text: QUADRATIC CHABAUTY AND RATIONAL POINTS I: p-ADIC HEIGHTS 41 Now using a slightly modiﬁed Mordell-Weil sieve on the sets X(Q ) , X(Q ) , and X(Q ) as described in Appendix , we ﬁnd that 41 U 73 U 101 U Q { ± ± ± ± ± } X( (i)) = ( 2 : 1 : 1), ( i : 4 : 1), (1 : 1 : 0) , or in other words, Theorem 8. We have X (37)(Q(i)) = {(±2i : ±1 : 1), (±1 : ±4 : 1), (i : ±1 : 0)}. 0 Remark 8. It is perhap√s interesting to note √that the computation of X(Q ) recovered the points (± −3, ±4) ∈ X (37)(Q( −3)) as well. 73 U 0 Acknowledgements It is a pleasure to thank Minhyong Kim for countless enlightening conversations, Ben Moonen and Michael Stoll for helpful suggestions, Harris Daniels and Álvaro § Lozano-Robledo for suggesting that we try the example in , and Steﬀen Müller for carrying out the Mordell-Weil sieve computation, as described in the appendix. Part of this paper builds on material in the thesis of the second author; he is very grateful to his examiners Guido Kings and Victor Flynn for several suggestions which have improved the present work and to the EPSRC for supporting this research. Appendix A. Applying the Mordell-Weil sieve, by J. Steffen Müller O Let K be a number ﬁeld with ring of integers and let X/K be a smooth ≥K projective geometrically irreducible curve of genus g 2 with Jacobian J/K of rank → r = rk(J/K). Fix an embedding ι : X (cid:44) J deﬁned over K. The Mordell-Weil sieve is a technique for obtaining information about rational points on X by combining information about the image of X(k ) inside J(k ) under ι for several primes v O v v of , where k is the residue ﬁeld at v. It was introduced by Scharaschkin [28]; furthKer informatvion on the case K = Q can be found, for instance, in [8] and [26]. Siksek [31] describes a variant of the Mordell-Weil sieve over number ﬁelds which is adapted to work well with his explicit Chabauty method over number ﬁelds introduced in loc. cit. The general idea of the Mordell-Weil sieve is as follows: Suppose for simplicity that there are no nontrivial K-torsion points on J (see [5, Remark 6.1] on how to remove this assumption). Also suppose that we know generators P , . . . , P of ⊂ 1 r J(K). Let M > 1 be an integer and let C J(K)/MJ(K) be a set of residue M classes c for which we want to show that the image of X(K) under ι does not map → to c under the canonical epimorphism π : J(K) J(K)/MJ(K). Let S be a ﬁnite O set of primes of such that X has good reduction at these primes and consider K the commutative diagram π◦ι (cid:47)(cid:47) X(K) J(K)/MJ(K) (cid:81) (cid:81) αS (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) ∈ X(k ) ∈ J(k )/MJ(k ) . v S v βS v S v v ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QUADRATIC

text: CHABAUTY

text: AND

text: RATIONAL

text: POINTS

text: I:

text: p-ADIC

text: HEIGHTS

text: Now

text: using

text: a

text: slightly

text: modiﬁed

text: Mordell-Weil

text: sieve

text: on

text: the

text: sets

text: X(Q

text: X(Q

text: and

text: X(Q

text: as

text: described

text: in

text: Appendix

text: we

text: ﬁnd

text: that

text: U

text: U

text: U

text: Q

text: X(

text: (i))

text: i

text: or

text: in

text: other

text: words,

text: Theorem

text: We

text: have

text: X

text: ±1

text: ±4

text: (i

text: ±1

text: Remark

text: It

text: is

text: perhap√s

text: interesting

text: to

text: note

text: √that

text: the

text: computation

text: of

text: X(Q

text: recovered

text: the

text: points

text: X

text: as

text: well.

text: U

text: Acknowledgements

text: It

text: is

text: a

text: pleasure

text: to

text: thank

text: Minhyong

text: Kim

text: for

text: countless

text: enlightening

text: conversations,

text: Ben

text: Moonen

text: and

text: Michael

text: Stoll

text: for

text: helpful

text: suggestions,

text: Harris

text: Daniels

text: and

text: Álvaro

text: Lozano-Robledo

text: for

text: suggesting

text: that

text: we

text: try

text: the

text: example

text: in

text: and

text: Steﬀen

text: Müller

text: for

text: carrying

text: out

text: the

text: Mordell-Weil

text: sieve

text: computation,

text: as

text: described

text: in

text: the

text: appendix.

text: Part

text: of

text: this

text: paper

text: builds

text: on

text: material

text: in

text: the

text: thesis

text: of

text: the

text: second

text: author;

text: he

text: is

text: very

text: grateful

text: to

text: his

text: examiners

text: Guido

text: Kings

text: and

text: Victor

text: Flynn

text: for

text: several

text: suggestions

text: which

text: have

text: improved

text: the

text: present

text: work

text: and

text: to

text: the

text: EPSRC

text: for

text: supporting

text: this

text: research.

text: Appendix

text: A.

text: Applying

text: the

text: Mordell-Weil

text: sieve,

text: by

text: J.

text: Steffen

text: Müller

text: O

text: Let

text: K

text: be

text: a

text: number

text: ﬁeld

text: with

text: ring

text: of

text: integers

text: and

text: let

text: X/K

text: be

text: a

text: smooth

text: ≥K

text: projective

text: geometrically

text: irreducible

text: curve

text: of

text: genus

text: g

text: with

text: Jacobian

text: J/K

text: of

text: rank

text: r

text: rk(J/K).

text: Fix

text: an

text: embedding

text: ι

text: X

text: J

text: deﬁned

text: over

text: K.

text: The

text: Mordell-Weil

text: sieve

text: is

text: a

text: technique

text: for

text: obtaining

text: information

text: about

text: rational

text: points

text: on

text: X

text: by

text: combining

text: information

text: about

text: the

text: image

text: of

text: X(k

text: inside

text: J(k

text: under

text: ι

text: for

text: several

text: primes

text: v

text: O

text: v

text: v

text: of

text: where

text: k

text: is

text: the

text: residue

text: ﬁeld

text: at

text: v.

text: It

text: was

text: introduced

text: by

text: Scharaschkin

text: furthKer

text: informatvion

text: on

text: the

text: case

text: K

text: Q

text: can

text: be

text: found,

text: for

text: instance,

text: in

text: and

text: Siksek

text: describes

text: a

text: variant

text: of

text: the

text: Mordell-Weil

text: sieve

text: over

text: number

text: ﬁelds

text: which

text: is

text: adapted

text: to

text: work

text: well

text: with

text: his

text: explicit

text: Chabauty

text: method

text: over

text: number

text: ﬁelds

text: introduced

text: in

text: loc.

text: cit.

text: The

text: general

text: idea

text: of

text: the

text: Mordell-Weil

text: sieve

text: is

text: as

text: follows:

text: Suppose

text: for

text: simplicity

text: that

text: there

text: are

text: no

text: nontrivial

text: K-torsion

text: points

text: on

text: J

text: (see

text: Remark

text: on

text: how

text: to

text: remove

text: this

text: assumption).

text: Also

text: suppose

text: that

text: we

text: know

text: generators

text: P

text: P

text: of

text: r

text: J(K).

text: Let

text: M

text: be

text: an

text: integer

text: and

text: let

text: C

text: J(K)/MJ(K)

text: be

text: a

text: set

text: of

text: residue

text: M

text: classes

text: c

text: for

text: which

text: we

text: want

text: to

text: show

text: that

text: the

text: image

text: of

text: X(K)

text: under

text: ι

text: does

text: not

text: map

text: to

text: c

text: under

text: the

text: canonical

text: epimorphism

text: π

text: J(K)

text: J(K)/MJ(K).

text: Let

text: S

text: be

text: a

text: ﬁnite

text: O

text: set

text: of

text: primes

text: of

text: such

text: that

text: X

text: has

text: good

text: reduction

text: at

text: these

text: primes

text: and

text: consider

text: K

text: the

text: commutative

text: diagram

text: π◦ι

text: X(K)

text: J(K)/MJ(K)

text: αS

text: X(k

text: J(k

text: )/MJ(k

text: v

text: S

text: v

text: βS

text: v

text: S

text: v

text: v

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Submitted to ’Chinese Physics C’ 1.0 cooling, the space charge eﬀect is restricted remarkably 0.8 in the high intensity case. The number of particles re- × 0.6 tained after 6 injections reaches 5.79 1011 ppp, corre- 0.4 sponding to a stacking eﬃciency of 96.5%. In future, the 0.2 space charge eﬀect in the stacking process will be further 0.0 investigated. -0.2 6 Acknowledgments -0.4 -0.6 The authors are very grateful for the valuable advice -0.8 from A. Sidorin (JINR), and help from T. Katayama -1.0 (Nihon University). -10 -8 -6 -4 -2 0 2 4 6 8 10 x (mm) Fig. 8. Particle distribution in horizontal phase space. References 10 8 1 J. C. Yang, J. W. Xia, G. Q. Xiao et al, Nuclear Instruments 6 and Methods in Physics Research Section B, 317: 263 (2013) 4 2 H. H. Gutbrod, I. Augustin, H. Eickhoﬀ et al, FAIR Baseline 2 Technical Report (Darmstadt: GSI, 2006), p. 429 3 J. E. Griﬃn, C. Ankenbrandt, J. A. MacLachlan et al, IEEE 0 Trans. Nucl. Sci., NS-30(4): 3502 (1983) -2 4 T. Katayama, C. Dimopoulou, F. Nolden et al, Simulation -4 study of barrier bucket accumulation with stochastic cooling -6 at GSI ESR, in Proceedings of COOL’11, edited by I. Meshkov -8 (Alushta: JINR, 2011), p. 136 5 http://betacool.jinr.ru/reports/Uppsala%20Uni 2007 March.pdf, -10 retrieved 22th March 2016 -10 -8 -6 -4 -2 0 2 4 6 8 10 6 M. Steck, C. Dimopoulou, B. Franzke et al, Demonstration of x (mm) Longitudinal Stacking in the ESR with Barrier Buckets and Stochastic Cooling, in Proceedings of COOL’11, edited by I. Fig. 9. Particle distribution in transverse real space. Meshkov (Alushta: JINR, 2011), p. 140 7 C. M. Bhat, Applications of barrier bucket RF systems at Fer- 0.10 milab, in PRIA2006, edited by K. Takayama (Tsukuba: KEK, 2006), p. 156 0.08 8 M. Fujieda, Y. Iwashita, A. Noda et al, Phys. Rev. ST Accel. Beams, 2: 122001 (1999) 9 http://www1.jinr.ru/Preprints/2013/eng 053(D9-2013- 0.06 53).pdf, retrieved 22th March 2016 10 H. Poth, Physics Report, 196: 135 (1990) 0.04 11 A. Hofmann, Tune shifts from self ﬁelds and images, in CAS - CERN Accelerator School: 5th General Accelerator Physics Course, edited by S. Turner (Jyva¨skyla¨: CERN, 1994), p. 0.02 329C348 12 A. V. Smirnov, I. Meshkov, A. O. Sidorin et al, Necessary Con- 0.00 dition for Beam Ordering, in Proceedings of COOL’07, edited -0.0006 -0.0004 -0.0002 0.0000 0.0002 0.0004 0.0006 by Volker RW Schaa (Bad Kreuznach: GSI, 2007), p. 87 dp/p 13 A. O. Sidorin, I. N. Meshkov, I. A. Seleznev et al, Nuclear In- struments and Methods in Physics Research Section A, 558: Fig. 10. Distribution of momentum deviations. 325 (2006) 14 http://betacool.jinr.ru/programs/Betacool%20Physics%20Guide.pdf, retrieved 22th March 2016 5 Conclusion 15 V. V. Parkhomchuk, Nuclear Instruments and Methods in Physics Research Section A, 441: 9 (2000) 16 L. J. Mao, J. C.Yang, J. W. Xia et al, Nuclear Instruments and The simulation and optimization of BB stacking in Methods in Physics Research Section A, 786: 91 (2015) the CRing at HIAF have been performed. The rela- 17 A. Piwinski, Synchro-betatron resonances, in CAS - CERN tions between stacking parameters and eﬃciency were Accelerator School: Advanced Accelerator Physics, edited by researched based on the simulation code BETACOOL. S. Turner (Oxford: CERN, 1987), p. 187C202 18 M. Martini, CERN preprint, CERN-PS-84-9-AA/27 The results demonstrate that BB stacking is an eﬀective 19 J.D. Bjorken, S. Mtingwa, Particle Accelerators, 13: 115 (1983) way to achieve intensity gain. With oﬀ-aligned electron 20 S. Abeyratne et al, arXiv:1209.0757 010201-5 )darm( ’x )mm( y rebmun elcitrap dezilamron ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Submitted

label: paragraph

text: to

text: ’Chinese

text: Physics

text: C’

font: default

text: cooling,

text: the

text: space

text: charge

text: eﬀect

text: is

text: restricted

text: remarkably

text: in

text: the

text: high

text: intensity

text: case.

text: The

text: number

text: of

text: particles

text: re-

text: tained

text: after

text: injections

text: reaches

text: ppp,

text: corre-

text: sponding

text: to

text: a

text: stacking

text: eﬃciency

text: of

text: In

text: future,

text: the

text: space

text: charge

text: eﬀect

text: in

text: the

text: stacking

text: process

text: will

text: be

text: further

text: investigated.

label: section

text: Acknowledgments

text: The

text: authors

text: are

text: very

text: grateful

text: for

text: the

text: valuable

text: advice

text: from

text: A.

text: Sidorin

text: (JINR),

text: and

text: help

text: from

text: T.

text: Katayama

text: (Nihon

text: University).

text: x

text: (mm)

text: Fig.

text: Particle

text: distribution

text: in

text: horizontal

text: phase

text: space.

text: References

text: J.

text: C.

text: Yang,

text: J.

text: W.

text: Xia,

text: G.

text: Q.

text: Xiao

text: et

text: al,

text: Nuclear

text: Instruments

text: and

text: Methods

text: in

text: Physics

text: Research

text: Section

text: B,

text: H.

text: H.

text: Gutbrod,

text: I.

text: Augustin,

text: H.

text: Eickhoﬀ

text: et

text: al,

text: FAIR

text: Baseline

text: Technical

text: Report

text: (Darmstadt:

text: GSI,

text: p.

text: J.

text: E.

text: Griﬃn,

text: C.

text: Ankenbrandt,

text: J.

text: A.

text: MacLachlan

text: et

text: al,

text: IEEE

text: Trans.

text: Nucl.

text: Sci.,

text: T.

text: Katayama,

text: C.

text: Dimopoulou,

text: F.

text: Nolden

text: et

text: al,

text: Simulation

text: study

text: of

text: barrier

text: bucket

text: accumulation

text: with

text: stochastic

text: cooling

text: at

text: GSI

text: ESR,

text: in

text: Proceedings

text: of

text: edited

text: by

text: I.

text: Meshkov

text: (Alushta:

text: JINR,

text: p.

text: March.pdf,

text: retrieved

text: March

text: M.

text: Steck,

text: C.

text: Dimopoulou,

text: B.

text: Franzke

text: et

text: al,

text: Demonstration

text: of

text: x

text: (mm)

text: Longitudinal

text: Stacking

text: in

text: the

text: ESR

text: with

text: Barrier

text: Buckets

text: and

text: Stochastic

text: Cooling,

text: in

text: Proceedings

text: of

text: edited

text: by

text: I.

text: Fig.

text: Particle

text: distribution

text: in

text: transverse

text: real

text: space.

text: Meshkov

text: (Alushta:

text: JINR,

text: p.

text: C.

text: M.

text: Bhat,

text: Applications

text: of

text: barrier

text: bucket

text: RF

text: systems

text: at

text: Fer-

text: milab,

text: in

text: edited

text: by

text: K.

text: Takayama

text: (Tsukuba:

text: KEK,

text: p.

text: M.

text: Fujieda,

text: Y.

text: Iwashita,

text: A.

text: Noda

text: et

text: al,

text: Phys.

text: Rev.

text: ST

text: Accel.

text: Beams,

text: retrieved

text: March

text: H.

text: Poth,

text: Physics

text: Report,

text: A.

text: Hofmann,

text: Tune

text: shifts

text: from

text: self

text: ﬁelds

text: and

text: images,

text: in

text: CAS

text: CERN

text: Accelerator

text: School:

text: General

text: Accelerator

text: Physics

text: Course,

text: edited

text: by

text: S.

text: Turner

text: (Jyva¨skyla¨:

text: CERN,

text: p.

text: A.

text: V.

text: Smirnov,

text: I.

text: Meshkov,

text: A.

text: O.

text: Sidorin

text: et

text: al,

text: Necessary

text: Con-

text: dition

text: for

text: Beam

text: Ordering,

text: in

text: Proceedings

text: of

text: edited

text: by

text: Volker

text: RW

text: Schaa

text: (Bad

text: Kreuznach:

text: GSI,

text: p.

text: dp/p

text: A.

text: O.

text: Sidorin,

text: I.

text: N.

text: Meshkov,

text: I.

text: A.

text: Seleznev

text: et

text: al,

text: Nuclear

text: In-

text: struments

text: and

text: Methods

text: in

text: Physics

text: Research

text: Section

text: A,

text: Fig.

text: Distribution

text: of

text: momentum

text: deviations.

text: retrieved

text: March

text: Conclusion

text: V.

text: V.

text: Parkhomchuk,

text: Nuclear

text: Instruments

text: and

text: Methods

text: in

text: Physics

text: Research

text: Section

text: A,

text: L.

text: J.

text: Mao,

text: J.

text: C.Yang,

text: J.

text: W.

text: Xia

text: et

text: al,

text: Nuclear

text: Instruments

text: and

text: The

text: simulation

text: and

text: optimization

text: of

text: BB

text: stacking

text: in

text: Methods

text: in

text: Physics

text: Research

text: Section

text: A,

text: the

text: CRing

text: at

text: HIAF

text: have

text: been

text: performed.

text: The

text: rela-

text: A.

text: Piwinski,

text: Synchro-betatron

text: resonances,

text: in

text: CAS

text: CERN

text: tions

text: between

text: stacking

text: parameters

text: and

text: eﬃciency

text: were

text: Accelerator

text: School:

text: Advanced

text: Accelerator

text: Physics,

text: edited

text: by

text: researched

text: based

text: on

text: the

text: simulation

text: code

text: BETACOOL.

text: S.

text: Turner

text: (Oxford:

text: CERN,

text: p.

text: M.

text: Martini,

text: CERN

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text: 1 Distributed Storage in Mobile Wireless Networks with Device-to-Device Communication Jesper Pedersen, Alexandre Graell i Amat, Senior Member, IEEE, Iryna Andriyanova, Member, IEEE, and Fredrik Bra¨nnstro¨m, Member, IEEE Abstract— We consider the use of distributed storage (DS) to devices to alleviate the wireless bottleneck has attracted a reduce the communication cost of content delivery in wireless signiﬁcant interest in the research community in the recent networks. Content is stored (cached) in a number of mobile years [5]–[8]. In all these works, simple content caching and/or devices using an erasure correcting code. Users retrieve content replication (i.e., a number of copies of a content are stored in from other devices using device-to-device communication or from the base station (BS), at the expense of higher communication the network) is considered. Additionally, the use of maximum cost. We address the repair problem when a device storing data distance separable (MDS) codes to facilitate decentralized leaves the cell. We introduce a repair scheduling where repair is random caching was investigated in [8]. performed periodically and derive analytical expressions for the A relevant problem in D2D-assisted mobile caching net- overall communication cost of content download and data repair works is the repairing of the lost data when a storage device as a function of the repair interval. The derived expressions are then used to evaluate the communication cost entailed by DS is unavailable, e.g., when a storage device fails or leaves the using several erasure correcting codes. Our results show that DS network. Repairing of the lost data was considered in [9], can reduce the communication cost with respect to the case where where the communication cost incurred by data download and content is downloaded only from the BS, provided that repairs are repair was analyzed for a caching scheme where data is stored performed frequently enough. If devices storing content arrive to in the mobile devices using replication and regenerating codes the cell, the communication cost using DS is further reduced and, for large enough arrival rate, it is always beneﬁcial. Interestingly, [10]. A strong assumption in [9] is that the repair of the lost we show that MDS codes, which do not perform well for classical content is performed instantaneously. As a result, content can DS, can yield a low overall communication cost in wireless DS. always be downloaded from the mobile devices. Under the Index Terms—Caching, content delivery, device-to-device com- assumption of instantaneous repair, the caching strategy that munication, distributed storage, erasure correcting codes. minimizes the overall communication cost is 2-replication. In this paper, we consider content caching in a wireless network scenario using erasure correcting codes. When using I. INTRODUCTION I erasure correcting codes to cache content, caching bears strong T IS predicted that the global mobile data trafﬁc will exceed ties with the concept of distributed storage (DS) for reliable 30 exabytes per month by 2020, nearly a tenfold increase data storage. Indeed, the set of mobile devices storing content compared to the trafﬁc in 2015 [1]. This dramatic increase can be seen as a distributed storage network. The fundamental threatens to completely congest the already burdened wireless difference with respect to DS for reliable data storage is networks. One popular approach to reduce peak trafﬁc is to that data download can be done not only from the storage store popular content closer to the end users, a technique nodes, but the base station (BS) can also assist to deliver known as caching. The idea is to deploy a number of access the data. Therefore, the strict guarantees on fault tolerance points (called helpers) with large storage capacity, but low-rate can be relaxed, which brings new and interesting degrees of wireless backhaul, and store data across them [2], [3]. Users freedom with respect to erasure-correcting coding for DS for can then download content from the helpers, resulting in a reliable data storage. Here, to avoid confusion with standard higher throughput per user. In [4] it was suggested to store (uncoded) caching, we will use the term wireless distributed content directly in the mobile devices, taking advantage of the storage, highlighting the resemblance with DS using erasure high storage capacity of modern smart phones and tablets. correcting codes for reliable data storage in, e.g., data centers. The requested content can then be directly retrieved from Similar to the scenario in [9], we consider a cellular system neighbouring mobile devices, using device-to-device (D2D) where mobile devices roam in and out of a cell according communication. This allows for a more efﬁcient content deliv- to a Poisson random process and request content at random ery at no additional infrastructure cost. Caching in the mobile times. The cell is served by a BS, which always has access to the content. Content is also stored across a limited number This paper was presented in part at the IEEE Information Theory Workshop, of mobile devices using an erasure correcting code. Our main Jeju Island, Korea, October 2015. focus is on the repair problem when a device that stores data This work was partially funded by the Swedish Research Council under grants 2011-5961 and 2011-5950, and by the European Research Council leaves the network. In particular, we introduce a more realistic under Grant No. 258418 (COOPNET). repair scheduling than the one in [9] where lost content is J. Pedersen, A. Graell i Amat, and F. Bra¨nnstro¨m are with the De- repaired (from storage devices using D2D communication or partment of Signals and Systems, Chalmers University of Technology, SE-41296 Gothenburg, Sweden (e-mail: {jesper.pedersen, alexandre.graell, from the BS) at periodic times. fredrik.brannstrom}@chalmers.se). We derive analytical, closed-form expressions for the overall I. Andriyanova is with the ETIS-UMR8051 group, EN- communication cost of content download and data repair as SEA/University of Cergy-Pontoise/CNRS, 95015 Cergy, France (e-mail: iryna.andriyanova@ensea.fr). a function of the repair interval. The derived expressions are

text: Distributed

label: title

text: Storage

text: in

text: Mobile

text: Wireless

text: Networks

text: with

text: Device-to-Device

text: Communication

text: Jesper

label: author

text: Pedersen,

text: Alexandre

text: Graell

text: i

text: Amat,

text: Senior

text: Member,

text: IEEE,

text: Iryna

text: Andriyanova,

text: Member,

text: IEEE,

text: and

text: Fredrik

text: Bra¨nnstro¨m,

text: Member,

text: IEEE

text: Abstract—

text: We

label: abstract

text: consider

text: the

text: use

text: of

text: distributed

text: storage

text: (DS)

text: to

text: devices

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text: alleviate

text: the

text: wireless

text: bottleneck

text: has

text: attracted

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text: communication

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text: interest

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text: the

text: research

text: community

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text: the

text: recent

text: networks.

text: Content

text: is

text: stored

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text: number

text: of

text: mobile

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text: erasure

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text: Users

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text: a

text: number

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text: other

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text: communication

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text: the

text: base

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text: (BS),

text: at

text: the

text: expense

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text: higher

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text: network)

text: is

text: considered.

text: Additionally,

text: the

text: use

text: of

text: maximum

text: cost.

text: We

text: address

text: the

text: repair

text: problem

text: when

text: a

text: device

text: storing

text: data

text: distance

text: separable

text: (MDS)

text: codes

text: to

text: facilitate

text: decentralized

text: leaves

text: the

text: cell.

text: We

text: introduce

text: a

text: repair

text: scheduling

text: where

text: repair

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text: random

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text: investigated

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text: periodically

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text: derive

text: analytical

text: expressions

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text: the

text: A

text: relevant

text: problem

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text: mobile

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text: overall

text: communication

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text: of

text: content

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text: data

text: repair

text: works

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text: the

text: repairing

text: of

text: the

text: lost

text: data

text: when

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text: storage

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text: as

text: a

text: function

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text: repair

text: interval.

text: The

text: derived

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text: communication

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text: e.g.,

text: when

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text: Our

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text: BS,

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text: analyzed

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text: caching

text: scheme

text: where

text: data

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text: stored

text: performed

text: frequently

text: enough.

text: If

text: devices

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text: mobile

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text: cell,

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text: strong

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text: MDS

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text: DS,

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text: mobile

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text: the

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text: content

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text: instantaneous

text: repair,

text: the

text: caching

text: strategy

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text: munication,

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text: codes.

text: minimizes

text: the

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text: In

text: this

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text: caching.

text: The

text: idea

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text: data.

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text: the

text: strict

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text: fault

text: tolerance

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text: with

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text: storage

text: capacity,

text: but

text: low-rate

text: can

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text: relaxed,

text: which

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text: new

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text: wireless

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text: caching,

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text: storage,

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text: access

text: to

text: the

text: content.

text: Content

text: is

text: also

text: stored

text: across

text: a

text: limited

text: number

text: This

text: paper

text: was

text: presented

text: in

text: part

text: at

text: the

text: IEEE

text: Information

text: Theory

text: Workshop,

text: of

text: mobile

text: devices

text: using

text: an

text: erasure

text: correcting

text: code.

text: Our

text: main

text: Jeju

text: Island,

text: Korea,

text: October

text: focus

text: is

text: on

text: the

text: repair

text: problem

text: when

text: a

text: device

text: that

text: stores

text: data

text: This

text: work

text: was

text: partially

text: funded

text: by

text: the

text: Swedish

text: Research

text: Council

text: under

text: grants

text: and

text: and

text: by

text: the

text: European

text: Research

text: Council

text: leaves

text: the

text: network.

text: In

text: particular,

text: we

text: introduce

text: a

text: more

text: realistic

text: under

text: Grant

text: No.

text: (COOPNET).

text: repair

text: scheduling

text: than

text: the

text: one

text: in

text: where

text: lost

text: content

text: is

text: J.

text: Pedersen,

text: A.

text: Graell

text: i

text: Amat,

text: and

text: F.

text: Bra¨nnstro¨m

text: are

text: with

text: the

text: De-

text: repaired

text: (from

text: storage

text: devices

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text: communication

text: or

text: partment

text: of

text: Signals

text: and

text: Systems,

text: Chalmers

text: University

text: of

text: Technology,

text: Gothenburg,

text: Sweden

text: (e-mail:

text: {jesper.pedersen,

text: alexandre.graell,

text: from

text: the

text: BS)

text: at

text: periodic

text: times.

text: fredrik.brannstrom}@chalmers.se).

text: We

text: derive

text: analytical,

text: closed-form

text: expressions

text: for

text: the

text: overall

text: I.

text: Andriyanova

text: is

text: with

text: the

text: group,

text: EN-

text: communication

text: cost

text: of

text: content

text: download

text: and

text: data

text: repair

text: as

text: SEA/University

text: of

text: Cergy-Pontoise/CNRS,

text: Cergy,

text: France

text: (e-mail:

text: iryna.andriyanova@ensea.fr).

text: a

text: function

text: of

text: the

text: repair

text: interval.

text: The

text: derived

text: expressions

text: are


text: 2.2 Projectivity: the consistency assumption A ranking model is a family of probability distributions that characterize the variability of a statistical population of rankings. In the case of full rankings, the statistical population is only composed of random permutations, and a ranking model reduces to one probability distribution p over the symmetric group S . But when one considers partial or incomplete rankings, they usually n are of various types, and the global variability of the statistical population is characterized by a family of probability distributions, one over the rankings of each type. In the case of top-k rankings − for instance, the number k usually varies from 1 to n 1 between observations, and the global variability of the statistical population is characterized by a family (P ) ≤ ≤ − where for each ∈ { − } k 1 k n 1 k 1, . . . , n 1 , P is a probability distribution over the set of k-tuples with distinct elements k (see Busse et al., 2007, for instance). Incomplete rankings are rankings on subsets of items. The varying parameter in a statistical population of incomplete rankings is thus the subset of items involved in each ranking. Let us ≤ ≤ introduce some notations. For distinct items a , . . . , a with 2 k n, we simply denote the (cid:31) · · · (cid:31) 1 k incomplete ranking a a by the expression π = a . . . a . Such an expression is called 1 k { }1 k an injective word, its content is the set c(π) = a , . . . , a and its length or size is the number | | ∈(cid:74) (cid:75) 1 k π = k. The rank of the item i c(π) in the ranking π is denoted by π(i). We denote by Γ the (cid:70) (cid:74) (cid:75) { ∈ |(cid:74) (cid:75) } n set of all incomplete rankings on n and by Γ(A) = π Γ c(π) = A the set of incomplete ∈ P n rankings with content A, for any A ( n ). Notice that Γ( n ) corresponds to S and that (cid:74) (cid:75) (cid:74) (cid:75) n Γ = ∈P Γ(A). Equipped with these notations, a ranking model for incomplete rankings is n A ( n ) (cid:74) (cid:75) ∈ P a family (P ) ∈P where for each A ( n ), P is a probability distribution over the set Γ(A) A A ( n ) A of rankings on A. Example 1. For n = 3, { } (cid:116) { } (cid:116) { } (cid:116) { } Γ = 12, 21 13, 31 23, 32 123, 132, 213, 231, 312, 321 3 { } { } { } { } ≡ Γ( 1, 2 ) Γ( 1, 3 ) Γ( 2, 3 ) Γ( 1, 2, 3 ) S 3 If there were no relationship between the diﬀerent probability distributions of a ranking model, the statistical analysis of partial and/or incomplete rankings would boil down to independent anal- yses for each type of ranking. Yet one should be able to transfer information from the observation of one type of ranking to another. In a context of top-k rankings analysis, if for instance item a appears very frequently in top-1 rankings, it is natural to expect that it be ranked in high position in top-k rankings with larger values of k, and reciprocally, if it is usually ranked high in top-k rankings, then its probability of being top-1 should be high. The same intuition holds for incom- plete rankings. If item a is usually preferred to item b in pairwise comparisons then rankings on { } a, b, c that place a before b should have higher probabilities than the others. Reciprocally if such rankings appear more frequently than the others, then item a should be preferred to item b with high probability in a pairwise comparison. The ranking literature thus relies on one fundamental assumption: the observed rankings in a statistical population of interest are induced by full rankings drawn from a single probability (cid:74) (cid:75) ∈ ≺ distribution p over S (see for instance Luce, 1977). Permutation σ S induces ranking or n ≺ ∈ (cid:31) n ⇒ equivalently is a linear extension of ranking if for all a, b n , a b σ(a) < σ(b). The (cid:88) ≺ probability that a random permutation Σ drawn from p induces a ranking is thus equal to P ∈ ≺ [Σ S ( )] = p(σ), (1) n ∈ ≺ σ Sn( ) 8

text: Projectivity:

text: the

text: consistency

text: assumption

text: A

text: ranking

text: model

text: is

text: a

text: family

text: of

text: probability

text: distributions

text: that

text: characterize

text: the

text: variability

text: of

text: a

text: statistical

text: population

text: of

text: rankings.

text: In

text: the

text: case

text: of

text: full

text: rankings,

text: the

text: statistical

text: population

text: is

text: only

text: composed

text: of

text: random

text: permutations,

text: and

text: a

text: ranking

text: model

text: reduces

text: to

text: one

text: probability

text: distribution

text: p

text: over

text: the

text: symmetric

text: group

text: S

text: But

text: when

text: one

text: considers

text: partial

text: or

text: incomplete

text: rankings,

text: they

text: usually

text: n

text: are

text: of

text: various

text: types,

text: and

text: the

text: global

text: variability

text: of

text: the

text: statistical

text: population

text: is

text: characterized

text: by

text: a

text: family

text: of

text: probability

text: distributions,

text: one

text: over

text: the

text: rankings

text: of

text: each

text: type.

text: In

text: the

text: case

text: of

text: top-k

text: rankings

text: for

text: instance,

text: the

text: number

text: k

text: usually

text: varies

text: from

text: to

text: n

text: between

text: observations,

text: and

text: the

text: global

text: variability

text: of

text: the

text: statistical

text: population

text: is

text: characterized

text: by

text: a

text: family

text: (P

text: where

text: for

text: each

text: k

text: k

text: n

text: k

text: n

text: P

text: is

text: a

text: probability

text: distribution

text: over

text: the

text: set

text: of

text: k-tuples

text: with

text: distinct

text: elements

text: k

text: (see

text: Busse

text: et

text: al.,

text: for

text: instance).

text: Incomplete

text: rankings

text: are

text: rankings

text: on

text: subsets

text: of

text: items.

text: The

text: varying

text: parameter

text: in

text: a

text: statistical

text: population

text: of

text: incomplete

text: rankings

text: is

text: thus

text: the

text: subset

text: of

text: items

text: involved

text: in

text: each

text: ranking.

text: Let

text: us

text: introduce

text: some

text: notations.

text: For

text: distinct

text: items

text: a

text: a

text: with

text: k

text: n,

text: we

text: simply

text: denote

text: the

text: k

text: incomplete

text: ranking

text: a

text: a

text: by

text: the

text: expression

text: π

text: a

text: a

text: Such

text: an

text: expression

text: is

text: called

text: k

text: k

text: an

text: injective

text: word,

text: its

text: content

text: is

text: the

text: set

text: c(π)

text: a

text: a

text: and

text: its

text: length

text: or

text: size

text: is

text: the

text: number

text: k

text: π

text: k.

text: The

text: rank

text: of

text: the

text: item

text: i

text: c(π)

text: in

text: the

text: ranking

text: π

text: is

text: denoted

text: by

text: π(i).

text: We

text: denote

text: by

text: Γ

text: the

text: n

text: set

text: of

text: all

text: incomplete

text: rankings

text: on

text: n

text: and

text: by

text: Γ(A)

text: π

text: Γ

text: c(π)

text: A

text: the

text: set

text: of

text: incomplete

text: P

text: n

text: rankings

text: with

text: content

text: A,

text: for

text: any

text: A

text: n

text: Notice

text: that

text: Γ(

text: n

text: corresponds

text: to

text: S

text: and

text: that

text: n

text: Γ

text: ∈P

text: Γ(A).

text: Equipped

text: with

text: these

text: notations,

text: a

text: ranking

text: model

text: for

text: incomplete

text: rankings

text: is

text: n

text: A

text: n

text: P

text: a

text: family

text: (P

text: ∈P

text: where

text: for

text: each

text: A

text: n

text: P

text: is

text: a

text: probability

text: distribution

text: over

text: the

text: set

text: Γ(A)

text: A

text: A

text: n

text: A

text: of

text: rankings

text: on

text: A.

text: Example

text: For

text: n

text: Γ

text: Γ(

text: Γ(

text: Γ(

text: Γ(

text: S

text: If

text: there

text: were

text: no

text: relationship

text: between

text: the

text: diﬀerent

text: probability

text: distributions

text: of

text: a

text: ranking

text: model,

text: the

text: statistical

text: analysis

text: of

text: partial

text: and/or

text: incomplete

text: rankings

text: would

text: boil

text: down

text: to

text: independent

text: anal-

text: yses

text: for

text: each

text: type

text: of

text: ranking.

text: Yet

text: one

text: should

text: be

text: able

text: to

text: transfer

text: information

text: from

text: the

text: observation

text: of

text: one

text: type

text: of

text: ranking

text: to

text: another.

text: In

text: a

text: context

text: of

text: top-k

text: rankings

text: analysis,

text: if

text: for

text: instance

text: item

text: a

text: appears

text: very

text: frequently

text: in

text: rankings,

text: it

text: is

text: natural

text: to

text: expect

text: that

text: it

text: be

text: ranked

text: in

text: high

text: position

text: in

text: top-k

text: rankings

text: with

text: larger

text: values

text: of

text: k,

text: and

text: reciprocally,

text: if

text: it

text: is

text: usually

text: ranked

text: high

text: in

text: top-k

text: rankings,

text: then

text: its

text: probability

text: of

text: being

text: should

text: be

text: high.

text: The

text: same

text: intuition

text: holds

text: for

text: incom-

text: plete

text: rankings.

text: If

text: item

text: a

text: is

text: usually

text: preferred

text: to

text: item

text: b

text: in

text: pairwise

text: comparisons

text: then

text: rankings

text: on

text: a,

text: b,

text: c

text: that

text: place

text: a

text: before

text: b

text: should

text: have

text: higher

text: probabilities

text: than

text: the

text: others.

text: Reciprocally

text: if

text: such

text: rankings

text: appear

text: more

text: frequently

text: than

text: the

text: others,

text: then

text: item

text: a

text: should

text: be

text: preferred

text: to

text: item

text: b

text: with

text: high

text: probability

text: in

text: a

text: pairwise

text: comparison.

text: The

text: ranking

text: literature

text: thus

text: relies

text: on

text: one

text: fundamental

text: assumption:

text: the

text: observed

text: rankings

text: in

text: a

text: statistical

text: population

text: of

text: interest

text: are

text: induced

text: by

text: full

text: rankings

text: drawn

text: from

text: a

text: single

text: probability

text: distribution

text: p

text: over

text: S

text: (see

text: for

text: instance

text: Luce,

text: Permutation

text: σ

text: S

text: induces

text: ranking

text: or

text: n

text: n

text: equivalently

text: is

text: a

text: linear

text: extension

text: of

text: ranking

text: if

text: for

text: all

text: a,

text: b

text: n

text: a

text: b

text: σ(a)

text: σ(b).

text: The

text: probability

text: that

text: a

text: random

text: permutation

text: Σ

text: drawn

text: from

text: p

text: induces

text: a

text: ranking

text: is

text: thus

text: equal

text: to

text: P

text: [Σ

text: S

text: p(σ),

text: n

text: σ

text: Sn(


text: 11 tonians, it is convenient to choose even particle numbers of Ψ , which can be represented by the antisymmetrized Hf in our bilayer mapping approach. Also, due to the prop- projections of the two degenerate Halperin (551) states erty of the antisymmetrized product state, AΨ6CFS⊗6CFS at Q = (N/2, N/2), where Ψ7/3[δV1(1)] occurs. In this can be constructed only when the particle number is a way, we might neglect many other degenerate Haﬀnian multiple of four [29]. states, which cannot be obtained by antisymmetrizing 1. Torus geometry the Halperin (551) state. Thus, potentially, our results in the torus geometry could seriously underestimate the (1) (1) Figure 2 shows the square of overlap between Ψ [δV ] true overlap between Ψ [δV ] and Ψ . 7/3 1 7/3 1 Hf and each of the four trial states, Ψ333, ΨHf, AΨ6CFS⊗6CFS, To circumvent this problem, we have also performed and CPHΨZ4 as a function of the Haldane pseudopoten- ED in the spherical geometry, where the ground-state tial variation δV (1)/V (1) for various particle numbers degeneracy issue does not occur. Postponing the de- 1 1 in the torus geometry. Note that the torus geometry tailed discussions to Sec. , it is suﬃcient for now to men- can accommodate diﬀerent parallelogram shapes for the tion that, fortunately, the overlap results obtained in the unit cell by continuously varying the angle between two spherical geometry are overall consistent with those in lateral vectors L and L , which can deform the unit the torus geometry reported below. 1 2 cell from square to hexagon [28]. As one can see, there (1) Figure 2 shows the overlap between Ψ [δV ] and 7/3 1 are somewhat wild ﬂuctuations in the behavior of the Ψ in the torus geometry, which is sizable at moderately Hf overlap across various particle numbers. Despite these (1) (1) negative δV /V for small particle numbers, say, N = ﬂuctuations, it is possible to extract the following overall 1 1 6 and 8. Unfortunately, the overlap decreases fast as behaviors of the overlap. the particle number increases to N = 10. In particular, (1) First, the overlap between Ψ [δV ] and Ψ shows 7/3 1 333 the overlap becomes negligibly small near the Coulomb (1) (1) the most stable behavior as a function of δV /V re- point at N = 10. We therefore conclude that Ψ has 1 1 Hf gardless of particle number. Speciﬁcally, the overlap is little chance of representing the exact 7/3 ground state close to unity for suﬃciently large positive δV (1)/V (1), around the Coulomb point in the thermodynamic limit. 1 1 but decreases fast as δV1(1)/V1(1) approaches the Coulomb Third, the overlap between Ψ7/3[δV1(1)] and AΨ6CFS⊗6CFS (1) (1) (1) point. At moderately negative δV /V , the overlap is somewhat similar to that between Ψ [δV ] and 1 1 7/3 1 becomes negligibly small. In contrast to the situation in (1) Ψ . That is, the overlap between Ψ [δV ] and Hf 7/3 1 the LLL, Ψ has a rather weak chance of representing 333 AΨ ⊗ is sizable at moderately negative δV (1)/V (1) the exact Coulomb ground state in the SLL. 6CFS 6CFS 1 1 at N = 8, but collapses almost completely at N = 12 Second, as for the fermionic Haﬀnian state, it is impor- (1) (1) (1) for the entire range of δV /V . Therefore, we also tant to properly deﬁne the overlap between Ψ [δV ] 1 1 7/3 1 conclude that AΨ ⊗ has little chance of repre- and Ψ . As mentioned in Sec. , Ψ has a multiple 6CFS 6CFS Hf Hf senting the exact 7/3 ground state around the Coulomb ground-stare degeneracy in the torus geometry, which point in the thermodynamic limit. diverges as a function of particle number [91]. Mean- (1) (1) Finally, let us discuss the overlap between Ψ [δV ] while, Ψ7/3[δV1 ] is uniquely obtained as the lowest en- 7/3 1 and C Ψ . It has been mentioned in Sec. that there ergy state of the Coulomb interaction, which generally PH Z4 is the double ground-state degeneracy for Ψ at Q = does not have any exact ground-state degeneracy (ex- Z4 (0, 0). After PH conjugation, C Ψ has the double cept that due to the center-of-mass shift). This leads PH Z4 ground-state degeneracy at Q = (N/2, N/2) and (0, 0) to an ambiguity regarding the deﬁnition of the overlap (1) for even and odd particle numbers, respectively, which between Ψ [δV ] and Ψ . 7/3 1 Hf can be taken as the zero physical momentum [67]. As Mathematically, one needs to properly deﬁne the over- explained above, we deﬁne the overlap in this situation lap between a non-degenerate target state, say |Ψ (cid:105), A (1) as the projected amplitude of Ψ [δV ] onto the de- and a trial state with multiple degenerate copies, say 7/3 1 generate Hilbert space expanded by the two degenerate {|Ψ (cid:105)}, where n is an index distinguishing between B,n copies of C Ψ . diﬀerent degenerate copies. One reasonable deﬁnition PH Z4 would be the projected amplitude of |Ψ (cid:105) onto the de- Figure 2 shows that, while ﬂuctuating somewhat (cid:80) A generate Hilbert space expanded by {|Ψ (cid:105)}. Math- across various particle numbers, the overlap between B,n ematically, the square of overlap deﬁned in this way Ψ [δV (1)] and C Ψ remains substantial around the 7/3 1 PH Z4 can be written as |O|2 = n |(cid:104)ΨA|ΨB,n(cid:105)|2. We use pre- Coulomb point. It is interesting to note that the overlap cisely this deﬁnition here and in the following subsection, between Ψ [δV (1)] and C Ψ seems to be peaked (1) 7/3 1 PH Z4 where the overlap between Ψ7/3[δV1 ] and CPHΨZ4 is around the Coulomb point, exactly where the Laugh- discussed. lin state loses its overlap signiﬁcantly. Therefore, we In the case of the fermionic Haﬀnian state, there is an conclude that C Ψ has a reasonably good chance PH Z4 additional serious problem that the ground-state degen- of representing the exact 7/3 ground state around the eracy diverges as a function of particle number. In this Coulomb point with its overlap being the highest among work, we focus only on supposedly the sub-Hilbert space the four trial states.

text: tonians,

text: it

text: is

text: convenient

text: to

text: choose

text: even

text: particle

text: numbers

text: of

text: Ψ

text: which

text: can

text: be

text: represented

text: by

text: the

text: antisymmetrized

text: Hf

text: in

text: our

text: bilayer

text: mapping

text: approach.

text: Also,

text: due

text: to

text: the

text: prop-

text: projections

text: of

text: the

text: two

text: degenerate

text: Halperin

text: states

text: erty

text: of

text: the

text: antisymmetrized

text: product

text: state,

text: at

text: Q

text: where

text: occurs.

text: In

text: this

text: can

text: be

text: constructed

text: only

text: when

text: the

text: particle

text: number

text: is

text: a

text: way,

text: we

text: might

text: neglect

text: many

text: other

text: degenerate

text: Haﬀnian

text: multiple

text: of

text: four

text: states,

text: which

text: cannot

text: be

text: obtained

text: by

text: antisymmetrizing

text: Torus

text: geometry

text: the

text: Halperin

text: state.

text: Thus,

text: potentially,

text: our

text: results

text: in

text: the

text: torus

text: geometry

text: could

text: seriously

text: underestimate

text: the

text: Figure

text: shows

text: the

text: square

text: of

text: overlap

text: between

text: Ψ

text: [δV

text: true

text: overlap

text: between

text: Ψ

text: [δV

text: and

text: Ψ

text: Hf

text: and

text: each

text: of

text: the

text: four

text: trial

text: states,

text: ΨHf,

text: To

text: circumvent

text: this

text: problem,

text: we

text: have

text: also

text: performed

text: and

text: as

text: a

text: function

text: of

text: the

text: Haldane

text: pseudopoten-

text: ED

text: in

text: the

text: spherical

text: geometry,

text: where

text: the

text: ground-state

text: tial

text: variation

text: δV

text: for

text: various

text: particle

text: numbers

text: degeneracy

text: issue

text: does

text: not

text: occur.

text: Postponing

text: the

text: de-

text: in

text: the

text: torus

text: geometry.

text: Note

text: that

text: the

text: torus

text: geometry

text: tailed

text: discussions

text: to

text: Sec.

text: it

text: is

text: suﬃcient

text: for

text: now

text: to

text: men-

text: can

text: accommodate

text: diﬀerent

text: parallelogram

text: shapes

text: for

text: the

text: tion

text: that,

text: fortunately,

text: the

text: overlap

text: results

text: obtained

text: in

text: the

text: unit

text: cell

text: by

text: continuously

text: varying

text: the

text: angle

text: between

text: two

text: spherical

text: geometry

text: are

text: overall

text: consistent

text: with

text: those

text: in

text: lateral

text: vectors

text: L

text: and

text: L

text: which

text: can

text: deform

text: the

text: unit

text: the

text: torus

text: geometry

text: reported

text: below.

text: cell

text: from

text: square

text: to

text: hexagon

text: As

text: one

text: can

text: see,

text: there

text: Figure

text: shows

text: the

text: overlap

text: between

text: Ψ

text: [δV

text: and

text: are

text: somewhat

text: wild

text: ﬂuctuations

text: in

text: the

text: behavior

text: of

text: the

text: Ψ

text: in

text: the

text: torus

text: geometry,

text: which

text: is

text: sizable

text: at

text: moderately

text: Hf

text: overlap

text: across

text: various

text: particle

text: numbers.

text: Despite

text: these

text: negative

text: δV

text: /V

text: for

text: small

text: particle

text: numbers,

text: say,

text: N

text: ﬂuctuations,

text: it

text: is

text: possible

text: to

text: extract

text: the

text: following

text: overall

text: and

text: Unfortunately,

text: the

text: overlap

text: decreases

text: fast

text: as

text: behaviors

text: of

text: the

text: overlap.

text: the

text: particle

text: number

text: increases

text: to

text: N

text: In

text: particular,

text: First,

text: the

text: overlap

text: between

text: Ψ

text: [δV

text: and

text: Ψ

text: shows

text: the

text: overlap

text: becomes

text: negligibly

text: small

text: near

text: the

text: Coulomb

text: the

text: most

text: stable

text: behavior

text: as

text: a

text: function

text: of

text: δV

text: /V

text: re-

text: point

text: at

text: N

text: We

text: therefore

text: conclude

text: that

text: Ψ

text: has

text: Hf

text: gardless

text: of

text: particle

text: number.

text: Speciﬁcally,

text: the

text: overlap

text: is

text: little

text: chance

text: of

text: representing

text: the

text: exact

text: ground

text: state

text: close

text: to

text: unity

text: for

text: suﬃciently

text: large

text: positive

text: δV

text: around

text: the

text: Coulomb

text: point

text: in

text: the

text: thermodynamic

text: limit.

text: but

text: decreases

text: fast

text: as

text: approaches

text: the

text: Coulomb

text: Third,

text: the

text: overlap

text: between

text: and

text: point.

text: At

text: moderately

text: negative

text: δV

text: /V

text: the

text: overlap

text: is

text: somewhat

text: similar

text: to

text: that

text: between

text: Ψ

text: [δV

text: and

text: becomes

text: negligibly

text: small.

text: In

text: contrast

text: to

text: the

text: situation

text: in

text: Ψ

text: That

text: is,

text: the

text: overlap

text: between

text: Ψ

text: [δV

text: and

text: Hf

text: the

text: LLL,

text: Ψ

text: has

text: a

text: rather

text: weak

text: chance

text: of

text: representing

text: AΨ

text: is

text: sizable

text: at

text: moderately

text: negative

text: δV

text: the

text: exact

text: Coulomb

text: ground

text: state

text: in

text: the

text: SLL.

text: at

text: N

text: but

text: collapses

text: almost

text: completely

text: at

text: N

text: Second,

text: as

text: for

text: the

text: fermionic

text: Haﬀnian

text: state,

text: it

text: is

text: impor-

text: for

text: the

text: entire

text: range

text: of

text: δV

text: /V

text: Therefore,

text: we

text: also

text: tant

text: to

text: properly

text: deﬁne

text: the

text: overlap

text: between

text: Ψ

text: [δV

text: conclude

text: that

text: AΨ

text: has

text: little

text: chance

text: of

text: repre-

text: and

text: Ψ

text: As

text: mentioned

text: in

text: Sec.

text: Ψ

text: has

text: a

text: multiple

text: Hf

text: Hf

text: senting

text: the

text: exact

text: ground

text: state

text: around

text: the

text: Coulomb

text: ground-stare

text: degeneracy

text: in

text: the

text: torus

text: geometry,

text: which

text: point

text: in

text: the

text: thermodynamic

text: limit.

text: diverges

text: as

text: a

text: function

text: of

text: particle

text: number

text: Mean-

text: Finally,

text: let

text: us

text: discuss

text: the

text: overlap

text: between

text: Ψ

text: [δV

text: while,

text: is

text: uniquely

text: obtained

text: as

text: the

text: lowest

text: en-

text: and

text: C

text: Ψ

text: It

text: has

text: been

text: mentioned

text: in

text: Sec.

text: that

text: there

text: ergy

text: state

text: of

text: the

text: Coulomb

text: interaction,

text: which

text: generally

text: PH

text: is

text: the

text: double

text: ground-state

text: degeneracy

text: for

text: Ψ

text: at

text: Q

text: does

text: not

text: have

text: any

text: exact

text: ground-state

text: degeneracy

text: (ex-

text: After

text: PH

text: conjugation,

text: C

text: Ψ

text: has

text: the

text: double

text: cept

text: that

text: due

text: to

text: the

text: center-of-mass

text: shift).

text: This

text: leads

text: PH

text: ground-state

text: degeneracy

text: at

text: Q

text: and

text: to

text: an

text: ambiguity

text: regarding

text: the

text: deﬁnition

text: of

text: the

text: overlap

text: for

text: even

text: and

text: odd

text: particle

text: numbers,

text: respectively,

text: which

text: between

text: Ψ

text: [δV

text: and

text: Ψ

text: Hf

text: can

text: be

text: taken

text: as

text: the

text: zero

text: physical

text: momentum

text: As

text: Mathematically,

text: one

text: needs

text: to

text: properly

text: deﬁne

text: the

text: over-

text: explained

text: above,

text: we

text: deﬁne

text: the

text: overlap

text: in

text: this

text: situation

text: lap

text: between

text: a

text: non-degenerate

text: target

text: state,

text: say

text: |Ψ

text: A

text: as

text: the

text: projected

text: amplitude

text: of

text: Ψ

text: [δV

text: onto

text: the

text: de-

text: and

text: a

text: trial

text: state

text: with

text: multiple

text: degenerate

text: copies,

text: say

text: generate

text: Hilbert

text: space

text: expanded

text: by

text: the

text: two

text: degenerate

text: {|Ψ

text: where

text: n

text: is

text: an

text: index

text: distinguishing

text: between

text: B,n

text: copies

text: of

text: C

text: Ψ

text: diﬀerent

text: degenerate

text: copies.

text: One

text: reasonable

text: deﬁnition

text: PH

text: would

text: be

text: the

text: projected

text: amplitude

text: of

text: |Ψ

text: onto

text: the

text: de-

text: Figure

text: shows

text: that,

text: while

text: ﬂuctuating

text: somewhat

text: A

text: generate

text: Hilbert

text: space

text: expanded

text: by

text: {|Ψ

text: Math-

text: across

text: various

text: particle

text: numbers,

text: the

text: overlap

text: between

text: B,n

text: ematically,

text: the

text: square

text: of

text: overlap

text: deﬁned

text: in

text: this

text: way

text: Ψ

text: [δV

text: and

text: C

text: Ψ

text: remains

text: substantial

text: around

text: the

text: PH

text: can

text: be

text: written

text: as

text: n

text: We

text: use

text: pre-

text: Coulomb

text: point.

text: It

text: is

text: interesting

text: to

text: note

text: that

text: the

text: overlap

text: cisely

text: this

text: deﬁnition

text: here

text: and

text: in

text: the

text: following

text: subsection,

text: between

text: Ψ

text: [δV

text: and

text: C

text: Ψ

text: seems

text: to

text: be

text: peaked

text: PH

text: where

text: the

text: overlap

text: between

text: and

text: is

text: around

text: the

text: Coulomb

text: point,

text: exactly

text: where

text: the

text: Laugh-

text: discussed.

text: lin

text: state

text: loses

text: its

text: overlap

text: signiﬁcantly.

text: Therefore,

text: we

text: In

text: the

text: case

text: of

text: the

text: fermionic

text: Haﬀnian

text: state,

text: there

text: is

text: an

text: conclude

text: that

text: C

text: Ψ

text: has

text: a

text: reasonably

text: good

text: chance

text: PH

text: additional

text: serious

text: problem

text: that

text: the

text: ground-state

text: degen-

text: of

text: representing

text: the

text: exact

text: ground

text: state

text: around

text: the

text: eracy

text: diverges

text: as

text: a

text: function

text: of

text: particle

text: number.

text: In

text: this

text: Coulomb

text: point

text: with

text: its

text: overlap

text: being

text: the

text: highest

text: among

text: work,

text: we

text: focus

text: only

text: on

text: supposedly

text: the

text: sub-Hilbert

text: space

text: the

text: four

text: trial

text: states.


text: 21 wave functions,” Phys. Rev. B 83, 241302 (2011). [85] E. Ardonne, F. J. M. van Lankvelt, A. W. W. Ludwig, and K. Schoutens, “Separation of spin and charge in paired [75] Z. Papi´c, “Solvable models for unitary and nonunitary topologicaslppinh-assiensg,”letPqhuyas.ntRuemv.HBal9l0s,ta0t7e5s3,”04P(h2y0s1.4R).ev. B 65, 041305 (2002). (cid:48) [76] A. Seidel and K. Yang, ”Halperin (m, m , n) bilayer quantum[8H6a]llMs.taBtaerskoenshtlhi ianndcyXlin.-dGe.rsW,”enP,h“yCs.laRsseivﬁ.cLateitotn. 1o0f 1A,b0e3li6a8n04and non-Abelian multilayer fractional quantum Hall states through (2008). the pattern of zeros,” Phys. Rev. B 82, 245301 (2010). [77] D. Green, “Strongly correlated states in low dimensions (The[8si7s]),S”.AGrXeriavede-tps,rinMts. (P2.0Z02a)le,taerl,XZiv.:cPoanpdi-´cm, aatn/d02R02.4S5.5.K. Mong, “Competing Abelian and non-Abelian topological orders in [78] S. H. Simon, E. H. Rezayi, and N. R. Cooper, “Generalized quaνnt=um1/H3 +all1p/3roqjeucatniotnumHaHmalilltboinlaiaynesr,s”,”PPhhyyss..RReevv..BB 7951,, 205139 (2015). 075318 (2007). [88] X. G. Wen, “Non-Abelian statistics in the fractional quantum Hall states,” Phys. Rev. Lett. 66, 802-805 (1991). [79] A. C. Archer, K. Park, and J. K. Jain, “Competing crystal[8p9h]aAse.sVianeztiheanldowMes.tBLaarnkdesahuli,le“vFeli,b”onPahcycsi.aRneyvo.nsLefrtot.m1A11b,elian bilayer quantum Hall states,” Phys. Rev. Lett. 113, 236804 146804 (2013). (2014). [80] J.-W. Rhim, J. K. Jain, and K. Park, “Analytical theory of str[9o0n]glBy.cAor.rBeleartneedvWigiagnnderNc.ryRsetganlsaiunltt,h“eAlonwaetostmLyaonfdAaubelelivaenl,”and non-Abelian fractional quantum Hall states,” Phys. Rev. Lett. Phys. Rev. B 92, 121103 (2015). 103, 206801 (2009). [81] E. Keski-Vakkuri and X.-G. Wen, “The ground state structu[9re1]aCnd. Hm.oLdeuel,arZ.trPaanpsfi´oc,rmanadtioRn.sTohf ofrmaactleio,n“aGleqoumaenttruimc cHonasltlruction of quantum Hall clustering Hamiltonians,” Phys. Rev. X states on a torus,” Int. J. Mod. Phys. B 7, 4227-4259 (1993). 5, 041003 (2015). [82] E. Ardonne, E. J. Bergholtz, J. Kailasvuori, and E. Wikber[g9,2]“DKe.gPeanrekra, cVy.oMf enloikn--AAlbaevleiardniaqnu,aNnt.uEm. HBoanllessttaeetel,saonndtJh.eK. Jain, “Possibility of p-wave pairing of composite fermions at torus: Domain walls and conformal ﬁeld theory,” J. Stat. Mech. Tνh=eo1r./2E,x”pP.h2y0s0. 8R,ePv0. 4B01568,(2R010081).67-R10170 (1998). [83] Z. Liu, A. Vaezi, K. Lee, and E.-A. Kim, “Non-Abelian phase[9s3i]nFt.wDo-.cMom. pHoanldenatneνa=nd2/E3. fHra.cRtieoznaayli,q“uFainntiutem-siHzealsltsutdaiteess:of the incompressible state of the fractionally quantized Hall eﬀect Emergence of Fibonacci anyons,” Phys. Rev. B 92, 081102 (2015)a.nd its excitations,” Phys. Rev. Lett. 54, 237-240 (1985). [84] B. A. Bernevig and F. D. M. Haldane, ”Model fractional qu[a9n4]tuSm. MH.alGl isrtvaitne,sAa.ndH.JMacakcpDoolnyanlodm, iaanlsd,”PP. Mhy.s.PRlaetvz.mLaent,t.“Collective-excitation gap in the fractional quantum Hall eﬀect,” 100, 246802 (2008). Phys. Rev. Lett. 54, 581-583 (1985).

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text: 8 degeneracy in the torus geometry, which can be under- more emphasis on the results obtained in the spherical stood in terms of its root conﬁgurations. Speciﬁcally, we geometry. In Sec. , we show the results for the overlap have found that there are four distinct root conﬁgura- between the exact 7/3 ground and the antisymmetrized tions for the Halperin (551) state: (i) |XX ◦ ◦ ◦ ◦ · · · (cid:105), Halperin (551) states in the spherical geometry. It turns (ii) | ↑ ◦ ↓ ◦ ◦ ◦ · · · (cid:105), (iii) | ↑ ◦◦ ↓ ◦ ◦ · · · (cid:105), and (iv) out, fortunately, that the overlap results obtained in the | ↑ ◦ ◦ ◦ ↓ ◦ · · · (cid:105), where ↑ and ↓ indicate a site occu- spherical geometry are overall consistent with those in pied by up and down spins, respectively, ◦ indicates an the torus geometry, as reported in Sec. . empty site, XX =↑↓ + ↓↑, and the ellipsis denotes the repetition of a given root conﬁguration. Each of these C. Antisymmetrized product state of two CF seas at 1/6 ﬁlling four root conﬁgurations has a six-fold degeneracy via the center-of-mass shift, altogether generating 24 degenerate states. Note that the root conﬁgurations of the Halperin Here, we consider the antisymmetrized product state (551) state can be constructed similarly to those of the of two CF seas at 1/6 ﬁlling, AΨ ⊗ . To this end, Halperin (331) state [76]. 6CFS 6CFS it is necessary to know how to obtain the CF sea state at After antisymmetrization, the root conﬁguration in (i) 1/6 ﬁlling, Ψ . Na¨ıvely, one may guess that Ψ is becomes | • • ◦ ◦ ◦ ◦ · · · (cid:105), where • indicate a ﬁlled site. 6CFS 6CFS obtained as the ground state of the Coulomb interaction Meanwhile, the root conﬁgurations in both (ii) and (iv) at ν = 1/6 in the LLL. Unfortunately, however, this guess reduce to a single identical state, | • ◦ • ◦ ◦ ◦ · · · (cid:105). Each is not correct since the actual ground state is likely to be of these two antisymmetrized root conﬁgurations, | • • ◦ the Wigner crystal of composite fermions rather than the ◦ ◦ ◦ · · · (cid:105) and | • ◦ • ◦ ◦ ◦ · · · (cid:105), generates six degenerate quantum Hall liquid state [79, 80]. In the torus geometry, copies via the center-of-mass shift. On the other hand, this manifests itself as the fact that the ground state the root conﬁguration in (iii) becomes | • ◦ ◦ • ◦ ◦ · · · (cid:105), occurs at odd pseudomomenta, indicating that it is not which has three degenerate copies since the period of the a uniform state. In this sense, the previously-mentioned center-of-mass shift is halved. As a consequence, the an- phase diagram in Fig. 1 (a) is in fact inaccurate. The tisymmetrized Halperin (551) state has altogether the actual 1/3 bilayer quantum Hall ground state at d/l (cid:29) 15-fold ground-state degeneracy in the torus geometry. B 1 is predicted to be the product state of two Wigner More importantly, it can be shown by counting the crystals of composite fermions at 1/6 ﬁlling. pseudomomentum of each root conﬁguration that there are two degenerate copies of the antisymmetrized Halperin For this reason, we instead construct Ψ as the ex- (551) state at Q = (N/2, N/2), where the exact 7/3 6CFS act ground state of the LLL FQH Hamiltonian in Eq. (4) ground state occurs. As mentioned in Sec. , however, (0) with V set equal to unity and all the other Hal- it is unclear at present whether the ground-state de- 1,3,5 dane pseudopotentials zero. It is important to note that generacy of the antisymmetrized Halperin (551) state is AΨ ⊗ does not necessarily describe a compress- the same as that of the fermionic Haﬀnian state in this 6CFS 6CFS ible phase even if Ψ ⊗ is compressible. The rea- momentum sector. 6CFS 6CFS son is in some sense similar to why the antisymmetrized Fortunately, the ground-state degeneracy issue does incompressible state is not necessarily incompressible, as not occur in the spherical geometry, where both fermionic shown in the example of Ψ = AΨ . Perhaps, a more Haﬀnian and Halperin (551) states are non-degenerate. Hf 551 directly relevant example is the previously-mentioned Speciﬁcally, in the spherical geometry, the fermionic antisymmetrized product state of two CF seas at quarter Haﬀnian state can be directly obtained as the exact, ﬁlling, AΨ ⊗ , which has a signiﬁcant square of non-degenerate, zero-energy ground state of the follow- 4CFS 4CFS overlap (over 90% when PH-symmetrized in the N = 12 ing three-b(cid:88)ody(cid:104)interaction Hamiltonian [77]: system) with the exact 5/2 ground state at the Coulomb point [29], which is known to be incompressible. It is not (cid:105) clear at this point whether AΨ ⊗ is indeed in- H = V P (3N /2 − 3) 6CFS 6CFS Hf 0 ijk φ (cid:54) (cid:54) compressible. Despite this uncertainty, we believe that it i=j=k is worthwhile to investigate how large overlap it can have + V P (3N /2 − 5) + V P (3N /2 − 6) , with the exact 7/3 ground state around the Coulomb 2 ijk φ 3 ijk φ point. In some sense, the overlap can provide us with a (20) hint for the compressibility of AΨ ⊗ . 6CFS 6CFS where P (L) is the projection operator onto the D. PH-conjugated Z parafermion state ijk 4 three-particle state at a given total angular momentum L. The V term is absent due to a symmetry reason [78]. 1 We have conﬁrmed that, in the spherical geometry, the Now, let us consider the PH-conjugated Z parafermion 4 ground state of H is indeed exactly identical to the state, C Ψ . As mentioned previously, the Z parafermion Hf PH Z4 4 antisymmetrized projection of the Halperin (551) state state can be obtained as the antisymmetrized projection up to machine accuracy. Based on this conﬁrmation, as of the Halperin (330) state [59, 60] with the Halperin far as the fermionic Haﬀnian state is concerned, we put (330) state deﬁned as

text: degeneracy

text: in

text: the

text: torus

text: geometry,

text: which

text: can

text: be

text: under-

text: more

text: emphasis

text: on

text: the

text: results

text: obtained

text: in

text: the

text: spherical

text: stood

text: in

text: terms

text: of

text: its

text: root

text: conﬁgurations.

text: Speciﬁcally,

text: we

text: geometry.

text: In

text: Sec.

text: we

text: show

text: the

text: results

text: for

text: the

text: overlap

text: have

text: found

text: that

text: there

text: are

text: four

text: distinct

text: root

text: conﬁgura-

text: between

text: the

text: exact

text: ground

text: and

text: the

text: antisymmetrized

text: tions

text: for

text: the

text: Halperin

text: state:

text: (i)

text: |XX

text: Halperin

text: states

text: in

text: the

text: spherical

text: geometry.

text: It

text: turns

text: (ii)

text: (iii)

text: and

text: (iv)

text: out,

text: fortunately,

text: that

text: the

text: overlap

text: results

text: obtained

text: in

text: the

text: where

text: and

text: indicate

text: a

text: site

text: occu-

text: spherical

text: geometry

text: are

text: overall

text: consistent

text: with

text: those

text: in

text: pied

text: by

text: up

text: and

text: down

text: spins,

text: respectively,

text: indicates

text: an

text: the

text: torus

text: geometry,

text: as

text: reported

text: in

text: Sec.

text: empty

text: site,

text: XX

text: and

text: the

text: ellipsis

text: denotes

text: the

text: repetition

text: of

text: a

text: given

text: root

text: conﬁguration.

text: Each

text: of

text: these

text: C.

text: Antisymmetrized

text: product

text: state

text: of

text: two

text: CF

text: seas

text: at

text: ﬁlling

text: four

text: root

text: conﬁgurations

text: has

text: a

text: six-fold

text: degeneracy

text: via

text: the

text: center-of-mass

text: shift,

text: altogether

text: generating

text: degenerate

text: states.

text: Note

text: that

text: the

text: root

text: conﬁgurations

text: of

text: the

text: Halperin

text: Here,

text: we

text: consider

text: the

text: antisymmetrized

text: product

text: state

text: state

text: can

text: be

text: constructed

text: similarly

text: to

text: those

text: of

text: the

text: of

text: two

text: CF

text: seas

text: at

text: ﬁlling,

text: AΨ

text: To

text: this

text: end,

text: Halperin

text: state

text: it

text: is

text: necessary

text: to

text: know

text: how

text: to

text: obtain

text: the

text: CF

text: sea

text: state

text: at

text: After

text: antisymmetrization,

text: the

text: root

text: conﬁguration

text: in

text: (i)

text: ﬁlling,

text: Ψ

text: Na¨ıvely,

text: one

text: may

text: guess

text: that

text: Ψ

text: is

text: becomes

text: where

text: indicate

text: a

text: ﬁlled

text: site.

text: obtained

text: as

text: the

text: ground

text: state

text: of

text: the

text: Coulomb

text: interaction

text: Meanwhile,

text: the

text: root

text: conﬁgurations

text: in

text: both

text: (ii)

text: and

text: (iv)

text: at

text: ν

text: in

text: the

text: LLL.

text: Unfortunately,

text: however,

text: this

text: guess

text: reduce

text: to

text: a

text: single

text: identical

text: state,

text: Each

text: is

text: not

text: correct

text: since

text: the

text: actual

text: ground

text: state

text: is

text: likely

text: to

text: be

text: of

text: these

text: two

text: antisymmetrized

text: root

text: conﬁgurations,

text: the

text: Wigner

text: crystal

text: of

text: composite

text: fermions

text: rather

text: than

text: the

text: and

text: generates

text: six

text: degenerate

text: quantum

text: Hall

text: liquid

text: state

text: In

text: the

text: torus

text: geometry,

text: copies

text: via

text: the

text: center-of-mass

text: shift.

text: On

text: the

text: other

text: hand,

text: this

text: manifests

text: itself

text: as

text: the

text: fact

text: that

text: the

text: ground

text: state

text: the

text: root

text: conﬁguration

text: in

text: (iii)

text: becomes

text: occurs

text: at

text: odd

text: pseudomomenta,

text: indicating

text: that

text: it

text: is

text: not

text: which

text: has

text: three

text: degenerate

text: copies

text: since

text: the

text: period

text: of

text: the

text: a

text: uniform

text: state.

text: In

text: this

text: sense,

text: the

text: previously-mentioned

text: center-of-mass

text: shift

text: is

text: halved.

text: As

text: a

text: consequence,

text: the

text: an-

text: phase

text: diagram

text: in

text: Fig.

text: (a)

text: is

text: in

text: fact

text: inaccurate.

text: The

text: tisymmetrized

text: Halperin

text: state

text: has

text: altogether

text: the

text: actual

text: bilayer

text: quantum

text: Hall

text: ground

text: state

text: at

text: d/l

text: ground-state

text: degeneracy

text: in

text: the

text: torus

text: geometry.

text: B

text: is

text: predicted

text: to

text: be

text: the

text: product

text: state

text: of

text: two

text: Wigner

text: More

text: importantly,

text: it

text: can

text: be

text: shown

text: by

text: counting

text: the

text: crystals

text: of

text: composite

text: fermions

text: at

text: ﬁlling.

text: pseudomomentum

text: of

text: each

text: root

text: conﬁguration

text: that

text: there

text: are

text: two

text: degenerate

text: copies

text: of

text: the

text: antisymmetrized

text: Halperin

text: For

text: this

text: reason,

text: we

text: instead

text: construct

text: Ψ

text: as

text: the

text: ex-

text: state

text: at

text: Q

text: where

text: the

text: exact

text: act

text: ground

text: state

text: of

text: the

text: LLL

text: FQH

text: Hamiltonian

text: in

text: Eq.

text: ground

text: state

text: occurs.

text: As

text: mentioned

text: in

text: Sec.

text: however,

text: with

text: V

text: set

text: equal

text: to

text: unity

text: and

text: all

text: the

text: other

text: Hal-

text: it

text: is

text: unclear

text: at

text: present

text: whether

text: the

text: ground-state

text: de-

text: dane

text: pseudopotentials

text: zero.

text: It

text: is

text: important

text: to

text: note

text: that

text: generacy

text: of

text: the

text: antisymmetrized

text: Halperin

text: state

text: is

text: AΨ

text: does

text: not

text: necessarily

text: describe

text: a

text: compress-

text: the

text: same

text: as

text: that

text: of

text: the

text: fermionic

text: Haﬀnian

text: state

text: in

text: this

text: ible

text: phase

text: even

text: if

text: Ψ

text: is

text: compressible.

text: The

text: rea-

text: momentum

text: sector.

text: son

text: is

text: in

text: some

text: sense

text: similar

text: to

text: why

text: the

text: antisymmetrized

text: Fortunately,

text: the

text: ground-state

text: degeneracy

text: issue

text: does

text: incompressible

text: state

text: is

text: not

text: necessarily

text: incompressible,

text: as

text: not

text: occur

text: in

text: the

text: spherical

text: geometry,

text: where

text: both

text: fermionic

text: shown

text: in

text: the

text: example

text: of

text: Ψ

text: AΨ

text: Perhaps,

text: a

text: more

text: Haﬀnian

text: and

text: Halperin

text: states

text: are

text: non-degenerate.

text: Hf

text: directly

text: relevant

text: example

text: is

text: the

text: previously-mentioned

text: Speciﬁcally,

text: in

text: the

text: spherical

text: geometry,

text: the

text: fermionic

text: antisymmetrized

text: product

text: state

text: of

text: two

text: CF

text: seas

text: at

text: quarter

text: Haﬀnian

text: state

text: can

text: be

text: directly

text: obtained

text: as

text: the

text: exact,

text: ﬁlling,

text: AΨ

text: which

text: has

text: a

text: signiﬁcant

text: square

text: of

text: non-degenerate,

text: zero-energy

text: ground

text: state

text: of

text: the

text: follow-

text: overlap

text: (over

text: when

text: PH-symmetrized

text: in

text: the

text: N

text: ing

text: Hamiltonian

text: system)

text: with

text: the

text: exact

text: ground

text: state

text: at

text: the

text: Coulomb

text: point

text: which

text: is

text: known

text: to

text: be

text: incompressible.

text: It

text: is

text: not

text: clear

text: at

text: this

text: point

text: whether

text: AΨ

text: is

text: indeed

text: in-

text: H

text: V

text: P

text: Hf

text: ijk

text: φ

text: compressible.

text: Despite

text: this

text: uncertainty,

text: we

text: believe

text: that

text: it

text: i=j=k

text: is

text: worthwhile

text: to

text: investigate

text: how

text: large

text: overlap

text: it

text: can

text: have

text: V

text: P

text: V

text: P

text: with

text: the

text: exact

text: ground

text: state

text: around

text: the

text: Coulomb

text: ijk

text: φ

text: ijk

text: φ

text: point.

text: In

text: some

text: sense,

text: the

text: overlap

text: can

text: provide

text: us

text: with

text: a

text: hint

text: for

text: the

text: compressibility

text: of

text: AΨ

text: where

text: P

text: (L)

text: is

text: the

text: projection

text: operator

text: onto

text: the

text: D.

text: PH-conjugated

text: Z

text: parafermion

text: state

text: ijk

text: three-particle

text: state

text: at

text: a

text: given

text: total

text: angular

text: momentum

text: L.

text: The

text: V

text: term

text: is

text: absent

text: due

text: to

text: a

text: symmetry

text: reason

text: We

text: have

text: conﬁrmed

text: that,

text: in

text: the

text: spherical

text: geometry,

text: the

text: Now,

text: let

text: us

text: consider

text: the

text: PH-conjugated

text: Z

text: parafermion

text: ground

text: state

text: of

text: H

text: is

text: indeed

text: exactly

text: identical

text: to

text: the

text: state,

text: C

text: Ψ

text: As

text: mentioned

text: previously,

text: the

text: Z

text: parafermion

text: Hf

text: PH

text: antisymmetrized

text: projection

text: of

text: the

text: Halperin

text: state

text: state

text: can

text: be

text: obtained

text: as

text: the

text: antisymmetrized

text: projection

text: up

text: to

text: machine

text: accuracy.

text: Based

text: on

text: this

text: conﬁrmation,

text: as

text: of

text: the

text: Halperin

text: state

text: with

text: the

text: Halperin

text: far

text: as

text: the

text: fermionic

text: Haﬀnian

text: state

text: is

text: concerned,

text: we

text: put

text: state

text: deﬁned

text: as