text: 6 (cid:16) (cid:17) ments. This research used resources of the National En- ergy Research Scientiﬁc Computing Center, which is sup- − − ported by the Oﬃce of Science of the U.S. Department Ψ = Ψ∞ + a be L/ξ /L, (A.6) L of Energy under Contract No. DE-AC02-05CH11231. (cid:54) where Ψ∞ = 0 allows for an exponential decay to a Appendix: Finite-Size Scaling Forms non-zero value, as is the case for ordered phases. At a critical point, the expected asymptotic scaling ∼ − Here we derive ﬁnite-size scaling forms for the order form becomes C(r) r η, so instead we ﬁnd (cid:90) parameter, which can be deﬁned as the structure factor (cid:2) (cid:3) per site (cid:88) − L R − − − S(0) = C + A r η + (L r) η dr, (A.7) 0 S(0) 1 R Ψ = = C((cid:126)r). (A.1) L N N (cid:126)r which after integration becomes (cid:104) (cid:105) This expression can be converted into an integral in → ∞ the continuum limit that i(cid:90)s obtained as L : A − − − − S(0) = C + − (L R)1 η R1 η . (A.8) 0 1 η L S(0) = C(r)dr. (A.2) Again, we cannot uniquely determine C and R, which 0 0 leaves the general scaling form Assuming that the correlation function approaches the ∼ − asymptotic form C(r) e r/ξ for r > R, we replace the − − S(0) = a + b (L R)1 η . (A.9) integral from 0 to R by a “core charge” C . This yields 0 a simple expression(cid:90) for th(cid:104)e structure factor,(cid:105) For small R/L this can be replaced by the simpler form − L R − − − S(0) = C + A e r/ξ + e (L r)/ξ dr, (A.3) 0 R S(0) = a + bL1−η. (A.10) where we take into account the periodicity of C(r) for ﬁnite simulation cells with periodic boundary conditions. Thus, we have used the following ﬁnite size scaling This gives (cid:104) (cid:105) form for the order parameter near the critical point: − − − − S(0) = C + Aξ e R/ξ e (L R)/ξ , (A.4) 0 Ψ = a/L + b/Lη. (A.11) L which ultimately is the same as The above ﬁnite-size scaling forms can also be de- rived for higher dimensions d > 1 by using the relation − − − S(0) = a be L/ξ, (A.5) Ψ = S(0)L d. In this case, the correlation function at L ∼ − a critical point is deﬁned as C(r) r2 (d+z+η), where z is the dynamic critical exponent for ground state phase i.e. we cannot uniquely determine C0 and R. This transitions. For the Gaussian phase transition separating leads to ﬁnite-size scaling form the Haldane and large-D phases, we expect z = 1.20 1 Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009). arXiv:1311.2946. 2 X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. 5 Y.-Z. You, Z. Bi, A. Rasmussen, K. Slagle, and C. Xu, B 87, 155114 (2013). Phys. Rev. Lett. 112, 247202 (2014). 3 F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, 6 A. W. Sandvik and J. Kurkija¨rvi, Phys. Rev. B 43, 5950 Phys. Rev. B 81, 064439 (2010). (1991). 4 A. Chandran, V. Khemani, and S. L. Sondhi (2013), 7 A. W. Sandvik, Journal of Physics A: Mathematical and ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: L/ξ

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font: default

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text: Ψ∞

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text: a

text: Appendix:

label: section

text: Finite-Size

text: Scaling

text: Forms

text: non-zero

text: value,

text: as

text: is

text: the

text: case

text: for

text: ordered

text: phases.

text: At

text: a

text: critical

text: point,

text: the

text: expected

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text: scaling

text: Here

text: we

text: derive

text: ﬁnite-size

text: scaling

text: forms

text: for

text: the

text: order

text: form

text: becomes

text: C(r)

text: r

text: η,

text: so

text: instead

text: we

text: ﬁnd

text: parameter,

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text: can

text: be

text: deﬁned

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text: structure

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text: site

text: L

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text: L

text: A

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text: L

text: C(r)dr.

text: Again,

text: we

text: cannot

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text: determine

text: C

text: and

text: R,

text: which

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text: the

text: general

text: scaling

text: form

text: Assuming

text: that

text: the

text: correlation

text: function

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text: the

text: asymptotic

text: form

text: C(r)

text: e

text: r/ξ

text: for

text: r

text: R,

text: we

text: replace

text: the

text: a

text: b

text: (L

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text: “core

text: charge”

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text: a

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text: R/L

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text: the

text: simpler

text: form

text: L

text: R

text: C

text: A

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text: r/ξ

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text: (L

text: r)/ξ

text: dr,

text: R

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text: where

text: we

text: take

text: into

text: account

text: the

text: periodicity

text: of

text: C(r)

text: for

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text: simulation

text: cells

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text: periodic

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text: Thus,

text: we

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text: size

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text: gives

text: form

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text: order

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text: critical

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text: C

text: Aξ

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text: e

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text: R)/ξ

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text: a/L

text: b/Lη.

text: L

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text: ultimately

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text: same

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text: The

text: above

text: ﬁnite-size

text: scaling

text: forms

text: can

text: also

text: be

text: de-

text: rived

text: for

text: higher

text: dimensions

text: d

text: by

text: using

text: the

text: relation

text: a

text: be

text: L/ξ,

text: Ψ

text: d.

text: In

text: this

text: case,

text: the

text: correlation

text: function

text: at

text: L

text: a

text: critical

text: point

text: is

text: deﬁned

text: as

text: C(r)

text: (d+z+η),

text: where

text: z

text: is

text: the

text: dynamic

text: critical

text: exponent

text: for

text: ground

text: state

text: phase

text: i.e.

text: we

text: cannot

text: uniquely

text: determine

text: and

text: R.

text: This

text: transitions.

text: For

text: the

text: Gaussian

text: phase

text: transition

text: separating

text: leads

text: to

text: ﬁnite-size

text: scaling

text: form

text: the

text: Haldane

text: and

text: large-D

text: phases,

text: we

text: expect

text: z

text: Z.-C.

text: Gu

text: and

text: X.-G.

text: Wen,

text: Phys.

text: Rev.

text: B

text: X.

text: Chen,

text: Z.-C.

text: Gu,

text: Z.-X.

text: Liu,

text: and

text: X.-G.

text: Wen,

text: Phys.

text: Rev.

text: Y.-Z.

text: You,

text: Z.

text: Bi,

text: A.

text: Rasmussen,

text: K.

text: Slagle,

text: and

text: C.

text: Xu,

text: B

text: Phys.

text: Rev.

text: Lett.

text: F.

text: Pollmann,

text: A.

text: M.

text: Turner,

text: E.

text: Berg,

text: and

text: M.

text: Oshikawa,

text: A.

text: W.

text: Sandvik

text: and

text: J.

text: Kurkija¨rvi,

text: Phys.

text: Rev.

text: B

text: Phys.

text: Rev.

text: B

text: A.

text: Chandran,

text: V.

text: Khemani,

text: and

text: S.

text: L.

text: Sondhi

text: A.

text: W.

text: Sandvik,

text: Journal

text: of

text: Physics

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text: Mathematical

text: and

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text: 6 analysis of the behavior of Foursquare users in this paper search results returned on mobile phones [14]. is only certain to be relevant to those Foursquare users in METHODS the data, who are by their nature as users of the service not representative of a general population, and similarly The Foursquare dataset consists of check-ins made by for the mobile phone users whose behavior is reﬂected by Foursquare users between November 2010 and Septem- the telecoms data. In particular, the Foursquare dataset ber 2011. The dataset is composed of information pub- contains check-ins that users shared publicly during the licly available on the Internet that can therefore be down- data collection period, and those users may have been to loaded and analyzed in such a way as we have here. We places where they did not check in, or kept certain check- downloaded all of the check-ins that users posted as non- ins private. It must be borne in mind that the likelihood private on Twitter, and which are thus publicly avail- of sharing a check-in at a given place may not be inde- able, through the oﬃcial Twitter search API by search- pendent of the place category; for example, users might ing for tweets containing ‘4sq’ and a link (to the check-in be less likely to share check-ins at home or at a train page). This resulted in a dataset estimated to account station than check-ins at entertainment venues. It is also for 20-25% of Foursquare check-ins made during this pe- possible that some of the patterns seen in Foursquare riod. We were also able to download venue information check-ins do extend outside the speciﬁcs of the services, including the venue location (latitude and longitude) and since building on Cho, Myers, and Leskovec’s observation category (College, Food, Professional, etc.) directly from of considerable similarity between the mobility patterns the Foursquare venue database, which is also publicly ac- in some LBSN datasets and in mobile telephone network cessible. data [9], we have shown similarities also between the mo- We analyze the set of 2,315,350 check-ins made to bility behavior of groups seen in the Foursquare dataset 109,314 venues in New York, by 104,266 users. We use analyzed in this work and the dataset from the mobile check-ins as the representation of user visits to places; network operator. check-ins are voluntary so they are an under-estimate Of course, mobile phone datasets also have their in- of the actual set of places visited by a user. However, herent biases, and in this case it could be that we under- the comparison between check-ins made with friends estimate colocation between users because they did not and alone shows considerable diﬀerences which may be happen to call one another around the time of meeting reﬂected in the general place visit behavior of users, (similarly to people not checking in on Foursquare) or since the mobility patterns apparent from LBSN check- because they were connected to diﬀerent cell towers with ins show strong similarity to those seen in other kinds of overlapping coverage areas, or that false positive coloca- mobility data including cellphone records [9]. In addition tions are inferred because people called one another while we downloaded the publicly available social networks of connected to the same cell tower despite not meeting face- the users, taking a pair to be friends when each follows to-face. We observe that the similarities in behavior that the other on Twitter. This social network has 99,725 we ﬁnd between the Foursquare and telecoms datasets nodes and 821,948 edges. provides some evidence in support of the diﬀerences not The telecoms dataset is a large anonymized set of being due to these eﬀects. However, the most reliable billing records for over one million mobile phone users in applications of results from the analysis of data such as Portugal, gathered over a twelve month period between these are most probably in areas directly relating to the 2006 and 2007. The dataset contains information about use of those services, whether the mobile phone network mobile phone calls, but does not contain text messages or online social networks. Taken in the context of the (SMS) or data usage (Internet). The dataset was ob- technological service concerned, that is, for applications tained directly from the operator and full permission was within such services, such as venue recommendation for granted for its use in our analysis. In order to preserve groups in LBSNs, the results are relevant and potentially privacy, individual phone numbers were anonymized by useful. the operator. Each user in the anonymized dataset is We argue that our ﬁndings have potential applications identiﬁed by a hashed ID. Each entry is a CDR (Call in location-aware technological services. For example, Detail Record), consisting of a timestamp, the IDs of the one important task in online location-based social net- caller and the callee, the call duration, and the cell tower works is that of venue recommendation: suggesting to IDs of the caller and callee towers. The dataset also the logged-in user places where that user might like to go. includes the latitude and longitude of the cell towers, Given that we have found diﬀerences between the places which allows us to study the relationship between the so- that people may visit with their friends and those where cial network constructed by placing edges between nodes they may go on their own, it may be that services such representing people whenever one person has called an- as Foursquare could beneﬁt from suggesting diﬀerent rec- other, and the physical location of the users. The social ommendations to a user based on knowledge of whether network has 1,954,188 nodes and 19,370,004 edges. they are with friends or alone. Similarly, these diﬀer- ACKNOWLEDGEMENTS ences in individual and group behavior could be used to improve mobility prediction, with possible uses in per- Chlo¨e Brown is a recipient of the Google Europe sonalized services such as location-aware advertising and Fellowship in Mobile Computing, and this research is

text: analysis

text: of

text: the

text: behavior

text: of

text: Foursquare

text: users

text: in

text: this

text: paper

text: search

text: results

text: returned

text: on

text: mobile

text: phones

text: is

text: only

text: certain

text: to

text: be

text: relevant

text: to

text: those

text: Foursquare

text: users

text: in

text: METHODS

text: the

text: data,

text: who

text: are

text: by

text: their

text: nature

text: as

text: users

text: of

text: the

text: service

text: not

text: representative

text: of

text: a

text: general

text: population,

text: and

text: similarly

text: The

text: Foursquare

text: dataset

text: consists

text: of

text: check-ins

text: made

text: by

text: for

text: the

text: mobile

text: phone

text: users

text: whose

text: behavior

text: is

text: reﬂected

text: by

text: Foursquare

text: users

text: between

text: November

text: and

text: Septem-

text: the

text: telecoms

text: data.

text: In

text: particular,

text: the

text: Foursquare

text: dataset

text: ber

text: The

text: dataset

text: is

text: composed

text: of

text: information

text: pub-

text: contains

text: check-ins

text: that

text: users

text: shared

text: publicly

text: during

text: the

text: licly

text: available

text: on

text: the

text: Internet

text: that

text: can

text: therefore

text: be

text: down-

text: data

text: collection

text: period,

text: and

text: those

text: users

text: may

text: have

text: been

text: to

text: loaded

text: and

text: analyzed

text: in

text: such

text: a

text: way

text: as

text: we

text: have

text: here.

text: We

text: places

text: where

text: they

text: did

text: not

text: check

text: in,

text: or

text: kept

text: certain

text: check-

text: downloaded

text: all

text: of

text: the

text: check-ins

text: that

text: users

text: posted

text: as

text: non-

text: ins

text: private.

text: It

text: must

text: be

text: borne

text: in

text: mind

text: that

text: the

text: likelihood

text: private

text: on

text: Twitter,

text: and

text: which

text: are

text: thus

text: publicly

text: avail-

text: of

text: sharing

text: a

text: check-in

text: at

text: a

text: given

text: place

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text: not

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text: inde-

text: able,

text: through

text: the

text: oﬃcial

text: Twitter

text: search

text: API

text: by

text: search-

text: pendent

text: of

text: the

text: place

text: category;

text: for

text: example,

text: users

text: might

text: ing

text: for

text: tweets

text: containing

text: and

text: a

text: link

text: (to

text: the

text: check-in

text: be

text: less

text: likely

text: to

text: share

text: check-ins

text: at

text: home

text: or

text: at

text: a

text: train

text: page).

text: This

text: resulted

text: in

text: a

text: dataset

text: estimated

text: to

text: account

text: station

text: than

text: check-ins

text: at

text: entertainment

text: venues.

text: It

text: is

text: also

text: for

text: of

text: Foursquare

text: check-ins

text: made

text: during

text: this

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text: possible

text: that

text: some

text: of

text: the

text: patterns

text: seen

text: in

text: Foursquare

text: riod.

text: We

text: were

text: also

text: able

text: to

text: download

text: venue

text: information

text: check-ins

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text: extend

text: outside

text: the

text: speciﬁcs

text: of

text: the

text: services,

text: including

text: the

text: venue

text: location

text: (latitude

text: and

text: longitude)

text: and

text: since

text: building

text: on

text: Cho,

text: Myers,

text: and

text: Leskovec’s

text: observation

text: category

text: (College,

text: Food,

text: Professional,

text: etc.)

text: directly

text: from

text: of

text: considerable

text: similarity

text: between

text: the

text: mobility

text: patterns

text: the

text: Foursquare

text: venue

text: database,

text: which

text: is

text: also

text: publicly

text: ac-

text: in

text: some

text: LBSN

text: datasets

text: and

text: in

text: mobile

text: telephone

text: network

text: cessible.

text: data

text: we

text: have

text: shown

text: similarities

text: also

text: between

text: the

text: mo-

text: We

text: analyze

text: the

text: set

text: of

text: check-ins

text: made

text: to

text: bility

text: behavior

text: of

text: groups

text: seen

text: in

text: the

text: Foursquare

text: dataset

text: venues

text: in

text: New

text: York,

text: by

text: users.

text: We

text: use

text: analyzed

text: in

text: this

text: work

text: and

text: the

text: dataset

text: from

text: the

text: mobile

text: check-ins

text: as

text: the

text: representation

text: of

text: user

text: visits

text: to

text: places;

text: network

text: operator.

text: check-ins

text: are

text: voluntary

text: so

text: they

text: are

text: an

text: under-estimate

text: Of

text: course,

text: mobile

text: phone

text: datasets

text: also

text: have

text: their

text: in-

text: of

text: the

text: actual

text: set

text: of

text: places

text: visited

text: by

text: a

text: user.

text: However,

text: herent

text: biases,

text: and

text: in

text: this

text: case

text: it

text: could

text: be

text: that

text: we

text: under-

text: the

text: comparison

text: between

text: check-ins

text: made

text: with

text: friends

text: estimate

text: colocation

text: between

text: users

text: because

text: they

text: did

text: not

text: and

text: alone

text: shows

text: considerable

text: diﬀerences

text: which

text: may

text: be

text: happen

text: to

text: call

text: one

text: another

text: around

text: the

text: time

text: of

text: meeting

text: reﬂected

text: in

text: the

text: general

text: place

text: visit

text: behavior

text: of

text: users,

text: (similarly

text: to

text: people

text: not

text: checking

text: in

text: on

text: Foursquare)

text: or

text: since

text: the

text: mobility

text: patterns

text: apparent

text: from

text: LBSN

text: check-

text: because

text: they

text: were

text: connected

text: to

text: diﬀerent

text: cell

text: towers

text: with

text: ins

text: show

text: strong

text: similarity

text: to

text: those

text: seen

text: in

text: other

text: kinds

text: of

text: overlapping

text: coverage

text: areas,

text: or

text: that

text: false

text: positive

text: coloca-

text: mobility

text: data

text: including

text: cellphone

text: records

text: In

text: addition

text: tions

text: are

text: inferred

text: because

text: people

text: called

text: one

text: another

text: while

text: we

text: downloaded

text: the

text: publicly

text: available

text: social

text: networks

text: of

text: connected

text: to

text: the

text: same

text: cell

text: tower

text: despite

text: not

text: meeting

text: face-

text: the

text: users,

text: taking

text: a

text: pair

text: to

text: be

text: friends

text: when

text: each

text: follows

text: to-face.

text: We

text: observe

text: that

text: the

text: similarities

text: in

text: behavior

text: that

text: the

text: other

text: on

text: Twitter.

text: This

text: social

text: network

text: has

text: we

text: ﬁnd

text: between

text: the

text: Foursquare

text: and

text: telecoms

text: datasets

text: nodes

text: and

text: edges.

text: provides

text: some

text: evidence

text: in

text: support

text: of

text: the

text: diﬀerences

text: not

text: The

text: telecoms

text: dataset

text: is

text: a

text: large

text: anonymized

text: set

text: of

text: being

text: due

text: to

text: these

text: eﬀects.

text: However,

text: the

text: most

text: reliable

text: billing

text: records

text: for

text: over

text: one

text: million

text: mobile

text: phone

text: users

text: in

text: applications

text: of

text: results

text: from

text: the

text: analysis

text: of

text: data

text: such

text: as

text: Portugal,

text: gathered

text: over

text: a

text: twelve

text: month

text: period

text: between

text: these

text: are

text: most

text: probably

text: in

text: areas

text: directly

text: relating

text: to

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text: dataset

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text: services,

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text: but

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text: Taken

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text: The

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text: 3 A. Ac susceptibility: Identiﬁcation of anomalous variation in critical current density. 1. Isoﬁeld ac susceptibility measurements Figure 1(a) shows the temperature dependences of the real part of the isoﬁeld ac susceptibility ( 0( )) in dc χ T ﬁelds (applied parallel to plane of the platelet shaped sample), as indicated. The 0( ) plots have been normal- χ T ized with respect to the saturated value of 0 recorded at χ = 2 K for = 0. The 0( ) curve for = 0 shows T H χ T H the saturated bdcehavior at low temperatures (dscee the inset (i) of Fig. 1(a)), revealing the characteristic full magnetic shielding eﬀect, typical of a superconductor. The super- conducting transition temperature, ( = 0) ≈ 7 1 K, T H . obtained via the onset of diamagneticc redscponse (see ex- panded plot in the inset (ii) of Fig. 1(a)) from this mea- surement agrees with the reported for this compound T in Ref. [47]. The 0( ) rcesponse for = 5 kOe in χ T H the main panel of Fig. 1(a) shows a much bdrcoader transi- tion to the normal state depicting the monotonically de- creasing value of 0( ) as the temperature is increased. χ T Since 0 is related to the critical current density, ( ) χ j H, T (( 0 ∼ − ) [10], where is the amplitudec of the χ βj /h h ac drive ancd acdepends on theascize and geometry of the β sample), the fall in | 0( )| reﬂects a decrease in ( ) χ T j H, T with increasing temperature. However, the 0c( ) re- χ T sponse for = 10 kOe shows an unusual deviation H from the smoodcth monotonic decrease prior to the super- conducting to normal transition, which in turn indicates a possible non-monotonic behavior of . This modula- Figure 2: (Color online) (a) Isothermal ac susceptibility ( 0( )) j χ H tion in 0( ) becomes more pronounced fcor = 15 kOe plotted against the dc magnetic ﬁeld at various temperatures, as χ T H (marked by arrow) and for = 20 kOe, budtc the same indicated. PE phenomenon could be observed for 5 5K. (b) characteristic could not beHiddcentiﬁed clearly at higher Portions of χ0(H) (in normalized units) and dχ0/dHT (<in a.rbitrary ﬁelds ( 25 kOe). In order to bring out the interest- units) plots at T = 4.5K. The ﬁrst zero crossing in dχ0/dH indi- ing chaHradcct>eristics of this modulation, we now examine cates the onset (Hon,ac) of the PE, whereas the second one marks p in Fig. 1(b), the behavior of 0( ) near the region, where the peak ﬁeld (Hac) of the PE. p χ T this modulation is observed for = 15 kOe. This fea- H ture can be more clearly discerneddicn an expanded plot of the derivative of 0( ) with respect to (see Fig. 1(c)). ﬁeld. The 0( ) response at each temperature initially The point of inﬂχectTion in 0( ), whereT the slope value decreases asχthHe magnetic ﬁeld is enhanced from zero ﬁeld χ T is minimum, identiﬁes the maximum in as a function value, revealing the usual fall in ( ) with increasing of temperature at a constant ﬁeld. Thjics temperature ﬁeld at a constant temperature. HjocwHever, an anomalous is marked as ( ) (∼ 4 56 K). After ( ), the slope dip in 0( ) response can be seen at higher ﬁeld values. 0 increaTspesHabruptly. , reaching aTmp aHximum value As theχteHmperature is increased, the onset ﬁeld values dχ /dT and then falling to a value close to zero. of this dip gradually shift towards lower ﬁeld values (cf. 2. Isothermal ac susceptibility measurements Fig. 2(a)), reaching closer to the respective critical ﬁeld values of superconducting-normal transition, which, in To further understand the details of the modula- turn, suggests that the observed anomaly is akin to the tion in ( ) reﬂected in the isoﬁeld ac susceptibility well documented peak eﬀect (PE) phenomenon [2, 3, 7– j H, T measuremc ents, we recorded isothermal ac susceptibility, 16, 25–38]. In Fig. 2(b), we show an expanded portion 0( ), data as well, where the above discussed anoma- of 0( ) response at = 4 5 K, where the onset ﬁeld χ H χ H T . lous trend could be observed more clearly over a wide ( ) of the peak eﬀect like feature and the peak ﬁeld Hon,ac range of temperatures ( ≤ 5 5 K), as shown in Fig. 2(a) ( p ) of the PE are indicated by arrows. We further T . Hac ( 0( ) values are normalized with respect to their sat- claprify the PE feature by plotting the derivative of 0 χ H χ urated value at = 0). The specimen was cooled in with respect to at = 4 5 K in the same panel. The H H T . a nominal zero ﬁeld down to a desired temperature and derivative curve ( 0 ) in Fig. 2(b) may appear on dχ /dH then the 0( ) data were recorded while increasing the dc ﬁrst examination to be analogous to the derivative plot χ H ##LTFigure##

text: A.

text: Ac

text: susceptibility:

text: Identiﬁcation

text: of

text: anomalous

text: variation

text: in

text: critical

text: current

text: density.

text: Isoﬁeld

text: ac

text: susceptibility

text: measurements

text: Figure

text: shows

text: the

text: temperature

text: dependences

text: of

text: the

text: real

text: part

text: of

text: the

text: isoﬁeld

text: ac

text: susceptibility

text: in

text: dc

text: χ

text: T

text: ﬁelds

text: (applied

text: parallel

text: to

text: plane

text: of

text: the

text: platelet

text: shaped

text: sample),

text: as

text: indicated.

text: The

text: plots

text: have

text: been

text: normal-

text: χ

text: T

text: ized

text: with

text: respect

text: to

text: the

text: saturated

text: value

text: of

text: recorded

text: at

text: χ

text: K

text: for

text: The

text: curve

text: for

text: shows

text: T

text: H

text: χ

text: T

text: H

text: the

text: saturated

text: bdcehavior

text: at

text: low

text: temperatures

text: (dscee

text: the

text: inset

text: (i)

text: of

text: Fig.

text: revealing

text: the

text: characteristic

text: full

text: magnetic

text: shielding

text: eﬀect,

text: typical

text: of

text: a

text: superconductor.

text: The

text: super-

text: conducting

text: transition

text: temperature,

text: K,

text: T

text: H

text: obtained

text: via

text: the

text: onset

text: of

text: diamagneticc

text: redscponse

text: (see

text: ex-

text: panded

text: plot

text: in

text: the

text: inset

text: (ii)

text: of

text: Fig.

text: from

text: this

text: mea-

text: surement

text: agrees

text: with

text: the

text: reported

text: for

text: this

text: compound

text: T

text: in

text: Ref.

text: The

text: rcesponse

text: for

text: kOe

text: in

text: χ

text: T

text: H

text: the

text: main

text: panel

text: of

text: Fig.

text: shows

text: a

text: much

text: bdrcoader

text: transi-

text: tion

text: to

text: the

text: normal

text: state

text: depicting

text: the

text: monotonically

text: de-

text: creasing

text: value

text: of

text: as

text: the

text: temperature

text: is

text: increased.

text: χ

text: T

text: Since

text: is

text: related

text: to

text: the

text: critical

text: current

text: density,

text: χ

text: j

text: H,

text: T

text: where

text: is

text: the

text: amplitudec

text: of

text: the

text: χ

text: βj

text: /h

text: h

text: ac

text: drive

text: ancd

text: acdepends

text: on

text: theascize

text: and

text: geometry

text: of

text: the

text: β

text: sample),

text: the

text: fall

text: in

text: reﬂects

text: a

text: decrease

text: in

text: χ

text: T

text: j

text: H,

text: T

text: with

text: increasing

text: temperature.

text: However,

text: the

text: re-

text: χ

text: T

text: sponse

text: for

text: kOe

text: shows

text: an

text: unusual

text: deviation

text: H

text: from

text: the

text: smoodcth

text: monotonic

text: decrease

text: prior

text: to

text: the

text: super-

text: conducting

text: to

text: normal

text: transition,

text: which

text: in

text: turn

text: indicates

text: a

text: possible

text: non-monotonic

text: behavior

text: of

text: This

text: modula-

text: Figure

text: (Color

text: online)

text: (a)

text: Isothermal

text: ac

text: susceptibility

text: j

text: χ

text: H

text: tion

text: in

text: becomes

text: more

text: pronounced

text: fcor

text: kOe

text: plotted

text: against

text: the

text: dc

text: magnetic

text: ﬁeld

text: at

text: various

text: temperatures,

text: as

text: χ

text: T

text: H

text: (marked

text: by

text: arrow)

text: and

text: for

text: kOe,

text: budtc

text: the

text: same

text: indicated.

text: PE

text: phenomenon

text: could

text: be

text: observed

text: for

text: (b)

text: characteristic

text: could

text: not

text: beHiddcentiﬁed

text: clearly

text: at

text: higher

text: Portions

text: of

text: (in

text: normalized

text: units)

text: and

text: (<in

text: a.rbitrary

text: ﬁelds

text: kOe).

text: In

text: order

text: to

text: bring

text: out

text: the

text: interest-

text: units)

text: plots

text: at

text: T

text: The

text: ﬁrst

text: zero

text: crossing

text: in

text: indi-

text: ing

text: chaHradcct>eristics

text: of

text: this

text: modulation,

text: we

text: now

text: examine

text: cates

text: the

text: onset

text: (Hon,ac)

text: of

text: the

text: PE,

text: whereas

text: the

text: second

text: one

text: marks

text: p

text: in

text: Fig.

text: the

text: behavior

text: of

text: near

text: the

text: region,

text: where

text: the

text: peak

text: ﬁeld

text: (Hac)

text: of

text: the

text: PE.

text: p

text: χ

text: T

text: this

text: modulation

text: is

text: observed

text: for

text: kOe.

text: This

text: fea-

text: H

text: ture

text: can

text: be

text: more

text: clearly

text: discerneddicn

text: an

text: expanded

text: plot

text: of

text: the

text: derivative

text: of

text: with

text: respect

text: to

text: (see

text: Fig.

text: ﬁeld.

text: The

text: response

text: at

text: each

text: temperature

text: initially

text: The

text: point

text: of

text: inﬂχectTion

text: in

text: whereT

text: the

text: slope

text: value

text: decreases

text: asχthHe

text: magnetic

text: ﬁeld

text: is

text: enhanced

text: from

text: zero

text: ﬁeld

text: χ

text: T

text: is

text: minimum,

text: identiﬁes

text: the

text: maximum

text: in

text: as

text: a

text: function

text: value,

text: revealing

text: the

text: usual

text: fall

text: in

text: with

text: increasing

text: of

text: temperature

text: at

text: a

text: constant

text: ﬁeld.

text: Thjics

text: temperature

text: ﬁeld

text: at

text: a

text: constant

text: temperature.

text: HjocwHever,

text: an

text: anomalous

text: is

text: marked

text: as

text: K).

text: After

text: the

text: slope

text: dip

text: in

text: response

text: can

text: be

text: seen

text: at

text: higher

text: ﬁeld

text: values.

text: increaTspesHabruptly.

text: reaching

text: aTmp

text: aHximum

text: value

text: As

text: theχteHmperature

text: is

text: increased,

text: the

text: onset

text: ﬁeld

text: values

text: dχ

text: /dT

text: and

text: then

text: falling

text: to

text: a

text: value

text: close

text: to

text: zero.

text: of

text: this

text: dip

text: gradually

text: shift

text: towards

text: lower

text: ﬁeld

text: values

text: (cf.

text: Isothermal

text: ac

text: susceptibility

text: measurements

text: Fig.

text: reaching

text: closer

text: to

text: the

text: respective

text: critical

text: ﬁeld

text: values

text: of

text: superconducting-normal

text: transition,

text: which,

text: in

text: To

text: further

text: understand

text: the

text: details

text: of

text: the

text: modula-

text: turn,

text: suggests

text: that

text: the

text: observed

text: anomaly

text: is

text: akin

text: to

text: the

text: tion

text: in

text: reﬂected

text: in

text: the

text: isoﬁeld

text: ac

text: susceptibility

text: well

text: documented

text: peak

text: eﬀect

text: (PE)

text: phenomenon

text: 7–

text: j

text: H,

text: T

text: measuremc

text: ents,

text: we

text: recorded

text: isothermal

text: ac

text: susceptibility,

text: In

text: Fig.

text: we

text: show

text: an

text: expanded

text: portion

text: data

text: as

text: well,

text: where

text: the

text: above

text: discussed

text: anoma-

text: of

text: response

text: at

text: K,

text: where

text: the

text: onset

text: ﬁeld

text: χ

text: H

text: χ

text: H

text: T

text: lous

text: trend

text: could

text: be

text: observed

text: more

text: clearly

text: over

text: a

text: wide

text: of

text: the

text: peak

text: eﬀect

text: like

text: feature

text: and

text: the

text: peak

text: ﬁeld

text: Hon,ac

text: range

text: of

text: temperatures

text: K),

text: as

text: shown

text: in

text: Fig.

text: p

text: of

text: the

text: PE

text: are

text: indicated

text: by

text: arrows.

text: We

text: further

text: T

text: Hac

text: values

text: are

text: normalized

text: with

text: respect

text: to

text: their

text: sat-

text: claprify

text: the

text: PE

text: feature

text: by

text: plotting

text: the

text: derivative

text: of

text: χ

text: H

text: χ

text: urated

text: value

text: at

text: The

text: specimen

text: was

text: cooled

text: in

text: with

text: respect

text: to

text: at

text: K

text: in

text: the

text: same

text: panel.

text: The

text: H

text: H

text: T

text: a

text: nominal

text: zero

text: ﬁeld

text: down

text: to

text: a

text: desired

text: temperature

text: and

text: derivative

text: curve

text: in

text: Fig.

text: may

text: appear

text: on

text: dχ

text: /dH

text: then

text: the

text: data

text: were

text: recorded

text: while

text: increasing

text: the

text: dc

text: ﬁrst

text: examination

text: to

text: be

text: analogous

text: to

text: the

text: derivative

text: plot

text: χ

text: H

text: ##LTFigure##


text: 4 III RESULTS decreasing 0( ) is present across nearly identical tem- χ T perature intervals in both the orientations. The anomaly is also present in isothermal 0( ) responses across simi- χ H lar ﬁeld intervals for both the orientations; typical curves are shown for = 4 K (Fig. 3(b)) and = 2 5 K (an in- T T . set in Fig. 3(b)). Thus, we may conclude that the above stated features observed in the ac susceptibility are char- acteristic of the pinned vortex matter in the bulk of the Ca Ir Sn crystal. The anomalous peak feature is some- 3 4 13 what less prominent for perpendicular to the plane of H the crystal, presumably due to the pinned vortex state being less spatially ordered prior to the PE in this ori- entation. The vortex phase boundaries determined from the characteristic features of the anomalous variation in are however found to be not signiﬁcantly dependent j c Figure 3: (Color online) (a) Temperature dependences of the ac susceptibility ( 0( )) for dc ﬁeld = 15kOe, applied perpendicular χ T as well as parallel to the plane of the platelet shaped crystal. The two orientations of the crystal (relative to ﬁeld) are drawn in this panel. (b) Isothermal 0( ) response (normalized) for these two χ H orientations of the crystal at = 4K (main panel) and = 2 5K T T . (inset panel). 0 at = 15 kOe (cf. Fig. 1(c)). However, it is to dχ /dT H be noted thadtcthe derivative of 0( ) with respect to χ H H in Fig. 2(b) has two zero crossings; the ﬁrst one is taken to imprint the onset of peak eﬀect like anomaly ( ) Hon,ac and the second one identiﬁes the peak position of thpe PE ( ). Hac 3p. Sample geometry and anomalous variation in current density The sample geometry and demagnetization factor can aﬀect the state of the spatial order of vortex matter in a Figure 4: (Color online) (a) Isothermal – scans (ﬁrst two given sample due to edge eﬀects [51, 52], which in turn M H quadrants) recorded at diﬀerent temperatures, as indicated. In- could inﬂuence the triggering of the anomalous variation set shows one typical ( ) curve recorded at = 2K with the in in isoﬁeld/isothermal scans. This motivated us to M H T mejacsure the ac susceptibility with ﬁeld applied normal to two peaks occurring in M(H) curve marked as I and II peak. (b) Expanded portions of the ( ) loops near an anomalous enhance- M H the plane of the Ca Ir Sn sample. We show in Fig. 3(a), ment in hysteresis width (∆ = ( ↑) − ( ↓)) obtained at 3 4 13 M M H M H a comparison of 0( ) in the two diﬀerent orientations: diﬀerent temperatures. The irreversibility ﬁeld ( ) and the up- parallel and peχrpeTndicular to the plane of the sample. per critical ﬁeld ( ) have been located in thHe irr( ) curve at H Hc2 M H It is apparent that a deviation from the monotonically = 2 5K (see inset). T . ##LTFigure## ##LTFigure##

text: III

text: RESULTS

text: decreasing

text: is

text: present

text: across

text: nearly

text: identical

text: tem-

text: χ

text: T

text: perature

text: intervals

text: in

text: both

text: the

text: orientations.

text: The

text: anomaly

text: is

text: also

text: present

text: in

text: isothermal

text: responses

text: across

text: simi-

text: χ

text: H

text: lar

text: ﬁeld

text: intervals

text: for

text: both

text: the

text: orientations;

text: typical

text: curves

text: are

text: shown

text: for

text: K

text: (Fig.

text: and

text: K

text: (an

text: in-

text: T

text: T

text: set

text: in

text: Fig.

text: Thus,

text: we

text: may

text: conclude

text: that

text: the

text: above

text: stated

text: features

text: observed

text: in

text: the

text: ac

text: susceptibility

text: are

text: char-

text: acteristic

text: of

text: the

text: pinned

text: vortex

text: matter

text: in

text: the

text: bulk

text: of

text: the

text: Ca

text: Ir

text: Sn

text: crystal.

text: The

text: anomalous

text: peak

text: feature

text: is

text: some-

text: what

text: less

text: prominent

text: for

text: perpendicular

text: to

text: the

text: plane

text: of

text: H

text: the

text: crystal,

text: presumably

text: due

text: to

text: the

text: pinned

text: vortex

text: state

text: being

text: less

text: spatially

text: ordered

text: prior

text: to

text: the

text: PE

text: in

text: this

text: ori-

text: entation.

text: The

text: vortex

text: phase

text: boundaries

text: determined

text: from

text: the

text: characteristic

text: features

text: of

text: the

text: anomalous

text: variation

text: in

text: are

text: however

text: found

text: to

text: be

text: not

text: signiﬁcantly

text: dependent

text: j

text: c

text: Figure

text: (Color

text: online)

text: (a)

text: Temperature

text: dependences

text: of

text: the

text: ac

text: susceptibility

text: for

text: dc

text: ﬁeld

text: applied

text: perpendicular

text: χ

text: T

text: as

text: well

text: as

text: parallel

text: to

text: the

text: plane

text: of

text: the

text: platelet

text: shaped

text: crystal.

text: The

text: two

text: orientations

text: of

text: the

text: crystal

text: (relative

text: to

text: ﬁeld)

text: are

text: drawn

text: in

text: this

text: panel.

text: (b)

text: Isothermal

text: response

text: (normalized)

text: for

text: these

text: two

text: χ

text: H

text: orientations

text: of

text: the

text: crystal

text: at

text: (main

text: panel)

text: and

text: T

text: T

text: (inset

text: panel).

text: at

text: kOe

text: (cf.

text: Fig.

text: However,

text: it

text: is

text: to

text: dχ

text: /dT

text: H

text: be

text: noted

text: thadtcthe

text: derivative

text: of

text: with

text: respect

text: to

text: χ

text: H

text: H

text: in

text: Fig.

text: has

text: two

text: zero

text: crossings;

text: the

text: ﬁrst

text: one

text: is

text: taken

text: to

text: imprint

text: the

text: onset

text: of

text: peak

text: eﬀect

text: like

text: anomaly

text: Hon,ac

text: and

text: the

text: second

text: one

text: identiﬁes

text: the

text: peak

text: position

text: of

text: thpe

text: PE

text: Hac

text: Sample

text: geometry

text: and

text: anomalous

text: variation

text: in

text: current

text: density

text: The

text: sample

text: geometry

text: and

text: demagnetization

text: factor

text: can

text: aﬀect

text: the

text: state

text: of

text: the

text: spatial

text: order

text: of

text: vortex

text: matter

text: in

text: a

text: Figure

text: (Color

text: online)

text: (a)

text: Isothermal

text: scans

text: (ﬁrst

text: two

text: given

text: sample

text: due

text: to

text: edge

text: eﬀects

text: which

text: in

text: turn

text: M

text: H

text: quadrants)

text: recorded

text: at

text: diﬀerent

text: temperatures,

text: as

text: indicated.

text: In-

text: could

text: inﬂuence

text: the

text: triggering

text: of

text: the

text: anomalous

text: variation

text: set

text: shows

text: one

text: typical

text: curve

text: recorded

text: at

text: with

text: the

text: in

text: in

text: isoﬁeld/isothermal

text: scans.

text: This

text: motivated

text: us

text: to

text: M

text: H

text: T

text: mejacsure

text: the

text: ac

text: susceptibility

text: with

text: ﬁeld

text: applied

text: normal

text: to

text: two

text: peaks

text: occurring

text: in

text: M(H)

text: curve

text: marked

text: as

text: I

text: and

text: II

text: peak.

text: (b)

text: Expanded

text: portions

text: of

text: the

text: loops

text: near

text: an

text: anomalous

text: enhance-

text: M

text: H

text: the

text: plane

text: of

text: the

text: Ca

text: Ir

text: Sn

text: sample.

text: We

text: show

text: in

text: Fig.

text: ment

text: in

text: hysteresis

text: width

text: obtained

text: at

text: M

text: M

text: H

text: M

text: H

text: a

text: comparison

text: of

text: in

text: the

text: two

text: diﬀerent

text: orientations:

text: diﬀerent

text: temperatures.

text: The

text: irreversibility

text: ﬁeld

text: and

text: the

text: up-

text: parallel

text: and

text: peχrpeTndicular

text: to

text: the

text: plane

text: of

text: the

text: sample.

text: per

text: critical

text: ﬁeld

text: have

text: been

text: located

text: in

text: thHe

text: irr(

text: curve

text: at

text: H

text: M

text: H

text: It

text: is

text: apparent

text: that

text: a

text: deviation

text: from

text: the

text: monotonically

text: (see

text: inset).

text: T

text: ##LTFigure##

text: ##LTFigure##


text: 8 V SUMMARY Table I: Superconducting parameters estimated for Ca Ir Sn using the GL theory at = 2 K 3 4 13 T (at 15 kOe) λ ξ Hc1 Hc2 κ Gi jc/j0 2712 Å 76 Å 80 Oe 56.4 kOe ∼ 35 ∼ 1 5 × 10−8 ∼ 10−5 . lines actually enhances at larger ﬁeld values as apparent The broad (H,T) region comprises per se two transitions, from Fig. 6. a Bragg glass (BG) to vortex glass (VG) transition com- V. SUMMARY mencing at Hon,ac(T) (as in Fig. 6 for Ca3Ir4Sn13) and the amorphizaption of the vortex matter commencing at To summarize, we have explored the vortex phase di- Hdc+(T) (as in Fig. 6). The Hac(T) line for Ca3Ir4Sn13 agram in a weakly-pinned single crystal of a low inpFig. 6 can be taken to markp the spinodal line of the T squupaetercoantdteuncttioorn, iCna3tIhre4Srne1c3e,ntwhliitcehratisuraettfroarctiitnsg naodveec-l sueecsopnrdogtrreasnssiivteiolyn.deTcrheeasgeaspabs eTtwaepepnroHacpdhc+esa4n5dKH, apacndvaalt- . physics. Table-1 contains various superconducting pa- 5 K, the said diﬀerence ceases to exist as the PE in the rameters evaluated (at = 2 K) for Ca Ir Sn , us- ac susceptibility subsumes the notion of SMP. T 3 4 13 ing the Ginzburg-Landau theory. The weak pinning na- In the end, we may point out that the vortex phase di- ture of this compound is reﬂected by the ratio of depin- agram (Fig. 6) of Ca Ir Sn also raises an important is- 3 4 13 ning and depairing current densities ( ) ~ 10−5. The sue regarding the source of diamagnetic response between smallness of this ratio may be attributjecd/jt0o the presence ∗ ∗ and . As per the G - L theory, the ex- of point defects and metallurgical source [57] of disor- Hpec/teTd upperHcrci2t/icTacl ﬁeld/temperature should have been der in such compounds. The ac susceptibility ( 0( )) the ( ∗ ∗) line in Fig. 6, the experimental values of χ H, T H , T measurements have revealed the occurrence of PE phe- the same are slightly enhanced to ( ) line. There is nomenon in this compound. On the other hand, the re- a signiﬁcant separation between thHect2w, Toclines (( ∗ ∗) H , T sults of dc magnetization measurements reﬂect features and ( )) even in an inﬁnitesimally small ﬁeld, as typically identiﬁed with the second magnetization peak they iHntce2r,sTecct the temperature axis at ∼ 6 85 K and T . anomaly. The phase boundaries constructed by both , respectively. It is also pertinent to note that the G - T the ac as well as the dc data prompt us to identify the Lctheory does not explicitly take into account the spatial metastable regions in the vortex phase diagram of this ordering of the vortex matter. We have witnessed a re- compound. The region lying between the onset lines of versible (amorphous) region (in the dc ( ) ( ) re- M H /M T the SMP ( ( )) in dc data and the PE like phe- sponse) that emerged from the disordered conﬁguration Hon,dc T nomenon ( SMP( )) in the ac data is taken as a super- (PE) of the vortex matter, wherein (i.e., above ( ∗ ∗) Hon,ac T H , T heated orderped vortex matter, which sees an enhance- line) the mean ﬁeld description does not hold. A possible ment in spatial disordering under the inﬂuence of an ac source for such an unusual observation could be the exis- ﬁeld. In contrast to this, the region lying between the tence of surface superconductivity [58] above the ∗ ∗ H /T plieesakexﬁheilbditosfetnhheaPncEem(Henpatc)inatnhdetohredeSrMinPg d(uHepdtco+t)haeneoﬀmeact- lsipnaecaen. dEsﬀuorrvtisvitnogeuxppltoorethtehHis cr2e/gTiocnlinoef vinorttheexHp–hTaspehdaisae- of an ac drive. Such an eﬀect ceases at higher tempera- gram are currently in progress [59], with some signiﬁcant tures ( 4 5 K) because of the absence of the SMP fea- success in sight. T > . ture there, only PE stands depicted at 4 5 K and for T > . Acknowledgments ﬁeld values lying between 5 kOe and15 kOe. The region Santosh Kumar would like to thank the Council of Scien- of the – phase space over which the anomalous varia- H T tiﬁc and Industrial Research, India for grant of the Senior tion in ( ) predominates, is very broad and it bears re- j H Research Fellowship. semblancce with anomalous data in several other low as T well as high superconductors, including the Cupractes. References T c [1] G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. 1242. Larkin, V. M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125, [6] T. Giamarchi, P. Le Doussal, Phys. Rev. B 55 (1997) and references therein. 6577. [2] M. J. Higgins and S. Bhattacharya, Physica C 257 (1996) [7] S. S. Banerjee, S. Saha, N. G. Patil, S. Ramakrishnan, A. 232 and references therein. K. Grover, S. Bhattacharya, G. Ravikumar, P. K. Mishra, [3] S. Bhattacharya, M. J. Higgins, Phys. Rev. Lett. 70 T. V. C. Rao, V. C. Sahni, C. V. Tomy, G. Balakrishnan, (1993) 2617. D. McK Paul, M. J. Higgins, Physica C 308 (1998) 25-32. [4] T. Giamarchi and P. Le Doussal, Phys. Rev. Lett. 72 [8] S. S. Banerjee, A. K. Grover, M. J. Higgins, Gutam (1994) 1530. I. Menon, P. K. Mishra, D. Pal, S. Ramakrishnan, T. [5] T. Giamarchi and P. Le Doussal, Phys. Rev. B 52 (1995), V. Chandrasekhar Rao, G. Ravikumar, V. C. Sahni, S. ##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: SUMMARY

text: Table

text: I:

text: Superconducting

label: caption

text: parameters

text: estimated

text: for

text: Ca

text: Ir

text: Sn

text: using

text: the

text: GL

text: theory

text: at

text: K

text: T

text: (at

label: table

text: kOe)

text: λ

text: ξ

text: κ

text: Gi

text: Å

text: Å

text: Oe

text: kOe

text: lines

text: actually

text: enhances

text: at

text: larger

text: ﬁeld

text: values

text: as

text: apparent

text: The

text: broad

text: (H,T)

text: region

text: comprises

text: per

text: se

text: two

text: transitions,

text: from

text: Fig.

text: a

text: Bragg

text: glass

text: (BG)

text: to

text: vortex

text: glass

text: (VG)

text: transition

text: com-

text: V.

text: SUMMARY

text: mencing

text: at

text: Hon,ac(T)

text: (as

text: in

text: Fig.

text: for

text: and

text: the

text: amorphizaption

text: of

text: the

text: vortex

text: matter

text: commencing

text: at

text: To

text: summarize,

text: we

text: have

text: explored

text: the

text: vortex

text: phase

text: di-

text: Hdc+(T)

text: (as

text: in

text: Fig.

text: The

text: Hac(T)

text: line

text: for

text: agram

text: in

text: a

text: weakly-pinned

text: single

text: crystal

text: of

text: a

text: low

text: inpFig.

text: can

text: be

text: taken

text: to

text: markp

text: the

text: spinodal

text: line

text: of

text: the

text: T

text: squupaetercoantdteuncttioorn,

text: naodveec-l

text: apacndvaalt-

text: physics.

text: contains

text: various

text: superconducting

text: pa-

text: K,

text: the

text: said

text: diﬀerence

text: ceases

text: to

text: exist

text: as

text: the

text: PE

text: in

text: the

text: rameters

text: evaluated

text: (at

text: K)

text: for

text: Ca

text: Ir

text: Sn

text: us-

text: ac

text: susceptibility

text: subsumes

text: the

text: notion

text: of

text: SMP.

text: T

text: ing

text: the

text: Ginzburg-Landau

text: theory.

text: The

text: weak

text: pinning

text: na-

text: In

text: the

text: end,

text: we

text: may

text: point

text: out

text: that

text: the

text: vortex

text: phase

text: di-

text: ture

text: of

text: this

text: compound

text: is

text: reﬂected

text: by

text: the

text: ratio

text: of

text: depin-

text: agram

text: (Fig.

text: of

text: Ca

text: Ir

text: Sn

text: also

text: raises

text: an

text: important

text: is-

text: ning

text: and

text: depairing

text: current

text: densities

text: The

text: sue

text: regarding

text: the

text: source

text: of

text: diamagnetic

text: response

text: between

text: smallness

text: of

text: this

text: ratio

text: may

text: be

text: the

text: presence

text: and

text: As

text: per

text: the

text: G

text: L

text: theory,

text: the

text: ex-

text: of

text: point

text: defects

text: and

text: metallurgical

text: source

text: of

text: disor-

text: Hpec/teTd

text: ﬁeld/temperature

text: should

text: have

text: been

text: der

text: in

text: such

text: compounds.

text: The

text: ac

text: susceptibility

text: the

text: line

text: in

text: Fig.

text: the

text: experimental

text: values

text: of

text: χ

text: H,

text: T

text: H

text: T

text: measurements

text: have

text: revealed

text: the

text: occurrence

text: of

text: PE

text: phe-

text: the

text: same

text: are

text: slightly

text: enhanced

text: to

text: line.

text: There

text: is

text: nomenon

text: in

text: this

text: compound.

text: On

text: the

text: other

text: hand,

text: the

text: re-

text: a

text: signiﬁcant

text: separation

text: between

text: Toclines

text: H

text: T

text: sults

text: of

text: dc

text: magnetization

text: measurements

text: reﬂect

text: features

text: and

text: even

text: in

text: an

text: inﬁnitesimally

text: small

text: ﬁeld,

text: as

text: typically

text: identiﬁed

text: with

text: the

text: second

text: magnetization

text: peak

text: they

text: the

text: temperature

text: axis

text: at

text: K

text: and

text: T

text: anomaly.

text: The

text: phase

text: boundaries

text: constructed

text: by

text: both

text: respectively.

text: It

text: is

text: also

text: pertinent

text: to

text: note

text: that

text: the

text: G

text: T

text: the

text: ac

text: as

text: well

text: as

text: the

text: dc

text: data

text: prompt

text: us

text: to

text: identify

text: the

text: Lctheory

text: does

text: not

text: explicitly

text: take

text: into

text: account

text: the

text: spatial

text: metastable

text: regions

text: in

text: the

text: vortex

text: phase

text: diagram

text: of

text: this

text: ordering

text: of

text: the

text: vortex

text: matter.

text: We

text: have

text: witnessed

text: a

text: re-

text: compound.

text: The

text: region

text: lying

text: between

text: the

text: onset

text: lines

text: of

text: versible

text: (amorphous)

text: region

text: (in

text: the

text: dc

text: re-

text: M

text: H

text: /M

text: T

text: the

text: SMP

text: in

text: dc

text: data

text: and

text: the

text: PE

text: like

text: phe-

text: sponse)

text: that

text: emerged

text: from

text: the

text: disordered

text: conﬁguration

text: Hon,dc

text: T

text: nomenon

text: SMP(

text: in

text: the

text: ac

text: data

text: is

text: taken

text: as

text: a

text: super-

text: (PE)

text: of

text: the

text: vortex

text: matter,

text: wherein

text: (i.e.,

text: above

text: Hon,ac

text: T

text: H

text: T

text: heated

text: orderped

text: vortex

text: matter,

text: which

text: sees

text: an

text: enhance-

text: line)

text: the

text: mean

text: ﬁeld

text: description

text: does

text: not

text: hold.

text: A

text: possible

text: ment

text: in

text: spatial

text: disordering

text: under

text: the

text: inﬂuence

text: of

text: an

text: ac

text: source

text: for

text: such

text: an

text: unusual

text: observation

text: could

text: be

text: the

text: exis-

text: ﬁeld.

text: In

text: contrast

text: to

text: this,

text: the

text: region

text: lying

text: between

text: the

text: tence

text: of

text: surface

text: superconductivity

text: above

text: the

text: H

text: /T

text: d(uHepdtco+t)haeneoﬀmeact-

text: lsipnaecaen.

text: dEsﬀuorrvtisvitnogeuxppltoorethtehHis

text: vinorttheexHp–hTaspehdaisae-

text: of

text: an

text: ac

text: drive.

text: Such

text: an

text: eﬀect

text: ceases

text: at

text: higher

text: tempera-

text: gram

text: are

text: currently

text: in

text: progress

text: with

text: some

text: signiﬁcant

text: tures

text: K)

text: because

text: of

text: the

text: absence

text: of

text: the

text: SMP

text: fea-

text: success

text: in

text: sight.

text: T

text: ture

text: there,

text: only

text: PE

text: stands

text: depicted

text: at

text: K

text: and

text: for

text: T

text: Acknowledgments

text: ﬁeld

text: values

text: lying

text: between

text: kOe

text: kOe.

text: The

text: region

text: Santosh

text: Kumar

text: would

text: like

text: to

text: thank

text: the

text: Council

text: of

text: Scien-

text: of

text: the

text: phase

text: space

text: over

text: which

text: the

text: anomalous

text: varia-

text: H

text: T

text: tiﬁc

text: and

text: Industrial

text: Research,

text: India

text: for

text: grant

text: of

text: the

text: Senior

text: tion

text: in

text: predominates,

text: is

text: very

text: broad

text: and

text: it

text: bears

text: re-

text: j

text: H

text: Research

text: Fellowship.

text: semblancce

text: with

text: anomalous

text: data

text: in

text: several

text: other

text: low

text: as

text: T

text: well

text: as

text: high

text: superconductors,

text: including

text: the

text: Cupractes.

text: References

text: T

text: c

text: G.

text: Blatter,

text: M.

text: V.

text: Feigel’man,

text: V.

text: B.

text: Geshkenbein,

text: A.

text: I.

text: Larkin,

text: V.

text: M.

text: Vinokur,

text: Rev.

text: Mod.

text: Phys.

text: T.

text: Giamarchi,

text: P.

text: Le

text: Doussal,

text: Phys.

text: Rev.

text: B

text: and

text: references

text: therein.

text: M.

text: J.

text: Higgins

text: and

text: S.

text: Bhattacharya,

text: Physica

text: C

text: S.

text: S.

text: Banerjee,

text: S.

text: Saha,

text: N.

text: G.

text: Patil,

text: S.

text: Ramakrishnan,

text: A.

text: and

text: references

text: therein.

text: K.

text: Grover,

text: S.

text: Bhattacharya,

text: G.

text: Ravikumar,

text: P.

text: K.

text: Mishra,

text: S.

text: Bhattacharya,

text: M.

text: J.

text: Higgins,

text: Phys.

text: Rev.

text: Lett.

text: T.

text: V.

text: C.

text: Rao,

text: V.

text: C.

text: Sahni,

text: C.

text: V.

text: Tomy,

text: G.

text: Balakrishnan,

text: D.

text: McK

text: Paul,

text: M.

text: J.

text: Higgins,

text: Physica

text: C

text: T.

text: Giamarchi

text: and

text: P.

text: Le

text: Doussal,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: S.

text: Banerjee,

text: A.

text: K.

text: Grover,

text: M.

text: J.

text: Higgins,

text: Gutam

text: I.

text: Menon,

text: P.

text: K.

text: Mishra,

text: D.

text: Pal,

text: S.

text: Ramakrishnan,

text: T.

text: T.

text: Giamarchi

text: and

text: P.

text: Le

text: Doussal,

text: Phys.

text: Rev.

text: B

text: V.

text: Chandrasekhar

text: Rao,

text: G.

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text: S.

text: Mortensen,

text: K.

text: Clausen,

text: C.

text: A.

text: Bolle

text: and

text: F.

text: de

text: la

text: Cruz,

text: Bhattacharya,

text: M.

text: J.

text: Higgins,

text: Z.

text: L.

text: Xiao,

text: E.

text: Y.

text: Andrei,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: L.

text: Gammel

text: and

text: D.

text: J.

text: Bishop,

text: Nature

text: (London)

text: C.

text: A.

text: Bolle,

text: F.

text: de

text: la

text: Cruz,

text: P.

text: L.

text: Gammel,

text: J.

text: V.

text: Waszczak

text: and

text: D.

text: J.

text: Bishop,

text: Phys.

text: Rev.

text: Lett.

text: C.

text: P.

text: Bean,

text: Rev.

text: Mod.

text: Phys.

text: A.

text: M.

text: Troyanovski,

text: M.

text: van

text: Hecke,

text: N.

text: Saha,

text: J.

text: Aarts

text: and

text: W.

text: A.

text: Fietz

text: and

text: W.

text: W.

text: Webb,

text: Phys.

text: Rev.

text: P.

text: H.

text: Kes,

text: Phys.

text: Rev.

text: Lett.

text: D.

text: Pal,

text: S.

text: Ramakrishnan,

text: A.

text: K.

text: Grover,

text: M.

text: J.

text: Higgins

text: M.

text: Tinkham,

text: Introduction

text: to

text: Superconductivity,

text: ed.,


text: 4 EUGEN KEIL We give a simple but crucial lemma from [3] rewritten in our nota- (cid:88) tion. ≤ ≤ ≤ Lemma 2.1. For any 1 K N and 1 y < z we have | | (cid:28) − b (n) 2 Ky1 k + N1/k log3(z). y,z − ≤ N K<n N The second summand on the right hand side is not necessary if K = N. Proof. We follow the argument of [3]. Insert the deﬁnition of b (n) (cid:88) (cid:88) (cid:88) y,z and open the square, to arrive at | | (3) b (n) 2 = µ(d)µ(e) 1. y,z − ≤ ≤ − ≤ N K<n N y d,e<z N K<n N | | dk n,ek n ≤ The inner sum is bounded by K/lcm(dk, ek) + 1 if lcm(d, e) N1/k an(cid:88)d is (cid:16)empty otherwise. Ther(cid:17)efore,(cid:88)when it (cid:88)is non-em(cid:88)pty, the r(cid:88)ight hand side of (3) bounded by K N1/k ≤ K N1/k + tk + t. lcm(dk, ek) lcm(d, e) dkek de ≤ ≤ | | y d,e<z y d,e t (d;e) d,e<z t (d;e) · The last inequality used the identity (d; e) lcm(d, e) = de with the trivial observation, that a sum of positive terms is always bigger then one(cid:88)of its summands. C(cid:88)hangi(cid:16)ng (cid:88)the orde(cid:17)r of summ(cid:88)ation(cid:16) in(cid:88)the abo(cid:17)ve sums, we ﬁnally obtain the upper bound | | ≤ − 2 − 2 b (n) 2 K (cid:88) tk d k + N1/k t(cid:88) (cid:16)d(cid:88)1 (cid:17) y,z − ≤ ≤ | | N K<n N t y d,t d t<z d<z,t d ≤ (cid:88) − (cid:88)− − − 2 K t k min(1, (t/y)k 1)2 + N1/k t 1 d 1 t t<z d<z (cid:28) − − − K tk 2y2 2k + K t k + N1/k log3 z ≤ t y t>y (cid:28) − Ky1 k + N1/k log3 z. In the case where K = N we get the bound K/lcm(dk, ek) for the inner sum in (3) instead of K/lcm(dk, ek) + 1, which removes the term (cid:3) N1/k log3 z from the ﬁnal estimate. ≤ Let H and h be integers satisfying the inequalities N1/k 2H < ≤ 2N1/k and N1/(k+1) < 2h 2N1/(k+1). Deﬁne by means of (2) ∗ b (n) = b − (n) and b (n) = b (n). i 2i 1,2i 2h,2H ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EUGEN

text: KEIL

text: We

text: give

text: a

text: simple

text: but

text: crucial

text: lemma

text: from

text: rewritten

text: in

text: our

text: nota-

text: tion.

text: Lemma

text: For

text: any

text: K

text: N

text: and

text: y

text: z

text: we

text: have

text: b

text: (n)

text: k

text: y,z

text: N

text: K<n

text: N

text: The

text: second

text: summand

text: on

text: the

text: right

text: hand

text: side

text: is

text: not

text: necessary

text: if

text: K

text: N.

text: Proof.

text: We

text: follow

text: the

text: argument

text: of

text: Insert

text: the

text: deﬁnition

text: of

text: b

text: (n)

text: y,z

text: and

text: open

text: the

text: square,

text: to

text: arrive

text: at

text: b

text: (n)

text: µ(d)µ(e)

text: y,z

text: N

text: K<n

text: N

text: y

text: d,e<z

text: N

text: K<n

text: N

text: dk

text: n,ek

text: n

text: The

text: inner

text: sum

text: is

text: bounded

text: by

text: K/lcm(dk,

text: ek)

text: if

text: lcm(d,

text: e)

text: is

text: otherwise.

text: it

text: the

text: hand

text: side

text: of

text: bounded

text: by

text: K

text: K

text: tk

text: t.

text: lcm(dk,

text: ek)

text: lcm(d,

text: e)

text: dkek

text: de

text: y

text: d,e<z

text: y

text: d,e

text: t

text: (d;e)

text: d,e<z

text: t

text: (d;e)

text: The

text: last

text: inequality

text: used

text: the

text: identity

text: (d;

text: e)

text: lcm(d,

text: e)

text: de

text: with

text: the

text: trivial

text: observation,

text: that

text: a

text: sum

text: of

text: positive

text: terms

text: is

text: always

text: bigger

text: then

text: its

text: summands.

text: of

text: sums,

text: we

text: ﬁnally

text: obtain

text: the

text: upper

text: bound

text: b

text: (n)

text: K

text: tk

text: d

text: k

text: y,z

text: N

text: K<n

text: N

text: t

text: y

text: d,t

text: d

text: t<z

text: d<z,t

text: d

text: K

text: t

text: k

text: (t/y)k

text: t

text: d

text: t

text: t<z

text: d<z

text: K

text: tk

text: K

text: t

text: k

text: z

text: t

text: y

text: t>y

text: k

text: z.

text: In

text: the

text: case

text: where

text: K

text: N

text: we

text: get

text: the

text: bound

text: K/lcm(dk,

text: ek)

text: for

text: the

text: inner

text: sum

text: in

text: instead

text: of

text: K/lcm(dk,

text: ek)

text: which

text: removes

text: the

text: term

text: z

text: from

text: the

text: ﬁnal

text: estimate.

text: Let

text: H

text: and

text: h

text: be

text: integers

text: satisfying

text: the

text: inequalities

text: and

text: Deﬁne

text: by

text: means

text: of

text: b

text: (n)

text: b

text: (n)

text: and

text: b

text: (n)

text: b

text: (n).

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 EUGEN KEIL multiplicative constant. Sections and reﬁne the above argument to obtain this improvement. 4. Upper Bounds for p > 1 + 1/k The ﬁrst observation to make is that we can restrict our attention ≤ ≥ to the range p 2 since the desired result follows for p 2 from the | | ≤ trivial estimate S (α) N. k Our task is to avoid the logarithmic factors from the estimates in Section . The ﬁrst logarithm can be dealt with by introducing a poly- nomial w(cid:90)eigh(cid:12) t(cid:88)before appl(cid:12)ying Ho¨l(cid:90)der’(cid:16)s (cid:88)(cid:48)in≥equal(cid:17)ity. (cid:88)Modify the ﬁrst calculation in(cid:12) Section as f(cid:12)ollows. Since p 2, we obtain (cid:12) (cid:12) (cid:48) 1 (cid:90) − p (cid:28) 1 − (cid:48) p/p | | I (p) = (cid:88) i 1iT (α) dα i p ip T (α) p dα k i p i 0 ≤ 0 ≤ ≤ i H i H i H (cid:28) 1 | | ip T (α) p dα. i ≤ 0 i H To avoid the logarithmic factor in the L1-mean, we interpolate in- − · stead between L1+δ and L2, for some small δ > 0. Write p = (1 θ) · − (cid:90)(1 + δ) + θ 2, which gives θ = p (−1+δ). By Lemma 2.2 and Lemma 2.3, 1 δ we obtain 1 | | (cid:28) − − − − · − − − − − T (α) p dα (2iNδ)1 θ(2 i(k 1)N)θ = Nδ(1 θ)+θ 2i(1 θ) i(k 1)θ = Np 12 iφ i δ 0 − − − − − − with φ = −1 ((p 1)k 1 δ(k 1)). The expression (p 1)k 1 is 1 δ positive, due to the condition p > 1 + 1/k and, therefore, when δ > 0 is small enough (dependent on p(cid:88)and k), we have φ > 0. We end up with (cid:28) − − (cid:28) − I (p) Np 1 ip2 iφ Np 1. k p p ≤ i H − Due to the exponential decay of 2 iφ our previously introduced factor of ip doesn’t cause any further problems and we are done with this case of the theorem as well. We now move towards establishing the lower bounds and leave the technically more demanding part p < 1 + 1/k for a later section. 5. Lower Bounds To deduce the lower bounds in the range p < 1+1/k we use Theorem 1.1. Observe that by Theorem 1.1 the L1-norm is bounded from below (cid:48) − by N1/(k+1). Applying Ho¨lder’s inequality to this with p = p/(p 1), ##LTLine## ##LTLine##

text: EUGEN

text: KEIL

text: multiplicative

text: constant.

text: Sections

text: and

text: reﬁne

text: the

text: above

text: argument

text: to

text: obtain

text: this

text: improvement.

text: Upper

text: Bounds

text: for

text: p

text: The

text: ﬁrst

text: observation

text: to

text: make

text: is

text: that

text: we

text: can

text: restrict

text: our

text: attention

text: to

text: the

text: range

text: p

text: since

text: the

text: desired

text: result

text: follows

text: for

text: p

text: from

text: the

text: trivial

text: estimate

text: S

text: (α)

text: N.

text: k

text: Our

text: task

text: is

text: to

text: avoid

text: the

text: logarithmic

text: factors

text: from

text: the

text: estimates

text: in

text: Section

text: The

text: ﬁrst

text: logarithm

text: can

text: be

text: dealt

text: with

text: by

text: introducing

text: a

text: poly-

text: nomial

text: the

text: ﬁrst

text: calculation

text: Section

text: as

text: Since

text: p

text: we

text: obtain

text: p

text: p/p

text: I

text: (p)

text: i

text: (α)

text: dα

text: i

text: p

text: ip

text: T

text: (α)

text: p

text: dα

text: k

text: i

text: p

text: i

text: i

text: H

text: i

text: H

text: i

text: H

text: ip

text: T

text: (α)

text: p

text: dα.

text: i

text: i

text: H

text: To

text: avoid

text: the

text: logarithmic

text: factor

text: in

text: the

text: we

text: interpolate

text: in-

text: stead

text: between

text: and

text: for

text: some

text: small

text: δ

text: Write

text: p

text: θ)

text: δ)

text: θ

text: which

text: gives

text: θ

text: p

text: By

text: Lemma

text: and

text: Lemma

text: δ

text: we

text: obtain

text: T

text: (α)

text: p

text: dα

text: i(k

text: θ)+θ

text: θ)

text: i(k

text: Np

text: iφ

text: i

text: δ

text: with

text: φ

text: −1

text: ((p

text: δ(k

text: The

text: expression

text: (p

text: is

text: δ

text: positive,

text: due

text: to

text: the

text: condition

text: p

text: and,

text: therefore,

text: when

text: δ

text: is

text: small

text: enough

text: (dependent

text: on

text: k),

text: we

text: have

text: φ

text: We

text: end

text: up

text: with

text: I

text: (p)

text: Np

text: iφ

text: Np

text: k

text: p

text: p

text: i

text: H

text: Due

text: to

text: the

text: exponential

text: decay

text: of

text: iφ

text: our

text: previously

text: introduced

text: factor

text: of

text: ip

text: doesn’t

text: cause

text: any

text: further

text: problems

text: and

text: we

text: are

text: done

text: with

text: this

text: case

text: of

text: the

text: theorem

text: as

text: well.

text: We

text: now

text: move

text: towards

text: establishing

text: the

text: lower

text: bounds

text: and

text: leave

text: the

text: technically

text: more

text: demanding

text: part

text: p

text: for

text: a

text: later

text: section.

text: Lower

text: Bounds

text: To

text: deduce

text: the

text: lower

text: bounds

text: in

text: the

text: range

text: p

text: we

text: use

text: Theorem

text: Observe

text: that

text: by

text: Theorem

text: the

text: is

text: bounded

text: from

text: below

text: by

text: Applying

text: Ho¨lder’s

text: inequality

text: to

text: this

text: with

text: p

text: p/(p

text: ##LTLine##

text: ##LTLine##


text: 10 EUGEN KEIL The ﬁrst term is at least of size N/(4rk) by the previous calculation. (cid:12) (cid:88) (cid:12) The second term(cid:12)can be bounded by(cid:12) a similar procedure as above. We (cid:12) (cid:12) have ≤ µ (n)e(an/rk) 2t/rk + rkt1/k. k ≤ n t By integration, the contribution of the integral on the right hand side is | | | | ≤ bounded above by 2π β N2/rk + β rkN1+1/k N/(10rk) by our choice (cid:3) of β and r, if N is big enough. Now we embark on the proof of the lower bound for the remaining ≤ case p = 1 + 1/k. Set R = N1/(5k). For r, s R we would have − ≥ | − | ≥ − ≥ − the contradiction (50N) 1 a/rk b/sk (rs) k N 1 if the two (cid:54) points a/rk = b/sk would have intersecting neighbourhoods of radius (cid:90) − (cid:90) (100N) 1. Therefore, the(cid:88)y are disjo(cid:88)int and we can use Lemma 5.1 to estimate 1 | | ≥ | | (cid:88)S (α) (k(cid:88)+1)/k dα µ (r) (cid:88) S (α) (k+1)/k dα k 2 k 0 ≤ |α−a/rk|≤(100N)−1 r R (a;rk)=1 ≥ − | | (cid:29) µ(cid:88)(r) (50N) 1 N/(10rk) (k+1)/k N1/k µ (r)ϕ(rk)/rk+1 2 2 ≤ ≤ r R (a;rk)=1 r R (cid:29) = N1/k µ (r)ϕ(r)/r2 N1/k log N, 2 ≤ r R (cid:88) where ϕ is Euler’s totient function and the last inequality follows by summation by parts from the estimate (cid:29) µ (r)ϕ(r)/r R, 2 (cid:88) (cid:88)≤ (cid:88) r R which can be deduced as follows. We have − − (cid:88) µ (r)ϕ(r)/r = ϕ(r)/r (1 µ (r))ϕ(r)/r 2 2 ≤ ≤ ≤ r R r R r R ≥ − − ≥ − (cid:29) (cid:80) R/ζ(2) + o(R) (1 µ (r)) (2/ζ(2) 1)R + o(R) R, 2 ≤ r R where we used the identities (1) and ϕ(r)/r = | µ(d)/d. d r 6. Upper Bounds for p < 1 + 1/k In this section we essentially repeat an argument from [3] in a more ≤ general setting. Since for p 1 the bound follows by application of Ho¨lder’s inequality using the upper bound for p > 1, we can restrict ourselves to the cases 1 < p < 1 + 1/k.

text: EUGEN

text: KEIL

text: The

text: ﬁrst

text: term

text: is

text: at

text: least

text: of

text: size

text: by

text: the

text: previous

text: calculation.

text: The

text: second

text: be

text: bounded

text: a

text: similar

text: procedure

text: as

text: above.

text: We

text: have

text: µ

text: (n)e(an/rk)

text: k

text: n

text: t

text: By

text: integration,

text: the

text: contribution

text: of

text: the

text: integral

text: on

text: the

text: right

text: hand

text: side

text: is

text: bounded

text: above

text: by

text: 2π

text: β

text: β

text: by

text: our

text: choice

text: of

text: β

text: and

text: r,

text: if

text: N

text: is

text: big

text: enough.

text: Now

text: we

text: embark

text: on

text: the

text: proof

text: of

text: the

text: lower

text: bound

text: for

text: the

text: remaining

text: case

text: p

text: Set

text: R

text: For

text: r,

text: s

text: R

text: we

text: would

text: have

text: the

text: contradiction

text: a/rk

text: b/sk

text: (rs)

text: k

text: N

text: if

text: the

text: two

text: points

text: a/rk

text: b/sk

text: would

text: have

text: intersecting

text: neighbourhoods

text: of

text: radius

text: Therefore,

text: are

text: and

text: we

text: can

text: use

text: Lemma

text: to

text: estimate

text: (α)

text: dα

text: µ

text: (r)

text: S

text: (α)

text: dα

text: k

text: k

text: r

text: R

text: µ

text: r

text: R

text: r

text: R

text: µ

text: log

text: N,

text: r

text: R

text: where

text: ϕ

text: is

text: Euler’s

text: totient

text: function

text: and

text: the

text: last

text: inequality

text: follows

text: by

text: summation

text: by

text: parts

text: from

text: the

text: estimate

text: µ

text: (r)ϕ(r)/r

text: R,

text: r

text: R

text: which

text: can

text: be

text: deduced

text: as

text: follows.

text: We

text: have

text: µ

text: (r)ϕ(r)/r

text: ϕ(r)/r

text: µ

text: (r))ϕ(r)/r

text: r

text: R

text: r

text: R

text: r

text: R

text: o(R)

text: µ

text: (r))

text: o(R)

text: R,

text: r

text: R

text: where

text: we

text: used

text: the

text: identities

text: and

text: ϕ(r)/r

text: µ(d)/d.

text: d

text: r

text: Upper

text: Bounds

text: for

text: p

text: In

text: this

text: section

text: we

text: essentially

text: repeat

text: an

text: argument

text: from

text: in

text: a

text: more

text: general

text: setting.

text: Since

text: for

text: p

text: the

text: bound

text: follows

text: by

text: application

text: of

text: Ho¨lder’s

text: inequality

text: using

text: the

text: upper

text: bound

text: for

text: p

text: we

text: can

text: restrict

text: ourselves

text: to

text: the

text: cases

text: p