text: 10 CHRISTOPHER PARK MOONEY relationship between τ -factorizations and τ-regular factorizations; however, there are a few reg subtle diﬀerences that cause some problems, especially with the deﬁnition of τ -very strongly reg atomic∼elements. In, [17], the author insisted that part of a being τ-very strongly atomic was that a = a. The fact that the very strongly associate relation need not be reﬂexive is the main reason there is not a perfect correspondence between the two approaches. We will see that τ -factorizations reg are simply very poorly behaved when it comes to τ -very strong atoms and rearrangement up reg to very strong associates. On the bright side, the τ-unreﬁnably irreducible element introduced in [19] will also behave quite nicely here. − Of course any non-trivial idempotent element, e, is a zero-diviso∼r since e(e 1) = 0. Fur- · (cid:54) thermore, since e = e2 = e e, with e not a unit, we see that e = e. This means that e is not very strongly atomic for any non-trivial idempotent element. On the other hand, since every non-trivial τ -factorization consists of a product of regular elements, we can have no reg non-trivial τ -factorizations of e. This means the only τ -factorizations of any zero-divisor, reg reg in particular e, are the trivial factorizations. Unfortunately, in the case of a non-trivial idem- potent, e, this means e is not a τ-very strong atom, and will never have a τ -very strongly reg atomic factorization. We demonstrate this in the following example. × × Example 4.1. Let K be an inﬁnite ﬁeld. R = K K with τ = R# R#. ∈ We consider the element (1, 0) Z(R). This ring has only elements which are strongly associate to idempotent elements and units. So the set of non-unit regular elements is empty − and our ring is vacuously a τ-r-UFR. On the other hand, we have (1, 0) = (µ 1, 1)(µ, 0), ∈ ∗ for any unit µ K , is the only type of τ -factorization of (1, 0),∼yet none of these are reg (cid:54) τ -very strongly atomic factorizations. The problem is that (µ, 0) = (µ, 0) since we have reg (µ, 0) = (1, 0)(µ, 0) and (1, 0) is not a unit. This shows we can have a τ-r-UFR which is not even τ -atomic. Moreover, each of these factorizations is non-very strongly associate. Let ∈ reg∗ − − µ, λ K . Then (1, 0) = (µ 1, 1)(µ, 0) = (λ 1, 1)(λ, 0) are tw∼o τ -factorizations of (1, 0), but − − (cid:54) reg (µ, 0) = (µλ 1, 0)(λ, 0) with (µλ 1, 0) not a unit shows (µ, 0) = (λ, 0). Since K is inﬁnite, there are inﬁnitely many τ -factorizations of (1, 0), none of which are very strongly associate. This reg leads us to the following results. Lemma 4.2. Let R be a commutative ring with 1 and let τ be a symmetric relation on R#. ∩ × Let τ := τ (Reg(R) Reg(R)). The collection of non-trivial τ-regular-factorizations and reg non-trivial τ -factorizations coincide. reg · · · ∈ Proof. Let a = λa a be a non-trivial τ-regular factorization. Then a Reg(R) by deﬁnition 1 n (cid:54) of τ-regular factorization, and a τa for all i = j. Since a is regular, and the set of regular |i j∈ ≤ ≤ ∈ elements is saturated, we have a a Reg(R) for each 1 i n, we know that a Reg(R) for ≤ ≤ i (cid:54) · · · i each 1 i n. This means a τ a for each i = j. Thus a = λa a is a τ -factorization. i r·e·g· j 1 n reg Conversely, suppose a = λa a is a non-trivial τ -factorization. Then a τ a for each (cid:54) 1 n∈ reg ≥ i reg j i = j. This means a τa and a , a Reg(R). In particular, since n 2, we can conclude · · · i j i j ∈ · · · that a a a is a product of regular elements, so a Reg(R). This means a = λa a is a 1 2 n 1 n (cid:3) τ-regular-factorization.

text: CHRISTOPHER

text: PARK

text: MOONEY

text: relationship

text: between

text: τ

text: -factorizations

text: and

text: τ-regular

text: factorizations;

text: however,

text: there

text: are

text: a

text: few

text: reg

text: subtle

text: diﬀerences

text: that

text: cause

text: some

text: problems,

text: especially

text: with

text: the

text: deﬁnition

text: of

text: τ

text: -very

text: strongly

text: reg

text: atomic∼elements.

text: In,

text: the

text: author

text: insisted

text: that

text: part

text: of

text: a

text: being

text: τ-very

text: strongly

text: atomic

text: was

text: that

text: a

text: a.

text: The

text: fact

text: that

text: the

text: very

text: strongly

text: associate

text: relation

text: need

text: not

text: be

text: reﬂexive

text: is

text: the

text: main

text: reason

text: there

text: is

text: not

text: a

text: perfect

text: correspondence

text: between

text: the

text: two

text: approaches.

text: We

text: will

text: see

text: that

text: τ

text: -factorizations

text: reg

text: are

text: simply

text: very

text: poorly

text: behaved

text: when

text: it

text: comes

text: to

text: τ

text: -very

text: strong

text: atoms

text: and

text: rearrangement

text: up

text: reg

text: to

text: very

text: strong

text: associates.

text: On

text: the

text: bright

text: side,

text: the

text: τ-unreﬁnably

text: irreducible

text: element

text: introduced

text: in

text: will

text: also

text: behave

text: quite

text: nicely

text: here.

text: Of

text: course

text: any

text: non-trivial

text: idempotent

text: element,

text: e,

text: is

text: a

text: zero-diviso∼r

text: since

text: e(e

text: Fur-

text: thermore,

text: since

text: e

text: e

text: e,

text: with

text: e

text: not

text: a

text: unit,

text: we

text: see

text: that

text: e

text: e.

text: This

text: means

text: that

text: e

text: is

text: not

text: very

text: strongly

text: atomic

text: for

text: any

text: non-trivial

text: idempotent

text: element.

text: On

text: the

text: other

text: hand,

text: since

text: every

text: non-trivial

text: τ

text: -factorization

text: consists

text: of

text: a

text: product

text: of

text: regular

text: elements,

text: we

text: can

text: have

text: no

text: reg

text: non-trivial

text: τ

text: -factorizations

text: of

text: e.

text: This

text: means

text: the

text: only

text: τ

text: -factorizations

text: of

text: any

text: zero-divisor,

text: reg

text: reg

text: in

text: particular

text: e,

text: are

text: the

text: trivial

text: factorizations.

text: Unfortunately,

text: in

text: the

text: case

text: of

text: a

text: non-trivial

text: idem-

text: potent,

text: e,

text: this

text: means

text: e

text: is

text: not

text: a

text: τ-very

text: strong

text: atom,

text: and

text: will

text: never

text: have

text: a

text: τ

text: -very

text: strongly

text: reg

text: atomic

text: factorization.

text: We

text: demonstrate

text: this

text: in

text: the

text: following

text: example.

text: Example

text: Let

text: K

text: be

text: an

text: inﬁnite

text: ﬁeld.

text: R

text: K

text: K

text: with

text: τ

text: R#

text: R#.

text: We

text: consider

text: the

text: element

text: Z(R).

text: This

text: ring

text: has

text: only

text: elements

text: which

text: are

text: strongly

text: associate

text: to

text: idempotent

text: elements

text: and

text: units.

text: So

text: the

text: set

text: of

text: non-unit

text: regular

text: elements

text: is

text: empty

text: and

text: our

text: ring

text: is

text: vacuously

text: a

text: τ-r-UFR.

text: On

text: the

text: other

text: hand,

text: we

text: have

text: (µ

text: for

text: any

text: unit

text: µ

text: K

text: is

text: the

text: only

text: type

text: of

text: τ

text: -factorization

text: of

text: of

text: these

text: are

text: reg

text: τ

text: -very

text: strongly

text: atomic

text: factorizations.

text: The

text: problem

text: is

text: that

text: (µ,

text: (µ,

text: since

text: we

text: have

text: reg

text: (µ,

text: and

text: is

text: not

text: a

text: unit.

text: This

text: shows

text: we

text: can

text: have

text: a

text: τ-r-UFR

text: which

text: is

text: not

text: even

text: τ

text: -atomic.

text: Moreover,

text: each

text: of

text: these

text: factorizations

text: is

text: non-very

text: strongly

text: associate.

text: Let

text: reg∗

text: µ,

text: λ

text: K

text: Then

text: (µ

text: (λ

text: are

text: tw∼o

text: τ

text: -factorizations

text: of

text: but

text: reg

text: (µ,

text: (µλ

text: with

text: (µλ

text: not

text: a

text: unit

text: shows

text: (µ,

text: (λ,

text: Since

text: K

text: is

text: inﬁnite,

text: there

text: are

text: inﬁnitely

text: many

text: τ

text: -factorizations

text: of

text: of

text: which

text: are

text: very

text: strongly

text: associate.

text: This

text: reg

text: leads

text: us

text: to

text: the

text: following

text: results.

text: Lemma

text: Let

text: R

text: be

text: a

text: commutative

text: ring

text: with

text: and

text: let

text: τ

text: be

text: a

text: symmetric

text: relation

text: on

text: R#.

text: Let

text: τ

text: τ

text: (Reg(R)

text: Reg(R)).

text: The

text: collection

text: of

text: non-trivial

text: τ-regular-factorizations

text: and

text: reg

text: non-trivial

text: τ

text: -factorizations

text: coincide.

text: reg

text: Proof.

text: Let

text: a

text: λa

text: a

text: be

text: a

text: non-trivial

text: τ-regular

text: factorization.

text: Then

text: a

text: Reg(R)

text: by

text: deﬁnition

text: n

text: of

text: τ-regular

text: factorization,

text: and

text: a

text: τa

text: for

text: all

text: i

text: j.

text: Since

text: a

text: is

text: regular,

text: and

text: the

text: set

text: of

text: regular

text: |i

text: j∈

text: elements

text: is

text: saturated,

text: we

text: have

text: a

text: a

text: Reg(R)

text: for

text: each

text: i

text: n,

text: we

text: know

text: that

text: a

text: Reg(R)

text: for

text: i

text: i

text: each

text: i

text: n.

text: This

text: means

text: a

text: τ

text: a

text: for

text: each

text: i

text: j.

text: Thus

text: a

text: λa

text: a

text: is

text: a

text: τ

text: -factorization.

text: i

text: r·e·g·

text: j

text: n

text: reg

text: Conversely,

text: suppose

text: a

text: λa

text: a

text: is

text: a

text: non-trivial

text: τ

text: -factorization.

text: Then

text: a

text: τ

text: a

text: for

text: each

text: n∈

text: reg

text: i

text: reg

text: j

text: i

text: j.

text: This

text: means

text: a

text: τa

text: and

text: a

text: a

text: Reg(R).

text: In

text: particular,

text: since

text: n

text: we

text: can

text: conclude

text: i

text: j

text: i

text: j

text: that

text: a

text: a

text: a

text: is

text: a

text: product

text: of

text: regular

text: elements,

text: so

text: a

text: Reg(R).

text: This

text: means

text: a

text: λa

text: a

text: is

text: a

text: n

text: n

text: τ-regular-factorization.


text: → − 2 (m, ν) (m, ν + − ) , (4.9) m 1 and Vertical Transformation (VT, relating cosmos of which perturbations with diﬀerent Hubble terms but same background time evolution, i.e. “iso-temporal transformation”), → − 2 (m, ν) (2 m + , ν) . (4.10) ν Under a HT, the two group of stable scale-invariant solutions, I and II , are respectively mapped to the two group of unstable scale-invariant solutions, III and IV , in the horizontal direction. Under a VT, I and II are mapped to IV and III respectively in vertical direction. − The solution of the slow-roll inﬂation, (ν, m) = ( 1, 3) , is connected horizontally to Wands’s Case, (ν, m) = (2, 3) . Clearly this duality, which connects Wands’s matter- dominated contraction and slow-roll inﬂation shown in [19] 13, is a special case of all possible HT’s with m = 3. On the other hand, under a VT, the Wands’s Case, (ν, m) = (2, 3) , is dual to CSBT cosmos, (ν, m) = (2, 0) . However, neither a HT nor a VT is a complete operation since each of them only maps a stable scale-invariant solution to an unstable scale-invariant solution and vice verse. We are looking for a complete duality transformation connecting a stable scale-invariant solution in an expansion phase to another stable and scale invariant solution in a contraction phase. The simplest way is to perform HT and VT consecutively, − −−→ − 2 −−→ − 2(m 1) − 2 HT V T (m, ν) (m, ν + − ) (2 m + − − , ν + − ) , (4.11) m 1 (m 1)ν + 2 m 1 as shown in Fig. 3. We call this transformation a complete duality transformation (HTVT in Fig. 3). It can connect all stable scale-invariant solutions in an expansion lying on Line I to Line II of all possible stable and scale invariant solutions in a contraction. − Interestingly the slow-roll inﬂation, (ν, m) = ( 1, 3) , is related to the CSTB cosmos, (ν, m) = (2, 0), in this way–both possess stable and scale invariant spectra with the former generated in an exponential expansion while the latter in a tachyon-matter-dominated contraction. Stability Analysis: Noting that both slow-roll inﬂation and CSTB cosmos can produce power spectra of density perturbations satisfying current cosmological constraints. And they are “dual” to each other in the (ν, m) parameter space. However hidden in their curvature perturbation spectra there is a signiﬁcant diﬀerence in ﬁne-tuning. The curvature perturba- tions of slow-roll inﬂation model are (cid:18) (cid:19) P H 2 P − = , (4.12) s ζ Φ˙ δΦ 13This duality is also discussed in various gauge choice with taking account in subdominated modes of perturbations [63, 64] – 14 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: (m,

text: ν)

text: (m,

text: ν

text: m

text: and

label: paragraph

text: Vertical

text: Transformation

text: (VT,

text: relating

text: cosmos

text: of

text: which

text: perturbations

text: with

text: diﬀerent

text: Hubble

text: terms

text: but

text: same

text: background

text: time

text: evolution,

text: i.e.

text: “iso-temporal

text: transformation”),

text: (m,

text: ν)

text: m

text: ν)

text: ν

text: Under

text: a

text: HT,

text: the

text: two

text: group

text: of

text: stable

text: scale-invariant

text: solutions,

text: I

text: and

text: II

text: are

text: respectively

text: mapped

text: to

text: the

text: two

text: group

text: of

text: unstable

text: scale-invariant

text: solutions,

text: III

text: and

text: IV

text: in

text: the

text: horizontal

text: direction.

text: Under

text: a

text: VT,

text: I

text: and

text: II

text: are

text: mapped

text: to

text: IV

text: and

text: III

text: respectively

text: in

text: vertical

text: direction.

text: The

text: solution

text: of

text: the

text: slow-roll

text: inﬂation,

text: (ν,

text: m)

text: is

text: connected

text: horizontally

text: to

text: Wands’s

text: Case,

text: (ν,

text: m)

text: Clearly

text: this

text: duality,

text: which

text: connects

text: Wands’s

text: matter-

text: dominated

text: contraction

text: and

text: slow-roll

text: inﬂation

text: shown

text: in

text: is

text: a

text: special

text: case

text: of

text: all

text: possible

text: HT’s

text: with

text: m

text: On

text: the

text: other

text: hand,

text: under

text: a

text: VT,

text: the

text: Wands’s

text: Case,

text: (ν,

text: m)

text: is

text: dual

text: to

text: CSBT

text: cosmos,

text: (ν,

text: m)

text: However,

text: neither

text: a

text: HT

text: nor

text: a

text: VT

text: is

text: a

text: complete

text: operation

text: since

text: each

text: of

text: them

text: only

text: maps

text: a

text: stable

text: scale-invariant

text: solution

text: to

text: an

text: unstable

text: scale-invariant

text: solution

text: and

text: vice

text: verse.

text: We

text: are

text: looking

text: for

text: a

text: complete

text: duality

text: transformation

text: connecting

text: a

text: stable

text: scale-invariant

text: solution

text: in

text: an

text: expansion

text: phase

text: to

text: another

text: stable

text: and

text: scale

text: invariant

text: solution

text: in

text: a

text: contraction

text: phase.

text: The

text: simplest

text: way

text: is

text: to

text: perform

text: HT

text: and

text: VT

text: consecutively,

text: HT

text: V

text: T

text: (m,

text: ν)

text: (m,

text: ν

text: m

text: ν

text: m

text: (m

text: m

text: as

text: shown

text: in

text: Fig.

text: We

text: call

text: this

text: transformation

text: a

text: complete

text: duality

text: transformation

text: (HTVT

text: in

text: Fig.

text: It

text: can

text: connect

text: all

text: stable

text: scale-invariant

text: solutions

text: in

text: an

text: expansion

text: lying

text: on

text: Line

text: I

text: to

text: Line

text: II

text: of

text: all

text: possible

text: stable

text: and

text: scale

text: invariant

text: solutions

text: in

text: a

text: contraction.

text: Interestingly

text: the

text: slow-roll

text: inﬂation,

text: (ν,

text: m)

text: is

text: related

text: to

text: the

text: CSTB

text: cosmos,

text: (ν,

text: m)

text: in

text: this

text: way–both

text: possess

text: stable

text: and

text: scale

text: invariant

text: spectra

text: with

text: the

text: former

text: generated

text: in

text: an

text: exponential

text: expansion

text: while

text: the

text: latter

text: in

text: a

text: tachyon-matter-dominated

text: contraction.

text: Stability

text: Analysis:

text: Noting

text: that

text: both

text: slow-roll

text: inﬂation

text: and

text: CSTB

text: cosmos

text: can

text: produce

text: power

text: spectra

text: of

text: density

text: perturbations

text: satisfying

text: current

text: cosmological

text: constraints.

text: And

text: they

text: are

text: “dual”

text: to

text: each

text: other

text: in

text: the

text: (ν,

text: m)

text: parameter

text: space.

text: However

text: hidden

text: in

text: their

text: curvature

text: perturbation

text: spectra

text: there

text: is

text: a

text: signiﬁcant

text: diﬀerence

text: in

text: ﬁne-tuning.

text: The

text: curvature

text: perturba-

text: tions

text: of

text: slow-roll

text: inﬂation

text: model

text: are

text: P

text: H

text: P

text: s

text: ζ

text: Φ˙

text: δΦ

text: duality

text: is

text: also

text: discussed

text: in

text: various

text: gauge

text: choice

text: with

text: taking

text: account

text: in

text: subdominated

text: modes

text: of

text: perturbations

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: References [1] WMAP Collaboration Collaboration, E. Komatsu et al., Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, Astrophys.J.Suppl. 192 (2011) 18, [arXiv:1001.4538]. [2] Planck collaboration Collaboration, P. Ade et al., Planck 2013 results. XV. CMB power spectra and likelihood, arXiv:1303.5075. [3] A. R. Liddle and D. Lyth, Cosmological inﬂation and large scale structure. Cambridge University Press, 2000. [4] S. Dodelson, Modern cosmology. Elsevier, 2003. [5] A. Borde and A. Vilenkin, Eternal inﬂation and the initial singularity, Phys.Rev.Lett. 72 (1994) 3305–3309, [gr-qc/9312022]. [6] R. Tolman, On the Theoretical Requirements for a Periodic Behaviour of the Universe, Phys.Rev. 38 (1931) 1758. [7] M. Novello and S. P. Bergliaﬀa, Bouncing Cosmologies, Phys.Rept. 463 (2008) 127–213, [arXiv:0802.1634]. [8] R. H. Brandenberger, The Matter Bounce Alternative to Inﬂationary Cosmology, arXiv:1206.4196. [9] B. A. Bassett, S. Tsujikawa, and D. Wands, Inﬂation dynamics and reheating, Rev.Mod.Phys. 78 (2006) 537–589, [astro-ph/0507632]. [10] J.-L. Lehners, Cosmic Bounces and Cyclic Universes, Class.Quant.Grav. 28 (2011) 204004, [arXiv:1106.0172]. [11] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, The Ekpyrotic universe: Colliding branes and the origin of the hot big bang, Phys.Rev. D64 (2001) 123522, [hep-th/0103239]. [12] J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt, and N. Turok, From big crunch to big bang, Phys.Rev. D65 (2002) 086007, [hep-th/0108187]. [13] Y.-F. Cai, T. Qiu, Y.-S. Piao, M. Li, and X. Zhang, Bouncing universe with quintom matter, JHEP 0710 (2007) 071, [arXiv:0704.1090]. [14] Y.-F. Cai, E. N. Saridakis, M. R. Setare, and J.-Q. Xia, Quintom Cosmology: Theoretical implications and observations, Phys.Rept. 493 (2010) 1–60, [arXiv:0909.2776]. [15] M. Gasperini and G. Veneziano, The Pre - big bang scenario in string cosmology, Phys.Rept. 373 (2003) 1–212, [hep-th/0207130]. [16] P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore, Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies, JHEP 0612 (2006) 080, [hep-th/0606090]. [17] P. Creminelli and L. Senatore, A Smooth bouncing cosmology with scale invariant spectrum, JCAP 0711 (2007) 010, [hep-th/0702165]. [18] L. R. Abramo and P. Peter, K-Bounce, JCAP 0709 (2007) 001, [arXiv:0705.2893]. [19] D. Wands, Duality invariance of cosmological perturbation spectra, Phys.Rev. D60 (1999) 023507, [gr-qc/9809062]. [20] F. Finelli and R. Brandenberger, On the generation of a scale invariant spectrum of adiabatic ﬂuctuations in cosmological models with a contracting phase, Phys.Rev. D65 (2002) 103522, [hep-th/0112249]. [21] R. Durrer and J. Laukenmann, The Oscillating universe: An Alternative to inﬂation, Class.Quant.Grav. 13 (1996) 1069–1088, [gr-qc/9510041]. – 17 –

text: References

label: reference

text: WMAP

text: Collaboration

text: Collaboration,

text: E.

text: Komatsu

text: et

text: al.,

text: Seven-Year

text: Wilkinson

text: Microwave

text: Anisotropy

text: Probe

text: (WMAP)

text: Observations:

text: Cosmological

text: Interpretation,

text: Astrophys.J.Suppl.

text: Planck

text: collaboration

text: Collaboration,

text: P.

text: Ade

text: et

text: al.,

text: Planck

text: results.

text: XV.

text: CMB

text: power

text: spectra

text: and

text: likelihood,

text: A.

text: R.

text: Liddle

text: and

text: D.

text: Lyth,

text: Cosmological

text: inﬂation

text: and

text: large

text: scale

text: structure.

text: Cambridge

text: University

text: Press,

text: S.

text: Dodelson,

text: Modern

text: cosmology.

text: Elsevier,

text: A.

text: Borde

text: and

text: A.

text: Vilenkin,

text: Eternal

text: inﬂation

text: and

text: the

text: initial

text: singularity,

text: Phys.Rev.Lett.

text: R.

text: Tolman,

text: On

text: the

text: Theoretical

text: Requirements

text: for

text: a

text: Periodic

text: Behaviour

text: of

text: the

text: Universe,

text: Phys.Rev.

text: M.

text: Novello

text: and

text: S.

text: P.

text: Bergliaﬀa,

text: Bouncing

text: Cosmologies,

text: Phys.Rept.

text: R.

text: H.

text: Brandenberger,

text: The

text: Matter

text: Bounce

text: Alternative

text: to

text: Inﬂationary

text: Cosmology,

text: B.

text: A.

text: Bassett,

text: S.

text: Tsujikawa,

text: and

text: D.

text: Wands,

text: Inﬂation

text: dynamics

text: and

text: reheating,

text: Rev.Mod.Phys.

text: J.-L.

text: Lehners,

text: Cosmic

text: Bounces

text: and

text: Cyclic

text: Universes,

text: Class.Quant.Grav.

text: J.

text: Khoury,

text: B.

text: A.

text: Ovrut,

text: P.

text: J.

text: Steinhardt,

text: and

text: N.

text: Turok,

text: The

text: Ekpyrotic

text: universe:

text: Colliding

text: branes

text: and

text: the

text: origin

text: of

text: the

text: hot

text: big

text: bang,

text: Phys.Rev.

text: J.

text: Khoury,

text: B.

text: A.

text: Ovrut,

text: N.

text: Seiberg,

text: P.

text: J.

text: Steinhardt,

text: and

text: N.

text: Turok,

text: From

text: big

text: crunch

text: to

text: big

text: bang,

text: Phys.Rev.

text: Y.-F.

text: Cai,

text: T.

text: Qiu,

text: Y.-S.

text: Piao,

text: M.

text: Li,

text: and

text: X.

text: Zhang,

text: Bouncing

text: universe

text: with

text: quintom

text: matter,

text: JHEP

text: Y.-F.

text: Cai,

text: E.

text: N.

text: Saridakis,

text: M.

text: R.

text: Setare,

text: and

text: J.-Q.

text: Xia,

text: Quintom

text: Cosmology:

text: Theoretical

text: implications

text: and

text: observations,

text: Phys.Rept.

text: M.

text: Gasperini

text: and

text: G.

text: Veneziano,

text: The

text: Pre

text: big

text: bang

text: scenario

text: in

text: string

text: cosmology,

text: Phys.Rept.

text: P.

text: Creminelli,

text: M.

text: A.

text: Luty,

text: A.

text: Nicolis,

text: and

text: L.

text: Senatore,

text: Starting

text: the

text: Universe:

text: Stable

text: Violation

text: of

text: the

text: Energy

text: Condition

text: and

text: Non-standard

text: Cosmologies,

text: JHEP

text: P.

text: Creminelli

text: and

text: L.

text: Senatore,

text: A

text: Smooth

text: bouncing

text: cosmology

text: with

text: scale

text: invariant

text: spectrum,

text: JCAP

text: L.

text: R.

text: Abramo

text: and

text: P.

text: Peter,

text: K-Bounce,

text: JCAP

text: D.

text: Wands,

text: Duality

text: invariance

text: of

text: cosmological

text: perturbation

text: spectra,

text: Phys.Rev.

text: F.

text: Finelli

text: and

text: R.

text: Brandenberger,

text: On

text: the

text: generation

text: of

text: a

text: scale

text: invariant

text: spectrum

text: of

text: adiabatic

text: ﬂuctuations

text: in

text: cosmological

text: models

text: with

text: a

text: contracting

text: phase,

text: Phys.Rev.

text: R.

text: Durrer

text: and

text: J.

text: Laukenmann,

text: The

text: Oscillating

text: universe:

text: An

text: Alternative

text: to

text: inﬂation,

text: Class.Quant.Grav.


text: 6 → − → − → − η¯ κa 3/2η¯, χ κa 3/2χ, χ¯ κ(cid:0)a 3/2χ¯, w(cid:1)e ﬁnd the Lagrangian (cid:0) (cid:1) (cid:0) (cid:1) √ √ L = −3aa˙2 + 3 a(cid:0)a˙ ψλ − ψ¯(cid:1)λ¯ + 3e(cid:0)ka + T˙2a(cid:1)3 − (cid:16)a3T˙ ψη(cid:17)− ψ¯η¯ + 3iT˙ λη¯ + λ¯η eκ2 √ eκ κ2 2eκ2 2eκ 2 + φ˙2a3 − a3φ˙ ψχ − ψ¯χ¯ + 3iφ˙ λχ¯ + λ¯χ + 3i λλ¯˙ + λ¯λ(cid:0)˙ − i (ηη(cid:1)¯˙ + η¯η˙) eκ2 eκ √ √ √ √ 2 − 3e kλλ¯ − 3e kηη¯ − 3e kχχ¯ 3i ak i (χχ¯˙ + χ¯χ˙) + + ψλ + ψ¯λ¯ √ a 2a a κ + 3eW2a3 −(cid:0) 3e ka2(cid:1)W − eWT2a3 (cid:0)− eWφ2a3(cid:1)− 9eWλλ¯(cid:0)+ 3eWη(cid:1)η¯ + 3eWχχ¯(cid:0) (cid:1) 4κ2 κ2 √ 2κ2 4κ2 2 4 √2 − 3ia3/2W(cid:0) ψλ +(cid:1)ψ¯λ¯ − i a3WT ψη + ψ¯η¯ + 3eWT λ¯η − λη¯ − i a3Wφ ψχ + ψ¯χ¯ 2κ 2κ 2 2κ 3eW − − − − 3eκ2 − 3 + φ λ¯χ λχ¯ eW ηη¯ + eW (χ¯η χη¯) eW χχ¯ + ηη¯χχ¯ ψψ¯λλ¯ 2 TT Tφ φφ 4a3 2e 1 1 + ψψ¯χχ¯ + ψψ¯ηη¯. 2e 4e (cid:16) (cid:17) Substituting the equations of motion o√f the auxiliary ﬁelds (21) into the supersymetry transformation of the fermions − · · · · · · · · · λ, η and χ from (8), we get δ λ = ζ¯ k aW/2 + , δη = ζ¯W + and δχ = ζ¯W + . Therefore, if any ζ T φ the ﬁelds on the r.h.s. of these equations has nonvanishing v.e.v., the corresponding fermion is a goldstino and supersymmetry is broken. In the case of λ, the breaking can be due to the cosmological constant or to a nonvanishing ∼ W. In fact, if the superpotential has the form W eφf(T), as in the examples in the next section, then W = W, (cid:54) φ i.e. χ contributes to the goldstino if W = 0. V. HAMILTONIAN ANALYSIS The canonical momenta are √ √ −6aa˙ − 3 aψ¯λ¯ 3 aψλ π = + , a eκ2 √ eκ √ eκ a3T˙ a3ψ¯η¯ − a3ψη 3a3√/2iλη¯ 3a3√/2iλ¯η π = + + + , T eκ2 √2eκ √2eκ 2 a3 2 a3 2a3φ˙ a3ψ¯χ¯ − a3ψχ 3a√3/2iλχ¯ 3a√3/2iλ¯χ π = + + + , φ eκ2 eκ eκ a3 a3 − − πλ = 3iλ¯, πλ¯ = 3iλ, i i π = η¯, π = η, η 2 η¯ 2 π = iχ¯, π = iχ. χ χ¯ As usual, we can see the appearence of the fermionic constraints Ωλ = πλ + 3iλ¯, Ωλ¯ = πλ¯ + 3iλ, − i − i Ω = π η¯, Ω = π η, (22) η η 2 η¯ η¯ 2 − − Ω = π iχ¯, Ω = π iχ. χ χ χ¯ χ¯ According to the Dirac formalism, the previous constraints are second class and the dynamics of the system is obtained when we impose the set of constraints (22) and introduce the Dirac brackets, and we obtain (cid:8) (cid:9) { } { } { } a, π = 1, φ, π = 1, T, π = 1, a D φ D T D − 1 { } − i { } − λ, λ¯ = , χ, χ¯ = , η, η¯ = i. (23) D 6i D 2 D ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: η¯

text: κa

text: χ

text: κa

text: χ¯

text: ﬁnd

text: the

text: Lagrangian

text: L

text: ψλ

text: ψ¯η¯

text: λη¯

text: λ¯η

text: eκ

text: κ2

text: a3φ˙

text: ψχ

text: ψ¯χ¯

text: 3iφ˙

text: λχ¯

text: λ¯χ

text: λλ¯˙

text: i

text: η¯η˙)

text: eκ

text: kλλ¯

text: kηη¯

text: kχχ¯

text: ak

text: i

text: (χχ¯˙

text: χ¯χ˙)

text: ψλ

text: ψ¯λ¯

text: a

text: a

text: κ

text: κ2

text: √2

text: ψλ

text: i

text: ψη

text: ψ¯η¯

text: λ¯η

text: λη¯

text: i

text: ψχ

text: ψ¯χ¯

text: 2κ

text: 2κ

text: 2κ

text: φ

text: λ¯χ

text: λχ¯

text: eW

text: ηη¯

text: eW

text: (χ¯η

text: χη¯)

text: eW

text: χχ¯

text: ηη¯χχ¯

text: ψψ¯λλ¯

text: TT

text: Tφ

text: φφ

text: ψψ¯χχ¯

text: ψψ¯ηη¯.

text: Substituting

text: the

text: equations

text: of

text: motion

text: o√f

text: the

text: auxiliary

text: ﬁelds

text: into

text: the

text: supersymetry

text: transformation

text: of

text: the

text: fermions

text: λ,

text: η

text: and

text: χ

text: from

text: we

text: get

text: δ

text: λ

text: ζ¯

text: k

text: δη

text: ζ¯W

text: and

text: δχ

text: ζ¯W

text: Therefore,

text: if

text: any

text: ζ

text: T

text: φ

text: the

text: ﬁelds

text: on

text: the

text: r.h.s.

text: of

text: these

text: equations

text: has

text: nonvanishing

text: v.e.v.,

text: the

text: corresponding

text: fermion

text: is

text: a

text: goldstino

text: and

text: supersymmetry

text: is

text: broken.

text: In

text: the

text: case

text: of

text: λ,

text: the

text: breaking

text: can

text: be

text: due

text: to

text: the

text: cosmological

text: constant

text: or

text: to

text: a

text: nonvanishing

text: W.

text: In

text: fact,

text: if

text: the

text: superpotential

text: has

text: the

text: form

text: W

text: eφf(T),

text: as

text: in

text: the

text: examples

text: in

text: the

text: next

text: section,

text: then

text: W

text: W,

text: φ

text: i.e.

text: χ

text: contributes

text: to

text: the

text: goldstino

text: if

text: W

text: V.

text: HAMILTONIAN

text: ANALYSIS

text: The

text: canonical

text: momenta

text: are

text: aψ¯λ¯

text: aψλ

text: π

text: a

text: eκ

text: eκ

text: a3ψ¯η¯

text: a3ψη

text: π

text: T

text: √2eκ

text: √2eκ

text: a3ψ¯χ¯

text: a3ψχ

text: π

text: φ

text: eκ

text: eκ

text: πλ

text: πλ¯

text: i

text: i

text: π

text: η¯,

text: π

text: η,

text: η

text: η¯

text: π

text: iχ¯,

text: π

text: iχ.

text: χ

text: χ¯

text: As

text: usual,

text: we

text: can

text: see

text: the

text: appearence

text: of

text: the

text: fermionic

text: constraints

text: Ωλ

text: πλ

text: Ωλ¯

text: πλ¯

text: i

text: i

text: Ω

text: π

text: η¯,

text: Ω

text: π

text: η,

text: η

text: η

text: η¯

text: η¯

text: Ω

text: π

text: iχ¯,

text: Ω

text: π

text: iχ.

text: χ

text: χ

text: χ¯

text: χ¯

text: According

text: to

text: the

text: Dirac

text: formalism,

text: the

text: previous

text: constraints

text: are

text: second

text: class

text: and

text: the

text: dynamics

text: of

text: the

text: system

text: is

text: obtained

text: when

text: we

text: impose

text: the

text: set

text: of

text: constraints

text: and

text: introduce

text: the

text: Dirac

text: brackets,

text: and

text: we

text: obtain

text: a,

text: π

text: φ,

text: π

text: T,

text: π

text: a

text: D

text: φ

text: D

text: T

text: D

text: i

text: λ,

text: λ¯

text: χ,

text: χ¯

text: η,

text: η¯

text: i.

text: D

text: D

text: D

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text: 7 Using the standard deﬁnition for the Hamiltonian and imposing the constraints (22), we can write the Hamiltonian of the theory as 1 − 1 H = NH + ψS ψ¯S¯, (24) 0 2 2 where (cid:0) (cid:1) (cid:0) (cid:1) H0 = −κ12√2πaa2 +(cid:0) κ22−aπ3T2 − (cid:1)3i2κa23πT λ(cid:0)η¯ + λ¯η +(cid:1) κ42aπ3φ2 − 3i2κ−a23πφ λχ¯ +−λ¯χ − 43κa32 W2 +3 ka2 W 3ka + a3 W2 + a3 W2 + 9Wλλ¯ 3Wηη¯ 3Wχχ¯ κ2 −κ2 2κ2 T 4−κ2 φ 2 4 2− (25) +32W−T λ√η¯ λ¯η +√ 32Wφ λ√χ¯ λ¯χ− + WTT ηη¯−+ WTφ (χη¯− χ¯η) +W χχ¯ 3 kλλ¯ + 3 kηη¯ + 3 kχχ¯ 9κ2 λλ¯χχ¯ 9κ2 λλ¯ηη¯ 3κ2 ηη¯χχ¯, φφ a 2a a 2a3 4a3 4a3 √ √ − S =√κ√πaa λ + κ√πaT3√η + √κπaφ3 χ 6i κakλ + 3i κa3 Wλ (26) +i a3 W η + i a3 W χ + 3iκ ληη¯ + 3iκ λχχ¯, κ T κ φ 2a3/2 a3/2 √ √ − S¯−=√κ√πaa λ¯ +−κ√πaT3√η¯ + √κπaφ3 χ−¯ + 6i κakλ¯ − 3i κa3 Wλ¯ (27) i a3 W η¯ i a3 W χ¯ 3iκ λ¯ηη¯ 3iκ λ¯χχ¯, κ T κ φ 2a3/2 a3/2 { } { } { } satisfy the Dirac algebra S, S¯ = 2H , H , S = H , S¯ = 0. D 0 0 D 0 D VI. SUPERPOTENTIAL SOLUTIONS From the H , (25), we identify the scalar potential 0 −3k U(a, φ, T) = + e2φV (T), (28) a2 which is related to the superpotential W(φ, T) by (cid:18) (cid:19) (cid:18) (cid:19) √ −3W2 3 k − 3k 1 ∂W 2 1 ∂W 2 U = + W + + . (29) 4 a a2 4 ∂φ 2 ∂T The form of the scalar potential (28) of a FRW geometry is consistent with k = 0, hence we restrict ourselves to this geometry, and we get the equation (cid:18) (cid:19) (cid:18) (cid:19) −3W2 1 ∂W 2 1 ∂W 2 e2φV (T) = + + . (30) 4 4 ∂φ 2 ∂T This suggest us a separation of variables of the form W(φ, T) = √1 eφf(T). With this ansatz we obtain the following 2 relation between the tachyon potential and the tachyonic component of the superpotential (cid:48) − (f )2 f2 = V (T), (31) where the prime denotes diﬀerentiation with respect to T. In order to ﬁnd solutions to this equation we must ﬁx the function V (T). For example if V (T) = 0, the solution is f(T) = eT . Further, for ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Using

text: the

text: standard

text: deﬁnition

text: for

text: the

text: Hamiltonian

text: and

text: imposing

text: the

text: constraints

text: we

text: can

text: write

text: the

text: Hamiltonian

text: of

text: the

text: theory

text: as

text: H

text: NH

text: ψS

text: ψ¯S¯,

text: where

text: λ¯η

text: λχ¯

text: +−λ¯χ

text: W

text: 9Wλλ¯

text: 3Wηη¯

text: 3Wχχ¯

text: κ2

text: −κ2

text: T

text: 4−κ2

text: φ

text: 2−

text: λ√η¯

text: λ¯η

text: λ√χ¯

text: λ¯χ−

text: WTT

text: ηη¯−+

text: WTφ

text: (χη¯−

text: χ¯η)

text: +W

text: χχ¯

text: kλλ¯

text: kηη¯

text: kχχ¯

text: λλ¯χχ¯

text: λλ¯ηη¯

text: ηη¯χχ¯,

text: φφ

text: a

text: a

text: S

text: =√κ√πaa

text: λ

text: κ√πaT3√η

text: √κπaφ3

text: χ

text: κakλ

text: Wλ

text: +i

text: W

text: η

text: i

text: W

text: χ

text: ληη¯

text: λχχ¯,

text: κ

text: T

text: κ

text: φ

text: S¯−=√κ√πaa

text: λ¯

text: +−κ√πaT3√η¯

text: √κπaφ3

text: χ−¯

text: κakλ¯

text: Wλ¯

text: i

text: W

text: η¯

text: i

text: W

text: χ¯

text: λ¯ηη¯

text: λ¯χχ¯,

text: κ

text: T

text: κ

text: φ

text: satisfy

text: the

text: Dirac

text: algebra

text: S,

text: S¯

text: H

text: S

text: H

text: S¯

text: D

text: D

text: D

text: VI.

text: SUPERPOTENTIAL

text: SOLUTIONS

text: From

text: the

text: H

text: we

text: identify

text: the

text: scalar

text: potential

text: U(a,

text: φ,

text: T)

text: (T),

text: which

text: is

text: related

text: to

text: the

text: superpotential

text: W(φ,

text: T)

text: by

text: k

text: ∂W

text: ∂W

text: U

text: W

text: a

text: ∂φ

text: ∂T

text: The

text: form

text: of

text: the

text: scalar

text: potential

text: of

text: a

text: FRW

text: geometry

text: is

text: consistent

text: with

text: k

text: hence

text: we

text: restrict

text: ourselves

text: to

text: this

text: geometry,

text: and

text: we

text: get

text: the

text: equation

text: ∂W

text: ∂W

text: (T)

text: ∂φ

text: ∂T

text: This

text: suggest

text: us

text: a

text: separation

text: of

text: variables

text: of

text: the

text: form

text: W(φ,

text: T)

text: √1

text: eφf(T).

text: With

text: this

text: ansatz

text: we

text: obtain

text: the

text: following

text: relation

text: between

text: the

text: tachyon

text: potential

text: and

text: the

text: tachyonic

text: component

text: of

text: the

text: superpotential

text: (f

text: V

text: (T),

text: where

text: the

text: prime

text: denotes

text: diﬀerentiation

text: with

text: respect

text: to

text: T.

text: In

text: order

text: to

text: ﬁnd

text: solutions

text: to

text: this

text: equation

text: we

text: must

text: ﬁx

text: the

text: function

text: V

text: (T).

text: For

text: example

text: if

text: V

text: (T)

text: the

text: solution

text: is

text: f(T)

text: eT

text: Further,

text: for

text: ##LTLine##

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text: 9 (cid:112) (cid:112) If we compute the superpotential corresponding to (37) following Appendix B, it can be shown that W (φ, 0) = − − T eφ 4(1 τ2) + τ4(A2 1)/τ2, where A = f(0). Further, at the minimum, i.e. at T = τ, we make the power − − expansion around this point and we get W (φ, τ) = eφ 4(1 τ2) + τ4(B2 2)/τ2, where B = f(τ). If we take the → ∞ T √ limit τ , in order to make the computation we set u = 1/T and ϑ = 1/τ and evaluate the result ﬁrst at u = 0 | − and then we set ϑ = 0, we get W (φ, u) = eφ C2 1, where C = f(u) . Thus√if we set C = 1, the T u=0,ϑ=0 → ∞ u=0,ϑ|=0 − minimum in this limit is supersymmetric. In the limit τ we have also W (φ, T) = eφ A2 1 T T=0 hence it does not vanish and supersymmetry is broken. Further, for T = τ we can make the power series ansatz in the neighborhood of this value, and we get W (φ, τ) = b, where b is another constant and it can be easily seen that T → ∞ if b = 0, the whole power series tends to zero when τ , i.e. in this limit W (φ, T) = 0 and supersymmetry is T conserved. An interesting question regards potentials with suitable supersymmetric properties, of the type of Sen conjectures. For instance the potential (cid:2) (cid:3) − V (T) = exp( nT) α + α T + α T2 + α T3 + α T4 , (38) 0 1 2 3 4 − in this case we have f(T) = (a + bT + cT2) exp ( nT/2) from which we obtain − W(T, φ) = (a + bT + cT2) exp (φ nT/2) , (39) the explicit coeﬃcients α depend on the free parameters a, n and V , in fact we demand the presence of a maximum i 0 for T = 0, this provides us with two equations which can be solved for b and c in (39) and with the condition V (0) < 0, details of the calculations are given in Appendix (), see Fig. 2. VT (cid:72) (cid:76) 6 5 4 3 2 1 1 2 3 4 5 T FIG. 2: Tachyon potential V(T) produced by f(T) in (C1) for a = 10, n = 3 and V0 = 5. (cid:54) (cid:54) At the maximum of this potential W (0, φ) = 0 and W (0, φ) = 0, hence supersymmetry is broken. Further, after T → φ → → ∞ condensation supersymmetry is restored because W 0 and W 0, when T . T φ VII. CONCLUSIONS We have studied a worldline supersymmetric theory in a FRW background, with a closed string tachyon. We have constructed the action in the formalism of the ‘new’-Θ variables in one dimension, which allows systematically to construct supergravity actions. We consider the solutions for the diﬀerential equation of the superpotential for given tachyonic potentials and we have obtained solutions with broken supersymmetry at the unstable, tachyonic, conﬁguration, and supersymmetric at the stable minimum. Furthermore, the superpotentials can have simple forms, but which correspond to complex fermionic masses. These superpotentials can be obtained as well by a power series ansatz, whose general solution depends on a real parameter which can be chosen such that the complex solutions can be mapped to real solutions. Some of these potentials have been considered in cosmological models, like in [14] where inﬂationary and big crunch sceneries are given, and it would be interesting to consider the supersymmetric versions. Acknowledgments ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: If

text: we

text: compute

text: the

text: superpotential

text: corresponding

text: to

text: following

text: Appendix

text: B,

text: it

text: can

text: be

text: shown

text: that

text: W

text: (φ,

text: T

text: eφ

text: where

text: A

text: Further,

text: at

text: the

text: minimum,

text: i.e.

text: at

text: T

text: τ,

text: we

text: make

text: the

text: power

text: expansion

text: around

text: this

text: point

text: and

text: we

text: get

text: W

text: (φ,

text: τ)

text: eφ

text: where

text: B

text: f(τ).

text: If

text: we

text: take

text: the

text: T

text: limit

text: τ

text: in

text: order

text: to

text: make

text: the

text: computation

text: we

text: set

text: u

text: and

text: ϑ

text: and

text: evaluate

text: the

text: result

text: ﬁrst

text: at

text: u

text: and

text: then

text: we

text: set

text: ϑ

text: we

text: get

text: W

text: (φ,

text: u)

text: eφ

text: where

text: C

text: f(u)

text: Thus√if

text: we

text: set

text: C

text: the

text: T

text: minimum

text: in

text: this

text: limit

text: is

text: supersymmetric.

text: In

text: the

text: limit

text: τ

text: we

text: have

text: also

text: W

text: (φ,

text: T)

text: eφ

text: T

text: hence

text: it

text: does

text: not

text: vanish

text: and

text: supersymmetry

text: is

text: broken.

text: Further,

text: for

text: T

text: τ

text: we

text: can

text: make

text: the

text: power

text: series

text: ansatz

text: in

text: the

text: neighborhood

text: of

text: this

text: value,

text: and

text: we

text: get

text: W

text: (φ,

text: τ)

text: b,

text: where

text: b

text: is

text: another

text: constant

text: and

text: it

text: can

text: be

text: easily

text: seen

text: that

text: T

text: if

text: b

text: the

text: whole

text: power

text: series

text: tends

text: to

text: zero

text: when

text: τ

text: i.e.

text: in

text: this

text: limit

text: W

text: (φ,

text: T)

text: and

text: supersymmetry

text: is

text: T

text: conserved.

text: An

text: interesting

text: question

text: regards

text: potentials

text: with

text: suitable

text: supersymmetric

text: properties,

text: of

text: the

text: type

text: of

text: Sen

text: conjectures.

text: For

text: instance

text: the

text: potential

text: V

text: (T)

text: exp(

text: nT)

text: α

text: α

text: T

text: α

text: α

text: α

text: in

text: this

text: case

text: we

text: have

text: f(T)

text: (a

text: bT

text: exp

text: from

text: which

text: we

text: obtain

text: W(T,

text: φ)

text: (a

text: bT

text: exp

text: (φ

text: the

text: explicit

text: coeﬃcients

text: α

text: depend

text: on

text: the

text: free

text: parameters

text: a,

text: n

text: and

text: V

text: in

text: fact

text: we

text: demand

text: the

text: presence

text: of

text: a

text: maximum

text: i

text: for

text: T

text: this

text: provides

text: us

text: with

text: two

text: equations

text: which

text: can

text: be

text: solved

text: for

text: b

text: and

text: c

text: in

text: and

text: with

text: the

text: condition

text: V

text: details

text: of

text: the

text: calculations

text: are

text: given

text: in

text: Appendix

text: see

text: Fig.

text: VT

text: T

text: FIG.

text: Tachyon

text: potential

text: V(T)

text: produced

text: by

text: f(T)

text: in

text: for

text: a

text: n

text: and

text: At

text: the

text: maximum

text: of

text: this

text: potential

text: W

text: φ)

text: and

text: W

text: φ)

text: hence

text: supersymmetry

text: is

text: broken.

text: Further,

text: after

text: T

text: φ

text: condensation

text: supersymmetry

text: is

text: restored

text: because

text: W

text: and

text: W

text: when

text: T

text: T

text: φ

text: VII.

text: CONCLUSIONS

text: We

text: have

text: studied

text: a

text: worldline

text: supersymmetric

text: theory

text: in

text: a

text: FRW

text: background,

text: with

text: a

text: closed

text: string

text: tachyon.

text: We

text: have

text: constructed

text: the

text: action

text: in

text: the

text: formalism

text: of

text: the

text: ‘new’-Θ

text: variables

text: in

text: one

text: dimension,

text: which

text: allows

text: systematically

text: to

text: construct

text: supergravity

text: actions.

text: We

text: consider

text: the

text: solutions

text: for

text: the

text: diﬀerential

text: equation

text: of

text: the

text: superpotential

text: for

text: given

text: tachyonic

text: potentials

text: and

text: we

text: have

text: obtained

text: solutions

text: with

text: broken

text: supersymmetry

text: at

text: the

text: unstable,

text: tachyonic,

text: conﬁguration,

text: and

text: supersymmetric

text: at

text: the

text: stable

text: minimum.

text: Furthermore,

text: the

text: superpotentials

text: can

text: have

text: simple

text: forms,

text: but

text: which

text: correspond

text: to

text: complex

text: fermionic

text: masses.

text: These

text: superpotentials

text: can

text: be

text: obtained

text: as

text: well

text: by

text: a

text: power

text: series

text: ansatz,

text: whose

text: general

text: solution

text: depends

text: on

text: a

text: real

text: parameter

text: which

text: can

text: be

text: chosen

text: such

text: that

text: the

text: complex

text: solutions

text: can

text: be

text: mapped

text: to

text: real

text: solutions.

text: Some

text: of

text: these

text: potentials

text: have

text: been

text: considered

text: in

text: cosmological

text: models,

text: like

text: in

text: where

text: inﬂationary

text: and

text: big

text: crunch

text: sceneries

text: are

text: given,

text: and

text: it

text: would

text: be

text: interesting

text: to

text: consider

text: the

text: supersymmetric

text: versions.

text: Acknowledgments

text: ##LTFigure##

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text: I. INTRODUCTION Graphite and diamond had been only two stable carbon structures found in nature until various types of low-dimensional carbon allotropes, including fullerenes, carbon nanotubes, and graphene, were recently discovered. For the past decades, these carbon-based nano- materials have attracted a great deal of attention because of their unique electronic and mechanical properties. Among them, graphene, a two-dimensional (2D) atomic layer of sp2- bonded carbon atoms, exhibits a Dirac cone structure near the Fermi surface, causing its charge carriers to behave as massless Dirac fermions, and possesses pseudospin degrees of freedom [1–3]. These features of graphene have prompted active investigation of its possible applications for novel nanoelectronic and spintronic devices [4, 5]. Because of its large sur- face area and high electrical conductivity, graphene has also been studied for a new energy storage material. While graphite, stacks of graphene sheets, is currently used as an anode material for Li-ion battery (LIB) because of its high Coulombic eﬃciency [6], there have been intensive studies, both theoretically and experimentally, on the potential of graphene as a high-capacity LIB anode material [7–14]. Some years ago, graphynes, sp-sp2 hybridized networks of carbon atoms, were proposed as possible 2D carbon structures with larger surface area than graphene [15]. Recent density- functional theory (DFT) calculations showed that graphynes could have some intriguing electronic properties similar to those of graphene, e.g., Dirac cones [16, 17] and very high carrier mobilities [18]. Furthermore, it was predicted that multilayer graphynes could be employed for high-capacity hydrogen storage materials [19] and for eﬀective LIB anode materials with higher Li diﬀusivity [20] and higher Li-storage capacity [21] than graphite, if they can be produced. The DFT methods have been successfully applied to understand electronic and mechan- ical characteristics of newly-synthesized nanomaterials and even to design novel materials with some intriguing physical properties without any experimental evidence. However, it has some well-known limitations; for example, the DFT calculations based on the exchange- correlation potential within the Local Density Approximation (LDA) tends to overestimate the cohesive energy of a solid. Even with much experimental eﬀort motivated by the promis- ing DFT-based predictions on graphynes, only ﬂakes or building blocks of ﬁnite-size gra- phynes have been synthesized so far [22–24] but there has been no successful report yet 2

text: I.

label: section

text: INTRODUCTION

text: Graphite

text: and

text: diamond

text: had

text: been

text: only

text: two

text: stable

text: carbon

text: structures

text: found

text: in

text: nature

text: until

text: various

text: types

text: of

text: low-dimensional

text: carbon

text: allotropes,

text: including

text: fullerenes,

text: carbon

text: nanotubes,

text: and

text: graphene,

text: were

text: recently

text: discovered.

text: For

text: the

text: past

text: decades,

text: these

text: carbon-based

text: nano-

text: materials

text: have

text: attracted

text: a

text: great

text: deal

text: of

text: attention

text: because

text: of

text: their

text: unique

text: electronic

text: and

text: mechanical

text: properties.

text: Among

text: them,

text: graphene,

text: a

text: two-dimensional

text: atomic

text: layer

text: of

text: bonded

text: carbon

text: atoms,

text: exhibits

text: a

text: Dirac

text: cone

text: structure

text: near

text: the

text: Fermi

text: surface,

text: causing

text: its

text: charge

text: carriers

text: to

text: behave

text: as

text: massless

text: Dirac

text: fermions,

text: and

text: possesses

text: pseudospin

text: degrees

text: of

text: freedom

text: These

text: features

text: of

text: graphene

text: have

text: prompted

text: active

text: investigation

text: of

text: its

text: possible

text: applications

text: for

text: novel

text: nanoelectronic

text: and

text: spintronic

text: devices

text: Because

text: of

text: its

text: large

text: sur-

text: face

text: area

text: and

text: high

text: electrical

text: conductivity,

text: graphene

text: has

text: also

text: been

text: studied

text: for

text: a

text: new

text: energy

text: storage

text: material.

text: While

text: graphite,

text: stacks

text: of

text: graphene

text: sheets,

text: is

text: currently

text: used

text: as

text: an

text: anode

text: material

text: for

text: Li-ion

text: battery

text: (LIB)

text: because

text: of

text: its

text: high

text: Coulombic

text: eﬃciency

text: there

text: have

text: been

text: intensive

text: studies,

text: both

text: theoretically

text: and

text: experimentally,

text: on

text: the

text: potential

text: of

text: graphene

text: as

text: a

text: high-capacity

text: LIB

text: anode

text: material

text: Some

text: years

text: ago,

text: graphynes,

text: hybridized

text: networks

text: of

text: carbon

text: atoms,

text: were

text: proposed

text: as

text: possible

text: carbon

text: structures

text: with

text: larger

text: surface

text: area

text: than

text: graphene

text: Recent

text: density-

text: functional

text: theory

text: (DFT)

text: calculations

text: showed

text: that

text: graphynes

text: could

text: have

text: some

text: intriguing

text: electronic

text: properties

text: similar

text: to

text: those

text: of

text: graphene,

text: e.g.,

text: Dirac

text: cones

text: and

text: very

text: high

text: carrier

text: mobilities

text: Furthermore,

text: it

text: was

text: predicted

text: that

text: multilayer

text: graphynes

text: could

text: be

text: employed

text: for

text: high-capacity

text: hydrogen

text: storage

text: materials

text: and

text: for

text: eﬀective

text: LIB

text: anode

text: materials

text: with

text: higher

text: Li

text: diﬀusivity

text: and

text: higher

text: Li-storage

text: capacity

text: than

text: graphite,

text: if

text: they

text: can

text: be

text: produced.

text: The

text: DFT

text: methods

text: have

text: been

text: successfully

text: applied

text: to

text: understand

text: electronic

text: and

text: mechan-

text: ical

text: characteristics

text: of

text: newly-synthesized

text: nanomaterials

text: and

text: even

text: to

text: design

text: novel

text: materials

text: with

text: some

text: intriguing

text: physical

text: properties

text: without

text: any

text: experimental

text: evidence.

text: However,

text: it

text: has

text: some

text: well-known

text: limitations;

text: for

text: example,

text: the

text: DFT

text: calculations

text: based

text: on

text: the

text: exchange-

text: correlation

text: potential

text: within

text: the

text: Local

text: Density

text: Approximation

text: (LDA)

text: tends

text: to

text: overestimate

text: the

text: cohesive

text: energy

text: of

text: a

text: solid.

text: Even

text: with

text: much

text: experimental

text: eﬀort

text: motivated

text: by

text: the

text: promis-

text: ing

text: DFT-based

text: predictions

text: on

text: graphynes,

text: only

text: ﬂakes

text: or

text: building

text: blocks

text: of

text: ﬁnite-size

text: gra-

text: phynes

text: have

text: been

text: synthesized

text: so

text: far

text: but

text: there

text: has

text: been

text: no

text: successful

text: report

text: yet


text: TABLE III: Carbon-carbon bond energiesa of sp-sp2 hybridized graphyne structure in units of eV which were determined through multiple-linear-regression ﬁts of the cohesive energies of graphene and three diﬀerent graphyne structures shown in Table II to Eq. (1). ≡ C-C C=C C C correlation coeﬃcient DMC 4.403(33) 6.143(42) 7.652(59) 0.99999 PBE-PP 4.520(17) 6.735(24) 8.762(32) 1.00000 PBE-PAW 4.537(23) 6.820(33) 8.900(43) 1.00000 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: III:

text: Carbon-carbon

label: caption

text: bond

text: energiesa

text: of

text: hybridized

text: graphyne

text: structure

text: in

text: units

text: of

text: eV

text: which

text: were

text: determined

text: through

text: multiple-linear-regression

text: ﬁts

text: of

text: the

text: cohesive

text: energies

text: of

text: graphene

text: and

text: three

text: diﬀerent

text: graphyne

text: structures

text: shown

text: in

text: Table

text: II

text: to

text: Eq.

label: table

text: C-C

text: C=C

text: C

text: C

text: correlation

text: coeﬃcient

text: DMC

text: PBE-PP

text: PBE-PAW

text: ##LTLine##

font: default

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: for fabrication of extended 2D graphynes. The experimental diﬃculty in the synthesis of graphynes leads us to speculate that the DFT prediction on graphyne energetics may be too optimistic as a result of an intrinsic limitation of the methodology. The relative energetic stabilities of these new carbon-based nanomaterials need to be analyzed through accurate numerical calculations which allow full incorporation of the electron-electron correlation. Here we performed quantum Monte Carlo (QMC) simulations to investigate ground- state energetics of various carbon structures such as diamond, graphene, graphynes, and carbynes, especially their cohesive energies. It is found that the cohesive energies of these carbon allotropes are signiﬁcantly overestimated in the DFT calculations when compared with the corresponding QMC energies, as many previous studies have reported [25, 26]. We also ﬁnd that the accuracy of the DFT cohesive energy depends on its carbon-carbon bonding nature; the discrepancy between the QMC cohesive energy and the DFT energy is the smallest for sp3-bonded diamond, and the largest for carbyne, one-dimensional (1D) chain of sp-bonded carbon atoms. In the case of graphene and graphynes, the magnitude of the overestimation of the DFT cohesive energy increases systematically as the ratio of the sp-bonded carbon atoms increases. According to our QMC calculations, the cohesive energy of graphene is 7.464(10) eV/atom, which is larger than the cohesive energy of γ-graphyne, energetically the most stable graphyne structure, by 0.698(12) eV/atom. II. METHODS AND COMPUTATIONAL DETAILS The QUANTUM ESPRESSO package [27] was used for the DFT calculations where we employed the exchange correlation potential with the generalized gradient approximation in the Perdew-Burke-Ernzerhof parameterization (PBE) scheme [28]. The energy-consistent pseudopotential (PP) proposed by Burkatzki, Filippi, and Dolg (BFD) [29, 30], was used for the valence-core partitioning in describing a carbon atom with the kinetic energy cutoﬀ of 120 Ry. The ﬁrst Brillouin zone integration was done using the Monkhorst-Pack scheme [31]. To avoid spurious interactions between image structures due to the periodic calculations, a vacuum layer of 10 ˚A was put in each of all non-periodic directions. The QMC calculations were performed using the QMCPACK package [32], where we employed trial many-body wave functions of the Slater-Jastrow (SJ) type with a single Slater determinant consisting of the Kohn-Sham orbitals obtained from the DFT calculations. A variant of the linear method of Umrigar and coworkers [33] was used in the variational Monte 3

text: for

text: fabrication

text: of

text: extended

text: graphynes.

text: The

text: experimental

text: diﬃculty

text: in

text: the

text: synthesis

text: of

text: graphynes

text: leads

text: us

text: to

text: speculate

text: that

text: the

text: DFT

text: prediction

text: on

text: graphyne

text: energetics

text: may

text: be

text: too

text: optimistic

text: as

text: a

text: result

text: of

text: an

text: intrinsic

text: limitation

text: of

text: the

text: methodology.

text: The

text: relative

text: energetic

text: stabilities

text: of

text: these

text: new

text: carbon-based

text: nanomaterials

text: need

text: to

text: be

text: analyzed

text: through

text: accurate

text: numerical

text: calculations

text: which

text: allow

text: full

text: incorporation

text: of

text: the

text: electron-electron

text: correlation.

text: Here

text: we

text: performed

text: quantum

text: Monte

text: Carlo

text: (QMC)

text: simulations

text: to

text: investigate

text: ground-

text: state

text: energetics

text: of

text: various

text: carbon

text: structures

text: such

text: as

text: diamond,

text: graphene,

text: graphynes,

text: and

text: carbynes,

text: especially

text: their

text: cohesive

text: energies.

text: It

text: is

text: found

text: that

text: the

text: cohesive

text: energies

text: of

text: these

text: carbon

text: allotropes

text: are

text: signiﬁcantly

text: overestimated

text: in

text: the

text: DFT

text: calculations

text: when

text: compared

text: with

text: the

text: corresponding

text: QMC

text: energies,

text: as

text: many

text: previous

text: studies

text: have

text: reported

text: We

text: also

text: ﬁnd

text: that

text: the

text: accuracy

text: of

text: the

text: DFT

text: cohesive

text: energy

text: depends

text: on

text: its

text: carbon-carbon

text: bonding

text: nature;

text: the

text: discrepancy

text: between

text: the

text: QMC

text: cohesive

text: energy

text: and

text: the

text: DFT

text: energy

text: is

text: the

text: smallest

text: for

text: diamond,

text: and

text: the

text: largest

text: for

text: carbyne,

text: one-dimensional

text: chain

text: of

text: sp-bonded

text: carbon

text: atoms.

text: In

text: the

text: case

text: of

text: graphene

text: and

text: graphynes,

text: the

text: magnitude

text: of

text: the

text: overestimation

text: of

text: the

text: DFT

text: cohesive

text: energy

text: increases

text: systematically

text: as

text: the

text: ratio

text: of

text: the

text: sp-bonded

text: carbon

text: atoms

text: increases.

text: According

text: to

text: our

text: QMC

text: calculations,

text: the

text: cohesive

text: energy

text: of

text: graphene

text: is

text: eV/atom,

text: which

text: is

text: larger

text: than

text: the

text: cohesive

text: energy

text: of

text: γ-graphyne,

text: energetically

text: the

text: most

text: stable

text: graphyne

text: structure,

text: by

text: eV/atom.

text: II.

text: METHODS

text: AND

text: COMPUTATIONAL

text: DETAILS

text: The

text: QUANTUM

text: ESPRESSO

text: package

text: was

text: used

text: for

text: the

text: DFT

text: calculations

text: where

text: we

text: employed

text: the

text: exchange

text: correlation

text: potential

text: with

text: the

text: generalized

text: gradient

text: approximation

text: in

text: the

text: Perdew-Burke-Ernzerhof

text: parameterization

text: (PBE)

text: scheme

text: The

text: energy-consistent

text: pseudopotential

text: (PP)

text: proposed

text: by

text: Burkatzki,

text: Filippi,

text: and

text: Dolg

text: (BFD)

text: was

text: used

text: for

text: the

text: valence-core

text: partitioning

text: in

text: describing

text: a

text: carbon

text: atom

text: with

text: the

text: kinetic

text: energy

text: cutoﬀ

text: of

text: Ry.

text: The

text: ﬁrst

text: Brillouin

text: zone

text: integration

text: was

text: done

text: using

text: the

text: Monkhorst-Pack

text: scheme

text: To

text: avoid

text: spurious

text: interactions

text: between

text: image

text: structures

text: due

text: to

text: the

text: periodic

text: calculations,

text: a

text: vacuum

text: layer

text: of

text: ˚A

text: was

text: put

text: in

text: each

text: of

text: all

text: non-periodic

text: directions.

text: The

text: QMC

text: calculations

text: were

text: performed

text: using

text: the

text: QMCPACK

text: package

text: where

text: we

text: employed

text: trial

text: many-body

text: wave

text: functions

text: of

text: the

text: Slater-Jastrow

text: (SJ)

text: type

text: with

text: a

text: single

text: Slater

text: determinant

text: consisting

text: of

text: the

text: Kohn-Sham

text: orbitals

text: obtained

text: from

text: the

text: DFT

text: calculations.

text: A

text: variant

text: of

text: the

text: linear

text: method

text: of

text: Umrigar

text: and

text: coworkers

text: was

text: used

text: in

text: the

text: variational

text: Monte


text: estimated to be -155.018(8) eV/atom for graphene and -155.057(1) eV/atom for diamond. With the same BFD pseudo-potential used here, the DMC energy of a single carbon atom was computed to be -147.554(3) eV. From this, the DMC cohesive energies of graphene and diamond were estimated to be 7.464(10) eV/atom and 7.503(3) eV/atom, respectively. We − tried a linear ﬁt of the graphene supercell energies as a function of N 5/4, which corresponds to the size dependence of supercell energies of a strictly two-dimensional electron system [36]. This also produced an excellent ﬁt with the extrapolated value of -155.037(4) eV/atom in − the bulk limit, lower only by 0.019(9) eV/atom than the corresponding value from the N 1 linear ﬁt. For further discussion, we made the ﬁnite-size analysis of the DMC supercell − energies with the ﬁt to a linear function of N 1 for all carbon structures, regardless of their dimensionalities, and ignored higher-order corrections to the volume [36]. The PBE and DMC cohesive energies of graphene and diamond, along with their zero- point energy (ZPE) corrections, are summarized in Table I, where the DMC cohesive energies of graphite were estimated through the addition of the previously-reported DMC interlayer binding energy [25] to the corresponding energies of a single graphene sheet. As seen in Table I, the PBE cohesive energies are signiﬁcantly larger than the DMC energies in both graphene and diamond, indicating that the DFT calculations based on the PBE exchange- correlation potential tend to overestimate the carbon binding as in LDA calculations. The DMC cohesive energy of graphite is found to be slightly larger than that of diamond, which is consistent with the fact that graphite is energetically the most stable carbon structure. The ZPE-corrected DMC cohesive energies of both graphite and diamond are in excellent agree- ment with the corresponding experimental values [43]. For the cohesive energy diﬀerence ∼ between graphite and diamond, the DMC result of 27 meV/atom is basically identical to the experimentally-reported value. This, largely due to the error cancellations, is fortuitous given that there are remaining systematic errors in our QMC calculations, e.g., ﬁxed-node errors and higher-order ﬁnite-size corrections. More comprehensive calculations beyond the SJ orbitals for larger supercells are called for to resolve the energy diﬀerences within 10 meV. Nevertheless, the results support the use of the QMC methods to compute accurate relative energetics of diﬀerent carbon allotropes. The DFT and the QMC calculations were performed for recently-proposed low- dimensional carbon allotropes of graphynes and carbyne, whose atomic structures are shown in Fig. 2(a)-(d), to analyze their energetic stabilities relative to that of graphene. The ge- 5

text: estimated

text: to

text: be

text: eV/atom

text: for

text: graphene

text: and

text: eV/atom

text: for

text: diamond.

text: With

text: the

text: same

text: BFD

text: pseudo-potential

text: used

text: here,

text: the

text: DMC

text: energy

text: of

text: a

text: single

text: carbon

text: atom

text: was

text: computed

text: to

text: be

text: eV.

text: From

text: this,

text: the

text: DMC

text: cohesive

text: energies

text: of

text: graphene

text: and

text: diamond

text: were

text: estimated

text: to

text: be

text: eV/atom

text: and

text: eV/atom,

text: respectively.

text: We

text: tried

text: a

text: linear

text: ﬁt

text: of

text: the

text: graphene

text: supercell

text: energies

text: as

text: a

text: function

text: of

text: N

text: which

text: corresponds

text: to

text: the

text: size

text: dependence

text: of

text: supercell

text: energies

text: of

text: a

text: strictly

text: two-dimensional

text: electron

text: system

text: This

text: also

text: produced

text: an

text: excellent

text: ﬁt

text: with

text: the

text: extrapolated

text: value

text: of

text: eV/atom

text: in

text: the

text: bulk

text: limit,

text: lower

text: only

text: by

text: eV/atom

text: than

text: the

text: corresponding

text: value

text: from

text: the

text: N

text: linear

text: ﬁt.

text: For

text: further

text: discussion,

text: we

text: made

text: the

text: ﬁnite-size

text: analysis

text: of

text: the

text: DMC

text: supercell

text: energies

text: with

text: the

text: ﬁt

text: to

text: a

text: linear

text: function

text: of

text: N

text: for

text: all

text: carbon

text: structures,

text: regardless

text: of

text: their

text: dimensionalities,

text: and

text: ignored

text: higher-order

text: corrections

text: to

text: the

text: volume

text: The

text: PBE

text: and

text: DMC

text: cohesive

text: energies

text: of

text: graphene

text: and

text: diamond,

text: along

text: with

text: their

text: zero-

text: point

text: energy

text: (ZPE)

text: corrections,

text: are

text: summarized

text: in

text: Table

text: I,

text: where

text: the

text: DMC

text: cohesive

text: energies

text: of

text: graphite

text: were

text: estimated

text: through

text: the

text: addition

text: of

text: the

text: previously-reported

text: DMC

text: interlayer

text: binding

text: energy

text: to

text: the

text: corresponding

text: energies

text: of

text: a

text: single

text: graphene

text: sheet.

text: As

text: seen

text: in

text: Table

text: I,

text: the

text: PBE

text: cohesive

text: energies

text: are

text: signiﬁcantly

text: larger

text: than

text: the

text: DMC

text: energies

text: in

text: both

text: graphene

text: and

text: diamond,

text: indicating

text: that

text: the

text: DFT

text: calculations

text: based

text: on

text: the

text: PBE

text: exchange-

text: correlation

text: potential

text: tend

text: to

text: overestimate

text: the

text: carbon

text: binding

text: as

text: in

text: LDA

text: calculations.

text: The

text: DMC

text: cohesive

text: energy

text: of

text: graphite

text: is

text: found

text: to

text: be

text: slightly

text: larger

text: than

text: that

text: of

text: diamond,

text: which

text: is

text: consistent

text: with

text: the

text: fact

text: that

text: graphite

text: is

text: energetically

text: the

text: most

text: stable

text: carbon

text: structure.

text: The

text: ZPE-corrected

text: DMC

text: cohesive

text: energies

text: of

text: both

text: graphite

text: and

text: diamond

text: are

text: in

text: excellent

text: agree-

text: ment

text: with

text: the

text: corresponding

text: experimental

text: values

text: For

text: the

text: cohesive

text: energy

text: diﬀerence

text: between

text: graphite

text: and

text: diamond,

text: the

text: DMC

text: result

text: of

text: meV/atom

text: is

text: basically

text: identical

text: to

text: the

text: experimentally-reported

text: value.

text: This,

text: largely

text: due

text: to

text: the

text: error

text: cancellations,

text: is

text: fortuitous

text: given

text: that

text: there

text: are

text: remaining

text: systematic

text: errors

text: in

text: our

text: QMC

text: calculations,

text: e.g.,

text: ﬁxed-node

text: errors

text: and

text: higher-order

text: ﬁnite-size

text: corrections.

text: More

text: comprehensive

text: calculations

text: beyond

text: the

text: SJ

text: orbitals

text: for

text: larger

text: supercells

text: are

text: called

text: for

text: to

text: resolve

text: the

text: energy

text: diﬀerences

text: within

text: meV.

text: Nevertheless,

text: the

text: results

text: support

text: the

text: use

text: of

text: the

text: QMC

text: methods

text: to

text: compute

text: accurate

text: relative

text: energetics

text: of

text: diﬀerent

text: carbon

text: allotropes.

text: The

text: DFT

text: and

text: the

text: QMC

text: calculations

text: were

text: performed

text: for

text: recently-proposed

text: low-

text: dimensional

text: carbon

text: allotropes

text: of

text: graphynes

text: and

text: carbyne,

text: whose

text: atomic

text: structures

text: are

text: shown

text: in

text: Fig.

text: to

text: analyze

text: their

text: energetic

text: stabilities

text: relative

text: to

text: that

text: of

text: graphene.

text: The

text: ge-


text: energy diﬀerences for sp-sp2 hybridized 2D graphynes and sp-bonded 1D carbyne are larger than the corresponding value of 0.458(10) eV/atom for graphene. As a matter of fact, the cohesive energy diﬀerence increases monotonically as η increases from zero (graphene) to one (carbyne). Note that η is 0.5, 0.67, and 0.75 for γ, β, and α-graphyne, respectively. This leads us to conclude that more correlation energies were missed in the PBE calculations for electrons participating in the sp bonds than for those in the sp2 bonds. The PBE- DMC cohesive energy diﬀerence shows the largest value in 1D sp-bonded carbyne while the 3D sp3-bonded diamond has the smallest energy diﬀerence (see the dashed horizontal line of Fig. 3(a)), indicating that the dimensionalities of carbon structures as well as their bonding nature aﬀect the accuracy of the PBE calculations for their cohesive energies. Figure 3(b) shows the PBE cohesive energy of a carbon allotrope relative to that of graphene, − ∆EPBE = EPBE(graphene) EPBE(allotrope), versus the relative DMC cohesive energy, coh coh coh − ∆EDMC = EDMC(graphene) EDMC(allotrope). As can be seen, the relative PBE cohesive coh coh coh energies are smaller than the corresponding DMC energies in all low-dimensional allotropes considered here. From this we conclude that these new carbon-based nanomaterials are energetically not as stable as the PBE calculations have predicted. The DMC cohesive energy diﬀerence between a low-dimensional carbon allotrope and graphene is shown in Fig. 4 as a function of η, the ratio of the sp-bonded carbon atoms. According to our DMC calculations, γ-graphyne (η=0.5), which is the most stable among new carbon structures presented in Fig. 4, has smaller cohesive energy than graphene by 0.698(12) eV/atom, while the cohesive energy of carbyne is smaller by 1.303(11) eV/atom than that of graphene. Experimental diﬃculty encountered in the synthesis of graphynes and carbynes could be explained by their signiﬁcantly smaller cohesive energies than graphene. It is seen that the cohesive energies of sp-sp2 hybridized graphynes decrease monotonically as η increases. This is consistent with the fact that the sp-bonded carbon atoms are bound only through two bonds, either one single bond and one triple bond or two double bonds, while the sp2-bonded atoms are bound through three bonds. Figure 4 gives a guideline for the energetic stability of a sp-sp2 hybridized 2D carbon network relative to graphene. As noted some years ago by Baugman et al. [15], one can design many diﬀerent types of graphynes that are 2D carbon phases involving acetylene bonds. Among them, (6,6,12)- graphyne with a rectangular symmetry has recently drawn much interest because it possesses direction-dependent Dirac cones to allow electron collimation transport without any external 7

text: energy

text: diﬀerences

text: for

text: hybridized

text: graphynes

text: and

text: sp-bonded

text: carbyne

text: are

text: larger

text: than

text: the

text: corresponding

text: value

text: of

text: eV/atom

text: for

text: graphene.

text: As

text: a

text: matter

text: of

text: fact,

text: the

text: cohesive

text: energy

text: diﬀerence

text: increases

text: monotonically

text: as

text: η

text: increases

text: from

text: zero

text: (graphene)

text: to

text: one

text: (carbyne).

text: Note

text: that

text: η

text: is

text: and

text: for

text: γ,

text: β,

text: and

text: α-graphyne,

text: respectively.

text: This

text: leads

text: us

text: to

text: conclude

text: that

text: more

text: correlation

text: energies

text: were

text: missed

text: in

text: the

text: PBE

text: calculations

text: for

text: electrons

text: participating

text: in

text: the

text: sp

text: bonds

text: than

text: for

text: those

text: in

text: the

text: bonds.

text: The

text: PBE-

text: DMC

text: cohesive

text: energy

text: diﬀerence

text: shows

text: the

text: largest

text: value

text: in

text: sp-bonded

text: carbyne

text: while

text: the

text: diamond

text: has

text: the

text: smallest

text: energy

text: diﬀerence

text: (see

text: the

text: dashed

text: horizontal

text: line

text: of

text: Fig.

text: indicating

text: that

text: the

text: dimensionalities

text: of

text: carbon

text: structures

text: as

text: well

text: as

text: their

text: bonding

text: nature

text: aﬀect

text: the

text: accuracy

text: of

text: the

text: PBE

text: calculations

text: for

text: their

text: cohesive

text: energies.

text: Figure

text: shows

text: the

text: PBE

text: cohesive

text: energy

text: of

text: a

text: carbon

text: allotrope

text: relative

text: to

text: that

text: of

text: graphene,

text: ∆EPBE

text: EPBE(graphene)

text: EPBE(allotrope),

text: versus

text: the

text: relative

text: DMC

text: cohesive

text: energy,

text: coh

text: coh

text: coh

text: ∆EDMC

text: EDMC(graphene)

text: EDMC(allotrope).

text: As

text: can

text: be

text: seen,

text: the

text: relative

text: PBE

text: cohesive

text: coh

text: coh

text: coh

text: energies

text: are

text: smaller

text: than

text: the

text: corresponding

text: DMC

text: energies

text: in

text: all

text: low-dimensional

text: allotropes

text: considered

text: here.

text: From

text: this

text: we

text: conclude

text: that

text: these

text: new

text: carbon-based

text: nanomaterials

text: are

text: energetically

text: not

text: as

text: stable

text: as

text: the

text: PBE

text: calculations

text: have

text: predicted.

text: The

text: DMC

text: cohesive

text: energy

text: diﬀerence

text: between

text: a

text: low-dimensional

text: carbon

text: allotrope

text: and

text: graphene

text: is

text: shown

text: in

text: Fig.

text: as

text: a

text: function

text: of

text: η,

text: the

text: ratio

text: of

text: the

text: sp-bonded

text: carbon

text: atoms.

text: According

text: to

text: our

text: DMC

text: calculations,

text: γ-graphyne

text: which

text: is

text: the

text: most

text: stable

text: among

text: new

text: carbon

text: structures

text: presented

text: in

text: Fig.

text: has

text: smaller

text: cohesive

text: energy

text: than

text: graphene

text: by

text: eV/atom,

text: while

text: the

text: cohesive

text: energy

text: of

text: carbyne

text: is

text: smaller

text: by

text: eV/atom

text: than

text: that

text: of

text: graphene.

text: Experimental

text: diﬃculty

text: encountered

text: in

text: the

text: synthesis

text: of

text: graphynes

text: and

text: carbynes

text: could

text: be

text: explained

text: by

text: their

text: signiﬁcantly

text: smaller

text: cohesive

text: energies

text: than

text: graphene.

text: It

text: is

text: seen

text: that

text: the

text: cohesive

text: energies

text: of

text: hybridized

text: graphynes

text: decrease

text: monotonically

text: as

text: η

text: increases.

text: This

text: is

text: consistent

text: with

text: the

text: fact

text: that

text: the

text: sp-bonded

text: carbon

text: atoms

text: are

text: bound

text: only

text: through

text: two

text: bonds,

text: either

text: one

text: single

text: bond

text: and

text: one

text: triple

text: bond

text: or

text: two

text: double

text: bonds,

text: while

text: the

text: atoms

text: are

text: bound

text: through

text: three

text: bonds.

text: Figure

text: gives

text: a

text: guideline

text: for

text: the

text: energetic

text: stability

text: of

text: a

text: hybridized

text: carbon

text: network

text: relative

text: to

text: graphene.

text: As

text: noted

text: some

text: years

text: ago

text: by

text: Baugman

text: et

text: al.

text: one

text: can

text: design

text: many

text: diﬀerent

text: types

text: of

text: graphynes

text: that

text: are

text: carbon

text: phases

text: involving

text: acetylene

text: bonds.

text: Among

text: them,

text: graphyne

text: with

text: a

text: rectangular

text: symmetry

text: has

text: recently

text: drawn

text: much

text: interest

text: because

text: it

text: possesses

text: direction-dependent

text: Dirac

text: cones

text: to

text: allow

text: electron

text: collimation

text: transport

text: without

text: any

text: external


text: ﬁeld [16]. Quantitative prediction for the cohesion energetics of a new graphyne structure can be made by analyzing the contribution of each carbon bond to their cohesive energies. There are three diﬀerent carbon-carbon bonds involved in a 2D hybridized graphyne structure, that ≡ is, a single bond (C-C), a double bond (C=C) and a triple bond (C C). Its cohesive energy can be assumed to be the sum of the bond energies; N E = ε N + ε N + ε N , (1) atom coh s s d d t t where E is the cohesive energy per atom and ε , ε and ε represent the single, the coh s d t double, and the triple bond energies, respectively. Here N , N , N , and N are the atom s d t numbers of the carbon atoms, the single bonds, the double bonds, and the triple bonds per unit cell, respectively. By ﬁtting the cohesive energies of graphene and three diﬀerent ≡ graphyne structures in Table II to Eq. (1), we obtained the C-C, the C=C, and the C C bond energies of 2D hybridized carbon networks. As shown in Table III, the multiple-linear- regression ﬁts of the DMC and both PBC data resulted in the correlations coeﬃcients very close to the unit value, justifying our assumption that the cohesive energy of a graphyne structure can be described by the sum of the carbon-carbon bond energies. With the PBE- PAW bond energies in Table III, the cohesive energy of (6,6,12)-graphyne is estimated to be 7.265(22) eV/atom, which is in excellent agreement with its PBE-PAW cohesive energy of 7.264 eV/atom. This strongly validates our approach to the cohesive energy with the carbon-carbon bond energies. We expect that with the DMC bond energies in Table III, one can make an accurate prediction for the cohesive energy of a newly-proposed graphyne. The predicted DMC cohesive energy of (6,6,12)-graphyne is 6.671(31) eV/atom. IV. CONCLUSIONS The cohesion energetics of various carbon allotropes have been studied using the QMC methods with full description of electron-electron correlation. We have found that the DFT calculations based on the PBE exchange-correlation potential signiﬁcantly overestimate the cohesive energies of these carbon-based materials. The degree of the PBE overestimation is found to increase as the ratio of sp-bonded carbon atoms increases. This suggests that the PBE predictions for the energetic stabilities of sp-sp2 hybridized 2D carbon networks such as graphynes, relative to that of sp2-bonded graphene, should be considered with a great deal 8

text: ﬁeld

text: Quantitative

text: prediction

text: for

text: the

text: cohesion

text: energetics

text: of

text: a

text: new

text: graphyne

text: structure

text: can

text: be

text: made

text: by

text: analyzing

text: the

text: contribution

text: of

text: each

text: carbon

text: bond

text: to

text: their

text: cohesive

text: energies.

text: There

text: are

text: three

text: diﬀerent

text: carbon-carbon

text: bonds

text: involved

text: in

text: a

text: hybridized

text: graphyne

text: structure,

text: that

text: is,

text: a

text: single

text: bond

text: (C-C),

text: a

text: double

text: bond

text: (C=C)

text: and

text: a

text: triple

text: bond

text: (C

text: C).

text: Its

text: cohesive

text: energy

text: can

text: be

text: assumed

text: to

text: be

text: the

text: sum

text: of

text: the

text: bond

text: energies;

text: N

text: E

text: ε

text: N

text: ε

text: N

text: ε

text: N

text: atom

text: coh

text: s

text: s

text: d

text: d

text: t

text: t

text: where

text: E

text: is

text: the

text: cohesive

text: energy

text: per

text: atom

text: and

text: ε

text: ε

text: and

text: ε

text: represent

text: the

text: single,

text: the

text: coh

text: s

text: d

text: t

text: double,

text: and

text: the

text: triple

text: bond

text: energies,

text: respectively.

text: Here

text: N

text: N

text: N

text: and

text: N

text: are

text: the

text: atom

text: s

text: d

text: t

text: numbers

text: of

text: the

text: carbon

text: atoms,

text: the

text: single

text: bonds,

text: the

text: double

text: bonds,

text: and

text: the

text: triple

text: bonds

text: per

text: unit

text: cell,

text: respectively.

text: By

text: ﬁtting

text: the

text: cohesive

text: energies

text: of

text: graphene

text: and

text: three

text: diﬀerent

text: graphyne

text: structures

text: in

text: Table

text: II

text: to

text: Eq.

text: we

text: obtained

text: the

text: C-C,

text: the

text: C=C,

text: and

text: the

text: C

text: C

text: bond

text: energies

text: of

text: hybridized

text: carbon

text: networks.

text: As

text: shown

text: in

text: Table

text: III,

text: the

text: multiple-linear-

text: regression

text: ﬁts

text: of

text: the

text: DMC

text: and

text: both

text: PBC

text: data

text: resulted

text: in

text: the

text: correlations

text: coeﬃcients

text: very

text: close

text: to

text: the

text: unit

text: value,

text: justifying

text: our

text: assumption

text: that

text: the

text: cohesive

text: energy

text: of

text: a

text: graphyne

text: structure

text: can

text: be

text: described

text: by

text: the

text: sum

text: of

text: the

text: carbon-carbon

text: bond

text: energies.

text: With

text: the

text: PBE-

text: PAW

text: bond

text: energies

text: in

text: Table

text: III,

text: the

text: cohesive

text: energy

text: of

text: is

text: estimated

text: to

text: be

text: eV/atom,

text: which

text: is

text: in

text: excellent

text: agreement

text: with

text: its

text: PBE-PAW

text: cohesive

text: energy

text: of

text: eV/atom.

text: This

text: strongly

text: validates

text: our

text: approach

text: to

text: the

text: cohesive

text: energy

text: with

text: the

text: carbon-carbon

text: bond

text: energies.

text: We

text: expect

text: that

text: with

text: the

text: DMC

text: bond

text: energies

text: in

text: Table

text: III,

text: one

text: can

text: make

text: an

text: accurate

text: prediction

text: for

text: the

text: cohesive

text: energy

text: of

text: a

text: newly-proposed

text: graphyne.

text: The

text: predicted

text: DMC

text: cohesive

text: energy

text: of

text: is

text: eV/atom.

text: IV.

text: CONCLUSIONS

text: The

text: cohesion

text: energetics

text: of

text: various

text: carbon

text: allotropes

text: have

text: been

text: studied

text: using

text: the

text: QMC

text: methods

text: with

text: full

text: description

text: of

text: electron-electron

text: correlation.

text: We

text: have

text: found

text: that

text: the

text: DFT

text: calculations

text: based

text: on

text: the

text: PBE

text: exchange-correlation

text: potential

text: signiﬁcantly

text: overestimate

text: the

text: cohesive

text: energies

text: of

text: these

text: carbon-based

text: materials.

text: The

text: degree

text: of

text: the

text: PBE

text: overestimation

text: is

text: found

text: to

text: increase

text: as

text: the

text: ratio

text: of

text: sp-bonded

text: carbon

text: atoms

text: increases.

text: This

text: suggests

text: that

text: the

text: PBE

text: predictions

text: for

text: the

text: energetic

text: stabilities

text: of

text: hybridized

text: carbon

text: networks

text: such

text: as

text: graphynes,

text: relative

text: to

text: that

text: of

text: graphene,

text: should

text: be

text: considered

text: with

text: a

text: great

text: deal