text: D α π 4 / G- m Γ α 8 m, and the gyrodamping αΓ scale with η2. They are therefore 1.5 80 proportional to the number of spins constituting the skyrmion 60 1 consistent with our numerical ﬁndings. When one does, how- 40 0.5 ever, change the external magnetic ﬁeld for ﬁxed strength λ 20 of the DM interaction, the internal structure of the skyrmion 0 0 0.03 0.035 0.04 0.03 0.035 0.04 and therefore also G(ω) change quantitatively in a way not B B z z predictable by scaling. In Fig. 10 we therefore show the B- D G 1 20 dependence of α , m, and αΓ. For increasing B the size of the skyrmion shrinks. From our previous analysis, it is there- 15 fore not surprising that both the effective mass m and the gy- 0.5 10 D G rodamping αΓ shrink substantially while α and are less 5 affected . Quantitatively, this change of m and αΓ is how- 0 0 0.03 0.035 0.04 0.03 0.035 0.04 ever not proportional to the number of spins constituting the B B z z skyrmion. D FIG. 10: Field dependence of the dissipative tensor α , the mass m, the gyrocoupling G and the gyrodamping αΓ for J = 1, λ = 0.18J. We have tested numerically the scaling properties for η = The main effect is that both m and αΓ shrink when the size of the 1.2 and ﬁnd that all features are quantitatively reproduced. skyrmion shrinks with increasing B . Small variations on the level of a few percent do, however, z occur reﬂecting the typical size of features arising from the discretization of the continuum theory. A conservative esti- The three terms scale with η0, η and η2, respectively. To ob- mate of such systematic discretization effects for the diffusive → tain a larger skyrmion, we therefore have to rescale λ λ/η motion is given by the error bars in Figs. 3 (all statistical er- → and B B/η2. This implies that the B term in the sLLG rors are smaller than the thickness of the line). For the ﬁeld- eﬀ equation scales with 1/η2 and therefore also the time axis has driven motion (Fig. 3 and Fig. 6) spatial discretization effects → to be rescaled, t η2t, implying that all time scales are a lead to a different source of errors. For very small ﬁeld gradi- factor of η2 longer and all frequencies a factor 1/η2 smaller. ents and high frequencies the displacement of the skyrmion is Similiarly, the driving current, i.e. v is reduced by a factor much smaller than the lattice spacing and the response is af- − s η 1 while the temperature remains unscaled. This implies that fected by a tiny pinning of the skyrmion to the discreet lattice. when M(r, t) is a solution for a given value of λ, B and v For larger gradients, however, nonlinear effects set in and for s and G(ω) the corresponding velocity-correlation function of small frequencies the skyrmion starts to approach the edge of the skyrmion, then M(r/η, t/η2) is a solution for λ/η, B/η2, the simulated area. In Fig. 3, we therefore used for the force- ∇ ∇ v /η with correlation function G(ωη2). driven motion B = 0.0005 for ω < 2ω and B = 0.0015 s → p Accordingly, the ω 0 limit and therefore the gyrocou- for ω > 2ω . Error bars have been estimated from variations G D p ∇ pling , the friction constant α and the diffusion constant of of the numerical values when B was varied from 0.0001 to the skyrmion are independent of η, consistent with the analyt- 0.0015. For the current-driven motion errors are so tiny that ical formulas eq. (6). In contrast, the mass of the skyrmion, they are not shown. ∗ Electronic address: \markcomm{footercolor}{nagaosa@ of skyrmions in a multiferroic material. Science 336, 198–201 riken.jp} (2012). 1 Hubert, A. & Scha¨fer, R. Magnetic Domains: The Analysis of 10 Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nature Magnetic Microstructures (Springer, Berlin, 1998). Nanotech. 8, 152–156 (2013). 2 Malozemoff, A. P. & Slonczewski, J.C. Magnetic Domain Walls 11 Lin, S. -Z., Reichhardt, C., Batista, C. D. & Saxena, A. Driven in Bubble Materials (Academic Press, New York, 1979). skyrmions and dynamical transitions in chiral magnets. Phys. Rev. 3 Skyrme, T. H. R. A Non-Linear Field Theory. Proc. Roy. Soc. Lett. 110, 207202 (2013). London A 260, 127–138 (1961). 12 Lin, S. -Z., Reichhardt, C., Batista, C. D. & Saxena, A. Particle 4 Skyrme, T. H. R. A uniﬁed ﬁeld theory of mesons and baryons. model for skyrmions in metallic chiral magnets: Dynamics, pin- Nuc. Phys. 31, 556–569 (1962). ning, and creep. Phys. Rev. B 87, 214419 (2013). 5 Bogdanov, A. N. & Yablonskii, D. A. Thermodynamically stable 13 Iwasaki, J., Mochizuki, M. & Nagaosa, N. Current-induced ”vortices” in magnetically ordered crystals. The mixed state of skyrmion dynamics in constricted geometries. Nature Nanotech. magnets. Sov. Phys. JETP 68, 101–103 (1989). 8, 742–747 (2013). 6 Mu¨hlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 14 Iwasaki, J., Koshibae, W. & Nagaosa, N. Colossal spin transfer 323, 915–919 (2009). torque effect on skyrmion along the edge. Nano letters 14 (8), 7 Yu, X. Z. et al. Real-space observation of a two-dimensional 4432–4437 (2014). skyrmion crystal. Nature 465, 901–904 (2010). 15 Jonietz, F. et al. Spin transfer torques in MnSi at ultralow current 8 Heinze, S. et al. Spontaneous atomic-scale magnetic skyrmion lat- densities. Science 330, 1648–1651 (2010). tice in two dimensions. Nature Phys. 7, 713–718 (2011). 16 Yu, X. Z. et al. Skyrmion ﬂow near room temperature in an ul- 9 Seki, S., Yu, X. Z., Ishiwata, S. & Tokura, Y. Observation tralow current density. Nat. Commun. 3, 988 (2012). ##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##

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text: numerical

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text: η,

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text: For

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text: motion

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text: formulas

text: eq.

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text: contrast,

text: the

text: mass

text: of

text: the

text: skyrmion,

text: they

text: are

text: not

text: shown.

text: Electronic

text: address:

text: \markcomm{footercolor}{nagaosa@

text: of

text: skyrmions

text: in

text: a

text: multiferroic

text: material.

text: Science

text: riken.jp}

text: Hubert,

text: A.

text: Scha¨fer,

text: R.

text: Magnetic

text: Domains:

text: The

text: Analysis

text: of

text: Fert,

text: A.,

text: Cros,

text: V.

text: Sampaio,

text: J.

text: Skyrmions

text: on

text: the

text: track.

text: Nature

text: Magnetic

text: Microstructures

text: (Springer,

text: Berlin,

text: Nanotech.

text: Malozemoff,

text: A.

text: P.

text: Slonczewski,

text: J.C.

text: Magnetic

text: Domain

text: Walls

text: Lin,

text: S.

text: -Z.,

text: Reichhardt,

text: C.,

text: Batista,

text: C.

text: D.

text: Saxena,

text: A.

text: Driven

text: in

text: Bubble

text: Materials

text: (Academic

text: Press,

text: New

text: York,

text: skyrmions

text: and

text: dynamical

text: transitions

text: in

text: chiral

text: magnets.

text: Phys.

text: Rev.

text: Skyrme,

text: T.

text: H.

text: R.

text: A

text: Non-Linear

text: Field

text: Theory.

text: Proc.

text: Roy.

text: Soc.

text: Lett.

text: London

text: A

text: Lin,

text: S.

text: -Z.,

text: Reichhardt,

text: C.,

text: Batista,

text: C.

text: D.

text: Saxena,

text: A.

text: Particle

text: Skyrme,

text: T.

text: H.

text: R.

text: A

text: uniﬁed

text: ﬁeld

text: theory

text: of

text: mesons

text: and

text: baryons.

text: model

text: for

text: skyrmions

text: in

text: metallic

text: chiral

text: magnets:

text: Dynamics,

text: pin-

text: Nuc.

text: Phys.

text: ning,

text: and

text: creep.

text: Phys.

text: Rev.

text: B

text: Bogdanov,

text: A.

text: N.

text: Yablonskii,

text: D.

text: A.

text: Thermodynamically

text: stable

text: Iwasaki,

text: J.,

text: Mochizuki,

text: M.

text: Nagaosa,

text: N.

text: Current-induced

text: ”vortices”

text: in

text: magnetically

text: ordered

text: crystals.

text: The

text: mixed

text: state

text: of

text: skyrmion

text: dynamics

text: in

text: constricted

text: geometries.

text: Nature

text: Nanotech.

text: magnets.

text: Sov.

text: Phys.

text: JETP

text: Mu¨hlbauer,

text: S.

text: et

text: al.

text: Skyrmion

text: lattice

text: in

text: a

text: chiral

text: magnet.

text: Science

text: Iwasaki,

text: J.,

text: Koshibae,

text: W.

text: Nagaosa,

text: N.

text: Colossal

text: spin

text: transfer

text: torque

text: effect

text: on

text: skyrmion

text: along

text: the

text: edge.

text: Nano

text: letters

text: Yu,

text: X.

text: Z.

text: et

text: al.

text: Real-space

text: observation

text: of

text: a

text: two-dimensional

text: skyrmion

text: crystal.

text: Nature

text: Jonietz,

text: F.

text: et

text: al.

text: Spin

text: transfer

text: torques

text: in

text: MnSi

text: at

text: ultralow

text: current

text: Heinze,

text: S.

text: et

text: al.

text: Spontaneous

text: atomic-scale

text: magnetic

text: skyrmion

text: lat-

text: densities.

text: Science

text: tice

text: in

text: two

text: dimensions.

text: Nature

text: Phys.

text: Yu,

text: X.

text: Z.

text: et

text: al.

text: Skyrmion

text: ﬂow

text: near

text: room

text: temperature

text: in

text: an

text: ul-

text: Seki,

text: S.,

text: Yu,

text: X.

text: Z.,

text: Ishiwata,

text: S.

text: Tokura,

text: Y.

text: Observation

text: tralow

text: current

text: density.

text: Nat.

text: Commun.

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text: In superconducting phase (cid:90) (cid:90) (cid:26)(cid:16) (cid:17) (cid:27) ∞ (cid:48) − − − − − SE = d3x dr 2rg(r)e χ/2 β2e χ/2 . (2.48) bulk r h (cid:90) (cid:90) After adding SE the total on-shell action is ﬁnite and reads bdy 1/T V SE = SE + SE = dτ dx dx W = 2 W , (2.49) bulk bdy 1 2 T 0 where we consider a homogeneous and equilibrium system. Thus V is the volume of the 2 system and the length of the Euclidean time circle is identiﬁed with inverse temperature(T ). W is a thermodynamic potential per unit volume: (cid:90) (cid:16) (cid:17) ∞ − − − − − − 2 (2 3η + η ) Φ(1)Φ(2) W = G(1) β2r β2 dr e χ/2 1 1 2 . (2.50) h L2 r h (cid:90) (cid:16) (cid:17) In our case, one of Φ(1) or Φ(2) is zero and W is simpliﬁed ∞ − − − − W = G(1) β2r β2 dr e χ/2 1 . (2.51) h r h − In normal sate, where G(1) = m (cid:0)and χ = 0, (2.51)(cid:1)becomes 0 r − − − W = h 4r2 µ2 2β2 . (2.52) 4 h (cid:10) (cid:11) With the relations for energy density((cid:15)), charge density (ρ), and entropy density(s) (cid:104) (cid:105) (cid:15) = T = 2m , ρ = Jt = µr , s = 4πr2 , (2.53) tt 0 h h derived from (A.10) and the deﬁnition of the Bekenstein-Hawking entropy, the equation (2.52) becomes the expression for the grand potential(Ω) − − W = (cid:15) T s µρ = Ω , (2.54) which agrees to [20]. (cid:10) (cid:11) In superconducting phase, (cid:104) (cid:105) − (cid:15) = T = 2G(1) , ρ = Jt = A(1) , s = 4πr2 , (2.55) tt h – 15 – ##LTLine## ##LTLine## ##LTLine##

text: In

label: paragraph

text: superconducting

text: phase

label: equation

text: SE

text: dr

text: bulk

text: r

text: h

text: After

text: adding

text: SE

text: the

text: total

text: on-shell

text: action

text: is

text: ﬁnite

text: and

text: reads

text: bdy

text: V

text: SE

text: SE

text: SE

text: dτ

text: dx

text: dx

text: W

text: W

text: bulk

text: bdy

text: T

text: where

text: we

text: consider

text: a

text: homogeneous

text: and

text: equilibrium

text: system.

text: Thus

text: V

text: is

text: the

text: volume

text: of

text: the

text: system

text: and

text: the

text: length

text: of

text: the

text: Euclidean

text: time

text: circle

text: is

text: identiﬁed

text: with

text: inverse

text: temperature(T

text: W

text: is

text: a

text: thermodynamic

text: potential

text: per

text: unit

text: volume:

text: 3η

text: η

text: W

text: β2

text: dr

text: e

text: h

text: r

text: h

text: In

text: our

text: case,

text: one

text: of

text: or

text: is

text: zero

text: and

text: W

text: is

text: simpliﬁed

text: W

text: β2

text: dr

text: e

text: h

text: r

text: h

text: In

text: normal

text: sate,

text: where

text: m

text: χ

text: r

text: W

text: h

text: µ2

text: h

text: With

text: the

text: relations

text: for

text: energy

text: charge

text: density

text: (ρ),

text: and

text: entropy

text: density(s)

text: T

text: ρ

text: Jt

text: µr

text: s

text: tt

text: h

text: h

text: derived

text: from

text: and

text: the

text: deﬁnition

text: of

text: the

text: Bekenstein-Hawking

text: entropy,

text: the

text: equation

text: becomes

text: the

text: expression

text: for

text: the

text: grand

text: potential(Ω)

text: W

text: T

text: s

text: µρ

text: Ω

text: which

text: agrees

text: to

text: In

text: superconducting

text: phase,

text: T

text: ρ

text: Jt

text: s

text: tt

text: h

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:104) (cid:105) (cid:2) (cid:3) 1 (cid:90) ddk (cid:104) A (cid:105) B O S− S(2) = J−a (k) + c( 1)i(k) Jb ren 2 ≥ (2π)d k ab ac i b k ω 0 (3.25) ≡ 1 ddk J−a GR Jb , 2 ≥ (2π)d k ab k ω 0 where the range of ω is chosen to be positive following the prescription in [46]. See also [29] for a careful derivation of the gauge invariant Green’s function matrix. Notice that for one ﬁeld case without mass term, this is the well known structure of the retarded A ∼ O S Green’s function: = 0 and GR / . In summary, to compute the retarded Green’s function. We need four square matrices A B S O A B of order N(the number of ﬁelds): , , , . and can be read oﬀ from the action (3.23), S after taking care of all divergences by counter terms. To construct regular matrices, and O , we solve a set of diﬀerential equations N times with independent initial conditions. The A B · O · S− retarded Green’s function is schematically + 1. There is one subtlety in our procedure. For our equations there is one subtlety caused by a symmetry of the system. Solving (I) the equations near horizon with the expansions (3.7)-(3.9) we ﬁnd that only two of a , x χ(I), and h(I) are free. Therefore, we cannot make a complete basis to construct a general tx Ja. It is due to the gauge ﬁxing g = 0. However, it turns out that there is a residual rx gauge transformation keeping g = 0, which is generated by the vector ﬁeld ξµ of which rx − S non-vanishing component is ξx = (cid:15)e iωt. So we may add a constant vector ( a) along the 0 residual gauge orbit. S a = (0, 1, iβ/ω)T , (3.26) 0 − S since L g = iωr2ξx and L ϕ = βξx. Notice that a satisﬁes the equations of motion ξ tx ξ 0 (2.4)-(2.7), since the residual gauge transformation leaves the linearised equation of motion invariant. Therefore, our procedure is equivalent to formally adding a constant ‘solutions’ {S } of the equations to the solution set a . S O i × With the matrices and , which is numerically computed, we may construct a 3 3 × matrix of the retarded Green’s function. We will focus on the 2 2 submatrix corresponding (0) (0) (0) (0) to a and h in (3.10). Since a is dual to U(1) current J and h is dual to energy- x tx (cid:32) x (cid:33) (cid:32) (cid:33)x tx momentum tensor T tx G G GR GR 11 12 = JxJx JxTtx . (3.27) G G GR GR 21 22 TtxJx TtxTtx From the linear response theory, we have the following relation between the response functions and the sources: – 20 – ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ddk

text: A

text: B

text: O

text: S−

text: J−a

text: (k)

text: c(

text: Jb

text: ren

text: k

text: ab

text: ac

text: i

text: b

text: k

text: ω

text: ddk

text: J−a

text: GR

text: Jb

text: k

text: ab

text: k

text: ω

text: where

text: the

text: range

text: of

text: ω

text: is

text: chosen

text: to

text: be

text: positive

text: following

text: the

text: prescription

text: in

text: See

text: also

text: for

text: a

text: careful

text: derivation

text: of

text: the

text: gauge

text: invariant

text: Green’s

text: function

text: matrix.

text: Notice

text: that

text: for

text: one

text: ﬁeld

text: case

text: without

text: mass

text: term,

text: this

text: is

text: the

text: well

text: known

text: structure

text: of

text: the

text: retarded

text: A

text: O

text: S

text: Green’s

text: function:

text: and

text: GR

text: In

text: summary,

text: to

text: compute

text: the

text: retarded

text: Green’s

text: function.

text: We

text: need

text: four

text: square

text: matrices

text: A

text: B

text: S

text: O

text: A

text: B

text: of

text: order

text: N(the

text: number

text: of

text: ﬁelds):

text: and

text: can

text: be

text: read

text: oﬀ

text: from

text: the

text: action

text: S

text: after

text: taking

text: care

text: of

text: all

text: divergences

text: by

text: counter

text: terms.

text: To

text: construct

text: regular

text: matrices,

text: and

text: O

text: we

text: solve

text: a

text: set

text: of

text: diﬀerential

text: equations

text: N

text: times

text: with

text: independent

text: initial

text: conditions.

text: The

text: A

text: B

text: O

text: S−

text: retarded

text: Green’s

text: function

text: is

text: schematically

text: There

text: is

text: one

text: subtlety

text: in

text: our

text: procedure.

text: For

text: our

text: equations

text: there

text: is

text: one

text: subtlety

text: caused

text: by

text: a

text: symmetry

text: of

text: the

text: system.

text: Solving

text: (I)

text: the

text: equations

text: near

text: horizon

text: with

text: the

text: expansions

text: we

text: ﬁnd

text: that

text: only

text: two

text: of

text: a

text: x

text: χ(I),

text: and

text: h(I)

text: are

text: free.

text: Therefore,

text: we

text: cannot

text: make

text: a

text: complete

text: basis

text: to

text: construct

text: a

text: general

text: tx

text: Ja.

text: It

text: is

text: due

text: to

text: the

text: gauge

text: ﬁxing

text: g

text: However,

text: it

text: turns

text: out

text: that

text: there

text: is

text: a

text: residual

text: rx

text: gauge

text: transformation

text: keeping

text: g

text: which

text: is

text: generated

text: by

text: the

text: vector

text: ﬁeld

text: ξµ

text: of

text: which

text: rx

text: S

text: non-vanishing

text: component

text: is

text: ξx

text: iωt.

text: So

text: we

text: may

text: add

text: a

text: constant

text: vector

text: a)

text: along

text: the

text: residual

text: gauge

text: orbit.

text: S

text: a

text: iβ/ω)T

text: S

text: since

text: L

text: g

text: and

text: L

text: ϕ

text: βξx.

text: Notice

text: that

text: a

text: satisﬁes

text: the

text: equations

text: of

text: motion

text: ξ

text: tx

text: ξ

text: since

text: the

text: residual

text: gauge

text: transformation

text: leaves

text: the

text: linearised

text: equation

text: of

text: motion

text: invariant.

text: Therefore,

text: our

text: procedure

text: is

text: equivalent

text: to

text: formally

text: adding

text: a

text: constant

text: ‘solutions’

text: {S

text: of

text: the

text: equations

text: to

text: the

text: solution

text: set

text: a

text: S

text: O

text: i

text: With

text: the

text: matrices

text: and

text: which

text: is

text: numerically

text: computed,

text: we

text: may

text: construct

text: a

text: matrix

text: of

text: the

text: retarded

text: Green’s

text: function.

text: We

text: will

text: focus

text: on

text: the

text: submatrix

text: corresponding

text: to

text: a

text: and

text: h

text: in

text: Since

text: a

text: is

text: dual

text: to

text: current

text: J

text: and

text: h

text: is

text: dual

text: to

text: energy-

text: x

text: tx

text: x

text: tx

text: momentum

text: tensor

text: T

text: tx

text: G

text: G

text: GR

text: GR

text: JxJx

text: JxTtx

text: G

text: G

text: GR

text: GR

text: TtxJx

text: TtxTtx

text: From

text: the

text: linear

text: response

text: theory,

text: we

text: have

text: the

text: following

text: relation

text: between

text: the

text: response

text: functions

text: and

text: the

text: sources:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [σ] [σ] [σ] Re Re Re 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 0.5 ω/μ 0.5 ω/μ 0.5 ω/β 0.0[0σ] 1 2 3 4 5 0.0[0σ] 1 2 3 4 5 0.00[.σ0] 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Im Im Im 3 3 3 2 2 2 1 ω/μ 1 ω/μ 1 ω/β -0 1 2 3 4 5 -0 1 2 3 4 5 -0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 1 1 → ∞ (a) β/µ = 0.1, T/T = 1.52, 1, (b) β/µ = 1, T/T = 3.2, 1, 0.89, (c) β/µ (µ = 0), T/T = c c c 0.94, 0.76, 0.37 (dotted, red, 0.66, 0.27 (dotted, red, orange, 13.2, 3.5, 1, 0.95, 0.7, 0.4, 0.25 orange, green, blue) green, blue) (dashed, dotted, red, orange, green, blue, purple) ∞ Figure 4. Electric conductivity(σ) for three cases β/µ = 0.1, 1 and (or µ = 0) (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) (cid:104) (cid:105) (0) J G G a (cid:104) x (cid:105) = 11 12 x . (3.28) Ttx G21 G22 h(tx0) We want to relate these Green’s functions to the electric (σ), thermal (κ¯), thermoelec- (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) tric (α, α¯) conductivities deﬁned as (cid:104) (cid:105) J σ αT E (cid:104) x (cid:105) = − ∇ x , (3.29) Q α¯T κ¯T ( T )/T x x ∇ where Q is the heat current, E is an electric ﬁeld and T is a temperature gradient. x x x As shown in [2, 3, 23], by taking into account a diﬀeomorphism invariance, (3.29) can be (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) expressed as (cid:104) (cid:105) (0) (0) J σ αT iω(a + µh ) (cid:104) (cid:105) −x (cid:104) (cid:105) = x tx . (3.30) Ttx µ Jx α¯T κ¯T iωh(tx0) The comparison of (3.28) and (3.30) yields (cid:32) (cid:33) (cid:32) (cid:33) − − σ αT iG11 i(G11µ G12) = −ω − − ω − − . (3.31) α¯T κ¯T i(G11µ G21) i(G22 G22(ω=0)+µ( G12 G21+G11µ)) ω ω 3.3 Electric/thermal/thermoelectric conductivites Figure 4 shows examples of electric optical conductivities(σ(ω)) for three cases of β/µ: – 21 – ##LTFigure## ##LTFigure## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: [σ]

font: default

text: [σ]

text: [σ]

text: Re

text: Re

text: Re

text: ω/μ

text: ω/μ

text: ω/β

text: Im

text: Im

text: Im

text: ω/μ

text: ω/μ

text: ω/β

text: (a)

text: β/µ

text: T/T

text: (b)

text: β/µ

text: T/T

text: (c)

text: β/µ

text: (µ

text: T/T

text: c

text: c

text: c

text: (dotted,

text: red,

text: (dotted,

text: red,

text: orange,

text: orange,

text: green,

text: blue)

text: green,

text: blue)

text: (dashed,

text: dotted,

text: red,

text: orange,

text: green,

text: blue,

text: purple)

text: Figure

text: Electric

text: conductivity(σ)

text: for

text: three

text: cases

text: β/µ

text: and

text: (or

text: µ

text: J

text: G

text: G

text: a

text: x

text: x

text: Ttx

text: We

text: want

text: to

text: relate

text: these

text: Green’s

text: functions

text: to

text: the

text: electric

text: (σ),

text: thermal

text: (κ¯),

text: thermoelec-

text: tric

text: (α,

text: α¯)

text: conductivities

text: deﬁned

text: as

text: J

text: σ

text: αT

text: E

text: x

text: x

text: Q

text: α¯T

text: κ¯T

text: T

text: )/T

text: x

text: x

text: where

text: Q

text: is

text: the

text: heat

text: current,

text: E

text: is

text: an

text: electric

text: ﬁeld

text: and

text: T

text: is

text: a

text: temperature

text: gradient.

text: x

text: x

text: x

text: As

text: shown

text: in

text: by

text: taking

text: into

text: account

text: a

text: diﬀeomorphism

text: invariance,

text: can

text: be

text: expressed

text: as

text: J

text: σ

text: αT

text: iω(a

text: µh

text: −x

text: x

text: tx

text: Ttx

text: µ

text: Jx

text: α¯T

text: κ¯T

text: The

text: comparison

text: of

text: and

text: yields

text: σ

text: αT

text: −ω

text: ω

text: α¯T

text: κ¯T

text: ω

text: ω

text: Electric/thermal/thermoelectric

text: conductivites

text: Figure

text: shows

text: examples

text: of

text: electric

text: optical

text: conductivities(σ(ω))

text: for

text: three

text: cases

text: of

text: β/µ:

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: (cid:34) (cid:35) (cid:90) (cid:88) √ − d 1 − −1 S = dd+1x g (∂ψ )2 . (2.3) ψ I 2 M I=1 The action S + S yields the equations of motion for matters5 HHH ψ (cid:0) (cid:1) ∇ ∗ − ∗ F MN + iq(Φ DNΦ ΦDNΦ ) = 0 , (2.4) M − D2 m2 Φ = 0 , (2.5) ∇ 2ψ = 0 , (2.6) I ∇ − where the covariant derivative is deﬁned by D Φ = ( iqA ) Φ, and the Ein- M M M stein’s equation (cid:32) (cid:33) (cid:88) − − d 1 − 1 d(d 1) − 1 − | | − | | − 1 R g R + F 2 DΦ 2 m2 Φ 2 (∂ψ )2 MN 2 MN L2 4 2 I I=1 (2.7) 1 1 1 ∗ ∗ = ∂ ψ ∂ ψ + F F Q + (D ΦD Φ + D ΦD Φ ) , M I N I MQ N M N N M 2 2 2 Since we are mainly interested in 2 + 1 dimensional systems, we will set d = 3 from now on. In order to construct a plane(x, y)-symmetric superconducting background, we take the following ansatz, − − dr2 r2 ds2 = G(r)e χ(r)dt2 + + (dx2 + dy2) , G(r) L2 (2.8) β A = A (r)dt , Φ = Φ(r) , ψ = β xi = δ xi , t I Ii L2 Ii where non-zero A (r) is introduced for a ﬁnite chemical potential or charge density t and non-zero Φ(r) will be a hair of black hole, yielding a ﬁnite superconducting order parameter. A special form of ψ is included for momentum relaxation, where β may be I interpreted as a strength of impurity. Plugging the ansatz (2.8) into the equations of motion (2.4)-(2.7), we have four equa- (cid:18) (cid:19) tions (2.9)-(2.12). Maxwell’s equation (2.4) yields (cid:48) (cid:48)(cid:48) χ 2 (cid:48) − 2q2Φ2 A + + A A = 0 , (2.9) t 2 r t G t where Φ can be taken to be real, since r component of Maxwell’s equation implies that the phase of Φ is constant. The complex scalar ﬁeld equation (2.5) becomes · · · · · · · · · 5Index convention: M, N, = 0, 1, 2, r, and µ, ν, = 0, 1, 2, and i, j, = 1, 2. – 5 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: d

text: −1

text: S

text: g

text: (∂ψ

text: ψ

text: I

text: M

text: The

text: action

text: S

text: S

text: yields

text: the

text: equations

text: of

text: motion

text: for

text: HHH

text: ψ

text: F

text: MN

text: iq(Φ

text: DNΦ

text: ΦDNΦ

text: M

text: Φ

text: 2ψ

text: I

text: where

text: the

text: covariant

text: derivative

text: is

text: deﬁned

text: by

text: D

text: Φ

text: iqA

text: Φ,

text: and

text: the

text: Ein-

text: M

text: M

text: M

text: stein’s

text: equation

text: d

text: d(d

text: R

text: g

text: R

text: F

text: DΦ

text: Φ

text: (∂ψ

text: MN

text: MN

text: I

text: ψ

text: ψ

text: F

text: F

text: Q

text: (D

text: ΦD

text: Φ

text: D

text: ΦD

text: Φ

text: M

text: I

text: N

text: I

text: MQ

text: N

text: M

text: N

text: N

text: M

text: Since

text: we

text: are

text: mainly

text: interested

text: in

text: dimensional

text: systems,

text: we

text: will

text: set

text: d

text: from

text: now

text: on.

text: In

text: order

text: to

text: construct

text: a

text: plane(x,

text: y)-symmetric

text: superconducting

text: background,

text: we

text: take

text: the

text: following

text: ansatz,

text: G(r)e

text: G(r)

text: β

text: A

text: A

text: (r)dt

text: Φ

text: Φ(r)

text: ψ

text: β

text: xi

text: δ

text: xi

text: t

text: I

text: Ii

text: Ii

text: where

text: non-zero

text: A

text: (r)

text: is

text: introduced

text: for

text: a

text: ﬁnite

text: chemical

text: potential

text: or

text: charge

text: density

text: t

text: and

text: non-zero

text: Φ(r)

text: will

text: be

text: a

text: hair

text: of

text: black

text: hole,

text: yielding

text: a

text: ﬁnite

text: superconducting

text: order

text: parameter.

text: A

text: special

text: form

text: of

text: ψ

text: is

text: included

text: for

text: momentum

text: relaxation,

text: where

text: β

text: may

text: be

text: I

text: interpreted

text: as

text: a

text: strength

text: of

text: impurity.

text: Plugging

text: the

text: ansatz

text: into

text: the

text: equations

text: of

text: motion

text: we

text: have

text: four

text: equa-

text: tions

text: Maxwell’s

text: equation

text: yields

text: χ

text: A

text: A

text: A

text: t

text: r

text: t

text: G

text: t

text: where

text: Φ

text: can

text: be

text: taken

text: to

text: be

text: real,

text: since

text: r

text: component

text: of

text: Maxwell’s

text: equation

text: implies

text: that

text: the

text: phase

text: of

text: Φ

text: is

text: constant.

text: The

text: complex

text: scalar

text: ﬁeld

text: equation

text: becomes

text: convention:

text: M,

text: N,

text: r,

text: and

text: µ,

text: ν,

text: and

text: i,

text: j,

text: ##LTLine##

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text: ∗ Hyperﬁne splitting in muonic hydrogen constrains new pseudoscalar interactions W.-Y. Keung1,3 and D. Marfatia2,3 † ‡ and 11 We constrain the possibility of a new pseudoscalar coupling between the muon and proton using a recent measurement of the 2S hyperﬁne splitting in muonic hydrogen. − Recent measurements of 2S 2P transition frequen- cies in the exotic atom constituted by a proton orbited by a muon [1, 2] ﬁnd the proton charge radius to be 7σ smaller than the 2010-CODATA [3] value obtained using − ordinary hydrogen and e p scattering. The 2S hyperﬁne splitting deduced from the same measurements shows ex- cellent agreement with predictions [1]. The discrepancy in the proton radius has generated a lot of interest, in- cluding the invocation of new fundamental interactions as an explanation. Here, we focus on the implications of the hyperﬁne splitting for new interactions between the muon and proton. Speciﬁcally, we consider the pos- sibility of a new pseudoscalar particle that couples to the muon and proton. Such an interaction is spin and ve- FIG. 1. Pseudoscalar exchange in muonic hydrogen. locity dependent and has a negligible eﬀect on the Lamb shift (which is used to extract the proton radius) in the nonrelativistic limit [4], but has a signiﬁcant eﬀect on the ∆E and r occurs for r = 1.045 fm and δE = − HFS Z Z a hyperﬁne splitting. 0.006 meV, and The measured value of the 2S hyperﬁne splitting (HFS) [1] − ≤ ≤ 0.018 meV δE 0.006 meV at 2σ. (4) a ± ∆E = 22.8089 0.0051 meV , (1) HFS We now compute δE , and subject it to the above 2σ a constraint. In the nonrelativistic (NR) limit, the pseu- is to be compared with the theoretical prediction [5] doscalar vertex becomes · · (cid:48) ± − ± J = u¯(p(cid:48))iγ u(p) −N→R iχ(cid:48)† σ pχ − iχ(cid:48)† σ p χ , ∆Eth = (22.9843 0.0030) (0.1621 0.0010)r +δE , 5 5 2m 2m HFS Z a (2) (cid:48) − where χ and χ are 2-component Pauli spinors. The µ p interaction in terms of the muon line (given by χ , σ ) µ µ in meV, where the Zemach radius [6] and the proton line (given by χ , σ ) is then (see Fig. 1), p p (cid:48)† · − (cid:48) (cid:48)† · − (cid:48) ± J J = i χ σ (p p )χ i χ σ (P P )χ , rZ = 1.045 0.004 fm , (3) 5,µ 5,p 2mµ µ µ µ 2mp p p p → (cid:48) (cid:48) and the NR scattering amplitude for p + P p + P − is obtained from e p scattering.1 δE is the con- is a tribution to HFS from the new pseudoscalar interaction. M − (cid:48) (cid:48) − Taking the experimental and theoretical uncertainties in i = ifµJ5,µ q2−im2 ifpJ5,p , with q = p p = P p . a quadrature, the best-ﬁt to the experimentally measured The couplings of the light pseudoscalar a of mass m to a the muon and to the proton are f and f , respectively. µ p 1 The use of the value of rZ obtained from e − p scattering is Then, anpewproµp−riapteintheerraectbioencaaursiesetshaetcloororpecotridoner.to rZ from using the M = − fµfp χ(cid:48)†σ · qχ χ(cid:48)†σ · qχ 1 4mµmp µ µ µ p p p q2+m2a ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Hyperﬁne

label: title

text: splitting

text: in

text: muonic

text: hydrogen

text: constrains

text: new

text: pseudoscalar

text: interactions

text: W.-Y.

label: author

text: and

text: D.

text: and

text: We

label: abstract

text: constrain

text: the

text: possibility

text: of

text: a

text: new

text: pseudoscalar

text: coupling

text: between

text: the

text: muon

text: and

text: proton

text: using

text: a

text: recent

text: measurement

text: of

text: the

text: hyperﬁne

text: splitting

text: in

text: muonic

text: hydrogen.

text: Recent

text: measurements

text: of

text: transition

text: frequen-

text: cies

text: in

text: the

text: exotic

text: atom

text: constituted

text: by

text: a

text: proton

text: orbited

text: by

text: a

text: muon

text: ﬁnd

text: the

text: proton

text: charge

text: radius

text: to

text: be

text: 7σ

text: smaller

text: than

text: the

text: value

text: obtained

text: using

text: ordinary

text: hydrogen

text: and

text: e

text: p

text: scattering.

text: The

text: hyperﬁne

text: splitting

text: deduced

text: from

text: the

text: same

text: measurements

text: shows

text: ex-

text: cellent

text: agreement

text: with

text: predictions

text: The

text: discrepancy

text: in

text: the

text: proton

text: radius

text: has

text: generated

text: a

text: lot

text: of

text: interest,

text: in-

text: cluding

text: the

text: invocation

text: of

text: new

text: fundamental

text: interactions

text: as

text: an

text: explanation.

text: Here,

text: we

text: focus

text: on

text: the

text: implications

text: of

text: the

text: hyperﬁne

text: splitting

text: for

text: new

text: interactions

text: between

text: the

text: muon

text: and

text: proton.

text: Speciﬁcally,

text: we

text: consider

text: the

text: pos-

text: sibility

text: of

text: a

text: new

text: pseudoscalar

text: particle

text: that

text: couples

text: to

text: the

text: muon

text: and

text: proton.

text: Such

text: an

text: interaction

text: is

text: spin

text: and

text: ve-

text: FIG.

text: Pseudoscalar

text: exchange

text: in

text: muonic

text: hydrogen.

text: locity

text: dependent

text: and

text: has

text: a

text: negligible

text: eﬀect

text: on

text: the

text: Lamb

text: shift

text: (which

text: is

text: used

text: to

text: extract

text: the

text: proton

text: radius)

text: in

text: the

text: nonrelativistic

text: limit

text: but

text: has

text: a

text: signiﬁcant

text: eﬀect

text: on

text: the

text: ∆E

text: and

text: r

text: occurs

text: for

text: r

text: fm

text: and

text: δE

text: HFS

text: Z

text: Z

text: a

text: hyperﬁne

text: splitting.

text: meV,

text: and

text: The

text: measured

text: value

text: of

text: the

text: hyperﬁne

text: splitting

text: (HFS)

text: meV

text: δE

text: meV

text: at

text: a

text: ∆E

text: meV

text: HFS

text: We

text: now

text: compute

text: δE

text: and

text: subject

text: it

text: to

text: the

text: above

text: 2σ

text: a

text: constraint.

text: In

text: the

text: nonrelativistic

text: (NR)

text: limit,

text: the

text: pseu-

text: is

text: to

text: be

text: compared

text: with

text: the

text: theoretical

text: prediction

text: doscalar

text: vertex

text: becomes

text: J

text: u(p)

text: −N→R

text: σ

text: pχ

text: σ

text: p

text: χ

text: ∆Eth

text: +δE

text: HFS

text: Z

text: a

text: where

text: χ

text: and

text: χ

text: are

text: Pauli

text: spinors.

text: The

text: µ

text: p

text: interaction

text: in

text: terms

text: of

text: the

text: muon

text: line

text: (given

text: by

text: χ

text: σ

text: µ

text: µ

text: in

text: meV,

text: where

text: the

text: Zemach

text: radius

text: and

text: the

text: proton

text: line

text: (given

text: by

text: χ

text: σ

text: is

text: then

text: (see

text: Fig.

text: p

text: p

text: J

text: J

text: i

text: χ

text: σ

text: (p

text: p

text: )χ

text: i

text: χ

text: σ

text: (P

text: P

text: )χ

text: rZ

text: fm

text: µ

text: µ

text: µ

text: p

text: p

text: p

text: and

text: the

text: NR

text: scattering

text: amplitude

text: for

text: p

text: P

text: p

text: P

text: is

text: obtained

text: from

text: e

text: p

text: δE

text: is

text: the

text: con-

text: is

text: a

text: tribution

text: to

text: HFS

text: from

text: the

text: new

text: pseudoscalar

text: interaction.

text: M

text: Taking

text: the

text: experimental

text: and

text: theoretical

text: uncertainties

text: in

text: i

text: with

text: q

text: p

text: p

text: P

text: p

text: a

text: quadrature,

text: the

text: best-ﬁt

text: to

text: the

text: experimentally

text: measured

text: The

text: couplings

text: of

text: the

text: light

text: pseudoscalar

text: a

text: of

text: mass

text: m

text: to

text: a

text: the

text: muon

text: and

text: to

text: the

text: proton

text: are

text: f

text: and

text: f

text: respectively.

text: µ

text: p

text: The

text: use

text: of

text: the

text: value

text: of

text: rZ

text: obtained

text: from

text: e

text: p

text: scattering

text: is

text: Then,

text: rZ

text: from

text: using

text: the

text: M

text: fµfp

text: qχ

text: qχ

text: µ

text: µ

text: µ

text: p

text: p

text: p

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 assumptions about our event generated Monte Carlo simulation and Sec. presents the boundaries on the neutrino mass. Sec. and Sec. present the discussion about our results and our main conclusions, respectively. II. NEUTRINOS FROM SUPERNOVAE As the shock evolves against the infalling dense matter of the outer core of the star, some of its energy is lost by the photodissociation of nuclei into protons and neutrons. At this moment, there is an abundant production of − → electronic neutrinos (ν ) via p + e n + ν . These electronic neutrinos accumulate in a very dense and opaque e e − environment behind the shock wave. When the shock reaches the zone with densities around 1011g cm 3 in a few miliseconds after the bounce the ν ’s are released. This process is called neutronization burst. It has a peak luminosity ∼ × − e∼ of 3.5 1053 erg s 1 and lasts 25 miliseconds. For a recent review of the current knowledge and status of the explosion mechanism, see [22]. At a distance d of the progenitor star, for each ﬂavor ν (β = e, e¯, x), the unoscillated sprectral number ﬂuxes β (F0 (E)) are: νβ L f (E) F0 (E) = νβ (cid:104)νβ (cid:105) , (1) νβ 4πd2 E νβ (cid:104) (cid:105) where L is the luminosity for the respective ﬂavor ν , E is the ν mean energy and f (E) is the quasi-thermal νβ β νβ β νβ spectrum written as follows [23]: (cid:18) (cid:19) E αβ − (cid:104) (cid:105) fνβ(E) = χβ (cid:104)E (cid:105) e (αβ+1)E/ Eνβ . (2) (cid:82) νβ (cid:104) (cid:105) (cid:104) (cid:105) The α is a parameter deﬁned by E2 / E 2 = (2 + α )/(1 + α ) and χ is the normalization constant factor: β νβ νβ β β (cid:104) β (cid:105) (cid:104) (cid:105) dEf (E) = 1. Several of these parameters, such as L in Eq. (1), and E2 and E 2 in Eq. (2), change their νβ νβ νβ νβ respective values with time after bounce and depend on the simulation of the star explosion [24]. After taking into account the eﬀect of neutrino oscillations, in the three neutrino families framework in a supernova environment, the ﬂuxes (F ) at Earth for each ﬂavor ν are [25]: νβ β − F = pF0 + (1 p)F0 , (3) νe νe νx − F = p¯F0 + (1 p¯)F0 (4) ν¯e ν¯e νx and − − 4F = (1 p)F0 + (1 p¯)F0 + (2 + p + p¯)F0 , (5) νx νe ν¯e νx where F0 are the primary unoscillated neutrino ﬂuxes at the production region, x = µ or τ and p(p¯) is the survival νβ probability of an electron (anti)neutrino after propagation through the SN mantle and the interestellar medium. For (cid:28) the actual oscillated parameters, p = 0 for normal hierarchy (NH) (m < m m ) and p = sin2 θ for inverted (cid:28) 1 2 3 12 hierarchy (IH) (m m < m ). 3 2 1 For massive neutrinos, the arrival time (t⊕) of the neutrino in the detector at Earth will be delayed [7]: t⊕ = t + d/c + ∆t, (6) i where t is the emission time at the source and c is the speed of light. The delay introduced by the neutrino mass i m in Eq. (6) is represented by ∆t and can be written as: ν ##LTLine## ##LTLine## ##LTLine##

text: assumptions

text: about

text: our

text: event

text: generated

text: Monte

text: Carlo

text: simulation

text: and

text: Sec.

text: presents

text: the

text: boundaries

text: on

text: the

text: neutrino

text: mass.

text: Sec.

text: and

text: Sec.

text: present

text: the

text: discussion

text: about

text: our

text: results

text: and

text: our

text: main

text: conclusions,

text: respectively.

text: II.

label: section

text: NEUTRINOS

text: FROM

text: SUPERNOVAE

text: As

text: the

text: shock

text: evolves

text: against

text: the

text: infalling

text: dense

text: matter

text: of

text: the

text: outer

text: core

text: of

text: the

text: star,

text: some

text: of

text: its

text: energy

text: is

text: lost

text: by

text: the

text: photodissociation

text: of

text: nuclei

text: into

text: protons

text: and

text: neutrons.

text: At

text: this

text: moment,

text: there

text: is

text: an

text: abundant

text: production

text: of

text: electronic

text: neutrinos

text: (ν

text: via

text: p

text: e

text: n

text: ν

text: These

text: electronic

text: neutrinos

text: accumulate

text: in

text: a

text: very

text: dense

text: and

text: opaque

text: e

text: e

text: environment

text: behind

text: the

text: shock

text: wave.

text: When

text: the

text: shock

text: reaches

text: the

text: zone

text: with

text: densities

text: around

text: cm

text: in

text: a

text: few

text: miliseconds

text: after

text: the

text: bounce

text: the

text: ν

text: ’s

text: are

text: released.

text: This

text: process

text: is

text: called

text: neutronization

text: burst.

text: It

text: has

text: a

text: peak

text: luminosity

text: e∼

text: of

text: erg

text: s

text: and

text: lasts

text: miliseconds.

text: For

text: a

text: recent

text: review

text: of

text: the

text: current

text: knowledge

text: and

text: status

text: of

text: the

text: explosion

text: mechanism,

text: see

text: At

text: a

text: distance

text: d

text: of

text: the

text: progenitor

text: star,

text: for

text: each

text: ﬂavor

text: ν

text: (β

text: e,

text: e¯,

text: x),

text: the

text: unoscillated

text: sprectral

text: number

text: ﬂuxes

text: β

text: (E))

text: are:

text: νβ

text: L

text: f

text: (E)

text: (E)

text: νβ

text: νβ

text: E

text: νβ

text: where

text: L

text: is

text: the

text: luminosity

text: for

text: the

text: respective

text: ﬂavor

text: ν

text: E

text: is

text: the

text: ν

text: mean

text: energy

text: and

text: f

text: (E)

text: is

text: the

text: quasi-thermal

text: νβ

text: β

text: νβ

text: β

text: νβ

text: spectrum

text: written

text: as

text: follows

text: E

text: αβ

text: fνβ(E)

text: χβ

text: e

text: Eνβ

text: νβ

text: The

text: α

text: is

text: a

text: parameter

text: deﬁned

text: by

text: E

text: α

text: α

text: and

text: χ

text: is

text: the

text: normalization

text: constant

text: factor:

text: β

text: νβ

text: νβ

text: β

text: β

text: β

text: dEf

text: (E)

text: Several

text: of

text: these

text: parameters,

text: such

text: as

text: L

text: in

text: Eq.

text: and

text: and

text: E

text: in

text: Eq.

text: change

text: their

text: νβ

text: νβ

text: νβ

text: νβ

text: respective

text: values

text: with

text: time

text: after

text: bounce

text: and

text: depend

text: on

text: the

text: simulation

text: of

text: the

text: star

text: explosion

text: After

text: taking

text: into

text: account

text: the

text: eﬀect

text: of

text: neutrino

text: oscillations,

text: in

text: the

text: three

text: neutrino

text: families

text: framework

text: in

text: a

text: supernova

text: environment,

text: the

text: ﬂuxes

text: (F

text: at

text: Earth

text: for

text: each

text: ﬂavor

text: ν

text: are

text: νβ

text: β

text: F

text: νe

text: νe

text: νx

text: F

text: ν¯e

text: ν¯e

text: νx

text: and

text: p

text: νx

text: νe

text: ν¯e

text: νx

text: where

text: are

text: the

text: primary

text: unoscillated

text: neutrino

text: ﬂuxes

text: at

text: the

text: production

text: region,

text: x

text: µ

text: or

text: τ

text: and

text: p(p¯)

text: is

text: the

text: survival

text: νβ

text: probability

text: of

text: an

text: electron

text: (anti)neutrino

text: after

text: propagation

text: through

text: the

text: SN

text: mantle

text: and

text: the

text: interestellar

text: medium.

text: For

text: the

text: actual

text: oscillated

text: parameters,

text: p

text: for

text: normal

text: hierarchy

text: (NH)

text: (m

text: m

text: m

text: and

text: p

text: θ

text: for

text: inverted

text: hierarchy

text: (IH)

text: (m

text: m

text: m

text: For

text: massive

text: neutrinos,

text: the

text: arrival

text: time

text: (t⊕)

text: of

text: the

text: neutrino

text: in

text: the

text: detector

text: at

text: Earth

text: will

text: be

text: delayed

text: t⊕

text: t

text: d/c

text: ∆t,

text: i

text: where

text: t

text: is

text: the

text: emission

text: time

text: at

text: the

text: source

text: and

text: c

text: is

text: the

text: speed

text: of

text: light.

text: The

text: delay

text: introduced

text: by

text: the

text: neutrino

text: mass

text: i

text: m

text: in

text: Eq.

text: is

text: represented

text: by

text: ∆t

text: and

text: can

text: be

text: written

text: as:

text: ν

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ACKNOWLEDGEMENTS The authors thank Scott Aaronson, Sarah Harrison, Stephen Jordan, Aleksander Kubica, John Preskill, and Thomas Vidick for conversations. A Virginia Gilloon Fellowship, an IQIM Fellowship, a Burke Fellowship, and NSF grant PHY-0803371 supported this work. The Institute for Quantum Information and Matter (IQIM) is an NSF Physics Frontiers Center supported by the Gordon and Betty Moore Foundation. NB thanks MIT, as well as QuICS at the U. of Maryland, for their hospitality during the completion of this work. Appendix A: Classical limit Our quantum voting scheme diﬀers from classical schemes for the following reasons. Quantum voters can superpose voting tactics, like prisoners who superpose tactics in the quantum Prisoner’s Dilemma. Quantum voters can submit mixtures of classical preferences, B whereas classical voters eﬀectively submit elements. (Superpositions and mixtures may i appeal to classical readers who agonize over choosing single classical preferences on Voting Day.) Voters can entangle quantum preferences, as discussed in Sec. . Quantum constitu- tions can output superpositions and mixtures of classical preferences. Finally, preference- basis measurements replace the determinism of classical voting with probabilistic maps. Elections could be quantized in many ways. The deﬁnitions in Sections and have merit because they depart minimally from their classical analogs. They depart, furthermore, only because (i) the classical deﬁnitions lack meaning in the context of superpositions and entanglement and (ii) quantum voting is modeled by a general quantum process, which consists of a preparation procedure, an evolution, and a measurement. We deﬁne as the classical limit any context in which every ρ is an element of tje [refer- B B i emce basos . If ρ is a element, society’s classical preference is determined (is not i soc soc B probabilistic). Even in the classical limit, ρ may be a superposition of elements or soc soc may be mixed. QMR on cyclic inputs does not reduce to classical majority rule. For this reason, QMR can violate the Quantum Arrow Conjecture. Appendix B: Properties of Quantum Majority Rule QMR satisﬁes the three postulates in the Quantum Arrow Conjecture. E Lemma 1. The QMR constitution respects quantum transitivity, quantum unanimity, and QIIA. E Proof. By the deﬁnition of “quantum constitution,” outputs a ρ that collapses to a B B soc preference-basis element under a measurement of . The element corresponds to a soc soc E classical preference, which obeys classical transitivity by deﬁnition. Hence obeys quantum transitivity. Step (2f) of Deﬁnition 6 ensures that QMR respects quantum unanimity. Be- E cause constructs ρ from the pairwise preferences calculated in Step (2b), QMR respects soc QIIA. ∗ [1] J. Eisert, M. Wilkens, and M. Lewenstein, Physical Review Letters 83, 3077 (1999), quant- ph/9806088. [2] A. P. Flitney and D. Abbott, eprint arXiv:quant-ph/0208069 (2002), quant-ph/0208069. 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ACKNOWLEDGEMENTS

text: The

text: authors

text: thank

text: Scott

text: Aaronson,

text: Sarah

text: Harrison,

text: Stephen

text: Jordan,

text: Aleksander

text: Kubica,

text: John

text: Preskill,

text: and

text: Thomas

text: Vidick

text: for

text: conversations.

text: A

text: Virginia

text: Gilloon

text: Fellowship,

text: an

text: IQIM

text: Fellowship,

text: a

text: Burke

text: Fellowship,

text: and

text: NSF

text: grant

text: supported

text: this

text: work.

text: The

text: Institute

text: for

text: Quantum

text: Information

text: and

text: Matter

text: (IQIM)

text: is

text: an

text: NSF

text: Physics

text: Frontiers

text: Center

text: supported

text: by

text: the

text: Gordon

text: and

text: Betty

text: Moore

text: Foundation.

text: NB

text: thanks

text: MIT,

text: as

text: well

text: as

text: QuICS

text: at

text: the

text: U.

text: of

text: Maryland,

text: for

text: their

text: hospitality

text: during

text: the

text: completion

text: of

text: this

text: work.

text: Appendix

text: A:

text: Classical

text: limit

text: Our

text: quantum

text: voting

text: scheme

text: diﬀers

text: from

text: classical

text: schemes

text: for

text: the

text: following

text: reasons.

text: Quantum

text: voters

text: can

text: superpose

text: voting

text: tactics,

text: like

text: prisoners

text: who

text: superpose

text: tactics

text: in

text: the

text: quantum

text: Prisoner’s

text: Dilemma.

text: Quantum

text: voters

text: can

text: submit

text: mixtures

text: of

text: classical

text: preferences,

text: B

text: whereas

text: classical

text: voters

text: eﬀectively

text: submit

text: elements.

text: (Superpositions

text: and

text: mixtures

text: may

text: i

text: appeal

text: to

text: classical

text: readers

text: who

text: agonize

text: over

text: choosing

text: single

text: classical

text: preferences

text: on

text: Voting

text: Day.)

text: Voters

text: can

text: entangle

text: quantum

text: preferences,

text: as

text: discussed

text: in

text: Sec.

text: Quantum

text: constitu-

text: tions

text: can

text: output

text: superpositions

text: and

text: mixtures

text: of

text: classical

text: preferences.

text: Finally,

text: preference-

text: basis

text: measurements

text: replace

text: the

text: determinism

text: of

text: classical

text: voting

text: with

text: probabilistic

text: maps.

text: Elections

text: could

text: be

text: quantized

text: in

text: many

text: ways.

text: The

text: deﬁnitions

text: in

text: Sections

text: and

text: have

text: merit

text: because

text: they

text: depart

text: minimally

text: from

text: their

text: classical

text: analogs.

text: They

text: depart,

text: furthermore,

text: only

text: because

text: (i)

text: the

text: classical

text: deﬁnitions

text: lack

text: meaning

text: in

text: the

text: context

text: of

text: superpositions

text: and

text: entanglement

text: and

text: (ii)

text: quantum

text: voting

text: is

text: modeled

text: by

text: a

text: general

text: quantum

text: process,

text: which

text: consists

text: of

text: a

text: preparation

text: procedure,

text: an

text: evolution,

text: and

text: a

text: measurement.

text: We

text: deﬁne

text: as

text: the

text: classical

text: limit

text: any

text: context

text: in

text: which

text: every

text: ρ

text: is

text: an

text: element

text: of

text: tje

text: [refer-

text: B

text: B

text: i

text: emce

text: basos

text: If

text: ρ

text: is

text: a

text: element,

text: society’s

text: classical

text: preference

text: is

text: determined

text: (is

text: not

text: i

text: soc

text: soc

text: B

text: probabilistic).

text: Even

text: in

text: the

text: classical

text: limit,

text: ρ

text: may

text: be

text: a

text: superposition

text: of

text: elements

text: or

text: soc

text: soc

text: may

text: be

text: mixed.

text: QMR

text: on

text: cyclic

text: inputs

text: does

text: not

text: reduce

text: to

text: classical

text: majority

text: rule.

text: For

text: this

text: reason,

text: QMR

text: can

text: violate

text: the

text: Quantum

text: Arrow

text: Conjecture.

text: Appendix

text: B:

text: Properties

text: of

text: Quantum

text: Majority

text: Rule

text: QMR

text: satisﬁes

text: the

text: three

text: postulates

text: in

text: the

text: Quantum

text: Arrow

text: Conjecture.

text: E

text: Lemma

text: The

text: QMR

text: constitution

text: respects

text: quantum

text: transitivity,

text: quantum

text: unanimity,

text: and

text: QIIA.

text: E

text: Proof.

text: By

text: the

text: deﬁnition

text: of

text: “quantum

text: constitution,”

text: outputs

text: a

text: ρ

text: that

text: collapses

text: to

text: a

text: B

text: B

text: soc

text: preference-basis

text: element

text: under

text: a

text: measurement

text: of

text: The

text: element

text: corresponds

text: to

text: a

text: soc

text: soc

text: E

text: classical

text: preference,

text: which

text: obeys

text: classical

text: transitivity

text: by

text: deﬁnition.

text: Hence

text: obeys

text: quantum

text: transitivity.

text: Step

text: of

text: Deﬁnition

text: ensures

text: that

text: QMR

text: respects

text: quantum

text: unanimity.

text: Be-

text: E

text: cause

text: constructs

text: ρ

text: from

text: the

text: pairwise

text: preferences

text: calculated

text: in

text: Step

text: QMR

text: respects

text: soc

text: QIIA.

text: J.

text: Eisert,

text: M.

text: Wilkens,

text: and

text: M.

text: Lewenstein,

text: Physical

text: Review

text: Letters

text: quant-

text: A.

text: P.

text: Flitney

text: and

text: D.

text: Abbott,

text: eprint

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: θ Θ R 4 (a) (b) ent region corresponds to the region where the natural 3.5 frequency exceeds the amplitude R(x) of the complex or- 2 3 der parameter. The boundaries between the coherent and 0 incoherent oscillators are thus determined by the relation | | 2.5 Ω + ω(x) = a cos(x) + (b + ib ) sin(x) . For ω = 0.02, −2 r i ≈ − 0 the left and right boundaries are thus x 0.4026 and 2 ≈ l −2 0x 2 −2 0x 2 xr 0.2724, respectively. These predictions are in good agreement with the values measured in direct numeri- (c) (d) −2.8 cal simulation (Fig. 2(a)). Figure 3(a) shows the overall 2 frequency Ω obtained by numerical continuation of the −2.9 solution of Eqs. (16) and (17) in the parameter ω , while 0 −3 0 Figs. 3(b,c) show the corresponding results for the frac- −3.1 −2 tion e of the domain occupied by the coherent oscillators −3.2 and the extent of the (ﬁrst) region of incoherence, i.e., the −2 0 2 −2 0 2 ≤ ≤ x x interval x x x , also as functions of ω . From the l r ≈0 ﬁgure we can see a clear transition at ω 0.0063 from 0 FIG. 2. (a) A snapshot of the phase−dis|tr|ibution θ(x, t) for a single domain-ﬁlling coherent stat≤e to ≤a “splay state G(x) = cos(x) and ω(x) = 0.02 exp( 2 x ). (b) The corre- with one incoherent cluster” in x x x , followed ≈l r sponding R(x). (c) The corresponding Θ(x). (d) The oscilla- by a subsequent transition at ω 0.065 from this state tor frequency θ¯ averaged over the time interval 0 < t < 1000 0 t to a “splay state with two incoherent clusters”. These (solid line). In the coherent region θ¯ coincides with the global − t transitions occur when the proﬁles of Ω + ω(x) and R(x) oscillation frequency Ω (open circles). All simulations are touch as ω increases and these points of tangency there- done with β = 0.05 and N = 512. 0 fore correspond to the locations where coherence is ﬁrst lost. Figure 4 shows that tangencies between Ω + ω(x) ≈ − and R(x) occur when ω 0.0063 and 0.065, imply- where φ(x) is a continuous function of x with φ(π) 0 − ing that intervals of incoherent oscillators appear ﬁrst at φ( π) = 2πq. We refer to this type of state as a near- x = 0 (i.e., the bump maximum) and subsequently at splay state. As ω becomes larger an incoherent region ≈ 0 x 1.83, as ω increases. These predictions are in ex- appears in the vicinity of x = 0, with width that in- 0 cellent agreement with the direct numerical simulations creases with increasing ω (Figs. 1(b,c)). We refer to 0 shown in Fig. 1. Moreover, the critical values of ω pre- this type of state as a chimera splay state. As ω in- 0 0 dicted by the self-consistency analysis are fully consistent creases further and exceeds a second threshold, a new with the simulation results when β is increased quasi- region of incoherence is born (Fig. 1(d)). Figure 2 pro- statically (not shown). In each case we repeated the vides additional information about the partially coherent simulations for decreasing β but found no evidence of near-splay state in Fig. 1(b). The ﬁgure shows the real order parameters R(x) and Θ(x) together with θ¯ , the lo- hysteresis in these transitions. t 2. Eﬀect on stationary chimera states cal rotation frequency averaged over a long time interval (Fig. 2(d)), and reveals that in the coherent region the − oscillation frequencies are identical (with θ¯ = Ω), with Chimera states with 2n evenly distributed coherent t an abrupt but continuous change in the frequency dis- clusters are readily observed when G(x) = cos(nx) and tribution within the incoherent region. The ﬁgures also ω is a constant. These states persist when a bump is in- reveals that as ω increases the incoherent region devel- troduced into the frequency distribution ω(x). Figure 5 0 ops a stronger and stronger asymmetry with respect to shows the phase distribution and the corresponding local x = 0, the bump maximum. This is a consequence of the order parameters−R|(x|) and Θ(x) for G(x) = cos(x) when asymmetry introduced by the direction of travel, i.e., the ω(x) = 0.1 exp( 2 x ). The ﬁgure shows that the t−wo sign of the frequency Ω in Eq. (18) as discussed further clusters persist, but are now always located near x = π 2 below. and π. This is a consequence of the fact that the presence 2 These states and the transitions between them can of the bump breaks the translation invariance of the sys- be explained within the framework of the self-consistent tem. Figures 5(b,c) show th−at the −local order parameter analysis. To compute the solution branches and the tran- Z(x) has the symmetry Z˜( x) = Z˜(x). This symme- sition thresholds, we numerically continue solutions of try implies Z˜ should take the form R exp (iΘ) = b sin(x), Eqs. (16) and (17) with respect to the parameter ω . where b = b is real. The corresponding self-consistency 0 r When ω0 = 0, the splay state (with positive slope) cor- equation tak(cid:28)es the form (cid:113) (cid:29) responds to a = π, b = 0, b = π, with Ω = π. The r i ± order parameter is therefore R exp (iΘ) = π exp ( ix). We use this splay state as the starting point for continu- − − − b2e iβ = Ω + ω(y) (Ω + ω(y))2 b2 sin2 y . ation. As ω increases, a region of incoherence develops 0 in the phase pattern in the vicinity of x = 0. From the (19) point of view of the self-consistent analysis, the incoher- ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: θ

text: Θ

text: R

text: (a)

text: (b)

text: ent

text: region

text: corresponds

text: to

text: the

text: region

text: where

text: the

text: natural

text: frequency

text: exceeds

text: the

text: amplitude

text: R(x)

text: of

text: the

text: complex

text: or-

text: der

text: parameter.

text: The

text: boundaries

text: between

text: the

text: coherent

text: and

text: incoherent

text: oscillators

text: are

text: thus

text: determined

text: by

text: the

text: relation

text: Ω

text: ω(x)

text: a

text: cos(x)

text: (b

text: ib

text: sin(x)

text: For

text: ω

text: −2

text: r

text: i

text: the

text: left

text: and

text: right

text: boundaries

text: are

text: thus

text: x

text: and

text: l

text: −2

text: −2

text: xr

text: respectively.

text: These

text: predictions

text: are

text: in

text: good

text: agreement

text: with

text: the

text: values

text: measured

text: in

text: direct

text: numeri-

text: (c)

text: (d)

text: cal

text: simulation

text: (Fig.

text: Figure

text: shows

text: the

text: overall

text: frequency

text: Ω

text: obtained

text: by

text: numerical

text: continuation

text: of

text: the

text: solution

text: of

text: Eqs.

text: and

text: in

text: the

text: parameter

text: ω

text: while

text: −3

text: Figs.

text: show

text: the

text: corresponding

text: results

text: for

text: the

text: frac-

text: −2

text: tion

text: e

text: of

text: the

text: domain

text: occupied

text: by

text: the

text: coherent

text: oscillators

text: and

text: the

text: extent

text: of

text: the

text: (ﬁrst)

text: region

text: of

text: incoherence,

text: i.e.,

text: the

text: −2

text: −2

text: x

text: x

text: interval

text: x

text: x

text: x

text: also

text: as

text: functions

text: of

text: ω

text: From

text: the

text: l

text: r

text: ≈0

text: ﬁgure

text: we

text: can

text: see

text: a

text: clear

text: transition

text: at

text: ω

text: from

text: FIG.

text: (a)

text: A

text: snapshot

text: of

text: the

text: phase−dis|tr|ibution

text: θ(x,

text: t)

text: for

text: a

text: single

text: domain-ﬁlling

text: coherent

text: stat≤e

text: to

text: ≤a

text: “splay

text: state

text: G(x)

text: cos(x)

text: and

text: ω(x)

text: exp(

text: x

text: (b)

text: The

text: corre-

text: with

text: one

text: incoherent

text: cluster”

text: in

text: x

text: x

text: x

text: followed

text: ≈l

text: r

text: sponding

text: R(x).

text: (c)

text: The

text: corresponding

text: Θ(x).

text: (d)

text: The

text: oscilla-

text: by

text: a

text: subsequent

text: transition

text: at

text: ω

text: from

text: this

text: state

text: tor

text: frequency

text: θ¯

text: averaged

text: over

text: the

text: time

text: interval

text: t

text: t

text: to

text: a

text: “splay

text: state

text: with

text: two

text: incoherent

text: clusters”.

text: These

text: (solid

text: line).

text: In

text: the

text: coherent

text: region

text: θ¯

text: coincides

text: with

text: the

text: global

text: t

text: transitions

text: occur

text: when

text: the

text: proﬁles

text: of

text: Ω

text: ω(x)

text: and

text: R(x)

text: oscillation

text: frequency

text: Ω

text: (open

text: circles).

text: All

text: simulations

text: are

text: touch

text: as

text: ω

text: increases

text: and

text: these

text: points

text: of

text: tangency

text: there-

text: done

text: with

text: β

text: and

text: N

text: fore

text: correspond

text: to

text: the

text: locations

text: where

text: coherence

text: is

text: ﬁrst

text: lost.

text: Figure

text: shows

text: that

text: tangencies

text: between

text: Ω

text: ω(x)

text: and

text: R(x)

text: occur

text: when

text: ω

text: and

text: imply-

text: where

text: φ(x)

text: is

text: a

text: continuous

text: function

text: of

text: x

text: with

text: φ(π)

text: ing

text: that

text: intervals

text: of

text: incoherent

text: oscillators

text: appear

text: ﬁrst

text: at

text: φ(

text: π)

text: We

text: refer

text: to

text: this

text: type

text: of

text: state

text: as

text: a

text: near-

text: x

text: (i.e.,

text: the

text: bump

text: maximum)

text: and

text: subsequently

text: at

text: splay

text: state.

text: As

text: ω

text: becomes

text: larger

text: an

text: incoherent

text: region

text: x

text: as

text: ω

text: increases.

text: These

text: predictions

text: are

text: in

text: ex-

text: appears

text: in

text: the

text: vicinity

text: of

text: x

text: with

text: width

text: that

text: in-

text: cellent

text: agreement

text: with

text: the

text: direct

text: numerical

text: simulations

text: creases

text: with

text: increasing

text: ω

text: (Figs.

text: We

text: refer

text: to

text: shown

text: in

text: Fig.

text: Moreover,

text: the

text: critical

text: values

text: of

text: ω

text: pre-

text: this

text: type

text: of

text: state

text: as

text: a

text: chimera

text: splay

text: state.

text: As

text: ω

text: in-

text: dicted

text: by

text: the

text: self-consistency

text: analysis

text: are

text: fully

text: consistent

text: creases

text: further

text: and

text: exceeds

text: a

text: second

text: threshold,

text: a

text: new

text: with

text: the

text: simulation

text: results

text: when

text: β

text: is

text: increased

text: quasi-

text: region

text: of

text: incoherence

text: is

text: born

text: (Fig.

text: Figure

text: pro-

text: statically

text: (not

text: shown).

text: In

text: each

text: case

text: we

text: repeated

text: the

text: vides

text: additional

text: information

text: about

text: the

text: partially

text: coherent

text: simulations

text: for

text: decreasing

text: β

text: but

text: found

text: no

text: evidence

text: of

text: near-splay

text: state

text: in

text: Fig.

text: The

text: ﬁgure

text: shows

text: the

text: real

text: order

text: parameters

text: R(x)

text: and

text: Θ(x)

text: together

text: with

text: θ¯

text: the

text: lo-

text: hysteresis

text: in

text: these

text: transitions.

text: t

text: Eﬀect

text: on

text: stationary

text: chimera

text: states

text: cal

text: rotation

text: frequency

text: averaged

text: over

text: a

text: long

text: time

text: interval

text: (Fig.

text: and

text: reveals

text: that

text: in

text: the

text: coherent

text: region

text: the

text: oscillation

text: frequencies

text: are

text: identical

text: (with

text: θ¯

text: Ω),

text: with

text: Chimera

text: states

text: with

text: evenly

text: distributed

text: coherent

text: t

text: an

text: abrupt

text: but

text: continuous

text: change

text: in

text: the

text: frequency

text: dis-

text: clusters

text: are

text: readily

text: observed

text: when

text: G(x)

text: cos(nx)

text: and

text: tribution

text: within

text: the

text: incoherent

text: region.

text: The

text: ﬁgures

text: also

text: ω

text: is

text: a

text: constant.

text: These

text: states

text: persist

text: when

text: a

text: bump

text: is

text: in-

text: reveals

text: that

text: as

text: ω

text: increases

text: the

text: incoherent

text: region

text: devel-

text: troduced

text: into

text: the

text: frequency

text: distribution

text: ω(x).

text: Figure

text: ops

text: a

text: stronger

text: and

text: stronger

text: asymmetry

text: with

text: respect

text: to

text: shows

text: the

text: phase

text: distribution

text: and

text: the

text: corresponding

text: local

text: x

text: the

text: bump

text: maximum.

text: This

text: is

text: a

text: consequence

text: of

text: the

text: order

text: parameters−R|(x|)

text: and

text: Θ(x)

text: for

text: G(x)

text: cos(x)

text: when

text: asymmetry

text: introduced

text: by

text: the

text: direction

text: of

text: travel,

text: i.e.,

text: the

text: ω(x)

text: exp(

text: x

text: The

text: ﬁgure

text: shows

text: that

text: the

text: t−wo

text: sign

text: of

text: the

text: frequency

text: Ω

text: in

text: Eq.

text: as

text: discussed

text: further

text: clusters

text: persist,

text: but

text: are

text: now

text: always

text: located

text: near

text: x

text: π

text: below.

text: and

text: π.

text: This

text: is

text: a

text: consequence

text: of

text: the

text: fact

text: that

text: the

text: presence

text: These

text: states

text: and

text: the

text: transitions

text: between

text: them

text: can

text: of

text: the

text: bump

text: breaks

text: the

text: translation

text: invariance

text: of

text: the

text: sys-

text: be

text: explained

text: within

text: the

text: framework

text: of

text: the

text: self-consistent

text: tem.

text: Figures

text: show

text: th−at

text: the

text: −local

text: order

text: parameter

text: analysis.

text: To

text: compute

text: the

text: solution

text: branches

text: and

text: the

text: tran-

text: Z(x)

text: has

text: the

text: symmetry

text: Z˜(

text: x)

text: Z˜(x).

text: This

text: symme-

text: sition

text: thresholds,

text: we

text: numerically

text: continue

text: solutions

text: of

text: try

text: implies

text: Z˜

text: should

text: take

text: the

text: form

text: R

text: exp

text: (iΘ)

text: b

text: sin(x),

text: Eqs.

text: and

text: with

text: respect

text: to

text: the

text: parameter

text: ω

text: where

text: b

text: b

text: is

text: real.

text: The

text: corresponding

text: self-consistency

text: r

text: When

text: ω0

text: the

text: splay

text: state

text: (with

text: positive

text: slope)

text: cor-

text: equation

text: the

text: form

text: responds

text: to

text: a

text: π,

text: b

text: b

text: π,

text: with

text: Ω

text: π.

text: The

text: r

text: i

text: order

text: parameter

text: is

text: therefore

text: R

text: exp

text: (iΘ)

text: π

text: exp

text: ix).

text: We

text: use

text: this

text: splay

text: state

text: as

text: the

text: starting

text: point

text: for

text: continu-

text: iβ

text: Ω

text: ω(y)

text: (Ω

text: y

text: ation.

text: As

text: ω

text: increases,

text: a

text: region

text: of

text: incoherence

text: develops

text: in

text: the

text: phase

text: pattern

text: in

text: the

text: vicinity

text: of

text: x

text: From

text: the

text: point

text: of

text: view

text: of

text: the

text: self-consistent

text: analysis,

text: the

text: incoher-

text: ##LTFigure##

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text: v = (0,−1) Ki Si Fig. 4: An example for 2-fold shadow cell v q K D 1 K 2 K 3 Fig. 5: A counter example contrast, the shape of stair polygons is much more simple. Therefore, when we study a k-fold translative packing of a triangle, it seems that using the concept of stair polygons is better than using the concept of shadow cells. 2 Normal k-Fold Translative Packing R Let D be a connected subset of 2 and K = {K , K , . . .} a family of convex 1 2 disks. Suppose that K is a k-fold packing of D. We say that K is normal provided K (cid:54)= K for all i (cid:54)= j. Wh(cid:102)en K is normal and K are translates i j i of a ﬁxed convex disk K, the corresponding family is called a normal k-fold translative packing of D with K. Let M(K, k, l) be the maximum number of convex disks in a normal k-fold translative packing of lI2 with K. Then, we 6 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: v

text: Ki

text: Si

text: Fig.

text: An

label: caption

text: example

text: for

text: shadow

text: cell

text: v

text: q

text: K

text: D

text: K

text: K

text: Fig.

text: A

text: counter

text: example

text: contrast,

text: the

text: shape

text: of

text: stair

text: polygons

text: is

text: much

text: more

text: simple.

text: Therefore,

text: when

text: we

text: study

text: a

text: k-fold

text: translative

text: packing

text: of

text: a

text: triangle,

text: it

text: seems

text: that

text: using

text: the

text: concept

text: of

text: stair

text: polygons

text: is

text: better

text: than

text: using

text: the

text: concept

text: of

text: shadow

text: cells.

text: Normal

text: k-Fold

text: Translative

text: Packing

text: R

text: Let

text: D

text: be

text: a

text: connected

text: subset

text: of

text: and

text: K

text: {K

text: K

text: a

text: family

text: of

text: convex

text: disks.

text: Suppose

text: that

text: K

text: is

text: a

text: k-fold

text: packing

text: of

text: D.

text: We

text: say

text: that

text: K

text: is

text: normal

text: provided

text: K

text: K

text: for

text: all

text: i

text: j.

text: K

text: is

text: normal

text: and

text: K

text: are

text: translates

text: i

text: j

text: i

text: of

text: a

text: ﬁxed

text: convex

text: disk

text: K,

text: the

text: corresponding

text: family

text: is

text: called

text: a

text: normal

text: k-fold

text: translative

text: packing

text: of

text: D

text: with

text: K.

text: Let

text: M(K,

text: k,

text: l)

text: be

text: the

text: maximum

text: number

text: of

text: convex

text: disks

text: in

text: a

text: normal

text: k-fold

text: translative

text: packing

text: of

text: with

text: K.

text: Then,

text: we

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTLine##

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