text: (cid:0) (cid:1) (cid:0) (cid:1) − − − + 350 2 π2 127 ln 2 + 1 π2 ln 2 + 5 ln2 2 1 ln3 2 δ2∗δ ∗ 27 9 9 24 2 1 2 (cid:0) (cid:1) − − − − − 3 7 π2 1 ζ(3) 3 ln 2 + 1 π2 ln 2 + 7 ln2 2 1 ln3 2 δ ∗δ2∗ 72 4 24 6 6 1 2 (cid:0) (cid:1) − − − − − 3 7 π2 1 ζ(3) 3 ln 2 + 1 π2 ln 2 + 7 ln2 2 1 ln3 2 δ ∗δ ∗δ ∗ 72 4 24 6 6 1 2 3 − 1 (cid:15) ∗(cid:15)2∗ + (cid:15) ∗(cid:15) ∗(cid:15) ∗ + 2δ ∗δ2∗ + 2δ ∗δ ∗δ ∗ ln w. (5.3) 9 2 3 2 3 4 1 2 1 2 3 As can be seen, the presence of terms involving ln w in r¯(3) given above makes the fourth-order part of r logarithmically divergent. 5.2 Spectral Index and Running of the Spectral Index The spectral index for scalar perturbation deﬁned as − ≡ d ln P s n 1 , (5.4) s d ln k can be computed from the derived expression for the power spectrum in Sec. given − by (4.13). The idea is that from the relation k = ν ∗/(η∗c ∗), telling us that η∗ is an − s s implicit function of k, one can determine d ln( η∗)/d ln k (see Refs. [13, 16]). The resulting expression can then be used to carry out the calculation of n as dictated by the deﬁnition s above. One may think that because the power spectrum under the current scheme involves logarithmic divergences starting at the third order, n may inherit the same bad feature. →s − However, η and k are independent in the limit as η 0 unless the turning point relating k and η∗ happens to be at η = η∗ = 0, which is a trivial scenario. Hence, even if we accept that the derived power spectrum containing logarithmically divergent parts is valid (which it is not), the resulting spectral index will still be ﬁnite. Let us try to see the ﬁniteness of the spectral indices under the current calculation scheme in a more rigorous way. We start with the deﬁnition of n above with P given by s s (4.1) and derive the appropriate working equation. The derivation of the working equation was done in Ref. [22] for the case c2 = 1 and we are here going to follow the same path. s Taking the logarithm of the power spectrum, w(cid:90)e ﬁnd (cid:112) d ln P d η s = 3 + 2k dτ g (k, τ), (5.5) s d ln k dk η∗ which implies upon using the deﬁnition of(cid:90)gs given by (2.20), (cid:112) − − η dτ c2 n 1 = 3 2k2 s . (5.6) s η∗ gs(k, τ) This is our sought-for working equation for the spectral index. References [22, 23, 41] discussed the square-root singularity at the turning point in the (cid:54) integrand above for the case where c2 = 1. For the case in hand, where c2 = constant, s s we have the same form of singularity and we wish to ﬁnd out using the expansion scheme – 25 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: π2

text: ln

text: π2

text: ln

text: δ2∗δ

text: π2

text: ln

text: π2

text: ln

text: δ

text: ∗δ2∗

text: π2

text: ln

text: π2

text: ln

text: δ

text: ∗δ

text: ∗δ

text: 2δ

text: ∗δ2∗

text: 2δ

text: ∗δ

text: ∗δ

text: ln

text: w.

text: As

text: can

text: be

text: seen,

text: the

text: presence

text: of

text: terms

text: involving

text: ln

text: w

text: in

text: given

text: above

text: makes

text: the

text: fourth-order

text: part

text: of

text: r

text: logarithmically

text: divergent.

text: Spectral

text: Index

text: and

text: Running

text: of

text: the

text: Spectral

text: Index

text: The

text: spectral

text: index

text: for

text: scalar

text: perturbation

text: deﬁned

text: as

text: d

text: ln

text: P

text: s

text: n

text: s

text: d

text: ln

text: k

text: can

text: be

text: computed

text: from

text: the

text: derived

text: expression

text: for

text: the

text: power

text: spectrum

text: in

text: Sec.

text: given

text: by

text: The

text: idea

text: is

text: that

text: from

text: the

text: relation

text: k

text: ν

text: ∗/(η∗c

text: telling

text: us

text: that

text: η∗

text: is

text: an

text: s

text: s

text: implicit

text: function

text: of

text: k,

text: one

text: can

text: determine

text: d

text: ln(

text: η∗)/d

text: ln

text: k

text: (see

text: Refs.

text: The

text: resulting

text: expression

text: can

text: then

text: be

text: used

text: to

text: carry

text: out

text: the

text: calculation

text: of

text: n

text: as

text: dictated

text: by

text: the

text: deﬁnition

text: s

text: above.

text: One

text: may

text: think

text: that

text: because

text: the

text: power

text: spectrum

text: under

text: the

text: current

text: scheme

text: involves

text: logarithmic

text: divergences

text: starting

text: at

text: the

text: third

text: order,

text: n

text: may

text: inherit

text: the

text: same

text: bad

text: feature.

text: →s

text: However,

text: η

text: and

text: k

text: are

text: independent

text: in

text: the

text: limit

text: as

text: η

text: unless

text: the

text: turning

text: point

text: relating

text: k

text: and

text: η∗

text: happens

text: to

text: be

text: at

text: η

text: η∗

text: which

text: is

text: a

text: trivial

text: scenario.

text: Hence,

text: even

text: if

text: we

text: accept

text: that

text: the

text: derived

text: power

text: spectrum

text: containing

text: logarithmically

text: divergent

text: parts

text: is

text: valid

text: (which

text: it

text: is

text: not),

text: the

text: resulting

text: spectral

text: index

text: will

text: still

text: be

text: ﬁnite.

text: Let

text: us

text: try

text: to

text: see

text: the

text: ﬁniteness

text: of

text: the

text: spectral

text: indices

text: under

text: the

text: current

text: calculation

text: scheme

text: in

text: a

text: more

text: rigorous

text: way.

text: We

text: start

text: with

text: the

text: deﬁnition

text: of

text: n

text: above

text: with

text: P

text: given

text: by

text: s

text: s

text: and

text: derive

text: the

text: appropriate

text: working

text: equation.

text: The

text: derivation

text: of

text: the

text: working

text: equation

text: was

text: done

text: in

text: Ref.

text: for

text: the

text: case

text: and

text: we

text: are

text: here

text: going

text: to

text: follow

text: the

text: same

text: path.

text: s

text: Taking

text: the

text: logarithm

text: of

text: the

text: power

text: spectrum,

text: ﬁnd

text: d

text: ln

text: P

text: d

text: η

text: s

text: dτ

text: g

text: (k,

text: τ),

text: s

text: d

text: ln

text: k

text: dk

text: η∗

text: which

text: implies

text: upon

text: using

text: the

text: deﬁnition

text: given

text: by

text: η

text: dτ

text: n

text: s

text: s

text: η∗

text: gs(k,

text: τ)

text: This

text: is

text: our

text: sought-for

text: working

text: equation

text: for

text: the

text: spectral

text: index.

text: References

text: discussed

text: the

text: square-root

text: singularity

text: at

text: the

text: turning

text: point

text: in

text: the

text: integrand

text: above

text: for

text: the

text: case

text: where

text: For

text: the

text: case

text: in

text: hand,

text: where

text: constant,

text: s

text: s

text: we

text: have

text: the

text: same

text: form

text: of

text: singularity

text: and

text: we

text: wish

text: to

text: ﬁnd

text: out

text: using

text: the

text: expansion

text: scheme

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Here, R is the Ricci scalar, g is the determinant of the metric g , and M is the µν Pl reduced Planck mass that we hereafter, take (the square) as unity for brevity. Such an action corresponds to the so-called k-inﬂation. [19, 20] The scope of this work involves slow-roll − k-inﬂation. When P = X V (φ), where V (φ) is the inﬂaton potential, the k-inﬂationary action above reduces to that of the canonical single ﬁeld inﬂation. The background geometry corresponding to the action above is described by the con- formally ﬂat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric given by − ds2 = dt2 a2(t)δ dxidxj. (2.2) ij With this metric and the equation for the action given above, the Friedmann and con- tinuity equations take the form 8πG H2 = E, 3 − E˙ = 3H (E + P ) , (2.3) ≡ respectively, where E is the energy density, H a˙/a is the Hubble constant, and an overdot denotes diﬀerentiation with respect to the coordinate time t. To study the perturbations about our otherwise, conformally ﬂat background, we must choose appropriate slicing and threading of spacetime. In connection to this, we ﬁnd it con- venient to use the Arnowitt-Deser-Misner (ADM) formalism [26] and decompose the (per- turbed) metric as − ds2 = N2dt2 h (dxi + Nidt)(dxj + Njdt), (2.4) ij where N and Ni are the lapse and shift functions respectively. Furthermore, we choose the uniform inﬂaton gauge (also called the comoving gauge) where the inﬂaton ﬂuctuation vanishes, i.e., δφ = 0, and the spatial components of the metric take the form (cid:0) (cid:1) R g = a2(η)e2 eγ , (∂iγ = γi = 0). (2.5) ij ij ij i R Here, and γ are the gauge-invariant scalar and tensor perturbations respectively. ij The tensor perturbations satisfy the auxiliary conditions that they be both transverse and traceless. R It is now possible to write down the equations of motion for the perturbations and γ . · · · ij For the scalar perturbation, one decomposes the action as S = S(0) + S(2) + S(3) + , where R S(n) means the nth-order action with respect to (see e.g., Ref. [27]). The background corresponds to the zeroth-order part while the interaction starts at n = 3. It is from the – 5 – ##LTLine##

text: Here,

label: paragraph

text: R

text: is

text: the

text: Ricci

text: scalar,

text: g

text: is

text: the

text: determinant

text: of

text: the

text: metric

text: g

text: and

text: M

text: is

text: the

text: µν

text: Pl

text: reduced

text: Planck

text: mass

text: that

text: we

text: hereafter,

text: take

text: (the

text: square)

text: as

text: unity

text: for

text: brevity.

text: Such

text: an

text: action

text: corresponds

text: to

text: the

text: so-called

text: k-inﬂation.

text: The

text: scope

text: of

text: this

text: work

text: involves

text: slow-roll

text: k-inﬂation.

text: When

text: P

text: X

text: V

text: (φ),

text: where

text: V

text: (φ)

text: is

text: the

text: inﬂaton

text: potential,

text: the

text: k-inﬂationary

text: action

text: above

text: reduces

text: to

text: that

text: of

text: the

text: canonical

text: single

text: ﬁeld

text: inﬂation.

text: The

text: background

text: geometry

text: corresponding

text: to

text: the

text: action

text: above

text: is

text: described

text: by

text: the

text: con-

text: formally

text: ﬂat

text: Friedmann-Lemaˆıtre-Robertson-Walker

text: (FLRW)

text: metric

text: given

text: by

label: equation

text: dxidxj.

text: ij

text: With

text: this

text: metric

text: and

text: the

text: equation

text: for

text: the

text: action

text: given

text: above,

text: the

text: Friedmann

text: and

text: con-

text: tinuity

text: equations

text: take

text: the

text: form

text: E,

text: E˙

text: (E

text: P

text: respectively,

text: where

text: E

text: is

text: the

text: energy

text: density,

text: H

text: a˙/a

text: is

text: the

text: Hubble

text: constant,

text: and

text: an

text: overdot

text: denotes

text: diﬀerentiation

text: with

text: respect

text: to

text: the

text: coordinate

text: time

text: t.

text: To

text: study

text: the

text: perturbations

text: about

text: our

text: otherwise,

text: conformally

text: ﬂat

text: background,

text: we

text: must

text: choose

text: appropriate

text: slicing

text: and

text: threading

text: of

text: spacetime.

text: In

text: connection

text: to

text: this,

text: we

text: ﬁnd

text: it

text: con-

text: venient

text: to

text: use

text: the

text: Arnowitt-Deser-Misner

text: (ADM)

text: formalism

text: and

text: decompose

text: the

text: (per-

text: turbed)

text: metric

text: as

text: h

text: (dxi

text: Nidt)(dxj

text: Njdt),

text: ij

text: where

text: N

text: and

text: Ni

text: are

text: the

text: lapse

text: and

text: shift

text: functions

text: respectively.

text: Furthermore,

text: we

text: choose

text: the

text: uniform

text: inﬂaton

text: gauge

text: (also

text: called

text: the

text: comoving

text: gauge)

text: where

text: the

text: inﬂaton

text: ﬂuctuation

text: vanishes,

text: i.e.,

text: δφ

text: and

text: the

text: spatial

text: components

text: of

text: the

text: metric

text: take

text: the

text: form

text: R

text: g

text: eγ

text: (∂iγ

text: γi

text: ij

text: ij

text: ij

text: i

text: R

text: Here,

text: and

text: γ

text: are

text: the

text: gauge-invariant

text: scalar

text: and

text: tensor

text: perturbations

text: respectively.

text: ij

text: The

text: tensor

text: perturbations

text: satisfy

text: the

text: auxiliary

text: conditions

text: that

text: they

text: be

text: both

text: transverse

text: and

text: traceless.

text: R

text: It

text: is

text: now

text: possible

text: to

text: write

text: down

text: the

text: equations

text: of

text: motion

text: for

text: the

text: perturbations

text: and

text: γ

text: ij

text: For

text: the

text: scalar

text: perturbation,

text: one

text: decomposes

text: the

text: action

text: as

text: S

text: where

text: R

text: S(n)

text: means

text: the

text: nth-order

text: action

text: with

text: respect

text: to

text: (see

text: e.g.,

text: Ref.

text: The

text: background

text: corresponds

text: to

text: the

text: zeroth-order

text: part

text: while

text: the

text: interaction

text: starts

text: at

text: n

text: It

text: is

text: from

text: the

text: ##LTLine##


text: − · · · the fact that η = 1/(aH)(1 + (cid:15) + ), the solutions of the Mukhanov-Sasaki equations 1 can be well approximated by Hankel functions [31] when the slow-roll parameters are nearly constant. Beyond this is a generalization to the case where the slow-roll parameters are not constant, inclusion of higher-order corrections in terms of these parameters, and a con- sideration of non-constant speed of sound. The important corrections may be captured by semi-analytical approaches utilizing Green’s function [32, 36], WKB approximation [37–39], the UA [13, 16, 23], etc. 2.3 Uniform Approximation and Power Spectra From (1.1) and the ﬁrst of (2.12), the Mukhanov-Sasaki equation for scalar perturbations in the canonical single-ﬁeld inﬂat(cid:0)ion take(cid:1)s the form (cid:48)(cid:48) − ≡ (cid:48)(cid:48) v + k2 U v = 0; U z /z (2.14) k s k s When the potential term is negligible in comparison to the wavenumber, the solution of this diﬀerential equation is simply a sum of two sinusoids. On the other hand, for long- (cid:48)(cid:48) − (cid:48)(cid:48) wavelength modes, the diﬀerential equation approaches a form given by v (z /z)v = 0. ∼ k R ∼ k It has the obvious solution v z which corresponds to the frozen modes constant k k | | ∼ involved in the power spectrum. More formally, for subhorizon modes wherein k/(aH) | | (cid:29) | | ∼ kη 1, we have a sinusoidal solution and for superhorizon modes wherein k/(aH) | | (cid:28) ∼ R ∼ kη 1, we have v z corresponding to constant. k k In the case of the more general k-inﬂation—the type of inﬂation we are interested here— the eﬀective potential depends on a chain of Hubble and sound ﬂow functions (cf. (2.13)) and as pointed out earlier, the resulting diﬀerential equation may not have a simple closed- form solution. However, the behaviour of the solution at extremal ends of the spectrum of ∼ kη k/(aH) remains the same; that is, we have two solutions corresponding to subhorizon and superhorizon modes and these two are connected somewhere between the two extreme values of kη. In principle, one can do a sort of matching typical of a square-well potential problems in one-dimensional quantum mechanics. In contrast, such a sort of matching dif- ferent solutions for diﬀerent regions is not explicitly existent in the UA. Instead, there is a single approximating solution that takes on all the burden of adjustment to ﬁt the boundary or initial conditions and the diﬀerential equation that it satisﬁes. In the case of the Mukhanov-Sasaki equation, the “diﬀering” behaviour of the solu- tion at extremal ends in the range of values of kη is taken into account by considering the turning-point. More speciﬁcally, one can rewrite the Mukhanov-Sasaki equation for scalar perturbation in the canonical single-ﬁel(cid:20)d inﬂation as(cid:21)5 (cid:48)(cid:48) 1 − v + g (η) = 0, (2.15) k 4η2 s where ν2 − ≡ 1 g (η) = s k2; ν2 + η2U . (2.16) s η2 s 4 s 5Analogous expressions hold for the tensor perturbations. – 8 – ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: fact

text: that

text: η

text: the

text: solutions

text: of

text: the

text: Mukhanov-Sasaki

text: equations

text: can

text: be

text: well

text: approximated

text: by

text: Hankel

text: functions

text: when

text: the

text: slow-roll

text: parameters

text: are

text: nearly

text: constant.

text: Beyond

text: this

text: is

text: a

text: generalization

text: to

text: the

text: case

text: where

text: the

text: slow-roll

text: parameters

text: are

text: not

text: constant,

text: inclusion

text: of

text: higher-order

text: corrections

text: in

text: terms

text: of

text: these

text: parameters,

text: and

text: a

text: con-

text: sideration

text: of

text: non-constant

text: speed

text: of

text: sound.

text: The

text: important

text: corrections

text: may

text: be

text: captured

text: by

text: semi-analytical

text: approaches

text: utilizing

text: Green’s

text: function

text: WKB

text: approximation

text: the

text: UA

text: etc.

label: section

text: Uniform

text: Approximation

text: and

text: Power

text: Spectra

text: From

text: and

text: the

text: ﬁrst

text: of

text: the

text: Mukhanov-Sasaki

text: equation

text: for

text: scalar

text: perturbations

text: in

text: the

text: canonical

text: single-ﬁeld

text: the

text: form

text: v

text: U

text: v

text: U

text: z

text: /z

text: k

text: s

text: k

text: s

text: When

text: the

text: potential

text: term

text: is

text: negligible

text: in

text: comparison

text: to

text: the

text: wavenumber,

text: the

text: solution

text: of

text: this

text: diﬀerential

text: equation

text: is

text: simply

text: a

text: sum

text: of

text: two

text: sinusoids.

text: On

text: the

text: other

text: hand,

text: for

text: long-

text: wavelength

text: modes,

text: the

text: diﬀerential

text: equation

text: approaches

text: a

text: form

text: given

text: by

text: v

text: (z

text: /z)v

text: k

text: R

text: k

text: It

text: has

text: the

text: obvious

text: solution

text: v

text: z

text: which

text: corresponds

text: to

text: the

text: frozen

text: modes

text: constant

text: k

text: k

text: involved

text: in

text: the

text: power

text: spectrum.

text: More

text: formally,

text: for

text: subhorizon

text: modes

text: wherein

text: k/(aH)

text: kη

text: we

text: have

text: a

text: sinusoidal

text: solution

text: and

text: for

text: superhorizon

text: modes

text: wherein

text: k/(aH)

text: R

text: kη

text: we

text: have

text: v

text: z

text: corresponding

text: to

text: constant.

text: k

text: k

text: In

text: the

text: case

text: of

text: the

text: more

text: general

text: k-inﬂation—the

text: type

text: of

text: inﬂation

text: we

text: are

text: interested

text: here—

text: the

text: eﬀective

text: potential

text: depends

text: on

text: a

text: chain

text: of

text: Hubble

text: and

text: sound

text: ﬂow

text: functions

text: (cf.

text: and

text: as

text: pointed

text: out

text: earlier,

text: the

text: resulting

text: diﬀerential

text: equation

text: may

text: not

text: have

text: a

text: simple

text: closed-

text: form

text: solution.

text: However,

text: the

text: behaviour

text: of

text: the

text: solution

text: at

text: extremal

text: ends

text: of

text: the

text: spectrum

text: of

text: kη

text: k/(aH)

text: remains

text: the

text: same;

text: that

text: is,

text: we

text: have

text: two

text: solutions

text: corresponding

text: to

text: subhorizon

text: and

text: superhorizon

text: modes

text: and

text: these

text: two

text: are

text: connected

text: somewhere

text: between

text: the

text: two

text: extreme

text: values

text: of

text: kη.

text: In

text: principle,

text: one

text: can

text: do

text: a

text: sort

text: of

text: matching

text: typical

text: of

text: a

text: square-well

text: potential

text: problems

text: in

text: one-dimensional

text: quantum

text: mechanics.

text: In

text: contrast,

text: such

text: a

text: sort

text: of

text: matching

text: dif-

text: ferent

text: solutions

text: for

text: diﬀerent

text: regions

text: is

text: not

text: explicitly

text: existent

text: in

text: the

text: UA.

text: Instead,

text: there

text: is

text: a

text: single

text: approximating

text: solution

text: that

text: takes

text: on

text: all

text: the

text: burden

text: of

text: adjustment

text: to

text: ﬁt

text: the

text: boundary

text: or

text: initial

text: conditions

text: and

text: the

text: diﬀerential

text: equation

text: that

text: it

text: satisﬁes.

text: In

text: the

text: case

text: of

text: the

text: Mukhanov-Sasaki

text: equation,

text: the

text: “diﬀering”

text: behaviour

text: of

text: the

text: solu-

text: tion

text: at

text: extremal

text: ends

text: in

text: the

text: range

text: of

text: values

text: of

text: kη

text: is

text: taken

text: into

text: account

text: by

text: considering

text: the

text: turning-point.

text: More

text: speciﬁcally,

text: one

text: can

text: rewrite

text: the

text: Mukhanov-Sasaki

text: equation

text: for

text: scalar

text: perturbation

text: in

text: the

text: canonical

text: inﬂation

text: v

text: g

text: (η)

text: k

text: s

text: where

text: ν2

text: g

text: (η)

text: s

text: ν2

text: s

text: η2

text: s

text: s

text: expressions

text: hold

text: for

text: the

text: tensor

text: perturbations.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≥ ∈ { } Lemma 14. Let S be a double-star with center vertices u and v, where d(u), d(v) 2. For k 2, 3 , let c be (cid:48) ∪ (cid:48) a k-coloring of the edges incident to u. There exists a 2-coloring c of the edges incident to v such that c c is { } color-blind distinguishing among u, v . (cid:54) (cid:48) (cid:48) ∈ Proof. If d(u) = d(v), then let c (vx) = c (uv) for all x N(v). Since no adjacent vertices in S have the same degree, ∪ (cid:48) c c is color-blind distinguishing. ∗ − ≤ (cid:98) (cid:99) If d(u) = d(v) = d, then consider the color-blind partition at u to be c (u) = (d i, i) for some i d/2 . If { } \ { } (cid:48) ∈ \ { } ∪ (cid:48) i = 0, then select a color a from 1, 2 c(uv) and color c (vx) = a for all x N(v) u . In this case c c is − (cid:48) color-blind distinguishing since the color-blind partition at v is (d 1, 1). Otherwise, i > 0 and set c (vx) = c(uv) for ∈ ∪ (cid:48) all x N(v). In this case, c c is color-blind distinguishing since the color-blind partition at v is (d, 0). Blocks isomorphic to cycles are a bit more complicated. We can start by partitioning the vertices of a cycle into parts based on the degrees of the vertices. If the cycle is uniform, then there is only one part; otherwise there are multiple paths where all vertices have the same degree. Between distinct parts the vertices change degree, so no conﬂicts can occur there. We can extend around the cycle using extensions along the paths, but at least one path needs to extend the coloring from a partial coloring on both endpoints. First, we consider a path of vertices of degree 3 and assume all edges incident to the ﬁrst vertex are colored. We extend the coloring using two colors to be color-blind distinguishing among the vertices on the path. ≥ ∈ { } ∈ { } Lemma 15. Let v v . . . v be a path in G where t 3 and d(v ) = 3 for all i 1, . . . , t . For k 2, 3 , let c be a 1 2 t i (cid:48) ∪ (cid:48) k-coloring of the edges incident to v . There exists a 2-coloring c of the edges incident to v , . . . , v such that c c {1 } 2 t is color-blind distinguishing among v , . . . , v . 1 t Proof. Let u , . . . , u − be the neighbors of v , . . . , v − that are not on the path. Let u and v be the neighbors 2 t 1 2 t 1 ∈ { } \ { } ∈ { }t\ { } t+1 of v that are not on the path. Let a = c(v v ) and select b 1, 2 a and d 1, 2 b ; observe that a = d if t (cid:54) 1 2 and only if a = 3. ∗ ∈ { } ∈ If c (v ) = (3), then color the edges v v with color b for all i 2, . . . , t , v u with color d for i { (cid:98) 1(cid:99)} i i+1∈ { (cid:98) − (cid:99)} ∗ ∈2i{2i } ∗ 1, . . . , t/2 , and v u with color b for i 1, . . . , (t 1)/2 . Observe c (v ) (2, 1), (1, 1, 1) , c (v ) = 2i+1 2i+1 ∗ ∪ (cid:48) 2 i (2, 1) when i > 2 is even that c (v ) = (3) when i is odd, so c c is color-blind distinguishing among v , . . . , v . ∗ i ∈ { } 1 t If c (v ) = (2, 1) and a = d, then color the edges v v with color a for all i 2, . . . , t , v u with color a for ∈ { 1(cid:98) (cid:99)} ∈ i{ i+1 (cid:98) − (cid:99)} ∗ 2i 2i i 1, . . . , t/2 , and v u with color b for i 1, . . . , (t 1)/2 . Observe c (v ) = (3) when i is even and ∗ 2i+1 2i+1∪ (cid:48) i c (v ) = (2, 1) when i is odd, so c c is color-blind distinguishing among v , . . . , v . i ∗ (cid:54) 1 ∈t{ } If c (v ) = (2, 1) and a = d, then color the edges v v with color b for all i 2, . . . , t , v u with color d for ∈ { 1(cid:98) (cid:99)} ∈ {i i+1 (cid:98) − (cid:99)} ∗ 2i 2i ∗ i 1, . . . , t/2 , and v u with color b for i 1, . . . , (t 1)/2 . Observe c (v ) = (1, 1, 1), c (v ) = (2, 1) 2∗i+1 2i+1 ∪ (cid:48) 2 i when i > 2 is even and c (v ) = (3) when i is odd, so c c is color-blind distinguishing among v , . . . , v . i 1 t ≥ Now, we consider a path of vertices that all have the same degree d 4, and assume all edges incident to the ﬁrst vertex are colored and the last vertex has one colored edge and extend the coloring using two colors to be color-blind distinguishing among the vertices on the path. ≥ ≥ ∈ { } ∈ { } Lemma 16. Let v v . . . v be a path in G where t 2 and d(v ) = d 4 for all i 1, . . . , t . For k 2, 3 , let c 1 2 t i (cid:48) be a k-coloring of the edges indicent to v and one edge incident to v other than v − v . There exists a 2-coloring c 1 ∪ (cid:48) t t 1 t { } of the edges incident to v , . . . , v − such that c c is a-blind distinguishing coloring among v , . . . , v . 2 t 1 1 t ∈ { } Proof. Let v be the vertex such that the colored edge incident to v is v v . For i 2, . . . , t let u , . . . , u − t+1 t t t+1 ∈ { − i},1 i,d 2 be the neighbors of v other than v − and v . Color all edges v v with color 1 for i 2, . . . , t 1 . ∗ i ∈ {i 1 − i+}1 {i i+}1 ∗ − If c (v ) = (d), then for all i 2, . . . , t 1 assign colors from 1, 2 to the edges v u such that c (v ) = (d 1, 1) 1 ∗ i i,j ∈ { } i when i is even and c (v ) = (d) when i is odd. Depending on the parity of t and if c(v v ) 1, 2 , assign colors { } i ∗ ∈ { − − − }t \t+{1∗ } from 1, 2 to the edges v u such that c (v ) (d), (d 1, 1), (d 2, 2), (d 2, 1, 1) c (v − ) . The resulting t t,j t t 1 coloring is color-blind distinguishing among v , . . . , v . ∗ (cid:54) ∈ { −1 } t { } ∗ If c (v ) = (d), then for all i 2, . . . , t 1 assign colors from 1, 2 to the edges v u such that c (v ) = (d) 1 ∗ − i i,j ∈ { }i when i is even and c (v ) = (d 1, 1) when i is odd. Depending on the parity of t and if c(v v ) 1, 2 , assign { } i ∗ ∈ { − − − t }t+\1 { ∗ } colors from 1, 2 to the edges v u such that c (v ) (d), (d 1, 1), (d 2, 2), (d 2, 1, 1) c (v − ) . The t t,j t t 1 resulting coloring is color-blind distinguishing among v , . . . , v . 1 t We now combine the above path-extension lemmas to demonstrate that we can extend a coloring to a cycle block when the edges incident to one vertex are colored. 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: Let

text: S

text: be

text: a

text: double-star

text: with

text: center

text: vertices

text: u

text: and

text: v,

text: where

text: d(u),

text: d(v)

text: For

text: k

text: let

text: c

text: be

text: a

text: k-coloring

text: of

text: the

text: edges

text: incident

text: to

text: u.

text: There

text: exists

text: a

text: c

text: of

text: the

text: edges

text: incident

text: to

text: v

text: such

text: that

text: c

text: c

text: is

text: color-blind

text: distinguishing

text: among

text: u,

text: v

font: default

text: Proof.

text: If

text: d(u)

text: d(v),

text: then

text: let

text: c

text: (vx)

text: c

text: (uv)

text: for

text: all

text: x

text: N(v).

text: Since

text: no

text: adjacent

text: vertices

text: in

text: S

text: have

text: the

text: same

text: degree,

text: c

text: c

text: is

text: color-blind

text: distinguishing.

text: If

text: d(u)

text: d(v)

text: d,

text: then

text: consider

text: the

text: color-blind

text: partition

text: at

text: u

text: to

text: be

text: c

text: (u)

text: (d

text: i,

text: i)

text: for

text: some

text: i

text: If

text: i

text: then

text: select

text: a

text: color

text: a

text: from

text: c(uv)

text: and

text: color

text: c

text: (vx)

text: a

text: for

text: all

text: x

text: N(v)

text: u

text: In

text: this

text: case

text: c

text: c

text: is

text: color-blind

text: distinguishing

text: since

text: the

text: color-blind

text: partition

text: at

text: v

text: is

text: (d

text: Otherwise,

text: i

text: and

text: set

text: c

text: (vx)

text: c(uv)

text: for

text: all

text: x

text: N(v).

text: In

text: this

text: case,

text: c

text: c

text: is

text: color-blind

text: distinguishing

text: since

text: the

text: color-blind

text: partition

text: at

text: v

text: is

text: (d,

text: Blocks

text: isomorphic

text: to

text: cycles

text: are

text: a

text: bit

text: more

text: complicated.

text: We

text: can

text: start

text: by

text: partitioning

text: the

text: vertices

text: of

text: a

text: cycle

text: into

text: parts

text: based

text: on

text: the

text: degrees

text: of

text: the

text: vertices.

text: If

text: the

text: cycle

text: is

text: uniform,

text: then

text: there

text: is

text: only

text: one

text: part;

text: otherwise

text: there

text: are

text: multiple

text: paths

text: where

text: all

text: vertices

text: have

text: the

text: same

text: degree.

text: Between

text: distinct

text: parts

text: the

text: vertices

text: change

text: degree,

text: so

text: no

text: conﬂicts

text: can

text: occur

text: there.

text: We

text: can

text: extend

text: around

text: the

text: cycle

text: using

text: extensions

text: along

text: the

text: paths,

text: but

text: at

text: least

text: one

text: path

text: needs

text: to

text: extend

text: the

text: coloring

text: from

text: a

text: partial

text: coloring

text: on

text: both

text: endpoints.

text: First,

text: we

text: consider

text: a

text: path

text: of

text: vertices

text: of

text: degree

text: and

text: assume

text: all

text: edges

text: incident

text: to

text: the

text: ﬁrst

text: vertex

text: are

text: colored.

text: We

text: extend

text: the

text: coloring

text: using

text: two

text: colors

text: to

text: be

text: color-blind

text: distinguishing

text: among

text: the

text: vertices

text: on

text: the

text: path.

text: Lemma

text: Let

text: v

text: v

text: v

text: be

text: a

text: path

text: in

text: G

text: where

text: t

text: and

text: d(v

text: for

text: all

text: i

text: t

text: For

text: k

text: let

text: c

text: be

text: a

text: t

text: i

text: k-coloring

text: of

text: the

text: edges

text: incident

text: to

text: v

text: There

text: exists

text: a

text: c

text: of

text: the

text: edges

text: incident

text: to

text: v

text: v

text: such

text: that

text: c

text: c

text: t

text: is

text: color-blind

text: distinguishing

text: among

text: v

text: v

text: t

text: Proof.

text: Let

text: u

text: u

text: be

text: the

text: neighbors

text: of

text: v

text: v

text: that

text: are

text: not

text: on

text: the

text: path.

text: Let

text: u

text: and

text: v

text: be

text: the

text: neighbors

text: t

text: t

text: }t\

text: of

text: v

text: that

text: are

text: not

text: on

text: the

text: path.

text: Let

text: a

text: c(v

text: v

text: and

text: select

text: b

text: a

text: and

text: d

text: b

text: observe

text: that

text: a

text: d

text: if

text: t

text: and

text: only

text: if

text: a

text: If

text: c

text: (v

text: then

text: color

text: the

text: edges

text: v

text: v

text: with

text: color

text: b

text: for

text: all

text: i

text: t

text: v

text: u

text: with

text: color

text: d

text: for

text: i

text: i

text: and

text: v

text: u

text: with

text: color

text: b

text: for

text: i

text: (t

text: Observe

text: c

text: (v

text: c

text: (v

text: i

text: when

text: i

text: is

text: even

text: that

text: c

text: (v

text: when

text: i

text: is

text: odd,

text: so

text: c

text: c

text: is

text: color-blind

text: distinguishing

text: among

text: v

text: v

text: i

text: t

text: If

text: c

text: (v

text: and

text: a

text: d,

text: then

text: color

text: the

text: edges

text: v

text: v

text: with

text: color

text: a

text: for

text: all

text: i

text: t

text: v

text: u

text: with

text: color

text: a

text: for

text: i{

text: i

text: and

text: v

text: u

text: with

text: color

text: b

text: for

text: i

text: (t

text: Observe

text: c

text: (v

text: when

text: i

text: is

text: even

text: and

text: i

text: c

text: (v

text: when

text: i

text: is

text: odd,

text: so

text: c

text: c

text: is

text: color-blind

text: distinguishing

text: among

text: v

text: v

text: i

text: ∈t{

text: If

text: c

text: (v

text: and

text: a

text: d,

text: then

text: color

text: the

text: edges

text: v

text: v

text: with

text: color

text: b

text: for

text: all

text: i

text: t

text: v

text: u

text: with

text: color

text: d

text: for

text: {i

text: i

text: and

text: v

text: u

text: with

text: color

text: b

text: for

text: i

text: (t

text: Observe

text: c

text: (v

text: c

text: (v

text: i

text: when

text: i

text: is

text: even

text: and

text: c

text: (v

text: when

text: i

text: is

text: odd,

text: so

text: c

text: c

text: is

text: color-blind

text: distinguishing

text: among

text: v

text: v

text: i

text: t

text: Now,

text: we

text: consider

text: a

text: path

text: of

text: vertices

text: that

text: all

text: have

text: the

text: same

text: degree

text: d

text: and

text: assume

text: all

text: edges

text: incident

text: to

text: the

text: ﬁrst

text: vertex

text: are

text: colored

text: and

text: the

text: last

text: vertex

text: has

text: one

text: colored

text: edge

text: and

text: extend

text: the

text: coloring

text: using

text: two

text: colors

text: to

text: be

text: color-blind

text: distinguishing

text: among

text: the

text: vertices

text: on

text: the

text: path.

text: Lemma

text: Let

text: v

text: v

text: v

text: be

text: a

text: path

text: in

text: G

text: where

text: t

text: and

text: d(v

text: d

text: for

text: all

text: i

text: t

text: For

text: k

text: let

text: c

text: t

text: i

text: be

text: a

text: k-coloring

text: of

text: the

text: edges

text: indicent

text: to

text: v

text: and

text: one

text: edge

text: incident

text: to

text: v

text: other

text: than

text: v

text: v

text: There

text: exists

text: a

text: c

text: t

text: t

text: t

text: of

text: the

text: edges

text: incident

text: to

text: v

text: v

text: such

text: that

text: c

text: c

text: is

text: a-blind

text: distinguishing

text: coloring

text: among

text: v

text: v

text: t

text: t

text: Proof.

text: Let

text: v

text: be

text: the

text: vertex

text: such

text: that

text: the

text: colored

text: edge

text: incident

text: to

text: v

text: is

text: v

text: v

text: For

text: i

text: t

text: let

text: u

text: u

text: t

text: t

text: i,d

text: be

text: the

text: neighbors

text: of

text: v

text: other

text: than

text: v

text: and

text: v

text: Color

text: all

text: edges

text: v

text: v

text: with

text: color

text: for

text: i

text: t

text: i

text: {i

text: {i

text: If

text: c

text: (v

text: (d),

text: then

text: for

text: all

text: i

text: t

text: assign

text: colors

text: from

text: to

text: the

text: edges

text: v

text: u

text: such

text: that

text: c

text: (v

text: (d

text: i

text: i,j

text: i

text: when

text: i

text: is

text: even

text: and

text: c

text: (v

text: (d)

text: when

text: i

text: is

text: odd.

text: Depending

text: on

text: the

text: parity

text: of

text: t

text: and

text: if

text: c(v

text: v

text: assign

text: colors

text: i

text: }t

text: from

text: to

text: the

text: edges

text: v

text: u

text: such

text: that

text: c

text: (v

text: (d),

text: (d

text: (d

text: (d

text: c

text: (v

text: The

text: resulting

text: t

text: t,j

text: t

text: t

text: coloring

text: is

text: color-blind

text: distinguishing

text: among

text: v

text: v

text: −1

text: t

text: If

text: c

text: (v

text: (d),

text: then

text: for

text: all

text: i

text: t

text: assign

text: colors

text: from

text: to

text: the

text: edges

text: v

text: u

text: such

text: that

text: c

text: (v

text: (d)

text: i

text: i,j

text: }i

text: when

text: i

text: is

text: even

text: and

text: c

text: (v

text: (d

text: when

text: i

text: is

text: odd.

text: Depending

text: on

text: the

text: parity

text: of

text: t

text: and

text: if

text: c(v

text: v

text: assign

text: i

text: t

text: colors

text: from

text: to

text: the

text: edges

text: v

text: u

text: such

text: that

text: c

text: (v

text: (d),

text: (d

text: (d

text: (d

text: c

text: (v

text: The

text: t

text: t,j

text: t

text: t

text: resulting

text: coloring

text: is

text: color-blind

text: distinguishing

text: among

text: v

text: v

text: t

text: We

text: now

text: combine

text: the

text: above

text: path-extension

text: lemmas

text: to

text: demonstrate

text: that

text: we

text: can

text: extend

text: a

text: coloring

text: to

text: a

text: cycle

text: block

text: when

text: the

text: edges

text: incident

text: to

text: one

text: vertex

text: are

text: colored.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: p a y y1 y2 x w w c w 1 1 2 r u v x x 2 b z z z 1 2 q { } { } { } { } D = a, b, c, x, y, z, w , M = pq, wr D = x , y , z , w , x , y , z , w , M = uv 1 1 1 1 2 2 2 2 p w r u v q 1-diamond reduction 2-diamonds reduction p r b1 b2 a1 a2 a a c u 1 2 v 1 c 2 c1 c2 b1 b2 q s { } { } { } { } D = b , c , b , c , M = ua , a v D = a , b , c , a , b , c , M = pq, rs 1 1 2 2 1 2 1 1 1 2 2 2 p r a a u 1 2 v q s 2-triangle reduction Sparse reduction Figure 3: The Reducible Conﬁgurations and their Reductions. 8 ##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##

text: p

text: a

text: y

text: x

text: w

text: w

text: c

text: w

text: r

text: u

text: v

text: x

text: x

text: b

text: z

text: z

text: z

text: q

text: D

text: a,

text: b,

text: c,

text: x,

text: y,

text: z,

text: w

text: M

text: pq,

text: wr

text: D

text: x

text: y

text: z

text: w

text: x

text: y

text: z

text: w

text: M

text: uv

text: p

text: w

text: r

text: u

text: v

text: q

text: reduction

text: reduction

text: p

text: r

text: a

text: a

text: c

text: u

text: v

text: c

text: q

text: s

text: D

text: b

text: c

text: b

text: c

text: M

text: ua

text: a

text: v

text: D

text: a

text: b

text: c

text: a

text: b

text: c

text: M

text: pq,

text: rs

text: p

text: r

text: a

text: a

text: u

text: v

text: q

text: s

text: reduction

text: Sparse

text: reduction

text: Figure

text: The

text: Reducible

text: Conﬁgurations

text: and

text: their

text: Reductions.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − in terms of the atomic displacements R R(0) where R(0) are the atomic positions in a k k k reference state, which will be determined later. The transition rate is then, (cid:12) (cid:12) (cid:88) (cid:12) (cid:12) (cid:12) (cid:12) wiBfO = 2(cid:126)π M(cid:88)eBO(cid:12)(cid:12)((cid:88){R(j0)}) 2 |(cid:104)Xf|Xi(cid:105)|2 δ(Θf − Θi + (cid:15)if) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) f (cid:12) (cid:12) (cid:12) 2 2π ∇ { } · (cid:104) | − | (cid:105) − + (cid:126) MBO( R(0) ) X (R R(0)) X δ(Θ Θ + (cid:15) ) + ...(.14) Rk e j f k k i f i if f k Here the cross terms are dropped because the zeroth order and ﬁrst order terms cannot have the same ﬁnal phonon wave functions – the number of phonons needed to ensure a nonzero overlap matrix element are different for the two cases. The ﬁrst term in this expansion represents a com- plete separation of the electron and phonon wave functions as if they are independent of each other and corresponds to the Frank-Condon approximation. The second term is the ﬁrst order correction to the Frank-Condon approximation arising from the BOA perturbation Hamiltonian HBO. 1 B. Non-adiabatic series According to Huang [20], the non-adiabatic matrix element deﬁned in Eq. (10) can be evaluated for linear phonon coupling, (cid:88) (cid:104) { } |∇ { } | { } (cid:105) · (cid:104) | − | (cid:105) Ψ ( R(0) ) H ( R(0) ) Ψ ( R(0) ) X (R R(0)) X , (15) f j Rk e j i j f k k i k where H is the electron part of the Hamiltonian. When electron-phonon coupling H = e ep { } − { } He( Rj ) He( R(j0) ) is introduced, th(cid:88)e electron wave functions are changed by a perturbation, (cid:104) | | (cid:105) | { } (cid:105) Ψ (cid:48) H Ψ | { } (cid:105) δΨ ( R ) = i −ep i Ψ (cid:48)( R(0) ) , (16) i j (cid:15) (cid:48) (cid:15) i j (cid:48)(cid:54) i i i =i (and a similar equation for the ﬁnal states). We write both the initial and ﬁnal states in the form, | { } (cid:105) | { } (cid:105) | (cid:105) Ψ ( R ) = Ψ ( R(0) ) + δΨ . (17) i(f) j i(f) j i(f) Substituting this into Eq (10) and keeping only the linear terms, 11 ##LTLine## ##LTLine## ##LTLine##

text: in

text: terms

text: of

text: the

text: atomic

text: displacements

text: R

text: where

text: are

text: the

text: atomic

text: positions

text: in

text: a

text: k

text: k

text: k

text: reference

text: state,

text: which

text: will

text: be

text: determined

text: later.

text: The

text: transition

text: rate

text: is

text: then,

text: wiBfO

text: δ(Θf

text: Θi

text: f

text: 2π

text: MBO(

text: X

text: (R

text: X

text: δ(Θ

text: Θ

text: Rk

text: e

text: j

text: f

text: k

text: k

text: i

text: f

text: i

text: if

text: f

text: k

text: Here

text: the

text: cross

text: terms

text: are

text: dropped

text: because

text: the

text: zeroth

text: order

text: and

text: ﬁrst

text: order

text: terms

text: cannot

text: have

text: the

text: same

text: ﬁnal

text: phonon

text: wave

text: functions

text: the

text: number

text: of

text: phonons

text: needed

text: to

text: ensure

text: a

text: nonzero

text: overlap

text: matrix

text: element

text: are

text: different

text: for

text: the

text: two

text: cases.

text: The

text: ﬁrst

text: term

text: in

text: this

text: expansion

text: represents

text: a

text: com-

text: plete

text: separation

text: of

text: the

text: electron

text: and

text: phonon

text: wave

text: functions

text: as

text: if

text: they

text: are

text: independent

text: of

text: each

text: other

text: and

text: corresponds

text: to

text: the

text: Frank-Condon

text: approximation.

text: The

text: second

text: term

text: is

text: the

text: ﬁrst

text: order

text: correction

text: to

text: the

text: Frank-Condon

text: approximation

text: arising

text: from

text: the

text: BOA

text: perturbation

text: Hamiltonian

text: HBO.

text: B.

text: Non-adiabatic

text: series

text: According

text: to

text: Huang

text: the

text: non-adiabatic

text: matrix

text: element

text: deﬁned

text: in

text: Eq.

text: can

text: be

text: evaluated

text: for

text: linear

text: phonon

text: coupling,

text: Ψ

text: H

text: Ψ

text: X

text: (R

text: X

text: f

text: j

text: Rk

text: e

text: j

text: i

text: j

text: f

text: k

text: k

text: i

text: k

text: where

text: H

text: is

text: the

text: electron

text: part

text: of

text: the

text: Hamiltonian.

text: When

text: electron-phonon

text: coupling

text: H

text: e

text: ep

text: He(

text: Rj

text: He(

text: is

text: introduced,

text: electron

text: wave

text: functions

text: are

text: changed

text: by

text: a

text: perturbation,

text: Ψ

text: H

text: Ψ

text: δΨ

text: R

text: i

text: −ep

text: i

text: Ψ

text: i

text: j

text: i

text: j

text: i

text: i

text: i

text: =i

text: (and

text: a

text: similar

text: equation

text: for

text: the

text: ﬁnal

text: states).

text: We

text: write

text: both

text: the

text: initial

text: and

text: ﬁnal

text: states

text: in

text: the

text: form,

text: Ψ

text: R

text: Ψ

text: δΨ

text: i(f)

text: j

text: i(f)

text: j

text: i(f)

text: Substituting

text: this

text: into

text: Eq

text: and

text: keeping

text: only

text: the

text: linear

text: terms,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:88) (cid:48) (cid:48) − 1 H = H Φ (cid:48) (cid:48)∆ u (cid:48) (cid:48) (27) f i N kl,k l k k l (cid:48) (cid:48) kl,k l Transforming to the normal-mode representation in terms of the generalized coordinates, (cid:88) √ √1 q = m u w , (28) j k kl j,kl N kl where w is the klth element of the eigenvector for mode j. Note that in this deﬁnition of the j,kl √ generalized coordinate(cid:88)qj, it has ab(cid:88)sorbed the mass(cid:88)factor(cid:88)mk. The Hamiltonian is expressed as, √ (cid:48) 1 1 − √1 H = q˙2 + ω2q2 q D (cid:48)(k )w (cid:48) m ∆ , (29) f 2 j 2 j j N j (cid:48) kk j jk k k j j j kk · where ω are the eigenfrequencies. A phase factor of the form exp(ik r (cid:48)), where k is the j j l j wave vector of mode j, from w (cid:48) (cid:48) is absorbed into the force constant matrix Φ yielding the j,k l (cid:48) dynamical matrix D, and reducing w (cid:48) (cid:48) to w (cid:48) (independent of l ). Since we assume that force j,k l jk constants remain the same after electron capture, (cid:88) D (cid:48)(k )w (cid:48) = ω2w (30) kk j jk j jk (cid:48) k The linear term causes a general coordinate displacement, (cid:88) √ −√1 δq = m ∆ w . (31) j k k jk N k We can express the normal coordinates of the lattice for the ﬁnal (f) state, qf, in terms of those j for the initial (i) state, q , j q = q + δq , (32) f,j j j so that the ﬁnal Hamiltonian is: 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: H

text: H

text: Φ

text: u

text: f

text: i

text: N

text: kl,k

text: l

text: k

text: k

text: l

text: kl,k

text: l

text: Transforming

text: to

text: the

text: normal-mode

text: representation

text: in

text: terms

text: of

text: the

text: generalized

text: coordinates,

text: √1

text: q

text: m

text: u

text: w

text: j

text: k

text: kl

text: j,kl

text: N

text: kl

text: where

text: w

text: is

text: the

text: klth

text: element

text: of

text: the

text: eigenvector

text: for

text: mode

text: j.

text: Note

text: that

text: in

text: this

text: deﬁnition

text: of

text: the

text: j,kl

text: generalized

text: it

text: has

text: the

text: The

text: Hamiltonian

text: is

text: expressed

text: as,

text: √1

text: H

text: q

text: D

text: )w

text: m

text: f

text: j

text: j

text: j

text: N

text: j

text: kk

text: j

text: jk

text: k

text: k

text: j

text: j

text: j

text: kk

text: where

text: ω

text: are

text: the

text: eigenfrequencies.

text: A

text: phase

text: factor

text: of

text: the

text: form

text: exp(ik

text: r

text: where

text: k

text: is

text: the

text: j

text: j

text: l

text: j

text: wave

text: vector

text: of

text: mode

text: j,

text: from

text: w

text: is

text: absorbed

text: into

text: the

text: force

text: constant

text: matrix

text: Φ

text: yielding

text: the

text: j,k

text: l

text: dynamical

text: matrix

text: D,

text: and

text: reducing

text: w

text: to

text: w

text: (independent

text: of

text: l

text: Since

text: we

text: assume

text: that

text: force

text: j,k

text: l

text: jk

text: constants

text: remain

text: the

text: same

text: after

text: electron

text: capture,

text: D

text: )w

text: kk

text: j

text: jk

text: j

text: jk

text: k

text: The

text: linear

text: term

text: causes

text: a

text: general

text: coordinate

text: displacement,

text: −√1

text: δq

text: m

text: w

text: j

text: k

text: k

text: jk

text: N

text: k

text: We

text: can

text: express

text: the

text: normal

text: coordinates

text: of

text: the

text: lattice

text: for

text: the

text: ﬁnal

text: (f)

text: state,

text: qf,

text: in

text: terms

text: of

text: those

text: j

text: for

text: the

text: initial

text: (i)

text: state,

text: q

text: j

text: q

text: q

text: δq

text: f,j

text: j

text: j

text: so

text: that

text: the

text: ﬁnal

text: Hamiltonian

text: is:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the ﬂat band approximation that is complemented with the requirement of ﬁnite spacing between the bands. We ﬁnally ﬁnd, (cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21) (cid:126) (cid:126) p ω − ω S F = exp j j S coth j I (cid:126) j , (42) j 2kT j 2kT pj sinh( ω /2kT ) j and   (cid:20) (cid:21) (cid:12) (cid:12) (cid:32) (cid:33)       (cid:12) (cid:88)  (cid:89) (cid:88)   (cid:12) (cid:20) (cid:21) (cid:12) S (cid:12) 1  M M  S Ipj+1 sinh((cid:126)ωj/2kT )  (cid:12)(cid:12) F = F p + (cid:126) j j D(ω ) , Ω j j sinh( ω /2kT ) S j (cid:80) k {pj} j=1 j=1 j Ipj sinh((cid:126)ωjj/2kT ) Mj=1 pj(cid:126)ωj+(cid:15)if=0 (43) where ω S = (cid:126)j Nδq2, (44) j 2 j and I is the modiﬁed Bessel function of order p. p B. Linear phonon matrix elements To evaluate the phonon matrix elements for the linear term, we rewrite it in terms of the normal mode coordinates q , (cid:12) j (cid:12) (cid:88) (cid:12)(cid:88) (cid:12) (cid:88) (cid:88) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2 (cid:104) | | (cid:105) | (cid:104) | | (cid:105)| (cid:12) M X q X = M(cid:12) X q X 2 (cid:12) (cid:12) j f j i (cid:88) j(cid:12) f (cid:89)j i (cid:12) (cid:12) (cid:12) (cid:12) f j f j (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2(cid:12) 1 ∂2 (cid:104) | | (cid:105) = X [1 + λM q exp(iφ )] X , (45) 2 ∂λ2 f j j j i f j λ=0 where φ is a random phase introduced to cancel out the cross terms, and, j (cid:104) | | (cid:105) − (cid:104) | (cid:105)(cid:104) | | (cid:105) M = Φ ∂ H Φ Φ Ψ Ψ ∂ H Ψ . (46) j f qj e i f i f qj e f 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: ﬂat

text: band

text: approximation

text: that

text: is

text: complemented

text: with

text: the

text: requirement

text: of

text: ﬁnite

text: spacing

text: between

text: the

text: bands.

text: We

text: ﬁnally

text: ﬁnd,

text: p

text: ω

text: ω

text: S

text: F

text: exp

text: j

text: j

text: S

text: coth

text: j

text: I

text: j

text: j

text: j

text: pj

text: sinh(

text: ω

text: j

text: and

text: S

text: M

text: M

text: S

text: F

text: F

text: p

text: j

text: j

text: D(ω

text: Ω

text: j

text: j

text: sinh(

text: ω

text: S

text: j

text: k

text: {pj}

text: j

text: Ipj

text: where

text: ω

text: S

text: j

text: j

text: and

text: I

text: is

text: the

text: modiﬁed

text: Bessel

text: function

text: of

text: order

text: p.

text: p

text: B.

text: Linear

text: phonon

text: matrix

text: elements

text: To

text: evaluate

text: the

text: phonon

text: matrix

text: elements

text: for

text: the

text: linear

text: term,

text: we

text: rewrite

text: it

text: in

text: terms

text: of

text: the

text: normal

text: mode

text: coordinates

text: q

text: j

text: M

text: X

text: q

text: X

text: X

text: q

text: X

text: j

text: f

text: j

text: i

text: f

text: i

text: f

text: j

text: f

text: j

text: ∂2

text: X

text: λM

text: q

text: exp(iφ

text: X

text: ∂λ2

text: f

text: j

text: j

text: j

text: i

text: f

text: j

text: where

text: φ

text: is

text: a

text: random

text: phase

text: introduced

text: to

text: cancel

text: out

text: the

text: cross

text: terms,

text: and,

text: j

text: M

text: Φ

text: H

text: Φ

text: Φ

text: Ψ

text: Ψ

text: H

text: Ψ

text: j

text: f

text: qj

text: e

text: i

text: f

text: i

text: f

text: qj

text: e

text: f

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: )2mc( 71-01 x )E( σ the energy bin is important. For capture cross sections, one is often interested only in the thresh- old value. The calculations presented here are a prelude to calculations of hot-electron inelastic multiphonon scattering, for which the energy dependence is important. The energy dependence is also important in luminescence curves, i.e., the classic Huang-Rhys problem that was treated in the single-phonon approximation in Ref. 23 (in the case of luminescence, MPPs dissipate only the relaxation energy of the defect, when one expects the phonon mode corresponding to the actual re- laxation to dominate; nevertheless, a fully convergent calculation would be needed to establish the degree of accuracy one obtains with the single-mode approximation). The accuracy of the calcu- 6.0 40 meV energy bin 50 meV energy bin 5.0 75 meV energy bin 4.0 3.0 2.0 1.0 0.0 0.3 0.4 0.5 0.6 0.7 Electron energy (eV) FIG. 4: Calculated full capture cross section using the electron matrix element from Fig. 1 and 12 phonon modes in the line-shape function. lation of the line shape function is controlled by the accuracy of the calculation of the generalized displacements. The latter depends on the accuracy of the calculation of the atomic displacements. We found that accuracy is enhanced signiﬁcantly if we allow the entire supercell to relax, which 27 ##LTFigure##

text: x

text: )E(

text: σ

text: the

text: energy

text: bin

text: is

text: important.

text: For

text: capture

text: cross

text: sections,

text: one

text: is

text: often

text: interested

text: only

text: in

text: the

text: thresh-

text: old

text: value.

text: The

text: calculations

text: presented

text: here

text: are

text: a

text: prelude

text: to

text: calculations

text: of

text: hot-electron

text: inelastic

text: multiphonon

text: scattering,

text: for

text: which

text: the

text: energy

text: dependence

text: is

text: important.

text: The

text: energy

text: dependence

text: is

text: also

text: important

text: in

text: luminescence

text: curves,

text: i.e.,

text: the

text: classic

text: Huang-Rhys

text: problem

text: that

text: was

text: treated

text: in

text: the

text: single-phonon

text: approximation

text: in

text: Ref.

text: (in

text: the

text: case

text: of

text: luminescence,

text: MPPs

text: dissipate

text: only

text: the

text: relaxation

text: energy

text: of

text: the

text: defect,

text: when

text: one

text: expects

text: the

text: phonon

text: mode

text: corresponding

text: to

text: the

text: actual

text: re-

text: laxation

text: to

text: dominate;

text: nevertheless,

text: a

text: fully

text: convergent

text: calculation

text: would

text: be

text: needed

text: to

text: establish

text: the

text: degree

text: of

text: accuracy

text: one

text: obtains

text: with

text: the

text: single-mode

text: approximation).

text: The

text: accuracy

text: of

text: the

text: calcu-

text: meV

text: energy

text: bin

text: meV

text: energy

text: bin

text: meV

text: energy

text: bin

text: Electron

text: energy

text: (eV)

text: FIG.

text: Calculated

text: full

text: capture

text: cross

text: section

text: using

text: the

text: electron

text: matrix

text: element

text: from

text: Fig.

text: and

text: phonon

text: modes

text: in

text: the

text: line-shape

text: function.

text: lation

text: of

text: the

text: line

text: shape

text: function

text: is

text: controlled

text: by

text: the

text: accuracy

text: of

text: the

text: calculation

text: of

text: the

text: generalized

text: displacements.

text: The

text: latter

text: depends

text: on

text: the

text: accuracy

text: of

text: the

text: calculation

text: of

text: the

text: atomic

text: displacements.

text: We

text: found

text: that

text: accuracy

text: is

text: enhanced

text: signiﬁcantly

text: if

text: we

text: allow

text: the

text: entire

text: supercell

text: to

text: relax,

text: which

text: ##LTFigure##


text:   (cid:88) ˆ   N N n + 1 ω S± = f ± = f ±dk = (cid:126)Nδq2, (5) k, k, Ω 2 k n k=1 where Ω is the volume of the reciprocal space Brillouin zone. In the last step we assumed that k the frequency and displacement do not change with k. In order to evaluated the last factor that includes the δ function, we note that each term in the summation over m has a different ωf , which spans the entire phonon band when m scans from 1 m to N. Thus as we convert the sum over m to integral over k, the argument ωf is also converted to m ω , (cid:40) (cid:32) (cid:33)(cid:41) k (cid:88) (cid:88) (cid:88) (cid:88) N (cid:126)(cid:32) (cid:126) − (cid:126) N (cid:126) − (cid:33) f δ ωf + ωf ωf + ni (ωf ωi) + (cid:15) ˆm,+ m ∈ (cid:88)− l ∈(cid:88) k (cid:88)j j j if m=1 l s+p 1 k s j=1 ≈ N (cid:12) (cid:126) (cid:126) − (cid:126) N (cid:126) − f (cid:12) δ ωf + ωf ωf + ni (ωf ωi) + (cid:15) dk Ω k,(cid:12)+ k l k j j j if k (cid:12) l∈s+p−1 k∈s j=1 D(ω) (cid:80) (cid:80) (cid:80) = S , (6) + Ωk (cid:126)ωf+ ∈ − (cid:126)ωf− ∈ (cid:126)ωf+ N ni(cid:126)(ωf−ωi)+(cid:15) =0 l s+p 1 l k s k j=1 j j j if where D(ω) is the phonon density of states. Combining the above equations and then setting all frequencies to ω, Eq. (4) n(cid:40)ow be(cid:41)comes, (cid:12) (cid:89) (cid:12) (cid:12) (cid:12) 1 N D(ω) (cid:80) t S−s Ss+p . (7) s!(s + p)! j=1 + Ωk p(cid:126)ω+ Nj=1 nij(cid:126)(ωjf−ωji)+(cid:15)if=0 But there is one such contribution for each ω or ω in the δ function, regardless of the sign of k l the frequency. For s modes subtracting a phonon and s + p modes adding a phonon there are total 2s + p such contributions. W(cid:40)e thus s(cid:41)um over all the t(cid:12)erms and obtain, (cid:89) (cid:12) (cid:12) (cid:12) 2s + p N D(ω) (cid:80) t S−s Ss+p . (8) s!(s(cid:81)+ p)! j=1 j + Ωk p(cid:126)ω+ Nj=1 nij(cid:126)(ωjf−ωji)+(cid:15)if=0 Finally, the factor N t is, (cid:89) (cid:12)(cid:12) j=1 j (cid:12)(cid:12) (cid:20) (cid:21) (cid:12)ˆ (cid:12) (cid:12) (cid:12) N 2 − (2n + 1)ω 2N − X (q )X (q + δq )dq = 1 (cid:126) δq2 = exp [ (S + S−)] . (9) nj j nj j j j 4 + j=1 39 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ˆ

text: N

text: N

text: n

text: ω

text: S±

text: f

text: f

text: ±dk

text: k,

text: k,

text: Ω

text: k

text: n

text: where

text: Ω

text: is

text: the

text: volume

text: of

text: the

text: reciprocal

text: space

text: Brillouin

text: zone.

text: In

text: the

text: last

text: step

text: we

text: assumed

text: that

text: k

text: the

text: frequency

text: and

text: displacement

text: do

text: not

text: change

text: with

text: k.

text: In

text: order

text: to

text: evaluated

text: the

text: last

text: factor

text: that

text: includes

text: the

text: δ

text: function,

text: we

text: note

text: that

text: each

text: term

text: in

text: the

text: summation

text: over

text: m

text: has

text: a

text: different

text: ωf

text: which

text: spans

text: the

text: entire

text: phonon

text: band

text: when

text: m

text: scans

text: from

text: m

text: to

text: N.

text: Thus

text: as

text: we

text: convert

text: the

text: sum

text: over

text: m

text: to

text: integral

text: over

text: k,

text: the

text: argument

text: ωf

text: is

text: also

text: converted

text: to

text: m

text: ω

text: k

text: N

text: N

text: f

text: δ

text: ωf

text: ωf

text: ωf

text: ni

text: (ωf

text: ωi)

text: ˆm,+

text: m

text: l

text: k

text: j

text: j

text: if

text: l

text: s+p

text: k

text: s

text: N

text: N

text: f

text: δ

text: ωf

text: ωf

text: ωf

text: ni

text: (ωf

text: ωi)

text: dk

text: Ω

text: k

text: l

text: k

text: j

text: j

text: j

text: if

text: k

text: k∈s

text: D(ω)

text: S

text: Ωk

text: N

text: l

text: s+p

text: l

text: k

text: s

text: k

text: j

text: j

text: j

text: if

text: where

text: D(ω)

text: is

text: the

text: phonon

text: density

text: of

text: states.

text: Combining

text: the

text: above

text: equations

text: and

text: then

text: setting

text: all

text: frequencies

text: to

text: ω,

text: Eq.

text: N

text: D(ω)

text: t

text: S−s

text: Ss+p

text: s!(s

text: p)!

text: Ωk

text: But

text: there

text: is

text: one

text: such

text: contribution

text: for

text: each

text: ω

text: or

text: ω

text: in

text: the

text: δ

text: function,

text: regardless

text: of

text: the

text: sign

text: of

text: k

text: l

text: the

text: frequency.

text: For

text: s

text: modes

text: subtracting

text: a

text: phonon

text: and

text: s

text: p

text: modes

text: adding

text: a

text: phonon

text: there

text: are

text: total

text: p

text: such

text: contributions.

text: thus

text: over

text: all

text: the

text: and

text: obtain,

text: p

text: N

text: D(ω)

text: t

text: S−s

text: Ss+p

text: p)!

text: j

text: Ωk

text: Finally,

text: the

text: factor

text: N

text: t

text: is,

text: j

text: N

text: X

text: (q

text: )X

text: (q

text: δq

text: )dq

text: exp

text: (S

text: S−)]

text: nj

text: j

text: nj

text: j

text: j

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: → 1 n (cid:126) − , (16) j exp( ω /kT ) 1 j (cid:18) (cid:19) (cid:126) n + 1 → ω j exp , (17) n kT j (cid:18) (cid:19) (cid:126) → ω 2n + 1 coth , (18) j 2kT and (cid:113) → 1 2 n (n + 1) (cid:126) , (19) j j sinh( ω/2kT ) we obtain Eqs. (42) and (43). [1] M.H. Evans, X.-G. Zhang, J.D. Joannopoulos, and S.T. Pantelides, Phys. Rev. Lett. 95, 106802 (2005). [2] G. Hadjisavvas, L. Tsetseris, and S.T. Pantelides, IEEE Electron Dev. Lett. 28, 1018 (2007). [3] O. D. Restrepo, K. Varga, and S. T. Pantelides, Appl. Phys. Lett. 94, 212103 (2009). [4] D.J. DiMaria, J. Appl. Phys. 86, 2100 (1999). [5] D.J. DiMaria, J. Appl. Phys. 87, 8707 (2000). [6] G. Meneghesso, G. Verzellesi, F. Danesin, F. Rampazzo, A. Tazzoli, M. Meneghini, and E. Zanoni, IEEE Trans. Dev. Mater. Reliab. 8, 332 (2008). [7] S.T. Pantelides, Y. Puzyrev, X. Shen, T. Roy, S. DasGupta, B.R. Tuttle, D.M. Fleetwood, and R.D. Schrimpf, Microelectron. Eng. 90, 3 (2012). [8] X. Shen, S. DasGupta, R. Reed, R. Schrimpf, D. Fleetwood, and S. Pantelides, J. Appl. Phys. 108, 114505 (2010). [9] M. Meneghini, U. Zehnder, B. Hahn, G. Meneghesso, and E. Zanoni, IEEE Electron Dev. Lett. 30, 1051 (2009). 42 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: n

text: j

text: exp(

text: ω

text: /kT

text: j

text: n

text: ω

text: j

text: exp

text: n

text: kT

text: j

text: ω

text: coth

text: j

text: and

text: n

text: (n

text: j

text: j

text: sinh(

text: we

text: obtain

text: Eqs.

text: and

label: reference

text: M.H.

text: Evans,

text: X.-G.

text: Zhang,

text: J.D.

text: Joannopoulos,

text: and

text: S.T.

text: Pantelides,

text: Phys.

text: Rev.

text: Lett.

text: G.

text: Hadjisavvas,

text: L.

text: Tsetseris,

text: and

text: S.T.

text: Pantelides,

text: IEEE

text: Electron

text: Dev.

text: Lett.

text: O.

text: D.

text: Restrepo,

text: K.

text: Varga,

text: and

text: S.

text: T.

text: Pantelides,

text: Appl.

text: Phys.

text: Lett.

text: D.J.

text: DiMaria,

text: J.

text: Appl.

text: Phys.

text: D.J.

text: DiMaria,

text: J.

text: Appl.

text: Phys.

text: G.

text: Meneghesso,

text: G.

text: Verzellesi,

text: F.

text: Danesin,

text: F.

text: Rampazzo,

text: A.

text: Tazzoli,

text: M.

text: Meneghini,

text: and

text: E.

text: Zanoni,

text: IEEE

text: Trans.

text: Dev.

text: Mater.

text: Reliab.

text: S.T.

text: Pantelides,

text: Y.

text: Puzyrev,

text: X.

text: Shen,

text: T.

text: Roy,

text: S.

text: DasGupta,

text: B.R.

text: Tuttle,

text: D.M.

text: Fleetwood,

text: and

text: R.D.

text: Schrimpf,

text: Microelectron.

text: Eng.

text: X.

text: Shen,

text: S.

text: DasGupta,

text: R.

text: Reed,

text: R.

text: Schrimpf,

text: D.

text: Fleetwood,

text: and

text: S.

text: Pantelides,

text: J.

text: Appl.

text: Phys.

text: M.

text: Meneghini,

text: U.

text: Zehnder,

text: B.

text: Hahn,

text: G.

text: Meneghesso,

text: and

text: E.

text: Zanoni,

text: IEEE

text: Electron

text: Dev.

text: Lett.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Occlusion Coherence: Detecting and Localizing Occluded Faces Golnaz Ghiasi, Charless C. Fowlkes University of California at Irvine, Irvine, CA 92697 Abstract The presence of occluders signiﬁcantly impacts object recognition accuracy. However, occlusion is typically treated as an unstruc- tured source of noise and explicit models for occluders have lagged behind those for object appearance and shape. In this paper we describe a hierarchical deformable part model for face detection and landmark localization that explicitly models part occlusion. The proposed model structure makes it possible to augment positive training data with large numbers of synthetically occluded instances. This allows us to easily incorporate the statistics of occlusion patterns in a discriminatively trained model. We test the model on several benchmarks for landmark localization and detection including challenging new data sets featuring signiﬁcant occlusion. We ﬁnd that the addition of an explicit occlusion model yields a detection system that outperforms existing approaches for occluded instances while maintaining competitive accuracy in detection and landmark localization for unoccluded instances. Keywords: Object Recognition, Face Detection, Occlusion, Deformable Part Model 1. Introduction Accurate localization of facial landmarks provides an im- portant building block for many applications including iden- tiﬁcation [1] and analysis of facial expressions [2]. Signiﬁ- cant progress has been made in this task, aided in part by the fact that faces have less intra-category shape variation and lim- ited articulation compared to other object categories of interest. However, feature point localization tends to break down when applied to faces in real scenes where other objects in the scene (hair, sunglasses, other people) are likely to occlude parts of the face. Fig. 1(a) depicts the output of a deformable part model [3] where the presence of occluders distorts the ﬁnal alignment of the model. A standard approach to handling occlusion in part-based models is to compete part feature scores against a generic back- ground model or ﬁxed threshold (as in Fig. 1(b)). However, setting such thresholds is fraught with difﬁculty since it is hard to distinguish between parts that are present but simply hard to detect (e.g., due to unusual lighting) and those which are gen- uinely hidden behind another object. Treating occlusions as an unstructured source of noise ig- (a) (b) (c) nores a key aspect of the problem, namely that occlusions are Figure 1: Occlusion impacts part localization performance. In panel (a) the induced by other objects and surfaces in the scene and hence output of a deformable part model [3] is distorted by the presence of occluders, should exhibit occlusion coherence. For example, it would disrupting localization even for parts that are far from the site of occlusion. seem very unlikely that every-other landmark along an object (b) Introducing independent occlusion of each part results in better alignment contour would happen to be occluded. Yet many occlusion but occlusion is treated as an outlier process and prediction of occlusion state is inaccurate. (c) The output of our hierarchical part model, which explicitly models make strong independence assumptions about occlu- models likely patterns of occlusion, shows improved localization as well as sion, making it difﬁcult to distinguish a priori likely from un- accurate prediction of which landmarks are occluded. likely patterns. Ultimately, an occluder should not be inferred simply by the lack of evidence for object features, but rather by positive evidence for the occluding object that explains away formable part model that encodes these principles for modeling the lack of object features. occlusion and achieves state-of-the-art performance on bench- The contribution of this paper is an efﬁcient hierarchical de- marks for occluded face localization and detection (depicted in Preprint submitted to Computer Vision and Image Understanding May 27, 2020 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: Occlusion

label: title

text: Coherence:

text: Detecting

text: and

text: Localizing

text: Occluded

text: Faces

text: Golnaz

label: author

text: Ghiasi,

text: Charless

text: C.

text: Fowlkes

text: University

text: of

text: California

text: at

text: Irvine,

text: Irvine,

text: CA

text: Abstract

text: The

label: abstract

text: presence

text: of

text: occluders

text: signiﬁcantly

text: impacts

text: object

text: recognition

text: accuracy.

text: However,

text: occlusion

text: is

text: typically

text: treated

text: as

text: an

text: unstruc-

text: tured

text: source

text: of

text: noise

text: and

text: explicit

text: models

text: for

text: occluders

text: have

text: lagged

text: behind

text: those

text: for

text: object

text: appearance

text: and

text: shape.

text: In

text: this

text: paper

text: we

text: describe

text: a

text: hierarchical

text: deformable

text: part

text: model

text: for

text: face

text: detection

text: and

text: landmark

text: localization

text: that

text: explicitly

text: models

text: part

text: occlusion.

text: The

text: proposed

text: model

text: structure

text: makes

text: it

text: possible

text: to

text: augment

text: positive

text: training

text: data

text: with

text: large

text: numbers

text: of

text: synthetically

text: occluded

text: instances.

text: This

text: allows

text: us

text: to

text: easily

text: incorporate

text: the

text: statistics

text: of

text: occlusion

text: patterns

text: in

text: a

text: discriminatively

text: trained

text: model.

text: We

text: test

text: the

text: model

text: on

text: several

text: benchmarks

text: for

text: landmark

text: localization

text: and

text: detection

text: including

text: challenging

text: new

text: data

text: sets

text: featuring

text: signiﬁcant

text: occlusion.

text: We

text: ﬁnd

text: that

text: the

text: addition

text: of

text: an

text: explicit

text: occlusion

text: model

text: yields

text: a

text: detection

text: system

text: that

text: outperforms

text: existing

text: approaches

text: for

text: occluded

text: instances

text: while

text: maintaining

text: competitive

text: accuracy

text: in

text: detection

text: and

text: landmark

text: localization

text: for

text: unoccluded

text: instances.

text: Keywords:

text: Object

text: Recognition,

text: Face

text: Detection,

text: Occlusion,

text: Deformable

text: Part

text: Model

text: Introduction

text: Accurate

text: localization

text: of

text: facial

text: landmarks

text: provides

text: an

text: im-

text: portant

text: building

text: block

text: for

text: many

text: applications

text: including

text: iden-

text: tiﬁcation

text: and

text: analysis

text: of

text: facial

text: expressions

text: Signiﬁ-

text: cant

text: progress

text: has

text: been

text: made

text: in

text: this

text: task,

text: aided

text: in

text: part

text: by

text: the

text: fact

text: that

text: faces

text: have

text: less

text: intra-category

text: shape

text: variation

text: and

text: lim-

text: ited

text: articulation

text: compared

text: to

text: other

text: object

text: categories

text: of

text: interest.

text: However,

text: feature

text: point

text: localization

text: tends

text: to

text: break

text: down

text: when

text: applied

text: to

text: faces

text: in

text: real

text: scenes

text: where

text: other

text: objects

text: in

text: the

text: scene

text: (hair,

text: sunglasses,

text: other

text: people)

text: are

text: likely

text: to

text: occlude

text: parts

text: of

text: the

text: face.

text: Fig.

text: depicts

text: the

text: output

text: of

text: a

text: deformable

text: part

text: model

text: where

text: the

text: presence

text: of

text: occluders

text: distorts

text: the

text: ﬁnal

text: alignment

text: of

text: the

text: model.

text: A

text: standard

text: approach

text: to

text: handling

text: occlusion

text: in

text: part-based

text: models

text: is

text: to

text: compete

text: part

text: feature

text: scores

text: against

text: a

text: generic

text: back-

text: ground

text: model

text: or

text: ﬁxed

text: threshold

text: (as

text: in

text: Fig.

text: However,

text: setting

text: such

text: thresholds

text: is

text: fraught

text: with

text: difﬁculty

text: since

text: it

text: is

text: hard

text: to

text: distinguish

text: between

text: parts

text: that

text: are

text: present

text: but

text: simply

text: hard

text: to

text: detect

text: (e.g.,

text: due

text: to

text: unusual

text: lighting)

text: and

text: those

text: which

text: are

text: gen-

text: uinely

text: hidden

text: behind

text: another

text: object.

text: Treating

text: occlusions

text: as

text: an

text: unstructured

text: source

text: of

text: noise

text: ig-

text: (a)

text: (b)

text: (c)

text: nores

text: a

text: key

text: aspect

text: of

text: the

text: problem,

text: namely

text: that

text: occlusions

text: are

text: Figure

text: Occlusion

text: impacts

text: part

text: localization

text: performance.

text: In

text: panel

text: (a)

text: the

text: induced

text: by

text: other

text: objects

text: and

text: surfaces

text: in

text: the

text: scene

text: and

text: hence

text: output

text: of

text: a

text: deformable

text: part

text: model

text: is

text: distorted

text: by

text: the

text: presence

text: of

text: occluders,

text: should

text: exhibit

text: occlusion

text: coherence.

text: For

text: example,

text: it

text: would

text: disrupting

text: localization

text: even

text: for

text: parts

text: that

text: are

text: far

text: from

text: the

text: site

text: of

text: occlusion.

text: seem

text: very

text: unlikely

text: that

text: every-other

text: landmark

text: along

text: an

text: object

text: (b)

text: Introducing

text: independent

text: occlusion

text: of

text: each

text: part

text: results

text: in

text: better

text: alignment

text: contour

text: would

text: happen

text: to

text: be

text: occluded.

text: Yet

text: many

text: occlusion

text: but

text: occlusion

text: is

text: treated

text: as

text: an

text: outlier

text: process

text: and

text: prediction

text: of

text: occlusion

text: state

text: is

text: inaccurate.

text: (c)

text: The

text: output

text: of

text: our

text: hierarchical

text: part

text: model,

text: which

text: explicitly

text: models

text: make

text: strong

text: independence

text: assumptions

text: about

text: occlu-

text: models

text: likely

text: patterns

text: of

text: occlusion,

text: shows

text: improved

text: localization

text: as

text: well

text: as

text: sion,

text: making

text: it

text: difﬁcult

text: to

text: distinguish

text: a

text: priori

text: likely

text: from

text: un-

text: accurate

text: prediction

text: of

text: which

text: landmarks

text: are

text: occluded.

text: likely

text: patterns.

text: Ultimately,

text: an

text: occluder

text: should

text: not

text: be

text: inferred

text: simply

text: by

text: the

text: lack

text: of

text: evidence

text: for

text: object

text: features,

text: but

text: rather

text: by

text: positive

text: evidence

text: for

text: the

text: occluding

text: object

text: that

text: explains

text: away

text: formable

text: part

text: model

text: that

text: encodes

text: these

text: principles

text: for

text: modeling

text: the

text: lack

text: of

text: object

text: features.

text: occlusion

text: and

text: achieves

text: state-of-the-art

text: performance

text: on

text: bench-

text: The

text: contribution

text: of

text: this

text: paper

text: is

text: an

text: efﬁcient

text: hierarchical

text: de-

text: marks

text: for

text: occluded

text: face

text: localization

text: and

text: detection

text: (depicted

text: in

text: Preprint

text: submitted

text: to

text: Computer

text: Vision

text: and

text: Image

text: Understanding

text: May

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: efﬁcients using examples from the other 9 folds. We used the standard evaluation protocol for this dataset and compared our method with the top published results avail- able on the FDDB website [40]. The continuous ROC curve for our method and leading methods are shown in Fig. 9 plotted on a semi-log scale. Our result is highly competitive with the top results. The model has better performance on the continuous ROC evaluation relative to other methods because it can pre- dict location of parts and compute accurate bounding ellipses around the faces. UCI Occluded Face Detection Dataset (UCI-OFD). In order to better measure the ability of our model to handle detection of occluded faces, we assembled a preliminary dataset for oc- cluded face detection. This dataset and benchmarking code are publicly available 5. It consists of 61 images from Flickr con- taining 766 labeled faces. Of the faces in these images, 430 include some amount of occlusion. Most of the faces are near frontal and vertical. Height (eyebrow to chin) of the smallest face is about 40 pixels. Precision/Recall curves of face detection of multi-resolution Figure 9: Face detection performance of HPM and state-of-the-art methods HPM, HPM, HPM-occ, DPM and Cascade DPM [41] are shown [40] on the continuous-ROC FDDB benchmark [5]. in Fig. 10(a). We further break down performance, plotting Pre- cision/Recall curves for the subset of faces with some amount based models have the elegant advantage of performing detec- of occlusion in (b) and fully visible in (c). Precision and recall tion and localization simultaneously. In this section, we com- for occluded subset of faces are calculated as below: pare the detection performance of our approach and other top tp tp methods on two datasets: FDDB [5] and our own Occluded Precision = o , Recall = o o tp + fp o tp + fn Face Detection (UCI-OFD) dataset. o o o Multi-resolution HPM. Since many face detection datasets such where tp and fn show number of correct detection and miss o o as FDDB contain many low-resolution faces, we trained a multi- detection of occluded faces, respectively. Our method signiﬁ- resolution variant of our model [34]. This model has a high and cantly outperforms other methods on the occluded subset and a low-resolution model for each viewpoint. The high resolution the performance of all of the methods are almost equal on the model has the same structure as our trained model for landmark visible subset. Fig. 11 shows example detection results pro- localization except that parts are represented as 3x3 HoG cells duced by the model on cluttered scenes containing many over- rather than 5x5. The low-resolution model has 7 parts (right lapping faces. eye, left eye, nose, mouth, chin, left jaw and right jaw) each of which is represented by 7x7 HoG cells with the spatial bin size 6. Discussion and Conclusion of 4. Each part has one shape mixture and 2 occlusion mix- tures (visible or occluded). The heights (eyebrow to chin) of Our experimental results demonstrate that adding coherent the large model and small model are about 100 and 60 pixels occlusion and hierarchical structure allows for substantial gains respectively. To detect even smaller images, we upsample input in performance for landmark localization and detection in part images by a factor of 2 to allow for detection of faces as small models. In images with relatively little occlusion, the HPM as 30 pixels. We trained this model using the same 1758 posi- gives similar detection and localization performance to other tive examples from HELEN68 and generated 8 virtual positive part-based approaches, e.g. DPM, but is signiﬁcantly more examples per example. For negative images we used 6000 im- robust to occlusion. Our results also suggest that when it is ages from the PASCAL VOC 2010 train-val set which do not useful to determine exactly which parts are occluded (e.g., for contain people. later use in face identiﬁcation), independent occlusion makes weaker predictions than our part occlusion mixtures which en- Detection on FDDB. We evaluated our multi-resolution model force coherence between neighboring landmarks. While not on the widely used FDDB dataset. This dataset contains 5171 speciﬁcally trained for landmark estimation, the ﬁnal HPM is faces in a set of 2845 images. Faces are annotated by ellipses competitive with pose regression techniques in terms of land- in this dataset and are as small as 20 pixels in height. To match mark localization accuracy on unoccluded faces (IBUG) and that, we map our predicted landmark locations to ellipses using a linear regression model. FDDB has 10 folds and the ROC curves are the average over the results of these folds. To com- 5https://github.com/golnazghiasi/ pute ellipses for each fold, we learned the linear regression co- hpm-detection-code/tree/master/UCI_OFD 11 ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: efﬁcients

text: using

text: examples

text: from

text: the

text: other

text: folds.

text: We

text: used

text: the

text: standard

text: evaluation

text: protocol

text: for

text: this

text: dataset

text: and

text: compared

text: our

text: method

text: with

text: the

text: top

text: published

text: results

text: avail-

text: able

text: on

text: the

text: FDDB

text: website

text: The

text: continuous

text: ROC

text: curve

text: for

text: our

text: method

text: and

text: leading

text: methods

text: are

text: shown

text: in

text: Fig.

text: plotted

text: on

text: a

text: semi-log

text: scale.

text: Our

text: result

text: is

text: highly

text: competitive

text: with

text: the

text: top

text: results.

text: The

text: model

text: has

text: better

text: performance

text: on

text: the

text: continuous

text: ROC

text: evaluation

text: relative

text: to

text: other

text: methods

text: because

text: it

text: can

text: pre-

text: dict

text: location

text: of

text: parts

text: and

text: compute

text: accurate

text: bounding

text: ellipses

text: around

text: the

text: faces.

text: UCI

text: Occluded

text: Face

text: Detection

text: Dataset

text: (UCI-OFD).

text: In

text: order

text: to

text: better

text: measure

text: the

text: ability

text: of

text: our

text: model

text: to

text: handle

text: detection

text: of

text: occluded

text: faces,

text: we

text: assembled

text: a

text: preliminary

text: dataset

text: for

text: oc-

text: cluded

text: face

text: detection.

text: This

text: dataset

text: and

text: benchmarking

text: code

text: are

text: publicly

text: available

text: It

text: consists

text: of

text: images

text: from

text: Flickr

text: con-

text: taining

text: labeled

text: faces.

text: Of

text: the

text: faces

text: in

text: these

text: images,

text: include

text: some

text: amount

text: of

text: occlusion.

text: Most

text: of

text: the

text: faces

text: are

text: near

text: frontal

text: and

text: vertical.

text: Height

text: (eyebrow

text: to

text: chin)

text: of

text: the

text: smallest

text: face

text: is

text: about

text: pixels.

text: Precision/Recall

text: curves

text: of

text: face

text: detection

text: of

text: multi-resolution

text: Figure

text: Face

text: detection

text: performance

text: of

text: HPM

text: and

text: state-of-the-art

text: methods

text: HPM,

text: HPM,

text: HPM-occ,

text: DPM

text: and

text: Cascade

text: DPM

text: are

text: shown

text: on

text: the

text: continuous-ROC

text: FDDB

text: benchmark

text: in

text: Fig.

text: We

text: further

text: break

text: down

text: performance,

text: plotting

text: Pre-

text: cision/Recall

text: curves

text: for

text: the

text: subset

text: of

text: faces

text: with

text: some

text: amount

text: based

text: models

text: have

text: the

text: elegant

text: advantage

text: of

text: performing

text: detec-

text: of

text: occlusion

text: in

text: (b)

text: and

text: fully

text: visible

text: in

text: (c).

text: Precision

text: and

text: recall

text: tion

text: and

text: localization

text: simultaneously.

text: In

text: this

text: section,

text: we

text: com-

text: for

text: occluded

text: subset

text: of

text: faces

text: are

text: calculated

text: as

text: below:

text: pare

text: the

text: detection

text: performance

text: of

text: our

text: approach

text: and

text: other

text: top

text: tp

text: tp

text: methods

text: on

text: two

text: datasets:

text: FDDB

text: and

text: our

text: own

text: Occluded

text: Precision

text: o

text: Recall

text: o

text: o

text: tp

text: fp

text: o

text: tp

text: fn

text: Face

text: Detection

text: (UCI-OFD)

text: dataset.

text: o

text: o

text: o

text: Multi-resolution

text: HPM.

text: Since

text: many

text: face

text: detection

text: datasets

text: such

text: where

text: tp

text: and

text: fn

text: show

text: number

text: of

text: correct

text: detection

text: and

text: miss

text: o

text: o

text: as

text: FDDB

text: contain

text: many

text: low-resolution

text: faces,

text: we

text: trained

text: a

text: multi-

text: detection

text: of

text: occluded

text: faces,

text: respectively.

text: Our

text: method

text: signiﬁ-

text: resolution

text: variant

text: of

text: our

text: model

text: This

text: model

text: has

text: a

text: high

text: and

text: cantly

text: outperforms

text: other

text: methods

text: on

text: the

text: occluded

text: subset

text: and

text: a

text: low-resolution

text: model

text: for

text: each

text: viewpoint.

text: The

text: high

text: resolution

text: the

text: performance

text: of

text: all

text: of

text: the

text: methods

text: are

text: almost

text: equal

text: on

text: the

text: model

text: has

text: the

text: same

text: structure

text: as

text: our

text: trained

text: model

text: for

text: landmark

text: visible

text: subset.

text: Fig.

text: shows

text: example

text: detection

text: results

text: pro-

text: localization

text: except

text: that

text: parts

text: are

text: represented

text: as

text: HoG

text: cells

text: duced

text: by

text: the

text: model

text: on

text: cluttered

text: scenes

text: containing

text: many

text: over-

text: rather

text: than

text: The

text: low-resolution

text: model

text: has

text: parts

text: (right

text: lapping

text: faces.

text: eye,

text: left

text: eye,

text: nose,

text: mouth,

text: chin,

text: left

text: jaw

text: and

text: right

text: jaw)

text: each

text: of

text: which

text: is

text: represented

text: by

text: HoG

text: cells

text: with

text: the

text: spatial

text: bin

text: size

text: Discussion

text: and

text: Conclusion

text: of

text: Each

text: part

text: has

text: one

text: shape

text: mixture

text: and

text: occlusion

text: mix-

text: tures

text: (visible

text: or

text: occluded).

text: The

text: heights

text: (eyebrow

text: to

text: chin)

text: of

text: Our

text: experimental

text: results

text: demonstrate

text: that

text: adding

text: coherent

text: the

text: large

text: model

text: and

text: small

text: model

text: are

text: about

text: and

text: pixels

text: occlusion

text: and

text: hierarchical

text: structure

text: allows

text: for

text: substantial

text: gains

text: respectively.

text: To

text: detect

text: even

text: smaller

text: images,

text: we

text: upsample

text: input

text: in

text: performance

text: for

text: landmark

text: localization

text: and

text: detection

text: in

text: part

text: images

text: by

text: a

text: factor

text: of

text: to

text: allow

text: for

text: detection

text: of

text: faces

text: as

text: small

text: models.

text: In

text: images

text: with

text: relatively

text: little

text: occlusion,

text: the

text: HPM

text: as

text: pixels.

text: We

text: trained

text: this

text: model

text: using

text: the

text: same

text: posi-

text: gives

text: similar

text: detection

text: and

text: localization

text: performance

text: to

text: other

text: tive

text: examples

text: from

text: and

text: generated

text: virtual

text: positive

text: part-based

text: approaches,

text: e.g.

text: DPM,

text: but

text: is

text: signiﬁcantly

text: more

text: examples

text: per

text: example.

text: For

text: negative

text: images

text: we

text: used

text: im-

text: robust

text: to

text: occlusion.

text: Our

text: results

text: also

text: suggest

text: that

text: when

text: it

text: is

text: ages

text: from

text: the

text: PASCAL

text: VOC

text: train-val

text: set

text: which

text: do

text: not

text: useful

text: to

text: determine

text: exactly

text: which

text: parts

text: are

text: occluded

text: (e.g.,

text: for

text: contain

text: people.

text: later

text: use

text: in

text: face

text: identiﬁcation),

text: independent

text: occlusion

text: makes

text: weaker

text: predictions

text: than

text: our

text: part

text: occlusion

text: mixtures

text: which

text: en-

text: Detection

text: on

text: FDDB.

text: We

text: evaluated

text: our

text: multi-resolution

text: model

text: force

text: coherence

text: between

text: neighboring

text: landmarks.

text: While

text: not

text: on

text: the

text: widely

text: used

text: FDDB

text: dataset.

text: This

text: dataset

text: contains

text: speciﬁcally

text: trained

text: for

text: landmark

text: estimation,

text: the

text: ﬁnal

text: HPM

text: is

text: faces

text: in

text: a

text: set

text: of

text: images.

text: Faces

text: are

text: annotated

text: by

text: ellipses

text: competitive

text: with

text: pose

text: regression

text: techniques

text: in

text: terms

text: of

text: land-

text: in

text: this

text: dataset

text: and

text: are

text: as

text: small

text: as

text: pixels

text: in

text: height.

text: To

text: match

text: mark

text: localization

text: accuracy

text: on

text: unoccluded

text: faces

text: (IBUG)

text: and

text: that,

text: we

text: map

text: our

text: predicted

text: landmark

text: locations

text: to

text: ellipses

text: using

text: a

text: linear

text: regression

text: model.

text: FDDB

text: has

text: folds

text: and

text: the

text: ROC

text: curves

text: are

text: the

text: average

text: over

text: the

text: results

text: of

text: these

text: folds.

text: To

text: com-

text: pute

text: ellipses

text: for

text: each

text: fold,

text: we

text: learned

text: the

text: linear

text: regression

text: co-

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##