text: Proceedings of Symposia in Pure Mathematics Mirror Symmetry in Flavored Aﬃne D-type Quivers Anindya Dey Abstract. We present non-trivial checks of three dimensional mirror sym- N metry for = 4, DˆN quiver gauge theories with unitary gauge groups using partition function on a round sphere. Type IIB (Hanany-Witten) realization of these theories and their mirror duals (as world volume gauge theories on coinci- dent D3 branes) involve 1/2-BPS boundary conditions associated with orbifold and orientifold 5-planes respectively, in addition to NS5 and D5 branes. We demonstrate that partition function for a given quiver in this class may be decomposed into distinct contributions from the aforementioned Type IIB in- gredients. As a byproduct of this computation, we ﬁnd a convenient way of deriving the mirror map for a given pair of dual theories. 1. Introduction N Mirror symmetry in = 4 supersymmetric aﬃne quiver gauge theories in three dimensions have been extensively studied in the literature [6, 7, 8]. In the Type IIB picture, these theories arise as world volume gauge theories on coincident D3 branes R × wrapping 2,1 L – L being a compact direction – with certain special 1/2-BPS boundary conditions at the two ends. In Type IIA, these can be obtained as world volume theories on coincident D2 branes with parallel D6 branes (and possibly a O6 plane) wrapping a transverse ALE (Asymptotically Locally Euclidean) space. The M-theory lift of this Type IIA scenario therefore involves coincident M2 branes probing a product of ALE spaces in the transverse direction with appropriate G- ﬂuxes. As is evident from the Type IIA description, Higgs branches of such theories arise as moduli spaces of instantons for classical gauge groups on ALE spaces [10]. Aside from other interesting features, such theories provide a rich laboratory for studying dualities in supersymmetric QFTs. Mirror symmetry, a particularly N important duality for = 4 theories in three dimensions, acts by simply exchanging the Coulomb and the Higgs branches of a pair of theories. It is a classic example of a IR duality which involves two or more theories with completely diﬀerent UV description ﬂowing to the same superconformal point in the IR. Key words and phrases. Localization,Mirror Symmetry, Dualities. The author would like to thank Jacques Distler for collaboration [1] on which much of this material is based. The author would also like to thank Peter Koroteev and Noppadol Mekareeya for some of the ﬁgures. This work (UTTG-01-14) is supported by the National Science Foundation under Grant Numbers PHY-1316033 and PHY-0969020. (cid:13) c 0000 (copyright holder) 1 ##LTLine##

text: Proceedings

text: of

text: Symposia

text: in

text: Pure

text: Mathematics

text: Mirror

text: Symmetry

text: in

text: Flavored

text: Aﬃne

text: D-type

text: Quivers

text: Anindya

text: Dey

text: Abstract.

text: We

text: present

text: non-trivial

text: checks

text: of

text: three

text: dimensional

text: mirror

text: sym-

text: N

text: metry

text: for

text: DˆN

text: quiver

text: gauge

text: theories

text: with

text: unitary

text: gauge

text: groups

text: using

text: partition

text: function

text: on

text: a

text: round

text: sphere.

text: Type

text: IIB

text: (Hanany-Witten)

text: realization

text: of

text: these

text: theories

text: and

text: their

text: mirror

text: duals

text: (as

text: world

text: volume

text: gauge

text: theories

text: on

text: coinci-

text: dent

text: branes)

text: involve

text: boundary

text: conditions

text: associated

text: with

text: orbifold

text: and

text: orientifold

text: respectively,

text: in

text: addition

text: to

text: and

text: branes.

text: We

text: demonstrate

text: that

text: partition

text: function

text: for

text: a

text: given

text: quiver

text: in

text: this

text: class

text: may

text: be

text: decomposed

text: into

text: distinct

text: contributions

text: from

text: the

text: aforementioned

text: Type

text: IIB

text: in-

text: gredients.

text: As

text: a

text: byproduct

text: of

text: this

text: computation,

text: we

text: ﬁnd

text: a

text: convenient

text: way

text: of

text: deriving

text: the

text: mirror

text: map

text: for

text: a

text: given

text: pair

text: of

text: dual

text: theories.

text: Introduction

text: N

text: Mirror

text: symmetry

text: in

text: supersymmetric

text: aﬃne

text: quiver

text: gauge

text: theories

text: in

text: three

text: dimensions

text: have

text: been

text: extensively

text: studied

text: in

text: the

text: literature

text: In

text: the

text: Type

text: IIB

text: picture,

text: these

text: theories

text: arise

text: as

text: world

text: volume

text: gauge

text: theories

text: on

text: coincident

text: branes

text: R

text: wrapping

text: L

text: L

text: being

text: a

text: compact

text: direction

text: with

text: certain

text: special

text: boundary

text: conditions

text: at

text: the

text: two

text: ends.

text: In

text: Type

text: IIA,

text: these

text: can

text: be

text: obtained

text: as

text: world

text: volume

text: theories

text: on

text: coincident

text: branes

text: with

text: parallel

text: branes

text: (and

text: possibly

text: a

text: plane)

text: wrapping

text: a

text: transverse

text: ALE

text: (Asymptotically

text: Locally

text: Euclidean)

text: space.

text: The

text: M-theory

text: lift

text: of

text: this

text: Type

text: IIA

text: scenario

text: therefore

text: involves

text: coincident

text: branes

text: probing

text: a

text: product

text: of

text: ALE

text: spaces

text: in

text: the

text: transverse

text: direction

text: with

text: appropriate

text: G-

text: ﬂuxes.

text: As

text: is

text: evident

text: from

text: the

text: Type

text: IIA

text: description,

text: Higgs

text: branches

text: of

text: such

text: theories

text: arise

text: as

text: moduli

text: spaces

text: of

text: instantons

text: for

text: classical

text: gauge

text: groups

text: on

text: ALE

text: spaces

text: Aside

text: from

text: other

text: interesting

text: features,

text: such

text: theories

text: provide

text: a

text: rich

text: laboratory

text: for

text: studying

text: dualities

text: in

text: supersymmetric

text: QFTs.

text: Mirror

text: symmetry,

text: a

text: particularly

text: N

text: important

text: duality

text: for

text: theories

text: in

text: three

text: dimensions,

text: acts

text: by

text: simply

text: exchanging

text: the

text: Coulomb

text: and

text: the

text: Higgs

text: branches

text: of

text: a

text: pair

text: of

text: theories.

text: It

text: is

text: a

text: classic

text: example

text: of

text: a

text: IR

text: duality

text: which

text: involves

text: two

text: or

text: more

text: theories

text: with

text: completely

text: diﬀerent

text: UV

text: description

text: ﬂowing

text: to

text: the

text: same

text: superconformal

text: point

text: in

text: the

text: IR.

text: Key

text: words

text: and

text: phrases.

text: Localization,Mirror

text: Symmetry,

text: Dualities.

text: The

text: author

text: would

text: like

text: to

text: thank

text: Jacques

text: Distler

text: for

text: collaboration

text: on

text: which

text: much

text: of

text: this

text: material

text: is

text: based.

text: The

text: author

text: would

text: also

text: like

text: to

text: thank

text: Peter

text: Koroteev

text: and

text: Noppadol

text: Mekareeya

text: for

text: some

text: of

text: the

text: ﬁgures.

text: This

text: work

text: is

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: under

text: Grant

text: Numbers

text: and

text: c

text: (copyright

text: holder)

text: ##LTLine##


text: time-dependent but spatially uniform. This produces the dynamical equation dm − ∂F − − = = 2am 4bm3 + h, (1) dt ∂m which is a more general form of Eq. (3) governing the spatially uniform solutions in Ref. [4]. It is known and straightforward to show that the equilibrium critical exponents for this mean-ﬁeld Ginzburg-Landau (MFGL) model are β = 1/2 and δ = 3. The dynamic critical exponents for this MFGL model at the critical period match the corresponding exponents for the equilibrium transition, as they do for the kinetic Ising model studied in Ref. [16]. We believe this result for the MFGL model has been demonstrated previously; at least the dynamical exponent β = 1/2 is established in Eq. (23) of Ref [4]. In any case, we establish the dynamic critical exponents β = 1/2 and δ = 3 numerically in Figs. 2 and 4, respectively, of this paper. √ √ − In implementing the MFGL, we choose parameters a = 3 3 and b = 3 3, which for 4 8 ± h = 0 yield minima of the free energy at m = 1. In a periodic applied ﬁeld h(t) = h(t+P ), we expect the dynamics to converge to limit cycle(s) of the form m(t) = m(t + P ). Setting ω = 2π/P , we can expand both h(t) and m(t) as complex Fourier series: (cid:88) (cid:88) h(t) = h eikωt ; m(t) = m eikωt (2) k k k k where here and for the remainder of this paper, a summation index without limits is −∞ ∞ ∗ understood to run from to + . Since h(t) and m(t) are real, it follows that h− = h k k ∗ and m− = m , so that h and m are real. The dynamic order parameter referred to as k k 0 0 Q in previous studies [5–7] is the real Fourier coeﬃcient m , while the component of the 0 conjugate ﬁeld identiﬁed in Ref. [16] – the period-averaged magnetic ﬁeld – is the real Fourier coeﬃcient h . 0 III. HIGHER-ORDER BIFURCATIONS In both mean-ﬁeld [1, 4] and kinetic Ising [5–7] models, above P there is a stable sym- c metric hysteresis loop with m = 0. Below P there are two stable asymmetric loops with 0 c ± opposite values m = m , as well as one unstable symmetric loop with m = 0. This 0 s 0 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: time-dependent

label: paragraph

text: but

text: spatially

text: uniform.

text: This

text: produces

text: the

text: dynamical

text: equation

text: dm

label: equation

text: ∂F

text: h,

text: dt

text: ∂m

text: which

text: is

text: a

text: more

text: general

text: form

text: of

text: Eq.

text: governing

text: the

text: spatially

text: uniform

text: solutions

text: in

text: Ref.

text: It

text: is

text: known

text: and

text: straightforward

text: to

text: show

text: that

text: the

text: equilibrium

text: critical

text: exponents

text: for

text: this

text: mean-ﬁeld

text: Ginzburg-Landau

text: (MFGL)

text: model

text: are

text: β

text: and

text: δ

text: The

text: dynamic

text: critical

text: exponents

text: for

text: this

text: MFGL

text: model

text: at

text: the

text: critical

text: period

text: match

text: the

text: corresponding

text: exponents

text: for

text: the

text: equilibrium

text: transition,

text: as

text: they

text: do

text: for

text: the

text: kinetic

text: Ising

text: model

text: studied

text: in

text: Ref.

text: We

text: believe

text: this

text: result

text: for

text: the

text: MFGL

text: model

text: has

text: been

text: demonstrated

text: previously;

text: at

text: least

text: the

text: dynamical

text: exponent

text: β

text: is

text: established

text: in

text: Eq.

text: of

text: Ref

text: In

text: any

text: case,

text: we

text: establish

text: the

text: dynamic

text: critical

text: exponents

text: β

text: and

text: δ

text: numerically

text: in

text: Figs.

text: and

text: respectively,

text: of

text: this

text: paper.

text: In

text: implementing

text: the

text: MFGL,

text: we

text: choose

text: parameters

text: a

text: and

text: b

text: which

text: for

text: h

text: yield

text: minima

text: of

text: the

text: free

text: energy

text: at

text: m

text: In

text: a

text: periodic

text: applied

text: ﬁeld

text: h(t)

text: h(t+P

text: we

text: expect

text: the

text: dynamics

text: to

text: converge

text: to

text: limit

text: cycle(s)

text: of

text: the

text: form

text: m(t)

text: m(t

text: P

text: Setting

text: ω

text: we

text: can

text: expand

text: both

text: h(t)

text: and

text: m(t)

text: as

text: complex

text: Fourier

text: series:

text: h(t)

text: h

text: eikωt

text: m(t)

text: m

text: eikωt

text: k

text: k

text: k

text: k

text: where

text: here

text: and

text: for

text: the

text: remainder

text: of

text: this

text: paper,

text: a

text: summation

text: index

text: without

text: limits

text: is

text: understood

text: to

text: run

text: from

text: to

text: Since

text: h(t)

text: and

text: m(t)

text: are

text: real,

text: it

text: follows

text: that

text: h−

text: h

text: k

text: k

text: and

text: m−

text: m

text: so

text: that

text: h

text: and

text: m

text: are

text: real.

text: The

text: dynamic

text: order

text: parameter

text: referred

text: to

text: as

text: k

text: k

text: Q

text: in

text: previous

text: studies

text: is

text: the

text: real

text: Fourier

text: coeﬃcient

text: m

text: while

text: the

text: component

text: of

text: the

text: conjugate

text: ﬁeld

text: identiﬁed

text: in

text: Ref.

text: the

text: period-averaged

text: magnetic

text: ﬁeld

text: is

text: the

text: real

text: Fourier

text: coeﬃcient

text: h

text: III.

text: HIGHER-ORDER

text: BIFURCATIONS

text: In

text: both

text: mean-ﬁeld

text: and

text: kinetic

text: Ising

text: models,

text: above

text: P

text: there

text: is

text: a

text: stable

text: sym-

text: c

text: metric

text: hysteresis

text: loop

text: with

text: m

text: Below

text: P

text: there

text: are

text: two

text: stable

text: asymmetric

text: loops

text: with

text: c

text: opposite

text: values

text: m

text: m

text: as

text: well

text: as

text: one

text: unstable

text: symmetric

text: loop

text: with

text: m

text: This

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: FIG. 1. (Color online) Dynamic phase diagram illustrating the bifurcation of Fourier coeﬃcients √ √ − m , m and m below P . Here a = 3 3, b = 3 3, and H = 1.5, for which it is found that 0 2 4 C 4 8 1 | | −| | P = 5.319357661995. Note that the values + m and m are displayed below P in Fig. 1 C 2j 2j c for simplicity; the two stable asymmetric loops actually have opposite complex Fourier components − m and m . 2j 2j behavior of m in the GL model deﬁned by Eq. (1) can be observed in Fig. 1. In addition, 0 the Fourier components m and m undergo a similar bifurcation at P . That is, above P 2 4 c c there is a stable symmetric hysteresis loop with m = m = 0. Below P there are two 2 4 c ± ± stable asymmetric loops with opposite values m = m and m = m , as well as one 2 s,2 4 s,4 unstable symmetric loop with m = 0. It was shown in Ref. [4] that m = 0 (for j integer) 0 2j above P , but the bifurcation of m and m below P has not been reported before to our c 2 4 c knowledge. A similar bifurcation occurs for all even Fourier components m . It is interesting 2j to note, however, that whereas the constant component m increases monotonically below 0 | | | | P , the amplitudes m and m increase over a limited range below P , and then decrease, c 2 4 c eventually approaching 0 as the period P decreases to 0. To within the numerical accuracy of our simulations, the bifurcation in all three com- ponents m , m and m occurred at the same critical value P . The critical period can be 0 2 4 c located numerically by applying the stability criterion (cid:88) | | − a m 2 = (3) k,c 6b k 4 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: (Color

label: caption

text: online)

text: Dynamic

text: phase

text: diagram

text: illustrating

text: the

text: bifurcation

text: of

text: Fourier

text: coeﬃcients

text: m

text: m

text: and

text: m

text: below

text: P

text: Here

text: a

text: b

text: and

text: H

text: for

text: which

text: it

text: is

text: found

text: that

text: C

text: P

text: Note

text: that

text: the

text: values

text: m

text: and

text: m

text: are

text: displayed

text: below

text: P

text: in

text: Fig.

text: C

text: c

text: for

text: simplicity;

text: the

text: two

text: stable

text: asymmetric

text: loops

text: actually

text: have

text: opposite

text: complex

text: Fourier

text: components

text: m

text: and

text: m

text: behavior

text: of

text: m

text: in

text: the

text: GL

text: model

text: deﬁned

text: by

text: Eq.

text: can

text: be

text: observed

text: in

text: Fig.

text: In

text: addition,

text: the

text: Fourier

text: components

text: m

text: and

text: m

text: undergo

text: a

text: similar

text: bifurcation

text: at

text: P

text: That

text: is,

text: above

text: P

text: c

text: c

text: there

text: is

text: a

text: stable

text: symmetric

text: hysteresis

text: loop

text: with

text: m

text: m

text: Below

text: P

text: there

text: are

text: two

text: c

text: stable

text: asymmetric

text: loops

text: with

text: opposite

text: values

text: m

text: m

text: and

text: m

text: m

text: as

text: well

text: as

text: one

text: unstable

text: symmetric

text: loop

text: with

text: m

text: It

text: was

text: shown

text: in

text: Ref.

text: that

text: m

text: (for

text: j

text: integer)

text: above

text: P

text: but

text: the

text: bifurcation

text: of

text: m

text: and

text: m

text: below

text: P

text: has

text: not

text: been

text: reported

text: before

text: to

text: our

text: c

text: c

text: knowledge.

text: A

text: similar

text: bifurcation

text: occurs

text: for

text: all

text: even

text: Fourier

text: components

text: m

text: It

text: is

text: interesting

text: to

text: note,

text: however,

text: that

text: whereas

text: the

text: constant

text: component

text: m

text: increases

text: monotonically

text: below

text: P

text: the

text: amplitudes

text: m

text: and

text: m

text: increase

text: over

text: a

text: limited

text: range

text: below

text: P

text: and

text: then

text: decrease,

text: c

text: c

text: eventually

text: approaching

text: as

text: the

text: period

text: P

text: decreases

text: to

text: To

text: within

text: the

text: numerical

text: accuracy

text: of

text: our

text: simulations,

text: the

text: bifurcation

text: in

text: all

text: three

text: com-

text: ponents

text: m

text: m

text: and

text: m

text: occurred

text: at

text: the

text: same

text: critical

text: value

text: P

text: The

text: critical

text: period

text: can

text: be

text: c

text: located

text: numerically

text: by

text: applying

text: the

text: stability

text: criterion

text: a

text: m

text: k,c

text: k

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: -2 10 -3 10 -4 10 k z -5 10 Cε1/2 10-6 zzzzzzzz01234567 -7 10 -10 -9 -8 -7 -6 -5 -4 10 10 10 10ε 10 10 10 FIG. 2. (Color online) Critical scaling of order parameter scaling variables z with respect to k the scaled period (cid:15). The plots for z , k = 0, ...7 are shown individually in the ﬁgure, along with a k reference line representing scaling with exponent 1/2. below the critical period P . Because we investigate deviations in various quantities at c and nearby a numerically determined critical period, this requires very accurate simulation, achieved using Cash-Karp Runge-Kutta integration in long double precision variables (ac- curate to twenty decimal places on the computer system used). The steady-state loops for a given ﬁeld period P above P were determined using a shooting method, which located the c initial magnetization values m(0) resulting in the same subsequent value m(t = P ) = m(0) at the end of the ﬁeld cycle. The use of the shooting method circumvents the issue of critical slowing down occurring near the critical period, in which the convergence time to the steady- state becomes inconveniently large during normal time evolution. For the scaling variables, − we use the scaled period (cid:15) = Pc P and Pc (cid:113) | | − | | z = m 2 m 2 (5) k k k,c | | Note the scaling variable z reduces to m for even k, since m = 0 in this case. k k k,c As shown in Fig. 2, the quantities z scale with respect to the scaled period (cid:15) with critical k 6 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: k

text: z

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text: (Color

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text: Critical

text: scaling

text: of

text: order

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text: nearby

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text: achieved

text: using

text: Cash-Karp

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text: integration

text: in

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text: variables

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text: P

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text: shooting

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text: same

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text: P

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text: end

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text: The

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text: the

text: shooting

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text: circumvents

text: the

text: issue

text: of

text: critical

text: slowing

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text: occurring

text: near

text: the

text: critical

text: period,

text: in

text: which

text: the

text: convergence

text: time

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text: the

text: steady-

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text: becomes

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text: For

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text: Pc

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text: z

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text: Note

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text: As

text: shown

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text: the

text: quantities

text: z

text: scale

text: with

text: respect

text: to

text: the

text: scaled

text: period

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text: exponent 1/2, for Fourier components k = 0 through k = 7. This agrees with the scaling exponent (β = 1/2) previously determined for m in the GL model (see Eq. (23) of Ref. [4]). 0 We have explicitly conﬁrmed the scaling with exponent β = 1/2 up to index k = 40. (We are conﬁdent the scaling continues with exponent β = 1/2 beyond k = 40. However, since z decreases with k, as seen in Fig. 2, the values of z decrease below the accuracy of our k k numerical simulation past k = 40.) − Deﬁning the deviation δm = m m , it is straightforward to show that the fact that k k k,c ∼ ∼ ∼ z (cid:15)1/2 implies that δm (cid:15) for k odd, and δm (cid:15)1/2 for k even. This scaling of the k k k deviations can be conﬁrmed analytically using a perturbation of the Fourier relation in Eq. (4) for frequency ω = ω + δω, i.e., just below the critical period. We insert m = m + δm c k k,c k into Eq. (4), expand and group terms, and then subtract Eq. (4) with the critical values m . Noting to ﬁrst order δω = (cid:15)ω , the result is k,c c (cid:88) − − − (cid:88) (cid:88) 0 = i(cid:15)ω km (iω k + 2a)δm 12b m m δm − − c k,c c k n1,c n2,c k n1 n2 n1,n2 − − 12b m − − δm δm 4b δm δm δm − −(6) k n1 n2,c n1 n2 n1 n2 k n1 n2 n1,n2 n1,n2  If we assume scaling relationships of the form     c (cid:15)p, for k odd k δm = (7) k c (cid:15)q, for k even k then the scaling exponents p = 1 and q = 1/2 can be determined from Eq. (6) as follows. − Considering Eq. (6) for odd k, for example k = 1, the ﬁrst term i(cid:15)ω m is linear in (cid:15). The c 1,c rest of the terms (to lowest order (cid:88)in the deviations for even and odd k) must then be linear in (cid:15) as well, in order that the equation obtained by inserting the scaling forms in Eq (7) is independent of (cid:15). The ﬁrst sum m m δm − − involves only odd δm , since n n1,c n2,c 1 n1 n2 (cid:88) k 1 n1,n2 and n must be odd in order that the term in the sum be nonzero. This establishes that 2 the scaling exponent p = 1 for the odd terms. The second sum m − − δm δm has 1 n1 n2,c n1 n2 n1,n2 non-zero terms with n and n either both odd or both even. If n and n are both odd, 1 2 1 2 the term scales as (cid:15)2 and can be neglected. The terms with both n and n even are the 1 2 7

text: exponent

text: for

text: Fourier

text: components

text: k

text: through

text: k

text: This

text: agrees

text: with

text: the

text: scaling

text: exponent

text: (β

text: previously

text: determined

text: for

text: m

text: in

text: the

text: GL

text: model

text: (see

text: Eq.

text: of

text: Ref.

text: We

text: have

text: explicitly

text: conﬁrmed

text: the

text: scaling

text: with

text: exponent

text: β

text: up

text: to

text: index

text: k

text: (We

text: are

text: conﬁdent

text: the

text: scaling

text: continues

text: with

text: exponent

text: β

text: beyond

text: k

text: However,

text: since

text: z

text: decreases

text: with

text: k,

text: as

text: seen

text: in

text: Fig.

text: the

text: values

text: of

text: z

text: decrease

text: below

text: the

text: accuracy

text: of

text: our

text: k

text: k

text: numerical

text: simulation

text: past

text: k

text: Deﬁning

text: the

text: deviation

text: δm

text: m

text: m

text: it

text: is

text: straightforward

text: to

text: show

text: that

text: the

text: fact

text: that

text: k

text: k

text: k,c

text: z

text: implies

text: that

text: δm

text: for

text: k

text: odd,

text: and

text: δm

text: for

text: k

text: even.

text: This

text: scaling

text: of

text: the

text: k

text: k

text: k

text: deviations

text: can

text: be

text: conﬁrmed

text: analytically

text: using

text: a

text: perturbation

text: of

text: the

text: Fourier

text: relation

text: in

text: Eq.

text: for

text: frequency

text: ω

text: ω

text: δω,

text: i.e.,

text: just

text: below

text: the

text: critical

text: period.

text: We

text: insert

text: m

text: m

text: δm

text: c

text: k

text: k,c

text: k

text: into

text: Eq.

text: expand

text: and

text: group

text: terms,

text: and

text: then

text: subtract

text: Eq.

text: with

text: the

text: critical

text: values

text: m

text: Noting

text: to

text: ﬁrst

text: order

text: δω

text: the

text: result

text: is

text: k,c

text: c

text: km

text: (iω

text: k

text: m

text: m

text: δm

text: c

text: k,c

text: c

text: k

text: k

text: m

text: δm

text: δm

text: δm

text: δm

text: δm

text: k

text: k

text: If

text: we

text: assume

text: scaling

text: relationships

text: of

text: the

text: form

text: c

text: for

text: k

text: odd

text: k

text: δm

text: k

text: c

text: for

text: k

text: even

text: k

text: then

text: the

text: scaling

text: exponents

text: p

text: and

text: q

text: can

text: be

text: determined

text: from

text: Eq.

text: as

text: follows.

text: Considering

text: Eq.

text: for

text: odd

text: k,

text: for

text: example

text: k

text: the

text: ﬁrst

text: term

text: m

text: is

text: linear

text: in

text: The

text: c

text: rest

text: of

text: the

text: terms

text: (to

text: lowest

text: order

text: the

text: deviations

text: for

text: even

text: and

text: odd

text: k)

text: must

text: then

text: be

text: linear

text: in

text: as

text: well,

text: in

text: order

text: that

text: the

text: equation

text: obtained

text: by

text: inserting

text: the

text: scaling

text: forms

text: in

text: Eq

text: is

text: independent

text: of

text: The

text: ﬁrst

text: sum

text: m

text: m

text: δm

text: involves

text: only

text: odd

text: δm

text: since

text: n

text: k

text: and

text: n

text: must

text: be

text: odd

text: in

text: order

text: that

text: the

text: term

text: in

text: the

text: sum

text: be

text: nonzero.

text: This

text: establishes

text: that

text: the

text: scaling

text: exponent

text: p

text: for

text: the

text: odd

text: terms.

text: The

text: second

text: sum

text: m

text: δm

text: δm

text: has

text: non-zero

text: terms

text: with

text: n

text: and

text: n

text: either

text: both

text: odd

text: or

text: both

text: even.

text: If

text: n

text: and

text: n

text: are

text: both

text: odd,

text: the

text: term

text: scales

text: as

text: and

text: can

text: be

text: neglected.

text: The

text: terms

text: with

text: both

text: n

text: and

text: n

text: even

text: are

text: the


text: Fitting of the data All S spectra were ﬁtted to the superposition of the complex χ components of the 31 yy   two magnetic susceptibilities χPy/NM and χCoFe with [32]   µ M µ + iωα(j) + iω χ(j) = 0 eﬀ,(j) 0 − γ(j) γ(j) , (5) D(j) iω µ (M +) + iωα(j) γ 0 eﬀ,(j) γ (j) (j) where (j) =Py/NM(cid:18)or CoFe and with (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) iωα iωα − ω 2 (j) (j) D = µ (M +) + µ + , (6) 0 eﬀ,(j) 0 γ γ γ (j) (j) (j) γ (j) α = µ ∆H , (7) (j) 0 (j) 2ω and M from eﬀ (cid:112) ω = µ γ (+M ) . (8) 0 eﬀ (cid:126) Here, γ = g/ is the gyromagnetic ratio with the spectroscopic g-factor g, and Eq. (8) is the Kittel equation for in-plane geometry with ω = 2πf. Equation (7) does not take inhomogeneous broadening into account. This does not inﬂuence our results, as we recover the inhomogeneous broadening from the ﬁtted ∆H as discussed below. The ﬁts to the complex S31 data are performed by(cid:88)a Levenberg-Marquardt optimization of S () = Z eiφ(j)χ(j)() + C + C . (9) 31 (j) yy 1 2 j Here, C and C are complex oﬀset and slope, respectively. They account for background 1 2 and drift in S (). In addition to C and C , we obtain one set of ﬁt parameters Z, φ, ∆H 31 1 2 and H for each resonance. res As detailed in Ref. [33], for the ﬁt of S to Eq. (9), we set g = 2. The g-factor is 31 recovered together with M in a second step, where we ﬁt the vs. f data to (f) extracted eﬀ 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fitting

label: section

text: of

text: the

text: data

text: All

text: S

text: spectra

text: were

text: ﬁtted

text: to

text: the

text: superposition

text: of

text: the

text: complex

text: χ

text: components

text: of

text: the

text: yy

text: two

text: magnetic

text: susceptibilities

text: χPy/NM

text: and

text: χCoFe

text: with

text: µ

text: M

text: µ

text: iωα(j)

text: iω

text: χ(j)

text: eﬀ,(j)

text: γ(j)

text: γ(j)

text: D(j)

text: iω

text: µ

text: (M

text: iωα(j)

text: γ

text: eﬀ,(j)

text: γ

text: (j)

text: (j)

text: where

text: (j)

text: CoFe

text: and

text: with

text: iωα

text: iωα

text: ω

text: (j)

text: (j)

text: D

text: µ

text: (M

text: µ

text: eﬀ,(j)

text: γ

text: γ

text: γ

text: (j)

text: (j)

text: (j)

text: γ

text: (j)

text: α

text: µ

text: ∆H

text: (j)

text: (j)

text: 2ω

text: and

text: M

text: from

text: eﬀ

text: ω

text: µ

text: γ

text: (+M

text: eﬀ

text: Here,

text: γ

text: g/

text: is

text: the

text: gyromagnetic

text: ratio

text: with

text: the

text: spectroscopic

text: g-factor

text: g,

text: and

text: Eq.

text: is

text: the

text: Kittel

text: equation

text: for

text: in-plane

text: geometry

text: with

text: ω

text: Equation

text: does

text: not

text: take

text: inhomogeneous

text: broadening

text: into

text: account.

text: This

text: does

text: not

text: inﬂuence

text: our

text: results,

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text: 1 2 0 a b P y 1 0 / C u 4 / A u 2 8 6 P y 1 0 / T a 5 9 0 (cid:176) 4 P y 1 0 / P t 5 60 P y 1 0 / C u 3 / P t 3 6 0 )/30 4 P y 5 / P t 1 0 eg 8 F r e q u1 2e n c y 1( G6 H z ) 2 0 d 0 7 eg) 3 0 (f 1· (d 2 f 0 Z D Z - 3 0 C o F e f C o F e 0 6 8 1 0 1 2 1 4 1 6 1 8 2 0 6 8 1 0 1 2 1 4 1 6 1 8 2 0 F r e q u e n c y ( G H z ) F r e q u e n c y ( G H z ) Figure 7. Measured Z and φ for the CoFe tabs. a Z extracted from ﬁts to S using CoFe 31 Eq. (9). Solid symbols are for > 0 and open symbols for < 0. Fit errors are smaller than symbol size. b Corresponding φ relative to φref obtained for the Py10/Cu4/Au2 sample with > 0. CoFe CoFe The inset shows the raw φref data (green diamonds) together with a linear ﬁt (blue line). CoFe with frequency, as demonstrated for the CoFe resonance of the Py10/Cu4/Au2 sample in the inset of Fig. 7b. This is expected due to the non-zero electrical length of the signal paths √ ◦ from P1 to P3 which results in a frequency dependent phase φ = 360 fl/c with c = c / ε , 0 r ≈ the relative permittivity ε 10 of the waveguides and the speed of light in vacuum c . r 0 ◦ The linear ﬁt shown in the inset of Fig. 7b has a slope of 241 /GHz corresponding to an ≈ electrical length of l 6.4 cm. This is in agreement with the physical length of the signal path from P1 to P3. ◦ From Fig. 7b we observe that ∆φ varies by up to 60 under inversion. This is attributed to contributions of d/dt component to the FMI. The corresponding component χ of the zy ◦ magnetic susceptibility tensor is odd under inversion and 90 out-of-phase with χ . Under yy this presumption we obtain the phase of the FMI of the component of the CoFe tab as = [φ (> 0) + φ (< 0)] /2. The phase of the FMI due to the component of the Py/NM CoFe CoFe tabs is then calculated (including the eﬀect of diﬀerent electrical lengths for signals passing − ◦ ◦ from P1 through either CoFe or Py/NM tabs to P3) as = 360 fd/c + 180 . Here, d = 325 µm is the center-to-center separation of the Py/NM and CoFe tab. The additional factor ◦ of 180 is obtained because of the symmetry of the arrangement of CoFe and Py/NM tabs 13 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 3 5 3 0 C o F e 1 5 P y 1 0 / P t 5 2 5 9 G H z I m [ S 3 1 ] Py5/Pt10 2 0 R e [ S ] (mT)11 05 5 0 H ( m 1T 0 ) 0 3 1 Py10/Pt5Py1P0/yC1u03/T/Pat53 H m 0 e x D0 5 = 0.006 m 0 Py10/C u4/Au2 a 0 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 F r e q u e n c y ( G H z ) Figure 3. Increased damping due to spin pumping. Linear ﬁts (solid lines) to the line width ∆H (symbols with error bars) of the Py/NM resonance extracted from S measurements shows 31 near zero inhomogeneous broadening (intercept of the linear ﬁts) for all samples. The increase of damping (slope of the linear ﬁts) relative to the intrinsic damping α = 0.006 (dashed line) is 0 attributed entirely to spin pumping. − − ◦ of i, i.e., a 90 phase shift. Furthermore, the ratio /is estimated to be in the order × − of unity for typical Pt/Py bilayers: using == 10 nm, == 3 106 (Ωm) 1, = 800 kA/m, × − ≈ = 3.5 1019 m 2, = 1 nm and = 0.1 as typical material parameters, we ﬁnd / 0.3. Thus, reliable separation of ac iSHE and FMI signals requires phase-sensitive detection. In Fig. 2b, we plot the imaginary part of S acquired simultaneously with S . In 31 11 S , signals of similar amplitude are observed at the resonance ﬁelds of both CoFe and 31 Py for all samples. As there is no reasonable expectation for a large ac iSHE in CoFe or Py10/Cu4/Au2, we presume that both resonance signals are due to FMI solely 1 [20]. While the CoFe FMI signal is very similar for all samples and both orientations of , the shape of the Py/NM resonance changes from a peak for Py10/Ta5 and Py10/Cu4/Au2 (dashed lines) to a dip for the samples with Pt caps (solid lines) 2. This is consistent with a phase shift ◦ of approximately 180 correlated with the presence of Pt in the NM stack. The behavior observed for Py10/Ta5 and Py10/Cu4/Au2 is consistent with the presumption that the ac iSHE signal is negligible in both cases. To quantify the ac iSHE eﬀect in our devices, we begin by ﬁtting the S data to a linear 31 1 The fact that an inductive signal is observed for a single CoFe layer is attributed to the non-uniform dynamic magnetization excitation through the ﬁlm thickness. 2 The apparent phase inversion of the Py10/Ta5 and Py10/Cu4/Au2 S peaks relative to the S signal is 31 11 the result of the inverted polarity for the Py/NM tabs relative to the CoFe tab in the dCPW as indicated in Fig. 1b 5 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 4 0 0 P y 1 0 / C u 4 / A u 2 3 5 0 P y 1 0 / T a 5 3 0 0 P y 1 0 / P t 5 2 5 0 P y 1 0 / C u 3 / P t 3 ) T P y 5 / P t 1 0 2 0 0 m ( res 1 5 0 H 0 1 0 0 m 5 0 8 1 2 1 6 2 0 F r e q u e n c y ( G H z ) Figure 6. Measured and Kittel ﬁts. Measured resonance ﬁeld for the Py/NM resonances of all samples (symbols with ﬁt errors) and corresponding Kittel ﬁts according to Eq. (8) (lines). − Sample M (kA/m) g µ ∆H (mT) α (1019 m 2) eﬀ 0 0 ± ± − ± ± Py10/Cu4/Au2 730 5 2.112 0.005 0.2 0.1 0.0077 0.0001 0.88 ± ± ± ± Py10/Ta5 742 7 2.121 0.007 0.1 0.2 0.0081 0.0002 1.08 ± ± ± ± Py10/Pt5 733 5 2.115 0.005 0.6 0.1 0.0126 0.0002 3.39 ± ± ± ± Py10/Cu3/Pt3 739 2 2.110 0.002 0.2 0.1 0.0088 0.0002 1.46 ± ± − ± ± Py5/Pt10 624 9 2.128 0.01 0.1 0.3 0.0221 0.0004 4.11 Table I. Fitted parameters from the Py/NM resonance for the samples used in this study. Numbers indicate layer thickness in nm. The eﬀective spin mixing conductance was estimated using Eq. (11) assuming α = 0.006. 0 | | losses in S in this frequency range are attributed mostly to the connection of the end 21 launches to the eCPW and signal attenuation due to the ground planes of the dCPW at | | distance δ to the eCPW. The crosstalk between port 1 and 3 of the device [ S (red 31 − line)] is below 28 dB in the frequency range of interest. This is expected for the symmetric arrangement of CoFe and Py/NM layers on our dCPW that results in destructive interference of any non-ferrous inductive cross-coupling between the eCPW and dCPW. The strong suppression of ﬁeld-independent coupling between the waveguides is reﬂected in the high signal-to-noise ratio observed in S -measurement in Fig. 2 in the main text. 31 10 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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