text: 10 Vulnerability of Deep Reinforcement Learning to Policy Induction Attacks Following the threat model presented in Section 1, this experiment considers an attacker capable of observing the states interactions between his target DQN and the game, but his domain of inﬂuence is limited to implementation of minor changes on the environment. Considering the visual representation of the envi- ronment in this setup, the minor changes incurred by attacker take the form of perturbing pixel values in the 4 consecutive frames of a given state. 5.1 Evaluation of Vulnerability to Adversarial Examples 100 FGSM JSMA 80 60 40 20 0 1.26 1.28 1.3 1.32 1.34 1.36 No. iterations x 1e6 Fig. 4: Success rate of crafting adversarial examples for DQN Successful implementations of the proposed policy induction attack mecha- nisms rely on the vulnerability of DQNs to targeted adversarial perturbations. To verify the existence of this vulnerability, the Qˆ networks of target were sam- pled at regular intervals during training in the game environment. In the next step, 100 observations comprised of a pair of consecutive states (s , s ) were t t+1 randomly selected from the experience memory of DQN, to ensure the possi- bility of their occurrence in the game. Considering s to be the variable that t+1 can be manipulated by the attacker, it is passed along with the model Qˆ to the adversarial example crafting algorithms. To study the extent of vulnerability, we evaluated the success rate of both FGSM and JSMA algorithms for each of the 100 random observations in inducing a random game action other than the ∗ current optimal a . The results, presented in Figure 4, verify that DQNs are t indeed vulnerable to adversarial example attacks. It is noteworthy that the suc- cess rate of FGSM with a ﬁxed perturbation limit decreases by one percent per 100000 observations as the number of observations increases. Yet, JSMA seems to be more robust to this eﬀect as it maintains a success rate of 100 percent throughout the experiment. 5.2 Veriﬁcation of Transferability To measure the transferability of adversarial examples between models, we trained another Q-network with a similar architecture on the same experience % etaR sseccuS ##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##

text: Vulnerability

text: of

text: Deep

text: Reinforcement

text: Learning

text: to

text: Policy

text: Induction

text: Attacks

text: Following

text: the

text: threat

text: model

text: presented

text: in

text: Section

text: this

text: experiment

text: considers

text: an

text: attacker

text: capable

text: of

text: observing

text: the

text: states

text: interactions

text: between

text: his

text: target

text: DQN

text: and

text: the

text: game,

text: but

text: his

text: domain

text: of

text: inﬂuence

text: is

text: limited

text: to

text: implementation

text: of

text: minor

text: changes

text: on

text: the

text: environment.

text: Considering

text: the

text: visual

text: representation

text: of

text: the

text: envi-

text: ronment

text: in

text: this

text: setup,

text: the

text: minor

text: changes

text: incurred

text: by

text: attacker

text: take

text: the

text: form

text: of

text: perturbing

text: pixel

text: values

text: in

text: the

text: consecutive

text: frames

text: of

text: a

text: given

text: state.

text: Evaluation

text: of

text: Vulnerability

text: to

text: Adversarial

text: Examples

text: FGSM

text: JSMA

text: No.

text: iterations

text: x

text: Fig.

text: Success

text: rate

text: of

text: crafting

text: adversarial

text: examples

text: for

text: DQN

text: Successful

text: implementations

text: of

text: the

text: proposed

text: policy

text: induction

text: attack

text: mecha-

text: nisms

text: rely

text: on

text: the

text: vulnerability

text: of

text: DQNs

text: to

text: targeted

text: adversarial

text: perturbations.

text: To

text: verify

text: the

text: existence

text: of

text: this

text: vulnerability,

text: the

text: Qˆ

text: networks

text: of

text: target

text: were

text: sam-

text: pled

text: at

text: regular

text: intervals

text: during

text: training

text: in

text: the

text: game

text: environment.

text: In

text: the

text: next

text: step,

text: observations

text: comprised

text: of

text: a

text: pair

text: of

text: consecutive

text: states

text: (s

text: s

text: were

text: t

text: randomly

text: selected

text: from

text: the

text: experience

text: memory

text: of

text: DQN,

text: to

text: ensure

text: the

text: possi-

text: bility

text: of

text: their

text: occurrence

text: in

text: the

text: game.

text: Considering

text: s

text: to

text: be

text: the

text: variable

text: that

text: can

text: be

text: manipulated

text: by

text: the

text: attacker,

text: it

text: is

text: passed

text: along

text: with

text: the

text: model

text: Qˆ

text: to

text: the

text: adversarial

text: example

text: crafting

text: algorithms.

text: To

text: study

text: the

text: extent

text: of

text: vulnerability,

text: we

text: evaluated

text: the

text: success

text: rate

text: of

text: both

text: FGSM

text: and

text: JSMA

text: algorithms

text: for

text: each

text: of

text: the

text: random

text: observations

text: in

text: inducing

text: a

text: random

text: game

text: action

text: other

text: than

text: the

text: current

text: optimal

text: a

text: The

text: results,

text: presented

text: in

text: Figure

text: verify

text: that

text: DQNs

text: are

text: t

text: indeed

text: vulnerable

text: to

text: adversarial

text: example

text: attacks.

text: It

text: is

text: noteworthy

text: that

text: the

text: suc-

text: cess

text: rate

text: of

text: FGSM

text: with

text: a

text: ﬁxed

text: perturbation

text: limit

text: decreases

text: by

text: one

text: percent

text: per

text: observations

text: as

text: the

text: number

text: of

text: observations

text: increases.

text: Yet,

text: JSMA

text: seems

text: to

text: be

text: more

text: robust

text: to

text: this

text: eﬀect

text: as

text: it

text: maintains

text: a

text: success

text: rate

text: of

text: percent

text: throughout

text: the

text: experiment.

text: Veriﬁcation

text: of

text: Transferability

text: To

text: measure

text: the

text: transferability

text: of

text: adversarial

text: examples

text: between

text: models,

text: we

text: trained

text: another

text: Q-network

text: with

text: a

text: similar

text: architecture

text: on

text: the

text: same

text: experience

text: etaR

text: sseccuS

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: Introduction The discovery of Fe-based superconductors (FBS) by Kamihara et al., [1] brought a huge impact to the physics community, since the compound consists of ferromagnetic Fe, which had been believed to be inevitably detrimental to the formation of Cooper pairs. To date, fundamental questions, such as mecha- nism of Cooper pairing and order parameter symmetry, are still under debate [2]. On the other hand, this material class is attractive for applications. For instance, AEFe As (AE: Alkali earth elements, AE- 2 2 122) and Fe(Se,Te) possess high upper critical ﬁelds (H ) exceeding 50 T and a low H anisotropy close c2 c2 to 1 at low temperature [3, 4], which is favourable for high-ﬁeld-magnet applications. Furthermore, Ba- 122 shows less deterioration of critical current across grain boundaries (GBs) [5, 6] than YBa Cu O − 2 3 7 δ (YBCO) and Bi-based cuprates. For Co-doped Ba-122, the critical GB misorientation angle (θ ), where ◦ − ◦ c J starts to fall off exponentially, has been reported to be 6 9 [5, 7]. Even high angle GBs do not im- c pede the current ﬂow very much in sintered K-doped Ba-122 wires and bulks, if clean and well-connected GBs are realised [8, 9]. Additionally, Co-doped Ba-122 exhibits a high tolerance for large densities of ﬂux pinning centres in the superconducting matrix, which leads to signiﬁcant increase in critical current density (J ) and irreversibility ﬁeld (H ) [10]. c irr Another advantage of Ba-122, in particular P-doped Ba-122, is its inherently high J . Putzke et al. c have reported on the enhancement of the vortex core energy of the ﬂux lines at the quantum critical point (QCP) of the antiferromagnetic phase [11]. Indeed, even microstructurally clean and optimally P-doped Ba-122 epitaxial thin ﬁlms, which were prepared by molecular beam epitaxy (MBE), exhibit a high self- ﬁeld J of over 6 MA/cm2 at 4.2 K [12]. Although excess magnetic Fe has been found to be harmful to c superconductivity in Fe(Se,Te) [13], Fe-rich P-doped Ba-122 thin ﬁlms showed a higher self-ﬁeld J of c over 10 MA/cm2 at 4.2 K, which is the highest value ever reported for FBS [6]. Whereas in the former case Fe is incorporated interstitially [14], in the latter case the Fe may form Fe-containing particles or regions with differing P-content, both acting as pinning centres [6]. Furthermore, the high J and low c anisotropy P-doped Ba-122 thin ﬁlms can be fabricated by tuning the processing conditions only, without any modiﬁcation of the target material used in pulsed laser deposition (PLD) [15]. The aforementioned advantages of P-doped Ba-122 are very suitable for high-ﬁeld-magnet applica- tions. Indeed, P-doped Ba-122 thin ﬁlms on technical substrates have been demonstrated as FBS coated conductors [16, 17]. To date, two kinds of technical substrates have been employed for FBS coated con- ductors: The cube-textured metal tapes with buffer layers (i.e., RABiTS) [18] and the Hastelloy tape on which biaxially textured buffer layers are prepared by ion-beam-assisted-deposition (IBAD) [19]. In contrast to Fe(Se,Te) coated conductors [20, 21], transport properties of P-doped Ba-122 coated con- ductors in the presence of extremely high magnetic ﬁelds have not yet been reported. Here, we report on in-ﬁeld transport properties of a P-doped Ba-122 thin ﬁlm grown by PLD on metal substrate with biaxially textured MgO template (IBAD-MgO) in a wide range of temperature and DC magnetic ﬁeld up to 35 T. We employ IBAD-MgO template with a relatively large in-plane full width at half maximum ◦ (FWHM) value (∆φ = 8 ), since it has been demonstrated by x-ray diffraction (XRD) and transmis- MgO sion electron microscopy (TEM) that the texture of MgO is transferred to the overlying P-doped Ba-122 ﬁlm, generating dislocation networks [17]. Such dislocation networks enhance the vortex pinning in P- ◦ doped Ba-122 [17], since θ is less than 9 [5]. Indeed, in-ﬁeld J properties of our P-doped Ba-122 on c ◦ c ◦ IBAD-MgO with ∆φ = 8 were superior to those of the ﬁlm on a template with ∆φ = 4 [17]. MgO (cid:107) MgO A high density of threading dislocations is very effective for improving J for H c in a wide range of c ◦− ◦ temperature and magnetic ﬁeld even close to H . Despite the relatively large θ of 6 9 for Ba-122, J irr c ◦c of our P-doped Ba-122 coated conductor with sharp FWHM values of both in-plane, ∆φ − = 5.7 , ◦ Ba 122 and out-of-plane misorientaion, ∆ω − = 1.2 (see Supplemental Fig. S1) is limited by the GBs in Ba 122 the low ﬁeld regime. However, at high ﬁeld, it exceeds a transport J of 105 A/cm2 at 15 T for ﬁeld c applied in both main crystallographic directions. Our P-doped Ba-122 coated conductor sample shows superior in-ﬁeld J properties over MgB and NbTi, and a comparable level to Nb Sn above 20 T. c 2 3 2

text: Introduction

label: section

text: The

label: paragraph

text: discovery

text: of

text: Fe-based

text: superconductors

text: (FBS)

text: by

text: Kamihara

text: et

text: al.,

text: brought

text: a

text: huge

text: impact

text: to

text: the

text: physics

text: community,

text: since

text: the

text: compound

text: consists

text: of

text: ferromagnetic

text: Fe,

text: which

text: had

text: been

text: believed

text: to

text: be

text: inevitably

text: detrimental

text: to

text: the

text: formation

text: of

text: Cooper

text: pairs.

text: To

text: date,

text: fundamental

text: questions,

text: such

text: as

text: mecha-

text: nism

text: of

text: Cooper

text: pairing

text: and

text: order

text: parameter

text: symmetry,

text: are

text: still

text: under

text: debate

text: On

text: the

text: other

text: hand,

text: this

text: material

text: class

text: is

text: attractive

text: for

text: applications.

text: For

text: instance,

text: AEFe

text: As

text: (AE:

text: Alkali

text: earth

text: elements,

text: AE-

text: and

text: Fe(Se,Te)

text: possess

text: high

text: upper

text: critical

text: ﬁelds

text: (H

text: exceeding

text: T

text: and

text: a

text: low

text: H

text: anisotropy

text: close

text: to

text: at

text: low

text: temperature

text: which

text: is

text: favourable

text: for

text: high-ﬁeld-magnet

text: applications.

text: Furthermore,

text: Ba-

text: shows

text: less

text: deterioration

text: of

text: critical

text: current

text: across

text: grain

text: boundaries

text: (GBs)

text: than

text: YBa

text: Cu

text: O

text: δ

text: (YBCO)

text: and

text: Bi-based

text: cuprates.

text: For

text: Co-doped

text: the

text: critical

text: GB

text: misorientation

text: angle

text: (θ

text: where

text: c

text: J

text: starts

text: to

text: fall

text: off

text: exponentially,

text: has

text: been

text: reported

text: to

text: be

text: Even

text: high

text: angle

text: GBs

text: do

text: not

text: im-

text: c

text: pede

text: the

text: current

text: ﬂow

text: very

text: much

text: in

text: sintered

text: K-doped

text: wires

text: and

text: bulks,

text: if

text: clean

text: and

text: well-connected

text: GBs

text: are

text: realised

text: Additionally,

text: Co-doped

text: exhibits

text: a

text: high

text: tolerance

text: for

text: large

text: densities

text: of

text: ﬂux

text: pinning

text: centres

text: in

text: the

text: superconducting

text: matrix,

text: which

text: leads

text: to

text: signiﬁcant

text: increase

text: in

text: critical

text: current

text: density

text: (J

text: and

text: irreversibility

text: ﬁeld

text: (H

text: c

text: irr

text: Another

text: advantage

text: of

text: in

text: particular

text: P-doped

text: is

text: its

text: inherently

text: high

text: J

text: Putzke

text: et

text: al.

text: c

text: have

text: reported

text: on

text: the

text: enhancement

text: of

text: the

text: vortex

text: core

text: energy

text: of

text: the

text: ﬂux

text: lines

text: at

text: the

text: quantum

text: critical

text: point

text: (QCP)

text: of

text: the

text: antiferromagnetic

text: phase

text: Indeed,

text: even

text: microstructurally

text: clean

text: and

text: optimally

text: P-doped

text: epitaxial

text: thin

text: ﬁlms,

text: which

text: were

text: prepared

text: by

text: molecular

text: beam

text: epitaxy

text: (MBE),

text: exhibit

text: a

text: high

text: self-

text: ﬁeld

text: J

text: of

text: over

text: at

text: K

text: Although

text: excess

text: magnetic

text: Fe

text: has

text: been

text: found

text: to

text: be

text: harmful

text: to

text: c

text: superconductivity

text: in

text: Fe(Se,Te)

text: Fe-rich

text: P-doped

text: thin

text: ﬁlms

text: showed

text: a

text: higher

text: self-ﬁeld

text: J

text: of

text: c

text: over

text: at

text: K,

text: which

text: is

text: the

text: highest

text: value

text: ever

text: reported

text: for

text: FBS

text: Whereas

text: in

text: the

text: former

text: case

text: Fe

text: is

text: incorporated

text: interstitially

text: in

text: the

text: latter

text: case

text: the

text: Fe

text: may

text: form

text: Fe-containing

text: particles

text: or

text: regions

text: with

text: differing

text: P-content,

text: both

text: acting

text: as

text: pinning

text: centres

text: Furthermore,

text: the

text: high

text: J

text: and

text: low

text: c

text: anisotropy

text: P-doped

text: thin

text: ﬁlms

text: can

text: be

text: fabricated

text: by

text: tuning

text: the

text: processing

text: conditions

text: only,

text: without

text: any

text: modiﬁcation

text: of

text: the

text: target

text: material

text: used

text: in

text: pulsed

text: laser

text: deposition

text: (PLD)

text: The

text: aforementioned

text: advantages

text: of

text: P-doped

text: are

text: very

text: suitable

text: for

text: high-ﬁeld-magnet

text: applica-

text: tions.

text: Indeed,

text: P-doped

text: thin

text: ﬁlms

text: on

text: technical

text: substrates

text: have

text: been

text: demonstrated

text: as

text: FBS

text: coated

text: conductors

text: To

text: date,

text: two

text: kinds

text: of

text: technical

text: substrates

text: have

text: been

text: employed

text: for

text: FBS

text: coated

text: con-

text: ductors:

text: The

text: cube-textured

text: metal

text: tapes

text: with

text: buffer

text: layers

text: (i.e.,

text: RABiTS)

text: and

text: the

text: Hastelloy

text: tape

text: on

text: which

text: biaxially

text: textured

text: buffer

text: layers

text: are

text: prepared

text: by

text: ion-beam-assisted-deposition

text: (IBAD)

text: In

text: contrast

text: to

text: Fe(Se,Te)

text: coated

text: conductors

text: transport

text: properties

text: of

text: P-doped

text: coated

text: con-

text: ductors

text: in

text: the

text: presence

text: of

text: extremely

text: high

text: magnetic

text: ﬁelds

text: have

text: not

text: yet

text: been

text: reported.

text: Here,

text: we

text: report

text: on

text: in-ﬁeld

text: transport

text: properties

text: of

text: a

text: P-doped

text: thin

text: ﬁlm

text: grown

text: by

text: PLD

text: on

text: metal

text: substrate

text: with

text: biaxially

text: textured

text: MgO

text: template

text: (IBAD-MgO)

text: in

text: a

text: wide

text: range

text: of

text: temperature

text: and

text: DC

text: magnetic

text: ﬁeld

text: up

text: to

text: T.

text: We

text: employ

text: IBAD-MgO

text: template

text: with

text: a

text: relatively

text: large

text: in-plane

text: full

text: width

text: at

text: half

text: maximum

text: (FWHM)

text: value

text: (∆φ

text: since

text: it

text: has

text: been

text: demonstrated

text: by

text: x-ray

text: diffraction

text: (XRD)

text: and

text: transmis-

text: MgO

text: sion

text: electron

text: microscopy

text: (TEM)

text: that

text: the

text: texture

text: of

text: MgO

text: is

text: transferred

text: to

text: the

text: overlying

text: P-doped

text: ﬁlm,

text: generating

text: dislocation

text: networks

text: Such

text: dislocation

text: networks

text: enhance

text: the

text: vortex

text: pinning

text: in

text: P-

text: doped

text: since

text: θ

text: is

text: less

text: than

text: Indeed,

text: in-ﬁeld

text: J

text: properties

text: of

text: our

text: P-doped

text: on

text: c

text: c

text: IBAD-MgO

text: with

text: ∆φ

text: were

text: superior

text: to

text: those

text: of

text: the

text: ﬁlm

text: on

text: a

text: template

text: with

text: ∆φ

text: MgO

text: MgO

text: A

text: high

text: density

text: of

text: threading

text: dislocations

text: is

text: very

text: effective

text: for

text: improving

text: J

text: for

text: H

text: c

text: in

text: a

text: wide

text: range

text: of

text: c

text: temperature

text: and

text: magnetic

text: ﬁeld

text: even

text: close

text: to

text: H

text: Despite

text: the

text: relatively

text: large

text: θ

text: of

text: for

text: J

text: irr

text: c

text: ◦c

text: of

text: our

text: P-doped

text: coated

text: conductor

text: with

text: sharp

text: FWHM

text: values

text: of

text: both

text: in-plane,

text: ∆φ

text: Ba

text: and

text: out-of-plane

text: misorientaion,

text: ∆ω

text: (see

text: Supplemental

text: Fig.

text: is

text: limited

text: by

text: the

text: GBs

text: in

text: Ba

text: the

text: low

text: ﬁeld

text: regime.

text: However,

text: at

text: high

text: ﬁeld,

text: it

text: exceeds

text: a

text: transport

text: J

text: of

text: at

text: T

text: for

text: ﬁeld

text: c

text: applied

text: in

text: both

text: main

text: crystallographic

text: directions.

text: Our

text: P-doped

text: coated

text: conductor

text: sample

text: shows

text: superior

text: in-ﬁeld

text: J

text: properties

text: over

text: MgB

text: and

text: NbTi,

text: and

text: a

text: comparable

text: level

text: to

text: Nb

text: Sn

text: above

text: T.

text: c


text: a) b) 107 H ||c YBCO 107 coated conductor 106 P-doped Ba-122 106 on MgO 105 Fone (RSAe,BTieT)S 105 NbTi Pon-d IoBpAeDd- BMag-O122 104 10 K 4.2 K 104 MgB2 Nb3Sn 103 15 K 103 102 20 K Jc,100=100 A/cm2 0 10 20 30 40 0 10 20 30 m0H (T) m0H (T) c) d) 1.0 1.0 H ||c H ||ab 4.2 K 4.2 K 0.8 10 K 0.8 10 K 15 K 15 K 20 K 20 K 0.6 0.6 0.4 point-like-pinning 0.4 0.2 0.2 GB pinning 0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 h=H/Hmax h=H/Hmax − FIG. 5. Field dependence of J and analysis of the ﬂux pinning density: a) J H properties of the P-doped c (cid:107) c Ba-122 coated conductor sample at 4.2K for H c in comparison to the data for P-doped Ba-122 on MgO single crystalline substrate[15], Fe(Se,Te) on RABiTS[21], YBCO coated conductor[40], MgB [41], NbTi[42, 43], and − 2 Nb Sn[44, 45]. b) J H properties of the P-doped Ba-122 coated conductor sample at various temperatures for both (cid:107)3 c (cid:107) H c (closed symbols) and H ab (open symbols). J is the criterion (100 A/cm2) used for determining H . c,100 − irr c) and d) The normalised pinning force f as a function of reduced ﬁeld h. The solid line is f = 25h0.5(1 h)2 p − p 16 5 for GB pinning, and the blue dotted line represent point-like pinning, f = 9h(1 h)2. p 4 3 15 )2mc/A( cJ xampF/pF=pf )2mc/A( cJ xampF/pF=pf ##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##

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text: − Linear presentation of E J curves at 4.2K a) b) 3.0 3.0 T=4.2 K 2.5 T=4.2 K 2.5 H || ab H || c 2.0 10 T 2.0 10 T 7.5 T 1.5 1.5 7.5 T 5 T 2.5 T 5 T 1.0 1.0 2.5 T 1 T 0.5 T 0.5 0.5 T 0.5 1 T 0.0 0.0 0.4 0.8 1.2 1.6 0.4 0.8 1.2 1.6 J (x106 A/cm2) J (x106 A/cm2) − − (cid:107) FIG. S3. E J curves: Linear presentation of the E J curves shown in Fig.4 up to 10T for a) H c and b) (cid:107) H ab. ∗ † Correspondence to:iida@nuap.nagoya-u.ac.jp [S1] Sato, H., Hiramatsu, H., Kamiya, T. & Hosono, H. High critical-current density with less anisotropy in BaFe (As,P) epitaxial ﬁlms: Effect of intentionally grown c-axis vortex-pinning centers. Appl. Phys. Lett. 2 2 104, 182603 (2014). [S2] Sato, H., Hiramatsu, H., Kamiya, T. & Hosono, H. Enhanced critical-current in P-doped BaFe As thin ﬁlms 2 2 on metal substrates arising from poorly aligned grain boundaries. Sci. Rep. 6, 36828 (2016). 3 )mc/V 4-01x( E )mc/V 4-01x( E ##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##

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text: T.

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text: epitaxial

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text: Effect

text: of

text: intentionally

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text: c-axis

text: vortex-pinning

text: centers.

text: Appl.

text: Phys.

text: Lett.

text: Sato,

text: H.,

text: Hiramatsu,

text: H.,

text: Kamiya,

text: T.

text: Hosono,

text: H.

text: Enhanced

text: critical-current

text: in

text: P-doped

text: BaFe

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text: plane crystallographic four-fold symmetry without in-plane rotational domains. The crystallinity of the ﬁlm was characterised on the basis of the full widths at half maximum (FWHMs) of the out-of-plane 004 (∆ω) and the in-plane 200 rocking curves (∆φ). The results of those XRD measurements can be found in Supplementary Information Fig. S1. The chemical composition was determined with an electron-probe microanalyser. The acceleration voltage of the electron beam was optimised while monitoring the Ni Kα spectrum to avoid the matrix effect from the Ni-containing Hastelloy metal-tapes. In-plane transport measurements A small bridge of 15 µm width and 500 µm length was patterned by photolithography, followed by ion-beam etching. Au electrodes with 50 nm thickness were formed by sputtering and lift-off. Transport properties using the resultant bridge were measured by a standard four-probe method. The temperature dependence of the resistivity of the P-doped Ba-122 coated conductor shows a T c,90 of 28.3 K (Fig. 1a), which is about 3 K lower than that of the optimally P-doped Ba-122 single crystals. T is deﬁned as the intersection between the steepest slope of the superconducting transition and a c,90 90% reduction of the ﬁt of the normal state resistivity using ρ = ρ + ATn. On the other hand, the n 0 onset T is deﬁned as the intersection between the ﬁt curve as stated above and the steepest slope of the c superconducting transition. The difference between T and the onset T is negligible. c,90 c The activation energy U (H) for vortex motion was evaluated by the temperature dependence of the 0 resistivity measurements in various ﬁeld strengths up to DC 35 T at the National High Magnetic Field Laboratory, Tallahassee, FL, USA. According to the model of thermally activated ﬂux ﬂow [26], the slope of linear ﬁt yields the pinning potential for vortex motion at given ﬁelds (Fig. 2c). On the assumption that − − U(T, H) = U (H)(1 T/T ), both equations, lnρ(T, H) = lnρ (H) U (H)/T and lnρ (H) = 0 c 0 0 0 lnρ + U (H)/T , are obtained, where ρ is a prefactor. 0f 0 c 0f In order to further understand the H anisotropy for a P-doped Ba-122 coated conductor sample, the c2 angular dependence of the magnetoresistivity was measured at 20 K. Using the same constant criterion ρ for evaluating H , the angular dependent upper critical ﬁeld [H (θ)] was derived (Fig. 3b). n,0.9 c2 c2 A criterion of 1 µV/cm was employed for evaluating J . In J measurement, the magnetic ﬁeld was c c always applied in the maximum Lorentz force conﬁguration. Low-ﬁeld measurements were performed in a Quantum Design physical property measurement system (PPMS) in magnetic ﬁelds up to 16 T. For high ﬁeld measurements up to DC 35 T, the experiments were conducted at the National High Magnetic Field Laboratory, Tallahassee, FL, USA. Acknowledgement A portion of this work was performed at the National High Magnetic Field Laboratory, which was supported by National Science Foundation Cooperative Agreement No. DMR-1157490, and the State of Florida. The work at Tokyo Institute of Technology was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) through Element Strategy Initiative to Form Core Research Center. K.I. acknowledges support by the Japan Society for the Promotion of Science (JSPS) Grant-in- Aid for Scientiﬁc Research (B) Grant Number 16H04646. H.Hi was also supported by JSPS for Young Scientists (A) Grant Number 25709058, JSPS Grant-in-Aid for Scientiﬁc Research on Innovative Areas Nano Informatics (Grant Number 25106007), and Support for Tokyotech Advanced Research (STAR). We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Karlsruhe Institute of Technology. Authors contribution K.I., C.T., J.H., H.S. and H.Hi. designed the study and wrote the manuscript together with J.J. and H.Ho. Thin ﬁlms preparation, structural characterisations and micro bridge fabrications were carried out by H.S. and H.Hi. K.I., C.T., J.H. and J.J. conducted high ﬁeld transport measurements. C.T., H.S. and H.Hi. performed low ﬁeld transport measurements. K.I., C.T., H.Hi, and H.Ho. supervised the projects. 7

text: plane

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text: The

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text: XRD

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text: The

text: chemical

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text: was

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text: with

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text: electron-probe

text: microanalyser.

text: The

text: acceleration

text: voltage

text: of

text: the

text: electron

text: beam

text: was

text: optimised

text: while

text: monitoring

text: the

text: Ni

text: Kα

text: spectrum

text: to

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text: matrix

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text: the

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text: Hastelloy

text: metal-tapes.

text: In-plane

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text: A

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text: µm

text: width

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text: followed

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text: etching.

text: Au

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text: Transport

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text: coated

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text: sample,

text: the

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text: was

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text: at

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text: Low-ﬁeld

text: measurements

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text: Quantum

text: Design

text: physical

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text: system

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text: Magnetic

text: Field

text: Laboratory,

text: Tallahassee,

text: FL,

text: USA.

text: Acknowledgement

text: A

text: portion

text: of

text: this

text: work

text: was

text: performed

text: at

text: the

text: National

text: High

text: Magnetic

text: Field

text: Laboratory,

text: which

text: was

text: supported

text: by

text: National

text: Science

text: Foundation

text: Cooperative

text: Agreement

text: No.

text: and

text: the

text: State

text: of

text: Florida.

text: The

text: work

text: at

text: Tokyo

text: Institute

text: of

text: Technology

text: was

text: supported

text: by

text: the

text: Ministry

text: of

text: Education,

text: Culture,

text: Sports,

text: Science

text: and

text: Technology

text: (MEXT)

text: through

text: Element

text: Strategy

text: Initiative

text: to

text: Form

text: Core

text: Research

text: Center.

text: K.I.

text: acknowledges

text: support

text: by

text: the

text: Japan

text: Society

text: for

text: the

text: Promotion

text: of

text: Science

text: (JSPS)

text: Grant-in-

text: Aid

text: for

text: Scientiﬁc

text: Research

text: (B)

text: Grant

text: Number

text: H.Hi

text: was

text: also

text: supported

text: by

text: JSPS

text: for

text: Young

text: Scientists

text: (A)

text: Grant

text: Number

text: JSPS

text: Grant-in-Aid

text: for

text: Scientiﬁc

text: Research

text: on

text: Innovative

text: Areas

text: Nano

text: Informatics

text: (Grant

text: Number

text: and

text: Support

text: for

text: Tokyotech

text: Advanced

text: Research

text: (STAR).

text: We

text: acknowledge

text: support

text: by

text: Deutsche

text: Forschungsgemeinschaft

text: and

text: Open

text: Access

text: Publishing

text: Fund

text: of

text: Karlsruhe

text: Institute

text: of

text: Technology.

text: Authors

text: contribution

text: K.I.,

text: C.T.,

text: J.H.,

text: H.S.

text: and

text: H.Hi.

text: designed

text: the

text: study

text: and

text: wrote

text: the

text: manuscript

text: together

text: with

text: J.J.

text: and

text: H.Ho.

text: Thin

text: ﬁlms

text: preparation,

text: structural

text: characterisations

text: and

text: micro

text: bridge

text: fabrications

text: were

text: carried

text: out

text: by

text: H.S.

text: and

text: H.Hi.

text: K.I.,

text: C.T.,

text: J.H.

text: and

text: J.J.

text: conducted

text: high

text: ﬁeld

text: transport

text: measurements.

text: C.T.,

text: H.S.

text: and

text: H.Hi.

text: performed

text: low

text: ﬁeld

text: transport

text: measurements.

text: K.I.,

text: C.T.,

text: H.Hi,

text: and

text: H.Ho.

text: supervised

text: the

text: projects.


text: 2.3 Examples Example 1. Considering the four-qubit GHZ and W states, i.e., |Ψ(cid:105) = √1 (|0000(cid:105)+|1111(cid:105)) 2 (cid:48) and |Ψ (cid:105) = 1(|0001(cid:105) + |0010(cid:105) + |0100(cid:105) + |1000(cid:105)). According to the partition (12)(34), we 2 have Ψ = (Ψ , Ψ , Ψ ) where (12)(34) u λ v (cid:32) (cid:33) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 0 0 1 0 √1 0 0 0 1 0 Ψu = ( 0 1 , 0 0 ) , Ψλ = 02 √1 , Ψv = ( 0 1 , 0 0 ) , (30) 2 (cid:48) and Ψ = (Ψ (cid:48), Ψ (cid:48), Ψ (cid:48)), where (12)(34) u λ v (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) (cid:18) (cid:19) (cid:18) (cid:19) 1 0 0 √1 √1 0 0 √1 1 0 Ψu(cid:48) = ( 0 0 , √1 02 ) , Ψλ(cid:48) = 02 √1 Ψv(cid:48) = ( √1 02 , 0 0 ) . (31) 2 2 2 (cid:48) According to Theorem 1 and Corollary 1, Ψ and Ψ are SLOCC equivalent if and only if (cid:101) (cid:101) (cid:48)† (cid:48)† rank[R(UP U )] = 1 , rank[R(V QV )] = 1 , (32) (cid:101) (cid:101) − where the submatrices P and Q in P and Q should further satisfy Q = Ψ 1P Ψ (cid:48). Eq. λ λ (cid:101) (cid:101) (32) induces only linear equations on the matrix elements, and we can easily ﬁnd that P = 0 and Q = 0, which indicates GHZ and W states are SLOCC inequivalent. Example 2. Considering the entangled states Ψ of the ﬁrst entanglement family in abcd Ref. [7], i.e. 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Examples

text: Example

text: Considering

text: the

text: four-qubit

text: GHZ

text: and

text: W

text: states,

text: i.e.,

text: √1

text: and

text: |Ψ

text: According

text: to

text: the

text: partition

text: we

text: have

text: Ψ

text: (Ψ

text: Ψ

text: Ψ

text: where

text: u

text: λ

text: v

label: equation

text: √1

text: Ψu

text: Ψλ

text: √1

text: Ψv

text: and

text: Ψ

text: (Ψ

text: Ψ

text: Ψ

text: where

text: u

text: λ

text: v

text: √1

text: √1

text: √1

text: √1

text: √1

text: √1

text: According

text: to

text: Theorem

text: and

text: Corollary

text: Ψ

text: and

text: Ψ

text: are

text: SLOCC

text: equivalent

text: if

text: and

text: only

text: if

text: rank[R(UP

text: U

text: rank[R(V

text: QV

text: where

text: the

text: submatrices

text: P

text: and

text: Q

text: in

text: P

text: and

text: Q

text: should

text: further

text: satisfy

text: Q

text: Ψ

text: Ψ

text: Eq.

text: λ

text: λ

text: induces

text: only

text: linear

text: equations

text: on

text: the

text: matrix

text: elements,

text: and

text: we

text: can

text: easily

text: ﬁnd

text: that

text: P

text: and

text: Q

text: which

text: indicates

text: GHZ

text: and

text: W

text: states

text: are

text: SLOCC

text: inequivalent.

text: Example

text: Considering

text: the

text: entangled

text: states

text: Ψ

text: of

text: the

text: ﬁrst

text: entanglement

text: family

text: in

text: abcd

text: Ref.

text: i.e.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48)† − † (V QV ) 1 . The solutions for Eq. (39) and Eq. (40) having rank 1 are              p11 p12 xp11 −xp12  p11 p12 αp11 αp12  P = p21 p22 xp21 −xp22 , P = p21 p22 βγ p21 βγ p22 , (41) −yp −yp −zp zp cβ p cβ p cγ p cγ p 11 12 11 12 a 11 a 12 a 11 a 12 −yp −yp −zp zp d p d p dαp dαp 21 22 21 22 bβ 21 bβ 22 bβ 21 bβ 22 where x, y, z and α, β, γ are nonzero parameters. A consistent solution of matrix P is γ cβ d cγ dα p = p = 0 , x = α = − , y = − = − , z = − = . (42) 12 21 β a bβ a bβ Eq. (42) predicts an invertible matrix P , and therefore |C(1)(cid:105) and |Ψ(2)(cid:105) are SLOCC equivalent. After getting P , we may also easily get (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 1 1 1 1 0 1 1 β p11 0 A = y , A = p11 − , A = , A = a , (43) 1 2 1 1 2 0 1 3 2 α γ 4 0 p22 x z p22 bβ which connect the matrices |C(1)(cid:105) = A ⊗A ⊗A ⊗A |Ψ(2)(cid:105). The SLOCC equivalence 1 2 3 4 of |C(1)(cid:105) and |Ψ(2)(cid:105) indicates that diﬀerent dimensional (1-D and 2-D) cluster states may be equivalent in realizing quantum computation tasks, which is important to the study of measurement-based quantum computation models. The above examples indicate that the new method works eﬀectively for ﬁnite dimen- sional four-partite systems. Diﬀerent choices of partitions, i.e. (13)(24) and (12)(34) for four-partite states, do not inﬂuence the application of the method. However, one must 17 ##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: (V

text: QV

text: The

text: solutions

text: for

text: Eq.

text: and

text: Eq.

text: having

text: rank

text: are

text: P

text: P

text: βγ

text: βγ

text: −yp

text: −yp

text: −zp

text: zp

text: cβ

text: p

text: cβ

text: p

text: cγ

text: p

text: cγ

text: p

text: a

text: a

text: a

text: a

text: −yp

text: −yp

text: −zp

text: zp

text: d

text: p

text: d

text: p

text: dαp

text: dαp

text: bβ

text: bβ

text: bβ

text: bβ

text: where

text: x,

text: y,

text: z

text: and

text: α,

text: β,

text: γ

text: are

text: nonzero

text: parameters.

text: A

text: consistent

text: solution

text: of

text: matrix

text: P

text: is

text: γ

text: cβ

text: d

text: cγ

text: dα

text: p

text: p

text: x

text: α

text: y

text: z

text: β

text: a

text: bβ

text: a

text: bβ

text: Eq.

text: predicts

text: an

text: invertible

text: matrix

text: P

text: and

text: therefore

text: and

text: are

text: SLOCC

text: equivalent.

text: After

text: getting

text: P

text: we

text: may

text: also

text: easily

text: get

text: β

text: A

text: y

text: A

text: A

text: A

text: a

text: α

text: γ

text: x

text: z

text: bβ

text: which

text: connect

text: the

text: matrices

text: A

text: ⊗A

text: ⊗A

text: ⊗A

text: The

text: SLOCC

text: equivalence

text: of

text: and

text: indicates

text: that

text: diﬀerent

text: dimensional

text: and

text: cluster

text: states

text: may

text: be

text: equivalent

text: in

text: realizing

text: quantum

text: computation

text: tasks,

text: which

text: is

text: important

text: to

text: the

text: study

text: of

text: measurement-based

text: quantum

text: computation

text: models.

text: The

text: above

text: examples

text: indicate

text: that

text: the

text: new

text: method

text: works

text: eﬀectively

text: for

text: ﬁnite

text: dimen-

text: sional

text: four-partite

text: systems.

text: Diﬀerent

text: choices

text: of

text: partitions,

text: i.e.

text: and

text: for

text: four-partite

text: states,

text: do

text: not

text: inﬂuence

text: the

text: application

text: of

text: the

text: method.

text: However,

text: one

text: must

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text:         ψ ψ · · · ψ  1111 1112 11I3I4  ψ ψ · · · ψ Ψ(I1I2)(I3I4) = 1...211 1...212 ... 12...I3I4 . (7) ψ ψ · · · ψ I1I211 I1I212 I1I2I3I4 The singular value decomposition of (7) goes as † Ψ = UΛV , (8) (I1I2)(I3I4) where Λ is a diagonal matrix of rank r; the unitary matrices U and V are composed by the left and right singular vectors of Ψ , i.e. U = ((cid:126)u , (cid:126)u , · · · ) and V = ((cid:126)v , (cid:126)v , · · · ), (I1I2)(I3I4) 1 2 1 2 † ∗ and the dimensions of vectors (cid:126)u and (cid:126)v are I · I and I · I respectively; V = (V )T is i j 1 2 3 4 the conjugate transpose of a matrix. It is legitimate to introduce the following triple-state set expression for the four-partite state based upon the partition (I I )(I I ): 1 2 3 4 Ψ = (Ψ , Ψ , Ψ ) . (9) (I1I2)(I3I4) u λ v C × Here, Ψ = (W((cid:126)u ), · · · , W((cid:126)u )) with W(u(cid:126) ) ∈ I1 I2, Ψ = (W((cid:126)v ), · · · , W((cid:126)v )) with u 1 r i v 1 r C × W(v(cid:126) ) ∈ I3 I4, and Ψ = Λ = diag{λ , · · · , λ } with λ > 0, i ∈ {1, · · · , r} being the i λ r 1 r i nonzero singular values in Λ. In this representation, Ψ and Ψ may be regarded as the u v tripartite states of r × I × I and r × I × I , and Ψ is a bipartite state of dimension 1 2 3 4 λ r × r, as shown in Figure 1. 6

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: The

text: singular

text: value

text: decomposition

text: of

text: goes

text: as

text: Ψ

text: UΛV

text: where

text: Λ

text: is

text: a

text: diagonal

text: matrix

text: of

text: rank

text: r;

text: the

text: unitary

text: matrices

text: U

text: and

text: V

text: are

text: composed

text: by

text: the

text: left

text: and

text: right

text: singular

text: vectors

text: of

text: Ψ

text: i.e.

text: U

text: and

text: V

text: and

text: the

text: dimensions

text: of

text: vectors

text: and

text: are

text: I

text: I

text: and

text: I

text: I

text: respectively;

text: V

text: (V

text: )T

text: is

text: i

text: j

text: the

text: conjugate

text: transpose

text: of

text: a

text: matrix.

text: It

text: is

text: legitimate

text: to

text: introduce

text: the

text: following

text: triple-state

text: set

text: expression

text: for

text: the

text: four-partite

text: state

text: based

text: upon

text: the

text: partition

text: (I

text: I

text: )(I

text: I

text: Ψ

text: (Ψ

text: Ψ

text: Ψ

text: u

text: λ

text: v

text: C

text: Here,

text: Ψ

text: with

text: Ψ

text: with

text: u

text: r

text: i

text: v

text: r

text: C

text: and

text: Ψ

text: Λ

text: diag{λ

text: λ

text: with

text: λ

text: i

text: r}

text: being

text: the

text: i

text: λ

text: r

text: r

text: i

text: nonzero

text: singular

text: values

text: in

text: Λ.

text: In

text: this

text: representation,

text: Ψ

text: and

text: Ψ

text: may

text: be

text: regarded

text: as

text: the

text: u

text: v

text: tripartite

text: states

text: of

text: r

text: I

text: I

text: and

text: r

text: I

text: I

text: and

text: Ψ

text: is

text: a

text: bipartite

text: state

text: of

text: dimension

text: λ

text: r

text: r,

text: as

text: shown

text: in

text: Figure


text: Here A are I × I submatrices. The realignment of the matrix A according to the ij 2 2 C × blocks A ∈ I2 I2 is deﬁned to be [14] ij (cid:0) (cid:1) T R(A) ≡ V(A ), · · · , V(A ), V(A ), · · · , V(A ), · · · , V(A ) , 11 I11 12 I12 I1I1 C · × · where R(A) ∈ I1 I1 I2 I2. By means of the complementary state, the following indis- pensable Lemma exits in following discussion. (cid:18) (cid:19) (cid:48) (cid:101) (cid:48) (cid:48) Lemma 1 Tripartite states Ψ = (Γ , · · · , Γ ) and Ψ = (Γ , · · · , Γ ) are SLOCC rI1I2 1 r rI1I2 1 r P Y equivalent if and only if there exist P = such that 0 P (cid:101) (cid:48)− rank[R(UP U 1)] = 1 . (13) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) Here U = (U , U ) with U = (V(Γ ), · · · , V(Γ )), U = (V(Γ ), · · · , V(Γ · )); 1 0 1 1 r 0 r+1 I1 I2 C × U = (U , U ) with U = (V(Γ ), · · · , V(Γ )), U = (V(Γ ), · · · , V(Γ · )); P ∈ r r, 1 0 1 1 r 0 r+1 I1 I2 C · − × · − and P ∈ (I1 I2 r) (I1 I2 r) are invertible matrices, and Y may be arbitrary. (cid:48) Proof: If Ψ is SLOCC equivalent to Ψ then rI1I2 rI1I2 − − (cid:48) (cid:48) (A 1 ⊗ A 1)(V(Γ ), · · · , V(Γ )) = (V(Γ ), · · · , V(Γ ))P , (14) 1 2 1 r 1 r 8 ##LTLine## ##LTLine##

text: Here

text: A

text: are

text: I

text: I

text: submatrices.

text: The

text: realignment

text: of

text: the

text: matrix

text: A

text: according

text: to

text: the

text: ij

text: C

text: blocks

text: A

text: is

text: deﬁned

text: to

text: be

text: ij

text: T

text: R(A)

text: V(A

text: V(A

text: V(A

text: V(A

text: V(A

text: C

text: where

text: R(A)

text: By

text: means

text: of

text: the

text: complementary

text: state,

text: the

text: following

text: indis-

text: pensable

text: Lemma

text: exits

text: in

text: following

text: discussion.

text: Lemma

text: Tripartite

text: states

text: Ψ

text: (Γ

text: Γ

text: and

text: Ψ

text: (Γ

text: Γ

text: are

text: SLOCC

text: r

text: r

text: P

text: Y

text: equivalent

text: if

text: and

text: only

text: if

text: there

text: exist

text: P

text: such

text: that

text: P

text: rank[R(UP

text: U

text: Here

text: U

text: (U

text: U

text: with

text: U

text: (V(Γ

text: V(Γ

text: U

text: (V(Γ

text: V(Γ

text: r

text: C

text: U

text: (U

text: U

text: with

text: U

text: (V(Γ

text: V(Γ

text: U

text: (V(Γ

text: V(Γ

text: P

text: r

text: r,

text: r

text: C

text: and

text: P

text: r)

text: r)

text: are

text: invertible

text: matrices,

text: and

text: Y

text: may

text: be

text: arbitrary.

text: Proof:

text: If

text: Ψ

text: is

text: SLOCC

text: equivalent

text: to

text: Ψ

text: then

text: (A

text: A

text: V(Γ

text: (V(Γ

text: V(Γ

text: ))P

text: r

text: r

text: ##LTLine##

text: ##LTLine##


text: (cid:82) (cid:32)(cid:82) (cid:33) Similarly, but this time using (58) also, (cid:82) (cid:82) | | | | | | 2 XjXiλ (cid:46) XjXiλ4 Xjλ4Xiλ4 (cid:46) 1 B(p,d/2) Jg 1 B(p,d/2) Jg + + , λ λ4 λ8 d2 J d2 J B(p,d/4) g B(p,d/4) g from which | | X X λ (cid:46) 1 j i λ d2 follows in the same manner. Lemma 5.7 gives estimates on the logarithmic derivatives of λ. We now begin working with η of (48). Write η = η . It follows from Lemma 5.7 part (i), g g | | (cid:46) η(p, q) + log[d(g(p), q)] 1, (59) K so that | | (cid:46) | | η(p, q) 1 + log[d(g(p), q)] . (60) K { Using Lemma 5.1, we have that X η, X X η exist and are continuous on U = (p, q) : (cid:54) − } i j i g p = g 1(q) , with | − | | | X λ(p, g 1(q)) (cid:46) 1 i X η(p, q) = − − , (61) i λ(p, g 1(q)) K d(p, g 1(q)) and | − | | − − | | | ≤ X X λ(p, g 1(q)) X λ(p, g 1(q))X λ(p, g 1(q)) (cid:46) 1 j i j i X X η(p, q) − + − − . j i λ(p, g 1(q)) λ(p, q 1(q))2 K d(p, g 1(q))2 (62) Moving on to φ˜ = φ˜ as in (49), we will need several regularity statements, and prefer g to break them into small pieces. Lemma 5.8. φ˜ is continuous on H × H. ∈ H×H (cid:54) − − (cid:54) Proof. If (p, q) is such that p = g 1(q) then, by Lemma 5.7 part (i), λ(p, g 1(q)) = 0, so φ˜ is continuous at (p, q) by the continuity of log away from 0. Suppose, therefore, that q = g(p), so that φ˜(p, q) = 0. Let (p , q ) be a sequence of points limiting on (p, q), and ∈ k k such that for all k, (p , q ) U (the presence of points outside U would not disturb the k k g g argument, as φ˜ is zero at these points, and we wish to show that φ˜(p , q ) converges to k k zero). Then, for all k, − − φ˜(p , q ) = η(p , q )(g 1(q ) 1p ) . k k k k k k 3 35 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Similarly,

text: but

text: this

text: time

text: using

text: also,

text: XjXiλ

text: Jg

text: Jg

text: λ

text: λ4

text: λ8

text: J

text: J

text: g

text: g

text: from

text: which

text: X

text: X

text: λ

text: j

text: i

text: λ

text: follows

text: in

text: the

text: same

text: manner.

text: Lemma

text: gives

text: estimates

text: on

text: the

text: logarithmic

text: derivatives

text: of

text: λ.

text: We

text: now

text: begin

text: working

text: with

text: η

text: of

text: Write

text: η

text: η

text: It

text: follows

text: from

text: Lemma

text: part

text: (i),

text: g

text: g

text: η(p,

text: q)

text: log[d(g(p),

text: q)]

text: K

text: so

text: that

text: η(p,

text: q)

text: log[d(g(p),

text: q)]

text: K

text: Using

text: Lemma

text: we

text: have

text: that

text: X

text: η,

text: X

text: X

text: η

text: exist

text: and

text: are

text: continuous

text: on

text: U

text: (p,

text: q)

text: i

text: j

text: i

text: g

text: p

text: g

text: with

text: X

text: λ(p,

text: g

text: i

text: X

text: η(p,

text: q)

text: i

text: λ(p,

text: g

text: K

text: d(p,

text: g

text: and

text: X

text: X

text: λ(p,

text: g

text: X

text: λ(p,

text: g

text: λ(p,

text: g

text: j

text: i

text: j

text: i

text: X

text: X

text: η(p,

text: q)

text: j

text: i

text: λ(p,

text: g

text: λ(p,

text: q

text: K

text: d(p,

text: g

text: Moving

text: on

text: to

text: φ˜

text: φ˜

text: as

text: in

text: we

text: will

text: need

text: several

text: regularity

text: statements,

text: and

text: prefer

text: g

text: to

text: break

text: them

text: into

text: small

text: pieces.

text: Lemma

text: φ˜

text: is

text: continuous

text: on

text: H

text: H.

text: H×H

text: Proof.

text: If

text: (p,

text: q)

text: is

text: such

text: that

text: p

text: g

text: then,

text: by

text: Lemma

text: part

text: (i),

text: λ(p,

text: g

text: so

text: φ˜

text: is

text: continuous

text: at

text: (p,

text: q)

text: by

text: the

text: continuity

text: of

text: log

text: away

text: from

text: Suppose,

text: therefore,

text: that

text: q

text: g(p),

text: so

text: that

text: φ˜(p,

text: q)

text: Let

text: (p

text: q

text: be

text: a

text: sequence

text: of

text: points

text: limiting

text: on

text: (p,

text: q),

text: and

text: k

text: k

text: such

text: that

text: for

text: all

text: k,

text: (p

text: q

text: U

text: (the

text: presence

text: of

text: points

text: outside

text: U

text: would

text: not

text: disturb

text: the

text: k

text: k

text: g

text: g

text: argument,

text: as

text: φ˜

text: is

text: zero

text: at

text: these

text: points,

text: and

text: we

text: wish

text: to

text: show

text: that

text: φ˜(p

text: q

text: converges

text: to

text: k

text: k

text: zero).

text: Then,

text: for

text: all

text: k,

text: φ˜(p

text: q

text: η(p

text: q

text: )(g

text: k

text: k

text: k

text: k

text: k

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:39) J e2Λh (68) f K h almost everywhere, with K(f) dependent on K only. Given the deﬁnition of h, we see that ◦ ◦ g = h δ(cid:107) (cid:107)− L − − . (69) q0 1 g 1(0) 1 If we deﬁne ◦ ◦ ◦ f = δ(cid:107) (cid:107) f δ(cid:107) (cid:107)− L − − , q0 h q0 1 g 1(0) 1 then f is quasiconformal, with essential dilatation equal to that of f . It is also true that, h at points of existence, − − J (p) = J (δ(cid:107) (cid:107)− (g 1(0) 1p)). (70) f fh q0 1 ◦ Given δ(cid:107) (cid:107)− L − − preserves sets of measure zero, it follows from (68) that q0 1 g 1(0) 1 Jfh(δ(cid:107)q0(cid:107)−1(g−1(0)−1p)) (cid:39) e2Λh(δ(cid:107)q0(cid:107)−1(g−1(0)−1p)) H (cid:39) at almost every p in . Using (69) and (70), this is seen to be equivalent to J e2Λ almost f everywhere, which is the conclusion of Theorem 1.1. We break the proof of Proposition 6.4 into three stages of increasing generality; ﬁrst ◦ ∈ ∞ ◦ with a quasilogarithmic potential of the form Λ g, ψ C (Proposition 6.5), then Λ g ψ 0 ◦ µ with µ compactly supported (Proposition 6.8), and ﬁnally Λ g with general admissible µ µ (which is Proposition 6.4). ∈ ∞ 6.3 dµ(q) = ψ(q) dq, with ψ C 0 To be precise, in this subsection, we prove the following. ≥ ◦ Proposition 6.5. Given K 1, there exists (cid:15) = (cid:15)(K) > 0 such that, if Λ = Λ g is a ∈ ∞ (cid:107) (cid:107) ∈ Q ψ quasilogarithmic potential, with ψ C , ψ < (cid:15), and g (K), then there exists a (cid:39) 0 1 0 quasiconformal mapping f with J e2Λ almost everywhere. The dilatation K(f) depends f K on K = K(g) only. In this case, identiﬁcation of the required (cid:15) > 0 comes from the following lemma, which is essentially Lemma 6.1 of [4]. ∈ Lemma 6.6. Suppose Fs, s [0, 1], is a f(cid:90)amily of quasiconform(cid:48)al mappings, G is a con- tinuous, positive, increasing, and locally Lipschitz function, and (cid:15) > 0 is such that ∞ (cid:48) 1 (cid:15) < (cid:15) = dσ. G(σ) 0 43 ##LTLine##

text: J

text: f

text: K

text: h

text: almost

text: everywhere,

text: with

text: K(f)

text: dependent

text: on

text: K

text: only.

text: Given

text: the

text: deﬁnition

text: of

text: h,

text: we

text: see

text: that

text: g

text: h

text: L

text: g

text: If

text: we

text: deﬁne

text: f

text: f

text: L

text: h

text: g

text: then

text: f

text: is

text: quasiconformal,

text: with

text: essential

text: dilatation

text: equal

text: to

text: that

text: of

text: f

text: It

text: is

text: also

text: true

text: that,

text: h

text: at

text: points

text: of

text: existence,

text: J

text: (p)

text: J

text: (g

text: f

text: fh

text: Given

text: L

text: preserves

text: sets

text: of

text: measure

text: zero,

text: it

text: follows

text: from

text: that

text: g

text: H

text: at

text: almost

text: every

text: p

text: in

text: Using

text: and

text: this

text: is

text: seen

text: to

text: be

text: equivalent

text: to

text: J

text: almost

text: f

text: everywhere,

text: which

text: is

text: the

text: conclusion

text: of

text: Theorem

text: We

text: break

text: the

text: proof

text: of

text: Proposition

text: into

text: three

text: stages

text: of

text: increasing

text: generality;

text: ﬁrst

text: with

text: a

text: quasilogarithmic

text: potential

text: of

text: the

text: form

text: Λ

text: g,

text: ψ

text: C

text: (Proposition

text: then

text: Λ

text: g

text: ψ

text: µ

text: with

text: µ

text: compactly

text: supported

text: (Proposition

text: and

text: ﬁnally

text: Λ

text: g

text: with

text: general

text: admissible

text: µ

text: µ

text: (which

text: is

text: Proposition

text: dµ(q)

text: ψ(q)

text: dq,

text: with

text: ψ

text: C

text: To

text: be

text: precise,

text: in

text: this

text: subsection,

text: we

text: prove

text: the

text: following.

text: Proposition

text: Given

text: K

text: there

text: exists

text: such

text: that,

text: if

text: Λ

text: Λ

text: g

text: is

text: a

text: Q

text: ψ

text: quasilogarithmic

text: potential,

text: with

text: ψ

text: C

text: ψ

text: and

text: g

text: (K),

text: then

text: there

text: exists

text: a

text: quasiconformal

text: mapping

text: f

text: with

text: J

text: almost

text: everywhere.

text: The

text: dilatation

text: K(f)

text: depends

text: f

text: K

text: on

text: K

text: K(g)

text: only.

text: In

text: this

text: case,

text: identiﬁcation

text: of

text: the

text: required

text: comes

text: from

text: the

text: following

text: lemma,

text: which

text: is

text: essentially

text: Lemma

text: of

text: Lemma

text: Suppose

text: Fs,

text: s

text: is

text: a

text: of

text: mappings,

text: G

text: is

text: a

text: con-

text: tinuous,

text: positive,

text: increasing,

text: and

text: locally

text: Lipschitz

text: function,

text: and

text: is

text: such

text: that

text: dσ.

text: G(σ)

text: ##LTLine##


text: Proposition 7.2 will be an obvious corollary to the lemmas that follow. Let us ﬁx ω and f as in the statement of the proposition. (cid:90) ⊂ H If U is Lebesgue measurable, we write ν(U) for the ω(p) dp measure of U, that is ν(U) = ω. U Now introduce the auxiliary function, 1 dω(p, q) := ν 4 (Bp,q) , ∪ where B := B(p, d(p, q)) B(q, d(q, p)). Despite the suggestive notation, this is not in p,q H general a metric, but only a quasimetric. The quasimetric space ( , d ) is called the David- ω H Semmes deformation of , a fascinating topic in its own right. We ﬁnd it convenient to (cid:39) (cid:39) ﬁrst prove ρ d , then d ρ . Before doing so, we note that f ω ω ω ∈ H Lemma 7.3. ν is doubling, that is, there exists C > 0 such that, for all p and r > 0, ≤ ν (B(p, 2r)) Cν (B(p, r)) . Proof. Using our assumed compara(cid:90)bility of wei(cid:90)ght and Jacobian, and the change of variable formula for quasiconformal mappings (27), (cid:46) | | ν (B(p, 2r)) = ω J = fB(p, 2r) . (80) f B(p,2r) B(p,2r) Let η be the quasisymmetric control function of f as in (23). With s deﬁned as the minimum of d(f(p), f(q)) over ∂B(p, r), and η := η(2), we have 2 ⊂ ⊂ ⊂ B(f(p), s) fB(p, r) fB(p, 2r) B(f(p), η s). 2 Using this, and that the Lebesgue measure is doubling with respect to the metric d, it follows | | ≤ | | (cid:46) | | (cid:46) | | (cid:46) fB(p, 2r) B(f(p), η s) B(f(p), s) fB(p, r) ν (B(p, r)) . (81) 2 Putting together (80) and (81), we have that ν is doubling as required. (cid:39) Lemma 7.4. d ρ . ω f ⊂ Proof. Using the inclusion B B(p, 2d(p, q)), and the doubling property of ν, p,q ≤ (cid:46) ν (B ) ν (B(p, 2d(p, q))) ν (B(p, d(p, q))) . p,q (cid:32) (cid:33) (cid:90) It follows that 1 (cid:39) 4 (cid:39) d (p, q) J d(f(p), f(q)), ω f B(p,d(p,q)) 52 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proposition

text: will

text: be

text: an

text: obvious

text: corollary

text: to

text: the

text: lemmas

text: that

text: follow.

text: Let

text: us

text: ﬁx

text: ω

text: and

text: f

text: as

text: in

text: the

text: statement

text: of

text: the

text: proposition.

text: H

text: If

text: U

text: is

text: Lebesgue

text: measurable,

text: we

text: write

text: ν(U)

text: for

text: the

text: ω(p)

text: dp

text: measure

text: of

text: U,

text: that

text: is

text: ν(U)

text: ω.

text: U

text: Now

text: introduce

text: the

text: auxiliary

text: function,

text: dω(p,

text: q)

text: ν

text: (Bp,q)

text: where

text: B

text: B(p,

text: d(p,

text: q))

text: B(q,

text: d(q,

text: p)).

text: Despite

text: the

text: suggestive

text: notation,

text: this

text: is

text: not

text: in

text: p,q

text: H

text: general

text: a

text: metric,

text: but

text: only

text: a

text: quasimetric.

text: The

text: quasimetric

text: space

text: d

text: is

text: called

text: the

text: David-

text: ω

text: H

text: Semmes

text: deformation

text: of

text: a

text: fascinating

text: topic

text: in

text: its

text: own

text: right.

text: We

text: ﬁnd

text: it

text: convenient

text: to

text: ﬁrst

text: prove

text: ρ

text: d

text: then

text: d

text: ρ

text: Before

text: doing

text: so,

text: we

text: note

text: that

text: f

text: ω

text: ω

text: ω

text: H

text: Lemma

text: ν

text: is

text: doubling,

text: that

text: is,

text: there

text: exists

text: C

text: such

text: that,

text: for

text: all

text: p

text: and

text: r

text: ν

text: (B(p,

text: Cν

text: (B(p,

text: r))

text: Proof.

text: Using

text: our

text: assumed

text: of

text: and

text: Jacobian,

text: and

text: the

text: change

text: of

text: variable

text: formula

text: for

text: quasiconformal

text: mappings

text: ν

text: (B(p,

text: ω

text: J

text: fB(p,

text: f

text: Let

text: η

text: be

text: the

text: quasisymmetric

text: control

text: function

text: of

text: f

text: as

text: in

text: With

text: s

text: deﬁned

text: as

text: the

text: minimum

text: of

text: d(f(p),

text: f(q))

text: over

text: ∂B(p,

text: r),

text: and

text: η

text: we

text: have

text: B(f(p),

text: s)

text: fB(p,

text: r)

text: fB(p,

text: B(f(p),

text: η

text: s).

text: Using

text: this,

text: and

text: that

text: the

text: Lebesgue

text: measure

text: is

text: doubling

text: with

text: respect

text: to

text: the

text: metric

text: d,

text: it

text: follows

text: fB(p,

text: B(f(p),

text: η

text: s)

text: B(f(p),

text: s)

text: fB(p,

text: r)

text: ν

text: (B(p,

text: r))

text: Putting

text: together

text: and

text: we

text: have

text: that

text: ν

text: is

text: doubling

text: as

text: required.

text: Lemma

text: d

text: ρ

text: ω

text: f

text: Proof.

text: Using

text: the

text: inclusion

text: B

text: B(p,

text: q)),

text: and

text: the

text: doubling

text: property

text: of

text: ν,

text: p,q

text: ν

text: (B

text: ν

text: (B(p,

text: q)))

text: ν

text: (B(p,

text: d(p,

text: q)))

text: p,q

text: It

text: follows

text: that

text: d

text: (p,

text: q)

text: J

text: d(f(p),

text: f(q)),

text: ω

text: f

text: B(p,d(p,q))

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 Warren et al. ally all work on afterglow emission shares one common inﬂuence the structure of the shock doing the accelerat- assumption: that the electrons responsible for the broad- ing. This nonlinear feedback loop results in a shock pre- band emission form a power law with a single spectral in- cursor, which acts to slightly slow the inﬂowing plasma dex p (this letter will also be used throughout this paper before it crosses the shock. More energetic particles thus to refer to momentum, but context should make it clear experience a larger velocity diﬀerence between the up- which deﬁnition applies at any particular instance). The stream and downstream limits of their scattering, which rare exceptions considered a Maxwellian thermal distri- in turn causes a harder spectrum at higher particle en- bution instead of (Pennanen et al. 2014) or in addition ergies (see, e.g., discussion in Berezhko & Ellison 1999; to (Giannios & Spitkovsky 2009) a power law tail.9 Blasi et al. 2005). The value for the electron spectral index p, starting While most work on nonlinear Fermi acceleration has from basic shock acceleration principles, is 2.23 (Kirk been done for ions accelerated in nonrelativistic shocks, et al. 2000; Keshet & Waxman 2005), in agreement with it is important to emphasize that nonlinear eﬀects must the results of Monte Carlo simulations (Ellison et al. occur in relativistic shocks as well, if acceleration is ef- 1990; Bednarz & Ostrowski 1998; Achterberg et al. 2001; ﬁcient. Furthermore, if no mechanism acts to transfer Ellison et al. 2013). Strictly speaking, this result ap- energy from ions to electrons, Fermi acceleration places plies only for ultra-relativistic shocks in the case where a large majority of the shock energy into ions. Ellison the upstream magnetic ﬁeld and shock normal are paral- et al. (2013) have shown that the nonlinear eﬀects re- lel, and then only in the limit where accelerated par- quired to ensure ﬂux conservation in relativistic shocks ticles carry negligible energy (the so-called “test par- may lead to orders of magnitude depression of electron ticle” regime). However, particle-in-cell (PIC) simula- acceleration eﬃciency (Warren et al. 2015). Handling tions show that self-generated magnetic ﬁeld turbulence the interaction between shocks and the particles they can overwhelm weak pre-existing ﬁelds (those for which accelerate is of paramount importance to properly un- the magnetization parameter of the inﬂowing plasma is derstanding GRB afterglows. (cid:46) − ≡ σ 10 5, with σ B2/(4πγ n m c2) for a shock mov- In addition to the electron spectral index p, GRB 0 0 0 p ing with Lorentz factor γ into plasma with mean mag- models use two parameters to handle the microphysics 0 netic ﬁeld B and proton number density n ) at rela- present at the shock: the fraction of energy density in 0 0 tivistic shocks (Sironi et al. 2013), and the results are electrons ((cid:15)e; see discussion surrounding Equation 5) and very similar to those at parallel or entirely unmagnetized magnetic ﬁeld ((cid:15)B). Applying the standard synchrotron shocks (e.g., Spitkovsky 2008). model for aft−erglows shows that the value of (cid:15)e in most Observations of GRB afterglows suggest softer spec- bursts is 0.1 0.3 (Santana et al. 2014). The value of tra than predicted by test-particle theory. Curran et al. (cid:15)B is much less constrained a−nd may−vary over several (2010) attempted to ﬁt X-ray observations both to a sin- orders of magnitude, from 10 7 to 10 3 (Santana et al. gle universal power law and to a distribution of p val- 2014; Wang et al(cid:46). 2015). If a particularly hard electron ues. Their most likely single value for p was 2.25, in spe−ctral index p 2.4 is chosen, (cid:15)B may even approach close agreement with theory, but statistical tests reject- 10 2. These ranges are signiﬁcantly larger th∼an th−e stan- ed a universal p value. When a Gaussian distribution dard interstellar medium (ISM) value of (cid:15)B 10 9, and of spectral indices was used instead, the best ﬁt was require ampliﬁcation of the magnetic ﬁeld by means of ± p = 2.36 0.59. The large uncertainty is almost entirely plasma instabilities. due to a long tail of softer—rather than harder—particle All of the preceding discussion of (cid:15)e and (cid:15)B rests on spectra. A search over a larger parameter space (varying, the assumption that the two parameters are constant e.g., explosion energy, electron spectral index, jet open- throughout the life of the afterglow. This is the de- ing angle, and energy density in electrons and magnetic fault state for afterglow modeling, though alternatives ﬁeld) found that GRB afterglows spanned the entire con- have been explored in a handful of cases. Rossi & Rees sidered range of spectral indices, as soft as p = 5 (Ryan (2003) explored a basic two-zone model for the exter- et al. 2015). For lack of a universal value for p stemming nal shock and afterglow, allowing for diﬀerent conditions from observations, the traditi−onal value used in studies at and downstream from the shock. Letting (cid:15)e and (cid:15)B of GRB afterglows is p = 2.4 2.5. vary with time has been proposed to explain both the It is well known that at non-relativistic speeds, ac- plateau phase of many X-ray light curves (Ioka et al. celerated particles may not form a simple power law. Ef- 2006) and the chromatic spectral breaks seen in many ﬁcient acceleration of particles10 moves such shocks be- afterglows (Panaitescu et al. 2006). The two parame- yond the test-particle regime, as the accelerated particles ters have been ﬁtted to decaying power laws in order to explore rebrightenings associated with a wind-ISM in- 9 The work of Eichler & Waxman (2005) also treated the pos- terface (Kong et al. 2010). The possibility that (cid:15)e and sibility of separate thermal and accelerated electron populations, (cid:15)B might increase with time was considered in Granot but used unreasonably low values for the thermal electron energy et al. (2006), as a means to reduce the γ-ray eﬃciency of because crucial numerical results had not yet been published. the prompt emission below the worryingly large values 10 We must be clear, here: if even 5% of the kinetic energy of − of 0.5 0.9. In each of these cases, however, the values inﬂowing particles (viewed in the frame where the shock is at rest) is transferred to a high-energy distribution of accelerated particles, for (cid:15)e and (cid:15)B were either chosen to match observations then the∼shock is beyond the test-particle regime. This is lower or given a speciﬁc functional form. than the 10% observed in long-term electron-ion PIC simulations No work of which we are aware has calculated (cid:15) (Sironi et al. 2013). e

text: Warren

text: et

text: al.

text: ally

text: all

text: work

text: on

text: afterglow

text: emission

text: shares

text: one

text: common

text: inﬂuence

text: the

text: structure

text: of

text: the

text: shock

text: doing

text: the

text: accelerat-

text: assumption:

text: that

text: the

text: electrons

text: responsible

text: for

text: the

text: broad-

text: ing.

text: This

text: nonlinear

text: feedback

text: loop

text: results

text: in

text: a

text: shock

text: pre-

text: band

text: emission

text: form

text: a

text: power

text: law

text: with

text: a

text: single

text: spectral

text: in-

text: cursor,

text: which

text: acts

text: to

text: slightly

text: slow

text: the

text: inﬂowing

text: plasma

text: dex

text: p

text: (this

text: letter

text: will

text: also

text: be

text: used

text: throughout

text: this

text: paper

text: before

text: it

text: crosses

text: the

text: shock.

text: More

text: energetic

text: particles

text: thus

text: to

text: refer

text: to

text: momentum,

text: but

text: context

text: should

text: make

text: it

text: clear

text: experience

text: a

text: larger

text: velocity

text: diﬀerence

text: between

text: the

text: up-

text: which

text: deﬁnition

text: applies

text: at

text: any

text: particular

text: instance).

text: The

text: stream

text: and

text: downstream

text: limits

text: of

text: their

text: scattering,

text: which

text: rare

text: exceptions

text: considered

text: a

text: Maxwellian

text: thermal

text: distri-

text: in

text: turn

text: causes

text: a

text: harder

text: spectrum

text: at

text: higher

text: particle

text: en-

text: bution

text: instead

text: of

text: (Pennanen

text: et

text: al.

text: or

text: in

text: addition

text: ergies

text: (see,

text: e.g.,

text: discussion

text: in

text: Berezhko

text: Ellison

text: to

text: (Giannios

text: Spitkovsky

text: a

text: power

text: law

text: Blasi

text: et

text: al.

text: The

text: value

text: for

text: the

text: electron

text: spectral

text: index

text: p,

text: starting

text: While

text: most

text: work

text: on

text: nonlinear

text: Fermi

text: acceleration

text: has

text: from

text: basic

text: shock

text: acceleration

text: principles,

text: is

text: (Kirk

text: been

text: done

text: for

text: ions

text: accelerated

text: in

text: nonrelativistic

text: shocks,

text: et

text: al.

text: Keshet

text: Waxman

text: in

text: agreement

text: with

text: it

text: is

text: important

text: to

text: emphasize

text: that

text: nonlinear

text: eﬀects

text: must

text: the

text: results

text: of

text: Monte

text: Carlo

text: simulations

text: (Ellison

text: et

text: al.

text: occur

text: in

text: relativistic

text: shocks

text: as

text: well,

text: if

text: acceleration

text: is

text: ef-

text: Bednarz

text: Ostrowski

text: Achterberg

text: et

text: al.

text: ﬁcient.

text: Furthermore,

text: if

text: no

text: mechanism

text: acts

text: to

text: transfer

text: Ellison

text: et

text: al.

text: Strictly

text: speaking,

text: this

text: result

text: ap-

text: energy

text: from

text: ions

text: to

text: electrons,

text: Fermi

text: acceleration

text: places

text: plies

text: only

text: for

text: ultra-relativistic

text: shocks

text: in

text: the

text: case

text: where

text: a

text: large

text: majority

text: of

text: the

text: shock

text: energy

text: into

text: ions.

text: Ellison

text: the

text: upstream

text: magnetic

text: ﬁeld

text: and

text: shock

text: normal

text: are

text: paral-

text: et

text: al.

text: have

text: shown

text: that

text: the

text: nonlinear

text: eﬀects

text: re-

text: lel,

text: and

text: then

text: only

text: in

text: the

text: limit

text: where

text: accelerated

text: par-

text: quired

text: to

text: ensure

text: ﬂux

text: conservation

text: in

text: relativistic

text: shocks

text: ticles

text: carry

text: negligible

text: energy

text: (the

text: so-called

text: “test

text: par-

text: may

text: lead

text: to

text: orders

text: of

text: magnitude

text: depression

text: of

text: electron

text: ticle”

text: regime).

text: However,

text: particle-in-cell

text: (PIC)

text: simula-

text: acceleration

text: eﬃciency

text: (Warren

text: et

text: al.

text: Handling

text: tions

text: show

text: that

text: self-generated

text: magnetic

text: ﬁeld

text: turbulence

text: the

text: interaction

text: between

text: shocks

text: and

text: the

text: particles

text: they

text: can

text: overwhelm

text: weak

text: pre-existing

text: ﬁelds

text: (those

text: for

text: which

text: accelerate

text: is

text: of

text: paramount

text: importance

text: to

text: properly

text: un-

text: the

text: magnetization

text: parameter

text: of

text: the

text: inﬂowing

text: plasma

text: is

text: derstanding

text: GRB

text: afterglows.

text: σ

text: with

text: σ

text: n

text: m

text: for

text: a

text: shock

text: mov-

text: In

text: addition

text: to

text: the

text: electron

text: spectral

text: index

text: p,

text: GRB

text: p

text: ing

text: with

text: Lorentz

text: factor

text: γ

text: into

text: plasma

text: with

text: mean

text: mag-

text: models

text: use

text: two

text: parameters

text: to

text: handle

text: the

text: microphysics

text: netic

text: ﬁeld

text: B

text: and

text: proton

text: number

text: density

text: n

text: at

text: rela-

text: present

text: at

text: the

text: shock:

text: the

text: fraction

text: of

text: energy

text: density

text: in

text: tivistic

text: shocks

text: (Sironi

text: et

text: al.

text: and

text: the

text: results

text: are

text: electrons

text: see

text: discussion

text: surrounding

text: Equation

text: and

text: very

text: similar

text: to

text: those

text: at

text: parallel

text: or

text: entirely

text: unmagnetized

text: magnetic

text: ﬁeld

text: Applying

text: the

text: standard

text: synchrotron

text: shocks

text: (e.g.,

text: Spitkovsky

text: model

text: for

text: aft−erglows

text: shows

text: that

text: the

text: value

text: of

text: in

text: most

text: Observations

text: of

text: GRB

text: afterglows

text: suggest

text: softer

text: spec-

text: bursts

text: is

text: (Santana

text: et

text: al.

text: The

text: value

text: of

text: tra

text: than

text: predicted

text: by

text: test-particle

text: theory.

text: Curran

text: et

text: al.

text: is

text: much

text: less

text: constrained

text: a−nd

text: may−vary

text: over

text: several

text: attempted

text: to

text: ﬁt

text: X-ray

text: observations

text: both

text: to

text: a

text: sin-

text: orders

text: of

text: magnitude,

text: from

text: to

text: (Santana

text: et

text: al.

text: gle

text: universal

text: power

text: law

text: and

text: to

text: a

text: distribution

text: of

text: p

text: val-

text: Wang

text: et

text: If

text: a

text: particularly

text: hard

text: electron

text: ues.

text: Their

text: most

text: likely

text: single

text: value

text: for

text: p

text: was

text: in

text: spe−ctral

text: index

text: p

text: is

text: chosen,

text: may

text: even

text: approach

text: close

text: agreement

text: with

text: theory,

text: but

text: statistical

text: tests

text: reject-

text: These

text: ranges

text: are

text: signiﬁcantly

text: larger

text: th∼an

text: th−e

text: stan-

text: ed

text: a

text: universal

text: p

text: value.

text: When

text: a

text: Gaussian

text: distribution

text: dard

text: interstellar

text: medium

text: (ISM)

text: value

text: of

text: and

text: of

text: spectral

text: indices

text: was

text: used

text: instead,

text: the

text: best

text: ﬁt

text: was

text: require

text: ampliﬁcation

text: of

text: the

text: magnetic

text: ﬁeld

text: by

text: means

text: of

text: p

text: The

text: large

text: uncertainty

text: is

text: almost

text: entirely

text: plasma

text: instabilities.

text: due

text: to

text: a

text: long

text: tail

text: of

text: softer—rather

text: than

text: harder—particle

text: All

text: of

text: the

text: preceding

text: discussion

text: of

text: and

text: rests

text: on

text: spectra.

text: A

text: search

text: over

text: a

text: larger

text: parameter

text: space

text: (varying,

text: the

text: assumption

text: that

text: the

text: two

text: parameters

text: are

text: constant

text: e.g.,

text: explosion

text: energy,

text: electron

text: spectral

text: index,

text: jet

text: open-

text: throughout

text: the

text: life

text: of

text: the

text: afterglow.

text: This

text: is

text: the

text: de-

text: ing

text: angle,

text: and

text: energy

text: density

text: in

text: electrons

text: and

text: magnetic

text: fault

text: state

text: for

text: afterglow

text: modeling,

text: though

text: alternatives

text: ﬁeld)

text: found

text: that

text: GRB

text: afterglows

text: spanned

text: the

text: entire

text: con-

text: have

text: been

text: explored

text: in

text: a

text: handful

text: of

text: cases.

text: Rossi

text: Rees

text: sidered

text: range

text: of

text: spectral

text: indices,

text: as

text: soft

text: as

text: p

text: (Ryan

text: explored

text: a

text: basic

text: two-zone

text: model

text: for

text: the

text: exter-

text: et

text: al.

text: For

text: lack

text: of

text: a

text: universal

text: value

text: for

text: p

text: stemming

text: nal

text: shock

text: and

text: afterglow,

text: allowing

text: for

text: diﬀerent

text: conditions

text: from

text: observations,

text: the

text: traditi−onal

text: value

text: used

text: in

text: studies

text: at

text: and

text: downstream

text: from

text: the

text: shock.

text: Letting

text: and

text: of

text: GRB

text: afterglows

text: is

text: p

text: vary

text: with

text: time

text: has

text: been

text: proposed

text: to

text: explain

text: both

text: the

text: It

text: is

text: well

text: known

text: that

text: at

text: non-relativistic

text: speeds,

text: ac-

text: plateau

text: phase

text: of

text: many

text: X-ray

text: light

text: curves

text: (Ioka

text: et

text: al.

text: celerated

text: particles

text: may

text: not

text: form

text: a

text: simple

text: power

text: law.

text: Ef-

text: and

text: the

text: chromatic

text: spectral

text: breaks

text: seen

text: in

text: many

text: ﬁcient

text: acceleration

text: of

text: moves

text: such

text: shocks

text: be-

text: afterglows

text: (Panaitescu

text: et

text: al.

text: The

text: two

text: parame-

text: yond

text: the

text: test-particle

text: regime,

text: as

text: the

text: accelerated

text: particles

text: ters

text: have

text: been

text: ﬁtted

text: to

text: decaying

text: power

text: laws

text: in

text: order

text: to

text: explore

text: rebrightenings

text: associated

text: with

text: a

text: wind-ISM

text: in-

text: The

text: work

text: of

text: Eichler

text: Waxman

text: also

text: treated

text: the

text: pos-

text: terface

text: (Kong

text: et

text: al.

text: The

text: possibility

text: that

text: and

text: sibility

text: of

text: separate

text: thermal

text: and

text: accelerated

text: electron

text: populations,

text: might

text: increase

text: with

text: time

text: was

text: considered

text: in

text: Granot

text: but

text: used

text: unreasonably

text: low

text: values

text: for

text: the

text: thermal

text: electron

text: energy

text: et

text: al.

text: as

text: a

text: means

text: to

text: reduce

text: the

text: γ-ray

text: eﬃciency

text: of

text: because

text: crucial

text: numerical

text: results

text: had

text: not

text: yet

text: been

text: published.

text: the

text: prompt

text: emission

text: below

text: the

text: worryingly

text: large

text: values

text: We

text: must

text: be

text: clear,

text: here:

text: if

text: even

text: of

text: the

text: kinetic

text: energy

text: of

text: of

text: In

text: each

text: of

text: these

text: cases,

text: however,

text: the

text: values

text: inﬂowing

text: particles

text: (viewed

text: in

text: the

text: frame

text: where

text: the

text: shock

text: is

text: at

text: rest)

text: is

text: transferred

text: to

text: a

text: high-energy

text: distribution

text: of

text: accelerated

text: particles,

text: for

text: and

text: were

text: either

text: chosen

text: to

text: match

text: observations

text: then

text: the∼shock

text: is

text: beyond

text: the

text: test-particle

text: regime.

text: This

text: is

text: lower

text: or

text: given

text: a

text: speciﬁc

text: functional

text: form.

text: than

text: the

text: observed

text: in

text: long-term

text: electron-ion

text: PIC

text: simulations

text: No

text: work

text: of

text: which

text: we

text: are

text: aware

text: has

text: calculated

text: (Sironi

text: et

text: al.

text: e