text: Lensing dispersion of supernova ﬂux: a probe of nonlinear structure growth 7 pute the cosmic shear power spectrum, and then changed the model verse in the box: multiscale simulations of cosmic structure”. We parameters (that in their case also include a redshift modulation of wish to thank an anonymous referee for insightful remarks on the the concentrations) in order to ﬁt the results of hydrodynamic sim- manuscript. ulations with different baryonic physical processes (Schaye et al. References 2010). By moving from an implementation featuring strong AGN Bartelmann M., Schneider P., 2001, Phys. Rep., 340, 291 winds to ones with negligible feedback of any kind they ﬁnd that Ben-Dayan I., Gasperini M., Marozzi G., Nugier F., Veneziano G., the best ﬁt normalization of the concentration-mass relation can be ∼ 2013a, J. Cosmology Astropart. Phys., 6, 2 moved by a factor ranging in a very substantial interval, from 0.3 ∼ − Ben-Dayan I., Gasperini M., Marozzi G., Nugier F., Veneziano G., up to 1.2 1.4. These numbers cannot be compared directly 2013b, Physical Review Letters, 110, 021301 with our Figure 5 because we do not have any redshift modulation, Ben-Dayan I., Kalaydzhyan T., 2013, ArXiv e-prints 1309.4771 and our pivot mass for the concentration-mass relation is different Bergstro¨m L., Goliath M., Goobar A., Mo¨rtsell E., 2000, A&A, from theirs. Yet, they provide a good indication that the lensing dis- 358, 13 persion will be very well able to substantially reduce the space of Bernardeau F., van Waerbeke L., Mellier Y., 1997, A&A, 322, 1 allowed baryonic physical models. Buote D. A., Gastaldello F., Humphrey P. J., Zappacosta L., Bul- The measurements by Jo¨nsson et al. (2010) were based on less lock J. S., Brighenti F., Mathews W. G., 2007, ApJ, 664, 123 than 200 SN Ia. The Union2.1 compilation (Suzuki et al. 2012) ∼ Cooray A., Sheth R., 2002, Phys. Rep., 372, 1 contains already 3 times as many objects (although not all com- Dodelson S., Vallinotto A., 2006, Phys. Rev. D, 74, 063515 ing from a single homogeneous dataset). In the future, Euclid will ∼ Duffy A. R., Schaye J., Kay S. T., Dalla Vecchia C., 2008, MN- be capable of observing the light curves of 3, 000 SN Ia (Laureijs RAS, 390, L64 et al. 2011), while the LSST will increase this number to a stagger- ∼ Fanizza G., Gasperini M., Marozzi G., Veneziano G., 2013, J. ing 100, 000 objects (LSST Science Collaboration et al. 2009). Cosmology Astropart. Phys., 11, 19 We can thus expect the measurements of the lensing-induced scat- Fedeli C., 2014, ArXiv e-prints 1401.2997 ter in the distance modulus of standard candles to become much more precise in forthcoming years (even signiﬁcantly more than Fedeli C., Dolag K., Moscardini L., 2012, MNRAS, 419, 1588 we assumed in our last bullet), and thus this method to emerge as Giocoli C., Bartelmann M., Sheth R. K., Cacciato M., 2010, MN- one of the principal probes of nonlinear structures. With increas- RAS, 408, 300 ing statistics the approach that has been exempliﬁed in the present Guy J. et al., 2007, A&A, 466, 11 paper can be improved to probe substructure and stochasticity of Guy J., Astier P., Nobili S., Regnault N., Pain R., 2005, A&A, dark matter halos (see Giocoli et al. 2010) as well as, directly, the 443, 781 effects of gas physics and star formation (thanks to a generalized Hamana T., Futamase T., 2000, ApJ, 534, 29 halo model that our group is developing, see Fedeli 2014). Hilbert S., Hartlap J., White S. D. M., Schneider P., 2009, A&A, The previous discussion does not take into account the pres- 499, 31 ence of unavoidable systematics in the measurements of the SN Ia Holz D. E., Linder E. V., 2005, ApJ, 631, 678 lensing ﬂux dispersion. It is likely that these systematics will not Jing Y. P., Zhang P., Lin W. P., Gao L., Springel V., 2006, ApJ, be reducible below a certain (still unknown) level, which is go- 640, L119 ing to obstruct to some extent the constraining power of SN Ia for Jo¨nsson J., Dahle´n T., Goobar A., Mo¨rtsell E., Riess A., 2007, J. the nonlinear matter clustering. The presence of this systematics Cosmology Astropart. Phys., 6, 2 ”ﬂoor” will need to be addressed properly when the statistics will Jo¨nsson J. et al., 2010, MNRAS, 405, 535 become large. Kainulainen K., Marra V., 2011, Phys. Rev. D, 84, 063004 The results of this work show not only how the lensing- Kronborg T. et al., 2010, A&A, 514, A44 induced dispersion of apparent magnitudes of SN Ia can lead to Laureijs R. et al., 2011, ArXiv e-prints 1110.3193 interesting constraints on nonlinear structure growth, but also how, LSST Science Collaboration et al., 2009, ArXiv e-prints, and because of this very fact, constraints on cosmological param- 0912.0201 eters based on this method (Dodelson & Vallinotto 2006; Marra, Ma C.-P., Fry J. N., 2000, ApJ, 543, 503 Quartin, & Amendola 2013; Quartin, Marra, & Amendola 2013; Marra V., Quartin M., Amendola L., 2013, Phys. Rev. D, 88, Ben-Dayan & Kalaydzhyan 2013) should be taken with grain of 063004 salt. Indeed, considering how sensitive this approach is to nonlinear McCarthy I. G., Schaye J., Bower R. G., Ponman T. J., Booth matter clustering, in the future it will be likely a crucial instrument C. M., Vecchia C. D., Springel V., 2011, MNRAS, 412, 1965 to study how galaxy formation physics contribute to shaping the McCarthy I. G. et al., 2010, MNRAS, 406, 822 LSS of the Universe. Me´nard B., Dalal N., 2005, MNRAS, 358, 101 Metcalf R. B., 1999, MNRAS, 305, 746 Metcalf R. B., 2001, MNRAS, 327, 115 Metcalf R. B., Silk J., 1999, ApJ, 519, L1 ACKNOWLEDGMENTS Metcalf R. B., Silk J., 2007, Physical Review Letters, 98, 071302 We are grateful to F. J. Nugier and R. B. Metcalf for inter- Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563 esting comments that allowed us to improve the presentation of Oguri M. et al., 2009, ApJ, 699, 1038 this work. CF has received funding from the European Commis- Peacock J. A., Dodds S. J., 1994, MNRAS, 267, 1020 sion Seventh Framework Programme (FP7/2007-2013) under grant Perlmutter S. et al., 1999, ApJ, 517, 565 ◦ agreement n 267251. LM acknowledges ﬁnancial contributions Planck Collaboration et al., 2013, ArXiv e-prints 1303.5076 from contract ASI/INAF I/023/12/0, from PRIN MIUR 2010-2011 Quartin M., Marra V., Amendola L., 2013, ArXiv e-prints ”The dark Universe and the cosmic evolution of baryons: from 1307.1155 current surveys to Euclid” and from PRIN INAF 2012 ”The Uni- Riess A. G. et al., 1998, AJ, 116, 1009 (cid:13) c 0000 RAS, MNRAS 000, 000–000

text: Lensing

text: dispersion

text: of

text: supernova

text: ﬂux:

text: a

text: probe

text: of

text: nonlinear

text: structure

text: growth

text: pute

text: the

text: cosmic

text: shear

text: power

text: spectrum,

text: and

text: then

text: changed

text: the

text: model

text: verse

text: in

text: the

text: box:

text: multiscale

text: simulations

text: of

text: cosmic

text: structure”.

text: We

text: parameters

text: (that

text: in

text: their

text: case

text: also

text: include

text: a

text: redshift

text: modulation

text: of

text: wish

text: to

text: thank

text: an

text: anonymous

text: referee

text: for

text: insightful

text: remarks

text: on

text: the

text: the

text: concentrations)

text: in

text: order

text: to

text: ﬁt

text: the

text: results

text: of

text: hydrodynamic

text: sim-

text: manuscript.

text: ulations

text: with

text: different

text: baryonic

text: physical

text: processes

text: (Schaye

text: et

text: al.

text: References

label: reference

text: By

text: moving

text: from

text: an

text: implementation

text: featuring

text: strong

text: AGN

text: Bartelmann

text: M.,

text: Schneider

text: P.,

text: Phys.

text: Rep.,

text: winds

text: to

text: ones

text: with

text: negligible

text: feedback

text: of

text: any

text: kind

text: they

text: ﬁnd

text: that

text: Ben-Dayan

text: I.,

text: Gasperini

text: M.,

text: Marozzi

text: G.,

text: Nugier

text: F.,

text: Veneziano

text: G.,

text: the

text: best

text: ﬁt

text: normalization

text: of

text: the

text: concentration-mass

text: relation

text: can

text: be

text: J.

text: Cosmology

text: Astropart.

text: Phys.,

text: moved

text: by

text: a

text: factor

text: ranging

text: in

text: a

text: very

text: substantial

text: interval,

text: from

text: Ben-Dayan

text: I.,

text: Gasperini

text: M.,

text: Marozzi

text: G.,

text: Nugier

text: F.,

text: Veneziano

text: G.,

text: up

text: to

text: These

text: numbers

text: cannot

text: be

text: compared

text: directly

text: Physical

text: Review

text: Letters,

text: with

text: our

text: Figure

text: because

text: we

text: do

text: not

text: have

text: any

text: redshift

text: modulation,

text: Ben-Dayan

text: I.,

text: Kalaydzhyan

text: T.,

text: ArXiv

text: e-prints

text: and

text: our

text: pivot

text: mass

text: for

text: the

text: concentration-mass

text: relation

text: is

text: different

text: Bergstro¨m

text: L.,

text: Goliath

text: M.,

text: Goobar

text: A.,

text: Mo¨rtsell

text: E.,

text: A&A,

text: from

text: theirs.

text: Yet,

text: they

text: provide

text: a

text: good

text: indication

text: that

text: the

text: lensing

text: dis-

text: persion

text: will

text: be

text: very

text: well

text: able

text: to

text: substantially

text: reduce

text: the

text: space

text: of

text: Bernardeau

text: F.,

text: van

text: Waerbeke

text: L.,

text: Mellier

text: Y.,

text: A&A,

text: allowed

text: baryonic

text: physical

text: models.

text: Buote

text: D.

text: A.,

text: Gastaldello

text: F.,

text: Humphrey

text: P.

text: J.,

text: Zappacosta

text: L.,

text: Bul-

text: The

text: measurements

text: by

text: Jo¨nsson

text: et

text: al.

text: were

text: based

text: on

text: less

text: lock

text: J.

text: S.,

text: Brighenti

text: F.,

text: Mathews

text: W.

text: G.,

text: ApJ,

text: than

text: SN

text: Ia.

text: The

text: compilation

text: (Suzuki

text: et

text: al.

text: Cooray

text: A.,

text: Sheth

text: R.,

text: Phys.

text: Rep.,

text: contains

text: already

text: times

text: as

text: many

text: objects

text: (although

text: not

text: all

text: com-

text: Dodelson

text: S.,

text: Vallinotto

text: A.,

text: Phys.

text: Rev.

text: D,

text: ing

text: from

text: a

text: single

text: homogeneous

text: dataset).

text: In

text: the

text: future,

text: Euclid

text: will

text: Duffy

text: A.

text: R.,

text: Schaye

text: J.,

text: Kay

text: S.

text: T.,

text: Dalla

text: Vecchia

text: C.,

text: MN-

text: be

text: capable

text: of

text: observing

text: the

text: light

text: curves

text: of

text: SN

text: Ia

text: (Laureijs

text: RAS,

text: et

text: al.

text: while

text: the

text: LSST

text: will

text: increase

text: this

text: number

text: to

text: a

text: stagger-

text: Fanizza

text: G.,

text: Gasperini

text: M.,

text: Marozzi

text: G.,

text: Veneziano

text: G.,

text: J.

text: ing

text: objects

text: (LSST

text: Science

text: Collaboration

text: et

text: al.

text: Cosmology

text: Astropart.

text: Phys.,

text: We

text: can

text: thus

text: expect

text: the

text: measurements

text: of

text: the

text: lensing-induced

text: scat-

text: Fedeli

text: C.,

text: ArXiv

text: e-prints

text: ter

text: in

text: the

text: distance

text: modulus

text: of

text: standard

text: candles

text: to

text: become

text: much

text: more

text: precise

text: in

text: forthcoming

text: years

text: (even

text: signiﬁcantly

text: more

text: than

text: Fedeli

text: C.,

text: Dolag

text: K.,

text: Moscardini

text: L.,

text: MNRAS,

text: we

text: assumed

text: in

text: our

text: last

text: bullet),

text: and

text: thus

text: this

text: method

text: to

text: emerge

text: as

text: Giocoli

text: C.,

text: Bartelmann

text: M.,

text: Sheth

text: R.

text: K.,

text: Cacciato

text: M.,

text: MN-

text: one

text: of

text: the

text: principal

text: probes

text: of

text: nonlinear

text: structures.

text: With

text: increas-

text: RAS,

text: ing

text: statistics

text: the

text: approach

text: that

text: has

text: been

text: exempliﬁed

text: in

text: the

text: present

text: Guy

text: J.

text: et

text: al.,

text: A&A,

text: paper

text: can

text: be

text: improved

text: to

text: probe

text: substructure

text: and

text: stochasticity

text: of

text: Guy

text: J.,

text: Astier

text: P.,

text: Nobili

text: S.,

text: Regnault

text: N.,

text: Pain

text: R.,

text: A&A,

text: dark

text: matter

text: halos

text: (see

text: Giocoli

text: et

text: al.

text: as

text: well

text: as,

text: directly,

text: the

text: effects

text: of

text: gas

text: physics

text: and

text: star

text: formation

text: (thanks

text: to

text: a

text: generalized

text: Hamana

text: T.,

text: Futamase

text: T.,

text: ApJ,

text: halo

text: model

text: that

text: our

text: group

text: is

text: developing,

text: see

text: Fedeli

text: Hilbert

text: S.,

text: Hartlap

text: J.,

text: White

text: S.

text: D.

text: M.,

text: Schneider

text: P.,

text: A&A,

text: The

text: previous

text: discussion

text: does

text: not

text: take

text: into

text: account

text: the

text: pres-

text: ence

text: of

text: unavoidable

text: systematics

text: in

text: the

text: measurements

text: of

text: the

text: SN

text: Ia

text: Holz

text: D.

text: E.,

text: Linder

text: E.

text: V.,

text: ApJ,

text: lensing

text: ﬂux

text: dispersion.

text: It

text: is

text: likely

text: that

text: these

text: systematics

text: will

text: not

text: Jing

text: Y.

text: P.,

text: Zhang

text: P.,

text: Lin

text: W.

text: P.,

text: Gao

text: L.,

text: Springel

text: V.,

text: ApJ,

text: be

text: reducible

text: below

text: a

text: certain

text: (still

text: unknown)

text: level,

text: which

text: is

text: go-

text: ing

text: to

text: obstruct

text: to

text: some

text: extent

text: the

text: constraining

text: power

text: of

text: SN

text: Ia

text: for

text: Jo¨nsson

text: J.,

text: Dahle´n

text: T.,

text: Goobar

text: A.,

text: Mo¨rtsell

text: E.,

text: Riess

text: A.,

text: J.

text: the

text: nonlinear

text: matter

text: clustering.

text: The

text: presence

text: of

text: this

text: systematics

text: Cosmology

text: Astropart.

text: Phys.,

text: ”ﬂoor”

text: will

text: need

text: to

text: be

text: addressed

text: properly

text: when

text: the

text: statistics

text: will

text: Jo¨nsson

text: J.

text: et

text: al.,

text: MNRAS,

text: become

text: large.

text: Kainulainen

text: K.,

text: Marra

text: V.,

text: Phys.

text: Rev.

text: D,

text: The

text: results

text: of

text: this

text: work

text: show

text: not

text: only

text: how

text: the

text: lensing-

text: Kronborg

text: T.

text: et

text: al.,

text: A&A,

text: induced

text: dispersion

text: of

text: apparent

text: magnitudes

text: of

text: SN

text: Ia

text: can

text: lead

text: to

text: Laureijs

text: R.

text: et

text: al.,

text: ArXiv

text: e-prints

text: interesting

text: constraints

text: on

text: nonlinear

text: structure

text: growth,

text: but

text: also

text: how,

text: LSST

text: Science

text: Collaboration

text: et

text: al.,

text: ArXiv

text: e-prints,

text: and

text: because

text: of

text: this

text: very

text: fact,

text: constraints

text: on

text: cosmological

text: param-

text: eters

text: based

text: on

text: this

text: method

text: (Dodelson

text: Vallinotto

text: Marra,

text: Ma

text: C.-P.,

text: Fry

text: J.

text: N.,

text: ApJ,

text: Quartin,

text: Amendola

text: Quartin,

text: Marra,

text: Amendola

text: Marra

text: V.,

text: Quartin

text: M.,

text: Amendola

text: L.,

text: Phys.

text: Rev.

text: D,

text: Ben-Dayan

text: Kalaydzhyan

text: should

text: be

text: taken

text: with

text: grain

text: of

text: salt.

text: Indeed,

text: considering

text: how

text: sensitive

text: this

text: approach

text: is

text: to

text: nonlinear

text: McCarthy

text: I.

text: G.,

text: Schaye

text: J.,

text: Bower

text: R.

text: G.,

text: Ponman

text: T.

text: J.,

text: Booth

text: matter

text: clustering,

text: in

text: the

text: future

text: it

text: will

text: be

text: likely

text: a

text: crucial

text: instrument

text: C.

text: M.,

text: Vecchia

text: C.

text: D.,

text: Springel

text: V.,

text: MNRAS,

text: to

text: study

text: how

text: galaxy

text: formation

text: physics

text: contribute

text: to

text: shaping

text: the

text: McCarthy

text: I.

text: G.

text: et

text: al.,

text: MNRAS,

text: LSS

text: of

text: the

text: Universe.

text: Me´nard

text: B.,

text: Dalal

text: N.,

text: MNRAS,

text: Metcalf

text: R.

text: B.,

text: MNRAS,

text: Metcalf

text: R.

text: B.,

text: MNRAS,

text: Metcalf

text: R.

text: B.,

text: Silk

text: J.,

text: ApJ,

text: ACKNOWLEDGMENTS

text: Metcalf

text: R.

text: B.,

text: Silk

text: J.,

text: Physical

text: Review

text: Letters,

text: We

text: are

text: grateful

text: to

text: F.

text: J.

text: Nugier

text: and

text: R.

text: B.

text: Metcalf

text: for

text: inter-

text: Navarro

text: J.

text: F.,

text: Frenk

text: C.

text: S.,

text: White

text: S.

text: D.

text: M.,

text: ApJ,

text: esting

text: comments

text: that

text: allowed

text: us

text: to

text: improve

text: the

text: presentation

text: of

text: Oguri

text: M.

text: et

text: al.,

text: ApJ,

text: this

text: work.

text: CF

text: has

text: received

text: funding

text: from

text: the

text: European

text: Commis-

text: Peacock

text: J.

text: A.,

text: Dodds

text: S.

text: J.,

text: MNRAS,

text: sion

text: Seventh

text: Framework

text: Programme

text: under

text: grant

text: Perlmutter

text: S.

text: et

text: al.,

text: ApJ,

text: agreement

text: n

text: LM

text: acknowledges

text: ﬁnancial

text: contributions

text: Planck

text: Collaboration

text: et

text: al.,

text: ArXiv

text: e-prints

text: from

text: contract

text: ASI/INAF

text: from

text: PRIN

text: MIUR

text: Quartin

text: M.,

text: Marra

text: V.,

text: Amendola

text: L.,

text: ArXiv

text: e-prints

text: ”The

text: dark

text: Universe

text: and

text: the

text: cosmic

text: evolution

text: of

text: baryons:

text: from

text: current

text: surveys

text: to

text: Euclid”

text: and

text: from

text: PRIN

text: INAF

text: ”The

text: Uni-

text: Riess

text: A.

text: G.

text: et

text: al.,

text: AJ,

text: c

text: RAS,

text: MNRAS


text: ∗ Pairing symmetry and dominant band in Sr RuO 2 4 Thomas Scaﬃdi, Jesper C. Romers, and Steven H. Simon † ‡ (Dated: May 26, 2020) We study the superconductivity pairing symmetry in Sr RuO in the limit of small interaction by 2 4 extending a renormalization group calculation developed by Raghu et al. [Phys. Rev. B 81, 224505 (2010)] to include spin-orbit coupling and multiband eﬀects. We show these eﬀects to be crucial to discriminate between the possible order parameters. In contrast to previous results and without the necessity of ﬁne-tuning, we obtain pseudospin-triplet gaps of the same order of magnitude on the two-dimensional γ band and the quasi-one-dimensional α and β bands. The ratio of the gap amplitude on the diﬀerent bands varies continuously with the interaction parameter. The favored pairing symmetry is shown to be chiral when γ is slightly dominant and helical when α and β are slightly dominant. Strontium ruthenate [1–3] is a layered perovskite ma- π γ terial exhibiting a transition at 1.5 K from a well-behaved α Fermi liquid to a superconducting phase. Strong experi- mental evidence points towards an odd-parity order pa- y β k rameter (OP) [4–7]. Based on multiple experiments[7– 12], the prevailing candidate for the symmetry of the OP has been the chiral p-wave state, d = (p ± ip )zˆ, x y −π which breaks time-reversal symmetry (TRS), hosts topo- −π π k logically protected chiral edge states and is analogous to x superﬂuid 3He-A [13, 14] (d is deﬁned below). On the other hand, this state is supposed to carry FIG. 1. Fermi surfaces for the tight-binding model H given in Eq. (1). edge currents at sample edges and domain walls, which have been elusive so far despite intense scrutiny [15, 16]. As a result, other OP symmetries have been considered on γ that is consistent with speciﬁc heat and scanning theoretically [17–20], including the helical states, d = tunneling microscopy (STM) measurements. p xˆ±p yˆ and d = p xˆ±p yˆ. These phases can be viewed x y y x In this Rapid Communication, we extend the renor- as time-reversal invariant versions of chiral superconduc- malization group (RG) scheme of Raghu et al.[30–34] by tors. Their edges host two counter-propagating Majo- including spin-orbit coupling[35, 36] and multiband ef- rana modes of opposite spin whose net charge current is fects. This enables us to study the orientation of d at zero. a microscopic level and determine the gap on the three Another controversy has arisen recently regarding the bands. We ﬁnd similarly sized gaps on the three bands band(s) on which the superconducting instability is dom- without the necessity of ﬁne-tuning. Depending on the inant. The Fermi surface (FS) of Sr RuO is made of 2 4 interaction parameter, we ﬁnd two OPs that are compat- three cylindrical sheets: The γ band is mainly derived ible with the thermodynamic data: either a chiral gap from the Ru 4d orbital and is fairly isotropic in the xy whose amplitude is slightly larger on γ, or a helical gap basal plane, while the α and β bands are mainly derived whose amplitude is slightly larger on α and β. from the Ru 4d and 4d orbitals and are quasi-one- xz yz The three bands of strontium ruthenate are reproduced dimensional (see Fig. 1). using the following tight-binding Hamiltonian for elec- The prevailing assumption in the ﬁeld has been that trons hopping on a s(cid:88)quare lattice [37, 38] γ is the active band, due to its proximity to a Van Hove singularity. This assumption was based on speciﬁc heat data [21] and backed by several calculations [22–26] that H = ψs†(k)Hˆs(k)ψs(k) (1) predicted a dominant gap on γ and a subdominant, near- k,s nodal gap on α and β. where ψ (k) = [c ; c ; c ]T with s = 1 (−1) This scenario was challenged recently. First, Raghu s k,A,s k,B,s k,C,−s et al.[27](see also [28]) showed that, in absence of band for up (down) spins. The matrix Hˆs(k) is given by 1 coupling and in the weak-coupling limit, α and β are the active bands. Second, Firmo et al.[29] reported a phe- nomenological model with a gap amplitude of similar size 1 See Supplemental Material for more details about the form of on the three bands but slightly larger on α and β than the spin-orbit coupling in the Ru atomic orbitals basis. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Pairing

label: title

text: symmetry

text: and

text: dominant

text: band

text: in

text: Sr

text: RuO

text: Thomas

label: author

text: Scaﬃdi,

text: Jesper

text: C.

text: Romers,

text: and

text: Steven

text: H.

text: Simon

text: (Dated:

text: May

label: date

text: We

label: abstract

text: study

text: the

text: superconductivity

text: pairing

text: symmetry

text: in

text: Sr

text: RuO

text: in

text: the

text: limit

text: of

text: small

text: interaction

text: by

text: extending

text: a

text: renormalization

text: group

text: calculation

text: developed

text: by

text: Raghu

text: et

text: al.

text: [Phys.

text: Rev.

text: B

text: to

text: include

text: spin-orbit

text: coupling

text: and

text: multiband

text: eﬀects.

text: We

text: show

text: these

text: eﬀects

text: to

text: be

text: crucial

text: to

text: discriminate

text: between

text: the

text: possible

text: order

text: parameters.

text: In

text: contrast

text: to

text: previous

text: results

text: and

text: without

text: the

text: necessity

text: of

text: ﬁne-tuning,

text: we

text: obtain

text: pseudospin-triplet

text: gaps

text: of

text: the

text: same

text: order

text: of

text: magnitude

text: on

text: the

text: two-dimensional

text: γ

text: band

text: and

text: the

text: quasi-one-dimensional

text: α

text: and

text: β

text: bands.

text: The

text: ratio

text: of

text: the

text: gap

text: amplitude

text: on

text: the

text: diﬀerent

text: bands

text: varies

text: continuously

text: with

text: the

text: interaction

text: parameter.

text: The

text: favored

text: pairing

text: symmetry

text: is

text: shown

text: to

text: be

text: chiral

text: when

text: γ

text: is

text: slightly

text: dominant

text: and

text: helical

text: when

text: α

text: and

text: β

text: are

text: slightly

text: dominant.

text: Strontium

text: ruthenate

text: is

text: a

text: layered

text: perovskite

text: ma-

text: π

text: γ

text: terial

text: exhibiting

text: a

text: transition

text: at

text: K

text: from

text: a

text: well-behaved

text: α

text: Fermi

text: liquid

text: to

text: a

text: superconducting

text: phase.

text: Strong

text: experi-

text: mental

text: evidence

text: points

text: towards

text: an

text: odd-parity

text: order

text: pa-

text: y

text: β

text: k

text: rameter

text: (OP)

text: Based

text: on

text: multiple

text: the

text: prevailing

text: candidate

text: for

text: the

text: symmetry

text: of

text: the

text: OP

text: has

text: been

text: the

text: chiral

text: p-wave

text: state,

text: d

text: (p

text: ip

text: )zˆ,

text: x

text: y

text: −π

text: which

text: breaks

text: time-reversal

text: symmetry

text: (TRS),

text: hosts

text: topo-

text: −π

text: π

text: k

text: logically

text: protected

text: chiral

text: edge

text: states

text: and

text: is

text: analogous

text: to

text: x

text: superﬂuid

text: (d

text: is

text: deﬁned

text: below).

text: On

text: the

text: other

text: hand,

text: this

text: state

text: is

text: supposed

text: to

text: carry

text: FIG.

text: Fermi

text: surfaces

text: for

text: the

text: tight-binding

text: model

text: H

text: given

text: in

text: Eq.

text: edge

text: currents

text: at

text: sample

text: edges

text: and

text: domain

text: walls,

text: which

text: have

text: been

text: elusive

text: so

text: far

text: despite

text: intense

text: scrutiny

text: As

text: a

text: result,

text: other

text: OP

text: symmetries

text: have

text: been

text: considered

text: on

text: γ

text: that

text: is

text: consistent

text: with

text: speciﬁc

text: heat

text: and

text: scanning

text: theoretically

text: including

text: the

text: helical

text: states,

text: d

text: tunneling

text: microscopy

text: (STM)

text: measurements.

text: p

text: xˆ±p

text: yˆ

text: and

text: d

text: p

text: xˆ±p

text: yˆ.

text: These

text: phases

text: can

text: be

text: viewed

text: x

text: y

text: y

text: x

text: In

text: this

text: Rapid

text: Communication,

text: we

text: extend

text: the

text: renor-

text: as

text: time-reversal

text: invariant

text: versions

text: of

text: chiral

text: superconduc-

text: malization

text: group

text: (RG)

text: scheme

text: of

text: Raghu

text: et

text: by

text: tors.

text: Their

text: edges

text: host

text: two

text: counter-propagating

text: Majo-

text: including

text: spin-orbit

text: and

text: multiband

text: ef-

text: rana

text: modes

text: of

text: opposite

text: spin

text: whose

text: net

text: charge

text: current

text: is

text: fects.

text: This

text: enables

text: us

text: to

text: study

text: the

text: orientation

text: of

text: d

text: at

text: zero.

text: a

text: microscopic

text: level

text: and

text: determine

text: the

text: gap

text: on

text: the

text: three

text: Another

text: controversy

text: has

text: arisen

text: recently

text: regarding

text: the

text: bands.

text: We

text: ﬁnd

text: similarly

text: sized

text: gaps

text: on

text: the

text: three

text: bands

text: band(s)

text: on

text: which

text: the

text: superconducting

text: instability

text: is

text: dom-

text: without

text: the

text: necessity

text: of

text: ﬁne-tuning.

text: Depending

text: on

text: the

text: inant.

text: The

text: Fermi

text: surface

text: (FS)

text: of

text: Sr

text: RuO

text: is

text: made

text: of

text: interaction

text: parameter,

text: we

text: ﬁnd

text: two

text: OPs

text: that

text: are

text: compat-

text: three

text: cylindrical

text: sheets:

text: The

text: γ

text: band

text: is

text: mainly

text: derived

text: ible

text: with

text: the

text: thermodynamic

text: data:

text: either

text: a

text: chiral

text: gap

text: from

text: the

text: Ru

text: orbital

text: and

text: is

text: fairly

text: isotropic

text: in

text: the

text: xy

text: whose

text: amplitude

text: is

text: slightly

text: larger

text: on

text: γ,

text: or

text: a

text: helical

text: gap

text: basal

text: plane,

text: while

text: the

text: α

text: and

text: β

text: bands

text: are

text: mainly

text: derived

text: whose

text: amplitude

text: is

text: slightly

text: larger

text: on

text: α

text: and

text: β.

text: from

text: the

text: Ru

text: and

text: orbitals

text: and

text: are

text: quasi-one-

text: xz

text: yz

text: The

text: three

text: bands

text: of

text: strontium

text: ruthenate

text: are

text: reproduced

text: dimensional

text: (see

text: Fig.

text: using

text: the

text: following

text: tight-binding

text: Hamiltonian

text: for

text: elec-

text: The

text: prevailing

text: assumption

text: in

text: the

text: ﬁeld

text: has

text: been

text: that

text: trons

text: hopping

text: on

text: a

text: lattice

text: γ

text: is

text: the

text: active

text: band,

text: due

text: to

text: its

text: proximity

text: to

text: a

text: Van

text: Hove

text: singularity.

text: This

text: assumption

text: was

text: based

text: on

text: speciﬁc

text: heat

text: data

text: and

text: backed

text: by

text: several

text: calculations

text: that

text: H

text: ψs†(k)Hˆs(k)ψs(k)

text: predicted

text: a

text: dominant

text: gap

text: on

text: γ

text: and

text: a

text: subdominant,

text: near-

text: k,s

text: nodal

text: gap

text: on

text: α

text: and

text: β.

text: where

text: ψ

text: (k)

text: [c

text: c

text: c

text: ]T

text: with

text: s

text: This

text: scenario

text: was

text: challenged

text: recently.

text: First,

text: Raghu

text: s

text: k,A,s

text: k,B,s

text: k,C,−s

text: et

text: also

text: showed

text: that,

text: in

text: absence

text: of

text: band

text: for

text: up

text: (down)

text: spins.

text: The

text: matrix

text: Hˆs(k)

text: is

text: given

text: by

text: coupling

text: and

text: in

text: the

text: weak-coupling

text: limit,

text: α

text: and

text: β

text: are

text: the

text: active

text: bands.

text: Second,

text: Firmo

text: et

text: reported

text: a

text: phe-

text: nomenological

text: model

text: with

text: a

text: gap

text: amplitude

text: of

text: similar

text: size

text: See

text: Supplemental

text: Material

text: for

text: more

text: details

text: about

text: the

text: form

text: of

text: on

text: the

text: three

text: bands

text: but

text: slightly

text: larger

text: on

text: α

text: and

text: β

text: than

text: the

text: spin-orbit

text: coupling

text: in

text: the

text: Ru

text: atomic

text: orbitals

text: basis.

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##

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: ##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: ##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: ##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: ##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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 Chiral Helical Chiral Helical ) 1 (a) (b) 1 (cid:8){ 8 nits 00..89 αβ u 0.7 γ 0.9 4 ry 0.6 ra 0.5 0.8 2 rbit 00..34 a ∆CC 0.7 1 R |∆( 00..12 0.6 Chiral ∆C/C | 00 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Helical ∆C/C θ/π θ/π Chiral R 0.5 0.5 Helical R (c) Phase(dz)/π (d) dxxˆ + dyyˆ 0.25 1 1 0.4 α 0.8 α 0 0.05 0.1 0.15 0.2 0.25 0.3 J/U θ2 00..46 π 0.2 FIG. 3. (Color online) Critical speciﬁc heat jump ∆CC and /y 0 ratio of the maxima of the gap amplitudes over the diﬀerent k γ -0.2 γ | | bands R = max ∆| α,β| . The vertical line separates the stability -0.4 max ∆γ -0.6 regions of the chiral and helical OPs. The curve for a given β -0.8 β OP is drawn in full width only in the OP’s stability region. 0 θ1 -1 The horizontal lines delimit the range of ∆C estimated from 0 kx/π 1 0 kx/π 1 ± C experiments: ∆C = 0.75 0.05 [29, 55]. The braces indicate the range of J/UC for which the prediction is in agreement with FIG. 4. (Color online) We represent the chiral OP d = dzzˆ experiments. for J/U = 0.06 (left) and the helical OP d = dxxˆ + dyyˆ for J/U = 0.08 (√right). Panels (a) and (b) show the gap | | ≡ · ∗ magnitude ∆ d d . In panel (c), the color code gives the complex phase of d around the three FSs. The width of z | | the curve is proportional to ∆ . In panel (d), the vectors are agreement with speciﬁc heat data have gaps of the same proportional to (d , d ), where d and d are real. The angle order on the three bands. The slightly dominant band x y x y θ refers to θ (θ ) in the case of β and γ (α). 1 2 is diﬀerent in the two cases: The chiral state has a gap approximately two times larger on γ than on α and β, while the ratio of the helical gap amplitude on γ over the one on α and β is approximately 0.7. We checked that with a dominant γ and the helical state with dominant both these states give rise to a T linear dependence of α and β lies in the anisotropy of the normal-state spin C/T below Tc, in agreement with experiments [55]. By dynamics. The chiral (helical) state has an out-of-plane tuning J/U towards smaller values, it is possible to ob- (in-plane) d and is therefore driven by in-plane (out-of- tain a largely dominant gap on γ like previously reported plane) magnetic ﬂuctuations. Due to SOC, the incom- [22–26]. mensurate peak Q is larger for the out-of-plane compo- As shown in Fig. 4, the gaps on α and β present near- nent of the susceptibility [57], thereby favoring a helical nodes near the direction [110] in both cases. The incom- state when the quasi-1D bands are dominant. On the mensurate peak Q in the antiferromagnetic ﬂuctuation other hand, the (ferromagnetic) long wavelength part of spectrum [56] of these bands is known to be responsible the spectrum is larger for the in-plane component, which for the appearance of these near-nodes [27, 29]. As its favors a chiral state when γ is dominant. ﬂuctuations are mostly ferromagnetic, the γ band has By a microscopic accounting of multiband and SOC been previously thought to host a fairly isotropic gap of eﬀects, our model reconciles the two distinct scenarios of the type dz = sin(kx) + i sin(ky), with only mild min- 2D superconductivity on γ versus quasi-1D superconduc- ima along [100] [22–26] and a complex phase increasing tivity on α and β inside one framework. As required by quasilinearly with θ (deﬁned in Fig. 4). Interestingly, we speciﬁc heat data[29] and in contrast to previous RG cal- ﬁnd gap minima on γ along [110], which shows that the culations [26, 27], similarly sized gaps on all three bands quasi-one-dimensional (quasi-1D) antiferromagnetic ﬂuc- are obtained and, depending on the interaction parame- tuations peak Q is a source of anisotropy on this band ter, the balance can be slightly tilted one way or another. as well. Besides, the complex phase of our solution for As shown in Fig. 3, this result is true for both the chiral dz in the chiral case [shown in Fig. 4(c)] is a highly non- and the helical state and is therefore robust regardless of trivial and non-monotonic function of θ. Likewise, the the favored pairing symmetry. in-plane orientation of d as a function of θ in the helical We now discuss experiments probing the relative size case [shown in Fig. 4(d)] is much more involved than for of the gaps on the diﬀerent bands. Recently, out-of-plane the archetypal function d = sin(kx)xˆ + sin(ky)yˆ. STM [29] has exhibited the presence of a near-nodal gap The rationale behind the association of the chiral state of 0.350 meV on α and β. We ﬁnd a position for the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Chiral

font: default

text: Helical

text: Chiral

text: Helical

text: (a)

text: (b)

text: nits

text: αβ

text: u

text: γ

text: ry

text: ra

text: rbit

text: a

text: ∆CC

text: R

text: Chiral

text: ∆C/C

text: Helical

text: ∆C/C

text: θ/π

text: θ/π

text: Chiral

text: R

text: Helical

text: R

text: (c)

text: Phase(dz)/π

text: (d)

text: dxxˆ

text: dyyˆ

text: α

text: α

text: J/U

text: θ2

text: π

text: FIG.

text: (Color

text: online)

text: Critical

text: speciﬁc

text: heat

text: jump

text: ∆CC

text: and

text: /y

text: ratio

text: of

text: the

text: maxima

text: of

text: the

text: gap

text: amplitudes

text: over

text: the

text: diﬀerent

text: k

text: γ

text: γ

text: bands

text: R

text: max

text: α,β|

text: The

text: vertical

text: line

text: separates

text: the

text: stability

text: max

text: ∆γ

text: regions

text: of

text: the

text: chiral

text: and

text: helical

text: OPs.

text: The

text: curve

text: for

text: a

text: given

text: β

text: β

text: OP

text: is

text: drawn

text: in

text: full

text: width

text: only

text: in

text: the

text: OP’s

text: stability

text: region.

text: θ1

text: The

text: horizontal

text: lines

text: delimit

text: the

text: range

text: of

text: ∆C

text: estimated

text: from

text: kx/π

text: kx/π

text: C

text: experiments:

text: ∆C

text: The

text: braces

text: indicate

text: the

text: range

text: of

text: J/UC

text: for

text: which

text: the

text: prediction

text: is

text: in

text: agreement

text: with

text: FIG.

text: (Color

text: online)

text: We

text: represent

text: the

text: chiral

text: OP

text: d

text: dzzˆ

text: experiments.

text: for

text: J/U

text: (left)

text: and

text: the

text: helical

text: OP

text: d

text: dxxˆ

text: dyyˆ

text: for

text: J/U

text: (√right).

text: Panels

text: (a)

text: and

text: (b)

text: show

text: the

text: gap

text: magnitude

text: d

text: d

text: In

text: panel

text: (c),

text: the

text: color

text: code

text: gives

text: the

text: complex

text: phase

text: of

text: d

text: around

text: the

text: three

text: FSs.

text: The

text: width

text: of

text: z

text: the

text: curve

text: is

text: proportional

text: to

text: In

text: panel

text: (d),

text: the

text: vectors

text: are

text: agreement

text: with

text: speciﬁc

text: heat

text: data

text: have

text: gaps

text: of

text: the

text: same

text: proportional

text: to

text: (d

text: d

text: where

text: d

text: and

text: d

text: are

text: real.

text: The

text: angle

text: order

text: on

text: the

text: three

text: bands.

text: The

text: slightly

text: dominant

text: band

text: x

text: y

text: x

text: y

text: θ

text: refers

text: to

text: θ

text: (θ

text: in

text: the

text: case

text: of

text: β

text: and

text: γ

text: (α).

text: is

text: diﬀerent

text: in

text: the

text: two

text: cases:

text: The

text: chiral

text: state

text: has

text: a

text: gap

text: approximately

text: two

text: times

text: larger

text: on

text: γ

text: than

text: on

text: α

text: and

text: β,

text: while

text: the

text: ratio

text: of

text: the

text: helical

text: gap

text: amplitude

text: on

text: γ

text: over

text: the

text: one

text: on

text: α

text: and

text: β

text: is

text: approximately

text: We

text: checked

text: that

text: with

text: a

text: dominant

text: γ

text: and

text: the

text: helical

text: state

text: with

text: dominant

text: both

text: these

text: states

text: give

text: rise

text: to

text: a

text: T

text: linear

text: dependence

text: of

text: α

text: and

text: β

text: lies

text: in

text: the

text: anisotropy

text: of

text: the

text: normal-state

text: spin

text: C/T

text: below

text: Tc,

text: in

text: agreement

text: with

text: experiments

text: By

text: dynamics.

text: The

text: chiral

text: (helical)

text: state

text: has

text: an

text: out-of-plane

text: tuning

text: J/U

text: towards

text: smaller

text: values,

text: it

text: is

text: possible

text: to

text: ob-

text: (in-plane)

text: d

text: and

text: is

text: therefore

text: driven

text: by

text: in-plane

text: (out-of-

text: tain

text: a

text: largely

text: dominant

text: gap

text: on

text: γ

text: like

text: previously

text: reported

text: plane)

text: magnetic

text: ﬂuctuations.

text: Due

text: to

text: SOC,

text: the

text: incom-

text: mensurate

text: peak

text: Q

text: is

text: larger

text: for

text: the

text: out-of-plane

text: compo-

text: As

text: shown

text: in

text: Fig.

text: the

text: gaps

text: on

text: α

text: and

text: β

text: present

text: near-

text: nent

text: of

text: the

text: susceptibility

text: thereby

text: favoring

text: a

text: helical

text: nodes

text: near

text: the

text: direction

text: in

text: both

text: cases.

text: The

text: incom-

text: state

text: when

text: the

text: bands

text: are

text: dominant.

text: On

text: the

text: mensurate

text: peak

text: Q

text: in

text: the

text: antiferromagnetic

text: ﬂuctuation

text: other

text: hand,

text: the

text: (ferromagnetic)

text: long

text: wavelength

text: part

text: of

text: spectrum

text: of

text: these

text: bands

text: is

text: known

text: to

text: be

text: responsible

text: the

text: spectrum

text: is

text: larger

text: for

text: the

text: in-plane

text: component,

text: which

text: for

text: the

text: appearance

text: of

text: these

text: near-nodes

text: As

text: its

text: favors

text: a

text: chiral

text: state

text: when

text: γ

text: is

text: dominant.

text: ﬂuctuations

text: are

text: mostly

text: ferromagnetic,

text: the

text: γ

text: band

text: has

text: By

text: a

text: microscopic

text: accounting

text: of

text: multiband

text: and

text: SOC

text: been

text: previously

text: thought

text: to

text: host

text: a

text: fairly

text: isotropic

text: gap

text: of

text: eﬀects,

text: our

text: model

text: reconciles

text: the

text: two

text: distinct

text: scenarios

text: of

text: the

text: type

text: dz

text: sin(kx)

text: i

text: sin(ky),

text: with

text: only

text: mild

text: min-

text: superconductivity

text: on

text: γ

text: versus

text: superconduc-

text: ima

text: along

text: and

text: a

text: complex

text: phase

text: increasing

text: tivity

text: on

text: α

text: and

text: β

text: inside

text: one

text: framework.

text: As

text: required

text: by

text: quasilinearly

text: with

text: θ

text: (deﬁned

text: in

text: Fig.

text: Interestingly,

text: we

text: speciﬁc

text: heat

text: and

text: in

text: contrast

text: to

text: previous

text: RG

text: cal-

text: ﬁnd

text: gap

text: minima

text: on

text: γ

text: along

text: which

text: shows

text: that

text: the

text: culations

text: similarly

text: sized

text: gaps

text: on

text: all

text: three

text: bands

text: quasi-one-dimensional

text: antiferromagnetic

text: ﬂuc-

text: are

text: obtained

text: and,

text: depending

text: on

text: the

text: interaction

text: parame-

text: tuations

text: peak

text: Q

text: is

text: a

text: source

text: of

text: anisotropy

text: on

text: this

text: band

text: ter,

text: the

text: balance

text: can

text: be

text: slightly

text: tilted

text: one

text: way

text: or

text: another.

text: as

text: well.

text: Besides,

text: the

text: complex

text: phase

text: of

text: our

text: solution

text: for

text: As

text: shown

text: in

text: Fig.

text: this

text: result

text: is

text: true

text: for

text: both

text: the

text: chiral

text: dz

text: in

text: the

text: chiral

text: case

text: [shown

text: in

text: Fig.

text: is

text: a

text: highly

text: non-

text: and

text: the

text: helical

text: state

text: and

text: is

text: therefore

text: robust

text: regardless

text: of

text: trivial

text: and

text: non-monotonic

text: function

text: of

text: θ.

text: Likewise,

text: the

text: the

text: favored

text: pairing

text: symmetry.

text: in-plane

text: orientation

text: of

text: d

text: as

text: a

text: function

text: of

text: θ

text: in

text: the

text: helical

text: We

text: now

text: discuss

text: experiments

text: probing

text: the

text: relative

text: size

text: case

text: [shown

text: in

text: Fig.

text: is

text: much

text: more

text: involved

text: than

text: for

text: of

text: the

text: gaps

text: on

text: the

text: diﬀerent

text: bands.

text: Recently,

text: out-of-plane

text: the

text: archetypal

text: function

text: d

text: sin(kx)xˆ

text: sin(ky)yˆ.

text: STM

text: has

text: exhibited

text: the

text: presence

text: of

text: a

text: near-nodal

text: gap

text: The

text: rationale

text: behind

text: the

text: association

text: of

text: the

text: chiral

text: state

text: of

text: meV

text: on

text: α

text: and

text: β.

text: We

text: ﬁnd

text: a

text: position

text: for

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 given by 0.05 < J/U < 0.065 and 0.075 < J/U < 0.085. 0.016 In Ref. [63] (see also references therein), a mean-ﬁeld (px + ipy)ˆz 0.014 pxxˆ + pyyˆ (MF) rotationally invariant slave bosons calculation was Singlet performed to study the impact of the Coulomb repulsion 0.012 | on the quasiparticle bands. From a survey of numer- λ | × 0.01 ous references, they located U in the region 1.5-3.1 eV 2 and J at 0.35 eV. Finally, Ref. [61] also reports a local U) 0.008 density approximation associated with a dynamical mean / t ( 0.006 ﬁeld theory (LDA+DMFT) calculation. They obtain an estimate of J = 0.4 eV by ﬁtting the predicted mass en- 0.004 hancement to the experimental value. This estimate is 0.002 somewhat larger than the previous ones. 0 0.05 0.1 0.15 0.2 0.25 0.3 J/U (cid:48) Ref. Method U U J J/U FIG. 5. RG eigenvalue λ for the chiral state, the most [60] cRPA 2.56 1.94 0.26 0.101 favoured helical state and the singlet state for the parameters − [61] cRPA 2.3 U 2J 0.25 0.108 given in the main text. [62] 1D RG 2.2 - 0.13-0.4 0.059-0.18 − [63] MF 1.5-3.1 U 2J 0.35 0.11-0.23 − [61] LDA+DMFT 2.3 U 2J 0.4 0.17 20 0.06 1 Chiral Helical × TABLE I. Interaction parameters (in eV) obtained by various 0.04 ) methods. |h C 0.02 λ | − 0 | e H -0.02 λ Pairing eigenvalue | ( -0.04 × pxxˆ + pyyˆ 2 pyxˆ − pxyˆ In Fig. 5, we show the pairing eigenvalue λ for diﬀerent U) -0.06 pxxˆ − pyyˆ / pyxˆ + pxyˆ pairing symmetries. The favoured state is the one with (t -0.080 0.05 0.1 0.15 0.2 0.25 0.3 the largest value of |λ|. We show the eigenvalue for two J/U odd-parity channels: one for the chiral state d = (p ± x ipy)zˆ and one for the most favoured helical state d = FIG. 6. Splitting of the RG eigenvalue λ between the four p xˆ + p yˆ. We also show the most favoured state in the diﬀerent helical states and the chiral state for the parameters x y even-parity (i.e. pseudo-spin singlet) channel. Except for given in the main text. very high J/U (> 0.29), the even-parity channel is never favoured. The splitting between the pairing eigenvalue of the he- to zero, from below in the former region and from above lical states and the chiral state is shown in Fig. 6. The in the latter. chiral state is favoured for J/U < 0.065 while the helical The ratio of the maxima of the gap amplitudes over state d = p xˆ + p yˆ takes over for J/U > 0.065. x y the diﬀerent bands R = max|∆α,β| for diﬀerent SOC pa- Magnitude of spin-orbit coupling max|∆γ| rameters η is shown in Fig. 8. Regardless of the value of η, the chiral state favoured at small J/U has a larger gap The impact of η on the splitting between the pair- magnitude on γ while the helical state at larger J/U has ing eigenvalue of the helical state and the chiral state a larger gap amplitude on α and β. is shown in Fig. 7. The curves are fairly similar for all In summary, the amplitude of η does not modify qual- values of η: a region of negative splitting (i.e. favoured itatively our ﬁndings of a favoured chiral state with a chiral state) at small J/U and a region of positive split- (slightly) dominant γ for small J/U and a favoured he- ting (i.e. favoured helical state) at large J/U. These two lical state with (slightly) dominant α and β for larger regions are linked by a cross-over at a certain value for J/U. J/U. As ﬁgured by the arrows in Fig. 7, increasing η does mostly two things: it increases the amplitude of the splitting (be it positive or negative) in the two aforemen- tionned regions and slightly increases the value of J/U ∗ at which the cross-over happens. It also makes the cross- † over smoother. In the limit η → 0, the splitting would go ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: given

text: by

text: J/U

text: and

text: J/U

text: In

text: Ref.

text: (see

text: also

text: references

text: therein),

text: a

text: mean-ﬁeld

text: (px

text: ipy)ˆz

text: pxxˆ

text: pyyˆ

text: (MF)

text: rotationally

text: invariant

text: slave

text: bosons

text: calculation

text: was

text: Singlet

text: performed

text: to

text: study

text: the

text: impact

text: of

text: the

text: Coulomb

text: repulsion

text: on

text: the

text: quasiparticle

text: bands.

text: From

text: a

text: survey

text: of

text: numer-

text: λ

text: ous

text: references,

text: they

text: located

text: U

text: in

text: the

text: region

text: eV

text: and

text: J

text: at

text: eV.

text: Finally,

text: Ref.

text: also

text: reports

text: a

text: local

text: U)

text: density

text: approximation

text: associated

text: with

text: a

text: dynamical

text: mean

text: t

text: ﬁeld

text: theory

text: (LDA+DMFT)

text: calculation.

text: They

text: obtain

text: an

text: estimate

text: of

text: J

text: eV

text: by

text: ﬁtting

text: the

text: predicted

text: mass

text: en-

text: hancement

text: to

text: the

text: experimental

text: value.

text: This

text: estimate

text: is

text: somewhat

text: larger

text: than

text: the

text: previous

text: ones.

text: J/U

text: Ref.

text: Method

text: U

text: U

text: J

text: J/U

text: FIG.

text: RG

text: eigenvalue

text: λ

text: for

text: the

text: chiral

text: state,

text: the

text: most

text: cRPA

text: favoured

text: helical

text: state

text: and

text: the

text: singlet

text: state

text: for

text: the

text: parameters

text: cRPA

text: U

text: given

text: in

text: the

text: main

text: text.

text: RG

text: MF

text: U

text: LDA+DMFT

text: U

text: Chiral

text: Helical

text: TABLE

text: I.

text: Interaction

text: parameters

text: (in

text: eV)

text: obtained

text: by

text: various

text: methods.

text: |h

text: C

text: λ

text: e

text: H

text: λ

text: Pairing

text: eigenvalue

text: pxxˆ

text: pyyˆ

text: pyxˆ

text: pxyˆ

text: In

text: Fig.

text: we

text: show

text: the

text: pairing

text: eigenvalue

text: λ

text: for

text: diﬀerent

text: U)

text: pxxˆ

text: pyyˆ

text: pyxˆ

text: pxyˆ

text: pairing

text: symmetries.

text: The

text: favoured

text: state

text: is

text: the

text: one

text: with

text: (t

text: the

text: largest

text: value

text: of

text: |λ|.

text: We

text: show

text: the

text: eigenvalue

text: for

text: two

text: J/U

text: odd-parity

text: channels:

text: one

text: for

text: the

text: chiral

text: state

text: d

text: (p

text: x

text: ipy)zˆ

text: and

text: one

text: for

text: the

text: most

text: favoured

text: helical

text: state

text: d

text: FIG.

text: Splitting

text: of

text: the

text: RG

text: eigenvalue

text: λ

text: between

text: the

text: four

text: p

text: xˆ

text: p

text: yˆ.

text: We

text: also

text: show

text: the

text: most

text: favoured

text: state

text: in

text: the

text: diﬀerent

text: helical

text: states

text: and

text: the

text: chiral

text: state

text: for

text: the

text: parameters

text: x

text: y

text: even-parity

text: (i.e.

text: pseudo-spin

text: singlet)

text: channel.

text: Except

text: for

text: given

text: in

text: the

text: main

text: text.

text: very

text: high

text: J/U

text: the

text: even-parity

text: channel

text: is

text: never

text: favoured.

text: The

text: splitting

text: between

text: the

text: pairing

text: eigenvalue

text: of

text: the

text: he-

text: to

text: zero,

text: from

text: below

text: in

text: the

text: former

text: region

text: and

text: from

text: above

text: lical

text: states

text: and

text: the

text: chiral

text: state

text: is

text: shown

text: in

text: Fig.

text: The

text: in

text: the

text: latter.

text: chiral

text: state

text: is

text: favoured

text: for

text: J/U

text: while

text: the

text: helical

text: The

text: ratio

text: of

text: the

text: maxima

text: of

text: the

text: gap

text: amplitudes

text: over

text: state

text: d

text: p

text: xˆ

text: p

text: yˆ

text: takes

text: over

text: for

text: J/U

text: x

text: y

text: the

text: diﬀerent

text: bands

text: R

text: max|∆α,β|

text: for

text: diﬀerent

text: SOC

text: pa-

text: Magnitude

text: of

text: spin-orbit

text: coupling

text: max|∆γ|

text: rameters

text: η

text: is

text: shown

text: in

text: Fig.

text: Regardless

text: of

text: the

text: value

text: of

text: η,

text: the

text: chiral

text: state

text: favoured

text: at

text: small

text: J/U

text: has

text: a

text: larger

text: gap

text: The

text: impact

text: of

text: η

text: on

text: the

text: splitting

text: between

text: the

text: pair-

text: magnitude

text: on

text: γ

text: while

text: the

text: helical

text: state

text: at

text: larger

text: J/U

text: has

text: ing

text: eigenvalue

text: of

text: the

text: helical

text: state

text: and

text: the

text: chiral

text: state

text: a

text: larger

text: gap

text: amplitude

text: on

text: α

text: and

text: β.

text: is

text: shown

text: in

text: Fig.

text: The

text: curves

text: are

text: fairly

text: similar

text: for

text: all

text: In

text: summary,

text: the

text: amplitude

text: of

text: η

text: does

text: not

text: modify

text: qual-

text: values

text: of

text: η:

text: a

text: region

text: of

text: negative

text: splitting

text: (i.e.

text: favoured

text: itatively

text: our

text: ﬁndings

text: of

text: a

text: favoured

text: chiral

text: state

text: with

text: a

text: chiral

text: state)

text: at

text: small

text: J/U

text: and

text: a

text: region

text: of

text: positive

text: split-

text: (slightly)

text: dominant

text: γ

text: for

text: small

text: J/U

text: and

text: a

text: favoured

text: he-

text: ting

text: (i.e.

text: favoured

text: helical

text: state)

text: at

text: large

text: J/U.

text: These

text: two

text: lical

text: state

text: with

text: (slightly)

text: dominant

text: α

text: and

text: β

text: for

text: larger

text: regions

text: are

text: linked

text: by

text: a

text: cross-over

text: at

text: a

text: certain

text: value

text: for

text: J/U.

text: J/U.

text: As

text: ﬁgured

text: by

text: the

text: arrows

text: in

text: Fig.

text: increasing

text: η

text: does

text: mostly

text: two

text: things:

text: it

text: increases

text: the

text: amplitude

text: of

text: the

text: splitting

text: (be

text: it

text: positive

text: or

text: negative)

text: in

text: the

text: two

text: aforemen-

text: tionned

text: regions

text: and

text: slightly

text: increases

text: the

text: value

text: of

text: J/U

text: at

text: which

text: the

text: cross-over

text: happens.

text: It

text: also

text: makes

text: the

text: cross-

text: over

text: smoother.

text: In

text: the

text: limit

text: η

text: the

text: splitting

text: would

text: go

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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##

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 1.0 f(a ) a 1 a well as F(ρ). When applied to the distribution Eq.(4) 00..20 0.5 1.0 1.5 the symmetry Eq.(5) imposes a functional constraint: (cid:45) 0.8 (cid:45)0.4 − 0.6 (cid:45) f(1 + α) = f(1 α) + α. (6) 0.6 (cid:45)10..08 f(a )- a (cid:45) Equation Eq.(6) is valid only for extended states. In- 0.4 deed, the ﬂuctuations of LDoS ρ of the localized states 0.2 are dominated by the ﬂuctuations of Γ and there is no n a a 0 a a reason for Eq.(5) to hold for P(x). 0.0 min max (cid:48) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 It follows from Eq.(6) that f (1) = 1, i.e. the only 2 power-law distribution P(x) (linear function f(α) ) com- FIG. 1: (Color online) Non-singular function f(α) obeying the − patible with Eq.(6) is: symmetry Eq.(6). In the inset: a generic function f(α) α. − P(x) = Ax 32 . (7) is natural to call ψ(i) localized if S is N-independent in ∝ ε As we will see, P(x) acquires the form Eq.(7) at the point this limit, while the scaling S N would suggest that ε of the Anderson transition on the BL. ψ(i) is both delocalized and ergodic. It is possible that → ∞ → → ∞ The function f(α) reaches its maximum f(α ) = 1 at Sε but the ratio Sε/N 0 at N . In such (cid:48) − 0 α = α , i.e. f (α ) = 0. Substituting α = α 1 into Eq. a case we will refer to ψ(i) as the non-ergodic extended 0 0 0 − (6) and its derivative over α we ﬁnd that for α = 2 α wave function. 1 0 | | holds: Below instead of ψ(i) 2 we will use variables x(i) = | | N ψ(i) 2 and deﬁne x = x(S ), so that x = → ε ε ε=0 x(S ) = x(N) 0. Typical values of x and S are (cid:48) ε=0 (cid:15) (cid:15) f(α ) = α , and f (α ) = 1. (8) determined by the(cid:90)distribution function P(x) as follows: 1 1 1 The straight line f(α) = α is thus a tangential to f(α) at ∞ − (cid:90) − the point α = α = 2 α . Therefore convexity of f(α) 1 0 (cid:48) x P(x) dx = 1 ε, (2) implies that α < 1 and α > 1, since 0 < f (1) = 1/2 < 1 0 x∞ε 1. In a similar way one may prove that the solutions α S (cid:48) q P(x) dx = ε . (3) to the equations f (αq) = q are ordered (αq > αq+1) and: N xε − 0 < qα f(α ) < α − . (9) q q q 1 III. THE MULTIFRACTAL ANSATZ FOR P(x) A. The support set and the function f(α) Consider the amplitude distribution function P(x) cor- responding to the multifractal statistics (a multifractal (cid:82) ansatz)14,16: For P(x) given by Eq.(4) one obtains from Eqs.(2),(3): (cid:82) A − ln x αε dα elnN f(α) P(x) = N x exp [ln N f(α(x))] , α(x) = 1 ln N . (4) Sε = ε ∞0dα elnN [f(α)−α] , (10) αε with the normalization constant A. The spectrum of (cid:82) where α is a solution to the equation: fractal dimensions f(α) deﬁned for α > 0 (i) is a con- ε (cid:82) vex function, and (ii) its maximal value equals to 1: ∞ − dα elnN [f(α) α] fmax = f(α0) =(cid:80)1. Besides that, for extended states α∞ε − = ε. (11) a number constraints for f(α) follow from the symme- dα elnN [f(α) α] 0 try of the distribution F(ρ) of the local density of states | | (LDoS) ρ((cid:15), i) = n ψn(i) 2 π [((cid:15)−(cid:15)Γnn)2+Γ2n]: AintatleadrgbeylnαNinaltlhtehveicininteitgyraolfs minaExiqms.a(1α0),(a1n1d) αareodfotmhe- − 0 1 − − F(ρ) = ρ 3 F(ρ 1). (5) functions f(α) and f(α) α, respectively, or the end points of the integration domain and can be done by the This symmetry was ﬁrst established for the localized saddle-point or the end-point approximations. states in strictly one-dimensional system with random B. Support set for a non-singular function f(α) potential17, and later derived from the nonlinear super- symmetric sigma-model18,19. The level widths Γ do not Consider now a generic non-singular function f(α) n ﬂuctuate much if the states are extended. Accordingly obeying the functional constraint Eq.(6). In this case − ﬂuctuations of ρ are mostly due to the ﬂuctuations of the maximal values 1 and 0 of f(α) and f(α) α are | | the wave-function amplitude ψ (i) 2, i.e. of x. There- reached at α = α and α = α such that α < α < n 0 1 min 1 fore P(x) should obey the symmetry relation Eq.(5) as 1 < α < α (see Fig. 1). 0 max ##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##

text: f(a

text: a

text: a

text: well

text: as

text: F(ρ).

text: When

text: applied

text: to

text: the

text: distribution

text: the

text: symmetry

text: imposes

text: a

text: functional

text: constraint:

text: α)

text: α)

text: α.

text: f(a

text: a

text: Equation

text: is

text: valid

text: only

text: for

text: extended

text: states.

text: In-

text: deed,

text: the

text: ﬂuctuations

text: of

text: LDoS

text: ρ

text: of

text: the

text: localized

text: states

text: are

text: dominated

text: by

text: the

text: ﬂuctuations

text: of

text: Γ

text: and

text: there

text: is

text: no

text: n

text: a

text: a

text: a

text: a

text: reason

text: for

text: to

text: hold

text: for

text: P(x).

text: min

text: max

text: It

text: follows

text: from

text: that

text: f

text: i.e.

text: the

text: only

text: power-law

text: distribution

text: P(x)

text: (linear

text: function

text: f(α)

text: com-

text: FIG.

text: (Color

text: online)

text: Non-singular

text: function

text: f(α)

text: obeying

text: the

text: patible

text: with

text: is:

text: symmetry

text: In

text: the

text: inset:

text: a

text: generic

text: function

text: f(α)

text: α.

text: P(x)

text: Ax

text: is

text: natural

text: to

text: call

text: ψ(i)

text: localized

text: if

text: S

text: is

text: N-independent

text: in

text: ε

text: As

text: we

text: will

text: see,

text: P(x)

text: acquires

text: the

text: form

text: at

text: the

text: point

text: this

text: limit,

text: while

text: the

text: scaling

text: S

text: N

text: would

text: suggest

text: that

text: ε

text: of

text: the

text: Anderson

text: transition

text: on

text: the

text: BL.

text: ψ(i)

text: is

text: both

text: delocalized

text: and

text: ergodic.

text: It

text: is

text: possible

text: that

text: The

text: function

text: f(α)

text: reaches

text: its

text: maximum

text: f(α

text: at

text: Sε

text: but

text: the

text: ratio

text: Sε/N

text: at

text: N

text: In

text: such

text: α

text: α

text: i.e.

text: f

text: (α

text: Substituting

text: α

text: α

text: into

text: Eq.

text: a

text: case

text: we

text: will

text: refer

text: to

text: ψ(i)

text: as

text: the

text: non-ergodic

text: extended

text: and

text: its

text: derivative

text: over

text: α

text: we

text: ﬁnd

text: that

text: for

text: α

text: α

text: wave

text: function.

text: holds:

text: Below

text: instead

text: of

text: ψ(i)

text: we

text: will

text: use

text: variables

text: x(i)

text: N

text: ψ(i)

text: and

text: deﬁne

text: x

text: x(S

text: so

text: that

text: x

text: ε

text: ε

text: x(S

text: x(N)

text: Typical

text: values

text: of

text: x

text: and

text: S

text: are

text: f(α

text: α

text: and

text: f

text: (α

text: determined

text: by

text: function

text: P(x)

text: as

text: follows:

text: The

text: straight

text: line

text: f(α)

text: α

text: is

text: thus

text: a

text: tangential

text: to

text: f(α)

text: at

text: the

text: point

text: α

text: α

text: α

text: Therefore

text: convexity

text: of

text: f(α)

text: x

text: P(x)

text: dx

text: ε,

text: implies

text: that

text: α

text: and

text: α

text: since

text: f

text: x∞ε

text: In

text: a

text: similar

text: way

text: one

text: may

text: prove

text: that

text: the

text: solutions

text: α

text: S

text: q

text: P(x)

text: dx

text: ε

text: to

text: the

text: equations

text: f

text: (αq)

text: q

text: are

text: ordered

text: (αq

text: and:

text: N

text: xε

text: qα

text: f(α

text: α

text: q

text: q

text: q

text: III.

text: THE

text: MULTIFRACTAL

text: ANSATZ

text: FOR

text: P(x)

text: A.

text: The

text: support

text: set

text: and

text: the

text: function

text: f(α)

text: Consider

text: the

text: amplitude

text: distribution

text: function

text: P(x)

text: cor-

text: responding

text: to

text: the

text: multifractal

text: statistics

text: (a

text: multifractal

text: For

text: P(x)

text: given

text: by

text: one

text: obtains

text: from

text: A

text: ln

text: x

text: αε

text: dα

text: elnN

text: f(α)

text: P(x)

text: N

text: x

text: exp

text: [ln

text: N

text: f(α(x))]

text: α(x)

text: ln

text: N

text: Sε

text: ε

text: ∞0dα

text: elnN

text: [f(α)−α]

text: αε

text: with

text: the

text: normalization

text: constant

text: A.

text: The

text: spectrum

text: of

text: where

text: α

text: is

text: a

text: solution

text: to

text: the

text: equation:

text: fractal

text: dimensions

text: f(α)

text: deﬁned

text: for

text: α

text: (i)

text: is

text: a

text: con-

text: ε

text: vex

text: function,

text: and

text: (ii)

text: its

text: maximal

text: value

text: equals

text: to

text: dα

text: elnN

text: [f(α)

text: α]

text: fmax

text: Besides

text: that,

text: for

text: extended

text: states

text: α∞ε

text: ε.

text: a

text: number

text: constraints

text: for

text: f(α)

text: follow

text: from

text: the

text: symme-

text: dα

text: elnN

text: [f(α)

text: α]

text: try

text: of

text: the

text: distribution

text: F(ρ)

text: of

text: the

text: local

text: density

text: of

text: states

text: (LDoS)

text: i)

text: n

text: ψn(i)

text: π

text: αareodfotmhe-

text: F(ρ)

text: ρ

text: F(ρ

text: functions

text: f(α)

text: and

text: f(α)

text: α,

text: respectively,

text: or

text: the

text: end

text: points

text: of

text: the

text: integration

text: domain

text: and

text: can

text: be

text: done

text: by

text: the

text: This

text: symmetry

text: was

text: ﬁrst

text: established

text: for

text: the

text: localized

text: saddle-point

text: or

text: the

text: end-point

text: approximations.

text: states

text: in

text: strictly

text: one-dimensional

text: system

text: with

text: random

text: B.

text: Support

text: set

text: for

text: a

text: non-singular

text: function

text: f(α)

text: and

text: later

text: derived

text: from

text: the

text: nonlinear

text: super-

text: symmetric

text: The

text: level

text: widths

text: Γ

text: do

text: not

text: Consider

text: now

text: a

text: generic

text: non-singular

text: function

text: f(α)

text: n

text: ﬂuctuate

text: much

text: if

text: the

text: states

text: are

text: extended.

text: Accordingly

text: obeying

text: the

text: functional

text: constraint

text: In

text: this

text: case

text: ﬂuctuations

text: of

text: ρ

text: are

text: mostly

text: due

text: to

text: the

text: ﬂuctuations

text: of

text: the

text: maximal

text: values

text: and

text: of

text: f(α)

text: and

text: f(α)

text: α

text: are

text: the

text: wave-function

text: amplitude

text: ψ

text: (i)

text: i.e.

text: of

text: x.

text: There-

text: reached

text: at

text: α

text: α

text: and

text: α

text: α

text: such

text: that

text: α

text: α

text: n

text: min

text: fore

text: P(x)

text: should

text: obey

text: the

text: symmetry

text: relation

text: as

text: α

text: α

text: (see

text: Fig.

text: max

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##

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: ##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: ##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: ##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##