text: 2 Cr1(β ) and Cr1-As2-Cr1(γ )/Cr2-As1-Cr2(γ ) angles are 2 1 2 also underestimated. Similar cases have been also noted in the studies of iron based superconductors[17, 27, 28] and it may be related to the strong spin ﬂuctuation, which is beyond DFT calculation. We adopt the experimental parameters in the following calculation unless otherwise speciﬁed. The band structure, the density of states(DOS) and Fermi surfaces for K Cr As have been calculated in Ref.29. Our 2 3 3 results as shown in Appendix A are similar to their calcula- z tions. In general, the 3d states of Cr are located from -2.5 eV to 2.0 eV and the states near the Fermi level are mainly at- x y tributed to Cr d , d and d − orbitals. The As 4p states z2 xy x2 y2 mainly lie 1.0 eV below the Fermi level and hybridize strongly J with the Cr 3d states. Along k direction, the bands are very 1 z dispersive due to the strong coupling between d oribials along , z direction. The bands attributed to Cr2 d and d − or- J xy x2 y2 1 J bitals are very close to half-ﬁlling. As the separation between 2 chains is large, the inplane band dispersion is very small but not negligible. However, we want to point out an important feature that is ignored in Ref.29. Cr1 ions have more d elec- trons than Cr2 ions, which is consistent with analysis from the crystal structure. We will show later that this difference also FIG. 1: Schematic view of the structure of A Cr As (a). (b) and results in different magnetic moments at Cr1 and Cr2 sites in 2 3 2 (c) show the all-in and in-out noncollinear magnetic states. The ex- magnetic states. (cid:48) change coupling parameters J , J and J are deﬁned in (d). The 1 1 2 The band structure of Rb Cr As , shown also in Appendix orange and purple spheres represent Cr1 and Cr2. 2 3 3 A, shares many common features with that of K Cr As , but 2 3 3 they also differ in some details. As the radius of Rb is big- TABLE I: Experimental and optimized structural parameters of ger than K, Cr1As layer can get more electrons and dxy and K Cr As using GGA in the paramagnetic phase. Deviations be- d − bands of Cr1 are fully occupied. Meanwhile, the d 2 3 3 x2 y2 xy tween the optimized and experimental values are given in parenthe- and d − bands of Cr2 are much less occupied. It leads to x2 y2 ses in % . great difference between the 3D Fermi surfaces of the two materials. Another difference is that there is an additional GGA EXP a(A˚ ) 10.113(+1.3) 9.983 electron Fermi surface near A(0, 0, π) point in Rb2Cr3As3, c(A˚ ) 4.147(-1.98) 4.230 which is attributed to dxy and dx2−y2 orbitals of Cr2. Finally Cr1-Cr1/Cr2-Cr2(A˚ ) 2.498; 2.588 2.615; 2.691 the d and d orbitals are much close to the Fermi level in xz yz Cr1-As1/Cr2-As2(A˚ ) 2.513; 2.494 2.51; 2.49 Rb Cr As . The calculated N(E ) in K Cr As (Rb Cr As ) 2 3 3 − F 2 3 3 2 3 3 Cr1-As2/Cr2-As1(A˚ ) 2.522; 2.506 2.516; 2.506 is 8.76(9.13) eV 1/f.u.. The calculated Pauli susceptibility ◦ × − α /α ( ) 59.6; 62.51 62.8;65.4 and speciﬁc heat coefﬁcient are χ = 2.83(2.95) 10 4 1 2 ◦ 0 β1/β2(◦) 60.8; 60.87 62.8;62.9 emu/mol and γ = 20.7(21.5) mJ/(K2 mol). The calculated γ γ /γ ( ) 110.6; 111.6 114.4;115.2 is about only one third of the experimental value in K Cr As , 1 2 2 3 3 suggesting strong correlation in these systems. Now we focus on the magnetic properties. We consider is 600 eV. Forces are minimized to less than 0.01 eV/A˚ in the four possible collinear magnetic states, the paramagnetic state structural relaxation. (PM), the ferromagnetic (FM) state, the interlayer antiferro- ↑↑↓↓ Due to the asymmetric distribution of alkali metal ions and magnetic (AFM) state and the up-up-down-down ( ) mag- the absence of inversion symmetry, there are two kinds of netic state. Due to strong magnetic frustration, we also con- Cr ions in K Cr As (Rb Cr As ): Cr1 and Cr2. The Cr1 sider two additional co-planar antiferromagnetic states: all-in 2 3 3 2 3 3 ions are surrounded by six inplane A ions and the inplane magnetic state (Fig.1(b)) and in-out magnetic state (Fig.1(c)). bondlength is 2.615(2.57)A˚ . The Cr2 ions are surrounded by We perform calculations with spin orbital coupling (SOC) and three inplane alkali ions and the inplane bondlength is a bit the calculated magnetic moments and the total relative ener- longer, 2.69(3.16)A˚ . From the number of A ions surrounding gies of above magnetic states are summarized in Table.II. The Cr ions, it is expected that the Cr1As layer will obtain more IOP state is the ground state in K Cr As (Rb Cr As ), with 2 2 2 2 2 2 electrons than the Cr2As layer. a large energy gain of 39(762) meV/cell relative to the PM The optimized and experimental structural parameters of state. The magnetic moments are 0.90(1.75) and 0.94(2.34) K Cr As are summarized in Table I. We ﬁnd that the lattice µ on Cr1 and Cr2 sites, respectively. The initial FM state 2 3 3 B constants are comparable with experimental values but the Cr- converges to a PM or AFM state, indicating that the inter- Cr bond lengths are underestimated by 0.1 A˚ . Thus, the Cr1- layer magnetic coupling J(cid:48) as indicated in Fig.1(d) is antifer- 1 As1-Cr1(α )/Cr2-As2-Cr2(α ), Cr1-As1-Cr2(β )/Cr2-As2- romagnetic and relatively strong. In the AFM state, the energy 1 2 1 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: angles

text: are

text: also

text: underestimated.

text: Similar

text: cases

text: have

text: been

text: also

text: noted

text: in

text: the

text: studies

text: of

text: iron

text: based

text: and

text: it

text: may

text: be

text: related

text: to

text: the

text: strong

text: spin

text: ﬂuctuation,

text: which

text: is

text: beyond

text: DFT

text: calculation.

text: We

text: adopt

text: the

text: experimental

text: parameters

text: in

text: the

text: following

text: calculation

text: unless

text: otherwise

text: speciﬁed.

text: The

text: band

text: structure,

text: the

text: density

text: of

text: states(DOS)

text: and

text: Fermi

text: surfaces

text: for

text: K

text: Cr

text: As

text: have

text: been

text: calculated

text: in

text: Our

text: results

text: as

text: shown

text: in

text: Appendix

text: A

text: are

text: similar

text: to

text: their

text: calcula-

text: z

text: tions.

text: In

text: general,

text: the

text: states

text: of

text: Cr

text: are

text: located

text: from

text: eV

text: to

text: eV

text: and

text: the

text: states

text: near

text: the

text: Fermi

text: level

text: are

text: mainly

text: at-

text: x

text: y

text: tributed

text: to

text: Cr

text: d

text: d

text: and

text: d

text: orbitals.

text: The

text: As

text: states

text: xy

text: mainly

text: lie

text: eV

text: below

text: the

text: Fermi

text: level

text: and

text: hybridize

text: strongly

text: J

text: with

text: the

text: Cr

text: states.

text: Along

text: k

text: direction,

text: the

text: bands

text: are

text: very

text: z

text: dispersive

text: due

text: to

text: the

text: strong

text: coupling

text: between

text: d

text: oribials

text: along

text: z

text: direction.

text: The

text: bands

text: attributed

text: to

text: d

text: and

text: d

text: or-

text: J

text: xy

text: J

text: bitals

text: are

text: very

text: close

text: to

text: half-ﬁlling.

text: As

text: the

text: separation

text: between

text: chains

text: is

text: large,

text: the

text: inplane

text: band

text: dispersion

text: is

text: very

text: small

text: but

text: not

text: negligible.

text: However,

text: we

text: want

text: to

text: point

text: out

text: an

text: important

text: feature

text: that

text: is

text: ignored

text: in

text: ions

text: have

text: more

text: d

text: elec-

text: trons

text: than

text: ions,

text: which

text: is

text: consistent

text: with

text: analysis

text: from

text: the

text: crystal

text: structure.

text: We

text: will

text: show

text: later

text: that

text: this

text: difference

text: also

text: FIG.

text: Schematic

text: view

text: of

text: the

text: structure

text: of

text: A

text: Cr

text: As

text: (a).

text: (b)

text: and

text: results

text: in

text: different

text: magnetic

text: moments

text: at

text: and

text: sites

text: in

text: (c)

text: show

text: the

text: all-in

text: and

text: in-out

text: noncollinear

text: magnetic

text: states.

text: The

text: ex-

text: magnetic

text: states.

text: change

text: coupling

text: parameters

text: J

text: J

text: and

text: J

text: are

text: deﬁned

text: in

text: (d).

text: The

text: The

text: band

text: structure

text: of

text: Rb

text: Cr

text: As

text: shown

text: also

text: in

text: Appendix

text: orange

text: and

text: purple

text: spheres

text: represent

text: and

text: A,

text: shares

text: many

text: common

text: features

text: with

text: that

text: of

text: K

text: Cr

text: As

text: but

text: they

text: also

text: differ

text: in

text: some

text: details.

text: As

text: the

text: radius

text: of

text: Rb

text: is

text: big-

text: TABLE

text: I:

text: Experimental

text: and

text: optimized

text: structural

text: parameters

text: of

text: ger

text: than

text: K,

text: layer

text: can

text: get

text: more

text: electrons

text: and

text: dxy

text: and

text: K

text: Cr

text: As

text: using

text: GGA

text: in

text: the

text: paramagnetic

text: phase.

text: Deviations

text: be-

text: d

text: bands

text: of

text: are

text: fully

text: occupied.

text: Meanwhile,

text: the

text: d

text: xy

text: tween

text: the

text: optimized

text: and

text: experimental

text: values

text: are

text: given

text: in

text: parenthe-

text: and

text: d

text: bands

text: of

text: are

text: much

text: less

text: occupied.

text: It

text: leads

text: to

text: ses

text: in

text: great

text: difference

text: between

text: the

text: Fermi

text: surfaces

text: of

text: the

text: two

text: materials.

text: Another

text: difference

text: is

text: that

text: there

text: is

text: an

text: additional

text: GGA

text: EXP

text: a(A˚

text: electron

text: Fermi

text: surface

text: near

text: π)

text: point

text: in

text: c(A˚

text: which

text: is

text: attributed

text: to

text: dxy

text: and

text: orbitals

text: of

text: Finally

text: the

text: d

text: and

text: d

text: orbitals

text: are

text: much

text: close

text: to

text: the

text: Fermi

text: level

text: in

text: xz

text: yz

text: Rb

text: Cr

text: As

text: The

text: calculated

text: N(E

text: in

text: K

text: Cr

text: As

text: (Rb

text: Cr

text: As

text: F

text: is

text: eV

text: The

text: calculated

text: Pauli

text: susceptibility

text: α

text: /α

text: and

text: speciﬁc

text: heat

text: coefﬁcient

text: are

text: χ

text: emu/mol

text: and

text: γ

text: mol).

text: The

text: calculated

text: γ

text: γ

text: /γ

text: is

text: about

text: only

text: one

text: third

text: of

text: the

text: experimental

text: value

text: in

text: K

text: Cr

text: As

text: suggesting

text: strong

text: correlation

text: in

text: these

text: systems.

text: Now

text: we

text: focus

text: on

text: the

text: magnetic

text: properties.

text: We

text: consider

text: is

text: eV.

text: Forces

text: are

text: minimized

text: to

text: less

text: than

text: eV/A˚

text: in

text: the

text: four

text: possible

text: collinear

text: magnetic

text: states,

text: the

text: paramagnetic

text: state

text: structural

text: relaxation.

text: (PM),

text: the

text: ferromagnetic

text: (FM)

text: state,

text: the

text: interlayer

text: antiferro-

text: Due

text: to

text: the

text: asymmetric

text: distribution

text: of

text: alkali

text: metal

text: ions

text: and

text: magnetic

text: (AFM)

text: state

text: and

text: the

text: up-up-down-down

text: mag-

text: the

text: absence

text: of

text: inversion

text: symmetry,

text: there

text: are

text: two

text: kinds

text: of

text: netic

text: state.

text: Due

text: to

text: strong

text: magnetic

text: frustration,

text: we

text: also

text: con-

text: Cr

text: ions

text: in

text: K

text: Cr

text: As

text: (Rb

text: Cr

text: As

text: and

text: The

text: sider

text: two

text: additional

text: co-planar

text: antiferromagnetic

text: states:

text: all-in

text: ions

text: are

text: surrounded

text: by

text: six

text: inplane

text: A

text: ions

text: and

text: the

text: inplane

text: magnetic

text: state

text: and

text: in-out

text: magnetic

text: state

text: bondlength

text: is

text: The

text: ions

text: are

text: surrounded

text: by

text: We

text: perform

text: calculations

text: with

text: spin

text: orbital

text: coupling

text: (SOC)

text: and

text: three

text: inplane

text: alkali

text: ions

text: and

text: the

text: inplane

text: bondlength

text: is

text: a

text: bit

text: the

text: calculated

text: magnetic

text: moments

text: and

text: the

text: total

text: relative

text: ener-

text: longer,

text: From

text: the

text: number

text: of

text: A

text: ions

text: surrounding

text: gies

text: of

text: above

text: magnetic

text: states

text: are

text: summarized

text: in

text: Table.II.

text: The

text: Cr

text: ions,

text: it

text: is

text: expected

text: that

text: the

text: layer

text: will

text: obtain

text: more

text: IOP

text: state

text: is

text: the

text: ground

text: state

text: in

text: K

text: Cr

text: As

text: (Rb

text: Cr

text: As

text: with

text: electrons

text: than

text: the

text: layer.

text: a

text: large

text: energy

text: gain

text: of

text: meV/cell

text: relative

text: to

text: the

text: PM

text: The

text: optimized

text: and

text: experimental

text: structural

text: parameters

text: of

text: state.

text: The

text: magnetic

text: moments

text: are

text: and

text: K

text: Cr

text: As

text: are

text: summarized

text: in

text: Table

text: I.

text: We

text: ﬁnd

text: that

text: the

text: lattice

text: µ

text: on

text: and

text: sites,

text: respectively.

text: The

text: initial

text: FM

text: state

text: B

text: constants

text: are

text: comparable

text: with

text: experimental

text: values

text: but

text: the

text: Cr-

text: converges

text: to

text: a

text: PM

text: or

text: AFM

text: state,

text: indicating

text: that

text: the

text: inter-

text: Cr

text: bond

text: lengths

text: are

text: underestimated

text: by

text: A˚

text: Thus,

text: the

text: layer

text: magnetic

text: coupling

text: as

text: indicated

text: in

text: is

text: antifer-

text: romagnetic

text: and

text: relatively

text: strong.

text: In

text: the

text: AFM

text: state,

text: the

text: energy

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 Abbas Askar, Mirek Giersz, Wojciech Pych, Arkadiusz Olech & Arkadiusz Hypki 2. Comparing MOCCA-COCOA Results & Observational Data The test star cluster evolved using MOCCA for 12 Gyrs was tidally underﬁlled, had a very high pri- mordial binary fraction (95%) and initially contained 553k objects. Using the sim2obs tool in COCOA, we generated a FITS image (Figure 1) from the MOCCA snapshot after 12 Gyrs of evolution. The simulated observation of the model cluster was made with a 2.5m optical telescope and the distance × to the cluster was set to 5 kpc. The FITS ﬁles (2 2 mosaic of the cluster) generated by sim2obs were used by observers to obtain photometric data for 49253 objects extending up to a radius of 465 arcsec. We compared the the surface brightness proﬁle from the photometry done by the observers with the SBP from MOCCA-COCOA results (Figure 1). Photometric data from observers was limited and the observational SBP extends to a smaller radius compared to all the data from MOCCA simulations. ∼ For this comparison, we limited the MOCCA-COCOA projected data to 700 arcsec. After photo- metric corrections, the SBP from the observational data and projected simulation data match closely. With COCOA we are able to ﬁt the SBP to the King Model. Table 1 shows a comparison of cluster parameters obtained from the SBP and King ﬁt for projected simulation and observation data. Figure 1. The left panel shows the grayscale synthetic image of the model cluster created using COCOA. The right panel shows the surface brightness proﬁle and King ﬁt of the cluster using projected MOCCA-COCOA data and observational data. Table 1. Comparison of cluster parameters obtained through projected simulation and observational data. These results may vary with diﬀerent projections of simulation data. Data King Scale Radius (pc) King Parameter W0 Half-light Radius (pc) MOCCA-COCOA 0.91 6.90 2.06 Observation 0.98 6.83 2.07 3. Acknowledgements AA, MG and AH were partially supported by the Polish National Science Centre through the grant DEC-2012/07/B/ST9/04412. References Giersz, M., Heggie, D. C., Hurley, J. R. & Hypki, A. 2013, MNRAS, 431, 2184 Hypki, A. & Giersz, M. 2013, MNRAS, 429, 1221 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Abbas

text: Askar,

text: Mirek

text: Giersz,

text: Wojciech

text: Pych,

text: Arkadiusz

text: Olech

text: Arkadiusz

text: Hypki

label: section

text: Comparing

text: MOCCA-COCOA

text: Results

text: Observational

text: Data

text: The

text: test

text: star

text: cluster

text: evolved

text: using

text: MOCCA

text: for

text: Gyrs

text: was

text: tidally

text: underﬁlled,

text: had

text: a

text: very

text: high

text: pri-

text: mordial

text: binary

text: fraction

text: and

text: initially

text: contained

text: objects.

text: Using

text: the

text: tool

text: in

text: COCOA,

text: we

text: generated

text: a

text: FITS

text: image

text: (Figure

text: from

text: the

text: MOCCA

text: snapshot

text: after

text: Gyrs

text: of

text: evolution.

text: The

text: simulated

text: observation

text: of

text: the

text: model

text: cluster

text: was

text: made

text: with

text: a

text: optical

text: telescope

text: and

text: the

text: distance

text: to

text: the

text: cluster

text: was

text: set

text: to

text: kpc.

text: The

text: FITS

text: ﬁles

text: mosaic

text: of

text: the

text: cluster)

text: generated

text: by

text: were

text: used

text: by

text: observers

text: to

text: obtain

text: photometric

text: data

text: for

text: objects

text: extending

text: up

text: to

text: a

text: radius

text: of

text: arcsec.

text: We

text: compared

text: the

text: the

text: surface

text: brightness

text: proﬁle

text: from

text: the

text: photometry

text: done

text: by

text: the

text: observers

text: with

text: the

text: SBP

text: from

text: MOCCA-COCOA

text: results

text: (Figure

text: Photometric

text: data

text: from

text: observers

text: was

text: limited

text: and

text: the

text: observational

text: SBP

text: extends

text: to

text: a

text: smaller

text: radius

text: compared

text: to

text: all

text: the

text: data

text: from

text: MOCCA

text: simulations.

text: For

text: this

text: comparison,

text: we

text: limited

text: the

text: MOCCA-COCOA

text: projected

text: data

text: to

text: arcsec.

text: After

text: photo-

text: metric

text: corrections,

text: the

text: SBP

text: from

text: the

text: observational

text: data

text: and

text: projected

text: simulation

text: data

text: match

text: closely.

text: With

text: COCOA

text: we

text: are

text: able

text: to

text: ﬁt

text: the

text: SBP

text: to

text: the

text: King

text: Model.

text: Table

text: shows

text: a

text: comparison

text: of

text: cluster

text: parameters

text: obtained

text: from

text: the

text: SBP

text: and

text: King

text: ﬁt

text: for

text: projected

text: simulation

text: and

text: observation

text: data.

text: Figure

text: The

text: left

text: panel

text: shows

text: the

text: grayscale

text: synthetic

text: image

text: of

text: the

text: model

text: cluster

text: created

text: using

text: COCOA.

text: The

text: right

text: panel

text: shows

text: the

text: surface

text: brightness

text: proﬁle

text: and

text: King

text: ﬁt

text: of

text: the

text: cluster

text: using

text: projected

text: MOCCA-COCOA

text: data

text: and

text: observational

text: data.

text: Table

text: Comparison

text: of

text: cluster

text: parameters

text: obtained

text: through

text: projected

text: simulation

text: and

text: observational

text: data.

text: These

text: results

text: may

text: vary

text: with

text: diﬀerent

text: projections

text: of

text: simulation

text: data.

text: Data

text: King

text: Scale

text: Radius

text: (pc)

text: King

text: Parameter

text: Half-light

text: Radius

text: (pc)

text: MOCCA-COCOA

text: Observation

text: Acknowledgements

text: AA,

text: MG

text: and

text: AH

text: were

text: partially

text: supported

text: by

text: the

text: Polish

text: National

text: Science

text: Centre

text: through

text: the

text: grant

text: References

text: Giersz,

text: M.,

text: Heggie,

text: D.

text: C.,

text: Hurley,

text: J.

text: R.

text: Hypki,

text: A.

text: MNRAS,

text: Hypki,

text: A.

text: Giersz,

text: M.

text: MNRAS,

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: FIG. 2. (Color online) Chern number dependence on the recoil momentum p in the case ε = J and J = 0. In the left panel we present the phase diagram of the lowest band Chern number c . 2 1 In the right panel we show corresponding phase diagram for the middle band. Since the sum of Chern numbers over all three bands is zero, the third band gives the same phase diagram as the ﬁrst one (c = c ). The green regions correspond to the Chern number zero. The yellow, red, cyan 1 3 and blue regions correspond to the Chern numbers 1, 2, -1 and -2 respectively. Nonzero Chern numbers are also displayed as labels. The hexagon represents the FBZ in the p-plane. → H → H corresponds to a gauge transformation. Thus there is a symmetry (p p + G, ). In the phase diagram (Fig. 2) we also show the FBZ in the p-plane, which is a hexagon with (cid:48) two inequivalent corners positioned at the points K and K . Now let us analyze eﬀects of the non-zero next-nearest neighbor coupling. For this we set J = 0.3J and ε = 2J. The phase diagrams of the Chern numbers are presented in the 2 { } Fig. 3. We can see regions with the Chern numbers corresponding to trivial phases 0, 0, 0 { ± ∓ } {± ∓ } and phases 0, 1, 1 and 1, 0, 1 . In the latter two types of regions we can ﬁnd points corresponding to non-zero band gaps ∆ > 0 and/or ∆ > 0 (Fig. 4). This shows that 12 23 there exist topological Chern insulating phases. For example at the point p = K, we have { − } the Chern numbers 0, 1, 1 , the band gap between the middle and highest bands being ≈ ∆ 0.26J. Band widths in this case are about 3J. By positioning the Fermi energy 23 in the gap between the second and third bands one arrives at the Chern insulating phase. {− } Another interesting point is p = 2K, which gives the Chern numbers 1, 0, 1 , the band 11 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

label: caption

text: online)

text: Chern

text: number

text: dependence

text: on

text: the

text: recoil

text: momentum

text: p

text: in

text: the

text: case

text: ε

text: J

text: and

text: J

text: In

text: the

text: left

text: panel

text: we

text: present

text: the

text: phase

text: diagram

text: of

text: the

text: lowest

text: band

text: Chern

text: number

text: c

text: In

text: the

text: right

text: panel

text: we

text: show

text: corresponding

text: phase

text: diagram

text: for

text: the

text: middle

text: band.

text: Since

text: the

text: sum

text: of

text: Chern

text: numbers

text: over

text: all

text: three

text: bands

text: is

text: zero,

text: the

text: third

text: band

text: gives

text: the

text: same

text: phase

text: diagram

text: as

text: the

text: ﬁrst

text: one

text: (c

text: c

text: The

text: green

text: regions

text: correspond

text: to

text: the

text: Chern

text: number

text: zero.

text: The

text: yellow,

text: red,

text: cyan

text: and

text: blue

text: regions

text: correspond

text: to

text: the

text: Chern

text: numbers

text: and

text: respectively.

text: Nonzero

text: Chern

text: numbers

text: are

text: also

text: displayed

text: as

text: labels.

text: The

text: hexagon

text: represents

text: the

text: FBZ

text: in

text: the

text: p-plane.

text: H

text: H

text: corresponds

text: to

text: a

text: gauge

text: transformation.

text: Thus

text: there

text: is

text: a

text: symmetry

text: (p

text: p

text: G,

text: In

text: the

text: phase

text: diagram

text: (Fig.

text: we

text: also

text: show

text: the

text: FBZ

text: in

text: the

text: p-plane,

text: which

text: is

text: a

text: hexagon

text: with

text: two

text: inequivalent

text: corners

text: positioned

text: at

text: the

text: points

text: K

text: and

text: K

text: Now

text: let

text: us

text: analyze

text: eﬀects

text: of

text: the

text: non-zero

text: next-nearest

text: neighbor

text: coupling.

text: For

text: this

text: we

text: set

text: J

text: and

text: ε

text: The

text: phase

text: diagrams

text: of

text: the

text: Chern

text: numbers

text: are

text: presented

text: in

text: the

text: Fig.

text: We

text: can

text: see

text: regions

text: with

text: the

text: Chern

text: numbers

text: corresponding

text: to

text: trivial

text: phases

text: and

text: phases

text: and

text: In

text: the

text: latter

text: two

text: types

text: of

text: regions

text: we

text: can

text: ﬁnd

text: points

text: corresponding

text: to

text: non-zero

text: band

text: gaps

text: and/or

text: (Fig.

text: This

text: shows

text: that

text: there

text: exist

text: topological

text: Chern

text: insulating

text: phases.

text: For

text: example

text: at

text: the

text: point

text: p

text: K,

text: we

text: have

text: the

text: Chern

text: numbers

text: the

text: band

text: gap

text: between

text: the

text: middle

text: and

text: highest

text: bands

text: being

text: Band

text: widths

text: in

text: this

text: case

text: are

text: about

text: By

text: positioning

text: the

text: Fermi

text: energy

text: in

text: the

text: gap

text: between

text: the

text: second

text: and

text: third

text: bands

text: one

text: arrives

text: at

text: the

text: Chern

text: insulating

text: phase.

text: Another

text: interesting

text: point

text: is

text: p

text: which

text: gives

text: the

text: Chern

text: numbers

text: the

text: band

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: FIG. 3. (Color online) Chern number dependence on the recoil momentum p in the case ε = 2J and J = 0.3J. In the left panel we present the Chern number c of the lowest band, while in 2 1 the right panel we show the Chern number c of the middle band. For the third band (not shown 2 − here) we have c = (c + c ). The green, yellow, red, cyan and blue regions correspond to the 3 1 2 Chern numbers 0, 1, 2, -1 and -2 respectively. Nonzero Chern numbers are also displayed as labels. A smaller hexagon shows the FBZ corresponding to the case J = 0. Since the introduction of 2 non-zero J changes the periodicity of the p-dependence, we also show a bigger hexagon, which is 2 now the FBZ in the p-plane. ≈ ≈ gaps ∆ 1.55J and ∆ 0.54J and the band widths of about 2J. The bottom and 12 23 top bands have non-zero Chern numbers, while it is zero for the middle band. Depending on the ﬁlling there are two types of topologically non-trivial phases. If the Fermi energy is positioned in one of the band gaps, we get a topological insulating phase. If the Fermi energy is situated within a band, the band is partially ﬁlled and supports the Chern metal phase. The discussed types of Chern number distributions over the bands are typical when J is non-zero and smaller than J and ε. 2 In the case of non-zero NNN coupling J the translation symmetry in the recoil momentum 2 p is smaller than in the case of zero NNN couplings: one has to shift the momentum by 2G rather than G. In the phase diagram presented in the Fig. 3 we show this by extending the FBZ, which is now a bigger hexagon. There are more types of Chern phases when the coupling J is larger than in the previous 2 12 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

text: online)

text: Chern

text: number

text: dependence

text: on

text: the

text: recoil

text: momentum

text: p

text: in

text: the

text: case

text: ε

text: and

text: J

text: In

text: the

text: left

text: panel

text: we

text: present

text: the

text: Chern

text: number

text: c

text: of

text: the

text: lowest

text: band,

text: while

text: in

text: the

text: right

text: panel

text: we

text: show

text: the

text: Chern

text: number

text: c

text: of

text: the

text: middle

text: band.

text: For

text: the

text: third

text: band

text: (not

text: shown

text: here)

text: we

text: have

text: c

text: (c

text: c

text: The

text: green,

text: yellow,

text: red,

text: cyan

text: and

text: blue

text: regions

text: correspond

text: to

text: the

text: Chern

text: numbers

text: and

text: respectively.

text: Nonzero

text: Chern

text: numbers

text: are

text: also

text: displayed

text: as

text: labels.

text: A

text: smaller

text: hexagon

text: shows

text: the

text: FBZ

text: corresponding

text: to

text: the

text: case

text: J

text: Since

text: the

text: introduction

text: of

text: non-zero

text: J

text: changes

text: the

text: periodicity

text: of

text: the

text: p-dependence,

text: we

text: also

text: show

text: a

text: bigger

text: hexagon,

text: which

text: is

text: now

text: the

text: FBZ

text: in

text: the

text: p-plane.

text: gaps

text: and

text: and

text: the

text: band

text: widths

text: of

text: about

text: The

text: bottom

text: and

text: top

text: bands

text: have

text: non-zero

text: Chern

text: numbers,

text: while

text: it

text: is

text: zero

text: for

text: the

text: middle

text: band.

text: Depending

text: on

text: the

text: ﬁlling

text: there

text: are

text: two

text: types

text: of

text: topologically

text: non-trivial

text: phases.

text: If

text: the

text: Fermi

text: energy

text: is

text: positioned

text: in

text: one

text: of

text: the

text: band

text: gaps,

text: we

text: get

text: a

text: topological

text: insulating

text: phase.

text: If

text: the

text: Fermi

text: energy

text: is

text: situated

text: within

text: a

text: band,

text: the

text: band

text: is

text: partially

text: ﬁlled

text: and

text: supports

text: the

text: Chern

text: metal

text: phase.

text: The

text: discussed

text: types

text: of

text: Chern

text: number

text: distributions

text: over

text: the

text: bands

text: are

text: typical

text: when

text: J

text: is

text: non-zero

text: and

text: smaller

text: than

text: J

text: and

text: ε.

text: In

text: the

text: case

text: of

text: non-zero

text: NNN

text: coupling

text: J

text: the

text: translation

text: symmetry

text: in

text: the

text: recoil

text: momentum

text: p

text: is

text: smaller

text: than

text: in

text: the

text: case

text: of

text: zero

text: NNN

text: couplings:

text: one

text: has

text: to

text: shift

text: the

text: momentum

text: by

text: rather

text: than

text: G.

text: In

text: the

text: phase

text: diagram

text: presented

text: in

text: the

text: Fig.

text: we

text: show

text: this

text: by

text: extending

text: the

text: FBZ,

text: which

text: is

text: now

text: a

text: bigger

text: hexagon.

text: There

text: are

text: more

text: types

text: of

text: Chern

text: phases

text: when

text: the

text: coupling

text: J

text: is

text: larger

text: than

text: in

text: the

text: previous

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: FIG. 6. (Color online) Bulk lattice spectrum projected along k for a number of diﬀerent k values y x − ≤ ≤ in the range K k K . Left: the spectrum for the recoil momentum p = K in the absence x x x of the NNN-coupling (J = 0) and for ε = J corresponding to the parameters used in the Fig. 2. 2 In that case there is no energy gap in the spectrum, but diﬀerent energy bands do not directly touch each other. A topological semi-metal phase is formed if the atoms ﬁll the ﬁrst energy band or the ﬁrst two bands. Right: the spectrum for the recoil momentum p = (2K , K ) in the case x y where ε = 0.5J and J = 0.5J corresponding to the phase diagram shown in Fig. 5. Now there are 2 ± ≈ two bands with non-zero Chern numbers 2 separated by a quasi-gap ∆ 1.35J containing a 13 middle band with a zero Chern number. the contour integral around the FBZ (the ﬁrst term on the r.h.s. of the above equation) is zero. Consequently the Chern number (17) can be calculated by integrating A around n each excluded singular point [44]: (cid:73) (cid:88) 1 · c = dk A , (21) n n 2π singul where the sum is over all singular points in the FBZ. Let us summarize our analytical results, details being presented in the Appendix. For the case where the recoil momentum coincides with the inverse lattice vector (p = G) we always have trivial phase with all three Chern numbers equal to zero. For the semi-metal case (Fig.2) with no NNN-hopping and p = K we ﬁnd two phases, depending on the mismatch ε 15 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: (Color

text: online)

text: Bulk

text: lattice

text: spectrum

text: projected

text: along

text: k

text: for

text: a

text: number

text: of

text: diﬀerent

text: k

text: values

text: y

text: x

text: in

text: the

text: range

text: K

text: k

text: K

text: Left:

text: the

text: spectrum

text: for

text: the

text: recoil

text: momentum

text: p

text: K

text: in

text: the

text: absence

text: x

text: x

text: x

text: of

text: the

text: NNN-coupling

text: (J

text: and

text: for

text: ε

text: J

text: corresponding

text: to

text: the

text: parameters

text: used

text: in

text: the

text: Fig.

text: In

text: that

text: case

text: there

text: is

text: no

text: energy

text: gap

text: in

text: the

text: spectrum,

text: but

text: diﬀerent

text: energy

text: bands

text: do

text: not

text: directly

text: touch

text: each

text: other.

text: A

text: topological

text: semi-metal

text: phase

text: is

text: formed

text: if

text: the

text: atoms

text: ﬁll

text: the

text: ﬁrst

text: energy

text: band

text: or

text: the

text: ﬁrst

text: two

text: bands.

text: Right:

text: the

text: spectrum

text: for

text: the

text: recoil

text: momentum

text: p

text: K

text: in

text: the

text: case

text: x

text: y

text: where

text: ε

text: and

text: J

text: corresponding

text: to

text: the

text: phase

text: diagram

text: shown

text: in

text: Fig.

text: Now

text: there

text: are

text: two

text: bands

text: with

text: non-zero

text: Chern

text: numbers

text: separated

text: by

text: a

text: quasi-gap

text: containing

text: a

text: middle

text: band

text: with

text: a

text: zero

text: Chern

text: number.

text: the

text: contour

text: integral

text: around

text: the

text: FBZ

text: (the

text: ﬁrst

text: term

text: on

text: the

text: r.h.s.

text: of

text: the

text: above

text: equation)

text: is

text: zero.

text: Consequently

text: the

text: Chern

text: number

text: can

text: be

text: calculated

text: by

text: integrating

text: A

text: around

text: n

text: each

text: excluded

text: singular

text: point

text: c

text: dk

text: A

text: n

text: n

text: 2π

text: singul

text: where

text: the

text: sum

text: is

text: over

text: all

text: singular

text: points

text: in

text: the

text: FBZ.

text: Let

text: us

text: summarize

text: our

text: analytical

text: results,

text: details

text: being

text: presented

text: in

text: the

text: Appendix.

text: For

text: the

text: case

text: where

text: the

text: recoil

text: momentum

text: coincides

text: with

text: the

text: inverse

text: lattice

text: vector

text: (p

text: G)

text: we

text: always

text: have

text: trivial

text: phase

text: with

text: all

text: three

text: Chern

text: numbers

text: equal

text: to

text: zero.

text: For

text: the

text: semi-metal

text: case

text: with

text: no

text: NNN-hopping

text: and

text: p

text: K

text: we

text: ﬁnd

text: two

text: phases,

text: depending

text: on

text: the

text: mismatch

text: ε

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: rewrite the matrix Hamiltonian in the state-vector notation as (cid:88) (cid:88) (cid:16) (cid:17) H | (cid:105) (cid:104) | | (cid:105) (cid:104) | (k) = s d (k) s + s g (k)eisαs(k) 0 + H. c. , (A1) s s ± ± s=0, 1 s= 1 where d (k) stands for the diagonal matrix elements: s − d (k) = sε + 2J f(k sp) . (A2) s 2 The oﬀ-diagonal matrix elements ∓ Jg(k p/2) = g±(k)eiα±(k) (A3) ≡ ≡ have been represented in terms of their amplitudes g± (k) g±(k) and phases α± (k) 1 1 α±(k). Since there is no coupling between the A and C sub-lattices, one can perform a k- dependent unitary transformation eliminating the phase factors | (cid:105) → | (cid:105) | (cid:105) ± s s, k = s eisαs(k), s = 1, | (cid:105) | (cid:105) | (cid:105) and leave the basis vector 0 unchanged ( 0 = 0, k ). In the new basis the Hamiltonian (A1) is characterized by real and symmetric matrix elements. Its eigenvectors can be cast in terms of these vectors with real coeﬃcients C (k): n,s (cid:88) (cid:88) | (cid:105) | (cid:105) ≡ | (cid:105) u = C (k) s, k s C (k)eisαs(k) , (A4) k,n n,s n,s ± ± s=0, 1 s=0, 1 Combining Eqs. (20) and (A4), one arrives at the following expression for the Berry connection (cid:88) − ∇ A (k) = sC2 (k) α (k). (A5) n ± n,s k s s= 1 This result together with Eq. (21) will be subsequently used in ﬁnding the Chern numbers. 17

text: rewrite

text: the

text: matrix

text: Hamiltonian

text: in

text: the

text: state-vector

text: notation

text: as

label: equation

text: H

text: (k)

text: s

text: d

text: (k)

text: s

text: s

text: g

text: (k)eisαs(k)

text: H.

text: c.

text: s

text: s

text: s=

text: where

text: d

text: (k)

text: stands

text: for

text: the

text: diagonal

text: matrix

text: elements:

text: s

text: d

text: (k)

text: sε

text: f(k

text: sp)

text: s

text: The

text: oﬀ-diagonal

text: matrix

text: elements

text: Jg(k

text: g±(k)eiα±(k)

text: have

text: been

text: represented

text: in

text: terms

text: of

text: their

text: amplitudes

text: g±

text: (k)

text: g±(k)

text: and

text: phases

text: α±

text: (k)

text: α±(k).

text: Since

text: there

text: is

text: no

text: coupling

text: between

text: the

text: A

text: and

text: C

text: sub-lattices,

text: one

text: can

text: perform

text: a

text: k-

text: dependent

text: unitary

text: transformation

text: eliminating

text: the

text: phase

text: factors

text: s

text: s,

text: k

text: s

text: eisαs(k),

text: s

text: and

text: leave

text: the

text: basis

text: vector

text: unchanged

text: k

text: In

text: the

text: new

text: basis

text: the

text: Hamiltonian

text: is

text: characterized

text: by

text: real

text: and

text: symmetric

text: matrix

text: elements.

text: Its

text: eigenvectors

text: can

text: be

text: cast

text: in

text: terms

text: of

text: these

text: vectors

text: with

text: real

text: coeﬃcients

text: C

text: (k):

text: n,s

text: u

text: C

text: (k)

text: s,

text: k

text: s

text: C

text: (k)eisαs(k)

text: k,n

text: n,s

text: n,s

text: Combining

text: Eqs.

text: and

text: one

text: arrives

text: at

text: the

text: following

text: expression

text: for

text: the

text: Berry

text: connection

text: A

text: (k)

text: (k)

text: α

text: (k).

text: n

text: n,s

text: k

text: s

text: s=

text: This

text: result

text: together

text: with

text: Eq.

text: will

text: be

text: subsequently

text: used

text: in

text: ﬁnding

text: the

text: Chern

text: numbers.


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text: (cid:104) (cid:16) (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17) d2ρ 1 − − − − − = e 3Ar2 2e2Ar2 3( 1 + eAr2) + Ar2 3 2Ar2 (20) dr2 (cid:16) c2k2 (cid:17) (cid:16) (cid:16) − − − − (cid:48)(cid:48) (cid:48) +2Ar2( 1 + 2Ar2) 8αnGn 3(n 1) 6G + G 6(1 + 3Ar2(1 + 2Ar2) (cid:16) (cid:17)(cid:17)(cid:17) (cid:16) (cid:16) − − (cid:48) (cid:48)(cid:48) − − − − 2 + n λ + 4r(1 + 3Ar2)λ r2λ(3) + eAr2(2(n 2) 3 + 2Ar2 3 (cid:16) (cid:17)(cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17) − − − − +2( B + Br2)r2 + 4Ar2 1 + 2(B Br2)r2 + 6 + 4Ar2 3 + 2(B Br2)r2 (cid:16) (cid:17) (cid:17)(cid:17) (cid:16) (cid:16) (cid:16) − (cid:48) − − − (cid:48)(cid:48) +4Ar2 1 + 2( B + Br2)r2 λ + 4r 1 + 2Ar2 1 + 2(B Br2)r2 λ (cid:17) (cid:17) (cid:16) (cid:17) (cid:17) − (cid:48)(cid:48) − − +r2 1 + 2( B + Br2)r2 λ(3) + r 2G 4 2n + 3Ar 3 + 4r 2nr (cid:16) (cid:16) (cid:16) (cid:16) (cid:48) − (cid:48)(cid:48) − − (cid:48) +6Ar2 + (2 + 6Ar2)λ rλ 4 + 2r nr + 12Ar2 + rλ G(3) + rG(4) (cid:17) (cid:16) (cid:17)(cid:17) (cid:16)(cid:16) (cid:16) (cid:48)(cid:48) − − − − − +eAr2 2G 3 + 2(n 2)r + 2Ar2 3 + 2r 2 + n + ( B + Br2)r (cid:17) (cid:16) (cid:17)(cid:17) (cid:16)(cid:16) (cid:16) − − − − − − − 2(B Br2)(n 2)r2 + 4Ar2 1 + 2(B Br2)r2 + r 2 + 4Ar2 1 (cid:17) (cid:16) (cid:16) (cid:17) (cid:17) − (cid:48) − (cid:48)(cid:48) − − +2(B Br2)r2 λ + r 1 + 2( B + Br2)r2λ + r 4 + (n 2)r 1 (cid:16) (cid:17) (cid:17) (cid:17)(cid:17)(cid:17)(cid:17)(cid:105) − − − (cid:48) +2(B Br2)r2 + 8Ar2 1 + 2( B + Br2)r2) + r 1 + 2( B + Br2)r2 λ G(3) − − +r( 1 + 2 B Br2 r2 G(4) (21) (cid:104) (cid:16) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17)(cid:17)(cid:17)(cid:105) d2p 1 − − − − r = 2e Ar2 3 1 + eAr2 + Ar2 3 + 4B + αGn(n 1)(2r22) dr2 k2 − − 2Ar2 1 + 4B + αGn(n 1) r2 (23) We observe that at center r = 0, our model provides that dρ dp r = 0, = 0 (24) dr dr d2ρ d2p r < 0, < 0. (25) dr2 dr2 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Ρ Ρ+pr 0.030 0.030 0.025 0.025 0.020 0.020 0.015 0.015 0.010 0.010 0.005 0.005 r r 0 2 4 6 8 10 0 2 4 6 8 10 Ρ+pt Ρ+pr+2pt 0.030 0.030 0.025 0.025 0.020 0.020 0.015 0.015 0.010 0.010 0.005 0.005 r r 0 2 4 6 8 10 0 2 4 6 8 10 Ρ-pr Ρ-pt 0.030 0.030 0.025 0.025 0.020 0.020 0.015 0.015 0.010 0.010 0.005 0.005 r r 0 2 4 6 8 10 0 2 4 6 8 10 Figure 6: Evolution of energy constraints at the stellar interior of strange star Her X-1. We ﬁnd that our model satisﬁes these conditions (see Figs.6-8)for spe- ciﬁc values of mass and radius which helps to ﬁnd the unknown parameters for diﬀerent strange stars. 4.2 Matching Conditions The intrinsic metric of the boundary surface will be the same whether it is derived from the internal or external geometry of the star. It conﬁrms that for any coordinate system the metric tensor components will be continuous across the boundary surface. for this purpose the matching conditions are required for the interior metric (??), so that the system of ﬁeld equations can be solved constrained to the boundary conditions that radial pressure p = 0 r at r = R. The interior metric (??) can be matched smoothly at r = R, to an exterior(cid:18)Schwarzs(cid:19)child me(cid:18)tric given(cid:19)by − − 2M − − 2M 1 − ds2 = 1 dt2 1 dr2 r2(dθ2 + sin2 θdϕ2). (27) r r 16 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: (cid:104) (cid:16) (cid:16) (cid:16) (cid:16) (cid:16) (cid:17) (cid:17) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17) − − − − V 2 V 2 = [ c2e2Ar2 2 1 eAr2 1 + 4B + αGn (n 1) r2 + eAr2 tv rv (cid:16) (cid:17) (cid:16) (cid:16) (cid:16) × − − − − − − − 1 + e Ar2 2Ae Ar2r2 + e Ar2 1 + eAr2 + 2Ar2 + Ae Ar2r2 (cid:17)(cid:17)(cid:17) (cid:16) (cid:16) (cid:17) (cid:17) − − − − − − 1 + eAr2 + 2Ar2 2Ae 2Ar2r2 1 + eAr2 1 + r 2Ar + 2Br ABr3 (cid:16) (cid:17)(cid:16) (cid:16) (cid:17) (cid:17) − − − − Br + B2r3 ) 2e 2Ar2 1 + eAr2 1 + Ar2 Br2 + B2r4 + Ar2 (cid:16) (cid:17)(cid:16) − − − − − − (cid:48) 2 − (cid:48) − (cid:48)(cid:48) 8αne 2Ar2Gn 3(n 1) 1 + eAr2( 1 + Ar2) (n 2) G G A G (cid:17) (cid:104) (cid:16) (cid:16) − − − − (cid:48) − − (cid:48) +16Aαne 3Ar2Gn 2(n 1)r2 1 + eAr2(1 + 2Br2) 2GarG (n 1)(G )2 (cid:17)(cid:17)(cid:105) (cid:104) (cid:16) (cid:16) (cid:17) (cid:17)(cid:105) − (cid:48)(cid:48) − − − − − − GG + 2αne 2Ar2Gn 3(n 1)r αGn(n 1)r2 2k2 1 + eAr2 (cid:104) (cid:16) (cid:17)(cid:16) (cid:16) (cid:17) − (cid:48)(cid:48) − − (cid:48)(cid:48) (cid:48) − (cid:48)(cid:48) − +2 GG + (B Br2)r2 AG + G 2(n 2)G A G(3) (cid:16) (cid:17)(cid:17)(cid:105) (cid:105) × − − − − − (cid:48) 4αne 3Ar2Gn 2(n 1)r 1 + eAr2(1 + 2Br2) G G(3) 2ArG (cid:104) − (cid:48) − (cid:48)(cid:48) − (cid:48) 1 +2G (n 1)G GArG (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) × − − − (2(e2Ar2( 1 + eAr2 + 2Ar2) + Ae2Ar2r2( 1 + eAr2 + 2Ar2) Ae2Ar2r2 (cid:16) (cid:17) (cid:16) (cid:17) − − − − 2 + eAr2 + 16AαnGn 3(n 1)r2 1 + eAr2( 1 + (B Br2)r2 (cid:16) (cid:17) (cid:16) (cid:16) (cid:48) − − (cid:48)(cid:48) − − − − − G ( 2 + n A) + G 8αnGn 3(n 1) 1 + eAr2 Ar2 (cid:17)(cid:17)(cid:16)(cid:16) (cid:17)(cid:17)(cid:17)(cid:105) (cid:48) − − (cid:48)(cid:48) − − − − G ( 2 + n A)G 4αnGn 3(n 1)r 1 + eAr2 1 + 2 − − − − (cid:48)(cid:48) − (cid:48) 1 (B Br2)r2 2 + n A)G G A + G(3) . | − |≤ From ﬁgure 6, we can see that υ2 υ2 1. This is used to check st sr whether local anisotropic matter distribution is stable or not. For purpose, we use the cracking concept introduced by Herrera (1992) which can examine that potentially stable region is such region where radial speed of sound is greater than the transverse speed of sound. Hence, our proposed compact star model is stable. 5 Conclusion 20

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text: we

text: can

text: see

text: that

text: υ2

text: υ2

text: This

text: is

text: used

text: to

text: check

text: st

text: sr

text: whether

text: local

text: anisotropic

text: matter

text: distribution

text: is

text: stable

text: or

text: not.

text: For

text: purpose,

text: we

text: use

text: the

text: cracking

text: concept

text: introduced

text: by

text: Herrera

text: which

text: can

text: examine

text: that

text: potentially

text: stable

text: region

text: is

text: such

text: region

text: where

text: radial

text: speed

text: of

text: sound

text: is

text: greater

text: than

text: the

text: transverse

text: speed

text: of

text: sound.

text: Hence,

text: our

text: proposed

text: compact

text: star

text: model

text: is

text: stable.

text: Conclusion


text: Arapoglu. A. S. , Deliduman. C. , Eksi. K. Y. , JCAP 1107, 020 (2011) [arXiv:1003.3179 [gr-qc]]. Astashenok. A. V. , Capozziello. S. , Odintsov. S. D. , arXiv:1401.4546 [gr-qc]. Astashenok. A. V. , Capozziello. S. , Odintsov. S. D. , JCAP 1312, 040 (2013) [arXiv:1309.1978 [gr-qc]]. Astashenok. A. V. , Capozziello. S. , Odintsov. S. D. , arXiv:1405.6663 [gr-qc]. Astashenok. A. V., Capozziello. S. , Odintsov. S. D. , arXiv:1408.3856 [gr-qc]. Bamba. K. , Nojiri. S. ’i. , Odintsov. S. D. , arXiv:1302.4831 [gr-qc]. Bamba. K. and Odintsov. S. D. , arXiv:1402.7114 [hep-th]. Bamba. K. , et al., Eur. Phys. J. C 67, 295 (2010) [arXiv:0911.4390 [hep-th]]. Bamba. K. , et al., Europhys. Lett. 89, 50003 (2010) [arXiv:0909.4397 [hep-th]]. Bamba. K., Nojiri. S. , Odintsov. S. D. , JCAP 0810 (2008) 045 [arXiv:0807.2575 [hep-th]]. Buchdahl. H. A., Mon. Not. Roy. Astron. Soc. 150, 1 (1970). Bhar, P., Rahaman, F., Ray, S., Chatterjee, V.: arXiv:1503.03439 Camenzind. M. , ”Compact Objects in Astrophysics”, Springer-Verlag Berlin Heidelberg (2007). Capozziello. S. and Laurentis. M. De , Phys. Rept. 509, 167 (2011) [arXiv:1108.6266 [gr-qc]]. Capozziello. S. et al., arXiv:1110.5026 [astro-ph.CO]. Capozziello. S. and Laurentis. M. De , Found. Phys. 40 (2010) 867 [arXiv:0910.2881 [hep-th]]. Capozziello. S. and Laurentis. M. De , Found. Phys. 40 (2010) 867 [arXiv:0910.2881 [hep-th]]. Capozziello. S. and Francaviglia. M. , Gen. Rel. Grav. 40 (2008) 357 [arXiv:0706.114 23

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text: Astashenok.

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text: Astashenok.

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text: D.

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text: Astashenok.

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text: Odintsov.

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text: Bamba.

text: K.

text: et

text: al.,

text: Eur.

text: Phys.

text: J.

text: C

text: [hep-th]].

text: Bamba.

text: K.

text: et

text: al.,

text: Europhys.

text: Lett.

text: [hep-th]].

text: Bamba.

text: K.,

text: Nojiri.

text: S.

text: Odintsov.

text: S.

text: D.

text: JCAP

text: [hep-th]].

text: Buchdahl.

text: H.

text: A.,

text: Mon.

text: Not.

text: Roy.

text: Astron.

text: Soc.

text: Bhar,

text: P.,

text: Rahaman,

text: F.,

text: Ray,

text: S.,

text: Chatterjee,

text: V.:

text: Camenzind.

text: M.

text: ”Compact

text: Objects

text: in

text: Astrophysics”,

text: Springer-Verlag

text: Berlin

text: Heidelberg

text: Capozziello.

text: S.

text: and

text: Laurentis.

text: M.

text: De

text: Phys.

text: Rept.

text: [gr-qc]].

text: Capozziello.

text: S.

text: et

text: al.,

text: [astro-ph.CO].

text: Capozziello.

text: S.

text: and

text: Laurentis.

text: M.

text: De

text: Found.

text: Phys.

text: [hep-th]].

text: Capozziello.

text: S.

text: and

text: Laurentis.

text: M.

text: De

text: Found.

text: Phys.

text: [hep-th]].

text: Capozziello.

text: S.

text: and

text: Francaviglia.

text: M.

text: Gen.

text: Rel.

text: Grav.


text: We assume that matter sector is ﬁlled by a ﬂuid with the non zero com- ponents of the anisotropic energy momentum tensor are given by T ν = − − − µ diag(ρc2, p , p , p ) where ρ is energy density, p radial pressure and r t t r p tangential component of the pressure. If we write down the equations of t motion given in (??), we obtain the following set of independent equations for (µ, ν) = (ct, ct) and (µ, ν) = (r, r): (cid:16) (cid:17)(cid:16) (cid:17) − 1 (cid:48) − − (cid:48)(cid:48) − (cid:48) (cid:48) (cid:48) 1 e2λ − (cid:48)(cid:48) (cid:48) (2rλ 2λ 1) + 8e 2λ f (G 2λ G ) + f (G 2 2(φ + φ 2) r2 GG GGG r2 − +(Gf f)e2λ = κ2ρc2e2λ (5) G 1 (cid:48) − − (2rφ 2λ + 1) (Gf f)e2λ = κ2p e2λ. (6) r2 G r Trace of (??) gives us the following auxiliary equation: ∇ ∇ − − − − − R + 8G ρ σf 4(Gf f) = κ2(ρc2 p 2p ). (7) ρσ G G r t If we expand it using the metric (??), we obtain: (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)− (cid:48) − (cid:48)(cid:48) (cid:48) − (cid:48) (cid:48) 2 (cid:48) − (cid:48) 1 e2λ − 2φ 1 e2λ φ + φ 2 φ λ + (φ λ ) + + 8e 2λ + r r2 r r2 × (cid:48)(cid:48) − (cid:48) (cid:48) (cid:48) − − − f (G 2λ G ) + f (G 2 + 4(Gf f)e2λ = κ2e2λ(ρc2 p 2p ). GG GGG G r t (8) If we specify the form of the metric functions, then we are able to calculate pressures and density using Eqs.(??)-(??). 3 Solutions for a viable f (G) model In this section, we shall try to solve (??,??,??) for a given viable model of f(G). For this reason, we consider the power law model f(G) = αGn 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: assume

text: that

text: matter

text: sector

text: is

text: ﬁlled

text: by

text: a

text: ﬂuid

text: with

text: the

text: non

text: zero

text: com-

text: ponents

text: of

text: the

text: anisotropic

text: energy

text: momentum

text: tensor

text: are

text: given

text: by

text: T

text: ν

text: µ

text: p

text: p

text: p

text: where

text: ρ

text: is

text: energy

text: density,

text: p

text: radial

text: pressure

text: and

text: r

text: t

text: t

text: r

text: p

text: tangential

text: component

text: of

text: the

text: pressure.

text: If

text: we

text: write

text: down

text: the

text: equations

text: of

text: t

text: motion

text: given

text: in

text: we

text: obtain

text: the

text: following

text: set

text: of

text: independent

text: equations

text: for

text: (µ,

text: ν)

text: (ct,

text: ct)

text: and

text: (µ,

text: ν)

text: (r,

text: r):

text: 2λ

text: 2λ

text: f

text: (G

text: 2λ

text: G

text: f

text: (G

text: φ

text: GG

text: GGG

text: +(Gf

text: G

text: 2λ

text: (Gf

text: G

text: r

text: Trace

text: of

text: gives

text: us

text: the

text: following

text: auxiliary

text: equation:

text: R

text: ρ

text: σf

text: f)

text: p

text: ρσ

text: G

text: G

text: r

text: t

text: If

text: we

text: expand

text: it

text: using

text: the

text: metric

text: we

text: obtain:

text: 2φ

text: φ

text: φ

text: φ

text: λ

text: (φ

text: λ

text: 2λ

text: r

text: r

text: f

text: (G

text: 2λ

text: G

text: f

text: (G

text: p

text: GG

text: GGG

text: G

text: r

text: t

text: If

text: we

text: specify

text: the

text: form

text: of

text: the

text: metric

text: functions,

text: then

text: we

text: are

text: able

text: to

text: calculate

text: pressures

text: and

text: density

text: using

text: Eqs.(??)-(??).

text: Solutions

text: for

text: a

text: viable

text: f

text: (G)

text: model

text: In

text: this

text: section,

text: we

text: shall

text: try

text: to

text: solve

text: for

text: a

text: given

text: viable

text: model

text: of

text: f(G).

text: For

text: this

text: reason,

text: we

text: consider

text: the

text: power

text: law

text: model

text: f(G)

text: αGn

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: proposed in (Cognola et al. 2006), where α is arbitrary constant and n > 0. ≤ If n 1, the f(G) term dominates on the Einstein term in the regime of 2 small curvature. We parameterize the metric as the following (Krori and Barua. 1975): 2λ = Ar2, 2φ = Br2 + C, where A, B and C are arbitrary constants to be evaluated by using some physical matching conditions(see Table 1). Indeed this solution was treated as a singularity free exact solution for astatic charged ﬂuid sphere. So our main motivation to introduce this set of functions for static metric goes back to the singularity free structure of our extended model for compact star. It is easy to show that neither densities nor curvature terms become singular in the vicinity of r = 0 with this appropriate set of metric functions. Energy density and pressure proﬁles are given by the following expres- sions: (cid:20) (cid:21) − e Ar2 − 1 − − p = (4r2B eAr2 + 1) + αGn(n 1) , (9) r κ2 r2 (cid:16) − (cid:32) − (cid:33) −e Ar2 − (cid:17) 4e 2Ar2 − − (cid:48)(cid:48) − (cid:48) ρ = (2r2A + eAr2 1) + 2 αn(n 1)Gn 3(G 2ArG ) (10) κ2r2 κ2 − − − − (cid:48) 1 eAr2 − − α − +αn(n 1)(n 2)Gn 3(G )2 2B(1 r2) + 2 Gn(n (111)), r2 2k2 (cid:16) − (cid:32)− (cid:33) −e Ar2 − (cid:17)4e 2Ar2 − − (cid:48)(cid:48) (cid:48) p = (2r2A + eAr2 1) + αn(n 1)Gn 3(G + ArG ) (12) t 2κ2r2 κ2 − − − − (cid:48) 1 eAr2 − − − 3α − +αn(n 1)(n 2)Gn 3(G )2 2(B Br2) (cid:16)Gn(n 1) r2 2κ2 − − − −e Ar2 (cid:17) − 2 − 1 eAr2 − 4e 2Ar2 2Br (B + B2r2 ABr2 + (Br Ar) + ) κ2 r r2 κ2 r − 1 eA(cid:20)r2 − − (cid:48)(cid:48) − (cid:48) (cid:21) − − − (cid:48) + (αn(n 1)Gn 1(G 2ArG ) + αn(n 1)(n 1)Gn 2(G )2) r2 − e Ar2 − 1 − − + (4r2B eAr2 + 1) + αGn(n 1) . (13) 2κ2 r2 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: proposed

text: in

text: (Cognola

text: et

text: al.

text: where

text: α

text: is

text: arbitrary

text: constant

text: and

text: n

text: If

text: n

text: the

text: f(G)

text: term

text: dominates

text: on

text: the

text: Einstein

text: term

text: in

text: the

text: regime

text: of

text: small

text: curvature.

text: We

text: parameterize

text: the

text: metric

text: as

text: the

text: following

text: (Krori

text: and

text: Barua.

text: 2λ

text: 2φ

text: C,

text: where

text: A,

text: B

text: and

text: C

text: are

text: arbitrary

text: constants

text: to

text: be

text: evaluated

text: by

text: using

text: some

text: physical

text: matching

text: conditions(see

text: Table

text: Indeed

text: this

text: solution

text: was

text: treated

text: as

text: a

text: singularity

text: free

text: exact

text: solution

text: for

text: astatic

text: charged

text: ﬂuid

text: sphere.

text: So

text: our

text: main

text: motivation

text: to

text: introduce

text: this

text: set

text: of

text: functions

text: for

text: static

text: metric

text: goes

text: back

text: to

text: the

text: singularity

text: free

text: structure

text: of

text: our

text: extended

text: model

text: for

text: compact

text: star.

text: It

text: is

text: easy

text: to

text: show

text: that

text: neither

text: densities

text: nor

text: curvature

text: terms

text: become

text: singular

text: in

text: the

text: vicinity

text: of

text: r

text: with

text: this

text: appropriate

text: set

text: of

text: metric

text: functions.

text: Energy

text: density

text: and

text: pressure

text: proﬁles

text: are

text: given

text: by

text: the

text: following

text: expres-

text: sions:

text: e

text: p

text: αGn(n

text: r

text: κ2

text: −e

text: ρ

text: αn(n

text: κ2

text: α

text: +αn(n

text: Gn(n

text: −e

text: p

text: αn(n

text: ArG

text: t

text: κ2

text: 3α

text: +αn(n

text: −e

text: (B

text: (Br

text: Ar)

text: κ2

text: r

text: κ2

text: r

text: (αn(n

text: αn(n

text: e

text: αGn(n

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: [27] Barber MJ, Faria M, Streit L and Strogan O (2008) Searching for communities in bipartite networks. Bernido CC, Carpio-Bernido, MV (Eds.), Proceedings of the 5th Jagna International Workshop: stochastic and quantum dynamics of biomolecular systems. January, 3rd - 5th, Bohol, Philippines. [28] Suzuki S and Wakita K (2009) Extracing multi-facet community structure from bipartite networks, Proc. of 2009 Int. Comf. on Computational Science and engineering (CSE 09), Vol.4, pp,312-319, August 2009. [29] Chang C and Tang C (2014) Community detection for networks with unipartite and bipartite structure, New Journal of Physics 16, 093001. [30] Scharge L, Optimization Modeling with LINGO. LINDO Systems Inc., 2004. [31] Murata T (2009) Detecting communities from bipartite networks based on bipartite modularities, Proc. of the 2009 International Conference on Computational Science and Engineering (CSE 09), Vol.4, pp. 50-57, August 2009. [32] Raghavan UN, Albert R and Kumara S. (2007) Near linear time algorithm for detect community structures in large scale networks, Physical Review E 76: 036106. [33] Barber MJ and Clark JW (2009) Detecting network communities by propagating labels under constraints, Phys. Rev. E 80: 026129. [34] Evans TS and Lambiotte R (2009) Line graphs, link partitions and overlapping communities. Phys Rev E 80: 016105. [35] Evans TS (2010) Clique graphs and overlapping communities. J Stat Mech: P12037. [36] Evans TS and Lambiotte R (2010) Line graphs of weighted networks for overlapping communities. Eur Phys J B 77: 265-272. [37] Zhang S, Liu HW, Ning XM and Zhang XS (2009) A hybrid graph-theoretic method for mining overlapping functional modules in large sparse protein interaction networks. International Journal of Data Mining and Bioinformatics 3: 68-84. [38] He DX, Liu D, Zhang W, Jin D and Yang B (2012) Discovering link communities in complex networks by exploiting link dynamics. J Stat Mech: P10015. [39] Chen BL, Chen L, Zhou SR and Xu XL (2013) Detecting community structure in bipartite networks based on matrix factorisation, Int. J. Wireless and Mobile Computing, 6(6): 599-607. [40] Leicht EA and Newman MEJ (2008) Community structure in directed networks, Phys. Rev. Lett. 100: 118703 [41] Kim S and Shi T (2012) Scalable spectral algorithm for community detection in directed networks, arXiv:1211.6807v1. [42] Zhang S, Wang RS and Zhang XS (2007) Identiﬁcation of overlapping community structure in complex networks using fuzzy c-means clustering. Physica A 374: 483-490. [43] Tasgin M and Bingol H (2006) Community detection in complex networks using genetic algorithm. http://arxiv.org/abs/0711.0491v1. [44] Gregory S. (2010) Finding overlapping communities in networks by label propagation. New Journal of Physics 12: 103018. [45] Lehmann S, Schwartz M and Hansen LK (2008) Biclique communities, Physical Review E 78: 016108. [46] Du N, Wang B, Wu B and Wang Y (2008) Overlapping community detection in bipartite networks, Proc. of the 2008 IEEE/WIC/ACM Int. Conf. on Web Intelligence and Intelligent Agent Technology (WI-IAT 08), Vol. 1, pp. 176-179, December 2008. [47] Miyauchi A and Sukegawa N (2013) Maximizing Barber’s bipartite modularity is also hard. arXiv:1310.4656v1. 13

text: Barber

text: MJ,

text: Faria

text: M,

text: Streit

text: L

text: and

text: Strogan

text: O

text: Searching

text: for

text: communities

text: in

text: bipartite

text: networks.

text: Bernido

text: CC,

text: Carpio-Bernido,

text: MV

text: (Eds.),

text: Proceedings

text: of

text: the

text: Jagna

text: International

text: Workshop:

text: stochastic

text: and

text: quantum

text: dynamics

text: of

text: biomolecular

text: systems.

text: January,

text: Bohol,

text: Philippines.

text: Suzuki

text: S

text: and

text: Wakita

text: K

text: Extracing

text: multi-facet

text: community

text: structure

text: from

text: bipartite

text: networks,

text: Proc.

text: of

text: Int.

text: Comf.

text: on

text: Computational

text: Science

text: and

text: engineering

text: (CSE

text: August

text: Chang

text: C

text: and

text: Tang

text: C

text: Community

text: detection

text: for

text: networks

text: with

text: unipartite

text: and

text: bipartite

text: structure,

text: New

text: Journal

text: of

text: Physics

text: Scharge

text: L,

text: Optimization

text: Modeling

text: with

text: LINGO.

text: LINDO

text: Systems

text: Inc.,

text: Murata

text: T

text: Detecting

text: communities

text: from

text: bipartite

text: networks

text: based

text: on

text: bipartite

text: modularities,

text: Proc.

text: of

text: the

text: International

text: Conference

text: on

text: Computational

text: Science

text: and

text: Engineering

text: (CSE

text: pp.

text: August

text: Raghavan

text: UN,

text: Albert

text: R

text: and

text: Kumara

text: S.

text: Near

text: linear

text: time

text: algorithm

text: for

text: detect

text: community

text: structures

text: in

text: large

text: scale

text: networks,

text: Physical

text: Review

text: E

text: Barber

text: MJ

text: and

text: Clark

text: JW

text: Detecting

text: network

text: communities

text: by

text: propagating

text: labels

text: under

text: constraints,

text: Phys.

text: Rev.

text: E

text: Evans

text: TS

text: and

text: Lambiotte

text: R

text: Line

text: graphs,

text: link

text: partitions

text: and

text: overlapping

text: communities.

text: Phys

text: Rev

text: E

text: Evans

text: TS

text: Clique

text: graphs

text: and

text: overlapping

text: communities.

text: J

text: Stat

text: Mech:

text: Evans

text: TS

text: and

text: Lambiotte

text: R

text: Line

text: graphs

text: of

text: weighted

text: networks

text: for

text: overlapping

text: communities.

text: Eur

text: Phys

text: J

text: B

text: Zhang

text: S,

text: Liu

text: HW,

text: Ning

text: XM

text: and

text: Zhang

text: XS

text: A

text: hybrid

text: graph-theoretic

text: method

text: for

text: mining

text: overlapping

text: functional

text: modules

text: in

text: large

text: sparse

text: protein

text: interaction

text: networks.

text: International

text: Journal

text: of

text: Data

text: Mining

text: and

text: Bioinformatics

text: He

text: DX,

text: Liu

text: D,

text: Zhang

text: W,

text: Jin

text: D

text: and

text: Yang

text: B

text: Discovering

text: link

text: communities

text: in

text: complex

text: networks

text: by

text: exploiting

text: link

text: dynamics.

text: J

text: Stat

text: Mech:

text: Chen

text: BL,

text: Chen

text: L,

text: Zhou

text: SR

text: and

text: Xu

text: XL

text: Detecting

text: community

text: structure

text: in

text: bipartite

text: networks

text: based

text: on

text: matrix

text: factorisation,

text: Int.

text: J.

text: Wireless

text: and

text: Mobile

text: Computing,

text: Leicht

text: EA

text: and

text: Newman

text: MEJ

text: Community

text: structure

text: in

text: directed

text: networks,

text: Phys.

text: Rev.

text: Lett.

text: Kim

text: S

text: and

text: Shi

text: T

text: Scalable

text: spectral

text: algorithm

text: for

text: community

text: detection

text: in

text: directed

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text: also

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