text: 2 with a probability q (being the only parameter of the 1.7 model). Despite its simplicity, QP models still arise con- troversies. The main concern is the question of existence 1.6 of the localization-delocalization transition in 2D models 1.5 for q > 0 (see [39–41] and references therein). In par- ticular, this issue might be important in the context of 1.4 application of QP models in the description of transport 1.3 properties in e.g. manganite ﬁlms [42], granular metals 1.2 [43] and doped semiconductors [44]. In the present paper we investigate eigenfunction local- 0.0 0.2 0.4 0.6 0.8 1.0 ization in random fractal lattices (RFL), i.e. in fractal p objects with random site connectivity in the absence of any diagonal disorder, Fig.1. We would like to stress that Figure 2: The plot illustrates the change of the spectral di- ± QP systems do not belong to fractal objects. That is, in mension of RFLs for the Hausdorﬀ dimension d = 1.75 0.02 H the QP case only an inﬁnite cluster of a percolated lat- when increasing the number of lattice links. The dimension- tice at the percolation threshold is a fractal object [45]. less parameter p is the fraction of links added to the minimal, Here, on the contrary, we consider a family of lattices single connected lattice. A parameter value p = 0 represents RFLs with the minimal number of links, i.e. no link can be with well deﬁned Hausdorﬀ dimension d [46]. Starting H removed without disconnecting a part of the lattice. Con- with the minimal, connected lattice (i.e. a lattice with- versely, p = 1 represents RFLs with the maximal number of out loops) and adding links between nearest neighbors links, i.e. no link can be added without creating an extra lat- wkeeepcatnhiencHreaaussedothrﬀe sdpiemcternasliodnimdeHnsﬁioxnedd.s oTfhaelraetftoircee, atnhde tpicweassitoeb. taTihneedvafrluome otfhteheexsppoencetnratlodfi∝mtheensniuomn bdesrfoofr daisgtiivnecnt main focus of this paper is to investigate the presence or sites visited by a random walker S nds/2, and averaging n absence of localization in RFLs while changing the spec- over 2000 independent RFLs and 500 realizations of random tral and Hausdorﬀ dimensions independently. It is worth walks of length 2500. noting that the theoretical model investigated here can be realized in ultra-cold atoms laboratories where lattice geometry can be nearly arbitrarily shaped [47–49]. clidean space and have their Hausdorﬀ and spectral di- This paper is organized as follows. In Section we de- mensions smaller than 2. A minimal RFL, i.e. a lattice scribe the growth algorithm of random fractal lattices with the smallest number of lattice links, is a single con- and how the spectral dimension changes when new lat- nected lattice generated by the growth algorithm deﬁned tice links are created. In Section we focus on localiza- (cid:48) (cid:48) in Ref. [46]. In a nutshell, a new lattice site (i , j ) is cho- tion properties of RFLs and on their dependence on the sen and linked to the existing lattice at site (i, j) with spectral and Hausdorﬀ dimensions. Particularly, we an- (cid:16) (cid:17) the probability alyze superlocalization resonances and formation of an (cid:80) energy gap which emerges in the system for small spec- → (cid:48) (cid:48) (φ (cid:48) (cid:48))η tral dimension. In Section we investigate transmission P (i, j) (i , j ) = i ,j , (1) (φ (cid:48) (cid:48))η probabilities through the system and quantum evolution i ,j of initially localized particles. Finally, in Section we where the summation goes over all possible choices, η is conclude. a free parameter and φ is a function fulﬁlling discrete i,j II. RANDOM FRACTAL LATTICES Laplace equation: 1 We consider a family of lattices that we call random φ = (φ + φ − + φ + φ − ) . (2) fractal lattices (RFLs), which ﬁrst arose in a model of di- i,j 4 i,j+1 i,j 1 i+1,j i 1,j electric breakdown [46]. The RFLs are lattices with ran- At start, we set φ = 0.5 everywhere. When a site dom site connectivity and with well-deﬁned Hausdorﬀ dH (i(cid:48), j(cid:48)) is being connie,jcted to the lattice, then the value of and spectral ds dimensions. The Hausdorﬀ dimension, or φi(cid:48),j(cid:48) is changed to zero. Before linking another lattice the capacity dimension, describes how the number of sites site, the values of φ in the neighborhood of (i, j) needs to scales with the system size. In other words, if a lattice be updated [typically in 5-20 iterations of Eq. (2)]. The in Fig. 1 is a fractal object, then the number of sites in- algorithm is stopped after reaching N lattice sites. The side a sphere o radius r is proportional to rdH, where in lattices grown in this manner have nonuniform geometry, general d is a noninteger exponent [45]. On the other both in the lattice sites and lattice links occurrence, as in H hand, the spectral dimension is related to a random walk Fig. 1. In particular, the two neighboring lattice sites do on the lattice. The number of distinct sites S covered not necessarily need to be connected and the closed loops n in n steps of a random walk is proportional to nds/2, if are forbidden, i.e. a minimal RFL is created. By adding d < 2. From a general analysis, the spectral dimension links between nearest neighbors to a given minimal RFL, s is never larger than the Hausdorﬀ dimension [32, 33]. one opens up new possibilities for a random walker to RFLs under consideration are embedded in the 2D Eu- explore and therefore increases the spectral dimension d s d s ##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##

text: with

text: a

text: probability

text: q

text: (being

text: the

text: only

text: parameter

text: of

text: the

text: model).

text: Despite

text: its

text: simplicity,

text: QP

text: models

text: still

text: arise

text: con-

text: troversies.

text: The

text: main

text: concern

text: is

text: the

text: question

text: of

text: existence

text: of

text: the

text: localization-delocalization

text: transition

text: in

text: models

text: for

text: q

text: (see

text: and

text: references

text: therein).

text: In

text: par-

text: ticular,

text: this

text: issue

text: might

text: be

text: important

text: in

text: the

text: context

text: of

text: application

text: of

text: QP

text: models

text: in

text: the

text: description

text: of

text: transport

text: properties

text: in

text: e.g.

text: manganite

text: ﬁlms

text: granular

text: metals

text: and

text: doped

text: semiconductors

text: In

text: the

text: present

text: paper

text: we

text: investigate

text: eigenfunction

text: local-

text: ization

text: in

text: random

text: fractal

text: lattices

text: (RFL),

text: i.e.

text: in

text: fractal

text: p

text: objects

text: with

text: random

text: site

text: connectivity

text: in

text: the

text: absence

text: of

text: any

text: diagonal

text: disorder,

text: We

text: would

text: like

text: to

text: stress

text: that

text: Figure

text: The

text: plot

text: illustrates

text: the

text: change

text: of

text: the

text: spectral

text: di-

text: QP

text: systems

text: do

text: not

text: belong

text: to

text: fractal

text: objects.

text: That

text: is,

text: in

text: mension

text: of

text: RFLs

text: for

text: the

text: Hausdorﬀ

text: dimension

text: d

text: H

text: the

text: QP

text: case

text: only

text: an

text: inﬁnite

text: cluster

text: of

text: a

text: percolated

text: lat-

text: when

text: increasing

text: the

text: number

text: of

text: lattice

text: links.

text: The

text: dimension-

text: tice

text: at

text: the

text: percolation

text: threshold

text: is

text: a

text: fractal

text: object

text: less

text: parameter

text: p

text: is

text: the

text: fraction

text: of

text: links

text: added

text: to

text: the

text: minimal,

text: Here,

text: on

text: the

text: contrary,

text: we

text: consider

text: a

text: family

text: of

text: lattices

text: single

text: connected

text: lattice.

text: A

text: parameter

text: value

text: p

text: represents

text: RFLs

text: with

text: the

text: minimal

text: number

text: of

text: links,

text: i.e.

text: no

text: link

text: can

text: be

text: with

text: well

text: deﬁned

text: Hausdorﬀ

text: dimension

text: d

text: Starting

text: H

text: removed

text: without

text: disconnecting

text: a

text: part

text: of

text: the

text: lattice.

text: Con-

text: with

text: the

text: minimal,

text: connected

text: lattice

text: (i.e.

text: a

text: lattice

text: with-

text: versely,

text: p

text: represents

text: RFLs

text: with

text: the

text: maximal

text: number

text: of

text: out

text: loops)

text: and

text: adding

text: links

text: between

text: nearest

text: neighbors

text: links,

text: i.e.

text: no

text: link

text: can

text: be

text: added

text: without

text: creating

text: an

text: extra

text: lat-

text: wkeeepcatnhiencHreaaussedothrﬀe

text: sdpiemcternasliodnimdeHnsﬁioxnedd.s

text: oTfhaelraetftoircee,

text: atnhde

text: tpicweassitoeb.

text: taTihneedvafrluome

text: bdesrfoofr

text: daisgtiivnecnt

text: main

text: focus

text: of

text: this

text: paper

text: is

text: to

text: investigate

text: the

text: presence

text: or

text: sites

text: visited

text: by

text: a

text: random

text: walker

text: S

text: and

text: averaging

text: n

text: absence

text: of

text: localization

text: in

text: RFLs

text: while

text: changing

text: the

text: spec-

text: over

text: independent

text: RFLs

text: and

text: realizations

text: of

text: random

text: tral

text: and

text: Hausdorﬀ

text: dimensions

text: independently.

text: It

text: is

text: worth

text: walks

text: of

text: length

text: noting

text: that

text: the

text: theoretical

text: model

text: investigated

text: here

text: can

text: be

text: realized

text: in

text: ultra-cold

text: atoms

text: laboratories

text: where

text: lattice

text: geometry

text: can

text: be

text: nearly

text: arbitrarily

text: shaped

text: clidean

text: space

text: and

text: have

text: their

text: Hausdorﬀ

text: and

text: spectral

text: di-

text: This

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Section

text: we

text: de-

text: mensions

text: smaller

text: than

text: A

text: minimal

text: RFL,

text: i.e.

text: a

text: lattice

text: scribe

text: the

text: growth

text: algorithm

text: of

text: random

text: fractal

text: lattices

text: with

text: the

text: smallest

text: number

text: of

text: lattice

text: links,

text: is

text: a

text: single

text: con-

text: and

text: how

text: the

text: spectral

text: dimension

text: changes

text: when

text: new

text: lat-

text: nected

text: lattice

text: generated

text: by

text: the

text: growth

text: algorithm

text: deﬁned

text: tice

text: links

text: are

text: created.

text: In

text: Section

text: we

text: focus

text: on

text: localiza-

text: in

text: Ref.

text: In

text: a

text: nutshell,

text: a

text: new

text: lattice

text: site

text: (i

text: j

text: is

text: cho-

text: tion

text: properties

text: of

text: RFLs

text: and

text: on

text: their

text: dependence

text: on

text: the

text: sen

text: and

text: linked

text: to

text: the

text: existing

text: lattice

text: at

text: site

text: (i,

text: j)

text: with

text: spectral

text: and

text: Hausdorﬀ

text: dimensions.

text: Particularly,

text: we

text: an-

text: the

text: probability

text: alyze

text: superlocalization

text: resonances

text: and

text: formation

text: of

text: an

text: energy

text: gap

text: which

text: emerges

text: in

text: the

text: system

text: for

text: small

text: spec-

text: (φ

text: tral

text: dimension.

text: In

text: Section

text: we

text: investigate

text: transmission

text: P

text: (i,

text: j)

text: (i

text: j

text: i

text: ,j

text: (φ

text: probabilities

text: through

text: the

text: system

text: and

text: quantum

text: evolution

text: i

text: ,j

text: of

text: initially

text: localized

text: particles.

text: Finally,

text: in

text: Section

text: we

text: where

text: the

text: summation

text: goes

text: over

text: all

text: possible

text: choices,

text: η

text: is

text: conclude.

text: a

text: free

text: parameter

text: and

text: φ

text: is

text: a

text: function

text: fulﬁlling

text: discrete

text: i,j

text: II.

text: RANDOM

text: FRACTAL

text: LATTICES

text: Laplace

text: equation:

text: We

text: consider

text: a

text: family

text: of

text: lattices

text: that

text: we

text: call

text: random

text: φ

text: (φ

text: φ

text: φ

text: φ

text: fractal

text: lattices

text: (RFLs),

text: which

text: ﬁrst

text: arose

text: in

text: a

text: model

text: of

text: di-

text: i,j

text: i,j

text: i

text: electric

text: breakdown

text: The

text: RFLs

text: are

text: lattices

text: with

text: ran-

text: At

text: start,

text: we

text: set

text: φ

text: everywhere.

text: When

text: a

text: site

text: dom

text: site

text: connectivity

text: and

text: with

text: well-deﬁned

text: Hausdorﬀ

text: dH

text: is

text: being

text: connie,jcted

text: to

text: the

text: lattice,

text: then

text: the

text: value

text: of

text: and

text: spectral

text: ds

text: dimensions.

text: The

text: Hausdorﬀ

text: dimension,

text: or

text: is

text: changed

text: to

text: zero.

text: Before

text: linking

text: another

text: lattice

text: the

text: capacity

text: dimension,

text: describes

text: how

text: the

text: number

text: of

text: sites

text: site,

text: the

text: values

text: of

text: φ

text: in

text: the

text: neighborhood

text: of

text: (i,

text: j)

text: needs

text: to

text: scales

text: with

text: the

text: system

text: size.

text: In

text: other

text: words,

text: if

text: a

text: lattice

text: be

text: updated

text: [typically

text: in

text: iterations

text: of

text: Eq.

text: The

text: in

text: Fig.

text: is

text: a

text: fractal

text: object,

text: then

text: the

text: number

text: of

text: sites

text: in-

text: algorithm

text: is

text: stopped

text: after

text: reaching

text: N

text: lattice

text: sites.

text: The

text: side

text: a

text: sphere

text: o

text: radius

text: r

text: is

text: proportional

text: to

text: rdH,

text: where

text: in

text: lattices

text: grown

text: in

text: this

text: manner

text: have

text: nonuniform

text: geometry,

text: general

text: d

text: is

text: a

text: noninteger

text: exponent

text: On

text: the

text: other

text: both

text: in

text: the

text: lattice

text: sites

text: and

text: lattice

text: links

text: occurrence,

text: as

text: in

text: H

text: hand,

text: the

text: spectral

text: dimension

text: is

text: related

text: to

text: a

text: random

text: walk

text: Fig.

text: In

text: particular,

text: the

text: two

text: neighboring

text: lattice

text: sites

text: do

text: on

text: the

text: lattice.

text: The

text: number

text: of

text: distinct

text: sites

text: S

text: covered

text: not

text: necessarily

text: need

text: to

text: be

text: connected

text: and

text: the

text: closed

text: loops

text: n

text: in

text: n

text: steps

text: of

text: a

text: random

text: walk

text: is

text: proportional

text: to

text: if

text: are

text: forbidden,

text: i.e.

text: a

text: minimal

text: RFL

text: is

text: created.

text: By

text: adding

text: d

text: From

text: a

text: general

text: analysis,

text: the

text: spectral

text: dimension

text: links

text: between

text: nearest

text: neighbors

text: to

text: a

text: given

text: minimal

text: RFL,

text: s

text: is

text: never

text: larger

text: than

text: the

text: Hausdorﬀ

text: dimension

text: one

text: opens

text: up

text: new

text: possibilities

text: for

text: a

text: random

text: walker

text: to

text: RFLs

text: under

text: consideration

text: are

text: embedded

text: in

text: the

text: Eu-

text: explore

text: and

text: therefore

text: increases

text: the

text: spectral

text: dimension

text: d

text: s

text: d

text: s

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text: 6 (a) (b) (c) (d) (e) (f) FIG. 5: Restricted wave functions for very low angular momenta for 1 minority and 7 majority particles. (a) - (c) show , ψ rW (d) - (f) show . ψ rZ this is the only possible simple state at = , i.e. seem to indicate that the vortex of winding appears L MN N no CF diagonalization is necessary. The restricted wave at = , but that the vortex is a coreless vortex L MN functions of some = states are shown in FIG. 7. in the majority component. On the other hand, (d) - L MN Again we see a very clear distinction between minority (f) show how evolves from a more or less Gaussian ψ rZ and majority species particles. The majority particles distribution in (d) to a situation where one node radi- are positioned on the vertices of a regular -gon, with ally approaches each majority particle from outside the M all the minority particles in the center. In (a) - (c) we see cloud. what looks like a single, double and triple vortex struc- V. OVERLAPS ture, respectively, but ﬁlled by the minority component. The qualitative behaviour of the amplitude contours is When working with trial wave functions like CF wave largely the same in the three plots. In particular, 7(a) functions, especially in a context for which CF was not looks remarkably similar to FIG. 4 except for the mi- originally intended, like the slowly rotating Bose systems nority particle in the middle. We will come back to this we are discussing, one should carefully compare the re- point in SEC. . sults obtained with ones obtained by other means. In It can be noted that, while the single vortex for a scalar the lowest Landau level, we are fortunate because the gas appears at = , the general multiply quantized Hilbert space of each sector is ﬁnite. Given enough L N L vortex of winding appears at ; for 20 par- computer resources, one can therefore in principle com- k L /N < k k ticles, the double and triple vortices appear at 1.8 pute the exact spectrum (at least to machine precision) L/N and 2.85 respectively [33]. The results presented here for a given , and compare the CF results to this. In L ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

label: paragraph

text: (a)

text: (b)

text: (c)

text: (d)

text: (e)

text: (f)

text: FIG.

text: Restricted

label: caption

text: wave

text: functions

text: for

text: very

text: low

text: angular

text: momenta

text: for

text: minority

text: and

text: majority

text: particles.

text: (a)

text: (c)

text: show

text: ψ

text: rW

text: (d)

text: (f)

text: show

text: ψ

text: rZ

text: this

text: is

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text: only

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text: at

text: i.e.

text: seem

text: to

text: indicate

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text: the

text: vortex

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text: winding

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text: L

text: MN

text: N

text: no

text: CF

text: diagonalization

text: is

text: necessary.

text: The

text: restricted

text: wave

text: at

text: but

text: that

text: the

text: vortex

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text: a

text: coreless

text: vortex

text: L

text: MN

text: functions

text: of

text: some

text: states

text: are

text: shown

text: in

text: FIG.

text: in

text: the

text: majority

text: component.

text: On

text: the

text: other

text: hand,

text: (d)

text: L

text: MN

text: Again

text: we

text: see

text: a

text: very

text: clear

text: distinction

text: between

text: minority

text: (f)

text: show

text: how

text: evolves

text: from

text: a

text: more

text: or

text: less

text: Gaussian

text: ψ

text: rZ

text: and

text: majority

text: species

text: particles.

text: The

text: majority

text: particles

text: distribution

text: in

text: (d)

text: to

text: a

text: situation

text: where

text: one

text: node

text: radi-

text: are

text: positioned

text: on

text: the

text: vertices

text: of

text: a

text: regular

text: -gon,

text: with

text: ally

text: approaches

text: each

text: majority

text: particle

text: from

text: outside

text: the

text: M

text: all

text: the

text: minority

text: particles

text: in

text: the

text: center.

text: In

text: (a)

text: (c)

text: we

text: see

text: cloud.

text: what

text: looks

text: like

text: a

text: single,

text: double

text: and

text: triple

text: vortex

text: struc-

text: V.

text: OVERLAPS

text: ture,

text: respectively,

text: but

text: ﬁlled

text: by

text: the

text: minority

text: component.

text: The

text: qualitative

text: behaviour

text: of

text: the

text: amplitude

text: contours

text: is

text: When

text: working

text: with

text: trial

text: wave

text: functions

text: like

text: CF

text: wave

text: largely

text: the

text: same

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text: the

text: three

text: plots.

text: In

text: particular,

text: functions,

text: especially

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text: a

text: context

text: for

text: which

text: CF

text: was

text: not

text: looks

text: remarkably

text: similar

text: to

text: FIG.

text: except

text: for

text: the

text: mi-

text: originally

text: intended,

text: like

text: the

text: slowly

text: rotating

text: Bose

text: systems

text: nority

text: particle

text: in

text: the

text: middle.

text: We

text: will

text: come

text: back

text: to

text: this

text: we

text: are

text: discussing,

text: one

text: should

text: carefully

text: compare

text: the

text: re-

text: point

text: in

text: SEC.

text: sults

text: obtained

text: with

text: ones

text: obtained

text: by

text: other

text: means.

text: In

text: It

text: can

text: be

text: noted

text: that,

text: while

text: the

text: single

text: vortex

text: for

text: a

text: scalar

text: the

text: lowest

text: Landau

text: level,

text: we

text: are

text: fortunate

text: because

text: the

text: gas

text: appears

text: at

text: the

text: general

text: multiply

text: quantized

text: Hilbert

text: space

text: of

text: each

text: sector

text: is

text: ﬁnite.

text: Given

text: enough

text: L

text: N

text: L

text: vortex

text: of

text: winding

text: appears

text: at

text: for

text: par-

text: computer

text: resources,

text: one

text: can

text: therefore

text: in

text: principle

text: com-

text: k

text: L

text: /N

text: k

text: k

text: ticles,

text: the

text: double

text: and

text: triple

text: vortices

text: appear

text: at

text: pute

text: the

text: exact

text: spectrum

text: (at

text: least

text: to

text: machine

text: precision)

text: L/N

text: and

text: respectively

text: The

text: results

text: presented

text: here

text: for

text: a

text: given

text: and

text: compare

text: the

text: CF

text: results

text: to

text: this.

text: In

text: L

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

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text: 9 1.00 1.000 0.95 Squaredoverlap 00..999905 Squaredoverlap 00..9905 Squaredoverlap 0.90 0.85 0.85 0.985 0 5 10 15 20 0 2 4 6 8 10 12 14 3 4 5 6 7 8 Number of majority particles Number of majority particles Number of majority particles (a) (b) (c) FIG. 8: The squared overlap | hΨ | Ψ i |2 between the numerically exact lowest-lying states, and the CF states (16), at CF exact = , for (a) = 1, (b) = 2, and (c) = 3, as function of . L MN N N N M We have that X X where we deﬁne ∆ = . z0 N N Proof. Our proof is based on induction. We ﬁrst deﬁne ∆ (p)Φ (α) = Φ (α + p e ). (A4) the subset The noZn-zeroZterms aσr∈eSnallZSlaterk=d1eteσrkmkinants. The PN,M,Λ|K = {p ∈ PN,M,Λ | pi = 0 ⇔ i > K} (A10) greatest determinant with respect to the ordering occurs and note that trivially, for all ∈ ⊃ , wtichuelnarσdoertdeerrmsitnhaentpaisrtiΦtio(n n(o)n)-daencdreawseinmglay.y Tthheisrepfoarre- there exists dp ∈ Q|PN,Λ,Λ| sucph thPaNt,Λe,qΛ.|1 (AP9N) ,hMo,Λld|1s. a p There is only one such p and ∆ ( ) = ∆ . Now, we write Z X assume as induction hypothesis Zthapt (A9)zhΛolds for all ∈ ≤ when . p PN,Λ∈,Λ|k k K ∆ (p)Φ (α) = n 0Φ (a(p)) (A5) Let p PN,Λ,Λ|K+1 anYd consider the polynomial Z Z pp Z 0≤ p p N for some integers npp0. It is important that ∆zpi. (A11) 6= 0 (A6) i=1 n . P pp It is a product of factors. Each factor is a sum of N It is in fact positive, since every permutation σ that gives N diﬀerentiation operators with respect to N diﬀerent tinhiisndcreetaersminignaonrdtelre.avIetsfothlleowelsemtheanttsfoorftαhe+smaNkl=le1stpσpkaerk- voaf rniaobnl-ezse.roTehxeporenseunlttsinagretetrhmossewthitaht tdhoenhoitghmeustltinpulymdbiefr- tition, = min( ), we have ferentiation operators for the same variable from several pmin PN,M,Λ − − diﬀerent factors ∆ . This is except forQthe last 1 ΦZ(a(pmin)) = n 1 ∆Z(p)ΦZ(α) (A7) ∆zpi as ∂z0i = ∂z0j zapnid any of the NN−K−1 combNinaKtions and in general pminpmin X gdievsecsritbheedsbaymXpe.erTmhuYistarteisotnrsicatesd part of Ni=1 ∆zpi can be 1 ΦZ(a(p)) = npp ∆Z(p)ΦZ(α) − p0<p nnpppp0 ΦZ(a(p)()A. 8) NN−K−1 ρ∈SK+1 Ki=+11 ∂ρpii = (NN−N−KK−−11)!∆Z(p), (A12) Eq. (A7) says that the Lemma holds for Φ (a) = and this means that Y Z Φ ( ( )) and eq. (A8) says that it holds for Φ ( ) = ΦZZ(aa(ppm))inif it holds for all p0 < p. It must therefoZreahold ∆ ( ) = (N − K − 1)! N ∆ + (A13) for all p. Z p NN−K−1 zpi D ∈ ∈ i=1 Lemma 2. For all , there exists Q|PN,Λ,Λ| such that p XPN,M,ΛY dp wwhhereereeaDch itseramphoalysnaotmmiaolstoKf dniﬀoner-eznertoiateixopnonoepnetrsa.toBrys the induction hypothesis, these can all be written on the N ∆ ( ) = ∆ (A9) form (A9). This implies that the hypothesis is also true Z p p˜∈PN,Λ,Λ dpp˜ i=1 zp˜i, for all k ≤ K + 1 and completes the proof. ##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##

font: default

text: Squaredoverlap

text: Squaredoverlap

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text: Number

text: of

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text: Number

text: of

text: majority

text: particles

text: Number

text: of

text: majority

text: particles

text: (a)

text: (b)

text: (c)

text: FIG.

text: The

text: squared

text: overlap

text: hΨ

text: Ψ

text: i

text: between

text: the

text: numerically

text: exact

text: lowest-lying

text: states,

text: and

text: the

text: CF

text: states

text: at

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text: exact

text: for

text: (a)

text: (b)

text: and

text: (c)

text: as

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text: MN

text: N

text: N

text: N

text: M

text: We

text: have

text: that

text: X

text: X

text: where

text: we

text: deﬁne

text: N

text: N

text: Proof.

text: Our

text: proof

text: is

text: based

text: on

text: induction.

text: We

text: ﬁrst

text: deﬁne

text: (p)Φ

text: (α)

text: Φ

text: (α

text: p

text: e

text: the

text: subset

text: The

text: noZn-zeroZterms

text: The

text: PN,M,Λ|K

text: {p

text: PN,M,Λ

text: pi

text: i

text: K}

text: greatest

text: determinant

text: with

text: respect

text: to

text: the

text: ordering

text: occurs

text: and

text: note

text: that

text: trivially,

text: for

text: all

text: n(o)n)-daencdreawseinmglay.y

text: Tthheisrepfoarre-

text: there

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text: dp

text: Q|PN,Λ,Λ|

text: sucph

text: a

text: p

text: There

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text: such

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text: Z

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text: p

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text: (p)Φ

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text: Let

text: p

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text: Z

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text: since

text: every

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text: voaf

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text: ΦZ(a(p))

text: npp

text: ∆Z(p)ΦZ(α)

text: ΦZ(a(p)()A.

text: ∂ρpii

text: Eq.

text: says

text: that

text: the

text: Lemma

text: holds

text: for

text: Φ

text: (a)

text: and

text: this

text: means

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text: Y

text: Z

text: Φ

text: and

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text: (N

text: K

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text: Lemma

text: For

text: all

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text: dp

text: wwhhereereeaDch

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text: the

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text: can

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text: Z

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text: for

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text: k

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text: and

text: completes

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text: Exploring pseudo-Nambu–Goldstone bosons by stimulated photon colliders in the mass range 0.1 eV to 10 keV Kensuke Homma1,2 and Yuichi Toyota1 1Graduate School of Science, Hiroshima University, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan 2International Center for Zetta-Exawatt Science and Technology, Ecole Polytechnique, Route de Saclay, F-91128 Palaiseau Cedex, France (Dated: May 29, 2020) Searching for pseudo-Nambu–Goldstone bosons (pNGBs) in weak-coupling domains is crucial for understanding the dark components in the universe. We propose searching for pNGBs coupled to two photons in the mass range from 0.1 eV to 10 keV. This could provide opportunities to test ∼ string-theory-based pNGBs beyond the GUT scale M 1016 GeV included in the weak coupling − proportional to M 1. We provide formulae that are applicable to photon–photon scattering via a pNGB resonance exchange with a stimulation process in an asymmetric head-on photon–photon collider by mixing three laser pulses in laboratory experiments. We discuss the quantum electrody- namic eﬀects on the pNGB exchange in the same mass–coupling domain as a background process from the standard model. We ﬁnd that a large unexplored mass–coupling domain is accessible by combining existing laser facilities, including free-electron lasers. I. INTRODUCTION has been reported in the photon energy spectra from a single galaxy and galaxy clusters [8, 9], and the argu- Spontaneous symmetry breaking could be one of the ments are still actively ongoing [10]. The possible inter- most robust guiding principles for general discussions of pretation of a pNGB decaying into two photons has been dark components in the universe. Whenever a global discussed [11]. Indeed, string theories predict pNGBs to symmetry is broken, a massless boson may appear as a be homogeneously distributed on a log scale in the mass Nambu–Goldstone boson (NGB). In nature, however, an range possibly up to 108 eV [3]. This situation motivates NGB emerges as a pseudo-NGB (pNGB) with a ﬁnite us to try to extend the same method upLto 10 keV in mass. Even if a pNGB is close to being massless, its de- general. Given the eﬀ→ectiv→e Lagrangian , we evaluate cay into lighter particles such as photons is kinematically the sensitivity to γγ φ γγ scattering by means of allowed. The neutral pion is such an example of a pNGB stimulation using coherent X-ray sources. However, we via chiral symmetry breaking. The eﬀective interaction emphasize that the formulae we provide here are never- Lagrangian is deﬁned as theless applicable to a wide mass range from 0.1 eV to 10 keV by combining diﬀerent types of coherent and inco- −L − 1 herent light sources. This mass range is yet to be probed = gM 1 F Fµνφ, (1) 4 µν intensively, and thus the proposed approach will open a window through which to explore string-theory-based where we assume a scalar-type ﬁeld φ for simplicity, and ∼ pNGBs beyond the GUT scale M 1016 GeV. M has the dimension of energy whereas g is a dimension- II. FORMULAE FOR STIMULATED less constant. ASYMMETRIC PHOTON–PHOTON There are several theoretical models that predict low- SCATTERING VIA PNGBS mass pNGBs such as axions [1], dilatons [2], and string- theory-based axion-like particles [3]. However, pinning To distinguish clearly between the photons in the col- down the physical mass of a pNGB by means of such liding beams and the signal photons whose frequencies models is commonly non-trivial. Therefore, experiments have been shifted by the scattering process, we consider are indispensable for investigating the physical mass as an asymmetric collision system (ACS) as illustrated in widely as possible in the lower mass range. Fig. 1. The ACS is obtained by boosting a center-of- We have previously advocated a novel method [4] for (cid:112) → → mass system (CMS) along the head-on collision axis (z stimulating γγ φ γγ scattering in a quasi-parallel axis) with a speed βc, where c is the speed of light (hence, collision system (QPS) via an s-channel resonant pNGB − the Lorentz factor is γ = 1/ 1 β2). Four-momenta in exchange by utilizing the coherent nature of laser ﬁelds. the ACS are deﬁned as Kinematically, this is analogous to four-wave mixing in − matter [5] if the nonlinear atomic process is replaced with p = ω (1, 0, 0, 1), p = ω (1, 0, 0, 1), (2) 1 1 2 2 the pNGB exchange. A QPS is intended for searching − p = ω (1, sin θ , 0, cos θ ), p = ω (1, sin θ , 0, cos θ ). for pNGBs in the sub-eV mass range. Indeed, we have 3 3 3 3 4 4 4 4 performed two previous laboratory searches [6, 7] based For later convenience, with an incident photon energy ω on this approach. in the CMS, we deﬁne the incident energies in the ACS as ∼ ≡ ≡ − Recently, an unidentiﬁed emission line, ω 3.5 keV, ω uω and ω u 1ω via the following relationships: 1 2 ##LTLine## ##LTLine##

text: Exploring

label: title

text: pseudo-Nambu–Goldstone

text: bosons

text: by

text: stimulated

text: photon

text: colliders

text: in

text: the

text: mass

text: range

text: eV

text: to

text: keV

text: Kensuke

label: author

text: and

text: Yuichi

text: School

text: of

text: Science,

text: Hiroshima

text: University,

text: Kagamiyama,

text: Higashi-Hiroshima,

text: Hiroshima

text: Japan

text: Center

text: for

text: Zetta-Exawatt

text: Science

text: and

text: Technology,

text: Ecole

text: Polytechnique,

text: Route

text: de

text: Saclay,

text: Palaiseau

text: Cedex,

text: France

text: (Dated:

text: May

label: date

text: Searching

label: abstract

text: for

text: pseudo-Nambu–Goldstone

text: bosons

text: (pNGBs)

text: in

text: weak-coupling

text: domains

text: is

text: crucial

text: for

text: understanding

text: the

text: dark

text: components

text: in

text: the

text: universe.

text: We

text: propose

text: searching

text: for

text: pNGBs

text: coupled

text: to

text: two

text: photons

text: in

text: the

text: mass

text: range

text: from

text: eV

text: to

text: keV.

text: This

text: could

text: provide

text: opportunities

text: to

text: test

text: string-theory-based

text: pNGBs

text: beyond

text: the

text: GUT

text: scale

text: M

text: GeV

text: included

text: in

text: the

text: weak

text: coupling

text: proportional

text: to

text: M

text: We

text: provide

text: formulae

text: that

text: are

text: applicable

text: to

text: photon–photon

text: scattering

text: via

text: a

text: pNGB

text: resonance

text: exchange

text: with

text: a

text: stimulation

text: process

text: in

text: an

text: asymmetric

text: head-on

text: photon–photon

text: collider

text: by

text: mixing

text: three

text: laser

text: pulses

text: in

text: laboratory

text: experiments.

text: We

text: discuss

text: the

text: quantum

text: electrody-

text: namic

text: eﬀects

text: on

text: the

text: pNGB

text: exchange

text: in

text: the

text: same

text: mass–coupling

text: domain

text: as

text: a

text: background

text: process

text: from

text: the

text: standard

text: model.

text: We

text: ﬁnd

text: that

text: a

text: large

text: unexplored

text: mass–coupling

text: domain

text: is

text: accessible

text: by

text: combining

text: existing

text: laser

text: facilities,

text: including

text: free-electron

text: lasers.

text: I.

text: INTRODUCTION

text: has

text: been

text: reported

text: in

text: the

text: photon

text: energy

text: spectra

text: from

text: a

text: single

text: galaxy

text: and

text: galaxy

text: clusters

text: and

text: the

text: argu-

text: Spontaneous

text: symmetry

text: breaking

text: could

text: be

text: one

text: of

text: the

text: ments

text: are

text: still

text: actively

text: ongoing

text: The

text: possible

text: inter-

text: most

text: robust

text: guiding

text: principles

text: for

text: general

text: discussions

text: of

text: pretation

text: of

text: a

text: pNGB

text: decaying

text: into

text: two

text: photons

text: has

text: been

text: dark

text: components

text: in

text: the

text: universe.

text: Whenever

text: a

text: global

text: discussed

text: Indeed,

text: string

text: theories

text: predict

text: pNGBs

text: to

text: symmetry

text: is

text: broken,

text: a

text: massless

text: boson

text: may

text: appear

text: as

text: a

text: be

text: homogeneously

text: distributed

text: on

text: a

text: log

text: scale

text: in

text: the

text: mass

text: Nambu–Goldstone

text: boson

text: (NGB).

text: In

text: nature,

text: however,

text: an

text: range

text: possibly

text: up

text: to

text: eV

text: This

text: situation

text: motivates

text: NGB

text: emerges

text: as

text: a

text: pseudo-NGB

text: (pNGB)

text: with

text: a

text: ﬁnite

text: us

text: to

text: try

text: to

text: extend

text: the

text: same

text: method

text: upLto

text: keV

text: in

text: mass.

text: Even

text: if

text: a

text: pNGB

text: is

text: close

text: to

text: being

text: massless,

text: its

text: de-

text: general.

text: Given

text: the

text: eﬀ→ectiv→e

text: Lagrangian

text: we

text: evaluate

text: cay

text: into

text: lighter

text: particles

text: such

text: as

text: photons

text: is

text: kinematically

text: the

text: sensitivity

text: to

text: γγ

text: φ

text: γγ

text: scattering

text: by

text: means

text: of

text: allowed.

text: The

text: neutral

text: pion

text: is

text: such

text: an

text: example

text: of

text: a

text: pNGB

text: stimulation

text: using

text: coherent

text: X-ray

text: sources.

text: However,

text: we

text: via

text: chiral

text: symmetry

text: breaking.

text: The

text: eﬀective

text: interaction

text: emphasize

text: that

text: the

text: formulae

text: we

text: provide

text: here

text: are

text: never-

text: Lagrangian

text: is

text: deﬁned

text: as

text: theless

text: applicable

text: to

text: a

text: wide

text: mass

text: range

text: from

text: eV

text: to

text: keV

text: by

text: combining

text: diﬀerent

text: types

text: of

text: coherent

text: and

text: inco-

text: −L

text: herent

text: light

text: sources.

text: This

text: mass

text: range

text: is

text: yet

text: to

text: be

text: probed

text: gM

text: F

text: Fµνφ,

text: µν

text: intensively,

text: and

text: thus

text: the

text: proposed

text: approach

text: will

text: open

text: a

text: window

text: through

text: which

text: to

text: explore

text: string-theory-based

text: where

text: we

text: assume

text: a

text: scalar-type

text: ﬁeld

text: φ

text: for

text: simplicity,

text: and

text: pNGBs

text: beyond

text: the

text: GUT

text: scale

text: M

text: GeV.

text: M

text: has

text: the

text: dimension

text: of

text: energy

text: whereas

text: g

text: is

text: a

text: dimension-

text: II.

text: FORMULAE

text: FOR

text: STIMULATED

text: less

text: constant.

text: ASYMMETRIC

text: PHOTON–PHOTON

text: There

text: are

text: several

text: theoretical

text: models

text: that

text: predict

text: low-

text: SCATTERING

text: VIA

text: PNGBS

text: mass

text: pNGBs

text: such

text: as

text: axions

text: dilatons

text: and

text: string-

text: theory-based

text: axion-like

text: particles

text: However,

text: pinning

text: To

text: distinguish

text: clearly

text: between

text: the

text: photons

text: in

text: the

text: col-

text: down

text: the

text: physical

text: mass

text: of

text: a

text: pNGB

text: by

text: means

text: of

text: such

text: liding

text: beams

text: and

text: the

text: signal

text: photons

text: whose

text: frequencies

text: models

text: is

text: commonly

text: non-trivial.

text: Therefore,

text: experiments

text: have

text: been

text: shifted

text: by

text: the

text: scattering

text: process,

text: we

text: consider

text: are

text: indispensable

text: for

text: investigating

text: the

text: physical

text: mass

text: as

text: an

text: asymmetric

text: collision

text: system

text: (ACS)

text: as

text: illustrated

text: in

text: widely

text: as

text: possible

text: in

text: the

text: lower

text: mass

text: range.

text: Fig.

text: The

text: ACS

text: is

text: obtained

text: by

text: boosting

text: a

text: center-of-

text: We

text: have

text: previously

text: advocated

text: a

text: novel

text: method

text: for

text: mass

text: system

text: (CMS)

text: along

text: the

text: head-on

text: collision

text: axis

text: (z

text: stimulating

text: γγ

text: φ

text: γγ

text: scattering

text: in

text: a

text: quasi-parallel

text: axis)

text: with

text: a

text: speed

text: βc,

text: where

text: c

text: is

text: the

text: speed

text: of

text: light

text: (hence,

text: collision

text: system

text: (QPS)

text: via

text: an

text: s-channel

text: resonant

text: pNGB

text: the

text: Lorentz

text: factor

text: is

text: γ

text: Four-momenta

text: in

text: exchange

text: by

text: utilizing

text: the

text: coherent

text: nature

text: of

text: laser

text: ﬁelds.

text: the

text: ACS

text: are

text: deﬁned

text: as

text: Kinematically,

text: this

text: is

text: analogous

text: to

text: four-wave

text: mixing

text: in

text: matter

text: if

text: the

text: nonlinear

text: atomic

text: process

text: is

text: replaced

text: with

text: p

text: ω

text: p

text: ω

text: the

text: pNGB

text: exchange.

text: A

text: QPS

text: is

text: intended

text: for

text: searching

text: p

text: ω

text: sin

text: θ

text: cos

text: θ

text: p

text: ω

text: sin

text: θ

text: cos

text: θ

text: for

text: pNGBs

text: in

text: the

text: sub-eV

text: mass

text: range.

text: Indeed,

text: we

text: have

text: performed

text: two

text: previous

text: laboratory

text: searches

text: based

text: For

text: later

text: convenience,

text: with

text: an

text: incident

text: photon

text: energy

text: ω

text: on

text: this

text: approach.

text: in

text: the

text: CMS,

text: we

text: deﬁne

text: the

text: incident

text: energies

text: in

text: the

text: ACS

text: as

text: Recently,

text: an

text: unidentiﬁed

text: emission

text: line,

text: ω

text: keV,

text: ω

text: uω

text: and

text: ω

text: u

text: 1ω

text: via

text: the

text: following

text: relationships:

text: ##LTLine##

text: ##LTLine##


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text: Accepted

text: for

text: ICRA

text: Final

text: version

text: will

text: be

text: available

text: at

text: ieeexplore.ieee.org

text: References

label: reference

text: R.

text: Achanta,

text: S.

text: Hemami,

text: F.

text: Estrada,

text: and

text: S.

text: Susstrunk.

text: Frequency-tuned

text: salient

text: region

text: detection.

text: In

text: Conf.

text: on

text: Computer

text: Vision

text: and

text: Pattern

text: Recognition

text: (CVPR),

text: pages

text: S.

text: Alpert,

text: M.

text: Galun,

text: A.

text: Brandt,

text: and

text: R.

text: Basri.

text: Image

text: segmentation

text: by

text: probabilis-

text: tic

text: bottom-up

text: aggregation

text: and

text: cue

text: integration.

text: Trans.

text: on

text: Pattern

text: Analysis

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text: Machine

text: Intelligence

text: (TPAMI),

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text: Kiranyaz,

text: and

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text: Automatic

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text: In

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text: Conv.

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text: Pattern

text: Recognition

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text: Bao,

text: Y.

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text: Y.

text: Cheng,

text: and

text: N.

text: Xi.

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text: indoor

text: scenes.

text: Sensors,

text: A.

text: Borji.

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text: a

text: salient

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text: A

text: dataset

text: and

text: a

text: baseline

text: model

text: for

text: salient

text: object

text: detection.

text: Trans.

text: on

text: Image

text: Processing

text: (TIP),

text: Feb

text: A.

text: Borji,

text: M.-M.

text: Cheng,

text: H.

text: Jiang,

text: and

text: J.

text: Li.

text: Salient

text: object

text: detection:

text: A

text: benchmark.

text: Trans.

text: on

text: Image

text: Processing

text: (TIP),

text: C.

text: Bringes,

text: Y.

text: Lin,

text: Y.

text: Sun,

text: and

text: R.

text: Alqasemi.

text: Determining

text: the

text: beneﬁt

text: of

text: human

text: input

text: in

text: human-in-the-loop

text: robotic

text: systems.

text: In

text: Robot

text: and

text: Human

text: Communication

text: (RO-MAN),

text: IEEE

text: International

text: Workshop

text: on,

text: pages

text: IEEE,

text: M.-M.

text: Cheng,

text: N.

text: J.

text: Mitra,

text: X.

text: Huang,

text: P.

text: H.

text: S.

text: Torr,

text: and

text: S.-M.

text: Hu.

text: Global

text: contrast

text: based

text: salient

text: region

text: detection.

text: Trans.

text: on

text: Pattern

text: Analysis

text: and

text: Machine

text: Intelligence

text: (TPAMI),

text: M.

text: Ciocarlie,

text: K.

text: Hsiao,

text: E.

text: G.

text: Jones,

text: S.

text: Chitta,

text: R.

text: B.

text: Rusu,

text: and

text: I.

text: A.

text: S¸ucan.

text: Towards

text: reliable

text: grasping

text: and

text: manipulation

text: in

text: household

text: environments.

text: In

text: Experimental

text: Robotics,

text: pages

text: Springer,

text: A.

text: Collet,

text: M.

text: Martinez,

text: and

text: S.

text: S.

text: Srinivasa.

text: The

text: moped

text: framework:

text: Object

text: recog-

text: nition

text: and

text: pose

text: estimation

text: for

text: manipulation.

text: The

text: International

text: Journal

text: of

text: Robotics

text: Research,

text: H.

text: Jiang,

text: J.

text: Wang,

text: Z.

text: Yuan,

text: Y.

text: Wu,

text: N.

text: Zheng,

text: and

text: S.

text: Li.

text: Salient

text: object

text: detection:

text: A

text: discriminative

text: regional

text: feature

text: integration

text: approach.

text: In

text: Conf.

text: on

text: Computer

text: Vision

text: and

text: Pattern

text: Recognition

text: (CVPR),

text: Jun

text: D.

text: A.

text: Klein,

text: D.

text: Schulz,

text: and

text: A.

text: B.

text: Cremers.

text: Realtime

text: hierarchical

text: clustering

text: based

text: on

text: boundary

text: and

text: surface

text: statistics.

text: In

text: Asian

text: Conf.

text: on

text: Computer

text: Vision

text: (ACCV),

text: ##LTLine##


text: Accepted for ICRA 2017 Final version will be available at ieeexplore.ieee.org D Figure 2: (a) Input image. (b) Ultrametric contour map (UCM) visualizing (P, Q) from [12]. (c) Proposed PaTS model. (d) Partitioning and corresponding tree for the ﬁrst seven binary splits. Note how relatively smaller segments enclosed by strong contours become salient. from an atomic partitioning generated from a color-gradient based watershed transform. Then, iteratively, those two segments P and Q neighboring in image space become merged into a parent node, which are globally most similar with respect to a sophisti- D cated distance measure (P, Q) (cf. [12], Eq. (2)). For saliency computations, we only annotate the areas A resp. A (in pixels), the length of the perimeter touching the P Q image boundary (where applicable), as well as their distance at the time of merging to the binary partition tree. The agglomerative clustering stops when the last two segments are merged to become root, representing the full scene. 2.2 Saliency Transform from Composition and Contrast We claim that saliency and grouping are the same fundamental process, but considered from two opposing viewpoints: in grouping, one focuses on which parts of the image are most similar and should belong together, whereas saliency tells us which ones are most contrasting and should stay separate. This motivated the idea to utilize the information already gained during hierarchical clustering to express an antagonal saliency. There are some important questions to answer when composing a contrast based saliency term from the hierarchical grouping similarities: ﬁrst of all, which of the two opposing segments is foreground and which one is background? Second, how should the image boundary be treated, could one model eﬀects of human peripheral vision to saliency? Finally, how is saliency inherited from object scale down to parts? Surprisingly, we found straightforward and well-performing solutions to all of these issues. Since our approach is working purely on the graph structure created from hierarchical grouping, we call it Partition Tree Saliency (PaTS). Figure 2 shows an example of the hierarchical grouping and the corresponding saliency. 4 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Accepted

text: for

text: ICRA

text: Final

text: version

text: will

text: be

text: available

text: at

text: ieeexplore.ieee.org

text: D

text: Figure

text: (a)

text: Input

text: image.

text: (b)

text: Ultrametric

text: contour

text: map

text: (UCM)

text: visualizing

text: (P,

text: Q)

text: from

text: (c)

text: Proposed

text: PaTS

text: model.

text: (d)

text: Partitioning

text: and

text: corresponding

text: tree

text: for

text: the

text: ﬁrst

text: seven

text: binary

text: splits.

text: Note

text: how

text: relatively

text: smaller

text: segments

text: enclosed

text: by

text: strong

text: contours

text: become

text: salient.

text: from

text: an

text: atomic

text: partitioning

text: generated

text: from

text: a

text: color-gradient

text: based

text: watershed

text: transform.

text: Then,

text: iteratively,

text: those

text: two

text: segments

text: P

text: and

text: Q

text: neighboring

text: in

text: image

text: space

text: become

text: merged

text: into

text: a

text: parent

text: node,

text: which

text: are

text: globally

text: most

text: similar

text: with

text: respect

text: to

text: a

text: sophisti-

text: D

text: cated

text: distance

text: measure

text: (P,

text: Q)

text: (cf.

text: Eq.

text: For

text: saliency

text: computations,

text: we

text: only

text: annotate

text: the

text: areas

text: A

text: resp.

text: A

text: (in

text: pixels),

text: the

text: length

text: of

text: the

text: perimeter

text: touching

text: the

text: P

text: Q

text: image

text: boundary

text: (where

text: applicable),

text: as

text: well

text: as

text: their

text: distance

text: at

text: the

text: time

text: of

text: merging

text: to

text: the

text: binary

text: partition

text: tree.

text: The

text: agglomerative

text: clustering

text: stops

text: when

text: the

text: last

text: two

text: segments

text: are

text: merged

text: to

text: become

text: root,

text: representing

text: the

text: full

text: scene.

text: Saliency

text: Transform

text: from

text: Composition

text: and

text: Contrast

text: We

text: claim

text: that

text: saliency

text: and

text: grouping

text: are

text: the

text: same

text: fundamental

text: process,

text: but

text: considered

text: from

text: two

text: opposing

text: viewpoints:

text: in

text: grouping,

text: one

text: focuses

text: on

text: which

text: parts

text: of

text: the

text: image

text: are

text: most

text: similar

text: and

text: should

text: belong

text: together,

text: whereas

text: saliency

text: tells

text: us

text: which

text: ones

text: are

text: most

text: contrasting

text: and

text: should

text: stay

text: separate.

text: This

text: motivated

text: the

text: idea

text: to

text: utilize

text: the

text: information

text: already

text: gained

text: during

text: hierarchical

text: clustering

text: to

text: express

text: an

text: antagonal

text: saliency.

text: There

text: are

text: some

text: important

text: questions

text: to

text: answer

text: when

text: composing

text: a

text: contrast

text: based

text: saliency

text: term

text: from

text: the

text: hierarchical

text: grouping

text: similarities:

text: ﬁrst

text: of

text: all,

text: which

text: of

text: the

text: two

text: opposing

text: segments

text: is

text: foreground

text: and

text: which

text: one

text: is

text: background?

text: Second,

text: how

text: should

text: the

text: image

text: boundary

text: be

text: treated,

text: could

text: one

text: model

text: eﬀects

text: of

text: human

text: peripheral

text: vision

text: to

text: saliency?

text: Finally,

text: how

text: is

text: saliency

text: inherited

text: from

text: object

text: scale

text: down

text: to

text: parts?

text: Surprisingly,

text: we

text: found

text: straightforward

text: and

text: well-performing

text: solutions

text: to

text: all

text: of

text: these

text: issues.

text: Since

text: our

text: approach

text: is

text: working

text: purely

text: on

text: the

text: graph

text: structure

text: created

text: from

text: hierarchical

text: grouping,

text: we

text: call

text: it

text: Partition

text: Tree

text: Saliency

text: (PaTS).

text: Figure

text: shows

text: an

text: example

text: of

text: the

text: hierarchical

text: grouping

text: and

text: the

text: corresponding

text: saliency.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: Zayas,

text: V.

text: G.

text: J.

text: Rodgers

text: and

text: C.

text: A.

text: Whiting,

text: “Type

text: IIB

text: supergravity

text: solutions

text: with

text: AdS

text: from

text: Abelian

text: and

text: non-Abelian

text: T

text: dualities,”

text: JHEP

text: [hep-th]].


text: × explaining the absence of Turing instabilities for 2 2 systems considered previously. Finally, we × perform a numerical existence/stability study for 3 3 example systems exhibiting Turing instability, answering in the aﬃrmative the fundamental question posed in [OZ03a, PSZ13] whether there can exist stable spatially periodic solutions of systems of conservation laws, at least at the level of numerical approximation. These studies suggest that, for supercritical Turing bifurcation, stable waves can emerge through the small-amplitude limit and persist up to rather large amplitudes. For sub-critical Turing bifurcations, all emerging waves are necessarily initially unstable, but appear in some cases to undergo secondary bifurcation to stability as amplitude is further increased. The numerically observed stability of intermediate-amplitude waves we regard as conclusive. Delicacy of numerical approximation as amplitude goes to zero, however, prevents us from obtaining a detailed stability diagram near the Turing bifurcation or even from making deﬁnitive conclusions about stability in that regime. Rigorous spectral stability analysis for conservation laws in this regime, analogous to those of [Mie95, Mie97, Sch96, SZJV16] in the reaction diﬀusion case, we regard therefore as a very interesting open problem. The studies in [MC00, Suk16] of reaction diﬀusion equations with an associated conservation law may oﬀer guidance in such an investigation. 2. Turing instability for conservation laws We begin by deﬁning a notion of Turing instability for systems of conservation laws (2.1) u + f(u; ε) = (Dεu ) , t x x x ∈ R u n, where ε is a bifurcation parameter and D for simplicity is taken constant. Linearizing (2.1) ≡ about a uniform state u(x, t) u yields the family of constant-coeﬃcient equations 0 − (2.2) u = L(ε)u := Aεu + Dεu t x xx ∈ − − ∈ R · with dispersion relations λ (ξ) σ( iξAε ξ2Dε), ξ , where σ( ) here and elsewhere denotes j spectrum of a matrix or linear operator. The state u is spectrally (hence nonlinearly) stable if 0 (cid:60) − − ≤ − | | (2.3) σ( iξAε ξ2Dε) θ ξ 2, θ > 0, ∈ R for all ξ [Kaw83]. Following the original philosophy applied by Turing [Tur52] to reaction diﬀusion systems, we seek a natural set of conditions guaranteeing low- and high-frequency stability– i.e., that (2.3) hold | | → ∞ | | (cid:54) ∞ for ξ 0, – but allowing instability at ﬁnite frequencies ξ = 0, . Should this be possible, then performing a homotopy in ε between stable and unstable states, we may expect generically to arrive at a special bifurcation point ε = ε∗, without loss of generality ε∗ = 0, for which (2.3) holds ± uniformly away from special points ξ = ξ∗, at which (cid:60) − − (2.4) max σ( iξAε∗ ξ2Dε∗) = 0 (cid:54) ξ=0 ± is achieved (note, by complex conjugate symmetry, that extrema appear in pairs) and for which (2.3) fails strictly as ε is further increased. We may then conclude, by standard bifurcation theory applied to the domain of periodic functions with period X := 2π/ξ∗ the appearance of nontrivial spatially periodic solutions with periods near X, similarly as in the reaction diﬀusion case [Mie95, Mie97, Sch96, SZJV16]. At ξ = 0, (2.3) yields that A is hyperbolic, in the sense that it has real semisimple eigenvalues. Without loss of generality, therefore, take A diagonal, with entries a , j = 1, . . . , n. In the sim- j plest case that A is strictly hyperbolic, in the sense that these a are distinct, we ﬁnd by spectral j perturbation expansion about ξ = 0 [Kaw83] that the corresponding eigenvalue expansions are − − λ (ξ) = ia ξ D ξ2 + O(ξ3), j j jj 2

text: explaining

text: the

text: absence

text: of

text: Turing

text: instabilities

text: for

text: systems

text: considered

text: previously.

text: Finally,

text: we

text: perform

text: a

text: numerical

text: existence/stability

text: study

text: for

text: example

text: systems

text: exhibiting

text: Turing

text: instability,

text: answering

text: in

text: the

text: aﬃrmative

text: the

text: fundamental

text: question

text: posed

text: in

text: whether

text: there

text: can

text: exist

text: stable

text: spatially

text: periodic

text: solutions

text: of

text: systems

text: of

text: conservation

text: laws,

text: at

text: least

text: at

text: the

text: level

text: of

text: numerical

text: approximation.

text: These

text: studies

text: suggest

text: that,

text: for

text: supercritical

text: Turing

text: bifurcation,

text: stable

text: waves

text: can

text: emerge

text: through

text: the

text: small-amplitude

text: limit

text: and

text: persist

text: up

text: to

text: rather

text: large

text: amplitudes.

text: For

text: sub-critical

text: Turing

text: bifurcations,

text: all

text: emerging

text: waves

text: are

text: necessarily

text: initially

text: unstable,

text: but

text: appear

text: in

text: some

text: cases

text: to

text: undergo

text: secondary

text: bifurcation

text: to

text: stability

text: as

text: amplitude

text: is

text: further

text: increased.

text: The

text: numerically

text: observed

text: stability

text: of

text: intermediate-amplitude

text: waves

text: we

text: regard

text: as

text: conclusive.

text: Delicacy

text: of

text: numerical

text: approximation

text: as

text: amplitude

text: goes

text: to

text: zero,

text: however,

text: prevents

text: us

text: from

text: obtaining

text: a

text: detailed

text: stability

text: diagram

text: near

text: the

text: Turing

text: bifurcation

text: or

text: even

text: from

text: making

text: deﬁnitive

text: conclusions

text: about

text: stability

text: in

text: that

text: regime.

text: Rigorous

text: spectral

text: stability

text: analysis

text: for

text: conservation

text: laws

text: in

text: this

text: regime,

text: analogous

text: to

text: those

text: of

text: in

text: the

text: reaction

text: diﬀusion

text: case,

text: we

text: regard

text: therefore

text: as

text: a

text: very

text: interesting

text: open

text: problem.

text: The

text: studies

text: in

text: of

text: reaction

text: diﬀusion

text: equations

text: with

text: an

text: associated

text: conservation

text: law

text: may

text: oﬀer

text: guidance

text: in

text: such

text: an

text: investigation.

text: Turing

text: instability

text: for

text: conservation

text: laws

text: We

text: begin

text: by

text: deﬁning

text: a

text: notion

text: of

text: Turing

text: instability

text: for

text: systems

text: of

text: conservation

text: laws

text: u

text: f(u;

text: ε)

text: (Dεu

text: t

text: x

text: x

text: x

text: R

text: u

text: n,

text: where

text: ε

text: is

text: a

text: bifurcation

text: parameter

text: and

text: D

text: for

text: simplicity

text: is

text: taken

text: constant.

text: Linearizing

text: about

text: a

text: uniform

text: state

text: u(x,

text: t)

text: u

text: yields

text: the

text: family

text: of

text: constant-coeﬃcient

text: equations

text: u

text: L(ε)u

text: Aεu

text: Dεu

text: t

text: x

text: xx

text: R

text: with

text: dispersion

text: relations

text: λ

text: (ξ)

text: σ(

text: iξAε

text: ξ

text: where

text: σ(

text: here

text: and

text: elsewhere

text: denotes

text: j

text: spectrum

text: of

text: a

text: matrix

text: or

text: linear

text: operator.

text: The

text: state

text: u

text: is

text: spectrally

text: (hence

text: nonlinearly)

text: stable

text: if

text: σ(

text: iξAε

text: θ

text: ξ

text: θ

text: R

text: for

text: all

text: ξ

text: Following

text: the

text: original

text: philosophy

text: applied

text: by

text: Turing

text: to

text: reaction

text: diﬀusion

text: systems,

text: we

text: seek

text: a

text: natural

text: set

text: of

text: conditions

text: guaranteeing

text: low-

text: and

text: high-frequency

text: stability–

text: i.e.,

text: that

text: hold

text: for

text: ξ

text: but

text: allowing

text: instability

text: at

text: ﬁnite

text: frequencies

text: ξ

text: Should

text: this

text: be

text: possible,

text: then

text: performing

text: a

text: homotopy

text: in

text: ε

text: between

text: stable

text: and

text: unstable

text: states,

text: we

text: may

text: expect

text: generically

text: to

text: arrive

text: at

text: a

text: special

text: bifurcation

text: point

text: ε

text: ε∗,

text: without

text: loss

text: of

text: generality

text: ε∗

text: for

text: which

text: holds

text: uniformly

text: away

text: from

text: special

text: points

text: ξ

text: ξ∗,

text: at

text: which

text: max

text: σ(

text: iξAε∗

text: ξ2Dε∗)

text: is

text: achieved

text: (note,

text: by

text: complex

text: conjugate

text: symmetry,

text: that

text: extrema

text: appear

text: in

text: pairs)

text: and

text: for

text: which

text: fails

text: strictly

text: as

text: ε

text: is

text: further

text: increased.

text: We

text: may

text: then

text: conclude,

text: by

text: standard

text: bifurcation

text: theory

text: applied

text: to

text: the

text: domain

text: of

text: periodic

text: functions

text: with

text: period

text: X

text: 2π/ξ∗

text: the

text: appearance

text: of

text: nontrivial

text: spatially

text: periodic

text: solutions

text: with

text: periods

text: near

text: X,

text: similarly

text: as

text: in

text: the

text: reaction

text: diﬀusion

text: case

text: At

text: ξ

text: yields

text: that

text: A

text: is

text: hyperbolic,

text: in

text: the

text: sense

text: that

text: it

text: has

text: real

text: semisimple

text: eigenvalues.

text: Without

text: loss

text: of

text: generality,

text: therefore,

text: take

text: A

text: diagonal,

text: with

text: entries

text: a

text: j

text: n.

text: In

text: the

text: sim-

text: j

text: plest

text: case

text: that

text: A

text: is

text: strictly

text: hyperbolic,

text: in

text: the

text: sense

text: that

text: these

text: a

text: are

text: distinct,

text: we

text: ﬁnd

text: by

text: spectral

text: j

text: perturbation

text: expansion

text: about

text: ξ

text: that

text: the

text: corresponding

text: eigenvalue

text: expansions

text: are

text: λ

text: (ξ)

text: ia

text: ξ

text: D

text: ξ2

text: j

text: j

text: jj