Gamma\]. Then, in view of the connection formula , we may write $$\begin{aligned} \label{odd-coefficients-sum}
\tilde \alpha_{2k+1}=& \frac{1}{4 \pi i}\int_{\Gamma_v}\frac{v(z)}{\cos{\frac{z}{4}}}\frac {dz} { z^{2k+1} }+
\frac{1}{4\pi i}\int_{\Gamma_l}\frac{v_{A}(z)}{\cos{\frac{z}{4}}}\frac {dz}{z^{2k+1}} \\
& -\frac{1}{4\pi i}\int_{\Gamma_l}\frac{\frac{2}{\pi}\varphi(z-2\pi)\ln(2\pi-z)}{\cos{\frac{z}{4}}}\frac {dz}{z^{2k+1}}
:=J_v+J_a+J_l, \end{aligned}$$ where the integration paths $\Gamma_v$ and $\Gamma_l$ are the same as in ; see Figure \[contour-Gamma\]. We note that the procedure in [@li-liu-xu-zhao2 Sec.3.3] applies here, with minor modifications. Case by case estimating gives $$\label{tlide-iv}
\left|J_v\right|\leq \left \{ \frac{M_v}{2 \pi} B\left (\frac{1}{4},\frac{1}{4}\right )\right\} \frac 1 {(3\pi)^{2k+1}}\approx\frac{3.1153\cdots}{(3\pi)^{2k+1}};$$see . Also, picking up the residue at $z=2\pi$ yields $$\label{tlide-ia}
J_a=\frac{2v_A(2\pi)}{(2\pi)^{2k+1}}=\frac{16\ln{2}}{\pi(2\pi)^{2k+1}},$$where $v_A(2\pi)=\frac {8 \ln 2} \pi$; see [@li-liu-xu-zhao2 (3.21)]. The dominant contribution comes from the last integral $J_l$. We follow the steps in [@li-liu-xu-zhao2 pp.299-301], and eventually obtain $$\label{tlide-il}
J_l=\frac{4\ln(2k+1) - (4\gamma+4\ln(2\pi))+\delta_{l,k} }{\pi(2\pi)^{2k+1}},$$where $\left|\delta_{l,k}\right|<2M_\varphi +\frac \pi k \tilde M_f +\frac 4 {k+\frac 1 2} e^{-k-\frac 1 2} $ for positive integers $k$ with $M_\varphi=\frac {e-1}{2e}$; see [@li-liu-xu-zhao2 (3.23)], and such that $$0< \frac {\varphi(x-2\pi)}{\sin\frac {x-2\pi} 4} -\frac 1 {\frac {x-2\pi} 4} =\left \{ \frac {\varphi(t)-1}{\sin\frac {t} 4} \right\} +\left\{ \frac 1 {\sin\frac {t} 4} -\frac 1 {\frac {t} 4}\right\}
\leq \tilde M_f=\sqrt 2 F\left (\frac{1}{2},\frac{1}{2};1;\frac{1}{2}\right ) -\frac 4\pi$$ for $x\in (2\pi, 3\pi]$, or, $t\in (0, \pi]$ for $t=x-2\pi$. Here use has been made of the fact that both terms in the curly braces are monotone increasing positive functions for $t\in (0, \pi]$; cf. the derivation of . Now substituting , and into yields $$\label{alpha-tidle-2k+1}
\tilde \alpha_{2k+1}=\frac{4\ln(2k+1)
+ ( 16\ln 2- 4\gamma-4\ln(2\pi))+\delta_{k}
}{\pi(2\pi)^{2k+1}},$$ where $\left|\delta_{k}\right|\leq 3.1153 \pi \left (\frac 2 3\right )^{2k+1}+ 2M_\varphi +\frac \pi k \tilde M_f +\frac 4 {k+\frac 1 2} e^{-k-\frac 1 2}$ for positive integers $k$, which is monotone decreasing in $k$. Straightforward calculation from yields for $k\geq 9$.
Thus, we complete the proof of Lemma \[lem-alpha-k-estimate\]. .3cm
For later use, we need the following corollary:
\[cor-ratio\] Assume that $\tilde \alpha_k$ are the positive constants in Lemma \[lem-alpha-k-estimate\]. Then we have $$\label{ratio-even-even}
\frac {\tilde\alpha_{2k}}{\tilde\alpha_{2k+2}} < \frac {254} 5=50.8~~\mbox{for}~~ k=5,6,\cdots,$$ $$\label{ratio-odd-odd}
\frac {\tilde\alpha_{2k+1}}{\tilde\alpha_{2k+3}} < 43~~\mbox{for}~~ k=0,1,2,\cdots,$$and $$\label{ratio-even-odd}
(2k+1) \frac {\tilde\alpha_{2k+2}}{\tilde\alpha_{2k+1}} < 0.12~~\mbox{and}~~(2k+1) \frac {\tilde\alpha_{2k+2}}{\tilde\alpha_{2k+3}} <3.7~~ \mbox{for}~~ k=0,1,2,\cdots.$$
The results follow accordingly from Lemma \[lem-alpha-k-estimate\] and Table \[tabel-little-k\]. To obtain one may have to evaluate the ratio $\frac {\tilde\alpha_{2k+1}}{\tilde\alpha_{2k+3}}$ up to $k=13$, such that $\frac {\tilde\alpha_{2k+1}}{\tilde\alpha_{2k+3}}= 38.578, 38.679, 38.762, 38.829, 38.886$ for $k=9, 10, 11, 12, 13$.
$k$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$
---------------------------------------------------------- ---------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
$\frac {\tilde\alpha_{2k}}{\tilde\alpha_{2k+2}} $ $56.305$ $60.184$ $57.345$ $53.150$ $49.797 $ $47.533 $ $46.044 $ $45.031 $ $44.303$
$\frac {\tilde\alpha_{2k-1}}{\tilde\alpha_{2k+1}} $ $21.333$ $30.720 $ $34.632 $ $36.358 $ $37.227 $ $37.730 $ $38.055 $ $38.282 $ $38.450 $
$ \frac { (2k-1)\tilde\alpha_{2k}}{\tilde\alpha_{2k-1}}$ $0.1041$ $0.1184$ $ 0.1007$ $0.0851$ $0.0749$ $0.0684$ $0.0642$ $0.0612$ $0.0590$
$ \frac { (2k-1)\tilde\alpha_{2k}}{\tilde\alpha_{2k+1}}$ $2.2222$ $3.6373 $ $3.4884 $ $3.0964 $ $2.7884 $ $2.5822 $ $2.4432 $ $2.3438 $ $2.2681 $
: The first few ratios. Calculation conducted using Maple, based on -.[]{data-label="tabel-little-k"}
Proof of Lemma \[lemma-R(N)\]
=============================
Now that we have proved Lemma \[lem-alpha-k-estimate\], we turn to the proof of Lemma \[lemma-R(N)\]..3cm
[**[Proof of Lemma \[lemma-R(N)\]]{}**]{}. To prove , the idea is as follows: First we show that $$\label{r-2l-2j}
(-1)^l r_{2l, 2j+2} >0~~\mbox{for}~~j=l+1,l+2,\cdots,~~l=0,1,2,\cdots,$$ and $$\label{r-2l-difference}
(-1)^{l+1} \left ( r_{2l, 2j+1} +\frac 1 2 r_{2l, 2j+2}\right ) >0~~\mbox{for}~~j=l+1,l+2,\cdots,~~l=0,1,2,\cdots.$$ Then follows immediately from and since $x=1/N\in (0, 1/2]$ for $N\geq 2$, and $$\begin{aligned}
\tilde R_{2l}(N) &= \sum^\infty_{j=l+1} (-1)^{l+1} \left (r_{2l, 2j+1}+ r_{2l, 2j+2}\; x \right ) x^{2j+1}\\
&\geq \sum^\infty_{j=l+1} (-1)^{l+1} \left (r_{2l, 2j+1}+ \frac 1 2 r_{2l, 2j+2} \right ) x^{2j+1}\\
& >0
\end{aligned}$$ for $N=n+1=2, 3, \cdots$, and $l=0,1,2,\cdots$..3cm
The above idea is simple, yet the verification of and is quite complicated. We begin with . First, a combination of and gives $$(-1)^{l} r_{2l, 2l+4}=d_{2l+2, 2l+4} \left \{(-1)^{l} \alpha_{2l+2} \right \}+ \left\{ -d_{2l+1, 2l+4}\right\} \left\{ (-1)^{l+1} \alpha_{2l+1}\right\} >0$$for $l=0,1,2,\cdots$. Here use has been made of , , and the facts that $d_{2l+2, 2l+4}>0$ and $d_{2l+1, 2l+4}<0$. Hence is true for $j=l+1$. Therefore, we need only to prove for $j=l+2, l+3, \cdots$. In view of , it suffices to show, by an induction argument, that $$\label{r-2l-2j-even-odd}
r_{l,j}^E:= (-1)^{l+1} \sum^l_{k=0} d_{2k, 2j+2} \alpha_{2k} >0~~\mbox{and}~~ r_{l,j}^O:=(-1)^{l+1} \sum^{l-1}_{k=0} d_{2k+1, 2j+2} \alpha_{2k+1} \geq 0$$for $j=l+2,l+3,\cdots$ and $l=0,1,2,\cdots$, where $\alpha_0=-1$. .3cm
Straightforward verification shows that the first inequality in holds for $l=0$: We see from and that $$r_{0,j}^E=r_{0, 2j+2}=\left ( \frac 1 {j+1}-\frac 1 {2j+1}\right ) +\frac 1 {8j} >0~~\mbox{for}~~j=2,3,\cdots.$$ Similarly, from and we have $r_{1,j}^E= \alpha_2 d_{2,2j+2}+\alpha_0 d_{0, 2j+2}$. Hence $$r_{1,j}^E= \frac 5 {192} \left [ 2j+2+\frac {2j-1} 4\right ] - \left [ \frac 1 {j+1}-\frac 1 {2j+1}+\frac 1 {8j}\right ]> \frac {5(j+1)}{96}-\frac 2 {3(j+1)} >0$$for $j=3,4,\cdots$. Thus the first inequality in is true for $l=1$.
Now assume for a non-negative integer $l$, then, replacing $l$ with $l+2$, we have $$\label{r-2l-2j-induction}
r_{l+2,j}^E = r_{l,j}^E + \tilde \alpha_{2l+4} \left [ (2l+3)! d_{2l+4, 2j+2} - \frac { \tilde \alpha_{2l+2}}{ \tilde \alpha_{2l+4} } (2l+1)! d_{2l+2, 2j+2}\right ] >0$$for $j=l+4,l+5,l+6,\cdots$. Indeed, if we write $$(2l+1)! d_{2l+2, 2j+2}= \frac {(2j+2l+2) (2j)! } {(2j-2l)!} + \frac { (2j-1)! } {4(2j-2l-2)!}:=A_{l}+B_{l},$$ Then, noting that for $l\geq 0$ and $j-l\geq 5$, in view of and Table \[tabel-little-k\], we have $$\begin{aligned}
& A_{l+1}+B_{l+1}-\frac{\tilde \alpha_{2l+2}}{ \tilde \alpha_{2l+4}}\left(A_{l}+B_{l}\right)\\
&\geq (2j-2l-1)(2j-2l) A_l + (2j-2l-3) (2j-2l-2)B_l- 61 \left( A_{l}+B_{l}\right)\\
&\geq 90A_l+56 B_l- 61 \left(A_{l}+B_{l}\right)\\
&>0,\end{aligned}$$ since $A_l> 4 B_l$ by straightforward verification. Alternatively, applying , for $l\geq 4$ and $j-l\geq 4$, we can modify the above inequalities to give $$A_{l+1}+B_{l+1}-\frac{\tilde \alpha_{2l+2}}{ \tilde \alpha_{2l+4}}\left(A_{l}+B_{l}\right)
\geq 56A_l+30 B_l- \frac {254} 5 \left(A_{l}+B_{l}\right)>0$$ The remaining cases, namely $j=l+4$ with $l=0,1,2,3$, can be justified by direct calculation: The values of $$(2l+3)! d_{2l+4, 2l+10} - \frac { \tilde \alpha_{2l+2}}{ \tilde \alpha_{2l+4} } (2l+1)! d_{2l+2, 2l+10}=
62.9,\; 1004.5,\; 0.66\times 10^6,\; 0.33\times 10^9,$$respectively for $l=0,1,2,3$. Summarizing all above, we see the validity of . Therefore, the first inequality in is true for $j=l+2,l+3,\cdots$ and $l=0,1,2,\cdots$. .3cm
The analysis of $r_{l,j}^O$ is similar to, and simpler than, that of the even terms $r_{l,j}^E$. First, for $l=0$, the sum in is empty and thus we understand that $r_{0,j}^O=0$ for all $j$. Also, it is readily seen that $r_{1,j}^O=d_{1, 2j+2} \alpha_1\equiv \frac 3 {16}$ for $j=3,4,\cdots$; cf. . Hence, the equality for $r_{l,j}^O$ in also holds for $l=1$.
Now assume that $r_{l,j}^O\geq 0$ for a non-negative integer $l$ and $j=l+2, l+3,\cdots$. From we may write $$\label{r-2l-2j-odd-induction}
r_{l+2,j}^O = r_{l,j}^O-\tilde \alpha_{2l+3}(2l+2)!\;d_{2l+3, 2j+2}
+\tilde\alpha_{2l+1}(2l)!\;d_{2l+1, 2j+2}:= r_{l,j}^O+c_+\Delta_{l,j}$$with a positive constant $c_+= \tilde \alpha_{2l+3} (2l)!\left | d_{2l+1, 2j+2}\right |$, and try to prove that $$\Delta_{l,j}:=\frac{(2l+2)!\;d_{2l+3, 2j+2}} {(2l)!\;d_{2l+1, 2j+2}} -\frac{\tilde\alpha_{2l+1}}{\tilde \alpha_{2l+3}}= \frac { 6j+2l+2 }{ 6j+2l } (2j-2l-1) (2j-2l) -\frac{\tilde\alpha_{2l+1}}{\tilde \alpha_{2l+3}} >0$$for $j=l+4, l+5,\cdots$. Here the last inequality comes from . Therefore, from and by induction, we have justified the validity of both inequalities in for all $j\geq l+2$ and $l\geq 0$. Accordingly, we have proved for all $j\geq l+1$ and $l\geq 0$, noting that the only exceptional case $j=l+1$ has been discussed earlier in this section. .3cm
In what follows we proceed to prove . First, taking into account the formulas and , we see that $r_{2l, 2l+3}+ \frac 1 2 r_{2l, 2l+4}$ can be represented as a linear combination of $\alpha_{2l+1}$ and $\alpha_{2l+2}$. More precisely, substituting in the coefficients $d_{k,s}$; see and , we have $$(-1)^{l+1}\left [ r_{2l, 2l+3}+ \frac 1 2 r_{2l, 2l+4}\right ]=(2l)! \; \tilde\alpha_{2l+1} \left [ \frac {(2l+1)(12l+5)} 8 -\frac {\tilde\alpha_{2l+2}} {\tilde\alpha_{2l+1}} \frac {(2l+2)^2(2l+1)} 2 \right ],$$which is positive for all $l$ since $\frac {\tilde\alpha_{2l+2}} {\tilde\alpha_{2l+1}} < \frac 1 {4(2l+1)}$ for $l\geq 0$; cf. . Thus is true for $j=l+1$, allowing us to just prove for $j=l+2,l+3,\cdots$ and $l=0,1,2,\cdots$.
For $l=0$, it is readily verified from and that $$-r_{0, 2j+1}- \frac 1 2 r_{0, 2j+2} =\left (\frac 1 {2j}-\frac 1 {2j+2} \right ) + \left ( \frac 1 {2(2j+1)}-\frac 1 {4(2j-1)}-\frac 1 {16j}\right )>0$$for $j=2,3,\cdots$. Here the right-hand side is the sum of positive numbers when $j\geq 3$, and equals to $\frac {11}{160}$ when $j=2$. Hence holds for $l=0$.
For $l=1$, recalling that $r_{2,s}=-\frac 5 {192} d_{2,s}+\frac 1 4 d_{1,s} +d_{0,s}$; cf. , from and we may write $$r_{2, 2j+1}+ \frac 1 2 r_{2, 2j+2} = \frac{ 5 j} {768} +\frac {281}{1536} - \left (\frac 1 {2j}-\frac 1 {2j+2} \right ) - \left ( \frac 1 {2(2j+1)}-\frac 1 {4(2j-1)}\right )+\frac 1 {16j}.$$Using the facts that $\frac 1 {2j} -\frac 1 { 2j+2 }\leq \frac 1 {8j}$ for $j\geq 3$, and $\frac 1 {2(2j+1)}-\frac 1 {4(2j-1)}< \frac 1 {8j}$ for $j\geq 1$, we have $r_{2, 2j+1}+ \frac 1 2 r_{2, 2j+2}> \frac{ 5 j} {768} +\frac {281}{1536} -\frac 3 {16j}>0$ for all $j=3,4,5,\cdots$. Hence holds for $l=1$.
Now assume for a non-negative integer $l$, then, from we have $$(-1)^{l+3}\left (r_{2l+4,2j+1}+\frac{1}{2}r_{2l+4,2j+2}\right )=(-1)^{l+1}\left (r_{2l,2j+1}+\frac{1}{2}r_{2l,2j+2}\right )+O_l+E_l.$$It suffices to show that $O_l+E_l >0$ for $j=l+4, l+5, \cdots$, where for $l=0,1,2,\cdots$, $$\label{O-l-def}
O_l:=(-1)^l \left[\alpha_{2l+3}\left(d_{2l+3, 2j+1}+\frac 1 2 d_{2l+3, 2j+2}\right)+\alpha_{2l+1}\left(d_{2l+1, 2j+1}+\frac 1 2 d_{2l+1, 2j+2}\right)\right]$$ and $$\label{E-l-def}
E_l:=(-1)^l \left[\alpha_{2l+4}\left(d_{2l+4, 2j+1}+\frac 1 2 d_{2l+4, 2j+2}\right)+\alpha_{2l+2}\left(d_{2l+2, 2j+1}+\frac 1 2 d_{2l+2, 2j+2}\right)\right] .$$
We may write $$\label{O-l-representation}
O_l =\tilde\alpha_{2l+3}(\tilde A_{l+1}+\tilde B_{l+1})-\tilde\alpha_{2l+1}(\tilde A_{l}+\tilde B_{l})= \tilde\alpha_{2l+3}\left [ \tilde A_{l+1}+\tilde B_{l+1} - \frac {\tilde\alpha_{2l+1}} {\tilde\alpha_{2l+3}}
(\tilde A_{l}+\tilde B_{l}) \right ]$$ with $\tilde A_l=\frac{(2j-1)!(5j+7l)}{4(2j-2l)!}$ and $\tilde B_l=\frac{(2j-2)!}{4(2j-2l-2)!}$. Observing that $\tilde A_{l+1} > (2j-2l)(2j-2l-1) \tilde A_{l}\geq 90 \tilde A_{l}$ and $\tilde B_{l+1} = (2j-2l-2)(2j-2l-3) \tilde B_{l}\geq 56 \tilde B_{l}$ for $j\geq l+5$ and $l\geq 0$, and recalling that $\frac {\tilde\alpha_{2l+1}} {\tilde\alpha_{2l+3}}< 43$ for $l\geq 0$, we have $$O_l \geq \tilde\alpha_{2l+3}\left [ \tilde A_{l+1}\left (1- \frac {43}{90}\right ) +\tilde B_{l+1} \left (1- \frac {43}{56} \right ) \right ] \geq \frac {47}{90} \tilde\alpha_{2l+3} \tilde A_{l+1},~~j\geq l+5, ~l\geq 0.$$
Now we turn to $E_l$. Similar to the discussion of $O_l$, we may also write $$E_l =\tilde \alpha_{2l+2} (\tilde C_l-\tilde D_l )- \tilde \alpha_{2l+4} (\tilde C_{l+1}- \tilde D_{l+1} ),$$where $\tilde C_l=\frac {(2j+1)!}{(2j-2l)!} -\frac {(2j-1)!}{(2j-2l-2)!}=\frac {(4l+2) (2j-l) (2j-1)!}{(2j-2l)!}$, and $\tilde D_l= \frac 1 2 \frac {(2j)!} {(2j-2l-1)!} - \frac 1 8 \frac {(2j-1)!} {(2j-2l-2)!} - \frac 1 4 \frac {(2j-2)!} {(2j-2l-3)!}$. It is readily verified that both constants are positive, and such that $ \frac 18 D_l < \tilde D_l< \frac 1 2 D_l$ for $j\geq l+4$ and $l\geq 0$ with $D_l=\frac {(2j)!} {(2j-2l-1)!}$. Therefore, we have for $j\geq l+5$ and $l\geq 0$ that $$\label{O+E-l-inequality}
O_l+E_l > \frac {47}{90} \tilde\alpha_{2l+3} \tilde A_{l+1} - \frac 1 2 \tilde \alpha_{2l+2} D_l - \tilde \alpha_{2l+4} \tilde C_{l+1}: =\frac {(2j-1)!\tilde\alpha_{2l+3}}{(2j-2l-1)!} \Omega ,$$where $$\begin{aligned}
\Omega&=(2j-2l-1)\left [ \frac {47}{360} (5j+7l+7) -\frac {2 (2l+3) \tilde \alpha_{2l+4}} {\tilde\alpha_{2l+3}} (2j-l-1)\right ]-
\frac {(2l+1) \tilde \alpha_{2l+2}} {\tilde\alpha_{2l+3}}\frac j{2l+1}\\
&\geq 9 \left [ \frac {47}{360} (5j+7l+7) -0.24 (2j-l-1)\right ]-
\frac {3.7} 3 j \\
&=\frac {193}{600}j +\frac {2077}{200} l+\frac {2077}{200},
\end{aligned}$$and thus is positive for $j\geq l+5$ and $l\geq 1$. Here use has been made of . For the special case $l=0$ and $j\geq 5$, taking Table \[tabel-little-k\] into account, again we have the positivity of $\Omega$: $$\Omega \geq (2j-1)\left [\frac {47}{360}(5j+ 7) - 0.24 (2j-1)\right ]-2.23 j=\frac {701}{100}+\frac {6049}{1800}(j-5) +\frac {311}{900}(j-5)^2$$
What remains is the case when $j=l+4$ with $l=0,1,2,\cdots$. Still we have . Since $\tilde A_{l+1}=\frac {(2j-1)! (12l+27)} { 4 \cdot 6!} > 56 \tilde A_{l}$ and $\tilde B_{l+1}=\frac {(2j-2)! } { 4 \cdot 4!} = 30 \tilde B_{l}$, in view of we have $$O_l \geq \tilde\alpha_{2l+3}\left [ \tilde A_{l+1}\left (1- \frac {43}{56}\right ) -\tilde B_{l+1} \left ( \frac {43}{30} -1 \right ) \right ] \geq \left [ \frac {13}{56}-\frac {13}{(2l+7)(12l+27)}\right ] \tilde\alpha_{2l+3} \tilde A_{l+1},$$ from which we see that $O_l>\frac 1 5 \tilde\alpha_{2l+3} \tilde A_{l+1}$ for $j= l+4$ with $l\geq 2$. As a results, we have a modified version of as $j=l+4$, $$O_l+E_l > \frac {1}{5} \tilde\alpha_{2l+3} \tilde A_{l+1} - \frac 1 2 \tilde \alpha_{2l+2} D_l - \tilde \alpha_{2l+4} \tilde C_{l+1}: =\frac {(2j-1)!\tilde\alpha_{2l+3}}{6!} \Omega_4 ,$$where $$\Omega_4 = \frac {12l+27} {20}- \frac {(2l+3) \tilde \alpha_{2l+4}} {\tilde\alpha_{2l+3}} (2l+14) - \frac {(2l+1) \tilde \alpha_{2l+2}} {\tilde\alpha_{2l+3}}\frac {l+4} {7(2l+1)}.$$From we readily see that $$\Omega_4\geq \frac {12l+27} {20}
-0.12 (2l+14) - {3.7}\times \frac 1 {7} = \frac {31}{140} + \frac 9 {25}(l-3),$$ and is positive for $l\geq 3$.
We fill the last gap by calculating from - that $ O_l+E_l=3.3236, ~ 1.9908, ~ 4.3827$, respectively for $l=0,1,2$, with $j=l+4$. Thus we complete the proof of , and hence of Lemma \[lemma-R(N)\].
Discussion
==========
We have proved the conjecture of Granath [@Granath], as stated in Theorem \[thm epsilon-N\] and Corollary \[thm-inequalities\], of which the results of Zhao [@zhao], Mortici [@Mortici] and Granath [@Granath] are special cases. The asymptotic expansion involved, namely , corresponds to the special case of Nemes’ expansion in descending powers of $n+h$, with $h=1$.
Earlier in [@li-liu-xu-zhao2], Li [*et al.*]{} consider the case $h=3/4$; cf. and . According to a result in [@li-liu-xu-zhao2], the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative $n$. As an application, we obtain optimal upper and lower bounds up to all orders, holding for all integers $n\geq 0$.
Then, a natural question may arise:
(Li, Liu, Xu and Zhao) Considering the general expansion in , for what $h$ do we have the “best” approximation in the sense of [@li-liu-xu-zhao2 Theorem1] (or, in the present paper), or in the sense of Theorem \[thm epsilon-N\] and Corollary \[thm-inequalities\]?
It is worth noting that the coefficients of the expansion possess a symmetric property, namely, $g_k(h)=(-1)^k g_k(3/2-h)$. Hence, if we take $h=1/2$, write $N=n+ 1/2$, and specify as $$\label{expansion-n+1/2}
\pi G_n\sim \ln(16N)+\gamma +\sum_{k=1} ^\infty \frac{\gamma_k}{N^k}.$$ Then it is readily seen that $\gamma_k=(-1)^k\alpha_k$, and hence $(-1)^{\frac {(l+1)(l+2)} 2} \gamma_{l+1} <0$ for nonnegative integers $l$, very similar to the result in Theorem \[coefficient\]. Naturally, analysis similar to what we have conducted in the present paper might lead to $
(-1)^{\frac {(l+1)(l+2)} 2} \epsilon_l(N) <0
$ for all $n=0,1,2,\cdots$ and $l=0,1,2,\cdots$, where $\epsilon_l(N)$ is the error of due to truncation at the $l$-th order term, with $N=n+ 1/2$.
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[^1]: Corresponding author ([*[E-mail address:]{}*]{} [stszyq@mail.sysu.edu.cn]{}). Investigation supported in part by the National Natural Science Foundation of China under grant numbers 10871212 and 11571375.
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abstract: 'Extensive muon spin relaxation ($\mu$SR) measurements have been performed to determine the magnetic field penetration depth $\lambda$ in high $T_{c}$ cuprate superconductors with simple hole doping, Zn-doping, overdoping, and formation of static SDW nano islands. System dependence of $n_{s}/m^{*}$ (superconducting carrier density / effective mass) reveals universal correlations between $T_{c}$ and $n_{s}/m^{*}$ in all these cases with/without perturbation. Evidence for spontaneous and microscopic phase separation into normal and superconducting regions was obtained in the cases with strong perturbation, i.e., Zn-doping (swiss cheese model), overdoping, and coexisting magnetic and superconducting states (SDW nano islands). The length scale of this heterogeneity is shown to be comparable to the in-plane coherence length. We discuss implication of these results on condensation mechanisms of HTSC systems, resorting to an analogy with pure $^{4}$He and $^{4}$He/$^{3}$He mixture films on regular and porous media, reminding essential features of Bose-Einstein, BCS and Kosterlitz-Thouless condensation/transition in 3-d and 2-d systems, and comparing models of BE-BCS crossover and phase fluctuations. Combining the $\mu$SR results on $n_{s}/m^{*}$ and the pseudo-gap behavior, we propose a new phase diagram for HTSC, characterized by: (1) the $T^{*}$ line that represents pair formation; (2) disappearance of this line above the critical hole concentration $x=x_{c}$; (3) in the underdoped region between $T_{c}$ and $T^{*}$, there exists another line $T_{dyn}$ which corresponds to the onset of dynamic superconductivity with superconducting phase fluctuations; and (4) the overdoped region being phase separated between hole-poor superfluid and hole-rich normal fermion metal regions. Finally, we elucidate anomalous reduction of superfluid spectral weight in the crossover from superconducting to metallic ground states found not only in overdoped HTSC cuprates but also in pressurized organic BEDT and A$_{3}$C$_{60}$ fulleride superconductors.'
address: 'Department of Physics, Columbia University, New York, NY 10027, USA'
author:
- 'Y.J. Uemura [^1]'
---
[**Superfluid density of high-$T_{c}$ cuprate systems: implication on condensation mechanisms, heterogeneity and phase diagram**]{}
Introduction
============
Muon spin relaxation ($\mu$SR) technique \[1-3\] has made significant contributions to studies of high-$T_{c}$ superconductors (HTSC). Measurements in transverse-fields (TF-$\mu$SR) provide a reliable way to determine the magnetic field penetration depth $\lambda$ of type-II superconductors and to infer details of the flux vortex lattice, while measurements in zero-filed (ZF-$\mu$SR) allow studies of static magnetism, leading to determination of the phase diagrams, local spin structures, and volume fraction of static magnetism. In this article, we provide a perspective view on development of these measurements in HTSC systems and their implications on condensation mechanisms, heterogeneity and phase diagrams, focusing on the behavior of $n_{s}/m^{*}$ (superconducting carrier density / effective mass). Since the parameter $n_{s}/m^{*}$ in HTSC systems plays a role similar to the superfluid density in superfluid He systems, we refer to it as “superfluid density” in the title and the text of this paper.
In the following section, Section II, we start with the results of the penetration depth measurements of hole-doped cuprate systems, which exhibit universal correlations between $T_{c}$ and $n_{s}/m^{*}$. We discuss these correlations in terms of energy scales of superconducting carriers. Then, we look into the cases involving heterogeneity, i.e., (Cu,Zn) substitution and overdoping in Section III, and systems with coexisting superconductivity and static magnetism, with the formation of magnetic nano-islands, in section IV. In all these cases, the superfluid density has a trade-off with the volume of non-superconducting regions created by perturbation and/or phase separation. We shall see that the universal correlations between $T_{c}$ and the superfluid density $n_{s}/m^{*}$ are followed not only by the simple hole doped cuprates but also by these HTSC systems with microscopic phase separation.
Thin films of superfluid $^{4}$He and $^{4}$He/$^{3}$He mixtures represent another set of systems where the superfluid transition temperature $T_{c}$ is strongly correlated with the superfluid density. In section V, we compare the results of HTSC systems with superfluid He films in normal and porous media.
Based on the correlations between $T_{c}$ and $n_{s}/m^{*}$ and the pseudogap phenomena in the underdoped region, some models/conjectures have been presented in terms of the condensation mechanisms of HTSC systems. To elucidate these models, in section VI, we will first consider differences and similarities among Bose-Einstein (BE) condensation, BCS condensation and Kosterlitz-Thouless (KT) transition. Then, we will compare models of BE-BCS crossover and phase fluctuations, taking into account relevance to the KT transition and distinction between pair-formation and dynamic superconductivity in the pseudo-gap regime. We will also introduce a phase diagram involving phase separation in the overdoped region in section VII, followed by a summary in section VIII.
Correlations between $T_{c}$ and $n_{s}/m^{*}$ in hole-doped HTSC
=================================================================
In type-II superconductors, the external magnetic field, between $H_{c1}$ and $H_{c2}$, forms a lattice of flux vortices, resulting in inhomogeneous internal field distributions. The width of this distribution is proportional to the muon spin relaxation rate $\sigma$ in TF-$\mu$SR, which is related to the penetration depth $\lambda$ and $n_{s}/m^{*}$ as $$\sigma \propto \lambda^{-2} = [4\pi n_{s}e^{2}/c^{2}] \times
[1/(1+\xi/{\it l\/})]\eqno{(1)},$$ as given by the London equation. In superconducting systems in the clean limit, where the coherence length $\xi$ is much shorter than the mean free path [*l*]{}, the relaxation rate $\sigma$ is proportional to $n_{s}/m^{*}$. The inhomogeneity of the internal field is due to partial screening of the external field by the supercurrent. So, it is quite natural that the field width and $\sigma$ are proportional to the “supercurrent density” $n_{s}/m^{*}$, in the same sense as the conductivity of normal metals is proportional to $n_{n}/m^{*}$, where $n_{n}$ denotes the normal state carrier density.
Generally, TF-$\mu$SR studies in type-II superconductors provide information on: (a) the temperature dependence of $\lambda$; (b) absolute values of $\lambda$ at $T \rightarrow 0$; (c) the coherence length $\xi$ via analyses of local field distribution; and (d) vortex lattice properties. Among these, use of single crystal specimens is essential in (a), (c) and (d). For example, early $\mu$SR results of $\lambda$ on ceramic samples of HTSC systems mostly exhibited behavior consistent with isotropic energy gap, and it is only after studies of high-quality single crystal specimens that the d-wave nature was confirmed in temperature dependence of $\lambda$. For details of (a), (c) and (d), readers are referred to refs. \[3-4\].
In contrast, the results on ceramic specimens have been very useful in comparing absolute values of $n_{s}/m^{*}$ in different systems. Ceramic specimens provide advantages over single crystal specimens in homogeneity of doped hole concentrations and in being less affected by crossover from the 3-d to 2-d vortex lattice. In this article, we shall focus on the system and doping dependence of the superfluid density, i.e., (b). In HTSC systems with a large anisotropy, the relaxation rate $\sigma$ measured in ceramic specimens is determined predominantly by the supercurrent flowing in the conducting ab-planes \[5\]. Thus, the results of $\sigma$ from ceramic specimens should be regarded as reflecting the in-plane penetration depth $\lambda_{ab}$ and the in-plane effective mass $m^{*}_{ab}$.
Figure 1 shows a plot of $\sigma(T\rightarrow 0)$ versus $T_{c}$ of various different HTSC systems \[6-11\]. In this figure, results from simple hole-doped HTSC systems \[6,7\] are shown with open symbols. With increasing hole concentration, $T_{c}$ increases linearly with $n_{s}/m^{*}$ in the underdoped region, and then shows a saturation. The slope of this linear relationship in the underdoped region is common to various different series of HTSC systems. These results have also been confirmed in several other $\mu$SR measurements \[12-16\]. This observation alone already implies that $n_{s}/m^{*}$ can be an important determining factor for $T_{c}$ in HTSC systems. We also find in Fig. 1 that the ratios of $T_{c}$ versus $n_{s}/m^{*}$ for some other superconductors \[17,18\] are not quite far from the value for HTSC systems.
If an independent estimate for the effective mass $m^{*}$ is available, the $\mu$SR measurements give the superconducting carrier density $n_{s}$. Then, one can calculate the number of carries existing in a region of coherence length squared on the conducting plane. For typical BCS superconductors, such as Sn or Pb, one finds more than 10,000 pairs overlapping with one another in the $\pi\xi^{2}$ area. For superfluid He, which is the well-known case for Bose-Einstein condensation, we find that each boson exist without overlapping with each other: i.e., one pair per $\pi\xi^{2}$. The systems following the linear relationship in Fig. 1 have 3 to 6 pairs overlapping within the $\pi\xi^{2}$ area, as illustrated in Fig. 2. This feature encourages us to consider HTSC and some novel superconductors in a crossover from BE to BCS condensation.
By knowing the average distance $c_{int}$ between the conducting CuO$_{2}$ planes, one can obtain $n_{s2d}/m^{*}$ where $n_{s2d}$ denotes the 2-dimensional carrier density. We remind here that the Fermi energy $\epsilon_{F}$ of 2-dimensional electron gas is proportional to $n_{n2d}/m^{*}$. Thus, we can consider $n_{s2d}/m^{*}$ as a parameter representing kinetic energy for translational motion of superconducting carries. For 3-d systems without phase separation, in which $n_{n}=n_{s}$, one can combine the muon relaxation rate $\sigma \propto n_{s}/m^{*}$ with the Sommerfeld constant $\gamma(T=T_{c})\propto n_{n}^{1/3}m^{*}$, or Pauli susceptibility $\chi(T=T_{c}) \propto n_{n}^{1/3}m^{*}$, to deduce $\epsilon_{F} \propto n_{s}^{2/3}/m^{*}$. The “effective” Fermi energy $\epsilon_{F}$ obtained in this way does not necessarily correspond to the real Fermi energy in band structure calculations. Since $1/\lambda^{2}$ corresponds to the Drude spectral weight in optical conductivity which condenses into a delta function at $\omega = 0$ below $T_{c}$, $\epsilon_{F}$ might also be called as a Drude energy scale.
Figure 3 shows a plot of $T_{c}$ versus $T_{F} = \epsilon_{F}/k_{B}$ thus obtained from the results of $n_{s}/m^{*}$ \[7\]. For A$_{3}$C$_{60}$ where a reliable value of [*l*]{} is not available, we used the clean-limit value of $n_{s}/m^{*}$ without corrections regarding $\xi$/[*l*]{} \[17,18\]. We find that HTSC, organic BEDT \[19\], and some other systems \[17-20\] have very high and nearly equal ratios of $T_{c}/T_{F}$. The $T_{B}$ line in this figure shows the Bose-Einstein condensation temperature for a non-interacting 3-d Bose gas having the boson density $n_{B}= n_{s}/2$ and mass $m_{B}=2m^{*}$. Compared to $T_{B}$, the actual transition temperatures $T_{c}$ of HTSC systems are reduced by a factor of 4 to 5. This reduction is natural in view of overlapping pairs which would reduce $T_{c}$, and in view of 2-dimensional character of HTSC systems. However the parallel behavior of $T_{B}$ and the observed results of $T_{c}$ suggests that the origin of the linear relationship between $T_{c}$ and $n_{s}/m^{*}$ could be deeply related to BE condensation. Figure 3 serves as an empirical way to classify various superconductors in a crossover from BE to BCS condensation.
HTSC Systems involving microscopic heterogeneity: Zn-doping and overdoping
==========================================================================
(Cu,Zn) substitution
--------------------
When a very small amount of Zn is substituted for Cu on the CuO$_{2}$ planes of HTSC systems, $T_{c}$ is reduced. Figure 4(a) shows the reduction of $\sigma(T\rightarrow 0)$ as a function of Zn concentration $c$ substituting in-plane Cu. The superfluid density $n_{s}/m^{*}$ decreases with increasing $c$. To account for this reduction, we proposed “Swiss Cheese Model” \[9\] where each Zn is assumed to destroy superconductivity in the surrounding region with the area $\pi\xi_{ab}^{2}$, as illustrated in Fig. 4(a). The solid lines in Fig. 4(a), which represent predictions of this model, are obtained just from $c$ and $\xi_{ab}$ deduced from the upper critical field $H_{c2}$, [*without any fitting*]{} to the data. Sufficiently good agreement between these lines and the observed results indicate that carriers within the $\pi\xi_{ab}^{2}$ area around each Zn no longer contribute to the superfluid density $n_{s}/m^{*}$. Subsequently, this picture was directly confirmed by Pan et al. \[21\] in Scanning Tunneling Microscopy studies, where the local density of states around Zn showed features characteristic to normal regions, as shown in Fig. 4(b). These $\mu$SR and STM results are consistent with the non-zero $\gamma$ term of the specific heat, which increases with increasing Zn concentration \[22\]. In the plot of $\sigma$ versus $T_{c}$ in Fig. 1, the results of Zn-doped systems (closed triangle and star symbols) follow the trajectory of simple hole doped HTSC systems (open symbols).
overdoping
----------
Tl$_{2}$Ba$_{2}$CuO$_{6+\delta}$ (Tl2201) systems have been extensively studied by various experimental methods as a prototype of overdoped HTSC systems. Tl2201 compounds have very small residual resistivity, which assures that the system lies well within the clean limit. To our surprise, with increasing overdoping from the nearly optimal $T_{c}$ = 85 K sample, the relaxation rate $\sigma(T\rightarrow 0)$ in TF-$\mu$SR decreased \[8,15\] as shown by the closed circle symbols in Fig. 1. This behavior is also shown in a plot of $\sigma(T\rightarrow 0)$ versus doping factor $\delta$ in Fig. 5(a). In view of no anomaly in $m^{*}$, the different behaviors of the normal state carriers $n_{n}$ and the superfluid density $n_{s}/m^{*}$ suggest that carriers are spontaneously separated into those which are involved in superconductivity and those which remain unpaired fermions, as illustrated in Fig. 5(a).
By defining the “gapped” and “ungapped” responses in the specific heat measurements, as illustrated in Fig. 5(a), we plotted the $\delta$ dependence of the gapped response obtained from the data of Loram [*et al.*]{} \[22\] in Tl2201 in Fig. 5(a). The good agreement between the $\mu$SR superfluid density $\sigma(T\rightarrow 0)$ (closed circles) and the “gapped” response in specific heat (open symbols) provides a support to our view \[8,23\] with spontaneous phase separation between superconducting and normal regions in overdoped HTSC systems.
The volume fraction of the superconducting region can also be estimated from the “specific heat jump” $\Delta C$. In BCS superconductors $\Delta C \propto C_{n} \propto \gamma_{n} T$, where $C_{n}$ denotes the normal-state specific heat, and $\gamma_{n}$ stands for the Sommerfeld constant derived in the normal state. Then, in the plot of $C/T$ versus $T$ as in the inset of Fig. 5(a), one would expect that the specific heat jump $\Delta C/T$ should be independent of $T_{c}$, for systems having a common $\gamma_{n}$ value. In Loram’s specific heat results of Tl2201 \[22\], $\gamma_{n}$ does not depend on doping, while the jump $\Delta C/T$ decreases with increasing doping. This observation further supports decreasing superfluid volume fraction with increasing doping (decreasing $T_{c}$) in Tl2201.
Residual normal response in the overdoped cuprate was also found in measurements of optical conductivity \[24\]. Furthermore, we calculated an expected doping dependence of $n_{s}/m^{*}$ based on a simple model assuming phase separation \[23\] (as described in section VII), and obtained a good agreement with the observed results in (Y,Ca)Ba$_{2}$Cu$_{3}$O$_{y}$ \[25\], as shown in Fig. 5(b). So far, there is no direct observation reported regarding the size of these phase-separated regions in overdoped HTSC.
HTSC systems with static SDW nano-islands
=========================================
Magnetic order of the parent compound La$_{2}$CuO$_{4}$ of the 214 cuprates was first confirmed by ZF-$\mu$SR measurements \[26\]. Figure 6(a) shows the time spectra obtained in the ZF measurements of antiferromagnetic La$_{2}$CuO$_{4}$ (AF-LCO) which has the N/’eel temperature $T_{N} > 250$ K. As in most other magnetic systems, we find that muon spin precession sets in below $T_{N}$, with the precession amplitude independent of temperature, while the frequency increasing with decreasing temperature as the sub-lattice magnetization builds up. In the La214 systems with the hole concentration near 1/8 per Cu, incommensurate static spin correlations have been found by neutron scattering \[27\]. Time spectrum of ZF-$\mu$SR in La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ (LBCO:0.125) \[28\] is shown in Fig. 6(b). In this case, we find a Bessel function line shape characteristic of ZF-$\mu$SR in incommensurate magnetic systems \[29,30\], such as the one observed in (TMTSF)$_{2}$PF$_{6}$ \[29\].
La$_{2}$CuO$_{4.11}$ (LCO:4.11) is a system with oxygen intercalated in a stage-4 structure. This system is superconducting with $T_{c}$ = 42 K, which is the highest among the La214 family, while also exhibiting static incommensurate magnetism below $T_{N}$ = 42 K \[31\] This incommensurate modulation has a very long correlation length ($\geq 600$ Å), as determined from a very sharp satellite magnetic Bragg peak in neutron scattering results. ZF-$\mu$SR spectra of this system \[10,32\], shown in Fig. 6(c), has a Bessel-function line shape, as expected for an incommensurate spin structure. The amplitude of this precession, however, increases gradually below $T_{N}$ with decreasing temperature, while the frequency is almost independent of temperature below $T_{N}$. Furthermore, the amplitude of precessing signal at $T\rightarrow 0$ is less than half of that in LBCO:0.125 (Fig. 6(b)), which indicates that the static magnetism exists in less than a half of the total volume.
In Fig. 7(a) and (b), we show temperature dependences of the volume fraction $V_{M}$ of muons in the region with static magnetic freezing, derived from the precession amplitude, and the frequency $\nu$ observed in LCO:4.11, (La$_{1.88}$Sr$_{0.12}$)CuO$_{4}$ (LSCO:0.12), LBCO:0.125 and (La$_{1.475}$Nd$_{0.4}$Sr$_{0.125}$)CuO$_{4}$ (LNSCO:0.125) \[33\]. In LCO:4.11 and LSCO:0.12, where superconductivity coexists with static magnetism, the static magnetism is confined to a partial volume fraction. As shown in Fig. 7(c), the temperature dependence of the neutron Bragg intensity in LCO:4.11 is consistent with the behavior of $V_{M}\nu^{2}$. Unlike usual magnetic systems, however, the $T$ dependence is mostly due to the change of the volume fraction $V_{M}$.
The internal field at the muon site in HTSC systems is due to dipolar field from neighbouring static Cu spins. Depending on the range of this field, the volume fraction of muons $V_{M}$ subject to static field is somewhat larger than the volume fraction $V_{Cu}$ of frozen Cu spins. Figure 8(a) shows our simulation results for the relationship between $V_{Cu}$ and $V_{M}$ for cases with incommensurate static magnetism in island regions having a radius $R$ \[10\]. Here we see that about 70-80% population of static Cu spins is enough to create observable static fields at the sites of all the muons. Thus, our previous result \[33\] on La$_{1.45}$Nd$_{0.4}$Sr$_{0.15}$CuO$_{4}$ (LNSCO:0.15), which is a superconductor with $T_{c} \sim 10 K$ with $V_{M} > 95$% is still compatible with a picture that superconductivity and static magnetism occur in mutually exclusive regions of CuO$_{2}$ plane.
Although the relationship between $V_{M}$ and $V_{Cu}$ is nearly independent of the radius $R$ of static SDW islands, the damping rate $\Lambda$ of the Bessel oscillation depends on $R$, as shown in Fig. 8(b). By converting observed $V_{M}$ into $V_{Cu}$ using Fig. 8(a), and then plotting observed $\Lambda$ in Fig. 8(b), we find that our ZF-$\mu$SR results in LCO:4.11 are consistent with the island size $R \sim$ 15 - 30 Å\[10\]. Note that this length scale is comparable to $\xi_{ab}$. This model of “static SDW nano islands” can be reconciled with the long range spin correlations found by neutron Bragg peaks, if we consider percolation of these islands via a help of interplaner correlations, as proposed in \[10\].
To study the relationship between the magnetic volume fraction $V_{Cu}$ and the superfluid density $n_{s}/m^{*}$, we performed $\mu$SR measurements in (La$_{1.85-y}$Eu$_{y}$Sr$_{0.15}$)CuO$_{4}$ (LESCO) \[11,34\] where the concentration of doped hole carriers is fixed to 0.15 per Cu, while the magnetic volume changes with increasing Eu concentration $y$. The volume fraction with static magnetism was determined by the ZF-$\mu$SR, while $n_{s}/m^{*}$ was measured in TF-$\mu$SR. The results shown in Fig. 9 clearly demonstrate a trade-off between $V_{Cu}$ and the superfluid density $n_{s}/m^{*}$. This implies that superconductivity does not exist in the volume which has static magnetism.
In the plot of $\sigma(T\rightarrow 0)$ versus $T_{c}$ in Fig. 1, the results of these systems with static stripe magnetism, shown by the striped square symbols, follow the trajectory of simple hole-doped and Zn-doped systems. This indicates that $n_{s}/m^{*}$ is again a determining factor of $T_{c}$ in HTSC systems where superconductivity and magnetism coexist. We also note that the results in overdoped Tl2201 can be viewed as following a monotonic relationship between $T_{c}$ and the superfluid density, with the slope roughly comparable to cuprates with/without other types of perturbations.
Analogy with superfluid He films
================================
Superfluid He films represent another set of systems where $T_{c}$ is strongly correlated with the 2-dimensional superfluid density $n_{s2d}$ at $T\rightarrow 0$. Using the published results, we made Fig. 10 \[35\] which shows correlations between $T_{c}$ and $n_{s2d}/m^{*}$ for $^{4}$He films on Mylar substrate (open circle symbol) \[36\], on Vycor Glass which represents a porous media (star symbol) \[37,38\], as well as for $^{4}$He/$^{3}$He mixtures adsorbed on fine alumina powders (closed diamond symbol) \[39\]. The results on Mylar films show linear relationship, consistent with the behavior expected for the KT transition \[40\] (see next section for details) as indicated by the solid line. With the lowest level of perturbation, this case corresponds to simple hole-doped HTSC systems in comparison between the cuprates and He films. The results on Vycor Glass is analogous to the Zn-doped cuprates: in both cases some normal regions are formed as a “healing region/layer”, while $T_{c}$ is still strongly correlated with the superfluid density despite non-trivial geometry of the superfluid.
Mixture of bosonic $^{4}$He and fermionic $^{3}$He liquids exhibits a phase diagram shown in the inset of Fig. 10. With increasing $^{3}$He fraction, $T_{c}$ decreases, being roughly proportional to the volume fraction of $^{4}$He. In bulk geometry, the mixture undergoes macroscopic phase separation into $^{4}$He-rich superfluid and $^{3}$He-rich normal fluid, the heavier superfluid existing underneath the lighter normal fluid in a container. This phase separation can be confined into a microscopic length scale by adsorbing the mixture onto porous media or fine powders \[39,41\], where superfluidity remains up to a large $^{3}$He fraction. The results for the $^{4}$He/$^{3}$He mixture in Fig. 10 exhibit a behavior very similar to that of overdoped Tl2201 in the cuprates in Fig. 1. In both the He mixture and the overdoped HTSC, inclusion of too many fermions (doped holes and $^{3}$He) results in microscopic phase separation and reduction of both $T_{c}$ and the superfluid density. Thus, we find a remarkable similarities \[35\] in the plot of $T_{c}$ versus superfluid density in cuprates (Fig. 1) and He films (Fig. 10) for the cases with/without perturbation.
Figure 1 shows correlations between $T_{c}$ and 3-dimensional (3-d) superfluid density while Fig. 10 shows $T_{c}$ versus 2-d superfluid density. By knowing the average interlayer distance $c_{int}$ between the CuO$_{2}$ planes, we can generate a plot of $T_{c}$ versus $n_{s2d}/m^{*}$ for the cuprate systems. The resulting plot, Fig. 11, clearly indicates that in HTSC systems, $T_{c}$ is higher for shorter $c_{int}$ for a given 2-d superfluid density $n_{s2d}/m^{*}$ \[42\]. This observation is consistent with the variation of $T_{c}$ reported for multilayer films with alternating layers of superconducting YBa$_{2}$Cu$_{3}$O$_{7}$ (YBCO) and insulating PrBa$_{2}$Cu$_{3}$O$_{7}$ (PBCO) \[43\], as shown in the inset of Fig. 11. These results indicate that the interlayer coupling is essential for obtaining higher $T_{c}$ in HTSC systems. The observed variation of $T_{c}$ cannot be explained by the simplest version of KT transition where $T_{c}$ should be determined solely by $n_{s2d}/m^{*}$.
Condensation mechanisms
=======================
BE and BCS condensation and KT transition
-----------------------------------------
Correlations between $T_{c}$ and the superfluid density shown in Fig. 1 and Fig. 11 help consideration of condensation mechanisms in HTSC systems. In BE condensation \[44\], the bosons are pre-formed at a very high “paring” temperature $T_{p}$. At a given temperature $T < T_{p}$ each boson has kinetic energy of $k_{B}T$, which defines the thermal wave length $\lambda_{th}$ representing the spread of the wave function of this boson due to the uncertainty principle. With decreasing $T$, $\lambda_{th}$ increases. When $\lambda_{th}$ becomes comparable to the inter-particle distance $n_{B}^{-1/3}$, the wave functions of neighbouring bosons start to overlap, as illustrated in Fig. 12(a). Thanks to the tendency of bosons to fall into the same state by building up phase coherence of wave functions, Bose condensation occurs at $T$ where $\lambda_{th} \sim n_{B}^{-1/3}$. From this, we have the BE condensation temperature $T_{B} \propto
n_{B}^{2/3}/m_{B}$, where $n_{B}$ and $m_{B}$ denote the density and mass of the boson. In this way, the superfluid density and $T_{c}$ is directly related in BE condensation.
In BCS condensation \[45\], the energy scale of attractive interaction determines the energy gap and $T_{c}$. The effective Fermi energy $\epsilon_{F} \propto n_{n}^{2/3}/m^{*}$ is much larger than $kT_{c}$ in BCS condensation, i.e., there is a sufficient density of fermions above $T_{c}$. However, the system stays in the normal state until the temperature is reduced to the pairing energy scale $T_{p}$, where bosons are formed. Once pairs are formed, their boson density is high enough at $T_{c}$, and thus the condensation occurs immediately at $T_{c} \sim T_{p}$. In BCS condensation, $T_{c}$ depends on carrier density through the density of states at the Fermi level which determines the strength of electron-phonon interaction. However, this dependence is rather indirect. Suppose we have a BCS superconductor with the carrier density $n_{s}$. If the Debye frequency is doubled, $T_{c}$ and the energy gap $\Delta$ would be doubled. However the carrier density stays unchanged, as can be found from the illustration of Fig. 12(b). This example shows that the superfluid density is not a direct determining factor for $T_{c}$ in BCS condensation.
Kosterlitz-Thouless transition \[40\] is a phenomenon in 2-dimensional superfluids / superconductors. The elementary excitation of superfluid He film is the formation of vortex anti-vortex pairs. The energy required for this is governed by the phase stiffness, which is proportional to the superfluid density. At the transition temperature $T_{KT}$, the thermal energy becomes sufficient to form un-bound vortices, which results in dissipation and destruction of superfluidity / phase coherence. Thermodynamic arguments lead to the universal relationship between the 2-d superfluid density at $T=T_{KT}$ and the value of $T_{c}$ as $kT_{KT}\propto n_{s2d}/m^{*}(T=T_{KT})$, which should be independent of material. This relationship was confirmed as the jump of the superfluid density at the superfluid transition of He films \[46\].
Evolution from 3-d to 2-d
-------------------------
Figure 13(a) illustrates evolution of various energy scales in superfluid He film and a thin film of BCS superconductor. The BE condensation temperature of He in the 3-d limit corresponds to the lambda temperature $T_{\lambda} = 2.2$ K, which is close to $T_{B}\propto n_{B}^{2/3}/m^{*}$, except for some reduction due to departure of real system from an ideal non-interacting Bose-gas limit. With decreasing film thickness $d$, $T_{c}$ starts to change when the thickness becomes comparable to a few times the coherence length $\xi \sim n_{B}^{-1/3}$. For $d \leq \xi$, $T_{c} \propto n_{s2d}/m^{*}$. In superfluid He film on Mylar, the superfluid density at $T\rightarrow 0$ is very close to that at just below $T_{c}$, as shown in Fig. 13(a). Thus the Kosterlitz-Thouless relation holds for $n_{s2d}$ at $T\rightarrow 0$, as can be seen in Fig. 10. Between $T_{c}$ and $T_{\lambda} = 2.2$ K, the order parameter exhibits phase fluctuations, i.e., superfluidity is dynamic. The pair formation energy scale is much higher than $T_{c}$ in He for any thickness value $d$.
In a thin film of a BCS superconductor, $T_{c}$ should show a similar reduction from the value $T_{c3d}$ in the bulk 3-d system, when the film thickness becomes comparable to or smaller than a few times $\xi$ in the clean limit (and [*l*]{} in the dirty limit), as shown in Fig. 13(b). The superfluid density $n_{s}/m^{*}$ at $T\rightarrow 0$ would follow a thickness dependence similar to that of $T_{c}$. Note that $n_{s}/m^{*}$ is very large, as it is related to $T_{F} \propto n_{n}^{2/3}/m^{*} \gg T_{c}$ in the 3-d limit. There should be difference between $n_{n}/m^{*}$ and $n_{s}/m^{*}$ in the region $d < \xi$ (or [*l*]{}), due to difficulty in forming pairs in a restricted geometry. On the other hand, the general argument of KT relationship should still hold. This is possible only if $n_{s2d}/m^{*}$ at $T=T_{c}$ for $d < \xi$ is much smaller than $n_{s2d}/m^{*}$ at $T\rightarrow 0$, as illustrated in Fig. 13(b). Therefore, the “jump” of the superfluid density at $T_{KT}$ is practically invisible in a BCS thin film. For $d \leq \xi$, the region $T_{c} < T < T_{c3d}$ is characterized by phase fluctuations. However, the pair formation occurs at $T \sim T_{c3d}$ for any thickness. Consequently, bosons exist only below $T_{c3d}$, as illustrated in Fig. 13(b).
BE-BCS Crossover and Phase Fluctuation Models
---------------------------------------------
In 1989 \[6\] and 1991 \[7\], we suggested the relevance of BE condensation to the universal relationship between $T_{c}$ and $n_{s}/m^{*}$ in HTSC systems shown in Fig. 1 \[6\]. By combining this phenomenon with the pseudogap behavior then observed by NMR and conductivity, we proposed a picture of BE to BCS crossover in 1993-94 \[47,48\], as illustrated in Fig. 14(a) \[42,47,48\]. General concept of the BE-BCS crossover has been considered earlier by several scientists \[49\], including Randeria and co-workers \[50\] who adopted this concept to the interpretation of the susceptibility/NMR results for the pseudo gap. Within our knowledge, however, our proposal was the first to combine the results of superfluid density with the pseudogap behavior in the underdoped cuprates. As shown in Fig. 14(a), we regarded the pseudogap temperature $T^{*}$ as the pair formation temperature $T_{p}$. The reduction of the c-axis dc conductivity below $T^{*}$ can be interpreted as resulting from reduced interplaner tunneling probability for paired bosonic carriers having 2e charges, while reduction of magnetic susceptibility can be attributed to the formation of a spin singlet bosonic (pairing) state in this picture.
In 1995, Emery and Kivelson (EK) \[51\] proposed a model based on phase fluctuations, as illustrated in Fig. 14(b). The BE-BCS model and the phase fluctuation model share many features in common. However, there are several important differences. Based on the linear relationship between $T_{c}$ and the superfluid density at $T\rightarrow 0$, EK presented arguments basically parallel to that for the KT transition in 2-d systems. They pointed out that $T_{c}$ of the underdoped cuprates is determined by the energy scale for the phase fluctuations. In their picture, the entire region of pseudogap below $T^{*}$ is characterized by phase fluctuations. They argued that the 2-dimensional aspect is essential for obtaining high $T_{c}$.
In order to calculate $T_{\theta}^{max}$ which denotes the energy scale for phase fluctuations to destroy superconductivity, EK multiplied $\lambda^{-2} \propto n_{s}/m^{*}$ to the interplaner distance $c_{int}$ for HTSC and some other 2-d systems. This is equivalent of obtaining $n_{s2d}/m^{*} \propto
T_{F}$ in 2-d. For 3-d systems, EK multiplied $n_{s}/m^{*}$ and the coherence length $\xi$, which leads to an energy scale much higher than $T_{F}$. This energy scale is unrealistically high, and irrelevant to condensation arguments. If one substitutes the interparticle distance $n_{s}^{-1/3}$ instead of $\xi$, we recover $T_{F}$ for 3-d systems. Then, Table 1 in ref \[51\] by EK becomes essentially equivalent to Fig. 3 of our 1991 paper \[7\], shown as a part of Fig. 3 of this article, where we plotted $T_{c}$ versus $T_{F}$. We consider that the distinction between 2-d and 3-d systems by EK is an artefact resulting from the overestimate of $T_{\theta}^{max}$ in 3-d systems. In general, the 2-dimensional aspect does not help increasing $T_{c}$ of any superconductor, as can be found in Fig. 13.
Figure 15 shows the current estimates of $T^{*}$ from various methods as plotted versus hole concentration. If the entire region below $T^{*}$ is supposed to have superconducting phase fluctuations, as conjectured by EK, the situation is rather similar to a thin-film BCS superconductor shown in Fig. 13(b). In contrast, if $T^{*}$ is solely representing the pair formation, as proposed in our BE-BCS crossover picture, and if there is a 2-dimensional aspect remaining in HTSC systems, there should be two different energy scales above $T_{c}$, as we discussed in ref \[35\]. They are the temperature $T_{dyn}$ at which the phase coherence of bosons completely disappear and dynamic superconductivity vanishes, in addition to the temperature $T_{p}$ at which pairs are dissolved into fermions. This situation is similar to the case of He films shown in Fig. 13(a). One of important differences between BE-BCS and EK conjectures lies in this point.
Pair formation and dynamic superconductivity
--------------------------------------------
Since high-$T_{c}$ cuprates have a highly 2-dimensional electronic structure, several experiments have detected the “dynamic superconductivity” existing above $T_{c}$: (1) In YBCO-PBCO films with a thick PBCO layers separating a single layer YBCO, $T_{c}$ is reduced to 15-20 K. Between this temperature and the $T_{c} \sim 90$ K of the bulk YBCO, one observes reduction of the normal state conductivity, which follows the predictions of the Kosterlitz-Thouless theory \[52\]. (2) Corson [*et al.*]{} \[53\] measured dynamic superfluid response in underdoped Bi2212, and found that the response depends on the measuring frequency $\omega$ above a certain “branch-off” temperature $T_{off}$, as shown in Fig. 16. Furthermore, the superfluid density $n_{s}/m^{*}$ at $T=T_{off}$ agrees well with the universal value $n_{s}/m^{*}(T=T_{KT})$ expected for the KT transition. The critical temperature $T_{c}$, at which $n_{s}/m^{*} =0$, increases with increasing $\omega$, suggesting that this phenomenon corresponds to the “dynamic superfluidity” expected above $T_{c}(\omega=0)$ and below $T_{dyn}$. It should be noted, however, that $T_{dyn}$ for the measuring frequency of 600 GHz is limited to $T\leq 100$ K.
These results indicate that certain cuprate systems with high anisotropy, such as underdoped Bi2212 and single-layer YBCO film, exhibit dynamic superconductivity as expected from the KT theory. However, these results do not necessarily provide explanation to the origin of the pseudo gap at $T=T^{*}$, since $T_{dyn}$ observed in Bi2212 is significantly lower than the pseudo-gap temperature $T^{*}$. Namely, the phase fluctuations alone cannot explain the entire pseudo-gap phenomena. Dependence to an external magnetic field should be quite different if these two distinct energy scales above $T_{c}$ correspond respectively to pairing and dynamic superconductivity. The former would not depend much on external fields, while the latter should be very sensitive to the field. Indeed, the pseudo gap below $T^{*}$ was found to be insensitive to the applied field in tunneling measurements \[54\]. Recently, Lavrov [*et al.*]{} \[55\] found that c-axis conductivity shows negative magneto-resistance upon cooling at a temperature well below $T^{*}$ but above $T_{c}$. This negative magneto-resistance may be due to the onset of dynamic superconductivity below $T_{dyn}$.
The results of the Nernst effect \[56\], which appear in the underdoped region of La214 systems above $T_{c}$ but well below $T^{*}$ can also be interpreted as possible evidence for dynamic superconductivity below $T_{dyn}$. Other phenomena potentially related to this energy scale include: (a) superfluid response of the ARPES coherence peak \[57\]; and (b) 41 meV neutron resonance mode \[58\].
Based on these considerations, we propose a new phase diagram, with two distinct lines of $T^{*}$ and $T_{dyn}$ in the underdoped region as shown in Fig. 17. Note that the pair formation at $T^{*}$ is necessary for superconductivity in cuprates, but is not enough to support any phase coherence. It is only when the thermal energy scale becomes less than the energy scale representing the number density of bosons that the phase coherence can set in. The onset temperature $T_{dyn}$ of dynamic superfluidity represents this energy scale. In highly 2-dimensional cases, one should further cool down below $T_{dyn}$ before achieving long-range phase coherence at $T_{c}$.
Evolution from superconductor to normal metal
=============================================
phase diagram for the overdoped cuprates
----------------------------------------
In BCS superconductors in the clean limit, all the carriers in the Fermi sphere contribute to superfluid. There should be no normal carriers remaining at $T \rightarrow 0$. Therefore, neither the BE-BCS nor the phase fluctuation pictures can explain the reduction of $n_{s}$ with increasing hole doping observed in several overdoped cuprate systems \[8,15,25\] described in section III-B.
Tallon, Loram and co-workers \[59\] have noticed that the $T^{*}$ line may be heading towards $T=0$ at a “critical hole concentration” around $x_{c} \sim 0.19$. $\mu$SR studies of Panagopoulos [*et al.*]{} \[60\] show that static magnetism, either spontaneously existing or induced by Zn doping, disappears at $x \geq x_{c}$. Generally, increasing hole doping would tend to destroy magnetic interactions in the cuprates.
Let us make the following three assumptions: (a) the paring in HTSC systems is due to a magnetic interaction; (b) the $T^{*}$ line represents the pair formation, and (c) the $T^{*}$ line disappears at $x = x_{c}$. Then, no genuine superconducting pairing exists in the overdoped region at $x \geq x_{c}$. However, if the energy loss for charge disproportionation is overcome by the gain of pairing and condensation energies, the system can spontaneously phase separate into a “hole-poor” superfluid with the local hole concentration $x \leq x_{c}$ and a “hole-rich” normal fermion region with $x > x_{c}$. If this phase separation remains microscopic with the length scale comparable to $\xi_{ab}$, we can expect superconductivity in the overdoped region, similarly to the case of $^{4}$He/$^{3}$He mixture.
The energy loss of charge disproportionation can be estimated following Coulomb blockade type calculation and/or formation of nano-islands serving as capacitor allays. Our crude estimate \[23\] shows that this energy becomes comparable to paring/condensation energy gain in the cuprates. Based on these considerations, we proposed a phase diagram shown in Fig. 17, with phase separation in the overdoped region. As shown in Fig. 5(b), the doping evolution of the superfluid density $n_{s}$, calculated using this model \[23\], shows a good qualitative agreement with the observed results. We also note that this model can reproduce a very sharp temperature dependence of $n_{s}/m^{*}$ and $H_{c2}$ at $T\rightarrow 0$ found in the heavily overdoped region of Tl2201 \[8,15,61\].
Tallon and Loram \[59\] presented a view that the existence of superconductivity in the overdoped cuprates at $x \geq x_{c}$, where the $T^{*}$ line does not exist, implies that the magnetic interaction below the $T^{*}$ line is not required for superconductivity but rather is a competing factor which weakens superconductivity. The present model with phase separation provides an alternative view where the pseudogap phenomena, as a necessary factor (pair formation) for superconductivity, can be compatible with superconductivity in the overdoped cuprates.
analogous cases in other systems
--------------------------------
In HTSC cuprates, as well as in 2-d organic (BEDT-TTF)$_{2}$-X and 3-d A$_{3}$C$_{60}$ superconductors, superconductivity appears in the vicinity of the insulator to metal transition. Superconductivity is taken over by a presumably Fermi-liquid metallic state by increasing carrier doping in cuprates, or by application of chemical and/or external pressure in BEDT and A$_{3}$C$_{60}$. In all these cases, we expect that the normal state spectral density (or Drude weight) $n_{n}/m^{*}$ to increase in the process towards a simple Fermi liquid. We showed that the superconducting spectral weight [*decreases*]{} in this process for the case of cuprates.
We performed magnetization measurements of the in-plane penetration depth in (BEDT-TTF)$_{2}$Cu(NCS)$_{2}$ under applied pressure \[62\], and found that $n_{s}/m^{*}$ decreases with increasing pressure as shown in Fig. 18(a), contrary to the behavior of $n_{n}/m^{*}$ found in quantum oscillation measurements \[63\]. This system is known to be well in the clean limit. These results indicate that not only in cuprates but also in BEDT, the crossover from superconducting to metallic ground state is associated with anomalous reduction of the superfluid spectral weight.
Similar behavior is also seen in A$_{3}$C$_{60}$ systems. As shown in Fig. 1 and Fig. 18(b), the muon relaxation rate $\sigma(T\rightarrow0)$ measured in A$_{3}$C$_{60}$ decreases with decreasing lattice constant \[17,18\], in the approach towards a simple metallic state. In view of rather large residual resistivity in the normal state, the results for these fullerides might be subject to correction related to the mean free path. However, the un-corrected raw data indicate that $1/\lambda^{2}$ again shows anomalous reduction when the system approaches presumably a simple metallic ground state.
These three systems exhibit similar phase diagrams: superconductivity appears in the evolution from magnetic and insulating ground state to presumably simple Fermi liquid state; in BEDT systems, a pseudo-gap like behavior near the magnetic phase was found in susceptibility. These features suggest a possibility that the anomalous results in the overdoped cuprates may be a generic behavior shared by a wider range of superconductors based on correlated electron systems. We note that all these systems follow the universal linear relationship in the plot of $T_{c}$ versus $n_{s}/m^{*}$, with approximately the same slope as shown in Fig. 1 and in ref. \[62\].
Summary
=======
In this paper, we showed that $T_{c}$ in HTSC cuprate systems exhibits universal correlations with the superfluid spectral weight $n_{s}/m^{*}$, for the cases of simple hole doping, as well as for more complicated cases with Zn-doping, overdoping and static SDW nano island formation, where the system undergoes spontaneous phase separation between superconducting and normal regions with the length scale comparable to the in-plane coherence length. Robustness of this relation for the case with/without perturbation is analogous to the case of superfluid $^{4}$He and $^{4}$He/$^{3}$He mixture films in non-porous and porous media. In all these cases, the superfluid density is the determining factor for $T_{c}$. This is the basic feature of BE condensation.
By slightly revising the BE-BCS crossover picture, we have proposed a new phase diagram for the cuprates which has (a) two separate lines of $T^{*}$ and $T_{dyn}$ above $T_{c}$ in the pseudogap state in the underdoped region, (b) disappearance of the $T^{*}$ line at the critical hole concentration $x_{c}$, and (c) phase separation in the overdoped region. In this model, the $T^{*}$ line represents pair formation, whereas the $T_{dyn}$ line corresponds to the onset of dynamic superconductivity. Low dimensionality prevents formation of long-range phase coherence between $T_{dyn}$ and $T_{c}$. We also noted anomalous reduction of the superfluid spectral weight $n_{s}/m^{*}$ in the overdoped cuprates as well as in the 2-d organic BEDT and 3-d fulleride superconductors, when the system approach simple metallic ground state.
acknowledgement
===============
The author would line to acknowledge collaboration with G.M. Luke, K.M. Kojima, S. Uchida, R.J. Birgeneau, K. Yamada and discussions with O. Tchernyshyov and M. Randeria. This work is supported by a grant from US NSF DMR-01-02752 and CHE-01-17752.
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\[Figure 15.\]
0.2 truein
\[Figure 16.\]
0.2 truein
\[Figure 17.\]
0.2 truein
\[Figure 18.\]
[^1]: E-mail: tomo@lorentz.phys.columbia.edu
|
---
author:
- Martin Hohenadler
- Wolfgang von der Linden
title: 'Lang-Firsov approaches to polaron physics: From variational methods to unbiased quantum Monte Carlo simulations'
---
Introduction {#sec:intro}
============
In the last decades, there has been substantial interest in simple models for electron-phonon (el-ph) interaction in condensed matter. Despite intensive theoretical efforts, it was not before the advent of numerical methods in the 1980’s that a thorough understanding on the basis of exact, unbiased results was achieved. Although at the present our knowledge of the rather simple cases of a single carrier (the [*polaron problem*]{}) or two carriers (the [*bipolaron problem*]{}) in Holstein and Fröhlich models is fairly complete, this is not true for arbitrary band fillings. There is still a major desire to develop more efficient simulation techniques to tackle strongly correlated many-polaron models, which are expected to describe several aspects of real materials currently under investigation, such as quantum dots and quantum wires, high-temperature superconductors or colossal-magnetoresistance manganites.
One of the principle problems in computer simulations of microscopic models is the limitation in both system size and parameter values. Whereas the former can be overcome for the polaron and the bipolaron problem in some cases, it is very difficult to obtain results of similar quality in the many-electron case. Moreover, many approaches still suffer from severe restrictions concerning the parameter regions accessible. For example, interesting materials such as the cuprates and manganites are characterized by small but finite phonon frequencies—as compared to the electronic hopping integral—and intermediate to strong el-ph interaction. Unfortunately, simulations turn out to be most difficult exactly for such parameters, and it is therefore highly desirable to improve existing simulation methods.
In this chapter, we shall mainly review different versions of a recently developed quantum Monte Carlo (QMC) method applicable to Holstein-type models with one, two or many electrons. The appealing advantages of QMC over other numerical methods include the accessibility of rather large systems, the exact treatment of bosonic degrees of freedom ([i.e.]{}, no truncation is necessary), and the possibility to consider finite temperatures to study phase transitions. The important new aspect here is the use of canonically transformed Hamiltonians, which permits the introduction of exact sampling for the phonon degrees of freedom, enabling us to carry out accurate simulations in practically all interesting parameter regimes.
Additionally, based on a generalization of the Lang-Firsov transformation, we shall present a simple variational approach to the polaron and the bipolaron problem which yields surprisingly accurate results.
The chapter is organized as follows. In section \[sec:model\], we present the general model Hamiltonian. Section \[sec:lf\] is devoted to a discussion of the Lang-Firsov transformation, and section \[sec:vpa\] contains the derivation of the variational approach. The QMC method is introduced in section \[sec:qmc\]. Section \[sec:results\] gives a selection of results for the cases of one, two and many electrons. Finally, we summarize in section \[sec:summary\].
Model {#sec:model}
=====
In this paper we focus on the [*extended Holstein-Hubbard model*]{} defined by $$\begin{aligned}
\label{eq:H}\nonumber
H
=
&-&
t\sum_{{\langle}ij{\rangle}{\sigma}}c^\dag_{i{\sigma}}c_{j{\sigma}}
+
U\sum_i {{\widehat{n}}}_{i{\uparrow}} {{\widehat{n}}}_{i{\downarrow}}
+
V\sum_{{\langle}ij{\rangle}} {{\widehat{n}}}_i {{\widehat{n}}}_j
\\
&+&
\frac{{\omega}_0}{2}\sum_i({{\widehat{p}}}^2_i+{{\widehat{x}}}^2_i)
-
g'\sum_i {{\widehat{n}}}_i {{\widehat{x}}}_i
\,.\end{aligned}$$ Here $c^\dag_{i{\sigma}}$ creates an electron with spin ${\sigma}$ at site $i$, and ${{\widehat{n}}}_i=\sum_{\sigma}{{\widehat{n}}}_{i{\sigma}}$ with ${{\widehat{n}}}_{i{\sigma}}=c^\dag_{i{\sigma}}c^{{\phantom{\dag}}}_{i{\sigma}}$. The phonon degrees of freedom at site $i$ are described by the momentum ${{\widehat{p}}}_i$ and coordinate (displacement) ${{\widehat{x}}}_i$ of a harmonic oscillator. The microscopic parameters are the nearest-neighbour (denoted by ${\langle}{\rangle}$) hopping amplitude $t$, the on-site (Hubbard–) repulsion $U$, the nearest-neighbour Coulomb repulsion $V$, the Einstein phonon frequency ${\omega}_0$ and the el-ph coupling $g'$.
This model neglects both long-range Coulomb and el-ph interaction, which is often a suitable approximation for metallic systems due to screening. Two simple limiting cases of the Hamiltonian (\[eq:H\]) are the Holstein model ($U=V=0$) and the Hubbard model ($g'=V=0$). In general, the physics of the model (\[eq:H\]) is determined by the competition of the various interactions. Depending on the choice of parameters and band filling, it describes fascinating phenomena such as (bi-)polaron formation, Mott– and Peierls quantum phase transitions or superconductivity. As we shall see below, the [*adiabaticity ratio*]{} $$\alpha={\omega}_0/t$$ permits us to distinguish two physically different regimes, namely the [*adiabatic regime*]{} $\alpha<1$ and the [*non-adiabatic regime*]{} $\alpha>1$.
We further define the dimensionless el-ph coupling parameter $\lambda=g'^2/({\omega}_0 W)$, where $W=4t\mathrm{D}$ is the bare bandwidth in D dimensions. Alternatively, $\lambda$ may also be written as $\lambda=2{E_{\mathrm{P}}}/W$, [i.e.]{}, the ratio of the polaron binding energy in the atomic limit $t=0$, ${E_{\mathrm{P}}}=g'^2/2{\omega}_0$, and half the bare bandwidth. A useful constant in the non-adiabatic regime is $g^2={E_{\mathrm{P}}}/{\omega}_0$. We exclusively consider hypercubic lattices with linear size $N$ and volume $N^\mathrm{D}$, and assume periodic boundary conditions in real space.
Lang-Firsov transformation {#sec:lf}
==========================
The cornerstone of the methods presented here is the canonical [*(extended) Lang-Firsov transformation*]{} of the Hamiltonian (\[eq:H\]). The original Lang-Firsov (LF) transformation [@LangFirsov] has been used extensively to study Holstein-type models. A well-known, early approximation is due to Holstein [@Ho59a], who replaced the hopping term by its expectation value in a zero-phonon state, neglecting emission and absorption of phonons during electron transfer. However, this approach yields reliable results only in the non-adiabatic strong-coupling (SC) limit. For $\lambda=\infty$ (or $t=0$), the LF transformation provides an exact solution of the single-site problem [@Ma90].
Whereas transformed Hamiltonians have been treated numerically before [@dMeRa97; @Robin97; @FeLoWe00], the first QMC method making use of the LF transformation has been proposed in [@HoEvvdL03].
We introduce the extended LF transformation by defining the unitary operator $$\label{eq:polaron:U}
{\widehat{\Phi}}
=
{\mathrm{e}}^S
\,,\quad
S = {\mathrm{i}}\sum_{ij}\gamma_{ij} {{\widehat{n}}}_i {{\widehat{p}}}_j$$ with real parameters $\gamma_{ij}$, $i,j=1,\dots,N^\mathrm{D}$. ${\widehat{\Phi}}$ as defined in equation (\[eq:polaron:U\]) has the form of a translation operator, and fulfills ${\widehat{\Phi}}^\dag={\widehat{\Phi}}^{-1}$. Given an electron at site $i$, ${\widehat{\Phi}}$ mediates displacements $\gamma_{ij}$ of the harmonic oscillators at all sites $j$. Hence, the extended transformation is capable of describing an extended phonon cloud, important in the large-polaron or bipolaron regime. We shall use this transformation for the variational approach. However, the standard (local) LF transformation will be expedient as a basis for unbiased QMC simulations, in which the transformed Hamiltonian is treated exactly.
Operators have to be transformed according to ${\widetilde{{\widehat{A}}}}=
{\widehat{\Phi}}{\widehat{A}}{\widehat{\Phi}}^\dag$. Defining the function $f(\eta) =
{\mathrm{e}}^{\eta{\widehat{S}}} {\widehat{A}} {\mathrm{e}}^{-\eta{\widehat{S}}}$ we obtain $$\label{eq:polaron:fprime}
f'(\eta)
=
{\mathrm{e}}^{\eta{\widehat{S}}}{[{\widehat{S}},{\widehat{A}}]}{\mathrm{e}}^{-\eta{\widehat{S}}}
\,,$$ where $f'\equiv\partial f/\partial \eta$. A simple calculation gives $${[{\widehat{S}},c_{i{\sigma}}]}
=
-{\mathrm{i}}\sum_l\gamma_{il}\,{{\widehat{p}}}_l\,c_{i{\sigma}}
\,,\quad
{[{\widehat{S}},c^\dag_{i{\sigma}}]}
=
{\mathrm{i}}\sum_l\gamma_{il}\,{{\widehat{p}}}_l\,c^\dag_{i{\sigma}}
\,.$$ Substitution in equation (\[eq:polaron:fprime\]), integration with respect to $\eta$ and setting $\eta=1$ results in $${\widetilde{c}}_{i{\sigma}}^\dag
=
c_{i{\sigma}}^\dag\, {\mathrm{e}}^{ {\mathrm{i}}\sum_j\gamma_{ij}{{\widehat{p}}}_j}
\,,\qquad
{\widetilde{c}}_{i{\sigma}}
=
c_{i{\sigma}}\, {\mathrm{e}}^{-{\mathrm{i}}\sum_j\gamma_{ij}{{\widehat{p}}}_j}
\,.$$ For phonon operators, the relation $${\widetilde{{\widehat{A}}}}
=
{\mathrm{e}}^{{\widehat{S}}} {\widehat{A}} {\mathrm{e}}^{-{\widehat{S}}}
=
{\widehat{A}} + {[{\widehat{S}},{\widehat{A}}]} + \frac{1}{2!}{[{\widehat{S}},{[{\widehat{S}},{\widehat{A}}]}]} + \cdots
\,,$$ yields $$\label{eq:polaron:trop}
{\widetilde{{{\widehat{x}}}}}_i
=
{{\widehat{x}}}_i + \sum_j \gamma_{ij} {{\widehat{n}}}_j
\,,\qquad
{\widetilde{{{\widehat{p}}}}}_i\;
=
{{{\widehat{p}}}}_i
\,.$$ Collecting these results, the transformation of the Hamiltonian (\[eq:H\]) leads to $$\begin{aligned}
\label{eq:LF-H-extended}\nonumber
\hspace{-0.75em}
{\widetilde{H}}
&=&
\underbrace{ -t\sum_{{\langle}ij{\rangle}{\sigma}}
c^\dag_{i{\sigma}}c^{{\phantom{\dag}}}_{j{\sigma}} {\mathrm{e}}^{{\mathrm{i}}\sum_l(\gamma_{il}-\gamma_{jl}){{\widehat{p}}}_l}}
_{{\widetilde{H}}_{\mathrm{kin}}}
+
\underbrace{ \frac{{\omega}_0}{2}\sum_{\phantom{{\langle}{\rangle}}i\phantom{{\langle}{\rangle}}}({{\widehat{p}}}^2_i+{{\widehat{x}}}^2_i)}_{{\widetilde{H}}_{\mathrm{ph}}\equiv {\widetilde{H}}_{\mathrm{ph}}^p+
{\widetilde{H}}_{\mathrm{ph}}^x}
+
\underbrace{ \sum_{ij} {{\widehat{n}}}_j
{{\widehat{x}}}_i({\omega}_0\gamma_{ij}-g'\delta_{ij})}
_{{\widetilde{H}}_{\mathrm{ep}}}
\\
&&
+
\underbrace{ \sum_{ij} {{\widehat{n}}}_i {{\widehat{n}}}_j\hspace{-0.25em}
\left(
\frac{{\omega}_0}{2}\sum_l\gamma_{lj}\gamma_{li}-g'\gamma_{ij} +
\frac{U}{2}\delta_{ij} +
V\delta_{{\langle}ij{\rangle}}
\right)
- \mbox{\small$\frac{1}{2}$}U\sum_i {{\widehat{n}}}_i}_{{\widetilde{H}}_{\mathrm{ee}}}
.\end{aligned}$$ Here the term ${\widetilde{H}}_{\mathrm{ep}}$ describes the coupling between electrons and phonons, whereas ${\widetilde{H}}_{\mathrm{ee}}$ represents an effective el-el interaction. Hamiltonian (\[eq:LF-H-extended\]) will be the starting point for the variational approach in section \[sec:vpa\].
For QMC simulations, it is more suitable to require that the el-ph terms in ${\widetilde{H}}_\mathrm{ep}$ cancel. This can be achieved by setting $\gamma_{ij} = \gamma\delta_{ij}$ with $$\label{eq:polaron:gamma}
\gamma
=
\sqrt{\frac{\lambda W}{{\omega}_0}}
\,.$$ The parameter $\gamma$ corresponds to the distortion which minimizes the potential energy of the shifted harmonic oscillator $E_{\mathrm{pot}}=\frac{\omega_0}{2} x^2 - g'x$. This leads us to the standard LF transformation $$\label{eq:polaron:LFop}
{\widehat{\Phi}}_0
=
{\mathrm{e}}^{S_0}
\,,\quad
S_0={\mathrm{i}}\gamma\sum_i {{\widehat{n}}}_i {{\widehat{p}}}_i
\,,$$ and the familiar results for the transformed operators $${\widetilde{c}}^\dag_{i{\sigma}}
=
c^\dag_{i{\sigma}}{\mathrm{e}}^{{\mathrm{i}}\gamma {{\widehat{p}}}_i}
\,,\qquad
{\widetilde{c}}_{i{\sigma}}
=
c_{i{\sigma}}{\mathrm{e}}^{-{\mathrm{i}}\gamma {{\widehat{p}}}_i}$$ and $$\label{eq:polaron:LFvariables}
{\widetilde{{{\widehat{x}}}}}_i
=
{{\widehat{x}}}_i + \gamma {{\widehat{n}}}_i
\,,\qquad
{\widetilde{{{\widehat{p}}}}}_i
=
{{\widehat{p}}}_i
\,.$$ In contrast to the non-local transformation (\[eq:polaron:U\]), only the oscillator at the site of the electron is displaced. The transformed Hamiltonian reads $$\begin{aligned}
\label{eq:LF-H-local}\nonumber
{\widetilde{H}}
&=&
\underbrace{ -t\sum_{{\langle}ij{\rangle}{\sigma}}c^\dag_{i{\sigma}}c^{{\phantom{\dag}}}_{j{\sigma}}
{\mathrm{e}}^{{\mathrm{i}}\gamma({{\widehat{p}}}_i-{{\widehat{p}}}_j)}}
_{{\widetilde{H}}_{\mathrm{kin}}}
+
\underbrace{ \frac{{\omega}_0}{2}\sum_{\phantom{{\langle}{\rangle}}i\phantom{{\langle}{\rangle}}}({{\widehat{p}}}^2_i+{{\widehat{x}}}^2_i)}_{{\widetilde{H}}_{\mathrm{ph}}}
\\
&&
+
\underbrace{ (\mbox{\small$\frac{1}{2}$}U-{E_{\mathrm{P}}})
\sum_i{{\widehat{n}}}_i^2
+
V \sum_{{\langle}ij{\rangle}}{{\widehat{n}}}_i {{\widehat{n}}}_j
-
\mbox{\small$\frac{1}{2}$}U
\sum_i {{\widehat{n}}}_i
}_{{\widetilde{H}}_{\mathrm{ee}}}
\,.\end{aligned}$$ As we shall discuss in detail in section \[sec:qmc\], the difficulties encountered in QMC simulations of the original Hamiltonian (\[eq:H\]) are to a certain extent related to (bi-)polaron effects, [i.e.]{}, to the dynamic formation of spatially rather localized lattice distortions which surround the charge carriers and follow their motion in the lattice.
For a single electron, the aforementioned [*Holstein-Lang-Firsov*]{} (HLF) approximation [@Ho59a] becomes exact in the non-adiabatic SC or small-polaron limit, and agrees qualitatively with exact results also in the intermediate-coupling (IC) regime [@ZhJeWh99]. Although it overestimates the shift $\gamma$ of the equilibrium position of the oscillator in the presence of an electron, and does not reproduce the retardation effects when the electron hops onto a previously unoccupied site, the approximation mediates the crucial impact of el-ph interaction on the lattice. Consequently, the transformed Hamiltonian (\[eq:LF-H-local\]) can be expected to be a good starting point for QMC simulations, which then merely need to account for the rather small fluctuations around the (shifted or unshifted) equilibrium positions. In principle, it would also be possible to develop a QMC algorithm based on the Hamiltonian (\[eq:LF-H-extended\])—the basis of our variational approach—with the parameters $\gamma_{ij}$ determined variationally, but the local LF transformation proves to be sufficient.
The Hamiltonian (\[eq:LF-H-local\]) does no longer contain a term coupling the electron density ${{\widehat{n}}}$ and the lattice displacement ${{\widehat{x}}}$. By contrast, the extended transformation does not eliminate the interaction term completely. On top of that, the hopping term involves all phonon momenta ${{\widehat{p}}}_i$ as well as the parameters $\gamma_{ij}$, and the el-el interaction becomes long ranged \[cf equation (\[eq:LF-H-extended\])\].
For spin dependent carriers with ${{\widehat{n}}}_i^2\neq{{\widehat{n}}}_i$, the interaction term ${\widetilde{H}}_{\mathrm{ee}}$ contains a Hubbard-like attractive interaction. Whereas the latter can be treated exactly in the case of two electrons (section \[sec:partition-function-two-electrons\]), the many-electron case requires the introduction of auxiliary fields which complicate simulations. However, no such difficulties arise for the spinless Holstein model considered in section \[sec:results\].
Variational approach {#sec:vpa}
====================
For simplicity, we shall restrict the following derivation to one dimension; an extension to $\mathrm{D}>1$ is straight forward. Furthermore, we only consider finite clusters with periodic boundary conditions, although infinite systems may also be treated. The results of this section have originally been presented in [@HoEvvdL03; @HovdL05].
One electron
------------
As noted before, the simple variational method presented here is based on the extended transformation (\[eq:polaron:U\]), leading to the Hamiltonian (\[eq:LF-H-extended\]). We treat the $\gamma_{ij}$ as variational parameters which are determined by minimizing the ground-state energy in a zero-phonon basis in which ${\langle}{\widetilde{H}}_\mathrm{ep}{\rangle}=0$.
For systems with translation invariance the [*displacement fields*]{} satisfy the condition $\gamma_{ij}=\gamma_{|i-j|}$. Together with $\sum_i{{\widehat{n}}}_i=1$ for a single electron we get ${\widetilde{H}}_{\mathrm{ee}}=\frac{{\omega}_0}{2}\sum_l\gamma^2_l-g'\gamma_0$.
The eigenvalue problem of the transformed Hamiltonian (\[eq:LF-H-extended\]) is solved by making the following ansatz for the one-electron basis states $$\label{eq:polaron:basis_vpa}
\left\{
{\left|l{\right{\rangle}}}
= c^\dag_{l{\sigma}}{\left|0{\right{\rangle}}}
\otimes
\prod_{\nu=1}^N {|\phi_0^{(\nu)}{\rangle}}
\,,\quad
l=1,\dots,N
\right\}
\,,$$ where ${|\phi_0^{(\nu)}{\rangle}}$ denotes the ground state of the harmonic oscillator at site $\nu$. The non-zero matrix elements of the hopping term are $$\begin{aligned}
\label{eq:polaron:K}\nonumber
{\lal\right|}{\widetilde{H}}_{\mathrm{kin}}{\left|l'{\right{\rangle}}}
&=&
-t\delta_{{\left{\langle}}ll'{\right{\rangle}}} \prod_\nu {{\langle}\phi^{(\nu)}_0|}
{\mathrm{e}}^{{\mathrm{i}}(\gamma_{l\nu}-\gamma_{l'\nu}){{\widehat{p}}}_\nu}
{|\phi^{(\nu)}_0{\rangle}}
\\\nonumber
&=&
-t\delta_{{\left{\langle}}ll'{\right{\rangle}}}\;\prod_\nu\;\int\,{\mathrm{d}}x\,
\phi(x+\gamma_{l\nu})\phi(x+\gamma_{l'\nu})
\\
&=&
-t\delta_{{\left{\langle}}ll'{\right{\rangle}}} {\mathrm{e}}^{-\frac{1}{4}\sum_\nu(\gamma_{\nu}-\gamma_{\nu+l-l'})^2}
\,,\end{aligned}$$ where $\phi(x)$ denotes the real-space wavefunction of the harmonic-oscillator ground state. The Kronecker symbol $\delta_{{\left{\langle}}ll'{\right{\rangle}}}$ forces $l$ and $l'$ to represent nearest-neighbor sites. A simple calculation gives for the other terms in equation (\[eq:LF-H-extended\]) $$\label{eq:polaron:matelem}
{\lal\right|}{\widetilde{H}}_{\mathrm{ph}}{\left|l'{\right{\rangle}}}
=
\delta_{ll'}\frac{{\omega}_0}{2}
\,,\,\,\,
{\lal\right|}{\widetilde{H}}_{\mathrm{ep}}{\left|l'{\right{\rangle}}}
=
0
\,,\,\,\,
{\lal\right|}{\widetilde{H}}_{\mathrm{ee}}{\left|l'{\right{\rangle}}}
=
\delta_{ll'}\left(
\frac{{\omega}_0}{2}\sum_l\gamma_l^2
-g'\gamma_0
\right)
.$$ In the zero-phonon subspace spanned by the basis states (\[eq:polaron:basis\_vpa\]), the eigenstates of Hamiltonian (\[eq:LF-H-extended\]) with momentum $k$ are $$\label{eq:polaron:eigen}
{|\psi_k{\rangle}}
=
c^\dag_{k{\sigma}}{\left|0{\right{\rangle}}}\otimes\prod_\nu{|\phi_0^{(\nu)}{\rangle}}$$ with eigenvalues $$\label{eq:polaron:E(k)}
E(k)
=
{E_{\mathrm{kin}}}+ \frac{N{\omega}_0}{2} + \frac{{\omega}_0}{2}\sum_l\gamma_l^2-g'\gamma_0$$ and the kinetic energy $$\label{eq:polaron-vpa:Ekin}
{E_{\mathrm{kin}}}=
-t\sum_{\delta=\pm1} {\mathrm{e}}^{{\mathrm{i}}k\delta}
{\mathrm{e}}^{-\frac{1}{4}\sum_\nu(\gamma_\nu-\gamma_{\nu+\delta})^2}
\,.$$ Defining the Fourier transform $$\label{eq:polaron:FTgamma}
\overline{\gamma}_q
=
\frac{1}{\sqrt{N}}\sum_l {\mathrm{e}}^{{\mathrm{i}}ql}\gamma_l$$ and using ($\gamma_l \in {\mathbb{R}}$) $$\sum_\nu \gamma_\nu \gamma_{\nu+\delta}
=
\sum_q\overline{\gamma}_q\overline{\gamma}_{-q} {\mathrm{e}}^{{\mathrm{i}}q\delta}
=
\sum_q\overline{\gamma}_q^2\cos q\delta
\,,$$ we may write $${E_{\mathrm{kin}}}=
-t\sum_\delta {\mathrm{e}}^{{\mathrm{i}}k\delta} {\mathrm{e}}^{-\frac{1}{2}
\sum_q(1-\cos q\delta)\overline{\gamma}_q^2}
=
\varepsilon_0(k) {\mathrm{e}}^{-\frac{1}{2} \sum_q(1-\cos q)\overline{\gamma}_q^2}
=
\varepsilon(k)$$ with the tight-binding dispersion $\varepsilon_0(k)=-2t\cos k$. Hence the ground-state energy becomes $$\label{eq:polaron:evs}
E(k)
=
\varepsilon(k) + \frac{N{\omega}_0}{2}
+\frac{{\omega}_0}{2}\sum_q\overline{\gamma}_q^2 -
\frac{g'}{\sqrt{N}}\sum_q\overline{\gamma}_q
\,.$$ The variational parameters $\overline{\gamma}_p$ are determined by requiring $$\label{eq:polaron:deriv}
\frac{\partial E}{\partial \overline{\gamma}_p}
=
-\overline{\gamma}_p \varepsilon(k)
(1-\cos p) + {\omega}_0 \overline{\gamma}_p -
\frac{g'}{\sqrt{N}}
\overset{!}{=}0
\,,$$ so that the optimal values $\overline{\gamma}_p$ can be obtained from $$\label{eq:polaron:gammap}
\overline{\gamma}_p
=
\frac{g'}{\sqrt{N}}
\frac{1}{{\omega}_0 +
\varepsilon(k)(1-\cos p)}
\,.$$ Since $\varepsilon(k)$ depends implicitly on the $\overline{\gamma}_p$, equation (\[eq:polaron:gammap\]) has to be solved self-consistently. It has the typical form of the random-phase approximation since a variational ansatz for the untransformed Hamiltonian may be written as $$\label{eq:polaron:rpa_type}
{\widehat{\Phi}}^\dag{|\psi_k{\rangle}}
=
\frac{1}{\sqrt{N}} \sum_j
{\mathrm{e}}^{{\mathrm{i}}k j} \,c^\dag_{j{\sigma}} \,
{\mathrm{e}}^{-{\mathrm{i}}\sum_l \gamma_{jl} {\widehat{p}}_l}\,
{|0{\rangle}}\otimes\prod_\nu {|\phi_0^{(\nu)}{\rangle}}
\,,$$ with ${\widehat{\Phi}}$ as defined in equation (\[eq:polaron:U\]).
We shall also calculate the quasiparticle spectral weight for momentum $k=0$, defined as $$\label{eq:polaron:def_z0}
\sqrt{z_0}
=
{{\left{\langle}}0\right|} {\widetilde{c}}_{k=0,{\sigma}}{\left|\psi_0{\right{\rangle}}}
\,.$$ Here ${\left|\psi_0{\right{\rangle}}}$ denotes the ground state with one electron of momentum $p=0$ and the oscillators in the ground state ${\left|\phi_0{\right{\rangle}}}$. Fourier transformation and the same manipulations as in equation (\[eq:polaron:K\]) lead to $$\begin{aligned}
\label{eq:polaron:z0}\nonumber
\sqrt{z_0}
&=&
\frac{1}{N}\sum_{ij}{{\left{\langle}}\phi_0\right|}{{\left{\langle}}0\right|}
{\widetilde{c}}^{\phantom{\dag}}_{i{\sigma}}c^\dag_{j{\sigma}}{\left|0{\right{\rangle}}}{\left|\phi_0{\right{\rangle}}}
\\\nonumber
&=&
\frac{1}{N}\sum_i {{\left{\langle}}\phi_0\right|} {\mathrm{e}}^{-{\mathrm{i}}\sum_k\gamma_{ik} {{\widehat{p}}}_k}{\left|\phi_0{\right{\rangle}}}
\\
&=&
{\mathrm{e}}^{-\frac{1}{4}\sum_q{\widetilde{\gamma}}_q^2}
\,.\end{aligned}$$ Just as the HLF approximation, the present variational method becomes exact in the non-interacting limit ($\lambda=0$) and in the non-adiabatic SC limit. Furthermore, it yields the correct results both for $\alpha=0$ (classical phonons) and $\alpha=\infty$, and also gives accurate results for large $\alpha$ and finite $\lambda$, since the displacements of the oscillators—only local and generally overestimated in the HLF approximation—are determined variationally.
Two electrons
-------------
As in the one-electron case, the use of a zero-phonon basis leads to ${\langle}{\widetilde{H}}_{\mathrm{ep}}{\rangle}=0$ and, neglecting the ground-state energy of the oscillators, we also have ${\langle}{\widetilde{H}}_{\mathrm{ph}}{\rangle}=0$. Hence, ${\widetilde{H}}={\widetilde{H}}_{\mathrm{kin}}+{\widetilde{H}}_{\mathrm{ee}}$ with the transformed hopping term $${\widetilde{H}}_{\mathrm{kin}}
=
-t_{\mathrm{eff}}\sum_{{\langle}ij{\rangle}{\sigma}}
c^\dag_{i{\sigma}}c^{{\phantom{\dag}}}_{j{\sigma}}
=
\sum_{k{\sigma}} \varepsilon(k)\; c^\dag_{k{\sigma}}c^{{\phantom{\dag}}}_{k{\sigma}}$$ and $\varepsilon(k)=- 2\; t_{\mathrm{eff}} \cos(k)$. Here the effective hopping $$\label{eq:bipolaron:teff}
t_{\mathrm{eff}}
=
\frac{1}{2}\sum_{\delta=\pm1}
{\mathrm{e}}^{-\frac{1}{4}\sum_l(\gamma_{l-\delta}-\gamma_{l})^2}
\,,$$ where rotational invariance has been exploited. For two electrons of opposite spin ([i.e.]{}, ${{\widehat{n}}}_{i\sigma}{{\widehat{n}}}_{j\sigma}=0$ for $i\ne j$) and $V=0$, ${\widetilde{H}}_\mathrm{ee}$ in equation (\[eq:LF-H-extended\]) reduces to $$\label{eq:bipolaron:Iee_vij}
{\widetilde{H}}_{\mathrm{ee}}
=
2 v_0 - U + 2 \sum_{ij} v_{ij} {{\widehat{n}}}_{i{\uparrow}} {{\widehat{n}}}_{j{\downarrow}}
\,,\quad
v_{ij}
=
\frac{{\omega}_0}{2}
\sum_l\gamma_{lj}\gamma_{li}-g'\gamma_{ij}
+ \mbox{\small$\frac{1}{2}$}\delta_{ij} U
\,.$$ The eigenstates of the two-electron problem have the form $$\label{eq:bipolaron:states_k}
{\left|\psi_k{\right{\rangle}}}
=
\sum_p \overline{d}^{{\phantom{\dag}}}_p c^\dag_{k-p{\downarrow}} c^\dag_{p{\uparrow}}{\left|0{\right{\rangle}}}
\,,$$ suppressing the phonon component \[cf equation (\[eq:polaron:eigen\])\], and may be written as $$\label{eq:bipolaron:psi_rs}
{\left|\psi_k{\right{\rangle}}}
=
\frac{1}{\sqrt{N}}\sum_i {\mathrm{e}}^{{\mathrm{i}}k x_i}
\sum_l d^{{\phantom{\dag}}}_l c^\dag_{i{\downarrow}} c^\dag_{i+l{\uparrow}}{\left|0{\right{\rangle}}}
\,,$$ with the Fourier transform $$\label{eq:bipolaron:FT}
{\boldsymbol{d}}
=
F \overline{{\boldsymbol{d}}}
\,,\quad
(F)_{lp} = {\mathrm{e}}^{{\mathrm{i}}x_l p} / \sqrt{N}
\,.$$ The normalization of equation (\[eq:bipolaron:states\_k\]) reads $${{\left{\langle}}\psi_k\right|}\psi_k{\rangle}=
\sum_p |d_p|^2\,.$$ Using equation (\[eq:bipolaron:states\_k\]), we find for the expectation value of ${\widetilde{H}}_{\mathrm{kin}}$ $$\begin{aligned}
\nonumber
{{\left{\langle}}\psi_k\right|}{\widetilde{H}}_{\mathrm{kin}}{\left|\psi_k{\right{\rangle}}}
&=&
\sum_{pp'}
\overline{d}_p^* \overline{d}_{p'}^{\phantom{*}}
\sum_{q} \varepsilon(q)
\\\nonumber
&&
\times
\bigg(
\underbrace{ {{\left{\langle}}0\right|}
c^{{\phantom{\dag}}}_{p{\uparrow}} c^{{\phantom{\dag}}}_{k-p{\downarrow}}
{{\widehat{n}}}^{{\phantom{\dag}}}_{q{\uparrow}}
c^{\dag }_{k-p'{\downarrow}} c^\dag_{p'{\uparrow}}{\left|0{\right{\rangle}}}
}
_{\delta_{p,p'}\delta_{q,p}}
+
\underbrace{ {{\left{\langle}}0\right|}
c^{{\phantom{\dag}}}_{p{\uparrow}} c^{{\phantom{\dag}}}_{k-p{\downarrow}}
{{\widehat{n}}}^{{\phantom{\dag}}}_{q{\downarrow}}
c^{\dag }_{k-p'{\downarrow}} c^\dag_{p'{\uparrow}}{\left|0{\right{\rangle}}}
}
_{\delta_{p,p'}\delta_{q,k-p}}
\bigg)
\\\nonumber
&=&
\sum_p |\overline{d}_p|^2 \left[\varepsilon(p) + \varepsilon(k-p)\right]
\\
&=&
-4\;t_{\mathrm{eff}}\,{\boldsymbol{d}}^\dag T_k {\boldsymbol{d}}
\,.\end{aligned}$$ In the last step we have introduced $T_k=\frac{1}{2}F
\,{\mathrm{diag}}[\cos (p) + \cos (k-p)\big]\,F^\dag$ and made use of equation (\[eq:bipolaron:FT\]).
The expectation value of the interaction term, best computed in the real-space representation (\[eq:bipolaron:psi\_rs\]), takes the form $$\begin{aligned}
\nonumber
{{\left{\langle}}\psi_k\right|}{\widetilde{H}}_{\mathrm{ee}}{\left|\psi_k{\right{\rangle}}}
&=&
(2v_0-U)\sum_l |d_l|^2
+ \frac{2}{N} \sum_{ij} v_{ij}
\sum_{j'j''}\sum_{ll'}
d_l^* d^{\phantom{*}}_{l'}
{\mathrm{e}}^{{\mathrm{i}}k (x_{l\phantom{'}}-x_{l'})}
\\\nonumber
&&
\qquad\qquad\qquad\qquad\qquad\quad
\times
\underbrace{
{{\left{\langle}}0\right|}
c^{{\phantom{\dag}}}_{j' + l{\uparrow}} c^{{\phantom{\dag}}}_{j'{\downarrow}}
{{\widehat{n}}}^{{\phantom{\dag}}}_{i{\uparrow}} {{\widehat{n}}}^{{\phantom{\dag}}}_{j{\downarrow}}
c^\dag_{j''{\downarrow}} c^\dag_{j'' + l'{\uparrow}}
{\left|0{\right{\rangle}}}}_{
\delta_{jj'}\delta_{jj''}\delta_{i,j+l}\delta_{l,l'}
}
\\\nonumber
&=&
(2v_0-U) \sum_l |d_l|^2
+ \frac{2}{N} \sum_{jl} v_{j+l,j} |d_{l}|^2
\\
&=&
(2v_0-U) {\boldsymbol{d}}^\dag {\boldsymbol{d}} + 2{\boldsymbol{d}}^\dag V {\boldsymbol{d}}
\,,\end{aligned}$$ with the diagonal matrix $V_{ij}=\delta_{ij} v_{i}$.
The minimization of the total energy with respect to ${\boldsymbol{d}}$ yields the eigenvalue problem $$\label{eq:bipolaron:minimize}
(-4t_{\mathrm{eff}} \;T_k + 2 V)\,{\boldsymbol{d}}
=
(E_0 - 2v_0 + U)\,{\boldsymbol{d}}
\,.$$ The vector of coefficients ${\boldsymbol{d}}$ and thereby the ground state are found by minimizing the ground-state energy $E_0$ through variation of the displacement fields $\gamma_{ij}$. Similar to the one-electron case, this procedure takes into account displacements of the oscillators not only at the same but also at surrounding sites of the two electrons, and is therefore capable of describing extended bipolaron states (see section \[sec:res-bipolaron\]). Note that the two-electron problem is diagonalized exactly without phonons ([i.e.]{}, for $\lambda=0$).
Quantum Monte Carlo {#sec:qmc}
===================
In this section, we present an overview of our recently developed QMC algorithms for Holstein-type models [@HoEvvdL03; @HoEvvdL05; @HovdL05; @HoNevdLWeLoFe04].
As mentioned before, in contrast to the variational approach, the QMC approaches discussed here, based on the local LF transformation (\[eq:LF-H-local\]) which does not contain any free parameters, are unbiased. They yield exact results with only statistical errors that can in principle be made arbitrarily small.
The motivation for the development of improved QMC schemes for Holstein models stems from the fact that calculations with existing methods often suffer from strong autocorrelations, [i.e.]{}, non-negligible statistical correlations between successive MC configurations [@wvl1992; @HoEvvdL03]. In fact, autocorrelations may render accurate simulations impossible within reasonable computing time. As discussed in [@HoEvvdL03], the problem becomes particularly noticeable for small phonon frequencies and low temperatures.
Whereas autocorrelations can be avoided to a large extent for one or two electrons by integrating out the phonons analytically, no efficient general schemes exist for finite charge-carrier densities (see discussion in [@HoEvvdL03]).
In the sequel, we present a general ([i.e.]{}, applicable for all densities) solution for this problem in several steps. First, the effects due to el-ph interaction are separated from the free lattice dynamics by means of the LF transformation (\[eq:LF-H-local\]). Since the latter contains the crucial impact of the electronic degrees of freedom on the lattice, simulations may be based only on the purely phononic part of the resulting action. The fermionic degrees of freedom can then be taken into account exactly by [*reweighting*]{} of the probability distribution. Consequently, we may completely ignore the electronic weights in the updating process, and thereby dramatically reduce the computational effort. The [*principal component representation*]{} of the phonon coordinates allows exact sampling and avoids any autocorrelations.
Partition function {#sec:partition-function}
------------------
We begin by deriving the partition function for the case of a single electron. Then we discuss the differences occurring in the cases of two or more carriers.
### One electron {#sec:partition-function-one-electron}
The partition function is defined as $${\mathcal{Z}}={{\mathrm{Tr}}\,}{\mathrm{e}}^{-\beta{\widetilde{H}}}$$ with ${\widetilde{H}}$ given by equation (\[eq:LF-H-local\]) and the inverse temperature $\beta=(k_{\mathrm{B}}T)^{-1}$. For a single electron, ${\widetilde{H}}_{\mathrm{ee}}=-{E_{\mathrm{P}}}$ becomes a constant which needs only to be considered in calculating the total energy.
Using the Suzuki-Trotter decomposition [@wvl1992], we obtain $$\label{eq:polaron:suzuki-trotter}
{\mathrm{e}}^{-\beta{\widetilde{H}}}
\approx
({\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{kin}}} {\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{ph}}^p} {\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{ph}}^x})^L
\equiv
{\mathcal{U}}^L
\,,$$ where ${\Delta\tau}=\beta/L\ll1$. Splitting up the trace into a bosonic and a fermionic part and inserting $L-1$ complete sets of oscillator momentum eigenstates we find the approximation $${\mathcal{Z}}_L
=
{{\mathrm{Tr}}\,}_{\mathrm{f}}\int\,{\mathrm{d}}p_1{\mathrm{d}}p_2\cdots{\mathrm{d}}p_L
{\lap_1\right|}{\mathcal{U}}{\left|p_2{\right{\rangle}}}\cdots{\lap_L\right|}{\mathcal{U}}{\left|p_1{\right{\rangle}}}$$ with ${\mathrm{d}}p_\tau\equiv\prod_i {\mathrm{d}}p_{i,\tau}$. Each matrix element can be evaluated by inserting a complete set of phonon coordinate eigenstates $\int{\mathrm{d}}x {|x{\rangle}}{\lasx|}$, since all $x$-integrals are of Gaussian form and can easily be carried out. The result is $${\lap_\tau\right|} {\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{ph}}^x}{\left|p_{\tau+1}{\right{\rangle}}}
=
C^{N^{{\mathrm{D}}}}
{\mathrm{e}}^{-\frac{1}{2{\omega}_0{\Delta\tau}}\sum_i\left(p_{i,\tau}-p_{i,\tau+1}\right)^2}
\,,\quad
C
=
\sqrt{\frac{2\pi}{{\omega}_0{\Delta\tau}}}
\,.$$ The normalization factor in front of the exponential has to be taken into account in the calculation of the total energy, but cancels when we measure other observables. With the abbreviation $\mathcal{D}p={\mathrm{d}}p_1{\mathrm{d}}p_2\cdots{\mathrm{d}}p_L$ the partition function finally becomes $$\label{eq:polaron:Z}
{\mathcal{Z}}_L
=
C^{N^{{\mathrm{D}}}L} \int\,\mathcal{D}p\,\, {w_{\mathrm{b}}}\,{w_{\mathrm{f}}}\,,$$ where $$\label{eq:polaron:omega}
{w_{\mathrm{b}}}=
{\mathrm{e}}^{-{\Delta\tau}S_{\mathrm{b}}}
, ~~
{w_{\mathrm{f}}}=
{{\mathrm{Tr}}\,}_{\mathrm{f}}\,\Omega
,~~
\Omega
=
\prod_{\tau=1}^L {\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{kin}}^{(\tau)}}
\,.$$ Here ${\widetilde{H}}_{\mathrm{kin}}^{(\tau)}$ corresponds to ${\widetilde{H}}_{\mathrm{kin}}$ with the phonon operators ${{\widehat{p}}}_i$, ${{\widehat{p}}}_j$ replaced by the momenta $p_{i,\tau}$, $p_{j,\tau}$ on the $\tau$th Trotter slice, and its exponential may be written as $$\label{eq:polaron:matrices}
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{kin}}^{(\tau)}}
=
D_\tau \kappa D_\tau^\dag
\,,\quad
\kappa_{jj'}
=
\left({\mathrm{e}}^{{\Delta\tau}t\,h^{\mathrm{tb}}}\right)_{jj'}
\,,\quad
(D_\tau)_{jj'}
=
\delta_{jj'}{\mathrm{e}}^{{\mathrm{i}}\gamma p_{j,\tau}}
\,,$$ where $h^{\mathrm{tb}}$ is the $N^{{{\mathrm{D}}}}\times N^{{{\mathrm{D}}}}$ tight-binding hopping matrix. To save some computer time, we employ the checkerboard breakup [@LoGu92] $$\label{eq:many-electrons:checker}
{\mathrm{e}}^{{\Delta\tau}t\sum_{{\langle}ij{\rangle}} c^\dag_i c^{\phantom{\dag}}_j}
\approx
\prod_{{\langle}ij{\rangle}}
{\mathrm{e}}^{{\Delta\tau}t c^\dag_i c^{\phantom{\dag}}_j}
\,.$$ Using equation (\[eq:many-electrons:checker\]), the numerical effort scales as $N^{2{{\mathrm{D}}}}$ instead of $N^{3{{\mathrm{D}}}}$ (see also section \[sec:qmc\_comparison\]), but the error due to this additional approximation is of the same order ${\Delta\tau}^2$ as the Trotter error in equation (\[eq:polaron:suzuki-trotter\]).
According to equation (\[eq:polaron:matrices\]), we have the same matrix $\kappa$ for every time slice, which is transformed by the diagonal unitary matrices $D_\tau$. The matrix $\Omega$ can be calculated in an efficient way by noting that the transformation matrices $D^\dag_\tau$ and $D_{\tau+1}$ at time slice $\tau$ may be combined to a diagonal matrix $$\label{eq:polaron:trickforM}
(D_{\tau,\tau+1})_{ij}
=
\delta_{ij}
{\mathrm{e}}^{{\mathrm{i}}\gamma(p_{i,\tau+1}-p_{i,\tau})}
\,.$$ Due to the cyclic invariance of the fermionic trace, $D_1$ can be shifted to the end of the product, where it combines with $D^\dag_L$ to $D_{L,1}$. Hence we can write $$\label{eq:polaron:omega2}
\Omega
=
\prod_{\tau=1}^L
\kappa \,D_{\tau,\tau+1}
\,,$$ with periodic boundary conditions in imaginary time. In the one-electron case, the fermionic weight ${w_{\mathrm{f}}}=\sum_n {\lan\right|}\Omega{\left|n{\right{\rangle}}}$ is given by the sum over the diagonal elements of the matrix representation of $\Omega$ in the basis of one-electron states (dropping unnecessary spin indices) $$\label{eq:polaron:oneelbasis}
{\left|n{\right{\rangle}}}
=
c^\dag_n{\left|0{\right{\rangle}}}
\,.$$ The bosonic action in equation (\[eq:polaron:omega\]) contains only classical variables: $$\label{eq:polaron:action}
S_{\mathrm{b}}
=
\frac{{\omega}_0}{2}\sum_{i,\tau}p_{i,\tau}^2 +
\frac{1}{2{\omega}_0{\Delta\tau}^2}\sum_{i,\tau}
\left(p_{i,\tau} - p_{i,\tau+1}\right)^2
\,,$$ where the indices $i=1,\dots,N^{{\mathrm{D}}}$ and $\tau=1,\dots,L$ run over all lattice sites and time slices, respectively, and $p_{i,L+1}=p_{i,1}$. It may also be written as $$\label{eq:polaron:action-w-matrix}
S_{\mathrm{b}}
=
\sum_i {\boldsymbol{p}}_i^{\mathrm{T}} A {\boldsymbol{p}}_i$$ with ${\boldsymbol{p}}_i=(p_{i,1},\dots,p_{i,L})$ and a [*periodic*]{}, tridiagonal $L\times
L$ matrix $A$ with non-zero elements $$\label{eq:polaron:matrixA}
(A)_{l,l}
=
\frac{{\omega}_0}{2}+\frac{1}{{\omega}_0{\Delta\tau}^2}
\,,\quad
(A)_{l,l\pm1}
=
-\frac{1}{2{\omega}_0{\Delta\tau}^2}
\,.$$ Since ${\mathcal{Z}}_L$ is a trace, it follows that $(A)_{1,L}=(A)_{L,1}=-(2{\omega}_0{\Delta\tau}^2)^{-1}$.
### Two electrons {#sec:partition-function-two-electrons}
In contrast to [@HovdL05], here we also take into account nearest-neighbour Coulomb repulsion $V$. For two electrons, the Hamiltonian (\[eq:LF-H-local\]) simplifies to $$\label{eq:LF-H-QMC-bipolaron}
{\widetilde{H}}
=
{\widetilde{H}}_{\mathrm{kin}}
+
{\widetilde{H}}_{\mathrm{ph}}
+
{\widetilde{H}}_{\mathrm{ee}}
-2{E_{\mathrm{P}}}\,,\quad
{\widetilde{H}}_{\mathrm{ee}}=
(U-2{E_{\mathrm{P}}})\sum_i{{\widehat{n}}}_{i{\uparrow}}{{\widehat{n}}}_{i{\downarrow}}
+
V\sum_{{\langle}ij{\rangle}} {{\widehat{n}}}_i{{\widehat{n}}}_j
\,.$$ Again, the constant shift can be neglected in the QMC simulation, but in contrast to the single-electron case, we have a non-trivial interaction term. The Suzuki-Trotter decomposition yields $$\label{eq:bipolaron:suzuki-trotter}
{\mathrm{e}}^{-\beta{\widetilde{H}}}
\approx
\left(
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{kin}}}
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{ph}}^p}
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{ph}}^x}
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{ee}}}
\right)^L \equiv {\mathcal{U}}^L\,.$$ Using the same steps as above we obtain $$\label{eq:bipolaron:omega}
{w_{\mathrm{b}}}=
{\mathrm{e}}^{-{\Delta\tau}S_{\mathrm{b}}}
\,,\quad
{w_{\mathrm{f}}}=
{{\mathrm{Tr}}\,}_{\mathrm{f}}\,\Omega
\,,
\quad
\Omega
=
\prod_{\tau=1}^L
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{kin}}^{(\tau)}}
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{ee}}}
\,,$$ with $S_{\mathrm{b}}$ given by equation (\[eq:polaron:action\]).
As pointed out in [@HovdL05], the numerical effort for two electrons increases substantially in higher dimensions. Therefore, we restrict ourselves to ${{\mathrm{D}}}=1$.
Previously, we only considered the case of two electrons of opposite spin (forming a singlet) [@HovdL05]. Here we shall also present results for the triplet state.
#### Singlet {#singlet .unnumbered}
In the singlet case we choose the two-electron basis states $$\label{eq:bipolaron:basis}
\left\{
{\left|l{\right{\rangle}}}\equiv
{\left|i,j{\right{\rangle}}}
\equiv
c^\dag_{i{\uparrow}}c^\dag_{j{\downarrow}}{\left|0{\right{\rangle}}}
\,,\quad
i,j = 1,\dots,N
\right\}
\,,$$ where we have used a combined index $l=1,\dots,N^2$. The tight-binding hopping matrix, denoted as $\kappa$, has dimension $N^2\times N^2$, and the corresponding exponential in equation (\[eq:bipolaron:omega\]) can again be written as ${\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{kin}}^{(\tau)}
}=D_\tau \kappa D_\tau^\dag$ \[cf equation (\[eq:polaron:omega\])\], where $$(D_\tau)_{ll'}
=
\delta_{ll'}
{\mathrm{e}}^{{\mathrm{i}}\gamma(p_{i,\tau}+p_{j,\tau})}$$ is diagonal in the basis (\[eq:bipolaron:basis\]).
The remaining contribution to $\Omega$ comes from the effective el-el interaction term ${\widetilde{H}}_{\mathrm{ee}}$ in terms of the sparse matrix $$({\mathcal{V}})_{ll'}
=
\sum_k
(\delta_{lk}\;{\mathrm{e}}^{-{\Delta\tau}(U-2{E_{\mathrm{P}}})\delta_{ij}})_{lk}
({\mathrm{e}}^{-{\Delta\tau}V\delta_{{\langle}ij{\rangle}}})_{kl'}
\,.$$ The momenta ${\boldsymbol{p}}$ merely enter the diagonal matrix $D$; the $N^2\times N^2$ matrices ${\mathcal{V}}$ and $\kappa$ are fixed throughout the entire MC simulation. Finally, we have $$\label{eq:bipolaron:matproduct}
\Omega
=
\prod_\tau D_\tau \kappa D^\dag_\tau {\mathcal{V}}\,,$$ and the fermionic trace can be calculated as the sum over the diagonal elements of the matrix $\Omega$ in the basis (\[eq:bipolaron:basis\]), [i.e.]{}, $${{\mathrm{Tr}}\,}_{\mathrm{f}}\,\Omega
=
\sum_{ij} {\lai,j\right|} \Omega {\left|i,j{\right{\rangle}}}
\,.$$
#### Triplet {#triplet .unnumbered}
For two electrons with parallel spin we use the basis states $$\label{eq:bipolaron:basis_triplet}
\left\{
{\left|l{\right{\rangle}}}\equiv
{\left|i,j{\right{\rangle}}}
\equiv
c^\dag_{i}c^\dag_{j}{\left|0{\right{\rangle}}}
\,,\quad
i = 1,\dots,N
\,,\,
j=i+1,\dots,N
\right\}
\,,$$ [i.e.]{}, double occupation of a site is not possible. Since we can further not distinguish between the states ${\left|i,j{\right{\rangle}}}$ and ${\left|j,i{\right{\rangle}}}$, the dimension of the electronic Hilbert space is reduced from $N^2$ (singlet case) to $N(N-1)/2$. Consequently, for the same system size, simulations for the triplet case will be much faster.
### Many-electron case {#sec:partition-function-many-electrons}
The one-electron QMC algorithm can easily be extended to the spinless Holstein model with many electrons. For the latter, assuming $V=0$, the interaction term in equation (\[eq:LF-H-local\]) reduces to ${\widetilde{H}}_{\mathrm{ee}}=-{E_{\mathrm{P}}}\sum_i
{{\widehat{n}}}_i$. Therefore, the grand-canonical Hamiltonian becomes $$\label{eq:many-electrons:Hspinless}
{\widetilde{{\mathcal{H}}}}
=
{\widetilde{H}} - \mu \sum_i {{\widehat{n}}}_i
=
\underbrace{ -t\sum_{{\langle}ij{\rangle}} c^\dag_i c^{{\phantom{\dag}}}_j {\mathrm{e}}^{{\mathrm{i}}\gamma({{\widehat{p}}}_i - {{\widehat{p}}}_j)}
}_{{\widetilde{H}}_{\mathrm{kin}}}
+
{\widetilde{H}}_{\mathrm{ph}}
-
\underbrace{ ({E_{\mathrm{P}}}+ \mu) \sum_{\phantom{{\langle}}i\phantom{{\rangle}}} {{\widehat{n}}}_i
}_{{\widetilde{H}}'_{\mathrm{ee}}}
\,,$$ where $\mu$ denotes the chemical potential. For half filling $n=0.5$ \[$N/2$ spinless fermions on $N$ sites, cf equation (\[eq:many-electrons:n\])\], the latter is given by $\mu=-{E_{\mathrm{P}}}$, whereas for $n\neq 0.5$, it has to be adjusted to yield the carrier density of interest.
The approximation to the partition function may again be cast into the form of equation (\[eq:polaron:Z\]), with ${w_{\mathrm{b}}}$ as defined by equations (\[eq:polaron:omega\]) and (\[eq:polaron:action\]), respectively. The fermionic weight is given by $${w_{\mathrm{f}}}=
{{\mathrm{Tr}}\,}_{\mathrm{f}}({{\widehat{B}}}_1{{\widehat{B}}}_{2}\cdots{{\widehat{B}}}_L)
\,,\quad
{{\widehat{B}}}_\tau
=
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}_{\mathrm{kin}}^{(\tau)}}
{\mathrm{e}}^{-{\Delta\tau}{\widetilde{H}}'_{\mathrm{ee}}}
\,.$$ Following Blankenbecler [[*et al.*]{}]{}[@BlScSu81], the fermion degrees of freedom can be integrated out exactly leading to $$\label{eq:many-electrons:omega}
{w_{\mathrm{f}}}=
\det (1 + B_1\,\cdots\,B_L)
\equiv
\det (1 + \Omega)\,,$$ where the $N^{{\mathrm{D}}}\times N^{{\mathrm{D}}}$ matrix $B_\tau$ is given by $$\label{eq:many-electrons:matprod}
B_\tau
=
D_\tau\,\kappa\,D^\dag_\tau\,{\mathcal{V}}\,.$$ Here $\kappa$ and $D_\tau$ are identical to equation (\[eq:polaron:matrices\]), and $$({\mathcal{V}})_{ij}
=
\delta_{ij}\,{\mathrm{e}}^{{\Delta\tau}({E_{\mathrm{P}}}+\mu)}
\,.$$
There is a close relation to the one-electron Green function $$G_{ij}
=
\underbrace{ {\langle}{\widetilde{c}}^{{\phantom{\dag}}}_i {\widetilde{c}}^\dag_j{\rangle}}_{G^{a}_{ij}}
+
\underbrace{ {\langle}{\widetilde{c}}^{\dag}_i {\widetilde{c}}^{{\phantom{\dag}}}_j{\rangle}}_{G^{b}_{ij}}
\,.$$ In real space and imaginary time, we have [@BlScSu81; @Hi85] $$\label{eq:many-electrons:GA}
G^{a}_{ij}
=
{\langle}{\widetilde{c}}^{{\phantom{\dag}}}_i {\widetilde{c}}^\dag_j{\rangle}=
(1+\Omega)^{-1}_{ij}
\,,\quad
G^{b}_{ij}
=
\delta_{ij} - G^{a}_{ij}
=
(\Omega\,G^{a})_{ji}
\,.$$
At this stage, with the above results for the partition function, a QMC simulation of the transformed Holstein model would proceed as follows. In each MC step, a pair of indices $(i_0,\tau_0)$ on the $N^{{\mathrm{D}}}\times L$ lattice of phonon momenta $p_{i,\tau}$ is chosen at random. At this site, a change $p_{i_0,\tau_0}\mapsto p_{i_0,\tau_0}+\Delta p$ of the phonon configuration is proposed. To decide upon the acceptance of the new configuration using the Metropolis algorithm [@wvl1992], the corresponding weights ${w_{\mathrm{b}}}{w_{\mathrm{f}}}$ and ${w_{\mathrm{b}}}'{w_{\mathrm{f}}}'$ have to be calculated. Due to the local updating process, the computation of the change of the bosonic weight $\Delta{w_{\mathrm{b}}}={w_{\mathrm{b}}}'/{w_{\mathrm{b}}}$ is very fast, which is not the case for the fermionic weight $\Delta{w_{\mathrm{f}}}={w_{\mathrm{f}}}'/{w_{\mathrm{f}}}$. By varying $\tau_0$ sequentially from 1 to $L$ instead of picking random values, the calculation of the ratio of the fermionic weights can be reduced to only two matrix multiplications.
It turns out that a local updating as described above does not permit efficient simulations for small phonon frequencies or low temperatures. Therefore, we shall introduce an alternative global updating in terms of principal components in section \[sec:qmc-pc\].
Observables {#sec:QMC_observables}
-----------
Using the transformed Hamiltonian (\[eq:LF-H-local\]), the expectation value $${\langle}O{\rangle}=
{\mathcal{Z}}^{-1}\,{{\mathrm{Tr}}\,}\, {\widehat{ O}} \,{\mathrm{e}}^{-\beta H}
=
{\mathcal{Z}}^{-1}\,{{\mathrm{Tr}}\,}\, {\widehat{{\widetilde{O}}}}\,{\mathrm{e}}^{-\beta {\widetilde{H}}}$$ of an observables $O$ is computed according to $${\langle}O{\rangle}= {\mathcal{Z}}^{-1}\,{{\mathrm{Tr}}\,}_{\mathrm{f}} \int\,{\mathrm{d}}p\,
{\lap\right|}{\widehat{{\widetilde{O}}}}\,{\mathrm{e}}^{-\beta{\widetilde{H}}}{\left|p{\right{\rangle}}}
\,.$$
As a result of the analytic integration over the phonon coordinates ${{\widehat{x}}}$, interesting observables such as the correlation function ${\langle}{{\widehat{n}}}_i
{{\widehat{x}}}_j{\rangle}$ are difficult to measure accurately. Other quantities such as the quasiparticle weight, and the closely related effective mass [@WeRoFe96], can be determined from the one-electron Green function at long imaginary times [@BrCaAsMu01], but results for one electron or two electrons would not be as accurate as in existing work ([e.g.]{}, [@JeWh98; @RoBrLi99III; @KuTrBo02]).
The situation is strikingly different in the many-electron case, for which many methods fail to produce results of high accuracy for large systems and physically relevant parameters. Moreover, other important observables, such as the one-electron Green function, can be calculated with our approach.
### One electron {#sec:obs-one-electron}
The electronic kinetic energy is defined as $$\label{eq:polaron:Ekin}
{E_{\mathrm{kin}}}=
{\langle}{\widetilde{H}}_{\mathrm{kin}} {\rangle}=
-t {\mathcal{Z}}^{-1}\,
\sum_{{\langle}ij{\rangle}}{\mathrm{Tr}}\;
\big( \,c_i^\dag c^{{\phantom{\dag}}}_j\,
{\mathrm{e}}^{{\mathrm{i}}\gamma({{\widehat{p}}}_i-{{\widehat{p}}}_j)} \,{\mathrm{e}}^{-\beta {\widetilde{H}}}\,
\big)
\,.$$ Repeating the steps used to derive the partition function, and noting that the additional phase factors in equation (\[eq:polaron:Ekin\]) again lead to the same matrix $\Omega$ as in equation (\[eq:polaron:omega2\]), we find $$\begin{aligned}
\nonumber
{E_{\mathrm{kin}}}&=&
-t {\mathcal{Z}}^{-1}_L\,\sum_{{\langle}ij{\rangle}}\int\,{\mathcal{D}}p\,{w_{\mathrm{b}}}\sum_n
{\lan\right|}\Omega c^\dag_i c^{{\phantom{\dag}}}_j{\left|n{\right{\rangle}}}
\\
&=&
-t {\mathcal{Z}}^{-1}_L\,\sum_{{\langle}ij{\rangle}}\int\,{\mathcal{D}}p\,{w_{\mathrm{b}}}{\laj\right|}\Omega{\left|i{\right{\rangle}}}\end{aligned}$$ with the one-electron states (\[eq:polaron:oneelbasis\]). Introducing the matrix elements $(\Omega)_{ij} = {\lai\right|}\Omega{\left|j{\right{\rangle}}}$ and the expectation value with respect to ${w_{\mathrm{b}}}$, $$\label{eq:polaron:Ob}
{\langle}O {\rangle}_{\mathrm{b}}
= \frac{\int\,{\mathcal{D}}p\,{w_{\mathrm{b}}}\; O(p)}{ \int\,{\mathcal{D}}p\,{w_{\mathrm{b}}}}$$ we obtain $$\label{eq:polaron:Ek}
{E_{\mathrm{kin}}}=
-t\; \frac{\sum_{{\langle}ij{\rangle}}\;{\langle}\Omega_{ji}{\rangle}_{\mathrm{b}}}
{
\sum_{i}\;{\langle}\Omega_{ii}{\rangle}_{\mathrm{b}}
}
\,.$$ Here we have anticipated the reweighting discussed in section \[sec:qmc-reweighting\].
The total energy can be obtained from $E=-\partial(\ln{\mathcal{Z}})/\partial\beta$ as $$\begin{aligned}
\label{eq:polaron:E0}\nonumber
E
&=&
{E_{\mathrm{kin}}}+ \frac{{\omega}_0}{2}\sum_i{\left{\langle}}p_i^2{\right{\rangle}}+ E'_{\mathrm{ph}} - {E_{\mathrm{P}}}\,,
\\
E'_{\mathrm{ph}}
&=&
\frac{N^{{\mathrm{D}}}}{2{\Delta\tau}}-\frac{1}{2{\omega}_0{\Delta\tau}^2L}
\sum_{i,\tau}{\left{\langle}}\left(p_{i,\tau}-p_{i,\tau+1}\right)^2{\right{\rangle}}\,.\end{aligned}$$ To compare with other work we subtract the ground-state energy of the phonons, $E_{0,{\mathrm{ph}}}=N^{{\mathrm{D}}}{\omega}_0/2$.
### Two electrons {#sec:obs-two-electrons}
For two electrons, exploiting spin symmetry, we have $${E_{\mathrm{kin}}}=
-t\sum_{{\langle}ij{\rangle}{\sigma}}{\langle}{\widetilde{c}}^\dag_{i{\sigma}}{\widetilde{c}}^{{\phantom{\dag}}}_{j{\sigma}}{\rangle}=
-2t \sum_{{\langle}ij{\rangle}}
{\langle}c^\dag_{i{\uparrow}} c^{{\phantom{\dag}}}_{j{\uparrow}} {\mathrm{e}}^{{\mathrm{i}}\gamma({{\widehat{p}}}_i-{{\widehat{p}}}_j)}
{\rangle}\,.$$ A simple calculation gives $${\langle}{\widetilde{c}}^\dag_{i{\uparrow}}{\widetilde{c}}^{{\phantom{\dag}}}_{j{\uparrow}}{\rangle}=
{\mathcal{Z}}_L^{-1}\int\,{\mathcal{D}}p\,w_{\mathrm{b}}
{\mathrm{e}}^{{\mathrm{i}}\gamma(p_{i,1}-p_{j,1})}
{{\mathrm{Tr}}\,}_{\mathrm{f}}
(
\Omega\, c^\dag_{i{\uparrow}}c^{{\phantom{\dag}}}_{j{\uparrow}}
)
\,.$$ Writing out explicitly the fermionic trace we obtain $$\begin{aligned}
\nonumber
{{\mathrm{Tr}}\,}_{\mathrm{f}}
(\Omega\, c^\dag_{i{\uparrow}}c^{{\phantom{\dag}}}_{j{\uparrow}})
&=&
\sum_{i'j'}
{\lai',j'\right|}
\Omega c^\dag_{i{\uparrow}}c^{{\phantom{\dag}}}_{j{\uparrow}}
{\left|i',j'{\right{\rangle}}}
\\
&=&
\sum_{j'}
{\laj,j'\right|}\Omega{\left|i,j'{\right{\rangle}}}
\,,\end{aligned}$$ and the kinetic energy finally becomes $${E_{\mathrm{kin}}}=
-2t{\mathcal{Z}}_L^{-1}\int\,{\mathcal{D}}p\,w_{\mathrm{b}}
\sum_{{\langle}ij{\rangle}}\sum_{j'}
{\mathrm{e}}^{{\mathrm{i}}\gamma(p_{i,1}-p_{j,1})}
{\laj,j'\right|}\Omega{\left|i,j'{\right{\rangle}}}
\,.$$ In addition to ${E_{\mathrm{kin}}}$, we shall also consider the correlation function $$\label{eq:bipolaron:rho}
\rho(\delta)
=
\sum_i {\langle}{{\widehat{n}}}_{i{\uparrow}} {{\widehat{n}}}_{i+\delta{\downarrow}}{\rangle}\,,
\quad
\delta = 0,1,\dots,N/2-1$$ depending on the distance $\delta$. We find $$\rho(\delta)
=
{\mathcal{Z}}_L^{-1}\int\,{\mathcal{D}}p\,w_{\mathrm{b}} \sum_i
{\lai,i+\delta\right|}\Omega{\left|i,i+\delta{\right{\rangle}}}
\,.$$
### Many-electron case {#sec:obs-many-electrons}
The calculation of observables within the formalism presented here is similar to the standard determinant QMC method [@BlScSu81; @Hi85; @LoGu92]. For an equal-time ([i.e.]{}, static) observable $O$ we have $${\langle}O {\rangle}_{\mathrm{b}}
=
\frac{\int\,{\mathcal{D}}p\,{w_{\mathrm{b}}}{w_{\mathrm{f}}}{{\mathrm{Tr}}\,}_{\mathrm{f}} ({\widehat{O}} {{\widehat{B}}}_1\cdots{{\widehat{B}}}_L)}
{\int\,{\mathcal{D}}p\,{w_{\mathrm{b}}}}
\,.$$ The carrier density $$\label{eq:many-electrons:n}
n
=
\frac{1}{N^{{{\mathrm{D}}}}}\sum_i{\langle}{{\widehat{n}}}_i {\rangle}$$ may be calculated from $G^{b}$ \[equation (\[eq:many-electrons:GA\])\] using ${\langle}{{\widehat{n}}}_i{\rangle}= {\langle}G^{b}_{ii}{\rangle}$.
Similarly, the modulus of the kinetic energy per site is given by $$\label{eq:many-electrons:Ek}
{{\overline{E}}_{\mathrm{kin}}}=
\frac{t}{N^{{{\mathrm{D}}}}} \sum_{{\langle}ij{\rangle}}{\langle}G^{b}_{ji}{\rangle}\,.$$ Equal-time two-particle correlation functions such as $$\rho(\delta)
=
\sum_i
{\langle}{{\widehat{n}}}_i {{\widehat{n}}}_{i+\delta}
{\rangle}$$ may be calculated in the same way as in [@BlScSu81; @Hi85]. For a given phonon configuration, Wick’s Theorem [@Ma90] yields $$\begin{aligned}
\nonumber
{\langle}{{\widehat{n}}}_i {{\widehat{n}}}_j{\rangle}_p
&=&
{\langle}c^\dag_i c^{{\phantom{\dag}}}_i c^\dag_j c^{{\phantom{\dag}}}_j{\rangle}_p
\\\nonumber
&=&
{\langle}c^\dag_i c^{{\phantom{\dag}}}_i{\rangle}_p {\langle}c^\dag_j c^{{\phantom{\dag}}}_j{\rangle}_p
+
{\langle}c^\dag_i c^{{\phantom{\dag}}}_j{\rangle}_p {\langle}c^{{\phantom{\dag}}}_i c^{\dag}_j{\rangle}_p
\\
&=&
G^{b}_{ii} G^{b}_{jj} + G^{b}_{ij} G^{a}_{ij}
\,,\end{aligned}$$ and ${\langle}{{\widehat{n}}}_i {{\widehat{n}}}_j{\rangle}$ is then determined by averaging over all phonon configurations.
The time-dependent one-particle Green function $$\label{eq:many-electrons:Gb_tau}
G^{b}({\boldsymbol{k}},\tau)
=
{\langle}c^{\dag}_{{\boldsymbol{k}}} (\tau)c^{{\phantom{\dag}}}_{{\boldsymbol{k}}} {\rangle}=
{\langle}{\mathrm{e}}^{\tau{\mathcal{H}}} c^{\dag}_{{\boldsymbol{k}}} {\mathrm{e}}^{-\tau{\mathcal{H}}} c^{{\phantom{\dag}}}_{{\boldsymbol{k}}} {\rangle}$$ is related to the momentum– and energy-dependent spectral function $$\label{eq:many-electrons:akwqmc}
A({\vec{k}},{\omega}-\mu)
=
-\frac{1}{\pi}
{{\rm Im}\ }G^b({\vec{k}},{\omega}-\mu)$$ through $$\label{eq:many-electrons:maxent}
G^{b}({\vec{k}},\tau)
=
\int_{-\infty}^\infty\,d {\omega}\,\frac{{\mathrm{e}}^{-\tau({\omega}-\mu)}A({\vec{k}},{\omega}-\mu)}
{1 + {\mathrm{e}}^{-\beta({\omega}-\mu)}}\,.$$ The inversion of the above relation is ill-conditioned and requires the use of the maximum entropy method [@HoNevdLWeLoFe04; @wvl1992; @JaGu96]. Fourier transformation leads to $$G^{b}({\vec{k}},\tau)
=
\frac{1}{N^\mathrm{D}}\sum_{ij} {\mathrm{e}}^{{\mathrm{i}}{\boldsymbol{k}}\cdot({\boldsymbol{r}}_i-{\boldsymbol{r}}_j)} G^b_{ij}(\tau)\,.$$ The allowed imaginary times are $\tau_l=l{\Delta\tau}$, with non-negative integers $0\leq l\leq L$. Within the QMC approach, we have [@BlScSu81; @Hi85] $$\label{eq:many-electrons:Gb_tau_QMC}
G^{b}_{ij}(\tau_l)
=
(
G^{a} B_1\cdots B_l
)_{ji}
\,.$$ The one-electron density of states is given by $$\label{eq:many-electrons:dosgen}
N({\omega}-\mu)
=
-\frac{1}{\pi} {{\rm Im}\ }G({\omega}-\mu)
\,,$$ where $G({\omega}-\mu)=(N^\mathrm{D})^{-1}\sum_{{\vec{k}}} G({\vec{k}},{\omega}-\mu)$. It may be obtained numerically via $$\label{eq:many-electrons:DOS}
N(\tau) = G^{b}_{ii}(\tau)\,,$$ and subsequent analytical continuation.
### Suzuki-Trotter error {#sec:obs-trotter}
The error associated with the approximation made in, [e.g.]{}, equation (\[eq:polaron:suzuki-trotter\]) can be systematically reduced by using smaller values of ${\Delta\tau}$. In practice, there are two strategies to handle this so-called [*Suzuki-Trotter error*]{}. Owing to the usually large numerical effort for QMC simulations, ${\Delta\tau}$ is often simply chosen such that the systematic error is smaller than the statistical errors for observables. A second, more satisfactory, but also more costly method is to run simulations at different values of ${\Delta\tau}$, and to exploit the ${\Delta\tau}^2$ dependence of the results to extrapolate to ${\Delta\tau}=0$.
For the results in section \[sec:results\], we have used a scaling toward ${\Delta\tau}=0$ based on typical values ${\Delta\tau}=0.1$, 0.075 and 0.05 to obtain the results for one and two electrons. In contrast, for the numerically more demanding calculations of dynamic properties in the many-electron case, ${\Delta\tau}=0.1$ has been chosen. This is justified by the uncertainties in the analytical continuation.
Reweighting {#sec:qmc-reweighting}
-----------
As pointed out at the end of section \[sec:partition-function\], the calculation of the change of the fermionic weight ${w_{\mathrm{f}}}$ represents the most time-consuming part of the updating process. Consequently, it would be highly desirable to avoid the evaluation of ${w_{\mathrm{f}}}$. This may be achieved by using only the bosonic weight ${w_{\mathrm{b}}}$ in the updating, and treating ${w_{\mathrm{f}}}$ as part of the observables. For the expectation value of an observable $O$, such a reweighting requires calculation of $$\label{eq:polaron:reweighting}
{\langle}O {\rangle}=
\frac{{\langle}O\, {w_{\mathrm{f}}}{\rangle}_{\mathrm{b}}}{{\langle}{w_{\mathrm{f}}}{\rangle}_{\mathrm{b}}}
\,,$$ where the subscript “b” indicates that the average is computed based on ${w_{\mathrm{b}}}$ only \[cf equation (\[eq:polaron:Ob\])\].
Reweighting of the probability distribution is frequently used in MC simulations if a minus-sign problem occurs [@wvl1992]. Here, the splitting into the configuration weight ${w_{\mathrm{b}}}$ and the observable $O {w_{\mathrm{f}}}$ is practicable provided the variance of both ${w_{\mathrm{f}}}$ and $O {w_{\mathrm{f}}}$ is small, which is the case after the LF transformation. Furthermore, we require a significant overlap of the two distributions, which may be quantified using the Kullback-Leibler number [@HoEvvdL03], in order to avoid prohibitive statistical noise. In fact, our calculations show that, in general, for the [*untransformed*]{} model the reweighting method cannot be applied. For a detailed discussion of this point in the one-electron case see [@HoEvvdL03]. Here we merely note that no problems arise when simulating the transformed model.
Apart from the significant advantage that the fermionic weight ${w_{\mathrm{f}}}$ only has to be calculated when observables are measured, the reweighting method becomes particularly effective in the present case when combined with the principal component representation introduced in section \[sec:qmc-pc\]. In this case, we will be able to perform an exact sampling of the phonons without any autocorrelations. For a reliable error analysis for observables calculated according to equation (\[eq:polaron:reweighting\]) the Jackknife procedure [@DavHin] is applied.
Principal components {#sec:qmc-pc}
--------------------
The reweighting method allows us, in principle, to skip enough sweeps between measurements to reduce autocorrelations to a minimum. However, even though a single phonon update requires negligible computer time compared to the evaluation of ${w_{\mathrm{f}}}$, for critical parameters, an enormous number of such steps will be necessary between successive measurements [@HoEvvdL03]. On top of that, reliable results require knowledge of the longest autocorrelation times, which have to be determined in separate simulations for each set of parameters.
Due to the structure of the bosonic action $S_{\mathrm{b}}$ \[see equation (\[eq:polaron:action\])\], even relatively small (local) changes to the phonon momenta lead to large variations in $S_\mathrm{b}$ and hence the weight ${w_{\mathrm{b}}}$. As a consequence, only minor changes may be proposed in order to reach a reasonable acceptance rate. Unfortunately, this strategy is the very origin of autocorrelations.
The problem can be overcome by a transformation to the normal modes of the phonons (along the imaginary time axis), so that we can sample completely uncorrelated configurations. As the fermion degrees of freedom are treated exactly, the resulting QMC method is then indeed free of any autocorrelations.
To find such a transformation, let us recall the form of the bosonic action, given by equation (\[eq:polaron:action-w-matrix\]), which we write as $$\label{eq:polaron:pc}
S_{\mathrm{b}}
=
\sum_i {\boldsymbol{p}}_i^{\mathrm{T}} A {\boldsymbol{p}}_i
=
\sum_i {\boldsymbol{p}}_i^{\mathrm{T}} A^{1/2} A^{1/2} {\boldsymbol{p}}_i
\equiv
\sum_i \vec{\xi}_i^{\mathrm{T}}\cdot\vec{\xi}_i$$ with the [*principal components*]{} $\vec{\xi}_i=A^{1/2}{\boldsymbol{p}}_i$, in terms of which the bosonic weight takes the simple Gaussian form $$\label{eq:polaron:action_quad}
{w_{\mathrm{b}}}=
{\mathrm{e}}^{-{\Delta\tau}\sum_i \vec{\xi}^{\mathrm{T}}_i\cdot\vec{\xi}_i}
\,.$$ The sampling can now be performed directly in terms of the new variables $\vec{\xi}$. To calculate observables we have to transform back to the physical momenta $\vec{p}$ using $A^{-1/2}$. Comparison with equation (\[eq:polaron:action-w-matrix\]) shows that instead of the ill-conditioned matrix $A$ we now have the ideal case that we can easily generate exact samples of a Gaussian distribution. With the new coordinates ${\boldsymbol{\xi}}$, the probability distribution can be sampled exactly, [e.g.]{}, by the Box-Müller method [@numrec_web]. In contrast to a standard Markov chain MC simulation, every new configuration is accepted and measurements can be made at each step, so that simulation times are significantly reduced.
From the definition of the principal components it is obvious that an update of a single variable $\xi_{i,\tau}$, say, actually corresponds to a change of all $p_{i,\tau'}$, $\tau'=1,\dots,L$. Thus, in terms of the original phonon momenta $\vec{p}$, the updating becomes non-local.
The principal component representation can be used for one, two and many electrons, since the bosonic action \[equation (\[eq:polaron:action\_quad\])\] is identical. This even holds for models including, [e.g.]{}, spin-spin interactions, as long as the phonon operators enter in the same form as in the Holstein model.
An important point is the combination of the principal components with the reweighting method. Using the latter, the changes to the original momenta ${\boldsymbol{p}}$, which are made in the simulation, do not depend in any way on the electronic degrees of freedom. Thus we are actually sampling a set of independent harmonic oscillators, as described by $S_{\mathrm{b}}$. The crucial requirement for the success of this method is the use of the LF transformed model, in which the (bi-)polaron effects are separated from the zero-point motion of the oscillators around their current equilibrium positions.
Finally, as there is no need for a warm-up phase, and owing to the statistical independence of the configurations, the present algorithm is perfectly suited for parallelization.
Minus-sign problem {#sec:sign}
------------------
The motivation for our development of a novel QMC approach to Holstein models was to improve on the performance of existing methods, especially in the many-electron case. As pointed out in [@HoEvvdL05], the LF transformation causes a sign problem even for the pure Holstein model which, in general, may significantly affect the applicability of the method. Therefore, we briefly discuss the resulting limitations, focussing on the many-electron case.
We shall see that there is a fundamental difference between simulations for one or two electrons—the carrier density being zero in the thermodynamic limit—and grand-canonical calculations at finite density $n>0$. Whereas for one or two carriers the sign problem turns out to be rather uncritical—the average sign approaches unity upon increasing system size, in contrast to the usual behaviour [@wvl1992]—restrictions are encountered in simulations of the many-electron case.
![\[fig:many-electrons:sign\_beta\_omega\] Average sign ${{\langle}{\mathrm{sign}}{\rangle}}$ in the many-electron case as a function of el-ph coupling $\lambda$ in ${{\mathrm{D}}}=1$ (a) for different inverse temperatures $\beta$, and (b) for different values of the adiabaticity ratio $\alpha$. Lines are guides to the eye, and errorbars are smaller than the symbols shown. The data presented in figures \[fig:many-electrons:sign\_beta\_omega\] and \[fig:many-electrons:sign\_n\_N\] are for ${\Delta\tau}=0.05$. \[Taken from [@HoNevdLWeLoFe04].\]](sign_beta.eps "fig:"){width="45.00000%"} ![\[fig:many-electrons:sign\_beta\_omega\] Average sign ${{\langle}{\mathrm{sign}}{\rangle}}$ in the many-electron case as a function of el-ph coupling $\lambda$ in ${{\mathrm{D}}}=1$ (a) for different inverse temperatures $\beta$, and (b) for different values of the adiabaticity ratio $\alpha$. Lines are guides to the eye, and errorbars are smaller than the symbols shown. The data presented in figures \[fig:many-electrons:sign\_beta\_omega\] and \[fig:many-electrons:sign\_n\_N\] are for ${\Delta\tau}=0.05$. \[Taken from [@HoNevdLWeLoFe04].\]](sign_omega.eps "fig:"){width="45.00000%"}
Since ${w_{\mathrm{b}}}$ is strictly positive, we define the average sign as
$$\label{eq:polaron:sign}
{{\langle}{\mathrm{sign}}{\rangle}}=
{\langle}{w_{\mathrm{f}}}{\rangle}_{\mathrm{b}} / {\langle}|{w_{\mathrm{f}}}|{\rangle}_{\mathrm{b}}
\,.$$
For simplicity, we first show results for $n=0.5$, while the effect of band filling will be discussed later. The choice $n=0.5$ is convenient since we know the chemical potential, and we shall see below that the sign problem is most pronounced for a half-filled band. Moreover, most existing QMC results for the spinless Holstein model are for half filling (see references in [@HoEvvdL03]).
Figure \[fig:many-electrons:sign\_beta\_omega\](a) shows the dependence of ${{\langle}{\mathrm{sign}}{\rangle}}$ on the el-ph coupling strength. It takes on a minimum near $\lambda=1$ (for $\alpha<1$) that becomes more pronounced with decreasing temperature. At weak coupling (WC) and SC, ${{\langle}{\mathrm{sign}}{\rangle}}\approx1$, so that accurate simulations can be carried out. These results are quite similar to the cases of one or two electrons [@Hohenadler04].
The dependence on phonon frequency \[figure \[fig:many-electrons:sign\_beta\_omega\](b)\] also bears a close resemblance to the polaron problem [@Hohenadler04]. Whereas ${{\langle}{\mathrm{sign}}{\rangle}}$ becomes very small for $\alpha\ll1$, it increases noticeably in the non-adiabatic regime $\alpha>1$, permitting efficient and accurate simulations.
As illustrated in figure \[fig:many-electrons:sign\_n\_N\](a), the average sign depends strongly on the band filling $n$. While it is close to one in the vicinity of $n=0$ or $n=1$ (equivalent to one or two electrons), a significant reduction is visible near half filling $n=0.5$. The minimum occurs at $n=0.5$, and the results display particle-hole symmetry as expected. Here we have chosen $\beta t=8$, $\alpha=0.4$ and $\lambda=1$, for which the sign problem is most noticeable according to figure \[fig:many-electrons:sign\_beta\_omega\].
![\[fig:many-electrons:sign\_n\_N\] Average sign in the many-electron case as a function of (a) band filling $n$, and (b) system size $N$.](sign_n.eps "fig:"){width="45.00000%"} ![\[fig:many-electrons:sign\_n\_N\] Average sign in the many-electron case as a function of (a) band filling $n$, and (b) system size $N$.](sign_size.eps "fig:"){width="45.00000%"}
In figure \[fig:many-electrons:sign\_n\_N\](b), we report the average sign as a function of system size, again for $n=0.5$. The dependence is strikingly different from the one-electron case. While in the latter ${{\langle}{\mathrm{sign}}{\rangle}}\rightarrow1$ as $N\rightarrow\infty$ [@HoEvvdL03; @Hohenadler04], here the average sign decreases nearly exponentially with increasing system size, a behaviour well-known from QMC simulations of Hubbard models [@wvl1992]. Obviously, this limits the applicability of our method. However, we shall see below that we can nevertheless obtain accurate results at low temperatures, small phonon frequencies, and over a large range of the el-ph coupling strength. Moreover, we would like to point out that for such parameters, other methods suffer strongly from autocorrelations, rendering simulations extremely difficult.
The dependence of the sign problem on the dimension of the system is again similar to the single-electron case [@Hohenadler04]. The minimum at intermediate $\lambda$ becomes more pronounced for the same parameters $N$, $\alpha$, $\beta t$ and $\lambda$ as one increases the dimension of the cluster.
To conclude with, we would like to point out that, in principle, the sign problem can be compensated by performing sufficiently long QMC runs, but we have to keep in mind that the statistical errors increase proportional to ${{\langle}{\mathrm{sign}}{\rangle}}^{-2}$ [@wvl1992], setting a practical limit to the accuracy.
Comparison with other approaches {#sec:qmc_comparison}
--------------------------------
The QMC method presented above seems to be most advantageous—as compared to other approaches—in the case of the spinless Holstein model with many electrons. For the latter, other methods are severely restricted by autocorrelations, rendering accurate simulations in the physically important adiabatic, IC regime virtually impossible even at moderately low temperatures. In contrast, the present method enables us to study the single-particle spectrum on rather large clusters and for a wide range of model parameters and band filling (see section \[sec:res-manypol\]). Unfortunately, the generalization to the spinful Hubbard-Holstein model suffers severely from the sign problem.
For the polaron and the bipolaron problem, our method requires more computer time than other QMC algorithms [@dRLa82; @deRaLa86; @Ko98; @Mac04]. However, we are able to consider practically all parameter regimes on reasonably large clusters in one (polaron and bipolaron problem) and two dimensions (polaron problem).
Finally, a discussion of the scaling of computer time with the system parameters can be found in [@HoEvvdL03; @HoEvvdL05; @HovdL05].
Selected results {#sec:results}
================
We now come to a selection of results obtained with the methods discussed so far, most of which have been published before [@HoEvvdL03; @HoEvvdL05; @HovdL05; @HoNevdLWeLoFe04]. Note that errorbars will be suppressed in the figures if smaller than the symbolsize. Moreover, lines connecting data points are guides to the eye only.
Small-polaron cross-over {#sec:res-polaron}
------------------------
The Holstein model with a single electron (for a review see [@FeAlHoWe06]) exhibits a cross-over from a large polaron (${{\mathrm{D}}}=1$) or a quasi-free electron (${{\mathrm{D}}}>1$) to a small polaron with increasing el-ph coupling strength.
### Quantum Monte Carlo {#quantum-monte-carlo}
To investigate the small-polaron cross-over, following previous work [@dRLa82; @dRLa83; @Ko97; @WeRoFe96; @JeWh98; @dMeRa97; @RoBrLi99; @RoBrLi99III; @KuTrBo02], we calculate the electronic kinetic energy ${E_{\mathrm{kin}}}$ given by equation (\[eq:polaron:Ek\]). As we shall compare results for different dimensions, we define the normalized quantity $$\label{eq:Ek}
{{\overline{E}}_{\mathrm{kin}}}=
{E_{\mathrm{kin}}}/(-2t\mathrm{D})$$ with ${{\overline{E}}_{\mathrm{kin}}}=1$ for $T=0$ and $\lambda=0$.
The inverse temperature will be fixed to $\beta t = 10$, low enough to identify the cross-over. Calculations at even lower temperatures can easily be done for $\alpha>1$, but $\alpha<1$ requires very large numbers of measurements to ensure satisfactorily small statistical errors. System sizes were 32 sites in 1D, a $12\times12$ cluster in 2D, and a $6\times6\times6$ lattice in 3D. In contrast to ${{\mathrm{D}}}=1$, 2, where results are well converged with respect to system size, non-negligible finite-size effects (maximal relative changes of up to 20 % between $N=5$ and $N=6$ for $\alpha\ll1$; much smaller changes otherwise) are observed in three dimensions. Moreover, for small $N$, effects due to thermal population of states with non-zero momentum ${\boldsymbol{k}}$—absent in ground-state calculations—are visible, as discussed below. Nevertheless, the main characteristics are well visible already for $N=6$. For a detailed study of finite-size and finite-temperature effects see [@Hohenadler04].
![\[fig:Ek\] Normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ \[equation (\[eq:Ek\])\] of the Holstein model with one electron from QMC as a function of el-ph coupling $\lambda$ for different adiabaticity ratios $\alpha$ and different dimensions D of the lattice ($N$ denotes the linear cluster size). Here and in subsequent figures, QMC data have been extrapolated to ${\Delta\tau}=0$ (see section \[sec:obs-trotter\]). \[Taken from [@HoEvvdL05].\]](polaron_qmc1d.eps "fig:"){width="45.00000%"}\
![\[fig:Ek\] Normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ \[equation (\[eq:Ek\])\] of the Holstein model with one electron from QMC as a function of el-ph coupling $\lambda$ for different adiabaticity ratios $\alpha$ and different dimensions D of the lattice ($N$ denotes the linear cluster size). Here and in subsequent figures, QMC data have been extrapolated to ${\Delta\tau}=0$ (see section \[sec:obs-trotter\]). \[Taken from [@HoEvvdL05].\]](polaron_qmc2d.eps "fig:"){width="45.00000%"}\
![\[fig:Ek\] Normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ \[equation (\[eq:Ek\])\] of the Holstein model with one electron from QMC as a function of el-ph coupling $\lambda$ for different adiabaticity ratios $\alpha$ and different dimensions D of the lattice ($N$ denotes the linear cluster size). Here and in subsequent figures, QMC data have been extrapolated to ${\Delta\tau}=0$ (see section \[sec:obs-trotter\]). \[Taken from [@HoEvvdL05].\]](polaron_qmc3d.eps "fig:"){width="45.00000%"}
Figure \[fig:Ek\] shows ${{\overline{E}}_{\mathrm{kin}}}$ as a function of the el-ph coupling $\lambda$ for different phonon frequencies varying over two orders of magnitude, in one to three dimensions. Generally, the kinetic energy is large at WC, where the ground state consists of a weakly dressed electron (${{\mathrm{D}}}>1$) or a large polaron (${{\mathrm{D}}}=1$). It reduces more or less strongly—depending on $\alpha$—in the SC regime, where a small, heavy polaron exists, defined as an electron surrounded by a lattice distortion essentially localized at the same site. The finite values of ${{\overline{E}}_{\mathrm{kin}}}$ even for large $\lambda$ are a result of undirected motion of the electron inside the surrounding phonon cloud. In contrast, the quasiparticle weight is exponentially reduced in the SC regime (see, [e.g.]{}, [@KuTrBo02]), whereas the effective mass becomes exponentially large.
In all dimensions, the phonon frequency has a crucial influence on the behaviour of the kinetic energy. While in the adiabatic regime $\alpha<1$ the small-polaron cross-over is determined by the condition $\lambda={E_{\mathrm{P}}}/2t{{\mathrm{D}}}>1$, the corresponding criterion for $\alpha>1$ is $g^2={E_{\mathrm{P}}}/{\omega}_0>1$. The former condition reflects the fact that the loss in kinetic energy of the electron has to be outweighed by a gain in potential energy in order to make small-polaron formation favourable. The latter condition expresses the increasing importance of the lattice energy for $\alpha>1$, since the formation of a “localized” state requires a sizable lattice distortion. As a consequence, for large phonon frequencies, the critical coupling shifts to $\lambda_\mathrm{c}>1$, whereas for $\alpha<1$ we have $\lambda_\mathrm{c}= 1$. Additionally, the decrease of ${{\overline{E}}_{\mathrm{kin}}}$ at $\lambda_\mathrm{c}$ becomes significantly sharper with decreasing phonon frequency.
Concerning the effect of dimensionality, figure \[fig:Ek\] reveals that, for fixed $\alpha$, the small-polaron cross-over becomes more abrupt in higher dimensions, with a very sharp decrease in 3D. Nevertheless, there is no real phase transition [@Loe88]. Figure \[fig:Ek\] also contains results for $N=6$ in one and two dimensions, [i.e.]{}, for the same linear cluster size as in 3D (dashed lines). Clearly, for such small clusters, the spacing between the discrete allowed momenta ${\boldsymbol{k}}$ is too large to permit substantial thermal population, so that results are closer to the ground state \[[e.g.]{}, ${{\overline{E}}_{\mathrm{kin}}}(\lambda=0)\approx 1$\], and exhibit a slightly more pronounced decrease near the critical coupling. However, the sharpening of the latter with increasing dimensionality is still well visible.
### Variational approach {#variational-approach}
To test the validity of the variational approach of section \[sec:vpa\] we have calculated the total energy \[equation (\[eq:polaron:evs\])\] and the quasiparticle weight \[equation (\[eq:polaron:z0\])\] on a cluster with $N=4$ for various values of $\alpha$. A comparison with exact diagonalization results [@Mars95] is depicted in figure \[fig:polaron:E0z0vpa\]. We only consider the regime $\alpha\geq1$ where the zero-phonon approximation is expected to be justified. The overall agreement is strikingly good. Minor deviations from the exact results increase with decreasing $\alpha$. For the smallest frequency shown, $\alpha=1$, the result of the HLF approximation is also reported. Clearly, the variational method represents a significant improvement over the HLF approximation, underlining the importance of taking into account non-local distortions. Similar conclusions can be drawn for larger system sizes (see figure 3 in [@HoEvvdL03]).
![\[fig:polaron:E0z0vpa\] Total energy $E$ (a) and quasiparticle weight $z_0$ (b) for $N=4$ as functions of the el-ph coupling $\lambda$ for different values of the adiabaticity ratio $\alpha$. Symbols correspond to variational results and full lines represent exact $T=0$ data obtained with the Lanczos method [@Mars95]. Dashed lines are results of the HLF approximation. \[Taken from [@HoEvvdL03].\]](vpa_E0.eps "fig:"){width="45.00000%"} ![\[fig:polaron:E0z0vpa\] Total energy $E$ (a) and quasiparticle weight $z_0$ (b) for $N=4$ as functions of the el-ph coupling $\lambda$ for different values of the adiabaticity ratio $\alpha$. Symbols correspond to variational results and full lines represent exact $T=0$ data obtained with the Lanczos method [@Mars95]. Dashed lines are results of the HLF approximation. \[Taken from [@HoEvvdL03].\]](vpa_z0.eps "fig:"){width="45.00000%"}
![\[fig:polaron:gammas\] Polaron-size parameter $\gamma_\delta$ for $N=16$ as a function of the el-ph coupling $\lambda$ for various distances $\delta$ in the (a) adiabatic and (b) anti-adiabatic regime. Also shown is the LF parameter $\gamma$ \[equation (\[eq:polaron:gamma\])\]. \[Taken from [@HoEvvdL03].\]](vpa_gamma_omega0.1.eps "fig:"){height="33.00000%"} ![\[fig:polaron:gammas\] Polaron-size parameter $\gamma_\delta$ for $N=16$ as a function of the el-ph coupling $\lambda$ for various distances $\delta$ in the (a) adiabatic and (b) anti-adiabatic regime. Also shown is the LF parameter $\gamma$ \[equation (\[eq:polaron:gamma\])\]. \[Taken from [@HoEvvdL03].\]](vpa_gamma_omega4.0.eps "fig:"){height="33.00000%"}
In figure \[fig:polaron:gammas\] we present results for the variational displacement fields $\gamma_\delta$, which provide a measure for the polaron size. For $\alpha=0.1$ we see an abrupt cross-over from a large to a small polaron at $\lambda\approx 1.2$. For smaller $\lambda$, the electron induces lattice distortions at neighboring sites even at a distance of more than three lattice constants. Above $\lambda\approx1.2$ we have a mobile small polaron extending over a single site only. In contrast, for the anti-adiabatic case $\alpha=4$, the cross-over is much more gradual, and $\gamma_1>0$ even for $\lambda\gg1$. The same behaviour has been found by Marsiglio [@Marsiglio95] who determined the correlation function ${\langle}{{\widehat{n}}}_i {{\widehat{x}}}_{i+\delta}{\rangle}$ by exact diagonalization; within the variational approach ${\langle}{{\widehat{n}}}_i {{\widehat{x}}}_{i+\delta}{\rangle}= \gamma_\delta$. Although in Marsiglio’s results the cross-over to a small polaron for $\alpha=0.1$ occurs at a smaller value of the coupling $\lambda\approx 1$, the simple variational approach reproduces the main characteristics.
Bipolaron formation in the extended Holstein-Hubbard model {#sec:res-bipolaron}
----------------------------------------------------------
In contrast to Cooper pairing of electrons with opposite momentum, two electrons may also form a bound state by travelling sufficiently close in real space. Bipolaron formation may be studied in the framework of the 1D extended Holstein-Hubbard model, and a brief review of previous work has been given in [@HoAivdL04; @HovdL05]. Here we merely note that depending on the choice of parameters, the ground state of the model may either consist of two polarons, a large bipolaron, an inter-site bipolaron or a small bipolaron (in the singlet case). A summary of the conditions on the model parameters is given in table \[tab:bipolaron:bipolaronconditions\]. Whereas existing work is almost exclusively concerned with the singlet case, here we shall also consider two electrons of the same spin. Triplet bipolarons are expected to play a role, [e.g.]{}, in the ferromagnetic state of the manganites [@AlBr99; @AlBr99_2; @David_AiP]. Furthermore, we are not aware of any previous work for $V>0$.
----------------- -- -- ----------------- ---------------------------- ---------------------------- -------------------------
Large bipolaron Small bipolaron Two Inter-site Small
polarons bipolaron bipolaron
$\lambda<0.5$ $\lambda>0.5$ $U>2{E_{\mathrm{P}}}$ (WC) $U<2{E_{\mathrm{P}}}$ (WC)
or and $U\ll2{E_{\mathrm{P}}}$
$g<0.5$ $g>0.5$ $U>4{E_{\mathrm{P}}}$ (SC) $U<4{E_{\mathrm{P}}}$ (SC)
----------------- -- -- ----------------- ---------------------------- ---------------------------- -------------------------
: \[tab:bipolaron:bipolaronconditions\]Conditions for the existence of different singlet bipolaron states in the one-dimensional Holstein-Hubbard model [@HovdL05].
### Quantum Monte Carlo {#sec:res_qmc}
Owing to the increased numerical effort compared to the one-electron case, we shall only present results for $N\leq12$ in one dimension. However, finite-size effects are small even for the most critical parameters [@HovdL05].
We define the effective kinetic energy of the two electrons as $$\label{eq:ekeff}
{{\overline{E}}_{\mathrm{kin}}}=
{E_{\mathrm{kin}}}/(-4t)
\,.$$ In figure \[fig:Ek\_lambda\_omega\](a) we depict ${{\overline{E}}_{\mathrm{kin}}}$ as a function of the el-ph coupling for different values of $\alpha$ and ${U/t}$, at $\beta t=10$, [i.e.]{}, much closer to the ground state than in some previous work [@deRaLa86].
Figure \[fig:Ek\_lambda\_omega\](a) reveals a strong decrease of ${{\overline{E}}_{\mathrm{kin}}}$ near $\lambda=0.5$ for $\alpha=0.4$ and ${U/t}=0$. With increasing $\alpha$, the cross-over becomes less pronounced, and shifts to larger values of $\lambda$. For the same value of $\alpha$, the cross-over to a small bipolaron is sharper than the small-polaron cross-over \[cf figure \[fig:Ek\](a)\]. For finite on-site repulsion ${U/t}=4$, ${{\overline{E}}_{\mathrm{kin}}}$ remains fairly large up to $\lambda\approx1$ (for $\alpha=0.4$), in agreement with the SC result $\lambda_{\mathrm{c}}=1$ for ${U/t}=4$ (see discussion in [@HoAivdL04]). At even stronger coupling, the Hubbard repulsion is overcome, and a small bipolaron is formed. Again, the critical coupling increases with phonon frequency. Finally, the kinetic energy in the triplet case (corresponding to $U/t=\infty$) is comparable to the results for $U/t=4$ up to $\lambda\approx1$, but significantly larger in the SC regime since on-site bipolaron formation is not possible.
The influence of nearest-neighour repulsion $V$ is revealed in figure \[fig:Ek\_lambda\_omega\](b), again for ${U/t}=4$. For all values of $\alpha$ shown, the cross-over sharpens noticeably for $V>0$. The reason is that $V>0$ suppresses the (more mobile) inter-site bipolaron state, leading to a direct cross-over from a large to a small bipolaron.
The nature of the bipolaron state is revealed by the correlation function $\rho(\delta)$ \[equation (\[eq:bipolaron:rho\])\], which gives the probability for the two electrons to be separated by a distance $\delta\geq0$, and provides a measure of the bipolaron size. The phonon frequency determines the degree of retardation of the el-ph interaction, and thereby limits the distance between the two electrons in a bound state. In the sequel, we shall focus on the most interesting case of small phonon frequencies, which has often been avoided in previous work for reasons outlined in section \[sec:qmc\].
![\[fig:Ek\_lambda\_omega\] Normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ \[equation (\[eq:ekeff\])\] from QMC as a function of the el-ph coupling $\lambda$ for different values of the adiabaticity ratio $\alpha$, the on-site repulsion $U$ and the nearest-neighbour repulsion $V$. \[(a) taken from [@HovdL05].\]](Ek_lambda_N12_beta5_omega.eps "fig:"){height="34.00000%"} ![\[fig:Ek\_lambda\_omega\] Normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ \[equation (\[eq:ekeff\])\] from QMC as a function of the el-ph coupling $\lambda$ for different values of the adiabaticity ratio $\alpha$, the on-site repulsion $U$ and the nearest-neighbour repulsion $V$. \[(a) taken from [@HovdL05].\]](Ek_V_omega.eps "fig:"){height="34.00000%"}
Starting with $U\ll{E_{\mathrm{P}}}$, a cross-over from a small to an inter-site bipolaron to two weakly bound polarons takes place upon increasing the Hubbard interaction [@BoKaTr00]. Since the latter competes with the retarded el-ph interaction, the phonon frequency is expected to be an important parameter. In figure \[fig:S0S1\], we show the kinetic energy and the correlation function $\rho(\delta)$ as a function of ${U/t}$ for IC $\lambda=1$. Starting from a small bipolaron for ${U/t}=0$, the kinetic energy increases with increasing Hubbard repulsion, equivalent to a reduction of the effective bipolaron mass [@BoKaTr00; @ElShBoKuTr03]. Although the cross-over is slightly washed out by the finite temperature in our simulations, there is a well-conceivable increase in ${{\overline{E}}_{\mathrm{kin}}}$ up to ${U/t}\approx4$, above which the kinetic energy begins to decrease slowly. The increase of ${{\overline{E}}_{\mathrm{kin}}}$ originates from the breakup of the small bipolaron, as indicated by the decrease of $\rho(0)$ in figure \[fig:S0S1\](b). Close to ${U/t}=4$, the curves for $\rho(0)$ and $\rho(1)$ cross, and it becomes more favourable for the two electrons to reside on neighboring sites. The inter-site bipolaron only exists below a critical Hubbard repulsion $U_{\mathrm{c}}$. The latter is given by $U_{\mathrm{c}}=2{E_{\mathrm{P}}}$ ([i.e.]{}, here $U_{\mathrm{c}}/t=4$) at weak el-ph coupling, and by $U_{\mathrm{c}}=4{E_{\mathrm{P}}}$ at SC. For an intermediate value $\lambda=1$ as in figure \[fig:S0S1\], the cross-over from the inter-site state to two weakly bound polarons is expected to occur somewhere in between, but is difficult to locate exactly from the QMC results.
![\[fig:S0S1\] (a) Normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ and (b) correlation functions $\rho(0)$, $\rho(1)$ from QMC as a function of the Hubbard repulsion ${U/t}$ for different values of the adiabaticity ratio $\alpha$. \[Taken from [@HovdL05].\]](Ek_U_lambda1_omega.eps "fig:"){width="45.00000%"} ![\[fig:S0S1\] (a) Normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ and (b) correlation functions $\rho(0)$, $\rho(1)$ from QMC as a function of the Hubbard repulsion ${U/t}$ for different values of the adiabaticity ratio $\alpha$. \[Taken from [@HovdL05].\]](rho_U_lambda1_omega.eps "fig:"){width="45.00000%"}
Figure \[fig:S0S1\] further illustrates that the cross-over becomes steeper with decreasing phonon frequency. In the adiabatic limit $\alpha=0$, it has been shown to be a first-order phase transition [@PrAu99], whereas for $\alpha>0$ retardation effects suppress any non-analytic behaviour. At the same ${U/t}$, ${{\overline{E}}_{\mathrm{kin}}}$ increases with $\alpha$ since for a fixed $\lambda$, the bipolaron becomes more weakly bound. For the same reason, the cross-over to an inter-site bipolaron—showing up in figure \[fig:S0S1\] as a crossing of $\rho(0)$ and $\rho(1)$—shifts to smaller values of ${U/t}$.
![\[fig:rho\_temp\] Correlation function $\rho(\delta)$ from QMC as a function of $\delta$ for different inverse temperatures $\beta$, $N=12$ and $\alpha=0.4$. \[Taken from [@HovdL05].\]](rho_delta_U0_lambda0.25_temperature.eps "fig:"){width="45.00000%"}\
![\[fig:rho\_temp\] Correlation function $\rho(\delta)$ from QMC as a function of $\delta$ for different inverse temperatures $\beta$, $N=12$ and $\alpha=0.4$. \[Taken from [@HovdL05].\]](rho_delta_U0_lambda1.0_temperature.eps "fig:"){width="45.00000%"}\
![\[fig:rho\_temp\] Correlation function $\rho(\delta)$ from QMC as a function of $\delta$ for different inverse temperatures $\beta$, $N=12$ and $\alpha=0.4$. \[Taken from [@HovdL05].\]](rho_delta_U4_lambda1.0_temperature.eps "fig:"){width="45.00000%"}
Let us now consider the effect of temperature on $\rho(\delta)$. To this end, we plot in figures \[fig:rho\_temp\](a)–(c) $\rho(\delta)$ at different temperatures, for parameters corresponding to the three regimes of a large, small and inter-site bipolaron, respectively.
For the parameters in figure \[fig:rho\_temp\](a) (${U/t}=0$, $\lambda=0.25$), the two electrons are most likely to occupy the same site, but the bipolaron extends over a distance of several lattice constants. Clearly, in this regime, the cluster size $N=12$ used here is not completely satisfactory, but still provides a fairly accurate description as can be deduced from calculations for $N=14$ (not shown). Nevertheless, on such a small cluster, no clear distinction between an extended bipolaron and two weakly bound polarons can be made. As the temperature increases from $\beta t=10$ to $\beta t = 1$, the probability distribution broadens noticeably, [i.e.]{}, it becomes more likely for the two electrons to be further apart. In particular, for the highest temperature shown, $\rho(0)$ has reduced by about 30 % compared to $\beta t=10$.
A different behaviour is observed for the small bipolaron, which exist at stronger el-ph coupling $\lambda=1$. Figure \[fig:rho\_temp\](b) reveals that $\rho(\delta)$ peaks strongly at $\delta=0$, but is very small for $\delta>0$ at low temperatures. Increasing temperature, $\rho(\delta)$ remains virtually unchanged up to $\beta t = 3$. Only at very high temperatures there occurs a noticeable transfer of probability from $\delta=0$ to $\delta>0$. At the highest temperature shown, $\beta t=0.5$, the two electrons have a non-negligible probability for traveling a finite distance $\delta>0$ apart, although most of the probability is still contained in the peak located at $\delta=0$.
Finally, we consider in figure \[fig:rho\_temp\](c) the inter-site bipolaron, taking ${U/t}=4$ and $\lambda=1$ (cf figure 2 in [@HovdL05]). At low temperatures, $\rho(\delta)$ takes on a maximum for $\delta=1$. For smaller values of $\beta t$, the latter diminishes, until at $\beta t=1$, the distribution is completely flat, so that all $\delta$ are equally likely.
The different sensitivity of the bipolaron states to changes in temperature found above can be explained by their different binding energies. The latter is given by $\Delta E_0=E_0^{(2)}-2 E_0^{(1)}$, where $E_0^{(1)}$ and $E_0^{(2)}$ denote the ground-state energy of the model with one and two electrons, respectively.
![\[fig:vpa\_Ek\] Variational results for the normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}$ as a function of the el-ph coupling $\lambda$, and for different adiabaticity ratios $\alpha$. Also shown are results of the HLF approximation. \[Taken from [@HovdL05].\]](Ek_lambda_U0_vpa.eps){width="45.00000%"}
Generally, the thermal dissociation is expected to occur at a temperature such that the thermal energy $k_{\mathrm{B}} T = (\beta T)^{-1}$ becomes comparable to $\Delta E_0$, in accordance with our numerical data. The large and the inter-site bipolaron are relatively weakly bound as a result of the rather small effective interaction $U_{\mathrm{eff}}\approx U-2{E_{\mathrm{P}}}$ [@HoAivdL04]. The binding energies are $\Delta
E_0\approx-(0.32\pm0.08) t$ and $-(0.28\pm0.08)t$, respectively, so that we expect a critical inverse temperature $\beta t\approx 2.5$–5, in agreement with figures \[fig:rho\_temp\](a) and (c). In contrast, the small bipolaron in figure \[fig:rho\_temp\](b) has a significantly larger binding energy $\Delta E\approx-(3.43\pm0.09)t$, and therefore remains stable up to $\beta t\approx 0.3$. Thermal dissociation of bipolarons occurs at even lower temperatures for $V>0$, especially in the triplet case, owing to the reduced binding energy.
### Variational approach {#sec:res_vpa}
Whereas the QMC approach is limited to finite temperatures and relatively small clusters, the variational method of section \[sec:vpa\] yields ground-state results on much larger systems. To scrutinize the quality of the variational method, we compare the ground-state energy for ${U/t}=0$ to the most accurate approach currently available in one dimension, namely the variational diagonalization [@BoKaTr00]. We find a good agreement over the whole range of $\lambda$. As expected from the nature of the approximation, slight deviations occur for $\alpha\lesssim1$, similar to the one-electron case.
Despite the success in calculating the total energy—being the quantity that is optimized—one has to be careful not to overestimate the validity of any variational method. To reveal the shortcomings of the current approach, we show in figure \[fig:vpa\_Ek\] the normalized kinetic energy ${{\overline{E}}_{\mathrm{kin}}}=t_{\mathrm{eff}}$ \[see equations (\[eq:bipolaron:teff\]) and (\[eq:ekeff\])\] as a function of el-ph coupling, and for different $\alpha$. We have chosen $N=25$ to ensure negligible finite-size effects. In principle, figure \[fig:vpa\_Ek\] displays a behaviour similar to the QMC data in figure \[fig:Ek\_lambda\_omega\](a). There is a jump-like decrease of ${{\overline{E}}_{\mathrm{kin}}}$ near $\lambda=0.5$ for $\alpha=0.4$, which becomes washed out and moves to larger $\lambda$ with increasing phonon frequency. For $\alpha=0.4$, the cross-over in the variational results is much too steep, regardless of the fact that the latter are for $T=0$, a common defect of variational methods. Moreover, for $\alpha=0.4$–2, the variational kinetic energy is too small above the bipolaron cross-over compared to the QMC data, whereas for $\alpha=4$, the decay of ${{\overline{E}}_{\mathrm{kin}}}$ with increasing $\lambda$ is too slow.
The reason for the failure is the absence of retardation effects, which play a dominant role in the formation of bipolaron states. The increased importance of the phonon dynamics—not included in the variational method—for the two-electron problem leads to a less good agreement with exact results than in the one-electron case. In particular, our variational results overestimate the position of the cross-over (figure \[fig:vpa\_Ek\]) compared to the value $\lambda_{\mathrm{c}}=0.5$ expected in the adiabatic regime. Nevertheless, the method represents a significant improvement over the simple HLF approximation, due to the variational determination of the parameters $\gamma_{ij}$. This is illustrated in figure \[fig:vpa\_Ek\], where we also show the HLF result ${{\overline{E}}_{\mathrm{kin}}}= {\mathrm{e}}^{-g^2}$ for $\alpha=0.4$ and 4.0. In contrast to the variational approach, the HLF approximation yields an exponentially decreasing kinetic energy for all values of the phonon frequency. Whereas such behaviour actually occurs in the anti-adiabatic limit $\alpha\to\infty$, the situation is different for small $\alpha$ \[see figures \[fig:Ek\_lambda\_omega\](a) and \[fig:vpa\_Ek\]\]. The variational method presented here accounts qualitatively for the influence of the phonon frequency on bipolaron formation.
![\[fig:results:QMC\_lambda0.1\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=0.1$. Here and in subsequent figures ${\Delta\tau}=0.1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_wc_a.eps "fig:"){width="45.00000%"} ![\[fig:results:QMC\_lambda0.1\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=0.1$. Here and in subsequent figures ${\Delta\tau}=0.1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_wc_b.eps "fig:"){width="45.00000%"}\
![\[fig:results:QMC\_lambda0.1\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=0.1$. Here and in subsequent figures ${\Delta\tau}=0.1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_wc_c.eps "fig:"){width="45.00000%"} ![\[fig:results:QMC\_lambda0.1\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=0.1$. Here and in subsequent figures ${\Delta\tau}=0.1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_wc_d.eps "fig:"){width="45.00000%"}
Many-polaron problem {#sec:res-manypol}
--------------------
We review recent results on the carrier-density dependence of photoemission spectra of many-polaron systems in the framework of the spinless Holstein model (\[eq:many-electrons:Hspinless\]) in one dimension. We shall see that the sensitivity to changes in $n$ strongly depends on the phonon frequency and el-ph coupling strength, with the most interesting physics being observed in the adiabatic, IC regime often realized experimentally. This regime is characterized by the existence of large polarons at low carrier density. At larger densities, a substantial overlap of the single-particle wavefunctions occurs, leading to a dissociation of the individual polarons and finally to a restructuring of the whole many-particle ground state. Note that the many-polaron problem has since been studied also by means of other methods [@LoHoFe06; @HoWeAlFe05; @WeBiHoScFe05], confirming the original findings of [@HoNevdLWeLoFe04].
### Weak coupling
For WC $\lambda=0.1$, the sign problem is not severe (section \[sec:sign\]) so that simulations can easily be performed for large lattices with $N=32$, making the dispersion of quasiparticle features well visible.
Figure \[fig:results:QMC\_lambda0.1\] shows the evolution of the one-electron spectral function $A(k,{\omega}-\mu)$ with increasing electron density $n$. At first sight, we see that the spectra bear a close resemblance to the free-electron case, [i.e.]{}, there is a strongly dispersive band running from $-2t$ to $2t$ which can be attributed to weakly dressed electrons. As expected, the height (width) of the peaks increases (decreases) significantly in the vicinity of the Fermi momentum $k_{\mathrm{F}}$, determined by the crossing of the band with the chemical potential. However, in contrast to the case of a rigid tight-binding band, we shall see below (figure \[fig:results:dos\_wc\]) that a significant redistribution of spectral weight occurs with increasing $n$.
![\[fig:results:dos\_wc\] One-electron density of states $N({\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$ and $\lambda=0.1$. \[Taken from [@HoNevdLWeLoFe04].\]](dos_wc.eps){width="60.00000%"}
We would like to point out that the apparent absence of any phonon signatures in figure \[fig:results:QMC\_lambda0.1\] is not a defect of the maximum entropy method, but results from the large scale of the $z$-axis chosen. As a consequence, the peaks running close to the bare band dominate the spectra and suppress any small phonon peaks present. At higher resolution, for all densities $n=0.1$–0.4, we observe the band flattening [@Stephan; @WeFe97; @HoAivdL03] at large wavevectors which originates from the intersection of the approximately free-electron dispersion with the bare phonon energy at ${\omega}-\mu={\omega}_0$.
To complete our discussion of the WC regime, we show in figure \[fig:results:dos\_wc\] the one-electron density of states (DOS) $N({\omega}-\mu)$ given by equation (\[eq:many-electrons:DOS\]). Clearly, for small $n$, there is a peak with large spectral weight at the Fermi level. In contrast, for large $n$, the tendency toward formation of a Peierls– (band–) insulating state at $n=0.5$ suppresses the DOS at the Fermi level, although we are well below the critical value of $\lambda$ at which the cross-over to the insulating state takes place at $T=0$ [@BuMKHa98; @HoWeBiAlFe06]. The additional small features separated from $\mu$ by the bare phonon energy ${\omega}_0$ will be discussed below.
### Strong coupling
![\[fig:results:QMC\_lambda2.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=2$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_sc_a.eps "fig:"){width="45.00000%"} ![\[fig:results:QMC\_lambda2.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=2$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_sc_b.eps "fig:"){width="45.00000%"}\
![\[fig:results:QMC\_lambda2.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=2$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_sc_c.eps "fig:"){width="45.00000%"} ![\[fig:results:QMC\_lambda2.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$, $N=32$, $\beta t=8$, $\alpha=0.4$, and $\lambda=2$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_sc_d.eps "fig:"){width="45.00000%"}
We now turn to the SC limit taking $\lambda=2$. At low density $n=0.1$ \[figure \[fig:results:QMC\_lambda2.0\](a)\], we expect the well-known, almost flat polaron band having exponentially reduced spectral weight (given by $e^{-g^2}$ in the single-electron, SC limit) which, nevertheless, can give rise to coherent transport at $T=0$. As discussed in [@HoNevdLWeLoFe04], such weak signatures are difficult to determine accurately using the maximum entropy method. Generally, it is known that the reliability of dynamic properties obtained by means of the maximum entropy method crucially depends on the size of statistical errors and the general structure of the spectra. A detailed discussion of this point has been given in [@HoNevdLWeLoFe04].
Besides, the spectrum consists of two incoherent features located above and below the chemical potential, which reflect the phonon-mediated transitions to high-energy electron states. Here, the maximum of the photoemission spectra (${\omega}-\mu>0$) follows a tight-binding cosine dispersion. The incoherent part of the spectra is broadened according to the phonon distribution.
For all band fillings, the chemical potential is expected to be located in a narrow polaron band with little spectral weight. There exists a finite gap to the photoemission (inverse photoemission) parts of the spectrum, so that the system typifies as a polaronic metal. We shall see below that a completely different behaviour is observed at IC. Notice that the incoherent inverse photoemission (photoemission) signatures are more pronounced at small (large) wavevectors.
Finally, for $n=0.4$ \[figure \[fig:results:QMC\_lambda2.0\](d)\], the incoherent features lie rather close to the Fermi level, thus being accessible by low-energy excitations. Now, the photoemission spectrum for $k<\pi/2$ is almost symmetric to the inverse photoemission spectrum for $k>\pi/2$ and already reveals the gapped structure which occurs at $n=0.5$ due to charge-density-wave formation accompanied by a Peierls distortion [@HoWeBiAlFe06].
As in the WC case discussed above, the properties of the system also manifest itself in the DOS, shown in figure \[fig:results:dos\_sc\]. Owing to the strong el-ph interaction, the spectral weight at the chemical potential is exponentially small for all fillings $n$. At half filling, the DOS exhibits particle-hole symmetry, and the system can be described as a Peierls insulator, consisting of a polaronic superlattice. In contrast to the WC case, the ground state is characterized as a polaronic insulator rather than as a band insulator.
![\[fig:results:dos\_sc\] One-electron density of states $N({\omega}-\mu)$ from QMC for different band fillings $n$ and cluster sizes $N$, $\beta t=8$, $\alpha=0.4$ and $\lambda=2$. \[Taken from [@HoNevdLWeLoFe04].\]](dos_sc.eps){width="60.00000%"}
### Intermediate coupling
As discussed in the introduction, a cross-over from a polaronic state to a system with weakly dressed electrons can be expected in the IC regime. Here we choose $\lambda=1$, which corresponds to the critical value for the small-polaron cross-over in the one-electron problem \[cf figure \[fig:Ek\](a)\]. Owing to the sign problem, which is particularly noticeable for $\lambda=1$ (see figure \[fig:many-electrons:sign\_beta\_omega\]), we have to decrease the system size as we increase the electron density $n$.
![\[fig:results:QMC\_lambda1.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$ and cluster sizes $N$, $\beta t=8$, $\alpha=0.4$, and $\lambda=1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_ic_a.eps "fig:"){width="45.00000%"} ![\[fig:results:QMC\_lambda1.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$ and cluster sizes $N$, $\beta t=8$, $\alpha=0.4$, and $\lambda=1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_ic_b.eps "fig:"){width="45.00000%"}\
![\[fig:results:QMC\_lambda1.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$ and cluster sizes $N$, $\beta t=8$, $\alpha=0.4$, and $\lambda=1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_ic_c.eps "fig:"){width="45.00000%"} ![\[fig:results:QMC\_lambda1.0\] One-electron spectral function $A(k,{\omega}-\mu)$ from QMC for different band fillings $n$ and cluster sizes $N$, $\beta t=8$, $\alpha=0.4$, and $\lambda=1$. \[Taken from [@HoNevdLWeLoFe04].\]](akw_ic_d.eps "fig:"){width="45.00000%"}
We shall see that the cross-over is rather difficult to detect from the QMC results only. However, the data presented here are perfectly consistent with more recent studies employing other methods such as exact diagonalization [@HoNevdLWeLoFe04], cluster perturbation theory [@HoWeAlFe05] or self-energy calculations [@LoHoFe06].
Figure \[fig:results:QMC\_lambda1.0\] shows the spectral function for $\lambda=1$ and increasing band filling. Owing to the overlap of large polarons in the IC regime, we start with a very low density $n=0.05$ \[figure \[fig:results:QMC\_lambda1.0\](a)\]. Compared to the behaviour for $\lambda=2$ \[figure \[fig:results:QMC\_lambda2.0\](a)\], we notice that the polaron band now lies much closer to the incoherent features, and that there is a mixing of these two parts of the spectrum at small values of $k$. Nevertheless, the almost flat polaron band is well visible for large $k$.
With increasing density, the polaron band merges with the incoherent peaks at higher energies, signaling the above-anticipated density-driven cross-over from a polaronic to a (diffusive) metallic state, with the broad main band crossing the Fermi level.
Further information about the density dependence can be obtained from the one-electron DOS. The latter is presented in figure \[fig:results:dos\_ic\] for different fillings $n=0.05$–0.5. As in figure \[fig:results:QMC\_lambda1.0\], the cluster size is reduced with increasing $n$ in order to cope with the sign problem. To illustrate the rather small influence of finite-size effects, figure \[fig:results:dos\_ic\] also contains results for $N=10$.
For low density $n=0.05$, the DOS in figure \[fig:results:dos\_ic\] lies in between the results for WC and SC discussed above. Although the spectral weight at the chemical potential is strongly reduced compared to $\lambda=0.1$, $N(0)$ is still significantly larger than for $\lambda=2$.
![\[fig:results:dos\_ic\] One-electron density of states $N({\omega}-\mu)$ from QMC for different band fillings $n$, cluster sizes $N$ and inverse temperatures $\beta$. Here $\alpha=0.4$ and $\lambda=1$. \[Taken from [@HoNevdLWeLoFe04].\]](dos_ic.eps){width="60.00000%"}
When the density is increased to $n=0.2$, the DOS at the chemical potential increases, as a result of the dissociation of polarons. Increasing $n$ further, a pseudogap begins to form at $\mu$, which is a precursor of the charge-density-wave gap at half filling and zero temperature.
In the case of half filling $n=0.5$, the DOS has become symmetric with respect to $\mu$. There are broad features located either side of the chemical potential, which take on maxima close to ${\omega}-\mu=\pm{E_{\mathrm{P}}}$. However, apart from the SC case, where the single-polaron binding energy is still a relevant energy scale, the position of these peaks is rather determined by the energy of the upper and lower bands, split by the formation of a Peierls state. The gap of size $\sim\lambda$ expected for the insulating charge-ordered state at $T=0$ is partially filled in due to the finite temperature considered here.
Furthermore, we find additional, much smaller features roughly separated from $\mu$ by the bare phonon frequency ${\omega}_0$, whose height decreases with decreasing temperature, as revealed by the results for $\beta t=10$ (figure \[fig:results:dos\_ic\]). These peaks—not present at $T=0$ [@SyHuBeWeFe04; @HoWeBiAlFe06]—arise from thermally activated transitions to states with additional phonons excited, and are also visible in figures \[fig:results:dos\_wc\] and \[fig:results:dos\_sc\]. While for WC ($\lambda=0.1$, figure \[fig:results:dos\_wc\]), the maximum of these features is almost exactly located at $|{\omega}-\mu|={\omega}_0$, it moves to $|{\omega}-\mu|\approx1.25{\omega}_0$ for IC ($\lambda=1$, figure \[fig:results:dos\_ic\]), and finally to $|{\omega}-\mu|\approx2.5{\omega}_0$ for SC ($\lambda=2$, figure \[fig:results:dos\_sc\]). Although the exact positions of the peaks are subject to uncertainties due to the maximum entropy method, this evolution reflects the shift of the maximum in the phonon distribution function with increasing coupling. The maximum entropy method yields an envelope of the multiple peaks separated by ${\omega}_0$.
### Anti-adiabatic regime
The comparison of the spectral functions for $n=0.1$ and $n=0.3$ in figure 10 of [@HoNevdLWeLoFe04] reveals that there is no density-driven cross-over of the system as observed in the adiabatic case even for the critical value $g^2=1$. In particular, owing to the large phonon energy, there are no low-energy excitations close to the polaron band, so that the latter remains well separated from the incoherent features even for $n=0.3$. Furthermore, the spectral weight of the polaron band also remains almost unchanged as we increase the density from $n=0.1$ to $n=0.3$. Consequently, almost independent small polarons are formed also at finite electron densities, in accordance with previous findings for small systems [@CaGrSt99].
Summary {#sec:summary}
=======
We have reviewed quantum Monte Carlo and variational approaches to Holstein models based on Lang-Firsov transformations of the Hamiltonian. The methods have been applied to investigate single polarons and bipolarons, respectively, as well as a many-polaron system.
The variational methods include displacements of the lattice at all lattice sites, which enables them to quite accurately describe large polaron or bipolaron states.
Using the transformed Hamiltonian, we have shown that quantum Monte Carlo simulation can be based on exact sampling without autocorrelations, which proves to be an enormous advantage for small phonon frequencies or low temperatures. Indeed, we have used a grand-canonical algorithm to obtain dynamical properties of many-polaron systems in all interesting parameter regimes. Such simulations are currently not possible with other Monte Carlo methods.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank A. R. Bishop, H. G. Evertz, H. Fehske, J. Loos, D. Neuber, W. von der Linden, G. Wellein for fruitful discussion.
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---
abstract: 'While many-particle entanglement can be found in natural solids and strongly interacting atomic and molecular gases, generating highly entangled states between weakly interacting particles in a controlled and scalable way presents a significant challenge. We describe here a one-step method to generate entanglement in a dilute gas of cold polar molecules. For molecules in optical traps separated by a few micrometers, we show that maximally entangled states can be created using the strong off-resonant pulses that are routinely used in molecular alignment experiments. We show that the resulting alignment-mediated entanglement can be detected by measuring laser-induced fluorescence with single-site resolution and that signatures of this molecular entanglement also appear in the microwave absorption spectra of the molecular ensemble. We analyze the robustness of these entangled molecular states with respect to intensity fluctuations of the trapping laser and discuss possible applications of the system for quantum information processing.'
address:
- 'Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA'
- 'Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford St., Cambridge, MA 02138, USA'
- 'Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA'
- 'Department of Chemistry, University of California, Berkeley, CA 94703, USA'
author:
- Felipe Herrera
- Sabre Kais
- 'K. Birgitta Whaley'
bibliography:
- 'ame-v2.bib'
title: Entanglement creation in cold molecular gases using strong laser pulses
---
The concept of entanglement has evolved from being regarded as a perplexing and even undesirable consequence of quantum mechanics in the early studies by Schrödinger and Einstein [@EPR:1935], to being now widely considered as a fundamental technological resource that can be harnessed in order to perform tasks that exceed the capabilities of classical systems [@Horodecki:2009review]. Besides its pioneering applications in secure communication protocols and quantum computing , entanglement has also been found to be an important unifying concept in the analysis of magnetism [@Ghosh:2003; @New-Kais1; @New-Kais2; @Amico:2008review], electron correlations [@New-Kais3] and quantum phase transitions [@Osborne:2002; @Osterloh:2002; @Amico:2008review]. Many properties and applications of entanglement have been demonstrated using a variety of physical systems including photons [@Aspect:1981; @Gisin:1998; @Zeilinger:1998; @Zhao:2004; @Peng:2005], trapped neutral atoms [@Mandel:2003; @Bloch:2008; @Urban:2009; @Wilk:2010; @Isenhower:2010], trapped ions [@Turchette:1998; @Haffner:2005; @Blatt:2008; @Jost:2009; @Moehring:2009], and hybrid architectures [@Blinov:2004; @Fasel:2005]. Entanglement has also been shown to persist in macroscopic [@Berkley:2003; @Yamamoto:2003; @Steffen:2006; @Lee:2011] and biological systems [@Engel:2007; @Sarovar:2010]. Despite this significant progress, the theory of quantum entanglement and its technological implications are still far from being completely understood [@Horodecki:2009review].
Trapped neutral atoms are regarded as a promising platform for applications of quantum entanglement due their relatively long coherence times [@Bloch:2008], which can exceed those of solid state and trapped ion architectures by orders of magnitude [@Ladd:2010]. Moreover, the sources of single-particle decoherence are well characterized in electromagnetic traps [@Bloch:2008], and can be compensated using standard state transfer techniques [@Bergmann:1998]. In order to address individual atoms in an optical trap for coherent state manipulation, it is necessary to separate the particles from each other by a distance comparable to optical wavelengths [@Bloch:2005; @Weitenberg:2011]. However, it is difficult to achieve entanglement between ground state atoms at such long distances, due to the short range nature of their mutual interaction. It is nevertheless is possible to enhance interactions between atoms in optical traps by either controlling the interatomic distance [@Jaksch:1999; @Duan:2003; @Hayes:2007], or exciting atoms to an internal state that supports long-range interactions [@Brennen:2000; @Deutsch:2005; @Jaksch:2000; @Lukin:2001; @Saffman:2005]. Using these methods, recent experiments have demonstrated the generation and characterization of entangled atomic states [@Mandel:2003; @Anderlini:2007; @Wilk:2010; @Isenhower:2010], which are the first steps towards the study of many-particle entanglement and the development of quantum technologies using optically trapped particles.
Quantum entanglement can also be studied using trapped polar molecules [@Carr:2009]. Arrays of polar molecules can be prepared in optical lattices with full control over the internal states including the hyperfine structure [@Ospelkaus:2006; @Ni:2008; @Ospelkaus:2010-hyperfine; @Chotia:2012]. Trapped molecules inherit the long coherence times of their atomic counterparts and the long-range dipole-dipole interaction between molecules offers a route for entanglement generation. Since the dipole moment of freely rotating molecules averages to zero, proposals for molecular entanglement creation have involved the application of DC electric fields to spatially orient the dipoles [@Yelin:2009]. One promising approach consists of placing the oriented dipoles in an ordered array using an optical lattice and performing entangling gate operations using microwave pulses, building on analogies with architectures for NMR quantum computation [@DeMille:2002; @Wei:2011; @Zhu:2013]. In order to overcome the complexity involved in controlling the “always-on” interaction between oriented dipoles, conditional transitions between weakly and strongly interacting states have also been proposed as a route to generation of intermolecular entanglement [@Yelin:2006; @Charron:2007; @Kuznetsova:2008]. This approach has recently been demonstrated experimentally for cold atoms [@Wilk:2010; @Isenhower:2010]. Theoretical work has shown that entanglement can also be generated by coupling internal states with collective motional states in strongly interacting molecular arrays [@Rabl:2007; @Ortner:2011], analogously to methods developed for trapped ions [@Soderberg:2010]. In addition to these approaches for the controlled generation of pairwise entanglement between molecules, many-particle entanglement is also expected to emerge in the pseudo-spin dynamics of an ensemble of polar molecules with tunable interactions [@Micheli:2006; @Herrera:2010; @Jesus:2010; @Gorshkov:2011prl; @Baranov:2012].
In contrast with previous approaches for generation of entanglement between dipolar molecules, the scheme proposed here does not involve the use of DC electric fields. Instead, we introduce here a method for deterministic generation of entanglement that uses strong optical laser pulses far-detuned from any vibronic transition. We consider closed-shell polar molecules in their ground rovibrational state, with each molecule individually confined in an optical trap in order to suppress collisional losses. We show that a single off-resonant laser pulse can mediate the entanglement of weakly interacting polar molecules separated by up to several micrometers. The degree of entanglement and the timescale of the entanglement operation are shown to have a well-defined dependence on experimental parameters such as the pulse intensity and duration. The laser parameters considered in this work are consistent with the technology developed to study molecular alignment in thermal gases [@Friedrich:1995; @Sakai:1999; @Stapelfeldt:2003; @Seideman:2005]. We note that entanglement of polar rigid rotors in strong laser fields has been considered before in the high-density regime [@Liao:2004; @Liao:2006], where the dipole-dipole interaction energy is comparable to the rotational constant. The approach presented here allows for the generation of laser-mediated entanglement of rotors in dilute gases for the first time.
The remainder of this paper is organized as follows. Section \[sec:ac fields\] reviews the rotational structure of closed-shell molecules in strong off-resonant optical fields. In Section \[sec:entanglement generation\] we analyze the generation of entanglement between two distant polar molecules due to the action of a single off-resonant laser pulse. The dependence of the degree of entanglement on experimental parameters is discussed in detail. In section \[sec:entanglement quantification\] we discuss two entanglement detection schemes, one based on Bell-type measurements for systems possessing single-molecule addressability and another scheme that employs microwave spectroscopy with only global addressing capability. In section \[sec:decoherence\] we investigate the effects of motional decoherence and show that entanglement in optical traps can be robust against this type of noise. We close with a summary and conclusions in Section \[sec:conclusions\].
Molecules in far-detuned optical fields {#sec:ac fields}
=======================================
We consider closed-shell diatomic molecules in the vibrational and electronic ground state. The state of the molecules in the absence of external fields is represented by ${| N,M_N \rangle}$, which is an eigenstate of the rigid rotor Hamiltonian $\hat H_{\rm R} = B_{\rm e}\hat N^2$ and $\hat N_Z$, where $\hat N$ is the rotational angular momentum operator and $\hat N_Z$ its component along the space-fixed $Z$-axis. $B_{\rm e}$ is the rotational constant. The interaction of a molecule with a monochromatic electromagnetic field $\mathbf{E}(\r,t)=\frac{1}{2}\left[ \hat\epsilon E(t)e^{i\omega t} + c.c. \right]$ whose frequency $\omega$ is far-detuned from any vibronic resonance can be described by the time-independent effective Hamiltonian [@Seideman:2005] $$\hat H_{\rm AC} = -\sum_{p,p'}\hat\alpha_{p,p'}E_{p}(\r)E^*_{p'}(\r),
\label{eq:ac space fixed}$$ where $E_p(\r)$ is the space-fixed $p$-component of the positive-frequency field in the spherical basis and $\hat \alpha_{p,p'}$ is the molecular polarizability operator. For diatomic molecules in a linearly polarized field, transforming the polarizability operator to the rotating body-fixed frame allows Eq. (\[eq:ac space fixed\]) to be rewritten as $$\hat H_{\rm{AC}} = -\frac{|E_0|^2}{4}\left\{\frac{1}{3}(\alpha_{\parallel}+2\alpha_{\perp})+\frac{2}{3}(\alpha_{\parallel}-\alpha_{\perp}) \mathcal{D}^{(2)}_{0,0}(\theta)\right\},
\label{eq:ac diatomic}$$ where $\mathcal{D}^{(2)}_{0,0}=(3\cos^2\theta -1)/2$ is an element of the Wigner rotation matrix [@Zare], $E_0$ is the field amplitude for the selected polarization and $\theta$ is the polar angle of the internuclear axis with respect to this. The polarizabilty tensor for diatomic molecules is parametrized by its parallel $\alpha_\parallel$ and perpendicular $\alpha_\perp$ components, with $\alpha_\parallel>\alpha_\perp$. The first term in Eq. (\[eq:ac diatomic\]) leads to a state-independent shift of the rotational levels and the second term induces coherences between rotational states ${| NM_N \rangle}$, according to the selection rules $\Delta N=0,\pm 2$ and $\Delta M_N = 0$. Therefore the parity of rotational states in the presence of a far-detuned field is conserved.
![Dimensionless rotational energy $E/B_{\rm e}$ of a molecule in the presence of a linearly-polarized CW far-detuned laser, as a function of the light-matter coupling strength $\Omega_{\rm{I}} = |E_0|^2\Delta\alpha/4B_{\rm e}$: (a) Energies of the first six states with $M_N=0$ (blue) and $|M_N|=1$ (red). The states of the lowest doublet ${| g \rangle}={| \tilde 0,0 \rangle}$ and ${| e \rangle}={| \tilde 1,0 \rangle}$ define a two-level subspace. $B_{\rm e}$ is the rotational constant, $\Delta\alpha$ is the polarizability anisotropy, and $|E_0|^2 = I/2\epsilon_0c$, where $I$ is the intensity of the laser. The notation ${| \tilde N,M_N \rangle}$ indicates that the rotational quantum number $N$ is not conserved for $\Omega_{\rm I}\neq 0$. $M_N$ is the projection of the rotational angular momentum along the laser polarization.[]{data-label="fig:ac energies"}](Figure1){width="70.00000%"}
Ignoring the state-independent light shift (which contributes with just an overall phase to the eigenstates) and expressing the energy in units of $B_{\rm e}$, the single-molecule Hamiltonian $\hat H = \hat H_{\rm R}+\hat H_{\rm AC}$ can then be written as $$\hat H = \hat N^2 - \frac{2}{3}\Omega_{\rm I} \mathcal{D}^{(2)}_{0,0}(\theta),
\label{eq:dimless ac}$$ where $\Omega_{\rm I} = {|E_0|^2(\alpha_{\parallel}-\alpha_{\perp})}/{4B_{\rm e}}$ is a dimensionless parameter that characterizes the strength of the light-matter interaction and is proportional to the field intensity $I_0=\frac{1}{2}c\epsilon_0|E_0|^2$. In Fig. \[fig:ac energies\] we plot the lowest eigenvalues of $\hat H$ as a function of $\Omega_{\rm I}$. The figure shows that for intense fields $\Omega_{\rm I}\gg 10$, the energy spectrum consists of closely spaced doublets, as first discussed in Ref. [@Friedrich:1995]. The lowest doublet states ${| g \rangle}$ and ${| e \rangle}$ correlate adiabatically with the states ${| g \rangle}\equiv{| 0,0 \rangle}$ and ${| e \rangle}\equiv{| 1,0 \rangle}$ in the limit $\Omega_{\rm I}\rightarrow0$. Since the eigenstates of Hamiltonian in Eq. ($\ref{eq:dimless ac}$) have well-defined parity, the induced dipole moments ${\langle g |}\mathbf{d}{| g \rangle}$ and ${\langle e |}\mathbf{d}{| e \rangle}$ vanish, but the transition dipole moment ${\langle e |}\mathbf{d}{| g \rangle}$ is finite for polar molecules, where $\mathbf{d}$ is the electric dipole operator.
The light-matter interaction term $\hat H_{\rm AC}$ in Eq. (\[eq:ac diatomic\]) has been widely used to describe the alignment of polar and non-polar molecules in intense off-resonant fields [@Friedrich:1995; @Bonin:1997; @Seideman:2005]. From a classical point of view, the electric field of a strong off-resonant optical field polarizes the molecular charge distribution, inducing an instantaneous dipole moment. The field then exerts a torque on the rotating dipole that changes the angular momentum of the molecule, favouring the alignment of the dipole axis along the field polarization direction. However, the orientation of the dipole is not well-defined in AC electric fields. The degree of alignment for diatomic molecules is typically measured by the expectation value $\mathcal{A} = \langle\cos^2\theta\rangle$ [@Seideman:2005; @Sakai:1999; @Stapelfeldt:2003], with $\theta$ defined in Eq. (\[eq:ac diatomic\]). $\mathcal{A}$ is close to unity for aligned molecules. Adiabatic alignment in the presence of strong off-resonant laser pulses has been extensively studied both experimentally and theoretically [@Stapelfeldt:2003; @Seideman:2005]. In adiabatic alignment experiments the laser pulse turn-on and turn-off times are long compared with the free rotational timescale $t_R\equiv \hbar/B_{\rm e}$. Under adiabatic conditions, the rotational motion of the molecules is described by the eigenstates of Eq. (\[eq:dimless ac\]) with adiabatically varying values of $\Omega_{\rm I}(t)$.
In this work we consider molecules driven by strong off-resonant pulses that are adiabatic with respect to the rotational timescales, but not necessarily adiabatic with respect to longer timescales such as the dipole-dipole interaction time between distant molecules (see below).
Dynamical entanglement generation using strong laser pulses {#sec:entanglement generation}
===========================================================
We now consider the dipole-dipole interaction between polar molecules in the presence of a strong off-resonant laser. The single-molecule Hamiltonian $\hat H = \hat H_{\rm R}+\hat H_{\rm AC}$ is given in Eq. (\[eq:dimless ac\]) with intensity-dependent eigenvalues shown in Fig. \[fig:ac energies\]. Using the two-level single-molecule subspace $\mathcal{S}_1 = \left\{{| g \rangle},{| e \rangle}\right\}$ the dipole-dipole interaction operator can be written as $$\hat V_{\rm dd} = \gamma(1-3\cos^2\Theta) U_{\rm dd}(R)\times\left\{{| g_1e_2 \rangle}{\langle e_1g_2 |}+{| e_1e_2 \rangle}{\langle g_1g_2 |}+{\textrm{ H.c.}}\right\},
\label{eq:exchange coupling}$$ where $\gamma = d^{-2}{\langle e | \hat d_0 | g \rangle}^2$ is a universal dimensionless parameter that depends on the external field strength and polarization, $U_{\rm dd}=d^2/R^3$ is the interaction energy scale, $R$ is the intermolecular distance, $\Theta$ is the polar angle of the intermolecular axis with respect to the laser polarization, $\hat d_0$ is the component of the electric dipole operator along the laser polarization and $d$ is the permanent dipole moment of the molecule. At distances such that $U_{\rm dd}/B_{\rm e}\ll 1$, the interaction operator $\hat V_{\rm dd}$ does not mix the states ${| g \rangle}$ and ${| e \rangle}$ with higher field-dressed rotational states.
The two-molecule Hamiltonian matrix $\mathcal{H} = \hat H_1+\hat H_2+\hat V_{\rm dd}$ in the subspace $\mathcal{S}_2=\left\{{| g_1g_2 \rangle},{| g_1,e_2 \rangle},{| e_1g_2 \rangle},{| e_1,e_2 \rangle}\right\}$ can be written in two equivalent forms (up to a constant energy shift) as $$\begin{aligned}
\mathcal{H} &=& \varepsilon_{\rm e}\left({\hat c^{\dagger}_{1}}{×}{\hat c_{1}}+{\hat c^{\dagger}_{2}}{×}{\hat c_{2}}\right)+J_{12}\left({\hat c^{\dagger}_{1}}+{\hat c_{1}}\right)\left({\hat c^{\dagger}_{2}}+{\hat c_{2}}\right)\nonumber\\
&=&\frac{\varepsilon_{\rm e}}{2}(\sigma_Z^1+\sigma_Z^2)+J_{12}\sigma_X^1\sigma_X^2,
\label{eq:H second-quantized}\end{aligned}$$ where the operator ${\hat c^{\dagger}_{i}}={| e_i \rangle}{\langle g_i |}$ creates a rotational excitation on the $i$-th molecule, with the states ${| g_i \rangle}$ and ${| e_i \rangle}$ equivalently represented by eigenstates of $\sigma_Z^i$ with eigenvalues $-1,+1$, respectively, where $\sigma_\alpha^i$ ($\alpha = X,Y,Z$) is a spin-1/2 Pauli matrix. $J_{12} \equiv {\langle e_1g_2 |}\hat V_{\rm dd}{| g_1e_2 \rangle}={\langle e_1e_2 |}\hat V_{\rm dd}{| g_1g_2 \rangle}$ is the exchange coupling energy, and $\varepsilon_{\rm e}$ is the splitting of the lowest doublet in Fig. \[fig:ac energies\]. The eigenstates of $\mathcal{H}$ involving the single excitation sector are the symmetric and antisymmetric Bell states $ {| \Psi_\pm \rangle}=2^{-1/2}\left\{{| g_1e_2 \rangle}\pm{| e_1g_2 \rangle}\right\}$ with the eigenvalues $E_\pm=\varepsilon_{\rm e}\pm J_{12}$. The ground and highest excited states can be written as $$\begin{array}{lcr}
{| \Phi_{-}(\alpha) \rangle}&=&\cos\alpha\,{| g_1g_2 \rangle}-\sin\alpha\,{| e_1e_2 \rangle}\\
{| \Phi_+(\alpha) \rangle}&=&\sin\alpha\,{| g_1g_2 \rangle}+\cos\alpha\,{| e_1e_2 \rangle}
\end{array},
\label{eq:adiabatic states}$$ with eigenvalues $E_{\pm}=\varepsilon_{\rm e}\pm K$, where $K=\sqrt{\varepsilon_{\rm e}^2+J_{12}^2}$. The states ${| \Phi_{\pm}(\alpha) \rangle}$ are linear combinations of the remaining Bell states ${| \Phi^\pm \rangle} = 2^{-1/2}\left\{{| g_1g_2 \rangle}\pm{| e_1e_2 \rangle}\right\}$. The mixing angle $\alpha$ is defined by $\tan(2\alpha)=J_{12}/\varepsilon_{\rm e}$. The states ${| \Phi_\pm(\alpha) \rangle}$ are separable in the limits $\alpha \rightarrow 0$ and $\alpha\rightarrow \pm\infty$. The ground state of the system is ${| \Phi_-(\alpha) \rangle}$ for all values of $\alpha$.
Since the eigenstates of this two-molecule Hamiltonian are entangled for any finite value of the ratio $J_{12}/\varepsilon_{\rm e}$, we may consider the possibility of tuning the degree of entanglement by manipulating the transition energy $\varepsilon_{\rm e}$ with a strong off-resonant field. This corresponds to varying the effective magnetic field $h = \varepsilon_{\rm e}/2$ for the spin chain Hamiltonian in Eq. (\[eq:H second-quantized\]). The possibility of preparing the states ${| \Phi_\pm(\alpha) \rangle}$ in Eq. (\[eq:adiabatic states\]) using strong continuous-wave (CW) off-resonant laser fields was first pointed out in Ref. [@Lemeshko:2012]. However, since in practice the achievable intensity of CW lasers is limited, we consider here an alternative dynamical preparation of molecular entanglement using pulsed lasers.
Polar molecules can be prepared in the rovibrational ground state ${| g \rangle}$ inside an optical trap [@Carr:2009]. A strong linearly polarized off-resonant field can then be used to bring the energy of the excited state ${| e \rangle}$ close to degeneracy with the ground state ${| g \rangle}$ by adiabatically following the energy level diagram in Fig. \[fig:ac energies\]. In the presence of a laser pulse, both the dipolar coupling $J_{12}(t)$ and the excitation energy $\varepsilon_{\rm e}(t)$ become time-dependent. We take the initial two-molecule wavefunction as ${| \Psi(0) \rangle}={| g_1g_2 \rangle}$. For this initial condition the state evolution is determined by the Hamiltonian sub-block $$\mathcal{H} =\left(\begin{array}{cc}
0 & J_{12}(t) \\
J_{12}(t) &2\varepsilon_{e}(t) \\
\end{array} \right),
\label{eq:two-level matrix}$$ with no participation of the single-excitation manifold since the Hamiltonian in Eq. (\[eq:H second-quantized\]) is block-diagonal. The state of the system is described by a superposition of the form $${| \Phi(t) \rangle}=a(t){| g_1g_2 \rangle}+b(t){| e_1e_2 \rangle}.
\label{eq:final state}$$ Expressing the energy in units of the rotational constant $B_{\rm e}$ and time in units of $t_{\rm R}=\hbar/B_{\rm e}$, we can write the equations of motion $i\dot a(\tau)=J(\tau)b(\tau)$ and $i\dot b(\tau)=J(\tau)a(\tau) + 2E(\tau)b(\tau)$, which we integrate numerically using a standard Runge-Kutta-Fehlberg method [@Pozrikidis-book]. We have defined here the dimensionless energies $J=J_{12}/B_{\rm e}$, $E = \epsilon_e/B_{\rm e}$, and time $\tau =t/t_{\rm R}$. The dipole-dipole interaction timescale $t_{\rm dd}=\hbar/U_{\rm dd}$ depends on the intermolecular distance. The ratio between the rotational and interaction timescales $t_{\rm dd}/ t_{\rm R}$ is larger than unity for distances larger than the characteristic dipole radius (in atomic units) $$R_0=\left(d^2/B_{\rm e}\right)^{1/3}.
\label{eq:R0}$$ We solve the time-dependent Schrödinger equation by evaluating the energies $E(t)$ and $J(t)$ at each time step using an intensity parameter of the form $\Omega_{\rm I}(t) = [f(t)]^2\Omega_0$, for a Gaussian electric field envelope $f(t)=\rme^{-(t/t_0)^2}$. We take $t_0\gg t_{\rm R}$ to ensure adiabaticity with respect to the rotational motion. Under this condition we may extract $E(t)$ from Fig. \[fig:ac energies\]. The exchange energy $J_{12}(t)$ is evaluated using the instantaneous eigenstates ${| g(t) \rangle}$ and ${| e(t) \rangle}$ of the single-molecule Hamiltonian in Eq. (\[eq:dimless ac\]). The parameter $\gamma$ varies in the range $1/3 \leq \gamma\leq 1$ as a function of $\Omega_{\rm I}$, increasing monotonically from its lower bound at $\Omega_{\rm I} =0$ and reaching unity asymptotically as $\Omega_{\rm I}$ increases. The presence of a weak DC electric field in addition to the time-dependent laser field significantly changes this simple behaviour. We discuss the effect of a DC field in detail in \[sec:dc fields\]. In the following we shall consider the evolution of the system in the absence of DC electric fields.
![ Evolution of the two-molecule concurrence $C(\rho)$ under the action of a Gaussian off-resonant laser pulse with intensity profile $\Omega_{\rm I}(t) = f^2(t)\Omega_0$, centered at $t = 0$. The intermolecular distance is $R = 10\,R_0$ and the pulsewidth $\tau_{\rm p}=t_{\rm dd}=10^3 t_{\rm R}$. Curves are labeled according to the value of the peak intensity $\Omega_0$. The dashed line shows the envelope function of the pulse $f(t)$. $t_{\rm dd}$ is the dipole-dipole interaction time and $t_{\rm R}=\hbar/B_{\rm e}$ is the rotational timescale.[]{data-label="fig:evolution"}](Figure2){width="70.00000%"}
Tuning entanglement with a single laser pulse
---------------------------------------------
We consider here pulses that are non-adiabatic with respect to the interaction timescale $t_{\rm dd}=(R/R_0)^3\,t_{\rm R}$. For a laser pulse that is adiabatic with respect to both $t_{\rm R}$ and $t_{\rm dd}$, an initial separable two-particle state would simply acquire a dynamical phase after the pulse is over and no net entanglement would be created in the system.
We define the entanglement radius $R_{\rm e}$ as the intermolecular separation at which the dipole-dipole interaction energy $U_{\rm dd}$ is equal to the energy of the transition ${| g_1g_2 \rangle}\rightarrow{| e_1e_2 \rangle}$, i.e., $$R_{\rm e}= \left(d^2/2\varepsilon_{\rm e}\right)^{1/3}.
\label{eq:entanglement radius}$$ For two molecules within this radius, mixing of the states ${| g_1g_2 \rangle}$ and ${| e_1e_2 \rangle}$ is energetically allowed in the presence of a strong laser pulse. In the absence of DC electric fields, on account of the exponentially decreasing splitting of the doublet states as a function of the intensity parameter $\Omega_{\rm I}$, the entanglement radius $R_{\rm e}$ increases exponentially with $\Omega_{\rm I}$. For concreteness, the value $\Omega_{\rm I}= 300$ corresponds to $R_{\rm e}\approx 3000 \,R_0$, which corresponds to distances of several micrometers between molecules (see \[sec:dc fields\]).
![Asymptotic two-molecule concurrence $C(\rho)$ as a function of the intermolecular distance $R$ (in units of $R_0$), long after the action of a Gaussian off-resonant laser pulse. For the distance $R=100\; R_0$, we choose the pulsewidth $\tau_{\rm p}=10^6\;t_{\rm R}$ (FWHM) and peak intensity $\Omega_0 = 270$, to obtain a maximally entangled state with $C(\rho)=1$. $t_{\rm dd}/t_{\rm R} = (R/R_0)^3$ is the dipole-dipole interaction time in units of the rotational timescale $t_{\rm R}=\hbar/B_{\rm e}$.[]{data-label="fig:distance dependence"}](Figure3){width="70.00000%"}
Let us consider a pair of polar molecules separated by a distance $R_0\ll R < R_e$, where both molecules are initially in their rotational ground states, i.e., ${| \Psi(0) \rangle}={| g_1g_2 \rangle}$. The evolution of this system in the presence of a single Gaussian laser pulse is given by Eq. (\[eq:final state\]) and depends on three independent parameters: the intermolecular distance $R$, the pulse peak intensity $\Omega_0$, and the pulsewidth $\tau_{\rm p}$ (FWHM). We use the binary concurrence $C(\rho) = 2|ab|$ to quantify the degree of entanglement of the time evolved state ${| \Phi(t) \rangle}$. The concurrence, which completely determines the degree of entanglement of pure binary states [@Horodecki:2009review; @Amico:2008review], vanishes for separable states and is unity for maximally entangled states. Fig. \[fig:evolution\] shows the evolution of concurrence for a pair of molecules separated by $R = 10\,R_0$ under the action of a strong off-resonant Gaussian pulse. The pulsewidth $\tau_{\rm p}$ is chosen equal to the dipole-dipole interaction time $t_{\rm dd}$, while the peak intensity $\Omega_0$ is varied. Figure \[fig:evolution\] shows that molecular entanglement is created in the presence of the laser pulse and reaches an asymptotic constant value when the pulse is over. We find that the qualitative behaviour of the system evolution is independent of $R$, $\Omega_0$ and $\tau_{\rm p}$, but that the actual value of the asymptotic concurrence depends strongly on the choice of these parameters.
Fig. \[fig:distance dependence\] shows how the asymptotic concurrence $C(\rho)$ depends on the intermolecular distance $R$, or equivalently on the interaction time $t_{\rm dd}$, for fixed pulse parameters $\tau_{\rm p}=10^6\,t_{\rm R}$ and $\Omega_0=270$. We have chosen the pulse parameters here to ensure that two molecules separated by $R=100\,R_0$ ($R<R_{\rm e}$) become maximally entangled ($C(\rho)=1$). For smaller distances $R\leq 100\,R_0$, the asymptotic concurrence has an oscillatory dependence on $R$. For such distances the pulsewidth $\tau_{\rm p}$ is longer than the corresponding interaction time $t_{\rm dd}$. The system undergoes Rabi-type oscillations between the states ${| g_1g_2 \rangle}$ and ${| e_1e_2 \rangle}$ while the pulse is on. The oscillation stops when the pulse is over, giving the asymptotic concurrence shown in Fig. \[fig:distance dependence\]. For larger distances $R>100 \,R_0$, the concurrence decays monotonically with $R$, and eventually for $R\gg R_{\rm e}$ there is no entanglement. In this case the pulsewidth $\tau_{\rm p}$ is smaller than $t_{\rm dd}$, and the state population does not have time to undergo a Rabi cycle. Our calculations show that the behaviour of the asymptotic concurrence in Fig. \[fig:distance dependence\] is independent of the choice of pulse parameters $\Omega_0$ and $\tau_{\rm p}$. The fast decay of the entanglement with distance is particularly useful for an array of molecules. By choosing the laser pulse parameters $\Omega_0$ and $\tau_{\rm p}$ appropriately, it is possible to prepare highly entangled states between nearest neighbours only.
![Asymptotic concurrence $C(\rho)$ as a function of the peak intensity parameter $\Omega_{\rm I}$, long after the action of a Gaussian off-resonant laser pulse. The intermolecular distance is $R = 100 \,R_0$. Data is shown for different pulsewidths (FWHM): $\tau_{\rm p} = t_{\rm dd}$ (circles), $\tau_{\rm p} = 3t_{\rm dd}/4$ (diamonds), $\tau_{\rm p} = t_{\rm dd}/2$ (triangles), and $\tau_{\rm p}=t_{\rm dd}/4$ (squares). $t_{\rm dd} = 10^6 t_{\rm R}$ is the dipole-dipole interaction time and $t_{\rm R}=\hbar/B_{\rm e}$ is the rotational timescale.[]{data-label="fig:intensity dependence"}](Figure4){width="70.00000%"}
The dependence of the asymptotic concurrence $C(\rho)$ on the laser pulse peak intensity $\Omega_0$ is shown in Fig. \[fig:intensity dependence\]. Data are shown for a fixed distance $R =100\,R_0$ and for different values of the pulsewidth $\tau_{\rm p}$. For all values of $\tau_{\rm p}$, the concurrence is negligibly small below an intensity threshold, here $\Omega_0\approx 70$, whose value depends on the intermolecular distance $R$. Independently of the pulsewidth, the asymptotic concurrence increases with the intensity above this threshold until it reaches the maximum value ($C(\rho) = 1$). For a given distance $R$, the maximum concurrence is achieved at smaller peak intensities $\Omega_0$ when the pulsewidth is equal to the dipole-dipole interaction time $t_{\rm dd}$. After reaching the maximum value, the concurrence decreases with intensity as the population of the doubly excited state ${| e_1e_2 \rangle}$ exceeds $|b(t)|^2=1/2$ in Eq. (\[eq:final state\]). In the strong field limit $\Omega_0\rightarrow\infty$, when $R$ and $\tau_{\rm p}=t_{\rm dd}$ are held constant, the population is completely transferred from ${| g_1g_2 \rangle}$ to ${| e_1e_2 \rangle}$, with no net entanglement creation.
The presence of an intensity threshold for the creation of molecular entanglement in Fig. \[fig:intensity dependence\] can be related to the notion of entanglement radius $R_{\rm e}$ described earlier. For molecules within this radius, the mixing of the ground state ${| g_1g_2 \rangle}$ with the two-excitation state ${| e_1e_2 \rangle}$ is energetically favourable since the energy ratio $J_{12}/2\varepsilon_{\rm e}=\gamma(1-3\cos^2\Theta) (R_{\rm e}/R)^3$ exceeds unity. When this energy ratio is less than unity, the state mixing is suppressed and the concurrence becomes negligible. For a given distance $R$ and pulsewidth $\tau_{\rm p}$, the intensity threshold thus occurs at values of $\Omega_0$ for which $R_{\rm e}/R\sim 1$. In Fig. \[fig:intensity dependence\], $R_{\rm e}\approx 100\,R_0$ for $\Omega_0 = 130$.
Example: alkali-metal dimers in optical lattices
------------------------------------------------
------ -- ------- ------------------------ ------------- --------------------- ------- -------------
$d$ $\Delta\alpha_{\rm V}$ $B_{\rm e}$ $I_0$ $R_0$ $t_{\rm R}$
(D) ($a_0^3$) (cm$^{-1}$) ($10^{8}$ W/cm$^2$) (nm) (ps)
RbCs 1.238 441 0.0290 0.4 6.4 1.15
KRb 0.615 360 0.0386 0.7 3.7 0.86
LiCs 5.529 327 0.1940 3.8 9.3 0.17
LiRb 4.168 280 0.2220 5.0 7.3 0.15
------ -- ------- ------------------------ ------------- --------------------- ------- -------------
: Molecular parameters for selected polar alkali-metal dimers: $I_0$ is the laser intensity corresponding to $\Omega_{\rm I} \equiv \left({4\pi}/{c}\right){I_0\Delta\alpha_{\rm V}}/{2B_{\rm e}}=1$. $R_0=(d^2/B_{\rm e})^{1/3}$ is the characteristic length of the dipole-dipole interaction and $t_{\rm R}=\hbar/B_{\rm e}$ is the timescale of the rotational motion. Values of the polarizability anisotropy $\Delta\alpha_{\rm V}$, dipole moment $d$ and rotational constant $B_{\rm e}$ are taken from Ref. [@deiglmayr:2008-alignment].
\[tab:intensities\]
Table \[tab:intensities\] lists the laser intensity $I_0$ of a traveling wave corresponding to a light-matter interaction parameter $\Omega_{\rm I}=1$ for selected polar alkali-metal dimers that have been optically trapped at ultracold temperatures [@Carr:2009; @Chotia:2012]. Predicted values for the polarizability anisotropy $\Delta\alpha_{\rm V}$ and rotational constants for the rovibrational ground state are taken from Ref. [@deiglmayr:2008-alignment]. For alkali-metal dimers, $I_0$ is on the order of $10^7-10^8$ W/cm$^2$. This is well within the realm of feasibility, since continuous-wave laser beams with frequencies in the mid-infrared region ($\lambda\sim 1\,\mu$m) can have intensities on the order of $10^8$ W/cm$^2$ when focused to micrometer size regions [@Sugiyama:2007; @Rungsimanon:2010], while intensities higher than $10^{10}$ W/cm$^2$ can be achieved using pulsed lasers. Strong laser pulses are routinely used in molecular alignment experiments, with pulse durations varying from less than a femtosecond to hundreds of nanoseconds [@Sakai:1999; @Seideman:2005].
We now consider the interaction of pairs of polar molecules with a strong off-resonant pulse when the molecules are trapped in individual sites of an optical lattice. Typical experimental lattice site separations are in the range $a_L=400 - 1000$ nm [@Bloch:2005; @Danzl:2009]. For most alkali-metal dimers in Table \[tab:intensities\], these distances correspond to $R \sim 10^2 \,R_0$. The results in Figs. \[fig:distance dependence\] and \[fig:intensity dependence\] therefore show that highly-entangled states of molecules in different lattice sites can be prepared using a single laser pulse. For example, two LiRb molecules separated by $a_L=730$ nm can be prepared in a maximally entangled state by using a single Gaussian pulse with peak intensity $I = 1.35\times 10^{11}$ W/cm$^2$ and pulsewidth $\tau_{\rm p}=t_{\rm dd} = 150$ ns. These laser parameters can be achieved using current technology [@Sakai:1999]. It is therefore possible to generate highly entangled states in currently available optical lattice realizations by choosing the appropriate combination of parameters $\Omega_0$ and $\tau_{\rm p}$, regardless of the molecular species.
Detection of molecular entanglement in optical traps {#sec:entanglement quantification}
====================================================
In this section we discuss how the alignment-mediated entanglement created between polar molecules in different sites of an optically trapped molecular array may be observed experimentally. We first show that the pairwise entanglement created in an ensemble of molecules as described in Sec. \[sec:entanglement generation\] gives rise to coherent oscillations in the microwave absorption line shape. Thus the global entanglement of the ensemble may already be detected by measurement of the linear spectral response as a function of frequency. We then outline how the time dependence of an initially entangled state generated by a strong laser pulse that subsequently evolves under the free rotational Hamiltonian may be tracked using correlations between local orientation measurements and a Bell inequality analysis [@Milman:2007; @Milman:2009]. For pairwise entanglement of a pure state, this allows a direct measurement of the concurrence measure of entanglement for the initially entangled state. This second entanglement detection scheme requires either single site addressing resolution in an optical lattice or individual trapping in separate dipole traps. Such addressability is now possible for trapped atoms [@Wilk:2010; @Isenhower:2010; @Weitenberg:2011] and is a subject of much experimental effort for trapped molecules. In contrast, the first approach is more amenable to current technology because it requires only global and not individual addressing.
To show how these two detection schemes work, we shall consider explicitly an ensemble of molecules trapped in individual sites of a double-well optical lattice. Such lattices can be prepared by superimposing standing waves with different periodicity [@Sebby:2006; @Anderlini:2006; @Sebby:2007; @Lee:2007; @Folling:2007]. When the distance between two neighbouring double wells is a few times longer than the separation between the double-well minima, the alignment-mediated entanglement operation described in Sec. \[sec:entanglement generation\] can be designed such that only molecules within a single double-well become entangled. Separability between neighboring pairs is ensured by increasing the distance between adjacent double wells. We consider identical independent molecular pairs here for simplicity. In practice, inhomogeneities in the entanglement preparation step would lead to a distribution of concurrence values throughout the array. In the remainder of this section we discuss the detection of entangled pairs initially prepared at time $t = 0$ by a strong laser pulse in the pure state ${| \Phi_0 \rangle} = a_0{| g_1g_2 \rangle}+b_0{| e_1e_2 \rangle}$ and show how we may measure the value of the initial concurrence, $C(\rho_0)=2|a_0b_0|$. For times $t>0$, each molecule of the pair evolves under the free rotational Hamiltonian $\hat H_R$ (Section \[sec:ac fields\]). The state component ${| e_1 e_2 \rangle}$ therefore acquires a relative dynamical phase which may modify time-dependent observables but does not change the concurrence. Our analysis will show that we can effectively extract the initial state concurrence $C(\rho_0)$ from both the linear absorption spectrum and orientational Bell inequality measurements.
Global entanglement measure in optical lattices {#sec:global}
-----------------------------------------------
It is well known that the macroscopic response of an ensemble of particles to an external field is affected by the presence of entanglement in the system [@Amico:2008review]. In particular, thermodynamic properties such the heat capacity and magnetic susceptibility have been established as entanglement witnesses for spin chains [@Amico:2008review; @Vedral:2008]. In this section we will identify the signatures of entanglement on the AC dielectric susceptibility of a gas sample of $\mathcal{N}$ identical molecules. For simplicity we consider an ensemble of identical entangled pairs but the results can readily be generalized to many-particle entangled states.
In the absence of DC or near resonant AC electric fields, an ensemble of rotating polar molecules is unpolarized. An applied electric field $\mathbf{E}(t)$ creates a polarization $\mathbf{P}(t)$. To lowest order in the field, this polarization is given by $$\frac{\mathbf{P}(t)}{\mathcal{N}} = \frac{i}{\hbar}\int_{-\infty}^t dt'\left\{ \langle \mathbf{d}(t')\mathbf{d}(t)\rangle_0 - \langle \mathbf{d}(t)\mathbf{d}(t')\rangle_0\right\}\cdot\mathbf{E}(t'),
\label{eq:Kubo}$$ where $\langle {\cdots} \rangle_0$ denotes an expectation value with respect to the state of the ensemble in the absence of the external field. Typically the system is in a thermal state $\hat \rho = \mathcal{Z}^{-1}(\beta)\rme^{-\beta \hat H_0}$, where $\hat H_0$ is the field-free Hamiltonian, $\mathcal{Z}(\beta)={\textrm{ Tr}}\{\rme^{-\beta \hat H_0}\}$ is the partition function and $\beta^{-1} = k_BT$. For equilibrium states the autocorrelation function $\langle \hat A(t)\hat B(t')\rangle_0$ depends only on the time difference $\tau = t-t'$. As noted above, for analysis of the entanglement after the strong laser pulse is switched off, the Hamiltonian $\hat H_0$ is given by the two-molecule Hamiltonian $\mathcal{H}$ in Eq. (\[eq:H second-quantized\]) with $\Omega_I = 0$.
Given the polarization, Eq. (\[eq:Kubo\]), the microwave susceptibility for a thermal ensemble can be written as [@Mukamel-book] $$\chi(\omega) = -\mathcal{N}P_0(\beta)\left(\frac{d^2}{3\hbar}\right)\frac{1}{\omega-\omega_{eg}+i\gamma_e},
\label{eq:chi MW thermal}$$ where $P_0(\beta)\leq 1$ is the thermal population of the rotational ground state ${| 0,0 \rangle}$, and $\gamma_e$ is decay rate of the rotational excited state ${| 1,0 \rangle}$. The absorption spectrum is given by $$A(\omega) = \mathcal{N} \frac{ P_0(\beta)(d^2/3) \Gamma_e}{\left[(\hbar\omega - 2B_{\rm e})^2+\Gamma_e^2\right]},
\label{eq:absorption_thermal}$$ where $A(\omega)\equiv {\rm Im}\{\chi(\omega)\}$ and $\Gamma_e = \hbar\gamma_e$ is the transition linewidth. Let us now consider the microwave susceptibility for an ensemble of entangled pairs initially prepared in the pure state ${| \Phi_0 \rangle} = a_0{| g_1g_2 \rangle}+b_0{| e_1e_2 \rangle}$. Unlike the thermal case, the corresponding density matrix $\rho_0 = {| \Phi_0 \rangle}{\langle \Phi_0 |}$ describes a non-stationary state, with coherences that evolve according to $\hat H_0$ (in the absence of external perturbations). In this case the response of the system to the field $\mathbf{E}(t)$ is given by Eq. (\[eq:Kubo\]) as for the thermal case, but the autocorrelation function ${\langle \Phi_0 |}\mathbf{d}(t)\mathbf{d}(t'){| \Phi_0 \rangle}$ now depends on the absolute values of the time arguments $t$ and $t'$, where these are defined with respect to a common initial time.
The eigenstates of the coupled pairs in the limit $J_{12}/2\varepsilon_{\rm e}\ll 1 $ are ${| \Phi_1 \rangle} = {| g_1g_2 \rangle}$ with energy $E_1=0$, ${| \Psi_A \rangle} = 2^{-1/2}\left[{| g_1e_2 \rangle} - {| e_1g_2 \rangle}\right]$ with energy $E_A = \varepsilon_{e}- J_{12}$, ${| \Psi_S \rangle} = 2^{-1/2}\left[{| g_1e_2 \rangle} + {| e_1g_2 \rangle}\right]$ with energy $E_S=\varepsilon_{e}+ J_{12}$, and ${| \Phi_4 \rangle} = {| e_1e_2 \rangle}$ with energy $E_4=2\varepsilon_{\rm e}$ (see Eq. (\[eq:adiabatic states\])). The energetic ordering of the states ${| \Psi_A \rangle}$ and ${| \Psi_B \rangle}$ depends on the sign of $J_{12}$. Using the non-stationary state $\Phi_0$ in the Kubo formula of Eq. (\[eq:Kubo\]), the microwave absorption spectra at frequencies $\omega\approx \omega_{S1} \equiv(E_S-E_1)/\hbar$ can be written as $$\begin{aligned}
A(\omega)&=&\mathcal{N}_{\rm P}\left(\frac{2d^2}{3\hbar}\right)\left[|a_0|^2\frac{\gamma_S}{(\omega_{S1}-\omega)^2+\gamma_S^2} \right.\nonumber\\
&&\left. +|a_0b_0|\frac{\mathcal{F}_\omega(t)}{(\omega_{S1}-\omega)^2+\gamma_S^2}\right] ,
\label{eq:absorption dimer}\end{aligned}$$ where $\mathcal{N}_{\rm P} = \mathcal{N}/2$ is the number of pairs, $\gamma_S$ is the decay rate of the state $\Psi_S$. In the derivation of Eq. (\[eq:absorption dimer\])we have used the transition dipole moments ${\langle \Psi_S |}\mathbf{d}{| \Phi_1 \rangle} = \sqrt{2}{\langle e |}\mathbf{d}{| g \rangle}={\langle \Phi_4 |}\mathbf{d}{| \Psi_S \rangle}$, and ${\langle \Psi_A |}\mathbf{d}{| \Phi_1 \rangle}=0={\langle \Phi_4 |}\mathbf{d}{| \Psi_A \rangle}$. The function $\mathcal{F}_\omega(t)$ contains the time dependence from the evolution of the entangled state under $\hat H_0$ and can be written as $$\mathcal{F}_\omega(t) = \rme^{-\gamma_{41}t}\left[(\omega_{S1}-\omega)\sin\phi_{41}(t)+\gamma_S \cos\phi_{41}(t)\right],
\label{eq:dynamical lineshape}$$ where $\phi_{41}(t) = \omega_{41}t-\theta_{ba}$ is the free phase evolution of the two-molecule coherence, $\theta_{ba}$ is the relative phase of the two components of the initial state, defined by $a_0^*b_0 = |a_0b_0|\rme^{i\theta_{ba}}$, and $\gamma_{41}$ is a decoherence rate introduced to account for dephasing channels. The amplitude of the time-dependent lineshape depends on the magnitude of the two-molecule coherence $|a_0b_0|=C(\rho_0)/2$. For a maximally entangled two-molecule state ${| \Phi_0 \rangle}$ with relative phase $\theta_{ba} = 0$, the peak absorption (per molecule) at the resonance frequency $\omega = \omega_{S1}$ is $$\frac{A(\omega_{S1})\Gamma_S}{\mathcal{N}} = \frac{d^2}{6}\left[1+\cos(2\omega_{eg}t)\right].$$ The presence of dynamical peaks in the absorption or emission spectra is a general feature of wavepacket evolution that has been widely studied for single atoms and molecules [@Mukamel-book]. More recently, the coherent oscillation of spectral peaks in the [*nonlinear*]{} optical response of molecular aggregates has been associated with entanglement between molecular units [@Sarovar:2010; @Ishizaki:2010]. Equation (\[eq:absorption dimer\]) shows that it is possible to identify entanglement in an ensemble of dipolar molecular pairs by measuring the linear absorption spectra. The procedure would be as follows. After preparing the system in an entangled state using a strong off-resonant laser pulse, a weak microwave field tuned near resonance with the lowest dipole-allowed transition would give an absorption spectrum whose line width shows damped oscillations at frequency $\omega_{41} = 4B_{\rm e}/\hbar$. The presence of oscillations serves as an entanglement witness. Eq. (\[eq:dynamical lineshape\]) shows that the amplitude of this linewidth oscillation is proportional to the concurrence $C(\rho_0)=2|a_0b_0|$ of the initially prepared state, while the decay of the oscillation depends on the decoherence rate $\gamma_{41}$. Measuring the amplitude of these oscillations can thus allow measurement of the pairwise entanglement between the dipolar molecules.
Bell’s inequality for orientation correlations
----------------------------------------------
Bell inequalities quantify the differences between quantum and classical correlations of measurements performed in different bases on quantum systems and provide critical tests of the incompatibility of quantum mechanics with local realism. Violation of a Bell inequality constitutes evidence of nonlocal quantum correlations such as entanglement between distant particles [@Laloe:2001]. Not all entangled bipartite states violate the inequality, although all separable states do satisfy the inequality [@Terhal:2000; @Werner:2001]. For the case of entangled molecules in the presence of DC electric fields, it was recently shown that violations of Bell inequalities can be established [@Milman:2007; @Milman:2009]. In the following we adapt and simplify the analysis in Ref. [@Milman:2007] to analize the orientational entanglement of polar molecules trapped in an optical double well lattice and prepared in the pure state $ {| \Phi_0 \rangle} = a_0{| g_1g_2 \rangle}+b_0{| e_1e_2 \rangle}$ by the action of a strong off- resonant laser pulse. We assume that the subsequent evolution is determined as in Sec. \[sec:global\] by the field-free rigid rotor Hamiltonian $\mathcal H_{\rm R}$, i.e., we neglect the small perturbation due to the trapping potential.
The degree of orientation of a single molecule is given by the expectation value of the operator $\hat O = \cos\theta$ [@Stapelfeldt:2003; @Seideman:2005], where $\theta$ is the polar angle of the internuclear axis with respect to the quantization axis. The orientation operator in the two-level basis $\mathcal{S}_1=\{{| g \rangle}\equiv{| 0,0 \rangle},{| e \rangle}\equiv{| 1,0 \rangle}\}$ can be written as $ \hat O = \sigma_X/\sqrt{3} $, with eigenvalues $\lambda_\pm = \pm 1/\sqrt{3}$, corresponding to the molecule being oriented parallel (plus sign) or antiparallel (minus sign) to the direction of the quantization axis. For our proposed realization with molecules trapped in double well optical lattices, orientation measurements can be performed in a using laser-induced fluoresence [@Orr-Ewing:1994] with single-site resolution.
We consider the two-time orientation correlation function for a molecular pair $ E(t_1,t_2)=\langle\hat{{O}}_1(t_1)\otimes\hat{{O}}_2(t_2)\rangle$, where $\hat{{O}}_i(t_i) = \hat U_i^{\dagger}(t_i)\hat O(0)\hat U_i(t_i)$ [@Milman:2007; @Milman:2009; @Lemeshko:2011]. The free evolution operator is given by $\hat U(t) = \rme^{-i\hat H_{\rm R}t/\hbar}$, where $\hat H_{\rm R}=B_{\rm e} \sigma_Z$ in the two-level basis. The orientation correlation vanishes for separable two-molecule states, but remains finite for entangled states. In particular for a pair of molecules initially in the state ${| \Phi_0 \rangle}=a_0{| g_1g_2 \rangle}+b_0{| e_1e_2 \rangle}$, the orientation correlation function is given by $$E(t_1,t_2) = \frac{1}{3}C(\rho_0)\cos\left(\omega_{eg}t_1+\omega_{eg}t_2+\theta_{ba}\right),
\label{eq:correlation function}$$ where $C(\rho_0)$ is the concurrence of the initial pure state $\rho_0={| \Phi_0 \rangle}{\langle \Phi_0 |}$, $\theta_{ba}$ the relative phase between the state components (see above) and the rotational frequency is $\omega_{eg} = 2B_{\rm e}/\hbar$. The correlation function is invariant under particle exchange and symmetric around $t_1=t_2=\pi/2$ for the relative phase $\theta_{ba}=n\pi$, with $n$ an integer.
Bell measurements can be divided into three steps [@Laloe:2001]. First is the preparation of a pair of particles, typically spins, in a repeatable way. Second, an experimental setting is chosen independently for each particle. The setting for spins corresponds to the orientation of a Stern-Gerlach apparatus that measures the spin projections of particles A and B along the directions $\vec{a}$ and $\vec{b}$, respectively. Finally, the correlation $E(\vec{a},\vec{b})$ between the measurement outcomes for different sets of directions $(\vec{a},\vec{b})$ are collected. For quantum correlation the Bell’s inequality in the Clauser-Horne-Shimony-Holt form [@CHSH:1969; @Horodecki:2009review] $$|E(\vec{a},\vec{b})+E(\vec{a},\vec{b}')+E(\vec{a}',\vec{b})-E(\vec{a}',\vec{b}')|\leq 2\lambda^2_{\rm max}
\label{eq:Bell inequality}$$ is violated, where $\lambda_{\rm max}$ is the maximum value of the measurement outcome. The quantum mechanical spin projection operator is $\vec{a}\cdot\vec{\sigma}$, with $\vec{\sigma}=(\sigma_X,\sigma_Y,\sigma_Z)$. For spin-$1/2$ particles $\lambda_{\rm max}=1$.
There is a one-to-one correspondence between Bell measurements based on spin orientations $\vec{a}$ and $\vec{b}$ and a scheme based on the free rotational evolution of molecules. In the two-state basis used here, the molecular orientation operator in the Heisenberg picture can be written as $ \hat O(\tau_a) = \frac{1}{\sqrt{3}}\rme^{i\hat{\sigma}_z\tau_a/2}\;\hat{\sigma}_X\;\rme^{-i\hat{\sigma}_z\tau_a/2}\equiv \vec{a}\cdot\vec{\sigma}$, where we have defined the orientation vector $\vec{a} = (1/\sqrt{3})(\cos\tau_a,-\sin\tau_a,0)$, and $\tau_a=2B_{\rm e}t_a/\hbar$. The time evolution of the orientation operator $\hat O(\tau_a)$ thus corresponds to a clockwise rotation of the orientation direction $\vec{a}$ from the positive $X$ axis by an angle $ \tau_a$ in the $XY$ plane. Therefore, choosing the time $t_a$ when to perform a molecular orientation measurement is equivalent to choosing the orientation of the Stern-Gerlach apparatus for the case of spin-$1/2$ particles. The two-time orientation correlator in Eq. (\[eq:correlation function\]) can thus be written as $E(t_a,t_b)=\langle\vec{a}\cdot\vec{\sigma}\otimes\vec{b}\cdot\vec{\sigma}\rangle$, which is the form of the correlation function for spin systems. Following the equivalence between spin orientation and rotational evolution, the magnitude of the quantity $$S = E(t_a,t_b)+E(t_a,t_b')+E(t_a',t_b)-E(t_a',t_b').
\label{eq:rotational inequality}$$ can then be used to test violations of Bell’s inequality. For our purposes it is sufficient to set $t_a=t_b=0$ and $t_a' = t_b' = t$ in Eq. (\[eq:rotational inequality\]) and evaluate the absolute value of $S_1(t) = E(0,0)+E(0,t)+E(t,0)-E(t,t)$ using Eq. (\[eq:correlation function\]). In Fig. \[fig7:Bell violation\] we plot $|S_1(t)|$ as a function of time for several parent states ${| \Phi_0 \rangle}=a_0{| g_1g_2 \rangle}+b_0{| e_1e_2 \rangle}$ with different concurrences $C(\rho)$ and relative phases $\theta_{ba}$. The upper bound imposed by Bell’s inequality over the $|S_1(t)|$ is $2\lambda_{\rm max}^2=2/3$. For the states shown in Fig. \[fig7:Bell violation\], this limit is violated over a wide range of times within a rotational period $T_{\rm R}=\pi t_{\rm R}$. The violation of the classical bound serves as an entanglement witness. Most importantly, the figure clearly shows that the degree of violation of Bell’s inequality depends on the concurrence $C(\rho_0)$ of the entangled state. Therefore, once the signal is calibrated it should be possible to use the magnitude of $S_1(t)$ at a chosen time to quantify the molecular entanglement.
![Violation of Bell’s inequality for molecular orientation correlations. The absolute value of $S_1(t)= E(0,0)+E(0,t)+E(t,0)-E(t,t)$ is plotted as a function of time for several states of the form ${| \Phi \rangle} = |a|{| g_1g_2 \rangle}+|b|\rme^{i\theta_{ba}}{| e_1e_2 \rangle}$. Each panel shows $|S_1|$ for three values of the concurrence: $C =1.0$ (black line), $C=0.9$ (red line), and $C=0.8$ (blue line). Panels (a) and (b) correspond to the relative phases $\theta_{ba}=0$ and $\theta_{ba}=\pi/4$, respectively. $E(t,t')$ is the two-time orientation correlation function. Time in is units of the rotational period $T_R = \pi\hbar/B_{\rm e}$.[]{data-label="fig7:Bell violation"}](Figure5){width="70.00000%"}
We close with some comments on experimental feasibility of these measurements. The preparation of entangled pairs can be done using the methods described in Sec. \[sec:entanglement generation\]. An ensemble of identical pairs can be prepared to enhance the sensitivity of the correlation measurements. Performing orientation measurements in individual sites with laser-induced fluorescence [@Orr-Ewing:1994] is significantly less destructive than femtosecond photodissociation measurements. Experimental violations of Bell’s inequality have been established in a large number of experiments using photons [@Freedman:1972; @Aspect:1981; @Zeilinger:1998; @Gisin:1998; @Gisin:2001], trapped atoms [@Rowe:2001], superconducting junctions [@Ansmann:2009], quantum dots [@Sun:2012], and even elementary particles [@Apostolakis:1998], but to the best of our knowledge it has not been established with molecules. Our analysis shows that it is possible with current technology to look for violations of Bell’s inequality for molecules in long-wavelength optical lattices or in separate dipole traps.
Robustness of entanglement against motional decoherence {#sec:decoherence}
=======================================================
Entanglement between distant molecules can be expected to decay in time due to relaxation and dephasing processes resulting from environmental perturbations. For entangled molecules in optical traps decoherence processes arise from their interaction with noisy external fields. Far-detuned optical traps, for example, are sensitive to laser intensity fluctuations and beam pointing noise, which can cause heating of the trapped atoms or molecules [@Savard:1997; @Gehm:1998]. Trap noise affects the precision of atomic clocks [@Takamoto:2005; @Ludlow:2006] and also the dynamics of strongly-correlated cold atomic ensembles [@Pichler:2012]. Additional sources of decoherence influence the dynamics of the system in the presence of static electric and magnetic fields [@Yu:2003]. In this Section we analyze the robustness of alignment-mediated entanglement of molecules trapped in optical lattices to fluctuations in the optical trapping laser fields. Our primary focus here is on motional decoherence in optical arrays, which is most sensitive to the effective lattice temperature.
For an array of interacting polar molecules, the fluctuation of the dipole-dipole interaction energy $U_{\rm dd}(R)$ with the motion of the molecules in the trapping potential represents a source of decoherence for the collective rotational state dynamics. The vibrational motion of the molecules in an optical lattice potential can be represented by phonons interacting with the coherent rotational excitation transfer between molecules in different sites. Following Ref. [@Herrera:2011] we write the Hamiltonian for a one-dimensional molecular array in the absence of static electric fields as $$\begin{aligned}
\mathcal{H} &=& \sum_i\epsilon_{eg} {\hat c^{\dagger}_{i}}{\hat c_{i}} + \sum_{i,j} J_{ij} {\hat c^{\dagger}_{i}}{\hat c_{j}} \nonumber\\
&&+ \sum_k \hbar\omega_k {\hat a^{\dagger}_{k}}{\hat a_{k}}+\sum_{i,j\neq i}\sum_k \lambda_{ij}^k {\hat c^{\dagger}_{i}}{\hat c_{j}}\left({\hat a_{k}} + {\hat a^{\dagger}_{k}}\right),
\label{eq:lattice Hamiltonian}\end{aligned}$$ where ${\hat a^{\dagger}_{k}}$ creates a phonon in the $k$-th normal mode with frequency $\omega_k$. The first and second terms determine the coherent state transfer between molecules in different sites, with site energy $\epsilon_{eg}=2B_{\rm e}$ and hopping amplitude $J_{ij}$ (evaluated at equilibrium distances). The third term describes the vibrational energy of the molecular center of mass in the trapping potential, which we assume harmonic as an approximation. In the absence of DC electric fields the phonon spectrum is dispersionless [@Herrera:2011], i.e., $\omega_k = \omega_0$. The last term represents the interaction between the internal and external molecular degrees of freedom, characterized by the energy scale $$\lambda_{ij}^{k}(\omega_0)=-3J_{12}\left[\frac{l_0(\omega_0)}{a_L}\right]f^k_{ij}\frac{(i-j)}{|i-j|^{5}},
\label{eq:lambda}$$ where $\omega_0$ is the trapping frequency of the optical lattice, $a_L$ is the lattice constant, $l_0 = \sqrt{\hbar/2m\omega_0}$ is the oscillator length, and $f^k_{ij}$ is a mode-coupling function that satisfies the relation $f_{ij}^k=-f_{ji}^k$.
We have omitted terms of the form $({\hat c^{\dagger}_{i}}{\hat c^{\dagger}_{j}} + {\textrm{ H.c}})$ in Eq. (\[eq:lattice Hamiltonian\]), since these only affect the dynamics of the system when $J_{12}/\epsilon_{eg}\sim 1$. As discussed in Section \[sec:entanglement generation\].1, this condition is satisfied only in the presence of a strong off-resonant pulse. However, the laser pulse width $\tau_p$ is orders of magnitude shorter than the timescale of the oscillation of molecules in the lattice potential ($\tau_{\rm p}\ll \omega_0^{-1}$). This separation of timescales allows us to neglect the coupling between internal and translational degrees of freedom under the action of a strong off-resonant laser pulse, even when $J/\epsilon_{eg}\sim 1$. After the pulse is over, the coupling to phonons can become important when the timescale for internal state evolution $h/J_{12}$ is comparable with $1/\omega_0$. Under this condition the molecular array evolves according to the Hamiltonian in Eq. (\[eq:lattice Hamiltonian\]) over a timescale shorter than the molecular trapping lifetime $\tau_{\rm trap}\sim 1 $ s [@Chotia:2012].
The Hamiltonian in Eq. (\[eq:lattice Hamiltonian\]) can be rewritten as $\mathcal{H} = \mathcal{H}_S+\mathcal{H}_B+\mathcal{H}_{SB}$ using the unitary transformation ${\hat c^{\dagger}_{\mu}} = \sum_i u_{i\mu}{\hat c^{\dagger}_{i}}$. The Hamiltonian $\mathcal{H}_S = \sum_\mu \varepsilon_\mu {\hat c^{\dagger}_{\mu}}{\hat c_{\mu}}$ describes the collective rotational states in terms of excitonic states ${| \mu \rangle} = {\hat c^{\dagger}_{\mu}}{| g \rangle}$ with energy $\varepsilon_\mu$. The second term $\mathcal{H}_B=\hbar\omega_0\sum_k{\hat a^{\dagger}_{k}}{\hat a_{k}}$ describes free lattice phonons, and the term $$\mathcal{H}_{SB} = \sum_{\mu\nu}\lambda_{\mu\nu}^k{\hat c^{\dagger}_{\mu}}{\hat c_{\nu}}({\hat a_{k}}+{\hat a^{\dagger}_{k}}),
\label{eq:system-bath}$$ describes the interaction of the excitonic system with the phonon environment. The interaction energy in the exciton basis is given by $\lambda^k_{\mu\nu} = \sum_{ij}u^*_{i\mu}u_{j\nu}\lambda_{ij}^k$. The internal state evolution of the excitonic system depends strongly on the characteristics of the phonon environment. For low phonon frequencies $\omega_0<J_{12}/h$ the interaction energy $\lambda^k$ can become the largest energy scale in the Hamiltonian, and non-Markovian effects in the evolution of the system density matrix $\rho(t)$ become important [@Breuer-Petruccione-book]. We assume here for simplicity that $\hbar\omega_0>J_{12}$, or more precisely $(l_0/a_L)^2(J_{12}/\hbar\omega_0)< 1$ [@Herrera:2012] so that we are in a weak coupling regime. Note that $\omega_0$ is determined by the trapping strength of the optical lattice and that both this and the dipolar interaction $J_{12}$ can be tuned in this system to a far greater extent than is possible for Hamiltonians describing excitonic energy transfer in molecular aggregates [@Agranovich:2008]. In this weak coupling regime, the system evolution can then be described by a quantum master equation in the Born-Markov and secular approximations [@Breuer-Petruccione-book][^1] as $\dot \rho(t) = -(i/\hbar)\left[ \mathcal{H}_S,\rho(t)\right]+\mathcal{D}\left(\rho(t)\right)$.
Let us consider the case of two interacting polar molecules coupled to a common phonon environment via the nonlocal term in Eq. (\[eq:system-bath\]). The dissipative dynamics of the system density matrix $\rho(t)$ is determined by $$\mathcal{D}(\rho(t)) = \gamma_0\mathcal{P}_1^{(-)}\rho(t) \mathcal{P}_1^{(-)}-\frac{1}{2}\gamma_0\{\mathcal{P}_1^{(+)},\rho(t)\},
\label{eq:dissipator nonlocal}$$ where $\mathcal{P}_1^{(\pm)}={| \Psi_S \rangle}{\langle \Psi_S |}\pm{| \Psi_A \rangle}{\langle \Psi_A |}$ are projection superoperators, $\gamma_0$ is the pure-dephasing rate, and $\{A,B\}$ denotes the anticommutator. The projection into the two-excitation eigenstate $\mathcal{P}_2={| e_1e_2 \rangle}{\langle e_1e_2 |}$ does not contribute in the absence of DC electric fields (see discussion in \[sec:dc fields\]). The single-excitation eigenstates are ${| \Psi_S \rangle}=2^{-1/2}({| e_1g_2 \rangle}+{| g_1e_2 \rangle})$ and ${| \Psi_A \rangle} = 2^{-1/2}({| e_1g_2 \rangle}-{| g_1e_2 \rangle})$. Equation (\[eq:dissipator nonlocal\]) shows that for a system prepared in the pure state ${| \Phi \rangle}=a{| g_1g_2 \rangle}+b{| e_1e_2 \rangle}$ we have $\mathcal D(\rho) = 0$. In other words, the two-molecule entangled states prepared using a strong laser pulse do not decohere due to the interaction with environmental phonons in the optical lattice, regardless of the strength of the coupling to the environment and the effective lattice temperature. This is a consequence of the nonlocal nature of the interaction with the phonon environment and implies that under these conditions, the states ${| \Phi_{\pm} \rangle}=\left[{| g_1g_2 \rangle}\pm{| e_1e_2 \rangle}\right]$ provide a basis for a decoherence-free subspace in which all pairwise entangled states may be defined.
We can understand the effects of motional decoherence on the entangled triparticle and many-particle states by estimating the full phonon decoherence rates, given by $\gamma_{\mu\nu,\mu'\nu'}(\omega)$, with $\mu, \nu$ indexing the excitonic states. In Eq. (\[eq:dissipator nonlocal\]) the pure dephasing rate is defined as $\gamma_0=\gamma_{AA,AA}(0)=\gamma_{SS,SS}(0)=-\gamma_{AA,SS}(0)=-\gamma_{SS,AA}$(0). In the Born-Markov and secular approximations, dephasing and relaxation processes that lead to decoherence and entanglement decay occur at the rate $\gamma_{\mu\nu,\mu'\nu'}(\omega) = (1/\hbar^2)\int_{-\infty}^\infty d\tau\rme^{i\omega\tau}\langle \hat B_{\mu\nu}(\tau)\hat B_{\mu'\nu'}(0)\rangle$, where $\langle \hat B_{\mu\nu}(\tau)\hat B_{\mu'\nu'}(0)\rangle$ is the bath correlation function with $\hat B_{\mu\nu} = \sum_k\lambda_{\mu\nu}^k({\hat a_{k}}+{\hat a^{\dagger}_{k}})$. In \[sec:spectral density\] we use a classical stochastic model to approximate the bath correlation function under the influence of random intensity fluctuations of the trapping laser. This procedure allows us to write the decoherence rates as $$\gamma_{\mu\nu,\mu'\nu'}(\omega) = \frac{1}{\hbar^2}\left[n(\omega)+1\right]\left[J^{\rm cl}_{\mu\nu,\mu'\nu'}(\omega) - J^{\rm cl}_{\mu\nu,\mu'\nu'}(-\omega)\right],
\label{eq:transition rate}$$ where $n(\omega) = (\rme^{\beta\hbar\omega}-1)^{-1}$ is the Bose distribution function and $$J^{\rm cl}_{\mu\nu,\mu'\nu'}(\omega) = \sum_k\lambda_{\mu\nu}^k\lambda_{\mu'\nu'}^k\left(\frac{\omega}{\omega_k}\right)\frac{\beta}{(\omega - \omega_k)^2+\beta^2},$$ is the semiclassical spectral density for optical lattice phonons. In \[sec:spectral density\] we show that the broadening parameter can be written as $\beta = \kappa \omega_0^2$, where the factor $\kappa>0$ is proportional to the strength of the laser intensity noise. The trapping noise causes damping of the correlation function as $\langle B_{\mu\nu}(t)B_{\mu\nu}(0)\rangle\propto\rme^{-\beta|t|}\cos(\omega't)$, where $\omega' = \sqrt{\omega_0^2-\beta^2}$. The bath autocorrelation time $\tau_c$ is order $\beta^{-1}$. The condition for the Markov approximation to hold is thus $\beta^{-1}\ll h/J_{12}$.
For fixed trapping parameters $\omega_0$, $a_L$ and $\beta$, this analysis shows that different molecular species can undergo very different open system dynamics, depending on the strength of the dipolar interaction between molecules in different sites. For instance, let us consider LiCs ($d=5.5$ D) and KRb ($d = 0.6$ D) species as examples of molecules with high and low permanent dipole moments, respectively. For an optical lattice with $a_L = 1\,\mu$m and noise-induced damping rate $\beta = 100$ Hz, the open system dynamics would have Markovian behaviour for KRb molecules ($J_{12}/h = 10 $ Hz), but for LiCs molecules ($J_{12}/h = 1.4$ kHz) the system dynamics can be expected to be non-Markovian. A very attractive feature of this trapped dipolar molecule array is that the transition between Markovian and non-Markovian dynamics can be studied experimentally for any molecular species by manipulating the laser intensity noise in order to tune the parameter $\beta$ as in Ref. [@DErrico:2012], or by changing the lattice spacing $a_L$ to manipulate $J_{12}$.
In the regime where the Markov and secular approximations are valid, we can estimate the phonon-induced decoherence rate $\gamma(\omega_S)$ in Eq. (\[eq:transition rate\]) (with state indices removed for simplicity) at the characteristic system frequency $\omega_{\rm S} = J_{12}/\hbar$. For a lattice temperature such that $\hbar\omega_S/k_{\rm b}T\ll 1$ the decoherence rate scales as $\gamma(\omega_S)\sim 4\pi^2(J_{12}/h)^2(l_0/a_L)^2 H(\omega_S)$, with $H(\omega) = (\omega/\omega_0)\beta/[(\omega-\omega_0)^2+\beta^2]$. For experimentally realizable parameters $\beta = 1$ kHz, $\omega_0 = 10$ kHz and $a_L = 500$ nm, the decoherence rate for KRb molecules ($\omega_S/2\pi=0.13$ kHz) is $\gamma(\omega_S)\sim 10^{-5}$ Hz, which is negligibly small compared with the typical loss rate of molecules from optical traps ($\gamma_{\rm trap}\sim 1$ Hz) due to incoherent Raman scattering of lattice photons. We conclude that the entangled states of polar molecules containing double excitations can be robust to phonon-induced decoherence in optical lattice settings for which the weak coupling condition $\hbar\omega_0/J_{12}\gg 1$ holds.
Conclusion {#sec:conclusions}
==========
In this work we present a scheme to generate entanglement in arrays of optically trapped polar molecules. Starting from an array of molecules prepared in their rovibrational ground state, a single strong off-resonant laser pulse can be used to generate entanglement between molecules in different sites of the array. The strong laser field induces the alignment of molecules along its polarization direction during the pulse. For such laser alignment of polar molecules interacting via a dipole-dipole term, the energy ratio between the coupling and site energies $J_{12}/\varepsilon_{\rm e}$ can be larger than unity, allowing generation of two-particle wavefunctions of the form ${| \Phi \rangle} = a{| g_1g_2 \rangle}+b{| e_1e_2 \rangle}$ in the presence of the strong laser field. For $|ab|\neq 0$, the laser alignment will thus induce entangled states, where the precise form of the resulting entangled state may be controlled by the duration and strength of the laser pulse. The subsequent evolution after the laser pulse is completed adds a dynamical phase to the entangled state but does not change the concurrence measure of the extent of entanglement. The proposed generation scheme does not depend on the number of coupled molecules and also holds for a many-particle system. Here for simplicity we have considered explicitly only the two-particle case.
We emphasize that this alignment-mediated entanglement involving double excitation states is not possible with static electric fields. The rotational structure of an aligned molecule is such that the transition energy $\varepsilon_{\rm e}$ between the lowest two rotational states ${| g \rangle}$ and ${| e \rangle}$ becomes comparable in magnitude with the dipole-dipole interaction energy $J_{ij} ={\langle g_ig_j |}\hat V_{\rm dd}{| e_ie_j \rangle}$, for molecules separated by distances of up to several micrometers. At such large distances the ratio $J_{12}/\varepsilon_{\rm e}$ is negligibly small in the absence of DC electric fields and double-excitation transitions of the type ${| g_1g_2 \rangle}\rightarrow{| e_1e_2 \rangle}$ are energetically suppressed.
We have demonstrated explicitly that the degree of entanglement in a molecular pair can be manipulated by tuning experimental parameters such as the laser pulse intensity and duration, as well as the intermolecular distance. We presented two methods to detect and measure entanglement in optical traps after the strong laser pulse is applied. The first approach requires only global microwave addressing of the molecular array. Here we showed that the linear microwave response of an ensemble of entangled pairs contains a contribution to the absorption lineshape that is proportional to the amount of pairwise entanglement and that oscillates in time at a frequency of order $B_{\rm e}/h$, where $B_{\rm e}$ is the rotational constant. Measuring the absorption peak oscillations over this timescale would then allow the concurrence of the state to be determined. The second approach is based on measurements of molecular orientation correlations to establish violations of Bell’s inequality. This method relies on the ability to optically address individual sites of a molecular array in order to perform laser-induced fluorescence measurements. Finally, we also analyzed the robustness of the strong field alignment-mediated molecular entanglement in optical arrays with respect to motional decoherence induced by fluctuations in the trapping lasers.
The results presented in this work for a molecular pair can readily be generalized to larger molecular arrays, as indicated in the text of the paper. In this context, it is useful to recognize that the system Hamiltonian can be mapped into a quantum-Ising model with a tunable magnetic field, a model that has been widely used in the study of quantum phase transitions [@Amico:2008review]. Furthermore, the form of Ising Hamiltonian describing the system is 2-local, which supports universal quantum computation when combined with the ability to implement arbitrary single-particle unitary transformations [@Lloyd:1995]. Therefore, an array of optically-trapped polar molecules driven by strong off-resonant laser pulses provides both a test-bed for studies of quantum entanglement in many-body systems and a novel platform for the development of quantum technologies.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Roman Krems for helpful comments on the manuscript. FH and SK were supported by the NSF CCI center “Quantum Information for Quantum Chemistry (QIQC)”, award number CHE-1037992. FH was also supported by NSERC Canada.
Molecules in combined off-resonant laser and DC electric fields {#sec:dc fields}
===============================================================
In this appendix we describe the dipole-dipole interaction between polar molecules in combined presence of DC electric fields and strong off-resonant pulsed laser fields. We discuss how the addition of a DC electric field affects the entanglement creation scheme described in Section \[sec:entanglement generation\].
Dipole-dipole interaction in combined fields {#dipole-dipole-interaction-in-combined-fields .unnumbered}
--------------------------------------------
Let us consider a polar molecule in its vibrational ground state, under the influence of a DC electric field and a CW far-detuned optical field. If the laser polarization is collinear with the direction of the DC electric field (space-fixed $Z$ axis), the dimensionless molecular Hamiltonian $\hat H = \hat H_{\rm R} + \hat H_{\rm DC} + \hat H_{\rm AC}$ can be written in analogy with Eq. (\[eq:dimless ac\]) as $$\hat H = \hat N^2 -\lambda\mathcal{D}^{(1)}_{0,0}-\frac{2}{3}\Omega_{\rm I}\mathcal{D}^{(2)}_{0,0},
\label{app:dimless ac/dc}$$ where $\lambda = dE_Z/B_{\rm e}$ parametrizes the strength of the DC electric field. $E_Z$ is the magnitude of the DC electric field and $d$ is the permanent dipole moment of the molecule. The rotational structure for $E_Z=0$ and large laser intensities $\Omega_{\rm I}$ consists of harmonically spaced tunneling doublets separated by an energy proportional to $\Omega_{\rm I}$ as shown in Fig. \[fig:ac energies\] of the main text. Each doublet is composed of states with opposite parity whose energy splitting decreases exponentially with $\Omega_{\rm I}$. Due to this near degeneracy, a very weak DC electric field strongly couples the field-dressed doublet states, splitting their energy levels linearly with $\lambda$ [@Friedrich:1999]. The two lowest doublet states ${| g \rangle}$ and ${| e \rangle}$ for $\lambda\ll 1$ correlate adiabatically with ${| g \rangle} \approx \sqrt{a}{| 0,0 \rangle}+\sqrt{b}{| 1,0 \rangle}$ and ${| e \rangle} \approx \sqrt{b}{| 0,0 \rangle} - \sqrt{a}{| 1,0 \rangle}$ as $\Omega_{\rm I}\rightarrow 0$, with $a\gg b$ and ${| NM_N \rangle}$ is an eigenstate of $\hat H_{\rm R}$.
In the absence of DC electric fields the dipole-dipole interaction operator $\hat V_{\rm dd}$ has only one non-zero matrix element $J_{ij} = {\langle e_ig_j |}\hat V_{\rm dd}{| g_ie_j \rangle}={\langle e_ie_j |}\hat V_{\rm dd}{| g_ig_j \rangle}$, defined in Eq. (\[eq:exchange coupling\]). In the presence of DC electric fields the parity of the rotational states is broken and the following matrix elements become finite: $V_{ij}^{gg} = {\langle g_ig_j |}\hat V_{\rm dd}{| g_ig_j \rangle}$, $V_{ij}^{ee} = {\langle e_ie_j |}\hat V_{\rm dd}{| e_ie_j \rangle}$, and $V_{ij}^{eg} = {\langle e_ig_j |}\hat V_{\rm dd}{| e_ig_j \rangle}$. The dipolar energies $\left\{J_{ij},V_{ij}^{gg},V_{ij}^{ee},V_{ij}^{eg}\right\}$ determine the dynamics of interacting polar molecules in the regime where the energy $\Delta \epsilon_{eg}$ for the transition ${| g \rangle}\rightarrow{| e \rangle}$ is much larger than the dipole-dipole energy $U_{\rm dd}=d^2/R^3$, where $R$ is the intermolecular distance. In the regime $\Delta\epsilon_{eg}\sim U_{\rm dd}$ two additional dipole-dipole transitions become important: $A_{ij} = {\langle e_ig_j |}\hat V_{\rm dd}{| g_ig_j \rangle}$ and $B_{ij} = {\langle e_ig_j |}\hat V_{\rm dd}{| e_ie_j \rangle}$. These matrix elements couple the single excitation manifold with the ground and doubly excited states, and vanish in the absence of DC electric fields.
![ Dipole-dipole interaction energies $J_{12}$, $D_{12} \equiv V_{12}^{eg}-V_{12}^{gg}$, $A_{12}$ and $B_{12}$ as a function of the intensity parameter $\Omega_{\rm I}$. Curves are labeled according to the DC electric field strength $\lambda=dE_Z/B_{\rm e}$. The DC and AC electric fields are collinear. Energy is in units of $U_{\rm dd}=d^2/R^3$ and the intermolecular axis is taken perpendicular to the orientation of the fields.[]{data-label="fig:dipole energies"}](Figure6){width="80.00000%"}
In analogy with the definition of $J_{ij}$ in Eq (\[eq:exchange coupling\]) we can write the dipole-dipole energies in units of $U_{\rm dd}(1-\cos^2\Theta)$ as $V^{gg}_{ij} = \mu_g^2$, $V^{eg}_{ij} = \mu_{e}\mu_g$, $V^{ee}_{ij} = \mu_{e}^2$, $A_{ij} = \mu_{eg}\mu_g$, and $B_{ij} = \mu_{eg}\mu_e$, where $\mu_{eg} = d^{-1}{\langle e |}\hat d_0{| g \rangle}$ is the dimensionless transition dipole, $\mu_e=d^{-1}{\langle e |}\hat d_0{| e \rangle}$ is the dimensionless dipole moment of the excited state and $\mu_g=d^{-1}{\langle g |}\hat d_0{| g \rangle}$ is the dipole moment of the ground state. For the choice of rotational states used here we have $\mu_{eg}>0$, $\mu_g>0$ and $\mu_e<0$, which give $A_{ij} = -B_{ij}>0$. It is convenient to define the differential dipolar shift $D_{ij} = V^{eg}_{ij}-V^{gg}_{ij}=\mu_g(\mu_e-\mu_g)<0$ to describe the single-excitation dynamics [@Herrera:2011]. We evaluate the dipole-dipole matrix elements using the eigenvectors of the single-molecule Hamiltonian in Eq. (\[app:dimless ac/dc\]). In Fig. \[fig:dipole energies\] we show the dependence of the dipole-dipole energies $J_{ij}$, $D_{ij}$, $A_{ij}$ and $B_{ij}$ on the laser intensity parameter $\Omega_{\rm I}$ and the DC field strength parameter $\lambda$. The figure shows that the exchange interaction energy $J_{ij}$ tends to zero at high intensities $\Omega_{\rm I}\gg 10$ in the presence of a perturbatively small DC electric field $\lambda\ll 1$. The energies $A_{ij}$ and $B_{ij}$ also vanish at high intensities. Only the diagonal dipolar shifts $V_{ij}^{eg}$, $V^{ee}_{ij}$ and $V_{ij}^{gg}$ are finite in the high intensity regime for any non-zero DC field strength.
![Entanglement radius $R_{\rm e}$ in units of $R_0$ (log scale), as a function of the laser intensity parameter $\Omega_{\rm I}$. Curves are labeled according to the electric field strength $\lambda = dE_Z/B_{\rm e}$. $R_0\equiv(d^2/B_{\rm e})^{1/3}$ is a characteristic dipolar radius.[]{data-label="fig:entanglement length"}](Figure7){width="70.00000%"}
Disadvantages for dynamical entanglement creation {#disadvantages-for-dynamical-entanglement-creation .unnumbered}
-------------------------------------------------
The presence of a DC electric field modifies the state evolution under the action of a strong off-resonant laser pulse in two ways. First, a static electric field strongly mixes the quasi-degenerate doublet states at high laser intensities (Fig. \[fig:ac energies\]), resulting in a linear DC Stark shift that increases the energy splitting $\varepsilon_{\rm e}$. The Stark splitting significantly modifies the entanglement radius $R_{\rm e} = (d^2/2\varepsilon_{\rm e})^{1/3}$, as shown in Fig. \[fig:entanglement length\]. The value of $R_{\rm e}$ increases exponentially with the laser intensity parameter $\Omega_{\rm I}$ in the absence of DC electric fields, but has an upper bound in combined fields. The bound depends on the DC field strength $\lambda = dE_Z/B_{\rm e}$, which determines the splitting of the states ${| g \rangle}$ and ${| e \rangle}$. For larger values of $\lambda$, the intermolecular distance at which the dipole-dipole interaction between molecules becomes comparable with the Stark splitting becomes smaller. For the molecular species used in Table \[tab:intensities\], $\lambda\sim 1$ corresponds to $E_Z\sim 1 $ kV/cm. For such large field strengths, $R_{\rm e}\approx R_0\sim 1$ nm for most alkali-metal dimers. Therefore, molecules in optical lattices with site separation $R\sim 10^2$ nm cannot be entangled using strong off-resonant fields when DC electric fields $E_Z\sim 1$ kV/cm are present. Figure \[fig:entanglement length\] however shows that in the presence of stray fields $E_Z\leq 1$ mV/cm ($\lambda\leq 10^{-6}$), alignment-mediated entanglement of alkali-metal dimers in optical lattices is still possible.
Second, breaking the parity symmetry of the rotational states results in additional contributions to the dipole-dipole interaction such as already discussed. The matrix elements $A_{ij}$ and $B_{ij}$ mix the subspaces $\mathcal{S}_1 = \left\{{| g_1e_2 \rangle}, {| e_1g_2 \rangle}\right\}$ and $\mathcal{S}_2 = \left\{{| g_1g_2 \rangle}, {| e_1e_2 \rangle}\right\}$, the two-molecule state for the initial condition ${| \Phi(0) \rangle} = {| g_1g_2 \rangle}$ is given by ${| \Phi(t) \rangle} = a(t){| g_1g_2 \rangle}+b(t){| e_1g_2 \rangle}+c(t){| g_1e_2 \rangle}+d(t){| e_1e_2 \rangle}$, with $|ad|\neq 0$ and $|bc|\neq 0$. Therefore, for intermolecular distances $R\leq R_{\rm e}$ the two-molecule state evolution in combined DC and off-resonant fields no longer follows the simple two-state dynamics described in Section \[sec:entanglement generation\].
Static electric fields also affect the dynamics of the entangled states after the laser pulse is over. Local system-environment coupling occurs in the presence of a static electric field [@Herrera:2011]. The local interaction of a pair of molecules with the phonon environment is described by $\hat H_{\rm int} = \kappa ({\hat c^{\dagger}_{1}}{\hat c_{1}}+{\hat c^{\dagger}_{2}}{\hat c_{2}})({\hat a_{}}+{\hat a^{\dagger}_{}})$, with $\kappa\propto D_{12}$. The associated dissipator can be written as $$\begin{aligned}
\mathcal{D}'(\rho(t)) &=& \gamma_0\mathcal{P}_1^{(+)}\rho(t) \mathcal{P}_1^{(+)}-\frac{1}{2}\gamma_0\{\mathcal{P}_1^{(+)},\rho(t)\}\nonumber\\
&& + 4\gamma'_0 \mathcal{P}_2 \rho(t) \mathcal{P}_2-2 \gamma'_0\left\{\mathcal{P}_2,\rho(t)\right\},
\label{eq:dissipator local} \end{aligned}$$ where $\mathcal P_2 = {| e_1e_2 \rangle}{\langle e_1e_2 |}$ is a Lindblad generator that induces dephasing of the doubly excited state. Therefore the two-molecule entangled state ${| \Phi \rangle} = a{| g_1g_2 \rangle}+b{| e_1e_2 \rangle}$ no longer belongs to a Decoherence-Free Subspace (DFS) with respect to the phonon environment, i.e. $\mathcal{D}'(\rho(t))\neq 0$. The decoherence rate $\gamma'_0$ would depend on the magnitude of the dipolar shift $D_{ij}$, which can be tuned by manipulating the strength of an applied static electric field and the intensity of the trapping laser. In addition to the phonon-induced fluctuations of the site energies in the presence of DC electric fields, the molecular energies also undergo fluctuations due to electric field noise, which acts as a global source of decoherence that can lead to entanglement decay as discussed for general bipartite and tripartite states in Refs. [@Yu:2003].
Model spectral density of optical lattice phonons {#sec:spectral density}
=================================================
In this appendix we derive the expression for the transition rate $\gamma_{\mu\nu,\mu'\nu'}(\omega)$ in Eq. (\[eq:transition rate\]) using a semiclassical model for the phonon environment in optical lattices. We start from the system-bath interaction operator in the exciton basis $ \hat H_{SB} = \sum_{\mu\nu}\sum_k \lambda_{\mu\nu}^k{\hat c^{\dagger}_{\mu}}{\hat c_{\nu}}\left({\hat a_{k}}+{\hat a^{\dagger}_{k}}\right)$ and define the time correlation function $C_{\mu\nu,\mu'\nu'}(t) = \langle \hat B_{\mu\nu}(t)\hat B_{\mu'\nu'}(0)\rangle$, where the bath operator $\hat B(t)$ in the interaction picture is given by $ \hat B_{\mu\nu}(t) = \sum_k \lambda_{\mu\nu}^k\left[{\hat a_{k}}(t)+{\hat a^{\dagger}_{k}}(t)\right]$.
The classical vibrational energy of the array can be written as $H = (1/2)\sum_k\dot Q_k^2 +\omega_k^2 Q_k^2$, where $Q_k = \sum_{j=1}^\mathcal{N} \alpha_{jk} \sqrt{m}\,x_j$ are the normal modes of vibration defined in terms of the displacements $x_j$ from equilibrium and the molecular mass $m$. Promoting normal coordinates to quantum operators as $\hat Q_k = \sqrt{\hbar/2\omega_k}\left({\hat a_{k}}+{\hat a^{\dagger}_{k}}\right)$ allows us to write the semiclassical bath operator $ B^{\rm cl}_{\mu\nu}(t) = \sum_k \lambda_{\mu\nu}^k \sqrt{\frac{2\omega_k}{\hbar}}Q^{\rm cl}_k(t)$. The [*classical*]{} bath correlation function can thus be written as $$C_{\rm cl}(t) = \sum_k \lambda_{\mu\nu}^k\lambda^k_{\mu'\nu'}\left(\frac{2\omega_k}{\hbar}\right)\langle Q_k(t)Q_k(0)\rangle_{\rm cl},
\label{app:classical TCF}$$ where we used the fact that different modes ($k'\neq k$) are uncorrelated. The classical bath correlation function is a real quantity, i.e., $C^*_{\rm cl}(t) = C_{\rm cl}(t)$.
The quantum bath correlation function (omitting system state indices) is defined as $C(\tau) = \langle \hat B(\tau)\hat B(0)\rangle$ and satisfies $C^*(t) = C(-t)$ [@Breuer-Petruccione-book]. The system transition rate is given by $\gamma(\omega) = G(\omega)/\hbar^2$ where $G(\omega) = \int_{-\infty}^\infty d\tau \rme^{i\omega\tau} C(\tau)$ is a real positive quantity. Using the detailed balance condition $G(-\omega) = \rme^{-\beta\hbar\omega}G(\omega)$, where $\beta = 1/k_{\rm b}T$, it is possible to write $$G(\omega) = \frac{2}{1-\rme^{-\beta\hbar\omega}}G_A(\omega),$$ where $G_A(\omega) = \int_{-\infty}^{\infty} d\tau \rme^{i\omega\tau}{\rm Im}\{C(\tau)\}$. We use this expression to obtain a semiclassical approximation to the quantum rate $\gamma(\omega)$.
The approximation scheme consists on relating the antisymmetric function $G_A(\omega)$ to the Fourier transform $G_{\rm cl}(\omega)=\int_{-\infty}^\infty \rme^{i\omega\tau}C_{\rm cl}(\tau)$ of the classical bath correlation function in Eq. (\[app:classical TCF\]). Following Ref. [@Egorov:1999], we use $G_A(\omega)\approx (\beta\hbar\omega/2)\,G_R(\omega)$, and postulate the semiclassical closure $C_R(t) = C_{\rm cl}(t)$. This procedure is known as the harmonic approximation. The approximate quantum transition rate is thus given by $$\gamma(\omega) = \frac{1}{\hbar^2} \frac{\beta\hbar\omega}{1-\rme^{\beta\hbar\omega}}G_{\rm cl}(\omega).
\label{app:semiclassical rate}$$
The next step is specific to the system considered here. It involves the evaluation of the correlation function $\langle Q_k(t)Q_k(0)\rangle_{\rm cl}$ from the classical equations of motion of a molecule in the optical lattice potential. For simplicity, we consider the potential to have the harmonic form $V(x) = \frac{1}{2}m\omega_k^2x^2$, where $\omega_k$ is the frequency of the normal mode $k$. The most general form of the mode frequency is $\omega_k = \omega_0 f(k)$, where $\omega_0 = (2/\hbar)\sqrt{V_LE_R}$ is the trapping frequency as determined by the lattice depth $V_L$ and the recoil energy $E_R$ of the molecule. The function $f(k)$ accounts for the dispersion of the phonon spectrum and is determined by the dipole-dipole interaction between ground state molecules in different lattice sites [@Herrera:2011]. In this work we consider molecules in the absence of static electric fields, therefore the induced dipole moment vanishes and the phonon spectrum is dispersionless. For any $k$, the mode frequency $\omega_k = \omega_0$ thus depends on the trapping laser intensity $I_L$ since $V_L \propto I_L$ [@Bloch:2005; @Carr:2009]. The laser intensity noise therefore modulates the phonon frequency $\omega_0$ and can lead to heating when the noise amplitude is large enough [@Savard:1997; @Gehm:1998]. The motion of a molecule in a fluctuating harmonic potential can be modeled by the equation of motion (for each $k$) $$\ddot Q_k +\omega_k^2(t) Q_k = 0,
\label{app:SEOM}$$ where $\omega_k^2 = \omega_0^2\left[1+\alpha\xi(t)\right]$, and $\alpha\xi(t)$ is proportional to the relative intensity noise, i.e, $\alpha\xi(t) \propto (I_L(t)-\langle I_0\rangle)/\langle I_0\rangle$.
The equation of motion in Eq. (\[app:SEOM\]) is a stochastic differential equation with multiplicative noise, for which no exact analytical solution exists [@VanKampen-book]. Using a cumulant expansion approach, the equation of motion for the correlation function $\langle Q(t)Q(0)\rangle$ can be written as [@VanKampen-book] $$\frac{d^2}{dt^2}\langle Q(t)Q(0)\rangle+2\beta\frac{d}{dt}\langle Q(t)Q(0)\rangle + \omega_0^{'2}\langle Q(t)Q(0)\rangle = 0,
\label{app:correlation EOM}$$ where $\beta = \alpha^2\omega_0^2c_2/4$ is an effective noise-induced damping coefficient and $\omega_0^{'2} = \omega_0^2(1-\alpha^2\omega_0c_1)$ is an effective oscillator frequency which includes a noise-induced shift from the deterministic value $\omega_0$. Equation (\[app:correlation EOM\]) is valid for all times provided $\alpha\tau_c\ll 1$, where $\tau_c$ is the noise autocorrelation time. The coefficients $c_1$ and $c_2$ are related to the noise autocorrelation function by $$\begin{aligned}
c_1 &=& \int_0^\infty \langle \xi(t)\xi(t-\tau)\rangle \sin(2\omega_0\tau)d\tau\\
c_2 &=& \int_0^\infty \langle \xi(t)\xi(t-\tau)\rangle [1-\cos(2\omega_0\tau)]d\tau.\end{aligned}$$ The effective damping constant can thus be written as $\beta = (\alpha^2\omega_0^2/8)[S(0)-S(2\omega_0)]$, where $S(\omega) = \int_{-\infty}^\infty\langle \xi(t)\xi(t-\tau)\rangle\rme^{-i\omega\tau}d\tau$ is the noise spectral density. The dependence of the damping coefficient on the spectral density at twice the natural frequency indicates that this is parametric dynamical process that can lead to heating ($\beta<0$) when $S(2\omega_0)>S(0)$. Here we assume that the static laser noise is dominant and use $\beta>0$, which is satisfied for trapping lasers with approximate $1/f$ noise as in Ref. [@Savard:1997].
The solution to Eq. (\[app:correlation EOM\]) is $\langle Q(t)Q(0)\rangle = \langle Q^2(0)\rangle\rme^{-\beta |t|}\cos(\omega' t)$, with $\omega' = \sqrt{\omega_0^2-\beta^2}$. We have assumed the oscillator is underdamped ($\omega_0>\beta$), and ignored the noise-induced frequency shift ($\omega_0' = \omega_0$). The mean square amplitude $\langle Q^2(0)\rangle$ can be obtained by averaging over initial conditions using Boltzmann statistics. For an ensemble of identical one-dimensional harmonic oscillators we have $\langle Q^2(0)\rangle = k_{\rm b}T/\omega_0^2$. Combining these results we can write the classical bath correlation function in Eq. (\[app:classical TCF\]) as $$C_{\rm cl}(t) = \sum_k\lambda_{\mu\nu}^k\lambda_{\mu'\nu'}^k\left(\frac{k_{\rm b}T}{\hbar\omega_k}\right)\rme^{-\beta|t|}\cos(\omega'_kt).
\label{app:classical TCF thermal}$$ By inserting the Fourier transform of Eq. (\[app:classical TCF thermal\]) into Eq. (\[app:semiclassical rate\]) we obtain the semiclassical transition rate $$\gamma_{\mu\nu,\mu'\nu'}(\omega) = \frac{1}{\hbar^2}\left[n(\omega)+1\right]\left[J^{\rm cl}_{\mu\nu,\mu'\nu'}(\omega)-J^{\rm cl}_{\mu\nu,\mu'\nu'}(-\omega)\right],
\label{app:rate vs SD}$$ where $n(\omega) = (\rme^{\beta\hbar\omega}-1)^{-1}$ is the Bose distribution function and we have defined the semiclassical phonon spectral density $$J^{\rm cl}_{\mu\nu,\mu'\nu'}(\omega) = \sum_k\lambda_{\mu\nu}^k\lambda_{\mu'\nu'}^k\left(\frac{\omega}{\omega_k}\right)\frac{\beta}{(\omega-\omega_k')^2+\beta^2}.
\label{app:spectral density}$$ This approximate expression for $J(\omega)$ should be compared with exact phonon spectral density for an ensemble of free quantum oscillators $J_{\mu\nu,\mu'\nu'}(\omega) = \omega^2\sum_k\lambda_{\mu\nu}^k\lambda_{\mu'\nu'}^k\delta(\omega-\omega_k)$, which also satisfies Eq. (\[app:rate vs SD\]).
References {#references .unnumbered}
==========
[^1]: Note that the secular approximation does not allow for coherence transfer [@Engel:2007]
|
---
abstract: 'In this work, we propose a passive foldable airframe as a protective mechanism for a small aerial robot. A foldable quadrotor is designed and fabricated using the origami-inspired manufacturing paradigm. Upon an accidental mid-flight collision, the deformable airframe is mechanically activated. The rigid frame reconfigures its structure to protect the central part of the robot that houses sensitive components from a crash to the ground. The proposed robot is fabricated, modeled, and characterized. The 51-gram vehicle demonstrates the desired folding sequence in less than 0.15 s when colliding with a wall when flying.'
author:
- 'Jing Shu and Pakpong Chirarattananon[^1][^2]'
bibliography:
- 'bibtex.bib'
title: ' **** '
---
Introduction
============
Recent rapid developments of aerial robots have shown promise. Following the advances in flight dynamics and control [@mulgaonkar2018robust], planning and localization [@qin2017vins; @chirarattananon2018direct], etc., there emerge numerous applications of these small flying robots that involve interactions of robots with objects or environments. These include, for instance, transportation of a suspended payload [@lee2018geometric], climbing on a vertical surface [@pope2017multimodal], perching on an overhang [@graule2016perching; @hsiao2018ceiling], aerial manipulation [@kim2018origami]. As the complexity of the tasks grows, it inevitably escalates the chance of failures. Despite attempts to circumvent an accident, it is still likely unforeseen circumstances would lead to an undesired collision that destabilizes the flight. The impact from a subsequent fall could lead to a destructive damage on the robot.
Researchers have proposed various strategies to deal with collisions. Thus far, the most common direction is to reduce the detrimental impact from collisions so that the robots retain the attitude stability. One solution is to incorporate a protective frame or is dynamically decoupled from the robot’s body [@briod2014collision; @sareh2018rotorigami]. With an appropriate damping mechanism, this allows the robot to continue flying as the influence on the attitude dynamics is drastically reduced. Another approach is to develop vehicles that are mechanically and dynamically robust. The flapping-wing robot in [@tu2019acting], for example, is capable of navigating tight space as it is resilient against collisions owing to the intrinsic compliance of its aerodynamic surfaces. Alternatively, in [@mintchev2017insect],, the authors opt to embrace the collisions and ensure the robot suffers no damage from the succeeding fall. The insect-inspired collision tolerance is accomplished with a deformable airframe that is held rigid by the magnetic joints [@mintchev2017insect] or prestretched elastomeric membranes [@mintchev2018bioinspired]. The energy absorbing property of the material shelters the central case from a violent impact.
This paper addresses the issue of destructive falls from collisions of aerial robots by taking an inspiration from the conglobating behavior of pill bugs [@smigel2008conglobation], Armadillidiidae and pill millipedes exhibit a defensive mechanism when triggered by stimuli by rolling into a ball. In conglobation, legs and sensitive ventral surfaces are wrapped and protected by the segmented dorsal exoskeleton. Herein, the observed defensive strategy is translated into a deformable multicopter in Fig. \[fig:robot\_prototype\] that folds the rigid airframe to safeguard central components when triggered by a collision. As a result, the delicate parts are shielded from the impact in the subsequent drop to the ground.
![Photo of a 51.2-gram foldable quadrotor fabricated by the origami-inspired method [@rus2018design; @zuliani2018minimally]. The fold is mechanically activated upon a mid-flight impact to a vertical surface.[]{data-label="fig:robot_prototype"}](Figs/robot_prototype.eps){width="8.4cm"}
Unlike the approach in [@mintchev2017insect],, which employs a synergic implementation of a dual-stiffness behavior and energy absorbing structures, we implement an origami-inspired method[@rus2018design; @zuliani2018minimally] to create a transformable airframe for collision tolerance. Upon a collision, our robot deforms by folding according to the predetermined features without the need of extra sensors or actuators. To achieve the desired mechanical intelligence, the foldable arms are built to be nominally rigid in the flight configuration. In addition, the interlocking mechanism for transitioning between the flight and folded states is purely mechanical. Both aspects potentially reduce the weight of the robot.
In the next section, we elaborate on the design principles and describe how the structural rigidity is attained from flat parts. Important flight components and the fabrication method are given. The section describes the fold kinematics and the mechanism for transitioning between the flying state and folded state. Section \[sec:benchtop\_experiments\] sets out the experimental characterization of the proposed robot and compares the results to the model predictions. Flight and mid-air collision are presented in Section \[sec:mid-flight\_collision\], followed by conclusion and discussion.
- no actual resilient part, uses passive method for energy storage. - no magnets for interlocking. - fold direction - “Stiffness is desirable in the design of foldable drones in order to prevent unwanted oscillations during flight” - Folding not only for deformation, but for energy absorbiona - “The magnets and the elastomer ring lock the frame in the deployed configuration by counteracting the forces and torques that are produced by the propellers during flight.”
Foldable Quadrotor Design and Model {#sec:design_and_model}
===================================
Overview
--------
![[]{data-label="fig:panel_folding_cad"}](Figs/panel_folding_cad.eps){width="8cm"}
![[]{data-label="fig:panel_folding_stages"}](Figs/panel_folding_stages.eps){width="8cm"}
In the mechanical design process, we aim to create an aerial robot that is mechanically resilience to withstand an impact. This must be achieved while preserving the structural rigidity and keeping the mass minimal as required for flight.
In the conglobation of pill bugs [@smigel2008conglobation], it can be seen that several rigid shells in multiple segments shift from an approximately planar alignment to form a circular arc. While each segment keeps its intrinsic stiffness, the deformation is globally achieved by small adjustments of the relative orientations. This is akin to how a large structural shift can be obtained from folding flat rigid structures. The observation provides a motivation to employ the origami-inspired design to realize the analogous robotic conglobation with aerial robots.
In contrast to the designs in [@sareh2018rotorigami; @zhao2018deformable], our foldable design is relatively simple and minimal. Each feature is clearly customized for the desired property. First, we take the minimum amount of folding joints to ensure the airframe is rigid in the desired directions when flying, yet foldable when triggered. Second, the design allows the folds to be mechanically activated upon a collision. In addition, all the folds are coupled as a one degree of freedom system. All arms fold in synchronization for the protection of the central body.
Fig \[fig:panel\_folding\_cad\] schematically demonstrates the folding mechanism of the proposed robot. The foldable airframe of the prototype consists of the ground tile, foldable arms, fold coupler, and fold triggers. In the flying state, the propeller’s thrust is nominally vertical. Each propeller generates an upward force, of which the resultant torque is countered and balanced by the joint limits, keeping the arms in the flight configuration. Upon a collision, the impact rotates the fold trigger to push part of the arms. .
Components and Fabrication
--------------------------
The origami-inspired quadrotor consists of standard electronic parts and the foldable airframe. We employ commercial off-the-shelf parts: 7$\times$20-mm brushed motors with a rated no-load speed of 54,000 RPM at 3.7V, 40-mm 3-blade propellers, and a single-cell 400-mAh Li-ion battery. Together, four propellers can generate over 0.65 N. A F3-EVO brushed flight controller is used for stabilization and receiving user’s commands through a mini radio receiver.
The origami-inspired airframe was manufactured from the planar fabrication paradigm [@rus2018design; @zuliani2018minimally]. Sheets of materials were cut and patterned using CO$_2$ laser (Epilog Mini 24). The laser-cut layer of polyimide film (25-$\mu$m Kapton, Dupont) was sandwiched between two structural layers (300-$\mu$m fiberglass). Parts and features were pin aligned and 170-$\mu$m double-sided pressure sensitive adhesives (300LSE, 3M) were used to compose the laminates as shown in Fig. \[fig:panel\_folding\_arm\]A. The laminates were released from the frame by the CO$_2$ laser in the final steps. The use of middle flexible layer enables us to create flexural (revolute) joints, while the structural rigidity is provided by the fiberglass. Folding the planar laminates gives rise to 3D assemblies.
The airframe was constructed from three laminates . The foldable airframe is then fitted with joint limits, motors, propellers, and flight electronics aided by 3D printed components (Black Resin, Form 2, Formlabs). In total, the robot’s mass is 51.2 g. The mass of the foldable airframe, including the mechanical ground, arms, and fold triggers, is 14.9 g.
Foldable Arms
-------------
![(A) (B) The conceptual fabrication and the joint kinematics of the foldable arm. (C) The added components on the airframe and the associated variables.[]{data-label="fig:panel_folding_arm"}](Figs/panel_folding_arm.eps){width="8cm"}
The airframe plays the most vital role in the folding mechanism. As in other flying robots, the flight condition requires the structure to be rigid in multiple directions. To satisfy this stringent restriction with the limitation of thin planar structures (which are relatively compliant in one direction as the stiffness is proportional to the cube of the thickness), we incorporate folds into the design.
As seen in Fig. \[fig:panel\_folding\_cad\], in the flying configuration, each foldable arm consists of pairs rigid tiles oriented approximately in perpendicular. In combination, this prevents each arm from bending up owing to the propeller’s thrust, or from deforming about the vertical axis due to the propeller’s yaw torque.
Each arm is composed of four rigid fiberglass tiles linked together by four flexural hinges as illustrated in Fig. \[fig:panel\_folding\_arm\]B. Tile is directly connected to the robot’s base and acts as a mechanical ground, whereas a motor is attached to the tip of tile . To achieve the desired kinematics, the exposed Kapton is folded radially (pleat fold) and adhered to the fiberglass. This permanently transforms the planar laminate into a three-dimensional component. The resultant arm forms a closed-link structure with four revolute joints, described by the angles $\theta_i$’s as indicated in Fig. \[fig:panel\_folding\_arm\]B (the rotation directions follow the right hand rule). The motion is constrained to one degree of freedom (DOF). Thanks to the symmetry, $\theta_2$ is always equal to $\theta_4$. The corresponding joint kinematics can be computed using the homogeneous transformation matrices. We define each angle to be zero when its two associated tiles are co-planar. Fig. \[fig:kinematics\]B shows how $\theta_1$ and $\theta_3$ are related to $\theta_2$ and $\theta_4$.
![Joint kinematics of the 1-DOF foldable airframe. Black lines represent the joint angles at various configurations. The red lines (associated with the red axis on the right) indicates the displacement of the fold trigger.[]{data-label="fig:kinematics"}](Figs/kinematics.eps){width="8cm"}
Folded and Flying States
------------------------
The most prominent characteristics of the foldable arm is that the primary fold angle $\theta_1$ has its minimum of $\approx10^\circ$ when $\theta_{2,4}=90^\circ$, or when tiles and are perpendicular. This condition is marked by the vertical dotted line in Fig. \[fig:kinematics\].
For flight, the motor and propeller are affixed to the far end of tile . The propeller’s thrust produces a positive torque about $\theta_1$. If $\theta_2 > 90^\circ$, this torque undesirably induces further positive rotation of both $\theta_1$ and $\theta_2$. To prevent an overrotation, we implement a joint limit to constrain the rotation of $\theta_2$ to the maximum of $\theta_{2,\text{max}}=110^\circ$ as illustrated in Fig. \[fig:panel\_folding\_arm\]C. This designates the equilibrium configuration of the robot in flight. With this design, we passively rely on the propeller’s thrust to keep the foldable arm extended when flying. In this configuration, tiles and , and, likewise, and , are steeply angled. Therefore, they provide the structural rigidity required for flight, impeding the arm from bending in two critical directions.
On the other hand, if we begin with $\theta_2 < 90^\circ$, the positive torque about $\theta_1$ from the thrust results in a negative torque on $\theta_2$. This unfolds $\theta_2$, or the joint between tiles and , and folds up $\theta_1$, rendering the arm to be in the folded state. Exploiting another surface of the joint limit located on tile , the upper limit of $\theta_1$ is physically confined to $70^\circ$. In this condition, the folded arm protects the flight controller or other components on the base from impact. The transition between the flight state and the folded state is achieved using the fold trigger as described later in Section \[sec:fold\_trigger\].
Fold Coupler
------------
![A schematic diagram demonstrating the kinematics of the fold coupler and the arm. For clarity, other arms are not shown.[]{data-label="fig:fold_coupler"}](Figs/fold_coupler.eps){width="6.0cm"}
To ensure all arms fold in synchronization when triggered, we incorporate a fold coupler into the airframe. The coupler is connected to tiles of all arms. In an ideal condition, this reduces the DOF of the whole airframe to one and the fold angles of all arms are always identical.
The coupling mechanism is implemented into the design of tile . As depicted in Fig. \[fig:fold\_coupler\], the inner edge of tile is tapered by having the top edge trimmed by $13^\circ$. In the folding motion, the trajectory of the top edge (highlighted in navy) forms a surface of an imaginary cone (dashed orange lines). The cone axis coincides with the joint axis of $\theta_4$. The apex of the cone is situated at the center of the airframe. From the top view, the top edge is always radially aligned towards the airframe’s center, with the exact position depending on $\theta_4$.
The coupler design leverages the resultant radial symmetry. The coupler is a symmetric tile with a vertical pin joint located at the center of the airframe. The coupler restricts the projected (top view) angles between the top edges of tiles from multiple arms to $90^\circ$. In the arm folding, all the top edges, and the coupler rotates together as seen from the top. Two additional flexural joints are implemented for each arm to satisfy the associated kinematic constraints in the three dimensional space.
Fold Trigger {#sec:fold_trigger}
------------
![(Left) A top view of the fold trigger and the robot’s arm when they collide with a vertical surface (a propeller not shown). The definitions of the yaw angle ($\psi$), the trigger joint ($\phi$), and the displacement ($x$) are given. (Right) An isometric view of the arm and the trigger.[]{data-label="fig:panel_fold_trigger"}](Figs/panel_fold_trigger.eps){width="8cm"}
The transition from the flight state to the folded state upon a collision is obtained via a fold trigger. Fig. \[fig:panel\_fold\_trigger\] presents the trigger as a mechanical extension of the arm from the motor mount and tile . The trigger, also fabricated by lamination, contains one flexural joint ($\phi$) and a tip that makes a contact against tile when $\phi>0^\circ$. In the nominal flight condition ($\theta_2=\theta_{2,\text{max}}=110^\circ$), $\phi=0^\circ$.
To describe a collision and the folding process, we define the yaw angle ($\psi$) of the robot to represent the heading, or the relative orientation between the robot and surface as shown in Fig. \[fig:panel\_fold\_trigger\]. Neglecting small pitch and roll rotations, $\psi=0$ corresponds to the scenario where the arm is perpendicular to the surface.
Upon impact, the surface causes a positive $\phi$ rotation. The linear displacement of the trigger in the direction perpendicular to the surface ($x$) is directly related to $\phi$. To simplify the analysis, we regard part of the trigger as a circular arc of radius $r=18$ mm as shown in Fig. \[fig:panel\_fold\_trigger\]. As a result, it can be shown that $${\label{eq:fold_trigger_displacement}}
x = r \left ( \sin(\psi)-\sin(\psi-\phi) \right ).$$ The rotation $\phi$ pushes the tip of the trigger against tile at the distance $h=5$ mm above the joint axis of $\theta_2$ . This loosely couples $\phi$ and $\theta_2$ such that, when the trigger’s tip and tile are in contact ($x>0$, $\phi>0^\circ$), their kinematics satisfy $${\label{eq:fold_trigger_arm_angle}}
l\sin(\phi) = h \left ( \cot(\theta_2)-\cot(\theta_{2,\text{max}}) \right ),$$ where $l=40$ mm is the distance between the contact point to the joint axis of $\phi$ as shown in Fig. \[fig:panel\_fold\_trigger\]. By combining equations and , we can get rid of $\phi$ and numerically obtain a direct relationship between $\theta_2$ and $x$. The result depends on the yaw angle $\psi$. The outcomes are presented in Fig. \[fig:kinematics\] for three representative yaw angles: $\psi=-30^\circ,0^\circ,30^\circ$. Owing to the symmetry of the robot, the range of $90^\circ$, i.e., $\psi\in(-45^\circ,45^\circ]$, covers all the possibilities.
![The ratio of balanced impact force to thrust ($F/T$) at different arm configurations ($\theta_2$, $\theta_4$) and yaw angels according to equation \[eq:trigger\_balanced\_torque\].[]{data-label="fig:force_ratio_plot"}](Figs/force_ratio_plot.eps){width="7.0cm"}
With all the kinematics determined, we proceed to evaluate the collision force required to activate the fold by equating the virtual work done by the propeller’s thrust against the force exerted by the surface on the trigger arm ($F$). Since all arms are coupled, the total thrust contributed by all propellers ($T$) must be considered. This thrust acts at the distance $d$ away from the axis of joint $\theta_1$ (with an offset angle $\gamma=10^\circ$, see Fig. \[fig:panel\_folding\_arm\]C). This yields $F\mathrm{d} x = -Td\cos\gamma\mathrm{d} \theta_1$ or $${\label{eq:trigger_balanced_torque}}
\frac{F}{T} = -d\cos\gamma \frac{\mathrm{d}\theta_1}{\mathrm{d}x},$$ where $d=40$ mm by design. From the joint kinematics used to produce Fig. \[fig:kinematics\] and the relationship between $\theta_2$, $x$, and $\psi$ obtained earlier, we numerically compute the force ratio ($F/T$) at various fold configurations and yaw angles. The results, representing the force required for activating the fold at different operating states, are shown in Fig. \[fig:force\_ratio\_plot\]. This reveals that the force ratio is an increasing function of $\theta_2$, irrespective of the yaw angle. The activation force $F$ required to initiate the fold when the robot is in the flight state ($\theta_2=110^\circ$) can be found at the upper limit of Fig \[fig:force\_ratio\_plot\] when $\theta_2=\theta_{2,\text{max}}$. As previously mentioned, without the pushing force from the trigger, the arm retains its torque equilibrium thanks to the counter torque provided by the joint limit.
In flight, upon the collision to the wall, the condition for the robot to fold depends on the total thrust. The magnitude of the corresponding impulsive force is determined by equation . In practice, the force is subject to the impact velocity, the surface properties (such as the coefficient of restitution), etc. Furthermore, we may calculate the amount of work required for activating the fold by integrating equation over $\mathrm{d}x$, starting from the hovering state ($\theta_2=110^\circ$) to the flipping point ($\theta_2=90^\circ$): $${\label{eq:trigger_energy}}
W = \int_{\theta_2=110^\circ}^{\theta_2=90^\circ}F\mathrm{d}x = \int_{\theta_2=110^\circ}^{\theta_2=90^\circ} -Td\cos\gamma \frac{\mathrm{d}}{\mathrm{d}x}\theta_1\mathrm{d}x.$$ This can be considered the minimum amount of kinetic energy needed in the collision for the robot to fold upon an inelastic collision. This number is independent of the yaw direction. In practice, however, with frictional or viscous losses, and structural compliance, we anticipate the robot would require more energy to overcome the folding process.
![Photo of the benchtop experimental setup showing the robot next to the artificial wall. The wall is fixed on a loadcell and a linear stage for force and distance measurements.[]{data-label="fig:benchtop_setup"}](Figs/bencthtop_setup.eps){width="8.0cm"}
Benchtop Force Measurements {#sec:benchtop_experiments}
===========================
In this section, we tested the fabricated robot on a benchtop platform to verify that i) all the arms fold as intended when one of the fold trigger is pushed; and ii) the pushing force and energy required for folding at different yaw angles follows the trend predicted by the models from Section \[sec:design\_and\_model\].
Experimental Setup
------------------
{width="17.0cm"}
![Example force measurements plotted against the location of the surface. Three representative data points (out of 68) from three yaw angles are shown. The dark solid lines are filtered measurements and the red dots are the maximum values.[]{data-label="fig:panel_raw_measurements"}](Figs/panel_raw_measurements.eps){width="7.0cm"}
![The maximum values of measured force from all 68 data points taken at different yaw angles (red dots) in the benchtop experiments, overlaid by the model prediction (black line).[]{data-label="fig:force_vs_yaw"}](Figs/force_vs_yaw.eps){width="6.4cm"}
![The empirical values of energy needed (work done) to fold the airframe in the benchtop experiments.[]{data-label="fig:work_vs_yaw"}](Figs/work_vs_yaw.eps){width="6.4cm"}
To measure the force, the robot is mounted on a 1-DOF rotational platform, enabling a quick and precise adjustment of the yaw angle as illustrated in Fig. \[fig:benchtop\_setup\]. We constructed a vertical rigid surface from an acrylic plate to simulate the collision to a wall. The acrylic plate is placed on top of a load cell (nano17, ATI) and mounted on a linear motorized stage (range 100 mm). The stage was driven by a microstepping driver (TB6600) to translate the surface towards the robot at 0.25 mm.s$^{-1}$ instead of moving the robot towards a static surface. As the force measurements were taken from the surface, not the robot, this reduces measurement noises caused by the vibration from the spinning propellers.
The signal generation for the microstepping driver and the data acquisition were carried out using a computer running the Simulink Real-Time system (Mathworks) with a DAQ (PCI-6229, National Instruments). The experiments were performed when the motors were supplied with 3.7V power. We used the same load cell on a similar setup and equipment to measure the total thrust generated by the robot with the same power supply in advance. The total thrust was found to be $T=0.52$ N, similar to the weight of the robot (0.50 N), or the expected thrust in stable flight.
We performed the force measurements at various yaw angles, ranging from $-35^\circ$ to $45^\circ$ at the increment of $5^\circ$. At each angle, four measurements were taken, making up 68 measurements in total. The missing angles ($-45^\circ$ and $-40^\circ$) are due to the actual geometry of the fold trigger. In this range, two arms are likely to contact the surface together (at slightly different time), complicating the measurement process.
{width="17.0cm"}
Measurement Results
-------------------
Fig. \[fig:panel\_benchtop\_fold\] shows an example of the folding process when the fold trigger was pushed by the moving surface as recorded by a camera at 240 Hz. The image sequence reveals all the arms started to fold at slightly different time and the whole fold completes in approximately $0.2$ s. The observed asynchronization is expected in practice due to the inherent structural compliance of parts and flexural hinges that were neglected in the analysis of the kinematics. Nevertheless, this verifies that the coupler functions as intended.
Three representative force measurements with respect to the wall position are plotted in Fig. \[fig:panel\_raw\_measurements\]. Raw force measurements are low-pass filtered (cut off frequency of 50 Hz) and the maximum filtered forces are marked by the red dots. The corresponding positions are labelled $x=0$ mm. It can be seen that, prior to this point, the force increases with $x$.
As the translating platform moves towards the robot, but no contact is made ($x$ is negative), the measured force is approximately zero. Upon contact, the surface pushes the fold trigger, producing the torque countering the torque from the thrust, resulting in non-zero force measurements. This incrementally and simultaneously replaces the counter torque contributed by the joint limit. The process occurs over a non-zero distance due to the unmodeled compliance of the airframe. At the maximum, the pushing force completely overcomes the thrust. It is reasonable to assume that the ratio of the maximum $F$ to $T$ corresponds to force ratio ($F/T$) when $\theta_2=\theta_{2,\text{max}}=110^\circ$ presented in Fig. \[fig:force\_ratio\_plot\].
The maximum measured forces from all experiments are plotted against the yaw angle in Fig. \[fig:force\_vs\_yaw\]. Also shown is the prediction from the model as given by equation , with $T=0.52$ N. The results show a reasonable agreement with the model for positive yaw angles. However, the measurements are up to $\approx30\%$ higher than the model prediction for some negative yaw angles. We believe the discrepancy is caused by several factors. One explanation is the simplification in the model that treats the geometry of the fold trigger as a circular arc. This could lead to an incorrect point of contact to the wall, contact angle, and different effective moment arms, all of which possibly contribute to the modeling errors. Another aspect is the inherent compliance of the structure and flexural joints, together with friction, structural vibration, and damping effects that are not considered in the model.
In addition to the force required to activate the fold, we compute the amount of energy or work exerted by the surface by numerically integrating the force over displacement (corresponding to the areas covered in Fig. \[fig:panel\_raw\_measurements\]) for all data points. The results are given in figure \[fig:work\_vs\_yaw\]. The plot reveals a similar trend to the measured maximum force in Fig. \[fig:force\_vs\_yaw\]. At most angles, the robot required $\approx 0.5-1$ mJ to fold, with the exception near $\psi\approx -45^\circ$. The obtained values are generally a few times larger than the theoretical prediction given by equation of $0.23$ mJ (when $T=0.52$ N). This is not surprising as the theoretical bound does not take into account frictional losses and compliance in the structure and flexural joints. It is likely that these unaccounted effects are more pronounced at negative yaw angles, resulting in large values of force and work as observed in Fig. \[fig:force\_vs\_yaw\] and \[fig:work\_vs\_yaw\].
In a flight scenario, if it is assumed that the kinetic energy is all taken up as work required for the robot to fold upon the collision to a wall, the amount 2 mJ (taken from Fig. \[fig:work\_vs\_yaw\]) equates to the impact speed of $\approx 30$ cm.s$^{-1}$ for the robot with 51.2g mass. It is conceivable that with other losses or if not all energy is converted, the minimum speed required for activating the fold could be higher.
Overall, the experiments verify that the proposed mechanism enables all four arms to fold when one arm is triggered. This is achieved over almost 360$^\circ$ of yaw angle in a static scenario (taking into account the symmetry of the robot). Moreover, the measurement results suggest that, depending on the yaw angle, the force required is in the same order of magnitude as the weight of the robot. In other words, no significant impact is needed for the fold activation.
Mid-Flight Collision Demonstration {#sec:mid-flight_collision}
==================================
With the battery, onboard electronics, and the commercial flight controler, the quadrotor is capable of stable flight. To demonstrate the fold activation from a mid-flight collision, the robot was remotely controlled to fly horizontally at the speed of $\approx$1.5 m.s$^{-1}$ (estimated from the video footage) towards a vertical surface (acrylic plate covered by paper for improved visibility). This speed is notably above the calculated bound of 30 cm.s$^{-1}$. Upon impact, the airframe completed the fold in less than $0.2$ s before crashing to the ground in the folded configuration. The flight was captured by a camera at 240 Hz. The video frames are shown in Fig. \[fig:panel\_flight\]. The foldable structure and onboard components were intact after the crash.
Conclusion
===========
This paper employs an origami-inspired strategy for a small multicopter to protect the delicate components residing in the center of the body from crashing to the ground after a collision. This is achieved by creating a foldable structure that is activated by impact, allowing the airframe to deform in a mid-flight collision such that the rigid structure shields the body part from the fall. The proposed design has been experimentally verified in both static measurements and actual flights.
.
In summary, the developed prototype makes use of an intelligent mechanical design to overcome the contradicting requirements on the structural stiffness. By aligning a pair of planar structures in a perpendicular direction, we obtain the desired rigidity. The interlocking mechanism, or joint limits, lets the robot use the thrust force to remain in the flight state. The fold is passively triggered as the impact force overcomes the thrust. The activation, as a result, does not necessitate an extra sensor or actuator. We believe the presented solution can be further extended for other applications, in particular, for small flying robots that severely suffer from restrictions in payload and power consumption.
[^1]: This work was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region of China (grant number CityU-21211315).
[^2]: J. Shu is with the Department of Mechanical Engineering and P. Chirarattananon is with the Department of Biomedical Engineering, City University of Hong Kong, Hong Kong SAR, China (email: pakpong.c@cityu.edu.hk).
|
---
abstract: 'The insulating state of magnetite (Fe$_{3}$O$_{4}$) can be disrupted by a sufficiently large dc electric field. Pulsed measurements are used to examine the kinetics of this transition. Histograms of the switching voltage show a transition width that broadens as temperature is decreased, consistent with trends seen in other systems involving “unpinning” in the presence of disorder. The switching distributions are also modified by an external magnetic field on a scale comparable to that required to reorient the magnetization.'
address:
- '$^{1}$Department of Chemistry, Rice University, 6100 Main St., Houston, TX 77005, USA'
- '$^{2}$CRANN, School of Physics, Trinity College, Dublin 2, Ireland'
- '$^{3}$Department of Physics and Astronomy, Rice University, 6100 Main St., Houston, TX 77005'
- '$^{4}$Department of Electrical and Computer Engineering, Rice University, 6100 Main St,.Houston, TX 77005'
author:
- 'A A Fursina$^{1}$[^1], R G S Sofin$^{2}$, I V Shvets$^{2}$, and D Natelson$^{3,4}$'
title: 'Statistical distribution of the electric field driven switching of the Verwey state in Fe$_{3}$O$_{4}$'
---
Introduction
============
Magnetite is an archetypal strongly correlated transition metal oxide, with properties not well described by single-particle band structure. Below 858 K, magnetite, which may be written as Fe$^{3+}_{\mathrm{A}}$(Fe$^{2+}$Fe$^{3+}$)$_{\mathrm{B}}$O$_{4}$, is ferrimagnetically ordered, with the A and B sublattices having oppositely directed magnetizations. The moments of the five unpaired $3d$ electrons of the tetrahedrally coordinated A-site Fe$^{3+}$ ions are compensated by those of the octahedrally coordinated B-site Fe$^{3+}$ ions. The net magnetization results from the octahedrally coordinated B-site Fe$^{2+}$ that have four unpaired $3d$ electrons [@McQueeney:2005]. Upon cooling, bulk magnetite undergoes a first-order phase transition from a moderately conducting high temperature state to a more insulating low temperature state at what is now called the Verwey[@Verwey:1939] temperature, $T_{\mathrm{V}}\approx 122$ K. The change in electronic properties is coincident with a structural transition from a high temperature cubic inverse spinel to a low temperature monoclinic unit cell. The nature of the ordered insulating state remains an active topic of current research[@Rozenberg:2006; @Piekarz:2006; @Schlappa:2008; @Subias:2009]. Experiments indicate the onset of multiferroicity[@Rado:1975] in magnetite below 40 K[@Alexe:2009], further highlighting the rich physics in this correlated system.
Recently, nanostructured electrodes have been used to apply strong electric fields in the plane of magnetite films[@Lee:2008; @Fursina:2008]. Below $T_{\mathrm{V}}$, a sufficient applied voltage triggers a breakdown of the comparatively insulating low-temperature state and a sudden increase in conduction [@Lee:2008; @Fursina:2008]. This is an example of electric field-driven breakdown of a gapped state in strongly correlated oxides[@Asamitsu:1997; @Oka:2005; @Sugimoto:2008] similar to Landau-Zener breakdown in classic semiconductors. The electric field-driven transition in magnetite is consistent with expectations[@Sugimoto:2008] based on such a mechanism (via geometric scaling [@Lee:2008; @Fursina:2008], lack of intrinsic hysteresis [@Fursina:2009], changes of both contact and bulk resistance at the transition [@Fursina:2010a; @Fursina:2010b]). These prior experiments examined films of various thicknesses, from 30 nm to 100 nm. No strong thickness dependence was observed in the switching properties, consistent with the applied lateral electric field at the sample surface acting as the driver of the breakdown (though thinner films showed a less pronounced Verwey transition in low-bias resistance vs. temperature measurements, consistent with expectations).
Here we report studies of the statistical variations of this electric field-driven transition in Fe$_{3}$O$_{4}$, as a function of temperature and magnetic field perpendicular to the film surface (out-of-plane). We find that there is a statistical distribution of switching voltages, $V_{\mathrm{SW}}$, that becomes more broad and shifts to higher voltages as $T$ is reduced. We discuss these trends in the context of switching kinetics in other systems that exhibit similar trends. The application of a magnetic field perpendicular to the plane of the Fe$_{3}$O$_{4}$ film alters the $V_{\mathrm{SW}}$, shifting the mean by several mV (several percent) and changing its shape, within a range of fields comparable to that required to reorient the magnetization out of plane.
Experimental Techniques
=======================
![(color online) Details of $V_{\mathrm{SW}}$ distribution experiment. (a and b) The fragments of $I$-$V$ curves in the vicinity of a transition demonstrating one (a) and three (b) switching events in a single pulsed $I$-$V$ cycle. (c) Typical SEM image of Ti/Au electrodes, patterned on magnetite film surface, separated by nanogap $<$ 100 nm. (d) An example of $V_{\mathrm{SW}}$ distribution histogram at 90 K.[]{data-label="fig1"}](Vsw_IV_single_multiple_v2_p_arxiv.eps){width="10cm"}
The 50 nm Fe$_{3}$O$_{4}$ (100) thin films used in the present study were grown on (100) oriented MgO single crystal substrates as described elsewhere [@Shvets_backscat; @Shvets_high_res_Xray]. Contact electrodes (2nm adhesion layer of Ti and 15 nm layer of Au) were patterned by e-beam lithography on the surface of the Fe$_{3}$O$_{4}$ film. As before [@Lee:2008; @Fursina:2009], $V_{\mathrm{SW}}$ scales linearly with the channel length, $L$ (the electrode spacing), implying an electric field-driven transition. Long channels ($L >$ 100 nm) required large switching voltages that would alter the electrode geometry over numerous switching cycles, distorting the shape of $V_{\mathrm{SW}}$ histograms. To minimize $V_{\mathrm{SW}}$, electrodes separated by 10-30 nm were patterned using a self-aligned technique [@Fursina:2008]. Electrical characterization of the devices was performed using a semiconductor parameter analyzer (HP 4155A). To minimize self-heating when in the conducting state, the voltage was applied as pulses 500 $\mu$s in duration with a 5 ms period [@Fursina:2009; @Fursina:2010a]. The samples were cooled below $T_{\mathrm{V}}$ with no magnetic field applied, and the distribution of $V_{\mathrm{SW}}$ was obtained by executing several thousand consecutive forward pulsed $I$-$V$ sweeps in the vicinity of the transition point (typically a 0.2-0.3 V range) at a fixed (to within 50 mK) temperature, and recording the number of switching events at each voltage.
Each voltage value is essentially an independent test to see if switching takes place under the pulse conditions. Hence, some sweeps show one (figure \[fig1\] a) or several (figure \[fig1\] b) switching events. Even if the system is switched to the conducting state at $V_{\mathrm{SW}}$(1), it may return to the Off state between pulses, and then switch to the On state at some higher voltage, $V_{\mathrm{SW}}$(2), and so on. The $V_{\mathrm{SW}}$ distribution at a particular temperature is built by recording all switching events over several thousands (3000-6000) of $I$-$V$ cycles and then counting the number of switchings at a certain $V_{\mathrm{SW}}$, to produce a “\# of counts” vs. $V_{\mathrm{SW}}$ histogram. A typical $V_{\mathrm{SW}}$ distribution at 90K is shown in figure \[fig1\] d. The distribution is a single peak, symmetrical around the most probable $V_{\mathrm{SW}}$ value.
Results and Discussion
======================
This procedure was repeated at each temperature below $T_{\mathrm{V}}$ ($\sim$ 110K for devices under test; see figure \[fig2\]b inset), down to $\sim$ 75 K. At $T < 75$ K, the high values of $V_{\mathrm{SW}}$ necessary led to irreversible alteration of the electrode geometry, resulting in asymmetric, distorted $V_{\mathrm{SW}}$ histograms. Near 80-90 K, $I$-$V$ cycles with multiple $V_{\mathrm{SW}}$ events (figure \[fig1\]b) were observed more frequently. Thus, the total number of switching events observed varied with $T$, even with a fixed number of $I$-$V$ cycles at each temperature. To compare $V_{\mathrm{SW}}$ distributions at different temperatures, the distributions were normalized, plotted as (\# of counts)/(max \# of counts) vs $V_{\mathrm{SW}}$, where “max \# of counts” is the number of events at the most probable $V_{\mathrm{SW}}$ and “\# of counts” is the number of events at a certain $V_{\mathrm{SW}}$. Figure \[fig2\]a is an example of normalized $V_{\mathrm{SW}}$ distributions in the 77 K-105 K temperature range. The measured widths of the $V_{\mathrm{SW}}$ distributions are not limited by temperature stability.
As has been discussed elsewhere[@Fursina:2009], the use of pulses is essential to minimize the role of self-heating once the system has been driven into the more conducting state. This self-heating and the short timescale[@Fursina:2009] required to raise the local temperature in the channel significantly makes it extremely challenging to determine directly whether the initial breakdown takesplace through the formation of a conducting filament or through a uniform switching; once a highly conducting path is formed, the whole channel rapidly becomes conducting through self-heating. The filamentary picture is certainly likely, based on other breakdown phenomena in solids, and the statistical variation in $V_{\mathrm{SW}}$ is consistent with the idea of a process involving run-to-run variability associated with *local* details rather than global material properties, but this is not definitive.
![(color online) (a) Normalized $V_{\mathrm{SW}}$ distributions at different temperatures (77 K - 105 K). (b) Temperature dependence of the mean switching voltage, $\bar{V}_{SW}$. Inset shows zero-bias $R$ vs $T$ plot demonstrating $T_{\mathrm{V}} \sim$ 110 K.(c) Temperature dependence of $V_{\mathrm{SW}}$ distribution width, $\sigma(V_{\mathrm{SW}})$ (black circles). []{data-label="fig2"}](Vsw_hist_3panels_p_no_P_arxiv.eps){width="10cm"}
The $V_{\mathrm{SW}}$ distribution at each temperature is characterized by two main parameters: the mean switching value $\bar{V}_{\mathrm{SW}}=(\sum_{i=1}^N V_{{\mathrm{SW}},i})/N$, where $N$ is the total number of switching events; and the width of distribution, calculated as a standard deviation: $\sigma(V_{\mathrm{SW}})=\sqrt{(\sum_{i=1}^N (V_{{\mathrm{SW}},i}-\bar{V}_{sw})^2)/{N-1}}$. As expected, the $\bar{V}_{\mathrm{SW}}(T)$ has the same $T$-dependence (see figure \[fig2\] b) as $V_{\mathrm{SW}}(T)$ in single $I$-$V$ experiments described in previous publications [@Lee:2008; @Fursina:2009]. More interesting is the $\sigma(V_{\mathrm{SW}})$ temperature dependence, showing broadening of the $V_{\mathrm{SW}}$ distribution as the temperature decreases (figure \[fig2\] c). We note a deviation from monotonous temperature dependence of $\sigma(V_{\mathrm{SW}})$ at 100 K, observed in several devices tested. This is a temperature well below $T_{\mathrm{V}} = 110~K$ (see figure \[fig2\]b inset), where several physical parameters (resistance, heat capacity and magnetoresistance) change abruptly.
This increase in $\sigma(V_{\mathrm{SW}})$ as temperature decreases is rather counter-intuitive. One might expect “freezing” of temperature fluctuations and decrease in thermal noise the temperature decreases and, thus, narrowing of $V_{\mathrm{SW}}$ distributions. The field-driven breakdown of the insulating state is an example of the “escape-over-barrier” problem, addressed generally by A. Garg [@Garg:1995]. Below $T_{\mathrm{V}}$, the (temperature-dependent) effective free energy of the electronic system is at a global minimum value in the insulating state, while the external electric field modifies the free energy landscape, lowering the free energy of another local minimum corresponding to the more conducting state. As the external field is increased beyond some critical value, the minimum corresponding to the more conducting state becomes the global minimum. The nonequilibrium transition to the conducting state then corresponds to some process that crosses the free energy barrier between these minima. At a sufficiently large value of the external field, the free energy has only one minimum, corresponding to the conducting state.
As Garg showed, one may consider thermal activation over the free energy barrier as well as the possibility of quantum escape. This free energy picture predicts a broadening of the transition driving force ($V_{\mathrm{SW}}$ here) distribution as the absolute value of the driving force increases, consistent with our observations. This free energy picture has proven useful in studying other nonequilibrium transitions, such as magnetization reversal in nanoparticles [@Wernsdorfer:1997] and nanowires [@Varga:2003; @Varga:2004]. Pinning due to local disorder is one way to find increasing distribution widths as $T \rightarrow 0$, as seen in investigations of field-driven magnetization reversal in nanowires [@Varga:2003; @Varga:2004]. Unfortunately, quantitative modeling in this framework requires several free parameters and is difficult without a detailed understanding of the underlying mechanism.
Qualitatively similar phenomenology (distribution of switching thresholds that broadens as $T$ is decreased) is also observed in the current-driven superconducting-normal transition in ultrathin nanowires[@Sahu:2009; @Pekker:2009]. In this latter case as in ours, self-heating in the switched state is of critical importance, as is the temperature variation of the local thermal path. Again, quantitative modeling using this self-heating approach would require the introduction of multiple parameters that are difficult to constrain experimentally, as well as detailed thermal modeling of the nanoscale local effective temperature distribution, and is beyond the scope of this paper.
We also examined the dependence of the switching distributions on applied out-of-plane magnetic field, $H$. $V_{\mathrm{SW}}$ distributions (3000 cycles each) were collected consecutively at 12 magnetic field values: 0 T (first) $\rightarrow$ 0.2 T $\rightarrow$ 0.4 T $\rightarrow$ 0.6 T $\rightarrow$ 0.8 T $\rightarrow$ 1 T $\rightarrow$ 2 T $\rightarrow$ 3 T $\rightarrow$ 4 T $\rightarrow$ 5 T $\rightarrow$ 6 T $\rightarrow$ 0 T(last). Figure \[Vsw\_H\_dep\] a shows the resultant $V_{\mathrm{SW}}$ distributions at 80 K at several selected magnetic fields. As can be seen, magnetic field shifts the $V_{\mathrm{SW}}$ peak to higher $V$ values and narrows the $V_{\mathrm{SW}}$ distributions. To reassure that the observed $\bar{V}_{\mathrm{SW}}$ shift is not from irreversible changes in the device, a control experiment returning to $H=0$ T was performed after experiments in all non-zero magnetic fields. The $V_{\mathrm{SW}}$ distributions at $H=0$ T initially \[$H=0$ T (first)\] and in the end \[$H=0$ T (last)\] are identical (see figure \[Vsw\_H\_dep\] a), meaning that observed changes in $V_{\mathrm{SW}}$ distributions (shift and narrowing) are indeed caused by the applied magnetic field. Figure \[Vsw\_H\_dep\] b quantifies the dependence of $\bar{V}_{\mathrm{SW}}$ and $\sigma(V_{\mathrm{SW}})$. It is clear that both parameters saturate as $H$ is increased beyond 1 T, [*i.e.*]{}; further increases of $H$ up to 6 T have no significant effect. Magnetic field of the opposite polarity (not shown) has exactly the same effect on the position and the width of $V_{\mathrm{SW}}$ distributions. Note that the shape of the distribution, in particular its asymmetry about the peak value of $V_{\mathrm{SW}}$, evolves nontrivially with magnetic field, becoming more symmetric in the high field limit.
![(a) Examples of $V_{\mathrm{SW}}$ distributions at selected magnetic fields (T=80 K).(b) Magnetic field dependence of the mean switching value, $\bar{V}_{\mathrm{SW}}$ (red squares), and the width of $V_{\mathrm{SW}}$ distributions, $\sigma(V_{\mathrm{SW}})$ (black circles).[]{data-label="Vsw_H_dep"}](Vsw_H_dependence_p_arxiv.eps){width="8.5cm"}
![Dependences of the resistance ($R/R(H=0T)$) (a) and the magnetization ($M/M_{\mathrm{s}}$) (b) on the out-of-plane magnetic field applied.[]{data-label="R_M_vs_H"}](R_and_M_vs_H_p_arxiv.eps){width="8.0cm"}
We consider whether this magnetic field dependence of the switching originates with some dependence of the bulk resistance or contact resistances, as this would alter the electric field distribution in the channel. Since $V_{\mathrm{SW}}$ scales with the channel length, $L$ [@Lee:2008; @Fursina:2009] [*i.e.*]{}, as does the resistance of the channel ($R \sim L$), one might expect an *increase* in $V_{\mathrm{SW}}$ could result from an increase in the device resistance with applied magnetic field. However, Fe$_3$O$_4$ has a negative magnetoresistance (MR) [@Sofin:2005; @deTeresa:2007; @Eerenstein:2002]. Figure \[R\_M\_vs\_H\] a shows an example of normalized resistance dependence, $R/R(H=0T)$, on the out-of-plane magnetic field at 80 K. The resistance remains effectively unchanged up to $\sim$0.4 T and then decreases as $|H|$ increases. Thus, when $\bar{V}_{\mathrm{SW}}$ and $\sigma(V_{\mathrm{SW}})$ experience the predominance of their changes upon $H$ application ($H<1T$, see figure \[Vsw\_H\_dep\]), the resistance of the device either stays constant or decreases. In the $H$ range when $R$ experiences significant changes (see figure \[R\_M\_vs\_H\] a), $\bar{V}_{\mathrm{SW}}$ and $\sigma(V_{\mathrm{SW}})$ remain essentially unchanged (figure \[Vsw\_H\_dep\]). Therefore, the shift of $\bar{V}_{\mathrm{SW}}$ in the presence of $H$ does not originate from the change in the resistance value of the device.
Another Fe$_3$O$_4$ film parameter effected by $H$ is the magnetization of the film. Figure \[R\_M\_vs\_H\] shows the normalized out-of-plane magnetization, $M/M_{\mathrm{s}}$, as a function of the out of plane $H$, where $M$ is the magnetization of the film and $M_{\mathrm{s}}$ is the saturated magnetization. This data is consistent with prior measurements on magnetite films[@Zhou:2004] While we do not know the microscopic arrangement of $\mathbf{M}$ in the film in the absence of an external $\mathbf{H}$, magnetostatic energy considerations mean that $\mathbf{M}$ under that condition lies in the plane of the film. The $H$ range over which $M$ is fully reoriented out of the plane (up to 1 T) matches the $H$ range of changes in the position of $\bar{V}_{\mathrm{SW}}$ and $\sigma(V_{\mathrm{SW}},T)$ (fig. \[Vsw\_H\_dep\] b). This suggests (though does not prove) that the switching kinetics parameters $\bar{V}_{\mathrm{SW}}$ and $\sigma(V_{\mathrm{SW}},T)$, and therefore the stability of the gapped, low temperature, insulating state is tied the magnetization direction of magnetite films.
This observation is intriguing because it is not clear how the nonequilibrium breakdown of the low temperature state would be coupled to the magnetization. Possible factors include magnetoelastic effects such as magnetostriction[@Tsuya:1977] ($\sim$ parts in $10^4$ per Tesla) affecting the tunneling matrix element between $B$-site iron atoms; and spin-orbit coupling playing a similar role[@Yamauchi:2010]. There have been reports of significant magnetoelectric and multiferroic effects in magnetite [@Rado:1975; @Alexe:2009], and a recent calculation [@Yamauchi:2010] argues that these originate through the interplay of orbital ordering and on-site spin-orbit interactions of the B-site electrons. In this picture, reorientation of the spin distorts the partially filled minority-spin orbitals occupied on the B-site (formally) Fe$^{2+}$ ions. Such a distortion would be a natural explanation for the observed correlation between $\mathbf{M}$ and the kinetics of the electric field-driven breakdown of the ordered state, which directly involves the motions of those charge carriers. It is unclear how this kind of spin-orbit physics would explain the evolution of the $V_{\mathrm{SW}}$ distribution, however. It would also be worth considering whether there is any correlation between the characteristics of the switching distributions reported here, and the recently observed glassy relaxor ferroelectric relaxations in bulk magnetite crystals[@Schrettle:2011].
Additional, detailed experiments as a function of directionality of $\mathbf{H}$, $\mathbf{M}$, and crystallographic orientation should be able to test these alternatives. With the existing (100) films, studies of $\bar{V}_{\mathrm{SW}}$ and $\sigma(V_{\mathrm{SW}},T)$ as a function of $H$ in the plane as well as perpendicular to the plane should be able to access the tensorial form of the $H$ dependence. Comparison with appropriately directed $M$ vs. $H$ data as a function of temperature would be a clear test of whether the observed agreement between $H$-field scales (in $V_{\mathrm{SW}}$ and reorientation of $\mathbf{M}$ is coincidental. Further measurements on films grown with different crystallographic orientations would serve as a cross-check. It is important to note, however, that the acquisition of such data is very time intensive due to the need to acquire many thousands of switching events. In turn, there is a companion requirement of extremely good device stability, to avoid irreversible changes in the metal configuration over the thousands of switching cycles.
Conclusions
===========
We have studied the statistical distribution of the electric field needed for breakdown of the low temperature state of Fe$_{3}$O$_{4}$. The distribution of critical switching voltages moves to higher voltages and broadens, as $T$ is reduced. This broadening is consistent with phenomenology in other nonequilibrium experimental systems incorporating disorder and thermal runaway effects. The breakdown distributions are altered by modest external magnetic fields normal to the film, suggesting a need for further experiments to understand the connection between magnetization and breakdown of the correlated state.
The authors acknowledge valuable conversations with Paul Goldbart and David Pekker. This work was supported by the US Department of Energy grant DE-FG02-06ER46337. DN also acknowledges the David and Lucille Packard Foundation and the Research Corporation. RGSS and IVS acknowledge the Science Foundation of Ireland grant 06/IN.1/I91.
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[^1]: Present address: Department of Chemistry, University of Nebraska-Lincoln, 525 HAH, Lincoln, NE 68588-0304, USA
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abstract: 'In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.'
address:
- |
Yau Mathematical Sciences Center\
Tsinghua University\
Beijing, P. R. China
- |
Mathematical Institute\
University of Oxford\
Andrew Wiles Building\
Radcliffe Observatory Quarter\
Woodstock Road\
Oxford\
OX2 6GG
author:
- 'Cindy (Sin Yi) Tsang'
- Stanley Yao Xiao
title: |
Binary quartic forms with bounded invariants\
and small Galois groups
---
Introduction {#Intro}
============
The problem of enumerating ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence classes of integral and irreducible binary forms of a fixed degree has a long history. The quadratic and cubic cases were solved in [@Gau; @Sieg] and [@Dav1; @Dav2], respectively, where the forms are ordered by the natural height, namely the discriminant $\Delta(-)$. The quartic case turns out to be much more challenging because while the ring of polynomial invariants for both binary quadratic and cubic forms is generated by $\Delta(-)$ as an algebra, that for binary quartic forms is generated by two independent invariants, usually denoted by $I(-)$ and $J(-)$. For $$\label{F generic}F(x,y) = a_4 x^4 + a_3 x^3y + a_2 x^2y^2 + a_1 xy^3 + a_0 y^4,$$ they are given by the explicit formulae $$\begin{aligned}
I(F)& = 12a_4 a_0 - 3a_3 a_1 + a_2^2,\\
J(F)& = 72a_4 a_2 a_0 + 9a_3 a_2 a_1 - 27a_4 a_1^2 - 27a_3^2 a_0 - 2a_2^3,\end{aligned}$$ which are of degrees two and three, respectively. In [@BhaSha], instead of using the discriminant, Bhargava and Shankar introduced the height function $$\label{BS height} H_{\mathrm{\tiny BS}}(F) = \max\{|I(F)|^3, J(F)^2/4\}.$$ For $X>0$, let us define $$\begin{aligned}
N_{{{\mathbb{Z}}}}(X) &= \#\{[F]:\mbox{integral and irreducible binary}\\ &\hspace{3cm}\mbox{quartic forms $F$ such that $H_{\mathrm{\tiny BS}}(F)\leq X$}\},\end{aligned}$$ where $[-]$ denotes ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence class. In [@BhaSha], they proved that $$\label{BS formula}N_{{\mathbb{Z}}}(X) = \frac{44 \zeta(2)}{135} X^{5/6} + O_\epsilon \left(X^{3/4 + \epsilon} \right)\mbox{ for any $\epsilon>0$}.$$ This is the first result ever obtained, and as far as we know, the only known result in the literature, for the quartic case.
Set-up and notation {#notation sec}
-------------------
In this paper, we shall also be interested in the quartic case, but only the integral and irreducible binary quartic forms $F$ with *small* Galois group ${\operatorname{Gal}}(F)$, which is defined to be the Galois group of the splitting field of $F(x,1)$ over ${{\mathbb{Q}}}$. We know that ${\operatorname{Gal}}(F)$ is isomorphic to one of the following: $$\begin{aligned}
S_4 & = \mbox{the symmetric group on four letters},\\
A_4 & = \mbox{the alternating group on four letters},\\
D_4 & = \mbox{the dihedral group of order eight},\\
C_4 & = \mbox{the cyclic group of order four},\\
V_4 & = \mbox{the Klein-four group}.\end{aligned}$$ We shall say that ${\operatorname{Gal}}(F)$ is *small* if it is isomorphic to $D_4,C_4$, or $V_4$. Recall that the *cubic resolvent of $F$* is defined by $${{\mathcal{Q}}}_F(x) = x^3 - 3I(F)x + J(F).$$ Then, equivalently, we have the classical characterization that for irreducible $F$ $${\operatorname{Gal}}(F)\mbox{ is small if and only if ${{\mathcal{Q}}}_F(x)$ is reducible}.$$ It turns out that whether ${\operatorname{Gal}}(F)$ is small or not may also be characterized in terms of binary quadratic forms and the following so-called *twisted action* of ${\operatorname{GL}}_2({{\mathbb{R}}})$.\
Given a complex binary form $\xi(x,y)$, let ${\operatorname{GL}}_2({{\mathbb{R}}})$ act on it via $$\xi_T(x,y) = \frac{1}{\det(T)^{\deg\xi/2}}\xi(t_1x+t_2y, t_3x+ t_4y) \mbox{ for }T= \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4\end{pmatrix}.$$ Observe that this is only an action up to sign when $\deg \xi$ is odd, in the sense that for $T_1,T_2\in{\operatorname{GL}}_2({{\mathbb{R}}})$, we only have $\xi_{T_1T_2} = \pm (\xi_{T_1})_{T_2}$ in general. Now, given a real binary quadratic form $f(x,y) = \alpha x^2 + \beta xy + \gamma y^2$ with $\Delta(f)\neq0$, write $$M_f = \begin{pmatrix} \beta & 2\gamma \\ -2\alpha & -\beta \end{pmatrix}$$ for its associated matrix in ${\operatorname{GL}}_2({{\mathbb{R}}})$. Its action on binary quartic forms clearly remain unchanged if we scale $f(x,y)$ by a constant in ${{\mathbb{R}}}^\times$. In [@X], the second-named author proved that for any real binary quartic form $F$ with $\Delta(F)\neq0$, elements of $$\{T\in{\operatorname{GL}}_2({{\mathbb{R}}}) : T\mbox{ is not a scalar multiple of }I_{2\times 2}{\text{ and }}F_T = F\}$$ all arise from binary quadratic forms in this way; see Proposition \[auto theorem\]. Recall that an integral binary quadratic form is called *primitive* if its coefficients are coprime. Using this result from [@X], in Section \[Galois gp sec\], we shall first show that:
\[small char thm\]Let $F$ be an integral binary quartic form with $\Delta(F)\neq0$. Then, the following are equivalent.
(1) ${{\mathcal{Q}}}_F(x)$ is reducible.
(2) $F_T = F$ for some $T\in{\operatorname{GL}}_2({{\mathbb{Q}}})$ which is not a scalar multiple of $I_{2\times 2}$.
(3) $F_{M_f} = F$ for an integral and primitive binary quadratic form $f$ with $\Delta(f)\neq0$.
Moreover, in the case that ${{\mathcal{Q}}}_F(x)$ is reducible:
(a) If $\Delta(F)\neq\square$, then there is a unique such $f$ up to sign.
(b) If $\Delta(F)=\square$, then there are exactly three such $f$ up to sign, among which one is definite and two are indefinite.
Given a real binary quadratic form $f(x,y)$ with $\Delta(f)\neq0$, let us further make the following definitions. First put $$\begin{aligned}
\label{VRf}
V_{{{\mathbb{R}}},f} &= \{\mbox{real binary quartic forms $F$ such that $F_{M_f} = F$}\},\\
V_{{{\mathbb{Z}}},f}& = \{\mbox{integral binary quartic forms $F$ such that $F_{M_f} = F$}\}. \notag\end{aligned}$$ Clearly $V_{{{\mathbb{R}}},f}$ is a vector space over ${{\mathbb{R}}}$ and $V_{{{\mathbb{Z}}},f}$ a lattice over ${{\mathbb{Z}}}$. A straightforward calculation shows that $\dim_{{{\mathbb{R}}}} V_{{{\mathbb{R}}},f}$ is three; see (\[abc family\]) and (\[abc family 2\]) below. Also, put $$V_{{{\mathbb{R}}},f}^0 = \{F\in V_{{{\mathbb{R}}},f}:\Delta(F)\neq0\}{\text{ and }}V_{{{\mathbb{Z}}},f}^0 = \{F\in V_{{{\mathbb{Z}}},f}:\Delta(F)\neq0\}.$$ For $F\in V_{{{\mathbb{R}}},f}^0$, we shall define two new invariants as follows. As we shall see in (\[in V char\]), there is a unique root $\omega_f(F)$ of ${{\mathcal{Q}}}_F(x)$ corresponding to $f$. Let $\omega'_f(F),\omega''_f(F)$ denote the other two roots of ${{\mathcal{Q}}}_F(x)$ and define $$\label{LK def}L_f(F) = \omega_f(F) {\text{ and }}K_f(F) = -\omega'_f(F)\omega''_f(F).$$ By Proposition \[explicit LK\] below, they have degrees one and two, respectively, in the coefficients of $F$. Following (\[BS height\]), let us define the *height of $F$ associated to $f$* by $$H_f(F) = \max\{L_f(F)^2, |K_f(F)|\}.$$ This is comparable to the height (\[BS height\]) because by comparing coefficients in $$x^3 - I(F)x + J(F) = (x-\omega_f(F))(x-\omega'_f(F))(x-\omega''_f(F)),$$ we easily deduce the relations $$\label{IJ family}
3I(F) = L_f(F)^2 + K_f(F) {\text{ and }}J(F) = L_f(F) K_f(F),$$ which in turn imply that $$\label{H compare} (H_f(F)/10)^3 \leq H_{\text{\tiny BS}}(F) \leq H_f(F)^3.$$ Let us note that $$\label{Delta LK}
\Delta(F) = \frac{4I(F)^3 - J(F)^2}{27} = \left(\frac{L_f(F)^2 + 4K_f(F)}{9}\right)\left(\frac{2L_f(F)^2 - K_f(F)}{9}\right)^2,$$ where the first equality is well-known, and the second equality holds by (\[IJ family\]). Also, our height $H_f(-)$ is an invariant in the sense that for any $T\in{\operatorname{GL}}_2({{\mathbb{R}}})$, we have $$H_{f_T}(F_T) = H_f(F),$$ as shown in Proposition \[LK invariant\] below. This implies that the map $$\label{V bijection} V_{{{\mathbb{R}}},f} \longrightarrow V_{{{\mathbb{R}}},f_T}; \hspace{1em}F\mapsto F_T,$$ which is a well-defined bijection because $M_{f_T} = T^{-1}M_fT$, is height-preserving when restricted to the forms of non-zero discriminant.\
Now, let us return to the integral and irreducible binary quartic forms with small Galois group. Write $V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}}$ for the set of all such forms and set $$V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},\dagger} = \{F\in V_{{\mathbb{Z}}}^{{\mathrm{\tiny sm}}}:{\operatorname{Gal}}(F)\not\simeq V_4\}.$$ By Theorem \[small char thm\], we know that $$\begin{aligned}
\label{union}V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}} & = \bigcup_{f\in{\mathfrak{F}}^*} \{F\in V_{{{\mathbb{Z}}},f}^0:F\mbox{ is irreducible}\}, \\\notag
V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},\dagger} & = \bigsqcup_{f\in{\mathfrak{F}}^*} \{F\in V_{{{\mathbb{Z}}},f}^0:F\mbox{ is irreducible and }{\operatorname{Gal}}(F)\not\simeq V_4\}, \end{aligned}$$ where ${\mathfrak{F}}^*$ denotes the set of all integral and primitive binary quadratic forms of non-zero discriminant, up to sign. In particular, given $F\in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},\dagger}$, there is a unique $f\in{\mathfrak{F}}^*$ such that $F\in V_{{{\mathbb{Z}}},f}^0$, and we may define the *height of $F$* by setting $$H(F) = H_f(F).$$ For $X>0$, let us define $$\begin{aligned}
N_{{\mathbb{Z}}}^\dagger(X) &= \#\{[F] : F\in V_{{\mathbb{Z}}}^{{\mathrm{\tiny sm}},\dagger}\mbox{ such that }H(F)\leq X\},\\
N_{{{\mathbb{Z}}},f}^\dagger(X) &= \#\{[F] : F\in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},\dagger}\cap V_{{{\mathbb{Z}}},f}^0\mbox{ such that } H(F)\leq X\}.\end{aligned}$$ Then, by (\[V bijection\]) and (\[union\]), we have $$N_{{\mathbb{Z}}}^\dagger(X) = \sum_{f\in{\mathfrak{F}}} N_{{{\mathbb{Z}}},f}^\dagger(X),$$ where ${\mathfrak{F}}$ denotes a set of representatives of the ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence classes on ${\mathfrak{F}}^*$. In Theorem \[Small Gal MT\], which is our main result, for $f\in{\mathfrak{F}}^*$, we shall determine the asymptotic formula for $N_{{{\mathbb{Z}}},f}^\dagger(X)$. In fact, we shall consider the finer counts $$\begin{aligned}
N_{{{\mathbb{Z}}},f}^{(D_4)}(X) = \# \{[F] : F \in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}} \cap V_{{{\mathbb{Z}}},f}^0 \mbox{ such that } {\operatorname{Gal}}(F) \simeq D_4 {\text{ and }}H(F) \leq X\},\\
N_{{{\mathbb{Z}}},f}^{(C_4)}(X) = \# \{[F] : F \in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}} \cap V_{{{\mathbb{Z}}},f}^0 \mbox{ such that } {\operatorname{Gal}}(F) \simeq C_4 {\text{ and }}H(F) \leq X\},\\
N_{{{\mathbb{Z}}},f}^{(V_4)}(X) = \# \{[F] : F \in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}} \cap V_{{{\mathbb{Z}}},f}^0 \mbox{ such that } {\operatorname{Gal}}(F) \simeq V_4 {\text{ and }}H_f(F) \leq X\},\end{aligned}$$ and show that the latter two are negligible compared to $N_{{{\mathbb{Z}}},f}^{(D_4)}(X)$. This means that most of the forms in $V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}}\cap V_{{{\mathbb{Z}}},f}^0$ have Galois group isomorphic to $D_4$. However, all of our error estimates depend upon $f$. Currently, we do not know how to control them in a uniform way, and so we are unable to obtain an asymptotic formula for $N_{{{\mathbb{Z}}}}^\dagger(X)$ by summing over $f\in{\mathfrak{F}}$.\
Finally, let us explain, for each $f\in{\mathfrak{F}}^*$, how counting forms in $V_{{\mathbb{Z}}}^{{\mathrm{\tiny sm}}}\cap V_{{{\mathbb{Z}}},f}^0$ may be reduced to counting lattice points. Write $f(x,y) = \alpha x^2 + \beta xy + \gamma y^2$ with $\alpha,\beta,\gamma\in{{\mathbb{Z}}}$. By (\[abc family\]) and (\[abc family 2\]), the set $V_{{{\mathbb{R}}},f}$ is a vector space isomorphic to ${{\mathbb{R}}}^3$ via $$\begin{aligned}
\Theta_1: a_4x^4 + a_3x^3y + a_2x^2y^2 + a_1xy^3 + a_0y^4&\mapsto (a_4,a_3,a_2)\hspace{1em}\mbox{if $\alpha\neq0$},\\\label{iso2}
\Theta_2: a_4x^4 + a_3x^3y + a_2x^2y^2 + a_1xy^3 + a_0y^4&\mapsto (a_4,a_2,a_0)\hspace{1em}\mbox{if $\beta,\beta^2+4\alpha\gamma\neq0$}.\end{aligned}$$ Recall that the subset $V_{{{\mathbb{Z}}},f}$ has the structure of a rank-three ${{\mathbb{Z}}}$-lattice, which may be identified with the lattices $$\label{Lambda def}\Lambda_{f,1} = \Theta_1(V_{{{\mathbb{Z}}},f}){\text{ and }}\Lambda_{f,2}=\Theta_2(V_{{{\mathbb{Z}}},f})$$ in ${{\mathbb{Z}}}^3$. Let us mention here that we shall use the isomorphism $$\Theta_{w(f)},\mbox{ where }w(f)=\begin{cases}
1 & \mbox{if $f$ is irreducible},\\
2 &\mbox{if $f$ is reducible}.
\end{cases}$$ Thus, the problem is reduced to counting points in $\Lambda_{f,1}$ or $\Lambda_{f,2}$, and then sieving out those which come from reducible forms. In turn, counting lattice points amounts to computing certain volumes by a result of Davenport [@Dav]; see Proposition \[Davenport\].
Statement of the main theorem
-----------------------------
It is clear that we may choose the set ${\mathfrak{F}}$ of representatives to be such that for all $f\in{\mathfrak{F}}$, the $x^2$-coefficient is positive, and $$\label{reducible f shape}
f(x,y) = \alpha x^2 + \beta xy,\mbox{ where }\gcd(\alpha,\beta)=1{\text{ and }}0<\alpha\leq \beta$$ when $f$ is reducible. Let $\sim$ denote ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence. Then, our main result is:
\[Small Gal MT\] Let $f(x,y)$ be an integral and primitive binary quadratic form of non-zero discriminant and with positive $x^2$-coefficient. Write $D_f = |\Delta(f)|$, and put $$s_f = \begin{cases}8&\text{if $D_f$ is odd},\\1&\text{if $D_f$ is even}.\end{cases}$$
(a) Suppose that $f$ is positive definite. Then, we have $$\hspace{0.5cm}N^{(D_4)}_{{{\mathbb{Z}}},f}(X) =
\dfrac{1}{s_fr_f}\dfrac{13 \pi }{27D_f^{3/2}} X^{3/2} + O_f(X^{1+\epsilon})\mbox{ for any }\epsilon>0,$$ where $$\hspace{0.5cm}r_f = \begin{cases} 6 & \text{if $f(x,y) \sim x^2 + xy + y^2$},\\
2 & \mbox{if $f(x,y)\sim ax^2 + cy^2$}\\ &\hspace{1em}\mbox{ or $f(x,y)\sim ax^2 + bxy + ay^2$ with $a\neq b$},\\
1 & \text{otherwise}.\end{cases}$$
(b) Suppose that $f$ is reducible and that $f$ has the shape (\[reducible f shape\]). Then, we have $$\hspace{0.5cm}N^{(D_4)}_{{{\mathbb{Z}}},f}(X)=
\dfrac{1}{s_fr_f}\dfrac{8}{9\beta^{3/2}}X^{3/2}\log X + O_f(X^{3/2}),$$ where $$\hspace{0.5cm}r_f =\begin{cases}
1& \text{if $\beta\nmid\alpha^2+1${\text{ and }}$\beta\nmid \alpha^2-1$},\\
2& \text{otherwise}.
\end{cases}$$
(c) Suppose that $f$ is indefinite and irreducible. Define $t_{D_f}\in{{\mathbb{R}}}$ to be such that $e^{t_{D_f}}$ is the fundamental unit of the quadratic order ${{\mathbb{Z}}}[(D_f+\sqrt{D_f})/2]$, or equivalently $$t_{D_f} = \log((u_{D_f}+v_{D_f}\sqrt{D_f})/2),$$ where $(u_{D_f},v_{D_f})\in{{\mathbb{N}}}^2$ is the least solution to $x^2-D_fy^2=\pm4$. Then, we have $$\hspace{0.5cm}N_{{{\mathbb{Z}}},f}^{(D_4)}(X)=\frac{1}{s_fr_f}\frac{32t_{D_f}}{9D_f^{3/2}}X^{3/2}+ O_f(X^{1+\epsilon}) \mbox{ for any }\epsilon>0,$$ where $$\hspace{0.5cm}r_f = \begin{cases}
2 & \mbox{if $f(x,y)\sim ax^2 + bxy - ay^2$}\\&\hspace{1em}\mbox{or $f(x,y)\sim ax^2 + b xy + cy^2$ with $a\mid b$},\\
1 & \text{otherwise}.\end{cases}$$
(d) In all three cases, for any $\epsilon>0$, we have $$N_{{{\mathbb{Z}}},f}^{(V_4)}(X) = O_{f,\epsilon}(X^{1+\epsilon}),$$ and also $$N_{{{\mathbb{Z}}},f}^{(C_4)}(X) = \begin{cases}O_{f,\epsilon}(X^{1/2+\epsilon})&\mbox{if $-\Delta(f)\neq\square$},\\O_f(X) &\mbox{if $-\Delta(f)=\square$}.\end{cases}$$
Notice that the error terms in Theorem \[Small Gal MT\] depend upon $f$. Hence, we are unable to obtain an asymptotic formula for $N_{{{\mathbb{Z}}}}^\dagger(X)$ by summing over $f\in{\mathfrak{F}}$. However, there are only three $f\in{\mathfrak{F}}$ that need to be considered if we restrict to the forms in $$V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},*} = \{F \in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}} : F_T = F\mbox{ for some }T\in{\operatorname{GL}}_2({{\mathbb{Z}}})\setminus\{\pm I_{2\times 2}\}\}.$$ This is because by Proposition \[auto theorem\] below, such a matrix $T$ must be of the shape $M_f$ or $M_f/2$ up to sign, where $f\in{\mathfrak{F}}^*$. From (\[union\]), we then deduce that $$\begin{aligned}
V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},*} &= \bigcup_{\substack{f\in{\mathfrak{F}}^*\\\Delta(f)\in\{-4,1,4\}}} \{F\in V_{{{\mathbb{Z}}},f}^0:F\mbox{ is irreducible}\},\\\notag
V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},*,\dagger} &= \bigsqcup_{\substack{f\in{\mathfrak{F}}^*\\\Delta(f)\in\{-4,1,4\}}} \{F\in V_{{{\mathbb{Z}}},f}^0:F\mbox{ is irreducible}{\text{ and }}{\operatorname{Gal}}(F)\not\simeq V_4\}.\end{aligned}$$ For $X>0$, let us put $$N_{{{\mathbb{Z}}}}^{*,\dagger}(X) = \#\{[F]: F \in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},*,\dagger}\mbox{ such that }H(F)\leq X\}.$$ Then, by (\[V bijection\]) and the above discussion, we have $$N_{{{\mathbb{Z}}}}^{*,\dagger}(X) = N_{{{\mathbb{Z}}},f^{(1)}}^{*,\dagger}(X)+N_{{{\mathbb{Z}}},f^{(2)}}^{*,\dagger}(X)+N_{{{\mathbb{Z}}},f^{(3)}}^{*,\dagger}(X),$$ where we may take $$f^{(1)}(x,y) = x^2+y^2,\, f^{(2)}(x,y) = x^2 + xy,\, f^{(3)}(x,y) = x^2 +2xy,$$ whose discriminants are $-4,1$, and $4$, respectively. It follows that:
\[corollary\]We have $$N_{{{\mathbb{Z}}}}^{*,\dagger}(X) = \frac{1}{9}X^{3/2}\log X + O(X^{3/2}).$$
Theorem \[Small Gal MT\] implies that $$N_{{{\mathbb{Z}}},f^{(1)}}^\dagger(X) = O(X^{3/2}){\text{ and }}N_{{{\mathbb{Z}}},f^{(i)}}^\dagger(X) = \frac{1}{18}X^{3/2}\log X + O(X^{3/2})\mbox{ for $i=2,3$}$$ Summing these terms up then yields the claim.
Finally, as a consequence of the proof of Theorem \[Small Gal MT\], we also have:
\[negative Pell\]Let $D = \beta^2 + 4\alpha^2$, where $\alpha,\beta\in{{\mathbb{N}}}$ are coprime and $D$ is not a square. Then, the negative Pell’s equation $x^2 - Dy^2 = -4$ has integer solutions if and only if the integral binary quadratic form $\alpha x^2 + \beta xy - \alpha y^2$ is ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent to a form of the shape $a x^2 + b xy + c y^2$ with $a$ dividing $b$.
We now discuss some potential applications of our Theorem \[Small Gal MT\] and Corollary \[corollary\].\
First, it is natural to ask whether the asymptotic formula (\[BS formula\]), which was proven using Proposition \[Davenport\], admits a secondary main term. From the arguments in [@BhaSha], we see that the error term arising from volumes of the lower dimensional projections in Proposition \[Davenport\] is only of order $O(X^{3/4})$. Thus, possibly $X^{3/4}$ is the order of a second main term, but it is dominated by another error term coming from $$N_{{{\mathbb{Z}}},{\mathrm{\tiny BS}}}^{*}(X) = \#\{[F]: F \in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},*}\mbox{ such that }H_{\mathrm{\tiny BS}}(F)\leq X\}.$$ In particular, it was shown in [@BhaSha Lemma 2.4] that $$N_{{{\mathbb{Z}}},{\mathrm{\tiny BS}}}^*(X) = O_{\epsilon}(X^{3/4+\epsilon})\mbox{ for any }\epsilon>0.$$ Our Corollary \[corollary\] removes this obstacle, because $$N_{{{\mathbb{Z}}}}^{*,\dagger}(X^{1/3}) \leq N_{{{\mathbb{Z}}},{\mathrm{\tiny BS}}}^*(X) \leq N_{{{\mathbb{Z}}}}^{*,\dagger}(10X^{1/3}) + O_{\epsilon}(X^{1/3+\epsilon})$$ by (\[H compare\]) and Theorem \[Small Gal MT\] (d), whence we have $$N_{{{\mathbb{Z}}},{\mathrm{\tiny BS}}}^*(X)\asymp X^{1/2}\log X.$$ This improvement potentially allows one to prove a secondary main term for (\[BS formula\]) by using similar methods from [@BhaShaTsi], where it was shown that the counting theorem in [@DH] for cubic fields has a secondary main term of order $X^{5/6}$; this latter fact was proven independently in [@TT] as well.\
Next, integral binary quartic forms are closely related to quartic orders, and maximal irreducible quartic orders may be regarded as quartic fields. More generally, by the construction of Birch-Merriman [@BM] or Nakagawa [@Nakagawa], any integral binary form $F$ gives rise to a ${{\mathbb{Z}}}$-order $Q_F$ whose rank is the degree of $F$, where ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence class of $F$ corresponds to isomorphism class of $Q_F$. By [@DF], it is well-known that all cubic orders come from integral binary cubic forms, which enabled the enumeration of cubic orders having a non-trivial automorphism as well as cubic fields by their discriminant; see [@BhaShn] and [@DH], respectively. But this is not true for orders of higher rank. Parametrizations of quartic and quintic orders were given by Bhargava in his seminal work [@HCL3] and [@HCL4]. In [@Wood], Wood further showed that the quartic orders arising from integral binary quartic forms are exactly those having a monogenic *cubic resolvent*; see [@HCL3] for the definition. This implies that the forms in $$V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},\star}= \{F\in V_{{\mathbb{Z}}}^{{\mathrm{\tiny sm}}}: Q_F\mbox{ is maximal}\}$$ correspond to quartic $D_4$-, $C_4$-, and $V_4$-fields whose ring of integers has a monogenic cubic resolvent. In our upcoming paper [@TX2], we shall enumerate ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence classes of forms in $V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},\star}$ with respect to a height corresponding to the conductor of fields, as motivated by [@ASVW]. In fact, we shall that show that $$\mbox{for all }f\in{\mathfrak{F}}^*:F\in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}},\star}\cap V_{{{\mathbb{Z}}},f}^0 \neq\emptyset \mbox{ if and only if }\Delta(f) \in \{-4,1,4\}.$$ Thus, our counting theorem in [@TX2] may be regarded as a refinement and an extension of Corollary \[corollary\] above.\
Last but not least, binary quartic forms are connected to elliptic curves as well. In particular, any integral binary quartic form $F$ gives rise to an elliptic curve $$E_F : y^2 = x^3 - \frac{I(F)}{3}x - \frac{J(F)}{27}$$ defined over ${{\mathbb{Q}}}$. In [@BhaSha], Bhargava and Shankar applied (\[BS formula\]) as well as a parametrization of 2-Selmer groups due to Birch and Swinnerton-Dyer to show that the average rank of elliptic curves over ${{\mathbb{Q}}}$, when ordered by a *naive* height analogous to (\[BS height\]), is at most $3/2$. This result is remarkable in that it is the first to show, unconditional on the BSD-conjecture and the Grand Riemann Hypothesis, boundedness of the average rank of large families of elliptic curves over ${{\mathbb{Q}}}$. Conditional bounds were obtained by Brumer [@Bru], Heath-Brown [@HB], and Young [@You] previously. Now, the relations in (\[IJ family\]) imply that for $F\in V_{{{\mathbb{Z}}}}^{{\mathrm{\tiny sm}}}\cap V_{{{\mathbb{Z}}},f}^0$ with $f\in{\mathfrak{F}}^*$, we have $$E_F: y^2 = \left(x+\frac{L_f(F)}{3}\right)\left(x^2 - \frac{L_f(F)}{3}x - \frac{K_f(F)}{9}\right),$$ which has a rational $2$-torsion point. Hence, our Theorem \[Small Gal MT\] potentially allows one to study arithmetic properties of elliptic curves with $2$-torsion over ${{\mathbb{Q}}}$. Let us remark that unlike a *large* family of elliptic curves over ${{\mathbb{Q}}}$, in the sense of [@BhaSha Section 3], the family consisting of those curves with a rational $2$-torsion exhibits a rather peculiar behaviour. Indeed, Klagsbrun and Lemke-Oliver [@K-LO] proved that the average size of the 2-Selmer groups in this family is unbounded, and they conjectured an asymptotic growth rate. One might be able to obtain such an asymptotic growth rate using our Theorem \[Small Gal MT\] and a sieve that detects local solubility; this line of inquiry is pursued in an upcoming paper due to D. Kane and Z. Klagsbrun.
Characterization of forms with small Galois groups {#Galois gp sec}
==================================================
Cremona covariants {#Cremona section}
------------------
Let $F$ be a real binary quartic form with $\Delta(F)\neq0$. As Cremona defined in [@Cre], we have three quadratic covariants ${\mathfrak{C}}_{F,\omega}(x,y)$, each of which is associated to a root $\omega$ of ${{\mathcal{Q}}}_F(x)$; see [@X Subsection 4.2] for the explicit definition. They satisfy the syzygy $$\label{cre square} {\mathfrak{C}}_{F,\omega}(x,y)^2=\frac{1}{3} \left(F_4(x,y) + 4 \omega F(x,y)\right),$$ where $F_4$ is the *Hessian covariant of $F$* and is given by $$\begin{aligned}
F_4(x,y) & = 3(a_3^2-8a_4a_2)x^4 + 4(a_3a_2-6a_4a_1)x^3y + 2(2a_2^2 - 24a_4a_0 - 3a_3a_1)x^2y^2 \\ & \hspace{6.25cm}+ 4(a_2a_1 - 6a_3a_0)xy^3 + (3a_1^2-8a_2a_0)y^4.\end{aligned}$$ We shall label the roots $\omega_1(F),\omega_2(F),\omega_3(F)$ of ${{\mathcal{Q}}}_F(x)$ such that $${\mathfrak{C}}_{F,\omega_i(F)}(x,y) = {\mathfrak{C}}_{F,i}(x,y)\mbox{ for all }i=1,2,3,$$ where ${\mathfrak{C}}_{F,i}(x,y)$ is defined as in [@X (4.6)]. Then, from (\[cre square\]) and the explicit expressions for ${\mathfrak{C}}_{F,\omega}(x,y)$ given in [@X], we have the following observations:
(1) For $\omega = \omega_1(F)$, the binary quadratic form ${\mathfrak{C}}_{F,\omega}(x,y)$ has real coefficients.
(2) For $\omega = \omega_2(F),\omega_3(F)$, we have:\
$\bullet$ If $\Delta(F)>0$, then $\lambda_\omega\cdot{\mathfrak{C}}_{F,\omega}(x,y)$ has real coefficients for some $\lambda_\omega\in\{1,\sqrt{-1}\}$.\
$\bullet$ If $\Delta(F)<0$, then $\lambda\cdot{\mathfrak{C}}_{F,\omega}(x,y)$ does not have real coefficients for all $\lambda\in{{\mathbb{C}}}^\times$.
Also, it is easy to check that $$\label{cre disc}
\Delta({\mathfrak{C}}_{F,\omega_1(F)}), \Delta({\mathfrak{C}}_{F,\omega_3(F)}) > 0 {\text{ and }}\Delta({\mathfrak{C}}_{F,\omega_2(F)})<0.$$ We shall require the following result by the second-named author in [@X].
\[auto theorem\] Let $F$ be a real binary quartic form with $\Delta(F)\neq0$. Then, a set of representatives for the quotient group $$\{T\in{\operatorname{GL}}_2({{\mathbb{R}}}): F_T = F\}/\{\lambda\cdot I_{2\times2}:\lambda\in{{\mathbb{R}}}^\times\}$$ is given by $$\begin{cases}\{I_{2\times2}, M_{f}: f \in\{{\mathfrak{C}}_{F,\omega_1(F)},\lambda_{\omega_2(F)}\cdot{\mathfrak{C}}_{F,\omega_2(F)},\lambda_{\omega_3(F)}\cdot{\mathfrak{C}}_{F,\omega_3(F)} \}&\text{if } \Delta(F) > 0,\\
\{I_{2\times 2},M_{f}: f \in\{{\mathfrak{C}}_{F,\omega_1(F)}\}\}&\text{if } \Delta(F) < 0.\end{cases}$$ Furthermore, the quadratic forms ${\mathfrak{C}}_{F,\omega_1(F)}(x,y),{\mathfrak{C}}_{F,\omega_2(F)}(x,y)$, and ${\mathfrak{C}}_{F,\omega_3(F)}(x,y)$, are pairwise non-proportional over ${{\mathbb{C}}}^\times$.
For the first statement, see [@X Proposition 4.6]. As for the second statement, since ${\mathfrak{C}}_{F,\omega_i(F)}(x,y)$ are covariants, replacing $F$ by a ${\operatorname{GL}}_2({{\mathbb{R}}})$-translate if necessary, we may assume that $F(x,y) = a_4x^4 + a_2x^2y^2 \pm a_4y^4$. In this special case, it is not hard to verify the claim using the explicit expressions for ${\mathfrak{C}}_{F,\omega_i(F)}(x,y)$ in [@X (4.6)].
Let $F$ be a real binary quartic form with $\Delta(F)\neq0$. Proposition \[auto theorem\] implies that for any real binary quadratic form $f$ with $\Delta(f)\neq0$, we have $F\in V_{{{\mathbb{R}}},f}$ if and only if $$\label{in V char}
f(x,y)\mbox{ is proportional to ${\mathfrak{C}}_{F,\omega}(x,y)$ for a root $\omega$ of ${{\mathcal{Q}}}_F(x)$}.$$ Moreover, this root $\omega$ is unique, and we shall denote it by $\omega_f(F)$. This was required in order to define the $L_f$- and $K_f$-invariants in (\[LK def\]).
Proof of Theorem \[small char thm\]
-----------------------------------
The key is the following lemma.
\[omega in Z\] Let $F$ be an integral binary quartic form with $\Delta(F)\neq0$ and let $\omega$ be a root of ${{\mathcal{Q}}}_F(x)$. Then, the quadratic form ${\mathfrak{C}}_{F,\omega}(x,y)$ is proportional over ${{\mathbb{C}}}^\times$ to a form with integer coefficients if and only if $\omega\in{{\mathbb{Z}}}$.
If $\omega\in{{\mathbb{Z}}}$, then we easily see from (\[cre square\]) that $\lambda\cdot {\mathfrak{C}}_{F,\omega}(x,y)$ has integer coefficients for some $\lambda\in{{\mathbb{C}}}^\times$. Conversely, if $\lambda\cdot {\mathfrak{C}}_{F,\omega}(x,y)$ has integer coefficients for some $\lambda\in{{\mathbb{C}}}^\times$, then consider the action of an element $\sigma\in{\operatorname{Gal}}(\overline{{{\mathbb{Q}}}}/{{\mathbb{Q}}})$, where $\overline{{{\mathbb{Q}}}}$ is an algebraic closure of ${{\mathbb{Q}}}$. It is clear from the definition of $ {\mathfrak{C}}_{F,\omega}(x,y)$ that $\lambda\in\overline{{{\mathbb{Q}}}}$. From (\[cre square\]), we have $$\frac{4}{3}(\omega - \sigma(\omega))F(x,y) = {\mathfrak{C}}_{F,\omega}(x,y)^2 - \sigma({\mathfrak{C}}_{F,\omega}(x,y)^2) =\left(1-\frac{\lambda^2}{\sigma(\lambda)^2}\right){\mathfrak{C}}_{F,\omega}(x,y)^2,$$ and this last binary quartic form has zero discriminant. This shows that $\omega - \sigma(\omega) = 0$ for all $\sigma\in{\operatorname{Gal}}(\overline{{{\mathbb{Q}}}}/{{\mathbb{Q}}})$. Thus, we have $\omega\in{{\mathbb{Q}}}$, and so $\omega\in{{\mathbb{Z}}}$ since ${{\mathcal{Q}}}_F(x)$ is monic.
The first claim in Theorem \[small char thm\] now follows from Proposition \[auto theorem\], Lemma \[omega in Z\], and (\[in V char\]). Note that $\Delta(F) = 27^2\Delta({{\mathcal{Q}}}_F)$, which means that ${{\mathcal{Q}}}_F(x)$ has three integer roots if and only if ${{\mathcal{Q}}}_F(x)$ is reducible and $\Delta(F)=\square$. The second claim then follows from this fact and (\[cre disc\]).
Basic properties of forms in $V_{{{\mathbb{R}}},f}$ of non-zero discriminant {#properties section}
============================================================================
Throughout this section, let $f(x,y) = \alpha x^2 + \beta xy + \gamma y^2$ be a real binary quadratic form with $\Delta(f)\neq0$. It is not hard to check, by a direct calculation, that $$\label{abc family}
V_{{{\mathbb{R}}},f} =\left\lbrace
\begin{array}{@{}c@{}c}
Ax^4 + Bx^3y + Cx^2 y^2 + \left(\dfrac{4 \beta \gamma A - (\beta^2 + 2 \alpha \gamma) B + 2 \alpha \beta C}{2\alpha^2} \right)xy^3 \\\\
+\left(\dfrac{4 \gamma(\beta^2 + 2\alpha \gamma) A - \beta(\beta^2 + 4 \alpha \gamma) B + 2\alpha \beta^2 C}{8\alpha^3}\right)y^4:
A,B,C\in{{\mathbb{R}}}\end{array}
\right\rbrace$$ if $\alpha\neq0$, and similarly that $$\label{abc family 2}
V_{{{\mathbb{R}}},f} =\left\lbrace
\begin{array}{@{}c@{}c}
Ax^4 +\left( \dfrac{\gamma(4 \beta^2 + 8 \alpha \gamma)A + 2 \alpha \beta^2 B - 8 \alpha^3 C}{\beta(\beta^2 + 4 \alpha \gamma)} \right)x^3 y + B x^2 y^2\\\\
- \left(\dfrac{8 \gamma^3 A - 2 \beta^2 \gamma B - \alpha(4 \beta^2 + 8 \alpha \gamma)C}{\beta(\beta^2 + 4 \alpha \gamma)} \right) xy^3 + Cy^4:
A,B,C\in{{\mathbb{R}}}\end{array}
\right\rbrace$$ if $\beta,\beta^2+4\alpha\gamma\neq0$. Below, we shall give some basic properties of $V_{{{\mathbb{R}}},f}^0$ and $V_{{{\mathbb{Z}}},f}^0$.
The two new invariants {#LK section}
----------------------
Recall the definitions of the $L_f$- and $K_f$-invariants given in (\[LK def\]). First, we shall show that they are indeed invariants under the twisted action of ${\operatorname{GL}}_2({{\mathbb{R}}})$ in the following sense.
\[LK invariant\]For all $F\in V_{{{\mathbb{R}}},f}^0$ and $T\in{\operatorname{GL}}_2({{\mathbb{R}}})$, we have $$L_{f_T}(F_T) = L_f(F){\text{ and }}K_{f_T}(F_T) = K_f(F).$$
Notice that ${{\mathcal{Q}}}_F(x) = {{\mathcal{Q}}}_{F_T}(x)$. For any root $\omega$ of ${{\mathcal{Q}}}_F(x)$, because ${\mathfrak{C}}_{F,\omega}(x,y)$ is a covariant up to sign by (\[cre square\]), if ${\mathfrak{C}}_{F,\omega}(x,y)$ is proportional to $f(x,y)$, then ${\mathfrak{C}}_{F_T,\omega}(x,y)$ is proportional to $f_T(x,y)$. It then follows from the definition that $ L_{f_T}(F_T) = L_f(F)$. Since $I(F_T) = I(F)$, we also have $K_{f_T}(F_T) = K_f(F)$ by the first equality in (\[IJ family\]).
We shall give explicit formulae for $L_f(-)$ and $K_f(-)$ in two special cases.
\[explicit LK\]The following holds.
(a) Assume that $\alpha\neq0$. Then, for all $F\in V_{{{\mathbb{R}}},f}^0$ as in (\[abc family\]), we have $$\begin{aligned}
\hspace{5mm} L_f(F) &= -(12 \gamma A - 3 \beta B + 2 \alpha C)/(2\alpha),\\
\hspace{5mm} K_f(F)&= (72 \beta^2 \gamma A^2 + 9 \alpha(\beta^2+4 \alpha \gamma)B^2 + 8 \alpha^3 C^2\\\notag
&\hspace{2em}- 18 \beta(\beta^2+4 \alpha \gamma)AB + 12 \alpha(3\beta^2-4 \alpha \gamma)AC - 24 \alpha^2 \beta BC)/(4
\alpha^3).\end{aligned}$$ Moreover, we have $$\hspace{5mm} \frac{4(L_f(F)^2 + 4K_f(F))}{9} = \frac{L_{f,1}(F)^2 - \Delta(f)L_{f,2}(F)^2}{\alpha^4},$$ where $$\hspace{5mm} L_{f,1}(F) = 4(\beta^2 - \alpha\gamma)A - 3\alpha\beta B + 2\alpha^2 C {\text{ and }}L_{f,2}(F) = 2(2\beta A - \alpha B).$$
(b) Assume that $\gamma=0$. Then, for all $F\in V_{{{\mathbb{R}}},f}^0$ as in (\[abc family 2\]), we have $$\begin{aligned}
\hspace{5mm} L_f(F) & = (2\beta^2B - 12\alpha^2C)/\beta^2,\\
\hspace{5mm} K_f(F) & = (-\beta^4 B^2 + 144\alpha^4C^2 + 36\beta^4AC - 24\alpha^2\beta^2BC)/\beta^4. \end{aligned}$$ Moreover, we have $$\hspace{5mm} \frac{4(L_f(F)^2 + 4K_f(F))}{9}=\frac{8C}{\beta^2}\left(8\beta^2A-8\alpha^2B+\frac{40\alpha^4}{\beta^2}C\right).$$
This may be verified by explicit computation.
We shall also need the following observation.
\[LK integers\]Assume that $f$ is integral. Then, for all $F\in V_{{{\mathbb{Z}}},f}^0$, we have $$L_f(F), K_f(F),(L_f(F)^2 + 4K_f(F))/9, (2L_f(F)^2 - K_f(F))/9\in{{\mathbb{Z}}}.$$ Moreover, when $f$ is primitive in addition, we have $$4(2L_f(F)^2 - K_f(F))/(9\Delta(f)) \in {{\mathbb{Z}}}.$$
We have $L_f(F)\in{{\mathbb{Z}}}$ by Lemma \[omega in Z\]. Since $I(F)\in{{\mathbb{Z}}}$, we deduce from the first equality in (\[IJ family\]) that $K_f(F)\in{{\mathbb{Z}}}$ holds as well. Observe that $$\begin{aligned}
I(F)+K_f(F)&=(L_f(F)^2+4K_f(F))/3,\\
2I(F)-K_f(F)&=(2L_f(F)^2-K_f(F))/3,\end{aligned}$$ both of which are integers. Since $\Delta(F)\in{{\mathbb{Z}}}$, we deduce from (\[Delta LK\]) that at least one of the above expressions is divisible by $3$. But again by (\[IJ family\]), we have $$3I(F)=(L_f(F)^2+4K_f(F))/3+(2L_f(F)^2-K_f(F))/3,$$ so in fact both expressions are divisible by $3$. This proves the first claim.\
Next, assume that $f$ is primitive in addition. In view of Proposition \[LK invariant\], by applying a ${\operatorname{GL}}_2({{\mathbb{Z}}})$-action on $f$ if necessary, we may assume that $\alpha\neq0$ and that $\alpha$ is coprime to $\Delta(f)$. Using Proposition \[explicit LK\] (a), we then compute that $$\label{2LKD} \frac{4(2L_f(F)^2 - K_f(F))}{9} = \Delta(f)\left(\frac{ \alpha(B^2-4AC) + 2A(\beta B - 4\gamma A)}{\alpha^3}\right).$$ This expression is an integer by the first claim, and hence must be divisible by $\Delta(f)$, because $\alpha$ is taken to be coprime to $\Delta(f)$. This proves the second claim.
Determinants of the two lattices {#det sec}
--------------------------------
In this subsection, assume that $f$ is integral and primitive. Let $\Lambda_{f,1}$ and $\Lambda_{f,2}$ denote the lattices defined in (\[Lambda def\]). Below, we shall compute their determinants in terms of the number $s_f$ as in Theorem \[Small Gal MT\].
\[det prop\]We have $\det(\Lambda_{f,1}) = s_f|\alpha|^3$ and $\det(\Lambda_{f,2}) = s_f|\beta(\beta^2+4\alpha\gamma)|/8$.
Observe that the linear transformation defined by the matrix $$\begin{pmatrix}1&0&0\\0&0&1\\ *&-{{\mathcal{B}}}&*\end{pmatrix},\text{ where }{{\mathcal{B}}}= \frac{\beta(\beta^2+4\alpha\gamma)}{8\alpha^3},$$ has determinant ${{\mathcal{B}}}$, and it sends $\Lambda_{f,1}$ to $\Lambda_{f,2}$. Thus, it suffices to prove the first claim. Recall from (\[abc family\]) that $\Lambda_{f,1}$ is the set of tuples $(A,B,C)\in{{\mathbb{Z}}}^3$ satisfying $$\begin{aligned}
4\beta\gamma A-(\beta^2+2\alpha\gamma)B+2\alpha\beta C&\equiv0\hspace{-2cm}&\pmod{2\alpha^2},\\
4\gamma(\beta^2+2\alpha\gamma)A-\beta(\beta^2+4\alpha\gamma)B+2\alpha\beta^2C&\equiv 0\hspace{-2cm}&\pmod{8\alpha^3}.\end{aligned}$$ If $\beta\gamma=0$, then it is easy to check that $\det(\Lambda_{f,1})=s_f|\alpha|^3$. If $\beta\gamma\neq0$, then we shall use the fact that $$\det(\Lambda_{f,1}) = \prod_{p}\det(\Lambda_{f,1}^{(p)})= \prod_{p\mid 2\alpha}\det(\Lambda_{f,1}^{(p)}),\mbox{ where }\Lambda_{f,1}^{(p)} = {{\mathbb{Z}}}_p\otimes_{{\mathbb{Z}}}\Lambda_{f,1},$$ and so $\det(\Lambda_{f,1})=s_f|\alpha|^3$ indeed holds by Lemma \[Lap det\] below.
\[Lap det\]Let $p$ be a prime dividing $2\alpha$ and let $p^k\| \alpha$. Then, we have $$\det(\Lambda_{f,1}^{(p)}) = s_f^{\epsilon_p}p^{3k}, \mbox{ where }\epsilon_p = \begin{cases} 1 &\mbox{if $p=2$},\\ 0 & \mbox{if $p\geq 3$}.\end{cases}$$
For brevity, write $$\alpha = p^k a{\text{ and }}\beta = p^\ell b,\mbox{ where $k,\ell,a,b\in{{\mathbb{Z}}}$ with $k,\ell\geq0$ and $p\nmid a,b$}.$$ Then, the claim may be restated as $$\det(\Lambda_{f,1}^{(p)}) = \begin{cases} p^{3k+3\epsilon_p} &\mbox{if $\ell=0$},\\ p^{3k} & \mbox{if $\ell\geq1$}.\end{cases}$$ By definition, the lattice $\Lambda_{f,1}^{(p)}$ is the set $(A,B,C)\in{{\mathbb{Z}}}_p^3$ of tuples satisfying $${{\mathcal{T}}}_1(A,B,C)\equiv0\hspace{-3mm}\pmod{p^{2k+\epsilon_p}}{\text{ and }}{{\mathcal{T}}}_2(A,B,C)\equiv0\hspace{-3mm}\pmod{p^{3k+3\epsilon_p}},$$ where $$\begin{aligned}
\label{A1 A2 p}
{{\mathcal{T}}}_1(A,B,C)&=p^\ell b(4\gamma A-p^\ell bB)-2p^ka\gamma B+2p^{k+\ell}abC,\\\notag
{{\mathcal{T}}}_ 2(A,B,C) &=(p^{2\ell}b^2+4p^ka\gamma)(4\gamma A-p^\ell bB) - 8p^ka\gamma^2 A+2p^{k+2\ell}ab^2C.\end{aligned}$$ Observe that we have the relation $$\label{A1 A2 p relation}
{{\mathcal{T}}}_2(A,B,C)-p^\ell b{{\mathcal{T}}}_1(A,B,C)=2p^ka\gamma(4\gamma A-p^\ell bB).$$ For $\ell=0$, we deduce from (\[A1 A2 p relation\]) that $\Lambda_{f,1}^{(p)}$ is defined solely by $${{\mathcal{T}}}_2(A,B,C)\equiv0\hspace{-3mm}\pmod{p^{3k+3\epsilon_p}}.$$ For $\ell\geq1$ and $\ell\geq k+2\epsilon_{p}$, it is easy to see that $\Lambda_{f,1}^{(p)}$ is in fact defined by $$A\equiv0\mbox{ (mod $p^{2k}$) and }B\equiv0\mbox{ (mod $p^k$)}.$$ For $\ell\geq1$ and $\ell\leq k+\epsilon_p$, we shall first show that $\Lambda_{f,1}^{(p)}$ is also defined by $$\label{congruences}\begin{cases}
A\equiv0&\pmod{p^{2\ell-2\epsilon_p}},\\
B\equiv0&\pmod{p^{\ell-\epsilon_p}},\\
(4\gamma A-p^\ell bB)/p^{2\ell-\epsilon_p}\equiv0&\pmod{p^{k-\ell+\epsilon_p}},\\
{{\mathcal{T}}}_2(A,B,C)/p^{k+2\ell+\epsilon_p}\equiv 0&\pmod{p^{2k-2\ell+2\epsilon_p}}.\end{cases}$$ If (\[congruences\]) is satisfied, then from (\[A1 A2 p relation\]), it is easy to see that $(A,B,C)\in\Lambda_{f,1}^{(p)}$. Conversely, if $(A,B,C)\in\Lambda_{f,1}^{(p)}$, then the assumption $\ell\leq k+\epsilon_p$ implies that $$\mbox{${{\mathcal{T}}}_1(A,B,C)\equiv0$ (mod $p^{k+\ell}$)} {\text{ and }}\mbox{${{\mathcal{T}}}_2(A,B,C)\equiv0$ (mod $p^{k+2\ell+\epsilon_p}$)},$$ while reducing (\[A1 A2 p relation\]) mod $p^{2k+\ell+\epsilon_p}$ also yields $$4\gamma A-p^\ell bB\equiv0\mbox{ (mod $p^{k+\ell}$)}.$$ From these three congruence equations, it follows that (\[congruences\]) is indeed satisfied. In all cases, we then see that $\det(\Lambda_{f,1}^{(p)})$ is as claimed.
Forms with abelian Galois groups {#abelian Gal sec}
--------------------------------
In this subsection, assume that $f$ is integral. Consider an irreducible form $F\in V_{{{\mathbb{Z}}},f}^0$. By Theorem \[small char thm\], we have ${\operatorname{Gal}}(F)\simeq D_4$, $C_4$, or $V_4$. To distinguish among these three possibilities, note that the *cubic resolvent polynomial of $F$*, defined by $$R_F(x) = a_4^3X^3 - a_4^2a_2X^2 + a_4(a_3a_1 - 4a_4a_0)X - (a_3^2a_0 + a_4a_1^2 - 4a_4a_2a_0)$$ when $F$ has the shape (\[F generic\]), is reducible since ${\operatorname{Gal}}(F)$ is small. Also, it has a unique root $r_F\in{{\mathbb{Q}}}$ precisely when $\Delta(F)\neq\square$, in which case we define $$\theta_1(F) = (a_3^2 - 4a_4(a_2 - r_Fa_4))\Delta(F){\text{ and }}\theta_2(F) = a_4(r_F^2a_4 - 4a_0)\Delta(F).$$ Then, we have the well-known criterion $$\begin{aligned}
{\operatorname{Gal}}(F)\simeq V_4 & \iff \Delta(F)=\square,\\
{\operatorname{Gal}}(F)\simeq C_4 & \iff \Delta(F) \neq\square \mbox{ and }\theta_1(F),\theta_2(F)=\square\mbox{ in ${{\mathbb{Q}}}$}.\end{aligned}$$ See [@Con] for example. We then deduce that:
\[abelian Gal prop\]Let $F\in V_{{{\mathbb{Z}}},f}^0$ be an irreducible form. Then, we have $${\operatorname{Gal}}(F)\simeq V_4\iff L_f(F)^2 + 4K_f(F)=\square,$$ as well as $${\operatorname{Gal}}(F)\simeq C_4\iff \begin{cases}L_f(F)^2+4K_f(F)\neq\square,\\(L_f(F)^2 + 4K_f(F))(2L_f(F)^2 - K_f(F))/\Delta(f)=\square.\end{cases}$$
Observe that by (\[Delta LK\]), we have $$\Delta(F)=\square\mbox{ if and only if }L_f(F)^2+4K_f(F)=\square.$$ The first claim is then clear. Next, suppose that $\Delta(F)\neq\square$. By Proposition \[LK invariant\], we may assume that $\alpha\neq0$. For $F$ in the shape as in (\[abc family\]), a direct computation yields $$r_F = (-4\gamma A + \beta B)/(2\alpha A).$$ Using Proposition \[explicit LK\] (a), we further compute that $$\begin{aligned}
\theta_1(F) & = 4\alpha^2(2L_f(F)^2 - K_f(F))\Delta(F)/(9 \Delta(f)),\\
\theta_2(F) & = \beta^2 (2L_f(F)^2-K_f(F))\Delta(F)/(9 \Delta(f)).\end{aligned}$$ By (\[Delta LK\]) and the criterion above, it follows that $\theta_1(F), \theta_2(F)$ are squares if and only if $(L_f(F)^2 + 4K_f(F))(2L_f(F)^2 - K_f(F))/\Delta(f)$ is a square, as desired.
Reducible forms
---------------
In this subsection, assume that $f$ is integral. We shall study the reducible forms in $V_{{{\mathbb{Z}}},f}^0$. Let us first make a definition and an observation.
\[reducible types def\]Let $F\in V_{{{\mathbb{Z}}},f}^0$ be a reducible form.
(1) We say that $F$ is of *type $1$* if $F = m\cdot pp_{M_f}$ for some $m\in{{\mathbb{Q}}}^\times$ and integral binary quadratic form $p$.
(2) We say that $F$ is of *type $2$* if $F = pq$ for some integral binary quadratic forms $p$ and $q$ satisfying $p_{M_f} = -p$ and $q_{M_f} = - q$.
\[type 1\]For all reducible forms $F\in V_{{{\mathbb{Z}}},f}^0$ of type $1$, we have $$L_f(F)^2 + 4K_f(F) = \square.$$
This may be verified by a direct computation.
Below, we shall show that the two reducibility types in Definition \[reducible types def\] are in fact the only possibilities. We shall require two further lemmas.
\[linear factor\]Let $\ell(x,y) = \ell_1x+\ell_0y$ be a non-zero complex binary linear form, and suppose that $\ell_{M_f} =\lambda\cdot \ell$ for some $\lambda\in{{\mathbb{C}}}^\times$. Then, we have $\lambda = \pm\sqrt{-1}$, with $$\lambda = \begin{cases}
-\sqrt{-1} &\mbox{if and only if } \ell_0 = (\beta +\sqrt{\Delta(f)})\ell_1/(2\alpha),\\
\sqrt{-1} &\mbox{if and only if } \ell_0 = (\beta - \sqrt{\Delta(f)})\ell_1/(2\alpha),
\end{cases}$$ in the case that $\alpha\neq0$.
The hypothesis implies that $$\frac{1}{\sqrt{-\Delta(f)}}\begin{pmatrix} \beta & - 2\alpha \\ 2\gamma & -\beta\end{pmatrix}\begin{pmatrix}\ell_1 \\ \ell_0\end{pmatrix} =
\lambda\begin{pmatrix} \ell_1 \\ \ell_0\end{pmatrix}.$$ Then, by computing the eigenvalues and eigenspaces of the $2\times2$ matrix above, we see that the claim holds.
\[quadratic factor\]Let $p(x,y) = p_2 x^2 + p_1xy + p_0y^2$ be a non-zero complex binary quadratic form, and suppose that $p_{M_f} = \lambda\cdot p$ for some $\lambda\in{{\mathbb{C}}}^\times$. Then, we have $\lambda = \pm1$, with $$\lambda = \begin{cases}-1&\text{if and only if $p_0 = (\beta p_1 - 2\gamma p_2)/(2\alpha)$},\\
1& \text{if and only if $p = (p_2/\alpha)f$},
\end{cases}$$ in the case that $\alpha\neq0$.
The hypothesis implies that $$\frac{1}{-\Delta(f)}\begin{pmatrix}
\beta^2 & -2\alpha & 4\alpha^2 \\ 4\beta\gamma & -(\beta^2+4\alpha\gamma) & 4\alpha\beta \\ 4\gamma^2 & -2\beta\gamma & \beta^2
\end{pmatrix}\begin{pmatrix} p_2 \\ p_1 \\ p_0\end{pmatrix} = \lambda\begin{pmatrix} p_2 \\ p_1 \\ p_0\end{pmatrix}.$$ Then, by computing the eigenvalues and eigenspaces of the $3\times3$ matrix above, it is not hard to check that the claim holds.
\[reducibility types\]Any reducible form $F\in V_{{{\mathbb{Z}}},f}^0$ is either of type $1$ or of type $2$.
Write $F = g^{(1)}g^{(2)}g^{(3)}g^{(4)}$, where the $g^{(k)}$ are complex binary linear forms, and are pairwise non-proportional because $\Delta(F)\neq0$. Since $F$ is reducible, by renumbering if necessary, we may assume that $$\begin{cases}
g^{(1)}, g^{(2)}g^{(3)}g^{(4)} &\mbox{when $F$ has exactly one rational linear factor},\\
g^{(1)},g^{(2)}, g^{(3)}g^{(4)} &\mbox{when $F$ has exactly two rational linear factors},\\
g^{(1)}g^{(2)},g^{(3)}g^{(4)} &\mbox{when $F$ has no rational linear factor},\\
g^{(1)}, g^{(2)},g^{(3)},g^{(4)} &\mbox{when $F$ has four rational linear factors},
\end{cases}$$ have integer coefficients and are irreducible. We have $M_f^2 = \Delta(f)\cdot I_{2\times2}$ and $F_{M_f}= F$ by definition. Hence, up to scaling, the matrix $M_f$ acts on the $g^{(k)}$ via a permutation $\sigma$ on four letters of order dividing two. This has two consequences.\
By (\[V bijection\]), without loss of generality, we may assume that $\alpha\neq0$. First, the form $F$ cannot have exactly one rational linear factor, for otherwise $$\sigma(1) = 1 {\text{ and }}\sigma(k_0) = k_0 \mbox{ for at least one $k_0\in\{2,3,4\}$}.$$ From Lemma \[linear factor\], it would follow that $\Delta(f)$ is a square and that $g^{(k_0)}$ is proportional to a form with integer coefficients, which is a contradiction. Second, when $F$ has four rational linear factors, by further renumbering if necessary, we may assume that $$\sigma\in\{(1), (12), (12)(34)\}.$$ Now, in all three of the possible cases for the factorization of $F$, define $$p = g^{(1)}g^{(2)}{\text{ and }}q = g^{(3)}g^{(4)},$$ which are integral binary quadratic forms by definition. We then deduce that $$(p_{M_f},q_{M_f}) = (\lambda\cdot q, \lambda^{-1}\cdot p) {\text{ or }}(p_{M_f},q_{M_f}) = (\lambda\cdot p, \lambda^{-1}\cdot q)$$ for some $\lambda\in{{\mathbb{Q}}}^\times$. In the former case, it is clear that $F$ is of type $1$. In the latter case, we have $\lambda=-1$ by Lemma \[quadratic factor\] and the fact that $\Delta(F)\neq0$, so $F$ is of type $2$.
Parametrizing forms in $V_{{{\mathbb{R}}},f}$ of non-zero discriminant
======================================================================
Throughout this section, let $f(x,y) = \alpha x^2 + \beta xy + \gamma y^2$ be a real binary quadratic form with $\Delta(f)\neq0$ and $\alpha>0$. We shall give an alternative parametrization of $V_{{{\mathbb{R}}},f}^0$, different from (\[abc family\]) and (\[abc family 2\]), in terms of the regions $$\begin{aligned}
\label{Omega def}
\Omega^0 & = \{(L,K)\in{{\mathbb{R}}}^2 \mid L^2 + 4K\neq0 {\text{ and }}2L^2 - K\neq0\},\\\notag
\Omega^+ & = \{(L,K) \in {{\mathbb{R}}}^2\mid L^2 + 4K>0 {\text{ and }}2L^2 - K \neq0\},\\\notag
\Omega^- & = \{(L,K) \in {{\mathbb{R}}}^2\mid L^2 + 4K<0 {\text{ and }}2L^2 - K>0\},\end{aligned}$$ corresponding to the $L_f$- and $K_f$-invariants, as well as a parameter $t\in{{\mathbb{R}}}$ arising from the *orthogonal group of $f$*, defined by $$O_f({{\mathbb{R}}}) = \{T\in{\operatorname{GL}}_2({{\mathbb{R}}}): \det(T) = \pm1 {\text{ and }}f_T = \pm f\}.$$ Note that by (\[Delta LK\]), for any $F\in V_{{{\mathbb{R}}},f}^0$, we have $$\begin{aligned}
(L_f(F),K_f(F))\in \Omega^+ &\iff \Delta(F)>0,\\
(L_f(F),K_f(F))\in \Omega^- &\iff \Delta(F)<0.\end{aligned}$$ First, we shall show that it suffices to consider $x^2+ y^2$ and $x^2-y^2$. It shall be helpful to recall (\[V bijection\]) as well as the isomorphisms $\Theta_1$ and $\Theta_2$ defined in Subsection \[notation sec\].
\[Psi lemma\]Define a matrix $$T_f=\begin{pmatrix} \delta_f^{-1/4} & 0 \\ 0 & \delta_f^{1/4} \end{pmatrix}\cdot \frac{1}{ 2\sqrt{ \alpha}} \begin{pmatrix} 2\alpha & \beta \\ 0 & 2\end{pmatrix},\mbox{ where }\delta_f = \frac{|\Delta(f)|}{4}$$ Then, we have a well-defined bijective linear map $$\begin{cases}\Psi_f:V_{{{\mathbb{R}}},x^2+y^2}\longrightarrow V_{{{\mathbb{R}}},f};\hspace{1em}\Psi_f(F) = F_{T_f}&\mbox{if $f$ is positive definite},\\\Psi_f:V_{{{\mathbb{R}}},x^2-y^2}\longrightarrow V_{{{\mathbb{R}}},f};\hspace{1em}\Psi_f(F) = F_{T_f}&\mbox{if $f$ is indefinite},\end{cases}$$ and we have $\det(\Psi_f) = 8\alpha^3|\Delta(f)|^{-3/2}$.
The first claim holds by (\[V bijection\]) and the fact $$\delta_f^{-1/2}\cdot f= \begin{cases} (x^2+y^2)_{T_f} &\mbox{if $f$ is positive definite},\\(x^2 - y^2)_{T_f}&\mbox{if $f$ is indefinite}.\end{cases}$$ Identifying $V_{{{\mathbb{R}}},x^2\pm y^2}$ and $V_{{{\mathbb{R}}},f}$ with ${{\mathbb{R}}}^3$ via $\Theta_1$, we see from (\[abc family\]) that $$\label{general parameter} \Psi_f: \begin{pmatrix}a_4\\a_3\\a_2\end{pmatrix}\mapsto\begin{pmatrix} \frac{\alpha^2}{\delta_f} && 0 && 0 \\[0.5ex]
\frac{2\alpha\beta}{\delta_f} && \frac{\alpha}{\sqrt{\delta_f}} && 0\\[0.5ex]
\frac{3\beta^2}{2\delta_f}&& \frac{3\beta}{2\sqrt{\delta_f}} && 1
\end{pmatrix}\begin{pmatrix}a_4\\a_3\\a_2\end{pmatrix},$$ from which the second claim follows.
In the subsequent subsections, we shall prove the following propositions.
\[Phi+\]There exists an explicit bijection $$\Phi:\Omega^+ \times [-\pi/4,\pi/4) \longrightarrow V_{{{\mathbb{R}}},{x^2+y^2}}^0,$$ defined as in (\[Phi+ def\]), such that
(a) we have $L_{x^2+y^2}(\Phi(L,K,t)) = L$ and $K_{x^2+y^2}(\Phi(L,K,t)) = K$,
(b) the Jacobian matrix of $\Theta_1\circ\Phi$ has determinant $-1/18$.
\[Phi-\]There exist explicit injections $$\Phi^{(1)},\Phi^{(2)} :\Omega^+\times {{\mathbb{R}}}\longrightarrow V_{{{\mathbb{R}}},x^2-y^2}^0{\text{ and }}\Phi^{(3)},\Phi^{(4)} :\Omega^-\times {{\mathbb{R}}}\longrightarrow V_{{{\mathbb{R}}},x^2-y^2}^0,$$ defined as in (\[Phi- def\]), with $$V_{{{\mathbb{R}}},x^2-y^2}^0 = \Phi^{(1)}(\Omega^+ \times{{\mathbb{R}}}) \sqcup \Phi^{(2)}(\Omega^+ \times{{\mathbb{R}}}) \sqcup \Phi^{(3)}(\Omega^- \times{{\mathbb{R}}}) \sqcup \Phi^{(4)}(\Omega^- \times{{\mathbb{R}}})$$ such that
(a) we have $L_{x^2-y^2}(\Phi^{(i)}(L,K,t)) = L$ and $K_{x^2-y^2}(\Phi^{(i)}(L,K,t)) = K$,
(b) the Jacobian matrix of $\Theta_1\circ\Phi^{(i)}$ has determinant $-1/18$,
for all $i=1,2,3,4$.
In view of (\[reducible f shape\]), we shall give another parametrization of $V_{{{\mathbb{R}}},f}$ when $\gamma=0$, which does not require reducing to the form $x^2 - y^2$ via Lemma \[Psi lemma\].
\[Phi f\]Suppose that $\gamma=0$. Then, there exist explicit injections $$\Phi_f^{(1)},\Phi_f^{(2)}:\Omega^0 \times{{\mathbb{R}}}\longrightarrow V_{{{\mathbb{R}}},f}^0,$$ defined as in (\[Phi f def\]), with $$V_{{{\mathbb{R}}},f}^0 = \Phi_f^{(1)}(\Omega^0\times{{\mathbb{R}}})\sqcup \Phi_f^{(2)}(\Omega^0\times{{\mathbb{R}}})$$ such that
(a) we have $L_{f}(\Phi^{(i)}(L,K,t)) = L$ and $K_{f}(\Phi^{(i)}(L,K,t)) = K$,
(b) the Jacobian matrix of $\Theta_2\circ\Phi_f^{(i)}$ has determinant $-1/18$,
for both $i=1,2$.
For $t\in {{\mathbb{R}}}$, we shall use the notation $$\label{T def}T^+(t) = \begin{pmatrix} \cos t & \sin t \\ - \sin t & \cos t \end{pmatrix}{\text{ and }}T^-(t) = \begin{pmatrix}\cosh t & \sinh t\\\sinh t & \cosh t\end{pmatrix},$$ which is an element of $O_{x^2+y^2}({{\mathbb{R}}})$ and $O_{x^2-y^2}({{\mathbb{R}}})$, respectively.
Positive definite case {#para pos def}
----------------------
Define $$\label{Phi+ def}\Phi:\Omega^+ \times [-\pi/4,\pi/4) \longrightarrow V_{{{\mathbb{R}}},{x^2+y^2}}^0;\hspace{1em}\Phi(L,K,t) = (F_{(L,K)})_{T^+(t)},$$ where $$F_{(L,K)}(x,y) = \frac{-3L + \sqrt{L^2 + 4K}}{24}x^4 + \frac{-L - \sqrt{L^2 + 4K}}{4} x^2 y^2 + \frac{-3L + \sqrt{L^2 + 4K}}{24} y^4.$$ The image of $\Phi$ lies in $V_{{{\mathbb{R}}},{x^2+y^2}}$ by (\[abc family\]) and (\[V bijection\]). Using Propositions \[LK invariant\] and \[explicit LK\] (a), it is easy to check that Proposition \[Phi+\] (a) holds.\
Now, by (\[abc family\]), an arbitrary $F\in V_{{{\mathbb{R}}},x^2+y^2}^0$ has the shape $$F(x,y) = a_4x^4 + a_3x^3y + a_2x^2y^2 - a_3xy^3 + a_4y^4.$$ Write $L = L_{x^2+y^2}(F)$ and $K = K_{x^2+y^2}(F)$. Note that $(L,K)\in\Omega^+$ because $\Delta(F)>0$ by (\[Delta LK\]). For $t\in{{\mathbb{R}}}$, a direct computation yields $$F_{T^+(t)}(x,y) = A(t)x^4 + B(t)x^3y + C(t)x^2y^2 - B(t)xy^3 + A(t)y^4,$$ where $$\begin{cases}A(t) = \dfrac{6a_4+a_2}{8} + \dfrac{2a_4-a_2}{8}\cos(4t) - \dfrac{a_3}{4}\sin(4t),\vspace{2mm}\\
B(t) = a_3\cos(4t) + \dfrac{2a_4-a_2}{2}\sin(4t),\vspace{2mm}\\
C(t) = \dfrac{6a_4+a_2}{4} - \dfrac{3(2a_4-a_2)}{4}\cos(4t) + \dfrac{3a_3}{2}\sin(4t).
\end{cases}$$ It is not hard to show that there exists a unique $t_0\in(-\pi/4,\pi/4]$ such that $B(t_0)=0$ and $2A(t_0)-C(t_0)>0$. Put $(A,C) = (A(t_0),C(t_0))$. Then, we have $$(L,K) = (L_{x^2+ y^2}(F_{T^+(t_0)}),K_{x^2+y^2}(F_{T^+(t_0)})) = (-6A - C, -2C(6A- C))$$ by Propositions \[LK invariant\] and \[explicit LK\] (a). We solve that $F_{T^+(t_0)}=F_{(L,K)}$, or equivalently $$F = (F_{(L,K)})_{T^+(-t_0)} = \Phi(L,K,-t_0).$$ Since $-t_0\in[-\pi/4,\pi/4)$ is uniquely determined by $F$, this shows that $\Phi$ is a bijection.\
Finally, the above calculation also yields $$(\Theta_1\circ\Phi)(L,K,t) =(\Phi_{1}(L,K,t),\Phi_2(L,K,t),\Phi_3(L,K,t)),$$ where $$\label{pos def para}
\begin{cases} \Phi_{1}(L,K,t) = -\dfrac{L}{8} + \dfrac{\sqrt{L^2 + 4K}}{24} \cos(4t),\vspace{2mm}\\
\Phi_{2}(L,K,t) = \dfrac{\sqrt{L^2 + 4K}}{6} \sin(4t),\vspace{2mm}\\
\Phi_{3}(L,K,t) = -\dfrac{L}{4} - \dfrac{\sqrt{L^2 + 4K}}{4} \cos(4t).
\end{cases}$$ By a direct computation, we then see that Proposition \[Phi+\] (b) holds.
Indefinite case {#geometric para indefinite}
---------------
Define $$\label{Phi- def}\begin{cases} \Phi^{(i)}: \Omega^+ \times {{\mathbb{R}}}\longrightarrow V_{{{\mathbb{R}}},x^2-y^2}^0; \hspace{1em}\Phi^{(i)}(L,K,t) = (F_{(L,K)}^{(i)})_{T^-(t)} & \mbox{for $i=1,2$},\\
\Phi^{(i)}: \Omega^- \times {{\mathbb{R}}}\longrightarrow V_{{{\mathbb{R}}},x^2-y^2}^0; \hspace{1em}\Phi^{(i)}(L,K,t) = (F_{(L,K)}^{(i)})_{T^-(t)} & \mbox{for $i=3,4$},
\end{cases}$$ where $$\begin{aligned}
F_{(L,K)}^{(i)}(x,y)&=\frac{3L + (-1)^{i}\sqrt{L^2 + 4K}}{24}x^4 + \frac{-L +(-1)^{i}\sqrt{L^2 + 4K}}{4} x^2 y^2 \\&\hspace{7.5cm}+\frac{3L + (-1)^{i}\sqrt{L^2 + 4K}}{24}y^4\end{aligned}$$ for $i=1,2$, and $$F_{(L,K)}^{(i)}(x,y)=\frac{(-1)^{i}\sqrt{2L^2-K}}{3}x^3y-Lx^2y^2+\frac{(-1)^{i}\sqrt{2L^2-K}}{3}xy^3$$ for $i=3,4$. The images of $\Phi^{(1)},\Phi^{(2)},\Phi^{(3)},\Phi^{(4)}$ lie in $V_{{{\mathbb{R}}},{x^2-y^2}}$ by (\[abc family\]) and (\[V bijection\]). Using Propositions \[LK invariant\] and \[explicit LK\] (a), it is easy to check that Proposition \[Phi-\] (a) holds.\
Now, by (\[abc family\]), an arbitrary $F\in V_{{{\mathbb{R}}},x^2-y^2}^0$ has the shape $$F(x,y) = a_4x^4 + a_3x^3y + a_2x^2y^2 + a_3xy^3 + a_4y^4.$$ Write $L = L_{x^2-y^2}(F)$ and $K = K_{x^2-y^2}(F)$. For $t\in{{\mathbb{R}}}$, a direct computation yields $$F_{T^-(t)}(x,y) = A(t)x^4 + B(t)x^3y + C(t)x^2y^2 + B(t)xy^3 + A(t)y^4,$$ where $$\begin{cases}
A(t) = \dfrac{6a_4-a_2}{8} + \dfrac{2a_4+a_2}{8}\cosh(4t) + \dfrac{a_3}{4}\sinh(4t),\vspace{2mm}\\
B(t) = a_3\cosh(4t) + \dfrac{2a_4+a_2}{2}\sinh(4t),\vspace{2mm}\\
C(t) = -\dfrac{6a_4-a_2}{4} + \dfrac{3(2a_4+a_2)}{4}\cosh(4t) + \dfrac{3a_3}{2} \sinh(4t).
\end{cases}$$ Note that $\frac{d}{dt}A(t) = \frac{1}{2}B(t)$. It is not hard to check that:
- If $\Delta(F)>0$, then there is a unique $t_0\in{{\mathbb{R}}}$ such that $B(t_0)=0$.
- If $\Delta(F)<0$, then $B(t)\neq0$ for all $t\in{{\mathbb{R}}}$, and there is a unique $t_0\in{{\mathbb{R}}}$ such that $A(t_0)=0$.
Put $(A,B,C) = (A(t_0),B(t_0),C(t_0))$. Then, we have $$\begin{aligned}
(L,K) & = (L_{x^2-y^2}(F_{T^-(t_0)}), K_{x^2-y^2}(F_{T^-(t_0)})) = \begin{cases} (6A - C, 2C(6A+C)) & \mbox{if $\Delta(F)>0$},\\
(-C, -9B^2 + 2C^2)&\mbox{if $\Delta(F)<0$}.\end{cases}\end{aligned}$$ by Propositions \[LK invariant\] and \[explicit LK\] (a). We solve that $F_{T^-(t_0)} = F_{(L,K)}^{(i)}$, or equivalently $$F = (F_{(L,K)}^{(i)})_{T^-(-t_0)} = \Phi^{(i)}(L,K,-t_0),\mbox{ for exactly one $i\in\{1,2,3,4\}$}.$$ Since $t_0$ is uniquely determined by $F$, this shows that $\Phi^{(1)},\Phi^{(2)},\Phi^{(3)},\Phi^{(4)}$ are all injections, and that the stated disjoint union holds.\
Finally, the above calculation also yields $$(\Theta_1\circ\Phi^{(i)})(L,K,t) = (\Phi_1^{(i)}(L,K,t), \Phi_2^{(i)}(L,K,t), \Phi_3^{(i)}(L,K,t)),$$ where $$\label{indef para 1}\begin{cases}
\Phi_1^{(i)}(L,K,t) = \dfrac{L}{8} + \dfrac{(-1)^i\sqrt{L^2+4K}}{24}\cosh(4t),\vspace{2mm} \\
\Phi_2^{(i)}(L,K,t) = \dfrac{(-1)^i\sqrt{L^2+4K}}{6}\sinh(4t),\vspace{2mm} \\
\Phi_3^{(i)}(L,K,t) = -\dfrac{L}{4} + \dfrac{(-1)^i\sqrt{L^2+4K}}{4}\cosh(4t),\\
\end{cases}$$ for $i=1,2$, and $$\label{indef para 2}\begin{cases}
\Phi_1^{(i)}(L,K,t) = \dfrac{L}{8} - \dfrac{L}{8}\cosh(4t) + \dfrac{(-1)^i\sqrt{2L^2-K}}{12}\sinh(4t),\vspace{2mm} \\
\Phi_2^{(i)}(L,K,t) = \dfrac{(-1)^i\sqrt{2L^2-K}}{3}\cosh(4t) - \dfrac{L}{2}\sinh(4t),\vspace{2mm} \\
\Phi_3^{(i)}(L,K,t) = -\dfrac{L}{4} - \dfrac{3L}{4}\cosh(4t) + \dfrac{(-1)^i\sqrt{2L^2-K}}{2}\sinh(4t),\\
\end{cases}$$ for $i=3,4$. By a direct computation, we then see that Proposition \[Phi-\] (b) holds.
Reducible case {#red para section}
--------------
Suppose $\gamma=0$. For $t\in{{\mathbb{R}}}$, put $$T(t) = \begin{pmatrix} e^{-t} & 0 \\[0.5ex] \dfrac{2\alpha \sinh t}{\beta} & e^{t} \end{pmatrix},$$ which is an element of $O_f({{\mathbb{R}}})$. Define $$\label{Phi f def}\Phi_f^{(i)}:\Omega^0 \times{{\mathbb{R}}}\longrightarrow V_{{{\mathbb{R}}},f}^0 ; \hspace{1em}\Phi_f^{(i)}(L,K,t) = (F_{f,(L,K)}^{(i)})_{T(t)}\hspace{1em}\mbox{for $i=1,2$},$$ where $$\begin{aligned}
F_{f,(L,K)}^{(i)}(x,y)&=\left(\frac{L^2+(-1)^i72\alpha^2L+4K+144\alpha^4}{(-1)^i144\beta^2}\right) x^4 + \left(\frac{\alpha L + (-1)^i 4 \alpha^3}{\beta}\right) x^3 y\\&\hspace{2.5cm}+ \left(\frac{L +(-1)^i 12\alpha^2}{2}\right)x^2 y^2 +(-1)^i 4 \alpha \beta xy^3 + (-1)^i\beta^2 y^4.\end{aligned}$$ The images of $\Phi_f^{(1)},\Phi_f^{(2)}$ lie in $V_{{{\mathbb{R}}},f}$ by (\[abc family 2\]) and (\[V bijection\]). Using Propositions \[LK invariant\] and \[explicit LK\] (b), it is easy to check that Proposition \[Phi f\] (a) holds.\
Now, by (\[abc family 2\]), an arbitrary $F\in V_{{{\mathbb{R}}},f}^0$ has the shape $$\label{reducible generic}F(x,y) = a_4x^4 + \left(\frac{2\alpha(\beta^2 a_2 - 4 \alpha^2a_0)}{\beta^3}\right)x^3 y + a_2x^2 y^2 +\left(\frac{4 \alpha a_0}{\beta}\right) xy^3 + a_0y^4.$$ Write $L = L_f(F)$ and $K= K_f(F)$. For $t\in{{\mathbb{R}}}$, a direct computation yields $$F_{T(t)}(x,y) = A(t)x^4 + (*)x^3y + B(t)x^2y^2 + (*)xy^3 + C(t)y^4,$$ where $$\label{red ABC} \begin{cases}
A(t) = e^{-4t}a_4 + \dfrac{\alpha^2}{\beta^2}(e^{4t}-1)e^{-4t}a_2 + \dfrac{\alpha^4}{\beta^4}(e^{4t}-1)(e^{4t}-5)e^{-4t}a_0\vspace{2mm},\\
B(t) = a_2 + \dfrac{6\alpha^2}{\beta^2}(e^{4t}-1)a_0,\vspace{2mm}\\
C(t) = e^{4t}a_0.
\end{cases}$$ Since $\Delta(F)\neq0$, we have $(-1)^i a_0 > 0$ for a unique $i\in\{1,2\}$, and there is a unique $t_{0}\in{{\mathbb{R}}}$ such that $C(t_{0}) = (-1)^i\beta^2$. Put $(A,B) = (A(t_0),B(t_0))$. Then, we have $$\begin{aligned}
(L,K) &= (L_f(F_{T(t_0)}) , K_f(F_{T(t_0)})) \\&= (2B - (-1)^i12\alpha^2, -B^2 + (-1)^i36\beta^2A - (-1)^i24\alpha^2B + 144\alpha^4),\end{aligned}$$ by Propositions \[LK invariant\] and \[explicit LK\] (b). We solve that $F_{T(t_0)} = F_{f,(L,K)}^{(i)}$, or equivalently $$F = (F_{f,(L,K)}^{(i)})_{T(-t_0)} = \Phi_f^{(i)}(L,K,-t_0).$$ Since $t_0$ and $i$ are uniquely determined by $F$, this shows that $\Phi_f^{(1)}$ and $\Phi_f^{(2)}$ are both injections, and that the stated disjoint union holds.\
Finally, the above calculation also yields $$(\Theta_2\circ\Phi_f^{(i)})(L,K,t) = (\Phi^{(i)}_{f,1}(L,K,t),\Phi_{f,2}^{(i)}(L,K,t),\Phi^{(i)}_{f,3}(L,K,t)),$$ where $$\label{red para}
\begin{cases} \Phi^{(i)}_{f,1}(L,K,t)=\dfrac{(-1)^i e^{-4t}}{144\beta^2}(L^2+4K) + \dfrac{\alpha^2}{2\beta^2}L
+ \dfrac{(-1)^i\alpha^4e^{4t}}{\beta^2}, \vspace{2mm}\\
\Phi_{f,2}^{(i)}(L,K,t)=\dfrac{L}{2}+(-1)^i6\alpha^2 e^{4t},\vspace{2mm}\\
\Phi^{(i)}_{f,3}(L,K,t)=(-1)^i\beta^2 e^{4t}.
\end{cases}$$ By a direct computation, we then see that Proposition \[Phi f\] (b) holds.
Definition of a bounded semi-algebraic set {#proof sec}
==========================================
Throughout this section, let $f(x,y) = \alpha x^2 + \beta xy + \gamma y^2$ be an integral and primitive binary quadratic form with $\Delta(f)\neq0$ and $\alpha>0$, in the shape (\[reducible f shape\]) whenever $f$ is reducible. As we have already explained in Subsection \[notation sec\], the proof of Theorem \[Small Gal MT\] is reduced to counting points in the lattices in (\[Lambda def\]), which in turn amounts to certain volume computations, by the result below.
\[Davenport\]Let ${{\mathcal{R}}}$ be a bounded semi-algebraic multi-set in ${{\mathbb{R}}}^n$ having maximum multiplicity $m$ and which is defined by at most $k$ polynomial inequalities, each having degree at most $\ell$. Then, the number of integral lattice points (counted with multiplicity) contained in the region ${{\mathcal{R}}}$ is $${\operatorname{Vol}}({{\mathcal{R}}}) + O(\max\{{\operatorname{Vol}}({\overline}{{{\mathcal{R}}}}), 1\}),$$ where ${\operatorname{Vol}}({\overline}{{{\mathcal{R}}}})$ denotes the greatest $d$-dimensional volume of any projection of ${{\mathcal{R}}}$ onto a coordinate subspace by equating $n-d$ coordinates to zero, with $1 \leq d \leq n-1$. The implied constant in the second summand depends only on $n,m,k,\ell$.
This is a result of Davenport [@Dav], and the above formulation is due to Bhargava and Shankar in [@BhaSha Proposition 2.6].
For $X>0$, define $$V_{{{\mathbb{R}}},f}^0(X) = \{F\in V_{{{\mathbb{R}}},f}^0 : H_f(F)\leq X\}{\text{ and }}V_{{{\mathbb{Z}}},f}^0(X) = \{F\in V_{{{\mathbb{Z}}},f}^0 : H_f(F)\leq X\}.$$ However, to prove Theorem \[Small Gal MT\], we cannot apply Proposition \[Davenport\] directly to $$\Theta_{w(f)}(V_{{{\mathbb{R}}},f}^0({{\mathbb{R}}})),\mbox{ where }w(f) = \begin{cases}
1 & \mbox{if $f$ is irreducible},\\ 2 & \mbox{if $f$ is reducible},
\end{cases}$$ as in Subsection \[notation sec\], to count the lattice points in $\Theta_{w(f)}(V_{{{\mathbb{Z}}},f})\subset\Lambda_{f,w(f)}$ because
(1) the set $\Theta_{w(f)}(V_{{{\mathbb{R}}},f}^0(X))$ is unbounded when $f$ is indefinite,
(2) distinct forms in $V_{{{\mathbb{Z}}},f}^0(X)$ might be ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent.
Recall (\[Omega def\]) and define $$\Omega^*(X) = \{(L,K)\in\Omega^*: \max\{L^2,|K|\}\leq X\}\mbox{ for }*\in\{0,+,-\}.$$ In the notation of Lemma \[Psi lemma\] as well as Propositions \[Phi+\], \[Phi-\], and \[Phi f\], we have $$\label{para} V_{{{\mathbb{R}}},f}^0(X) = \begin{cases}
(\Psi_f\circ\Phi)(\Omega^+(X)\times [-\pi/4,\pi/4)),\\
\bigsqcup\limits_{i=1}^{2}(\Psi_f\circ\Phi^{(i)})(\Omega^+(X)\times {{\mathbb{R}}})\sqcup\bigsqcup\limits_{i=3}^{4}(\Psi_f\circ\Phi^{(i)})(\Omega^-(X)\times {{\mathbb{R}}}),\\
\bigsqcup\limits_{i=1}^{2}\Phi_f^{(i)}(\Omega^0(X)\times{{\mathbb{R}}}),
\end{cases}$$ respectively, if $f$ is positive definite, indefinite, and reducible. We shall overcome the two issues above by restricting the values for $t\in{{\mathbb{R}}}$.\
For brevity, in this section, write $$D_f = |\Delta(f)|{\text{ and }}\delta_f = D_f/4,$$ as in Theorem \[Small Gal MT\] and Lemma \[Psi lemma\], respectively.
\[S(X) def\] If $f$ is positive definite, define $${{\mathcal{S}}}_f(X) = (\Psi_f\circ\Phi)(\Omega^+(X) \times [-\pi/4,\pi/4)).$$ If $f$ is reducible, define $${{\mathcal{S}}}_f(X) = \bigsqcup_{i=1}^{2}\Phi_f^{(i)}(\Omega^0(X)\times [t_{f,1},t_{f,2}]) \mbox{ for }t_{f,1} = -\frac{\log 8}{4} {\text{ and }}t_{f,2} =\frac{\log(5X/18)}{4}.$$ If $f$ is indefinite and irreducible, define $${{\mathcal{S}}}_f(X) = \bigsqcup_{i=1}^2(\Psi_f\circ\Phi^{(i)})(\Omega^+(X)\times [0,t_{D_f})) \sqcup \bigsqcup_{i=3}^4(\Psi_f\circ\Phi^{(i)})(\Omega^-(X)\times [0,t_{D_f})),$$ where $t_{D_f}$ is defined as in Theorem \[Small Gal MT\] (c).
The goal of this section to prove the following preliminary results and estimates:
\[issue1 prop\] The set $\Theta_{w(f)}(S_f(X))$ is bounded, semi-algebraic, and definable by an absolutely bounded number of polynomial inequalities whose degrees are absolutely bounded.
\[issue2 prop\]The following statements hold.
(a) A form in $V_{{{\mathbb{Z}}},f}^0(X)$ is ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent to at least one form in ${{\mathcal{S}}}_f(X)$.
(b) A form in $V_{{{\mathbb{Z}}},f}^0(X)$ for which $\Delta(F)\neq\square$ is ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent to exactly $r_f$ forms in ${{\mathcal{S}}}_f(X)$, where $r_f$ is defined as in Theorem \[Small Gal MT\].
Alternative description
-----------------------
First, we shall give an alternative description of the set ${{\mathcal{S}}}_f(X)$ in terms of the coefficients of the forms in $V_{{{\mathbb{R}}},f}^0(X)$.
\[Sf positive definite\]If $f$ is positive definite, then ${{\mathcal{S}}}_f(X) = V_{{{\mathbb{R}}},f}^0(X)$.
This is clear from (\[para\]).
\[Sf reducible\] If $f$ is reducible, then $${{\mathcal{S}}}_f(X) = \{F\in V_{{{\mathbb{R}}},f}^0(X) : \beta^2/8 \leq |C_F|\leq 5\beta^2X/18\},$$ where $C_F$ denotes the $y^4$-coefficient of $F$.
For $i=1,2$ and for any $F=\Phi_f^{(i)}(L,K,t)$, we have $C_{F} = (-1)^i\beta^2 e^{4t}$ by (\[red para\]), and the claim is then clear from (\[para\]).
\[Sf indefinite\]If $f$ is an indefinite and irreducible, then $${{\mathcal{S}}}_f(X) = \{F\in V_{{{\mathbb{R}}},f}^0(X): 1\leq E_{f,1}(F)Z_f(F)/E_{f,2}(F)< e^{8t_{D_f}}\},$$ where in the notation of Proposition \[explicit LK\] (a), we define $$E_{f,1}(F) = L_{f,1}(F) - \sqrt{D_f}L_{f,2}(F) {\text{ and }}E_{f,2}(F) = L_{f,1}(F) + \sqrt{D_f}L_{f,2}(F),$$ and for $F$ in the image of $\Psi_f\circ\Phi^{(i)}$, we define $$Z_f(F) = \begin{cases} 1 &\mbox{for $i=1,2$},\\ \dfrac{L_f(F)^2 + 4K_f(F)}{(4L_f(F) - (-1)^i2\sqrt{2L_f(F)^2 - K_f(F)})^2} &\mbox{for $i=3,4$}.\end{cases}$$
For $i=1,2,3,4$, consider $F = (\Psi_f\circ\Phi^{(i)})(L,K,t)$. For $k=1,2$, we have $$E_{f,k}(F) = \begin{cases} (-1)^i 2\alpha^2\sqrt{L_f(F)^2+4K_f(F)}e^{(-1)^{k+1}4t}/3 & \mbox{if $i=1,2$},\\ -2\alpha^2(3L_f(F) + (-1)^{k+i}2\sqrt{2L_f(F)^2-K_f(F)})e^{(-1)^{k+1}4t}/3 &\mbox{if $i=3,4$},\end{cases}$$ by a direct computation using (\[general parameter\]), (\[indef para 1\]), and (\[indef para 2\]). We then see that $$E_{f,1}(F)Z_f(F)/E_{f,2}(F) = e^{8t},$$ from which the claim follows.
Proof of Proposition \[issue1 prop\]
------------------------------------
From (\[pos def para\]), (\[indef para 1\]), (\[indef para 2\]), and (\[red para\]), it is clear that the set ${{\mathcal{S}}}_f(X)$ is bounded. Thus, it remains to show that ${{\mathcal{S}}}_f(X)$ is a semi-algebraic set definable by an absolutely bounded number of polynomial inequalities whose degrees are absolutely bounded.
### The case when $f$ is positive definite or reducible
The claim follows immediately from Lemmas \[Sf positive definite\] and \[Sf reducible\] as well as Proposition \[explicit LK\].
### The case when $f$ is indefinite and irreducible
The only problem is that $Z_f(F)$ is not a polynomial in the $x^4$, $x^3y$, and $x^2y^2$-coefficients of $F$. We shall resolve this issue in Lemma \[poly ineq\] below. The claim then follows from Lemma \[Sf indefinite\] and Proposition \[explicit LK\].
\[poly ineq\]For $i=3,4$, let $F\in(\Psi_f\circ\Phi^{(i)})(\Omega^-\times{{\mathbb{R}}})$. Then, the condition $$1\leq E_{f,1}(F)Z_f(F)/E_{f,2}(F) < e^{8t_{D_f}}$$ is equivalent to an absolutely bounded number of polynomial inequalities in the variables $L_f(F),K_f(F),E_{f,1}(F),E_{f,2}(F)$ whose degrees are absolutely bounded.
For brevity, define $$\begin{aligned}
Y_{f,1}(F) &= - E_{f,1}(F)(L_f(F)^2 + 4K(F)) + E_{f,2}(F)(17L_f(F)^2 - 4K_f(F)),\\
Y_{f,2}(F) & = - E_{f,1}(F)(L_f(F)^2 + 4K_f(F)) + e^{8t_{D_f}}E_{f,2}(F)(17L_f(F)^2 - 4K_f(F)),\end{aligned}$$ as well as write $$(L,K, E_1,E_2,Z, Y_1,Y_2) = (L_f(F), K_f(F), E_{f,1}(F), E_{f,2}(F),Z_f(F), Y_{f,1}(F), Y_{f,2}(F)).$$ Note that $L^2+4K<0$ by (\[Delta LK\]) because $\Delta(F)<0$. This implies that $Z<0$ and so the stated condition may be rewritten as $$\label{remove abs}\begin{cases}E_2 \leq E_1Z < e^{8t_{D_f}}E_2&\text{if $E_2 > 0$, which is equivalent to $i=3$},\\
E_2 \geq E_1Z > e^{8t_{D_f}}E_2&\text{if $E_2<0$, which is equivalent to $i=4$}.\end{cases}$$ By rearranging, we may further rewrite the above as $$\label{xmp ineq}12E_2 L\sqrt{2L^2 - K}\leq (-1)^iY_1 {\text{ and }}12e^{8t_{D_f}}E_2L\sqrt{2L^2 - K} > (-1)^iY_2.$$ From here, we shall consider the different possibilities for the signs of $E_2$, $L$, $Y_1,Y_2$. For example, when $E_2 > 0$ and $L \geq 0$, the above is equivalent to $Y_1\leq 0$ and $$\begin{cases}
(12E_2 L)^2(2L^2 - K) \leq Y_1^2&\text{if $Y_2 > 0$},\\
(12E_2 L)^2(2L^2 - K) \leq Y_1^2{\text{ and }}(12e^{8t_{D_f}}E_2L)^2(2L^2 - K) > Y_2^2&\text{if $Y_2 \leq 0$}.
\end{cases}$$ The other cases are analogous. We then see that the claim holds.
Integral orthogonal groups
--------------------------
We shall require an explicit description of $$O_f({{\mathbb{Z}}}) = O_f({{\mathbb{R}}})\cap {\operatorname{GL}}_2({{\mathbb{Z}}}).$$ In the notation of Lemma \[Psi lemma\], observe that $$\label{O relation}
O_f({{\mathbb{R}}}) = \begin{cases} T_f^{-1}(O_{x^2+y^2}({{\mathbb{R}}}))T_f &\mbox{if $f$ is positive definite},\\ T_f^{-1}(O_{x^2-y^2}({{\mathbb{R}}})) T_f &\mbox{if $f$ is indefinite}.\end{cases}$$ Moreover, it is well-known that $$\begin{aligned}
\label{O principal} O_{x^2+y^2}({{\mathbb{R}}}) &= \{J_k T^+(t) : k\in\{1,4\}{\text{ and }}t\in{{\mathbb{R}}}\},\\\notag
O_{x^2-y^2}({{\mathbb{R}}}) & = \{\pm J_kT^-(t) : k\in\{1,2,3,4\} {\text{ and }}t\in{{\mathbb{R}}}\},\end{aligned}$$ where $T^+(t)$ and $T^-(t)$ are defined as in (\[T def\]), and $$\label{J def}J_1 = \begin{pmatrix} 1&0 \\ 0&1\end{pmatrix},\, J_2 = \begin{pmatrix} 0&1 \\ 1&0\end{pmatrix},\,J_3 = \begin{pmatrix} 0&1 \\ -1&0\end{pmatrix},\, J_4 = \begin{pmatrix} 1&0 \\ 0&-1\end{pmatrix}.$$ We shall need the following lemma.
\[Dickson ambiguous\]Suppose that $T\in O_f({{\mathbb{Z}}})\setminus\{\pm I_{2\times 2}\}$ has finite order. Then, the form $f$ is ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent to a form of the shape $$\begin{cases}
x^2 + y^2,\, x^2 + xy + y^2, {\text{ or }}ax^2 + bxy - ay^2 & \mbox{if $\det(T) = 1$},\\
xy,\, x^2-y^2,\,ax^2+cy^2,{\text{ or }}ax^2 + bxy + ay^2&\mbox{if $\det(T)=-1$},
\end{cases}$$ for some integers $a,b$, and $c$.
By [@Newman Chapter IX], for example, a finite cyclic subgroup of ${\operatorname{GL}}_2({{\mathbb{Z}}})$ not contained in $\{\pm I_{2\times2}\}$ is conjugate to the subgroup generated by one of the following: $$\begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix},\,
\begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix},\,
\begin{pmatrix}0 & -1 \\ 1 & 1 \end{pmatrix},\,
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\,
\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}.$$ We then deduce that there exists $P\in{\operatorname{GL}}_2({{\mathbb{Z}}})$ such that $Q = P^{-1}TP$ is equal to one of the following matrices up to sign: $$\begin{pmatrix} 0&1 \\ -1& -1\end{pmatrix},\,
\begin{pmatrix} -1&-1 \\ 1& 0\end{pmatrix},\,
\begin{pmatrix} 0&1 \\ -1& 0\end{pmatrix},\,
\begin{pmatrix} 1&0 \\ 0& -1\end{pmatrix},\,
\begin{pmatrix} 0&1 \\ 1& 0\end{pmatrix}.$$ Since $f$ is primitive with $\alpha>0$ by assumption and $(f_P)_Q = \pm f_P$, we then check that $f_P$ must have one of the stated shapes.
\[Of pos def\]Suppose that $f$ is positive definite. Then, we have $$O_f({{\mathbb{Z}}}) = \{\pm I_{2\times2}\}$$ if $f$ is not ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent to the forms below, and the group $O_f({{\mathbb{Z}}})$ is equal to $$\begin{cases}
\{\pm I_{2\times 2},\pm\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right),\pm\left(\begin{smallmatrix} 1& 0\\ 0 & -1 \end{smallmatrix}\right),\pm\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)\}&\text{\hspace{-2.5mm}if $f(x,y) = x^2+y^2$},\\[0.5ex]
\{\pm I_{2\times2}\pm\left(\begin{smallmatrix}1&1\\-1&0\end{smallmatrix}\right),
\pm\left(\begin{smallmatrix}0&-1\\1&1\end{smallmatrix}\right),&\text{\hspace{-2.5mm}if $f(x,y) = x^2 + xy + y^2$},\\\hspace{1.25cm}
\pm\left(\begin{smallmatrix} 1& 1\\ 0 & -1 \end{smallmatrix}\right),\pm\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right),\pm\left(\begin{smallmatrix} -1& 0\\ 1 & 1 \end{smallmatrix}\right)\}&\\[0.5ex]
\{\pm I_{2\times2},\pm\left(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right)\}& \text{\hspace{-2.5mm}if $f(x,y) = \alpha x^2 + \gamma y^2$ for $\alpha \neq\gamma$},\\[0.5ex]
\left\{\pm I_{2 \times 2}, \pm \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right) \right\} & \text{\hspace{-2.5mm}if $f(x,y) = \alpha x^2 + \beta xy + \alpha y^2$ for $\beta\notin\{0,\alpha\}$}.
\end{cases}$$
Elements in $O_f({{\mathbb{Z}}})$ have finite order by (\[O relation\]) and so the first claim follows from Lemma \[Dickson ambiguous\]. Using (\[O relation\]), we compute that elements in $O_f({{\mathbb{R}}})$ are of the forms $$\begin{pmatrix}
\phi_t + \frac{\beta \psi_t}{2\sqrt{\delta_f}}& \frac{\gamma\psi_t}{\sqrt{\delta_f}}\vspace{1mm}\\
-\frac{\alpha\psi_t}{\sqrt{\delta_f}} & \phi_t - \frac{\beta\psi_t}{2\sqrt{\delta_f}}
\end{pmatrix}
{\text{ and }}\begin{pmatrix}
\phi_t - \frac{\beta\psi_t}{2\sqrt{\delta_f}}&\frac{\beta}{\alpha}\left(\phi_t- \frac{\beta\psi_t}{2\sqrt{\delta_f}}\right) +\frac{\gamma\psi_t}{\sqrt{\delta_f}}\vspace{2mm}\\
\frac{\alpha\psi_t}{\sqrt{\delta_f}} & - \phi_t - \frac{\beta\psi_t}{2\sqrt{\delta_f}}
\end{pmatrix},$$ where $t\in{{\mathbb{R}}}$ and $(\phi_t,\psi_t) = (\cos t,\sin t)$. With the help of the proof of Lemma \[Dickson ambiguous\], it is not hard to check that $O_f({{\mathbb{Z}}})$ is as claimed.
\[Of reducible\]Suppose that $f$ is reducible. Then, the group $O_f({{\mathbb{Z}}})$ is equal to $$\begin{cases}
\left \{ \pm I_{2\times2}\right \} &\text{if $\beta\nmid\alpha^2+1$ and $\beta\nmid \alpha^2-1$},\\[0.5ex]
\left\{\pm I_{2\times 2},\pm\left(\begin{smallmatrix}\alpha&&\beta\\-\frac{\alpha^2+1}{\beta}&&-\alpha\end{smallmatrix}\right)\right\}&\text{if $\beta\mid\alpha^2+1$ and $\beta\nmid \alpha^2-1$},\\[1ex]
\left\{\pm I_{2\times2},\pm\left(\begin{smallmatrix}\alpha&&\beta\\-\frac{\alpha^2-1}{\beta}&&-\alpha\end{smallmatrix}\right)\right\}&\text{if $\beta\nmid\alpha^2+1$ and $\beta\mid \alpha^2-1$},\\[1ex]
\left\{\pm I_{2\times2}, \pm \left(\begin{smallmatrix} -1 & 0 \\2 & 1 \end{smallmatrix}\right),\pm\left(\begin{smallmatrix} 1 & 1 \\ -2 & -1\end{smallmatrix}\right),\pm\left(\begin{smallmatrix}1 & 1 \\ 0 & -1\end{smallmatrix}\right)\right\}&\text{if }f(x,y) = x^2 + xy,\\[1ex]
\left\{ \pm I_{2\times 2}, \pm\left(\begin{smallmatrix} -1 & 0 \\ 1 & 1 \end{smallmatrix}\right),\pm\left(\begin{smallmatrix} 1 & 2 \\ -1 & -1 \end{smallmatrix}\right),\left(\begin{smallmatrix} 1 & 2 \\ 0 & -1 \end{smallmatrix}\right)\right\}&\text{if }f(x,y) = x^2 + 2xy.
\end{cases}$$
Using (\[O relation\]), we compute that elements in $O_f({{\mathbb{R}}})$ are of the forms $$\pm\begin{pmatrix} \phi_t - \psi_t & 0 \vspace{2mm}\\ \frac{2\alpha\psi_t}{\beta} & \phi_t + \psi_t\end{pmatrix}
{\text{ and }}\pm\begin{pmatrix} \phi_t + \psi_t & \frac{\beta}{\alpha}(\phi_t +\psi_t) \vspace{2mm}\\ -\frac{2\alpha\psi_t}{\beta} & -\phi_t- \psi_t\end{pmatrix},$$ where $t\in{{\mathbb{R}}}$ and $(\phi_t,\psi_t)\in\{(\cosh t,\sinh t),(\sinh t,\cosh t)\}$. For the matrix on the left to have integer entries, necessarily $$2\cosh t,2\sinh t\in{{\mathbb{Z}}}\mbox{ so }(2\cosh t, 2\sinh t) = (2,0).$$ Similarly, for the matrix on the right to have integer entries, necessarily $$2\alpha\cosh t,2\alpha\sinh t,(\cosh t +\sinh t)/\alpha\in {{\mathbb{Z}}}\mbox{ so }(2\alpha\cosh t,2\alpha\sinh t) = (\alpha^2+1,\alpha^2-1).$$ We then deduce that $$O_f({{\mathbb{Z}}}) = \left\{\pm I_{2\times2},\pm\left(\begin{smallmatrix} -1&&0\\2\alpha/\beta&&1\end{smallmatrix}\right), \left(\begin{smallmatrix}\alpha&&\beta\\-(\alpha^2\pm1)/\beta&&-\alpha\end{smallmatrix}\right)\right\}\cap {\operatorname{GL}}_2({{\mathbb{Z}}}).$$ Since $f$ has the shape (\[reducible f shape\]) by assumption, we have $$\beta\mid \alpha^2 + 1{\text{ and }}\beta\mid \alpha^2-1\iff \alpha=1{\text{ and }}\beta\in\{1,2\},$$ and we see that the claim indeed holds.
\[Of indefinite\]Suppose that $f$ is indefinite and irreducible. Define $$G_f({{\mathbb{Z}}}) = \{\pm T_{D_f}^n : n\in{{\mathbb{Z}}}\},\mbox{ where }T_{D_f}=\begin{pmatrix} \frac{u_{D_f} - \beta v_{D_f}}{2} & - \gamma v_{D_f} \vspace{2mm}\\ \alpha v_{D_f} & \frac{u_{D_f} + \beta v_{D_f}}{2} \end{pmatrix}$$ and $(u_{D_f},v_{D_f})\in{{\mathbb{N}}}^2$ is the least solution to $x^2 - D_fy^2 = \pm4$. Then, we have $$O_f({{\mathbb{Z}}}) = G_f({{\mathbb{Z}}})$$ if $f$ is not ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalence to the forms below, and the group $O_f({{\mathbb{Z}}})$ is equal to $$\begin{cases}
G_f({{\mathbb{Z}}})\sqcup G_f({{\mathbb{Z}}})\left(\begin{smallmatrix}1&\beta/\alpha \\ 0 & -1 \end{smallmatrix}\right) &\mbox{if $f(x,y) = \alpha x^2 + \beta xy + \gamma y^2$ with $\alpha\mid \beta$},\\
G_f({{\mathbb{Z}}})\sqcup G_f({{\mathbb{Z}}})\left(\begin{smallmatrix}0&1\\-1&0 \end{smallmatrix}\right)&\mbox{if $f(x,y) = \alpha x^2 + \beta xy - \alpha y^2$}.
\end{cases}$$
By (\[O relation\]), elements in $O_f({{\mathbb{R}}})$ of infinite order are of the shape $$\pm \begin{pmatrix} \phi_t - \frac{\beta\psi_t}{2 \sqrt{\delta_f}} & -\frac{\gamma\psi_t}{\sqrt{\delta_f}} \vspace{2mm}\\ \frac{\alpha\psi_t}{\sqrt{\delta_f}} & \phi_t + \frac{\beta\psi_t}{2\sqrt{\delta_f}}\end{pmatrix},$$ where $t\in{{\mathbb{R}}}$ and $(\phi_t,\psi_t) \in\{(\cosh t,\sinh t),(\sinh t,\cosh t)\}$. We then see that $$G_f({{\mathbb{Z}}}) = \{\pm I_{2\times 2}\}\sqcup\{T\in O_f({{\mathbb{Z}}}) : T\mbox{ has infinite order}\}.$$ Hence, the first claim follows from Lemma \[Dickson ambiguous\] and the fact that $ax^2 + bxy + ay^2$ is ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent to the form $$\label{ambiguous transformation} (2a-b)x^2 + (2a-b)xy + ay^2\mbox{ via } \begin{pmatrix}-1&-1\\1 & 0\end{pmatrix}.$$ Now, again by (\[O relation\]), elements in $O_f({{\mathbb{R}}})$ of finite order have the shape $$\label{indefinite finite order}\begin{pmatrix} \frac{-\beta}{\sqrt{D_f}} & - \frac{2\gamma}{\sqrt{D_f}} \vspace{2mm}\\ \frac{2\alpha}{\sqrt{D_f}} & \frac{\beta}{\sqrt{D_f}}\end{pmatrix} {\text{ and }}\begin{pmatrix} \phi_t + \frac{\beta \psi_t}{2 \sqrt{\delta_f}} & \frac{\beta}{\alpha} \left(\phi_t + \frac{\beta\psi_t}{2 \sqrt{\delta_f}} \right) - \frac{\gamma\psi_t}{\sqrt{\delta_f}} \vspace{2mm}\\ - \frac{\alpha\psi_t}{\sqrt{\delta_f}} & - \phi_t - \frac{\beta\psi_t}{2 \sqrt{\delta_f}}, \end{pmatrix}$$ where $t\in{{\mathbb{R}}}$ and $(\phi_t,\psi_t) \in\{(\cosh t,\sinh t),(\sinh t,\cosh t)\}$. Notice that the matrix on the left cannot lie in ${\operatorname{GL}}_2({{\mathbb{Z}}})$ because $D_f$ is not square when $f$ is irreducible. Using the description of $O_{x^2-y^2}({{\mathbb{R}}})$, it is then not hard to check that $[O_f({{\mathbb{Z}}}):G_f({{\mathbb{Z}}})]\leq 2$, from which the second claim follows.
Proof of Theorem \[negative Pell\]
----------------------------------
Suppose that $f(x,y) = \alpha x^2 + \beta xy - \alpha y^2$ and that $D_f$ is not a square. In the notation of Proposition \[Of indefinite\], we have $$\mbox{$x^2 - D_fy^2 = -4$ has integer solutions} \mbox{ if and only if }\det(T_{D_f}) = -1$$ by definition. But Proposition \[Of indefinite\] also implies that $\det(T_{D_f}) = -1$ is equivalent to $$O_f({{\mathbb{Z}}})\mbox{ has an element of finite order and negative determinant}.$$ The theorem now follows from Lemma \[Dickson ambiguous\] and (\[ambiguous transformation\]).
Proof of Proposition \[issue2 prop\]
------------------------------------
We shall need the following lemma.
\[key corollary\]For all $F\in V_{{{\mathbb{Z}}},f}^0$ with $\Delta(F)\neq\square$ and $T\in {\operatorname{GL}}_2({{\mathbb{Z}}})\setminus\{\pm I_{2\times 2}\}$, we have
(a) $F_T \in V_{{{\mathbb{Z}}},f}^0$ if and only if $T\in O_f({{\mathbb{Z}}})$,
(b) $F_T = F$ if and only if $T = \pm D_f^{-1/2}M_f$.
Note that $F_T\in V_{{{\mathbb{Z}}},f_T}^0$ by (\[V bijection\]). By Theorem \[small char thm\] (a), we then have $F_T\in V_{{{\mathbb{Z}}},f}^0$ if and only if $f_T = \pm f$, whence part (a) holds. By Theorem \[small char thm\] (a) and Proposition \[auto theorem\], we have $F_T = F$ if and only if $T$ is proportional to $M_f$, from which part (b) follows since $\det(T)=\pm1$.
### The case when $f$ is positive definite or reducible
Let us first observe that:
\[V in S lemma\]We have $V_{{{\mathbb{Z}}},f}^0(X)\subset{{\mathcal{S}}}_f(X)$.
Let $F\in V_{{{\mathbb{Z}}},f}^0(X)$ be given. If $f$ is positive definite, then clearly $F\in {{\mathcal{S}}}_f(X)$ by Lemma \[Sf positive definite\]. If $f$ is reducible, then recall Lemma \[Sf reducible\], and we have $F\in {{\mathcal{S}}}_f(X)$ since $$\frac{8C_F}{\beta^2}\in{{\mathbb{Z}}}{\text{ and }}\left\lvert\frac{8C_F}{\beta^2}\right\rvert\leq \left\lvert\frac{4(L_f(F)^2 + 4K_f(F))}{9}\right\rvert \leq \frac{20X}{9}$$ by (\[reducible generic\]) and Proposition \[explicit LK\] (b), respectively.
Lemma \[V in S lemma\] implies that part (a) holds. Together with Lemma \[key corollary\] (a, it further implies that for $F\in V_{{{\mathbb{Z}}},f}^0(X)$ with $\Delta(F)\neq\square$, the number of forms in ${{\mathcal{S}}}_f(X)$ which are ${\operatorname{GL}}_2({{\mathbb{Z}}})$-equivalent to $F$ is equal to $$[O_f({{\mathbb{Z}}}) : \mbox{Stab}_{O_f({{\mathbb{Z}}})}(F)].$$ By Lemma \[key corollary\] (b), we in turn have $$\textstyle[O_f({{\mathbb{Z}}}) : \mbox{Stab}_{O_f({{\mathbb{Z}}})}(F)] = [O_f({{\mathbb{Z}}}) : O_f({{\mathbb{Z}}})\cap\{\pm I_{2\times 2},\pm D_f^{-1/2}M_f\}],$$ which may be verified to be equal to $r_f$ using Propositions \[Of pos def\] and \[Of reducible\].
### The case when $f$ is indefinite and irreducible
We shall use the notation from Lemma \[Psi lemma\], Proposition \[Of indefinite\], (\[T def\]), and (\[J def\]). Then, by definition, we have $$T_{D_f} = T_f^{-1}J_{k(f)} T^-(t_{D_f})T_f,\mbox{ where }k(f) = \begin{cases}1 &\mbox{if }u_{D_f}^2 - D_fv_{D_f}^2 = -4,\\
2&\mbox{if }u_{D_f}^2 - D_fv_{D_f}^2 = 4,\end{cases}$$ Now, by (\[para\]) and (\[Phi- def\]), a form in $V_{{{\mathbb{Z}}},f}^0(X)$ is of the shape $$F = (F_{(L,K)}^{(i)})_{T^-(t)T_f},\mbox{ where }(L,K,t)\in\Omega^0(X)\times{{\mathbb{R}}}{\text{ and }}i\in\{1,2,3,4\}.$$ Observe that $J_1$ and $J_2$ commute with $T^-(t)$ as well as fix the forms in $V_{{{\mathbb{R}}},x^2-y^2}$. For any $n\in{{\mathbb{Z}}}$, we then deduce that $$F_{T_{D_f}^n} = (F_{(L,K)}^{(i)})_{T^-(t)J_{k(f)}^nT^-(nt_{D_f})T_f} = (F_{(L,K)}^{(i)})_{T^-(t+nt_{D_f})T_f}.$$ Let $n_1\in{{\mathbb{Z}}}$ be the unique integer such that $0\leq t+n_1t_{D_f} < t_{D_f}$. The existence of $n_1$ then implies part (a).\
Next, suppose that $\Delta(F)\neq\square$, in which case $$\mbox{for $T\in{\operatorname{GL}}_2({{\mathbb{Z}}})$}: F_T\in V_{{{\mathbb{Z}}},f}^0\mbox{ if and only if }T\in O_f({{\mathbb{Z}}})$$ by Lemma \[key corollary\] (a). If $O_f({{\mathbb{Z}}}) = G_f({{\mathbb{Z}}})$, then part (b) holds by the uniqueness of $n_1$. If $O_f({{\mathbb{Z}}}) \neq G_f({{\mathbb{Z}}})$, then recall from Proposition \[Of indefinite\] that $$O_f({{\mathbb{Z}}}) = G_f({{\mathbb{Z}}}) \sqcup G_f({{\mathbb{Z}}})M,\mbox{ where $M$ has finite order}.$$ From (\[O relation\]), we see that $$M = \pm T_{f}^{-1}J_{k_0}T^-(t_0)T_f,\mbox{ where $t_0\in{{\mathbb{R}}}$ and $k_0\in\{3,4\}$}.$$ Then, for any $n\in{{\mathbb{Z}}}$, it is straightforward to verify that $$\begin{aligned}
F_{T_{D_f}^nM} & = (F_{(L,K)}^{(i)})_{T^-(t+nt_{D_f})J_{k_0} T^-(t_0)T_f} \\&=
\begin{cases}
(F_{(L,K)}^{(i)})_{T^-(-(t+nt_{D_f})+t_0)T_f} & \mbox{for }i\in\{1,2\},\\
(F_{(L,K)}^{(j)})_{T^-(-(t+nt_{D_f})+t_0)T_f} & \mbox{for $i\in\{3,4\}$, where $j\in\{3,4\}\setminus\{i\}$}.
\end{cases}\end{aligned}$$ There is a unique $n_2\in{{\mathbb{Z}}}$ such that $0\leq -(t+n_2t_{D_f}) + t_0< t_{D_f}$. Observe that $$F_{T_{D_f}^{n_1}} = F_{T_{D_f}^{n_2}M} \mbox{ would imply }F_{T_{D_f}^{n_1}}= (F_{T_{D_f}^{n_1}})_{T_{D_f}^{n_2-n_1}M}.$$ But $T_{D_f}^{n_2-n_1}M$ has finite order, and so it cannot proportional to $M_f$ by (\[indefinite finite order\]), which is a contradiction by Lemma \[key corollary\] (b). Then, we conclude from Proposition \[Of indefinite\] that part (b) indeed holds.
Error estimates and the main theorem
====================================
Throughout this section, let $f(x,y) = \alpha x^2 + \beta xy + \gamma y^2$ be an integral and primitive binary quadratic form with $\Delta(f)\neq0$ and $\alpha>0$, in the shape (\[reducible f shape\]) whenever $f$ is reducible. Let $D_f,r_f$ and $s_f$ be as in Theorem \[Small Gal MT\].\
In Subsections \[error sec1\] and \[error sec2\], respectively, we shall first prove:
\[error prop1\]For any $\epsilon>0$, we have $$\#\{F\in {{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0:L_f(F)^2 + 4K_f(F)=\square\} = O_{f,\epsilon}(X^{1+\epsilon}),$$ and $$\begin{aligned}
&\#\{F\in {{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0:(L_f(F)^2 + 4K_f(F))(2L_f(F) - K_f(F))/\Delta(f) =\square\\ &\hspace{8.5cm}{\text{ and }}L_f(F)\neq0\} = O_f(X^{1/2+\epsilon}).\end{aligned}$$ Further, the number $$\#\{F \in {{\mathcal{S}}}_f(X) \cap V_{{{\mathbb{Z}}},f}^0 : - 4K_f(F)/\Delta(f) = \square \text { and } L_f(F) = 0\}$$ is equal to zero if $-\Delta(f) \ne \square$. and is bounded by $O_f(X)$ otherwise.
Propositions \[error prop1\], \[abelian Gal prop\], and \[issue2 prop\] then imply part d) of Theorem \[Small Gal MT\].\
The reader should compare the last claim above with [@X2 Theorem 1.4].
\[error prop2\] We have $$\#\{F\in {{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0 : F\mbox{ is reducible}\} = \begin{cases} O_f(X(\log X)^2) & \mbox{if $f$ is irreducible},\\
O_f(X(\log X)^3)&\mbox{if $f$ is reducible}.\end{cases}$$
Now, from Propositions \[issue2 prop\], \[error prop1\], and \[error prop2\], we also easily see that $$\label{N1}N_{{{\mathbb{Z}}},f}^{(D_4)}(X) = \frac{1}{r_f}\#(S_f(X)\cap V_{{{\mathbb{Z}}},f}^0) + O_{f,\epsilon}(X^{1+\epsilon}) \mbox{ for any }\epsilon >0.$$ Let ${{\mathcal{L}}}_{f,w(f)}$ be a linear transformation on ${{\mathbb{R}}}^3$ which takes $\Lambda_{f,w(f)}$ to ${{\mathbb{Z}}}^3$, and define $${{\mathcal{R}}}_f(X) = ({{\mathcal{L}}}_{f,w(f)}\circ\Theta_{w(f)})({{\mathcal{S}}}_f(X)), \mbox{ where }w(f) = \begin{cases}
1 & \mbox{if $f$ is irreducible},\\ 2 & \mbox{if $f$ is reducible},
\end{cases}$$ as before. Observe that then $$\#({{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0) = \#(\Theta_{w(f)}({{\mathcal{S}}}_f(X))\cap \Lambda_{f,w(f)}) = \#({{\mathcal{R}}}_f(X)\cap {{\mathbb{Z}}}^3).$$ By Proposition \[issue1 prop\], we may apply Proposition \[Davenport\] to obtain $$\begin{aligned}
\label{N2}
\#(S_f(X)\cap V_{{{\mathbb{Z}}},f}^0) &= {\operatorname{Vol}}({{\mathcal{R}}}_f(X)) + O(\max\{{\operatorname{Vol}}(\overline{{{\mathcal{R}}}_f(X)}),1\}) \\\notag
&= \frac{1}{\det(\Lambda_{f,w(f)})}{\operatorname{Vol}}(\Theta_{w(f)}({{\mathcal{S}}}_f(X))) \\\notag
& \hspace{3cm}+ O_f(\max\{{\operatorname{Vol}}(\overline{\Theta_{w(f)}({{\mathcal{S}}}_f(X))},1\}),\end{aligned}$$ where by Proposition \[det prop\], we know that $$\det(\Lambda_{f,w(f)}) = \begin{cases} s_f\alpha^3 &\mbox{if $f$ is irreducible},\\
s_f\beta^3/8&\mbox{if $f$ is reducible}.
\end{cases}$$ Hence, it remains to compute the above volumes, which we shall do in Subsection \[proof sec\].
Proof of Proposition \[error prop1\] {#error sec1}
------------------------------------
Recall the notation from Proposition \[explicit LK\]. By definition and Proposition \[LK integers\], we then have a well-defined map $$\iota:V_{{{\mathbb{Z}}},f}^0 \longrightarrow {{\mathbb{Z}}}^3;\hspace{1em} \iota(F)= (L_f(F),L_{f,1}(F),L_{f,2}(F)).$$ Using Proposition \[explicit LK\], it is easy to verify that $\iota$ is in fact injective. We shall also need the following result due to Heath-Brown [@HB1].
\[HB lemma\]Let $\xi(x_1,x_2,x_3)$ be a ternary quadratic form such that its corresponding matrix $M_\xi$ has non-zero determinant. For $B_1,B_2,B_3>0$, let $N_\xi(B_1,B_2,B_3)$ denote the number of tuples $(x_1,x_2,x_3)\in{{\mathbb{Z}}}^3$ such that $$|x_1|\leq B_1,\, |x_2|\leq B_2,\, |x_3|\leq B_3,\, \gcd(x_1,x_2,x_3)=1,\, \xi(x_1,x_2,x_3)=0.$$ Then, we have $$N_\xi(B_1,B_2,B_3) \ll_{\epsilon} \left(1+\left(B_1B_2B_3\cdot \frac{\det_0(M_\xi)^2}{|\det(M_\xi)|} \right)^{1/3+\epsilon}\right)d_3(|\det(M_\xi)|),$$ where $\det_0(M_\xi)$ denotes the greatest common divisor of the $2\times2$ minors of $M_\xi$, and $d_3(|\det(M_\xi)|)$ is the number of ways to write $|\det(M_\xi)|$ as a product of three positive integers.
See [@HB1 Corollary 2].
In what follows, consider $F\in{{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0$, and for brevity, write $$(L,K,L_1,L_2) = (L_f(F),K_f(F),L_{f,1}(F),L_{f,2}(F)).$$ Since $\iota$ is injective, it is enough to estimate the number of choices for $(L,L_1,L_2)$. To that end, let us put ${{\mathcal{D}}}_f = \Delta(f)$. Recall from Propositions \[explicit LK\] and \[LK integers\] that $$L,K,L_1,L_2\in{{\mathbb{Z}}},\mbox{ as well as }L_1^2 - {{\mathcal{D}}}_fL_2^2 = 4\alpha^4(L^2 + 4K)/9,$$ which is non-zero by (\[Delta LK\]). By the definition of our height, we also have $$\label{bounds} \begin{cases}L = O_f(X^{1/2}){\text{ and }}K = O_f(X) &\mbox{in all cases},\\ L_1 = O_f(X^{1/2}){\text{ and }}L_2 = O_f(X^{1/2})&\mbox{if $f$ is irreducible}.\end{cases}$$ The latter estimate holds by $$\begin{cases}
(\ref{pos def para}), (\ref{general parameter}) &\mbox{if $f$ is positive definite},\\(\ref{indef para 1}), (\ref{indef para 2}), (\ref{general parameter}),\mbox{ and $0\leq t < t_{D_f}$} &\mbox{if $f$ is indefinite and irreducible},
\end{cases}$$ as well as the fact that $L_1$ and $L_2$ are linear in the coefficients of $F$. Finally, we shall write $d(-)$ for the divisor function.
Suppose that $L^2 + 4K=\square$. Then, we have $$L_1^2 - {{\mathcal{D}}}_f L_2^2 = U^2,\mbox{ where }U\in{{\mathbb{N}}}\mbox{ is such that }U = O_f(X^{1/2}).$$ If $f$ is reducible, then ${{\mathcal{D}}}_f=\square$ and so clearly there are $$O_f\left(\sum_{U=1}^{X^{1/2}}d(U^2)\right) = O_{f,\epsilon}\left(\sum_{U=1}^{X^{1/2}}X^\epsilon\right) = O_{f,\epsilon}(X^{1/2+\epsilon})$$ choices for the pair $(L_1,L_2)$. If $f$ is irreducible, then note that $$(L_1/n)^2 - {{\mathcal{D}}}_f(L_2/n)^2 = (U/n)^2,\mbox{ where } n = \gcd(L_1,L_2,U),$$ and applying Lemma \[HB lemma\] to the ternary quadratic form $\xi$ with matrix $$M_\xi = \begin{pmatrix}1 & 0 & 0 \\ 0 & -{{\mathcal{D}}}_f& 0 \\ 0 & 0 & -1\end{pmatrix},\mbox{ with }\begin{cases}\det(M_\xi) = {{\mathcal{D}}}_f,\\\det_0(M_\xi) = 1,\end{cases}$$ we deduce from (\[bounds\]) that there are $$O_f\left(\sum_{n=1}^{X^{1/2}}N_\xi\left(\frac{X^{1/2}}{n}, \frac{X^{1/2}}{n},\frac{X^{1/2}}{n}\right)\right) = O_{f,\epsilon}\left(\sum_{n=1}^{X^{1/2}} \left(1+\frac{X^{1/2+\epsilon}}{n^{1+\epsilon}}\right)\right)= O_{f,\epsilon}(X^{1/2 + \epsilon})$$ choices for the pair $(L_1,L_2)$. In both cases, we see that there are $$O_f(X^{1/2})\cdot O_{f,\epsilon}(X^{1/2+\epsilon}) = O_{f,\epsilon}(X^{1+\epsilon})$$ choices for $(L,L_1,L_2)$ in total, whence the claim.
Suppose that $(L^2+4K)(2L^2-K)/{{\mathcal{D}}}_f =\square$. By Proposition \[LK integers\], we may write $$\gcd(L^2 + 4K, 4(2L^2 - K)/{{\mathcal{D}}}_f) = 9ma^2,\mbox{ where }m,a\in{{\mathbb{N}}}{\text{ and }}m\mbox{ is square-free}.$$ From the hypothesis, we then easily see that $$L^2 + 4K = 9mU^2{\text{ and }}4(2L^2-K)/{{\mathcal{D}}}_f =9mV^2,\mbox{ where }U,V \in {{\mathbb{N}}},$$ as well as that $m$ divides $L$. In particular, a simple calculation yields $$L^2 = m(U^2 + {{\mathcal{D}}}_fV^2),\mbox{ whence } mW^2 = U^2 + {{\mathcal{D}}}_fV^2,\mbox{ where } W\in{{\mathbb{Z}}}\mbox{ with }L = mW.$$ Now, suppose also that $L\neq0$, in which case $m = O_f(X^{1/2})$ by (\[bounds\]). Note also that $$m(W/n)^2 = (U/n)^2 + {{\mathcal{D}}}_f(V/n)^2,\mbox{ where }n = \gcd(W,U,V).$$ Applying Lemma \[HB lemma\] to the ternary quadratic form $\xi_m$ with matrix $$M_{\xi_m} = \begin{pmatrix}m & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -{{\mathcal{D}}}_f\end{pmatrix},\mbox{ with }\begin{cases}\det(M_{\xi_m}) = m{{\mathcal{D}}}_f,\\\det_0(M_{\xi_m}) = \gcd(m,{{\mathcal{D}}}_f) \leq |{{\mathcal{D}}}_f|,\end{cases}$$ we then see from (\[bounds\]) that there are $$\begin{aligned}
O_f\left(\sum_{n=1}^{X^{1/2}/m}N_{\xi_m}\left(\frac{X^{1/2}}{mn}, \frac{X^{1/2}}{m^{1/2}n},\frac{X^{1/2}}{m^{1/2}n}\right)\right) &= O_{f,\epsilon}\left(\sum_{n=1}^{X^{1/2}/m}\left(1+\frac{X^{1/2+\epsilon}}{(mn)^{1+\epsilon}}\right)m^\epsilon\right)\\
& = O_{f,\epsilon}\left( \frac{X^{1/2}}{m^{1-\epsilon}} + \frac{X^{1/2+\epsilon}}{m}\right)\end{aligned}$$ choices for $(x,u,v)$ when $m$ is fixed. It follows that we have $$O_{f,\epsilon}\left(\sum_{m=1}^{X^{1/2}}\left(\frac{X^{1/2}}{m^{1-\epsilon}} + \frac{X^{1/2+\epsilon}}{m}\right)\right)=O_{f,\epsilon}\left(X^{1/2+\epsilon}\right)$$ choices for $(m,x,u,v)$ and hence for $(L,K)$.\
Next, regard $(L,K)$ as being fixed, and recall that $$L_1^2 - {{\mathcal{D}}}_fL_2^2 = T, \mbox{ where }T = 4\alpha^4(L^2+4K)/9.$$ We claim that there are $O_f(d(T))$ choices for $(L_1,L_2)$. If $f$ is positive definite or if $f$ is reducible, then this is clear. If $f$ is indefinite and irreducible, then by Definition \[S(X) def\] as well as Propositions \[LK invariant\] and \[Phi-\], we have $$F = (\Psi_f\circ\Phi^{(i)})(L,K,t),\mbox{ where }0\leq t<t_{D_f}{\text{ and }}i\in\{1,2,3,4\}.$$ Since ${{\mathcal{D}}}_f>0$, we must have $L^2 + 4K>0$ by the hypothesis, and so in fact $i\in\{1,2\}$. From the proof of Lemma \[Sf indefinite\], we know that $$L_1 - \sqrt{D_f}L_2 = (-1)^i\sqrt{T}e^{4t}{\text{ and }}L_1 + \sqrt{D_f}L_2 = (-1)^i\sqrt{T}e^{-4t},$$ which implies that $$L_1 = (-1)^i\sqrt{T}\cosh(4t){\text{ and }}L_2 = (-1)^i\sqrt{T}\sinh(4t)/\sqrt{D_f}.$$ Since $t = O_f(1)$, we then deduce that indeed there are $O_f(d(T))$ choices for $(L_1,L_2)$. Using the bound $d(T) = O_{\epsilon}(T^\epsilon) = O_{f,\epsilon}(X^{\epsilon})$, we conclude that there are $$O_{f,\epsilon}(X^{1/2+\epsilon})\cdot O_{f,\epsilon}(X^{\epsilon}) = O_{f,\epsilon}(X^{1/2+\epsilon})$$ choices for $(L,L_1,L_2)$ in total, whence the claim.
Suppose that $L=0$ and that $F$ is in the shape as in (\[abc family\]). Using Proposition \[explicit LK\], we then deduce that $$C = (-12 \gamma A + 3 \beta B)/(2 \alpha),\mbox{ and so }K = -9{{\mathcal{D}}}_f(\alpha B^2 - 4\beta AB + 16\gamma A^2)/(4\alpha^3).$$ Hence $$(L^2 + 4K)(2L^2 - K)/81{{\mathcal{D}}}_f = -4 {{\mathcal{D}}}_f (\alpha B^2 - 4 \beta AB + 16 \gamma A^2)^2/(4 \alpha^3)^2,$$ from which it follows that the above expression is a square if and only if $-{{\mathcal{D}}}_f$ is a square. This also follows immediately from the observation that the above product is equal to $-4K^2/{{\mathcal{D}}}_f$ in this case.\
We now suppose that $-\Delta(f) = \square$, so in particular $f$ is positive definite. $F$ is then determined by $(A,B)\in{{\mathbb{Z}}}^2$, and that $|K|\leq X$ implies $$\left\lvert \left(B-\frac{2\beta}{\alpha}A\right)^2 - \frac{4{{\mathcal{D}}}_f}{\alpha^2}A^2\right\rvert\ll_f X.$$ Hence there are $O_f(X)$ choices for $(A,B)$. It follows that the claim holds.
Proof of Proposition \[error prop2\] {#error sec2}
------------------------------------
By Lemma \[type 1\] and Proposition \[error prop1\], we have $$\label{type1 estimate} \#\{F\in {{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0: F\mbox{ is reducible of type $1$}\} = O_{f,\epsilon}(X^{1+\epsilon}),$$ whence it is enough to consider the reducible forms in ${{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0$ of type 2; recall Definition \[reducible types def\]. By definition, such a form has the shape $$F(x,y) = p_2q_2 x^4 + (p_2q_1 + p_1q_2)x^3y + (p_2q_0 + p_1q_1 + p_0q_2)x^2y^2 + (*)xy^3 + (*)y^4,$$ where $p_2,p_1,p_0,q_2,q_1,q_0\in{{\mathbb{Z}}}$, and we have $$p_0 = (\beta p_1 - 2\gamma p_2)/(2\alpha){\text{ and }}q_0 = (\beta q_1 - 2\gamma q_2)/(2\alpha)$$ by Lemma \[quadratic factor\]. We have the condition $$\begin{aligned}
\label{pq condition1}
&|(\alpha p_1^2 - 2\beta p_1p_2 + 4\gamma p_2^2)/\alpha|,|(\alpha q_1^2 - 2 \beta q_1 q_2 + 4 \gamma q_2^2)/\alpha|, \\\notag
&\hspace{4.5cm}|p_2|,|\alpha p_1 - \beta p_2|,|q_2|,\lvert \alpha q_1 - \beta q_2|\geq1\end{aligned}$$ since the above numbers are all integers. Using Proposition \[explicit LK\] (a), we compute that $$\dfrac{L_f(F)^2 + 4K_f(F)}{9} = \dfrac{\alpha p_1^2 - 2 \beta p_1 p_2 + 4 \gamma p_2^2}{\alpha}\cdot\dfrac{\alpha q_1^2 - 2 \beta q_1 q_2 + 4 \gamma q_2^2}{\alpha}.$$ Now, by the definition of our height, we clearly have $$\label{pq condition2}
|(\alpha p_1^2 - 2\beta p_1p_2 + 4\gamma p_2^2)/\alpha|,|(\alpha q_1^2 - 2 \beta q_1 q_2 + 4 \gamma q_2^2)/\alpha|\leq X.$$ Observe also that $$\label{pq condition3}p_2q_2,p_2q_1 + p_1q_2,p_1q_1 = O_f(X^{1/2})\mbox{ if $f$ is indefinite and irreducible}$$ by (\[indef para 1\]), (\[indef para 2\]), (\[general parameter\]), and the bound $0\leq t < t_{D_f}$. We then deduce that $$\label{reducible type 2 bound}
\#\{F\in {{\mathcal{S}}}_f(X)\cap V_{{{\mathbb{Z}}},f}^0: F\mbox{ is reducible of type $2$}\}
\leq \#({{\mathcal{R}}}_f'(X)\cap {{\mathbb{Z}}}^4),$$ where we define $${{\mathcal{R}}}_f'(X) = \{(p_2,p_1,q_2,q_1)\in{{\mathbb{R}}}^4: (\ref{pq condition1}),\,(\ref{pq condition2}),{\text{ and }}(\ref{pq condition3})\}.$$ It is clear that this set is bounded and semi-algebraic. Hence, we may apply Proposition \[Davenport\] to estimate the number of integral points it contains.
### The case when $f$ is irreducible
Let us define $${{\mathcal{R}}}_f''(X) ={{\mathcal{L}}}_{D_f}({{\mathcal{R}}}'_f(X)),\mbox{ where }{{\mathcal{L}}}_{D_f} = \left(\begin{smallmatrix}\sqrt{D_f}&0&0&0\\-\beta & \alpha & 0 & 0 \\ 0 & 0& \sqrt{D_f} & 0\\ 0 & 0 & -\beta & \alpha \end{smallmatrix}\right).$$ Applying Proposition \[Davenport\], we then obtain $$\begin{aligned}
\#({{\mathcal{R}}}_f'(X)\cap{{\mathbb{Z}}}^4) & = {\operatorname{Vol}}({{\mathcal{R}}}_f'(X)) + O(\max\{{\operatorname{Vol}}(\overline{{{\mathcal{R}}}_f(X)},1\})\\
& = \frac{1}{\det({{\mathcal{L}}}_{D_f})}{\operatorname{Vol}}({{\mathcal{R}}}_f''(X)) + O_f(\max\{{\operatorname{Vol}}(\overline{{{\mathcal{R}}}_f''(X)}),1\})\end{aligned}$$ For any $(u_2,u_1,v_2,v_1) \in {{\mathcal{R}}}_f''(X)$, from (\[pq condition1\]) and (\[pq condition2\]), we deduce that $$|u_2|,|u_1|,|v_2|,|v_1|\geq 1$$ as well as that $$\label{uv condition}
\begin{cases}
1\leq|u_1^2+ u_2^2|,|v_1^2+v_2^2|\leq \alpha^4X&\mbox{if $f$ is positive definite},\\
1\leq|u_1^2 - u_2^2|,|v_1^2-v_2^2|\leq \alpha^4X&\mbox{if $f$ is indefinite}.
\end{cases}$$ This, together with (\[pq condition3\]), implies that in fact $$1\leq |u_2|,|u_1|,|v_2|,|v_1|, |u_2v_2|,|u_1v_1|\ll_f X^{1/2}.$$ We then compute that $$\begin{aligned}
{\operatorname{Vol}}({{\mathcal{R}}}_f''(X)) &= O_f\left( \prod_{i=1}^{2}\int_{1}^{X^{1/2}/v_i} du_i dv_i\right)= O_f(X(\log X)^2),\\
{\operatorname{Vol}}(\overline{{{\mathcal{R}}}_f''(X)}) &= O_f(X\log X).\end{aligned}$$ The claim now follows from (\[type1 estimate\]) and (\[reducible type 2 bound\]).
### The case when $f$ is reducible
Let us define $${{\mathcal{R}}}''_f(X) = {{\mathcal{L}}}_{0,D_f}({{\mathcal{R}}}'_f(X)),\mbox{ where }{{\mathcal{L}}}_{0,D_f} =\left(\begin{smallmatrix}1 &1&0&0\\[0.75ex]-1&1&0&0 \\[0.75ex] 0&0&1&1\\[0.75ex] 0&0&-1&1 \end{smallmatrix}\right) \left(\begin{smallmatrix}\sqrt{D_f}&0&0&0\\-\beta & \alpha & 0 & 0 \\ 0 & 0& \sqrt{D_f} & 0\\ 0 & 0 & -\beta & \alpha \end{smallmatrix}\right).$$ Since $D_f=\square$ in this case, we see that $${{\mathcal{L}}}_{0,D_f}({{\mathcal{R}}}_f'(X)\cap{{\mathbb{Z}}}^4)\subset {{\mathcal{R}}}_f''(X)\cap{{\mathbb{Z}}}^4\mbox{ and so }\#({{\mathcal{R}}}_f'(X)\cap {{\mathbb{Z}}}^4) \leq \#({{\mathcal{R}}}_f''(X)\cap{{\mathbb{Z}}}^4).$$ Now, applying Proposition \[Davenport\], we have $$\#({{\mathcal{R}}}_f''(X)\cap{{\mathbb{Z}}}^4) = {\operatorname{Vol}}({{\mathcal{R}}}_f''(X)) + O(\max\{{\operatorname{Vol}}(\overline{{{\mathcal{R}}}_f''(X)}),1\}).$$ For any $(z_1,z_2,z_3,z_4)\in{{\mathcal{R}}}_f''(X)$, the conditions (\[pq condition1\]) and (\[pq condition2\]) imply that $$|z_1|,|z_2|,|z_3|,|z_4|\geq 1{\text{ and }}|z_1z_2z_3z_4|\leq \alpha^4X,$$ which is analogous to (\[uv condition\]). We then compute that $$\begin{aligned}
{\operatorname{Vol}}({{\mathcal{R}}}_f''(X)) &= O_f\left(\int_1^{X} \int_1^{\frac{X}{z_4}} \int_{1}^{\frac{X}{z_3 z_4}} \int_1^{\frac{X}{z_2 z_3 z_4}} d z_1 dz_2 dz_3 dz_4\right) = O_f(X(\log X)^3),\\
{\operatorname{Vol}}(\overline{{{\mathcal{R}}}_f''(X)}) &= O_f(X(\log X)^2).\end{aligned}$$ The claim now follows from (\[type1 estimate\]) and (\[reducible type 2 bound\]).
Proof of Theorem \[Small Gal MT\] {#proof sec}
---------------------------------
We have already proven part (d). To prove parts (a) through (c), it remains to compute the volumes in (\[N2\]).
### The case when $f$ is positive definite
We have $${\operatorname{Vol}}(\Theta_{1}({{\mathcal{S}}}_f(X))) = \frac{8\alpha^3}{D_f^{3/2}}\cdot\frac{1}{18}\cdot{\operatorname{Vol}}(\Omega^+(X)\times[-\pi/4,\pi/4))$$ by Lemma \[Psi lemma\] and Proposition \[Phi+\] (b), as well as $${\operatorname{Vol}}(\Omega^+(X)\times[-\pi/4,\pi/4)) = \int_{-X^{1/2}}^{X^{1/2}} \int_{-L^2/4}^X \frac{\pi}{2} dK dL
=\frac{13\pi}{12}X^{3/2}.$$ Observe also that $${\operatorname{Vol}}(\overline{\Theta_{1}({{\mathcal{S}}}_f(X))}) = O_f(X)$$ because $\Theta_{1}({{\mathcal{S}}}_f(X))$ lies in the cube centered at the origin of side length $O_f(X^{1/2})$ by (\[pos def para\]) and (\[general parameter\]). We then deduce part (a) from (\[N1\]) and (\[N2\]).
### The case when $f$ is reducible
We have $${\operatorname{Vol}}(\Theta_2({{\mathcal{S}}}_f(X))) =\frac{1}{18}\cdot 2\cdot {\operatorname{Vol}}(\Omega^0(X)\times [t_{f,1},t_{f,2}])$$ by Proposition \[Phi f\], as well as $${\operatorname{Vol}}(\Omega^0(X)\times [t_{f,1},t_{f,2}])= \int_{-X^{1/2}}^{X^{1/2}}\int_{-X}^X \frac{1}{4}\log\left(\frac{20X}{9}\right)dKdL= X^{3/2}\log(20X/9).$$ We then deduce part (b) from Lemma \[S bar\] below as well as (\[N1\]) and (\[N2\]).
\[S bar\]We have ${\operatorname{Vol}}(\overline{\Theta_{2}({{\mathcal{S}}}_f(X))}) = O_f(X^{3/2})$.
By Definition \[S(X) def\], an element in $\Theta_{2}({{\mathcal{S}}}_f(X))$ takes the form $$(A,B,C) = (\Theta_2\circ\Phi_f)(L,K,t), \mbox{ where }(L,K,t)\in \Omega^0(X)\times [t_{f,1},t_{f,2}].$$ Let us recall that $$\label{red bounds}|L|\leq X^{1/2},\, |K|\leq X,\, 4t_{f,1} =-\log 8,\, 4t_{f,2} = \log(5X/18).$$ Then, from (\[red para\]), we see that $1$-dimensional projections of $\Theta_{2}({{\mathcal{S}}}_f(X))$ have lengths of order $O_f(X)$. As for the $2$-dimensional projections, note that (\[para\]) and (\[red bounds\]) yield $$|C| = \beta^2 e^{4t}{\text{ and }}1\ll_f |C|\ll_f X,$$ as well as the estimates $$\left\lvert B - \frac{6\alpha^2C}{\beta^2}\right\rvert \leq \frac{1}{2}X^{1/2}{\text{ and }}\left\lvert A -\frac{\alpha^4C}{\beta^4} \right\rvert \leq\frac{5}{144|C|}X + \frac{\alpha^2}{2\beta^2}X^{1/2}.$$ Hence, the projections of $\Theta_2({{\mathcal{S}}}_f(X))$ onto the $BC$-plane and $AC$-plane, respectively, have areas bounded by $$O_f\left(\int_1^{X} X^{1/2} dC\right) {\text{ and }}O_f\left(\int_1^{X}\left(\frac{1}{C}X+ X^{1/2}\right)dC\right).$$ Similarly, from (\[para\]) and (\[red bounds\]), we deduce that $$|2B-L| = 12\alpha^2 e^{4t},\, 1\ll_f |2B-L|\ll_f X,\, |B|\ll_f X,$$ as well as the estimate $$\left\lvert A - \frac{\alpha^2B}{6\beta^2}\right\rvert
\leq \frac{5\alpha^2}{12\beta^2}\left(\frac{1}{|2B-L|} X+ X^{1/2}\right).$$ Note that $|L|\leq X^{1/2}$ also implies that $$|2B - L| \geq |2|B| - |L|| \geq 2|B| - X^{1/2} \mbox{ when }|B|\geq X^{1/2}/2.$$ Hence, the projection of $\Theta_2({{\mathcal{S}}}_f(X))$ onto the $AB$-plane has area bounded by $$O_f\left(\int_{0}^{1+X^{1/2}/2}(X+X^{1/2})dB + \int_{1+X^{1/2}/2}^{X}\left(\frac{1}{2B-X^{1/2}}X+X^{1/2}\right)dB \right).$$ It follows that all of the $2$-dimensional projections of $\Theta_2({{\mathcal{S}}}_f(X))$ have areas of order $O_f(X^{3/2})$, and this proves the lemma.
### The case when $f$ is indefinite and irreducible
We have $${\operatorname{Vol}}(\Theta_{1}({{\mathcal{S}}}_f(X))) = \frac{8\alpha^3}{D_f^{3/2}}\cdot\frac{1}{18}\cdot 2\cdot\left({\operatorname{Vol}}(\Omega^+(X)\times [0,t_{D_f})) + {\operatorname{Vol}}(\Omega^-(X)\times [0,t_{D_f}))\right)$$ by Lemma \[Psi lemma\] and Proposition \[Phi-\], as well as $$\begin{aligned}
{\operatorname{Vol}}(\Omega^+(X)\times [0,t_{D_f})) &= \int_{-X^{1/2}}^{X^{1/2}} \int_{-L^2/4}^X t_{D_f} dK dL
=\frac{13t_{D_f}}{6}X^{3/2},\\[0.5ex]
{\operatorname{Vol}}(\Omega^-(X)\times [0,t_{D_f})) &= \int_{-X^{1/2}}^{X^{1/2}} \int_{-X}^{-L^2/4} t_{D_f} dK dL = \frac{11 t_{D_f}}{6}X^{3/2},\end{aligned}$$ Observe also that $${\operatorname{Vol}}(\overline{\Theta_{1}({{\mathcal{S}}}_f(X))}) = O_f(X)$$ because $\Theta_{1}({{\mathcal{S}}}_f(X))$ lies in the cube centered at the origin of side length $O_f(X^{1/2})$ by (\[indef para 1\]), (\[indef para 2\]), (\[general parameter\]), and the bound on $t$. We then deduce part (c) from (\[N1\]) and (\[N2\]).
Acknowledgments
===============
The first-named author was partially supported by the China Postdoctoral Science Foundation Special Financial Grant (grant number: 2017T100060). We would like to thank the referee for many useful suggestions which helped improve the exposition of the paper significantly.
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---
abstract: 'In this article we show that there are at most two integers up to $2(n-k)$, which can occur as the degrees of nonzero Stiefel-Whitney classes of vector bundles over the Stiefel manifold $V_k({\mathbb{R}}^n)$. In the case when $n> k(k+4)/4$, we show that if $w_{2^q}(\xi)$ is the first nonzero Stiefel-Whitney class of a vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$ then $w_t(\xi)$ is zero if $t$ is not a multiple of $2^q.$ In addition, we give relations among Stiefel-Whitney classes whose degrees are multiples of $2^q$.'
address:
- 'Stat-Math Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bangalore 560059, INDIA.'
- 'Stat-Math Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bangalore 560059, INDIA.'
author:
- Prateep Chakraborty
- Ajay Singh Thakur
title: 'On Stiefel-Whitney Classes of vector bundles over real Stiefel Manifolds'
---
[^1]
Introduction
============
The real Stiefel manifold $V_k({\mathbb{R}}^n)$ is the set of all orthonormal $k$-frames in ${\mathbb{R}}^n$ and it can be identified with the homogeneous space $SO(n)/SO(n-k)$. The main aim of this article is to study Stiefel-Whitney classes of vector bundles over a real Stiefel manifold.
Recall that the degree of the first nonzero Stiefel-Whitney class of a vector bundle over a CW-complex $X$ is a power of $2$ (cf., for example, [@milnor page 94]). In the case when $X$ is a $d$-dimensional Sphere $S^d$, it is a theorem of Atiyah-Hirzebruch [@atiyah Theorem 1] that $d$ can occur as the degree of a nonzero Stiefel-Whitney class of a vector bundle over $S^d$ if and only if $d=1,2,4,8.$ The possible Stiefel-Whitney classes of vector bundles over Dold manifold and stunted real projective space are completely determined by Stong [@stong] and Tanaka [@tanaka], respectively. In this article we shall deal with case $X = V_k({\mathbb{R}}^n)$ and derive certain results on Stiefel-Whitney classes.
In [@knt], it was observed that for a vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$, $n>k$, the Stiefel-Whitney class $w_{n-k}(\xi) = 0$ if $n-k \neq 1,2,4,8$ and $w_{n-k+1}(\xi) = 0$ if $n-k = 2,4,8$. We extend this observation to get the following theorem where we show that there are at most two integers up to $2(n-k)$, which can occur as the degrees of nonzero Stiefel-Whitney classes of any vector bundle over $V_k({\mathbb{R}}^n)$.
\[tanaka\] Let $\xi$ be a vector bundle over $V_k({\mathbb{R}}^n), n> k$. Let $i$ be a positive integer with $i\leq 2(n-k)$. Then $w_i(\xi)=0$ if one of the following conditions is satisfied.
1. $n-k \neq 1,2,4,8$ and $i \neq 2^{\varphi(n-k-1)}$
2. $n-k = 1,2,4,8$ and $i \neq n-k, 2(n-k)$.
In the above theorem, $\varphi(m)$, for a non-negative integer $m$, is the number of integers $l$ such that $0<l\leq m$ and $l\equiv 0,1,2,4 \pmod 8$.
From Theorem \[tanaka\], we observe that if $i$ is the first nonzero Stiefel-Whitney class of a vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$ and $i\leq2(n-k)$, then $i$ is of the form $2^{\varphi(n-k-1)}.$ Now in the next theorem, for a vector bundle over $V_k({\mathbb{R}}^n)$, we derive vanishing of certain Stiefel-Whitney classes whose degrees depend on the degree of the first nonzero Stiefel-Whitney class.
\[main\] Let $n > k(k+4)/4$. Let $\xi$ be a vector bundle over $V_k(\mathbb{R}^n)$ with first non-zero Stiefel Whitney class in degree $2^q$. If $i$ is a multiple of $2^q$ and is written as $i = 2^{q+t_1}+2^{q+t_2}+\cdots + 2^{q+t_m} \mbox{ with } t_j\geq0 \mbox{ and } t_j<t_{j+1}$, then $w_i(\xi) = w_{2^{q+t_1}}(\xi)\cdot w_{2^{q+t_2}}(\xi)\cdots w_{2^{q+t_m}}(\xi)$. Further, if $i$ is not a multiple of $2^q$, then $w_i(\xi) = 0$.
Recall ([@naolekar]) that $\mbox{ucharrank}(X)$ of $X$ is the maximal degree up to which every cohomology class of $X$ is a polynomial in the Stiefel-Whitney classes of a vector bundle over $X$. The *ucharrank* of $V_k({\mathbb{R}}^n)$ was computed in [@knt], except for the cases $n-k = 4,8$, in which cases it was shown that $\mbox{ucharrank}(V_k({\mathbb{R}}^n))$ is bounded above by $n-k$. In Example \[exmp\], we construct a vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$, when $n-k = 4,8$, such that $w_{n-k}(\xi) \neq 0$ and hence improve the result in [@knt] to obtain $\mbox{ucharrank}(V_k({\mathbb{R}}^n)) = n-k$.
To prove our results we need the Steenrod algebra action on the mod-2 cohomology ring $H^*(V_k({\mathbb{R}}^n);{\mathbb Z}_2)$. Recall [@borel Proposition 9.1 and 10.3] that the cohomology ring $H^*(V_k(\mathbb R^n);{\mathbb Z}_2)$ has a simple system of generators $a_{n-k}, a_{n-k+1},\ldots , a_{n-1}$, where $a_i\in H^i(V_k(\mathbb R^n)$ with the following relations: $$a_i^2= \left \{\begin{array}{cl} a_{2i} & \mbox{ if } 2i \leq n-1 \\0 & \mbox{ otherwise. } \end{array}\right.$$ The action of Steenrod algebra is completely determined by knowing that (see [@borel], Remarque 2 in §10): $$Sq^i(a_j)=\left\{\begin{array}{cl}
\displaystyle{j \choose i}a_{j+i} & \mbox{if $ j+i\leq n-1$,}\\
& \\
0 & \mbox{otherwise.}
\end{array}\right.$$
Notations {#notations .unnumbered}
---------
In this article we shall only consider real Stiefel manifold. The cohomology ring will always be with ${\mathbb Z}_2$-coefficients, unless specified otherwise.
Proof of Theorem \[tanaka\]
===========================
We first recall the description of Stiefel-Whitney classes of vector bundles over stunted real projective space, due to Tanaka [@tanaka]. For $n>k$, let $P_{n,k}$ be the stunted real projective space obtained from $\mathbb{RP}^{n-1}$ by collapsing the subspace $\mathbb{RP}^{n-k-1}$ to a point. Consider the following cofibration sequence $$\mathbb{RP}^{n-k-1}\longrightarrow\mathbb{RP}^{n-1}\stackrel{g}\longrightarrow P_{n,k}.$$ The induced map in cohomology $g^*:H^j(P_{n,k})\rightarrow H^j(\mathbb{RP}^{n-1})$ is an isomorphism when $n-k\leq j\leq n-1$. Therefore, for any vector bundle $\xi$ over $P_{n,k}$, the Stiefel-Whitney class $w_j(\xi)\neq0$ if and only if $w_j(g^*(\xi))\neq0$. From [@adams] (also cf. [@tanaka]), we know that the image $g^*:{\widetilde}{KO}(P_{n,k})\rightarrow {\widetilde}{KO}(\mathbb{RP}^{n-1})$ is generated by $2^{\varphi(n-k-1)}\gamma$, where $\gamma$ is the canonical line bundle over $\mathbb{RP}^{n-1}$ and for a non-negative integer $m$, $\varphi(m)$ is as defined in the Introduction. If we denote the generator of $H^*(\mathbb{RP}^{n-1})$ by $t$, then for any integer $d$, the total Stiefel-Whitney class of the element $d2^{\varphi(n-k-1)}\gamma$ in the image of $g^*$, is given as $$w(d2^{\varphi(n-k-1)}\gamma)=(1+t)^{d2^{\varphi(n-k-1)}}=(1+t^{2^{\varphi(n-k-1)}})^d.$$ Therefore, the nonzero Stiefel-Whitney classes of any vector bundle $\xi$ over $P_{n,k}$ can occur only in degrees $r2^{\varphi(n-k-1)}$ for some integer $r$.
To prove Theorem \[tanaka\], we shall use the following observation. For a non-negative integer $m$, we note that if $m\equiv 1,2,3,4,5\pmod 8$, then $\varphi(m)=[m/2]+1$ and if $m\equiv 0,6,7\pmod 8$, then $\varphi(m)=[m/2]$. From here we can conclude that for a positive integer $m$, we have $2^{\varphi(m-1)} \geq m$ and the equality holds only if $m = 1,2,4 \mbox{ and } 8$.
Recall (cf. [@james]) that there is a cellular embedding $f:P_{n,k}\hookrightarrow V_{k}({\mathbb{R}}^n)$ such that the cellular pair $(V_{k}({\mathbb{R}}^n), P_{n,k})$ is $2(n-k)$ connected (cf. [@james Proposition 1.3]). Hence, the induced map in cohomology $f^*:H^j(V_{k}({\mathbb{R}}^n)) \rightarrow H^j(P_{n,k})$ is injective for $j\leq 2(n-k)$. Therefore, for a vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$, the Stiefel-Whitney class $w_j(\xi)\neq0$ if and only if $w_j(f^*(\xi))\neq0$ when $n-k\leq j\leq 2(n-k).$ By the description of Stiefel-Whitney classes of vector bundles over $P_{n,k}$, as discussed above, it follows that $w_j(\xi)=0$ if $n-k\leq j\leq \min\{n-1,2(n-k)\}$ and $j\neq r2^{\varphi(n-k-1)}$ for any integer $r$. As, $2^{\varphi(n-k-1)} \geq (n-k)$ and the equality holds only if $n-k = 1,2,4$, and $8$, the only multiples of $2^{\varphi(n-k-1)}$ that can occur in between $(n-k)$ and $2(n-k)$ are $2^{\varphi(n-k-1)}, 2^{\varphi(n-k-1)+1}$. Moreover, both these multiples will occur in this range only when $n-k = 1,2,4,8$. Now the proof of the theorem follows if $2(n-k) \leq n-1$. If $n-1 <2(n-k)$ then the injectivity of the map $f^*$ gives $H^j(V_{n,k}) = 0$, and hence $w_j(\xi) = 0$, for $n-1 < j \leq 2(n-k)$. This completes the proof.
If we assume $n \geq 2k$ then $n-1 < 2(n-k)$. Then the proof of the following corollary follows from Theorem \[tanaka\].
\[stunted\] Let $V_k({\mathbb{R}}^n)$ be a Stiefel manifold with $n \geq 2k$. Then $w_i(\xi) = 0$ for $i \leq n-1$ and $i \neq 2^{\varphi(n-k-1)}$ for any vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$.
If we fix $k$ and vary $n$ then we have the following corollary.
Let $k$ be fixed. Then except for finitely many values of $n$, the Stiefel-Whitney classes $w_i(\xi)=0$ for $i\leq n-1$ and any vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$.
The proof follows from Corollary \[stunted\] by using the fact that $n-1<2^{\varphi(n-k-1)}$ except for finitely many values of $n.$
In view of Theorem \[tanaka\], it will be interesting to know whether there exists a vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$ such that $w_{2^{\varphi(n-k-1)}}(\xi) \neq 0$. We have complete answer when $2^{\varphi(n-k-1)} = n-k$. We observed in the above proof that $2^{\varphi(n-k-1)}=n-k$ if and only if $n-k=1,2,4,8.$ In the case when $n-k = 1,2$, the existence of a vector bundle $\xi$ such that $w_{n-k}(\xi) \neq 0$ is a consequence of the fact that $H^1(V_k({\mathbb{R}}^{k+1});{\mathbb Z}_2) \neq 0$ and the mod-2 reduction map $H^2(V_k({\mathbb{R}}^{k+2}); {\mathbb Z}) \rightarrow H^2(V_k({\mathbb{R}}^{k+2});{\mathbb Z}_2)$ is the projection map ${\mathbb Z}\rightarrow {\mathbb Z}_2$ (cf. [@knt]). In the following example, when $n-k=4,8$ we construct a vector bundle $\xi$ over $V_k({\mathbb{R}}^n)$ such that $w_{n-k}(\xi)\neq 0.$
\[exmp\] **
Let $\alpha: Spin(n) \rightarrow V_k({\mathbb{R}}^n)$ be the principal $Spin(n-k)$-bundle over $V_k({\mathbb{R}}^n)=Spin(n)/Spin(n-k)$. If ${\widetilde}{RO}Spin(n-k)$ and ${\widetilde}{R}Spin(n-k)$ are the reduced real and complex representation rings respectively, then we have the following commutative diagram:
$$\label{diag1}
\xymatrix{
{\widetilde}{RO}Spin(n-k) \ar[r]\ar[d] & {\widetilde}{KO}(Spin(n)/Spin(n-k)) \ar[ld]^{f^*} \\
{\widetilde}{KO}(Spin(n-k+1)/Spin(n-k)) &
}$$
Here $f: S^{n-k} = Spin(n-k+1)/Spin(n-k) \rightarrow Spin(n)/Spin(n-k)$ is the natural inclusion. In the case when $n-k = 8$, the map ${\widetilde}{RO}Spin(8) \rightarrow {\widetilde}{KO}(S^8)$ in Diagram \[diag1\] is surjective (cf. p.195, [@husemoller]) and hence the map $f^*$ is surjective If $[\xi] \in {\widetilde}{KO}(\mathbb{S}^8)$ is the class of the Hopf bundle over $S^8$ then there exists a bundle $\eta$ over $V_k({\mathbb{R}}^n)$ such that $f^*([\eta]) = [\xi]$. As $w_8(\xi) \neq 0$, we have $w_8(\eta) \neq 0$.
Next when $n-k =4$, we use the following diagram: $$\label{diag2}
\xymatrix{
{\widetilde}{R}Spin(n-k) \ar[r]\ar[d] & {\widetilde}{K}(Spin(n)/Spin(n-k)) \ar[ld]^{f^*} \\
{\widetilde}{K}(Spin(n-k+1)/Spin(n-k)) &
}$$ The map ${\widetilde}{R}Spin(4) \rightarrow {\widetilde}{K}(S^4)$ in Diagram \[diag2\] is surjective (cf. p.195, [@husemoller]). Using the fact that the Hopf bundle $\xi$ over $S^4$ is a complex vector bundle with $w_4(\xi) \neq 0$, we proceed as above to conclude that there exists a complex vector bundle $\eta$ over $V_k({\mathbb{R}}^n)$ such that the Stiefel-Whitney class $w_4(\eta_{{\mathbb{R}}})$ of the underlying real bundle $\eta_{{\mathbb{R}}}$ is nonzero.
Proof of Theorem \[main\]
=========================
Recall the description of the cohomology ring $H^*(V_k({\mathbb{R}}^n))$ as in the Introduction. Because of the relations among the generators $a_{n-k}, a_{n-k+1}, \cdots, a_{n-1}$, we can write any nonzero cohomology class $x \in H^j(V_{k}({\mathbb{R}}^n))$ as $$x = \sum a_{i_1}\cdot a_{i_2}\cdots a_{i_r}$$ such that $i_{t} < i_{t+1}$. If a monomial $a_{i_1}\cdot a_{i_2}\cdots a_{i_r}$ in the above summand represents a nonzero cohomology class then we have $$(n-k) +(t-1) \leq \deg a_{i_t} \leq n-1 -r +t.$$ This implies that $$\sum_{t =1}^{r} (n-k) +(t-1) \leq \sum_{t=1}^{r} a_{i_t} \leq \sum_{t=1}^r n -1 -r +t.$$ Hence, $r(n-k) + r(r-1)/2 \leq j \leq r(n-1) - r(r-1)/2$.
For $0\leq p\leq k$, we define $T_p$ as the set $\{j\in \mathbb{N}: p(n-k) + p(p-1)/2 \leq j\leq p(n-1) - p(p-1)/2\}$. Therefore, by the above discussion we have the following lemma.
\[monomial\]If $x = a_{i_1}\cdot a_{i_2}\cdots a_{i_r}$ with $i_{t} < i_{t+1}$ represents a nonzero cohomology class of $V_k({\mathbb{R}}^n)$, then $\deg x \in T_r$.
If we assume $n > k(k+4)/4$, then in the following lemma we give an upper bound for the length of each $T_p$.
\[difference\] Let $n > k(k+4)/4$. Then $|r_1 -r_2| < n-k$ for any $p$ and $r_1, r_2 \in T_p$.
For any $r_1,r_2\in T_p$, we have $|r_1-r_2|\leq p(n-1)-p(p-1)/2-p(n-k)-p(p-1)/2=p(k-p).$ The maximum value of the set $\{p(k-p):1\leq p\leq k\}$ is $k^2/4$ if $k$ is even and $(k^2-1)/4$ if $k$ is odd. Since $n>k(k+4)/4$ if and only if $n-k>k^2/4,$ we have $|r_1-r_2|<n-k.$
In the following lemma, we derive some results involving binomial coefficients which we shall use in the proof of Theorem \[main\].
\[binomial\] Let $s$ be an odd number and $r \leq 2^t$. Then the binomial coefficients
1. ${2^ts+r-1\choose r}$ is even if and only if $r\neq 0,2^{t}$.
2. 3. ${2^ts-1\choose 2^{t+1}}$ is odd if $s \equiv 3 \pmod 4$.
To prove Statement (1), we note that if $r \neq 0$ then $${2^{t}s+r-1\choose r} = \left(\frac{2^ts }{r}\right)\left( \prod_{l=1}^{[(r-1)/2]}\frac{2^ts +2}{2l}\right) \left( \prod_{l=1}^{[r/2]}\frac{2^ts +2l-1}{2l-1}\right)$$Now it is easy to see that the second and third products in the right hand side of the above equality can be written as ratios of two odd integers. Further, $(2^ts/r)$ can be written as a ratio of two odd integers if and only if $r = 2^t$. From here we conclude Statement (1).
Next we prove Statement (2). We first note that $${2^ts-1\choose2^{t+1}}= \left(\prod_{l=1}^{2^t}\frac{2^ts-2l}{2l} \right) \left(\prod_{l=1}^{2^t}\frac{2^ts-2l -1}{2l-1} \right)$$ Now if $ l \neq 2^{t-1} \mbox{ or } 2^t$, then $\frac{2^ts-2l}{2l}$ can be written as a ratio of two odd integers. On the other hand if $l = 2^{t-1} \mbox { and } 2^t$ then the product $$\left(\frac{2^ts-2^t}{2^t}\right)\left(\frac{2^ts-2^{t+1}}{2^{t+1}}\right)=(s-1)(s-2)/2,$$ which is an odd number as $s \equiv 3 \pmod 4$. This completes the proof of Statement (2).
We now prove Theorem \[main\].
Let $i = 2^{q+t_1}+2^{q+t_2}+ \cdots+2^{q+t_m}$ with $t_j\geq0$ and $t_j<t_{j+1}$. If $i$ is a power of 2 (i.e., when $m =1$) or $H^{i}(V_k({\mathbb{R}}^n))=0$, then the first statement of the theorem follows easily. Next we assume that $m>1$ and $H^{2^qr}(V_k({\mathbb{R}}^n))\neq0$. By Wu’s formula we get $$\begin{array}{ccl}
Sq^{2^{q+t_1}}(w_{i-2^{q+t_1}}(\xi))& = &\displaystyle\sum_{r=0}^{2^{q+t_1}}{i-2^{q+t_1+1}+r-1\choose r}w_{2^{q+t_1}-r}(\xi)\cdot w_{i-2^{q+t_1}+r}(\xi)\\\\
& =& w_{2^{q+t_1}}(\xi)\cdot w_{i-2^{q+t_1}}(\xi)+w_{i}(\xi).
\end{array}$$ The last equality above follows by Lemma \[binomial\](1). Next we prove that the left hand side of the above equation is zero. For this it is enough to prove that if $x = a_{i_1} \cdot a_{i_2} \cdots a_{i_{p}}$, with $i_j < i_{j+1}$, is a nonzero cohomology class of degree $i-2^{q+t_1}$, then the Steenrod square $Sq^{2^{q+t_1}}(x)=0$. For this first note that $$Sq^{2^{q+t_1}}(x) =Sq^{2^{q+t_1}}(a_{i_1} \cdot a_{i_2} \cdots a_{i_{p}})=\sum_{l_1 + \cdots + l_{p} = 2^{q+t_1}} Sq^{l_1}(a_{i_1}) \cdots Sq^{l_{p}}(a_{i_{p}}).$$ We shall show that each summand in the right hand side of the above equation is zero. As the monomial $a_{i_1} \cdot a_{i_2} \cdots a_{i_{p}}$ represents a nonzero cohomology class, it follows by Lemma \[monomial\] that its degree, $i-2^{q+t_1} \in T_{p}$. If a summand $ Sq^{l_1}(a_{i_1}) \cdots Sq^{l_{p}}(a_{i_{p}})$ is nonzero then for all $j$ we have $l_j + i_j \leq n-1$, $Sq^{l_j}(a_{i_j}) = a_{{i_j} + l_j}$. Moreover, as $n\geq 2k$, we have $a_{i_j}^2 =0$ for all $j$ and this will imply that $l_{j_1} +i_{j_1} \neq l_{j_2} +i_{j_2}$ for $j_1 \neq j_2$. Hence, $$p(n-k) +p(p-1)/2 \leq \sum_{j=1}^{p} i_j + l_j = i \leq p(n-1) - p(p-1)/2.$$ This implies that $i \in T_{p}$. Since, $i-2^{q+t_1}$ also belongs to $T_p$, the difference, $i - (i-2^{q+t_1})= 2^{q+t_1} \geq 2^q \geq n-k$, gives a contradiction to Lemma \[difference\] and hence, we conclude that $Sq^{2^{q+t_1}}(x)=0$. This proves that $w_{i}(\xi)= w_{2^{q+t_1}}(\xi)\cdot w_{i-2^{q+t_1}}(\xi).$ Now the proof of the first statement follows by induction on $m$.
Now we prove the last statement of the theorem by applying induction on the set $\{i:i \text{ is not a multiple of }2^q\}$. If $i<2^q$, then $w_i(\xi)=0$ by hypothesis. Next assume that $i > 2^q$, $H^i(V_k({\mathbb{R}}^n) \neq 0$ and $i$ is not a multiple of $2^q$. We can write $i$ as $i=2^ts$ where $s$ is odd, $s\geq3$ and $t<q$. Applying Lemma \[binomial\](1) on Wu’s formula we get $$Sq^{2^t}(w_{2^t(s-1)}(\xi))=w_i(\xi).$$ If $2^t(s-1)$ is not a multiple of $2^q$ or $H^{2^t(s-1)}(V_k({\mathbb{R}}^n))=0$, then by induction we have $w_i(\xi)=0.$ Now assume that $H^{2^t(s-1)}(V_k({\mathbb{R}}^n))\neq0$ and $2^t(s-1)$ is a multiple of $2^q$. Let $2^t(s-1) = 2^{q+t_1} + 2^{q+t_2} + \cdots + 2^{q+t_m}$ with $t_j < t_{j+1}$. We have the following two cases:
**Case $m>1$:** As, $2^t(s-1)$ is a multiple of $2^q$, by the first statement of the theorem, we have $$\label{eqn2}
\begin{array}{ccl}w_{i}(\xi) &= &Sq^{2^t}(w_{2^{q+t_1}}(\xi)\cdot w_{2^{q+t_2}}(\xi)\cdots w_{2^{q+t_m}}(\xi))\\
& &\\
&=& \underset{l_1+\cdots l_m=2^t}{\sum}Sq^{l_1}(w_{2^{q+t_1}}(\xi))\cdots Sq^{l_m}(w_{2^{q+t_m}}(\xi)).\end{array}$$ Now observe that for each $j$, we have $l_j \leq 2^{t} < 2^q$, and hence, the Steenrod square, $$Sq^{l_j}(w_{2^{q+t_j}}(\xi))={2^{q+t_j}-1\choose l_j}w_{2^{q+t_j}+l_j}(\xi).$$ As $m>1$, we have $2^{q+t_j}< 2^t(s-1)$. Therefore $2^{q+t_j}+l_j<2^ts$. Further if for some $j$, we have $l_j>0$, then $2^{q+t_j}+l_j$ is not a multiple of $2^q$ and hence by induction, $w_{2^{q+t_j}+l_j}(\xi)=0.$ Now observe that in each summand in the right hand side of Equation \[eqn2\], there is at least one $l_j$ such that $l_j>0.$ Therefore, $$w_{2^ts}(\xi)=\sum_{l_1+\cdots l_m=2^t}Sq^{l_1}(w_{2^{q+t_1}}(\xi))\cdots Sq^{l_m}(w_{2^{q+t_m}}(\xi))= 0.$$
**Case $m=1$:** In this case, $2^t(s-1) = 2^{q+t_1}$ for some $t_1 \geq 0$. Thus $s-1$ is a power of $2.$ First we consider the case when $s\geq 5.$ Here we observe that $2^{t+1} < 2^ts - 2^{t+1}.$ Since, $s-1 \equiv 0 \pmod 4$, then by Lemma \[binomial\](2), we have $$Sq^{2^{t+1}}(w_{2^ts-2^{t+1}}(\xi))=w_{2^{t+1}}(\xi)\cdot w_{2^ts-2^{t+1}}(\xi)+w_{2^ts}(\xi).$$ As $2^ts-2^{t+1}=2^t(s-1) -2^t = 2^{q+t_1} -2^t = 2^t(2^{q-t+t_1}-1)$, we have that $2^ts-2^{t+1}$ is not a multiple of $2^q$ and hence, by induction, $w_{2^ts-2^{t+1}}(\xi) = 0$. Therefore, $w_{2^ts}(\xi) = 0$. Next we deal with the case $s = 3$. We observe that in this case, $t = q-1$ and $t_1 =0$. Thus, we obtain $$Sq^{2^{q-1}}(w_{2^q}(\xi))=w_{2^{q-1}3}(\xi) = w_i(\xi).$$ If $2^q \in T_1$, then $w_{2^q}(\xi) = a_j$ for some $j$, as $n \geq 2k$. Therefore $w_i(\xi) = Sq^{2^{q-1}}(a_j)$. So $w_i(\xi) = 0$ if $i > n-1$. If $i \leq n-1$, then $w_i(\xi)$ is again zero, as $w_{2^q}(\xi)$ is the only nonzero Stiefel-Whitney class up to degree $n-1$ (cf. Corollary \[stunted\]). Next we assume that $2^q \not \in T_1$. Since $w_{2^q}(\xi) \neq 0$, we have that $2^q \in T_p$ for some $p$ such that $p \geq 2$. To prove $w_{2^{q-1}3}(\xi)=0$, we consider the Steenrod square operation, $$\label{eqn}
Sq^{2^{q-1}}(a_{i_1} \cdot a_{i_2} \cdots a_{i_{p}})=\sum_{l_1 + \cdots + l_{p} = 2^{q-1}} Sq^{l_1}(a_{i_1}) \cdots Sq^{l_{p}}(a_{i_{p}})$$ on a monomial $a_{i_1} \cdot a_{i_2} \cdots a_{i_{p}}$ such that $i_j<i_{j+1}$, which represents a nonzero cohomology class of degree $2^q$. By Lemma \[monomial\], the degree of the monomial, $2^q \in T_p$. If a summand $ Sq^{l_1}(a_{i_1}) \cdots Sq^{l_{p}}(a_{i_{p}})$ is nonzero then for all $j$ we have $l_j + i_j \leq n-1$, $Sq^{l_j}(a_{i_j}) = a_{{i_j} + l_j}.$ Moreover, as $n\geq 2k$, we have $l_{j_1} +i_{j_1} \neq l_{j_2} +i_{j_2}$ for $j_1 \neq j_2$. Hence, $$p(n-k) +p(p-1)/2 \leq \sum_{j=1}^{p} i_j + l_j = 2^{q-1}3 \leq p(n-1) -p(p-1)/2.$$ This implies that $2^{q-1}3 \in T_{p}$. As $p \geq 2$, we have $2^q \geq 2(n-k)+1$ and this implies that $2^{q-1} > n-k$. Therefore, the difference $2^{q-1}3 - 2^{q} = 2^{q-1}>n-k$. This is a contradiction to Lemma \[difference\]. This shows that each summand in the right hand side of Equation \[eqn\] is zero. Hence $w_{2^{q-1}3}(\xi) =0$ if $p \geq 2$.
Acknowledgment: {#acknowledgment .unnumbered}
---------------
The authors thank Aniruddha C. Naolekar and Parameswaran Sankaran for their valuable suggestions and comments.
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[^1]: The research of first author is supported by NBHM postdoctoral fellowship. The research of second author is supported by DST-Inspire Faculty Scheme (IFA-13-MA-26)
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---
abstract: 'Correctly identifying sleep stages is important in diagnosing and treating sleep disorders. This work proposes a joint classification-and-prediction framework based on convolutional neural networks (CNNs) for automatic sleep staging, and, subsequently, introduces a simple yet efficient CNN architecture to power the framework[^1]. Given a single input epoch, the novel framework jointly determines its label (classification) and its neighboring epochs’ labels (prediction) in the contextual output. While the proposed framework is orthogonal to the widely adopted classification schemes, which take one or multiple epochs as contextual inputs and produce a single classification decision on the target epoch, we demonstrate its advantages in several ways. First, it leverages the dependency among consecutive sleep epochs while surpassing the problems experienced with the common classification schemes. Second, even with a single model, the framework has the capacity to produce multiple decisions, which are essential in obtaining a good performance as in ensemble-of-models methods, with very little induced computational overhead. Probabilistic aggregation techniques are then proposed to leverage the availability of multiple decisions. To illustrate the efficacy of the proposed framework, we conducted experiments on two public datasets: Sleep-EDF Expanded (Sleep-EDF), which consists of 20 subjects, and Montreal Archive of Sleep Studies (MASS) dataset, which consists of 200 subjects. The proposed framework yields an overall classification accuracy of 82.3% and 83.6%, respectively. We also show that the proposed framework not only is superior to the baselines based on the common classification schemes but also outperforms existing deep-learning approaches. To our knowledge, this is the first work going beyond the standard single-output classification to consider multitask neural networks for automatic sleep staging. This framework provides avenues for further studies of different neural-network architectures for automatic sleep staging.'
author:
- 'Huy Phan$^*$, Fernando Andreotti, Navin Cooray, Oliver Y. Chén, and Maarten De Vos[^2] [^3]'
bibliography:
- 'bibliography.bib'
title: Joint Classification and Prediction CNN Framework for Automatic Sleep Stage Classification
---
sleep stage classification, joint classification and prediction, convolutional neural network, multi-task.
Introduction {#sec:intro}
============
Identifying the sleep stages from overnight Polysomnography (PSG) recordings plays an important role in diagnosing and treating sleep disorders, which affects millions of people [@Krieger2017; @Redmond2006]. Traditionally, this task has been done manually by experts via visual inspection which is tedious, time-consuming, and is prone to subjective error. Automatic sleep stage classification [@Aboalayon2016], that performs as well as manual scoring, can help to ease this task tremendously, therefore facilitating home monitoring of sleep disorders [@Kelly2012].
\[!t\] ![Illustration of (a) the standard classification approach, (b) the common classification approach with the contextual input of three epochs, and (c) the joint classification and prediction with the contextual output of three epochs proposed in this work.[]{data-label="fig:contextualinput_vs_output"}](figure1.eps "fig:"){width="0.9\linewidth"}
The guiding principle of automatic sleep staging is to split the signal into a sequence of epochs, each of which is usually 30 seconds long, and the classification is then performed epoch-by-epoch. In order to uncover a sleep stage at each epoch, proper features need to be derived from the signal, such as electroencephalography (EEG). Traditionally, many features have been designed based on prior knowledge of sleep. These hand-crafted features range from time-domain features [@Krakovska2011; @Koley2012; @Redmond2006] to frequency-domain features [@Phan2013; @Susmakova2008; @Koley2012; @Fell1996], via features derived from nonlinear processes [@Kim2000; @Lee2012; @Susmakova2008; @Zhang2001]. Using these features, the classification goal is often accomplished by conventional machine learning algorithms, such as Support Vector Machine (SVM) [@Alickovic2018; @Koley2012], $k$-nearest neighbors ($k$-NN) [@Phan2013], Random Forests [@Memar2018; @Boostania2017; @Imtiaz2015b].
The advent of deep learning and its astonishing progress in numerous domains have stimulated interest in applying them for automatic sleep staging. The power of deep networks lies in their great capability of automatic feature learning from data, thus avoiding the reliance on hand-crafted features. Significant progress on results obtained from different sleep staging benchmark using various deep learning techniques have been reported [@Stephansen2017; @Mikkelsen2018; @Zhang2017; @Tsinalis2016; @Tsinalis2016b; @Supratak2017; @Dong2017; @Phan2018c; @Phan2018d], mirroring a relentless trend where learned features ultimately outperform and displace long-used hand-crafted features. CNN [@LeCun2015; @Lecun1989], the cornerstone of deep learning techniques, has been frequently employed for the task [@Mikkelsen2018; @Zhang2017; @Tsinalis2016]. The weight sharing mechanism at the convolutional layers forces the shift-invariance of the learned features and greatly reduces the model’s complexity, consequently leading to improvement of the model’s generalization [@LeCun2015]. Other network variants, such as Deep Belief Networks (DBNs) [@Laengkvist2012], Auto-encoder [@Tsinalis2016b], Deep Neural Networks (DNNs) [@Dong2017], have also been explored. Moreover, Recurrent Neural Networks (RNNs), e.g. Long Short-Term Memory (LTSM) [@Hochreiter1997], which are capable of sequential modelling, have been found efficient in capturing long-term sleep stage transition and are usually utilized to complement other network types, such as CNNs [@Supratak2017; @Stephansen2017] and DNNs [@Dong2017]. Standalone RNNs have also been exploited for learning sequential features of sleep [@Phan2018d; @Koch2018a; @Koch2018b]. The classification is usually performed therein by the networks in an end-to-end fashion [@Mikkelsen2018; @Zhang2017; @Tsinalis2016]; a separate classifier, such as SVM, can be used alternatively [@Phan2018d; @Ansari2018].
Motivation and Contributions
============================
Motivation
----------
Sleep is a temporal process with slow stage transitions, implying continuity of sleep stages and strong dependency between consecutive epochs [@Iber2007; @Liang2011; @Sousa2015]. For instance, out of 228,870 epochs in the entire MASS dataset [@Oreilly2014] used in this work, $83.3\%$ pairs of adjacent epochs have the same label. The ratio is still as high as $79.3\%$, when two epochs are separated by one epoch. This nature of sleep has inspired a widely adopted practice in neural-network-based sleep staging systems, namely the use of *contextual input* that augments a target epoch by its surrounding epochs (*many-to-one*) in the classification task [@Stephansen2017; @Mikkelsen2018; @Tsinalis2016; @Supratak2017]. Common input context size is of three and five epochs [@Chambon2018; @Mikkelsen2018; @Tsinalis2016; @Tsinalis2016b]. This classification scheme can also be interpreted as an extension of the standard classification setup, i.e. determining the sleep stage corresponding to a single epoch of input signals (*one-to-one*) [@Phan2018c; @Phan2018d; @Andreotti2018]. Figure \[fig:contextualinput\_vs\_output\] (b) provides a schematic presentation of contextual input of three epochs in comparison with the standard one-to-one classification approach in Figure \[fig:contextualinput\_vs\_output\] (a). While multiple-epoch input does not always provide performance gains, as shown in our experiments, it poses a problem of inherent *modelling ambiguity*. That is, when training a network with contextual input, such as three epochs illustrated in Figure \[fig:contextualinput\_vs\_output\] (b), it remains unclear whether the network is truly modelling the class distribution of the target epoch at the center or that of the left and right neighbor. In our experiment, such a network (i.e. the many-to-one baseline in Section \[ssec:baseline\]) achieves an accuracy of $82.1\%$ in determining the labels of the center epochs. However, when aligning the network output with those labels of the left and right neighbor, the accuracy is just marginally lower, reaching $81.1\%$ and $80.8\%$, respectively. Last but not least, the contextual input causes the network’s computational complexity increase at a linear scale due to the enlarged input size.
\[!t\] {width="0.85\linewidth"}
In this work, we formulate sleep staging as a joint classification and prediction problem. In other words, this is equivalent to a *one-to-many* problem, which is an extension of the standard one-to-one classification scheme while being orthogonal to the common many-to-one classification scheme. With this new formulation, given a single target epoch as input, our objective is to simultaneously determine its label (classification) and its neighboring epochs’ labels (prediction) in the *contextual output*, as demonstrated in Figure \[fig:contextualinput\_vs\_output\] (c). By classification, we mean determining the label of an epoch given its information. In contrast, prediction implies determining the label of an epoch without knowing its information. The rationale behind this idea is that, given the strong dependency of consecutive epochs, using information of an epoch, we should be able to infer the label of its neighbors. The major benefit of the joint classification and prediction formulation are two-fold. First, with the single-epoch input, the employed model does not experience the modelling ambiguity and the computational overhead induced by the large contextual input as previously discussed. Second, the employed model can produce an ensemble of decisions, which is the key in our obtained state-of-the-art performance, with a negligible induced computational cost. Ensemble of models [@Hinton2015; @Dietterich2000], a well-established method to improve the performance of a machine learning algorithm, has been found generalizable to automatic sleep staging, evidenced by conventional methods [@Alickovic2018; @Bhuiyan2016; @Koley2012] and recently developed deep neural networks [@Stephansen2017]. However, building many different models on the same data for model fusion is cumbersome and costly. Opposing to ensemble of models [@Stephansen2017; @Alickovic2018; @Bhuiyan2016; @Koley2012], in our joint classification and prediction formulation, the ensemble of decisions is produced with a *single* multi-task model. Afterwards, an aggregation method can be used to fuse the ensemble of decisions to produce a reliable final decision.
We further proposed a CNN framework to deal with the joint problem. Although the proposed framework is generic in the sense that any CNN can fit in, we employ a simple CNN architecture with time-frequency image input. The efficiency of this architecture for automatic sleep staging was demonstrated in our previous work [@Phan2018c]. To suit the task of joint classification and prediction, we replace the CNN’s canonical softmax layer with a *multi-task softmax* layer and introduce the *multi-task loss* function for network training. Without confusion, we will refer to the proposed framework as multi-task framework, joint classification and prediction framework, and one-to-many framework interchangeably throughout this article.
Contributions
-------------
The main contributions of this work are as follows.
\(i) We formulate automatic sleep staging as a joint classification and prediction problem. The new formulation avoids the shortcomings of the common classification scheme while improving modelling performance.
\(ii) A CNN framework is then proposed for the joint problem. To that end, we present and employ a simple and efficient CNN coupled with a multi-task softmax layer and the multi-task loss function to conduct joint classification and prediction.
\(iii) We further propose two probabilistic aggregation methods, namely additive and multiplicative voting, to leverage ensemble of decisions available in the proposed framework.
\(iv) Performance-wise, we demonstrate experimentally good performance on two publicly available datasets: Sleep-EDF [@Kemp2000; @Goldberger2000] with 20 subjects and MASS [@Oreilly2014], a large sleep dataset with 200 subjects.
Evaluation Datasets {#sec:datasets}
===================
We used two public datasets: Sleep-EDF Expanded (Sleep-EDF) and Montreal Archive of Sleep Studies (MASS) in this work and conducted analyses under both unimodal (i.e. single-channel EEG) and multimodal conditions (i.e combinations of EEG, EOG, and EMG channels). It should be noted that even though we selected the typical EEG, EOG, and EMG channels in our analyses, the proposed framework, however, can be used straightforwardly to study other signal modalities.
Sleep-EDF Expanded (Sleep-EDF)
------------------------------
Sleep-EDF dataset [@Kemp2000; @Goldberger2000] consists of two subsets: (1) Sleep Cassette (*SC*) subset consisting of 20 subjects aged 25-34 aiming at studying the age effects on sleep in healthy subjects and (2) Sleep Telemetry (*ST*) subject consisting of 22 Caucasian subjects for study temazepam effects on sleep. We adopted the *SC* subset in this study. PSG recordings, sampled at 100 Hz, of two subsequent day-night periods are available for each subject, except for one subject (subject 13) who has only one-night data. Each 30-second epoch of the recordings was manually labelled by sleep experts according to the R&K standard [@Hobson1969] into one of eight categories {W, N1, N2, N3, N4, REM, MOVEMENT, UNKNOWN}. Similar to previous works [@Tsinalis2016; @Tsinalis2016b; @Supratak2017], N3 and N4 stages were merged into a single stage N3. MOVEMENT and UNKNOWN were excluded. Since full EMG recordings are not available, we only used the Fpz-Cz EEG and the EOG (horizontal) channels in our experiments. Only the in-bed parts (from *lights off* time to *lights on* time) of the recordings were included as recommended in [@Imtiaz2014; @Imtiaz2015; @Tsinalis2016; @Tsinalis2016b].
Montreal Archive of Sleep Studies (MASS)
----------------------------------------
MASS comprises whole-night recordings from 200 subjects (97 males and 103 females with an age range of 18-76 years). These recordings were pooled from different hospital-based sleep laboratories. The available cohort 1 was divided into five subsets of recordings, SS1 - SS5. As stated in the seminal work [@Oreilly2014], heterogeneity between subsets is expected. Opposing to the majority of previous works which targeted only one homogeneous subset of the cohort [@Dong2017; @Supratak2017], we experimented with all five subsets. Each epoch of the recordings was manually labelled by experts according to the AASM standard [@Iber2007] (SS1 and SS3) and the R&K standard [@Hobson1969] (SS2, SS4, and SS5). We converted them into five sleep stage {W, N1, N2, N3, and REM} as suggested in [@Imtiaz2014; @Imtiaz2015]. Those recordings with 20-second epochs were converted into 30-second ones by including 5-second segments before and after each epoch. We adopted and studied combinations of the C4-A1 EEG, an average EOG (ROC-LOC), and an average EMG (CHIN1-CHIN2) channels in our experiments. The signals, originally sampled at 256 Hz, were downsampled to 100 Hz.
Joint Classification and Prediction CNN Framework {#sec:framework}
=================================================
Overview {#ssec:overview}
--------
The proposed framework, with a schematic illustration shown in Figure \[fig:overview\], can be described in a stage-wise fashion. The raw signals of a certain epoch index $n$ are first transformed into log-power spectra. The spectra are then preprocessed for frequency smoothing and dimension reduction using frequency-domain filter banks. The resulting channel-specific images are then stacked to form a multi-channel time-frequency image, denoted as $\mathbf{X}_n$. Subsequently, a multi-task CNN is exercised on the multi-channel time-frequency image for joint classification and context prediction. The former task is to maximize the conditional probability $P(y_n\,|\, \mathbf{X}_n)$ which characterizes the likelihood of a sleep stage $y_n \in \mathcal{L}=\{1,2,\ldots,Y\}$, where $\mathcal{L}$ denotes the label set of $Y$ sleep stages. The latter one is to maximize the conditional probabilities $(P(y_{n-\tau}\,|\, \mathbf{X}_n), \ldots, P(y_{n-1}\,|\, \mathbf{X}_n), P(y_{n+1}\,|\, \mathbf{X}_n), \ldots, $ $P(y_{n+\tau}\,|\, \mathbf{X}_n))$ of the neighboring epochs in the output context size of $2\tau + 1$. The labels of the epochs in the output context, where $(y_{n-\tau}, \ldots, y_{n}, \ldots, y_{n+\tau})$, can be obtained by probability maximization.
Formally, under this joint classification and prediction formulation, the CNN performs the one-to-many mapping $$\begin{aligned}
\mathcal{\hat{F}}: \mathbf{X}_n \mapsto (y_{n-\tau},\ldots,y_{n}, \ldots, y_{n+\tau}) \in \mathcal{L}^{2\tau+1}.
\label{eq:joint_class_pred}\end{aligned}$$ Note that the order of the epochs in the neighborhood is encoded by the order of the output labels. This formulation is orthogonal to the common classification one with contextual input of size $2\tau + 1$, in which a network performs the many-to-one mapping $$\begin{aligned}
\mathcal{F}: (\mathbf{X}_{n-\tau}, \ldots, \mathbf{X}_n, \ldots, \mathbf{X}_{n+\tau}) \mapsto y_n \in \mathcal{L}.
\label{eq:class_contextual_input}\end{aligned}$$
Both formulations (\[eq:joint\_class\_pred\]) and (\[eq:class\_contextual\_input\]) can be interpreted as different extensions of the standard one-to-one classification scheme [@Phan2018c; @Phan2018d; @Andreotti2018]. They will reduce to the standard one when $\tau=0$. However, with our joint classification and prediction formulation, at a certain epoch index $n$ there exists an ensemble of exact $2\tau + 1$ decisions, wherein one classification decision made by itself (i.e. $\mathbf{X}_n$) and $2\tau$ prediction decisions made by its neighbors $(\mathbf{X}_{n-\tau}, \ldots, \mathbf{X}_{n-1}, \mathbf{X}_{n+1}, \ldots, \mathbf{X}_{n+\tau})$. These decisions can be aggregated to form the final decision that is generally better that any individual ones.
Time-Frequency Image Representation {#ssec:representation}
-----------------------------------
Given a 30-second signal epoch (i.e. EEG, EOG, or EMG), we firstly transform it into a power spectrum using short-time Fourier transform (STFT) with a window size of two seconds and 50% overlap. Hamming window and 256-point Fast Fourier Transform (FFT) are used. The spectrum is then converted to logarithm scale to produce a log-power spectrum image of size $F \times T$, where $F=129$ and $T=29$.
For frequency smoothing and dimension reduction, the spectrum is filtered by a frequency-domain filter bank. Any frequency-domain filter bank, such as the regular triangular one [@Phan2018c], could serve this purpose. However, it is more favorable to learn the filter bank specifically for the task at hand. Our recent works in [@Phan2018c; @Phan2018d] demonstrated that a filter bank learned by a DNN in a discriminative fashion is more competent than the regular one in automatic sleep staging. The learned filter bank is expected to emphasize the subbands that are more important for the task and attenuate those less important. Hence, we use the filter bank pretrained with a DNN for preprocessing here. One such filter bank with $M=20$ filters is learned for each EEG, EOG, and EMG channel. Filtering the log-power spectrum image reduces its size to $M \times T$. When multiple channels are used, we obtain one such time-frequency image for each channel. For generalization, we denote the time-frequency image as $\mathbf{X} \in \mathbb{R}^{P \times M \times T}$ where $P$ denotes the number of channels. $P=1, 2, 3$ is equivalent to the cases when {EEG}, {EEG, EOG}, and {EEG, EOG, EMG} are employed, respectively.
Multi-Task CNN for Joint Classification and Prediction {#ssec:cnn}
------------------------------------------------------
Our recent work [@Phan2018c] presented a simple CNN architecture that was shown efficient for sleep staging. We adapt this architecture here by tailoring the last layer, i.e. the multi-task softmax layer, to perform joint classification and prediction. The proposed CNN architecture is illustrated in Figure \[fig:cnn\]. Opposing to typical deep CNNs [@Stephansen2017; @Mikkelsen2018; @Tsinalis2016; @Supratak2017], the proposed CNN consists of only three layers: one over-time convolutional layer, one pooling layer, and one *multi-task softmax* layer. This simple architecture has three main characteristics. First, similar to those in [@Phan2017; @Phan2016; @Supratak2017], its convolutional layer simultaneously accommodates convolutional kernels with varying sizes, and is therefore able to learn features at different resolutions. Second, the exploited *1-max* pooling strategy at the pooling layer is more suitable for capturing the *shift-invariance* property of temporal signals than the common subsampling pooling since a particular feature could occur at any temporal position rather than in a local region of the input signal [@Kim2014; @Phan2016; @Phan2017]. Third, opposing to the canonical softmax, the multi-task softmax layer is adapted to suit the joint classification and prediction. Furthermore, the multi-task loss is introduced for network training.
Assume that we obtain a training set $\mathcal{S}~=~\left\{\left(\mathbf{X}^{(i)}_{n_i}, (\mathbf{y}^{(i)}_{n_i - \tau}, \ldots, \mathbf{y}^{(i)}_{n_i}, \ldots, \mathbf{y}^{(i)}_{n_i + \tau})\right)\right\}^N_{i=1}$ of size $N$ from the training data. An epoch $i$ is represented by the multi-channel time-frequency image $\mathbf{X}^{(i)}_{n_i} \in \mathbb{R}^{P \times M \times T}$ as described in \[ssec:representation\] and $n_i$ denotes the corresponding index of the epoch in the original signal. Each epoch $i$ is associated with the sequence of one-hot encoding vectors $(\mathbf{y}^{(i)}_{n_i - \tau}, \ldots, \mathbf{y}^{(i)}_{n_i}, \ldots, \mathbf{y}^{(i)}_{n_i + \tau})$ which represent the sleep stages of the epochs in the context $[n_i-\tau, n_i+\tau]$ of size $2\tau+1$. We use this training set to train the multi-task CNN for joint classification and context prediction.
\[!t\] ![Illustration of the proposed multi-task CNN architecture. The convolution layer of the CNN consists of two filter sets with temporal widths $w=3$ and $w=5$. Each filter set has two individual filters. The colors of the output layer indicate different subtasks jointly modelled by the network.[]{data-label="fig:cnn"}](figure3.eps "fig:"){width="0.85\linewidth"}
\[!t\] {width="0.8\linewidth"}
### Over-Time Convolutional Layer
Each 3-dimensional filter $\mathbf{w} \in \mathbb{R}^{P \times M \times w}$ of the convolutional layer has the temporal size of $w < T$ while the frequency and channel size entirely cover the frequency and channel dimension of a multi-channel time-frequency image input. The filter is convolved with the input image over time with a stride of $1$. ReLU activation [@Nair2010] is then applied to the feature map.
The CNN is designed to have $R$ filter sets with different temporal widths $w$ to capture features at multiple temporal resolutions. Each filter consists of $Q$ different filters of the same temporal width to allow the CNN to learn multiple complementary features. As a result, the total number of filters is $Q \times R$.
### 1-Max Pooling Layer
We employ 1-max pooling function [@Kim2014; @Phan2016] on a feature map produced by convolving a filter over an input image to retain the most prominent feature. Pooling all feature maps of $Q \times R$ filters results in a feature vector of size $Q \times R$.
With the over-time convolution layer coupled with the 1-max pooling layer, the CNN functions as a template learning and matching algorithm. The convolutional filters play the role of time-frequency templates that are tuned for the task at hand. Convolving a filter through time can be interpreted as template matching operation, resulting in a feature map that indicates how well the template is matched to different parts of the input image. In turn, 1-max pooling retains a single maximum value, i.e. the maximum matching score, of the feature map as the final feature.
### Multi-Task Softmax Layer
Opposing to a classification network that typically uses the canonical softmax layer for classification, we propose a multi-task softmax layer to suit joint classification and prediction. The idea is that the network should be penalized for both misclassification and misprediction on a training example. The classification and prediction errors on a training example $i$ is computed as the sum of the cross-entropy errors on the individual subtasks: $$\begin{aligned}
E^{(i)}(\bm{\theta}) = \sum_{n=n_i - \tau}^{n_i +\tau}\mathbf{y}^{(i)}_n\log\left(\hat{\mathbf{y}}^{(i)}_n(\bm{\theta})\right),
\label{eq:multitask_cross_entropy}\end{aligned}$$ where $\bm{\theta}$ and $\hat{\mathbf{y}}$ denote the network parameters and the probability distribution outputted by the CNN, respectively.
The network is trained to minimize the multi-task cross-entropy error over $N$ training samples: $$\begin{aligned}
E(\bm{\theta}) = -\frac{1}{N}\sum_{i=1}^{N}E^{(i)}(\bm{\theta}) + \frac{\lambda}{2}\|\bm{\theta}\|_2^2.
\label{eq:multitask_loss}\end{aligned}$$ Here, $\lambda$ denotes the hyper-parameter that trades off the error terms and the $\ell_2$-norm regularization term. For further regularization, *dropout* [@Srivastava2014] is also employed. The network training is performed using the *Adam* optimizer [@Kingma2015].
Ensemble of Decisions and Aggregation {#ssec:aggregation}
-------------------------------------
As previously mentioned, one major advantage of the proposed framework is the capacity to produce multiple decisions on a certain epoch even with a single model (the multi-task CNN in this case). Practically, the classification and prediction outputs on a certain epoch may be inconsistent as in ensemble-of-models methods [@Hinton2015; @Dietterich2000]; aggregation of these multi-view decisions is necessary to derive a more reliable one. To that end, we study two probabilistic aggregation schemes: additive and multiplicative voting.
Let $P(y_n\,|\,\mathbf{X}_i)$ denote the estimated probability output on the sleep stage $y_n \in \mathcal{L}$ at the epoch index $n$ given the epoch $\mathbf{X}_i$ in the neighborhood $[n-\tau, n+\tau]$, i.e. $n-\tau \le i \le n+\tau$, as illustrated in Figure \[fig:context\_smoothing\]. The likelihood $P(y_n)$ obtained by additive and multiplicative voting is given by $$\begin{aligned}
P(y_n) = \frac{1}{2\tau+1} \sum_{i=n-\tau}^{n+\tau} P(y_n\,|\,\mathbf{X}_i), \label{eq:additive_smoothing} \\
P(y_n) = \frac{1}{2\tau+1} \prod_{i=n-\tau}^{n+\tau} P(y_n\,|\,\mathbf{X}_i), \label{eq:multiplicative_smoothing}\end{aligned}$$ respectively. Eventually, the predicted label $\hat{y}_n$ is determined by likelihood maximization: $$\begin{aligned}
\hat{y}_n = \operatorname*{arg\,max}_{y_n}P(y_n), \text{~for~} y_n \in \mathcal{L}. \label{eq:likelihood_maximization}\end{aligned}$$
Between the two aggregation schemes, the multiplicative one favors likelihoods of categories with consistent decisions and suppresses likelihoods of those categories with diverged decisions stronger than the additive counterpart [@phan2017c].
Experiments {#sec:experiment}
===========
We aim at achieving several goals in the conducted experiments. Firstly, we prove empirically the feasibility of predicting labels of the neighboring epochs in the output context concurrently with classifying the current one. Secondly, we demonstrate the advantages of the joint classification and prediction (i.e. many-to-one) formulation over the commonly adopted many-to-one scheme as well as the standard one-to-one classification scheme. Thirdly, we provide performance comparison with various developed baseline systems as well as other deep-learning approaches recently proposed for sleep staging to illustrate the proposed framework’s efficiency.
Experimental Setup
------------------
For Sleep-EDF, we conducted leave-one-subject-out cross validation. At each iteration, 19 training subjects were further divided into 15 subjects for training and 4 subject for validation. For MASS, we performed 20-fold cross validation on the MASS dataset. At each iteration, 200 subjects were split into training, validation, and test set with 180, 10, and 10 subjects, respectively. The sleep staging performance over 20 folds will be reported for both datasets.
[|>m[1.0in]{}|>m[0.75in]{}|]{} & [**Value**]{}\
\[0ex\] Filter width $w$ & $\{3, 5, 7\}$\
\[0ex\] Number of filters $Q$ & varied\
\[0ex\] Output context size & 3\
\[0ex\] Dropout & $0.2$\
\[0ex\] $\lambda$ for regularization & $10^{-3}$\
\[0ex\]
\[tab:cnn\_param\]
[|>m[0.25in]{}|>m[0.3in]{}|>m[0.25in]{}|>m[0.45in]{}|>m[0.5in]{}|>m[0.01in]{} @m[0pt]{}@]{} Layer & Size & \#Fmap & Activation & Dropout &\
\[0ex\] conv1 & 3 $\times$ 3 & 96 & ReLU & - &\
\[0ex\] pool1 & 2 $\times$ 1 & - & - & 0.2 &\
\[0ex\] conv2 & 3 $\times$ 3 & 96 & ReLU & - &\
\[0ex\] pool2 & 2 $\times$ 2 & - & - & 0.2 &\
\[0ex\] fc1 & 1024 & - & ReLU & 0.2 &\
\[0ex\] fc2 & 1024 & - & ReLU & 0.2 &\
\[0ex\]
\[tab:cnn\_baseline\]
Parameters
----------
The parameters associated with the proposed CNN are given in Table \[tab:cnn\_param\]. We varied the number of convolutional filters $Q$ of the CNN in the set {100, 200, 300, 400, 500, 1000} to investigate its influence. Furthermore, we experimented with the output context size of $3$ (equivalent to $\tau = 1$). Influence of this parameter will be further discussed in Section \[sec:discussion\].
The network implementation was based on *Tensorflow* framework [@Abadi2016]. Graphic card NVIDIA GTX 1080 Ti was used for network training. The network was trained for 200 epochs with a batch size of 200. The learning rate was set to $10^{-4}$ for the *Adam* optimizer. During training, the network that yielded the best overall accuracy on the validation set was retained for evaluation. Furthermore, we always randomly generated a data batch to have an equal number of samples for all sleep stages to mitigate the class imbalance issue commonly seen in sleep data.
Baseline Systems {#ssec:baseline}
----------------
To manifest the advantages offered by the proposed frameworks, we constructed two baseline frameworks for comparison:
- One-to-one: this baseline complies with the standard classification setup, taking a single epoch as input and producing a single decision on its label.
- Many-to-one: this baseline conforms to the commonly adopted scheme with contextual input and outputs a single decision on a target epoch. We fixed the contextual input size to 3, i.e. we augmented a target epoch with two nearest neighbors on its left- and right-hand side.
Both baseline frameworks were designed to maintain common experimental settings as those of the proposed one-to-many framework, i.e. the CNN architecture, the learned filter bank, etc. However, it is necessary to use the canonical softmax layer and the standard cross-entropy loss for their classification-only purpose.
We also developed and repeated the experiments with a typical deep CNN architecture as an alternative to the proposed CNN described in Section \[ssec:cnn\]. This deep CNN baseline consists of 6 layers (2 convolutional layers, 2 subsampling layers, and 2 fully connected layers) with their parameters characterized in Table \[tab:cnn\_baseline\]. For simplicity, we refer to our proposed CNN as 1-max CNN to distinguish from the deep CNN baseline. With these experiments, our goal is to show the generalizability of the proposed framework regardless the network base as well as the efficacy of the 1-max CNN in comparison to a typical deep CNN architecture.
\[!t\] {width="0.9\linewidth"}
\[!t\] ![The overall classification accuracy (a)-(c) and the amount of training time (b)-(d) of the proposed framework in comparison with those of the one-to-one, and many-to-one schemes on the first cross-validation fold. We commonly set $Q=1000$ while $P=2$ for Sleep-EDF and $P=3$ for MASS.[]{data-label="fig:inputcontext_vs_outputcontext"}](figure6.eps "fig:"){width="1\linewidth"}
Experimental Results
--------------------
### Classification vs prediction accuracy
In this experiment, we seek to empirically validate the proposed framework by demonstrating the feasibility of context prediction. Since we employed the output context size of $3$, without confusion, let us refer to the network’s subtasks as *classification*, *left prediction*, and *right prediction*, which correspond to decisions on the input epoch, its left neighbor, and its right neighbor.
We show in Figure \[fig:predictionvsclassification\] the accuracy rates of classification, left prediction, and right prediction subtasks obtained by the 1-max CNN (with varying number of convolutional filters $Q$) and the deep CNN baseline with the different number of input modalities $P$. Unlike the classification subtask, the CNNs do not have access to the signal information of the left and right neighboring epochs. As a result, inference for their labels relies solely on their dependency with the input epoch. It can be expected that the accuracy rates of the left and right prediction subtasks are lower than that of the classification subtask in most of the cases. Nevertheless, overall both CNNs maintain a good accuracy level in prediction relative to the classification accuracy, especially in multimodal cases (e.g. $P=2$ for Sleep-EDF and $P=3$ for MASS). More specifically, averaging over all $Q$ and $P$, the left and right prediction accuracies of the 1-max CNN are only $2.9\%$ and $1.4\%$ lower than the classification accuracy on Sleep-EDF whereas the respective gaps of $2.2\%$ and $1.3\%$ are seen in MASS. Similar patterns can also be seen with the deep CNN baseline with the graceful degradation of $4.0\%$ and $2.5\%$ in Sleep-EDF and $3.3\%$ and $2.2\%$ in MASS correspondingly. These results strengthen the assumption about the dependency between neighboring PSG epochs and consolidate the feasibility of joint classification and prediction modelling.
### Advantages of the joint classification and prediction
Figure \[fig:predictionvsclassification\] also highlights the performance improvements obtained by the joint classification and prediction framework after the aggregation step in comparison to individual subtasks. Averaging over all $P$ and $Q$, the 1-max CNN with additive voting leads to $2.8\%$ and $4.5\%$ absolute accuracy gains over the classification subtask’s accuracy on Sleep-EDF and MASS, respectively. The gains yielded by the multiplicative voting are even better, reaching $3.0\%$ and $4.7\%$, respectively. Accordingly, the deep CNN baseline produces $2.2\%$ and $2.5\%$ absolute gains with additive voting and $2.6\%$ and $2.8\%$ with multiplicative voting on the two datasets. Between two voting schemes, the performance gain of the multiplicative one is slightly better than that of the additive counterpart with a difference around $0.2 - 0.3\%$ on both Sleep-EDF and MASS.
To demonstrate the advantages of the proposed framework over the common classification schemes, we further compare its performance and computational complexity with the one-to-one and many-to-one baseline schemes described in Section \[ssec:baseline\]. For simplicity, we utilized all available modalities (i.e. $P=3$) in this experiment and made use of multiplicative-voting aggregation in the proposed framework. Additionally, we set the number of convolutional filters $Q=1000$ when the 1-max CNN was employed.
Figure \[fig:inputcontext\_vs\_outputcontext\] depicts the overall accuracy obtained by the three frameworks and their computational complexity in terms of the training time. Note that we only included the training time of the first cross-validation fold as a representative here and the training time was expected to scale linearly with the amount of training data. Four important points should be noticed from the figure. Firstly, contextual input does not always help as the many-to-one baseline with the 1-max CNN experiences a performance drop of $0.6\%$ absolute compared to the one-to-one on MASS compared to the one-to-one on MASS although it improves accuracy rates in other cases. Secondly, the proposed one-to-many framework consistently outperforms its counterparts. Adopting the 1-max CNN as the base, our framework outperforms the one-to-one and many-to-one opponents with $2.5\%$ and $0.2\%$ absolute in Sleep-EDF and $1.0\%$ and $1.6\%$ absolute in MASS, respectively. Similar gains of $2.5\%$ and $1.0\%$ in Sleep-EDF; $1.7\%$ and $0.3\%$ in MASS are achieved when the deep CNN baseline is used. Thirdly, between the network bases, the 1-max CNN surpasses the deep CNN baseline with an improvement of $2.6\%$ absolute in Sleep-EDF and $0.9\%$ absolute in MASS although its architecture is much simpler. Fourthly, concerning the computational complexity, three times larger input of the many-to-one baseline roughly triples the training time compared to that of the one-to-one. For instance, 4.0 hours versus 1.36 hours in MASS can be seen with the 1-max CNN. Differently, with the training time of $1.6$ hours. Using the same network, the proposed framework only increases computing time by as small as $0.2$ hours. The training time of the deep CNN baseline also exposes similar patterns.
[|>m[0.1in]{}|>m[1.6in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0.3in]{}|>m[0in]{} @m[0pt]{}@]{} & & & &\
\[0ex\] & Acc. & $\kappa$ & MF1 & Sens. & Spec. & Acc. & $\kappa$ & MF1 & Sens. & Spec. & Acc. & $\kappa$ & MF1 & Sens. & Spec. &\
\[0ex\] & ***One-to-many + 1-max CNN*** & $\bm{81.9}$ & $\bm{0.74}$ & $\bm{73.8}$ & $\bm{73.9}$ & $\bm{95.0}$ & $\bm{82.3}$ & $\bm{0.75}$ & $74.7$ & $74.3$ & $\bm{95.1}$ & &\
\[0ex\] & *One-to-one + 1-max CNN* & $79.8$ & $0.72$ & $72.0$ & $72.4$ & $94.6$ & $79.7$ & $0.72$ & $72.2$ & $72.8$ & $94.6$ & &\
\[0ex\] & *Many-to-one + 1-max CNN* & $80.9$ & $0.73$ & $73.6$ & $74.2$ & $94.9$ & $82.1$ & $0.75$ & $75.4$ & $75.4$ & $95.1$ & &\
\[0ex\] & *One-to-many + deep CNN baseline* & $79.3$ & $0.71$ & $69.7$ & $70.2$ & $94.2$ & $79.7$ & $0.71$ & $71.2$ & $70.9$ & $94.3$ & &\
\[0ex\] & *One-to-one + deep CNN baseline* & $76.7$ & $0.67$ & $67.6$ & $68.6$ & $93.7$ & $77.1$ & $0.68$ & $69.3$ & $69.8$ & $93.8$ & &\
\[0ex\] & *Many-to-one + deep CNN baseline* & $78.3$ & $0.69$ & $70.7$ & $71.1$ & $94.1$ & $78.7$ & $0.70$ & $71.8$ & $72.4$ & $94.2$ & &\
\[0ex\] & ***One-to-many + 1-max CNN*** & $\bm{78.6}$ & $\bm{0.70}$ & $\bm{70.6}$ & $71.2$ & $\bm{94.1}$ & $\bm{82.5}$ & $\bm{0.75}$ & $\bm{76.1}$ & $75.8$ & $\bm{95.0}$ & $\bm{83.6}$ & $\bm{0.77}$ & $77.9$ & $77.4$ & $\bm{95.3}$\
\[0ex\] & *One-to-one + 1-max CNN* & $75.9$ & $0.67$ & $69.6$ & $71.1$ & $93.7$ & $80.7$ & $0.73$ & $74.9$ & $75.5$ & $94.8$ & $82.7$ & $0.75$ & $77.6$ & $77.8$ & $95.1$\
\[0ex\] & *Many-to-one + 1-max CNN* & $76.3$ & $0.67$ & $69.8$ & $71.3$ & $93.8$ & $80.9$ & $0.73$ & $75.1$ & $75.5$ & $94.8$ & $82.1$ & $0.75$ & $76.6$ & $76.9$ & $95.0$\
\[0ex\] & *One-to-many + deep CNN baseline* & $78.0$ & $0.69$ & $69.8$ & $70.1$ & $93.8$ & $81.9$ & $0.74$ & $75.2$ & $74.7$ & $94.8$ & $82.7$ & $0.75$ & $76.9$ & $76.3$ & $95.0$\
\[0ex\] & *One-to-one + deep CNN baseline* & $74.5$ & $0.65$ & $68.4$ & $70.0$ & $93.4$ & $79.2$ & $0.71$ & $73.5$ & $74.3$ & $94.4$ & $81.0$ & $0.73$ & $76.4$ & $77.4$ & $94.9$\
\[0ex\] & *Many-to-one + deep CNN baseline* & $77.4$ & $0.68$ & $71.6$ & $72.8$ & $94.0$ & $81.2$ & $0.73$ & $76.0$ & $76.4$ & $94.8$ & $82.4$ & $0.75$ & $78.2$ & $78.9$ & $95.2$\
\[0ex\]
\[tab:performance\]
[|>m[1.2in]{}|>m[0.9in]{}|>m[0.8in]{}|>m[0.5in]{}|>m[0.6in]{}|>m[0.5in]{}|>m[0.4in]{}|>m[0.4in]{}|>m[0in]{} @m[0pt]{}@]{} & Method & Input channel & Feature type & Subjects & Independent testing & In-bed data only & Overall accuracy &\
\[0ex\] **This work & Multitask 1-max CNN & Fpz-Cz + hor. EOG & learned & 20 SC & yes & yes & $82.3$ &\
\[0ex\] **This work & Multitask 1-max CNN & Fpz-Cz & learned & 20 SC & yes & yes & $81.9$ &\
\[0ex\] Phan *et al.* [@Phan2018c] & 1-max CNN & Fpz-Cz & learned & 20 SC & yes & yes & $79.8$ &\
\[0ex\] Phan *et al.* [@Phan2018d] & Attentional RNN & Fpz-Cz & learned & 20 SC & yes & yes & $79.1$ &\
\[0ex\] Andreotti *et al.* [@Andreotti2018] & ResNet & Fpz-Cz + hor. EOG & learned & 20 SC & yes & yes & $76.8$ &\
\[0ex\] Tsinalis *et al.* [@Tsinalis2016b] & Deep auto-encoder & Fpz-Cz & hand-crafted & 20 SC & yes & yes & $78.9$ &\
\[0ex\] Tsinalis *et al.* [@Tsinalis2016] & Deep CNN & Fpz-Cz & learned & 20 SC & yes & yes & $74.8$ &\
\[0ex\] Supratak *et al.* [@Supratak2017] & Deep CNN + RNN & Fpz-Cz & learned & 20 SC & yes & no & $82.0$ &\
\[0ex\] Alickovic & Subasi [@Alickovic2018] & Ensemble SVM & Pz-Oz & hand-crafted & 10 SC + 10 ST & yes & no & $91.1$ &\
\[0ex\] Sanders *et al.* [@Sanders2014] & Decision trees & Fpz-Cz & hand-crafted & 10 ST & yes & no & $75.0$ & &\
\[0ex\] Dimitriadis *et al.* [@Dimitriadis2018] & $k$-NN & Fpz-Cz & hand-crafted & 20 SC & yes & no & $94.4$ &\
\[0ex\] Mikkelsen & De Vos [@Mikkelsen2018] & Deep CNN & Fpz-Cz + hor. EOG & learned & 20 SC & no & yes & $84.0$ &\
\[0ex\] Imtiaz *et al.* [@Imtiaz2015b] & Ensemble SVM & Fpz-Cz + Pz-Oz & hand-crafted & 20 SC + 22 ST & no & yes & $78.9$ &\
\[0ex\] Munk *et al.* [@Munk2018] & GMM & Pz-Oz & hand-crafted & 19 SC & no & no & $73.2$ &\
\[0ex\] Rodríguez-Sotelo *et al.* [@Rodriguez-Sotelo2014] & $k$-NN & Fpz-Cz + Pz-Oz & hand-crafted & 20 SC & no & no & $80.0$ &\
\[0ex\] Aboalayon *et al.* [@Aboalayon2016] & Decision trees & Fpz-Cz + Pz-Oz & hand-crafted & 20 SC & no & no & $93.1$ &\
\[0ex\]****
\[tab:performance\_comparison\_EDF\]
[|>m[0.9in]{}|>m[0.9in]{}|>m[1.5in]{}|>m[0.5in]{}|>m[0.6in]{}|>m[0.5in]{}|>m[0.4in]{}|>m[0in]{} @m[0pt]{}@]{} & Method & Input channel & Feature type & Subjects & Independent testing & Overall accuracy &\
\[0ex\]
**This work & Multitask 1-max CNN & C4-A1 + ROC-LOC + CHIN1-CHIN2 & learned & 200 & yes & $83.6$ &\
\[0ex\] **This work & Multitask 1-max CNN & C4-A1 + ROC-LOC & learned & 200 & yes & $82.5$ &\
\[0ex\] **This work & Multitask 1-max CNN & C4-A1 & learned & 200 & yes & $78.6$ &\
\[0ex\] *Chambon et al.*^2^ [@Chambon2018] & Deep CNN & C4-A1 + ROC-LOC + CHIN1-CHIN2 & learned & 200 & yes & $79.9$ &\
\[0ex\] *DeepSleepNet1*^2^ [@Supratak2017] & Deep CNN & C4-A1 + ROC-LOC + CHIN1-CHIN2 & learned & 200 & yes & $80.7$ &\
\[0ex\] *Tsinalis et al.*^2^ [@Tsinalis2016] & Deep CNN & C4-A1 + ROC-LOC + CHIN1-CHIN2 & learned & 200 & yes & $77.9$ &\
\[0ex\] Andreotti *et al.* [@Andreotti2018] & ResNet & C4-A1 + ROC-LOC + CHIN1-CHIN2 & learned & 200 & yes & $79.4$ &\
\[0ex\] Chambon *et al.* [@Chambon2018] & Deep CNN & 6 EEG + 2 EOG + 3 EMG & learned & 61 (SS3 only) & yes & $83.0$ &\
\[0ex\] Supratak *et al.* [@Supratak2017] & Deep CNN & F4-EOG (left) & learned & 62 (SS3 only) & yes & $81.5$ &\
\[0ex\] Dong *et al.* [@Dong2017] & DNN & F4-EOG (left) & learned & 62 (SS3 only) & yes & $81.4$ &\
\[0ex\] Dong *et al.* [@Dong2017] & Random Forests & F4-EOG (left) & hand-crafted & 62 (SS3 only) & yes & $81.7$ &\
\[0ex\] Dong *et al.* [@Dong2017] & SVM & F4-EOG (left) & hand-crafted & 62 (SS3 only) & yes & $79.7$ &\
\[0ex\]******
\[tab:performance\_comparison\_MASS\]
### Performance comparison {#sssec:performance_comparison}
Table \[tab:performance\] provides a comprehensive performance comparison on the experimental dataset using different metrics, including overall accuracy, kappa index $\kappa$, average specificity, average sensitivity, and average macro F1-score (MF1). The comparison covers all combinations of different frameworks (i.e. the proposed and the baselines) and network bases (i.e. the proposed 1-max CNN and the deep CNN baseline). As can be seen, the proposed one-to-many framework powered by the 1-max CNN ([**one-to-many + 1-max CNN**]{}) outperforms other combinatorial systems presented in this work on both datasets and over different combinations of modalities. There are occasional exceptions where using the 1-max CNN in the baseline frameworks yields marginally better average MF1 and Sensitivity than [**one-to-many + 1-max CNN**]{}, such as on MASS with $P=3$; however, [**one-to-many + 1-max CNN**]{} remains optimal on other metrics.
To see an overall picture, in Tables \[tab:performance\_comparison\_EDF\] and \[tab:performance\_comparison\_MASS\] we relate the proposed method’s accuracy to those reported by previous works on the two datasets. With this comprehensive comparison, we also aim at providing a benchmark for future work.
As can be seen from Table \[tab:performance\_comparison\_EDF\], the results on Sleep-EDF vary noticeably due to the lack of standardization in experimental setup. We observe two factors that greatly affects performance on this dataset: (1) independent/dependent testing and (2) whether or not using only *in-bed* parts of the recordings as recommended in [@Imtiaz2015; @Imtiaz2015b; @Tsinalis2016; @Tsinalis2016b]. Dependent testing happens when data of a test subject is also involved in training, such as in Aboalayon *et al.* [@Aboalayon2016], and biases the evaluation results. In addition to in-bed parts (i.e. from *lights off* time to *lights on* time [@Imtiaz2015]), many previous studies also included other parts, such as in Supratak *et al.* [@Supratak2017], or even entire recordings, such as in Dimitriadis *et al.* [@Dimitriadis2018] and in Alickovic & Subasi [@Alickovic2018], into their experiments. These add-on data, which are mainly *Wake* epochs, often boost the performance as *Wake*, in general, is easier to be recognized than other sleep stages. Therefore, the performance comparison is improper unless two methods use a similar experimental setup. With respect to this, the proposed method outperforms other competitors that commonly used independent testing and in-bed data only. It should be noted that these results do not cover a large body of studies on the early version of Sleep-EDF dataset [@Kemp2000; @Goldberger2000] which consists of only 8 PSG recordings.
A few recent attempts has evaluated automatic sleep staging on a subset [@Supratak2017; @Dong2017; @Chambon2018] rather than the entire 200 subjects of the MASS dataset. The discrepancy in data makes a direct comparison between their results and ours inappropriate. To avoid possible mismatch in experimental setup, we re-implemented DeepSleepNet [@Supratak2017] and the deep CNN architecture proposed by Chambon *et al.* [@Chambon2018], both of which recently reported the state-of-the-art results on the MASS subset SS3, for a compatible comparison. Note that we experimented with DeepSleepNet1 (CNN) in [@Supratak2017] here, and will leave DeepSleepNet2 (CNN combined with RNN for long-term context modelling) for future work. In addition, we also implemented the deep CNN proposed by Tsinalis *et al.* [@Tsinalis2016] which demonstrated good performance on Sleep-EDF. While our developed baselines (cf. Table \[tab:performance\]) are more efficient than these networks under the common experimental setup used in this work, the improvements by the proposed multitask 1-max CNN are most prominent, as can be seen from Table \[tab:performance\_comparison\_MASS\]. More specifically, compared to the best opponent, DeepSleepNet [@Supratak2017], a margin of $2.9\%$ on overall accuracy is obtained when all three adopted channels ($P=3$) were used.
For completeness, we show in Table \[tab:confusion\_matrix\] the confusion matrices and class-wise performance in terms of sensitivity and selectivity [@Imtiaz2014] obtained by the proposed one-to-many framework with the 1-max CNN base. Particularly, one may notice modest performance on N1 stage, which has been proven challenging to be correctly recognized [@Phan2018c; @Phan2018d; @Supratak2017; @Tsinalis2016] due to its similarities with other stages and its infrequency. Possibilities for improvement would be to over-sample the under-present class during training and to explore weighting schemes for a network’s loss [@phan2018e; @Koch2018b] so that the network is penalized stronger if making errors on this infrequent class than other ones. We further provide alignment of ground-truth and system-output hypnograms for one subject of the MASS dataset in Figure \[fig:hypnogram\_mass\].
[|>m[0.25in]{}|>m[0.25in]{}|>m[0.3in]{}|>m[0.3in]{}|>m[0.3in]{}|>m[0.3in]{}|>m[0.3in]{}|>m[0.25in]{}|>m[0.25in]{}|>m[0in]{} @m[0pt]{}@]{} & & &\
\[0ex\] & W & N1 & N2 & N3 & REM & & &\
\[0ex\] & W & 3403 & 322 & 230 & 32 & 522 & 75.5 & 79.3 &\
\[0ex\] & N1 & 441 & 880 & 725 & 9 & 707 & 31.9 & 55.7 &\
\[0ex\] & N2& 230 & 263 & 15263 & 795 & 1026 & 86.8 & 88.1 &\
\[0ex\] & N3 & 65 & 0 & 658 & 4850 & 18 & 86.7& 85.3 &\
\[0ex\] & REM & 154 & 114 & 457 & 3 & 6983 & 90.6 & 75.4 &\
\[0ex\] & W & 26261 & 2148 & 1450 & 72 & 1112 & 84.6 & 86.3 &\
\[0ex\] & N1 & 2924 & 7948 & 5498 & 22 & 2965 & 41.1 & 55.2 &\
\[0ex\] & N2& 759 & 3429 & 95486 & 4849 & 3395 & 88.5 & 86.9 &\
\[0ex\] & N3 & 30 & 13 & 6098 & 24223 & 18 & 79.7 & 83.0 &\
\[0ex\] & REM & 466 & 872 & 1353 & 6 & 37473 & 93.3 & 83.3 &\
\[0ex\]
\[tab:confusion\_matrix\]
\[!t\] {width="0.85\linewidth"}
Discussion {#sec:discussion}
==========
In this section, we investigate the causes of the proposed framework’s performance improvement over the baseline ones. Furthermore, the proposed framework encompasses several influential factors, such as the number of convolutional filters $Q$ of the 1-max CNN, the number of input modalities $P$, and the output context size. We will discuss and elucidate their effects on the framework’s performance. The multitask framework will also be contrasted against an equivalent ensemble method to shed light on their similar behaviour.
Investigating the Causes of Improvement
---------------------------------------
To accomplish this goal, we divided the dataset into a *non-transition* and *transition* set and explored how different frameworks perform on them. Considering MASS for this investigation, the former set is the major one ($83.4\%$ epochs in total) consisting of epochs with the same label as their left and right neighbors. The latter, which is the minor set ($16.6\%$ epochs in total), comprises those epochs at stage transitions, i.e. their labels differ from those of their left/right neighbors or both.
The overall accuracy on these sets are shown in Table \[tab:performance\_transition\_nontransition\]. On one hand, the downgrading accuracy on the transition set reflects the fact that manual labelling of sleep stages if of low accuracy near stage transitions [@Rosenberg2014]. Since a 30-second epoch likely contains the signal information of two transitioning stages while only one label is assigned to such an epoch, up to half of the epoch may not match the assigned label. More often than not, the labels assigned to these epochs are subjective to the scorer. The accuracy of the one-to-one baseline framework on this small subset, which is above the chance level, is likely due to the bias towards the scorer’s subjectivity. The chance-level accuracy of the many-to-one and one-to-many frameworks, on the other hand, can be explained by the fact that taking into account the left and right neighboring epochs has balanced the contribution of the two transitioning stages.
Disregarding the ambiguous transition set, the cause of performance improvement turns out to be depending upon the accuracy on the major non-transition set. As can be seen, the proposed framework outperforms the other two with a gap of $2.7\%$ and $1.3\%$ on this set, respectively. Further investigation on this set reveals a substantial level of label agreement between the proposed framework and the one-to-one baseline, up to $91.0\%$. However, for the remaining $9.0\%$ epochs on which their labels disagree, the proposed framework yields an accuracy of $60.4\%$, roughly doubling that obtained by the baseline ($30.5\%$). Analogously, in comparison with the many-to-one baseline, the label agreement is as high as $92.0\%$ whereas an accuracy gap of $15.2\%$ is seen on the dissenting subset with $52.4\%$ of the proposed framework compared to $37.2\%$ of the baseline.
[|>m[0.7in]{}|>m[0.65in]{}|>m[0.65in]{}|>m[0in]{} @m[0pt]{}@]{} & Non-transition (Size $83.4\%$) & Transition (Size $16.6\%$) &\
\[0ex\] One-to-many & $89.5$ & $53.3$ &\
\[0ex\] One-to-one & $86.8$ & $62.0$ &\
\[0ex\] Many-to-one &$88.2$ & $51.1$ &\
\[0ex\]
\[tab:performance\_transition\_nontransition\]
Influence of the Number of Convolutional Filters
------------------------------------------------
In general, more features can be learned by the proposed 1-max CNN with the increasing number of convolutional filters $Q$ and one can expect improvement on the performance. However, influence of $Q$ on the framework’s performance is very modest as can be seen from Figure \[fig:context\_smoothing\]. For instance, on Sleep-EDF, fixing $P=2$ and multiplicative voting, using $Q=1000$ only brings up $0.5\%$ absolute accuracy gain over the case of $Q=100$ even though the number of filters is ten times larger. A similar finding can also be drawn for MASS ($P=3$) with a modest improvement of $0.4\%$. The slight influence of the number of filters $Q$ suggests that we can maintain a very good performance even with a much smaller network.
Benefits of Multimodal Input
----------------------------
Single-channel EEG has been found prevalent in literature [@Koley2012; @Kuo2011; @Supratak2017; @Tsinalis2016; @Phan2018c; @Phan2018d] mainly due to its simplicity. However, apart from brain activities, sleep also involves eye movements and muscular activities at different levels. For instance Rapid Eye Movement (REM) stage usually associates with rapid eye movements and high muscular activities are usually seen during the Awake stage. As a result, EOG and EMG are valuable additional sources, complementing EEG in multimodal automatic sleep staging systems [@Chambon2018; @Lajnef2015; @Huang2014; @Mikkelsen2018; @Andreotti2018; @Stephansen2017], not to mention their importance in manual scoring rules [@Hobson1969; @Iber2007].
Figure \[fig:context\_smoothing\] reveals and demonstrates the benefit of using EOG and EMG to complement EEG in the proposed framework. Consistent improvements on overall accuracy can be seen on both Sleep-EDF and MASS. Taking MASS for example, averaging over spectrum of $Q$, as compared to the single-channel EEG, coupling EEG and EOG leads to an absolute gain of $4.1\%$ and is further boosted by another $1.1\%$ with the compound of EEG, EOG, and EMG.
The Trade-off Problem with the Output Context Size
--------------------------------------------------
It is straightforward to extend the output context in the proposed framework. Doing so, we are able to increase the number of decisions in an ensemble, which is expected to enhance the classification performance [@Hinton2015]. However, extending the output context confronts us with a trade-off problem. A large context weakens the link between the input epoch and the far-away neighbors in the output context. Oftentimes, this deteriorates the prediction decisions on these epochs and, as a consequence, reduces the quality of individual decisions in the ensemble. The low quality of these prediction decisions may outweigh the benefits of the increased cardinality, worsening the performance instead collectively.
To support our argument, we increased the output context size to 5 (i.e. $\tau=2$) and repeated the experiment in which we set $Q=1000$ for the 1-max CNN and used $P=2$ for Sleep-EDF and $P=3$ for MASS. Figure \[fig:influence\_contextsize\] shows the obtained performance alongside those obtained with the output context size of {1, 3} (i.e. $\tau = \{0,1\}$). Note that, with the context size of 1, the framework is reduced to the one-to-one baseline framework described in Section \[ssec:baseline\]. With the context size of $5$ the proposed framework still maintains its superiority over the standard classification setup, however, a graceful degradation compared to the context size of $3$ can be observed. Specifically, the accuracy rates obtained by both additive and multiplicative voting schemes slightly decline by $0.1\%$ on Sleep-EDF while the respective accuracy losses of $0.3\%$ and $0.2\%$ can be seen on MASS.
To remedy the weak links between the input epoch and far-away epochs, one possibility is to combine multiple epochs into the input to form the contextual input. In addition, it would be worth exploring incorporation of long-term context (i.e. in order of dozens of epochs), for example using RNNs as in [@Supratak2017; @Stephansen2017]. However, a detailed study of the proposed frame work in these many-to-many settings is out of the scope of this article and is left for future work.
Multitask vs Ensemble
---------------------
To examine the comparability between the proposed multitask framework with its ensemble equivalence, we repeated the experiments with the ensemble consisting of three separate CNNs for individual subtasks: left prediction, classification, and right prediction. We studied both the 1-max CNN and the deep CNN baseline here. Again, we set $Q=1000$, and $P=2$ for Sleep-EDF and $P=3$ for MASS when 1-max CNN was used. The results obtained with the ensemble models and the proposed multitask models are contrasted *vis-à-vis* in Figure \[fig:multitask\_vs\_ensemble\].
Our analyses show that the separate CNNs of an ensemble model perform better than its corresponding multitask model on the individual subtasks. This is due to the fact that the multitask model needs to deal with a harder modelling task which combines all the subtasks as a whole. However, after aggregation, their differences become negligible as can be seen over all CNN architectures and datasets. More importantly, on both datasets, the proposed multitask 1-max CNN outperforms the deep CNN baseline in its both forms, namely multitask and ensemble.
\[!t\] ![Influence of the output context size to the overall accuracy of the proposed framework. The results obtained with a common $Q=1000$, in addition, $P=2$ (Sleep-EDF) and $P=3$ (MASS).[]{data-label="fig:influence_contextsize"}](figure8.eps "fig:"){width="1\linewidth"}
\[!t\] ![Performance comparison of the proposed multitask 1-max CNN with its equivalent ensemble model. The results are obtained with $Q=1000$, $P=2$ with Sleep-EDF and $P=3$ for MASS.[]{data-label="fig:multitask_vs_ensemble"}](figure9.eps "fig:"){width="1\linewidth"}
Conclusions {#sec:conclusion}
===========
This work introduced a joint classification and prediction formulation wherein a multi-task CNN framework is proposed for automatic sleep staging. Motivated by the dependency nature of sleep epochs, the framework’s purpose is to jointly perform classification of an input epoch and prediction of the labels of its neighbors in the context output. While being orthogonal to the widely adopted many-to-one classification scheme relying on contextual input, we argued that the proposed framework avoids the shortcomings experienced by the many-to-one approach, such as the inherent modelling ambiguity and the induced computational overhead due to large contextual input. More importantly, due to multitasking, the framework is able to conveniently produce multiple decisions on a certain epoch thereby forming the reliable final decision via aggregation. We demonstrated the generalizability of the framework on two public datasets, Sleep-EDF and MASS.
Acknowledgment {#acknowledgment .unnumbered}
==============
The research was supported by the NIHR Oxford Biomedical Research Centre and Wellcome Trust under Grant 098461/Z/12/Z.
[^1]: The source code and the relevant experimental setup are available at\
<http://github.com/pquochuy/MultitaskSleepNet> for reproducibility.
[^2]: H. Phan, F. Andreotti, N. Cooray, O. Y. Chén, and M. De Vos are with the Institute of Biomedical Engineering, University of Oxford, Oxford OX3 7DQ, United Kingdom.
[^3]: $^*$Corresponding author: [huy.phan@eng.ox.ac.uk]{}
|
---
abstract: 'Using ideas of Ramakrishnan, we consider the icosahedral analogue of the theorems of Sarnak and Brumley on Hecke–Maass newforms with Fourier coefficients in a quadratic order. Although we are unable to conclude the existence of an associated Galois representation in this case, we show that one can deduce some implications of such an association, including weak automorphy of all symmetric powers and the value distribution of Fourier coefficients predicted by the Chebotarev density theorem.'
address: 'School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom'
author:
- 'Andrew R. Booker'
bibliography:
- 'icos.bib'
title: A note on Maass forms of icosahedral type
---
[^1]
Introduction
============
In [@sarnak], Sarnak showed that a Hecke–Maass newform with integral Fourier coefficients must be associated to a dihedral or tetrahedral Artin representation. Brumley [@brumley] later generalized this to Galois-conjugate pairs of forms with coefficients in the ring of integers of ${\mathbb{Q}}(\sqrt{d})$ for a fundamental discriminant $d\ne5$, which are associated to dihedral, tetrahedral or octahedral representations. In this note we consider the remaining case of nondihedral forms with coefficients in ${\mathbb{Z}}\bigl[\frac{1+\sqrt5}2\bigr]$, which are predicted to correspond to icosahedral representations.
The results of Sarnak and Brumley depend crucially on the existence and cuspidality criteria of the symmetric cube and symmetric fourth power lifts from $\operatorname{GL}(2)$, as established by Kim and Shahidi [@KS1; @kim; @KS2]. For the icosahedral case, in order to conclude the existence of an associated Artin representation we would need to know the expected cuspidality criterion for the symmetric sixth power lift (which is not yet known to be automorphic). Appealing to ideas and results of Ramakrishnan [@R1; @R2; @R3], we show that one can nevertheless derive some of the consequences entailed by the existence of an associated icosahedral representation, including weak automorphy of all symmetric powers and the value distribution of Fourier coefficients predicted by the Chebotarev density theorem. Our precise result is as follows.
\[thm:main\] Let ${\mathbb{A}}$ be the adèle ring of ${\mathbb{Q}}$, and put $A=\{0,\pm1,\pm2,\pm\varphi,\pm\varphi^\tau\}$, where $\varphi=\frac{1+\sqrt5}2$ and $\tau$ denotes the nontrivial automorphism of ${\mathbb{Q}}(\varphi)$. Let $\pi=\bigotimes\pi_v$ and $\pi'=\bigotimes\pi_v'$ be nonisomorphic unitary cuspidal automorphic representations of $\operatorname{GL}_2({\mathbb{A}})$ with normalized Hecke eigenvalues $\lambda_\pi(n)$ and $\lambda_{\pi'}(n)$, respectively. Assume that $\pi$ and $\pi'$ are not of dihedral Galois type, and suppose that $\lambda_\pi(n)$ and $\lambda_{\pi'}(n)$ are elements of ${\mathbb{Z}}[\varphi]$ satisfying $\lambda_{\pi'}(n)=\lambda_\pi(n)^\tau$ for every $n$. Then:
1. $\pi$ corresponds to a Maass form of weight $0$ and trivial nebentypus character.
2. For any place $v$, $\pi_v$ is tempered if and only if $\pi_v'$ is tempered.
3. If $S$ denotes the set of primes $p$ at which $\pi_p$ is not tempered, then
1. $\#\{p\in S:p\le X\}\ll X^{1-\delta}$ for some $\delta>0$;
2. $\pi_v\cong\pi_v'$ for all $v\in S\cup\{\infty\}$;
3. $\lambda_\pi(p)\in A$ for every prime $p\notin S$;
4. for each $k\ge0$, there is a unique isobaric automorphic representation $\Pi_k=\bigotimes\Pi_{k,v}$ of $\operatorname{GL}_{k+1}({\mathbb{A}})$ satisfying $\Pi_{k,p}\cong\operatorname{sym}^k\pi_p$ for all primes $p\notin S$ at which $\pi_p$ is unramified;
5. if $S$ is finite or $\operatorname{sym}^5\pi$ is automorphic then $S=\emptyset$ and $\pi_\infty$ is of Galois type (so that $\pi$ corresponds to a Maass form of Laplace eigenvalue $\frac14$).
4. For each $\alpha\in A$, the set of primes $p$ such that $\lambda_\pi(p)=\alpha$ has a natural density, depending only on the norm $\alpha\alpha^\tau$, as follows:
$\alpha\alpha^\tau$ $0$ $1$ $4$ $-1$
--------------------- ----------- ----------- --------------- --------------
$\mathrm{density}$ $\frac14$ $\frac16$ $\frac1{120}$ $\frac1{10}$
Preliminaries {#sec:prelim}
=============
Before embarking on the proof of Theorem \[thm:main\], we first recall some facts about the analytic properties of the standard and Rankin–Selberg $L$-functions associated to isobaric automorphic representations. We refer to [@R1 §1] for essential background and terminology.
Let $\pi=\sigma_1\boxplus\cdots\boxplus\sigma_n=\bigotimes\pi_v$ be an isobaric automorphic representation of $\operatorname{GL}_d({\mathbb{A}})$ for some $d\ge1$, and assume that the cuspidal summands $\sigma_i$ have finite-order central characters. Then for any finite set of places $S\supseteq\{\infty\}$, the partial $L$-function $L^S(s,\pi)=\prod_{v\notin S}L(s,\pi_v)$ converges absolutely for $\Re(s)>1$ and continues to an entire function, apart from a possible pole at $s=1$ of order equal to the number of occurrences of the trivial character among the $\sigma_i$. Furthermore, $L^S(s,\pi)$ has no zeros in the region $\{s\in{\mathbb{C}}:\Re(s)\ge1\}$.
Given two such isobaric representations, $\pi_1$ and $\pi_2$, we can form the irreducible admissible representation $\pi_1\boxtimes\pi_2=\bigotimes(\pi_{1,v}\boxtimes\pi_{2,v})$, where for each place $v$, $\pi_{1,v}\boxtimes\pi_{2,v}$ is the functorial tensor product defined by the local Langlands correspondence. Then for any set $S$ as above, $L^S(s,\pi_1\boxtimes\pi_2)
=\prod_{v\notin S}L(s,\pi_{1,v}\boxtimes\pi_{2,v})$ agrees with the partial Rankin–Selberg $L$-function $L^S(s,\pi_1\times\pi_2)$, which again converges absolutely for $\Re(s)>1$, continues to an entire function apart from a possible pole at $s=1$, and does not vanish in $\{s\in{\mathbb{C}}:\Re(s)\ge1\}$. The order of the pole is characterized by the facts that (i) it is bilinear with respect to isobaric sum, and (ii) if $\pi_1$ and $\pi_2$ are cuspidal then $L^S(s,\pi_1\boxtimes\pi_2)$ has a simple pole if $\pi_1\cong\pi_2^\vee$ and no pole otherwise.
Given an irreducible admissible representation $\Pi$ of $\operatorname{GL}_d({\mathbb{A}})$, let $\operatorname{cond}(\Pi)$ denote its conductor, and let $\{c_n(\Pi)\}_{n=1}^\infty$ be the unique sequence of complex numbers satisfying $$-\frac{L'}{L}(s,\Pi)=\sum_{n=1}^\infty\frac{\Lambda(n)c_n(\Pi)}{n^s}
\quad\text{and}\quad c_n(\Pi)=0\text{ whenever }\Lambda(n)=0.$$ Then $c_n(\Pi)$ is multiplicative in $\Pi$, in the sense that if $\pi_1$ and $\pi_2$ are isobaric representations as above then $c_n(\pi_1\boxtimes\pi_2)=c_n(\pi_1)c_n(\pi_2)$ for all $n$ coprime to $\gcd(\operatorname{cond}(\pi_1),\operatorname{cond}(\pi_2))$.
Similarly, for any isobaric representation $\pi$ of $\operatorname{GL}_d({\mathbb{A}})$ and any $k\ge0$, we can form the irreducible admissible representation $\operatorname{sym}^k\pi=\bigotimes\operatorname{sym}^k\pi_v$. If $d=2$ and $\pi$ has trivial central character then for all $n$ coprime to $\operatorname{cond}(\pi)$ we have $$c_n(\operatorname{sym}^k\pi)=P_k(c_n(\pi))$$ for certain polynomials $P_k\in{\mathbb{Z}}[x]$; in particular, $$P_0=1,\;
P_1=x,\;
P_2=x^2-1,\;
P_3=x^3-2x,\;
P_4=x^4-3x^2+1,\;
P_5=x^5-4x^3+3x.$$
Finally, we recall some standard tools from analytic number theory.
\[lem:positive\] Let $\{c_n\}_{n=1}^\infty$ be a sequence of nonnegative real numbers satisfying $c_n\ll n^\sigma$ for some $\sigma\ge0$, and put $$D(s)=\sum_{n=1}^\infty\frac{c_n}{n^s}
\quad\text{for }\Re(s)>\sigma+1.$$ Suppose that $(s-1)D(s)$ has analytic continuation to an open set containing $\{s\in{\mathbb{C}}:\Re(s)\ge1\}$, and set $r=\operatorname{Res}_{s=1}D(s)$. Then
1. $\sum_{n\le X}c_n=rX+o(X)
\quad\text{as }X\to\infty.$
2. If $r=0$ then there exists $\delta>0$ such that $\sum_{n=1}^\infty c_n/n^{1-\delta}<\infty.$
These are the Wiener–Ikehara theorem [@tenenbaum Ch. II.7, Thm. 11] and Landau’s theorem [@tenenbaum Ch. II.1, Cor. 6.1], respectively.
Proof of Theorem \[thm:main\] {#sec:proof}
=============================
We are now ready for the proof. Our argument is sequential, but to aid the reader we have separated it into six main steps, as follows.
Initial observations
--------------------
Let $\pi$ and $\pi'$ be as in the statement of the theorem. If $\pi_p$ and $\pi_p'$ are both tempered for some prime $p$, then $\max\{|\lambda_\pi(p)|,|\lambda_\pi(p)^\tau|\}\le2$, which holds if and only if $\lambda_\pi(p)\in A$. Thus, conclusion (2) of the theorem implies conclusion (3c). Moreover, the equality $\lambda_{\pi'}(n)=\lambda_\pi(n)^\tau$ for $n\in\{p,p^2\}$ implies that the $L$-factors $L(s,\pi_p)$ and $L(s,\pi_p')$ have the same degree, and thus $\pi_p$ is ramified if and only if $\pi_p'$ is ramified. We set $N=\gcd(\operatorname{cond}(\pi),\operatorname{cond}(\pi'))$.
Since $\lambda_\pi(n)\in{\mathbb{R}}$ for every $n$, $\pi$ must be self dual. Suppose that $\pi$ is the automorphic induction of a Hecke character of infinite order. Then the value distribution of $\lambda_\pi(p)$ for primes $p$ is the sum of a point mass of weight $\frac12$ at $0$ and a continuous distribution. In particular, the set $\{p:\lambda_\pi(p)\in A\}$ has density $\frac12$. On the other hand, by [@KS2 Theorem 4.1], the set of $p$ at which $\pi_p'$ is tempered has lower Dirichlet density at least $34/35$, and as observed above, $\lambda_\pi(p)\in A$ for any such $p$. This is a contradiction, so $\pi$ cannot be induced from a Hecke character of infinite order. (See [@sarnak] for an alternative proof in the Maass form case, based on transcendental number theory.) By hypothesis, $\pi$ is also not of dihedral Galois type, and it follows that $\pi$ has trivial central character.
Suppose that $\pi_\infty$ is a discrete series representation of weight $k\ge2$. Since $\pi$ has trivial central character, $k$ must be even. Then $\pi$ corresponds to a holomorphic newform with Fourier coefficients $\lambda_\pi(n)n^{(k-1)/2}$, which must lie in a fixed number field. Considering primes $n=p$, since $\lambda_\pi(p)\in{\mathbb{Z}}[\varphi]$ that is only possible if $\lambda_\pi(p)=0$ for all but finitely many $p$, contradicting the fact that $L(s,\pi\boxtimes\pi)$ has a pole at $s=1$. Thus, $\pi_\infty$ must be a principal or complementary series representation of weight $0$, which establishes (1).
Next suppose that $\pi$ is of tetrahedral or octahedral type. Then $\operatorname{Ad}(\pi)\cong\operatorname{sym}^2\pi$ corresponds to an irreducible $3$-dimensional Artin representation with Frobenius traces $\lambda_\pi(p)^2-1$ for all primes $p\nmid N$, and image isomorphic to $A_4$ or $S_4$, respectively. In the tetrahedral case, from the character table of $A_4$ we see that $\lambda_\pi(p)^2-1\in\{3,-1,0\}$, so that $\lambda_\pi(p)\in{\mathbb{Z}}$; by strong multiplicity one, that contradicts the hypothesis that $\pi\not\cong\pi'$. In the octahedral case, from the character table of $S_4$ and the Chebotarev density theorem, $\lambda_\pi(p)^2-1=1$ for a positive proportion of primes $p$; that contradicts the hypothesis that $\lambda_\pi(p)\in{\mathbb{Z}}[\varphi]$.
In summary, we have shown that $\pi$ corresponds to a Maass form of weight $0$ and trivial character, is not in the image of automorphic induction and is not of solvable polyhedral type. By symmetry these conclusions apply to $\pi'$ as well. Moreover, by Atkin–Lehner theory, if there is a prime $p\mid N$ with $p^2\nmid N$ then $\lambda_\pi(p)^2=\lambda_{\pi'}(p)^2=1/p$. That contradicts the hypothesis that $\lambda_\pi(n)$ and $\lambda_{\pi'}(n)$ are algebraic integers, so for every $p\mid N$ we must have $p^2\mid N$ and $\lambda_\pi(p)=\lambda_{\pi'}(p)=0$.
Equivalence of $\operatorname{sym}^3\pi$ and $\operatorname{sym}^3\pi'$ {#sec:equiv}
-----------------------------------------------------------------------
By the seminal works of Gelbart–Jacquet [@GJ], Ramakrishnan [@R0], Kim–Shahidi [@KS1] and Kim [@kim], we know that the representations $\operatorname{sym}^k\pi$ and $\operatorname{sym}^k\pi'$ for $k\le4$, $\pi\boxtimes\pi'$, $\pi\boxtimes\operatorname{sym}^2\pi'$ and $\pi'\boxtimes\operatorname{sym}^2\pi$ are all automorphic. Moreover, since $\pi$ and $\pi'$ are not in the image of automorphic induction and are not of solvable polyhedral type, $\operatorname{sym}^k\pi$ and $\operatorname{sym}^k\pi'$ are cuspidal for $k\le4$.
For brevity of notation, we set $$a_n=c_n(\pi)\quad\text{and}\quad b_n=a_n^\tau=c_n(\pi').$$ Note that $a_p=\lambda_\pi(p)$ for all primes $p$, and $a_n=0$ whenever $(n,N)>1$. For $f\in{\mathbb{R}}[x,y]$, let $$D_f(s)=\sum_{n=1}^\infty
\frac{\Lambda(n)f(a_n,b_n)}{n^s}
=\sum_{(n,N)=1}\frac{\Lambda(n)f(a_n,b_n)}{n^s}
+f(0,0)\sum_{p\mid N}\frac{\log{p}}{p^s-1}.$$ For any $f$ such that $(s-1)D_f(s)$ has an analytic continuation to an open set containing $\{s\in{\mathbb{C}}:\Re(s)\ge1\}$, we define $$r(f)=\operatorname{Res}_{s=1}D_f(s).$$ In particular, by the properties of Rankin–Selberg $L$-functions described in §\[sec:prelim\], $r(P_i(x)P_j(y))$ is defined for $i,j\le 4$. Note also that $r(f)$ is linear in $f$.
Consider $$\label{eq:F}
\begin{aligned}
F&=(x-y)^2((x-y)^2-5)\\
&=P_4(x)-4P_3(x)y+6P_2(x)P_2(y)-4xP_3(y)+P_4(y)+4P_2(x)-6xy+4P_2(y).
\end{aligned}$$ Note that for $u,v\in{\mathbb{Z}}$ with $u\equiv v\pmod2$, we have $$F\biggl(\frac{u+v\sqrt5}2,\frac{u-v\sqrt5}2\biggr)
=25v^2(v^2-1)\ge0.$$ Since $\operatorname{sym}^k\pi$ and $\operatorname{sym}^k\pi'$ are cuspidal for $k\le4$ and $\pi\not\cong\pi'$, we have $r(F)=6r(P_2(x)P_2(y))$.
Suppose that $\operatorname{sym}^2\pi\cong\operatorname{sym}^2\pi'$. Then $a_n=\pm b_n$ for all $n$; writing $a_n=\frac{u_n+v_n\sqrt5}2$ as above, it follows that $2\mid v_n$, so that $$F(a_n,b_n)=25v_n^2(v_n^2-1)\ge75v_n^2=15(a_n-b_n)^2.$$ This implies $$6=r(F)\ge15r((x-y)^2)=30,$$ which is absurd. Hence, $\operatorname{sym}^2\pi\not\cong\operatorname{sym}^2\pi'$ and $r(F)=0$.
By [@wang Theorem B], this in turn implies that $\pi\boxtimes\operatorname{sym}^2\pi'$ and $\pi'\boxtimes\operatorname{sym}^2\pi$ are cuspidal. Also, in view of the identity $$(xy)^2=(P_2(x)+1)(P_2(y)+1)=P_2(x)P_2(y)+P_2(x)+P_2(y)+1,$$ we have $r((xy)^2)=1$, so that $\pi\boxtimes\pi'$ is cuspidal.
Since $F(a_n,b_n)$ is nonnegative, by Lemma \[lem:positive\](2) there exists $\varepsilon>0$ such that $$\sum_{n=1}^\infty\frac{\Lambda(n)F(a_n,b_n)}{n^{1-\varepsilon}}<\infty.$$ Applying Cauchy–Schwarz and the inequality $$(x-y)^4F(x,y)=(x-y)^6((x-y)^2-5)\le(x-y)^8\le128(x^8+y^8),$$ we have $$\begin{aligned}
\biggl(\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{1-\varepsilon/3}}
(a_n-b_n)^2F(a_n,b_n)\biggr)^2
&\le\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{1-\varepsilon}}
F(a_n,b_n)\cdot\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{1+\varepsilon/3}}
(a_n-b_n)^4F(a_n,b_n)\\
&\le128\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{1-\varepsilon}}F(a_n,b_n)
\cdot\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{1+\varepsilon/3}}
(a_n^8+b_n^8).\end{aligned}$$ Noting that $x^8=(P_4(x)+3P_2(x)+2)^2$, the final sum on the right-hand side converges, by Rankin–Selberg. Thus, the series defining $D_{(x-y)^2F}(s)$ converges absolutely for $\Re(s)\ge1-\varepsilon/3$, so that $r((x-y)^2F)=0$.
Next, we compute that $$\begin{aligned}
20P_3(x)P_3(y)&=20-(x-y)^2F(x,y)+P_2(x)P_4(x)-6P_3(x)\cdot yP_2(x)+15P_4(x)P_2(y)\\
&\quad+15P_2(x)P_4(y)-6xP_2(y)\cdot P_3(y)+P_2(y)P_4(y)+14P_4(x)-38P_3(x)y\\
&\quad+60P_2(x)P_2(y)-38xP_3(y)+14P_4(y)+38P_2(x)-48xy+38P_2(y).\end{aligned}$$ Evaluating $r$ of both sides and using that $r((x-y)^2F)=0$, we see that $r(P_3(x)P_3(y))=1$, whence $\operatorname{sym}^3\pi\cong\operatorname{sym}^3\pi'$. Similarly, we have $$xP_2(y)\cdot yP_2(x)=P_3(x)P_3(y)+xP_3(y)+yP_3(x)+xy,$$ from which it follows that $\pi\boxtimes\operatorname{sym}^2\pi'\cong\pi'\boxtimes\operatorname{sym}^2\pi$. Also, from $$P_4(x)-P_4(y)=(x+y)(P_3(x)-P_3(y)+xP_2(y)-yP_2(x)),$$ we get $P_4(a_n)=P_4(b_n)$, so that $\operatorname{sym}^4\pi_p\cong\operatorname{sym}^4\pi_p'$ for all $p\nmid N$. By strong multiplicity one, $\operatorname{sym}^4\pi\cong\operatorname{sym}^4\pi'$.
Nontempered and archimedean places
----------------------------------
In view of the identity $$x^2(P_3(x)-P_3(y))+(x^2+xy-1)(xP_2(y)-yP_2(x))
=(x-y)(x^2-x-1)(x^2+x-1),$$ for every $n$ we have either $a_n=b_n\in{\mathbb{Z}}$ or $a_n\in\{\pm\varphi,\pm\varphi^\tau\}$. For any prime $p\nmid N$, it follows that if either of $\pi_p$, $\pi_p'$ is nontempered then $\pi_p\cong\pi'_p$.
Next we show that this conclusion holds for ramified and archimedean places as well. If $\pi_p$ is nontempered then, as explained in [@MR Remark 1], $\pi_p$ is a twist of an unramified complementary series representation, i.e. $\pi_p\cong(|\cdot|_p^s\boxplus|\cdot|_p^{-s})\otimes\chi$ for some $s>0$ and unitary character $\chi$ of ${\mathbb{Q}}_p^\times$. Since $\operatorname{sym}^3\pi_p\cong\operatorname{sym}^3\pi_p'$, $\pi_p'$ must also be nontempered, so we similarly have $\pi_p'\cong(|\cdot|_p^{s'}\boxplus|\cdot|_p^{-s'})\otimes\chi'$ for some $s'>0$ and unitary character $\chi'$. Thus, $$\operatorname{sym}^3(|\cdot|_p^s\boxplus|\cdot|_p^{-s})\otimes\chi^3
\cong\operatorname{sym}^3(|\cdot|_p^{s'}\boxplus|\cdot|_p^{-s'})\otimes(\chi')^3,$$ from which it follows that $s=s'$ and $\chi^3=(\chi')^3$. Comparing central characters, we deduce that $\chi=\chi'$, whence $\pi_p\cong\pi_p'$. Running through this argument again with the roles of $\pi$ and $\pi'$ reversed, we obtain conclusions (2) and (3b) of the theorem for finite places.
Similarly, we have $\pi_\infty=(|\cdot|_{\mathbb{R}}^s\boxplus|\cdot|_{\mathbb{R}}^{-s})\otimes\operatorname{sgn}^\epsilon$ for some $\epsilon\in\{0,1\}$ and $s\in i{\mathbb{R}}\cup(0,\frac12)$, and comparing the parameters of $\operatorname{sym}^3\pi_\infty$ and $\operatorname{sym}^3\pi_\infty'$, we conclude that $\pi_\infty\cong\pi_\infty'$.
Value distribution of $\lambda_\pi(p)$
--------------------------------------
Making use of the isomorphism $\pi\boxtimes\operatorname{sym}^2\pi'\cong\pi'\boxtimes\operatorname{sym}^2\pi$, if $0<j<i\le 8-j$ then $$\begin{aligned}
r(x^iy^j)&=r(x^{i-2}y^{j-1}(yP_2(x)+y))
=r(x^{i-2}y^{j-1}(xP_2(y)+y))\\
&=r(x^{i-1}y^{j+1})+r(x^{i-2}y^j)-r(x^{i-1}y^{j-1}),\end{aligned}$$ and similarly with the roles of $x$ and $y$ reversed. By systematic application of this rule and linearity, we reduce the computation of $r(x^iy^j)$ for $i+j\le8$ to that of $r(P_i(x)P_j(y))$, $r(P_i(x)P_j(x))$ and $r(P_i(y)P_j(y))$ for $i,j\le4$, all of which are determined from the conclusions obtained in §\[sec:equiv\]. After some computation we arrive at the following table of values for $r(x^iy^j)$:
$i\setminus j$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
---------------- ------ ----- ----- ----- ----- ----- ----- ----- ------
$0$ $1$ $0$ $1$ $0$ $2$ $0$ $5$ $0$ $14$
$1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$2$ $1$ $0$ $1$ $0$ $2$ $0$ $6$
$3$ $0$ $0$ $0$ $1$ $0$ $4$
$4$ $2$ $0$ $2$ $0$ $5$
$5$ $0$ $0$ $0$ $4$
$6$ $5$ $0$ $6$
$7$ $0$ $1$
$8$ $14$
This enables us to compute $r(f)$ for any $f$ of total degree at most $8$ without having to work out the full expansion as in .
Consider $$H=(xy+1)x^2(x^2-1)(x^2-4).$$ Then $H(a_n,b_n)\ge0$ for all $n$, and for $p\nmid N$, $\pi_p$ and $\pi_p'$ are tempered if and only if $H(a_p,b_p)=0$. We verify by the above that $r(H)=0$, so by Lemma \[lem:positive\](2) there exists $\delta\in(0,1]$ such that $\sum_{n=1}^\infty\Lambda(n)H(a_n,b_n)/n^{1-\delta}<\infty$. Thus, with $S$ as in the statement of the theorem, we have $$\label{eq:Hestimate}
\sum_{\substack{p\in S\\p\le X}}(\log{p})a_p^8
\le\sum_{\substack{n\le X\\a_n\notin A}}\Lambda(n)a_n^8
\ll\sum_{n\le X}\Lambda(n)H(a_n,b_n)
\le X^{1-\delta}\sum_{n=1}^\infty\frac{\Lambda(n)H(a_n,b_n)}{n^{1-\delta}}
\ll X^{1-\delta}.$$ Including the possible contribution from ramified primes, which are finite in number, we see that $\pi_p$ and $\pi_p'$ are tempered for all but $O(X^{1-\delta})$ primes $p\le X$, which proves (3a).
Next, for each $\alpha\in A$ we define a polynomial $f_\alpha\in{\mathbb{R}}[x,y]$ of total degree at most $6$, as follows:
$\alpha$ $f_\alpha$ $r(f_\alpha)/f_\alpha(\alpha,\alpha^\tau)$
------------------------------ ------------------------------------- --------------------------------------------
$0$ $(xy+1)(x^2-1)(x^2-4)$ $\frac14$
$\pm1$ $(x^2+y^2-3)x(x+\alpha)(x^2-4)$ $\frac16$
$\pm2$ $(xy+1)x(x^2-1)(x+\alpha)$ $\frac1{120}$
$\pm\varphi,\pm\varphi^\tau$ $(x-y)(1\pm x\pm y)(x-\alpha^\tau)$ $\frac1{10}$
In each case, we have $f_\alpha(\beta,\beta^\tau)\ge0$ for all $\beta\in{\mathbb{Z}}\cup A$, with $f_\alpha(\beta,\beta^\tau)=0$ for $\beta\in A\setminus\{\alpha\}$ and $f_\alpha(\alpha,\alpha^\tau)>0$. Also, since $\deg f_\alpha\le 6$, we have $f_\alpha(a_n,b_n)\ll a_n^6\le a_n^8$ whenever $a_n\notin A$. Thus, by and Lemma \[lem:positive\](1), $$\begin{aligned}
\sum_{\substack{p\le X\\\lambda_\pi(p)=\alpha}}\log{p}
&=O(X^{1/2})+\sum_{\substack{n\le X\\a_n=\alpha}}\Lambda(n)
=O(X^{1/2}+X^{1-\delta})+\frac1{f_\alpha(\alpha,\alpha^\tau)}
\sum_{n\le X}\Lambda(n)f_\alpha(a_n,b_n)\\
&=\frac{r(f_\alpha)}{f_\alpha(\alpha,\alpha^\tau)}X+o(X)
\quad\text{as }X\to\infty.\end{aligned}$$ By partial summation, it follows that $\{p:\lambda_\pi(p)=\alpha\}$ has natural density $r(f_\alpha)/f_\alpha(\alpha,\alpha^\tau)$, whose values are shown in the table. This proves (4).
Weak automorphy of symmetric powers
-----------------------------------
Let $G=\operatorname{SL}_2({\mathbb{F}}_5)$, which is the smallest group supporting a $2$-dimensional icosahedral representation [@wang §2]. Then $G$ has nine irreducible representations, with dimensions $1$, $2$, $2$, $3$, $3$, $4$, $4$, $5$ and $6$. Their characters all take values in ${\mathbb{Z}}[\varphi]$ and can be written as $$\begin{aligned}
\chi_0&=1,\quad\chi_1=\chi,\quad\chi_2=\chi^\tau,\quad\chi_3=P_2(\chi),\quad\chi_4=P_2(\chi^\tau),\\
\chi_5&=\chi\chi^\tau,\quad\chi_6=P_3(\chi),\quad\chi_7=P_4(\chi),\quad\chi_8=\chi P_2(\chi^\tau),\end{aligned}$$ where $\chi$ is one of the characters of dimension $2$ and $\chi^\tau$ is its Galois conjugate. (Our numbering scheme is more or less arbitrary, and was made for notational convenience below.) The character table is as follows:
$\begin{psmallmatrix}1&0\\0&1\end{psmallmatrix}$ $\begin{psmallmatrix}4&0\\0&4\end{psmallmatrix}$ $\begin{psmallmatrix}3&2\\4&3\end{psmallmatrix}$ $\begin{psmallmatrix}2&2\\4&2\end{psmallmatrix}$ $\begin{psmallmatrix}2&0\\0&3\end{psmallmatrix}$ $\begin{psmallmatrix}4&1\\0&4\end{psmallmatrix}$ $\begin{psmallmatrix}4&2\\0&4\end{psmallmatrix}$ $\begin{psmallmatrix}1&1\\0&1\end{psmallmatrix}$ $\begin{psmallmatrix}1&2\\0&1\end{psmallmatrix}$
---------- -------------------------------------------------- -------------------------------------------------- -------------------------------------------------- -------------------------------------------------- -------------------------------------------------- -------------------------------------------------- -------------------------------------------------- -------------------------------------------------- --------------------------------------------------
$\chi_0$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$
$\chi_1$ $2$ $-2$ $1$ $-1$ $0$ $\varphi$ $\varphi^\tau$ $-\varphi$ $-\varphi^\tau$
$\chi_2$ $2$ $-2$ $1$ $-1$ $0$ $\varphi^\tau$ $\varphi$ $-\varphi^\tau$ $-\varphi$
$\chi_3$ $3$ $3$ $0$ $0$ $-1$ $\varphi$ $\varphi^\tau$ $\varphi$ $\varphi^\tau$
$\chi_4$ $3$ $3$ $0$ $0$ $-1$ $\varphi^\tau$ $\varphi$ $\varphi^\tau$ $\varphi$
$\chi_5$ $4$ $4$ $1$ $1$ $0$ $-1$ $-1$ $-1$ $-1$
$\chi_6$ $4$ $-4$ $-1$ $1$ $0$ $1$ $1$ $-1$ $-1$
$\chi_7$ $5$ $5$ $-1$ $-1$ $1$ $0$ $0$ $0$ $0$
$\chi_8$ $6$ $-6$ $0$ $0$ $0$ $-1$ $-1$ $1$ $1$
Let $\langle\;,\;\rangle$ denote the inner product on $L^2(G)$. For each $k\ge0$ and $i\in\{0,\ldots,8\}$, let $$m_{k,i}=\langle P_k(\chi),\chi_i\rangle$$ be the multiplicity of $\chi_i$ in the $k$th symmetric power of $\chi$, so that $$P_k(\chi)=\sum_{i=0}^8 m_{k,i}\chi_i.$$ In view of the character table, for any $\alpha\in A$ we may choose $g\in
G$ with $\chi(g)=\alpha$ and evaluate both sides of the above at $g$ to get $$P_k(\alpha)=\sum_{i=0}^8 m_{k,i}h_i(\alpha,\alpha^\tau),$$ where we write $$\begin{aligned}
h_0&=1,\quad h_1=x,\quad h_2=y,\quad h_3=P_2(x),\quad h_4=P_2(y),\\
h_5&=xy,\quad h_6=P_3(x),\quad h_7=P_4(x),\quad h_8=xP_2(y).\end{aligned}$$
On the automorphic side, we make the parallel definitions $$\begin{aligned}
\sigma_0&=1,\quad\sigma_1=\pi,\quad\sigma_2=\pi',\quad\sigma_3=\operatorname{sym}^2\pi,\quad\sigma_4=\operatorname{sym}^2\pi',\\
\sigma_5&=\pi\boxtimes\pi',\quad\sigma_6=\operatorname{sym}^3\pi,\quad\sigma_7=\operatorname{sym}^4\pi,\quad
\sigma_8=\pi\boxtimes\operatorname{sym}^2\pi',\end{aligned}$$ and we set $$\Pi_k={
\mathop{
\vphantom{\bigoplus}
\mathchoice
{\vcenter{\hbox{\resizebox{\widthof{$\displaystyle\bigoplus$}}{!}{$\boxplus$}}}}
{\vcenter{\hbox{\resizebox{\widthof{$\bigoplus$}}{!}{$\boxplus$}}}}
{\vcenter{\hbox{\resizebox{\widthof{$\scriptstyle\oplus$}}{!}{$\boxplus$}}}}
{\vcenter{\hbox{\resizebox{\widthof{$\scriptscriptstyle\oplus$}}{!}{$\boxplus$}}}}
}\displaylimits
}_{i=0}^8(
\underbrace{\sigma_i\boxplus\cdots\boxplus\sigma_i}_{m_{k,i}\text{ times}})
\quad\text{for }k\ge0.$$ By construction, if $p\notin S$ and $\pi_p$ is unramified, then for every power $n=p^j$ we have $$c_n(\operatorname{sym}^k\pi)=P_k(a_n)=\sum_{i=0}^8 m_{k,i}h_i(a_n,b_n)
=\sum_{i=0}^8 m_{k,i}c_n(\sigma_i)=c_n(\Pi_k).$$ Thus, $\operatorname{sym}^k\pi_p\cong\Pi_{k,p}$, as claimed.
It remains to prove the uniqueness of $\Pi_k$. Suppose that $\Pi_k'$ is another such isobaric representation. Then for any $i\in\{0,\ldots,8\}$, $$\sum_{(n,N)=1}\frac{\Lambda(n)c_n(\Pi_k'\boxtimes\sigma_i)}{n^s}
=\sum_{(n,N)=1}\frac{\Lambda(n)c_n(\Pi_k\boxtimes\sigma_i)}{n^s}
+\sum_{(n,N)=1}\frac{\Lambda(n)(c_n(\Pi_k')-c_n(\Pi_k))c_n(\sigma_i)}{n^s}.$$ Since $c_n(\Pi_k')=c_n(\operatorname{sym}^k\pi)=c_n(\Pi_k)$ whenever $(n,N)=1$ and $a_n\in A$, by Cauchy–Schwarz we have $$\begin{aligned}
\biggl(\sum_{(n,N)=1}&\frac{\Lambda(n)|(c_n(\Pi_k')-c_n(\Pi_k))c_n(\sigma_i)|}{n^{1-\delta/3}}\biggr)^2\\
&\qquad\le\sum_{(n,N)=1}\frac{\Lambda(n)|c_n(\Pi_k')-c_n(\Pi_k)|^2}{n^{1+\delta/3}}
\cdot\sum_{\substack{(n,N)=1\\a_n\notin A}}\frac{\Lambda(n)c_n(\sigma_i)^2}{n^{1-\delta}}.\end{aligned}$$ Since $\Pi_{k,p}'\cong\operatorname{sym}^k\pi_p$ is tempered for all unramified $p\notin S$, the cuspidal summands of $\Pi_k'$ must be unitary, so the first sum on the right-hand side converges by Rankin–Selberg. As for the second, for $a_n\notin A$ we have $c_n(\sigma_i)^2\ll H(a_n,b_n)$, so it converges as well. Therefore, $$\operatorname{Res}_{s=1}\sum_{(n,N)=1}\frac{\Lambda(n)c_n(\Pi_k'\boxtimes\sigma_i)}{n^s}
=\operatorname{Res}_{s=1}\sum_{(n,N)=1}\frac{\Lambda(n)c_n(\Pi_k\boxtimes\sigma_i)}{n^s}
=m_{k,i},$$ so $\sigma_i$ occurs as a summand of $\Pi_k'$ with multiplicity $m_{k,i}$. Since $\Pi_k$ and $\Pi_k'$ are both representations of $\operatorname{GL}_{k+1}({\mathbb{A}})$, we have $\Pi_k'\cong\Pi_k$, as desired. This establishes (3d).
Temperedness and Galois type at $\infty$
----------------------------------------
Suppose that $\operatorname{sym}^5\pi$ is automorphic. Then it agrees with an isobaric representation at all unramified finite places. By the uniqueness of $\Pi_5$, we must have $\operatorname{sym}^5\pi_p\cong\Pi_{5,p}=\pi_p\boxtimes\operatorname{sym}^2\pi_p'$ for all $p\nmid N$. When $\lambda_\pi(p)=\lambda_{\pi'}(p)$, this implies the relation $$\lambda_\pi(p)^5-4\lambda_\pi(p)^3+3\lambda_\pi(p)
=\lambda_\pi(p)(\lambda_\pi(p)^2-1),$$ so that $\lambda_\pi(p)\in\{0,\pm1,\pm2\}$. Thus, $\pi_p$ is tempered for all $p\nmid N$. In particular, $S$ is finite.
Finally, suppose that $S$ is finite. Then, by what we have already shown, $\pi$ is *s-icosahedral* in the sense of Ramakrishnan [@R2]. Appealing to [@R2 Theorem A], we conclude that $\pi$ is tempered and $\pi_\infty$ is of Galois type. This establishes (3e) and concludes the proof.
[^1]: The author was partially supported by EPSRC Grant `EP/K034383/1`. No data were created in the course of this study.
|
---
abstract: 'We present the results of hydrodynamic simulations of the interaction between a $10$ Jupiter mass planet and a red or asymptotic giant branch stars, both with a zero-age main sequence mass of $3.5~{\rm M_\odot}$. Dynamic in-spiral timescales are of the order of few years and a few decades for the red and asymptotic giant branch stars, respectively. The planets will eventually be destroyed at a separation from the core of the giants smaller than the resolution of our simulations, either through evaporation or tidal disruption. As the planets in-spiral, the giant stars’ envelopes are somewhat puffed up. Based on relatively long timescales and even considering the fact that further in-spiral should take place before the planets are destroyed, we predict that the merger would be difficult to observe, with only a relatively small, slow brightening. Very little mass is unbound in the process. These conclusions may change if the planet’s orbit enhances the star’s main pulsation modes. Based on the angular momentum transfer, we also suspect that this star-planet interaction may be unable to lead to large scale outflows via the rotation-mediated dynamo effect of Nordhaus and Blackman. Detectable pollution from the destroyed planets would only result for the lightest, lowest metallicity stars. We furthermore find that in both simulations the planets move through the outer stellar envelopes at Mach-3 to Mach-5, reaching Mach-1 towards the end of the simulations. The gravitational drag force decreases and the in-spiral slows down at the sonic transition, as predicted analytically.'
title: Hydrodynamic Simulations of the Interaction between Giant Stars and Planets
---
Introduction
============
An increasing number of planets is being discovered at intermediate distances from their host stars [@udry07]. @Villaver2009, @Mustill2012 and @Nordhaus2013 among others calculated that expanding giants, whether red giant branch (RGB) stars or asymptotic giant branch (AGB) stars could engulf planets orbiting 2-4 times the maximum radius attained by the star. Such an interaction would likely result in the destruction of the planet producing an observational signature as well as long-lasting and observable evolutionary effects on stars.
Observational clues of star-planet interactions are in the form of putative planets discovered around post-main sequence stars, close enough that an interaction must have taken place when the star was in its giant phase in the recent past. Examples are the $1.25~{\rm M_J}$ planet (where ${\rm M_J}$ is the mass of Jupiter) orbiting 0.116 au from a horizontal branch star [@setiawan10], two Earth-sized objects orbiting a subdwarf B star at a separation of 0.0060 and 0.0076 au [@charpinet11], or three earth-sized planets orbiting a subdwarf B pulsator [@Silvotti2014]. These planets must have been engulfed in the envelope of the giant star that became the subdwarf star today. @charpinet11 and @passy12planet showed how planets may have been much more massive initially and lost much of their mass in the CE phase. However, they could not determine how the core of the planets survived the interaction instead of plunging into the core of the giant.
A second type of planet around post-giant stars consists of one or more planets detected at au-distance from post-CE binaries, rather than single stars. Some of those finds have been debated because the planets would not be in stable orbits [@Horner2013] or the data could be as easily explained with alternative, non-planet scenarios. However, other data is more convincingly, though not conclusively, explained by the presence of planetary systems [e.g., NN Serpentis; @Parsons2014]. These planets, contrary to the planets at sub-au orbital separations from [*single*]{} post-giant stars, are unlikely to have been involved in the CE that created today’s close binary, but they must have been impacted by the ejection of the giant’s envelope. It has been speculated that they may have formed in the aftermath of the CE binary interaction [@Beuermann2011], similarly to how the planets around pulsar PSR1257+12 [@Wolszczan1992] were formed after the supernova explosion.
Theoretically @soker98 suggested that star-planet interactions could generate blue horizontal branch stars by enhancing the RGB mass loss rate and decreasing the envelope mass of the red giant star. @carlberg09 calculated instead the extent to which interactions (tidal interactions and/or mergers) between giants and planets would spin up giants. @nelemans98 found that companions with masses more than $20-25~{\rm M_J}$ could survive a common envelope with a $1~{\rm M_\odot}$ red giant and expel the envelope. On the other hand, using the formalism from @nelemans98, @villaver07 found that companions less massive than $120~{\rm
M_J}=0.11~{\rm M_\odot}$ would evaporate inside the envelope of a $5~{\rm
M_\odot}$ AGB star. Both these studies rely on the uncertain and highly debated efficiency of the common envelope ejection formalism [@DeMarco2011].
@nordhaus06 investigated analytically $3~{\rm M_\odot}$ RGB and AGB stars interacting with a low mass companion (planet, brown dwarf, or low mass main-sequence star). They found that envelope ejection in the RGB case is unlikely, but possible for AGB stars from low-mass main sequence stars. While planets may have too low a mass to eject the AGB star’s envelope, they found that a planet may induce differential rotation mediated dynamo that can eject material. Furthermore, the planet may tidally disrupt, creating a disc inside the envelope that can lead to a disc driven outflow. @metzger12 investigated mergers between hot Jupiters and their host main sequence stars and predicted that prior to merger, as the planet penetrates the star’s atmosphere, an EUV/X-ray transient is produced in the hot wake following the planet. The merger would also drive an outflow and hydrogen recombination in the outflow would cause an optical transient. They argued that the galactic rate of mergers between hot Jupiters and their host stars should be $0.1-1~{\rm yr^{-1}}$ and that should be similar to the rate observed for planets and giant stars.
Despite past efforts, many questions still remain. What is the fate of the planet in the CE interaction? Does it survive, or is it destroyed by ablation or tidal disruption? Whatever the fate of the planet, will the interaction lead to an alteration of the star and its subsequent evolution, such as spin-up, mass loss, or a change in the surface abundances?
Presumably, once a planet is tidally brought closer to an expanding giant star, the star would fill its Roche lobe and transfer mass to the planet. Since the planet is much less massive than the star, it is likely that the planet would be engulfed by the giant’s extended envelope and have a common envelope (CE) interaction [@Ivanova2013]. CE interactions are thought to happen also between giants and stellar mass companions [@Paczynski1976] and give rise to compact evolved binaries. CE simulations using a variety of techniques [e.g. @sandquist98; @ricker12; @passy12; @nandez14; @staff15] have a range of uncertainties and shortcomings [e.g., @Nandez2015], but can be used as starting points to determine the nature of star-planet interactions. By running hydrodynamic simulations of the CE interaction between a $10~{\rm M_J}$ planet and an RGB or an AGB star, we start addressing numerically aspects of the interaction such as the timescale of the interaction, the final separation or the extent to which the stellar envelope is spun-up.
We also exploit the lightness of the planetary companion relative to the stellar envelope to carry out a study of gravitational drag experienced by a body in a common envelope simulation [@Ricker2008]. This is much more difficult when the companion is more massive because the gas is stirred considerably and it is difficult to extract some of the quantities needed to carry out the calculation.
We describe the numerical method that we use in section \[methodssection\]. Then in section \[resultssection\] we present our results including an appraisal of how numerical considerations impact our conclusions. In Section \[dragforcesubsection\] we assess the drag forces acting on the planet and we exploit the relative composure of the envelope gas to compare these forces to their analytically-derived equivalent. We finally discuss our results in Section \[summarysection\].
Method {#methodssection}
======
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We used a modified version of the grid-based hydrodynamics code <span style="font-variant:small-caps;">enzo</span> [@oshea04; @passy12; @bryan14] to run the hydrodynamics simulations. The calculations were performed on a $256^3$ grid in the adiabatic approximation with outflow boundary conditions. In addition, we also performed the same simulations on a grid with $512^3$ resolution, to test if the resolution affects the results.
The structure of the giant stars were calculated using the stellar evolution code Modules for Experiments in Stellar Astrophysics [<span style="font-variant:small-caps;">mesa</span> ; @paxton11; @paxton13]. We used two stellar structures, evolved from the same 3.5-[M$_\odot$]{}, zero-age main sequence, solar metallicity model. The first stellar structure was that of the model 283 million years after joining the zero-age man sequence. At this time the star had reached the RGB, having lost 0.01 [M$_\odot$]{}. It had a radius of 55 [R$_\odot$]{} (approximately the maximum radius that this type of star reaches on the RGB), a luminosity of 680 [L$_\odot$]{}, and an effective temperature of 3960 K.
The second stellar structure was taken 330 million years after the zero-age main sequence. At this time the star had reached the thermally-pulsating AGB, had a mass of 3.05 [M$_\odot$]{} a radius of 470 [R$_\odot$]{}, a luminosity of $1.4\times10^4~{\rm L_\odot}$ and an effective temperature of 2920 K (the structure was taken between two thermal pulses). This is the same stellar structure used for the simulations of @staff15. Stars more massive than $\sim2~{\rm M_\odot}$ grow a lot larger on the AGB than on the RGB, providing for an opportunity for planets that had not interacted on the RGB to do so during the AGB. This is the reason why we chose to use a star more massive than $2~{\rm M_\odot}$. Stars less massive than approximately $2~{\rm M_\odot}$ have similar maximum RGB and AGB radii and this means that they have most of their interactions on the RGB. For the more luminous low mass ($M < 2~{\rm M_\odot}$) RGB stars, which can attain radii close to $200~{\rm R_\odot}$, the nature of the star-planet interaction will be intermediate between the RGB and AGB cases considered here.
We mapped the 1-D <span style="font-variant:small-caps;">mesa</span> model into the <span style="font-variant:small-caps;">enzo</span> computational domain. <span style="font-variant:small-caps;">mesa</span> models have much higher resolution compared to the linear resolution of the 3-D Cartesian <span style="font-variant:small-caps;">enzo</span> grid that we use. The size of the simulation box used for the simulation with the smaller RGB star was $3\times10^{13}~{\rm cm}$ (2 au), such that each cell in $256^3$ cell domain had a size of $1.2\times10^{11}~{\rm cm}$ (1.7 [R$_\odot$]{}). In the simulation with the larger AGB star, the simulation box was $2.2\times10^{14}~{\rm cm}$ (15 au), and each cell in $256^3$ resolution had a size of $8.6\times10^{11}~{\rm cm}$ (12 [R$_\odot$]{}). The cell size was half these values for the $512^3$ resolution simulation. The cores of the giant stars, where much mass is concentrated, as well as the planet companion cannot be resolved. Instead, they are approximated by point masses, with a smoothed gravitational potential as discussed in @passy12 and by @staff15. We used a smoothing length of 3 cells, instead of 1.5, which, as was discussed in @staff15, results in better energy conservation.
<span style="font-variant:small-caps;">mesa</span> takes microphysics into account, while we use an ideal gas equation of state with an adiabatic pressure-density relation ($\gamma=5/3$) in <span style="font-variant:small-caps;">enzo</span>. Because of this and the addition of a point mass with a smoothed gravitational potential, the star is not in perfect hydrostatic equilibrium in <span style="font-variant:small-caps;">enzo</span>. Following the method described by @passy12, we force the starting model to hydrostatic equilibrium by dampening the velocities by a factor of 3 for each time step after mapping the stellar structures into the computational domain. We then check the stability by running the simulations without damping the velocities for 4 dynamical times (the dynamical times are 0.07 years and 1.8 years for the RGB and AGB stars, respectively). The simulation volume not occupied by stellar gas is filled with a hot medium, which has a density four orders of magnitude lower than the giant star’s least dense point and a high temperature so as to balance the pressure at the surface of the giant star. Despite this, the outer layers of the star tend to diffuse out somewhat [see @staff15]. The 3D star constructed in this way tends to be slightly larger than it was initially. For both models the post-stabilization radius was $\approx$5 per cent larger (2.5 [R$_\odot$]{} and 23 [R$_\odot$]{} larger for the RGB and AGB models, respectively, at a density one order of magnitude less than the initial lowest density in the star).
Once the giant star is stabilized, we insert a planet with a mass of $10~{\rm M_J}$ at $1.1$ times the radius of the <span style="font-variant:small-caps;">mesa</span> model ($R_{\rm
star}$), in a circular orbit. In both simulations this initial configuration results in the giant stars massively overflowing their Roche lobe radii. This is the case with many CE simulations [e.g., @sandquist98; @passy12] and may have some effect on the CE outcome (Iaconi et al., 2016, in preparation). However, in the case of planetary companions, it is likely that the effect of starting close to the surface is minimal: companions as far as 2-3 stellar radii are likely to be captured [@Villaver2009; @Mustill2012], but the angular momentum of the orbit transferred to the primary would confer to it only a relatively minor surface velocity of $1.1-1.3~{\rm
km~s^{-1}}$ for the AGB star and $3.2-3.9~{\rm km~s^{-1}}$ for the RGB star (this range was found assuming that all the orbital angular momentum of the planet at a distance of 2-3 stellar radii is transferred to the envelope of the giant, and that this envelope rotates rigidly), not too different from our non-rotating initial models.
Results: the in-spiral {#resultssection}
======================
![The unbound mass in the computational domain as a function of time for the RGB star (upper panel), and the AGB star (lower panel) orbited by a $10~{\rm M_J}$ planet (in both cases the figures are from the low resolution simulations).[]{data-label="rgbmunb"}](3.5rgb510mj1.1munb.eps "fig:"){width="45.00000%"} ![The unbound mass in the computational domain as a function of time for the RGB star (upper panel), and the AGB star (lower panel) orbited by a $10~{\rm M_J}$ planet (in both cases the figures are from the low resolution simulations).[]{data-label="rgbmunb"}](3.5agb10mj1.1munb.eps "fig:"){width="45.00000%"}
With our starting conditions ($a_0 = 1.1 R_{\rm star} = 61~{\rm
R_\odot}=0.28~{\rm au}$, and an orbital period of 26 days for the RGB star and $a_0 =1.1~R_{\rm star}=520~{\rm R_\odot}=2.4~{\rm au}$ and an initial period of 1.9 years for the AGB star), the planet is rapidly engulfed by stellar envelope gas. We show the evolution of the density for the RGB star in Fig. \[rgbrhopanels\] and for the AGB star in Fig. \[agbrhopanels\]. As the planet in-spirals, the giant star’s envelope expands. This puffed-up envelope has typical densities of $\sim10^{-10}~{\rm g~cm^{-3}}$ in the RGB simulation and $\sim10^{-12}~{\rm
g~cm^{-3}}$ in the AGB simulation. As the stellar envelope is puffed-up due to the interaction, the photosphere is likely to be located near the edge of this expanding gas. Due to the high temperature ambient medium, this low-density puffed up gas may be artificially heated. Therefore we cannot accurately determine the temperature of the photosphere, nor how fast it would cool off radiatively and therefore recede. Especially in the RGB simulation, where the interaction is reasonably quick, it seems likely that a significant increase in the photospheric radius could be achieved.
Some low density gas is lost from the domain, $\lesssim0.01~{\rm M_\odot}$ in both simulations. Of the mass lost from the simulation box, $\lesssim10$ per cent ($\approx10^{-3}~{\rm M_\odot}\approx1~{\rm M_J}$) is unbound in the RGB simulation, and $\lesssim30$ per cent ($\approx3\times 10^{-3}~{\rm
M_\odot} \approx3~{\rm M_J}$) is unbound in the AGB simulation (see Fig. \[rgbmunb\]). We note that initially, a larger amount of the ambient medium is unbound in the AGB simulation compared with the RGB simulation, which in part explains why the AGB simulation unbinds more mass. Pre-empting our discussion on energy conservation in Section 3.1, we note that there is considerable uncertainty on the mass unbinding. The change in the planet’s orbital energy as it spirals through the RGB star layers, that can lead to unbinding of envelope, is $\sim3\times10^{45}~{\rm erg}$, an order of magnitude smaller than the artificial growth in the total energy in the box for that simulation. This artificial growth in energy may therefore be the main driver for the meagre mass unbinding observed, making our estimate for the RGB case an upper limit.
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![The velocity of the gas in the equatorial plane at the end of the low resolution simulations. [*Top panel:*]{} RGB star, [*bottom panel:*]{} AGB star. The plotted velocity is $\sqrt{v_x^2+v_y^2}$, with the arrows showing the direction. The pink circle indicates the size of the giant star prior to the interaction. It is only the puffed up, low density matter at larger radii that gains significant rotational velocity, while the high density interior of the stars have very low rotational velocity.[]{data-label="rotvelfig"}](rgbplanetrotvelbw.eps "fig:"){width="45.00000%"} ![The velocity of the gas in the equatorial plane at the end of the low resolution simulations. [*Top panel:*]{} RGB star, [*bottom panel:*]{} AGB star. The plotted velocity is $\sqrt{v_x^2+v_y^2}$, with the arrows showing the direction. The pink circle indicates the size of the giant star prior to the interaction. It is only the puffed up, low density matter at larger radii that gains significant rotational velocity, while the high density interior of the stars have very low rotational velocity.[]{data-label="rotvelfig"}](agbplanetrotvelbw.eps "fig:"){width="45.00000%"}
Shown in Fig. \[multifigure\] are a series of quantities, which we describe and compare in detail below, for both low and high resolution, RGB and the AGB simulations. In the top panel we show the separation between the planet and the core of the giant star as a function of time, then we show the planet’s Mach number ($M_{\rm p}$) as it moves through the stellar envelope; third is its velocity with respect to the Keplerian value ($v_{\rm Kep}$), then its velocity with respect to the grid, followed by the envelope density surrounding the planet, and, finally, the gravitational drag force acting on the planet in the low resolution simulation (sixth panel) and the high resolution simulation (seventh or bottom panel). The last two panel rows will be exhaustively discussed in Section \[dragforcesubsection\]. The sound speed used to compute the Mach number is just the sound speed in the cell in which the particle is located. Likewise, the density and velocity of the gas surrounding the particle are the values of the cell where the particle is located.
The overall behaviour of the separation is a gradual decrease over 2-3 years in the RGB simulation, and over approximately 60-80 years in the AGB simulation (faster for the higher resolution simulations), after which we cannot follow the evolution because the separation approaches 0.05 and 0.4 au for the RGB and AGB simulations, respectively, which is close to two smoothing lengths (one smoothing length in the lower resolution simulations is $3.5\times10^{11}~{\rm cm}=0.023~{\rm au}$ for the RGB star and $2.6\times10^{12}~{\rm cm}=0.17~{\rm au}$ for the AGB star), at which point the smoothing of the potential may begin to impact the results (see also Section \[numconssection\]).
The oscillatory behavior seen in the separation plot for the low resolution RGB star in Fig. \[multifigure\] has a period of $\approx25$ days (similar to the planet’s initial orbital period) and is due to the development of an eccentricity, typically observed during the fast in-spiral phase of CE simulations and ascribed to the non symmetric distribution of gas [see e.g. @passy12]. Between $\sim100$ and $\sim300$ days in the RGB simulation, the orbital separation and the planet’s velocity remain approximately constant. Following this, the planet speeds up as the separation decays. In the higher resolution RGB simulation, the snapshots from the hydrodynamics simulation were produced less frequently, with a frequency of $0.1$ years (which is larger than the oscillatory period), and this oscillatory behaviour is therefore partly hidden in Fig. \[multifigure\]. In the AGB simulation, the separation also remains approximately constant for the first $\sim 30$ years and the orbit develops a lower eccentricity than for the RGB case. After this, the separation decays, and between 50 and 60 years there is a rapid decrease in the separation. Although we observe a period of faster in-spiral between 700 and 800 days in the RGB simulation, this is not as prominent as in past CE simulations or in the CE between the planet and the AGB star. It is however similar to the behavior of a 0.01 [M$_\odot$]{} companion plunging into the 0.88 [M$_\odot$]{}, 85 [R$_\odot$]{} RGB star (De Marco et al. 2012).
During the interaction the outer layers of the puffed up envelope gain rotation. At densities lower than the initial photospheric density ($\rho<8\times10^{-9}~{\rm g~cm^{-3}}$ for the RGB star and $\rho<1\times10^{-9}~{\rm g~cm^{-3}}$ for the AGB star), rotational velocities of $\gtrsim20~{\rm km~s^{-1}}$ are found in the RGB simulation, and $\gtrsim5~{\rm km~s^{-1}}$ are found in the AGB simulation (see Fig. \[rotvelfig\]). But since the planet has a low angular momentum due to its low mass, the planet is unable to noticeably spin up the higher density, more massive, layers of the giant stars. At higher densities than the photospheric density of the initial model the star is therefore not rotating.
The velocity of this puffed-up envelope is, however, small compared to the planet’s velocity around the giant star, and the planet’s orbital velocity with respect to the grid is therefore similar to the velocity relative to the surrounding gas in both simulations. The planet’s velocity early in the simulation is seen to oscillate between $100$ and $120~{\rm km~s^{-1}}$ in the RGB case, while in the AGB simulation the orbital velocity of the planet varies less around a value of $\approx 35~{\rm km~s^{-1}}$. During the fast in-spiral, at approximately 600 days in the RGB simulation, the velocity continues to oscillate and increases up to $\approx130~{\rm km~s^{-1}}$, while in the AGB simulation the velocity is seen to oscillate more towards the end of the simulation, reaching a maximum of $\approx50~{\rm km~s^{-1}}$. As the planet in-spirals, its velocity is found to remain approximately Keplerian throughout both simulations.
Once the planet becomes submerged in the stellar envelope, the Mach number jumps to 4 or 5 in both simulations (see Fig. \[multifigure\]). During the rapid in-spiral phase, the Mach number decreases as the sound speed grows deeper inside the giant stars. At the end of both simulations, the planet’s velocity is approximately the same as the sound speed. In Sec. \[dragforcesubsection\] we will discuss the drag force in relation to the Mach number of the particles.
Numerical considerations {#numconssection}
------------------------
To test whether the resolution affects our results, we have performed both simulations with a higher resolution of $512^3$ cells. The results are qualitatively similar. The main differences are that the orbital separation tends to a lower value in the higher resolution simulation, and that the in-spiral is faster (see Fig. \[multifigure\]). In both cases, following the slow-down in orbital decay, the in-spiral continues at a slower pace until we stop the simulation. We also found that about half as much mass becomes unbound in the higher resolution simulation ($\approx5\times10^{-4}~{\rm M_\odot}$ vs. $\approx9\times10^{-4}~{\rm M_\odot}$ for the lower resolution RGB simulation), despite the fact that at higher resolution more orbital energy is delivered as the higher resolution allows us to follow the inspiral further. The star remains somewhat more compact in the higher resolution simulation, which is evident in the steeper increase in $V_{\rm p}$ as the planet approaches the core, particularly in the RGB simulation. The accretion radius of the planet is not resolved in any of our simulations. We discuss the implications in Section \[dragforcesubsection\].
Our simulations conserve energy reasonably well. We find that the total energy on the grid in the RGB simulation increases by $\approx3\times10^{46}~{\rm erg}$ over the course of the simulation (see Fig. \[energiesrgb\]). This is $\approx3$ per cent of the initial gravitational potential energy of the gas on the grid, which was $\approx1\times10^{48}~{\rm erg}$. It is $\sim$10 times the change in the planet’s potential energy, and $\sim$100 times the change in the planet’s kinetic energy. However, over the same time, $8.8\times10^{-3}~{\rm M_\odot}$ are lost from the grid. If all this lost mass carried the thermal energy of the initial low density ambient medium, this mass-loss from the domain would remove $6.6\times10^{46}~{\rm erg}$ from the grid[^1]. We therefore estimate that the total energy has increased by up to $9.6\times10^{46}~{\rm erg}$, corresponding to $10$ per cent of the initial potential energy of the star. We compare the energy gained due to non-conservation to the potential energy of the star[^2] instead of the total energy in the box. The latter quantity is meaningless, because the total energy in the box can be made arbitrarily high and close to zero by the addition of an arbitrary quantity of hot “vacuum".
We also emphasize that 10 per cent is an upper limit to the non-conservation, because most of the mass lost from the grid has a lower thermal energy than the initial hot ambient medium. We find that much of the gas leaving the simulation box has a specific thermal energy of $\sim10^{13}~{\rm erg~g^{-1}}$. Assuming that this is representative for all the gas leaving the box, we can determine a lower limit to the energy non-conservation. Then $\approx2\times10^{44}~{\rm erg}$ would be lost from the box (i.e. a factor $\gtrsim100$ less than the above estimate), and hence the energy non-conservation over the course of the simulation would be approximately $3$ per cent. In assuming that the gas that leaves the box removes $2\times10^{44}~{\rm erg}$ we omitted accounting for its kinetic and potential energies. However, kinetic energy and gravitational potential energy of the stellar envelope material lost from the grid have opposite signs and are of the same order of magnitude as the thermal energy. Hence, it will not significantly change our estimate. Therefore, the energy non-conservation in the RGB simulation is between 3 and 10 per cent, likely closer to 3 per cent. The conservation is slightly better in the higher resolution simulation, but still has a lower limit of roughly 3 per cent.
In the AGB simulation, we find that the total energy decreases by $3\times10^{45}~{\rm erg}$ over 60 years. However, the total potential energy in the AGB simulation is $\approx8\times10^{46}~{\rm erg}$, approximately a factor ten less than in the RGB simulation, since the AGB star is less tightly bound. The specific thermal energy of the ambient medium is similar to that in the RGB simulation, and therefore the level of non-conservation is even more sensitive to how much thermal energy is carried away by the lost mass. We find that $\sim1\times10^{31}~{\rm g}$ was lost from the grid over the course of the simulation. If all of this mass had a specific thermal energy of $\sim10^{13}~{\rm erg~g^{-1}}$, we find that the total energy should have dropped $\sim1\times10^{44}~{\rm erg}$, which is small compared to the actual drop of $3\times10^{45}~{\rm erg}$. This way, we find that the energy is conserved to within $4$ per cent in the AGB simulation. This is, however, an estimate for the energy conservation based on a lower estimate for the energy lost from the grid associated with mass loss. It is likely that the lost mass has taken out a larger amount of thermal energy, which would make the conservation better, unless the lost mass has removed more than $\sim6\times10^{45}~{\rm erg}$ (corresponding to a specific thermal energy of more than $\sim6\times10^{14}~{\rm erg~g^{-1}}$). We therefore expect the energy to be conserved to within a few per cent also in this simulation.
Drag forces {#dragforcesubsection}
===========
The torque acting on the planet dictates the rate of in-spiral. Determining whether simulations represent the drag forces with sufficient accuracy is an important step when determining whether results of simulations are reliable. Below we consider both gravitational and hydrodynamic drag components and compare what should be going on in nature, expressed by analytical approximations, with what is going on inside the simulation, calculated from the quantities that are output from the code.
Gravitational vs. Hydrodynamic Drag
-----------------------------------
In nature, a planet in a CE phase with its host star would experience a drag force composed of gravitational and hydrodynamic components. The hydrodynamic drag force is due to the ram pressure on the planet from the surrounding gas, and this force can be estimated: $$F_{\rm hydro,drag}\sim \rho v_{\rm p}^2~\pi R_{\rm p}^2,
\label{hydrodrageq}$$ where $v_{\rm p}$ is the planet’s relative velocity with respect to the surrounding gas, $R_{\rm p}$ is the radius of the planet, and $\rho$ is the density of the envelope gas surrounding the planet.
The gravitational drag is instead due to gravitational forces between the gas flowing past the planet and the planet itself. Although there is no accurate expression for the gravitational drag in the presence of a density gradient [@MacLeod2015], an approximate expression can be found in @iben93 and @passy12: $$F_{\rm grav,drag}\sim \zeta \rho v_{\rm p}^2~\pi R_{\rm A}^2,
\label{gravdrageq}$$ where $\zeta$ is a numerical factor that depends on the Mach number (it is larger than 2 for supersonic motion and less than unity for subsonic motion [@shima85]), and $R_{\rm A}$ is the accretion radius given by [@iben93]: $$R_{\rm A}=\frac{2GM_{\rm p}}{v_{\rm p}^2+c_s^2},
\label{rabondi}$$ for subsonic and sonic speeds, when pressure effects are included [@bondi52]. This tends to $$R_{\rm A}=\frac{2GM_{\rm p}}{v_{\rm p}^2},
\label{rahoyle}$$ for high Mach numbers [@hoyle39]. In Eq. \[rabondi\], $c_s$ is the sound speed. For simplicity we assume $\zeta=1$ always, which means that we will underestimate the gravitational drag force in the supersonic regime, and overestimate it in the subsonic one.
Our simulations do not reproduce the hydrodynamical drag, because the planet is approximated by a point particle and has no surface. Some hydrodynamic drag may be felt by the planet in the simulations due to the fact that some gas gathers in the potential well of the planet moving with it and in so doing it collides with surrounding gas. However, because of the relatively low mass of the planet, this effect is small in the simulations.
Ricker and Taam (2008) predicted that the hydrodynamic drag should be much weaker than the gravitational drag in common envelope simulations with stellar-mass companions. However, planets have much lower mass and a correspondingly weaker gravitational drag. As we show in Fig. \[multifigure\], towards the end of the RGB simulation the hydrodynamic drag should be comparable to or even dominate the gravitational drag including pressure effects (which is the relevant gravitational drag force at that time). At this point the simulations misrepresent the force on the planet, and we stop them. This does not happen in the AGB simulations, where the hydrodynamic drag should always be negligible compared to the gravitational drag force.
A Comparison Between Numerical and Analytical Expressions of the Gravitational Drag
-----------------------------------------------------------------------------------
In order to compare the analytical estimates of the gravitational drag (Eqs. 2, 3 and 4) with the actual gravitational drag experienced by the planet in the simulations, we need to device a way to extract this information from the simulation outputs. We calculate the difference in the planet’s energy (kinetic plus gravitational potential energies) between two successive snapshots from the simulation. This difference is due to the gravitational drag force, which does work ($W$) on the planet. This force is approximately anti-parallel with the planet’s motion, and its magnitude is therefore given by: $$F_{\rm drag,code}=W/s,$$ where $s$ is the distance travelled by the planet between two snapshots. We estimate $s$ by taking the velocity of the planet at the first snapshot and multiplying it by the time between the snapshots. The resulting drag force is plotted alongside the other relevant quantities in Figs. \[fitsfigure\] .
This estimate is approximate but reasonably accurate. We checked that this estimate of the force is similar to what would result from determining the orbit-averaged radial position of the planet at each time step, thereby determining the force by calculating the second differential of that radial distance and multiplying by the planet’s mass. Another method is to read the total acceleration on the planet from the code output. This method is more noisy because the total value of the acceleration includes the dominating centripetal value, which needs to be subtracted from the total.
The values of the orbital energy of the planet and of its velocity vary between one output frame and the next (see Fig. \[fitsfigure\]). The planet’s total energy decreases but occasionally it grows slightly between two snapshots, which results in a drag force that is instantaneously negative. In addition, the planet’s velocity can vary by up to $25$ per cent between snapshots for the AGB simulation (see Fig. \[fitsfigure\], where we plotted the planet’s velocity with respect to the surrounding gas, in contrast to in Fig. \[multifigure\], where we plotted the planet’s velocity relative to the grid).
To eliminate the oscillations we fitted the total energy, as well as the planet’s velocity and use the fitted curves to determine the value of the gravitational drag force. The force curve has an upturn at the end of the curve, which is an edge effect inherited by the fit to the planet’s velocity (middle panel in Fig. \[fitsfigure\]). In this figure, we only show values for the lower resolution simulations. More details about the fits are provided in the appendix.
The gravitational drag force values for both the lower and higher resolution simulations calculated with the method we have explained above, are also shown in the bottom two panels of Fig. \[multifigure\] (black curve), where they are compared to the hydrodynamic drag force (blue curve) from Eq. \[hydrodrageq\], and to the analytically-derived gravitational drag force (supersonic case: solid red curve, or including pressure effects: dashed red curve). We plot the gravitational drag force both including and excluding pressure effects, because the planet is supersonic until $\sim$800 days in the lower resolution RGB simulation and $\sim$55 years in the lower resolution AGB simulation (see the second panel in Fig. \[multifigure\]).
Both the expressions in Eqs. \[hydrodrageq\] and \[gravdrageq\], depend on the density surrounding the planet and the velocity of the planet with respect to the surrounding gas. These are plotted in the 4th and 5th rows of Fig. \[multifigure\]. Initially, the planet starts just outside the star at $1.1 R_{\rm star}$, and the density surrounding it is the low ambient background density. However, as the giant star expands, the planet finds itself embedded in higher density material. The few oscillations seen in the density surrounding the planet at $\sim$200 days in the RGB simulation and $\sim$20 years in the AGB simulation are due to the planet acquiring a slight eccentricity, or because the star in our simulations is not entirely spherical at this point in time, and so the planet may encounter different densities even if it is in a circular orbit. As the planet in-spirals through the star’s envelope, the density gradually increases, to reach a maximum at the end of the simulation of $\approx
10^{-4}~{\rm g~cm^{-3}}$ after $\sim$1000 days, in the RGB simulation, and $\approx 10^{-6}~{\rm g~cm^{-3}}$ after $\sim$80 years, in the AGB simulation.
The gravitational drag force calculated from the simulations follows closely the pace of the in-spiral from which it is calculated. It increases during the fast in-spiral phase, between 600 and 800 days to $\sim
1-2\times10^{31}~{\rm dyne}$ in the RGB simulation, and between 30 and 50 years to $\sim 1-2\times10^{29}~{\rm dyne}$ in the AGB simulation. This leads to an acceleration of the planet due to the drag of approximately $-1~{\rm cm~s^{-2}}$ in the RGB case, and $-0.01~{\rm cm~s^{-2}}$ in the AGB case. The difference in drag force in the RGB and the AGB simulations is primarily due to the different densities encountered by the planet. The peak force in the higher resolution simulations is approximately a factor of two larger than in the lower resolution simulations (see Fig. \[multifigure\]).
Looking at the drag forces in the last panel of Fig. \[multifigure\] – for the high resolution simulations, we see that the computationally-derived force is 2-3 times larger than the analytically-calculated [*supersonic*]{} gravitational drag force for both RGB (at around 600 days) and AGB (between 40 and 50 years) simulations, due either to the use of $\zeta = 1$ in Eq. 4, or to an actual effect of the density gradient exerting an added component to the force [@MacLeod2015]. Following this increase, the drag force then decreases becoming the same as the expression for the gravitational drag including pressure effects at 820 days and 65 years for the RGB and AGB simulations, respectively. The force values peak when the planet’s Mach numbers are slightly larger than unity in both simulations. We interpret this by looking at Eq. \[gravdrageq\] and noticing that in the supersonic regime, $R_{\rm A}$, and the gravitational drag force, can be seen to increase with decreasing velocity (smaller Mach numbers). However, when the force transitions to the sonic case, $\zeta$ is smaller, and hence the force decreases. Although a quantitative comparison cannot be carried out, this behaviour is that predicted by @Ostriker1999: the force is largest at, or slightly above the sonic point and drops dramatically just below Mach-1. In these simulations we conclude this is what causes the sudden decrease in drag force that makes the in-spiral slow down (see Fig. \[multifigure\]).
We finally note that the particle representing the core of the giant also experiences a drag force. However, this particle moves very slowly, much slower than the local sound speed. As the particle is very massive, it affects its surroundings significantly, for instance by attracting mass. It may therefore be difficult to accurately determine the relative velocity between the particle and its surroundings. We have not made any attempts to calculate the force acting on it.
The dependence of simulated gravitational drag on resolution
------------------------------------------------------------
Presumably the difference between lower and higher resolution simulations has something to do with the strength of the interaction which takes place in the vicinity of the planet. It is likely that the strength of the interaction is not well represented, for example, when $R_{\rm A}$ is not well resolved. In our higher resolution simulations $R_{\rm A}$ is a factor 3-8 smaller than the cell size, while in the lower resolution simulations it is a factor 5-15 smaller than the cell size (the range is due to the fact that $R_{\rm A}$ varies, while the cell size is fixed). In principle a convergence test carried out over multiple resolutions with a range of smoothing lengths could identify a problem, but due to the computational expense of these simulations, only limited resolution tests are carried out. When limited tests are carried out several effects can counter each other and confuse the issue of whether the gravitational drag is well represented.
We have performed several test simulations of a point mass moving through a constant density medium in order to eliminate the density gradient, which can complicate matters, and to reduce the computational expense of the tests. In these test simulations we have varied the density, thermal energy (and hence the sound speed), and the particle’s velocity. In this way we have been varying $R_{\rm A}$ so as to make it larger or smaller than the cell size. We found that when $R_{\rm A}$ is under-resolved, the drag force acting on the particle in the simulation tends to be overestimated compared to the analytical expression.
Contrary to this expectation, we find that the peak drag force is a factor $\approx2$ [*higher*]{} in the high resolution simulations than in the low resolution simulations. It is possible that this may be due to the density structure of the 1D [mesa]{} model in the 3D grid being more compact and have higher density in the higher resolution (Fig. \[multifigure\] fifth panel from the top), compared to the lower resolution simulations. The more compact and dense giant star in the higher resolution simulation means that not only the peak drag force, but the drag force acting on the planet in general is larger than at lower resolution. This may make up for the possible force under-estimation due to not resolving $R_A$.
{width="45.00000%"} {width="45.00000%"}
Discussion {#summarysection}
==========
The intensity, timescale and frequency of planet merger transients
------------------------------------------------------------------
We start by examining the interaction timescales. We find that the planet in-spirals relatively fast (few years for RGB stars and $\sim$100 years for the AGB case), although this is slow compared to CE interactions with more massive companions [e.g., @ricker12; @passy12; @DeMarco2012]. The hydrodynamic drag was not modelled, but we found that it could play a role at the end of the RGB interaction and this could shorten the in-spiral timescale somewhat. Pre-empting our discussion in Section \[ssec:destroying\] the planet will be eventually destroyed on a timescale that is likely of the same order of magnitude as the one characterising the initial in-spiral.
During the time of the in-spiral, the photosphere expands and the star likely brightens. Using the values of the density at the photosphere from the <span style="font-variant:small-caps;">mesa</span> models ($\approx9\times10^{-9}~{\rm g~cm^{-3}}$ and $1.6\times10^{-9}~{\rm g~cm^{-3}}$ for the RGB and AGB stars, respectively) we find that the interaction with the planet has caused the RGB star to expand by $\approx 40$ per cent over 3 years, and the AGB star to expand by $\approx 20$ per cent over 80 years. If the temperature of the photosphere remains constant, this would indicate a modest increase in luminosity by a factor of two for the RGB star, and $\approx40$ per cent for the AGB star. The temperature however, may decreases somewhat, as is demonstrated by Mira stars that can double their radius and halve their effective temperature over pulsation cycles of a few $\times$100 days (e.g., $o$ Cet; @Ireland2008 [@Ireland2011]). Additional cooling of the photosphere may be expected in the case of the AGB star. If we accounted for a decreasing temperature linearly inverse to the increase in radius then the luminosity would actually drop. It is possible, on the other hand, that the photosphere would be farther out than we have considered because of the low density material that readily expands out. Unfortunately we cannot integrate the optical depth of the material because its temperature is affected by the artificially large “vacuum" temperature used in [enzo]{}, making these tenuous outer layers more optically thick than they should be.
The average thermal timescales of the stars are 7600 and 30 years for the RGB and AGB stars, respectively. The RGB simulation ends at $\approx$4 [R$_\odot$]{}, and this is likely an upper limit. It is entirely possible that with a higher resolution, the “destruction depth" of $\sim 1$ [R$_\odot$]{}(Sec. \[ssec:destroying\]) would be reached within similar timescales. For the AGB star this is less likely. However its thermal timescale is much shorter and of the order of the in-spiral timescale. It is therefore possible that the AGB star would contract on the same timescales as it is expanding because of the injected orbital energy. If this happened, it is likely that the in-spiral would continue. We posit therefore that both interactions would result in the planet destruction within a timescale that is of the same order of magnitude of the in-spiral timescale.
Assuming that there are $10^{10}$ stars in the Galaxy that are able to evolve off the main sequence over the age of the universe, and that they have an average lifetime of 10 billion years, then we would have $10^7$ RGB and $10^6$ AGB stars at any given time in the Galaxy (using RGB and AGB lifetimes of 10 and 1 million years, respectively - see @Moe2006 for references to this back of the envelope calculation). Given the planet-swallowing timescales determined in this work, this would mean that one RGB star in a million would be undergoing an interaction with a companion if all RGB star went through one such interaction in their lives. For the AGB it would be one star in 10000 if they too went through one such interaction in their lives. This would mean that 10 RGB stars and 100 AGB stars in the Galaxy would be going through such interaction at any one time. These predictions are similar to what could be surmised by the considerations of @metzger12 who discuss that the rate of planet-main sequence star merger should be similar to the rate of planet-giant star merger, both approximately a few per year.
Given the long brightening and dimming timescales (relatively to any survey timescale) and the relatively small variation amplitude predicted, these phenomena may not be observable, unless a more powerful outburst could be triggered [@soker91; @Bear2011]. This would be quite different from the case of a planet-main sequence star merger discussed by @metzger12. Alternatively, as discussed in @nordhaus06, if the planet is tidally disrupted, it can form a disc deep inside the star which can lead to a disc driven outflow.
Very little mass is unbound from the system. Energy loss due to non-conservation may have decreased the mass-loss rate from the AGB star somewhat, but this could not be said of the RGB star for which the total energy slightly increased due to lack of perfect conservation. Additionally, non-simulated effects that could increase the mass-loss rate may be the interference, particularly for the case of the AGB star, of the orbital period with the fundamental pulsation period of the star. If little or no mass is ejected from the system due to the interaction, it is likely that the stars will settle back into an equilibrium stage after radiating their excess energies over their thermal timescales of 7600 and 30 years, for the RGB and AGB stars, respectively.
The interaction caused the puffed-up, low density, outer layers of the star to rotate, with velocities $>20~{\rm km~s^{-1}}$ in the RGB star and $>5~{\rm km~s^{-1}}$ in the AGB star (the extra angular momentum transported by a planet captured tidally would only change these values slightly). At higher densities, we found no significant rotation. This could indicate that the differential rotation mediated dynamo effect suggested in @nordhaus06 will not lead to large scale outflows. We expect that as the interaction ends and the star settles back into its original configuration, and the angular momentum is redistributed in the star, the surface rotation would slow down. Hence, an apparently relatively fast spinning giant star for a brief period could be an indication of a recent CE interaction between the star and a giant planet.
@carlberg09 investigated the ability of planet accretion to spin up stars, and found that in some cases RGB stars could become rapid rotators due to merger with a companion planet although they found that fast rotation was more likely to be achieved if the planet was captured by a sub-giant, as stronger mass loss from giants can remove angular momentum from the envelope preventing the rapid rotation. If this happened, the giant would be slowly rotating or not rotating at all. Based on our simulations we suggest that the CE event is still capable of causing a rapid rotation in the outer puffed up envelope, as the CE interaction is fast and mass loss therefore can not remove angular momentum sufficiently fast to prevent the spin-up.
Destroying Planets and Polluting Giant Stars {#ssec:destroying}
--------------------------------------------
RGB star AGB star
--------------------------------- ---------- ----------
Mass ([M$_\odot$]{}) 3.5 3.0
Envelope mass ([M$_\odot$]{}) 3.0 2.5
Hydrogen mass ([M$_\odot$]{}) 2.1 1.75
Enrichment@\[Fe/H\]$_\odot$ (%) 0.8 1
Enrichment@\[Fe/H\]=-1.7 (%) 43 50
: Increase in the mass fraction of iron assuming that the destroyed planet has a core made of iron with a mass of 10 M$_\oplus$.[]{data-label="table1"}
We assume that at some point the planet will be destroyed in the envelope of the giant star. During the in-spiral there are competing processes that try to disrupt the planet. These act on different timescales and vary with depth. The planet can be (i) disrupted by shear between its outer layers and the stellar ambient density, (ii) it can he ablated by heating and (iii) it can be tidally disrupted. We find that the planet is stable against Kelvin-Helmholtz and Rayleigh-Taylor instabilities caused by shear [discussed in @passy12planet] for the conditions prevailing during our simulations. We find that a $10~{\rm
M_J}$ planet will be ablated by heating when the separation between the planet and the core of the giant star is $\sim1~{\rm R_\odot}$ [@soker98]. This is also the distance from the stellar core at which the planet will overflow its Roche lobe. This is a much smaller separation than the values of 10 and 85 [R$_\odot$]{} reached at the end of our RGB and AGB simulations, respectively. Therefore we presume that this event has not yet taken place, but will in time (a time possibly commensurate with the time for the early in-spiral).
Next we ask whether massive planets such as those we have simulated, once destroyed at $\sim$1 [R$_\odot$]{} can alter the giant composition in an observable way. The masses and compositions of the cores of massive exoplanets are poorly known [especially for hot Jupiters; for a recent review, see @spiegel14]. We assume that a $10~{\rm M_J}$ planet consists mainly of an atmosphere of hydrogen and helium in solar proportions, and of a core with an iron mass of $m_{\rm Fe}=10~{\rm M_\oplus}=3\times10^{-5}~{\rm
M_\odot}$ [@guillot99]. The base of the convective region in our <span style="font-variant:small-caps;">mesa</span> RGB model is at $\approx0.4~{\rm R_\odot}$ while for the AGB star it is at $\approx0.2~{\rm
R_\odot}$; both are deeper than the location at which we predicted the planet to be destroyed. The disrupted planet mass will therefore quickly be mixed into the giant stars’ envelopes due to convection.
In Table \[table1\] we list the RGB and AGB star masses, envelope masses, and hydrogen masses for a hydrogen mass fraction of 70 per cent. For a Solar metallicity [$\epsilon_{\rm Fe}=7.47$ for the Sun or $m_{\rm Fe}/m_{\rm H}\approx 0.0017$; @scott15] we therefore see that the added iron from the planet increases the envelope metallicity too little to be observed.
If we assumed that the iron mass fraction has to grow by at least a factor of 1.5 to be discerned from the base metallicity of the star, then the base metallicity of the star should be \[Fe/H\]$<-1.7$ in the AGB case. A giant with a mass of $\sim$1 [M$_\odot$]{} and an envelope mass of 0.5 [M$_\odot$]{} would enable us to detect the pollution more readily at higher, but still sub-solar metallicities (\[Fe/H\]$<-1.3$).
Since there appears to be a correlation between a planet’s metal fraction (i.e., core mass in a gas giant) and the metallicity of the host star [@guillot06], it may be that such low metallicity stars cannot harbour metal rich planets. On the other hand, there may also be considerable variability in the metal content of planets. For instance, the planet HD 149026b is thought to contain $60-93~{\rm M_\oplus}$ of heavy elements [@fortney06], much more than we have considered above. However, even such large core mass would not be able to noticeably alter the observed metallicity of a Solar metallicity star.
Another possibility for getting metal enrichment in AGB stars was discussed by @soker92, who studied common envelope interactions between AGB stars and brown dwarfs, and suggested that for separations between $3-10~{\rm
R_\odot}$, the brown dwarf would excite gravity waves that could lead to a spin up of the inner envelope. This could also lead to mixing near the core, causing extra dredge-up of core material into the envelope. Hence, if this process happened near the last stages of mass loss, the wind of the AGB star would be enriched in heavier elements. However, this star would be a much more evolved AGB star than the one we have considered in this work, and this mechanism requires that the companion enters the common envelope only at the very late stages of AGB evolution.
Summary
=======
We have simulated the CE interaction between a $10~{\rm M_J}$ planet and a 3.5 [M$_\odot$]{} RGB star or a 3.05 [M$_\odot$]{} AGB star using the grid code Enzo with a uniform, cubic grid with a maximum resolution of 512 cells on a side. These simulations have several limitations, but can give order-of-magnitude quantitative information.
The limited resolution in our simulation affects the final separation of our simulations, and some of the results from late times in our simulations may not be accurate. Another effect of the resolution is that the accretion radius is not resolved, which can lead to an overestimate of the force. However, in lower resolution simulations the star diffuses out more leading to lower densities which can cause an underestimate of the force, somewhat counteracting the overestimate from not resolving the accretion radius. Future simulations using an adaptive mesh refinement simulation code, or possibly a smoothed particle hydrodynamics code may be able to overcome some of these limitations. We nevertheless found that:
- Plunge-in times of the order of years to decades are seen in our simulations for the RGB and AGB cases, respectively. The plunge-in times of low mass companions such as planets in the envelopes of giants are relatively longer than for more massive, stellar companions, with the longer times being witnessed for the more evolved, lower density primaries.
- We concluded that the planets should not be disrupted during the simulated phase. We cannot tell with precision how much longer the planets will take to reach a depth where disruption takes place.
- Destroyed planets will pollute the envelopes of giant stars, but the effect is likely to be witnessed only in the lowest mass giants with the lowest metallicity, if these stars can have planets with suitably massive metal cores.
- Only a very small amount of the primary star’s envelope mass is unbound by the planet in our simulation. It is possible that if the planet interacts with the star’s pulsation this may trigger further unbinding, or, if the planet is tidally disrupted it can form a disc inside the giant star from which a disc driven outflow can form.
- The expanding giant’s luminosity may increase by a modest factor over a relatively short timescale of the early in-spiral (though still long compared to survey timescales). This effect would likely be relatively rare and difficult to observe.
- In line with other studies we find that the penetration of the planets into the giants will stimulate faster rotation. However as this rotation is limited to the outer layers, it is not clear in what timescales the angular momentum will re-distribute into the entire envelope and what the final rotation rate of the giants will be.
- Analytically, it is predicted that the gravitational drag force would peak at the sonic point and greatly diminish for sub-sonic regimes. In our simulations the slowing down of the in-spiral takes place at such a transition. The overall force experienced by the planets in our simulations is larger than calculated analytically and is larger for higher resolution. This may simply be due to us assuming $\zeta=1$ in Eq. \[gravdrageq\]. It is also possible that the presence of a density gradient may enhance the intensity of the gravitational drag. We leave further comparisons between numerical and analytical gravitational drag to future work.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for constructive comments which helped improve the paper. We thank B. Pandey and M. Wardle for constructive discussions during this work, and C. O’Neill for helpful input. J.E.S acknowledges support from the Australian Research Council Discovery Project (DP12013337) program. O.D. gratefully acknowledges support from the Australian Research Council Future Fellowship grant FT120100452. J.-C.P. acknowledges funding from the Alexander-von-Humboldt Foundation. This research was undertaken, in part, on the NCI National Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government. Computations described in this work were performed using the Enzo code (http://enzo- project.org), which is the product of a collaborative effort of scientists at many universities and national laboratories.
[0]{}
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Fitting results
===============
For both of the simulations, we fit the planet’s velocity and the planet’s total energy, to get smooth curves. We have not attempted to estimate the $\chi^2$ of the fits. Instead, we have shown in Fig. \[fitsfigure\] the data and the fitted curves from the low resolution simulations, and limit ourself to stating that qualitatively the fits look reasonable. In this appendix we show the details of the fits and the results for the two low resolution simulations.
RGB case
--------
We fit the planet’s velocity with a fifth order polynomial ($ax^5+bx^4+cx^3+dx^2+ex+f$) over the entire data-range from 0 to 980 days. The result of the fit is: $a=9.85028\times10^{-8}$, $b=-0.000237$, $c=0.18886$, $d=-56.0575$, $e=7229.48$, and $f=1.04192\times10^7$. The planet’s negative total energy was fit with two different curves. To ensure a reasonably continous fit with a reasonably continuous first derivative, we fit the curves over a larger range than we plot them. From 0 to 560 days we use a third order polynomial ($f_1(x)=a_1x^3+b_1x^2+c_1x+d_1$) where we found the coefficients to be $a_1=1.93253\times10^{36}$, $b_1=3.2207\times10^{38}$, $c_1=2.28898\times10^{39}$, and $d_1=1.07075\times10^{45}$. This was used to plot the curve from 0 to 510 days. Then from 400 days to 980 days we used a sixth order polynomial ($f_2(x)=a_2x^6+b_2x^5+c_2x^4+d_2x^3+e_2x^2+f_2x+g_2$), where we found the coefficients to be $a_2=4.21665\times10^{30}$, $b_2=-1.73393\times10^{34}$, $c_2=2.89879\times10^{37}$, $d_2=-2.51962\times10^{40}$, $e_2=1.20186\times10^{43}$, $f_2=-2.98564\times10^{45}$, and $g_2=3.03351\times10^{47}$. This was plot from 510 days to 980 days.
AGB case
--------
We fit the planet’s velocity to a fifth order polynomial ($ax^5+bx^4+cx^3+dx^2+ex+f$) over the entire data-range from 0 to 77.5 years. The result of the fit is: $a=0.00832347$, $b=-1.52515$, $c=96.5916$, $d=-2287.46$, $e=22336.7$, and $f=3.35088\times10^6$, with 150 degrees of freedom. The planet’s negative total energy we fit with three different curves. To ensure a reasonably continuous fit with a reasonably continuous first derivative, we fit the curves over a larger range than we plot them. From 0 to 35 years we fit a first order polynomial ($f_1(x)=ax+b$) and results in $a=5.28612\times10^{41}$ and $b=1.05581\times10^{44}$. This we used to plot from 0 to 27 years. From 5 to 70 years we fit a sixth order polynomial ($f_2(x)=ax^6+bx^5+cx^4+dx^3+ex^2+fx+g$) which results in $a=-1.22317\times10^{35}$, $b=2.12089\times10^{37}$, $c=-1.30396\times10^{39}$, $d=3.64203\times10^{40}$, $e=-4.51527\times10^{41}$, $f=2.24252\times10^{42}$, and $g=1.07388\times10^{44}$, which was plotted from 27 years to 55.5 years. Finally, from 45 to 77.5 years we fit an eight order polynomial ($f_3(x)=ax^8+bx^7+cx^6+dx^5+ex^4+fx^3+gx^2+hx+i$), which was fitted from 45 to 77.5 years and results in: $a=3.45742\times10^{31}$, $b=-8.18603\times10^{33}$, $c=7.09231\times10^{31}$, $d=-2.71983\times10^{37}$, $e=4.27398\times10^{38}$, $f=5.97299\times10^{22}$, $g=1.12244\times10^{21}$, $h=2.26162\times10^{19}$, $i=1.24344\times10^{16}$. This was plotted from 55.5 years to 77.5 years.
[^1]: The kinetic energy and the gravitational potential energy of the low density ambient medium are negligible compared to the thermal energy.
[^2]: The total energy of the star is almost identical to its potential energy, because kinetic and internal components are not very large.
|
---
abstract: 'Investigating the atmospheres of rocky exoplanets is key to performing comparative planetology between such worlds and the terrestrial planets that reside in the inner Solar System. Terrestrial exoplanet atmospheres exhibit weak signals and attempting to detect them pushes at the boundaries of what is possible for current instrumentation. We focus on the habitable zone terrestrial exoplanet LHS 1140b. Given its 25-day orbital period and 2-hour transit duration, capturing transits of LHS 1140b is challenging. We observed two transits of this object, approximately one year apart, which yielded four data sets thanks to our simultaneous use of the IMACS and LDSS3C multi-object spectrographs mounted on the twin Magellan telescopes at Las Campanas Observatory. We present a jointly fit white light curve, as well as jointly fit 20 nm wavelength-binned light curves from which we construct a transmission spectrum. Our median uncertainty in $R_p^2/R_s^2$ across all wavelength bins is 140 ppm, and we achieve an average precision of 1.28$\times$ the photon noise. Our precision on $R_p^2/R_s^2$ is a factor of two larger than the feature amplitudes of a clear, hydrogen-dominated atmosphere, meaning that we are not able to test realistic models of LHS 1140b’s atmosphere. The techniques and caveats presented here are applicable to the growing sample of terrestrial worlds in the *TESS* era, as well as to the upcoming generation of ground-based giant segmented mirror telescopes (GSMTs).'
author:
- 'Hannah Diamond-Lowe'
- 'Zachory Berta-Thompson'
- David Charbonneau
- Jason Dittmann
- 'Eliza M.-R. Kempton'
bibliography:
- 'MasterBibliography.bib'
title: |
Simultaneous Optical Transmission Spectroscopy of a Terrestrial, Habitable-Zone Exoplanet\
with Two Ground-Based Multi-Object Spectrographs
---
Introduction {#sec:intro}
============
Planetary atmospheres hold clues about surface processes, formation histories, and the potential for habitability for the planets they surround. Under the right circumstances, they can also reveal the presence of life on other worlds via biomarker gases. In the Solar System we see a great diversity of atmospheres, from the puffy hydrogen and helium envelopes around Jupiter and Saturn to the heavy carbon dioxide layer around Venus and the nitrogen-rich sky of Titan.
The terrestrial bodies of the Solar System boast a wide variety of atmospheric compositions and masses, but all are secondary, high mean molecular weight atmospheres. Results from the *Kepler* mission, combined with statistical and empirical follow-up, reveal that such worlds also exist in abundance outside our solar system, along with a completely new kind of terrestrial planet that has retained a hydrogen- and helium-dominated envelope [@Fressin2013]. For planets with radii $<10\ R_{\oplus}$, those with radii $>1.6\ R_{\oplus}$ have low bulk densities and likely host puffy hydrogen and helium envelops captured from the stellar nebula, while those with radii $<1.6\ R_{\oplus}$ are rocky in nature and likely host high mean molecular weight secondary atmospheres , though given the difficulties in detecting secondary atmospheres around small planets, we have not yet spectroscopically characterized any. The $1.6\ R_{\oplus}$ mark is not a hard cut-off. Another way to look at this is that planets with bulk densities less than that of rock ($2.5-3.0$ g/cm$^3$) must have significantly large envelopes of hydrogen and helium in order to explain their low masses relative to their radii, whereas planets with bulk densities at or above that of rock are likely compositionally similar to the terrestrial objects found in the Solar System.
To understand the rocky exoplanets we must probe their atmospheres and determine their compositions. In this paper we focus on the technique of transmission spectroscopy, whereby observations of a planet’s transit across its star, taken over a range of wavelengths, can reveal the planet’s atmospheric composition, since different molecules absorb stellar light at different wavelengths. Within the limits of current instrumentation we begin the exploration of small planet atmospheres by looking for small planets that orbit the small stars closest to us. This is a simple function of the planet-to-star radius ratio $R_p/R_s$ (the larger the ratio, the easier it is to detect the planet) and the need for high signal-to-noise measurements to differentiate the planet radius at one wavelength from another (the closer the star, the more photons can be collected per observation).
Before the launch of the *Transiting Exoplanet Survey Satellite* [*TESS*; @Ricker2015], the ground-based transit surveys MEarth and TRAPPIST [@Nutzman2008; @Gillon2013; @Irwin2015] discovered a handful of small planets around three small, nearby stars: GJ 1132, TRAPPIST-1, and LHS 1140, which follow-up observations by the *Spitzer Space Telescope* and *K2* confirmed and, for the TRAPPIST-1 and LHS 1140 systems, bolstered with additional planet discoveries [@Berta-Thompson2015; @Gillon2017a; @Dittmann2017a; @Dittmann2017b; @Ment2019]. Follow-up by radial velocity instruments such as the High Accuracy Radial-velocity Planet Searcher [HARPS; @Pepe2004] provided masses for planets in the GJ 1132 and LHS 1140 systems, thereby confirming their rocky natures. HAPRS also discovered an additional non-transiting planet in the GJ 1132 system [@Bonfils2018]. In the case of the two nearest terrestrial planets HD 219134 b,c [@Gillon2017b], their presence was detected via radial velocities from HARPS-North [@Cosentino2012], and were later found to transit by Spitzer. The dimness of TRAPPIST-1 makes radial velocity measurements challenging, so masses for the TRAPPIST-1 planets are instead estimated using transit timing variations [TTVs; @Wang2017], revealing that some of the TRAPPIST-1 planets may have bulk densities comparable to that of water. Now in the era of *TESS* the sample of small planets orbiting small ($<0.3\ R_{\odot}$), nearby ($<15$ pc) stars is growing, with LHS 3844b and LTT 1445Ab added recently [@Vanderspek2018; @Winters2019]
Though the presence of these small planets provides a tantalizing opportunity for atmospheric follow-up, the most we are able to do with current instrumentation is rule out the lowest mean molecular weight atmospheres dominated by hydrogen and helium, which confirms the aforementioned work on *Kepler* planets with radii $<10\ R_{\oplus}$. So far cloud-free low mean molecular weight atmospheres are ruled out for TRAPPIST-1b,c,d,e,f and for GJ 1132b [@deWit2016; @deWit2018; @Diamond-Lowe2018].
With the goal of eventually detecting atmospheric biomarkers on habitable zone worlds, we designed a project to determine the capabilities and limitations of current instrumentation to detect and characterize the atmosphere LHS 1140b [@Dittmann2017a], to date the only known habitable zone terrestrial exoplanet orbiting a star bright enough for us to employ the technique of transmission spectroscopy. Since the planet’s discovery, Data Release 2 of the Gaia mission [@GaiaMission2016; @GaiaDR22018] moved LHS 1140 farther away, to $14.993 \pm 0.015$ pc. This means that the stellar radius of LHS 1140 is larger than initially thought, which in turn increases the derived planet radius. With this new information, we find that LHS 1140b has a radius of $1.727 \pm 0.032\ R_{\oplus}$ and a mass of $6.98 \pm 0.89\ M_{\oplus}$, making its density of $7.5 \pm 1.0$ g/cm$^3$ consistent with a terrestrial composition [@Ment2019]. The planet’s surface gravity is $23.7 \pm 2.7$ m/s$^2$ with an estimated effective temperature ($T_{eff} = 235 \pm 5$ K), assuming an albedo of zero. The atmospheric scale height of a planet is directly proportional to the planet’s temperature, and inversely proportional to its surface gravity. In the case of LHS 1140b, its atmospheric scale height, and therefore the amplitudes of its atmospheric features, are below what it detectable from our observations.
We note that when we began this project, we assumed a lower surface gravity for the planet. This came about because the initial mass and radius estimates of LHS 1140b gave a bulk density consistent a composition of more than $50$% iron, which is implausible and in stark defiance of conventional planetary formation scenarios [@Zeng2016; @Dittmann2017a]. As such, it seemed likely that the mass of LHS 1140b would be refined and lowered in a subsequent season of radial velocity measurements (Figure 2 of @Morley2017 provides an illustration of this thinking). Once additional measurements were made, the surface gravity of LHS 1140b decreased, but not by as much as we had predicted.
Despite the difficulty involved in detecting the atmosphere of LHS 1140b, it is one of the few terrestrial planets orbiting a nearby M star for which liquid water could potentially exist on the planet surface. However, equilibrium temperature is not the sole determinant for habitability. M stars like LHS 1140 spend more time in the pre-main sequence phase than G stars like the Sun before settling onto the main sequence branch [@Baraffe2002; @Baraffe2015]. This means that M stars have longer periods of high energy activity which can strip the atmospheres of the planets orbiting them [@Luger2015]. However, some high-energy flux, particularly in the near-ultraviolet (NUV), may be necessary to jump-start life [@Ranjan2017].
@Spinelli2019 use the UV and X-ray capabilities of the space-based *Swift* observatory to investigate the high-energy nature of LHS 1140. They find that while LHS 1140 exhibits low levels of UV activity, its relatively high ratio of far-ultraviolet (FUV) to NUV flux could produce O$_2$ and H$_2$O abiotically through the dissociation of CO$_2$. The low amounts of NUV received by LHS 1140b (2% the amount that Earth receives) may not provide enough of a spark for abiogenesis. However, these are current measurements of LHS 1140 do not represent its past levels of UV radiation. Detecting the atmosphere around LHS 1140b would provide a clue to past behavior of LHS 1140, and vice versa.
In this work, we are not ultimately able to investigate the atmosphere of LHS 1140b, illustrating the limitations of current instrumentation and the need for better habitable zone terrestrial planet targets. LHS 1140b has an orbital period of $24.736959 \pm 0.000080$ days and a transit duration of 2.12 hours [@Ment2019], making transits of this object rare and difficult to observe due to the 6 hours of observing time necessary to capture both the transit and adequate baseline on either side from which to measure the depth. *Spitzer* observed transits of LHS 1140b in its 4.5 $\mu$m broadband photometric bandpass [DDT Program 13174, PI Dittmann; @Ment2019 Dittmann, et. al, *in prep*]; this infrared point complements the optical observations we undertake here.
In this paper we present our observing program in Section \[sec:obs\]. We detail our data extraction process, along with an illustrative diagram in Section \[sec:extract\]. We then detail the analysis of our extracted spectra in Section \[sec:analysis\]. The results of this work, along with a discussion of their implications, are presented in Section \[sec:results\], followed by our conclusions in Section \[sec:conclusion\].
Observations {#sec:obs}
============
Given the period (24.7 days) and transit duration (2.1 hours) of LHS 1140b, opportunities to observe a complete transit of this object from Las Campanas Observatory in Chile, where we could also capture data before and after transit, were rare. However the 2-hour transit duration offers the advantage that a single transit observation yields a high signal-to-noise measurement of the transit depth. In 2017 and 2018, there was one opportunity per year to observe a complete transit of LHS 1140b, along with baseline before and after transit.
We were awarded two nights on the Magellan I (Baade) and Magellan II (Clay) telescopes through the Center for Astrophysics $\vert$ Harvard & Smithsonian (PI Diamond-Lowe) to simultaneously observe the 2017 and 2018 transits of LHS 1140b with both telescopes. We used the IMACS and LDSS3C multi-object spectrographs on Baade and Clay, respectively, to observe the transit across the optical and near infrared spectrum. We were able to capture both transits, yielding a total of four data sets (two with IMACS and two with LDSS3C) for our project. The details of these observations are presented in Table \[tab:obs\].
[ccccccC]{} Data Set & Date & Exposure Time & Duty Cycle & Number of & Minimum &\
(Instrument, Year) & (UTC) & (s) & (%) & Exposures & Airmass &\
IMACS 2017\* & 2017-10-27, 00:37:10 – 07:09:02 & 15 & 32.3 & 510 & 1.029 & 0.60\
LDSS3C 2017 & 2017-10-27, 00:28:15 – 07:14:14 & 15 & 46.9 & 766 & 1.029 & 0.80\
IMACS 2018 & 2018-11-02, 00:34:47 – 07:15:41 & 15 & 32.3 & 508 & 1.029 & 0.50\
LDSS3C 2018 & 2018-11-02, 01:10:51 – 07:11:45 & 15 & 46.9 & 686 & 1.029 & 0.40\
When designing these observations we wanted to keep as many aspects in common as possible between the LDSS3C and IMACS instruments so as to minimize the systematic differences between the two. The field of view of LDSS3C is 8.3$'$, while the f/2 camera on IMACS has a field of view of 30$'$. The field of LHS 1140 is relatively sparse. Fortunately, there is a comparison star, 2MASS J00450309-1518437, located 145.34$''$ away (Figure \[fig:ds9regions\]). This main-sequence G-type star is non-variable in the MEarth photometry down to the 1 mmag level (Jonathan Irwin, *priv. comm.*), and is brighter than LHS 1140. To compare, $T = 11.2991$ for LHS 1140, while $T = 10.5629$ for the comparison star, where $T$ stands for $TESS$ magnitude [@Stassun2019]. The TESS bandpass ranges from 600 - 1000 nm, which is exactly the bandpass in which we make our observations; the TESS magnitudes are therefore a useful basis of comparison between these stars of different spectral types. Because the comparison star is brighter, we are limited by the photon noise of LHS 1140, not the comparison.
To get the same wavelength coverage for LHS 1140 and the comparison star, we ideally want to orient our mask such that the two stars are aligned in the cross-dispersion (spatial) direction. However, there is a background star that was 16.5$''$ away from LHS 1140 during the observations. Lining up LHS 1140 with the comparison star would have placed this background star within a few arcseconds of the edge of the slit. To ensure that this background star did not contaminate the LHS 1140 spectrum by peaking in and out of the slit during observations, we oriented the LDSS3C and IMACS masks such that the spectra of LHS 1140 and the background star are dispersed parallel to each other, with the comparison star almost aligned in the cross-dispersion direction (Figure \[fig:ds9regions\]). Because LHS 1140 is a high proper motion star it will be necessary to re-check its position with respect to any background stars in future observations.
![On sky projection of the LHS 1140 field from the Dark Sky Survey (DSS), which is available in `SAOImageDS9` [@Joye2003]. The solid grey circle and square outlines are the mask and detector footprint, respectively, of the IMACS instrument. The dashed grey circle and rectangle outlines are the mask and detector footprint, respectively, of the LDSS3C instrument. Light blue rectangles are the IMACS science slits; the LHS 1140 and comparison star slits are marked. LHS 1140 is a high proper motion star. The yellow circle in the LHS 1140 slit shows the position of LHS 1140 at the time of the 2018 observations (Table \[tab:obs\]). We observe four other comparison stars with IMACS but do not use them in the analysis in order to minimize the difference between the IMACS and LDSS3C observations. Orange squares indicate the IMACS alignment holes. There is at least 50$''$ separation in the cross-dispersion direction between the IMACS alignment holes and the science slits in case we needed to model out-of-slit flux (see Section \[sec:analysis\]). For LDSS3C, the sizes of the LHS 1140 and comparison star slits are slightly shorter in the cross-dispersion direction than shown (see text, Section \[subsec:LDSS3Cobs\]). For clarity, we do not show the alignment star holes for LDSS3C. The grey-filled strip on the LDSS3C detector indicates a region of bad pixels where slits should not be placed.[]{data-label="fig:ds9regions"}](Figure_RegionsDS9_Labels_Rotated.pdf){width="46.50000%"}
Target Comparison
--------------- -------------- -------------------------
Name LHS 1140 2MASS J00450309-1518437
RA 00:44:59.33 00:45:03.09
Dec -15:16:17.54 -15:18:43.87
$V$ mag 14.15 11.01
$T$ mag\* 11.2219 10.5629
$J$ mag 9.612 9.975
Spectral type M4.5 G3
: Stars used in this work \[tab:stars\]
Magellan I (Baade) IMACS Observations {#subsec:IMACSobs}
-------------------------------------
The Inamori-Magellan Areal Camera & Spectrograph (IMACS) can perform both imaging and spectroscopy. Its detector is made up of eight CCDs which produce an 8192$\times$8192 pixel mosaic, or $27.5'\times27.5'$ ([IMACS User Manual](http://www.lco.cl/telescopes-information/magellan/instruments/imacs/user-manual/the-imacs-user-manual)). We use the f/2 camera, which has a 30$'$ field-of-view diameter. With this field-of-view we are able to capture five comparison stars, but we only use 2MASS J00450309-1518437 (Table \[tab:stars\]) in the analysis in order to be consistent with the LDSS3C observations.
Between the 2017 and 2018 observations we discovered large instrument systematics that led us to redesign our 2018 mask. These systematics and potential solutions are discussed in detail in Section \[subsec:IMACSextract\], but we present this new mask in Figure \[fig:ds9regions\]. The key improvements to the 2018 mask are 1) slits that are 70$''$ long in the cross-dispersion direction in order to estimate the sky background outside of the extended point-spread-function of the stellar spectra, and 2) ensuring that the area on either side (in the cross-dispersion direction) of the slits has no alignment holes in case we need to model and remove out-of-slit flux. The slit widths in the dispersion direction are 10$''$ to avoid light losses. We recommend that future users of IMACS for similar observations adopt these features when designing their masks. We also cut a calibration mask which is identical to the science mask except with slit widths in the dispersion direction of 0.5$''$.
For our detector settings we use `2\times2` binning and a `Fast` readout speed. These settings allow for a readout time of 15.6 seconds, making the duty cycle for these observations 49%. Gains and readout noise levels for each of the eight IMACS chips can be found in the [IMACS](http://www.lco.cl/telescopes-information/magellan/instruments/imacs/user-manual/the-imacs-user-manual#Mosaic_CCD_Cameras) user manual. During the afternoon prior to observations we use the science mask to take biases, darks, and quartz flats, and we use the 0.5$''$-slit calibration mask to take helium, neon, and argon arcs. During nighttime observations, we take a non-dispersed reference image of the LHS 1140 field with the science mask before and after the science observations. After the nighttime observations we take another set of biases and darks. The 16-bit analog-to-digital converter (ADC) has a saturation limit of 65,535 analog-to-digital units (ADUs), which we do not surpass for all pixels used in the data analysis. We note that with IMACS, the overscan region is sufficient for bias-level subtraction and dark current adds only a few e$^-$/hour. While biases and darks do not greatly affect our data reduction, taking enough flats is crucial. We were careful to collect at least as many photons in our quartz flats as we do in-transit photons of LHS 1140 in order to not be noise-limited by the flats.
For all observations requiring a disperser (i.e., flats, arcs, and science spectra), we use the Gri-300-26.7 grism (300 lines/mm with a blaze angle of 26.7$^{\circ}$). This grism has a wavelength range of 500-900 nm and a central wavelength of 800 nm. This gives a dispersion of 0.125 nm/pixel. With this grism we use the WBP 5694-9819 order blocking filter to mitigate any blue light that could cause second-order contamination in our spectra.
Magellan II (Clay) LSDD3C Observations {#subsec:LDSS3Cobs}
--------------------------------------
The Low Dispersion Survey Spectrograph (LDSS3C) has gone through several upgrades to make it more sensitive at redder wavelengths. The instrument has a f/11 focal ratio and a single CCD detector made up of 2048$\times$4096 pixels or $6.4'\times13'$ ([LDSS3C User Manual](http://www.lco.cl/Members/gblanc/ldss-3/ldss-3-user-manual-tmp)). The 8.3$'$ diameter field of view radius of LDSS3C means that 2MASS J00450309-1518437 (Table \[tab:stars\]) is the only comparison star we are able to observe simultaneously with LHS 1140. We cut our slits 10$''$ wide in the dispersion direction to avoid light losses as seeing and airmass change throughout the night. We cut the comparison star slit 20$''$ long in the cross-dispersion direction in order to capture enough photons to remove the sky background. We cut the LHS 1140 slit 30$''$ longer on one side to account for the background star near LHS 1140. We also cut a mask for wavelength calibrations, which is identical to the science mask except with slit widths of 0.5$''$ in the dispersion direction.
We present the alignment of our science mask on the sky in Figure \[fig:ds9regions\]. The LDSS3C detector suffers from some hot pixels, which can saturate and ruin a spectrum. We mark these pixels with a grey-filled rectangle over the LDSS3C detector.
Our detector settings are as follows: `2\times2` detector binning, `Fast` readout speed, and `Low` gain. We find that this allows for a 15.6 s readout time, bringing the duty cycle to 49%. Gains and readout noise can be found in the [LDSS3C](http://www.lco.cl/Members/gblanc/ldss-3/ldss-3-user-manual-tmp#section-12) user manual. Note that the `Low` gain setting actually refers to the inverse gain, and therefore allows for longer exposure times than the `High` gain setting. The full well depth of the detector is 200,000 e$^-$, with a linear pixel response up to 177,000 e$^-$ [@Stevenson2016a]. Like IMACS, the 16-bit analog-to-digital converter (ADC) of LDSS3C has a saturation limit of 65,535 analog-to-digital units (ADUs), which we do not surpass for all pixels used in the data analysis.
Using the science mask we take biases, darks, and quartz flats during the afternoon prior to observations. We also take helium, neon, and argon arcs using the 0.5$''$ calibration mask. During nighttime observations, we take a non-dispersed reference image of the LHS 1140 field with the science mask before and after the science observations. After the nighttime observations we take another set of biases and darks.
For all observations that require a disperser (i.e., flats, arcs, and science spectra) we use the VPH-Red grism which provides a wavelength coverage of 640-1040 nm (see @Stevenson2016a for details). The VPH-Red grism has a high throughput at redder wavelengths where LHS 1140, an M-star, is brightest. We use the OG590 order-blocking filter to mitigate order contamination introduced to the spectra by the VPH-Red grism.
Data Extraction {#sec:extract}
===============
In this section we discuss how we turn the raw IMACS and LDSS3C data – a time-series of `FITS` files containing 2D stellar spectra – into a time-series of 1D stellar spectra, for both LHS 1140 and the comparison star. This final product of the extraction will be the starting point of the data analysis (Section \[sec:analysis\]), where we investigate the planet radius of LHS 1140b at different wavelengths.
The process for extracting the IMACS and LDSS3C spectra of LHS 1140 and the comparison star is identical. We use the custom pipeline [`mosasaurus`](http://github.com/zkbt/mosasaurus) to perform the extraction. This pipeline has evolved from earlier versions [e.g., @Diamond-Lowe2018] and is now generalized for IMACS and LDSS3C. Though still specialized, this code is modular and may be useful to others performing multi-object transmission spectroscopy of exoplanets.
`mosasaurus` extraction steps {#subsec:mosasaurus}
-----------------------------
Turning raw images into a time-series of wavelength-calibrated 1D spectra is a long process. Here we outline the steps of our pipeline. A visual representation of the steps can be see in Figure \[fig:extract\].
{width="100.00000%"}
1. **Set-up** We read in the `FITS` files we need for the extraction. These are the darks, biases, quartz flats, arcs (helium, neon, and argon), undispersed reference images, and science images. Following the prescription of @Eastman2010, we convert the UTC time stamps recorded in the headers of these images into a single BJD$_{TDB}$ time stamp marking the middle of the exposure.
2. **Master images** For each type of image we stitch the raw `FITS` files together to create a coherent image for each of the input files. For IMACS, this results in a 4096$\times$4096 pixel image, and for LDSS3C, a 1024$\times$2048 pixel image (recall that we used 2$\times$2 binning on each instrument). In the process of stitching, we trim the bias overscan regions from each CCD chip (eight for IMACS, two for LDSS3C) and subtract their median in the cross-dispersion direction from the rest of the image. We then take an average of each image type to create the master images. We do this by comparing all of the images of a type and rejecting outliers that deviate by 5$\times$ the median absolute deviation (MAD), and then taking the mean of the images. We refer to this rejection of outliers and averaging of the images as “stacking.” Depending on the image type, we perform extra calibrations:
1. *Biases* We simply stitch and stack all bias images to make the master bias image.
2. *Darks* We stitch each dark image, and then subtract the master bias. Then, we stack the dark images to create the master dark image.
3. *Flats, arcs, reference images, science images* In the process of stitching these files together, we multiply each CCD chip by the appropriate gain listed in the IMACS and LDSS3C user manuals. After stitching, we subtract the master bias and master dark from each image, and then stack each image type to create the master flat, arc, reference, and science images. In Figure \[fig:extract\] we show a master reference image, with red $\times$’s marking LHS 1140 and the comparison star in their slits. From the master flat we also create a bad pixel mask.
3. **Extraction rectangles** Using an interactive plotting tool developed for `mosasaurus`, we indicate which stars on the master reference image we wish to extract. `mosasaurus` then cuts out a rectangle around each of the desired spectra on the master science image, and a corresponding rectangle from the master flat and arc. The extraction rectangle for LHS 1140 spectrum is shown in red in Figure \[fig:extract\].
4. **Stellar spectra and sky-background** Using the rectangle cut from the master science image, we use an interactive plotting tool to indicate the spectral traces of LHS 1140 (purple line, Figure \[fig:extract\]) and the comparison star. An extraction region is defined as a set number of pixels away from the center of the stellar trace (purple band). We also indicate portions of sky-background on either side of the spectral trace (light blue bands). These are used to fit and remove the sky-background flux from the stellar flux during extraction.
5. **Normalized flat for each star** We use the extraction rectangles cut from the master flat to create a normalized flat for each star (flat for LHS 1140 shown in Figure \[fig:extract\]). The normalized flat is made by dividing each column of pixels in the cross-dispersion direction by the median value of that column. When making the median filter we only use portions of the flat extraction rectangle that correspond to pixels that are included in the spectral extraction, i.e., the stellar extraction region, the sky-background regions, and any intervening regions. We divide the extraction rectangles for each science exposure by the corresponding normalized flat.
6. **Extract spectra** We cycle through the science exposures and extract spectra of LHS 1140 and the comparison star in the following steps:
1. *Sky background* For each column of pixels in the cross-dispersion direction of an extraction rectangle we use the sky-background regions (designated in Step 4) to make a 2^nd^-order polynomial fit to the pixel column. This makes a 2D, polynomial-smoothed estimate of the sky background in the extraction rectangles of each exposure (Figure \[fig:extract\]). We note that a median of the sky-background pixels can also be used, with similar results.
2. *Sky in stellar extraction region* We take the portion of the 2D sky background that covers the stellar extraction region designated in Step 4 (purple) and sum in the cross-dispersion direction, creating a 1D estimate of the sky background (light blue spectrum in Figure \[fig:extract\]).
3. *Extracted spectrum* We divide the extraction rectangle (Step 4) by the normalized flat (Step 5) and sum the stellar extraction region in the cross-dispersion direction (purple spectrum in Figure \[fig:extract\]). We then subtract the 1D sky background estimate (blue spectrum) to get the extracted spectrum (red spectrum).
7. **Rough wavelength calibration** We need to create a wavelength solution to convert the extracted spectra from flux vs. pixel to flux vs. wavelength. Using another interactive plotting tool, we take the arc extraction rectangles for each star and mark the helium, neon, and argon lines. We then compare where our marked wavelengths are in pixel space to a template of lines for the grisms we used with the [LDSS3C](http://www.lco.cl/telescopes-information/magellan/instruments/ldss-3/atlas-of-comparsion-lamp-spectra-for-ldss-3/lamp-spectra) and [IMACS](https://github.com/zkbt/mosasaurus/blob/master/data/IMACS/gri-300-26.7/HeNeAr.txt) detectors. We use a polynomial to fit the marked arc lines to the template lines, and apply this wavelength solution to each of the extracted spectra. Finally, we re-sample each spectrum so that they are on a common, uniform wavelength grid; we ensure that flux is conserved in this process. The result works reasonably well, but there are visible mismatches in spectral features between LHS 1140 and the comparison star, and also between exposures taken at different times throughout the night (zoomed-in inset, Figure \[fig:extract\]). This rough wavelength calibration aligns the spectra to within 0.5 nm for IMACS spectra, and 1.0 nm for LDSS3C spectra (0.2 and 0.4 pixels, respectively). We will eventually bin these spectra into 20 nm wavelength bins, and this slight misalignment can introduce additional noise.
8. **Fine wavelength calibration** For a single spectrum we isolate prominent telluric and stellar spectral features – the O$_2$ doublet (760.5 nm), the Ca triplet (849.8, 854.2, and 866.2 nm), and the water line forest (930-980 nm) – and cross-correlate them with the same features in all other spectra in a data set. Our stars are close enough (in the Sun’s local moving group) and our spectral resolution low enough (upper limits of 250 km/s/pixel for IMACS and 165/km/s/pixel for LDSS3C) that comparing telluric O$_2$ and H$_2$O features to stellar Ca features is not introducing error in to our wavelength calibration. After the cross-correlation, we re-run the flux-conserving re-sampling routine to reflect the new wavelength grid for each spectrum. With this technique we align our spectra to within 0.25 nm (or 0.10 pixels; zoomed-in inset, Figure \[fig:extract\]). We use multiple data sets for this work so we also wavelength calibrate between the data sets.
We note that one improvement to our pipeline would be to change the extraction region around the stellar spectra such that it evolves over the time-series. This would entail re-tracing the stellar spectra in every exposure [@Jordan2013; @Rackham2017; @May2018] or utilizing an optimal extraction routine [@Stevenson2016a; @Bixel2019]. Systematics introduced by using a fixed aperture are decorrelated against during analysis (Section \[sec:analysis\]), and do not alter the results of this work.
{width="100.00000%"}
Issues with Magellan I (Baade) IMACS data {#subsec:IMACSextract}
-----------------------------------------
The 2017 IMACS data set exhibited anomalies that led us to perform a deep exploration of this data set, and ultimately decide not to include it in our analysis. The ACCESS collaboration [@ACCESSCollab2014] noticed similar systematics, which are thoroughly outlined in @Espinoza2017[^1].
We find that the source of these anomalies is an excess of light scattered by the IMACS instrument that occurs when the disperser is in place [Chapter 3, @Espinoza2017]. Figure \[fig:IMACSextract\] shows that this excess light adds non-negligible flux in portions of the detector which should be masked. We call this excess flux, unimaginatively, “mask flux.” We also see an excess of flux in the wings the stellar profile in the cross-dispersion direction. In Figure \[fig:IMACSextract\] we show the extraction rectangle of the comparison star from the 2017 IMACS data set, as well as a cut across the extraction rectangle in the cross-dispersion direction, to demonstrate the excess flux that we see. We compare these to the same figures for the 2018 IMACS data set, which does not exhibit excess flux.
@Espinoza2017 outlines a process to model and remove the mask flux. We were able to remove the mask flux from the comparison star spectra, however due to the alignment star holes near the LHS 1140 slits and the closeness of LHS 1140 to the edge of the slit, the flux profile in the cross-dispersion direction is difficult to model for this star. We therefore do not include the 2017 IMACS data set in our analysis.
For the 2018 IMACS data set we made significant changes to our mask (see Section \[subsec:IMACSobs\]) to ensure that we captured the full PSF of LHS 1140 and the comparison star, and were able to model and remove the mask flux. The 2018 observations occurred on a dark night (no moon) and we did not see the same excess mask flux in these data. The extra-long slits in the cross-dispersion direction did help us to capture the full PSF of LHS 1140 and the comparison star, along with enough sky background to do the extraction.
Data Analysis {#sec:analysis}
=============
In Section \[sec:extract\] we turned the raw `FITS` files that we collected during our observations into time-series of 1D wavelength-calibrated spectra of LHS 1140 and the comparison star. These time-series spectra exhibit two types of systematic trends which we address before constructing a transmission spectrum: 1) instrument systematics derived from the Magellan telescopes and the IMACS and LDSS3C spectrographs, and 2) telluric systematics derived from the Earth’s atmosphere, which we peer through as we observe. So as to not tamper with the transit information buried in the time-series, we model the systmatics at the same time as we model the transit properties of LHS 1140. We ultimately want to simultaneously analyze the spectra from each data set in order to construct the transmission spectrum.
We built a custom data analysis pipeline that picks up where `mosasaurus` left off. The pipeline, named [`decorrasaurus`](https://github.com/hdiamondlowe/decorrasaurus/releases/tag/v1.0), is built to take in IMACS and LDSS3C data cubes from `mosasaurus` and return decorrelated light curves that can be turned into transmission spectra.
`decorrasaurus` decorrelation steps {#subsec:decorrasaurus}
-----------------------------------
[cp[10.5cm]{}Cccc]{} Parameter & & &\
Name & & & L17 & I18 & L18\
**airmass** & From the header files, the average airmass of the field recorded at each exposure during observation. & t & & &\
**rotation angle** & From the header files, rotation angle of the instrument recorded at each exposure. This can be correlated with changes in illumination or flexure during observation. & t & & &\
**centroid** & Derived during extraction, the stellar centroid measured in the cross-dispersion direction. This is the median of the centroids across all wavelengths for each star in each exposure. & t, s & & &\
**width** & Derived during extraction, the width of the spectral trace in the cross-dispersion direction. This is the median of the measured widths across all wavelengths for each star in each exposure. & t, s & & &\
**peak** & Derived during extraction, the brightness of the brightest pixel in the cross-dispersion direction measured at every wavelengths for each star in each exposure. This is summed in wavelength space for each wavelength bin. & t, s, & & &\
**shift** & Derived during extraction, the linear change in the dispersion direction needed to align the spectra with each other. This is calculated for each star in each exposure. & t, s& & &\
**stretch** & Derived during extraction, the multiplicative change in the dispersion direction needed to align the spectra with each other. This is calculated for each star in each exposure. & t, s & & &\
**polynomial** & Specified during analysis, the degree of the polynomial component of the model. & t & 3 & 1 & 3\
Turning time-series of wavelength-calibrated 1D spectra into decorrelated light curves and a transmission spectrum is also a lengthy process. Here we outline the steps of our pipeline.
1. **Set-up** We read in the `mosasaurus` data cubes that we wish to analyze. `decorrasaurus` can work with a single data set, or multiple data sets simultaneously if parameters are to be jointly fit across multiple data sets. We also specify which parameters should be fixed or varied and how to bin the light curves in wavelength-space.
2. **Make light curves** Here we transform a time-series of wavelength-calibated 1D spectra of LHS 1140 and the comparison star into a time-series of normalized fluxes, or a light curve. This requires summing up each the spectra in a given wavelength bin. We chop the spectra in wavelength space (recall that all spectra were interpolated onto a common wavelength grid in Step 8 of Section \[subsec:mosasaurus\]) in order to make the wavelength bins. If necessary, we take fractions of pixels in order to meet the chosen wavelength cut-offs; we ensure that flux is conserved in this process. We normalize each wavelength-binned time-series of fluxes by the median flux for that time-series. We then divide the LHS 1140 time-series by the comparison star time-series to make the light curve.
3. **Make a model** The model that we fit to the data has two components:
1. *Systematics* The systematics component of the model $\mathcal{S}(t)$ is comprised of a polynomial specified during set-up and physical parameters recorded from the data extraction. Table \[tab:sysparams\] lists the parameters used in the systematics model, along with explanations. This model component can be described as: $$\mathcal{S}(t) = 1 + \sum_{n=1}^{N_{\mathrm{poly}}}c_nt^n + \sum_{m=1}^{M_{\mathrm{phys}}}c_mP_m(t,^*\!\!\lambda)$$ where $t$ is the time-array covered by the light curve, $P_m(t,^*\!\!\lambda)$ are the physical parameters derived from the extraction (they are all functions of time $t$ but some also have a wavelength $\lambda$ dependency), and $c_n,c_m$ are the coefficients we fit for.
2. *Transit* The transit component of the model $\mathcal{T}(t)$ is made with the `batman` package [@Kreidberg2015]. Table \[tab:sysparams\] explains which transit parameters we fix or vary for each fit we perform.
The complete model $\mathcal{M}(t)$ that we fit to the light curve is: $$\mathcal{M}(t) = \mathcal{S}(t)\mathcal{T}(t)$$ In steps 5 & 6 we fit for the systematics coefficients and the transit parameters simultaneously to achieve the best fit to the light curve data.
4. **Limb-darkening coefficients** The parameters that describe the opacity at the stellar limb are crucial for constructing an accurate transit model. We use the `Limb Darkening ToolKit` [`LDTk;` @Parviainen2015] to interpolate stellar models from the `PHOENIX` library [@Husser2013] and calculate the quadratic limb-darkening coefficients for the wavelength range of interest. Because these parameters are highly correlated, we use the formulation $2u_0 + u_1$ and $u_0 - 2u_1$, were $u_0$ and $u_1$ are the limb-darkening coefficients returned by `LDTk`, to decorrelate the coefficients for the fit [@Holman2006]. Hereafter we refer to the uncorrelated quadratic limb-darkening coefficients simply as limb-darkening parameters. We fix the limb-darkening parameters in the Levenberg-Marquardt fit (Step 5), but allow them to vary with a Gaussian prior for the full sampling of the parameter space (Step 6).
5. **Least-squares fitting** We perform three iterations of a Levenberg-Marquardt least-squares fitting routine using the `lmfit` package [@Newville2016]. The advantage of this fit is that it is fast, which means we can test different model parameters and choose the best ones to marginalize over.
1. *Iteration 1* We use the calculated photon noise for each data set to weight the residuals in the least-squares minimization.
2. *Iteration 2* We clip any points that are 5$\times$ the median absolute deviation of the residuals. We again use the calculated photon noise to weight the residuals.
3. *Iteration 3* We calculate the standard deviation of the residuals for each data set in the fit. We use this calculated error to weight the residuals in the least-squares minimization.
After the three iterations we use the best-fit values to compare different models to each other, employing both the Bayesian Information Criterion (BIC) and the Akaike information criterion (AIC), which penalize excessive model parameters. When we have found the best model parameters to decorrelated against, we continue to the next step where we more fully sample the parameter space in order to estimate the uncertainties of the free parameters.
6. **Dynamic nested sampling** To estimate the uncertainties in our free parameters we need to perform a more complete exploration of the parameter space. This has frequently been done with a Markov-Chain Monte Carlo algorithm, such as the one used in the `emcee` package [@Foreman-Mackey2013]. Here, due to the large number of free parameters we fit, we employ `dynesty` [@Speagle2019], an open-source dynamic nested sampling routine. `dynesty` requires priors on all of its parameters. We use the $1\sigma$ uncertainties derived from the Levenberg-Marquardt fits (Step 5) to set the priors for `dynesty`. For all free parameters we use flat priors bounded at 10$\times$ the Levenberg-Marquardt uncertainties, except for the limb-darkening parameters where we use a Gaussian prior with the standard deviation equal to 1$\times$ the uncertainty calculated by `LDTk` (Step 4). We also add a parameter $s$ to marginalize over. This parameter re-scales the $\chi^2$ value to unity, and we include it in our log-likelihood function such that it multiplies the theoretical uncertainty associated with each data point [@Berta2012].
Two data analyses {#subsec:3analyses}
-----------------
------------------------------------------- ------------------- ------------------------------- ------------------- -------------------------------
White light curve $\lambda$-binned light curves White light curve $\lambda$-binned light curves
Mid-transit time difference, $\Delta t_0$ Free Fixed Free Fixed
Period, $P$ Fixed Fixed Fixed, joined Fixed, joined
Inclination, $i$ Free Fixed Free, joined Fixed, joined
Scaled semi-major axis, $a/R_s$ Free Fixed Free, joined Fixed, joined
Limb-darkening, $2u_0 + u_1, u_0 - 2u_1$ Fixed Fixed Free, joined Free, joined
Planet-to-star radius ratio, $R_p/R_s$ Free Free Free, joined Free, joined
Scaling factor, $s$ — — Free Free
------------------------------------------- ------------------- ------------------------------- ------------------- -------------------------------
[lcccchhc]{} & Initial value & Fitted value & Priors & Fitted value & Value used in & & Compare to\
& for Step 5 & from Step 5 & for Step 6 & from Step 6 & $\lambda$-bin fits & & @Ment2019\
$\Delta t_0$, L17 (days)& 0.0 & -0.0027 & \[-0.0040, -0.0013\] & **-0.0027** $\pm$ 0.0002 & & This work & —\
$\Delta t_0$, I18 (days)& 0.0 & 0.0019 & \[0.0003, 0.0035\] & **0.0019** $\pm$ 0.0001 & & This work & —\
$\Delta t_0$, L18 (days)& 0.0 & 0.0021 & \[0.0008, 0.0034\] & **0.0021** $\pm$ 0.0001 & & This work & —\
$P$ (days) & 24.736959 & 24.736959 & — & — & 24.736959 & Ment, et al. (2019) & **24.736959** $\pm$ 0.000080\
$i$ (degrees) & 89.89 & 89.8345 & \[89.7921, 89.8768\] & 89.851$^{+0.0186}_{-0.0285}$ & & This work & **89.89**$_{-0.03}^{+0.05}$\
$a/R_s$& 95.34 & 92.5481 & \[90.5089, 94.5874\] & 93.3738$^{+0.6671}_{-1.1317}$ & & This work& **95.34** $\pm$ 1.06\
$2u_0 + u_1$ & 1.0423 & 1.0423 & $\mathcal{N}(\mu=1.0423, \sigma=0.0194)$ & 1.0356$^{+0.0179}_{-0.0178}$ & & This work & —\
$u_0 - 2u_1$ & -0.2439 & -0.2439 & $\mathcal{N}(\mu=-0.2439, \sigma=0.0894)$ & -0.1853$^{+0.0847}_{-0.0873}$ & & This work & —\
$R_p/R_s$ & 0.0739 & 0.0744 & \[0.0657, 0.0832\] & 0.0754 $\pm$ 0.0005 & & This work & 0.07390 $\pm$ 0.00008\
$s$, L17 & — & 1 & \[0.01, 10\] & 2.5430$^{+0.0872}_{-0.0851}$ & & This work $---$\
$s$, I18 & — & 1 & \[0.01, 10\] & 3.3716$^{+0.1044}_{-0.1004}$ & & This work $---$\
$s$, L18 & — & 1 & \[0.01, 10\] & 3.4593$^{+0.1095}_{-0.1034}$ & & This work $---$\
In order to marginalize over the appropriate parameters, we build up to the transmission spectrum by first analyzing our data sets separately, and then analyzing them jointly. Each analysis involves constructing both a white light curve and a set of wavelength-binned light curves. Table \[tab:transitparams\] lists the transit parameters and whether they are free or fixed in each fit. The breakdown of the white light curves into wavelength-binned light curves (20 nm bins) is shown in Figure \[fig:spectrabinned\]. For all analyses we assume a circular orbit for LHS 1140b.
![Representative spectra of LHS 1140 (solid lines) and the comparison star (dotted lines) from the three data sets we analyze in this work. Grey vertical lines indicate the 20 nm wavelength bins that we use to construct the transmission spectrum.[]{data-label="fig:spectrabinned"}](Figure_SpectraBinned.pdf){width="50.00000%"}
### Data sets fit independently {#subsubsec:individual}
We first treat our data sets independently. The main purpose of this step is to decide which systematic model parameters should be used to decorrelate each data set. Table \[tab:sysparams\] lists all possible systematic parameters along with an explanation of how they are constructed, and which are used to decorrelated each data set. We use the wavelength-binned light curves to determine the best decorrelation parameters (Step 5 of Section \[subsec:decorrasaurus\]). We first test out parameters on a single 200 nm wavelength bin from 750-850 nm, which is mostly free of spectral features for both LHS 1140 and the comparison star. Once the best parameters are found for this test bin, we expand the analysis to all of the 200 nm wavelength bins in the data set. Since some wavelength bins are more correlated with certain parameters than others, we add decorrelation parameters as necessary. Once determined, we use the same decorrelation parameters for each wavelength bin in a data set, and also for the white light curve.
### Data sets fit jointly {#subsubsec:joint}
Once the decorrelation parameters for each data set are determined by analyzing them separately, we then use those decorrelation parameters to perform a joint fit across all three data sets. The raw, decorrelated, and time-binned white light curves are shown in Figure \[fig:whitelightcurve\]. The parameters we use or derive from the joint white light curve fit are presented in Table \[tab:whitelcvalues\], along with their priors. We compare our results to those of @Ment2019 and find them to be in agreement. The @Ment2019 analysis included high cadence (2 second integrations) *Spitzer* data, we adopt the period $P$, inclination $i$, and semi-major axis $a/R_s$ derived from that work as the fixed parameters in the wavelength-binned light curve analysis.
![*Panel a* Raw light curves from the three data sets used in this work. Over-plotted grey lines are the fitted models to each data set. *Panel b* Light curves with the systematics component of the model divided out, leaving just the data and the transit model. *Panel c* All of the data are combined, and binned into 3-minute time bins. *Panel d* Residuals of Panel c.[]{data-label="fig:whitelightcurve"}](Figure_WhiteLightCurve.pdf){width="50.00000%"}
We then bin the light curves into 20 nm bins. We analyze the three data sets jointly in each wavelength bin, but the wavelength bins are independently analyzed from each other. We fix the orbital parameters that are common to all wavelength bins to their fitted values or literature values. We show the parameters fit in each wavelength bin graphically in Figure \[fig:parameters\].
We present the wavelength-binned data along with the best light curve fits in Figure \[fig:wavebinlightcurve\]. Table \[tab:transitdepths\] provides the measured values of $R_p^2/R_s^2$, along with the RMS for each wavelength bin and how close we were able to get to the photon noise limit. Across all 20 wavelength bins we achieve an average uncertainty in $R_p^2/R_s^2$ of 0.014% (140 ppm) and an average RMS value of 1.28$\times$ the photon noise.
![Results from the wavelength-binned joint fit. The radius ratio and scaled limb-darkening coefficients are shared across all three data sets, and so there is only one resulting value in each wavelength bin (green points with error bars). The rest of the parameters are fit simultaneously, but separately for each data set (colors correspond to the same data sets as in Figure \[fig:whitelightcurve\]. We do not see any obvious correlations between any of the parameters and the resulting measurement of $R_p^2/R_s^2$.[]{data-label="fig:parameters"}](Figure_Parameters.pdf){width="48.00000%"}
{width="100.00000%"}
------------ ------------------- ------- -------------------
Wavelength $R_p^2/R_s^2$ RMS $\times$ Expected
(nm) (%) (ppm) Noise
610-630 0.559 $\pm$ 0.034 2348 1.23
630-650 0.515 $\pm$ 0.022 1436 1.17
650-670 0.511 $\pm$ 0.019 1242 1.28
670-690 0.534 $\pm$ 0.021 1386 1.24
690-710 0.542 $\pm$ 0.016 1012 1.21
710-730 0.521 $\pm$ 0.018 1151 1.37
730-750 0.512 $\pm$ 0.013 837 1.33
750-770 0.527 $\pm$ 0.013 840 1.26
770-790 0.518 $\pm$ 0.013 845 1.28
790-810 0.549 $\pm$ 0.012 784 1.36
810-830 0.545 $\pm$ 0.011 725 1.24
830-850 0.523 $\pm$ 0.011 778 1.31
850-870 0.537 $\pm$ 0.013 810 1.35
870-890 0.531 $\pm$ 0.012 778 1.33
890-910 0.584 $\pm$ 0.012 828 1.33
910-930 0.553 $\pm$ 0.012 817 1.26
930-950 0.559 $\pm$ 0.014 1120 1.27
950-970 0.517 $\pm$ 0.016 799 1.31
970-990 0.535 $\pm$ 0.016 747 1.18
990-1010 0.577 $\pm$ 0.021 1019 1.30
------------ ------------------- ------- -------------------
: Best fit $R_p^2/R_s^2$\[tab:transitdepths\]
Results & Discussion {#sec:results}
====================
From our observations we produce a transmission spectrum and compare it to models. In doing so we demonstrate the limits of the ground-based transmission spectroscopy technique employed here to investigate the atmosphere of LHS 1140b.
Planetary atmospheric detection {#subsec:atmodetection}
-------------------------------
Complete derivations of the properties of planetary atmospheres can be found in textbooks, but for the purposes of transmission spectroscopy, we are interested in the scale height $H$ of a planet’s atmosphere, or how extended the atmosphere is, and what kinds of features it produces. The scale height is calculated by
$$\label{eqn:H}
H = \frac{k_BT}{\mu g}$$
where $k_B$ is the Boltzmann constant, $T$ is the planet’s mean atmospheric temperature, $\mu$ is the mean molecular weight of the planet’s atmosphere, and $g$ is the planet’s surface gravity. We do not know the mean temperature of LHS 1140b’s atmosphere, but we can estimate a temperature-pressure profile using its equilibrium temperature.
An estimate of the amplitude of features in the transmission spectrum of an atmosphere is given by
$$\label{eqn:deltad}
\begin{aligned}
\Delta \delta &= \left(\frac{R_p + NH}{R_s}\right)^2 - \left(\frac{R_p}{R_s}\right)^2\\
&= \frac{2R_pNH + NH^2}{R_s^2}
\end{aligned}$$
where $N$ is the number of scale heights we can observe before the atmosphere becomes optically thick (when optical depth $\tau = 1$). The last term of $NH^2/R_s^2$ is negligible.
Model transmission spectrum {#subsec:exotransmit}
---------------------------
The relative feature amplitudes of a planetary atmosphere observed over a range of wavelengths can be compared to models in order to reveal the presence of an atmosphere and its composition. We construct a model transmission spectrum for LHS 1140b using the open-source code `Exo-Transmit` [@Miller-RicciKempton2012; @Kempton2017]. The code inputs are a temperature-pressure profile, an equation-of-state specific to the atmospheric composition, the 1-bar planet radius and surface gravity, and the stellar radius.
Following the same procedures outlined in [@Miller-Ricci2009] and @Miller-Ricci2010, we use custom double-grey temperature-pressure profiles for the LHS 1140b atmosphere. (The default temperature-pressure profiles that come with `Exo-Transmit` are isothermal). The equation-of-state files corresponding to the atmospheres we test in this work are readily available in `Exo-Transmit`. Since we do not know the 1-bar planet radius exactly, we adjust it until the model transmission atmosphere best fits the data. This adjustment changes both the absolute depth of the model as well as the amplitude of the features.
Observed transmission spectrum {#subsec:transmission}
------------------------------
From the wavelength-binned jointly fitted $R_p/R_s$ values, we construct a transmission spectrum. In Figure \[fig:transmission\] we present the final transmission spectrum and compare it to model transmission spectra calculated for the system using `Exo-Transmit` [@Kempton2017].
Given the small atmospheric features of the LHS 1140b atmosphere, we are not able to rule out even the lowest mean molecular weight cases. We therefore only present these cases – clear 1$\times$ and 10$\times$ solar metallicity atmospheres – and do not address models of higher mean molecular weight atmospheres. We do not expect a terrestrial planet like LHS 1140b to possess such light atmospheres , but these end-member compositions are the first atmospheres to rule out.
Atmospheric detection limits
----------------------------
To explore the limits of our observed transmission spectrum we can perform simple calculations using Equations \[eqn:H\] and \[eqn:deltad\]. LHS 1140b’s surface gravity ($23.7 \pm 2.7$ m/s$^2$) and cool equilibrium temperature [$235 \pm 5$ K, assuming a Bond albedo of 0 and a planet-wide energy distribution; @Ment2019] combine to make the scale height of this planet’s atmosphere $40.9 \pm 4.7$ km for the lowest mean molecular weight case ($\mu = 2$) for the unrealistic, pure light H$_2$ atmosphere.
For the LHS 1140 system where $R_p = 1.727 \pm 0.032\ R_{\oplus}$ and $R_s = 0.2139 \pm 0.0014\ R_{\odot}$, the amplitude of the transmission features for the lowest mean molecular weight case is $65.3 \pm 7.7$ ppm, assuming we can see down 1.6 scale heights. (We estimate $N = 1.6$ from the 20 nm wavelength-binned model transmission spectrum). This is a factor of two below the median precision we are able to achieve in this project. For a more realistic atmosphere dominated by CH$_4$, H$_2$O, O$_2$, or CO$_2$, the feature amplitudes are at the level of 8 ppm or lower, a factor of 17 below our precision.
{width="100.00000%"}
Future instruments {#subsec:discussion}
------------------
The heavily anticipated *James Webb Space Telescope* will be capable of robust detections of planetary atmospheres with instruments capable of performing transmission spectroscopy across a broad wavelength range. @Morley2017 simulated JWST observations with NIRSpec/G235M and the F170LP filter for several nearby planets orbiting small stars, assuming equilibrated atmospheres derived from Titan, Earth, and Venus elemental compositions. The authors conclude that it will not be possible to detect an atmosphere around LHS 1140b with JWST due to an unrealistic amount of observing time. This conclusion is still valid, despite a refinement some of the LHS 1140 system parameters [@Ment2019].
The next generation of ground-based optical telescopes – the *Giant Magellan Telescope*, the *Thirty Meter Telescope*, and the *European Extremely Large Telescope* – will be larger than any we currently have. All three have planned multi-object spectrographs as either first-light or second generation instruments. Exposure time calculators are not yet available for these modes, but a simple scaling to the larger collecting areas of these telescopes reveals that the low mean molecular weight atmospheres tested in this study could be ruled out on LHS 1140b with as few as seven transits with GMT or five transits with TMT. These ground-based observatories will still have to contend with LHS 1140b’s infrequent transits. High resolution spectroscopy [@Snellen2013; @Birkby2018] will be possible with the GSMTs, but LHS 1140b will still likely be below the detection thresholds for this technique.
Perhaps the most promising avenue for detecting the atmospheres of habitable-zone terrestrial exoplanets is to find more amenable targets. As *TESS* continues to discover new worlds around our closest stellar neighbors, we are likely to find planets with more accessible atmospheres than that of LHS 1140b. If a star like LHS 1140 were discovered at 5 pc away (instead of 15) with a transiting habitable-zone terrestrial exoplanet, we would be able to detect or rule out a low mean molecular weight atmosphere around the planet from 10 transits with both Magellan telescopes (i.e., the observational set-up used in this work), or two transits with *GMT*. *TESS* has already grown the sample of nearby ($<15$ pc) terrestrial exoplanets, including prime targets such as LHS 3844b [@Vanderspek2018] and LTT 1445Ab [@Winters2019], though neither planet resides in the habitable zone.
Conclusion {#sec:conclusion}
==========
LHS 1140b orbits in the habitable zone of its host M dwarf. This world is at the upper end of the radius regime that defines terrestrial planets [@Fulton2017], but we know from radius and mass measurements that it is rocky in nature [@Ment2019]. However, given the high surface gravity and cool equilibrium temperature of LHS 1140b, its atmosphere is not readily accessible to transmission spectroscopy.
With this work we set out to ambitiously capture two transits of LHS 1140b while also exploring the synergy between the IMACS and LDSS3C spectrographs. Because LHS 1140b transits infrequently, ground-based opportunities for observation are rare. We designed a multi-year program that employed both Magellan I/IMACS & II/LDSS3C, though LDSS3C is preferred for M dwarf observations because its red observing mode collects more than twice as many photons as IMACS at the wavelengths where M dwarfs emit the bulk of their photons (Figure \[fig:spectrabinned\]).
Though we are not able to investigate the atmosphere of LHS 1140b in this work, we detail our extraction and analysis pipelines in order to illustrate how we convert raw spectroscopic information into wavelength-calibrated time series. We construct both a white light curve and 20 nm wavelength-binned light curves by jointly fitting our data sets. Across all of the wavelength-binned light curves we achieve an average uncertainty in $R_p^2/R_2^2$ of 14% and an average precision of 1.28$\times$ the photon noise. We will employ the techniques laid out in this work for ground-based transmission spectroscopy studies of the recently discovered terrestrial worlds LHS 3844b and LTT 1445Ab [Vanderspek2018,Winters2019]{} with Magellan II (Clay)/LDSS3C (PI Diamond-Lowe). These worlds do not reside in the habitable zones of their systems, but they are more amenable to atmospheric follow-up.
Finally, in the *TESS* era, we emphasize the need for robust mass measurements to accompany the detected radii of newly discovered transiting exoplanets. Without knowledge of the bulk densities of these worlds we will under- or over-estimate our ability to detect their atmospheres.
This paper includes data gathered with both of the 6.5m Magellan Telescopes (Baade & Clay) located at Las Campanas Observatory, Chile. We thank the contributors to the IMACS and LDSS3C projects, the telescope operators and staff at Las Campanas Observatory, and the writers and contributors of the open-source software used in this work. We especially thank members of the ACCESS collaboration Néstor Espinoza, Benjamin Rackham, David Osip, and Mercedes Lopez-Morales for in-depth conversations about the workings of the IMACS multi-object spectrograph. We also thank Robin Wordsworth, Dimitar Sasselov, and Laura Kreidberg for helpful comments and conversations. We thank Erik Strand for assistance during the 2017 observations. H.D.-L. recognizes support from the National Science Foundation Graduate Research Fellowship Program (grant number DGE1144152). J.A.D. would like to acknowledge support from the Heising-Simons Foundation for their support, whose 51 Pegasi b Postdoctoral Fellowship program has enabled this work. The work of E.M.-R.K. was supported by the National Science Foundation under Grant No. 1654295 and by the Research Corporation for Science Advancement through their Cottrell Scholar program. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed here are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.\
, `SAOImageDS9` [@Joye2003]
[^1]: [repositorio.uc.cl/handle/11534/21313](https://repositorio.uc.cl/handle/11534/21313)
|
---
abstract: 'We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is $\{0,1,2,3,4\}$. This positively answers two conjectures from a previous paper by the last two authors.'
address:
- ' Department of Mathematics, College of Saint Benedict and Saint John’s University, 37 College Avenue South, Saint Joseph, MN 56374-5011, USA '
- ' Department of Mathematics and Statistics, Northern Arizona University PO Box 5717, Flagstaff, AZ 86011-5717, USA '
author:
- 'Bret J. Benesh'
- 'Dana C. Ernst'
- Nándor Sieben
bibliography:
- 'game.bib'
title: |
The spectrum of nim-values for achievement games\
for generating finite groups
---
Introduction
============
Anderson and Harary [@anderson.harary:achievement] introduced two impartial games *Generate* and *Do Not Generate* in which two players alternately take turns selecting previously unselected elements of a finite group $G$. The first player who builds a generating set for the group from the jointly-selected elements wins the *achievement game* $\gen(G)$. The first player who cannot select an element without building a generating set loses the *avoidance game* $\dng(G)$. The outcomes of both games were studied for some of the more familiar finite groups, including abelian, dihedral, and symmetric groups in [@anderson.harary:achievement; @Barnes].
A fundamental problem in the theory of impartial combinatorial games [@albert2007lessons; @SiegelBook] is determining the nim-value of a game. The nim-value determines the outcome of the game, and it also allows for the easy calculation of the nim-values of game sums. In [@ErnstSieben], Ernst and Sieben used structure digraphs for studying the nim-values of both the achievement and avoidance games, which they applied in the context of certain finite groups including cyclic, abelian, and dihedral. Loosely speaking, a structure digraph is a quotient of the game digraph by an equivalence relation called *structure equivalence*. Structure equivalence respects the nim-values of the positions of the game and drastically simplifies the calculation of the nim-values. The *type* of a structure class is a triple that encodes the nim-values of the positions. Ernst and Sieben [@ErnstSieben Proposition 3.20] determined the spectrum of types for the avoidance game $\dng(G)$, which in turn allowed them to determine that the spectrum of nim-values for $\dng(G)$ is $\{0,1,3\}$.
The goal of this paper is to determine the spectrum of nim-values for the achievement game $\gen(G)$. Our approach is very similar to that of the avoidance game, but the required calculations are significantly more difficult for groups of even order. One reason for the increased difficulty is that the game digraph of the avoidance game is a subgraph of the the game digraph of the achievement game, and hence the achievement game has more positions than the avoidance game. As a result, the structure digraphs for achievement games can be more complex. Moreover, the types associated to structure classes no longer suffice since types contain insufficient information to be closed under type calculus. To overcome this apparent shortcoming, we introduce the *extended type* of a structure class, which adds a fourth component to the existing type. To analyze the behavior of the structure digraphs together with the associated extended types, we develop several type restrictions and then rely on computer calculations to handle the large number of cases. We prove that the spectrum of nim-values for the achievement game $\gen(G)$ is $\{0,1,2,3,4\}$, which positively answers Conjectures 4.8 and 4.9 from [@ErnstSieben].
The structure of the paper is as follows. We start with some preliminaries from [@BeneshErnstSiebenSymAlt; @BeneshErnstSiebenDNG; @BeneshErnstSiebenGeneralizedDihedral; @ErnstSieben], and follow with a short characterization of the spectrum of $\gen(G)$ for $G$ of odd order. The bulk of the work is spent on characterizing the spectrum of $\gen(G)$ for $G$ of even order.
Preliminaries
=============
We now give a more precise description of our game. We also recall some definitions and results from [@BeneshErnstSiebenGeneralizedDihedral; @ErnstSieben]. The positions of $\gen(G)$ are the possible sets of jointly selected elements. The starting position is the empty set. The options of a nonterminal position $P$ are of the form $P\cup\{g\}$ for some $g\in G\setminus P$. The set of options of $P$ is denoted by $\opt(P)$.
The *nim-value* of a position $P$ is recursively defined by $$\nim(P)=\mex\{\nim(Q)\mid Q\in \opt(P)\},$$ where the *minimum excludant* $\mex(S)$ is the smallest nonnegative integer missing from $S$. The terminal positions of the game have no options, and so their nim-value is $\mex(\emptyset)=0$. The winning positions for the player who is about to move (N-positions) are those with nonzero nim-value. The winning strategy always moves the opponent into a position with zero nim-value.
Type calculus
-------------
The set $\mathcal{M}$ of maximal subgroups of $G$ plays an important role in this game. For a position $P$ we let $$\lceil P \rceil := \bigcap\{M\in\mathcal{M} \mid P\subseteq M\}.$$ We use the simplified notation $\lceil P,g_1,\ldots, g_n \rceil$ for $\lceil P\cup \{g_1,\ldots, g_n\} \rceil$. If $P$ is a terminal position of the game, then $P$ is a generating set of $G$, and so $\lceil P \rceil=\bigcap\emptyset=G$. Note that $\lceil \emptyset \rceil=\bigcap \mathcal{M}$ is the Frattini subgroup $\Phi(G)$.
Two positions $P$ and $Q$ are *structure equivalent* if $\lceil P \rceil=\lceil Q \rceil$. Structure equivalence is an equivalence relation. The maximum element of the equivalence class of $P$ is $\lceil P \rceil$, so we denote the *structure class* of $P$ by $X_I$ where $I=\lceil P \rceil$. The set of equivalence classes is denoted by $\mathcal{D}$.
The option relationship between positions is compatible with structure equivalence [@ErnstSieben Corollary 4.3], so we say $X_J$ is an option of $X_I$ if $Q\in \opt(P)$ for some $P\in X_I$ and $Q\in X_J$. The set of options of $X_I$ is denoted by $\opt(X_I)$.
The vertices of the *structure digraph* are the structure classes. The arrows of this digraph connect structure classes to their options.
The parity of an integer $n$ is $\pty(n):=n\text{ mod }2$. The *type* of the structure class $X_I$ is $$\type(X_I):=(\pty(|I|), \nim(P),\nim(Q)),$$ where $P,Q\in X_I$ with $\pty(|P|)=0$ and $\pty(|Q|)=1$. This is well-defined by [@ErnstSieben Proposition 4.4]. Note that $\type(X_G)=(\pty(|G|),0,0)$ and $\type(X_I)$ is an element of $\mathbb{T}:=\{0,1\} \times \mathbb{W} \times \mathbb{W}$, where $\mathbb{W}:=\mathbb{N} \cup \{0\}$. Additionally, the second component of $\type(X_{\Phi(G)})$ is the nim-value of $\gen(G)$, since the starting position $\emptyset$ is in $X_{\Phi(G)}$. We say that the *parity of the structure class* $X_I$ is the parity of $|I|$. The sets of even and odd structure classes are denoted by $\mathcal{E}$ and $\mathcal{O}$, respectively. Thus, $\mathcal{D}=\mathcal{E} \dot\cup \mathcal{O}$.
Let $\pi_i:\mathbb{W}^n\to\mathbb{W}$ and $\tilde\pi_i:\mathbb{W}^n\to\mathbb{W}^{n-1}$ denote projection functions defined by $$\begin{aligned}
\pi_i(x_1,\ldots,x_n) & :=x_i, \\
\tilde\pi_i(x_1,\ldots,x_n) & :=(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n).\end{aligned}$$ We are going to use the standard image notation $f(A):=\{f(a)\mid a\in A\}$ if $A$ is a subset of the domain of $f$.
For $T \subseteq \mathbb{T}$, define $E_T:=\pi_2(T)$, $O_T:=\pi_3(T)$, $e_T:=\mex(O_T)$, and $o_T:=\mex(E_T)$. We also define $$\begin{aligned}
\mex_0(T) &:=(0,e_T,\mex(E_T\cup\{e_T\})) \\
\mex_1(T) &:=(1,\mex(O_T\cup\{o_T\}),o_T).\end{aligned}$$ We refer to this computation as *type calculus*.
The following consequence of [@ErnstSieben Corollary 4.3, Proposition 4.4] is our main tool to compute nim-values.
\[prop:TypeCalculation\] If $X_I\in\mathcal{D}$, then $$\type(X_I)=
\begin{cases}
\mex_0(\type(\opt(X_I)), & \text{$|I|$ is even} \\
\mex_1(\type(\opt(X_I)), &\text{$|I|$ is odd}. \\
\end{cases}$$
Let $I$ have odd order and $X_I$ have options with types $(0,1,2)$ and $(1,4,3)$. Then $E_T=\{1,4\}$ and $O_T=\{2,3\}$. So $$\type(X_I)=\mex_1(\{(0,1,2),(1,4,3)\})=(1,1,0)$$ since the odd positions in $X_I$ have nim-value $o_T=\mex(\{1,4\})=0$, while the even positions in $X_I$ have nim-value $\mex(\{2,3,o_T\})=1$.
The *deficiency* of a subset $S$ of $G$ is the minimum size $\delta_G(P)$ of a subset $Q$ of $G$ such that $\langle P\cup Q \rangle=G$. Structure equivalent positions have equal deficiencies [@BeneshErnstSiebenGeneralizedDihedral Proposition 3.2]. We define $$\begin{aligned}
&\mathcal{D}_k:=\{X_I\in \mathcal{D}\mid \delta_G(I)=k\},
&&\mathcal{E}_k:=\mathcal{E}\cap \mathcal{D}_k,
&&\mathcal{O}_k:=\mathcal{O}\cap \mathcal{D}_k,
\\
&\mathcal{D}_{\geq k}:=\bigcup\{\mathcal{D}_i \mid i \geq k\},
&&\mathcal{E}_{\geq k}:= \mathcal{E} \cap\mathcal{D}_{\geq k},
&&\mathcal{O}_{\geq k}:= \mathcal{O} \cap\mathcal{D}_{\geq k}.\end{aligned}$$ We will write $\mathcal{D}_k(G)$ when we want to emphasize the dependence on $G$. We recursively define $$\begin{aligned}
\mathcal{D}_{k,0} & :=\{X_I\in \mathcal{D}_k \mid \opt(X_I) \subseteq \mathcal{D}_{k-1} \} \\
\mathcal{D}_{k,l} & :=\{X_I\in \mathcal{D}_k \mid \opt(X_I) \subseteq \mathcal{D}_{k-1} \cup \mathcal{D}_{k, l-1} \}\end{aligned}$$ for $k,l\ge 1$. It is easy to check that the union of the nested collection $\mathcal{D}_{k,0}\subseteq \mathcal{D}_{k,1}\subseteq \mathcal{D}_{k,2}\subseteq \cdots$ is $\mathcal{D}_k$.
![ \[fig:sds\] Structure diagram symbols with $p$ denoting the parity of the structure class.](DefOfEven.pdf "fig:") ![ \[fig:sds\] Structure diagram symbols with $p$ denoting the parity of the structure class.](DefOfOdd.pdf "fig:") ![ \[fig:sds\] Structure diagram symbols with $p$ denoting the parity of the structure class.](DefOfEvenOdd.pdf "fig:") ![ \[fig:sds\] Structure diagram symbols with $p$ denoting the parity of the structure class.](DefOfMidarrow.pdf "fig:") ![ \[fig:sds\] Structure diagram symbols with $p$ denoting the parity of the structure class.](DefOfDoublearrow.pdf "fig:") ![ \[fig:sds\] Structure diagram symbols with $p$ denoting the parity of the structure class.](DefOfArrow.pdf "fig:")
We visualize the structure digraph of $\gen(G)$ with a structure diagram. In a *structure diagram*, vertices are denoted by triangles or circles. A structure class with even or odd parity is represented by a triangle with a flat bottom or flat top, respectively. A structure class with an unknown or unimportant parity is represented by a circle. We use several arrow types to indicate whether a change in deficiency occurs between a structure class and its option. A summary of these symbols is shown in Figure \[fig:sds\]. Note that Proposition \[prop:Rule0EverythingHasALowerDeficiencyOption\] justifies that no other arrow types are necessary.
Extended type calculus
----------------------
A further complication is that some of our restrictions require information about the even options of $X_I$, so we need to include this information in our type calculus. This motivates the following.
For $X_{I}\in\mathcal{D}_{k}$, the *smoothness* of $X_I$ is $$\smo(X_I)=\begin{cases}
2 & \text{if } \pty(X_I)=0\\
1 & \text{if } \pty(X_I)=1 \text{ and } \opt(X_{I})\cap\mathcal{E}_{k}\neq\emptyset\\
0 & \text{otherwise}.
\end{cases}$$ We say that $X_I$ is *smooth* if $\smo(X_I)\ge 1$ and *rough* otherwise.
Note that an even structure class is always smooth, while the smoothness of an odd structure class depends on whether it has an even option with the same deficiency. The smoothness of an even structure class plays no role in our computations. We only define it to make the extended type in the next definition always a quadruple. This simplifies our formulas.
The *extended type* of $X_I$ is $\etype(X_{I}):=(\type(X_I),\smo(X_I))$.
Note that $\etype(X_I)$ is an element of $\mathbb{E}:=\mathbb{T} \times \{0,1,2\}$, although we will typically write extended types flattened as a quadruple $(p,e,o,s)$.
![ \[esds\] Extended structure diagram symbols for structure classes. For odd structure classes, we use a double solid boundary if $X_I$ is smooth ($s=1$), a single dotted boundary if $X_I$ is rough ($s=0$), and single solid boundary if the smoothness is unknown or unimportant.](DefOf2.pdf "fig:") ![ \[esds\] Extended structure diagram symbols for structure classes. For odd structure classes, we use a double solid boundary if $X_I$ is smooth ($s=1$), a single dotted boundary if $X_I$ is rough ($s=0$), and single solid boundary if the smoothness is unknown or unimportant.](DefOfDotted.pdf "fig:") ![ \[esds\] Extended structure diagram symbols for structure classes. For odd structure classes, we use a double solid boundary if $X_I$ is smooth ($s=1$), a single dotted boundary if $X_I$ is rough ($s=0$), and single solid boundary if the smoothness is unknown or unimportant.](DefOfDouble.pdf "fig:") ![ \[esds\] Extended structure diagram symbols for structure classes. For odd structure classes, we use a double solid boundary if $X_I$ is smooth ($s=1$), a single dotted boundary if $X_I$ is rough ($s=0$), and single solid boundary if the smoothness is unknown or unimportant.](DefOfSingle.pdf "fig:")
In an *extended structure diagram*, we also indicate the smoothness of the structure classes. Smooth odd structure classes are drawn with a double solid boundary while rough odd structure classes are drawn with a single dotted boundary. A summary of these symbols is shown in Figure \[esds\].
For $(A,B)\in\mathcal{P}(\mathbb{E})\times\mathcal{P}(\mathbb{E})$ we define $$\begin{aligned}
\emex_0(A,B) &:= (\mex_0(\tilde\pi_4(A \cup B)),2), \\
\emex_1(A,B) &:= (\mex_1(\tilde\pi_4(A \cup B)),1-\min(\pi_1(A))).\end{aligned}$$ We refer to this computation as *extended type calculus*.
We think of these two functions as ways of finding the extended type of $X_I \in \mathcal{D}_n$, either real or hypothetical. The first input $A$ consists of the extended types of the options of $X_I$ in $\mathcal{D}_n$, while the second input $B$ consists of the extended types of the options of $X_I$ in $\mathcal{D}_{n-1}$.
![\[fig:Z6\]Extended structure diagram for $\gen(\mathbb{Z}_6)$. The quadruples insides the triangles are the corresponding extended types.](Z6.pdf)
Extended type calculus allows us to recursively compute the extended types of every structure class, starting from the terminal structure class.
Figure \[fig:Z6\] depicts the extended structure diagram for $\gen(\mathbb{Z}_6)$. The maximal subgroups are $\langle 2 \rangle$ and $\langle 3 \rangle$. The structure classes are $X_{\langle 1 \rangle}\in\mathcal{E}_0$, $X_{\langle 3 \rangle}\in\mathcal{E}_1$, and $X_{\langle 2 \rangle},X_{\langle 0 \rangle}\in\mathcal{O}_1$. Note that $\mathcal{D}_{1,0}=\{X_{\langle 3 \rangle},X_{\langle 2 \rangle}\}$ and $\mathcal{D}_{1,1}=\{X_{\langle 3 \rangle},X_{\langle 2 \rangle},X_{\langle 0 \rangle}\}$. Extended type calculus can be used, for example, to compute $$\begin{aligned}
\etype(X_{\langle 0 \rangle})
&=\emex_1(\etype(\{X_{\langle 2 \rangle},X_{\langle 3 \rangle}\}),\etype(\{X_{\langle 1 \rangle}\})) \\
&=\emex_1(\{(1,2,1,0),(0,1,2,2)\},\{(0,0,0,2)\}) \\ &=(1,4,3,1).\end{aligned}$$ The structure class $X_{\langle 0 \rangle}$ is smooth while $X_{\langle 2 \rangle}$ is rough. The nim-value of the game is $$\nim(\gen(\mathbb{Z}_6))=\nim(\emptyset)=\pi_2(\etype(X_{\lceil \emptyset \rceil}))=\pi_2(\etype(X_{\langle 0 \rangle}))=\pi_2(1,4,3,1)=4.$$
Some known option-type restrictions
-----------------------------------
The following three results follow from [@BeneshErnstSiebenGeneralizedDihedral Proposition 3.8], Lagrange’s Theorem, and [@BeneshErnstSiebenGeneralizedDihedral Proposition 3.9], respectively.
\[prop:Rule0EverythingHasALowerDeficiencyOption\] If $X_{I}\in\mathcal{D}_{k}$ for some $k\ge1$, then $\opt(X_{I})\subseteq\mathcal{D}_{k-1}\cup \mathcal{D}_{k}$ and $\opt(X_{I})\cap\mathcal{D}_{k-1}\ne\emptyset$.
The statement is depicted in Figure \[fig:sds\]. It essentially restricts the possible arrow types between structure classes.
\[prop:Rule1EvenOptionsOnlyHaveEvenOptions\] If $X_{I}\in\mathcal{E}$ then $\opt(X_{I})\subseteq\mathcal{E}$.
This means that an even structure class has only even options, as shown in Figure \[rules\](a).
\[prop:Rule2EverythingHasAnEvenOption\] If $G$ is a group of even order and $X_I$ has an option, then $X_I$ has an even option.
The statement is depicted in Figure \[rules\](b).
Groups of odd order
===================
The type of a structure class can be determined relatively easily if $G$ has odd order. The following theorem is an extension of [@ErnstSieben Theorem 4.7] and has a proof that is very similar to the proof of [@BeneshErnstSiebenGeneralizedDihedral Proposition 3.10]. Note that we implicitly use Proposition \[prop:TypeCalculation\] in the following proof, as well as throughout the rest of the paper.
\[prop:OrderGIsOdd\] If $G$ is a group of odd order, then $$\type(X_I) =
\begin{dcases}
(1,0,0), & X_I \in \mathcal{O}_0\\
(1,2,1), & X_I \in \mathcal{O}_1\\
(1,2,0), & X_I \in \mathcal{O}_2\\
(1,1,0), & X_I \in \mathcal{O}_{\geq 3}.
\end{dcases}$$
We will use structural induction on the structure classes. By Proposition \[prop:Rule0EverythingHasALowerDeficiencyOption\] and Lagrange’s Theorem, $X_I\in\mathcal{O}_m$ for $m\ge 1$ implies $\opt(X_I)\subseteq\mathcal{O}_{m}\cup\mathcal{O}_{m-1}$ and $\mathcal{O}_{m-1}\cap\opt(X_I)\ne\emptyset$.
If $X_I \in \mathcal{O}_0$, then $\type(X_I)=(1,0,0)$ since $I=G$. If $X_I \in \mathcal{O}_1$, then $\type(X_I)=(1,2,1)$ since $$\type(\opt(X_I))
=
\begin{dcases}
\{(1,0,0)\} & \text{if } \opt(X_I)\subseteq\mathcal{O}_0 \\
\{(1,0,0),(1,2,1)\} & \text{otherwise}
\end{dcases}$$ by induction. If $X_I \in \mathcal{O}_2$, then $\type(X_I)=(1,2,0)$ since $$\type(\opt(X_I))
=
\begin{dcases}
\{(1,2,1)\} & \text{if } \opt(X_I)\subseteq\mathcal{O}_1 \\
\{(1,2,1),(1,2,0)\} & \text{otherwise}
\end{dcases}$$ by induction. If $X_I \in \mathcal{O}_{3}$, then $\type(X_I)=(1,1,0)$ since $$\type(\opt(X_I))
=
\begin{dcases}
\{(1,2,0)\} & \text{if } \opt(X_I)\subseteq\mathcal{O}_2 \\
\{(1,2,0),(1,1,0)\} & \text{otherwise}
\end{dcases}$$ by induction. If $X_I \in \mathcal{O}_{\geq 4}$, then $\type(X_I)=(1,1,0)$, since every option of $X_I$ has type $(1,1,0)$ by induction.
Groups of even order
====================
Our main goal in this section is to compute the possible nim-values of $\gen(G)$ for a group $G$ of even order. Our approach is similar to that of Proposition \[prop:OrderGIsOdd\]. We want to recursively build all possible types of structure classes with a given deficiency from the already-computed types with lower deficiency. Unfortunately this simple approach is not sufficient to complete this computation, because it quickly becomes unwieldy for groups of even order as it yields an infinite number of potential types. However, we can use group theory to impose restrictions on the type calculations, which will reduce the number of potential types by eliminating many types that are not possible. We already have three of these restrictions: Propositions \[prop:Rule0EverythingHasALowerDeficiencyOption\], \[prop:Rule1EvenOptionsOnlyHaveEvenOptions\], and \[prop:Rule2EverythingHasAnEvenOption\]. In this section, we develop additional restrictions involving smoothness, which is the reason why we introduced extended types. We then use these restrictions to carry out the computation on extended types using the algorithm in Subsection \[subsec:algorithm\].
Additional option-type restrictions
-----------------------------------
In this subsection we present two option type restrictions that involve smoothness. A diagrammatic depiction of the statements are shown in Figures \[rules\](c) and \[rules\](d), respectively.
\[prop:ForcedEvenOptions1\] Let $X_I, X_J \in \mathcal{O}_n$ such that $X_J$ is an option of $X_I$. If $X_J$ is smooth, then so is $X_I$.
Suppose that $X_J$ has an option in $\mathcal{E}_n$, as shown in Figure \[proofFigs\](a). Then there is a $g \in G$ such that $X_{\lceil J,g \rceil} \in \mathcal{E}_n$. By Cauchy’s Theorem, there is an element $t$ in $\lceil J,g \rceil$ of order $2$. Since $X_I\in\mathcal{O}_n$, $t\notin I$. Then $\lceil I,t\rceil$ has even order, so $X_I$ has an option $X_{\lceil I,t\rceil}$ in $\mathcal{E}$. Since $I \leq \lceil I,t \rceil \leq \lceil J,t \rceil \leq \lceil J,g \rceil$ with both $X_I$ and $X_{\lceil J,g \rceil}$ in $\mathcal{D}_n$, we conclude that $X_{\lceil I,t \rceil} \in \mathcal{E}_n$. Thus, $X_I$ has an option $X_{\lceil I,t \rceil}$ in $\mathcal{E}_n$.
---------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------
![ \[proofFigs\] Figures for Propositions \[prop:ForcedEvenOptions1\] and \[prop:GeneralizedEvenNonContainment\].](Rule9detailed.pdf "fig:") ![ \[proofFigs\] Figures for Propositions \[prop:ForcedEvenOptions1\] and \[prop:GeneralizedEvenNonContainment\].](Rule10detailed.pdf "fig:")
(a) (b)
---------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------
\[prop:GeneralizedEvenNonContainment\] Let $G$ be a group of even order and assume that $X_I \in \mathcal{O}_n$ and $X_J \in \mathcal{O}_{n-1}$ such that $X_J$ is an option of $X_I$. If $X_J$ is rough, then so is $X_I$.
Assume $X_L$ is an even option of $X_I$. We will show that $X_L$ is in $\mathcal{E}_{n-1}$. Since $L$ has even order, it contains an element $t$ of even order. Let $K:=\lceil I,t\rceil$, as shown in Figure \[proofFigs\](b). Note that $X_K \in \mathcal{E}_n \cup \mathcal{E}_{n-1}$ by Proposition \[prop:Rule0EverythingHasALowerDeficiencyOption\] since $t$ has even order and $X_K$ is an option of $X_I$. By Lagrange’s Theorem, $X_{\lceil J,t\rceil} \in \mathcal{E}$. Since $X_J$ is rough, we have $X_{\lceil J,t \rceil} \not\in \mathcal{E}_{n-1}$. Hence $X_{\lceil J,t \rceil} \in \mathcal{E}_{n-2}$ by Proposition \[prop:Rule0EverythingHasALowerDeficiencyOption\]. We have $J=\lceil I,g \rceil$ for some $g \in G$. Since $X_{\lceil K,g \rceil} = X_{\lceil I,g,t \rceil} = X_{\lceil J,t \rceil} \in \mathcal{E}_{n-2}$, we conclude that $X_K \in \mathcal{E}_{n-1}$ by Proposition \[prop:Rule0EverythingHasALowerDeficiencyOption\]. Since $K$ is a subgroup of $L$, $X_L\in\mathcal{E}_{n-1}$, as well.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ \[rules\] Diagrams for option type restrictions. As with commutative diagrams, solid arrows are assumed to exist, indicating premises, while the dashed arrows are guaranteed to exist, indicating conclusions. A crossed-out dashed arrow is guaranteed not to exist. ](Rule1.pdf "fig:") ![ \[rules\] Diagrams for option type restrictions. As with commutative diagrams, solid arrows are assumed to exist, indicating premises, while the dashed arrows are guaranteed to exist, indicating conclusions. A crossed-out dashed arrow is guaranteed not to exist. ](Rule2.pdf "fig:") ![ \[rules\] Diagrams for option type restrictions. As with commutative diagrams, solid arrows are assumed to exist, indicating premises, while the dashed arrows are guaranteed to exist, indicating conclusions. A crossed-out dashed arrow is guaranteed not to exist. ](Rule9.pdf "fig:") ![ \[rules\] Diagrams for option type restrictions. As with commutative diagrams, solid arrows are assumed to exist, indicating premises, while the dashed arrows are guaranteed to exist, indicating conclusions. A crossed-out dashed arrow is guaranteed not to exist. ](Rule10.pdf "fig:")
\(a) Proposition \[prop:Rule1EvenOptionsOnlyHaveEvenOptions\] \(b) Proposition \[prop:Rule2EverythingHasAnEvenOption\] \(c) Proposition \[prop:ForcedEvenOptions1\] \(d) Proposition \[prop:GeneralizedEvenNonContainment\]
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Spectrum of extended types {#subsec:algorithm}
--------------------------
The next definition introduces the spectrum of extended types for groups of even order.
For $k\ge 0$ we let $$E_k :=\bigcup\{ \etype(\mathcal{D}_k(G)) \mid G \text{ is a group of even order}\}.$$ We also define $E:=\bigcup_k E_k$ to be the spectrum of extended types of groups of even order.
For $k,l\ge 0$ we let $E_k:=\bigcup_l E_{k,l}$, where $$\begin{aligned}
E_{k,l} & :=\bigcup\{ \etype(\mathcal{D}_{k,l}(G)) \mid G \text{ is a group of even order}\}.\end{aligned}$$ We also define $E:=\bigcup_k E_k$ to be the spectrum of extended types of groups of even order.
Finding this spectrum appears to be difficult, so we define the set of feasible extended types. The set of feasible extended types is easier to compute and turns out to be a superset of the spectrum of extended types. Recall that the nim-value of a game occurs as the second component of some extended type. This larger set of feasible extended types does not introduce extraneous nim-values because Example \[ex:nimExamplesParity\] demonstrates that we can find examples of groups with each of the nim-values.
The next definition reformulates Propositions \[prop:Rule0EverythingHasALowerDeficiencyOption\] and \[prop:Rule1EvenOptionsOnlyHaveEvenOptions\] in the language of extended types.
A pair $(A,B)$ in $\mathcal{P}(\mathbb{E})\times\mathcal{P}(\mathbb{E})$ is *$0$-feasible* if $B\neq \emptyset$ and $1\not\in\pi_1(A\cup B)$.
The four criteria in the following definition are reformulations of Propositions \[prop:Rule0EverythingHasALowerDeficiencyOption\], \[prop:Rule2EverythingHasAnEvenOption\], \[prop:ForcedEvenOptions1\], and \[prop:GeneralizedEvenNonContainment\], respectively.
A pair $(A,B)$ in $\mathcal{P}(\mathbb{E})\times\mathcal{P}(\mathbb{E})$ is *$1$-feasible* if it satisfies the following conditions:
1. $B\not =\emptyset$.
2. $0\in\pi_1( A\cup B )$.
3. $1\in\pi_4(A)$ implies $0\in\pi_1(A)$.
4. $0\in\pi_4(B)$ implies $0\not\in\pi_1(A)$.
We are ready to define our approximation to the $E_k$.
\[Ebardef\] We let $\bar{E}_{0} := \{(0,0,0,2)\}$. For $k,l\ge 1$, we recursively define $$\begin{aligned}
\bar{E}_{k,0} & :=\{\emex_p(\emptyset,B) \mid p\in\{0,1\}, B\in\mathcal{P}(\bar E_{k-1}), (\emptyset,B) \text{ is $p$-feasible} \}, \\
\bar{E}_{k,l} & :=\{\emex_p(A, B) \mid p\in\{0,1\}, (A,B)\in\mathcal{P}(\bar E_{k,l-1})\times\mathcal{P}(\bar E_{k-1}), (A,B) \text{ is $p$-feasible} \}, \\
\bar{E}_{k} & :=\bigcup\{ \bar{E}_{k,l} \mid l\in\mathbb{W} \}.\end{aligned}$$ We also let $\bar E:=\bigcup_k \bar{E}_k$ to be the *feasible spectrum* of extended types of groups of even order.
The reason why we are distinguishing between $E_k$ and $\bar{E}_k$ is that $\bar{E}_k$ may contain extended types that cannot exist for an actual group.
For $k\in\mathbb{N}$, $\bar{E}_{k,0}\subseteq \bar{E}_{k,1}\subseteq \bar{E}_{k,2}\subseteq\cdots \subseteq \bar{E}_{k}$.
Fix $k\in\mathbb{N}$. We will prove that $\bar{E}_{k,l} \subseteq \bar{E}_{k,l+1}$ by induction on $l$, and it is clear that $\bar{E}_{k,l} \subseteq \bar{E}_k$ by definition of $\bar{E}_k$.
Let $t \in \bar{E}_{k,0}$. Then $t=\emex_p(\emptyset,B)$ for some $p \in \{0,1\}$ and $B \in \mathcal{P}(\bar{E}_{k-1})$ such that $(\emptyset,B)$ is $p$-feasible. Since $\emptyset \in \mathcal{P}(\bar{E}_{k,0})$, we have $t=\emex_p(\emptyset, B) \in \bar{E}_{k,1}$.
Now suppose that $\bar{E}_{k,l-1} \subseteq \bar{E}_{k,l}$, and let $r \in \bar{E}_{k,l}$. Then $r=\emex_p(A, B)$ for some $p \in \{0,1\}$, $A \in \mathcal{P}(\bar{E}_{k,l-1})$, and $B \in \mathcal{P}(\bar{E}_{k-1})$ such that $(A,B)$ is $p$-feasible. Since $\bar{E}_{k,l-1} \subseteq \bar{E}_{k,l}$ by induction, we have $\mathcal{P}(\bar{E}_{k,l-1}) \subseteq \mathcal{P}(\bar{E}_{k,l})$ and so $A \in \mathcal{P}(\bar{E}_{k,l})$. Then $r=\emex_p(A, B) \in \bar{E}_{k,l+1}$, and we conclude that $\bar{E}_{k,l} \subseteq \bar{E}_{k,l+1}$.
The next result shows that every extended type that actually occurs in a group is a feasible extended type. This is no surprise. Both $E_k$ and $\bar E_k$ are recursively computed in the same way with the $\emex$ function, although the construction of the extended types that actually occur may have additional restrictions on the input than the construction of the feasible extended types. Thus, the creation of $E_k$ is a possibly more restrictive process, so it must be a subset of $\bar E_k$.
For all $k\in \mathbb{W}$, $E_{k} \subseteq \bar{E}_{k}$.
For a contradiction, assume there is a least $k$ such that $E_k \not\subseteq \bar E_k$. Since $E_0 = \{(0,0,0,2)\}=\bar E_0$, we may assume that $k \geq 1$. Then there must be a least $l$ such that $E_{k,l} \not\subseteq \bar E_{k,l}$, and let $t \in E_{k,l} \setminus \bar E_{k,l}$. Since $t \in E_k$, there is a finite group $G$ of even order and structure class $X_I \in \mathcal{D}_{k,l}(G)$ such that $t=\etype(X_I)$. Then $t = \emex_p(A,B)$ where $A:=\etype(\opt(X_I) \cap \mathcal{D}_{k,l-1}(G))$, $B:=\etype(\opt(X_I) \cap \mathcal{D}_{k-1})$, and $p:=\pty(|I|)$. By Propositions \[prop:Rule0EverythingHasALowerDeficiencyOption\], \[prop:Rule1EvenOptionsOnlyHaveEvenOptions\], \[prop:Rule2EverythingHasAnEvenOption\], \[prop:ForcedEvenOptions1\], and \[prop:GeneralizedEvenNonContainment\], we have that $(A,B)$ is $p$-feasible. Additionally, $$B \subseteq \etype(\mathcal{D}_{k-1}(G)) \subseteq E_{k-1} \subseteq \bar E_{k-1},$$ by the choice of $k$.
If $l=0$ then $t=\emex_p(A,B)=\emex_p(\emptyset,B) \in \bar E_{k,0}$, a contradiction. Thus, we may assume that $l \geq 1$. Then $$A =\etype(\opt(X_I) \cap \mathcal{D}_{k,l-1}(G)) \subseteq \etype(\mathcal{D}_{k,l-1}(G)) \subseteq E_{k,l-1} \subseteq \bar E_{k,l-1},$$ by the choice of $l$. Thus, $t=\etype(X_I)=\emex_p(A,B) \in \bar E_{k,l}$, a contradiction.
etype in $\bar E_k$ $k$
--------------------- -------
$(0,0,0,2)$ $0$
$(0,1,2,2)$ $1$
$(1,1,2,1)$ $1,2$
$(1,2,1,0)$ $1$
$(1,4,3,1)$ $1$
: \[table:eTypeTable\] The elements of $\bar E_k$ for each deficiency $k$. We found instances of every extended type with each deficiency using a computer search except for the five with a $\fbox{\text{box}}$ around them.
etype in $\bar E_k$ $k$
--------------------- ---------------------
$(0,0,2,2)$ $2$
$(1,0,1,1)$ $\fbox{2},3,\ldots$
$(1,0,2,1)$ $2,3$
$(1,1,0,0)$ $2$
$(1,3,0,0)$ $2$
: \[table:eTypeTable\] The elements of $\bar E_k$ for each deficiency $k$. We found instances of every extended type with each deficiency using a computer search except for the five with a $\fbox{\text{box}}$ around them.
etype in $\bar E_k$ $k$
--------------------- --------------
$(1,3,2,1)$ $\fbox{2}$
$(1,4,0,0)$ $2$
$(1,4,1,1)$ $\fbox{2}$
$(1,4,2,1)$ $\fbox{2}$
$(0,0,1,2)$ $3,4,\ldots$
: \[table:eTypeTable\] The elements of $\bar E_k$ for each deficiency $k$. We found instances of every extended type with each deficiency using a computer search except for the five with a $\fbox{\text{box}}$ around them.
etype in $\bar E_k$ $k$
--------------------- --------------
$(1,0,1,0)$ $3,4,\ldots$
$(1,0,2,0)$ $\fbox{3},4$
$(1,1,2,0)$ $3$
$(1,3,1,0)$ $3$
$(1,3,2,0)$ $3$
: \[table:eTypeTable\] The elements of $\bar E_k$ for each deficiency $k$. We found instances of every extended type with each deficiency using a computer search except for the five with a $\fbox{\text{box}}$ around them.
\[prop:BigProp\] The elements of $\bar E$ are the extended types shown in Table \[table:eTypeTable\].
Using Definition \[Ebardef\], we computed $\bar E$ using a GAP [@GAP] program. The code and its output are available on the companion web page [@WEB3]. The results show that the computation of each $\bar E_k$ finishes in finitely many iterations. This is indicated by the equality of $\bar E_{k,l}$ and $\bar E_{k,l+1}$ for some $l$. The results also show that $\bar E_5=\bar E_6$. Hence $\bar E=\bigcup_{k=0}^5\bar E_k$ and the whole computation finishes in finitely many steps.
Even though we computed $\bar E$ with a computer, we also verified the output by hand. Note that a human can eliminate many of the large number of cases that the computer checked.
We demonstrate the computation of $\bar E_1=\bar E_{1,1}$. We have $$\begin{aligned}
\bar E_0&=\{(0,0,0,2)\}, \\
\bar E_{1,0}&=\{(0,1,2,2),(1,2,1,0)\}, \\
\bar E_{1,1}&=\{(0,1,2,2),(1,1,2,1),(1,2,1,0),(1,4,3,1)\}, \\
\bar E_{1,2}&=\bar E_{1,1}.\end{aligned}$$ For example $(0,1,2,2)=\emex_0(\emptyset,\bar E_0)$ and $(1,4,3,1)=\emex_1(\bar E_{1,0},\bar E_0)$. Note that $(\emptyset,\bar E_0)$ is 0-feasible and $(\bar E_{1,0},\bar E_0)$ is 1-feasible. Also, note that $\bar E_1=\bar E_{1,1}$ since $\bar E_{1,2}=\bar E_{1,1}$.
We found examples of the extended types with every deficiency shown in Table \[table:eTypeTable\] using a computer search except for the five listed with a box around them. For instance, it is possible that $(1,0,2,0) \in \bar E_3 \setminus E_3$, but we have not found such an example. However, we have verified that $(1,0,2,0) \in E_4$ by looking at subgroups of `SmallGroup(500,48)` in GAP’s [@GAP] SmallGroup database.
Spectrum of nim-values
======================
We are now ready to determine the spectrum of nim-values of $\gen(G)$. If the order of $G$ is odd, then Proposition \[prop:OrderGIsOdd\] implies that the spectrum of nim-values of $\gen(G)$ is a subset of $\{0,1,2\}$. If the order of $G$ is even, then Proposition \[prop:BigProp\] and the containment $$\{\pi_2(\type(X_{\Phi(G)}))\mid G\text{ is an even group}\}\subseteq \pi_2(E)\subseteq\pi_2(\bar E).$$ shows that the spectrum of nim-values of $\gen(G)$ is a subset of $\{0,1,2,3,4\}$. The next example verifies that we have equality in both cases.
\[ex:nimExamplesParity\] The nim-values for the following odd and even-ordered groups were computed in [@ErnstSieben]. The groups listed in the table have the smallest possible order for the given parity and nim-value.
=0ex =0ex
$G$ $\mathbb{Z}_{1}$ $\mathbb{Z}_{3}^{3}$ $\mathbb{Z}_{3}$ $\mathbb{Z}_{2}^{3}$ $\mathbb{Z}_{4}$ $\mathbb{Z}_{2}$ $S_{3}$ $\mathbb{Z}_{6}$
----------------- ------------------ ---------------------- ------------------ ---------------------- ------------------ ------------------ --------- ------------------
$\nim(\gen(G))$ $0$ $1$ $2$ $0$ $1$ $2$ $3$ $4$
The discussion above together with Example \[ex:nimExamplesParity\] immediately implies the following result.
The spectrum of nim-values of $\gen(G)$ for groups with odd order is $\{0,1,2\}$. The spectrum of nim-values of $\gen(G)$ for groups with even order is $\{0,1,2,3,4\}$.
Now we have our main result.
The spectrum of nim-values of $\gen(G)$ for a finite group $G$ is $\{0,1,2,3,4\}$.
Open Problems and Conjectures
=============================
We close with a handful of open problems and conjectures.
1. One can find examples of all of the extended types for each deficiency in Table \[table:eTypeTable\] except for the boxed $(1,0,1,1)$, $(1,3,2,1)$, $(1,4,1,1)$, $(1,4,2,1)$ types from $\bar E_2$ and $(1,0,2,0)$ from $\bar E_3$. Do these five extended types actually occur with the appropriate deficiencies?
2. Computer experimentation shows that adding a type restriction corresponding to the following conjecture eliminates all but $(1,0,2,0)$ of the five boxed extended types from Table \[table:eTypeTable\]:
> *If $G$ is even and $k\ge 0$, then every $X_I\in\mathcal{O}_{k+1}$ has an option $X_L\in\mathcal{E}_k$.*
In fact, proving this conjecture for the special case when $X_I \in \mathcal{O}_2$ would be sufficient since the remaining four boxed extended types are all in $\mathcal{O}_2$. However, we were not able to prove this conjecture. A natural idea for a proof of this conjecture would be to prove the stronger statement:
> *If $G$ is even and $k\ge 0$, then for every $X_I \in \mathcal{O}_{k+1}$ there is a $t$ of even order such that $X_{\lceil I,t \rceil} \in \mathcal{E}_k$.*
Unfortunately, this statement is not true. In private correspondence, Marsden Conder provided a counterexample: `SmallGroup(240,191)` in GAP’s [@GAP] SmallGroup database, which is isomorphic to $\mathbb{Z}_2^4 \rtimes \mathbb{Z}_{15}$. However, this is not a counterexample for the original conjecture.
3. Does a type give algebraic information about the corresponding subgroup? For instance, does the type characterize what kind of maximal subgroups contain the subgroup?
4. In [@BeneshErnstSiebenDNG], the authors provide a checklist in terms of maximal subgroups for determining the nim-value of $\dng(G)$. Is there an analogous set of criteria for determining the nim-value of $\gen(G)$?
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Bob Guralnick and Marston Conder for giving us thoughtful examples to consider.
|
---
author:
- 'A. Shulevski'
- 'R. Morganti'
- 'P. D. Barthel'
- 'J. J. Harwood'
- 'G. Brunetti'
- 'R. J. van Weeren'
- 'H. J. A. Röttgering'
- 'G. J. White'
- 'C. Horellou'
- 'M. Kunert-Bajraszewska'
- 'M. Jamrozy'
- 'K. T. Chyzy'
- 'E. Mahony'
- 'G. Miley'
- 'M. Brienza'
- 'L. Bîrzan'
- 'D. A. Rafferty'
- 'M. Brüggen'
- 'M. W. Wise'
- 'J. Conway'
- 'F. de Gasperin'
- 'N. Vilchez'
bibliography:
- 'aa.bib'
title: 'AGN duty cycle estimates for the ultra-steep spectrum radio relic VLSS J1431.8+1331'
---
[Steep spectrum radio sources associated with active galactic nuclei (AGN) may contain remnants of past AGN activity episodes. Studying these sources gives us insight into the AGN activity history. Novel instruments like the LOw Frequency ARray (LOFAR) are enabling studies of these fascinating structures to be made at tens to hundreds of MHz with sufficient resolution to analyse their complex morphology.]{} [Our goal is to characterize the integrated and resolved spectral properties of VLSS J1431+1331 and estimate source ages based on synchrotron radio emission models, thus putting constraints on the AGN duty cycle.]{} [Using a broad spectral coverage, we have derived spectral and curvature maps, and used synchrotron ageing models to determine the time elapsed from the last time the source plasma was energized. We used LOFAR, Giant Metrewave Radio Telescope (GMRT) and Jansky Very Large Array (VLA) data.]{} [We confirm the morphology and the spectral index values found in previous studies of this object. Based on our ageing analysis, we infer that the AGN that created this source currently has very low levels of activity or that it is switched off. The derived ages for the larger source component range from around 60 to 130 Myr, hinting that the AGN activity decreased or stopped around 60 Myr ago. We observe that the area around the faint radio core located in the larger source component is the youngest, while the overall age of the smaller source component shows it to be the oldest part of the source.]{} [Our analysis suggests that VLSS J1431.8+1331 is an intriguing, two-component source. The larger component seems to host a faint radio core, suggesting that the source may be an AGN radio relic. The spectral index we observe from the smaller component is distinctly flatter at lower frequencies than the spectral index of the larger component, suggesting the possibility that the smaller component may be a shocked plasma bubble. From the integrated source spectrum, we deduce that its shape and slope can be used as tracers of the activity history of this type of steep spectrum radio source. We discuss the implications this conclusion has for future studies of radio sources having similar characteristics.]{}
Introduction {#c4:intro}
============
Active galactic nuclei (AGN) are cosmic powerhouses that produce prodigious amounts of energy and deposit it into the interstellar and the intergalactic medium (ISM, IGM). To estimate the total energy output of an AGN over cosmic time, it is necessary to determine its duty cycle, defined as the ratio of the time intervals during which the AGN is active and shut down. This is important, since we have a limited insight into the AGN duty cycle [@RefWorks:34; @RefWorks:178], while we have some knowledge of the energy output of an AGN during the duration of the active phase [@RefWorks:42; @RefWorks:43]. Evidence of multiple episodes in radio AGN activity has been growing steadily [@RefWorks:28; @RefWorks:262; @RefWorks:6; @RefWorks:2]. [@RefWorks:96] have shown that the period of time that an AGN is active is $ 10^{7} - 10^{8} $ years in their source sample of several AGN radio relics, while the time between the active phases is an order of magnitude shorter. The missing pieces of information are important for various reasons. The (total) energy budget is important for studying galactic evolution because the AGN energy output influences the cooling of gas and galactic assembly. Another important area of astrophysics that is affected is the star formation history in the host galaxy, since AGN “feedback” can potentially quench star formation through influencing the ISM. The AGN fuelling [@RefWorks:105; @RefWorks:143] can potentially also be interrupted.
An observational signature of a “switched off” AGN is the steep spectrum of the associated radio emission at high frequencies ($ \alpha < -1.5 $)[^1], leading to brighter emission at low frequencies. The reason for this is that to first order (neglecting other energy loss mechanisms), the plasma ejected from the AGN during its active phase will mainly lose energy through synchrotron radiation and inverse Compton (IC) scattering of electrons off cosmic microwave background (CMB) photons. As a consequence, high energy particles that radiate the most at high radio frequencies lose their energy fastest. This produces a characteristic synchrotron radio spectrum that evolves over time in such a way that most of the particles that still radiate after the AGN has shut down are low energy particles, and the radio emission is strongest at low frequencies. Thus, low frequency radio surveys are the best tool for detecting such sources.
The organization of the paper is as follows. In section \[c4:intro:target\] we describe our target in some detail and outline the goals of our study. Section \[c4:obsdata\] describes our data set and gives an overview of the data reduction procedures. In Section \[c4:res\] we present the results of the data analysis, and we discuss our findings in Section \[c4:disc\]. We finish by presenting our conclusions in Section \[c4:fin\].
VLSS J1431.8+1331 {#c4:intro:target}
=================
Several steep spectrum sources were studied by [@RefWorks:54; @RefWorks:145], identified by matching the VLSS and NVSS[^2] catalogues. One of these, VLSS J1431.8+1331, was identified as being potentially an AGN radio relic. These structures are radio remnants of a past AGN activity episode, not to be confused with radio halos and relics, which are a consequence of particle acceleration and ensuing radio synchrotron emission during galaxy cluster mergers. The object has an ultra steep ($ \alpha \simeq -2 $) spectrum at high frequencies with a radio morphology pointing towards a complex activity history. It presents two distinct source regions, a larger one and a smaller one (henceforth labelled A and B, respectively), connected by a faint bridge of radio emission (Figure \[c4:SDSS\_LOFAR\]). Region A appears to contain a faint radio core and has a steep spectrum showing little spectral curvature compared to Region B whose spectrum flattens out at low frequencies.
The host galaxy of VLSS J1431.8+1331 is the brightest cluster galaxy (BCG) of the galaxy cluster maxBCG J217.95869+13.53470 [@RefWorks:144], located at a redshift[^3] of $ z = 0.1599 $. The cluster has been observed in X-rays by [@RefWorks:86] using XMM Newton. They derived temperature excesses to the north and west of the BCG and a slight increase in entropy coinciding with the south-west component of radio emission. Based on these observations, they propose that maxBCG J217.95869+13.53470 has interacted with a smaller group of galaxies located to the north-east of the BCG. According to the analysis presented, the smaller galaxy group came from the south-east, and we are observing the system a short while after the closest approach. Shocks related to this process might have compressed a bubble of plasma expelled by the AGN and revived its synchrotron emission, thus giving rise to the smaller region of radio emission to the south-west.
![LOFAR HBA image contours (red) of VLSS J1431.8+1331 (centred on 148 MHz and with a bandwidth of 48 MHz) superposed on a Sloan Digital Sky Survey (SDSS) greyscale image of the galaxy cluster maxBCG J217.95869+13.53470. Eleven log-spaced contour levels are shown in red, spanning the surface brightness interval between $ -10 $ and $ 100 $ [mJy beam$^{-1}$]{}, with a beam size of $ 10\farcs5 \times 8 \arcsec $. The white cross marks the position of the bright cluster galaxy (BCG) AGN host, and the source components are labelled.[]{data-label="c4:SDSS_LOFAR"}](J1431_SDSS_cnt_nw.pdf){width="90mm"}
In this work, we aim to investigate whether Region B represents a merger shock rejuvenated synchrotron emission region. We also want to derive more information about the physical nature of the various source components of VLSS J1431.8+1331. The co-existence of a (possible) rejuvenated plasma bubble and an AGN relic provides us with a serendipitous chance to compare their spectral properties (at high resolution and low frequencies). Using observations made with the LOFAR telescope’s [@RefWorks:157] high band antennas, we perform spectral index mapping down to around $ 140 $ MHz with a resolution comparable to that of the Giant Metrewave Radio Telescope (GMRT) and the Jansky Very Large Array (VLA) study made by [@RefWorks:54] at higher frequencies. The resulting broad-band data set has enabled us to study the ageing of the plasma in detail across the source. By doing this, we can put constraints on the timescales involved in the past activity of the AGN responsible for the radio emission, and compare our findings with the results of previous studies.
Observations and data reduction {#c4:obsdata}
===============================
The target was observed on the night of February 17, 2013 for a total on source time with LOFAR’s high band antennas (HBA) of 5.7 hours. The HBA observation was taken in interleaved mode, i.e. 3C 295 was observed as a calibrator source for two minutes, followed by a scan of the target of 11-minute duration with a one-minute gap between calibrator and target scans, allowing for beam forming and target re-acquisition. The 325 sub-bands were recorded covering 64 MHz of bandwidth between 116 MHz and 180 MHz. Each sub-band has 64 frequency channels spanning a bandwidth of 200 kHz. The integration time was set to two seconds for both calibrator and target. Four polarizations were recorded. The HBA station field of view (FoV) spans around four degrees full width at half maximum (FWHM).
--------------------------- ---------------------------------------
Central frequency \[MHz\] 140
Bandwidth \[MHz\] 64
Integration time 2 seconds
Observation duration 6 hours
Polarization Full Stokes
UV range $ 0.25 k\lambda \, - \, 20 k\lambda $
--------------------------- ---------------------------------------
: LOFAR observation configuration[]{data-label="c4:table:1"}
The data were pre-processed by the observatory pipeline [@RefWorks:180] as follows. Each sub-band was automatically flagged for RFI using the AOFlagger [@RefWorks:133] and averaged in time to ten seconds per sample and in frequency by a factor of 16, which gives us four channels per sub-band in the output data. The calibrator data were used to derive amplitude solutions for each station using the BlackBoard Self calibration - BBS [@RefWorks:182] tool that takes the time and frequency varying LOFAR station beams into account. The calibrator flux density scale was set according to [@RefWorks:181]. The obtained (complex) station gain solutions were used to fix the target amplitude scale.
Only the Dutch LOFAR stations were used. The amplitude-corrected target visibilities were phase-(self)calibrated incrementally, using progressively longer baselines to get to the final resolution (Vilchez et al. in prep.). The initial phase calibration model was derived from the VLSS catalogue covering the FoV out to the first null of the station beam. It included spectral index information for each source. Before initializing the calibration, we have concatenated the data into 4 MHz (20 SB) groups previously averaging each sub-band to one frequency channel. We have chosen this setup to maximize the S/N while maintaining frequency dependent ionospheric phase rotation to a manageable level. We did not perform any directional solving and did not explicitly correct for ionospheric (direction dependent) effects.
The station beams are complex-valued and time-, frequency-, and direction-dependent, and are not the same for all of the stations. The imaging was done by using the LOFAR imager [@RefWorks:183], which incorporates the LOFAR beam and uses the A-projection [@RefWorks:184] algorithm to image the entire FoV. The imager does not (at this stage) implement spectral index correction when it does the multi-frequency imaging. We used Briggs weights [@RefWorks:185] with the robustness parameter set to $ -2 $ (uniform weights) and performed a UV plane selection to include all of the baselines to achieve the highest possible resolution. Discarding 8 MHz from either edge of the band, we have imaged the data so as to obtain a final HBA data set of six images of 8 MHz bandwidth each. We averaged these images together to obtain one broadband image. Deconvolution artefacts are present around the sources in the image, mostly due to the residual calibration errors caused by the ionosphere. The observation was performed at night when the ionospheric conditions are expected to be most favourable. Based on our inspection of the phase calibration solutions for the target field, we judged the ionospheric conditions to be predominantly calm for the duration of the observing run.
Given the low declination of the target, some uncertainties still exist about the accuracy of the LOFAR in-band spectral index. Thus, we have decided to use only two LOFAR HBA images (given in table \[c4:table:2\]) in our spectral analysis.
In addition to our LOFAR data, we use GMRT and VLA data kindly supplied by RvW to constrain the spectral index and curvature. The properties of the dataset are listed in Table \[c4:table:2\]. For the purposes of our work, we list the properties of the smoothed GMRT and VLA images (adapted to the LOFAR HBA synthesized beam size). The r.m.s. noise values for each of the flux measurements were derived according to [@RefWorks:141], taking the background noise away from bright sources and scaling for the contribution of the size of the measurement region used on the target into consideration. To account for the flux uncertainties stemming from the 3C 295 flux scale and the imperfections of the beam model, we added 20% of the measured flux density value in quadrature to the derived noise in the case of the LOFAR measurements. The same was done for the GMRT measurements, but the flux density value added was 5% [@RefWorks:186]. The details of the GMRT and VLA data sets and the reduction procedure can be found in [@RefWorks:54; @RefWorks:145].
Instrument $ \nu $ \[MHz\] $ \Delta \nu $ \[MHz\] $ \sigma $ \[[mJy beam$^{-1}$]{}\]
------------ ----------------- ------------------------ ------------------------------------
LOFAR 144 48 0.5
LOFAR 135 8 2.3
LOFAR 145 8 1.8
GMRT 325 32 0.1
GMRT 610 32 0.1
VLA 1425 100 0.05
VLA 1425 100 0.02
: Details about the images used in our analysis.[]{data-label="c4:table:2"}
Results {#c4:res}
=======
Radio morphology {#c4:morph}
----------------
{width="\textwidth"}
{width="\textwidth"}
Figure \[c4:tgt\_regions\] shows the LOFAR image of the target spanning the HBA band. We can discern two different source regions [described in detail by @RefWorks:145], the larger and brighter one (A) to the north-east, and a smaller region (B) to the south-west connected by a faint “bridge” of radio emission. Both regions seem to be slightly curved to the north-east. This is more noticeable for Region B in the higher resolution image, outlined in contours in the right-hand panel. A hint of a faint radio core is visible in the high resolution VLA image of [@RefWorks:145], while the brightest part of the target in the LOFAR image is the south-eastern part of Region A.
The right-hand panel in Figure \[c4:tgt\_regions\] shows the flux density measurement regions used in the subsequent analysis. The measurement areas have been chosen to be of similar size to the synthesized LOFAR beam (around 24.5 kpc across). We have measured the flux density in each area using our maps (LOFAR, GMRT, and VLA), thus covering about a factor of 10 in frequency range (144 to 1425 MHz). All of the maps were convolved to the same restoring beam size as that of the HBA beam of $ 10\farcs5 \times 8\arcsec $.
Spectral analysis and radiative ages {#c4:spec}
------------------------------------
In what follows, we use the LOFAR measurements to extend the spectral index analysis to the lowest frequencies ever for this object. Our goal is to trace the lowest energy particle population to better constrain the radio spectra at low frequencies. We also map the radiative ages over the source surface by fitting synchrotron ageing models to our data.
### Spectral index and curvature {#c4:specidxcrv}
We have derived spectral index and spectral curvature ($ \mathrm{SPC} = \alpha_{\mathrm{low}} - \alpha_{\mathrm{high}} $) maps of the target. In Figure \[c4:spix\_curv\_maps\] we present the spectral index derived from the LOFAR HBA, GMRT, and VLA maps and the corresponding spectral curvature across the source. The spectral indices were derived by modified[^4] least squares fitting of a power law to the data for each pixel of the maps. Pixels were blanked that had a value below $ 15 \sigma $ for each map when deriving the spectral index and $ 3 \sigma $ when deriving the spectral curvature. This was mainly done to avoid the artefacts present around the target in the LOFAR HBA images. The maximum UV distance of the LOFAR data of 20 $ k\lambda $ corresponds to the one of the GMRT and VLA of 22 $ k\lambda $. The minimum baseline length of the used data sets was 0.25 $ k\lambda $ (the VLA data were taken in the A + B + C array configuration); i.e., the largest spatial scale that can be detected in the images is 13.7 .
Region A shows a relatively steep low frequency spectral index of $ \alpha \, \sim \, -1.2 $ around the radio core and the western edge. The spectral index steepens to around $ \alpha \, \sim \, -2 $ going towards the edges. The spectral curvature map indicates that for this source component, the spectrum is flat around the radio core, with breaks at higher frequencies developing at the western edge and in the south-east.
### Ages {#c4:ages}
Synchrotron spectral ageing theory was established by [@RefWorks:126]. Later, these foundations were expanded by [@RefWorks:187; @RefWorks:125], [@RefWorks:278; @RefWorks:188; @RefWorks:82], and [@RefWorks:34] among others. Appendix A gives an overview of the ageing models used in this work.
When the AGN is active, it accelerates charged particles, resulting in their having a power law energy distribution. The particles lose energy mainly by radiating synchrotron radiation and through IC scattering off of CMB photons. The radio spectrum has a spectral index $ \alpha_{0} $ at low frequencies, which depends on the energy distribution of the radiating particles and which is commonly known as the injection (spectral) index. Typical values for the injection spectral index are $ \alpha_{0} \in [-0.6, -0.8] $. At high frequencies, due to the preferential cooling of high energy electrons through synchrotron radiation and IC scattering, the radio spectrum steepens, and a spectral break develops. After the AGN shuts down, the active radio regions stop being replenished with energetic particles, and the spectrum exhibits a steep drop-off at high frequencies.
To get an estimate on the ages of the different regions, we fitted synchrotron ageing models to the data. The fitting was performed using the Kapteyn package [@RefWorks:245] utilizing a Python based code that implements the models (see Appendix A for details). We used models with a Jaffe-Perola energy-loss term, assuming instantaneous particle injection and continuous particle injection followed by ageing (JP and KGJP, respectively) to estimate the ages. We also tried a continuous-injection-only (CI) model for the source area containing the radio core (Area 4) and found it to be inconsistent with the data (Figure \[c4:reg\_ages\]).
Determining the radiative ages requires knowledge about the magnetic field. We calculated it according to [@RefWorks:14] using the assumption of equipartition between the energy contained in the field and relativistic particles, using values of $ 10 $ MHz and $ 10 $ GHz for the cutoff frequencies and an electron-to-proton ratio equal to unity. The calculation was done for Regions A and B separately, taking the path length through the regions to equal the average of their major and minor axes as projected on the sky. We set the spectral index to $ \alpha_{144}^{1425} \, = \, -1.9 $ for both regions for the purposes of this calculation [estimated from our spectral index map and the ones given in @RefWorks:145]. Using the average value of the derived magnetic field for both regions, we find that $ B \, = \, 4.4 \, \mu $G.
This equipartition value for the magnetic field needs to be corrected, taking a cutoff in the particle energy into account instead of a cutoff in the frequency of the radiation [@RefWorks:272]. We adopted the method proposed by [@RefWorks:279] [see also @RefWorks:273]. The ratio of energy between protons and electrons is assumed be unity. Furthermore, we used spectral index values of $ -1 $ and $ -1.8 $ over the entire energy band, as well as two different values for the low energy cutoff, expressed via the electron Lorentz factor $ \gamma $. The derived magnetic field values are given in Table \[c4:table:mag\]. Different parameter assumptions change the magnetic field by a factor of 3. We adopt a magnetic field value of $ 4 \, \mu $G in our further analysis, which agrees with what is found for similar sources [@RefWorks:34].
[p[2cm]{} p[2cm]{} p[2cm]{} p[1cm]{}]{}\
$ \alpha $ & $ \gamma_{\mathrm{min}} $ & $ B_{\mathrm{eq}} $\[$\mu$G\] & $ B_{\mathrm{eq}}^{\mathrm{cor}} $\[$\mu$G\]\
\
-1 & 100 & 2.18 & 3.86\
-1 & 700 & 2.18 & 2.37\
-1.8 & 100 & 4.40 & 11.26\
-1.8 & 700 & 4.40 & 3.925\
[p[0.8cm]{} p[1.9cm]{} p[1.9cm]{} p[1.8cm]{} p[0.6cm]{}]{}\
$ N^{o}_{=} $ & $ t_{\mathrm{on}} $ \[Myr\] & $ t_{\mathrm{off}} $ \[Myr\] & $ \alpha_{0} $ & $ \chi^{2} $\
\
\
\
3 & $ 20.4 \pm 0.0 $ & $ 108.4 \pm 2.8 $ & $ -0.88^{+0.28}_{-0.38} $ & 0.17\
4 & - & $ 60.0 \pm 3.5 $ & $ -1.12 \pm 0.38 $ & 0.72\
6 & $ 22.9 \pm 6.3 $ & $ 117.4 \pm 0.0 $ & $ -0.74^{+0.14}_{-0.33} $ & 0.47\
7 & - & $ 48.6 \pm 5.0 $ & $ -1.50^{+0.47} $ & 0.23\
8 & - & $ 91.3 \pm 9.3 $ & $ -1.50^{+0.47} $ & 1.72\
9 & - & $ 113.2 \pm 3.5 $ & $ -1.03^{+0.38}_{-0.33} $ & 0.01\
10 & - & $ 104.7 \pm 2.7 $ & $ -0.93^{+0.33}_{-0.28} $ & 0.01\
\
\
\
11 & - & $ 69.8 \pm 10.9 $ & $ -1.50^{+0.43} $ & 2.72\
12 & - & $ 97.2 \pm 0.0 $ & $ -1.50^{+0.66} $ & 1.91\
13 & - & $ 60.6 \pm 13.9 $ & $ -1.50^{+0.81} $ & 1.25\
\
\
\
14 & - & $ 121.4 \pm 5.9 $ & $ -0.60_{-0.29} $ & 1.49\
15 & - & $ 131.6 \pm 13.5 $ & $ -0.60_{-0.66} $ & 0.51\
16 & $ 7.8 \pm 7.0 $ & $ 132.4 \pm 2.8 $ & $ -0.6 $ & 4.98\
17 & - & $ 123.1 \pm 7.5 $ & $ -0.6_{-0.14} $ & 2.34\
We fitted the JP and KGJP models to our data for each area of the source using the following procedure. The free parameters of the model fit were the active and switched-off times, as well as a flux scaling factor, while we kept the magnetic field and injection spectral index fixed. To understand the injection spectral index dependence, we took $ 20 $ different values for the injection spectral index evenly distributed between $ -0.6 $ and $ -1.5 $ and performed the model fitting for each source area separately using different injection spectral index values.
We find that the best fit to the data is obtained for the models that use the JP assumption for the energy loss. Most of the source areas are best fitted with a model that assumes an infinitesimally short duration particle acceleration phase and ageing (JP), while some are best fitted with a model that assumes a period of continuous injection and then ageing (KGJP). Our best-fit parameter values are given in Table \[c4:table:3\], and the derived ages for the various source areas are presented in Figure \[c4:reg\_ages\]. The confidence intervals for the injection spectral index parameter were obtained by considering all of the injection spectral index values for which the model was accepted (irrespective of the $ \chi^{2} $ value of the fit). The adopted best-fit value of the injection spectral index was the value used in the model fit with the lowest $ \chi^{2} $ value. The ages and their corresponding confidence intervals were the ones obtained by the model fit using the best-fit injection spectral index value.
For Regions 7, 8, 11, 12, and 13, the best-fit injection index was the steepest in the range, and we could only place an upper limit on it, while the inverse was true for Regions 14, 15, and 17. For Region 16, the only model fit that was not rejected was the one with an injection index of $ \alpha_{0} = -0.6 $. With all this in mind, judging from the regions for which we have constrained the injection index, we deduce that the youngest part of the source is located in Region A, in the area around the faint radio core (Region 4), while the oldest source region is Region B. The derived ages range from 60 to 130 Myr. The models indicate that for Region B the injection spectral index has a value of $ \alpha_{0} \, = \, -0.6 $, while for Region A, the best fit models are those with the injection index of around $ \alpha_{0} \, = \, -1 $. We discuss the implications of these findings further in Section \[c4:disc\].
### Colour-colour representation and shift diagrams {#c4:colour-shift}
The peculiar morphology of the source suggests that complex processes are shaping the emission regions and influencing their energy balance. We investigate whether the energy loss is predominantly due to synchrotron ageing and IC losses.
To do this and to gain more insight into the plasma properties across the source, we employed the colour-colour representation described by [@RefWorks:189]. Such plots provide a simple overview of the measurements compared to different ageing models and allow for easier determination of which theoretical models describe the measurements best. Also, the outlined shapes in this representation are conserved with respect to changes in the magnetic field and adiabatic compression or expansion.
We have constructed two spectral indices from our data, low and high. The low spectral index was obtained by fitting a first-order polynomial through the $ 135 $, $ 145 $, and $ 325 $ MHz data points for each region, and the high frequency spectral index was obtained by a first-order polynomial fit through the $ 325 $, $ 610 $, and $ 1425 $ MHz data points. The resulting colour-colour plots for a representative set of source regions are shown in Figure \[c4:colour-colour\]. This procedure is similar to the spectral curvature derivation described previously, with the curvature being the difference between the low and high spectral indices. Plotting these two indices against each other is another way of distinguishing source regions with different properties that can give additional insight into the physical processes at work in the emission regions.
In the colour-colour plot for source Region A we have taken only the measurement areas into account for which we have a good constraint on the injection index. We did not perform such a pre-selection for the colour-colour plot for Region B, since in that case we have a more limited number of measurement areas.
We can see that Area $ 4 $ is close to a power law, but that it is best described by KGJP or JP models having a steep injection index (determined by the intersection point of the ageing models and the power law line on the diagram) of $ \alpha_{0} = -1.5 $. Measurement areas $ 3 $, $ 6 $, $ 9 $, and $ 10 $ are best described by a KGJP model with an injection index of $ \alpha_{0} = -1 $. The areas in the source Region B show larger scatter and are not represented well with a single ageing model; a KGJP or JP model with injection index of $ \alpha_{0} < -0.6 $ is indicated by the measurements and supported by the derived injection indices and ages in Section \[c4:ages\].
Following the reasoning of [@RefWorks:190], [@RefWorks:191; @RefWorks:192], and [@RefWorks:193], we performed a spectral shift analysis on Regions $ 3 $, $ 6 $, $ 9 $, and $ 10 $ to gain further insight into our target and test the assumptions we made during the spectral analysis. We have chosen these regions since they outline a single curve in the colour-colour space that is consistent with a particular ageing model.
The idea behind the shift technique is as follows. Assuming that the energy loss of the particles is predominantly through synchrotron and IC mechanisms and that the magnetic field does not vary significantly across the source, then the spectra for all of the regions are self-similar, the only difference between them being the position of the break frequency. Moreover, all of the spectra should align by shifting them in the frequency-flux density plane if the dominant energy loss mechanism is synchrotron radiation and IC scattering and if our line of sight samples similar source regions. We chose Region $ 9 $ as reference (it has the best model fit) and have shifted the spectra for Regions $ 3 $, $ 6 $, and $ 10 $ so that their break frequencies matched that of Region $ 9 $. The break frequencies for the KGJP model were determined using [@RefWorks:34]:
$$\label{c4:ageq}
\nu_{\mathrm{b}}^{\mathrm{low}} = \left(\frac{1590}{(B^{2} + B_{\mathrm{CMB}}^{2})t_{\mathrm{s}}}\right)^{2} \cdot \frac{B}{(1 + z)}$$
and
$$\nu_{\mathrm{b}}^{\mathrm{high}} = \nu_{\mathrm{b}}^{\mathrm{low}}\left( \frac{t_{\mathrm{s}}}{t_{\mathrm{off}}} \right)^{2}$$
where $ t_{s} \, = \, t_{on} \, + \, t_{off} $. For the JP model, we set $ t_{on} \, = \, 0 $. The procedure has produced the shift diagram for these regions, shown in the left-handed panel of Figure \[c4:shifts\]. It shows by how much the spectrum of each region was shifted in the $ \log(\nu) - \log(S) $ plane. The data points for all of the regions after the shift, producing the global spectrum, are shown in the right-handed panel of Figure \[c4:shifts\]. We also fitted JP and KGJP models (Appendix A) through all of the shifted data points to show that indeed the shifted spectra are self-similar and taken together produce a spectrum that is consistent with radiative ageing and IC scattering being the dominant energy loss mechanism of the electrons.
The shifts made to the individual spectra to line them up can give us information about the physical parameters of the source [@RefWorks:190; @RefWorks:193]. Specifically, shifts in the $ log(\nu) $ axis relate to $ \gamma^{2}B $, while shifts along the $ log(S) $ axis depend on $ N_{\mathrm{tot}}B $, where $ N_{\mathrm{tot}} $ is the total number of energetic particles in the source volume for a given beam size.
We can fit a straight line to the shifts in the $ log(\nu) - log(S) $ plane (Figure \[c4:shifts\]) for the measurement areas belonging to source component A. This means that the spectra of individual regions are self-similar, meaning that there are no significant variations in either particle energy or magnetic field value from one area to another, and that the spectral shapes are the result of synchrotron ageing and IC losses. Moreover, the slope of the fit indicates the injection index.
That the shifted data are fitted so well using radiative ageing models shows that, indeed, synchrotron ageing and IC scattering are the dominant energy loss mechanisms. The values we get for the injection index and that they match what we had used for $ \alpha_{0} $ previously also serve to strengthen the inner consistency of our analysis.
### The steepness of the injection index {#c4:steepinj}
The derived results on the ages of source Region A are consistent with it being a relic from a past AGN activity episode; this claim is mainly supported by the derived age of Area $ 4 $ compared to its surroundings and its identification with the faint radio core. There is consistency throughout our model fits: for Region A, the models with steeper injection index ($ \alpha_{0} \in [-0.9, -1.5] $) than what is usually accepted as standard value ($ \alpha_{0} \in [-0.6, -0.8] $) are the ones that best fit the data. We discuss the implications of the steep injection index further in Section \[c4:disc\].
Interestingly, [@RefWorks:251] find larger-than-expected injection indices for their sample of FRII radio galaxies.
### The integrated flux density spectrum {#c4:intspec}
The integrated spectrum is slightly curved and steep. To see how spectral ageing in different regions of the source can affect the integrated spectrum, we fitted a KGJP model to the integrated flux density measurements, and compared the results with the region fits done previously.
![JP and KGJP model fits for Regions 4 (blue) and 16 (red), along with the corresponding data points (LOFAR HBA, GMRT, and VLA). The sum of the fits for all regions is represented by the dashed green line. The integrated flux density data points (VLSS, LOFAR HBA, GMRT, and VLA) are best fitted by a KGJP model (black line) with an injection index of $ \alpha_{0} \, = \, -1.23 \pm 0.37 $ and $ t_{\mathrm{on}} \, = \, 0.1 $ Myr, $ t_{\mathrm{off}} \, = \, 77.7 $ Myr. The measurement regions do not cover the source completely at lower frequencies, so the sum of the fits does not precisely follow the integrated flux density. A CI model that is rejected by the integrated flux data is represented with a green line.[]{data-label="c4:int_spec"}](J1431_int_regions_fit.pdf){width="75mm"}
In Figure \[c4:int\_spec\] we show the integrated flux density measurements of VLSS J1431.8+1331 from our LOFAR HBA, GMRT, and VLA data sets along with the VLSS catalogue value. We can see that the integrated flux density follows the sum of the fitted flux densities for the different regions; the offset can be explained by the fact that our measurement regions do not perfectly cover the entire surface of the target. The spectra of different regions have breaks at different frequencies and have different curvatures. When they combine, the effect is that the integrated spectrum has smaller curvature. This demonstrates that we should be careful when interpreting integrated spectral shapes of complex sources. The activity history, coupled with the detailed source morphology and the relative contribution of different source components can influence the shape of the integrated spectrum. We discuss the implications of this further in Section \[c4:disc\].
Discussion {#c4:disc}
==========
The spectral index maps we derived using LOFAR data, which extend a factor of two lower in frequency than previous studies of this object, confirm that there is a frequency break in the spectrum of source Region B, while the larger one, Region A, still retains a steep spectral index at lower frequencies.
Our spectral mapping is in good agreement with the findings of [@RefWorks:145]. It confirms that the spectral index of Region B derived at lower (LOFAR HBA) frequencies continues to flatten out and reaches injection values. We are observing the particles that still have kept their energy from the last episode of acceleration. The data suggest a spectral break in the spectrum of this region at relatively low frequencies. This conclusion is supported by the spectral curvature map in Figure \[c4:spix\_curv\_maps\].
Our synchrotron ageing models indicate that the youngest radio plasma in the target (around 60 Myr old) is located in the vicinity of the fading AGN core (Area 4 in Figure \[c4:reg\_ages\]). This is what would be expected if the AGN was diminishing in activity over time for a FRI-like radio source morphology. The rest of Region A (apart from Areas 4, 7, and 8) is older (around 100 Myr). The oldest part of the source is Region B, with ages ranging from 120 Myr to 130 Myr. What we are seeing, then, could be a signature of an AGN hosted by the BCG of the cluster that was active around 120 Myr ago when it produced source Region B and the outer part of Region A. Then, the AGN switched to very low levels of activity around 60 Myr ago.
We can use morphological and kinematic arguments as an independent check of the timescales involved. The relative speed of maxBCG J217.95869+13.53470 with respect to the group of galaxies to the north-east is around 3200 [$\,$km$\,$s$^{-1}$]{} [@RefWorks:86]. It is suspected of being a separate smaller cluster; one of its galaxies hosts a spectrally confirmed, currently active AGN. We assume that this value is equally distributed between the two clusters and that the position of the BCG of maxBCG J217.95869+13.53470 at the beginning of the outburst was at the half-way point between Regions A and B. Then, the activity began to decrease as the BCG followed the cluster motion from south-west to north-east. The motion of the Intra Cluster Medium (ICM) halo gas (centred on the cluster core) has bent the radio emission, giving it its current appearance. In this scenario, the distance from the assumed position where the AGN in the host galaxy started its previous outburst of activity to the present position of the galaxy is 104 [kpc]{}. Moving at the previously mentioned speed, it would take 32 Myr for the AGN host galaxy to arrive to its present position, which is around four times shorter than the estimated age of Region B. Large uncertainties in the cluster velocity estimates propagate into the kinematic timescale, and we should keep this in mind.
What does the difference in the observed injection spectral indices between the two source regions mean? If we assume that Region A was produced as a result of an AGN activity, the steeper injection spectral indices we infer for its plasma broadly agree with what was observed for several radio galaxies by [@RefWorks:251] (The radio galaxies in their sample are FRIIs, while J1431.8+1331 was probably an FRI radio galaxy.) Region B would be plasma released in a previous episode of AGN activity that has faded, losing its energy through radiation and expansion. [@RefWorks:86] estimate that it would take Region B around 300 Myr to get to its present position, rising as a buoyant bubble. It could have been compressed by a merger shock produced by the interaction of the two galaxy clusters (the BCG of one that hosts J1431.8+1331). The shock compression would cause the plasma bubble to gain energy, resulting in a boost in its radio emission.
The compression of the plasma in Region B would lead to the formation of a sheath-like radio emission region, which as time passes, would evolve into a filamentary structure [@RefWorks:146]. When looking at the radio morphology of Region B, this is a possible scenario if we assume that we see it nearly edge on.
In relation to the compression shock scenario, another explanation of the source spectral properties presents itself. Its plausibility increases if we look at the injection spectral index values for the different measurement regions given in Table \[c4:table:3\]. The regions for which we only have upper limits to the injection index and which have the steepest injection index are situated in Region A and in between Regions A and B. This may point to the possibility that what we are actually observing is the spectrum above the break frequency for Region A, meaning that the ages we derive for it are lower limits to the true age. In this context, Region B can be even older, and the shock compression could have not only revived its radio emission but also shifted its break frequency to the higher values that we observe [@RefWorks:274].
We can estimate the maximum electron ages using Eq. (\[c4:ageq\]) by taking the minimum value of the magnetic field of $ B_{\mathrm{min}} = \frac{B_{\mathrm{CMB}}}{\sqrt{3}} = 2.52 \, \mu $G. Assuming a value for the break frequency of 60 MHz, we get maximum ages of around $ 350 $ Myr, which make the scenario mentioned above plausible. Under the same assumptions, but using a magnetic field value of $ 10 \, \mu $G, we find an age of $ 160 $ Myr, which is more than a factor of two smaller. As can be expected, with everything else being equal, uncertainties in the magnetic field determination (Table \[c4:table:mag\]) can affect the derived ages.
As we have shown in Section \[c4:disc\], the integrated flux density spectrum is steep and curved. This is caused by superposition of different source regions that have spectral breaks at different frequencies. It is similar to the spectra of the cluster sources observed by [@RefWorks:34]. The integrated source spectrum contains information about the detailed spectral properties, depending on which source region is dominating the radio emission. In the case of J1431.8+1331, the steep-spectrum, larger region is the dominant one, so the overall spectrum is very steep at higher frequencies, showing characteristics of a fading radio source spectrum.
Conclusions {#c4:fin}
===========
We have observed the steep-spectrum radio source VLSS J1431.8+1331 using LOFAR, adding to an already existing data set taken with the GMRT and the VLA by [@RefWorks:145]. Using the resulting broad band frequency coverage and high resolution imaging, we were able to study the spectral properties of different regions across the source.
Based on our analysis, we conclude that the AGN in Region A shut down around 60 Myr years ago.
We found that different source regions have different spectral properties and ages. Our results are consistent with the results of [@RefWorks:86], who suggest that source Region B might be a shock compressed plasma bubble.
The integrated flux density spectrum can be used as a classifying tool to identify (unresolved) sources in surveys that are comprised of multiple regions resulting from different stages of AGN activity.
LOFAR, the Low Frequency Array designed and constructed by ASTRON, has facilities in several countries, which are owned by various parties (each with their own funding sources), and which are collectively operated by the International LOFAR Telescope (ILT) foundation under a joint scientific policy.\
RM gratefully acknowledges support from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Advanced Grant RADIOLIFE-320745.\
This research made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.\
This research made use of APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com\
RJvW acknowledges the support by NASA through the Einstein Postdoctoral grant number PF2-130104 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.\
We thank the anonymous referee for the constructive comments that have improved the manuscript.
Appendix A: Synchrotron emission models {#c4:appb .unnumbered}
=======================================
The fundamentals of the formalism describing the synchrotron emission mechanisms were given by [@RefWorks:195] and later expanded by [@RefWorks:126] in his seminal paper discussing various particle injection and energy loss processes, including synchrotron radiation. [@RefWorks:187] and [@RefWorks:125] expanded their analysis further.
Assuming that the radiating particles have energies described by a power law with a spectral index $ \gamma $: $ E^{-\gamma} $, they produce a radio spectrum that is also described by a power law with a spectral index $ \alpha = -\frac{\gamma - 1}{2} $. As time passes, these particles lose energy through synchrotron radiation and inverse Compton (IC) scattering off the CMB photons. These losses can be represented by an energy loss term, depending on the model used. The Kardashev - Pacholczyk (KP) model assumes that the pitch angle of the radiating particles (the angle between the instantaneous velocity vector of the particle and the magnetic field lines) remains constant over time, giving a loss term:
$$b = c_{2} B^{2}\left[sin^{2} \theta + \left(\frac{B_{\mathrm{IC}}}{B}\right)^{2}\right]$$
where $ \theta $ is the pitch angle, $ B_{\mathrm{IC}} = \sqrt{\frac{2}{3}}B_{\mathrm{CMB}} $ is the effective IC magnetic field, and $ B_{\mathrm{CMB}} = 3.25(1 + z)^{2} $ is the equivalent CMB magnetic field [@RefWorks:196]. Here, $ c_{2} = 2.37\cdot10^{-3} $ (CGS units) is a constant defined in Pacholczyk (1971), and $ B $ is the magnetic field in the radiating region. In the Jaffe - Perola (JP) model, it is assumed that the pitch angle is isotropized on timescales much shorter than the radiation timescale, and the loss term becomes
$$b = c_{2} B^{2}\left[\frac{2}{3} + \left(\frac{B_{\mathrm{IC}}}{B} \right)^{2}\right]$$
For the particle energy we have $ E = c_{1} \sqrt{\nu/x} $ where $ c_{1} = 4\cdot10^{-10}/\sqrt{B} $ (CGS units) is a constant, $ \nu $ the observing frequency, and $ x = \nu/\nu_{\mathrm{b}} $ is the scaled frequency, depending on $ \nu_{\mathrm{b}} $, the frequency of the observed spectral break.
We need to find the particle distribution function, which depends on the energy of the particles. There are several separate cases we should consider. In the simplest case, there is a particle injection episode which lasts for an infinitesimally short time (the injection happens at $ t=0 $), and afterwards the particles radiate away their energy [@RefWorks:126]:
$$N(t_{\mathrm{OFF}},b,\gamma,E) = \left\{ \begin{array}{ccc} E^{-\gamma}(1 - bEt_{\mathrm{OFF}})^{\gamma - 2} & \mbox{for} & E < \frac{1}{bt_{\mathrm{OFF}}} \\ \\ 0 & \mbox{for} & E \geqslant \frac{1}{bt_{\mathrm{OFF}}} \end{array} \right.$$
We label this model JP or KP depending on the treatment of the loss term, $ b $, we adopt. Here, $ t_{\mathrm{OFF}} $ is the time elapsed since the injection or, in this case, the source age $ t_{s} = t_{\mathrm{OFF}} $. When there is AGN activity (or shock particle acceleration) that is ongoing, then the distribution function becomes
$$N(t_{\mathrm{ON}},b,\gamma,E) = \left\{ \begin{array}{ccc} E^{-\gamma}t_{\mathrm{ON}} & \mbox{for} & E < \frac{1}{bt_{\mathrm{ON}}} \\ \frac{E^{-(\gamma + 1)}}{b(\gamma - 1)} & \mbox{for} & E \geqslant \frac{1}{bt_{\mathrm{ON}}} \end{array} \right.$$
This is a continuous injection model, and we label it CIJP or CIKP. In this case, the source age $ t_{s} = t_{ON} $. Finally, if there is a period of injection of energetic particles followed by a cessation of the activity, we have [@RefWorks:188; @RefWorks:54]
$$N(t_{\mathrm{OFF}},t_{\mathrm{ON}},b,\gamma,E) =$$ $$\left\{ \begin{array}{ccc} \frac{E^{-(\gamma + 1)}}{b(\gamma - 1)((1 - bE(t_{\mathrm{OFF}} - t_{\mathrm{ON}}))^{\gamma - 1} - (1 - bEt_{\mathrm{OFF}})^{\gamma - 1})} & \mbox{for} & E < \frac{1}{bt_{\mathrm{ON}}} \\ \frac{E^{-(\gamma + 1)}}{b(\gamma - 1)(1 - bE(t_{\mathrm{OFF}} - t_{\mathrm{ON}}))^{\gamma - 1}} & \mbox{for} & \frac{1}{bt_{\mathrm{OFF}}} \leqslant E \leqslant \frac{1}{b(t_{\mathrm{OFF}} - t_{\mathrm{ON}})} \\ 0 & \mbox{for} & E > \frac{1}{b(t_{\mathrm{OFF}} - t_{\mathrm{ON}})} \end{array} \right.$$
This continuous injection model with a switch off we label KGJP or KGKP [@RefWorks:54]. Now, we can calculate the observed flux density as
$$S(\nu) = S_{0}\sqrt{\nu}\int F(x)x^{-1.5}N(x)dx$$
for the JP case, and
$$S(\nu) = S_{0}\sqrt{\nu}\int_{0}^{\pi} \sin^{2}\theta \int F(x)x^{-1.5}N(x,\theta)d\theta dx$$
in the KP case, where $ S_{0} $ is a flux density scaling factor, and $ F(x) = x\int_{x}^{\infty} K_{5/3}(z)dz $ with $ K_{5/3} $ being the modified Bessel function.
[^1]: We use the $ S \propto \nu ^{\alpha} $ definition for the spectral index $ \alpha $ throughout this work.
[^2]: VLSS is the VLA Low frequency Sky Survey carried out at 74 MHz [@RefWorks:128]. NVSS stands for the NRAO VLA Sky Survey carried out at a frequency of 1.4 GHz [@RefWorks:139]
[^3]: The adopted cosmology in this work is $ H_{0} = {70.5} $ [$\,$km$\,$s$^{-1}$]{}Mpc$ ^{-1} $, $ \Omega_{M} = 0.27 $, $ \Omega_{\Lambda} = 0.73 $. At the redshift of VLSS J1431+1331.8, $ 1\arcsec \, = \, 2.747 $ kpc [@RefWorks:155].
[^4]: We drew 100 samples from the interval around the flux density points bounded by the error bars assuming uniform probability distribution. We then fitted for the spectral index; the mean and standard deviation of the fits for the sample are shown in the corresponding maps.
|
---
abstract: 'This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as their logarithmic limit sets are computed. The projective solution space of spun-normal surface theory is related to the space of incompressible surfaces and to the unit ball of the Thurston norm. It is shown that all boundary curves of the Whitehead link complement are strongly detected by its character variety. The specific results obtained can be used to study the geometry and topology of the Whitehead link complement and its Dehn surgeries. The methods can be applied to any cusped hyperbolic 3–manifold of finite volume.'
address: |
Stephan Tillmann\
School of Mathematics and Statistics F07, The University of Sydney, NSW 2006 Australia\
[stephan.tillmann@sydney.edu.au]{}
author:
- Stephan Tillmann
subtitle: The Whitehead link complement
title: 'Tropical varieties associated to ideal triangulations:'
---
Introduction
============
The study of the topology and the geometry of 3–manifolds via affine algebraic sets, which are related to representations of the fundamental group into $\SL$ and $\PSL$, has led to many deep results about 3–manifolds including the proofs of the Smith conjecture [@Mo], the cyclic surgery theorem [@CGLS] and the finite surgery theorem [@BZ96]. At the heart is a construction due to Culler and Shalen [@cs], which associates essential surfaces in the manifold to the finitely many ideal points of a curve in the character variety via actions on trees. Work of Morgan and Shalen [@ms1] associates, more generally, transversely measured codimension-one laminations in the manifold to the ideal points of a component of the character variety.
For any ideal triangulation of a non-compact, orientable, topologically finite 3–manifold there is a so-called deformation variety [@defo], which is equipped with an algebraic map into (but possibly not onto) the character variety. The deformation variety is a version of Thurston’s parameter space [@t]; both depend on the choice of ideal triangulation and are therefore not unique (not even up to birational equivalence). The deformation variety is compactified in [@defo] using the *logarithmic limit set* due to Bergman [@gb], and it is shown that each point in the logarithmic limit set is associated with a transversely measured singular codimension-one foliation of $M,$ formalising a relationship originally observed by Thurston. The logarithmic limit set of the deformation variety can be computed with the software package , which resulted from work of Bogart, Jensen, Speyer, Sturmfels and Thomas [@bjsst]. The ramification of this is an algorithmic approach to the character variety techniques of Culler, Morgan and Shalen in general, and to computing boundary curves strongly detected by the character variety in particular.
This paper illustrates this algorithmic approach to the character variety techniques of Culler, Morgan and Shalen by applying it to the Whitehead link complement (with its standard triangulation). To give a complete and unified picture, all relevant varieties and the maps between them are computed for the Whitehead link complement, and the spun-normal surfaces and ideal points are analysed in detail. The resulting information recovers the computation of the space of incompressible surfaces by Floyd and Hatcher [@fh], as well as the boundary curve space computed by Lash [@la] and Hoste and Shanahan [@hs]. It is shown that all non-compact spun-normal surfaces are essential and dual to ideal points of the character variety. However, it is also shown that there are *fake ideal points* of the deformation variety: there is a trivial, closed normal surface associated to an ideal point of the deformation variety of the Whitehead link complement.
The contents of this paper is as follows:
[**Section \[sec:prel\_varieties\] (Character varieties, ideal points and essential surfaces)**]{} This section summarises background material pertaining to various varieties associated to representations of 3–manifold groups into $\SL$ and $\PSL.$ It serves as a quick trip through Culler-Shalen theory and discusses Bergman’s logarithmic limit set [@gb] and the eigenvalue variety [@tillus_ei]. The eigenvalue variety is a multi–cusped analogue of the $A$–polynomial of Cooper, Culler, Gillet, Long and Shalen [@ccgls].
[**Section \[sec:tillus\_defo\] (The deformation variety and normal surfaces)**]{} This section describes the relationships between the deformation variety, the character and the eigenvalue varieties exhibited in [@defo]. The required elements of normal surface theory of an ideally triangulated 3–manifold are outlined, and the projective admissible solution space is defined and related to the logarithmic limit set of the deformation variety.
None of the material up to this point is new; it is mostly collated from [@gb; @sh1; @tillus_ei; @tillmann08-finite; @defo] and included for convenience of the reader.
[**Section \[sec:whl\] (The Whitehead link)** ]{} The ideal triangulation $\tri_\whl$ of the complement $\whl$ of the Whitehead link that is used for all computations in this paper is described, and differences between the right-handed and the left-handed Whitehead link, as well as different choices of meridians and longitudes are discussed.
[**Section \[whl:associated varieties\] (Deformation and Eigenvalue varieties)** ]{} The deformation variety $\D(\tri_\whl)$ and the Dehn surgery component $\PEi_0(\whl)$ of the $\PSL$–eigenvalue variety are computed. It turns out that $\D(\tri_\whl)$ and $\PEi_0(\whl)$ are birationally equivalent; this is shown via a variety $\PTa(\whl)$, which parameterises certain representations of $\pi_1(\whl)$ into $\PSL$ and which can be viewed as a de-singularisation of $\D(\tri_\whl).$ These computations were used by Indurskis [@gi] to determine Culler-Shalen semi-norms. The $\SL$– and $\PSL$–character varieties of $\whl$, and in particular their respective Dehn surgery components $\X_0 (\whl )$ and $\PX_0 (\whl )$, are computed in Appendix \[whl:Character varieties\].
All maps constructed are summarised in below commutative diagram. The horizontal maps are birational isomorphisms, and all other maps are generically 4–to–1 and correspond to actions by the Klein four group. The bar in the notation indicates that a variety is associated to representations into $\PSL.$
\(m) \[matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em\] [ & \_0() & \_0()\
(\_) & \_0() & \_0()\
& \_0 () &\
]{}; (m-1-2) edge node \[left\] [$q_1$]{} (m-2-2) (m-2-2) edge node \[right\] [$$]{} (m-3-2) (m-1-3) edge node \[right\] [$q_3$]{} (m-2-3) (m-2-1) edge node \[above\] [$\edefo$]{} (m-2-2) (m-1-2) edge node \[above\] [$\operatorname{e}$]{} (m-1-3) (m-2-2) edge node \[above\] [$\operatorname{\overline{e}}$]{} (m-2-3) (m-2-1) edge node \[below\] [$\chidefo$]{} (m-3-2);
[**Section \[whl:embedded normal surfaces\] (Embedded surfaces)** ]{} The projective admissible solution space $\N(\tri_\whl)$ of spun-normal surface theory is computed, and a criterion due to Dunfield is used to determine which spun-normal surfaces are essential. The unit ball of the Thurston norm is derived using this information. Results by Walsh [@wa] and Kang and Rubinstein [@KR-2015] on the normalisation of essential surfaces are used to determine the space $\FH$ of all essential surfaces and the boundary curve space $\BC (\whl).$ This computation is compared with work of Floyd and Hatcher [@fh], Lash [@la] and Hoste and Shanahan [@hs].
[**Section \[whl:surfaces arising from degenerations\] (Logarithmic limit sets)** ]{} The logarithmic limit set $\D_\infty(\tri_\whl)$ of the deformation variety $\D(\tri_\whl)$ is homeomorphic to a closed subset of $\N(\tri_\whl)$ according to [@defo]. Both $\D_\infty(\tri_\whl)$ and the logarithmic limit set of the eigenvalue variety $\PEi_0(\whl)$ are determined. The computation in particular shows that all strict boundary curves of $\whl$ are detected by the Dehn surgery component of the character variety. Since the boundary curves of fibres are detected by reducible characters, this shows that all boundary slopes of the Whitehead link are strongly detected. In addition, it is shown that $\D_\infty(\tri_\whl)$ has a fake ideal point.
[**Acknowledgements.**]{} The author thanks Debbie Yuster for the calculation in Appendix \[sec:log lim debbie\], and the referee for suggestions that improved the exposition of this paper. This research has benefitted from the use of the following software packages: by Marc Culler, Nathan Dunfield, Matthias Goerner and Jeff Weeks [@snappy], by Ben Burton [@bab], and by Anders Jensen [@gfan]. The author thanks Craig Hodgson, Diane Maclagan, Hyam Rubinstein, Saul Schleimer and Bernd Sturmfels for helpful discussions. Research of the author is currently supported by an Australian Research Council Future Fellowship (project number FT170100316).
Character varieties, ideal points and essential surfaces {#sec:prel_varieties}
========================================================
This section summarises background material pertaining to various varieties associated to representations of 3–manifold groups into $\SL$ and $\PSL.$ The first four subsections outline the bread and butter of Culler-Shalen theory; the reader is referred to Shalen [@sh1] for details. This is followed up with a description of Bergman’s logarithmic limit set [@gb] in §\[combinatorics:Logarithmic limit set\], and the eigenvalue varieties [@tillus_ei] in §§\[boundary:Eigenvalue variety\]–\[boundary:PSL-Eigenvalue variety\].
Character variety {#sec:character variety}
-----------------
Let $\G$ be a finitely generated group. The *representation variety* of $\G$ is the set $\R (\G) = \operatorname{Hom}(\G, \SL )$, which is regarded as an affine algebraic set (see [@cs]). Two representations are *equivalent* if they differ by an inner automorphism of $\SL$. A representation is *irreducible* if the only subspaces of $\C^2$ invariant under its image are trivial. This is equivalent to saying that the representation cannot be conjugated to a representation by upper triangular matrices. Otherwise a representation is *reducible*.
Each $\gamma \in \G$ defines a *trace function* $I_\gamma : \R (\G) \to
\C$ by $I_\gamma(\rho) = \operatorname{tr}\rho(\gamma)$, which is an element of the coordinate ring $\C [\R (\G)]$. For each $\rho \in \R (\G)$, its *character* is the function $\chi_\rho : \Gamma \to \C$ defined by $\chi_\rho (\gamma ) = \operatorname{tr}\rho (\gamma )$. Irreducible representations are determined by characters up to equivalence, and the reducible representations form a closed subset $\Red (\G)$ of $\R (\G)$. The collection of characters $\X (\G)$ can be regarded as an affine algebraic set, which is called the *character variety*. There is a regular map $\operatorname{t}: \R (\G) \to \X (\G)$ taking representations to characters. If $\G$ is the fundamental group of a topological space $M$, then $\R (M)$ and $\X (M)$ denote $\R (\G)$ and $\X
(\G)$ respectively.
There also is a character variety arising from representations into $\PSL$, see Boyer and Zhang [@bozh], and the relevant objects are denoted by placing a bar over the previous notation. The natural map $\operatorname{q}: \X (\G) \to
\PX (\G)$ is finite–to–one, but in general not onto. As with the $\SL$–character variety, there is a surjection $\operatorname{\overline{t}}: \PR (\G) \to \PX
(\G)$, which is constant on conjugacy classes, and with the property that if $\prho$ is an irreducible representation, then $\operatorname{\overline{t}}^{-1}(\operatorname{\overline{t}}(\prho ))$ is the orbit of $\prho$ under conjugation.
Dehn surgery component {#Dehn surgery component}
----------------------
Suppose $M$ is the interior of an orientable, compact, connected 3–manifold $\overline{M}$ with non-empty boundary consisting of a pairwise disjoint union of tori. If $M$ admits a complete hyperbolic structure of finite volume, then there is a discrete and faithful representation $\pi_1(M) \to \PSL$. This representation is necessarily irreducible, as hyperbolic geometry otherwise implies that $M$ has infinite volume. The following result (see [@t; @sh1]) will be relied on:
\[thurston thm\] Let $M$ be a complete hyperbolic 3–manifold of finite volume with $h$ cusps, and let $\prho_0 : \pi_1(M) \to \PSL$ be a discrete and faithful representation associated to the complete hyperbolic structure. Then ${\prho_0}$ admits a lift $\rho_0$ into $\SL$ which is still discrete and faithful. The (unique) irreducible component $\X_0$ in the $\SL$–character variety containing the character $\chi_0$ of $\rho_0$ has (complex) dimension $h$.
Furthermore, if $T_1, \ldots , T_h$ are the boundary tori of a compact core of $M$, and if $\gamma_i$ is a non–trivial element in $\pi_1(M)$ which is carried by $T_i$, then $\chi_0 ( \gamma_i) = \pm 2$ and $\chi_0$ is an isolated point of the set $$\X^* = \{ \chi \in \X_0 \mid I^2_{\gamma_1} = \ldots = I^2_{\gamma_h} = 4 \}.$$
The respective irreducible components containing the so–called *complete representations* $\rho_0$ and ${\prho_0}$ are denoted by $\R_0(M)$ and $\PR_0(M)$ respectively. In particular, $\operatorname{t}(\R_0) = \X_0$ and $\operatorname{\overline{t}}(\PR_0)
= \PX_0$ are called the respective *Dehn surgery components* of the character varieties of $M$, since the holonomy representations of hyperbolic manifolds or orbifolds obtained by performing high order Dehn surgeries on $M$ are near $\prho_0$ (see [@t]). We remark that Dehn surgery components are generally not unique, even if one considers representations into $\PSL$ (see [@CaLuTi] for examples). The point is that there are two conjugacy classes of discrete and faithful representations into $\PSL.$ They correspond to the two different choices of orientation of $M$ and are related by complex conjugation.
Essential surfaces
------------------
An *(embedded) surface* $S$ in the topologically finite 3–manifold $M = \text{int} (\overline{M})$ will always mean a 2–dimensional PL submanifold of $M$ with the property that its closure $\overline{S}$ in $\overline{M}$ is *properly embedded* in $\overline{M},$ that is, a closed subset of $\overline{M}$ with $\partial \overline{S} = \overline{S} \cap \partial \overline{M}.$ A surface $S$ in $M$ is said to be *essential* if its closure $\overline{S}$ is essential in $\overline{M}$ as described in the following definition:
[@sh1] A surface $\overline{S}$ in a compact, irreducible, orientable 3–manifold $\overline{M}$ is said to be essential if it has the following five properties:
1. $\overline{S}$ is bicollared;
2. the inclusion $\pi_1(\overline{S}_i) \to \pi_1(\overline{M})$ is injective for every component $\overline{S}_i$ of $\overline{S}$;
3. no component of $\overline{S}$ is a 2–sphere;
4. no component of $\overline{S}$ is boundary parallel;
5. $\overline{S}$ is nonempty.
The boundary curves of $\overline{S},$ $\overline{S} \cap \partial\overline{M},$ are also called the *boundary curves* of $S.$
Ideal points, and valuations {#sec:cs-theory}
----------------------------
In case that $V$ is a 1–dimensional irreducible variety, there is a unique non–singular projective variety $\tilde{V}$ which is birationally equivalent to $\overline{J(V)}$. $\tilde{V}$ is called the *smooth projective completion* of $V$, and the *ideal points* of $\tilde{V}$ are the points of $\tilde{V}$ corresponding to $\overline{J(V)} - J(V)$ under the birational equivalence. Moreover, the function fields of $V$ and $\tilde{V}$ are isomorphic.
Let $C \subset \X(M)$ be a curve, i.e. a 1–dimensional irreducible subvariety, and denote its smooth projective completion by $\tilde{C}$. We will refer to the ideal points of $\tilde{C}$ also as ideal points of $C$. The function fields of $C$ and $\tilde{C}$ are isomorphic. Denote them by $K$. Any ideal point $\xi$ of $\tilde{C}$ determines a (normalised, discrete, rank 1) valuation $ord_\xi$ of $K$, by [$$ord_\xi (f) =
\begin{cases}k & \text{if $f$ has a zero of order $k$ at $\xi$} \\
\infty & \text{if $f=0$} \\
-k & \text{if $f$ has a pole of order $k$ at $\xi$}
\end{cases}$$ ]{}Note that $ord_\xi (z)=0$ for all non-zero constant functions $z \in \C$. In the language of algebraic geometry, the valuation ring $\{ f \in K \mid ord_\xi (f) \ge 0\}$ of $ord_\xi$ is the local ring at $\xi$.
Culler and Shalen [@cs; @sh1] associate essential surfaces in a 3–manifold $M$ to *ideal points of curves* in the character variety $\X (M)$. The ingredients are as follows:
1. A curve in $\X (M)$ yields a field $F$ with a discrete valuation at each ideal point and a representation $\mathcal{P} : \pi_1(M) \to \text{SL}_2(F)$.
2. The group $\text{SL}_2(F)$ acts on the associated Bruhat-Tits tree $\tree_v$ and using $\mathcal{P}$ we can pull this back to an action of $\pi_1(M)$ on $\tree_v$ that is non–trivial and without inversions.
3. A non–trivial action of $\pi_1(M)$ without inversions on a trees gives essential surfaces in $M$ via a construction due to Stallings.
Logarithmic limit set {#combinatorics:Logarithmic limit set}
---------------------
Let $$\C[X^{\pm}]=\C[X_1^{\pm 1}, \ldots , X_n^{\pm 1}].$$ Given an ideal $I\subset \C[X^{\pm}],$ let $V = V(I)$ be the corresponding subvariety of $(\C -\{ 0\})^n.$ Bergman gives the following three descriptions of a *logarithmic limit set* in [@gb]:
*1. The tropical variety:* Define $V_{\operatorname{val}}$ as the set of $n$–tuples $$\label{log lim def2}
(-v(X_1), \ldots , -v(X_n))$$ as $v$ runs over all real–valued valuations on $\C[X^{\pm}]/I$ satisfying $\sum v(X_i)^2 = 1.$
*2. A geometric construction:* Define the support $s(f)$ of an element $f \in \C[X^{\pm}]$ to be the set of points $\alpha = (\alpha_1,{\ldots} ,\alpha_n) \in \Z^n$ such that $X^\alpha = X_1^{\alpha_1} \cdots X_n^{\alpha_n}$ occurs with non–zero coefficient in $f$. Then define $V_{\operatorname{sph}}$ to be the set of $\xi \in
S^{n-1}$ such that for all non–zero $f \in I$, the maximum value of the dot product $\xi \cdot \alpha$ as $\alpha$ runs over $s(f)$ is assumed at least twice. Geometrically, this is: $$V_{\operatorname{sph}} = \bigcap_{0\neq f \in I} \operatorname{Sph}(\operatorname{Newt}(f)),$$ where $\operatorname{Newt}(f)$ is the Newton polytope and $\operatorname{Sph}(\operatorname{Newt}(f))$ its spherical dual. The latter consists of all outward pointing unit normal vectors to the support planes of $\operatorname{Newt}(f)$ which meet $\operatorname{Newt}(f)$ in more than one point.
Bergman shows in [@gb] that $V_{\operatorname{val}}=V_{\operatorname{sph}}.$
*3. The logarithmic limit set:* Define $V_{\log}$ as the set of limit points on $S^{n-1}$ of the set of real $n$–tuples in the interior of $B^n$: $$\label{log lim def1}
\bigg\{ \frac{ (\log |x_1|, \ldots , \log |x_n|)}
{\sqrt{1 + \sum (\log |x_i|)^2}}
\mid x \in V \bigg\} .$$
Bergman shows in [@gb] that we have $V_{\log} \subset V_{\operatorname{val}}=V_{\operatorname{sph}}.$ It follows from work by Bieri and Groves [@BG1984] that all sets in fact coincide:
We have $V_{\log} = V_{\operatorname{val}}=V_{\operatorname{sph}}.$
Let $V_\infty=V_{\log}$ be the logarithmic limit set of the variety $V$. The ideal generated by a set of polynomials $\{ f_i\}_i$ shall be denoted by $I(f_i)_i$, and the variety generated by the ideal $I$ is denoted by $V(I)$ or $V(f_i)_i$, if $I = I(f_i)_i$. The following fact completely determines the tropical variety of a principal ideal:
Let $f \in \C[X^\pm ]$. If $f \ne 0$, then $V(f)_\infty = \operatorname{Sph}(\operatorname{Newt}(f))$.
In general, we have the following:
\[Bergman-Bieri-Groves\] The logarithmic limit set $V_\infty$ of an algebraic variety $V$ in $(\C -\{
0\})^n$ is a finite union of rational convex spherical polytopes, all having the same dimension, namely $$(\dim_\sC V) - 1.$$
The following is of interest to applications:
[@tillus_ei]\[curve finding lemma\] Let $V$ be an algebraic variety in $(\C -\{ 0\})^m$, and $\xi \in V_\infty$ be a point with rational coordinate ratios. Then there is a curve $C$, i.e. an irreducible subvariety of complex dimension one, in $V$ such that $\xi
\in C_\infty$.
Eigenvalue variety {#boundary:Eigenvalue variety}
------------------
Let $M$ be an orientable, irreducible, compact 3–manifold with non–empty boundary consisting of a disjoint union of $h$ tori, denoted $T_1, \ldots, T_h.$
Given a presentation of $\pi_1(M)$ with a finite number, $n$, of generators, $\gamma_1,{\ldots} ,\gamma_n$, introduce four affine coordinates (representing matrix entries) for each generator, which are denoted by $g_{ij}$ for $i=1,{\ldots} ,n$ and $j=1,2,3,4$. View $\R (M)$ as an affine algebraic set in $\C^{4n}$ defined by an ideal $J$ in $\C[g_{11},{\ldots} ,g_{n4}]$. There are elements $I_\gamma \in \C[g_{11},{\ldots} ,g_{n4}]$ for each $\gamma \in
\pi_1(M)$ such that $I_\gamma (\rho) = \operatorname{tr}\rho (\gamma)$ for each $\rho \in
\R (M)$.
Identify $\m_i$ and $\l_i$ with generators of $\operatorname{im}(\pi_1(T_i)\to\pi_1(M))$. Thus, $\m_i$ and $\l_i$ are words in the generators for $\pi_1(M)$. In the ring $\C[g_{11},{\ldots},g_{n4}, m_1^{\pm 1},l_1^{\pm 1},{\ldots},m_h^{\pm 1}, l_h^{\pm 1}]$, we define the following polynomial equations: $$\begin{aligned}
\label{ev:cood m} I_{\m_i} &= m_i + m_i^{-1}, \\
\label{ev:cood l} I_{\l_i} &= l_i + l_i^{-1}, \\
\label{ev:cood ml} I_{\m_i\l_i} &= m_il_i + m_i^{-1}l_i^{-1},\end{aligned}$$ for $i=1,{\ldots} ,h$. Let $\R_E(M)$ be the variety in $\C^{4n} \times
(\C-\{0\})^{2h}$ defined by $J$ together with the above equations. For each $\rho \in \R(M)$, the equations (\[ev:cood m\]–\[ev:cood ml\]) have a solution since commuting elements of $\SL$ always have a common invariant subspace. The natural projection $p_1: \R_E(M) \to \R(M)$ is therefore onto, and $p_1$ is a dominating map.
If $(a_1,{\ldots} ,a_{4n}) \in \R (M)$, then there is an action of $\Z_2^{h}$ on the resulting points $$(a_1,{\ldots} ,a_{4n}, m_1, l_1,{\ldots} ,m_h, l_h) \in \R_E(M)$$ by inverting both entries of a tuple $(m_i, l_i)$ to $(m_i^{-1}, l_i^{-1})$. The group $\Z_2^{h}$ acts transitively on the fibres of $p_1$. The map $p_1$ is therefore finite–to–one with degree $\le 2^h$. The maximal degree is in particular achieved when the interior of $M$ admits a complete hyperbolic structure of finite volume, since Theorem \[thurston thm\] implies that there are points in $\X_0(M) - \cup_{i=1}^h \{ I^2_{\m_i} = 4\}$.
The *eigenvalue variety* $\Ei (M)$ is the closure of the image of $\R_E(M)$ under projection onto the coordinates $(m_1,{\ldots} ,l_h)$. It is therefore defined by an ideal of the ring $\C [m_1^{\pm 1}, l_1^{\pm 1},
\ldots , m_h^{\pm 1}, l_h^{\pm 1}]$ in $(\C - \{ 0\})^{2h}$.
Note that this construction factors through a variety $\X_E(M)$, which is the character variety with its coordinate ring appropriately extended, since coordinates of $\X (M)$ can be chosen such that $I_\gamma$ (as a function on $\X (M)$) is an element of the coordinate ring of $\X (M)$ for each $\gamma
\in \pi_1(M)$. Let $\operatorname{t}_E : \R_E(M) \to \X_E(M)$ be the natural quotient map, which is equal to $\operatorname{t}: \R (M) \to \X (M)$ on the first $4n$ coordinates, and equal to the identity on the remaining $2h$ coordinates.
There also is a restriction map $r: \X (M) \to \X(T_1) \times {\ldots} \times
\X(T_h)$, which arises from the inclusion homomorphisms $\pi_1(T_i) \to
\pi_1(M)$. Therefore denote the map $\X_E(M) \to \Ei (M)$ by $r_E$. Denote the closure of the image of $r$ by $\X_\partial (M)$. There is the following commuting diagram of dominating maps:
$
\begin{CD}
\R_E(M) @>\operatorname{t}_E >> \X_E(M) @>r_E >> \Ei (M) \\
@Vp_1VV @Vp_2VV @Vp_3VV \\
\R(M) @>\operatorname{t}>> \X(M) @>r >> \X_\partial(M)
\end{CD}
$
Note that the maps $p_1,p_2,t$ and $t_E$ have the property that every point in the closure of the image has a preimage, and that the maps $p_1, p_2$ and $p_3$ are all finite–to–one of the same degree.
Recall the construction by Culler and Shalen. Starting with a curve $C \subset
\X (M)$ and an irreducible component $R_C$ in $\R (M)$ such that $\operatorname{t}(R_C) =
C$, one obtains the tautological representation $\P : \pi_1(M) \to SL_2(\C
(R_C))$. Let $R'_C$ be an irreducible component of $\R_E(M)$ with the property that $p_1(R'_C) = R_C$. Then $\C (R'_C)$ is a finitely generated extension of $\C(R_C)$, and $\P$ may be thought of as a representation $\P : \pi_1(M) \to
SL_2(\C (R'_C))$.
If $\overline{r_E\operatorname{t}_E(R'_C)}$ contains a curve $E$, then $\C (R'_C)$ is a finitely generated extension of $\C (E)$, and essential surfaces can be associated to each ideal point of $E$ using $\P$. Since the eigenvalue of at least one peripheral element blows up at an ideal point of $E$, the associated surfaces necessarily have non–empty boundary (see [@ccgls]).
Thus, if there is a closed essential surface associated to an ideal point of $C$, then either $\overline{r_E\operatorname{t}_E(R'_C)}$ is 0–dimensional, or there is a neighbourhood $U$ of an ideal point $\xi$ of $R'_C$ such that there are (finite) points in $\overline{r_E\operatorname{t}_E(U)} - r_E\operatorname{t}_E(U)$. The later are called *holes in the eigenvalue variety*, examples of which are given in [@tillus_kino].
The action of $\Z_2^{h}$ on the eigenvalue variety $\Ei (M)$ induces an action of $\Z_2^{h}$ on its logarithmic limit set $\Ei_\infty (M)\subset S^{2h -1}$. The action is generated by taking $\xi = (x_1,{\ldots} ,x_{2i-1}, x_{2i}, {\ldots} ,x_{2h})
\in\Ei_\infty(M)$ to $(x_1,{\ldots} ,-x_{2i-1}, -x_{2i}, {\ldots} ,x_{2h})
\in\Ei_\infty(M)$ for $i=1,{\ldots} ,h$. Factoring by these symmetries, a quotient of $\Ei_\infty (M)$ is obtained in the space $\RR
P^{2h-1}/\Z_2^{h-1} \cong S^{2h-1}$. The quotient map extends to a map $\varphi :
S^{2h -1} \to S^{2h -1}$ of spheres, which has degree $2^h$. Let $$P = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ and let $P_h$ be the block diagonal matrix with $h$ copies of $P$ on its diagonal. Then $P_h$ is orthogonal, and its restriction to $S^{2h -1}$ is a map of degree one. The main result of [@tillus_ei] is:
\[boundary slopes lemma\] Let $M$ be an orientable, irreducible, compact 3–manifold with non–empty boundary consisting of a disjoint union of $h$ tori. If $\xi \in \Ei_\infty
(M)$ is a point with rational coordinate ratios, then there is an essential surface with boundary in $M$ whose projectivised boundary curve coordinate is $\varphi (P_h \xi)$. Moreover, the image $\varphi(P_h \Ei_\infty(M))$ is a closed subset of $\BC (M)$.
$\mathbf{\PSL}$–eigenvalue variety {#boundary:PSL-Eigenvalue variety}
----------------------------------
Analogous to $\Ei(M)$, one can define a $\PSL$–eigenvalue variety $\PEi(M)$, since the function $I^2_\gamma : \PX (M) \to \C$ defined by $I^2_\gamma
(\prho) = (\operatorname{tr}\prho(\gamma))^2$ is regular (i.e. polynomial) for all $\gamma\in\pi_1(M)$ (see [@bozh]). Thus, a variety $\PR_E(M)$ is constructed using the relations $$\begin{aligned}
I^2_{\m_i} &= M_i + 2 + M_i^{-1}, \\
I^2_{\l_i} &= L_i + 2+L_i^{-1}, \\
I^2_{\m_i\l_i} &= M_iL_i + 2+M_i^{-1}L_i^{-1},\end{aligned}$$ for $i=1,{\ldots} ,h$.
Consider the representation of $\Z \oplus \Z$ into $\PSL$ generated by the images of $$\label{four group}
\begin{pmatrix} i&0 \\ 0&-i \end{pmatrix}
\quad \text{and} \quad
\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix}.$$ In $\PSL$, the image of this representation is isomorphic to $\Z_2 \oplus
\Z_2$, but the image of any lift to $\SL$ is isomorphic to the quaternion group $Q_8$ (in Quaternion group notation). If $\prho \in \PR (M)$ restricts to such an irreducible abelian representation on a boundary torus $T_i$, then the equations $M_i + 2 + M_i^{-1}=0$, $L_i + 2+L_i^{-1}=0$, $M_iL_i +
2+M_i^{-1}L_i^{-1}=0$ have no solution. In particular, the projection $\overline{p}: \PR_E(M) \to \PR(M)$ may not be onto.
The closure of the image of $\PR_E(M)$ onto the coordinates $(M_1,L_1,{\ldots},M_h ,L_h)$ is denoted by $\PEi (M)$ and called the *$\PSL$-eigenvalue variety*. The relationship between ideal points of $\PEi (M)$ and strongly detected boundary curves is the same as for the $\SL$–version, since the proof of Lemma \[boundary slopes lemma\] applies without change:
\[psl boundary slopes lemma\] Let $M$ be an orientable, irreducible, compact 3–manifold with non–empty boundary consisting of a disjoint union of $h$ tori. If $\xi \in
\PEi_\infty (M)$ is a point with rational coordinate ratios, then there is an essential surface with boundary in $M$ whose projectivised boundary curve coordinate is $\varphi (P_h \xi)$.
The deformation variety and normal surfacess {#sec:tillus_defo}
============================================
This section summarises the approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry given in [@defo]. Throughout this section, $M$ is the interior of an orientable, compact, connected 3–manifold with non-empty boundary consisting of a pairwise disjoint union of tori.
Ideal triangulations and shape parameters {#combinatorics:Deformation variety}
-----------------------------------------
An ideal (topological) triangulation $\tri$ of $M$ is a triple $\tri = (\widetilde{\Delta}, \Phi, h),$ where
1. $\widetilde{\Delta} = \bigcup_{k=1}^{n} \widetilde{\Delta}_k$ is a pairwise disjoint union of oriented standard Euclidean 3–simplices,
2. $\Phi$ is a collection of orientation reversing Euclidean isometries between the 2–simplices in $\widetilde{\Delta};$ termed *face pairings*,
3. $h\co (\widetilde{\Delta} \setminus \widetilde{\Delta}^{(0)} )/ \Phi \to M$ is a homeomorphism, where the domain is given the quotient topology.
We write $p \co (\widetilde{\Delta} \setminus \widetilde{\Delta}^{(0)} ) \to M$ for the composition of the natural quotient map with $h,$ and note that $M$ is determined by $\tri.$
The associated *pseudo-manifold* (or *end-compactification* of $M$) is $P = \widetilde{\Delta} / \Phi$. We denote the quotient map with the same letter, $p\co \widetilde{\Delta} \to P$. Denote $\Sigma^k$ the set of all (possibly singular) $k$–simplices in $P.$ The degree $\deg(\sigma^k)$ of a $k$–simplex in $\Sigma^k$ is the number of $k$–simplices in its preimage in $\widetilde{\Delta}.$
An *ideal $k$–simplex* is a $k$–simplex with its vertices removed. The vertices of the $k$–simplex are termed the *ideal vertices* of the ideal $k$–simplex. Similar for singular simplices. The standard terminology of (ideal) edges, (ideal) faces and (ideal) tetrahedra will be used for the (ideal) simplices of dimensions one, two, and three respectively.
Let $\Delta^3$ be the standard 3–simplex with a chosen orientation. Suppose the edges $\Delta^3$ are labeled by $z,$ $z'$ and $z''$ so that opposite edges have the same labeling and all labels are used. Then the cyclic order of $z,$ $z'$ and $z''$ viewed from each vertex depends only on the orientation of the 3–simplex. It follows that, up to orientation preserving symmetries, there are two possible labelings, and we fix one of these labelings.
Suppose $\Sigma^3 = \{ \sigma_1, \ldots, \sigma_n\}.$ Since $M$ is orientable, the 3–simplices in $\Sigma^3$ may be oriented coherently. For each $\sigma_i \in \Sigma^3,$ fix an orientation preserving simplicial map $f_i\co \Delta^3 \to \sigma_i.$ An Euler characteristic argument shows that $|\Sigma^3| = |\Sigma^1|.$ Let $\Sigma^1 = \{ e_1, \ldots, e_n\},$ and let $a^{(k)}_{ij}$ be the number of edges in $f_i^{-1}(e_j)$ which have label $z^{(k)}.$
For each $i \in \{1,\ldots, n\},$ define $$\label{eq:para}
p_i = z_i (1 - z''_i) - 1,\quad
p'_i = z'_i (1 - z_i) - 1,\quad
p''_i = z''_i (1 - z'_i) - 1,$$ and for each $j \in \{1,\ldots, n\},$ let $$\label{eq:glue}
g_j = \prod_{i=1}^n z_i^{a_{ij}} {(z'_i)}^{a'_{ij}} {(z''_i)}^{a''_{ij}}
-1.$$ Setting $p_i=p'_i=p''_i = 0$ gives the *parameter equations*, and setting $g_j=0$ gives the *hyperbolic gluing equations*. For a discussion and geometric interpretation of these equations, see [@t; @nz]. The parameter equations imply that $z_i^{(k)} \neq 0,1.$
Suppose $\tri$ is an ideal triangulation of $M.$ The *deformation variety $\D (\tri)$* is the variety in $(\C-\{0 \})^{3n}$ defined by the parameter equations and the hyperbolic gluing equations.
Developing maps, characters and eigenvalues {#sec: developing}
-------------------------------------------
\[sec: Holonomies and eigenvalue variety\]
The following facts can be found in [@y §5]; see also [@ST] for a more detailed discussion.
Given $Z \in \D (\tri),$ each ideal tetrahedron in $M$ has edge labels which can be lifted equivariantly to $\widetilde{M}.$ Following [@y], we define a continuous map $\operatorname{dev}_Z\co \widetilde{M} \to \H^3,$ which maps every ideal tetrahedron $\sigma$ in $\widetilde{M}$ to an ideal hyperbolic 3–simplex $\Delta(\sigma),$ such that the labels carried forward to the edges of $\Delta(\sigma)$ correspond to the shape parameters of $\Delta(\sigma)$ determined by its hyperbolic structure; see [@t] for the geometry of hyperbolic ideal tetrahedra. If each shape parameter has positive imaginary part, then the associated hyperbolic ideal tetrahedron is positively oriented and $\operatorname{dev}_Z$ is a developing map for a (possibly incomplete) hyperbolic structure on $M.$
For each $Z \in \D (\tri),$ the Yoshida map $\operatorname{dev}_Z$ can be used to define a representation $\rho_Z \co \pi_1(M) \to \PSL$ as follows (see [@y]). A representation into $\PSL$ is an action on $\H^3,$ and this is the unique representation which makes $\operatorname{dev}_Z$ $\pi_1(M)$–equivariant: $\operatorname{dev}_Z(\gamma \cdot x) = \rho_Z(\gamma) \operatorname{dev}_Z(x)$ for all $x \in \widetilde{M},$ $\gamma \in \pi_1(M).$ Thus, $\rho_Z$ is well–defined up to conjugation, since it only depends upon the choice of the embedding of the initial tetrahedron $\sigma.$ This yields a well–defined map $\chi_{\tri}\co \D (\tri) \to \PX (N)$ from the deformation variety to the $\PSL$–character variety. It is implicit in [@nr] that $\chi_{\tri}$ is algebraic; see [@ac] for details using a faithful representation of $\PSL \to SL(3, \C).$ Note that the image of each peripheral subgroup under $\rho_Z$ has at least one fixed point on the sphere at infinity. We summarise this discussion in the following lemma:
Let $M$ be the interior of a compact, connected, orientable 3–manifold with non-empty boundary consisting of a pairwise disjoint union of tori, and let $\tri$ be an ideal triangulation of $M.$ For each $Z \in \D(\tri),$ there exists a representation $\prho_Z \co \pi_1(M) \to \PSL$ and a $\prho_Z$–equivariant, continuous map $D_Z \co \widehat{P} \to \overline{\H}^3$ with the property that for each ideal tetrahedron $\sigma \subset \widetilde{M},$ the image $D_Z(\sigma)$ is a hyperbolic ideal tetrahedron with edge invariants determined by $Z.$ Moreover, $\prho_Z$ is well-defined up to conjugacy and the well-defined map $$\chi_\tri \co \D(\tri) \to \PX (M)$$ is algebraic.
For cusped hyperbolic 3–manifolds, there are a number of special facts that are well-known to follow from work in [@t; @nz]. The following highlights the fact that the deformation variety facilitates the study of the Dehn surgery components in the character variety, i.e.the components containing the characters of discrete and faithful representations.
Let $M$ be an orientable, connected, cusped hyperbolic 3–manifold. Let $\tri$ be an ideal triangulation of $M$ with the property that all edges are essential. Then there exists $Z \in \D(\tri),$ such that $\prho_Z\co \pi_1(M)\to \PSL$ is a discrete and faithful representation. Moreover, the whole [Dehn surgery component]{} $\PX_0(M)$ containing the character of $\prho_Z$ is in the image of $\chi_\tri.$
Denote $C$ a compact core of $M;$ this is obtained by removing a small open regular neighbourhood from each vertex in $P,$ such that $C$ inherits a decomposition into truncated tetrahedra. Each boundary torus $T_i,$ $i=1,{\ldots} ,h,$ of $C$ inherits a triangulation $\tri_i$ induced by $\tri.$ Let $\gamma$ be a closed simplicial path on $T_i.$ In [@nz], the *holonomy* $\mu(\gamma)$ is defined as $(-1)^{|\gamma|}$ times the product of the moduli of the triangle vertices touching $\gamma$ on the right, where $|\gamma|$ is the number of 1–simplices of $\gamma,$ and the moduli asise from the corresponding edge labels.
At $Z \in \D (\tri),$ evaluating $\mu(\gamma)$ gives a complex number $\mu_Z(\gamma) \in \C \setminus\{0\}.$ It is stated in [@t; @nz], that $\mu_Z(\gamma)$ is the square of an eigenvalue; one has: $$(\operatorname{tr}\prho_Z (\gamma))^2 = \mu_Z(\gamma) + 2 + \mu_Z(\gamma)^{-1}.$$ This can be seen by putting a common fixed point of $\prho_Z (\pi_1(T_i))$ at infinity in the upper–half space model, and writing $\prho_Z(\gamma)$ as a product of Möbius transformations, each of which fixes an edge with one endpoint at infinity and takes one face of a tetrahedron to another.
Choose a basis $\{\m_i, \l_i\}$ of $\pi_1(T_i)\cong H_1(T_i)$ for each boundary torus $T_i.$ Since $\mu_Z\co \pi_1(T_i) \to \C\setminus \{0\}$ is a homomorphism for each $i=1,{\ldots} ,h,$ there is a well–defined rational map: $$\operatorname{\overline{e}}\co \D (\tri) \to (\C-\{0\})^{2h} \qquad
\operatorname{\overline{e}}(Z) = (\mu_Z (\m_1),{\ldots} ,\mu_Z (\l_h )).$$ The closure of its image is contained in the *$\PSL$–eigenvalue variety* $\PEi (M)$ of [@tillus_ei].
Ideal points {#sec:Ideal points and normal surfaces}
------------
Since $\D (\tri)$ is a variety in $(\C \setminus\{ 0\})^{3n},$ Bergman’s construction in [@gb] can be used to define its set of ideal points. Let $$\begin{aligned}
Z &= (z_1, z'_1, z''_1, z_2, {\ldots} ,z''_n),\\
\log |Z| &= (\log |z_1|, \ldots , \log |z''_n|),\\
u(Z) &= \frac{1}{\sqrt{1 + (\log |z_1|)^2 + {\ldots} + (\log |z''_n|)^2}}.\end{aligned}$$ The map $\D (\tri) \to B^{3n}$ defined by $Z \to u(Z)\log |Z|$ is continuous, and the *logarithmic limit set* $\D_\infty (\tri)$ is the set of limit points on $S^{3n-1}$ of its image. Thus, for each $\xi \in \D_{\infty}(M)$ there is a sequence $\{ Z_i\}$ in $\D (M)$ such that $$\lim_{i\to \infty} u(Z_i) \log |Z_i| = \xi.$$ The sequence $\{ Z_i\}$ is said to *converge* to $\xi,$ written $Z_{i} \to \xi,$ and $\xi$ is called an ideal point of $\D (\tri).$ Whenever an edge invariant of a tetrahedron converges to one, the other two edge invariants “blow up”. Thus, an ideal point of $\D (\tri)$ is approached if and only if a tetrahedron degenerates.
Since the Riemann sphere is compact, there is a subsequence, also denoted by $\{ Z_i\},$ with the property that each shape parameter converges in $\C \cup \{ \infty \}.$ In this case $\{ Z_i\}$ is said to *strongly converge* to $\xi.$ If $\xi$ has rational coordinate ratios, a strongly convergent sequence may be chosen on a curve in $\D (\tri)$ according to Lemma \[curve finding lemma\].
Normal surface theory
---------------------
A *normal surface* in a (possibly ideal) triangulation $\tri$ is a properly embedded surface which intersects each tetrahedron of $\tri$ in a pairwise disjoint collection of *triangles* and *quadrilaterals*, as shown in Figure \[normaldiscs\]. These triangles and quadrilaterals are called *normal discs*. In an ideal triangulation of a non-compact 3–manifold, a normal surface may contain infinitely many triangles; such a surface is called *spun-normal* [@tillmann08-finite]. A normal surface may be disconnected or empty.
![The seven types of normal disc in a tetrahedron.[]{data-label="normaldiscs"}](normaldiscs)
We now describe an algebraic approach to normal surfaces. The key observation is that each normal surface contains finitely many quadrilateral discs, and is uniquely determined (up to normal isotopy) by these quadrilateral discs. Here a *normal isotopy* of $M$ is an isotopy that keeps all simplices of all dimensions fixed. Let $\square$ denote the set of all normal isotopy classes of normal quadrilateral discs in $\tri$. A normal quadrilateral disc is up to normal isotopy uniquely determined by a pair of opposite edges of a tetrahedron. Hence $|\square| = 3n$ where $n$ is the number of tetrahedra in $\tri$. These normal isotopy classes are called quadrilateral *types*.
We identify $\RR^\square$ with $\RR^{3n}.$ Given a normal surface $S,$ let $x(S) \in \RR^\square = \RR^{3n}$ denote the integer vector for which each coordinate $x(S)(q)$ counts the number of quadrilateral discs in $S$ of type $q \in \square$. This *normal $Q$–coordinate* $x(S)$ satisfies the following two algebraic conditions.
First, $x(S)$ is admissible. A vector $x \in \RR^\square$ is *admissible* if $x \ge 0$, and for each tetrahedron $x$ is non-zero on at most one of its three quadrilateral types. This reflects the fact that an embedded surface cannot contain two different types of quadrilateral in the same tetrahedron.
Second, $x(S)$ satisfies a linear equation for each interior edge in $M,$ termed a *$Q$–matching equation*. Intuitively, these equations arise from the fact that as one circumnavigates the earth, one crosses the equator from north to south as often as one crosses it from south to north. We now give the precise form of these equations. To simplify the discussion, we assume that $M$ is oriented and all tetrahedra are given the induced orientation; see [@tillmann08-finite Section 2.9] for details. These are easiest to describe with respect to a model for the neighbourhood of $e$ in $M,$ called its *abstract neighbourhood*.
The *degree of an edge* $e$ in $P,$ $\deg(e),$ is the number of 1–simplices in $\widetilde{\Delta}$ which map to $e.$ Given the edge $e$ in $P,$ there is an associated *abstract neighbourhood $B(e)$* of $e.$ This is a ball triangulated by $\deg (e)$ 3–simplices, having a unique interior edge $\widetilde{e},$ and there is a well-defined simplicial quotient map $p_{e}\co B(e)\to P$ taking $\widetilde{e}$ to $e.$ This abstract neighbourhood is obtained as follows.
If $e$ has at most one pre-image in each 3–simplex in $\widetilde{\Delta},$ then $B(e)$ is obtained as the quotient of the collection $\widetilde{\Delta}_{e}$ of all 3–simplices in $\widetilde{\Delta}$ containing a pre-image of $e$ by the set $\Phi_{e}$ of all face pairings in $\Phi$ between faces containing a pre-image of $e.$ There is an obvious quotient map $b_{e}\co B(e)\to P$ which takes into account the remaining identifications on the boundary of $B(e).$
If $e$ has more than one pre-image in some 3–simplex, then multiple copies of this simplex are taken, one for each pre-image. The construction is modified accordingly, so that $B(e)$ again has a unique interior edge and there is a well defined quotient map $b_{e}\co B(e)\to P.$ Complete details can be found in [@tillus_normal], Section 2.3.
Let $\sigma$ be a tetrahedron in $B(e)$. The boundary square of a normal quadrilateral of type $q$ in $\sigma$ meets the equator of $\partial
B(e)$ if and only it has a vertex on $e$. In this case, it has a slope $\pm1$ of a well–defined sign on $\partial B(e)$ which is independent of the orientation of $e$. Refer to Figures \[fig:matchingquadpos\] and \[fig:matchingquadneg\], which show quadrilaterals with *positive* and *negative slopes* respectively.
Given a quadrilateral type $q$ and an edge $e,$ there is a *total weight* $\operatorname{wt}_e(q)$ of $q$ at $e,$ which records the sum of all slopes of $q$ at $e$ (we sum because $q$ might meet $e$ more than once, if $e$ appears as multiple edges of the same tetrahedron). If $q$ has no corner on $e,$ then we set $\operatorname{wt}_e(q)=0.$ Given edge $e$ in $M,$ the $Q$–matching equation of $e$ is then defined by $0 = \sum_{q\in \square}\; \operatorname{wt}_e(q)\;x(q)$.
\[thm:admissible integer solution gives normal\] For each $x\in \RR^\square$ with the properties that $x$ has integral coordinates, $x$ is admissible and $x$ satisfies the $Q$–matching equations, there is a (possibly non-compact) normal surface $S$ such that $x = x(S).$ Moreover, $S$ is unique up to normal isotopy and adding or removing vertex linking surfaces, i.e., normal surfaces consisting entirely of normal triangles.
This is related to Hauptsatz 2 of [@haken61-knot]. For a proof of Theorem \[thm:admissible integer solution gives normal\], see [@kang05-spun Theorem 2.1] or [@tillmann08-finite Theorem 2.4]. The set of all $x\in \RR^{\square}$ with the property that (i) $x\ge 0$ and (ii) $x$ satisfies the $Q$–matching equations is denoted $\mathcal{Q}(\tri).$ This naturally is a polyhedral cone, but the set of all admissible $x\in \RR^{\square}$ typically meets $\mathcal{Q}(\tri)$ in a non-convex set $\mathcal{F}(\tri).$
The *projectivised solution space* $\mathcal{PQ}(\tri)$ and the *projectivised admissible solution space* $\N(\tri)$, are obtained as the quotient spaces under the multiplication action of the positive real numbers. These are usually identified via radial projection with compact subsets of either the standard simplex or a sphere centred at the origin. For the purpose of this paper, we view $\N(\tri)$ as a compact subset of a sphere centred at the origin.
The relationship between gluing and matching equations {#subsec:relationship between gluing and matching}
------------------------------------------------------
The deformation variety $\D (\tri)$ is not defined by a principal ideal, hence its logarithmic limit set $\D_\infty(\tri)$ is in general not directly determined by its defining equations (see [@tillus_ei] for details). However, it is contained in the intersection of the spherical duals of the Newton polytopes of its defining equations. The Newton polytope of a polynomial is the convex hull of its exponent vectors. The spherical dual of a polytope is the set of unit normal vectors to its supporting hyperplanes. For a polynomial $p,$ we write $\operatorname{Sph}(p)$ for the spherical dual of the Newton polytope of $p.$ See [@gb; @tillus_ei] for details. From and , we have $$\label{comb:tent log lim}
\D_\infty(\tri) \subseteq \D_{\text{pre-}\infty}(\tri) =
\overset{n}{\underset{i=1}{\bigcap}}
\big( \operatorname{Sph}(g_i) \cap \operatorname{Sph}(p_i) \cap \operatorname{Sph}(p'_i) \cap \operatorname{Sph}(p''_i)
\big).$$ The set $\D_{\text{pre-}\infty} (\tri)$ is termed a *tropical pre-variety*.
\[comb:homeo\] Let $M$ be the interior of an orientable, connected, compact 3–manifold with non-empty boundary consisting of tori, and $\tri$ be an ideal triangulation of $M.$ The set $\D_{\text{pre-}\infty}(\tri)$ is homeomorphic with the projective admissible solution space $\N(\tri)$ of normal surface theory. In particular, $\D_\infty (\tri)$ is homeomorphic with a closed subset of $\N (\tri).$
The proof of the above result in [@defo] gives a canonical homeomorphism $N \co \D_\infty (\tri) \to \N (\tri)$ as follows. At its heart is the following simple relationship between gluing equations and matching equations. Recall the description of the hyperbolic gluing equation (\[eq:glue\]) of $e_j$: $$1 = \prod_{i=1}^n z_i^{a_{ij}} {(z'_i)}^{a'_{ij}}
{(z''_i)}^{a''_{ij}}.$$ Denoting $q_i^{(j)}$ the quadrilateral type separating the pair of edges with label $z_i^{(j)}$ in tetrahedron $\sigma_i$, it follows that the the Q–matching equation of $e_j$ is: $$0 = \sum_{i=1}^n (a''_{ij} - a'_{ij}) q_i
+ (a_{ij} - a''_{ij}) q'_i
+ (a'_{ij} - a_{ij}) q''_i$$ (see [@tillmann08-finite], Section 2.9). Let $$C_1 = \begin{pmatrix}
0 & 1 & -1 \\
-1 & 0 & 1 \\
1 & -1 & 0
\end{pmatrix},$$ and let $C_n$ be the $(3n \times 3n)$ block diagonal matrix with $n$ copies of $C_1$ on its diagonal. Hence letting $A$ be the exponent matrix of the hyperbolic gluing equations and $B$ the coefficient matrix of the Q–matching equations, we have $AC_n = B.$ The desired natural homeomorphism is given by letting $N(\xi)$ be the unique normal $Q$–coordinate such that $\xi = C_n^T N(\xi)$ for a given $\xi \in \D_\infty (\tri).$
If $\xi$ has rational coordinate ratios, one can associate a unique spun-normal surface $S(\xi)$ to it as follows. By assumption, there is $r>0$ such that $rN(\xi)$ is an integer solution, and hence corresponds to a unique spun-normal surface without vertex linking components according to Theorem \[thm:admissible integer solution gives normal\]. One then requires $r$ to be minimal with respect to the condition that the surface is 2–sided.
Essential surfaces and boundary curves
--------------------------------------
Morgan and Shalen [@ms1] compactified the character variety of a 3–manifold by identifying ideal points with certain actions of $\pi_1(M)$ on $\RR$–trees (given as points in an infinite dimensional space), and dual to these are codimension–one measured laminations in $M.$ The work in [@defo] takes a different but related approach by compactifying the deformation variety with certain transversely measured singular codimension–one foliations (a finite union of convex rational polytopes), and dual to these are (possibly trivial) actions on $\RR$–trees. The relationship between these compactifications is investigated in Section 6 of [@defo], with focus on the original consideration of ideal points of curves and surfaces dual to Bass–Serre trees due to Culler and Shalen [@cs]. A surface which can be reduced to an essential surface that is detected by the character variety in the sense of [@tillus_mut] will be called *weakly dual to an ideal point of a curve in the character variety*. The following result combines statements of Theorem 1.2 and Proposition 4.3 in [@defo]:
[@defo]\[comb:essential prop\] Let $M$ be the interior of a compact, connected, orientable, irreducible 3–manifold with non-empty boundary consisting of a disjoint union of tori, and let $\tri$ be an ideal triangulation of $M.$ Let $\xi \in \D_\infty(\tri)$ be a point with rational coordinate ratios, and assume that the spun-normal surface $S(\xi)$ is not closed.
Then $S(\xi)$ is non-trivial and (weakly) dual to an ideal point of a curve in the character variety of $M$. Moreover, every essential surface detected by this ideal point has the same boundary curves as $S(\xi).$ In particular, the boundary curves of $S(\xi)$ are strongly detected.
It is an open question whether there is an ideal point of the deformation variety that is associated with a closed essential surface but with the property that any associated sequence of characters does not approach an ideal point of the character variety.
For a more general statement pertaining to all points in $\D_\infty(\tri)$, and thus completing the description of its relationship with the Morgan–Shalen compactification, see Theorem 1.2 in [@defo].
The Whitehead link {#sec:whl}
==================
The right–handed Whitehead link is shown in the figure on the right. Throughout this paper, $\whl = \whl^+$ denotes its complement in $S^3.$ This is a hyperbolic 3–manifold isometric with the same orientation
[r]{}[44mm]{}
{width="3.8cm"}
to the manifold $m129$ in ’s census. The complement of a left–handed Whitehead link is oppositely oriented, and denoted by $\whl^-.$
The standard triangulation $\tri_\whl$ with four ideal tetrahedra of $\whl$ will be considered throughout; the triangulation is given in Figure \[fig:whl\_triangulation\] and Table \[tab:Face pairings\], and is derived from ’s manifold data (see Appendix \[sec:snap data\]). We remark that $\whl$ has three inequivalent minimal triangulations with four tetrahedra; they arise from the three diffent ways one can choose a “diagonal" in an octahedron; the triangulation used here is shipped as $m129 : \#2$ in ’s cusped orientable manifold census. There are four tetrahedra in $\tri_\whl,$ shown in Figure \[fig:whl\_triangulation\], and hence four edges, which will be called red, green, black and blue. The black edge is labelled with one arrow, the blue with two, the green with three and the red with four. One of the cusps corresponds to the ideal endpoints of the red edge, and the other to the ideal endpoints of the green edge. The cusps are therefore referred to as the red cusp (also:cusp 0) and the green cusp (also:cusp 1) respectively. The face pairings are described in Table \[tab:Face pairings\], where the face of tetrahedron $\sigma_i$ opposite vertex $j$ is denoted by $F_{i,j}.$
![Triangulation $\tri_\whl$ of $\whl$[]{data-label="fig:whl_triangulation"}](whl_triangulation.png){width="14cm"}
----------------------- ----------------------- ----------------------- -----------------------
$F_{0,0} \to F_{2,0}$ $F_{1,0} \to F_{0,1}$ $F_{2,0} \to F_{0,0}$ $F_{3,0} \to F_{2,1}$
$F_{0,1} \to F_{1,0}$ $F_{1,1} \to F_{3,1}$ $F_{2,1} \to F_{3,0}$ $F_{3,1} \to F_{1,1}$
$F_{0,2} \to F_{1,3}$ $F_{1,2} \to F_{2,3}$ $F_{2,2} \to F_{3,3}$ $F_{3,2} \to F_{0,3}$
$F_{0,3} \to F_{3,2}$ $F_{1,3} \to F_{0,2}$ $F_{2,3} \to F_{1,2}$ $F_{3,3} \to F_{2,2}$
----------------------- ----------------------- ----------------------- -----------------------
: Face pairings[]{data-label="tab:Face pairings"}
[r]{}[58mm]{}
{width="5.5cm"}
The convention for orientations of the peripheral elements is shown on the right, where the right–hand rule is applied to a link projection and the induced triangulation of the cusp cross–section is viewed from outside the manifold.
The induced triangulations of the cusp cross–sections of $\whl$ are given in Figure \[fig:whl\_cusps\]. There is a standard meridian, and two choices for longitudes are considered, which are termed *geometric* and *topological* and denoted by $\l^g$ and $\l^t$ respectively. The geometric longitudes are chosen by for $m129.$ They have the following property. Let $\whl (p,q)$ be the manifold obtained from hyperbolic Dehn filling on one of the cusps. The coefficients $(p,q) \in \RR^2$ are called *exceptional* if $\whl (p,q)$ does not admit a complete hyperbolic metric. If the longitude is geometric, then the set of exceptional coefficients is contained in the rectangle with vertices $\pm (2, 1),$ $\pm (2, -1)$ (see [@hmw]). If a null–homologous longitude is chosen, the set of exceptional coefficients is contained in a parallelogram with vertices $\pm (-4,1),$ $\pm (0,1)$ for $\whl^-$ (see [@nr]), and $\pm (4,1),$ $\pm (0,1)$ for $\whl^+.$ The natural linear maps between the Dehn surgery coefficients with respect to the different peripheral systems are given in Table \[tab:surg coeff\].
$\{\m , \l^t\}$ for $\whl^-$ $\{\m , \l^g\}$ for $\whl^+$ and $\whl^-$ $\{\m , \l^t\}$ for $\whl^+$
------------------------------ ------------------------------------------- ------------------------------
$(p+2q, -q)$ $(p,q)$ $(p+2q,q)$
: Surgery coefficients with respect to the peripheral systems[]{data-label="tab:surg coeff"}
![Triangulation of cusp cross–section and peripheral elements[]{data-label="fig:whl_cusps"}](whl_cusps.png){width="12cm"}
Affine algebraic sets {#whl:associated varieties}
=====================
This section determines the various affine algebraic sets associated with the Whitehead link complement, as well as the maps between them. Differences between result below and results in Neumann-Reid [@nr] arise from the fact that the latter use a left-handed link and have a different convention regarding shape parameters.
Deformation variety {#sec:defo equations}
-------------------
The shape parameters are assigned in Figure \[fig:whl\_triangulation\], and the following hyperbolic gluing equations can be read off from this figure: $$\begin{aligned}
\tag{\text{red \& green}} 1 &= w x y z \\
\tag{\text{black}} 1 &= w'x'y'z'(w'')^2 (x'')^2\\
\tag{\text{blue}} 1 &= w'x'y'z'(y'')^2 (z'')^2.\end{aligned}$$ The deformation variety $\D(\tri_\whl)$ is defined by these equations and the parameter relations: $$\begin{aligned}
&1=w (1 - w''), &&1=x (1 - x''), &&1=y (1 - y''), &&1=z (1 - z''),\label{eq:whl:z} \\
&1=w' (1 - w), &&1=x' (1 - x), &&1=y' (1 - y), &&1=z' (1 - z), \label{eq:whl:z'}\\
&1=w'' (1 - w'), &&1=x'' (1 - x'), &&1=y'' (1 - y'), &&1=z'' (1 - z').\label{eq:whl:z''}\end{aligned}$$ Using the fact that equations of the form $ww'w''=-1$ result from the parameter relations, it is not hard to see that $\D(\tri_\whl)$ can be defined by the parameter relations in (\[eq:whl:z\]) and (\[eq:whl:z’\]) together with the following two equations: $$\label{whl: defo simple relations}
1 = w x y z \qquad\text{and}\qquad w''x'' = y''z''.$$ Then $\D(\tri_\whl)$ is mapped into $(\C-\{ 0\})^4$ by the projection $\varphi (w,w',{\ldots} ,z'') = (w,x,y,z).$ The closure of the resulting image is a variety defined by: $$\begin{aligned}
\label{whl:param1} 0 &= 1 - wxyz,\\
\label{whl:param2} 0 &= wx(1 - y)(1 - z) - (1 - w)(1 - x)yz.\end{aligned}$$ This variety is the *parameter space* of [@t; @nz], and is denoted by $\D'(\tri_\whl).$ For any $w,x,y,z\in\C-\{0,1\}$ subject to (\[whl:param1\]) and (\[whl:param2\]), there is a unique point on $\D(\tri_\whl).$ Thus, $\D(\tri_\whl)$ and $\D'(\tri_\whl)$ are birationally equivalent, and the inverse map: $$\begin{aligned}
&\varphi^{-1}(w,x,y,z)\\
= &\bigg(w,\frac{1}{1-w}, \frac{w-1}{w},
x,\frac{1}{1-x}, \frac{x-1}{x},
y,\frac{1}{1-y}, \frac{y-1}{y},
z,\frac{1}{1-z}, \frac{z-1}{z}\bigg)\end{aligned}$$ is not regular at the intersection of $\D'(\tri_\whl)$ with the collection of hyperplanes $$\{ w=1 \} \cup \{ x=1 \} \cup \{ y=1 \} \cup \{ z=1 \}$$ in $(\C-\{0\})^4.$ If $\D'(\tri_\whl)$ intersects any one of these hyperplanes, then it intersects either exactly two or four of them.
One of the variables, say $w,$ can be eliminated from the system of equations (\[whl:param1\]) & (\[whl:param2\]), and hence there is a map $\varphi' : \D' (\whl ) \to (\C-\{0\})^3$ with the closure of its image defined by a single irreducible equation: $$0= g(x,y,z) = x - xy - xz + yz - xy^2z^2 + x^2y^2z^2.$$ Again, $\varphi'$ is a birational isomorphism onto its image, and this in particular shows that $\D(\tri_\whl)$ is irreducible. Whence $\D(\tri_\whl)$ is an irreducible variety in $(\C-\{0\})^{12}$ defined by the $10$ equations in (\[eq:whl:z\]), (\[eq:whl:z’\]) and (\[whl: defo simple relations\]), and so:
The deformation variety $\D(\tri_\whl)$ is a complete intersection variety of dimension two.
Symmetries
----------
There are symmetries in the defining equations of the deformation variety $\D(\tri_\whl),$ which descend to symmetries in (\[whl:param1\]) and (\[whl:param2\]). Consider the following involutions: $$\begin{aligned}
\label{whl:klein1} &\tau_1 (w,x,y,z) = (z,y,x,w) &\text{and} &&\tau_2 (w,x,y,z) = (y,z,w,x), \\
&\tau_3 (w,x,y,z) = (x,w,z,y), & &&\\
\label{whl:klein2} &\tau_4 (w,x,y,z) = (w,x,z,y) &\text{and} &&\tau_5 (w,x,y,z) = (x,w,y,z).\end{aligned}$$ Then $\tau_1\tau_2=\tau_3=\tau_4\tau_5,$ and each of the pairs (\[whl:klein1\]) and (\[whl:klein2\]) generates a Klein four group. The elements $\tau_6 = \tau_1 \tau_4$ and $\tau_7 = \tau_2 \tau_4$ have order four, and the group $D_4$ generated by all involutions is a dihedral group. For any $p \in \D'(\tri_\whl)$ and any $\tau \in D_4,$ $\tau p \in \D'(\tri_\whl),$ and if $\varphi^{-1} : \D'(\tri_\whl) \to \D(\tri_\whl)$ is regular at $p,$ then it is regular at $\tau p.$
Holonomies {#subsec:holonomies}
----------
Each cusp cross section inherits a triangulation induced by $\tri_\whl.$ Let $\gamma$ be a closed simplicial path on a cusp cross section. In [@nz], the *derivative of the holonomy* $H'(\gamma)$ is defined as $(-1)^{|\gamma|}$ times the product of the moduli of the triangle vertices touching $\gamma$ on the right, where $|\gamma|$ is the number of 1–simplices of $\gamma,$ and the moduli asise from the corresponding edge labels.
The derivatives of the holonomies can this be read off from Figure \[fig:whl\_cusps\], and simplify to: $$\begin{aligned}
\label{whl:holo0}
H'(\m_0 ) &= \frac{z}{w'y''} =\frac{x'z''}{y}, && H'(\l^t_0)= x^2 y^2,
&& H'(\l^g_0) = \frac{(w'')^2}{(y'')^2} =\frac{(z'')^2}{(x'')^2}, \\
\label{whl:holo1}
H'(\m_1 ) &= \frac{w'z''}{y} = \frac{x}{w''z'}, && H'(\l^t_1)= w^2 y^2,
&& H'(\l^g_1) = \frac{(z'')^2}{(w'')^2}=\frac{(x'')^2}{(y'')^2}.\end{aligned}$$ The completeness equations for the holonomies yield that the complete hyperbolic structure is attained at $(w,x,y,z) = (i,i,i,i).$ The action of $D_4$ on the holonomies is described in Table \[tab:action on hol\], where $(m_i,l_i) = (H'(\m_i ), H'(\l^t_i)).$ The relationship between elements of $D_4$ and elements of the symmetry group of $\whl$ can be deduced from this table.
$\tau_1$ $\tau_2$ $\tau_3$ $\tau_4$ $\tau_5$ $\tau_6$ $\tau_7$
------------- ----------------------- ----------------------- ----------------------- ----------------------- ------------- ----------------------- -----------------------
$(m_0,l_0)$ $(m_0,l_0)$ $(m_0^{-1},l_0^{-1})$ $(m_0^{-1},l_0^{-1})$ $(m_1^{-1},l_1^{-1})$ $(m_1,l_1)$ $(m_1,l_1)$ $(m_1^{-1},l_1^{-1})$
$(m_1,l_1)$ $(m_1^{-1},l_1^{-1})$ $(m_1,l_1)$ $(m_1^{-1},l_1^{-1})$ $(m_0^{-1},l_0^{-1})$ $(m_0,l_0)$ $(m_0^{-1},l_0^{-1})$ $(m_0,l_0)$
: Action of $D_4$ on holonomies[]{data-label="tab:action on hol"}
Fundamental group {#subsec:Fundamental group}
-----------------
An abstract presentation of $\pi_1(\whl)$ can be computed from the triangulation using the set-up in the next section: $$\pi_1(\whl )=\langle\a,\b,\c,\d \mid \d = \a\b, \b\c\d = \c\d\c, \a\c = \b\a\rangle,$$ and the peripheral elements are the following words in the generators: $$\begin{aligned}
&\m_0 = \b^{-1}, &&\l_0^g = \a \d^{-1} \c^{-1},
&&\l_0^t = \m_0^{-2} \l_0^g,\\
&\m_1 = \d, &&\l_1^g = \a^{2}\b\c,
&&\l_1^t = \m_1^{-2} \l_1^g.\end{aligned}$$ Thus, $\pi_1(\whl )$ can be generated by two meridians, and one obtains a single relation: $$\begin{aligned}
\label{whl:fundamental group}
\pi_1(\whl ) = \langle \m_0, \m_1 \mid &
\m_1\m_0\m_1 \m_0^{-1}\m_1^{-1}\m_0^{-1}\m_1\m_0\\
&\notag\quad = \m_0\m_1 \m_0^{-1}\m_1^{-1}\m_0^{-1}\m_1\m_0\m_1 \rangle\end{aligned}$$
Developing map
--------------
Given a point $Z = (w,w',{\ldots} ,z'') \in \D(\tri_\whl),$ there are (1) a continuous map $\operatorname{dev}_Z \co \widetilde{\whl} \to \H^3$ such that each ideal tetrahedron in $\widetilde{\whl}$ (with the lifted ideal triangulation) is mapped to an ideal hyperbolic tetrahedron of the specified shape, and (2) a unique representation $\prho_Z \co \pi_1(\whl) \to \PSL$ which makes $\operatorname{dev}_Z$ $\pi_1(\whl)$–equivariant (see Section 2.5 in [@defo]). We start with an embedding of (a lift of) $\sigma_0$ with vertices at the points $0,1,\infty, w,$ and develop around the edge $[0, \infty]$ according to the parameters given by $Z.$ This results in one lift of each ideal tetrahedron to $\H^3.$ The resulting ideal vertices of the tetrahedra are indicated in Figure \[fig:whl\_triang\_dev\]. Some tetrahedra may intersect or be “inverted”. If all tetrahedra are positively oriented, then the resulting fundamental domain is an ideal octahedron.
Indeed, one obtains a combinatorial ideal octahedron, if one only applies the face pairings shown in Table \[tab:pairings for fundom\]. The Whitehead link complement is then obtained by applying the remaining face pairings, which match the boundary faces of the ideal octahedron.
----------------------- ----------------------- ----------------------- -----------------------
$F_{0,2} \to F_{1,3}$ $F_{1,2} \to F_{2,3}$ $F_{2,2} \to F_{3,3}$ $F_{3,2} \to F_{0,3}$
$F_{0,3} \to F_{3,2}$ $F_{1,3} \to F_{0,2}$ $F_{2,3} \to F_{1,2}$ $F_{3,3} \to F_{2,2}$
----------------------- ----------------------- ----------------------- -----------------------
: Face pairings used for fundamental domain[]{data-label="tab:pairings for fundom"}
![Developing map[]{data-label="fig:whl_triang_dev"}](whl_triang_dev.png){width="16cm"}
Applied to the developed ideal tetrahedra in $\H^3,$ the remaining face pairings are: $$\begin{aligned}
\rho_Z(\a)\co & [0,1,wyz]\to [\infty, w, 1], \qquad \rho_Z(\b)\co [w, \infty, wz] \to [1, \infty, wyz],\\
\rho_Z(\c)\co & [0,1,w] \to [0,wyz,wz], \qquad \rho_Z(\c\d)\co [w,0,wz] \to [wz,\infty,wyz],\end{aligned}$$ This determines a representation of $\pi_1(\whl)$ into $\PSL$: $$\begin{aligned}
\rho_Z(\a) &= \sqrt{\frac{w'}{x'}}
\begin{pmatrix}
w+\frac{x'}{w'} & -\frac{x'}{w'} \\
1 & 0\\
\end{pmatrix} ,
&&
\rho_Z(\b)= \sqrt{\frac{x'z''}{y}}
\begin{pmatrix}
\frac{y}{x'z''} & 1 - x''z' \\
0 & 1\\
\end{pmatrix} ,
\\
\rho_Z(\c) &= \sqrt{\frac{w''y'}{x}}
\begin{pmatrix}
1 & 0 \\
xw'(1-wy)& \frac{x}{w''y'}\\
\end{pmatrix},
&&
\rho_Z(\d) = \sqrt{\frac{y}{w'z''}}
\begin{pmatrix}
(1-wx)w' & \frac{y''}{w''} \\
\frac{w'}{x'} & \frac{1 - xz}{w''}\\
\end{pmatrix}.\end{aligned}$$ We note: $$\operatorname{tr}\rho_Z(\b) = \sqrt{\frac{x'z''}{y}} + \sqrt{\frac{y}{x'z''}} = \sqrt{H'(\m_0 )} + \frac{1}{\sqrt{H'(\m_0 )}}.$$ A calculation using the identity $wxyz=1$ and the standard identities for shape parameters (such as $ww'w''=-1$ and $w'(1-w)=1$) gives the following (where we have shown every second step): $$\begin{aligned}
\operatorname{tr}\rho_Z(\d) &= \sqrt{\frac{y}{w'z''}} \Big(w'-ww'x+\frac{1}{w''}-\frac{xz}{w''}\Big) = \sqrt{\frac{y}{w'z''}} \Big(1+\frac{w'(z-1)}{yz}\Big)
=\sqrt{\frac{y}{w'z''}} \Big(1+\frac{w'z''}{y}\Big)\\ &= \sqrt{H'(\m_1 )} + \frac{1}{\sqrt{H'(\m_1 )}}.\end{aligned}$$ This algebraically confirms the consistency of the choice of meridians (up to orientation) in §\[subsec:holonomies\] and §\[subsec:Fundamental group\].
At the complete structure, the peripheral elements have to be parabolic. This forces: $$\begin{aligned}
\a_0 &= \begin{pmatrix} 1+i & -1 \\ 1 & 0 \end{pmatrix},
&&\b_0 = \begin{pmatrix} 1 & 1-i \\ 0 & 1 \end{pmatrix},\\
\c_0 &= \begin{pmatrix} 1 & 0 \\ -1+i & 1 \end{pmatrix},
&&\d_0 = \begin{pmatrix} 1+i & 1 \\ 1 & 1-i \end{pmatrix}.\end{aligned}$$ There is a well–defined map $\chidefo=\chi_{\tri_\whl}\co \D(\tri_\whl)\to \PX (\whl),$ and it is shown in the next section that $\chidefo$ has degree 4. This corresponds to the observation that the manifolds $\whl (p_0,q_0,p_1,q_1),$ $\whl (-p_0,-q_0,p_1,q_1),$ $\whl (p_0,q_0,-p_1,-q_1),$ and $\whl (-p_0,-q_0,-p_1,-q_1)$ are geometrically distinguished (they “spiral in different directions into the cusps”), whilst they have isomorphic fundamental groups.
Face pairings {#whl:tautologicum}
-------------
A variety $\Ta (\whl )$ parameterising representations into $\SL$ is computed as follows. Putting: $$\label{whl:taut}
\rho(\m_0 ) = \begin{pmatrix} s & c \\ 0 & s^{-1} \end{pmatrix} \text{ and }
\rho(\m_1 ) = \begin{pmatrix} u & 0 \\ 1 & u^{-1} \end{pmatrix},$$ one obtains a single equation using (\[whl:fundamental group\]): $$\begin{aligned}
\label{whl:taut_rel}
0&= f(s,u,c) \\
\notag&= (s-s^{-1})(u-u^{-1})
+ c(s^{-2}u^{-2}-u^{-2}-s^{-2}+4-s^2-u^2+s^2u^2)\\
\notag &\qquad +c^2(2s^{-1}u^{-1}-su^{-1}-s^{-1}u+2su) + c^3
\in \C[s^{\pm 1}, u^{\pm 1}, c].\end{aligned}$$ Then $\Ta (\whl )$ is an irreducible hypersurface in $(\C - \{ 0\})^2 \times \C,$ and its intersection with $c=0$ is the collection of lines $\{ s^2 = 1, c=0\} \cup \{ u^2 = 1, c=0\},$ which parametrises reducible representations. Moreover, $\Ta (\whl )$ is a cover of the Dehn surgery component $\X_0(\whl)$ of the character variety, since any irreducible representation of $\pi_1(\whl )$ into $\SL$ is conjugate to an element of $\Ta (\whl ),$ and it is a 4–to–1 branched cover of $\X_0(\whl)$ since $\Ta (\whl )$ is not contained in the union of hypersurfaces $s^2 = 1$ and $u^2 = 1.$
If $f(s,u,c)=0,$ then $f(-s,u,-c) = f(s,-u,-c)=f(-s,-u,c)=0,$ and the four solutions correspond to the action of $\operatorname{Hom}(\pi_1(\whl), \Z_2)$ on $\Ta(\whl ).$ To obtain a description of the corresponding quotient map $\Ta (\whl ) \to \PTa (\whl ),$ and hence a parametrisation of the associated variety $\PTa
(\whl)$ of representations into $\PSL,$ note that (\[whl:taut\]) can be adjusted by a conjugation and rewritten in the form: $$\begin{aligned}
\label{whl:good form}
&\rho(\m_0 ) = \frac{1}{s} \begin{pmatrix} s^2 & c s u\\ 0 & 1 \end{pmatrix}
\quad\text{and}\quad
\rho(\m_1 ) = \frac{1}{u} \begin{pmatrix} u^2 & 0 \\ 1 & 1 \end{pmatrix},\end{aligned}$$ which is still subject to $0= f(s,u,c).$ The map $\operatorname{q}_1: (\C-\{0\})^2\times \C \to (\C-\{0\})^2\times \C$ defined by $\operatorname{q}_1 (s,u,c) = (s^2, u^2, c s u)$ can be identified with the natural quotient map $\Ta (\whl ) \to \PTa (\whl ),$ and the defining equation for $\PTa (\whl )$ can be derived from this relationship. Thus, $\PTa (\whl )$ can be viewed as a variety of representations into $GL_2(\C)$: $$\begin{aligned}
\label{whl:ptaut}
& \prho_{GL}(\m_0 ) = \begin{pmatrix} \s & d \\ 0 & 1 \end{pmatrix}
\quad\text{and}\quad
\prho_{GL}(\m_1 ) = \begin{pmatrix} \u & 0 \\ 1 & 1 \end{pmatrix}
\quad \text{subject to}\\
0&= \s\u (\s -1)(\u-1)
+ d(1 - \s- \u + 4 \s \u - \s^2\u - \s\u^2 + \s^2\u^2)\\
\notag
&\quad + d^2 ( 2 - \s- \u + 2\s\u)+ d^3 \in \C[\s^{\pm 1}, \u^{\pm 1},d].\end{aligned}$$ For each $\prho_{GL}$ as above, there is a unique $\PSL$–representation $\prho$ such that $\prho_{GL}$ and $\prho$ are identical as representations into the group of projective transformations of $\C P^1.$ Hence there is a bijection between the set of all of the above $GL_2(\C )$–representations of $\pi_1(\whl )$ with $d\neq 0$ and the set of all irreducible $\PSL$–representations of $\pi_1(\whl )$ (up to conjugation) which lift to $\SL.$
There is a birational isomorphism $\edefo : \D(\tri_\whl) \to \PTa (\whl ).$
To construct the map $\edefo: \D(\tri_\whl) \to \PTa (\whl ),$ conjugate the face pairings $\rho_Z(\a),$ …, $\rho_Z(\d)$ by a *suitably chosen* matrix $A$ to obtain a form analogous to (\[whl:good form\]), and then adjust the resulting representation by multiplication: $$\begin{aligned}
\prho'_Z(\m_0) &:=
\sqrt{\frac{x}{w''y'}}\ A \ \rho_Z(\b^{-1}) \ A^{-1} =
\begin{pmatrix} \frac{w''y'}{x} & z -1 \\ 0 & 1 \end{pmatrix}
\\
\text{and}\quad
\prho'_Z(\m_1 ) &:=
\sqrt{\frac{w''z'}{x}}\ A \ \rho_Z(\d)\ A^{-1} =
\begin{pmatrix} \frac{x}{w''z'} & 0 \\ 1 & 1 \end{pmatrix}.\end{aligned}$$ It can be verified that $\prho'_Z$ defines a $GL_2(\C)$–representation for each $Z \in \D(\tri_\whl).$ This induces a map $\edefo : \D(\tri_\whl) \to \PTa (\whl )$ defined by: $$\begin{aligned}
\edefo (w,w',{\ldots} ,z'') =
\bigg( \frac{w''y'}{x} , \frac{x}{w''z'}, z-1 \bigg).\end{aligned}$$ An elementary calculation shows that this map is 1–1 and that its image is dense in $\PTa (\whl ).$ A rational inverse is given by the following map $\PTa (\whl ) \to \D'(\tri_\whl)$: [$$\begin{aligned}
(\s, \u, d) \to
\bigg(
\frac{d + d^2 - d\s + \s\u + d\s\u}{(1 + d)(d + \s\u)},
\frac{\s\u}{d + d^2 - d\s + \s\u + d\s\u},
1 + \frac{d}{\s\u},
1 + d \bigg),\end{aligned}$$ ]{}which composed with $\varphi: \D'(\tri_\whl) \to \D(\tri_\whl)$ gives a map $\edefo^{-1} : \PTa (\whl ) \to \D(\tri_\whl).$ Composition of the maps $\edefo$ and $\edefo^{-1}$ induces the identity on both $\D(\tri_\whl)$ and $\PTa (\whl ),$ and this proves the lemma.
The only singularity of $\D(\tri_\whl)$ is at infinity; at the ideal point where all of $w,x,y,z$ tend to one, and the only singularity of $\D' (\whl )$ is at the corresponding point $(w,x,y,z)=(1,1,1,1).$ The variety $\PTa (\whl )$ is an irreducible variety without singularities, and the above proof shows that it parametrises developing maps when thought of as a variety in $(\C-\{0\})^3.$ It is shown in Subsection \[whl:eigenvalues at (1,1,1,1)\] that the intersection of $\PTa (\whl )$ with $c=0$ corresponds to the ideal point of $\D(\tri_\whl)$ where $w,x,y,z \to 1.$ Thus, it is a natural de-singularisation of $\D(\tri_\whl).$
$\chidefo\co \D(\tri_\whl) \to \PX_0 (\whl )$ is generically 4–to–1 and onto.
Note that $\chidefo\co \D(\tri_\whl) \to \PX (\whl )$ factors through $\PTa (\whl),$ and hence it is enough to show that the map $\PTa (\whl) \to \PX_0 (\whl )$ is generically 4–to–1 and onto. Firstly, $\PTa (\whl)$ is irreducible, and hence its image in $\PX (\whl)$ is irreducible. Secondly, the above lemma implies that $\PTa (\whl)$ contains a discrete and faithful representation, and hence maps to the Dehn surgery component. Since there are $\prho \in \PTa (\whl)$ such that $(\operatorname{tr}\prho (\m_0))^2 \ne 4 \ne (\operatorname{tr}\prho (\m_1))^2,$ there are generically four elements of each conjugacy class of representations contained in $\PTa (\whl).$ Hence, the degree of $\chidefo$ is four.
Eigenvalue maps {#whl:Eigenvalue maps}
---------------
As in [@tillus_ei], denote the respective eigenvalue varieties by $\Ei (\whl )$ and $\PEi (\whl ).$ The aim of this section is to show that the subvariety $\PEi_0 (\whl)$ corresponding to $\PX_0(\whl )$ is birationally equivalent to $\D(\tri_\whl).$ The following lemma establishes suitable affine coordinates and maps needed for this.
\[whl: eigenvalue maps lem\] There are a quotient map $\operatorname{q}_3\co \Ei (\whl ) \to \PEi (\whl )$ corresponding to the action of $\operatorname{Hom}(\pi_1(\whl ), \Z_2),$ and *eigenvalue maps* $\operatorname{e}\co \Ta (\whl ) \to \Ei (\whl )$ and $\operatorname{\overline{e}}\co \PTa (\whl ) \to \PEi (\whl )$ such that the following diagram commutes:
$
\begin{CD}
\Ta(\whl ) @>\operatorname{e}>> \Ei_0(\whl ) \\
@V\operatorname{q}_1 VV @VV\operatorname{q}_3 V\\
\PTa (\whl ) @>\operatorname{\overline{e}}>> \PEi_0 (\whl )
\end{CD}
$
Let $\vartheta_\gamma \co \Ta (\whl ) \to \C$ be the holomorphic map which takes $\rho$ to the upper left entry of $\rho (\gamma).$ Then define $\operatorname{e}\co \Ta (\whl ) \to \Ei (\whl )$ to be the map $$\label{whl:eigenvalue map}
\operatorname{e}(\rho ) = (\vartheta_{\m_0}(\rho ), \vartheta_{\l^t_0}(\rho ),
\vartheta_{\m_1}(\rho ), \vartheta_{\l^t_1}(\rho )).$$ Since every $\rho \in \Ta (\whl )$ is triangular on the peripheral subgroups, it follows that this map is well–defined. Denote the affine coordinates of $\Ei (\whl )$ by $(s,t,u,v),$ so they correspond to eigenvalues of $\m_0,$ $\l^t_0,$ $\m_1$ and $\l^t_1.$
Denote the map which takes $\prho_{GL}$ to the upper left entry of $\prho_{GL}(\gamma )$ by $\vartheta_\gamma$ as well, and let $\operatorname{\overline{e}}\co \PTa (\whl ) \to \PEi (\whl )$ be the map $$\operatorname{\overline{e}}(\prho_{GL} ) = (\vartheta_{\m_0}(\prho_{GL} ),
\vartheta_{\l^t_0}(\prho_{GL} ),
\vartheta_{\m_1}(\prho_{GL} ), \vartheta_{\l^t_1}(\prho_{GL} )).$$ To verify that this map has the right range, note that the upper left entries of $\prho_{GL} (\m_0)$ and $\prho_{GL} (\m_1)$ are the squares of the eigenvalues of matrices representing the associated $\PSL$–representation. The longitudes are the following words in the meridians: $$\begin{aligned}
\l^t_0 &= \m_0^{-1}\m_1\m_0\m_1^{-1}\m_0^{-1}\m_1^{-1}\m_0\m_1\\
\l^t_1 &= \m_1^{-1}\m_0\m_1\m_0^{-1}\m_1^{-1}\m_0^{-1}\m_1\m_0,\end{aligned}$$ and hence $\prho_{GL} (\l^t_i) = \prho (\l^t_i)$ for any $\prho_{GL}$ and its corresponding unique $\PSL$–representation $\prho.$ In particular, $\vartheta_{\l^t_i} (\prho_{GL})$ is an eigenvalue of $\prho_{GL} (\l^t_i) = \prho (\l^t_i)$ and it is independent of the choice of the signs of matrices representing $\prho (\m_0)$ and $\prho (\m_1).$ It follows that $\PEi (\whl )$ can be given the affine coordinates $(\s, t, \u, v).$
The natural quotient map which makes the above diagram commute is therefore $\operatorname{q}_3 \co \Ei (\whl ) \to \PEi (\whl )$ defined by $\operatorname{q}_3 (s,t,u,v) = (s^2,t,u^2,v).$ This map clearly corresponds to the action of $\operatorname{Hom}(\pi_1(\whl ), \Z_2)$ on $\Ei (\whl ).$
Since any irreducible $\SL$–representation is conjugate to a representation in $\Ta(\whl),$ the closure of the image of $\operatorname{e}$ is the component of $\Ei(\whl)$ corresponding to the Dehn surgery component $\X_0(\whl).$ It follows that the closure of the image of $\operatorname{\overline{e}}$ corresponds to $\PX_0(\whl).$ In particular the composite mapping $\Psi= \operatorname{\overline{e}}\circ \edefo \co \D(\tri_\whl) \to \PEi_0 (\whl )$ is onto.
From the face pairing $\rho_Z\co \pi_1(M)\to \PSL,$ one computes $\vartheta_{\m_0} (\prho_Z) = \frac{x'z''}{y},$ $\vartheta_{\l^t_0} (\prho_Z) = xy,$ $\vartheta_{\m_1} (\prho_Z) = \frac{w'z''}{y}$ and $\vartheta_{\l^t_1} (\prho_Z) = wy.$ The map $\Psi \co \D(\tri_\whl) \to \PEi (\whl)$ with respect to the chosen coordinates is therefore given by $$\begin{aligned}
\label{whl:holo map}
\Psi (w,w',w'',x,x',x'',y,y',y'',z,z',z'') =
\big( \frac{x'z''}{y},x y, \frac{w'z''}{y}, w y \big)\end{aligned}$$ We have $\Psi\varphi^{-1}(i,i,i,i) = (1,-1,1,-1)$ at the complete structure.
\[whl:degree one lemma\] The map $\Psi \co \D(\tri_\whl) \to \PEi_0 (\whl)$ is a birational isomorphism.
Since $\Psi$ is a regular map and $\D(\tri_\whl)$ is irreducible, it follows that the closure of its image, which has been identified as $\PEi_0(\whl ),$ is irreducible. Thus, $\Psi \co \D(\tri_\whl) \to\PEi_0 (\whl )$ is a regular map of irreducible varieties. It remains to show that it has degree one.
Assume that $\Psi\varphi^{-1}(w_0,x_0,y_0,z_0)=\Psi\varphi^{-1}(w_1,x_1,y_1,z_1),$ where $(w_i,x_i,y_i,z_i)$ are two regular points of $\varphi$ on $\D'(\tri_\whl).$ An elementary calculation shows that the points are either identical or we have $w_0=x_0$ and $w_1=x_1.$ Thus, all points on which $\Psi$ does not have degree one are contained on the hypersurface $w=x.$ Since for any $w \in\C-\{0,\pm 1\},$ the point $\varphi^{-1} (w,-w^{-1},w,-w^{-1})$ is contained in $\D(\tri_\whl),$ $\D(\tri_\whl)$ is not contained in this hypersurface.
An inverse $\PEi_0(\whl ) \to \D(\tri_\whl)$ taking $(\s,t,\u,v) \to (w,w',{\ldots} ,z'')$ is determined by the following map $\PEi_0(\whl ) \to \D' (\whl )$ which can be computed from (\[whl:holo map\]): $$(\s,t,\u,v) \to
\Bigg(
\frac{v(\s - \u)}{\s t - \u v},
\frac{t(\s - \u)}{\s t - \u v},
\frac{\s t-\u v}{\s-\u},
\frac{\s t-\u v}{tv(\s-\u)}
\Bigg).$$ This map is not regular on a 1–dimensional subvariety of $\PEi_0(\whl),$ which is defined by the following three equations: $$\begin{aligned}
\label{whl: inverse not defined}
\s = \u, && t = v, && 0 = \s - t + \s t + \s^2 t - \s^2 t^2 - \s^3 t^2 + \s^4
t^2 - \s^3 t^3.\end{aligned}$$ See Subsection \[whl:compute eigen\] for a computation of the eigenvalue varieties and this subvariety.
Embedded surfaces {#whl:embedded normal surfaces}
=================
This section gives a complete description of the space $\FH$ of all essential surfaces in the Whitehead link complement, and deduces the associated boundary curve space $\BC (\whl)$ and the unit ball of the Thurston norm. This is compared with previous work of Floyd and Hatcher [@fh], Lash [@la] and Hoste and Shanahan [@hs].
The main tools used to analyse incompressible surfaces in the projective admissible solution space $\N(\tri_\whl)$ of spun-normal surface theory are a criterion due to Dunfield that determines which spun-normal surfaces are essential, and results by Walsh [@wa] and Kang and Rubinstein [@KR-2015] on the normalisation of essential surfaces.
Complete description of the projective admissible solution space
----------------------------------------------------------------
The convention for quadrilateral coordinates for orientable manifolds in [@tillmann08-finite] is used. Table \[tab:whl:quad types\] indicates the position of the quadrilaterals in the tetrahedra. For reference, we also add the shape parameters, where we write $w = z_0,$ $x = z_1,$ $y = z_2,$ and $z = z_3.$ The notation $ij/kl$ means that the particular quadrilateral type separates the vertices $i$ and $j$ from the vertices $k$ and $l.$
Quadrilateral $q_i$ $q'_i$ $q''_i$
--------------- --------- --------- ---------
Separates $01/23$ $03/12$ $02/13$
Edge label $z_i$ $z'_i$ $z''_i$
: Quadrilateral types[]{data-label="tab:whl:quad types"}
The $Q$–matching equations can be worked out directly from the triangulation or by using their relationship with the gluing equations described in §\[subsec:relationship between gluing and matching\]. There are the following three equations: $$\begin{aligned}
\tag{\text{red \& green}} 0 &= q'_0-q''_0 + q'_1-q''_1 + q'_2-q''_2 + q'_3-q''_3, \\
\tag{\text{black}} 0 &= q_0-2q'_0+q''_0 + q_1-2q'_1+q''_1-q_2+q''_2-q_3+q''_3
\\
\tag{\text{blue}} 1 &= -q_0+q''_0-q_1+q''_1+q_2-2q'_2+q''_2+q_3-2q'_3+q''_3\end{aligned}$$ These are equivalent to the following two, which correspond to : $$\begin{aligned}
\label{whl:Q-match1} 0 &= q'_0-q''_0 + q'_1-q''_1 + q'_2-q''_2 + q'_3-q''_3, \\
\label{whl:Q-match2} 0 &= q_0-q'_0 + q_1-q'_1 - q_2+q'_2 - q_3+q'_3. \end{aligned}$$ To obtain an admissible solution, one now sets two quadrilateral coordinates from each tetrahedron equal to zero and solves the above equations subject to this constraint. Hence the set $\N(\tri_\whl)$ can be computed by solving $3^4 = 81$ systems of linear equations. This is automated by feeding the triangulation to [Regina]{}. Alternatively, given its description as a tropical pre-variety in Proposition \[comb:homeo\], one can use [gfan ]{}to compute $\N(\tri_\whl)$ from the defining equation of $\D (\tri_\whl)$ using the command `tropical_intersection`.
![Projective admissible solution space $\N(\tri_\whl)$: The labels of the nodes correspond to the vertex solutions; the shown PL arcs between them can be realised as geodesics in $S^{11}$ with pairwise disjoint interior, and the quadrilateral as a convex spherical quadrilateral.[]{data-label="fig:whl_admissible"}](whl_admissible-eps-converted-to.pdf){width="12cm"}
The set $\N(\tri_\whl)$ is a finite union of convex spherical polytopes in $S^{11}\subset \RR^{12}.$ It turns out that there are 28 geodesic arcs and one geodesic quadrilateral, spanned by a total of 20 vertices. These are indicated in Figure \[fig:whl\_admissible\], where geodesic arcs are represented as PL arcs.
The vertices are denote by $V_1,{\ldots} ,V_{20},$ and called *vertex solutions*. These correspond to extremal solutions to the linear equations. Each vertex solution $V_i$ is rescaled by a positive real to obtain a (minimal) integer solution, and the resulting normal surface is denoted by $F_i.$ Hence the normal $Q$–coordinate $N(F_i)$ is a multiple of $V_i.$ These normal surfaces are described in Table \[table:white\_normal\] and the information given in the table is as follows:
Vertex $q_0$ $q'_0$ $q''_0$ $q_1$ $q'_1$ $q''_1$ $q_2$ $q'_2$ $q''_2$ $q_3$ $q'_3$ $q''_3$ type class $\partial$–curves $\nu(\l^t_0),$ $-\nu(\m_0),$ $\nu(\l^t_1),$ $-\nu(\m_1)$
------------ ------------ ------------- -------------- ------------ ------------- -------------- ------------ ------------- -------------- ------------ ------------- -------------- ------- -------- ------------------- ---------------- --------------- ---------------- --------------
1 1 0 0 0 0 0 1 0 0 0 0 0 $T_1$ $N'_1$ $0/1$ 0, 0, 0, –1
2 0 0 0 1 0 0 1 0 0 0 0 0 $T_1$ $N_1$ $1/0$ 0, –1, 0, 0
3 0 0 0 1 0 0 0 0 0 1 0 0 $T_1$ $N'_1$ $0/1$ 0, 0, 0, 1
4 1 0 0 0 0 0 0 0 0 1 0 0 $T_1$ $N_1$ $1/0$ 0, 1, 0, 0
5 1 0 0 0 1 0 0 0 0 0 0 1 $S_3$ $N'_2$ $2/1$ –2, 0, 0, –1
6 0 0 0 0 0 1 1 0 0 0 1 0 $S_3$ $N'_2$ $2/1$ 2, 0, 0, –1
7 0 0 1 0 0 0 1 0 0 0 1 0 $S_3$ $N_2$ $1/2$ 0, –1, 2, 0
8 0 1 0 1 0 0 0 0 0 0 0 1 $S_3$ $N_2$ $1/2$ 0, –1, –2, 0
9 0 1 0 1 0 0 0 0 1 0 0 0 $S_3$ $N'_2$ $2/1$ 2, 0, 0, 1
10 0 0 1 0 0 0 0 1 0 1 0 0 $S_3$ $N'_2$ $2/1$ –2, 0, 0, 1
11 0 0 0 0 0 1 0 1 0 1 0 0 $S_3$ $N_2$ $1/2$ 0, 1, –2, 0
12 1 0 0 0 1 0 0 0 1 0 0 0 $S_3$ $N_2$ $1/2$ 0, 1, 2, 0
13 0 0 0 0 1 0 0 1 0 0 0 2 $R_2$ $N'_3$ $1/1$ –4, –1, –2, –1
14 0 1 0 0 0 2 0 0 0 0 1 0 $R_2$ $N'_3$ $1/1$ 4, 1, –2, –1
15 0 0 2 0 1 0 0 0 0 0 1 0 $R_2$ $N_3$ $1/1$ –2, –1, 4, 1
16 0 1 0 0 0 0 0 1 0 0 0 2 $R_2$ $N_3$ $1/1$ –2, –1, –4, –1
17 0 1 0 0 0 0 0 0 2 0 1 0 $R_2$ $N'_3$ $1/1$ 4, 1, 2, 1
18 0 0 2 0 1 0 0 1 0 0 0 0 $R_2$ $N'_3$ $1/1$ –4, –1, 2, 1
19 0 1 0 0 0 2 0 1 0 0 0 0 $R_2$ $N_3$ $1/1$ 2, 1, –4, –1
20 0 0 0 0 1 0 0 0 2 0 1 0 $R_2$ $N_3$ $1/1$ 2, 1, 4, 1
Angle $\alpha_0$ $\alpha'_0$ $\alpha''_0$ $\alpha_1$ $\alpha'_1$ $\alpha''_1$ $\alpha_2$ $\alpha'_2$ $\alpha''_2$ $\alpha_3$ $\alpha'_3$ $\alpha''_3$
$\alpha^+$ 0 0 $\pi$ $\pi$ 0 0 0 0 $\pi$ $\pi$ 0 0
$\alpha^-$ 0 $\pi$ 0 $\pi$ 0 0 0 $\pi$ 0 $\pi$ 0 0
$\beta^+$ $\pi$ 0 0 0 0 $\pi$ 0 0 $\pi$ $\pi$ 0 0
$\beta^-$ $\pi$ 0 0 0 $\pi$ 0 0 $\pi$ 0 $\pi$ 0 0
$\gamma^+$ $\pi$ 0 0 0 0 $\pi$ $\pi$ 0 0 0 0 $\pi$
$\gamma^-$ $\pi$ 0 0 0 $\pi$ 0 $\pi$ 0 0 0 $\pi$ 0
$\delta^+$ 0 0 $\pi$ $\pi$ 0 0 $\pi$ 0 0 0 0 $\pi$
$\delta^-$ 0 $\pi$ 0 $\pi$ 0 0 $\pi$ 0 0 0 $\pi$ 0
First, the normal $Q$–coordinate $N(F_i)$ is given, then the topological type of $F_i,$ where $T_1$ stands for a once–punctured torus, $S_3$ for a thrice–punctured sphere, $R_2$ for a twice–punctured $\RR P^2.$ The abbreviations $K_i$ and $T_i$ will be used in subsequent figures for an $i$–punctured Klein bottle and an $i$–punctured torus respectively, and $G_2$ denotes a (closed) genus two surface.
The column *class* specifies the equivalence class (defined in Subsection \[whl:Equivalence classes\]) that the projective normal $Q$–coordinate $V_i$ of $F_i$ belongs to.
The column *$\partial$–curves* encodes the number of boundary components on the respective cusps; if there are $i$ boundary components on the (red) cusp 0, and $j$ on the (green) cusp 1, this is written as $i/j.$
Last, the corresponding boundary curves are given as they are computed from the chosen (oriented, topological) peripheral system. The boundary curves are determined by the signed intersection numbers of the peripheral elements with the spun-normal surface. We here summarise how this is computed; the details can be found in [@defo §4.2] and [@tillmann08-finite §3.1]. Let $\gamma$ be a closed simplicial path on a cusp cross section with respect to the triangulation induced by $\tri.$ The $Q$–modulus of a vertex with label $z_i$ is $q''_i-q'_i$, of a vertex with label $z'_i$ is $q_i-q''_i$, and of a vertex with label $z''_i$ is $q'_i-q_i$. The linear functional $\nu(\gamma)$ is defined to be the sum of the $Q$–moduli of all vertices of triangles touching $\gamma$ to the right. For our generators, we obtain the following linear functionals: $$\begin{aligned}
\nu (\m_0) &= q_1-q''_1+q'_2-q''_2-q_3+q'_3
&&\nu (\l_0^t) = -2q'_1+2q''_1 -2q'_2+2q''_2\\
\nu (\m_1) &= q_0-q'_0-q'_1+q''_1-q_3+q''_3
&&\nu (\l_1^t) = -2q'_0+2q''_0 -2q'_2+2q''_2\end{aligned}$$ The signs in the table respect the transverse orientations, which are relevant when the boundary curves of surfaces corresponding to linear combinations of the $N(F_i)$ are computed.
Some spun-normal surfaces
-------------------------
Using the explicit description of normal surfaces, one can work out the topological type and the position of normal surfaces in the manifold. The pictures of the gluing pattern of some surfaces are shown in Figure \[fig:whl\_surfaces\], where the quadrilaterals and finitely many triangles are used to obtain compact surfaces, along whose boundary components infinite normal annuli have to be attached. The boundary components are drawn in the colour of the corresponding cusp. Triangle coordinates are labelled by numbering the vertices of the four tetrahedra from $0$ to $15$.
\
\
Equivalence classes {#whl:Equivalence classes}
-------------------
A homeomorphism $N\co \D_{\text{pre-}\infty}(\whl)\to \N(\tri_\whl)$ is given in [@defo] between $\N(\tri_\whl)$ and the tropical pre-variety $\D_{\text{pre-}\infty}(\whl)$ obtained from the canonical defining equations. There is an induced action of the group $D_4$ of symmetries of $\D(\whl)$ on $\D_{\text{pre-}\infty}(\whl).$ Since the elements of $D_4$ interchange coordinates, the induced action corresponds to interchanging coordinate triples of elements in $\D_\infty(\whl).$ Moreover, there is an induced action of $D_4$ on $\N(\tri_\whl)$ via the homeomorphism $N$ which again corresponds to interchanging coordinate triples of elements.
Indeed, $D_4$ can be identified with a group of symmetries of the triangulation. There is a Klein four group $K_f,$ identified with $\langle\tau_1, \tau_2\rangle,$ which stabilises the cusps, and orbits of the action of $K_f$ on $\N(\tri_\whl)$ give six equivalence classes amongst the vertex solutions in $\N(\tri_\whl)$: $$\begin{aligned}
N_1 &= \{ V_2,V_4 \},
&& N'_1 = \{ V_1,V_3 \},\\
N_2 &= \{ V_7,V_8,V_{11},V_{12}\},
&& N'_2 = \{ V_5,V_6,V_{9},V_{10}\},\\
N_3 &= \{ V_{15},V_{16},V_{19},V_{20}\},
&& N'_3 = \{ V_{13},V_{14},V_{17},V_{18} \}.\end{aligned}$$
The classification up to isotopy is given in §\[sec:Incompressible normal surfaces\] and §\[incompressible after FH\]. The orbit under $K_f$ corresponds to the different ways a surface can “spin into the cusps”. In particular, the (two or four) normal coordinates in each of the classes $N_k$, $N'_k$ correspond to isotopic surfaces.
Members of the classes $N_k$ and $N'_k$ are interchanged by symmetries interchanging the cusps. These symmetries correspond to the remaining elements of $D_4.$ One can visualise the action of $D_4$ on $\N(\tri_\whl)$ by considering the action of the dihedral group on Figure \[fig:whl\_admissible\] induced by its standard action on a square. The quotient by the action of $K_f$ is pictured in Figure \[fig:whl\_results\](a), where the topological types of minimal representatives for some points are indicated.
Note that there are arcs in $\N(\tri_\whl)$ connecting elements of $N_3,$ e.g.the vertices $V_{16}$ and $V_{19}.$ The geometric sum $F_{16}+F_{19}$ is a twice–punctured torus. However, $\frac{1}{2}(N(F_{16})+N(F_{19}))$ is also an admissible integer solution, and the corresponding normal surface is a once-punctured Klein bottle. The corresponding equivalence class (i.e.$K_f$ orbit) is denoted by $N_4.$ Similarly, for arcs in $\N(\tri_\whl)$ joining elements of $N'_3$ we obtain an equivalence class $N'_4$ whose elements are midpoints of these arcs.
The surface determined by a minimal integer solution corresponding to the point $V_0:= \frac{1}{2}(V_1+V_3) = \frac{1}{2}(V_2+V_4)$ in the “centre" of $\N(\tri_\whl)$ is a genus two surface. This point is fixed by all symmetries, and we denote its equivalence class by $N_0.$ The quadrilateral spanned by $V_1, V_2, V_3, V_4$ is called the *centre square*.
The $D_4$ orbit of a point $P \in \N(\tri_\whl)$ is now analysed. If $P=V_0,$ then its equivalence class only contains one element. If $P$ is contained in the square $[V_1,V_2,V_3,V_4]$ but not equal to its centre, then its $D_4$ orbit contains exactly four elements. Note that different elements in the square can have the same *$\partial$–coordinate* $$(\nu_P(\l_0),-\nu_P(\m_0),\nu_P(\l_1),-\nu_P(\m_1)).$$ This is true for instance for $V_1$ and $\frac{3}{4}V_1 + \frac{1}{4}V_3.$
However, elements of the same $D_4$ orbit are distinguished by their $\partial$–coordinates. If $P$ is contained in $N_4$ or $N'_4,$ or is the midpoint of an arc $[V,W]$ in $\N(\tri_\whl)$ with $V \in N_i,$ $W \in N'_i$ and $i=2$ or $3,$ then its $D_4$ orbit contains exactly four elements. Moreover, the elements of the orbit are distinguished by their $\partial$–coordinates. If $P \in \N(\tri_\whl)$ is not contained in any of the sets considered above, then its $D_4$ orbit contains exactly eight elements, and all these elements are distinguished by their $\partial$–coordinates.
Moreover, the elements of an equivalence class of a point in $\N(\tri_\whl)$ have the same projectivised (i.e.unoriented) $\partial$–coordinate. Thus, the set of projectivised $\partial$–coordinates arising from $\N(\tri_\whl)$ can be computed using the incidence structure amongst the equivalence classes, and the result is shown in Figure \[fig:whl\_results\](b). This shows that each equivalence class is uniquely determined by its projectivised $\partial$–coordinate unless its elements are contained in the centre square of $\N(\tri_\whl).$
\[whl:lem:slopes det surf\] An embedded spun-normal surface in $\whl$ is uniquely determined by its *transversely oriented* boundary curves if its projectivised normal $Q$–coordinate is not contained in the centre square in $\N(\tri_\whl)$.
Let $S$ be an embedded spun-normal surface. Then there is a unique $\alpha>0$ and a unique point $P \in \N(\tri_\whl)$ such that $N(S)=\alpha P.$ If $P$ is not contained in the centre square, then there is no other point in $\N(\tri_\whl)$ with the same $\partial$–coordinate. Thus, $S$ is uniquely determined by $$\begin{aligned}
&\alpha (\nu_P(\l_0),-\nu_P(\m_0),\nu_P(\l_1),-\nu_P(\m_1))\\
=& (\nu_{N(S)}(\l_0),-\nu_{N(S)}(\m_0),\nu_{N(S)}(\l_1),-\nu_{N(S)}(\m_1)).\end{aligned}$$ This proves the lemma.
![Isotopic once-punctured tori. The reader will find it a pleasant exercise in normal surface theory to show that the once-punctured tori normalise to the same normal surface in the complement of a *normal* thrice-punctured disc in $\tri_\whl.$[]{data-label="fig:whl-isotopy"}](whl-isotopy.pdf){width="14cm"}
Criteria for incompressibility {#sec:Dunfield}
------------------------------
Dunfield and Garoufalidis [@DuGa] give a simple criterion for a vertex surface to be incompressible, namely that it be a vertex surface with non-empty boundary which has a quadrilateral in each tetrahedron. However, this criterion does not apply to any of the vertex normal surfaces in the given ideal triangulation of the Whitehead link complement.
A certificate for incompressibility can be given using a different result, which the author learned from Nathan Dunfield. This is only summarised here, and complete details will be given in [@splittings]. A *semi-angle structure* is an assignment $q \mapsto \alpha(q)$ of a non-negative real number (called angle) to each quadrilateral type $q$ in an ideal triangulation such that the sum of the angles associated to the three distinct quadrilateral types supported by each tetrahedron equals $\pi,$ and the sum of all angles of the quadrilateral types facing an edge in the 3–manifold equals $2\pi.$ See [@LuTi] for a detailed discussion of this viewpoint, We view $\alpha\in [0,\pi]^{\square} = [0,\pi]^{3n},$ where $\square$ is the set of all isotopy classes of quadrilateral discs. We remark that if $\alpha\in (0,\pi)^{\square}$, then it is termed an *angle structure*. This can be viewed as a linear hyperbolic structure and we refer the reader to the excellent exposition in [@FuGe] for a history and applications of angle structures.
An embedded normal surface $S$ has normal $Q$–coordinate $N(S) \in \NN^\square = \NN^{3n}.$ Then $S$ is said to be *dual to the semi-angle structure $\alpha$*, if $\alpha \cdot N(S) = 0,$ where the standard Euclidean inner product is taken. In other words, for each quadrilateral type $q$ that has non-zero weight in $S,$ we have $\alpha(q)=0$.
Let $M$ be the interior of a compact 3–manifold with boundary a non–empty disjoint union of tori. Let $S$ be an embedded normal surface (possibly non–compact) in $(M; \tri)$ without any boundary parallel components. If $S$ is dual to a semi-angle structure of $(M; \tri)$, then it is essential.
Here is a sketch of the proof. Suppose $S$ had a compression disc $D.$ Then $D$ can be put into a normal form relative to the triangulation and the normal surface, and inherits an induced semi-angle structure. The combinatorial Gauß-Bonnet formula then implies that $D$ has non-positive Euler characteristic, a contradiction. Hence the surface is incompressible. The proof is concluded with an observation by Hatcher that incompressible implies boundary incompressible for an embedded surface with boundary only on the torus boundary components of a 3-manifold. A proof of Dunfield’s theorem is given in [@splittings].
The above theorem has the following simple corollary:
\[cor:dunfield criterion\] Let $M$ be the interior of a compact 3–manifold with boundary a non–empty disjoint union of tori, $\tri$ be an ideal triangulation of $M$, and $\alpha\in [0,\pi]^{\square}$ be a semi-angle structure. If $S$ and $F$ are compatible normal surfaces (possibly non–compact) that are both dual to $\alpha$, then each 2–sided normal surface that is a Haken sum of $S$ and $F$ is essential.
Essential normal surfaces {#sec:Incompressible normal surfaces}
-------------------------
Listed in Table \[tab:whl:quad types\] are a number of semi-angle structures on $\tri_\whl,$ as well as the vertex surfaces that are dual to these. In particular, each of the 20 vertex surfaces is essential. It follows from the information in the table and Corollary \[cor:dunfield criterion\] that each 2–sided normal surface in $\whl$ is essential except possibly those whose projectivised normal coordinates lie on the sides or the interior of the central square spanned by $V_1, V_2, V_3, V_4.$
Each surface $F_1, F_2, F_2, F_4$ is a once-punctured torus and can be given a natural orientation arising from the transverse orientation of its unique boundary curve. In this way, it represents an element $[F_i] \in H_2(\whl^c, \partial \whl^c; \Z),$ where $\whl^c$ is a compact core of $\whl.$ Note that $[F_3] = - [F_1]$ and $[F_4] = - [F_2].$ As in [@thu-norm], one can now argue that the surfaces $S_i$ are norm minimising, and that each normal surface with projectivised normal coordinate along the boundary of the central square is norm minimising, and hence essential. In particular, we have the following:
The boundary of the central square naturally corresponds to the boundary of the unit ball of the Thurston norm.
It is also shown in [@thu-norm] that every essential surface homologous to a fibre is isotopic to a fibre. Whence every fibre arises as a normal surface with respect to $\tri_\whl.$ This verifies a result of Kang and Rubinstein [@KR-2015].
We now claim that every normal surfaces not in the boundary of the central square is not a fibre. This can be seen either by considering their boundary slopes, or by determining their image in homology. All classes are mapped to zero except for the surfaces along segments between thrice punctured spheres and once-punctured tori. These cannot be fibres since they only evaluate non-zero on one cusp. We have thus established:
Every fibre in $\whl$ normalises in precisely two ways. A spun-normal surface in $\whl$ is a fibre if and only if its projectivised normal coordinate lies in the interior of an edge of the central square.
It is also not difficult to describe the once–punctured tori corresponding to the vertices of the central square. The surfaces $F_1$ and $F_3$ have identical projectivised boundary slopes, and so do $F_2$ and $F_4$. Constructing the corresponding normal surfaces explicitly, one can see that for each of the solutions, there are four normal triangles which form an annulus on the surface and cap the quadrilaterals off at one of the cusps. The union of the two respective annuli on $F_1$ and $F_3$ is a boundary parallel torus. Similar for $F_2$ and $F_4$. It follows that each pair is obtained from a thrice punctured sphere that meets one of the boundary components in two meridians and the other in a longitude by tubing the two meridional boundary components together. The two choices of annulus give the two surfaces in each pair. However, these are in fact isotopic surfaces. The isotopy is indicated in Figure \[fig:whl-isotopy\].
The central surface, $G_2,$ is a closed genus two surface linking the red and green edge, and hence compressible. To one side, it compresses to a torus linking the red cusp, and to the other to a torus linking the green cusp. Its complements are therefore compression bodies. The surface $G_2$ is made of up four quadrilaterals and no triangles. Any surface $F$ whose projectivised normal coordinate lies in the interior of the central square is a Haken sum of the form $F = n_1 S_i + n_2 S_j + n_3 G_2,$ where $S_i$ and $S_j$ correspond to vertices of an edge of the central square. In particular, this has the same boundary slope as the surface $n_1 S_i + n_2 S_j$. From the local structure around the green and red edges, we see that each such surface is compressible, namely there are $n_3$ compression discs, and after the compressions we have a surface in same homology class as $n_1 S_i + n_2 S_j$ and with same Euler characteristic, and hence it is a fibre and in fact isotopic to this. At this point, we may record the following:
A spun-normal surface in $\whl$ is essential if and only if its projectivised normal coordinate does not lie in the interior of the central square.
This completes the classification of projectivised essential normal surfaces up to normal isotopy.
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Essential surfaces {#incompressible after FH}
------------------
It follows from the main result of Walsh [@wa] that every essential surface in $\whl$ that is not a fibre or a semi-fibre can be normalised with respect to $\tri_\whl.$ A semi-fibre is a separating surface that lifts to a fibre in a double cover. An inspection of the three double covers of $\whl$ shows that there is no such surface in $\whl.$ It is well known to experts that every non-compact essential surface that is not a fibre can be put into spun-normal form by choosing, for each cusp on which the surface has a cusp, a direction of spinning. A proof of this was recently announced by Kang and Rubinstein [@KR-2015]. It follows that the space $\FH$ of projectivised classes of essential surfaces in $\whl$ is precisely the orbit space of the action of $K_f$ on the space of projectivised essential normal surfaces. This is shown in Figure \[fig:whl\_fh\_normal\]. Here, the labels of the vertices are with reference to Figure \[fig:whl\_fh\_surfs\], which is taken from an earlier version of [@fh], and the remaining surfaces can be determined from the symmetry interchanging the components.
Up to permuting the link components, the surfaces in Figure \[fig:whl\_fh\_surfs\] have the following normalisations. The pair of once-punctured tori $\{S_7, S_8\}$ normalises to the pair $\{F_2, F_4\}$; the twice-punctured tori $S_2$ and $S_3$ normalise to the pairs $\{ F_1+F_4, F_2+F_3\}$ and $\{F_1+F_2, F_3+F_4\};$ the thrice-punctured sphere $S_5$ normalises to the surfaces $\{F_7, F_8, F_{11}, F_{12}\}$; the twice-punctured projective plane $S_4$ normalises to the surfaces $\{F_{15}, F_{16}, F_{19}, F_{20}\}$; and the twice punctured torus $S_6$ normalises to the surfaces $\{F_{15}+F_{20}, F_{16}+F_{19}\}$.
The space of all incompressible surfaces in $\whl$ was described by Floyd and Hatcher [@fh] using branched surfaces. They give an algorithm to compute a complex with rational barycentric coordinates that carries all essential surfaces that are *meridional incompressible.* The condition for an essential surface $S \subset S^3 \setminus L$ to be meridional incompressible is as follows. If there is a disc $D\subset S^3$ with $D\cap S = \partial D$ and $D$ meeting $L$ transversely in one point in the interior of $D$, then there is a disc $D' \subset S\cup L$ with $\partial D' = \partial D,$ $D'$ also meeting $L$ transversely in one point. Geometrically, this corresponds to the existence of accidental parabolics. The complex carrying all meridional incompressible surfaces in the Whitehead link complement is shown in Figure \[fig:whl\_fh\_mi\] — in [@fh Figure 5.4] this is the 1–complex with label \[2,1,2\] (the meaning of this label is irrelevant here).
The vertices of the 1–complex in Figure \[fig:whl\_fh\_mi\] are labelled $1, 2, 3, 4, 4', 5, 5', 6, 6'$, and we denote the 1–simplex with vertices $i$ and $j$ by $[i, j].$ It follows from [@fh Proposition 6.1] that the interior of each union $[5, 2] \cup [2, 5']$ and $[5, 3]\cup [3, 5']$ consists of fibres, and hence corresponds to a cone over a fibred face of the Thurston norm ball (after choosing coherent orientations of the surfaces representing the vertices).
An explanation of how to modify the algorithm of [@fh] to obtain the essential surfaces that do not satisfy the meridional incompressibility condition is given in §8 of [@fh]. The resulting complex carrying all essential surfaces in the Whitehead link complement according to [@fh] is the 2–complex shown in Figure \[fig:whl\_fh\_ai\]. This does not agree with our argument in §\[sec:Incompressible normal surfaces\] that the space $\FH$ of essential surfaces is as shown in Figure \[fig:whl\_fh\_normal\]. A possible explanation is that the isotopy between the once-punctured tori $S_7$ and $S_8$ was missed in [@fh], because they are represented as non-isotopic surfaces in Figure \[fig:whl\_fh\_ai\].
Boundary curve space
--------------------
The boundary curve space can be computed from the complex $\FH.$ The surfaces carried by the segment $[5,3]$ in Figure \[fig:whl\_fh\_mi\] have the same boundary curves are those carried by $[5,2].$ Similar for $[3, 5']$ and $[2, 5'].$ All other surfaces are uniquely determined by their boundary curves. Moreover, the projectivised curves match up as shown in Figure \[fig:whl\_fh\_bcn\]. The dotted blue segments correspond to boundary curves of surfaces that are not meridionally incompressible.
Using the notation of Lash [@la], the boundary curve space is described explicitly as: $$\begin{aligned}
& \{ (-2, t, -2t, 1) \mid t \in [0,\infty] \} && a \\
\cup & \{ (0, t, 0, 1) \mid t \in [0,\infty] \} && b\\
\cup & \{ (2t, t, 4, 1) \mid t \in (0,1) \} && c\\
\cup & \{ (4t, t, 2, 1) \mid t \in (1,\infty) \} && d\\
\cup & \{ (2t+2, t, 2t, 1) \mid t \in (0,1) \} && e\\
\cup & \{ (2, t, 2t+2, 1) \mid t \in (1,\infty) \} && f\\
\cup & \{ (-t+3, 1, t+3, 1) \mid t \in [-1,1 \} && g\\
\cup & \{ (-2t, 0, 0, 1) \mid t \in [0,1] \} && h\\
\cup & \{ (0, 1, -2t, 0) \mid t \in [0,1] \} && i\\\end{aligned}$$ Lash [@la] only lists segments $a$–$g$ and we include $h$ and $i$ to account for the surfaces that are not meridional incompressible. We also changed the sign of the meridian coordinates in [@la] to account for a difference in conventions. We remark that Lash [@la] and Hoste-Shanahan [@hs] appear to compute only the space of projectivised boundary slopes of essential surfaces that are meridional incompressible, following only the main algorithm of Floyd and Hatcher. This explains why their spaces are not connected.
Surfaces and boundary curves arising from degenerations {#whl:surfaces arising from degenerations}
=======================================================
The logarithmic limit set of the deformation variety $\D(\tri_\whl)$ and the logarithmic limit set of the eigenvalue variety $\PEi_0(\whl)$ are determined in this section. Together with the results on essential surfaces of the previous section, these computations show that all strict boundary curves of $\whl$ are detected by the Dehn surgery component of the character variety. Since the boundary curves of fibres are detected by reducible characters, this shows that all boundary slopes of the Whitehead link are strongly detected. In addition, it is shown that $\D_\infty(\tri_\whl)$ has a fake ideal point, and some interesting degenerations are analysed in detail.
The logarithmic limit sets
--------------------------
The tropical variety $\D_\infty(\whl)$ is determined in Appendix \[sec:log lim debbie\] and shown in Figure \[fig:whl\_log\_lim-D\]. The above analysis of the rational maps and the normal surfaces together with the results concerning strongly detected boundary curves in [@defo] also determines the logarithmic limit set of $\PEi_0(\whl)$, which is shown in Figure \[fig:whl\_log\_lim-E\]. As discussed above, the isotopy relation is given by the action of the Klein four group $K_f,$ and the respective quotient spaces are shown in Figure \[fig:whl\_detected\].
It follows from this calculation and Theorem \[comb:essential prop\] that all non-fibre essential surfaces in $\whl$ are detected by the deformation variety and the character varieties. Moreover, no fibre is detected by the deformation variety nor the geometric component in the character variety. We also note that
1. there is a *fake ideal point*, namely the centre $V_0$ of the central square, which corresponds to an ideal point of the deformation variety that maps to a finite point of the character variety and the associated surface compresses to two boundary parallel tori;
2. each ideal point that is in the interior of the segments $[V_1,V_0]$, $[V_2,V_0]$, $[V_3,V_0]$ or $[V_4,V_0]$ detects a surface that is not incompressible but can be compressed to an essential dual surface.
Detected slopes
---------------
The relationship between ideal points of the $\PSL$–eigenvalue variety $\PEi(\whl)$ and ideal points of the deformation variety $\D(\whl)$ has already been discussed, and it follows that the logarithmic limit set of $\PEi_0(\whl)$ is as pictured in Figure \[fig:whl\_log\_lim\](b). In particular, all these slopes arise from sequences of irreducible representations. This was already shown for the slopes of meridional incompressible surfaces by Lash [@la], and the present computation extends this to the boundary slopes of all essential surfaces that are not fibres.
Let $S \subset \whl$ be an essential surface that is not a fibre. Then the boundary slopes of $S$ are strongly detected by a sequence of irreducible representations.
It is well known that the boundary slopes of fibres are strongly detected by sequences of reducible representations. The slopes detected by both reducible are irreducible representations are precisely the slopes of the once-punctured tori in $N_1$ and $N'_1.$ There are sequences of shape parameters which converge to ideal points corresponding to elements of $N_1$ and $N'_1$ giving rise to sequences of irreducible representations in the character variety since the eigenvalue of a longitude is constant equal to $-1$ throughout the degeneration. There also are sequences of reducible representations detecting the same slopes: $$\rho_m (\m_i) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
\qquad \text{and} \qquad
\rho_m(\m_j) = \begin{pmatrix} m & 0 \\ 0 & m^{-1} \end{pmatrix},$$ where $m \to 0$ or $m \to \infty.$
Some interesting degenerations
------------------------------
Recall that $\D'(\tri_\whl)$ is defined by $$\begin{aligned}
\label{whl:det det}
1 = wxyz \quad \text{and }\quad
wx(1 - y)(1 - z) = (1 - w)(1 - x)yz,\end{aligned}$$ and that for any $w,x,y,z \in \C -\{0, 1\}$ subject to these equations, there is a unique point on $\D (\whl )$. We will now describe sequences of points in $\D'(\tri_\whl)$ which correspond to well–defined sequences in $\D (\whl )$ approaching the desired ideal points.
1\. The point $(w,x,y,z) = (w, -w^{-1},w,-w^{-1})$ satisfies equations (\[whl:det det\]) for any $w \in \C-\{0\}$, and if $w$ has positive imaginary part, it defines a positively oriented triangulation of $\whl$, and in particular an incomplete hyperbolic structure on $\whl$. This implies that the triangulation can be deformed through positively oriented tetrahedra from the complete structure where $w=i$ to the ideal points where $w\to 0$ or $w\to \pm 1$. These points correspond to $V_1$ and $V_3$ for $w\to 1$ and $w\to -1$ respectively, and to $\frac{1}{2}(V_{16}+V_{19})$ for $w \to 0$, since the growth rates of all parameters are equal.
We give one example, where we scale to integer coordinates. As $w \to 0$, the ideal point $$\xi = (-1,0,1,1,-1,0,-1,0,1,1,-1,0)$$ is approached. The corresponding normal $Q$–coordinate is $$\begin{gathered}
N(\xi) = (0,1,0,0,0,1,0,1,0,0,0,1),\\
\text{whilst}\quad
\frac{1}{2}(N(S_{16})+N(S_{19})) = (0,1,0,0,0,1,0,1,0,0,0,1),\end{gathered}$$ which we can rescale to $\frac{1}{2}(V_{16}+V_{19})$.
Moreover, one easily obtains the limiting eigenvalues. As $w\to 0$ there are the following power series expansions for the resulting eigenvalues: $$\begin{aligned}
\s &= 1,
&& t = -1,\\
\u &= \frac{1+w}{w(1-w)} = \frac{1}{w} + 2 \sum_{i=0}^\infty w^i,
&& v = w^2.\end{aligned}$$ The first two equations reflect the fact that one cusp is complete, and comparing the growth rates at the second cusp (remembering that $\u$ is the square of an eigenvalue, whilst $v$ is an eigenvalue) gives a point in boundary curve space with coordinates $[0,0,4,1]$.
2\. For any $w \in \C -\{0, \pm 1\}$, the point $$(w,x,y,z) = (w,\frac{1-w}{1+w}, -\frac{1+w}{1-w} ,-w^{-1}) \in \D'(\tri_\whl)$$ gives a point on $\D (\whl )$. Note that the associated triangulation involves either only flat tetrahedra or both positively and negatively oriented ones. One may use power series expansions to show that a detected surface for $w\to 0$ is $S_8$: $$\begin{aligned}
x = \frac{1-w}{1+w} = 1 + 2 \sum_{i=1}^\infty (-1)^i w^i,&&
y = -\frac{1+w}{1-w} = -1 - 2 \sum_{i=1}^\infty w^i,&&
z = -\frac{1}{w}.\end{aligned}$$ Thus, as $w\to 0$, the ideal point $$\xi = (-1,0,1,0,1,-1,0,0,0,1,-1,0)$$ is approached. The corresponding normal $Q$–coordinate is $$N(\xi) = (0,1,0,1,0,0,0,0,0,0,0,1),$$ which coincides with the normal $Q$–coordinate of $S_8$, and may be rescaled to $V_8$.
3\. Detecting elements of $N_3 \cup N'_3$ is a little more involved. Let $$\begin{aligned}
w &= \frac{-1 - \eps^3 + \sqrt{5 - 4 \eps + 4 \eps^2 - 2 \eps^3 +
\eps^6}}{2(1+\eps^2)}
&& x = \eps\\
y &= -\eps \frac{2(1+\eps^2)}{-1 - \eps^3 + \sqrt{5 - 4 \eps + 4 \eps^2 - 2
\eps^3 + \eps^6}}
&& z = -\eps^{-2}\end{aligned}$$ Formal substitution of these assignments in the defining equations of the deformation variety shows that whenever the expressions are defined for any small, non–zero $\eps$, they determine a point of $\D (\whl )$. Since $w$ is differentiable at $\eps=0$ and takes the value $w(0) = \frac{-1\pm \sqrt{5}}{2}$, it follows that $w$ and $y$ have converging power series expansions at $\eps=0$, and $y$ has a zero of order one at $\eps=0$. The first few terms are: $$\begin{aligned}
w(\eps) &= \frac{-1\pm \sqrt{5}}{2} - \frac{1}{\sqrt{5}}\eps +
\big(1 - \frac{17}{5\sqrt{5}}\big) \eps^2
+ \big(-3 + \frac{123}{25\sqrt{5}}\big)\eps^3
+{\ldots} \\
y(\eps) &= - \frac{2}{-1+\sqrt{5}} \eps -
\frac{8}{\sqrt{5}(-1+\sqrt{5})^2}\eps^2 +
{\ldots}\end{aligned}$$ Thus, as $w\to 0$, we approach $$\xi = (0,0,0,-1,0,1,-1,0,1,2,-2,0).$$ A detected surface is therefore $S_{13}$.
4\. Let $(w,x,y,z) = (w, w^{-1},w^{-1}, w)$. Then $w \to 1$ approaches an ideal point corresponding to $V_0$. The chosen degeneration is through triangulations involving positively and negatively oriented tetrahedra, or triangulations which are entirely flat. In fact, there cannot be a degeneration through positively oriented tetrahedra to this ideal point, since the hyperbolic gluing equation $1 = w x y z$ would imply that throughout this degeneration $\arg (w) + \arg (x) + \arg (y) + \arg (z) = 2 \pi$, whilst for each parameter the argument converges to zero.
Experimentation with [SnapPy ]{}suggests that elements from all but the fourth set can be approached through degenerations only involving positively oriented tetrahedra. Using the projection of the right–handed Whitehead link, the following surgery coefficients can be approached through degenerations involving positively oriented triangulations:
- $(\infty) ,(4,1)$. Four tetrahedra degenerate and the splitting surface is a 1–sided Klein bottle.
- $(\infty) ,(0,1)$. Two tetrahedra degenerate and two become flat with parameters equal to $-1$. The splitting surface is a 2–sided torus.
- $(4,0),(0,2)$. Three tetrahedra degenerate, the remaining tetrahedron has shape $\frac{1}{2} + \frac{1}{2} i$. The limiting orbifold has volume approximately $0.9159$. ([SnapPy ]{}does not compute a splitting surface if all cusps have been filled.)
- $(8,2),(4,2)$. Three tetrahedra degenerate, the remaining tetrahedron has shape $1+i$, and the limiting orbifold has volume approximately $0.9159$. (Dito concerning the splitting surface.)
These examples will be put in a general framework in [@splittings].
Appendices {#appendices .unnumbered}
==========
Manifold data {#sec:snap data}
=============
The “gluing and completeness” data obtained from is given in Table \[tab:snap data whl\]. The relationship to the shape parameters used here is: $w = z_1',$ $x = z_2'',$ $y = z_3''$ and $z = z_4''.$ From this, one can verify that the holonomies of the peripheral elements given by are the inverses of the holonomies of the meridians and geometric longitudes given above, where ’s cusp $0$ corresponds to the green cusp (here:cusp 1), and cusp $1$ to the red cusp (here:cusp 0).
$z_1$ $z_2$ $z_3$ $z_4$ $1-z_1$ $1-z_2$ $1-z_3$ $1-z_4$
------------ ------- ------- ------- ------- --------- --------- --------- --------- ----
$H'(\m_0)$ 1 0 –1 0 –1 0 1 1 0
$H'(\m_1)$ 0 0 0 1 1 –1 0 0 0
$H'(\l_0)$ 1 0 0 0 0 1 –1 1 0
$H'(\l_1)$ –1 0 0 0 0 –1 –1 1 0
$e_1$ 1 1 1 1 1 –2 0 0 –1
$e_2$ 0 –1 –1 –1 –1 1 1 1 1
$e_3$ –1 1 1 1 1 0 –2 –2 –1
$e_4$ 0 –1 –1 –1 –1 1 1 1 1
: ’s gluing matrix for $m129$[]{data-label="tab:snap data whl"}
Eigenvalue variety {#sec:Eigenvalue variety}
==================
Eigenvalue variety {#whl:eigenvalue variety}
------------------
\[whl:compute eigen\]
Denote the component in $\Ei (\whl )$ arising from reducible representations by $\Ei^r (\whl ).$ We have $\Ei_0 (\whl ) = \overline{\Ei (\whl ) - \Ei^r (\whl )},$ where the overline denotes the Zariski closure. With respect to the chosen affine coordinates, we have $\Ei^r (\whl ) = \{ t = v = 1\},$ which is 2–dimensional. $\Ei_0 (\whl )$ is 2–dimensional since it is the closure of the image of the eigenvalue map. From this, it also follows that the intersection of $\Ei_0 (\whl )$ and $\Ei^r (\whl )$ is a union of two lines: $$\Ei^r(\whl ) \cap \Ei_0(\whl ) = \{ s = t = v = 1\} \cup \{ t = u = v = 1\}.$$ The following calculations use well known elimination and extension theorems which can be found in [@clo]. Consider $\Ta (\whl)$ and the eigenvalue map $\operatorname{e}$ (see equation (\[whl:eigenvalue map\])). The rational functions determining the eigenvalues of the longitudes are: $$\begin{aligned}
t =&\vartheta_{\l^t_0}(\rho )
=s^{-2}-s^{-2}u^2+u^2
+ c(2 s^{-1}u + s^{-3}u^{-1}-s^{-3}u) + c^2 s^{-2},\\
v =&\vartheta_{\l^t_1}(\rho )
=u^{-2}-u^{-2}s^2+s^2
+ c(2 u^{-1}s + u^{-3}s^{-1}-u^{-3}s) + c^2 u^{-2}.\end{aligned}$$ Then $s^2 t - u^2 v = s^2-u^2 + c(su^{-1}-s^{-1}u).$ Since $\Ta (\whl )$ is not contained in the hyperplane $s^2=u^2,$ this shows that the map $\operatorname{e}: \Ta (\whl ) \to \Ei_0 (\whl)$ has degree one, and also determines an inverse mapping: $$\label{whl:eigen to taut}
(s,t,u,v) \to \bigg(s,u, \frac{s^2(t -1) + u^2 (1-v)}{su^{-1}-s^{-1}u}\bigg).$$
Recall the defining equation (\[whl:taut\_rel\]) of $\Ta (\whl)$: $$\begin{aligned}
f_1 &= (s-s^{-1})(u-u^{-1})
+ c(s^{-2}u^{-2}-u^{-2}-s^{-2}+4-s^2-u^2+s^2u^2)\\
&\qquad +c^2(2s^{-1}u^{-1}-su^{-1}-s^{-1}u+2su) + c^3.\end{aligned}$$ Two additional polynomials $f_2 = s^3 u t + {\ldots} $ and $f_3 = s u^3 v + {\ldots} $ are obtained from the above expressions for $t$ and $v.$ The only variable to be eliminated is $c,$ and the leading coefficients of $c$ in $f_1,$ $f_2$ and $f_3$ are monomials in $\C[s,u].$ The elimination is done using resultants (see [@clo] for details). Since $s$ and $u$ are units, it follows that the eigenvalue variety is defined by $Res(f_1,f_2,c) = Res(f_1,f_3,c) = Res(f_2,f_3,c)=0.$ Eliminating redundant factors from the resultants gives the following set of defining equations for $\Ei_0 (\whl )$: $$\begin{aligned}
h_1 &=t-s^2t+s^2t^2-s^4t^2-u^2-2s^2tu^2+s^4tu^2-t^2u^2+2s^2t^2u^2\\
&\qquad +s^4t^3u^2+tu^4-s^2tu^4+s^2t^2u^4-s^4t^2u^4,\\
h_2 &= s^2-v-s^4v+u^2v+2s^2u^2v+s^4u^2v-s^2u^4v+s^2v^2-u^2v^2\\
&\qquad -2s^2u^2v^2-s^4u^2v^2+u^4v^2+s^4u^4v^2-s^2u^4v^3,\\
h_3 &= s^4t-s^6t-s^2tu^2+s^4tu^2+s^6t^2u^2-s^2u^2v\\
&\qquad +u^4v+s^2u^4v-2s^4tu^4v-u^6v+s^2u^6v^2.\end{aligned}$$
The mapping (\[whl:eigen to taut\]) is not defined when $s^2=u^2.$ One may now compute the defining equations of the subvariety of $\Ei_0(\whl)$ on which it is not defined: $$\begin{aligned}
s^2 = u^2, && t = v, &&
0 = -t + s^2(1+t)+s^4t(1-t)-s^6t^2(1+t)+s^8t^2.\end{aligned}$$
With a little more effort, one can compute the following inverse mappings defined on two open sets, which cover all but eight points of the eigenvalue variety: $$\begin{aligned}
\label{whl:ev: inv sl 1}\operatorname{e}^{-1}_1: \Ei_0(\whl ) \to \Ta (\whl ) &&
(s,t,u,v) &\to \Bigg(s,u,\frac{(s^2-v)(1-u^2)}{su(1+v)}\Bigg)\\
\label{whl:ev: inv sl 2}\operatorname{e}^{-1}_2: \Ei_0(\whl )\to \Ta (\whl ) &&
(s,t,u,v) &\to \Bigg(s,u,\frac{(u^2-t)(1-s^2)}{su(1+t)}\Bigg)\end{aligned}$$ The points corresponding to the complete structure are always singularities of the eigenvalue variety — here determined by the points where $t=v=-1$ and $s^2 = u^2 =1.$ The other points of $\Ei_0 (\whl )$ where neither of the above maps are defined are subject to $t=v=-1$ and $s^2 = u^2 = -1.$
The defining equations for the $\PSL$–eigenvalue variety can be worked out from $\PTa (\whl )$ similarly to the above, or from the results of the previous subsection using Lemma \[whl: eigenvalue maps lem\]. In particular, one has: $$\begin{aligned}
t =&\vartheta_{\l^t_0}(\prho_{GL} )
=\s^{-1}-\s^{-1}\u+\u
+ d(2 \s^{-1} + \s^{-2}\u^{-1}-\s^{-2}) + d^2 \s^{-2}\u^{-1},\\
v =&\vartheta_{\l^t_1}(\prho_{GL} )
=\u^{-1}-\u^{-1}\s+\s
+ d(2 \u^{-1} + \u^{-2}\s^{-1}-\u^{-2}) + d^2 \u^{-2}\s^{-1}.\end{aligned}$$ Then $\s\u^2 v - \s^2\u t = \s\u^2-\s^2\u+d(\u-\s)$ implies that the degree of the map $\PTa (\whl ) \to \PEi (\whl )$ is equal to one. It follows from the above discussion that there are only two points where neither of the following inverse maps is not defined: $$\begin{aligned}
\label{whl:ev: inv psl 1}\operatorname{\overline{e}}^{-1}_1: \PEi_0(\whl ) \to \PTa (\whl ) &&
(\s,t,\u,v) &\to \Bigg(\s,\u,\frac{(\s-v)(1-\u)}{(1+v)}\Bigg)\\
\label{whl:ev: inv psl 2}\operatorname{\overline{e}}^{-1}_2: \PEi_0 (\whl )\to \PTa (\whl ) &&
(\s,t,\u,v) &\to \Bigg(\s,\u,\frac{(\u-t)(1-\s)}{(1+t)}\Bigg).\end{aligned}$$
The results of this and the previous subsections are summarised in the following lemma. This provides an alternative proof of Lemma \[whl:degree one lemma\].
The varieties $\Ta (\whl )$ and $\Ei_0 (\whl )$ are birationally equivalent, and so are the varieties $\PTa (\whl )$ and $\PEi_0 (\whl ).$
Dehn fillings on one cusp
-------------------------
If one of the cusps is assumed to be complete, then the resulting subvariety, which parametrises hyperbolic Dehn fillings on the other cusp, is defined by: $$\begin{aligned}
0 &= 1 - t + 4\s t - \s^2 t + \s^2 t^2 &&\text{when } \u = 1,v = -1,\\
0 &= 1 - v + 4\u v - \u^2v + \u^2v^2 &&\text{when } \s= 1, t = -1.\end{aligned}$$ The boundary curves $(0,0,0,1),$ $(0,0,4,1)$ and $(0,1,0,0),$ $(4,1,0,0)$ are detected by these curves respectively. The above equations may be written as the following well–defined trace relations: $$4 = \operatorname{tr}\prho(\m_i^2) - \operatorname{tr}\prho(\m_i^2\l_i^t)
= \operatorname{tr}\prho(\m_i^2) - \operatorname{tr}\prho(\l_i^g) \text{ where } i\in\{0,1\}.$$ Curves in $\D(\tri_\whl)$ corresponding to these curves in the eigenvalue variety can readily be determined. Consider points in $\PEi_0 (\whl )$ of the form $(1,-1,\u, v).$ The preimage under $\edefo$ of such a point is of the form $\varphi^{-1} (w,-w^{-1},w,-w^{-1})$ for some $w\in \C-\{0,\pm 1\},$ and hence $H'(\m_1) = \frac{1+w}{w(1-w)}$ and $H'(\l^t_1) = w^2.$
Eigenvalues at $\mathbf{(w,x,y,z) = (1,1,1,1)}$ {#whl:eigenvalues at (1,1,1,1)}
-----------------------------------------------
Using the implicit function theorem, the eigenvalues at the ideal point $$\xi = (0,1,-1,0,1,-1,0,1,-1,0,1,-1)$$ where $w,x,y,z \to 1$ can be computed. Note that this point maps to a singularity of $\D'(\tri_\whl).$ We use the projection $(w,x,y,z) \to (w,y,z),$ with image defined by: $$0= g(w,y,z) = w - wy - wz + yz - wy^2z^2 + w^2y^2z^2.$$ The holonomies of meridians take the form $$H'(\m_0) = - yz \frac{1-w}{1-y},
\qquad
H'(\m_1) = - \frac{1-z}{yz(1-w)},$$ and as $\xi$ is approached, the eigenvalues of the longitudes converge to one. Since the growth rates of all parameters are equal, let $$\begin{aligned}
w = 1 + \eps, && y = 1 - \s^{-1} \eps + \eps Y && z = 1 - \u \eps,\end{aligned}$$ and we are seeking $Y(\eps)$ such that $g(w(\eps),y(\eps),z(\eps))=0$ for all $\eps\in B_\delta(0)$ and $Y(0)=0$ for some $\s,\u \in \C-\{0\}.$ Substitution yields a polynomial $F(\eps,Y),$ and the implicit function theorem applies if $(\s-1)(\u-1)=0.$ We have $$H'(\m_0) = \frac{(1-\u \eps)(1-\s^{-1} \eps+\eps Y)}{\s^{-1}+Y}
\quad\text{and}\quad
H'(\m_1) = \frac{\u}{(1-\u \eps)(1+\eps)}.$$ Thus, for any $\s,\u \in \C-\{0\},$ one can obtain the limiting eigenvalues $(1,1,\u,1)$ and $(\s,1,1,1)$ as $\xi$ is approached. These eigenvalues are precisely the eigenvalues of the reducible representations in $\PTa(\whl).$
Character varieties {#whl:Character varieties}
===================
A complete description of the character varieties associated to $\whl$ is now given.
Since $\Ta (\whl )$ is irreducible, it follows that the $\SL$–character variety is of the form $\X(\whl) = \X_0(\whl) \cup \X^r(\whl),$ where $\X_0(\whl)$ is the Dehn surgery component and $\X^r(\whl)$ is the set of reducible characters. A defining equation for the $\SL$–Dehn surgery component can be worked out from $\Ta (\whl ).$ Let $X = \operatorname{tr}\rho (\m_0 ), Y = \operatorname{tr}\rho (\m_1 )$ and $Z = \operatorname{tr}\rho (\m_0\m_1 ),$ then $\X_0(\whl )$ is defined by $$\label{eq:SL_Dehn surgery component}
0 = F (X,Y,Z) = XY + (2 - X^2 - Y^2)Z + XYZ^2 - Z^3.$$ The set of reducible characters is parametrised by $\operatorname{tr}\rho[\m_0,\m_1]=2,$ which is equivalent to: $$4 = X^2 + Y^2 + Z^2 - XYZ.$$
The form of the relator in (\[whl:fundamental group\]) implies that a $\PSL$–representation lifts to $\SL$ if and only if the relator is equal to the identity in $\SL$ for any assignment of matrices representing the $\PSL$–representation. The Dehn surgery component of the $\PSL$–character variety can be worked out from (\[eq:SL\_Dehn surgery component\]). With $\pX = X^2, \pY = Y^2$ and $\pZ = Z^2$ one obtains: $$\begin{aligned}
0 &= \pF (\pX ,\pY ,\pZ ) \\
&=
-\pX \pY + (4 - 4 \pX + \pX^2 - 4\pY + \pY^2)\pZ
- (4 - 2\pX - 2\pY + \pX\pY)\pZ^2 + \pZ^3.\end{aligned}$$ Note that $\pF (X^2, Y^2, Z^2) = F (X,Y,Z) F(-X,Y,Z) = F (X,Y,Z) F(X,-Y,Z).$
Computation of $\PSL$–representations which do not lift to $\SL$ reveals that there is only a finite set, parametrised by $(\pX, \pY, \pZ) = (0,0,2 \pm \sqrt{2}).$ These points do not satisfy $\pF,$ and the corresponding representations are irreducible on the peripheral subgroups, with image isomorphic to $\Z_2 \oplus \Z_2.$ In particular, they do not contribute any points to the $\PSL$–eigenvalue variety. Thus, all boundary curves which are detected by $\PEi_0(\whl)$ are also detected by $\Ei_0(\whl).$
The Logarithmic limit set {#sec:log lim debbie}
=========================
The following calculation was first done by Debbie Yuster. The ideal formed by the hyperbolic gluing equations and parameter relations given in Section \[sec:defo equations\] is: $$\begin{aligned}
I=&\langle wxyz-1, w'x'y'z'w''^2x''^2-1, w'x'y'z'y''^2z''^2-1, \\
&w-ww''-1, w'-ww'-1, w''-w''w'-1, \\
& x-xx''-1, x'-x'x-1, x''-x''x'-1,\\
& y-yy''-1, y'-y'y-1, y''-y''y'-1, \\
& z-zz''-1, z'-z'z-1, z''-z''z'-1\rangle.\end{aligned}$$ Solving the first three expressions for $z,$ $z',$ and $z''$ respectively gives: $$z= \frac{1}{wxy}, \qquad
z'=\frac{1}{w'x'y'w''^2x''^2}, \qquad
z''=\frac{w''x''}{y''}.$$ The sign in the last equation is determined by equations (\[whl: defo simple relations\]). Substitute these values back into the 15 ideal generators above. The first three generators become $0,$ and we obtain an ideal with 12 generators in the variables $w,w',w'',x,x',x'',y,y',$ and $y''.$ The last three of these generators, respectively, are: $$\begin{aligned}
&\frac{1}{wxy}-\frac{w''x''}{wxyy''}-1,\\
&\frac{1}{w'x'y'w''^2x''^2}-\frac{1}{w'x'y'w''^2x''^2wxy}-1,\\
&\frac{w''x''}{y''}-\frac{1}{w''x''y''w'x'y'}-1.\end{aligned}$$ We multiply each of these by the least common multiple of its denominators, thus clearing denominators. This gives the ideal: $$\begin{aligned}
J=&\langle w-ww''-1, w'-ww'-1, w''-w''w'-1, x-xx''-1, x'-x'x-1, \\
& x''-x''x'-1, y-yy''-1, y'-y'y-1, y''-y''y'-1, y''-w''x''-wxyy'', \\
& w'w''^2x'x''^2y'-1-y''w''x''w'x'y', wxy-1-w'w''^2x'x''^2y'wxy\rangle .\end{aligned}$$ Since $ww'w''+1,$ $xx'x''+1,$ and $yy'y''+1$ are in $J,$ we make the following substitutions: $$w''= \frac{-1}{ww'},\qquad
x''=\frac{-1}{xx'},\qquad
y''=\frac{-1}{yy'}$$ and clear denominators as before. After deleting repeated elements, we are left with four generators: $$K=\langle ww'+1-w', xx'+1-x', yy'+1-y', yy'-w^2x^2yw'x'+ww'xx'\rangle.$$
We relabel the variables so they are alphabetically contiguous, starting from the beginning of the alphabet, as follows: $w\rightarrow a,$ $w'\rightarrow b,$ $x\rightarrow d,$ $ x'\rightarrow e,$ $ y\rightarrow c,$ $ y'\rightarrow f.$
We now perform a few operations to make computing the tropical variety easier. Namely, we homogenise the ideal by first homogenising the generators and then saturating with respect to the homogenising variable. For ideals $I_0$ and $J_0,$ the saturation is the set of all elements $f$ in the ambient ring such that $J_0^Nf$ is contained in $I_0$, for some integer $N$. Saturating an ideal by a principal monomial ideal does not alter the tropical variety.
We homogenise using the variable $g$ and call the resulting ideal $L$: $$L=\langle ab+g^2-bg, de+g^2-eg, cf+g^2-fg, cfg^5-a^2d^2cbe+abdeg^3\rangle.$$ We saturate $L$ with respect to $\langle g\rangle$ and find minimal generators, using `Macaulay 2` [@M2]. The final ideal is: $$\begin{aligned}
&\langle de-eg+g^2, cf-fg+g^2, ab-bg+g^2,\\
&acd-ace+acg-bcd+bce-bcg-beg+bg^2+cdg-ceg+cg^2+eg^2-fg^2\rangle.\end{aligned}$$ Next we run `Gfan`’s commands `tropical_starting_cone` and `tropical_traverse` on these generators, and we obtain a tropical variety isomorphic to the tropical variety of $I$ with the isomorphism determined by the above substitutions. The result is shown in Figure \[fig:whl\_log\_lim-D\].
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---
abstract: 'This paper proposes a new way to do event generation and analysis in searches for new physics at the LHC. An abstract notation is used to describe the new particles on a level which better corresponds to detector resolution of LHC experiments. In this way the SUSY discovery space can be decomposed into a small number of eigenmodes each with only a few parameters, which allows to investigate the SUSY parameter space in a model-independent way. By focusing on the experimental observables for each process investigated the Bottom-Up Approach allows to systematically study the boarders of the experimental efficiencies and thus to extend the sensitivity for new physics.'
author:
- |
Claus Horn\
*SLAC National Accelerator Laboratory, Stanford University, CA\
*
title: 'A [Bottom-Up Approach]{}to SUSY Analyses'
---
[Introduction]{} To be prepared for the potential discovery of all phenomenological manifestations of the 120 dimensional parameter space of the MSSM poses challenges to computing and manpower even in modern day collaborations. Experimental approaches in the past mostly focused on a small number of theoretically motivated benchmark points or generated grids of points in simplified models with less parameters but also reduced phenomenological coverage. Newer ideas generate large numbers (order of 1000) of random points in higher dimensional spaces.
Former approaches to general searches for new physics tried to look for all possible combinations of final state particles [@Knuteson] and did not take into account that these signatures are the result of an underlying structure induced by the fundamental interactions of the particle types of the MSSM which can be classified as presented here. These structures (the elementary mass spectra of Sect. \[sec:elemetarymassspectra\]) are the main source of the observed correlations and degeneracies in the mapping between SUSY parameters and observable signatures. Therefore, solving the inverse problem [@inverseProblem] does not start with inclusive measurements of any given final state but it starts with doing the analyses according to these structures.
The approach presented here will argue that the only experimentally relevant parameters are the masses of the new particles, resulting in a very small number of parameters that have to be considered in the analysis of each eigenmode (the dominant observable channels). The aim of each analysis in this approach is to map out the detector efficiency as a function of these parameters. The mapping from observables to the parameters of a theoretical model can then be factored out to the generator level, omitting the time consuming generation and analysis of events for each parameter point. As a result, the experimental findings can be interpreted in many different theoretical models in many dimensions and with high precision.
The idea to focus directly on the parameters of the new particles instead of the abstract parameters of a particular SUSY model has been applied to specific problems in the past (see Sect. \[sec:examples\]) and has inspired the proposal for an investigation of simplified SUSY models [@simplifiedModels]. Unfortunately, it has not been applied in LHC analyses so far.
The point made in this paper is that searches for new physics in general could benefit significantly from a Bottom-Up analysis approach. In particular, it is argued that an application of the [Bottom-Up Approach]{}would allow to cover the SUSY discovery space in a model independent way.
[Observables]{} New SUSY particles are characterized by a number of quantum numbers, of which most, like e.g. the spin, are fixed for a given particle type and do not vary with the SUSY model parameters. The only quantities that may vary are the masses and effective couplings of the new particles.
While special SUSY breaking models pose restrictions on the mass ratios that can be investigated and therefore cover only parts of the observable space. The [Bottom-Up Approach]{}allows to investigate the complete observable space accessible to experiment at once and in a systematic way by directly varying the masses of the new particles in the events generated.
On the other hand, the couplings only change the branching ratios and thus have no effect on the efficiency of each elementary mass spectrum analysis. Similarly, all theoretical distinctions in the particles, for instance between $u_R$ and $u_L$, and theoretical parameters which only affect the branching ratios, like the trilinear couplings $A_i$, are irrelevant for the event generation and analysis step and can be factored out to a separate step of theoretical interpretation. The experimentally relevant parameters that have to be considered in event generation are [^1]: $$m(\tilde{g}), \{ m(\tilde{q}), m(\tilde{b}), m(\tilde{t}) \}, m(\tilde{\chi_1}), m(\tilde{\chi_2}), \{ m(\tilde{e}), m(\tilde{\mu}), m(\tilde{\tau}), m(\tilde{\nu}) \}.$$ However, by decomposing the analysis according to their eigenmodes, only up to five of them are relevant at once for the analysis of each mode (as indicated by the brackets).
In addition, the experiments will deliver measurements of the effective cross sections for each channel investigated. These can be compared to determine the branching fractions and thus indirectly provide measurements for the couplings of the new particles and finally the composition of the gauginos which might shed some light on the nature of the underlying SUSY model.
[Elementary Mass Spectra of the MSSM]{} \[sec:elemetarymassspectra\] Each complex mass spectrum (comprising all sparticles found at a given SUSY model point) can be decomposed into a small number of elementary mass spectra by applying the following transformations:
- Convert it to Abstract Notation [@myThesis] by replacing all squarks with $\tilde{q}$, all gauginos with $\tilde{\chi}$ and all sleptons with $\tilde{l}$;[^2]
- Identify the decay channels with the highest value in ${\rm BR} \frac{\epsilon_S}{\epsilon_B}$.[^3]
As a result one recovers always the same small number of elementary mass spectra for all MSSM parameter points[^4].
To see which are the elementary mass spectra of the MSSM one may attempt to construct them from scratch. In the construction the following constraints have to be fulfilled:
- To result in a high cross section, sparticles have to be produced via strong couplings, i.e. via $\tilde{q}\tilde{q}$, $\tilde{q}\tilde{g}$ or $\tilde{g}\tilde{g}$ production;
- To ensure a cold dark matter candidate the LSP can be only $\tilde{\chi}_1^0$ or $\tilde{G}$;
- Due to phase space constraints, the branching ratios for longer decay chains are very small. The only exceptions that have to be considered are:
- Wino like higher gauginos, or
- Intermediate sleptons which increase the selection efficiency.
{width="12cm"}
A set of elementary mass spectra that can be derived in this way is shown in Figure \[fig:elementarymassspectraMSSM\]. Together they form the MSSM discovery space. Since experiments allow to differentiate between light, heavy-flavor and top quarks as well as different types of leptons they may be subdivided into different cases as indicated in the figure [^5]. Note that Fig. \[fig:elementarymassspectraMSSM\] provides a compact overview of possible SUSY signatures, not all of which are covered by current LHC analyses. One observes, for instance, that isolated, high energy photons are very important and appear in combination with many other final states.
[Analysis of pMSSM Points]{} \[sec:pmssmana\] As a cross check for the sensibility of the results obtained in the last section 1000 randomly generated and not yet excluded pMSSM [@pMSSMparameters] parameter points (see [@JoAnneTom] for a description of the experimental constraints applied) were investigated with respect to their elementary mass spectra content. The steps performed were:
- PYTHIA was used to derive the complex mass spectra and decay tables for all sparticles for each pMSSM parameter point[^6];
- Starting with squarks and gluino, all the decay products were recorded and were followed iteratively until the LSP, resulting typically in ten to forty possible decay chains for each pMSSM point;
- Each decay chain was translated into abstract notation and the corresponding elementary mass spectra were extracted, summing the contributing branching ratios, resulting typically in about one to four elementary mass spectra for each pMSSM point;
- The average and maximum branching ratios were calculated for all 1000 pMSSM points.
The result is shown in Table \[tab:elementaryMassSpectra\]. Note that the pMSSM points only include spectra where the $\chi_1^0$ is the LSP. The elementary mass spectra were not distinguished with respect to the number of jets. Only decays with ${\rm BR}\ge{1\%}$ were considered for each sparticle.
Elementary Mass Spectra avg BR \[%\] max BR \[%\]
-------------------------------------------------------------------------------- -------------- --------------
$(\tilde{q},\tilde{g}), \tilde{\chi}$ 86.4477 100
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\chi}$ 9.8834 100
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\nu}, \tilde{\chi}$ 1.2841 67.1513
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\tau}, \tilde{\chi}$ 1.1207 42.1953
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{e}, \tilde{\chi}$ 0.6217 29.5350
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\mu}, \tilde{\chi}$ 0.5578 25.7138
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\chi}, \tilde{\chi}$ 0.0840 14.4386
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\chi}, \tilde{\nu}, \tilde{\chi}$ 0.0006 0.2469
One observes that the number of different elementary mass spectra in the pMSSM is indeed quite small and in agreement with the arguments given in Section \[sec:elemetarymassspectra\].[^7] In contrast to mSUGRA based expectations, the importance of the slepton mode is quite small, while jet-only and boson modes seem to deserve some extra attention.
[Event Generation]{} While SUSY analyses traditionally start by choosing a parameter point in a specific SUSY model, the [Bottom-Up Approach]{}starts from the elementary mass spectra derived in Section \[sec:elemetarymassspectra\]. This makes it independent of theoretical assumptions of special SUSY models (like the physics at the GUT scale, in a hidden-sector, etc.), which are out of the reach of LHC experiments. Since the new particles can be characterized on an abstract level just by their fundamental interactions, the results may be interpreted on much more general grounds (beyond the existence of supersymmetry).
While traditionally, an analysis is performed on the basis of a complex mass spectrum at some SUSY model point, the aim in the [Bottom-Up Approach]{}is the measurement of the effective cross section for a given eigenmode. Each eigenmode is analyzed separately. The combinations of the results from analyzing the different elementary mass spectra then allows to reconstruct the complex mass spectrum realized by nature.
Hence, for the generation of events the mass spectrum is directly specified in the generator (like e.g. PYTHIA) and only decays to the next lighter sparticle in the spectrum are allowed (For others the branching ratio is set to zero.), so that the total branching ratio for the longest decay channel, passing through all sparticles in the spectrum is $100\%$.
From each PYTHIA input file corresponding to a given eigenmode events can then be generated with several different kinematics by varying the specified mass values (see Section \[sec:analysis\] for more details.).
The decomposition of the analyses according to the elementary mass spectra can be expected to be very helpful for the determination of the total sparticle mass spectrum since it separates out the different eigenmodes while the traditional approach only looks at the convoluted spectrum at different SUSY model points[^8].
[Analysis]{} \[sec:analysis\] The aim of any search for new particles is to claim discovery, which can be done if the number of expected signal events, $N_{\rm exp}$, exceeds some limit which depends on the number of expected background (B) and observed data events ($N_{\rm obs}$): $$N_{\rm exp} > N_{\rm lim}(B, N_{\rm obs})$$ Traditionally, $N_{\rm exp}$ is estimated by generating events for a specific SUSY model point, $\vec{\theta}$, and determining the efficiency at that point, $\epsilon(\vec{\theta})$: $$N_{\rm exp}(\vec{\theta}) = \mathcal{L} \sum_{\rm ch} \sigma_{\rm ch}(\vec{\theta}) \times \epsilon_{\rm ch}(\vec{\theta})$$ where $\sigma_{\rm ch}$ is the effective production cross section for a given analysis channel, ch, and $\mathcal{L}$ the integrated luminosity of the data investigated. In order to get a good sampling of $N_{\rm exp}$ this requires the analysis of a large number of channels (one for every combination of final states) in a large dimensional space (e.g. ${\rm dim}(\vec{\theta})=19$ in the pMSSM).
Instead, the decomposition proposed here is: $$N_{\rm exp}(\vec{\theta}) = \mathcal{L} \sum_{\rm em} {\sigma}_{\rm em}(\vec{\theta}) \times \epsilon_{\rm em}[\vec{m}_{\rm em}(\vec{\theta})]$$ with a small number of eigenmodes, em (see Sect. \[sec:eigenmodes\]), and ${\rm dim}(\vec{m}_{\rm em})$ = 2 to 5.
Since it starts from the masses the [Bottom-Up Approach]{}separates the event generation and analysis step from the interpretation of the results. In consequence, it allows to profit from the fact that the dominant factor in the calculation of $N_{\rm exp}$, $\sigma_{\rm em}$, which varies exponentially with the masses, can be calculated quickly and in high resolution at the generator level. Therefore, $N_{\rm exp}$ can be determined easily in many dimensions of model parameters while only a few mass spectra variations have to be generated to determine the effect on the detector efficiency, $\epsilon_{\rm em}(\vec{m}_{\rm em})$, which usually varies only by a few percent.
Thus, the aim of each analysis is to map out the detector sensitivity by generating specific efficiency benchmark points in the observable space which are driven by experimental constraints. This allows the analysis to explicitly focus on the kinematically challenging regions of its specific efficiency space. Typical examples for such experimental extremes are signatures with very low [${\rm E}_{\rm T}^{\rm miss}$]{}and jets only (if the LSP is at its lower boarder), decay products with very low $p_T$ (if the mass difference between two sparticles is small), boosted decay products (if the mass difference is large) and quasi stable particles (if a sparticle’s only decay mode offers very little phase space, i.e. the two sparticles have similar masses). These borders of efficiency were often neglected in the past resulting in holes in the observable parameter spaces (see e.g. [@JoAnneTom]).
For the investigation of the mass parameter space it is important to notice that the number of necessary points is relatively small since the hierarchy has to be conserved. For instance, the kinematically extreme low/high mass cases of mass spectra with three new particles can be studied by considering just $4$ scenarios (instead of $2^3=8$), as illustrated in Fig \[fig:massspectraexample\].[^9]
{width="6cm"}
At the level of each analysis it will typically be possible to choose a more effective parameterization of the mass space (e.g. by considering mass differences) which allows to concentrate on the most important parameters of the detector efficiency.
A variation of the SUSY model parameters will typically result in:
- No changes of the masses; these parameters are thus irrelevant for analysis, or
- Changes of several masses at the same time, thus changing the detector response in a complex way.
A direct variation of the masses, on the other hand, will produce a well understood change in detector response (like the variation of a jet’s $p_T$) and interpolation in the mass space is thus in general straight forward. On the other hand, the change in sensitivity due to changes in model parameters is dominated by the effect of changing branching ratios of the different eigenmodes. Finally, a study of the jet kinematics resulting from variations of the produced squark and gluino masses can be factored out since it is common to all elementary mass spectra (see Sect. \[sec:susyjets\]).
[Example Applications]{}\[sec:examples\] This section describes three brief examples how the [Bottom-Up Approach]{}may be applied to specific analyses.
[Search for Gravitinos]{} A search for gravitinos resulting form the GMSB decay $\tilde{\chi} \rightarrow \gamma \tilde{G}$ was performed with HERA data where the neutralinos may be produced by the exchange of a virtual slepton [@myThesis].
A traditional approach of considering different GMSB parameter points would not allow for the investigation of the complete observable space (due to mass correlations imposed by the GMSB model) and only allow the analysis of a few GMSB parameter points. Instead, events were generated for different sparticle masses in agreement with the [Bottom-Up Approach]{}. Since the selection efficiency was measured as a function of the neutralino mass (see Figure \[fig:myEfficiency\]) the results could be interpreted in the complete GMSB parameter space by applying the following steps:
- For each parameter point, the effective signal cross section and neutralino mass was calculated and the kinematic region determined;
- The neutralino mass was used to determine the signal efficiency;
- The efficiency, the effective cross section and the investigated data luminosity were used to calculate the number of expected signal events (similarly for the number of expected background events);
- The number of detected data events in the given kinematic region was compared to the number of expected signal and background events to calculate the confidence level for the given parameter point.
Note that different selection criteria may be used for different kinematic regions[^10]. Thus, once the detector efficiency is measured in terms of the relevant parameters all the calculations above only take a few seconds for each parameter point.
{width="8cm"}
[**Summary:**]{} While traditionally, the generation and analysis of events for all values of GMSB parameters seems impossible. Following the [Bottom-Up Approach]{}, the complete GMSB parameter space could be investigated by analyzing 6 different neutralino mass points.
[SUSY Jets]{}\[sec:susyjets\] Traditionally, SUSY discovery reach is often described in the two dimensional mSUGRA plan of $m_0$ and $\,m_{\frac{1}{2}}$. This misses the fact that a general description of jet kinematics in SUSY production at the LHC requires three parameters: the masses of the gluino, the lightest squark and the next lightest gaugino. However, for different mass combinations a different of the three strong sparticle pair-production processes dominates:
- If $m(\tilde{q}) \gg m(\tilde{g})$: $\tilde{g}\tilde{g}$-production dominates leading to events with four parton level jets. Jet kinematics depend only on $m(\tilde{g})-m(\tilde{\chi})$.
- If $m(\tilde{q}) = m(\tilde{g})$: $\tilde{g}\tilde{q}$-production dominates leading to events with three parton level jets.
- If $m(\tilde{q}) \ll m(\tilde{g})$: $\tilde{q}\tilde{q}$-production dominates leading to events with two parton level jets. Jet kinematics depend only on $m(\tilde{q})-m(\tilde{\chi})$.
The first case has been investigated in detail and compared to results from ${\rm D\O}$ [@jayGluinoStudy]. Since the ${\rm D\O}$ analysis was performed in the mSUGRA plane, the reach for a generic $m(\tilde{g})$-$m(\tilde{\chi}_1^0)$ point is unknown (in mSUGRA the gaugino mass ratio is fixed to: $m(\tilde{g})/m(\tilde{\chi}_1^0) = 6$). Unfortunately, this restricted view hides the fact that different selection criteria would be optimal for different kinematic configurations. The improved selection criteria proposed in [@jayGluinoStudy] are:
- Optimized cuts for different bins in [${\rm E}_{\rm T}^{\rm miss}$]{}and ${\rm H}_{\rm T}$ ($=\sum_{\rm jets} E_T$), and
- In the case where $\tilde{g}$ and $\tilde{\chi}_1^0$ are nearly degenerate, considering cases with initial and final state radiation will result in signatures with increased [${\rm E}_{\rm T}^{\rm miss}$]{}and thus sensitivity.
The result of this study is shown in Figure \[fig:gluinoStudy\].
{width="8cm"}
It was shown that by optimizing the selection criteria for different mass regions the search reach can be significantly extended.
By adding one more parameter ($m(\tilde{q})$) to the previous example it would be possible to investigate SUSY production at the LHC in a model-independent way. Thereto, the other two cases ($\tilde{q}\tilde{q}$ and $\tilde{g}\tilde{q}$) should be analyzed in a similar fashion as done for the $\tilde{g}\tilde{g}$ case. This would allow to determine the sensitivity for any combination of the three masses (i.e. for any SUSY parameter point)[^11].
More complex analysis with additional sparticles in the spectrum, can then focus on the borders of the so mapped out efficiency space to see where it can be extended due to the additionally radiated particles.
[**Summary:**]{} The analysis of jet kinematics in the mSUGRA plane leaves holes in the observable parameter space and does not consider all kinematic configurations. Following the [Bottom-Up Approach]{}, a complete study of the SUSY jet kinematics is possible independent of the underlying SUSY model. Additionally, it would be possible to define a few jet-benchmark points at the boarder of the trigger efficiency which could serve as a reference for more complex analyses (see Sect. \[sec:eigenmodes\]).
[Boosted Neutralinos]{} The ATLAS collaboration has performed a study to identify boosted LSPs decaying into three jets via RPV $\lambda^{"}$ couplings by analyzing the jet substructure [@boostedNeutralino]. In order to investigate the sensitivity within some SUSY model space one would traditionally analyze a number of benchmark points.
However, following the [Bottom-Up Approach]{}the efficiency can be determined in a model-independent way by applying the following steps:
- Consider the simplest elementary mass spectrum only consisting of a squark and a neutralino.
- Generate events for a few different mass points to map out the efficiency as a function of neutralino boost and mass.
- Write a function which returns the distribution of neutralino boosts (using a generator) as a function of the masses of the sparticles in a given decay chain.
- Chose theoretical parameters to interpret the results in, run a generator to calculate the mass spectrum for each parameter point, determine the neutalino boost and calculate the sensitivity for that parameter point[^12].
A conservative estimate for the trigger efficiencies can be taken from the studies proposed in the last example. [**Summary:**]{} While traditionally, the generation and analysis of events for all pMSSM parameters would be impossible. Following the [Bottom-Up Approach]{}, the effect of changing kinematics can be determined in a model-independent way and the sensitivity for any pMSSM parameter point can be calculated at generator level.
The above are just a few examples of possible benefits of the [Bottom-Up Approach]{}. One can easily imagine how it could profitably be applied to many other analyses and especially the eigenmode analyses described in the next section.
[Eigenmodes]{}\[sec:eigenmodes\] One additional complication not addressed so far is that the elementary mass spectra may appear in different combinations since sparticles are pair-produced resulting in two separate decay chains per event[^13].
With the aim to investigate the sensitivity for a given elementary mass spectrum, the question arises which of the combinations with the other spectra is prevailing. For the pMSSM points investigated in Sect. \[sec:pmssmana\] the answer is shown in Table \[tab:prevailance\].
$(\tilde{q},\tilde{g}), \tilde{\chi}$ $(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\chi}$ $(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{l}, \tilde{\chi}$
---------------------------------------------------------------- --------------------------------------- ----------------------------------------------------- ----------------------------------------------------------------
$(\tilde{q},\tilde{g}), \tilde{\chi}$ 75% 17% 6%
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{\chi}$ 17% 0.9% 0.7%
$(\tilde{q},\tilde{g}), \tilde{\chi}, \tilde{l}, \tilde{\chi}$ 6% 0.7% 0.1%
As one might expect, the contribution of each spectrum is biggest when combined with the shortest mode, $(\tilde{q},\tilde{g}), \tilde{\chi}$. It seems thus reasonable to focus on the three classes of eigenmodes illustrated in Figure \[fig:theeigenmodes\].
{width="9cm"}
Together they cover $98\%$ of the SUSY branching fraction. Hence, finding supersymmetry can be reduced, in first order, to the investigation of these three combinations of elementary mass spectra.
In order to investigate them, one has to take into account all the different cases shown in Figure \[fig:elementarymassspectraMSSM\]. This gives rise to the eigenmodes listed in Table \[tab:theeigenmodes\] which also shows the corresponding final states.
Eigenmode Eigenmode Class Possible Signatures
----------- ----------------- ------------------------------------------------------------------
Aq A [${\rm E}_{\rm T}^{\rm miss}$]{}+ (2jets, 3jets, 4jets)
Ab A [${\rm E}_{\rm T}^{\rm miss}$]{}+ (2 b-jets, 3 b-jets, 4 b-jets)
At A [${\rm E}_{\rm T}^{\rm miss}$]{}+ (2 top, 3 top, 4 top)
BZ B [${\rm E}_{\rm T}^{\rm miss}$]{}+ jets + (ll, $\nu\nu$, qq, bb )
BW B [${\rm E}_{\rm T}^{\rm miss}$]{}+ jets + (qq, l$\nu$, bt )
C0l C [${\rm E}_{\rm T}^{\rm miss}$]{}+ jets
C1l C [${\rm E}_{\rm T}^{\rm miss}$]{}+ jets + 1l
C2l C [${\rm E}_{\rm T}^{\rm miss}$]{}+ jets + 2l
Note that while all different cases of eigenmodes have to be considered to be sensitive to all kinds of mass spectra that nature may have chosen, not all possible decay channels of the radiated standard model bosons necessarily have to be analyzed. For the eigenmodes of class B for instance, one can start by considering the leptonic decays of W and Z. Adding additional sub-channels will increase the sensitivity. A subsequent separation of the Higgs cases compared to W/Z may be achieved by considering invariant mass distributions and ratios of the different decay modes.
Note that the mass parameters can be chosen in a way that their effects on the efficiency are independent over most of the parameter space, as for instance $m(\tilde{q})-m(\tilde{\chi}_2)$ (controlling the jet $p_T$) and $m(\tilde{l})-m(\tilde{\chi}_1^0)$ (controlling the lepton $p_T$).
In addition to the delta-m variables, for the full analysis there is another independent parameter, $m_{\rm SUSY} = {\rm min}(m(\tilde{g}),m(\tilde{q}))$, which can be investigated by shifting the whole spectrum in mass. It factors out the main tradeoff between missing energy and production cross section.
When investigating the jet kinematics for the eigenmodes of classes B and C ($\epsilon_{\rm jets_{BC}}$) a complication compared to the situation for class A (as discussed in Sect. \[sec:susyjets\]) arises from to the fact that the first gauginos of the two chains now have different masses giving rise to an additional parameter $m(\tilde{\chi}_2)$ which has to be considered[^14]. For alternative trigger paths the efficiency may be written as a sum of terms which depend dominantly on only a subset of the mass parameters. For instance, $\epsilon_{\rm qq} = \epsilon_{jet100}({\rm max}(m(\tilde{q})-m(\tilde{\chi}_1),m(\tilde{q})-m(\tilde{\chi}_2))) + \epsilon_{2jet50}(min(m(\tilde{q})-m(\tilde{\chi}_1),m(\tilde{q})-m(\tilde{\chi}_2)))$, corresponding to the trigger requirement: one jet with $p_T>100$ or two jets with $p_T>50$.[^15]
For the class B eigenmodes there is then only one more parameter to be varied: $m(\tilde{\chi}_2)-m(\tilde{\chi}_1)$. While large values will lead to boosted bosons, going to smaller values will make the channels of the heavier bosons disappear first and finally lead to quasi stable long lived particles. All these details are difficult to study in a coherent way by looking at convoluted SUSY model points.
Similarly, for the class C eigenmodes there are two new parameters $m(\tilde{\chi}_2)-m(\tilde{l})$ and $m(\tilde{l})-m(\tilde{\chi}_1)$ which control the momenta of the radiated leptons. One may expect the efficiency to factorizes, in first order, for different final states. For instance: $\epsilon_{\rm C2l} = \epsilon_{\rm jets_{BC}} \times \epsilon_{l}(m(\tilde{\chi}_2)-m(\tilde{l})) \times \epsilon_{l}(m(\tilde{l})-m(\tilde{\chi}_1))$. Second order effects resulting form jet-electron and electron-jet fake rates as well as isolation effects can be studied in addition, they can be expected to be independent of $m_{SUSY}$ for instance (for the same jet and lepton kinematics).
[Analysis Channels]{} The last question that remains to be answered is which analyses should be performed to investigate the eigenmodes of Table \[tab:theeigenmodes\]. Their final states can be grouped together as shown in Table \[tab:analyzeschanels\]. As one observes, a small number of analyzes suffice to investigate the efficiency for all the eignemodes[^16]. This gives a total of 9 final states which can be analyzed in a model-independent way as described above, i.e. events should be generated for the contributing eigenmodes[^17] and the efficiencies determined as a function of the masses. The sensitivity in the pMSSM e.g. can then be determined at generator level separately for each analysis by comparing it with the standard model background. Thereto, each eigenmode is weighted with its branching fraction at a given parameter point[^18], resulting in a conservative estimate in sensitivity. When several eigenmodes are analyzed the total sensitivity for a given parameter point increases as bigger branching fractions are being covered[^19].
Once this has been done, one may consider special SUSY parameter points at the border of the experimental efficiency space which may profit from the additional investigation of other combinations of the elementary mass spectra.
Signature Contributing Eigenmodes
--------------------------------------------- ------------------------- --
[${\rm E}_{\rm T}^{\rm miss}$]{}+ jets Aq, BZ, BW, C0l
[${\rm E}_{\rm T}^{\rm miss}$]{}+ b-jets Ab, (BZ, BW)
[${\rm E}_{\rm T}^{\rm miss}$]{}+ top At, (BW)
[${\rm E}_{\rm T}^{\rm miss}$]{}+ jets + 1l BW, C1l
[${\rm E}_{\rm T}^{\rm miss}$]{}+ jets + 2l BZ, C2l
[Conclusions]{} The [Bottom-Up Approach]{}may be helpful for any given analysis, and provides a coherent and experiment driven approach for searches for new physics at the LHC.
Since the [Bottom-Up Approach]{}separates the analysis and interpretation steps it provides an interface between experiment and theory, which allows both sides to focus on their specific tasks. Since the [Bottom-Up Approach]{}allows to calculate ${\rm N}_{\rm exp}$ in many dimensions with high precision it would allow to choose the pMSSM as a standard for the interpretation of LHC results.
A comparison of the main features of the traditional approach and the [Bottom-Up Approach]{}is given in Table \[tab:comparizon\].
Feature Traditional Approach [Bottom-Up Approach]{}
----------------------------------------- ------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------
Aim Analysis of SUSY model points Analysis of elementary mass spectra
Dependent on SUSY model assumptions Yes No
Determination of mass spectrum Difficult from convoluted spectrum Favored by direct analysis of eigenmodes
Benchmarks points Theoretically motivated Motivated by detector efficiency
Analysis focus Changing efficiency due to varying contribution from different modes for different SUSY model points Investigation of detector efficiency as function of new particle masses for each mode
Coverage of full observable phase space No Yes
Interpretation of results Within chosen model, limited by number of SUSY model points that can be generated and analyzed Within complete MSSM, covering full phase space accessible to detector sensitivity
Precision in ${\rm N}_{\rm exp}$ Low High
If this approach is applied for SUSY analysis at the LHC it would profit from adjusted interfaces, both to the generator programs to allow the generation of events for a specific eigenmode, as well as the SUSY model generator to facilitate the mapping from observable space to SUSY model spaces.
Although the particles of the MSSM cover most ways that new particles could interact given the known standard model couplings, one could imagine that a similar decomposition into eigenmodes could be performed based on more general BSM models, which then could be analyzed following the [Bottom-Up Approach]{}. A basis for the description of such models could be provided by On Shell Effective Theory [@Marmoset].
In focusing on a systematic investigation of the reach in detector efficiency in terms of the physical observables of the new particles (and thus extending the reach in SUSY parameter space) the application of the [Bottom-Up Approach]{}could increase the chances for discovery of supersymmetry at the LHC as well as facilitate the interpretation of the results.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like the thank the following people for insightful discussions and comments:\
Dong Su, Michael Peskin, JoAnne Hewett and Thomas Rizzo (special thanks for providing the pMSSM parameter points), Jay Wacker, Giacomo Polesello and Paul de Jong.
[99]{} CDF Collaboration, [*Model-Independent Global Search for New High-$p_T$ Physics at CDF*]{}, arXiv:0712.2534v2 N. Arkani-Hamed, G. Kane, J. Thaler, L.-T. Wang, [*Supersymmetry and the LHC Inverse Problem*]{}, arXiv:0512190v1 J. Alwall, P. Schuster, N. Toro, [*SimpliÞed Models for a First Characterization of New Physics at the LHC*]{}, arXiv:0810.3921v1 C. Horn, [*Search for Gravitinos in R-parity violating Supersymmetry at HERA*]{}, Ph.D. thesis A. Djouadi, J. Kneur, G. Moultaka, [*A Fortran Code for the Supersymmetric and Higgs Particle Spectrum in the MSSM*]{}, arXiv:0211331v2 C.F. Berger, J.S. Gainer, J.L. Hewett, T.G. Rizzo, [*Supersymmetry without Prejudice*]{}, arXiv:0812.0980v3 M. Nojiri, G. Polesello, D. Tovey, [*A hybrid method for determining particle masses at the Large Hadron Collider with fully identiÞed cascade decays*]{}, arXic:0712.2718v2 J. Alwall, M.-P. Le, M. Lisanti, J. G. Wacker, [*Searching for Directly Decaying Gluinos at the Tevatron*]{}, arXiv:0803.0019v2 Sky French for the ATLAS Collaboration, [*Discovering heavy particles decaying into single boosted jets with substructure using the $k_T$ algorithm*]{}, to appear in proceedings of the 7th International Conference on Supersymmetry and the Unification of Fundamental Interactions N. Arkani-Hamed, P. Schuster, N. Toro, J. Thaler, L.-T. Wang, B. Knuteson, S. Mrenna, [*MARMOSET: The Path from LHC Data to the New Standard Model via On-Shell Effective Theories*]{}, arXiv:0703088v1
[^1]: This may be extended by considering a third neutralino $\chi_3$. However, the findings in Sect. \[sec:pmssmana\] indicate that its contribution is probably very small.
[^2]: Note that this level of abstraction preserves the two main features of the new particles: Their spins (which determine angular distributions) and their types of interactions (which determine the types of standard model particles they may decay into).
[^3]: BR denotes the branching ratio and $\epsilon_S$, $\epsilon_B$ the selection efficiencies for signal and background events, respectively.
[^4]: One complex mass spectrum may comprise several versions of the same elementary mass spectrum.
[^5]: Note that some longer spectra, like the three gaugino case, could eventually be added but, as will be seen in Sect. \[sec:pmssmana\], they can be expected to have minor effects on the discovery potential. Also note that additional subdivisions are possible but they are considered internal to an analysis and do not affect the outcome of the discussion.
[^6]: The use of PYTHIA might not always give the exact result and it was not used in [@JoAnneTom]. The point of the discussion here is just a qualitative one. For more detailed and more exact results about the pMSSM see [@JoAnneTom].
[^7]: Note that contrary to common rumor SUSY decay chains prefer to be short.
[^8]: Usually, much information about the masses of the new particles can already be inferred by computing invariant masses and kinematic edges. A dedicated analysis taking into account all kinematic and topological variables and their correlations, however, will lead to improved mass measurements (see e.g. [@improvingMassWithEventInfo]).
[^9]: Note that also mass configurations where some of the intermediate sparticles are off-shell should be considered since they lead to the same final states and have different kinematics. For these cases the number of relevant mass parameters is reduced (see the example in Sect. \[sec:susyjets\]).
[^10]: In this analysis a multivariate discrimination technique was used which allows for an automatic optimization of the selection criteria for changing signal distributions, see [@myThesis] for more details.
[^11]: The total efficiency is the weighted sum of the efficiencies of the three cases, where the weights are the corresponding production cross sections: $\epsilon_{\rm jets_A} = \frac{1}{\sigma^{\rm gg} + \sigma^{\rm qq} + \sigma^{\rm gq}} (\sigma^{\rm gg}\epsilon_{\rm gg} + \sigma^{\rm qq}\epsilon_{\rm qq} + \sigma^{\rm gq}\epsilon_{\rm gq})$.
[^12]: The total efficiency is a weighted sum over all decay chains, where the weights are the branching ratios.
[^13]: Although this section focuses on the neutralino LSP case, the extension to gravitino LSPs should be straight forward.
[^14]: In the cases with additional jets from hadronic decays of radiated standard model bosons one may take the scenario discussed so far as a reference and investigate where the additional jets are able to extend the efficiency borders.
[^15]: Note that doing the analysis in this way naturally includes the search for mono-jets in the regions where only one of the delta-m values is sufficiently large.
[^16]: Note that their separation is not needed for discovery. However, the measurement of the relative branching ratios of the two spectra contributing to the two lepton channel, for instance, could provide one of the few handles to determine the composition of the gauginos.
[^17]: Note that the separate specification of decays for the two produced sparticles probably needs some adjustment to the generator programs currently available.
[^18]: If a complex spectrum contains several realizations of the same elementary spectrum the contributions are summed up, weighted by the efficiencies for the different mass configurations.
[^19]: Note that there is no interference of the different eigenmodes since each single event results from only one specific eigenmode.
|
---
abstract: 'Propensity score methods are widely used in observational studies for evaluating marginal treatment effects. The generalized propensity score (GPS) is an extension of the propensity score framework, historically developed in the case of binary exposures, for use with quantitative or continuous exposures. In this paper, we proposed variance estimators for treatment effect estimators on continuous outcomes. Dose-response functions (DRF) were estimated through weighting on the inverse of the GPS, or using stratification. Variance estimators were evaluated using Monte Carlo simulations. Despite the use of stabilized weights, the variability of the weighted estimator of the DRF was particularly high, and none of the variance estimators (a bootstrap-based estimator, a closed-form estimator especially developped to take into account the estimation step of the GPS, and a sandwich estimator) were able to adequately capture this variability, resulting in coverages below to the nominal value, particularly when the proportion of the variation in the quantitative exposure explained by the covariates was large. The stratified estimator was more stable, and variance estimators (a bootstrap-based estimator, a pooled linearized estimator, and a pooled model-based estimator) more efficient at capturing the empirical variability of the parameters of the DRF. The pooled variance estimators tended to overestimate the variance, whereas the bootstrap estimator, which intrinsically takes into account the estimation step of the GPS, resulted in correct variance estimations and coverage rates. These methods were applied to a real data set with the aim of assessing the effect of maternal body mass index on newborn birth weight.'
author:
- |
Valérie Garès[^1]\
Guillaume Chauvet[^2]\
David Hajage[^3]
nocite: '[@austin2018assessing]'
title: 'Closed-form variance estimators for weighted and stratified dose-response function estimators using generalized propensity score'
---
Introduction {#intro}
============
In observational cohort studies, confounding may occur when the distribution of baseline covariates differs between treated and control subjects. The propensity score is one of the methods that helps in reducing or minimizing this confounding to get valid inferences on treatment effects. It was first developed for binary or categorical exposures ([@ROSENBAUM1983]). In this setting, the propensity score is defined as the probabiblity of being exposed conditionally on baseline characteristics. Different propensity score methods have been proposed to estimate the treatment effects: covariate adjustment using the propensity score ([@austin_conditioningpropensityscore_2007]), stratification on the propensity score ([@rosenbaum_reducingbiasobservational_1984; @lunceford_stratificationweightingpropensity_2004]), propensity-score matching ([@austin_methodspropensityscorematching_2009; @rubin_matchingusingestimated_1996; @abadie_matchingestimatedpropensity_2016]) and propensity score weighting ([@rosenbaum_modelbaseddirectadjustment_1987; @austin_performancedifferentpropensity_2013; @li_weighting_2013; @li_balancing_2018]).
In many studies, the exposure of interest is continuous rather than binary. For example, we may not only know whether an individual is a smoker or not, but also the pack-years of cigarettes smoked, or the duration of smoking. Another example is the body mass index, which may be more informative as a continuous variable than if reduced to a dummy variable indicating obesity ([@zhang2016causal; @austin2018assessing]). Considering this type of exposure variable, one may be interested in estimating the dose-response function. If this term may evoke the dose of a medication, we will use it regardless of the nature of the exposure as long as it is quantitative. The propensity score has been generalized into a propensity function for quantitative exposures which is known as the generalized propensity score (GPS) ([@hirano2004propensity; @imai2004causal; @bia2008stata; @zhang2016causal; @austin2018assessing]). Similarly to the binary case, different propensity score methods have been proposed to estimate the treatment effects on outcomes using the GPS: covariate adjustment ([@austin2018assessing]), stratification ([@zhang2016causal]) and inverse probability of treatment weighting (IPTW) ([@zhang2016causal; @austin2018assessing]).
In the case of binary exposure, several authors have proposed valid closed-form variance estimators adapted to each treatment effect estimators: adjustment ([@zou_variance_2016]), stratification ([@williamson_variancereductionrandomised_2014]), matching ([@abadie_matchingestimatedpropensity_2016]) and weighting ([@lunceford_stratificationweightingpropensity_2004; @lunceford_stratificationweightingpropensity_2017; @hajage2018closed]). Note that all these estimators take into account the fact that the theoretical propensity score value of an individual is unknown, and is estimated from the data in the first-stage of analysis. To our knowledge, variance estimation for treatment effect estimated using the GPS framework has received little attention. In this work, we develop and evaluate closed-form variance estimators for stratified and weighted treatment effect estimators using the influence function linearization technique ([@deville_varianceestimationcomplex_1999]). These variance estimators are also compared to bootstrap-based variance estimators.
The paper is organized as follows. In Section \[notations\], we introduce some notations. In Section \[weight\], we describe the weighted treatment effect estimator based on the GPS. In Section \[strat\], we describe the stratified treatment effect estimator. In Section \[variance\], we describe the variance estimators developed in this study. In Section \[simul\], the performances of the models are assessed on a benchmark of simulated databases. They are applied in Section \[example\] on a real example extracted from the PreCARE cohort study, with the aim of assessing the effect of maternal Body Mass Index (BMI) on newborn birth weight. Finally, we discuss in Section \[conclusion\] the pros and cons of the different estimation methods, and we describe areas for future research.
Treatment effect estimator using the generalized propensity score {#effect}
=================================================================
Notations and assumptions {#notations}
-------------------------
Let $T$ denote the level of a quantitative exposure which is a continuous variable, and $Z$ a set of $p$ baseline measured covariates. Let $Y(t),~t \in \Psi$, denote a set of potential outcomes which is assumed to exist under Rubin’s framework for causal inference. More precisely, we assume that $T$ is a continuous exposure ([i.e.]{}, $\Psi$ is a subset of $\mathbb{R}$) and that $Y_i(t)$ is the outcome that would be observed for subject $i$ if he/she received (maybe contrary to the reality) the level of exposure $T = t$. In practice, we only observe one level of exposure for each subject $i$ and the corresponding outcome. The observed data consists of $(Z_i, T_i, Y_i)$ for subjects $i=1,\ldots,n$.\
We are interested in estimating the dose-response function $$\begin{aligned}
\label{muT}
\mu(t) & = & {\mathbb{E}}[Y_i(t)],\end{aligned}$$ which corresponds to the average response if all subjects were exposed to the level $T = t$.
In randomized studies, it can be assumed that $Y(t)$ is independent of $T$, which is denoted as $Y(t) {\protect\mathpalette{\protect\independenT}{\perp}}T , \quad \forall t \in \Psi$. In this work, we only assume that $Y(t)$ is independent of $T$ given $Z$, which is known as the weak unconfoundedness assumption and denoted as $$\begin{aligned}
\label{weak:ass}
Y(t) {\protect\mathpalette{\protect\independenT}{\perp}}T | Z, \quad \forall t \in \Psi.\end{aligned}$$ This assumption means that any association between the actual exposure and the potential outcomes is explained by a set of baseline covariates $Z$ ([@hirano2004propensity; @zhang2016causal]). Note that this assumption cannot be checked from the data.\
Let us denote by $$\begin{aligned}
\label{r:tz}
r(t \mid z) & \equiv & f_{T|Z}(t|z),\end{aligned}$$ the conditional density of exposure variable $T$ given the covariates, which is called the generalized propensity score (GPS) ([@hirano2004propensity]). We make the positivity assumption, namely $$\begin{aligned}
\label{pos:ass}
r(t \mid z)>0 & \textrm{ for any } & t \in \Psi \textrm{ and for any } z.\end{aligned}$$ This means that any level of exposure $T=t$ is possible for any subject, whatever his/her baseline characteristics. A violation of this assumption may lead to biased estimators, or estimators with a large variability ([@moore2012causal]). Note that this assumption may and should be checked from the data. In the case of a binary exposure, assessing the positivity assumption may involve examining the overlap between the distribution of the estimated propensity score for the exposed and the exposed samples ([@mccaffrey_tutorial_2013]), or by examining the distribution of the estimated weights used for inverse probability of treatment weighting, looking for extreme values ([@austin_moving_2015]). In the context of the generalized propensity score and to our knowledge, diagnostics for assessing the positivity assumption have not yet received much attention, even though the estimation of the proportion of the variation in the continuous exposure explained by the covariates seems a promising approach ([@austin2018assessing]).
We focus on two approaches for the estimation of the dose-response function: inverse probability of treatment weighting, and stratification. Both approaches are presented in [@zhang2016causal] and are briefly described in Sections \[weight\] and \[strat\].
Weighted treatment effect estimator {#weight}
-----------------------------------
The first estimator is obtained by fitting a generalized linear regression model between the dose-response function and the exposure, used as the sole dependent variable. Focusing on the case of a linear dose-response function, our model is $$\begin{aligned}
\label{mod1}
\mu(T) & = & \beta_0 + \beta_1~T + \epsilon,\end{aligned}$$ where $\beta_0$ is the average response observed in case of null exposure ($T = 0$), and $\beta_1$ is the average response change if the level of exposure is increased by one unit. Other dose-response functions ([e.g.]{}in case of non-linear relationship) and/or other link functions may be better suited for other types of outcome ([e.g.]{}, a binomial link function for a binary outcome), and may therefore be alternatively used.\
The parameter $\beta=(\beta_0,\beta_1)^{\top}$ is estimated by weighted least squares, which leads to $$\begin{aligned}
\label{betahat:w}
\hat{\beta}_{w} & \equiv & (\hat{\beta}_{w0},\hat{\beta}_{w1})^{\top} = \left(\sum_{i=1}^n \hat{w}_i \tilde{T}_i \tilde{T}_i^{\top} \right)^{-1} \left(\sum_{i=1}^n \hat{w}_i \tilde{T}_i Y_i \right)\end{aligned}$$ where $\tilde{T}_i=(1,T_i)^{\top}$, and the weights $\hat{w}_i$ that we use are presented thereafter. This leads to the first estimator $$\begin{aligned}
\label{est:1}
\hat{\mu}_{w}(t) & = & \hat{\beta}_{w0} + \hat{\beta}_{w1}~t\end{aligned}$$ for the dose-response function.\
The weights used in equation (\[betahat:w\]) are computed as follows. We first introduce the theoretical Generalized Propensity Score (GPS) weights, defined as $$\begin{aligned}
\label{gps:weight}
w_i(\gamma) & = & \frac{W(T_i|\gamma)}{r(T_i|Z_i,\gamma)},\end{aligned}$$ where $r(t|z,\gamma)$ is the conditional density of the exposure variable defined in (\[r:tz\]), and where $W(\cdot|\gamma)$ is a stabilization factor. As is currently done in the literature, we use $W(t|\gamma) \equiv f_T(t|\gamma)$ the marginal density of the exposure variable. Note that the weights depend on some unknown vector of parameters $\gamma$, which needs to be estimated.\
We suppose that $T_i$ follows a normal distribution, both conditionally on $Z_i$ and non conditionally. We may therefore write $$\begin{aligned}
f_T(t|\gamma) & = & \frac{1}{\sqrt{2\pi \sigma_T^2}} \exp\left\{-\frac{1}{2 \sigma_T^2} \left(T_i-\mu_T\right)^2 \right\}, \label{f_T:norm} \\
r(t|z,\gamma) & = & \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left\{-\frac{1}{2 \sigma^2} \left(T_i-\alpha^{\top} \tilde{Z} _i \right)^2 \right\}, \label{r_T:norm}\end{aligned}$$ with $\tilde{Z}_i^{\top}=(1,Z_i^{\top})$ and $\gamma=(\mu_T,\sigma_T^2,\alpha^{\top},\sigma^2)^{\top} $. The parameters $\mu_T$ and $\sigma_T^2$ in equation (\[f\_T:norm\]) are estimated by $$\begin{aligned}
\label{est:param:fT}
\widehat{\mu}_{T}=\frac{1}{n}\sum_{i=1}^n T_i & \textrm{and} & \widehat{\sigma}^2_{T}=\frac{1}{n-1}\sum_{i=1}^n (T_i-\widehat{\mu}_{T})^2.\end{aligned}$$ By fitting a linear regression model between the exposure variable and the covariates, namely $$\begin{aligned}
\label{prop:mod}
T_i & = & \tilde{Z_i}^{\top} \alpha + \eta_i,\end{aligned}$$ the parameters $\alpha$ and $\sigma^2$ in equation (\[r\_T:norm\]) are estimated by $$\begin{aligned}
\label{est:param:rT}
\hat{\alpha} & = & \left(\sum_{i=1}^n \tilde{Z}_i \tilde{Z}_i^{\top} \right)^{-1} \left(\sum_{i=1}^n \tilde{Z}_i T_i \right), \\
\hat{\sigma}^2 & = & \frac{1}{n-p-1} \sum_{i=1}^n (T_i-\widehat{\alpha}^T \tilde{Z}_i)^2. \nonumber\end{aligned}$$ This leads to the estimator $\hat{\gamma}=(\hat{\mu}_T,\hat{\sigma}_T^2,\hat{\alpha}^{\top},\hat{\sigma}^2)^{\top}$. By plugging this estimator in (\[gps:weight\]), we obtain the estimated weights $\hat{w}_i \equiv w_i(\hat{\gamma})$ used in equation (\[betahat:w\]). The model (\[prop:mod\]) is called the propensity model in the remainder of this paper.
Stratified treatment effect estimator {#strat}
-------------------------------------
The weighted estimator of the dose-response function considered in equation (\[est:1\]) of Section \[weight\] proceeds through a linear regression on the whole sample, using weights to adjust for possible imbalance in the covariates.\
An alternative approach consists in partitioning the sample into $L$ strata, in such a way that the units inside a given stratum are somewhat similar with respect to the covariates. This may be done by fitting the propensity model in (\[prop:mod\]), ordering the units in the sample with respect to the prediction $Z_i^{\top} \widehat{\alpha}$, and using the quantiles as cut-off points ([@zhang2016causal]).\
Inside any stratum $l=1,\ldots,L$, we fit the regression model $$\begin{aligned}
\label{mod:st}
\mu(T) & = & \beta_{l0} + \beta_{l1}~T + \epsilon_l,\end{aligned}$$ and by estimating the parameter $\beta_l=(\beta_{l0},\beta_{l1})^{\top}$ by ordinary least squares, we obtain $$\begin{aligned}
\label{betahat:st:1}
\hat{\beta}_{l} & \equiv & (\hat{\beta}_{l0},\hat{\beta}_{l1})^{\top} = \left(\sum_{i \in S_l} \tilde{T}_i \tilde{T}_i^{\top} \right)^{-1} \left(\sum_{i \in S_l} \tilde{T}_i Y_i \right),\end{aligned}$$ with $S_l$ the subset of sampled units which belong to the stratum $l$. The stratified estimator of the parameter $\beta$ in (\[mod1\]) is obtained by pooling these $L$ estimators, which leads to $$\begin{aligned}
\label{betahat:st}
\widehat{\beta}_{st} & \equiv & (\hat{\beta}_{st0},\hat{\beta}_{st1})^{\top}
= \sum_{l=1}^L\frac{n_l}{n} \hat{\beta}_{l},\end{aligned}$$ with $n_l$ the number of sampled units in the stratum $S_l$. Note that if the quantiles are used as cut-off points, we have (up to rounding) $n_l=n/L$, and $\widehat{\beta}_{st}$ is the simple mean of the estimators $\hat{\beta}_{l},~l=1,\ldots,L$.\
This leads to the second estimator $$\begin{aligned}
\label{est:2}
\hat{\mu}_{st}(t) & = & \hat{\beta}_{st0} + \hat{\beta}_{st1}~t\end{aligned}$$ for the dose-response function. Again, ordinary least squares may be replaced by a generalized linear model and appropriate link function to fit other types of outcome.
Closed form variance estimators {#variance}
===============================
In this Section, our objective is to develop closed-form variance estimators for the estimators of the dose-response function presented in equations (\[est:1\]) and (\[est:2\]). Without loss of generality, we focus on variance estimation for the estimated coefficients of regression $\hat{\beta}_{w}$ and $\widehat{\beta}_{st}$.\
We follow the influence function linearization technique developped by Deville ([@deville_varianceestimationcomplex_1999]), see also [@hajage2018closed]. For an estimator $\hat{\beta}$, this technique consists in finding a so-called estimated linearized variable $\hat{I}_i$, summarizing the variability in the estimation of the parameter. Ideally, the linearized variable should account for all the estimation steps which lead to the estimator $\hat{\beta}$.\
The proposed variance estimator for the weighted estimator $\hat{\beta}_{w}$ presented in Section \[weight\] is given in Section \[evar:weight\]. The proposed variance estimator for the stratified estimator $\hat{\beta}_{st}$ presented in Section \[strat\] is given in Section \[evar:strat\].
Weighted treatment effect estimator {#evar:weight}
------------------------------------
The variance estimator for $\hat{\beta}_{w}$ is obtained by observing that the coefficient of regression is estimated in a two-step process, involving two estimating equations. First, the unknown parameter $\gamma$ used to compute the weights is obtained by solving the system of estimating equations $$\begin{aligned}
\label{Fn:gamma}
F_n (\gamma) \equiv \frac{1}{n} \sum_{i=1}^n F_i(\gamma) & = & 0,\end{aligned}$$ where $$\begin{aligned}
\label{Fi:gamma}
F_i(\gamma) & = & \left(
\begin{array}{l}
T_i-\mu_T \\
(T_i-\mu_T)^2-\frac{n-1}{n} \sigma_T^2 \\
(T_i-\tilde{Z}_i^{\top} \alpha) \tilde{Z}_i \\
(T_i-\tilde{Z}_i^{\top} \alpha)^2-\frac{n-p-1}{n}\sigma^2
\end{array}
\right).\end{aligned}$$ Then, the estimator $\hat{\beta}_{w}$ is obtained as the solution of the estimating equation $$\begin{aligned}
\label{ee:beta}
H_n (\hat{\gamma},\beta) \equiv \frac{1}{n} \sum_{i=1}^n w_i(\hat{\gamma}) H_i(\beta) = 0 & \textrm{with} & H_i(\beta) = \tilde{T}_i (Y_i-\tilde{T}_i^{\top} \beta).\end{aligned}$$
After some algebra, this leads to the following linearized variable for $\hat{\beta}_{w}$: $$\begin{aligned}
\label{lin:weight:1}
\hat{I}_{1i} & = & \hat{A}^{-1} \left\{w_i(\hat{\gamma})H_i(\hat{\beta})+\hat{B} \hat{C}^{-1}F_i(\hat{\gamma}) \right\},\end{aligned}$$ with $$\begin{aligned}
\hat{A} & = & \frac{1}{n} \sum_{i=1}^n w_i(\hat{\gamma}) \tilde{T}_i \tilde{T}_i^{\top}, \nonumber \\
\hat{B} & = & \frac{1}{n} \sum_{i=1}^n H_i(\hat{\beta}) \nabla w_i(\hat{\gamma})^{\top}, \label{lin:weight:2} \\
\hat{C} & = & \left(
\begin{array}{cccc}
1 & 0 & 0_{1K} & 0 \\
0 & \frac{n-1}{n} & 0_{1K} & 0 \\
0_{K1} & 0_{K1} & \frac{1}{n} \sum_{i=1}^n \tilde{Z}_i \tilde{Z}_i^{\top} & 0_{K1} \\
0 & 0 & 0 & \frac{n-p-1}{n}
\end{array}
\right), \nonumber\end{aligned}$$ and where $0_{\bullet \diamond}$ stands for the null matrix with $\bullet$ rows and $\diamond$ columns. The computation details are given in Appendix \[appen1\].\
The resulting variance estimator is $$\begin{aligned}
\label{v:lin:weight}
\hat{V}_{lin}(\hat{\beta}_{w}) = \frac{1}{n(n-1)} \sum_{i=1}^n (\hat{I}_{1i}-\bar{I}_1)(\hat{I}_{1i}-\bar{I}_1)^{\top}
& \textrm{ with } & \bar{I}_1 = \frac{1}{n} \sum_{i=1}^n \hat{I}_{1i}.\end{aligned}$$
Stratified treatment effect estimator {#evar:strat}
-------------------------------------
Inside each stratum $\ell=1,\ldots,L$, the intermediary estimators $\hat{\beta}_l$ are estimated by solving the estimating equations $$\begin{aligned}
\label{strat:est:eq:1}
\Phi(\beta_{\ell}) & \equiv & \frac{1}{n_\ell} \sum_{i \in S_\ell} \Phi(Y_i,T_i,\beta_{\ell}) = 0,\end{aligned}$$ with $$\begin{aligned}
\label{strat:est:eq:2}
\Phi(Y_i,T_i;\beta_{\ell}) & = & \left(
\begin{array}{l}
Y_i - \tilde{T}_i^{\top} \beta_{\ell} \\
T_i(Y_i - \tilde{T}_i^{\top} \beta_{\ell})
\end{array}
\right).\end{aligned}$$
After some algebra, the linearized variable of $\hat{\beta}_l$ is $$\begin{aligned}
\label{strat:est:eq:3}
\hat{I}_{2l,i} & = & \frac{1}{\hat{\sigma}_{\ell,T}^2}
\left(
\begin{array}{cc}
\hat{\sigma}_{\ell,T}^2+\hat{m}_{\ell,T}^2 & -\hat{m}_{\ell,T} \\
-\hat{m}_{\ell,T} & 1
\end{array}
\right) \times
\left(
\begin{array}{l}
Y_i - \tilde{T}_i^{\top} \hat{\beta}_{\ell} \\
T_i(Y_i - \tilde{T}_i^{\top} \hat{\beta}_{\ell})
\end{array}
\right),\end{aligned}$$ where $$\begin{aligned}
\label{strat:est:eq:4}
\hat{m}_{\ell,T} = \frac{1}{n_\ell} \sum_{i \in S_\ell} T_i
& \textrm{and} &
\hat{\sigma}_{\ell,T}^2 = \frac{1}{n_\ell-1} \sum_{i \in S_\ell} (T_i-\hat{m}_{\ell,T})^2.\end{aligned}$$ The computation details are given in Appendix \[appen2\]. This leads to the pooled variance estimator $$\begin{aligned}
\label{pool:vest}
\hat{V}(\hat{\beta}_{st}) & = & \sum_{\ell=1}^L
\frac{(p_\ell)^2}{n_\ell(n_\ell-1)} \sum_{i \in S_l} (\hat{I}_{2l,i}-\bar{I}_{2l})(\hat{I}_{2l,i}-\bar{I}_{2l})^{\top} \\
\textrm{with } \bar{I}_{2l} & = & \frac{1}{n_l} \sum_{i \in S_l} \hat{I}_{2l,i}. \nonumber\end{aligned}$$
Note that the strata are built by using the quantiles of the predicted $\tilde{Z}_i^{\top} \hat{\alpha}$ given by the propensity model, and the strata boundaries are therefore estimated rather than known. This is not accounted for in the variance estimator proposed in equation (\[pool:vest\]). Taking this estimation into account could possibly be performed by following the approach in [@williamson_varianceestimationstratified_2012], but this would require fully specifying the joint distribution between the outcome, the exposure and the covariates.\
An advantage of the variance estimator given in (\[pool:vest\]) is its robustness to the misspecification of the model linking the dose-response function and the exposure. Alternatively, a model-based variance estimator could be derived.
Simulations {#simul}
===========
Data-generating process {#ss-data}
-----------------------
We adapt the method described in [@hajage2018closed]. First, we randomly generate $p+1$ normally distributed covariates $Z_1, \dots, Z_k \dots, Z_K, Z_U$ from the following multivariate normal distribution:
$$[Z_1, \dots, Z_K, Z_U] \sim \mathcal{N}(0;\Sigma) \textrm{ with } \Sigma =
\begin{pmatrix}
1 & 0 & \dots & 0 & \sigma_{1} \\
0 & 1 & \dots & 0 & \sigma_{2} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \dots & 1 & \sigma_{K} \\
\sigma_{1} & \sigma_{2} & \dots & \sigma_{K} & 1 \\
\end{pmatrix}.$$
Thus, $Z_1, \dots, Z_K$ are mutually independent following a standard normal distribution, but are each correlated to a standard normal variable $Z_U$ through covariance parameters $\sigma_{k}$, $k=1,\ldots,K$.
A covariate $U$ is then computed by applying the following transformation to $Z_U$: $U = F_{Z_U}(u) = {\mathbb{P}}(Z_U < u)$ ([i.e.]{}$U$ is the cumulative distribution function of $Z_U$). By construction, $U$ follows a uniform distribution $\mathcal{U}(0,1)$ which is still correlated to other covariates $Z_1, \dots, Z_L$.\
The treatment allocation $T$ is drawn from a linear model where: $$\label{eq:T}
T = \alpha_0 + \sum_{k=1}^K \alpha_k Z_k + \eta,$$ with $\eta \sim \mathcal{N}(0,\sigma^2)$. The parameter $\sigma^2$ is linked to the coefficient of determination $R^2$ which measures the proportion of the variance (of the exposure) explained by the regression model, and is defined as: $$\begin{aligned}
R^2&=&\frac{\mathrm{var}(\sum_{k=1}^K \alpha_k Z_k)}{\mathrm{var}(Y)}\\
&=&\frac{\sum_{k=1}^K \alpha_k^2 }{\sum_{k=1}^K \alpha_k^2 +\sigma^2}. \label{link}\end{aligned}$$
$R^2$ is bounded between 0 and 1. This simple parameter (classic in linear regression) allows to easily control the degree of confounding in the simulated samples ([@austin2018assessing]). $R^2$ close to 0 corresponds to weak confounding, $R^2$ close to 1 corresponds to strong confounding.
The continuous outcome is then generated from $U$ as $$\label{eq:Y}
Y = \beta_0 + \beta_1 T + \sigma^2_Y U,$$ and therefore $Y \sim \mathcal{N}(\mu,\sigma^2_Y)$ where $\mu= \beta_0 + \beta_1 T$. The key mechanism by which this algorithm generates confounding in the estimation of the dose response function is the way in which the exposure $T$ and the outcome $Y$ depend both on $U$. Figure \[fig:dag\] represent the directed acyclic graph corresponding to this data-generating process. Confounding is due to $U$ being a common ancestor of $T$ and $Y$. $Z_1, \dots, Z_K$ are sufficient to adjust for confounding, because $T$ is independent of $U$ given $Z_1, \dots, Z_K$ ([@havercroft_simulatingmarginalstructural_2012]). Thus, unlike Austin (2018, equation 2, page 1877), the association between the confounding factors and the outcome is not induced by including these covariates with the exposure $T$ in a conditional equation. By directly setting the vector of parameters $\beta$ of the marginal dose-response function at desired theoretical value, our data generating algorithm allows to evaluate and compare the performance of different analytical methods by their ability to estimate $\beta$ and the variability of this estimation.
Simulation parameters {#ss-param}
---------------------
We fixed $K=10$, and the true parameters $\alpha_k$ and $\sigma_k$ were set to values presented in Table \[tab:coef\] inspired from [@austin2018assessing]. Coefficients $\alpha_0$ and $\beta_0$ are fixed to 0 in all scenarios.
Several scenarios were considered, defined by:
1. the sample size: $n \in \{500, 1000, 2000\}$;
2. the degree of confounding tuned by the coefficient of determination $R^2 \in \{0.2,0.4,0.6,0.8\}$
3. the residual variance in the outcome model $\sigma_Y^2 \in \{0.25,0.5\}$
4. the treatment effect: $\beta_1 \in \{0,1,2\}$.
A total of $B=1000$ datasets were generated for each scenario.
[200pt]{}[|cc|cc|]{} **Parameter** & **Value** & **Parameter** & **Value**\
$\alpha_1$ & $1$ & $\sigma_1$ & $0.2$\
$\alpha_2$ & $1.5$ & $\sigma_2$ & $0.3$\
$\alpha_3$ & $2$ & $\sigma_3$ & $-0.4$\
$\alpha_4$ & $3$ & $\sigma_4$ & $-0.3$\
$\alpha_5$ & $-2$ & $\sigma_5$ & $-0.2$\
$\alpha_6$ & $-2$ & $\sigma_6$ & $0.15$\
$\alpha_7$ & $1$ & $\sigma_7$ & $0.2$\
$\alpha_8$ & $1.5$ & $\sigma_8$ & $-0.2$\
$\alpha_9$ & $2$ & $\sigma_9$ & $-0.2$\
$\alpha_{10}$& $3$ & $\sigma_9$ & $0.2$\
Estimation of the parameters of the dose-response function and their variance {#ss-methods}
-----------------------------------------------------------------------------
All statistical methods estimating the dose-response function described in Section \[effect\] were applied to each simulated dataset, and compared to the naive (unweighted) maximum likelihood estimator.
Three different variance estimators were associated with the weighted estimator of the dose-response function. First, we evaluated the sandwich variance estimator previously used in [@zhang2016causal]. This estimator takes into account the lack of independence in the weighted sample (*e.i.* the ’duplication’ of subjects in the analysis generated by the weights), but not the fact that the GPS used to derive the weights was estimated rather than known with certainty ([@austin_varianceestimationwhen_2016]). The linearized variance estimator proposed in Section \[evar:weight\] was also applied. Finally, a bootstrap variance estimator based on $N_{\text{boot}} = 200$ bootstrap samples was also used. The weights defined in Equation \[gps:weight\] and the parameters of the dose-response function ($\hat{\beta}_{0b}$ and $\hat{\beta}_{1b}$, $b \in 1,\dots,N_{\text{boot}}$) were reestimated in each bootstrap sample. The bootstrap variance estimator was computed as the empirical variance of the estimated regression coefficients associated with dose-response function across the $N_{\text{boot}}$ bootstrap samples.
The stratified estimator of the dose-response function was used with strata defined according to the deciles of the linear predictors of the propensity model. We also considered three variance estimators: the pooled linearized variance estimator given in equation , the pooled variance estimator using model-based variance estimator from the maximum likelihood estimator in each stratum, and the boostrap-based variance estimator based on $N_{\text{boot}} = 200$ bootstrap samples. Again, the propensity model and the parameters of the dose-response function were reestimated within each bootstrap samples.
The evaluated methods are summarized in table \[tab:methods\].
---------------------------- -------------------- ---------------------------------
**Dose-response function** **Variance** **GPS estimation**
**estimator** **estimator** **taken into account**
**in the variance estimator ?**
Naive Model-based
Weighted Sandwich
Weighted Linearized
Weighted Bootstrap
Stratified Pooled model-based
Stratified Pooled linearized
Stratified Bootstrap
---------------------------- -------------------- ---------------------------------
: Methods for dose-response function and variance estimation[]{data-label="tab:methods"}
Performance criteria {#ss-perf}
--------------------
Results were assessed in terms of the following criteria:
1. Bias of the treatment effect estimation: $\frac{1}{B} \sum_{b=1}^B (\widehat{\beta}_b - \beta)$;
2. Root mean square error (RMSE): $\sqrt{\frac{1}{B} \sum_{b=1}^B (\widehat{\beta}_b - \beta)^2}$;
3. Variability ratio of the treatment effect, defined as: $\frac{\frac{1}{B}\sum_{b=1}^{B}\widehat{\mathrm{SE}}(\widehat{\beta}_b)}
{\sqrt{\frac{1}{B^\prime-1}\sum_{b^\prime=1}^{B^\prime}(\widehat{\beta}_{b^\prime}-\bar{\widehat{\beta}})^2}}$, where $\widehat{\mathrm{SE}}(\widehat{\beta})$ is the estimated standard error of treatment effect $\widehat{\beta}$. It allows evaluating the performance of the variance estimators: a ratio $> 1$ (or $< 1$) suggests that standard errors overestimate (respectively, underestimate) the variability of the estimate of treatment effect. The denominator is the empirical Monte-Carlo standard deviation of the treatment effect, estimated over $B^\prime= 10000$ random samples independent from the samples used in the numerator.
4. Finally, the coverage evaluates if the procedure for constructing the 95% confidence interval achieves the advertised nominal level. Ninety-five percent confidence intervals were constructed by as $\widehat{\beta} \pm 1.96\widehat{\mathrm{SE}}(\widehat{\beta})$ (where $\widehat{\mathrm{SE}}(\widehat{\beta})$ depends on the variance estimation method used). Coverage is defined by the proportion of times $\beta$ is included in the 95% confidence interval of $\beta$ estimated from the model.
Software {#ss-software}
--------
All simulations involved the use of `R` 3.5.[@r] Sandwich variance estimation were computed with the `svyglm` function from the `survey` package ([@lumley_analysis_2004]). The `boot` package was used for boostrap sampling ([@boot_package]). Graphics were generated using the `ggplot2` package ([@wickham_ggplot2elegantgraphics_2009]).
Results
-------
For each of the two parameters of the dose-response function ($\beta_0$ and $\beta_1$), Figure \[fig:bias\] displays the bias of the estimates for different values of $R^2$ and $\sigma_Y^2$. In this figure, the theoretical values of $\beta_0$ (the intercept coefficient of the dose-response function) and $\beta_1$ (the slope coefficient) are 0 and 1 respectively. Boxplots were plotted to allow the graphical assessment of the variability of the estimates. For the estimation of $\beta_1$ (lower panel), the performance of all methods was highly influenced by the value of $R^2$, with a (negative) bias which increased with increasing $R^2$. This may be explained by the fact that the more $R^2$ increases, the less the positivity assumption is respected. Also, the bias increases with the value of $\sigma_Y^2$. Overall, the stratification method gave the smallest bias, while the naive method gave the largest bias. The weighted method gave acceptable bias only for value of $R^2\leq 0.4$. All methods seemed to give approximately unbiased estimates of $\beta_0$ (upper panel), but the graphical evaluation of the bias is made difficult by the very high variability of the estimates. In fact, the same trends as previously reported for $\beta_1$ were observed for $\beta_0$ estimates, except that the bias was positive instead of negative. The variability of the estimates increased for all methods as $R^2$ and $\sigma_Y^2$ increased. The variability associated with the weighted estimator seemed much larger than with the stratified or the naive estimator, particularly for the estimation of $\beta_1$. The combination of a significant bias and a very high variability for large $R^2$ values led to the highest RMSE values being observed for the weighted method (Figure \[fig:RMSE\]). On the contrary, the stratification method was associated with the lowest RMSE, regardless of the $R^2$ and $\sigma_Y^2$ values for the estimation of both $\beta_0$ and $\beta_1$.
Figure \[fig:stdr\] displays the boxplots of the standard errors of the dose-response function parameters ($\beta_0$ in the upper panel, $\beta_1$ in the lower panel) estimated with all methods listed in Table \[tab:methods\]. The empirical Monte-Carlo standard deviation associated with each dose-response function parameter estimator in each evaluated scenario was indicated by a red horizontal line. In all scenarios, the highest empirical standard deviation were observed with the weighted estimator of the dose-response function parameters, especially for large values of $R^2$. The empirical standard deviation estimates associated with the stratified estimator of the dose-response function were higher than those associated with the naive estimator, especially for an $R^2$ value of 0.8. The value of $\sigma_Y^2$ had less effect than $R^2$ on empirical standard deviations. For the weighted estimator of the dose-response function, the standard errors estimated with boostrap, linearized and sandwich methods underestimated the empirical standard deviation. The variability of these standard error estimates was also very large, and this phenomenon increased with $R^2$. Among variance estimation methods associated with the stratified estimator of the dose-response function, the bootstrap (which take into account the GPS estimation step) produced the closest estimation of the empirical standard deviation of $\hat{\beta_1}$ coefficient. The two other estimators (pooled linearized and pooled model-based estimators) overestimated the standard deviation of the $\hat{\beta_1}$ coefficient. All variance methods associated with the stratified estimator produced reasonably good estimates of the variance of $\hat{\beta_0}$. Finally, the model-based variance estimator of the naive dose-response function estimator had good performance in all scenarios.
The patterns of over or underestimation of the empirical standard deviation are more precisely observable in Figure \[fig:var\_ratio\], which illustrates the ratio of the average standard error and empirical standard deviation of the intercept (upper panel) and the slope (lower panel) estimated coefficients for different values of parameters $R^2$ and $\sigma_Y^2$. Overall, sandwich, bootstrap and linearized variance estimators of the weighted estimator of dose-response function resulted in similar values of variability ratio and were negatively biased, except for the sandwich variance estimator of $\beta_1$ with $R^2 = 0.2$. This underestimation of the empirical standard deviation increased with $R^2$, for the two coefficients of the dose-response function. Overall, the variance estimators of the stratified estimator performed well for the intercept parameter of the dose-response function, and tended to overestimate the empirical variance for the slope parameter. The pooled linearized variance method gave slightly lower variability ratios than the pooled model-based variance method. The boostrap variance method gave ratio values very close to 1 for the slope coefficient, whereas the two others estimators clearly led to an overestimation of the empirical variance. The performance of the three variance estimator of the stratified estimator stayed stable as the $R^2$ increased. Finally, the value of $\sigma_Y^2$ did not affect substantially the previous description of the results for all estimators.
Finally, coverage rates for dose-response function estimates are reported on Figure \[fig:Coverage\] for different values of parameters $R^2$ and $\sigma_Y^2$. Overall, results were consistent with those described previously. For the intercept coefficient of the dose-response function (upper panel), confidence intervals based on weighted estimator of the dose-response function were the most unconservative. Their coverage rate decreased as the $R^2$ increased, whereas the performance of other estimators performed well in all scenarios. Pooled model-based and pooled linearized variance estimator of the stratified estimator of the dose-response function were too conservative for the estimation of $\beta_1$, while boostrap-based method gave approximately correct coverage rates. Again, the performance of the weighted estimators was greatly influenced by $R^2$ values, with coverage rates deteriorating while $R^2$ increased.
Supplementary simulations showed that these different results were not affected by a change in the theoretical value of $\beta_1$ (see Supplementary materials, Section 1). The effect of the sample size $n$ had also been studied and showed that the bias and variability associated with the different estimators increased as the sample size decreased (see Supplementary materials, Section 2).
Finally, we also studied the effect of different number of strata for the stratified method on the different estimations (see Supplementary materials, Section 3). The different estimations of $\beta_0$ were better for a few number of strata, while the different estimations for $\beta_1$ were better for a large number of strata. The variance estimation of $\widehat{\beta_0}$ and $\widehat{\beta_1}$ were better for a few number of strata.\
Real data application {#example}
=====================
The different dose-response function estimation methods and associated variance estimation methods have been applied on a real cohort extracted from the PreCARE study. PreCARE is a prospective multicenter cohort study of pregnant women aiming to examine the association between socioeconomic exposure and adverse maternal or neonatal outcomes ([@gonthier_association_2017]). It included all consecutive women registered to deliver or who delivered in 4 public teaching hospitals in northern Paris (France) between October 2010 and May 2012. Women were included at the beginning of their pregnancy during their first visit at 1 of the 4 facilities and were followed until hospital discharge after delivery. Overall, 10,419 women and their newborns were included. The objective of this analysis was to study the relationship between pre-pregnancy maternal body mass index (BMI) on newborn birth weight. This analysis was based on the 8,775 women for whom information about BMI, birth weight and confounding factors included in the propensity score model was available. The list of co-variables included in the propensity score model was maternal age, parity, history of pre-eclampsia, history of preterm delivery (ie, before 37 weeks’ gestational age), the presence of a social deprivation (social deprivation index $\geq 1$) ([@kantor_socioeconomic_2017]), and maternal birth place (France vs other). The $R^2$ value associated with the propensity model was 0.05.
All the statistical methods described in Section \[ss-methods\] were applied. The estimated parameters of the dose-response function are reported in Table \[tab:real\]. In this Table, $\hat{\beta}_0$ represents the estimated mean birth weights (in grams) when maternal BMI is equal to 10, and $\hat{\beta}_1$ represents the increment of the estimated mean birth weight when maternal BMI increases of 1 unit. All $\hat{\beta}_1$ values had qualitatively similar values indicating the positive association between maternal BMI and birth weight. Of note, the graphical inspection of the relationship between maternal BMI and birth weight may suggest a ’plateau’ effect for the highest (and rarest) BMI values ([@froslie_categorisation_2010]). For the sake of simplicity, this eventual deviation from the linearity hypothesis of the relationship between maternal BMI and birth weight has been neglected.
As in the simulation study, standard errors associated with the weighted dose-response function estimation method were higher than those associated with the stratified estimation method, indicating a higher variability of maternal BMI effect estimate. Among variance estimation methods associated with the weighted estimator, the methods which take into account the GPS estimation step (linearized and bootstrap) produce lower standard errors than the sandwich method. The same was observed among the variance estimation methods associated with the stratified estimator: bootstrap standard error was lower than the standard errors estimated with the two other methods which do not take into account the GPS estimation step. Overall, these results were consistent with the results observed in the simulation study.
----------------------------------- --------- --------
Method
Naive 3065.41 12.78
Model-based standard error (20.62) (1.42)
Weighted 3116.06 8.98
Sandwich standard error (32.02) (2.31)
Linearized standard error (29.18) (2.10)
Bootstrap standard error (31.17) (2.25)
Stratified 3081.81 11.61
Pooled model-based standard error (20.09) (1.33)
Pooled linearized standard error (21.17) (1.45)
Bootstrap standard error (17.32) (1.13)
----------------------------------- --------- --------
: Estimated dose-response function in the case study.[]{data-label="tab:real"}
$\widehat{\mathrm{SE}}(\hat{\beta}_p)$: estimated standard error
Conclusions and perspectives {#conclusion}
============================
GPS-based methods have been proposed as a generalization of the propensity score framework for assessing the marginal effect of a quantitative exposure on an outcome of interest, through the estimation of a dose-response function. This research focuses on the variance estimation of the dose-response function parameters, in the case of continuous outcome. We have considered different dose-response function estimation methods and different variance estimation for each methods.
Experimental tests on simulated databases show that the stratification method gives the best estimation of the parameters of the dose-response function, and the boostrap method gives the best estimation of the associated variance. The pooled variance estimators (using linearized or maximum likelihood model-based estimator) of the stratified estimator overestimate the variance, resulting in estimated 95% confidence intervals whose empirical coverage rates are substantially higher than the nominal level. This phenomenon is related to what was already reported in the case of binary exposure: the use of a variance estimator that does not account for the fact that the propensity score is estimated rather than known with certainty leads to an overestimation of the variability of the estimate of treatment effect.
At the beginning of this research project, our main objective was to develop of a closed-form estimator of the variance of the coefficients of the dose-response function estimated using GPS-weighting, taking into account the weights estimation step. But after the evaluation of the performance of GPS-weighting and of the proposed variance estimator, as well as the performance of the bootstrap estimator (already used in [@zhang2016causal]), we had to admit that our enthusiasm about GPS-weighting was dampened. This study shows that GPS-weighting adds up three important issues: a greater bias than the stratified method as the $R^2$ increases, a high variability of the estimates, and the failure of different variance estimators to correctly capture this variability, even though similar approaches have been successfully used in the context of propensity score weighting for binary exposure ([@williamson_variancereductionrandomised_2014; @austin_varianceestimationwhen_2016; @hajage2018closed]). Even if the bias observed in simulations was relatively limited and became really significant for large values of $R^2$, the high variability of estimates led to RMSE values equal to or greater than those observed without any adjustment. Moreover, the underestimation of the variance with all variance estimators led to coverage rates well below the nominal level. These shortcomings lead us to not recommend the use of GPS-weighting for assessing the effect of a quantitative exposure in observational studies, and to prefer more efficient alternative methods like the stratification method that was also evaluated in this study, or covariate adjustment using the GPS which also seems to provide more accurate estimates than GPS-weighting, as shown by [@austin2018assessing] for a binary outcome.
Given the performance of the stratified estimator combined with bootstrap variance estimation, a future research may focus on the development of a closed-form variance estimator which takes into account the fact that strata boundaries are estimated rather than known. This could be particularly useful for the analysis of very large datasets (such as healthcare-administrative databases), for which the repeated calculations required for the bootstrap methods could be an issue (see Supplementary materials, Figure 16, for a comparison of execution times recorded with each variance estimation method). Another topic of research could seek to improve the performance of the weighted estimator. As suggested by Austin, ’large value of $R^2$ results in some subjects having large weights, resulting in estimates with high variability’ ([@austin2018assessing]). The use of the marginal density of the quantitative exposure to stabilize the estimated weights was already shown to significantly improve the performance of GPS-weighting compared to the use of unstabilized weights ([@zhao2020propensity]). Nevertheless, our study shows that this stabilization fails to make the method competitive with the simple alternative that is stratification. Perhaps a different choice for the numerator of the GPS-weights could help to reduce even more the unstability and improve the overall performance. Another research perspective would be to study more complex dose-response functions. Indeed, our research was deliberately limited to the study of simple linear response models with only two parameters (the intercept and the slope), because the main objective of our study was not to compare different approaches for estimating a complex function, but to study the ability of various variance estimators to capture the variability of the estimates. While the inclusion of polynomial terms in the weighted model does not raise any particular difficulty, studying more complex models, including smooth coefficients or non-parametric modelization of the dose response function ([@zhao2020propensity]) would be interesting in order to get closer to real clinical situations in whom dose-response functions are not always linear. But the development and evaluation of variance estimators (including the comparison to the empirical variance estimation) adapted to these situations is not simple, and was beyond the scope of this work.
Acknowledgments {#acknowledgments .unnumbered}
===============
The French Ministry of Health funded the PreCARE study. The authors thank Elie Azria for his permission to use the data from the PreCARE cohort, and his insightful comments on the analysis of this case study.
Data availability statement {#data-availability-statement .unnumbered}
===========================
The data from the case study are available on request from the corresponding author upon reasonable request. The data are not publicly available due to privacy or ethical restrictions.
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The authors declare no potential conflict of interests.
Supporting information {#supporting-information .unnumbered}
======================
Additional simulation results may be found in the online version of this article at the publisher’s web site.
Figures {#figures .unnumbered}
=======
![Directed acyclic graph corresponding to the data-generating algorithm[]{data-label="fig:dag"}](dag.pdf){width="4cm"}
![Boxplots of the difference between $\widehat{\beta}$ and $\beta$ for different values of $R^2 \in \{0.2,0.4,0.6,0.8\}$ and $\sigma_Y^2 \in \{0.25,0.5\}$, $\beta_0=0$, $\beta_1=1$ and $n =1000$.[]{data-label="fig:bias"}](bias.pdf){width="15cm" height="15cm"}
![Root mean square error (RMSE) of $\widehat{\beta}$ for different values of $R^2 \in \{0.2,0.4,0.6,0.8\}$ and $\sigma_Y^2 \in \{0.25,0.5\}$, $\beta_0=0$, $\beta_1=1$ and $n =1000$.[]{data-label="fig:RMSE"}](rmse.pdf){width="15cm" height="15cm"}
![Boxplots of the standard deviation estimates of $\widehat{\beta}$ for different values of $R^2 \in \{0.2,0.4,0.6,0.8\}$ and $\sigma_Y^2 \in \{0.25,0.5\}$, $\beta_0=0$, $\beta_1=1$ and $n =1000$. The red line corresponds to the Monte Carlo standard deviation estimate.[]{data-label="fig:stdr"}](sdsr.pdf){width="15cm" height="18.75cm"}
![Variability ratio for different values of $R^2 \in \{0.2,0.4,0.6,0.8\}$ and $\sigma_Y^2 \in \{0.25,0.5\}$, $\beta_0=0$, $\beta_1=1$ and $n =1000$.[]{data-label="fig:var_ratio"}](vratio.pdf){width="15cm" height="15cm"}
![Coverage rate for different values of $R^2 \in \{0.2,0.4,0.6,0.8\}$ and $\sigma_Y^2 \in \{0.25,0.5\}$, $\beta_0=0$, $\beta_1=1$ and $n =1000$.[]{data-label="fig:Coverage"}](coverage.pdf){width="15cm" height="15cm"}
Variance estimator for the weighted dose-response function estimator {#appen1}
====================================================================
We write $$\begin{aligned}
\label{lin:eq1}
H_n (\hat{\gamma},\hat{\beta})-H_n (\gamma,\beta) & = & \frac{1}{n} \sum_{i=1}^n w_i(\hat{\gamma}) \{H_i(\hat{\beta})-H_i(\beta)\} \\
& + & \frac{1}{n} \sum_{i=1}^n \left\{w_i(\hat{\gamma})-w_i(\gamma)\right\} H_i(\beta). \nonumber\end{aligned}$$ We first consider the first term in the right-hand side of (\[lin:eq1\]), denoted as $\Delta_1$. Making use of a first-order Taylor expansion, we obtain $$\begin{aligned}
\label{lin:eq2}
\Delta_1 & = & \frac{1}{n} \sum_{i=1}^n w_i(\hat{\gamma}) \{\nabla H_i(\beta)^{\top}(\hat{\beta}-\beta)+o_p(n^{-0.5})\} \nonumber \\
& = & \left(\frac{1}{n} \sum_{i=1}^n w_i(\gamma) \nabla H_i(\beta) \right) (\hat{\beta}-\beta) + o_p(n^{-0.5}) \nonumber \\
& = & -A (\hat{\beta}-\beta) + o_p(n^{-0.5}),\end{aligned}$$ where $$\begin{aligned}
\label{lin:eq3}
A & = & E[-w_i(\gamma) \nabla H_i(\beta)].\end{aligned}$$ In view of the system of estimating equations (\[ee:beta\]), we have $$\begin{aligned}
\nabla H_i(\beta) & = & - \tilde{T}_i \tilde{T}_i^{\top},\end{aligned}$$ which leads to $$\begin{aligned}
\label{lin:eq5}
A & = & E[w_i(\gamma) \tilde{T}_i \tilde{T}_i^{\top}].\end{aligned}$$
We now consider the second term in the right-hand side of (\[lin:eq1\]), denoted as $\Delta_2$. Making use of a first-order Taylor expansion, we obtain $$\begin{aligned}
\label{lin:eq4}
\Delta_2 & = & \frac{1}{n} \sum_{i=1}^n \left\{\nabla w_i(\gamma))^{\top} (\hat{\gamma}-\gamma) + o_p(n^{-0.5})\right\} H_i(\beta) \nonumber \\
& = & \left(\frac{1}{n} \sum_{i=1}^n H_i(\beta) \nabla w_i(\gamma)^{\top} \right) (\hat{\gamma}-\gamma) + o_p(n^{-0.5}) \nonumber \\
& = & B (\hat{\gamma}-\gamma) + o_p(n^{-0.5}),\end{aligned}$$ where $$\begin{aligned}
\label{lin:eq6}
B & = & E[H_i(\beta) \nabla w_i(\gamma)^{\top}].\end{aligned}$$ Making use of equation (\[gps:weight\]), we obtain after some algebra $$\begin{aligned}
\label{eq:nabla:wi}
\nabla w_i(\gamma) & = & w_i(\gamma)
\left(
\begin{array}{l}
\frac{T_i-\mu_T}{\sigma_T^2} \\
\frac{1}{2\sigma_T^2} \left\{\frac{(T_i-\mu_T)^2}{\sigma_T^2}-1\right\} \\
-\tilde{Z}_i \frac{(T_i-\tilde{Z}_i^{\top} \alpha)}{\sigma^2} \\
-\frac{1}{2\sigma^2} \left\{\frac{(T_i-\tilde{Z}_i^{\top} \alpha)^2}{\sigma^2}-1\right\}
\end{array}
\right).\end{aligned}$$
Since $\gamma$ is estimated by solving the estimating equation (\[Fn:gamma\]), we also have $$\begin{aligned}
F_n(\hat{\gamma})-F_n(\gamma) & = & \nabla F_n(\gamma)^{\top} \left\{\hat{\gamma}-\gamma \right\}+o_p(n^{-0.5}),\end{aligned}$$ and since $F_n(\hat{\gamma})=0$, this leads to $$\begin{aligned}
\hat{\gamma}-\gamma & = & -\left\{\nabla F_n(\gamma)\right\}^{-1} \times F_n(\gamma) + o_p(n^{-0.5}) \nonumber \\
& = & C^{-1} \times F_n(\gamma) + o_p(n^{-0.5})\end{aligned}$$ where $$\begin{aligned}
C & = & E[-\nabla F_n(\gamma)].\end{aligned}$$ From the definition of $F_n$ given in equation (\[Fn:gamma\]), we obtain after some algebra $$\begin{aligned}
C & = & \left(
\begin{array}{cccc}
1 & 0 & 0_{1K} & 0 \\
0 & \frac{n-1}{n} & 0_{1K} & 0 \\
0_{K1} & 0_{K1} & E(\tilde{Z}_i \tilde{Z}_i^{\top}) & 0_{K1} \\
0 & 0 & 0 & \frac{n-p-1}{n}
\end{array}
\right).\end{aligned}$$
By gathering equations (\[lin:eq1\]), (\[lin:eq2\]), (\[lin:eq4\]) and (\[lin:eq6\]), and since $H_n (\hat{\beta})=0$, we obtain $$\begin{aligned}
-H_n (\gamma,\beta) & = & -A (\hat{\beta}-\beta) + B C^{-1} F_n(\gamma) + o_p(n^{-0.5}),\end{aligned}$$ which finally gives $$\begin{aligned}
\hat{\beta}-\beta & = & \frac{1}{n} \sum_{i=1}^n I_{1i}+o_p(n^{-0.5}), \\
\textrm{where } I_{1i} & = & A^{-1} \left\{w_i(\gamma)H_i(\beta)+BC^{-1}F_i(\gamma) \right\}.\end{aligned}$$ The variable $I_{1i}$ is the theoretical linearized variable of $\hat{\beta}$. It involves unknown parameters, which need to be estimated for variance estimation. This leads to the estimated linearized variable $\hat{I}_{1i}$ given in equation (\[lin:weight:1\]).
Variance estimator for the stratified dose-response function estimator {#appen2}
======================================================================
Recall that the intermediary estimator $\hat{\beta}_{\ell},~\ell=1,\ldots,L,$ is obtained by solving the estimating equation (\[strat:est:eq:1\]). We have $$\begin{aligned}
\label{app2:eq1}
\Phi(\widehat{\beta}_{\ell})-\Phi(\beta_{\ell}) & = & -\Phi(\beta_{\ell}) \nonumber \\
& \simeq & E\left\{ \nabla \Phi(\beta_{\ell})\right\} \times \{\widehat{\beta}_{\ell}-\beta_\ell\}.\end{aligned}$$ This leads to $$\begin{aligned}
\label{app2:eq2}
\widehat{\beta}_{\ell}-\beta_\ell & \simeq & - E\left\{ \nabla \Phi(\beta_{\ell})\right\}^{-1} \times \Phi(\beta_{\ell}) \nonumber \\
& = & \frac{1}{\sigma_{\ell,T}^2} \left(
\begin{array}{cc}
\sigma_{\ell,T}^2+m_{\ell,T}^2 & -m_{\ell,T} \\
-m_{\ell,T} & 1
\end{array}
\right) \times \frac{1}{n_\ell} \sum_{i \in S_\ell} \left(
\begin{array}{l}
Y_i - \tilde{T}_i^{\top} \beta_{\ell} \\
T_i(Y_i -\tilde{T}_i^{\top} \beta_{\ell})
\end{array}
\right) \nonumber \\
& = & \frac{1}{n_\ell} \sum_{i \in S_\ell} I_{2l,i}\end{aligned}$$ with $$\begin{aligned}
\label{app2:eq3}
I_{2l,i} & = & \frac{1}{\sigma_{\ell,T}^2}
\left(
\begin{array}{cc}
\sigma_{\ell,T}^2+m_{\ell,T}^2 & -m_{\ell,T} \\
-m_{\ell,T} & 1
\end{array}
\right) \times
\left(
\begin{array}{l}
Y_i - \tilde{T}_i^{\top} \beta_{\ell} \\
T_i(Y_i - \tilde{T}_i^{\top} \beta_{\ell})
\end{array}
\right),\end{aligned}$$ and where $m_{\ell,T}=E\left\{T_i \mathbf{1}_{\left\{i \in S_\ell\right\}}\right\}$ and $\sigma_{\ell,T}^2=V\left\{T_i \mathbf{1}_{\left\{i \in S_\ell\right\}}\right\}$.\
The variable $I_{2l,i}$ is the theoretical linearized variable of $\hat{\beta}_{\ell}$. Replacing the unknown parameters by suitable estimators leads to the estimated linearized variable $\hat{I}_{2l,i}$ given in (\[strat:est:eq:3\]).
R code for the different variance estimators {#appen3}
============================================
######################################################################
# This code is provided for illustrative purposes only and comes with
# absolutely NO WARRANTY.
######################################################################
library(survey)
library(boot)
######################################################################
# Weight estimation
######################################################################
# Fit the propensity model. Trt is the exposure, Z1 to Z10 are the covariates
modT <- lm(Trt ~ Z1 + Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9 + Z10, data = data)
# Linear predictor
data$m <- m <- modT$fitted
# Computation of the estimated weights
n <- nrow(data)
s <- sqrt(sum(modT$residuals^2)/(n-length(modT$coef)))
wd <- dnorm(data$Trt, m, s)
mu <- mean(data$Trt)
su <- sd(data$Trt)
wn <- dnorm(data$Trt, mu, su)
data$w <- w <- wn/wd
######################################################################
# Weighted estimator - sandwich standard error
######################################################################
mod <- svyglm(Y ~ Trt, design = svydesign(id = ~1, weights = ~ w, data = data),
family = gaussian)
summary(mod)
######################################################################
# Weighted estimator - linearized standard error
######################################################################
coefs.ipw <- mod$coefficients
variables <- names(data)[grep("^Z", names(data))]
Z <- as.matrix(data[, variables])
Ztilde <- cbind(1, Z)
dw <- w*cbind(
(data$Trt - mu)/(su^2),
(((data$Trt - mu)/su)^2 - 1)/(2*su^2),
-as.vector((data$Trt - data$m)/(s^2))*Ztilde,
-(((data$Trt - data$m)/s)^2 - 1)/(2*s^2)
)
Ttilde <- cbind(1, data$Trt)
tmp <- cbind(Ttilde, data$Trt, data$Trt^2)
A <- matrix(colMeans(tmp * w), 2, 2)
sA <- solve(A)
H <- Ttilde*as.vector((data$Y - mod$fitted.values))
F <- cbind(
data$Trt - mu,
(data$Trt - mu)^2 - ((n-1)/n)*(su^2),
as.vector((data$Trt - data$m))*Ztilde,
((data$Trt - data$m)^2) - ((n-length(modT$coef))/n)*(s^2)
)
B <- crossprod(H, dw)/n
mZZ <- crossprod(Ztilde, Ztilde)/n
C <- diag(length(variables) + 1 + 3)
C[2, 2] <- (n-1)/n
C[3:(length(variables)+3), 3:(length(variables)+3)] <- mZZ
C[nrow(C), ncol(C)] <- ((n-length(modT$coef))/n)
sC <- solve(C)
I <- t(sA%*%t((w*H + t(B %*% sC%*%t(F)))))
sds.ipw.lin <- sqrt(apply(I, 2, var)/n)
names(sds.ipw.lin) <- names(coefs.ipw)
print(coefs.ipw)
print(sds.ipw.lin)
######################################################################
# Weighted estimator - bootstrap standard error
######################################################################
f.boot.ipw <- function(data, i) {
df <- data[i, ]
modT <- lm(Trt ~ Z1 + Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9 + Z10, data = df)
m <- modT$fitted
n <- nrow(df)
s <- sqrt(sum(modT$residuals^2)/(n-length(modT$coef)))
wd <- dnorm(df$Trt, m, s)
mu <- mean(df$Trt)
su <- sd(df$Trt)
wn <- dnorm(df$Trt, mu, su)
df$w <- wn/wd
lm.wfit(cbind(rep(1, nrow(df)), df$Trt), df$Y, df$w)$coef
}
rcoefs <- boot(data, f.boot.ipw, R = 200)$t
sds.ipw.boot <- apply(rcoefs, 2, sd)
names(sds.ipw.boot) <- names(coefs.ipw)
print(sds.ipw.boot)
######################################################################
# Stratified estimator - Pooled model-based standard error
######################################################################
cl <- 10 # number of strata
data$Tcl <- cut(data$m, breaks = quantile(data$m, probs = seq(0, 1, 1/cl)),
include.lowest = TRUE)
W1 <- apply(data.frame(levels(data$Tcl)), MARGIN = 1, function(x) {
data2 <- subset(data, data$Tcl == x)
nk <- nrow(data2)
pk <- nk/n
mod <- glm(Y ~ Trt, data = data2, family = gaussian)
coefs <- mod$coef
sds <- (summary(mod)$coefficients[,2])^2
return(c(pk*coefs, pk^2*sds))
})
coefs.strat <- apply(W1[1:2,], MARGIN = 1, sum)
sds.strat.pool1 = sqrt(apply(W1[3:4,],MARGIN = 1, sum))
print(coefs.strat)
print(sds.strat.pool1)
######################################################################
# Stratified estimator - Pooled linearized standard error
######################################################################
W2 <- apply(data.frame(levels(data$Tcl)), MARGIN = 1, function(x) {
data2 <- subset(data,data$Tcl == x)
nk <- dim(data2)[1]
pk <- nk/n
mod <- glm(Y ~ Trt, data = data2, family = gaussian)
coefs <- mod$coef
mhat <- mean(data2$Trt)
shat <- var(data2$Trt)
uhat <- rep((1/shat), nk)*as.vector(rep(shat+mhat^2,nk) -
mhat*data2$Trt)*as.vector(data2$Y-coefs[1]-coefs[2]*data2$Trt)
uhat2 <- rep((1/shat), nk)*as.vector(data2$Trt-mhat)
*as.vector(data2$Y-coefs[1]-coefs[2]*data2$Trt)
ubar <- mean(uhat)
ubar2 <- mean(uhat2)
sds1 <- 1/(nk*(nk-1))*sum((uhat-ubar)^2)
sds2 <- 1/(nk*(nk-1))*sum((uhat2-ubar2)^2)
return(c(pk*coefs, pk^2*sds1, pk^2*sds2))
})
sds.strat.pool2 = sqrt(apply(W2[3:4,], MARGIN = 1, sum))
print(sds.strat.pool2)
######################################################################
# Stratified estimator - Bootstrap standard error
######################################################################
f.boot.strat <- function(data, i) {
df <- data[i, ]
modT <- lm(Trt ~ Z1 + Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9 + Z10, data = df)
df$m <- modT$fitted
df$Tcl <- cut(df$m, breaks = quantile(df$m, probs = seq(0, 1, 1/cl)),
include.lowest = TRUE)
W <- apply(data.frame(levels(df$Tcl)), MARGIN = 1, function(x) {
df2 <- subset(df, df$Tcl == x)
nk <- nrow(df2)
pk <- nk/n
coefs <- lm.fit(cbind(rep(1, nrow(df)), df$Trt), df$Y)$coef
return(c(pk*coefs))
})
apply(W[1:2,], MARGIN = 1, sum)
}
rcoefs <- boot(data, f.boot.strat, R = 200)$t
sds.strat.boot <- apply(rcoefs, 2, sd)
names(sds.strat.boot) <- names(coefs.strat)
print(sds.strat.boot)
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[^1]: Univ Rennes, INSA, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France, valerie.gares@insa-rennes.fr
[^2]: Univ Rennes, ENSAI, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France, chauvet@ensai.fr
[^3]: Sorbonne Université, INSERM, Institut Pierre Louis d’Epidémiologie et de Santé Publique, AP-HP, Hôpital Pitié-Salpêtrière, Département de Santé Publique, Centre de Pharmacoépidémiologie, Paris, France
|
---
abstract: |
In this paper we present an inexact stepsize selection for the Diluted $R \rho R$ algorithm [@rehacek2007], used to obtain the maximum likelihood estimate to the density matrix in quantum state tomography. We give a new interpretation for the diluted $R \rho R$ iterations that allows us to prove the global convergence under weaker assumptions. Thus, we propose a new algorithm which is globally convergent and suitable for practical implementation.\
\
PACS number(s): 03.65.Wj
author:
- 'D. S. Gonçalves'
- 'M. A. Gomes-Ruggiero'
- 'C. Lavor'
bibliography:
- 'steplength\_diluted\_rrhor.bib'
title: 'Global convergence of diluted iterations in maximum-likelihood quantum tomography '
---
Introduction
============
In quantum state tomography, the aim is to find an estimate for the density matrix associated to the ensemble of identically prepared quantum states, based on measurement results [@hradil1997; @parisqse2004; @dgoncalves2013]. This is an important procedure in quantum information and computation, for example, to verify the fidelity of the prepared state [@ncbook; @clavor2011] or in quantum process tomography [@maciel2012].
Besides the experimental design to get a tomographically complete set of measurements, post processing routines are required to recover information from the measurement results. Some approaches are based on direct inversion of the data while others rest in statistical based methods. For a survey, the reader can see [@parisqse2004].
Among the statistical based methods, the Maximum Likelihood estimation (ML) [@parisqse2004; @hradil2004] has been often used by experimentalists [@james2001]. The Maximum Likelihood estimate for the density matrix is that one which maximizes the probability of the observed data. In [@hradil1997; @hradil2004], it was proposed an iterative procedure to solve the problem of the maximum likelihood estimation for density matrices. We refer to this procedure as the $R \rho R$ algorithm. The main properties of the $R \rho R$ algorithm are: keeping the positivity and unit trace of the iterates and its low computational cost, involving only matrix products at each iteration.
Although in practice the $R \rho R$ method works in most of the cases, there is no theoretical guarantee of convergence, regardless the dataset and the initial point. In [@rehacek2007], the authors presented an example where the method gets into a cycle. In the same work, they proposed some kind of relaxation of the $R \rho R$ iterations, controlling the step size at each iteration by a positive parameter $t$. They called this kind of iterations as Diluted $R \rho R$ iterations. It was proved that the Diluted $R \rho R$ method converges to the maximum likelihood solution if, at each iteration, the optimal value of the step size $t$ is chosen.
However, to find the optimal value of $t$ means to solve another optimization problem at each iteration, which represents an undesirable additional computational cost in practice. This issue was remarked in [@rehacek2007], where it was suggested some heuristics in order to get some reasonable guess for the step size $t$ in practical implementations, but loosing the convergence warranty.
In this work we propose a new stepsize selection procedure which is reliable and feasible in practice. We give a new interpretation to the Diluted $R \rho R$ iteration, where the search direction is a combination of two ascent directions controlled by the step size $t$. This allow us to apply an inexact line search to determine the step length. Instead of the optimal value of $t$, at each iteration, it is enough to find a value which ensures a sufficient improvement in the likelihood function in order to prove the global convergence. We propose an algorithm, using an Armijo-like condition [@bertsekas1999; @nocedal1999] and a backtracking procedure, and prove that it is globally convergent and also computationally practicable.
This paper is organized as follows. Section \[theoryrrhor\] reviews the theory of the $R \rho R$ algorithms for quantum tomography. The concepts of nonlinear optimization used to prove the convergence of the Diluted $R \rho R$ are presented in Section \[globaltheory\]. Section \[main\] presents the proof of global convergence of the Diluted $R \rho R$ algorithm under line search and Armijo condition. Examples illustrating the differences and similarities of our proposal to the traditional fixed step length are presented in Section \[examples\]. Section \[final\] closes this work with some final considerations.
$R \rho R$ iterations for quantum tomography {#theoryrrhor}
============================================
In this section we address the theory and motivation behind the $R \rho R$ and the Diluted $R \rho R$ algorithms, following the references [@hradil1997; @hradil2004; @rehacek2007].
Here we consider measurements described by a POVM set ${\left\{E_i\right\}}_i$, where $E_i$ are semidefinite positive operators which sum to the identity. The relation between the density matrix $\rho$ and the probability outcomes is given by the Born’s rule [@parisqse2004]: $$p_i(\rho) = {\mbox{tr}\left( E_i \rho \right)}.$$ Linear inversion methods equate the predicted probabilities ${\left\{p_i(\rho)\right\}}_i$ with the experimental data ${\left\{f_i\right\}}_i$: $$f_i = {\mbox{tr}\left( E_i \rho \right)}, \forall i$$ and the inversion of these linear equations gives an estimate of $\rho$. The main problem with this approach is that, in general, the frequencies are noisy and this fact can leads to a matrix $\rho$ outside the density matrix space (the Hermitian semidefinite positive trace one matrices).
Among the statistical based methods, the Maximum Likelihood estimation [@parisqse2004; @hradil2004] has been often used by experimentalists [@james2001]. Let us denote $\rho^{\dagger}$ the conjugate transpose of $\rho$ and $\rho \succeq 0$ to say that the Hermitian matrix $\rho$ is semidefinite positive (or $\rho \succ 0$ for a strictly positive matrix). The ML estimation searches within the density matrix space: $${\cal S} = {\left\{\rho\ |\ \rho = \rho^{\dagger}, \ \rho \succeq 0, {\mbox{tr}\left( \rho \right)}=1\right\}},$$ that one which maximizes the likelihood function. The likelihood function is the probability of getting the observed data given the density matrix $\rho$. A common used likelihood [@hradil1997; @hradil2004], for a given data set ${\left\{f_i\right\}}$, is $${\cal L}(\rho) \propto \prod_i p_i(\rho)^{N f_i},$$ and since the log-likelihood is more tractable, our goal is to find $\rho$ that solves the problem $$\label{mlprob}
\begin{aligned}
\max_{\rho} & \ \ \sum_i f_i \log p_i(\rho) & \equiv \ F(\rho) \\
\mbox{s.t} & \ \ {\mbox{tr}\left( \rho \right)} = 1 \\
\ & \ \ \rho \succeq 0.
\end{aligned}$$ The maximization of the objective function $F(\rho)$ in [(\[mlprob\])]{} is constrained to the density matrix space ${\cal S}$ which is the intersection of the semidefinite positive cone $\rho \succeq 0$ with the affine subspace ${\mbox{tr}\left( \rho \right)}=1$. The constraints may motivate one to try semidefinite programming (SDP) methods [@klerkbook2002] for solving [(\[mlprob\])]{}, but efficient solvers [@sdpt3; @sedumi] are available only for linear and quadratic objective functions.
Other methods are based on the reparameterization [@james2001; @goncalves2012] of the matrix variable $\rho = \rho(\theta)$ in order to automatically fulfill the constraints and then to solve an unconstrained maximization problem in the new variable $\theta$. However, generic numerical optimization methods are often slow when the number of parameters $d^2$ ($d$ is the dimension of the Hilbert space) is large.
Here, we study an alternative algorithm, proposed in [@hradil2004], which takes advantage of the structure of the problem [(\[mlprob\])]{} and has good convergence properties.
Consider the gradient of the objective function $F(\rho)$, given by $$\label{gradr}
\nabla F(\rho) = \sum_i \frac{f_i}{{\mbox{tr}\left( E_i \rho \right)}} E_i \equiv R(\rho),$$ and let ${\mbox{int}\left({\cal S}\right)}$ be the interior of ${\cal S}$, that is $${\mbox{int}\left({\cal S}\right)} = {\left\{\rho \in {\cal S}\ |\ \rho \succ 0\right\}}.$$ As it was shown in [@hradil2004], a matrix $\rho \in {\mbox{int}\left({\cal S}\right)}$ solves [(\[mlprob\])]{} if it satisfies the extremal equation $$\label{extremal}
R(\rho) \rho = \rho,$$ or equivalently $$\label{extremal2}
R(\rho) \rho R(\rho) = \rho.$$ If the density matrix $\rho$ is restricted to diagonal matrices, the equation [(\[extremal\])]{} can be solved by the expectation-maximization (EM) algorithm [@vardi1993]. The EM algorithm is guaranteed to increase the likelihood at each step and converges to a fixed point of [(\[extremal\])]{}. However, the EM algorithm cannot be applied to the quantum problem, because without the diagonal constraint it does not preserve the positivity of the density matrix. In [@hradil2004], it was proposed an iterative procedure based on the equation [(\[extremal2\])]{} instead. Let $k$ be the iteration index, and so, $\rho^k$ the current approximation to the solution. An iteration of the $R \rho R$ algorithm is given by: $$\rho^{k+1} = {\cal N}\,R(\rho^k) \rho^k R(\rho^k),$$ where ${\cal N}$ is the normalization constant which ensures unit trace.
Notice that the positivity is explicitly preserved at each step. Another remarkable property of the $R \rho R$ algorithm is its computational cost: at each iteration, it is just required to compute a matrix-matrix product. This is a quite cheap iteration in contrast with the iteration of an semidefinite programming method.
Although the $R \rho R$ algorithm is a generalization of the EM algorithm, its convergence is not guaranteed in general. In [@rehacek2007], it was presented a counterexample where the method produces a cycle. For this reason, in that work was proposed the diluted iteration of the $R \rho R$ algorithm, or simply “Diluted $R \rho R$”.
The idea is to control each iteration step by mixing the operator $R(\rho)$ with the identity operator: $$\label{dilutedit}
\rho^{k+1} = {\cal N} \left[ \frac{I + t R(\rho^k)}{1 + t} \right] \rho^k \left[ \frac{I + t R(\rho^k)}{1 + t} \right],$$ where $t>0$ and ${\cal N}$ is the normalization constant. It is important to observe that as $t \rightarrow \infty$, the iteration tends to the original $R \rho R$ iteration. Moreover, when $t>0$ is sufficient small, it was proved that the likelihood function is strictly increased, whenever $R(\rho)\rho \ne \rho$. It was also shown that the “Diluted $R \rho R$” is convergent to the ML density matrix, if the initial approximation is the maximally mixed state $\rho^0=(1/d)I$ and the optimal value of $t$: $$\label{els}
t = \mbox{arg}\max_{t>0} F(\rho^{k+1}(t)),$$ is used at each iteration.
In nonlinear optimization [@bertsekas1999; @nocedal1999], this is called exact line search. Though the convergence can be achieved using this procedure, in general, solving [(\[els\])]{} may be computationally demanding. Albeit in [@rehacek2007] the authors proved the convergence with the exact line search, they suggest that, in practice, one could use an ad hoc scheme to determine the “best” value of the steplength $t$ to be used through all iterations.
Here, instead of [(\[els\])]{}, we propose an inexact line search to determine the steplength $t$ in each iteration [(\[dilutedit\])]{}. We do not search the best possible $t>0$, but one that ensures a sufficient improvement in the log-likelihood. We prove that this procedure is well-defined and that the iterations [(\[dilutedit\])]{} converge to a solution of [(\[mlprob\])]{}, from any positive initial matrix $\rho^0$. The implementation of the inexact line search is straightforward and we also present some examples showing the improvements, against an ad hoc fixed $t$ strategy.
Global convergence theory for ascent direction methods {#globaltheory}
======================================================
The purpose of this section is to expose some basic concepts of nonlinear optimization which are necessary to prove the global convergence of the Diluted $R \rho R$ algorithm under an inexact line search scheme. These concepts are classical for the optimization community and are detailed in [@nocedal1999; @bertsekas1999]. To make it easier, we have adapted these concepts using the quantum tomography notation.
Consider the following maximization problem over the set of Hermitian matrices ${\cal H}$: $$\label{maxprob}
\begin{aligned}
\max_{\rho} & \ \ F(\rho) \\
\mbox{s.t} & \ \ \rho \in \Omega,
\end{aligned}$$ where $f: {\cal H} \rightarrow {\mathbb R}$ is a continuously differentiable function and $\Omega \subset {\cal H}$ is a convex set.
Given an approximation $\rho^k$ for the solution of problem [(\[maxprob\])]{}, ascent direction methods try to improve the current objective function value generating an ascent direction $D^k$ and updating the iterate $$\label{iteration}
\rho^{k+1} = \rho^k + t_k D^k,$$ where $t_k$ is called stepsize or steplength.
A direction $D^k$ is an [*ascent direction*]{} at the iterate $\rho^k$ if $${\mbox{tr}\left( \nabla F(\rho^k) D^k \right)} > 0,$$ and this ensures that, for a sufficient small $t_k>0$, the function value is increased. An ascent direction $D^k$ is [*feasible*]{} if $\rho^{k+1}$, belongs to $\Omega$ for $t_k \in (0,{\varepsilon})$, for some ${\varepsilon}>0$.
One of the insights of this work is that the diluted $R \rho R$ iteration [(\[dilutedit\])]{} can be written as an ascent direction iteration [(\[iteration\])]{} and so, using the theory of this section, we can prove the global convergence.
The iteration [(\[iteration\])]{} can be repeated while there exists a feasible ascent direction. If at some point $\rho^*$ there is no feasible ascent direction, then $\rho^*$ is a [*stationary point*]{}. It is well known that every local maximizer is a stationary point, but the converse is not true in general. If the function $f$ is concave on the convex set $\Omega$, then a stationary point is also a maximizer.
A maximization algorithm for the problem [(\[maxprob\])]{} is called [*globally convergent*]{} [@bertsekas1999; @nocedal1999] if every limit point of the sequence generated by the algorithm is a stationary point, regardless the initial approximation $\rho^0$. Although feasible ascent directions ensure that, for a sufficient small $t_k > 0$, we can increase the function value, this is not enough to ensure the global convergence. The reason is that a simple increase in the objective function, $F(\rho^{k+1})>F(\rho^k)$, along an ascent direction is a too modest objective. In order to achieve local maximizers, or at least stationary points, a [*sufficient*]{} increase at each iteration is required.
Of course that a natural choice for the steplength $t_k$, along the direction $D^k$, is the solution of the problem: $$\label{exls}
t_k = \mbox{argmax}_{t} \ F(\rho^k + t D^k),$$ that is called [*exact line search*]{}. However, finding the global maximizer of $f$ along the direction $D^k$ is itself a hard problem, and unless the function $f$ has a special structure such as a quadratic function, for instance, the computational effort is considerable.
To avoid the considerable computational effort in the exact line search [(\[exls\])]{}, an inexact line search can be performed. A natural scheme is to consider successive stepsize reductions. Since the search is on a ascent direction, eventually for a small $t_k$, we can obtain $F(\rho^k + t_k D^k) > F(\rho^k)$. But, this simple increase can not eliminate some convergence difficulties. One possible strategy is the use of the [*Armijo rule*]{}, which asks for a steplength $t$ such that a sufficient improvement in the objective function is obtained: $$\label{armijo}
F(\rho^k + t D^k) > F(\rho^k) + \gamma t \ {\mbox{tr}\left( \nabla F(\rho^k) D^k \right)},$$ where $\gamma \in (0,1)$. We can decrease the steplength $t$ until the condition [(\[armijo\])]{} is verified. There are other alternatives to the successive stepsize reduction, for instance, strategies based on quadratic or cubic interpolation [@nocedal1999].
Besides the steplength selection, requirements on the ascent directions $D^k$ are also necessary to avoid certain problems. For example, it is not desirable to have directions $D^k$ with small norm when we are far from the solution. It is also necessary to avoid that the sequence of directions ${\left\{D^k\right\}}$ become orthogonal to the gradient of $f$, because, in this case, we are in directions of almost zero variation where too small or none improvement on the objective function can be reached. A general condition that avoid such problems is called [*gradient related*]{} condition [@bertsekas1999].
A sequence of directions ${\left\{D^k\right\}}$ is gradient related if for any subsequence ${\left\{\rho^k\right\}}_{k \in {\cal K}}$ that converges to a nonstationary point, the corresponding subsequence ${\left\{D^k\right\}}_{k \in {\cal K}}$ is bounded and satisfies $$\lim_{k \rightarrow \infty} \inf_{k \in {\cal K}} {\mbox{tr}\left( \nabla F(\rho^k) D^k \right)} > 0.$$
This condition means that ${\left\VertD^k\right\Vert}$ does not become ’too small’ or ’too large’ relative to ${\left\Vert\nabla F(\rho^k)\right\Vert}$, and that the $D^k$ and $\nabla F(\rho^k)$ do not become orthogonal.
If an algorithm generates ascent directions satisfying the gradient related condition and the stepsizes are selected according to the Armijo rule, then it is possible to prove the global convergence [@bertsekas1999].
\[teogc\] Let ${\left\{\rho^k\right\}}$ be a sequence generated by a feasible ascent directions method $\rho^{k+1} = \rho^k + t_k D^k$, and assume that ${\left\{D^k\right\}}$ is gradient related and $t_k$ is chosen by the Armijo rule. Then, every limit point of ${\left\{D^k\right\}}$ is a stationary point.
See [@bertsekas1999 Proposition 2.2.1].
Convergence of the Diluted $R \rho R$ {#main}
=====================================
Sections \[theoryrrhor\] and \[globaltheory\] gave us the necessary background to show the convergence of the diluted $R \rho R$ iterations using an inexact line search to determine the stepsize $t$. In this section, firstly we show that the diluted iteration [(\[dilutedit\])]{} can be written as an ascent direction iteration [(\[iteration\])]{}. So, we give a geometrical interpretation and prove that the corresponding sequence of directions ${\left\{D^k\right\}}$ is gradient related. Finally, using the Armijo condition and a backtracking procedure, we present an algorithm which is globally convergent following the Theorem \[teogc\].
From now on, we will use the notation $\nabla F(\rho)$ instead of $R(\rho)$. So, the equation [(\[dilutedit\])]{} becomes: $$\rho^{k+1} = {\cal N} \left[ \frac{I + t \nabla F(\rho^k)}{1 + t} \right] \rho^k \left[ \frac{I + t \nabla F(\rho^k)}{1 + t} \right]$$ or $$\rho^{k+1} = \frac{(I + t \nabla F(\rho^k) )\, \rho^k \,(I + t \nabla F(\rho^k) )}{{\mbox{tr}\left( (I + t \nabla F(\rho^k) )\, \rho^k \,(I + t \nabla F(\rho^k) ) \right)}} \equiv G(\rho^k).$$
The expression above can be seen as a fixed point iteration. Expanding that expression, we obtain $$\label{gdef}
G(\rho) = \frac{\rho + t\,(\nabla F(\rho) \rho + \rho \nabla F(\rho) ) + t^2\, \nabla F(\rho)\rho \nabla F(\rho)}{1 + 2t + t^2 {\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}}.$$
Notice that $\rho^*$ is a fixed point of $G(\rho)$, $G(\rho^*)=\rho^*$, for $t>0$, if the following conditions are satisfied $$\begin{aligned}
\rho^* & = & \nabla F(\rho^*) \rho^* \nabla F(\rho^*) = \nabla F(\rho^*) \rho^*.\end{aligned}$$ If the above conditions are verified at a positive definite trace one matrix $\bar{\rho}$, then the [*optimality conditions*]{} [@bertsekas1999; @nocedal1999] for the problem [(\[mlprob\])]{} are satisfied and $\rho^*$ is the maximum likelihood estimate.
The following two lemmas are useful when concerning the $R \rho R$ iterations.
\[lemtrgradrho\] For all $\rho \in {\mbox{int}\left({{\cal S}}\right)}$, we have $${\mbox{tr}\left( \nabla F(\rho)\rho \right)} = 1.$$
Directly from [(\[gradr\])]{}.
\[lemtrgradrhograd\] If $\rho \in {\mbox{int}\left({{\cal S}}\right)}$, then $${\mbox{tr}\left( \nabla F(\rho)\rho \nabla F(\rho) \right)} \ge 1,$$ with equality if and only if $\rho = \nabla F(\rho) \rho$.
From Lemma \[lemtrgradrho\], $$1 = {\mbox{tr}\left( \nabla F(\rho) \rho \right)} = {\mbox{tr}\left( \nabla F(\rho) \rho^{1/2} \rho^{1/2} \right)},$$ and from the Cauchy-Schwarz inequality, we obtain $$1 = {\left\vert{\mbox{tr}\left( \nabla F(\rho) \rho^{1/2} \rho^{1/2} \right)}\right\vert}^2 \le {\mbox{tr}\left( \nabla F(\rho)\rho \nabla F(\rho) \right)}{\mbox{tr}\left( \rho \right)} = {\mbox{tr}\left( \nabla F(\rho)\rho \nabla F(\rho) \right)}.$$ The equality in Cauchy-Schwarz occurs when $\nabla F(\rho) \rho^{1/2} = \alpha \rho^{1/2}$, or equivalently, when $\nabla(\rho) \rho = \rho$.
Let us simplify the expression [(\[gdef\])]{}, defining for some $\rho$, $$q(t) = 1 + 2t + t^2 {\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}.$$ Since ${\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)} \ge 1$, for any density matrix $\rho$, we have that $q(t) \ge 1$ for all $t \ge 0$. Furthermore, if $\rho \in {\cal S}$, the set of density matrices, $G(\rho) \in {\cal S}$ as well, for any $t \ge 0$. Thus, $G(\rho)$ defines a path on the density matrices space ${\cal S}$, parameterized by $t$ such that, when $t \rightarrow 0$, $G(\rho) \rightarrow \rho$, and when $t \rightarrow \infty$, $G(\rho) \rightarrow \tilde{\rho}$, where $$\tilde{\rho} = \frac{\nabla F(\rho) \rho \nabla F(\rho)}{{\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}},$$ as in the original $R \rho R$ algorithm [@hradil1997].
Let us also define the point $$\label{rhobar}
\bar{\rho} = \frac{\nabla F(\rho) \rho + \rho \nabla F(\rho)}{2}.$$ Unlike the point $\tilde{\rho}$, the point $\bar{\rho}$, in general, is not in the set ${\cal S}$.
Now, rewriting the expression [(\[gdef\])]{}, we obtain $$\label{gexp}
G(\rho) = \frac{1}{q(t)} \rho + \frac{2t}{q(t)} \left( \frac{\nabla F(\rho) \rho + \rho \nabla F(\rho) }{2} \right) + \frac{t^2 {\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)} }{q(t)} \frac{\nabla F(\rho) \rho \nabla F(\rho)}{{\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}},$$ that is $$G(\rho) = \frac{1}{q(t)} \rho + \frac{2t}{q(t)} \bar{\rho} + \frac{t^2 {\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)} }{q(t)} \tilde{\rho}.$$ Therefore, we have a convex combination of the points $\rho,\ \bar{\rho}$, $\tilde{\rho}$, and the path defined by $t$ is in the convex set whose extreme points are $\rho,\ \bar{\rho}$ and $\tilde{\rho}$, as we can see in Figure \[figgeo\].\
(0,0) circle (80pt); (40pt,-30pt) circle (2pt) node\[below\][$\rho^k$]{}; (100pt,0pt) circle (2pt) node\[below\][$\bar{\rho}^k$]{}; (60pt,30pt) circle (2pt) node\[above\][$\tilde{\rho}^k$]{}; (40pt,-30pt) – (100pt,0pt) – (60pt,30pt) – (40pt,-30pt); (40pt,-30pt) – (98pt,-1pt); (40pt,-30pt) – (59pt,27pt); (40pt,-30pt) .. controls (100pt,0pt) and (70pt,22pt) .. (60pt,30pt); (75pt,-15pt) node\[below\][$\bar{D}^k$]{}; (40pt,20pt) node\[below\][$\tilde{D}^k$]{}; (-170pt,-100pt) rectangle (170pt,100pt); (-160pt,-90pt)\[right\]node [${\mbox{tr}\left( \rho \right)}=1$]{}; (50pt,40pt) – (30pt,50pt) node\[left\][[$(t \rightarrow \infty)$]{}]{}; (30pt,-35pt) – (20pt,-40pt) node\[left\][[$(t = 0)$]{}]{}; (75pt,15pt) – (95pt,35pt) node\[right\][[$\rho^{k+1}(t)$]{}]{}; (0pt,0pt) node [${\cal S}$]{};
Finally, defining the directions $$\begin{aligned}
\bar{D} & = & \bar{\rho} - \rho = \frac{\nabla f (\rho) \rho + \rho \nabla F(\rho) }{2} - \rho, \label{dbar} \\
\tilde{D} & = & \tilde{\rho} - \rho = \frac{\nabla F(\rho) \rho \nabla F(\rho)}{{\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}} - \rho, \label{dtilde}\end{aligned}$$ and using [(\[gexp\])]{}, we obtain $$G(\rho) = \rho + \frac{2t}{q(t)}\bar{D} + \frac{t^2 {\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}}{q(t)} \tilde{D},$$ which provides us an iteration like [(\[iteration\])]{} $$\hat{\rho} = G(\rho) = \rho + t D,$$ where $$\label{direction}
D = \frac{2}{q(t)}\bar{D} + \frac{t \, {\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}}{q(t)} \tilde{D}.$$ The search direction $D$ is a combination of the directions $\bar{D}$ and $\tilde{D}$ controlled by the parameter $t$. From Figure \[figgeo\], we can see that as $t \rightarrow \infty$, $D$ goes to the direction $\tilde{D}$, whereas $t \rightarrow 0$, $D$ becomes parallel to $\bar{D}$. It is worth to prove that these are feasible ascent directions.
The directions $\bar{D}$ and $\tilde{D}$ are feasible ascent directions for any nonstationary point $\rho$.
To prove that $\bar{D}$ is an ascent direction, we need to show that ${\mbox{tr}\left( \nabla F(\rho) \bar{D} \right)}>0$. Using the definition of $\bar{D}$, we get $$\bar{D} = \bar{\rho} - \rho = \frac{\nabla F(\rho) \rho + \rho \nabla F(\rho)}{2} - \rho.$$ For a nonstationary $\rho$ ($\nabla F(\rho) \rho \ne \rho$), $${\mbox{tr}\left( \nabla F(\rho) \bar{D} \right)} = {\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)} - {\mbox{tr}\left( \nabla F(\rho) \rho \right)} =$$ $${\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)} - 1 > 0,$$ by the Cauchy-Schwarz inequality, which implies that $\bar{D}$ is an ascent direction. If $\rho \in {\mbox{int}\left({\cal S}\right)}$, then there exists $t>0$ such that $\rho + t \bar{D} \in {\cal S}$, so the direction is feasible.
In a similar way for $\tilde{D}$, $${\mbox{tr}\left( \nabla F(\rho) \tilde{D} \right)} = \frac{ {\mbox{tr}\left( \nabla F(\rho) \nabla F(\rho) \rho \nabla F(\rho) \right)} }{{\mbox{tr}\left( \nabla F(\rho) \rho \nabla F(\rho) \right)}} - 1 > 0.$$ Using the fact that $\tilde{\rho} \in {\cal S}$, for $t \in (0,1]$, we get $\rho + t \tilde{D} \in {\cal S}$ as well.
Since the direction $D$ is a positive combination of feasible ascent directions, it is also a feasible ascent direction.
Now, if we can show that the sequence of directions ${\left\{D^k\right\}}$ is gradient related, then we can prove the global convergence under an inexact line search scheme. First, we present some technical lemmas which are useful to show the desired result.
\[lemarhobar\] For $\rho^k \succ 0$ and ${\mbox{tr}\left( \rho^k \right)}=1$, the matrix $\bar{\rho}^k$, defined in [(\[rhobar\])]{}, is the solution of the problem $$\label{problemarhobar}
\begin{aligned}
\max_{\rho} & \ \ \ {\mbox{tr}\left( \nabla F(\rho^k) (\rho - \rho^k) \right)} - \frac{1}{2}{\mbox{tr}\left( (\rho - \rho^k)({\rho^k})^{-1}(\rho - \rho^k) \right)} \\
\mbox{s.t} & \ \ \ {\mbox{tr}\left( \rho \right)} = 1.
\end{aligned}$$
Consider the optimality conditions for [(\[problemarhobar\])]{}: $$\label{kkt1}
-\nabla F(\rho^k) + \frac{1}{2}\left[(\rho - \rho^k)(\rho^k)^{-1} + (\rho^k)^{-1}(\rho - \rho^k)\right] + \lambda_0 I = 0,$$ $$\label{kkt2}
{\mbox{tr}\left( \rho \right)} = 1.$$ In equation [(\[kkt1\])]{}, multiplying at the right by $\rho^k$ and taking the trace, we have $$-{\mbox{tr}\left( \nabla F(\rho^k) \rho^k \right)} + {\mbox{tr}\left( \rho - \rho^k \right)} + \lambda_0 {\mbox{tr}\left( \rho^k \right)} = 0,$$ which implies that $\lambda_0 = 1$. So, from $$-\nabla F(\rho^k) + \frac{1}{2}\left[(\rho - \rho^k)(\rho^k)^{-1} + (\rho^k)^{-1}(\rho - \rho^k)\right] + I = 0,$$ we obtain $$\rho (\rho^k)^{-1} + (\rho^k)^{-1} \rho = \nabla F(\rho^k).$$ Using the symmetry of the solution $\rho$, the symmetry of $\rho^k$, and $\nabla F(\rho^k)$, we conclude that $$\rho = \nabla F(\rho^k) \rho^k = \rho^k \nabla F(\rho^k) = \frac{\nabla F(\rho^k) \rho^k + \rho^k \nabla F(\rho^k)}{2} = \bar{\rho}^k.$$
\[lemagr\] The sequence of directions ${\left\{\bar{D}^k\right\}}$, used to define the sequence ${\left\{\rho^k\right\}}$ by $$\rho^{k+1} = \rho^k + t\, \bar{D}^k,$$ satisfies $$\lim_{k \rightarrow \infty} \inf_{k \in {\cal K}} {\mbox{tr}\left( \nabla F(\rho^k)(\bar{\rho}^k - \rho^k) \right)} > 0,$$ for all subsequence ${\left\{\rho^k\right\}}_{k \in {\cal K}}$ that converges to a non-stationary point $\rho'$.
Suppose there is a subsequence ${\left\{\rho^k\right\}}_{k \in {\cal K}}$ that converges to a non-stationary point $\rho'$. Lemma \[lemarhobar\] tell us that $\bar{\rho}^k$ is the solution of [(\[problemarhobar\])]{}. Thus, at $\bar{\rho}^k$, the gradient of the objective function of [(\[problemarhobar\])]{} is orthogonal to the hyperplane ${\mbox{tr}\left( \rho \right)}=1$, that is $${\mbox{tr}\left( \left[\nabla F(\rho^k) - \frac{1}{2}\left( (\bar{\rho}^k - \rho^k)(\rho^k)^{-1} + (\rho^k)^{-1} (\bar{\rho}^k - \rho^k) \right) \right] (\rho - \bar{\rho}^k) \right)} = 0,$$ $\forall \rho \mbox{ \ such that \ } {\mbox{tr}\left( \rho \right)}=1$. Since the feasible set of [(\[problemarhobar\])]{} contains ${\cal S}$, we have $${\mbox{tr}\left( \left[\nabla F(\rho^k) - \frac{1}{2}\left( (\bar{\rho}^k - \rho^k)(\rho^k)^{-1} + (\rho^k)^{-1} (\bar{\rho}^k - \rho^k) \right) \right] (\rho - \bar{\rho}^k) \right)} = 0, \ \forall \rho \in {\cal S}.$$ Expanding the last expression, we obtain $${\mbox{tr}\left( \nabla F(\rho^k)(\rho - \bar{\rho}^k) \right)} = - \frac{1}{2}\left[ {\mbox{tr}\left( (\rho^k - \bar{\rho}^k) (\rho^k)^{-1} (\rho - \bar{\rho}^k) \right)} + {\mbox{tr}\left( (\rho - \bar{\rho}^k) (\rho^k)^{-1} (\rho^k - \bar{\rho}^k) \right)} \right],$$ $\noindent \forall \rho \in {\cal S}$. In particular, for $\rho = \rho^k$, $$\label{keyp}
{\mbox{tr}\left( \nabla F(\rho^k)(\bar{\rho}^k - \rho^k) \right)} = {\mbox{tr}\left( (\rho^k - \bar{\rho}^k) (\rho^k)^{-1} (\rho^k - \bar{\rho}^k) \right)} = {\left\Vert\rho^k - \bar{\rho}^k\right\Vert}_{(\rho^k)^{-1}}^2.$$ Using the continuity of the solution given by Lemma [(\[lemarhobar\])]{}, we have $$\lim_{k \rightarrow \infty, \ k \in {\cal K}} \bar{\rho}^k = \bar{\rho} = \frac{\nabla F(\rho')\rho' + \rho' \nabla F(\rho')}{2}.$$ Taking limits in [(\[keyp\])]{}, we obtain $$\lim_{k \rightarrow \infty} \inf_{k \in {\cal K}} {\mbox{tr}\left( \nabla F(\rho^k)(\bar{\rho}^k - \rho^k) \right)} = {\left\Vert\rho' - \bar{\rho}\right\Vert}_{(\rho')^{-1}}^2 > 0.$$ Since $\rho'$ is non-stationary, the right hand side of the above inequality is strictly positive and this completes the proof.
Finally, using the previous lemmas, we can prove the main assertion of this section.
\[main\_prop\] The sequence of directions ${\left\{D^k\right\}}$ is gradient related.
First, let us show that ${\left\{D^k\right\}}$ is bounded. In fact, $\rho^{k+1}(t_k) = \rho^k + t_k D^k = G(\rho^k)$ is in ${\cal S}$, since $\rho^k \succ 0$ and $t_k \ge 0$, by definition. In particular, for $t_k=1$, we have $\rho^{k+1}(1) = \rho^k + D^k \in {\cal S}$, and since ${\cal S}$ is bounded, then ${\left\{D^k\right\}}$ is also bounded.
Now, let ${\left\{\rho^k\right\}}_{k \in {\cal K}}$ be a subsequence of the sequence ${\left\{\rho^k\right\}}$ generated by the iterations $\rho^{k+1} = \rho^k + t_k D^k$. Suppose ${\left\{\rho^k\right\}}_{k \in {\cal K}}$ converges to a nonstationary point $\rho'$. Using the definition of $D^k$, we obtain $${\mbox{tr}\left( \nabla F(\rho^k) D^k \right)} = \frac{2}{q(t_k)} {\mbox{tr}\left( \nabla F(\rho^k)\bar{D}^k \right)} + \frac{t_k {\mbox{tr}\left( \nabla F(\rho^k) \rho^k \nabla F(\rho^k) \right)}}{q(t_k)} {\mbox{tr}\left( \nabla F(\rho^k) \tilde{D}^k \right)}.$$ The second term in the right hand side is nonnegative, then $${\mbox{tr}\left( \nabla F(\rho^k) D^k \right)} \ge \frac{2}{q(t_k)} {\mbox{tr}\left( \nabla F(\rho^k)\bar{D}^k \right)}.$$ Considering $t_k \in (0,t_{max}]$, we have $${\mbox{tr}\left( \nabla F(\rho^k) D^k \right)} \ge \frac{2}{q(t_{max})} {\mbox{tr}\left( \nabla F(\rho^k)\bar{D}^k \right)}.$$ Taking the limit for a subsequence converging to a nonstationary point, $$\lim_{k \rightarrow \infty} \inf_{k \in {\cal K}} {\mbox{tr}\left( \nabla F(\rho^k) D^k \right)} \ge \frac{2}{q(t_{max})} \lim_{k \rightarrow \infty} \inf_{k \in {\cal K}} {\mbox{tr}\left( \nabla F(\rho^k)\bar{D}^k \right)},$$ and since ${\left\{\bar{D}^k\right\}}$ is gradient related, by Lemma \[lemagr\], $$\lim_{k \rightarrow \infty} \inf_{k \in {\cal K}} {\mbox{tr}\left( \nabla F(\rho^k) D^k \right)} > 0,$$ which implies that ${\left\{D^k\right\}}$ is gradient related.
Thus, choosing the step size $t_k$ at each iteration, such that the Armijo condition [(\[armijo\])]{} is fulfilled, we obtain a globally convergent algorithm following the Theorem \[teogc\].\
\
In this way, we can define the steps of a globally convergent algorithm that uses an inexact line search as the following:\
\
[**Algorithm 1**]{}\
\
[**Step 0.**]{} Given $\rho^0 \succ 0$ such that ${\mbox{tr}\left( \rho^0 \right)}=1$, $t_{max} > 0$ and $0<\alpha_0<\alpha_1<1$, set $k=0$ and $t_0 = t_{max}$.\
\
[**Step 1.**]{} If some stopping criterion is verified, stop. Otherwise, compute the directions $\bar{D}^k$ and $\tilde{D}^k$, defined in [(\[dbar\])]{} and [(\[dtilde\])]{}. Set $t=\max {\left\{1,\ t_{k-1}\right\}}$.\
\
[**Step 2.**]{} Set $$D = \left( \frac{2}{q(t)}\bar{D}^k + \frac{t \, {\mbox{tr}\left( \nabla F(\rho^k) \rho^k \nabla F(\rho^k) \right)}}{q(t)} \tilde{D}^k \right).$$ If $$F(\rho^k + t D) \le F(\rho^k) + \gamma\,t\, {\mbox{tr}\left( \nabla F(\rho^k) D \right)},$$ choose $t \in [\alpha_0\,t, \ \alpha_1\,t]$ and go to Step 2.\
\
[**Step 3.**]{} Set $t_k = t$, $D^k = D$ and $\rho^{k+1} = \rho^k + t_k\, D^k$. Go to the step 1.\
\
The Theorem \[mainteo\] states the desired result, that is, any limit point of the sequence generated by Algorithm 1 is a stationary point, regardless the initial approximation. Since the problem [(\[mlprob\])]{} is convex, then a stationary problem is also a solution.
\[mainteo\] Every limit point $\rho^*$ of a sequence ${\left\{\rho^k\right\}}$, generated by the Algorithm 1, is a stationary point, that is, $\nabla F(\rho^*) \rho^* = \rho^* = \rho^* \nabla F(\rho^*)$.
Using the Proposition \[main\_prop\], we have that ${\left\{D^k\right\}}_k$, used in Algorithm 1, is gradient related. Since the step selection in Algorithm 1 satisfies the Armijo condition, then we can apply the Theorem \[teogc\] to obtain the claimed result.
In the step 2 of Algorithm 1, instead of successive reductions of the steplength $t$, one could use, for instance, a quadratic or cubic interpolation [@nocedal1999] to estimate $t$ that maximizes $F(\rho^k + t D^k)$, in order to turn the search more effective.
Illustrative examples {#examples}
=====================
In this section we selected two illustrative examples to show that Algorithm 1 outperforms the Diluted $R \rho R$ algorithm with fixed stepsize [@rehacek2007]. Besides Algorithm 1 converges in problems where the original $R \rho R$ does not, it also reduces the number of iterations when compared to the fixed stepsize version of the Diluted $R \rho R$, without harming the convergence behavior in cases where the last one works.
First, we consider the counterexample where the pure $R \rho R$ method gets into a cycle [@rehacek2007]. Suppose we made three measurements on a qubit with the apparatus described by $\Pi_0 = {\vert0\rangle}{\langle0\vert}$ and $\Pi_1 = {\vert1\rangle}{\langle1\vert}$, detecting ${\vert0\rangle}$ once and ${\vert1\rangle}$ twice. We used the completely mixed state as starting point and considered convergence when the distance between two consecutive iterates is small enough (less than $10^ {-7}$). For each $t$ fixed in the Diluted $R \rho R$, we define $t_{max}=t$ in the algorithm that uses line search. We also used $\gamma=10^{-4}$ and $\alpha_0=\alpha_1=0.5$ in the Algorithm 1.
\
The Figure \[fig1\] brings the comparison between the version with fixed step size (stars) against the one with line search (circles), described in the previous section. In the left panel, we can see that the number of iterations grows up as the stepsize $t$ increases, for the “fixed $t$” strategy. This was expected because as $t \rightarrow \infty$, the iterations tend to be pure $R \rho R$ iterations, and in this limit case, there is no convergence. Conversely, the line search strategy keeps the number of iterations bounded, regardless the value of $t_{max}$.
The right panel is a zoomed version of the left one, in order to show the behavior for small values of $t$. As expected, although the Diluted $R \rho R$ guarantees the monotonic increase of the likelihood for sufficient small steps, repeating too small steps leads to more iterations of the method. The Algorithm 1 ensures a substantial increase of the likelihood through the line search procedure. To avoid extremely small steps, at each iteration of the Algorithm 1, the first trial for $t_k$ is at least one.
Second, we consider as data the theoretical probabilities for the W state. The Figure \[fig2\] presents the number of iterations for different values of $t$ (log-scale). Again, fixed small values of $t$ will produce a higher number of iterations. It is also important to note, in this example, that the behavior of the line search version is the same as the “fixed $t$” one, as the suggested step length $t_{max}$ increases. This means that in the Algorithm 1, the full step $t_k = t_{max}$ was accepted (fulfills the Armijo condition) in every iteration.
In [@rehacek2007], the authors claim that one should first try a larger value for the step size $t$ and perform Diluted $R \rho R$ iterations with the same value of $t$. If the iterations do not converge, then try a smaller value of $t$. This ad hoc procedure was motivated because the pattern of the Figure \[fig2\] often occurs in practice, and then larger $t$ means less iterations. However, what should be a good guess for a larger value of $t$ in order to ensure few iterations? And if the convergence does not occur, how to choose a smaller value of $t$ to guarantee the convergence? These issues could result in a lot of re-runs until a good value of $t$ can be found, which can change from one dataset to another.
These examples illustrate that the Armijo line search procedure represents an improvement on the Diluted $R \rho R$ algorithm, adjusting the step length $t$ just when necessary, and show that the convergence does not depend on a specific choice of a fixed step length or the starting point.
Final remarks {#final}
=============
We proved the global convergence of the Diluted $R \rho R$ algorithm under a line search procedure with Armijo condition. The inexact line search is a weaker assumption than the exact line search used in convergence proofs of a previous work [@rehacek2007]. Moreover, the proposed globalization by line search does not depend on the guess of a fixed step length for all iterations. Instead, as usual in nonlinear optimization, the step length is adjusted just when necessary in order to ensure a sufficient improvement in the likelihood at each iteration. Thus, the Armijo line search procedure is a reliable globalization and represents a practical improvement in the Diluted $R \rho R$ algorithm for quantum tomography.
Acknowledgements {#acknowledgements .unnumbered}
================
We thanks to the Brazilian research agencies FAPESP, CNPq and INCT-IQ (National Institute for Science and Technology for Quantum Information). DG also thanks the Brittany Region (France) and INRIA for partial financial support.
|
---
author:
- 'Susarla Raghuram[^1] '
- Anil Bhardwaj
date: '****'
title: 'Photochemistry of atomic oxygen green and red-doublet emissions in comets at larger heliocentric distances'
---
.\
[In comets the atomic oxygen green (5577 Å) to red-doublet (6300, 6364 Å) emission intensity ratio ([G/R ratio]{}) of 0.1 has been used to confirm H$_2$O as the parent species producing forbidden oxygen emission lines. The larger ($>$0.1) value of [G/R ratio]{} observed in a few comets is ascribed to the presence of higher CO$_2$ and CO relative abundances in the cometary coma.]{} [We aim to study the effect of CO$_2$ and CO relative abundances on the observed [G/R ratio]{}in comets observed at large ($>$2 au) heliocentric distances by accounting for important production and loss processes of [O($^1$S)]{} and [O($^1$D)]{} atoms in the cometary coma.]{} [Recently we have developed a coupled chemistry-emission model to study photochemistry of [O($^1$S)]{} and [O($^1$D)]{} atoms and the production of green and red-doublet emissions in comets [Hyakutake]{} and [Hale-Bopp]{}. In the present work we applied the model to six comets where green and red-doublet emissions are observed when they are beyond 2 au from the Sun.]{} [The collisional quenching of [O($^1$S)]{} and [O($^1$D)]{} can alter the [G/R ratio]{} more significantly than that due to change in the relative abundances of CO$_2$ and CO. In a water-dominated cometary coma and with significant ($>$10%) CO$_2$ relative abundance, photodissociation of H$_2$O mainly governs the red-doublet emission, whereas CO$_2$ controls the green line emission. If a comet has equal composition of CO$_2$ and H$_2$O, then $\sim$50% of red-doublet emission intensity is controlled by the photodissociation of CO$_2$. The role of CO photodissociation is insignificant in producing both green and red-doublet emission lines and consequently in determining the [G/R ratio]{}. Involvement of multiple production sources in the [O($^1$S)]{} formation may be the reason for the observed higher green line width than that of red lines. The [G/R ratio]{} values and green and red-doublet line widths calculated by the model are consistent with the observation.]{} [Our model calculations suggest that in low gas production rate comets the [G/R ratio]{} greater than 0.1 can be used to constrain the upper limit of CO$_2$ relative abundance provided the slit-projected area on the coma is larger than the collisional zone. If a comet has equal abundances of CO$_2$ and H$_2$O, then the red-doublet emission is significantly ($\sim$50%) controlled by CO$_2$ photodissociation and thus the [G/R ratio]{} is not suitable for estimating CO$_2$ relative abundance.]{}
Introduction
============
Green (5577 Å) and red-doublet (6300, 6364 Å) emissions are due to the electronic transition of oxygen atoms from metastable $^1$S and $^1$D states, respectively, to the ground $^3$P state. Since [O($^1$S)]{} and [O($^1$D)]{} are metastable states, resonance fluorescence by solar photons is not an effective excitation mechanism for populating these states. Dissociative excitation of O-bearing neutrals by photons and photoelectrons, and thermal recombination of atomic oxygen constituted ions are the sources of these metastable states in the cometary coma [@Bhardwaj02; @Bhardwaj12; @Raghuram13]. Most of the comets observed around heliocentric distance of 1 au have H$_2$O as the principal constituent in the cometary coma [@Bockelee04]. Based on the theoretical work of [@Festou81] green to red-doublet emission intensity ratio (hereafter [G/R ratio]{}) of 0.1 has customarily been used as a benchmark to confirm the parent source of these prompt emissions as H$_2$O in several comets observed around 1 au from the Sun [@Cochran84; @Cochran08; @Morrison97; @Zhang01; @Cochran01; @Furusho06; @Capria05; @Capria08; @Capria10]. The observed [G/R ratio]{} of more than 0.1 has been attributed to higher relative abundances of CO$_2$ and CO [@Furusho06; @Capria10; @McKay12; @Decock13]. Since no experimental cross section or yield for the production of [O($^1$S)]{}from H$_2$O is available in the literature, the calculated photorates of [@Festou81] have been questioned by [@Huestis06]. In a H$_2$O-dominated cometary coma, the red-doublet emission intensity is determined by formation and destruction rates of [O($^1$D)]{} [@Bhardwaj02; @Bhardwaj12; @Raghuram13]. Since the red-doublet emission is mainly governed by photodissociation of H$_2$O, the observed intensity of 6300 Å has been used to estimate the production rate as well as to understand the spatial distribution of H$_2$O in the cometary coma [e.g. @Delsemme76; @Delsemme79; @Fink84; @Schultz92; @Morgenthaler01; @Furusho06].
We have recently developed a coupled chemistry-emission model for the production of green and red-doublet emissions by accounting for important production and loss mechanisms of [O($^1$S)]{} and [O($^1$D)]{} atoms. The model has been applied to comets [Hyakutake]{} [@Bhardwaj12] and [Hale-Bopp]{} [@Raghuram13]. Our model calculations showed that in a H$_2$O-dominated cometary coma more than 90% of the [O($^1$D)]{} is populated via photodissociative excitation of H$_2$O and the rest through photodissociation of CO$_2$ and CO. We also demonstrated that the [G/R ratio]{} depends not only on the photochemistry involved in the formation of [O($^1$D)]{} and [O($^1$S)]{}, but also on the projected area observed on the comet, which is a function of slit dimension and geocentric distance of the comet [@Bhardwaj12]. The model calculations on comets [Hyakutake]{} and [Hale-Bopp]{} showed that the intensity of the \[OI\] 6300 Å line is largely governed by photodissociation of H$_2$O, whereas the \[OI\] 5577 Å emission line is mainly controlled by the photodissociation of both H$_2$O and CO$_2$. It is also suggested that CO$_2$ can produce [O($^1$S)]{} more efficiently than H$_2$O. The calculated mean excess energy profiles in various photodissociation processes showed that the photodissociation of CO$_2$ can produce [O($^1$S)]{} with higher excess velocity compared to the photodissociation of H$_2$O [@Raghuram13]. All these calculations are carried out at $\sim$1 au.
At larger heliocentric distances the cometary coma is composed of larger proportions of CO and CO$_2$ than at 1 au [@Meech04; @Crovisier99; @Biver97; @Biver99; @Bockelee04; @Bockelee10]. At heliocentric distances of more than 2 au the prompt emissions of atomic oxygen are observed in several comets, viz. [C/2007 Q3 (Siding Spring)]{}, [C/2006 W3 (Christensen)]{}, [C/2009 P1 (Garradd)]{}, [C/2001 Q4 (NEAT)]{}, [116P/Wild 4]{}, and [C/2003 K4 (LINEAR)]{}[@Furusho06; @McKay12; @Decock13]. Assuming that CO$_2$ and CO are the main sources of green and red-doublet emissions, the observed [G/R ratio]{} in comets at large heliocentric distances ($>$ 2 au) has been used to estimate the CO$_2$ abundance in comets [@Decock13; @McKay12].
The present study is aimed at studying the photochemistry of [O($^1$S)]{} and [O($^1$D)]{} atoms and associated green and red-doublet emission production mechanisms in the above mentioned six comets at larger heliocentric distance ($>$ 2 au) where the gas production rate of CO can be equal to that of H$_2$O. One of the objectives of the study is to verify whether the [G/R ratio]{} value can be used to infer the CO$_2$ relative abundance, with respect to H$_2$O, in the comets that are observed at larger heliocentric distances. In this study we have shown that even at large heliocentric distances, the photodissociation of CO is only a minor source of [O($^1$S)]{} and [O($^1$D)]{} atoms, and its impact on the [G/R ratio]{} is negligible. The red-doublet emission intensity is mainly governed by H$_2$O, while the green line emission intensity is controlled by CO$_2$. We have also demonstrated that collisional quenching can significantly change the observed [G/R ratio]{} and that its impact on the [G/R ratio]{} is much greater than that due to variation in the CO$_2$ and H$_2$O abundances.
Model
=====
The details of model and the photochemical reactions considered in the model are presented in our previous works [@Bhardwaj12; @Raghuram13]. Here we present the input parameters that have been used in the model for the calculation of green and red-doublet emission intensities for the observed conditions of six comets (viz. [116P/Wild 4]{}, [C/2003 K4 (LINEAR)]{}, [C/2007 Q3 (Siding Spring)]{}, [C/2006 W3 (Christensen)]{}, [C/2009 P1 (Garradd)]{}, [C/2001 Q4 (NEAT)]{}). The photochemical reaction network and cross sections of photon and electron impact processes that have been considered in our previous work remain the same for the present calculation. The photoelectron impact excitation reactions are accounted for by degrading solar extreme ultraviolet (EUV) generated photoelectrons and electron impact cross sections in the cometary coma using the analytical yield spectrum (AYS) technique which is based on the Monte Carlo method. Details of the AYS approach and the method for calculating photoelectron flux and excitation rates are given in our earlier papers and references therein [@Bhardwaj99a; @Haider05; @Bhardwaj11; @Raghuram12; @Bhardwaj12]. The production and loss mechanisms for the [O($^1$S)]{} and [O($^1$D)]{} considered in the model calculations are presented in our previous papers [@Bhardwaj12; @Raghuram13]. Only the dominant O-bearing neutral species H$_2$O, CO$_2$, and CO are considered in the present model.
The neutral gas production rates used in the model calculations for different comets during observation periods of oxygen emission lines are tabulated in Table \[tab-comet\]. In some comets these gas production rates are not measured, and so we have made a reasonable approximation to incorporate CO$_2$ and CO in the model. However, we vary the CO$_2$ and CO relative abundances on these comets to assess the impact on the green and red-doublet emission intensities and subsequently on the [G/R ratio]{}.
\[1\]
\[tab-comet\]
------------------------------- ------ ---------- ---------------------- ------------------------ -------- ----- ------ ----------------- -------------- -- -- -- -- -- -- --
Comet $r$ $\Delta$ Slit dimension Q(H$_2$O) CO$_2$ CO Reference
(au) (au) ($''$ $\times$ $''$) (s$^{-1}$) (%) (%) cal. obs.
[116P/Wild 4]{} 2.40 1.4 8 $\times$ 1 1 $\times$ 10$^{27}$ 10 20 0.09 0.15 [@Furusho06]
[C/2003 K4 (LINEAR)]{} 2.60 2.36 0.80 $\times$ 11 1 $\times$ 10$^{29}$ 10 25 0.09 0.09 [@Decock13]
[C/2007 Q3 (Siding Spring)]{} 2.96 2.48 3.20 $\times$ 1.6 4 $\times$ 10$^{27}$ 17 10 0.12 0.20 [@McKay12]
[C/2006 W3 (Christensen)]{} 3.13 2.35 3.20 $\times$ 1.6 2.0 $\times$ 10$^{28}$ 42 98 0.18 0.24 $\pm$ 0.08 [@McKay12]
[C/2009 P1 (Garradd)]{} 3.25 3.50 0.44 $\times$ 12 2.3 $\times$ 10$^{27}$ 25 100 0.14 0.21 [@Decock13]
[C/2001 Q4 (NEAT)]{} 3.70 3.40 0.45 $\times$ 11 3.8 $\times$ 10$^{27}$ 75 100 0.23 0.33 [@Decock13]
comet X 3.70 3.40 0.45 $\times$ 11 4 $\times$ 10$^{27}$ 100 100 0.25 – –
------------------------------- ------ ---------- ---------------------- ------------------------ -------- ----- ------ ----------------- -------------- -- -- -- -- -- -- --
[@Furusho06] observed the forbidden oxygen lines in comet [116P/Wild 4]{} when it was at 2.4 au from the Sun. Using the infrared satellite AKARI, [@Ootsubo12] measured the H$_2$O production rate in this comet as $\sim$1 $\times$ 10$^{27}$ s$^{-1}$ and abundance of CO$_2$ was found to be 10% relative to the water at heliocentric distance of 2.22 au. [@Ootsubo12] also determined the upper limit for CO abundance in this comet as 20% relative to water. In our model we have used these measured gas production rates and relative abundances as input assuming that these values did not vary significantly in this comet from 2.2 au to 2.4 au.
Using the SPITZER space telescope, [@Woodward07] measured the H$_2$O production rate in comet [C/2003 K4 (LINEAR)]{} as 2.43 $\times$ 10$^{29}$ s$^{-1}$ when the comet was at 1.76 au from the Sun during pre-perihelion. [@Decock13] observed atomic oxygen forbidden lines in this comet when it was at 2.6 au heliocentric distance. Since the H$_2$O production rate was not measured at 2.6 au we scaled the [@Woodward07] measured H$_2$O production rate to heliocentric distance of 2.6 au assuming that it varies as the inverse square of heliocentric distance. However, we evaluate the impact of the estimated H$_2$O production rate on the calculated G/R by decreasing its value by a factor of 2. Since CO$_2$ and CO are not observed in this comet we have assumed their abundances to be 10% and 25% relative to H$_2$O, respectively. We show that CO does not play a significant role in determining green and red-doublet emission line intensities, whereas the CO$_2$ abundance is important in determining the [G/R ratio]{}.
In comet [C/2007 Q3 (Siding Spring)]{}, only the \[OI\] 6300 Å emission line was observed and the intensity of \[OI\] 5577 Å was estimated with a 3$\sigma$ upper limit when it was at a heliocentric distance of 2.96 au [@McKay12]. The AKARI satellite detected both H$_2$O and CO$_2$ in comet [C/2007 Q3]{} during its pre-perihelion period and measured the CO$_2$ relative abundance as 17% relative to H$_2$O production rate when the comet was at a heliocentric distance of 3.3 au [@Ootsubo12]. Assuming that the photodissociation of H$_2$O is the major source for the observed \[OI\] 6300 Å emission, [@McKay12] inferred the H$_2$O production rate in comet [C/2007 Q3]{} as 1.8 $\times$ 10$^{27}$ s$^{-1}$ which is smaller by a factor of 2 than the [@Ootsubo12] measurement. [@Ootsubo12] observation covers a larger (43$''$ $\times$ 43$''$) projected area on the coma which can account for most of the H$_2$O produced from extended distributed sources like icy grains in the coma in the observed field of view compared to that of the [@McKay12] observation (3.2$'' \times$ 1.62$''$). Hence, we have used the [@Ootsubo12] measured gas production rates in the model. We have taken the H$_2$O production rate on comet [C/2007 Q3]{} as 4 $\times$ 10$^{27}$ s$^{-1}$ with 17% and 10% relative abundances of CO$_2$ and CO with respect to water, respectively, in our model.
By making radio observations on comet [C/2006 W3 (Christensen)]{}, [@Bockelee10] derived H$_2$O and CO production rates as 4.2 $\times$ 10$^{28}$ and 3.9 $\times$ 10$^{28}$ s$^{-1}$, respectively. During this measurement the comet was at a heliocentric distance of 3.2 au. These values are higher by a factor of 2 compared to the infrared satellite observed values reported by [@Ootsubo12] which were derived when the comet was nearly at the same heliocentric distance. [@Ootsubo12] reported 42% and 98% of CO$_2$ and CO abundances relative to H$_2$O, respectively, in this comet when it was at 3.13 au. During the green and red-doublet emission observation, comet [C/2006 W3]{} was at a heliocentric distance of 3.13 au [@McKay12]. For this comet we have used the [@Ootsubo12] measured H$_2$O production rate as well as CO$_2$ and CO relative abundances in our model.
The H$_2$O production rate in comet [C/2009 P1 (Garradd)]{} beyond 2 au has been reported by various workers [@Paganini12; @Villanueva12; @Bodewits12b; @Combi13; @Farnham12; @Feaga12]. Using the SWIFT satellite, [@Bodewits12b] observed the OH 3080 Å emission line in comet [C/2009 P1]{} and derived the H$_2$O production rate when it was between 2 au and 4 au heliocentric distances. We have taken H$_2$O production at 3.25 au from the Sun as 2.3 $\times$ 10$^{27}$ s$^{-1}$ by linearly interpolating the [@Bodewits12b] derived production rates between 3 au and 3.5 au heliocentric distances. [@Decock13] used the observed [G/R ratio]{} at 3.25 au and estimated that around 25% CO$_2$ abundance relative to H$_2$O was present in this comet. We assumed 25% CO$_2$ relative abundance in the coma of comet [C/2009 P1]{} in the model. Since CO is highly volatile and the comet is at a large heliocentric distance we assumed that the gas production rates for H$_2$O and CO are equal in this comet.
In comet [C/2001 Q4 (NEAT)]{}, the H$_2$O production rate is measured by [@Biver09] and [@Combi09] at different heliocentric distances using hydrogen Ly-$\alpha$ (1216 Å) and radio (557 GHz) emissions, respectively. [@Combi09] fitted the observed H$_2$O production rate as a function of heliocentric distance (r$_h$) as 3.5 $\times$ 10$^{29}$ $\times$ r$_h^{-1.7}$ s$^{-1}$. We used this expression to calculate the H$_2$O production rate in this comet at 3.7 au where the green and red-doublet emissions were observed [@Decock13]. Since the comet is at a large heliocentric distance we assumed that the CO and H$_2$O abundances are equal. Based on the observed [G/R ratio]{} on this comet, [@Decock13] suggested that the CO$_2$ relative abundance in this comet could be between 60% and 80% with respect to H$_2$O. In our model we have assumed the CO$_2$ relative abundance at 3.7 au heliocentric distance to be 75%.
To evaluate the individual contributions of major O-bearing species in producing green and red-doublet emissions and their affect on the [G/R ratio]{} we have made a case study for a hypothetical comet X in which we assumed equal gas production of H$_2$O, CO$_2$, and CO in the comet. This is similar to the observation of [@Ootsubo12] on comet [C/2006 W3]{} in which it is found that CO$_2$ and H$_2$O gas production rates are equal ($\sim$8 $\times$ 10$^{27}$ s$^{-1}$); however, the CO production rate is around 3 times higher when the comet was at 3.7 au from the Sun.
The solar flux, which is required to calculate photorates of different species, is taken from the SOLAR2000 (S2K) v.2.36 model of [@Tobiska04] at 1 au and scaled according to the observed heliocentric distance of different comets. The electron temperature that determines the dissociative recombination rates of ions is taken as constant 300 K in the cometary coma. The effect of this constant temperature profile on the model calculation is discussed later. The yield of [O($^1$S)]{} at solar H Ly-$\alpha$ in the photodissociation of H$_2$O is taken as 0.5%. The impact of this assumption was discussed in our previous work [@Bhardwaj12]. The photodissociative excitation cross section for CO$_2$ producing [O($^1$D)]{} is taken from [@Jain12b]. The photorate for the production of [O($^1$S)]{} from the photodissociation of CO has been taken from the theoretically estimated value of [@Festou81] and scaled to the observed heliocentric distance.
Results
=======
Since these comets have different water production rates (varying from 10$^{27}$ to 10$^{29}$ s$^{-1}$), as well as different CO$_2$ and CO relative abundances with respect to H$_2$O, we present calculations in comet [C/2006 W3]{} which is followed by a discussion on the calculated results for other comets.
Production processes of [O($^1$S)]{} and [O($^1$D)]{}
-----------------------------------------------------
The calculated production rates for the [O($^1$S)]{} from different processes in comet [C/2006 W3]{} are presented in Fig. \[fig:o1s-prod\]. The major production source of oxygen atoms in the $^1$S metastable state is photodissociation of CO$_2$ followed by photodissociation of CO and H$_2$O. The contribution from the photoelectron impact excitation reactions is smaller compared to photodissociative excitation processes. Above 10$^3$ km, the dissociative recombination of CO$_2^+$ also contributes significantly. The solar flux in wavelength bin 955–1165 Å is the main source that dissociates CO$_2$ and produces atomic oxygen in the $^1$S state. Since the yield for photodissociation of CO$_2$ in this wavelength bin is almost unity, the absorption of solar photons of this wavelength bin by CO$_2$ leads to the formation of O($^1$S) and CO [@Raghuram13; @Bhardwaj12].
![Calculated volumetric [O($^1$S)]{} production rate profiles for major production mechanisms in comet [C/2006 W3 (Christensen)]{} with H$_2$O production rate of 2 $\times$ 10$^{28}$ s$^{-1}$ and 42% CO$_2$ and 98% CO abundances relative to H$_2$O in the cometary coma when the comet was at 3.13 au heliocentric distance. h$\nu$: solar photon; e$_{ph}$: photoelectron; and e$_{th}$: thermal electron.[]{data-label="fig:o1s-prod"}](fig1){width="22pc"}
The calculated [O($^1$D)]{} production rate profiles for different mechanisms are shown in Fig. \[fig:1d-prod\]. The major production of [O($^1$D)]{} is via photodissociation of H$_2$O, but close to the nucleus ($<$30 km) photodissociation of CO$_2$ is also a significant [O($^1$D)]{} production process, and above 30 km the radiative decay of [O($^1$S)]{} becomes a more important source of [O($^1$D)]{} than the former. The photodissociation of CO and OH are minor production sources of [O($^1$D)]{}. Most of the [O($^1$D)]{} production ($>$95%) is due to photodissociation of H$_2$O by solar H Ly-$\alpha$ photon flux.
![Calculated volumetric [O($^1$D)]{} production rate profiles for major production mechanisms in comet [C/2006 W3 (Christensen)]{} with H$_2$O production rate of 2 $\times$ 10$^{28}$ s$^{-1}$ and 42% CO$_2$ and 98% CO relative abundances with respect to H$_2$O in the cometary coma when the comet was at 3.13 au from the Sun. h$\nu$: solar photon.[]{data-label="fig:1d-prod"}](fig2){width="22pc"}
Loss processes of [O($^1$S)]{} and [O($^1$D)]{}
-----------------------------------------------
The calculated [O($^1$S)]{} and [O($^1$D)]{} destruction rate profiles in comet [C/2006 W3]{} are presented in Fig. \[fig:1ds-los\]. Since this comet has a low neutral gas production rate, the collisional quenching is a dominant [O($^1$S)]{} destructive mechanism only close to the nucleus ($<$30 km). The radiative decay which produces photons at 5577 Å and 2972 Å is the major loss process for [O($^1$S)]{} throughout the coma. The calculated loss rate profiles of [O($^1$D)]{} by various processes are also presented in the same figure. Below 300 km, quenching by H$_2$O and CO$_2$ are the dominant loss mechanisms of the [O($^1$D)]{}. Above 300 km, the radiative decay, which results in the emission of photons at 6300 Å and 6364 Å, is a major loss process for [O($^1$D)]{}. Quenching by CO is a minor loss process for [O($^1$D)]{} which is not shown in the figure.
![Calculated radial loss rate profiles for major loss mechanisms of the [O($^1$D)]{} and [O($^1$S)]{} in comet [C/2006 W3 (Christensen)]{} with H$_2$O production rate of 2 $\times$ 10$^{28}$ s$^{-1}$ and 42% CO$_2$ and 98% CO abundances relative to H$_2$O in the cometary coma when the comet was at 3.13 au from the Sun.[]{data-label="fig:1ds-los"}](fig3){width="22pc"}
The calculated number density profiles of [O($^1$S)]{} and [O($^1$D)]{} in comet [C/2006 W3]{} along with parent species H$_2$O, CO$_2$, and CO are presented in Fig. \[fig:nden\]. Close to the cometary nucleus the flatness in the calculated [O($^1$S)]{} and [O($^1$D)]{} number density profiles is due to collisional quenching by cometary species (mainly H$_2$O) and depends on the neutral gas production rate of the comet.
![Calculated number density profiles of [O($^1$S)]{}, [O($^1$D)]{}, and OH, along with those of H$_2$O, CO, and CO$_2$ in comet [C/2006 W3 (Christensen)]{} with H$_2$O production rate of 2 $\times$ 10$^{28}$ s$^{-1}$ and 42% CO$_2$ and 98% CO relative abundances with respect to H$_2$O in the cometary coma when the comet was at 3.13 au from the Sun.[]{data-label="fig:nden"}](fig4){width="22pc"}
\[OI\] green to red-doublet emission intensity ratio and line widths
--------------------------------------------------------------------
The calculated number density profiles shown in Fig. \[fig:nden\] are multiplied with Einstein emission transition probabilities [see Table 1 in @Raghuram13] to obtain emission rates. By integrating these emission rates along the line of sight we calculated the emission intensities of green and red-doublet lines as a function of projected distance. The calculated surface brightness profiles for \[OI\] 5577 Å and red-doublet (6300 + 6364 Å) emissions are shown in Fig. \[fig:rg-inten\] with solid curves. It can be seen in this figure that close to the nucleus (below 40 km projected distance) the green line emission is more intense than the red-doublet emission, which is mainly due to the [O($^1$S)]{} emission rate (1.26 s$^{-1}$) being higher by about two orders of magnitude compared to that of [O($^1$D)]{} (8.59 $\times$ 10$^{-3}$ s$^{-1}$). The calculated [G/R ratio]{} in comet [C/2006 W3]{}, which is shown with a dashed curve (”with CO”) in Fig. \[fig:rg-inten\], varies between 1.8 and $\sim$0.2. In the same figure the calculated [G/R ratio]{} profiles for different cases are also presented. Since there is an uncertainty in the photo-rate of CO in producing [O($^1$S)]{}, which is discussed later, we also did calculations for the [G/R ratio]{} neglecting this source mechanism which is shown in Fig. \[fig:rg-inten\] with dotted curve (”without CO”). In this case the calculated [G/R ratio]{} varies between 1.6 and 0.18. Since comet [C/2006 W3]{} has a very low gas production rate the collisional quenching may be less important. To assess the effect of collisional quenching on the green and red-doublet emissions, we calculated the [G/R ratio]{} without considering collisional destruction mechanisms of [O($^1$S)]{} and [O($^1$D)]{}. In this case the calculated [G/R ratio]{} is a constant value of 0.18 throughout the coma which is represented with the dash-dotted line in Fig. \[fig:rg-inten\].
Similarly, all these calculations have been carried out on other comets. Considering both collisional quenching and photodissociation of CO, the calculated [G/R ratio]{} profiles as a function of projected distance in six comets are presented in Fig. \[fig:grat-prj\]. This figure shows that close to the nucleus in comets [C/2006 W3]{} and [C/2001 Q4]{}, the calculated [G/R ratio]{} value is more than one which is due to higher CO$_2$ relative abundances and strong collisional quenching of [O($^1$D)]{} by cometary species, whereas in other comets this value is always less than one throughout the coma. In comets [C/2006 W3]{} and [C/2001 Q4]{}, the CO$_2$ abundances are very large (see Table \[tab-comet\]) and no significant collisional quenching of [O($^1$S)]{}. Thus, the green line intensity throughout the coma is determined by CO$_2$ and subsequently the [G/R ratio]{}governed by the quenching of [O($^1$D)]{} depending on the H$_2$O production rate. In other comets the [G/R ratio]{} is small because of lower CO$_2$ abundances compared to former comets.
![Calculated \[OI\] red-doublet (6300+6364 Å) and 5577 Å line brightness profiles (with solid curves) along the cometocentric projected distances on comet [C/2006 W3 (Christensen)]{} with H$_2$O production rate of 2 $\times$ 10$^{28}$ s$^{-1}$ and 42% CO$_2$ and 98% CO relative abundances with respect to H$_2$O in the cometary coma when the comet was at 3.13 au from the Sun. The calculated G/R ratio profiles considering CO, without considering CO, and without quenching are plotted with dashed, dotted, and dash-dotted curves, respectively, on the right y-axis.[]{data-label="fig:rg-inten"}](fig5){width="22pc"}
![The calculated radial profiles of the [G/R ratio]{} in different comets. The input parameters used to calculate the [G/R ratio]{} are tabulated in Table \[tab-comet\]. It can be seen that in comets [C/2006 W3]{} and [C/2001 Q4]{}, due to substantial collisional quenching of [O($^1$S)]{} and [O($^1$D)]{} with other cometary species, the calculated [G/R ratio]{} is more than 1 closer to the nucleus, whereas above 400 km projected distances it is a constant.[]{data-label="fig:grat-prj"}](fig6){width="22pc"}
We calculated the average [G/R ratio]{} over the observed projected area on each comet. The projected area on a comet changes with the dimension of slit used for observation and the geocentric distance of comet. The calculated averaged G/R ratios on different comets are tabulated in Table \[tab-comet\] along with the values derived from observations. Our calculated [G/R ratio]{} values are comparable (within a factor of two) with the observations on different comets observed at different heliocentric and geocentric distances.
The percentage contributions for various production processes involved in the formation of [O($^1$S)]{} and [O($^1$D)]{} in these comets at three different projected distances are given in Table \[tabprj\]. These calculations suggest that in all these comets, the photodissociation of H$_2$O and CO$_2$ together produce 50–80% of [O($^1$S)]{}, whereas, irrespective of CO$_2$ and CO relative abundances, the major ($\sim$50 to 80%) source for the formation of [O($^1$D)]{} is photodissociation of H$_2$O followed by radiative decay of [O($^1$S)]{} (10–15%). At larger projected distances ($>$10$^3$ km), dissociative recombination processes of H$_2$O$^+$ and CO$_2^+$ ions are also important production sources of [O($^1$S)]{} (30–40%) and [O($^1$D)]{} ($\sim$20%).
\[0.9\]
\[tabprj\]
[@lcccccccccccccccccccc@]{}
&\
Comet& & & & & &\
\[2pt\] &10$^2$&10$^3$&10$^4$&10$^2$&10$^3$&10$^4$&10$^2$&10$^3$&10$^4$&10$^2$&10$^3$&10$^4$ &10$^2$&10$^3$&10$^4$&10$^2$&10$^3$&10$^4$\
\[5pt\] [116P/Wild 4]{}& 43 & 33 & 25 & 33 & 25 & 20 & 9 & 7 & 5 & & & & 5 & 15 & 22 & 5 & 15 & 17\
& (83)& (71) & (59) & (3) & (2) & (2) & (0.5) & (0.5) & (0.5) &(8)&(9)&(9) & (4) & (14) & (22) & (0.1) & (0.7)& (2)\
\[8pt\] [C/2003 K4 (LINEAR)]{}& 46 & 42 & 32 & 35 & 32 & 25 & 11 & 11 & 9 & & & & 1 & 4 & 13 & 1 & 5 & 14\
& (88) & (83) & (71) & (3) & (3) & (2)& (1) & (1)& (1) &(7)&(8)& (9) & (0.5) & (3) & (13) & (0.5) & (0.5) & (1)\
\[8pt\] [C/2007 Q3 (Siding Spring)]{}& 35 & 29 & 21 & 46 & 37 & 28 & 4 & 3 & 2 & & & & 2 & 18 & 17 & 4 & 15 & 23\
& (82) & (73) & (60) & (4) & (4) & (3)& (0.5) & (0.5)& (0.5) &(9)&(10)&(11) & (2) & (9) & (20) & (0.5) & (2) & (3)\
\[8pt\] [C/2006 W3 (Christensen)]{}& 18 & 15 & 11 & 55 & 47 & 37 & 17 & 15 & 11 & & & & 0.5 & 3 & 8 & 3 & 15 & 28\
& (69) & (61) & (48) & (9) & (8)& (7)& (3) & (2)& (2) & (15) & (16) & (17) & (1) & (6) & (15) & (0.5) & (3) & (5)\
\[8pt\] [C/2009 P1 (Garradd)]{}& 22 & 17 & 14 & 41 & 32 & 28 & 22 & 17 & 15 & & & & 3 & 9 & 13 & 8 & 20 & 23\
& (71) & (58) & (47) & (6) & (5) & (4) & (3) & (3)& (2) & (13) &(14) & (13) & (4) & (14) & (19) & (1) & (3) & (3)\
\[8pt\] [C/2001 Q4 (NEAT)]{}& 11 & 9 & 7 & 63 & 50 & 40 & 11 & 9 & 7 & & & & 1 & 4 & 6 & 7 & 23 & 34\
& (57) & (47) & (37) & (13) & (11) & (9)& (2) & (2)& (2) & (21) & (22) & (22) & (2) & (8) & (14) & (1) & (5) & (8)\
\[10pt\]
comet X& 8 & 6 & 5 & 63 & 47 & 41 & 8 & 6 & 5 & & & & 1 & 3 & 5 & 12 & 30 & 37\
& (48) & (38) & (31) & (15) & (12) & (10)& (2) & (1)& (1) & (23) & (25) & (24) & (3) & (10) & (13) & (3) & (8) & (10)\
The calculated percentage contributions of different production processes in the total intensity of \[OI\] emissions over the observed coma on these comets are tabulated in Table \[tab-slit\]. These calculations suggest that photodissociation of CO$_2$ and H$_2$O together contribute 50–70% to the green line emission and the remaining contribution is through dissociative recombination of H$_2$O$^+$ and CO$_2^+$ ions. In the case of red-doublet emission, photodissociation of H$_2$O and radiative decay of [O($^1$S)]{}together produce 70–90% and contributions from other sources are very small.
In the case of hypothetical comet X which has equal H$_2$O, CO$_2$, and CO, gas production rates, $\sim$80% of green line emission intensity is governed by CO$_2$ (via photodissociation of CO$_2$ and dissociative recombination of CO$_2^+$), whereas photodissociation of H$_2$O and CO together contribute around 10%. Dissociative recombination of CO$_2^+$ is the second important source and contributes around 30% to the total green line emission. In this case around 35% of red-doublet emission is produced via H$_2$O photodissociation. The production of [O($^1$D)]{} via CO$_2$ photodissociation is around 10% of the total while it is $\sim$25% via radiative decay of [O($^1$S)]{}, which is also essentially produced from CO$_2$. In this case both CO$_2$ and H$_2$O play equally important roles in producing red-doublet emission.
We also calculated the mean excess energy released in these photodissociative excitation reactions. The maximum excess energy in photodissociation of H$_2$O producing [O($^1$S)]{} by solar Ly-$\alpha$ photons is 1.27 eV, whereas the mean excess energy in the photodissociation of CO$_2$ forming [O($^1$S)]{} is 2.55 eV. Mean excess energies in photodissociative excitation of H$_2$O, CO$_2$, and CO producing [O($^1$D)]{} are 2.12, 4.46, and 2.54 eV, respectively. We assumed that most of these excess energies will result in kinetic motion of daughter products. Thus, the excess velocities of [O($^1$S)]{} in photodissociative excitation of H$_2$O and CO$_2$ are 1.3 km s$^{-1}$ and 4.4 km s$^{-1}$, respectively. Similarly, the calculated excess velocities of [O($^1$D)]{} in photodissociation of H$_2$O, CO$_2$, and CO are 1.6, 5.8, and 3.6 km s$^{-1}$, respectively.
Considering only photoreactions and using the calculated contributions of each process over the cometary coma (see Table \[tab-slit\]) we calculated the mean excess energies of [O($^1$S)]{} and [O($^1$D)]{}. Our calculated mean velocities of [O($^1$S)]{} and [O($^1$D)]{} atoms on these comets are tabulated in Table \[tab-slit\] along with the derived velocities based on the observed line widths. In comets having large CO$_2$ relative abundances the width of the green line, which is a function of mean [O($^1$S)]{} velocity, is mainly determined by photodissociation of CO$_2$. Since the mean excess energy released in photodissociation of CO$_2$ is higher, the width of the green line would be greater compared to the red-doublet emission line width (which is mainly determined by photodissociation of H$_2$O). Our calculated green line widths in different comets, which are presented in Table \[tab-slit\], are higher than the calculated red-doublet emission line widths, which is consistent with the observations.
\[0.95\]
\[tab-slit\]
--------------------------------------- --------- -------- --------- ------ --------- --------- ------ ----------- ------ ----------- -- -- -- -- -- -- -- -- -- -- -- -- -- --
\[2pt\] Cal Obs Cal Obs
[116P/Wild 4]{} 34 (72) 26 (2) 7 (0.5) (9) 14 (13) 14 (1) 1.70 – 1.32 –
\[2pt\] [C/2003 K4 (LINEAR)]{} 40 (81) 31 (3) 10 (1) (8) 6 (5) 7 (0.5) 2.05 2.38–2.76 1.87 1.81–2.12
\[2pt\] [C/2007 Q3 (Siding Spring)]{} 28 (72) 37 (4) 3 (0.5) (10) 9 (10) 15 (1) 2.04 – 1.44 –
\[2pt\] [C/2006 W3 (Christensen)]{} 14 (61) 47 (8) 14 (3) (16) 3 (6) 15 (3) 2.48 – 1.58 –
\[2pt\] [C/2009 P1 (Garradd)]{} 16 (55) 31 (4) 17 (3) (13) 10 (15) 21 (3) 1.85 2.16–2.54 1.25 1.25–1.67
\[2pt\] [C/2001 Q4 (NEAT)]{} 8 (44) 47(11) 8 (2) (22) 4 (9) 26 (6) 2.30 2.31–2.55 1.65 2.39–2.75
\[2pt\] comet X 6 (36) 46(11) 6 (1) (25) 4 (10) 31 (8) 2.35 – 1.65 –
--------------------------------------- --------- -------- --------- ------ --------- --------- ------ ----------- ------ ----------- -- -- -- -- -- -- -- -- -- -- -- -- -- --
Depending on the composition and activity of the nucleus, comets have different gas production rates at different heliocentric distances. In order to appraise the collisional quenching of [O($^1$S)]{} and [O($^1$D)]{} with increase in H$_2$O production rate, we calculated the radiative efficiencies of [O($^1$S)]{} and [O($^1$D)]{} for different water production rates. The calculated radiative efficiency profiles are shown in Fig. \[fig:rad-eff\]. This calculation shows that for a given water production rate the [O($^1$D)]{} is always much more quenched than that of [O($^1$S)]{}. This is mainly because the lifetime of [O($^1$D)]{} ($\sim$120 s) is larger by two orders of magnitude than that of [O($^1$S)]{} ($\sim$0.8 s).
![The calculated radiative efficiency profiles of [O($^1$S)]{} (green lines) and [O($^1$D)]{} (red lines) for different production rates with 5% CO$_2$ and 10% CO relative abundances with respect to H$_2$O. The circles, squares, triangles, and cross symbols represent the calculated radiative efficiencies for the water production rates of 10$^{26}$, 10$^{28}$, 10$^{30}$, and 10$^{31}$, respectively.[]{data-label="fig:rad-eff"}](fig7){width="22pc"}
Since CO$_2$ is a potentially important source of [O($^1$S)]{} we have also carried out model calculations of the [G/R ratio]{} on different water production rates by varying its relative abundance from 0% to 100% with respect to H$_2$O. The calculations presented in Fig. \[fig:gr-co2\] suggests that by increasing CO$_2$ relative abundance in a comet the [G/R ratio]{} increases almost monotonically.
![The calculated [G/R ratio]{} as a function of CO$_2$ abundance for different water production rates, with 10% CO abundance relative to H$_2$O in the coma. These calculations are done at heliocentric and geocentric distances of 1 au using a square slit of side 5$''$.[]{data-label="fig:gr-co2"}](fig8){width="22pc"}
Discussion
==========
For comets close to 1 au from the Sun the dominant species in the cometary coma is H$_2$O. Because of lower ice-sublimation temperatures of CO and CO$_2$, at large heliocentric distances the cometary coma is dominantly composed of CO$_2$ and CO [@Meech04; @Crovisier99; @Biver97; @Biver99; @Bockelee04; @Bockelee10]. Owing to strong absorption of cometary H$_2$O infrared emission lines by terrestrial water molecules, it is difficult to detect H$_2$O in the coma for ground-based observations, but the spatial profiles of water can be easily derived in comets from ground-based observatories by observing infrared H$_2$O non-resonance fluorescence emissions [@Mumma95; @Mumma96; @Russo00]. Since H$_2$O does not have any transitions in the visible region the emissions of its daughter products have been used as tracers to understand the spatial distribution of water in the cometary coma. The observed atomic oxygen visible emissions (viz. O\[I\] 5577, 6300, and 6364 Å) have been used to quantify the H$_2$O production rate in several comets around 1 au [@Delsemme76; @Delsemme79; @Fink84; @Schultz92; @Morgenthaler01]. Since CO$_2$ and CO can also produce these metastable oxygen atoms, based on the theoretical work of [@Festou81], the [G/R ratio]{} of 0.1 has been used as the benchmark to confirm H$_2$O as the parent species for these oxygen emission lines. The available theoretical and experimental cross sections for the production of [O($^1$S)]{} and [O($^1$D)]{} from different O-bearing species have been reviewed in our previous work [@Bhardwaj12]. Our coupled chemistry-emission model, which has been applied to comets [Hyakutake]{} and [Hale-Bopp]{}, suggested that the observed [G/R ratio]{} not only depends on the relative abundances of CO$_2$ and CO, but also on the projected area observed on the comet [@Bhardwaj12].
Since CO$_2$ does not emit ultraviolet or visible photons we cannot detect this molecule directly in the cometary ultraviolet or visible spectra. Moreover, CO$_2$ is a symmetric molecule with no permanent dipole moment and so it is difficult to observe this molecule even in radio range from the ground [@Ootsubo12]. Thus, this molecule is probed using indirect methods using the emissions of its dissociative products, like the CO Cameron band (a$^3\Pi$–X$^1\Sigma^+$) in ultraviolet [@Weaver94; @Feldman97] and visible atomic oxygen green and red-doublet emissions [@Furusho06; @McKay12; @Decock13]. Our earlier works [@Bhardwaj11; @Raghuram12] have shown that CO Cameron band emission is not suitable for measuring CO$_2$ abundances in comets since this emission is mainly governed by photoelectron impact excitation of CO rather than the photodissociation of CO$_2$.
Assuming that the green line emission is governed by photodissociation of CO$_2$ while the red-doublet emission is controlled by photodissociation of H$_2$O, the observed [G/R ratio]{} has been used to quantify CO$_2$ relative abundance in the comets [@McKay12; @Decock13]. At larger heliocentric distances CO$_2$ and CO are the dominant O-bearing species in the coma which can produce green and red-doublet emissions. In several comets the observed [G/R ratio]{} at large ($>$2 au) heliocentric distances is more than 0.1 [@Decock13; @McKay12; @Furusho06].
Impact of CO on the [G/R ratio]{} {#sec:roco}
---------------------------------
At larger heliocentric distances, although CO abundance is substantial in the cometary comae, the photodissociation of CO is not a potential source of [O($^1$S)]{} and [O($^1$D)]{} atoms. This is mainly due to the proximity in the threshold energies of photodissociative excitation and photoionization of CO molecules. The threshold energies for dissociation of CO into [O($^1$D)]{} and [O($^1$S)]{} states are 14.35 and 16.58 eV, respectively, whereas it is 14 eV for ionization. Moreover, the branching ratio of ionization for the photons having energy more than 14 eV is $\sim$0.98 [@Huebner92]. Since the ionization energy is smaller than energy required in the formation of [O($^1$S)]{} and [O($^1$D)]{}, most of the photons ($>$90%) having energy $>$14 eV ionize the CO molecule rather than causing photodissociative excitation. Based on [@Huebner79] compiled cross sections, [@Festou81] estimated that the photodissociation of CO produces [O($^1$S)]{} and [O($^1$D)]{} with nearly equal rates. To evaluate the role of CO we also did calculations in comet [C/2006 W3]{} by discarding photodissociation of CO as a source mechanism of both [O($^1$S)]{} and [O($^1$D)]{} (see Fig. \[fig:rg-inten\]). Though CO production rate is equal to that of H$_2$O in this comet (see Table \[tab-comet\]), by removing CO contribution the calculated [G/R ratio]{} decreased by a maximum of about 10%. Similarly, our calculated percentage contribution over the observed coma on different comets, which is presented in Table \[tab-slit\], also suggests that the role of CO is very small ($<$20%) in producing [O($^1$S)]{} and [O($^1$D)]{} atoms and subsequently in determining the red-doublet emission intensity. Even without considering photodissociation of CO in the model the calculated [G/R ratio]{} values are in agreement with the observations. Based on these calculations we can suggest that the photodissociation of CO is an insignificant source of [O($^1$S)]{} and [O($^1$D)]{}. Hence, the photodissociation of CO has almost no impact on the [G/R ratio]{}.
Impact of CO$_2$ on the [G/R ratio]{}
-------------------------------------
The relative abundance of CO$_2$ with respect to H$_2$O, is very important in determining the [G/R ratio]{}. This can be understood from the calculated [G/R ratio]{} profiles on comets [116P]{}, [C/2007 Q3]{}, and [C/2001 Q4]{}, which have nearly same H$_2$O production rates (1–4 $\times$ 10$^{28}$ s$^{-1}$), but different relative abundances of CO$_2$ and CO, and are observed at different heliocentric distances (see Fig. \[fig:grat-prj\]). As discussed in Section \[sec:roco\], the CO abundance does not have any appreciable impact on the [G/R ratio]{}. Hence, the change in the calculated [G/R ratio]{} on these comets can be ascribed mainly to the difference in CO$_2$ relative abundances. The calculated [G/R ratio]{} profiles on these comets are shown in Fig. \[fig:grat-prj\], which shows that by increasing CO$_2$ the [G/R ratio]{}increases. In comet [C/2001 Q4]{}, due to higher (75%) CO$_2$ relative abundance, the calculated [G/R ratio]{} value is more than one close to the cometary nucleus. Similar behaviour is seen for comet [C/2006 W3]{} which is due to larger ($\sim$40%) CO$_2$ relative abundance with respect to H$_2$O, and also due to significant collisional quenching of [O($^1$D)]{} (see Fig. \[fig:rg-inten\]). We found that by doubling CO$_2$ relative abundance the [G/R ratio]{} changes by $\sim$25%, whereas collisional quenching alone can vary its value by an order of magnitude. The model calculated [G/R ratio]{} values in comets [116P]{}, [C/2007 Q3]{}, and [C/2009 P1]{} are smaller by a factor of around 1.5 compared to the observations.
The detection of CO$_2$ molecules in the coma has been carried out using several infrared satellites by observing its fundamental vibrational band emission ($\nu_3$) at 4.26 $\mu$m [@Crovisier96; @Crovisier97; @Crovisier99; @Colangeli99; @Reach10; @Ootsubo12]. The quantification of CO$_2$ abundance based on the observed infrared emission intensity is subjected to opacity of the cometary coma. Since the CO$_2$ fluorescence efficiency factor ($g-$factor) is larger compared to that of H$_2$O and CO [see Table 2 of @Ootsubo12], these emission lines are optically thick in the inner coma, which can result in underestimation of CO$_2$ abundance if proper treatment of radiative transfer is not accounted for in the analysis. The optical depth effects in the inner coma may cause the surface brightness profile of these emissions to be much flatter and resemble the presence of extended sources in the coma. In comet Hale-Bopp, [@Bockelee10] have shown that the observed broad extent of infrared CO brightness is due to optical depth effects of the emitted radiation and not because of extended sources. Since these comets are observed at larger heliocentric distances and have low gas production rates, the collision dominated coma size is only a few hundred kilometers. Thus, the opacity effects of these IR emissions can be significant close to the nucleus and can influence the derivation of CO$_2$ production rate based on the observed flux over the field of view. The discrepancies between the [@Ootsubo12] derived production rates and other observations might be due to opacity of the cometary comae or may be due to assumed input parameters in the derivation of gas production rates. Under assumed optically thin condition the [@Ootsubo12] derived gas production rates in several comets can be regarded as lower limits. Considering these observational facts we varied CO$_2$ abundances in the model to assess our model calculated [G/R ratio]{} with the observations. By increasing CO$_2$ abundances in these comets by a factor of 3 we could achieve better agreement with the observed [G/R ratio]{}.
Similarly, the calculations presented in Fig. \[fig:gr-co2\] demonstrate that for a constant H$_2$O production rate, the [G/R ratio]{} increases with increasing CO$_2$ relative abundance. This figure suggests that for a constant CO$_2$ relative abundance, by increasing the H$_2$O production rate, the collisional quenching of [O($^1$S)]{} and [O($^1$D)]{} can increase the [G/R ratio]{}. Thus, the observation of a larger [G/R ratio]{} value need not be always due to higher CO$_2$ abundances.
In the case of hypothetical comet X, which has CO$_2$ abundance equal to that of H$_2$O, the calculated percentage contributions of different processes to red-doublet emissions presented in Table \[tab-slit\] suggest that the red-doublet emission intensity is equally controlled by CO$_2$ and H$_2$O. If a comet has equal abundances of CO$_2$ and H$_2$O, which is the case for comet [C/2006 W3]{} observed by [@Ootsubo12] at 3.7 au from the Sun, deriving the water production rates based on the observed red-doublet emission intensity may result in over estimation of H$_2$O. In this case the derivation of CO$_2$ abundances using the observed [G/R ratio]{} also leads to improper estimation. This calculation suggests that in a comet having high CO$_2$ abundance, the red-doublet emission intensity is not suitable for measuring H$_2$O rates. Similarly, our model calculations on comet [C/2001 Q4]{}, which has 75% CO$_2$ relative abundance, suggest that around 30% of red-doublet emissions are governed by both photodissociation of CO$_2$ and radiative decay of [O($^1$S)]{}, which is comparable to the contribution from H$_2$O ($\sim$45%) (see Table \[tab-slit\]).
Impact of collisional quenching of [O($^1$S)]{} and [O($^1$D)]{} on the [G/R ratio]{}
-------------------------------------------------------------------------------------
The [G/R ratio]{} at a given projected distance mainly depends on the formation and destruction processes of excited oxygen atoms in the cometary coma along the line of sight. The abundances of O-bearing species and solar flux governs the formation rate of these metastable species while the chemical lifetime and collisional quenching by other cometary species determines the destruction rate. In a comet having moderate H$_2$O production rate of 4 $\times$ 10$^{28}$ s$^{-1}$, the radius of the H$_2$O collisional zone is around 1000 km [@Whipple76]. When the comet is at a larger heliocentric distance, a lower gas evaporation rate results in a smaller collisional coma. Discarding the collisional quenching effect the observed [G/R ratio]{} has been used to infer CO$_2$ production rate in comets observed at large heliocentric distances [@McKay12; @Decock13]. Our calculated [G/R ratio]{} values as a function of projected distance on different comets (cf. Figures \[fig:rg-inten\] and \[fig:grat-prj\]) have shown that the collisional quenching of [O($^1$S)]{} and [O($^1$D)]{} can result in larger (even $>$1) [G/R ratio]{} values. Since the [G/R ratio]{} is averaged over the observed large projected distances, the collisional quenching may not influence the average value. In this case the observed [G/R ratio]{} is mainly determined by photochemical reactions of H$_2$O and CO$_2$ in producing red and green emissions, respectively. Hence, the observed [G/R ratio]{} value can be used to estimate the upper limit of CO$_2$ abundance relative to the H$_2$O production rate. But in the case of observations over smaller projected distances the collisional zone can predominantly affect the observed [G/R ratio]{} value which eventually can lead to the estimation of higher CO$_2$ abundances. Since the comets considered in this study are observed over large projected distances the effect of collisional quenching is small on the averaged [G/R ratio]{}. In such cases the observed [G/R ratio]{} value can be effective in constraining the upper limit of the CO$_2$ relative abundance.
Green and red-doublet emission line widths
------------------------------------------
[@Cochran08] made high-resolution observations on different comets and found that the green line width is higher than both red-doublet emission lines. The observation of these forbidden lines made on 12 comets have also shown the same feature [@Decock13]. The wider green line implies higher mean velocity distribution of O($^1$S) atoms in the cometary coma. The high velocity of [O($^1$S)]{} atoms in the cometary coma could be due to a parent source other than H$_2$O or could be due to involvement of high energy photons in H$_2$O dissociation. Our model calculations on comet [Hale-Bopp]{} showed that CO$_2$ photodissociation is a potentially more important source than that of H$_2$O in producing [O($^1$S)]{} atoms with high excess velocity [@Raghuram13].
From the calculations presented in Table \[tab-slit\], it can be understood that both CO$_2$ and H$_2$O are the important sources of [O($^1$S)]{}, whereas [O($^1$D)]{} is mainly sourced from H$_2$O. Since high energy photons (955–1165 Å) mainly dissociate CO$_2$ and produce [O($^1$S)]{}, the mean excess energy released in this reaction is larger ($\sim$2.5 eV) compared to that of H$_2$O ($\sim$1.2 eV). This results in the production of [O($^1$S)]{} atoms with large velocities (4.3 km s$^{-1}$) in cometary coma. The calculations presented in Table \[tab-slit\] show that above 10$^4$ km projected distances, the thermal recombination of H$_2$O$^+$ and CO$_2^+$ ions together results in the production of 15–40% of [O($^1$S)]{} and around 20% of [O($^1$D)]{}. [@Rosen00] and [@Seiersen03] experimentally determined the excess energies and branching ratios for the dissociative products in dissociative recombination of H$_2$O$^+$ and CO$_2^+$ ions, respectively. Based on these measured branching fractions and by theoretical estimation, we calculated the excess velocities of [O($^1$S)]{} and [O($^1$D)]{} and green and red line widths by incorporating the dissociative recombination reactions. We found an increase in our calculated green and red line widths by a factor of 1.2–1.7 and 1.1–2.2, respectively. However, without accounting for dissociative recombination reactions in our model, the calculated [G/R ratio]{} values (see Table \[tab-comet\]) and line widths (see Table \[tab-slit\]) are consistent with the observations. In comet [C/2001 Q4]{} our calculated red line width is smaller than the observed value. It can be noticed that in this comet both green and red-doublet line widths are nearly the same and the red line widths are higher compared to those on other comets, which is difficult to explain based on the model calculations.
Our calculations show that the dissociative recombination of H$_2$O$^+$ and CO$_2^+$ ions are an important source of [O($^1$S)]{} in the outer coma. In the model calculations we assumed a constant electron-ion recombination temperature of 300 K. Since comets are observed at large heliocentric distances the temperature values can be less than 300 K. To study the effect of electron temperature on the calculated [G/R ratio]{} and line widths we decreased the temperature to 200 K. We did not find any noticeable change in the calculated [G/R ratio]{} values or line widths. Since most of the green and red-doublet emission intensities are determined by photodissociation reactions in the inner coma the contribution of thermal recombination of ions on the averaged [G/R ratio]{} is rather small.
Several observations beyond 2 au have shown that the H$_2$O production rate in comets does not vary as a function of the inverse square of heliocentric distance [@Biver97; @Biver99; @Biver07; @Bodewits12b]. Hence, extrapolation of the H$_2$O production rate based on an approximation of the inverse square of heliocentric distance may be inappropriate. We evaluated the implications of this extrapolation in comet [C/2003 K4]{}by decreasing the H$_2$O production rate by a factor of 2. No significant change (decrease by $\le$ 5%) is observed in the model calculated [G/R ratio]{} and the calculated line widths.
Effect of atmospheric seeing
----------------------------
For small (0.6–2$''$) slit observations the differential atmospheric seeing can be an issue while determining the [G/R ratio]{} based on the observed atomic oxygen green and red-doublet line emission fluxes. The work carried out by [@McKay14] suggests that the differential refraction is potentially important for the near UV (e.g. CN 3870 Å) compared to the observation in the wavelength region 5000–6500 Å. They calculated the effect of differential refraction to be around 5% or less. In order to estimate the atmospheric seeing effect in determining slit-averaged [G/R ratio]{} we convolved our model calculated green and red-doublet emission fluxes with Gaussian function with full width at half maximum of seeing value 1.0$''$. All these comets, except [116P]{}, considered in the present work have been observed at larger ($>$2 au) geocentric distances; hence, the projected area on these comets would be larger.
In comet [C/2001 Q4]{}, which was observed at r = 3.7 au and $\Delta$ = 3.4 au, the model calculated G/R ratio varies between 2.2 and 0.2 below 100 km projected distance (cf. Figures \[fig:grat-prj\] and \[fig:gr-neat\]). After incorporating the seeing effect we found that the calculated [G/R ratio]{} is a constant throughout the projected distances with a value of about 0.2 as shown in Fig. \[fig:gr-neat\]. However, we do not find any change in the slit-averaged [G/R ratio]{} after accounting for the seeing effect in the model. Since the slit-averaged [G/R ratio]{} is over a much larger projected area, while the seeing effect is confined to distances close to the nucleus (cf. Fig. \[fig:grat-prj\]). We also assessed the output by changing the seeing value (from 1.0$''$ to 0.5$''$). No appreciable change in the modelled [G/R ratio]{} is observed. This suggests that the atmospheric seeing effect does not influence the [G/R ratio]{} for the comets observed at larger ($>$2 au) geocentric distances. Detailed analysis of the atmospheric seeing on different comets observed at different geocentric distances ($<$ 2 au) are being carried out and will be presented in our next paper (Decock et al. 2014, in preparation).
![The model calculated [G/R ratio]{} on comet [C/2001 Q4 (NEAT)]{} at r = 3.7 au and $\Delta$ = 3.4 au. The black dashed line is the calculated [G/R ratio]{} after convolving with a Gaussian function of full width at half maximum of 1.0$''$ seeing value. []{data-label="fig:gr-neat"}](fig9){width="22pc"}
Summary and conclusion
======================
The observation of green and red-doublet emission lines in comets at larger ($>$ 2 au) heliocentric distances suggest that the [G/R ratio]{} value is larger than 0.1. Moreover, the high-resolution observation reports that the green line is wider than the red-doublet lines, which is difficult to explain based on the single parent source for these oxygen emission lines [@Decock13]. We have developed a coupled chemistry-emission model for atomic oxygen visible prompt emissions and applied it on six comets, (viz. [116P/Wild 4]{}, [C/2003 K4 (LINEAR)]{}, [C/2007 Q3 (Siding Spring)]{}, [C/2006 W3 (Christensen)]{}, [C/2009 P1 (Garradd)]{}, [C/2001 Q4 (NEAT)]{}) which are observed at heliocentric distances greater than 2 au. By accounting for important chemical reactions in the model we calculated the [G/R ratio]{} values and widths of green and red-doublet emission lines on these comets. It is found that CO$_2$ is potentially more important than H$_2$O in [O($^1$S)]{} production while [O($^1$D)]{} is mainly controlled by H$_2$O. The photodissociation of CO is an insignificant source of metastable oxygen atoms. The observed large green line width in several comets is due to higher velocity of [O($^1$S)]{} atoms that are essentially produced via photodissociation of CO$_2$ by higher energy (955-1165 Å) photons. We have shown that the collisional quenching of [O($^1$S)]{} and [O($^1$D)]{} by H$_2$O can lead to a larger [G/R ratio]{} value and that its impact on the [G/R ratio]{} is larger than the change in CO$_2$ relative abundance. Hence, the larger [G/R ratio]{} value need not always be linked to larger CO$_2$ abundances. In a comet having large ($>$50%) CO$_2$ abundances, the photodissociation of CO$_2$ plays a significant role in producing both green and red-doublet emissions; thus, this process should also be accounted for while deriving the H$_2$O production rate based on the red-doublet emission intensity. When a comet is observed over a larger projected distance where the collisional zone is less resolvable, the collisional quenching does not affect the observed [G/R ratio]{}. At larger heliocentric distances, due to smaller gas production rates, the radius of a collisional coma is smaller; hence, the [G/R ratio]{} observed over larger projected distances can be used to constrain the CO$_2$ relative abundance. However, if the slit-projected area on the comet is smaller (with respect to the collisional zone), the derived CO$_2$ abundance based on the [G/R ratio]{} would be overestimated. Our model calculated [G/R ratio]{} and line widths of green and red-doublet emission are in agreement with the observation.
S. Raghuram was supported by the ISRO Senior Research Fellowship during the period of this work. Solar Irradiance Platform historical irradiances are provided courtesy of W. Kent Tobiska and Space Environment Technologies. These historical irradiances have been developed with partial funding from the NASA UARS, TIMED, and SOHO missions. We thank Alice Decock, Jeffrey P. Morgenthaler, and Adam McKay for the helpful discussions on the seeing effect on observation of comets.
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[^1]: Corresponding author : raghuramsusarla@gmail.com
|
---
author:
- |
Abhishek Kumar\
MIT-IBM Watson AI Lab, IBM Research\
`abhishk@us.ibm.com`\
Prasanna Sattigeri\
MIT-IBM Watson AI Lab, IBM Research\
`psattig@us.ibm.com`\
Kahini Wadhawan\
MIT-IBM Watson AI Lab, IBM Research\
`kahini.wadhawan@ibm.com`\
Leonid Karlinsky\
MIT-IBM Watson AI Lab, IBM Research\
`leonidka@il.ibm.com`\
Rogerio Feris\
MIT-IBM Watson AI Lab, IBM Research\
`rsferis@us.ibm.com`\
William T. Freeman\
MIT\
`billf@mit.edu`\
Gregory Wornell\
MIT\
`gww@mit.edu`\
bibliography:
- 'ml.bib'
title: 'Co-regularized Alignment for Unsupervised Domain Adaptation'
---
|
---
abstract: 'It is known that, for each real number $\xi$ such that $1,\xi,\xi^2$ are linearly independent over ${\mathbb{Q}}$, the uniform exponent of simultaneous approximation to $(1,\xi,\xi^2)$ by rational numbers is at most $(\sqrt{5}-1)/2 \cong 0.618$ and that this upper bound is best possible. In this paper, we study the analogous problem for ${\mathbb{Q}}$-linearly independent triples $(1,\xi,\xi^3)$, and show that, for these, the uniform exponent of simultaneous approximation by rational numbers is at most $2(9+\sqrt{11})/35 \cong 0.7038$. We also establish general properties of the sequence of minimal points attached to such triples that are valid for smaller values of the exponent.'
address: |
Département de Mathématiques\
Université d’Ottawa\
585 King Edward\
Ottawa, Ontario K1N 6N5, Canada
author:
- Stéphane Lozier
- Damien ROY
title: Simultaneous approximation to a real number and to its cube by rational numbers
---
[^1]
Introduction {#sec:intro}
============
In order to construct approximations to real numbers by algebraic integers of bounded degree, H. Davenport and W. M. Schmidt were led to study, through a duality argument, the problem of uniform approximation by rational numbers to consecutive powers of real numbers [@DSb]. To describe their result, although in a slightly weaker form, fix a positive integer $n$ and a point $\Xi=(\xi_0,\dots,\xi_n) \in {\mathbb{R}}^{n+1}$ with $\xi_0\neq 0$. We say that a real number $\lambda\ge 0$ is a *uniform exponent of approximation* to $\Xi$ (by rational numbers) if there exists a constant $c=c(\Xi)>0$ such that the system of inequations $$|x_0| \le X
{\quad\text{and}\quad}\max_{1\le i\le n} |x_0\xi_i-x_i\xi_0| \le cX^{-\lambda}$$ admits a non-zero solution ${\mathbf{x}}=(x_0,\dots,x_n)\in{\mathbb{Z}}^{n+1}$ for each real number $X\ge 1$. Let $\lambda(\Xi)$ denote the supremum of these exponents $\lambda$. Then, Theorems 1a, 2a and 4a of [@DSb] can essentially be summarized as follows:
Let $n\ge 2$ be an integer and let $\xi\in{\mathbb{R}}$ such that the point $\Xi=(1,\xi,\dots,\xi^n) \in {\mathbb{R}}^{n+1}$ has ${\mathbb{Q}}$-linearly independent coordinates. Then, we have $$\label{intro:eq:DS}
\lambda(\Xi)
\le
\begin{cases}
1/\gamma \cong 0.618 &\text{if $n=2$,}\\
1/2 &\text{if $n=3$,}\\
[n/2]^{-1} &\text{if $n\ge 4$,}
\end{cases}$$ where $\gamma=(1+\sqrt{5})/2$ denotes the golden ratio, and $[n/2]$ stands for the integer part of $n/2$.
The problem remains to determine, for each $n\ge 1$, the supremum $\lambda_n$ of $\lambda(1,\xi,\dots,\xi^n)$ as $\xi$ runs through all real numbers which are not algebraic over ${\mathbb{Q}}$ of degree $\le n$. By Dirichlet’s theorem on simultaneous approximation [@Sc Ch. II, Thm. 1A], we know that $\lambda_n\ge 1/n$ for each $n\ge 1$. When $n=1$, this estimate is sharp: we have $\lambda_1=1$ since $\lambda(1,\xi)=1$ for each $\xi\in{\mathbb{R}}\setminus{\mathbb{Q}}$. However, it is shown in [@Rb] that $\lambda_2=1/\gamma >1/2$. So, is optimal for $n=2$. For larger integers $n$, the value of $\lambda_n$ is unknown, but there have been some recent improvements upon . In [@La], M. Laurent proved that $\lambda_n\le \lceil n/2 \rceil^{-1}$ for each $n\ge 3$, where $\lceil n/2 \rceil$ denotes the smallest integer greater than or equal to $n/2$. Moreover, it is shown in [@Rc] that $\lambda_3\le (1 + 2\gamma - \sqrt{1+4\gamma^2})/2 \cong 0.4245$.
One goal of the present paper is to prove the following result of similar nature.
\[intro:thm:main\] Let $\xi\in{\mathbb{R}}$ such that $1,\xi,\xi^3$ are linearly independent over ${\mathbb{Q}}$. Then, we have $$\lambda(1,\xi,\xi^3) \le \mu := \frac{2(9+\sqrt{11})}{35} \cong 0.7038.$$
This estimate refines the upper bound $\lambda(1,\xi,\xi^3) \le 5/7 \cong 0.714$ established by the first author in [@Lo Thm. 10.5], but it is not best possible neither. The method that we present in this paper is capable of lowering it, possibly down to $(1+3\sqrt{5})/11 \cong 0.7007$ but we have not been able to go so far.
Before saying a word on this method, we mention two “generic” consequences of Theorem \[intro:thm:main\]. The first one follows from a simple adaptation of the arguments of Davenport and Schmidt in [@DSb §2]. Upon defining the *height* $H(\alpha)$ of an algebraic number $\alpha$ as the largest absolute value of the coefficients of its irreducible polynomial in ${\mathbb{Z}}[T]$, it reads as follows.
Let $\xi\in{\mathbb{R}}$ such that $1,\xi,\xi^3$ are linearly independent over ${\mathbb{Q}}$ and let $\tau < 1+1/\mu \cong 2.421$. Then, there exists infinitely many algebraic integers $\alpha$ which are roots of polynomials of the form $T^4+aT^3+bT+c$ in ${\mathbb{Z}}[T]$ and satisfy $|\xi-\alpha|\le H(\alpha)^{-\tau}$.
The second consequence is a version of Gel’fond’s transcendence criterion for lacunary polynomials. It follows from a direct application of Jarník’s transference principle [@Ja Thm. 1], and is in fact equivalent to Theorem \[intro:thm:main\].
Let $\xi\in{\mathbb{R}}$ and let $\tau > 1/(1-\mu) \cong 3.376$. Suppose that, for each sufficiently large real number $X$, there exists a non-zero polynomial $P(T) = aT^3+bT+c \in {\mathbb{Z}}[T]$ with $\max\{|a|,|b|,|c|\} \le X$ and $|P(\xi)|\le X^{-\tau}$. Then $1,\xi,\xi^3$ are linearly dependent over ${\mathbb{Q}}$.
The search of an optimal upper bound for the values $\lambda(1,\xi,\xi^3)$ from Theorem \[intro:thm:main\] fits in the following general framework. Let ${\mathcal{C}}$ be a closed algebraic subset of ${\mathbb{P}}^n({\mathbb{R}})$ of dimension one defined by homogeneous polynomials of ${\mathbb{Q}}[x_0,\dots,x_n]$, and let ${{\mathcal{C}}^{\textit{li}}}$ denote the set of points $P$ of ${\mathcal{C}}$ whose representatives $\Xi = (\xi_0,\dots,\xi_n) \in {\mathbb{R}}^{n+1}$ have ${\mathbb{Q}}$-linearly independent coordinates. Since $\lambda(a\Xi)=\lambda(\Xi)$ for each $a\in{\mathbb{R}}^*$, we may define $\lambda(P)=\lambda(\Xi)$ independently of the choice of $\Xi$. Then, the question is to determine the least upper bound $\lambda({\mathcal{C}})$ of the numbers $\lambda(P)$ with $P\in{{\mathcal{C}}^{\textit{li}}}$.
For example, for a fixed integer $k\ge 2$, let ${\mathcal{C}}_{1,k}$ denote the zero locus of the polynomial $x_0^{k-1}x_2-x_1^k$ in ${\mathbb{P}}^2({\mathbb{R}})$. Then ${{\mathcal{C}}^{\textit{li}}}_{1,k}$ consists of the points of ${\mathbb{P}}^2({\mathbb{R}})$ having a set of ${\mathbb{Q}}$-linearly independent homogeneous coordinates of the form $(1,\xi,\xi^k)$, and so $\lambda({\mathcal{C}}_{1,k})$ is the supremum of the numbers $\lambda(1,\xi,\xi^k)$ with $\xi\in{\mathbb{R}}$ and $1,\xi,\xi^k$ linearly independent over ${\mathbb{Q}}$. Then, for $k=2$, we have $\lambda({\mathcal{C}}_{1,2})=1/\gamma\cong 0.618$ by [@Rb Thm. 1.1], while for $k=3$, the above Theorem \[intro:thm:main\] gives $\lambda({\mathcal{C}}_{1,3})\le \mu \cong 0.7038$. In this context, it would be interesting to know if there exist curves ${\mathcal{C}}$ for which $\lambda({\mathcal{C}})$ is arbitrarily close to 1.
The proof of Theorem \[intro:thm:main\] goes first by attaching to the triple $(1,\xi,\xi^3)$ a sequence of minimal points $({\mathbf{x}}_i)_{i\ge 1}$ from ${\mathbb{Z}}^3$, as in [@DSb]. A simple but crucial property of this sequence is that ${\mathbf{x}}_{i-1}$, ${\mathbf{x}}_i$ and ${\mathbf{x}}_{i+1}$ are linearly independent for infinitely many indices $i\ge 2$. For such $i$, let $j$ be the next integer with the same property. Then, the points ${\mathbf{x}}_i,{\mathbf{x}}_{i+1},\dots,{\mathbf{x}}_j$ all lie in the same $2$-dimensional subspace of ${\mathbb{R}}^3$. Initially and for a long time, we tried to construct explicit auxiliary polynomials $P$ with integer coefficients vanishing at triples or even quadruples of these points, including points coming before ${\mathbf{x}}_i$ or after ${\mathbf{x}}_j$, but this soon became very complicated. We will not mention these constructions here (except for the polynomial $g$ in Section \[sec:F\]) because we discovered that it is in fact much more efficient to deal simply with the pairs $({\mathbf{x}}_i,{\mathbf{x}}_j)$, provided that we take into account the content of their cross products ${\mathbf{x}}_i\wedge{\mathbf{x}}_j$, namely the gcd of its coordinates, denoted $|q_i|$ for an integer $q_i$ defined in Section \[sec:search\]. Assuming a lower bound $\lambda(1,\xi,\xi^3)>\lambda_0$, the idea is to construct polynomials $P\in{\mathbb{Z}}[{\mathbf{x}},{\mathbf{y}}]$ for which the integer $|P({\mathbf{x}}_i,{\mathbf{x}}_j)|$ is relatively small for analytic reasons, and divisible by a certain power $q_i^k$ of $q_i$ for algebraic reasons. If it happens that $|P({\mathbf{x}}_i,{\mathbf{x}}_j)|<|q_i|^k$, then we conclude that $P({\mathbf{x}}_i,{\mathbf{x}}_j)=0$. On the other hand, if we can show that $P({\mathbf{x}}_i,{\mathbf{x}}_j)\neq 0$ by some arithmetic argument, then we obtain $|q_i|^k \le |P({\mathbf{x}}_i,{\mathbf{x}}_j)|$ which imposes constrains on the growth of the points ${\mathbf{x}}_i$ and ${\mathbf{x}}_j$. The details concerning the construction of such polynomials are explained in Section \[sec:search\].
The most basic polynomial in this respect is $\varphi({\mathbf{x}})=x_0^2x_2-x_1^3$, which defines the curve ${\mathcal{C}}_{1,3}$. In Section \[sec:prelim\], we show that, if $\lambda(1,\xi,\xi^3) > 2/3$, then $\varphi({\mathbf{x}}_i)\neq 0$ for each sufficiently large $i$. Then, imitating the proof of Theorem 1a from [@DSb], we conclude, as a first approximation, that $\lambda(1,\xi,\xi^3)\le \sqrt{3}-1 \cong 0.732$. The next most important polynomial is $F$ introduced and studied in Section \[sec:F\]. Then come $D^{(2)}$, $D^{(3)}$ and $D^{(6)}$ introduced in Section \[sec:search\], with the property that $D^{(k)}({\mathbf{x}}_i,{\mathbf{x}}_j)$ is divisible by $q_i^k$ for each pair $(i,j)$ as above and $k=2,3,6$. They are the simplest polynomials that we found. Assuming $\lambda(1,\xi,\xi^3)>0.6985$, it appears that, for $i$ large enough, none of them vanishes at the point $({\mathbf{x}}_i,{\mathbf{x}}_j)$. This is proved for $F$ in Section \[sec:NVF\], for $D^{(2)}$ in Section \[sec:NVD2\], and for both $D^{(3)}$ and $D^{(6)}$ in Section \[sec:NVD3D6\]. Then, the lower bound for $\lambda(1,\xi,\xi^3)$ given by Theorem \[intro:thm:main\] is proved in Section \[sec:NVD3D6\] on the basis of these non-vanishing results.
In Section \[sec:pol\], we show that, if $\lambda(1,\xi,\xi^3) > (1+3\sqrt{5})/11 \cong 0.7007$, then there exists a non-zero polynomial $P$ which vanishes at $({\mathbf{x}}_i,{\mathbf{x}}_j)$ for infinitely many pairs $(i,j)$ as above. This non-explicit construction suggests that we probably have $\lambda(1,\xi,\xi^3) \le (1+3\sqrt{5})/11$ because, if such a polynomial relation exists, we would expect it to be relatively simple, but we already ruled out the simplest ones.
All polynomials that we construct come from a graded factorial ring ${\mathcal{R}}\subset {\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ defined in Section \[sec:search\]. When $\lambda(1,\xi,\xi^3)>2/3$, it is shown in Section \[sec:search:remark\] that any two relatively prime homogeneous elements of ${\mathcal{R}}$ have only finitely many common zeros of the form $({\mathbf{x}}_i,{\mathbf{x}}_j)$. This suggests a natural way to avoid the delicate non-vanishing results. In Section \[sec:special\], we give an example where this strategy applies. However, in the general situation of Section \[sec:pol\] we have not been able to put it in practice.
Preliminaries {#sec:prelim}
=============
From now on, we fix a real number $\xi$ such that $1,\xi,\xi^3$ are linearly independent over ${\mathbb{Q}}$. For each ${\mathbf{x}}=(x_0,x_1,x_2)\in{\mathbb{R}}^3$, we set $$\|{\mathbf{x}}\| = \max\{|x_0|,|x_1|,|x_2|\},
\quad
L({\mathbf{x}}) = L_\xi({\mathbf{x}}) = \max\{|x_1-x_0\xi|,|x_2-x_0\xi^3|\}.$$ We also fix choices of $\lambda>0$ and $c>0$ such that, for each sufficiently large real number $X$, there exists a non-zero point ${\mathbf{x}}\in{\mathbb{Z}}^3$ with $$\label{prelim:mainhyp}
\|{\mathbf{x}}\| \le X {\quad\text{and}\quad}L({\mathbf{x}}) \le c X^{-\lambda}.$$ Our goal is to show that $\lambda\le 2(9+\sqrt{11})/35$. We proceed in several steps. In what follows, whenever we use the Vinogradov symbols $\gg$, $\ll$, or their conjunction $\asymp$, the implied constants depend only on $\xi$, $\lambda$ and $c$.
We first note that an argument similar to that of Davenport and Schmidt in [@DSb §3] shows the existence of a sequence of non-zero points $({\mathbf{x}}_i)_{i\ge 1}$ in ${\mathbb{Z}}^3$ such that, upon writing $$X_i = \|{\mathbf{x}}_i\| {\quad\text{and}\quad}L_i = L({\mathbf{x}}_i)$$ for each $i\ge 1$, we have
- $X_1 < X_2 < X_3 < \cdots$,
- $1/2 > L_1 > L_2 > L_3 > \cdots$,
- if $L({\mathbf{x}}) < L_i$ for some ${\mathbf{x}}\in{\mathbb{Z}}\setminus\{0\}$ and some $i\ge 1$, then $\|{\mathbf{x}}\|\ge X_{i+1}$.
Such a sequence is unique up to its first terms and up to multiplication of each of its terms by $\pm 1$ because, for ${\mathbf{x}},{\mathbf{y}}\in{\mathbb{Z}}$ with $L({\mathbf{x}})=L({\mathbf{y}})<1/2$, we have ${\mathbf{y}}=\pm {\mathbf{x}}$. We call it a sequence of *minimal points* for the triple $(1,\xi,\xi^3)$. The construction of Davenport and Schmidt is slightly different in that they use the absolute value of the first coordinate of each point instead of its norm, but it can be checked that the resulting sequences are the same modulo the above equivalence relation.
Fix $({\mathbf{x}}_i)_{i\ge 1}$, $(X_i)_{i\ge 1}$ and $(L_i)_{i\ge 1}$ as above. Then, our main hypothesis is equivalent to $$\label{prelim:hypLiXip}
L_i \le c X_{i+1}^{-\lambda}$$ for each sufficiently large $i$. Moreover, the points ${\mathbf{x}}_i$ are primitive and so, they are two by two linearly independent over ${\mathbb{Q}}$. We write their coordinates in the form $${\mathbf{x}}_i = (x_{i,0},x_{i,1},x_{i,2})\,.$$ Then, for each $i\ge 1$, the condition $L_i<1/2$ from b) implies that $x_{i,0}\neq 0$.
For each $i\ge 1$, we denote by $W_i = \langle {\mathbf{x}}_i,{\mathbf{x}}_{i+1}\rangle_{\mathbb{R}}$ the two-dimensional subspace of ${\mathbb{R}}^3$ generated by ${\mathbf{x}}_i$ and ${\mathbf{x}}_{i+1}$.
Following [@Sc Ch. I, §4], when $W$ is a subspace of ${\mathbb{R}}^n$ generated by elements of ${\mathbb{Q}}^n$, we define its *height* $H(W)$ as the covolume (or determinant) of the lattice $W\cap {\mathbb{Z}}^n$ in $W$. The next lemma is a first step in estimating the height of the subspaces $W_i$ of ${\mathbb{R}}^3$.
\[prelim:lemma:HWi\] For each $i\ge 1$, the set $\{{\mathbf{x}}_i,{\mathbf{x}}_{i+1}\}$ is a basis of $W_i\cap {\mathbb{Z}}^3$ and we have $H(W_i) \asymp \|{\mathbf{x}}_i\wedge{\mathbf{x}}_{i+1}\|$.
The first assertion follows by a direct adaptation of the arguments of Davenport and Schmidt in their proof of [@DSa Lemma 2] (see [@Rb Lemma 4.1]). It implies that the cross product ${\mathbf{x}}_i\wedge{\mathbf{x}}_{i+1}$ is a primitive point of ${\mathbb{Z}}^3$ and that $H(W_i)$ is its Euclidean norm.
The proof of the next lemma is similar to that of [@Rb Lemma 4.1] but requires some adjustments as its scope is more general.
\[prelim:lemma:xjwedgexi\] For any integers $1\le i<j$, we have $\|{\mathbf{x}}_i\wedge{\mathbf{x}}_j\| \asymp X_jL_i$.
The estimate $\|{\mathbf{x}}_i\wedge{\mathbf{x}}_j\| \ll X_jL_i$ is a well-known fact which follows, for example, from [@Rb Lemma 3.1(i)]. We claim that, for $1\le i<j$ with $j$ large enough, we have $\|{\mathbf{x}}_i\wedge{\mathbf{x}}_j\| \ge |x_{j,0}|L_i/3$. As $|x_{j,0}|\asymp X_j$, this will suffice to complete the proof of the lemma. To prove this claim, we set $\Xi=(1,\xi,\xi^3)$ and define $\Delta_\ell={\mathbf{x}}_\ell-x_{\ell,0}\Xi$ for each $\ell\ge 1$. We also assume, as we may, that $x_{\ell,0}> 0$ for each $\ell\ge 1$. Then, the last two coordinates of $x_{i,0}\Delta_j-x_{j,0}\Delta_i$ coincide, up to sign, with coordinates of ${\mathbf{x}}_i\wedge{\mathbf{x}}_j$ and so we have $$\label{prelim:lemma:xjwedgexi:eq}
\|{\mathbf{x}}_i\wedge{\mathbf{x}}_j\| \ge \|x_{i,0}\Delta_j-x_{j,0}\Delta_i\|.$$ Now, suppose that $\|{\mathbf{x}}_i\wedge{\mathbf{x}}_j\| < x_{j,0}L_i/3$. Since $\|\Delta_\ell\| = L_\ell$ for each $\ell\ge 1$, we deduce from that $x_{i,0}L_j > (2/3)x_{j,0}L_i$, and so $x_{j,0}< (3/2)x_{i,0}$. Assuming that $j$ is large enough, this implies that $\|{\mathbf{x}}_j-{\mathbf{x}}_i\| < \|{\mathbf{x}}_i\|$. Then, since ${\mathbf{x}}_i$ is a minimal point, we conclude that $L_i < L({\mathbf{x}}_j-{\mathbf{x}}_i) = \|\Delta_j-\Delta_i\|$ and the inequality yields $$x_{j,0}L_i/3
> \| x_{i,0}(\Delta_j-\Delta_i) - (x_{j,0}-x_{i,0})\Delta_i \|
> x_{i,0}L_i - (x_{j,0}-x_{i,0})L_i$$ in contradiction with $x_{j,0}< (3/2)x_{i,0}$. Thus our hypothesis forces $j$ to be bounded.
Combining the two previous results, we get:
\[prelim:cor:HWi\] For any $i\ge 1$, we have $H(W_i) \asymp X_{i+1}L_i$.
We denote by $I$ the set of all integers $i\ge 2$ such that ${\mathbf{x}}_{i-1},{\mathbf{x}}_i,{\mathbf{x}}_{i+1}$ are linearly independent.
The same argument as in the proof of [@DSb Lemma 5] shows that $I$ is an infinite set. We order its elements by increasing magnitude.
A result of Schmidt [@Sc Ch. I, Lemma 8A] shows that $H(S\cap T)H(S+T) \le H(S)H(T)$ for any pair of subspaces $S$ and $T$ of ${\mathbb{R}}^n$ generated by elements of ${\mathbb{Q}}^n$. Applying it to the present situation, like in [@Rc §3], we obtain:
\[prelim:prop:height\] For any pair of consecutive elements $i<j$ in $I$, we have $$X_j \ll (X_{i+1}X_{j+1})^{1-\lambda}
{\quad\text{and}\quad}1 \ll X_{j+1}L_jL_{j-1}.$$
For such $i$ and $j$, we have $W_i\cap W_j = \langle {\mathbf{x}}_j\rangle_{\mathbb{R}}$ and $W_i+W_j = {\mathbb{R}}^3$. Since $H(\langle {\mathbf{x}}_j \rangle_{\mathbb{R}})$ is the Euclidean norm of ${\mathbf{x}}_j$, we deduce from the inequality of Schmidt recalled above that $X_j \ll H(W_i) H(W_j)$. The conclusion then follows from Corollary \[prelim:cor:HWi\] using $L_i\ll X_{i+1}^{-\lambda}$ and $L_j\ll X_{j+1}^{-\lambda}$ to get the first estimate, and using $W_i = W_{j-1}$ to get the second one.
An alternative proof of the second estimate goes by observing that the determinant of the three points ${\mathbf{x}}_{j-1},{\mathbf{x}}_j,{\mathbf{x}}_{j+1}$ is non-zero and by estimating from above its absolute value as in [@DSb Lemma 4].
\[prelim:cor:approximable\] We have $X_j^{\lambda/(1-\lambda)} \ll X_{j+1}$ and $L_j \ll X_j^{-\lambda^2/(1-\lambda)}$ for any $j\in I$. In particular, if $\xi$ or $\xi^3$ is badly approximable, then we must have $\lambda \le (\sqrt{5}-1)/2\cong 0.618$.
We may assume that $j$ is not the first element of $I$. Then, upon denoting by $i$ the preceding element of $I$, we have $X_{i+1}\le X_j$ and the first estimate of the proposition leads to $X_j^\lambda \ll X_{j+1}^{1-\lambda}$, thus $L_j \ll X_{j+1}^{-\lambda} \ll X_j^{-\lambda^2/(1-\lambda)}$. This proves the first assertion. Now, if $\xi$ or $\xi^3$ is badly approximable, then we also have $L_j\gg X_j^{-1}$ for each $j\ge 1$. In particular, this holds for each $j\in I$, thus $\lambda^2/(1-\lambda) \le 1$ and so $\lambda \le (\sqrt{5}-1)/2$.
\[prelim:cor:XjLj\] Suppose that $\lambda\ge 2/3$. Then, for each $j\in I$, we have $X_j^2\ll X_{j+1}$ and $L_j\ll L_{j-1}^2$.
The first estimate follows directly from the preceding corollary. Since $X_{j+1} \ll L_j^{-1/\lambda} \le L_j^{-3/2}$, the second estimate of Proposition \[prelim:prop:height\] yields $1\ll L_j^{-1/2}L_{j-1}$ and so $L_j\ll L_{j-1}^2$.
Up to now, all of the above applies not only to the triple $(1,\xi,\xi^3)$ but also to any ${\mathbb{Q}}$-linearly independent triple of real numbers of the form $(1,\xi,\eta)$. The polynomials that we now introduce are specific to the present study.
Let ${\mathbf{x}}=(x_0,x_1,x_2)$, ${\mathbf{y}}=(y_0,y_1,y_2)$ and ${\mathbf{z}}=(z_0,z_1,z_2)$ be triples of indeterminates. We set $$\varphi({\mathbf{x}}) = x_0^2x_2-x_1^3
{\quad\text{and}\quad}\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{z}}) = x_0y_0z_2 + x_0y_2z_0 + x_2y_0z_0 - 3x_1y_1z_1.$$
The cubic form $\varphi$ satisfies $\varphi(1,\xi,\xi^3)=0$ and for that reason plays in the present context a role which is analog to that of the quadratic form $x_0x_2-x_1^2$ in [@DSb §3]. The polynomial $\Phi$ is the symmetric trilinear form for which $$\Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{x}}) = 3 \varphi({\mathbf{x}}).$$ The next result, analogous to Lemma 2 of [@DSb], is the main result of this section.
\[prelim:thm:phi\] Suppose that $\lambda>2/3$. Then we have $\varphi({\mathbf{x}}_i)\neq 0$ for each sufficiently large index $i$.
Suppose that $\varphi({\mathbf{x}}_i)=0$ for some integer $i\ge 2$. Then, ${\mathbf{x}}_i$ takes the form ${\mathbf{x}}_i=(p^3,p^2q,q^3)$ for some non-zero coprime integers $p$ and $q$. The vector ${\mathbf{n}}= {\mathbf{x}}_{i-1}\wedge{\mathbf{x}}_i$ is non-zero and orthogonal to ${\mathbf{x}}_i$. However, as $p^2q\neq 0$, the vector ${\mathbf{x}}_i$ is not orthogonal to $(0,1,0)$. So the first or the third coordinate of ${\mathbf{n}}$ is non-zero. As the first coordinate of ${\mathbf{n}}$ is an integer multiple of $q$ and the third an integral multiple of $p^2$, this implies that $\|{\mathbf{n}}\| \ge \min\{|q|,p^2\} \gg X_i^{1/3}$. On the other hand, Lemma \[prelim:lemma:xjwedgexi\] gives $\|{\mathbf{n}}\| \asymp X_i L_{i-1} \ll X_i^{1-\lambda}$. Combining these estimates gives $X_i^{1-\lambda} \gg X_i^{1/3}$ and so $i$ is bounded from above.
We set $\Xi = (1,\xi,\xi^3)$ and, for each $i\ge 1$, we put $$\delta({\mathbf{x}}_i) = \Phi({\mathbf{x}}_i,\Xi,\Xi) = 2x_{i,0}\xi^3 - 3x_{i,1}\xi^2 + x_{i,2}
{\quad\text{and}\quad}\delta_i = |\delta({\mathbf{x}}_i)|.$$
These quantities $\delta({\mathbf{x}}_i)$ are useful in dealing with limited developments of polynomials involving the function $\Phi$ as the next result illustrates.
\[prelim:cor:est\_varphi\] Suppose that $\lambda >2/3$. For each sufficiently large $i$, we have $$|\varphi({\mathbf{x}}_i)| \asymp X_i^2\delta_i
{\quad\text{and}\quad}X_i^{-2} \ll \delta_i \ll L_i.$$
Put $\Delta_i={\mathbf{x}}_i-x_{i,0}\Xi$, so that $L_i=\|\Delta_i\|$. By the multilinearity of $\Phi$ and the fact that $\Phi(\Xi,\Xi,\Xi)=3\varphi(\Xi)=0$, we find $$\begin{aligned}
\delta({\mathbf{x}}_i)
&= \Phi(\Delta_i,\Xi,\Xi) = {\mathcal{O}}(L_i),\\
\varphi({\mathbf{x}}_i)
&= x_{i,0}^2\Phi(\Delta_i,\Xi,\Xi) + x_{i,0}\Phi(\Delta_i,\Delta_i,\Xi) + \varphi(\Delta_i)
= x_{i,0}^2\delta({\mathbf{x}}_i) + {\mathcal{O}}(X_iL_i^2).
\end{aligned}$$ For all sufficiently large values of $i$, Theorem \[prelim:thm:phi\] gives $\varphi({\mathbf{x}}_i)\neq 0$. As $\varphi({\mathbf{x}}_i)$ is an integer and as $X_iL_i^2\ll X_i^{1-2\lambda}=o(1)$, we deduce that $|\varphi({\mathbf{x}}_i)| \asymp X_i^2\delta_i$ for each $i$ with $\varphi({\mathbf{x}}_i)\neq 0$ and so, $\delta_i\gg X_i^{-2}$ for the same values of $i$.
\[prelim:cor:first\_est\_lambda\] We have $\lambda \le \sqrt{3}-1 \cong 0.732$.
By Corollary \[prelim:cor:est\_varphi\], we have $1 \ll X_j^2\delta_j \ll X_j^2L_j$ for each large enough index $j$. As $L_j\ll X_{j+1}^{-\lambda}$, this gives $X_{j+1}^{\lambda} \ll X_j^2$. When $j\in I$, Corollary \[prelim:cor:approximable\] also gives $X_j^\lambda \ll X_{j+1}^{1-\lambda}$. Combining the two estimates and letting $j$ go to infinity within $I$, we conclude that $\lambda^2\le 2(1-\lambda)$ and so $\lambda \le \sqrt{3}-1$.
As we will see this upper bound is not optimal and we will improve it in what follows. We end this section with a general estimate which will be useful for this purpose.
\[prelim:prop:est\_Phi\] Suppose that $\lambda>2/3$. For any choice of integers $i\le j\le k$ with $i\in I$ large enough, we have $\Phi({\mathbf{x}}_i,{\mathbf{x}}_j,{\mathbf{x}}_k)\neq 0$ and $$|\Phi({\mathbf{x}}_i,{\mathbf{x}}_j,{\mathbf{x}}_k)| \asymp X_jX_k\delta_i \asymp \frac{X_jX_k}{X_i^2} |\varphi({\mathbf{x}}_i)|.$$
In view of Corollary \[prelim:cor:est\_varphi\], it suffices to show that $|\Phi({\mathbf{x}}_i,{\mathbf{x}}_j,{\mathbf{x}}_k)| \asymp X_jX_k\delta_i$ for all triples of integers $i\le j\le k$ with $i\in I$ large enough. Using the multilinearity of the function $\Phi$ as in the proof of Corollary \[prelim:cor:est\_varphi\], we find $$\Phi({\mathbf{x}}_i,{\mathbf{x}}_j,{\mathbf{x}}_k)
= x_{j,0}x_{k,0}\delta({\mathbf{x}}_i) + x_{i,0}x_{k,0}\delta({\mathbf{x}}_j) + x_{i,0}x_{j,0}\delta({\mathbf{x}}_k) + {\mathcal{O}}(X_kL_iL_j),$$ assuming only $1\le i\le j\le k$. In the right hand side of this equality, the first three summands may not be distinct. To conclude, we simply need to show that, when $i\in I$, we have $X_i\delta_j = o(X_j\delta_i)$ if $i<j$, $X_i\delta_k = o(X_k\delta_i)$ if $i<k$, and $L_iL_j = o(X_j\delta_i)$. The estimates for $\delta_i,\delta_j,\delta_k$ provided by Corollary \[prelim:cor:est\_varphi\] reduce the problem to showing $$X_iL_j = o(X_jX_i^{-2}) \ \text{if $i<j$,}
\ \
X_iL_k = o(X_kX_i^{-2}) \ \text{if $i<k$,}
\ \
L_iL_j = o(X_jX_i^{-2}).$$ The third estimate is clear because $L_iL_j \le L_i^2 \ll X_i^{-4/3}$ and $X_jX_i^{-2} \ge X_i^{-1}$. To prove the first estimate, we note that, by Corollary \[prelim:cor:XjLj\], we have $X_i^2 \ll X_{i+1}$ and $L_i\ll L_{i-1}^2 \ll X_i^{-4/3}$ since $\lambda>2/3$ and $i\in I$. Thus, when $i<j$, we obtain $X_iL_j \le X_iL_i = o(1)$ while $X_jX_i^{-2} \ge X_{i+1}X_i^{-2} \gg 1$. The proof of the second estimate is the same.
The polynomial $F$ and the point $\psi$ {#sec:F}
=======================================
We introduce two new actors in the present study.
For triples of indeterminates ${\mathbf{x}}$ and ${\mathbf{y}}$, we set: $$\begin{aligned}
F({\mathbf{x}},{\mathbf{y}})
&= \Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{y}})^2 - 4\varphi({\mathbf{x}})\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{y}}),\\
\psi({\mathbf{x}},{\mathbf{y}})
&= \Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{y}}){\mathbf{x}}-2\varphi({\mathbf{x}}){\mathbf{y}}.\end{aligned}$$
This section is devoted to estimates for $F$. In particular, we will show that $|F({\mathbf{x}}_i,{\mathbf{x}}_j)|$ is a relatively small integer for each pair of consecutive indices $i<j$ in $I$, and that, if it is non-zero for infinitely many such pairs, then $\lambda>5/7$. We also provide estimates for $\psi$ which we view as an analog of the point $[{\mathbf{x}},{\mathbf{x}},{\mathbf{y}}]$ defined in [@Rb §2], which plays a crucial role in the study [@Rb] of simultaneous approximation to a real number and to its square (see [@Rb Cor. 5.2]). The polynomial $F$ however has no analog in [@Rb].
Recalling that $\Xi=(1,\xi,\xi^3)$ and using the notation of Section 2 for the coordinates of points, we first establish the following formulas.
\[F:lemma:Fpsi\] We have the identities
- $F({\mathbf{x}},\Xi) = F({\mathbf{x}}-x_0\Xi,\Xi)$,
- $\varphi(\psi({\mathbf{x}},{\mathbf{y}})) = -\varphi({\mathbf{x}})\Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{y}})F({\mathbf{x}},{\mathbf{y}}) - 8\varphi({\mathbf{x}})^3\varphi({\mathbf{y}})$.
Note that $\varphi(\psi({\mathbf{x}},{\mathbf{y}}))$ simplifies to $- 8\varphi({\mathbf{x}})^3\varphi({\mathbf{y}})$ for points ${\mathbf{x}},{\mathbf{y}}\in{\mathbb{Z}}^3$ with $F({\mathbf{x}},{\mathbf{y}})=0$, a fact that is interesting to compare with [@Rb Lemma 2.1(i)].
By the multilinearity of $\Phi$, we have, for indeterminates $a$ and $b$, $$\varphi(a{\mathbf{x}}+b{\mathbf{y}}) = a^3\varphi({\mathbf{x}}) + a^2b\Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{y}}) + ab^2\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{y}}) + b^3\varphi({\mathbf{y}}).$$ Substituting $\Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{y}})$ for $a$ and $-2\varphi({\mathbf{x}})$ for $b$ in this identity yields (ii). We also note that $F({\mathbf{x}},\Xi)$ is the discriminant of $$\varphi({\mathbf{x}}+T\Xi) = \varphi({\mathbf{x}}) + \Phi({\mathbf{x}},{\mathbf{x}},\Xi)T + \Phi({\mathbf{x}},\Xi,\Xi)T^2$$ viewed as a polynomial in $T$. Then (i) follows from the fact that the discriminant of a polynomial $p(T)$ stays invariant under the change of variable $T\mapsto T-x_0$.
In the course of this research, we were also lead to work with polarized versions of $F$. In particular the polynomial $$g({\mathbf{x}},{\mathbf{u}},{\mathbf{y}})
= \Phi({\mathbf{x}},{\mathbf{u}},{\mathbf{y}})\Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{y}}) - \Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{u}})\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{y}}) - \varphi({\mathbf{x}})\Phi({\mathbf{u}},{\mathbf{y}},{\mathbf{y}}),$$ involving a third triple of indeterminates ${\mathbf{u}}$, was playing a central role until we discovered the approach that will be presented in the next section. We simply mention its existence in case it comes back in future investigations.
\[F:prop:est\_F\] For any ${\mathbf{x}},{\mathbf{y}}\in{\mathbb{Z}}^3$, we have $$\begin{aligned}
F({\mathbf{x}},{\mathbf{y}}) =
&-4x_0^3y_0\delta({\mathbf{x}})\delta({\mathbf{y}}) + {\mathcal{O}}\big( \|{\mathbf{y}}\|^2L({\mathbf{x}})^4 + \|{\mathbf{x}}\|^4L({\mathbf{y}})^2 \big),\\
\|\psi({\mathbf{x}},{\mathbf{y}})\|
&\ll \|{\mathbf{x}}\|\/\|{\mathbf{y}}\|\/L({\mathbf{x}})^2 + \|{\mathbf{x}}\|^3L({\mathbf{y}}).
\end{aligned}$$
Write ${\mathbf{y}}=y_0\Xi+\Delta{\mathbf{y}}$. Since $F$ is quadratic in its second argument, we find $$\label{F:prop:est_F:eq1}
F({\mathbf{x}},{\mathbf{y}}) = y_0^2F({\mathbf{x}},\Xi) + 2y_0A + F({\mathbf{x}},\Delta{\mathbf{y}}),$$ where $$A = \Phi({\mathbf{x}},{\mathbf{x}},\Xi)\Phi({\mathbf{x}},{\mathbf{x}},\Delta{\mathbf{y}})-4\varphi({\mathbf{x}})\Phi({\mathbf{x}},\Delta{\mathbf{y}},\Xi).$$ To estimate $A$, we write ${\mathbf{x}}=x_0\Xi+\Delta{\mathbf{x}}$ and expand it as a polynomial in $x_0$. Since $$\begin{aligned}
\Phi({\mathbf{x}},{\mathbf{x}},\Xi) &= 2x_0\delta({\mathbf{x}})+{\mathcal{O}}(L({\mathbf{x}})^2),\\
\Phi({\mathbf{x}},{\mathbf{x}},\Delta{\mathbf{y}}) &= x_0^2\delta({\mathbf{y}})+{\mathcal{O}}(\|{\mathbf{x}}\|\/L({\mathbf{x}})L({\mathbf{y}})),\\
\varphi({\mathbf{x}}) &= x_0^2\delta({\mathbf{x}})+{\mathcal{O}}(\|{\mathbf{x}}\|\/L({\mathbf{x}})^2),\\
\Phi({\mathbf{x}},\Delta{\mathbf{y}},\Xi) &=x_0\delta({\mathbf{y}})+{\mathcal{O}}(L({\mathbf{x}})L({\mathbf{y}})),
\end{aligned}$$ we obtain $$A = -2x_0^3\delta({\mathbf{x}})\delta({\mathbf{y}}) + {\mathcal{O}}(\|{\mathbf{x}}\|^2L({\mathbf{x}})^2L({\mathbf{y}})).$$ By Lemma \[F:lemma:Fpsi\] (i), we also have $F({\mathbf{x}},\Xi) = F(\Delta{\mathbf{x}},\Xi) = {\mathcal{O}}(L({\mathbf{x}})^4)$, while it is clear that $F({\mathbf{x}},\Delta{\mathbf{y}}) = {\mathcal{O}}(\|{\mathbf{x}}\|^4L({\mathbf{y}})^2)$. Substituting these estimates into yields $$\begin{aligned}
F({\mathbf{x}},{\mathbf{y}}) = &-4x_0^3y_0\delta({\mathbf{x}})\delta({\mathbf{y}})\\
&+ {\mathcal{O}}\big(\|{\mathbf{y}}\|^2L({\mathbf{x}})^4 + \|{\mathbf{x}}\|^2\|{\mathbf{y}}\|L({\mathbf{x}})^2L({\mathbf{y}}) + \|{\mathbf{x}}\|^4L({\mathbf{y}})^2\big).\end{aligned}$$ Finally, we may omit the middle term in the error estimate as it is the geometric mean of the other two. The estimate for $\|\psi({\mathbf{x}},{\mathbf{y}})\|$ is proved along similar lines and we leave this task to the reader.
\[F:cor:est\_Fij\] Suppose that $\lambda>2/3$. For any pair of consecutive integers $i<j$ in $I$, we have $$F({\mathbf{x}}_i,{\mathbf{x}}_j)
= - 4 x_{i,0}^3 x_{j,0} \delta({\mathbf{x}}_i) \delta({\mathbf{x}}_j) + {\mathcal{O}}( X_j^2 L_i^4).$$
Since $\lambda>2/3$, Corollary \[prelim:cor:XjLj\] gives $X_i^2\ll X_{i+1}\le X_j$ and $L_j \ll L_{j-1}^2 \le L_i^2$, so $X_i^4L_j^2 \ll X_j^2L_i^4$ and thus we may omit the product $X_i^4L_j^2$ in the error term from the preceding proposition.
If we assume furthermore that $F({\mathbf{x}}_i,{\mathbf{x}}_j)=0$, then the above estimate yields $X_i^3 X_j\delta_i \delta_j \ll X_j^2 L_i^4 \ll X_j^2 X_{i+1}^{-4\lambda}$ and, in view of Corollary \[prelim:cor:est\_varphi\], this has the following consequence.
\[F:cor:Fzero\] Suppose that $\lambda>2/3$. Then, for any pair of consecutive elements $i<j$ in $I$ with $F({\mathbf{x}}_i,{\mathbf{x}}_j) = 0$, we have $$|\varphi({\mathbf{x}}_i)\varphi({\mathbf{x}}_j)| \ll X_i^{-1} X_{i+1}^{-4\lambda} X_j^3.$$
The next result deals with the complementary case where $F({\mathbf{x}}_i,{\mathbf{x}}_j)\neq 0$.
\[F:prop:Fnotzero\] Suppose that $\lambda>(5-\sqrt{13})/2 \cong 0.697$. Then, for consecutive elements $i<j$ in $I$, we have $|\varphi({\mathbf{x}}_i)\varphi({\mathbf{x}}_j)| = o(X_i^{-1}X_j)$ as $i\to\infty$. For the pairs $(i,j)$ with $F({\mathbf{x}}_i,{\mathbf{x}}_j)\neq 0$, we also have $$1 \le |F({\mathbf{x}}_i,{\mathbf{x}}_j)| \ll X_j^2L_i^4 {\quad\text{and}\quad}X_j \gg X_{i+1}^{2\lambda}.$$
We first note that the ratio $\theta=(1-\lambda)/\lambda$ satisfies $2\theta\ge \lambda$ since, by Corollary \[prelim:cor:first\_est\_lambda\], we have $\lambda\le \sqrt{3}-1$. Then, for consecutive elements $i<j$ in $I$, we obtain $$\begin{aligned}
{2}
\frac{X_i}{X_j}|\varphi({\mathbf{x}}_i)\varphi({\mathbf{x}}_j)|
&\asymp X_i^3X_j\delta_i\delta_j
\ll X_i^3 &&X_{i+1}^{-\lambda} X_j X_{j+1}^{-\lambda}
\quad\text{by Corollary \ref{prelim:cor:est_varphi},} \notag\\
&\ll X_{i+1}^{3\theta-\lambda} X_j X_{j+1}^{-\lambda}
&&\ \text{since $X_i\ll X_{i+1}^\theta$ by Corollary \ref{prelim:cor:approximable},} \notag\\
&\le X_j^{3\theta+1-\lambda} X_{j+1}^{-\lambda}
&&\ \text{since $X_{i+1}\le X_j$ and $3\theta-\lambda>0$,} \notag\\
&\ll X_{j+1}^{3\theta^2+\theta-1}
&&\ \text{since $X_j\ll X_{j+1}^\theta$ by Corollary \ref{prelim:cor:approximable}.} \notag\end{aligned}$$ As $3\theta^2+\theta-1 = (\lambda^2-5\lambda+3)/\lambda^2 <0$, this proves our first assertion. It also shows that $X_i^3X_j\delta_i\delta_j=o(1)$ as $i\to \infty$. Thus Corollary \[F:cor:est\_Fij\] yields $1\le |F({\mathbf{x}}_i,{\mathbf{x}}_j)| \ll X_j^2L_i^4$ when the integer $F({\mathbf{x}}_i,{\mathbf{x}}_j)$ is non-zero and $i$ is large enough. In that case, using $L_i\ll X_{i+1}^{-\lambda}$, we find $X_j \gg X_{i+1}^{2\lambda}$.
The non-vanishing of $F$ has important consequences. The next lemma provides useful estimates that we will need repeatedly.
\[F:lemma:tech\_alpha\_beta\] Suppose that $\lambda>(5-\sqrt{13})/2 \cong 0.697$. Then, for any pair of consecutive elements $i<j$ in $I$ with $F({\mathbf{x}}_i,{\mathbf{x}}_j)\neq 0$, we have $$X_{i+1}^{2\lambda} \ll X_j\ll X_{j+1}^\alpha,
\quad
X_{j+1}\ll X_j^{2/\lambda} \ll X_{i+1}^\beta$$ where $\alpha=2\lambda(1-\lambda)/(3\lambda-1)$ and $\beta=2(1-\lambda)/(3\lambda-2)$.
For such pairs $(i,j)$, Proposition \[F:prop:Fnotzero\] gives $X_{i+1}^{2\lambda} \ll X_j$. Combining this with the estimate $X_j \ll (X_{i+1}X_{j+1})^{1-\lambda}$ from Proposition \[prelim:prop:height\], we obtain $X_j \ll X_{j+1}^\alpha$. On the other hand, by Corollary \[prelim:cor:est\_varphi\], we have $X_j^{-2} \ll L_j \ll X_{j+1}^{-\lambda}$, and so $X_{j+1} \ll X_j^{2/\lambda}$. Combining this with the same estimate from Proposition \[prelim:prop:height\] yields $X_j^{2/\lambda} \ll X_{i+1}^\beta$.
\[F:cor:upperbound\_lambda\] Suppose that $F({\mathbf{x}}_i,{\mathbf{x}}_j)\neq 0$ for infinitely many pairs of consecutive integers $i<j$ in $I$. Then, we have $\lambda \le 5/7 \cong 0.714$.
Assuming, as we may, that $\lambda > (5-\sqrt{13})/2$, Lemma \[F:lemma:tech\_alpha\_beta\] gives $X_{i+1}^{2\lambda} \ll X_j$ and $X_j \ll X_{i+1}^{\lambda\beta/2}$ for each of these pairs $(i,j)$. Therefore, we must have $4\le \beta$ and so $\lambda\le 5/7$.
Search for algebraic relations {#sec:search}
==============================
The basic search
----------------
Given triples of indeterminates ${\mathbf{x}}=(x_0,x_1,x_2)$ and ${\mathbf{y}}=(y_0,y_1,y_2)$, our aim is to find non-zero polynomials $P\in{\mathbb{Z}}[{\mathbf{x}},{\mathbf{y}}]$ which vanish at the point $({\mathbf{x}}_i,{\mathbf{x}}_j)\in{\mathbb{Z}}^3\times{\mathbb{Z}}^3$ for infinitely many pairs of consecutive elements $i<j$ in $I$. Such vanishing should derive simply from the integral nature of the points ${\mathbf{x}}_k$, the general growth of $\|{\mathbf{x}}_k\|$, and the inequality $$\label{search:eqL}
L_\xi({\mathbf{x}}_k) \ll \|{\mathbf{x}}_{k+1}\|^{-\lambda},$$ independently of the value of the implied constant. As the latter condition remains satisfied if we replace the sequence $({\mathbf{x}}_k)_{k\ge 1}$ by $(a_k{\mathbf{x}}_k)_{k\ge 1}$ for bounded non-zero integers $a_k$, we deduce that the polynomials $P(a{\mathbf{x}},b{\mathbf{y}})$ should share the same vanishing for any choice of non-zero integers $a$ and $b$. As a consequence, that vanishing applies to each of the bi-homogeneous components of $P$ and so we may restrict our search to bi-homogeneous polynomials, namely polynomials that are separately homogeneous in ${\mathbf{x}}$ and in ${\mathbf{y}}$. Similarly the condition is preserved if we replace the number $\xi$ by $a\xi$ for some non-zero integer $a$ and replace each ${\mathbf{x}}_k$ by its image under the polynomial map $\theta_a({\mathbf{x}}) = (x_0, ax_1, a^3x_2)$. Thus, the polynomials $P(\theta_a({\mathbf{x}}),\theta_a({\mathbf{y}}))$ should also have the same vanishing for any non-zero integer $a$. In particular, that vanishing applies to each of the homogeneous components of $P$ for the *weight*, upon defining the weight of a monomial $x_0^{e_0}x_1^{e_1}x_2^{e_2}y_0^{f_0}y_1^{f_1}y_2^{f_2}$ as $e_1+3e_2+f_1+3f_2$. So, we may further restrict our search to weight-homogeneous polynomials.
Note that the polynomials $$\label{search:eq:STUV}
S=\varphi({\mathbf{x}}),\quad
T=\Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{y}}),\quad
U=\Phi({\mathbf{x}},{\mathbf{y}},{\mathbf{y}}),\quad
V=\varphi({\mathbf{y}})$$ obtained by polarization of $\varphi({\mathbf{x}})$ are bi-homogeneous as well as homogeneous for the weight, of weight $3$ equal to their total degree. So, any polynomial in $S,T,U,V$ which is bihomogeneous as a polynomial in ${\mathbf{x}}$ and ${\mathbf{y}}$ is automatically homogeneous for the weight. This makes the subring ${\mathbb{Q}}[S,T,U,V]$ of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ particularly pleasant to work with. It is within that ring that we will search for polynomials. Its generators $S$, $T$, $U$ and $V$ are algebraically independent over ${\mathbb{Q}}$, as a short computation shows that their images under the specialization $x_0\mapsto 0$ and $y_0\mapsto 1$ are so. Thus, ${\mathbb{Q}}[S,T,U,V]$ can be viewed as a ring of polynomials in $4$ variables.
The bi-degree and the weight give rise to an ${\mathbb{N}}^3$-grading on the ring ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$. But since we will work in the subring ${\mathbb{Q}}[S,T,U,V]$, it is only the ${\mathbb{N}}^2$-grading given by the bi-degree that will matter. So, we simply consider the ${\mathbb{Q}}$-vector space decomposition $${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}] = \bigoplus_{(m,n)\in{\mathbb{N}}^2} {\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]_{(m,n)}$$ where ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]_{(m,n)}$ stands for the bi-homogeneous part of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ of bi-degree $(m,n)$.
Forcing divisibility
--------------------
For each $i\in I$, we denote by a subscript $i$ the values of the polynomials at the point $({\mathbf{x}}_i,{\mathbf{x}}_j)$, where $j$ stands for the successor of $i$ in $I$. Thus, we have $$S_i=\varphi({\mathbf{x}}_i),\quad
T_i=\Phi({\mathbf{x}}_i,{\mathbf{x}}_i,{\mathbf{x}}_j),\quad
U_i=\Phi({\mathbf{x}}_i,{\mathbf{x}}_j,{\mathbf{x}}_j),\quad
V_i=\varphi({\mathbf{x}}_j).$$ We also note that $V_i=S_j$. Since ${\mathbf{x}}_i$ and ${\mathbf{x}}_{i+1}$ form a basis of the integer points in $W_i$ and since ${\mathbf{x}}_j$ is a primitive point of ${\mathbb{Z}}^3$, we can write $$\label{search:eqpq}
{\mathbf{x}}_j = p_i{\mathbf{x}}_i+q_i{\mathbf{x}}_{i+1}$$ for relatively prime integers $p_i$ and $q_i$ with $q_i\neq 0$. The next proposition gathers several properties of the integers $q_i$.
\[search:prop:qi\] For each pair of consecutive elements $i<j$ in $I$, we have
- $|q_i| \asymp X_j/X_{i+1}$,
- $T_i\equiv 3p_iS_i,\ U_i\equiv 3p_i^2S_i,\ V_i\equiv p_i^3S_i \mod q_i$,
- $\gcd(q_i,S_i)=\gcd(q_i,V_i)$.
Taking the exterior product of both sides of with ${\mathbf{x}}_i$, we find ${\mathbf{x}}_i\wedge{\mathbf{x}}_j = q_i{\mathbf{x}}_i\wedge{\mathbf{x}}_{i+1}$. Then, applying Lemma \[prelim:lemma:xjwedgexi\] separately to each product yields $X_jL_i\asymp |q_i|X_{i+1}L_i$ and (a) follows. The equality also yields ${\mathbf{x}}_j\equiv p_i{\mathbf{x}}_i \mod q_i$ and thus $T_i \equiv \Phi({\mathbf{x}}_i,{\mathbf{x}}_i,p_i{\mathbf{x}}_i) = 3p_iS_i \mod q_i$. The other two congruences from (b) are proved in the same way. Finally (c) is an immediate consequence of the congruence $V_i\equiv p_i^3S_i \mod q_i$ together with the fact that $p_i$ is prime to $q_i$.
In particular, the above congruences imply that the integer $T_i^2-3S_iU_i$ is congruent to $0$ modulo $q_i$ and thus divisible by $q_i$. The next observation is crucial for the present work and will allow us to reach higher divisibility properties.
\[search:lemma:equiv\_cond\] Let $p$ and $q$ be indeterminates over the ring ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$, let $k\in{\mathbb{N}}$ and let $P({\mathbf{x}},{\mathbf{y}})$ be a bihomogeneous element of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$. Then the following assertions are equivalent
- $q^k$ divides $P({\mathbf{x}},{\mathbf{x}}+q{\mathbf{y}})$ in ${\mathbb{Q}}[q,{\mathbf{x}},{\mathbf{y}}]$,
- $q^k$ divides $P({\mathbf{x}},p{\mathbf{x}}+q{\mathbf{y}})$ in ${\mathbb{Q}}[p,q,{\mathbf{x}},{\mathbf{y}}]$.
It is clear that (ii) implies (i). To prove the converse, suppose that $P({\mathbf{x}},{\mathbf{x}}+q{\mathbf{y}})=q^kQ(q,{\mathbf{x}},{\mathbf{y}})$ for some $Q\in {\mathbb{Q}}[q,{\mathbf{x}},{\mathbf{y}}]$. Substituting $p{\mathbf{x}}$ for ${\mathbf{x}}$ in this equality and denoting by $d$ the degree of $P$ in ${\mathbf{x}}$, we obtain $$p^d P({\mathbf{x}},p{\mathbf{x}}+q{\mathbf{y}}) = P(p{\mathbf{x}},p{\mathbf{x}}+q{\mathbf{y}}) = q^k Q(q,p{\mathbf{x}},{\mathbf{y}}).$$ Thus $q^k$ divides $p^d P({\mathbf{x}},p{\mathbf{x}}+q{\mathbf{y}})$ in ${\mathbb{Q}}[p,q,{\mathbf{x}},{\mathbf{y}}]$, and (ii) follows.
\[search:def:Jk\] For each $k\in{\mathbb{N}}$, we denote by $J^{(k)}$ the ideal of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ generated by the bihomogeneous elements of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ which satisfy the equivalent conditions of the lemma.
It can be shown that $J:=J^{(1)}$ is the ideal of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ generated by the coordinates of the exterior product ${\mathbf{x}}\wedge {\mathbf{y}}$ in the standard basis, and that $J^{(k)} = J^k$ is the $k$-th power of $J$ for each integer $k\ge 1$. However, we will not need this fact here and this is why we adopt a different notation. Our interest in these ideals is motivated by the following result.
\[search:prop:divJk\] Let $k\in{\mathbb{N}}^*$ and let $P$ be a bi-homogeneous element of $J^{(k)}\cap {\mathbb{Z}}[{\mathbf{x}},{\mathbf{y}}]$. Then, for each pair of consecutive elements $i<j$ of $I$, the integer $P({\mathbf{x}}_i,{\mathbf{x}}_j)$ is divisible by $q_i^k$. In particular, when $P({\mathbf{x}}_i,{\mathbf{x}}_j)\neq 0$, we have $|q_i|^k \le |P({\mathbf{x}}_i,{\mathbf{x}}_j)|$.
Since $P\in J^{(k)}$, we have $P({\mathbf{x}},p{\mathbf{x}}+q{\mathbf{y}}) = q^k Q(p,q,{\mathbf{x}},{\mathbf{y}})$ for some polynomial $Q$ with coefficients in ${\mathbb{Q}}$. Since $P$ has integer coefficients, the same is true of $Q$. The first assertion follows by specializing $p$, $q$, ${\mathbf{x}}$ and ${\mathbf{y}}$ in $p_i$, $q_i$, ${\mathbf{x}}_i$ and ${\mathbf{x}}_{i+1}$ respectively.
Let $\rho$ denote the automorphism of the ${\mathbb{Q}}$-algebra ${\mathbb{Q}}[q,{\mathbf{x}},{\mathbf{y}}]$ which fixes $q$ and ${\mathbf{x}}$ but maps ${\mathbf{y}}$ to ${\mathbf{x}}+q{\mathbf{y}}$. According to Definition \[search:def:Jk\], a bi-homogeneous element $P$ of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ belongs to $J^{(k)}$ for some integer $k\ge 0$ if and only if $q^k$ divides $\rho(P)$ in ${\mathbb{Q}}[q,{\mathbf{x}},{\mathbf{y}}]$. When $P\neq 0$, there exists a largest integer $k$ with that property. We call it the *$J$-valuation* of $P$ and denote it $v_J(P)$. Clearly it satisfies $v_J(PQ)=v_J(P)+v_J(Q)$ for any pair of non-zero bi-homogeneous elements $P$ and $Q$ of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$.
A quick computation shows that $$\label{search:eq:rhoSTUV}
\begin{aligned}
\rho(S)&=\varphi({\mathbf{x}})=S,\\
\rho(T)&=\Phi({\mathbf{x}},{\mathbf{x}},{\mathbf{x}}+q{\mathbf{y}})=3S+qT,\\
\rho(U)&=\Phi({\mathbf{x}},{\mathbf{x}}+q{\mathbf{y}},{\mathbf{x}}+q{\mathbf{y}})=3S+2qT+q^2U, \\
\rho(V)&=\varphi({\mathbf{x}}+q{\mathbf{y}})=S+qT+q^2U+q^3V.
\end{aligned}$$ From this we deduce that the polynomials $$A:=T^2-3SU \in {\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]_{(4,2)},
\quad
B:=T^3-3TA-27S^2V \in {\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]_{(6,3)}$$ satisfy $$\label{search:eq:rhoAB}
\begin{aligned}
\rho(A)&=(3S+qT)^2-3S(3S+2qT+q^2U)=q^2A,\\
\rho(B)&=(3S+qT)^3-3(3S+qT)q^2A-\dots =q^3B.
\end{aligned}$$ Therefore, they belong respectively to $J^{(2)}$ and $J^{(3)}$. More precisely, $A$ and $B$ have respective $J$-valuations $2$ and $3$, while $S$, $T$, $U$, $V$ have valuation $0$.
According to Proposition \[search:prop:divJk\], the fact that $A$ belongs to $J^{(2)}$ implies that $q_i^2$ divides $A_i:=T_i^2-3S_iV_i$ for each $i\in I$, strengthening the remark made just after Proposition \[search:prop:qi\]. In this context, we note that $$F:=F({\mathbf{x}},{\mathbf{y}})=T^2-4SU=(4A-T^2)/3.$$ So, if for consecutive elements $i<j$ in $I$ the integer $F_i:=F({\mathbf{x}}_i,{\mathbf{x}}_j)$ vanishes, then $T_i^2=4A_i$ is divisible by $4q_i^2$ and thus $q_i$ divides $T_i$. Taking into account the congruences of Proposition \[search:prop:qi\] (b), we deduce that $q_i$ also divides $3p_iS_i$, $U_i$ and $3V_i$. Since $q_i$ is relatively prime to $p_i$, this proves the first part of the following proposition.
\[search:prop:Fzero\] For each $i\in I$ such that $F_i=0$, the integer $q_i$ divides $3S_i$, $T_i$, $U_i$ and $3V_i$. In particular, if $\lambda>2/3$, we have $|q_i| \le 3|S_i|$ for all such large enough indices $i$.
The second part follows from the fact that, when $\lambda>2/3$, the integers $S_i$ are all non-zero except for finitely many indices $i$. In the next section, we will analyze the consequences of this result and show that, if $F_i=0$ for infinitely many $i\in I$, then $\lambda\le (5-\sqrt{13})/2\cong 0.697$.
The ring ${\mathcal{R}}$
------------------------
From now on, we restrict our attention to the graded ring $${\mathcal{R}}=\bigoplus_{\ell\ge 0}{\mathcal{R}}_\ell
\quad \text{where} \quad
{\mathcal{R}}_\ell := {\mathbb{Q}}[S,T,U,V] \cap {\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]_{(2\ell,\ell)} \quad (\ell\ge 0),$$ which contains $F$ but also $T$, $A$, $B$ and $S^2V$. We will say that an element $P$ of ${\mathcal{R}}$ is *homogeneous of degree $\ell$* if it belongs to ${\mathcal{R}}_\ell$ (which is equivalent to asking that $P$ is homogeneous of degree $\ell$ in ${\mathbf{y}}$). Thus, $T$, $A$, $F$, $B$ and $S^2V$ are homogeneous of respective degrees 1, 2, 2, 3 and 3. The next result provides two presentations of ${\mathcal{R}}$ as a weighted polynomial ring in three variables.
\[search:prop:R\] We have ${\mathcal{R}}={\mathbb{Q}}[T,F,S^2V]={\mathbb{Q}}[T,A,B]$. For each $\ell\ge 0$, one basis of the vector space ${\mathcal{R}}_\ell$ over ${\mathbb{Q}}$ consists of the products $T^{\ell-2m-3n}F^m(S^2V)^n$ where $(m,n)$ runs through all pairs of non-negative integers $m$ and $n$ with $2m+3n\le \ell$. Another basis consists of the products $T^{\ell-2m-3n}A^mB^n$ for the same pairs $(m,n)$.
We first note that, for each $a,b,c,d\in {\mathbb{N}}$, the monomial $S^aT^bU^cV^d$ is a bihomogeneous element of ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$ of bidegree $(3a+2b+c,b+2c+3d)$. So it belongs to ${\mathcal{R}}$ if and only if $3a+2b+c=2(b+2c+3d)$, a condition that amounts to $a=c+2d$. Since $S$, $T$, $U$ and $V$ are algebraically independent over ${\mathbb{Q}}$, this implies that the products $$S^{c+2d}T^bU^cV^d = T^b (SU)^c (S^2V)^d \quad (b,c,d\in{\mathbb{N}})$$ form a basis of ${\mathcal{R}}$ as a vector space over ${\mathbb{Q}}$, so ${\mathcal{R}}={\mathbb{Q}}[T,SU,S^2V]$ is a polynomial ring in $3$ variables, and the first assertion of the proposition is easily verified. The other assertions follow from this in view of the degrees of $T$, $A$, $B$, $F$ and $S^2V$.
For each $\ell\in{\mathbb{Z}}$, we denote by $\tau(\ell)$ the number of pairs $(m,n)\in{\mathbb{N}}^2$ with $2m+3n\le \ell$.
In particular, this gives $\tau(\ell)=0$ when $\ell<0$. With this notation at hand, we can now state and prove the main result of this section.
\[search:thm:dim\] For each choice of integers $k,\ell\ge 0$, we have $$\dim_{\mathbb{Q}}{\mathcal{R}}_\ell = \tau(\ell)
{\quad\text{and}\quad}\dim_{\mathbb{Q}}\frac{{\mathcal{R}}_\ell}{{\mathcal{R}}_\ell\cap J^{(k)}}
= \begin{cases}
\tau(k-1) &\text{if $k\le \ell$,}\\
\tau(\ell) &\text{if $k > \ell$.}
\end{cases}$$
The first formula $\dim_{\mathbb{Q}}{\mathcal{R}}_\ell = \tau(\ell)$ is an immediate consequence of the previous proposition. It tells us that, in order to prove the second one, we may restrict to $k\le \ell+1$ because, for $k=\ell+1$, the combination of the two formulas implies that ${\mathcal{R}}_\ell\cap J^{(\ell+1)}=0$ and therefore ${\mathcal{R}}_\ell\cap J^{(k)}=0$ whenever $k>\ell$.
For any $(m,n)\in{\mathbb{N}}^2$ with $2m+3n\le \ell$, the formulas and give $$\label{search:thm:dim:eq1}
\rho(T^{\ell-2m-3n}A^mB^n) = q^{2m+3n}(3S+qT)^{\ell-2m-3n}A^mB^n,$$ and so $T^{\ell-2m-3n}A^mB^n$ belongs to $J^{(k)}$ if $2m+3n\ge k$. In view of the preceding proposition, this means that the quotient ${\mathcal{R}}_\ell/({\mathcal{R}}_\ell\cap J^{(k)})$ is generated, as a ${\mathbb{Q}}$-vector space, by the classes of the products $T^{\ell-2m-3n}A^mB^n$ where $(m,n)$ runs through the elements of ${\mathbb{N}}^2$ with $2m+3n<k$ (recall that $k\le \ell+1$). So, it remains to prove that these classes are linearly independent over ${\mathbb{Q}}$ and for this, we may further assume that $k\ge 1$.
Suppose on the contrary that there exist rational numbers $a_{m,n}$ not all zero, indexed by pairs $(m,n)\in{\mathbb{N}}^2$ with $2m+3n < k$, such that $$\sum_{2m+3n<k} a_{m,n}T^{\ell-2m-3n}A^mB^n \in J^{(k)}.$$ Let $r=\min\{2m+3n\,;\, a_{m,n}\neq 0\}$. By the above, we obtain that $$P:=\sum_{2m+3n=r} a_{m,n}T^{\ell-r}A^mB^n \in J^{(r+1)}.$$ However, the formula implies that $$\begin{aligned}
\rho(P)
&= \sum_{2m+3n=r} a_{m,n}q^r(3S+qT)^{\ell-r}A^mB^n \\
&\equiv q^r(3S)^{\ell-r} \sum_{2m+3n=r} a_{m,n}A^mB^n \mod q^{r+1}.\end{aligned}$$ Thus $\rho(P)$ is not divisible by $q^{r+1}$, and so $P\notin J^{(r+1)}$, a contradiction.
Examples
--------
Besides $S=\varphi$ and $F$, the proof of Theorem \[intro:thm:main\] uses three more auxiliary polynomials $D^{(2)}$, $D^{(3)}$ and $D^{(6)}$ that we now introduce in the form of examples that illustrate the above considerations. Their superscript refers to their $J$-valuation.
Since $\dim_{\mathbb{Q}}({\mathcal{R}}_6/({\mathcal{R}}_6\cap J^{(2)})) = \tau(1)=1$, any pair of elements of ${\mathcal{R}}_6$ are ${\mathbb{Q}}$-linearly dependent modulo $J^{(2)}$. Since $$F = \frac{1}{3}(4A-T^2) \equiv -\frac{T^2}{3}
{\quad\text{and}\quad}S^2V = \frac{1}{27}(T^3-3TA-B) \equiv \frac{T^3}{27} \mod J^{(2)},$$ we obtain $$\label{search:eq:D2}
D^{(2)} := F^3+27(S^2V)^2 \in {\mathcal{R}}_6\cap J^{(2)}.$$
Since $\dim_{\mathbb{Q}}({\mathcal{R}}_6/({\mathcal{R}}_6\cap J^{(3)})) = \tau(2)=2$, the products $F^3$, $TF(S^2V)$ and $(S^2V)^2$ are ${\mathbb{Q}}$-linearly dependent modulo $J^{(3)}$. We find that $$F^3 \equiv \frac{12T^4A-T^6}{27},
\quad
TF(S^2V) \equiv \frac{7T^4A-T^6}{81},
\quad
(S^2V)^2 \equiv \frac{T^6-6T^4A}{27^2}$$ modulo $J^{(3)}$, and thus $$\label{search:eq:D3}
D^{(3)} := F^3-18TF(S^2V)-135(S^2V)^2 \in {\mathcal{R}}_6\cap J^{(3)}.$$
\[search:example:D6\] Since $\dim_{\mathbb{Q}}({\mathcal{R}}_9/({\mathcal{R}}_9\cap J^{(6)})) = \tau(5)=5$, the six products $TF^4$, $F^3(S^2V)$, $T^2F^2(S^2V)$, $TF(S^2V)^2$, $T^3(S^2V)^2$ and $(S^2V)^3$ are ${\mathbb{Q}}$-linearly dependent modulo $J^{(6)}$. Explicitly, this yields $$\label{search:eq:D6}
\begin{aligned}
D^{(6)} := TF^4&+10F^3(S^2V)-11T^2F^2(S^2V)\\
&-180TF(S^2V)^2-T^3(S^2V)^2-675(S^2V)^3 \in {\mathcal{R}}_9\cap J^{(6)}.
\end{aligned}$$ Instead of checking this relation by a direct computation, it is simpler and more useful to derive it from an alternative formula for $D^{(6)}$. To this end, we define $$\label{search:eq:MN}
M:=F^2-3TS^2V,
\quad
N:=D^{(3)}=F^3-18TFS^2V-135(S^2V)^2.$$ It is easy to verify that $M\in{\mathcal{R}}_4\cap J^{(2)}$, and we already know that $N\in{\mathcal{R}}_6\cap J^{(3)}$. Thus any ${\mathbb{Q}}$-linear combination of $M^3$ and $N^2$ belongs to ${\mathcal{R}}_{12}\cap J^{(6)}$. On the other hand, since $$M\equiv F^2 {\quad\text{and}\quad}N\equiv F^3 \mod S^2V,$$ we find that $S^2V$ divides $M^3-N^2$. Since $S^2V$ has $J$-valuation $0$, the quotient is an element of ${\mathcal{R}}_9\cap J^{(6)}$. Expanding the expression $M^3-N^2$, we find that this quotient is $27D^{(6)}$, namely $$\label{search:eq:D6bis}
M^3-N^2 = 27S^2V D^{(6)}.$$
We will need the following consequence of these constructions.
\[search:prop:D2D3D6\] Suppose that $\lambda > (5-\sqrt{13})/2\cong 0.697$. For each $i\in I$, we have
- $|q_i|^2 \ll |F_i|^3 + |S_i^2V_i|^2$ if $D^{(2)}_i\neq 0$,
- $|q_i|^3 \ll |F_i|^3 + |T_iF_iS_i^2V_i|$ if $F_i\neq 0$ and $D^{(3)}_i\neq 0$,
- $|q_i|^6 \ll |T_iF_i^4| + |T_i^3(S_i^2V_i)^2|$ if $F_i\neq 0$ and $D^{(6)}_i\neq 0$.
The statement (a) is a direct consequence of Proposition \[search:prop:divJk\] applied to the polynomial $D^{(2)}$, and does not require any hypothesis on $\lambda$. To prove (b) and (c), we first note that, by Propositions \[prelim:prop:est\_Phi\] and \[F:prop:Fnotzero\] we have $|T_i|\asymp (X_j/X_i)|S_i|$ and $|S_iV_i| = o(X_j/X_i)$ for $i<j$ running through all pairs of consecutive elements of $I$ with $i\to\infty$. Therefore $$|S_i^2V_i| = o(|T_i|)=o(|T_iF_i|)$$ as $i$ goes to infinity through elements of $I$ with $F_i\neq 0$ (assuming, as we may, that there are infinitely many such $i$). Since all monomials that compose $D^{(6)}$ can be obtained by multiplying $TF^4$, $T^2F^2(S^2V)$ and $T^3(S^2V)^2$ by appropriate powers of $S^2V/(TF)$, we deduce that, for the same values of $i$, we have $$|D_i^{(6)}| \ll |T_iF_i^4| + |T_i^2F_i^2S_i^2V_i| + |T_i^3(S_i^2V_i)^2| \ll |T_iF_i^4| + |T_i^3(S_i^2V_i)^2|$$ (in the second estimate, we dropped the middle term as it is the geometric mean of the other two). Similarly, we find that $|D_i^{(3)}| \ll |F_i|^3 + |T_iF_iS_i^2V_i|$. Then, (b) and (c) follow from Proposition \[search:prop:divJk\].
An additional remark {#sec:search:remark}
--------------------
Since ${\mathcal{R}}$ is a polynomial ring over ${\mathbb{Q}}$ in the variables $T$, $F$ and $S^2V$, it is a unique factorization domain and it makes sense to talk about irreducible elements of ${\mathcal{R}}$ although, a priori, such polynomials may not remain irreducible in the ring ${\mathbb{Q}}[{\mathbf{x}},{\mathbf{y}}]$. Similarly we can talk about relatively prime elements of ${\mathcal{R}}$. One can show that the polynomials $D^{(2)}$, $D^{(3)}$ and $D^{(6)}$ constructed above are irreducible elements of ${\mathcal{R}}$. Clearly, $F$ is another one. Therefore the next proposition implies that, for each sufficiently large $i\in I$, at most one of the integers $F_i$, $D^{(2)}_i$, $D^{(3)}_i$ or $D^{(6)}_i$ is zero.
\[search:prop:remark\] Suppose that $\lambda>2/3$, and let $P$, $Q$ be relatively prime homogeneous elements of ${\mathcal{R}}={\mathbb{Q}}[T,F,S^2V]$. Then there are only finitely many $i\in I$ such that $P$ and $Q$ both vanish at the point $(T_i,F_i,S_i^2V_i)$.
The proof uses the following estimate that we will also need later for other purposes.
\[search:lemma:oTcube\] Suppose that $\lambda>2/3$. Then, $|S_i^2V_i| = o(|T_i|^3)$ for $i\in I$.
For consecutive elements $i<j$ in $I$ with $i$ large enough so that $S_i$, $T_i$ and $V_i$ are non-zero, Proposition \[prelim:prop:est\_Phi\] gives $|T_i| \asymp (X_j/X_i)|S_i|$. Then, as $|S_i|\ge 1$, this yields $|S_i^2V_i/T_i^3| \asymp (X_i/X_j)^3 |V_i/S_i| \le (X_i/X_j)^3 |V_i|$. By Corollary \[prelim:cor:est\_varphi\], we also have $|V_i|\ll X_j^2X_{j+1}^{-\lambda} \le X_j^{2-\lambda} \le X_j^{4/3}$, while Corollary \[prelim:cor:XjLj\] gives $X_i^2 \ll X_{i+1}\le X_j$. Combining these estimates, we conclude that $|S_i^2V_i/T_i^3| \ll (X_i/X_j)^3 X_j^{4/3} \ll X_j^{-1/6}=o(1)$.
The hypothesis implies that the de-homogenized polynomials $\bar{P}=P(1,F/T^2,S^2V/T^3)$ and $\bar{Q}=Q(1,F/T^2,S^2V/T^3)$ are relatively prime elements of ${\mathbb{Q}}[F/T^2,S^2V/T^3]$ viewed as polynomials in $2$ variables. Therefore, as such, they have at most finitely many common zeros in ${\bar{{\mathbb{Q}}}}^2$.
Since $\lambda>2/3$, it follows from Proposition \[prelim:prop:est\_Phi\] that there exists an index $i_0$ such that $S_i$, $T_i$ and $V_i$ are all non-zero for each $i\in I$ with $i\ge i_0$. For those $i$, the ratio $S_i^2V_i/T_i^3$ is a non-zero rational number and, by the previous lemma, it tends to $0$ as $i\to\infty$. Thus, there are only finitely many values of $i\ge i_0$ for which $(F_i/T_i^2,S_i^2V_i/T_i^3)$ is a common zero of $\bar{P}$ and $\bar{Q}$, and so there are only finitely many $i\in I$ such that $P$ and $Q$ vanish at $(T_i,F_i,S_i^2V_i)$.
Non-vanishing of F {#sec:NVF}
==================
The main goal of this section is to show that $F_i\neq 0$ for any sufficiently large $i\in I$ if $\lambda >(5-\sqrt{13})/2$. The following result is a first step.
\[NVF:prop:allF0\] Suppose that $F_i=0$ for all but finitely many $i\in I$. Then $\lambda\le 2/3$.
We proceed by contradiction assuming, on the contrary, that $\lambda>2/3$. Then, according to Theorem \[prelim:thm:phi\], we have $S_i\neq 0$ for all but finitely many indices $i$ and so there exists an integer $i_0$ such that $F_i=0$ and $S_i\neq 0$ for all $i\ge i_0$. Put $\epsilon=\lambda-2/3$. For each pair of consecutive elements $i<j$ in $I$ with $i\ge i_0$, Corollary \[F:cor:Fzero\] together with the estimate $|q_i|\asymp X_j/X_{i+1}$ of Proposition \[search:prop:qi\] yields $$|S_iS_j| \ll X_i^{-1} X_{i+1}^{-8/3-4\epsilon} X_j^3
\asymp |q_i|^3 (X_{i+1}^{1/3}/X_i) X_{i+1}^{-4\epsilon}.$$ As $|S_i| \ll X_i^2X_{i+1}^{-\lambda} \ll (X_i/X_{i+1}^{1/3})^2$ (by Corollary \[prelim:cor:est\_varphi\]), we deduce that $$|S_iS_j| \ll |q_i|^3 |S_i|^{-1/2}X_{i+1}^{-4\epsilon}.$$ By Proposition \[search:prop:Fzero\], we also have $|q_i|\le 3|S_i|$ and $|q_j|\le 3|S_j|$. Thus the above estimate yields $$\label{NVF:prop:allF0:eq1}
|q_j| \ll |q_i|^{3/2}X_{i+1}^{-4\epsilon}.$$ In particular, for $i$ large enough, we have $$\log |q_j| \le \frac{3}{2} \log |q_i|.$$ On the other hand, since $\lambda\ge 2/3$, we also have $X_i^2 \ll X_{i+1} \le X_j$ by Corollary \[prelim:cor:XjLj\], thus $\log X_j \ge (2+o(1))\log X_i$ as $i$ goes to infinity in $I$, and so $$\frac{\log |q_j|}{\log X_j} \le \left(\frac{3}{4}+o(1)\right) \frac{\log |q_i|}{\log X_i}$$ showing that the ratio $\log |q_i|/\log X_i$ tends to $0$ as $i$ goes to infinity in $I$. In particular, we must have $1\le |q_i|\le X_i^\epsilon$ for each sufficiently large $i\in I$, in contradiction with .
\[NVF:thm\] Suppose that $\lambda>(5-\sqrt{13})/2 \cong 0.697$. Then, we have $F_i\neq 0$ for each sufficiently large element $i$ of $I$.
Assume, on the contrary, that $F_i=0$ for infinitely many values of $i$. Then, by the previous proposition, there exist arbitrarily large elements $i$ of $I$ such that, upon denoting by $j$ the next element of $I$, we have $F_i\neq0$ and $F_j=0$. We may further assume, by Theorem \[prelim:thm:phi\], that $S_j\neq 0$ and $S_k\neq 0$ where $k$ is the next element of $I$ after $j$. Since $F_j=0$, Corollary \[F:cor:Fzero\] gives $$\label{NVF:thm:eq1}
|S_j| \le |S_jS_k| \ll X_j^{-1}X_{j+1}^{-4\lambda}X_k^3.$$ By Propositions \[search:prop:qi\] and \[search:prop:Fzero\], we also have $X_k \asymp |q_j| X_{j+1} \ll |S_j| X_{j+1}$. Substituting this upper bound for $X_k$ in the previous estimate and then using the standard upper bound $|S_j|\ll X_j^2X_{j+1}^{-\lambda}$ from Corollary \[prelim:cor:est\_varphi\], we obtain $$1 \ll |S_j|^2 X_j^{-1} X_{j+1}^{3-4\lambda} \ll X_j^3 X_{j+1}^{3-6\lambda},$$ and so $X_j\gg X_{j+1}^{2\lambda-1}$. On the other hand, since $F_i\neq 0$, Lemma \[F:lemma:tech\_alpha\_beta\] yields $X_j\ll X_{j+1}^\alpha$ where $\alpha=2\lambda(1-\lambda)/(3\lambda-1)$. As $j$ can be made arbitrarily large, we deduce that $2\lambda -1 \le \alpha$ and so $\lambda \le (7+\sqrt{17})/16 \cong 0.695$, a contradiction.
\[NVF:cor:oT\] Suppose that $\lambda>(5-\sqrt{13})/2 \cong 0.697$. Then, we have $|F_iS_i^2V_i| = o(|T_i|)$ as $i$ goes to infinity in $I$.
According to Proposition \[prelim:prop:est\_Phi\], we have $T_i\neq 0$ and $|T_i|\asymp (X_j/X_i) |S_i|$ for each large enough $i\in I$. By the theorem, we also have $F_i\neq 0$. Then, upon denoting by $j$ the successor of $i$ in $I$, Proposition \[F:prop:Fnotzero\] gives $|F_i|\ll X_j^2L_i^4$. Combining this with the estimates of Corollary \[prelim:cor:est\_varphi\] for $|S_i|$ and $|V_i|=|S_j|$, we obtain $$\left|\frac{F_iS_i^2V_i}{T_i}\right|
\asymp \frac{X_i}{X_j}|F_iS_iV_i|
\ll \frac{X_i}{X_j}(X_j^2L_i^4)(X_i^2L_i)(X_j^2L_j)
\ll X_i^3X_{i+1}^{-5\lambda}X_j^3X_{j+1}^{-\lambda}.$$ Assuming $i$ large enough, we also have $F_h\neq 0$ where $h$ stands for the predecessor of $i$ in $I$. Then Lemma \[F:lemma:tech\_alpha\_beta\] gives $X_i\ll X_{i+1}^\alpha$ (since $F_h\neq 0$) and $X_{j+1}\ll X_{i+1}^\beta$ (since $F_i\neq 0$). We first use the estimate $X_j \ll (X_{i+1}X_{j+1})^{1-\lambda}$ from Proposition \[prelim:prop:height\] to eliminate $X_j$ from our upper bound for $|F_iS_i^2V_i/T_i|$, and then we use the preceding estimates to eliminate $X_i$ and $X_{j+1}$. This yields $|F_iS_i^2V_i/T_i| \ll X_{i+1}^\tau$ where $$\tau = 3\alpha+(3-8\lambda)+(3-4\lambda)\beta
= \frac{\lambda(2\lambda-1)(23-33\lambda)}{(3\lambda-1)(3\lambda-2)}$$ is negative since $\lambda >23/33=0.\overline{69}$.
\[NVF:cor:oq\] Suppose that $\lambda\ge 0.6985$. Then, $|S_i^2V_i| = o(|q_i|)$ for $i\in I$.
The proof is similar to that of Corollary \[NVF:cor:oT\]. Using the estimate $|q_i| \asymp X_j/X_{i+1}$ from Proposition \[search:prop:qi\], we find $$\left|\frac{S_i^2V_i}{q_i}\right|
\ll \frac{X_{i+1}}{X_j}(X_i^2L_i)^2(X_j^2L_j)
\ll X_i^4X_{i+1}^{1-2\lambda}X_jX_{j+1}^{-\lambda}
\ll X_i^4X_{i+1}^{2-3\lambda}X_{j+1}^{1-2\lambda},$$ and so $|S_i^2V_i/q_i| \ll X_{i+1}^\tau$ with $$\tau = 4\alpha+(2-3\lambda)+(1-2\lambda)\frac{2\lambda}{\alpha}
= -\frac{\lambda^3+13\lambda^2-11\lambda+1}{(3\lambda-1)(1-\lambda)}
< 0.$$ The difference with the proof of Corollary \[NVF:cor:oT\] is that we used the lower bound $X_{j+1} \gg X_{i+1}^{2\lambda/\alpha}$ of Lemma \[F:lemma:tech\_alpha\_beta\] to eliminate $X_{j+1}$, as it appears with the negative exponent $1-2\lambda$.
The last result below motivates the non-vanishing results for $D^{(2)}$ that we prove in the next section.
\[NVF:prop:D2nonzero\] Suppose that $\lambda\ge 0.6985$. For any pair of consecutive elements $i<j$ of $I$ satisfying $F_i^3+27(S_i^2V_i)^2\neq 0$, with $i$ large enough so that $F_i\neq 0$, we have $$|q_i|^2 \ll |F_i|^3
{\quad\text{and}\quad}X_{i+1} \ll X_j^{2/(6\lambda-1)} \ll X_{j+1}^{(2-2\lambda)/(8\lambda-3)}.$$ If there are infinitely many such pairs $(i,j)$, then $\lambda < 0.709$.
The inequality $|q_i|^2 \ll |F_i|^3$ follows immediately from Proposition \[search:prop:D2D3D6\] (a) combined with the preceding Corollary. Using $|q_i| \asymp X_j/X_{i+1}$ (Proposition \[search:prop:qi\]) and $|F_i| \ll X_j^2L_i^4$ (Proposition \[F:prop:Fnotzero\]), it yields $X_{i+1}\ll X_j^{2/(6\lambda-1)}$. Then, using $X_j\ll (X_{i+1}X_{j+1})^{1-\lambda}$ (Proposition \[prelim:prop:height\]) to further eliminate $X_{i+1}$, we find $X_j^{2/(6\lambda-1)} \ll X_{j+1}^{(2-2\lambda)/(8\lambda-3)}$. This proves the first assertion of the proposition. Moreover, the combination of $X_{i+1}\ll X_j^{2/(6\lambda-1)}$ with $X_j \ll X_{i+1}^{\lambda(1-\lambda)/(3\lambda-2)}$ (Lemma \[F:lemma:tech\_alpha\_beta\]) yields $(6\lambda-1)(3\lambda-2) \le 2\lambda(1-\lambda)$ if there are infinitely many such pairs $(i,j)$, and the second assertion follows.
Non-vanishing of $D^{(2)}$ {#sec:NVD2}
==========================
The main result of this section is that $D_i^{(2)}\neq 0$ for each sufficiently large $i\in I$ if $\lambda \ge 0.6985$. We start by establishing algebraic consequences of a possible vanishing.
\[NVD2:prop\] Suppose that $D^{(2)}_i=F_i^3+27S_i^4V_i^2=0$ for some $i\in I$. Then, there exists a unique integer $R_i$ for which $F_i=-3R_i^2$ and $S_i^2V_i=R_i^3$. This integer satisfies $$\label{NVD2:prop:eq1}
\gcd(q_i,R_i)=\gcd(q_i,S_i).$$ Moreover, $q_i^6$ divides $cS_i^7(T_i-3R_i)^2$ for some integer constant $c>0$ not depending on $i$.
We rewrite the hypothesis in the form $(-F_i/3)^3=(S_i^2V_i)^2$. Since $S_i^2V_i\in{\mathbb{Z}}$, the first assertion of the proposition follows. Then, using the equality $\gcd(q_i,S_i)=\gcd(q_i,V_i)$ from Proposition \[search:prop:qi\] (c), we deduce that $$\gcd(q_i,R_i)^3 = \gcd(q_i^3,S_i^2V_i) = \gcd(q_i,S_i)^3$$ which yields . We also find that $$\begin{aligned}
4A_i &= T_i^2+3F_i = (T_i-3R_i)(T_i+3R_i),\\
4B_i &= T_i^3-9T_iF_i-108S_i^2V_i = (T_i-3R_i)(T_i^2+3R_iT_i+36R_i^2).
\end{aligned}$$ Since $q_i^2\mid A_i$ and $q_i^3\mid B_i$, this means that $q_i^6$ divides both $$(T_i-3R_i)^3(T_i+3R_i)^3 {\quad\text{and}\quad}(T_i-3R_i)^2(T_i^2+3R_iT_i+36R_i^2)^2$$ and therefore $q_i^6$ divides $cR_i^7(T_i-3R_i)^2$ where $c$ denotes the resultant of the polynomials $(x-3y)(x+3y)^3$ and $(x^2+3xy+36y^2)^2$. This integer $c$ is non-zero since these polynomials have no common zero in ${\mathbb{P}}^1({\mathbb{C}})$. Using , we conclude that $q_i^6$ divides $$c\gcd(q_i^7,R_i^7)(T_i-3R_i)^2 = c\gcd(q_i^7,S_i^7)(T_i-3R_i)^2$$ and so it divides $cS_i^7(T_i-3R_i)^2$.
By a similar method, one can show that $$q_i \mid 3S_i(T_i-3R_i),
\quad
q_i^2 \mid 9S_i^2(T_i-3R_i)
{\quad\text{and}\quad}q_i^3 \mid 162S_i^4(T_i-3R_i),$$ but these divisibility relations can also be derived, up to the value of the constant, from the one of the proposition.
\[NVD2:prop:cor\] Suppose that $\lambda>(5-\sqrt{13})/2 \cong 0.697$. Then, for all pairs of consecutive elements $i<j$ of $I$ with $F_i^3+27S_i^4V_i^2=0$ and $i$ large enough so that $F_i\neq 0$, we have $$X_i^2X_j^4 \ll X_{i+1}^6 |S_i|^9
{\quad\text{and}\quad}|S_i^2S_j| \ll X_{i+1}^{-6\lambda}X_j^3.$$
For these pairs $(i,j)$, the integer $R_i$ defined in the proposition satisfies $R_i^2\asymp |F_i|$. Since Corollary \[NVF:cor:oT\] gives $|F_i|=o(|T_i|)=o(T_i^2)$, we deduce that $|T_i-3R_i| \asymp |T_i| \asymp (X_j/X_i) |S_i|$, where the last estimate comes from Proposition \[prelim:prop:est\_Phi\]. In particular, if $i$ is large enough, the integer $cS_i^7(T_i-3R_i)^2$ is non-zero and, as it is divisible by $q_i^6$, we obtain $$q_i^6 \le c|S_i|^7 |T_i-3R_i|^2 \asymp (X_j/X_i)^2 |S_i|^9.$$ Since $|q_i| \asymp X_j/X_{i+1}$ (by Proposition \[search:prop:qi\]), this yields the first estimate. The second one follows directly from the upper bound $|F_i| \ll X_j^2L_i^4$ of Proposition \[F:prop:Fnotzero\] together with $S_i^4S_j^2 = |F_i|^3/27$.
\[NVD2:thm\] Suppose that $\lambda\ge 0.6985$. Then, we have $F_i^3+27S_i^4V_i^2\neq 0$ for any large enough $i\in I$.
Suppose on the contrary that the set $I_2:=\{i\in I\,;\, F_i^3+27S_i^4V_i^2=0\}$ is infinite. As a first step, suppose also that $I\setminus I_2$ is infinite. Then, there exists infinitely many triples of consecutive elements $i<j<k$ of $I$ with $i\in I\setminus I_2$ and $j\in I_2$. Choosing them large enough, we may further assume that $F_j\neq 0$. Since $F_j^3+27S_j^4V_j^2=0$, we have $S_j\neq 0$, $S_k=V_j\neq 0$ and the previous corollary gives $$\label{NVD2:thm:eq1}
X_j^2X_k^4 \ll X_{j+1}^6 |S_j|^9
{\quad\text{and}\quad}|S_j^2S_k| \ll X_{j+1}^{-6\lambda}X_k^3,$$ while Corollary \[prelim:cor:est\_varphi\] gives $$\label{NVD2:thm:eq2}
|S_j| \ll X_j^2X_{j+1}^{-\lambda}.$$ Eliminating $|S_j|$ between and , and using the lower bound $|S_k|\ge 1$ to further eliminate $|S_k|$ from the resulting estimates, we obtain $$X_{j+1}^{9\lambda-6}X_k^4 \ll X_j^{16}
{\quad\text{and}\quad}X_j^4 X_{j+1}^{54\lambda-12} \ll X_k^{19}.$$ Then, eliminating $X_k$, we find $X_{j+1}^{43\lambda-18} \ll X_j^{32}$. On the other hand, since $F_i^3+27S_i^4V_i^2\neq 0$, Proposition \[NVF:prop:D2nonzero\] gives $X_j \ll X_{j+1}^{(6\lambda-1)(1-\lambda)/(8\lambda-3)}$. Combining the latter two estimates and noting that they hold for infinitely many $j$, we conclude that $(43\lambda-18)(8\lambda-3) \le 32(6\lambda-1)(1-\lambda)$, in contradiction with our hypothesis that $\lambda\ge 0.6985$.
Thus $I\setminus I_2$ is a finite set and so, there exists an integer $i_0$ such that $F_i\neq 0$ and $F_i^3+27S_i^4V_i^2 = 0$ for each $i\in I$ with $i\ge i_0$. Suppose now that the sequence $(|S_i|)_{i\in I}$ is unbounded. Then, there exist infinitely many triples of consecutive elements $i<j<k$ of $I$ with $i\ge i_0$ and $|S_j|\le |S_k|$. For these triples, the inequalities and are again satisfied. We use the hypothesis $|S_j|\le |S_k|$ to eliminate $|S_k|$ from the second inequality of and then eliminate $|S_j|$ from the resulting three inequalities. This yields $$X_{j+1}^{9\lambda-6}X_k^4 \ll X_j^{16}
{\quad\text{and}\quad}X_j^2X_{j+1}^{18\lambda-6} \ll X_k^5.$$ Then, eliminating $X_k$, we find $X_{j+1}^{13\lambda-6} \ll X_j^{8}$. Since Lemma \[F:lemma:tech\_alpha\_beta\] also gives $X_j\ll X_{j+1}^{\alpha}$ and since $j$ can be taken arbitrarily large, we conclude that $13\lambda-6 \le 8\alpha$ which again contradicts our hypothesis on $\lambda$. This means that the sequence $(|S_i|)_{i\in I}$ is bounded and so, the first inequality of yields $X_j^2X_k^4 \ll X_{j+1}^6$ for infinitely many pairs of consecutive elements $i<j$ of $I$. Then using the estimates $X_{j+1}^{\lambda/2}\ll X_j$ and $X_{j+1}^{2\lambda} \ll X_k$ coming from Lemma \[F:lemma:tech\_alpha\_beta\], we conclude that $X_{j+1}^{9\lambda} \ll X_{j+1}^6$ for the same pairs $(i,j)$ and so $\lambda\le 2/3$, a contradiction.
\[NVD2:thm:cor\] Suppose that $\lambda\ge 0.7034$. Then, $|T_iF_iS_i^2V_i|=o(|q_i|^3)$ for $i\in I$.
Let $h<i<j$ be consecutive elements of $I$. Applying the usual estimates for $|q_i|$, $|T_i|$, $|F_i|$, $|S_i|$ and $|V_i|=|S_j|$, as in the proofs of Corollaries \[NVF:cor:oT\] and \[NVF:cor:oq\], we find $$|q_i|^{-3}|T_iF_iS_i^2V_i|
\asymp X_i^{-1}X_{i+1}^3X_j^{-2} |F_iS_i^3V_i|
\ll X_i^5 X_{i+1}^{3-7\lambda} X_j^2 X_{j+1}^{-\lambda}.$$ Using the upper bounds $X_i\ll (X_{h+1}X_{i+1})^{1-\lambda}$ and $X_j \ll (X_{i+1}X_{j+1})^{1-\lambda}$ of Proposition \[prelim:prop:height\] to eliminate $X_i$ and $X_j$, we obtain $$|q_i|^{-3}|T_iF_iS_i^2V_i|
\ll X_{h+1}^{5-5\lambda} X_{i+1}^{10-14\lambda} X_{j+1}^{2-3\lambda}.$$ By Proposition \[NVF:prop:D2nonzero\] and Theorem \[NVD2:thm\], we also have $$X_{h+1} \ll X_{i+1}^{(2-2\lambda)/(8\lambda-3)}
{\quad\text{and}\quad}X_{i+1}^{(8\lambda-3)/(2-2\lambda)} \ll X_{j+1}.$$ As $5-5\lambda>0>2-3\lambda$, we conclude that $|q_i|^{-3}|T_iF_iS_i^2V_i| \ll X_{i+1}^\tau$ with $$\tau = (5-5\lambda)\frac{2-2\lambda}{8\lambda-3} + (10-14\lambda) + (2-3\lambda)\frac{8\lambda-3}{2-2\lambda}
< 0.
\qedhere$$
We conclude this section with the following consequence of the above corollary.
\[NVD2:prop:D3D6\] Suppose that $\lambda\ge 0.7034$. Then, for each $i\in I$ large enough so that $T_i\neq 0$ and $F_i\neq 0$, we have $$\begin{aligned}
|q_i| &\ll |F_i| &&\text{if $D^{(3)}_i\neq 0$,}\\
|q_i|^6 &\ll |T_iF_i^4| &&\text{if $D^{(3)}_i\neq 0$ and $D^{(6)}_i\neq 0$.}
\end{aligned}$$
Suppose that $D_i^{(3)}\neq 0$. Then, the estimate $|T_iF_iS_i^2V_i|=o(|q_i|^3)$ of the previous corollary combined with Proposition \[search:prop:D2D3D6\] (b) yields $|q_i| \ll |F_i|$. From this we deduce that $|T_iF_iS_i^2V_i| = o(|F_i|^3)$, so $|T_iS_i^2V_i|=o(|F_i|^2)$ and thus $|T_i^3(S_i^2V_i)^2| = o(|T_iF_i^4|)$. Combining the latter estimate with Proposition \[search:prop:D2D3D6\] (c) yields $|q_i|^6 \ll |T_iF_i^4|$ if $D^{(6)}_i\neq 0$.
Non-vanishing of $D^{(3)}$ and $D^{(6)}$ {#sec:NVD3D6}
========================================
We first prove non-vanishing results for $D^{(3)}$ and $D^{(6)}$, and then prove Theorem \[intro:thm:main\] as a consequence of these and of the above Proposition \[NVD2:prop:D3D6\].
\[NVD3D6:propD3\] Suppose that $\lambda>(5-\sqrt{13})/2 \cong 0.697$. Then $D_i^{(3)}\neq 0$ and $D_i^{(6)}\neq 0$ for each sufficiently large $i\in I$.
Suppose that $D_i^{(3)}=0$ for some $i\in I$ large enough so that $S_i\neq 0$ and $V_i\neq 0$. Then, we have $F_i(F_i^2-18T_iS_i^2V_i)=135S_i^4V_i^2$, so $F_i$ is a non-zero divisor of $135S_i^4V_i^2$ and therefore $1\le |F_i| \ll S_i^4V_i^2$. On the other hand, Corollary \[NVF:cor:oT\] shows that $|F_iS_i^2V_i|=o(|T_i|)$. This yields $$\max\{|F_i|^3,\,|S_i^4V_i^2|\} \ll |F_i^2S_i^4V_i^2| = o(|T_iF_iS_i^2V_i|)$$ and therefore $0 = D_i^{(3)} = -(18+o(1))T_iF_iS_i^2V_i$. Thus $i$ must be bounded from above. This proves the non-vanishing assertion for $D^{(3)}$.
For $D^{(6)}$, we proceed by contradiction assuming, on the contrary, that $I_6:=\{i\in I\,;\, D_i^{(6)}=0\}$ is an infinite set. For each large enough $i\in I_6$, the integers $F_i$, $S_i$, $T_i$ and $V_i$ are all non-zero and the formula of Example \[search:example:D6\] yields $0=M_i^3-N_i^2$ where $$M_i=F_i^2-3T_iS_i^2V_i
{\quad\text{and}\quad}N_i=D_i^{(3)}=F_i^3-18T_iF_iS_i^2V_i-135S_i^4V_i^2.$$ Then, we can write $M_i=R_i^2$ and $N_i=R_i^3$ for a unique integer $R_i$ and the above formulas become $$\label{NVD3D6:propD6:eq1}
\begin{aligned}
F_i^2-R_i^2 &=3T_iS_i^2V_i, \\
F_i^3-R_i^3 &=18T_iF_iS_i^2V_i+135S_i^4V_i^2.
\end{aligned}$$ In particular, we have $F_i-R_i\neq 0$. Since Corollary \[NVF:cor:oT\] shows that $S_i^2V_i=o(|T_i/F_i|)=o(|T_iF_i|)$, we deduce that $$F_i^3-R_i^3 = (18+o(1))T_iF_iS_i^2V_i = (6+o(1))F_i(F_i^2-R_i^2)$$ which, after division by $F_i^2(F_i-R_i)$, yields $$(R_i/F_i)^2-(5+o(1))(R_i/F_i)-(5+o(1))=0.$$ Thus the sequence of ratios $(R_i/F_i)_{i\in I_6}$ has at most two irrational accumulation points, and so $|R_i|\asymp |F_i| \asymp |F_i-R_i| \asymp |F_i+R_i|$ for each large $i\in I_6$. The first equality in then gives $$\label{NVD3D6:propD6:eq2}
F_i^2 \asymp |T_iS_i^2V_i|.$$ The equalities also imply that $F_i-R_i$ divides $135S_i^4V_i^2$ and therefore $$|F_i| \asymp |F_i-R_i| \le 135 |S_i^4V_i^2|.$$ Substituting this upper bound for $|F_i|$ into , we obtain $$|T_i| \ll |S_i^6V_i^3|.$$ Viewing this as a lower bound for $|S_i^2V_i|$, we also deduce from that $$|T_i|^{4/3} \ll F_i^2.$$ Let $j$ denote the successor of $i$ in $I$. Applying the usual estimates for $|F_i|$, $|S_i|$, $|T_i|$ and $|V_i|=|S_j|$, as in the proof of Corollary \[NVF:cor:oT\], the last two inequalities yield $$\begin{aligned}
1 &\ll |T_i|^{-1}|S_i^6V_i^3|
\asymp \frac{X_i}{X_j}|S_i|^5|V_i|^3
\ll \frac{X_i}{X_j}(X_i^2L_i)^5(X_j^2L_j)^3
= X_i^{11} X_j^5 L_i^5 L_j^3, \\
1 &\ll |T_i|^{-1}|F_i|^{3/2}
\asymp \frac{X_i}{X_j |S_i|} |F_i|^{3/2}
\ll \frac{X_i}{X_j}(X_j^2L_i^4)^{3/2}
= X_i X_j^2 L_i^6.\end{aligned}$$ Then, using $L_i\ll X_{i+1}^{-\lambda}$, $L_j\ll X_{j+1}^{-\lambda}$ and $X_j\ll (X_{i+1}X_{j+1})^{1-\lambda}$ to eliminate $L_i$, $L_j$ and $X_j$ from the above upper bounds, we obtain $$1 \ll X_i^{11} X_{i+1}^{5-10\lambda} X_{j+1}^{5-8\lambda}
{\quad\text{and}\quad}1 \ll X_i X_{i+1}^{2-8\lambda} X_{j+1}^{2-2\lambda}.$$ For $i$ large enough, we also have $F_h\neq 0$ where $h$ denotes the predecessor of $i$ of $I$, and so Lemma \[F:lemma:tech\_alpha\_beta\] gives $X_i \ll X_{i+1}^\alpha$ where $\alpha = 2\lambda(1-\lambda)/(3\lambda-1)$. Applying this to eliminate $X_i$ from the preceding estimates, we obtain $$X_{j+1}^{8\lambda-5} \ll X_{i+1}^{11\alpha+5-10\lambda}
{\quad\text{and}\quad}X_{i+1}^{8\lambda-2-\alpha} \ll X_{j+1}^{2-2\lambda}$$ and so $(8\lambda-2-\alpha)(8\lambda-5) \le (11\alpha+5-10\lambda)(2-2\lambda)$. This is the required contradiction.
In view of the above result, we may rewrite Proposition \[NVD2:prop:D3D6\] as follows.
\[NVD3D6:cor\] Suppose that $\lambda\ge 0.7034$. Then, we have $|q_i|\ll |F_i|$ and $|q_i|^6 \ll |T_iF_i^4|$ for each $i\in I$ large enough so that $T_i\neq 0$ and $F_i\neq 0$.
Suppose on the contrary that $\lambda >\mu=2(9+\sqrt{11})/35$. Applying the usual estimates, as in the proof of Corollary \[NVD2:thm:cor\], the above corollary yields, for each large enough $i\in I$, $$1 \ll |q_i|^{-6}|T_iF_i^4|
\ll X_i X_{i+1}^{6-17\lambda} X_j^3
\ll X_{h+1}^{1-\lambda} X_{i+1}^{10-21\lambda} X_{j+1}^{3(1-\lambda)}$$ where $h$ denotes the predecessor of $i$ in $I$ and $j$ its successor. We also have $$1 \le |S_j|
\ll X_j^2X_{j+1}^{-\lambda}
\ll X_{i+1}^{2(1-\lambda)} X_{j+1}^{2-3\lambda}.$$ Since $\lambda>\mu$, these estimates remain valid if we substitute $\mu$ for $\lambda$ in both of them. More precisely, for each large enough $i\in I$, we have $$1 \le X_{h+1}^{1-\mu} X_{i+1}^{10-21\mu} X_{j+1}^{3(1-\mu)}
{\quad\text{and}\quad}1 \le X_{i+1}^{2(1-\mu)}(2X_{j+1})^{2-3\mu}.$$ Put $$\nu = \frac{2(1-\mu)}{3\mu-2} = 2+\sqrt{11}.$$ Then, the last two inequalities are respectively equivalent to $$\left(\frac{X_{i+1}^\nu}{X_{j+1}}\right)^{3\nu} \le \frac{X_{h+1}^\nu}{X_{i+1}}
{\quad\text{and}\quad}2 \le \frac{X_{i+1}^\nu}{X_{j+1}}\,.$$ This is impossible since the second inequality shows that, for all large enough pairs of consecutive elements $i<j$ of $I$, the ratios $X_{i+1}^\nu/X_{j+1}$ are bounded below by $2$, while the first implies that they decrease to $1$ as $i$ goes to infinity in $I$.
A general family of auxiliary polynomials {#sec:pol}
=========================================
Let $d\in {\mathbb{N}}$. For each non-empty subset $E$ of the set $${\mathcal{T}}_d:=\{ (m,n)\in{\mathbb{N}}^2 \,;\, 2m+3n\le d \},$$ we choose a non-zero polynomial $P_E$ of ${\mathcal{R}}_d$ of the form $$P_E = \sum_{(m,n)\in E} a_{m,n} T^{d-2m-3n}F^m(S^2V)^n$$ whose $J$-valuation is maximal (i.e. which lies in $J^{(\ell)}$ for a largest possible $\ell\in {\mathbb{N}}$). We will not need its precise $J$-valuation but just the fact that, by Theorem \[search:thm:dim\], we have $P_E\in J^{(k+1)}$ if $|E|>\tau(k)$ for some integer $k\ge -1$. We denote by ${\mathcal{P}}_d$ the finite set of all polynomials $P_E$ as $E$ runs through the non-empty subsets of ${\mathcal{T}}_d$. The goal of this section is to prove the following result.
\[pol:thm\] Suppose that $\lambda >\lambda_0:=(1+3\sqrt{5})/11 \cong 0.7007$. Then there exists a positive integer $d$ and a polynomial $P\in{\mathcal{P}}_d$ such that $P({\mathbf{x}}_i,{\mathbf{x}}_j)=0$ for infinitely many pairs of consecutive elements $i<j$ in $I$.
We start with two simple lemmas.
\[pol:lemma:E\] Let $a,b,c>0$ and $E=\{(m,n)\in{\mathbb{N}}^2 \,;\, am+bn\le c \}$. Then, $$\frac{c^2}{2ab} \le |E| \le \frac{(a+b+c)^2}{2ab}.$$ In particular, for each $d\in {\mathbb{N}}$, we have $$\frac{d^2}{12} \le \tau(d)=|{\mathcal{T}}_d| \le \frac{(d+5)^2}{12}.$$
For each $z\in{\mathbb{R}}$, set ${\mathcal{T}}(z):=\{(x,y)\in{\mathbb{R}}^2\,;\, x,y\ge 0 \text{ and } ax+by\le z \}$. Then, $${\mathcal{T}}(c) \subseteq \bigcup_{(m,n)\in E} \Big( (m,n)+[0,1]^2 \Big) \subseteq {\mathcal{T}}(a+b+c)$$ and the estimates for $|E|$ follow by computing the area of the three regions. The estimates for $\tau(d)$ correspond to the choice of parameters $a=2$, $b=3$, $c=d$, since for these we have $|E|=\tau(d)$.
\[pol:lemma:est\] Suppose that $\lambda>(5-\sqrt{13})/2\cong 0.697$. For each sufficiently large element $i$ of $I$, we have $$1\le |F_iS_i^2V_i| \le |T_i|
{\quad\text{and}\quad}|q_i|\le |T_i|\le |q_i|^3.$$
The first series of inequalities is a direct consequence of Corollary \[NVF:cor:oT\] together with the fact that $F_i$, $S_i$ and $V_i$ do not vanish for each sufficiently large $i$ (by Theorems \[prelim:thm:phi\] and \[NVF:thm\]). For the second series, we first recall that, by Proposition \[prelim:prop:est\_Phi\], we have $|T_i| \asymp X_iX_j\delta_i$ for each pair of consecutive elements $i<j$ in $I$ with $i$ large enough. Combining this with the estimates $X_i^{-2} \ll \delta_i \ll X_{i+1}^{-\lambda}$ of Corollary \[prelim:cor:est\_varphi\], and $X_i\ll X_{i+1}^\alpha$ of Lemma \[F:lemma:tech\_alpha\_beta\], we obtain $$X_{i+1}^{-\alpha} X_j \ll |T_i| \ll X_{i+1}^{\alpha-\lambda} X_j.$$ Since $|q_i|\asymp X_j/X_{i+1}$ and $\alpha < 0.4$, this yields $|q_i| =o(|T_i|)$. Using the estimate $X_j\gg X_{i+1}^{2\lambda}$ from Proposition \[F:prop:Fnotzero\], it also yields $$|T_i| \ll |q_i|^3 X_{i+1}^{3.4-\lambda} X_j^{-2}
\ll |q_i|^3 X_{i+1}^{3.4-5\lambda}
=o(|q_i|^3).
\qedhere$$
The next result provides the main tool in the proof of Theorem \[pol:thm\].
\[pol:prop\] Suppose that $\lambda>(5-\sqrt{13})/2\cong 0.697$ and let $\epsilon>0$. Then, there exist integers $d=d(\epsilon)$ and $i_0=i_0(\epsilon)$ with the following property. For each pair of consecutive elements $i<j$ in $I$ with $i\ge i_0$ such that $$\label{pol:prop:eq1}
\log\left|\frac{T_i^2}{F_i}\right| \log\left|\frac{T_i^3}{S_i^2V_i}\right|
\ge
6(\log|T_i|)^2 - (6-\epsilon)(\log|q_i|)^2,$$ there exists a polynomial $P\in {\mathcal{P}}_d$ such that $P({\mathbf{x}}_i,{\mathbf{x}}_j)=0$.
Fix $i\in I$ satisfying , with $i$ large enough so that we have $|q_i|\ge 3$ and that all the inequalities of Lemma \[pol:lemma:est\] are satisfied. Then, none of the integers $F_i$, $S_i$, $T_i$, $V_i$ is zero, and we may define $$f=\frac{\log|F_i|}{\log|q_i|},
\quad
s=\frac{\log|S_i^2V_i|}{\log|q_i|},
\quad
t=\frac{\log|T_i|}{\log|q_i|}
{\quad\text{and}\quad}\sigma=(2t-f)(3t-s).$$ More precisely, we have $|F_i|\ge 1$, $|S_i^2V_i| \ge 1$, and so Lemma \[pol:lemma:est\] yields $$\label{pol:prop:eqfst}
0 \le f \le t, \quad 0 \le s\le t {\quad\text{and}\quad}1 \le t\le 3,$$ while the hypothesis becomes $$\label{pol:prop:eqsigma}
\sigma \ge 6t^2-(6-\epsilon).$$ Let $d\in{\mathbb{N}}^*$ and let $E$ denote the set of points $(m,n)\in{\mathcal{T}}_d$ satisfying $$\label{pol:prop:eq2}
\left| T_i^{d-2m-3n} F_i^m (S_i^2V_i)^n \right| \le |q_i|^k,
\quad\text{with}\quad
k:=\left[\frac{6td}{\sigma+6}\right].$$ We claim that we can choose $d$ so that $|E| > \tau(k)$ when $i$ is large enough. If we take this for granted, then we have $P_E\in J^{(k+1)}$ and so $q_i^{k+1}$ divides $P_E({\mathbf{x}}_i,{\mathbf{x}}_j)$, where $j$ denotes the successor of $i$ in $I$. On the other hand, the conditions imply that $|P_E({\mathbf{x}}_i,{\mathbf{x}}_j)| \le c(d) |q_i|^k$ with a constant $c(d)$ depending only on $d$. Thus, for $i$ large enough, we have $|P_E({\mathbf{x}}_i,{\mathbf{x}}_j)| < |q_i|^{k+1}$ and so $P_E({\mathbf{x}}_i,{\mathbf{x}}_j)=0$, as requested by the theorem.
To prove the claim, we first note that the condition is equivalent to $$(d-2m-3n)t+mf+ns \le k {\quad\Longleftrightarrow\quad}(2t-f)m +(3t-s)n \ge dt-k.$$ Applying Lemma \[pol:lemma:E\] with $a=2t-f$, $b=3t-s$, $c=dt-k$, and noting that these values of $a$, $b$, $c$ are positive in view of and , we find that the set of points $(m,n)\in{\mathbb{N}}^2$ which do not satisfy this condition has cardinality at most $(dt-k+5t)^2/(2\sigma)$. So we obtain $$|E| \ge |{\mathcal{T}}_d|-\frac{(dt-k+5t)^2}{2\sigma}.$$ The same lemma also gives $|{\mathcal{T}}_d| = \tau(d) \ge d^2/12$ and $\tau(k) \le (k+5)^2/12$. Therefore the condition $|E| > \tau(k)$ is fulfilled if $$\label{pol:prop:eq3}
\frac{d^2}{12}-\frac{(dt-k+5t)^2}{2\sigma} > \frac{(k+5)^2}{12}.$$ We need to show that this inequality holds as soon as $d$ is large enough, independently of the values of $f$, $s$, $t$ and $\sigma$ satisfying and , as these depend on $i$. By , we have $2t^2\le \sigma\le 6t^2$ and $1\le t\le 3$. Thus, holds if $$d^2 - \frac{6}{\sigma}\left(td-\frac{6td}{\sigma+6}\right)^2 -\left(\frac{6td}{\sigma+6}\right)^2
\ge c_1d+c_2,$$ for some absolute constants $c_1$ and $c_2$. After simplifications, this becomes $$\left(\frac{\sigma-6t^2+6}{\sigma+6}\right) d^2 \ge c_1d+c_2.$$ Using and the crude estimate $\sigma+6\le 6t^2+6\le 60$, we find that the latter inequality holds if $(\epsilon/60) d^2 \ge c_1d+c_2$. So it holds as soon as $d$ is sufficiently large in terms of $\epsilon$ only.
Suppose on the contrary that no such pair $(d,P)$ exists. Then, for each integer $d\ge 1$, there are only finitely many pairs of consecutive elements $i<j$ of $I$ for which $({\mathbf{x}}_i,{\mathbf{x}}_j)$ is a zero of at least one of the polynomials in the finite set ${\mathcal{P}}_d$. Therefore, the preceding proposition shows that, for each $\epsilon >0$ and each $i\in I$ with $i\ge i_0(\epsilon)$, we have $$\label{pol:thm:eq:main}
\log\left|\frac{T_i^2}{F_i}\right| \log\left|\frac{T_i^3}{S_i^2V_i}\right|
< 6(\log|T_i|)^2 - (6-\epsilon)(\log|q_i|)^2.$$ Our first goal is to replace this condition by an inequality involving only $\log X_{h+1}$, $\log X_{i+1}$ and $\log X_{j+1}$ where, as usual, $h$ denotes the element of $I$ that comes immediately before $i$, and $j$ the one that comes immediately after. To this end, we first note that the hypothesis $\lambda>\lambda_0=(1+3\sqrt{5})/11$ implies that $L_i = o(X_{i+1}^{-\lambda_0})$ and $L_j = o(X_{j+1}^{-\lambda_0})$. So, for $i$ is large enough, we have $$\begin{aligned}
&|T_i| \le {\overline{T}}_i:=X_iX_jX_{i+1}^{-\lambda_0}
&&\text{by Proposition \ref{prelim:prop:est_Phi}},
\label{pol:thm:eq:majT}\\
&|F_i| \le {\overline{F}}_i:=X_j^2X_{i+1}^{-4\lambda_0}
&&\text{by Proposition \ref{F:prop:Fnotzero}},
\label{pol:thm:eq:majF}\\
&|S_i^2V_i| \le {\overline{G}}_i:=(X_i^2X_{i+1}^{-\lambda_0})^2(X_j^2X_{j+1}^{-\lambda_0})
&&\text{by Corollary \ref{prelim:cor:est_varphi}},
\label{pol:thm:eq:majSV}\\
&X_i \le X_{h+1}^{1-\lambda}X_{i+1}^{1-\lambda_0},
\quad
X_j \le (X_{i+1}X_{j+1})^{1-\lambda_0}
&&\text{by Proposition \ref{prelim:prop:height}}.
\label{pol:thm:eq:majX}\end{aligned}$$ Moreover, the estimates of Lemma \[F:lemma:tech\_alpha\_beta\] combined with $|q_i| \asymp X_j/X_{i+1}$ from Proposition \[search:prop:qi\] imply that $$\log |q_i| =\log(X_j/X_{i+1})+{\mathcal{O}}(1) \asymp \log X_{i+1} \asymp \log X_{h+1}.$$ So, there is a constant $c>0$, independent of the choice of $\epsilon>0$, such that, for $i$ large enough, we have $$\label{pol:thm:eq:minq}
(6-\epsilon)(\log |q_i|)^2 \ge 6(\log(X_j/X_{i+1}))^2-c\epsilon(\log X_{h+1})^2.$$ Assume that $i$ is large enough so that – hold. If we subtract the left hand side of from its right hand side and expand the resulting expression as a polynomial in $\log |T_i|$, we find a linear polynomial whose coefficient of $\log |T_i|$ is $\log |F_i^3(S_i^2V_i)^2| \ge 0$. Therefore remains true if we replace everywhere $|T_i|$ by the upper bound ${\overline{T}}_i$ given by . By and , we also have $$\label{pol:thm:eq:factors}
\frac{{\overline{T}}_i^2}{|F_i|} \ge \frac{{\overline{T}}_i^2}{{\overline{F}}_i} = X_i^2X_{i+1}^{2\lambda_0}
{\quad\text{and}\quad}\frac{{\overline{T}}_i^3}{|S_i^2V_i|} \ge \frac{{\overline{T}}_i^3}{{\overline{G}}_i} = X_i^{-1}X_{i+1}^{-\lambda_0}X_jX_{j+1}^{\lambda_0}.$$ As both of these lower bounds are greater than $1$, they have a positive logarithm. So, we may further replace $|F_i|$ by ${\overline{F}}_i$ and $|S_i^2V_i|$ by ${\overline{G}}_i$, and thus $$\log\frac{{\overline{T}}_i^2}{{\overline{F}}_i} \log\frac{{\overline{T}}_i^3}{{\overline{G}}_i}
< 6(\log{\overline{T}}_i)^2 -(6-\epsilon)(\log |q_i|)^2.$$ Using , and , this yields $$\begin{aligned}
0
&< -\log(X_i^2X_{i+1}^{2\lambda_0})\log(X_i^{-1}X_{i+1}^{-\lambda_0}X_jX_{j+1}^{\lambda_0}) \\
&\qquad + 6 \big(\log(X_iX_jX_{i+1}^{-\lambda_0})\big)^2
- 6 \big(\log(X_jX_{i+1}^{-1})\big)^2 + c\epsilon (\log X_{h+1})^2 \\
&= 2(\log X_i)\log \big( X_i^4 X_{i+1}^{-4\lambda_0} X_j^5 X_{j+1}^{-\lambda_0} \big) \\
&\qquad + 2(\log X_{i+1})\log \big( X_{i+1}^{4\lambda_0^2-3} X_j^{6-7\lambda_0} X_{j+1}^{-\lambda_0^2} \big)
+ c\epsilon (\log X_{h+1})^2.
\end{aligned}$$ Note that, in this last expression, the first product is positive for $i$ large enough because, using $X_i\gg X_{i+1}^{\lambda_0/2}$ and $X_j\gg X_{j+1}^{\lambda_0/2}$ (Lemma \[F:lemma:tech\_alpha\_beta\]), we find that $$X_i^4 X_{i+1}^{-4\lambda_0} X_j^5 X_{j+1}^{-\lambda_0}
\gg X_{i+1}^{-2\lambda_0} X_j^3
\ge X_j^{3-2\lambda_0}$$ tends to infinity with $i$. This allows us to use to eliminate both $X_i$ and $X_j$. After simplifications, this yields $$\label{pol:thm:eq:mainbis}
\begin{aligned}
0 < &(4(1-\lambda)^2+c\epsilon/2)(\log X_{h+1})^2 \\
&+ (1-\lambda)(13-17\lambda_0)(\log X_{h+1})(\log X_{i+1})\\
&+ (1-\lambda)(5-6\lambda_0)(\log X_{h+1})(\log X_{j+1})\\
&+ (12-35\lambda_0+24\lambda_0^2)(\log X_{i+1})^2\\
&+ (11-24\lambda_0+12\lambda_0^2)(\log X_{i+1})(\log X_{j+1})
\end{aligned}$$ Now, put $$\epsilon=\frac{8(1-\lambda_0)^2-8(1-\lambda)^2}{c},\quad
\rho_i=\frac{\log X_{i+1}}{\log X_{h+1}}, \quad
\rho_j=\frac{\log X_{j+1}}{\log X_{i+1}},$$ thus fixing the choice of $\epsilon>0$. We substitute this value of $\epsilon$ into and note that the resulting inequality remains valid if we replace $\lambda$ by $\lambda_0$. After dividing both sides by $(\log X_{h+1})^2$, it yields $$\label{pol:thm:eq:mainter}
\begin{aligned}
0 < 4(1-\lambda_0)^2
&+ (1-\lambda_0)(13-17\lambda_0)\rho_i + (1-\lambda_0)(5-6\lambda_0)\rho_i\rho_j\\
&+ (12-35\lambda_0+24\lambda_0^2)\rho_i^2 + (11-24\lambda_0+12\lambda_0^2)\rho_i^2\rho_j.
\end{aligned}$$
Suppose that there are arbitrarily large pairs of consecutive elements $i<j$ in $I$ with $\rho_j<\rho_i$. Then, holds with $\rho_j$ replaced by $\rho_i$ because in the right hand side of this inequality all terms involving $\rho_j$ have positive coefficients. This means that $$\label{pol:thm:eq:rho}
\begin{aligned}
0 < 4(1-\lambda_0)^2
&+ (1-\lambda_0)(13-17\lambda_0)\rho_i\\
&+ (17-46\lambda_0+30\lambda_0^2)\rho_i^2 + (11-24\lambda_0+12\lambda_0^2)\rho_i^3.
\end{aligned}$$ On the other hand, Lemma \[F:lemma:tech\_alpha\_beta\] gives $\rho_i \le \beta+o(1)$ with $\beta=2(1-\lambda)/(3\lambda-2)$ and so, for $i$ large enough, we have $$\rho_i < \beta_0:=\frac{2(1-\lambda_0)}{3\lambda_0-2}=\frac{5+3\sqrt{5}}{2}.$$ This is a contradiction because can be rewritten in the form $$\label{pol:thm:eq:factor}
0 < (\rho_i-\beta_0)(a\rho_i^2-b\rho_i-c)
\quad
\text{with $a>b>c>0$}$$ while it follows from Corollary \[prelim:cor:XjLj\] that $\rho_i\ge 2+o(1)$ and so $a\rho_i^2-b\rho_i-c>0$ for each sufficiently large $i$.
Thus we have $\rho_i\le \rho_j$ for each sufficiently large pairs of consecutive elements $i<j$ in $I$. Then $(\rho_i)_{i\in I}$ converges to a limit $\rho$ with $2\le \rho\le \beta < \beta_0$ and, by continuity, the inequality holds with $\rho_i$ and $\rho_j$ replaced by $\rho$. So holds with $\rho_i$ replaced by $\rho$. Again, this is impossible.
\[pol:remark:rel\_prime\] As the proof of Proposition \[pol:prop\] shows, when holds, we get several polynomials $P$ satisfying $P({\mathbf{x}}_i,{\mathbf{x}}_j)=0$ for the same pairs $(i,j)$ by varying the integer $d$. If we could make these relatively prime as a set, this would contradict Proposition \[search:prop:remark\], meaning that holds for any given $\epsilon>0$ and any sufficiently large $i\in I$. Then, the above argument would yield $\lambda \le (1+3\sqrt{5})/11$.
Assuming that $\lambda = (1+3\sqrt{5})/11$, we also note that holds for any given $\epsilon>0$ and any sufficiently large $i\in I$ when $$X_j \asymp X_i^{(5+3\sqrt{5})/2}, \quad
X_{i+1} \asymp X_i^{(3\sqrt{5}-1)/2}, \quad
L_i \asymp X_{i+1}^{-\lambda} \asymp X_i^{-2},$$ where $j$ stands for the successor of $i$ in $I$. Then, one finds that $|S_i|\asymp |V_i|\asymp 1$, $|T_i| \asymp X_i^{3(1+\sqrt{5})/2}$, $|F_i|\asymp X_i^{3(\sqrt{5}-1)}$ and $|q_i|\asymp X_i^3$.
A special family of auxiliary polynomials {#sec:special}
=========================================
In view of Remark \[pol:remark:rel\_prime\] above, we would get $\lambda \le (1+3\sqrt{5})/11$ if we could prove for example that the polynomials of ${\mathcal{P}}_d$ are irreducible for arbitrarily large values of $d$. This is probably too much to ask. Nevertheless, this can be done for certain polynomials of the type $P_E$ as the next result illustrates.
\[special:thm\] Let $d=12\ell+2$ for some $\ell\in{\mathbb{N}}$, and let $$E_\ell=\left\{ (m,n)\in{\mathbb{N}}^2\,;\, 2m+3n\le d, \ \frac{m}{6\ell+1}+\frac{n}{3\ell}\ge 1 \right\}.$$ Then the set of polynomials of ${\mathcal{R}}_d \cap J^{(6\ell+2)}$ of the form $$\label{special:thm:eq:P}
\sum_{(m,n)\in E_\ell} a_{m,n} T^{d-2m-3n}F^m(S^2V)^n$$ is a one-dimensional vector space generated by an irreducible polynomial $P_\ell$ of ${\mathcal{R}}$ which has $a_{6\ell+1,0}\neq 0$ and $a_{0,3\ell}\neq 0$.
Before going into its proof, we deduce the following consequence.
\[special:cor\] Suppose that $\lambda>2/3$, and let $\epsilon>0$. For each $i\in I$ large enough so that $S_iT_iV_i\neq 0$, we have $|F_i|\gg |q_i|$ or $|T_iS_i^2V_i|\gg |q_i|^{2-\epsilon}$ with implied constants depending only on $\xi$ and $\epsilon$.
Each $E_\ell$ is the set of integral points in the triangle with vertices $(6\ell+1,0)$, $(0,3\ell)$ and $(0,4\ell+(2/3))$. So, for consecutive $i<j$ in $I$, we have $$\begin{aligned}
|P_\ell({\mathbf{x}}_i,{\mathbf{x}}_j)|
&\ll_{\xi,P} |F_i|^{6\ell+1} + |T_i|^{3\ell+2}|S_i^2V_i|^{3\ell} + |S_i^2V_i|^{4\ell+(2/3)}\\
&\ll |F_i|^{6\ell+1} + |T_iS_i^2V_i|^{3\ell+2}\end{aligned}$$ where the second inequality uses the estimate $|S_i^2V_i|=o(|T_i|^3)$ from Lemma \[search:lemma:oTcube\]. If $P_\ell({\mathbf{x}}_i,{\mathbf{x}}_j)\neq 0$, Proposition \[search:prop:divJk\] also gives $|q_i|^{6\ell+2}\le |P_\ell({\mathbf{x}}_i,{\mathbf{x}}_j)|$, thus $$\label{special:cor:eq}
|F_i| \gg_{\xi,\ell} |q_i|^{(6\ell+2)/(6\ell+1)} \ge |q_i|
\quad\text{or}\quad
|T_iS_i^2V_i| \gg_{\xi,\ell} |q_i|^{(6\ell+2)/(3\ell+2)}.$$ Choose $\ell$ to be the smallest positive integer such that $(6\ell+2)/(3\ell+2) \ge 2-\epsilon$. Since $P_\ell$ and $P_{\ell+1}$ are irreducible of distinct degrees, they are relatively prime and Proposition \[search:prop:remark\] shows that at least one of them does not vanish at the point $({\mathbf{x}}_i,{\mathbf{x}}_j)$ for $i$ sufficiently large. Then holds with the given value of $\ell$ or with $\ell$ replaced by $\ell+1$, and the result follows.
As a first step towards the proof of Theorem \[special:thm\], we first note that the irreducibility of $P_\ell$ derives simply from the non-vanishing of its coefficients of indices $(6\ell+1,0)$ and $(0,3\ell)$.
\[special:lemma:irred\] With the notation of Theorem \[special:thm\], let $P$ be a non-zero polynomial of the form with non-zero coefficients of indices $(6\ell+1,0)$ and $(0,3\ell)$. Then $P$ is an irreducible element of ${\mathcal{R}}$.
Suppose on the contrary that $P$ is not irreducible. Then it factors as a product $P=AB$ where $A$ and $B$ are homogeneous elements of ${\mathcal{R}}$ of degree less than $d$. For the present purpose, we define the index of each monomial $T^kF^m(S^2V)^n$ with $(k,m,n)\in{\mathbb{N}}^3$ as $\imath(m,n)=m/(6\ell+1)+n/(3\ell)$. We also define the index $\imath(Q)$ of an arbitrary non-zero element $Q$ of ${\mathcal{R}}$ as the *smallest* index of its monomials. Then, we have $\imath(P)=\imath(A)+\imath(B)$ and the homogeneous part $P_0$ of $P$ of smallest index $\imath(P)$ is the product of the homogeneous parts $A_0$ of $A$ and $B_0$ of $B$ with smallest index. However, the function $\imath$ is injective on the set of pairs $(m,n)\in{\mathbb{N}}^2$ with $2m+3n<12\ell+2$ because its kernel on ${\mathbb{Z}}^2$ is the subgroup generated by $(6\ell+1,-3\ell)$. Thus $A_0$ and $B_0$ are monomials. This is impossible because the hypothesis implies that $\imath(P)=1$ with $P_0$ involving the monomials associated with $(6\ell+1,0)$ and $(0,3\ell)$.
The polynomials $F$, $G:=TS^2V$, $H:=S^4V^2$ are homogeneous elements of ${\mathcal{R}}$ of respective degrees $2$, $4$, $6$. They generate a graded subalgebra of ${\mathcal{R}}$, $${\mathcal{S}}:={\mathbb{Q}}[F,G,H]=\bigoplus_{\ell=0}^\infty {\mathcal{S}}_{2\ell},$$ where ${\mathcal{S}}_{2\ell}={\mathcal{S}}\cap{\mathcal{R}}_{2\ell}$ admits, as a basis, the products $F^kG^mH^n$ with $(k,m,n)\in{\mathbb{N}}^3$ satisfying $k+2m+3n=\ell$. In particular, we have $$\label{special:eq:dimS}
\dim_{\mathbb{Q}}{\mathcal{S}}_{2\ell}=\tau(\ell).$$ Moreover, the formulas show that ${\mathcal{S}}={\mathbb{Q}}[F,M,N]$. As $F$, $M$ and $N$ are homogeneous of respective degrees $2$, $4$ and $6$, it follows that the products $F^kM^mN^n$ with $(k,m,n)\in{\mathbb{N}}^3$, $k+2m+3n=\ell$, form another basis of ${\mathcal{S}}_{2\ell}$ over ${\mathbb{Q}}$. The connection with the current situation is the following.
\[specvail:lemma:decomp\] With the notation of Theorem \[special:thm\], the set of polynomials of the form constitutes the vector space ${\mathcal{V}}_\ell:={\mathcal{S}}_{12\ell+2} \oplus \langle G^{3\ell}T^2 \rangle_{\mathbb{Q}}$.
For any $\ell\in {\mathbb{N}}$, the vector space ${\mathcal{S}}_{2\ell}$ is generated by the products $T^kF^m(S^2V)^n$ with $(k,m,n)\in{\mathbb{N}}^3$ satisfying $k+2m+3n=2\ell$ and $k\le n$. So, equivalently, it is generated by the products $T^{2\ell-2m-3n}F^m(S^2V)^n$ with $(m,n)\in{\mathbb{N}}^2$ satisfying $2m+3n\le 2\ell$ and $\ell \le m+2n$. Let ${\mathcal{T}}_{2\ell}^*$ denote this subset of ${\mathbb{N}}^2$. Then, the conclusion follows by observing that we have $E_\ell = {\mathcal{T}}_{12\ell+2}^* \cup \{(0,3\ell)\}$.
The proof of the next result is very similar to that of Theorem \[search:thm:dim\], based on the fact that $M$ and $N$ have respective $J$-valuations $2$ and $3$, so we omit its proof.
\[special:lemma:dimS\] For each choice of integers $k,\ell\ge 0$, we have $$\dim_{\mathbb{Q}}{\mathcal{S}}_{2\ell} = \tau(\ell)
{\quad\text{and}\quad}\dim_{\mathbb{Q}}\frac{{\mathcal{S}}_{2\ell}}{{\mathcal{S}}_{2\ell}\cap J^{(k)}}
= \begin{cases}
\tau(k-1) &\text{if $k\le \ell$,}\\
\tau(\ell) &\text{if $k > \ell$.}
\end{cases}$$ Moreover, if $k\le \ell$, a basis of ${\mathcal{S}}_{2\ell}\cap J^{(k)}$ over ${\mathbb{Q}}$ is given by the products $F^{\ell-2m-3n}M^mN^n$ with $(m,n)\in{\mathbb{N}}^2$ satisfying $k\le 2m+3n\le \ell$.
In view of Lemmas \[special:lemma:irred\] and \[specvail:lemma:decomp\], we need to show that ${\mathcal{V}}_\ell \cap J^{(6\ell+2)}$ is one-dimensional generated by a polynomial $P_\ell$ whose coefficients of $F^{6\ell+1}$ and $G^{3\ell}T^2$ are both non-zero.
By the formulas of Lemma \[special:lemma:dimS\], we have ${\mathcal{S}}_{2\ell}\cap J^{(\ell+1)}=\{0\}$ for each $\ell \ge 0$. In particular, this gives ${\mathcal{S}}_{12\ell+2}\cap J^{(6\ell+2)}=\{0\}$, and so $\dim_{\mathbb{Q}}({\mathcal{V}}_\ell\cap J^{(6\ell+2)})\le 1$. Moreover, since ${\mathcal{V}}_\ell$ has dimension $\tau(6\ell+1)+1$, Theorem \[search:thm:dim\] implies that it contains a non-zero element of $J^{(6\ell+2)}$. From this, we conclude that ${\mathcal{V}}_\ell\cap J^{(6\ell+2)}$ is one-dimensional generated by a polynomial $P_\ell$ outside of ${\mathcal{S}}_{12\ell+2}$. So, the coefficient of $G^{3\ell}T^2$ in $P_\ell$ is non-zero, and it remains to show the same for the coefficient of $F^{6\ell+1}$.
For this purpose, we note that, since $HT^2=G^2$, we have $H{\mathcal{V}}_\ell \subseteq {\mathcal{S}}_{12\ell+8}$ and thus $HP_\ell$ belongs to ${\mathcal{S}}_{12\ell+8} \cap J^{(6\ell+2)}$. By Lemma \[special:lemma:dimS\], the latter vector space is generated by the products $F^{6\ell+4-2m-3n}M^mN^n$ with $(m,n)\in{\mathbb{N}}^2$ satisfying $6\ell+2\le 2m+3n \le 6\ell+4$, thus $$\label{special:thm:eq:HP}
HP_\ell =
\sum_{k=0}^\ell (r_kFN + s_kF^2M + t_kM^2)M^{3k}N^{2\ell-2k}$$ with $r_k,s_k,t_k\in {\mathbb{Q}}$ ($0\le k\le \ell$) not all zero. In order to fix the choice of $P_\ell$ up to multiplication by $\pm 1$, we request that these coefficients are relatively prime integers. Since $P_\ell \in {\mathcal{S}}_{12\ell+2}+\langle G^{3\ell}T^2 \rangle_{\mathbb{Q}}$, they must satisfy $$\sum_{k=0}^\ell (r_kFN + s_kF^2M + t_kM^2)M^{3k}N^{2\ell-2k} \equiv a G^{3\ell+2} \mod H,$$ for some $a\in{\mathbb{Z}}$. A priori, this is a congruence in ${\mathcal{S}}$ but we may view it as a congruence in ${\mathbb{Z}}[F,G,H]$ because both sides belong to that ring. Let $(2,H)$ denote the ideal of ${\mathbb{Z}}[F,G,H]$ generated by $2$ and $H$. Since $M$ and $N$ are respectively congruent to $F^2+G$ and $F^3$ modulo $(2,H)$, this yields $$\begin{aligned}
\sum_{k=0}^\ell
\Big(r_kF^{6\ell-6k+4}(F^2+G&)^{3k}
+ s_kF^{6\ell-6k+2}(F^2+G)^{3k+1}\\
&+ t_kF^{6\ell-6k}(F^2+G)^{3k+2}\Big) \equiv aG^{3\ell+2} \mod (2,H).\end{aligned}$$ Substituting $G+F^2$ for $G$ in this congruence, it becomes $$\begin{aligned}
\sum_{k=0}^\ell
\Big(r_kF^{6\ell-6k+4}G^{3k}
&+ s_kF^{6\ell-6k+2}G^{3k+1}\\
&+ t_kF^{6\ell-6k}G^{3k+2}\Big) \equiv a(G+F^2)^{3\ell+2} \mod (2,H),\end{aligned}$$ which by comparing coefficients on both sides yields $$r_k\equiv a\binom{3\ell+2}{3k},\quad
s_k\equiv a\binom{3\ell+2}{3k+1},\quad
t_k\equiv a\binom{3\ell+2}{3k+2}
\mod 2.$$ As $r_k,s_k,t_k$ are not all even, $a$ must be odd and the above congruences determine the parity of all coefficients. We also observe that $M$ and $N$ are respectively congruent to $F^2$ and $F^3+H$ modulo $(2,G)$, and so $$HP_\ell
\equiv
\sum_{k=0}^\ell (r_k(F^3+H) + s_kF^3 + t_kF^3)F^{6k+1}(F^3+H)^{2\ell-2k} \mod (2,G).$$ In particular, the coefficient $b$ of $F^{6\ell+1}$ in $P_\ell$ is an integer with $$b
\equiv \sum_{k=0}^\ell \big(r_k(2\ell-2k+1) + (s_k+t_k)(2\ell-2k)\big)
\equiv \sum_{k=0}^\ell r_k
\mod 2.$$ On the other hand, it is known that $\sum_{k=0}^\ell \binom{3\ell+2}{3k}=(2^{3\ell+2}-(-1)^\ell)/3$. Thus, $b$ is odd and therefore non-zero.
[9]{}
H. Davenport, W. M. Schmidt, Approximation to real numbers by quadratic irrationals, *Acta Arith. ***13** (1967), 169-176. H. Davenport, W. M. Schmidt, Approximation to real numbers by algebraic integers, *Acta Arith. ***15** (1969), 393–416. V. Jarník, Zum Khintchineschen Übertragungssatz, [*Trudy Tbilisskogo mathematicheskogo instituta im. A. M. Razmadze = Travaux de l’Institut mathématique de Tbilissi*]{} [**3**]{} (1938), 193–212. M. Laurent, Simultaneous rational approximation to the successive powers of a real number, [*Indag. Math. (N.S.)*]{} [**11**]{} (2003), 45–53. S. Lozier, On simultaneous approximation to a real number and its cube, M.Sc. thesis, U. of Ottawa, 2010, 86 pp. D. Roy, Approximation to real numbers by cubic algebraic integers I, *Proc. London Math. Soc. ***88** (2004), 42–62. D. Roy, On simultaneous rational approximations to a real number, its square, and its cube, *Acta Arith. ***133** (2008), 185–197. W. M. Schmidt, *Diophantine Approximations and Diophantine equations*, Lecture Notes in Math., vol. 1467, Springer-Verlag, 1991.
[^1]: Research partially supported by NSERC
|
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author:
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Johannes Walcher\
*Departments of Physics and Mathematics, McGill University,\
*Montréal, Québec, Canada**
date: January 2012
title: |
On the Arithmetic of D-brane Superpotentials.\
Lines and Conics on the Mirror Quintic
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Introduction
============
The purpose of this paper is to continue pushing the limit of the calculation of D-brane superpotentials using the methods developed in [@mowa; @newissues]. The object of study is the value $$\eqlabel{critical}
\calw(z) = \calw(u;z)|_{\del_u\calw=0}$$ of the spacetime superpotential, at the critical point in the open string direction, compactly denoted by $u$, as a function of the closed string moduli, $z$. Note right away that does not depend on the ambiguous off-shell parameterization of the open string moduli space, and is as such a true holomorphic invariant of the underlying D-brane configuration.[^1]
As in [@open; @newissues], we have in mind a comparison between three different points of view on $\calw(z)$. The mathematically best defined framework is the B-model. In that context, $z\in M$ is the complex structure parameter of a family of Calabi-Yau threefolds $\caly\to M$. We denote the manifold of modulus $z$ by $Y_z$, or simply $Y$ when $z$ is generic. In standard cases, the D-brane is associated with a family of holomorphic vector bundles $\cale$ over $\caly$, or more generally an object in the derived category $D^b(Y)$ varying appropriately with $z$. The off-shell superpotential $\calw(u;z)$ which measures the obstructions to deforming $\cale$ in the open string directions, $u$, as a function of the closed string moduli, $z$, is given by the holomorphic Chern-Simons functional, or an appropriate extension or dimensional reduction thereof for more general objects of $D^b(Y)$. A general effective description of the on-shell superpotential involves the (truncated) normal function $\nu_\calc(z)$ associated to a family of algebraic cycles $\calc$ that if required can be obtained as the algebraic second Chern class of $\cale$. See [@mowa] for detailed explanations.
Near a singular point of maximal unipotent monodromy of the family $\caly$, one can obtain a dual, A-model, point of view on $\calw$. The D-brane there is manufactured starting from a Lagrangian submanifold, $L$, of the mirror Calabi-Yau $X$, together with a flat connection. While classically the deformations of $L$ are unobstructed, worldsheet instanton corrections may induce a superpotential that lifts the D-brane’s moduli space. The critical points of the superpotential can be identified with unobstructed objects of the Fukaya category. The difficult problem is to properly count the holomorphic discs with boundary on the Lagrangian that give rise to that superpotential.
The third point of view, developed by Ooguri and Vafa [@oova], comes from embedding into the type IIA/B superstring compactified on $X$/$Y$ for A/B-model respectively. One considers the effective two-dimensional theory living on a D-brane partially wrapped on $L$ or $\cale$, and extended along a two-dimensional subspace $\reals^2
\subset\reals^4$. According ref. [@oova], the superpotential $\calw$ not only controls the supersymmetric vacua, but is also related, via its expansion in the appropriate limits, to the BPS content of the two-dimensional theory. Mirror symmetry relates the choice of $(X,L)$ and $(\caly,\cale)$ and implies that the superpotentials computed in A- and B-model are identical. Duality with M-theory explains the relation to the BPS content of the two-dimensional theory.
The essence of the mirror correspondence is that while the calculations from A-model or spacetime perspective are forbiddingly difficult in general, the B-model is relatively straightforward. On the other hand, the interpretation in terms of novel geometric invariants is best (though by no means completely) understood in the A-model, and most interesting, from the spacetime perspective. Interesting mathematics is everywhere. In this paper, we will present new B-model calculations whose successful A-model and space-time interpretation could force a significant extension of the reach of these models, into number theory.
Detailed calculations in open string mirror symmetry are now available in a variety of situations. Motivated and guided by a number of works involving non-compact manifolds [@agva; @akv; @mayr; @lmw], a quantitative mirror correspondence involving D-branes on the quintic was established in [@open]. Further works involving compact manifolds include [@krwa; @knsc; @joso; @ghkk; @ahmm; @agbe; @japs1; @japs2; @latest]. In all these examples, the underlying manifold was selected from the beginning of the long and well-known list of complete intersections in toric varieties, for instance hypersurfaces in weighted projective spaces. (The noteworthy exception is [@japs2], which deals with Pfaffian Calabi-Yau manifolds.) The choice of cycle in the B-model followed by exploiting divisibility properties of weights of specific monomials. In a sense, these D-branes were as close as they could be, to the “toric branes” that are customarily studied in the context of non-compact examples. Somewhat by accident, a subset of those cycles turned out to be relevant as the mirror of real slices of the A-model manifold.
In this work, we return to the quintic Calabi-Yau, with a somewhat different rationale for selecting the D-branes. As far as A-model is concerned, methods for constructing Lagrangian submanifolds of compact Calabi-Yau manifolds are in short supply. As far as the B-model, holomorphic vector bundles are much easier to produce, perhaps surpassed in simplicity only by matrix factorizations. The simplest constructions, however, pull vector bundles back from projective space, which results in rather boring superpotentials (at least on-shell). Matrix factorizations have the additional disadvantage that they do not come with a readily usable version of holomorphic Chern-Simons. Since to obtain an interesting holomorphic invariant, what we really need is a non-trivial algebraic cycle class. So we might as well construct the representative cycle $\calc$, directly, and then calculate as in [@mowa]. Although this seems like a reasonable approach to finding new D-branes, it is not systematically developed. So a large initial portion of this work is concerned with identifying appropriate $\calc$.
There are several motivations for pursuing this direction. First of all, these methods will definitely take us further away from the set of toric branes, and we can prepare ourselves for unexpected new phenomena. (As mentioned above, the calculations in [@japs2] are outside the toric realm. However, the complication there is introduced in the bulk, , at the level of the Calabi-Yau, while the D-branes follow the simpler pattern, conjecturally related to the real A-branes.) In due course, these results will shape expectations in investigating A-branes and their invariants.
Another broad motivation for this work is a more systematic exploration of the set of all possible D-branes for fixed closed string background, from the holomorphic point of view. This is related on the one hand to speculations about background independence in this context [@newa]. On the other hand, a better overview over the set of all D-branes might also be important for realizing open/closed string correspondence on compact manifolds. In the context of the topological string, the invariant holomorphic information contained in the superpotential could be the minimal amount necessary.
To organize the advance, we recall again that on general grounds, all possible on-shell superpotentials in the sense of are realized geometrically as truncated normal functions. On families of Calabi-Yau threefolds, we are looking for the image of the Chow group ${\rm CH}^2(Y)$ of algebraic cycles modulo rational equivalence, under the Abel-Jacobi map. A natural filtration on ${\rm CH}^2(Y)$ is the minimal degree of a curve representing a given cycle class. This minimal degree is our organizing principle. In this paper, we will proceed up to degree 1 and 2 on the mirror quintic, which as it turns out are already immensely interesting.
We begin in section \[vangeemen\] with a review of what is known about lines on the mirror quintic, and then proceed to calculate the inhomogeneous Picard-Fuchs equation associated with van Geemen lines. The dissection of conics in section \[conics\] is perhaps hard to follow, so we have attempted a shorter summary in section \[sofar\]. The first part of section \[largecomplex\] is warmly recommended, as well as a glance at eq. . Section \[largevolume\] contains the main results, and section \[discussion\] is a best attempt at an interpretation.
Lines on the Mirror Quintic {#vangeemen}
===========================
Investigations into low degree curves on the quintic have a long history reaching back before the beginning of mirror symmetry, and provided important information regarding the latter’s enumerative predictions. Instead of the generic quintic, we are here concerned with curves on the special one-parameter family of mirror quintics. This family is related to the vanishing locus of the polynomial $$\eqlabel{WW}
W= \frac {x_1^5}5 + \frac{x_2^5}5+\frac{x_3^5}5 + \frac{x_4^5}5
+ \frac{x_5^5}5 - \psi x_1x_2x_3x_4x_5$$ in five homogeneous complex coordinates $(x_1,\ldots,x_5)$, and the one parameter, $\psi$. We denote by $Y$ the generic quintic in $\projective^4$. By $Y_\psi$ we denote the member of the Dwork family $\caly\to M$ for fixed $\psi$, given by $\{W=0\}\subset\projective^4$. The actual mirror quintic is of course the resolution of the quotient of $Y_\psi$ by $(\zet_5)^3$. This is useful to keep in mind, but will play only a minor role in the present discussion.
2875 = 375 + 2500
-----------------
The space of lines on the one-parameter family of mirror quintics has been investigated thoroughly by Mustaţǎ [@mustatathesis], building on the earlier work [@albanokatz]. The main results of [@mustatathesis] is the following: for fixed generic $\psi$, the quintic $Y_\psi$ contains precisely $375$ isolated lines, and $2$ (isomorphic) families of lines, each parameterized by a genus $626$ curve. One of the isolated lines is the coordinate line $$\eqlabel{coordinate}
x_1+x_2=0 \,,\qquad
x_3+x_4=0\,,\qquad
x_5=0\,,$$ while the others are obtained by either permuting the $(x_1,\ldots,x_5)$, or inserting a fifth root of unity in the first two equations. This leads to the count $5!/2^3\cdot 5^2
=375$.
Special members of the families can easily be written down. If $\omega$ is a non-trivial third root of unity, and $(a,b)$ satisfy the equations $$\eqlabel{mostly}
a^5+b^5=27\,,\qquad \psi ab = 6$$ then the line $$\eqlabel{perhaps}
\begin{split}
x_1+\omega x_2+\omega^2 x_3 &=0 \\
a(x_1+x_2+x_3) - 3 x_4 &=0\\
b(x_1+x_2+x_3)-3 x_5 &=0
\end{split}$$ lies on the quintic $Y_\psi$. (This is easiest to see by parameterizing solutions of as $$(x_1,x_2,x_3,x_4,x_5) = (u+v,u+\omega v,u+\omega^2 v,au,bu)$$ where $(u,v)$ are homogeneous coordinates on $\projective^1$. Then plugging this into , and using $1+\omega+\omega^2=0$, gives $u^5(3+a^5+b^5-5\psi ab)+
u^2 v^3(30-5\psi ab)=0$ which directly yields .)
Taking into account the phase and permutation symmetries, one obtains a set of $5000$ lines, called van Geemen lines. This being more that the number of lines on the generic quintic threefold, which is $2875$, was historically important because it allowed the conclusion that there exist families of lines on the generic member $Y_\psi$ of the family . The structure of the families at fixed $\psi$, as mentioned above, was worked out only more recently, and consists of two curves of genus $626$.
Anticipating results of our Abel-Jacobi calculations in the next subsection, we note how it will distinguish the two families of lines: exchanging $a$ and $b$ is equivalent to exchanging $x_4$ and $x_5$, from which the holomorphic three-form and hence the normal function, and superpotential, pick up a minus sign. In a slightly different way, changing the choice of third root of unity, , the transformation $$\eqlabel{firstgalois}
\omega\mapsto\omega^2$$ is equivalent to exchanging $x_2$ and $x_3$, and hence also inverts the Abel-Jacobi image.
The global structure of the families of lines, with varying $\psi$, was also worked out in [@mustatathesis]: the curves parameterizing the families containing the van Geemen lines fit together to a single smooth irreducible surface, whose Stein factorization (, the collapse of the connected components in the fibers) gives a double cover of $\psi$-space, with branch points at $\psi=0$, and $\psi^5=\frac{128}3$. This is the discriminant locus of the equations . In particular, the two families that are distinguished for fixed $\psi$ are exchanged as one moves around in the complex structure moduli space. Quite importantly however, $\psi=\infty$ is not a branch point, so in particular, the choice of third root of unity $\omega$ is a good invariant to distinguish classes of D-branes, in the large complex structure limit.
Let us record this as the first instance of an intriguing observation: the mapping is nothing but the Galois group of the number field generated by $\omega$ (which is the imaginary quadratic number field $\rationals(\sqrt{-3})$). The statement about Abel-Jacobi means that [*the space-time superpotential furnishes a non-trivial representation of the Galois group of the number field over which the D-brane is defined.*]{} Following our sober discussion, this might not seem so very surprising. But it has some astonishing consequences that we will explore later on.
To close, we repeat here the count of lines which shows that the isolated lines and the two families account for all rational curves of degree $1$ on the family of mirror quintics. (That is, for generic values of $\psi$. At $\psi=0$, for instance, all lines belong to families, as explained in [@albanokatz], and exploited frequently.) The families contributing with the Euler characteristic of their parameter space gives, $$2\cdot (2\cdot 626-2)+375 = 2500 +375=2875$$
Inhomogeneous Picard-Fuchs equation
-----------------------------------
We need to recall a minimum of material from [@mowa; @newissues]: if $Y$ is a Calabi-Yau threefold, and $C\subset Y$ a holomorphic curve, it makes a contribution to the superpotential [@witten] $$\eqlabel{understanding}
\calw(z) = \int^C\Omega$$ This depends on the complex structure parameter $z$ via the choice of a holomorphic three-form $\Omega$ on $Y$, which is to be integrated over a three-chain $\Gamma$ ending on $C$. We have written with the understanding that the actual physical invariant quantities are the tensions of BPS domain walls (or masses of BPS solitons), which are given by the difference of superpotential values at the critical points, so $\Gamma$ is then the three-chain interpolating between two homologous holomorphic curves.
The reason that makes sense even when $C$ is non-trivial in homology is that we calculate $\calw$ as a solution of the inhomogeneous Picard-Fuchs equation, $$\call\calw(z) = f(z)$$ where $\call$ is the Picard-Fuchs differential operator of the family $(\caly,\Omega)$. Since $\call\Omega=d\beta$ is an exact form, the inhomogeneity $f(z)$ originates from integrating $\int_C\beta$, together with some contribution from differentiating $C$. Both are clearly local and meaningfully associated to $C$, whether homologically trivial or not.
With respect to the standard choice of $\Omega$, the Picard-Fuchs operator of the quintic mirror has the form $$\eqlabel{standard}
\call=\theta^4-5z(5\theta+1)(5\theta+2)(5\theta+3)(5\theta+4)$$ (where $z=(5\psi)^{-5}$ and $\theta=\frac{d}{d\ln z}$.) The inhomogeneity $f(z)$ was calculated in [@mowa] for the Deligne conics $C_{\pm}$ given by $x_5^2=\pm\sqrt{5\psi} x_1x_3$, within the plane $P=
\{x_1+x_2=0\,, x_3+x_4=0\}$, with the result $$\eqlabel{deligne}
f_\pm(z) = \pm \frac{15}{32\pi^2} \sqrt{z}$$ Since the line is residual to those conics in the intersection of $P$ with $Y$, and since, on general grounds, the inhomogeneity associated to $P\cap Y$ vanishes, we can conclude immediately that $f(z)=0$ for any of the $375$ isolated lines.
Another general Hodge theoretic result is that curves that can be holomorphically deformed into each other give rise to the same normal function. Mathematically, this is the statement that “algebraic equivalence implies Abel-Jacobi equivalence” (a statement valid for curves on Calabi-Yau threefolds). Physically, finite holomorphic deformations correspond to open string moduli, which are flat directions of the superpotential.
Given this, we conclude that the two families of lines containing the van Geemen lines map under Abel-Jacobi each to a single point in the intermediate Jacobian. It is therefore sufficient to calculate just for the van Geemen lines. Also, as anticipated above, the images of the two families differ just by a sign.
To calculate $f(z)$ for the van Geemen lines, we may proceed as in [@mowa; @newissues]. The key feature to exploit is that any of the lines is part of the intersection of $Y$ with a plane, and that the calculation of $\int_C\beta$ localizes to the intersection points of $C$ with the residual quartic in that plane. The difference to [@mowa; @newissues] is that here there is actually a two-parameter family of planes containing any given $C$, so we can make any choice that seems convenient.
The details are straightforwardly executed, and we obtain the inhomogeneity associated with the van Geemen lines, $$\eqlabel{vginh}
f_{\rm van\; Geemen}(z) = \frac{1+2\omega}{4\pi^2}\,\cdot\,
\frac{32}{45}\,\cdot\,
\frac{\frac{63}{\psi^{5}}+\frac{1824}{\psi^{10}}-\frac{512}{\psi^{15}}}
{\Bigl(1-\frac{128}{3\psi^5}\Bigr)^{5/2}}$$ ($z=(5\psi)^{-5}$). Notice that as $\omega$ is a non-trivial third root of unity, the inhomogeneity has an overall factor $\sqrt{-3}$ multiplying a function with a power series expansion around $\psi=\infty$ with rational coefficients. The main theme of this paper is to investigate irrationalities in the expansion of the solutions of inhomogeneous Picard-Fuchs equations. As an overall factor, the irrationality might seem rather mild in the present case. This is however dictated by the anticipated sign change under the Galois action $\sqrt{-3}
\mapsto-\sqrt{-3}$. Later examples will be more complicated, and also the solutions of the inhomogeneous Picard-Fuchs equation associated to will already be quite illuminating, see section \[largevolume\].
Conics on the Mirror Quintic {#conics}
============================
The basic framework to search for conics on the quintic is easy to describe, following Katz [@katz]: the moduli space of conics in $\projective^4$ is fibered over the Grassmannian $G(3,5)$ of projective planes in $\projective^4$. The fiber over a plane $A\cong\projective^2\subset\projective^4$ is given by the conics in a fixed $\projective^2$, spanned by the monomials of degree $2$ in three homogeneous coordinates on $\projective^2$, and isomorphic to a copy of $\projective^5$. We denote a conic in a fixed $\projective^2$ by $B$. The conic $B\subset A\subset \projective^4$ is contained in the quintic $Y$ precisely if $$\eqlabel{practice}
Y\cap A = B\cup C$$ where $C$ is a cubic curve in $A$.
Overview
--------
In practice, the equation means the following: the plane $A$ is defined as the vanishing locus of two linearly independent linear equations in the five homogeneous coordinates, $x_1,\ldots,x_5$ of $\projective^4$. Up to taking linear combinations of those two equations, there are $6$ independent parameters entering these equations, which are just (local) coordinates on $G(3,5)$. The equations for $A$ being linear, and non-degenerate, they can be solved for two of the five $x_i$’s, say $x_4$ and $x_5$. The result can be substituted in the quintic polynomial defining $Y$, yielding a quintic polynomial, $p_5$, in $3$ variables. Note that there are $21$ different quintic monomials in $3$ variables.
The conic $B\subset A$ is given as the vanishing locus of a quadratic polynomial $p_2$ in three variables, say $(x_1,x_2,x_3)$, and depends on $6$ homogeneous parameters. Likewise, the residual cubic is given by a cubic polynomial $p_3$, and depends on $10$ parameters. The equation then is the vanishing of the coefficients of the $21$ independent quintic monomials in $$\eqlabel{linearly}
p_5 - p_2\cdot p_3$$ Since we may fix the scale of either $p_2$ or $p_3$ arbitrarily, there are $6+6+10-1=21$ independent parameters entering those $21$ equations. (Note that the $10$ parameters for $C$ enter linearly in , which may therefore a priori be reduced to a system of $11$ equations in $11$ variables. This is the more standard dimensionality of the counting problem.) Generically then, we expect a finite number of solutions. This is in fact true, and there are $609250$ conics on the generic quintic [@katz].
For special quintics, for example a member $Y_\psi$ of the one-parameter family of quintics , there will be some number of isolated solutions, and some number of continuous families. There can also be conics that are reducible to two intersecting lines. When counted appropriately, all these will add up to $609250$. In this paper, our main focus is not on counting solutions, but on performing calculations for particular conics that deform with $Y_\psi$ as $\psi$ is varied. Conics that are isolated for fixed $\psi$ will deform to one-parameter families, while families that exist at fixed $\psi$ can either deform as families or be lifted to isolated solutions. Globally these local branches of solutions will fit together to various components of the “relative Hilbert scheme” of conics $\calh_{\rm conics}\to M$ on the one-parameter family of quintics , $\caly\to M$.
The goal in this section is to identify an interesting subset of components of $\calh_{\rm
conics}$. In the next section, we will study the branch structure around the large complex structure limit, $\psi\to\infty$. To simplify our life, we will neglect obstructed families of conics, avoid the singular loci, and all other phenomena that occur at special values of $\psi$.
A fair number of solutions of can be found by exploiting the symmetries of the problem. The full symmetry group $G$ of consists of the phase symmetries multiplying the $x_i$’s and $\psi$ by fifth roots of unity, and the symmetric group $S^5$ that acts by permuting the $x_i$’s, $$\eqlabel{symmetries}
(\zet_5)^4 \to G \to S^5$$ The two subgroups play a somewhat different role in the problem. To construct the mirror quintic, we are ultimately interested in dividing out by the subgroup $(\zet_5)^3
\subset(\zet_5)^4$ fixing $\psi$. This means that we should be looking at orbits of curves under the group $(\zet_5)^3$, and a non-trivial stabilizer contributes an additional factor at the very end of the calculation. On the other hand, no subgroup of $S^5$ will be gauged, and a curve with non-trivial stabilizer in $S^5$ is not special in any other way.
At a more practical level, the phase symmetries act diagonally on the variables parameterizing $A$, $B$, and $C$, and dividing out by them does not reduce the dimensionality of the problem, but merely the degree (which is quite helpful anyways, of course!). The permutation symmetries act non-diagonally, and can reduce both the dimensionality and the degree. It is a good idea to keep track whether the subgroup of interest acts with unit determinant on the $x_i$’s or not. If it does, one might expect the solutions of the reduced problem to still be isolated, although this is neither necessary nor sufficient in general. Also, we may point out that a conic that is isolated as a solution invariant under a particular symmetry could in fact sit in a family of conics the generic member of which breaks that symmetry.
We will return to pointing out these, and many more, features of the space of conics after we have presented a few explicit solutions.
S3-invariant conics
-------------------
To begin with, one may look for conics that are invariant under permutation of three of the five homogeneous coordinates of $\CP^4$, which we choose to be $x_1,x_2,x_3$, see [@mustata]. (If our concern were counting conics, we would of course have to account for that choice.) We parameterize the equations for the plane as follows $$\eqlabel{plane1}
A : \left\{ \begin{array}{c}
a_1(x_1+x_2 + x_3) + x_4\, \\
a_2(x_1+x_2+x_3)+x_5
\end{array}\right\}$$ and solve for $x_4$ and $x_5$. Note that this means in principle that we are working in a specific open patch of the full moduli space. One can check that the solutions in the other patches precisely serve to compactify the families that we shall write down below. The conic $B\subset A$ is given by $$\eqlabel{conic1}
B : \{ x_1^2 +x_2^2 +x_3^2 + b_1(x_1x_2+x_1x_3+x_2x_3) \}$$ where we have gauged the coefficient of $x_1^2+x_2^2+x_3^2$ to $1$. This is again only an open patch, but it captures all solutions. For once, we display the residual cubic: $$\eqlabel{cubic1}
C : \{c_1\,(x_1^3+x_2^3+x_3^3) + c_2\,(x_1^2x_2+ x_1^2 x_3+x_2^2x_1+x_2^2x_3+x_3^2x_1+x_3^2x_2)
+ c_3\, x_1x_2x_3 \}$$ which depends on three parameters. Finally, there are five symmetric polynomials of degree 5 in three variables, giving rise to as many equations for the six variables $a_1,a_2,b_1,c_1,c_2,
c_3$. Thus we see that we generically expect a one-parameter family of solutions (for fixed $\psi$). Writing out those equations explicitly, we see that four of them can be solved linearly for $c_1, c_2, c_3$, and $b_1$ in terms of $a_1$ and $a_2$. For example, $$b_1 = \frac{1-2\psi a_1a_2}{1-\psi a_1a_2}$$ The remaining equation is $$\eqlabel{S3invariant}
1- a_1^5 - a_2^5 + 5 \psi^2 a_1^2 a_2^2 - 5 \psi a_1a_2$$ Thus, for fixed $\psi$, picking any solution of , the intersection of the quintic $Y_\psi$ with the plane decomposes as the union of the conic and the cubic . This is the solution found by Mustaţǎ [@mustata].
For completeness, and anticipating a stratagem that will be relevant later, we note that the invariant ansatz is not the only way to produce an $S^3$-invariant plane. Indeed, the two equations defining $A$ might also transform in the two-dimensional irreducible representation of $S^3$, , $A$ might be given by $\{x_1-x_2,x_2-x_3\}$. This eliminates any free parameters in $A$, while bringing back those in $B$ and $C$ to $16-1=15$, and the number of equations to $21$. In the present case, allowing the equations to transform non-trivially under the symmetry group does not uncover any new solutions. In later examples it will.
Z2xZ2-invariant conics
----------------------
The next case of interest is the subgroup $\zet_2\times\zet_2\subset S^5$, with generators acting by exchanging $(x_1,x_3)$ and $(x_2,x_4)$ respectively. [^2] Assuming the equations for the plane to be invariant leads to the ansatz $$\eqlabel{plane2}
A : \left\{ \begin{array}{c}
x_1+x_3 + a_1 x_5\, \\
x_2+x_4 + a_2 x_5
\end{array}\right\}$$ We here see immediately that for any value of $a_1,a_2$, the plane contains, at $x_5=0$, one of the $375$ isolated lines discussed in the previous section. Therefore, if the quintic has any conic in such a plane, the residual cubic in eq. will be reducible, so that there are then, actually, two conics in that plane. A priori, we do not expect to find any such conic at all, since the quartic curve residual to the line in the plane would need to develop four nodes where we only have two parameters at our disposal to move the plane. The symmetries help, however, as we shall see presently.
We solve the equations in for $x_1$ and $x_2$, and make the ansatz $$\eqlabel{conic2}
b_6 x_3^2+ b_5 x_3 x_4 + b_4 x_3 x_5 + b_3 x_4^2 + b_2 x_4 x_5+ b_1 x_5^2$$ for the equation defining the conic. This is invariant under $x_3\to x_1=-x_3-a_1 x_5$ and under $x_4\to x_2= - x_4- a_2 x_5$, precisely if $$\eqlabel{impose}
b_5=0\,,\qquad b_2=a_2 b_3\,,\qquad b_4 = a_1 b_6$$ Eliminating the cubic, we find that the solution of is, in the gauge $b_6=1$, reduced to the three equations $$\eqlabel{Z2Z2invariant}
\begin{split}
1-a_1^5-a_2^5+5 a_1^3 b_1-5 a_1 b_1^2 &=0\\
a_2^3-\psi b_1-a_1^3 b_3+2 a_1 b_1 b_3 &=0 \\
a_2-\psi b_3+a_1 b_3^2 &=0
\end{split}$$ for the four variables $a_1,a_2,b_1,b_3$. We see that this describes two one-parameter families of conics for each $\psi$: the first equation admits two solutions for $b_1$, the middle equation then determines $b_3$ uniquely, while the third equation relates $a_2$ and $a_1$. The two families share the planes, but not any conics. As in the previous subsection, the eq. describes only an open patch of the families. Below, we will see a bit of the compactification, as dictated by the embedding in the moduli space of conics in $\projective^4$. As an example, one might verify the symmetry under exchange of $a_1$ and $a_2$.
Taking advantage of phase symmetries
------------------------------------
Going slowly enough over the previous discussion reveals an option for finding further solutions: not all of the $\zet_2\times\zet_2$ symmetry group under which the plane is invariant need to fix the two conics in that plane individually. Instead, the two conics might be exchanged by one generator, and fixed by the other. In particular, we can choose the diagonal $\zet_2^+\subset\zet_2\times\zet_2$ to fix the two conics, and the exchange of $(x_1,x_3)$ to exchange them. The generator of $\zet_2^+$ acts as $$\eqlabel{Z2p}
(x_1,x_2,x_3,x_4,x_5)\mapsto (x_3,x_4,x_1,x_2,x_5)$$ This relaxes the constraint to $$\eqlabel{relax}
b_2= a_2b_3 + \frac 12 a_1 b_5 \,,\qquad\qquad
b_4 = a_1 b_6 + \frac 12 a_2 b_5$$ To simplify the equations further, we employ a device that will be useful also later: note that the ansatz is covariant under a $(\zet_5)^2$ subgroup of the group of phase symmetries, provided we act in a particular way on the coefficients of conic and cubic. So we may absorb those phase symmetries by appropriate variable substitutions, and thereby reduce the degree of the equations. In the present case, we first solve for $b_1,b_2,b_3,b_4$, and $a_2$ linearly. (Doing this excludes the families above, on which the rank of the equations is reduced.) Still working in the gauge $b_6=1$, we then substitute $$\psi = \tilde \psi a_1$$ and the remaining equations depend only on $a_1^5$. Introducing $\tilde a_1=a_1^5$ as a new variable reduces the degree sufficiently to be able to fully understand the equations. Indeed, $\tilde a_1$ appears only linearly, and the remaining relations for $b_5$ and $\tilde \psi$ boil down to $$-64 \tilde\psi+2 \tilde\psi^6-32 b_5^2+
11 \tilde\psi^5 b_5^2+
25 \tilde\psi^4 b_5^4+
30 \tilde\psi^3 b_5^6+
20 \tilde\psi^2 b_5^8+
7 \tilde\psi b_5^{10}+
b_5^{12} = p_1 p_2 p_3$$ with $$\eqlabel{split}
\begin{split}
p_1 &= -2+\tilde\psi+b_5^2 \\
p_2 &=
16+8 \tilde\psi+4 \tilde\psi^2+2 \tilde\psi^3+\tilde\psi^4+
(8+8 \tilde\psi+6 \tilde\psi^2+4 \tilde\psi^3) b_5^2 +
\\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+(4+6 \tilde\psi+6 \tilde\psi^2) b_5^4+
(2+4 \tilde\psi) b_5^6+
b_5^8 \\
p_3 &=2 \tilde\psi+b_5^2
\end{split}$$ We may then substitute back $a_1$ and $\psi$, to obtain the relations in a way that will be useful later on. As an example, we write the equations corresponding to $p_1$: $$\eqlabel{p1}
\begin{split}
64+5 \psi^2 a_1^3-40 \psi a_1^4+12 a_1^5 &=0 \\
\psi-2 a_1+a_1 b_5^2 &=0
\end{split}$$ The two solutions for $b_5$ correspond, as it should be, to the two conics that exist in the plane determined by the first equation. The remaining parameters of our ansatz are given by $$\eqlabel{remaining1}
\begin{array}{c}
a_2 = a_1\,, \qquad
b_1 = \frac{1}{8} (-\psi a_1+6 a_1^2+2 a_1^2 b_5 )\,, \\[.2cm]
b_2 = a_1+\frac12 a_1 b_5 \,,\qquad
b_3 = 1 \,,\qquad
b_4 = a_1+ \frac12 a_1 b_5 \,,\qquad
b_6 = 1
\end{array}$$ For future reference, we note that the conics corresponding to are, in addition to $\zet_2^+$, invariant under the group $\zet_2^-$ whose generator acts as $$\eqlabel{Z2m}
\zet_2^-: (x_1,x_2,x_3,x_4,x_5)\mapsto (x_2,x_1,x_4,x_3,x_5)$$ Indeed, when $a_2=a_1$, the equations for the plane are exchanged under that $\zet_2^-$, while the conic is invariant as $b_6=b_3$ and $b_4=b_2$.
For $p_3$, the relations analogous to are: $$\eqlabel{p3}
\begin{split}
\psi^{10}+4096 a_1^5-160 \psi^5 a_1^5+1024 a_1^{10} &=0 \\
2 \psi+a_1 b_5^2 &=0
\end{split}$$ and the explicit solution is $$\eqlabel{remaining3}
\begin{array}{c}
\displaystyle
a_2 = \frac{\psi^2}{4 a_1} \,,\qquad
b_1 = - \frac{\psi^5}{64 a_1^3} + \frac{a_1^2}{2} + \frac{1}{16}\psi^2 b_5\\[.2cm]
\displaystyle
b_2 = -\frac{\psi^3}{8 a_1^2}+ \frac{1}{2} a_1 b_5\,,\qquad
b_3 = -\frac{\psi}{2a_1}\,,\qquad
b_4 = a_1+\frac{\psi^2 b_5}{8 a_1} \,,\qquad
b_6 =1
\end{array}$$ These solutions are invariant only under $\zet_2^+$.
Anticipating some of the later discussion, we note that the conic is reducible when the $b_i$’s take the values in , provided $b_5$ satisfies the condition in . So we see that in fact, the plane with $4 a_2 a_1=\psi^2$ and $a_1$ satisfying the first equation in meets the quintic in a collection of five lines.
We have not discussed in detail the conics corresponding to the factor $p_2$ in . In fact, that solution arises from , simply by the phase symmetry acting on $x_2$ and $x_4$. For instance, one may check that instead of $a_2=a_1$, merely $a_2^5=a_1^5$ holds on that solution.
For completeness, we take a brief look at solutions with $b_6=b_3=0$ (which was excluded by our above choice of gauge). We find that the only such solutions are $$\eqlabel{completeness}
a_1=a_2= b_2=b_3=b_4=b_6=0 \,,\qquad\qquad
b_5^2 = 5 \psi b_1^2$$ so that we recover the conics studied in [@mowa].
Z2-invariant conics
-------------------
An important feature of the previous subsection was that the group $\zet_2^+$ (see eq. ) acts with unit determinant on $(x_1,x_2,x_3,x_4,x_5)$, and the number of equations matched the number of variables also in the reduced problem. We were able to fully reduce those equations, and thereby isolate the components of $\calh_{\rm conic}$ invariant under $\zet_2^+$.
As an ultimate possibility, we now study conics that are invariant under the action $$\eqlabel{finalsym}
\zet_2^-: (x_1,x_2,x_3,x_4,x_5)\mapsto (x_2,x_1,x_4,x_3,x_5)$$ As symmetries of $\projective^4$, the groups $\zet_2^+$ and $\zet_2^-$ are of course equivalent. But, as it turns out, we get a new class of solutions if we modify the ansatz for the plane $A$ containing the conic, and take one equation to be invariant, and the other to transform with a sign (this being equivalent to the way acted on ): $$\eqlabel{plane3}
A:\left\{
\begin{array}{c}
a_1(x_1+x_2)+a_2(x_3+x_4) + x_5 \\
(x_1-x_2) + a_3 (x_3-x_4)
\end{array}
\right\}$$ The equation for the conic has to be invariant (we eliminate only $x_5$ in order to make the symmetry manifest): $$\eqlabel{conic3}
B: \{ b_1 (x_1+x_2)^2 + b_2 (x_1+x_2)(x_3+x_4) + b_3 (x_3+x_4)^2
+ b_4 (x_3-x_4)^2 \}$$ Now absorb the phase symmetries by substituting $$\eqlabel{substitute}
\begin{array}{c}
a_1\to a_1^{-1/5}\,,\qquad
a_2\to a_2^{-1/5}\,, \qquad
a_3\to a_3\, a_1^{1/5} a_2^{-1/5}\,,\qquad
\psi\to\psi\, a_1^{-2/5}a_2^{-2/5} \\[.5cm]
b_1\to b_1\, a_1^{-2/5}\,,\qquad
b_2\to b_2\, a_1^{-1/5} a_2^{-1/5} \,,\qquad
b_3\to b_3\, a_2^{-2/5} \,,\qquad
b_4\to b_4\, a_2^{-2/5}
\end{array}$$ where we denote the new variables by the same letters as the old ones. Then, in the patch $b_2=1$, the $a_1,a_2,a_3^2,b_4$ can be solved for linearly. When substituted back, we remain with two equations of relatively high degree involving $b_1, b_3$, and $\psi$, $$\eqlabel{equations}
q_1(b_1,b_3,\psi) = q_2(b_1,b_3,\psi) =0$$ These can be further reduced if we exploit the inherent symmetry of exchanging $b_1$ and $b_3$, and substitute $b_1-b_3=u$, $b_1b_3=v$. Then, we eliminate $\psi$ by computing the resultant of those two equations, to decompose the set of conics invariant under the $\zet_2^-$-symmetry as much as possible into constituents: $$\eqlabel{resultant}
{\rm Resultant}(q_1,q_2;\psi)(u,v) \propto (9-12u+16v)\cdot (1-4v) \cdot (u^2-4v)
\cdot Q_m$$ Here we have excluded factors that do not lead to a solution of the original system, because the equations for $a_1,a_2,a_3,b_4$ that we solved earlier actually became singular. To each factor of , there corresponds a component[^3] of $\calh_{\rm conics}$ that can be reconstructed in the following way: given a pair $(u,v)$ for which that factor vanishes, we find a $\psi$ solving the equations (the existence of a common root of $q_1$ and $q_2$ being precisely the characterization of the resultant), and then a unique set of $a_1,a_2,a_3^2,b_4$ solving the equations for a conic on the quintic under the $\zet_2^-$-invariant ansatz , , after absorbing the $\zet_5\times\zet_5$ phase symmetries as in . Undoing that substitution introduces 2 fifth roots of unity, one of which corresponds to the phase of $\psi$ that originally parameterized the family of quintics, while the other is a genuine label of a conic contained therein. We thus obtain various branches of conics for each factor of the resultant , depending on which solution we choose. These branches will interact in various ways as (the original) $\psi$ (appearing in ) is varied around the moduli space. The different factors of might split further under this procedure (but they will not mix). An obvious splitting arises when we remember that the map $(b_1,b_3)\to (u,v)$ is actually two-to-one. For example, $$\eqlabel{realized}
9-12u+16 v = (3-4b_1)(3-4b_3)$$ so that the first factor in actually describes two sets of conics in that sense. Also, although we did not bother pointing this out, it is clear that the equations are invariant under $a_3\to-a_3$, so we also need to choose a sign for $a_3$ when we go back.
The main component of $\calh_{\rm conics}$ that we found is characterized by the factor, $$\begin{gathered}
\eqlabel{monster}
Q_m = \\
\begin{array}[t]{l}
\scriptscriptstyle
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Relationships
-------------
We now describe how the solutions of that we have found so far by imposing certain symmetries fit together as components of the Hilbert scheme, $\calh_{\rm conics}$, of conics on the one-parameter family of quintics . (As emphasized before, we do not claim that we have identified all components, nor that all components that we have found are irreducible.)
### $\zet_2\times\zet_2$ meets $S^3$
First of all, we point out that the family of $S^3$-invariant conics found by Mustaţǎ and the family of $\zet_2\times\zet_2$-invariant conics meet. A common member occurs in the first family if we put $a_1=1$, $a_2=0$ in , where the conic acquires some additional symmetry, and in particular the $\zet_2\times\zet_2$ symmetry manifest in . (Note that this solves and that the conic is also invariant because $b_1=1$.) To, conversely, exhibit that conic on the $\zet_2\times\zet_2$-invariant family, we first need to go to a different patch of the moduli space. We note that the plane in is equivalent to $$\eqlabel{equivalent}
\left\{\begin{array}{c}
x_1+x_3+ \tilde a_1(x_2+x_4) \\
\tilde a_2(x_2+x_4) + x_5
\end{array}
\right\}$$ where $\tilde a_1=-a_1/a_2,\tilde a_2=a_2^{-1}$. If we now put $\tilde a_1=1,\tilde
a_2=0$, we recover the plane invariant under both $S^3$ and $\zet_2\times\zet_2$ symmetry that we just discussed. The quintic meets that plane in a conic plus three lines. One of those is the line common to all the planes, the irreducible conic is the $S^4$-enhancement in one of the families of $\zet_2\times\zet_2$-invariant conics, while the remaining two lines represent the second family.
Another example of a common member of the two types of families occurs when $\tilde a_1=0, \tilde a_2=1$ in . This conic appears on the family invariant under permutation of $(x_2,x_4,x_5)$ in the limit $a_2=a_1$, $a_1^{-1}=0$ in the appropriate version of the plane .
It is a useful exercise to write the equations for the families of $\zet_2\times\zet_2$-invariant conics in the patch with coordinates $\tilde a_1,\tilde a_2$. The parameterization was valid as long as we could eliminate $x_1$ and $x_2$—this is not possible when $\tilde a_2\to 0$ in . So eliminating $x_5$ instead of $x_2$, we write for the conic $$\tilde b_1 x_2^2+\tilde a_1 \tilde b_3 x_2 x_3+
\tilde b_3 x_3^2+\tilde b_4 x_2 x_4+\tilde a_1 \tilde b_3 x_3 x_4+
\tilde b_1 x_4^2$$ Then, in the patch $\tilde b_1=1$, the equations for the family are $$\eqlabel{transform}
\begin{split}
-5 \tilde a_1+5 \tilde a_1^3 \tilde b_3+(1-\tilde a_1^5-\tilde a_2^5) \tilde b_3^2 &=0 \\
-1-4 \tilde a_1^5-4 \tilde a_2^5+(1+4 \tilde a_1^5+4 \tilde a_2^5) \tilde b_4
+(1-\tilde a_1^5-\tilde a_2^5) \tilde b_4^2 &=0 \\
-10 \tilde a_1^3-5 \psi \tilde a_2+5 \tilde a_1^3 \tilde b_4+
\tilde b_3 (1+4 \tilde a_1^5+4 \tilde a_2^5)+
2\tilde b_3\tilde b_4(1- \tilde a_1^5 -\tilde a_2^5) &=0
\end{split}$$ These equations have a structure comparable to that of , and in the relevant open patches the two systems are completely equivalent.
### Reducible conics
Secondly, we record that the $\zet_2\times\zet_2$-invariant families contain reducible conics. We have seen this in the discussion we just had at $\tilde a_1=1$, $\tilde a_2=0$. Another example is $\tilde a_1=\tilde a_2=0$, which plane, $x_1+x_3=x_5=0$, meets the quintic in the union of five of the $375$ isolated lines that we discussed in the previous section. Yet another example is $a_1=1$, $a_2=0$ in .
### $\zet_2^-$ meets $\zet_2^+$.
Thirdly, we study in more detail the locus $u^2-4v=(b_1-b_3)^2=0$ corresponding to the vanishing of the third factor of in the set of $\zet_2^-$-invariant conics. We find that the relevant planes have $a_2=a_1$, $a_3=-1$ in , and after eliminating the cubic, the resulting equations split in two components. One solution may be written as $$\eqlabel{remaining4}
a_1=a_2=\frac 1\psi \,,\qquad a_3=-1\,,\qquad
b_2=1\,,\qquad b_3=b_1\,,\qquad b_4=-1+2b_1$$ with $b_1$ satisfying $$\eqlabel{p4}
-16+\psi^5+64 b_1+\psi^5 b_1-64 b_1^2+4 \psi^5 b_1^2=0$$ The other is characterized by the vanishing of $$\eqlabel{p5}
3-20 \psi a_1+5 \psi^2 a_1^2+512 a_1^5$$ After $a_1\to 1/(2a_1)$, we may recognize this equation as being equivalent to the first line in . Indeed, when $a_2=a_1$, and $a_3=-1$, the ansatz is also invariant under $\zet_2^+$ from , and with appropriate substitutions, the solution given by is nothing but that in , . One can also check that the solution , is contained in the family at $a_1=a_2=\psi/2$.
### A family of reducible conics
Next, we discuss the conics associated with the vanishing of the second factor in , $$\eqlabel{lach}
1-4v=1-4 b_1b_3$$ We see quite rapidly that under this condition, the conic is reducible (remember that we work with $b_2=1$). The two components must be lines on the mirror quintic, which are completely understood as reviewed in the previous section. Since it is clear that the relevant lines are not on the list of $375$ isolated lines, they must belong to the families containing the van Geemen lines. This indeed makes sense: for generic $\psi$, we have two distinct families of lines in $Y_\psi$ that meet in a curve $K_\psi \subset Y_\psi$. Each point in $K_\psi$ is the intersection point of two lines, one from each family, which together can be properly viewed as a reducible conic. (The part of the Clemens conjecture stating that rational curves are generically disjoint obviously does not hold here.) In other words, each $Y_\psi$ contains a family of reducible conics parameterized by $K_\psi$. At certain isolated points in $K_\psi$, the reducible conic acquires the $\zet_2^-$ symmetry we have discussed, and shows up on our list.
As a further check, one may start from the $\zet_2^-$-invariant solutions with $1-4v=0$ and verify that it indeed deforms as a one-parameter family of reducible conics, which generically break the $\zet_2^-$ symmetry.
Moreover, we can now remember the $\zet_2^+$-invariant solution , , which conics were also reducible with components not on the list of $375$. These must also belong to the family of reducible conics parameterized by $K_\psi$. In fact, they must correspond to singular points on $K_\psi$ since we actually have four such lines (, two reducible conics) in the corresponding planes.
### Another coincidence
Finally, we note that the conics associated with the vanishing of the first factor in , which according to can be realized for instance at $b_1=3/4$, actually belong to the Mustaţǎ family of $S^3$-invariant conics. This can easily be checked.
Summary So Far {#sofar}
==============
We have seen that the generic $Y_\psi$ contains (at least) three types of families of conics, and we have identified a number of isolated conics. We have not attempted to enumerate the solutions, mostly because we did not work out the global description of all the families. Numerical methods indicate that these might in fact be all solutions: elementary search algorithms (such as those of Mathematica) return only solutions on one of our families, or isolated solutions with a non-trivial symmetry group. (This has to be taken with a dose of skepsis, because such algorithms have a higher chance of finding solutions with symmetry or those occuring in families.) We have also checked that there are no solutions with other types of symmetry enhancements than those we have discussed.
There is first of all the family with $S^3$ symmetry, parameterized by solutions of , and originally found by Mustaţǎ [@mustata]. Secondly, we found two families of conics invariant under the action of a $\zet_2\times\zet_2$ symmetry group. These are parameterized by solutions of , and have the interesting feature of sharing their planes. The $\zet_2\times\zet_2$ invariant families meet the $S^3$-invariant families in various ways, and contain reducible conics. Thirdly, there is a family consisting entirely of reducible conics. This family can be obtained by intersecting the van Geemen families of lines. We have not written down the equations describing that family globally, but identified two members, one with $\zet_2^+$ symmetry and one with $\zet_2^-$ symmetry .
Turning to the isolated conics, we have first of all those studied in [@mowa], see eq. . Secondly, we have the $\zet_2^+\times\zet_2^-$-invariant conics of , (their $\zet_5$-orbit was also discussed around there). One may check (for instance numerically), that these conics are in fact isolated also in the space of conics without any discrete symmetries. Finally, we have $\zet_2^-$-invariant conics associated with solutions of , which are also isolated forgetting the symmetry. As it stands, is not particular useful of course. It is somewhat unfortunate that we have not been able to make further progress on those equations, for instance with the purpose of checking whether the corresponding part of $\calh_{\rm conics}$ is in fact irreducible or not. In the next section, we will retreat to studying the expansion in the large complex structure limit.
We now switch to the main topic of interest in this paper, namely the Abel-Jacobi image of $\calh_{\rm conics}$. We first of all dispose of the families: recall that the isolated lines have a vanishing Abel-Jacobi image (in the sense described in the introduction, and in section \[vangeemen\]), and the van Geemen lines give rise to the inhomogeneity . As described above, the families are all algebraically equivalent to some combination of those lines, and therefore they do not give rise to any new inhomogeneity.
We only remain with the isolated conics. We shall denote the family of conics (component of $\calh_{\rm conics}$) studied in [@mowa], $\cali_0$. That associated with will be called “the first component”, $\cali_1$, and that of , (which might still be reducible), the “main component”, $\cali_2$.
Expansion in Large Complex Structure Limit {#largecomplex}
==========================================
We realized in the previous section that the expression is too large to allow writing down explicitly all coefficients determining the “main component”, $\cali_2$, of the space of $\zet_2^-$-invariant conics, at least not without significantly increased computing power. Progress is still possible, however.
Newton-Puiseux expansion {#newtonpuiseux}
------------------------
The main idea is easy to describe: instead of reducing the equations satisfied by the parameters in , algebraically, we expand those parameters in (fractional) power series around large complex structure point $\psi=\infty$, and determine the expansion coefficients recursively from the equations. This is in principle sufficient to calculate the expansion of $\calw(z)$ for the purpose of testing mirror symmetry. In practice, the calculation is limited to the first few orders in the expansion. The method will also not allow to easily calculate monodromies around complex structure moduli space, which would be desirable in order to fix the solution of the homogeneous Picard-Fuchs equation in $\calw$.
For what we’ll call the “first component” of $\calh_{\rm conics}$, $\cali_1$, we are able to calculate the inhomogeneity exactly, see . This allows expansion to much higher order, calculation of monodromies, and is also a useful cross-check on the calculations around $\cali_2$.
But let’s first be a bit more general and take $v$ as any one of the parameters entering an ansatz for the curve $\calc\subset \caly$ under consideration (, one of the $a_i$ or $b_j$ in , ). (We could also imagine $\caly$ to be a more general family of algebraic varieties than the mirror quintic, degenerating in some way, and $\calc$ to be some general algebraic cycle.) For a one-parameter family, $v$ will, as a function of $\psi$, satisfy a parameter-dependent polynomial equation $$\eqlabel{Pvpsi}
P(v,\psi)=0$$ (obtained in the example by projecting eq. onto $(v,\psi)$, and generally at least as complicated as ). Let’s assume that $P$ is irreducible.
Following Newton, we can study the behavior of the roots of as $\psi\to\infty$, by looking at the polygon spanned by monomials with non-zero coefficients in $P(v,\psi)$: say $$\eqlabel{say}
P(v,\psi) = \sum_{m,n} p_{m,n} v^m \psi^n\,,$$ and let $\Pi$ be the convex hull of points $(m,n)\in\zet^2$ with $p_{m,n}\neq 0$. Because we are looking at $\psi\to\infty$, the interesting part of $\Pi$ actually is its [*upper boundary*]{}, $\widehat\Pi$, which is the set of point $(m,n)\in\Pi$ such that $(m,n')\notin \Pi$ for $n'>n$. This $\widehat\Pi$ consists of a finite sequence of segments of decreasing slope and varying length.
Let $v_k(\psi)$ (with $1\le k\le E$) be one of the $E$ roots of $P(v,\psi)=0$ for fixed $\psi$. (Here, $E:={\max}\{m,p_{m,n}\neq 0 \text{ for some $n$} \}$ is the $v$-degree of $P$.). There is then a rational number $\alpha_k$ such that $$c_0 := \lim_{\psi\to\infty} \psi^{-\alpha_k} v_k(\psi) \notin \{ 0, \infty \}$$ In other words, we are making an ansatz of the form $$\eqlabel{newtonansatz}
v_k(\psi) = \psi^{\alpha_k} \bigl( c_0 + \calo(\psi^{-\beta}) \bigr) \quad \text{with
$c_0\neq 0$ and $\beta>0$}$$ plug this into , and collect terms of same order in $\psi$: $$\eqlabel{collect}
P(v_k(\psi),\psi) = \psi^{\alpha_{P}} \bigl( P_0(c_0) + \calo(\psi^{-\beta})\bigr)$$ Then, $$\eqlabel{alphaP}
\alpha_{P} = \max\{\alpha_k m+ n, p_{m,n}\neq 0 \}$$ and $$\eqlabel{P0}
P_0(c_0) = \sum_{\alpha_k m+n = \alpha_{P}} p_{m,n} c_0^m$$ For $P_0(c_0)=0$ to have a non-zero solution, there must be at least two non-zero terms in $P_0(c_0)$. This shows that (i) $\alpha_k$ must be the negative slope of one of the upper edges of $\Pi$ (, of a segment of $\widehat\Pi$), and (ii) $P_0(c_0)$ is the sum of $p_{m,n}c_0^m$ along that segment. In particular, the degree of $P_0(c_0)$ is the length of (the projection onto the $m$-axis of) that segment.
Having found the lowest order term, we can proceed with the expansion to higher order. Several things can happen: for example, the polynomial $P_0(c_0)$ might be reducible, or the sub-leading terms in might vanish together with $P_0(c_0)$. The next term in that does not vanish after imposing the leading order equation will determine the exponent $\beta$ in . Past this, and if the original $P$ is irreducible, we are set to calculate the coefficients in $$\eqlabel{fractional}
v_k(\psi) = \psi^{\alpha_k} \sum_{d=0}^\infty c_d \psi^{-\beta d}$$ recursively from the $\psi$-expansion . It is in the nature of things that the coefficients $c_d$ for $d>1$ are finite algebraic expressions in $c_0$, modulo $P_0(c_0)=0$.
In a somewhat more formal language, we can view $v$ as generator of an algebraic extension of the field of rational functions on $\psi$-space, of degree $E$. When localizing that extension at $\psi=\infty$, it splits into extensions of the local field of power (Laurent) series in $\psi^{-1}$, of degree $e_k$ given by the length of the corresponding segment of $\widehat\Pi$. (Observe that $\sum e_k = E$.) The generators of these local extensions are precisely the Puiseux series . Geometrically, we think of an algebraic curve as an $E$-fold cover of $\psi$-space, and what we are doing is simply parameterizing the various branches at $\psi=\infty$.
An underlying piece of structure is hidden in the following fact: the coefficients $p_{m,n}$ are, generally speaking, algebraic combinations of the coefficients entering the definition of $\caly$ (and, possibly, the ansatz for $\calc$). In other words, the $p_{m,n}$ live in the field $K$ over which the underlying algebraic variety (and, possibly, the ansatz for $\calc$) is defined. As we have seen above, to any branch of is associated a polynomial $P_0\in K[c_0]$ whose vanishing determines the leading coefficient $c_0$, and all other coefficients are algebraic combinations of $c_0$. Thus, each branch belongs to a certain algebraic extension $K(c_0)$ of the residue field $K$ at $\psi=\infty$.
Given this structure, we now think of the branches of solutions of with the same leading exponent and extension of residue field (splitting $P_0$ into irreducible factors if necessary) together in one group. It should be kept in mind, however, that $P_0$ does not fully characterize the local extension because as discussed above, sub-leading terms in $P(v,\psi)$ could play a significant role. And the global extension determined by $P(v,\psi)$ itself of course is not visible in the local expansion of any given group.
When we do not have the power to reduce to an equation of the type explicitly, we can still obtain the Puiseux expansion of the various parameters by studying the original larger system of equations . Assuming the underlying family to be one-dimensional, the discussion is similar, with Newton polygon replaced by Newton polyhedron, and vectors of leading exponents. The comments about importance of sub-leading terms in the expansion of the equations however become more acute. Indeed, with more equations, there are more ways in which they can be degenerate, and the actual extension of $K$ governing each group of solutions might only be determined at higher order in the expansion.
There are several more possibilities with higher-dimensional families: for example, a family of cycles could define a transcendental extension of the parameter space, or require an additional blowup on top of the extension.
We also mention an interesting alternative point of view on the expansion we have discussed: so far, we started with curves embedded in $\projective^4$, and imagined solving the equations that determined which of those would lie in the quintic hypersurface, then passing to the limit $\psi\to\infty$. Instead, we might also first go to large complex structure, and note that the mirror quintic degenerates there into the union of 5 copies of $\projective^3$. While curves on the quintic are virtually rigid, curves in projective space have a large number of parameters. So the problem to study is the lifting of these moduli spaces under the perturbation away from large complex structure. This description is more intrinsic and presumably better suited to understand the mirror symmetry.
Finally, we note that in our example, the initial family $\caly$ is defined of course over $K=\rationals$. In fact, it is not entirely clear how to imagine families of Calabi-Yau manifolds defined intrinsically over a number field other than the rationals, other than by specializing parameters of a higher-dimensional family.[^4] With a D-brane (in the form of an algebraic cycle) on top of $\caly$, extensions of $\rationals$ are forced on us.
The first component
-------------------
Let’s warm-up to Newton-Puiseux expansions on the first component $\cali_1$ of $\calh_{\rm conics}$, for which we can write down the equations globally in $\psi$. We begin by writing in the form $$\begin{aligned}
\eqlabel{curveid1}
64+5 a^3 \psi^2 -40 a^4 \psi+12 a^5 &=0\\
\eqlabel{curveid2}
-128 -5 a^2 \psi^3+40 a^3 \psi^2-12 a^4 \psi+64 b^2 &=0\,,\end{aligned}$$ with the abbreviation $a\equiv a_1=a_2$, $b\equiv b_5$. Recall again that the vanishing of the first equation selects a plane intersecting the quintic in (a line and) two conics determined by , , and distinguished by the choice of root in .
The Newton polygon of is shown in Fig. \[newton\].
We see that in the limit $\psi\to\infty$, the $5$ branches of solutions of split into 2 groups, with asymptotic exponent for $a$ given by the negative slope of the two upper segments of the Newton polygon: the first group has asymptotic behavior $$a = \psi^{-2/3} a_0 + \cdots$$ with $$64 + 5 a_0^3 = 0$$ while the second, $$a = \psi a_0 + \cdots$$ with $a_0$ one of the roots of the equation $$\eqlabel{under}
12 a_0^2 -40 a_0 + 5 = 0$$ To determine the sub-leading terms, and the expansion of $b$, we plug the leading order solution back into , . We find that for the first group $$b = b_0\psi^{5/6} + \cdots$$ with $$64 b_0^2-5 a_0^2 = 0$$ and the local expansion parameter is $\psi^{-5/3}$.
For the second group, with local parameter $\psi^{-5}$, the leading order terms in vanish under the condition . Therefore, we have to determine the leading order term in $$b = b_0 + \cdots$$ from the sub-leading terms in : $$64+5 a_0 a_1-60 a_0^2 a_1+24 a_0^3 a_1-32 b_0^2 =0$$ Since this involves the coefficient $a_1$ in the expansion of $a$: $$a = a_0 \psi + a_1 \psi^{-4} + \cdots$$ we first have to first solve to that order. We find $$a_1 = \frac{128(2729-852 a_0)}{425}$$ and then the equation $$\eqlabel{additional}
30-12 a_0+5 b_0^2 = 0$$ In terms of the residue field at $\psi=\infty$, this is a second quadratic extension on top of .
To write the expansion in a more compact form that we will use later on, we introduce the more convenient $$w = \frac{1}{5\psi} = z^{1/5}$$ Then, on the first group of branches of , , we have $$\eqlabel{expansion1}
\begin{split}
a&=\textstyle
-\frac{4\lambda^2}{5} w^{2/3}+\frac{128\lambda^4}{15} w^{7/3}-\frac{492800}{3} w^4+
\frac{534732800\lambda^2}{81} w^{17/3}-\frac{73034301440\lambda^4}{243} w^{22/3}+
\cdots \\
b&=\textstyle
\frac{\lambda^5}{1250} w^{-5/6}+\frac{14\lambda}{3} w^{5/6}-\frac{1132\lambda^3}{15}w^{5/2}
+\frac{1035344\lambda^5}{405} w^{25/6}-\frac{16285375600\lambda}{243} w^{35/6}+ \cdots
\end{split}$$ with $\lambda$ one of the roots of the equation, $$\eqlabel{oneof}
\lambda^6 = 5^4$$ We might record here a typical feature of these expansions: the equation means that $\lambda$ is, up to a root of unity, equal to $5^{2/3}$. A third root of unity is equivalent to the phase of the local expansion parameter, $w^{5/3}$. The additional choice of sign is associated with the choice of root in . The local monodromy $w\to\ee^{2\pi\ii}w$ permutes those branches cyclically. (As one might expect, the better local variable is actually $z=w^5$. The concomitant $5$-th roots of unity in $a$ will cancel out in .)
For the second group, $$\eqlabel{expansion2}
\begin{split}
a &=\textstyle \frac{6+\lambda^2}{12} w^{-1}
-\frac{3200(-599+355 \lambda^2)}{17} w^4
+\frac{76800000(-23778566+14088349 \lambda^2)}{289} w^9- \cdots \\
b&=\textstyle \lambda+\frac{2000 (-74708 \lambda+44263 \lambda^3)}{51} w^5
-\frac{1600000 (-326587981456 \lambda+193497180065 \lambda^3)}{2601} w^{10} +\cdots
\end{split}$$ where $\lambda$ is one of the roots of $$\eqlabel{twoof}
5\lambda^4+20\lambda^2-48=0$$ Note that while again the choice of sign for $\lambda$ originates from , the local monodromy $w\to\ee^{2\pi\ii}w$ acts trivially.
Before leaving this family of cycles for a while, we show the result of the computation of the inhomogeneous Picard-Fuchs equation. The algorithm of [@mowa; @newissues] can be applied without much change. The main complication is that one has to keep the parameters $a$ and $b$ implicit throughout. Since the line residual to the two conics has a vanishing superpotential (see section \[vangeemen\]), the inhomogeneity should be odd under $b\to-b$. With standard conventions, such as reviewed in section \[vangeemen\], we find the Picard-Fuchs inhomogeneity associated with conics in $\cali_1$ to be: $$\eqlabel{ffirst}
\begin{array}[t]{l}
\displaystyle
\qquad\qquad\call\int^C\Omega = f(z) \\[.5cm]
f(z) = \frac{1}{4\pi^2} \,
\frac{b}{320\,(-128 + 3 \psi^5)^3\, (-5308416 + 26104832\psi^5 + 459\psi^{10})^3}
\; \cdot\\
\scriptscriptstyle
\cdot[-3529208202219460015329116160 -
5917959309462446377556508672\, a^4 \psi -
24080174251679112693326807040\, a^3 \psi^2
\\[-.3cm]\scriptscriptstyle
- 37102979749413690361774080000\, a^2\psi^3 +
5322140674208202106664386560\, a\psi^4
\\[-.3cm]\scriptscriptstyle
+ 377013614277474642973792665600\,\psi^5
- 223673316478788106348117622784\, a^4\psi^6 +
231620425022730366652294103040\, a^3\psi^7
\\[-.3cm]\scriptscriptstyle
+ 577173365083785450174157946880\, a^2\psi^8 +
1161971462867073400022583214080\, a\psi^9
\\[-.3cm]\scriptscriptstyle
+ 1138625829170016488325937889280\,\psi^{10}
- 162426814061060730487566237696\, a^4\psi^{11} +
462200036747394287493017763840\, a^3\psi^{12}
\\[-.3cm]\scriptscriptstyle
+ 196861662250863298084696227840\, a^2\psi^{13} -
198567289143941889876285194240\, a\psi^{14}
\\[-.3cm]\scriptscriptstyle
+
385678957625260010043531591680\,\psi^{15}
- 188475902674373195063233609728\, a^4\psi^{16} +
397300557436660139725013647360\, a^3\psi^{17}
\\[-.3cm]\scriptscriptstyle
+ 468813519263945326185655828480\, a^2\psi^{18} +
479723528675140620247262822400\, a\psi^{19}
\\[-.3cm]\scriptscriptstyle
+ 352752475928491530510768537600\,\psi^{20}
- 39263076586488037778065981440\, a^4\psi^{21} +
110777498597321397283848192000\, a^3\psi^{22}
\\[-.3cm]\scriptscriptstyle
+ 42233632645599612642734899200\, a^2\psi^{23} +
16695932913990986817444249600\, a\psi^{24}
\\[-.3cm]\scriptscriptstyle
+ 5506564481958675778539356160\,\psi^{25}
- 279092702543449176793939968\, a^4\psi^{26} +
884770078321237750123069440\, a^3\psi^{27}
\\[-.3cm]\scriptscriptstyle
+ 34251597272406042397900800\, a^2\psi^{28} -
12180273406238980319477760\, a\psi^{29}
\\[-.3cm]\scriptscriptstyle
- 7891860706457745044275200\,\psi^{30}
+557447463014026659692544\, a^4\psi^{31} -
1763923787950883886858240\, a^3\psi^{32}
\\[-.3cm]\scriptscriptstyle
- 71223763050638247444480\, a^2\psi^{33} +
4711857482247092305920\, a\psi^{34}
\\[-.3cm]\scriptscriptstyle
+ 1639504965244195307520\,\psi^{35}
- 34139433836832735744\, a^4\psi^{36} +
110844573279392655360\, a^3\psi^{37}
\\[-.3cm]\scriptscriptstyle
- 4645064401757907840\, a^2\psi^{38} -
375748813003714560\, a\psi^{39}
\\[-.3cm]\scriptscriptstyle
- 14770116391956480\,\psi^{40}
+ 66315921005988\, a^4\psi^{41} -
220588897640760\, a^3\psi^{42} +
26084392488495 a^2 \psi^{43} +
193405158000 a\psi^{44} ]
\end{array}$$ We will not say much here about the structure of that result, just as we skipped the detailed discussion of the geometry of $\cali_1$. Note however that the factor $-128+3\psi^5$ in the denominator indicates an interesting interaction of $\cali_1$ with the van Geemen lines (, eq. ). It is easy to check that the conics in $\cali_1$ become reducible there (although not only there). The other factor in the denominator is the discriminant of .
In the expansion , becomes $$\eqlabel{firstgroup}
4\pi^2 f(z) = \textstyle
\frac{25\lambda}{54} z^{1/6}-\frac{2003 \lambda^3}{8} z^{1/2}
+\frac{18846875\lambda^5}{486} z^{5/6}-
\frac{6020738135875 \lambda}{2187} z^{7/6}+\cdots$$ and on the second group of branches, , we have $$\eqlabel{secondgroup}
4\pi^2 f(z) = \textstyle
10000 (-7624 \lambda+4517 \lambda^3) z-\frac{4000000
(-520331498984 \lambda+308286536785 \lambda^3)}{51} z^2 + \cdots$$
The main component
------------------
We now turn to Puiseux expansions of the solutions of that satisfy $Q_m=0$ (see eq. ). The ansatzs for plane and conic are (see eqs. , ), $$\begin{split}
A&:\left\{
\begin{array}{c}
a_1(x_1+x_2)+a_2(x_3+x_4) + x_5 \\
(x_1-x_2) + a_3 (x_3-x_4)
\end{array}
\right\}
\\
B&: \{ b_1 (x_1+x_2)^2 + b_2 (x_1+x_2)(x_3+x_4) + b_3 (x_3+x_4)^2
+ b_4 (x_3-x_4)^2 \}
\end{split}$$ For completeness, we display the ansatz for the cubic, $$\begin{gathered}
C:\{c_1 (x_1+x_2)^3 + c_2 (x_1+x_2)^2(x_3+x_4) + c_3(x_1+x_2)(x_3+x_4)^2
+c_4(x_3+x_4)^3 \\+ c_5 (x_1+x_2) (x_3-x_4)^2 + c_6 (x_3+x_4)(x_3-x_4)^2\}\end{gathered}$$ as well as the full set of relations, $$\eqlabel{fullset}
\begin{array}{lcl}
\frac1{80}-\frac{a_1^5}5 - b_1c_1&\qquad\qquad
& -\frac{\psi a_1}{16}+\frac{a_3^2}{8}-b_4c_1-b_1c_5\\
-a_1^4 a_2-b_2 c_1-b_1c_2 & &-\frac{\psi a_2}{16}- b_4c_2-b_2c_5-b_1c_6\\
\frac{\psi a_1}{16} -2 a_1^3 a_2^2-b_3c_1-b_2c_2-b_1 c_3 &&
-\frac{\psi a_1 a_3^2}{16}-b_4c_3-b_3c_5-b_2c_6\\
\frac{\psi a_2}{16} - 2a_1^2 a_2^3-b_3c_2-b_2c_3-b_1 c_4 & &
\frac{1}{8}-\frac{\psi a_2a_3^2}{16}-b_4 c_4-b_3c_6\\
-a_1a_2^4-b_3c_3-b_2c_4 & &
\frac{\psi a_1a_3^2}{16}+\frac{a_3^4}{16}-b_4c_5\\
\frac1{80}-\frac{a_2^5}5-b_3c_4 & &
\frac{1}{16}+\frac{\psi a_2a_3^2}{16}-b_4 c_6
\end{array}$$ After scaling one of the $b_j$’s to $1$, we have $12$ equations for $12$ variables, in addition to $\psi$, which we want to turn into a local expansion parameter. For each of our variables $v_i$ ($i=1,\ldots, 12$), we make an ansatz of the form $$v_i = \sum_{d=0} (v_i)_d \psi^{\alpha_i - \beta d}$$ with rational $\alpha_i$, $\beta$, plug into those equations, and solve order by order in $\psi$.[^5]
Remembering the warnings emitted in subsection \[newtonpuiseux\], we have a little bit of extra work to do at low order: the equations at lowest order might not determine all $(v_i)_0$ immediately. They could also split into several pieces that lie on separate components of $\calh_{\rm conics}$. For the latter issue, we keep only those that lie on $\cali_2$, , which satisfy . For the former, we continue to higher order. This determines the local expansion parameter $\psi^{-\beta}$, and eventually, all obstructions are lifted, and we can mechanically solve the recursion. We identify (a power of) one of the $(v_i)_0$ as generator of the number field associated with the corresponding group of branches. The information on the various groups belonging to $\cali_2$ is collected in Table \[groups\].
[|l|l|l|l|]{} & exponents, in order & local & generator of number field and\
\# & $(a_1,a_2,a_3,b_1,b_2,b_3,b_4)$ & par. & minimal polynomial\
1 & $(-1,-1,0,0,0,0,0^*)$ & $\psi^{-5}$ & $\lambda= (a_3^{10})_0$;
-------------------------------------------------------------------------------------
$\scriptstyle \lambda^{10}-243 \lambda^9+27675 \lambda^8-1529140 \lambda^7$
\[-.3cm\] $\scriptstyle +49599473 \lambda^6+221079468 \lambda^5+49599473 \lambda^4$
\[-.3cm\] $\scriptstyle -1529140 \lambda^3+27675 \lambda^2-243 \lambda+1$
-------------------------------------------------------------------------------------
: Groups of branches of $\zet_2^-$-invariant conics. Some choices capture fairly obvious symmetries of the ansatz: $a_3\mapsto-a_3$ corresponds to exchange of $x_3$ and $x_4$. Multiplication of $(x_1,x_2)$ and $(x_3,x_4)$ by opposite fifth roots of unity can also be absorbed without touching the local expansion parameter. The exchange $a_1\leftrightarrow a_2, a_3\leftrightarrow 1/a_3$ produces further groups, but leaves the first invariant (this is related to the symmetry $\lambda\to 1/\lambda$). In each group, $0^{*}$ is the exponent of the variable that we have found convenient to scale to $1$.[]{data-label="groups"}
\
2 & $(0,0,-\frac12,1,1,\frac 12,0^*)$ & $\psi^{-1/2}$ & $\lambda=(b_3)_0$ ; $\lambda^{10}-62208$\
3 & $(\frac 17,0,-\frac 17,\frac 27,-\frac 47,-\frac 57,0^*)$ & $\psi^{-5/7}$ & $\lambda=(2a_3^{2}a_2)_0$; $\lambda^{14}-5\lambda^7+5$\
4 &$(0,\frac12,\frac12,-\frac12,0^*,\frac12,\frac12)$ & $\psi^{-5/2}$ & $\lambda=(32b_3^5)_0$; $\lambda^4+11\lambda^2-1$\
To illustrate the complexity, we give some of the lowest order terms in the expansion of the fourth group in the table: $$\begin{split}
a_1 a_2 & = \textstyle
\frac{-8\lambda-\lambda^3}{20} \psi^{1/2}
+\frac{-47-4\lambda^2}{20} \psi^{-2}
+ \frac{130229\lambda+11743\lambda^3}{40} \psi^{-9/2} + \cdots \\
a_2^5 &= \textstyle
\frac{\lambda^3}{32} \psi^{5/2} + \frac{3+2\lambda^2}{32} +
\frac{-1105\lambda-101\lambda^3}{64} \psi^{-5/2} +\cdots\\
a_3^2/a_2^4 &= \textstyle
- \frac{16(21+2\lambda^2)}{15} \psi^{-1} + \frac{32(5024\lambda
+453\lambda^3)}{15} \psi^{-7/2} - \frac{16(71099+6411\lambda^2)}{3} \psi^{-6}
+\cdots
\end{split}$$ We have also ventured into the calculation of the inhomogeneous Picard-Fuchs equation for these cycles. Working order by order in the residue calculus of [@mowa], we obtain for the third group in Table \[groups\]: $$\begin{gathered}
\eqlabel{solving}
f(z) = \textstyle\frac{\ii\lambda^{1/2}}{4\pi^2}
\Bigl[
\frac{25(5\lambda^5-\lambda^{12})}{343} z^{1/7}
+\frac{500(94-17\lambda^7)}{2401} z^{2/7}
+\frac{225(70585\lambda^2-31748\lambda^9)}{16807} z^{3/7}\\
\textstyle
+\frac{400(2394125\lambda^4-191028\lambda^{11})}{117649} z^{4/7}
+\frac{3875(245997065\lambda^6-63500311\lambda^{13})}{823543} z^{5/7}
+\cdots
\Bigr]\end{gathered}$$ (as usual, $z=(5\psi)^{-5}$). This illustrates again the general structure we have been discussing: the seventh root of unity is the phase of the local expansion parameter $z^{1/7}$. The additional square-root in originates from the choice of sign of $a_3$ in : the exchange of $x_3$ and $x_4$ changes the cycle class by a sign. The remaining irrationality is intrinsic to the group of algebraic cycles under consideration.
Expansion in Large Volume Limit {#largevolume}
===============================
We are now ready to study the A-model expansion of the space-time superpotential. The main focus is the so-called multi-cover formula that relates the A-model expansion to the BPS content of the supersymmetric space-time theory.
Schematically, the general prediction of ref. [@oova] was that a single BPS state of charge $\beta$ should make a contribution to the space-time superpotential of the form $$\eqlabel{oovast}
\calw_\beta(q) \sim {\rm Li}_2(q^\beta) \sim \sum_k \frac{q^{\beta k}}{k^2}$$ where $t=\log q$ is the complex scalar in the supermultiplet coupling to $\beta$, and ${\rm Li}_2$ is the standard Euler’s di-logarithm function. The sum over $k$ originates as the Laplace transform of the D0-brane charge in the M-theory derivation of . In the context of [@oova] one assumes a local A-model setup with a non-compact Lagrangian as D-brane, where $t$ represents Kähler moduli as well as freely adjustable D-brane moduli.
If $n_\beta$ is the degeneracy of BPS states of charge $\beta$, the total superpotential is $$\eqlabel{degeneracy}
\calw = \sum_\beta n_\beta\calw_\beta \,.$$ This superpotential, together with its higher-derivative generalizations in the context of the open topological string, is equivalently computable from a sum over world-sheet instantons with boundary on the background D-brane. Disentangling the contributions in the various charge sectors, see, , [@lmv], leads to the customary relations between open Gromov-Witten invariants and BPS (Ooguri-Vafa) invariants.
For example, for the standard (“inner”) brane on the (resolved) conifold at zero framing, there are two BPS states of charge $(0,1)$ and $(1,-1)$, respectively, with a space-time superpotential: $$\eqlabel{conifold}
\calw (t,u) = \sum_{k=1}^\infty\Bigl( \frac{\ee^{k u}}{k^2} + \frac{\ee^{k(t-u)}}{k^2}\Bigr)$$ for the Kähler modulus $t$, and the open string (D-brane) modulus $u$. In the worldsheet computation of , one counts holomorphic maps from the disk to the conifold, with boundary mapping to the Lagrangian submanifold wrapped by the D-brane. The sum over $k$ originates from those maps that factor via degree $k$ multi-coverings of the disk by itself, such as $$\eqlabel{degreek}
z \mapsto z^k$$
It has been noted in [@open; @newissues; @ahmm] that these multi-cover formulas are not suitable in the context of compact manifolds. The main physical reason is that anomalies prevent a full separation of open and closed string moduli, while open Gromov-Witten invariants are not defined in general. The basic conundrum is already implicit in [@oova], where the masses of 2-d BPS solitons are determined by the critical values of the superpotential, which however is only generated by integrating out those very solitons. The issue could be resolved if it were possible to make sense of $\calw$ off-shell, , away from its critical points, ideally without additional information from the Kähler potential. In non-compact situations, certain natural choices are suggested by the symmetries of the asymptotic geometry [@agva; @akv]. In compact situations, one class of off-shell choices was proposed in [@joso; @ahmm], and a somewhat different one in [@ghkk].
Our way to deal with the ambiguities is to consider the critical points of the superpotential in the $u$-direction (, ). For the conifold, $$\begin{split}
\del_u\calw(t,u) & = -\log (1-\ee^{u}) +\log(1-\ee^{t-u}) = 0 \\
\Rightarrow & \ee^u = \pm \ee^{t/2}
\end{split}$$ Then the difference of critical values is given by $$\eqlabel{elevate}
\calw(t,u_+)- \calw(t,u_-) = 4 \sum_{k\;{\rm odd}} \frac{\ee^{tk/2}}{k^2}$$ This on-shell superpotential encodes less information than , but depends on fewer choices. If the inner brane on the conifold as a local model captures enough of the global geometry, one can elevate to a multi-cover formula instead of . $$\eqlabel{thisversion}
\calw (q) = \sum_{d\;{\rm odd}}\tilde n_d q^{d/2} =
\sum_{d,k\;{\rm odd}} n_d \frac{q^{dk/2}}{k^2}$$ Indeed, this modification of was found in [@open] to relate the rational open Gromov-Witten invariants $\tilde n_d$ of the real quintic to integer invariants $n_d$, that fit into a larger framework of real enumerative geometry.
In [@newissues; @ahmm], other modifications of the di-logarithm were identified, such as $$\eqlabel{triple}
\sum_{3\nmid k} \frac{q^{d k/3}}{k^2}$$ albeit without a description of either local or global A-model geometry.
Through the examples of the present paper, we will see that and are just the simplest versions of a much more elaborate class of “multi-cover” formulas. The relevance of certain arithmetic functions in these new multi-cover formulas is rather intriguing, and indicative of deeper connections between mirror symmetry and number theory that we hope to explore elsewhere.
Van Geemen lines
----------------
We first return to the van Geemen families of lines. Their inhomogeneous Picard-Fuchs equation was calculated in section \[vangeemen\], $$\call\calw_B(z) = f_{\rm van\;Geemen}(z)$$ where $$\call= \theta^4 - 5 z(5\theta+1)(5\theta+2)(5\theta+3)(5\theta+4)$$ $\theta=\frac{d}{d\ln z}$, $z=(5\psi)^{-5}$, and, with $1+\omega+\omega^2=0$, $$f_{\rm van\; Geemen}(z) = \frac{1+2\omega}{4\pi^2}\,\cdot\,
\frac{32}{45}\,\cdot\,
\frac{\frac{63}{\psi^{5}}+\frac{1824}{\psi^{10}}-\frac{512}{\psi^{15}}}
{\Bigl(1-\frac{128}{3\psi^5}\Bigr)^{5/2}}$$ It is straightforward to solve this equation in a power series around $z=0$, and apply the usual mirror map to obtain $$\eqlabel{scared}
\begin{split}
\widehat{\calw}_A(q)& =4\pi^2\calw_A(q) = \frac{4\pi^2\calw_B}{\varpi_0}(z(q)) \\
&=\sqrt{-3}\,\bigl(\textstyle 140000 q+\frac{11148100000}{3} q^2+
\frac{5015947794500000}{27} q^3+
\frac{330137902935872500000}{27} q^4 \\
&\textstyle+\frac{76015582693256843498840000}{81} q^5+
\frac{57929080529317310275946498060000}{729} q^6+\cdots \bigr)
\end{split}$$ Instead of being scared away by the growth of the numerators of the expansion coefficients, let us look at the denominators. We define $\tilde n_d$ as the coefficient of $q^d$: $$\eqlabel{firstexample}
\widehat{\calw}_A(q) = \sum_{d=1}^\infty \tilde n_d q^d$$ Remarkably, the $\tilde n_d$ are not rational numbers, in distinction to all previous examples in the literature. From expansion to large order, we observe that the denominator of $\tilde n_d$ grows as $3^d$, but otherwise contains at most a factor of $d^2$, , we have $$\eqlabel{discou}
d^2 3^d \frac{\tilde n_d}{\sqrt{-3}} \in \zet$$ Given previous experience, in which the $d^2 \tilde n_d$ were always (rational) integers, the result could seem a bit disappointing. On the other hand, the denominators are remarkably smaller than those in $\calw_B$. Roughly speaking, the mirror map reduces $(d!)^2$ to $d^2$. It is natural to expect that the factors of $d^2$ in the denominator can be removed by an appropriate multi-cover formula. It is remarkable that such a formula indeed exists!
On expanding $$\eqlabel{firstD}
\widehat{\calw}_A(q) = \sum_{d=1}^\infty n_d \sum_{k=1}^\infty \frac{\chi(k)}{k^2} q^{dk}$$ where $\chi(k)$ depends on the residue of $k\bmod 3$, $$\eqlabel{dirichlet}
\chi(k) = \begin{cases} 0 & k\equiv 0\bmod 3 \\
1 & k\equiv 1\bmod 3\\
-1 &k\equiv 2\bmod 3
\end{cases}$$ we find $$3^d \frac{n_d}{\sqrt{-3}} \in \zet$$ The first few $n_d$ are[^6] $$\begin{array}{rcl}
n_1 & = & \sqrt{-3}\, 140000 \\
n_2 & = & \sqrt{-3}\, \frac{11148205000}{3}\\
n_3 & = & \sqrt{-3}\,\frac{5015947794500000}{27}\\
n_4 & = & \sqrt{-3}\,\frac{330137902960955725000}{27} \\
\vdots & & \qquad\quad \vdots
\end{array}$$
The function $\chi(k)$ in is, of course, just the quadratic character modulo 3, which is the non-trivial Dirichlet character of order 3, one of the standard arithmetic functions of algebraic number theory. $$\chi(k) = \left(\frac{k}{3}\right)$$ Incidentally, we may now recognize and as having a rather similar form, with $\chi(k)$ replaced by the trivial (principal) Dirichlet character of order 2, and $3$, respectively.
The D-logarithm
---------------
The results so far motivate us to introduce more general twists of the di-logarithm, which we will call the D-logarithm, of the form $$\eqlabel{Dlog}
{\rm Li}_2^{\rm D}(x) = \sum_{k=1}^\infty \frac{a_k}{k^2} x^k$$ where $(a_k)$ are sequences of numbers that we will specify (see subsections \[definition\] and \[redefinition\]).
The purpose of the D-logarithm is to serve as a refinement of the multi-cover formula for general D-brane superpotentials. The notation and terminology, however, is suggested by the special case that $a_k=\chi(k)$ is a Dirichlet character, and in which we write, $${\rm Li}_2^{(\chi)}(x) = \sum_{k=1}^\infty \frac{\chi(k)}{k^2} x^k$$ When $\chi$ is a trivial Dirichlet character, we recover the formulas of [@open; @newissues], while $\chi(k)=\left(\frac{k}{3}\right)$ is relevant for the van Geemen lines.
The original occurrence of twists of this type of course is in Dirichlet L-functions, $$L(s;\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ In fact, given the relation of special values $$L(2;\chi) = {\rm Li}_2^{(\chi)}(1)$$ one might view the D-logarithm as a natural alternative analytic continuation of those special values. We will see below however that the coefficients $(a_k)$ relevant for the D-logarithm are in general different from those occuring in typical L-series.
First component, first group
----------------------------
So let us consider now the A-model expansion of the superpotential for the family of conics that we have been calling $\cali_1$, and whose inhomogeneous Picard-Fuchs equation is calculated in .
As discussed in section \[largecomplex\], the $10$ branches of $\cali_1$ over the complex structure moduli space fall into two groups in the large volume limit. The first group contains $6$ branches that are in correspondence with the solutions of ; the second has $4$ branches and is governed by . We do not have an A-model interpretation of this structure, but we can calculate the superpotential as a solution of . This will allow us to delineate the definition of the D-logarithm.
In the expansion , we find after the standard mirror map: $$\eqlabel{facilitate}
\begin{split}
\widehat{\calw}_A&(q) = \textstyle
600\cdot 5^{2/3}\cdot q^{1/6}- 100150 q^{1/2}
+\frac{30155000}{3} \cdot 5^{1/3} \cdot q^{5/6}
-\frac{1965946252000}{1323}\cdot 5^{2/3}\cdot q^{7/6}\\
\textstyle
+ &\textstyle \frac{12016906931000}{9} q^{3/2}
-\frac{7939808112480350000}{29403}\cdot 5^{1/3}\cdot q^{11/6}
+\frac{21851198476716995185000}{369603} \cdot 5^{2/3}\cdot q^{13/6} \\
-&\textstyle\frac{205546823520516323768}{3} q^{5/2}
+\frac{94301250909743023365521125000}{5688387}\cdot 5^{1/3}\cdot q^{17/6}\\
-&\textstyle
\frac{795791304680793507631175999075000}{191850201} \cdot 5^{2/3}\cdot q^{19/6}
+
\frac{783657804098608936611454866250}{147} q^{7/2}
+\cdots
\end{split}$$ We have here substituted $\lambda=5^{2/3}$ in order to facilitate the following observations. Denoting the coefficient of $q^{d/6}$ by $\tilde n_d$, we have (all of what follows is confirmed to rather high order, up to $d \gtrsim 600$) when $2\mid d$, $\tilde n_d=0$ when $3\mid d$, $d^2\tilde n_d\in\zet$ when $(d,6)=1$, $d^2 3^{\lfloor\frac{3d}{4}\rfloor}\tilde n_d\in 5^{\frac{2}{3}}\zet$ when $d=1\bmod 3$ and $\in 5^{\frac{1}{3}}\zet$ when $d=2\bmod 3$.
There are some similarities, but also noticeable differences to : The irrationality of the $\tilde n_d$ is not just an overall factor for $\widehat\calw_A$. The denominators of the irrational $\tilde n_d$ grow with $d$ as a power of $3$, although slightly less rapidly than . The $\tilde n_d$ for $3\mid d$ are integer up to a factor of $d^2$.
As for the van Geemen lines , our goal now is to describe the D-logarithm such that via $$\eqlabel{want}
\widehat\calw_A(q) = \sum_d\tilde n_d q^{d/6} = \sum_{d} n_d {\rm Li}_2^{\rm D}(q^{d/6})
= \sum_{d} n_d \sum_k \frac{a_k}{k^2} q^{dk/6}$$ the $n_d$ will have the remaining $d^2$ dropped from their denominators. Some important points are clear at this stage already. The $n_d$ will remain irrational, and we are prepared to live with a growing denominator. The $a_k$ appearing in the D-logarithm will also be irrational. The most natural way to capture the symmetries of is that the $a_k$ are rational up to a factor of $5^{2d(k-1)/3}$. This means in particular that the $a_k$ must depend on $d$. Given the symmetries, one expects that the $a_k$ will depend on $d\bmod 3$.
Before looking for a solution to these constraints, we add the following piece of information: in a putative physics interpretation, to be further discussed below, the $n_d$ should be related to a degeneracy of appropriate BPS states that multiplies those states’ contribution to the space-time superpotential, see eq. . Looking back at eqs. , , and , we see that the variations of the multi-cover formula are related to the vacuum structure of the associated D-brane configuration, and more precisely to the action of the symmetry group of the algebraic equation determining that structure, the equation being $\varphi^2=1$, $\varphi^3=1$, and $\omega^2+\omega+1=0$ in those three cases, respectively. Together with the present data, this indicates that the correct version of the Ooguri-Vafa formula depends on the [*arithmetic properties*]{} of the “BPS degeneracies” $n_d$, and more specifically on the action of the Galois group of the relevant number field.
A first hint that the result fits into the general framework comes from the part of the expansion with only rational coefficients. As remarked above, the $\tilde n_{3d}$ are rational, with denominator $(3d)^2$. And with the simple twist by the principal character of order 2, $$\eqlabel{firsthint}
\sum_{d\;{\rm odd}} \tilde n_{3d} q^{d/2} = \sum_{d,k\;{\rm odd}} \frac{n_{3d}}{k^2} q^{dk/2}$$ one finds that the $n_{3d}$ are indeed integer. This is a quite non-trivial check that there is no simple mistake in .
The irrational part of is governed by the number field $K=\rationals(5^{1/3})$, with Galois completion $L=\rationals(5^{1/3},\sqrt{-3})$. A crucial observation that sets us onto the right track is the following. Whereas $\tilde n_7$ and $\tilde n_{11}$ do not have an obvious congruence with $\tilde n_1$ (it is difficult to ascertain any statements about $d=5$ or multiples thereof, because $5$ divides the discriminant of the number field), one finds that in the combination $$\eqlabel{crucial}
\tilde n_{13} - \frac{\tilde n_1}{13^2} = 5^{2/3}\cdot
\frac{129297032406609431200}{3^7}$$ the $13^2$ in the denominator cancels out. So we learn that $a_{13}=1$. Invoking rudimentary knowledge of elementary algebraic number theory, we understand that indeed $13$ is special with respect to our number theoretic situation: it is a prime that splits completely in the field extension $L/\rationals$. Actually, already in $K$, $$\eqlabel{splitcompletely}
x^3 - 5 = (x+2)(x+5)(x+6) \bmod 13$$ That is not a coincidence can be confirmed by checking the next primes with the same property: $67,127,\ldots$.
Now let’s again pause and compare with the van Geemen lines: when $k=p$ is prime, $a_p=\left(\frac{p}{3}\right)=1$ when $p$ is a quadratic residue $\bmod 3$. By reciprocity, this is the case precisely when $p$ splits completely in $\rationals(\sqrt{-3})$, , $x^2+3=0$ has two solutions in $\zet/p$. This is clearly consistent with the observations around . When $p$ does not split in $\rationals(\sqrt{-3})$, $a_p=-1$. More generally we have the multiplicativity $a_{k_1k_2}=a_{k_1}a_{k_2}$.
Adding knowledge of all previous cases as well as gives us a very clear understanding of when $a_p=1$: either there is no field extension, or $p$ splits completely in the extension field. But how do we generalize the non-trivial $\left(\frac{p}{3}\right)$ when $K=\rationals(5^{1/3})$, or some other number field? Since $L=K(\sqrt{-3})$ is a non-abelian extension, with Galois group $G=S^3$ the full permutation group, consultation of the literature might suggest to try characters of higher-dimensional representations of $G$. This however is incompatible with the expectation that the $a_p$ should themselves be general irrationals, and seems in fact impossible to realize. Nevertheless, some detective’s work reveals that the correct answer indeed involves the structure of $G={\rm Gal}(L/\rationals)$, especially its action at the primes. We explain this next.
D-logarithm mod k2 {#definition}
------------------
In the previous subsections, we have listed various constraints that we expect the D-logarithm to satisfy and collected hints that its underlying sequence $(a_k)$ is determined by the arithmetic properties of the invariants $n_d$. The for us characteristic property, namely that the formula clear the $d^2$ from the denominators, can be used to determine the $a_k$ as algebraic numbers (integers) $\bmod k^2$. This is what we do here.[^7]
Based on the cases involving Dirichlet characters, we believe that in general there are distinguished representatives for $a_k$, which then define the D-logarithm ${\rm Li}_2^{\rm D}(x)$ as an analytic function of $x$. These distinguished representatives should be obtained either from a proper physics derivation of , or an appropriate mathematical interpretation of the D-logarithm as a multi-cover formula.
We first make explicit that the D-logarithm depends on $n_d$ by rewriting as $$\eqlabel{want2}
\widehat\calw_A(q) = \sum_d\tilde n_d q^{d/6} = \sum_{d} n_d \sum_k
\frac{a_k^{(d)}}{k^2} q^{dk/6}$$ with the understanding that $a_k^{(d)}$ depends on the upper index $d \bmod 3$. We know that $a_k^{(0)}=1$ for all odd $k$ and $a_p^{(d)}=1$ when $p$ is a prime that splits completely in $K=\rationals(5^{1/3})$.[^8]
Plugging into we extract the following values for $a_k^{(d)}\bmod k^2$ for $k=p$ the first few primes, $d=1,2$: (We are continuing to write $\lambda=5^{2/3}$ but of course other cube roots of $5^2$ would do as well.) $$\begin{tabular}{c|c|c|}
$p$ & $a_p^{(1)} \bmod p^2$ & $a_p^{(2)} \bmod p^2$\\\hline
7 & 30 & 18 \\
11 & $82\cdot 5^{2/3}$ & $103 \cdot 5^{1/3}$ \\
13 & 1 & 1 \\
17 & $247 \cdot 5^{2/3}$ & 150 $\cdot 5^{1/3}$ \\
19 & 68 & 292 \\
23 & $59 \cdot 5^{2/3}$ & $477 \cdot 5^{1/3}$ \\
29 & $538 \cdot 5^{2/3}$ & $700\cdot 5^{1/3}$ \\
31 & 521 & 439 \\
\vdots & \vdots & \vdots
\end{tabular}$$ The structure is fairly obvious: when $p\equiv 1\bmod 3$ (but doesn’t split completely), $a_p^{(1)}$ and $a_p^{(2)}$ are the two roots $\bmod p^2$ of the equation $$\eqlabel{cuberoot}
a^2 + a + 1 = 0$$ When $p\equiv 2\bmod 3$, we find that $b^{(1)}= a_p^{(1)}/5^{2/3}$ is the root $\bmod p^2$ of the equation $$\eqlabel{cr2}
25 b^3 = 1$$ while $b^{(2)}=a_p^{(2)}/5^{1/3}$ is the root $\bmod p^2$ of the equation $$\eqlabel{cr3}
5 b^3 = 1$$ The solutions to and are unique, but how do we discriminate between the two roots of ? It turns out that the relevant information comes from the Frobenius automorphism of the residue field extension at $p$: the reduction of $a_p^{(d)}\bmod p$ should agree with $\bigl(5^{2d/3}\bigr)^{p-1}$, $$a_p^{(d)} = \bigl(5^{2d/3}\bigr)^{p-1} \bmod p$$ which then uniquely selects the solution of . (For example $30=2=
(5^{2/3})^6=5^4\bmod 7$.)
This reduction also holds for $p\equiv 2\bmod 3$: $$\frac{a_p^{(1)}}{5^{2/3}} = \frac{\bigl(5^{2/3}\bigr)^{p-1}}{5^{2/3}} \bmod p$$ and a similar equation holds for $d=2 \bmod 3$.
The distinction between $p=1$, or $2$, $\bmod3$ lies in the order of the Frobenius automorphism. It is equal to $3$, or $2$, respectively, which corresponds exactly to the two non-trivial cycle classes of the Galois group $G=S^3$.
To understand that these findings are in agreement with the conditions we knew for $a_p^{(d)}=1$, we recall that when $p$ splits completely, or $n_d$ is rational, the Frobenius automorphism acts trivially. This is true in the first situation because the local field extension is trivial, and in the latter can also be viewed as a consequence of Fermat’s little theorem, $a^p=a\bmod p$ for any $a\in \zet$. This theorem also implies that the Frobenius acts on $n_d$ in the same fashion as on $5^{2d/3}$. Thus, we may summarize: $$\eqlabel{pprime}
\text{
\framebox{\parbox{\textwidth-5.5cm}{
\centering{{\em For $p$ prime, $a_p^{(d)}$ is the lift $\bmod p^2$ of the
Frobenius automorphism at $p$ acting on $n_d$.}}}}}$$
What about $a_k$ when $k$ is [*not*]{} prime? A basic expectation is a multiplicative structure relating $a_{k_1 k_2}$ to $a_{k_1}\cdot a_{k_2}$. It is clear however that this has to be refined because we have obtained $a_p$ only $\bmod p^2$. Moreover, the condition that $a_k$ should be rational up to a factor $5^{2d(k-1)/3}$ (see page ) is not naively compatible with a multiplicative structure for $a_k^{(d)}$.
When $k=p_1p_2$ is the product of two distinct primes, one could imagine fixing representatives of $a_{p_1}$ and $a_{p_2}$ and require that $a_{p_1p_2}$ coincide with $a_{p_1}a_{p_2}$ $\bmod (p_1p_2)^2$. But this is not sufficient when $k$ is divisible by a higher prime power $p^e$, $e>1$. To deal with this situation we introduce $e$ as an additional index and let $$\eqlabel{lift}
\text{
\framebox{\parbox{\textwidth-4.5cm}{
\centering{$a_{p,e}^{(d)}$ be the lift $\bmod p^{2e}$ of the Frobenius at $p$ acting on $n_d$.}}}}$$ For instance, when $p=1\bmod 3$, $a_{p,e}^{(d)}$ is the unique solution of $\bmod p^{2e}$ that agrees with the Frobenius $\bmod p$. We agree that $a_{p}^{(d)}=
a_{p,1}^{(d)}$.
The issue with the irrationality of the multiplicative structure can be resolved by mixing the two sequences $a_k^{(1)}$ and $a_k^{(2)}$.
Here then is the explicit algorithm that allows the recursive calculation of all $a_k^{(d)}
\bmod k^2$. For any prime $p$ dividing $k$, we denote by $e_p$ the largest power of $p$ dividing $k$, and we require $$\eqlabel{multiplicative}
\text{
\framebox{\parbox{\textwidth-10cm}{
\centering{$a_k^{(d)} = a_{p,e_p}^{(d k/p)} \cdot a_{k/p}^{(d)} \bmod p^{2e_p}$}}}}$$ Imposing these conditions for all primes dividing $k$ determines $a_k^{(d)}$ uniquely $\bmod k^2$.
An interesting aspect of eq. is that the value of $a_k^{(d)}$ depends recursively on the representatives chosen for $a_{k/p}^{(d)}$ $\bmod k^2$ (which is previously determined only $\bmod (k/p)^2$). The structure of however is such that those choices do not affect the integrality properties of the resulting $n_d$. This means that the representatives for $a_k^{(d)}\bmod k^2$ are not independent from one another, and is good evidence that more distinguished representatives should exist.
How this really works in detail is, of course, clearest in the examples. We have seen above that $a_7^{(1)}=30\bmod 7^2$, and easily find $a_{7,2}^{(1)}=1353\bmod 7^4$, so that $a_{49}^{(1)} = 40590$. Stripping the $\bmod$s and solving , we find $$\begin{split}
n_1&=\tilde n_1 = 5^{2/3} \cdot 600\\
n_7 &= \tilde n_7 - \frac{30}{7^2} n_1 = -5^{2/3} \cdot\frac{40121362000}{3^3} \\
n_{49} &= \tilde n_{49} - \frac{30}{7^2} n_7 - \frac{40590}{7^4} n_1 =
5^{2/3} \cdot \frac{392867\cdots993000}{3^{33}}
\end{split}$$ Plugging one into the other, we see that $$\begin{split}
n_{49} &= \tilde n_{49} - \frac{a_7^{(1)}}{7^2} \tilde n_7 -
\Bigl(\frac{a_{49}^{(1)}}{7^4}- \frac{(a_{7}^{(1)})^2}{7^4}\Bigr) n_1 \qquad\qquad\qquad \\
&= \tilde n_{49} - \frac{a_7^{(1)}}{7^2} \tilde n_7 -
\frac{\bigl(a_{7,2}^{(1)}- a_{7}^{(1)}\bigr)a_7^{(1)}}{7^4} n_1 \\
&= \tilde n_{49} - \frac{a_{7,2}^{(1)}}{7^2}\tilde n_7
+\frac{a_{7,2}^{(1)}-a_7^{(1)}}{7^2}\Bigl(\tilde n_7-\frac{a_7^{(1)}}{7^2} n_1\Bigr)
\end{split}$$ Since $a_{7,2}^{(1)}=a_7^{(1)}\bmod 7^2$, this shows that we may change $a_7^{(1)}$ by multiples of $7^2$ without affecting the fact that the denominator of $n_{49}$ is not divisible by $7$.
We have checked up to $d\gtrsim 600$ that using the three formulas , , and in returns $n_d$ with no $d^2$ in the denominator.
First component, second group {#redefinition}
-----------------------------
The definition of the D-logarithm $\bmod k^2$ that we have given in the previous subsection was slanted towards the example . To see that the prescriptions , , and make sense, and are correct, in more generality, we here study the second group of branches of the first component $\cali_1$ of conics on the mirror quintic.
The A-model expansion in this example begins $$\eqlabel{thisexample}
\begin{split}
\widehat{\calw}_A(q) =&
10000 \lambda (-7624+4517 \lambda^2) q -\textstyle\frac{3200000\lambda
(-40650831529+24084846092 \lambda^2)}{51} q^2 \\
&\textstyle\quad + \frac{250000
\lambda (-5248611469517402890552+3109702672077500263451 \lambda^2}{3^3\cdot 17^2} q^3 \\
&\qquad\scriptstyle -\frac{2500000 \lambda (-781124731396525415521048504088+
462801576865994098449442008739 \lambda^2)}{3^3\cdot 17^3 }q^4 + \cdots
\end{split}$$ where $\lambda$ is a root of $$\eqlabel{minimal}
5\lambda^4+20 \lambda^2-48 = 0$$ The Galois completion of $K=\rationals(\lambda)$ is $L=K(\sqrt{-3/5})$, with Galois group the dihedral group $D_4$. This can be visualized by arranging the four roots of in a square, $$\eqlabel{square}
\begin{array}{ccc}
\textstyle\sqrt{-2+2\sqrt{\frac{17}{5}}} & \line(1,0){40} &\sqrt{-2-2\sqrt{\frac{17}{5}}} \\
\line(0,1){40} & & \line(0,1){40} \\
\textstyle-\sqrt{-2-2\sqrt{\frac{17}{5}}} & \line(1,0){40} &
-\sqrt{-2+2\sqrt{\frac{17}{5}}}
\end{array}$$ The discriminant of $K$ is $-3\cdot 5^3\cdot 17^2$, and we may check that the denominators in behave as $d^2 3^d 17^d$. We want to remove the $d^2$ using the appropriate D-logarithm.
The symmetries in this example do not completely constrain the irrationality of the $\tilde n_d$, which are general linear combinations of $\lambda$ and $\lambda^3$. In other words, only the behavior of $\widehat\calw_A$ under $\pi$-rotation of is fixed. As a consequence, the D-logarithm has a more severe dependence on $d$ than in the previous example, leading to even more intricate checks of the formalism.
So let us explain how is implemented in the present example. We assume that $n_d$ is an algebraic number with denominator vanishing at most at the discriminant of $K$, and want to determine $a_p$ when $p$ does not divide the discriminant.
As one learns in algebraic number theory, the structure of $K/\rationals$ at the (rational) prime $p$ is related to the factorization $\bmod p$ of the minimal polynomial of an integral generator of $K$. In our examples, we could choose $\mu=5\lambda$, with minimal polynomial $$\eqlabel{PPP}
P = \mu^4 + 100\mu^2 -6000$$ Then to each factor of $P\bmod p$ is associated a prime (ideal) $\mathfrak{p}_i$ in $K$ “lying over $p$”, and the degree of that factor is the degree of the associated residue field extension $(\zet_L/\mathfrak{p}_i)/(\zet/p)$. (As above, $L$ is the Galois completion of $K$, and $\zet_L$ is the ring of integers in $L$.) This being an extension of a finite field, it has cyclic Galois group generated by a single element, called the Frobenius element $\sigma_i=
\sigma(\mathfrak{p_i}/p)$, which acts as $y\mapsto y^p$ in the residue field. In the Galois extension $L$, the Frobenius elements associated with different factors of $p$ would all be conjugate to each other, so determine a conjugacy class in the Galois group $G={\rm Gal}(L/\rationals)$. In our case, $K/\rationals$ is not Galois, so we work with $\sigma_i$ that act as definite elements of $G$ on the roots of .
Now given $n_d$ as a (non-zero) algebraic number in $K$ we consider, for each $\mathfrak{p}_i$ dividing $p$, the algebraic number $$z_i = \frac{\sigma_i(n_d)}{n_d}$$ These $z_i$ themselves live in $L$, but not generally in $K$, and moreover, depend on $i$. So what do we mean in by “$a_p$ is given by the action of Frobenius at $p$”? The underlying idea, familiar in algebraic number theory, is to work “locally around $p$”, , approximate numbers $\bmod p$ (or, more generally, $\bmod p^e$ for $e>1$). In the local approximation, we can both find representatives for $z_i$, and interpolate between the different $z_i$, as $\mathfrak{p}_i$ varies over $p$. Moreover, given an approximation to order $p$, we can lift it $\bmod p^2$, and this is our definition of $a_p$. The lifts $\bmod p^{2e}$, needed in , are then obtained in a straightforward continuation of this procedure.
In formulas, $a_p$ is the number in $\zet_K/p^2$ that agrees with $z_i$ at each prime $\mathfrak{p}_i$ dividing $p$, $$a_p = z_i \bmod \mathfrak{p}_i^2$$ As in the previous examples, we have found this procedure (augmented with and for $k$ not prime) such that via $$\widehat{\calw}_A = \sum_d\tilde n_d q^d = \sum_d n_d \sum_k \frac{a_k}{k^2} q^{dk}$$ it returns invariants $n_d$ with no $d^2$ in the denominator, up to some significant order $d$.
To make the procedure easier to follow, we discuss as an example, the results for $$n_1 = 10000 \lambda (-7624+4517 \lambda^2)$$ (For incidental reasons, we revert here to the non-integral generator $\lambda$. This does not change the results.) Clearly, under $\lambda\mapsto-\lambda$, $n_1\mapsto -n_1$, so $z=-1$. For any element $g\in G$ of the Galois group that reverses the sign under the square-roots in (namely, rotation by $\pi/2$ (order 4) or horizontal or vertical flip (order 2)), the resulting $z= g(n_1)/n_1$ is a root of the polynomial $$\eqlabel{galpo}
583443+135146154523047386 z^2+583443 z^4$$ Because $583443=3^5 \cdot 7^4$, but $7$ does not divide the discriminant of $K$, that prime will require a bit of a special treatment.
For the first few non-trivial primes, we find the following table $$\begin{array}{c|c|c|c}
p& P/5^3\bmod p & f_p:= (n_1)^{p-1}\bmod p & a_p \bmod p^2\\ \hline
7 & (2+\lambda) (5+\lambda) (1+\lambda^2) & 2+3 \lambda^2 &40+20\lambda^2\\
11 & 8+4 \lambda^2+\lambda^4 & 2+7 \lambda^2 &24+62\lambda^2\\
13 & 6+4 \lambda^2+\lambda^4 & 4+12 \lambda^2 &43+25\lambda^2\\
17 & (7+\lambda)^2 (10+\lambda)^2 & {\it ramifies} & \\
19 & (7+\lambda^2) (16+\lambda^2) & 18 & -1 \\
23 & (1+\lambda) (8+\lambda) (15+\lambda) (22+\lambda) & 1 & 1 \\
29 & 2+4 \lambda^2+\lambda^4 & 19+11 \lambda^2 & 48+417\lambda^2\\
31 & (3+8 \lambda+\lambda^2) (3+23 \lambda+\lambda^2) & 18+11 \lambda^2 &235+445\lambda^2\\
37 & (14+\lambda) (23+\lambda) (15+\lambda^2)& 3+20 \lambda^2 & 780+390\lambda^2 \\
\vdots & \vdots & \vdots & \vdots
\end{array}$$ How did we find the last column? From the degree of the factors of $P\bmod p$, we may read off the order of the various Frobenius elements. In most cases, this determines them completely: for inert primes such as $11$, $13$, $29$, the Frobenius has order $4$, so must be rotation by $\pi/2$ in . For primes that split completely such as $p=23$, the Frobenius is trivial (as we were happy to learn some time ago!). For primes with one quadratic, and two linear factors, such as $7,37$, the Frobenius must be a diagonal flip. The only ambiguous cases are those with two quadratic factors, which could correspond to horizontal/vertical flip, or rotation by $\pi$.
It is easy to check that for $p=11,13,29,31$, $f_p$ solves $\bmod p$, and $a_p$ is simply the lift of that solution $\bmod p^2$. In particular, the Frobenius at $31$ must be horizontal/vertical flip.
For $p=19$, $f_{19}=-1$ at both factors, so Frobenius must be rotation by $\pi$. We keep $a_{19}=-1\bmod 19^2$, just as we use $a_{23}=1$ since $23$ splits completely.
For $p=37$, $f_{37}=1$ at the linear factors, and $f_{37}=-1$ at the quadratic factor. In other words, $f_{37}$ is a solution of $z^2-1=0\bmod 37$, and $a_{37}$ is the lift of that solution $\bmod 37^2$.
What happened at $p=7$? From the structure of the factorization, it should be in the same class as $p=37$. However, $(f_7)^2-1\neq 0\bmod 7$ …… Some reflection reveals that the denominator of $z$ in being divisible by $7$ is due to the fact that $n_1$ vanishes at the two linear factors of $7$, so the action of the Frobenius automorphism as $(n_1)^{p-1}$ becomes completely ambiguous there. Independently however, we have known that the Frobenius should restrict to $1$ at the linear factors, and to $-1$ at the quadratic factor. This can be used to determine that $a_p = 5+6\lambda^2\bmod 7$, which as a solution of $z^2-1=0$ may then safely be lifted $\bmod 7^2$.
We may summarize the A-model discussion by stating that once again the Ooguri-Vafa multi-cover formula has proven to be basically correct, but that it needs a significant refinement in arithmetically non-trivial situations, which we have encountered here for the first time. The refinement is provided by the D-logarithm, which we conjecture is an analytic function attached to individual algebraic numbers $n_d$. The sequences $(a_k)$ defining the D-logarithm are specified $\bmod k^2$ by studying the action of the Galois group on $n_d$. It remains to be seen whether this remarkable structure can be sharpened and explained more fundamentally, and how it ties in with the rest of our subject. In the remaining section \[discussion\], we will present some initial thoughts that make this not impossible.
Main component
--------------
The only purpose of this subsection is to point out that it is possible to calculate the large volume expansion of the superpotential also on the main component of $\calh_{\rm conics}$, which we called $\cali_2$ in section \[largecomplex\]. Consider the third group in Table \[groups\]. We solve the Picard-Fuchs equation with inhomogeneity , apply the mirror map, and obtain $$\eqlabel{difficult}
\begin{split}
\frac{4\pi^2}{(-\lambda)^{1/2}}\calw_A = &
\textstyle
- 175 \lambda^5 (-5+\lambda^7)q^{1/7}- \frac{125 (-94+17 \lambda^7)}{4}q^{2/7}
-\frac{25 \lambda^2 (-70585+31748 \lambda^7)}{63}q^{3/7} \\
&\textstyle -\frac{25\lambda^4 (-2394125+191028 \lambda^7)}{2^4\cdot 7^2} q^{4/7}
- \frac{31 \lambda^6 (-245997065+63500311 \lambda^7}{5\cdot 7^3} q^{5/7} \\
&\textstyle
-\frac{5 \lambda (-8907388019619+2655707519021 \lambda^7)}{2^2 \cdot 3^3\cdot 7^4} q^{6/7}
- \frac{5\lambda^3 (-3595649177+980861072 \lambda^7)}{7} q\\
&\textstyle
-\frac{ \lambda^5 (-1905271484195274460+512248788482392343 \lambda^7)}{2^7 \cdot 7^7}q^{8/7}
+\cdots \end{split}$$ where $\lambda$ is the algebraic number with minimal polynomial $$\lambda^{14}-5\lambda^7+5 = 0$$ It is difficult to obtain convincing tests of our general formalism from , but the first few orders in the expansion are encouraging: $p=2,5,7$ divide the discriminant, so are allowed in the denominator. Subtraction of $\tilde n_1$ from $\tilde n_3$ works as expected, with $a_3^{(1)}=8 \lambda^4+2
\lambda^{11}$. I have no explanation for the apparent anomaly at $d=6$ (which has $3^3$ in the denominator).
Discussion
==========
In this work, we have studied families of algebraic cycles on the mirror quintic represented by curves of low degree. We have obtained a fairly complete picture of those conics that deform with the mirror quintic. We have seen how the Newton-Puiseux expansion around large complex structure limit splits the algebraic cycle into groups, each governed by an algebraic number field. We have then calculated the truncated normal function (up to an additive constant) by solving the inhomogeneous Picard-Fuchs equation. The irrationality does not disappear after application of the mirror map, confirming a long-standing expectation. To exhibit the underlying (algebraic) integrality[^9] of the expansion, we have introduced the D-logarithm as an arithmetic twist of the di-logarithm. This formalism generalizes all previously known cases and we might expect that it is complete. Indeed, we formulate the main computational result of this paper as follows:
[**Conjecture:**]{}
*The A-model ($q$-)expansion of the truncated normal function associated with an algebraic cycle takes the form $$\eqlabel{takestheform}
\widehat \calw_A(q) = \sum_d \tilde n_d q^{d/r}$$ were $r\in \zet_{>0}$, and the $\tilde n_d$ live in an algebraic number field $K$, with $d^2 \tilde n_d$ singular at most at the discriminant of $K$.*
In the expansion $$\eqlabel{singular}
\widehat\calw_A(q) = \sum_{d} n_d \,{\rm Li}_2^{\rm D} (q^{d/r})$$ the $n_d$ themselves are singular at most at the discriminant of $K$. Here, the D-logarithm $${\rm Li}_2^{\rm D}(x) = \sum_{k=1}^\infty \frac{a_k}{k^2} x^k$$ is an analytic function that may be attached to any such algebraic number $n_d$. The coefficients $a_k$ are determined $\bmod k^2$ by studying the action of the Galois group on $n_d$.
(We did not state all assumptions explicitly, such as that we are on a family of Calabi-Yau threefolds, expand around a large complex structure limit, [*etc.*]{}. The allowed singularities include denominators whose order at the discriminant grows (say linearly) with $d$. It should also be clear that we expect the same to work for higher-dimensional moduli spaces. The statements defining the D-logarithm are , , .)
It is possible that several ingredients for a proof of the above statements are contained in the work of Vologodsky, Schwarz and Kontsevich, see [@scvo].
Turning to possible interpretations of the result, we recall that the truncated normal function gives the contribution to the space-time superpotential of a D-brane configuration that specifies the algebraic cycle, in the B-model. By mirror symmetry, there should be an A-model setup that calculates the expansion directly. In our examples, such a setup would contain a Lagrangian submanifold $L\subset X$ of the quintic threefold, and the $q$-expansion should be the expansion in worldsheet instantons of disc topology.
We lack the tools to exhibit such A-branes directly, but we can nevertheless try to understand whether there is room for the various ingredients: a number field $K$ governing the vacuum structure of $L$, an action of the Galois group $G$, and instanton contributions that evaluate to algebraic numbers $\tilde n_d\in K$.
\(i) The irrationality of the instanton contribution is insofar surprising as it has not been seen in any previous example. If anything, $\tilde n_d$ should be open Gromov-Witten invariants counting holomorphic maps $(D,\del D)\to (X,L)$. In all cases studied so far, such invariants always evaluated to rational numbers. Mathematically, the counts are given by intersection theory on moduli spaces $\overline{\calm}$ of stable maps as integrals against the virtual fundamental class, $$\eqlabel{letalone}
\tilde n_d \overset{?}{\sim} \#\{u:(D,\del D)\to (X,L)\}
\overset{?}{\sim}
\int_{[\overline{\calm}]^{\rm virt}} {\bf 1}$$ In general however, open Gromov-Witten invariants have not actually been defined, let alone does there exist a formula like . The two exceptions are toric manifolds [@kali] and anti-holomorphic involutions [@jake]. The main obstacle to doing this in general has long been recognized to be the presence in $\overline{\calm}$ of boundaries in real co-dimension one. It is not clear therefore whether there actually exists a good invariant intersection theory on these spaces.
\(ii) From the world-sheet point of view, formulas such as arise as the reduction of the path-integral to the finite-dimensional space of zero-modes: roughly speaking, because of the vanishing of the fermion kinetic term in the action, one has to pull down the four fermion interaction involving the curvature. In many, favorable, situations, the resulting bosonic integrals have a cohomological interpretation in terms of intersection theory. However, except perhaps with large amounts of supersymmetry, there is no [*a priori*]{} reason why this should happen. It is quite conceivable that in the presence of boundaries, we do not have a strict intersection theoretic interpretation, but the integral still makes sense, and calculates a kind of “volume” of the moduli space. Such a volume could very well evaluate to an algebraic number.
\(iii) For a related thought, we recall that ordinary Gromov-Witten invariants are in general not integer, but rational, numbers because of the presence of certain kinds of orbifold singularities in the moduli space. The denominators are the orders of the corresponding identification groups. In the context of open Gromov-Witten theory, the moduli spaces could have other kinds of singularities, such as boundaries and corners, and in particular the latter could potentially make arbitrary irrational contributions.
\(iv) Of course, those two options, (ii) and (iii), assume that open Gromov-Witten theory exists in general, and defines actual invariants with an “enumerative” meaning. An alternative attitude is that any such definition will depend on arbitrary choices (a common examples being the framing ambiguity of ref. [@akv; @kali]). With a superpotential interpretation for the open Gromov-Witten invariants, this would mirror the issue, discussed in the introduction, that only the on-shell values of the space-time superpotential have an invariant meaning independent for example of field redefinitions. In this interpretation, the invariants $\tilde n_d$ would be irrational because they are [*on-shell*]{} and [*invariant*]{}, whereas the actual (rational) counts of discs would happen [*off-shell*]{}, and [*not be invariant*]{}.
\(v) This way of looking at the situation is perhaps best suited for explaining how the field extension could arise in the A-model. At the beginning, the underlying Lagrangian $L$ might have non-trivial topology and deformations, which get lifted by those very worldsheet instantons that we are trying to count. Intuitively, the critical points of the superpotential correspond to points in the moduli space of $L$ at which the worldsheet instantons are “balanced” against each other. If there are sufficiently many discs of comparable area, then these critical point conditions will select some general irrational points in the classical moduli space of $L$. If finitely many discs are relevant for this problem, and off-shell counts are rational, then we should be dealing with a finite algebraic extension of the rationals. (At the moment, I do not see how to get infinite, or transcendental extensions in the B-model.)
\(vi) Ideally, one would like to understand this in a suitable local model. For customary toric Lagrangian branes however, there are at most two discs determining the critical points, as for the conifold . A local model realizing a non-trivial field extension is therefore not likely to be toric (and in a sense would not be fully local since disc instantons ending in different places on $L$ would be relevant).
\(vii) A local model would also be desirable in order to understand the structure of the multi-cover formula . Otherwise, we have comparably little to offer for interpreting the invariants $n_d$. From the previous discussion in ref. [@oova], we expect a relation to the spectrum of appropriate BPS states (solitons) interpolating between the supersymmetric vacua. Given that the latter are in correspondence with roots of a polynomial equation, the Galois group of the relevant number field will act also on those BPS states. An irrational “dimension” could be part of the package of these Galois representations. The $a_k$ would then be other traces, and the formula could perhaps be understood by revisiting the derivation in [@oova] in light of such results.
For a very brief sampling of other recent works on various ways to relate geometry and physics of Calabi-Yau threefolds with number theory, see [@gmoore; @candelas; @vanstraten; @noriko; @vangeemen; @yang]
I am grateful to Henri Darmon, Hans Jockers, Sheldon Katz, Matt Kerr, Josh Lapan, Wolfgang Lerche, Greg Moore, David Morrison, and Noriko Yui for valuable discussions, comments, and encouragement. I thank Anca Mustaţǎ for sharing the results of [@mustata]. I would like to thank the KITP in Santa Barbara for sunny hospitality during the tedious writing of section \[conics\]. Some of the results presented here were also announced at the BIRS Workshop on “Number theory and physics at the crossroads”, May 8–13, 2011. Special thanks to Henri Darmon for help in unraveling the D-logarithm. This research is supported in part by an NSERC discovery grant and a Tier II Canada Research Chair. This research was supported in part by DARPA under Grant No. HR0011-09-1-0015 and by the National Science Foundation under Grant No. PHY05-51164.
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[^1]: For a perhaps not over-simplified way to see arithmetic arising in this context, imagine that $\calw(u)$ is a polynomial with integer coefficients, and the superpotential of a supersymmetric theory with 4 supercharges. Then the supersymmetric vacua and the [*critical values*]{} $\calw|_{\del\calw(u)=0}$, which are the actual holomorphic invariants encoded in $\calw(u)$, generically belong to a finite algebraic extension of the rationals. The statement of arithmeticy is somewhat different in the context of attractors in supergravity [@gmoore], where one looks at critical points of [*transcendental functions*]{} (periods).
[^2]: $\zet_2\times\zet_2$-invariant conics at $\psi=0$ have also been studied in [@mustata].
[^3]: We are not claiming here that all of those components are irreducible. It’s just the best we can do at this point.
[^4]: I thank Ron Donagi for a helpful discussion on this issue.
[^5]: Implementing this requires more diligence and patience than is appropriate to perhaps explain.
[^6]: The first convincing case is $\tilde n_{11}
= \sqrt{-3}\,\frac{5195025975738748330135719454410630564027766563867792882680000}{3^{10} 11^2}$, compared with $n_{11} = \tilde n_{11}+\tilde n_1/11^2=
\sqrt{-3}\,\frac{42934098973047506860625780614963888958907161684857860740000}{3^{10}}$
[^7]: We give here a bottom-up presentation of the relevant results, following more or less the path along which we came to them. A more straight-forward mathematical definition will be written up elsewhere.
[^8]: We will not make statements for $k$ that are not co-prime with the discriminant of the number field, which for our given $K$ is equal to $-3^3\cdot 5^2$.
[^9]: One should work with a notion of integrality that requires a non-negative valuation at all primes except those that ramify.
|
---
abstract: 'The excitation mechanism of low degree acoustic modes is investigated through the analysis of the stochastic time variations of their energy. The correlation between the energies of two different modes is interpreted as the signature of the occurrence rate of their excitation source. The different correlations determined by Foglizzo (1998) constrain the physical properties of an hypothetical source of excitation, which would act in addition to the classical excitation by the turbulent convection. Particular attention is drawn to the effect of coronal mass ejections. The variability of their occurrence rate with the solar cycle could account for the variation of the correlation between IPHIR and GOLF data. Such an interpretation would suggest that the mean correlation between low degree acoustic modes is at least $0.2\%$ at solar minimum.'
author:
- 'T. Foglizzo'
title: 'Are solar granules the only source of acoustic oscillations ?'
---
Introduction
============
The theoretical mechanism of excitation of solar acoustic modes by the turbulent convection is well documented (Goldreich & Keeley 1977; Goldreich & Kumar 1988; Kumar & Goldreich 1989; Balmforth 1992a,b,c; Goldreich 1994). High frequency oscillations were interpreted in different ways (Kumar & Lu 1991; Goode 1992; Restaino 1993; Vorontsov 1998), in order to address the question of the depth of the excitation source. Local observations led Rimmele (1995) and Espagnet (1996) to identify the excitation of 5-minute oscillations with acoustic events occurring in the downflowing intergranular regions, rather than overshooting granules.\
Considering granules of $1000$ km diameter renewed on a time scale comparable with their turnover time ($8$ min), there are about $5\times 10^{9}$ excitations by solar granules per damping time ($3.7$ days). Even if the number of efficient excitations is smaller by a factor 100 (Brown 1991), each low degree mode is stochastically excited so many times per damping time, that modes with different frequencies are expected to have uncorrelated energies. In particular, the fitting procedures used to determine the frequency, linewidth, and splitting of p modes often rely on the statistical independence of neighbouring modes (Appourchaux 1998).\
Baudin (1996), however, noticed a possible correlation between low degree p modes in IPHIR data. By measuring the mean correlation of p modes in IPHIR and GOLF data, and determining their statistical significance, Foglizzo (1998) (hereafter F98) suggested the existence of an additional source of excitation during the IPHIR period (end of 1988).\
The goal of this study is to use the observed correlations as a constraint on the physical properties of this hypothetical additional source. The theoretical relationship between the occurrence rate of this source, its contribution to the energy of low degree p modes, and their correlation coefficient, is established in Sect. 2. Available observational constraints are reviewed in Sect. 3. The possible role of comets, X-ray flares and coronal mass ejections (CMEs) is discussed in Sect. 4.
Correlation coefficient between two modes excited by a single mechanism\[seccorr\]
==================================================================================
The energy of a mode seen as a random walk\[secrw\]
---------------------------------------------------
Let us consider impulsive excitations, distributed over the solar surface. The exciting events, indexed by $k$, occur at the random poissonian time $t_{k}$. They correspond to a radial velocity perturbation $v_{k}({\bf r})$ localized in a cone of angle $\alpha_{k}$ around the random direction ($\theta_{k},\phi_{k}$), with a radial extension $\delta r_{k}$. $v_{k}({\bf r})$ is described by an amplitude $v_{k}$ and dimensionless shape functions $g_{k}$ and $h_{k}$ as follows: $$\begin{aligned}
v_{k}({\bf r})&\equiv& g_{k}(\alpha) h_{k}(r)v_{k},\label{defv1}\\
h_{k}(r)&\sim& 1\;\;{\rm for}\;\;R_{\star}>r>R_{\star}-\delta r_{k},
\label{defv2}\\
g_{k}(\alpha)&\sim& 1\;\;
{\rm for}\;\;\alpha<\alpha_{k}. \label{defv3}\end{aligned}$$ $\alpha(\theta,\theta_{k},\phi-\phi_{k})\ge 0$ is the angle made by the direction $\theta,\phi$ with the direction $\theta_{k},\phi_{k}$. Let $M_{k}$ be the mass of gas in the volume defined by the functions $g_{k}$ and $h_{k}$, so that the kinetic energy of the perturbation is ${\cal E}_{k}=M_{k}v_{k}^{2}/2$. The mean interpulse time is defined as $\Delta t\equiv <t_{k+1}-t_{k}>$.\
We assume that each impulsive event triggers free oscillations of the set of p modes. Each perturbation is projected onto the basis of eigenvectors. The damping time $\tau_{d}$ depends in principle on the mode considered, but is nearly constant ($\sim~3.7$ days) for the modes considered by F98, in the plateau region between $2.5$mHz and $3.5$mHz (linewidth of $ 1 \mu$Hz).\
In appendix A it is shown that the total energy of the oscillations associated to the frequency $\omega_{nl}+m\Omega$ is: $$\begin{aligned}
E_{nlm}^{+}(t)&=&\left|\sum_{t_{k}<t}
\e^{-{t-t_{k}\over\tau_{d}}} a_{nlm}^{k}
\e^{i\Phi^{k}_{nlm}}\right|^{2},\label{enerstoc}\\
a_{nlm}^{k}&\equiv&
g_{lm}^{k}(\theta_{k})h_{nl}^{k}M_{k}^{1\over2}v_{k},\\
\Phi^{k}_{nlm}&\equiv& (\omega_{nl}+m\Omega)t_{k}+m\phi_{k}
\label{defphi}.\end{aligned}$$ where the real function $g_{l,m}^{k}=g_{l,-m}^{k}$ depends on the angular shape of the excitation projected onto the spherical harmonics, and $h_{nl}^{k}$ depends on the radial shape of the excitation projected onto the radial part $v_{nl}^{r}(r)$ of the eigenfunction: $$\begin{aligned}
g_{lm}^{k}(\theta_{k})&\equiv &q_{lm}\int_{0}^{{\alpha_{k}\over
2}}\d\phi\int_{0}^{\pi}\d(\cos\theta)
P_{l}^{|m|}(\cos\theta)\nonumber\\
&\times&\cos(m\phi)g_{k}(\theta,\theta_{k},\phi),\label{glm}\\
h_{nl}^{k}&\equiv&{1\over M_{k}^{1\over2}}
\int_{R_{\star}-\delta r_{k}}^{R_{\star}}
\rho r^{2}v_{nl}^{r}(r)h_{k}(r)\d r.
\label{alphanlm}\end{aligned}$$ The relationship between the spherical harmonics and the Legendre associated functions $P_{l}^{m}$ through the constant $q_{lm}$ is recalled in Eqs. (\[spha1\])-(\[spha2\]). The exponential damping in Eq. (\[enerstoc\]) can be schematized as a term selecting the finite set of excitations which occurred within one damping time before the time $t$: $$E_{nlm}^{+}(t)\sim \left | \sum_{0<t-t_{k}<\tau_{d}}
a_{nlm}^{k}\e^{i\Phi^{k}_{nlm}}\right |^{2}.\label{enlmap}$$ The series of phases $\omega_{nl}t_{k}$ and $\phi_{k}$ are independent random variables uniformly distributed in the interval $[0,2\pi]$. The series $\Phi^{k}_{nlm}$ defined by Eq. (\[defphi\]) is therefore also uniformly distributed in $[0,2\pi]$. Ignoring the academic case where the ratio of the frequencies is a simple rational number, $\Phi^{k}_{nlm}$ and $\Phi^{k}_{n'l'm'}$ can be considered independent for two different realistic modes.\
As a consequence, the energy of the wave is interpreted as the squared length of a random walk in the complex plane. Each step of this random walk is defined by an amplitude $a_{nlm}^{k}$, and a phase $\Phi^{k}_{nlm}$ (negative values of the amplitude can be converted into an increment of the phase). The number of steps $N$ is the number of excitations, over the whole surface of the sun, within one damping time: $$N\equiv{\tau_{d}\over\Delta t}.$$ The central limit theorem ensures that the real and imaginary parts of the complex sum (\[enlmap\]) converge towards independent normal distributions if $N$ is large enough, resulting in an exponential distribution of energy (see also Kumar 1988).\
According to this random walk interpretation, two different modes excited by the same series of events must have correlated energies. This correlation should approach $100\%$ if the number of steps of this random walk is small, if the interpulse mean time $\Delta t$ is longer than the damping time $\tau_{d}$.\
The independence of the phases $\Phi^{k}_{nlm}$ and $\Phi^{k}_{n'l'm'}$, however, makes the correlation decrease to zero when the number of excitations increases.\
In particular, the random walks associated with the two components $\omega_{nl}\pm m\Omega$ of a mode $n,l$ have the same series of amplitudes, but have two independent series of phases. This leads us to expect that waves travelling in opposite directions do not have the same instantaneous total energy (except on average), although they are excited by the same events.
Theoretical correlation between the energies of two modes \[seccor\]
--------------------------------------------------------------------
The mean energy $<E_{nlm}>$ of the mode $nlm$ is directly proportional to the mean energy $<e_{nlm}>$ received from each excitation (appendix B): $$<E_{nlm}>= {N\over2}<e_{nlm}>.\label{azerty}$$ The correlation ${\cal C}_{nlm}^{n'l'm'}$ between two modes $nlm$, $n'l'm'$ excited by the same source described by Eq. (\[enerstoc\]) is derived in appendix B: $$\begin{aligned}
{\cal C}_{nlm}^{n'l'm'}={<e_{nlm}e_{n'l'm'}>\over
<e_{nlm}^{2}>^{1\over2}<e_{n'l'm'}^{2}>^{1\over2}}\nonumber\\
\times\left(
1+N{<e_{nlm}>^{2}\over<e_{nlm}^{2}>}\right)^{-{1\over2}}
\left(
1+N{<e_{n'l'm'}>^{2}\over<e_{n'l'm'}^{2}>}\right)^{-{1\over2}}.
\label{corgen1}\end{aligned}$$
Latitudinal distribution and sizes of the excitations\[size\]
-------------------------------------------------------------
The ratio $<e_{nlm}^{2}>/<e_{nlm}>^{2}$, related to the spread of the distribution, is called kurtosis (it is sometimes defined with an additional constant $-3$ which we omit here). $${<e_{nlm}^{2}>\over<e_{nlm}>^{2}}=
{<g_{lm}^{4}h_{nl}^{4}{\cal E}^{2}>
\over <g_{lm}^{2}h_{nl}^{2}{\cal E}>^{2}}.$$ The kurtosis of $e_{nlm}$ is partly produced by the kurtosis of ${\cal E}_{k}$ (independent of $nlm$), but also by the projection of the spatial distribution of excitations ($\theta_{k},\alpha_{k},\delta r_{k}$) onto the eigenfunctions $nlm$.\
The random variations of the size $\alpha_{k},\delta r_{k}$ of the excitation play a negligible role as long as it is smaller than the wavelength of the mode $nlm$. For the modes $l=0$ and $l=1$ analysed by F98, we restrict ourselves to perturbations such that $\alpha_{k}<\pi$, and $\delta r_{k}$ is smaller than the depth of the first radial node of the eigenfunction $v_{nl}$, and thus neglect the random variations of $\alpha_{k},\delta r_{k}$.\
Let us estimate the kurtosis of the distribution $g_{lm}(\theta_{k})$, due to the projection of the latitudinal distribution of excitations on the mode $nlm$. From Eq. (\[glm\]), the function $g_{lm}(\theta_{k})$ is independent of $\theta_{k}$ for the radial mode $l=0$: $${<g_{0,0}^{4}>\over<g_{0,0}^{2}>^{2}}=1.$$ By contrast, the mode $l=1,m=\pm1$ is more excited by equatorial events than polar ones. For small scale excitations ($\alpha_{k}\equiv 0$) distributed uniformly over the sphere, Eq. (\[glm\]) implies: $${<g_{1,\pm1}^{4}>\over<g_{1,\pm1}^{2}>^{2}}={6\over5}.\label{rat11}$$ The kurtosis of $g^{2}_{1,\pm1}$ should reach a value even closer to unity if the excitations are distributed in an equatorial region, like CMEs at solar minimum or big flares.\
Let us approximate the kurtosis of $e_{nlm}$ as the product of the kurtosis of ${\cal E}_{k}$ by the kurtosis of $g^{2}_{lm}(\theta_{k})$. $${<e_{nlm}^{2}>\over<e_{nlm}>^{2}}\sim
{<{\cal E}^{2}>\over <{\cal E}>^{2}}\times{<g_{lm}^{4}>
\over <g_{lm}^{2}>^{2}}.$$ Using this approximation in Eq. (\[corgen1\]), the correlation between low degree modes $l=0,m=0$ and $l=1,m=\pm1$ excited in the random direction $(\theta_{k},\phi_{k})$ by a small scale excitation ($\alpha_{k}\equiv 0$) is: $$\begin{aligned}
{\cal C}_{n,0,0}^{n',0,0}&=&\left(1+N_{\rm eff}\right)^{-1},
\label{c00}\\
{\cal C}_{n,0,0}^{n',1,\pm1}&=&\left({5\over6}\right)^{1\over2}
\left(1+N_{\rm eff}\right)^{-{1\over2}}
\left(1+{5N_{\rm eff}\over6}\right)^{-{1\over2}},
\label{c01}\\
{\cal C}_{n,1,\pm1}^{n',1,\pm1}&=&
\left(1+{5N_{\rm eff}\over6}\right)^{-1},
\label{c11}\end{aligned}$$ where we have incorporated the kurtosis of ${\cal E}_{k}$ into the definition of the effective number $N_{\rm eff}$ of excitations per damping time: $$\begin{aligned}
\beta&\equiv&{<{\cal E}^{2}>\over <{\cal E}>^{2}}\ge1,\\
N_{\rm eff}&\equiv& {N\over\beta}.\end{aligned}$$
The correlation displayed in Fig. \[fitnum\] appears to be hardly sensitive to the geometrical content of the function $g_{lm}$, since all curves are similar within $10\% $. Error bars of the observationally measured correlations being typically larger than $0.1$, Eq. (\[c00\]) shall be considered accurate enough for the purpose of this study: $${\cal C}_{n,l,m}^{n',l',m'}\sim\left(1+N_{\rm eff}\right)^{-1}.
\label{fitcor1}$$ This is equivalent to approximating the kurtosis of $e_{nlm}$ by the kurtosis of ${\cal E}$.
Case of two excitation mechanisms superimposed\[secsup\]
--------------------------------------------------------
F98 used a one parameter model of two sources of excitation, such that each mode contains a fraction $\lambda$ of energy common to all the modes. They assumed that both sources produce exponential distributions of energy. This simplification led to the simple relation $\lambda={\cal C}^{1/2}$. This simplification, however, does not apply to the case of a correlation produced by rare impulsive events. If $N\ll1$, the energy is damped to zero except during isolated pulses of duration $\sim\tau_{\rm d}/2$. The distribution $E$ thus follows a Bernoulli statistics, with ${\rm Var}(E)/<E>^{2}\gg1$ (Eq. \[vaava\]) instead of $1$ for an exponential distribution. The occurrence rate of the excitation mechanism therefore appears to be a key parameter which cannot be neglected. Let us consider two excitations mechanisms acting simultaneously on two modes $n,l,m$ and $n',l',m'$:
\(i) excitation by granules, with a high occurrence rate ($N_{1}\gg1$).
\(ii) excitation by additional events with a total acoustic energy ${\cal E}_{\XX}^{k}$, occurring on average $N_\XX$ times per damping time, and contributing to a fraction $\lambda_{nlm}$ of the power of the mode $nlm$.\
We show in appendix C (Eq. \[corgenlamb\]) that the correlation between the energies of two oscillators excited by these sources is the same as in Eq. (\[corgen1\]), but multiplying the kurtosis of $e_{nlm}$ by $\lambda_{nlm}^{2}$. The latitudinal distribution and sizes of the excitations play a relatively small role according to Sect. \[size\]. The effective number $N_{\rm eff}$ of excitations per damping time depends on the kurtosis $\beta_{\XX}$ of ${\cal E}_{\XX}$: $$\begin{aligned}
N_{\rm eff}\equiv {N_{\XX}\over\beta_{\XX}}.\end{aligned}$$ Making the additional assumption that the fraction $\lambda_{nlm}=\lambda$ varies little among the modes considered by F98. The mean correlation between these modes is: $$\begin{aligned}
{\cal C}\sim
\left(1+{N_{\rm eff}\over\lambda^{2}}\right)^{-1}.
\label{coraprox}\end{aligned}$$ A significant correlation can thus be produced by a source representing only a small fraction $\lambda$ of the total power of each mode if $N_{\rm eff}$ is small enough.\
Interpreting the observed correlation ${\cal C}$ as a consequence of an additional source of excitation therefore requires it to contribute to a fraction $\lambda$ of the total power, deduced from Eq. (\[coraprox\]): $$\lambda\sim
\left\lbrace{{\cal C}\;N_{\rm eff}\over1-{\cal
C}}\right\rbrace^{1\over2}. \label{apllic}$$ According to Eq. (\[azerty\]), the fraction $\lambda$ of power coming from the additional source is related to the mean acoustic energy input $<e_\XX>$ into the mode $nlm$: $$\lambda={N_\XX\over2}{<e_\XX>\over <E>}. \label{frala}$$ Eq. (\[apllic\]) becomes: $$\begin{aligned}
<e_\XX>&=&
{2\over \beta_{\XX}}
{{\cal C}\over1-{\cal C}}
{<E>\over \lambda},\label{elam}\\
&=&{2\over\beta_{\XX}^{1\over2}}
\left\lbrace{{\cal C}\over1-{\cal C}}\right\rbrace^{1\over2}
{<E>\over N_\XX^{1\over2}}.
\label{apllic11}\end{aligned}$$ Let us assume that the occurrence rate $N_\XX$ varies from $N_{\rm min}$ to $N_{\rm max}$ with the solar cycle, while the properties of the exciting events ($<e_\XX>,\beta_\XX$) remain unchanged. We use Eq. (\[apllic11\]) to express the relationship between the minimum and maximum correlations ${\cal C}_{\rm min},{\cal C}_{\rm max}$ : $${\cal C}_{\rm min}=\left\lbrace
1+{N_{\rm max}\over N_{\rm min}}
{1-{\cal C}_{\rm max}\over{\cal C}_{\rm max}}
\left({<E>_{\rm min}\over<E>_{\rm max}}\right)^{2}
\right\rbrace^{-1}.\label{minmax}$$
Efficiency of the generation of acoustic waves\[effic\]
-------------------------------------------------------
Since the distribution of acoustic energy ${\cal E}_\XX$ produced by the additional source is not directly observable, we are led to assume that it resembles the observed distribution of total energy ${\cal E}_{\rm T}$ of the additional source. Let $p_{\rm T}({\cal E})$ be the density of probability of the source of energy ${\cal E}_{\rm min}\le{\cal E}_{\rm T}\le{\cal E}_{\rm max}$, occurring $N_{\rm T}$ times per damping time. Let us assume that the excitation of low degree modes is efficient only in the range ${\cal E}_{1}\le{\cal E}_{\rm T}\le{\cal E}_{2}$, inside which the efficiency $f_{\XX}\equiv {\cal E}_\XX/{\cal E}_{\rm T}$ is constant: $$\begin{aligned}
{\cal E}_\XX &=& f_{\XX} {\cal E}_{\rm T}\;\;
{\rm for }\;{\cal E}_{1}\le{\cal E}_{\rm T}\le{\cal E}_{2},\\
{\cal E}_\XX &=& 0\;\;{\rm otherwise}.\end{aligned}$$ In appendix C, Eq. (\[apllic11\]) is rewritten as the fraction of the mean acoustic energy $<{\cal E}_\XX>$ which must be injected into each mode $nlm$ in order to produce the observed correlation: $${<e_\XX>\over<{\cal E}_\XX>}
={2\over f_{\XX}
\left(
\int_{{\cal E}_{1}}^{{\cal E}_{2}}{\cal E}^{2}p_{\rm T}({\cal E})
\d {\cal E}
\right)^{1\over2}}
\left\lbrace{{\cal C}\over1-{\cal C}}\right\rbrace^{1\over2}
{<E>\over N_{\rm T}^{1\over2}}.\label{filtre}$$ This fraction is therefore minimal when the range of efficient excitations ${\cal E}_{1},{\cal E}_{2}$ contains the range of energies where the product ${\cal E}_{\rm T}^{2}p({\cal E}_{\rm T})$ is maximal. Using in Eq. (\[apllic11\]) the occurrence rate $N_{\rm T}$ and kurtosis $\beta_{\rm T}$ of the distribution of total energy ${\cal E}_{\rm T}$ instead of the distribution of acoustic energy ${\cal E}_{\XX}$ leads to a lower bound of the ratio ${<e_\XX>/<{\cal E}_\XX>}$: $${<e_\XX>\over<{\cal E}_\XX>}\ge
{1\over f_{\XX}}{2\over N_{\rm T}^{1\over2}\beta_{\rm T}^{1\over2}}
\left\lbrace{{\cal C}\over1-{\cal C}}\right\rbrace^{1\over2}
{<E>\over <{\cal E}_{\rm T}>}.
\label{apllic22}$$ For a given excitation mechanism one can estimate the number ${\cal N}$ of p modes with a wavelength longer than the size of each exciting event, which receive a comparable amount of energy from the source. Considering that the low degree modes in F98 belong to this set, $$<{\cal E}_{\XX}>=\sum_{nlm} <e_{nlm}>\ge{\cal N}<e_\XX>.\label{calN}$$ From Eqs. (\[apllic22\])-(\[calN\]) we deduce a constraint on observable quantities, which shall be useful in Sect. \[fcme\] in order to discriminate between possible excitation mechanisms: $${2\over N_{\rm T}^{1\over2}\beta_{\rm T}^{1\over2}}
\left\lbrace{{\cal C}\over1-{\cal C}}\right\rbrace^{1\over2}
{<E>\over <{\cal E}_{\rm T}>}\le {f_{\XX}\over{\cal N}}<{1\over{\cal N}}.
\label{apllic33}$$
Observational constraints
=========================
Exponential distribution of individual p modes energy\[dis1\]
-------------------------------------------------------------
The distribution of energy of low degree p modes agrees reasonably well with an exponential distribution (Chaplin 1995, 1997 for BiSON data, F98 for IPHIR and GOLF data). Chaplin (1995, 1997), however, noticed significant deviations in the high energy tail of the distribution, which could be due to an additional excitation mechanism. They estimated that the probability that such deviations occur by chance during the period of observation is only $0.1\% $. Among $22512$ events covering $18331$ hours of BiSON data from 1987 to 1994, Chaplin (1997) found $51$ events above $6.5$ times the mean energy, whereas less than $41$ would be expected in $90\% $ of the cases for an exponential distribution. Crudely speaking, about $10$ events are unexpected in the distribution, suggesting $N_\XX \ge 0.05$.
Solar cycle variations of the total power of low degree p modes \[dis2\]
------------------------------------------------------------------------
The typical energy of a low degree p mode in the range $2.7{\rm
mHz}\le\nu\le 3.4{\rm mHz}$ considered by F98 is $<E>\sim 8\times
10^{27}$ ergs (Chaplin 1998). The velocity damping time being $\tau_{d}\sim 3.7$ days, the flux of energy required to excite this p mode is $2<E>/\tau_{d}\sim 5\times 10^{22}\es$.\
According to Libbrecht (1986), the total energy in all the p modes (about $10^{7}$) is $10^{34}$ ergs within a factor $10$.\
Anguera Gubau (1992) and Elsworth (1993) measured a global $30\%$ decrease of the power of low degree p modes at solar maximum. This decrease seems to preclude a high value of the fraction $\lambda$ of the total power which some additional source could contribute to. Nevertheless, the physical mechanisms by which the p mode power might decrease, such as damping by active regions, or modification of the properties of the convection, have not yet been quantitatively estimated. This might be efficient enough to dominate the energy input due to an additional source. Moreover, the measurement of the amplitude of global p modes is influenced by the presence of active regions covering a significant fraction of the solar surface at solar maximum (Cacciani & Moretti, 1997). Given these uncertainties, no firm constraint can be deduced from these observations. We shall consider “likely” a fraction $\lambda\le 30\% $.
Observed correlations\[obsc\]
-----------------------------
F98 determined the mean correlations between the energies of low degree p modes at two different epochs: 160 days in 1988 near solar maximum using IPHIR data ($l=0$, $19\le n\le 23$ and $l=1$, $18\le
n\le 23$), and 310 days in 1996-97 near solar minimum using GOLF data ($l=0$ and $l=1$, $17\le n\le 25$). The mean correlation coefficient ${\cal C}$ they measured is:
\(i) ${\cal C}=10.7\pm 5.9\%$ in 1988 (IPHIR data),
\(ii) ${\cal C}<0.6\%$ in 1996-97 (GOLF data).\
According to F98, the probability that the correlation measured from IPHIR data could occur by chance is $0.7\%$ if the modes were independent. F98 also rejected the possibility of an instrumental artefact by checking that the noise of IPHIR data at different frequencies is not correlated.\
If the granules were the only source of p-mode excitation, the correlation would be less than $10^{-5}\%$ according to Eq. (\[fitcor1\]), well below the detection limit. The actual limit of detection, of order $0.6\% $ (F98), corresponds to at least one exciting event every 29 minutes, on average. If due to a single mechanism of excitation, the correlation measured with IPHIR data would correspond to an exciting event every $12\pm 7.2$ hours.\
Fig. \[fig2\] shows the relationship between $N_{\rm eff}$ and $\lambda$ required by the correlations observed in IPHIR data, and the upper bound set by GOLF data.\
The correlation measured by F98 in 160 days of IPHIR data imposes that $N_\XX\ge \tau_{\rm d}/160=0.02$ in the IPHIR period, which is coherent with the fraction of abnormal events in BiSON data.\
Applying Eq. (\[apllic\])-(\[elam\]) to a typical mode of energy $8\times 10^{27}$ ergs and damping time $\tau_{\rm d}=3.7$ days, the constraint ${\cal C}\ge 4.8\% $ leads us to look for a mechanism occurring at most a few times per day in 1988, with an acoustic energy input of at least $10^{26}$ ergs per mode: $$\begin{aligned}
0.02\le N_\XX&\le &
7.1\times \left({\beta_\XX\over4}\right)
\left({\lambda\over0.3}\right)^{2},\\
<e_\XX>&\ge &6.7\times\left({4\over \beta_\XX}\right)
\left({0.3\over\lambda}\right)
\times 10^{26}{\rm ergs}.\end{aligned}$$
Some candidates: comet impacts, X-ray flares and coronal mass ejections\[fcme\]
===============================================================================
[cccc]{} & comet & flare & CME\
kinetic energy (ergs) & & &\
average & - & $\sim 10^{30}$ & $6.7\times 10^{30}$\
maximum & $2\times 10^{32}$ & $\sim 10^{33}$ & $4\times 10^{32}$\
kurtosis $\beta$ & - & $\sim 33$ & 4-20\
momentum (g cm s$^{-1}$) & & &\
average & - & $\sim {}10^{22}$ & $\sim 10^{23}$\
maximum& $6\times 10^{24}$ & - & $\sim 10^{24}$\
occurrence rate (day$^{-1}$) & & &\
solar minimum & $< 0.1$ & 1 & 0.2\
solar maximum & $< 0.1$ & 19 & 3\
\[table2\]
Comet impacts
-------------
Sungrazing comets of the Kreutz group are small remnants of an earlier passage of a larger progenitor comet. Among the 10 sungrazers observed by SMM, 4 appeared during the 160 days IPHIR campaign (MacQueen & St. Cyr 1991). A total of 59 sungrazers were observed by LASCO aboard SOHO. Even with an occurrence rate of $0.1$ per day, ${\cal C}\ge 4.8\% $ in Eq. (\[apllic11\]) would require an energy input of $6 \beta_\XX^{-1/2}\times 10^{27} $ ergs per event and per mode. Given the small scale of the comet, all $10^{7}$ p modes should receive the same energy input as low degree modes, which corresponds to a total acoustic energy of $\sim 10^{34}$ ergs per event. This already exceeds by two orders of magnitude the kinetic energy of a comet with the mass as Halley’s hitting the solar surface; not to mention the variability issue between IPHIR and SOHO periods, nor the difficult question of the efficiency $f$ of the energy transfer (addressed by Gough 1994; Kosovichev & Zharkova 1995).
X-ray flares
------------
### Observations of waves generated by flares
Although sunspots are known to absorb high degree acoustic waves (Braun 1987, 1988), observations of the waves generated by a large solar flare on 24th April 1984 by Haber (1988a,b) showed a $19\%$ increase of power in outward travelling wave, dominating the sunspot absorption. Nevertheless, Braun & Duvall (1990) also observed the wave emission from a flare on 10th March 1989, and found an upper bound of $10\%$ power increase, if any.\
Very recently, Kosovichev & Zharkova (1998) analysed the shock wave produced by the “fairly moderate” flare of 9th July 1996, observed by MDI aboard SOHO. The wave amplitude is associated to a momentum a factor 30 smaller than the one expected from their theoretical model. This unexpectedly high efficiency led them to conclude that the seismic flare source might be located in subsurface layers.\
The effect of flares on low degree modes is however less clear. The high energy events of BiSON data, appearing above the exponential energy distribution of low degree modes, do not seem to be correlated neither with the sunspot number, nor with the strength of X-ray flares (Chaplin 1995). Using the sum of normalized energies of low degree modes in IPHIR data, Gavryusev & Gavryuseva (1997) found an anticorrelation between big pulses in p modes and the mean solar magnetic field but no correlation with the sunspot number.
### Theoretical estimates of p mode excitation by flares
The theoretical estimate of the energy transmitted from a flare to p modes can follow three different approaches:
\(i) the region surrounding the flare is heated and expands vertically on a time scale short enough to communicate momentum to the atmosphere below it (Wolff 1972). This process could extract $10^{28}$ ergs of acoustic energy from a $10^{32}$ ergs flare ($f\sim 10^{-4}$). According to Wolff (1972), the error bar in this estimate is at least a factor $10$.
\(ii) the plasma flows down towards the foot points of the field lines, and hits the solar surface, thus communicating momentum to it. This approach was followed by Kosovichev & Zharkova (1995) who found a smaller effect than that observed by Haber (1988a).
\(iii) the pressure perturbation associated with the restructuring of the magnetic field in the flare region might be more effective, as suggested by Kosovichev & Zharkova (1995).
### Energy distribution of flares
The total energy released by a flare (less than $10^{27}$ ergs to $10^{33}$ ergs) is usually computed assuming that the observed hard X-rays are produced by bremsstrahlung from a distribution of accelerated electrons impinging upon a thick target plasma. Using the Hard X-Ray Burst Spectrometer on the Solar Maximum Mission satellite from 1980 to 1989, Crosby (1993) showed that the flaring rate varies by about a factor 20 between the solar maximum and minimum, while the energy distribution remains constant. Applying Eq. (\[minmax\]) to the correlation observed by IPHIR, with the $30\% $ variation of the p mode total power, suggests the following correlation ${\cal C}_{\rm min}$ at solar minimum: $$0.15\%\le{\cal C}_{\rm min}\le 0.59\%,\label{cminflare}$$ which is compatible with the upper bound obtained from GOLF data.\
The occurrence rate of flares follows a power law distribution $p({\cal E})\sim {\cal E}^{-\gamma}$ against the total flare energy with a slope $\gamma<2$, indicating that the largest flux of energy occurs in rare big energy events. Let ${\cal E}_{\rm min}\ll {\cal E}_{\rm max}$ be the range of energies over which the power law distribution is observed, its mean energy $<{\cal E}_{\rm F}>$ and kurtosis $\beta_{\rm F}$ are: $$\begin{aligned}
<{\cal E}_{\rm F}>&\sim & {\gamma-1\over2-\gamma}
\left({{\cal E}_{\rm max}\over
{\cal E}_{\rm min}}\right)^{2-\gamma}{\cal E}_{\rm min},
\label{Eflare}\\
\beta_{\rm F}&\sim
&{(2-\gamma)^{2}\over(3-\gamma)(\gamma-1)} \left({{\cal E}_{\rm max}\over
{\cal E}_{\rm min}}\right)^{\gamma-1}. \label{sigflare}\end{aligned}$$ According to Crosby (1993), about $13$ flares occurred per day in 1980-1982 (solar maximum), with a slope of the energy distribution $\gamma\sim1.5$ in the range $10^{28}\le {\cal E}_{\rm F}\le 10^{32}$ ergs. Two thirds of the flares considered by Crosby (1993) fall in this range according to their Fig. 6, so that the occurrence rate of these flares can be taken as $8.7$ per day ($N_{\rm cme}\sim 32.2$). Eqs. (\[Eflare\])-(\[sigflare\]) imply $<{\cal E}_{\rm F}>\sim 10^{30}$ ergs and $\beta_{\rm F}\sim 33$. Assuming that the efficiency $f$ does not depend on the energy, the distributions of total energy and of energy input per mode are identical, and $N_{\rm eff}= 1.0$. Eqs. (\[apllic\]) and (\[apllic22\]) imply: $$\begin{aligned}
22.4\% &\le & \lambda \le 44.6\%\label{fl1}\\
<e_{\rm F}>&\ge & 10^{26} \;{\rm ergs}\;{\rm event}^{-1}\;{\rm mode}^{-1},
\label{fl2}\\
{<{\cal E}_{\rm F}>\over <e_{\rm F}>} &\le & 10^{4} \;{\rm modes}.
\label{fl3}\end{aligned}$$ With ${\cal E}^{2}p({\cal E})\sim {\cal E}^{1/2}$, Eq. (\[filtre\]) indicates that Eq. (\[fl3\]) would still be correct if the energy dependence of the efficiency $f({\cal E})$ were to select only the most energetic events.\
The size of the flare region viewed from earth is no more than a few arcminutes, so that all p modes with a degree $l\le l_{\rm max}$ must receive an equal amount of energy. With $l_{\rm max}=37$ according to Wolff (1972), this corresponds to a set of ${\cal N}\ge 10^{4}$ modes, which contradicts Eq. (\[apllic33\]). In conclusion, the total acoustic energy input required from X-ray flares to produce the observed correlation exceeds their total energy.
Coronal Mass Ejections
----------------------
### Geometry and timing of CMEs
Coronal mass ejections correspond to the release of $3.3\times10^{15}$ g of matter on average, with a mean velocity of $350\kms$ (SMM data from 1980 and 1984-1989, Hundhausen 1997). The footpoints of the outer loop are separated on average by about 45 degrees in latitude, much larger than an active region or flare (Harrison 1986, Hundhausen 1993). This large scale structure favours low degree modes. According to Hundhausen (1994), the origin of the CME comes from a gradual build up and storage of energy in a pre-ejection structure (driven by the spreading of magnetic field lines, the emergence of magnetic flux, or the shear of field lines). A finite quantity of this energy is then released in a catastrophic breakdown in the stability or equilibrium of the stressed structure. When the CME is associated with an X-ray flare, the CME kinetic energy seems to be uncorrelated to the flare peak intensity (Hundhausen 1997).\
In Hundhausen (1994), the acceleration of the CME observed by SMM on 23rd August, 1988 was measured. Its launching precedes the flare event (see also Harrison 1986), and the acceleration to a velocity of $1000$ km s$^{-1}$ occurs in less than 10 min (see the CMEs of 17th August, 1989 and 16th February, 1986 for similar examples in Hundhausen 1994).\
Note that a CME was associated to the flare of 9th July, 1996 analysed by Kosovichev & Zharkova (1998).
### Solar cycle variability
While the average mass of a CME varies little from year to year (Hundhausen 1994b), the annual variation of their mean velocity does not follow the solar cycle (Hundhausen 1994a). It would therefore be interesting to check the correlation between the high energy events noticed by Chaplin (1995, 1997) and this new indicator.\
According to Webb & Howard (1994), the occurrence rate of CME varies from 0.2 per day at solar minimum to 3 per day at solar maximum. As for flares, the variation of the occurrence rate is enough to account for the observed variation of the correlation. Eq. (\[minmax\]) implies: $$0.20\%\le{\cal C}_{\rm min}\le 3.3\%.\label{cmincme}$$
### Energy distribution of CMEs
Hundhausen (1994a,b) showed that the distribution of CME velocities ($10\kms$ to $2000\kms$) is more widely spread than their distribution of masses ($10^{14}$ to $10^{16}$ g). As a consequence, the distribution of kinetic energy is spread over a much wider range than the potential energy.\
Hundhausen (1997) selected a sample of 249 CMEs measured by SMM in 1984-1989, associated with X-ray flares. The spread of their distribution of kinetic energy (from $10^{28}$ to $10^{33}$ ergs), measured from Fig. 1 of Hundhausen (1997) is such that $\beta_{\rm cme}\sim 20.2$. Note that a smaller value ($\beta_{\rm cme}\sim 3.9$) is obtained when considering the distribution of squared velocities of $109$ CMEs during the 160 days IPHIR period in the catalogue of Burkepile & St. Cyr (1993). The apparent contradiction may come partly from the distribution of masses, but mostly from the high sensitivity of $\beta_{\rm cme}$ to the high energy events which occurred in 1989 at solar maximum. Moreover, fast CMEs are underrepresented in those statistics because the velocity is measured only when the CME is visible on more than one coronograph image (Hundhausen 1994a). As for flares, the distribution of acoustic energy input might be different from the distribution of CMEs kinetic energy, for example, if only a subclass of CMEs excite acoustic waves. In particular, some CMEs are known to be slowly accelerated with the solar wind, while others are much faster than the solar wind (MacQueen & Fisher 1983).\
With $\beta_{\rm cme}=20$ and an occurrence rate of $3$ CMEs per day ($N_{\rm cme}=11.1$, $N_{\rm eff}=0.6$), IPHIR correlations in Eqs. (\[apllic\]) and (\[apllic22\]) imply (see Fig. \[figevent\]): $$\begin{aligned}
16.7\%\le \lambda &\le& 33.2\%,\label{cme1}\\
<e_{\rm cme}>&\ge& 2.4\times 10^{26}\;
{\rm ergs}\;{\rm event}^{-1}\;{\rm mode}^{-1},\label{cme2}\\
{<{\cal E}_{\rm cme}>\over <e_{\rm cme}>}&\le& 2.8\times 10^{4} \;
{\rm modes}.
\label{cme3}\end{aligned}$$ Estimates of both $\lambda$ and the acoustic energy input per mode $<e_{\rm cme}>$ would be multiplied by a factor 2.2 if the value $\beta_{\rm cme}=4$ were chosen. Although the kinetic energy distribution of CMEs is less accurately known than for flares, one can infer from the published ones (Hundhausen 1994b, Hundhausen 1997) that the product ${\cal E}^{2}p({\cal E})$ is maximum at high energy. As for flares, Eq. (\[cme3\]) would not be changed if the efficiency of acoustic wave generation were highest at high energy.\
Although Eqs. (\[cme1\])-(\[cme3\]) resemble Eqs. (\[fl1\])-(\[fl3\]) obtained for flares, the following important differences should be noted:
\(i) the estimate for CMEs is based on their kinetic energy only, which is certainly much smaller than the energy of the mechanism responsible for their ejection.
\(ii) the large scale of CMEs might favour the excitation of low degree modes, especially if their ejection mechanism is rooted in the convection zone (${\cal N}\le 100$).
\(iii) although the range of energy of CMEs (kinetic + potential $\sim 8.5\times 10^{30}$ ergs on average, Hundhausen 1994b) is the same as for big flares, their momentum can be one hundred times larger than the momentum carried by downflowing electrons produced by flares (see Table \[table2\]). This is favourable to a higher efficiency of the energy transfer to acoustic modes.
Conclusion
==========
We have investigated the consequences of interpreting the observed correlations in terms of an hypothetical excitation mechanism, in addition to the well established excitation by the granules. In particular, this interpretation requires impulsive events occurring no more than a few times per day in the IPHIR period. This has drawn our attention to the largest X-ray flares and CMEs at solar maximum. The variation of their typical energy and occurrence rate with the solar cycle could account for the variation of the correlation between the IPHIR and GOLF observations.\
If due to solar flares, the correlation determined from IPHIR data requires that at least $10^{-4}$ of the energy of each X-ray flares is injected into each low degree mode at solar maximum. Given the number of modes (at least $10^{4}$ due to the small scale of flares) which should receive the same energy as low degree modes, we are inclined to exclude X-ray flares regardless of the efficiency of the acoustic wave emission by each event. This reasoning, however, does not exclude the possibility of more energetic processes associated to flares, such as the restructuring of the magnetic field in the flare region mentioned by Wolf (1972) and Kosovichev & Zharkova (1995).\
If the correlation is due to CMEs, at least $3.6\times 10^{-5}$ of the [*kinetic*]{} energy of each CME should be injected into each low degree mode at solar maximum. This leaves two possibilities open:
\(i) CMEs may correlate only a few tens of low degree modes by injecting a few per cents of the CME kinetic energy into these modes. Higher degree modes ($l\ge 10$) would not be correlated in this case.
\(ii) CMEs could correlate more modes if significantly more than the observed mechanical energy of CMEs can be extracted from their energy reservoir and injected into acoustic modes.\
If all CMEs participate to acoustic emission with an efficiency $f$ independent of their energy, they should be responsible for a fraction $16.7\%\le \lambda\le 33.2\%$ of low degree p mode total power at solar maximum. Nevertheless, this fraction might be significantly smaller if only a subset of high energy CMEs participates to acoustic emission. The contribution of CMEs to the low degree p mode power has to be reconciled with the observed $30\%$ decrease of their power at solar maximum. A better theoretical understanding of the efficiency of the energy transfer from a CME event to acoustic modes is needed.\
If our interpretation of IPHIR correlations is correct, we should be able to make the following observational tests:
- detect the effect of the largest CMEs on the energy of low degree p modes, even at solar minimum,
- measure the correlation ${\cal C}_{\rm min}\ge 0.2 \% $ at solar minimum, using longer time series and more modes than F98,
- confirm with SOHO, during the next solar maximum, the correlations determined from IPHIR data, measure them with a better accuracy, and determine whether higher degree modes are also correlated.\
Observations of the influence of CMEs on acoustic modes could improve our understanding of both the excitation of acoustic modes and the ejection mechanism of CMEs.
The author thanks T. Amari, M. Tagger, D. Gough and D. Saundby for helpful discussions.
Energy of a mode excited stochastically\[app1\]
===============================================
Let us consider a spherically symmetric, adiabatic model of the sun. Any perturbation of velocity $v$ can be projected onto the basis of orthogonal eigenfunctions ${\bf v}_{nlm}$, associated to the (supposedly real) eigenvalues $\omega_{nl}$. The structure of the equations allows us to write each component of the velocity vector as the product of a function of $r$ only and a function of $\theta,\phi$ (see Unno 1989). This function is a spherical harmonic for the radial velocity $v^r$. $$\begin{aligned}
{\bf v}({\bf r},t)&=& \sum_{nlm} A_{nlm}(t) {\bf v}_{nlm}({\bf r}),\\
{\bf v}_{nlm}({\bf r})&\equiv&\left\lbrack
v_{nl}^r(r),v_{nl}^{h}(r){\p\over\p\theta},
v_{nl}^{h}(r){\p\over\sin\theta\p\phi}
\right\rbrack Y_{l}^m(\theta,\phi).\end{aligned}$$ The eigenvectors are normalized as follows: $$\begin{aligned}
\int_{0}^{M_\star}|{\bf v}_{nlm}|^{2}\d M_{r}=1,\label{normv}\\
=\int_{0}^{R_{\star}} \rho\left\lbrack
|v_{nl}^r|^{2}+l(l+1)|v_{nl}^{h}|^{2}\right\rbrack r^{2}\d r. \end{aligned}$$ The spherical harmonics are written in terms of Legendre associated functions $P_{l}^{m}$: $$\begin{aligned}
Y_{l}^{m}(\theta,\phi)&=&(-1)^{m+|m|\over2}q_{lm}P_{l}^{|m|}(\cos\theta)
\e^{im\phi},\label{spha1}\\
q_{lm}&\equiv &
\left\lbrack{2l+1\over4\pi}{(l-|m|)!\over(l+|m|)!}\right\rbrack^{1\over2} .
\label{spha2}\end{aligned}$$ The radial velocity perturbation $v_{k}({\bf r})$ is described by Eqs. (\[defv1\])-(\[defv3\]). The angle $\alpha\ge 0$ made by the direction $\theta,\phi$ with the direction $\theta_{k},\phi_{k}$ is defined by: $$\begin{aligned}
\cos\alpha(\theta,\theta_{k},\phi-\phi_{k})\equiv \nonumber\\
\cos\theta\cos\theta_{k}&+&\sin\theta\sin\theta_{k}\cos(\phi-\phi_{k}).\end{aligned}$$ By projecting the perturbation onto the basis of eigenvectors, we obtain: $$\begin{aligned}
A_{nlm}(t>t_{k})&=&\cos\omega_{nl}(t-t_{k})
\int_{0}^{M_{\star}}{\bf v}_{k}({\bf
r})\cdot{\bf v}_{nlm}^{\star}({\bf r})\d M_{r},\\
&=&2(-1)^{m+|m|\over2}a_{nlm}^{k}\e^{-im\phi_{k}}
\cos\omega_{nl}(t-t_{k}),\label{anlmreel} \\
a_{nlm}^{k}&\equiv &h_{nl}^{k}g_{lm}^{k}(\theta_{k})M_{k}^{1\over2}v_{k} \end{aligned}$$ where the real dimensionless functions $g_{lm}^{k}(\theta_{k})$ and $h_{nl}^{k}$ are defined as follows: $$\begin{aligned}
g_{lm}^{k}(\theta_{k})&\equiv&
{q_{lm}\over 2}\int_{0}^{2\pi}\d\phi\int_{0}^{\pi}\d(\cos\theta)
P_{l}^{|m|}(\cos\theta)\nonumber\\
&\times&\e^{-im(\phi-\phi_{k})}g_{k}(\theta,\theta_{k},\phi-\phi_{k}),\\
&=&q_{lm}\int_{0}^{{\alpha_{k}\over
2}}\d\phi\int_{0}^{\pi}\d(\cos\theta)
P_{l}^{|m|}(\cos\theta)\nonumber\\
&\times&\cos(m\phi)g_{k}(\theta,\theta_{k},\phi),\\
h_{nl}^k&\equiv&{1\over M_{k}^{1\over2}}
\int_{0}^{R_{\star}} \rho r^{2}v_{nl}^{r}(r)h_{k}(r)\d r. \end{aligned}$$ Note that the function $g_{l,m}^{k}=g_{l,-m}^{k}$ is real because of the cylindrical symmetry of the impulsion.\
Eq. (\[anlmreel\]) corresponds to initial conditions with zero displacement at $t=t_{k}$, suitable to describe the transfer of impulsion due to a shock.\
Assuming that the eigenfrequencies $\omega_{nl}$ are real, we add by hand a damping term, with a time scale $\tau_{d}$. After a series of excitations indexed by $k$, and neglecting the transient velocities present during each event, the linearity of the equations allows us to write the velocity as follows: $$\begin{aligned}
A_{nlm}(t)&=&2(-1)^{m+|m|\over2}\nonumber\\
&\times&\sum_{t_{k}<t}\e^{-{t-t_{k}\over\tau_{d}}}a_{nlm}^{k}
\e^{-im\phi_{k}}\cos\omega_{nl}(t-t_{k}).\label{anlmdamp} \end{aligned}$$ Denoting by ${\bf \xi}_{nlm}({\bf r},t)$ and ${\bf V}_{nlm}({\bf
r},t)$ the displacement and velocity associated to the mode $(n,l,m)$, the acoustic energy $E_{nlm}$ of each mode is the sum of the kinetic and potential energies: $$E_{nlm}\equiv
{1\over2}\int_{0}^{M_{\star}}(|{\bf V}_{nlm}|^{2}+
\omega_{nl}^{2}|{\bf \xi}_{nlm}|^{2})\d M_{r}.$$ Neglecting the slow damping compared to the fast oscillations ($\omega_{nl}\tau_{d}\sim 7\times 10^{3}\gg 1$ for the p modes we consider), and using the normalization (\[normv\]), we may write the energy as follows: $$E_{nlm}(t)=
{1\over2}\left(|A_{nlm}|^{2}+
{1\over \omega_{nl}^{2}}\left|{\d A_{nlm}\over \d
t}\right|^{2}\right). \label{enlmanlm}$$ Using Eq. (\[anlmdamp\]) and some algebra, we separate the two contributions from the waves propagating azimuthally in opposite directions: $$E_{nlm}(t)=E_{nlm}^{+}(t)+E_{nlm}^{-}(t),
\label{decompe}$$ with $$\begin{aligned}
E_{nlm}^{+}&\equiv&
\left|\sum_{t_{k}<t}
\e^{-{t-t_{k}\over\tau_{d}}}a_{nlm}^{k}
\e^{i(\omega_{nl}t_{k}+m\phi_{k})}\right|^{2},\label{enlmdg} \\
E_{nlm}^{-}&\equiv&E_{nl-m}^{+}.\end{aligned}$$ Although Eqs. (\[anlmdamp\]) and (\[enlmanlm\]) imply $E_{nlm}=E_{nl-m}$, the two components $E_{nlm}^{+}$ and $E_{nlm}^{-}$, are not equal, except on average.\
This separation of the components of the energy is important since the rotation enables us to separate these two components. Let us restrict ourselves to the simplest case of a solid body rotation, and neglect the Coriolis forces. This is equivalent to replacing in Eq. (\[enlmdg\]) the azimuthal angle $\phi$ by $\phi+\Omega t$ and $\phi_{k}$ by $\phi_{k}+\Omega t_{k}$, and thus we obtain Eqs. (\[enerstoc\])-(\[defphi\]). $E_{nlm}^{+}$ is the energy associated to the frequency $\omega_{nl}+m\Omega$, while $E_{nlm}^{-}$ is associated to the frequency $\omega_{nl}-m\Omega$.\
The average energy input $<e_{nlm}>$ of a single random excitation onto the mode $n,l,m$ is proportional to the total kinetic energy ${\cal E}$ of the perturbation: $$<e_{nlm}>\equiv <2h_{nl}^{2}g_{lm}^{2}{\cal E}>
.\label{squav}$$
Correlation produced by a single excitation mechanism
=====================================================
Case $N\ll1$
------------
In order to treat the general case, we need to establish first some properties of the case $N\ll1$. The index $nlm$ of the energy is omitted in what follows, for the sake of clarity The mean energy $<E>$ of the damped oscillator is directly proportional to the mean energy input $<e>$ of each impulsive event: $$\begin{aligned}
{1\over\Delta t}\int_{0}^{\Delta t}(\e^{-{t\over\tau_{\rm d}}})^{2}\d
t&\sim&{N\over2},\\
<E>&\sim&{N\over 2}<e>\label{meanE},\\
<E^{2}>&\sim&{N\over 4}<e^{2}>.\label{meanE2}\end{aligned}$$ The energies $E,E'$ of two modes $nlm$, $n'l'm'$ satisfy: $${<EE'>\over <E><E'>}\sim {1\over N}
{<ee'>\over <e><e'>}.
\label{meanEE}$$
General case
------------
The correlation in the general case is derived using the following theorem: if a continuous function $f(x)$ satisfies $f(2x)=f(x)$ for any value of $x$, and $\lim_{x\to0}f(x)$ exists, then $f$ is constant.\
Let us divide the random series of excitation into two independent subset of same statistical characteristics, with an occurrence rate $N/2$. Denoting by $E(N/2)$ the energy of the mode excited by only one subset, one can check from Eq. (\[enerstoc\]) that: $$<E(N)>=2<E(N/2)>,$$ and conclude that $f(N)\equiv E/N$ is a constant, which we deduce from the case $N\ll1$ (Eq. \[meanE\]): $$<E>\equiv {N\over2}<e>.
\label{energen}$$ The variance of the energy can also be written as follows: $${\Var E\over <E>^{2}}(N)={1\over 2}{\Var E\over <E>^{2}}(N/2)+{1\over 2}.$$ This implies that $f(N)\equiv N(\Var E/<E>^{2} -1)$ is a constant, which we deduce from the case $N\ll1$ (Eq. \[meanE2\]): $${\Var E\over <E>^{2}}=
1+{1\over N}{<e^{2}>\over <e>^{2}}.
\label{vaava}$$ The correlation ${\cal C}$ between two modes $nlm$, $n'l'm'$ is defined as: $${\cal C}\equiv {<EE'>-<E><E'>\over
(\Var E)^{1\over2}(\Var E')^{1\over2}}.\label{defcor}$$ Using the fact that $${<EE'>\over <E><E'>}(N)={1\over2}{<EE'>\over <E><E'>}(N/2)+{1\over2},$$ we conclude that $f(N)\equiv N(<EE'>/(<E><E'>)-1)$ is a constant, which we deduce from the case $N\ll1$ (Eq. \[meanEE\]). The correlation expressed by Eq. (\[corgen1\]) is then derived from Eqs. (\[vaava\])-(\[defcor\]).
Correlation produced by two excitation mechanisms\[Aseveral\]
=============================================================
Let us consider two independent series of impulsive events due to two different excitation mechanisms superimposed, indexed by $k$, characterized by the series of velocity perturbations $v_{k}^{(1)},v_{k}^{(2)}$, and mean interpulse times $\Delta t^{(1)},\Delta t^{(2)}$. The energy of a mode $(nlm)$ can be decomposed into two random walks of $N_{1}$ and $N_{2}$ steps. $$\begin{aligned}
E_{nlm}^{+}(t)=\nonumber\\
\left |
\sum_{t_{k}^{(1)}<t} \e^{-{t-t_{k}^{(1)}\over\tau_{d}}}
a_{nlm}^{k(1)}\e^{i\Phi_{nlm}^{k(1)}}
+
\sum_{t_{k}^{(2)}<t} \e^{-{t-t_{k}^{(2)}\over\tau_{d}}}
a_{nlm}^{k(2)}\e^{i\Phi_{nlm}^{k(2)}}
\right |^{2}.\end{aligned}$$ Let us write this equation for two modes $(nlm)$ and $(n'l'm')$. Hereafter we simply denote the quantities specific to the mode $(n'l'm')$ with a prime, for the sake of clarity. $E_{i}$ and $E'_{i}$ are the energies of the modes $nlm$ and $n'l'm'$ excited by the excitation mechanism $(i)$ only ($i=1,2$). The series of phases $\Phi^{k(1)},\Phi^{k(2)},\Phi^{k(1)'},\Phi^{k(2)'}$ are independent. $$\begin{aligned}
<E>&=&<E_{1}>+<E_{2}>,\\
{\rm Var}(E)&=&{\rm Var}(E_{1})+{\rm Var}(E_{2})+2<E_{1}><E_{2}>,
\label{varsom}\\
{\rm Var}(E)\;{\cal C}&=&{\rm Var}(E_{1}){\cal C}_{1}+
{\rm Var}(E_{2}){\cal C}_{2}\label{corsom},\end{aligned}$$ where ${\cal C}_{i}$ is the correlation between $E_{i}$ and $E'_{i}$. Let us define the fractions $\lambda,\lambda'$ of the total power of each mode contributed by the second mechanism as follows: $$\begin{aligned}
\lambda&\equiv& {<E_{2}>\over <E>},\\
\lambda'&\equiv &{<E_{2}>\over <E>}.\end{aligned}$$ Let convection be the first excitation mechanism ($C_{1}= 0$), and let us use the index $({\XX})$ for the properties $N_{\XX},e_{\XX}$ of the second excitation mechanism. Using Eqs. (\[vaava\]), (\[corgen1\]) and (\[varsom\]) in (\[corsom\]), we obtain: $$\begin{aligned}
{\cal C}={<ee'>\over
<e^{2}>^{1\over2}<e'^{2}>^{1\over2}}\nonumber\\
\times\left(
1+{N_{\XX}\over\lambda^{2}}
{<e>^{2}\over<e^{2}>}\right)^{-{1\over2}}
\left(
1+{N_{\XX}\over\lambda'^{2}}
{<e'>^{2}\over<e'^{2}>}\right)^{-{1\over2}}.
\label{corgenlamb}\end{aligned}$$ Let $p_{\rm T}({\cal E})$ be the density of probability of the additional source of energy as defined in Sect. \[effic\]. The number of efficient excitations per damping time is independent of $f_{\XX}$: $$N_\XX=N_{\rm T}\int_{{\cal E}_{1}}^{{\cal E}_{2}}
p_{\rm T}({\cal E})
\d {\cal E}.\label{ap1}$$ The mean energy and kurtosis of the distribution of efficient events are: $$\begin{aligned}
<{\cal E}_\XX>&=& f_{\XX}
{\int_{{\cal E}_{1}}^{{\cal E}_{2}}{\cal E}p_{\rm T}({\cal E})
\d {\cal E}
\over
\int_{{\cal E}_{1}}^{{\cal E}_{2}}p_{\rm T}({\cal E})\d {\cal E}},
\label{ap2}\\
\beta_\XX&=&{\int_{{\cal E}_{1}}^{{\cal E}_{2}}p_{\rm T}({\cal E})
\d {\cal E}
\int_{{\cal E}_{1}}^{{\cal E}_{2}}{\cal E}^{2}p_{\rm T}({\cal E})
\d {\cal E}
\over
\left(\int_{{\cal E}_{1}}^{{\cal E}_{2}}{\cal E}p_{\rm T}({\cal E})
\d {\cal E}
\right)^{2}}
.\label{ap3}\end{aligned}$$ The kurtosis is also independent of $f_{\XX}$. Eq. (\[filtre\]) is obtained by incorporating Eqs. (\[ap1\])-(\[ap3\]) into Eq. (\[apllic11\]).
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\[section\] \[section\] \[section\] \[section\]
\[section\]
addtoreset[equation]{}[section]{}
****
A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes
[Guanhua Li]{}\
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.\
e-mail: `206674@swansea.ac.uk`
[Eugene Lytvynov]{}\
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.\
e-mail: `e.lytvynov@swansea.ac.uk`
2010 [*AMS Mathematics Subject Classification:*]{} 60F99, 60J60, 60J75, 60J80, 60K35
[*Keywords:*]{} Birth-and-death process; Continuous system; Permanental point process; Glauber dynamics; Kawasaki dynamics
Introduction
============
Let $X$ be a locally compact Polish space and let $\nu$ be a Radon non-atomic measure on it. Let $\Gamma=\Gamma_{X}$ denote the space of all locally finite subsets (configurations) in $X$.
A Glauber dynamics (a birth-and-death process of an infinite system of particles in $X$) is a Markov process on $\Gamma$ whose formal (pre-)generator has the form $$\label{1}
\begin{aligned}
(L_{\mathrm G}F)(\gamma)&=\sum_{x\in{\gamma}}d(x,\gamma\setminus x)(F(\gamma\setminus
x)-F(\gamma))
\\
& +\int_{X}\nu(dx)\,b(x, \gamma)\left( F(\gamma\cup
x)-F(\gamma)\right), \quad \gamma\in\Gamma.
\end{aligned}$$ Here and below, for simplicity of notation we write $x$ instead of $\{x\}$. The coefficient $d(x,\gamma\setminus x)$ describes the rate at which particle $x$ of configuration $\gamma$ dies, while $b(x,\gamma)$ describes the rate at which, given configuration $\gamma$, a new particle is born at $x$.
A Kawasaki dynamics (a dynamics of hopping particles) is a Markov process on $\Gamma$ whose formal (pre-)generator is $$\label{2}
(L_{\mathrm K}F)(\gamma)=\sum_{x\in \gamma}c(x,y,\gamma\setminus
x)(F(\gamma\setminus x\cup y)-F(\gamma)), \quad \gamma\in\Gamma.$$ The coefficient $c(x, y, \gamma\setminus x)$ describes the rate at which particle $x$ of configuration $\gamma$ hops to $y$, taking the rest of the configuration, $\gamma\setminus x$, into account.
Equilibrium Glauber and Kawasaki dynamics which have a standard Gibbs measure as symmetrizing (and hence invariant) measure were constructed in [@KL; @KLR]. In [@LO], this construction was extended to the case of an equilibrium dynamics which has a determinantal (fermion) point process as invariant measure, For further studies of equilibrium and non-equilibrium Glauber and Kawasaki dynamics, we refer to [@BCC; @FK; @FKK; @FKL; @FKO; @HS; @G1; @G2; @KKL; @KKM; @KKP; @KKZ; @KMZ; @P] and the references therein.
The aim of this note is to show that general criteria of existence of Glauber and Kawasaki dynamics which were developed in [@LO] are appliable to a wide class of manental ($\alpha\in \mathbb{N}$) point processes, proposed by Shirai and Takahashi [@ST]. This class includes classical permanental (boson) point processes, see e.g. [@DVJ; @ST]. We will also consider a diffusion approximation for the Kawasaki dynamics at the level of Dirichlet forms (compare with [@KKL]). This will lead us to an equilibrium dynamics of interacting Brownian particles for which an $\alpha$-permanental point process is a symmetrizing measure. As a by-product of our considerations, we will also extend the result of [@ST] on the existence of $\alpha$-permanental point process.
Equilibrium Glauber and Kawasaki dynamics – general results
===========================================================
Let $X$ be a locally compact Polish space. We denote by $\mathcal{B}(X)$ the Borel $\sigma$-algebra on $X$, and by $\mathcal{B}_{0}(X)$ the collection of all sets from $\mathcal{B}(X)$ which are relatively compact. We fix a Radon, non-atomic measure on $(X,\mathcal{B}(X))$. (For most applications, the reader may think of $X$ as $\mathbb{R}^{d}$ and $\nu$ as the Lebegue measure.)
The configuration space $\Gamma$ over $X$ is defined as the set of all subsets of $X$ which are locally finite $$\Gamma:=\big\{\gamma\subset X : \,
|\gamma_\Lambda|<\infty\text{ for each }\Lambda\in{\mathcal B}_0(X
)\big\},$$ where $|\cdot|$ denotes the cardinality of a set and $\gamma_\Lambda:= \gamma\cap\Lambda$. One can identify any $\gamma\in\Gamma$ with the positive Radon measure $\sum_{x\in\gamma}\varepsilon_x$, where $\varepsilon_x$ is the Dirac measure with mass at $x$ and $\sum_{x\in\varnothing}\varepsilon_x{:=}$zero measure. The space $\Gamma$ can be endowed with the vague topology, i.e., the weakest topology on $\Gamma$ with respect to which all maps $$\Gamma\ni\gamma\mapsto{\langle}\varphi,\gamma{\rangle}:=\int_{X}
\varphi(x)\,\gamma(dx) =\sum_{x\in\gamma}\varphi(x),\quad
\varphi\in C_0(X),$$ are continuous. Here, $C_0(X)$ is the space of all continuous, real-valued functions on $X$ with compact support. We denote the Borel $\sigma$-algebra on $\Gamma$ by $\mathcal{B}(\Gamma)$. A point process in $X$ is a probability measure on $(\Gamma, \mathcal{B}(\Gamma))$.
We fix a point process $\mu$ which satisfies the so-called condition $(\Sigma'_{\nu})$ [@DVJ; @MWM], i.e., there exist a measurable function $r: X\times \Gamma\rightarrow [0, +\infty]$, called the Papangelou intensity of $\mu$, such that $$\label{3}
\int_{\Gamma}\mu(d\gamma)\int_{X}\gamma(dx)F(x,
\gamma)=\int_{\Gamma}\mu(d\gamma)\int_{X}\nu(dx)\,r(x,\gamma)F(x,\gamma\cup
x)$$ for any measurable function $F: X\times \Gamma\rightarrow
[0,+\infty]$. The condition $(\Sigma'_{\nu})$ can be thought of as a kind of weak Gibbsianess of $\mu$. Intuitively, we may treat the Papangelou intensity as $$\label{7}
r(x,\gamma)=\exp[-E(x,\gamma)],$$ where $E(x,\gamma)$ is the relative energy of interaction between particle $x$ and configuration $\gamma$.
To define an equilibrium Glauber dynamics for which $\mu$ is a symmetrizing measure, we fix a death coefficient as a measurable function $d:X\times \Gamma\rightarrow [0, +\infty]$, and then define a birth coefficient $b: X\times\Gamma\rightarrow [0,
+\infty]$ by $$\label{5}
b(x,\gamma)=d(x,\gamma)r(x,\gamma), \quad (x,\gamma)\in X\times
\Gamma .$$ To define a Kawasaki dynamics, we fix a measurable function $c:X^{2}\times \Gamma^2\rightarrow [0,+\infty]$ which satisfies $$\label{4}
r(x,\gamma)c(x,y,\gamma)=r(y,\gamma)c(y,x,\gamma), \quad
(x,y,\gamma)\in X^{2}\times \Gamma.$$ Formulas and are called the balance conditions [@G1; @G2]. We will also assume that the function $c(x,y,\gamma)$ vanishes if at least one of the functions $r(x,\gamma)$ and $r(y,\gamma)$ vanishes, i.e., $$\begin{aligned}
\label{6}
c(x,y,\gamma)=c(x,y,\gamma)\chi_{\{r>0\}}(x,\gamma)\chi_{\{r>0\}}(y,\gamma).\end{aligned}$$ Here, for a set $A$, $\chi_{A}$ denotes the indicator function of $A$. We refer to [@LO Remark 3.1] for a justification of this assumption, which involves the interpretation of $r(x,\gamma)$ as in , see also Remark \[ghdstsgcf\] below.
We denote by $\mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)$ the space of all functions of the form $$\label{hdtrse}\Gamma\ni\gamma\mapsto F(\gamma)=g(\langle \varphi_{1},\gamma\rangle,\dots,\langle \varphi_{N},\gamma\rangle),$$ where $N\in \mathbb{N}$, $\varphi_{1},\dots, \varphi_{N}\in C_{0}(X)$ and $g\in C_{\mathrm b}(\mathbb{R}^{N})$. Here, $C_{\mathrm b}(\mathbb{R}^{N})$ denotes the space of all continuous bounded functions on $\mathbb{R}^{N}$. We assume that, for each $\Lambda\in \mathcal{B}_{0}(X)$, $$\begin{gathered}
\label{8}
\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\gamma(dx)\,d(x,\gamma\setminus
x)<\infty,\\
\label{9}
\int_{\Gamma}\mu(d\gamma)\int_{X}\gamma(dx)\int_{X}\nu(dy)\,c(x,y,\gamma\setminus
x)(\chi_{\Lambda}(x)+\chi_{\Lambda}(y))<\infty.\end{gathered}$$ As easily seen, conditions and are sufficient in order to define bilinear forms $$\begin{aligned}
{\mathcal{E}}_{\mathrm G}(F,G):&=\int_{\Gamma}\mu(d\gamma)\int_{X}\gamma(dx)\,d(x,\gamma\setminus
x)(F(\gamma\setminus x)-F(\gamma))(G(\gamma\setminus x)-G(\gamma)),
\\
{\mathcal{E}}_{\mathrm K}(F,G):&=\frac{1}{2}\int_{\Gamma}\mu(d\gamma)\int_{X}\gamma(dx)\int_{X}\nu(dy)\,c(x,y,\gamma\setminus
x)(F(\gamma\setminus x\cup y)-F(\gamma))
\\
&
\times
(G(\gamma\setminus x\cup y)-G(\gamma)),\end{aligned}$$ where $F,G\in \mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)$.
For the construction of the Kawasaki dynamics, we will also assume that the following technical assumptions holds: $$\label{10}
\begin{aligned}
& \exists u,v\in \mathbb{R}\quad \forall \Lambda\in\mathcal B_0(X):
\\
& \qquad \int_{\Lambda}\gamma(dx)\int_{\Lambda}\nu(dy)\,r(x,\gamma\setminus
x)^{u}r(y,\gamma\setminus x)^{v}c(x,y,\gamma\setminus y)\in
L^{2}(\Gamma, \mu)<\infty.
\end{aligned}$$ Note that in formula and below, we use the convention $\frac{0}{0}:=0$.
The following theorem was essentially proved in [@LO].
\[asd\] (i) Assume that a point process $\mu$ satisfies . Assume that conditions , , respectively , , , and are satisfied. Let $\sharp=\mathrm G, \mathrm K$. Then the bilinear form $(\mathcal{E}_{\sharp},
\mathcal{F}C_{\mathrm b}(C_{0}(x),\Gamma))$ is closable in $L^{2}(\Gamma,
\mu)$ and its closure will be denoted by $(\mathcal{E}_{\sharp},
D(\mathcal{E}_{\sharp}))$. Further there exists a conservative Hunt process (Glauber, respectively Kawasaki dynamics) $$\begin{aligned}
M^{\sharp}=\left(\Omega^{\sharp},
\mathcal{F}^{\sharp},(\mathcal{F}^{\sharp}_{t})_{t\geq
0},(\Theta_{t}^{\sharp})_{t\geq 0}, (X^{\sharp}(t))_{t\geq 0},
(P^{\sharp}_{\gamma})_{\gamma\in \Gamma}\right)\end{aligned}$$ on $\Gamma$ which is properly associated with $(\mathcal{E}_{\sharp}, D(\mathcal{E}_{\sharp}))$, i.e., for all ($\mu$-version of) $F\in L^{2}(\Gamma, \mu)$ and $t>0$ $$\Gamma\ni
\gamma\mapsto
p^{\sharp}_{t}F(\gamma):=\int_{\Omega^\sharp}F(X^{\sharp}(t))\, dP_{\gamma}^{\sharp}$$ is an $\mathcal{E}^{\sharp}$-quasi continuous version of $\exp(tL_{\sharp})F$, where $(-L_{\sharp}, D(L_{\sharp}))$ is the generator of $(\mathcal{E}_{\sharp}, D(\mathcal{E}_{\sharp}))$. $M^{\sharp}$ is up-to $\mu$-equivalence unique. In particular, $M^{\sharp}$ is $\mu$-symmetric and has $\mu$ as invariant measure.
\(ii) $M^\sharp$ from (i) is up to $\mu$-equivalence unique between all Hunt processes $$M'=\left(\Omega',
\mathcal{F}',(\mathcal{F}'_{t})_{t\geq 0}, (\Theta'_{t})_{t\geq 0},
(X'(t))_{t\geq 0}, (P'_{\gamma})_{\gamma\in \Gamma}\right)$$ on $\Gamma$ having $\mu$ as invariant measure and solving a martingale problem for $(L_{\sharp}, D(L_{\sharp}))$, i.e., for all $G\in
D(H_\sharp)$ $$\widetilde G({X}'(t))-\widetilde G({X}'(0))-\int_0^t (L_\sharp
G)({X}'(s))\,ds,\quad t\ge0,$$ is an $({\mathcal F}_t')$-martingale under ${P}_\gamma'$ for ${\mathcal E}_\sharp$-q.e. $\gamma\in\Gamma$. Here, $\widetilde G$ denotes an ${\mathcal E}_\sharp$-quasi-continuous version of $G$.
\(iii) Further assume that, for each $\Lambda\in \mathcal{B}_{0}(X)$, $$\label{12}
\int_\Lambda \gamma(dx)\, d(x,\gamma\setminus x)\in L^2(\Gamma,\mu),\quad
\int_\Lambda \nu(dx)\,b(x,\gamma)\in L^2(\Gamma,\mu),$$ in the Glauber case, and $$\label{13}
\int_X\gamma(dx)\int _X \nu(dy)\, c(x,y,\gamma\setminus
x)(\chi_\Lambda(x)+\chi_\Lambda(y))\in L^2(\Gamma,\mu)$$ in the Kawasaki case. Then $\mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)\subset D(L_\sharp)$, and for each $F\in \mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)$, $L_{\sharp}F$ is given by formulas and , respectively.
We refer to [@MR] for an explanation of notions appearing in Theorem \[asd\], see also a brief explanation of them in [@LO].
The statement follows from Theorems 3.1 and 3.2 in [@LO]. Note that, although these theorems are formulated for determinantal point processes only, their proof only uses the $(\Sigma'_{\nu})$ property of these point processes. Note also that condition is formulated in [@LO] only for $v=1$, however the proof of Lemma 3.2 in [@LO] admits a straightforward generalization to the case of an arbitrary $v\in \mathbb{R}$.
Part (iii) of Theorem \[asd\] states that the operator $(-L_{\sharp}, D(L_{\sharp}))$ is the Friedrichs’ extention of the operator $(-L_{\sharp}, \mathcal{F}C_{\mathrm b}(C_{0}(X), \Gamma))$ defined by formulas , , respectively.
Let us fix a parameter $s\in [0,1]$ and define $$\begin{aligned}
\label{14}
d(x,\gamma):&=r(x,\gamma)^{s-1}\chi_{\{r>0\}}(x,\gamma),\quad (x,
\gamma)\in X\times \Gamma,\\
\label{15}
b(x,\gamma):&=r(x,\gamma)^{s}\chi_{\{r>0\}}(x,\gamma), \quad (x,
\gamma)\in X\times \Gamma,
$$ $$\label{16}
\begin{gathered}
c(x,y,\gamma):=a(x,y)r(x,\gamma)^{s-1}r(y,\gamma)^{s}\chi_{\{r>0\}}(x,\gamma)\chi_{\{r>0\}}(y,\gamma),
\\
(x,y,\gamma)\in{X^{2}\times \Gamma}.
\end{gathered}$$ Here the function $a:X^{2}\rightarrow [0,+\infty)$ is bounded, measurable, symmetric (i.e., $a(x,y)=a(y,x)$), and satisfies $$\label{17}
\sup_{x\in X}\int_{X}a(x, y)\,\nu(dy)<\infty.$$ Note that the balance conditions and are satisfied for these coefficients, and so is condition .
\[tesa4wa\] Note that, if $X=\mathbb R^d$ and $a(x,y)$ has the form $a(x-y)$ for a function $a:\mathbb R^d\to[0,\infty)$, then condition means that $a\in L^1(\mathbb R^d,dx)$. (Here and below, in the case $X=\mathbb R^d$, we use an obvious abuse of notation.)
\[ghdstsgcf\] Using representation , we can rewrite formulas – as follows: $$\begin{aligned}
d(x,\gamma\setminus x)&=\exp[(1-s)E(x, \gamma\setminus
x)]\chi_{\{E<+\infty\}}(x,\gamma\setminus x),\\
b(x,\gamma\setminus x)&=\exp[-sE(x,\gamma\setminus
x)]\chi_{\{E<+\infty\}}(x,\gamma\setminus x),\\
c(x,y,\gamma\setminus x)&=a(x,y)\exp[(1-s)E(x,\gamma\setminus
x)-sE(y,\gamma\setminus x)]\\
& \times\chi_{\{E<+\infty\}}(x,\gamma\setminus x)\chi_{\{E<+\infty\}}(y, \gamma\setminus x).\end{aligned}$$ So, if the corresponding dynamics exist, one can give the following heuristic description of them: Both dynamics are concentrated on configurations $\gamma\in \Gamma$ such that, for each $x\in
\gamma$, the relative energy of interaction between $x$ and the rest of configuration, $\gamma\setminus x$, is finite; those particles tend to die, respectively hop, which have a high energy of interaction with the rest of the configuration, while it is more probable that a new particle is born at $y$, respectively $x$ hops to $y$, if the energy of interaction between $y$ and the rest of the configuration is low.
Let us assume that the point process $\mu$ satisfies: $$\forall\Lambda\in \mathcal{B}_{0}(X):\quad
\int_{\Lambda}\gamma(dx)\in{L^{2}(\Gamma, \mu)}.$$ Then, by choosing $u=1-s$ and $v=-s$ in , we conclude that the coefficient $c$ given by satisfies .
We will construct below a class of point processes $\mu$ for which the coefficients $d$, $b$ and $c$ given above satisfy the other conditions of Theorem \[asd\].
Permanental point processes and corresponding equilibrium dynamics
==================================================================
Let $K$ be a linear, bounded, self-adjoint operator on the real space $L^{2}(X,\nu)$. Further assume that $K\geq 0$ and $K$ is locally of trace class, i.e., $\operatorname{Tr}(P_{\Lambda}KP_{\Lambda})<\infty$ for all $\Lambda\in
\mathcal{B}_{0}(X)$, where $P_{\Lambda}$ denotes the operator of multiplication by $\chi_{\Lambda}$. Hence, each operator $P_{\Lambda}\sqrt{K}$ is of Hilbert–Schmidt class. Following [@LM] (see also [@GY Lemma A.4]), we conclude that $\sqrt{K}$ is an integral operator whose integral kernel, $\varkappa(x,y)$, satisfies $$\label{31}
\int_{\Lambda}\int_X\nu(dx)\nu(dy)\varkappa(x,y)^{2}<\infty \quad \text{for all
}\quad \Lambda\in \mathcal{B}_{0}(X).$$ In particular, $$\label{32}
\varkappa(x,\cdot)\in L^{2}(X,\nu) \quad \text{for $\nu$-a.a.\quad $x\in X$.}$$ Hence, $K$ is an integral operator whose integral kernel can be chosen as $$\label{33}
\begin{aligned}
k(x,y) & =\int_{X}\varkappa(x,z)\varkappa(z,y)\nu(dz)
\\
& =\int_{X}\varkappa(x,z)\varkappa(y,z)\nu(dz)
=(\varkappa(x,\cdot),\varkappa(y,\cdot))_{L^2(X,\nu)}.
\end{aligned}$$ We also have, for each $\Lambda\in \mathcal{B}_{0}(X)$, $$\label{34}
\begin{aligned}
\operatorname{Tr}(P_{\Lambda}KP_{\Lambda})&=\|\sqrt{K}P_{\Lambda}\|^{2}_{\mathrm{HS}}
\\
&=\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\varkappa(x,y)^{2}
=\int_{\Lambda}k(x,x)\,\nu(dx),
\end{aligned}$$ where $\|\cdot\|_{\mathrm{HS}}$ denotes the Hilbert–Schmidt norm.
\[37\] There exists a random field $(Y(x))_{x\in X}$ on a probability space $(\Omega, \mathcal{A}, P)$ such that the mapping $$\label{35}
X\times \Omega \ni (x,\omega)\mapsto Y(x, \omega)$$ is measurable, and for $\nu$-a.a. $x\in X$, $Y(x)$ is a Gaussian random variable with mean $0$ and such that $$\label{36}
\mathbb{E}\left(Y(x)Y(y)\right)=k(x,y) \quad \text{for
$\nu^{\otimes 2}$-a.a.\ $(x,y)\in X^{2}$ and $\nu$-a.a.\ $x=y\in X$}.$$
\[cfdxrbv\] The statement of Proposition \[37\] is well-known if the integral kernel of the operator $K$ admits a continuous version (see e.g.Theorem 1.8 and p. 456 in [@ST]). In the latter case, $(Y(x))_{x\in X}$ is a Gaussian random field and formula holds for all $(x,y)\in X^{2}$.
Consider a standard triple of real Hilbert spaces $$H_{+}\subset
H_{0}=L^{2}(X,\nu)\subset H_{-}\,.$$ Here the Hilbert space $H_{+}$ is densely and continuously embedded into $H_{0}$, the inclusion operator $H_{+}\hookrightarrow H_{0}$ is of Hilbert–Schmidt class, and the Hilbert space $H_{-}$ is the dual space of $H_{+}$ with respect to the center space $H_{0}$ (see e.g. [@BK]).
Let $\mathbb{P}$ be the standard Gaussian measure on $H_{-}$, i.e., the probability measure on the Borel $\sigma$-algebra $\mathcal{B}(H_{-})$ which has Fourier transform $$\int_{H_{-}}e^{i\langle \omega,
f\rangle}\,\mathbb{P}(d\omega)=\exp\Big[-\frac{1}{2}\|f\|
^{2}_{H_{0}}\Big],\quad f\in H_{+}\,,$$ where $\langle \omega, f\rangle$ denotes the dual pairing between $\omega\in H_{-}$ and $f\in
H_{+}$. Then the mapping $H_{+}\ni f\rightarrow
\langle \cdot, f\rangle$ can be extended by continuity to an isometry $$\label{38}
I: H_{0}\rightarrow L^{2}(H_{-},\mathbb{P}).$$ For any $f\in H_{0}$ we denote $\langle \cdot,
f\rangle:=If$. Thus, for each $f\in H_{0}$, $\langle
\cdot, f\rangle$ is a (complex) Gaussian random variable with mean $0$ and for any $f,g\in H_{0}$ $$\label{39}
\int_{H_{-}}\langle \omega,f\rangle\langle
\omega,g\rangle\,\mathbb P(d\omega)=(f,g)_{L^2(X,\nu)}.$$ Thus, by , we set for $\nu$-a.a. $x\in X$, $\widetilde{Y}(x,\omega):=\langle\omega, k(x,\cdot)\rangle$. Hence $\widetilde{Y}(x)$ is a Gaussian random variable and by and , holds.
Hence, it remains to prove that there exists a random field $Y=(Y(x))_{x\in X}$ for which the mapping is measurable and such that $Y(x,\omega)=\widetilde{Y}(x,\omega)$ for $\nu\otimes\mathbb{P}$-a.a. $(x,\omega)$. To this end, we fix any $\Lambda\in \mathcal{B}_{0}(X)$ and denote by $\mathcal{B}(\Lambda)$ the trace $\sigma$-algebra of $\mathcal{B}(X)$ on $\Lambda$. We define a set $\mathcal{D}_{\Lambda}$ of the functions $u:\Lambda\times X\rightarrow \mathbb{R}$ of the form $$\label{40}
u(x,y)=\sum^{n}_{i=1}\chi_{\Delta_{i}}(x)f_{i}(y),$$ where $\Delta_{i}\in \mathcal{B}(\Lambda)$, $f_{i}\in
H_{+}$, $i=1,\dots, n$. Define a linear mapping $$\begin{aligned}
\label{41}
I_{\Lambda}:\mathcal{D}_{\Lambda}\rightarrow
L^{2}(\Lambda\times H_{-}, \nu\otimes \mathbb{P})\end{aligned}$$ by setting, for each $u\in \mathcal{D}_{\Lambda}$ of the form , $$(I_{\Lambda}u)(x,\omega)=\sum^{n}_{i=1}\chi_{\Delta_{i}}(x)\langle\omega,f_{i}\rangle,\quad
(x,\omega)\in \Lambda\times H_{-}\,.$$ Clearly, $I_{\Lambda}$ can be extended to an isometry $$I_{\Lambda}: L^{2}(\Lambda\times X, \nu^{\otimes
2})\rightarrow L^{2}(\Lambda\times H_{-}, \nu\otimes
\mathbb{P}),$$ and we have $I_{\Lambda}=\mathbf 1_{\Lambda}\otimes I$, where $\mathbf 1_{\Lambda}$ is the identity operator in $L^{2}(\Lambda, \nu)$ and the operator $I$ is as in .
Fix any $u\in L^{2}(\Lambda\times X, \nu^{\otimes
2})$. As easily seen, there exist a sequence $(u_{n})_{n=1}^{\infty}\subset \mathcal{D}_{\Lambda}$ such that $u_n\to u$ in $L^2(\Lambda\times X,
\nu^{\otimes2})$ and for $\nu$-a.a. $x\in \Lambda$, $u_{n}(x,\cdot)\rightarrow u(x,\cdot)$ in $L^{2}(X, \nu)$ Hence, for $\nu$-a.a. $x\in \Lambda$, $I_{\Lambda}u_{n}(x,\cdot)\rightarrow I_{\Lambda}u(x,\cdot)$ in $L^{2}(H_{-},\mathbb{P})$, which implies $$\label{gh}
(I_{\Lambda}u)(x,\omega)=\langle \omega, u(x,\cdot)\rangle
\quad\text{for $\mathbb{P}$-a.a.\ $\omega\in H_{-}$\,.}$$
Now, denote by $\varkappa_{\Lambda}$ the restriction of $\varkappa$ to the set $\Lambda\times X$. For $\nu$-a.a. $x\in \Lambda$, we define $Y_{\Lambda}(x):=(I_{\Lambda}\varkappa_{\Lambda})(x,\cdot)$. Hence, by , for $\nu$-a.a. $x\in \Lambda$, $Y_{\Lambda}(x)=\widetilde{Y}(x)$ $\mathbb{P}$-a.e. Finally, let $(\Lambda_{n})_{n=1}^{\infty}\subset \mathcal{B}_{0}(X)$ be such that $\Lambda_{n}\cap \Lambda_{m}=\emptyset$ if $n\neq m$ and $\bigcup^{\infty}_{n=1}\Lambda_{n}=X$. Setting $Y(x):=Y_{\Lambda_{n}}(x)$ for $\nu$-a.a. $x\in \Lambda_{n}, n\in
\mathbb{N}$, we conclude the statement.
Let $Y$ be a random field as in Proposition \[37\]. For each $\Lambda\in \mathcal{B}_{0}(X)$, we have $$\begin{aligned}
\mathbb{E}\left(\int_{\Lambda}Y(x)^{2}\,\nu(dx)\right)&=\int_{\Lambda}\mathbb E(Y(x)^{2})\,\nu(dx)\\
&=\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\varkappa(x,y)^{2}<\infty.\end{aligned}$$ In particular, the function $Y(x)^{2}$ is locally $\nu$-integrable $\mathbb P$-a.s. Let $l\in\mathbb{N}$ and let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space on which $l$ independent copies $Y_{1},
Y_{2},\ldots, Y_{l}$ of a random field $Y$ as in Proposition \[37\] are defined. Denote by $\mu^{(l)}$ the Cox point process on $X$ with random intensity $g^{(l)}(x)=\sum^{l}_{i=1}Y_{i}(x)^{2}$, which is locally $\nu$-integrable $\mathbb{P}$-a.s. Thus, $\mu^{(l)}$ is the probability measure on $(\Gamma,
\mathcal{B}(\Gamma))$ which satisfies $$\label{44}
\int_{\Gamma}\mu^{(l)}(d\gamma)F(\gamma)=\int_{\Omega}\mathbb{P}(d\omega)\int_{\Gamma}\pi_{{g}^{(l)}(x,\omega)\nu(dx)}(d\gamma)F(\gamma)$$ for each measurable function $F:\Gamma\rightarrow [0,+\infty]$. Here, for a locally $\nu$-integrable function $g: X\rightarrow
[0,+\infty)$, we denote by $\pi_{g(x)\nu(dx)}$ the Poisson point process in $X$ with intensity measure $g(x)\nu(dx)$, see e.g [@DVJ]. This is the unique point process in $X$ which satisfies the Mecke identity $$\label{45}
\int_{\Gamma}\pi_{g(x)\nu(dx)}(d\gamma)\,\int_{X}\gamma(dx)F(x,\gamma)=\int_{\Gamma}\pi_{g(x)\nu(dx)}(d\gamma)\int_{X}\nu(dx)\,g(x)F(x,\gamma\cup
x)$$ for each measurable $F: X\times\Gamma \rightarrow [0,+\infty]$. By and (compare with e.g. [@MW]), for each $l\in \mathbb{N}$, the point process $\mu^{(l)}$ satisfies condition $(\Sigma'_{\nu})$ and its Papangelou intensity is given by $$\begin{aligned}
r^{(l)}(x,\gamma)=\widetilde{\mathbb{E}}(g^{(l)}(x)\mid\mathcal{F})(\gamma)
=\widetilde{\mathbb{E}}\Big(\sum^{l}_{i=1}Y_{i}(x)^{2}\mid\mathcal{F}\Big)(\gamma).\label{46}\end{aligned}$$ Here $\widetilde{\mathbb{E}}$ denotes the (conditional) expectation with respect to the probability measure $$\begin{aligned}
\label{47}
\widetilde{\mathbb{P}}(d\omega,d\gamma)=\widetilde{\mathbb{P}}(d\omega)\,\pi_{g^{(l)}(x,\omega)\nu(dx)}(d\gamma)\end{aligned}$$ on $\Omega\times\Gamma$, while $\mathcal{F}$ denotes the $\sigma$-algebra on $\Omega\times\Gamma$ generated by the mappings $$\Omega\times\Gamma\ni(\omega,\gamma)\rightarrow F(\gamma)\in
\mathbb{R},$$ where $F: \Gamma\rightarrow \mathbb{R}$ is measurable.
Recall that a point process $\mu$ in $X$ is said to have correlation functions if, for each $n\in\mathbb{N}$, there exist a non-negative, measurable, symmetric function $k_{\mu}^{(n)}$ on $X^{n}$ such that, for any measurable, symmetric function $f^{n}:
X^{n}\rightarrow [0,+\infty]$, $$\label{48}
\begin{aligned}
&\int_{\Gamma}\sum_{\{x_{1},\dots,x_{n}\}\subset
\gamma}f^{(n)}(x_{1},\dots,x_{n})\,\mu(d\gamma)
\\
&\quad=\frac{1}{n!}\int_{X^{n}}f^{(n)}(x_{1},\dots,x_{n})k_{\mu}^{(n)}(x_{1},\dots,x_{n})\nu(dx_{1})\dotsm\nu(dx_{n}).
\end{aligned}$$ As well known (e.g. [@DVJ]), for a locally $\nu$-integrable function $g: X\rightarrow [0,+\infty)$, the Poisson point process $\pi_{g(x)\nu(dx)}$ has correlation functions $$\begin{aligned}
\label{49}
k^{(n)}_{\mu}(x_{1},\ldots,x_{n})=g(x_{1})\dotsm g(x_{n}).\end{aligned}$$
Let us recall the notion of $\alpha$-permanent [@VJ], called $\alpha$-determinant in [@ST]. For a square matrix $A=(a_{ij})^{n}_{i,j=1}$ and $\alpha\in\mathbb{R}$, we set $$\operatorname{per}_\alpha A:=\sum_{\sigma\in S_{n}}\alpha^{n-m(\sigma)}\prod^{n}_{i=1}a_{i\sigma(i)},$$ where $S_{n}$ is the group of all permutations of $\{1, \dots, n\}$ and $m(\sigma)$ denotes the number of cycles in $\sigma$. In particular, $\operatorname{per}_{1}A$ is the usual permanent of $A$, while $\operatorname{per}_{-1}A$ is the usual determinant of $A$. Analogously to [@ST subsec. 6.4], we conclude from , and that the point process $\mu^{(l)}$ has correlation functions $$\label{50}
k_{\mu^{(l)}}^{(n)}(x_{1}, \dots,
x_{n})=\operatorname{per}_{\frac{l}{2}}(lk(x_{i},x_{j}))^{n}_{i,j=1}\quad
\text{for $\nu^{\otimes n}$-a.a.\ $(x_{1}, \dots, x_{n})\in X^{n}$.}$$ For $l=2$, the point process $\mu^{(2)}$ is often called a boson point process, see e.g. [@DVJ; @LM]. Thus, we have proved the following
\[as\] For each $l\in \mathbb{N}$, there exists a point process $\mu^{(l)}$ in $X$ whose correlation functions are given by . The $\mu^{(l)}$ satisfies condition $(\Sigma'_{\nu})$ and its Papangelou intensity is given by .
Recall that in [@ST], under the same assumptions on the operator $K$, the existence of a point process with correlation functions was proved for even $l\in \mathbb{N}$, and for odd $l\in
\mathbb{N}$ the statement of Proposition \[as\] was proved under the additional assumption of continuity of the integral kernel $k(\cdot,\cdot)$.
We will now prove that, for a point process $\mu^{(l)}$ as in Proposition \[as\], Glauber and Kawasaki dynamics with coefficients , and , respectively exist.
\[asdfg\] (i) For each point process $\mu^{(l)}$ as in Proposition \[as\], the coefficients $d(x,\gamma)$ and $b(x,\gamma)$ defined by and , satisfy conditions and and so statements (i) and (ii) of Theorem \[asd\] hold, in particular, a corresponding Glauber dynamics exists.
\(ii) Assume additionally that $k(x,x)$ is bounded outside a set $\Delta\in\mathcal{B}_0(X)$. Then for a point process $\mu^{(l)}$ as in Proposition \[as\], the coefficient $c(x,y,\gamma)$ defined by , satisfies , , and , and so statements (i) and (ii) of Theorem \[asd\] hold, in particular, a corresponding Kawasaki dynamics exists.
We start with the following
\[serts\] For each $n\in\mathbb N$ and for $\nu$-a.a. $x\in X$ $$\label{trstr}
\int_\Gamma r(x,\gamma)^n\,\mu(d\gamma)\le\frac{(2n)!}{2^n\,n!}\,k(x,x)^n.$$
Using Jensen’s inequality for conditional expectation and the formula for moments of a Gaussian measure (see e.g. [@BK Chapter 2, Section 2, Lemma 2.1]), we have $$\begin{aligned}
\int_\Gamma r(x,\gamma)^n\,\mu(d\gamma)&=
\widetilde{\mathbb E}(\widetilde{\mathbb E}(Y(x)^2\mid\mathcal F)^n)
\le\widetilde{\mathbb E}(\widetilde{\mathbb E}(
Y(x)^{2n}\mid\mathcal F))
\\
&=\widetilde{\mathbb E}(Y(x)^{2n})
\le\frac{(2n)!}{2^n\,n!}\,\|\varkappa(x,\cdot)\|^{2n}_{L^2(X,\nu)}
=\frac{(2n)!}{2^n\,n!}\,k(x,x)^n\end{aligned}$$ for $\nu$-a.a. $x\in X$.
We will only prove statement (ii) of Theorem \[asdfg\], as the proof of statement (i) is similar and simper. Also, for simplicity of notation, we will only consider the case $l=1$ (for $l>1$ the proof being similar). We will also omit the upper index (1) from our notation. By we have, for each $\Lambda\in \mathcal{B}_{0}(X)$, $$\label{55}
\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\int_{X}\gamma(dx)\int_{X}\nu(dy)\,c(x,y,\gamma\setminus
x)(\chi_{\Lambda}(x)+\chi_{\Lambda}(y))
\\
&\quad=\int_{\Gamma}\mu(d\gamma)\int_{X}\nu(dx)\int_{X}\nu(dy)\,r(x,\gamma)c(x,y,\gamma)(\chi_{\Lambda}(x)
+\chi_{\Lambda}(y))
\\
&\quad=\int_{\Gamma}\mu(d\gamma)\int_{X}\nu(dx)\int_{X}\nu(dy)\,a(x,y)r(x,\gamma)^{s}r(y,\gamma)^{s}
\chi_{\{r>0\}}(x,\gamma)
\\
&\quad\times\chi_{\{r>0\}}(y,\gamma)(\chi_{\Lambda}(x)+\chi_{\Lambda}(y))
\\
& \quad \leq\int_{\Gamma}\mu(d\gamma)\int_{X}\nu(dx)\int_{X}\nu(dy)\,a(x,y)r(x,\gamma)^{s}r(y,\gamma)^{s}(\chi_{\Lambda}(x)
+\chi_{\Lambda}(y))
\\
& \quad =2\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)r(x,\gamma)^{s}r(y,\gamma)^{s}
\\
&\quad \leq2\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)(1+r(x,\gamma))(1+r(y,\gamma)).
\end{aligned}$$ By $$\begin{aligned}
\label{56}
\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)<\infty.\end{aligned}$$ Below, $C_{i}, i=1,2,3,\dots$, will denote positive constants whose explicit values are not important for us. We have, by $$\label{57}
\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)r(x,\gamma)
\\
&\quad=\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\,r(x,\gamma)\Big(\int_{X}\nu(dy)\,a(x,y)\Big)
\\
&\quad\leq C_{1}\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)r(x,\gamma)
\\
&\quad=C_{1}\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\gamma(dx)=C_{1}\int_{\Lambda}k(x,x)\,\nu(dx)<\infty.
\end{aligned}$$ Next, by $$\label{58}
\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)r(y,\gamma)
\\
&\quad =\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)\int_{\Gamma}\mu(d\gamma)r(y,\gamma)
\\
&\quad =\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)k(y,y)
\\
&\quad =\int_{\Lambda}\nu(dx)\int_{\Delta}\nu(dy)\,a(x,y)k(y,y)+
\int_{\Lambda}\nu(dx)\int_{\Delta^c}\nu(dy)\,a(x,y)k(y,y)
\\
&\quad\le C_2\int_{\Lambda}\nu(dx)\int_{\Delta}\nu(dy)k(y,y)+C_3\int_{\Lambda}\nu(dx)\int_{\Delta^c}\nu(dy)\,a(x,y)<\infty,
\end{aligned}$$ where we used that the function $a$ is bounded and $k(y,y)$ is bounded on ${\Delta}^{c}$. Analogously, using Lemma \[serts\], we have $$\label{rtser5s}
\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)a(x,y)r(x,\gamma)r(y,\gamma)
\\
&\quad\leq\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)\|r(x,\cdot)\|_{L^{2}(\mu)}\,\|r(y,\cdot)\|_{L^{2}(\mu)}
\\
&\quad\leq C_{4}\int_{\Lambda}\nu(dx)\int_{X}\nu(dy)\,a(x,y)k(x,x)k(y,y)
\\
&\quad\leq
C_{5}\int_{\Lambda}\nu(dx)\,k(x,x)\int_{\Delta}\nu(dy)\,k(y,y)
\\
&\quad+C_{6}\int_{\Lambda}\nu(dx)k(x,x)\int_{{\Delta}^{c}}\nu(dy)\,a(x,y)<\infty.
\end{aligned}$$ Thus, by –, the theorem is proven.
\[sdisd\] (i) Let $s\in\left[\frac12,1\right]$, and let the conditions of Theorem \[asdfg\] (i) be satisfied. Then the coefficients $d(x,\gamma)$ and $b(x,\gamma)$ defined by and , satisfy condition . Thus, $\mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)\subset D(L_{\mathrm G})$, and for each $F\in \mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)$, $L_{\mathrm G}F$ is given by formula .
\(ii) Let $s\in\left[\frac12,1\right]$, and let the conditions of Theorem \[asdfg\] (ii) be satisfied. Further assume that either $$\label{fdygsdet}
\forall\Lambda\in\mathcal B_0(X)\ \exists\Lambda'\in\mathcal B_0(X)\ \forall x\in\Lambda\ \forall y\in(\Lambda')^c:\quad a(x,y)=0,$$ or $$\label{fdre65}
\int_\Delta k(x,x)^2\,\nu(dx)<\infty,$$ where $\Delta$ is as in Theorem \[asdfg\] (ii). Then the coefficient $c(x,y,\gamma)$ defined by , satisfies condition . Thus, $\mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)\subset D(L_{\mathrm K})$, and for each $F\in \mathcal{F}C_{\mathrm b}(C_{0}(X),\Gamma)$, $L_{\mathrm K}F$ is given by formula .
If $X=\mathbb R^d$ and the function $a$ is as in Remark \[tesa4wa\], then condition means that the function $\tilde a$ has a compact support.
We again prove only the part related to Kawasaki dynamics and only in the case $l=1$, omitting the upper index (1) from our notation. We first assume that is satisfied. Since the function $a$ is bounded and satisfies , it suffices to show that, for each $\Lambda\in \mathcal{B}_{0}(X)$, $$\label{asgasdg}
\int_{\Lambda}\gamma(dx)\int_{\Lambda}\nu(dy)r(x,\gamma\setminus
x)^{s-1}r(y,\gamma\setminus x)^{s}\chi_{\{r>0\}}(x,\gamma\setminus
x)\chi_{\{r>0\}}(y,\gamma\setminus x)\in L^{2}(\mu).$$ We note that, for $s\in\left [\frac{1}{2},1\right]$, $2s-1\in [0,1]$. Therefore, by the Cauchy inequality, we have $$\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\Big(\int_{\Lambda}\gamma(dx)\,r(x,\gamma\setminus
x)^{s-1}\chi_{\{r>0\}}(x,\gamma\setminus
x)\nonumber
\\
& \quad \times \int_{\Lambda}\nu(dy)\,r(y,\gamma\setminus
x)^{s}\chi_{\{r>0\}}(y,\gamma\setminus x)\Big)^{2}\nonumber
\\
&\quad\leq\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\gamma(dx)\,r(x,\gamma\setminus
x)^{2(s-1)}\chi_{\{r>0\}}(x,\gamma\setminus
x)\notag\\
&\quad\times\Big(\int_{\Lambda}\nu(dy)\,r(y,\gamma\setminus
x)^{s}\chi_{\{r>0\}}(y,\gamma\setminus x)\Big)^{2} \gamma(\Lambda)\nonumber
\\
&\quad=\int_{\Gamma}\mu(d\gamma)\int_{\Lambda}\nu(dx)\,r(x,\gamma)^{2s-1}\chi_{\{r>0\}}(x,\gamma)\notag
\\
&\quad\times
\Big(\int_{\Lambda}\nu(dy)r(y,\gamma)^{s}\chi_{\{r>0\}}(y,\gamma)\Big)^{2}(\gamma(\Lambda)+1)\nonumber
\\
&\quad\leq
\int_{\Gamma}\mu(d\gamma)\Big(\int_{\Lambda}\nu(dx)(1+r(x,\gamma))\Big)^{3}(\gamma(\Lambda)+1)\nonumber
\\
&\quad\leq\Big(\int_{\Gamma}\mu(d\gamma)\Big(\int_{\Lambda}\nu(dx)(1+r(x,\gamma))\Big)^{6}\Big)^{1/2}
\Big(\int_{\Gamma}\mu(d\gamma)(\gamma(\Lambda)+1)^{2}\Big)^{1/2}.\label{ghsghsdf}\end{aligned}$$
By Lemma \[serts\], we have, for each $n\in\mathbb{N}$, $$\label{hjkhjfd}
\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\Big(\int_{\Lambda}\nu(dx)\,r(x,\gamma)\Big)^{n}
\\
&\quad=\int_{\Lambda}\nu(dx_{1})\dotsm\int_{\Lambda}\nu(dx_{n})\int_{\Gamma}\mu(d\gamma)\,
r(x_1,\gamma)\dotsm
r(x_{n},\gamma)\
\\
&\quad\leq\int_{\Lambda}\nu(dx_{1})\dotsm\int_{\Lambda}\nu(dx_{n})\|r(x_{1},\cdot)\|_{L^{n}(\mu)}\dotsm\|r(x_{n},\cdot)
\|_{L^{n}(\mu)}
\\
&\quad\leq\frac{(2n)!}{2^{n}n!}\Big(\int_{\Lambda}\nu(dx)k(x,x)\Big)^{n}<\infty
\end{aligned}$$ Now, follows from and .
Next, we assume that is satisfied. We fix $\Lambda\in\mathcal{B}_{0}(X)$ and denote $$u(x,y):=a(x,y)(\chi_{\Lambda}(x)+\chi_{\Lambda}(x)).$$ Then, by the Cauchy inequality, $$\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\Big(\int_{X}\gamma(dx)\int_{X}\nu(dy)\,u(x,y)r(x,\gamma\setminus
x)^{s-1}\chi_{\{r>0\}}(x,\gamma\setminus x)
\\
&\quad
\times r(y,\gamma\setminus
x)^{s}\chi_{\{r>0\}}(y,\gamma\setminus x)\Big)^{2}
\\
&\quad\leq
\int_{\Gamma}\mu(d\gamma)\int_{X}\gamma(dx)\int_{X}\nu(dy)\,u(x,y)r(x,\gamma\setminus
x)^{2(s-1)}\chi_{\{r>0\}}(x,\gamma\setminus x)
\\
&\quad\times r(y,\gamma\setminus
x)^{2s} \chi_{\{r>0\}}(y,\gamma\setminus
x)\int_{X}\gamma(dx')\int_{X}\nu(dy')\,u(x',y')
\\
&\quad =\int_{\Gamma}\mu(d\gamma)\int_{X}\nu(dx)\int_{X}\nu(dy)\,
u(x,y)r(x,\gamma)^{2s-1}\chi_{\{r>0\}}(x,\gamma)
\\
&\quad\times r(y,\gamma)^{2s}\chi_{\{r>0\}}
(y,\gamma)\int_{X}(\gamma+\varepsilon_{x})(dx')\int_{X}\nu(dy')\,u(x',y')
\\
&\quad\leq
\int_{\Gamma}\mu(d\gamma)\int_{X}\nu(dx)\int_{X}\nu(dy)\,u(x,y)(1+r(x,\gamma))(1+r(y,\gamma)^{2}
\\
&\quad\times\Big(\int_{X}\gamma(dx')\int_{X}\nu(dy')u(x',y')+\int_{X}\nu(dy')u(x,y')\Big).\end{aligned}$$ By , it suffices to prove that $$\begin{gathered}
\label{aisv}
\int_{\Gamma}\mu(d\gamma)\Big(\int_{X}\nu(dx)\int_{X}\nu(dy)\,u(x,y)(1+r(x,\gamma))
(1+r(y,\gamma)^{2})\Big)^{2}<\infty,\\
\label{dfgsdfh}
\int_{\Gamma}\mu(d\gamma)\Big(\int_{X}\gamma(dx)\int_{X}\nu(dy)\,u(x,y)\Big)^{2}<\infty.\end{gathered}$$ We first to prove . We have, by Proposition \[as\], $$\begin{aligned}
&\int_{\Gamma}\Big(\int_{X}\gamma(dx)\int_{X}\nu(dy)u(x,y)\Big)^{2}
\\
&\quad =\int_{X}\nu(dy)\int_{X}\nu(dy')\int_{\Gamma}\mu(d\gamma)\int_{X}\gamma(dx)\int_{X}\gamma(dx')
\,u(x,y)u(x',y')
\\
&\quad=\int_{X}\nu(dy)\int_{X}\nu(dy')\int_{\Gamma}\mu(d\gamma)\Big(\int_{X}\gamma(dx)\,u(x,y)u(x,y')
\\
&\quad
+\int_{X}\gamma(dx)\int_{X}(\gamma-\varepsilon_{x})(dx')\,u(x,y)u(x',y')\Big)
\\
&\quad=\int_{X}\nu(dy)\int_{X}\nu(dy')\Big(\int_{X}\nu(dx)\,k(x,x)u(x,y)u(x,y')
\\
& \quad
+\int_{X}\nu(dx)\int_{X}\nu(dx')\Big(\frac12\,k(x,x')^{2}+k(x,x)k(x',x')\Big)u(x,y)u(x',y')\Big)
\\
&\quad\leq\int_{X}\nu(dy)\int_{X}\nu(dy')\Big(\int_{X}\nu(dx)\,k(x,x)u(x,y)u(x,y')
\\
&\quad
+\int_{X}\nu(dx)\int_{X}\nu(dx')\,\frac32\,k(x,x)k(x',x')u(x,y)u(x',y')\Big)
\\
&\quad=\int_{X}\nu(dy)\int_{X}\nu(dy')\int_{X}\nu(dx)\,k(x,x)u(x,y)u(x,y')
\\
&\quad
+\frac32\Big(\int_{X}\nu(dy)\int_{X}\nu(dx)\,k(x,x)u(x,y)\Big)^{2}
\\
&\quad\leq\int_{\Delta}\nu(dx)\,k(x,x)\Big(\int_{X}\nu(dy)\,u(x,y)\Big)^{2}
\\
&\quad\text{}+C_{7}\int_{X}\nu(dy)\int_{X}\nu(dy')\int_{X}\nu(dx)\,u(x,y)u(x,y')
\\
&\quad\text{}+\frac32\Big(\int_{\Delta}\nu(dx)\,k(x,x)\int_{X}\nu(dy)\,u(x,y)+C_{7}\int_{X}\nu(dy)
\int_{X}\nu(dx)\,u(x,y)
\Big)^{2}<\infty.\end{aligned}$$ Next, we prove . By Lemma \[serts\] and , we have $$\begin{aligned}
&\int_{\Gamma}\mu(d\gamma)\Big(\int_{X}\nu(dx)\int_{X}\nu(dy)\,u(x,y)(1+r(x,\gamma))(1+r(y,\gamma)^{2})
\Big)^{2}
\\
&\quad=\int_{X}\nu(dx)\int_{X}\nu(dx')\int_{X}\nu(dy)\int_{X}\nu(dy')\,u(x,y)u(x',y')
\\
&\quad\times\int_{\Gamma}\mu(d\gamma)(1+r(x,\gamma))(1+r(x',\gamma))(1+r(y,\gamma)^{2})(1+r(y',\gamma)^{2})
\\
&\quad\leq\int_{X}\nu(dx)\int_{X}\nu(dx')\int_{X}\nu(dy)\int_{X}\nu(dy')\,u(x,y)u(x',y')\left(1+\|r(x,\cdot)
\|_{L^{4}(\mu)}\right)
\\
&\quad\times\left(1+\|r(x',\cdot)\|_{L^{4}(\mu)}\right)\left(1+\|r(y,\cdot)^2\|_{L^{4}(\mu)}\right)
\left(1+\|r(y',\cdot)^2\|_{L^{4}(\mu)}\right)\\
&\quad\leq
C_{8}\Big(\int_{X}\nu(dx)\int_{X}\nu(dy)\,u(x,y)(1+k(x,x))(1+k(y,y)^{2})\Big)^{2}<\infty.\end{aligned}$$ Thus, the theorem is proven.
Diffusion approximation
=======================
From now on, we set $X=\mathbb R^d$, $d\in\mathbb N$, and $\nu$ to be Lebesgue measure. We will show that, under an appropriate scaling, the Dirichlet form of the Kawasaki dynamics converges to a Dirichlet form which identifies a diffusion process on $\Gamma$ having a permanental point process $\mu^{(l)}$ as a symmetrizing measure. The way we scale the Kawasaki dynamics will be similar to the ansatz of [@KKL].
We denote by $\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma)$ the space of all functions of the form where $N\in \mathbb{N}$, $\varphi_{1},\dots, \varphi_{N}\in C^\infty_{0}(\mathbb R^d)$ and $g\in C^\infty_{\mathrm b}(\mathbb{R}^{N})$. Here, $C^\infty_{0}(\mathbb R^d)$ denotes the space of smooth functions on $\mathbb R^d$ with compact support, and $C^\infty_{\mathrm b}(\mathbb{R}^{N})$ denotes the space of all smooth bounded functions on $\mathbb{R}^{N}$ whose all derivatives are bounded. Clearly, $$\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma)\subset \mathcal{F}C_{\mathrm b}(C_{0}(\mathbb R^d),\Gamma),$$ and the set $\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma)$ is a core for the Dirichlet form $(\mathcal E_{\mathrm K}, D(\mathcal E_{\mathrm K}))$.
We fix $s=1/2$. Let us assume that the function $a(x,y)$ is as in Remark \[tesa4wa\]. Thus, the coefficient $c(x,y,\gamma)$ has the form $$\label{sersa} c(x,y,\gamma)=a(x-y)r(x,\gamma)^{-1/2}r(y,\gamma)^{1/2}\chi_{\{r>0\}}(x,\gamma)\chi_{\{r>0\}}(y,\gamma)
.$$ Note that $y-x$ describes the change of the position of a particle which hops from $x$ to $y$. We now scale the function $a$ as follows: for each $\varepsilon>0$, we denote $$\label{drh}a_\varepsilon(x):=\varepsilon^{-d-2}a(x/\varepsilon),\quad x\in\mathbb R^d.$$ The Dirichlet form $(\mathcal E_{\mathrm K}, D(\mathcal E_{\mathrm K}))$ which corresponds to the choice of function $a$ as in will be denoted by $(\mathcal E_{\varepsilon},D(\mathcal E_{\varepsilon}))$.
\[trsres\] Assume that the function $a$ has compact support, and the value $a(x)$ only depends on $|x|$, i.e., $a(x)=\tilde a(|x|)$ for some function $\tilde a:[0,\infty)\to\mathbb R$. Further assume that the function $\varkappa(x,y)$ has the form $\varkappa(x-y)$ for some $\varkappa:\mathbb R^d\to\mathbb C$, and $$\label{tsrs5r}
\lim_{y\to0}\int_{\mathbb R^d} (\varkappa(x)-\varkappa(x+y))^2\,dx=0.$$ For each $l\in\mathbb N$, define a bilinear form $(\mathcal E_0,\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma))$ by $$\label{sawawe} \mathcal E_0(F,G):=c\int_\Gamma \mu^{(l)}(d\gamma)\int_{\mathbb R^d}dx\, r(x,\gamma)
{\langle}\nabla_x F(\gamma\cup x),\nabla_x G(\gamma\cup x){\rangle}.$$ Here $$c:=\frac{1}{2}\int_{\mathbb R^d}a(x)x_1^2\,dx$$ ($x_1$ denoting the first coordinate of $x\in\mathbb R^d$), $\nabla_x$ denotes the gradient in the $x$ variable, and ${\langle}\cdot,\cdot{\rangle}$ stands for the scalar product in $\mathbb R^d$. Then, for any $F,G\in \mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma)$, $$\mathcal E_{\varepsilon}(F,G)\to\mathcal E_0(F,G)\quad \text{as }{\varepsilon}\to0.$$
\[gdctsjh\] Assume that the function $\varkappa$ is differentiable on $\mathbb R^d$. Denote $$K(x,\delta):=\sup_{y\in B(x,\delta)}|\nabla \varkappa(y)|, \quad x\in\mathbb R^d,\quad \delta>0.$$ Here $B(x,\delta)$ denotes the closed ball in $\mathbb R^d$ centered at $x$ and of radius $\delta$. Assume that, for some $\delta>0$, $$\label{sraserw}
K(\cdot,\delta)\in L^2(\mathbb R^d,dx).$$ Then condition is clearly satisfied. Note that condition is slightly stronger than the condition $|\nabla\varkappa|\in L^2(\mathbb R^d,dx)$.
Again we will only present the proof in the case $l=1$, omitting the upper index $(1)$. We start with the following
\[ufytcfdf\] Fix any $\Lambda\in\mathcal B_0(\mathbb R^d)$ and $\alpha\in(0,1]$. Then, under the conditions of Theorem \[trsres\], $$r(x+{\varepsilon}y,\gamma)^\alpha\to r(x,\gamma)^\alpha \quad \text{in}\quad L^2(\Gamma\times\Lambda\times\mathbb R^d,\mu(d\gamma)
\,dx\,dy\, a(y))\quad \text{as}\quad {\varepsilon}\to0.$$
We first prove the statement for $\alpha=1$. Thus, equivalently we have to prove that $$\label{ctdstrmbgy}
r(x+{\varepsilon}y,\gamma)\to r(x,\gamma)\quad \text{in} \quad L^2(\Omega\times\Gamma
\times\Lambda\times\mathbb R^d,\tilde{\mathbb P}(d\omega,d\gamma)\,dx\,dy\, a(y))\quad \text{as}\quad {\varepsilon}\to0.$$ We have, using Jensen’s inequality for conditional expectation, $$\label{xsersaw}
\begin{aligned}
&\int_\Lambda dx\int_{\mathbb R^d}dy\,a(y)\int_{\Omega\times\Gamma}\tilde{\mathbb P}(d\omega,d\gamma)
\,(r(x+{\varepsilon}y)-r(x,\gamma))^2
\\
&\quad =\int_\Lambda dx\int_{\mathbb R^d}dy\,a(y)\int_{\Omega\times\Gamma}\tilde{\mathbb P}(d\omega,d\gamma)
\tilde{\mathbb E}(Y(x+{\varepsilon}y)^2-Y(x)^2\mid\mathcal F)^2
\\
&\quad \le\int_\Lambda dx\int_{\mathbb R^d}dy\,a(y)\int_{\Omega\times\Gamma}\tilde{\mathbb P}(d\omega,d\gamma)
(Y(x+{\varepsilon}y)^2-Y(x)^2)^2
\\
&\quad =\int_\Lambda dx\int_{\mathbb R^d}dy\,a(y)\int_{\Omega}d{\mathbb P}\, (Y(x+{\varepsilon}y)^4+Y(x)^4
-2Y(x+{\varepsilon}y)^2Y(x)^2).
\end{aligned}$$ Using the formula for moments of a Gaussian measure, we have $$\label{dserazaz}
\begin{aligned}
&\int_{\Omega} Y(x+{\varepsilon}y)^4\,d{\mathbb P}
\\
&\quad=
3\Big(\int_{\mathbb R^d}
\varkappa(x+{\varepsilon}y-u)^2\,du\Big)^2
\\
&\quad=3\Big(\int_{\mathbb R^d}
\varkappa(x-u)^2\,du\Big)^2
\\
&\quad=\int_{\Omega} Y(x)^4\,d{\mathbb P}.
\end{aligned}$$ Analogously, using condition and the dominated convergence theorem, we get $$\label{drtdsrfts}
\begin{aligned}
&\int_\Lambda dx\int_{\mathbb R^d}dy\,a(y)\int_{\Omega}d{\mathbb P}\,Y(x+{\varepsilon}y)^2Y(x)^2
\\
&\quad=\int_\Lambda dx\int_{\mathbb R^d}dy\,a(y)\bigg[
\int_{\mathbb R^d}\varkappa(x+{\varepsilon}y-u)^2 \,du\cdot \int_{\mathbb R^d} \varkappa(x-u')^2\,du'
\\
&\quad\text{}+2\Big(\int_{\mathbb R^d} \varkappa(x+{\varepsilon}y-u)\varkappa(x-u)\,du\Big)^2\bigg]
\\
&\quad\to\int_\Lambda dx\int_{\mathbb R^d}dy\,a(y)\int_{\Omega}d{\mathbb P}\, Y(x)^4\quad \text{as}\quad {\varepsilon}\to0.
\end{aligned}$$ By –, statement follows.
To prove the result for $\alpha\in(0,1)$, it is now sufficient to show the following
[*Claim.*]{} Let $(\mathbf A,\mathcal A,m)$ be a measure space and let $m(A)<\infty$. Let $f_{\varepsilon}\in L^2(m)$, $f_{\varepsilon}\ge0$, ${\varepsilon}\in[-1,1]$, and let $f_{\varepsilon}\to f_0$ in $L^2(m)$ as ${\varepsilon}\to0$. Then, for each $\alpha\in(0,1)$, $f_{\varepsilon}^\alpha\to f_0^\alpha$ in $L^2(m)$ as ${\varepsilon}\to0$.
By e.g. [@Bauer Theorems 21.2 and 21.4], $f_{\varepsilon}\to f_0$ in $L^2(m)$ implies that
- $f_{\varepsilon}\to f_0$ in measure;
- $\displaystyle\sup_{{\varepsilon}\in[-1,1]} \int f_{\varepsilon}^2\,dm<\infty$;
- For each $\theta>0$ there exist $h\in L^1(m)$ and $\delta>0$ such that, for all $0<|{\varepsilon}|\le1$ and for each $A\in\mathcal A$ $$\int_ A h\,dm\le \delta \Rightarrow \int _Af_{\varepsilon}^2\,dm\le \theta.$$
Hence, for $\alpha\in(0,1)$, we get
- $f_{\varepsilon}^\alpha\to f_0^\alpha$ in measure;
- $\displaystyle\sup_{{\varepsilon}\in[-1,1]} \int f_{\varepsilon}^{2\alpha}\,dm\le \sup_{{\varepsilon}\in[-1,1]} \int (1+f_{\varepsilon}^2)\,dm<\infty;$
- Let $\theta$, $h$, and $\delta$ be as in (iii). Set $h':=h+\frac{\delta}{\theta}$. Clearly, $h\in L^1(m)$. Assume that, for some $A\in\mathcal A$, $\int_A h'\,dm\le\delta$. Hence $\int_A h\,dm\le\delta$, and therefore $\int_A f_{\varepsilon}^2\,dm\le\delta$ for all $0<|{\varepsilon}|\le1$. Furthermore, we get $\int_A\frac{\delta}{\theta}\,dm\le\delta$, and therefore $m(A)\le \theta$. Now $$\int_A f_{\varepsilon}^{2\alpha}\,dm\le \int_A(1+f_{\varepsilon}^2)\,dm\le2\theta.$$
Applying again [@Bauer Theorems 21.2 and 21.4], we conclude the claim.
Fix any $F \in\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma)$. We have $$\begin{aligned}
&\mathcal E_{\varepsilon}(F,F)\\
& =\frac{1}{2}\int_\Gamma\mu(d\gamma)\int_{\mathbb R^d}dx\int_{\mathbb R^d}dy\,{\varepsilon}^{-d-2}
a((x-y)/{\varepsilon})r(x,\gamma)^{1/2} r(y,\gamma)^{1/2}(F(\gamma\cup x)-F(\gamma\cup y))^2\\
&=\frac{1}{2}\int_\Gamma\mu(d\gamma)\int_{\mathbb R^d}dx\int_{\mathbb R^d}dy\, a(y)r(x+{\varepsilon}y,\gamma)^{1/2}
r(x,\gamma)^{1/2}\left(\frac{F(\gamma\cup\{x+{\varepsilon}y\})-F(\gamma\cup x)}{{\varepsilon}}\right)^2.\end{aligned}$$ Assume that $0<|{\varepsilon}|\le1.$ Noting that the function $F$ is local (i.e., there exists $\Delta\in\mathcal B_0(\mathbb R^d)$ such that $F(\gamma)=F(\gamma_\Delta)$ for all $\gamma\in \Gamma$) and that the function $a$ has a compact support, we conclude that there exists $\Lambda\in\mathcal B_0(\mathbb R^d)$ such that
$$\begin{gathered}
\mathcal E_{\varepsilon}(F,F)\\=\frac{1}{2}\int_\Gamma\mu(d\gamma)\int_{\Lambda}dx\int_{\mathbb R^d}dy\, a(y)r(x+{\varepsilon}y,\gamma)^{1/2}
r(x,\gamma)^{1/2}\left(\frac{F(\gamma\cup\{x+{\varepsilon}y\})-F(\gamma\cup x)}{{\varepsilon}}\right)^2.\label{gsaw4ty}\end{gathered}$$
$$\label{gsaw4ty}
\begin{aligned}
\mathcal E_{\varepsilon}(F,F)
& =\int_\Gamma\mu(d\gamma)\int_{\Lambda}dx\int_{\mathbb R^d}dy\, a(y)r(x+{\varepsilon}y,\gamma)^{1/2}
r(x,\gamma)^{1/2}
\\
& \times \left(\frac{F(\gamma\cup\{x+{\varepsilon}y\})-F(\gamma\cup x)}{{\varepsilon}}\right)^2.
\end{aligned}$$
By the dominated convergence theorem $$\label{dse5ry}r(x,\gamma)^{1/2}\left(\frac{F(\gamma\cup\{x+{\varepsilon}y\})-F(\gamma\cup x)}{{\varepsilon}}\right)^2\to r(x,\gamma)^{1/2}\langle \nabla_x F(\gamma\cup x),y{\rangle}^2$$ in $L^2(\Gamma\times\Lambda\times\mathbb R^d,\mu(d\gamma)\,dx\,dy\, a(y))$ as ${\varepsilon}\to0$. By Lemma \[ufytcfdf\] with $\alpha=1/2$, and $$\label{hdstred}\mathcal E_{\varepsilon}(F,F)\to \frac{1}{2}\int_\Gamma\mu(d\gamma)\int_{\Lambda}dx\int_{\mathbb R^d}dy\, a(y) r(x,\gamma)\langle \nabla_x F(\gamma\cup x),y{\rangle}^2.$$ Since $a(y)=\tilde a(|y|)$, for any $i,j\in\{1,\dots,d\}$, $i\ne j$, we have $$\int_{\mathbb R^d}a(y)y_iy_j\,dy=0$$ and $$c=\frac{1}{2}\int_{\mathbb R^d}a(y)y_i^2\,dy,\quad i=1,\dots,d.$$ Therefore, by , $$\mathcal E_{\varepsilon}(F,F)\to c\int_\Gamma\mu(d\gamma)\int_{\mathbb R^d}dx\, r(x,\gamma)|\nabla_xF(\gamma\cup x)|^2.$$ From here the theorem follows by the polarization identity.
We will now show that the limiting form $(\mathcal E_0,\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),
\Gamma))$ is closable and its closure identifies a diffusion process.
In what follows, we will assume that the conditions of Theorem \[trsres\] are satisfied. We have $$\begin{aligned}
k(x,y)&=\int_{\mathbb R^d}\varkappa(x-u)\varkappa(y-u)\,du
\\
&=\int_{\mathbb R^d}\varkappa(u-y)\varkappa(u-x)\,du
=\int_{\mathbb R^d}\varkappa(u)\varkappa(u+y-x)\,du.\end{aligned}$$ Hence, by , the function $k(x,y)$ is continuous on $(\mathbb R^d)^2$. Thus, by Remark \[cfdxrbv\], $(Y(x))_{x\in X}$ is a Gaussian random field and formula holds for all $(x,y)\in (\mathbb R^d)^2$.
Consider the semimetric $$\label{gdtrdjjjv}
\begin{aligned}
D(x,y):&=\frac12\Big(\int_\Omega(Y(x)-Y(y))^2\,d\mathbb P\Big)^{1/2}
\\
&=\frac12\big(k(x,x)+k(y,y)
-2k(x,y)\big)^{1/2}
\\
&=\Big( \int_{\mathbb R^d}\varkappa(u)\big(\varkappa(u)-\varkappa(u+y-x)\big)
\,du \Big)^{1/2},
\quad x,y\in\mathbb R^d.
\end{aligned}$$ The associated metric entropy $H(D,\delta)$ is defined as $H(D,\delta):=\log N(D,\delta)$, where $N(D,\delta)$ is the minimal number of points in a $\delta$-net in $B(0,1)=\{x\in\mathbb R^d\mid |x|\le1\}$ with respect to the semimetric $D$, i.e., points $x_i$ such that the open balls centered at $x_i$ and of radius $\delta$ (with respect to $D$) cover $B(0,1)$. The expression $$J(D):=\int_0^1 \sqrt{H(D,\delta)}\,d\delta$$ is called the Dudley integral. The following result holds, see e.g. [@Bogachev Corollary 7.1.4] and the references therein.
\[gtsre\] Assume that $J(D)<\infty$. Then the Gaussian random field $(Y(x))_{x\in\mathbb R^d}$ has a continuous modification.
\[cdtrstrh\] Let $\varkappa$ be as in Remark \[gdctsjh\]. Then, by , for any $x,y\in B(0,1)$ $$\begin{aligned}
D(x,y)^{2}&\le\|\varkappa(\cdot)\|_{L^2(\mathbb R^d,dx)}\Big(\int_{\mathbb R^d}(\varkappa(u)-
\varkappa(u+y-x))^2\,du\Big)^{1/2}\\
&\le \|\varkappa(\cdot)\|_{L^2(\mathbb R^d,dx)}\|K(\cdot,2)\|_{L^2(\mathbb R^d,dx)}|y-x|,\end{aligned}$$ where we assumed that $K(\cdot,2)\in L^2(\mathbb{R}^d,dx)$. Then $J(D)<\infty$, see e.g. [@Bogachev Example 7.1.5].
Denote by $\ddot\Gamma$ the space of all multiple configurations in $\mathbb R^d$. Thus, $\ddot\Gamma$ is the set of all Radon $\mathbb Z_+\cup\{+\infty\}$-valued measures on $\mathbb R^d$, In particular, $\Gamma\subset\ddot\Gamma$. Analogously to the case of $\Gamma$, we define the vague topology on $\ddot\Gamma$ and the corresponding Borel $\sigma$-algebra $\mathcal B(\ddot\Gamma)$.
\[dse5duyd\] Let $\varkappa(x,y)$ be of the form $\varkappa(x-y)$ for some $\varkappa\in L^2(\mathbb R^d,dx)$. Let $J(D)<\infty$. Let $l\in \mathbb N$ and $c>0$. Then
\(i) The bilinear form $(\mathcal E_0,\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma))$ defined by is closable on $L^2(\Gamma,\mu^{(l)})$ and its closure will be denoted by $(\mathcal E_0,D(\mathcal E_0))$.
\(ii) There exists a conservative diffusion process $$\begin{aligned}
M^{0}=\left(\Omega^{0},
\mathcal{F}^{0},(\mathcal{F}^{0}_{t})_{t\geq
0},(\Theta_{t}^{0})_{t\geq 0}, (X^{0}(t))_{t\geq 0},
(P^{0}_{\gamma})_{\gamma\in \ddot\Gamma}\right)\end{aligned}$$ on $\ddot\Gamma$ which is properly associated with $(\mathcal{E}_{0}, D(\mathcal{E}_{0}))$. In particular, $M^{0}$ is $\mu^{(l)}$-symmetric and has $\mu^{(l)}$ as invariant measure. In the case $d\ge2$, the set $\ddot\Gamma\setminus\Gamma$ is $\mathcal E^0$-exceptional, so that $\ddot\Gamma$ may be replaced by with $\Gamma$ in the above statement.
We again discuss only the case $l=1$, omitting the upper index $(1)$. By , for any $F,G\in \mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma)$, $$\begin{aligned}
\mathcal E_0(F,G)&=c\int_{\Omega\times\Gamma}\tilde{\mathbb P}(d\omega,d\gamma)\int_{\mathbb R^d}
dx\, \tilde {\mathbb E}(Y(x,\omega)^2\mid\mathcal F) {\langle}\nabla_x F(\gamma\cup x),\nabla_x G(\gamma\cup x){\rangle}\\
&=\int_{\Omega\times\Gamma}\tilde{\mathbb P}(d\omega,d\gamma)\int_{\mathbb R^d}
dx\, Y(x,\omega)^2
\\
&\times {\langle}\nabla_x (F(\gamma\cup x)-F(\gamma)),\nabla_x (G(\gamma\cup x)-G(\gamma)){\rangle}.
\end{aligned}$$ Fix $(\omega,\gamma)\in\Omega\times\Gamma$. Denote $$f(x):=F(\gamma\cup x)-F(\gamma), \quad g(x):=G(\gamma\cup x)-G(\gamma).$$ Clearly, $f,g\in C_0^\infty(\mathbb R^d)$. In view of Theorem \[gtsre\], $Y(x,\omega)^2$ is a continuous function of $x\in\mathbb R^d$. Hence, by [@dSKR Theorem 6.2], the bilinear form $$\mathcal E(f,g):=\int_{\mathbb R^d} {\langle}\nabla f(x),\nabla g(x){\rangle}Y(x,\omega)^2dx,
\quad f,g\in C_0^\infty(\mathbb R^d),$$ is closable on $L^2(\mathbb R^d, |Y(x,\omega)|^2\,dx)$. Now the closability of $(\mathcal E_0,\mathcal{F}C^\infty_{\mathrm b}(C^\infty_{0}(\mathbb R^d),\Gamma))$ on $L^2(\Gamma,\mu^{(l)})$ follows by a straightforward generalization of the proof of [@dSKR Theorem 6.3]. Part (ii) of the theorem can be shown completely analogously to [@MR2; @RS], see also [@KLR].
Heuristically, the generator of $(\mathcal E_0,D(\mathcal E_0))$ has the form $$(LF)(\gamma)=\sum_{x\in\gamma}\Big(\Delta_xF(\gamma)+\Big{\langle}\frac{\nabla_xr(x,\gamma
\setminus x)}{r(x,\gamma\setminus x)}\,,\nabla_x F(\gamma)\Big{\rangle}\Big).$$ Here, for $x\in\gamma$, $\nabla_xF(\gamma):=\nabla_yF(\gamma\setminus x\cup y)\big|_{y=x}$ and analogously $\Delta_x$ is defined. However, we should not expect that $r(x,\gamma)$ is differentiable in $x$.
[*Acknowledgments.*]{} We are grateful to Alexei Daletskii, Dmitri Finkelshtein, Yuri Kondratiev, and Olexandr Kutoviy for many useful discussions. EL acknowledges the financial support of the SFB 701 “Spectral structures and topological methods in mathematics”, Bielefeld University. EL was partially supported by the International Joint Project grant 2008/R2 of the Royal Society and by the PTDC/MAT/67965/2006 grant, University of Madeira.
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abstract: 'In this paper we study the Novikov-Veselov equation and the related modified Novikov-Veselov equation in certain Sobolev spaces. We prove local well-posedness in $H^s ({\mathbb{R}}^2)$ for $s > \frac{1}{2}$ for the Novikov-Veselov equation, and local well-posedness in $H^s ({\mathbb{R}}^2)$ for $s > 1$ for the modified Novikov-Veselov equation. Finally we point out some ill-posedness issues for the Novikov-Veselov equation in the supercritical regime.'
address: 'Department of Mathematics, University of Toronto, Toronto, On, Canada'
author:
- Yannis Angelopoulos
title: 'Well-Posedness And Ill-Posedness Results For The Novikov-Veselov Equation'
---
Introduction
============
The Novikov-Veselov equation was introduced by Novikov and Veselov in [@NV1], [@NV2]. It has the following form:
$$\label{1}
\left\{\begin{aligned}
\partial_t u + \partial^3 u +\bar{\partial}^3 u + NL_1 (u) + NL_2 (u) = 0 \\
u(0,x,y) = \phi(x,y)\
\end{aligned}
\right.$$
and as it is indicated by the arguments in the functions above, it is an equation posed on ${\mathbb{R}}^2$. Moreover by convention, both the initial data $\phi : {\mathbb{R}}^2 \rightarrow {\mathbb{R}}$ and the solution $u : {\mathbb{R}}\times {\mathbb{R}}^2 \rightarrow {\mathbb{R}}$ are taken to be real-valued. The operators $\partial$ and $\bar{\partial}$ are given by the formulas $\partial = \frac{1}{2} (\partial_x - i \partial_y )$, $\bar{\partial} = \frac{1}{2} (\partial_x + i \partial_y )$, and the nonlinear part is given by: $$NL_1 (u) = \frac{3}{4} \partial (u \bar{\partial}^{-1} \partial u), \quad \quad NL_2 (u) = \frac{3}{4} \bar{\partial} (u \partial^{-1} \bar{\partial} u)$$ This equation has the remarkable property of being completely integrable. The form of its linear and its nonlinear parts suggest similarities with Korteweg-de-Vries type equations. To our knowledge, it hasn’t been proven so far that possesses solutions for data in any “reasonable” space (where by “reasonable” here we mean a standard Sobolev space), although many results have been obtained in other directions (see [@P] and the references therein).
There is plenty of literature around this equation through different methods (the inverse scattering method for instance), and in different formulations (at non zero energy, in our case the energy is 0). The interested reader can look at [@K], [@KN1], [@KN2], [@KN3], [@G], [@GM], [@LMS], [@LMSS], [@N] for different results on a variety of problems concerning .
For the Novikov-Veselov equation we use the Fourier restriction norm method of Bourgain (see [@B], and also [@B1] for a result on an equation in dimension 2) in order to prove local well-posedness for initial data in $H^s ({\mathbb{R}}^2)$ where $s> \frac{1}{2}$. The way that this method is implemented here follows closely the lines of the preprint of Molinet and Pilod [@MP], where a similar result is proved for the Zakharov-Kuznetsov equation whose linear part resembles the one of . It is based on a bilinear estimate for high-low frequency interactions (the analogue of (3.16) of [@MP]) and a Strichartz-type estimate given by a harmonic analysis result of Carbery, Kenig and Ziesler [@CKZ].
A closely related equation to is the modified Novikov-Veselov equation (we use here the formulation given by Perry in [@P], see also [@Bo]): $$\label{mnv}
\left\{\begin{aligned}
\partial_t u + \partial^3 u +\bar{\partial}^3 u + mNL_1 (u) + mNL_2 (u) + mNL_3 (u) + mNL_4 (u) = 0 \\
u(0,x,y) = \phi(x,y)\
\end{aligned}
\right.$$ where $$mNL_1 (u) = \frac{3}{4} \partial u \bar{\partial}^{-1} \partial (|u|^2 ), \quad mNL_2 (u) = \frac{3}{4} \bar{\partial} u \partial^{-1} \bar{\partial} (|u|^2 )$$ $$mNL_3 (u) = \frac{3}{4} u \bar{\partial}^{-1} [\partial (\bar{u} \partial u )], \quad mNL_4 (u) = \frac{3}{4} u \partial^{-1} [\bar{\partial} (\bar{u} \bar{\partial} u )]$$
For we use the same techniques to obtain new well-posedness results. Instead of a bilinear estimate, we use two trilinear ones for different frequency interactions and we rely again on Strichartz estimates in order to prove the crucial trilinear $X^{s,b}$ estimate.
It should be noted that for the modified Novikov-Veselov equation, the existence of global solutions was proven by Perry in [@P] for data in the space $$H^{2,1} ({\mathbb{R}}^2) \cap L^1 ({\mathbb{R}}^2) \mbox{ where }$$ $$H^{m,n} ({\mathbb{R}}^2) = \{ u \in L^2 ({\mathbb{R}}^2) | (I-\Delta)^{m/2} u, (1 + |\cdot|)^n u(\cdot) \in L^2 ({\mathbb{R}}^2) \}$$
Finally we prove that for data in $\dot{H}^{s} ({\mathbb{R}}^2)$ where $s < -1$ it is impossible to prove the existence of solutions using the fixed-point method, no matter which subspace of $\dot{H}^{s} ({\mathbb{R}}^2)$ you choose. Ill-posedness results of this type were first demonstrated by Bourgain (see [@B2] for example). Here we follow the lines of the article of Molinet, Saut and Tzvetkov [@MST1] (see also the nice and short expositions [@MST2] and [@Tz], the first one by the same team of authors for the Benjamin-Ono equation, and the second one by Tzvetkov for the KdV equation – which is in some sense more similar to ours), where a much stronger result was proven for the Kadomtsev-Petviashvili I equation.
#### **Overview of the article.**
In the following section (Section 2) we introduce some notational conventions that will be used throughout the article.
In Section 3, we record the dispersive estimates and the Strichartz estimates that follow from them. We also state the important $H^{-1/4}_{x,y} \rightarrow L^4_{t,x,y}$ Strichartz-type inequality for the linear propagator that was mentioned before, which is derived from a result of Carbery, Kenig and Ziesler.
In Section 4, we prove the bilinear and trilinear estimates (following the related section of the paper of Molinet and Pilod [@MP]) that will be used in the proofs of the main well-posedness results of the article. These results will be proven in sections 5 and 6.
To be more precise, in Section 5 we show the following well-posedness theorem for where the fixed-point arguments takes place in the Bourgain-type $X^{s,b}$ spaces for the Novikov-Veselov equation.
\[wpnv\] The Novikov-Veselov equation is locally well-posed in $H^s ({\mathbb{R}}^2)$ for any $s > \frac{1}{2}$.
And in Section 6 we will prove the analogue for , but with a different regularity threshold.
\[wpmnv\] The modified Novikov-Veselov equation is locally well-posed in $H^s ({\mathbb{R}}^2)$ for any $s > 1$.
In both theorems, by “locally well-posed in $H^s ({\mathbb{R}}^2)$” we mean that there is some time ${\delta}= {\delta}(\| \phi \|_{H^s ({\mathbb{R}}^2)} )$ such that there exists a unique solution to either or satisfying $u(0,x,y) = \phi(x,y)$ and $$u \in C([0,{\delta}]; H^s ({\mathbb{R}}^2)) \cap X^{s,b}_{{\delta}} \mbox{ for some $b > 1/2$}$$ Moreover since we show this by a fixed-point argument, the data-to-solution map is smooth in a neighborhood of the initial data in $H^s ({\mathbb{R}}^2)$, in any time interval of the form $[0,{\delta}']$ for ${\delta}' \in (0,{\delta})$.
It should be noted that from the point of view of scaling (and hence, of criticality) there is plenty of room for improvement in both cases.
Finally in Section 7, we prove a result that is more or less expected, that a fixed-point argument can’t give us a solution in the supercritical regime for , following the exposition of Molinet, Saut and Tzvetkov in [@MST1]. The actual formulation of our result is the following one:
\[ipnv\] Fix any $s,T \in \mathbb{R}$, $s<-1$, $T > 0$. Then there is no continuously embedded subspace $X_T$ of $C([0,T]; \dot{H}^s (\mathbb{R}^2 ))$ where the following inequalities hold: $$\| e^{itNV} \phi \|_{X_T} \lesssim \| \phi \|_{H^s (\mathbb{R}^2 )} \quad \forall \phi \in H^s (\mathbb{R}^2 ), t\in [0,T]$$ $$\left\| \int_0^t e^{i(t-s)NV} [NL_1 (u)(s) + NL_2 (u)(s)] ds \right\|_{X_T} \lesssim \| u \|_{X_T}^2 \quad \forall t\in[0,T]$$
Such estimates would be needed for a fixed-point argument, so in the end we conclude that for data in $\dot{H}^s ({\mathbb{R}}^2)$, there is no proper subspace to run the contraction scheme for the Duhamel formula.
#### **Acknowledgments.**
This work is part of the PhD thesis research of the author at the University of Toronto. The author would like to thank Professors James Colliander and Peter Perry for their encouragement and for many interesting conversations about this work.
The author would also like to thank Daniel Egli and Arick Shao for reading parts of this paper and for the many useful discussions about it.
Notation
========
For a function $f$ we denote by $\hat{f}$ its Fourier transform in space and by $\check{f}$ the inverse Fourier transform, and by $\tilde{f}$ its space-time Fourier transform. We use the symbols $\| \cdot \|_{L^p}$ and $\| \cdot \|_{H^s}$ for the Lebesgue and Sobolev norms of a function, respectively. For two quantities $A$, $B$, we use the relation $A \lesssim B$ to indicate that there is some positive constant $C > 0$ such that $A {\leqslant}CB$, and we use the relation $A \approx B$ to indicate that there is a (possibly different) constant $C > 0$ such that $C^{-1} B {\leqslant}A {\leqslant}CB$.
Moreover we introduce spectral cut-offs in space and in space and time. We define a function $\chi \in \mathcal{S} ({\mathbb{R}})$ such that $supp(\hat{\chi}) \subset [-2,2]$ and $\hat{\chi} = 1$ in $[-1,1]$. First we define $\tilde{\chi}$ by $$\hat{\tilde{\chi}} (\xi) = \hat{\chi} (\xi) - \hat{\chi} (2\xi)$$ Now we define ${\varphi}$ and $\psi$ as: $$\hat{{\varphi}} (\xi,\mu) = \hat{\tilde{\chi}} (|(\xi,\mu)|) \mbox{ and } \hat{\psi}(\tau, \xi,\mu) = \hat{\tilde{\chi}} \left(\tau - \frac{1}{4} \xi^3 + \frac{3}{4} \xi\mu^2 \right)$$ For $k\in \mathbb{N}$, $k{\geqslant}1$ we define further the functions ${\varphi}_k$ and $\psi_k$: $$\hat{{\varphi}}_k (\xi,\mu) = \hat{{\varphi}} \left( \frac{(\xi,\mu)}{2^k} \right) \mbox{ and } \hat{\psi}_k (\tau,\xi,\mu) = \hat{\psi}\left( \frac{(\tau, \xi,\mu)}{2^k} \right)$$ and finally we define also ${\varphi}_0$ and $\psi_0$: $$\hat{{\varphi}}_0(\xi,\mu) = \hat{\chi} (|(\xi,\mu)|) \mbox{ and } \hat{\psi}_0 (\tau,\xi,\mu) = \hat{\chi} \left(\tau -\frac{1}{4} \xi^3 - \frac{3}{4} \xi\mu^2 \right)$$ These cut-offs in frequency form partitions of unity ($\{ \hat{{\varphi}}_k \}_{k=0}^{\infty}$ and $\{ \hat{\psi}_k \}_{k=0}^{\infty}$) and give rise to the following Littlewood-Paley operators: $$\widehat{P_k f} (\xi,\mu) = \hat{{\varphi}}_k (\xi,\mu) \hat{f} (\xi,\mu), \quad \quad \widetilde{Q_k u} (\tau,\xi,\mu) = \hat{\psi}_k (\tau,\xi,\mu) \tilde{u} (\tau,\xi,\mu) \mbox{ for $k{\geqslant}0$}$$
Dispersive and Strichartz Estimates
===================================
A computation shows that $$\partial^3 + \bar{\partial}^3 = \frac{1}{4} \partial_{xxx}^3 - \frac{3}{4} \partial_{xyy}^3$$ which turns into the following equation: $$\label{2}
\left\{\begin{aligned}
\partial_t u + \frac{1}{4} \partial_{xxx}^3 - \frac{3}{4} \partial_{xyy}^3 + NL_1 (u) + NL_2 (u) = 0 \\
u(0,x) = \phi(x)\
\end{aligned}
\right.$$ Finally we investigate some of the dispersive properties of this equation. We consider first the linear part of the equation, taking the Fourier transform in space. This gives us the following: $$\hat{u} (t, \xi , \mu ) = e^{it \left( \frac{1}{4} \xi^3 -\frac{3}{4} \xi \mu^2 \right)} \hat{\phi} (\xi , \mu)$$ From this formula it becomes clear that we have $$\| e^{itNV} \phi \|_{L^2_x} = \| \phi \|_{L^2_x}$$ where $e^{itNV}$ is the propagator of the linear part of the Novikov-Veselov equation. On the other hand we can define the following measure in $\mathbb{R}^{2+1}$ based on the solution given by the Fourier inversion formula above: $$\int_{\mathbb{R}^{2+1}} F(\xi, \mu, \tau) d\rho(\xi, \mu, \tau) = \int_{\mathbb{R}^2} F(\xi, \mu, P(\xi, \mu)) d\xi$$ where we define $P(\xi, \mu) = \frac{1}{4} \xi^3 - \frac{3}{4} \xi \mu^2$. Computing the Hessian of $P(\xi,\mu)$ (denote it by $HP$) we can see that $detHP(\xi,\mu) = 0 $ only for $\xi = \mu = 0$, so away from 0 the hypersurface defined by this measure has non-vanishing Gaussian curvature. Following the usual proof technique of Strichartz estimates (see for example [@MuSc]) we can prove using the above remark for the measure $\rho $ that for any function $\phi = P_N \phi$ that is frequency localized in $\{ 1/2 {\leqslant}|(\xi, \mu)| {\leqslant}2 \}$ (or in some other dyadic block) we have by the standard stationary phase theorem $$\| e^{itNV} P_N \phi \|_{L^{\infty}_x} \lesssim \dfrac{1}{\langle t \rangle} \| P_N \phi \|_{L^1_x}$$ Scaling considerations for the linear equation show that from a solution $u$ we can consider another solution: $$u_{\lambda} (t,x , y) = u \left( \frac{t}{\lambda^3}, \frac{x}{\lambda}, \frac{y}{\lambda} \right)$$
Using this,the usual $TT^{*}$ argument and the Hardy-Littlewood-Sobolev inequality we finally get an $L^2 \rightarrow L^p_t L^q_x$ estimate for $(p,q)$ satisfying $\frac{3}{p} + \frac{2}{q} = 1$, so that in the end we have for any function $\phi$ that $$\| e^{itNV} \phi \|_{L^p_t L^q_x} \lesssim \| \phi \|_{L^2_x} \mbox{ where } \frac{3}{p} + \frac{2}{q} = 1, \quad 3 < p {\leqslant}\infty, 2 {\leqslant}q < \infty$$ Note that the diagonal Strichartz pair is the $L^5_{t,x}$ one. Also note that just by rescaling we have the following more general estimate: $$\| e^{itNV} \phi \|_{L^p_t L^q_x} \lesssim \| \phi \|_{\dot{H}^{\gamma}_x} \mbox{ where } \frac{3}{p} + \frac{2}{q} = 1 - \gamma, \gamma {\geqslant}0$$ Again, we don’t consider the case of endpoints for any $\gamma$.
Note also that the Strichartz admissibility condition would give us that there is an $\dot{H}^{-1/4}_{x,y} \rightarrow L^4_{t,x,y}$ mapping property for the group $e^{itNV}$. Of course this doesn’t follow directly from the proof of Strichartz estimates (the admissibility condition doesn’t normally hold for negative Sobolev spaces), but it can be established differently using the following result of Carbery, Kenig and Ziesler [@CKZ] – see section 3 of the Molinet-Pilod work [@MP] for the formulation given here.
Let $Q(\xi,\mu)$ be a homogeneous polynomial of degree ${\geqslant}2$, and let $K_Q (\xi, \mu) = detHQ(\xi,\mu)$. Also let $Q(D)$ and $|K_Q (D)|^{1/8}$ be the multipliers associated to $Q(\xi,\mu)$ and $|K_Q (\xi,\mu)|^{1/8}$ respectively. Then for any $f\in L^2 (\mathbb{R}^2 )$ we have that $$\| |K_Q (D)|^{1/8} e^{itQ} f \|_{L^4_{t,x,y}} \lesssim \| f \|_{L^2 (\mathbb{R}^2 )}$$
This result applies for $K(\xi,\mu) = P(\xi,\mu)$. But in this case notice that we have $$detHP(\xi,\mu) = -\frac{3}{4} (\xi^2 + \mu^2)$$ So $K(D)$ is actually $\Delta$ up to a constant. So for we have the following estimate: $$\label{l4}
\| |D|^{1/4} e^{itNV} \phi \|_{L^4_{t,x,y}} \lesssim \| \phi \|_{L^2 (\mathbb{R}^2 )}$$
For the similar computation concerning the Zakharov-Kuznetsov equation, see again section 3 of [@MP].
Multilinear Estimates
=====================
In this section – which follows closely the section on Bilinear Estimates of [@MP] – we prove bilinear and trilinear estimates that will be needed in the proofs of bilinear and trilinear estimates in $X^{s,b}$ spaces. Such estimates can be viewed as refinements of the Strichartz inequalities when the functions involved interact in a specific way with respect to their frequency localizations. Here we follow [@MP], and we write these estimates in an “$X^{s,b}$ manner”. But they can be written also as bilinear and trilinear estimates for properly frequency localized linear solutions $e^{itNV} \cdot$ (by the transference principle for $X^{s,b}$ spaces).
First let us make the remark, that since we are asking for refinements of the Strichartz estimates, we can’t rely entirely on Hölder’s inequality and we have to take into consideration the interactions of the functions involved. In the $X^{s,b}$ formulations of multilinear estimates, this can be understood better by looking at the so called “resonant” function which is defined in the following way: we consider initially the functions $$w(\tau, \xi, \mu) = \tau - \frac{1}{4} \xi^3 + \frac{3}{4} \xi\mu^2, \quad w_1 (\tau_1, \xi_1, \mu_1) = w(\tau_1, \xi_1, \mu_1),$$ $$w_2 (\tau, \tau_1, \xi, \xi_1, \mu, \mu_1) = w(\tau-\tau_1, \xi-\xi_1, \mu-\mu_1)$$ and then we take a certain difference of these three function to arrive at the definition of the “resonant” function $$R(\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1) = w -w_1 - w_2 =$$ $$= \frac{3}{4} \xi_1 \xi (\xi-\xi_1) - \frac{3}{4} \xi_1 (\mu-\mu_1)^2 - \frac{3}{4} (\xi-\xi_1)\mu_1^2 - \frac{3}{2} \xi\mu_1 (\mu-\mu_1)$$ which is related of course to the $Q$ localizations that were introduced in the notational section.
Before moving into the actual estimates let us state (without proof) two basic facts that will be used in the upcoming proofs. The first one is a version of the mean value theorem.
\[ap1\] Let $I, J \subset {\mathbb{R}}$ be two intervals and $f : J \rightarrow {\mathbb{R}}$ be a smooth function. Then the following holds: $$|\{ x \in J | f(x) \in I \}| {\leqslant}\dfrac{|I|}{\inf_{y \in J} |f'(y)|}$$
The second one is another measure theoretic tool and can be found in [@MST].
\[ap2\] In this lemma we use the notation $(\xi, \mu) \in {\mathbb{R}}\times {\mathbb{R}}$, i.e. the “first” axis corresponds to $\xi$ and the “second” one to $\mu$. Let $J \subset {\mathbb{R}}\times {\mathbb{R}}$. Assume also that the projection in $\mu$ is contained in some set $I\subset {\mathbb{R}}$ and that there exists some constant $C > 0$ such that $\forall \mu_0 \in I$ it holds that $$|J \cap \{(\xi, \mu_0)\}| {\leqslant}C$$ Then we have that $$| J | {\leqslant}C|I|$$
Now we can state and prove the bilinear estimate.
\[lem1\] Let $k_f , k_g $ be such that $$k_f {\geqslant}2, k_g {\leqslant}k_f - 2$$ which can be seen as $k_g << k_f$.
Then we have: $$\label{bl}
\| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \lesssim \frac{2^{\frac{k_g}{2}}}{2^{k_f}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}}$$
Let $(\tau_1 , \xi_1 , \mu_1 )$ denote the variables corresponding to $\widetilde{P_{k_f} f}$ and $(\tau-\tau_1 , \xi-\xi_1 , \mu-\mu_1 )$ the ones corresponding to $\widetilde{P_{k_g} g}$. By our assumptions we have that $$|(\xi_1 , \mu_1 )| \approx 2^{k_f} >> 2^{k_g} \approx |(\xi-\xi_1 , \mu-\mu_1 )|$$ By applying successively Plancherel’s identity, Young’s inequality, Cauchy-Schwarz and Plancherel again we have: $$\| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} = \| \widetilde{P_{k_f} Q_{l_f} f} \ast \widetilde{P_{k_g} Q_{l_g} g} \|_{L^2_{\tau,\xi,\mu}} {\leqslant}$$ $${\leqslant}\sup_{\tau,\xi,\mu} |A_{\tau,\xi,\mu}|^{1/2} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}}$$ where the set $A_{\tau,\xi,\mu}$ is defined as $$A_{\tau,\xi\mu} = \left\{ (\tau,\xi,\mu) | |(\xi_1 , \mu_1 )| \approx 2^{k_f}, |(\xi-\xi_1 ,\mu-\mu_1 )| \approx 2^{k_g}, \right.$$ $$\left. |\tau_1 - P_1 | \approx 2^{l_f}, |\tau-\tau_1 - P_2 | \approx 2^{l_g} \right\}$$ where $P_1 (\xi_1 , \mu_1 ) = P(\xi_1 , \mu_1)$ and $P_2 (\xi,\xi_1, \mu,\mu_1) = P(\xi-\xi_1, \mu-\mu_1)$.
Applying the triangle inequality we get: $$|A_{\tau,\xi,\mu}| {\leqslant}\min(2^{l_f}, 2^{l_g}) |B_{\tau,\xi,\mu}|$$ where $$B_{\tau,\xi,\mu} = \{ (\xi_1, \mu_1) | |(\xi_1 , \mu_1 )| \approx 2^{k_f}, |(\xi-\xi_1 ,\mu-\mu_1 )| \approx 2^{k_g}, |\tau + P - R| \lesssim \max(2^{l_f}, 2^{l_g}) \}$$ for $R = R(\xi,\xi_1,\mu,\mu_1)$ the “resonant” function.
Now we consider three different cases by taking into account the interaction between $\xi_1$ and $\mu_1$.\
**Subcase 1: $|\xi_1| >> |\mu_1|$** In this situation we follow step-by-step the proof of estimate (3.16) as it given in pages 8 and 9 of [@MP] (see the remark after the proof for the reasoning).
We apply Theorem \[ap1\] for the set $B_{\tau,\xi,\mu}$ where we fix $\mu_1$ (we call this $B_{\tau,\xi,\mu} (\mu_1 )$) after computing the following derivative: $$| \partial_{\xi_1} R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1) | = \left| \frac{3}{4} (\xi_1^2 - \mu_1^2) - \frac{3}{4}[ (\xi - \xi_1 )^2 - (\mu - \mu_1 )^2] \right|$$ Taking into account that $|\xi_1| >> |\mu_1|$ we have: $$| \partial_{\xi_1} R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1) | \gtrsim 2^{2k_f}$$ This gives us that: $$|B_{\tau,\xi,\mu} (\mu_1 )| {\leqslant}\dfrac{\max (2^{l_f}, 2^{l_g})}{2^{2k_f}}$$ Applying Lemma \[ap2\] we further get that: $$|B_{\tau,\xi,\mu}| {\leqslant}\dfrac{\max (2^{l_f}, 2^{l_g}) 2^{k_g}}{2^{2k_f}}$$ which in the end gives us that $$|A_{\tau,\xi,\mu}| {\leqslant}\dfrac{2^{k_g}}{2^{2k_f}}\max (2^{l_f}, 2^{l_g}) \min (2^{l_f}, 2^{l_g})$$\
**Subcase 2: $|\xi_1| << |\mu_1|$** This reduces to Subcase1 since we have again the same bound $$| \partial_{\xi_1} R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1) | \gtrsim 2^{2k_f}$$\
**Subcase 3: $|\xi_1| \approx |\mu_1|$** Now the argument used in Subcases 1 and 2 can’t work, since we are considering the case where $|\partial_{\xi_1} R|$ is obviously no longer bounded below by $2^{2k_f}$. We compute first the derivative of $R$ with respect to $\mu_1$. $$| \partial_{\mu_1} R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1) | = \left| -\frac{3}{2} (\xi - \xi_1 )\mu_1 + \frac{3}{2} \xi_1 (\mu-\mu_1) - \frac{3}{2} \xi \mu + 3 \xi \mu_1 \right| \Rightarrow$$ $$\label{mu}
\begin{aligned}
\Rightarrow | \partial_{\mu_1} R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1) | = \\ = \left| \frac{3}{2} \xi \mu_1 - \frac{3}{2} \xi (\mu-\mu_1 )- \frac{3}{2} (\xi - \xi_1 )\mu_1 + \frac{3}{2} \xi_1 (\mu-\mu_1) \right|
\end{aligned}$$ Since $|(\xi_1 , \mu_1 )| \approx 2^{k_f}$ and $|\xi_1| \approx |\mu_1|$, this implies that $$\label{xi}
|\xi_1| \approx |\mu_1| \approx 2^{k_f}$$ Moreover since $k_f >> k_g$ and as $|(\xi-\xi_1 , \mu-\mu_1 )| \approx 2^{k_g}$, we have by that $$\label{xi2}
|\xi| \approx |\mu| \approx 2^{k_f}$$ Then, going back to , we observe that by and , the first term is of order $2^{2k_f}$ while everything else is just of order $2^{k_g}$, so we conclude in this situation that we have the following estimate: $$| \partial_{\mu_1} R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1) | \gtrsim 2^{2k_f}$$ We prove now the required estimate by following again the proof given in [@MP], just by interchanging the roles of $\xi_1$ and $\mu_1$. To be a bit more specific, in this situation we fix $\xi_1$ in $B$, i.e. we consider the set $B_{\tau,\xi,\mu}(\xi_1)$ and repeat the analysis of Subcase 1.
Note that in Subcases 1 and 2 of Lemma \[lem1\] the estimate is identical to the similar situation in the Zakharov-Kuznetsov equation. The partial derivative of the “resonant” function with respect to $\xi_1$ in that case has the form: $$\left| \frac{3}{4} (\xi_1^2 + \mu_1^2) - \frac{3}{4}[ (\xi - \xi_1 )^2 + (\mu - \mu_1 )^2] \right|$$ and one can see that this gives us the desired bound for frequencies of different sizes.
Now we turn to the trilinear estimates that we will need for the modified Novikov-Veselov equation . Again we consider only specific frequency interactions. These trilinear estimates are actually based on the bilinear estimate (Lemma \[lem1\]) that we just showed.
\[lem2\] 1) Consider dyadic numbers $k_f, k_g, k_h$ with the property $$k_f {\geqslant}2, k_g {\geqslant}k_f + 2, k_f - 1 {\leqslant}k_h {\leqslant}k_f + 1$$ Then we have the estimate $$\| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim \dfrac{2^{\frac{3k_f}{2}}}{2^{k_g}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} 2^{\frac{l_h}{2}} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}}$$ 2) Consider dyadic numbers $k_f, k_g, k_h$ with the property $$k_f {\geqslant}2, k_g {\leqslant}k_f - 2, k_f - 1 {\leqslant}k_h {\leqslant}k_f + 1$$ Then we have the estimate $$\| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim 2^{\frac{k_g}{2}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} 2^{\frac{l_h}{2}} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}}$$
1\) Our condition on the frequencies tells us that $f$ and $h$ are localized at roughly the same level, while $g$ is localized at a bigger one. We don’t assume any condition for the localizations $l_f, l_g$ and $l_h$.
First we apply Plancherel and we get $$\| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} = \| \widetilde{P_{k_f} Q_{l_f} f} \ast \widetilde{P_{k_g} Q_{l_g} g} \ast \widetilde{P_{k_h} Q_{l_h} h} \|_{L^2_{\tau,\xi,\mu}}$$ Now we apply Young’s inequality (and Plancherel for $P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g$) and we have as in Lemma \[lem1\] that the above quantity is bounded by the following $$\lesssim \sup_{\tau,\xi,\mu} |C_{\tau,\xi,\mu}|^{1/2} \| \widetilde{P_{k_h} Q_{l_h} h} \|_{L^2_{\tau,\xi,\mu}} \| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g\|_{L^2_{t,x,y}}$$ where $$C_{\tau,\xi,\mu} = \left\{ (\tau,\xi,\mu) | |(\xi_1 , \mu_1 )| \approx 2^{k_h}, |(\xi-\xi_1 ,\mu-\mu_1 )| \approx 2^{k_f} + 2^{k_g}, \right.$$ $$\left. |\tau_1 - P_1 | \approx 2^{l_h}, |\tau-\tau_1 - P_2 | \approx 2^{l_f} + 2^{l_g} \right\}$$ This is a similar definition to the one of $A_{\tau,\xi,\mu}$ in Lemma \[lem1\]. But just note here that the variables $(\xi-\xi_1,\mu-\mu_1)$ correspond to the convolution $\widetilde{P_{k_f} Q_{l_f} f} \ast \widetilde{P_{k_g} Q_{l_g} g}$ and that’s where the sizes of $|(\xi-\xi_1,\mu-\mu_1)|$ and $|\tau - \tau_1 - P_2 |$ come from.
At this point we apply Lemma \[lem1\] to the second term and Plancherel to the first: $$\sup_{\tau,\xi,\mu} |C_{\tau,\xi,\mu}|^{1/2} \frac{2^{\frac{k_f}{2}}}{2^{k_g}} 2^{\frac{l_f}{2}} 2^{\frac{l_h}{2}} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}}$$ We use a crude bound for the set $C_{\tau,\xi,\mu}$ and we have $$|C_{\tau,\xi,\mu}|^{1/2} \lesssim \min(2^{\frac{l_h}{2}}, \max(2^{\frac{k_f}{2}}, 2^{\frac{k_f}{2}})) 2^{k_h}$$ which in the end gives us the desired inequality $$\| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim \dfrac{2^{\frac{3k_f}{2}}}{2^{k_g}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} \min(2^{\frac{l_h}{2}}, \max(2^{\frac{k_f}{2}}, 2^{\frac{k_f}{2}})) \times$$ $$\times \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim \dfrac{2^{\frac{3k_f}{2}}}{2^{k_g}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} 2^{\frac{l_h}{2}} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}}$$
The last inequality is just a crude bound, if $l_h$ is smaller than $l_f$ or $l_g$ we just have equality, if $l_h$ is the largest of all three then we have actually a better estimate, but we don’t really need it for our purposes.
2\) In this situation $f$ and $h$ are frequency localized at roughly the same level, while $g$ is frequency localized at a smaller one.
For the proof of the estimate we follow the exact same method as in part 1 of this lemma, and again we will consider one case for the space-time localizations, namely that $l_h {\leqslant}l_f, l_g$. We estimate $C_{\tau,\xi,\mu}$ in the same way, but now we take into consideration that $$\max(k_f, k_g, k_h) = k_f \mbox{ or } k_h \mbox{ (and $k_f \approx k_h$)}$$ which means that $$|C_{\tau,\xi,\mu}|^{1/2} \| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim \dfrac{2^{\frac{k_g}{2}} 2^{k_h}}{2^{k_f}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} \min(2^{\frac{l_h}{2}}, \max(2^{\frac{k_f}{2}}, 2^{\frac{k_f}{2}})) \times$$ $$\times \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} \approx$$ $$\approx 2^{\frac{k_g}{2}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} \min(2^{\frac{l_h}{2}}, \max(2^{\frac{k_f}{2}}, 2^{\frac{k_f}{2}})) \times$$ $$\times \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim 2^{\frac{k_g}{2}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}} 2^{\frac{l_h}{2}} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}}$$ The last inequality follows as in 1).
Well-Posedness for the Novikov-Veselov equation
===============================================
In this section our goal is to prove Theorem \[wpnv\]. We introduce first the Bourgain-type spaces where the contraction scheme will be implemented. The $X^{s,b}$ space associated to the Novikov-Veselov equation is the following: $$X^{s,b} = \left\{ u \in L^2_{t,x} | \langle (\xi, \mu) \rangle^s \left\langle \tau - \frac{1}{4}\xi^3 + \frac{3}{4}\xi \mu^2 \right\rangle^b \tilde{u} (\tau, \xi, \mu) \in L^2_{\tau, \xi, \mu} \right\} \mbox{ with norm }$$ $$\| u \|_{X^{s,b}} = \left\| \langle (\xi, \mu) \rangle^s \left\langle \tau - \frac{1}{4} \xi^3 + \frac{3}{4} \xi \mu^2 \right\rangle^b \tilde{u} (\tau, \xi, \mu) \right\|_{L^2_{\tau, \xi, \mu}}$$
Bilinear Estimates
------------------
In order to be able to apply Bourgain’s machinery (see for example [@B], [@T]), we will need the following proposition.
\[propbl\] The following inequality holds true: $$\| NL_1 (u,v) \|_{X^{s,-1/2+2{\varepsilon}}} \lesssim \| u \|_{X^{s,1/2+{\varepsilon}}} \| v \|_{X^{s,1/2+{\varepsilon}}}$$ for any $s > \frac{1}{2}$ and any ${\varepsilon}> 0$.
The proof of this proposition applies in the same way for $NL_2 (u,v) = \frac{3}{4} \bar{\partial} (u \partial^{-1} \bar{\partial} u)$. This shows that for the nonlinear part of the Novikov-Veselov equations, the following bilinear estimate holds: $$\label{nl}
\| NL(u) \|_{X^{s,-1/2+2{\varepsilon}}} = \| NL_1 (u,u) + NL_2 (u,u) \|_{X^{s,-1/2+2{\varepsilon}}} \lesssim \| u \|_{X^{s,1/2+{\varepsilon}}}^2$$
We will use the functions $w, w_1, w_2$ that were defined in the previous section.
A standard duality argument transforms the estimate as shown below: $$\| NL_1 (u,v) \|_{X^{s,-1/2+2{\varepsilon}}} \lesssim \| u \|_{X^{s,1/2+{\varepsilon}}} \| v \|_{X^{s,1/2+{\varepsilon}}} \Rightarrow$$ $$\Rightarrow \sup_{\| h' \|_{X^{-s, 1/2-2{\varepsilon}}} =1} \left| \int_{\mathbb{R}^{2+1}} h' (t,x,y) NL(u,v) (t,x,y) dt dx dy \right| \lesssim$$ $$\lesssim \| u \|_{X^{s,1/2+{\varepsilon}}} \| v \|_{X^{s,1/2+{\varepsilon}}}$$ We want to eliminate the $X^{s,b}$ norms and be left only with $L^2$ ones. First we apply Plancherel to the left-hand side and we rewrite the right-hand side: $$\sup_{\| \tilde{h} \|_{X^{-s, 1/2-2{\varepsilon}}} =1} \left| \int_{\mathbb{R}^{2+1}} \tilde{h'} (\tau, \xi, \mu) \widetilde{NL}(u,v) (\tau, \xi, \mu) d\tau d\xi d\mu \right| \lesssim$$ $$\lesssim \| \langle (\xi , \mu) \rangle^s \langle w \rangle^{1/2+{\varepsilon}} \tilde{u} \|_{L^2_{\tau, \xi,\mu}} \| \langle (\xi , \mu) \rangle^s \langle w \rangle^{1/2+{\varepsilon}} \tilde{v} \|_{L^2_{\tau, \xi,\mu}}$$
We make the following definitions: $$\tilde{f} (\tau_1,\xi_1,\mu_1) = |\tilde{u} (\tau_1,\xi_1,\mu_1) \langle w_1 \rangle^{1/2+{\varepsilon}} \langle(\xi_1,\mu_1)\rangle^s |,$$ $$\tilde{g} (\tau-\tau_1, \xi-\xi_1, \mu-\mu_1) = |\tilde{v}(\tau-\tau_1, \xi-\xi_1, \mu-\mu_1) \langle w_2 \rangle^{1/2+{\varepsilon}} \langle(\xi-\xi_1, \mu-\mu_1) \rangle^s$$ $$\tilde{h} (\tau, \xi, \mu) = |\tilde{h'} (\tau,\xi,\mu) \langle w \rangle^{1/2-2{\varepsilon}} \langle(\xi,\mu) \rangle^{-s}|$$
Computing the convolutions coming from $\widetilde{NL}(u,v)$ we restate our estimate once more: $$I = \int_{\mathbb{R}^6} K \tilde{f} (\tau_1,\xi_1,\mu_1) \tilde{g} (\tau-\tau_1, \xi-\xi_1, \mu-\mu_1) \tilde{h} (\tau,\xi,\mu) d\tau d\tau_1 d\xi d\xi_1 d\mu d\mu_1 \lesssim$$ $$\lesssim \| f \|_{L^2_{\tau,\xi,\mu}} \| g \|_{L^2_{\tau,\xi,\mu}} \| h \|_{L^2_{\tau,\xi,\mu}}$$ where the function $K$ is defined as $$K(\tau,\tau_1,\xi,\xi_1,\mu.\mu_1) =$$ $$\dfrac{|i\xi+\mu||i(\xi-\xi_1) + (\mu-\mu_1)|\langle (\xi,\mu) \rangle^s}{|-i(\xi-\xi_1) + (\mu-\mu_1)| \langle w \rangle^{1/2-{\varepsilon}} \langle (\xi_1,\mu_1) \rangle^s \langle w_1 \rangle^{1/2+{\varepsilon}} \langle (\xi-\xi_1, \mu-\mu_1 ) \rangle^s \langle w_2 \rangle^{1/2+{\varepsilon}} } =$$ $$= \dfrac{|(\xi,\mu)|\langle (\xi,\mu) \rangle^s}{\langle w \rangle^{1/2-{\varepsilon}} \langle (\xi_1,\mu_1) \rangle^s \langle w_1 \rangle^{1/2+{\varepsilon}} \langle (\xi-\xi_1, \mu-\mu_1 ) \rangle^s \langle w_2 \rangle^{1/2+{\varepsilon}} }$$ In the rest of the proof, we’ll localize $f, g, h$ in certain frequencies (i.e. restrict the range of $(\xi,\mu), (\xi_1 , \mu_1)$ and $(\xi-\xi_1 , \mu-\mu_1)$). When this happens, we’ll call the frequencies corresponding to $\tilde{f}$ by $k_f$, and we use the same notation for $\tilde{g}, \tilde{h}$. Then $I$ restricted to these frequencies will be called $I_{k_f , k_g , k_h}$, specifically we’ll have: $$I_{k_f , k_g , k_h} = \int_{{\mathbb{R}}^6} K_{k_f , k_g , k_h} \widetilde{f} (\tau_1,\xi_1,\mu_1) \widetilde{g} (\tau-\tau_1, \xi-\xi_1, \mu-\mu_1) \widetilde{h} (\tau,\xi,\mu) d\tau d\tau_1 d\xi d\xi_1 d\mu d\mu_1$$
We’ll break the proof into several cases dealing with the interactions between different frequencies (with respect to space and not time).
#### **Case 1: Low-Low-Low Frequencies**
First we consider the case where $\tilde{f}, \tilde{g}, \tilde{h}$ have their $(\xi, \mu)$ supports in approximately the same region, a dyadic shell of size $2^k$ for $k$ small. We consider the case where $$k_f , k_g , k_h {\leqslant}1$$ This case is trivial for $s {\geqslant}1$ since in such a situation we have that $K \lesssim 1$ and we use Cauchy-Schwarz to throw $h$ in $L^2$ and then use the $X^{s,b}$ version of which reads as: $$\label{x4}
\left\| \mathcal{F}^{-1}_{t,x,y} \left( \frac{|\xi|^{1/4} \tilde{f}}{\langle w \rangle^{1/2+{\varepsilon}}} \right) \right\|_{L^4_{t,x,y}} \lesssim \| f \|_{L^2_{t,x,y}}$$ for $f$ and $g$. Notice that this gives us a gain of $1/4$ of derivative, so the range of $s$ can be improved. In more detail (and using that $\langle w \rangle^{1/2+{\varepsilon}} \gtrsim 1$ and similarly for $w_1$, $w_2$) we have: $$I_{LLL} = \sum_{k_f , k_g , k_h {\leqslant}1} I_{k_f , k_g , k_h } \lesssim$$ $$\lesssim \sum_{k_f , k_g , k_h {\leqslant}1} \dfrac{2^{k_h (s+1)}}{2^{k_f s} 2^{k_g s}} \| P_{k_h} h \|_{L^2_{t,x,y}} \times$$ $$\times \frac{2^{\frac{k_f}{4}}}{2^{\frac{k_f}{4}}} \left\| \mathcal{F}^{-1}_{t,x,y}\left( \frac{\tilde{f}}{\langle w \rangle^{1/2+{\varepsilon}}} \right) \right\|_{L^4_{t,x,y}} \frac{2^{\frac{k_f}{4}}}{2^{\frac{k_f}{4}}}\left\| \mathcal{F}^{-1}_{t,x,y}\left( \frac{\tilde{f}}{\langle w \rangle^{1/2+{\varepsilon}}} \right) \right\|_{L^4_{t,x,y}} \lesssim$$ $$\lesssim \sum_{k_f , k_g , k_h {\leqslant}1} \dfrac{2^{k_h (s+1)}}{2^{k_f (s+1/4)} 2^{k_g (s+1/4)}} \| P_{k_h} h \|_{L^2_{t,x,y}} \| P_{k_f} f \|_{L^2_{t,x,y}} \| P_{k_g} g \|_{L^2_{t,x,y}}$$ Now since $k_f \approx k_g \approx k_h$, we’ll be able to apply Cauchy-Schwarz in all the frequencies for $$s+1 - 2s -\frac{1}{2} < 0 \Rightarrow s > \frac{1}{2}$$ as follows for some ${\varepsilon}' > 0$: $$\sum_{k_f \approx k_g \approx k_h} 2^{-{\varepsilon}' k_g} \| P_{k_h} h \|_{L^2_{t,x,y}} \| P_{k_f} f \|_{L^2_{t,x,y}} \| P_{k_g} g \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim \left(\sum_{k_g} \| P_{k_g} g \|_{L^2_{t,x,y}}^2 \right)^{1/2} \sum_{k_f \approx k_h} \| P_{k_h} h \|_{L^2_{t,x,y}} \| P_{k_f} f \|_{L^2_{t,x,y}} \left( \sum_{k_g} 2^{-2{\varepsilon}' k_g} \right)^{1/2} \lesssim$$ $$\lesssim \| g \|_{L^2_{t,x,y}} \sum_k \| P_k h \|_{L^2_{t,x,y}} \| P_k f \|_{L^2_{t,x,y}} \lesssim$$ $$\lesssim \| g \|_{L^2_{t,x,y}} \left( \sum_k \| P_k f \|_{L^2_{t,x,y}}^2 \right)^{1/2} \left( \sum_k \| P_k h \|_{L^2_{t,x,y}}^2 \right)^{1/2} \lesssim$$ $$\lesssim \| g \|_{L^2_{t,x,y}} \| f \|_{L^2_{t,x,y}} \| h \|_{L^2_{t,x,y}}$$
#### **Case 2: High-High-High Frequencies**
Again all three frequencies are comparable. But now we consider the following set: $$\{ k_f , k_g , k_h | k_f , k_g {\geqslant}2, k_g - 1 {\leqslant}k_f {\leqslant}k_g + 1, k_f - 1 {\leqslant}k_h {\leqslant}k_f + 1, k_g - 1 {\leqslant}k_h {\leqslant}k_g + 1 \}$$ which we call $J$.
We note that for each triplet $(k_f , k_g , k_h )$ we can follow the exact same proof as in Case1. The fact that all three frequencies are roughly the same allows us to add them in the same way, so again for $s > \frac{1}{2}$ (and just $s meg 1$ as stated in the theorem) we have the desired bound for $$I_{HHH} = \sum_{J} I_{k_f , k_g , k_h }$$
#### **Case 3: High-High-Low Frequencies**
We consider the interaction between high frequencies for $f$ and $g$ and low ones for $h$ (which is possible by the convolution in $I$). The set of frequencies is the following: $$\{ k_f , k_g , k_h | k_f {\geqslant}2, k_h {\leqslant}k_f - 2, k_f - 1 {\leqslant}k_g {\leqslant}k_f + 1 \} = J$$ There is a symmetric case where the roles of $k_f$ and $k_g$ are interchanged, but the estimates are the same. The same method as in Cases1 and 2 can be used although the situation is even better since each term $I_{k_f , k_g , k_h }$ has the biggest frequencies in the denominator. Taking $s > \frac{1}{2}$ as before we can prove the desired estimate.
#### **Case 4: High-Low-High Frequencies**
This is the most interesting among all the cases, where we’ll need to use more tools and not just the Cauchy-Schwarz inequality and the Strichartz estimates for . Unlike Case 3, the fact that $h$ is frequency localized at a high level makes the handling of the derivative that is introduced by the nonlinearity problematic (so this case is not symmetric to the previous one). The set of frequencies is the following: $$\{ k_f , k_g , k_h | k_f {\geqslant}2, k_g {\leqslant}k_f - 2, k_f - 1 {\leqslant}k_h {\leqslant}k_f + 1 \} = J$$ By just applying the Cauchy-Schwarz inequality and applying the Strichartz estimates, we can’t eliminate the derivative that was introduced by the nonlinearity. We have to use the functions $w, w_1 , w_2$ to counterbalance the loss of derivative in the numerator.
We will use Lemma \[lem1\] to deal with the $H-L-H$ case as it is done in the bottom half of page 11 of [@MP], we include the argument here for completeness. We first express $I_{HLH}$ (which the form of $I$ for this case) with respect to the required localizations: $$I_{HLH} = \sum_{k_f, k_g, k_h \in J} I_{k_f, k_g, k_h}$$ We further decompose each term in this sum with respect to the $Q_l$ operators: $$I_{k_f, k_g, k_h} = \sum_{l_f, l_g, l_h} I_{k_f, k_g, k_h}^{l_f, l_g, l_h}$$ Applying Cauchy-Schwarz and Plancherel, for each term in the left-hand side we have the following bound taking into consideration the localizations that we are imposing: $$I_{k_f, k_g, k_h}^{l_f, l_g, l_h} \lesssim \frac{2^{k_f}}{2^{s k_g}} 2^{(-1/2 + 2{\varepsilon}) l_h} 2^{(-1/2 - {\varepsilon}) l_f} 2^{(-1/2 - {\varepsilon}) l_g} \times$$ $$\times \| P_{k_f} Q_{l_f} f P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}}$$ Now we apply Lemma \[lem1\] and bound the above quantity by the following: $$\lesssim 2^{-(s-1/2) k_g} 2^{(-1/2 + 2{\varepsilon}) l_h} 2^{-{\varepsilon}l_f} 2^{-{\varepsilon}l_g} \| P_{k_f} Q_{l_f} f \|_{L^2_{t,x,y}} \| P_{k_g} Q_{l_g} g \|_{L^2_{t,x,y}} \| P_{k_h} Q_{l_h} h \|_{L^2_{t,x,y}}$$ We are ready now to sum over all the indices. Clearly all terms for $l_f, l_g, l_h$ are summed easily, and then we apply Cauchy-Schwarz for $k_g$ to get $$I_{HLH} \lesssim \| g \|_{L^2_{t,x,y}} \sum_{k_f, k_h \in J} \| P_{k_f} f \|_{L^2_{t,x,y}} \| P_{k_h} h \|_{L^2_{t,x,y}}$$ which by the fact that $k_f \approx k_h$ is actually $$\| g \|_{L^2_{t,x,y}} \sum_{k_f} \| P_{k_f} f \|_{L^2_{t,x,y}} P_{k_f} h \|_{L^2_{t,x,y}}$$ and by applying Cauchy-Schwarz with respect to $k_f$ we finally have that: $$I_{HLH} \lesssim \| g \|_{L^2_{t,x,y}} \left( \sum_{k_f} \| P_{k_f} f \|_{L^2_{t,x,y}}^2 \right)^{1/2} \left( \sum_{k_f} \| P_{k_f} h \|_{L^2_{t,x,y}}^2 \right)^{1/2} \Rightarrow$$ $$\Rightarrow I_{HLH} \lesssim \| f \|_{L^2_{t,x,y}} \| g \|_{L^2_{t,x,y}} \| h \|_{L^2_{t,x,y}}$$ which is the required estimate.
#### **Case 5: Low-High-High Frequencies**
This last case can be treated in the exact same way as Case 4, with the roles of $k_f$ and $k_g$ interchanged.
The Fixed-Point Argument
------------------------
Proposition \[propbl\] of the previous subsection is the main tool for the fixed-point method in $X^{s,b}$ spaces.
Let us recall some basic facts about these spaces from [@T]. First, we define a variation of them, we call them $X^{s,b}_{{\delta}}$ for some $0 {\leqslant}{\delta}{\leqslant}1$ through the norm: $$\| u \|_{X^{s,b}_{{\delta}}} = \inf_{v(t) = u(t), t\in [0,{\delta}]} \| v \|_{X^{s,b}}$$ We have the following theorem.
\[xsbthm\] The $X^{s,b}_{{\delta}}$ spaces (for some ${\delta}\in (0,1)$) have the following properties: $$1. \quad \| \chi(t) e^{itNV} f \|_{X^{s,b}_{{\delta}}} \lesssim \| f \|_{H^s ({\mathbb{R}}^2)} \mbox{ for any $s,b \in {\mathbb{R}}$}$$ $$2. \quad \left\| \chi(t) \int_0^t e^{i(t-t')NV} F(t') dt' \right\|_{X^{s,b}_{{\delta}}} \lesssim \| F \|_{X^{s,b-1}_{{\delta}}}$$ for any $s \in {\mathbb{R}}$ and $-\frac{1}{2} < b-1 {\leqslant}0$. $$3. \quad \| u \|_{X^{s,b'}_{{\delta}}} \lesssim {\delta}^{b-b'} \| u \|_{X^{s,b}_{{\delta}}} \mbox{ for any $s\in {\mathbb{R}}$ and $-\frac{1}{2} < b' < b < \frac{1}{2} $}$$ where $\chi$ is a $C^{\infty}_0 ({\mathbb{R}})$ function that is equal to 1 in $[-1,1]$.
Now we are ready to give the proof of Theorem \[wpnv\]
We write Duhamel’s formula for : $$u(t,x,y) = \chi(t) e^{itNV} {\varphi}(x,y) + \chi(t) \int_0^t e^{i(t-t')NV} NL(u)(t' , x,y) dt'$$ with a function $\chi$ as before. We evaluate $u$ in the $X^{s, 1/2 + {\varepsilon}}_{{\delta}}$ norm for $s > \frac{1}{2}$, some ${\varepsilon}> 0$, and some $0 < {\delta}< 1$ small (to be chosen later), and we apply successively 1, 2, 3 of Theorem \[xsbthm\] and, finally, the estimate (which is a consequence of Proposition \[propbl\]): $$\| u \|_{X^{s, 1/2 + {\varepsilon}}_{{\delta}}} \lesssim \| \phi \|_{H^s({\mathbb{R}}^2)} + \left\| \chi(t) \int_0^t e^{i(t-t')NV} NL(u)(t' ,x,y) dt' \right\|_{X^{s,1/2+{\varepsilon}}_{{\delta}}} \lesssim$$ $$\lesssim \| \phi \|_{H^s({\mathbb{R}}^2)} + \| NL(u) \|_{X^{s, -1/2 + {\varepsilon}}_{{\delta}}} \lesssim$$ $$\lesssim \| \phi \|_{H^s({\mathbb{R}}^2)} + {\delta}^{{\varepsilon}} \| NL(u) \|_{X^{s, -1/2 + 2{\varepsilon}}_{{\delta}}} \lesssim$$ $$\lesssim \| \phi \|_{H^s({\mathbb{R}}^2)} + {\delta}^{{\varepsilon}} \| u \|_{X^{s, 1/2 + {\varepsilon}}_{{\delta}}}^2$$ Since ${\varepsilon}> 0$, we can apply the fixed-point argument by choosing an appropriate ${\delta}= {\delta}(\| \phi \|_{H^s ({\mathbb{R}}^2)})$ and this finishes the proof of the Theorem.
Well-Posedness for the modified Novikov-Veselov equation
========================================================
We will follow the same lines for the the proof of Theorem \[wpmnv\]. The fixed-point argument will take place in the same $X^{s,b}$ spaces.
Trilinear Estimates
-------------------
In this situation we will need a trilinear estimate in $X^{s,b}$ spaces. It has the following form:
\[proptrl\] Let $s > 1$ and define the quantity $$mNL_1 (u,v,w) = \frac{3}{4} \partial u \bar{\partial}^{-1} \partial (v \bar{w} )$$ Then we have the inequality: $$\label{xsbm}
\| mNL_1 (u,v,w) \|_{X^{s,-1/2+2{\varepsilon}}} \lesssim \| u \|_{X^{s,1/2+{\varepsilon}}} \| v \|_{X^{s,1/2+{\varepsilon}}} \| w \|_{X^{s,1/2+{\varepsilon}}}$$ for any ${\varepsilon}> 0$.
As before, the proof should apply as well to the other nonlinearities (defined analogously) $mNL_{2,3,4} (u,v,w)$, so that we’ll have in the end $$\| mNL_1 (u) + mNL_2 (u) + mNL_3 (u) + mNL_4 (u) \|_{X^{s,-1/2+2{\varepsilon}}} \lesssim \| u \|_{X^{s,1/2+{\varepsilon}}}^3$$
As in the bilinear situation, we’ll break the proof into several cases by taking again Fourier localizations (with respect to the space variables) for the different functions involved.
First we rewrite the estimate in its dual form as before. So is equivalent to $$II = \int_{\mathbb{R}^9} K(\tau,\tau_1,\tau_2,\xi,\xi_1,\xi_2,\mu,\mu_1,\mu_2) \tilde{e} (\tau_1,\xi_1,\mu_1) \tilde{f} (\tau_2-\tau_1, \xi_2-\xi_1, \mu_2-\mu_1) \times$$ $$\times\tilde{g} (\tau-\tau_2,\xi-\xi_2,\mu-\mu_2) \tilde{h}(\tau,\xi,\mu) d\tau d\tau_1 d\tau_2 d\xi d\xi_1 d\xi_2 d\mu d\mu_1 d\mu_2 \lesssim$$ $$\lesssim \| e \|_{L^2_{\tau,\xi,\mu}} \| f \|_{L^2_{\tau,\xi,\mu}} \| g \|_{L^2_{\tau,\xi,\mu}} \| h \|_{L^2_{\tau,\xi,\mu}}$$ where $$K(\tau,\tau_1,\tau_2,\xi,\xi_1,\xi_2,\mu,\mu_1,\mu_2) =$$ $$\dfrac{|(\xi,\mu)|\langle (\xi,\mu) \rangle^s}{\langle w \rangle^{\frac{1}{2}-2{\varepsilon}} \langle (\xi_1,\mu_1) \rangle^s \langle w_1 \rangle^{\frac{1}{2}+{\varepsilon}} \langle (\xi-\xi_1, \mu-\mu_1 ) \rangle^s \langle w_{3} \rangle^{\frac{1}{2}+{\varepsilon}} \langle (\xi_2 - \xi_1, \mu_2 -\mu_1 ) \rangle^s \langle w_{4} \rangle^{\frac{1}{2}+{\varepsilon}} }$$ for $$w_{3} = w(\tau_2 - \tau_1, \xi_2-\xi_1,\mu_2-\mu_1), \quad \quad w_{4} = w(\tau-\tau_2, \xi-\xi_2,\mu-\mu_2)$$ and $\tilde{e}, \tilde{f}, \tilde{g}, \tilde{h}$ defined in an analogous way to the bilinear case.
Again we use $k_e, k_f, k_g, k_h$ to denote the frequencies corresponding to the functions $e,f,g,h$.
#### **Case 1: High-High-High-High and Low-Low-Low-Low Interactions**
These two situations can be treated in the same as for the bilinear estimate. Assume that $$k_e \approx k_f \approx k_g \approx k_h \approx k$$ where $k$ is either ${\leqslant}1$ or $>>1$. We apply Cauchy-Schwarz and the $L^6$ Strichartz estimate (which ends up in a $1/6$ loss of derivative) in its $X^{s,b}$ form: $$\left\| \mathcal{F}^{-1} \left( \dfrac{\widetilde{P_{k_f} f}}{\langle w \rangle^{1/2+{\varepsilon}}} \right) \right\|_{L^6_{t,x,y}} \lesssim 2^{\frac{k_f}{6}} \| f \|_{L^2_{t,x,y}}$$ We have: $$II_{k_e, k_f, k_g, k_h} \lesssim \dfrac{2^{(s+1)k}}{2^{3sk}} \| P_{k_h} h \|_2 \times$$ $$\times \left\| \mathcal{F}^{-1} \left( \dfrac{\widetilde{P_{k_e} e}}{\langle w \rangle^{1/2+{\varepsilon}}} \right) \mathcal{F}^{-1} \left( \dfrac{\widetilde{P_{k_f} f}}{\langle w \rangle^{1/2+{\varepsilon}}} \right) \mathcal{F}^{-1} \left( \dfrac{\widetilde{P_{k_g} g}}{\langle w \rangle^{1/2+{\varepsilon}}} \right) \right\|_2 \lesssim$$ $$\lesssim \dfrac{2^{(s+3/2)k}}{2^{3sk}} \| e \|_2 \| f \|_2 \| g \|_2 \| h \|_2$$ Now we sum all the pieces in the High-High-High-High case and we have: $$II_{HHHH} = \sum_{k >> 1} II_{k_f, k_g, k_h} \sum_{k >> 1} \dfrac{2^{(s+3/2)k}}{2^{3sk}} \| P_{k_e} e \|_2 \| P_{k_f} f \|_2 \| P_{k_g} g \|_2 \| P_{k_h} h \|_2$$ and by assuming that $$s+\frac{3}{2} - 3s < 0 \Rightarrow s > \frac{3}{4}$$ we can apply Cauchy-Schwarz for every piece and conclude that: $$II_{HHHH} \lesssim \| e \|_2 \| f \|_2 \| g \|_2 \| h \|_2$$ Note that in this particular case we have a better bound for $s$ than the one stated in the Theorem.
#### **Case 2: Low-High-High-High and High-Low-High-High Interactions**
We will treat only the case of High-Low-High-High Interactions since the case of Low-High-High-High Interactions is easier. This is because $h$ is localized at a frequency level that is smaller than all other functions, so the derivative coming from the nonlinearity can be easily balanced.
As before, we decompose $II$ with respect to the $Q$ operator as well and we have: $$II_{k_e, k_f, k_g, k_h} = \sum_{l_e, l_f, l_g, l_h} II_{k_e, k_f, k_g, k_h}^{l_e, l_f, l_g, l_h}$$ We assume the following relation for the frequencies: $$k_h \approx k_e \approx k_f >> k_g$$ We apply Cauchy-Schwarz and 2) of Lemma \[lem2\] with $e$ in the role of $h$, and we have that $$II_{k_e, k_f, k_g, k_h}^{l_f, l_g, l_h} \lesssim$$ $$\lesssim \dfrac{2^{(s+1)k_h} 2^{\frac{k_g}{2}}}{2^{s k_e} 2^{s k_f} 2^{s k_g}} \dfrac{2^{\frac{l_e}{2}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}}}{2^{(1/2-2{\varepsilon})l_h} 2^{(1/2+{\varepsilon})l_e} 2^{(1/2+{\varepsilon})l_f} 2^{(1/2+{\varepsilon})l_g}} \times$$ $$\times \| P_{k_e} Q_{l_e} e \|_2 \| P_{k_f} Q_{l_f} f \|_2 \| P_{k_g} Q_{l_g} g \|_2 \| P_{k_h} Q_{l_h} h \|_2$$ By our condition on the frequencies we have that $$\dfrac{2^{(s+1)k_h} 2^{\frac{k_g}{2}}}{2^{s k_e} 2^{s k_f} 2^{s k_g}} \approx 2^{(s+1-2s) k_h} 2^{(1/2-s) k_g}$$ We impose the following condition on $s$: $$s+1-2s < 0 \Rightarrow s > 1$$ And now we can sum first all $l$ frequencies using Cauchy-Schwarz as: $$\sum_{l_e, l_f, l_g, l_h} II_{k_e, k_f, k_g, k_h}^{l_e, l_f, l_g, l_h} \lesssim$$ $$\lesssim \sum_{l_e, l_f, l_g, l_h} 2^{(s+1-2s) k_h} 2^{(1/2-s) k_g} \dfrac{1}{2^{(1/2-2{\varepsilon})l_h} 2^{{\varepsilon}l_e} 2^{{\varepsilon}l_f} 2^{{\varepsilon}l_g}} \times$$ $$\times \| P_{k_e} Q_{l_e} e \|_2 \| P_{k_f} Q_{l_f} f \|_2 \| P_{k_g} Q_{l_g} g \|_2 \| P_{k_h} Q_{l_h} h \|_2 \lesssim$$ $$\lesssim 2^{(s+1-2s) k_h} 2^{(1/2-s) k_g} \| P_{k_e} e \|_2 \| P_{k_f} f \|_2 \| P_{k_g} g \|_2 \| P_{k_h} h \|_2$$ Using our condition on $s$ we can apply Cauchy-Schwarz for the $k$ frequencies as well and finally conclude that: $$II_{HLHH} = \sum_{k_h \approx k_e \approx k_f >> k_g} \sum_{l_e, l_f, l_g, l_h} II_{k_e, k_f, k_g, k_h}^{l_e, l_f, l_g, l_h} \lesssim$$ $$\lesssim \sum_{k_h \approx k_e \approx k_f >> k_g} 2^{(s+1-2s) k_h} 2^{(1/2-s) k_g} \| P_{k_e} e \|_2 \| P_{k_f} f \|_2 \| P_{k_g} g \|_2 \| P_{k_h} h \|_2 \Rightarrow$$ $$\Rightarrow II_{HLHH} \lesssim \| e \|_2 \| f \|_2 \| g \|_2 \| h \|_2$$ There are more situations where the High-Low-High Interactions can be imposed differently on $e,f,g$ but they are all symmetric, so the same proof works.
#### **Case 3: Low-High-Low-High and High-Low-High-Low Interactions**
Again as in Case 2, we won’t deal with the Low-High-Low-High Interactions since they can be treated in the same way as the interactions in Case 2 (they are actually easier since the derivative coming from the nonlinearity can be easily balanced as it is on the “low level”).
For the High-Low-High-Low interactions we consider the following condition on frequencies: $$k_h \approx k_f >> k_e \approx k_g$$ We use again the same decomposition as in Case 2: $$II_{k_e, k_f, k_g, k_h} = \sum_{l_e, l_f, l_g, l_h} II_{k_e, k_f, k_g, k_h}^{l_e, l_f, l_g, l_h}$$ and apply Cauchy-Schwarz for each piece, but now combined with 1) of Lemma \[lem1\]: $$II_{k_e, k_f, k_g, k_h}^{l_e, l_f, l_g, l_h} \lesssim$$ $$\lesssim \dfrac{2^{(s+1)k_h} 2^{\frac{3 k_g}{2}}}{2^{s k_e} 2^{(s+1) k_f} 2^{s k_g}} \dfrac{2^{\frac{l_e}{2}} 2^{\frac{l_f}{2}} 2^{\frac{l_g}{2}}}{2^{(1/2-2{\varepsilon})l_h} 2^{(1/2+{\varepsilon})l_e} 2^{(1/2+{\varepsilon})l_f} 2^{(1/2+{\varepsilon})l_g}} \times$$ $$\times \| P_{k_e} Q_{l_e} e \|_2 \| P_{k_f} Q_{l_f} f \|_2 \| P_{k_g} Q_{l_g} g \|_2 \| P_{k_h} Q_{l_h} h \|_2$$ Using the condition on the $k$ frequencies we have: $$\dfrac{2^{(s+1)k_h} 2^{\frac{3 k_g}{2}}}{2^{s k_e} 2^{(s+1) k_f} 2^{s k_g}} \approx 2^{(3/2-2s) k_g}$$ Imposing the condition $$\frac{3}{2} - 2s < 0 \Rightarrow s > \frac{3}{4}$$ we can apply Cauchy-Schwarz to add up all the $k$ and $l$ frequency pieces as in Case 2 and get again the required estimate: $$II_{HLHL} \lesssim \| e \|_2 \| f \|_2 \| g \|_2 \| h \|_2$$ Note that in this case as well we have a better bound for $s$ than the one stated in the Theorem.
Note that there is no High-Low-Low-Low interactions case. Also note that there are other more complicated cases with interactions on three different levels but they can be treated in the similar ways as the interactions above.
The Fixed-Point Argument
------------------------
Applying Proposition \[proptrl\] on a solution of \[mnv\] given by the Duhamel formula, and using the properties of the $X^{s,b}$ spaces from Theorem \[xsbthm\] we have the estimate: $$\| u \|_{X^{s, 1/2 + {\varepsilon}}_{{\delta}}} \lesssim \| \phi \|_{H^s({\mathbb{R}}^2)} + {\delta}^{{\varepsilon}} \| u \|_{X^{s, 1/2 + {\varepsilon}}_{{\delta}}}^3$$ We can see then from this that the proof of Theorem \[wpmnv\] can be given in the exact same manner as the one of Theorem \[wpnv\], so it won’t be repeated here.
Ill-Posedness Issues for the Novikov-Veselov equation
=====================================================
In this final section we will prove Theorem \[ipnv\]. This will be done by proving a failure of differentiability at the origin of the data-to-solution map of .
\[dtsm\] For any $s < -1$, there is no $T > 0$ such that the data-to-solution map of $$NV(t) : \phi \rightarrow u(t), \quad t\in [0,T]$$ is $C^2$ at 0 as a map from $\dot{H}^s (\mathbb{R}^2 )$ to $\dot{H}^s (\mathbb{R}^2 )$.
Deriving Theorem \[ipnv\] from Theorem \[dtsm\] is a standard fact and won’t be shown here, the interested reader can take a look at the proof of the analogous Theorem 5.2 for the KP-I equation in [@MST1].
Before giving a proof of Theorem \[dtsm\] though, we will record the failure of a bilinear estimate for the (related to the homogeneous $\dot{H}^s$ spaces) $\dot{X}^{s,b}$ spaces that are defined as $$\dot{X}^{s,b} = \left\{ u \in L^2_{t,x} | |(\xi, \mu)|^s \left\langle \tau - \frac{1}{4}\xi^3 + \frac{3}{4}\xi \mu^2 \right\rangle^b \tilde{u} (\tau, \xi, \mu) \in L^2_{\tau, \xi, \mu} \right\} \mbox{ with norm }$$ $$\| u \|_{\dot{X}^{s,b}} = \left\| |(\xi, \mu)|^s \left\langle \tau - \frac{1}{4} \xi^3 + \frac{3}{4} \xi \mu^2 \right\rangle^b \tilde{u} (\tau, \xi, \mu) \right\|_{L^2_{\tau, \xi, \mu}}$$
\[fbl\] The inequality $$\| NL_1 (u,v) \|_{\dot{X}^{s,-b'}} \lesssim \| u \|_{\dot{X}^{s,b}} \| v \|_{\dot{X}^{s,b}}$$ fails for any $b,b' \in {\mathbb{R}}$ and $s < -1$.
The same holds true for $NL_2 (u,v)$ which has as a result the failure of the control in $X^{s,b}$ of the nonlinear part. Note also that this result is natural since scaling considerations indicate that the critical Sobolev space for is $\dot{H}^{-1} ({\mathbb{R}}^2)$. To be a bit more precise a scaled solution for the equation is the following one: $$u_{\lambda} (t,x,y) = \frac{1}{\lambda^2} u \left( \frac{t}{\lambda^3}, \frac{x}{\lambda}, \frac{y}{\lambda} \right)$$ and then we notice that we have: $$\phi_{\lambda} (x,y) = \frac{1}{\lambda^2} \phi \left( \frac{x}{\lambda}, \frac{y}{\lambda} \right) \Rightarrow \| \phi_{\lambda} \|_{\dot{H}^{-1} ({\mathbb{R}}^2)} = \| \phi \|_{\dot{H}^{-1} ({\mathbb{R}}^2)}$$
It should be expected that a similar result is true for the modified Novikov-Veselov equation . In this situation we should have that the data-to-solution map fails to be $C^3$ at the origin with respect to the topology of $\dot{H}^s ({\mathbb{R}}^2)$ for any $s < 0$, as we can see according to scaling considerations. The solution of remains invariant under the following transformation: $$u_{\lambda} (t,x,y) = \frac{1}{\lambda} u \left( \frac{t}{\lambda^3}, \frac{x}{\lambda}, \frac{y}{\lambda} \right)$$ and in this case we have: $$\phi_{\lambda} (x,y) = \frac{1}{\lambda} \phi \left( \frac{x}{\lambda}, \frac{y}{\lambda} \right) \Rightarrow \| \phi_{\lambda} \|_{L^2 ({\mathbb{R}}^2)} = \| \phi \|_{L^2 ({\mathbb{R}}^2)}$$
Proposition \[fbl\] won’t imply directly Theorem \[dtsm\], but the counterexample that will cause the failure of the bilinear $\dot{X}^{s,b}$ estimate will be used to cause the failure of the differentiability at the origin of the data-to-solution map.
We employ again the dual formulation of the $\dot{X}^{s,b}$ estimate. We rewrite it here for the convenience of the reader: $$\int_{\mathbb{R}^6} K_0(\tau,\tau_1,\xi,\xi_1,\mu.\mu_1) \tilde{f} (\tau_1,\xi_1,\mu_1) \tilde{g} (\tau-\tau_1, \xi-\xi_1, \mu-\mu_1) \tilde{h} (\tau,\xi,\mu) d\tau d\tau_1 d\xi d\xi_1 d\mu d\mu_1$$ $$\lesssim \| f \|_{L^2_{\tau,\xi,\mu}} \| g \|_{L^2_{\tau,\xi,\mu}} \| h \|_{L^2_{\tau,\xi,\mu}}$$ where the function $K_0$ is defined as $$K_0(\tau,\tau_1,\xi,\xi_1,\mu.\mu_1) = \dfrac{|(\xi,\mu)||(\xi,\mu)|^s}{\langle w \rangle^{b'} |(\xi_1,\mu_1)|^s \langle w_1 \rangle^{b} |(\xi-\xi_1, \mu-\mu_1 )|^s \langle w_2 \rangle^{b} }$$
We have to note that the definition of $K_0$ is not strictly correct since we would like to bound the quantity defined above by below. The fraction that shows up because of the nonlinearity is the following: $$\dfrac{(i\xi+\mu)(i(\xi-\xi_1) + (\mu-\mu_1))}{i(\xi-\xi_1) - (\mu-\mu_1)}$$ We can make the following changes now: $$i\xi + \mu = |i\xi + \mu| e^{iH(\xi,\mu)}$$ for some function $H$. Moreover by noticing that $$\left| \dfrac{i(\xi-\xi_1) + (\mu-\mu_1)}{i(\xi-\xi_1) - (\mu-\mu_1)} \right| = 1$$ we can write in the end: $$\dfrac{(i\xi+\mu)(i(\xi-\xi_1) + (\mu-\mu_1))}{i(\xi-\xi_1) - (\mu-\mu_1)} = |i\xi + \mu| e^{iH(\xi,\mu)} e^{i\tilde{H}(\xi-\xi_1, \mu-\mu_1)}$$ for some other function $\tilde{H}$. These two new terms are just phase changes that can be absorbed in the definitions of the functions $h$ and $g$ respectively. The final outcome is not affected since they don’t influence in any significant way the $L^2$ norms of these functions. So finally, our definition of $K_0$ is good enough for our purposes.
The functions $f,g,h$ are given as before as follows (note that the absolute values are there as before, but in this case too we can argue as in the remark above): $$\tilde{f} (\tau_1,\xi_1,\mu_1) = |\tilde{u} (\tau_1,\xi_1,\mu_1) \langle w_1 \rangle^{b} |(\xi_1,\mu_1)|^s |,$$ $$\tilde{g} (\tau-\tau_1, \xi-\xi_1, \mu-\mu_1) = |\tilde{v}(\tau-\tau_1, \xi-\xi_1, \mu-\mu_1) \langle w_2 \rangle^{b} |(\xi-\xi_1, \mu-\mu_1)|^s$$ $$\tilde{h} (\tau, \xi, \mu) = |\tilde{h'} (\tau,\xi,\mu) \langle w \rangle^{b'} |(\xi,\mu)|^{-s}|$$
We define now the functions $f,g,h$ that will show the failure of the bilinear estimate.
The function $f,g,h$ that will be used from now are defined as follows $$\widetilde{f}(\tau_1, \xi_1, \mu_1) = \begin{cases} 1 \mbox{ for $\xi_1 \in [-N-c, N], \mu_1 \in [-2c, -c], w_1 \in [0,1]$} \\
0 \mbox{ otherwise}
\end{cases}$$
$$\widetilde{g}(\tau', \xi', \mu' ) = \begin{cases} 1 \mbox{ for $\xi' \in [N+2c, N+3c], \mu' \in [3c, 4c], |w_2| {\leqslant}2 + Cc N^2$} \\
0 \mbox{ otherwise}
\end{cases}$$ where $\tau' = \tau-\tau_1, \xi' = \xi-\xi_1, \mu' = \mu-\mu_1$
$$\widetilde{h}(\tau, \xi,\mu) = \begin{cases} 1 \mbox{ for $\xi \in [c, 3c]. \mu \in [c, 3c], w \in [0,1]$} \\
0 \mbox{ otherwise}
\end{cases}$$ for some $0 < c \approx \frac{1}{N^2} << 1$, i.e. $N >> 1$, and any ${\varepsilon}> 0$.
Note that with this choice of functions, the “resonant” function satisfies the following bound in their supports: $$|R(\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1)| \approx cN^2 \approx const.$$ by the choice of $c$. We can now compute for $(\tau, \xi, \mu) \in supp(\widetilde{h})$: $$\left( \dfrac{\widetilde{f}}{\langle w \rangle^b} \ast \dfrac{\widetilde{g}}{\langle w \rangle^b} \right) (\tau, \xi, \mu) = \int_{supp(\widetilde{f})} \dfrac{1}{\langle w_1 \rangle^b \langle w_2 \rangle^b} d\tau_1 d\xi_1 d\mu_1 =$$ $$= \int_{-N-c}^{-N} \int_{-2c}^{-c} \int_{w_1 \in [0,1]} \dfrac{d\tau_1 d\xi_1 d\mu_1}{\langle w_1 \rangle^b \langle w - w_1 + R\rangle^b} \gtrsim$$ $$\gtrsim\int_{-N-c}^{-N} \int_{-2c}^{-c} \int_{w_1 \in [0,1]} \dfrac{d\tau_1 d\xi_1 d\mu_1}{\langle w_1 \rangle^b \langle 1 + R\rangle^b} \gtrsim$$ $$\gtrsim \int_{-N-c}^{-N} \int_{-2c}^{-c} \dfrac{d\xi_1 d\mu_1}{\langle 1 + R\rangle^b} \gtrsim c^2$$ where the first from the inequalities holds as $|w - w_1| \lesssim 1$, the second one by Fubini and the last one by noticing that since we work in the support of $f$ and $g$, we have that $$R \approx cN^2 \approx 1 \Rightarrow \dfrac{1}{1 + |R|^b} \approx \dfrac{1}{1 + |cN^2 |^b } \approx const.$$
On the other hand we have the following: $$K_0(\tau,\tau_1,\xi,\xi_1,\mu.\mu_1) \gtrsim \dfrac{N^{-4s}}{N^2 \langle w \rangle^{b'} \langle w_1 \rangle^b \langle w_2 \rangle^b} =$$ $$= \dfrac{N^{-4s-2}}{\langle w \rangle^{b'} \langle w_1 \rangle^b \langle w_2 \rangle^b} =$$ So finally using all previous estimates, we note now that the integral of interest is bounded below by: $$c^2 N^{4s-2} \int_{c}^{3c} \int_{c}^{3c} \int_{w\in[0,1]} \dfrac{1}{ \langle w \rangle^{b'}} d\tau d\xi d\mu \gtrsim c^4 N^{-4s-2}$$
By the definitions of $f,g,h$ we can compute directly their $L^2$ norms: $$\| f \|_{L^2} = c, \quad \| g \|_{L^2} = c (2 + Cc N^2 )^{1/2}, \quad \| h \|_{L^2} = c$$ So if the inequality that was initially stated was true, we would have: $$c^4 N^{-4s-2}\lesssim c^3 (2 + Cc N^2 )^{1/2}$$ and since we assume that $cN^2 \approx 1$ we would actually have that: $$N^{-4s-4} \lesssim 1$$ which gives us a contradiction assuming that $$-4s - 4 > 0 \Leftrightarrow s < -1$$ since $N >> 1$.
We are now ready to give the proof of Theorem \[dtsm\].
We consider a parameter ${\varepsilon}> 0$ and the equation $$\label{4}
\left\{\begin{aligned}
\partial_t u + \partial^3 u +\bar{\partial}^3 u + NL_1 (u) + NL_2 (u) = 0 \\
u(0,x) = {\varepsilon}\phi(x)\
\end{aligned}
\right.$$ For a standard computation shows that $$\left.\dfrac{\partial^2 u}{\partial {\varepsilon}}\right|_{{\varepsilon}= 0} (t,x,y) := u_2 (t,x,y) = \int_0^t e^{i(t-s)NV} NL(e^{isNV} \phi , e^{isNV} \phi) ds$$ which is the second derivative of the data-to-solution map $NV$ for evaluated at 0, where $NL = NL_1 + NL_2$. Note that we’ll show the computations only for $NL_1$, the case of $NL_2$ is similar.
If $NV$ was $C^2$ at the origin, then we would have the following inequality for $u_2$: $$\label{5}
\| u_2 (t,x,y) \|_{\dot{H}^s (\mathbb{R}^2 )} \lesssim \| \phi \|^2_{\dot{H}^s (\mathbb{R}^2 )}$$ Hence our goal in order to complete the proof of Theorem \[ipnv\] is to show the failure of . We proceed by making a choice for $\phi$ based on the computations in the proof of Proposition \[fbl\]. Specifically we choose a function such that it consists of a part that is supported $supp(\widetilde{f})$ and another part that is supported in $supp(\widetilde{h})$ (as they were given in the proof of Proposition \[fbl\]).
From now on, the function $\phi$ will have the following form (we define it through its Fourier transform): $$\hat{\phi} (\xi ,\mu) = c^{-1} N^{-s} \chi_1 (\xi,\mu) + c^{-1} N^{-s} \chi_2 (\xi , \mu)$$ $$\mbox{ where $\chi_1$ is the indicator function of $D_1 = [-N-c , -N] \times [-2c, -c]$ }$$ $$\mbox{ and $\chi_2$ of $D_2 = [N+2c, N+3c] \times [3c, 4c] $ }$$
As before we choose $$cN^2 \approx 1$$
In order to use this definition appropriately we present some formulas first. The analogue of Lemma 4, page 376 of [@MST1] reads as follows for the Novikov-Veselov equation (its proof is the same): $$\int_0^t e^{i(t-s)PV} F(s, x, y) ds =$$ $$= const. \int_{\mathbb{R}^3} e^{it(\xi^3 - \xi^2 \mu) + ix\xi + iy\mu} \dfrac{e^{it(\tau - \xi^3 + \xi^2 \mu)} - 1}{\tau - \frac{1}{4}\xi^3 + \frac{3}{4}\xi^2 \mu} \tilde{F} (\tau,\xi,\mu) d\tau d\xi d\mu$$
After some computations we arrive at the following formula: $$u_2 (t,x,y) = const. \int_{\mathbb{R}^4} (i\xi + \mu) e^{it(\xi^3 - \xi^2 \mu) + ix\xi + iy\mu} \dfrac{e^{itR (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1)} - 1}{R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1)} \times$$ $$\quad \quad \quad \quad \quad \times \dfrac{i(\xi - \xi_1 ) + (\mu -\mu_1 )}{i(\xi - \xi_1 ) - (\mu - \mu_1 )} \hat{\phi} (\xi_1 , \mu_1 ) \hat{\phi} (\xi - \xi_1 , \mu - \mu_1 ) d\xi d\mu d\xi_1 d\mu_1$$
By the definition of $\phi$, the formula for $u_2$ can be broken into three parts, according to where $(\xi_1,\mu_1)$ and $(\xi-\xi_1, \mu-\mu_1)$ belong to. In two of these parts $(\xi_1 ,\mu_1)$ and $(\xi-\xi_1, \mu-\mu_1)$ belong to the same set and in the third one they belong to different sets. Denoting the function inside the integral for $u_2$ by $\Phi$ we have that $$u_2 (t,x,y) = I(t,x,y) + II(t,x,y) + III(t,x,y) \mbox{ where }$$ $$I(t,x,y) = \frac{const.}{c^2 N^{2s}} \int_{(\xi_1,\mu_1) \in D_1, (\xi-\xi_1, \mu-\mu_1) \in D_1} \Phi d\xi d\mu d\xi_1 d\mu_1$$ $$II(t,x,y) = \frac{const.}{c^2 N^{2s}} \int_{(\xi_1,\mu_1) \in D_2, (\xi-\xi_1, \mu-\mu_1) \in D_2} \Phi d\xi d\mu d\xi_1 d\mu_1$$ $$III(t,x,y) = \frac{const.}{c^2 N^{2s} } \int_{(\xi_1,\mu_1) \in D_1, (\xi-\xi_1, \mu-\mu_1) \in D_2} \Phi d\xi d\mu d\xi_1 d\mu_1 +$$ $$+ \frac{const.}{cdN^s } \int_{(\xi_1,\mu_1) \in D_2, (\xi-\xi_1, \mu-\mu_1) \in D_1} \Phi d\xi d\mu d\xi_1 d\mu_1$$ First we give upper bounds for $I$. We take absolute values inside and we get rid of the exponential, also we note that for $(\xi_1 ,\mu_1), (\xi-\xi_1, \mu-\mu_1) \in D_1$ we have that: $$|\xi| \approx |\xi-\xi_1| \approx |\xi_1| \approx N, \quad |\mu| \approx |\mu-\mu_1| \approx |\mu_1| \approx c$$ which implies according to our choice of $c$ that $$|R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1)| \approx N^3$$ Now we can compute the following: $$\| I \|_{\dot{H}^s ({\mathbb{R}}^2)} \lesssim \dfrac{N^{s+1} c}{N^{2s} N^3} = N^{-s-4}$$ Similarly we can get the following bound for $II$: $$\| II \|_{\dot{H}^s ({\mathbb{R}}^2)} \lesssim \dfrac{N^{s+1} c}{N^{2s} N^3} = N^{-s-4}$$ We now turn to $III$ and for convenience we break it into two parts, $IV$ and $V$. Taking the Fourier transform of $IV$ as $(x,y) \rightarrow (\xi,\mu)$ we note that $$\mathcal{F} IV (t,\xi,\mu) = \dfrac{const. (i\xi + \mu) e^{it(\xi^3 - \xi^2 \mu)}}{cdN^s} \int_{(\xi_1,\mu_1) \in D_1, (\xi-\xi_1, \mu-\mu_1) \in D_2} \times$$ $$\times \dfrac{e^{itR (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1)} - 1}{R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1)} \dfrac{i(\xi - \xi_1 ) + (\mu -\mu_1 )}{i(\xi - \xi_1 ) - (\mu - \mu_1 )} d\xi_1 d\mu_1$$
For $R$ we use the computations of the previous section to show that for $(\xi_1,\mu_1) \in D_1$ and $(\xi-\xi_1, \mu-\mu_1) \in D_2$ we have the estimate: $$|R (\xi_1, \xi-\xi_1, \mu_1, \mu-\mu_1)| \approx cN^2 \approx 1$$ Using our definition of $c$ we can see that $$\left| \dfrac{e^{itR} - 1}{R} \right| \gtrsim 1$$
Note that we have to deal also with the other fraction inside $\mathcal{F} IV$, but this is not really a problem, since once we take the modulus of it (as we will in order to compute the $\dot{H}^s$ norm of $IV$) we note that this fraction has modulus 1, so it can be seen as a phase change which won’t affect in any significant way the computation of the measure of the set that we are interested in (for more on this, see the related remark in the proof of Proposition \[fbl\]).
So finally we have: $$\| IV \|_{\dot{H}^s (\mathbb{R}^2)} \gtrsim \dfrac{ccc^s}{N^{2s}} \approx N^{-4s-4}$$ The same estimate holds for $V$, so in the end we can state that: $$\| III \|_{\dot{H}^s (\mathbb{R}^2)} \gtrsim \dfrac{ccc^s}{N^{2s}} \approx N^{-4s-4}$$ If was true, then as $\| \phi \|_{\dot{H}^s ({\mathbb{R}}^2)} \approx 1$, we would have that: $$1 \approx \| \phi \|_{\dot{H}^s ({\mathbb{R}}^2)} \gtrsim \| u_2 (t,x,y) \|_{\dot{H}^s (\mathbb{R}^2 )} \gtrsim$$ $$\gtrsim \| III \|_{\dot{H}^s (\mathbb{R}^2)} - \| II \|_{\dot{H}^s (\mathbb{R}^2)} - \| I \|_{\dot{H}^s (\mathbb{R}^2)} \gtrsim N^{-4s-4} - N^{-s-4}$$ $$\Rightarrow N^{-4s-4} \lesssim 1 + N^{-s-4}$$ We use now that $s < -1$. We consider two cases.
**(i) $-4 {\leqslant}s < -1$:** In this case we have that $-s-4 {\leqslant}0$. This implies that $N^{-4s-4} \lesssim 1$ which is a contradiction since $N >> 1$ and $-4s-4 > 0$.
**(ii) $s < -4$:** In this case $-s-4 > 0$ which implies that $$N^{-4s-4} \lesssim N^{-s-4} \Rightarrow N^{-3s} \lesssim 1$$ a contradiction again because $N >> 1$ and $-3s > 0$.
[100]{} L.V. Bogdanov, The Veselov-Novikov equation as a natural generalization of the Korteweg de Vries equation, Teoret. Mat. Fiz. 70, no. 2, 309–314, 1987. English translation: Theoret. and Math. Phys. 70, no. 2, 219–223, 1987. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV equation, GAFA, 107-156, 3, 1993. J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, GAFA, 315-341, 3, 1993. J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Sel. Math. New. Ser. 3, 115-159, 1997. A. Carbery, C.E. Kenig, S. Ziesler, Restriction For Homogeneous Polynomial Surfaces In $\mathbb{R^3}$, arXiv:1108.4123 P. G. Grinevich, The scattering transform for the two-dimensional Schrdinger operator with a potential that decreases at infinity at fixed nonzero energy, (Russian), Uspekhi Mat. Nauk, 55, no. 6(336), 3-70, 2000; Russian Math. Surveys, 55, no. 6, 1015Ð1083, 2000. P. G. Grinevich and S. V. Manakov, Inverse scattering problem for the two-dimensional Schrdinger operator, the $\bar{\partial}$-method and nonlinear equations, Funktsional. Anal. i Prilozhen. 20:2, 14Ð24, 1986; English transl., Functional Anal. Appl. 20, 94Ð103, 1986. A. Kazeykina, Solitons and large time asymptotics of solutions for the Novikov-Veselov equation, PhD thesis at École Polytechnique, 2012. A. Kazeykina, R. G. Novikov, Large time asymptotics for the Grinevich-Zakharov potentials. Bulletin des Sciences Mathématiques. 135, 374-382, 2011. A. Kazeykina, R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy. Nonlinearity. 24, 1821-1830, 2011. A. Kazeykina, R. G. Novikov, A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with nonsingular scattering data. Inverse Problems, 28(5), 055017, 2012. M. Lassas, J. L. Mueller, S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two, Comm. Partial Differential Equations 32, no. 4-6, 591Ð610, 2007. M. Lassas, J. L. Mueller, S. Siltanen, A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I: Analysis, Phys. D 241 no. 16, 1322Ð1335, 2012. L. Molinet, D. Pilod, Bilinear Strichartz Estimates For The Zakharov-Kuznetsov Equation And Applications, arXiv:1302.2933v1 L. Molinet, J.-C. Saut, N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Mathematical Journal, Vol. 115, No. 2, 353-384, 2002. L. Molinet, J.-C. Saut, N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33, 982 Ð 988, 2001. L. Molinet, J.-C. Saut, N. Tzvetkov, Global well-posedness for the KPII equation on the background of non localized solution, Ann. Inst. H. Poincaré Anal. Non Linéaire 28, no.5, 653-676, 2011. C. Muscalu, W. Schlag, Classical and Multilinear Harmonic Analysis, Vol. 1, Cambridge Studies In Advanced Mathematics 137, 2013. R. G. Novikov, Absence of exponentially localized solitons for the NovikovÐVeselov equation at positive energy. Phys. Lett. A 375, 1233Ð1235, 2011. S.P. Novikov, A.P. Veselov, Finite-gap two-dimensional potential Schršodinger operators. Explicit formulas and evolution equations. Dokl. Akad. Nauk SSSR 279, no. 1, 20â24, 1984. English translation: Soviet Math. Dokl. 30, no. 3, 588â591, 1984. S.P. Novikov, A.P. Veselov, Two-dimensional Schršodinger operator: inverse scattering transform and evolutional equations. Solitons and coherent structures (Santa Barbara, Calif., 1985). Phys. D 18, no. 1-3, 267â273, 1986. P. Perry, Miura Maps and Inverse Scattering for the Novikov-Veselov Equation, preprint at arXiv:1201.2385, to appear in Analysis & PDE. T. Tao, Nonlinear Dispersive Equations: Local And Global Analysis, Regional Conference Series In Mathematics, 106, AMS, Providence, RI, 2006. N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C.R. Acad. Sci. Paris Sér. I Math. 329, 1043-1047, 1999.
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---
abstract: 'Star formation in elliptical galaxies (Es) was and is mostly dominated by mergers and accretions with many suggestive examples seen among local galaxies. Present day star formation in Es is easily measurable in of Es and appears bursty in character. Direct age determinations from integrated light indicate real age scatter. If one assumes the oldest-looking galaxies are a Hubble time old, the light weighted mean ages of the rest spread to 0.5 of a Hubble time, with scatterlings at very young ages. Larger Es and Es in clusters have less age scatter than smaller or field Es. The size trend is clear. The environment trend needs to be rechecked with better data even though it agrees with high redshift field/cluster results.'
address: |
Astronomy Department, University of Michigan\
Ann Arbor, MI 48109-1090
author:
- 'Guy Worthey[^1]'
title: 'Star Formation History of Elliptical Galaxies from Low-Redshift Evidence'
---
Introduction {#introduction .unnumbered}
============
The most appealing picture of star formation in elliptical galaxies for cosmologists is one in which Es formed very early in the universe and have been quiescent ever since. If such galaxies exist, they potentially measure the curvature parameter $q_0$ since one could predict their size and luminosity fairly well and use them as standard rulers or candles. This picture is supported by the superficial uniformity of ellipticals in appearance and the existence of scaling relations between observed parameters (stellar velocity dispersion \[$\sigma$\], surface brightness, size, luminosity, colors, and line strengths), some of which scale with very small scatter.
The hope for using Es as cosmological tracers is alive and well, but when nearby ellipticals are examined in detail they exhibit a large variety of morphological and kinematic peculiarities and strong evidence for star formation much less than a Hubble time ago, leading to a picture of elliptical formation in which galaxy-galaxy mergers and accretion events are the dominant formation mechanism, and that we still witness the “tail end” of E formation today in the form of observed galaxy mergers and Es that display current star formation. I think we should view elliptical formation as a [*process*]{} rather than an [*event*]{}.
=3.0in
\[fig1\]
Figure 1 does not show the variety of dynamical or morphological changes that could be taking place in the history of an elliptical, but concentrating on star formation alone, a bursting scheme like that of Fig. 1 fits the facts outlined in this article. Note in Fig. 1 a difference in mass-weighted age between cluster and field ellipticals and the bursty star formation with quiescent periods during which ellipticals will look superficially normal if the intraburst period is longer than the stellar population fade time.
This fade time is illustrated in Fig. 2 using the commonly used and easily measured line index Mg$_2$, which has the additional advantage that it participates in the Mg$_2$-$\sigma$ scaling relation (discussed later) so that there is a relatively clear definition of “normal:” the gaussian scatter from this relation. Fig. 2 shows that, if only a few percent by mass of gas is consumed in a starburst, the elliptical galaxy will appear “normal” after less than 0.5 Gyr. Therefore a population of Es will always look “normal” except for a few oddballs caught in the aftermath of starburst. The rest of this article should show that this is a workable scheme.
=2.5in
\[fig2\]
Local evidence for merging {#local-evidence-for-merging .unnumbered}
==========================
I will differentiate between “merging” and “accretion” events. Accretion events are small, involving less than 10% of the galaxy’s mass, while mergers are larger. Each event can also be partially or completely stellar, so that star formation may not result. Theoretical expectations are that gaseous accretions will rapidly, dissipatively sink to galaxy center, forming stars when it becomes dense enough. Stellar accretions survive as kinematic substructure that disappears if there is time to phase mix (e.g. Kormendy 1984; Balcells & Quinn 1990; Balcells 1991). Large, violent mergers are characterized by the ejection of tidal tails and a somewhat protracted series of starbursts as the two galaxies violently relax to an $r^{1/4}$ profile, followed by a steady drizzle of gas that can last as long as a few Gyr (Hibbard & Mihos 1995). The end state of such violent encounters has long been postulated to end in the formation of a bulge-dominated elliptical-like galaxy, with “correct” kinematics and surface brightness profile (e.g. Barnes 1992, Hernquist 1993, Heyl et al. 1994) if a little gas is involved. Note that purely stellar mergers give very large cores that are not observed, so a certain amount of gaseous dissipation is required.
[**Ongoing disk-disk mergers:**]{} Many IRAS galaxies and Arp (1966) peculiar galaxies appear to be disk-disk mergers in progress: still recognizable as two spiral galaxies, but blatantly interacting. Toomre (1977) counted 11 such mergers with NGC numbers. Assuming a constant merger rate and a typical duration of 0.5 Gyr for this morphological phase found that $\sim$250 NGC galaxies would have formed this way over a Hubble time. Compare this number to the fraction of the 6032 (my estimate) galaxies in the original NGC catalog that are Es: about 13% = 780 objects.
[**Qualitative post-merger signatures:**]{} Both morphological and kinematic peculiarities are predicted outcomes of large accretions or mergers. Morphological “fine structure,” including ripples, jet features, boxiness, and X structure at large radius was measured by Schweizer et al. (1990) and Schweizer & Seitzer (1992). The galaxies with the most such morphological complexity have systematically bluer colors, weaker metallic line strengths, and stronger H$\beta$ line strength, indicating a connection between recent merger status and the mean age of the stellar populations (the other alternative is that morphologically disturbed galaxies are preferentially metal-poor, but this idea is generally dismissed).
Kinematic peculiarities such as minor axis rotation, cores that rotate counter to the rotation of the galaxy outskirts, and kinematic discontinuities with radius are observed in more than half of all ellipticals (Bender 1996). Many of these peculiarities are dynamically long-lived, and do not constitute evidence for or against [*recent*]{} merging, but do argue strongly for merging as a process, since a single radial collapse can not produce such peculiar motions. Kinematic discontinuities are often mirrored by cospatial line strength gradient discontinuities, adding further weight to the merger interpretation (Bender & Surma 1992). In the case of high surface brightness elliptical NGC 1700, Statler et al. (1996) argue from an array of isophotal shape and kinematic data that at least 3 or more stellar subsystems must have merged 2–4 Gyr ago to explain the observed substructure, but the inner regions have phase-mixed to uniformity. Very few galaxies have been studied to this level of detail.
[**Accretion in Non-Es:**]{} A number of galaxies illustrate that multi-episode gaseous accretion/merger events occur. A note of thanks to Kennicutt (1996) for pointing out most of the examples I describe. First, we catch an accretion event in progress in the Milky Way as the Saggitarius dwarf spheroidal is in the process of being tidally disrupted (Ibata et al. 1995) and is strung out over at least $\sim$40 degrees of arc (M. Mateo, private communication). More evidence for the accretion of spheroidal-sized, metal-poor, but not always old galaxies into the Galactic halo comes from young stars on halo orbits, the presence of galactic streams, and age spreads in globular clusters as summarized in Freeman (1996). Some workers now start with the assumption that the entire metal poor halo was built by accretion events over the Galaxy’s lifetime, while the bulge may have been built from disk/bar instabilities. It does seem likely that bulges and disks evolved together since their scale lengths are always in nearly the same ratio (Courteau et al. 1996).
Accretion of gas is observed in external spiral galaxies from H [I]{} in the form of high velocity clouds of up to $\sim 10^8 M_\odot$ (Kamphius 1993), or in the form of blatant tidal disruption prior to merging as in the case of NGC 3359 and NGC 4565 (Sancisi et al. 1990). H[I]{} observations of E galaxies NGC 4472, UGC 7636, NGC 3656, NGC 5128, and NGC 2865, among others, show evidence for ongoing or recent gas accretion (Sancisi 1996). Spiral NGC 4826 (M64) is a rare example of gas-gas counter-rotation in which the sense of rotation switches at a radius of $\sim$1 kpc. This galaxy appears to have accreted a gas-rich companion in the recent past, with the inner gas disk the remnant of the original disk (Braun et al. 1992; Rubin 1994; Rix et al. 1995).
Many star-gas counter-rotations are seen. Bertola et al. (1992) list 9 examples of S0 galaxies with gas-star counterrotation. In NGC 4526 (Bettoni et al. 1991) and NGC 3626 (Cirri et al. 1995) the counterrotation extends over the entire disk, with gas masses of $10^8
- 10^9 M_\odot $. More easily seen are polar ring galaxies in which a (usually) spiral galaxy is ringed by gas, stars, and/or dust at a nearly perpendicular angle. From their catalog of ($\sim$70) polar ring galaxies, Whitmore et al. (1990) estimate that about 5% of S0 galaxies went through a polar ring phase. Even more spectacular than gas-star is star-star counterrotation. The edge-on S0 galaxy NGC 4550 is seen to have two cospatial counterrotating disks with nearly identical masses and scale lengths and with velocity dispersions of 45 and 50 km s$^{-1}$ (Rubin et al. 1992; Rix et al. 1992), indicating two separate epochs of galaxy formation. In Sb NGC 7217 20-30% of the the disk stars are in a retrograde cold disk (Merrifield & Kuijken 1994). In spirals, such oddities are rare probably because spirals are relatively fragile compared to Es.
The evidence from spirals indicates that accretion is a common phenomenon. Further, although galaxies like NGC 4550 must be rare, they reveal a fabulous wealth of possibilities for galaxy formation. Before you knew about 4550, what odds would you have assigned to its existence?
[**Current E star formation:**]{} Most Es have readily measurable nebular emission (Goudfrooij et al. 1994, Gonzàlez 1993) probably indicative of star formation. The Goudfrooij narrow band imaging indicates a variety of morphologies of the ionized gas, from disk-like to nuclear to diffuse and filamentary. Fig. 3 shows reprocessed Gonzàlez (1993) spectroscopic data on Es and Kennicutt (1983) H$\alpha$ data on spiral galaxies with the same conversion from H$\alpha$ flux to star formation rate. Fig. 3 shows the fraction of the galaxy assembled over 10 Gyr assuming a constant star formation rate. I corrected roughly for $M/L$ changes as a function of Hubble type, but assumed zero extinction correction. A correction will push the star formation rate to higher values.
=3.0in
\[fig3\]
There are two points to notice about the elliptical galaxies in relation to spirals. (1) The observed SFR is nonzero in of the Es, but 100 times smaller than late-type spirals. (2) Fig. 3 might almost be an illustration of binomial statistics in which Sc galaxies have large numbers of star formation events and are thus distributed in an almost gaussian way, while E galaxies have small numbers of star formation events and are thus distributed in a way that resembles Poisson statistics: [*fundamentally bursty*]{} in character.
Age from integrated light {#age-from-integrated-light .unnumbered}
=========================
It is now possible to measure a light-weighted mean age using integrated light indices for old stellar populations. The “light-weighted” part means that young stars, because they are brighter, can heavily influence the mean age that one obtains and that it is easy to obscure older generations of stars. The mean ages are derived from plotting Balmer indices versus hand-picked metal indices that are more metallicity sensitive than average. The spectra from which these measurements are taken need to have good S/N and need to have careful accounting for systematics like instrumental resolution and galaxy velocity dispersion so that the observations transform to the same system as the models.
Such pickiness is needed because easier measurements, such as broad-band colors, D4000, or Mg$_2$ are largely degenerate with age and metallicity along a null spectral change slope of d log($Z$) = $-2/3$ d log(age) along which colors and most line strengths stay the same (Worthey 1994). This implies that if one wants a 15% age estimate, one must know the metal abundance to 10%. By picking spectral indices that are preferentially sensitive to age and arraying them against indices that are preferentially sensitive to metal abundance the age-metal degeneracy is largely broken in a [*differential*]{} sense: the age zeropoint is still quite uncertain, but relative mean age changes are readily detectable.
=3.5in
\[fig4\]
Figure 4 tells us several things. First, the spread in mean age is real in the sense of being well beyond observational error. Regardless of age zeropoint, many galaxies have ages less than half a Hubble time, and several appear very young. At present we can’t tell if these ages are the real formation ages of the galaxies or the effect of small, recent bursts of star formation. Second, there is a distinct trend for the younger large galaxies to be more metal rich, a tendency which will be discussed below. Third, there are dependencies on size and probably on field/cluster environment. Smaller galaxies have a larger spread in mean age than the large ones, as well as a tendency to be slightly more metal poor. Field galaxies appear to be more volatile than cluster galaxies (note the lack of large cluster Es younger than 3 Gyr). In this sample there do not appear to be any cluster Es that are both small and young. Note that the sample was not chosen in an intelligent way, and a proper volume-limited sample may show somewhat different trends.
The real scatter of ages tells us that Es have had a fairly complex history, and that they are not quite finished forming. It does not tell us the relative importance of mergers versus accretions.
[**Aside: Age from abundance ratio variations?**]{} Worthey et al. (1992) plotted Mg$_2$ versus iron indices to find that large Es deviated from solar-neighborhood \[Mg/Fe\] in the sense of enhanced Mg relative to Fe by about a factor of two. Smaller Es have a nearly solar mixture, so the amount of enrichment from Type II supernovae relative to Type I gets larger in larger Es. Unfortunately, the mechanism for varying this ratio is not known. It could either be a variation in formation timescale or a (mild) variation in upper IMF strength as a function of galaxy size. The implications for E formation are different for those two cases and so, skipping the details, we can’t really constrain E formation until we know for sure what mechanism causes the Type II to Type I shift.
Slipperiness of tight scaling relations {#slipperiness-of-tight-scaling-relations .unnumbered}
=======================================
As mentioned above, colors and line-strengths usually scale with structural parameters like brightness, size, $\sigma$, and combinations of these quantities. The tightest are the Mg$_2$-$\sigma$ relation and the fundamental plane (in $\mu$, $R$, $\sigma$ space). The interesting thing about these “tightness relations” is the small scatter observed. The small scatter is interesting because significant spread in velocity anisotropy, density structures, and stellar ages could reasonably be expected to raise the scatter to much higher levels than is observed (e.g. Djorgovski et al. 1996). More than one conspiracy must be operating to thus limit the range of E properties.
Empirically, somewhat more than half the scatter from the fundamental plane can be fairly unambiguously tied to stellar population changes because the scatter correlates with color and line strength residuals such that blue colors correspond to high surface brightness, as one would expect from the presence of a younger subpopulation (Prugniel & Simien 1996; J[ø]{}rgensen & Franx 1996). If that is so, then the tightness of the fundamental plane places a restriction on the age scatter that is allowed for ellipticals. As a first cut, one can take the observed scatter in cluster Es and convert that to a scatter in age or metallicity. Bender et al. (1993) find, analyzing the Mg$_2$-$\sigma$ relationship, an allowed age (or metallicity) scatter of 15% RMS at a given $\sigma$ with a non-gaussian blue tail. This seems to say that of Es were formed in the first of the universe. This interpretation is probably misleading because (1) the scatter does not look one-sided – it looks like there really are galaxies on the redder (older or more metal rich) side of the mean relation, and (2) there appears to be a rough trend for younger Es to be more metal rich (Figure 4; Worthey et al. 1996). Point (1) means that either metallicity plays a significant/dominant role, or that the mean age for Es is substantially younger than that of the universe, and what we are seeing is scatter about a “mean history” of formation. Point (2) implies that younger (bluer) populations will be more metal rich (redder), and this will do a lot to artificially tighten the Mg$_2$-$\sigma$ and the fundamental plane relations, allowing more age spread than one might otherwise have guessed.
Summary and high-redshift remarks {#summary-and-high-redshift-remarks .unnumbered}
=================================
The tightness relations can also be tracked with redshift. If Es are pure passive evolvers, there is a clear prediction for bluer Mg$_2$ and brighter $M/L$ with redshift. If we think of E formation as a decaying process of continued activity, we can predict a lot less clearly what is going on. In fact, if an ongoing accretion [*process*]{} dominates the line strengths and colors, we may see [*less*]{} evolution than purely passive. If large bursts of star formation at fairly late times are important we should see [*more*]{} than passive evolution. Work on both the fundamental plane and the Mg$_2$ sigma relation in cluster Es out to redshifts of 0.5 or so (see Dressler, this volume; Bender et al. 1996) indicates that the galaxies identified as Es tend to evolve as passively evolving populations that were formed before $z = 2$.
In the field, however, two different redshift surveys show that the population of red galaxies declines too fast to be consistent with passive evolution of old luminous galaxies with extrememly high confidence (Kauffmann et al. 1997). These high redshift results lend confidence that the Fig. 1 picture is roughly correct. In that picture, field and cluster Es form by some mixture of merging and accretion in a bursty manner which allows lots of time for blue colors to fade and for the galaxy to look “normal” between bursts. Active star formation should be seen in some ellipticals. Merging events and merger remnants should exist. Tight scaling relations are preserved until star formation gets really messy, probably before $z=2$. The Balmer-metal age-diagnostic diagrams should show a spread in age. The Fig. 1 picture seems to summarize the observed situation fairly well.
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[^1]: Hubble Fellow
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abstract: 'For an arbitrary finite monoid $M$ and subgroup $K$ of the unit group of $M$, we prove that there is a bijection between irreducible representations of $M$ with nontrivial $K$-fixed space and irreducible representations of $\mathcal{H}_K$, the convolution algebra of $K\times K$-invariant functions from $M$ to $F$, where $F$ is a field of characteristic not dividing $|K|$. When $M$ is reductive and $K = B$ is a Borel subgroup of the group of units, this indirectly provides a connection between irreducible representations of $M$ and those of $F[R]$, where $R$ is the Renner monoid of $M$. We conclude with a quick proof of Frobenius Reciprocity for monoids for reference in future papers.'
author:
- 'Jared Marx-Kuo,'
- 'Vaughan McDonald,'
- 'John M. O’Brien,'
- Alexander Vetter
bibliography:
- 'references.bib'
nocite: '[@*]'
title: Convolution Algebras for Finite Reductive Monoids
---
Introduction
============
Motivation
----------
Let $M$ be a reductive monoid over a finite field. Let $G(M)$ be the unit group of $M$, a connected reductive group, with maximal torus $T$ contained in Borel subgroup $B$. Recall that $M$ has the Renner decomposition $M = \bigsqcup_{r\in R} B\underline{r} B,$ where $R$, the Renner monoid of $M$, plays the role of the Weyl group of a connected reductive group. It is well-known that $\mathcal{H}(M,B)$, the $B\times B$-invariant convolution algebra of functions from $M$ to $\mathbb{C}$, is isomorphic to the monoid algebra $\mathbb{C}[R]$ of $R$, just as the equivalent convolution algebra $\mathcal{H}(G,B)$ of a connected reductive group is isomorphic to the group algebra of the Weyl group.
In the group case, the Borel-Matsumoto theorem implies a bijection between the irreducible representations $(\pi, V)$ of $G$ with nonzero Borel-fixed space $V^B = \{v\in V: \pi(b)v = v\;\forall b\in B\}$ and irreducible representations of $\mathcal{H}(G,B)$. Since the representation theory of $\mathcal{H}(G,B)$ reflects the representation theory of the Weyl group of $G$, the Borel-Matsumoto Theorem classifies many irreducible representations of $G$.
In this paper, we prove an analogous result to the Borel-Matsumoto theorem for finite monoids. We prove that, for $K$ a subgroup of $G(M)$, there is a bijection between irreducible representations of $M$ with nonzero $K$-fixed subspaces and representations of the convolution algebra of $K\times K$-invariant functions from $M$ to $F$, where $F$ is of characteristic not dividing $|K|$. When $M$ is reductive, $K = B$, and $F = \mathbb{C}$, we get the desired connection between representation theory of $M$ and that of its Renner monoid via $\mathcal{H}(M,B)$.
We hope to extend the result to the case of $p$-adic reductive monoids. For $p$-adic reductive groups, a nearly identical proof replacing summation with integration with respect to a Haar measure works. However, subtleties related to the nature of smooth representations of monoids prevented a direct extension of the proof from that of finite monoids. In a future paper, we hope to find an alternative proof.
A Borel-Matsumoto Theorem for Finite Monoids
============================================
Let $M$ be a finite monoid, $G(M)$ the group of units of $M$, $K$ a subgroup of $G(M)$, and $F$ a field of characteristic not dividing $|K|$.
For $\phi, \psi: M\rightarrow F$, Godelle [@godelle] defines the convolution product $\phi * \psi$ by $$(\phi*\psi)(m) = \sum\limits_{yz=m}\phi(y)\psi(z)$$
Similarly, for $(\pi, V)$ a representation of M and $\phi$ as above define $\pi(\phi)$ by
$$\pi(\phi)v = \sum\limits_{x\in M}\phi(x)\pi(x)v$$
For $\phi, \psi\in \mathcal{H}$, $\pi(\phi*\psi) = \pi(\phi)\circ\pi(\psi).$
Consider $\pi(\phi)\circ\pi(\psi).$ We have the following:
$$\begin{aligned}
(\pi(\phi)\circ\phi(\psi))v &= \sum\limits_{x\in M}\phi(x)\pi(x)\sum\limits_{y\in M}\psi(y)\pi(y)v) \\
&=\sum\limits_{x,y\in M} \phi(x)\psi(y)\pi(x)\pi(y)v \\
&=\sum\limits_{x,y\in M} \phi(x)\psi(y)\pi(xy)v \\
&=\sum\limits_{z\in M}\sum\limits_{xy = z} \phi(x)\psi(y)\pi(z)v \\
&=\sum\limits_{z\in M}(\phi*\psi)(z)\pi(z)v \\
&= \pi(\phi*\psi)v\end{aligned}$$
Thus $\pi(\phi)\circ\pi(\psi) = \pi(\phi*\psi)$.
Let $\mathcal{H}$ be the $F$-algebra of functions from $M$ to $F$ under addition and convolution. Define, for $v\in V$,
$$\mathcal{H}v=\{\pi(\phi)v: \phi\in\mathcal{H}\}.$$
Define an action of $M$ on $\mathcal{H}v$ by $m \cdot (\pi(\phi)v) = \pi(m)\pi(\phi)v$. Define $f_m: M\rightarrow F$ by $f_m(m) = 1, f_m(x) = 0$ for $x\neq m$. Since $\pi(f_m)v = \pi(m)v$, then $\mathcal{H}v$ is closed under action by $M$. Thus, it is a subrepresentation.
Similarly, let $\mathcal{H}_K$ be the $F$-algebra of functions from $M$ to $F$ under convolution that are constant on double-cosets of K; i.e. $\phi: M\rightarrow F$ such that $\phi(m)=\phi(k_1mk_2)$ for all $k_1, k_2 \in K$. Furthermore, let $V^K = \{v \in V \; | \; \pi(k) v = v \quad \forall k \in K\}$.
Let $(\pi,V)$ be an irreducible representation of M with $V^K\neq\{0\}$. Then $V^K$ is irreducible as an $\mathcal{H}_K$-module.
We follow Bump’s proof of the group case closely [@bump]. We claim that, for all nonzero $u\in V^K$, that $\mathcal{H}_Ku$:= {$\pi(\phi)u: \phi \in \mathcal{H}_K$} equals $V^K$. In other words, we wish to show that, for all $v\in V^K$ there exists $\phi\in\mathcal{H}_K$ such that $\pi(\phi)u = v$.
Since $(\pi,V)$ is an irreducible representation of M, there are no proper non-trivial subrepresentations in V. Because there is an M-action on $\mathcal{H}u\neq\{0\}$, then $\mathcal{H}u = V$. Thus there exists $\psi\in\mathcal{H}$ such that $\pi(\psi)u = v$.
Define $\phi\in\mathcal{H}$ by, for $x\in M$ $$\phi(x) = \frac{1}{|K|^2}\sum\limits_{k_1,k_2\in K}\psi(k_1xk_2)$$ Since $\phi$ must be invariant over left and right cosets of K, $\phi$ lies in $\mathcal{H}_K$. Now consider the following:
$$\begin{aligned}
\pi(\phi)u = \frac{1}{|K|^2}\sum\limits_{k_1,k_2\in K}\sum\limits_{x\in M}\psi(k_1xk_1)\pi(x)u\end{aligned}$$
Notice that $x\mapsto k_1^{-1}xk_2^{-1}$ is a bijection from M to M, as it has an inverse $x\mapsto k_1xk_2$. Thus we can make the following change of variables:
$$\begin{aligned}
\pi(\phi)u &= \frac{1}{|K|^2}\sum\limits_{k_1,k_2\in K}\sum\limits_{x\in M}\psi(x)\pi(k_1^{-1}xk_2^{-1})u \\
&= \frac{1}{|K|^2}\sum\limits_{k_1,k_2\in K}\sum\limits_{x\in M}\psi(x)\pi(k_1)^{-1}\pi(x)\pi(k_2)^{-1}u.\end{aligned}$$
Since $u\in V^K$, we have that $\pi(k_2)^{-1}u = u$. Thus,
$$\begin{aligned}
\pi(\phi)u &= \frac{1}{|K|}\sum\limits_{k_1\in K}\sum\limits_{x\in M}\psi(x)\pi(k_1)^{-1}\pi(x)u \\
&= \frac{1}{|K|}\sum\limits_{k_1\in K}\pi(k_1)^{-1}\sum\limits_{x\in M}\psi(x)\pi(x)u \\
&=\frac{1}{|K|}\sum\limits_{k_1\in K}\pi(k_1)^{-1}\pi(\psi)u.\end{aligned}$$
Since $\pi(\psi)u = v$ and $v\in V^K$,
$$\begin{aligned}
\pi(\phi)u &= \frac{1}{|K|}\sum\limits_{k_1\in K}\pi(k_1)^{-1}v = v.\end{aligned}$$
Thus, for all $v\in V^K$ there exists $\phi\in\mathcal{H}_K$ such that $\pi(\phi)u = v$. Thus, $V^K$ is irreducible as an $\mathcal{H}_K$-module.
Denote, for $(\pi,V)$ a representation of M, let $(\pi|_G,V)$ be the restricted representation of $G(M)$ defined by $\pi|_G(g)=\pi(g)$ for $g\in G(M)$. Define the contragredient representation of $G(M)$ $(\hat{\pi}|_G, \hat{V})$ by $\langle\pi|_G(g)v, \hat{v}\rangle = \langle v, \hat{\pi}|_G(g^{-1})\hat{v}\rangle$ for all $g\in G(M)$.
Let l: $V^K \rightarrow F$ be a linear functional. Then there exists $\hat{v}\in\hat{V}^K$ such that for all $v\in V^K$, l(v) = $\langle v,\hat{v}\rangle$. [@bump]
Let $\hat{v}_0$ be a linear functional on V that restricts to l on $V^K$.
Define $\hat{v} = \frac{1}{|K|}\sum_{k\in K}\hat{\pi}|_G(k)\hat{v}_0$. For $v\in V^K$, then, we have the following equalities:
$$\begin{aligned}
\langle v, \hat{v}\rangle &= \frac{1}{|K|}\sum\limits_{k\in K}\langle v, \hat{\pi}|_G(k)\hat{v}_0\rangle \\
&=\frac{1}{|K|}\sum\limits_{k\in K}\langle\pi|_G(k)^{-1}v, \hat{v}_0\rangle \\
&= \frac{1}{|K|}\sum\limits_{k\in K}\langle\pi(k)^{-1}v, \hat{v}_0\rangle \\
&= \frac{1}{|K|}\sum\limits_{k\in K}\langle v, \hat{v}_0\rangle \\
&= l(v)
\end{aligned}$$
If $V^K\neq 0$ then $\hat{V}^K\neq 0$. [@bump]
Let $R$ be an algebra over $F$, and $N_1, N_2$ simple $R$-modules that are finite-dimensional as vector spaces over F. If there exist linear functionals $L_i: N_i \rightarrow F$ and $n_i\in N_i$ such that $L_i(n_i)\neq 0$ and $L_1(rn_1) = L_2(rn_2)$ for all $r\in R$, then $N_1\cong N_2$ as R-modules. [@bump]
We particularly care about the case when two representations $(\pi_i,V_i)$ share matrix coefficients $\langle \pi_i(m)v, \hat{v}_0 \rangle$ for all $m\in M$.
Let $(\pi,V)$ and $(\sigma, W)$ be two irreducible representations of $M$ with nonzero matrix coefficients $\langle\pi(m)v, \hat{v}_0\rangle = \langle\sigma(m)w, \hat{w}_0\rangle$ for some v, $v_0$, w, $w_0$, and all $m\in M$. Then $(\pi,V)\cong(\sigma,W)$.
Define actions of $F[M]$ on $V$ and $W$ by letting $mv = \pi(m)v$ and $mw = \sigma(m)w$ for all $v\in V, w\in W,$ and $m\in M$ respectively and then extending by linearity. Thus $V$ and $W$ become $F[M]$-modules. Because the representations are each irreducible, $V$ and $W$ are simple as $F[M]$-modules. Since $\langle mv, \hat{v}_0\rangle = \langle mw, \hat{w}_0\rangle$ for all $m\in M$ are two equal linear functionals on $V$ and $W$, then $V\cong W$ as $F[M]$-modules by Lemma 4. Equivalently, $(\pi,V)\cong(\sigma,W)$.
Now we prove the second half of the Borel-Matsumoto Theorem.
If $(\pi, V)$ and $(\sigma, W)$ are two irreducible representations of $M$ with $V^K$ and $W^K$ nonzero and isomorphic as $\mathcal{H}_K$-modules, then $(\pi,V)\cong(\sigma,W)$.
Let $\lambda: V^K\rightarrow W^K$ be an isomorphism of $\mathcal{H}_K$-modules and $l: W^K\rightarrow F$ be a linear functional not equal to zero. Then there exist $\hat{v}\in\hat{V}^K$ and $\hat{w}\in\hat{W}^K$ such that $(l\circ\lambda)(v) = \langle v, \hat{v}\rangle$ and $l(w) = \langle w, \hat{w}\rangle$ for all $v\in\hat{V}^K, w\in\hat{W}^K$. Furthermore, there exist $w_0\in W^K, v_0\in V^K$ such that $\langle w_0, w\rangle\neq 0$ since $l$ is nontrivial and $v_0 = \lambda^{-1}(w_0)$ since $\lambda$ is an isomorphism.
Then for $\phi\in\mathcal{H}_K$, we have that
$$\label{eq1}
\langle\sigma(\phi)w_0,\hat{w}\rangle = \langle\sigma(\phi)\lambda(v_0), \hat{w}\rangle = \langle\lambda(\pi(\phi)v_0), \hat{w}\rangle = (l\circ\lambda)(\pi(\phi)v_0) = \langle\pi(\phi)v_0, \hat{v}\rangle.$$
We show that equation \[eq1\] holds for all $\phi\in\mathcal{H}$ as well as $\mathcal{H}_K$. For $\phi\in\mathcal{H}$, define $\phi_K\in\mathcal{H}_K$ by
$$\phi_K(x) = \frac{1}{|K|^2}\sum\limits_{k_1, k_2\in K}\phi(k_1xk_2)$$
for all $x\in M$. By equation \[eq1\], then $\langle\pi(\phi_K)v_0, \hat{v}\rangle = \langle\sigma(\phi_K)w_0, \hat{w}\rangle$. Furthermore, we have that
$$\begin{aligned}
\langle \pi(\phi_K)v_0, \hat{v}\rangle &= \langle\frac{1}{|K|^2}\sum\limits_{k_1, k_2\in K}\sum\limits_{x\in M}\phi(k_1xk_2)\pi(x)v_0, \hat{v}\rangle \\
&= \frac{1}{|K|^2}\langle\sum\limits_{k_1, k_2\in K}\sum\limits_{x\in M}\phi(x)\pi(k_1)^{-1}\pi(x)\pi(k_2)^{-1}v_0, \hat{v}\rangle \\
&= \frac{1}{|K|^2}\langle\sum\limits_{k_1, k_2\in K}\pi(k_1)^{-1}\circ(\sum\limits_{x\in M}\phi(x)\pi(x))\circ\pi(k_2)^{-1}v_0, \hat{v}\rangle \\
&=\frac{1}{|K|^2}\langle\sum\limits_{k_1, k_2\in K}\pi(k_1)^{-1}\pi(\phi)\pi(k_2)^{-1}v_0, \hat{v}\rangle \\
&=\frac{1}{|K|^2}\sum\limits_{k_1, k_2\in K}\langle\pi(k_1)^{-1}\pi(\phi)\pi(k_2)^{-1}v_0, \hat{v}\rangle \\
&=\frac{1}{|K|^2}\sum\limits_{k_1, k_2\in K}\langle\pi|_G(k_1)^{-1}\pi(\phi)\pi|_G(k_2)^{-1}v_0, \hat{v}\rangle \\
&= \frac{1}{|K|^2}\sum\limits_{k_1, k_2\in K}\langle\pi(\phi)\pi|_G(k_2)^{-1}v_0, \hat{\pi}|_G(k_1)\hat{v}\rangle.\\
&= \frac{1}{|K|^2} \sum_{k_1, k_2 \in K} \langle\pi(\phi)v_0, \hat{v}\rangle \\
&= \langle\pi(\phi)v_0, \hat{v}\rangle.\end{aligned}$$
since $v_0\in V^K$ and $\hat{v}\in\hat{V}^K$.
Thus $\langle\pi(\phi_K)v_0, \hat{v}\rangle = \langle\pi(\phi)v_0, \hat{v}\rangle$ for all $\phi\in\mathcal{H}$. Similarly, $\langle\sigma(\phi_K)w_0, \hat{w}\rangle = \langle\sigma(\phi)w_0, \hat{w}\rangle$. With this information, then, we have that $\langle\pi(\phi_K)v_0,\hat{v}\rangle = \langle\sigma(\phi_K)w_0,\hat{w}\rangle$ implies that $\langle\pi(\phi)v_0,\hat{v}\rangle = \langle\sigma(\phi)w_0, \hat{w}\rangle$.
Let $\phi_m\in\mathcal{H}$ for all $m\in M$ be the function that sends all x in M with $x\neq m$ to 0 and m to 1. Then $\pi(\phi_m)v = \pi(m)v$ and $\sigma(\phi_m)w = \sigma(m)w$.
Thus, we have that $\langle\pi(m)v_0, \hat{v}\rangle = \langle\sigma(m)w_0, \hat{w}\rangle$ for all $m\in M$. By Lemma 5, then, $(\pi,V)$ and $(\sigma,W)$ are equivalent.
Frobenius Reciprocity
=====================
Although Doty alluded to the fact that Frobenius Reciprocity holds for monoids [@doty], he left it without proof. For completeness, we give an explicit proof.
Let $M$ be a finite monoid, $G(M)$ its group of units, $N$ a submonoid of $M$, $G(N)$ its group of units, and $(\pi, V)$ a representation of $M$. Define the vector space $Ind_N^M V$ as follows:
$$\Ind_N^{M} V = \{f : M \rightarrow V \; | \; f(nm) = \pi(n)f(m)\, \quad \forall n\in N, \; m\in M\}$$
Define $(\pi^M, Ind_N^M V)$ by $\pi^M(m)f(x) = f(xm)$ for all m.
The pair $(\pi^M, Ind_N^M V)$ is a representation of M.
First, we check that $Ind_N^M V$ is closed under the action of $\pi^M(m)$. Trivially, if $f(nx)=\pi(n)f(x)$ then $\pi^M(m)f(nx) = f(nxm) = \pi(n)f(xm)$ for all $m\in M, n\in N$.
We check that $\pi^M(m)$ is linear for all $m$. $$\forall z \in F, \; \forall f,g \in Ind_N^M \quad z \pi^M(m) f(x) = z f(xm) = \pi^M(m) (zf)(x)$$ $$\pi^M(m)(f + g)(x) = (f + g)(xm) = \pi^M(m)(f)(x) + \pi^M(m)(g)(x)$$ Now, we check that $\pi^M$ is a homomorphism of monoids. Let $m, x, y\in M$. Then $\pi^M(mx)f(y) = f(ymx) = \pi^M(x)f(ym) = \pi^M(m)\pi^M(x)f(y)$. Finally, $\pi^M(1)f(x) = f(x)$, implying that $\pi^M$ maps the identity to the identity. Clearly, then, $\pi^M(mx)=\pi^M(m)\pi^M(x)$, and $(\pi^M, Ind_N^M)$ is a representation of M.
Thus we can call $(\pi^M, Ind_N^M V)$ the induced representation of M. We have that
If $(\pi, V)$ is a representation of N, a submonoid of M, and $(\sigma, W)$ a representation of M, then $\Hom_M(W,Ind_N^M V)\cong \Hom_N(W,V)$ as vector spaces.
For $\phi\in \Hom_M(W,Ind_N^M V)$, define $F: \Hom_M(W,Ind_N^M V)\rightarrow \Hom_N(W,V)$ by $F(\phi)$, such that $F(\phi)(w) = \phi(w)(1)$, 1 being the identity element of $M$. We first show that $F(\phi)$ is linear. Because $\phi$ is linear, $$F(\phi)(w + w_0) = \phi(w + w_0)(1) = \phi(w)(1) + \phi(w_0)(1) = F(\phi)(w) + F(\phi)(w_0)$$ and for $z\in F$, $$F(\phi)(zw) = \phi(zw)(1) = z\phi(w)(1) = zF(\phi)(w)$$ We now claim that $F(\phi)$ is a morphism of $N$-modules For $n\in N$, $$\begin{aligned}
F(\phi)(\sigma(n)w) & = \phi(\sigma(n)w)(1) = \pi^M(n)\phi(\sigma(1)w)(1) \\
& = \phi(w)(n) = \pi(n)\phi(w)(1) = \pi(n)F(\phi)(w)\end{aligned}$$ Thus $F(\phi)$ is an $N$-module homomorphism from $W$ to $V$. Since $$F(\phi + \psi) (w) = (\phi + \psi)(w)(1) = \phi(w)(1) +\psi(w)(1) = F(\phi)(w) + F(\psi)(w)$$ and $F(z \cdot \phi)(w) = (z\phi)(w)(1) = zF(\phi)(w)$, then F is a vector space homomorphism. For $\tau\in \Hom_N(W,V)$, let $G: \Hom_N(W,V)\rightarrow \Hom_M(W,Ind_N^M V)$ such that $$(G(\tau)(w))(m) = G(\tau)(w)(m) = \tau(\sigma(m)w)$$ then, $\tau(\sigma(nm)w) = \tau(\sigma(n)\sigma(m)w) = \pi(n)\tau(\sigma(m)w)$, so $G(\tau)(w)$ is in $Ind_N^M$. We check that $G(\tau)(-)(m)$ is linear. This follows from the definition: $$\begin{aligned}
G(\tau)(w + w_0)(m)
& = \tau(\sigma(m)(w + w_0)) = \tau(\sigma(m)w + \sigma(m)w_0) \\
& = \tau(\sigma(m)w) + \tau(\sigma(m)w_0) = G(\tau)(w)(m) + G(\tau)(w_0)(m)\end{aligned}$$ and for $z\in F$, we have $$G(\tau)(zw)(m) = \tau(\sigma(m)(zw)) = z\tau(\sigma(m)(w))$$ Next, we check that $G(\tau)$ respects M. We have that for $x\in M$, $$\begin{aligned}
G(\tau)(\sigma(x)w)(m)
& = \tau(\sigma(m)\sigma(x)w) = \tau(\sigma(mx)w) \\
& = (\pi^M(x)\circ\tau)(\sigma(m)w) = \pi^M(x)(G(\tau)(w)(m))\end{aligned}$$ Thus $G(\tau)\in \Hom_M(W, Ind_N^M V)$. Finally, we check that G itself is linear: $$\begin{aligned}
G(\tau + \eta)(w)(m)
& = (\tau + \eta)(\sigma(m)w) \\
& = \tau(\sigma(m)w) + \eta(\sigma(m)w) = G(\tau)(w)(m) + G(\eta)(w)(m)\end{aligned}$$ and for $k\in K, \quad G(k\tau)(w)(m) = k(\tau(\sigma(w)m)) = k \cdot G(\tau)(w)(m)$. Thus G is a homomorphism of vector spaces.
Now, we show that F and G are inverses. First, we check the mapping $G\circ F: Hom_M(W,Ind_N^M)\rightarrow Hom_M(W,Ind_N^M)$. Let $\phi\in Hom_M(W,Ind_N^M)$. Then $G\circ F(\phi)$ works as follows. Since $F(\phi)$ is the map sending w to $\phi(w)(1)$, $$\begin{aligned}
(G\circ F)(\phi)(w)(m)
& = G(F(\phi))(w)(m) = F(\phi)(\sigma(m)w) \\
& = F(\phi)(\sigma(1*m)w) = \pi^M(m)F(\phi)(w) \\
& = \pi^M(m)\phi(w)(1) = \phi(w)(m)\end{aligned}$$ by definition of the induced representation. Since $(G\circ F)(\phi)(w)(m) = \phi(w)(m)$, $G\circ F$ is the identity morphism on $Hom_M(W,Ind_N^W)$.
Next, we check $F\circ G: Hom_N(V,W)\rightarrow Hom_N(V,W)$. Let $\tau\in Hom_N(V,W)$. Then $$\begin{aligned}
(F \circ G)(\tau)(w)(n) & = F(G(\tau))(w)(n) \\
& = G(\tau)(w)(1 \cdot n) = \tau(\pi(n)w) = \sigma(n) \tau(w) = \tau(w)(n)\end{aligned}$$ Thus $F\circ G$ is the identity morphism on $Hom_N(V, W)$. Since we have that both $G\circ F$ and $F\circ G$ are identity morphisms on their respective domains, they are inverses. Thus, we have that $Hom_M(W, Ind_N^M)\cong Hom_N(W,V)$ as vector spaces over $F$.
Further directions
==================
In a possible sequel, we would like to study smooth representations of $p$-adic reductive monoids. In the group case, the Borel-Matsumoto theorem extends easily to smooth representations of $p$-adic reductive groups [@bump]. The proof is virtually identical, with summation replaced by integration over a Haar measure. A similar result may hold for $p$-adic reductive monoids; however, the authors ran into some difficulty defining a suitable measure. Several subtle differences between the properties of smooth representations of $p$-adic reductive monoids and those of $p$-adic reductive groups prevented an immediate extension of the proof for the group case. A better description of smooth representations of $p$-adic reductive monoids may enable an alternative proof.
Also, for a finite reductive monoid $M$ with Borel subgroup $B$, we would like to explore reconstructing the irreducible representations of $M$ with nonzero $B$-fixed space from those of $\mathcal{H}(M,B)$. The Borel-Matsumoto theorem guarantees the existence of a bijection between irreducible representations of $M$ with nonzero $B$-fixed space and irreducible representations of $\mathcal{H}(M,B)$; however, it does not explicitly construct the bijection. In the finite reductive group case, Deligne and Lusztig used $\ell$-adic cohomology of certain varieties associated with $G$ to construct irreducible representations of $G$ [@deligne-lusztig]. We believe that a similar technique could work in the monoid case.
Acknowledgements
================
This research was conducted at the 2018 University of Minnesota-Twin Cities REU in Algebraic Combinatorics. Our research was supported by NSF RTG grant DMS-1745638. We would like to thank Benjamin Brubaker and Andy Hardt for their help and support during our time in Minnesota.
Jared Marx-Kuo, <span style="font-variant:small-caps;">Department of Mathematics, The University of Chicago, Chicago, IL 60637</span>
*E-mail address*: `jmarxkuo@uchicago.edu`
Vaughan McDonald, <span style="font-variant:small-caps;">Department of Mathematics, Harvard University, Cambridge, MA 02138</span>
*E-mail address*: `vmcdonald@college.harvard.edu`
John M. O’Brien, <span style="font-variant:small-caps;">Department of Mathematics, Kansas State University, Manhattan, KS 66506</span>
*E-mail address*: `colbyjobrien@ksu.edu`
Alexander Vetter, <span style="font-variant:small-caps;">Department of Mathematics and Statistics, Villanova University, Villanova, PA 19085</span>
*E-mail address*: `avetter@villanova.edu`
|
---
title: 'Heavy quark spin selection rules in S-wave meson-antimeson states'
---
By studying the spin structure of an S-wave meson-antimeson system in the heavy quark limit, we find two selection rules for the $c\bar{c}$ spin $J_{c\bar{c}}$ being only 1: (a) $J=min(J_\ell)-1$ or $J=max(J_\ell)+1$; (b) $J^C=1^+,2^-,3^+,\cdots$ if the two mesons are different but belong to the same doublet. Here $J_\ell$ is the total angular momentum of the light degree of freedom. The rules may constrain decay channels of a meson-antimeson molecule or resonance. For the recently observed $Z_c(3900)$ state, it in principle can decay into $J_{c\bar{c}}=0$ charmonium channels if it is a $D\bar{D}^*$ molecule or resonance. The non-observation of $h_c\pi$ decay might imply that this state is a tetraquark.
Introduction
============
In recent years, many unexpected exotic states, which are called X, Y, or Z, are observed and parts of them have been confirmed. All the hidden charm XYZ mesons are above the $D\bar{D}$ threshold (see Fig. \[xyz\]), which makes their properties difficult to understand in the conventional quark model. In QCD, exotic configurations such as hadron-antihadron molecule, tetraquark, and hybrid, are allowed. It is obvious that these XYZ states are all near-threshold resonances, which can be understood in the popular molecule picture. The observation of these XYZ states provides us a good opportunity to search for meson-antimeson molecule candidates.
![Charmonium-like spectra and various thresholds. Dotted lines are meson-antimeson thresholds. Dashed lines are $c\bar{c}$ mesons predicted in quark model [@Isgur]. Solid lines are observed charmonia and solid dots are newly observed XYZ mesons. Red color means that the structure was observed by at least two experiments.[]{data-label="xyz"}](speccharm13.eps){width="5.5in"}
After the observation of three hidden-charm charged mesons $Z(4430)$, $Z_1(4050)$, and $Z_2(4250)$ by Belle [@charged-belle], BESIII recently announced the charged meson $Z_c(3900)$ in Ref. [@Z3900-bes] and then Belle [@Z3900-belle] and an analysis from CLEO-c data [@Z3900-cleo] confirmed its existence. In the following study, BESIII reports two more charged structures $Z_c(4025)$ [@Z4025] and $Z_c(4020)$ [@Z4020] in different decay channels. The status of the spectra implies that more near-threshold heavy quark states would be observed in the future and molecules might be identified in them. It is necessary to understand the meson-antimeson picture deeper.
We here discuss a feature for meson-antimeson system: heavy quark spin selection rule, which was first noticed in Ref. [@Voloshin] by Voloshin. He found that the $c\bar{c}$ spin $J_{c\bar{c}}$, in the heavy quark limit, is only 1 if $X(3872)$ is a $D^0\bar{D}^{*0}$ molecule with $J^{PC}=1^{++}$. Since $X(3872)$ is also probably a charmonium with $J_{c\bar{c}}=1$, the mixing between a charmonium and a molecule is not suppressed by heavy quark mass. However, the decays into $J_{c\bar{c}}=0$ charmonia, e.g. $X(3872)\to\pi\pi\eta_c$, are suppressed. For the purpose of future investigation on molecules, we extend the study to a general meson-antimeson system. Voloshin used interpolating current and Fierz transformation to get the selection rule. We explore the problem in a different framework.
Formalism
=========
We want to extract information for $J_{c\bar{c}}\neq0$ in the heavy quark limit. For our purpose, we construct spin wave function for a meson-antimeson system at quark level and use the re-coupling scheme to get the spin wave function of the $c\bar{c}$ pair [@ours].
In the heavy quark limit, the total spin of a $c\bar{q}$ meson $j$ may be divided into two parts: the heavy quark spin $j_c=\frac12$ and the rest angular momentum $j_\ell$. $j_\ell$ is the total angular momentum of the light quark spin $s_{\bar{q}}$ and the orbital angular momentum of the system. One obviously has $\vec{j}=\vec{j}_c+\vec{j}_\ell$. In the heavy quark limit, both $j_c$ and $j_\ell$ are “good” quantum numbers. It is convenient for us to treat the system as an S-wave meson of a charm quark and an effective light antiquark $\bar{\tilde{q}}$. Then the wave function for a $D$ meson is $$\begin{aligned}
D_j&\sim [c_{j_1}\bar{\tilde q}_{j_2}-(-1)^{j-j_1-j_2}\bar{\tilde q}_{j_2}c_{j_1}],\end{aligned}$$ where subscripts are spins and the factor $(-1)^{j-j_1-j_2}$ comes from the exchange of two fermions. For an antimeson, we use the wavefunction $$\begin{aligned}
\bar{D}_j&\sim [\bar{c}_{j_1}{\tilde q}_{j_2}-(-1)^{J-j_1-j_2}{\tilde q}_{j_2}\bar{c}_{j_1}].\end{aligned}$$ Here, we have assumed the C-parity transformation to be $$\hat{C}: D_j\leftrightarrow \bar{D}_j .$$ Now we discuss the spin wave function of the system formed by a meson $A$ ($B$) and an antimeson $\bar{B}$ ($\bar{A}$). There are two combinations $A\bar{B}+B\bar{A}$ and $A\bar{B}-B\bar{A}$, which follows the fact that $A$ and $\bar{B}$ do not have definite C-parity while their combinations may have. Considering the exchange of two states, one gets the wave function for the system with a given C-parity $C_X$ $$\begin{aligned}
\label{WF-G}
X_J&\sim&[A\bar{B}+(-1)^{J-J_A-J_B}\bar{B}A]+C_X[(-1)^{J-J_A-J_B}B\bar{A}+\bar{A}B]\nonumber\\
&\to&\Big\{(c_{j_1}\bar{q}_{\bar{j}_2})_{J_{12}} (\bar{c}_{\bar{j}_3}\tilde{q}_{j_4})_{J_{34}}+C_X(-1)^{J-J_{12}-J_{34}}(c_{j_3}\bar{\tilde{q}}_{\bar{j}_4})_{J_{34}}
(\bar{c}_{\bar{j}_1}q_{j_2})_{J_{12}}\Big\},\end{aligned}$$ where the factor $(-1)^{J-J_A-J_B}$ ($(-1)^{J-J_{12}-J_{34}}$) is from the exchange of two bosons. We have added a bar to the spin of the antiquark in order to make it identifiable just from the symbol $\bar{j}$ in the spin wave function. It is enough for us to consider only the following part $$\begin{aligned}
\label{wf-chi}
\chi_J&=&\frac{1}{\sqrt2}\Big\{(j_1\bar{j}_2)_{J_{12}} (\bar{j}_3j_4)_{J_{34}}+C_X(-1)^{J-J_{12}-J_{34}}(j_3\bar{j}_4)_{J_{34}}
(\bar{j}_1j_2)_{J_{12}}\Big\}\nonumber\\
&=&\frac{1}{\sqrt2}\sum_{J_{13},J_{24}}(j_1\bar{j_3})_{J_{13}}[(\bar{j_2}j_4)_{J_{24}}-C_X(-1)^{j_2+j_4+J_{13}+J_{24}}(\bar{j_4}j_2)_{J_{24}}]\nonumber\\
&&\times\sqrt{(2J_{12}+1)(2J_{34}+1)(2J_{13}+1)(2J_{24}+1)}\left\{\begin{array}{ccc}\frac12&j_2&J_{12}\\\frac12&j_4&J_{34}\\J_{13}&J_{24}&J\end{array}\right\}.\end{aligned}$$ Here, $(j_3\bar{j_1})_{J_{13}}=(j_1\bar{j_3})_{J_{13}}$ is the spin wave function of the $c\bar{c}$ pair and the part in the brackets is the spin wave function of $\bar{q}\tilde{q}$ with the C-parity $c_q=C_X(-1)^{J_{13}}$. One should note the case for $A=B$, where an additional factor $\frac{1}{\sqrt2}$ is needed and $(\bar{j_2}j_4)_{J_{24}}=(\bar{j_4}j_2)_{J_{24}}$. With this formula, it is not difficult to find the $c\bar{c}$ spin and corresponding amplitudes.
There are also suggestions that the XYZ mesons are baryon-antibaryon bound states [@qiao]. The above formalism may also be applied to this case. The difference lies in the light degree of freedom. One may consult details given in Ref. [@ours].
Heavy quark spin selection rules
================================
When one applies the above formula to the $D\bar{D}^*$ system, we have $$\begin{aligned}
\chi_J=\left\{\begin{array}{lll}
(j_1\bar{j}_3)_1(\bar{j}_2j_4)_1, &&(C_X=+)\\
\frac{1}{\sqrt2}(j_1\bar{j}_3)_0(\bar{j}_2j_4)_1-\frac{1}{\sqrt2}(j_1\bar{j}_3)_1(\bar{j}_2j_4)_0, &&(C_X=-)
\end{array}\right..\end{aligned}$$ Therefore the $c\bar{c}$ spin is only 1 for the case $J^C=1^+$, which is the result obtained in Ref. [@Voloshin]. By applying the formula to various hidden charm meson-antimeson systems, one finds cases for $J_{c\bar{c}}\neq0$, which are presented in Table \[HQSR\]. These results imply two selection rules:
\(a) If $J=|j_2-j_4|-1$ or $J=j_2+j_4+1$, $J_{c\bar{c}}\neq0$;
\(b) If $A$ and $B$ are different but belong to the same doublet, $J_{c\bar{c}}\neq0$ when $J^C=1^+,2^-,3^+,\cdots$.
These rules are not difficult to understand by exploring the resulting $J^C$ of the system from the two cases $J_{c\bar{c}}=1$ and $J_{c\bar{c}}=0$. The latter case gives less $J^C$ combinations. The reason that the C-parity appears in the second rule is that the light degree part has a definite C-parity.
------------------------------------------- ----------------------- ---------- ----------------------------------------
State $J^{C}$ $J_{24}$ Selection rule for $S_{c\bar{c}}\neq0$
$D\bar{D}^*/D_0^*\bar{D}_1^\prime$ $1^{\pm}$ $0,1$ $J^C=1^+$
$D^*\bar{D}^*/D_1^\prime\bar{D}_1^\prime$ $0^+,1^-,2^+$ $0,1$ $J=2$
$D^*\bar{D}_1^\prime$ $0^\pm,1^\pm,2^\pm$ $0,1$ $J=2$
$D^*\bar{D}_1/D_1^\prime \bar{D}_1$ $0^\pm,1^\pm,2^\pm$ $1,2$ $J=0$
$D^*\bar{D}^*_2/D_1^\prime\bar{D}^*_2$ $1^\pm,2^\pm,3^\pm$ $1,2$ $J=3$
$D_1\bar{D}^*_2$ $1^\pm,2^\pm,3^\pm$ $0\sim3$ $J^C=1^+,2^-,3^+$
$D^*_2\bar{D}^*_2$ $0^+,1^-,2^+,3^-,4^+$ $0\sim3$ $J=4$
------------------------------------------- ----------------------- ---------- ----------------------------------------
: S-wave meson-antimeson states, quantum numbers, and selection rules for $J_{c\bar{c}}\neq0$.[]{data-label="HQSR"}
Constraints for strong decays
=============================
In the heavy quark limit, the spin-flip of the $c\bar{c}$ pair in a meson-antimeson state is suppressed, i.e. the $c\bar{c}$ spin $J_{c\bar{c}}$ is conserved in the strong decay of the system. If it contains only $J_{c\bar{c}}=1$ part, one in principle cannot observe decay products involving spin-singlet charmonium. Therefore, decay channels containing charmonium may reflect the spin structure of the initial meson.
For the interesting $D\bar{D}^{*}$ state, the possibility $I^G(J^{PC})=0^+(1^{++})$ is related with $X(3872)$ while the case $I^G(J^{PC})=1^+(1^{+-})$ is related with the newly observed $Z_c(3900)$. If $Z_c(3900)$ is a $D\bar{D}^{*}$ molecule or resonance, there is no special selection for the $c\bar{c}$ spin. Its decays into $\chi_{c0,1,2}\pi\pi$, $\eta_c\rho$, and $h_c\pi$ are all expected. However, in a recent search for $Z_c(3900)$ in the process $e^+e^-\to\pi\pi h_c$ at BESIII [@Z4020], no significant signal was observed. This fact implies that $Z_c(3900)$ might contain only $J_{c\bar{c}}=1$ component and might be a compact tetraquark. If it is the case, the decay into $\eta_c\rho$ should also be suppressed. More experimental information is needed to get a stronger conclusion.
For the $D^*\bar{D}^*$ system, the allowed $J^{PC}$ are $0^{++}$, $1^{+-}$, and $2^{++}$. Here we concentrate only on isovector case which is related with charged states around the threshold. The possible decay channels for S-wave $D^*\bar{D}^*$ molecule or resonance are $0^{++}\to \eta_c\pi,\,J/\psi\rho,\,\chi_{c1}\pi$, $1^{+-}\to \eta_c\rho,\,h_c\pi,\,J/\psi\pi$, and $2^{++}\to J/\psi\rho, \, \chi_{c1}\pi,\,\chi_{c2}\pi$. BESIII observed a charged $Z_c(4020)$ in the invariant mass of $h_c\pi$ [@Z4020]. If it is a $D^*\bar{D}^*$ molecule, its $J^{PC}=1^{+-}$ are favored and its decay into both $\eta_c\rho$ and $J/\psi\pi$ channels should be observed. If $J/\psi\pi$ channel cannot be observed, the state might contain only $J_{c\bar{c}}=0$ component and not be a molecule. In either case, the channel $\eta_c\rho$ is expected.
Interesting observations also exist for decay properties of other meson-antimeson states. For example, if a $J^{PC}=1^{--}$ $D^*\bar{D}_2^*$ molecule or resonance exists, its branching ratios for $h_c\eta$ ($h_c\pi$) and $\eta_c\omega$ ($\eta_c\rho$) channels are expected to be large since the coefficient for the $J_{c\bar{c}}=0$ spin component in the wave function is $\frac{\sqrt{10}}{4}$.
To summarize, we discussed in what condition the $c\bar{c}$ spin is only 1 for an S-wave meson-antimeson state. We obtain two selection rules in the heavy quark limit. Since the $c\bar{c}$ spin is conserved in the strong decay, the rules are helpful to understand the structure of the initial meson from their decay channels. In reality, the mass of heavy quark is finite and the selection rules are violated. The strength of violation needs to be investigated in the future.
This project was supported by the National Natural Science Foundation of China (No. 11275115), SRF for ROCS, SEM, and Independent Innovation Foundation of Shandong University.
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|
---
abstract: |
Proteins are a matter of dual nature. As a physical object, a protein molecule is a folded chain of amino acids with multifarious biochemistry. But it is also an instantiation along an evolutionary trajectory determined by the function performed by the protein within a hierarchy of interwoven interaction networks of the cell, the organism and the population. A physical theory of proteins therefore needs to unify both aspects, the biophysical and the evolutionary. Specifically, it should provide a model of how the DNA gene is mapped into the functional phenotype of the protein.
We review several physical approaches to the protein problem, focusing on a mechanical framework which treats proteins as evolvable condensed matter: Mutations introduce localized perturbations in the gene, which are translated to localized perturbations in the protein matter. A natural tool to examine how mutations shape the phenotype are Green’s functions. They map the evolutionary linkage among mutations in the gene (termed epistasis) to cooperative physical interactions among the amino acids in the protein. We discuss how the mechanistic view can be applied to examine basic questions of protein evolution and design.
author:
- 'Jean-Pierre Eckmann $^{1,2}$, Jacques Rougemont$^{1}$, Tsvi Tlusty $^{3,4}$'
bibliography:
- 'green.bib'
title: 'Proteins: the physics of amorphous evolving matter'
---
Sections marked with $^*$ contain more technical material and can be omitted at first reading.
The protein problem: a theoretical physics perspective {#sec:proteinproblem}
======================================================
The macromolecules that make living matter – lipids, hydrocarbons, nucleic acids, and in particular proteins – are among the most studied objects of Nature. Proteins comprise the central nano-machinery of the cell, whose numerous functions include the formation of structural elements, catalyzing metabolic reactions and conveying biochemical signals [@alberts1998cell; @Fersht1999; @Howard2001; @Goodsell2009; @Whitford2013a]. For their significance in life, proteins and the genes that encode them have been extensively investigated using various experimental methods, such as crystallography, biochemical assays, mass spectrometry, fluorescence imaging, electron microscopy, directed evolution and deep sequencing [@Rambo2013; @Cohen2001; @Collins2011; @Mandala2018; @Barrera2011; @Mehmood2015; @Ha2012; @Mardis2013; @Chapman2011; @Fernandez-Leiro2016]. In parallel, sophisticated computational models, such as molecular dynamics, have been developed to predict the structure, function and folding of proteins [@Karplus2002; @Karplus2005; @Adcock2006; @Dror2018; @Scheraga2018; @Isralewitz2001].
These experiments and simulations provide valuable data on protein structure, dynamics and genetics. However, there remain two inherent challenges: (i) Sparsity of data – the protein is the outcome of long evolutionary search in a high-dimensional space of gene sequences, which is impossible to sample, even by high-throughput experiments. (ii) Complexity of interactions – The function of a protein arises from collective many-body interactions in the heterogeneous amino acid matter, which are hard to probe and model.
In light of these challenges, we focus in this colloquium on a complementary theoretical approach that links the protein problem to the realm of condensed matter physics. Rather than using realistic simulations predicting the dynamics and function of concrete proteins, we shall discuss minimal models that allow, under several simplifying assumptions, to examine basic questions of protein evolution, especially how the collective physical interactions within the protein direct its evolution.[^1]
The structure of many proteins is known at a resolution of a few angstroms and there are detailed computational models of the forces between the amino acids. Here, however, the protein will be examined at a coarse grained level, in the spirit of lattice [@Lau1989; @Shakhnovich1991] and network [@Chennubhotla2005] models. The protein will be described as a connected network whose nodes represent the amino acids. Furthermore, from this conceptual point of view, it suffices to assume that there are only two types of amino acids instead of the usual twenty.[^2] (for example the classical HP model [@Lau1989] used in , in which amino acids can only be either hydrophobic ($\aH$) or polar ($\aP$)). In a real protein, forces between the amino acids are a complicated combination depending for example on their polarity, hydrophobicity, charge and shape. At the coarse grained level, it again suffices to consider instead just simple springs between pairs of neighboring sites. This is akin to using harmonic approximations in mechanics, which provide a generic understanding and a good physical insight.
With this kind of simplifications, one can translate certain questions of biology to analogous questions in the physics of amorphous networks. Among the rich set of methods in this classical subject of physics, some tools seem particularly well adapted to the protein problem. The approach is based on the dual nature of the protein; it is a physical object whose formation and physical interactions are also represented in the ‘dual’ gene, a sequence of symbols from a four-letter alphabet of the DNA bases, ‘A’, ‘C’, ‘G’, ‘T’. Evolution progresses by introducing mutations, that is, permanent modifications of this sequence. There are local mutations (nucleotide substitutions, short insertions and deletions) besides larger scale modifications (translocations, inversions, duplications). A natural approach to study protein evolution is to model the effect of mutations on the physical properties of the amino acids network.
Local mutations amount to short jumps between neighboring sequences in the genotype space, differing by one letter only, while large-scale mutations are equivalent to longer jumps. Both classes of mutations can be described in terms of alterations of the mechanical properties of the amino acid network. However, we shall focus on the class of local mutations. Practically, local mutations are easy to treat with classical techniques of condensed matter, for instance via Green’s functions, since they induce localized perturbations in the spring network. More importantly, it is possible to statistically sample the genotype space with continuous trajectories progressing by consecutive local mutations. This will be the main axis of this colloquium. Along the evolutionary trajectory, mutations come in three flavors: The ones leading to some sort of functional catastrophe or significant disadvantage, and therefore get eliminated by selection; others which improve the properties of a protein and finally, the large ‘neutral’ majority which do not induce any significant change in the function of the protein [@Kimura1983; @Neher2011]. In this manner, the ‘learning’ evolutionary process reduces the problem of improving a protein from an exhaustive combinatorial search approach into a biased random walk. This drastically reduces the dimension of the space which one needs to explore.
The condensed-matter approach to the protein problem may be viewed as an example of a potentially general framework that may be used to examine other strongly-coupled biological systems. For example, one may analyze metabolic and genetic networks in terms of localized perturbations and Green’s functions. Such analysis may suggest common underlying principles. It might as well turn out that biology is more contingent and depends on the history of the evolutionary process, but at least the few examples we describe give us hope that a rational approach, based on the laws of physics, may be useful in some cases.
Biological molecules are far from being the spring networks we use as a model. Still, similar abstractions proved successful in many areas of physics. For example, the dynamical systems of the so-called Axiom A class [@Eckmann1985] are systems with a very special, yet simple, structure. And although most systems do not belong to the Axiom A class, it proved very useful to consider that they behave ‘as if’.
There is a long history of studies in similar spirit of abstraction and simplification, starting with the conformal maps of D’Arcy Thompson [@Thompson1942], through the morphogenetic studies based on the theory of catastrophes by René Thom [@Thom2018]. In the 21st century researchers have much more data available on biological systems. This allows to test hypotheses against measurements, infer from the data other questions to investigate, and suggest possible experiments to confirm or refute the theory. We close the introduction by two citations which reflect the general outlook of this colloquium:
**Misha Gromov**, in [@Gromov2013 Abstract]
> When you read a textbook on molecular/cellular biology you are enchanted by the logical beauty of biological structures. You want to share your excitement with your colleagues, but…you find out you are unable to do it: there is no language in the 21st century mathematics that can express this beauty. You feel there must be a new world of mathematical structures shadowing what we see in Life, a new language we do not know yet, something in the spirit the ‘language’ of calculus we use when describing physical systems.
**Giovanni Jona-Lasinio**, in [@Jonalasinio2012]:
> Theoretical physics was recognized as an independent field of research only at the end of the 19th century, shortly before the great conceptual revolutions of relativity and quantum mechanics. Today theoretical physics has multiple facets. I think that the time has come for a more precise characterization of the research field of theoretical biology, and for an assessment of its scope. \[Translated from Italian\]
We are convinced that such outlooks are important and our work should be viewed as an attempt in this general direction, in the hope that readers will be encouraged to proceed along this path.
Biology as a challenge to theorists {#sec:challenge}
===================================
Biological research has been extremely active in the past decades and experimental results have flourished to vastly improve our understanding of living matter. The challenge for theorists is to find subtopics which are at a stage where theoretical abstraction can be fruitful.
Here we focus on the relation between genes and the functions of proteins: genes (in DNA) code for amino acid chains that fold into the three-dimensional configurations of functional proteins. This sequence-to-function map is hard to decrypt since it links the collective physical interactions inside the protein to the corresponding evolutionary forces acting on the genome [@Koonin2002; @Xia2004; @Dill2012; @Zeldovich2008; @Liberles2012]. Furthermore, evolution selects the tiny fraction of functional sequences in an enormous, high-dimensional space [@Povolotskaya2010; @Keefe2001; @Koehl2002], which implies that proteins form non-generic, information-rich matter, outside the scope of standard statistical methods. Therefore, although the structure and physical forces within a protein have been extensively studied, the fundamental question of how a functional protein originates from a linear DNA sequence still provides research challenges, in particular how functionality constrains the accessible DNA sequences.
To examine the geometry of the sequence-to-function map, we devise below a mechanical model of proteins as amorphous evolving matter.[^3] Rather than simulating concrete proteins, we construct models which describe the hallmarks of the genotype-to-phenotype map (the translation of the gene to the protein). These models are sufficiently simple so that large-scale simulations can be performed, which allow to average over stochastic noise inherent to evolutionary dynamics. Furthermore, we restrict our approach to models in which the function of a protein arises from large-scale conformational changes, where big chunks of the protein move with respect to each other. These motions are central to certain functions [@Koshland1958; @Henzler-Wildman2007; @Savir2007; @Schmeing2009; @Savir2010a; @Huse2002; @Savir2013], For example, allosteric proteins are a type of ‘mechanical transducers’ that transmit regulatory signals between distant sites [@Perutz1970; @Goodey2008; @Lockless1999; @Ferreon2013].
We end this section by mentioning a few papers which have dealt with similar issues, and which highlight the increasing interest in connecting biological questions with methods from solid state physics.
Common to these studies is a mechanical perspective on protein function. The motivation originates from many observations of proteins whose functions involve collective patterns of forces and coordinated displacements of their amino acids [@Daniel2003; @Bustamante2004; @Hammes-Schiffer2006; @Boehr2006; @Karplus2002; @Henzler-Wildman2007; @Huse2002; @Eisenmesser2005; @Goodey2008; @Savir2010a]. In particular, the mechanisms of allostery [@Monod1965; @Perutz1970; @Cui2008; @Daily2008; @Motlagh2014; @Thirumalai2018; @Koshland1966], induced fit [@Koshland1958], and conformational selection [@Grant2010] often involve global conformational changes by hinge-like rotations, twists or shear-like sliding of protein subdomains [@Gerstein1994; @Mitchell2016; @Mitchell2017].
A now-standard approach to examine the link between function and motion is to model proteins as elastic networks of amino acids connected by spring-like bonds. Early studies that apply this class of models are from the 1980s and 90s [@Levitt1985; @Tirion1996], and in the last two decades the methods have been further developed and applied to many proteins [@Chennubhotla2005; @Bahar2010; @Lopez-Blanco2016]. Decomposing the dynamics of the network into normal modes revealed that low-frequency ‘soft’ modes capture functionally relevant large-scale motion [@Tama2001; @Bahar2010a; @Haliloglu2015], especially in allosteric proteins [@Ming2005; @Zheng2006; @Hawkins2006; @Arora2007; @Tehver2009; @Wrabl2011; @Greener2015].
Recent work associates the soft modes of protein conformations with the emergence of weakly connected regions as described above, but also ‘cracks’ [@Miyashita2003], ‘shear bands’ or ‘channels’ [@Mitchell2016; @Mitchell2017; @Tlusty2016; @Tlusty2017; @Dutta2018; @Rocks2019] that enable low-energy viscoelastic motion [@Qu2013; @Joseph2014]. Such contiguous domains evolve in models of allosteric proteins [@Hemery2015; @Flechsig2017; @Tlusty2017].
A source of inspiration for linking proteins to the physics of amorphous matter are the papers by the late Shlomo Alexander, especially [@Alexander1998; @Alexander1982]. In these works, Alexander highlighted the essential role of ‘floppy modes’ in the mechanical spectrum of amorphous solids. Also relevant are studies by Thorpe and Phillips on constraint theory and rigidity percolation in glasses, such as [@Thorpe1985; @Phillips1985]. Those works highlighted the ability to control the rigidity and accessible zero-energy modes of mechanical networks by balancing the number of degrees of freedom and the number of constraints, as was noted by Maxwell in 1864 [@Maxwell1864].
The link between the dynamical spectra of proteins and amorphous matter has been further explored in a recent series of works on mechanical metamaterials. The emergence of long-range allosteric response was used in [@Rocks2017] as a design principle for ‘programmable’ metamaterial made of amorphous spring networks [@Rocks2018]. A similar random network approach was applied in [@Yan2017] to design elastic materials with tailored mechanical response. These works suggest that tunable amorphous materials have the flexibility required to produce elaborate designs, as recently demonstrated by mimicking the cyclical conformational motion of protein motors [@Flechsig2018]. These promising approaches to metamaterial design are discussed elsewhere, for example in [@Ronellenfitsch2018; @Kim2018; @Rocklin2017; @Baardink2018].
The present Colloquium focuses on a different aspect: understanding fundamental properties of the protein evolution – in particular the genotype-to-phenotype map – within the framework of condensed matter theory.
Proteins as information machines {#sec:informationmachines}
================================
The building plan of a protein is determined by its corresponding gene, via the genetic code. The gene is a 1-dimensional string in an alphabet of 4 letters: the nucleotides ‘A’ (adenine), ‘C’ (cytosine), ‘G’ (guanine), and ‘T’ (thymine) (see [@Alberts2017], Ch. 6). The protein is a (folded) chain of amino acids (AA) which is translated from the gene according to the genetic code: each three successive letters (each non-overlapping triplet, called a codon) maps to a single AA. In principle, this would allow for $4^3$ possibilities, but in general there are only 20 different AA’s, making the code redundant, as we shall discuss in .
We view the gene, the 1-dimensional string of letters as the tape of a Turing machine [@Turing1936; @Herken1992; @Condon2018]. Since any alphabet can be recoded in binary (for example, each of the 4 nucleotides can be recoded as a 2-bits number), one can always think of it as a string of ‘0’s and ‘1’s. The proteins (and the transcription-translation machinery, which is itself made of proteins) would be the computer, which is able to read and interpret the string.
This particular machine is an example of a self-reproducing Turing machine [@Neumann1966], since the replication of the genome can be achieved by genome-encoded proteins. In addition, these machines are evolving when the genes are mutated. In other words, the machine can modify its own tape (see also [@Tlusty2016]). A further study in this direction is [@Dyson1970], but there are many more, see [@Freitag2004].
Handling reading errors {#sec:readingerrors}
-----------------------
Translation of the gene into its corresponding string of amino acids requires a specialized machinery, which includes the ribosome [@Alberts2017].[^4] The translation machinery ‘reads’ the code through chemical affinity, and might therefore mis-read the tape. Most amino acids are encoded by more than one codon, and this hard-coded redundancy of the genetic code helps to reduce the impact of such misreadings (see [@Tlusty2007; @Tlusty2008; @Tlusty2008a; @Tlusty2008b; @Tlusty2010] for a theoretical study and [@Eckmann2008] for an illustration).
As noted above this system allows for $64=4^3$ different codons (number of triplets from an alphabet of 4 nucleotides), but they generate only 21 different symbols.[^5] The geometric aspects of this arrangement of 21 among 64 possibilities can be understood in graph-theoretical terms: One presents the 64 codons as the nodes of a codon-graph, and two nodes are connected by a link if the corresponding codons differ in only one symbol. Note that swapping ‘C’ and ‘T’ in the codon’s third position always results in the same AA () and we can therefore reduce the graph to $48=4^2\cdot 3$ nodes.[^6] In the codon-graph, each amino acid is coded as a simply connected region, as shown in , with the exception of Serine (ser) (Arginin (arg) is disconnected in the 2D table, but not in the graph). Such an arrangement minimizes the ratio of surface by area for each region. This reduces the probability of coding the wrong AA, under the assumption that most reading errors involve only one-letter differences.
![A representation of the genetic code, as function of the measured polarity of each codon (values from [@Haig1991; @Woese1965]). The smoothness of the landscape shows that moving from one AA to its neighbor does not change the polarity too abruptly.[]{data-label="fig:landscape"}](ert_fig01.jpg){width="1.0\columnwidth"}
Additionally, amino acids with similar chemical properties (for example polarity) tend to be neighbors in this graph. This can be visualized by plotting the measured polarity as a function of the codon, which produces a relatively smooth landscape. The smoothness manifests the chemical similarity between neighboring amino acids, and implies that most misreadings change the polarity of AA only moderately. We note that, unlike the 2D landscape of , an ideal representation should wrap the surface so that each AA would have 8 neighbors (and can therefore be embedded only in high dimension).
For the connection between the numbers 21 and 48, an inequality can be given in terms of the genus of the codon-graph [@Tlusty2007; @Tlusty2007a; @Tlusty2010] (this uses results from [@Colin1993; @Banchoff1965]). Without going into further detail we conclude: *The optimal code must balance contradicting needs for tolerance to errors (with the smoothness of the mapping between codons and chemical space) and chemical diversity, which is essential for the versatility of protein function*.
Folding
-------
Having translated the gene into a linear chain of amino acids (the backbone, see ) via the genetic code (and modulo translation errors), this chain will spontaneously fold into a 3-dimensional shape which gives rise to its function. How this folding proceeds is an important and difficult question, which we shall not address here. Instead we will assume that a certain folding pattern is preserved (see [@Petsko2004] for a discussion of these issues). This assumption is practical, as we shall be mostly interested in how the function of the protein changes under point mutations of the gene, bit flips of the code in the tape. Such mutations often do not seriously affect the overall shape of the protein (see also [@Bussemaker1997]).
![Schematic illustration of protein folding: Proteins are polypeptides, linear heteropolymers of AAs (colored spheres), linked by covalent peptide bonds (red sticks), which form the protein backbone. These peptide bonds are much stronger than the non-covalent interactions among the AAs (side chains) and do not change when the protein mutates.[]{data-label="fig:backbone"}](ert_fig02top.jpg "fig:"){width=".9\columnwidth"} ![Schematic illustration of protein folding: Proteins are polypeptides, linear heteropolymers of AAs (colored spheres), linked by covalent peptide bonds (red sticks), which form the protein backbone. These peptide bonds are much stronger than the non-covalent interactions among the AAs (side chains) and do not change when the protein mutates.[]{data-label="fig:backbone"}](ert_fig02bottom.jpg "fig:"){width="0.5\columnwidth"}
We can next model the function of this folded amino acid chain and we will show that there is yet another level of redundancy besides the redundancy of the genetic code and the robustness of the folding. we shall see that there are many mutations which have no effect on performance. *Namely, there is high redundancy in the AA sequences that are mapped to the same or similar enough protein function*. we shall quantify this property in terms of dimension [@Grassberger1983; @Eckmann1992].
Mechanical views on protein evolution {#sec:views}
=====================================
Consider a protein interacting with a small molecule. Presence of the latter often induces a conformational change at some distance from the interaction site. One important example is the class of allosteric proteins for which an active site is regulated by binding at another site, resulting in a reconfiguration of the active site. More specifically, we shall examine the role of large-scale, functionally-relevant dynamical modes, and their link to long-range genetic correlations.
Before reviewing the literature on this issue, we illustrate such a mechanical effect on a particular example: human glucokinase (which is involved in sugar metabolism), see . The data were obtained from crystallographic structure of two conformations of that protein: the first (PDB[^7] accession 1v4s) corresponds to the binding of glucose to its active site and is compared to the conformation in the absence of glucose (PDB 1v4t) [@Kamata2004].
The backbone, see , is shown as a light blue curled tube, and the arrows indicate the displacement from one shape to the other (any Galilean motion between the two is eliminated). The color of the arrows indicates up/down motion relative to a horizontal plane. The red coloring in the twisted tube shows the high shear region separating two low-shear domains that move as rigid bodies (shear calculated by the method of [@Mitchell2016; @Rougemont2099]).
On a conceptual level, one can simplify the figure as shown in . The protein seems to have a central shear band and two external flaps which perform a rotating motion when a ligand attaches to the protein. This kind of mechanical phenomenology is accessible to the language of physics.
Large-scale motions take part in several basic biological functions and mechanisms. For example, in the induced fit [@Koshland1958] and conformational selection [@Bahar2007; @Grant2010] mechanisms, the presence of a substrate induces reshaping of the enzyme to properly align the catalytic groups in the active site. Such reshaping is a dynamic mechanism of *specific recognition* that allows the selection of a target ligand among similar competing molecules [@Savir2007; @Savir2013]. In *allostery*, reconfiguration of the active site is regulated by binding at a secondary, allosteric site, often via long-range mechanical interactions [@Motlagh2014; @Thirumalai2018]. In this Colloquium, we describe simple physical models for the emergence of these mechanisms via evolutionary tuning of the protein’s mechanical response.
![The motion and deformation between two states of glucokinase in state 1v4s and 1v4t. The arrows are scaled up for better visibility. They are colored green, resp. red depending on whether they move down or up relative to a plane passing horizontally through the center of the protein. Galilean motions have been eliminated. The red coloring of the tube corresponds to concentration of shear and is the same as in the leftmost panel of . See .[]{data-label="fig:gluco"}](ert_fig03.jpg){width="0.8\columnwidth"}
![A schematic interpretation of . Emphasis is given to the two moving pieces, with a hinge between them. This kind of hinge will be called the ‘fluid channel’ or ‘shear band’.[]{data-label="fig:glucosymbolic"}](ert_fig04.jpg){width="0.8\columnwidth"}
Like their dynamic phenotypes, proteins’ genotypes (their gene sequences), as explained in , are remarkably collective. The history of protein evolution can be traced by gathering evolutionary related proteins in different species (homologous proteins) and aligning their sequences. Genes of these proteins sometimes display long-range correlations [@Goebel1994; @Marks2011; @Jones2012; @Lockless1999; @Suel2003; @Hopf2017; @Poelwijk2017; @Halabi2009; @Tesileanu2015; @Juan2013]. The correlations indicate epistasis, the compensatory mutations that take place among residues linked by physical forces or common function. As an example [@Rougemont2099], consider again glucokinase. We aligned about 120 variants of this molecule and asked where along the gene have mutations preferentially occurred ().
{width="\x\columnwidth"}{width="\x\columnwidth"}{width="\x\columnwidth"}
Still, the relationship between sequence correlation, epistasis and selection pressure are not fully understood. As discussed in , the two main challenges are the intricacy of the physical forces among the amino acids, and the high dimensionality of the of the genotype-to-phenotype map [@Koonin2002; @Povolotskaya2010; @Liberles2012]. These inherent difficulties motivated the development of complementary approaches which utilized simplified coarse-grained models, such as lattice proteins [@Lau1989; @Shakhnovich1991] or elastic networks [@Chennubhotla2005]. Network and lattice models have been recently used to study the evolution of allostery in proteins and in biologically-inspired allosteric matter [@Tlusty2016; @Tlusty2017; @Hemery2015; @Flechsig2017; @Rocks2017; @Yan2017]. Our aim here is different: to construct a simplified condensed-matter model in terms of how the mechanical interactions within the protein shape its evolution.
Condensed-matter theory of proteins
===================================
This section will review a theory of proteins in terms of evolvable condensed matter. we shall discuss the conceptual roots of this approach in the physics of amorphous matter (mainly glasses) and spectral theory. We will introduce the basic setting of modeling proteins as evolving amino acid networks. The emergence of function is associated with the evolution of a weakly connected region, which enables a low-energy ‘floppy’ mode to appear. This minimal network approach allows one to examine basic questions of protein evolution.
we shall discuss two different models in this review. One will be called the ‘cylinder-model’ and the other the ‘HP-model’. The first model is simpler, but the second comes somewhat closer the biological reality. Before distinguishing the two models, we describe their common features.
Lattice models {#sec:lattice}
--------------
Our protein is modeled by a finite (regular) lattice in 2 (or 3) dimensions. We assume that the lattice forms a cylinder (periodic boundary conditions) or an open rectangle (open boundary conditions) of width $w$ and height $h$ (see the examples in –\[sec:pnasmodel\]).
It is important to note that $w$ and $h$ are *finite* while otherwise quite arbitrary. This is so because the protein should not be viewed as a problem of thermodynamic limits, but rather in the context of small amorphous objects. This being said, other aspects of the geometry seem less important. The points may also be chosen as lying on small perturbations from the regular lattice to avoid effects of lattice symmetry on the spectrum. The number of AAs should typically be in the range 200–2000, corresponding to the typical size of the protein.
Amino acids interact via electrostatic forces, van der Waals forces, hydrogen bonds, disulfide bonds and hydrophobicity [@Petsko2004; @Fersht1999]. All these are short range interactions, which amount to *local* coupling between lattice points. we shall therefore assume that each AA interacts with its nearest and next nearest neighbors. For example on a hexagonal lattice, with nearest and next-nearest neighbors linked, the number of connections (the node’s degree) is at most 12; all nodes in the protein interior have 12 links (bonds) while those at the boundary have fewer (but at least 3), see .
Finally, the coupling itself is modeled by harmonic springs carried by each graph link [@Born1954; @Alexander1998; @Chennubhotla2005; @Tirion1996]. Its strength is determined only by the types of AAs at each end of the link.
The lattice Laplacian {#subsec:lattice}
---------------------
The lattice and its links may be viewed as an abstract graph. This means that one can define gradients and Laplacians [@Biggs1993; @Chung1997].[^8] In the graph, there are $n_a=w\times h$ amino acid nodes, indexed by Roman letters, and $n_b$ bonds, indexed by Greek letters.[^9]
First, one endows every bond in the graph with an arbitrary but fixed orientation, and then the incidence matrix of a graph is the $n_b\times n_a$ matrix defined by $$\nabla_{\alpha i} =\left\{
\begin{array}{@{}rl@{}}
-1~, & \text{if $i$ is the initial vertex of edge $\alpha$}~,\\
1~, & \text{if $i$ is the final vertex of edge $\alpha$}~,\\
0~, & \text{if $i$ is not in $\alpha$}~.\\
\end{array}\right .$$ Remark that for any function $f$ on the vertices, the map $f \mapsto \nabla f$ is the co-boundary mapping of the graph, namely $$\nabla f (\alpha )= f(j)-f(i)~,$$ where $\alpha$ is the link connecting $i$ to $j$.
As in the continuum case, the Laplace operator $\Delta$ is the product $\Delta = \nabla^\T\nabla$, where $^\T$ denotes the adjoint. The non-diagonal elements $\Delta_{ij}$ are $-1$ if $i$ and $j$ are connected and $0$ otherwise. The diagonal part of $\Delta$ is the degree $\Delta_{ii}= z_i$, the number of nodes connected to $i$. Note that this is a discrete graph Laplacian, and no coordinates are involved so far.
We next embed the graph in a Euclidean space $\mathbb{R}^d$ ($d=2$), by assigning positions $r_i \in \mathbb{R}^d$ to each AA, to each lattice point $i=1,\dots,n_a$. This is coded as a $n_a\times d$ real matrix $\rb$. Finally, to each bond $\alpha $ we assign a spring with constant $k_{\alpha}$ which we view as the diagonal elements of an $n_b
{\times} n_b$ matrix $\Kb$: $$\Kb_{\alpha \beta}\,=\, k_\alpha\delta_{\alpha\beta}~.$$ This defines a deformable spring network which has an internal energy, an equilibrium configuration.
To account for the energy cost of deformations in the lattice protein, one considers the elastic tensor $\HH$ (or Hamiltonian) which we now describe in detail [@Chung1992 pp. 618–619]. The quantity $\HH$ is a tensor because the deformations are not scalars, but vectors in $\mathbb{R}^d$. We first denote by $\nb_{\alpha}$ the (normalized) direction vectors for each bond $\alpha=(i,j)$: $\nb_{\alpha}=\left(\rb_i - \rb_j\right)/\left\vert\rb_i -\rb_j\right\vert$. Then, we define the ‘embedded’ gradient tensor $\Db$ (of size $n_b\times n_a\times d$) which is obtained by multiplying each element of the graph gradient $\nabla$ by the corresponding vector $\nb$:
\_[i]{}= \_\^\_[i]{} , \[eq:mba\]
namely each projection of $\Db$ on a bond $\alpha$, $\Db_{\alpha, :}$, is a $n_a\times d$ matrix containing only $2d$ non-zero entries in rows $i$ and $j$, which correspond to the components of the unit vector along the bond $\alpha=(i,j)$.
Let $\ub_i$ be the displacement vectors of each vertex from $\rb_i$ to $\rb_i+\ub_i$, therefore $\ub$ is a $n_a\times d$ matrix. The elastic energy of such a perturbation is
\[eq:Hamiltonian\] = \^ = \_[ij]{}\_[k]{}\_[ik]{} (\_[ij]{})\_[k]{} \_[j]{} ,
where the Hamiltonian tensor is defined as
&=\^\
&=( \_\_[ik]{}\_\_[j]{} )\_[i,j=1,…,n\_a; k,=1,…,d]{} .
The $d\times d$ off-diagonal components are:
\_[ij]{} =& \_\_[i]{}\_ \_ \_\^\_[j]{}\
=&\_[ij]{} k\_[(i,j)]{}\_[(i,j)]{}\_[(i,j)]{}\^ ,
which we complete with the diagonal blocks ($i=j$) so that rows and columns sum to zero: $\HH_{ii}=-\sum_{j \ne i}{\HH_{ij}}$.
In this construction, we have assumed that the equilibrium configuration of the network (described by the vectors $\rb_i$) is such that all springs are at their equilibrium length, disregarding the possibility of ‘internal stress’ [@Alexander1998], hence the initial elastic energy is $0$. The extension of the theory to networks that are initially frustrated, where some springs are stretched or squeezed is a difficult subject. A paper which studies the conjectures by Alexander on internally stressed networks is [@Kustanovich2003]. The spring constant is therefore the derivative of the interaction at the equilibrium length of the spring.
We next consider only small deviations of the AAs from their equilibrium, the linear mechanical response of the protein to an applied force. While this approximation cannot account for plastic and non-affine deformations that often occur in real proteins, it certainly simplifies the analysis, in contrast to the inherent difficulties of studying fully nonlinear systems.
Given a prescribed ‘protein fold’ (the lattice positions $\rb_i$, $i=1,\dots,n_a$), a gene first determines the spring constants via a ‘genetic code’ which maps codons to AAs on the lattice and thereby determines the interaction strength between neighbors on the lattice. This in turn defines a phenotype of the protein, namely its mechanical response under deformations. Each choice of the gene, the set of codons $\cb=\{c_i\}$, defines a Hamiltonian $\HH=\HH(\cb)$. Component-wise, each $d {\times} d$ block $\HH_{ij}$ ($i \neq j$) depends only on the codons $c_i$ and $c_j$: $\HH_{ij}(\cb)=\HH_{ij}(c_i, c_j)$.
In summary, note that $\HH$ depends on three things:
The position $\rb_i$ of each amino acid, $i=1,\dots,n_a$,
The type $c_i$ of each amino acid, $i=1,\dots, n_a$,
The spring constants $k(c,c')$ representing the interaction strengths between amino-acids types $c$ and $c'$.
This definition is clearly versatile enough to be generalized to other systems, such as proteins made of the standard 20 AAs with specific interaction constants for each of the possible AA pair.
Hooke’s law
-----------
We have now a map from genes $\cb$ to Hamiltonians $\HH(\cb)$, and we want to study the deformability of the network as a function of $\cb$. In the linear regime of relatively moderate deformations, one can use Hooke’s law (see [@Alexander1998]) to relate a (small) deformation $\ub$ to the force by
= ,
where $\fb$ is a force vector field. We are interested in the inverse relation, since we want to know the deformation of the network (protein) as a function of the applied force $\fb$. This inverse will be described by Green’s function $\GG$:
\[eq:gf\] = .
Simulating evolution {#sec:simulating}
====================
Next, we consider modeling evolution, for a general ‘genetic code’. As described above, to each gene $\cb$ there is a natural Hamiltonian $\HH(\cb)$ associated with it. This is the mechanical genotype-to-phenotype map. We assume that a fitness function $F$ is given, mapping every $\HH$ to its fitness score $F(\HH)\in\real$. The observable we take later as $F$ will be an expectation value for some components of the force field $\fb$. The evolutionary process alters this fitness, by mutating individual random positions in the gene $\cb$ (the collection of $c_i$). This is realized by a Metropolis algorithm [@Metropolis1953]:
In an evolution simulation, one exchanges a randomly selected codon with another one (at the same position), while demanding that the fitness change $\delta F$ is positive or non-negative. We call $\delta F >0$ a beneficial mutation, whereas $\delta F=0$ corresponds to a neutral one. Deleterious mutations, $\delta F<0$, are generally rejected.
As in statistical physics, variants of this algorithm can be envisaged, for example, by asking for an increase of $F$ by a minimal factor $|F|\to |F|\cdot (1+\epsilon )$ with $\epsilon >0$, for a step to be accepted. Other possibilities include the introduction of ‘temperature’, accepting or rejecting even deleterious mutations, $\delta F<0$, with some probability. The rationale behind using these variants of Metropolis algorithms lies in the nature of natural mutations. For a review of the role of deleterious mutations, see [@Kondrashov2017]. Details of $F$ will be given when we discuss various models in .
Green’s function as a link between the theory of amorphous solids and living matter {#sec:green}
===================================================================================
In the previous section, the ground was prepared for studying the connection between the genes and the mechanical properties of the proteins they code. we shall use the mapping from the gene $\cb$ to the Hamiltonian $\HH(\cb)$ introduced above. One of the questions to be examined is how the protein reacts to forces applied to it, and how this response is encoded in the gene. Such forces occur when a small ligand molecule attaches to a binding site on the protein’s surface, inducing a mechanical response in other regions of the protein.
Intuitively, this means that we are looking for a relatively strong reaction to a weak signal. Such phenomena are captured by soft modes. Such modes are given by zero eigenvalues of the Hamiltonian $\HH$, and the corresponding deformations are described by the eigenvector $\ub$ of displacements of $\rb$ (corresponding to the zero eigenvalue).
Among the many approaches to the zero eigenvalue problem, we use the methods of Green’s functions, which are well adapted to the emergence of soft modes in protein evolution. Green’s function (also called the resolvent, matrix inverse) is useful here because of the following observation: Consider a mutation that alters just one $c_i$. Given the short-range nature of $\HH$, this implies that only a small number of terms in will change, independently of the size of the system. For example, for the hexagonal lattice in , no more than 12 terms change with each mutation.
Since, by Hooke’s law, $\fb=\HH(\cb)\ub$, the response of the system to an external force is given by the inverse relation $\ub=\GG(\cb)\fb$, where $\GG$ is Green’s function, the inverse of $\HH$. So, $\GG$ maps the genotype $\cb$ to the reaction of the protein to an external force $\fb$. A typical example of such a stimulus appears when $\fb$ ‘pinches’ 2 neighboring AAs towards each other; we would like to measure the effect of the pinch on another AA pair (usually on the opposite side of the protein).
In dimension $d{=}2$, the Hamiltonian $\HH$ has always $d(d-1)/2=3$ zero eigenvalues, owing to the rigid Galilean transformations (2 translations and 1 rotation) of the lattice as a whole. Therefore, since $\HH$ is bound to be singular, it lacks a proper inverse. instead, one may compute the inverse on the subspace of $\real^{n_a}\times\real^{d}$ in the complement of the 3 Galilean directions. This is called the pseudo-inverse [@Penrose1955] and is usually denoted by $\GG(\cb)=\HH(\cb)^{\pseudo}$.
Let $\PP$ be the projection on the subspace spanned by the generators of the Galilean transformations, then
()= ((-) () (-))\^[-1]{}()\^ .
It is easy to verify that if $\ub$ is orthogonal to the zero modes, $\ub=(1-\PP)\ub$, then $\ub = \GG \HH\ub$. The pseudo-inverse obeys the four requirements: (i) $\HH \GG \HH = \HH$ , (ii) $\GG \HH \GG = \GG$ , (iii) $(\HH \GG)^\T = \HH \GG$ , and (iv) $(\GG \HH)^\T = \GG \HH$ .
The projection onto the complement of the 0-space commutes with the action of mutations, since changing the AA at a site does not change the Galilean invariance of the lattice. Therefore, the pseudo-inverse can be used for our purposes just like the standard inverse.
Woodbury’s formula
------------------
When one changes a gene $\cb$ to some $\cb'$, then the change in the Hamiltonian is $\DH=\HH(\cb')-\HH(\cb)$ and correspondingly the changes in Green’s function from $\GG(\cb)$ to $\GG(\cb')$. The Woodbury formula [@Woodbury1950; @Deng2011] relates $\DG=\GG(\cb')-\GG(\cb)$ to $\DH$ as follows: First, one notes that the rank of the change tensor $\DH$ is equal to the number $r$ of bonds altered by the mutation. $\DH$ can therefore be written as $$\DH = \MM \Bb \MM^\T~,$$ where the $r {\times} r$ diagonal matrix $\Bb$ records the strength change of the bonds (from weak to strong or vice versa). $\MM$ is a $r\times n_a\times d$ tensor which is the restriction of $\Db$ (see ) to bonds which were changed. This allows one to easily calculate changes in Green’s function:
\[eq:Woodbury\] = -(\^[-1]{} + \^)\^\^ .
The reader who is not familiar with can compare it to the resolvent formula (in the commutative, scalar case): $$\frac{1}{x+y} -\frac{1}{x}=
- \frac{1}{x}\left(\frac{1}{\frac{1}{x}+\frac{1}{y}}\right)\frac{1}{x}~,$$ with $x^{-1}$ corresponding to $\GG$, and $y$ to $\Bb$ (and $\DH$) .
The Woodbury formula is especially useful since one has to invert only square matrices of size $r$ ($\Bb$ and the term in brackets in ), instead of inverting the larger tensor $\HH$ of size $d\times n_a$ [@Henderson1981]. For point mutations, this difference is dramatic, since the rank $r$ can be at most $z$, the number of neighbors of the mutated AA, implying that $r \ll n_a \times d$. For example, in the hexagonal model , $r \le z = 12$ while $n_a\times d = 1080$.
Dyson’s formula
---------------
Another useful (and more common) identity is Dyson’s formula [@Dyson1949; @Dyson1949a; @Abrikosov1963]. It can be obtained by applying the resolvent identity to $\GG'=\GG+\DG$, leading to
\[eq:Dyson\] ’ = - ’ .
Since $\GG'$ appears on the r.h.s., one may successively iterate this identity to get the Dyson series,
\[eq:dysonseries\] =’-= - + - .
The series is widely used in potential scattering, and is interpreted there as expansion in multiple scattering. The first term is usually called the Born term. We will interpret this identity in terms of multiple mutations and this will be another contact of methods known from the physics literature with questions in evolution.
Models: Protein as an evolving machine {#sec:models}
======================================
After introducing the basic principles of our approach, we now discuss how to apply them in specific models. As in any simplifying model, there is an intrinsic conflict: On the one hand, one would like to keep the model as simple as possible, because the goal is to tests basic principles, not specific proteins. On the other hand, there should still be some connection to real proteins. As mentioned before, one cannot apply the thermodynamic limits of standard statistical mechanics (infinite number of particles, long range potentials, and the like), since the protein boundary plays an important role. So the protein is treated as a finite, amorphous system.
A model with very simple structure (Cylinder-model) {#sec:cylinder}
---------------------------------------------------
This model, introduced in [@Tlusty2017], assumes that the coupling between nodes will only depend on one of the two AAs linked by a bond. Although we reformulate the problem differently, it is in fact equivalent to a lattice model as described above (), in the limit of infinite spring constants (bonds are solid rods). To get somewhat closer to the standard genetic code with its 20 AAs, we introduce $2^5=32$ species of AAs; each AA is coded by a 5-bit codon written in a binary alphabet of 0s and 1s.[^10] The geometry of the model is a square lattice with periodic boundary conditions in the horizontal direction, forming a cylinder. One realization is shown in (right) where the blue region corresponds to the shear band. This should be compared to where the shear band is between the red and green arrows (in the shear band is shown in red). we shall see later that the motion around shear bands in the models is similar in nature to the one of .
![ **The main features of the cylinder model**: Left: the mapping from the binary gene to the connectivity of the amino acid (AA) network that makes a functional protein. The color of the AAs represents their rigidity state as determined by the connectivity according to the algorithm of . Each AA can be in one of three states: rigid (gray) or fluid (non-rigid), which are divided between shearable (blue) and non-shearable (red).\
Right: the AAs in the model protein are arranged in the shape of a cylinder, in this case with a fluid channel (blue region). Such a configuration can transduce a mechanical signal of shear or hinge-like motion along the fluid channel.[]{data-label="fig:fig1"}](ert_fig06.pdf){width="\columnwidth"}
### The cylindrical amino acid network
We now define the model in further detail: We consider a geometry with height $h=18$, the number of layers in the $z$ direction, and width $w=30$, the circumference of the cylinder. The row and column coordinates of an AA are $(r,q)$, with $r$ for the row $(1,\dots,h)$ and $q$ for the column $(1,\dots,w)$. The cylindrical periodicity is realized by taking the horizontal coordinate $q$ modulo $w$, $q \rightarrow \textrm{ mod}_w(q-1) + 1$.
Each AA in row $r$ can connect to any of its five nearest neighbors in the next row below it. This defines $2^5 = 32$ effective species of amino acids that differ by their ‘chemistry’, by the pattern of their bonds. An AA at $(r,q)$ is encoded in the gene as a $5$-letter binary codon $\ell_{\rck}$, $k=-2,\dots,2$, where the $k$-th letter denotes the existence ($= 1$) or absence ($ = 0$) of the $k$-th bond. The full gene is therefore a binary sequence of length $2700 = 5\cdot w\cdot h$. Each of its $w\cdot h=540$ codons specifies which of the 5 bonds are present or absent. The effective size of the problem is only $n_s=2550 = w\cdot(h-1)\cdot 5$ because the bonds of the bottom row are never used and do not affect the configuration of the protein and the resulting dynamical modes.
### Evolution searches for a mechanical function
We next define the evolutionary fitness of the cylindrical protein as following: To become functional, the protein has to evolve a configuration of AAs and bonds that can transduce a mechanical signal from a prescribed input at the bottom of the cylinder to a prescribed output at its top. This signal is a large-scale, low-energy deformation where one domain moves rigidly with respect to another in a shear or hinge-like motion, which is facilitated by the presence of a fluidized, ‘floppy’ channel separating the rigid domains [@Alexander1983; @Phillips1985; @Alexander1998].
The definition of the fluid channel is described in detail in , but can be summarized by two features of amino acids in the channel (): (A) Fluidity – these AAs are not part of rigid sub-networks in the protein. Locally, this means that fluid AAs cannot be linked to too many rigid neighbors. (B) Shearability – the AAs in the channel should have enough fluid AAs around them to sustain low-energy shear motion.
The fluidity/rigidity and shearability propagate in a manner reminiscent of percolation. Note that, while the system ‘learns’, through mutations, to form a fluid channel, this learning is not by presenting it with many inputs, but by only checking the quality of the output under random mutations.
In [@Tlusty2016], Figure 8, the author imagined a feedback of the following type: a protein can evolve the ability to activate its own transcription in response to a stimulus, which is the first step towards cellular regulatory networks, see [@Lee2002] for how this appears in the biological context, and [@Djordjevic2003; @Laessig2007; @Tlusty2016 Fig. 8], for theoretical studies.
### Rigidity propagation algorithm$^*$ {#sec:rigidity}
The aim of this subsection is to define a model in which some local rigidity rules, spelled out below, are able to transmit deformability from the bottom of the cylinder to the top. There are many ways in which this can be realized, and the rules we give are a compromise between simplicity and the ability to fulfil this aim. We have tested other variants with similar outcome.
The large-scale deformations are governed by the rigidity pattern of the protein, which is determined by the connectivity of the AA network via a simple majority rule (, \[fig:st\]), as follows. These large scale deformations could in principle change the ability of the protein to bind a target, and in this way implement the response trigger in the feedback loop mentioned above. Each AA position will have two binary properties which define its state: *rigidity* $\sigma$ (an AA is either *solid*, $\sigma=1$, or *fluid*, $\sigma=0$) and *shearability* $s$ (an AA is either *shearable*, $s=1$, or *non-shearable*, $s=0$). Only 3 of the 4 possible combinations are allowed:
non-shearable and solid (yellow): $\sigma = 1; s = 0$,
non-shearable and fluid (red): $\sigma = 0; s = 0$,
shearable and fluid (blue): $\sigma = 0; s = 1$.
Non-shearable protein domains tend to move as rigid bodies (via translation or rotation), whereas shearable regions are easy to deform. The non-shearable domains are mostly rigid, but can still have pockets of fluid AAs.
![ Illustration of the percolation rules for shearability and rigidity states. Note that site $(r,q)$ was turned solid because it is attached to 2 solid sites below it. Also note that the red site above it is fluid, because it is attached to less than 2 solid sites below it. But it is not shearable because it does not connect to a shearable site below it. On the other hand, the top right site is shearable and fluid, since it is attached to only one solid site (namely $(r,q)$) and no others on the invisible part of the structure (as seen by its blue connections), and it is also connected to the blue site at $(r,q+2)$.[]{data-label="fig:st"}](ert_fig07.pdf){width="1.0\columnwidth"}
Given a fixed sequence and an input state in the bottom row of the cylinder, $\{\sigma_{1,q}$, $s_{1,q} \}$ the state of the cylinder is completely determined by ‘percolation’ of the two properties, rigidity/fluidity and shearability, through the network, as follows.
In a first sweep through the rows, we establish the rigidity property $\sigma$. The rigidity of AAs in row $r = 1$ are prescribed initially. In all other rows ($r = 2,\dots,h$) the bonds determine the value of the rigidity of $(r,q)$ through a majority rule:
\[eq:1\] \_[r,q]{}= ( \_[k=-2]{}\^2 \_ \_[r-1,q+k]{} - \_0 ),
where $\theta$ is the step function ($\theta(x \ge 0) =1$, $\theta(x<0)=0$)). The parameter $\sigma_0 = 2$ is the minimum number of rigid AAs from the $r-1$ row required to rigidly support an AA: In 2D each AA has two coordinates which are constrained if it is connected to two or more static AAs. In this way, the rigidity property of being pinned in place propagates through the lattice as a function of the initial row and of the bonds as encoded in the gene.
We next address the shearability property $s$ which is determined by the rigidity as follows: We assume that all fluid AAs in row $r=1$ are also shearable (blue: ($\sigma = 0; s = 1$)). A fluid node $(r,q)$ in row $r$ will be shearable if any of its neighbors at $(r-1,q)$ or $(r-1,q\pm1)$ is shearable:
\[eq:2\] s\_[r,q]{}= (1-\_[r,q]{}) ( \_[k=-1]{}\^1 [s\_[r-1,q+k]{}]{} - s\_0 ),
where $s_0 = 1$. The first factor on the r.h.s. ensures that a solid AA is never shearable.
### Fitness and mutations
As explained before, evolution searches for a functional protein which can transfer forces. The simulation of this search starts from a random sequence (of 2550 codons), and from an initial state (input) in the bottom row of the cylinder. For most simulations, this initial state consisted of only solid beads except a stretch 5 consecutive shearable beads, as shown in .
![ **Evolution of a mechanical function:**\
A configuration (left) with a prescribed input (black ellipse at bottom) and random connectivity pattern eventually evolved to form a fluid channel (right). The initial state has 6 fluid points (in black ellipse), our fitness requires 5 fluid points at the top (black ellipse).[]{data-label="fig:fig1b"}](ert_fig08.jpg){width="\columnwidth"}
We next define a fitness function which will direct the evolutionary process. The state with maximal fitness (the ‘target’) is a chain of $w$ values, fluid and shearable ($\sigma = 0; s = 1$) or solid ($\sigma = 1; s = 0$), in the top row, which the protein should yield as an output: we call it $x^*\equiv\{\sigma^*_{q}$, $s^*_{q} \}_{q=1,\dots,w}$. Given
a gene sequence $\cb$, which determines the connectivity $\ell_{\rck}$,
the input state, $\{\sigma_{1,q}$,$s_{1,q} \}_{q=1,\dots,w}$,
the algorithm described above uniquely defines the output state in the top row, $\{\sigma_{h,q}$, $s_{h,q} \}_{c=1,\dots,w}$. At each step of evolution, the output state is compared to the fixed target by measuring the Hamming distance[^11] to the target $x^*$:
\[eq:3\] F = -w+\_[q=1]{}\^w(1-s\_[h,q]{} - s\_[q]{}\^\* )(1-\_[h,q]{} - \_[q]{}\^\* ) .
This is the fitness function $F$ of .
*Remark*: It is an important feature of this model that the fitness of the network is only measured at the target line. This corresponds to the biological fact that the protein can only interact with the outside world through its surface (in our case, the ends of the cylinder). One of the major outcomes of the model is that this fitness still has a strong influence on the connectivity deep inside the interior of the protein. While similar, the propagation of fluidity should not be confused with learning in neural networks: In the learning case, the system is presented with several inputs and learns to recognize others, while here there is a fixed task, and the connections are only driven by the target function $F$.
### Simulation of evolutionary dynamics
Thanks to the simplicity of the model, one can easily perform $10^6$ simulations in a short time, and gain much better statistical insight than is possible with typical bioinformatic data (of course, at the price of disregarding many biochemical details). We present results for one specific fitness: the input at the bottom is a fluid region of length 6 and output target at the top is a fluid region of length 5. For other variants of this model, *cf*. [@Tlusty2017].
We study 200 independent initial states (genes), starting from a random sequence with about $90\%$ of the bonds present at the start. Given a sequence, we sweep according to the rules of –(\[eq:2\]) through the net, and measure the Hamming distance $F$ () between the last row and the desired target.
Solutions are then searched by successive mutations, with a Metropolis algorithm, [@Metropolis1953]. At each iteration, a randomly drawn digit in the gene is flipped, that is, the values of 0 and 1 are exchanged. This corresponds to erasing or creating a randomly chosen link of a randomly chosen AA. After each flip, a sweep is performed, and the new output at the top row is again compared to the target. If $F$ (which is negative) decreases, we backtrack and flip another randomly chosen bond. This procedure is repeated until optimal fitness is reached $(F=0)$. This will happen with probability 1 if such a state exists, and typically requires $10^3$-$10^5$ mutations.
![Following the progress of evolution during a typical run. It is a sequence of mostly neutral steps, a fraction of deleterious ones, and rare beneficial steps. The vertical axis is the accumulated number of steps of each type.[]{data-label="fig:deleterious"}](ert_fig09.png){width="0.7\columnwidth"}
Although the functional sequences are extremely sparse among the $32^{510}=2^{2550}$ possible sequences, the small bias for getting closer to the target in configuration space directs the search rather quickly.
Once a maximal $F$ is reached, we move away from it by further mutations and then look again for a new optimum. Reaching a state with $F = 0$ takes around 11.2 beneficial mutations on average. Getting from an initial sequence to a maximizer is called a ‘generation’. For each of the 200 initial random genes, we followed 5000 generations, finding a total of $10^6$ optima. The typical length of a generation between two maxima is about 1500 mutations (most of them neutral, see ), similar to the time it takes starting from a totally random gene. We also simulated $1$-generation paths starting from $10^6$ random genes. The two cases are very similar, but the destruction-reconstruction simulations show some correlations between consecutive generations, which disappear after about 4 generations.
A model with more realistic interactions (HP-model) {#sec:pnasmodel}
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This model differs from the cylinder-model in several respects:
Geometry – The lattice is hexagonal with open boundary conditions,
Two-body interactions – the bonds depend on the nature of the AA at *both* ends of the link,
Amino acids species – There are only 2 species, $\aH$ (hydrophobic) or $\aP$ (polar).
The two amino acids species are encoded in a binary genetic alphabet and a codon size of 1: each AA chain is encoded in a gene $\cb$ of the same length $n_a$, where $c_i=1$ encodes $\aH$ and $c_i=0$ encodes $\aP$, $i=1,\dots, n_a$ (in other words, the genetic code is the identity map).
We give next details of how the model is constructed. The lattice has width $h=20$ and height $w=10$, and therefore $n_a=200$ AAs (see ). The bonds stretch over the 12 nearest and next-nearest neighbors of an AA (see , right panel, for the connectivity and any of the panels in for the global arrangement of the lattice).
![ The protein is made of two species of AAs, polar ($\aP$, red) and hydrophobic ($\aH$, blue) whose sequence is encoded in a gene. Each AA forms weak or strong bonds with its $12$ nearest neighbors on the hexagonal lattice (right) according to the interaction rule in the table (left).[]{data-label="fig:1"}](ert_fig10left.png "fig:"){width="0.6\columnwidth"} ![ The protein is made of two species of AAs, polar ($\aP$, red) and hydrophobic ($\aH$, blue) whose sequence is encoded in a gene. Each AA forms weak or strong bonds with its $12$ nearest neighbors on the hexagonal lattice (right) according to the interaction rule in the table (left).[]{data-label="fig:1"}](ert_fig10right.jpg "fig:"){width="0.33\columnwidth"}
\[fig:lattice\]
The strength of the springs is given by the HP model [@Lau1989] according to the rule:
k\_[(i,j)]{} = + (-) c\_i c\_j ,
where $c_i$ and $c_j$ are the (binary) codons of the AA connected by bond $\alpha=(i,j)$, with $c_i=1$ corresponding to $\aH$ and $c_i=0$ to $\aP$. This implies that a strong $\aH{-}\aH$ bond has $k_\alpha=\ks $, whereas the other bonds $\aP{-}\aP$, $\aH{-}\aP$, and $\aP{-}\aH$ are weak with $k_\alpha=\kw $. In our simulations below we used $\ks =1$ and $\kw =0.01$.
### Pinching the network {#sec:pinch}
The network is subjected to an external ‘pinch’ which is a localized force $\fb$ applied at the boundary of the network. This force acts in the complement of the subspace of Galilean invariance, $\PP\fb=0$ (see ). The pinch acts on a pair of neighboring boundary vertices, $p'$ and $q'$, in the direction $\nb_{p'q'}=\rb_{p'}-\rb_{q'}$ which is parallel to the boundary (the L face of the network). Thus $\fb$ is a force dipole, two opposing forces, $\fb_{q'}=-\fb_{p'}=f\cdot \nb_{p'q'}$, which can be related to the deformation $\ub$ of the network by Green’s function . The pinch stimulus may represent localized interactions, for example a ligand biding at a specific binding site ().
The biological fitness is specified by how well the response of the network fits to a prescribed deformation vector $\vb$. This vector is zero except at $p$ and $q$ which are on the opposite side of the network (R face). The fitness function $F$ is therefore
\[eq:fitness\] F===\_p\_p+\_q\_q .
Note that is a specific way of defining $F$, adapted to the phenomenology of building a fluid channel that can transmit allosteric interaction between two specific sites. Other choices of $\fb$ and $\ub$ could be treated similarly, such as multi-site force patterns and multi-domain dynamical modes. For example, if the response can occur at any $(p,q)$ site at the R face, the model may describe the emergence of induced fit or conformational selection mechanisms (). To model the emergence of specific recognition, one sets as target strong response to a stimulus $\fb$, but hardly any response to a similar ‘competitor’ stimulus $\tilde{\fb}$.
![Illustration of the deformation field, $\ub(\cb)=\GG(\cb)\fb$, for three choices of $\cb$. The force $\fb=f\cdot\nb_{p'q'}$ is applied on the left of the lattice. The three panels show, from left to right, how the response $\ub$ (shown in small arrows) evolve as the network fitness $F=\vb^\T\,\ub(\cb)$, increases. The choice of $\vb$ corresponds to the vertical separation of the two central points on the right.[]{data-label="fig:pinch"}](ert_fig11.png){width="1.0\columnwidth"}
In the spirit of the Metropolis algorithm used in the cylinder model, one exchanges randomly AAs between $\aH$ and $\aP$ while looking for changes in the fitness $F$. A gene $\cbs$ is considered a solution if $F$ exceeds a certain large value.[^12] illustrates the vector field $\ub$ of the deformation for three genes $\cb$, along an evolutionary trajectory, improving the fitness value $F$ from left to right.
### The protein backbone {#sec:backbone}
One hallmark of proteins is that they are made from a long chain of amino acids connected by strong covalent bonds, called a backbone (see ). This backbone is then folded in an intricate way to form the protein, but the chain is not broken. Here, we assume that the folding process is just given and that the mutations we consider are moderate enough so that they do not change the general folding. Given this restriction, one still can ask whether the existence of the backbone affects such studies. From a conceptual point of view, having a backbone just means that some springs in the lattice are much stronger than the others, and therefore, it is not surprising that adding a backbone does not change the general picture.
![**Illustration of the backbone.** The backbone is shown as solid black serpentine curve. AAs in neighboring sites along the backbone tend to move together. We show two configurations: parallel to the channel (left) and perpendicular to the channel (right). Parallel: The backbone favors the formation of a narrow channel along the fold (compared to ). Perpendicular: The formation of the channel is ‘dispersed’ by the backbone.[]{data-label="fig:6"}](ert_fig12.png){width="1\columnwidth"}
In we show two extreme cases, a serpentine backbone either parallel to the shear band or perpendicular to it. The presence of the backbone does not interfere with the emergence of a low-energy mode of the protein whose flow pattern (displacement field) is similar to the backbone-less case with two eddies moving in a hinge-like fashion. In the parallel configuration, the backbone constrains the channel formation to progress along the fold (, left). The resulting channel is narrower than in the model without backbone (). In the perpendicular configuration, the evolutionary progression of the channel is much less oriented (, right).
We expect that, in a realistic 3D geometry, the backbone will have a weaker effect than what we observed in 2D networks, since the extra dimension adds more options to avoid the backbone constraint.
### Pathologies and broken networks$^*$
As our criterion for evolution is the floppiness (large eigenvalue of Green’s function), there is of course the trivial case where the network is just broken in two disjoint pieces or into pieces with dangling ends. Such broken networks exhibit floppy modes owing to the low energies of the relative motion of the disjoint domains with respect to each other. Any evolutionary search might end up in such non-functional unintended modes. The common pathologies one observes are:
isolated nodes at the boundary that become weakly connected via $\aH {\to }\aP$ mutations,
‘sideways’ channels that terminate outside the target region (which typically include around $8{-}10$ sites),
channels that start and end at the target region without connecting to the binding site.
All these are floppy modes that can vibrate independently of the pinch and cause the response to diverge (${>}F_\textrm{crit}$) without producing a functional mode. To avoid such pathologies, we apply the pinch force symmetrically: pinch the binding site on face L and look at responses on face R and vice versa. Thereby we not only look for the transmission of the pinch from the left to right but also from right to left. The basic algorithm is modified to accept a mutation only if it does not weaken the two-way response and enables hinge motion of the protein. This prevents the vibrations from being localized at isolated sites or unwanted channels. Of course, the presence of a backbone (see and ) will make disconnection of the network more difficult. This is also a more realistic model.
One may also impose a stricter minimum condition, $\DF \ge \varepsilon\ F$ with a small positive $\varepsilon$, say $1\%$. An alternative, stricter criterion would be the demand that each of the terms in $F$, $\vb_{\ui}\ub_{\ui}$ and $\vb_{\uj}\ub_{\uj} $, increases separately.
Connecting the models to biological concepts
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The theoretical methods introduced earlier lead to a family of easily implementable numerical simulations. We discuss these simulations and show that they can explain several basic observations from the biology literature. They also suggest new connections to be explored. The main idea is that mechanical properties of the protein constrain the genetics in multiple ways.
Each subsection introduces a technique of analysis, application to one of the models described earlier, and an interpretation in terms of biological questions.
Dimension of the solution set in the genotype and phenotype spaces {#sec:dimension}
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We describe the set of solutions for the cylinder-model of . The (genotype) sequence space is $\{0,1\}^{2550}$ (see , bottom left). One can view this space as a 2550-dimensional hypercube, with $2^{2550}$ corners, and any flip of a digit will move along an axis from one corner to another (the dimension of a hypercube equals the number of directions in which one can move from a corner). The space of configurations (phenotypes) is the arrangements of colors (see , top left), which is $\{0,1,2\}^{540}$, since there are 3 colors (red, blue, yellow).
![ **Dimensional reduction of the genotype-to-phenotype map:**\
Dimension measurement from $10^6$ independent configurations (phenotypes) for the cylinder-model. The dimension of the configuration space is about 9 (red curve), while in sequence space it is basically infinite (blue curve). All pairs seem to have the same distance, namely $1275=2550/2$, which is the typical distance between two random sequences. []{data-label="fig:fig2"}](ert_fig13.png){width="1.0\columnwidth"}
The set $\mathcal S$ of solutions to the mutation problem is a subset of this hypercube (). To determine the dimension of experimental data, with large sample size, it is convenient to use the box-counting algorithm [@Grassberger1983]. First, one counts the number $N(\rho)$ of pairs of points in $\mathcal S$ at Hamming distances $\le \rho$, with not more than $\rho $ changes. One then plots $\log N(\rho)$ vs $\log \rho$ and the dimension is the slope in this log-log plot, as indicated by black lines in . We see that the dimension in the space of configurations (phenotypes) is about 8-9, while, in the space of sequences (genotypes), the dimension is basically ‘infinite’, namely just limited by the maximal slope one can obtain [@Procaccia1988] from the $10^6$ simulations.[^13] It would be interesting to discuss this problem as a special case of the problem of hitting times of small sets in hypercubes (these hitting times are usually exponentially distributed). The novelty in the current context is the use of a very small drift, namely, we do not allow steps which increase the distance to the set $\mathcal S$.[^14]
The dramatic dimensional reduction in mapping genotypes to phenotypes stems from the different constraints that shape them [@Savir2010; @Savir2013; @Kaneko2015; @Friedlander2015]. In the phenotype space, most of the protein is rigid, and only a small number of shear motions are low-energy modes, which can be described by a few degrees-of-freedom. In the genotype space, in contrast, there are many neutral mutations which do not affect the motion of the protein.
The biological interpretation is that **the gene is much more random than the phenotype of the protein it forms**. However, we shall see below, in particular in , that the gene still needs to be quite precise in certain well-defined positions. In the case of allosteric proteins these critical positions are the hinges and other locations of strong stress.
Expansion of the protein universe {#sec:expansion}
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Here, we test the cylinder-model against the ideas of [@Povolotskaya2010]. Our results will give some insight about the nature of the set of solutions, genes of functional proteins. In [@Povolotskaya2010], the authors consider any two solutions with gene sequences $\textbf{s}_1$ and $\textbf{s}_2$. They ask how much the solution $\textbf{ s}_3$, one generation after $\textbf{ s}_2$, differs from $\textbf{ s}_1$, and define the following observable:
Let $x_i=(2s_{1,i}-1)\cdot (s_{3,i}-s_{2,i})$ (since $s_{1,i}\in\{0,1\}$, $x_i>0$ if the change between $s_{3,i}$ and $s_{2,i}$ is towards $\textbf{ s}_{1}$ and $x_i<0$ otherwise). Finally, $N_{\textrm{ away}}=\#\{i\,:\,x_i<0\}$ and $N_{\textrm{ towards}}=\#\{i\,:\,x_i>0\}$. In , the ratio $N_{\textrm{ towards}}/N_{\textrm{ away}}$ is plotted as a function of the distance $D$ between $\textbf{s}_1$ and $\textbf{s}_2$, normalized by the diameter $d_{\max}=2550$. The interested reader will notice the similarity to Fig. 3 in [@Povolotskaya2010]: In their case, because of the small number of experimental samples, they only see the low-$D$ region of , far from the diameter of the ‘protein universe’.
![ **Distribution of solutions in the sequence universe:**\
A measure for the expansion of functional genes in the sequences universe is the backward/forward ratio, the fraction of point mutations that make two sequences closer vs. the ones that increase the distance [@Povolotskaya2010]. The Hamming distances $D$ (normalized by the universe diameter $d_{\max}= 2550$) show that most sequences reach the edge of the universe, where no further expansion is possible. The black curve, $D/(1-D)$, is the backward/forward ratio of purely random mutations. Given the overwhelming number of samples near the maximal $D$, the Gaussian distribution is well visible (in the vertical direction). []{data-label="fig:fig2b"}](ert_fig14.pdf){width="1.0\columnwidth"}
The set of solutions is a very dilute, but complex, subset $\mathcal S$ of the hypercube. The search for a good gene corresponds to a slightly biased random walk along path of monotonically increasing fitness ($\delta F\ge0$). While we do not have a good mathematical description of such intricate walks, we can compare them to the null model of purely random walks. In this case, one gets a simple expression for the towards/away ratio, as a function of $D$, the normalized Hamming distance: $D/(1-D)$, which is shown as black curve ($D$ is the proportion of sites which differ between the pair of solutions $\textbf{ s}_1$ and $\textbf{ s}_2$). The good fit shows that the fitness-constrained evolutionary paths expand *as if one performed a random walk on the full cube*.
It is interesting to note: First, that this result must be intimately connected to the high dimension of the problem, since for low dimensional hypercubes it does not hold. Second, most samples are near the edge $D=1$ of the universe, where the Hamming distances among the sequences are close to the typical distance between any two random sequences. To conclude: [**While maintaining functionality, the divergence of acceptable gene sequences has all aspects of a random walk (on a hypercube)**]{}. This conclusion is close to the ‘expansion of the protein universe’ (in honor of E. Hubble), described in [@Povolotskaya2010].
Spectrum in phenotype and genotype spaces {#sec:spectrum}
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Another useful method to analyze large sets of solutions is by spectral analysis in terms of Singular Value Decompositions (SVD). For the cylinder model, we have $10^6$ binary vectors with $n=2550$ components each. To find the typical correlation spectrum of the solution, one forms a matrix $W$ of size $m\times n=10^6\times 2550$. The SVD of this matrix is a generalization of the spectral decomposition of positive (semi-definite) square matrices: $W$ is decomposed as $U\cdot D\cdot V^\T$, where $U$ is $m\times m$, $V$ is $n\times n$ and $D$ is an $m\times n$ diagonal matrix (only the elements $D_{ii}$ with $i=1,\dots,n$ are nonzero). In our case, $m \gg n$, which is required to obtain good statistics of the random process. The singular values $\lambda^{G}_i = D_{ii}$ are in general positive and in this case the decomposition is unique. The columns of $V$ are the (generalized) eigenvectors of $W$, the first few of which are shown in .
{width="70.00000%"}
The singular values $\lambda^{G}_i$ are the square roots of the spectrum of the covariance matrix $W^\textrm{ T}W$,[^15] which has the same eigenvectors as $W$. Therefore, the high values correspond to the principal covariance components, the directions with maximal variation in the solution set. Mutatis mutandis, we perform the same SVD for the configurations, using the 540 $s$-values (that is, of the shearability ) of vectors of the configurations.
Figure \[fig:fig3\] illustrates the difference between the configuration space (phenotype) and the sequence space (genotype):
*Configuration space* (The eight figures on the bottom left): The first mode is proportional to the average configuration. The next modes reflect the basic deviations of the solution around this average. For example, the second mode is left-to-right shift, the third mode is expansion-contraction etc. Since, the shearable/non-shearable interface can move at most one AA sideways between consecutive rows, the modes are constrained to diamond-shaped areas in the center of the protein. This is the overlap of the influence zones of the input and output rows.
*Sequence space* (The eight figures on the bottom right): The first eigenvector is the average bond occupancy in the $10^6$ solutions. The higher eigenvalues reflect the structure in the many-body correlations among the bonds. The typical pattern is that of ‘diffraction’ or ‘oscillations’ around the fluid channel. This pattern mirrors the biophysical constraint of constructing a rigid shell around the shearable region. Higher modes exhibit more stripes, until they become noisy, after about the tenth eigenvalue.
The sequence-spectrum, top right in has some outliers, which correspond to the localized modes shown in the eight panels below. Apart from that, the majority of the eigenvalues seem to obey the Marčenko-Pastur formula, see [@Marcenko1967]. If the matrix is $m\times n$, $m>n$, then the support of the spectrum is $\frac{1}{2}(\sqrt{m}\pm\sqrt{n})$. In our case, since we have a $10^6\times 2550$ matrix, one expects (if the matrices were really random) to find the spectrum at $\frac{1}{2}(\sqrt{10^6}\pm \sqrt{2550})$, which is close to the simulations, and confirms that most of the bonds are just randomly present or absent. The slight enlargement of the spectrum is attributed to memory effects between generations in the same branch. This corresponds to phylogenetic correlations among descendants in the same tree [@Felsenstein1985].[^16] We conclude: **The small number of discrete eigenvalues shows that a small number of parameters characterizes both the phenotype and the non-random part of genotype of proteins**.
### Geometry of the genotype and phenotype solution spaces$^*$
The $10^6$ genotype vectors form a ‘cloud’ of points in a 2550-dimensional space. The geometry of the cloud can be explored by plotting projections along the axes defined by the eigenvectors $\cb_i$, $i=1,2,\dots,2550$, . Consider for example the projection of the cloud onto the subspace spanned by $\cb_2$, $\cb_3$ and $\cb_{100}$. The variation along the $3$ axes is of comparable size. However, the equivalent projection of the 540-dimensional phenotypes along their eigenvectors $\ub_2$, $\ub_3$, and $\ub_{100}$ shows very small variation along the vertical ($\ub_{100}$) axis, similar to the projections of a flat ellipsoid.
The projections reflect the differing shapes of the solution clouds: in the genotype space the cloud is a $2550$-dimensional spheroid object, while in the phenotype space it is a flat discoid of dimension $\sim 10$.
![The projections of the set of $10^6$ solutions as 2550-dimensional gene sequences (left column) and corresponding 540-dimensional phenotypes (right column) onto their SVD eigenvectors $\cb$ and $\ub$. Top row shows axes 2 (horizontal) and 100 (vertical), bottom row shows axes 3 (horizontal) and 100 (vertical). Note that the phenotypes have larger variation along their 2 and 3-components than their 100-component, unlike the genotypes which appear evenly distributed in all directions.[]{data-label="fig:figure5b"}](ert_fig16.jpg){width="\columnwidth"}
Stability of the mechanical phenotype under mutations {#sec:stability}
-----------------------------------------------------
Any protein is the outcome of a long evolutionary trajectory starting from a distant ancestor. It is likely to find other descendants of the same ancestor in closely related species. In practice, one fetches the pairwise most similar proteins from a collection of species and aligns their sequences to identify homologous amino acids (all descendants of a given amino acid in the ancestor protein) [@Karlin1993]. Once this has been accomplished, one can study mutation patterns in this multiple sequence alignment. There are two questions of interest here:
Which positions in the gene are *conserved*, they encode the same AA?
Which pairs of positions are *co-varying*: a mutation at one position is frequently compensated by a mutation at the other position?
![The sensitivity of solutions to a single mutation, as a function of the mutation position. []{data-label="fig:mutateone"}](ert_fig17.jpg){width="1.0\columnwidth"}
The second question, about genetic correlations, will be discussed in detail in . In this subsection, we discuss the first question, regarding AA conservation, in the context of the cylinder-model. To produce one takes $10^6$ sample solutions and mutates, for each of them, every possible position in the AA network. One then asks which mutations destroy the solution. At every position, the intensity of the blue color is proportional to the probability that a mutation at that position destroys the solution. The end of the channel is very sensitive, but also the boundaries between the channel and the bulk. This should be compared to (center) where the analogous question was asked for glucokinase [@Rougemont2099], and answered by aligning 122 homologs of glucokinase. We conclude: **Sensitivity to mutations is localized near mechanically critical regions**.
{width="70.00000%"}
Shear modes in the amino acid network {#sec:shear}
-------------------------------------
We focus here on the HP-model, although similar results hold for the cylinder-model. In we have shown a pinch stimulus $\fb$ leads to a deformation field $\ub=\GG\fb$, see . As the fitness $F$ of improves, the system forms a fluid channel and the response field $\ub$ shows a hinge-like rotation, as is visible in . This should be compared to , showing the experimentally measured deformation of glucokinase. Opening a hinge in a network will not only move the two sides of the hinge but also shear the connecting bonds, especially at the hinge itself. Furthermore, remaining links near the opening of the hinge (at the opposite side of the protein) will be stretched as well. These observations can be quantified by measuring the shear. The shear $s$ at any (lattice) point is the symmetrized derivative of the displacement field $\ub$, which is computed as follows.
First, the displacement vector at the point $\xb\in\real^d$ is $\ub(\xb)$ with $u_i(x)=x_i'(x)-x_i$, the difference between the ‘new’ points $\xb'(\xb)$ and the ‘old’ points $\xb$ in $\real^d$. The local deformation matrix $\DD(\xb)$ is then given by
D\_[ij]{} = ,
so that $\nabla \ub(\xb) = \DD(\xb)-\id$. (This matrix is also called ‘compatibility matrix’ in the review [@Lubensky2015].) The shear matrix $\epsilonb$ is
\_[ij]{}()= (++ \_[k=1]{}\^d ) .
In short notation
\[eq:stress2\] &=((())\^+ () +(())\^() )\
&= (()-) ,
with $\CC(\xb)=\DD(\xb)^\T \DD(\xb)$, which is the metric of the coordinate transformation.
As a measure of the magnitude of the shear one can use
\[eq:sx\] s() &= (\^2) - (())\^2\
&=(- ())\^2 ,
which is the square of the Frobenius norm ($L^2$) of the traceless part of $\epsilonb $.[^17] The trace of $\epsilonb$ is related to the isotropic dilation, which in the protein is a smaller effect than the shear, and therefore will not be further considered.[^18]
### Implementation in the case of protein structure data$^*$
As proteins are discrete objects, we replace the derivatives by difference operators [@Gullett2008; @Mitchell2016].
We consider the crystallographic data of two conformations of a given protein (for example the PDB structures 1v4s and 1v4t in [@Kamata2004]). To produce from these data [@Rougemont2099], we take a ball of radius $\rho$ of about 10 around each atom $X$ in the protein.[^19] This will encompass $m =m(X)$ other atoms, at positions $\rb_i\in\real^3$, $i=1,\dots,m$. Let $\rb_0$ be the coordinates of $X$, and let $\AAA(X)$ be the $m\times 3$ matrix of the $m$ distance vectors $\rb_i-\rb_0$, in the first configuration (one of the PDB structures). Let $\BB(X)$ be the analogous matrix for the second configuration and compute
\[eq:q\] =(\^)\^[-1]{} \^ (\^- \^) (\^)\^[-1]{} .
The matrix $\QQ$ is an approximation of $\epsilonb$ and is obtained by observing that $\BB=\AAA\DD^\T$:
=\^= (\^)\^[-1]{} \^ \^(\^)\^[-1]{} .
By substituting this into , we verify that holds. With this approximate shear tensor $\epsilonb$ one computes its magnitude $s(\rb_0)$ using .
The discrete approximation of the shear field in glucokinase is shown in the leftmost panel of (with $m$ typically around 50). The strain is found to be large in the hinge of the protein, and also at a somewhat loose outer surface.[^20] In , the corresponding shear magnitude fields $s(\xb)$ are shown for the HP-model, using a statistical average over many solutions (see for more details). One again observes strong shear in the hinge. We conclude: **[shear is critical in the hinges among moving domains of the protein]{}**.
### Details of shear computation$^*$ {#sec:computation}
We describe here in detail the procedure of calculating the shear in the HP-model, leading to . This figure shows averages over many realizations of the random process, in the following sense. One starts with $10^6$ solutions.[^21] Each solution $\cb_*$ together with the (fixed) pinch $\fb$, defines 3 vectors
the gene of the functional protein, $\cbs$, (a vector of length $n_a=200$ codons),
the flow field (displacement), $\ub(\cbs)=\GG(\cbs) \fb$, (a vector of length $n_d =400$ of the $x$ and $y$ velocity components),
the shear field $\sb(\cb_*)$ (a vector of length $n_a=200$).
The $10^6$ solutions are then written as three corresponding matrices $W_C$, $W_U$ and $W_S$, of size $200\times 10^6$ resp, $400\times
10^6$, where each row of these matrices is one of $\cbs$, $\ub(\cbs)$, and $\sb(\cbs)$.
Next, one calculates the singular eigenvalues and corresponding eigenvectors of the three matrices (using SVD, as in ) and isolates the leading eigenvalues. The central row in shows that the flow field can be decomposed into successively weaker motions $\Uk$, with the strongest being a rotating hinge motion around the fluid channel. One should note that evolution in this case did not impose this *global* rotation, but only the *localized* response to a pinch on the left side of the sample.
Similarity of gene and shear {#sec:similarity}
----------------------------
The results for the HP-model reveal a tight relation between the gene fields $\Ck$ and the shear intensities $\Sk$, as shown in . Comparing the top and bottom rows, one observes a similar structure of the corresponding eigenfunctions.[^22] A similar relation is visible in for the cylinder-model. The functions of many proteins are known to involve large-scale motions of the amino acid network, such as hinge rotation, shear sliding, or twists [@Gerstein1994]. Recently, the strain that occurs during such conformations changes was computed in several proteins by comparing structures obtained from X-ray and NMR studies [@Mitchell2016]. However, the tight correspondence between the shear tensor and the genetic correlations, that we observe here () has not yet been measured in real protein. In principle, one would need to follow a procedure similar to the one presented here: first, to calculate the mechanical shear using the methods of [@Gullett2008; @Mitchell2016], and then to compare it to the genetic correlations from sequence alignment [@Rougemont2099].
Point mutations are localized mechanical perturbations {#sec:scattering}
------------------------------------------------------
A mutation in the HP-model may vary the strength of no more than $z=12$ bonds around the mutated AA (). The corresponding perturbation of the Hamiltonian $\DH$ is therefore localized, akin to a defect in a crystal [@Tewary1973; @Elliott1974]. The mechanics of mutations can be further explored by examining perturbations of Green’s function, $\Gs = \GG +\DG$. They obey the Dyson equation and the Dyson series, –(\[eq:dysonseries\]). This series has a straightforward physical interpretation as a sum over multiple scatterings (B): As a result of the mutation, the elastic force field is no longer balanced by the imposed force $\fb$, leaving a residual force field $\dfb = \DH\, \ub = \DH \, \GG\, \fb$. The first scattering term in the series balances $\dfb$ by the deformation $\dub = \GG \,\dfb = \GG\, \DH \,\GG \fb$. Similarly, the second scattering term accounts for further deformation induced by $\dub$, and so forth.[^23] We conclude: [**Standard expansions of Green’s functions correspond to hierarchical organization of the effects of mutations in terms of multiple scattering**]{}.
Mechanical function emerges as a sharp transition
-------------------------------------------------
As the evolution reaches a solution gene $\cbs$, there emerges a new (almost) zero energy-mode, $\ubs$, in addition to the Galilean symmetry modes (which we already projected away). As the other eigenvalues of $\GG(\cbs)$ remain typically distant from this small eigenvalue $\lams$, there will be a gap between $\lambda_*$ and the rest of the spectrum. While we do not have a proof of that such a gap should appear, this is found to be the generic case in the models described here. The response to a pinch will be mostly through this soft mode, as we show now.
Consider a sequence of mutations $\cb_k$ which converges to $\cbs$ (in the Hamming distance) as $k\to k_*$. The corresponding sequence of fitness values is $F_k=F(\cb_k)$. For the HP-model with the pinch introduced earlier, the fitness is ():
F\_k =(\_k) =(\_k) .
When $\cb_k$ gets closer to $\cbs$, the almost-zero eigenvalue $\lambda _k$ of $\HH(\cb_k)$ will dominate Green’s function, $\GG(\cb_k) = \HH(\cb_k)^\pseudo$
(\_k) \~ .
The fitness sequence is therefore
\[eq:divergence\] F\_k .
![Progression of the fitness $F$ corresponding to the evolution of (black). The fitness trajectory averaged over ${\sim} 10^6$ runs $\langle F \rangle$ is shown in red. Shown are the last 16 beneficial mutations towards the formation of the channel. The contribution of the emergent low-energy mode $\langle
F_k \rangle$ alone, shown in blue, dominates the fitness (according to ). []{data-label="fig:3a"}](ert_fig19.png){width="1.0\columnwidth"}
On average, the fitness increases exponentially with the number of beneficial mutations as shown in . The growth of the fitness follows the formation of the channel and the narrowing of the remaining rigid ‘neck’, in the middle of the channel. We lack, however, a quantitative explanation for the generic exponential dependence, which is probably related to the structure of the Hamiltonian $\HH$.[^24] In the particular instance of the two models considered here, one can argue that the mutations which improve the channel, all act multiplicatively on the fitness $F$. Since this discussion is ‘spectral,’ we expect it to hold for models with more colors (AAs) than the HP model.
As noted in , beneficial mutations are rare, and are separated by long stretches of neutral mutations. One may ask where the neutral mutations take place, and this is illustrated in , which shows that, in most sites, the effect of mutations is practically neutral.
![Landscape of the fitness change $\DF = \vbT \DG\, \fb~$, averaged over $ 10^6$ solutions, for all 200 possible positions of point mutations at a solution. Underneath, the average AA configuration of the protein is shown in shades of red ($\aP$) and blue ($\aH$). In most sites, mutations are neutral, while mutations in the channel are, on average, deleterious (blue, below the flat surface). []{data-label="fig:3b"}](ert_fig20.png){width="1.0\columnwidth"}
The vanishing of the spectral gap, $\lambda _k \to 0$, can further be viewed as a topological transition in the system: the AA network is being divided into two domains that can move independently of each other at low energetic cost. The relative motion of the domains defines the emergent soft mode and the collective degrees-of-freedom, for example the rotation of a hinge or the shear angle.
![The mean shear in the protein in a single run (black) and averaged over $10^6$ samples (red) as a function of the fraction $p$ of $\aP$-amino acids. The values of $p$ are shifted by the position of the jump, $p_\textrm{ c}$. Inset: Distribution of $p_c$.[]{data-label="fig:figure3d"}](ert_fig21.png){width="1.0\columnwidth"}
The soft mode appears at a dynamical transition, where the average shear in the protein jumps abruptly as the channel is formed and the protein can easily deform in response to the force probe (). The trajectories are plotted as a function of $p$, the fraction of AAs of type $\aP$. The distribution of the critical values $p_c$ is rather wide owing to the random initial conditions and finite-size effects.
Another connection is provided by the Kirchhoff matrix-tree theorem (see [@Tutte1948; @Chaiken1982]). Let $\MM$ be an $n$-by-$n$ graph Laplacian, with links $i\leftrightarrow j$ given by $\MM_{ij}=-1$ and $\sum_j \MM_{ij}=0$ for all $i$. The matrix $\MM$ has an eigenvalue 0, with eigenvector $(1,1,\dots,1)$. Take the submatrix $\MM_0$ where one row and one column are omitted. In analogy with the weighted links of the HP-model, assume now that $\det \MM_0=0$, that a second eigenvalue vanishes. Then, by the Kirchhoff theorem, the number of spanning trees of the full graph is equal to 0. In other words, the graph is disconnected, in analogy to the formation of the fluid channel.
Correlation and alignment {#sec:correlation}
-------------------------
As the shear band (fluid channel) is taking shape, the correlation among codons builds up. To see this, we align genes from the $10^6$ simulations, in analogy to sequence alignment of real protein families [@Goebel1994; @Marks2011; @Jones2012; @Lockless1999; @Suel2003; @Hopf2017; @Poelwijk2017; @Halabi2009; @Tesileanu2015; @Juan2013]. At each time step we calculate the two-codon correlation $Q_{ij}$ between all pairs of codons $c_i$ and $c_j$ ,
\[eq:correlation\] Q\_[ij]{} c\_i c\_j-c\_i c\_j ,
where brackets denote ensemble averages. One finds that most of the correlation is concentrated in the region where the channel will form. In one sees that the average correlation is tenfold larger in the channel than in the whole protein. Within the channel, the correlation is long-range, and propagates from side to side in the protein (see [@Dutta2018] for a figure).
![The average magnitude of the two-codon correlation $|Q_{ij}|$ as a function of the number of beneficial mutations, $t$. The red curve shows $|Q_{ij}|$ in the shear band (AAs in rows $7{-}13$, of ). The black curve shows $|Q_{ij}|$ for the whole protein. The correlations in the channel are clearly larger.[]{data-label="fig:3c"}](ert_fig22.png){width="1.0\columnwidth"}
Analogous correlated domains containing functionally-related amino acids that co-evolve appear in real protein families [@Lockless1999; @Suel2003; @Halabi2009; @Tesileanu2015], as well as in coarse-grained models of protein allostery [@Hemery2015; @Flechsig2017; @Tlusty2016; @Tlusty2017] and allosteric matter [@Rocks2017; @Yan2017].
We conclude: **Genetic correlations are significantly larger in the mechanically important regions**.
Conserved amino acids {#sec:conservation}
---------------------
In this section, we discuss single mutations (and the lack thereof), while in the next, we discuss the case where one mutation is ‘compensated’ by another (this is called epistasis). Both phenomena are intimately related to those sites on the protein which matter for the function: These are the sites which are mechanically important.
In the cylinder model (), mutations near the top edge and the boundary of the fluid channel have the most deleterious effect on the mechanical function (dark regions in the figure). Therefore, to preserve the functionality of the protein in this model, these sensitive amino acids are also conserved more than average among the solutions.
In comparison, the center panel of highlights the most conserved positions among the 122 aligned homologs of the real protein glucokinase. Similar to the model, here also conservation appears to be correlated with mechanical importance, as measured by the magnitude of the shear (left panel of ). We conclude: **Mechanically critical regions of a protein are sensitive to mutation**. If, however, a mutation does occur at such a sensitive amino acid, then it should be compensated by another one (or a few). This is dealt with in the next section.
Epistasis links protein mechanics to genetic correlations {#sec:epistasis}
---------------------------------------------------------
The correlations among amino acids in the gene exhibit tight correspondence to the pattern of the shear field (see and ). We now discuss how to link these genetic correlations among mutations to the physical interaction in the amino acid network. The procedure will be similar to how the effect of a single mutation was interpreted in terms of a scattering expansion of Green’s function ().
In genetics, the term epistasis refers to departure of fitness from additivity in the effect of combined mutations owing to *inter-genetic* interaction. For example, the phenotypic effect of one gene may be masked by a different gene [@Cordell2002; @Phillips2008; @Mackay2014]. In analogy, on the smaller scale of a single gene described here, *intra-genetic* epistasis is non-additivity of protein fitness owing to the non-linear interaction among its amino acids [@Breen2012; @Ortlund2007; @Harms2013; @Clark1997]. For example, one mutation can be compensated by another one in order to keep the protein functional. This second mutation can be far away on the gene sequence. The use of Green’s functions allows for a calculable definition of epistasis in terms of the Dyson series .
![**Force propagation, mutations and epistasis.** (A) Green’s function $\GG$ measures the propagation the mechanical signal across the protein (blue) from the force source $\fb$ (pinch) to the response site $\vb$, depicted as a ‘diffraction wave’. (B) A mutation $\DHi$ deflects the propagation of force. The effect of the mutation on the propagator $\DG$ can be described as a series of multiple scattering paths . The diagram shows the first scattering path, $\GG\,\DHi \GG$. (C) Epistasis is the departure from additivity of the combined fitness change of two mutations. The epistasis between two mutations, $\DHi$ and $\DHj$, is equivalent to a series of multiple scattering paths . The diagram shows the path $\GG\, \DHi \GG\,\DHj \GG$.[]{data-label="fig:2"}](ert_fig23.pdf){width="1.0\columnwidth"}
Algorithmically, one takes one functional solution obtained from the evolution algorithm and mutates one AA at a site $i$. This mutation induces a change $\DGi$ as the difference of the new and the old Green’s function. Then,
\_i =
is the change of the observable fitness $F$ (which can be computed by ). One can similarly perform another, independent mutation at a site $j\ne i$, producing a second deviation, $\DGj$ and $\DFj$ respectively. Finally, starting again from the original solution, one mutates both $i$ and $j$ simultaneously, with combined effects $\DGij$ and $\DFij$. It is then natural to define the epistasis $\epij$ as the departure of the double mutation from additivity of two single mutations,
\[eq:epistasis\] - - .
{width="85.00000%"}
The epistasis $\epij$ is simply the inner product value of this nonlinearity with the pinch and the response,
\[eq:curvature\] = ( - - ) .
shows how epistasis is directly related to mechanical forces among mutated AAs.
To evaluate the average epistatic interaction among amino acids in the HP-model, we perform the double mutation calculation for all $ 10^6$ solutions and take the ensemble average $\EE_{ij}=\langle \epij \rangle$. Landscapes of $\EE_{ij}$ show significant epistasis in the channel (). AAs outside the high shear region show only small epistasis, since mutations in the rigid domains hardly change the elastic response. The epistasis landscapes (A-C) are mostly positive since the mutations in the channel interact antagonistically [@Desai2007]: after a strongly deleterious mutation, a second mutation has a smaller effect.
In the gene, epistatic interactions are manifested in codon correlations [@Hopf2017; @Poelwijk2017] shown in D, which depicts two-codon correlations $Q_{ij}$ of from the alignment of $ 10^6$ functional genes $\cbs$. We find a tight correspondence between the mean epistasis $\EE_{ij}=\langle \epij \rangle$ and the codon correlations $Q_{ij}$. Both patterns exhibit strong correlations in the channel region with a period equal to channel’s length, 10 AAs. The similarity in the patterns of $Q_{ij}$ and $\EE_{ij}$ indicates that [**a major contribution to the long-range, strong correlations observed among aligned protein sequences stems from the mechanical interactions propagating through the amino acid network**]{}.
Epistasis as a sum over scattering paths {#sec:epistasis2}
----------------------------------------
One can classify epistasis according to the interaction range. Neighboring AAs exhibit *contact epistasis* [@Goebel1994; @Marks2011], because two adjacent perturbations, $\DHi$ and $\DHj$, interact nonlinearly via the ‘and’ gate of the interaction table of , $\DDHij \equiv \DHij - \DHi - \DHj \neq 0$ (where $\DHij$ is the perturbation by both mutations). In the case of contact epistasis, the leading term in the Dyson series of $\DDGij$ is a single scattering from an effective perturbation with an energy $\DDHij$, which yields the epistasis $$\epij = -\vbT \left( \GG\, \DDHij \GG \right) \fb+\dots ~.$$
*Long-range epistasis* among non-adjacent, non-interacting perturbations ($\DDHij=0$) is observed along the channel (). In this case, expresses the nonlinearity $\DDGij$ as a sum over multiple scattering paths which include both $i$ and $j$ (C),
= ( + ) - . \[eq:multiple\]
The perturbation expansion further links long-range epistasis to shear deformation: Near the transition at which the function emerges, Green’s function is dominated by the single soft mode, $\GG \simeq \ubs\ubsT/\lams$, with fitness $F$ given by . From and , one deduces a simple expression for the mechanical epistasis as a function of the shear,
F (+- ) . \[eq:antagonistic\]
The factor $h_i \equiv \ubsT\DHi\ubs/\lams$ in is the ratio of the change in the shear energy due to mutation at $i$ (the expectation value of $\DHi$) and the energy $\lams$ of the soft mode, and similarly for $h_j$. Thus, $h_i$ and $h_j $ are significant only in and around the shear band, where the bonds varied by the perturbations are deformed by the soft mode.
When both sites are outside the channel, $h_i,h_j \ll 1$, the epistasis is small, $\epij \simeq 2 h_i h_j F$. It remains negligible even if one of the mutations, $i$, is in the channel, $h_j \ll 1 \ll h_i$, and $\epij \simeq h_j F$. Epistasis can only be long-ranged along the channel when both mutations are significant, $h_i\gg 1$ and $ h_j \gg 1$, and $\epij \simeq 1$. It follows that can be roughly approximated as $$\epij \simeq F \cdot \min{(1,h_i)} \cdot \min{(1,h_j)}~.
\label{eq:rank1}$$
We conclude that epistasis is maximal when both sites are at the start or end of the channel, as illustrated in . The nonlinearity of the fitness function gives rise to antagonistic epistasis since the combined effect of two deleterious mutations is non-additive as either mutation is enough to diminish the fitness.
As evident from , the epistasis matrix is approximately a rank-one tensor $\epij \sim \ket{\mathbf{e}}\bra{\mathbf{e}}$, with a single dominant eigenvector, $\mathbf{e}_i \sim \min{(1,h_i)}$. The eigenvector $\ket{\mathbf{e}}$ is localized in and around the shear band. As a result, the epistasis matrix exhibits a ‘checkered’ pattern visible in D. The rank-one nature of the $\epij$ is verified numerically by spectral decomposition of the epistasis matrix obtained from the simulation. Interestingly, the genetic correlation matrix () is also approximately a rank-one tensor, $Q_{ij} \sim \ket{\mathbf{q}}\bra{\mathbf{q}}$, with a dominant eigenvector $\ket{\mathbf{q}}$ localized in the channel. This explains the striking similarity of the genetic correlation $Q_{ij}$ and the epistasis $\epij$ in D.
Again, comparing to the real protein glucokinase, the rightmost panel of shows that the correlation of mutations is concentrated in the mechanically critical regions of the protein (left panel). Mutations away from these spots seem more independent and need not be corrected for other mutations. We conclude: **mutations correlate near mechanically critical positions**.
Multilocus epistasis$^*$
------------------------
So far, we examined the interaction between two mutations in terms of the non-linearity of the double-mutation fitness function $\epij$ . This two-body interaction can be seen as the change in the effect of mutation $j$ in the presence of another mutation $i$. As an isolated mutation, $j$ has a fitness effect, $\DFj$, whereas in the presence of $i$ the effect of $j$ is $\DFij - \DFi$, and the difference defines $\epij
\equiv (\DFij - \DFi )- \DFj~$.
Higher-order epistasis, involving more than two mutations, has a significant role in shaping the fitness landscape [@Weinreich2013; @Poelwijk2017]. This motivates us to generalize the methodology of to many-body interactions. For example, the three-loci epistasis, $\epijk$, measures the change in the two-loci epistasis $\epij$ of the double $i,j$ mutation, induced by the presence of a third mutation, $k$ [@Horovitz1990]:
\[eq:epistasis3\] - (+ + ) + + + ,
where $\DFijk$ is the phenotypic effect of a triple $i,j,k$ mutation.
In a similar fashion, one derives the general $N^\mathrm{th}$-order epistasis, among mutations at positions $i_1,\dots,i_N$,
\[eq:epistasisN\] e\_[i\_1,i\_2,…, i\_N]{} \_[q=1]{}\^N(-1)\^[N-q]{} \_[i\_1<…<i\_q]{}[\_[i\_1,…, i\_q]{}]{} ,
where $\delta\mathbf{F}_{i_1,\dots, i_q}$ is the fitness effect of the $q$-site mutation at positions $i_1,\dots,i_q$. and are the second- and third-order epistasis terms ($N=2,3$), while the first-order epistasis ($N=1$) is the mutation effect itself, $e_i \equiv \DFi$. Summing over all orders of epistasis interactions () up to order $N$, one obtains the $N$-site mutation effect
\_[i\_1,i\_2,…, i\_N]{} = \_[q=1]{}\^N \_[i\_1<…<i\_q]{}[e]{}\_[i\_1,…, i\_q]{} .
To link the multi-locus epistasis to protein mechanics and deformation, we follow the derivation of . Near the transition at which the function emerges, we use the Dyson series , and the resulting $N$-site mechanical epistasis is: $$\label{eq:antagonisticN}
e_{i_1,\dots, i_N} =
-F \cdot\sum_{q=1}^N{\left(-1\right)^{N-q} \sum_{i_1<\dots <i_q}
{ \frac{\sum_{p=1}^q{h_{i_p}}}{1 + \sum_{p=1}^q{h_{i_p}}}}}~,$$ where elastic factor $h_{i_p} \equiv \ubsT\delta \mathbf{H}_{i_p}\ubs/\lams$ is the ratio of the change in the shear energy due to mutation at $i_p$ and the energy $\lams$ of the soft mode.
One concludes from that the $N$-order epistasis is significant only within and around the shear band, where the bonds are stretched and compressed by the soft mode. In this region, where all the elastic factors are large. $h_{i_p} \gg 1$, all orders of epistasis are relevant and are of the same magnitude, $$e_{i_1,i_2,\dots, i_N} \simeq F \cdot \left(-1\right)^N~.$$ We conclude: **the mechanically critical regions are strongly coupled with many-body epistatic interactions among the mutations.**
Outlook
=======
This colloquium has described a method which relates biological questions and concepts regarding protein evolution to the techniques of theoretical physics. Our purpose is to make this approach accessible to a wide community. While we made an effort to cite some of the current literature, there are certainly works which we have incompletely cited. We hope that this colloquium will encourage others to build bridges between other biological questions and the long tradition of physics and mathematics.
We thank Albert Libchaber for inspiring discussions and his essential participation in our work on protein. We thank Sandipan Dutta who participated in the work on Green’s functions. We thank Stanislas Leibler, Michael R. Mitchell, Elisha Moses, Giovanni Zocchi, and Olivier Rivoire for helpful discussions and encouragement. We are grateful to Karsten Kruse, Alberto Morpourgo, Pierre Collet, and the referees, for constructive comments on the manuscript. JPE was supported by an ERC advanced grant ‘Bridges’, and TT by the Institute for Basic Science IBS-R020 and the Simons Center for Systems Biology of the Institute for Advanced Study, Princeton.
[^1]: The main body of this colloquium is based on, and expands, ideas from papers [@Tlusty2007; @Eckmann2008; @Tlusty2008a; @Tlusty2010; @Tlusty2016; @Tlusty2017; @Dutta2018].
[^2]: 21, when counting the rare pyrrolysine [@Hao2002; @Srinivasan2002].
[^3]: In his book “What is Life?” [@schrodinger1944], Schrödinger uses the term ‘aperiodic crystal’ to describe material which contains genetic information. This is of course a very interesting forethought, but since the advent of quasiperiodic crystals, the term ‘amorphous’ leads to a more precise classification.
[^4]: In addition to the ribosome, the machinery includes two sets of molecules, tRNAs, which carry the amino acids, and aminoacyl-tRNA synthetases, which charge the tRNAs with amino acids. The translation is preceded by a transcription step in which the DNA gene (a segment of the genome) is copied into a mRNA (a single molecule).
[^5]: Some terminology: The individual symbols (A, C, G, T) refer to nucleotides. The triplets of 3 nucleotides form the 64 codons. The 64 codons code for 20 AA and the stop symbol (which does not generate an AA). One of the AA is Methionine (codon ATG) which marks the start of a protein.
[^6]: This graph is difficult to draw, as each node has $8=3+3+2$ neighbors which differ in exactly one position. So a representation would have to be in 8 dimensions. Recall that in a cube in 3-dimensions, every corner has 3 neighbors.
[^7]: PDB = protein data bank, https://www.rcsb.org.
[^8]: The first book is more combinatorial, and the second introduces more spectral concepts.
[^9]: If a bond $\alpha$ connects nodes $i$ and $j$ ($i\ne j)$, we also write $\alpha =(i,j)$.
[^10]: So there are 2 nucleotides and 32 non-redundant codons.
[^11]: The Hamming distance between two sequences is the number of indices where they differ.
[^12]: It is not reasonable to ask for $F=\infty$, but it suffices to look for $F>F_\textrm{crit}$. In our case, $F_\textrm{crit}=5$ is a good choice since in general, the channel will have already formed, and increasing $F_\textrm{ crit}$ will only enlarge the channel somewhat.
[^13]: For explanation of the flat pieces of the graph, see [@Eckmann1985 p. 647].
[^14]: We thank G. Ben Arous for helpful discussions on this point.
[^15]: The definition of the covariance requires to subtract the mean. Instead we project out the first eigenvalue.
[^16]: The continuous part of the sequence spectrum, which is not quite of the standard form, could in principle be studied by taking into account the known correlations. However, even the techniques of [@Guhr1998] seem difficult to implement.
[^17]: As all norms on finite dimensional spaces are equivalent, other norms amount basically just to a rescaling.
[^18]: There are many variants of the shear calculation see, [@continuum].
[^19]: It usually suffices to look at atoms N, C, O, along the backbone, while one may also include sidechains.
[^20]: The dilatation (the trace) is much smaller.
[^21]: The characteristics we are looking for do not show cleanly unless there are at least $10^5$ samples.
[^22]: One could in principle measure the distance between the corresponding pairs using an $L^2$ norm over the whole area.
[^23]: In problems of this local nature, calculating a mutated Green’s function using the Woodbury formula accelerates the computation by a factor of ${\sim}10^4$ as compared to standard matrix inversion.
[^24]: Note that the exponential increase is much stronger than what could by explained by the choice of the factor $\DF>\epsilon F $ of .
|
---
abstract: 'Finding independent sets of maximum size in fixed graphs is well known to be an NP-hard task. Using scaling limits, we characterize the asymptotics of sequential degree-greedy explorations and provide sufficient conditions for this algorithm to find an independent set of asymptotically optimal size in large sparse random graphs with given degree sequences.'
author:
- 'Matthieu Jonckheere, Manuel Sáenz'
bibliography:
- 'main.bib'
title: 'Asymptotic optimality of degree-greedy discovering of independent sets in Configuration Model graphs'
---
Introduction
============
Given a graph $G=(V,E)$, an *independent set* is a subset of vertices $A \subseteq V$ where no pair of vertices are connected to each other (i.e., for every pair $x, y \in A$ we have that $\{ x, y \} \notin E$). In the sequel, the number of vertices will be denoted by $n$.
Independent sets are relevant in the study of diverse physical and communication models. For example in physics, where dynamics that generate independent sets are used to study the deposition of particles in surfaces [@cadilhe2007random; @evans1993random] as well as to model the number of excitations in ultracold gases [@sanders2015sub]. In the context of communications, similar stochastic processes where used [@bermolen2014estimating; @dhara2016generalized] for theoretically modelling the possible number of simultaneous transmissions of information within a WiFi network.
An independent set is said to be *maximal* if it is not strictly contained in another independent set; and is said to be *maximum* if there is no other independent set of greater size. Given a (finite undirected) graph $G$, the size of the maximum independent set(s) is called the *independence number* and is usually represented with $\alpha(G)$. Finding a maximum independent set (or its size) in a general graph is known to be NP-hard[@frieze1997algorithmic]. For sparse random graphs, characterizing the size of maximum independent sets and defining algorithms that find independent sets of (asymptotically, i.e., as $n$ diverges) maximum size are two questions that have received a lot of attention in the last decades but remain largely open. Before explaining our main results, we review the extended body of literature on this problem.
Mainly two types of approaches have been followed to obtain maximum size independent sets in random graphs: on the one hand, using reversible stochastic dynamics (usually Glauber dynamics); and on the other hand, using sequential algorithms. The Glauber dynamics consists of a reversible Markov dynamics on graphs where vertices become occupied (at a fixed rate called the activation rate) when none of its neighbors are. And, if occupied, become unoccupied at rate $1$. When the rate of activation tends to infinity, this dynamics are easily shown to concentrate on configurations being independent sets of maximum size. Though this is a very useful property, and not unlike for many other discrete optimization problems, these dynamics might not be helpful in practice as the convergence towards a maximum size configuration can be extremely slow when the activation rate is large. In some special cases, the mixing time has been theoretically characterized. For example[@vigoda2001note], when the degree distribution is bounded by $\Delta \geq 0$ and the activation rate is small ($\beta < \frac{2}{\Delta - 2}$) the mixing time is $\mathcal{O}(n \log(n))$. In the case of a bipartite regular graph, it was shown [@galvin2006slow] that for $\beta$ large enough the mixing time is actually exponential in $n$. Finally, this method does not allow to theoretically characterize the (asymptotical) independence number.
A completely different approach consists in defining algorithms that explore the graph sequentially (and hence terminate the exploration in less than $n$ steps). For this, at each step $k\geq0$ the vertex set is partitioned in three: the *unexplored vertices* $\mathcal{U}_k$, the *active vertices* $\mathcal{A}_k$ and the *blocked vertices* $\mathcal{B}_k$. A typical sequential algorithm works as follows. Initially, it sets $\mathcal{U}_0 = V$, $\mathcal{A}_0 = \emptyset$ and $\mathcal{B}_0 = \emptyset$. To explore the graph, at the $k+1$-th step it selects a vertex $v_{k+1}\in\mathcal{U}_k$ (possibly taking into account its current or past degree towards other unexplored vertices), and changes its state into active. After this, it takes all of its unexplored neighbors, i.e. the set $\mathcal{N}_{v_{k+1}} = \{ w \in \mathcal{U}_k | (v_{k+1},w) \in E \}$, and changes their states into blocked. This means that, if in the $k+1$-th step vertex $v_{k+1}$ is selected, the resulting set of vertices will be given by $\mathcal{U}_{k+1} = \mathcal{U}_k \backslash \{ v_{k+1} \cup \mathcal{N}_{v_{k+1}} \}$, $\mathcal{A}_{k+1} = \mathcal{A}_{k+1} \cup \{ v_{k+1} \}$ and $\mathcal{B}_{k+1} = \mathcal{B}_k \cup \mathcal{N}_{v_{k+1}}$. Note that at each step, the set of active vertices defines an independent set. The algorithm keeps repeating this procedure until the step $k_n^*$ in which all vertices are either active or blocked (or equivalently $\mathcal{U}_{k_n^*} = \emptyset$). The set of active vertices at step $k_n^*$ then defines a maximal independent set.
In the *greedy algorithm*, during the $k$-th step the activating vertex $v_k$ is selected uniformly from the subgraph of remaining vertices $G_k$ (i.e., the subgraph formed by the unexplored vertices). If $G$ is a graph, we will call $\sigma_{Gr}(G)$ the size of the independent set obtained by the greedy algorithm ran on $G$. This algorithm has been extensively studied, specially in the context of Erdös-Renyi graphs. There are many ways of approaching the problem of determining the asymptotic value of the independent set obtained by the greedy algorithm in a sparse Erdös-Renyi graph. For example, there have been [@grimmett1975colouring; @karp1976probabilistic] combinatorial analysis of the problem. More akin to the rest of the paper, in [@Bermolen2017] an hydrodinamic limit for a one-dimensional Markov process associated to the algorithm was proved. The description of this exploration process in a Configuration Model cannot be described as a one-dimensional Markov process, as the unexplored vertices have degrees (towards other unexplored) that are not interchangeable and that depend in a complicated way on the evolution of the process. This makes the analysis much more involved that in the case of an Erdös-Renyi graph. There have been two works in the literature that describe an hydrodinamic limit for this process. In [@bermolen2017jamming], an hydrodinamic limit for the degree distribution of the remaining graph was obtained. While in [@brightwell2016greedy], a similar hydrodinamic limit was proved but for a modified dynamics that allows for a simplification of the limiting differential equations. Through these limits, the size of the independent set is determined.
The *degree-greedy* algorithm is a variation of the greedy algorithm that takes into account the degree of the vertices in the remaining graph. During the $k$-th step an unexplored vertex $v_k$ is selected uniformly from the vertices of *minimum degree* (towards other unexplored vertices) within the remaining subgraph $G_k$. If $G$ is a graph, we will denote by $\sigma_{DG}(G)$ the (possibly random) proportion of vertices in the independent set obtained by the degree-greedy algorithm ran on $G$. Although having been studied in the computer science community (for example, in [@halldorsson1997greed]), there are few exact mathematical results that characterize or bound the independent set found by this algorithm. A remarkable exception can be found in [@karp1981maximum; @aronson1998maximum] which, although being works about maximum matchings, imply that the degree-greedy algorithm is asymptotically optimal for Erdös-Renyi graphs when the mean degree is $\lambda < e$. Results by Wormald [@wormald1995differential] also describe an hydrodynamic limit for the process generated by the degree-greedy algorithm when run on a $d$-regular graph but without discussing asymptotic optimality.
#### Characterization of maximum independent sets.
In the case of Erdös-Renyi graphs, their independence numbers are better understood in the case of large connection probabilities. In this case, a second moment argument[@bollobas1998random] yields its convergence in probability towards $- \frac{2 \log np}{\log q}$. In this same context, the relationship between the independence number and the asymptotic size of the independent set obtained by the greedy algorithm is also known to be $1/2$. Where in the case of sparse Erdös-Renyi graphs, as a consequence of the results proved in [@bollobas1976cliques; @frieze1990independence], similar (but weaker) results can be shown. This shows that the greedy algorithm has, for $G$ a sparse Erdös-Renyi graph of mean degree $\lambda$, a performance ratio $\sigma(G)/\alpha(G)$ that is $1/2$ asymptotically in $n$ for large $\lambda$. As mentioned before, it follows from [@karp1981maximum; @aronson1998maximum] that the degree-greedy algorithm finds a.a.s. maximum independent sets in Erdös-Renyi graphs of mean degree smaller than $e$.
The proof of the existence of a limiting independence ratio for random $d$-regular graphs was given in [@bayati2010combinatorial]. In [@lauer2007large], Wormald proved a lower bound for the independence number of a $d$-regular graph. While in the same work, Wormald (and Gamarnik and Goldberg independently in [@gamarnik2010randomized]) showed that the proportion of vertices in the independent set found by a Greedy algorithm in a $d$-regular graph is (for $d\geq3$), asymptotically (in probability) when $n\rightarrow\infty$ and the girth[^1] $g \rightarrow\infty$, given by a certain function of $d$. Moreover, bounds were also proved in [@bollobas1981independence; @mckay1987lnl] and an alternative lower bounds in [@frieze1992independence]. Finally, in a recent work[@ding2016maximum], the exact large number law for the independence ratio of regular graphs of sufficiently large $d$ was established as the solution of a polynomial equation.
#### Contribution.
Both characterization of independence numbers and algorithms to discover independent sets in (deterministic) polynomial times for large sparse random graphs, e.g. in Configuration Models, are still open problems for most cases (the only exceptions being the characterizations of the independence numbers of regular graphs of large enough degree and the optimality of degree-greedy for sparse Erdös-Renyi graphs with $\lambda < e$).
Using scaling limits on sequential explorations that select only degree 1 vertices, we decompose the exploration in different steps and we show that these steps can be described as a combination of two maps acting on the degree-distribution of the graph. Using these results, we show that for a large class of sparse random graphs, the asymptotic optimality of degree-greedy is actually verified. We first give a sufficient condition, which can be easily verified in practice, for this exploration to be optimal *in one step*. We then show how to generalize this sufficient condition by characterazing the remaining graph after *several steps*.
We now state rigorously our results and give various examples.
Main results
============
We first state lemmas for deterministic graphs. Afterwads, we enunciate optimality results for random graphs when the hydrodynamic limit of the degree-greedy sequential exploration selects only degree 1 vertices. We then show that this property is actually satisfied by a large class of random graphs (including strictly sub-critical graphs, in the connectivity sense). We finally proceed to characterize this class of graphs.
First characterization of degree-greedy asymptotic optimality
-------------------------------------------------------------
#### Criterion for deterministic graphs {#criterion-for-deterministic-graphs .unnumbered}
In Lemma \[lemma:optimal\] below, we show that any algorithm that selects at each step of its implementation a vertex of degree 1 (or 0) is optimal in the sense that it finds a maximum independent set. An analogous version of this lemma for matchings can be found in [@karp1981maximum].
We introduce now some definitions to state this more precisely.
Given a graph $G = (V,E)$, we call a finite sequence $W=\{w_1,w_2,\ldots,w_m\}$ ($m\leq n$) of distinct vertices of $V$ a **selection sequence** (of $G$) if no vertex in $W$ is neighbor to another vertex in $W$ and every vertex in $V$ is either in $W$ or neighbor of a vertex in $W$.
Note that the conditions in this definition ensure that the vertices in $W$ define a maximal independent set. By definition, sequential algorithms define random selection sequences.
\[def:remaining\] Let $W=\{w_1,...,w_m\}$ be a selection sequence. Then, for every $1 \leq i \leq m$, we denote by **$i$-th remaining subgraph**, the subgraph formed by the vertices that are neither in $\{w_1,...,w_i\}$ nor neighbors of any of them. We denote it by $G_{i-1}$.
When there is no ambiguity to which value of $i$ the remaining graph corresponds to, we just call it the *remaining graph*. When analyzing the degree-greedy algorithm, the remaining graphs will refer to the remaining graphs with respect to the selection sequence defined by the algorithm. Of course, as a selection sequence $W = \{ w_1,...,w_m\}$ always determines a maximal independent set, $G_m = \emptyset$.
The degree-greedy algorithm run on a finite graph $G$ can be thought of as a random selection sequence $W_{DG}$, built inductively in the following manner: given $\{w_1,...,w_k\}$ the first $k \geq 1$ vertices of $W_{DG}$, $w_k$ is a vertex chosen uniformly from the lowest degree vertices of $G_k$.
Let $W$ be a selection sequence. We say that $W$ **has the property $T_1$** if for every $1\leq i \leq m$ the degree of $w_i$ in $G_{i-1}$ is equal or less than $1$.
Then, a selection sequence has the property $T_1$ if at each step it *selects* a vertex that has degree either 0 or 1 in the corresponding remaining graph.
We are now in a position to state our first lemma.
\[lemma:optimal\] Let $G$ be a finite graph and $W$ be a selection sequence of $G$. Then, if $W$ has the property $T_1$, $|W| = \alpha(G)$.
#### $T_1$ property and asymptotic optimality of the degree-greedy exploration. {#t_1-property-and-asymptotic-optimality-of-the-degree-greedy-exploration. .unnumbered}
From now on, we use the usual *big $\mathcal{O}(\cdot)$ and little $o(\cdot)$* notation to describe the asymptotic behaviour of functions of the graph size $n$. We also use the *probabilistic big $\mathcal{O}_{\mathbb{P}}(\cdot)$ and little $o_{\mathbb{P}}(\cdot)$* notation in the following sense:
- A sequence of random variables $X_n$ is $\mathcal{O}_{\mathbb{P}}(f_n)$ (for some function $f_n : \mathbb{N} \rightarrow \mathbb{R}_{>0}$) if for every $\epsilon > 0$ there exists $M > 0$ and $N \in \mathbb{N}$ s.t. $\mathbb{P}(|X_n / f_n| > M ) < \epsilon$ for every $n \geq N$.
- Conversely, a sequence of random variables $X_n$ is $o_{\mathbb{P}}(f_n)$ if $X_n/f_n \xrightarrow{\mathbb{P}} 0$ as $n \rightarrow \infty$.
We will also say that an event holds *with high probability* (w.h.p.) whenever its probability is some function $1+o(1)$ of the graph size.
Throughout the paper we will consider random graphs with given degrees, (a.k.a., Configuration Models [@van2016random]). In this contruction, given a *degree sequence* $\bar d^{(n)} \in \mathbb{N}_0^n$ (which could either be a fixed sequence or a collection of i.i.d. variables), we will form an $n$-sized *multi-graph*[^2] with degrees $\bar d^{(n)}$ in the following manner: first assign to each vertex $v \in \{1,...,n\}$ a number $d_v$ of half-edges, then sequentially match uniformly each half-edge with another unmatched one, and finally for every pair of vertices in the multi-graph establish an edge between them for every pair of matched half-edges they share. The distribution of the random multi-graphs generated according to this matching procedure will be denoted by $\mbox{CM}_n(\bar d^{(n)})$. Because all the matchings are made uniformly, the resulting multi-graph is equally distributed no matter in which order the half-edges are chosen to be paired [@van2016random]. This fact allows, when analyzing a process in the graph, for the matching to be incorporated into the dynamics in question.
In the remaining of the paper we will consider Configuration Model graphs that obey the following assumption:
**Convergence assumption (CA):** *when dealing with sequences of graphs $(G_n)_{n\geq1}$ with $G_n \sim \mbox{CM}_n(\bar d^{(n)})$, we will always assume that $D^{(n)} \xrightarrow{\mathbb{P}} D$ and $\mathbb{E}(D^{(n)2}) \xrightarrow{n\rightarrow\infty} \mathbb{E}(D^2)$.*
Where $D^{(n)}$ is the r.v. that gives the degree of a uniformly chosen vertex in $G_n$ and $D$ is a random variable with finite second moment. We will also use $(p^{(n)}_k)_{k\geq0}$ to refer to the distribution of $D^{(n)}$ and $(p_k)_{k\geq0}$ to the one of the asymptotic degree random variable $D$. When there is no ambiguity, we will omit the subindex in $G_n$.
Although this construction results in a multi-graph rather than a *simple graph*[^3], as shown in [@janson2009probability], this is not a problem because under the (CA) there is asymptotically a probability bounded away from 0 of obtaining a simple graph. This means that any event that has been showed to hold w.h.p. for $G \sim \mbox{CM}_n(\bar d^{(n)})$, can be automatically showed to hold w.h.p. for the construction conditioned to result in a simple graph.
Another important feature of the model is that its largest connected component asymptotically contains a positive proportion of the vertices of the graph iff $\nu := \mathbb{E}(D(D-1))/\mathbb{E}(D) > 1$ [@molloy1998size; @janson2009new]. When this condition holds, we will say that the graph is *supercritical*; and when it does not, that it is *subcritical*.
We can now state a sufficient condition for the degree-greedy algorithm to find w.h.p. an independent set that asymptotically contains the same proportion of vertices as a maximum one:
\[prop:optimalidadCM\] Let $G \sim \mbox{CM}_n(\bar d^{(n)})$ be a sequence of graphs distributed according to the CM, and assuming that the limiting $(p_k)_{k\in\mathbb{N}}$ is $\mathcal{O}(e^{-\gamma k})$ for some $\gamma > 0$. If the degree-greedy algorithm defines w.h.p. a selection sequence that selects only vertices of degree 1 or 0 until the remaining graph is subcritical, then (for every $\alpha > 0$) $\sigma_{DG}(G) = \alpha(G) + \mathcal{O}_{\mathbb{P}}(n^\alpha)$.
This is so because a subcritical graph looks like (up to sufficiently small differences) a collection of trees. We can then couple the algorithm running in the subcritical graph with one running in the collection of spanning trees of its components. This coupling will only differ in the components that are not trees which, as shown in Proposition \[prop:2coreChico\], contain (for every $\alpha > 0$) $\mathcal{O}_{\mathbb{P}}(n^\alpha)$ vertices if the limiting degree distribution has an exponentialy thin tail. Therefore, both algorithms find an independent set of roughly the same size.
Further characterizations of asymptotic optimality
--------------------------------------------------
We now give explicit criteria to show that a given limiting distribution meets the hypothesis in Proposition \[prop:optimalidadCM\]. We first state a criterion that can be easy to handle in practice. We then refine this criterion and give a general way of characterizing the degree distributions in question.
#### One application of the map $M_1$. {#sec:oneapplication .unnumbered}
To characterize when the degree-greedy algorithm does only select vertices of degree 1 or 0 until the remaining subgraph is subcritical, it will be useful to break the evolution of the process into discrete intervals of time for which we know for sure that the only vertices selected have these degrees.
The key observation is that if the graph initially has $n p_1 + o_\mathbb{P}(n)$ vertices of degree 1[^4], then the degree-greedy will select vertices of degree 1 at least until an equivalent number of degree 1 vertices have been explored. Then we define the map $M_1^{(n)} : \mathbb{R}_{\geq0} \longrightarrow \mathbb{R}_{\geq0}$ as the map that when evaluated in a degree distribution of an $n$ sized graph $(p_k)_{k\geq0}$ gives the resulting degree distribution of unexplored vertices after $n p_1 + o_{\mathbb{P}}(n)$ vertices of degree 1 have been activated or blocked (and their neighbors blocked). This is in principle a stochastic map but, as we will prove, the degree-greedy exploration converges to a deterministic limit which implies that $M_1^{(n)}(\cdot)$ also converges to a deterministic limit $M_1(\cdot)$.
In this section, we determine when the degree distribution obtained after one application of the map $M_1^{(n)}$ is w.h.p. subcritical, and therefore the hypothesis of Proposition \[prop:optimalidadCM\] are met. Our main result here is the following theorem:
\[thm:oneapplication\] Given the $G \sim \mbox{CM}_n(\bar d^{(n)})$ where the $(CA)$ holds towards a limiting degree distribution $(p_k)_{k\geq0}$ of mean $\lambda > 0$ and finite second moment. If
$$\tilde \nu : = G_D''(Q) / \lambda < 1$$
where $Q:=(1 - p_1/\lambda)$ and $G_D(z)$ is the generating function of the asymptotic degree r.v. $D$; then, (for every $\alpha > 0$) $\sigma_{DG}(G) = \alpha(G) + \mathcal{O}_\mathbb{P}(n^\alpha)$.
#### Further applications of $M_1$. {#sec:manyapplications .unnumbered}
Theorem \[thm:oneapplication\] establishes an asymptotic condition for the remaining graph obtained after one application of the map $M_1^{(n)}(\cdot)$ to be subcritical, and thus for the degree-greedy to be quasi-optimal (in the sense that it finds, asymptotically, an independent set with the same proportion of vertices as a maximum one). Here we compute the asymptotic degree distribution of the remaining graph after one application of $M_1^{(n)}(\cdot)$ and in doing so we allow for the study of further applications of $M_1^{(n)}(\cdot)$. This can be used to establish more general conditions that determine the quasi-optimality of the degree-greedy algorithm. For doing so, we determine an hydrodynamic limit for the second phase of $M_1^{(n)}(\cdot)$ and solve the obtained equations.
\[thm:furtherapp\] Define (for every $i, k \geq 1$) $\eta_k(i) := (-1)^{j-i} \tilde Q^j \binom{j}{i} \mathbb{1}_{i\leq j}$. Then, under the same assumptions of Theorem \[thm:oneapplication\] and if we call $(a_j)_{j\in\mathbb{N}}$ the components of the asymptotic distribution $(p_k)_{k\in\mathbb{N}}$ in the base $\{\eta_j(\cdot)\}_{j\in\mathbb{N}}$, we have that the remaining graph after one application of the map is a Configuration Model graph of $\tilde n := n [1 - \lambda ( 1 - Q^2) - p_0] + o_\mathbb{P}(n)$ vertices and asymptotic degree distribution given by
$$M^{(n)}_1\left(p^{(n)}_k\right)(i) \xrightarrow{\mathbb{P}} \sum_{j \geq i} a_j (-1)^{j-i} \tilde Q^j \binom{j}{i}, \ \mbox{for } i \geq 1$$
$$\mbox{and } M^{(n)}_1\left(p^{(n)}_k\right)(0) \xrightarrow{\mathbb{P}} 1 - \lambda(1-Q^2) - p_0 - \sum_{j=1}^\infty M^{(n)}_1\left(p^{(n)}_k\right)(j)$$
where $\tilde Q := \sum_{i\geq2}i Q^i p_i / Q^2 \lambda$.
As mentioned above, this result gives a way of generalizing the condition for quasi-optimality of Theorem \[thm:oneapplication\]:
**General criterion for quasi-optimality:** *given a degree distribution, if after a finite number of applications of the map $M_1(\cdot)$ the degree distribution obtained is subcritical, then the degree-greedy is quasi-optimal for a Configuration Model graph with that limiting distribution.*
The proof of this criterion is a direct consequence of Proposition \[prop:optimalidadCM\]. Then, Theorem \[thm:furtherapp\] can be used to verify it. In the next section we give numerical computations of distributions that meet the criterion.
Examples and applications {#sec:exandapp}
-------------------------
We apply our results to specific important distributions for which Theorems \[thm:oneapplication\] and \[thm:furtherapp\] can be applied. Furthermore, we explain how to compute the independence ratios for this distributions and we prove upper bounds for the independence ratios of distributions for which these theorems cannot be used.
#### Poisson distributions.
We first analyze the case of Poisson distributions. They are of particular importance because, by the results in [@kim2007poisson], they allow for the study of Erdös-Rényi ramdom graphs.
We first establish for which values of the mean degree $\lambda$, Poisson distributions meet the inequality in Theorem \[thm:oneapplication\]. Because the generating function for a Poisson distribution of mean $\lambda$ is given by $e^{\lambda(z-1)}$, we have that:
$$\tilde \nu (\lambda) = \frac{ (e^{\lambda(z-1)})''|_{Q(\lambda)} }{\lambda} = \lambda e^{-\lambda e^{-\lambda}}$$
Where we used that $Q(\lambda) = 1 - e^{-\lambda}$. Then, by Theorem \[thm:oneapplication\] we have that the degree-greedy algorithm is quasi-optimal if $\tilde \nu(\lambda) <1$, which happens when $\lambda$ is smaller than some $\lambda_0 \sim 1.41$. This means that there are non trivial (supercritical) values of the mean degree for Poisson distributions where the degree-greedy algorithm achieves an independent set with asymptotically the same proportion of vertices of a maximum one.
Moreover, because of the well known equality for Bernstein polynomials[@lorentz2012bernstein]
$$\binom{n}{i} x^i (1-x)^{n-i} = \sum_{j=i}^n \binom{n}{j} x^j \eta_j(i)$$
the expansion in the base $\{\eta_k(\cdot)\}_{k\in\mathbb{N}}$ can be explicitly computed for binomial distributions, which in turn gives the transformation for Poisson distributions taking the usual limit. Using this, it can be easily seen that after $i$ applications of the map $M_1(\cdot)$ to a Poisson distribution, the resulting distribution is a linear combination of a Poisson distribution of mean $\lambda_i$ and a term $\delta_1(\cdot)$ (we here ignore degree 0 vertices as they do not play a role in the dynamics), with respective coefficients $A_i$ and $B_i$.
This transformation can be used to derive the following recursion relation for $\lambda_i$, $A_i$ and $B_i$:
$$\begin{cases}
\lambda_{i+1} = \tilde Q_i Q_i \lambda_i \\
A_{i+1} = e^{-(1-Q_i)\lambda_i} A_i \\
B_{i+1} = - A_i \tilde Q_i Q_i \lambda_i e^{- \lambda_i}
\end{cases}$$
Where $Q_i$ and $\tilde Q_i$ are the corresponding coefficients defined in Theorems \[thm:oneapplication\] and \[thm:furtherapp\] for the distribution after the $i$-th application. Then, the degree-greedy algorithm will find a quasi-optimal independent set for a sequence of graphs with limiting Poisson distribution with mean $\lambda$ if for some $i \geq 1$, the parameters $A_i$, $B_i$ and $\lambda_i$ given by the recursion above define a subcritical distribution. According to the results in [@karp1981maximum; @aronson1998maximum], this should happen iff $\lambda < e$.
#### Power-law distributions.
Here we look at the case where the degree distribution obeys a power law of parameter $\alpha > 3$. Because the generating function of a power law distribution $p_k = C_\alpha k^{-\alpha}$ is given by $C_\alpha \mbox{Li}_\alpha(z)$ (where $\mbox{Li}_\alpha(z)$ is the polylogarithm function of order $\alpha$):
$$\tilde \nu(\alpha) = \frac{(C_\alpha \mbox{Li}_\alpha(z))''|_{Q(\alpha)}}{\sum_{i\geq1} i C_\alpha i^{-\alpha}} = \frac{\mbox{Li}_{\alpha-2}(1-\zeta(\alpha-1)^{-1})-\mbox{Li}_{\alpha-1}(1-\zeta(\alpha-1)^{-1})}{\zeta(\alpha-1)}$$
Where $\zeta(z)$ is the Riemman zeta function and in the last line we used that $Q(\alpha) = 1 - \zeta(\alpha-1)^{-1}$ and that $\mbox{Li}_{\alpha}(z)' = \mbox{Li}_{\alpha-1}(z)/z$. This last expression can be seen to be smaller than $1$ for every $\alpha > 3$; which means that for every power law distribution of finite second moment, the degree-greedy algorithm is quasi-optimal. In particular, whenever it has finite second moment and $\zeta(\alpha-2) > 2 \zeta(\alpha-1)$ (or $3 < \alpha \lesssim 3,478$), this distribution will be supercritial but nevertheless the degree-greedy will be quasi-optimal.
#### Computing independence ratios.
As a consequence of Theorem \[thm:furtherapp\], whenever a sequence of graphs is under its hypothesis and after a finite number of applications of the map $M_1(\cdot)$ on its asymptotic degree distribution a subcritical distribution is obtained, the asymptotic independence number can be obtained by computing the number of vertices in the independent set constructed by the degree-greedy algorithm. For every two vertices removed by the algorithm, one is added to the independent set. Also, at the beginning of the dynamics all degree 0 vertices are selected at once. Here we will denote by $r$ the number of degree 0 vertices generated during the application of map $M_1(\cdot)$. Then, by Theorem \[thm:furtherapp\],
$$\label{eq:indnumestimation}
\alpha(G) = n \left( \sum_{i=1}^\infty r_i + \frac{\lambda_i(1-Q_i^2)}{2} \prod_{j=1}^{i-1} \left[1-\lambda_j (1-Q_j^2)-r_j\right] \right) + o_\mathbb{P}(n).$$
Where $\lambda_i$, $Q_i$ and $r_i$ are the corresponding parameters of the distribution obtained after $i$ applications of the map $M_1(\cdot)$ over the limiting degree distribution of the graph sequence.
#### Upper bounds for independence ratios.
For sequences of graphs where the quasi-optimality condition does not hold, one can nevertheless use Theorem \[thm:oneapplication\] to construct upper bounds on the limiting independence number.
To do this, we may construct a new graph sequence $\tilde G$ defined as the graphs where the degrees of a certain proportion of vertices in the original graphs $G$ were changed to 1 so that the inequality in Theorem \[thm:oneapplication\] holds. The resulting graphs will be distributed as the original ones with some of its edges removed. Then, because the independence number is monotonic on edge removal, we will have that $\alpha(G) \leq \alpha(\tilde G)$. Finally, using equation , $\alpha(\tilde G)$ can be computed and thus the desired upper bound is obtained.
Proofs
======
In this Section, we provide proofs for all our results.
Proof of Lemma \[lemma:optimal\] on deterministic graphs
--------------------------------------------------------
We will prove Lemma \[lemma:optimal\] by induction on the graph size $|G|$. For $|G|=1$ and $|G|=2$ the statement is trivially true.
Now assume that it holds for $|G|=n$, we will show that it is thus also true for $|G|=n+1$. Suppose that it is not the case that $|W| = \alpha(G)$, then there is an independent set $A$ such that $|W|<|A|=\alpha(G)$. Calling $W'=(w_i)_{i=2}^{|W|}$, by the induction hypothesis (because $W'$ has the $T_1$ property in $G_1$) we know that $|W'|=\alpha(G_1)$.
Calling $n_1$ the vertex adjacent to $w_1$,[^5] because the independent set $A$ is of maximum size, it has to contain either $w_1$ or $n_1$ (if not, one could construct an even larger independent set by adding $w_1$ to the vertices in $A$, which would be a contradiction). This implies that $$|A|= |A \backslash \{w_1,n_1\}| + 1 > |W| = |W'| + 1,$$ which means that $|A \backslash \{w_1,n_1\}| > |W'|$ which is a contradiction because $A \backslash \{w_1,n_1\}$ defines an independent set of $G_1$ and by hypothesis we have that the independent set defined by $W'$ is an independent set of $G_1$ of maximum size. We then have that $W$ defines an independent set of maximum size of $G$, advancing in this way the induction.
\[coro:arbol\] If $H$ is a collection of trees, then the degree-greedy algorithm run on $H$ finds a.s. a maximum independent set.
Because $H$ is a collection of trees, for every leaf removed by the degree-greedy algorithm, further leafs (or isolated vertices) will be created. The algorithm will have the property $T_1$ as it will only select leafs (or isolated vertices). The conclusion is then reached by Lemma \[lemma:optimal\].
Proof of Proposition \[prop:optimalidadCM\]
-------------------------------------------
We first deal with the case where $G$ is a subcritical graph.
To show the condition for optimality of Proposition \[prop:optimalidadCM\], we will study the number of times a breadth-first exploration process of the components joins two already explored vertices (for more details on the breadth-first exploration of components see [@van2016random]) forming a loop. Here we will call $N(C(u))$ the number of times this happens during the exploration of the connected component associated to the vertex $u \in V$ and $N$ the total number of times it happens in the exploration of all the components in the graph. For every component $C(u)$ we will have that if $N(C(u)) = 0$, then the component is exactly a tree (as no loops are formed during the exploration process). The idea will be to prove that in a subcritical graph with degrees with an exponentially thin tail, almost every vertex is in a component that is a tree. Note that this is not a consequence of the results in [@janson2007simple], as it is not enough to show that the 2-core[^6] is $o_\mathbb{P}(n)$ to conclude this.\
\
By calling $v(i)$ the vertex visited during the $i$-th step of the exploration process of the connected component of vertex $1$ and $T$ the stopping time in which the process finishes, we can write $$N\left( C(1) \right) = \sum_{i=1}^{T} \sum_{j>i}^{T}\mathbb{1}_{\{v(i)=v(j)\}}$$
We can now state the following bound on the first moment of $N(C(1))$ that we will use to prove our lemma. Here we will work with the CM with i.i.d. degrees, the proofs for the CM with fixed degrees are analogous.
\[lemma:cotacond\] Let $G \sim \mbox{CM}_n(\overline{d})$ where the $(CA)$ holds towards a limiting degree distribution $(p_k)_{k\geq0}$ of mean $\lambda > 0$ and criticality parameter $0 < \nu < \infty$. If $C(1)$ is the connected component of vertex $1$, then conditional to the number $T$ of edges in the component and the total number $l_n$ of edges in the graph, we have that
$$\mathbb{E}\left(N(C(1))|T,l_n\right) \leq \frac{(\nu_n + 1) l_n T(T-1)}{2 (l_n -T)^2}$$
where $\nu_n := \sum_{u\in V} d_u(d_u-1) / l_n$.
The bound will be derived making use of the corresponding bound for the probability that (conditional on $T$ and $l_n$) a particular vertex is visited by the exploration process at time $i$. So, firstly we want to show that for $u \in V$ and $i \leq T$
$$\label{eq:desprobcond}
\mathbb{P}(v(i)=u|T,l_n) \leq \frac{d_u}{l_n-T}$$
Where $d_u(i)$ is the number of unmatched half-edges of $u$ by the step $i$ of the exploration. This can be shown using that the random variables $l_n$ and $T$ are a.s. positive and bounded by $n^2$. Then for any set $A$ $\sigma(T,l_n)$-measurable we have that $$\begin{aligned}
\mathbb{E}(\mathbb{1}_A \mathbb{1}_{\{ v(i) = u \}}) & = \mathbb{P}(A, v(i)=u) \\
& = \sum_{l \leq n^2} \sum_{t \leq n^2} \mathbb{P}(A, v(i)=u|T=t,l_n=l) \mathbb{P}(T=t,l_n=l)\\
& = \sum_{l \leq n^2} \sum_{t \leq n^2} \mathbb{P}(A|v(i)=u,T=t,l_n=l) \mathbb{P}(v(i)=u|T=t,l_n=l) \mathbb{P}(T=t,l_n=l)\\
& \leq \sum_{l \leq n^2} \sum_{t \leq n^2} \mathbb{1}_A(t,l) \frac{d_u}{l-t}\mathbb{P}(T=t,l_n=l)\\
& = \mathbb{E}\left(\mathbb{1}_A \frac{d_u}{l_n-T}\right)\end{aligned}$$
Where in the forth line we have used that $\mathbb{P}(A|v(i)=u,T=t,l_n=l) = \mathbb{1}_A(t,l)$ because $A$ is $\sigma(T,l_n)$-measurable and that because the matching is done uniformly between all the unmatched half-edges at time $i$ we have that $\mathbb{P}(v(i)=u|T=t,l_n=l) = \frac{d_u(i)}{l-(i-1)} \leq \frac{d_u}{l-t}$.\
\
Now using that $$\begin{aligned}
N\left( C(1) \right) & = \sum_{i=1}^{T} \sum_{j>i}^{T}\mathbb{1}_{\{v(i)=v(j)\}}\\
& = \sum_{i=1}^{T} \sum_{j>i}^{T} \sum_{u\in V} \mathbb{1}_{\{v(i)=u\}} \mathbb{1}_{\{v(j)=u\}}\end{aligned}$$
And taking conditional expectation on $T$ and $l_n$ over both sides of the equality and using these inequalities, we get that
$$\begin{aligned}
\mathbb{E}\left( N\left( C(1) \right) | T, l_n \right) & = \sum_{i=1}^{T} \sum_{j>i}^{T} \sum_{u\in V} \mathbb{P}(v(i)=u,v(j)=u|T,l_n)\\
& \leq \sum_{i=1}^{T} \sum_{j>i}^{T} \sum_{u\in V} \left(\frac{d_u}{l_n-T}\right)^2\\
& = \frac{l_n}{(l_n-T)^2} \left( \sum_{i=1}^{T} \sum_{j>i}^{T} 1 \right) \left( \sum_{u\in V} d_u \frac{d_u}{l_n}\right)\\
& = \frac{(\nu_n + 1) l_n T(T-1)}{2 (l_n -T)^2}\end{aligned}$$
Where for the inequality in the second line we used that for all $i < j$ $$\begin{aligned}
\mathbb{P}(v(i)=u;v(j)=u|T,l_n) & = \mathbb{P}(v(j)=u|v(i)=u;T,l_n) \mathbb{P}(v(i)=u|T,l_n) \\ & \leq \mathbb{P}(v(j)=k|T,l_n) \mathbb{P}(v(i)=k|T,l_n)\end{aligned}$$
By means of this lemma, if we denote by $B$ the number of *bad vertices* that are in components that are not trees, we can prove that (under certain assumptions) it grows slower than any positive power of the graph size $n$.
\[prop:2coreChico\] Let $G \sim \mbox{CM}_n(\overline{d})$ where the $(CA)$ holds towards a limiting degree distribution $(p_k)_{k\geq0}$ of mean $0 < \lambda < \infty$, criticality parameter $0 < \nu < 1$ and such that there exists $\gamma >0$ where $p_k = \mathcal{O}(e^{-\gamma k})$. Then, for every $0<\alpha<1$ we have that $B = \mathcal{O}_{\mathbb{P}}(n^{\alpha})$.
The proof will apply Lemma \[lemma:cotacond\] and a coupling, presented by S. Janson et al. in [@janson2008largest], which we will describe in the following.\
\
Suppose we are in the step $i \leq \sqrt{n}$ (the exact power of $n$ is in fact irrelevant, it only needs to be $o(n)$ and of an order smaller than the components sizes) of the exploration process. Then the probability of finding a vertex of degree $k$ (excluding the vertices already explored) will be $$\frac{nk \mathbb{P}(D_n =k)}{ n \lambda_n- \mathcal{O}(\sqrt{n})} = \frac{k \mathbb{P}(D_n =k)}{ \lambda_n } ( 1 + \mathcal{O}(\sqrt{n}) )$$
Where $\lambda_n := \mathbb{E}(D_n)$. Defining a random variable distributed according to
$$\label{eq:treecoupling}
\mathbb{P}(X \geq x ) = \mbox{min}\left( 1, \frac{\nu'}{\nu} \sum_{k \geq x} \frac{(k+1) p_{k+1}}{\lambda} \right)$$
For $\nu' = \nu + \epsilon'$ fixed and with $0 < \epsilon' < 1 - \nu$, then for every $n$ large enough $X-1$ stochastically dominates the step size of the random walk associated to the exploration process. As a consequence, we will have that if $\tilde T$ is the hitting time of $0$ of a random walk starting from $1$ and with step size $X-1$, $T \leq \tilde T$ a.s. whenever $\tilde T \leq \sqrt{n}$. The variable $\tilde T$ can also be thought of as the total progeny of a Galton-Watson process with progeny law given by i.i.d. copies of $X-1$. By summing over the expression in (\[eq:treecoupling\]) we get that $\mathbb{E}(X-1) \leq \frac{\nu'}{\nu}\nu <1$, which means that the associated Galton-Watson process is sub-critical.
Making use of this coupling, we can now prove the proposition. Taking expectation to $N(C(1))$ and separating according to different values of $T$ and $l_n$ we obtain that $$\begin{aligned}
\mathbb{E}(N(C(1))) & = \mathbb{E}(N(C(1)) \mathbb{1}_{\{T\leq n^\beta\}} \mathbb{1}_{\{|l_n - n \lambda_n| < rn\}})\\
& + \mathbb{E}(N(C(1)) \mathbb{1}_{\{T\leq n^\beta\}} \mathbb{1}_{\{|l_n - n \lambda_n| \geq rn\}})\\
& + \mathbb{E}(N(C(1)) \mathbb{1}_{\{T > n^\beta\}} )\end{aligned}$$ Where $r < \lambda/2$ and $0<\beta<1/2$ (but its exact value will be fixed later on). By the upper bound on the conditional expectation in Lemma \[lemma:cotacond\] we get that the first term is
$$\begin{aligned}
\mathbb{E}(& N(C(1)) \mathbb{1}_{\{T\leq n^\beta\}} \mathbb{1}_{\{|l_n - n \lambda_n| < rn\}})\\
& \leq \frac{(\nu_n+1)}{2}\frac{(\lambda_n+r) n^{\beta+1} ( n^\beta-1)}{[(\lambda_n+r)n -n^\beta]^2} \mathbb{P}(T\leq n^\beta,|l_n - n \lambda_n| < rn) \\
& \leq \frac{(\nu_n+1)}{2}\frac{(\lambda_n+r) n^{\beta+1} ( n^\beta-1)}{[(\lambda_n+r)n -n^\beta]^2}\end{aligned}$$
Because by hypothesis $\lambda_n \rightarrow \lambda$ and $\nu_n \rightarrow \nu$, the right hand is then $\mathcal{O}(n^{2\beta-1})$. For the second term, bounding $N(C(1))$ by $T$, we get that
$$\begin{aligned}
\mathbb{E}(N(C(1)) \mathbb{1}_{\{T \leq n^\beta\}} \mathbb{1}_{\{|l_n - n \lambda_n| \geq rn\}}) & \leq n^\beta \mathbb{P}(T \leq n^\beta, |l_n - n \lambda_n| \geq rn) \\
& \leq n^\beta \mathbb{P}( |l_n - n \lambda_n| \geq rn) \end{aligned}$$
Using that by hypothesis the second moment of the degree random variable converges a.s. to a finite limit and by Chebyshev’s inequality, we get that this term is $\mathcal{O}(n^{\beta-1})$.\
\
For the last term, we will again use that $N(C(1)) \leq T$ a.s. to obtain that $$\mathbb{E}(N(C(1)) \mathbb{1}_{\{T > n^\beta\}} ) \leq \mathbb{E}(T \mathbb{1}_{\{T > n^\beta\}} ) \leq n \mathbb{P}(T > n^\beta)$$
Because of the coupling described above, we will also have that $\mathbb{P}(T > n^\beta) \leq \mathbb{P}(\tilde T > n^\beta)$ (because $n^\beta < \sqrt{n}$), where $\tilde T$ is the total progeny of a sub-critical Galton-Watson process with progeny $X-1$, where $X$ is distributed according to . By hypothesis, for some $\gamma' > 0$, $\mathbb{P}(X = k) = \mathcal{O}(e^{-\gamma' k})$. Then, $X$ will have finite exponential moment and by Theorem 2.1 in [@nakayama2004finite], so will $\tilde T$. Therefore, by Markov’s inequality, we will have that for every $\theta > 0$
$$\mathbb{P}( \tilde T > n^\beta ) \leq \frac{\mathbb{E}( \tilde T^\theta )}{n^{\beta \theta}} = \mathcal{O}(n^{\beta \theta})$$
Putting the three bounds together we obtain that (for $\theta$ large enough)
$$\mathbb{E}(N(C(1))) = \mathcal{O}(n^{2\beta-1}) + \mathcal{O}(n^{\beta-1}) +\mathcal{O}(n^{1-\beta\theta}) = \mathcal{O}(n^{2\beta-1})$$
Now, fix $0< \alpha < 1$. Using this and Markov’s inequality, we obtain that
$$\begin{aligned}
\mathbb{P}\left(N > n^{\alpha/2} \right) & \leq \frac{\mathbb{E}(N)}{n^{\alpha/2}}\\ & \leq \frac{\mathbb{E}(\sum_{u\in V} N(C(k)))}{n^{\alpha/2}} \\ & = \frac{n \mathbb{E}(N(C(1)))}{n^{\alpha/2}}\\ & = \mathcal{O}(n^{-\delta_1})\end{aligned}$$
Where $\delta_1 := \alpha/2 - 2 \beta$. By taking $\beta < \alpha / 4$ we have that $\delta_1 > 0$.
On the other hand, if we call $C_{max}$ the largest component of the graph, by Theorem 1.1 in [@janson2008largest][^7] we also have that $$\mathbb{P}(|C_{max}| > n^{\alpha/2}) = \mathcal{O}(n^{-\delta_2})$$ for some $\delta_2 > 0$. Then, taking $\delta = \min(\delta_1,\delta_2)>0$, we have that $$\mathbb{P}( B > n^\alpha ) \leq \mathbb{P}(N |C_{max}| > n^\alpha) \leq \mathbb{P}(N > n^{\alpha/2})+\mathbb{P}(|C_{max}| > n^{\alpha/2}) = \mathcal{O}(n^{-\delta})$$
This last proposition shows that the number of vertices in components that are not trees is (for every $\alpha>0$) $\mathcal{O}_{\mathbb{P}}(n^\alpha)$. Now, define $T_G$ as a graph formed by spanning trees of each of the components of $G$ (it is not important which specific ones are chosen). We call $W_{DG}(G)$ and $W_{DG}(T_G)$ the selection sequences defined by the degree-greedy algorithm run in $G$ and $T_G$ respectively. Observe that the components that are trees look exactly the same in $G$ and $T_G$.
We can then couple the realizations of the degree-greedy algorithm in $G$ and $T_G$ to make them coincide in these components in the following way:
- Call the connected components of $G$ as $C_1$, $C_2$,..., $C_l$ and the ones of $T_G$ as $C_1'$, $C_2'$,..., $C_l'$.
- Run a degree-greedy algorithm in each of the components. This generates the selection sequences $W_1$, $W_2$,..., $W_l$ for the components of $G$ and $W_1'$, $W_2'$,..., $W_l'$ for the ones of $T_G$. If for some $j \leq l$ the component $C_j$ is a tree, then $C_j = C_j'$ and the respective runs of the degree-greedy algorithm can be trivially coupled to give $W_j = W_j'$. Couple in this manner all the selection sequences of all the components that are trees.
- Now, construct $W_{DG}(G)$ inductively as follows: in each step $i \geq 1$, count the number of minimum degree vertices in each component $j \leq l$ of $G_{i-1}$ and call this number $d_j^{(i)}$. Select component $j \leq l$ with probability $d_j^{(i)}/\sum_{k=1}^m d_k^{(i)}$. Set $w_i$ (the $i$-th vertex of $W_{DG}(G)$) as the first vertex of $W_j$ not already in $\{w_1,...,w_{i-1}\}$.
- Finally, construct $W_{DG}(T_G)$ in an analogous way but using selection sequences $W_1'$, $W_2'$,..., $W_l'$.
It is straight forward that this construction gives a run of the degree-greedy algorithm in each of both graphs.
The construction was made in such a way that for every component that is a tree, the same vertices end up in $W_{DG}(G)$ as in $W_{DG}(T_G)$. This means that, at most, $|W_{DG}(G)|$ and $|W_{DG}(T_G)|$ will differ in size by $\mathcal{O}_\mathbb{P}(n^\alpha)$ (the number of vertices in components that are not trees). Besides, $T_G$ is a collection of trees and therefore, by Corollary \[coro:arbol\], the degree-greedy ran on it will define a selection sequence with the property $T_1$, and then by Lemma \[lemma:optimal\] $|W_{DG}(T_G)| = \alpha(T_G)$. But all the edges in $T_G$ are also present in $G$ and so $T_G$ will have a bigger maximum independent set than $G$[^8]. This implies that
$$|W_{DG}(T_G)| = |W_{DG}(G)| + \mathcal{O}_\mathbb{P}(n^\alpha) = \alpha(T_G) \geq \alpha(G)$$
which in term implies that (because $|W_{DG}(G)|= \sigma_{DG}(G)$)
$$|\alpha(G)-|W_{DG}(G)|| \leq \mathcal{O}_\mathbb{P}(n^\alpha)$$
proving the proposition.
We now give a proof for the case in which $G$ is a supercritical graph. Call $W_{DG}$ the selection sequence defined by the degree-greedy algorithm ran on $G$. We want to show that $|W_{DG}|= \alpha(G) + \mathcal{O}_\mathbb{P}(n^\alpha)$. W.h.p. we have that this sequence selects vertices of degree $1$ or $0$ at least until the remaining graph is subcritical. Suppose this is so, then there exists some value $k_0 \geq 1$ s.t. for every $k \geq k_0$ the remaining graph $G_k$ (see definition \[def:remaining\]) is subcritical and for every $l \leq k_0$ the vertex $W_{DG}(l)$ has degree either $1$ or $0$ in $G_l$.
The idea is to define a selection sequence $\tilde W$ that is similar to $W_{DG}$, has roughly the same size and for which $|\tilde W| \geq \alpha(G)$. Calling $T_{G_{k_0}}$ the graph formed by the spanning trees of $G_{k_0}$, we define the graph $\tilde G$ as a copy of $G$ in which the subgraph $G_{k_0}$ has been replaced by $T_{G_{k_0}}$. We can then define a selection sequence $\tilde W$ for $\tilde G$ that coincides with $W_{DG}$ until step $k_0$. For $k \geq k_0$, because $G_{k_0}$ is subcritial and the remaining graph of $\tilde G$ is a collection of spanning trees of $G_{k_0}$ ($T_{G_{k_0}}$), we can make $\tilde W$ to have the property $T_1$ and to differ in at most $\mathcal{O}_\mathbb{P}(n^\alpha)$ vertices from $W_{DG}$ in exactly the same way as in the subcritical case. Because $\tilde W$ has the property $T_1$, using Lemma \[lemma:optimal\], $|\tilde W| = \alpha(\tilde G)$. Furthermore, because all the edges present in $\tilde G$ are present in $G$ we have that $\alpha(\tilde G) \geq \alpha(G)$. Then,
$$|\tilde W| = |W_{DG}| + \mathcal{O}_\mathbb{P}(n^\alpha) = \alpha(\tilde G) \geq \alpha(G)$$
which means that
$$|\alpha(G)-\sigma_{DG}(G)| \leq \mathcal{O}_\mathbb{P}(n^\alpha)$$
as we wanted to show.
Hydrodynamic limit results {#sec:fluidres}
--------------------------
In this section we present the results we will use in the remaining of the paper to show the convergence of processes and stopping times towards deterministic limits. These results are not necessarily presented in full generality but rather in the most convenient form for the applications we intend.
Given a continuous time Markov jump process $A_t\in D[0,\infty)$ we will define its associated *Dynkin’s martingales*[^9] by
$$M_t : = A_t - A_0 - \int_0^t \delta[A_s] ds$$
where $\delta(\cdot)$ is the drift of $A_t$.
The following lemma will be our main tool to prove convergence of stochastic processes towards solutions of differential equations. It is an abstraction of the reasoning behind the proof of the limits in the main theorem of [@brightwell2016greedy].
\[lema:limitefluido\] Let $(A_t^{(n)}(1),A_t^{(n)}(2),...) \in D[0,\infty)^\mathbb{N}$ be a sequence of countable continuous time Markov jump processes where (for each $k\in\mathbb{N}$) $A_t^{(n)}(k)$ has drift $\delta_k(A_t^{(n)}(1),A_t^{(n)}(2),...)$. If (for every $k\in\mathbb{N}$):
(i) $\delta_k(A_t^{(n)}(1),A_t^{(n)}(2),...)/n = \sum_{i\geq1}^{i_k} \alpha_i A_t^{(n)}(i)/n$, where $i_k\in \mathbb{N}$
(ii) $\delta_k(A_t^{(n)}(1),A_t^{(n)}(2),...)/n$ are uniformly bounded
(iii) $A_0^{(n)}(k)/n \xrightarrow{n\rightarrow\infty} a_0(k)$ (for some constant $a_0(k)$)
(iv) the associated Dynkin’s martingales $M_t^{(n)}(k)$ have quadratic variation of order $o_{\mathbb{P}}(n^2)$
then, if the system of integral equations defined by
$$a_t(1) = a_0(1) + \int_0^t \delta_1(a_s(1),a_s(2),...) ds$$
$$a_t(2) = a_0(2) + \int_0^t \delta_2(a_s(1),a_s(2),...) ds$$
$$. \ . \ .$$
has a unique solution, the processes $A_t^{(n)}(1)$, $A_t^{(n)}(2)$,... converge in probability towards the continuous functions $a_t(1)$, $a_t(2)$ that are solution to the system.
Dividing by $n$ Dynkin’s formula we have that
$$\frac{A_t^{(n)}(1)}{n} = \frac{A_0^{(n)}(1)}{n} + \int_0^t \frac{ \delta_1^{(n)}[A_s^{(n)}(1),A_s^{(n)}(2),...]}{n}ds + \frac{M_t^{(n)}(1)}{n}$$
$$\frac{A_t^{(n)}(2)}{n} = \frac{A_0^{(n)}(2)}{n} + \int_0^t \frac{ \delta_2^{(n)}[A_s^{(n)}(1),A_s^{(n)}(2),...]}{n}ds + \frac{M_t^{(n)}(2)}{n}$$
$$. \ . \ .$$
Because the associated Dynkin’s martingales have quadratic variation of order $o_\mathbb{P}(n^2)$ by Doob’s inequality we have that, for each $k\in\mathbb{N}$, $\sup_{s\leq t}|M_t^{(n)}|/n$ converges uniformly in distribution in $C[0,\infty)$ towards 0. Since the $\delta_k^{(n)}(\cdot)$ are uniformly bounded, for every $k\in\mathbb{N}$ and $T>0$, the sequences of processes $(A_t^{(n)}(k)-M_t^{(n)})/n$ are uniformly Lipschitz and uniformly bounded in $[0,T]$. Then, by Arzela-Ascoli’s Theorem, for every $T >0$ these families of processes are tight in $C[0,T]$. This implies that for every subsequence there exists subsubsequences such that
$$\frac{A_t^{(n)}(1)-M^{(n)}_t(1)}{n} \rightarrow a_t(1)$$
$$\frac{A_t^{(n)}(2)-M^{(n)}_t(2)}{n} \rightarrow a_t(2)$$
$$. \ . \ .$$
uniformly in distribution in compact sets, for some continuous functions $a_t(1)$, $a_t(2)$, etc. Since the processes are countable, we may take a common subsubsequence[^10] where all this convergences hold. Furthermore, by Skorohod’s Representation Theorem, there exists an (abstract) probability space s.t. all these limits and the convergence (for every $k\in\mathbb{N}$) of $\sup_t |M^{(n)}_t(k)|/n$ towards $0$ hold uniformly and almost surely in compact sets. Then, for this subsubsequence
$$\frac{A_t^{(n)}(1)}{n} \xrightarrow{a.s.} a_t(1)$$
$$\frac{A_t^{(n)}(2)}{n} \xrightarrow{a.s.} a_t(2)$$
$$. \ . \ .$$
By hypothesis $\delta_k^{(n)}[A_s^{(n)}(1),A_s^{(n)}(2),...]/n = \sum_{\geq i}^{i_k} \alpha_i^{(k)} A_t^{(n)}(k)/n$, then $\lim_n \delta_k(A_t^{(n)}(1),A_t^{(n)}(2),...)/n = \delta_k(a_t(1),a_t(2),...)$. And because all the drifts are uniformly bounded, taking limit over this subsubsequence and using that $A_t^{(n)}(k)/n\rightarrow a_0(k)$ and dominated convergence yields
$$a_t(1) = a_0(1) + \int_0^t \delta_1(a_s(1),a_s(2),...) ds$$
$$a_t(2) = a_0(2) + \int_0^t \delta_2(a_s(1),a_s(2),...) ds$$
$$. \ . \ .$$
The convergence towards the solution of this system of integral equations is well defined as, by hypothesis, it has a unique solution. We now need to prove that this convergence is not only in this subsubsequence but rather in the whole original sequence. For this, note that since the limits are continuous and deterministic, this convergence is equivalent to convergence in distribution in the Skorohod topology on $D[0,\infty)$. But because every subsequence has a subsubsequence that converges to the same limit (because by hypothesis it is unique), the original sequence converges in distribution to it. Moreover, as the limit is deterministic, the convergence can equivalently be taken to be in probability.
In the following lemma, we will establish convergence criteria for the stopping times of sequences of decreasing processes that converge towards an hydrodynamic limit.
\[lema:convtiempo\] Let $(A_t^{(n)}(1),A_t^{(n)}(2),...) \in D[0,\infty)^\mathbb{N}$ be a sequence of countable continuous time Markov jump processes with $A^{(n)}_t(1)$ a decreasing process of transition matrix $Q^{(n)}_{ij}$ and define the stopping times $T^{(n)}:=\inf \{ t\geq0 : A_t^{(n)}(1) = 0 \}$ and the deterministic time $T:=\inf\{s\geq0: a_s(1)=0\}$. Under the same hypothesis of previous lemma and further assuming that:
(i) For every $t \leq T^{(n)}$ and (if $A_t^{(n)}(1) = i \geq 0$), $\sum_{j\leq i}Q^{(n)}_{ij}(A_t^{(n)}(2),...) \geq C^{(n)}n$ with $C^{(n)} \xrightarrow{n\rightarrow\infty} C > 0$
(ii) The function $a_t(1)$ is continuously differentiable with $\dot a_t(1) \leq - C'$ (for some $C' >0$ and $t\leq T$)
then, $T^{(n)} \xrightarrow{\mathbb{P}}T$.
Given $\delta > 0$, we want to show that the probability of $\{|T-T^{(n)}|\geq\delta\}$ goes to 0. For this, suppose that $T > T^{(n)}$. Now, suppose that the event $\{\sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\leq \delta C'\}$ holds. Then, $a_{T^{(n)}}(1)$ is at most $\delta/C'$. By hypothesis we have that for every $t\leq T$ the derivative of $a_t(1)$ is less than $-C'$, then $a_{T^{(n)}+\delta}(1) \leq a_{T^{(n)}}(1) - C' \delta \leq 0$. Therefore, $\{T-T^{(n)} \geq \delta\} \subseteq \{\sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)| \geq \delta C'\}$. And because $A_t^{(n)}(1)/n\xrightarrow{\mathbb{P}}a_t(1)$, the probability of this last event tends to 0.
On the other hand, now suppose that $T^{(n)} > T$. If the event $\{\sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\leq C \delta / 4\}$ holds, the value of $A_{T}^{(n)}/n$ will be at most $C \delta / 4$. Because by hypothesis $A_t^{(n)}(1)$ is decreasing and if $A_t^{(n)}(1)=i\geq0$ the process has transitions to lower states with rate at least $n C^{(n)} = n C + o(n)$. If we define $(Z_t)_{t\geq T}$ to be a pure death process with initial value $Z_T = C \delta / 4$ and death rate $n C / 2$, we can then couple (for $n$ larger than certain $n_0\geq0$) the process $A_t^{(n)}$ to be larger than $Z_t$ for $t \geq T$. Then, for $n \geq n_0$, $\{T^{(n)} - T \geq \delta\} \subseteq \{ Z_{T+\delta} \geq 0 \}$. But, defining $X\sim \mbox{Pois}(n C \delta / 2)$, the probability of this last event is equal to $\mathbb{P}(X \leq n C \delta / 4)$ and by Chebychev’s inequality we have that
$$\begin{array}{cl}
\mathbb{P}( X \leq n C \delta / 4 ) & = \mathbb{P}( nC\delta / 2 - X \geq n C \delta /4) \\
& \leq \mathbb{P}( | X - nC\delta / 2| \geq n C \delta /4) \\
& \leq \frac{n C \delta/2}{n^2 C^2 \delta^2 / 16} \\
& = \frac{8}{n C \delta}
\end{array}$$
Summarizing,
$$\begin{array}{cl}
\mathbb{P}( |T^{(n)}-T| \geq \delta ) & = \mathbb{P}( T - T^{(n)} > \delta) \\
& \ \ + \mathbb{P}( \sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\leq C \delta / 4, T^{(n)} - T > \delta ) \\
& \ \ + \mathbb{P}( \sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\geq C \delta / 4, T^{(n)} - T > \delta ) \\
& \leq \mathbb{P}(\sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\geq \delta C' ) \\
& \ \ + \mathbb{P}( \sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\leq C \delta / 4, T^{(n)} - T > \delta ) \\
& \ \ + \mathbb{P}( \sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\geq C \delta / 4 ) \\
& \leq \mathbb{P}(\sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\geq \delta C' ) + \mathbb{P}( X \leq n C \delta / 4 ) \\
& \ \ + \mathbb{P}( \sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\geq C \delta / 4 ) \xrightarrow{n\rightarrow\infty} 0
\end{array}$$
where the last term $\mathbb{P}( \sup_{t\leq T}|A_t^{(n)}(1)/n-a_t(1)|\geq C \delta / 4 )$ goes to $0$ because we are under the hypothesis of the previous lemma.
We now need to establish the convergence of the coordinates of countable Markov jump processes at specific stopping times towards corresponding values of its hydrodynamic limit. For this, we will use the following lemma.
\[coro:convcoord\] Let $(A_t^{(n)}(1),A_t^{(n)}(2),...) \in D[0,\infty)^\mathbb{N}$ be as in previous lemma. Then, for every $k\in\mathbb{N}$ we will have that $A^{(n)}_{T^{(n)}}(k)/n \xrightarrow{\mathbb{P}}a_T(k)$.
We have that
$$|A_{T^{(n)}}(k)/n-a_T(k)| \leq |A_{T^{(n)}}(k)/n-a_{T^{(n)}}|+|a_{T^{(n)}}-a_T(k)|$$
where the first term in the r.h.s. goes w.h.p. to 0 because by Lemma \[lema:limitefluido\] $A_t^{(n)}(k)/n\xrightarrow{\mathbb{P}}a_t(k)$ uniformly on compact sets (and since $T^{(n)}\xrightarrow{\mathbb{P}}T$, the $T^{(n)}$ are w.h.p. uniformly bounded); and the second one because $T^{(n)}\xrightarrow{\mathbb{P}}T$ by Lemma \[lema:convtiempo\] and $a_t(k)$ is continuous.
Proof of Theorem \[thm:oneapplication\]
---------------------------------------
For the analysis of the effect of $M_1^{(n)}(\cdot)$ we will break the degree-greedy dynamics in two. In the *first phase*, we will only connect the vertices of degree $1$ and we will accumulate the number of free half-edges stemming from the blocked vertices in a variable $B^{(n)}_t$. We will do this until we have explored $n p_1^{(n)}$ vertices of degree $1$. While in the *second phase*, we will take the final value of $B_t^{(n)}$ from phase $1$ and we will sequentially match each of this half-edges and remove the edges formed from the graph.
Because the Configuration Model is not sensitive to the order in which the matching of half-edges is done, this will not affect the final degree distribution obtained (which will be the same as the one obtained applying the map $M_1^{(n)}(\cdot)$) but will nevertheless make the analysis of the resulting limit easier. As we will see, for the proof of Theorem \[thm:oneapplication\] it will only be necessary to analyze the first of these two phases.
The proof will consist of four parts. First we define a stochastic process to describe the first phase of the dynamics and we prove that this process is Markov. Then, we establish hydrodynamic limits for the number of unpaired half-edges in the graph and the number of degree $1$ vertices. Afterward, we will prove a concentration result for the number of vertices not connected to any degree $1$ vertex after the first phase is over. Finally, these limits combined with an observation from percolation for Configuration Models, allow us to establish the criteria given by Theorem \[thm:oneapplication\].
*(i) Stochastic description of the first phase:* the stochastic process used to model the first phase of the matching of the degree 1 vertices will be similar in spirit to the one used in [@brightwell2016greedy] to study the greedy algorithm. Here, we also keep track of the number of unpaired blocked half-edges in the random variable $B_t^{(n)}$. The other variables used to describe the process will be: the number $U_t^{(n)}$ of unpaired half-edges, the number $A_t^{(n)}$ of remaining degree 1 vertices to match and (for $k \in \mathbb{N}$) the number $\mu_t^{(n)}(k)$ of unexplored degree $k$ vertices.
Then, at each time $t \geq 0$ the state of the process will be described by the infinite dimensional vector $(U_t^{(n)},A_t^{(n)},B_t^{(n)},\mu_t^{(n)}(2),\mu_t^{(n)}(3),...)$. The process in question evolves as follows: at each time $t \geq 0$ every degree 1 vertex in $A_t^{(n)}$ will have an exponential clock with rate $U_t^{(n)}/A_t^{(n)}$ (and, when $A_t^{(n)}=0$, we define the transition rate as $0$). Here we chose to fix the rates of the clocks in this way so as to simplify the differential equations of the hydrodynamic limit. When one of the clocks of some of these vertices rings, the vertex is removed from $A_t^{(n)}$ and its edge is uniformly paired to another unpaired edge. The state of vertex with the half-edge selected for the pairing is declared *blocked* and its free half-edges are added to $B_t^{(n)}$. The process will go on until there are no more degree 1 vertices to pair. The process then transitions with rate $U_t^{(n)}$. Because it is defined by well-behaved transition rates, it is direct that the resulting process is Markovian.\
\
*(ii) Hydrodynamic limit for $U_t^{(n)}$ and $A_t^{(n)}$:* we will now establish the convergence of this process towards the solutions of a set of differential equations. Because the different limits for this two coordinates of the process follow very analogous reasoning, we will present them in parallel. For this, we will first find the drift associated to each one of the coordinates of the state vector:
- With rate $U_t^{(n)}$, the clock of one of the degree 1 vertices rings, at which point its free half-edge is paired to another free half-edge. So, $U_t^{(n)}$ has a drift given by $\delta(U_t^{(n)}):= - 2 U_t^{(n)}$.
- With rate $U_t^{(n)}$, the clock of one of the degree 1 vertices rings, at which point two things can happen: with probability $\frac{A_t^{(n)}}{U_t^{(n)}}$ the degree 1 vertex is matched to another degree 1 vertex and therefore $A_t^{(n)}$ is reduced by $2$; or with probability $\frac{U_t^{(n)}-A_t^{(n)}}{U_t^{(n)}}$ is matched to a non-degree 1 vertex and therefore $A_t^{(n)}$ is reduced by $1$. So, $A_t^{(n)}$ has a drift given by $\delta'(A_t^{(n)},E_t^{(n)}):= -(A_t^{(n)} + U_t^{(n)})$.
Fix some $\delta > 0$, then the sequence of processes $\frac{U_t^{(n)}}{n}$ and $\frac{A_t^{(n)}}{n}$ are (for $n$ large enough) uniformly bounded by $\lambda + \delta$ and $p_1+ \delta$, respectively. They are then uniformly bounded.
By the (CA) that the Configuration Model meets and the definition of the process, we have that $U_0^{(n)}/n \rightarrow \lambda$ and $A_0^{(n)}/n \rightarrow p_1$.
Here we denote the Dynkin’s martingales associated to $U_t^{(n)}$ and $A_t^{(n)}$ by $M_t^{(n)}$ and $M'^{(n)}_t$, respectively. These are all martingales of locally finite variation. Therefore, their quadratic variation will be given by
$$[M^{(n)}_t]_t = \sum_{0\leq s \leq t}(\Delta M_s^{(n)})^2 = \sum_{0\leq s \leq t}(\Delta U_s^{(n)})^2 \leq \sum_{s \geq 0}(\Delta U_s^{(n)})^2 \leq 4 p_1^{(n)} n = \mathcal{O}(n)$$
$$[M'^{(n)}_t]_t = \sum_{0\leq s \leq t}(\Delta M'^{(n)}_s)^2 = \sum_{0\leq s \leq t}(\Delta A_s^{(n)})^2 \leq \sum_{s \geq 0}(\Delta A_s^{(n)})^2 \leq 2 p_1^{(n)} n = \mathcal{O}(n)$$
Both quadratic variations are then $o_\mathbb{P}(n^2)$.
Now, the corresponding system of integral equations (as presented in Lemma \[lema:limitefluido\]) is given by:
$$\label{eq:eqdifU}
u_t = \lambda - \int_0^t 2 u_s ds$$
$$\label{eq:eqdifA}
a_t = p_1 - \int_0^t \left(a_s + u_s\right) ds$$
These equations can be directly integrated (with initial conditions $(\lambda, p_1)$) to give
$$\label{eq:solU1}
u_t = \lambda e^{-2 t}$$
$$\label{eq:solA}
a_t = \lambda e^{-2 t} - (\lambda - p_1) e^{-t}$$
Where the solutions are clearly unique. Then, we are under the conditions of Lemma \[lema:limitefluido\] and we will therefore have that, uniformly in compact sets, $U_t^{(n)}/n\xrightarrow{\mathbb{P}}u_t$ and $A_t^{(n)}/n\xrightarrow{\mathbb{P}}a_t$.
Moreover, if we define the stopping times $T_1^{(n)} := \inf\{t\geq0:A_t^{(n)}=0\}$ and the deterministic time $T_1 := \inf\{t\geq0:a_t=0\}$, it can be proved that the conditions of Lemma \[lema:convtiempo\] hold. For this, first note that $A_t^{(n)}$ is in fact decreasing (it only transitions to states of lower value) and that for every $i\in\mathbb{N}$ if $A_t^{(n)} = i$ then it transitions to lower states with rate $\sum_{j\leq i} Q_{ij} = U_t^{(n)}$.
If $2 p_1 < \lambda$, then we will have that $U_t^{(n)}$ will be strictly positive for every $t\leq T_1^{(n)}$ because it is greater of equal to $\sum_{k\geq1} i p_i^{(n)} - 2 p_1^{(n)}$ which divided by $n$ converges (owing to the (CA)) to $\lambda - 2 p_1 > 0$. The conclusion still holds for any value of $p_1$ because of the Lemma \[lemma:caemitad\] proved in Appendix \[app:urn\], where it is proved that for this kind of matching process the proportion of degree $1$ vertices will w.h.p. (for any initial value) drop below $1/2$ at a time where $U_t^{(n)}$ is still a positive proportion of $n$.
Furthermore, $T_1$ can be explicitly found to be given by $T_1 = \log( 1 + p_1/\lambda)$. And since (for every $t\leq T$)
$$\dot a_t = - 2 (p_1+\lambda) e^{-2t} + \lambda e^{-t} \leq - 2 (p_1+\lambda) e^{-2T_1} + \lambda = -2 \frac{\lambda^2}{p_1 + \lambda}+\lambda,$$
where the r.h.s. is strictly negative, then we can apply Lemma \[lema:convtiempo\] to show that $T^{(n)}\xrightarrow{\mathbb{P}}T$. And by Corollary \[coro:convcoord\], we will have that
$$U_{T_1^{(n)}}/n \xrightarrow{\mathbb{P}} u_{T_1} = Q^2 \lambda$$
*(iii) Convergence of the measure of unexplored vertices:* if $\{v_j \not\leftrightarrow (1)\}$ represents the event that the vertex $v_j$ is not connected to a degree $1$ vertex, then (for $i\geq2$) the random variable $Z_i^{(n)} := \mu_{T_1^{(n)}}^{(n)}(i) = \sum_{j=1}^{n} \mathbb{1}_{\{v_j \not\leftrightarrow (1)\}} \mathbb{1}_{\{d_j=i\}}$ gives the number of degree $i$ vertices not connected to any degree $1$ vertix at the end of the first phase. In other words, $Z_i^{(n)}$ gives the number of vertices of degree $i$ that remain after the first phase of the map $M_1^{(n)}(\cdot)$. We will here prove that these variables converge in probability towards their means.
We first compute their corresponding mean values:
$$\mathbb{E}\left(Z_i^{(n)}\right) = \mathbb{E}\left( \sum_{j=1}^{n} \mathbb{1}_{\{v_j \not\leftrightarrow (1)\}} \mathbb{1}_{\{d_j=i\}} \right)$$
Which, because vertices are interchangeable, is equal to $n \mathbb{P}(v_1 \not\leftrightarrow (1),d_1=i)$. This last probability is easy to compute and gives $p_i^{(n)} \prod_{l=1}^i \left(1 - \frac{n p_1^{(n)}}{n \sum_{i\geq1}i p_i^{(n)}-(2l-1)} \right)$. Which converges as $n\rightarrow\infty$ to $(1-p_1/\lambda)^i p_i$.
We then have that
$$\label{eq:convmedia}
\mathbb{E}\left(Z_i^{(n)}\right) = n (1-p_1/\lambda)^i p_i + o(n)$$
We will now bound the variance of these variables:
$$\label{eq:varZi1}
\mbox{Var}\left( Z_i^{(n)} \right) = \mathbb{E}\left[ \left(Z_i^{(n)}\right)^2\right] - \mathbb{E}\left(Z_i^{(n)}\right)^2$$
Where the first term in will be given by
$$\mathbb{E}\left( \sum_{j\neq k}^n \mathbb{1}_{\{v_j \not\leftrightarrow (1)\}} \mathbb{1}_{\{v_k \not\leftrightarrow (1)\}} \mathbb{1}_{\{d_j=d_k=i\}} \right) + \mathbb{E}\left( Z_i^{(n)} \right)$$
Then, defining $A := \mathbb{E}\left( \sum_{j\neq k}^n \mathbb{1}_{\{v_j \not\leftrightarrow (1)\}} \mathbb{1}_{\{v_k \not\leftrightarrow (1)\}} \mathbb{1}_{\{d_j=d_k=i\}} \right)$, $B:=\mathbb{E}\left( Z_i^{(n)} \right)$ and $C:= \mathbb{E}\left(Z_i^{(n)}\right)^2$, we have that $\mbox{Var}\left( Z_i^{(n)} \right) = A + B - C$.
Now, the term $A$ can be bounded above by
$$\label{eq:cotasupvar}
n(n-1) \left[ \mathbb{P}(v_j \not\leftrightarrow v_k,v_j \not\leftrightarrow (1),v_k \not\leftrightarrow (1),d_j=d_k=i) + \mathbb{P}(v_j \leftrightarrow v_k, d_j=d_k=i) \right]$$
Where the events $\{v_j \leftrightarrow v_k\}$ and $\{v_j \not\leftrightarrow v_k\}$ represent the events where $v_j$ is connected to $v_k$ and its negation, respectively. These two probabilities can be explicitly calculated. The first one is equal to
$$\mathbb{P}(v_j \not\leftrightarrow (1),v_k \not\leftrightarrow (1)|v_j \not\leftrightarrow v_k,d_j=d_k=i)\mathbb{P}(v_j \not\leftrightarrow v_k|d_j=d_k=i)\mathbb{P}(d_j=d_k=i)$$
Furthermore, simple computations lead to
$$\begin{array}{cl}
\mathbb{P}(v_j \not\leftrightarrow (1),v_k \not\leftrightarrow (1)|v_j \not\leftrightarrow v_k,d_j=d_k=1) = \prod_{l=1}^i & \left(1 - \frac{n p_1^{(n)}}{n \sum_{i\geq1}i p_i^{(n)}-(i+2l-1)} \right) \\
& \times \left(1 - \frac{n p_1^{(n)}}{n \sum_{i\geq1}i p_i^{(n)}-(2i+2l-1)} \right),
\end{array}$$
$$\mathbb{P}(v_j \not\leftrightarrow v_k|d_j=d_k=1) = \prod_{l=1}^i \left( 1 - \frac{i}{n\sum_{j\geq1} j p_j^{(n)}-(2l-1)} \right)$$
and
$$\mathbb{P}(d_j=d_k=1) = p_i^{(n)} \frac{n p_i^{(n)}-1}{n}.$$
On the other hand, the second probability gives
$$\mathbb{P}(v_j \leftrightarrow v_k, d_j=d_k=i) = \left[ 1 - \prod_{l=1}^i \left( 1 - \frac{i}{n\sum_{j\geq1} j p_j^{(n)}-(2l-1)} \right) \right] p_i^{(n)} \frac{n p_i^{(n)}-1}{n}$$
Putting all this together shows that $A \leq n(n-1) \left[ (1-p_1/\lambda)^{2i}p_i^2 + o(1) \right]$. Finally, by , $B = n (1-p_1/\lambda)^i p_i + o(n)$ and $C = n^2 (1-p_1/\lambda)^{2i} p_i^2 +o(n^2)$. Which proves that the variance of $Z_i^{(n)}$ is $o(n^2)$.
By Chebychev’s inequality this implies that
$$\mathbb{P}\left(|Z_i^{(n)}-\mathbb{E}(Z_i^{(n)})| > \epsilon n\right) \leq \frac{\mbox{Var}\left(Z_i^{(n)}\right)}{\epsilon^2 n^2} \rightarrow 0$$
Which in turn means that for any initial asymptotic degree distribution that is bounded, all the $Z_i^{(n)}/n$ will converge jointly in probability to $(1-p_1/\lambda)^i p_i$. But because $\left(Z_i^{(n)}\right)_{n\geq0}$ is bounded by $(p_i^{(n)})_{n\geq0}$ and this last sequence is eventually uniformly summable, then
$$\left(Z_2^{(n)}/n,...,Z_i^{(n)}/n,...\right) \xrightarrow{\mathbb{P}} \left( Q^2 p_2,...,Q^ip_i,...\right)$$
Finally, because $\sum_{k\geq1} k^2 p_k^{(n)}$ converges in probability towards $\sum_{k\geq1} k^2 p_k$, then these sums are uniformly summable. And because, for every $k\geq2$, we have that $\mu_{T^{(n)}}^{(n)}(k)/n \leq p_k^{(n)}$, the sums $\sum_{k\geq2}k^2 \mu^{(n)}_{T^{(n)}}/n$ will also be uniformly summable. This implies that
$$\sum_{k\geq2} k^2 p_k^{(n)}\mu^{(n)}_{T_1^{(n)}}/n\xrightarrow{\mathbb{P}} \sum_{k\geq2} k^2 \mu_{T_1}(k)$$
We then recover the (CA) for the degree distribution of the remaining graph after the first phase. This means that, if we regard the unexplored blocked half-edges as degree $1$ vertices, the graph obtained when the first phase is finished can be treated as a Configuration Model[^11] with limiting degree distribution $\tilde p_1 = b_{T_1}/Z = (u_{T_1}-\sum_{i\geq2}\mu_{T_1}(i))/Z$, and (for $k\geq2$) $\tilde p_k = \mu_{T_1}(k)/Z$ (where $Z$ is just a normalization constant).\
\
*(iv) Criteria for subcriticality:* here we will establish under which circumstances the graph obtained after applying the map $M_1^{(n)}(\cdot)$ is w.h.p. subcritical. Note that, to do so, one in principle has to analyze what happens to the degree distribution during the second phase of the dynamics and then determine if the distribution obtained is subcritical or not. But the second phase of the dynamics is equivalent to matching $B_{T_1^{(n)}}^{(n)}$ degree $1$ vertices and then removing these vertices and the edges so formed. In the same way as was discussed in [@janson2009percolation], matching degree $1$ vertices and then removing them and their edges from the graph does not modify the criticality of a Configuration Model graph with finite second moment[^12]. We can then establish the subcriticality of the graph after applying $M_1^{(n)}(\cdot)$ just by computing the criticality parameter $\tilde \nu$ of a graph of limiting degree distribution $(\tilde p_k)_{k\geq1}$.
By explicitly computing the criticality parameter we obtain that
$$\tilde \nu = \frac{\sum_{i\geq2}i (i-1) \tilde p_i}{\sum_{i\geq1}i \tilde p_i} = \frac{1}{\lambda} \sum_{i\geq2} i(i-1) Q^{i-2} p_i = G_D''(Q)/\lambda$$
By Theorem 2.3 [@janson2009new], the obtained graph will be subcritical when this parameter is strictly less than $1$. The conclusion then follows by Proposition \[prop:optimalidadCM\][^13].
Proof of Theorem \[thm:furtherapp\]
-----------------------------------
For the convergence of the number of vertices still in the graph after phase 2, note that the number of half-edges matched during phase 1 (by the proof of Theorem \[thm:oneapplication\]) will be $(1-Q^2) \lambda n + o_\mathbb{P}(n)$. But during phase 1, for each half-edge matched, a vertex is removed from the graph. Then, at the end of phase 1 there will be $[1-(1-Q^2)\lambda]n + o_\mathbb{P}(n)$ vertices left. Also, the $n p_0+o_\mathbb{P}(n)$ vertices of degree 0 can be thought to be removed at the beginning of phase 1. But during phase 2, no vertex is removed from the graph, and therefore the conclusion.
The structure of the remaining of the proof is similar to that of Theorem \[thm:oneapplication\]. We will first give the description of the stochastic process associated to the *second phase* of the dynamics. After this, we use the results in \[sec:fluidres\] to establish hydrodynamic limits for this process. Finally, we use this limits to prove the statement of the theorem.\
\
*(i) Stochastic description of the second phase:* this phase of the dynamics consists of sequentially matching the half-edges of blocked vertices and removing the edges so formed. This is done until no more free blocked half-edges remain. In this phase, the states of vertices are not changed, only their degrees; and so, no vertex is added to the independent set.
Initially, we have that graph has $B_{T_1^{(n)}}^{(n)}$ free blocked half-edges and (for $k\geq2$) $\mu^{(n)}_{T_1^{(n)}}(k)$ unexplored vertices of degree $k$. Because of the results of step (iii) of the proof of Theorem \[thm:oneapplication\] we have that $B_{T_1^{(n)}}^{(n)}/n\xrightarrow{\mathbb{P}}Q^2 \lambda -\sum_{i\geq2}iQ^ip_i$, (for every $k\geq2$) $\mu_{T_1^{(n)}}^{(n)}(k)/n\xrightarrow{\mathbb{P}} Q^k p_k$ and $\mu_{T_1^{(n)}}^{(n)}(0)=\mu_{T_1^{(n)}}^{(n)}(1)=0$.
Then, at each time $t \geq 0$ the state of the process will be described by the infinite dimensional vector $(U_t^{(n)},B_t^{(n)},\mu_t^{(n)}(0),\mu_t^{(n)}(1),...)$. The process in question will evolve as follows: at each time $t \geq 0$ every free blocked half-edge will have an exponential clock with rate $U_t^{(n)}/B_t^{(n)}$ (and, when $B_t^{(n)}=0$, we define the transition rate as $0$). When one of the clocks of some of this half-edges rings, it is uniformly matched to another free half-edge and the edge formed is removed from the graph. The process will go on until there are no more free blocked half-edges to pair. Because it is defined by well-behaved transition rates, it is direct that the resulting process is Markovian.\
\
*(ii) Hydrodynamic limit of the second phase:* here we will establish the convergence of the process asociated to the second phase of the dynamics towards the solutions of a set of differential equations. As before, we will first find the drifts associated to each one of the coordinates of the state vector:
- With rate $U_t^{(n)}$, the clock of one of the free blocked half-edges rings, at which point it is paired to another free half-edge. So, $U_t^{(n)}$ has a drift given by $\delta(U_t^{(n)}):= - 2 U_t^{(n)}$.
- With rate $U_t^{(n)}$, the clock of one of the free bolcked half-edges rings, at which point two things can happen: with probability $\frac{B_t^{(n)}}{U_t^{(n)}}$ the half-edge is matched to another free blocked half-edge and therefore $B_t^{(n)}$ is reduced by $2$; or with probability $\frac{U_t^{(n)}-B_t^{(n)}}{U_t^{(n)}}$ is matched to a half-edge belonging to an unexplored vertex and $B_t^{(n)}$ is only reduced by $1$. So, $B_t^{(n)}$ has a drift given by $\delta'(U_t^{(n)},B_t^{(n)}):= -( B_t^{(n)} + U_t^{(n)} )$.
- Finally, with rate $U_t^{(n)}$, the clock of one of the free blocked half-edges rings and if matched to an unexplored half-edge, an unexplored vertex is selected according to the size-biased distribution and has one of its half-edges removed. This means that (for each $k\geq0$) with probability $\frac{k \mu_t^{(n)}(k)}{U_t^{(n)}}$ a degree $k$ vertex is selected to be matched and therefore $\mu_t^{(n)}(k)$ is reduced by $1$ and $\mu_t^{(n)}(k-1)$ is increased by $1$. So, $\mu_t^{(n)}(k)$ has a drift given by $\delta_k(\mu_t^{(n)}(k)) := -k \mu_t^{(n)}(k) + (k+1) \mu_t^{(n)}(k+1)$.
Fix some $\delta > 0$, then the sequence of processes $\frac{U_t^{(n)}}{n}$, $\frac{B_t^{(n)}}{n}$ and (for every $k\geq 0$) $\frac{\mu_t^{(n)}(k)}{n}$, are (for $n$ large enough) uniformly bounded by $\lambda + \delta$. They are then uniformly bounded. This, in turn, means that all the drifts associated to the coordinates of $(U_t^{(n)},B_t^{(n)},\mu_t^{(n)}(0),\mu_t^{(n)}(1),...)$ are uniformly bounded.
By the results shown in the proof of Theorem \[thm:oneapplication\], $U_0^{(n)}/n \rightarrow \lambda Q^2$, $B_0^{(n)} \rightarrow \lambda Q^2 - \sum_{i\geq2} i Q^i p_i$ and (for every $k\geq0$) $\mu_0^{(n)}(k)/n \rightarrow Q^k p_k \mathbb{1}_{k\geq2}$.
Here we will denote the Dynkin’s martingales associated to $U_t^{(n)}$, $B_t^{(n)}$ and (for each $k\geq0$) $\mu_t^{(n)}(k)$ by $M_t^{(n)}$, $M'^{(n)}_t$ and $N_t^{(n)}(k)$, respectively. These are all martingales of locally finite variation. Therefore, their quadratic variation will be given by
$$[M^{(n)}_t]_t = \sum_{0\leq s \leq t}(\Delta M_s^{(n)})^2 = \sum_{0\leq s \leq t}(\Delta U_s^{(n)})^2 \leq \sum_{s \geq 0}(\Delta U_s^{(n)})^2 \leq 4 B_0^{(n)} n = \mathcal{O}(n)$$
$$\label{eq:quadvarB2}
[M'^{(n)}_t]_t = \sum_{0\leq s \leq t}(\Delta M'^{(n)}_s)^2 = \sum_{0\leq s \leq t}(\Delta B_s^{(n)})^2 \leq \sum_{s \geq 0}(\Delta B_s^{(n)})^2 \leq 2 B_0^{(n)} n = \mathcal{O}(n)$$
$$[N^{(n)}_t(k)]_t = \sum_{0\leq s \leq t}(\Delta N^{(n)}_s(k))^2 = \sum_{0\leq s \leq t}(\Delta \mu_s^{(n)}(k))^2 \leq \sum_{s \geq 0}(\Delta \mu_s^{(n)}(k))^2 \leq Q^k p_k^{(n)} n = \mathcal{O}(n)$$
Now, the corresponding system of integral equations (as presented in Lemma \[lema:limitefluido\]) will be given by:
$$\label{eq:eqdifU2}
u_t = \lambda - \int_0^t 2 u_s ds$$
$$\label{eq:eqdifB2}
b_t = b_0 - \int_0^t (b_s + u_s) ds$$
$$\label{eq:eqdifMu2}
\mu_t(k) = Q^k p_k \mathbb{1}_{\{k\geq2\}} + \int_0^t (k+1) \mu_s(k+1) - k \mu_s(k) ds$$
The first two equations can be directly integrated to give
$$\label{eq:solU2}
u_t = \lambda Q^2 e^{-2 t}$$
$$\label{eq:solB}
b_t = Q^2 \lambda e^{-2t} - e^{-t} \sum_{i\geq2} i Q^i p_i$$
While for equations with $k\geq1$ ($\mu_t(0)$ can then be obtained noting that $\sum_{j=0}^\infty \mu_t(j)$ is a constant equal to $\tilde n /n$), they can be seen to have normal modes given by
$$\eta_k(i) = (-1)^{k-i} \binom{k}{i} \mathbb{1}_{\{i\leq k\}}$$
where $i\geq1$ and with associated eigenvalues $\omega_k = -k$. The uniqueness of the solutions can be proved by decoupling the system by writing it in the base of the normal modes. This results in a countable number of independent equations with Lipschitz derivatives and the uniqueness then follows by standard ODE theory.
We are thus under the conditions of Lemma \[lema:limitefluido\] and we will therefore have that, uniformly in compact sets, $U_t^{(n)}/n\xrightarrow{\mathbb{P}}u_t$, $B_t^{(n)}/n\xrightarrow{\mathbb{P}}b_t$ and (for every $k\geq0$) $\mu_t^{(n)}(k)/n\xrightarrow{\mathbb{P}}\mu_t$.
Moreover, if we define the stopping times $T_2^{(n)} := \inf\{t\geq0: B_t^{(n)}=0\}$ and the deterministic time $T_2 := \inf\{t\geq0:b_t=0\}$, it can be proved that the conditions of Lemma \[lema:convtiempo\] hold. For this, first note that $B_t^{(n)}$ is in fact decreasing (it only transitions to states of lower value) and that for every $i\in\mathbb{N}$ if $B_t^{(n)} = i$ then the transition rate to lower states is given by
$$\sum_{j\leq i} Q_{ij} = E_t^{(n)} + B_t^{(n)} = U_t^{(n)}$$
Because for every $t \leq T_2^{(n)}$ we have that $U_t^{(n)} \geq U_0^{(n)} - 2 B_0^{(n)}$, if the initial proportion of blocked vertices if smaller than $1/2$ then $U_t^{(n)}$ will be uniformly lower bounded by a positive number. But because of Lemma \[lemma:caemitad\] of Appendix \[app:urn\], the proportion of blocked vertices will drop w.h.p. (for any initial value) below $1/2$ at a time where $U_t^{(n)}$ is still a positive proportion of $n$. And so, the strict positivity of $U_t^{(n)}/n$ will still be true.
Furthermore, if we define $\tilde \lambda := \sum_{i\geq2} i Q^i p_i$, $T_2$ can be explicitly found to be given by $\log( Q^2 \lambda / \tilde \lambda) = - \log \tilde Q$. Since (for every $t\leq T_2$)
$$\dot b_t = -2 Q^2 \lambda e^{-2t} + e^{-t} \tilde \lambda \leq -2 Q^2 \lambda e^{-T_2} + \tilde \lambda = \tilde \lambda \left[ 1 - \frac{2 \tilde \lambda}{Q^2 \lambda} \right]$$
where the r.h.s. is strictly negative, then we can apply Lemma \[lema:convtiempo\] to show that $T_2^{(n)}\xrightarrow{\mathbb{P}}T_2$. And by Corollary \[coro:convcoord\], we will have that
$$U_{T_2^{(n)}}/n \xrightarrow{\mathbb{P}} u_{T_2}$$
$$B_{T_2^{(n)}}/n \xrightarrow{\mathbb{P}} b_{T_2}$$
$$\mu_{T_2^{(n)}}(0)/n \xrightarrow{\mathbb{P}} \mu_{T_2}(0)$$
$$\mu_{T_2^{(n)}}(1)/n \xrightarrow{\mathbb{P}} \mu_{T_2}(1)$$
$$. \ . \ .$$
Finally, because $\sum_{k\geq1} k^2 p_k^{(n)}$ converges in probability towards $\sum_{k\geq1} k^2 p_k$, then these sums are uniformly summable. And because by Lemma \[lemma:colaexp\], we have that $\mu_{T_2^{(n)}}^{(n)}(k)/n$ is $\mathcal{O}_\mathbb{P}(e^{-\gamma k})$ (for some $\gamma > 0$), the sums $\sum_{k\geq2}k^2 \mu^{(n)}_{T_2^{(n)}}/n$ will be w.h.p. eventually uniformly summable. This implies that
$$\sum_{k\geq1} k^2 p_k^{(n)}\mu^{(n)}_{T_2^{(n)}}/n\xrightarrow{\mathbb{P}} \sum_{k\geq2} k^2 \mu_{T_2}(k)$$
We then recover the (CA) for the degree distribution of the remaining graph after the second phase. This means that the graph obtained when the second phase is finished can be treated as a Configuration Model with limiting degree distribution (for $k\geq0$) $\tilde p_k = \mu_{T_2}(k)/Z$, where $Z$ is just a normalization constant.
Finally, we show that the condition of exponentially thin tails in Proposition \[prop:optimalidadCM\] is not really restrictive, as every initial distribution results in a distribution with this property after one application of the map $M^{(n)}_1(\cdot)$.
\[lemma:colaexp\] Under the same hypothesis of Theorem \[thm:oneapplication\] and further assuming that $p_1 > 0$, then $M_1(p_i)(k)$ is $\mathcal{O}(e^{-\gamma k})$ with $\gamma := - log( Q ) > 0$.
Recall that $M_1(p_i)(k)$ is given by the $k$-th coordinate of the solution of the system at time $T_2$. It can then be seen to be smaller or equal to the $k$-th coordinate of the solution of the modified system:
$$\begin{cases}
\tilde \mu_t(i) = Q^i p_i + \int_0^t (i+1) \tilde \mu_s(i+1) - i \tilde \mu_s(i) ds, & \mbox{if } i > k \\
\tilde \mu_t(i) = Q^i p_i + \int_0^t (i+1) \tilde \mu_s(i+1) ds, & \mbox{if } i = k \\
\tilde \mu_t(i) = Q^i p_i, & \mbox{if } 1 \leq i < k \\
\end{cases}$$
at time $T_2$.
Furthermore, the $k$-th coordinate of this system can be easily shown to converge monotonically $\tilde \mu_t(k) \nearrow \sum_{i=k}^\infty \tilde \mu_0(i)$ as $t \rightarrow \infty$. Then,
$$\mu_{T_2}(k) \leq \tilde \mu_{T_2}(k) \leq \sum_{i=k}^\infty \tilde \mu_0(i) = \sum_{i=k}^\infty \mu_0(i) \leq C \sum_{i=k}^\infty Q^k = \mathcal{O}(Q^k)$$
where $C>0$ is some constant.
Possible extensions
===================
In this work we showed that, for a random graph with given degrees, if the degree-greedy algorithm selects only degree 1 or 0 vertices until the remaining graph is subcritical, then the independent set obtained by it is of the same size as a maximum one up to an error term smaller than any positive power of the graph size. We then characterized for which asymptotic degree distributions this happens and gave a way of computing their independence ratio.
It is still an open issue to show if the independent set found is always maximum a.a.s. as in the Erdös-Renyi case; and if not, under which conditions it is.
In Section \[sec:exandapp\] we explained how, changing higher degree vertices by degree 1 vertices, upper bounds can be obtained for the independence number of general graphs. It would be possible, in principle, to obtain tighter bounds by finding an optimal way of dominating the studied graphs by a graph in which the degree-greedy algorithm is quasi-optimal.
Furthermore, Lemma \[lemma:colaexp\] seems to suggest that the pairing of degree 1 vertices quickly generates an exponential tail in the resulting degree distribution. We then conjecture that the condition of finite second moment in Theorems \[thm:oneapplication\] and \[thm:furtherapp\] could in fact be avoided, extending the result to heavy-tailed distributions.
Finally, the Glauber dynamics’ invariant measure is known (under certain limits) to concentrate around maximum independent sets. Nevertheless, when characterized in this limit, the mixing times are exponential in the graph size. The results of this work might help in showing that the mixing time could be reduced by starting the dynamics from an independent set found by a degree-greedy algorithm.
Appendix: Pairing urn model {#app:urn}
===========================
Suppose we have the following urn problem, which we will refer to as the *Pairing urn model*. Initially we have an urn with $k$ red balls and $n-k$ white balls. At each step of the process, a red ball is removed and after that a second ball is chosen uniformly from the urn and is also removed. The process is continued until there are no more red balls left. What we will prove here is that, if $k(n)= X_0 n + o_{\mathbb{P}}(n)$ (for some $0<X_0<1$), then the proportion of red balls drops w.h.p. below $1/2$ in a time where there are still a positive proportion of the balls still in the urn. We will denote by $(R_i)_{i\in\mathbb{N}}$ the process that gives at each step $i\geq1$ the number of red balls removed so far. We will also define the stopping time $T : = \inf \{ i \geq0 : (k-R_i) / (n- 2i) < 1/2 \}$ when the proportion of red vertices drops below $1/2$.
\[lemma:caemitad\] Let $0 < X_0 < 1$ be the asymptotic initial proportion of red balls in a pairing urn process of urn size $n\in\mathbb{N}$, then w.h.p. $T < n/2$.
We will compare $(R_i)_{i\in\mathbb{N}}$ to a second process $(\tilde R_i(l))_{i\in\mathbb{N}}$, where $l(n)$ is some function of $n$ to be fixed later. At each step $i\geq1$, $\tilde R_i(l)$ will give the total number of red balls removed so far from an urn without replacement with initially $k-l$ red and $n-k$ white balls. We will denote by $X_i$ and $\tilde X_i(l)$ the corresponding proportion of red balls in the urns for both processes.
At each time $j\geq1$ the probability of drawing a red ball for the pairing urn is given by a Bernoulli r.v. of parameter $X_j=(k-R_j)/(n-2j)$ and the corresponding probability for the urn without replacement will also be a Bernoulli r.v. of parameter $\tilde X_j(l)=(k-l-\tilde R_j(l))/(n-l-j)$. We can then couple both selection probabilities by the usual coupling for two Bernoulli variables. Defining $T_l := \inf\{ i\geq1 : X_i \leq \tilde X_i(l) \}$, we will then have that, for every $i \leq T_l$, $R_i - i \geq \tilde R_i(l)$. Now, suppose that $T_l > l$, then at step $l$ we will have that $R_l - l \geq \tilde R_l(l)$ which implies that the proportion of vertices obeys
$$X_l = \frac{k - R_l}{n-2l} \leq \frac{k - l - \tilde R_l(l)}{n - 2l} = \tilde X_l(l)$$
Which contradicts the hypothesis that $T_l > l$. We will then have that $T_l \leq l$ a.s. At each step $i\geq1$, the corresponding value of $\tilde X_i(l)$ will be distributed according to
$$\mathbb{P}(\tilde R_i(l) = n-l-m) = \frac{\binom{k-i}{(k-i)-m}\binom{n-k}{m-(k-2i)}}{\binom{n-i}{i}}$$
By [@hoeffding1963probability] we will have that for $i \leq l$, the probability that the proportion of vertices deviating $\delta > 0$ from its mean (which is its initial value $\tilde X_0(l) = (k-l)/(n-l)$) will be upper bounded by
$$\label{eq:ldurna}
\mathbb{P}( \tilde X_i(l) - \tilde X_0(l) > \delta ) \leq \left[ \left( \frac{\tilde X_0(l)}{\tilde X_0(l)+q(n,\delta)} \right)^{\tilde X_0(l)+q(n,\delta)} \left( \frac{1-\tilde X_0(l)}{1- \tilde X_0(l)-q(n,\delta)} \right)^{1-\tilde X_0(l)-q(n,\delta)} \right]^n$$
Where $q(n,\delta) := (1 - 2l/n) \delta$. In particular, if asymptotically $2 l(n) < n$, we will then have that
$$\mathbb{P}\left( \sup_{i\leq l} \tilde X_i - \tilde X_0 \geq \delta \right) \xrightarrow{n\rightarrow\infty} 0$$
And because $T_l \leq l$, fixing $l = 2 k + \delta' - n$ (where $\delta'>0$), gives $\tilde X_0(l) < 1/2$; which implies that the proportion of red balls for the first process will drop in a finite time below $1/2$. Further more, if there exists a $\delta'$ s.t. $2 l < n$, the number of remaining balls in the urn at time $l$ will be a positive proportion of $n$ (i.e., $T < n/2$) as in each step exactly two balls are removed. It is easy to check that there will exist such $\delta'$ whenever $X_0 < 3/4$.
Lets see that when the initial proportion of red vertices is higher than $3/4$ our claim is still true. To prove this, we will work inductively. Call $a_1 := 1/2$ and (for every $i \geq 2$) $a_i := 1 - (2/3)^{i}$. We will suppose that initially $X_0 \in [a_j,a_{j+1})$. The process then arrives at the interval $[a_{j-1},a_j)$ in a finite time and with a positive proportion of $n$ of balls left in the urn. This will follow from the same argument as above by fixing $l(n) = ((X_0 - a_j) /(1- a_j) + \delta'') n$, for some $\delta''> 0$ small enough. This value of $l(n)$ will then give that $\tilde X_0(l) < a_j$ and $l(n) < (1/3+ \delta'') n$, assuring that the same reasoning as before can be used. Because these intervals are a partition of $(1/2,1)$, if the pairing urn starts with any initial proportion of red balls higher than $1/2$, it will eventually (after going through a finite number of intervals) drop below $1/2$ at a time where there are still a positive proportion of balls in the urn. Which is equivalent to say that $T < n/2$.
[^1]: Length of the shortest cycle.
[^2]: That is, a graph where there are posibly edges between a vertex and itself (*selfedges*) and multiple edges between a pair of vertices (*multiedges*).
[^3]: One with no self nor multi edges.
[^4]: For simplicity, vertices of degree 0 will be omitted from the analysis because, when selected, they block no vertices and then do not modify the number of unexplored vertices of other degrees. We can think that the algorithm selects them immediately after they are produced. If, on the other hand, one is interested in the size of the independent set produced, one would have to keep track of their number. In any case, there is no difficulty in doing so.
[^5]: Here we will assume that $d_{w_1}=1$. The proof is very similar in the case where $w_1$ is an isolated vertex.
[^6]: The 2-core of a graph is the maximum subgraph with minimum degree 2.
[^7]: Here we use that the tails of the degree distribution are exponentially thin and therefore $\mathcal{O}(k^{-\gamma''})$ for every $\gamma'' > 0$.
[^8]: This is because the independence number is monotonically decreasing when adding edges to a graph.
[^9]: Which are martingales because of Dynkin’s formula.
[^10]: Which can be constructed by a diagonal argument.
[^11]: For this we also need the number of remaining vertices to tend to infinity when $n\rightarrow\infty$, but this can be easily checked to be the case.
[^12]: This is so because, for each degree $1$ vertex removed, the largest connected component’s size is reduced at most by $1$. This is not true for larger degree vertices.
[^13]: In principle, one could only apply this proposition for distributions with asymptotically exponentially thin tails. But, as shown in Lemma \[lemma:colaexp\], this will be true for every degree distribution after applying the map $M^{(n)}_1(\cdot)$ one time.
|
---
abstract: 'We prove that the metric completion of a canonical Ricci-flat Kähler metric on the nonsingular part of a projective Calabi-Yau variety $X$ with ordinary double point singularities, is a compact metric length space homeomorphic to the projective variety $X$ itself. As an application, we prove a conjecture of Candelas and de la Ossa for conifold flops and transitions.'
address: 'Department of Mathematics, Rutgers University, Piscataway, NJ 08854'
author:
- Jian Song
title: On a conjecture of Candelas and de la Ossa
---
[^1]
**Introduction** {#section1}
================
Yau’s solution to the Calabi conjecture in [@Ya1] gives the existence of a unique Ricci-flat Kähler metric in any given Kähler class. Calabi-Yau manifolds and Ricci-flat Kähler metrics play a central role in the study of string theory. A natural problem in both mathematics and physics is to understand how Calabi-Yau manifolds of distinct topological types can be connected via algebraic, analytic and geometric processes. In the algebraic aspect, this is exactly the well-known Reid’s fantasy [@Re] built on deep works of Clemens [@Cl], Friedman [@Fr], Hirzebruch [@Hi] and many others (cf. [@T1; @CaO; @CGH; @T2; @Gro1; @Gro2; @GW2; @Ro]). A geometric transition is an algebraic notion of connectedness for the moduli space of Calabi-Yau threefolds, which involves with a birational contraction and a complex smooth deformation. It can be considered as the three dimensional analogue of analytic deformations among $K3$ surfaces. A conifold transition is a special geoemtric transition, where the contracted variety has only ordinary double points as singularity. The first physical interpretation of a conifold transition is given by Strominger [@Str]. In geometric and analytic aspect, the geometric transition should be considered for Calabi-Yau varieties coupled with canonical metrics such as Ricci-flat Kähler metrics. In [@CaO], Candelas and de la Ossa conjecture that an algebraic conifold transition should be also analytic and geometric, i.e., the transition should be continuous in suitable geometric sense (Conjecture \[conj\]). The recent work of Rong and Zhang [@RZ] proves a version of their conjecture by showing that an algebraic geometric transition is indeed continuous in Gromov-Hausdorff topology. The goal of the paper is to establish a strong version of Canelas and de la Ossa’s conjecture for conifold transitions. In general, a geometric transition is not necessarily projective or even Kähler [@Fr; @T2]. The balanced metrics on non-Kähler Calabi-Yau threefolds are proposed in the direction to study non-Kähler geometric transitions [@FLY].
We now state the main results of the paper. Let $f: X\rightarrow Y$ be a small contraction morphism of a smooth projective Calabi-Yau threefold $X$ such that $Y$ is a normal Calabi-Yau variety with only ordinary double points as singularities. Let $\mathcal{L}_0$ be an ample line bundle over $Y$ and $\alpha$ be a Kähler class on $X$. Then there also exists a unique Ricci-flat Kähler metric $g(t)\in c_1(\alpha+ t[\pi^*\mathcal{L}_0])$ for $t\in (0, 1]$ by Yau’s theorem. There exists a unique singular Ricci-flat Kähler metric $g_Y$ associated to its Kähler current $\omega_Y\in c_1(\mathcal{L}_0)$ obtained in [@EGZ]. In particular, $\omega_Y$ has bounded local potentials on $Y$ and $g_Y$ is a smooth Kähler metric on $Y_{reg}$, the nonsingular part of $Y$ [@EGZ].
\[main1\] The metric completion of $(Y_{reg}, g_Y)$ is a compact length space homeomorphic to the projective variety $Y$ itself, denoted by $(Y, d_Y)$. Furthermore, $(X, g(t))$ converges to $( Y, d_Y)$ in Gromov-Hausdorff topology, as $t\rightarrow 0$.
Theorem \[main1\] shows that the algebraic small contraction can be realized by a continuous deformation of smooth Calabi-Yau Kähler metrics in Gromov-Hausdorff topology. In fact, much stronger estimates are obtained in section 4 for degeneration of the Calabi-Yau metrics near the exceptional rational curves of $f$. Theorem \[main1\] can be also applied to conifold flops and transitions as stated in the following corollaries. We also remark that the convergence is in fact smooth outside the exceptional rational curves as shown in [@To1].
\[main2\] Let
$$\begin{diagram}\label{conflop}
\node{X} \arrow{se,b,}{f} \arrow[2]{e,t,..}{ } \node[2]{X'} \arrow{sw,r}{f'} \\
\node[2]{Y}
\end{diagram}$$
be a conifold flop between two smooth projective Calabi-Yau threefolds $X$ and $X'$. Let $(Y, d_Y)$ be the compact metric length space induced by the singular Ricci-flat Kähler metric $g_Y$ as in Theorem \[main1\]. Then there exist a smooth family of smooth Ricci-flat Kähler metrics $g(t)$ of $X$ and a smooth family of smooth Ricci-flat Kähler metrics $g'(s)$ of $X'$ for $t, s \in (0, 1]$, such that $(X, g(t))$ and $(X', g'(s))$ converge to $(Y, d_Y)$ in Gromov-Hausdorff topology as $t, s\rightarrow 0$.
It shows that any discrete conifold flop between two Calabi-Yau threefolds can be connected by a continuous path of Calabi-Yau metrics in Gromov-Hausdorff topology.
Theorem \[main1\] can also be applied to conifold transitions of Calabi-Yau threefolds. An algebraic geometric transition (cf. [@Ro; @RZ]) is a triple $T(X, Y, Y_s)$ connecting Calabi-Yau threefolds of different topological types, where $Y$ is a singular Calabi-Yau variety obtained from $X$ by a birational contraction morphism and $Y_s$ is a smooth complex deformation of $Y$. A conifold transition is a geometric transition such that the contracted singular Calabi-Yau variety $Y$ has only ordinary double points as singularities. The precise definitions are given in Section 2. The following corollary shows that an algebraic conifold transition is also a diffeo-geometric transition via continuous families of Ricci-flat Kähler metrics.
\[main3\] Let $T(X, Y, Y_s)$ be a conifold transition of projective Calabi-Yau threefolds $$\begin{diagram}\label{geotran}
\node{X} \arrow[2]{e,t}{f } \node[2]{Y } \node[2]{Y_s} \arrow[2]{w}\\
\end{diagram}$$ for $s\in \Delta$, where $\Delta$ is the unit disc in ${\mathbb{C}}$. Then there exist a smooth family of smooth Ricci-flat Kähler metrics $g(t)$ of $X$ for $t \in (0, 1]$ and a smooth family of smooth Ricci-flat Kähler metrics $g_{Y_s}$ of $Y_s$ for $s\in \Delta^*$, such that $(X, g(t))$ and $(Y_s, g_{Y_s})$ converge to $(Y, d_Y)$ in Gromov-Hausdorff topology as $t, s\rightarrow 0$. Here $(Y, d_Y)$ is the compact length metric space given in Theorem \[main1\].
Corollary \[main3\] proves a conjecture of Candelas and de la Ossa (Conjecture \[conj\]) for conifold transitions by combining Theorem \[main1\] and the results of Rong and Zhang [@RZ] (Theorem \[RZ\]). If there exists an algebraic conifold transition between Calabi-Yau threefolds of distinct topology, it can be also constructed as a continuous transition for algebraic Calabi-Yau varieties coupled with canonical Ricci-flat Kähler metrics in the Gromov-Hausdorff “moduli space”. The convergence in Corollary \[main2\] and Corollary \[main3\] is stronger than Gromov-Hausdorff convergence and in fact, it is in local $C^\infty$-topology outside the exceptional locus as shown in [@To1].
The organization of this article is as follows: In section 2, we give the background of conifold transitions, singular Ricci-flat Kähler metrics and complex Monge-Ampère equations. In section 3, we review the Calabi symmetry and construct various local ansatz near an exceptional rational curve. In section 4, we obtain various uniform estimates for a degenerate family of Ricci-flat Kähler metrics. Finally, we prove the main results in section 5 with some discussion on generalizations to higher dimensional conifold transitions and to canonical surgery by the Kähler-Ricci flow.
**Conifold transitions and complex Monge-Ampère equations** {#section2}
===========================================================
In this section, we give a brief introduction on conifold transitions and canonical Ricci-flat Kähler metrics on singular Calabi-Yau varieties..
Let $X$ and $X'$ be two smooth Calabi-Yau threefolds with $f: X \rightarrow Y$ and $f' : X' \rightarrow Y$ being small contraction morphisms. Then the following diagram is called a flop between $X$ and $X'$
$$\begin{diagram}\label{flop}
\node{X} \arrow{se,b,}{f} \arrow[2]{e,t,..}{ } \node[2]{X'} \arrow{sw,r}{f'} \\
\node[2]{Y}
\end{diagram}$$
$Y$ in (\[flop\]) is a normal variety with canonical singularities and trivial canonical divisor. There exists a unique Ricci-flat Kähler metric in any given polarization by the work of Essydieux, Guedj and Zeriahi [@EGZ]. This is shown by solving degenerate complex Monge-Ampère equations related to constant scalar curvature metrics (cf. [@EGZ; @Z; @Kol2; @PhS; @PhS2; @PSS]) . Let $\mathcal{L}_0$ be an ample line bundle over $Y$ and so it induces an embedding morphism $Y \hookrightarrow {\mathbb{P}}^N$ into some big projective space. Let $\alpha$ be a Kähler class on $X$ and we define $\alpha_t= \alpha+ t [f^* \mathcal{L}_0]$. Obviously $\alpha_t$ is a Kähler class on $X$ whenever $t>0$. Let $\theta\in [\mathcal{L}_0]$ be a multiple of the pullback of the Fubini-Study metric on ${\mathbb{P}}^N$, $\omega_0\in [\alpha]$ a Kähler metric on $X$ and $\omega_t = \theta + t\omega_0$. Let $\Omega_{CY}$ be a smooth volume form on $X$ such that ${\sqrt{-1}\partial{\overline{\partial}}}\log \Omega_{CY} =0$. Then the solution of the following complex Monge-Ampère equation $$\label{maeqn}
(\omega_t + {\sqrt{-1}\partial{\overline{\partial}}}\varphi_t)^3 = c(t) \Omega_{CY}, ~~~ \sup_{X} \varphi_t = 0
$$ gives rise to a Ricci flat Kähler metric $g(t)$ associated to the Kähler form $$\omega(t) = \omega_t + {\sqrt{-1}\partial{\overline{\partial}}}\varphi_t,
$$ where $c(t)$ is a family of constants in $t$ determined by $\alpha_t ^3 = c(t) \int_X \Omega_{CY}$.
The following deep estimates are obtained in [@EGZ; @Z] built on techniques of Kolodziej [@Kol1].
\[EGZ1\] There exists $C>0$ such that for all $t\in (0, 1]$, $$||\varphi_t ||_{L^\infty(X)} \leq C.
$$
The local $C^\infty$ regularity follows from the $L^\infty$ estimate, Tsuji’s trick [@Ts] and the general linear theory (cf. [@EGZ; @ST1; @To1]).
\[EGZ2\] Let $D$ be the exceptional locus of $f: X \rightarrow Y$ and $X^{\circ} = X\setminus D$. For any compact set $K$ of $X^{\circ} $ and $k \geq 0$, there exists $C_{K, k}>0$ such that for all $t\in (0, 1]$, $$||\varphi_t ||_{C^k(K)} \leq C_{K, k}.
$$
From the above uniform estimates, we obtain a unique solution for the limiting degenerate complex Monge-Ampère equation [@EGZ; @To1]. It is shown in [@To1; @ZY] that the diameter of $(X, g(t))$ is uniformly bounded for $t\in (0,1]$. The following corollary follows from Theorem \[EGZ1\] and Proposition \[EGZ2\] by letting $t\rightarrow 0$.
\[EGZ3\] Let $Y_{reg}$ be the nonsingular part of $Y$. There exists a unique $\varphi_0 \in PSH(Y, \theta) \cap L^\infty(Y) \cap C^\infty (Y_{reg})$ such that $\sup_X \varphi_0 = 0$ and $$(\theta + {\sqrt{-1}\partial{\overline{\partial}}}\varphi_0)^3 = c_\theta \Omega_{CY}, ~~ \int_X \theta^3 = c_\theta \int_X \Omega_{CY}.
$$
The Kähler current $$\omega_Y = \theta + {\sqrt{-1}\partial{\overline{\partial}}}\varphi_0
$$ induces the unique singular Ricci-flat Kähler metric on $Y$ in $c_1(\mathcal{L}_0)$.
By the general theory in Riemannian geometry [@C; @CC1; @CC2; @CCT], one can always take the Gromov-Hausdorff limit for the family of $(X, g(t))$ with $t\in (0, 1]$. On the other hand, $g(t)$ converges to $g_Y$ in $C^\infty(X^\circ)$. Naturally, one would ask if the intrinsic limit of $(X, g(t))$ in Gromov-Hausdorff topology is homeomorphic to $Y$ as a projective variety, and if it coincides with its extrinsic limit.
We now introduce the notion of geometric transitions for Calabi-Yau threefolds (cf. [@Ro; @RuZ; @RZ]).
\[defgt\] Let $X$ be a Calabi-Yau threefold and $f: X \rightarrow Y$ be a contraction morphism from $X$ to a normal Calabi-Yau variety $Y$. Suppose that $Y$ admits a smooth projective deformation $\pi: \mathcal{M} \rightarrow \Delta$ over the unit disc $\Delta \in {\mathbb{C}}$ such that $K_{\mathcal{M}/\Delta} = {\mathcal{O}}_{\mathcal{M}}$ with smooth fibres $Y_s=\pi^{-1} (s)$ of Calabi-Yau three folds for $s \neq 0$ and $Y=Y_0$.
Then the following diagram is called a geometric transition $T(X, Y, Y_s)$ $$\begin{diagram}\label{geotran2}
\node{X} \arrow[2]{e,t}{f } \node[2]{Y } \node[2]{Y_s} \arrow[2]{w}\\
\end{diagram}.$$
\[defct\] A geometric transition $T(X, Y, Y_s)$ is called a conifold transition if $Y$ admits only ordinary double points as singularity.
\[defcflop\] A flop between two Calabi-Yau threefolds $X$ and $X'$ as in (\[flop\])is called a conifold flop if $Y$ admits only ordinary double points as singularity.
A local model for conifold singularities is given by $$\{ z\in {\mathbb{C}}^4~|~z_1^2 + z_2^2 + z_3^2 + z_4^2 =0 \}.$$ A well-known example for a conifold transition is given in [@GMS]. Let $Y$ be the hypersurface in ${\mathbb{P}}^4$ defined by $$z_3 g(z_0, ..., z_4)+ z_4 h(z_0, ..., z_4)=0$$ with generic homogeneous polynomials $g, h$ of degree $4$ in $[z_0, z_1, ..., z_4] \in {\mathbb{P}}^4$. The singular locus of $Y$ is given by $\{ z_3=z_4=g(z)=h(z)=0\} $, which consists of $16$ ordinary double points. The small resolution of the singularities of $Y$ gives rise to a smooth Calabi-Yau threefold $X$ and $Y$ can also be smoothed to generic smooth quintic threefolds in ${\mathbb{P}}^4$.
Let $T(X, Y, Y_s)$ be a geometric transition associated to a smoothing $\mathcal{M} \rightarrow \Delta$. Let $\mathcal{L}$ be an ample line bundle over $\mathcal{M}$ and $\mathcal{L}_s = \mathcal{L}|_{Y_s}$ . Then there exists a unique smooth Calabi-Yau Kähler metric $g_{Y_s} \in c_1 (\mathcal{L}_s)$ for $s \in \Delta^*$. When $s=0$, there exists a unique singular Ricci-flat Käher metric $g_Y$ by Theorem \[EGZ3\] such that the associated Kähler current $\omega_Y \in c_1(\mathcal{L}_0)$ has bounded local potential and $\omega_Y^3$ is a Calabi-Yau volume form on $Y$. In fact, $g_Y$ is smooth on $Y_{reg}$, the nonsingular part of $Y$.
Let $\alpha$ be a Kähler class of $X$. Then $\alpha_t = \alpha + t [\mathcal{L}_0]$ is a Kähler class of $X$ for $t\in (0, 1]$ and there exists a unique smooth Calabi-Yau Kähler metric $g(t) \in c_1 (\alpha_t)$. The following is a natural mathematical formulation for a conjecture of Candelas and de la Ossa [@RZ].
\[conj\] Let $T(X, Y, Y_s)$ be a conifold transition. The metric completion of $(Y_{reg}, g_Y)$ is a compact length metric space homeomorphic to $Y$ as a projective variety. If we denote such a metric space by $(Y, d_Y)$, then $(X, g(t))$ and $(Y_s, g_{Y_s} )$ converge to $(Y, d_Y)$ in Gromov-Hausdorff topology as $t, s \rightarrow 0$ $$\begin{diagram}\label{diagcand}
\node{(X, g(t) )} \arrow[2]{e,t,b}{d_{GH} } \node[2]{ (Y, d_Y) } \node[2]{(Y_s, g_{Y_s} )} \arrow[2]{w,t}{d_{GH}}
\end{diagram}, ~~ ~ ~ t, s \rightarrow 0.$$
The following theorem of Rong and Zhang [@RZ] proves a version of the above conjecture for general geometric transitions.
\[RZ\] Let $T(X, Y, Y_s)$ be a geometric transition. The metric completion of $(Y_{reg}, g_Y)$ is a compact length metric space and we denote it by $(Y', d_{Y'})$. Then $(X, g(t))$ and $(Y_s, g_{Y_s} )$ converge to $(Y', d_{Y'})$ in Gromov-Hausdorff topology as $t, s \rightarrow 0$.
The contribution of the paper is to give uniform estimates near the exceptional rational curves in the case of conifold transitions and prove the metric completion $(Y', d_{Y'})$ is homeomorphic to the projective variety $Y$ itself (cf. Theorem \[main1\]).
The algebraic structure of conifold transitions are rather well understood (cf. [@Ro]). Let $T(X, Y, Y_s)$ be a conifold transition. $Y$ is then a normal Calabi-Yau threefold with isolated conifold singularities $y_1, y_2, ..., y_d$. If $f: X \rightarrow Y$ is a minimal resolution of $Y$ at $y_1, y_2, ..., y_d$, each component of the exceptional locus $D_j = f^{-1}(y_j)$ is a smooth rational curve ${\mathbb{P}}^1$ with normal bundle $$N_{{\mathbb{P}}^1} = {\mathcal{O}}_{{\mathbb{P}}^1} (-1) \oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-1).$$ We define $X^{\circ} = X \setminus ( D_1\cup ... \cup D_d)$ and obviously $X^\circ$ is isomorphic to $Y_{reg}$. We will try to understand the local structure near these exceptional rational curves analytically in the next section.
**Local ansatz** {#section3}
================
In this section, we will apply the Calabi ansatz introduced by Calabi [@C1] (also see [@Li; @SY]) to understand the small contraction near the exceptional locus.
[*Calabi ansatz.*]{} Let $E= {\mathcal{O}}_{{\mathbb{P}}^n}(-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^n}(-1) \oplus ... \oplus {\mathcal{O}}_{{\mathbb{P}}^n}(-1)={\mathcal{O}}_{{\mathbb{P}}^n}(-1)^{\oplus (n+1)}$ be the holomorphic bundle over ${\mathbb{P}}^n$ of rank $n+1$. Let $z=(z_1, z_2, ..., z_n)$ be a fixed set of inhomogeneous coordinates for ${\mathbb{P}}^n$. and $$\omega_{FS}= {\sqrt{-1}\partial{\overline{\partial}}}\log (1+|z|^2) \in {\mathcal{O}}_{{\mathbb{P}}^n}(1)$$ be the Fubini-Study metric on ${\mathbb{P}}^n$ and $h$ be the hermitian metric on $\mathcal{O}_{\mathbb{P}^n} (-1)$ such that $Ric(h) = -\omega_{FS}$. This induces a hermtian metric $h_E$ on $E$ is given by $$h_E= h^{\oplus (n+1)}.$$ Under a local trivialization of $E$, we write $$e^\rho = h_\xi (z) |\xi|^2, ~~~ \xi = (\xi_1, \xi_2, ..., \xi_{n+1}),$$ where $h_\xi(z)$ is a local representation for $h_E $ with $h_\xi (z)= (1+|z|^2).$
Now we are going to define a family of Kähler metrics on $E$ as below $$\label{metricrep1}
\omega = a \omega_{FS} + {\sqrt{-1}\partial{\overline{\partial}}}u(\rho)
$$ for an appropriate choice of convex smooth function $u= u(\rho)$. In fact, we have the following criterion due to Calabi [@C1] for the above form $\omega$ to be Kähler.
\[kacon\]
$\omega$ defined as above, extends to a global Kähler form on $E$ if and only if
1. $a>0$,
2. $u'>0$ and $u''>0$ for $\rho\in (-\infty, \infty)$,
3. $U_0 (e^\rho) = u(\rho)$ is smooth on $(-\infty, 0]$ with $U_0' (0)>0$.
Straightforward calculations show that $$\label{metricrep2}
\omega = (a + u'(\rho)) \omega_{FS} + h_\xi e^{-\rho} ( u' \delta_{\alpha \beta} + h_\xi e^{-\rho} ( u'' - u') \xi^{\bar \alpha} \xi^{\beta} ) \nabla \xi^\alpha \wedge \nabla \xi^{\bar \beta}.
$$ Here, $$\nabla \xi^\alpha = d \xi^\alpha + h_\xi^{-1} \partial h_\xi \xi^\alpha$$ and $\{ dz^i, \nabla \xi^\alpha\}_{i=1, ..., n, \alpha=1, 2, ..., n+1}$ is dual to the basis
$$\nabla_{z^i} = \frac{\partial}{\partial z^i} - h_\xi ^{-1} \frac{\partial h_\xi }{\partial z^i} \sum_\alpha \xi^\alpha \frac{\partial }{\partial \xi^\alpha}, ~~~~~~~ \frac{\partial}{\partial \xi ^\alpha}.
$$
Let $L={\mathcal{O}}_{{\mathbb{P}}^n}(-1)$. Then $E= L^{\oplus (n+1)}$. Let $p_\alpha: E \rightarrow L$ be the projection from $E$ to its $\alpha$th component, and let $e^{\rho_\alpha} = (1+|z|^2) |\xi_\alpha|^2$ and so $e^{\rho} = \sum_{\alpha=1}^{n+1} e^{\rho_\alpha}$.
[*A local conifold flop.*]{} From now on, we assume that $n=1$ and so $E$ is a rank two bundle over ${\mathbb{P}}^1$. Let $P_0$ be the zero section of $ E$ which is a rational curve with normal bundle ${\mathcal{O}}_{{\mathbb{P}}^1}(-1) \oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-1)$. Then by contracting $P_0$, one obtains the variety $\hat E$ with only one isolated double point as singularity. We can now define a flop for $E$ by letting $E'= {\mathcal{O}}_{{\mathbb{P}}^1} (-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^1} (-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-1)$. $\hat E$ is isomorphic to $E$ with a similar local trivialization $(w, \eta_1, \eta_2)$. Then the flop between $E$ and $E'$ $$\begin{diagram}\label{diag1}
\node{E} \arrow{se,b,}{f} \arrow[2]{e,t,..}{\check{f} } \node[2]{E'} \arrow{sw,r}{f'} \\
\node[2]{\hat E}
\end{diagram},$$ can be viewed as change of coordinates as below $$z = \frac{\eta_2}{\eta_1}, ~~\xi_1 = \eta_1, ~~\xi_2 = w\eta_1,$$ or $$w= \frac{\xi_2}{\xi_1}, ~~ \eta_1=\xi_1, ~~\eta_2 = z\xi_1.$$ We also have the following relation $$e^\rho= (1+|z|^2) (|\xi_1|^2 + |\xi_2|^2) = (1+ |w|^2)( |\eta_1|^2 + |\eta_2|^2).$$
Let $\pi: E\rightarrow {\mathbb{P}}^1$ and $\pi': E' \rightarrow {\mathbb{P}}^1$. Then for each fixed $w\in {\mathbb{P}}^1$ the proper transformation of $(\pi') ^{-1}(w)$ via $\check{f} ^{-1}$ is the hypersurface of $E$ given by $$\xi_2 = w \xi_1.$$ Such a hypersurface is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^1}(-1)$ or simply ${\mathbb{C}}^2$ blow-up at one point and we denote it by $L_w$. Hence we obtained a meromorphic family of isomorphic surfaces $L_w$ in $E$ parametrized by $w\in {\mathbb{P}}^1 = \pi'(E')$. We can also view $E$ as a meromorphic fibration of ${\mathbb{C}}^2$ blow-up at one point over ${\mathbb{P}}^1$. The following lemma can be obtained by explicit calculations.
For each $w\in {\mathbb{P}}^1$, $L_w$ is isomorphic to ${\mathbb{C}}^2$ blow-up at one point. Furthermore, for $w_1\neq w_2$, $$L_{w_1} \cap L_{w_2} = P_0,$$ where $P_0$ is the zero section of $E$.
[*Local forms.*]{} We will define two reference forms on $E$. We first fix a Kähler form $\hat \omega$ on $E$ by $$\begin{aligned}
&&\hat\omega \\
&=& \omega_{FS} + {\sqrt{-1}\partial{\overline{\partial}}}e^{\rho}\\
&= &(1+ (1+|z|^2)e^\rho) \omega_{FS} + \sqrt{-1} z\bar \xi d\xi\wedge d\bar z + \sqrt{-1} \bar z \xi dz \wedge d\bar \xi + \sqrt{-1} (1+|z|^2) d\xi \wedge d\bar \xi.
$$ Then we choose a smooth closed nonnegative real $(1,1)$ form $\tau$ defined by $$\begin{aligned}
&&\tau\\
&=& {\sqrt{-1}\partial{\overline{\partial}}}e^{\rho_1}\\
& =& \sqrt{-1} |\xi_1|^2 dz\wedge d\bar z + \sqrt{-1}z\bar \xi_1 d\xi_1 \wedge d\bar z + \sqrt{-1}\bar z \xi_1 dz \wedge d\bar \xi_1 + \sqrt{-1}(1+|z|^2) d\xi_1 \wedge d\bar \xi_1\\
& =& \sqrt{-1} (1+|z|^2) e^{\rho_1 }\omega_{FS} + \sqrt{-1} z\bar \xi_1 d\xi_1 \wedge d\bar z + \sqrt{-1}\bar z \xi_1 dz \wedge d\bar \xi_1 + \sqrt{-1}(1+|z|^2) d\xi_1 \wedge d\bar \xi_1.
\end{aligned}$$ Although $\tau$ is not big, it defines a flat degenerate Kähler form on $L_w$ for each $w$.
\[locm1\] Let $\nu_1 = z \xi_1$, $\nu_2 = \xi_1$, and $\nu=(\nu_1, \nu_2)$. Then $e^{\rho_1} = |\nu|^2$ and $$\tau = \sqrt{-1} \left( d\nu_1 \wedge d\overline \nu_1 + d\nu_2\wedge d\overline \nu_2 \right)$$ is the pullback of the flat Euclidean metric on ${\mathbb{C}}^2$. Hence $\tau$ is flat on $L_w \setminus P_0$ for each $w\in {\mathbb{P}}^1$.
The restriction of $\hat \omega$ on $L_w$ for fixed $w$ is given by $$\begin{aligned}
\hat \omega &=& \omega_{FS} + {\sqrt{-1}\partial{\overline{\partial}}}e^\rho =\omega_{FS} + (1+ |w|^2) {\sqrt{-1}\partial{\overline{\partial}}}e^{\rho_1}\\
&=& \left(1+ (1+ |w|^2 )(1+|z|^2)e^{\rho_1} \right) \omega_{FS} \\
&&+ \sqrt{-1} (1+ |w|^2 ) \left( z\bar \xi_1 d\xi_1 \wedge d\bar z + \bar z \xi_1 dz \wedge d\bar \xi_1 + (1+|z|^2) d\xi_1 \wedge d\bar \xi_1 \right).
$$ We are only interested the local behavior of these forms near the zero section $P_0$, so we define $$\label{defome}\Omega= \{ \rho < 0\} \subset E,
$$ or equivalently in local coordinates, $$e^\rho= (1+|z|^2)|\xi|^2= |\nu|^2 + (1+|\nu_1/\nu_2|^2)|\xi_2|^2\leq 1.$$ Then $$\label{compp}
L_w \cap \Omega= \{ (z, \xi_1, \xi_2)~|~ \xi_2= w \xi_1, ~e^{\rho_1} \leq (1+|w|^2)^{-1} \} .$$ We now compare $\hat\omega$ and $\tau$ on each meromorphic fibre $L_w\simeq {\mathcal{O}}_{{\mathbb{P}}^1}(-1)$.
\[loccom\] For all $w \in {\mathbb{C}}$, $$\label{loccom2}
\tau |_{L_w\cap \Omega } \leq \hat \omega |_{L_w\cap \Omega } \leq 2 e^{-\rho_1} \tau |_{L_w\cap \Omega} .
$$
The lower bound for $\hat\omega$ is trivial because $$\hat\omega\geq {\sqrt{-1}\partial{\overline{\partial}}}e^\rho\geq{\sqrt{-1}\partial{\overline{\partial}}}e^{\rho_1} = \tau.$$ Restricted on each $L_w \cap \Omega$, $${\sqrt{-1}\partial{\overline{\partial}}}e^\rho = (1+|w|^2){\sqrt{-1}\partial{\overline{\partial}}}e^{\rho_1} \leq e^{-\rho_1} \tau,$$ because $1+|w|^2 \leq e^{-\rho_1}$ by (\[compp\]). We also have on $L_w$, $$e^{\rho_1} \omega_{FS} \leq \frac{ \sqrt{-1} |\nu_2|^2}{ 1+ |\nu_1/\nu_2|^2} d\left(\frac{\nu_1}{\nu_2} \right) \wedge d \overline{ \left(\frac{\nu_1}{\nu_2} \right)} \leq {\sqrt{-1}\partial{\overline{\partial}}}|\nu|^2 = \tau.$$ The lemma follows immediately as $$\hat\omega = \omega_{FS} + {\sqrt{-1}\partial{\overline{\partial}}}e^{\rho} \leq 2e^{-\rho_1} \tau$$ restricted on $L_w\cap \Omega$.
We also remark that the estimate (\[loccom2\]) still holds if one changes the trivialization by $U(2)$ action on $\xi=(\xi_1, \xi_2)$, because $\hat \omega$ is invariant by $U(2)$-action and the bounding coefficients do not depend on the choice of local trivialization as long as they are equivalent by $U(2)$-action.
[*A local model.*]{} We will now construct a family of complete Ricci-flat Kähler metrics on $E$. Such metrics are given in [@CaO] and here we give the calculations in terms of the Calabi ansatz. Let $\omega_E(t)=t \omega_{FS} + {\sqrt{-1}\partial{\overline{\partial}}}u$ be a Kähler metric with Calabi symmetry defined on $E$ for $t\in (0, 1]$. Then the Ricci curvature of $\omega_E(t)$ is given by $$Ric(\omega_E(t) ) = -{\sqrt{-1}\partial{\overline{\partial}}}\left(\log (t+u')u'u'' - 2\rho\right).$$ The vanishing Ricci curvature is equivalent the following equation $$\left((t+u')u'u''\right)' = e^{2\rho},$$ and then by integration twice, we have $$\label{cubic}
2(u')^3 + 3 t (u')^2 - 3 e^{2\rho} = 0.
$$ For each $t>0$, equation (\[cubic\]) can be explicitly solved for $u'$ by the cubic formula and it is asymptotically of order $t^{-1/2} e^\rho$ near $\rho= -\infty$.
when $t=0$, equation (\[cubic\]) becomes $(u')^3 = 3e^{2\rho}/2$ and the solution is explicitly given by $$u_{\hat E}' = (3/2)^{1/3} e^{2\rho/3}, ~~u_{\hat E}''= (2/3)^{2/3} e^{2\rho/3}.$$ Such $u_{\hat E}$ induces a complete Ricci-flat Kähler metric $$\label{limcy}
\omega_{CY, \hat E} = {\sqrt{-1}\partial{\overline{\partial}}}u_{\hat E}
$$ on $\hat E$ with an isolated cone singularity.
**Estimates** {#section4}
=============
From now on, we consider the small contraction morphism $$\pi: X \rightarrow Y$$ from a smooth Calabi-Yau threefold $X$ to a conifold $Y$. Without loss of generality, we assume that $y_1, ..., y_d$ are all the ordinary double points of $Y$ with $D_i = \pi^{-1} (y_j)$ for $j=1, ..., d$.
Due to the estimates in Proposition \[EGZ2\] away from the exceptional rational curves $D_1$, ..., $D_d$, it suffices to prove a uniform estimate for the degenerating family of Calabi-Yau metrics in a small neighborhood of each exceptional rational curve. Without loss of generality, we localize the problem by looking at a neighborhood of a fixed irreducible rational curve $D$ isomorphic to $$\Omega= E\cap \{\rho <0\}$$ with $D =\{ \rho=-\infty\} $ as defined in (\[defome\]).
Let $\omega(t)$ be the Ricci-flat Kähler metric on $X$ defined in (\[maeqn\]) for $t\in (0, 1]$ with the same assumptions. We will restrict all the metrics and apply estimates to $\Omega$. Then for each $t\in (0, 1]$, $\omega(t)$ is equivalent to $\hat \omega$ on $\Omega$.
The goal in the section is to obtain a second order estimate for the local potential of $\omega(t)$. The usual method in [@Ya1; @SW2] does not quite work in this case as there does not exist a good reference metric with admissible curvature properties, in particular, some component in the curvature tensor of $\hat\omega$ tends $-\infty$ near $P_0$. The geometric interpretation of such difficulty is that the degenerate locus for the complex Monge-Ampère equation (\[maeqn\]) has codimension greater than one. Since $E$ admits a meromorphic family of ${\mathbb{C}}^2$ blow-up at one point as shown in section 3, we consider a partial 2nd order estimate by bounding the metric along each meromorphic fibre. We therefore take advantage of the geometric flop structure and apply the maximum principle by a meromorphic slicing, so that the exceptional locus restricted to each meromorphic fibre has codimension one and we can apply ideas in [@SW2]. More precisely, for any Kähler form $\omega$ on $E$, we can take the fibre-wise trace of $\omega$ with respect to $\tau$ along each $L_w$.
We define for $t\in (0, 1]$,
$$H (t, \cdot)= tr_{\tau|_{L_w\cap \Omega}} (\omega (t) |_{L_w\cap \Omega}).$$
Here $\tau$ and $\omega(t)$ are restricted to $L_w$ as smooth real closed $(1,1)$-forms. $H$ can also be expressed as $$H(t, \cdot) = \frac{ \omega(t) \wedge \tau \wedge dw\wedge d\bar w}{\tau^2 \wedge dw \wedge d\bar w}.$$
\[H1\] $H \in C^\infty( \overline{\Omega} \setminus S)$ for all $t\in (0, 1]$, where $S=\{ \rho_1=-\infty \}$. Furthermore,
1. for all $t\in (0, 1]$, $$\sup_{\Omega} e^{\rho_1} H(t, \cdot) < \infty ;$$
2. there exists $C>0$ such that for all $t\in (0, 1]$,
$$\sup_{\partial \Omega} e^{\rho_1} H (t, \cdot) \leq C.$$
For each fixed $t\in (0,1]$, $\omega(t)$ is equivalent to $\hat\omega$ on $\Omega$ and so $(a)$ follows immediately from Lemma \[loccom\]. By Proposition \[EGZ2\], there exists $C>0$ such that for all $t\in (0,1]$ and $p \in \partial \Omega$, $$\omega(t) \leq C \hat\omega.$$ Therefore for all $t\in (0, 1]$, $$\sup_{\partial \Omega} e^{\rho_1} H(t, \cdot) \leq C \sup_{\partial \Omega} e^{\rho_1} tr_{\tau|_{L_w\cap\Omega}} (\hat\omega|_{L_w \cap \Omega}) \leq 2C$$ and it proves $(b)$.
\[H2\] Let $\Delta_t$ be the Laplace operator associated to the Ricci-flat Kähler metric $g(t)$ for $t\in (0, 1]$. Then
$$\Delta_t \log H \geq 0.
$$
We define $$I = \log H$$ and break the proof into the following steps.
[*Step 1.*]{} We first make a choice of special coordinates. On $\Omega$, we have the standard local coordinates with Calabi symmetry as defined in the previous section, i.e., for each $p \in \Omega$, we have at $p$, $ (z(p), \xi_1(p), \xi_2(p))$. Once we fix $p$, there exists a unique $w\in {\mathbb{P}}^1$ such that $p\in L_w$.
1. Near $p \in \Omega$, we first choose the coordinates $(\nu_1, \nu_2, w)$, where $\nu_1= \xi_1$ and $\nu_2=z\xi_1$ as defined in Lemma \[locm1\] . We will apply a linear transformation to $(\nu_1, \nu_2, w)$ such that $$x=(x_1, x_2, x_3)^T= A^{-1} (\nu_1, \nu_2, w)^T.$$ We assume that $A$ is in the form of $$\left(
\begin{array}{cc}
A' & a \\
0 & 1 \\
\end{array}
\right),$$ where $A'$ is a $2\times 2$ matrix and $a$ is a $2\times 1$ vector. Immediately, we have $$x_3 = w.$$
2. Suppose $g(t)$ at $(t, p)$ is given by the following hermitian matrix with respect to coordinates $(\nu_1, \nu_2, w)$
$$G= \left(
\begin{array}{cc}
B & b \\
\overline{b}^T & c \\
\end{array}
\right),$$ where $B$ is a $2\times 2$ hermitian matrix, $b$ a $2\times 1$ vector. Then under the new coordinates $x$, $g(t)$ at $p$ is given by the following hermitian matrix
$$\begin{aligned}
\bar A^T G A &=& \left(
\begin{array}{cc}
\overline{A'}^T B A' & \overline{A'}^T B a + \overline{A'}^T b \\
\overline{a}^T B A' + \overline{b}^T A' & \overline{a}^T B a + \overline{a}^T b + \overline{b}^T a + c \\
\end{array}
\right) \\
&=& \left(
\begin{array}{cc}
\overline{A'}^T B A' & \overline{A'}^T (B a + b) \\
(\overline{a}^T B + \overline{b}^T) A' & \overline{a}^T B a + 2Re(\overline{a}^T b )+ c \\
\end{array}
\right).
\end{aligned}$$
3. We choose a unitary matix $A'$ such that $\overline{A'}^T B A'$ is diagonalized, i.e., $$\left(
\begin{array}{cc}
\lambda_1 & 0 \\
0 & \lambda_2 \\
\end{array}
\right)$$ and choose $a$ such that $$Ba= -b$$ since $B$ has rank $2$. Therefore under the coordinates $x$, at $(t, p)$,
$$g= \left(
\begin{array}{ccc}
\lambda_1 & 0 & 0 \\
0 & \lambda_2 & 0 \\
0 & 0 & \lambda_3 \\
\end{array}
\right),$$ where $\lambda_3= c - \overline{a}^TB a.$ The matrix representation of $\tau$ under the coordinates $(\nu_1, \nu_2, w)$ is given by $$\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
\end{array} \right),$$ and so its matrix representation under the coordinates $X$ at $(t, p)$ is given by $$\tau= \overline{A}^T \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
\end{array}
\right) A= \left(
\begin{array}{ccc}
I_{2\times 2} & \overline{A'}^T a \\
\overline{a}^T A' & \overline{a}^T a \\
\end{array}
\right)$$ since $A'$ is unitary. Since $x_3=w$ and on $L_w$, $x_3$ is constant and $\omega|_{L_w\cap \Omega} =\sqrt{-1} \sum_{i, j=1}^2 g_{i\bar j} dx^i \wedge d\overline{x^j}$. Then at $(t, p)$, $$g|_{L_w} = \left(
\begin{array}{cc}
\lambda_1 & 0 \\
0 & \lambda_2 \\
\end{array}
\right), ~~~~~~~\tau |_{L_w} = \left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right).$$ Finally, we arrive at $$I(t, p) = \log \sum_{i, j=1, 2}(\tau |_{L_w}) ^{i\bar j} (g|_{L_w})_{i\bar j} = \log (\lambda_1 + \lambda_2).$$
[*Step 2.*]{} Now we calculate $\Delta_t I$ at $(t, p)$ under the coordinates $x$. Notice that $\tau|_{L_w}$ is a constant form $\sqrt{-1} (dx_1\wedge d \overline x_1 + dx_2 \wedge d\overline x_2) $ and so all derivatives of $\tau|_{L_w}$ vanish.
We now apply the Laplace operator $\Delta_t$ to $H$. $$\begin{aligned}
&&\Delta_t H \\
&=& \sum_{k, l=1}^3g^{k\bar l} \left( \sum_{i, j=12} \left( \tau|_{L_w} \right)^{i\bar j} g_{i\bar j} \right)_{k\bar l}\\
&=&\sum_{k, l=1}^3 \sum_{i, j=1, 2} g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j} g_{i\bar j, k\bar l} - \sum_{k, l=1}^3\sum_{i, j, p, q=1, 2}g^{k \bar l}g_{i\bar j} \left( \tau|_{L_w} \right)^{i\bar q} \left( \tau|_{L_w} \right)^{ p\bar j} \tau_{p\bar q, k\bar l} \\
&=& - \sum_{k, l=1}^3 \sum_{i, j=1,2} g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j}R_{i\bar j k\bar l} + \sum_{k, l, p, q=1}^3\sum_{i, j =1,2}g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j} g^{p\bar q} g_{p \bar j, \bar l} g_{i\bar q, k}\\
&=& - \sum_{i, j=1, 2} \left( \tau|_{L_w} \right)^{i\bar j} R_{i\bar j} + \sum_{k, l, p,q=1}^3\sum_{i, j=1, 2}g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j} g^{p\bar q} g_{p \bar j, \bar l} g_{i\bar q, k}\\
&=& \sum_{k,l, p, q=1}^3 \sum_{i, j=1, 2}g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j}g^{p\bar q} g_{p \bar l, \bar j} g_{k\bar q, i}.
$$ Then $$\begin{aligned}
&& \Delta_t I \\
&=&( H )^{-1} \sum_{k, l=1}^3 \sum_{i, j=1, 2; p, q=1,2,3}g^{k\bar l}\left( \tau|_{L_w} \right)^{i\bar j} g^{p\bar q} g_{p \bar j, \bar l} g_{i\bar q, k} - (H )^{-2} |\nabla I |^2_g\\
&=& (H )^{-2}\sum_{k, l=1}^3\left( H \sum_{i, j=1, 2; p, q=1,2,3}g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j} g^{p\bar q} g_{p \bar j, \bar l} g_{i\bar q, k} - g^{k\bar l} (\sum_{i, j=1, 2} \left( \tau|_{L_w} \right)^{i\bar j} g_{i\bar j, k}) (\sum_{i, j=1,2}\left( \tau|_{L_w} \right)^{j\bar i} g_{j\bar i, \bar l}) \right)\\
&=& (H )^{-2} \sum_{k, l=1}^3 \left( H \sum_{i, j=1, 2; p, q=1,2,3}g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j}g^{p\bar q} g_{p \bar l, \bar j} g_{k\bar q, i} - g^{k\bar l} (\sum_{i, j=1, 2} \left( \tau|_{L_w} \right)^{i\bar j} g_{i\bar j, k}) (\sum_{i, j=1, 2} \left( \tau|_{L_w} \right)^{j\bar i} g_{j\bar i, \bar l}) \right).
$$
[*Step 3.*]{} The proof of the proposition is now reduced to show that $$\sum_{k, l=1}^3 \left(H \sum_{i, j=1, 2; p, q=1,2,3}g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j} g^{p\bar q} g_{p \bar l, \bar j} g_{k\bar q, i} - g^{k\bar l} (\sum_{i, j=1, 2} \left( \tau|_{L_w} \right)^{i\bar j} g_{i\bar j, k}) (\sum_{i, j=1, 2} \left( \tau|_{L_w} \right)^{j\bar i} g_{j\bar i, \bar l}) \right)\geq 0.$$ Note that $\tau_{i\bar j}=\delta_{ij}$ for $i, j=1, 2$ and $g= (\lambda_1, \lambda_2, \lambda_3)$. Then $$\begin{aligned}
&&\sum_{k, l=1}^3 g^{k\bar l} \left(\sum_{i, j=1, 2} \left( \tau|_{L_w} \right)^{i\bar j} g_{i\bar j, k} \right) \left(\sum_{i, j=1,2} \left( \tau|_{L_w} \right)^{j\bar i} g_{j\bar i, \bar l} \right)\\
&=&\sum_{k=1, 2,3} \lambda_k^{-1} |\sum_{i=1, 2}g_{i\bar i, k}|^2\\
&\leq& \sum_{i, j=1,2} \left( \sum_{k=1, 2,3} \lambda_k^{-1} |g_{i\bar i, k}|^2\right)^{1/2} \left(\sum_{k=1, 2,3} \lambda_k^{-1} |g_{j \bar j, k}|^2 \right)^{1/2}\\
&=&\left( \sum_{i=1,2} \left(\sum_{k=1, 2,3} \lambda_k^{-1} |g_{i\bar i, k}|^2 \right)^{1/2} \right)^2\\
&=&\left( \sum_{i=1,2} \lambda_i^{1/2} \left( \sum_{k=1, 2,3} \lambda_k^{-1}\lambda_i^{-1} |g_{i\bar i, k}|^2 \right)^{1/2} \right)^2\\
&\leq& \left(\sum_{i=1, 2} \lambda_i \right) \left( \sum_{i=1, 2;k=1, 2,3} \lambda_k^{-1}\lambda_i^{-1} |g_{i\bar i, k}|^2 \right) \\
&\leq& H \left( \sum_{k, l=1, 2, 3; i=1, 2} \lambda_k^{-1}\lambda_l^{-1} |g_{i\bar l, k}|^2 \right) \\
&=& H \sum_{i, j=1,2; k, l, p,q=1,2, 3} g^{k\bar l} \left( \tau|_{L_w} \right)^{i\bar j} g^{p\bar q} g_{p\bar j, \bar l} g_{i\bar q, k} .
$$
This completes the proof of the proposition.
There exists $C>0$ such that on $\Omega$, for all $t\in (0, 1]$, $$ H \leq C e^{-\rho_1}.
$$
Let $$I_{\epsilon} = \log H + (1+\epsilon) \rho_1$$ for $\epsilon>0$. Let $S= \{\rho_1=-\infty\} $. Then for all $\epsilon>0$, $\limsup_{p \rightarrow S} I_\epsilon = -\infty$ by Lemma \[H1\], and on $\Omega\setminus S$, $$\Delta_t I_\epsilon >0,$$ because of Proposition \[H2\] and the fact that $\Delta_t \rho_1= \Delta_t \log (1+|z|^2)|\xi_1|^2 = tr_{\omega}(\omega_{FS})>0$ on $\Omega\setminus S$.
Applying the maximum principle for $I_\epsilon$, we know that the maximum of $I_\epsilon$ has to be achieved on $\partial \Omega$. Then by Lemma \[H1\], there exists $C>0$ such that for all $\epsilon\in (0,1]$ and $t\in(0,1]$, $$\sup_{\Omega\setminus S} I_\epsilon = \sup_{\partial \Omega} I_\epsilon \leq \sup_{\partial \Omega} I_0 \leq C.$$ The corollary is then proved by letting $\epsilon \rightarrow 0$.
We define a holomorphic vector $V$ on $\Omega$ by $$V= \xi_1 \frac{\partial}{\partial \xi_1} + \xi_2 \frac{\partial}{\partial \xi_2}.$$ $V$ vanishes along $P_0$ and $$|V|^2_{\hat \omega} = e^\rho.$$ We also consider the normalized vector field $$W= \frac{V}{|V|_{\hat \omega}} = e^{-\rho/2} \sum_{\alpha=1,2} \xi_\alpha \frac{\partial}{\partial \xi_\alpha}.$$
Now we can obtain uniform bounds for the degenerating Ricci-flat Kähler metrics $g(t)$ near the exceptional curves.
\[keyest\] There exists $C>0$ such that for all $t\in (0, 1]$ and on $\Omega$, $$\label{tanest}
C^{-1}\omega_{\hat E} \leq \omega(t) \leq Ce^{-\rho} \omega_{\hat E},
$$ and $$\label{verest}
|W|^2_{g(t)} \leq Ce^{-\rho/2},
$$ where $\omega_{\hat E}= {\sqrt{-1}\partial{\overline{\partial}}}e^\rho$.
We break the proof into the following steps.
[*Step 1.*]{} We apply the similar argument in the proof of Schwarz lemma [@Y2; @ST1]. Notice that $\omega (t) = \omega_t + {\sqrt{-1}\partial{\overline{\partial}}}\varphi$ with $\varphi\in C^\infty(X)$ uniformly bounded in $L^\infty(X)$ for $t\in (0, 1]$. Also there exists $C_1>0$ such that for all $t\in (0,1]$ and on $\Omega$, $\omega_t\geq C_1 \omega_{\hat E}$ on $\Omega.$ Then we consider the quantity $$L = \log tr_{\omega} (\omega_{\hat E}) - \varphi.$$ $\omega_{\hat E}$ restricted to $\Omega$ is in fact the pullback of a flat metric on ${\mathbb{C}}^4$ given by a local morphism $(\xi_1, \xi_2, z\xi_1, z\xi_2)$. Then straightforward calculations give $$\Delta_t L \geq tr_{\omega}(\omega_t) - 3 \geq tr_\omega(\omega_{\hat E}) - 3.$$ Applying the maximum principle, we have $$tr_\omega(\omega_{\hat E}) \leq \sup_{\partial \Omega} tr_{\omega}(\omega_{\hat E}) +3.$$ Note that $tr_{\omega(t)}(\omega_{\hat E}) $ is uniformly bounded on $\partial \Omega$. Hence $tr_{\omega}(\omega_{\hat E})$ is uniformly bounded above and so there exists $C_1>0$ such that $$\label{sch1}
\omega\geq C_1 \omega_{\hat E}.
$$
[*Step 2.*]{} Since $\omega^3$ is uniformly equivalent to $\hat \omega^3$ and $e^{-\rho} \omega_{\hat E}^3$ in $\Omega$, there exists $C_2>0$ such that $$\label{sch2}
\omega^3 \leq C_2 e^{-\rho} \omega_{\hat E}^3.
$$ By the estimates (\[sch1\]) and (\[sch2\]), there exists $C_3>0$ such that $$tr_{\omega_{\hat E}} (\omega) \leq C_3 e^{-\rho}$$ and so $\omega\leq C_3 e^{-\rho} \omega_{\hat E}.$ This completes the proof for estimate (\[tanest\]).
[*Step 3.*]{} Let $V_1 = \xi_1 \frac{\partial}{\partial \xi_1}$ be the holomorphic vector field on $\Omega$. Then $V_1$ vanishes on $\rho_1=-\infty$ and $ |V_1|^2_{\hat \omega} = (1+|z|^2) |\xi_1|^2= e^{\rho_1}.$ Using the normal coordinates for $\omega$, we can show that
$$\Delta_t |V_1|^2_g = - (V_1)^i (V_1)^{\bar j} R_{i\bar j} + g^{k\bar l} g_{i\bar j} \left((V_1)^{i})_k ((V_1)^{\bar j} \right)_{\bar l} = |\partial V|^2_g$$ and so $$\Delta_t \log |V_1|^2_g = (|V_1|^2_g)^{-2} \left( |V_1|^2_g |\partial V_1|^2_g - |\nabla_t |V_1|_g^2|^2 \right)\geq 0.$$
We now define $$G_\epsilon = \log \left( e^{\epsilon \rho_1}|V_1|^2_\omega tr_{\tau|_{L_w}} (\omega|_{L_w}) \right) = I + \log |V_1|_\omega^2 + \epsilon \rho_1$$ for $\epsilon\in (0, 1]$. For each $t\in (0, 1]$, $G_\epsilon$ is smooth in $\Omega$ away from $ \rho_1 = -\infty $, and it tends to $-\infty$ near $\rho_1=-\infty $ for all $\epsilon \in (0,1]$ by Lemma \[H1\]. Furthermore, there exists $C_4>0$ such that for all $\epsilon\in (0,1]$ and $t\in (0, 1]$, $$\sup_{\partial \Omega} G_\epsilon \leq C_4.$$ On the other hand, $$\Delta_t G_\epsilon= \Delta_t I + \Delta_t \log |V_1|^2_\omega + \epsilon \Delta_t \rho_1> 0.$$ By the maximum principle, $$\sup_{\Omega} G_\epsilon \leq \sup_{\partial \Omega } G_\epsilon \leq C_4.$$ Then by letting $\epsilon$ tend to $0$, we have $$\label{dirc}
|V_1|^2_\omega tr_{\tau|_{L_w}} (\omega|_{L_w}) \leq C_4.
$$
[*Step 4.*]{} We will apply the estimate (\[dirc\]) to prove (\[verest\]). Under the coordinates $(w, \nu_1, \nu_2)$, we have $$\left( \tau|_{L_w} \right)_{\nu_i \bar\nu_j }= \delta_{ij}, ~~~V_1 = \nu_1\frac{\partial }{\partial \nu_1} + \nu_2 \frac{\partial}{\partial \nu_2} - w\frac{\partial }{\partial w}, ~~~|V_1|_{\tau}^2 = |\nu|^2= e^{\rho_1},$$ At any $p\in L_w$ with $w=0$, we have $$V_1 = V, ~~\rho_1 = \rho,$$ $$\begin{aligned}
|V_1|^2_g &=& |\nu_1|^2 g_{\nu_1 \bar \nu_1} + 2Re (\nu_1 \bar \nu_2 g_{\nu_1\bar\nu_2}) + |\nu_2|^2 g_{\nu_2 \bar \nu_2} \\
& \leq & (|\nu_1|^2 + |\nu_2|^2) \left( g_{\nu_1 \bar \nu_1} + g_{\nu_2 \bar \nu_2 } \right) \\
&=& e^{\rho_1} tr_{\tau|_{L_0}} (\omega|_{L_0}) .
$$ Combined with (\[dirc\]), there exists $C_4>0$ such that for all $t\in (0, 1]$ and on $L_0 \cap \Omega$, $$|V_1|_\omega^2 \leq C_4 e^{ \rho_1/2}, ~ or ~ |V|_\omega^2 \leq C_4 e^{ \rho/2}.$$ Equivalently, we have, $$\left( |W|^2_\omega \right) |_{L_0\cap \Omega} \leq C_7 e^{-\rho/2}|_{L_0\cap \Omega}.$$ Notice that $W$, $V$, $\rho$ are $U(2)$-invariant in terms of $\xi$ and all the bounds we have derived do not depend on the choice of trivialization differing by $U(2)$-action. Therefore we have $$|W|^2_\omega \leq C_7 e^{-\rho/2}$$ uniformly for $t\in (0, 1]$ and $\Omega$. This completes the proof of the proposition.
The uniform bound on $diam(X, g(t))$ is already known to the general Calabi-Yau degeneration due to [@To1]. The following corollary follows from Proposition \[keyest\].
There exists $C>0$ such that for all $t\in (0, 1]$,
$$diam(X, g(t)) \leq C, ~~~~diam(X\setminus \{ D_1, ..., D_d\}, g(t)) \leq C.
$$
Since $|W|_{g(t)}^2 \leq Ce^{-\rho/2}$, any point $p=(z, \xi)$ in $\Omega$ can be connected by a radial path $\gamma_{p}$ defined by $$(z, s \xi), ~ s \in \left[0, \frac{1}{(1+|z|^2)|\xi|^2} \right]$$ to $P_0$ and $\partial \Omega$ with $p' = \gamma \left(\frac{1}{(1+|z|^2)|\xi|^2} \right)\in \partial \Omega$. Then the arc length of $\gamma_{p}$ with respect to $g(t)$ is uniformly bounded, i.e., there exists $C'>0$ such that for all $t\in (0, 1]$ and for all $p\in \Omega$, $$|\gamma_p |_{g(t)} \leq C'.$$ On the other hand, $g(t)$ is uniformly equivalent to $\hat \omega$ on $\partial \Omega$. Given any two points $p, q\in \Omega$, we can joint $p, q$ by $\gamma_p$, $\gamma_q$ and a smooth geodesic path $\gamma_{p', q'}$ with respect to $\hat \omega$ joining $p'$ and $q'$ in $\partial \Omega$. Therefore, both $diam(\Omega, g(t))$ and $diam(\Omega\setminus P_0, g(t))$ are uniformly bounded and this completes the proof of the corollary.
The following corollary shows that the restriction of $g(t)$ to the exceptional rational curve is uniformly bounded above.
There exists $C>0$ such that for all $t\in (0, 1]$ ,
$$\omega(t)|_{P_0} \leq C \omega_{FS}|_{P_0}.
$$
By (\[tanest\]) in Proposition \[keyest\], there exist $C_1, C_2>0$ such that for all $t\in (0, 1]$ and on $\Omega$, $$\frac{\omega\wedge \omega_{\hat E}^2}{\omega_{FS}\wedge \omega_{\hat E}^2} \leq C_1e^{-\rho} \frac{\omega_{\hat E}^3}{\omega_{FS}\wedge \omega_{\hat E}^2} \leq C_2.$$ For any point $p\in P_0$, there exist $e_1 \in T_p P_0$ and $e_2, e_3 \in T_p E$ such that they form an orthonormal basis of $T_p E$ with respect to $\hat \omega$. Obviously, $$\omega_{\hat E} (e_1, \cdot ) = 0.$$ Then $$tr_{\omega_{FS} |_{P_0}} ( \omega|_{P_0}) = \frac{\omega (e_1\wedge \overline{e_1})}{\omega_{FS} (e_1\wedge \overline{e_1})} =\frac{ \omega\wedge\omega_{\hat E}^2 (e_1\wedge \overline{e_1}\wedge \cdot\cdot\cdot \wedge e_3\wedge \overline{e_3} ) } { \omega_{FS}\wedge \omega_{\hat E}^2 (e_1\wedge \overline{e_1}\wedge \cdot\cdot\cdot \wedge e_3\wedge \overline{e_3} ) } \leq C_2.$$
In fact, the following proposition shows that exceptional rational curve become extinct as $t\rightarrow 0$.
\[diam2\] There exists $C>0$ such that for all $t\in (0, 1]$ such that $$diam(P_0, g(t)|_{P_0}) \leq C t^{1/3}.
$$
There exists $C>0$ such that for all $t\in (0, 1]$, $$\int_{P_0} \omega(t) = P_0 \cdot [\alpha_t] = t P_0 \cdot [\alpha] \leq C t.$$ Then the proposition is proved by the same argument in the proof of Lemma 3.2 in [@SW2].
We define for $r>0$, $$\Omega_{ r } = \{ (z, \xi) \in E~|~ e^{\rho} = (1+|z|^2)|\xi|^2 \leq r^2 \} .
$$
Then we have the following proposition.
For any $\epsilon>0$, there exist $\varepsilon>0$ and $\delta>0$ such that for all $t\in (0, \varepsilon)$, $$diam(\Omega_\delta, g(t)) < \epsilon.
$$
Given any two points $p, q \in \Omega_\delta$, there exist $p', q'\in P_0$ such that $p$ and $q$ can be connected to $p'$ and $q'$ by radial paths $\gamma_{p, p'}$ and $\gamma_{q, q'} $. For any $\epsilon>0$, we choose $\delta >0 $ such that the arc length of $\gamma_{p, p'}$ and $\gamma_{q, q'} $ is smaller than $\epsilon/3$ with respect to $g(t)$ for all $t\in (0, 1]$ by applying (\[verest\]) in Proposition \[keyest\]. By Proposition \[diam2\], we can choose $\varepsilon>0$ sufficiently small such that for $t\in (0, \varepsilon)$, $$diam(P_0, g(t)|_{P_0}) \leq \frac{\epsilon}{3}.$$ Then $$dist_{g(t)} (p, q) \leq dist_{g(t)} (p, p') + dist_{g(t)} (q, q') + diam(P_0, g(t)|_{P_0}) < \epsilon$$ and the proposition follows.
**Proof of Theorem \[main1\] and its generalizations** {#section5}
======================================================
Let $T(X, Y, Y_s)$ be a conifold transition. Let $g_Y$ be the unique singular Calabi-Yau Kähler metric associated to the Kähler current on $Y$ as defined in section 2. Note that $g_Y$ is smooth in $Y_{reg}=Y\setminus \{ y_1, ..., y_d \} $ and so we define a similar distance function on $Y$ as in Definition 5.1 in [@SW2].
\[defndy\] We extend $g_Y$ on $Y_{reg}$ to a nonnegative (1,1)-tensor $\tilde{g}_Y$ on the whole space $Y$ by setting $\tilde{g}_Y|_{y_j}( \cdot, \cdot) =0$ for $j=1, ..., d$. Of course, $\tilde{g}_Y$ may be discontinuous at $y_1$, ..., $y_d$. Define a distance function $d_Y: Y \times Y \rightarrow \mathbb{R}$ by $$d_Y(y, y') = \inf_{\gamma} \int_0^1 \sqrt{ g_Y ( \gamma'(s), \gamma'(s))} ds,
$$ where the infimum is taken over all piecewise smooth paths $\gamma: [0,1] \rightarrow Y$ with $\gamma(0)=y$, $\gamma(1)=y'$.
The goal is to show that such a metric space is exactly the Gromov-Hausdorff limit of $(X, g(t))$ as $t\rightarrow 0$ and it is isomorphic to the metric completion of $(Y_{reg}, g_Y)$.
\[proof2\] $(Y, d_Y)$ is a compact metric space homeomorphic to the projective variety $Y$ itself. Furthermore, $(X, g(t))$ converges to $(Y, d_Y)$ in Gromov-Hausdorff topology as $t\rightarrow 0$.
The same argument in section 3 in [@SW2] can be applied to prove the proposition with uniform estimates from Proposition \[keyest\] and Proposition \[diam2\].
\[proof1\] Let $(\tilde Y, d_{\tilde Y} )$ be the metric completion of $(Y_{reg}, g_Y)$. Then $(\tilde Y, d_{\tilde Y} )$ is isomorphic to $(Y, d_Y)$.
There are two ways to complete the proof of the theorem. The first approach is purely analytic. We can modify the argument in [@SW3] to show that $(\tilde{Y}, d_{\tilde Y} )$ is homeomorphic to $Y$ as a projective variety and indeed, $(X, g(t))$ converges to $(Y, d_Y)$ in Gromov-Hausdorff topology. The details can also be found in [@SW3] and in [@S] for higher codimensional surgeries by the Kähler-Ricci flow. This approach does not make use of the general theory on Riemannian manifolds with bounded Ricci curvature [@C; @CC1; @CC2; @CCT] and it gives explicit estimates to understand the analytic and geometric contractions.
The second approach relies on the results in [@RZ]. By Theorem \[RZ\], $(X, g(t))$ converges to $(\tilde Y, d_{\tilde Y} )$ in Gromov-Hausdorff topology, and hence by the uniqueness of the limiting metric space, $$(Y, d_Y)=(\tilde Y, d_{\tilde Y} ).$$ The theorem follows from Theorem \[proof1\].
[*Proof of main results.*]{} Theorem \[main1\] follows immediately from Theorem \[proof1\] and Theorem \[proof2\]. Corollary \[main2\] is a straightforward consequence of Theorem \[main1\]. Corollary is proved by combining Theorem \[main1\] and Theorem \[RZ\].
[*Discussions.*]{} We now discuss some generalizations of Theorem \[main1\] and future questions. First of all, Theorem \[main1\], Corollary \[main2\] and \[main3\] can be generalized to higher dimensional conifold small contractions, flops and transitions with little modification. The same estimates can be applied to high codimensional surgery by the Kähler-Ricci flow if the exceptional locus is ${\mathbb{P}}^n$ with normal bundle ${\mathcal{O}}_{{\mathbb{P}}^n} (-a_1)\oplus ... \oplus {\mathcal{O}}_{{\mathbb{P}}^n} (-a_{m+1})$ for $a_i \in \mathbb{Z}^+$. Consequentially, it is shown in [@S] that the Kähler-Ricci flow performs certain family of flipping contractions and resolution, in Gromov-Hausdorff topology. The blow-up limit of Type I singularities of the Ricci flow is a complete shrinking Ricci soliton [@H1; @EMT]. In the case of the Kähler-Ricci flow, finite time singularity arises from contraction of special rational curves. It is natural to conjecture that the curvature tensor blows up at the same rate as the extinction rate of such rational curves. We now make the following conjecture.
With the same assumptions in Theorem \[main1\], there exists $C>0$ such that for all $t\in (0, 1]$, such that the curvature tensor $Rm(t)$ of $g(t)$ is bounded as below $$\sup_X |Rm(t)|_{g(t)} \leq C t^{-1}.
$$ Furthermore, the rescaled Ricci-flat Kähler metrics $\tilde g(t)$ of $g(t)$ converge to the Ricci-flat Kähler metric $g_{CY, \hat E}$ on $\hat E$ with an isolated cone singularity given by (\[limcy\]), in pointed Gromov-Hausdorff topology.
In particular, the metric singularity of $g_Y$ near the ordinary double point should be asymptotically close to the local model given by (\[limcy\]).
[**Acknowledgements**]{} The author would like to thank Yuguang Zhang for a number of enlightening discussions and Valentino Tosatti for many helpful suggestions.
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[^1]: Research supported in part by National Science Foundation grant DMS-0847524 and a Sloan Foundation Fellowship.
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abstract: 'In the Orientiworld framework the Standard Model fields are localized on D3-branes sitting on top of an orientifold 3-plane. The transverse 6-dimensional space is a non-compact orbifold (or a more general conifold). The 4-dimensional gravity on D3-branes is reproduced due to the 4-dimensional Einstein-Hilbert term induced at the quantum level. The orientifold 3-plane plays a crucial role, in particular, without it the D3-brane world-volume theories would be conformal due to the tadpole cancellation. We study non-perturbative gauge dynamics in various ${\cal N}=1$ supersymmetric orientiworld models based on the ${\bf Z}_3$ as well as ${\bf Z}_5$ and ${\bf Z}_7$ orbifold groups. Our discussions illustrate that there is a rich variety of supersymmetry preserving dynamics in some of these models. On the other hand, we also find some models with dynamical supersymmetry breaking.'
address: |
C.N. Yang Institute for Theoretical Physics\
State University of New York, Stony Brook, NY 11794\
and\
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855
author:
- 'Zurab Kakushadze[^1]'
date: 'March 3, 2002'
title: On Gauge Dynamics and SUSY Breaking in Orientiworld
---
=10000 epsf
Introduction
============
Extra dimensions naturally arise in superstring theory (or M-theory), which is believed to be a consistent theory of quantum gravity. However, in order to model the real world with critical string theory (or M-theory), one must address the question of why the extra dimensions have not been observed. One way to make extra dimensions consistent with observation is to assume that they are compact with small enough volume. If the Standard Model gauge and matter fields propagate in such extra dimensions (as is the case in, say, weakly coupled heterotic string theory), then their linear sizes should not be larger than about inverse TeV [@anto]. On the other hand, in the Brane World scenario the Standard Model gauge and matter fields are assumed to be localized on branes (or an intersection thereof), while gravity lives in a larger dimensional bulk of space-time. Such a scenario with compact extra dimensions can, for instance, be embedded in superstring theory via Type I$^\prime$ compactifications. Then the extra dimensions transverse to the branes can have sizes as large as about a tenth of a millimeter [@TeV].
To begin with considering compact (or, more generally, finite volume) extra dimensions was motivated by the requirement that at the distance scales for which gravity has been measured one should reproduce 4-dimensional gravity. However, as was pointed out in [@DGP; @DG], 4-dimensional gravity can be reproduced even in theories with infinite-volume extra dimensions. In particular, according to [@DG] 4-dimensional gravity can be reproduced on a 3-brane in infinite-volume bulk (with 6 or more space-time dimensions) up to ultra-large distance scales. Thus, in these scenarios gravity is almost completely localized on a brane (which is almost $\delta$-function-like) with ultra-light modes penetrating into the bulk. As was explained in [@DG], this dramatic modification of gravity in higher codimension models with infinite volume extra dimensions is due to the Einstein-Hilbert term on the brane, which is induced via loops of non-conformal brane matter [@DGP; @DG].
In [@orient] we described an explicit string theory framework for embedding models with infinite-volume extra dimensions. In this framework, which we refer to as Orientiworld, the Standard Model gauge and matter fields are localized on (a collection of) D3-branes embedded in infinite-volume extra space. In particular, we consider unoriented Type IIB backgrounds in the presence of some number of D3-branes as well as an orientifold 3-plane embedded in an orbifolded space-time. The D3-brane world-volume theory in this framework is non-conformal (at least for some backgrounds with at most ${\cal N}=1$ supersymmetry). At the quantum level we have the Einstein-Hilbert term induced on the branes, which leads to almost complete localization of gravity on the D3-branes. In particular, as was discussed in [@orient], (at least in some backgrounds) up to an ultra-large cross-over distance scale the gravitational interactions of the Standard Model fields localized on D3-branes are described by 4-dimensional laws of gravity.
The orientiworld framework appears has a rich structure for model building. In particular, since the extra dimensions have infinite volume, the number of D3-branes is arbitrary. Moreover, the number of allowed orbifold groups is infinite. Thus, [*a priori*]{} the orbifold group can be an arbitrary[^2] subgroup of $Spin(6)$, or, if we require ${\cal N}=1$ supersymmetry to avoid bulk tachyons, of $SU(3)$. To obtain a finite string background, we still must impose [*twisted*]{} tadpole cancellation conditions. However, twisted tadpoles must also be canceled in compact Type IIB orientifolds. Then the number of consistent solutions of the latter type is rather limited [@class] as we can only have a finite number of D3-branes, and, moreover, the number of allowed orbifold groups is also finite as they must act crystallographically on the compact space. On the other hand, as we already mentioned, in the orientiworld framework the number of consistent solutions is [*infinite*]{}, which is encouraging for phenomenologically oriented model building.
This richness of the orientiworld framework can be exploited to construct various models for phenomenological applications. Thus, in [@orient] we gave a construction of an ${\cal N}=1$ supersymmetric 3-generation Pati-Salam model in the orientiworld framework. The purpose of this paper is to explore non-perturbative gauge dynamics in orientiworld models. In particular, we discuss ${\cal N}=1$ orientiworld models based on ${\bf Z}_3$ as well as ${\bf Z}_5$ and ${\bf Z}_7$ orbifolds. The examples we study illustrate that various non-perturbative phenomena can be expected in orientiworld models including dynamical supersymmetry breaking.
The rest of this paper is organized as follows. In section II we review the orientiworld framework. In section III we discuss various couplings in unoriented Type IIB backgrounds. In section IV we discuss the ${\bf Z}_3$ models. In section V we give the ${\bf Z}_5$ and ${\bf Z}_7$ examples. We give some concluding remarks in section VI.
Orientiworld Framework
======================
In this section we review the orientiworld framework. First we describe the underlying oriented Type IIB orbifold backgrounds. We then consider their orientifolds. Parts of our discussion here will closely [@BKV; @orient1; @orient].
Oriented Backgrounds
--------------------
Consider Type IIB string theory with $N$ parallel coincident D3-branes where the space transverse to the D-branes is ${\cal M}={\bf R}^6/\Gamma$. The orbifold group $\Gamma= \left\{ g_a \mid a=1,\dots,|\Gamma| \right\}$ ($g_1=1$) must be a finite discrete subgroup of $Spin(6)$ (it can be a subgroup of $Spin(6)$ and not $SO(6)$ as we are dealing with a theory containing fermions). If $\Gamma\subset SU(3)$ ($SU(2)$), we have ${\cal N}=1$ (${\cal N}=2$) unbroken supersymmetry, and ${\cal N}=0$, otherwise.
Let us confine our attention to the cases where type IIB on ${\cal M}$ is a modular invariant theory[^3]. The action of the orbifold on the coordinates $X_i$ ($i=1,\dots,6$) on ${\cal M}$ can be described in terms of $SO(6)$ matrices: $g_a:X_i\rightarrow \sum_j (g_a)_{ij} X_j$. (The action of $g_a$ on the world-sheet superpartners of $X_i$ is the same.) We also need to specify the action of the orbifold group on the Chan-Paton charges carried by the D3-branes. It is described by $N\times N$ matrices $\gamma_a$ that form a representation of $\Gamma$. Note that $\gamma_1$ is an identity matrix and ${\mbox {Tr}}(\gamma_1)=N$.
The D-brane sector of the theory is described by an oriented open string theory. In particular, the world-sheet expansion corresponds to summing over oriented Riemann surfaces with arbitrary genus $g$ and arbitrary number of boundaries $b$, where the boundaries of the world-sheet correspond to the D3-branes. In [@BKV] it was shown that the one-loop massless (and, in non-supersymmetric cases, tachyonic) tadpole cancellation conditions require that $$\label{tadpole}
{\mbox {Tr}}(\gamma_a)=0~~~\forall a\not=1~.$$ In [@BKV] it was also shown that this condition implies that the Chan-Paton matrices $\gamma_a$ form an $n$-fold copy of the [*regular*]{} representation of $\Gamma$. The regular representation decomposes into a direct sum of all irreducible representations ${\bf r}_i$ of $\Gamma$ with degeneracy factors $n_i=|{\bf r}_i|$. The gauge group is ($N_i\equiv nn_i$) $$G=\otimes_i U(N_i)~.$$ The matter consists of Weyl fermions and scalars transforming in bifundamentals $({\bf N}_i,{\overline {\bf N}}_j)$ (see [@LNV] for details). The overall center-of-mass $U(1)$, which is inherited from the parent ${\cal N}=4$ supersymmetric $U(N)$ gauge theory and is always present in such models, is free - matter fields are not charged under this $U(1)$. We do, however, have matter charged under the rest of the $U(1)$’s, which we will refer to as non-trivial $U(1)$’s. If $\Gamma\subset SO(3)$, then all the non-trivial $U(1)$’s are anomaly free, in fact, in these cases gauge theories are necessarily non-chiral. As was discussed in [@orient], these $U(1)$’s acquire masses at the one-loop level via couplings to the corresponding twisted R-R two-forms for which there are induced kinetic terms on D3-branes. If $\Gamma \subset SU(3)$ but $\Gamma\not\subset SO(3)$, then we have chiral gauge theories and some of the non-trivial $U(1)$ factors are actually anomalous (in particular, we have $U(1)_k SU(N_l)^2$ mixed anomalies), and are broken (that is, acquire masses) at the tree-level via a generalized Green-Schwarz mechanism [@IRU; @Poppitz; @orient]. If in these cases we also have anomaly-free non-trivial $U(1)$’s, then the latter acquire masses at the one-loop level just as in the $\Gamma\subset SO(3)$ cases [@orient]. So the non-trivial $U(1)$ factors decouple in the infra-red. As to the non-Abelian parts of the gauge theories, it was shown in [@BKV] that they are conformal in the large $N$ limit[^4], including in the non-supersymmetric cases. The key reason for this conformal property is the tadpole cancellation condition (\[tadpole\]), which, as was explained in [@BKV], implies that all planar diagrams[^5] with external lines corresponding to non-Abelian gauge as well as matter fields reduce to those of the parent ${\cal N}=4$ theory, which is conformal. At finite $N$ conformality of the non-Abelian parts of the ${\cal N}=2$ gauge theories becomes evident from vanishing of the one-loop $\beta$-function (as ${\cal N}=2$ gauge theories perturbatively are not renormalized beyond one loop), and can also be argued for ${\cal N}=1$ cases [@BKV]. In non-supersymmetric cases, however, we always have twisted closed string sectors tachyons, which prevent one from considering finite $N$ cases[^6].
Unoriented Backgrounds
----------------------
Let us now consider a generalization of the above setup by including orientifold planes. In the following we will mostly be interested in finite $N$ theories, so let us focus on theories with at least ${\cal N}=1$ unbroken supersymmetry. Thus, consider Type IIB string theory on ${\cal M}={\bf C}^3/\Gamma$ where $\Gamma\subset SU(3)$. Consider the $\Omega (-1)^{F_L}J$ orientifold of this theory, where $\Omega$ is the world-sheet parity reversal, $F_L$ is the fermion number operator, and $J$ is a ${\bf Z}_2$ element ($J^2=1$) acting on the complex coordinates $z_i$ ($i=1,2,3$) on ${\bf C}^3$ such that the set of points in ${\bf C}^3$ fixed under the action of $J$ has real dimension $\Delta=0$ or $4$.
If $\Delta=0$ then we have an orientifold 3-plane. If $\Gamma$ has a ${\bf Z}_2$ subgroup, then we also have an orientifold 7-plane. If $\Delta=4$ then we have an orientifold 7-plane. We may also have an orientifold 3-plane depending on whether $\Gamma$ has an appropriate ${\bf Z}_2$ subgroup. Regardless of whether we have an orientifold 3-plane, we can [*a priori*]{} introduce an arbitrary number of D3-branes[^7]. On the other hand, if we have an orientifold 7-plane we must introduce 8 of the corresponding D7-branes to cancel the corresponding R-R charge appropriately. (The number 8 of D7-branes is required by the corresponding untwisted tadpole cancellation conditions.)
We need to specify the action of $\Gamma$ on the Chan-Paton factors corresponding to the D3- and D7-branes (if the latter are present, which is the case if we have an orientifold 7-plane). Just as in the previous subsection, these are given by Chan-Paton matrices which we collectively refer to as $\gamma^\mu_a$, where the superscript $\mu$ refers to the corresponding D3- or D7-branes. Note that ${\mbox{Tr}}(\gamma^\mu_1)=n^\mu$ where $n^\mu$ is the number of D-branes labelled by $\mu$.
Now the world-sheet expansion contains oriented as well as unoriented Riemann surfaces. The unoriented Riemann surfaces contain handles and boundaries as well as cross-caps. The latter are the (coherent Type IIB) states that describe the familiar orientifold planes. The presence of the cross-caps modifies the twisted one-loop tadpole cancellation conditions, which can now be written as: $$B_a+\sum_\mu C^\mu_a {\mbox{Tr}}(\gamma^\mu_a)=0~,~~~a\not=1~.$$ These should be contrasted with (\[tadpole\]) in the oriented case. In particular, in certain cases some $B_a$, which correspond to contributions due to cross-caps, need not vanish. This makes possible (albeit does not guarantee - see below) non-conformal gauge theories on D3-branes in the orientiworld context [@orient1; @orient2; @orient].
Thus, let us see what kind of orientiworld models we can have. For definiteness let us focus on the cases where we do have an orientifold 3-plane (that is, $\Delta=0$). If there are no orientifold 7-planes (that is, if $\Gamma$ does not contain a ${\bf Z}_2$ element), then the orientifold projection $\Omega$ can be either of the $SO$ or the $Sp$ type: the corresponding orientifold 3-plane is referred to as O3$^-$ or O3$^+$, respectively. That is, before orbifolding, if we place $2N$ D3-branes on top of the O3$^-$-plane (O3$^+$-plane), we have the ${\cal
N}=4$ super-Yang-Mills theory with the $SO(2N)$ ($Sp(2N)$) gauge group[^8]. (We are using the convention where $Sp(2N)$ has rank $N$.) After the orbifold projections the 33 (that is, the D3-brane) gauge group is a subgroup of $SO(2N)$ ($Sp(2N)$), which can contain $U(N_k)$ factors as well as $SO$ ($Sp$) subgroups. The 33 matter can contain bifundamentals in any of these subgroups as well as rank-2 antisymmetric (symmetric) representations in the unitary subgroups. Next, if we have an O7-plane, the orientifold projection $\Omega$ must always be of the $SO$ type on the D7-branes - this is required by the tadpole cancellation condition. This, in particular, implies that the 33 and 77 matter cannot contain rank-2 symmetric representations. Note that we also have 37 matter in bifundamentals of the 33 and 77 gauge groups. Finally, note that there is no overall center-of-mass $U(1)$ in these models (the parent theory is an $SO$ or $Sp$ gauge theory), so all $U(1)$’s (if present) are non-trivial in the sense of the previous subsection.
If $\Gamma\subset SO(3)$, then the corresponding gauge theories are necessarily non-chiral. If we have $U(1)$ factors, they acquire masses at the one-loop level via the mechanism discussed in the previous subsection [@orient]. As to the non-Abelian parts of the corresponding gauge theories, as was discussed in [@orient], they are always conformal despite the fact that some twisted $B_a$ can be non-zero in such models (see [@orient] for a detailed explanation of why this is so)[^9]. The situation is very different in the cases where $\Gamma\subset SU(3)$ but $\Gamma\not\subset SO(3)$. All such theories are non-conformal [@orient]. In fact, generically they are chiral with a few (essentially trivial) exceptions. Thus, in some cases the twisted tadpole cancellation conditions allow a choice such that all twisted Chan-Paton matrices are trivial (that is, they are identity matrices). In such a case the gauge theory is a pure $SO$ or $Sp$ ${\cal N}=1$ super-Yang-Mills theory (that is, we have no chiral matter supermultiplets). In all other cases we have chiral matter. More precisely, there is one possible exception where the gauge group is $SU(4)\otimes U(1)_A$ with matter transforming in ${\bf 6}(+2)$ (the $U(1)_A$ charge is given in parenthesis). Such a theory is conformal as the matter is non-trivially charged under the anomalous $U(1)_A$, but the latter is broken at the tree level via a generalized Green-Schwarz mechanism, and the resulting non-Abelian gauge theory turns out to be non-conformal (as ${\bf 6}$ of $SU(4)$ is a real representation). In general, if we have chiral matter, we have at least one anomalous $U(1)$. Such anomalous $U(1)$’s acquire masses at the tree level [@IRU; @Poppitz; @orient]. If in these cases we also have anomaly-free $U(1)$’s, they acquire masses at the one-loop level as in the $\Gamma\subset
SO(3)$ cases [@orient].
Various Couplings In Orientiworld
=================================
For our subsequent discussions it will be useful to understand some couplings in the orientifold backgrounds. For simplicity we will focus on the cases with O3-planes but without O7-planes. In fact, we will specialize on orbifold groups $\Gamma={\bf Z}_p$ such that $\Gamma\subset SU(3)$ but $\Gamma\not\subset SO(3)$, where $p$ is a prime. The action of the generator $\theta$ or ${\bf Z}_p$ on the complex coordinates $z_\alpha$, $\alpha=1,2,3$, on ${\bf C}^3/\Gamma$ is given by $$\theta z_\alpha=\omega^{\ell_\alpha} z_\alpha~,$$ where $\omega\equiv \exp(2\pi i/p)$, and $\sum_{\alpha=1}^3 \ell_\alpha=p$.
Oriented Backgrounds
--------------------
Before we turn to the unoriented backgrounds, let us recall some facts about the oriented theories. Thus, consider Type IIB on ${\bf R}^{1,3}\times
({\bf C}^3/{\bf Z}_p)$. The closed string sector has $p-1$ twisted sectors $\theta^k$, $k=1,\dots,p-1$. In each twisted sector we have a complex NS-NS scalar $\phi_k$ and a complex R-R two-form $C_k$, which satisfy the reality condition [@orbifold] $$\phi_{p-k}=\phi_k^*~,$$ and similarly for $C_k$.
Next, consider $N$ D3-branes placed at the orbifold fixed point in $({\bf C}^3/{\bf Z}_p)$. Let ${\cal F}$ be the ($N\times N$ matrix valued) D3-brane gauge field strength, which satisfies the orbifold projection $\gamma_k {\cal F}\gamma_k^{-1}={\cal F}$ (recall that we have $N=np$ and $\gamma_k={\rm diag}(I_n, \omega I_n, \omega^2 I_n,\dots,\omega^{p-1}I_n)$, where $I_n$ is the $n\times n$ identity matrix). We have a Chern-Simons coupling of the following form [@Doug; @DM; @DGM]: $$\label{CS}
S_{\rm \small{CS}}={1\over 2\pi\alpha^\prime}\sum_{k=1}^{p-1}
\int_{\rm D3} C_k
\wedge {\rm Tr}\left(\gamma_k ~e^{2\pi\alpha^\prime{\cal F}}\right)~.$$ In particular, the term linear in ${\cal F}$ $$\label{mixing}
\sum_{k=1}^{p-1} \int_{\rm D3} C_k
\wedge {\rm Tr}\left(\gamma_k ~{\cal F}\right)$$ describes the mixing between the $p-1$ twisted two-forms $C_k$ and $p-1$ anomalous $U(1)$’s (note that we have $p$ $U(1)$’s, but one of them is an overall center-of-mass $U(1)$ which does not couple to the twisted two-forms $C_k$). Since the fields $C_k$ have non-vanishing kinetic terms supported at the orbifold fixed point (that is, they propagate in ${\bf
R}^{1,3}$ that coincides with the D3-brane world-volumes), the anomalous $U(1)$’s are actually massive already at the tree level.
Now consider the supersymmetric completion of the couplings (\[mixing\]), which gives the corresponding Fayet-Iliopoulos (FI) couplings: $$S_{\rm {\small FI}}=\sum_{k=1}^{p-1} \int_{\rm D3} \phi_k
{\rm Tr}\left(\gamma_k ~{\cal D}\right)~,$$ where ${\cal D}$ is the (matrix valued) auxiliary field corresponding to ${\cal F}$. The FI term for a given non-trivial $U(1)_j$, $j=1,\dots,p-1$, therefore reads: $$\xi_{{\rm{\small FI}},j}=\sum_{k=1}^{p-1} \phi_k {\rm Tr}\left(\gamma_k ~
\lambda_j\right)~,$$ where $\lambda_j$ is the Chan-Paton matrix corresponding to this $U(1)_j$. Note that at the orbifold point the D-terms give masses to the twisted NS-NS scalars $\phi_k$, which are now part of the massive $U(1)$ gauge supermultiplets.
Next, consider the terms in (\[CS\]) quadratic in ${\cal F}$: $$\pi \alpha^\prime\sum_{k=1}^{p-1} \int_{\rm D3} C_k
\wedge {\rm Tr}\left(\gamma_k ~{\cal F}^2\right)~.$$ The supersymmetric completion of this coupling gives a coupling proportional to $$\label{gauge}
\sum_{k=1}^{p-1} \int_{\rm D3} \phi_k {\rm Tr}
\left(\gamma_k ~{\cal F}^2\right)~.$$ That is, the twisted NS-NS scalars contribute to the gauge couplings, while the twisted R-R scalars (dual to the twisted two-forms) contribute to the corresponding $\theta$-angles (that is, the gauge kinetic function is given by $f=S+f_1$, where $S$ is the untwisted sector dilaton supermultiplet, while $f_1$ is the contribution which depends on the twisted closed string sector moduli) [@LNV; @IRU].
Unoriented Backgrounds
----------------------
Once we add an O3-plane, the above discussion is modified as follows. First, note that the twisted Chan-Paton matrices now have the form: $\gamma_k={\rm diag}(I_{n_0},\omega I_{n_1},\dots, \omega^{p-1} I_{n_{p-1}})$, where $\sum_{k=0}^{p-1} n_k=N$, and the integers $n_k$ satisfy the reality condition $n_{p-k}=n_k$, $k=1,\dots,p-1$, and otherwise are no longer identical but are determined by the twisted tadpole cancellation conditions [@KaSh; @orient1; @orient2; @IRU] (here $\eta=-1$ for the O3-$^{-}$ plane, while $\eta=1$ for the O3$^+$-plane): $${\rm Tr}\left(\gamma_{2k}\right)=-4\eta\prod_{\alpha=1}^3
\left(1+\omega^{k\ell_\alpha}\right)^{-1}~.$$ In particular, some $n_k$ can actually vanish. The gauge group is now $SO(n_0)$ or $Sp(n_0)$ (depending on whether we choose an O3$^-$- or O3$^+$-plane) times $\bigotimes_{k=1}^{(p-1)/2} U(n_k)$ (if any of the $n_k$ vanishes, we simply delete the corresponding subgroup). Second, the orientifold projection removes the real parts of the complex fields $\phi_k$ and $C_k$, while the imaginary parts ${\rm Im}(\phi_k)$ and ${\rm Im}(C_k)$, $k=1,\dots,(p-1)/2$, are combined (after dualizing the two-forms $C_k$ to scalars ${\widetilde\phi}_k$) into $(p-1)/2$ twisted chiral supermultiplets [@DGM; @CVET]. If we actually have an anomalous $U(1)_{n_j}$ factor coming from the $U(n_j)$ subgroup (that is, if the corresponding $n_j\not=0$), then this $U(1)_{n_j}$ becomes massive at the tree level via the Chern-Simons coupling of the corresponding field strength ${\cal F}_{n_j}$ with the fields ${\rm Im}(C_k)$. This can be seen from the part of (\[mixing\]) surviving the orientifold projection (it is not difficult to see that the Chan-Paton matrix $\lambda_{n_j}$ for this $U(1)_{n_j}$ factor is a matrix block-diagonal w.r.t. partitions of $N$ into $n_0,\dots,n_{p-1}$ integers with only non-vanishing entries being $n_j$ 1’s and $n_{p-j}(=n_j)$ $-1$’s): $$-2 \sum_{k=1}^{(p-1)/2}\sum_{j=1}^{(p-1)/2} {\rm Im}\left({\rm Tr}
\left[\gamma_k\lambda_{n_j}\right]\right) \int_{\rm D3} {\rm Im}(C_k) \wedge
{\cal F}_{n_j}~.$$ Similarly, the Fayet-Iliopoulos terms are given by: $$\label{FI}
\xi_{{\rm{\small FI}},n_j}=-2\sum_{k=1}^{(p-1)/2} {\rm Im}(\phi_k)~
{\rm Im}\left({\rm Tr}
\left[\gamma_k\lambda_{n_j}\right]\right)~.$$ These couplings imply that, since we have $m_A=\sum_{j=1}^{(p-1)/2}\left[1-
\delta_{n_j,0}\right]$ anomalous $U(1)$’s, precisely $m_A$ linear combinations of the $(p-1)/2$ chiral superfields (whose lowest components are given by complex scalars $\Phi_k={\rm Im}(\phi_k)+i~{\rm Im}({\widetilde \phi}_k)$) become part of the massive $U(1)$ superfields at the orbifold point.
Finally, let us see what happens to the correction $f_1$ to the gauge kinetic function due to the twisted moduli $\Phi_k$. In the unoriented backgrounds these corrections actually [*vanish*]{} [@CVET]. Thus, consider the part of (\[gauge\]) surviving the orientifold projection. For the $SO(n_0)/Sp(n_0)$ part of the gauge group (if present) the corresponding part of the trace ${\rm Tr}\left(\gamma_k {\cal F}^2\right)$ is always real, so due to the reality condition $\phi_{p-k}=\phi_k^*$ only the real parts of $\phi_k$ can contribute. But it is precisely the real parts of $\phi_k$ that are removed by the orientifold projection. As to the $U(n_j)$ subgroups, note that up to the appropriate normalization factors the corresponding corrections to $f_1$ are the same for the non-Abelian parts $SU(n_j)$ as those for the Abelian parts $U(1)_{n_j}$. The latter are given by $$\sum_{k=1}^{p-1}\sum_{j=1}^{(p-1)/2}{\rm Tr}
\left(\gamma_k\lambda_{n_j}^2\right)
\int_{\rm D3}\phi_k{\cal F}_{n_j}^2~.$$ Note that the traces ${\rm Tr}\left(\gamma_k\lambda_{n_j}^2\right)$ are all real, so only the real parts of $\phi_k$ can contribute. This implies that in the orientifold backgrounds the gauge couplings (and the corresponding $\theta$-angles) do not receive twisted moduli dependent corrections at the tree level[@CVET].
The ${\bf Z}_3$ Models
======================
The simplest choice of the orbifold group $\Gamma$ in the present context is $\Gamma={\bf Z}_3=\{1,\theta,\theta^2\}$, where the generator $\theta$ of ${\bf Z}_3$ acts on the complex coordinates $z_\alpha$, $\alpha=1,2,3$, on ${\bf C}^3$ as follows: $$\theta z_\alpha=\omega z_\alpha~,$$ where $\omega\equiv\exp(2\pi i/3)$. At the origin of ${\bf C}^3/\Gamma$ we can place an O3$^-$- or O3$^+$-plane. Let $\eta=-1$ in the former case, while $\eta=+1$ in the latter case. The twisted tadpole cancellation requires that [@orient1] $${\rm Tr}(\gamma_\theta)=4\eta~.$$ We can then place $(3N+4\eta)$ D3-branes on top of the O3-plane, and choose $$\gamma_\theta={\rm diag}(\omega I_N,\omega^{-1} I_N, I_{N+4\eta})~.$$ The twisted closed string sector gives rise to a single massless chiral supermultiplet corresponding to the orbifold blow-up mode. The massless open string spectrum gives rise to the gauge theory on the D3-branes. For $\eta=1$ and $N=0$ we have pure $Sp(4)$ super-Yang-Mills theory[^10]. For $\eta=1$ and $N\in 2{\bf N}$ we have $SU(N)\otimes Sp(N+4)\otimes
U(1)_A$ gauge theory with matter in the chiral supermultiplets $\Phi_\alpha=3\times ({\bf S},{\bf 1})(+2)$ and $Q_\alpha=3\times ({\overline {\bf N}},
{\bf N+4})(-1)$, where the anomalous $U(1)_A$ charges are given in parentheses, and ${\bf S}$ stands for the two-index $N(N+1)/2$ dimensional symmetric representation of $SU(N)$. For $\eta=-1$ and $N=4$ we have $SU(4)\otimes U(1)_A$ gauge theory with matter in the chiral supermultiplets $\Phi_\alpha=3\times {\bf 6}(+2)$. For $\eta=-1$ and $N=5$ we have $SU(5)\otimes
U(1)_A$ gauge theory with matter in the chiral supermultiplets $\Phi_\alpha=3\times {\bf 10}(+2)$ and ${\bf Q}_\alpha=3\times
{\overline {\bf 5}}(-1)$. For $\eta=-1$ and $N\geq 6$ we have $SU(N)\otimes SO(N-4)\otimes
U(1)_A$ gauge theory with matter in the chiral supermultiplets $\Phi_\alpha=3\times ({\bf A},{\bf 1})(+2)$ and ${\bf Q}_\alpha=3\times
({\overline {\bf N}},
{\bf N-4})(-1)$, where ${\bf A}$ stands for the two-index $N(N-1)/2$ dimensional antisymmetric representation of $SU(N)$. (Note that in the $\eta=-1$ and $N=6$ case the $SO$ part of the gauge group is actually Abelian.) In the cases where we have the $Q_\alpha$ matter, we have the following tree-level superpotential: $$\label{tree}
{\cal W}_{\rm{\small tree}}=y \epsilon_{\alpha\beta\gamma}
\Phi_\alpha Q_\beta Q_\gamma+\dots~,$$ where $y$ is the corresponding Yukawa coupling, and the ellipses stand for non-renormalizable couplings. Note that at the renormalizable level we have an $SO(3)$ global symmetry[^11], and the subscript $\alpha$ in $\Phi_\alpha$ and $Q_\alpha$ corresponds to the triplet of $SO(3)$.
The $Sp$ Theories
-----------------
Some examples (more precisely, their compact versions) of the above theories with $\eta=-1$ (that is, with the $SO$ orientifold projection) were discussed in [@Sagnotti; @KaSh; @KST; @LPT; @CVET]. Here we will focus on the theories with $\eta=1$ (that is, with the $Sp$ orientifold projection). Let us note that the $\eta=1$ models are actually simpler to discuss then their $\eta=-1$ counterparts. This is because in the $\eta=-1$ cases the $SU$ part of the gauge theory is asymptotically free (while the $SO$ part is not asymptotically free), so in the infra-red we have to deal with a strongly coupled chiral gauge theory[^12]. In contrast, in the $\eta=1$ case it is the $SU$ part that is not asymptotically free, while the $Sp$ part is, so in the infra-red the non-perturbative dynamics reduces to that of a strongly coupled $Sp(N_c=N+4)$ gauge theory with $N_f=3N/2$ flavors (by one flavor of $Sp(N_c)$ we mean a pair of chiral supermultiplets in $N_c$ of $Sp(N_c)$). And these theories are well understood [@IP].
First, consider the model with $\eta=1$ and $N=0$. The gauge theory on the D3-branes is pure $Sp(4)$ super-Yang-Mills theory. Note that the twisted closed string sector gives rise to a single chiral supermultiplet, which plays no role in the gauge dynamics (this follows from our discussion in the previous section). Non-perturbatively we have gaugino condensate in $Sp(4)$ with confinement and chiral symmetry breaking. This model is interesting as it provides a simple setup where one gets pure ${\cal N}=1$ superglue on D3-branes. In particular, it would be interesting to use this setup to identify BPS domain walls in the $Sp(4)$ super-Yang-Mills theory, but this is outside of the scope of this paper.
Let us now discuss the $\eta=1$ and $N\in 2{\bf N}$ models. For presentation purposes we will discuss the $N=2$ model after we discuss all the other cases.
[*The $SU(4)\otimes Sp(8)\otimes U(1)_A$ Model*]{}
This is the $\eta=1$ and $N=4$ model. The gauge group is $SU(4)\otimes Sp(8)\otimes U(1)_A$, the chiral matter is given by $\Phi_\alpha=3\times ({\bf 10},{\bf 1})(+2)$ and $Q_\alpha=3\times ({\overline {\bf 4}},
{\bf 8})(-1)$, and the tree-level superpotential is given by (\[tree\]). As we have already mentioned, the $SU(4)$ gauge coupling is weak in the infra-red, while the $Sp(8)$ gauge coupling becomes strong. The low energy degrees of freedom are given by mesons (note that there are no baryons in $Sp$ theories [@IP]) $$\begin{aligned}
&&{\cal M}_{[\alpha\beta]}=3\times({\overline {\bf 10}},{\bf 1})(-2)~,\\
&&{\cal M}_{\{\alpha\beta\}}=6\times({\overline {\bf 6}},{\bf 1})(-2)~.\end{aligned}$$ (Note that for $SU(4)$ we actually have ${\overline {\bf 6}}={\bf 6}$.) There is no non-perturbative superpotential in this theory, and at the origin of the meson moduli space we have confinement without chiral symmetry breaking [@IP]. Note that due to the tree-level superpotential (\[tree\]) the mesons ${\cal M}_{[\alpha\beta]}$ pair up with the fields $\Phi_\alpha$ and acquire masses: $${\cal W}_{\rm{\small tree}}=y\epsilon_{\alpha\beta\gamma}\Phi_\alpha
{\cal M}_{[\beta\gamma]}~.$$ So at low energies we have the $SU(4)$ gauge theory with matter chiral supermultiplets in $P_I=6\times {\overline {\bf 6}}(-2)$, $I=1,\dots,6$. Actually, so far we have been ignoring the anomalous $U(1)_A$. The corresponding D-term is given by $$D=-2P^2+\xi_{\rm{\small FI}}~.$$ From (\[FI\]) it follows that $\xi_{\rm{\small FI}}$ is negative for positive values of ${\rm Im}(\phi_1)$ (which corresponds to the size of the blow-up). This implies that all the fields $P_I$ as well as ${\rm Im}(\phi_1)$ have vanishing expectation values. That is, the blow-up mode is frozen and the orbifold [*cannot*]{} be blown up in this model. As we discussed in the previous section, at the orbifold point the twisted chiral superfield becomes part of the massive $U(1)_A$ gauge supermultiplet. The low energy theory is therefore $SO(6)$ gauge theory with 6 vectors. Note that this theory is not infra-red free. However, it has a dual magnetic description [@IS] in terms of $SO(4)\sim SU(2)_L\otimes SU(2)_R$ gauge theory with 6 flavors of quarks $q^I$ ($I=1,\dots,6$) in the $4\sim (2,2)$ dimensional representation along with $6(6+1)/2=21$ gauge singlets $M_{IJ}$ and the superpotential $${\cal W}_{\rm{\small magnetic}}=M_{IJ}q^I q^J~.$$ Note that the magnetic theory is actually free in the infra-red.
[*The $SU(N)\otimes Sp(N+4)\otimes U(1)_A$ Models with $N>4$*]{}
Before we discuss the $\eta=1$ and $N=6$ model, we would like to discuss the $\eta=1$ and $N>6$ models. In this case the $Sp(N+4)$ theory has a dual magnetic description (albeit the magnetic theory is also strongly coupled) [@IP]. In this dual picture the gauge group is $SU(N)\otimes Sp(2N-8)
\otimes U(1)_A$, the matter is given by $$\begin{aligned}
&&\Phi_\alpha=3\times ({\bf S},{\bf 1})(+2)~,\\
&&{\cal M}_{[\alpha\beta]}=3\times ({\overline {\bf S}},{\bf 1})(-2)~,\\
&&{\cal M}_{\{\alpha\beta\}}=6\times ({\overline{\bf A}},{\bf 1})(-2)~,\\
&&q_\alpha=3\times ({\bf N},{\bf 2N-8})(+1)~, \end{aligned}$$ and the superpotential is given by $${\cal W}_{\rm{\small magnetic}}=y\epsilon_{\alpha\beta\gamma}\Phi_\alpha
{\cal M}_{[\beta\gamma]}+{\cal M}_{\{\alpha\beta\}}q_\alpha q_\beta~.$$ The mesons ${\cal M}_{[\alpha\beta]}$ pair up with the fields $\Phi_\alpha$ and acquire masses, so at low energies we have the mesons ${\cal M}_{\{\alpha\beta\}}$ and the quarks $q_\alpha$. As we have already mentioned, the $Sp(2N-8)$ gauge coupling is strong in the infra-red. On the other hand, the $SU(N)$ gauge coupling remains weak (even after integrating out the fields ${\cal M}_{[\alpha\beta]}$ and $\Phi_\alpha$). Moreover, the orbifold blow-up mode is frozen in this model. This can be seen as follows. The $U(1)_A$ D-term reads: $$D=-2M_{\{\alpha\beta\}}^2 +q^2+\xi_{\rm{\small FI}}~,$$ where $\xi_{\rm{\small FI}}<0$ for positive ${\rm Im}(\phi_1)$. We must also ensure D-flatness for the $SU$ and $Sp$ subgroups. The D-flat directions are in one-to-one correspondence with chiral gauge invariant operators. As far as the $Sp$ part is concerned, such operators can only contain the following combinations: $$\begin{aligned}
&&\Sigma_{[\alpha\beta]}=
q_{[\alpha}q_{\beta]}=3\times ({\bf S},{\bf 1})(+2)~,\\
&&\Sigma_{\{\alpha\beta\}}=q_{\{\alpha}q_{\beta\}}=
6\times ({\bf A},{\bf 1})(+2)~.\end{aligned}$$ Note, however, that $\Sigma_{[\alpha\beta]}$ cannot enter as we cannot construct $SU(N)$ gauge invariant operators from $\Sigma_{[\alpha\beta]}$, $\Sigma_{\{\alpha\beta\}}$ and $M_{\{\alpha\beta\}}$. On the other hand, the F-flatness conditions imply that $\Sigma_{\{\alpha\beta\}}=0$. It then follows that the FI term must vanish along with the vacuum expectation values of ${\cal M}_{\{\alpha\beta\}}$ and $q_\alpha$.
Let us now discuss the $N=6$ case. The above discussion is essentially unmodified in this case except that in the dual magnetic theory the $Sp(4)$ subgroup is actually weakly coupled. As to the $SU(6)$ gauge theory, its one-loop $\beta$-function coefficient vanishes (after integrating out the the fields ${\cal M}_{[\alpha\beta]}$ and $\Phi_\alpha$), but it is still free in the infra-red.
[*The $SU(2)\otimes Sp(6)\otimes U(1)_A$ Model*]{}
This is the $\eta=1$ and $N=2$ model. This model is interesting as supersymmetry is dynamically broken in this model. The gauge group is $SU(2)\otimes Sp(6)\otimes U(1)_A$, the chiral matter is given by $\Phi_\alpha=3\times ({\bf 3},{\bf 1})(+2)$ and $Q_\alpha=3\times ({\bf 2},
{\bf 6})(-1)$, and the tree-level superpotential is given by (\[tree\]). The $U(1)_A$ D-term is given by $$D=2\Phi^2-Q^2+\xi_{\rm{\small FI}}~,$$ where $\xi_{\rm{\small FI}}$ is negative for positive values of ${\rm Im}(\phi_1)$.
First consider the case where $\Phi_\alpha=0$. Then the above D-term vanishes only if $Q_\alpha=0$ and ${\rm Im}(\phi_1)=0$. The $SU(2)$ gauge coupling is weak in the infra-red, so as far as the $Sp$ part of the gauge theory is concerned we have $Sp(N_c=6)$ with $N_f=3$ flavors. In general, the $Sp(N_c)$ theory with $N_f\leq N_c/2$ flavors (that is, $2N_f$ fields $Q_i$, $i=1,\dots,2N_f$, in $N_c$ of $Sp(N_c)$) has a dynamically generated superpotential [@IP]: $${\cal W}_{\rm{\small non-pert}}\sim
\left({\Lambda^{3(N_c/2+1)-N_f}\over {\rm Pf}(M)}\right)^{1/(N_c/2+1-N_f)}~,$$ where $M_{ij}=-M_{ji}=Q_iQ_j$ are the meson fields, and $\Lambda$ is the dynamically generated scale. For $N_f=N_c/2$ the gauge group is completely broken for ${\rm Pf}(M)\not=0$, and this superpotential is generated by an instanton in the broken $Sp(N_c)$. For $N_f<N_c/2$ the superpotential is associated with gaugino condensation in the unbroken $Sp(N_c-2N_f)$. In particular, for $0<N_f\leq N_c/2$ the above superpotential has a runaway behavior w.r.t. the vacuum expectation values of $M_{ij}$ (that is, there is no supersymmetric vacuum for finite values of $M_{ij}$).
In our case the mesons are ${\cal M}_{[\alpha\beta]}=3\times({\bf 3},
{\bf 1})(-2)$ and ${\cal M}_{\{\alpha\beta\}}=6\times({\bf 1},
{\bf 1})(-2)$. At first it might seem that we have global supersymmetry breaking as the $U(1)_A$ D-term grows with non-vanishing vacuum expectation values of the mesons. However, recall that we have assumed that $\Phi_\alpha=0$. But there is nothing stopping $\Phi_\alpha$ from being non-vanishing. Suppose some or all $\Phi_\alpha\not=0$. Then the D-term can [*a priori*]{} be set to zero. Nonetheless, we still have no supersymmetric vacuum. Thus, in this case due to the tree-level superpotential (\[tree\]) two of the fields $Q_\alpha$ acquire masses (this is independent of a particular configuration of the vacuum expectation values of $\Phi_\alpha$ as long as at least one of the fields $\Phi_\alpha\not=0$). So as far as the $Sp(6)$ part of the gauge theory is concerned, at low energies we have $Sp(6)$ with one flavor. As we already mentioned above, in this theory we have a dynamically generated runaway (in the corresponding meson field) superpotential with no supersymmetric vacuum. The $D$-term, however, can now vanish, so there is no stable vacuum with broken global supersymmetry.
Even so, as was discussed in detail in [@local], generically we do expect to have a stable vacuum with broken [*local*]{} supersymmetry. In particular, if in the context of global supersymmetry we have a runaway superpotential with the runaway directions corresponding to charged matter fields, then in the context of supergravity the runaway directions are generically expected to be stabilized due to contributions coming from the K[ä]{}hler potential (see [@local] for details). Note that this mechanism is four-dimensional, and we indeed have four-dimensional supergravity on D3-branes via the mechanism of [@DG].
Comments on the $\eta=-1$ Models
--------------------------------
In this subsection we would like to comment on some properties of the ${\bf Z}_3$ models with $\eta=-1$, that is, those where we have an O3$^-$-plane. As we have already mentioned, compact versions of some of these models were discussed in [@Sagnotti; @KaSh; @KST; @LPT; @CVET].
[*The $SU(4)\otimes U(1)_A$ Model*]{}
This is the $\eta=-1$ and $N=4$ model. The gauge group is $SU(4)\otimes
U(1)_A$, the chiral matter is given by $\Phi_\alpha=3\times {\bf 6}(+2)$, and there is no tree-level superpotential. The $U(1)_A$ D-term is given by $$D=2\Phi^2+\xi_{\rm{\small FI}}~,$$ where $\xi_{\rm{\small FI}}$ is negative for positive values of ${\rm Im}(\phi_1)$.
The $SU(4)$ gauge theory is strongly coupled in the infra-red. We can view this theory as $SO(6)$ with 3 vectors. In this theory there are two distinct branches [@IS]. The first branch has a dynamically generated runaway superpotential in the vacuum expectation values of the mesons ${\cal M}_{\{\alpha\beta\}}=6\times {\bf 1}(+2)$. On this branch supersymmetry is broken via the mechanism mentioned at the end of the previous subsection. The second branch has a vanishing superpotential, so supersymmetry is intact. Note that in this model the orbifold blow-up mode can be non-zero. In fact, on the first branch the non-supersymmetric vacuum has a non-zero blow-up mode, while on the second branch it depends on the mesons ${\cal M}_{\{\alpha\beta\}}$ (in both cases the blow-up mode is fixed from the requirement that the $U(1)_A$ D-term vanish).
[*The $SU(5)\otimes U(1)_A$ Model*]{}
This is the $\eta=-1$ and $N=5$ model. The gauge group is $SU(5)\otimes
U(1)_A$, the chiral matter is given by $\Phi_\alpha=3\times {\bf 10}
(+2)$ and $Q_\alpha=3\times {\overline {\bf 5}}(-1)$, and the tree-level superpotential is given by (\[tree\]). The $U(1)_A$ D-term in this model is given by $$D=2\Phi^2-Q^2+\xi_{\rm{\small FI}}~,$$ where $\xi_{\rm{\small FI}}$ is negative for positive values of ${\rm Im}(\phi_1)$.
To analyze the gauge dynamics in this model, let us first consider the model with the gauge group $SU(5)$, chiral matter in $\Phi_\alpha=
3\times {\bf 15}$ and $Q_\alpha=3\times {\overline {\bf 5}}$, and [*no*]{} tree-level superpotential. This theory is $s$-confining [@schmaltz][^13]. A simple way of understanding this is as follows. Consider the following gauge invariant operator: $$\Sigma_{\{\alpha\beta\}}\equiv\Phi_{\{\alpha}\Phi_{\beta\}}
\epsilon_{\gamma\delta\eta}\Phi_\gamma\Phi_\delta\Phi_\eta=6\times {\bf 1}~.$$ So we have a D-flat direction corresponding to turning on non-vanishing vacuum expectation values of $\Phi_\alpha$. The original $SU(5)$ gauge group can be broken down to $SU(2)$ along this direction. To see this, consider the branching of ${\bf 10}$ of $SU(5)$ under the breaking $SU(5)\supset
SU(3)\otimes SU(2)\otimes U(1)$: $$\begin{aligned}
&&{\bf 5}=({\bf 3},{\bf 1})(-2)+({\bf 1},{\bf 2})(+3)~,\\
&&{\bf 10}=({\bf 1},{\bf 1})(+6)+({\overline {\bf 3}},{\bf 1})(-4)+
({\bf 3},{\bf 2})(+1)~. \end{aligned}$$ Now let us turn on non-zero vacuum expectation values for $({\bf 1},{\bf 1})(+6)$ in all three $\Phi_\alpha$, and also for $({\overline {\bf 3}},{\bf 1})(-4)$ in at least one of the $\Phi_\alpha$. This is consistent with the flat directions $\Sigma_{\{\alpha\beta\}}$. The gauge group is broken down to $SU(2)$, and the left-over charged matter consists of three ${\bf 2}$’s coming from $\Phi_\alpha$ as well as three ${\bf 2}$’s coming from $Q_\alpha$. That is, we have $SU(2)$ with 3 flavors, which is $s$-confining.
Let us now include the tree-level superpotential (\[tree\]). Then two of the three ${\bf 2}$’s coming from $Q_\alpha$ acquire masses, and we have $SU(2)$ with 2 flavors. In this theory we have quantum modification of the moduli space [@Seiberg], but supersymmetry is unbroken[^14]. Finally, note that the anomalous $U(1)_A$ D-term can also be canceled in this model.
[*The $SU(6)\otimes SO(2)\otimes U(1)_A$ Model*]{}
This is the $\eta=-1$ and $N=6$ model. The gauge group is $SU(6)\otimes
SO(2)\otimes
U(1)_A$, the chiral matter is given by $\Phi_\alpha=3\times ({\bf 15},{\bf 1})
(+2)$ and $Q_\alpha=3\times ({\overline {\bf 6}},{\bf 2})(-1)$, and the tree-level superpotential is given by (\[tree\]). (Note that $SO(2)\sim
U(1)$, and the doublet ${\bf 2}$ of $SO(2)$ refers to the states with opposite $U(1)$ charges.) The $U(1)_A$ D-term is given by $$D=2\Phi^2-Q^2+\xi_{\rm{\small FI}}~,$$ where $\xi_{\rm{\small FI}}$ is negative for positive values of ${\rm Im}(\phi_1)$. Note that $$\Sigma\equiv\epsilon_{\alpha\beta\gamma}\Phi_\alpha\Phi_\beta\Phi_\gamma=
({\bf 1},{\bf 1})(+6)~.$$ Thus, $\Sigma$ is a chiral gauge invariant operator w.r.t. $SU(6)\otimes
SO(2)$. So we have a D-flat direction, which is also F-flat, corresponding to turning on non-vanishing expectation values of $\Phi_\alpha$ (the $U(1)_A$ D-term can be canceled by appropriately turning on ${\rm Im}(\phi_1)$). It is not difficult to see that at generic points along this flat direction the $SU(6)$ subgroup is broken down to a $U(1)$, so the resulting gauge group is $U(1)\otimes SO(2)\sim
U(1)\otimes U(1)$. In the process of this Higgsing two of the original three $Q_\alpha$ fields become massive, and we have total of 12 chiral supermultiplets charged under this $U(1)\otimes U(1)$. These supermultiplets can be used to Higgs the remaining Abelian gauge group completely.
The above discussion suggests that there is no dynamically generated superpotential in this model. Another way of arriving at the same conclusion is as follows. Consider giving vacuum expectation values to the fields $\Phi_\alpha$ so that $\Phi_1$ breaks $SU(6)$ down to $Sp(6)$, $\Phi_2$ breaks $Sp(6)$ down to $Sp(4)\otimes Sp(2)$, and finally $\Phi_3$ breaks $Sp(4)\otimes Sp(2)$ down to $Sp(2)\otimes Sp(2)\otimes Sp(2)$ (this Higgsing is consistent with the flat direction $\Sigma$). The resulting gauge group is $SU(2)\otimes SU(2)\otimes SU(2)\otimes SO(2)$ (the anomalous $U(1)_A$ is not shown), and the charged chiral matter is given by $({\bf 2},{\bf 2},{\bf 1},{\bf 1})$, $({\bf 2},{\bf 1},{\bf 2},{\bf 1})$, $({\bf 1},{\bf 2},{\bf 2},{\bf 1})$, $({\bf 2},{\bf 1},{\bf 1},{\bf 2})$, $({\bf 1},{\bf 2},{\bf 1},{\bf 2})$, $({\bf 1},{\bf 1},{\bf 2},{\bf 2})$. We can further break the gauge group down to $SU(2)\otimes U(1)$ by giving non-vanishing expectation values to the last two fields. The resulting matter is given by $\chi^{\pm}_i={\bf 2}(\pm q_i)$, $i=1,2,3$, where $\pm q_i$ are the $U(1)$ charges. As far as the $SU(2)$ subgroup is concerned, we have $SU(2)$ with three flavors of quarks in the fundamental of $SU(2)$. In this theory we have confinement without chiral symmetry breaking (at the origin of the moduli space), and there is no non-perturbative superpotential [@Seiberg]. This suggests that the $SU(6)\otimes SO(2)$ theory with the chiral matter $\Phi_\alpha$ and $Q_\alpha$ [*and*]{} the tree-level superpotential (\[tree\]) is an $s$-confining theory. (Such theories [*without*]{} a tree-level superpotential were classified in [@schmaltz].)
[*The $SU(N)\otimes SO(N-4)\otimes U(1)_A$ Models with $N\geq 7$*]{}
These are the $\eta=-1$ and $N\geq 7$ models. The $SO(N-4)$ gauge invariant operators are given by mesons $$\begin{aligned}
&&{\cal M}_{[\alpha\beta]}=3\times ({\overline {\bf A}},{\bf 1})(-2)~,\\
&&{\cal M}_{\{\alpha\beta\}}=6\times ({\overline {\bf S}},{\bf 1})(-2)~, \end{aligned}$$ and baryons $$\left({\cal B}_{\alpha_1\dots\alpha_{N-4}}\right)_{A_1\dots A_{N-4}}=
Q_{\alpha_1 A_1 i_1}\cdots Q_{\alpha_{N-4} A_{N-4} i_{N-4}}
\epsilon_{i_1\dots i_{N-4}}~,$$ where $A_m$ is the ${\overline {\bf N}}$ index, while $i_m$ is the ${\bf N-4}$ index. (Note that the $U(1)_A$ charge of the baryon operators is $-(N-4)$.) There are no $SU(N)$ gauge invariant operators involving ${\cal M}_{\{\alpha\beta\}}$. So all the $SU(N)$ gauge invariant operators must be constructed from $\Phi_\alpha$, $\Theta_\alpha\equiv\epsilon_{\alpha
\beta\gamma}{\cal M}_{[\beta\gamma]}$ and the baryons ${\cal B}_{\alpha_1\dots\alpha_{N-4}}$. If $N$ is odd we have no gauge invariant operators involving totally antisymmetrized products of only $\Phi_\alpha$ or only $\Theta_\alpha$. On the other hand, if $N$ is even we have no such operators as $N/2>3$ (and the index $\alpha$ takes only three values). So the building blocks for the gauge invariant operators must be $\Phi_\alpha\Theta_\beta$ and $\Phi_{\alpha_1}\cdots\Phi_{\alpha_n}{\cal B}_{\beta_1\dots\beta_{N-4}}$, where $n=(N-4)/2$ if $N$ is even, and $n=N-2$ is $N$ is odd. This implies that if $N$ is even all gauge invariant operators have zero $U(1)_A$ charge. On the other hand, if $N$ is odd the gauge invariant operators must have $U(1)_A$ charges which are positive multiples of $N$.
Now, if $\Lambda$ is the dynamically generated scale of the $SU(N)$ theory, then $\Lambda^{\beta_0}$ ($\beta_0=3N-{1\over 2}\times 3 \times (N-2)-
{1\over 2}\times 3\times (N-4)=9$ is the one-loop $\beta$-function coefficient for $SU(N)$) has the $U(1)_A$ charge $$q_A=(+2)\times{N(N-1)\over 2}+(-1)\times N\times (N-4)=3N~.$$ This together with the above arguments indicates that in theories with even $N$ we cannot have a non-perturbative superpotential (as the superpotential must have vanishing $U(1)_A$ charge). For odd $N$, however, this argument does not rule out a non-perturbative superpotential.
To understand the odd $N$ cases in more detail, let us use the standard $U(1)_R$ symmetry arguments. We will assign $+1$ $U(1)_R$ charges to the gauginos of the $SU(N)$ as well as $SO(N-4)$ gauge groups, and $U(1)_R$ charges $q_\Phi$ and $q_Q$ to the fields $\Phi_\alpha$ respectively $Q_\alpha$. Then the requirement that the $U(1)_R SU(N)^2$ and $U(1)_R SO(N-4)^2$ anomalies vanish gives the following values: $$\begin{aligned}
&&q_\Phi={{2N-12}\over 3N}~,\\
&&q_Q={{2N+6}\over 3N}~.\end{aligned}$$ This implies that the $(U(1)_A,U(1)_R)$ charges read: $$\begin{aligned}
&&\Lambda^3:~~~(N,0)~,\\
&&\Phi_\alpha\Theta_\beta:~~~(0,2)~,\\
&&\Phi_{\alpha_1}\cdots\Phi_{\alpha_{N-2}}
{\cal B}_{\beta_1\dots\beta_{N-4}}:~~~(N,2(2N-9)/3)~.\end{aligned}$$ The superpotential must have $U(1)_R$ charge $+2$. We do have combinations with $U(1)_R$ charge $+2$ (and $U(1)_A$ charge 0), which can schematically be written as $$\left({\Phi_{\alpha_1}\cdots\Phi_{\alpha_{N-2}}
{\cal B}_{\beta_1\dots\beta_{N-4}}\over \Lambda^3}\right)^{3\over{2N-9}}=
\left({\left[\Phi_{\alpha_1}\cdots\Phi_{\alpha_{N-2}}
{\cal B}_{\beta_1\dots\beta_{N-4}}\right]^3\over
\Lambda^9}\right)^{1\over{2N-9}}~.$$ These combinations, however, are not holomorphic in the gauge invariant operators for odd $N\geq 7$ (in particular, we have a branch point at the origin). We therefore conclude that for these values of $N$ we do not have a non-perturbative superpotential either.
Next, recall that the tree-level superpotential is given by $${\cal W}_{\rm{\small tree}}=y\Phi_\alpha\Theta_\alpha~.$$ The F-flatness conditions then imply that $\Theta_\alpha=0$ and $\epsilon_{\alpha\beta\gamma}\Phi_{\beta[A_1A_2]} Q_{\gamma A_2 i}=0$. In particular, in all cases at hand the gauge invariant operators involving the $SO(N-4)$ mesons $\Theta_\alpha$ must vanish.
On the other hand, in the odd $N$ cases the gauge invariant operators involving the $SO(N-4)$ baryons can be written as $$\left(\Phi_{\alpha_1}\Phi_{\alpha_2}\right)
\left(\left(\Phi_{\beta_1}Q_{\gamma_1 i_1}\right)\cdots
\left(\Phi_{\beta_{N-4}}Q_{\gamma_{N-4} i_{N-4}}\right)\right)
\epsilon_{i_1\dots i_{N-4}}~.$$ Due to the aforementioned F-flatness conditions the $\beta_m$ and $\gamma_m$ indices must be symmetrized pairwise. Each such symmetrization gives 6 of the $SO(3)$ global symmetry (note that $6=5+1$). So we have $(N-4)$ 6’s of $SO(3)$ completely antisymmetrized. However, at most 3 6’s of $SO(3)$ can be completely antisymmetrized without vanishing. This implies that for $N>7$ the above gauge invariant operators should vanish to be compatible with the F-flatness conditions, and the blow-up mode is frozen at its vanishing value. So in the odd $N>7$ cases the $SO(N-4)$ gauge subgroup is unbroken, and therefore so is the $SU(N)$ gauge subgroup.
A similar analysis can be performed in the even $N$ cases. Here the gauge invariant operators involving the $SO(N-4)$ baryons can be written as $$\left(Q_{\alpha_1 i_1}\cdots Q_{\alpha_{(N-4)/2} i_{(N-4)/2}}\right)
\left(\left(\Phi_{\beta_1}Q_{\gamma_1 j_1}\right)\cdots
\left(\Phi_{\beta_{(N-4)/2}}Q_{\gamma_{(N-4)/2} j_{(N-4)/2}}\right)\right)
\epsilon_{i_1\dots i_{(N-4)/2}j_1\dots j_{(N-4)/2}}~.$$ In this case we therefore have $(N-4)/2$ 6’s of $SO(3)$ completely antisymmetrized. This implies that for $N>10$ these gauge invariant operators must vanish to be compatible with the F-flatness conditions, and the blow-up mode is frozen at its vanishing value. So in the even $N>10$ cases the $SO(N-4)$ gauge subgroup is unbroken, and therefore so is the $SU(N)$ gauge subgroup.
Now, in the $N=7$ and $N=10$ cases we have 3 6’s of $SO(3)$ completely antisymmetrized. Since 6 is reducible ($6=5+1$), and a totally antisymmetric product of 3 5’s vanishes, we have $1\cdot 5 \cdot 5$ with the two 5’s antisymmetrized. In particular, the singlet $\Phi_\alpha Q_\alpha$ must be non-vanishing, that is, for at least one value of $\alpha$ we must have $\Phi_\alpha\not=0$ and $Q_\alpha\not=0$. Without loss of generality we can choose $\Phi_1\not=0$ and $Q_1\not=0$. The F-flatness conditions then imply that either $\Phi_2=Q_2=0$ or $\Phi_3=Q_3=0$. But then the gauge invariant operators containing $SO(N-4)$ baryons also vanish in the $N=7$ and $N=10$ cases.
Finally, consider the $N=8$ case, where we have 2 6’s of $SO(3)$ antisymmetrized. In this case the relevant products are $1\cdot 5$ and $5\cdot 5$ (the latter is antisymmetrized). So in this case the above argument does not apply. However, for $N=8$ we have $SO(N-4)=SO(4)\sim
SU(2)_L\otimes SU(2)_R$, and the matter is $$\begin{aligned}
&&\Phi_\alpha=3\times ({\bf 28},{\bf 1},{\bf 1})(+2)~,\\
&&L_\alpha=3\times ({\overline {\bf 8}},{\bf 2},{\bf 1})(-1)~,\\
&&R_\alpha=3\times ({\overline {\bf 8}},{\bf 1},{\bf 2})(-1)~.\end{aligned}$$ The basic $SU(2)_L\otimes SU(2)_R$ gauge invariant operators are $$\begin{aligned}
&&{\cal L}_\alpha=\epsilon_{\alpha\beta\gamma}L_\beta L_\gamma=3\times
({\overline {\bf 28}},{\bf 1},{\bf 1})(-2)~,\\
&&{\cal L}_{\{\alpha\beta\}}=L_{\{\alpha} L_{\beta\}}=6\times
({\overline {\bf 36}},{\bf 1},{\bf 1})(-2)~,\\
&&{\cal R}_\alpha=\epsilon_{\alpha\beta\gamma}R_\beta R_\gamma=3\times
({\overline {\bf 28}},{\bf 1},{\bf 1})(-2)~,\\
&&{\cal R}_{\{\alpha\beta\}}=R_{\{\alpha} R_{\beta\}}=6\times
({\overline {\bf 36}},{\bf 1},{\bf 1})(-2)~\end{aligned}$$ The mesons ${\cal L}_{\{\alpha\beta\}}$ and ${\cal R}_{\{\alpha\beta\}}$ cannot enter the $SU(8)$ gauge invariant operators, while the mesons ${\cal L}_\alpha={\cal R}_\alpha=0$ due to the F-flatness conditions.
The above arguments indicate that in the $N\geq 7$ models the gauge group is unbroken, and the blow-up mode is zero (so that the $U(1)_A$ D-term vanishes). At low energies the theory flows into an interacting fixed point. As was pointed out in [@orient1], in the large $N$ limit this superconformal field theory is actually the same as the parent ${\cal N}=4$ supersymmetric $SO(3N-4)$ gauge theory (which, in turn, in the large $N$ limit is the same as the parent ${\cal N}=4$ supersymmetric $SU(3N-4)$ gauge theory).
Other Examples
==============
In this section we would like to briefly mention other interesting ${\bf Z}_p$ examples. In particular, we will discuss a ${\bf Z}_5$ example and a ${\bf Z}_7$ example.
A ${\bf Z}_5$ Example
---------------------
Consider the ${\bf Z}_5$ orbifold group whose generator $\theta$ has the following action on the complex coordinates $z_\alpha$ ($\omega\equiv
\exp(2\pi i/5)$): $$\theta z_{1,2}=\omega z_{1,2}~,~~~\theta z_3=\omega^3 z_3~.$$ The tadpole cancellation conditions have the following solution (see subsection B of section III): $${\rm Tr}\left(\gamma_{\theta^2}\right)=-4\eta
{1\over (1+\omega)^2(1+\omega^3)}=-4\eta(\omega+
\omega^4)~.$$ This then implies that $$\gamma_\theta={\rm diag}(I_N,\omega I_N,\omega^2 I_{N-4\eta},\omega^3
I_{N-4\eta},\omega^4 I_N)~.$$ In particular, consider the $\eta=-1$ and $N=0$ case. We then have $SU(4)\otimes U(1)_A$ gauge group with a chiral supermultiplet in ${\bf 6}(+2)$. As far as the non-Abelian part of the gauge group is concerned, we can view this model as $SO(6)$ with one vector. In this theory we have a runaway superpotential [@IS], so supersymmetry is broken in this model.
A ${\bf Z}_7$ Example
---------------------
Consider the ${\bf Z}_7$ orbifold group whose generator $\theta$ has the following action on the complex coordinates $z_\alpha$ ($\omega\equiv
\exp(2\pi i/7)$): $$\theta z_1=\omega z_1~,~~~\theta z_2=\omega^2 z_2~~~\theta z_3=\omega^4 z_3~.$$ The tadpole cancellation conditions have the following solution (see subsection B of section III): $$\gamma_{\theta}=-4\eta~.$$ This then implies that $$\gamma_\theta={\rm diag}(I_{N-4\eta},\omega I_N,\omega^2 I_N,\omega^3
I_N,\omega^4 I_N,\omega^5 I_N,\omega^6 I_N)~.$$ In particular, consider the $\eta=-1$ and $N=0$ case. We then have pure $SO(4)\sim SU(2)_L\otimes SU(2)_R$ super-Yang-Mills theory. It would be interesting to use this setup to identify BPS domain walls in this gauge theory.
Concluding Remarks
==================
The above discussions illustrate that the orientiworld framework has a rich variety of non-perturbative phenomena that can arise in the orientiworld models. In particular, we can have dynamical supersymmetry breaking as well as various interesting supersymmetry preserving phenomena such as confinement, domain walls, [*etc*]{}.
The orientiworld framework gives a consistent embedding of non-conformal gauge theories in the Type IIB string theory context. Generalizing the gauge/string theory correspondence of [@malda; @GKP; @witten] to such theories would be very interesting, but it is also expected to be non-trivial as the corresponding supergravity solutions are often expected to be singular. Solving the problem of such singularities might also shed light on non-supersymmetric cases, which would be interesting in the context of the cosmological constant problem along the lines of [@DGS].
This work was supported in part by the National Science Foundation and an Alfred P. Sloan Fellowship. I would like to thank the New High Energy Theory Center at Rutgers University for their kind hospitality while parts of this work were completed. I would also like to thank Albert and Ribena Yu for financial support.
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[^1]: E-mail: zurab@insti.physics.sunysb.edu
[^2]: More precisely, there is a mild restriction on allowed orbifold groups if we require modular invariance of the closed string sector in the corresponding oriented Type IIB background.
[^3]: This is always the case if $\Gamma\subset SU(3)$. For non-supersymmetric cases this is also true provided that $\not\exists{\bf Z}_2\subset\Gamma$. If $\exists{\bf Z}_2\subset\Gamma$, then modular invariance requires that the set of points in ${\bf R}^6$ fixed under the ${\bf Z}_2$ twist has real dimension 2.
[^4]: In this limit we take the string coupling $g_s\rightarrow
0$ together with $N\rightarrow \infty$ while keeping $Ng_s$ fixed.
[^5]: In the large $N$ limit non-planar diagrams are suppressed by powers of $1/N$.
[^6]: In the large $N$ limit the closed twisted sector tachyons are harmless as the string coupling $g_s$ goes to zero. Also, in this limit all non-trivial $U(1)$’s decouple in the infra-red.
[^7]: In general, codimension-3 and higher objects (that is D-branes and orientifold planes) do not introduce untwisted tadpoles.
[^8]: Note that we can also place $2N+1$ D3-branes on top of the O3$^-$-plane to obtain the $SO(2N+1)$ gauge group.
[^9]: The $\Gamma\subset SU(2)$ orientifolds were originally discussed in [@orient1; @FS; @PU].
[^10]: Recall that in our conventions $Sp(N)$ with $N\in 2{\bf N}$ has rank $N/2$.
[^11]: Non-renormalizable couplings suppressed by powers of $M_s$ break this global $SO(3)$ symmetry to its discrete subgroup subsumed in the discrete ${\bf Z}_3$ gauge symmetry.
[^12]: One exception is the $\eta=-1$ and $N=4$ model, where the non-Abelian gauge group is $SU(4)$, the matter consists of 3 chiral supermultiplets in ${\bf 6}$ of $SU(4)$, and there is no tree-level superpotential. This theory can be thought of as $SO(6)$ with 3 flavors, which is well understood [@IS].
[^13]: The simplest example of an $s$-confining theory is $SU(N_c)$ with $N_f=N_c+1$ flavors. In this theory we have confinement without chiral symmetry breaking at the origin of the meson and baryon moduli space.
[^14]: In [@LPT] it was argued that supersymmetry is broken in this model once we include the tree-level superpotential. In particular, the non-perturbative superpotential is known for the $SU(5)$ theory with $3\times{\bf 10}$ and $3\times{\overline{\bf 5}}$ and no tree-level superpotential [@schmaltz]. If we now add the tree-level superpotential and write the total superpotential in terms of the $SU(5)$ gauge invariant degrees of freedom, we will find that this total superpotential has a runaway behavior in terms of the gauge invariant degrees of freedom. This, however, does not necessarily imply that supersymmetry is broken. Thus, for vacuum expectation values of gauge invariant operators larger than the dynamically generated scale $\Lambda$ in the original $SU(5)$ theory the description in terms of the $SU(5)$ gauge invariant operators is no longer valid. Now, if supersymmetry were broken, the corresponding non-supersymmetric vacuum would have various vacuum expectation values stabilized at $\sim M_s$ (as this stabilization is due to the contributions coming from the K[ä]{}hler potential as we discussed at the end of the previous section), while in the case of a weakly coupled background ($g_s\ll 1$) we have $\Lambda\ll M_s$. In this case we must therefore consider the low energy theory after Higgsing (and not the other way around) as we did above where we saw that supersymmetry is intact. However, if $g_s$ is somewhat large (the one-loop $\beta$-function coefficient for the $SU(5)$ theory is 9), then we could have supersymmetry breaking along the lines of [@LPT].
|
---
abstract: 'A neutron scattering investigation of the magnetoelectric coupling in PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$ (PFN) has been undertaken. Ferroelectric order occurs below 400 K, as evidenced by the softening with temperature and subsequent recovery of the zone center transverse optic phonon mode energy ($\hbar \Omega_{0}$). Over the same temperature range, magnetic correlations become resolution limited on a terahertz energy scale. In contrast to the behavior of nonmagnetic disordered ferroelectrics (namely Pb(Mg,Zn)$_{1/3}$Nb$_{2/3}$O$_{3}$), we report the observation of a strong deviation from linearity in the temperature dependence of $(\hbar \Omega_{0})^{2}$. This deviation is compensated by a corresponding change in the energy scale of the magnetic excitations, as probed through the first moment of the inelastic response. The coupling between the short-range ferroelectric and antiferromagnetic correlations is consistent with calculations showing that the ferroelectricity is driven by the displacement of the body centered iron site, illustrating the multiferroic nature of magnetic lead based relaxors in the dynamical regime.'
author:
- 'C. Stock'
- 'S.R. Dunsiger'
- 'R. A. Mole'
- 'X. Li'
- 'H. Luo'
title: 'Coupled magnetic and ferroelectric excitations in PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$'
---
*Introduction* – Creating materials with a strong coupling between ferroelectric and magnetic order has been a central goal of research in $3d$ transition metal compounds [@Cheong07:6; @Kimura03:426; @Scott12:222]. Fundamentally, this phenomenon is startling, since typically ferroelectricity results from relative shifts of negative and positive ions with closed electronic configurations, while magnetism is related to ordering of spins of electrons in incomplete ionic shells. While multiferroicity has now been reported in a variety of compounds, as typified by compounds like BiFeO$_{3}$ and BiMnO$_{3}$, coupling between the magnetic and polar orders is often weak and occurs on widely disparate temperature scales [@Santos02:122]. Magnon-phonon interactions are typically also very weak in improper multiferroics like TbMnO$_{3}$, where only tiny shifts of $\sim$ 0.1 meV in phonon frequencies due to magnetic ordering have been measured [@Rovillain10:81]. By contrast, we demonstrate a strong intrinsic coupling in a disordered ferroelectric between short range ferroelectric and magnetic orders by measuring the magnetic and lattice fluctuations. Recently, the multiferroic nature of disordered systems has attracted considerable attention and the correlation between short range ferroelectric and antiferromagnetic orders has been reported for several different systems [@r1; @r2; @r3]. For example, in the magnetic relaxor ${(1-x)}$BiFeO$_{3}$-$x$BaTiO$_3$, elastic neutron diffraction studies suggest the polar nanoregions are identical to short range magnetic nanodomains [@r3].
Some of the most studied nonmagnetic ferroelectric compounds are the lead based relaxors with the chemical form PbBO$_{3}$, where the B site is a mixture between two different ions [@Park97:82; @Ye98:81; @Bokov06:41]. PbMg$_{1/3}$Nb$_{2/3}$O$_{3}$ (PMN) and PbZn$_{1/3}$Nb$_{2/3}$O$_{3}$ (PZN) are prototypical relaxors in which the random occupancy of the B site results in a suppression of a classical ferroelectric phase transition, replaced by short range ferroelectric order. These polar nanoregions manifest as a strong temperature dependent neutron and x-ray diffuse scattering cross-section which has a significant dynamic component. The latter component tracks the frequency dependence of the dielectric constant [@Stock10:81; @Hirota02:65; @Xu04:70; @Stock07:76; @Gehring09:79]. A dynamic signature of ferroelectricity in these compounds is a soft and transverse optic phonon mode located at the Brillouin zone center. The dielectric constant is related to the soft phonon energy ($\hbar \Omega_{0}$) via the Lyndane-Sachs-Teller relation- $1/\epsilon \propto (\hbar \Omega_{0})^{2}$ [@Lyddane41:59]. PbTiO$_{3}$ displays a conventional soft ferroelectric mode, where $(\hbar \Omega)^{2}$ softens to a minimum energy at the structural transition and then recovers linearly below the phase transition [@Shirane70:2; @Kempa06:79]. Even though a long-range ferroelectric ground state is absent in relaxors PMN and PZN, a soft zone center transverse optic mode is present and recovers at the same temperature where static short range polar correlations are onset [@Wakimoto02:65; @Stock04:69; @Cowley11:60].
The disordered compound PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$ (PFN) is also based on a perovskite structure with the B site a mixture of nonmagnetic Nb$^{5+}$ and magnetic Fe$^{3+}$ ($d^{5}$, $S$=5/2) ions. Two structural transitions are reported, the cubic unit cell transforming to tetragonal ($a/c=0.9985$) at 376 K, followed by a monoclinic structure ($\beta$=89.94$^{\circ}$) below 355 K [@Lampis99:11]. The dielectric constant is peaked near 375 K, however is broad in temperature as well as frequency dependent, indicative of short range ferroelectric order [@Majumder06:99]. A weak cusp at $\sim 150$ K and history dependence below $\sim 20$ K has been observed in measurements of the bulk magnetization [@bokov; @blinc; @bhat]. Neutron diffraction measurements have observed short-range three dimensional antiferromagnetic correlations peaked at a wave vector $\vec{Q}$=(1/2,1/2,1/2) coexisting with a magnetic Bragg peak indicative of a long-range magnetically ordered component. The presence of two distinct lineshapes for the magnetic diffraction has led to models involving two phases defined by different Fe$^{3+}$ clustering sizes [@Kleemann10:105].
Motivated by dielectric measurements suggestive of a coupling between the magnetic and ferroelectric orders (Ref. ) we present a neutron scattering study of the lattice and magnetic fluctuations in PFN performed at the PUMA spectrometer (FRM2 reactor) on a 1 cm$^{3}$ sample. We show that the temperature dependent lattice dynamics are dominated by a soft transverse optic mode $(\hbar\Omega)$ measured near the nuclear zone center. Although $(\hbar\Omega)^{2}$ recovers, it deviates from the linear response observed in other ferroelectrics (namely PbTiO$_{3}$, PMN, and PZN). An investigation of the first moment of the magnetic inelastic response suggests that this is compensated by a corresponding change in the energy scale of magnetic excitations, indicative of magnetoelectric coupling [*in the dynamical regime*]{}. These measurements are complementary to Raman scattering studies [@correa; @garcia], where the zone center phonon response is also explored, typically at higher characteristic energies. While Correa [*et al.*]{} [@correa] report anomalous shifts with sublattice magnetization of the Fe-O phonon mode frequency centered around 87 meV, the shifts are very small ($<3$ cm$^{-1}$ or 0.37 meV). The so called $F_{2g}$ mode associated with Pb localization around 65 cm$^{-1}$ (8 meV) is completely unaffected by the antiferromagnetic transition observed in their sample. We show that the magnetoelastic effects observed using inelastic neutron scattering in the low energy regime are much more dramatic. More generally, using Raman techniques, the assignment of phonon modes in the cubic state of the relaxor ferroelectric is controversial [@correa], making neutron scattering invaluable.
![\[mag\_elastic\] a) the elastic ($\hbar \omega$=0) magnetic intensity as a function of temperature at $\vec{Q}$=(1/2,1/2,1/2). b) the inverse of the correlation length as a function of temperature derived from the constant energy scans displayed in panel e). The static magnetization is shown in panels $c)$ and $d)$ under the application of a 100 Gauss field along the \[100\] axis.](mag_elastic.eps){width="8.2cm"}
$\textit{Static magnetic properties}$ – Figure \[mag\_elastic\] plots the static magnetic properties of PFN measured through the use of elastic neutron scattering (with an energy resolution $\delta E$=1.25 meV=0.30 THz) and static bulk magnetization. Panel $a)$ shows a plot of the elastic magnetic intensity as a function of temperature at $\vec{Q}$=(1/2,1/2,1/2), located at the peak of the magnetic intensity (see panel $e$). It shows only a smooth growth of intensity and no anomaly indicative of a well defined phase transition. The increase in elastic magnetic intensity is mimicked by the static magnetization (panel $c)$ which shows a deviation from Curie-Weiss behavior over a similar temperature range. The combination of the half-integer position in momentum and its broad lineshape indicates the origin of the scattering is from clusters of Fe$^{3+}$ ions. A Lorentzian squared lineshape was used to model the data, motivated by random fields [@Birgeneau83:28]. Such random fields are thought to be the underlying cause of the avoided long range magnetic ordering, which would be characterized by a Bragg peak. Similar ideas have been proposed to understand the ferroelectric properties of disordered PMN and PZN [@Westphal92:68; @Fisch03:67]. Given the large vertical resolution on the PUMA spectrometer (Ref. ), we integrated over the vertical direction analytically, giving the ${3/2}$ power to the momentum dependence shown below. The magnetic correlation lengths (panel $b$) were thus extracted from fits to a resolution convolution of the following lineshape (examples illustrated in $e$): $$\begin{aligned}
I(\vec{Q})=C {(\gamma r_{0})^2 \over 4} g^{2} f^2(Q) m^{2} e^{-\langle u^{2} \rangle Q^{2}} {V^{*} {\xi^{3}/\pi^{2}} \over [1+(|\vec{Q}-\vec{Q}_{0}|\xi)^{2}]^{3/2}},
\nonumber\end{aligned}$$ where $C$ is the calibration constant, $(\gamma r_{0})^{2}/4$ is 73 mbarns sr$^{-1}$, $g=2$ is the Land[é]{} factor, $f^{2}(Q)$ is the Fe$^{3+}$ magnetic form factor, $m$ is the magnetic moment size, $V^{*}$ is the volume of the Brillouin zone, $\xi$ the correlation length, $e^{-\langle u^{2} \rangle Q^{2}}$ is the Debye-Waller factor, and $a$ is the lattice constant. Performing scans along all high symmetry directions (\[111\], \[110\], and \[001\]), the magnetic correlation length is found to be spatially isotropic within error. As demonstrated in Fig. \[mag\_elastic\] $b)$, this quantity never diverges, but saturates at the small value of $\xi$=17Å with an inflection point around $\sim$ 80 K. The inflection point in the temperature dependent correlation length is mimicked by the static magnetization presented in Fig. \[mag\_elastic\] $d)$, albeit at much lower temperatures of $\sim$ 25 K. The divergence between field-cooled (FC) and zero-field cooled (ZFC) magnetization reflects the development of a spin-glass-like state at low temperatures and is close to the $\sim$ 20 K anomaly observed using muon spin relaxation [@Rotaru09:79]. While there is apparent sample dependence evidenced by the different values quoted in the literature for this divergence in the magnetization (27.6 K [@Kumar08:93],$\sim$20 K [@Rotaru09:79], and 10.6 K [@Kleemann10:105]), as pointed out in Ref. , the exact value may be strongly technique and field dependent. The differing onset temperatures of spin glass-like ordering produced by different techniques (neutrons - 80 K compared to static magnetization - 25 K) which probe different timescales is similar to results published on canonical spin glasses and frustrated magnets, where the magnetic correlations are dominated by slow fluctuations over a broad frequency range, sampled with differing energy resolutions [@Stock10:105; @Murani78:41].
The high temperature fit of the static magnetization yields a Curie constant of $k_{B}\Theta_{CW}$=-240 $\pm$ 10 K which is a measure of the average Fe$^{3+}$-Fe$^{3+}$ exchange coupling. The negative sign indicates antiferromagnetic coupling between the spins. Within the mean-field approximation $k_{B}\Theta_{CW}=-{1\over3} z J S(S+1)$ (with $z$=3 neighbors and $S={5\over2}$) gives an estimate of the average $J=2.4 \pm 0.4$ meV $\sim$ 28 K. This implies a magnetic band width of the order of 10 meV ($\sim 2SJ$). The energy scale of the magnetic coupling is consistent with the presence of strong low energy magnetic spectral weight at high temperatures, within the spectrometer resolution (Fig. \[mag\_elastic\]$e$).
![\[phonon\] a) the phonon dispersion for both the T$_{1}$ and T$_{2}$ modes. b)-d) representative constant-Q scans taken at $\vec{Q}$=(2.15, 1.85, 0) e) constant-Q scans at the zone center at 150 K (filled circles) and 3 K (open circles) f) the frequency squared of the soft transverse optic mode as a function of wavevector squared near the zone center.](figure_phonons.eps){width="8.2cm"}
$\textit{Ferroelectric properties}$ – The lattice dynamics are similar in structure and energy scale to PMN and PbTiO$_{3}$ [@Swainson09:79; @Tomeno06:73; @Hlinka06:73]. The phonons are described by a transverse optic mode which is gapped at the nuclear zone center and a lower energy acoustic mode which is gapless. The dispersion near the nuclear zone center is shown in Fig. \[phonon\] $a)$ and illustrative constant-$\vec{Q}$ scans are shown in panels $b-d)$. The constant-$Q$ scans show there is little change in frequency in the low-energy acoustic mode. However, the higher energy optic mode gradually hardens as the temperature is decreased, as expected for the recovery from a structural transition. In a similar manner to the case of PMN, we observed the optic mode is over damped in energy near the nuclear zone center, plotted in panel $e)$, where a very broad and unresolvable peak was observed at 150 K. A more well defined peak at $\sim$ 11 meV is observed at 3 K [@Stock12:86]. To extract the zone center soft mode energy, we rely on the optic mode frequencies at finite $q$ away from the zone center and fit the results to $(\hbar \Omega(q))^{2}=(\hbar \Omega_{\circ})^{2}+\alpha q^{2}$. $\Omega_{\circ}$ and $\alpha$ are respectively the optic mode frequency at the zone center and a temperature independent measure of the curvature near the zone center. Representative results from this analysis are plotted in panel $f)$ and where the zone center frequency could be measured, good agreement was observed. This method has been applied before to the relaxors and found to be in good agreement with zone center scans, as well as Raman data [@Cao08:78; @Shirane70:2].
$\textit{Magnetic dynamics}$ – The temperature dependent magnetic dynamics are illustrated in Fig. \[mag\_inelastic\] through constant $\vec{Q}$=(1/2,1/2,1/2) scans (panels $a-c$) and constant $\hbar\omega$= 2 meV scans (panels $d-f$). The fluctuations are dominated by a peak at $\vec{Q}$=(1/2, 1/2, 1/2) which is both momentum and energy broadened, characteristic of short range and glass-like dynamics. The dynamic magnetic response is not dispersive and is overdamped for all temperatures investigated. Constant $\vec{Q}$=(1/2,1/2,1/2) scans were fit to a resolution convolved damped harmonic oscillator as described in the supplementary information.
![\[mag\_inelastic\] $a-c)$ are constant-$\vec{Q}$=(1/2,1/2,1/2) scans at several temperatures. The solid curves represent a fit to the simple harmonic oscillator described in the text. $d-f)$ are constant $\hbar \omega$=2 meV scans at the sample temperatures. The resolution full-widths are represented by the solid bars.](figure_mag_fluctuations.eps){width="9.0cm"}
![\[param\] $a)$ the soft transverse optic phonon frequency squared as a function of temperature for both PFN and PMN. $b)$ and $c)$ illustrates the magnetic fitting parameters described in the text. $b)$ shows the line width as a function of temperature and $c)$ the susceptibility. $d)$ the zeroeth moment for the static and dynamic components on the THz timescale. $e)$ shows the change in the first moment with temperature.](param_figure.eps){width="8.0cm"}
The temperature variation of $(\hbar\Omega_{0})^{2}$ (proportional to $1/\epsilon$), the timescale of the magnetic fluctuations ($2\Gamma \sim 1/\tau$), and the susceptibility ($\chi_{0}$) at $\vec{Q}$=(1/2, 1/2, 1/2) are illustrated in Fig. \[param\] $a-c)$. The energy of the soft optic mode in PMN is also plotted in Fig. \[param\] $a)$, showing a linear recovery down to low temperatures [@Stock05:74; @Wakimoto02:65]. For comparison, the data from PMN has been scaled by a factor of 1.4 to agree with PFN at 400 K. While at temperatures above $\sim$ 100 K the zone center energy tracks the measured response in prototypical relaxors (such as PMN), a significant deviation from the linear recovery is observed at low temperatures.
The linewidth of the magnetic fluctuations measured at $\vec{Q}$=(1/2,1/2,1/2) is plotted in Fig. \[param\] $b)$ and illustrates a linear decrease towards saturation at $\sim$ 80 K. The dashed line is given by $2\Gamma=k_{B}T$, demonstrating that the energy scale of the magnetic fluctuations at high temperatures is set by $k_{B}T$. Figure \[param\] $c)$ illustrates the susceptibility $\chi_{0}$ derived from fits shown in Fig. \[mag\_inelastic\]. The dashed line is a high temperature fit to $1/(T-\Lambda)$ with a deviation at $\sim$ 80 K - the same temperature as the inflection point in the elastic correlation length (Fig. \[mag\_elastic\] $b$) and the saturation of the inelastic linewidth (Fig. \[param\] $b$).
Given the broad nature of the magnetic response in momentum and energy of our data and the ambiguity over its functional form, it is important to ensure all the spectral weight is accounted for. In general, the total integrated magnetic intensity over all momentum and energy transfers is a conserved quantity satisfying the zeroeth moment sum rule. We calculated the zeroeth moment sum (Fig. \[param\] $d$) by integrating the magnetic intensity over the $Q$ and $E$ range presented in Fig.\[mag\_inelastic\]. The data were normalized against the known cross section of an acoustic phonon. The dashed line is the expected value based on a $S={5\over2}$ moment for Fe$^{3+}$, indicating the experiment indeed probed all of the magnetic scattering. The filled points are the total spectral weight summing over the elastic and inelastic channels, while the open circles represent the inelastic component. The difference at low temperatures corresponds to an estimate of the total spectral weight measured to be static, or within resolution limits (0.3 THz). The decrease of the spectral weight with increasing temperature indicates the magnetic response extends to higher energies not directly probed.
The deviation of $(\hbar\Omega)^{2}$ from linearity (Fig. \[param\]$a$) differs from nonmagnetic counterparts PMN and PZN and is suggestive of a coupling to another energy scale. Given the magnetic response is strongly damped (Fig. \[mag\_inelastic\] ), we have investigated the magnetic energy scale through a study of the first moment $\langle \mathcal{H} \rangle$ calculated from the Hohenberg Brinkman (Ref. ) sum rule. The sum rule is a general result for isotropic correlations and is applicable to the case where no sharp peak is observed in a constant $Q$ scan: $$\begin{aligned}
\label{first_moment} \langle \mathcal{H} \rangle=-{3\over4}{{\int_{-\infty}^{\infty}dE (E\chi''(Q,E))}\over{(1-\cos(\vec{Q}\cdot\vec{d}))}}.
\nonumber\end{aligned}$$ The integration was performed at $\vec{Q}$=(1/2,1/2,1/2) and $d$ is the distance between nearest neighbors. The change in the first moment with temperature is plotted in Fig. \[param\] $e)$, showing a substantial reduction with decreasing temperature. The change is of order the expected change in energy between the soft optic phonon in PFN and PMN, indicating a coupling between the two orders.
$\textit{Summary and Conclusions}$ – Our results illustrate that in the presence of short-range magnetic order, the soft phonon dynamics are directly altered from the linear recovery in $(\hbar \Omega)^{2}$ observed in classic and disordered ferroelectrics. The first moment, a measure of the magnetic energy scale, illustrates the change in energy is taken up by the magnetic spin terms demonstrating coupling between the two orders. Such a coupling may be expected given the local bonding environment in PFN. The ferroelectric order in compounds of the form ABO$_{3}$ has been found to be determined by the condensation of predominately the Last and Slater phonon modes, with significant contributions from shifts in the A and O sites [@Harada70:A26]. In the fully ordered case of PFN, the exchange interaction between two Fe$^{3+}$ ions would involve orbitals from Pb and also O. Therefore, the hybridization associated with ferroelectric order would be expected to alter the exchange pathways coupling magnetic ions. Such a scenario has been suggested to exist in fully site ordered EuTiO$_{3}$ [@Katsufuji01:64]. An alternate explanation is proposed as a result of calculations (Ref. ) which suggest that the ferroelectricity in PFN predominately originates from the displacement of the Fe$^{3+}$ site, the ratio of the displacement of Fe$^{3+}$ to Nb$^{5+}$ being greater than 10. Such a large difference in displacement could provide a route for explaining the strong coupling between the two orders observed here.
Strong magnetoelectric coupling appears to be favored in disordered systems where the symmetry contraints of the lattice are relaxed, as in compounds like pervoskite (Sr,Mn)TiO$_{3}$ and nonperovskite (Ni,Mn)TiO$_{3}$ [@r1; @r2]. However, enhanced dielectric constants have been reported previously in a number of candidate materials for multiferroicity, which were later shown to arise from nanoscale disorder [@zhu07; @ruff12]. Using a microscopic technique robust against such extrinsic effects, we have found evidence of coupling between the short range magnetic and ferroelectric orders in the archetypal ferroelectric PFN. The temperature dependence of $(\hbar \Omega_{0})^{2}$ associated with the soft transverse optic mode, sensitive to ferroelectric correlations, deviates strongly from the linear recovery observed in classic ferroelectrics as well as prototypical nonmagnetic relaxors Pb(Mg,Zn)$_{1/3}$Nb$_{2/3}$O$_{3}$. We are grateful for funding from EU-NMI3, the Carnegie Trust for the Universities of Scotland, STFC, and Deutsche Forschungsgemeinschaft Grant TRR 80.
Appendix
========
Here we present supplementary information regarding the experimental details, spectrometer calibration, and sum rules used in the main text. Supplementary data regarding the momentum dependence of the magnetic scattering are also presented. The data demonstrate the absence of well-defined spin waves and show that the magnetic excitations are represented by strongly overdamped fluctuations characteristic of the short range magnetic order.
Experimental details for the neutron scattering and susceptibility measurements
-------------------------------------------------------------------------------
Neutron scattering measurements were performed on the PUMA thermal triple-axis spectrometer located at the FRM2 reactor (Garching, Germany). Two sets of measurements were performed with the 1 cm$^{3}$ (with lattice constant $a$=4.01 Å) sample to measure both the magnetic and lattice fluctuations. To measure the phonon dispersion curves and temperature dependence, the sample was oriented in the (HK0) scattering plane. The magnetic scattering was investigated with the sample mounted in the (HHL) plane. In both sets of measurements, the sample was cooled in a closed cycle refrigerator. A PG(002) vertically focused monochromator was used to select an incident energy E$_{i}$ and the final energy was fixed at E$_{f}$=14.8 meV using a PG(002) flat analyzer crystal. The energy transfer was then defined as $\hbar \omega$=E$_{i}$-E$_{f}$. To measure the phonon curves, it was desirable to obtain a high count rate at the expense of momentum resolution and therefore horizontal focusing was used on both the monochromator and the analyser. The horizontal focusing, on both the incident and scattered sides was removed for studies of the static and fluctuating magnetic response. Higher order contamination was reduced through the use of a pyrolytic graphite filter in the scattered beam. The counting time was determined by a low efficiency monitor placed in the incident beam and was corrected for variable contamination by higher order scattering from the monochromator using the same calibration described elsewhere. [@Stock04:69]
Magnetization measurements were performed using a Quantum Design Materials Properties Measurement System (MPMS) on a small 7.0 mg piece of PFN taken from the same crystal growth batch. A field of 100 Gauss was applied along the $a$ axis and measurements under field cooled (FC) and zero-field cooled conditions were performed.
Spectrometer calibration constant derived from acoustic phonons
---------------------------------------------------------------
To calculate the zeroeth and first moments of the magnetic scattering, we have put the magnetic intensities on an absolute scale. The calibration constant for the experiment was obtained by measuring a low-energy acoustic phonon. In the long-wavelength (low $q$) limit, it can be assumed that we are in the hydrodynamic regime where only the center of mass is moving and the structure factor for the acoustic phonon will match that of the nearby nuclear Bragg peak. In the setup used on PUMA with an incident beam monitor with an efficiency $\propto {1\over{\sqrt{E}}}$, the measured energy integrated intensity takes the form $$\begin{aligned}
I(\vec{Q})=A\left({\hbar \over {2 \omega_{0}} } \right) [1+n(\omega_{0})]|F_{N}|^{2} {{Q^{2} \cos^{2}(\beta)} \over {M}}e^{-2W}.\end{aligned}$$ where $A$ is the spectrometer calibration constant, $\hbar \omega_{0}$ is the acoustic phonon frequency, $[1+n(\omega_{0})]$ is the Bose factor, $|F_{N}|^{2}$ the structure factor of the nuclear Bragg peak, $M$ the mass of the unit cell, and $e^{-2W} \sim 1$ is the Debye Waller factor. For the measured magnetic scattering in the (HHL) scattering plane, we have used an acoustic phonon measured at $\vec{Q}$=(0.15,0.15,2) and T=300 K as a reference with a horizontally flat monochromator and analyzer.
Lineshape- inelastic magnetic response
--------------------------------------
To describe the broad overdamped lineshape characterizing the magnetic dynamics, a relaxational form determined by a single energy scale $\Gamma\propto1/\tau$ described by $\chi''(Q,E)\propto \chi_{0}(Q)E\Gamma/(E^{2}+\Gamma^{2})$ was initially fit to the constant $Q$ scans shown in Fig. 3 of the main text. Using this form for the magnetic dynamics, $\chi_{0}$ is related to the real part of the susceptibility. While this line shape described the data well at high temperatures, it failed to fit the response below $\sim$ 100 K. To correct this, following Ref. , we fit the following damped harmonic oscillator lineshape to all temperatures. $$\begin{aligned}
\label{SHO} S(E)= {\chi_{0}} [n(E)+1] \times \\
\left( {{1}\over {[1+
{{\left(E- E_{0}\right)^{2}}\over {\Gamma^{2}}} }]} - {{1}\over {[1+
{{\left(E+ E_{0}\right)^{2}}\over {\Gamma^{2}}} }]} \right),
\nonumber\end{aligned}$$ where $\chi_{0}$ is a measure of the strength of the magnetic scattering, $[n(E)+1]$ is the thermal population (or Bose) factor, $\hbar \Omega$ the mode position, and $\Gamma$ is the half-width. To account for elastic scattering from static (defined by our resolution width) correlations, we have included a Gaussian in the fit centered at the elastic energy position. The parameter E$_{0}$ was found to be temperature independent with E$_{0}$=0.5 $\pm$ 0.2 meV and can be physically interpreted as a magnetic anisotropy energy scale.
![\[dispersion\] The momentum dependence of the scattering at T=3 K. $a-b)$ illustrate constant energy scans at E=6 and 2 meV. $c-e)$ show constant-Q scans taken at positions close to $\vec{Q}$=(${1\over 2}, {1\over 2}, {1\over 2}$)](dispersion.eps){width="8.2cm"}
Zeroeth moment sum rule for magnetic scattering
-----------------------------------------------
The total integrated magnetic intensity over all momentum and energy transfers is a conserved quantity satisfying the zeroeth moment sum rule. Accounting for the orientation factor in magnetic neutron scattering and the fact that there is ${1 \over 2}$ a Fe$^{3+}$ site per unit cell, the integral over $S(\vec{Q},E)$ is, $$\begin{aligned}
\langle \mu_{eff}^{2} \rangle = \int dE \int d^{3}Q S(\vec{Q},E)=...\nonumber \\
{1\over \pi} \int dE \int d^{3}Q [1+n(E)]\chi''(\vec{Q},E)=... \nonumber \\
{2 \over 3} g^{2} \mu_{B}^{2} S(S+1) \times {1 \over 2}.\end{aligned}$$ Setting $S={5 \over 2}$ gives a total expected integral of 11.7 $\mu_{B}^{2}$. This is in agreement with the total integrated moment discussed in the main text.
$\vec{Q}$-E dependence
----------------------
The zeroeth and first moment analysis outlined in the main text relies on a knowledge of the momentum dependence of the magnetic scattering with energy transfer. Fig. \[dispersion\] illustrates the momentum dependence of the magnetic scattering through both constant energy (panels $a-b)$ and also constant-Q scans (panels $c-e)$). The constant energy scans have been fitted to a Gaussian centered at $\vec{Q}$=(${1\over 2}, {1\over 2}, {1\over 2}$) and show only a single central peak. The constant-Q scans also do not display any sign of spin waves or dispersion of the magnetic excitations. The first moment analysis has used the fact that the excitations are peaked only near $\vec{Q}$=(${1\over 2}, {1\over 2}, {1\over 2}$) and this is substantiated by the results.
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abstract: 'We analyze allowed region of muon anomalous magnetic moment (muon $g-2$), satisfying lepton flavor violations, $Z$ boson decays, and collider physics, in a framework of multi-charged particles. Then we explore the typical size of the muon $g-2$, and discuss which mode dominantly affects muon $g-2$.'
author:
- Takaaki Nomura
- Hiroshi Okada
title: 'Muon anomalous magnetic moment, $Z$ boson decays, and collider physics in multi-charged particles'
---
Introduction
============
Muon anomalous magnetic moment (muon $g-2$) is one of the promising phenomenologies to confirm the new physics. Therefore it still remains discrepancy between the standard model (SM) and new physics [@Hagiwara:2011af]; $$\begin{aligned}
\Delta a_\mu=(26.1\pm8)\times10^{-10},\label{eq:damu}\end{aligned}$$ where the $3.3\sigma$ deviation from the SM prediction with a positive value; recent theoretical analysis further indicates 3.7$\sigma$ deviation [@Keshavarzi:2018mgv]. Furthermore, several upcoming experiments such as Fermilab E989 [@e989] and J-PARC E34 [@jpark] will provide the result with more precise manner. In theoretical point of view, several mechanisms have been historically proposed through, e.g., gauge contributions [@Altmannshofer:2014pba; @Mohlabeng:2019vrz; @Abdallah:2011ew], Yukawa contributions at one-loop level [@Lindner:2016bgg], and Barr-Zee contributions [@bz] at two-loop level. Especially, when one supposes the muon $g-2$ would be related to the other phenomenologies such as neutrino masses and dark matter candidate, Yukawa contributions at one-loop level would be likely to be promising candidates [@Ma:2001mr; @Okada:2013iba; @Baek:2014awa; @Okada:2014nsa; @Okada:2014qsa; @Okada:2015hia; @Okada:2016rav; @Nomura:2016rjf; @Ko:2016sxg; @Baek:2016kud; @Nomura:2016ask; @Lee:2017ekw; @Chiang:2017tai; @Das:2017ski; @Nomura:2017ezy; @Nomura:2017tzj; @Cheung:2017kxb; @Cheung:2018itc; @Cai:2017jrq; @Chakrabarty:2018qtt; @CarcamoHernandez:2019xkb; @Chen:2019nud; @Nomura:2017ohi; @Baumholzer:2018sfb]. In this case, one has to simultaneously satisfy several constraints of lepton flavor violations (LFVs) such as; $\ell_i\to \ell_j\gamma$, $\ell_i\to\ell_j\ell_k\bar\ell_\ell$($i,j,k,\ell=(e,\mu,\tau)$), and lepton flavor conserving(violating) $Z$ boson decays such as $Z\to\ell\bar\ell'$, $Z\to\nu\bar\nu'$ [@pdg]. Particularly, $\ell_\mu\to \ell_e\gamma$ gives the most stringent constraint, and the current branching ratio should be less than $4.2\times10^{-13}$ [@TheMEG:2016wtm], and its future bound will reach at $6\times10^{-14}$ [@Renga:2018fpd]. Also $Z$ boson decays will be tested by a future experiment such as CEPC [@cepc].
In this paper, we introduce several multi-charged fields (bosons and fermions) with general $U(1)_Y$ hypercharges to get positive muon $g-2$, and we estimate the allowed region to satisfy all constraints of the muon $g-2$, LFVs, and $Z$ boson decays. Also, we consider the constraint of collider physics, since multi-charged fields are severely restricted by the Large Hadron Collider (LHC). We discuss the necessity of extra charged scalar in order to make exotic charged leptons decay into the SM particles and decay chains of exotic charged particles. Then the signature of exotic charged particles are explored and we consider an allowed scenario accommodating muon $g-2$ and collider constraints.
This paper is organized as follows. In Sec. II, we review the model and formulate LFVs, muon $g-2$, $Z$ boson decays, and renormalization group for $g_Y$. In Sec. III, we estimate the allowed region for each $N$, comparing to collider physics. We conclude in Sec. IV.
Model setup and Constraints with common part
============================================
$L_L$ $e_R$ $L'_{L/R}$ $H$ $h^{+n}$
----------- ------------ ---------- --------------- ----------- ---------------
$SU(2)_L$ $\bm{2}$ $\bm{1}$ $\bm{2}$ $\bm{2}$ $\bm{1}$
$U(1)_Y$ $-\frac12$ $-1$ $-\frac{N}2$ $\frac12$ $\frac{N-1}2$
: Charge assignments of fields under $SU(2)_L\times U(1)_Y$, where $n\equiv\frac{N-1}{2}$ with $N=3,5,\cdots$, and all the new fields are color singlet.[]{data-label="tab:1"}
In our set up of the model, we introduce an isospin doublet fermion $L'_a\equiv[\psi^{-n}_a,\psi^{-n-1}_a]^T\ (a=1)$ for simplicity [^1], and a new boson $h^{+n}$ with $n\equiv\frac{N-1}2$ ($N=3,5,\cdots$), as shown in Table \[tab:1\]. Notice here that $N$ is defined by odd number, where $N=1$ is not considered because $L'_L$ cannot be discriminated from $L_L$. The valid Lagrangian is given by $$\begin{aligned}
-\mathcal{L}^n_{Y} &= f_{ia} \bar L_{L_i} L'_{R_a} h^{n} + {\rm h.c.}{\nonumber}\\
&=f_{ia}[\bar \nu_{L_i} \psi^{-n}_a h^{n}+\bar\ell_i\psi^{-n-1}_a h^n]+ {\rm h.c.},
\label{Eq:lag-yukawa} \end{aligned}$$ where $i=1-3,\ a=1$ are generation indices. The Yukawa Lagrangian $y_{\ell_{ii}}\bar L_{L_i}e_{R_i}H$ provides masses for the charged leptons [$(m_{\ell_i}\equiv y_{\ell_{ii}}v/\sqrt2$)]{} by developing a nonzero vacuum expectation value (VEV) of $H$, which is denoted by $\langle H\rangle\equiv v/\sqrt2$. The exotic lepton $L'$ has vector-like mass and new scalar field $h^{\pm n}$ does not develop a VEV. We denote mass of $L'$ and $h^{\pm n}$ by $m_\psi$ and $m_h$ respectively.
Lepton flavor violations and muon anomalous magnetic moment {#lfv-lu}
-----------------------------------------------------------
The Yukawa terms of ($f,g$) give rise to $\ell_i\to\ell_j\gamma$ processes at one-loop level. The branching ratio is given by $$\begin{aligned}
B(\ell_i\to\ell_j \gamma)
\approx
\frac{48\pi^3 \alpha_{\rm em}}{{G_{\rm F}^2 m_{\ell_i}^2} } C_{ij}\left( |a_{L_{ij}}|^2 + |a_{R_{ij}}|^2\right),\end{aligned}$$ where $G_{\rm F}\approx1.166\times 10^{-5}$ GeV$^{-2}$ is the Fermi constant, $\alpha_{\rm em}(m_Z)\approx {1/128.9}$ is the fine-structure constant [@pdg], $C_{21}\approx1$, $C_{31}\approx0.1784$, and $C_{32}\approx0.1736$. $a_{L/R}$ is formulated as $$\begin{aligned}
& a_{L_{ij}}
\approx
- m_{\ell_i} \sum_{a=1-3} \frac{f_{ja} f^\dag_{ai} }{(4\pi)^2}
\left[ n F(\psi^{-n-1}_a , h^n) + (n+1) F(h^n,\psi^{-n-1}_a)\right], \label{eq:lfv-L}\\
&a_{R_{ij}}
\approx
- m_{\ell_j} \sum_{a=1-3} \frac{f_{ja} f^\dag_{ai} }{(4\pi)^2}
\left[ n F(\psi^{-n-1}_a , h^n) + (n+1) F(h^n,\psi^{-n-1}_a)\right], \label{eq:lfv-R}\\
&F_{}(1,2)\approx \frac{(m_1^2-m_2^2)\{5 m_1^2m_2^2-m_2^4(1+3n)+m_1^4(2+3n)\}- 12m_1^2 m_2^2
\{-nm_2^2+(1+n)m_1^2\}
\ln\left[\frac{m_1}{m_2}\right]}
{12(m_1^2-m_2^2)^4},
\label{eq:lfv-lp}
$$ where $m_{\psi^{-n-1}}\equiv m_\psi$, and $m_{h^n}\equiv m_h$. The current experimental upper bounds are given by [@TheMEG:2016wtm; @Aubert:2009ag] $$\begin{aligned}
B(\mu\rightarrow e\gamma) &\leq4.2\times10^{-13}(6\times10^{-14}),\quad
B(\tau\rightarrow \mu\gamma)\leq4.4\times10^{-8}, \quad
B(\tau\rightarrow e\gamma) \leq3.3\times10^{-8}~,
\label{expLFV}
\end{aligned}$$ where parentheses of $\mu\to e\gamma$ is a future reach of MEG experiment [@Renga:2018fpd].
[*The muon anomalous magnetic moment*]{} ($\Delta a_\mu$): We can also estimate the muon anomalous magnetic moment through ${a_{L,R}}$, which is given by $$\begin{aligned}
\Delta a_\mu\approx -m_\mu (a_L+a_R)_{22}.\label{eq:damu}\end{aligned}$$ The $3.3\sigma$ deviation from the SM prediction is $\Delta a_\mu=(26.1\pm8)\times10^{-10}$ [@Hagiwara:2011af] with a positive value.
![Feynman diagrams for $Z\to \ell_i\bar\ell_j$ and $Z \to \nu_i \bar \nu_j$, where upper diagrams represent contribution to $Z \bar \ell \ell$ while the down ones are for $Z \bar \nu \nu$.[]{data-label="fig:zto2ell"}](Z_to2ell.eps){width="100mm"}
Flavor-Conserving(Changing) Leptonic $Z$ Boson Decays {#subsec:Zll}
-----------------------------------------------------
Here, we consider the $Z$ boson decay into two leptons through the Yukawa terms $f$ at one-loop level [@Chiang:2017tai]. Since some components of $f$ are expected to be large so as to obtain a sizable $\Delta a_\mu$, the experimental bounds on $Z$ boson decays could be of concern at one loop level. First of all, the relevant Lagrangian is given by [^2] $$\begin{aligned}
{\cal L}&\sim
\frac{g_2}{c_w} \left[\bar\ell\gamma^\mu \left(-\frac12 P_L+s_W^2\right)\ell
+\frac12\bar\nu\gamma^\mu P_L\nu
\right] Z_\mu{\nonumber}\\
&+ \frac{g_2}{c_w} \left[
\left(-\frac12 P_L+n s_W^2\right)\bar\psi^{n}\gamma^\mu \psi^{-n}
+ \left(-\frac12 P_L+(n+1) s_W^2\right)\bar\psi^{n+1}\gamma^\mu \psi^{-n-1}
\right] Z_\mu{\nonumber}\\
& +in\frac{g_2 s_W^2}{c_W}(h^n\partial^\mu h^{-n} - h^{-n}\partial^\mu h^{n})Z_\mu,\end{aligned}$$ where $s(c)_W\equiv\sin(\cos)\theta_W\sim0.23$ stands for the sine (cosine) of the Weinberg angle. The decay rate of the SM at tree level is then given by $$\begin{aligned}
&{\rm \Gamma}(Z\to\ell^-_i\ell^+_j)_{SM}
\approx
\frac{m_Z}{12\pi} \frac{g_2^2}{c_W^2} \left(s_W^4-\frac{s_W^2}2 + \frac18 \right)\delta_{ij},\\
&{\rm \Gamma}(Z\to\nu_i\bar\nu_j)_{SM}
\approx
\frac{m_Z}{96\pi} \frac{g_2^2}{c_W^2} \delta_{ij}.\end{aligned}$$ Combining all the diagrams in Fig. \[fig:zto2ell\], the ultraviolet divergence cancels out and only the finite part remains [@Chiang:2017tai]. The resulting form is given by $$\begin{aligned}
&\Delta{\rm \Gamma}(Z\to\ell^-_i\ell^+_j)
\approx
\frac{m_Z}{12\pi} \frac{g_2^2}{c_W^2}
\left[ \frac{|B_{ij}^{\ell}|^2}{2} - {\rm Re}[A_{ij} (B^{\ell})^*_{ij}]
-\left(-\frac{s_W^2}2 + \frac18\right)\delta_{ij}\right] \label{eq:Zll},\\
&\Delta{\rm \Gamma}(Z\to\nu_i\bar\nu_j)
\approx
\frac{m_Z}{24\pi} \frac{g_2^2}{c_W^2}
\left[ {|B_{ij}^{\nu}|^2}
-\frac{\delta_{ij}}{4}\right] \label{eq:Znunu},\end{aligned}$$ where $$\begin{aligned}
&A_{ij}\approx s^2_W\delta_{ij},
\quad B^{\ell}_{ij}\approx \frac{\delta_{ij}}{2} - \frac{f_{ia} f^\dag_{aj}}{(4\pi)^2} G^{\ell}(\psi,h),
\quad B^{\nu}_{ij}\approx \frac{\delta_{ij}}{2} + \frac{f_{ia} f^\dag_{aj}}{(4\pi)^2} G^{\nu}(\psi,h),
\\
&G^{\ell}(\psi,h)\approx-ns^2_W\left(-\frac12 +s_w^2\right) H_1(\psi,h)
-\left(-\frac12 +s_w^2\right)^2 H_2(\psi,h)
+\left(-\frac12 +(n+1) s_w^2\right) H_3(\psi,h),\\
&G^{\nu}(\psi,h)\approx-ns^2_W\left(-\frac12 +s_w^2\right) H_1(\psi,h)
- \frac12 H_2(\psi,h)
+\left(-\frac12 +n s_w^2\right) H_3(\psi,h),
\\
&H_1(1,2)= \frac{m_1^4-m_2^4+4 m_1^2 m_2^2\ln\left[\frac{m_2}{m_1}\right]}{2(m_1^2-m_2^2)^2},\\
&H_2(1,2)= \frac{m_2^4 - 4m_1^2 m_2^2 +3m_1^4 - 4 m_2^2(m_2^2-2m_1^2)\ln[m_2]-4m_1^4\ln[m_1]}{4(m_1^2-m_2^2)^2},\\
&H_3(1,2)=m_1^2\left( \frac{m_1^2-m_2^2 + 2 m_2^2\ln\left[\frac{m_2}{m_1}\right]}{(m_1^2-m_2^2)^2}\right).\end{aligned}$$ 0 G\^[/]{}(m\_,m\_h)&-ns\^2\_W(-12 +s\_w\^2)\_3\
&-(-12 +s\_w\^2)\^2(\_3 - \_2)\
&+(-12 +(n+1) s\_w\^2) m\^2\_, Notice here that the upper index of $B$ represents $\psi\equiv \psi^{-n-1}$ for cahrged-lepton final state, while $\psi\equiv \psi^{-n}$ for the neutrino final state. One finds the branching ratio by dividing the total $Z$ decay width $\Gamma_{Z}^{\rm tot} = 2.4952 \pm 0.0023$ GeV [@pdg]. The current bounds on the lepton-flavor-(conserving)changing $Z$ boson decay branching ratios at 95 % CL are given by [@pdg]: $$\begin{aligned}
& \Delta {\rm BR}(Z\to {\rm Invisible})\approx \sum_{i,j=1-3}\Delta {\rm BR}(Z\to\nu_i\bar\nu_j)< \pm5.5\times10^{-4} ,
\label{eq:zmt-con}\\
& \Delta {\rm BR}(Z\to e^\pm e^\mp) < \pm4.2\times10^{-5} ~,\
\Delta {\rm BR}(Z\to \mu^\pm\mu^\mp) < \pm6.6\times10^{-5} ~,\
\Delta {\rm BR}(Z\to \tau^\pm\tau^\mp) < \pm8.3\times10^{-5} ~,\label{eq:zmt-con}\\
& {\rm BR}(Z\to e^\pm\mu^\mp) < 7.5\times10^{-7} ~,\
{\rm BR}(Z\to e^\pm\tau^\mp) < 9.8\times10^{-6} ~,\
{\rm BR}(Z\to \mu^\pm\tau^\mp) < 1.2\times10^{-5} ~,\label{eq:zmt-cha}
$$ where $\Delta {\rm BR}(Z\to f_i\bar f_j)$ ($i= j$) is defined by $$\begin{aligned}
\Delta {\rm BR}(Z\to f_i \bar f_j)\approx
\frac{{\rm \Gamma}(Z\to f_i \bar f_j)- {\rm \Gamma}(Z\to f_i \bar f_j)_{SM}}
{\Gamma_{Z}^{\rm tot}}.\end{aligned}$$ We consider these constraints in our global analyses below.
0
Oblique parameters
-------------------
In order to estimate the testability via collider physics, we have to consider the oblique parameters that restrict the mass hierarchy between each of component $\psi^{-n}$ and $\psi^{-n-1}$.
Here we focus on the new physics contributions to the $S$ and $T$ parameters in the case of $\Delta U=0$. Then $\Delta S$ and $\Delta T$ are defined as $$\begin{aligned}
\Delta S&={16\pi} \frac{d}{dq^2}[\Pi_{33}(q^2)-\Pi_{3Q}(q^2)]|_{q^2\to0},\quad
\Delta T=\frac{16\pi}{s_{W}^2 m_Z^2}[\Pi_{\pm}(0)-\Pi_{33}(0)],\end{aligned}$$ where $s_{W}^2\approx0.23$ is the Weinberg angle and $m_Z$ is the $Z$ boson mass. The loop factors $\Pi_{33,3Q,\pm}(q^2)$ are calculated from the one-loop vacuum-polarization diagrams for $Z$ and $W^\pm$ bosons, which are respectively given by [@Cheung:2016fjo; @Cheung:2017lpv] $$\begin{aligned}
\Pi_{33}(q^2)&=\frac1{2(4\pi)^2}
\left[...\right], \label{eq:pi33}\\
\Pi_{3Q}(q^2)&=
\frac1{(4\pi)^2}
\left[...\right],
\\
\Pi_{\pm}(q^2)&=
\frac1{(4\pi)^2}
\left[...\right],\\
G(q^2,m1,m2)&\equiv...,
$$ where .... Fixing $\Delta U=0$, the experimental bounds on $\Delta S$ and $\Delta T$ are given by [@Baak:2012kk] $$\begin{aligned}
\Delta S = (0.05 \pm 0.09), \quad
\Delta T = (0.08 \pm 0.07),
\end{aligned}$$ with a correlation coefficient of $+0.91$. The $\Delta \chi^2$ can be calculated as [@Dawson:2009yx] $$\begin{aligned}
\Delta \chi^2=\sum_{(i,j)=1,2}(\Delta S-0.05,\Delta T-0.08)
\left[
\begin{array}{cc}
718.19 & -840.28 \\
-840.28 & 1187.2 \\
\end{array}\right]
\left[
\begin{array}{c}
\Delta S-0.05 \\ \Delta T-0.08
\end{array}\right],
\end{aligned}$$ and we impose the 99% confidence level limit that corresponds to $\Delta \chi^2=9.210$ in our numerical analysis.
Beta function of $g_Y$ {#beta-func}
----------------------
Here we estimate the effective energy scale by evaluating the Landau pole for $g_Y$ in the presence of new exotic fields with nonzero multiple hypercharges. Each contribution of the new beta function of $g_Y$ from one $SU(2)_L$ doublet fermion with $-N/2$ hypercharge is given by [@Ko:2016sxg] $$\begin{aligned}
\Delta b^f_Y={\frac{3}{5}\times}\frac{4}{3}\times\left(\frac{N}2\right)^2 \ .
$$ Similarly, the contribution to the beta function from one $SU(2)_L$ singlet boson with $(N-1)/2$ hypercharge is given by $$\begin{aligned}
\Delta b^b_Y={\frac{3}{5}\times}\frac{1}{3}\times\left(\frac{N-1}2\right)^2 ,\end{aligned}$$ [where $3/5$ is the rescaled coefficient.]{} Then one finds the energy evolution of the gauge coupling $g_Y$ as [@Kanemura:2015bli] $$\begin{aligned}
\frac{1}{g^2_Y(\mu)}&=\frac1{g_Y^2(m_{in.})}-\frac{b^{SM}_Y}{(4\pi)^2}\ln\left[\frac{\mu^2}{m_{in.}^2}\right]
-\theta(\mu-m_{thres.}) \frac{(\Delta b^f_Y+\Delta b^b_Y)}{(4\pi)^2}\ln\left[\frac{\mu^2}{m_{thres.}^2}\right]
,\label{eq:rge_gy}\end{aligned}$$ where $\mu$ is a reference energy scale, and we assume that $m_{in.}(=m_Z)<m_{thres.}=$500 GeV, where $m_{in.}$ $m_{thres.}$ are initial and threshold mass, respectively. The resulting running of $g_Y(\mu)$ versus the scale $\mu$ is shown in Fig. \[fig:rge\] for each of $N=3,5,7,9,11,13$.
![The running of $g_Y$ in terms of a reference energy of $\mu$, depending on each of $N=3,5,7,9,11,13$.[]{data-label="fig:rge"}](Landau-rge.eps){width="13cm"}
Muon $g-2$ and physics of Each $N$
==================================
In this section we estimate muon $g-2$ taking into account constraints from LFVs and Z decays and discuss constraint and prospect for collider physics in some number of $N$. In addition to the Yukawa interaction explaining muon $g-2$, we need extra particles and/or interactions to make exotic particles decay into SM ones. Here we summarize extensions for the cases of $N=3$, $N=5$ and $N=7$ as follows. [^3]\
[**(1) $N=3$**]{}: In this case, we have interaction term $$\mathcal{L}_{ex 1} = g_{ij} \bar L^c_{L_i} L_{L_j}h^+ + h.c. \ ,
\label{EQ:extra_int_1}$$ without introducing extra particle. Then all exotic particles can eventually decay into the SM particles.\
[**(2) $N=5$**]{}: in this case, we have interaction term $$\mathcal{L}_{ex 2} = g'_{ij} \bar e^c_{R_i} e_{R_j}h^{++} + h.c. \ ,
\label{EQ:extra_int_2}$$ without introducing extra particle, and exotic particles can decay into the SM particles as in the $N=3$ case. Here it is also worthwhile mentioning the we can explain the active neutrino sector at two-loop level, if both extra terms are introduced with extra doubly(singly) charged particle for $N=3(5)$ cases. This is called Zee-Babu model [@zee; @babu].\
[**(3) $N=7$**]{}: In this case, we need to introduce $h^{\pm}$ and $h^{\pm \pm}$ in addition to $h^{\pm \pm \pm}$ in order to make it decay into the SM particles. We then have interactions $\mathcal{L}_{ex 1(2)}$ and new interaction in scalar potential: $$V_{ex} = \mu_X h^{+++} h^{--} h^{-} + c.c. \ ,
\label{EQ:extra_int_3}$$ with which triply charged scalar can decay into the SM particles through doubly and singly charged scalar decay by $\mathcal{L}_{ex 1(2)}$ interaction. Note that new Yukawa interactions affect LFVs, muon $g-2$, and Z decays. Especially, these terms contribute to the muon $g-2$ negatively. Therefore, we require these terms are enough small to satisfy the sizable muon $g-2$.
![Muon $g-2$ as a function of $L'$ mass obtained from parameter scan for $N=3$, $N=5$ and $N=7$ where red, green, yellow, and blue color points respectively correspond to those with $\ell_i \to \ell_j \gamma$ constraints, $\ell_i \to \ell_j \gamma$ plus $Z\to\nu_i\bar\nu_j$, $\ell_i \to \ell_j \gamma$ plus $Z\to\mu\bar\mu$, and $\ell_i \to \ell_j \gamma$ plus all of the $Z\to f_i\bar f_j$.[]{data-label="fig:mg2-I"}](figN=3.eps "fig:"){width="7cm"} ![Muon $g-2$ as a function of $L'$ mass obtained from parameter scan for $N=3$, $N=5$ and $N=7$ where red, green, yellow, and blue color points respectively correspond to those with $\ell_i \to \ell_j \gamma$ constraints, $\ell_i \to \ell_j \gamma$ plus $Z\to\nu_i\bar\nu_j$, $\ell_i \to \ell_j \gamma$ plus $Z\to\mu\bar\mu$, and $\ell_i \to \ell_j \gamma$ plus all of the $Z\to f_i\bar f_j$.[]{data-label="fig:mg2-I"}](figN=5.eps "fig:"){width="7cm"} ![Muon $g-2$ as a function of $L'$ mass obtained from parameter scan for $N=3$, $N=5$ and $N=7$ where red, green, yellow, and blue color points respectively correspond to those with $\ell_i \to \ell_j \gamma$ constraints, $\ell_i \to \ell_j \gamma$ plus $Z\to\nu_i\bar\nu_j$, $\ell_i \to \ell_j \gamma$ plus $Z\to\mu\bar\mu$, and $\ell_i \to \ell_j \gamma$ plus all of the $Z\to f_i\bar f_j$.[]{data-label="fig:mg2-I"}](figN=7.eps "fig:"){width="7cm"}
![Muon $g-2$ as a function of $L'$ and $h^{n}$ masses (left and right plots) obtained from parameter scan imposing all the constraints as discussed in Fig. \[fig:mg2-I\], where black and pink points respectively correspond to cases of $N=5$ and $N=7$. Note here that there are not any allowed points for $N=3$.[]{data-label="fig:mg2-II"}](M-amu.eps "fig:"){width="7cm"} ![Muon $g-2$ as a function of $L'$ and $h^{n}$ masses (left and right plots) obtained from parameter scan imposing all the constraints as discussed in Fig. \[fig:mg2-I\], where black and pink points respectively correspond to cases of $N=5$ and $N=7$. Note here that there are not any allowed points for $N=3$.[]{data-label="fig:mg2-II"}](mk-amu.eps "fig:"){width="7cm"}
Muon $g-2$ and flavor constraints for each case
-----------------------------------------------
In this subsection, we scan Yukawa coupling in Eq. (\[Eq:lag-yukawa\]) and estimate muon $g-2$ taking into account constraints from LFV charged lepton decay as well as $Z \to \ell_i^+ \ell^-_j$ processes discussed in previous section. Here we universally scan $f_{i1}$ in the range of $$f_{i1} \in [10^{-6}, \sqrt{4 \pi}],$$ where the upper bound is requirement from perturbativity. Firstly we take wide mass range of $\{m_\psi, m_h \} \in [100, 5000]$ GeV in our parameter scan where $m_\psi$ and $m_h$ are respectively mass of $L'$ and $h^n$. In Fig. \[fig:mg2-I\], we show the value of muon $g-2$ as a function of exotic lepton mass for $N=3$, $N=5$ and $N=7$ where red, green, yellow, and blue color points respectively correspond to those with $\ell_i \to \ell_j \gamma$ constraints, $\ell_i \to \ell_j \gamma$ plus $Z\to\nu_i\bar\nu_j$, $\ell_i \to \ell_j \gamma$ plus $Z\to\mu\bar\mu$, and $\ell_i \to \ell_j \gamma$ plus all of the $Z\to f_i\bar f_j$. We see that $Z\to\mu\bar\mu$ and $Z\to\nu_i\bar\nu_j$ constraints severely exclude the parameter region, and exotic particle masses are preferred to be relatively light as $m_{\psi, h} \lesssim 500$ GeV. Then we focus on light mass region which can accommodate with muon $g-2$. The left and right plots in Fig. \[fig:mg2-II\] show the value of muon $g-2$ as a function of $L'$ and $h^n$ masses respectively imposing all the constraints as discussed in Fig. \[fig:mg2-I\], where black and pink points respectively correspond to cases of $N=5$ and $N=7$. Furthermore we show contour plot for $\Delta a_\mu$ and $\Delta BR_{\mu \mu} \equiv \Delta BR (Z \to \mu^+ \mu^-)$ on $\{M(=m_h = m_\psi), f_{21} \}$ plane where we take only $f_{21}$ to be non-zero and other $f_{ij}$ to be zero. In the plots, the (light-)yellow region is $(2 \sigma) 1 \sigma$ region for muon $g-2$ and shaded region is excluded by $\Delta BR (Z \to \mu^+ \mu^-)$. Thus one find that the mass scale is constrained by $\Delta BR_{\mu \mu}$ even if only $f_{21}$ is non-zero. We thus find that $L'$ mass should be relatively light as $m_\psi \sim (150, 200, 250 )$ GeV for $N=(3, 5,7)$ to explain muon $g-2$ within $1 \sigma$ while charged scalar mass $m_h$ can be heavier than $m_\psi$.
![Contours of $\Delta a_\mu$ and $\Delta BR_{\mu \mu} \equiv \Delta BR (Z \to \mu^+ \mu^-)$ on $\{M(=m_h = m_\psi), f_{21} \}$ plane where we take only $f_{21}$ to be non-zero and other $f_{ij}$ to be zero. The (light-)yellow region is $(2 \sigma) 1 \sigma$ region for muon $g-2$ and shaded region is excluded by $\Delta BR (Z \to \mu^+ \mu^-)$.[]{data-label="fig:contour"}](amudeltabrn3.eps "fig:"){width="7cm"} ![Contours of $\Delta a_\mu$ and $\Delta BR_{\mu \mu} \equiv \Delta BR (Z \to \mu^+ \mu^-)$ on $\{M(=m_h = m_\psi), f_{21} \}$ plane where we take only $f_{21}$ to be non-zero and other $f_{ij}$ to be zero. The (light-)yellow region is $(2 \sigma) 1 \sigma$ region for muon $g-2$ and shaded region is excluded by $\Delta BR (Z \to \mu^+ \mu^-)$.[]{data-label="fig:contour"}](amudeltabrn5.eps "fig:"){width="7cm"} ![Contours of $\Delta a_\mu$ and $\Delta BR_{\mu \mu} \equiv \Delta BR (Z \to \mu^+ \mu^-)$ on $\{M(=m_h = m_\psi), f_{21} \}$ plane where we take only $f_{21}$ to be non-zero and other $f_{ij}$ to be zero. The (light-)yellow region is $(2 \sigma) 1 \sigma$ region for muon $g-2$ and shaded region is excluded by $\Delta BR (Z \to \mu^+ \mu^-)$.[]{data-label="fig:contour"}](amudeltabrn7.eps "fig:"){width="7cm"}
Collider physics and constraints
--------------------------------
In explaining muon $g-2$ by the interaction Eq. (\[Eq:lag-yukawa\]), the mass scale of exotic lepton doublet $L'$ is required to be less than $\sim 300$ GeV. Thus exotic charged lepton can be produced at the LHC with sizable production cross section and we should take into account collider constraints to explore if the mass scale for explaining muon $g-2$ is allowed. In our study, we focus on the exotic charged lepton with the highest electric charge since it has the largest pair production cross section and provide the most stringent constraint.
Firstly, we estimate the pair production cross section of the highest charged leptons for each case. These charged leptons can be pair produced by Drell-Yan(DY) process, $q \bar q \to Z/\gamma \to \psi^{+n} \psi^{-n}$, and also by photon fusion(PF) process $\gamma \gamma \to \psi^{+n} \psi^{-n}$ [@Babu:2016rcr; @Ghosh:2017jbw; @Ghosh:2018drw]. Here we estimate the cross section applying [MADGRAPH/MADEVENT5]{} [@Alwall:2014hca], where the necessary Feynman rules and relevant parameters of the model are implemented using FeynRules 2.0 [@Alloul:2013bka] and the [NNPDF23LO1]{} PDF [@Deans:2013mha] is adopted. In Fig. \[fig:CX\] we show the cross sections including both DY and PH processes at the LHC 8(13) TeV for left(right) plots. We thus find that cross section is large when electric charge is increased where PF process highly enhance the cross section.
![The pair production cross section of the exotic charged leptons with the highest electric charged for each case at the LHC 8(13) TeV for left(right) plots.[]{data-label="fig:CX"}](CX8.eps "fig:"){width="7cm"} ![The pair production cross section of the exotic charged leptons with the highest electric charged for each case at the LHC 8(13) TeV for left(right) plots.[]{data-label="fig:CX"}](CX13.eps "fig:"){width="7cm"}
Secondly we list the decay chain of the highest charged lepton for each case.\
[**(1) $N =3$**]{}: The decay chain of $E^{\pm \pm}$ is $$\psi^{\pm \pm} \to \ell^\pm_i h^{\pm (*)} \to \ell^\pm_i \ell^\pm_j \nu,$$ where charged scalar can be ether on-shell or off-shell. Thus, $\psi^{++} \psi^{--}$ pair production process gives four charged leptons with missing transverse energy. The singly charged scalar with $m_{h^+} > 100$ GeV is allowed by collider experiment and we require the mass is heavier than 100 GeV [@pdg].\
[**(2) $N =5$**]{}: The decay chain of $\psi^{\pm \pm \pm}$ is $$\psi^{\pm \pm \pm} \to \ell^\pm_i h^{\pm \pm (*)} \to \ell^\pm_i \ell^\pm_j \ell^\pm_k \left[\to \ell_i^\pm h^\pm h^\pm \to \ell^\pm_i \ell^\pm_j \ell^\pm_k \nu \right],$$ where charged scalar can be ether on-shell or off-shell as previous case, and process in square bracket can be induced introducing singly charged scalar with interaction Eq. (\[EQ:extra\_int\_1\]). Thus, $\psi^{+++} \psi^{---}$ pair production process gives six charged leptons. Note that doubly charged scalar mass is constrained by the LHC data as $m_{h^{\pm \pm}} \gtrsim 700-800$ GeV and $m_{h^{\pm \pm}} \gtrsim 400$ GeV when $h^{\pm \pm}$ decay into $e^\pm e^\pm (\mu^\pm \mu^\pm)$ and $\tau^\pm \tau^\pm$ respectively [@CMS:2017pet; @Aaboud:2017qph]. The constraint is looser as $m_{h^{\pm \pm} } \gtrsim 200$ GeV when $h^{\pm \pm}$ dominantly decay via $h^{\pm \pm} \to h^\pm h^\pm \to \ell^\pm_i \ell^\pm_j \nu \nu$ process. To explain muon $g-2$, we require $h^{\pm \pm}$ to dominantly decay into singly charged scalars [@Primulando:2019].\
[**(3) $N =7$**]{}: The decay chain of $\psi^{\pm \pm \pm \pm}$ is $$\psi^{\pm \pm \pm \pm} \to \ell^\pm_i h^{\pm \pm \pm (*)} \to \ell^\pm_i h^{\pm \pm (*)} h^{\pm (*)} \to h^{\pm (*)} h^{\pm (*)} h^{\pm (*)} \to \ell^\pm_i \ell^\pm_j \ell^\pm_k \ell^\pm_l \nu \nu \nu,$$ where triply charged scalar decays via interaction in Eq. (\[EQ:extra\_int\_3\]). Also as in the previous case, we require doubly charged scalar decay into same sign singly charged scalar pair. Thus, $\psi^{++++} \psi^{----}$ pair production process gives [eight]{} charged leptons with missing transverse energy. In general, constraint on mass of triply charged scalar is weaker than that on $\psi^{\pm \pm \pm \pm}$ and we will not explicitly discuss the constraint.
Finally, let us discuss collider constraints on our scenario to explain muon $g-2$. We note that the highest charged lepton dominantly decay into $\psi^{\pm n} \to \mu^\pm h^{\pm n-1}$ since $f_{21}$ coupling is required to be large for explaining muon $g-2$. In addition to the conditions discussed above we classify benchmark scenarios as follows:\
[**(a)**]{} singly charged scalar decay into $\ell = e, \mu$ in decay chain and exotic charged lepton has sufficiently short decay length,\
[**(b)**]{} exotic charged leptons have long decay length and pass through detector,\
[**(c)**]{} singly charged scalar decays into $\tau \nu$ mode and the highest charged scalar mass is slightly lighter than that of the highest charged lepton.
For scenario [**(a)**]{}, inclusive multi-lepton search constrains the cross section where upper bound of the cross section is $\sim 1$ fb at the LHC 8 TeV for the signal in which number of charged lepton $N_\ell$ ($\ell = e, \mu$) is $N_{\ell} \geqslant 3$ [@Aad:2014hja]. Comparing the cross section for 8 TeV in Fig. \[fig:CX\], the charged lepton masses are required to be $m_\psi \gtrsim (650, 900, 1100)$ GeV. In this scenario, the region explaining muon $g-2$ in 1$\sigma$ is excluded for all $N$ and the largest value of muon $g-2$ is roughly $\Delta a_\mu \sim 10^{-10}$ for each case. Scenario [**(b)**]{} can be realized when charged scalar in decay chains is off-shell and extra couplings in Eq. (\[EQ:extra\_int\_1\])-(\[EQ:extra\_int\_3\]) are sufficiently small. For long-lived charged particle, upper bound of the cross section is given in ref. [@Aaboud:2019trc] for the LHC 13 TeV. Comparing the result for chargino, we find the upper limit is less than $1$fb, and since we have multiply charged leptons the constraint will be stronger. Thus the collider constraint in this scenario is stronger than the scenario (a) and we cannot expect sizable muon $g-2$. For scenario [**(c)**]{}, the decay chain provides signature for each case such that case (1) gives low energy muon with missing transverse energy, and case (2) and (3) give multi-tau lepton signature with low energy muon since we require mass difference between $\psi^{-n-1}$ and $h^{\pm n}$ is small and $h^{\pm n}$ is on-shell. [In Fig. \[fig:distPT\], we show the event ratio for the distribution of transverse momentum of muon, $\mu$, in $\psi^{\pm \pm \pm \pm} \to h^{\pm \pm \pm} \mu^\pm \to h^{\pm \pm} \mu^\pm \tau^\pm \nu \to h^\pm \mu^\pm \tau^\pm \tau^\pm \nu \nu \to \mu^\pm \tau^\pm \tau^\pm \tau^\pm \nu \nu \nu$ decay chain at the LHC 13 TeV for different values of [$\Delta M$]{} indicating mass difference between $\psi^{\pm \pm \pm \pm}$ and $h^{\pm \pm \pm}$ where the behaviors are similar if we change colliding energy from 13 TeV to 8 TeV or 14 TeV; here we consider case (3) but we will have similar results for the other cases. The masses of $h^{\pm \pm}$ and $h^{\pm}$ are also fixed to be $m_{h^{\pm \pm}} = 150$ GeV and $m_{h^\pm} = 100$ GeV. In Fig. \[fig:distPT2\], we also show the event ratios for the distribution of transverse momentum of $\tau$ in the same process where three $\tau$ leptons are distinguished by transverse momentum as $p_T(\tau_3) < p_T(\tau_2) < p_T(\tau_1)$ for each event. It is found that transverse momentum of some $\tau$ leptons are generally sizable and they can be detected at detector. On the other hand transverse momentum of $\mu$ tends to be small for $\Delta M \lesssim 10$ GeV and it will be missed by event trigger. For multi-lepton search in ref. [@Aad:2014hja], they require one muon or electron should have $p_T > 26$ GeV and $p_T > 15$ GeV from the second muon(electron). As a result number of dimuon signal events becomes less than $\sim 0.1 \%$ after trigger for $\Delta M = 10$ GeV. We thus see that if $\Delta M \lesssim 10$ GeV most of events are missed by event trigger and we can escape experimental bound. Therefore the scenario [**(c)**]{} with small $\Delta M$ still can be allowed since analysis of multi-tau signature is more difficult and explicit bound is not given.]{} Thus we conclude that to obtain sizable muon $g-2$ by interaction Eq. (\[Eq:lag-yukawa\]) we should rely on this specific scenario. Therefore multi-tau signature is important to test the mechanism to explain muon $g-2$ although analysis of it is challenging.
![The distribution of transverse momentum of muon $\mu$ in $\psi^{\pm \pm \pm \pm} \to h^{\pm \pm \pm} \mu^\pm \to h^{\pm \pm} \mu^\pm \tau^\pm \nu \to h^\pm \mu^\pm \tau^\pm \tau^\pm \nu \nu \to \mu^\pm \tau^\pm \tau^\pm \tau^\pm \nu \nu \nu$ decay at the LHC 13 TeV (neutrino and anti-neutrino are not distinguished here); the vertical axis shows event ration defined by $N_{\rm bin}/N_{\rm total}$ where $N_{\rm total}$ and $N_{\rm bin}$ indicate number of events in total and those inside corresponding bins. Here [$\Delta M$]{} indicate mass difference between $\psi^{\pm \pm \pm \pm}$ and $h^{\pm \pm \pm}$.[]{data-label="fig:distPT"}](DistPT.eps){width="7cm"}
![The distribution of transverse momentum of muon $\tau$ leptons for the same process as Fig. \[fig:distPT\] where three $\tau$ leptons are distinguished by transverse momentum as $p_T(\tau_3) < p_T(\tau_2) < p_T(\tau_1)$ for each events. []{data-label="fig:distPT2"}](PTtau1.eps "fig:"){width="5cm"} ![The distribution of transverse momentum of muon $\tau$ leptons for the same process as Fig. \[fig:distPT\] where three $\tau$ leptons are distinguished by transverse momentum as $p_T(\tau_3) < p_T(\tau_2) < p_T(\tau_1)$ for each events. []{data-label="fig:distPT2"}](PTtau2.eps "fig:"){width="5cm"} ![The distribution of transverse momentum of muon $\tau$ leptons for the same process as Fig. \[fig:distPT\] where three $\tau$ leptons are distinguished by transverse momentum as $p_T(\tau_3) < p_T(\tau_2) < p_T(\tau_1)$ for each events. []{data-label="fig:distPT2"}](PTtau3.eps "fig:"){width="5cm"}
Conclusions
===========
We have analyzed muon $g-2$, LFVs, and $Z$ decays including collider physics in multi-charged particles. We have found LFVs do not restrict the allowed region of muon $g-2$, while $Z\to\nu_i\bar\nu_j$ invisible decay and $Z\to\mu\bar\mu$ give stringent constraints and the allowed region is drastically disappeared. Also, larger $N$ increases the allowed region of muon $g-2$. [*However once we consider the constraint of collider physics, the typical size of muon $g-2$ is of the order $10^{-10}$*]{}, depending on the benchmark scenarios in (a,b,c). To obtain sizable muon $g-2$ of $\mathcal{O}(10^{-9})$, we have found that the specific scenario is required for decay chain of the charged particles in which the mass of $L'$ is slightly heavier than $h^{\pm n}$ and charged scalar bosons decay into mode only including $\tau$ and neutrinos. Therefore analysis of multi-tau lepton signature is important to fully test the scenario to explain muon $g-2$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government. This was also supported by the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City (H.O.). H. O. is sincerely grateful for KIAS and all the members.
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[^1]: When $L'$ provides a flavor structure of neutrino mass matrix, we minimally need two families to satisfy the neutrino oscillation data.
[^2]: We neglect one-loop contributions in the SM.
[^3]: For $N$ is more than 7, model is rather complicated because some fields have to be introduced in order to make new fields into the SM. Thus, we consider $N=3,5,7$.
|
---
abstract: 'This paper introduces a new measure-conjugacy invariant for actions of free groups. Using this invariant, it is shown that two Bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy. This answers a question of Ornstein and Weiss.'
author:
- |
Lewis Bowen[^1]\
University of Hawaii
title: 'A measure-conjugacy invariant for free group actions'
---
[**Keywords**]{}: Ornstein’s isomorphism theorem, Bernoulli shifts, measure conjugacy.\
[**MSC**]{}:37A35\
Introduction
============
This paper is motivated by an old and central problem in measurable dynamics: given two dynamical systems, determine whether they are measurably-conjugate, i.e., isomorphic. Let us set some notation.
A [**dynamical system**]{} (or system for short) is a triple $(G,X,\mu)$ where $(X,\mu)$ is a probability space and $G$ is a group acting by measure-preserving transformations on $(X,\mu)$. We will also call this a [**dynamical system over $G$**]{}, a [**$G$-system**]{} or an [**action of $G$**]{}. In this paper, $G$ will always be a discrete countable group. Two systems $(G,X,\mu)$ and $(G,Y,\nu)$ are [**isomorphic**]{} (i.e., [**measurably conjugate**]{}) if and only if there exist conull sets $X' \subset X, Y' \subset Y$ and a bijective measurable map $\phi:X'\to Y'$ such that $\phi^{-1}$ is measurable, $\phi_*\mu=\nu$ and $\phi(gx)=g\phi(x) \forall g\in G, x \in X'$.
A special class of dynamical systems called Bernoulli systems or Bernoulli shifts has played a significant role in the development of the theory as a whole; it was the problem of trying to classify them that motivated Kolmogorov to introduce the mean entropy of a dynamical system over $\Z$ \[Ko58, Ko59\]. That is, Kolmogorov defined for every system $(\Z,X,\mu)$ a number $h(\Z,X,\mu)$ called the [**mean entropy**]{} of $(\Z,X,\mu)$ that quantifies, in some sense, how “random” the system is. See \[Ka07\] for a historical survey.
Bernoulli shifts also play an important role in this paper, so let us define them. Let $(K,\kappa)$ be a standard Borel probability space. For a discrete countable group $G$, let $K^G = \prod_{g \in G} K$ be the set of all functions $x: G \to K$ with the product Borel structure and let $\kappa^G$ be the product measure on $K^G$. $G$ acts on $K^G$ by $(gx)(f)=x(g^{-1}f)$ for $x \in K^G$ and $g,f \in G$. This action is measure-preserving. The system $(G,K^G,\kappa^G)$ is the [**Bernoulli shift over $G$ with base $(K,\kappa)$**]{}. It is nontrivial if $\kappa$ is not supported on a single point.
Before Kolmogorov’s seminal work \[Ko58, Ko59\], it was unknown whether all nontrivial Bernoulli shifts over $\Z$ were measurably conjugate to each other. He proved that $h(\Z,K^\Z,\kappa^\Z)=H(\kappa)$ where $H(\kappa)$, the [**entropy of $\kappa$**]{} is defined as follows. If there exists a finite or countably infinite set $K' \subset K$ such that $\kappa(K')=1$ then $$H(\kappa)= - \sum_{k\in K'} \mu(\{k\}) \log( \mu(\{k\}) )$$ where we follow the convention $0\log(0)=0$. Otherwise, $H(\kappa)=+\infty$. Thus two Bernoulli shifts over $\Z$ with different base measure entropies cannot be measurably conjugate.
The converse was proven by D. Ornstein in the groundbreaking papers \[Or70a, Or70b\]. That is, he proved that if two Bernoulli shifts $(\Z, K^\Z, \kappa^\Z), (\Z,L^\Z,\lambda^\Z)$ are such that $H(\kappa)=H(\lambda)$ then they are isomorphic.
In \[Ki75\], Kieffer proved a generalization of the Shannon-McMillan theorem to actions of a countable amenable group $G$. In particular, he extended the definition of mean entropy from $\Z$-systems to $G$-systems. This leads to the generalization of Kolmogorov’s theorem to amenable groups.
In the landmark paper \[OW87\], Ornstein and Weiss extended most of the classical entropy theory from $\Z$-systems to $G$-systems where $G$ is any countable amenable group. In particular, they proved that if two Bernoulli shifts $(G, K^G, \kappa^G)$, $(G,L^G,\lambda^G)$ over a countably infinite amenable group $G$ are such that $H(\kappa)=H(\lambda)$ then they are isomorphic. Thus Bernoulli shifts over $G$ are completely classified by base measure entropy.
Now let us say that a group $G$ is [**Ornstein**]{} if $H(\kappa)=H(\lambda)$ implies $(G, K^G, \kappa^G)$ is isomorphic to $(G,L^G,\lambda^G)$ whenever $(K,\kappa)$ and $(L,\lambda)$ are standard Borel probability spaces. By the above, all countably infinite amenable groups are Ornstein. Stepin proved that any countable group that contains an Ornstein subgroup is itself Ornstein \[St75\]. It is unknown whether every countably infinite group is Ornstein.
In \[OW87\], Ornstein and Weiss asked whether all Bernoulli shifts over a nonamenable group are isomorphic. The next result shows that the answer is ‘no’:
\[thm:free\] Let $G=\langle s_1,\ldots,s_r \rangle$ be the free group of rank $r$. If $(K_1,\kappa_1), (K_2,\kappa_2)$ are standard probability spaces with $H(\kappa_1)+H(\kappa_2)<\infty$ then $(G,K^G_1,\kappa_1^G)$ is measurably conjugate to $(G,K^G_2,\kappa_2^G)$ if and only if $H(\kappa_1)=H(\kappa_2)$.
The reason Ornstein and Weiss thought the answer might be ‘yes’ is due to a curious example presented in \[OW87\]. It pertains to a well-known fundamental property of entropy: it is nonincreasing under factor maps. Let $(G,X,\mu)$ and $(G,Y,\nu)$ be two systems. A map $\phi:X \to Y$ is a factor if $\phi_*\mu=\nu$ and $\phi(gx)=g\phi(x)$ for a.e. $x\in X$ and every $g\in G$. If $G$ is amenable then the mean entropy of a factor is less than or equal to the mean entropy of the source. This is essentially due to Sinai. So if $K_n=\{1,\dots,n\}$ and $\kappa_n$ is the uniform probability measure on $K_n$ then $(G,K_2^G,\kappa_2^G)$, which has entropy $\log(2)$, cannot factor onto $(G,K_4^G,\kappa_4^G)$, which has entropy $\log(4)$.
The argument above fails if $G$ is nonamenable. Indeed, let $G=\langle a,b \rangle$ be a rank 2 free group. Identify $K_2$ with the group $\Z/2\Z$ and $K_4$ with $\Z/2\Z \times \Z/2\Z$. Then $$\phi(x)(g) := \big(x(g)+x(ga), x(g) + x(gb) \big) ~\forall x \in K_2^G, g\in G$$ is a factor map from $(G,K_2^G,\kappa_2^G)$ onto $(G,K_4^G,\kappa_4^G)$. This is Ornstein-Weiss’ example. It is the main ingredient in the proof of the next theorem, which will appear in a separate paper.
Let $G$ be any countable group that contains a nonabelian free subgroup. Then every nontrivial Bernoulli shift over $G$ factors onto every Bernoulli shift over $G$.
To prove theorem \[thm:free\], the following invariant is introduced. Let $(X,\mu)$ be any probability space on which $G=\langle s_1,\ldots,s_r\rangle$, the rank $r$ free group, acts by measure-preserving transformations. Let $\alpha=\{A_1,\ldots,A_n\}$ be a partition of $X$ into finitely many measurable sets. Let $B(e,n) \subset G$ denote the ball of radius $n$ centered at the identity element with respect to the word metric induced by $S=\{s_1^{\pm 1}, \ldots, s_r^{\pm 1}\}$. The join of two partitions $\alpha, \beta$ of $X$ is defined by $\alpha \vee \beta = \{A \cap B ~|~ A \in \alpha, B \in \beta\}$. Let $$\begin{aligned}
H(\alpha) &:=& -\sum_{A \in \alpha} \mu(A)\log(\mu(A)),\\
F(\alpha) &:=& (1-2r)H(\alpha)+ \sum_{i =1}^r H(\alpha \vee s_i\alpha),\\
\alpha^n &:=& \bigvee_{g\in B(e,n)} g\alpha,\\
f(\alpha) &:=& \inf_{n} F(\alpha^n).\end{aligned}$$ A partition $\alpha$ is [**generating**]{} if the smallest $G$-invariant $\sigma$-algebra containing $\alpha$ is the $\sigma$-algebra of all measurable sets (up to sets of measure zero). The main theorem of this paper is:
\[thm:main\] Let $G=\langle s_1,\ldots,s_r\rangle$. Let $(G,X,\mu)$ be a system. If $\alpha$ and $\beta$ are finite measurable generating partitions of $X$ then $f(\alpha)=f(\beta)$. Hence this number, denoted $f(G,X,\mu)$, is a measure-conjugacy invariant.
Theorem \[thm:bernoulli\] below implies that if $|K|<\infty$ then $f(G,K^G,\kappa^G)=H(\kappa)$. This and Stepin’s theorem proves theorem \[thm:free\]. A simple exercise reveals that if $r=1$, then $f(G,X,\mu)=h(G,X,\mu)$ is Kolmogorov’s entropy.
Here is a brief outline of the paper. In the next section, standard entropy-theory definitions are presented. In §\[sec:topological\], an equivalence relation, called combinatorial equivalence, is introduced on the space of finite partitions of $X$, where $(X,\mu)$ is a standard probability space on which a countable group $G$ acts. We prove that the combinatorial equivalence class of a finite generating partition is dense in the space of all generating partitions. In §\[sec:splittings\], we introduce an operation on partitions called splitting and show that any two combinatorially equivalent partitions have a common splitting. This culminates in a condition sufficient for a function from the space of partitions to $\R$ to induce a measure-conjugacy invariant. In §\[sec:f\], this condition is shown to hold for the function $F$ defined above. This proves theorem \[thm:main\]. Then we prove theorem \[thm:bernoulli\] (that $f(G,K^G,\kappa^G)=H(\kappa)$ if $|K|<\infty$) and conclude theorem \[thm:free\].
[**Acknowledgments**]{}: I am grateful to Russell Lyons for suggesting this problem, for encouragements and for many helpful conversations. I would like to thank Dan Rudolph for pointing out errors in a preliminary version. I would like to thank Benjy Weiss for many comments that helped me to greatly improve the exposition of this paper and simplify several arguments.
Some Standard Definitions {#sec:standard}
=========================
For the rest of this section, fix a standard probability space $(X,\mu)$.
A [**partition**]{} $\alpha=\{A_1,\ldots,A_n\}$ is a pairwise disjoint collection of measurable subsets $A_i$ of $X$ such that $\cup_{i=1}^n A_i = X$. The sets $A_i$ are called the [**partition elements**]{} of $\alpha$. Alternatively, they are called the [**atoms**]{} of $\alpha$. Unless stated otherwise, all partitions in this paper are finite (i.e., $n<\infty$).
If $\alpha$ and $\beta$ are partitions of $X$ then we write $\alpha=\beta$ a.e. if for all $A\in \alpha$ there exists $B\in\beta$ with $\mu(A \Delta B)=0$. Let $\sP$ denote the set of all a.e.-equivalence classes of finite partitions of $X$. By a standard abuse of notation, we will refer to elements of $\sP$ as partitions.
If $\alpha, \beta \in \sP$ then the [**join**]{} of $\alpha$ and $\beta$ is the partition $\alpha \vee \beta=\{A \cap B\,|\,A \in \alpha, ~B\in \beta\} $.
Let $\sF$ be a $\sigma$-algebra contained in the $\sigma$-algebra of all measurable subsets of $X$. Given a partition $\alpha$, define the [**conditional information function**]{} $I(\alpha|\sF):X \to \R$ by $$I(\alpha|\sF)(x) = -\log\big(\mu(A_x|\sF)(x)\big)$$ where $A_x$ is the atom of $\alpha$ containing $x$. Here $\mu(A_x|\sF):X \to \R$ is the conditional expectation of $\chi_{A_x}$, the characteristic function of $A_x$, with respect to the $\sigma$-algebra $\sF$. The [**conditional entropy of $\alpha$ with respect to $\sF$**]{} is defined by $$H(\alpha|\sF) = \int_X I(\alpha | \sF)(x) \, d\mu(x).$$
If $\beta$ is a partition then, by abuse of notation, we can identify $\beta$ with the $\sigma$-algebra equal to the set of all unions of partition elements of $\beta$. Through this identification, $I(\alpha|\beta)$ and $H(\alpha|\beta)$ are well-defined. Let $I(\alpha)=I(\alpha|\{\emptyset,X\})$ and $H(\alpha)=H(\alpha|\{\emptyset,X\})$.
\[lem:relative\] For any two partitions $\alpha, \beta$ and for any two $\sigma$-algebras $\sF_1, \sF_2$ with $\sF_1 \subset \sF_2$, $$H(\alpha \vee \beta) = H(\alpha) + H(\beta|\alpha),$$ $$H(\alpha | \sF_2) \le H(\alpha | \sF_1).$$
This is well-known. For example, see \[Gl03, Proposition 14.16, page 255\].
Define $d:\sP \times \sP \to \R$ by $$d(\alpha,\beta) = H(\alpha|\beta) + H(\beta|\alpha) = 2H(\alpha \vee \beta) - H(\alpha) - H(\beta).$$ By \[Pa69, theorem 5.22, page 62\] this defines a distance function on $\sP$. If $G$ is a group acting by measure-preserving transformations on $(X,\mu)$ then the action of $G$ on $\sP$ is isometric. I.e., if $g \in G$, $\alpha, \beta \in \sP$ then $d(g\alpha, g\beta) = d(\alpha,\beta)$. From now on, we consider $\sP$ with the topology induced by $d(\cdot,\cdot)$.
Let $G$ be a group acting on $(X,\mu)$. Let $\alpha$ be a partition of $X$. Let $\Sigma_\alpha$ be the smallest $G$-invariant $\sigma$-algebra containing $\alpha$. Then $\alpha$ is [**generating**]{} if for any measurable set $A \subset X$ there exists a set $A' \in \Sigma_\alpha$ such that $\mu(A \Delta A')=0$. Let $\sP_{gen} \subset \sP$ denote the set of all generating partitions.
Combinatorially Equivalent Partitions {#sec:topological}
=====================================
For this section, fix a countable group $G$ and an action of $G$ on a standard probability space $(X,\mu)$ by measure-preserving transformations.
Given $\alpha \in \sP$ and $F \subset G$ finite, let $\alpha^F = \bigvee_{f \in F} f\alpha$.
If $\alpha, \beta \in \sP$ are such that for all $A \in \alpha$ there exists $B \in \beta$ with $\mu(A-B)=0$ then we say that $\alpha$ [**refines $\beta$**]{} and denote it by $\alpha \ge \beta$. Equivalently, $\beta$ is a [**coarsening**]{} of $\alpha$.
\[def:top\] Let $\alpha, \beta \in \sP$. We say that $\alpha$ is [**combinatorially equivalent**]{} to $\beta$ if there exist finite sets $L,M \subset G$ such that $\alpha \le \beta^L$ and $\beta \le \alpha^M$. Let $\sP_{eq}(\alpha) \subset \sP$ denote the set of partitions combinatorially equivalent to $\alpha$
The goal of this section is to prove theorem \[thm:dense\] below: if $\alpha$ is a generating partition then $\sP_{eq}(\alpha)$ is dense in the subspace of all generating partitions.
\[lem:approx1\] Let $\alpha$ be a generating partition and $\beta=\{B_1,\ldots,B_m\} \in \sP$. Let $\epsilon>0$. Then there exists a partition $\beta'=\{B'_1,\ldots,B'_m\}$ and a finite set $L \subset G$ such that $\alpha^L \ge \beta'$ and for all $i=1\ldots m$, $\mu(B_i \Delta B'_i) \le \epsilon$.
Since $\alpha$ is generating, there exists a finite set $L \subset G$ such that for every $i\in\{1,\ldots,m\}$, there is a set $B''_i$, equal to a finite union of atoms of $\alpha^L$, such that $\mu(B_i \Delta B''_i) < \frac{\epsilon}{m}$. For $i=1\ldots m-1$, let $$B'_i := B''_i - \bigcup_{j\ne i} B''_j.$$ $$B'_m := X - \bigcup_{i < m} B'_i = B''_m \cup \bigcup_{i\ne j} B''_i \cap B''_j.$$ Observe that for all $i=1\ldots m$, $$B_i - \bigcup_j B''_j \Delta B_j \subset B'_i \subset B_i \cup \bigcup_j B''_j \Delta B_j.$$ Thus $$\mu(B'_i \Delta B_i) \le m\Big(\frac{\epsilon}{m}\Big) = \epsilon.$$ By construction, $\beta'=\{B'_1,\ldots,B'_m\} \le \alpha^L$.
Let $\alpha=\{A_1,\ldots,A_n\} \in \sP$ and $\beta \in \sP_{gen}$. Let $\epsilon>0$. Then there exists a finite set $M\subset G$ such that for all finite $L \subset G$ with $M \subset L$, the partition elements $\{B^L_1,\ldots,B^L_{m_L}\}$ of $\beta^L$ can be ordered so that there exists an $r\in \{1,\ldots,m_L\}$ and a function $f:\{1,2,\ldots r\} \to \{1,2,\ldots,n\}$ so that for all $i \in \{1,\ldots,r\}$, $$\frac{\mu(B^L_i \cap A_{f(i)})}{\mu(B^L_i)} \ge 1-\epsilon$$ and $$\mu\Big( \bigcup_{i > r} B^L_i\Big) < \epsilon.$$
Let $\delta>0$ be such that $\delta < \Big(\frac{\epsilon}{n}\Big)^2$. By the previous lemma, there exists a partition $\alpha' =\{A'_1,\ldots,A'_n\} \in \sP$ and a finite set $M \subset G$ such that $\alpha' \le \beta^M$ and $\mu(A'_i\Delta A_i)<\delta$ for all $i$. Let $L$ be any finite subset of $G$ with $M \subset L$.
Let $\beta^L=\{B^L_1,\ldots,B^L_{m_L}\}$ and let $f:\{1,\ldots,{m_L}\} \to \{1,\ldots,n\}$ be the function $f(i)=j$ if $\mu(B^L_i -A'_j)=0$. This is well-defined since $\beta^L$ refines $\alpha'$.
After reordering the partition elements of $\beta^L=\{B^L_1,\ldots,B^L_{m_L}\}$ if necessary, we may assume that there is an $r \in \{0,\ldots,{m_L}\}$ such that, if $r>0$ then for all $ i \le r$, $$\frac{\mu(B^L_i \cap A_{f(i)})}{\mu(B^L_i)} \ge 1-\sqrt{\delta},$$ and if $i>r$ then $$\frac{\mu(B^L_i \cap A_{f(i)})}{\mu(B^L_i)} < 1-\sqrt{\delta}.$$ So if $i>r$ then $$\mu(B^L_i \cap A_{f(i)})< (1-\sqrt{\delta})\mu(B^L_i).$$ So $$\begin{aligned}
\mu(B^L_i) &=& \mu(B^L_i - A_{f(i)}) + \mu(B^L_i \cap A_{f(i)})\\
&<& \mu(B^L_i - A_{f(i)})+ (1-\sqrt{\delta})\mu(B^L_i).\end{aligned}$$ Solve for $\mu(B^L_i)$ to obtain $$\begin{aligned}
\mu(B^L_i) &<& \frac{1}{\sqrt{\delta}} \mu(B^L_i - A_{f(i)}).\end{aligned}$$ Since the atoms $B^L_i$ are pairwise disjoint, it follows that $$\begin{aligned}
\mu\Big(\bigcup_{i>r} B^L_i\Big) &<& \frac{1}{\sqrt{\delta}} \mu\Big(\bigcup_{i>r} B^L_i - A_{f(i)}\Big).\end{aligned}$$ Since $\mu(B^L_i - A'_{f(i)})=0$, it must be that $B^L_i - A_{f(i)} \subset A'_{f(i)} - A_{f(i)}$, up to a set of measure zero. So, $$\begin{aligned}
\mu\Big( \bigcup_{i>r} B^L_i \Big) &\le & \frac{1}{\sqrt{\delta}} \mu\Big(\bigcup_{i>r} A'_{f(i)} - A_{f(i)}\Big)\\
&\le& n\sqrt{\delta}< \epsilon.\end{aligned}$$
\[thm:dense\] If $\alpha$ is a generating partition then $$\sP_{gen} \subset \overline{ \sP_{eq}(\alpha)}.$$ I.e., the subspace of partitions combinatorially equivalent to $\alpha$ is dense in the space of all generating partitions.
Let $\alpha=\{A_1,\ldots,A_n\}$ and $\beta =\{B_1,\ldots,B_m\} \in \sP_{gen}$. Without loss of generality, we assume that $\mu(A_i)>0$ for all $i=1\ldots n$. Let $\epsilon>0$. By the previous lemma, there exists a finite set $L \subset G$ such that the atoms of $\beta^L=\{B^L_1,\ldots,B^L_{m_L}\}$ can be ordered so that there exists an $r\in \{1,\ldots,m_L\}$ and a function $f:\{1,2,\ldots r\} \to \{1,2,\ldots,n\}$ so that for all $i \in \{1,\ldots,r\}$, $$\frac{\mu(B^L_i \cap A_{f(i)})}{\mu(B^L_i)} \ge 1-\epsilon$$ and $$\begin{aligned}
\label{eqn:3}
\mu\Big( \bigcup_{i > r} B^L_i\Big) < \epsilon.\end{aligned}$$ By choosing $\epsilon$ small enough (if necessary) we may assume that $f$ is onto (for example, by choosing $\epsilon$ to be smaller than $\frac{1}{2}\mu(A_j)$ over all $j=1\ldots n$).
By definition of $\beta^L$, $m_L \le m^{|L|}$. If necessary, we may assume that $m_L=m^{|L|}$ after modifying $\beta^L$ by adding to it several copies of the empty set. That is, for some $i$, it may occur that $B^L_i =\emptyset$.
Let $\delta>0$ be such that $\delta<\epsilon$. By lemma \[lem:approx1\] there exists a partition $\gamma=\{C_1,\ldots,C_m\}$ and a finite set $M\subset G$ such that $\gamma \le \alpha^M$ and $\mu(C_i \Delta B_i) < \delta$ for all $i$. By choosing $\delta$ small enough we may assume the following. Let $\gamma^L=\{C^L_1,\ldots,C^L_{m_L}\}$. Then, after reordering the atoms of $\gamma^L$ if necessary, $$\begin{aligned}
\label{eqn:1}
\mu\Big(\bigcup_{j=1}^{m_L} C^L_j - B_j^L \Big) \le \epsilon.\end{aligned}$$ Let $$\begin{aligned}
C'_i &=& \{x \in C_i \, |\, \textrm{ if } x \in C^L_j \textrm{ for some } j \textrm{ then } x \in A_{f(j)}\}\\
&=& \bigcup_{j=1}^{m_L}\, C_i \cap C^L_j \cap A_{f(j)}.\end{aligned}$$
Let $C_{i,j} = C_i \cap A_j - C'_i$. Let $$\gamma_1=\{C'_i\,|\, i=1\ldots m\} \cup \{C_{i,j}\,|\, 1 \le i,j\le m\}.$$
Note that $\gamma_1 \le (\alpha^M)^L = \alpha^{LM}$ where $LM=\{lm~|~l \in L, m\in M\}$. We claim that $\gamma_1$ is combinatorially equivalent to $\alpha$. Let $\Sigma_1$ be the smallest $G$-invariant collection of subsets of $X$ that is closed under finite intersections and unions and contains the atoms of $\gamma_1$. It suffices to show that every atom of $\alpha$ is in $\Sigma_1$. Observe that, for each $i$, $C_i=C'_i \cup \bigcup_{j=1}^m C_{i,j}$. Hence, $C_i \in \Sigma_1$. Therefore the atoms of $\gamma^L$ are also in $\Sigma_1$. Since $f$ is onto, the definition of $C'_i$ implies $$C'_i\cap A_{p} = \cup\{ C'_i \cap C^L_j ~|~ f(j)=p\}.$$ So $C'_i \cap A_{p}$ is in $\Sigma_1$ for all $i,p$. Now $C_i \cap A_p = C_{i,p} \cup (C'_i \cap A_p)$. So $C_i\cap A_p \in \Sigma_1$ for all $i,p$. Since $$A_{p} = \bigcup_{i=1}^m C_i \cap A_{p},$$ $A_p \in \Sigma_1$. Since $p$ is arbitrary, this proves the claim. Thus $\gamma_1 \in \sP_{eq}(\alpha)$.
We claim that $\mu(C'_i \Delta C_i) \le 3\epsilon$ for all $i$. By definition, $$C'_i\Delta C_i = C_i - C'_i \subset \bigcup_{j=1}^{m_L} C^L_j - A_{f(j)}.$$ For each $j$, $$C^L_j - A_{f(j)} \subset (C^L_j - B^L_j) \cup (B^L_j - A_{f(j)}).$$ Thus, $$\begin{aligned}
\label{eqn:0}
C'_i \Delta C_i \subset \bigcup_{j=1}^{m_L}(C^L_j - B^L_j) \cup\bigcup_{j=1}^{r}(B^L_j - A_{f(j)})\cup\bigcup_{j>r}(B^L_j - A_{f(j)}).\end{aligned}$$ If $j\le r$, then by definition of $r$, $$\begin{aligned}
\frac{\mu(B^L_j \cap A_{f(j)})}{\mu(B^L_j)} \ge 1-\epsilon.\end{aligned}$$ This implies $$\begin{aligned}
\mu(B^L_j - A^L_{f(j)}) \le \epsilon \mu(B^L_j).\end{aligned}$$ Thus $$\begin{aligned}
\label{eqn:2}
\mu\Big(\bigcup_{j=1}^{r} B^L_j - A^L_{f(j)} \Big) \le \sum_j \epsilon \mu(B^L_j) \le \epsilon.\end{aligned}$$ Equations \[eqn:0\], \[eqn:1\], \[eqn:2\] and \[eqn:3\] imply the claim.
Since $\delta<\epsilon$ and $\mu(C_i \Delta B_i) < \delta$ for all $i$, the above claim implies that $\mu(C'_i\Delta B_i) \le 4\epsilon$ for all $i$. Thus we have shown that for every $\epsilon>0$, there exists a partition $\gamma_1=\{C'_1,\ldots,C'_m,\ldots\}$, combinatorially equivalent to $\alpha$, containing at most $m+m^2$ partition elements and such that $\mu(C'_i \Delta B_i) < 4\epsilon$ for $i=1\ldots m$. This implies that $\beta$ is in the closure of $\sP_{eq}(\alpha)$. Since $\beta$ is arbitrary this implies the theorem.
Splittings {#sec:splittings}
==========
In this section, $G$ can be any finitely generated group with finite symmetric generating set $S$. Let $(X,\mu)$ be a standard probability space on which $G$ acts by measure-preserving transformations.
\[defn:splitting\] Let $\alpha$ be a partition. A [**simple splitting**]{} of $\alpha$ is a partition $\sigma$ of the form $\sigma=\alpha \vee s\beta$ where $s\in S$ and $\beta$ is a coarsening of $\alpha$.
A [**splitting**]{} of $\alpha$ is any partition $\sigma$ that can be obtained from $\alpha$ by a sequence of simple splittings. In other words, there exist partitions $\alpha_0, \alpha_1,\ldots,\alpha_m$ such that $\alpha_0=\alpha$, $\alpha_m = \sigma$ and $\alpha_{i+1}$ is a simple splitting of $\alpha_i$ for all $1 \le i < m$.
If $\sigma$ is a splitting of $\alpha$ then $\alpha$ and $\sigma$ are combinatorially equivalent. The splitting concept originated from Williams’ work \[Wi73\] in symbolic dynamics.
The [**Cayley graph**]{} $\Gamma$ of $(G,S)$ is defined as follows. The vertex set of $\Gamma$ is $G$. For every $s \in S$ and every $g \in G$ there is a directed edge from $g$ to $gs$ labeled $s$. There are no other edges.
The [**induced subgraph**]{} of a subset $F \subset G$ is the largest subgraph of $\Gamma$ with vertex set $F$. A subset $F \subset G$ is [**connected**]{} if its induced subgraph in $\Gamma$ is connected.
\[lem:splittings\] If $\alpha, \beta \in \sP$, $\alpha$ refines $\beta$ and $F \subset G$ is finite, connected and contains the identity element $e$ then $$\alpha \vee \bigvee_{f \in F^{-1}} f\beta$$ is a splitting of $\alpha$.
We prove this by induction on $|F|$. If $|F|=1$ then $F=\{e\}$ and the statement is trivial. Let $f_0 \in F-\{e\}$ be such that $F_1=F-\{f_0\}$ is connected. To see that such an $f_0$ exists, choose a spanning tree for the induced subgraph of $F$. Let $f_0$ be any leaf of this tree that is not equal to $e$.
By induction, $\alpha_1 := \alpha \vee \bigvee_{f \in F^{-1}_1} f\beta $ is a splitting of $\alpha$. Since $F$ is connected, there exists an element $f_1 \in F_1$ and an element $s_1 \in S$ such that $f_1s_1=f_0$. Since $f_1 \in F_1$, $\alpha_1$ refines $(f^{-1}_1\beta)$. Thus $$\alpha \vee \bigvee_{f \in F^{-1}} f\beta = \alpha_1 \vee f_0^{-1}\beta= \alpha_1 \vee s_1^{-1}(f_1^{-1}\beta)$$ is a splitting of $\alpha$.
\[prop:ball splittings\] Let $\alpha,\beta$ be two combinatorially equivalent generating partitions. Then there is an $n \ge 0$ such that $$\alpha^n = \bigvee_{g \in B(e,n)} \, g\alpha$$ is a splitting of $\beta$. Here $B(e,n)$ is the ball of radius $n$ centered at the identity element in $G$ with respect to the word metric induced by $S$. Of course, $\alpha^n$ is also a splitting of $\alpha$.
This proposition is a variation of a result that is well-known in the case $G=\Z$ in the context of subshifts of finite-type. For example, see \[LM95, theorem 7.1.2, page 218\]. It was first proven in \[Wi73\].
Let $L,M \subset G$ be finite sets such that $\alpha \le \beta^L$ and $\beta \le \alpha^M$. Let $l, m \in \N$ be such that $L \subset B(e,l)$ and $M \subset B(e,m)$. So $\alpha \le \beta^l$ and $\beta \le \alpha^m$. Since balls are connected and $\alpha \le \beta^l$, the previous lemma implies $\beta^l \vee \alpha^{m+l}$ is a splitting of $\beta^l$, and therefore, is a splitting of $\beta$. But $\beta^l \vee \alpha^{m+l} = (\beta \vee \alpha^m)^l = \alpha^{m+l}$.
\[thm:invariant\] Let $\Phi:\sP \to \R$ be any continuous function. Suppose that $\Phi$ is monotone decreasing under splittings; i.e., if $\sigma$ is a splitting of $\alpha$ then $\Phi(\sigma) \le \Phi(\alpha)$. Define $\phi:\sP \to \R$ by $$\phi(\alpha)=\lim_{n \to \infty} \Phi(\alpha^n)=\inf_n \Phi(\alpha^n).$$
Then, for any two finite generating partitions $\alpha_1$ and $\alpha_2$, $\phi(\alpha_1)=\phi(\alpha_2)$. So we may define $\phi(G,X,\mu)=\phi(\alpha)$ for any finite generating partition $\alpha$. The number $\phi(G,X,\mu)$ is a measure-conjugacy invariant.
Let $\alpha$ and $\beta$ be two combinatorially equivalent finite partitions. We claim that $\phi(\alpha)=\phi(\beta)$. To see this, suppose, for a contradiction, that $\phi(\alpha)<\phi(\beta)$. Then there exists an $n\ge 0$ such that $\Phi(\alpha^n) <\phi(\beta)$. But by the previous proposition, there is an $m\ge 0$ such that $\beta^m$ is a splitting of $\alpha^n$ which implies $\Phi(\alpha^n) \ge \Phi(\beta^m)\ge \phi(\beta)$, a contradiction. So $\phi(\alpha)=\phi(\beta)$.
For $n\ge 0$ and $\alpha \in \sP$, let $\Phi_n(\alpha)=\Phi(\alpha^n)$. Since $\Phi$ is continuous and the map $\alpha \mapsto \alpha^n$ is also continuous, it follows that $\Phi_n$ is continuous. Since $\phi(\alpha) = \inf_n \Phi_n(\alpha)$, it follows that $\phi$ is upper semi-continuous, i.e., if $\{\beta_n\}$ is a sequence of partitions converging to $\alpha$ then $\limsup_n \phi(\beta_n) \le \phi(\alpha)$.
Now let $\alpha, \beta$ be two finite generating partitions. By theorem \[thm:dense\], there exist finite partitions $\{\beta_n\}_{n=1}^\infty$ combinatorially equivalent to $\beta$ such that $\{\beta_n\}_{n=1}^\infty$ converges to $\alpha$. So $\phi(\beta) = \limsup_n \phi(\beta_n) \le \phi(\alpha)$. Similarly, $\phi(\alpha)\le \phi(\beta)$. So $\phi(\alpha)=\phi(\beta)$.
The $f$-invariant {#sec:f}
=================
In this section, $G=\langle s_1,\ldots,s_r \rangle$. Let $(X,\mu)$ be a standard probability space on which $G$ acts by measure-preserving transformations and let $S=\{s_1^{\pm 1},\ldots,s_r^{\pm 1}\}$. Note $|S|=2r$. Let $F:\sP \to \R$ be defined as in the introduction.
\[prop:monotone\] Let $\alpha \in \sP$. If $\sigma$ is a splitting of $\alpha$ then $F(\sigma)\le F(\alpha)$.
By induction, it suffices to prove the proposition in the special case in which $\sigma$ is a simple splitting of $\alpha$. So let $\sigma=\alpha \vee t\beta$ for some $t\in S$ and coarsening $\beta$ of $\alpha$. For any $s \in S$, $$\begin{aligned}
H(\sigma \vee s\sigma)&=&H(\alpha \vee s\alpha) + H(\sigma \vee s\sigma|\alpha \vee s\alpha)\\
&=& H(\alpha \vee s\alpha) + H(s\sigma|\alpha \vee s\alpha) +H(\sigma|\alpha \vee s\alpha \vee s\sigma)\\
&\le &H(\alpha \vee s\alpha) + H(\sigma|\alpha \vee s^{-1}\alpha) +H(\sigma|\alpha \vee s\alpha).\end{aligned}$$ The last inequality occurs because $\mu$ is $G$-invariant, so $H(s\sigma|\alpha \vee s\alpha)=H(\sigma|\alpha \vee s^{-1}\alpha)$.
Since $H(\sigma)=H(\alpha) + H(\sigma|\alpha)$, the above implies $$\begin{aligned}
F(\sigma) &\le& (1-2r)\big(H(\alpha) + H(\sigma|\alpha)\big) + \sum_{i=1}^r H(\alpha \vee s\alpha) + H(\sigma|\alpha \vee s^{-1}\alpha) +H(\sigma|\alpha \vee s\alpha)\\
&=& F(\alpha) + (1-2r)H(\sigma|\alpha) + \sum_{s\in S} H(\sigma|\alpha \vee s\alpha).\end{aligned}$$ Since $\sigma \le \alpha \vee t\alpha$, $H(\sigma | \alpha \vee t\alpha)=0$. Hence $$\begin{aligned}
F(\sigma) -F(\alpha) &\le& (1-2r)H(\sigma|\alpha) + \sum_{s\in S-\{t\}} H(\sigma|\alpha \vee s\alpha)\\
&=& \sum_{s\in S-\{t\}}\Big( H(\sigma|\alpha \vee s\alpha) - H(\sigma|\alpha) \Big) \le 0.\end{aligned}$$
Theorem \[thm:main\] now follows from the proposition above and theorem \[thm:invariant\].
Let $K$ be a finite set and $\kappa$ a probability measure on $K$. Let $K^G$ be the product space with the product measure $\kappa^G$. The system $(G,K^G,\kappa^G)$ is called the [**Bernoulli shift**]{} over $G$ with base measure $\kappa$.
Let $A_k= \{x \in K^G\, |\, x(e)=k\}$ where $e$ denotes the identity element in $G$. Then $\alpha=\{A_k~|~k\in K\}$ is the [**Bernoulli partition**]{} associated to $K$. It is generating and $H(\kappa)=H(\alpha)$, by definition.
\[thm:bernoulli\] Let $G=\langle s_1,\ldots,s_r \rangle$ be the free group of rank $r$. Let $K$ be a finite set and $\kappa$ a probability measure on $K$. Then $$f(G,K^G,\kappa^G) = H(\kappa).$$
Let $\alpha$ be the Bernoulli partition associated to $K$. Let $g_1,\ldots,g_n$ be $n$ distinct elements of $G$. It follows from the Bernoulli condition that the collection $\{g_i\alpha\}_{i=1}^n$ of partitions is independent. This means that if $j:\{1,\ldots,n\} \to K$ is any function then $$\kappa^G\Big( \bigcap_{i=1}^n ~g_iA_{j(i)} \Big) = \prod_{i=1}^n ~\kappa^G(A_{j(i)}).$$ It is well-known that this implies $$H\Big( \bigvee_{i=1}^n g_i\alpha \Big) = \sum_{i=1}^n H(g_i\alpha) = nH(\alpha).$$ See, for example, \[Gl03, prop. 14.19, page 257\]. So for any $k\ge 1$, $$\begin{aligned}
F(\alpha^k) &=& \Big(\frac{1}{2}\sum_{s \in S} H(\alpha^k \vee s\alpha^k)\Big) - (|S|-1)H(\alpha^k)\\
&=& \Big(\frac{1}{2} \sum_{s \in S} |B(e,k) \cup B(s,k)|H(\alpha)\Big) - (|S|-1)|B(e,k)|H(\alpha).\end{aligned}$$ Suppose $r >1$. Then, since $G=\langle s_1,\ldots,s_r \rangle$ is free, it is a short exercise to compute: $$|B(e,k)| = 1 + |S|\frac{(|S|-1)^k -1}{|S|-2},$$ $$|B(e,k)\cup B(s,k)| = 2\frac{(|S|-1)^{k+1} -1}{|S|-2}$$ for all $s\in S$. Thus, $$\begin{aligned}
F(\alpha^k) &=& H(\alpha)\Big(|S|\frac{(|S|-1)^{k+1} -1}{|S|-2} - (|S|-1) - (|S|-1) |S|\frac{(|S|-1)^k -1}{|S|-2}\Big)\\
& =& H(\alpha).\end{aligned}$$ If $r=1$ then $|B(e,k)|=2k+1$ and $|B(e,k)\cup B(s,k)| = 2k+2$. So it follows in a similar way that $F(\alpha^k)=H(\alpha)$. So $f(G,X,\mu)=\lim_{k \to\infty} F(\alpha^k)=H(\alpha)=H(\kappa)$.
According to Stepin’s theorem \[St75\], if $(K_1,\kappa_1), (K_2,\kappa_2)$ are standard Borel probability spaces with $H(\kappa_1)=H(\kappa_2)$ then $(G,K_1^G,\kappa_1^G)$ is measurably conjugate to $(G,K_2^G,\kappa_2^G)$.
Now suppose $(K_1,\kappa_1)$, $(K_2,\kappa_2)$ are Borel probability spaces such that $(G,K_1^G,\kappa_1^G)$ is measurably conjugate to $(G,K_2^G,\kappa_2^G)$. Let $(L_1,\lambda_1), (L_2,\lambda_2)$ be probability spaces with $|L_1| + |L_2|<\infty$ and $H(\lambda_i) = H(\kappa_i)$ for $i=1,2$. By Stepin’s theorem, $(G,L_i^G,\lambda_i^G)$ is measurably conjugate to $(G,K_i^G,\kappa_i^G)$. By the above theorem, $f(G,L_i^G,\lambda_i^G)=H(\lambda_i)$. Since $f$ is a measure-conjugacy invariant, $H(\kappa_1)=H(\kappa_2)$.
[10]{}
E. Glasner. *Ergodic theory via joinings.* Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. xii+384 pp.
A. Katok *Fifty years of entropy in dynamics: 1958–2007*. J. Mod. Dyn. 1 (2007), no. 4, 545–596.
J. C. Kieffer. *A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space*. Ann. Probability 3 (1975), no. 6, 1031–1037.
A. N. Kolmogorov. *A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces*. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 119 1958 861–864.
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D. Lind and B. Marcus. *An introduction to symbolic dynamics and coding*. Cambridge University Press, Cambridge, 1995. xvi+495 pp.
D. Ornstein. *Bernoulli shifts with the same entropy are isomorphic*. Advances in Math. 4 1970 337–352.
D. Ornstein. *Two Bernoulli shifts with infinite entropy are isomorphic*. Advances in Math. 5 1970 339–348.
D. Ornstein and B. Weiss. *Entropy and isomorphism theorems for actions of amenable groups*. J. Analyse Math. 48 (1987), 1–141.
W. Parry. *Entropy and generators in ergodic theory*. W. A. Benjamin, Inc., New York-Amsterdam 1969 xii+124 pp.
A. M. Stepin. *Bernoulli shifts on groups*. (Russian) Dokl. Akad. Nauk SSSR 223 (1975), no. 2, 300–302.
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[^1]: email:lpbowen@math.hawaii.edu
|
---
abstract: 'We carry out a numerical study of the quantum Hall ferromagnetism in a two-subband system using a set of experimental parameters in a recently experiment \[X. C. Zhang, I. Martin, and H. W. Jiang, Phys. Rev. B **74**, 073301 (2006)\]. Employing the self-consistence local density approximation for growth direction wave function and the Hartree-Fock theory for the pseudospin anisotropy energy, we are able to account for the easy-axis and easy-plane quantum Hall ferromagnetism observed at total filling factor $\nu = 3$ and $\nu= 4$, respectively. Our study provides some insight of how the anisotropy energy, which highly depends upon the distribution of growth direction wave functions, determines the symmetry of the quantum Hall ferromagnets.'
author:
- 'Xiao-Jie Hao$^{(1)}$'
- 'Tao Tu$^{(1)}$'
- 'Yong-Jie Zhao$^{(1)}$'
- 'Guang-Can Guo$^{(1)}$'
- 'H. W. Jiang$^{(2)}$'
- 'Guo-Ping Guo$^{(1)}$'
title: 'Numerical Studies of Quantum Hall Ferromagnetism in Two-Subband Systems'
---
Introduction {#sec:introdunction}
============
Multi-component quantum Hall systems have exhibited a collection of interesting phenomena which are the manifestation of electronic correlations [@review]. Such correlations become particularly prominent when two or more sets of Landau levels (LLs) with different layer, subband, valley, spin, or Landau level indices are brought into degeneracy [@Eisenstein1994; @KunYang1994; @Wescheider; @Shayegan]. In experimental systems, different LLs can be tuned to cross by varying gate voltage, charge density, magnetic field or the magnetic field tilted angle to the sample. One of the attractions is the formation of quantum Hall ferromagnets (QHFs) due to the exchange interactions of the two subbands states, termed as pseudospins [@QHF; @MacDonald1990]. Self-consistent local density approximation (SCLDA) and Hartree-Fock mean field method can be performed on the calculation of ground state energy and quasi-particle energy gap [@Wescheider; @MacDonald2000; @Shayegan1998; @MacDonald1990].
Recent experiments in single quantum well with two-subband occupied systems [@Hirayama; @Jiang2006], showed evidence of QHFs when two LLs were brought into degeneracy. The QHFs can either be easy-axis or easy-plane, depending on the details of LL crossing configurations [@Hirayama; @Jiang2006]. In this paper we follow the theoretical framework of Jungwirth and MacDonald [@MacDonald2000] and numerically calculate the pseudospin anisotropy energy using the sample parameters in the experiment of Zhang *et al.* [@Jiang2006]. The result confirms the QHFs taking place at total filling factor $\nu =3$ and $\nu =4$ are expected to be easy-plane and easy-axis QHFs, respectively. As we will discuss in this paper, the gate bias voltage, which affects the spatial distributions of the wave functions of the two subbands, plays a leading role in the formation of easy-plane and easy-axis QHFs.
Pseudospin Quantum Hall Ferromagnets {#sec:pseud-quant-hall}
====================================
The pseudospin representation is used to describe the valence LLs degenerated in the Fermi level [@MacDonald1990; @MacDonald2000]. Following the theoretic studies of pseudospin QHFs [@MacDonald1990; @MacDonald2000], here we focus on a two-subband two dimensional electron system, in which the LLs are labeled by $(\xi ,n,s)$, where $\xi =S/A$ is first/second subband (in the no biased quantum well, also called symmetry/antisymmetry subband) index, $n=0,1,\cdots $ is Landau level in-plane orbit radius quantum number, and $s=\pm \frac{1}{2}$ represents real spin. When two LLs are brought close to degeneracy but still sufficiently far from other LLs, one of them can be labeled as pseudospin up ($\sigma =\Uparrow $) and the other as pseudospin down ($\sigma =\Downarrow $). In the experimental work of Zhang *et al.* [@Jiang2006], as shown in Fig. \[fig:Landau\], we can label pseudospin up ($\sigma =\Uparrow $) as $(S,1,\frac{1}{2})$ and pseudospin down ($\sigma =\Downarrow $) as $(A,0,\frac{1}{2})$ at filling factor $\nu =3
$. At filling factor $\nu =4$, we label pseudospin up ($\sigma =\Uparrow $) as $(S,1,\mp \frac{1}{2})$ and pseudospin down ($\sigma =\Downarrow $) as $%
(A,0,\pm \frac{1}{2})$. Here the upper and lower signs before spin index refer to the crossing point at lower and higher magnetic field, respectively.
![Landau level diagram of two-subband system. Four Landau levels labeled by $(\protect\xi, n, s)$ are crossing to each other during the increasing of the magnetic field (see text for details). Three arrows point out the crossing point of Landau levels at filling factor $\protect\nu=3$ and $\protect\nu=4$. []{data-label="fig:Landau"}](fig1_LL.eps){width="0.7\columnwidth"}
At this moment, we would emphasize the essential similarity between two-subband system and ordinary bilayer system (double quantum well). On one hand, due to a separation of charges to opposite sides of the well originating from the Coulomb repulsion, a wide single quantum well can be modeled as an effective bilayer system [Wescheider,MacDonald2000,Abolfath]{}. Therefore in a single quantum well two-subband structure, the states of electrons can be characterized by two parameters similar to a bilayer system: the tunneling gap $\Delta _{SAS}$ and effective layer separation $d$. The $\Delta _{SAS}$ is chosen as equal to the difference between the lowest two subband energy levels and $d$ is given by the distance between two centers of the charge distribution in this single quantum well. On the other hand, let us assume the two dimensional electron gas is in the $x-y$ plane in the following calculation and discussion, so the sample growth direction is aligned with the $z$ axis. In a no biased bilayer system, although the layer wave function in $z$ direction is spatially separated in two layers, once included in the effect of tunneling between two layers $\Delta _{t}$ and rediagonalized the $z$ direction Hamiltonian, it will be just like the two-subband system which has symmetry and antisymmetry subbands. And the symmetry-antisymmetry gap $%
\Delta _{SAS}$ is equal to the inter-layer tunneling $\Delta _{t}$. Even when a bias voltage between two layers exists, the circumstance is more or less the same. Since they are theoretically identical, all the pseudospin language theory used in bilayer system is available in the two-subband system.
While pseudospin up and pseudospin down LLs are degenerate but the number of electrons is not enough to fill all the two LLs, electrons will stay in a broken-symmetry ground state. Actually, the state electrons choosing is a linear combination of two pseudospin LLs which minimizes the system total energy. Typically, the many-body ground state can be written as follows [MacDonald2000]{}: $$\label{eq:many-body-groud}
\left\vert \Psi\lbrack \hat{m}\rbrack \right\rangle =\prod_{k=1}^{N_{\phi
}}c_{\hat{m},k}^{\dagger}|0\rangle,$$ where $c_{\hat{m},k}^{\dagger }$ creates the single-particle state oriented in a certain unit vector $\hat{m}=(\sin{\theta}\cos{\varphi}, \sin{\theta}%
\sin{\varphi},\cos{\theta})$ with wave function: $$\label{eq:groud-state}
\psi _{\hat{m},k}(\vec{r})=\cos \big(\frac{\theta }{2}\big)\psi _{\Uparrow
,k}(\vec{r})+\sin \big(\frac{\theta }{2}\big)e^{i\varphi }\psi _{\Downarrow
,k}(\vec{r}).$$ Here $\psi _{\sigma ,k}(\vec{r})$ is the single-particle state wave function which contains growth direction subband wave function $\lambda_{\xi}(z)$ and in-plane LL wave function $L_{n, s, k}(x, y)$: $$\psi _{\sigma ,k}(\vec{r}) =\lambda _{\xi }(z)L_{n,s,k}(x, y).
\label{eq:single-particle}$$
Then the Hartree-Fock energy of the system can be obtained as following [MacDonald2000]{}: $$E_{HF}(\hat{m})\equiv \frac{\langle \Psi \lbrack \hat{m}]|H|\Psi \lbrack
\hat{m}]\rangle }{N_{\phi }}=-\sum_{i=x,y,z}\left( E_{i}-\frac{1}{2}U_{%
\mathbf{1},i}-\frac{1}{2}U_{i,\mathbf{1}}\right) m_{i}+\frac{1}{2}%
\sum_{i,j=x,y,z}U_{i,j}m_{i}m_{j}. \label{eq:HF_energy}$$Whether the ground state is easy-axis or easy-plane only depends on the quadratic coefficient in pseudospin magnetization $m_{i}$ (i.e. the pseudospin anisotropy energy $U_{xx}$, $U_{yy}$ and $U_{zz}$ ): $$U_{ij}=\frac{1}{4}\int \frac{d^{2}\vec{q}}{(2\pi )^{2}}v_{ij}(0)-\frac{1}{4}%
\int \frac{d^{2}\vec{q}}{(2\pi )^{2}}v_{ij}(\vec{q}).
\label{eq:anisotropy_energy}$$The first $\vec{q}=0$ term in Eq. (\[eq:anisotropy\_energy\]) is the Hartree term and the second term is the Fock term. $v_{ij}(\vec{q})$ can be expanded to sum of several pseudospin matrix elements $v_{\sigma
_{1}^{\prime },\sigma _{2}^{\prime },\sigma _{1},\sigma _{2}}(\vec{q})$, which are products of subband and the in-plane parts [@MacDonald2000]. If $U_{zz}<U_{xx}=U_{yy}$, the system is in easy-axis QHF, which means the pseudospin magnetization $\hat{m}$ is aligned either up or down; and if $%
U_{zz}>U_{xx}=U_{yy}$, the system is in easy-plane QHF, which is a coherent superposition of the two pseudospin LLs.
Growth Direction Wave Function Calculation Using SCLDA {#sec:lda}
======================================================
We numerically calculate the growth direction ($z$ direction) wave function $\lambda_{\xi}(z)$ in Eq. (\[eq:single-particle\]) using SCLDA to compare the pseudospin anisotropy energy $U_{xx}$ and $%
U_{zz}$ in Eq. (\[eq:HF\_energy\]) [@LDA]. In the SCLDA method, wave function $\lambda(z)$ is described by the Schrödinger equation: $$\label{eq:schrodinger}
\left(-\frac{1}{2m^{*}} \frac{\partial^{2}}{\partial z^{2}} + V_{b}(z) +
V_{gate}(z) + V_{xc}(z) + V_{H}(z) \right) \lambda_{i}(z) =
\varepsilon_{i}\lambda_{i}(z).$$ Here $m^{*}$ is the effective electron mass in GaAs, $V_{b}$ corresponds to the conduction band discontinuity, $V_{gate}$ is the bias potential caused by the difference of front and back gate voltage $\left\vert\Delta
V_{g}\right\vert$, $V_{xc}$ refers to the exchange-correlation potential related to the electron charge distribution $n(z)$ (We use the form of $%
V_{xc}$ given by Hedin and Lundqvist [@ExchangeCorrelation]). The Hartree term $V_{H}$ due to electrostatic potential is given in the Poisson equation: $$\label{eq:poisson}
V_{H}(z)=-\frac{2\pi e^{2}}{\epsilon} \int dz^{\prime}\left\vert
z-z^{\prime}\right\vert n(z^{\prime}).$$
The subband energies $\varepsilon_{\xi}$, wave functions $\lambda_{\xi}(z)$ and the electron charge distributions $n_{\xi}(z)$ of both subbands can be calculated by solving the Schrödinger equation Eq. (\[eq:schrodinger\]) and the Poisson equation Eq. (\[eq:poisson\]) simultaneously [Algorithm]{}.
Numerical Result and Discussion {#sec:numer-result-dissc}
===============================
Taking the unit of energy as $e^{2}/\epsilon
l_{B}$ ($l_{B}$ is the magnetic length), we calculate the pseudospin anisotropy energy $U_{xx}$, $U_{yy}$ and $U_{zz}$ at filling factor $\nu=3$ and $\nu=4$ in Zhang *et al.*’s work [@Jiang2006]. At total filling factor $\nu=3$, $U_{xx} \equiv U_{yy}=-0.370<U_{zz}=0.017$, so the system will stay in easy-plane QHF. At one degenerate point of total filling factor $\nu=4$, $U_{xx} \equiv U_{yy}=0>U_{zz}=-0.1266$, and at the other degenerate point $U_{xx} \equiv U_{yy}=0>U_{zz}=-0.1293$. Both of the energy differences of $U_{zz}-U_{xx}$ at $\nu=4$ indicate easy-axis QHF ground states.
To illustrate the evolution from easy-axis QHF to easy-plane QHF, we calculate the phase diagrams of $U_{zz}-U_{xx}$ as a function of bias gate voltage $\left\vert\Delta Vg\right\vert$ and total density $n$ at filling factor $\nu=3$ (Fig. \[fig:v3\_phase\]) and $\nu=4$ (Fig. \[fig:v4\_phase\]). In the following discussion, in order to make consistency, we choose $%
e^{2}/\epsilon l_{0}$ ($l_{0}=10 nm$) as the unit of energy. In the phase diagram (Fig. \[fig:phase\]), we label the density and bias voltage $%
\left\vert\Delta Vg\right\vert$ position where the crossing occurs in the experimental work of Zhang *et al.* [@Jiang2006]. In the left/blue (right/red) parts of each figure (Fig. \[fig:v3\_phase\] and Fig. \[fig:v4\_phase\]), where $U_{zz}< (>) U_{xx}$, easy-axis (easy-plane) has the lower electron interaction energy. From Fig. \[fig:phase\], we find that the anisotropy energy difference $U_{zz}-U_{xx}$ is very sensitive to the bias voltage $\left\vert\Delta Vg\right\vert$. If we could vary the gate voltage across the black line labeled $U_{xx}=U_{zz}$ from left to right in a determined density, a quantum phase transition from easy-axis to easy-plane QHF will happen.
![Energy components of anisotropy energy $U_{xx}$ and $U_{zz}$ v.s. bias voltage $\vert\Delta Vg\vert$. Except for $U_{xx}\equiv 0$ at $\protect%
\nu=4$, all the other terms are same at filling factor $\protect\nu=3$ and $%
\protect\nu=4$. $Uzz_{H}$ and $Uzz_{F}$ correspond to the Hartree term and Fock term in the Hartree-Fock calculation of $U_{zz}$. []{data-label="fig:energy"}](fig3_energy.eps){width="0.7\columnwidth"}
In order to make a comparison of the effect of each term in anisotropy energy, we plot them in Eq. (\[eq:anisotropy\_energy\]) as a function of $%
\left\vert\Delta Vg\right\vert$ at a certain density of $8.5%
\times10^{11}/cm^{2}$ in Fig. \[fig:energy\]. Here we have to point out that all the formulas at $\nu=3$ and $4$ are the same, except that $U_{xx}$ vanishes at $\nu=4$. So the Fig. \[fig:energy\] is applicable to both filling factor $\nu=3$ and $\nu=4$ only if in the $\nu=4$ situation the $%
U_{xx}$ term is set to a constant zero. The reason for the different behavior of $U_{xx}$ at filling factor $\nu=3$ and $\nu=4$ will be discussed in the following. Note that the dominated term in $U_{zz}-U_{xx}$ is the Hartree term $Uzz_{H}$, which is due to the electrostatic potential of electrons. It shows that the $Uzz_{H}$ term increases immediately with the increasing bias voltage $\left\vert\Delta Vg\right\vert$.
![(Color online) Effective subband separation $\left\vert\Delta
d\right\vert$ as a function of bias gate voltage $\left\vert\Delta
V_{g}\right\vert$ at the density of $8.5\times10^{11}/cm^{2}$. Three insets show the well profiles (black dashed lines) with first (blue solid lines) and second (red dash-dot lines) subbands wave functions at different bias voltage: $\left\vert\Delta V_{g}\right\vert = 0.0 V$, $0.3 V$, and $1.0 V$.[]{data-label="fig:d_Vg"}](fig4_d_Vg.eps){width="0.7\columnwidth"}
To examine what the role the bias voltage plays in the determination of easy-plane or easy-axis QHF, we give some SCLDA results in the Fig. [fig:d\_Vg]{}. We define an effective subband separation $\Delta d$ to describe the distance between the cores of first and second subbands wave functions. The $\left\vert \Delta d\right\vert $ as a function of bias voltage is plotted in Fig. \[fig:d\_Vg\], also with some demonstrations of quantum well configurations and subbands wave functions in different bias voltages ($%
\left\vert \Delta V_{g}\right\vert =0.0V$, $0.3V$, and $1.0V$). The data in Fig. \[fig:d\_Vg\] are all selected from the same density $8.5\times
10^{11}/cm^{2}$ as in Fig. \[fig:energy\]. It is obvious that the bias gate voltage changes the effective separation of two subbands while changing the well potential $V_{gate}(z)$ in Eq. (\[eq:schrodinger\]). The larger the bias voltage is added, the farther the lowest two subbands are separated. Since at a well separated $z$ direction wave function configuration, all the electrons near the Fermi level filling in the same subband, i.e. the same pseudospin level, will raise a larger electrostatic energy, the easy-axis QHF is not favorable. Thus an easy-plane QHF can save more Hartree energy in the larger bias voltage situation. On the other hand, when the potential of the quantum well maintains a good symmetry in the small bias voltage limit, electrons can pick up one of the two pseudospin levels to keep the Hartree energy minimal and avoid the energetic penalty from inter-subband tunneling as well.
As mentioned before, the difference of $U_{xx}$ between total filling factor $\nu=3$ and $\nu=4$ could be explained in the similar way. The anisotropy energy $U_{xx}$ or $U_{yy}$ is constituted of Hartree part and Fock part (see Eq. (\[eq:anisotropy\_energy\])). In our numerical calculation, we find that the Hartree terms of $U_{xx}$ and $U_{yy}$ are always zero at both $\nu=3$ and $\nu=4$. And the only non-zero term in $U_{xx}$ or $U_{yy}$ is the Fock term $Uxx_{F} \equiv Uyy_{F}<0$ at filling factor $\nu=3$, which owes to the exchange interaction. It implies that at filling factor $\nu=4$, where pseudospin up $\sigma=\Uparrow(S, 1, \mp\frac{1}{2})$ and pseudospin down $\sigma=\Downarrow(A, 0, \pm\frac{1}{2})$ have opposite real spins, the easy-plane anisotropy, in which the electrons stay in both subband equally, would cost much more exchange energy. But at $\nu=3$ the pseudospin up $%
\sigma=\Uparrow (S, 1, \frac{1}{2})$ and pseudospin down $\sigma=\Downarrow
(A, 0, \frac{1}{2})$ have the same spin. Then there is no such problem need to be considered. Therefore, the easy-axis QHF is more likely to happen at total filling factor $\nu=4$ than $\nu=3$.
On the basis of above discussion, we may summarize as following. The bias gate voltage added to the sample changes the quantum well profile in the growth direction as well as the spacial separation of the lowest two subbands wave functions. For a larger bias gate voltage, the potential of the well is much skewer, so the two subbands wave functions locate in the opposite side of the quantum well (inset of Fig. \[fig:d\_Vg\]). As a result, the Hartree energy will arise if all the electrons stay in one narrow subband or pseudospin level. Thus a easy-plane QHF, in which the electrons fill the two pseudospin levels equally, is more energetically favorable at a large bias gate voltage. In addition, a state with opposite real spins will expend more exchange energy, so the easy-plane QHF is more easily to form at a pseudospin configuration in which the two pseudospin level have the same real spin (filling factor $\nu =3$ in this paper).
Conclusion {#sec:conclusion}
==========
Using self-consistent local density approximation method and Hartree-Fock mean field theory, we calculated wave function in the growth direction and the anisotropy energy in the two-subband quantum Hall system. The data shows great consistent with the observed easy-plane and easy-axis quantum hall ferromagnets at filling factor $\nu =3$ and $\nu =4$ in the experiment of Zhang *et al.* [@Jiang2006] . Also, by analyzing the numerical result, we give an easy-to-understand explanation about the anisotropy occuring at these filling factors.
Acknowledgments {#sec:acknowlegements .unnumbered}
===============
This work was funded by National Fundamental Research Program, the Innovation funds from Chinese Academy of Sciences, NCET-04-0587, and National Natural Science Foundation of China (Grant No. 60121503, 10574126, 10604052).
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