text: 6 N. Kleeorin and I. Rogachevskii f(Ω,χ) caused by either rotation and stratiﬁcation in helical turbulence or rotation and ij (cid:20) (cid:28) (cid:21) inhomogeneous kinetic helicity for a slow rotation, Ω τ 1: 0 − (q 1) 4 ∇ ∇ f(Ω,χ) = ρ τ (cid:96)2 Ω λ + Ω λ + (Ω + Ω ) χ. (3.4) ij 2q 0 0 0 i j j i 15 i j j i (cid:29) For a fast rotation, Ω τ 1, the contribution to the Reynolds stress f(Ω,χ) caused 0 ij by either rotation and stratiﬁcation in helical turbulence or rotation and inhomogeneous kinetic helicity is given by (cid:90) (cid:104) (cid:105) ∇ f(Ω,χ) = τ L + Lλ f(0,χ) dk. (3.5) ij ijmn ijmn mn After integration in k space in Eq(cid:110). (3.5), we obtain (cid:104) (cid:105)(cid:111) − (q 1) · ∇ f(Ω,χ) = C ρ (cid:96)2 Ωˆ λ + Ωˆ λ + Ωˆ Ωˆ Ωˆ (λ + ) χ, (3.6) ij Ω 4q 0 0 i j j i i j where Ωˆ = Ω/Ω. In the derivation of Eq. (3.6) we take into account that the turbulent (cid:29) time τ for a fast rotation, Ω τ 1 is determined by Eq. (3.2). To integrate over the Ω 0 angles in k-space for a fast rotation we use the integrals given at the end of Appendix . 3.3. Formation of the mean velocity shear Let us consider the case when the angular velocity, Ω = (0, Ω, 0), is perpendicular to the density stratiﬁcation axes, λ = (λ, 0, 0). For simplicity, also consider the case when ∇ ∇ the gradient of the kinetic helicity is parallel to λ, i.e., χ = ( χ, 0, 0). In this case, f(Ω,χ)(x) is only one nonzero contribution to the Reynolds stress f(Ω,χ) caused by either xy ij rotation and stratiﬁcation in helical turbulence or rotation and inhomogeneous kinetic (cid:28) (cid:18) (cid:19) helicity for Ω τ 1: 0 − (q 1) 4 ∇ f(Ω,χ)(x) = f(Ω,χ)(x) = ρ (x) (Ωτ )(cid:96)2 λ χ + χ . (3.7) xy yx 2q 0 0 0 15 The steady-state solution of the momentum equation for the y-component of the mean (S) (cid:104) (cid:16) (cid:17) (cid:105) velocity U reads: ∇ ∇ − ρ (x) ν U(S) f(Ω,χ)(x) = 0, (3.8) x 0 T x y yx where ν = (q + 3)u (cid:96) /30 is the turbulent viscosity (Elperin et al. 2002) and we T 0 0 take into account that the gradient of the mean pressure along Ω vanishes. Integrating Eq. (3.8) over x, we determine the formed large-scale s(cid:18)hear: (cid:19) − S ≡ ∇ U(S) = fy(Ωx,χ) = 15(q 1)Ωτ2 λ χ + 4 ∇χ . (3.9) x y ρ ν 2q(q + 3) 0 15 0 T It follows from Eq. (3.9) that the large-scale shear is produced in rotating turbulence due to either inhomogeneous kinetic helicity or a combined action of a density stratiﬁed ﬂows and uniform kinetic helicity. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N.

text: Kleeorin

text: and

text: I.

text: Rogachevskii

text: f(Ω,χ)

text: caused

text: by

text: either

text: rotation

text: and

text: stratiﬁcation

text: in

text: helical

text: turbulence

text: or

text: rotation

text: and

text: ij

text: inhomogeneous

text: kinetic

text: helicity

text: for

text: a

text: slow

text: rotation,

text: Ω

text: τ

text: (q

text: f(Ω,χ)

text: ρ

text: τ

text: Ω

text: λ

text: Ω

text: λ

text: (Ω

text: Ω

text: χ.

text: ij

text: i

text: j

text: j

text: i

text: i

text: j

text: j

text: i

text: For

text: a

text: fast

text: rotation,

text: Ω

text: τ

text: the

text: contribution

text: to

text: the

text: Reynolds

text: stress

text: f(Ω,χ)

text: caused

text: ij

text: by

text: either

text: rotation

text: and

text: stratiﬁcation

text: in

text: helical

text: turbulence

text: or

text: rotation

text: and

text: inhomogeneous

text: kinetic

text: helicity

text: is

text: given

text: by

text: f(Ω,χ)

text: τ

text: L

text: Lλ

text: dk.

text: ij

text: ijmn

text: ijmn

text: mn

text: After

text: integration

text: in

text: k

text: space

text: in

text: we

text: obtain

text: (q

text: f(Ω,χ)

text: C

text: ρ

text: Ωˆ

text: λ

text: Ωˆ

text: λ

text: Ωˆ

text: Ωˆ

text: Ωˆ

text: (λ

text: χ,

text: ij

text: Ω

text: i

text: j

text: j

text: i

text: i

text: j

text: where

text: Ωˆ

text: Ω/Ω.

text: In

text: the

text: derivation

text: of

text: Eq.

text: we

text: take

text: into

text: account

text: that

text: the

text: turbulent

text: time

text: τ

text: for

text: a

text: fast

text: rotation,

text: Ω

text: τ

text: is

text: determined

text: by

text: Eq.

text: To

text: integrate

text: over

text: the

text: Ω

text: angles

text: in

text: k-space

text: for

text: a

text: fast

text: rotation

text: we

text: use

text: the

text: integrals

text: given

text: at

text: the

text: end

text: of

text: Appendix

text: Formation

text: of

text: the

text: mean

text: velocity

text: shear

text: Let

text: us

text: consider

text: the

text: case

text: when

text: the

text: angular

text: velocity,

text: Ω

text: Ω,

text: is

text: perpendicular

text: to

text: the

text: density

text: stratiﬁcation

text: axes,

text: λ

text: (λ,

text: For

text: simplicity,

text: also

text: consider

text: the

text: case

text: when

text: the

text: gradient

text: of

text: the

text: kinetic

text: helicity

text: is

text: parallel

text: to

text: λ,

text: i.e.,

text: χ

text: χ,

text: In

text: this

text: case,

text: f(Ω,χ)(x)

text: is

text: only

text: one

text: nonzero

text: contribution

text: to

text: the

text: Reynolds

text: stress

text: f(Ω,χ)

text: caused

text: by

text: either

text: xy

text: ij

text: rotation

text: and

text: stratiﬁcation

text: in

text: helical

text: turbulence

text: or

text: rotation

text: and

text: inhomogeneous

text: kinetic

text: helicity

text: for

text: Ω

text: τ

text: (q

text: f(Ω,χ)(x)

text: f(Ω,χ)(x)

text: ρ

text: (x)

text: (Ωτ

text: λ

text: χ

text: χ

text: xy

text: yx

text: The

text: steady-state

text: solution

text: of

text: the

text: momentum

text: equation

text: for

text: the

text: y-component

text: of

text: the

text: mean

text: (S)

text: velocity

text: U

text: reads:

text: ρ

text: (x)

text: ν

text: U(S)

text: f(Ω,χ)(x)

text: x

text: T

text: x

text: y

text: yx

text: where

text: ν

text: (q

text: is

text: the

text: turbulent

text: viscosity

text: (Elperin

text: et

text: al.

text: and

text: we

text: T

text: take

text: into

text: account

text: that

text: the

text: gradient

text: of

text: the

text: mean

text: pressure

text: along

text: Ω

text: vanishes.

text: Integrating

text: Eq.

text: over

text: x,

text: we

text: determine

text: the

text: formed

text: large-scale

text: S

text: U(S)

text: fy(Ωx,χ)

text: λ

text: χ

text: ∇χ

text: x

text: y

text: ρ

text: ν

text: T

text: It

text: follows

text: from

text: Eq.

text: that

text: the

text: large-scale

text: shear

text: is

text: produced

text: in

text: rotating

text: turbulence

text: due

text: to

text: either

text: inhomogeneous

text: kinetic

text: helicity

text: or

text: a

text: combined

text: action

text: of

text: a

text: density

text: stratiﬁed

text: ﬂows

text: and

text: uniform

text: kinetic

text: helicity.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 FIG. 1. Cell size at birth in simulations (a,b) and in experiments (c) as a discrete time-series, with insets showing a zoomed in view of one particular trial of simulation (b) and the data (c). (a,b) Multiple trials (diﬀerent colors) of numerical simulations − of the DSM in Eq. 9 with the simple, one-line pseudo-code: vi+1 = (2 (1 α) vi + v0) 2ξt/2, where vi denote cell size at birth in the i-th generation, ξ is a normally distributed random variable with zero mean and variance σ2, and v is a constant that t (cid:104) (cid:105) t 0 sets the mean cell size. Here, σ = 0.22 and v = 1. (a) α = 0 leads to unconﬁned diﬀusion and a divergent distribution. (b) t b Any 0 < α < 2, here α = 0.5, has the necessary feedback to achieve a stationary distribution. (c) Data from Ref. [17]. v in (cid:104) (cid:105) b this case represents cell length at birth and is normalized so that v = 1. b where x and ξ(i) respectively denote the value of x mean and scaling according to Eq. 14, as seen in experi- i t and ξ at the i-th generation. The ﬁrst term approaches ments [5, 25]. t → ∞ zero as n if 0 < α < 2. If ξ is normally distributed The Pearson correlation coeﬃcients (CC) between two t · · with variance σ2, then the variance σ2 of x will be the variables (denoted by C ( , )) can also be obtained. Since t x sum of the variances in the series in the second term. The CCs are not aﬀected by addition or multiplication by a geometric series converges for 0 < α < 2, and can readily constant, v can be replaced by x in the following calcu- b be evaluated to give the variance σ2 as lations. The CC between cell size at birth of a mother x cell and that of the daughter cell is therefore [20] λ2σ2 σ2 = −t . (13) (cid:104) (cid:48)(cid:105) − (cid:104) (cid:105) x α (2 α) (cid:48) (cid:48) xx x 2 C (v , v ) = C (x, x ) = . (15) b b σ2 x Furthermore, since x is a sum of normal variables, it n ≤ ≥ will also be normally distributed. If α 0 or α 2, the Substituting in Eq. 11, sum of the series diverges, and hence there is no station- ary distribution. The case α = 0 produces unconﬁned (cid:48) − diﬀusion and is analogous to the case where the strength C (vb, vb) = 1 α. (16) of the restoring force is zero (k = 0) in an OU process, as seen in Eq. 5. Fig. 1ab demonstrates the diﬀerence be- ≈ Importantly, the value of α 1/2 extracted via the tween time-series generated by α = 0 and by 0 < α < 2. ratio of CVs in Eq. 14 also predicts the CC between size The variance of t can be obtained similarly. d at birth and at division, in agreement with experiments It is not obvious a priori whether the widths of the dis- [5, 8, 26]. tributions of interdivision time and cell size are related. · Similarly, using Eq. 8 and 10, the interdivision time It turns out that the two CVs (denoted by CV ( )) are can be written as related by a dimensionless quantity [20]. The log-size is ≡ related to the actual size by x = ln (v /v ) ln (1 + δv ). − ≈b 0 −b − Since δv = v /v 1 is small, x δv = v /v 1. ln 2 αx b b 0 ≈ b b 0 t = + ξ . (17) Therefore, CV (vb) σx. Calculating CV (td) in a simi- d λ t lar manner leads to The CC between the interdivision times of a mother- CV (v ) ≈ √ln 2 daughter pair can then be shown to be [5, 27] b . (14) CV (td) 2α (cid:48) − C (t , t ) = α/2. (18) d d Eq. 14 allows to extract the parameter α from CVs that can be accurately measured. Since x and t are d both distributed normally, the model predicts that these That this CC is non-zero has implications for the pop- distributions can be collapsed after normalizing by the ulation growth rate, which we review later. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

label: paragraph

text: FIG.

text: Cell

label: caption

text: size

text: at

text: birth

text: in

text: simulations

text: (a,b)

text: and

text: in

text: experiments

text: (c)

text: as

text: a

text: discrete

text: time-series,

text: with

text: insets

text: showing

text: a

text: zoomed

text: in

text: view

text: of

text: one

text: particular

text: trial

text: of

text: simulation

text: (b)

text: and

text: the

text: data

text: (c).

text: (a,b)

text: Multiple

text: trials

text: (diﬀerent

text: colors)

text: of

text: numerical

text: simulations

text: of

text: the

text: DSM

text: in

text: Eq.

text: with

text: the

text: simple,

text: one-line

text: pseudo-code:

text: α)

text: vi

text: where

text: vi

text: denote

text: cell

text: size

text: at

text: birth

text: in

text: the

text: i-th

text: generation,

text: ξ

text: is

text: a

text: normally

text: distributed

text: random

text: variable

text: with

text: zero

text: mean

text: and

text: variance

text: and

text: v

text: is

text: a

text: constant

text: that

text: t

text: t

text: sets

text: the

text: mean

text: cell

text: size.

text: Here,

text: σ

text: and

text: v

text: (a)

text: α

text: leads

text: to

text: unconﬁned

text: diﬀusion

text: and

text: a

text: divergent

text: distribution.

text: (b)

text: t

text: b

text: Any

text: α

text: here

text: α

text: has

text: the

text: necessary

text: feedback

text: to

text: achieve

text: a

text: stationary

text: distribution.

text: (c)

text: Data

text: from

text: Ref.

text: v

text: in

text: b

text: this

text: case

text: represents

text: cell

text: length

text: at

text: birth

text: and

text: is

text: normalized

text: so

text: that

text: v

text: b

text: where

text: x

text: and

text: ξ(i)

text: respectively

text: denote

text: the

text: value

text: of

text: x

text: mean

text: and

text: scaling

text: according

text: to

text: Eq.

text: as

text: seen

text: in

text: experi-

text: i

text: t

text: and

text: ξ

text: at

text: the

text: i-th

text: generation.

text: The

text: ﬁrst

text: term

text: approaches

text: ments

text: t

text: zero

text: as

text: n

text: if

text: α

text: If

text: ξ

text: is

text: normally

text: distributed

text: The

text: Pearson

text: correlation

text: coeﬃcients

text: (CC)

text: between

text: two

text: t

text: with

text: variance

text: then

text: the

text: variance

text: σ2

text: of

text: x

text: will

text: be

text: the

text: variables

text: (denoted

text: by

text: C

text: can

text: also

text: be

text: obtained.

text: Since

text: t

text: x

text: sum

text: of

text: the

text: variances

text: in

text: the

text: series

text: in

text: the

text: second

text: term.

text: The

text: CCs

text: are

text: not

text: aﬀected

text: by

text: addition

text: or

text: multiplication

text: by

text: a

text: geometric

text: series

text: converges

text: for

text: α

text: and

text: can

text: readily

text: constant,

text: v

text: can

text: be

text: replaced

text: by

text: x

text: in

text: the

text: following

text: calcu-

text: b

text: be

text: evaluated

text: to

text: give

text: the

text: variance

text: σ2

text: as

text: lations.

text: The

text: CC

text: between

text: cell

text: size

text: at

text: birth

text: of

text: a

text: mother

text: x

text: cell

text: and

text: that

text: of

text: the

text: daughter

text: cell

text: is

text: therefore

text: λ2σ2

text: σ2

text: −t

text: x

text: α

text: α)

text: xx

text: x

text: C

text: (v

text: v

text: C

text: (x,

text: x

text: b

text: b

text: σ2

text: x

text: Furthermore,

text: since

text: x

text: is

text: a

text: sum

text: of

text: normal

text: variables,

text: it

text: n

text: will

text: also

text: be

text: normally

text: distributed.

text: If

text: α

text: or

text: α

text: the

text: Substituting

text: in

text: Eq.

text: sum

text: of

text: the

text: series

text: diverges,

text: and

text: hence

text: there

text: is

text: no

text: station-

text: ary

text: distribution.

text: The

text: case

text: α

text: produces

text: unconﬁned

text: diﬀusion

text: and

text: is

text: analogous

text: to

text: the

text: case

text: where

text: the

text: strength

text: C

text: (vb,

text: vb)

text: α.

text: of

text: the

text: restoring

text: force

text: is

text: zero

text: (k

text: in

text: an

text: OU

text: process,

text: as

text: seen

text: in

text: Eq.

text: Fig.

text: demonstrates

text: the

text: diﬀerence

text: be-

text: Importantly,

text: the

text: value

text: of

text: α

text: extracted

text: via

text: the

text: tween

text: time-series

text: generated

text: by

text: α

text: and

text: by

text: α

text: ratio

text: of

text: CVs

text: in

text: Eq.

text: also

text: predicts

text: the

text: CC

text: between

text: size

text: The

text: variance

text: of

text: t

text: can

text: be

text: obtained

text: similarly.

text: d

text: at

text: birth

text: and

text: at

text: division,

text: in

text: agreement

text: with

text: experiments

text: It

text: is

text: not

text: obvious

text: a

text: priori

text: whether

text: the

text: widths

text: of

text: the

text: dis-

text: tributions

text: of

text: interdivision

text: time

text: and

text: cell

text: size

text: are

text: related.

text: Similarly,

text: using

text: Eq.

text: and

text: the

text: interdivision

text: time

text: It

text: turns

text: out

text: that

text: the

text: two

text: CVs

text: (denoted

text: by

text: CV

text: are

text: can

text: be

text: written

text: as

text: related

text: by

text: a

text: dimensionless

text: quantity

text: The

text: log-size

text: is

text: related

text: to

text: the

text: actual

text: size

text: by

text: x

text: ln

text: (v

text: /v

text: ln

text: δv

text: ≈b

text: −b

text: Since

text: δv

text: v

text: /v

text: is

text: small,

text: x

text: δv

text: v

text: /v

text: ln

text: αx

text: b

text: b

text: b

text: b

text: t

text: ξ

text: Therefore,

text: CV

text: (vb)

text: σx.

text: Calculating

text: CV

text: (td)

text: in

text: a

text: simi-

text: d

text: λ

text: t

text: lar

text: manner

text: leads

text: to

text: The

text: CC

text: between

text: the

text: interdivision

text: times

text: of

text: a

text: mother-

text: CV

text: (v

text: √ln

text: daughter

text: pair

text: can

text: then

text: be

text: shown

text: to

text: be

text: b

text: CV

text: (td)

text: 2α

text: C

text: (t

text: t

text: d

text: d

text: Eq.

text: allows

text: to

text: extract

text: the

text: parameter

text: α

text: from

text: CVs

text: that

text: can

text: be

text: accurately

text: measured.

text: Since

text: x

text: and

text: t

text: are

text: d

text: both

text: distributed

text: normally,

text: the

text: model

text: predicts

text: that

text: these

text: That

text: this

text: CC

text: is

text: non-zero

text: has

text: implications

text: for

text: the

text: pop-

text: distributions

text: can

text: be

text: collapsed

text: after

text: normalizing

text: by

text: the

text: ulation

text: growth

text: rate,

text: which

text: we

text: review

text: later.

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text: 9 As an example, we review below a model of cell cycle in the same spirit as the problem of cell size regulation regulation in E. coli, whose cell divisions appear to follow [31, 48–50]. In fact, for a constitutively expressed pro- a constant time T after the initiation of DNA replication tein, the distributions of protein numbers at birth can for a broad range of mean growth rates [2, 47]. The time be described by a DSM [49]. Analysis analogous to that T can be larger than the mean doubling time τ, in which in Fig. 2, but for the copy number of a constitutively case the cells maintain multiple ongoing rounds of DNA expressed protein in E. coli in the same data set [17], replication. The tight coupling between initiation and reveals that the partial CC is also zero in this case. How- division implies that the cell size at birth v is ever, it is unclear how protein number and cell size are d simultaneously regulated. One way to investigate this question is via a multi- vd = vieλ(T+ξT), (34) dimensional, or vector, AR model. An AR(1) vector model in M dimensions can be written as where v is the cell size at initiation, λ = ln 2/τ is the i (cid:48) growth rate, and ξ describes ﬂuctuations with magni- (cid:126)x = A(cid:126)x +(cid:126)b + ξ(cid:126), (36) T tude σ in the time between initiation and division. At T a coarse-grained level, the initiator accumulation model where (cid:126)x is a vector of the abundances at birth of the can be described as [11, 20, 40] M cellular components, which can include cell size, and (cid:48) × (cid:126)x is the vector at the next generation. A is a M M (cid:48) v˜ = (v + Ov ) eλξt, (35) matrix representing the regulatory interactions between i i 0 components, (cid:126)b is a vector representing the basal syn- thesis level between generations, and ξ(cid:126) is a vector of (cid:48) where v˜ is the total cell size of the daughter cells (typ- noise terms uncorrelated between generations but may i ically two) at the next initiation, O is the number of ori- be cross-correlated at the same generation. gins of replication (i.e. the site along the chromosome at For the one-dimensional case, the condition for station- which DNA replication initiates), and v is a constant. arity is that 2 > α > 0 so that the variance of x in Eq. 0 As in the division-centric model, regulation is subject to 13 is ﬁnite. This condition is equivalent to that the zero − − a time-additive noise ξ with magnitude σ . of 1 (1 α) z lie outside the unit circle. In the multi- t t Analysis and simulations of the initiation-centric model dimensional case, the condition is similarly that all the − of Eqs. 34-35 show that it produces emergent adder cor- zeros of det (I Az) lie outside the unit circle [28]. Given relations at division [13, 40], as long as the magnitude a stable AR(1) vector model, the multi-dimensional ana- of the ﬂuctuations in the coordination between initia- logue of the Yule-Walker equations can be used to es- tion and division is much less than that in the control timate by maximum likelihood the regulatory matrix A (cid:29) of initiation (σ σ ). This is indeed the case in ex- from measurem(cid:18)ents [51]. For the data set disc(cid:19)ussed above t T periments for fast-growing bacteria, although the picture [17], this method results in appears diﬀerent for slow-growing bacteria [21], which ± − ± 0.50 0.02 0.02 0.01 we discuss later. The model also generates cell size and A = − ± ± , 0.16 0.02 0.60 0.02 interdivision time distributions whose CVs only depend on the magnitudes λσ and λσ of the ﬂuctuations, and where the ﬁrst and second components are respectively t T the regulation strength α. The distributions therefore cell size and protein number at birth (both normalized collapse after scaling by the mean if these parameters are by their means), and plus-minus shows the standard error constant across growth conditions. At the bulk-level, the in the estimate. This novel result suggests that the copy initiation-centric model produces the observed exponen- number of this constitutively expressed protein does not tial scaling of mean cell size with mean growth rate, as aﬀect cell size regulation, while cell size does aﬀect the discussed in the Introduction, without requiring param- regulation of this protein number. It is unknown whether eters to depend on mean growth rate [40]. These results, this result holds for all constitutively expressed proteins, together with previous results regarding the universality and how this result will change for proteins that are not of cell size distributions, suggest that the initiator accu- constitutively expressed. mulation model may be a robust molecular mechanism To better understand cross-correlations between cell that produces adder correlations, and that models of size and protein numbers from a mechanistic perspective, cell cycle regulation can continue to shed light on the recent works have investigated a dynamical model in the underlying biology. form d(cid:126)x/dt = A(cid:126)x, where (cid:126)x is now the abundances of Regulation of protein numbers the cellular components during the cell cycle, and A now describes the regulatory interactions in time [50]. This Recent works have begun investigating the statistics model leads to the components growing as a sum of expo- of the copy numbers of proteins at the single-cell level nentials that can be approximated as a single exponential

text: As

text: an

text: example,

text: we

text: review

text: below

text: a

text: model

text: of

text: cell

text: cycle

text: in

text: the

text: same

text: spirit

text: as

text: the

text: problem

text: of

text: cell

text: size

text: regulation

text: regulation

text: in

text: E.

text: coli,

text: whose

text: cell

text: divisions

text: appear

text: to

text: follow

text: In

text: fact,

text: for

text: a

text: constitutively

text: expressed

text: pro-

text: a

text: constant

text: time

text: T

text: after

text: the

text: initiation

text: of

text: DNA

text: replication

text: tein,

text: the

text: distributions

text: of

text: protein

text: numbers

text: at

text: birth

text: can

text: for

text: a

text: broad

text: range

text: of

text: mean

text: growth

text: rates

text: The

text: time

text: be

text: described

text: by

text: a

text: DSM

text: Analysis

text: analogous

text: to

text: that

text: T

text: can

text: be

text: larger

text: than

text: the

text: mean

text: doubling

text: time

text: τ,

text: in

text: which

text: in

text: Fig.

text: but

text: for

text: the

text: copy

text: number

text: of

text: a

text: constitutively

text: case

text: the

text: cells

text: maintain

text: multiple

text: ongoing

text: rounds

text: of

text: DNA

text: expressed

text: protein

text: in

text: E.

text: coli

text: in

text: the

text: same

text: data

text: set

text: replication.

text: The

text: tight

text: coupling

text: between

text: initiation

text: and

text: reveals

text: that

text: the

text: partial

text: CC

text: is

text: also

text: zero

text: in

text: this

text: case.

text: How-

text: division

text: implies

text: that

text: the

text: cell

text: size

text: at

text: birth

text: v

text: is

text: ever,

text: it

text: is

text: unclear

text: how

text: protein

text: number

text: and

text: cell

text: size

text: are

text: d

text: simultaneously

text: regulated.

text: One

text: way

text: to

text: investigate

text: this

text: question

text: is

text: via

text: a

text: multi-

text: vd

text: vieλ(T+ξT),

text: dimensional,

text: or

text: vector,

text: AR

text: model.

text: An

text: vector

text: model

text: in

text: M

text: dimensions

text: can

text: be

text: written

text: as

text: where

text: v

text: is

text: the

text: cell

text: size

text: at

text: initiation,

text: λ

text: ln

text: is

text: the

text: i

text: growth

text: rate,

text: and

text: ξ

text: describes

text: ﬂuctuations

text: with

text: magni-

text: T

text: tude

text: σ

text: in

text: the

text: time

text: between

text: initiation

text: and

text: division.

text: At

text: T

text: a

text: coarse-grained

text: level,

text: the

text: initiator

text: accumulation

text: model

text: where

text: is

text: a

text: vector

text: of

text: the

text: abundances

text: at

text: birth

text: of

text: the

text: can

text: be

text: described

text: as

text: M

text: cellular

text: components,

text: which

text: can

text: include

text: cell

text: size,

text: and

text: is

text: the

text: vector

text: at

text: the

text: next

text: generation.

text: A

text: is

text: a

text: M

text: M

text: v˜

text: (v

text: Ov

text: eλξt,

text: matrix

text: representing

text: the

text: regulatory

text: interactions

text: between

text: i

text: i

text: components,

text: is

text: a

text: vector

text: representing

text: the

text: basal

text: syn-

text: thesis

text: level

text: between

text: generations,

text: and

text: is

text: a

text: vector

text: of

text: where

text: v˜

text: is

text: the

text: total

text: cell

text: size

text: of

text: the

text: daughter

text: cells

text: (typ-

text: noise

text: terms

text: uncorrelated

text: between

text: generations

text: but

text: may

text: i

text: ically

text: two)

text: at

text: the

text: next

text: initiation,

text: O

text: is

text: the

text: number

text: of

text: ori-

text: be

text: cross-correlated

text: at

text: the

text: same

text: generation.

text: gins

text: of

text: replication

text: (i.e.

text: the

text: site

text: along

text: the

text: chromosome

text: at

text: For

text: the

text: one-dimensional

text: case,

text: the

text: condition

text: for

text: station-

text: which

text: DNA

text: replication

text: initiates),

text: and

text: v

text: is

text: a

text: constant.

text: arity

text: is

text: that

text: α

text: so

text: that

text: the

text: variance

text: of

text: x

text: in

text: Eq.

text: As

text: in

text: the

text: division-centric

text: model,

text: regulation

text: is

text: subject

text: to

text: is

text: ﬁnite.

text: This

text: condition

text: is

text: equivalent

text: to

text: that

text: the

text: zero

text: a

text: time-additive

text: noise

text: ξ

text: with

text: magnitude

text: σ

text: of

text: α)

text: z

text: lie

text: outside

text: the

text: unit

text: circle.

text: In

text: the

text: multi-

text: t

text: t

text: Analysis

text: and

text: simulations

text: of

text: the

text: initiation-centric

text: model

text: dimensional

text: case,

text: the

text: condition

text: is

text: similarly

text: that

text: all

text: the

text: of

text: Eqs.

text: show

text: that

text: it

text: produces

text: emergent

text: adder

text: cor-

text: zeros

text: of

text: det

text: (I

text: Az)

text: lie

text: outside

text: the

text: unit

text: circle

text: Given

text: relations

text: at

text: division

text: as

text: long

text: as

text: the

text: magnitude

text: a

text: stable

text: vector

text: model,

text: the

text: multi-dimensional

text: ana-

text: of

text: the

text: ﬂuctuations

text: in

text: the

text: coordination

text: between

text: initia-

text: logue

text: of

text: the

text: Yule-Walker

text: equations

text: can

text: be

text: used

text: to

text: es-

text: tion

text: and

text: division

text: is

text: much

text: less

text: than

text: that

text: in

text: the

text: control

text: timate

text: by

text: maximum

text: likelihood

text: the

text: regulatory

text: matrix

text: A

text: of

text: initiation

text: (σ

text: σ

text: This

text: is

text: indeed

text: the

text: case

text: in

text: ex-

text: from

text: For

text: the

text: data

text: set

text: above

text: t

text: T

text: periments

text: for

text: fast-growing

text: bacteria,

text: although

text: the

text: picture

text: this

text: method

text: results

text: in

text: appears

text: diﬀerent

text: for

text: slow-growing

text: bacteria

text: which

text: we

text: discuss

text: later.

text: The

text: model

text: also

text: generates

text: cell

text: size

text: and

text: A

text: interdivision

text: time

text: distributions

text: whose

text: CVs

text: only

text: depend

text: on

text: the

text: magnitudes

text: λσ

text: and

text: λσ

text: of

text: the

text: ﬂuctuations,

text: and

text: where

text: the

text: ﬁrst

text: and

text: second

text: components

text: are

text: respectively

text: t

text: T

text: the

text: regulation

text: strength

text: α.

text: The

text: distributions

text: therefore

text: cell

text: size

text: and

text: protein

text: number

text: at

text: birth

text: (both

text: normalized

text: collapse

text: after

text: scaling

text: by

text: the

text: mean

text: if

text: these

text: parameters

text: are

text: by

text: their

text: means),

text: and

text: plus-minus

text: shows

text: the

text: standard

text: error

text: constant

text: across

text: growth

text: conditions.

text: At

text: the

text: bulk-level,

text: the

text: in

text: the

text: estimate.

text: This

text: novel

text: result

text: suggests

text: that

text: the

text: copy

text: initiation-centric

text: model

text: produces

text: the

text: observed

text: exponen-

text: number

text: of

text: this

text: constitutively

text: expressed

text: protein

text: does

text: not

text: tial

text: scaling

text: of

text: mean

text: cell

text: size

text: with

text: mean

text: growth

text: rate,

text: as

text: aﬀect

text: cell

text: size

text: regulation,

text: while

text: cell

text: size

text: does

text: aﬀect

text: the

text: discussed

text: in

text: the

text: Introduction,

text: without

text: requiring

text: param-

text: regulation

text: of

text: this

text: protein

text: number.

text: It

text: is

text: unknown

text: whether

text: eters

text: to

text: depend

text: on

text: mean

text: growth

text: rate

text: These

text: results,

text: this

text: result

text: holds

text: for

text: all

text: constitutively

text: expressed

text: proteins,

text: together

text: with

text: previous

text: results

text: regarding

text: the

text: universality

text: and

text: how

text: this

text: result

text: will

text: change

text: for

text: proteins

text: that

text: are

text: not

text: of

text: cell

text: size

text: distributions,

text: suggest

text: that

text: the

text: initiator

text: accu-

text: constitutively

text: expressed.

text: mulation

text: model

text: may

text: be

text: a

text: robust

text: molecular

text: mechanism

text: To

text: better

text: understand

text: cross-correlations

text: between

text: cell

text: that

text: produces

text: adder

text: correlations,

text: and

text: that

text: models

text: of

text: size

text: and

text: protein

text: numbers

text: from

text: a

text: mechanistic

text: perspective,

text: cell

text: cycle

text: regulation

text: can

text: continue

text: to

text: shed

text: light

text: on

text: the

text: recent

text: works

text: have

text: investigated

text: a

text: dynamical

text: model

text: in

text: the

text: underlying

text: biology.

text: form

text: where

text: is

text: now

text: the

text: abundances

text: of

text: Regulation

text: of

text: protein

text: numbers

text: the

text: cellular

text: components

text: during

text: the

text: cell

text: cycle,

text: and

text: A

text: now

text: describes

text: the

text: regulatory

text: interactions

text: in

text: time

text: This

text: Recent

text: works

text: have

text: begun

text: investigating

text: the

text: statistics

text: model

text: leads

text: to

text: the

text: components

text: growing

text: as

text: a

text: sum

text: of

text: expo-

text: of

text: the

text: copy

text: numbers

text: of

text: proteins

text: at

text: the

text: single-cell

text: level

text: nentials

text: that

text: can

text: be

text: approximated

text: as

text: a

text: single

text: exponential


text: MNRAS 000, 1–16 (2017) Preprint 30 May 2020 Compiled using MNRAS LATEX style ﬁle v3.0 The sdA problem I: Physical Properties Ingrid Pelisoli1,2(cid:63), S. O. Kepler1, D. Koester3, 1Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, 91501-900 Porto-Alegre, RS, Brazil 2Department of Physics, University of Warwick, Coventry CV4 7AL, UK 3Institut fu¨r Theoretische Physik und Astrophysik, Universit¨at Kiel, D-24098 Kiel, Germany Accepted XXX. Received YYY; in original form ZZZ ABSTRACT The so-called sdA stars are deﬁned by having H-rich spectra and surface gravities similar to hot subdwarf stars, but eﬀective temperature below the zero-age horizontal branch. Their evolutionary history is an enigma: their surface gravity is too high for main sequence stars, but too low for single evolution white dwarfs. They are most likely byproducts of binary evolution, including blue-stragglers, extremely-low mass white dwarf stars (ELMs) and their precursors (pre-ELMs). A small number of ELMs with similar properties to sdAs is known. Other possibilities include metal-poor A/F dwarfs, second generation stars, or even stars accreted from dwarf galaxies. In this work, we analyse colours, proper motions and spacial velocities of a sample of sdAs from the Sloan Digital Sky Survey to assess their nature and evolutionary origin. We deﬁne a probability of belonging to the main sequence and a probability of being a (pre-)ELM based on these properties. We ﬁnd that 7 per cent of the sdAs are more likely to be (pre-)ELMs than main sequence stars. However, the spacial velocity distribution suggests that over 35 per cent of them cannot be explained as single metal-poor A/F stars. Key words: subdwarfs – white dwarfs – binaries: general – stars: evolution – stars: kinematics and dynamics 1 INTRODUCTION et al. 2016) or as single stars resulting from merger events (Brown et al. 2016b). Of remarkable interest are the The evolution of single stars is a fairly well under- (cid:46) (cid:12) extremely-low mass white dwarfs (ELMs, M 0.20 M , stood process. There are of course many uncertainties con- see e.g. the ELM Survey: Brown et al. 2010; Kilic et al. 2011; cerning speciﬁc phases (e.g. the asymptotic giant branch, Brown et al. 2012; Kilic et al. 2012b; Brown et al. 2013; Gi- Miller Bertolami 2016) and processes (e.g. convection, Bres- anninas et al. 2015; Brown et al. 2016a), which can only be san et al. 2013; Tremblay et al. 2013) throughout the evolu- formed in interacting binary systems within a Hubble time. tion, but relating an initial mass and metallicity to a ﬁnal The evolution of single stars can lead to white dwarfs with outcome can be done with reasonable precision. Only a very masses down to 0.30–0.45 solar masses (Kilic et al. 2007), small amount of objects, much less than 5 per cent of all but main-sequence progenitors that would end up as lower stars in the Galaxy, will end their lives as neutron stars or mass white dwarfs have main sequence lifetimes exceeding black holes — those with initial masses larger than about (cid:12) (cid:12) the age of the Universe. Objects with 0.30 < M < 0.45 M 7–10.6 M (e.g. Woosley & Heger 2015). The remaining are usually referred to as low-mass white dwarfs. Their bi- over 95 per cent stars will become white dwarf stars, whose nary fraction is still high, about 30 per cent (Brown et al. evolution can be approximated by a slow cooling process. 2011), because some form of enhanced mass-loss is needed Hence, as they are not only abundant but also long-lived, to form them, and binarity is the easiest way to achieve this. white dwarfs are very useful in obtaining information of all Severe mass loss in the ﬁrst ascent giant branch, attributed galactic populations (Isern et al. 2001; Liebert et al. 2005; to high metallicity (Hansen 2005), can also lead to single low Bono et al. 2013; Tremblay et al. 2016; Kilic et al. 2017; mass white dwarfs (Kilic et al. 2007). The binary fraction Cojocaru et al. 2017). − (cid:12) below 0.2 0.3 M , on the other hand, could be up to 100 White dwarf stars are also one possible result of binary per cent (Brown et al. 2016a), encompassing not only the evolution, both in binary systems (e.g. Rebassa-Mansergas ELMs but also the pre-ELMs, which still have not reached the white dwarf cooling track (Maxted et al. 2014). (cid:63) E-mail: ingrid.pelisoli@ufrgs.br Less than a hundred ELMs are known to date, making (cid:13) c 2017 The Authors

text: MNRAS

text: Preprint

text: May

text: Compiled

text: using

text: MNRAS

text: LATEX

text: style

text: ﬁle

text: The

label: title

text: sdA

text: problem

text: I:

text: Physical

text: Properties

text: Ingrid

label: author

text: S.

text: O.

text: D.

text: de

text: F´ısica,

text: Universidade

text: Federal

text: do

text: Rio

text: Grande

text: do

text: Sul,

text: Porto-Alegre,

text: RS,

text: Brazil

text: of

text: Physics,

text: University

text: of

text: Warwick,

text: Coventry

text: UK

text: fu¨r

text: Theoretische

text: Physik

text: und

text: Astrophysik,

text: Universit¨at

text: Kiel,

text: Kiel,

text: Germany

text: Accepted

label: date

text: XXX.

text: Received

text: YYY;

text: in

text: original

text: form

text: ZZZ

text: ABSTRACT

text: The

label: abstract

text: so-called

text: sdA

text: stars

text: are

text: deﬁned

text: by

text: having

text: H-rich

text: spectra

text: and

text: surface

text: gravities

text: similar

text: to

text: hot

text: subdwarf

text: stars,

text: but

text: eﬀective

text: temperature

text: below

text: the

text: zero-age

text: horizontal

text: branch.

text: Their

text: evolutionary

text: history

text: is

text: an

text: enigma:

text: their

text: surface

text: gravity

text: is

text: too

text: high

text: for

text: main

text: sequence

text: stars,

text: but

text: too

text: low

text: for

text: single

text: evolution

text: white

text: dwarfs.

text: They

text: are

text: most

text: likely

text: byproducts

text: of

text: binary

text: evolution,

text: including

text: blue-stragglers,

text: extremely-low

text: mass

text: white

text: dwarf

text: stars

text: (ELMs)

text: and

text: their

text: precursors

text: (pre-ELMs).

text: A

text: small

text: number

text: of

text: ELMs

text: with

text: similar

text: properties

text: to

text: sdAs

text: is

text: known.

text: Other

text: possibilities

text: include

text: metal-poor

text: A/F

text: dwarfs,

text: second

text: generation

text: stars,

text: or

text: even

text: stars

text: accreted

text: from

text: dwarf

text: galaxies.

text: In

text: this

text: work,

text: we

text: analyse

text: colours,

text: proper

text: motions

text: and

text: spacial

text: velocities

text: of

text: a

text: sample

text: of

text: sdAs

text: from

text: the

text: Sloan

text: Digital

text: Sky

text: Survey

text: to

text: assess

text: their

text: nature

text: and

text: evolutionary

text: origin.

text: We

text: deﬁne

text: a

text: probability

text: of

text: belonging

text: to

text: the

text: main

text: sequence

text: and

text: a

text: probability

text: of

text: being

text: a

text: (pre-)ELM

text: based

text: on

text: these

text: properties.

text: We

text: ﬁnd

text: that

text: per

text: cent

text: of

text: the

text: sdAs

text: are

text: more

text: likely

text: to

text: be

text: (pre-)ELMs

text: than

text: main

text: sequence

text: stars.

text: However,

text: the

text: spacial

text: velocity

text: distribution

text: suggests

text: that

text: over

text: per

text: cent

text: of

text: them

text: cannot

text: be

text: explained

text: as

text: single

text: metal-poor

text: A/F

text: stars.

text: Key

text: words:

text: subdwarfs

text: white

text: dwarfs

text: binaries:

text: general

text: stars:

text: evolution

text: stars:

text: kinematics

text: and

text: dynamics

text: INTRODUCTION

text: et

text: al.

text: or

text: as

text: single

text: stars

text: resulting

text: from

text: merger

text: events

text: (Brown

text: et

text: al.

text: Of

text: remarkable

text: interest

text: are

text: the

text: The

text: evolution

text: of

text: single

text: stars

text: is

text: a

text: fairly

text: well

text: under-

text: extremely-low

text: mass

text: white

text: dwarfs

text: (ELMs,

text: M

text: M

text: stood

text: process.

text: There

text: are

text: of

text: course

text: many

text: uncertainties

text: con-

text: see

text: e.g.

text: the

text: ELM

text: Survey:

text: Brown

text: et

text: al.

text: Kilic

text: et

text: al.

text: cerning

text: speciﬁc

text: phases

text: (e.g.

text: the

text: asymptotic

text: giant

text: branch,

text: Brown

text: et

text: al.

text: Kilic

text: et

text: al.

text: Brown

text: et

text: al.

text: Gi-

text: Miller

text: Bertolami

text: and

text: processes

text: (e.g.

text: convection,

text: Bres-

text: anninas

text: et

text: al.

text: Brown

text: et

text: al.

text: which

text: can

text: only

text: be

text: san

text: et

text: al.

text: Tremblay

text: et

text: al.

text: throughout

text: the

text: evolu-

text: formed

text: in

text: interacting

text: binary

text: systems

text: within

text: a

text: Hubble

text: time.

text: tion,

text: but

text: relating

text: an

text: initial

text: mass

text: and

text: metallicity

text: to

text: a

text: ﬁnal

text: The

text: evolution

text: of

text: single

text: stars

text: can

text: lead

text: to

text: white

text: dwarfs

text: with

text: outcome

text: can

text: be

text: done

text: with

text: reasonable

text: precision.

text: Only

text: a

text: very

text: masses

text: down

text: to

text: solar

text: masses

text: (Kilic

text: et

text: al.

text: small

text: amount

text: of

text: objects,

text: much

text: less

text: than

text: per

text: cent

text: of

text: all

text: but

text: main-sequence

text: progenitors

text: that

text: would

text: end

text: up

text: as

text: lower

text: stars

text: in

text: the

text: Galaxy,

text: will

text: end

text: their

text: lives

text: as

text: neutron

text: stars

text: or

text: mass

text: white

text: dwarfs

text: have

text: main

text: sequence

text: lifetimes

text: exceeding

text: black

text: holes

text: those

text: with

text: initial

text: masses

text: larger

text: than

text: about

text: the

text: age

text: of

text: the

text: Universe.

text: Objects

text: with

text: M

text: M

text: M

text: (e.g.

text: Woosley

text: Heger

text: The

text: remaining

text: are

text: usually

text: referred

text: to

text: as

text: low-mass

text: white

text: dwarfs.

text: Their

text: bi-

text: over

text: per

text: cent

text: stars

text: will

text: become

text: white

text: dwarf

text: stars,

text: whose

text: nary

text: fraction

text: is

text: still

text: high,

text: about

text: per

text: cent

text: (Brown

text: et

text: al.

text: evolution

text: can

text: be

text: approximated

text: by

text: a

text: slow

text: cooling

text: process.

text: because

text: some

text: form

text: of

text: enhanced

text: mass-loss

text: is

text: needed

text: Hence,

text: as

text: they

text: are

text: not

text: only

text: abundant

text: but

text: also

text: long-lived,

text: to

text: form

text: them,

text: and

text: binarity

text: is

text: the

text: easiest

text: way

text: to

text: achieve

text: this.

text: white

text: dwarfs

text: are

text: very

text: useful

text: in

text: obtaining

text: information

text: of

text: all

text: Severe

text: mass

text: loss

text: in

text: the

text: ﬁrst

text: ascent

text: giant

text: branch,

text: attributed

text: galactic

text: populations

text: (Isern

text: et

text: al.

text: Liebert

text: et

text: al.

text: to

text: high

text: metallicity

text: (Hansen

text: can

text: also

text: lead

text: to

text: single

text: low

text: Bono

text: et

text: al.

text: Tremblay

text: et

text: al.

text: Kilic

text: et

text: al.

text: mass

text: white

text: dwarfs

text: (Kilic

text: et

text: al.

text: The

text: binary

text: fraction

text: Cojocaru

text: et

text: al.

text: below

text: M

text: on

text: the

text: other

text: hand,

text: could

text: be

text: up

text: to

text: White

text: dwarf

text: stars

text: are

text: also

text: one

text: possible

text: result

text: of

text: binary

text: per

text: cent

text: (Brown

text: et

text: al.

text: encompassing

text: not

text: only

text: the

text: evolution,

text: both

text: in

text: binary

text: systems

text: (e.g.

text: Rebassa-Mansergas

text: ELMs

text: but

text: also

text: the

text: pre-ELMs,

text: which

text: still

text: have

text: not

text: reached

text: the

text: white

text: dwarf

text: cooling

text: track

text: (Maxted

text: et

text: al.

text: E-mail:

text: ingrid.pelisoli@ufrgs.br

text: Less

text: than

text: a

text: hundred

text: ELMs

text: are

text: known

text: to

text: date,

text: making

text: c

text: The

text: Authors


text: 16 I. Pelisoli et al. Figure A1. Comparison between our estimate of T (DK model) and both pipelines, as well as the comparison between the pipelines. eff Grey dots are outside of the area covered by SDSS grids, while black dots are within it. The red dashed line represents equality. The average diﬀerence for estimates within both grids is shown in each plot. It is clear that there is no discrepancy in the region covered by both grids. Our higher temperatures are backed up by both the GALEX ultraviolet ﬂux, and the SDSS classiﬁcation as type O and B, which is not coherent with their own estimated temperature. Figure A2. Comparison between our log g and that of the pipelines. Colours are the same as in Fig. A1. The spread is larger and our ﬁt favours, in average, slightly higher log g values, given the extension of our grid to higher values. Figure A3. Absolute diﬀerence in log g between our models and Figure B1. Comparison between the radial velocity estimated the SSPP determination, averaged every 100 points, as a function from SDSS spectra (vR), and the transversal velocity derived from of the metallicity [Fe/H] given by SSPP. The red dashed line the GPS1 proper motion given the estimated distance (vT), both shows the overall average. There seems to be no strong depen- assuming a main sequence (grey dots) and an ELM radius (red dence of the diﬀerence between surface gravity on the metallicity. crosses) for sample B only. The spacial motion is dominated by the tangential velocity if a main sequence radius is assumed, with many objects showing vT much above the escape velocity. That does not happen when an ELM radius is assumed, with very few exceptions. MNRAS 000, 1–16 (2017) ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: I.

text: Pelisoli

text: et

text: al.

text: Figure

text: Comparison

text: between

text: our

text: estimate

text: of

text: T

text: (DK

text: model)

text: and

text: both

text: pipelines,

text: as

text: well

text: as

text: the

text: comparison

text: between

text: the

text: pipelines.

text: eff

text: Grey

text: dots

text: are

text: outside

text: of

text: the

text: area

text: covered

text: by

text: SDSS

text: grids,

text: while

text: black

text: dots

text: are

text: within

text: it.

text: The

text: red

text: dashed

text: line

text: represents

text: equality.

text: The

text: average

text: diﬀerence

text: for

text: estimates

text: within

text: both

text: grids

text: is

text: shown

text: in

text: each

text: plot.

text: It

text: is

text: clear

text: that

text: there

text: is

text: no

text: discrepancy

text: in

text: the

text: region

text: covered

text: by

text: both

text: grids.

text: Our

text: higher

text: temperatures

text: are

text: backed

text: up

text: by

text: both

text: the

text: GALEX

text: ultraviolet

text: ﬂux,

text: and

text: the

text: SDSS

text: classiﬁcation

text: as

text: type

text: O

text: and

text: B,

text: which

text: is

text: not

text: coherent

text: with

text: their

text: own

text: estimated

text: temperature.

text: Figure

text: Comparison

text: between

text: our

text: log

text: g

text: and

text: that

text: of

text: the

text: pipelines.

text: Colours

text: are

text: the

text: same

text: as

text: in

text: Fig.

text: The

text: spread

text: is

text: larger

text: and

text: our

text: ﬁt

text: favours,

text: in

text: average,

text: slightly

text: higher

text: log

text: g

text: values,

text: given

text: the

text: extension

text: of

text: our

text: grid

text: to

text: higher

text: values.

text: Figure

text: Absolute

text: diﬀerence

text: in

text: log

text: g

text: between

text: our

text: models

text: and

text: Figure

text: Comparison

text: between

text: the

text: radial

text: velocity

text: estimated

text: the

text: SSPP

text: determination,

text: averaged

text: every

text: points,

text: as

text: a

text: function

text: from

text: SDSS

text: spectra

text: (vR),

text: and

text: the

text: transversal

text: velocity

text: derived

text: from

text: of

text: the

text: metallicity

text: [Fe/H]

text: given

text: by

text: SSPP.

text: The

text: red

text: dashed

text: line

text: the

text: proper

text: motion

text: given

text: the

text: estimated

text: distance

text: (vT),

text: both

text: shows

text: the

text: overall

text: average.

text: There

text: seems

text: to

text: be

text: no

text: strong

text: depen-

text: assuming

text: a

text: main

text: sequence

text: (grey

text: dots)

text: and

text: an

text: ELM

text: radius

text: (red

text: dence

text: of

text: the

text: diﬀerence

text: between

text: surface

text: gravity

text: on

text: the

text: metallicity.

text: crosses)

text: for

text: sample

text: B

text: only.

text: The

text: spacial

text: motion

text: is

text: dominated

text: by

text: the

text: tangential

text: velocity

text: if

text: a

text: main

text: sequence

text: radius

text: is

text: assumed,

text: with

text: many

text: objects

text: showing

text: vT

text: much

text: above

text: the

text: escape

text: velocity.

text: That

text: does

text: not

text: happen

text: when

text: an

text: ELM

text: radius

text: is

text: assumed,

text: with

text: very

text: few

text: exceptions.

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 18 I. Pelisoli et al. Table C1. Objects for which our ﬁt indicated T > 20000 K eff and log g < 6.5, which are possible hot subdwarfs not listed in Geier et al. (2017). SDSS J P-M-F 000138.22+282731.9 2824-54452-0443 003739.05+000842.3 3588-55184-0616 013940.56+003626.6 1907-53265-0339,1907-53315-0335 021419.64-084505.3 4395-55828-0478 062336.21+642730.0 2301-53712-0627 063659.39+832025.0 2548-54152-0076 064846.81+373614.3 2700-54417-0079 073225.82+153729.7 2713-54400-0172 073546.72+410350.5 2701-54154-0541 073654.60+280923.4 4456-55537-0728 075029.26+181749.5 2729-54419-0428 075640.08+071806.6 4843-55860-0525 080520.22+000944.5 4745-55892-0147 081544.25+230904.7 4469-55863-0004 082606.25+113913.8 4508-55600-0855 083830.10+135117.6 4500-55543-0263 090141.48+345924.4 4645-55623-0456 091301.01+305119.8 2401-53768-0389 091721.87+283656.0 5797-56273-0546 091914.64+480306.0 5813-56363-0709 093946.04+065209.4 1234-52724-0304 100233.49+164500.5 5326-56002-0106 100442.16+132711.6 5328-55982-0828 104159.64+192254.5 5886-56034-0628 104437.73+145213.3 5350-56009-0768 105847.65+203917.4 5876-56042-0729 111917.41+050617.3 4769-55931-0682 112121.98+453955.5 6648-56383-0750 113312.12+010824.8 2877-54523-0369 121123.36+611203.8 0954-52405-0567,6972-56426-0278 121910.44+230020.7 5979-56329-0474 132138.31+133200.1 5427-56001-0746 132210.94+421216.9 6622-56365-0473 133540.82+014725.2 4045-55622-0014 134531.00-005314.4 4043-55630-0180 140603.27+374216.6 4711-55737-0208 141055.68+374340.6 4712-55738-0466 141810.60-020038.8 4032-55333-0532,4035-55383-0990 150421.06-005613.8 4020-55332-0950 151404.97+065352.7 2927-54621-0268 151940.51+014457.1 4011-55635-0310 152729.40+222448.5 3954-55680-0244 153049.60+321425.4 4723-56033-0639 155105.64+452134.0 3454-55003-0204 155241.28+045428.7 4877-55707-0254 160612.98+521919.3 2187-54270-0224 161430.90+041843.4 2189-54624-0633 164014.57+320325.2 5202-55824-0900 165237.92+240302.8 4181-55685-0236 165406.79+271117.7 4185-55469-0985 165634.74+231256.4 3290-54941-0378 170126.69+204620.5 4175-55680-0810 172037.66+534009.3 0359-51821-0273 191837.28+370917.5 2821-54393-0203 204403.97-051135.6 0635-52145-0360 204802.45+002753.1 1116-52932-0512 211716.97-005401.6 4192-55469-0376 213301.81-004914.7 4194-55450-0178 215014.24+233039.1 5953-56092-0185 224145.04+292426.0 6585-56479-0883 225654.02+074449.8 2325-54082-0376 232810.27-084156.3 3145-54801-0371 MNRAS 000, 1–16 (2017) ##LTLine## ##LTLine## ##LTLine##

text: I.

text: Pelisoli

text: et

text: al.

text: Table

text: Objects

text: for

text: which

text: our

text: ﬁt

text: indicated

text: T

text: K

text: eff

text: and

text: log

text: g

text: which

text: are

text: possible

text: hot

text: subdwarfs

text: not

text: listed

text: in

text: Geier

text: et

text: al.

text: SDSS

label: table

text: J

text: P-M-F

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: APS/123-QED ∗ Magnetic AC control of the spin textures in a helical Luttinger liquid G. Dolcetto1,2,3, F. Cavaliere1,2, M. Sassetti1,2 † ‡ (Dated: May 26, 2020) We demonstrate the possibility to induce and control peculiar spin textures in a helical Luttinger liquid, by means of a time-dependent magnetic scatterer. The presence of a perturbation that breaks the time-reversal symmetry opens a gap in the spectrum, inducing single-particle backscattering and a peculiar spin response. We show that in the weak backscattering regime asymmetric spin textures emerge at the left and right side of the scatterer, whose spatial oscillations are controlled by the ratio between the magnetization frequency and the Fermi energy and by the electron interaction. This peculiar spin response marks a strong diﬀerence between helical and non-helical liquids, which are expected to produce symmetric spin textures even in the AC regime. PACS numbers: 71.10.Pm, 73.21.-b, 75.30.Fv I. INTRODUCTION be achieved by proximity coupling one edge of a QSH system with a narrow ferromagnetic insulator whose magnetization is precessing30,31. This setup has already Helical Luttinger liquids represent a new paradigm been considered, focusing mainly on the current pumped of the one-dimensional world1. They are character- in the topological insulator in the absence of an applied ized by spin-momentum locking, where the spin of the bias voltage19,30,31. The magnetic ordering acquired by electrons is correlated with their momentum. Recently the interacting helical system due to the nearby AC fer- helical and quasi-helical systems aroused great inter- romagnet has not been investigated, and this is the gap est as platforms for the search of the elusive Majo- we want to ﬁll. rana fermions2–4. Furthermore the possibility of gen- For a static magnetic impurity, anisotropies between in- erating and controlling spin currents by pure electrical plane and out-of plane components of the magnetization means has made these systems attractive for spintronics vector represent a signature of the helical regime21,22,32, applications5–14. A well known implementation of helical with 2k -oscillations in the in-plane magnetization21–23. liquids is represented by the edge states of quantum spin F These are present also in non-helical systems, but are Hall (QSH) systems15,16. HgTe/CdTe quantum wells17 usually suppressed by non-oscillating contributions21. and InAs/GaSb heterostructures18 have been shown ex- In this paper we show that an even richer scenario prerimentally to behave as two-dimensional topological emerges in the presence of a time-dependent magnetic insulators. If an even number of Kramer’s doublets exists scatterer. The in-plane components show a pure AC on the edge, the helical Luttinger liquid remains gapless response, contrary to the pure DC response of the out- in the presence of time-reversal (TR) invariant perturba- of-plane component. Furthermore, by tuning the ratio tions such as disorder, leading to ballistic transport and between the frequency of the AC ﬁeld and the Fermi en- quantized conductance, whose measure represents one of ergy, a further marked asymmetry between the regions to the experimental signatures of the helical regime. On the left and to the right of the scatterer is induced. This the other hand perturbations breaking the TR, such as represents a strong signature of the helical regime, and magnetic ﬁeld, may open a gap in the Dirac spectrum is not expected to occur in ordinary spinful liquids. We and lead to non-quantized conductance. Many other in- demonstrate that the spin textures are also inﬂuenced teresting signatures of the helical nature can be deduced by the strength of the electron interactions, so that an in this case. Fractional charge in antiparallel magnetic observation of left/right asymmetric spin patterns could domains19,20 and the emergence of spin-density waves in principle allow to estimate the stregth of the interacta- in the presence of magnetic impurities21–23 represent a tions, an important information for the helical Luttinger strong signature of helical systems. Also the peculiar liquid. Kondo screening in the presence of impurity spins has been extensively discussed1,24–29. The extreme sensitiv- ity to TR breaking couplings and the foundamental role played by the spin degree of freedom require a careful study of the spin response of helical Luttinger liquids to The paper is organized as follows. In Sec. we present magnetic perturbations. the model describing the helical Luttinger liquid in the presence of a time-dependent magnetic scatterer. In Sec. In this work we consider a helical Luttinger liquid per- we evaluate the spin response both in the weak and in the turbed by a time-dependent magnetic exchange coupling strong backscattering regime, and we discuss the results. which acts as a scatterer for incoming electrons. This can Section is devoted to the conclusions.

text: Magnetic

text: AC

text: control

text: of

text: the

text: spin

text: textures

text: in

text: a

text: helical

text: Luttinger

text: liquid

text: G.

text: F.

text: M.

text: (Dated:

text: May

text: We

text: demonstrate

text: the

text: possibility

text: to

text: induce

text: and

text: control

text: peculiar

text: spin

text: textures

text: in

text: a

text: helical

text: Luttinger

text: liquid,

text: by

text: means

text: of

text: a

text: time-dependent

text: magnetic

text: scatterer.

text: The

text: presence

text: of

text: a

text: perturbation

text: that

text: breaks

text: the

text: time-reversal

text: symmetry

text: opens

text: a

text: gap

text: in

text: the

text: spectrum,

text: inducing

text: single-particle

text: backscattering

text: and

text: a

text: peculiar

text: spin

text: response.

text: We

text: show

text: that

text: in

text: the

text: weak

text: backscattering

text: regime

text: asymmetric

text: spin

text: textures

text: emerge

text: at

text: the

text: left

text: and

text: right

text: side

text: of

text: the

text: scatterer,

text: whose

text: spatial

text: oscillations

text: are

text: controlled

text: by

text: the

text: ratio

text: between

text: the

text: magnetization

text: frequency

text: and

text: the

text: Fermi

text: energy

text: and

text: by

text: the

text: electron

text: interaction.

text: This

text: peculiar

text: spin

text: response

text: marks

text: a

text: strong

text: diﬀerence

text: between

text: helical

text: and

text: non-helical

text: liquids,

text: which

text: are

text: expected

text: to

text: produce

text: symmetric

text: spin

text: textures

text: even

text: in

text: the

text: AC

text: regime.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: be

text: achieved

text: by

text: proximity

text: coupling

text: one

text: edge

text: of

text: a

text: QSH

text: system

text: with

text: a

text: narrow

text: ferromagnetic

text: insulator

text: whose

text: magnetization

text: is

text: This

text: setup

text: has

text: already

text: Helical

text: Luttinger

text: liquids

text: represent

text: a

text: new

text: paradigm

text: been

text: considered,

text: focusing

text: mainly

text: on

text: the

text: current

text: pumped

text: of

text: the

text: one-dimensional

text: They

text: are

text: character-

text: in

text: the

text: topological

text: insulator

text: in

text: the

text: absence

text: of

text: an

text: applied

text: ized

text: by

text: spin-momentum

text: locking,

text: where

text: the

text: spin

text: of

text: the

text: bias

text: The

text: magnetic

text: ordering

text: acquired

text: by

text: electrons

text: is

text: correlated

text: with

text: their

text: momentum.

text: Recently

text: the

text: interacting

text: helical

text: system

text: due

text: to

text: the

text: nearby

text: AC

text: fer-

text: helical

text: and

text: quasi-helical

text: systems

text: aroused

text: great

text: inter-

text: romagnet

text: has

text: not

text: been

text: investigated,

text: and

text: this

text: is

text: the

text: gap

text: est

text: as

text: platforms

text: for

text: the

text: search

text: of

text: the

text: elusive

text: Majo-

text: we

text: want

text: to

text: ﬁll.

text: rana

text: Furthermore

text: the

text: possibility

text: of

text: gen-

text: For

text: a

text: static

text: magnetic

text: impurity,

text: anisotropies

text: between

text: in-

text: erating

text: and

text: controlling

text: spin

text: currents

text: by

text: pure

text: electrical

text: plane

text: and

text: out-of

text: plane

text: components

text: of

text: the

text: magnetization

text: means

text: has

text: made

text: these

text: systems

text: attractive

text: for

text: spintronics

text: vector

text: represent

text: a

text: signature

text: of

text: the

text: helical

text: A

text: well

text: known

text: implementation

text: of

text: helical

text: with

text: -oscillations

text: in

text: the

text: in-plane

text: liquids

text: is

text: represented

text: by

text: the

text: edge

text: states

text: of

text: quantum

text: spin

text: F

text: These

text: are

text: present

text: also

text: in

text: non-helical

text: systems,

text: but

text: are

text: Hall

text: (QSH)

text: HgTe/CdTe

text: quantum

text: usually

text: suppressed

text: by

text: non-oscillating

text: and

text: InAs/GaSb

text: have

text: been

text: shown

text: ex-

text: In

text: this

text: paper

text: we

text: show

text: that

text: an

text: even

text: richer

text: scenario

text: prerimentally

text: to

text: behave

text: as

text: two-dimensional

text: topological

text: emerges

text: in

text: the

text: presence

text: of

text: a

text: time-dependent

text: magnetic

text: insulators.

text: If

text: an

text: even

text: number

text: of

text: Kramer’s

text: doublets

text: exists

text: scatterer.

text: The

text: in-plane

text: components

text: show

text: a

text: pure

text: AC

text: on

text: the

text: edge,

text: the

text: helical

text: Luttinger

text: liquid

text: remains

text: gapless

text: response,

text: contrary

text: to

text: the

text: pure

text: DC

text: response

text: of

text: the

text: out-

text: in

text: the

text: presence

text: of

text: time-reversal

text: (TR)

text: invariant

text: perturba-

text: of-plane

text: component.

text: Furthermore,

text: by

text: tuning

text: the

text: ratio

text: tions

text: such

text: as

text: disorder,

text: leading

text: to

text: ballistic

text: transport

text: and

text: between

text: the

text: frequency

text: of

text: the

text: AC

text: ﬁeld

text: and

text: the

text: Fermi

text: en-

text: quantized

text: conductance,

text: whose

text: measure

text: represents

text: one

text: of

text: ergy,

text: a

text: further

text: marked

text: asymmetry

text: between

text: the

text: regions

text: to

text: the

text: experimental

text: signatures

text: of

text: the

text: helical

text: regime.

text: On

text: the

text: left

text: and

text: to

text: the

text: right

text: of

text: the

text: scatterer

text: is

text: induced.

text: This

text: the

text: other

text: hand

text: perturbations

text: breaking

text: the

text: TR,

text: such

text: as

text: represents

text: a

text: strong

text: signature

text: of

text: the

text: helical

text: regime,

text: and

text: magnetic

text: ﬁeld,

text: may

text: open

text: a

text: gap

text: in

text: the

text: Dirac

text: spectrum

text: is

text: not

text: expected

text: to

text: occur

text: in

text: ordinary

text: spinful

text: liquids.

text: We

text: and

text: lead

text: to

text: non-quantized

text: conductance.

text: Many

text: other

text: in-

text: demonstrate

text: that

text: the

text: spin

text: textures

text: are

text: also

text: inﬂuenced

text: teresting

text: signatures

text: of

text: the

text: helical

text: nature

text: can

text: be

text: deduced

text: by

text: the

text: strength

text: of

text: the

text: electron

text: interactions,

text: so

text: that

text: an

text: in

text: this

text: case.

text: Fractional

text: charge

text: in

text: antiparallel

text: magnetic

text: observation

text: of

text: left/right

text: asymmetric

text: spin

text: patterns

text: could

text: and

text: the

text: emergence

text: of

text: spin-density

text: waves

text: in

text: principle

text: allow

text: to

text: estimate

text: the

text: stregth

text: of

text: the

text: interacta-

text: in

text: the

text: presence

text: of

text: magnetic

text: represent

text: a

text: tions,

text: an

text: important

text: information

text: for

text: the

text: helical

text: Luttinger

text: strong

text: signature

text: of

text: helical

text: systems.

text: Also

text: the

text: peculiar

text: liquid.

text: Kondo

text: screening

text: in

text: the

text: presence

text: of

text: impurity

text: spins

text: has

text: been

text: extensively

text: The

text: extreme

text: sensitiv-

text: ity

text: to

text: TR

text: breaking

text: couplings

text: and

text: the

text: foundamental

text: role

text: played

text: by

text: the

text: spin

text: degree

text: of

text: freedom

text: require

text: a

text: careful

text: study

text: of

text: the

text: spin

text: response

text: of

text: helical

text: Luttinger

text: liquids

text: to

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: we

text: present

text: magnetic

text: perturbations.

text: the

text: model

text: describing

text: the

text: helical

text: Luttinger

text: liquid

text: in

text: the

text: presence

text: of

text: a

text: time-dependent

text: magnetic

text: scatterer.

text: In

text: Sec.

text: In

text: this

text: work

text: we

text: consider

text: a

text: helical

text: Luttinger

text: liquid

text: per-

text: we

text: evaluate

text: the

text: spin

text: response

text: both

text: in

text: the

text: weak

text: and

text: in

text: the

text: turbed

text: by

text: a

text: time-dependent

text: magnetic

text: exchange

text: coupling

text: strong

text: backscattering

text: regime,

text: and

text: we

text: discuss

text: the

text: results.

text: which

text: acts

text: as

text: a

text: scatterer

text: for

text: incoming

text: electrons.

text: This

text: can

text: Section

text: is

text: devoted

text: to

text: the

text: conclusions.


text: 4 dimensions in a system of two fermion ﬂavors. Below spinor can be expanded in the basis of two eigenfunctions ± we describe the exact nonperturbative solutions for zero of the internal chirality operator Γ given in Eq. (14), int modes beyond the weak-ﬁeld approximation (28). corresponding to the states moving up and down along III. EXACT ZERO MODES the longitudinal direction. To write the condensate in a convenient way, we use an In this section we ﬁnd a most general solution to antiholomorphic function ω(z¯) deﬁned in Eq. (8) so that Eq. (27) without imposing boundary conditions which the condensate W(z, z¯), Eq. (7), satisﬁes the equation of are closely related to the topological properties of the motion (4). One way to −obtain(cid:16)sim(cid:17) ilar periodic behavior zero modes. The boundary conditions –which param- for µ(z) while keeping ψ ﬁnite, is to put 1 eterize both independent solutions of Eq. (27) –will be z discussed in the next section. µ(z) = ω , (33) A. First solution 2 at the expense of reducing the periodicity and quadru- To ﬁnd a ﬁrst solution, we will construct a zero-mode pling the surface of a unit cell. Equation (33) is a minimal solution for a reduced version of the system (27) by set- − − single-valued choice for the function µ consistent with ting one of the components, either ψ or ψ , to zero. − − 1 2 condition (32). This choice implies that ψ has one dou- We conclude quickly that ψ = 0 immediately implies 1 2 ble zero per four unit cells, or equivalently, has a branch the triviality of the full solution. Indeed, the W conden- point behaving like z 12 in each cell. This is, of course, in sate (7) is a nonvanishing function almost everywhere so Z − direct correspondence to the behavior of each vortex. 2 that Eq. (27) implies ψ = 0. 1 To keep the analytical expression manageable, however, One can alternatively set the upper component to zero, − it is useful to group four cells and avoid fractional branch ψ1(I), = 0. Then(cid:16)the lower(cid:17)component should satisfy the cuts inside the cells. following equation: We now are left with the calculation of the lower com- − ∂z¯ − π4z ψ2(I),− = 0 , (29) preownreintte oitf uthsiengsetchoend(aznetrio-)-hmoolodme osropluhticionfunψc2(tIiIo),ns. ωW(z¯e) and µ(z), and a new function f(z, z¯): implying that the ﬁrst set of solutions of Eq. (27) has the − following form: ψ2(II), (z, z¯) = f(z, z¯)eπ4 zz¯µ2(z). (34) − − − ψ1(I), = 0 , ψ2(I), (z, z¯) = W(z, z¯) 12 , (30) The remaining equation in (27) becomes then where the function W(z, z¯) is given in Eqs. (7) and (8). − i√g − This solution has a large degeneracy as it can be mul- ∂ f(z, z¯) = ω(z¯)e πzz¯ . (35) z¯ 2 2 tiplied by an arbitrary holomorphic function h (z) and 1 it still will be a zero-mode solution. It is important to In the Appendix we derive a convenient expansion (A1) notice that the holomorphic nature of the function h is 1 of the Jacobi θ function which enters the function ω in a severe constraint. Rather than being a free function in 1 (cid:88) Eq. (8). Using this expansion we set each point of the transversal plane, it is determined by its behavior on a ﬁnite line, say at the boundary of the − | | condensate and by its behavior around the poles. f(z, z¯) = e π2 Xm,n 2fm,n(z, z¯), (36) ∈Z B. Second solution m,n so that the equation (35) becomes, term by term, as fol- We will now look for a second solution to Eq. (27) con- − lows: sidering a nonzero function ψ (z, z¯) from the very begin- 1 ning. From Eqs. (4) and (27) we see that the functions − − gN√C − − − W and ψ1 should obey a similar Abrikosov equation. We ∂z¯fm,n = i 2 20 ei8π (z¯ X¯m,n)e πz¯(z Xm,n). (37) deﬁne (cid:18) (cid:19) − − ψ1(II), (z, z¯) = χµ2(z)e π4 zz¯, (31) We ﬁnd now for all integer n and m: ig√NC 1 | − | wishhienrge aµt(ze)veanspcaolsaitriohnosloomf othrpehvicorftuenxcctoiornesw(1h2ic)h: is van- fm,n(z, z¯) = 2 2π0 ei8π π + z Xm,n 2 − − ∈ Z ×e πz¯(z Xm,n) µ(Xm,n) = 0 , m, n . (32) − , (38) (z X )2 m,n In Eq. (31) χ is a constant polarization two-component spinor. The exact form of the polarization spinor χ is and to avoid divergences we need to set in Eq. (38) the irrelevant for our discussion of this transversal two di- integration constant, which is an arbitrary holomorphic mensional problem, and therefore we omit it below. This function, to zero. From Eqs. (36) and (34) we then get ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: dimensions

text: in

text: a

text: system

text: of

text: two

text: fermion

text: ﬂavors.

text: Below

text: spinor

text: can

text: be

text: expanded

text: in

text: the

text: basis

text: of

text: two

text: eigenfunctions

text: we

text: describe

text: the

text: exact

text: nonperturbative

text: solutions

text: for

text: zero

text: of

text: the

text: internal

text: chirality

text: operator

text: Γ

text: given

text: in

text: Eq.

text: int

text: modes

text: beyond

text: the

text: weak-ﬁeld

text: approximation

text: corresponding

text: to

text: the

text: states

text: moving

text: up

text: and

text: down

text: along

text: III.

label: section

text: EXACT

text: ZERO

text: MODES

text: the

text: longitudinal

text: direction.

text: To

text: write

text: the

text: condensate

text: in

text: a

text: convenient

text: way,

text: we

text: use

text: an

text: In

text: this

text: section

text: we

text: ﬁnd

text: a

text: most

text: general

text: solution

text: to

text: antiholomorphic

text: function

text: ω(z¯)

text: deﬁned

text: in

text: Eq.

text: so

text: that

text: Eq.

text: without

text: imposing

text: boundary

text: conditions

text: which

text: the

text: condensate

text: W(z,

text: z¯),

text: Eq.

text: satisﬁes

text: the

text: equation

text: of

text: are

text: closely

text: related

text: to

text: the

text: topological

text: properties

text: of

text: the

text: motion

text: One

text: way

text: to

text: ilar

text: periodic

text: behavior

text: zero

text: modes.

text: The

text: boundary

text: conditions

text: –which

text: param-

text: for

text: µ(z)

text: while

text: keeping

text: ψ

text: ﬁnite,

text: is

text: to

text: put

text: eterize

text: both

text: independent

text: solutions

text: of

text: Eq.

text: –will

text: be

text: z

text: discussed

text: in

text: the

text: next

text: section.

text: µ(z)

text: ω

text: A.

text: First

text: solution

text: at

text: the

text: expense

text: of

text: reducing

text: the

text: periodicity

text: and

text: quadru-

text: To

text: ﬁnd

text: a

text: ﬁrst

text: solution,

text: we

text: will

text: construct

text: a

text: zero-mode

text: pling

text: the

text: surface

text: of

text: a

text: unit

text: cell.

text: Equation

text: is

text: a

text: minimal

text: solution

text: for

text: a

text: reduced

text: version

text: of

text: the

text: system

text: by

text: set-

text: single-valued

text: choice

text: for

text: the

text: function

text: µ

text: consistent

text: with

text: ting

text: one

text: of

text: the

text: components,

text: either

text: ψ

text: or

text: ψ

text: to

text: zero.

text: condition

text: This

text: choice

text: implies

text: that

text: ψ

text: has

text: one

text: dou-

text: We

text: conclude

text: quickly

text: that

text: ψ

text: immediately

text: implies

text: ble

text: zero

text: per

text: four

text: unit

text: cells,

text: or

text: equivalently,

text: has

text: a

text: branch

text: the

text: triviality

text: of

text: the

text: full

text: solution.

text: Indeed,

text: the

text: W

text: conden-

text: point

text: behaving

text: like

text: z

text: in

text: each

text: cell.

text: This

text: is,

text: of

text: course,

text: in

text: sate

text: is

text: a

text: nonvanishing

text: function

text: almost

text: everywhere

text: so

text: Z

text: direct

text: correspondence

text: to

text: the

text: behavior

text: of

text: each

text: vortex.

text: that

text: Eq.

text: implies

text: ψ

text: To

text: keep

text: the

text: analytical

text: expression

text: manageable,

text: however,

text: One

text: can

text: alternatively

text: set

text: the

text: upper

text: component

text: to

text: zero,

text: it

text: is

text: useful

text: to

text: group

text: four

text: cells

text: and

text: avoid

text: fractional

text: branch

text: should

text: satisfy

text: the

text: cuts

text: inside

text: the

text: cells.

text: following

text: equation:

text: We

text: now

text: are

text: left

text: with

text: the

text: calculation

text: of

text: the

text: lower

text: com-

text: ∂z¯

text: preownreintte

text: oitf

text: uthsiengsetchoend(aznetrio-)-hmoolodme

text: ωW(z¯e)

text: and

text: µ(z),

text: and

text: a

text: new

text: function

text: f(z,

text: z¯):

text: implying

text: that

text: the

text: ﬁrst

text: set

text: of

text: solutions

text: of

text: Eq.

text: has

text: the

text: following

text: form:

text: (z,

text: z¯)

text: f(z,

text: (z,

text: z¯)

text: W(z,

text: z¯)

text: The

text: remaining

text: equation

text: in

text: becomes

text: then

text: where

text: the

text: function

text: W(z,

text: z¯)

text: is

text: given

text: in

text: Eqs.

text: and

text: i√g

text: This

text: solution

text: has

text: a

text: large

text: degeneracy

text: as

text: it

text: can

text: be

text: mul-

text: f(z,

text: z¯)

text: ω(z¯)e

text: πzz¯

text: z¯

text: tiplied

text: by

text: an

text: arbitrary

text: holomorphic

text: function

text: h

text: (z)

text: and

text: it

text: still

text: will

text: be

text: a

text: zero-mode

text: solution.

text: It

text: is

text: important

text: to

text: In

text: the

text: Appendix

text: we

text: derive

text: a

text: convenient

text: expansion

text: notice

text: that

text: the

text: holomorphic

text: nature

text: of

text: the

text: function

text: h

text: is

text: of

text: the

text: Jacobi

text: θ

text: function

text: which

text: enters

text: the

text: function

text: ω

text: in

text: a

text: severe

text: constraint.

text: Rather

text: than

text: being

text: a

text: free

text: function

text: in

text: Eq.

text: Using

text: this

text: expansion

text: we

text: set

text: each

text: point

text: of

text: the

text: transversal

text: plane,

text: it

text: is

text: determined

text: by

text: its

text: behavior

text: on

text: a

text: ﬁnite

text: line,

text: say

text: at

text: the

text: boundary

text: of

text: the

text: condensate

text: and

text: by

text: its

text: behavior

text: around

text: the

text: poles.

text: f(z,

text: z¯)

text: e

text: π2

text: Xm,n

text: z¯),

text: ∈Z

text: B.

text: Second

text: solution

text: m,n

text: so

text: that

text: the

text: equation

text: becomes,

text: term

text: by

text: term,

text: as

text: fol-

text: We

text: will

text: now

text: look

text: for

text: a

text: second

text: solution

text: to

text: Eq.

text: con-

text: lows:

text: sidering

text: a

text: nonzero

text: function

text: ψ

text: (z,

text: z¯)

text: from

text: the

text: very

text: begin-

text: ning.

text: From

text: Eqs.

text: and

text: we

text: see

text: that

text: the

text: functions

text: gN√C

text: W

text: and

text: ψ1

text: should

text: obey

text: a

text: similar

text: Abrikosov

text: equation.

text: We

text: ∂z¯fm,n

text: i

text: (z¯

text: X¯m,n)e

text: πz¯(z

text: Xm,n).

text: deﬁne

text: (z,

text: z¯)

text: π4

text: zz¯,

text: We

text: ﬁnd

text: now

text: for

text: all

text: integer

text: n

text: and

text: m:

text: ig√NC

text: wishhienrge

text: aµt(ze)veanspcaolsaitriohnosloomf

text: is

text: van-

text: fm,n(z,

text: z¯)

text: π

text: z

text: Xm,n

text: Z

text: ×e

text: πz¯(z

text: Xm,n)

text: µ(Xm,n)

text: m,

text: n

text: (z

text: X

text: m,n

text: In

text: Eq.

text: χ

text: is

text: a

text: constant

text: polarization

text: two-component

text: spinor.

text: The

text: exact

text: form

text: of

text: the

text: polarization

text: spinor

text: χ

text: is

text: and

text: to

text: avoid

text: divergences

text: we

text: need

text: to

text: set

text: in

text: Eq.

text: the

text: irrelevant

text: for

text: our

text: discussion

text: of

text: this

text: transversal

text: two

text: di-

text: integration

text: constant,

text: which

text: is

text: an

text: arbitrary

text: holomorphic

text: mensional

text: problem,

text: and

text: therefore

text: we

text: omit

text: it

text: below.

text: This

text: function,

text: to

text: zero.

text: From

text: Eqs.

text: and

text: we

text: then

text: get

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text: ##LTLine##

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