text: 14 AKHIL MATHEW AND VESNA STOJANOSKA ⊂ Proof. Let W V be any subspace. We show that the nerve of Sub≤ (W) r (i.e., the left Kan extension of a point from Sub≤ (V ), evaluated on W) has the − r homotopy type of a wedge of (r 1)-spheres, following an argument of Folkman ⊂ presented in [Qui78, Proposition 8.6]: for each one-dimensional space L W, we ≤ have a subposet Sub≤ (W) consisting of those r-dimensional subspaces of W r,L that contain L. The intersection of any r of the subposets Sub≤ (W) contains r,L an initial object (i.e., minimal element) and is thus weakly contractible. Therefore, − Sub≤ (W) is (r 2)-connected. However, the nerve of Sub≤ (W) has dimension ≤ −r r r 1 by ?? . Putting these together, it follows that the nerve of Sub≤ (W) has − r the homotopy type of a wedge of (r 1)-spheres. Now it follows that T → is pointwise a wedge of r-spheres, and it is Subr(V ) Sub(V ) ≤ weakly contractible when evaluated on any subspace of dimension r. Since the \ − − poset Sub(V ) Sub (V ) has dimension n r 1, the same argument as in ?? goes r (cid:3) through. • Let X be a pointed cosimplicial space, and consider the homotopy spectral sequence ⇒ • Es,t = π − TotX , 2 t s → − which comes from the Tot tower, and the diﬀerentials d : Es,t Es+r,t+r 1. − r r r If t s > 0, the entire spectral sequence is a shift of the spectral sequence • • for ΩX . All the diﬀerentials mapping out of Es,t are determined by ΩX if − →r − t s > 0. But there are also diﬀerentials d : Es,s Es+r,s+r 1 which measure the •r r r • obstructions to ﬁnding points in π TotX . These cannot be determined by ΩX , 0 as they contribute to π . Nonetheless, the Es,s terms for s > 0 are determined by • 0 2 ΩX . • → [eorem]lmTheorem Given any cosimplicial space X , any diﬀerential d : Es,s − • ≤ − r r Es+r,s+r 1 depends only on the cosimplicial space ΩX for r s 1. r ≤ Proof. The diﬀerentials d , k r starting in ﬁltration s are precisely the ob- k • → • • → structions to lifting an element in ﬁb(Tot (X ) Tot − (X )) to ﬁb(Tot (X ) • s s 1 r+s Tot − (X )). In particular, if s 1 − − ≥ 2(s 1) (r + s) + 2 1, • (cid:3) then, by ?? , this obstruction problem is determined entirely by ΩX . • [rollary]lmCorollary Suppose X is a pointed cosimplicial space with the properties: • ≤ o . (1) Any element in π TotX has ﬁltration N. 0 (2) Any class in Es,s for s > N is either killed at a ﬁnite stage or supports a 2 ≤ − diﬀerential d for r s 1. r • Then if Y is a path-connected cosimplicial space such that there exists an • (cid:39) • equivalence of pointed cosimplicial spaces ΩX ΩY , the same conditions above • are valid for the totalization of Y .

text: AKHIL

text: MATHEW

text: AND

text: VESNA

text: STOJANOSKA

text: Proof.

text: Let

text: W

text: V

text: be

text: any

text: subspace.

text: We

text: show

text: that

text: the

text: nerve

text: of

text: Sub≤

text: (W)

text: r

text: (i.e.,

text: the

text: left

text: Kan

text: extension

text: of

text: a

text: point

text: from

text: Sub≤

text: (V

text: evaluated

text: on

text: W)

text: has

text: the

text: r

text: homotopy

text: type

text: of

text: a

text: wedge

text: of

text: (r

text: following

text: an

text: argument

text: of

text: Folkman

text: presented

text: in

text: Proposition

text: for

text: each

text: one-dimensional

text: space

text: L

text: W,

text: we

text: have

text: a

text: subposet

text: Sub≤

text: (W)

text: consisting

text: of

text: those

text: r-dimensional

text: subspaces

text: of

text: W

text: r,L

text: that

text: contain

text: L.

text: The

text: intersection

text: of

text: any

text: r

text: of

text: the

text: subposets

text: Sub≤

text: (W)

text: contains

text: r,L

text: an

text: initial

text: object

text: (i.e.,

text: minimal

text: element)

text: and

text: is

text: thus

text: weakly

text: contractible.

text: Therefore,

text: Sub≤

text: (W)

text: is

text: (r

text: However,

text: the

text: nerve

text: of

text: Sub≤

text: (W)

text: has

text: dimension

text: −r

text: r

text: r

text: by

text: Putting

text: these

text: together,

text: it

text: follows

text: that

text: the

text: nerve

text: of

text: Sub≤

text: (W)

text: has

text: r

text: the

text: homotopy

text: type

text: of

text: a

text: wedge

text: of

text: (r

text: Now

text: it

text: follows

text: that

text: T

text: is

text: pointwise

text: a

text: wedge

text: of

text: r-spheres,

text: and

text: it

text: is

text: Subr(V

text: Sub(V

text: weakly

text: contractible

text: when

text: evaluated

text: on

text: any

text: subspace

text: of

text: dimension

text: r.

text: Since

text: the

text: poset

text: Sub(V

text: Sub

text: (V

text: has

text: dimension

text: n

text: r

text: the

text: same

text: argument

text: as

text: in

text: goes

text: r

text: through.

text: Let

text: X

text: be

text: a

text: pointed

text: cosimplicial

text: space,

text: and

text: consider

text: the

text: homotopy

text: spectral

text: sequence

text: Es,t

text: π

text: TotX

text: t

text: s

text: which

text: comes

text: from

text: the

text: Tot

text: tower,

text: and

text: the

text: diﬀerentials

text: d

text: Es,t

text: Es+r,t+r

text: r

text: r

text: r

text: If

text: t

text: s

text: the

text: entire

text: spectral

text: sequence

text: is

text: a

text: shift

text: of

text: the

text: spectral

text: sequence

text: for

text: ΩX

text: All

text: the

text: diﬀerentials

text: mapping

text: out

text: of

text: Es,t

text: are

text: determined

text: by

text: ΩX

text: if

text: →r

text: t

text: s

text: But

text: there

text: are

text: also

text: diﬀerentials

text: d

text: Es,s

text: Es+r,s+r

text: which

text: measure

text: the

text: •r

text: r

text: r

text: obstructions

text: to

text: ﬁnding

text: points

text: in

text: π

text: TotX

text: These

text: cannot

text: be

text: determined

text: by

text: ΩX

text: as

text: they

text: contribute

text: to

text: π

text: Nonetheless,

text: the

text: Es,s

text: terms

text: for

text: s

text: are

text: determined

text: by

text: ΩX

text: [eorem]lmTheorem

text: Given

text: any

text: cosimplicial

text: space

text: X

text: any

text: diﬀerential

text: d

text: Es,s

text: r

text: r

text: Es+r,s+r

text: depends

text: only

text: on

text: the

text: cosimplicial

text: space

text: ΩX

text: for

text: r

text: s

text: r

text: Proof.

text: The

text: diﬀerentials

text: d

text: k

text: r

text: starting

text: in

text: ﬁltration

text: s

text: are

text: precisely

text: the

text: ob-

text: k

text: structions

text: to

text: lifting

text: an

text: element

text: in

text: ﬁb(Tot

text: (X

text: Tot

text: (X

text: to

text: ﬁb(Tot

text: (X

text: s

text: s

text: r+s

text: Tot

text: (X

text: In

text: particular,

text: if

text: s

text: (r

text: s)

text: then,

text: by

text: this

text: obstruction

text: problem

text: is

text: determined

text: entirely

text: by

text: ΩX

text: [rollary]lmCorollary

text: Suppose

text: X

text: is

text: a

text: pointed

text: cosimplicial

text: space

text: with

text: the

text: properties:

text: o

text: Any

text: element

text: in

text: π

text: TotX

text: has

text: ﬁltration

text: N.

text: Any

text: class

text: in

text: Es,s

text: for

text: s

text: N

text: is

text: either

text: killed

text: at

text: a

text: ﬁnite

text: stage

text: or

text: supports

text: a

text: diﬀerential

text: d

text: for

text: r

text: s

text: r

text: Then

text: if

text: Y

text: is

text: a

text: path-connected

text: cosimplicial

text: space

text: such

text: that

text: there

text: exists

text: an

text: equivalence

text: of

text: pointed

text: cosimplicial

text: spaces

text: ΩX

text: ΩY

text: the

text: same

text: conditions

text: above

text: are

text: valid

text: for

text: the

text: totalization

text: of

text: Y


text: FIBERS OF PARTIAL TOTALIZATIONS OF A POINTED COSIMPLICIAL SPACE 15 Proof. This follows from the above theorem. Concretely, any element of the • → • ∗ ﬁber ﬁb (Tot (X ) Tot − (X )) either maps to the connected component at in •s → s 1• ﬁb (Tot (X ) Tot − − (X )) for some t > 0, which is detected by a diﬀerential s s t 1 • that we see from the spectral sequence of ΩX (i.e., not a “fringed” diﬀerential in • the spectral sequence for X ), or it supports a “fringed” diﬀerential and fails to lift • → • ≤ − to ﬁb(Tot (X ) Tot − (X )), where r s 1. But in this range, everything s+r s• 1 (cid:3) only depends only on ΩX by ?? . A natural example of the type of situation in ?? arises from a descendable map → • of E∞-ring spectra S R. Denote by R the cosimplicial E∞-ring spectrum which → is the cobar construction on the map S R, also known as the Amitsur complex. • Descendability implies that the spectrum S is equivalent to the totalization of R . ∞ • • Under mild conditions, the spaces B(Ω R ) and BGL (R ) satisfy the hypotheses 1 (cid:39) of ?? , since for an E -ring spectrum R, the space GL (R) ΩBGL (R) is a 1 ∞ 1 1 union of connected components of ΩB(Ω R). We discuss this example further in [MS14]: it was the motivating example and application for us. References [ADL08] Gregory Z. Arone, William G. Dwyer, and Kathryn Lesh, Loop structures in Taylor towers, Algebr. Geom. Topol. 8 (2008), no. 1, 173–210. MR 2377281 (2008m:55025) [Ber07] Julia E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2043–2058. MR 2276611 (2007i:18014) [BK72] A. K. Bousﬁeld and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573 (51 #1825) [BWW05] Anders Bj¨orner, Michelle L. Wachs, and Volkmar Welker, Poset ﬁber theorems, Trans. Amer. Math. Soc. 357 (2005), no. 5, 1877–1899. MR 2115080 (2006a:05172) [Coo78] George Cooke, Replacing homotopy actions by topological actions, Trans. Amer. Math. Soc. 237 (1978), 391–406. MR 0461544 (57 #1529) A [DI04] Daniel Dugger and Daniel C. Isaksen, Topological hypercovers and 1-realizations, Math. Z. 246 (2004), no. 4, 667–689. MR 2045835 (2005d:55026) [DK84a] W. G. Dwyer and D. M. Kan, An obstruction theory for diagrams of simplicial sets, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 2, 139–146. MR 749527 (86c:55010c) [DK84b] , Realizing diagrams in the homotopy category by means of diagrams of simpli- cial sets, Proc. Amer. Math. Soc. 91 (1984), no. 3, 456–460. MR 744648 (86c:55010b) [DKS86] W. G. Dwyer, D. M. Kan, and J. H. Smith, An obstruction theory for simplicial cate- gories, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 2, 153–161. MR 849715 (87m:55023) [Dug] Daniel Dugger, A primer on homotopy colimits, Preprint available at http://math. uoregon.edu/~ddugger/hocolim.pdf. [Lur09] Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659 (2010j:18001) [Lur12] , Higher algebra, Preprint available at http://www.math.harvard.edu/~lurie, 2012. [MS14] Akhil Mathew and Vesna Stojanoska, The Picard group of topological modular forms via descent theory, Preprint available at http://arxiv.org/abs/1409.7702, 2014. [Qui78] Daniel Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. in Math. 28 (1978), no. 2, 101–128. MR 493916 (80k:20049) ##LTLine## ##LTLine##

text: FIBERS

label: paragraph

text: OF

text: PARTIAL

text: TOTALIZATIONS

text: OF

text: A

text: POINTED

text: COSIMPLICIAL

text: SPACE

text: Proof.

text: This

text: follows

text: from

text: the

text: above

text: theorem.

text: Concretely,

text: any

text: element

text: of

text: the

text: ﬁber

text: ﬁb

text: (Tot

text: (X

text: Tot

text: (X

text: either

text: maps

text: to

text: the

text: connected

text: component

text: at

text: in

text: •s

text: s

text: 1•

text: ﬁb

text: (Tot

text: (X

text: Tot

text: (X

text: for

text: some

text: t

text: which

text: is

text: detected

text: by

text: a

text: diﬀerential

text: s

text: s

text: t

text: that

text: we

text: see

text: from

text: the

text: spectral

text: sequence

text: of

text: ΩX

text: (i.e.,

text: not

text: a

text: “fringed”

text: diﬀerential

text: in

text: the

text: spectral

text: sequence

text: for

text: X

text: or

text: it

text: supports

text: a

text: “fringed”

text: diﬀerential

text: and

text: fails

text: to

text: lift

text: to

text: ﬁb(Tot

text: (X

text: Tot

text: (X

text: where

text: r

text: s

text: But

text: in

text: this

text: range,

text: everything

text: s+r

text: s•

text: only

text: depends

text: only

text: on

text: ΩX

text: by

text: A

text: natural

text: example

text: of

text: the

text: type

text: of

text: situation

text: in

text: arises

text: from

text: a

text: descendable

text: map

text: of

text: E∞-ring

text: spectra

text: S

text: R.

text: Denote

text: by

text: R

text: the

text: cosimplicial

text: E∞-ring

text: spectrum

text: which

text: is

text: the

text: cobar

text: construction

text: on

text: the

text: map

text: S

text: R,

text: also

text: known

text: as

text: the

text: Amitsur

text: complex.

text: Descendability

text: implies

text: that

text: the

text: spectrum

text: S

text: is

text: equivalent

text: to

text: the

text: totalization

text: of

text: R

text: Under

text: mild

text: conditions,

text: the

text: spaces

text: B(Ω

text: R

text: and

text: BGL

text: (R

text: satisfy

text: the

text: hypotheses

text: of

text: since

text: for

text: an

text: E

text: -ring

text: spectrum

text: R,

text: the

text: space

text: GL

text: (R)

text: ΩBGL

text: (R)

text: is

text: a

text: union

text: of

text: connected

text: components

text: of

text: ΩB(Ω

text: R).

text: We

text: discuss

text: this

text: example

text: further

text: in

text: it

text: was

text: the

text: motivating

text: example

text: and

text: application

text: for

text: us.

text: References

text: Gregory

text: Z.

text: Arone,

text: William

text: G.

text: Dwyer,

text: and

text: Kathryn

text: Lesh,

text: Loop

text: structures

text: in

text: Taylor

text: towers,

text: Algebr.

text: Geom.

text: Topol.

text: no.

text: MR

text: Julia

text: E.

text: Bergner,

text: A

text: model

text: category

text: structure

text: on

text: the

text: category

text: of

text: simplicial

text: categories,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: no.

text: MR

text: A.

text: K.

text: Bousﬁeld

text: and

text: D.

text: M.

text: Kan,

text: Homotopy

text: limits,

text: completions

text: and

text: localizations,

text: Lecture

text: Notes

text: in

text: Mathematics,

text: Vol.

text: Springer-Verlag,

text: Berlin-New

text: York,

text: MR

text: Anders

text: Bj¨orner,

text: Michelle

text: L.

text: Wachs,

text: and

text: Volkmar

text: Welker,

text: Poset

text: ﬁber

text: theorems,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: no.

text: MR

text: George

text: Cooke,

text: Replacing

text: homotopy

text: actions

text: by

text: topological

text: actions,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: MR

text: A

text: Daniel

text: Dugger

text: and

text: Daniel

text: C.

text: Isaksen,

text: Topological

text: hypercovers

text: and

text: Math.

text: Z.

text: no.

text: MR

text: W.

text: G.

text: Dwyer

text: and

text: D.

text: M.

text: Kan,

text: An

text: obstruction

text: theory

text: for

text: diagrams

text: of

text: simplicial

text: sets,

text: Nederl.

text: Akad.

text: Wetensch.

text: Indag.

text: Math.

text: no.

text: MR

text: Realizing

text: diagrams

text: in

text: the

text: homotopy

text: category

text: by

text: means

text: of

text: diagrams

text: of

text: simpli-

text: cial

text: sets,

text: Proc.

text: Amer.

text: Math.

text: Soc.

text: no.

text: MR

text: W.

text: G.

text: Dwyer,

text: D.

text: M.

text: Kan,

text: and

text: J.

text: H.

text: Smith,

text: An

text: obstruction

text: theory

text: for

text: simplicial

text: cate-

text: gories,

text: Nederl.

text: Akad.

text: Wetensch.

text: Indag.

text: Math.

text: no.

text: MR

text: [Dug]

text: Daniel

text: Dugger,

text: A

text: primer

text: on

text: homotopy

text: colimits,

text: Preprint

text: available

text: at

text: Jacob

text: Lurie,

text: Higher

text: topos

text: theory,

text: Annals

text: of

text: Mathematics

text: Studies,

text: vol.

text: Princeton

text: University

text: Press,

text: Princeton,

text: NJ,

text: MR

text: Higher

text: algebra,

text: Preprint

text: available

text: at

text: Akhil

text: Mathew

text: and

text: Vesna

text: Stojanoska,

text: The

text: Picard

text: group

text: of

text: topological

text: modular

text: forms

text: via

text: descent

text: theory,

text: Preprint

text: available

text: at

text: Daniel

text: Quillen,

text: Homotopy

text: properties

text: of

text: the

text: poset

text: of

text: nontrivial

text: p-subgroups

text: of

text: a

text: group,

text: Adv.

text: in

text: Math.

text: no.

text: MR

text: ##LTLine##

text: ##LTLine##


text: 10 AKHIL MATHEW AND VESNA STOJANOSKA attaching simplices ∆p, along the boundary ∂∆p, whose initial vertex is (0, 0). In particular, to ﬁnd the lift in (6), it suﬃces to contemplate a problem of the form (cid:47)(cid:47) C ∂∆p (cid:61)(cid:61) , (cid:15)(cid:15) ∆p C where the initial vertex of ∂∆p maps to an initial object of . But by the deﬁnition of an initial object of a quasi-category [Lur09, Deﬁnition 1.2.12.3], such lifting (cid:3) problems can always be solved. 4. Specialization to totalizations In this section, we will prove our main result on k-fold deloopability for total- izations (which is a restatement of ?? ). ≤ ≤ ≤ ≤ ⊂ ≤ − [eorem]lmTheorem Let 1 n m 2n+1. Then ∆ n ∆ m is (2n m+2)- fold deloopable. In order to prove ?? , we will follow the outline already laid out in the previous sections. Namely, we will write down the functor T introduced in ?? explicitly; it will be pointwise a wedge of spheres. Then we will show, using the obstruction theory, that it can be desuspended the right number of times. ≤ Our ﬁrst problem is, however, that that the nerve N(∆ n) is not a ﬁnite- dimensional simplicial set, and it will be convenient to replace it by one. [ﬁnition]lmDeﬁnition As before, let ∆ be the simplex category, of ﬁnite nonempty ⊂ totally ordered sets and order-preserving maps between them. We let ∆inj ∆ be the subcategory with the same objects but injective order-preserving maps. We let ≤ ≤ ∆inj, n and ∆ n denote the analogs where we only consider totally ordered sets of ≤ cardinality n + 1. ⊂ It is classical that the inclusion ∆inj ∆ is (homotopy) right coﬁnal; see for instance [Rie14, Example 8.5.12]. The advantage of the category ∆inj is that ≤ the nerve of a skeleton of ∆inj, n is n-dimensional, as any composable sequence → → · · · → ≤ x x x of arrows in a skeleton of ∆inj, n contains an identity. 0 1 n+1 ≤ ⊂ ≤ (Compare ?? .) However, the inclusion ∆inj, n ∆ n is no longer homotopy ≤ ≤ ﬁnal: ∆ n is weakly contractible (as it has a terminal object) while ∆inj, n fails to be weakly contractible (for instance for n = 1). Recall that any right coﬁnal map of quasi-categories is in particular a weak homotopy equivalence of underlying simplicial sets [Lur09, Proposition 4.1.1.3]. ≤ Let ∆inj, n be the overcategory; it is equivalent to the partially ordered set of /[n] { } ≤ → all nonempty subsets of [n] = 0, 1, . . . , n . We have a natural functor ∆inj, n ≤ → ≤ /[n] ∆inj, n ∆ n. Now we use: ≤ → ≤ [oposition]lmProposition[] The composite ∆inj, n ∆ n is right coﬁnal. /[n] ##LTLine## ##LTLine##

text: AKHIL

text: MATHEW

text: AND

text: VESNA

text: STOJANOSKA

text: attaching

text: simplices

text: ∆p,

text: along

text: the

text: boundary

text: ∂∆p,

text: whose

text: initial

text: vertex

text: is

text: In

text: particular,

text: to

text: ﬁnd

text: the

text: lift

text: in

text: it

text: suﬃces

text: to

text: contemplate

text: a

text: problem

text: of

text: the

text: form

font: default

text: C

text: ∂∆p

text: ∆p

text: C

text: where

text: the

text: initial

text: vertex

text: of

text: ∂∆p

text: maps

text: to

text: an

text: initial

text: object

text: of

text: But

text: by

text: the

text: deﬁnition

text: of

text: an

text: initial

text: object

text: of

text: a

text: quasi-category

text: Deﬁnition

text: such

text: lifting

text: problems

text: can

text: always

text: be

text: solved.

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text: Namely,

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text: /[n]

text: ##LTLine##

text: ##LTLine##


text: 11 0.04 −3 −The fermion operators can be written as f1 ∼ e−iφ1, f−∼1 ∼ 0.035 x)⟩| −4 y = − 2.23*x − 7.41 e∂ φiφ−−1. ∂ Tφh−e, sapnind density operator is given as Sz + x 1 x 1 0.03 xQ(r −5 +x)⟩ 0.025 xn|⟨Q(r) −−76 Q± = 1(Qx ± iQy) ∼ eiφ±1−iφ∓1. (C11) l 2 r xQ( 0.02 −8 −1.5 −1 −0.5 (cid:18) (cid:19) xQ(r) 0.015 ln[sin(πx/L)] If C1 = 1, C0 = −−1, C−1 = 1, then the K matrix is ⟨ 0 1 MC data given as K = − , and from (C10) and (C11), we 0.01 1 0 power law fitting obtain the scaling law 0.005 (cid:104) − (cid:105) ∼ − Q+(r)Q (r + x) x 2, (cid:18) (cid:19) 01 2 3 4 5 6 7 8 9 10 which is veriﬁed by the numerical result given in section x . (cid:18) (cid:19) n Furthermore, for the above K matrix, since l = FIG. 9. (Color online) Correlation function of the boundary 0 in an uniform Zeeman ﬁeld Bx = 0.4. Power law decaying cor- 0 relation function on the boundary shows that the edge states or l = n (n is an integer) satisﬁes are still gapless. Inset: Log-log ﬁtting, which shows that the − decaying power is approximately 2.23. − lT K 1l = 0, (C12) (cid:18) (cid:19) (cid:80) is the “charge vector” coupling to this probe ﬁeld. For (cid:18) (cid:19) the Higgs term24 that may gap out the boundary is (cid:80) 1 the Sz conservation symmetry, q = − , while for cos(nφ ) or cos(nφ− ). Equation (C12) is the gapping i i 1 1 1 condition for the perturbations. For instance, the pairing 1 the (Sz)2 conservation symmetry, q = . The Hall perturbation discussed in section satisﬁes the gapping i i 1 − condition. On the other hand, if this condition is not conductance is given by σH = 21πqT K 1q. satisﬁed for some perturbation, for example, a Zeeman Since (C8) is not gauge invariant if the system has a ﬁeld coupling boundary, we need to introduce a boundary action to (cid:48) ∼ recover the gauge invariance, H = B S cos(2φ + φ− ) + cos(2φ− + φ ) x x 1 1 1 1 which does not contain the Higgs term, then the bound- L − i − ary remains gapless even the symmetry is explicitly bro- boundary = 4π KIJ∂τφI∂xφJ VIJ∂xφI∂xφJ,(C9) ken. This is veriﬁed by our numerica(cid:104)l result shown in(cid:105) Fig. 9, where the correlation function Qx(r)Qx(r + x) remains power law if we add a Zeeman ﬁeld B = 0.4 (in x − units of t ) to the whole system. where I, J = 1, 1, and the ﬁeld φ only exist on the ij (cid:18) (cid:19) I Finally, we give the Chern-Simons theory of the CLS boundary and is deﬁned such that a = ∂ φ . The φ C C C Iµ µ I I state where = = − = 1. Form (C8), the K matrix ﬁeld satisﬁes the Kac-Moody algebra, 1 0 1 2 1 of the CSL is given as K = . Since det K = 3, 1 2 − − The ground state degeneracy of CSL on a torus is 3. [∂ φ , ∂ φ ] = 2πi(K 1) ∂ δ(x y). (C10) x I y J IJ x Furthermore, since the gapping condition (C12) has no solutions, the boundary can not be gapped out by small local perturbations. ∗ 7 P. W. Anderson, Science 235, 1196 (1987). 1 X. G. Wen, Phys. Rev. B 40, 7387 (1989). 8 L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937). 2 X. G. Wen and Q. Niu, Phys. Rev. B 41, 9377 (1990). 9 V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). 3 X. G. Wen, International Journal of Modern Physics B 04, 23910(1X9.9C0)h.en, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 82, 155138 (2010). 4 D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Let1t1. A48. ,K1i5ta5e9v(1a9n8d2)J.. Preskill, Phys. Rev. Lett. 96, 110404 (2006). 5 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 12 M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006). 6 P. Anderson, Materials Research Bulletin 8, 153 (1973). 13 Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009). ##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##

text: −3

text: −The

text: fermion

text: operators

text: can

text: be

text: written

text: as

text: f−∼1

text: x)⟩|

text: −4

text: y

text: e∂

text: φiφ−−1.

text: Tφh−e,

text: sapnind

text: density

text: operator

text: is

text: given

text: as

text: Sz

text: x

text: x

text: xQ(r

text: −5

text: +x)⟩

text: xn|⟨Q(r)

text: −−76

text: Q±

label: equation

text: iQy)

text: l

text: r

text: xQ(

text: −8

text: −1

text: xQ(r)

text: ln[sin(πx/L)]

text: If

text: −−1,

text: then

text: the

text: K

text: matrix

text: is

text: MC

text: data

text: given

text: as

text: K

text: and

text: from

text: and

text: we

text: power

text: law

text: fitting

text: obtain

text: the

text: scaling

text: law

text: Q+(r)Q

text: (r

text: x)

text: x

text: which

text: is

text: veriﬁed

text: by

text: the

text: numerical

text: result

text: given

text: in

text: section

text: x

text: n

text: Furthermore,

text: for

text: the

text: above

text: K

text: matrix,

text: since

text: l

text: FIG.

text: (Color

text: online)

text: Correlation

text: function

text: of

text: the

text: boundary

text: in

text: an

text: uniform

text: Zeeman

text: ﬁeld

text: Bx

text: Power

text: law

text: decaying

text: cor-

text: relation

text: function

text: on

text: the

text: boundary

text: shows

text: that

text: the

text: edge

text: states

text: or

text: l

text: n

text: (n

text: is

text: an

text: integer)

text: satisﬁes

text: are

text: still

text: gapless.

text: Inset:

text: Log-log

text: ﬁtting,

text: which

text: shows

text: that

text: the

text: decaying

text: power

text: is

text: approximately

text: lT

text: K

text: is

text: the

text: “charge

text: vector”

text: coupling

text: to

text: this

text: probe

text: ﬁeld.

text: For

text: the

text: Higgs

text: that

text: may

text: gap

text: out

text: the

text: boundary

text: is

text: the

text: Sz

text: conservation

text: symmetry,

text: q

text: while

text: for

text: cos(nφ

text: or

text: cos(nφ−

text: Equation

text: is

text: the

text: gapping

text: i

text: i

text: condition

text: for

text: the

text: perturbations.

text: For

text: instance,

text: the

text: pairing

text: the

text: conservation

text: symmetry,

text: q

text: The

text: Hall

text: perturbation

text: discussed

text: in

text: section

text: satisﬁes

text: the

text: gapping

text: i

text: i

text: condition.

text: On

text: the

text: other

text: hand,

text: if

text: this

text: condition

text: is

text: not

text: conductance

text: is

text: given

text: by

text: σH

text: K

text: satisﬁed

text: for

text: some

text: perturbation,

text: for

text: example,

text: a

text: Zeeman

text: Since

text: is

text: not

text: gauge

text: invariant

text: if

text: the

text: system

text: has

text: a

text: ﬁeld

text: coupling

text: boundary,

text: we

text: need

text: to

text: introduce

text: a

text: boundary

text: action

text: to

text: recover

text: the

text: gauge

text: invariance,

text: H

text: B

text: S

text: φ−

text: φ

text: x

text: x

text: which

text: does

text: not

text: contain

text: the

text: Higgs

text: term,

text: then

text: the

text: bound-

text: L

text: i

text: ary

text: remains

text: gapless

text: even

text: the

text: symmetry

text: is

text: explicitly

text: bro-

text: boundary

text: 4π

text: KIJ∂τφI∂xφJ

text: ken.

text: This

text: is

text: veriﬁed

text: by

text: our

text: result

text: shown

text: Fig.

text: where

text: the

text: correlation

text: function

text: Qx(r)Qx(r

text: x)

text: remains

text: power

text: law

text: if

text: we

text: add

text: a

text: Zeeman

text: ﬁeld

text: B

text: (in

text: x

text: units

text: of

text: t

text: to

text: the

text: whole

text: system.

text: where

text: I,

text: J

text: and

text: the

text: ﬁeld

text: φ

text: only

text: exist

text: on

text: the

text: ij

text: I

text: Finally,

text: we

text: give

text: the

text: Chern-Simons

text: theory

text: of

text: the

text: CLS

text: boundary

text: and

text: is

text: deﬁned

text: such

text: that

text: a

text: φ

text: The

text: φ

text: C

text: C

text: C

text: Iµ

text: µ

text: I

text: I

text: state

text: where

text: Form

text: the

text: K

text: matrix

text: ﬁeld

text: satisﬁes

text: the

text: Kac-Moody

text: algebra,

text: of

text: the

text: CSL

text: is

text: given

text: as

text: K

text: Since

text: det

text: K

text: The

text: ground

text: state

text: degeneracy

text: of

text: CSL

text: on

text: a

text: torus

text: is

text: φ

text: φ

text: δ(x

text: y).

text: x

text: I

text: y

text: J

text: IJ

text: x

text: Furthermore,

text: since

text: the

text: gapping

text: condition

text: has

text: no

text: solutions,

text: the

text: boundary

text: can

text: not

text: be

text: gapped

text: out

text: by

text: small

text: local

text: perturbations.

text: P.

text: W.

text: Anderson,

text: Science

text: X.

text: G.

text: Wen,

text: Phys.

text: Rev.

text: B

text: L.

text: D.

text: Landau,

text: Phys.

text: Z.

text: Sowjetunion

text: X.

text: G.

text: Wen

text: and

text: Q.

text: Niu,

text: Phys.

text: Rev.

text: B

text: V.

text: L.

text: Ginzburg

text: and

text: L.

text: D.

text: Landau,

text: Zh.

text: Eksp.

text: Teor.

text: Fiz.

text: X.

text: G.

text: Wen,

text: International

text: Journal

text: of

text: Modern

text: Physics

text: B

text: Z.-C.

text: Gu,

text: and

text: X.-G.

text: Wen,

text: Phys.

text: Rev.

text: B

text: D.

text: C.

text: Tsui,

text: H.

text: L.

text: Stormer,

text: and

text: A.

text: C.

text: Gossard,

text: Phys.

text: Rev.

text: Preskill,

text: Phys.

text: Rev.

text: Lett.

text: R.

text: B.

text: Laughlin,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Levin

text: and

text: X.-G.

text: Wen,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Anderson,

text: Materials

text: Research

text: Bulletin

text: Z.-C.

text: Gu

text: and

text: X.-G.

text: Wen,

text: Phys.

text: Rev.

text: B

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text: 8 0.15 0.2 0.1 MC data PG|1-11⟩ 0.1 MC data PG|111⟩ 0.05 exponential ﬁtting ⟩ power law ﬁtting ⟩ x x + xQr+ 0 xQr 0 xQr −0.05 xQr −0.1 ⟨ ⟨ −0.1 −0.2 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 x x 0.15 0.2 0.1 MC data PG|1-11⟩ 0.1 MC data PG|111⟩ ⟩ 0.05 exponential ﬁtting ⟩ power law ﬁtting zSr+x 0 zSr+x 0 zSr −0.05 zSr ⟨ ⟨ −0.1 −0.1 −0.2 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 x x FIG. 7. (Color online) Boundary of SPT phase can be gapped FIG. 8. (Color online) Boundary of CSL phase is robust when symmetry is explicitly broken. against all perturbations. | (cid:105) Our numerical result is shown in Fig. 7, where the topologically ordered chiral spin liquid state P 111 (cid:104) (cid:105) (cid:104) (cid:105) G correlation function SzSz and QxQx are both whose gapless boundary excitations are robust against r r+x r r+x exponentially decaying as expected. all local perturbations. Our data imply that the bound- We also calculate the boundary correlation function of ary of the SPT state is non-chiral while the boundary the CSL undergoing the same perturbation. The results of the chiral spin liquid is chiral. Noticing that at the | − (cid:105) | (cid:105) in Fig. 8 show that the boundary remains gapless under mean-ﬁeld level both 1 11 and 111 are chiral, it is the perturbation. This comparison give strong evidence remarkable that after Gutzwiller projection (or due to | − (cid:105) that the boundary of the state P 1 11 is non-chiral strong interactions) the former becomes non-chiral. This | (cid:105) G while the CSL state P 111 is chiral, as predicted by indicates that the physical properties of some mean-ﬁeld G Chern-Simons theory (see Appendix ). states might be dramatically changed after Gutzwiller V. CONCLUSION AND DISCUSSION projection. Our Gutzwiller approach can be applied to study SPT states protected by other symmetry groups, such as In summary, using the Monte Carlo method we stud- SU(2) or SO(3) symmetry, and so on. It can be also ied the physical properties of Gutzwiller-projected wave | − (cid:105) used to study symmetry enriched topological phases, functions. We especially studied the state P 1 11 − G where symmetry interplays with topological order result- (where 1, 1, 1 are the mean-ﬁeld Chern numbers of the fermions f , f , f− , respectively), including its spin Hall ing in an enriched phase diagram. 1 0 1 conductance, correlation function of the gapless edge Finally, we give some remarks about the Hamiltonians states, ground-state degeneracy and topological entan- of the Gutzwiller-projected states that we constructed glement entropy, and nonrobustness of the gapless edge above. In principle, for each Gutzwiller wavefunction | − (cid:105) states. All these evidences show that P 1 11 is a one can always ﬁnd a parent Hamiltonian of which the × G U(1) U(1) symmetry protected topological state. Our Gutzwiller wavefunction is the ground state. However, work may shed some light on simple lattice models and that Hamiltonian is generally very complicated and is experimental realization of SPT phases. diﬃcult to identify. Nevertheless, approximate Hamilto- nians can be constructed. For instance, in Ref. 37 a spin The spin Hall conductance is calculated by measuring Hamiltonian containing three-body interactions was pro- the spin pump in the Gutzwiller wave function caused posed via perturbation to onsite Hubbard interactions. by inserting symmetry ﬂux through the cylinder to the On the other hand, the reduced density matrix method mean-ﬁeld Hamiltonian. We ﬁnd that the internal gauge introduced in Ref. 40 also provides some hint of possible ﬁeld plays an important role since external symmetry ﬂux interactions that may stabilize the Gutzwiller wavefunc- will induce a nonzero background internal gauge ﬂux (see tions. also Ref. 40). Our observation indicates that in general the internal gauge ﬁeld cannot be ignored in studying the ACKNOWLEDGEMENTS physical response of Gutzwiller projected wave functions. | − (cid:105) We also compared the SPT state P 1 11 with the We thank Y.-M. Lu, Y. Zhang, Z.-C. Gu, H. Yao, H.- G ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: MC

text: data

text: MC

text: data

text: exponential

text: ﬁtting

text: power

text: law

text: ﬁtting

text: x

text: x

text: xQr+

text: xQr

text: xQr

text: xQr

text: x

text: x

text: MC

text: data

text: MC

text: data

text: exponential

text: ﬁtting

text: power

text: law

text: ﬁtting

text: zSr+x

text: zSr+x

text: zSr

text: zSr

text: x

text: x

text: FIG.

text: (Color

text: online)

text: Boundary

text: of

text: SPT

text: phase

text: can

text: be

text: gapped

text: FIG.

text: (Color

text: online)

text: Boundary

text: of

text: CSL

text: phase

text: is

text: robust

text: when

text: symmetry

text: is

text: explicitly

text: broken.

text: against

text: all

text: perturbations.

text: Our

text: numerical

text: result

text: is

text: shown

text: in

text: Fig.

text: where

text: the

text: topologically

text: ordered

text: chiral

text: spin

text: liquid

text: state

text: P

text: G

text: correlation

text: function

text: SzSz

text: and

text: QxQx

text: are

text: both

text: whose

text: gapless

text: boundary

text: excitations

text: are

text: robust

text: against

text: r

text: r+x

text: r

text: r+x

text: exponentially

text: decaying

text: as

text: expected.

text: all

text: local

text: perturbations.

text: Our

text: data

text: imply

text: that

text: the

text: bound-

text: We

text: also

text: calculate

text: the

text: boundary

text: correlation

text: function

text: of

text: ary

text: of

text: the

text: SPT

text: state

text: is

text: non-chiral

text: while

text: the

text: boundary

text: the

text: CSL

text: undergoing

text: the

text: same

text: perturbation.

text: The

text: results

text: of

text: the

text: chiral

text: spin

text: liquid

text: is

text: chiral.

text: Noticing

text: that

text: at

text: the

text: in

text: Fig.

text: show

text: that

text: the

text: boundary

text: remains

text: gapless

text: under

text: mean-ﬁeld

text: level

text: both

text: and

text: are

text: chiral,

text: it

text: is

text: the

text: perturbation.

text: This

text: comparison

text: give

text: strong

text: evidence

text: remarkable

text: that

text: after

text: Gutzwiller

text: projection

text: (or

text: due

text: to

text: that

text: the

text: boundary

text: of

text: the

text: state

text: P

text: is

text: non-chiral

text: strong

text: interactions)

text: the

text: former

text: becomes

text: non-chiral.

text: This

text: G

text: while

text: the

text: CSL

text: state

text: P

text: is

text: chiral,

text: as

text: predicted

text: by

text: indicates

text: that

text: the

text: physical

text: properties

text: of

text: some

text: mean-ﬁeld

text: G

text: Chern-Simons

text: theory

text: (see

text: Appendix

text: states

text: might

text: be

text: dramatically

text: changed

text: after

text: Gutzwiller

text: V.

text: CONCLUSION

text: AND

text: DISCUSSION

text: projection.

text: Our

text: Gutzwiller

text: approach

text: can

text: be

text: applied

text: to

text: study

text: SPT

text: states

text: protected

text: by

text: other

text: symmetry

text: groups,

text: such

text: as

text: In

text: summary,

text: using

text: the

text: Monte

text: Carlo

text: method

text: we

text: stud-

text: or

text: symmetry,

text: and

text: so

text: on.

text: It

text: can

text: be

text: also

text: ied

text: the

text: physical

text: properties

text: of

text: Gutzwiller-projected

text: wave

text: used

text: to

text: study

text: symmetry

text: enriched

text: topological

text: phases,

text: functions.

text: We

text: especially

text: studied

text: the

text: state

text: P

text: G

text: where

text: symmetry

text: interplays

text: with

text: topological

text: order

text: result-

text: (where

text: are

text: the

text: mean-ﬁeld

text: Chern

text: numbers

text: of

text: the

text: fermions

text: f

text: f

text: f−

text: respectively),

text: including

text: its

text: spin

text: Hall

text: ing

text: in

text: an

text: enriched

text: phase

text: diagram.

text: conductance,

text: correlation

text: function

text: of

text: the

text: gapless

text: edge

text: Finally,

text: we

text: give

text: some

text: remarks

text: about

text: the

text: Hamiltonians

text: states,

text: ground-state

text: degeneracy

text: and

text: topological

text: entan-

text: of

text: the

text: Gutzwiller-projected

text: states

text: that

text: we

text: constructed

text: glement

text: entropy,

text: and

text: nonrobustness

text: of

text: the

text: gapless

text: edge

text: above.

text: In

text: principle,

text: for

text: each

text: Gutzwiller

text: wavefunction

text: states.

text: All

text: these

text: evidences

text: show

text: that

text: P

text: is

text: a

text: one

text: can

text: always

text: ﬁnd

text: a

text: parent

text: Hamiltonian

text: of

text: which

text: the

text: G

text: symmetry

text: protected

text: topological

text: state.

text: Our

text: Gutzwiller

text: wavefunction

text: is

text: the

text: ground

text: state.

text: However,

text: work

text: may

text: shed

text: some

text: light

text: on

text: simple

text: lattice

text: models

text: and

text: that

text: Hamiltonian

text: is

text: generally

text: very

text: complicated

text: and

text: is

text: experimental

text: realization

text: of

text: SPT

text: phases.

text: diﬃcult

text: to

text: identify.

text: Nevertheless,

text: approximate

text: Hamilto-

text: nians

text: can

text: be

text: constructed.

text: For

text: instance,

text: in

text: Ref.

text: a

text: spin

text: The

text: spin

text: Hall

text: conductance

text: is

text: calculated

text: by

text: measuring

text: Hamiltonian

text: containing

text: three-body

text: interactions

text: was

text: pro-

text: the

text: spin

text: pump

text: in

text: the

text: Gutzwiller

text: wave

text: function

text: caused

text: posed

text: via

text: perturbation

text: to

text: onsite

text: Hubbard

text: interactions.

text: by

text: inserting

text: symmetry

text: ﬂux

text: through

text: the

text: cylinder

text: to

text: the

text: On

text: the

text: other

text: hand,

text: the

text: reduced

text: density

text: matrix

text: method

text: mean-ﬁeld

text: Hamiltonian.

text: We

text: ﬁnd

text: that

text: the

text: internal

text: gauge

text: introduced

text: in

text: Ref.

text: also

text: provides

text: some

text: hint

text: of

text: possible

text: ﬁeld

text: plays

text: an

text: important

text: role

text: since

text: external

text: symmetry

text: ﬂux

text: interactions

text: that

text: may

text: stabilize

text: the

text: Gutzwiller

text: wavefunc-

text: will

text: induce

text: a

text: nonzero

text: background

text: internal

text: gauge

text: ﬂux

text: (see

text: tions.

text: also

text: Ref.

text: Our

text: observation

text: indicates

text: that

text: in

text: general

text: the

text: internal

text: gauge

text: ﬁeld

text: cannot

text: be

text: ignored

text: in

text: studying

text: the

text: ACKNOWLEDGEMENTS

text: physical

text: response

text: of

text: Gutzwiller

text: projected

text: wave

text: functions.

text: We

text: also

text: compared

text: the

text: SPT

text: state

text: P

text: with

text: the

text: We

text: thank

text: Y.-M.

text: Lu,

text: Y.

text: Zhang,

text: Z.-C.

text: Gu,

text: H.

text: Yao,

text: H.-

text: G

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text: Contents 1 Introduction 1 2 Review of massive (bi-)gravity 4 2.1 Massive (bi-)gravity & allowed potentials 4 2.2 Covariant coupling to matter 5 3 Coupling to two metrics 6 3.1 One-loop detuning of the potential 6 3.2 Classical ghost in the mini-superspace 8 3.3 Classical ghost in the decoupling limit 8 3.3.1 Coupling to only one metric: λ = 0 9 g (cid:54) 3.3.2 Minimal coupling to both metrics: λ = 0 11 g 4 Eﬀective metric 12 4.1 One–loop requirement 12 4.2 Coupling to other ﬁelds 13 4.3 Absence of ghost in the mini-superspace 13 4.4 Absence of ghost in decoupling limit 14 4.5 Re-emergence of the ghost 15 5 Consequences for Cosmology—FLRW solutions 16 5.1 Exact FLRW solutions 16 5.2 Linear Hamiltonian analysis about the exact FLRW solution 18 6 Other consistent couplings to matter 19 6.1 Mini-superspace 20 6.2 Decoupling limit 20 6.3 Quantum corrections 20 6.4 FLRW solutions 21 7 Summary 21 A FLRW solutions with coupling to the eﬀective metric 23 A.1 Single specie coupled to the eﬀective metric 24 A.2 Multiple species coupled to the eﬀective metric 26 A.3 Fully generic matter sector 27 – 1 – ##LTLine## ##LTLine##

text: Contents

text: Introduction

text: Review

text: of

text: massive

text: (bi-)gravity

text: Massive

text: (bi-)gravity

text: allowed

text: potentials

text: Covariant

text: coupling

text: to

text: matter

text: Coupling

text: to

text: two

text: metrics

text: One-loop

text: detuning

text: of

text: the

text: potential

text: Classical

text: ghost

text: in

text: the

text: mini-superspace

text: Classical

text: ghost

text: in

text: the

text: decoupling

text: limit

text: Coupling

text: to

text: only

text: one

text: metric:

text: λ

text: g

text: Minimal

text: coupling

text: to

text: both

text: metrics:

text: λ

text: g

text: Eﬀective

text: metric

text: One–loop

text: requirement

text: Coupling

text: to

text: other

text: ﬁelds

text: Absence

text: of

text: ghost

text: in

text: the

text: mini-superspace

text: Absence

text: of

text: ghost

text: in

text: decoupling

text: limit

text: Re-emergence

text: of

text: the

text: ghost

text: Consequences

text: for

text: Cosmology—FLRW

text: solutions

text: Exact

text: FLRW

text: solutions

text: Linear

text: Hamiltonian

text: analysis

text: about

text: the

text: exact

text: FLRW

text: solution

text: Other

text: consistent

text: couplings

text: to

text: matter

text: Mini-superspace

text: Decoupling

text: limit

text: Quantum

text: corrections

text: FLRW

text: solutions

text: Summary

text: A

text: FLRW

text: solutions

text: with

text: coupling

text: to

text: the

text: eﬀective

text: metric

text: Single

text: specie

text: coupled

text: to

text: the

text: eﬀective

text: metric

text: Multiple

text: species

text: coupled

text: to

text: the

text: eﬀective

text: metric

text: Fully

text: generic

text: matter

text: sector

text: ##LTLine##

text: ##LTLine##


text: ﬁeld cannot minimally couple to both metrics g and f simultaneously without leading to a ghost at an unacceptable low scale. We now build on the lessons learnt here and propose a diﬀerent way to couple a single matter ﬁeld to both metrics simultaneously via a single eﬀective metric. 4 Eﬀective metric 4.1 One–loop requirement In principle, one should start by looking for couplings to matter which are free of the BD ghost at the classical level. However, to connect with our previous derivations, we start here instead by asking a slightly diﬀerent question: For what kind of coupling would the one-loop corrections from virtual matter ﬁelds take the form of the allowed potential ? As we shall see, this is a much simpler and straightforward requirement to satisfy. This will allow us to ﬁnd an eﬀective metric and further argue that it is free of the BD ghost (at least in some limits). Let us consider a single matter ﬁeld χ (taken to be a scalar ﬁeld for concreteness, but the exact nature of that ﬁeld is irrelevant to this argument) minimally coupled to an eﬀective (cid:0) (cid:1) metric g as follows, eﬀ √ L −1 − = g gµν∂ χ∂ χ + M2χ2 , (4.1) matter 2 eﬀ eﬀ µ ν where the eﬀective metric geﬀ is built out of g and f . As shown previously, the one-loop µν µν µν eﬀective action from matter loops will lead to a contribution going as √ L − − = M4 g + curvature corrections . (4.2) 1 loop eﬀ √ − If g is built out of both metrics g and f , the one√-loop contribution g will typically eﬀ µν µν − eﬀ lead to a ghost at an unacceptable low scale, unless g happens to take the form of one eﬀ of the allowed potentials introduced in section . Such an acceptable class of eﬀective metrics is for instance given by (cid:112) geﬀ = α2g + 2αβ g Xα + β2f , (4.3) µν µν µα ν µν − (cid:112) where we recall that X = g 1f and α and β are two arbitrary real dimensionless parameters. While the quantity Xα is not symmetric, we can show that the tensor Y Xα that enters ν ≡ −µα ν ≡ the eﬀective metric is symmetric. Working in matrix notation with X g 1f and Y gX, we have − Y = gX = gXg 1g . (cid:112) (cid:112) (4.4) − gXg 1 is nothing other than a similarity transformation on the matrix X, and by properties of − − − − the similarity transformations, it follows that gXg 1g = gg 1fg 1 = fg 1 = XT (using the fact that both g and f are symmetric). This implies that Y itself is symmetric, Y = XT g = XT gT = Y T . (4.5) – 13 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ﬁeld

text: cannot

text: minimally

text: couple

text: to

text: both

text: metrics

text: g

text: and

text: f

text: simultaneously

text: without

text: leading

text: to

text: a

text: ghost

text: at

text: an

text: unacceptable

text: low

text: scale.

text: We

text: now

text: build

text: on

text: the

text: lessons

text: learnt

text: here

text: and

text: propose

text: a

text: diﬀerent

text: way

text: to

text: couple

text: a

text: single

text: matter

text: ﬁeld

text: to

text: both

text: metrics

text: simultaneously

text: via

text: a

text: single

text: eﬀective

text: metric.

text: Eﬀective

text: metric

text: One–loop

text: requirement

text: In

text: principle,

text: one

text: should

text: start

text: by

text: looking

text: for

text: couplings

text: to

text: matter

text: which

text: are

text: free

text: of

text: the

text: BD

text: ghost

text: at

text: the

text: classical

text: level.

text: However,

text: to

text: connect

text: with

text: our

text: previous

text: derivations,

text: we

text: start

text: here

text: instead

text: by

text: asking

text: a

text: slightly

text: diﬀerent

text: question:

text: For

text: what

text: kind

text: of

text: coupling

text: would

text: the

text: one-loop

text: corrections

text: from

text: virtual

text: matter

text: ﬁelds

text: take

text: the

text: form

text: of

text: the

text: allowed

text: potential

text: As

text: we

text: shall

text: see,

text: this

text: is

text: a

text: much

text: simpler

text: and

text: straightforward

text: requirement

text: to

text: satisfy.

text: This

text: will

text: allow

text: us

text: to

text: ﬁnd

text: an

text: eﬀective

text: metric

text: and

text: further

text: argue

text: that

text: it

text: is

text: free

text: of

text: the

text: BD

text: ghost

text: (at

text: least

text: in

text: some

text: limits).

text: Let

text: us

text: consider

text: a

text: single

text: matter

text: ﬁeld

text: χ

text: (taken

text: to

text: be

text: a

text: scalar

text: ﬁeld

text: for

text: concreteness,

text: but

text: the

text: exact

text: nature

text: of

text: that

text: ﬁeld

text: is

text: irrelevant

text: to

text: this

text: argument)

text: minimally

text: coupled

text: to

text: an

text: eﬀective

text: metric

text: g

text: as

text: follows,

text: eﬀ

text: L

text: −1

text: g

text: gµν∂

text: χ∂

text: χ

text: matter

text: eﬀ

text: eﬀ

text: µ

text: ν

text: where

text: the

text: eﬀective

text: metric

text: geﬀ

text: is

text: built

text: out

text: of

text: g

text: and

text: f

text: As

text: shown

text: previously,

text: the

text: one-loop

text: µν

text: µν

text: µν

text: eﬀective

text: action

text: from

text: matter

text: loops

text: will

text: lead

text: to

text: a

text: contribution

text: going

text: as

text: L

text: g

text: curvature

text: corrections

text: loop

text: eﬀ

text: If

text: g

text: is

text: built

text: out

text: of

text: both

text: metrics

text: g

text: and

text: f

text: the

text: one√-loop

text: contribution

text: g

text: will

text: typically

text: eﬀ

text: µν

text: µν

text: eﬀ

text: lead

text: to

text: a

text: ghost

text: at

text: an

text: unacceptable

text: low

text: scale,

text: unless

text: g

text: happens

text: to

text: take

text: the

text: form

text: of

text: one

text: eﬀ

text: of

text: the

text: allowed

text: potentials

text: introduced

text: in

text: section

text: Such

text: an

text: acceptable

text: class

text: of

text: eﬀective

text: metrics

text: is

text: for

text: instance

text: given

text: by

text: geﬀ

text: 2αβ

text: g

text: Xα

text: µν

text: µν

text: µα

text: ν

text: µν

text: where

text: we

text: recall

text: that

text: X

text: g

text: and

text: α

text: and

text: β

text: are

text: two

text: arbitrary

text: real

text: dimensionless

text: parameters.

text: While

text: the

text: quantity

text: Xα

text: is

text: not

text: symmetric,

text: we

text: can

text: show

text: that

text: the

text: tensor

text: Y

text: Xα

text: that

text: enters

text: ν

text: −µα

text: ν

text: the

text: eﬀective

text: metric

text: is

text: symmetric.

text: Working

text: in

text: matrix

text: notation

text: with

text: X

text: g

text: and

text: Y

text: gX,

text: we

text: have

text: Y

text: gX

text: gXg

text: gXg

text: is

text: nothing

text: other

text: than

text: a

text: similarity

text: transformation

text: on

text: the

text: matrix

text: X,

text: and

text: by

text: properties

text: of

text: the

text: similarity

text: transformations,

text: it

text: follows

text: that

text: gXg

text: gg

text: fg

text: XT

text: (using

text: the

text: fact

text: that

text: both

text: g

text: and

text: f

text: are

text: symmetric).

text: This

text: implies

text: that

text: Y

text: itself

text: is

text: symmetric,

text: Y

text: XT

text: g

text: XT

text: gT

text: Y

text: T

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where T µν is the stress-energy tensor of χ. Here again, this equation for χ involves higher g eﬀ derivatives of π, but is a well-deﬁned equation for χ that can be solved by setting only two initial conditions for the ﬁeld χ. (cid:0) (cid:1) Next, the contribution from (4.1) to the equation of motion with respect to π is given by √ E − 1 − − matter = ∂ ∂ g T µα ((α + β)δν βΠν ) . (4.15) π Λ3 µ ν eﬀ geﬀ α α Now proceeding precisely as in section and us(cid:2)ing the relation(cid:3)(4.14) in the previous expression E for matter, we ﬁnd similarly as before, π √ E − − matter = ∂ g T bc Ra , (4.16) π a eﬀ g bc eﬀ where the tensor Ra is now deﬁned as bc ≡ − Ra Γα ((α + β)δa βΠa) + β∂aΠ . (4.17) bc bc α α bc Where the Γα are the Christoﬀel symbols associated with the eﬀective metric g . They satisfy bc eﬀ the relation − − Γα ((α + β)δa βΠa) = β∂aΠ , (4.18) bc α α bc which ensures the following identity ≡ Ra 0 . (4.19) bc So just like in the case where the ﬁeld χ only coupled to one metric, we see here that the contribution from the matter Lagrangian to the equation of motion for π vanishes identically, E ≡ matter 0 and the equation of motion for π is thus the standard one without the contribution π from the matter ﬁeld, which is manifestly second order. As a result, there is no ghost in the decoupling limit when the ﬁeld χ couples to both metrics simultaneously via the eﬀective metric g . eﬀ This is an important result by itself. Even if a careful constraint analysis ended up signaling the existence of a ghost, the absence of BD ghost in the decoupling limit means that the scale associated with the ghost is higher than Λ and therefore the theory is consistent at that scale. The existence of a ghost in the full theory would inevitably set a cut-oﬀ scale Λ c larger than Λ. 4.5 Re-emergence of the ghost Performing a more careful ADM analysis in all generality (i.e. beyond the mini-superspace approximation performed previously), we ﬁnd, after integrating out the shift in the Hamilto- nian, an operator of the form H ⊃ 1 α2β2 (∂ χ)2p2 N2 , (4.20) m2M2 (α + β)2 i χ Pl which is quadratic in the lapse and destroys the constraint that would otherwise project out the BD ghost, (see Ref. [99] for further details.) The mass of the associated ghost depends on the – 16 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: T

text: µν

text: is

text: the

text: stress-energy

text: tensor

text: of

text: χ.

text: Here

text: again,

text: this

text: equation

text: for

text: χ

text: involves

text: higher

text: g

text: eﬀ

text: derivatives

text: of

text: π,

text: but

text: is

text: a

text: well-deﬁned

text: equation

text: for

text: χ

text: that

text: can

text: be

text: solved

text: by

text: setting

text: only

text: two

text: initial

text: conditions

text: for

text: the

text: ﬁeld

text: χ.

text: Next,

text: the

text: contribution

text: from

text: to

text: the

text: equation

text: of

text: motion

text: with

text: respect

text: to

text: π

text: is

text: given

text: by

text: E

text: matter

text: g

text: T

text: µα

text: ((α

text: β)δν

text: βΠν

text: π

text: Λ3

text: µ

text: ν

text: eﬀ

text: geﬀ

text: α

text: α

text: Now

text: proceeding

text: precisely

text: as

text: in

text: section

text: and

text: the

text: in

text: the

text: previous

text: expression

text: E

text: for

text: matter,

text: we

text: ﬁnd

text: similarly

text: as

text: before,

text: π

text: E

text: matter

text: g

text: T

text: bc

text: Ra

text: π

text: a

text: eﬀ

text: g

text: bc

text: eﬀ

text: where

text: the

text: tensor

text: Ra

text: is

text: now

text: deﬁned

text: as

text: bc

text: Ra

text: Γα

text: ((α

text: β)δa

text: βΠa)

text: β∂aΠ

text: bc

text: bc

text: α

text: α

text: bc

text: Where

text: the

text: Γα

text: are

text: the

text: Christoﬀel

text: symbols

text: associated

text: with

text: the

text: eﬀective

text: metric

text: g

text: They

text: satisfy

text: bc

text: eﬀ

text: the

text: relation

text: Γα

text: ((α

text: β)δa

text: βΠa)

text: β∂aΠ

text: bc

text: α

text: α

text: bc

text: which

text: ensures

text: the

text: following

text: identity

text: Ra

text: bc

text: So

text: just

text: like

text: in

text: the

text: case

text: where

text: the

text: ﬁeld

text: χ

text: only

text: coupled

text: to

text: one

text: metric,

text: we

text: see

text: here

text: that

text: the

text: contribution

text: from

text: the

text: matter

text: Lagrangian

text: to

text: the

text: equation

text: of

text: motion

text: for

text: π

text: vanishes

text: identically,

text: E

text: matter

text: and

text: the

text: equation

text: of

text: motion

text: for

text: π

text: is

text: thus

text: the

text: standard

text: one

text: without

text: the

text: contribution

text: π

text: from

text: the

text: matter

text: ﬁeld,

text: which

text: is

text: manifestly

text: second

text: order.

text: As

text: a

text: result,

text: there

text: is

text: no

text: ghost

text: in

text: the

text: decoupling

text: limit

text: when

text: the

text: ﬁeld

text: χ

text: couples

text: to

text: both

text: metrics

text: simultaneously

text: via

text: the

text: eﬀective

text: metric

text: g

text: eﬀ

text: This

text: is

text: an

text: important

text: result

text: by

text: itself.

text: Even

text: if

text: a

text: careful

text: constraint

text: analysis

text: ended

text: up

text: signaling

text: the

text: existence

text: of

text: a

text: ghost,

text: the

text: absence

text: of

text: BD

text: ghost

text: in

text: the

text: decoupling

text: limit

text: means

text: that

text: the

text: scale

text: associated

text: with

text: the

text: ghost

text: is

text: higher

text: than

text: Λ

text: and

text: therefore

text: the

text: theory

text: is

text: consistent

text: at

text: that

text: scale.

text: The

text: existence

text: of

text: a

text: ghost

text: in

text: the

text: full

text: theory

text: would

text: inevitably

text: set

text: a

text: cut-oﬀ

text: scale

text: Λ

text: c

text: larger

text: than

text: Λ.

text: Re-emergence

text: of

text: the

text: ghost

text: Performing

text: a

text: more

text: careful

text: ADM

text: analysis

text: in

text: all

text: generality

text: (i.e.

text: beyond

text: the

text: mini-superspace

text: approximation

text: performed

text: previously),

text: we

text: ﬁnd,

text: after

text: integrating

text: out

text: the

text: shift

text: in

text: the

text: Hamilto-

text: nian,

text: an

text: operator

text: of

text: the

text: form

text: H

text: α2β2

text: (α

text: i

text: χ

text: Pl

text: which

text: is

text: quadratic

text: in

text: the

text: lapse

text: and

text: destroys

text: the

text: constraint

text: that

text: would

text: otherwise

text: project

text: out

text: the

text: BD

text: ghost,

text: (see

text: Ref.

text: for

text: further

text: details.)

text: The

text: mass

text: of

text: the

text: associated

text: ghost

text: depends

text: on

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: special structure of the graviton potential were explored in Ref. [33]. While graviton loops typically destabilise the structure of the graviton potential, they do so at a scale which is harmless and eﬀectively inconsequential for the theory. An analogous eﬀect is found in scalar theories which rely on derivative interactions, such as P (X) and galileon models [17]: in some limits these theories are also stable against quantum eﬀects within the regime of validity of their eﬀective ﬁeld theory [34–39]. As for the matter loops, the contributions to the graviton potential are identical to those in GR as long as matter couples covariantly to the dynamical metric. The resulting quantum corrections are thus nothing other than the standard cosmological constant which does not aﬀect the special structure of the graviton potential and is thus harmless (from what concerns the BD ghost1). This result is maintained at all loops and in this work we shall show that it trivially generalises to bi-gravity, as long as the matter ﬁeld only couples directly to one metric. Within the context of bi- and multi-gravity, more general couplings to matter have been explored in the literature, some of which can lead to an interesting new phenomenology [46– 50] see also [51] recent work directly in the vielbein formalism. In bi-gravity, both metrics are put on an equal footing, and coupling matter to both metrics simultaneously might thus appear natural at ﬁrst sight. In this manuscript we shall see that the same matter ﬁeld cannot have two kinetic terms (one with respect to each metric) without reintroducing the BD ghost. We show that using three diﬀerent methods, two classical and one quantum. Whilst these results were derived explicitly for bi-gravity it is straight-forward to show that they hold for any multi-gravity theory where the matter ﬁeld couples covariantly to more than one metric. To prove these results we show in the decoupling limit and in the mini-superspace ap- proximation that if a matter ﬁeld couples covariantly to both metrics in bi-gravity, the BD ghost re-emerges at an unacceptable low-scale, ruling out any predictions of the theory. We also show that the quantum corrections from the matter ﬁeld would destabilise the structure of the potential, and this also at an unacceptable low scale. As a result, while bi-gravity may enjoy a one-to-one correspondence between its two metrics, this correspondence is usually broken by the matter sector for a consistent theory. As we present in this manuscript, there may be exceptional cases where the matter sector couples to an eﬀective metric for which the one-to-one correspondence between the two metrics is maintained. However even in the exceptional case presented here, a ghost is expected to appear below the Planck scale. It is then unclear whether the new physics that ought to enter at or below the mass of the would- be-ghost can maintain the one-to-one correspondence between the two metrics. This would be interesting to understand more carefully. 1Massive gravity does not prevent the vacuum energy from acquiring large quantum corrections. If massive gravity was ever to tackle the old cosmological constant problem it would more likely be via the idea of degravitation [40–44] where the vacuum energy or cosmological constant could be large but have a small eﬀect on the geometry. See also Ref. [45] for recent developments. – 3 – ##LTLine##

text: special

text: structure

text: of

text: the

text: graviton

text: potential

text: were

text: explored

text: in

text: Ref.

text: While

text: graviton

text: loops

text: typically

text: destabilise

text: the

text: structure

text: of

text: the

text: graviton

text: potential,

text: they

text: do

text: so

text: at

text: a

text: scale

text: which

text: is

text: harmless

text: and

text: eﬀectively

text: inconsequential

text: for

text: the

text: theory.

text: An

text: analogous

text: eﬀect

text: is

text: found

text: in

text: scalar

text: theories

text: which

text: rely

text: on

text: derivative

text: interactions,

text: such

text: as

text: P

text: (X)

text: and

text: galileon

text: models

text: in

text: some

text: limits

text: these

text: theories

text: are

text: also

text: stable

text: against

text: quantum

text: eﬀects

text: within

text: the

text: regime

text: of

text: validity

text: of

text: their

text: eﬀective

text: ﬁeld

text: theory

text: As

text: for

text: the

text: matter

text: loops,

text: the

text: contributions

text: to

text: the

text: graviton

text: potential

text: are

text: identical

text: to

text: those

text: in

text: GR

text: as

text: long

text: as

text: matter

text: couples

text: covariantly

text: to

text: the

text: dynamical

text: metric.

text: The

text: resulting

text: quantum

text: corrections

text: are

text: thus

text: nothing

text: other

text: than

text: the

text: standard

text: cosmological

text: constant

text: which

text: does

text: not

text: aﬀect

text: the

text: special

text: structure

text: of

text: the

text: graviton

text: potential

text: and

text: is

text: thus

text: harmless

text: (from

text: what

text: concerns

text: the

text: BD

text: This

text: result

text: is

text: maintained

text: at

text: all

text: loops

text: and

text: in

text: this

text: work

text: we

text: shall

text: show

text: that

text: it

text: trivially

text: generalises

text: to

text: bi-gravity,

text: as

text: long

text: as

text: the

text: matter

text: ﬁeld

text: only

text: couples

text: directly

text: to

text: one

text: metric.

text: Within

text: the

text: context

text: of

text: bi-

text: and

text: multi-gravity,

text: more

text: general

text: couplings

text: to

text: matter

text: have

text: been

text: explored

text: in

text: the

text: literature,

text: some

text: of

text: which

text: can

text: lead

text: to

text: an

text: interesting

text: new

text: phenomenology

text: see

text: also

text: recent

text: work

text: directly

text: in

text: the

text: vielbein

text: formalism.

text: In

text: bi-gravity,

text: both

text: metrics

text: are

text: put

text: on

text: an

text: equal

text: footing,

text: and

text: coupling

text: matter

text: to

text: both

text: metrics

text: simultaneously

text: might

text: thus

text: appear

text: natural

text: at

text: ﬁrst

text: sight.

text: In

text: this

text: manuscript

text: we

text: shall

text: see

text: that

text: the

text: same

text: matter

text: ﬁeld

text: cannot

text: have

text: two

text: kinetic

text: terms

text: (one

text: with

text: respect

text: to

text: each

text: metric)

text: without

text: reintroducing

text: the

text: BD

text: ghost.

text: We

text: show

text: that

text: using

text: three

text: diﬀerent

text: methods,

text: two

text: classical

text: and

text: one

text: quantum.

text: Whilst

text: these

text: results

text: were

text: derived

text: explicitly

text: for

text: bi-gravity

text: it

text: is

text: straight-forward

text: to

text: show

text: that

text: they

text: hold

text: for

text: any

text: multi-gravity

text: theory

text: where

text: the

text: matter

text: ﬁeld

text: couples

text: covariantly

text: to

text: more

text: than

text: one

text: metric.

text: To

text: prove

text: these

text: results

text: we

text: show

text: in

text: the

text: decoupling

text: limit

text: and

text: in

text: the

text: mini-superspace

text: ap-

text: proximation

text: that

text: if

text: a

text: matter

text: ﬁeld

text: couples

text: covariantly

text: to

text: both

text: metrics

text: in

text: bi-gravity,

text: the

text: BD

text: ghost

text: re-emerges

text: at

text: an

text: unacceptable

text: low-scale,

text: ruling

text: out

text: any

text: predictions

text: of

text: the

text: theory.

text: We

text: also

text: show

text: that

text: the

text: quantum

text: corrections

text: from

text: the

text: matter

text: ﬁeld

text: would

text: destabilise

text: the

text: structure

text: of

text: the

text: potential,

text: and

text: this

text: also

text: at

text: an

text: unacceptable

text: low

text: scale.

text: As

text: a

text: result,

text: while

text: bi-gravity

text: may

text: enjoy

text: a

text: one-to-one

text: correspondence

text: between

text: its

text: two

text: metrics,

text: this

text: correspondence

text: is

text: usually

text: broken

text: by

text: the

text: matter

text: sector

text: for

text: a

text: consistent

text: theory.

text: As

text: we

text: present

text: in

text: this

text: manuscript,

text: there

text: may

text: be

text: exceptional

text: cases

text: where

text: the

text: matter

text: sector

text: couples

text: to

text: an

text: eﬀective

text: metric

text: for

text: which

text: the

text: one-to-one

text: correspondence

text: between

text: the

text: two

text: metrics

text: is

text: maintained.

text: However

text: even

text: in

text: the

text: exceptional

text: case

text: presented

text: here,

text: a

text: ghost

text: is

text: expected

text: to

text: appear

text: below

text: the

text: Planck

text: scale.

text: It

text: is

text: then

text: unclear

text: whether

text: the

text: new

text: physics

text: that

text: ought

text: to

text: enter

text: at

text: or

text: below

text: the

text: mass

text: of

text: the

text: would-

text: be-ghost

text: can

text: maintain

text: the

text: one-to-one

text: correspondence

text: between

text: the

text: two

text: metrics.

text: This

text: would

text: be

text: interesting

text: to

text: understand

text: more

text: carefully.

text: gravity

text: does

text: not

text: prevent

text: the

text: vacuum

text: energy

text: from

text: acquiring

text: large

text: quantum

text: corrections.

text: If

text: massive

text: gravity

text: was

text: ever

text: to

text: tackle

text: the

text: old

text: cosmological

text: constant

text: problem

text: it

text: would

text: more

text: likely

text: be

text: via

text: the

text: idea

text: of

text: degravitation

text: where

text: the

text: vacuum

text: energy

text: or

text: cosmological

text: constant

text: could

text: be

text: large

text: but

text: have

text: a

text: small

text: eﬀect

text: on

text: the

text: geometry.

text: See

text: also

text: Ref.

text: for

text: recent

text: developments.

text: ##LTLine##


text: [doi:10.1103/PhysRevD.89.027502]. [74] M. S. Volkov, Exact self-accelerating cosmologies in the ghost-free bigravity and massive gravity, Phys.Rev. D86 (2012) 061502, [arXiv:1205.5713], [doi:10.1103/PhysRevD.86.061502]. [75] D. Comelli, F. Nesti, and L. Pilo, Cosmology in general massive gravity theories, JCAP 1405 (2014) 036, [arXiv:1307.8329], [doi:10.1088/1475-7516/2014/05/036]. [76] L. Heisenberg, R. Kimura, and K. Yamamoto, Cosmology of the proxy theory to massive gravity, Phys.Rev. D89 (2014) 103008, [arXiv:1403.2049], [doi:10.1103/PhysRevD.89.103008]. [77] G. D’Amico, G. Gabadadze, L. Hui, and D. Pirtskhalava, Quasidilaton: Theory and cosmology, Phys.Rev. D87 (2013), no. 6 064037, [arXiv:1206.4253], [doi:10.1103/PhysRevD.87.064037]. [78] G. D’Amico, G. Gabadadze, L. Hui, and D. Pirtskhalava, On Cosmological Perturbations of Quasidilaton, Class.Quant.Grav. 30 (2013) 184005, [arXiv:1304.0723], [doi:10.1088/0264-9381/30/18/184005]. [79] G. Gabadadze, R. Kimura, and D. Pirtskhalava, Selfacceleration with quasidilaton, Phys.Rev. D90 (2014) 024029, [arXiv:1401.5403], [doi:10.1103/PhysRevD.90.024029]. [80] A. De Felice and S. Mukohyama, Towards consistent extension of quasidilaton massive gravity, Phys.Lett. B728 (2014) 622–625, [arXiv:1306.5502], [doi:10.1016/j.physletb.2013.12.041]. [81] A. De Felice, A. Emir Gu¨mru¨kc¸u¨o˘glu, and S. Mukohyama, Generalized quasidilaton theory, Phys.Rev. D88 (2013), no. 12 124006, [arXiv:1309.3162], [doi:10.1103/PhysRevD.88.124006]. [82] Z. Haghani, H. R. Sepangi, and S. Shahidi, Curvature perturbations of quasidilaton non-linear massive gravity, Phys.Rev. D87 (2013) 124014, [arXiv:1303.2843], [doi:10.1103/PhysRevD.87.124014]. [83] M. Fasiello and A. J. Tolley, Cosmological stability bound in massive gravity and bigravity, JCAP 1312 (2013) 002, [arXiv:1308.1647], [doi:10.1088/1475-7516/2013/12/002]. [84] D. Comelli, M. Crisostomi, and L. Pilo, FRW cosmological perturbations in massive bigravity, arXiv:1403.5679. [85] F. Koennig, A. Patil, and L. Amendola, Viable cosmological solutions in massive bimetric gravity, JCAP 1403 (2014) 029, [arXiv:1312.3208], [doi:10.1088/1475-7516/2014/03/029]. [86] A. De Felice, T. Nakamura, and T. Tanaka, Possible existence of viable models of bi-gravity with detectable graviton oscillations by gravitational wave detectors, PTEP 2014 (2014), no. 4 043E01, [arXiv:1304.3920], [doi:10.1093/ptep/ptu024]. [87] A. De Felice, A. E. Gu¨mru¨kc¸u¨o˘glu, S. Mukohyama, N. Tanahashi, and T. Tanaka, Viable cosmology in bimetric theory, JCAP 1406 (2014) 037, [arXiv:1404.0008], [doi:10.1088/1475-7516/2014/06/037]. [88] A. R. Solomon, Y. Akrami, and T. S. Koivisto, Cosmological perturbations in massive bigravity: I. Linear growth of structures, arXiv:1404.4061. [89] A. Schmidt-May, Consistent bimetric theory and its application to cosmology, arXiv:1405.1208. – 34 –

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