text: 26 ALAN CAREY, HARALD GROSSE, AND JENS KAAD 8. Second approximation of the homological index In this Section we will recollect what we have obtained thus far. Let us therefore → recall that A : H H is a pseudodiﬀerential operator of order 0 with respect 2 →2 R → R to ∆ := D2 : D(∆ ) H and that h : is a smooth bounded function 2 2 R → R2 2 such that dh : has compact support. Recall furthermore that A satisﬁes the dt relative Schatten class conditions described in Assumption 1.1 with respect to ∆ ∈ ∞ → R 2 and a ﬁxed number p (0, ). We let D : D(D ) L2( ) denote the closure 0 ∞ R → 1 R 1 of the unbounded operator i d : C ( ) L2( ). As usual we will also apply the dt c notation D and D for the closures of 1 2 ⊗ ⊗ → R ⊗ R ⊗ → R D 1 : D(D ) H L2( , H ) and 1 D : L2( ) D(D ) L2( , H ) 1 1 2 2 2 2 2 ∈ N Let m with m > p /2 be given. The main theorem of this Section expresses 0 the homological index in degree m of the closed unbounded operator − ∩ → R D := iD + D + hA : D(D ) D(D ) L2( , H ) + 1 2 1 2 2 as a power-series expansion. Each term in this expansion is a product where the ﬁrst term only contains a multiplication operator in the ﬂow parameter and the second term only contains operators on the ‘inner’ Hilbert space H and diﬀerentiation 2 with respect to the ﬂow parameter (there are no multiplication operators in the ﬂow parameter in the second term). This will make it possible to compute the trace in the ﬂow direction which will take the form of an integral over the real line. This computation will be carried out in the next Section. ∈ N Theorem 8.1. Let m with(cid:88)m > p0/(cid:0)2 be given. Then there exists a(cid:12) con(cid:1)stant (cid:12) C > 0 such that ∞ · · 1 · · · dl − − H-Indm(D ) = m λm lim Tr F (ε) hl (∆/ + β + λ) m 1 λ + ε(cid:38)0 l! ε dz z=0 l=0 ≥ ∈ for all λ C and where the sum converges absolutely for all ε (0, 1]. (cid:0) (cid:1) Proof. By Theorem 5.1 there exists a constant C > 0 such that 0 · · · − − H-Indm(D ) = m λm lim Tr F (ε) (∆/ + α (ε) + λ) m 1 λ + (cid:38) 0 ε 0 ≥ (cid:0) (cid:1) for all λ C . 0 By Proposition 6.1 there exists a constant C > 0 such that · · (cid:88)· (cid:0) − −1 (cid:12) (cid:1) m λm Tr F (ε) (∆/ + α (ε) + λ) m 1 (cid:12) 0 ∞ · · 1 · · dl · − − = m λm Tr F (ε) (∆/ + α (ε, ) + λ) m 1 l! dz 0 z=0 l=0 ∈ ≥ where the sum converges absolutely for all ε (0, 1] and all λ C . 1 B(cid:13)y Proposition 7.7(cid:0)there exists a constant C > 0 such that (cid:1)(cid:12) (cid:13) (cid:13) 2 (cid:12) (cid:13) · dl · − − − · − − (∆/ + 1)m+1 (∆/ + α (ε, ) + λ) m 1 hl (∆/ + β + λ) m 1 dz 0 ε z=0 (8.1) ≤ · · Θ(ε) l! (1/2)l ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ALAN

text: CAREY,

text: HARALD

text: GROSSE,

text: AND

text: JENS

text: KAAD

text: Second

text: approximation

text: of

text: the

text: homological

text: index

text: In

text: this

text: Section

text: we

text: will

text: recollect

text: what

text: we

text: have

text: obtained

text: thus

text: far.

text: Let

text: us

text: therefore

text: recall

text: that

text: A

text: H

text: H

text: is

text: a

text: pseudodiﬀerential

text: operator

text: of

text: order

text: with

text: respect

text: →2

text: R

text: R

text: to

text: D(∆

text: H

text: and

text: that

text: h

text: is

text: a

text: smooth

text: bounded

text: function

text: R

text: such

text: that

text: dh

text: has

text: compact

text: support.

text: Recall

text: furthermore

text: that

text: A

text: satisﬁes

text: the

text: dt

text: relative

text: Schatten

text: class

text: conditions

text: described

text: in

text: Assumption

text: with

text: respect

text: to

text: R

text: and

text: a

text: ﬁxed

text: number

text: p

text: We

text: let

text: D

text: D(D

text: denote

text: the

text: closure

text: R

text: R

text: of

text: the

text: unbounded

text: operator

text: i

text: d

text: C

text: As

text: usual

text: we

text: will

text: also

text: apply

text: the

text: dt

text: c

text: notation

text: D

text: and

text: D

text: for

text: the

text: closures

text: of

text: R

text: R

text: R

text: D

text: D(D

text: H

text: H

text: and

text: D

text: D(D

text: H

text: N

text: Let

text: m

text: with

text: m

text: p

text: be

text: given.

text: The

text: main

text: theorem

text: of

text: this

text: Section

text: expresses

text: the

text: homological

text: index

text: in

text: degree

text: m

text: of

text: the

text: closed

text: unbounded

text: operator

text: R

text: D

text: iD

text: D

text: hA

text: D(D

text: D(D

text: H

text: as

text: a

text: power-series

text: expansion.

text: Each

text: term

text: in

text: this

text: expansion

text: is

text: a

text: product

text: where

text: the

text: ﬁrst

text: term

text: only

text: contains

text: a

text: multiplication

text: operator

text: in

text: the

text: ﬂow

text: parameter

text: and

text: the

text: second

text: term

text: only

text: contains

text: operators

text: on

text: the

text: ‘inner’

text: Hilbert

text: space

text: H

text: and

text: diﬀerentiation

text: with

text: respect

text: to

text: the

text: ﬂow

text: parameter

text: (there

text: are

text: no

text: multiplication

text: operators

text: in

text: the

text: ﬂow

text: parameter

text: in

text: the

text: second

text: term).

text: This

text: will

text: make

text: it

text: possible

text: to

text: compute

text: the

text: trace

text: in

text: the

text: ﬂow

text: direction

text: which

text: will

text: take

text: the

text: form

text: of

text: an

text: integral

text: over

text: the

text: real

text: line.

text: This

text: computation

text: will

text: be

text: carried

text: out

text: in

text: the

text: next

text: Section.

text: N

text: Theorem

text: Let

text: m

text: be

text: given.

text: Then

text: there

text: exists

text: C

text: such

text: that

text: dl

text: H-Indm(D

text: m

text: λm

text: lim

text: Tr

text: F

text: (ε)

text: hl

text: β

text: λ)

text: m

text: λ

text: l!

text: ε

text: dz

text: for

text: all

text: λ

text: C

text: and

text: where

text: the

text: sum

text: converges

text: absolutely

text: for

text: all

text: ε

text: Proof.

text: By

text: Theorem

text: there

text: exists

text: a

text: constant

text: C

text: such

text: that

text: H-Indm(D

text: m

text: λm

text: lim

text: Tr

text: F

text: (ε)

text: α

text: (ε)

text: λ)

text: m

text: λ

text: ε

text: for

text: all

text: λ

text: C

text: By

text: Proposition

text: there

text: exists

text: a

text: constant

text: C

text: such

text: that

text: −1

text: m

text: λm

text: Tr

text: F

text: (ε)

text: α

text: (ε)

text: λ)

text: m

text: dl

text: m

text: λm

text: Tr

text: F

text: (ε)

text: α

text: (ε,

text: λ)

text: m

text: l!

text: dz

text: where

text: the

text: sum

text: converges

text: absolutely

text: for

text: all

text: ε

text: and

text: all

text: λ

text: C

text: Proposition

text: exists

text: a

text: constant

text: C

text: such

text: that

text: dl

text: α

text: (ε,

text: λ)

text: m

text: hl

text: β

text: λ)

text: m

text: dz

text: ε

text: Θ(ε)

text: l!

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 28 ALAN CAREY, HARALD GROSSE, AND JENS KAAD ∈ N Theorem 9.1. Let m wi(cid:88)th m > p0/2 be(cid:16)give(cid:16)n. Th(cid:0)ere exists a constan(cid:1)t C > 0 such that ∞ · · 1 · dl · (cid:17)(cid:12) (cid:17) · − − H-Indm(D ) = λm C (cid:0) Tr (cid:1)A λ +(cid:12) (D + r A )2 m 1/2 λ + m+1/2 (l + 1)! dr + (cid:12) 2 + l=0 − · · − − (cid:82) A− λ + (D + r A−)2 m 1/2 2 r=0 ≥ where the sum converges absolutely for all λ C and the constant is given by · ∞ − − C := m −∞(1 + η2) m 1 dη. m+1/2 π The starting point for proving Theorem 9.1 is the main Theorem 8.1 which we proved in Section . This (cid:16)latter theorem provides a power se(cid:17)ries expansion of the (cid:12) (cid:12) homological index where each entry in the sum has the form 1 · · · dl − − Tr F (ε) hl (∆/ + β + λ) m 1 l! ε dr r=0 ∈ N ∈ for l and ε (0, 1]. Note that we have replaced the complex parameter 0 z by a real parameter r. It is therefore now our main interest to compute the operator trace appearing in the above expression. This computation will rely on the factorization of the operator trace associated to a tensor product of Hilbert spaces. This factorization result is described in Appendix . For the convenience of the reader we recall that · dh · R → R ∈ F (ε) = 2 ε A : L2( , H ) L2( , H ) for all ε (0, 1] 2 2 dt and that β(r) is deﬁned as the closure of · · R ∞ → R r (D A + AD ) + r2 A2 : L2( , H ) L2( , H ) 2 2 2 2 ∈ R for all r . We remark that β(r) is a selfadjoint unbounded operator and further- more that → R β(r) + ∆/ = ∆ + (D + rA)2 : D(∆/ ) L2( , H ) 1 2 2 ∈ R → is a positive unbounded operator for all r . Recall here that ∆ : D(∆ ) R 1 1 L2( , H ) is deﬁned as the closure of the Laplacian 2 −d2 ⊗ ∞ R ⊗ → R 1 : C ( ) H L2( , H ) dt c 2 2 Let us also introduce the unbounded selfadjoint operators β (r) and β−(r) deﬁned + as the closures of · ∞ → r (D A + A D ) + r2A2 : H H and 2 + + 2 + 2 2 · ∞ → r (D A− + A−D ) + r2A2− : H H 2 2 2 2 ∈ R respectively, for all r , where we recall that A = h A and A− = h−A. ∞ ⊆ + + R ∞ ⊆ R Notice that the dense subspaces H H and L2( , H ) L2( , H ) are 2 2 2 2 deﬁned by H∞ := ∩∞ D(∆k/2) and L2(R, H )∞ := ∩∞ D(∆/ k/2) 2 k=0 2 2 k=0 respectively. ∈ N We will in this Section only consider a ﬁxed m with m > p /2. 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: ALAN

text: CAREY,

text: HARALD

text: GROSSE,

text: AND

text: JENS

text: KAAD

text: N

text: Theorem

text: Let

text: m

text: m

text: exists

text: a

text: C

text: such

text: that

text: dl

text: H-Indm(D

text: λm

text: C

text: Tr

text: λ

text: (D

text: r

text: A

text: m

text: λ

text: (l

text: dr

text: A−

text: λ

text: (D

text: r

text: m

text: where

text: the

text: sum

text: converges

text: absolutely

text: for

text: all

text: λ

text: C

text: and

text: the

text: constant

text: is

text: given

text: by

text: C

text: m

text: −∞(1

text: m

text: dη.

text: π

text: The

text: starting

text: point

text: for

text: proving

text: Theorem

text: is

text: the

text: main

text: Theorem

text: which

text: we

text: proved

text: in

text: Section

text: This

text: theorem

text: provides

text: a

text: power

text: expansion

text: of

text: the

text: homological

text: index

text: where

text: each

text: entry

text: in

text: the

text: sum

text: has

text: the

text: form

text: dl

text: Tr

text: F

text: (ε)

text: hl

text: β

text: λ)

text: m

text: l!

text: ε

text: dr

text: N

text: for

text: l

text: and

text: ε

text: Note

text: that

text: we

text: have

text: replaced

text: the

text: complex

text: parameter

text: z

text: by

text: a

text: real

text: parameter

text: r.

text: It

text: is

text: therefore

text: now

text: our

text: main

text: interest

text: to

text: compute

text: the

text: operator

text: trace

text: appearing

text: in

text: the

text: above

text: expression.

text: This

text: computation

text: will

text: rely

text: on

text: the

text: factorization

text: of

text: the

text: operator

text: trace

text: associated

text: to

text: a

text: tensor

text: product

text: of

text: Hilbert

text: spaces.

text: This

text: factorization

text: result

text: is

text: described

text: in

text: Appendix

text: For

text: the

text: convenience

text: of

text: the

text: reader

text: we

text: recall

text: that

text: dh

text: R

text: R

text: F

text: (ε)

text: ε

text: A

text: H

text: H

text: for

text: all

text: ε

text: dt

text: and

text: that

text: β(r)

text: is

text: deﬁned

text: as

text: the

text: closure

text: of

text: R

text: R

text: r

text: (D

text: A

text: AD

text: H

text: H

text: R

text: for

text: all

text: r

text: We

text: remark

text: that

text: β(r)

text: is

text: a

text: selfadjoint

text: unbounded

text: operator

text: and

text: further-

text: more

text: that

text: R

text: β(r)

text: (D

text: D(∆/

text: H

text: R

text: is

text: a

text: positive

text: unbounded

text: operator

text: for

text: all

text: r

text: Recall

text: here

text: that

text: D(∆

text: R

text: H

text: is

text: deﬁned

text: as

text: the

text: closure

text: of

text: the

text: Laplacian

text: R

text: R

text: C

text: H

text: H

text: dt

text: c

text: Let

text: us

text: also

text: introduce

text: the

text: unbounded

text: selfadjoint

text: operators

text: β

text: (r)

text: and

text: β−(r)

text: deﬁned

text: as

text: the

text: closures

text: of

text: r

text: (D

text: A

text: A

text: D

text: H

text: H

text: and

text: r

text: (D

text: A−

text: A−D

text: H

text: H

text: R

text: respectively,

text: for

text: all

text: r

text: where

text: we

text: recall

text: that

text: A

text: h

text: A

text: and

text: A−

text: h−A.

text: R

text: R

text: Notice

text: that

text: the

text: dense

text: subspaces

text: H

text: H

text: and

text: H

text: H

text: are

text: deﬁned

text: by

text: H∞

text: and

text: H

text: D(∆/

text: respectively.

text: N

text: We

text: will

text: in

text: this

text: Section

text: only

text: consider

text: a

text: ﬁxed

text: m

text: with

text: m

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON A SPECTRAL FLOW FORMULA FOR THE HOMOLOGICAL INDEX 29 (cid:90) ∈ N ∈ R ∈ (cid:0) (cid:1) Lemma 9.1. Let l , r , λ > 0, and ε (0, 1] be given. Then 0 ∞ · · − − 1 · · · · − F (ε) hl (∆/ + β(r) + λ) m 1 = F (ε) hl sm exp s(∆/ + β(r) + λ) ds ε m! ε 0 where the integral converges absolutely in trace norm. − − Proof. We start by rewriting the pow(cid:90)er of the res(cid:0)olvent (∆/ + β(r) + λ(cid:1)) m 1 using the Laplace transform, thus ∞ − − 1 · · − (∆/ + β(r) + λ) m 1 = sm exp s(∆/ + β(r) + λ) ds m! 0 Notice now that the operator · · − − R → R F (ε) hl (∆/ + β(r) + λ) m 1+δ : L2( , H ) L2( , H ) ε 2 2 ∈ (cid:90) is of trace class for all δ (0, 1/2), see Proposition 2.2 and the results of Appendix (cid:0) (cid:1) (recall here that m > p /2). It therefore suﬃces to show that the integral 0 ∞ · − · − sm (∆/ + β(r) + λ)m+1 δ exp s(∆/ + β(r) + λ) ds 0 ∈ converges absolutely in operator norm for some δ (0, 1/2). But this follows by an (cid:98) (cid:3) application of the spectral theorem for selfadjoint unbounded operators. (cid:98) R R ⊗ Recall that L2( , H ) may be regarded as L2( ) H . We may thus apply the ⊗2 R ⊗ → 2 bounded operator Tr 1 : L 1(L2( ) H ) L 1(H ) to any trace class operator R → R 2 2 T : L2( , H ) L2( , H ), see Appendix . We are particularly interested in the 2 2 trace class operator which we analyzed in Lemma 9.1. (cid:0) (cid:1) ∈ N ∈ R ∈ Lemma 9.2. Let l , r , λ > 0, and ε (0, 1] be given. Then 0 (cid:90) ⊗ · · − − (Tr 1) F (ε) hl (∆/ + β(r) + λ) m 1 ε − · ∞ (hl+1 hl−+1) A · − − = + · (λ + (D + rA)2 + ξ2) m 1 dξ 2 π (l + 1) −∞ (cid:0) (cid:1) Proof. By Lemma 9(cid:90).1 and Proposition C.2 we have that (cid:16) (cid:17) ⊗ · · − − (cid:0) (cid:1) (Tr 1) F (ε) hl (∆/ + β(r) + λ) m 1 ε ∞ 1 · · ⊗ · · − = sm (Tr 1) F (ε) hl exp s(∆/ + β(r) + λ) ds m! ε 0 ∈ ∞ (cid:0) (cid:1) Let now s (0, ) be given. Then (cid:16) (cid:17) (cid:0) (cid:1) ⊗ · · − (Tr 1) F (ε) hl exp( s(∆/ + β(r) + λ)) ε · − · dh · · − (9.1) = 2A exp s(λ + (D + rA)2) Tr ε hl exp( s∆ ) 2 dt ε 1 (cid:0) (cid:1) (cid:0) (cid:1) where we have applied the identity − − · − exp s(∆/ + β(r) + λ) = exp s(λ + (D + rA)2) exp( s∆ ) 2 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: A

text: SPECTRAL

text: FLOW

text: FORMULA

text: FOR

text: THE

text: HOMOLOGICAL

text: INDEX

text: N

text: R

text: Lemma

text: Let

text: l

text: r

text: λ

text: and

text: ε

text: be

text: given.

text: Then

text: F

text: (ε)

text: hl

text: β(r)

text: λ)

text: m

text: F

text: (ε)

text: hl

text: sm

text: exp

text: s(∆/

text: β(r)

text: λ)

text: ds

text: ε

text: m!

text: ε

text: where

text: the

text: integral

text: converges

text: absolutely

text: in

text: trace

text: norm.

text: Proof.

text: We

text: start

text: by

text: rewriting

text: the

text: of

text: the

text: β(r)

text: m

text: using

text: the

text: Laplace

text: transform,

text: thus

text: β(r)

text: λ)

text: m

text: sm

text: exp

text: s(∆/

text: β(r)

text: λ)

text: ds

text: m!

text: Notice

text: now

text: that

text: the

text: operator

text: R

text: R

text: F

text: (ε)

text: hl

text: β(r)

text: λ)

text: m

text: H

text: H

text: ε

text: is

text: of

text: trace

text: class

text: for

text: all

text: δ

text: see

text: Proposition

text: and

text: the

text: results

text: of

text: Appendix

text: (recall

text: here

text: that

text: m

text: p

text: It

text: therefore

text: suﬃces

text: to

text: show

text: that

text: the

text: integral

text: sm

text: β(r)

text: δ

text: exp

text: s(∆/

text: β(r)

text: λ)

text: ds

text: converges

text: absolutely

text: in

text: operator

text: norm

text: for

text: some

text: δ

text: But

text: this

text: follows

text: by

text: an

text: application

text: of

text: the

text: spectral

text: theorem

text: for

text: selfadjoint

text: unbounded

text: operators.

text: R

text: R

text: Recall

text: that

text: H

text: may

text: be

text: regarded

text: as

text: H

text: We

text: may

text: thus

text: apply

text: the

text: ⊗2

text: R

text: bounded

text: operator

text: Tr

text: L

text: H

text: L

text: to

text: any

text: trace

text: class

text: operator

text: R

text: R

text: T

text: H

text: H

text: see

text: Appendix

text: We

text: are

text: particularly

text: interested

text: in

text: the

text: trace

text: class

text: operator

text: which

text: we

text: analyzed

text: in

text: Lemma

text: N

text: R

text: Lemma

text: Let

text: l

text: r

text: λ

text: and

text: ε

text: be

text: given.

text: Then

text: (Tr

text: F

text: (ε)

text: hl

text: β(r)

text: λ)

text: m

text: ε

text: A

text: (λ

text: (D

text: m

text: dξ

text: π

text: (l

text: Proof.

text: By

text: Lemma

text: and

text: Proposition

text: we

text: have

text: that

text: (Tr

text: F

text: (ε)

text: hl

text: β(r)

text: λ)

text: m

text: ε

text: sm

text: (Tr

text: F

text: (ε)

text: hl

text: exp

text: s(∆/

text: β(r)

text: λ)

text: ds

text: m!

text: ε

text: Let

text: now

text: s

text: be

text: given.

text: Then

text: (Tr

text: F

text: (ε)

text: hl

text: exp(

text: s(∆/

text: β(r)

text: λ))

text: ε

text: dh

text: exp

text: s(λ

text: (D

text: Tr

text: ε

text: hl

text: exp(

text: s∆

text: dt

text: ε

text: where

text: we

text: have

text: applied

text: the

text: identity

text: exp

text: s(∆/

text: β(r)

text: λ)

text: exp

text: s(λ

text: (D

text: exp(

text: s∆

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON A SPECTRAL FLOW FORMULA FOR THE HOMOLOGICAL INDEX 31 (cid:16) (cid:90) (cid:17)(cid:12) (cid:0) (cid:1) (cid:12) To prove the proposition it therefore suﬃces to verify(cid:12)that ∞ · dl (cid:90) − − (cid:12) hl± λ + (D + rA)2 +(cid:0)ξ2 m 1 dξ (cid:1) (cid:12) dr −∞ 2 (cid:12) r=0 ∞ − − · dl − − = (1 + η2) m 1 dη λ + (D + rA±)2 m 1/2 2 −∞ dr r=0 (cid:3) But this is the content of the next lemma. (cid:16) (cid:90) (cid:17)(cid:12) ∈ N(cid:0) ∈ R (cid:1) (cid:12) Lemma 9.4. Let l , λ > 0 and µ be given. Th(cid:12)en we have the identity 0 ∞ · dl (cid:90) − − (cid:12) µl λ + (D + rA)2 + ξ(cid:0)2 m 1 dξ (cid:1) (cid:12) dr −∞ 2 (cid:12) r=0 ∞ − − · dl − − = (1 + η2) m 1 dη λ + (D + µrA)2 m 1/2 2 −∞ dr r=0 (cid:90) (cid:90) (cid:0) (cid:1) (cid:0) (cid:1) Proof. By a change of variables we obtain that ∞ ∞ − − − − · − − λ + (D + rA)2 + ξ2 m 1 dξ = (1 + η2) m 1 dη λ + (D + rA)2 m 1/2 2 2 −∞ −∞ Furthermore, by Lemma A.4 and Proposition B.3 we conclude that the map R → (cid:55)→ − − f : L (H ) f : r (λ + (D + rA)2) m 1/2 2 2 is smooth in operator norm. It therefore follows by an application of the chain rule that dlf dlf µl = µ dr dr R → · ∈ R where the smooth map f : L (H) is deﬁned by f (r) := f(µ r) for all r . µ µ (cid:3) This proves the lemma. The main Theorem 9.1 of this Section is now clearly a consequence of Proposition 9.3, Proposition C.2 and Theorem 8.1. 10. The integral trace formula for the homological index We are now ready to prove the main theorem of this paper. We need to start by proving a version of the main theorem where we assume that the scaling parameter λ > 0 is large. It is necessary to apply this assumption to begin with since our resolvent expansions only exist for large λ > 0. We will then be able to prove the general version of our theorem by noticing that both sides of the integral trace C \ −∞ formula have analytic extensions to the open subset ( , 0] of the complex plane. ∈ N (cid:90) Proposition 10.1. Let m with m > (cid:16)p /2 be given. There exists a constant 0 C > 0 such that · · 1 · − (cid:17)− H-Indm(D ) = λm C Tr A (λ + (D + r A )2) m 1/2 λ + m+1/2 + 2 + 0 − · − − A−(λ + (D + r A−)2) m 1/2 dr 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: A

text: SPECTRAL

text: FLOW

text: FORMULA

text: FOR

text: THE

text: HOMOLOGICAL

text: INDEX

text: To

text: prove

text: the

text: proposition

text: it

text: therefore

text: suﬃces

text: to

text: dl

text: hl±

text: λ

text: (D

text: m

text: dξ

text: dr

text: dl

text: m

text: dη

text: λ

text: (D

text: m

text: dr

text: But

text: this

text: is

text: the

text: content

text: of

text: the

text: next

text: lemma.

text: R

text: Lemma

text: Let

text: l

text: λ

text: and

text: µ

text: be

text: given.

text: we

text: have

text: the

text: identity

text: dl

text: µl

text: λ

text: (D

text: m

text: dξ

text: dr

text: dl

text: m

text: dη

text: λ

text: (D

text: m

text: dr

text: Proof.

text: By

text: a

text: change

text: of

text: variables

text: we

text: obtain

text: that

text: λ

text: (D

text: ξ2

text: m

text: dξ

text: m

text: dη

text: λ

text: (D

text: m

text: Furthermore,

text: by

text: Lemma

text: and

text: Proposition

text: we

text: conclude

text: that

text: the

text: map

text: R

text: f

text: L

text: (H

text: f

text: r

text: (λ

text: (D

text: m

text: is

text: smooth

text: in

text: operator

text: norm.

text: It

text: therefore

text: follows

text: by

text: an

text: application

text: of

text: the

text: chain

text: rule

text: that

text: dlf

text: dlf

text: µl

text: µ

text: dr

text: dr

text: R

text: R

text: where

text: the

text: smooth

text: map

text: f

text: L

text: (H)

text: is

text: deﬁned

text: by

text: f

text: (r)

text: f(µ

text: r)

text: for

text: all

text: r

text: µ

text: µ

text: This

text: proves

text: the

text: lemma.

text: The

text: main

text: Theorem

text: of

text: this

text: Section

text: is

text: now

text: clearly

text: a

text: consequence

text: of

text: Proposition

text: Proposition

text: and

text: Theorem

text: The

text: integral

text: trace

text: formula

text: for

text: the

text: homological

text: index

text: We

text: are

text: now

text: ready

text: to

text: prove

text: the

text: main

text: theorem

text: of

text: this

text: paper.

text: We

text: need

text: to

text: start

text: by

text: proving

text: a

text: version

text: of

text: the

text: main

text: theorem

text: where

text: we

text: assume

text: that

text: the

text: scaling

text: parameter

text: λ

text: is

text: large.

text: It

text: is

text: necessary

text: to

text: apply

text: this

text: assumption

text: to

text: begin

text: with

text: since

text: our

text: resolvent

text: expansions

text: only

text: exist

text: for

text: large

text: λ

text: We

text: will

text: then

text: be

text: able

text: to

text: prove

text: the

text: general

text: version

text: of

text: our

text: theorem

text: by

text: noticing

text: that

text: both

text: sides

text: of

text: the

text: integral

text: trace

text: C

text: formula

text: have

text: analytic

text: extensions

text: to

text: the

text: open

text: subset

text: of

text: the

text: complex

text: plane.

text: N

text: Proposition

text: Let

text: m

text: with

text: m

text: be

text: given.

text: There

text: exists

text: a

text: constant

text: C

text: such

text: that

text: H-Indm(D

text: λm

text: C

text: Tr

text: A

text: (λ

text: (D

text: r

text: A

text: m

text: λ

text: A−(λ

text: (D

text: r

text: m

text: dr

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON A SPECTRAL FLOW FORMULA FOR THE HOMOLOGICAL INDEX 35 × → Lemma A.4. Let α : U X OP1(∆) be holomorphic in supremum norm. Suppose → that ∆ + α(z, x) : D(∆) H is invertible as a closed unbounded operator for each ∈ × (z, x) U X. Then the map − × → − (cid:55)→ − (∆ + α) 1 : U X OP 2(∆) (z, x) (∆ + α(z, x)) 1 is holomorphic in supremum norm. ⊆ ∈ × Proof. Remark ﬁrst of all that D(∆1/2) D(α(z, x)) for all (z, x) U X. To ∈ { } ∞ → see this, let ξ D(∆1/2) and choose a sequence ξ in H with ξ ξ and → → n n ∆1/2ξ ∆1/2ξ. Then α(z, x)ξ α(z, x)(ξ). Indeed, this follows since n n (cid:0) (cid:1) − α(z, x)(ξ ) = (1 + ∆1/2) 1(1 + ∆1/2)α(z, x)(ξ ) n n − · = (1 + ∆1/2) 1 α(z, x) + δ(α(z, x)) + α(z, x)∆1/2 (ξ ) n → The unbounded operator ∆ + α(z, x) : D(∆) H is thus well-deﬁned for all ∈ × → (z, x) U X. The fact that ∆ + α(z, x) : D(∆) H is closed is a consequence of the Kato-Rellich theorem, see [Kat95, Chapther IV, Theorem 1.1]. It is now straightforward to verify that the map × → (cid:55)→ ∆ + α : U X OP2(∆) (z, x) ∆ + α(z, x) is holomorphic in supremum norm. ∈ × − ∈ − Let (z, x) U X and let us continue by showing that (∆+α(z, x)) 1 OP 2(∆). To this end, it is clearly suﬃcient to prove that − ∈ (∆ + 1)(∆ + α(z, x)) 1 OP0(∆). We therefore notice that the linear map − → β(z, x) := (∆ + α(z, x))(∆ + 1) 1 : H H ∈ is bijective. Furthermore, we have that β(z, x) OP0(∆). Since OP0(∆) is closed under holomorphic function(cid:0)al calculus (as a unital(cid:1)subalgebra of L (H)), see [CGK13, Lemma 2.3], we may conclude that − − − ∈ (∆ + 1)(∆ + α(z, x)) 1 = (∆ + α(z, x))(∆ + 1) 1 1 OP0(∆). To see that × → − (cid:55)→ − U X OP 2(∆) (z, x) (∆ + α(z, x)) 1 is holomorphic in supremum norm it is now enough to check that the map − × → (cid:55)→ − β 1 : U X OP0(∆) (z, x) (∆ + 1)(∆ + α(z, x)) 1 is holomorphic in supremum norm. We remark that it follows by Lemma A.3 that the map × → (cid:55)→ − β : U X OP0(∆) (z, x) (∆ + α(z, x))(∆ + 1) 1 is holomorphic in supremum norm. ∈ N − We now prove by induction on k that β 1 satisﬁes the conditions (1) and 0 (2) in Deﬁnition A.2. For k = 0 this follows from the identity − − − − − − − β(z, x) 1 β(w, y) 1 = β(z, x) 1(β(z, x) β(w, y))β(w, y) 1 ∈ × which is valid for all (z, x), (w, y) U X.

text: ON

text: A

text: SPECTRAL

text: FLOW

text: FORMULA

text: FOR

text: THE

text: HOMOLOGICAL

text: INDEX

text: Lemma

text: Let

text: α

text: U

text: X

text: be

text: holomorphic

text: in

text: supremum

text: norm.

text: Suppose

text: that

text: α(z,

text: x)

text: D(∆)

text: H

text: is

text: invertible

text: as

text: a

text: closed

text: unbounded

text: operator

text: for

text: each

text: (z,

text: x)

text: U

text: X.

text: Then

text: the

text: map

text: α)

text: U

text: X

text: OP

text: (z,

text: x)

text: α(z,

text: x))

text: is

text: holomorphic

text: in

text: supremum

text: norm.

text: Proof.

text: Remark

text: ﬁrst

text: of

text: all

text: that

text: D(α(z,

text: x))

text: for

text: all

text: (z,

text: x)

text: U

text: X.

text: To

text: see

text: this,

text: let

text: ξ

text: and

text: choose

text: a

text: sequence

text: ξ

text: in

text: H

text: with

text: ξ

text: ξ

text: and

text: n

text: n

text: Then

text: α(z,

text: x)ξ

text: α(z,

text: x)(ξ).

text: Indeed,

text: this

text: follows

text: since

text: n

text: n

text: α(z,

text: x)(ξ

text: x)(ξ

text: n

text: n

text: α(z,

text: x)

text: δ(α(z,

text: x))

text: α(z,

text: (ξ

text: n

text: The

text: unbounded

text: operator

text: α(z,

text: x)

text: D(∆)

text: H

text: is

text: thus

text: well-deﬁned

text: for

text: all

text: (z,

text: x)

text: U

text: X.

text: The

text: fact

text: that

text: α(z,

text: x)

text: D(∆)

text: H

text: is

text: closed

text: is

text: a

text: consequence

text: of

text: the

text: Kato-Rellich

text: theorem,

text: see

text: Chapther

text: IV,

text: Theorem

text: It

text: is

text: now

text: straightforward

text: to

text: verify

text: that

text: the

text: map

text: α

text: U

text: X

text: (z,

text: x)

text: α(z,

text: x)

text: is

text: holomorphic

text: in

text: supremum

text: norm.

text: Let

text: (z,

text: x)

text: U

text: X

text: and

text: let

text: us

text: continue

text: by

text: showing

text: that

text: (∆+α(z,

text: x))

text: OP

text: To

text: this

text: end,

text: it

text: is

text: clearly

text: suﬃcient

text: to

text: prove

text: that

text: α(z,

text: x))

text: We

text: therefore

text: notice

text: that

text: the

text: linear

text: map

text: β(z,

text: x)

text: α(z,

text: x))(∆

text: H

text: H

text: is

text: bijective.

text: Furthermore,

text: we

text: have

text: that

text: β(z,

text: x)

text: Since

text: is

text: closed

text: under

text: holomorphic

text: calculus

text: (as

text: a

text: of

text: L

text: (H)),

text: see

text: Lemma

text: we

text: may

text: conclude

text: that

text: α(z,

text: x))

text: α(z,

text: x))(∆

text: To

text: see

text: that

text: U

text: X

text: OP

text: (z,

text: x)

text: α(z,

text: x))

text: is

text: holomorphic

text: in

text: supremum

text: norm

text: it

text: is

text: now

text: enough

text: to

text: check

text: that

text: the

text: map

text: β

text: U

text: X

text: (z,

text: x)

text: α(z,

text: x))

text: is

text: holomorphic

text: in

text: supremum

text: norm.

text: We

text: remark

text: that

text: it

text: follows

text: by

text: Lemma

text: that

text: the

text: map

text: β

text: U

text: X

text: (z,

text: x)

text: α(z,

text: x))(∆

text: is

text: holomorphic

text: in

text: supremum

text: norm.

text: N

text: We

text: now

text: prove

text: by

text: induction

text: on

text: k

text: that

text: β

text: satisﬁes

text: the

text: conditions

text: and

text: in

text: Deﬁnition

text: For

text: k

text: this

text: follows

text: from

text: the

text: identity

text: β(z,

text: x)

text: β(w,

text: y)

text: β(z,

text: x)

text: x)

text: β(w,

text: y))β(w,

text: y)

text: which

text: is

text: valid

text: for

text: all

text: (z,

text: x),

text: (w,

text: y)

text: U

text: X.


text: ON A SPECTRAL FLOW FORMULA FOR THE HOMOLOGICAL INDEX 9 are bounded and positive. We remark also that these operators commute and that − − − they also both commute with (∆/ + 1) 1, (1 + ∆ ) 1 and (1 + ∆ ) 1. We ﬁnally 2 1 notice the relations · − − · − G (1 + ∆ ) 1 = (∆/ + 1) 1 = G (1 + ∆ ) 1 2 2 1 1 Let now s := s/(1 + p ) s := (sp )/(1 + p ) r := r/(1 + p ) r := (rp )/(1 + p ) 1 0 2 0 0 1 0 2 0 0 Since s + s = s and r + r = r we obtain that 1 2 1 2 − − · − − (∆/ + 1) r/2gA(∆/ + 1) s/2 = Gr1/2Gr2/2 (1 + ∆ ) r2/2A(1 + ∆ ) s2/2 (cid:0) (cid:1) 1 2 (cid:0) 2 (cid:1) 2 · − − · (1 + ∆ ) r1/2g(1 + ∆ ) s1/2 Gs2/2Gs1/2 1 1 2 1 R ⊗ ⊆ R Recall next that L q L2( ) L q(H ) L q L2( , H ) . It therefore suﬃces to 2 2 show that (cid:0) (cid:1) − − ∈ (1 + ∆ ) r2/2A(1 + ∆ ) s2/2 L q(H ) and 2 2 2 − − ∈ R (1 + ∆ ) r1/2g(1 + ∆ ) s1/2 L q L2( ) 1 1 (cid:3) But this follows from the above since p /(s +r ) = (p +1)/(s+r) = 1/(s +r ) 0 2 2 0 1 1 3. Existence of the homological index Throughout this Section, the assumptions and the notation of Section will remain (cid:0) (cid:1) in eﬀect. The following proposition shows that we have a well-deﬁned homological index: · − − − H-Indm(D ) := λm Tr (λ + D−D ) m (λ + D D−) m λ + + + ∈ N − for all λ > 0 and all m with m > (p 1)/2. The main purpose of this paper 0 is to compute this homological index in terms of an integral of a path of operators acting on the ‘inner’ Hilbert space H . 2 ∈ N − Proposition 3.1. Let λ > 0 and m with m > (p 1)/2 be given. Then the 0 diﬀerence − − − (λ + D−D ) m (λ + D D−) m + + is of trace class. Proof. We are interested in applying [CGK13, Theorem 3.1]. We recall that this − − − theorem shows that the diﬀerence (λ + D−D ) m (λ + D D−) m is of trace class + + provided that the following conditions are satisﬁed: D D → R (1) / : D( / ) L2( , H ) is a normal unbounded operator. + + 2 R → R (2) The bounded operator hA : L2( , H ) L2( , H ) is a pseudodiﬀerential 2 D D 2 operator of order 0 with respect to ∆/ := / − / . + (3) The sum of commutators D · · D D → R [ / , h A] + [h A, / −] : D( / ) L2( , H ) + + 2 R → R extends to a bounded operator F : L2( , H ) L2( , H ). 2 2

text: ON

text: A

text: SPECTRAL

text: FLOW

text: FORMULA

text: FOR

text: THE

text: HOMOLOGICAL

text: INDEX

text: are

text: bounded

text: and

text: positive.

text: We

text: remark

text: also

text: that

text: these

text: operators

text: commute

text: and

text: that

text: they

text: also

text: both

text: commute

text: with

text: and

text: We

text: ﬁnally

text: notice

text: the

text: relations

text: G

text: G

text: Let

text: now

text: s

text: p

text: s

text: (sp

text: p

text: r

text: p

text: r

text: (rp

text: p

text: Since

text: s

text: s

text: s

text: and

text: r

text: r

text: r

text: we

text: obtain

text: that

text: R

text: R

text: Recall

text: next

text: that

text: L

text: q

text: L

text: q(H

text: L

text: q

text: H

text: It

text: therefore

text: suﬃces

text: to

text: show

text: that

text: L

text: q(H

text: and

text: R

text: L

text: q

text: But

text: this

text: follows

text: from

text: the

text: above

text: since

text: p

text: /(s

text: +r

text: (p

text: +r

text: Existence

text: of

text: the

text: homological

text: index

text: Throughout

text: this

text: Section,

text: the

text: assumptions

text: and

text: the

text: notation

text: of

text: Section

text: will

text: remain

text: in

text: eﬀect.

text: The

text: following

text: proposition

text: shows

text: that

text: we

text: have

text: a

text: well-deﬁned

text: homological

text: index:

text: H-Indm(D

text: λm

text: Tr

text: (λ

text: D−D

text: m

text: (λ

text: D

text: D−)

text: m

text: λ

text: N

text: for

text: all

text: λ

text: and

text: all

text: m

text: with

text: m

text: (p

text: The

text: main

text: purpose

text: of

text: this

text: paper

text: is

text: to

text: compute

text: this

text: homological

text: index

text: in

text: terms

text: of

text: an

text: integral

text: of

text: a

text: path

text: of

text: operators

text: acting

text: on

text: the

text: ‘inner’

text: Hilbert

text: space

text: H

text: N

text: Proposition

text: Let

text: λ

text: and

text: m

text: with

text: m

text: (p

text: be

text: given.

text: Then

text: the

text: diﬀerence

text: (λ

text: D−D

text: m

text: (λ

text: D

text: D−)

text: m

text: is

text: of

text: trace

text: class.

text: Proof.

text: We

text: are

text: interested

text: in

text: applying

text: Theorem

text: We

text: recall

text: that

text: this

text: theorem

text: shows

text: that

text: the

text: diﬀerence

text: (λ

text: D−D

text: m

text: (λ

text: D

text: D−)

text: m

text: is

text: of

text: trace

text: class

text: provided

text: that

text: the

text: following

text: conditions

text: are

text: satisﬁed:

text: D

text: D

text: R

text: D(

text: H

text: is

text: a

text: normal

text: unbounded

text: operator.

text: R

text: R

text: The

text: bounded

text: operator

text: hA

text: H

text: H

text: is

text: a

text: pseudodiﬀerential

text: D

text: D

text: operator

text: of

text: order

text: with

text: respect

text: to

text: The

text: sum

text: of

text: commutators

text: D

text: D

text: D

text: R

text: h

text: A]

text: [h

text: A,

text: D(

text: H

text: R

text: R

text: extends

text: to

text: a

text: bounded

text: operator

text: F

text: H

text: H


text: suidar egrahc 3 2.5 2 1.5 1 0.5 experiment α=0.95 α=0.0 0 0 1 2 3 4 5 6 7 8 9 baryon number Figure 10: Isoscalar charge radii in the lightly bound model (α = 0.95), the standard Skyrme model (α = 0), and from experimental nuclear physics. The α = 0.95 curve passes through the points corresponding to global energy minima. experimental value of 138MeV. Most calibrations of the standard Skyrme model result in abnormally low values of the pion decay constant. Although this problem seems to be exacerbated in the lightly bound model, we expect for reasons to be explained below that the value of F would increase in a π quantised treatment of the model. A better value for the pion mass could be obtained by reducing the value of the parameter m in the model, but we have not explored this possibility. Reducing the value of m would probably reduce soliton energies, and thereby also increase the value of F . π The binding energy per nucleon is plotted alongside experimental values in ﬁgure 9. The theoretical binding energy per nucleon is calculated from − the formula E /B E , in which E represents the global minimimum of B 1 B the energy in the charge B sector. The corresponding experimental values are the binding energies for the lightest isotope at each mass number B. The theoretical curve closely follows the experimental curve for most values of B in the range considered, except that the anomalies at B = 4 and 8 are less pronounced in the theoretical curve than in the experimental curve. This 18 ##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##

text: suidar

font: default

text: egrahc

text: experiment

text: baryon

text: number

text: Figure

text: Isoscalar

label: caption

text: charge

text: radii

text: in

text: the

text: lightly

text: bound

text: model

text: (α

text: the

text: standard

text: Skyrme

text: model

text: (α

text: and

text: from

text: experimental

text: nuclear

text: physics.

text: The

text: α

text: curve

text: passes

text: through

text: the

text: points

text: corresponding

text: to

text: global

text: energy

text: minima.

text: experimental

text: value

text: of

text: Most

text: calibrations

text: of

text: the

text: standard

text: Skyrme

text: model

text: result

text: in

text: abnormally

text: low

text: values

text: of

text: the

text: pion

text: decay

text: constant.

text: Although

text: this

text: problem

text: seems

text: to

text: be

text: exacerbated

text: in

text: the

text: lightly

text: bound

text: model,

text: we

text: expect

text: for

text: reasons

text: to

text: be

text: explained

text: below

text: that

text: the

text: value

text: of

text: F

text: would

text: increase

text: in

text: a

text: π

text: quantised

text: treatment

text: of

text: the

text: model.

text: A

text: better

text: value

text: for

text: the

text: pion

text: mass

text: could

text: be

text: obtained

text: by

text: reducing

text: the

text: value

text: of

text: the

text: parameter

text: m

text: in

text: the

text: model,

text: but

text: we

text: have

text: not

text: explored

text: this

text: possibility.

text: Reducing

text: the

text: value

text: of

text: m

text: would

text: probably

text: reduce

text: soliton

text: energies,

text: and

text: thereby

text: also

text: increase

text: the

text: value

text: of

text: F

text: π

text: The

text: binding

text: energy

text: per

text: nucleon

text: is

text: plotted

text: alongside

text: experimental

text: values

text: in

text: ﬁgure

text: The

text: theoretical

text: binding

text: energy

text: per

text: nucleon

text: is

text: calculated

text: from

text: the

text: formula

text: E

text: /B

text: E

text: in

text: which

text: E

text: represents

text: the

text: global

text: minimimum

text: of

text: B

text: B

text: the

text: energy

text: in

text: the

text: charge

text: B

text: sector.

text: The

text: corresponding

text: experimental

text: values

text: are

text: the

text: binding

text: energies

text: for

text: the

text: lightest

text: isotope

text: at

text: each

text: mass

text: number

text: B.

text: The

text: theoretical

text: curve

text: closely

text: follows

text: the

text: experimental

text: curve

text: for

text: most

text: values

text: of

text: B

text: in

text: the

text: range

text: considered,

text: except

text: that

text: the

text: anomalies

text: at

text: B

text: and

text: are

text: less

text: pronounced

text: in

text: the

text: theoretical

text: curve

text: than

text: in

text: the

text: experimental

text: curve.

text: This

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: if any progress is to be made with this model. Despite these diﬃculties, we have found clear evidence that solitons in this model do not resemble the ansatz proposed in [5, 8, 9, 10, 11]. In this light it would be worthwhile therefore to revisit some of the results obtained therein. Acknowledgements This work was supported by EPSRC through the grant “Geometry, Holog- raphy and Skyrmions.” We acknowledge valuable conversations with Christoph Adam, Andrzej Wereszczynski, Paul Sutcliﬀe, and Nick Manton. The major- ity of simulations were performed using computer code originally developed in collaboration with Juha J¨aykka¨ and utilised the ARC High Performance Computing facilities at the University of Leeds. References [1] E. Witten, Nucl. Phys. B223 (1983) 433. [2] R.A. Battye et al., Phys.Rev. C80 (2009) 034323, arXiv:0905.0099. [3] R. Battye, N.S. Manton and P. Sutcliﬀe, Proc.Roy.Soc.Lond. A463 (2007) 261, arXiv:hep-th/0605284. [4] P. Sutcliﬀe, Journal of High Energy Physics 4 (2011) 45, 1101.2402. [5] C. Adam, J. Sa´nchez-Guill´en and A. Wereszczyn´ski, Phys. Lett. B691 (2010) 105, arXiv:1001.4544. [6] D. Harland, Phys. Lett. B 728 (2014) 518, arXiv:1311.2403. [7] P. Salmi and P. Sutcliﬀe, J.Phys. A48 (2015) 035401, 1409.8176. [8] C. Adam, J. Sa´nchez-Guill´en and A. Wereszczyn´ski, Phys. Rev. D 82 (2010) 085015, arXiv:1007.1567. [9] C. Adam et al., (2013), arXiv:1306.6337. [10] E. Bonenfant and L. Marleau, Phys. Rev. D 82 (2010) 054023, arXiv:1007.1396. [11] E. Bonenfant, L. Harbour and L. Marleau, Phys. Rev. D 85 (2012) 114045, arXiv:1205.1414. [12] K. Arthur et al., J. Math. Phys. 37 (1996) 2569. [13] J.M. Speight, (2014), arXiv:1406.0739. 21

text: if

text: any

text: progress

text: is

text: to

text: be

text: made

text: with

text: this

text: model.

text: Despite

text: these

text: diﬃculties,

text: we

text: have

text: found

text: clear

text: evidence

text: that

text: solitons

text: in

text: this

text: model

text: do

text: not

text: resemble

text: the

text: ansatz

text: proposed

text: in

text: In

text: this

text: light

text: it

text: would

text: be

text: worthwhile

text: therefore

text: to

text: revisit

text: some

text: of

text: the

text: results

text: obtained

text: therein.

text: Acknowledgements

label: section

text: This

text: work

text: was

text: supported

text: by

text: EPSRC

text: through

text: the

text: grant

text: “Geometry,

text: Holog-

text: raphy

text: and

text: Skyrmions.”

text: We

text: acknowledge

text: valuable

text: conversations

text: with

text: Christoph

text: Adam,

text: Andrzej

text: Wereszczynski,

text: Paul

text: Sutcliﬀe,

text: and

text: Nick

text: Manton.

text: The

text: major-

text: ity

text: of

text: simulations

text: were

text: performed

text: using

text: computer

text: code

text: originally

text: developed

text: in

text: collaboration

text: with

text: Juha

text: J¨aykka¨

text: and

text: utilised

text: the

text: ARC

text: High

text: Performance

text: Computing

text: facilities

text: at

text: the

text: University

text: of

text: Leeds.

text: References

text: E.

text: Witten,

text: Nucl.

text: Phys.

text: R.A.

text: Battye

text: et

text: al.,

text: Phys.Rev.

text: R.

text: Battye,

text: N.S.

text: Manton

text: and

text: P.

text: Sutcliﬀe,

text: Proc.Roy.Soc.Lond.

text: P.

text: Sutcliﬀe,

text: Journal

text: of

text: High

text: Energy

text: Physics

text: C.

text: Adam,

text: J.

text: Sa´nchez-Guill´en

text: and

text: A.

text: Wereszczyn´ski,

text: Phys.

text: Lett.

text: D.

text: Harland,

text: Phys.

text: Lett.

text: B

text: P.

text: Salmi

text: and

text: P.

text: Sutcliﬀe,

text: J.Phys.

text: C.

text: Adam,

text: J.

text: Sa´nchez-Guill´en

text: and

text: A.

text: Wereszczyn´ski,

text: Phys.

text: Rev.

text: D

text: C.

text: Adam

text: et

text: al.,

text: E.

text: Bonenfant

text: and

text: L.

text: Marleau,

text: Phys.

text: Rev.

text: D

text: E.

text: Bonenfant,

text: L.

text: Harbour

text: and

text: L.

text: Marleau,

text: Phys.

text: Rev.

text: D

text: K.

text: Arthur

text: et

text: al.,

text: J.

text: Math.

text: Phys.

text: J.M.

text: Speight,


text: this indicates that our numerical solution is not reliable, and we discard it. In all the simulations reported here, we ﬁnd that D(ε) remains small ∈ (less than 1.5%) for all ε [0.2, 1] but then explodes shortly below ε = 0.2 when B > 1. Simultaneously to the bad violation of D(ε) = 0, we ﬁnd that the numerical solutions develop spike-like singularities, indicating the development of spatial structures smaller than the resolution of our lattice. Once this occurs, we can no longer track the solution curve. We conclude, then, that the model is numerically inaccessible (at least with our methods) ≤ for 0 ε < 0.2. We will see shortly that, for B = 1 and B = 2, other methods can be used that allow us to get much closer to ε = 0, but these rely on the enhanced symmetry of these cases. Let us denote by E (B, ε) the energy of the global minimizer of E of min ε charge B (assuming that such exists). It is clear from the deﬁnition of E that ε this is a monotonically increasing function of ε, and that E (B, 0) = 2π2B. min It is bounded above by the total energy of any charge B BPS solution, for example, a superposition of B hedgehogs of disjoint support. Hence ≤ ≈ E (B, ε) B(2π2 + εE(2)(ϕ )) (19.74 + 72.36ε)B, (2.15) min H → so E (B, ε) converges to 2π2B as ε 0. Figure 1 shows a plot of E (B, ε) min ≤ ≤ min for B = 1, . . . , 8, 0.2 ε 1, as predicted by our numerical data. The energy value for B = 8 is taken from the minimum energy conﬁguration at each sampled value of ε. Energy and baryon number data for ε = 1, 0.2 can be found in appendix , table 2. Figure 2 presents, for each degree from B = 1 to B = 8, a pair of energy minimizers, for ε = 1 (left image) and ε = 0.2 (right image). In each case, a surface of constant baryon density is shown, and the colouring of each point represents the orientation of ϕ = (ϕ , ϕ , ϕ ) there. Since the B = 1 mini- 1 2 3 mizers are within the hedgehog ansatz, their pictures specify the colouring {| | | | | |} scheme: we radially project ϕ onto the unit cube max ϕ , ϕ , ϕ = 1 ± 1 2 3 ± and colour each face of the cube: where ( 1, 0, 0) are yellow/cyan, (0, 1, 0) ± are red/green and (0, 0, 1) are white/black. Except for B = 6, the ε = 1 minimizers are qualitatively similar to those of the usual Skyrme model with a pion mass: in both cases, the potential strongly penalizes ﬁelds where φ − is close to v over large regions, and hence, disfavours the hollow shell-like minimizers found in the Skyrme model without a potential, preferring the formation of chain-like structures. (The B = 6 skyrmion suggests that this eﬀect becomes important at lower B for our choice of potential than for the standard pion mass potential.) Swapping the quartic Skyrme term for the sextic term does not seem to have made a qualitative diﬀerence to the skyrmions. A similar observation was made by Floratos and Piette [16], who 6

text: this

text: indicates

text: that

text: our

text: numerical

text: solution

text: is

text: not

text: reliable,

text: and

text: we

text: discard

text: it.

text: In

text: all

text: the

text: simulations

text: reported

text: here,

text: we

text: ﬁnd

text: that

text: D(ε)

text: remains

text: small

text: (less

text: than

text: for

text: all

text: ε

text: but

text: then

text: explodes

text: shortly

text: below

text: ε

text: when

text: B

text: Simultaneously

text: to

text: the

text: bad

text: violation

text: of

text: D(ε)

text: we

text: ﬁnd

text: that

text: the

text: numerical

text: solutions

text: develop

text: spike-like

text: singularities,

text: indicating

text: the

text: development

text: of

text: spatial

text: structures

text: smaller

text: than

text: the

text: resolution

text: of

text: our

text: lattice.

text: Once

text: this

text: occurs,

text: we

text: can

text: no

text: longer

text: track

text: the

text: solution

text: curve.

text: We

text: conclude,

text: then,

text: that

text: the

text: model

text: is

text: numerically

text: inaccessible

text: (at

text: least

text: with

text: our

text: methods)

text: for

text: ε

text: We

text: will

text: see

text: shortly

text: that,

text: for

text: B

text: and

text: B

text: other

text: methods

text: can

text: be

text: used

text: that

text: allow

text: us

text: to

text: get

text: much

text: closer

text: to

text: ε

text: but

text: these

text: rely

text: on

text: the

text: enhanced

text: symmetry

text: of

text: these

text: cases.

text: Let

text: us

text: denote

text: by

text: E

text: (B,

text: ε)

text: the

text: energy

text: of

text: the

text: global

text: minimizer

text: of

text: E

text: of

text: min

text: ε

text: charge

text: B

text: (assuming

text: that

text: such

text: exists).

text: It

text: is

text: clear

text: from

text: the

text: deﬁnition

text: of

text: E

text: that

text: ε

text: this

text: is

text: a

text: monotonically

text: increasing

text: function

text: of

text: ε,

text: and

text: that

text: E

text: (B,

text: min

text: It

text: is

text: bounded

text: above

text: by

text: the

text: total

text: energy

text: of

text: any

text: charge

text: B

text: BPS

text: solution,

text: for

text: example,

text: a

text: superposition

text: of

text: B

text: hedgehogs

text: of

text: disjoint

text: support.

text: Hence

text: E

text: (B,

text: ε)

text: min

text: H

text: so

text: E

text: (B,

text: ε)

text: converges

text: to

text: as

text: ε

text: Figure

text: shows

text: a

text: plot

text: of

text: E

text: (B,

text: ε)

text: min

text: min

text: for

text: B

text: ε

text: as

text: predicted

text: by

text: our

text: numerical

text: data.

text: The

text: energy

text: value

text: for

text: B

text: is

text: taken

text: from

text: the

text: minimum

text: energy

text: conﬁguration

text: at

text: each

text: sampled

text: value

text: of

text: ε.

text: Energy

text: and

text: baryon

text: number

text: data

text: for

text: ε

text: can

text: be

text: found

text: in

text: appendix

text: table

text: Figure

text: presents,

text: for

text: each

text: degree

text: from

text: B

text: to

text: B

text: a

text: pair

text: of

text: energy

text: minimizers,

text: for

text: ε

text: (left

text: image)

text: and

text: ε

text: (right

text: image).

text: In

text: each

text: case,

text: a

text: surface

text: of

text: constant

text: baryon

text: density

text: is

text: shown,

text: and

text: the

text: colouring

text: of

text: each

text: point

text: represents

text: the

text: orientation

text: of

text: ϕ

text: (ϕ

text: ϕ

text: ϕ

text: there.

text: Since

text: the

text: B

text: mini-

text: mizers

text: are

text: within

text: the

text: hedgehog

text: ansatz,

text: their

text: pictures

text: specify

text: the

text: colouring

text: scheme:

text: we

text: radially

text: project

text: ϕ

text: onto

text: the

text: unit

text: cube

text: max

text: ϕ

text: ϕ

text: ϕ

text: and

text: colour

text: each

text: face

text: of

text: the

text: cube:

text: where

text: are

text: yellow/cyan,

text: are

text: red/green

text: and

text: are

text: white/black.

text: Except

text: for

text: B

text: the

text: ε

text: minimizers

text: are

text: qualitatively

text: similar

text: to

text: those

text: of

text: the

text: usual

text: Skyrme

text: model

text: with

text: a

text: pion

text: mass:

text: in

text: both

text: cases,

text: the

text: potential

text: strongly

text: penalizes

text: ﬁelds

text: where

text: φ

text: is

text: close

text: to

text: v

text: over

text: large

text: regions,

text: and

text: hence,

text: disfavours

text: the

text: hollow

text: shell-like

text: minimizers

text: found

text: in

text: the

text: Skyrme

text: model

text: without

text: a

text: potential,

text: preferring

text: the

text: formation

text: of

text: chain-like

text: structures.

text: (The

text: B

text: skyrmion

text: suggests

text: that

text: this

text: eﬀect

text: becomes

text: important

text: at

text: lower

text: B

text: for

text: our

text: choice

text: of

text: potential

text: than

text: for

text: the

text: standard

text: pion

text: mass

text: potential.)

text: Swapping

text: the

text: quartic

text: Skyrme

text: term

text: for

text: the

text: sextic

text: term

text: does

text: not

text: seem

text: to

text: have

text: made

text: a

text: qualitative

text: diﬀerence

text: to

text: the

text: skyrmions.

text: A

text: similar

text: observation

text: was

text: made

text: by

text: Floratos

text: and

text: Piette

text: who


text: Figure 2: Level sets of baryon density for skyrmions in the sextic model with ε = 1 (left) and ε = 0.2 (right), degrees B = 1 to B = 8. On each surface B {B } = 0.5 max (x) . U Also plotted is the (square root) potential density for ε = 0.2, suggesting → that ϕ is converging, as ε 0, to a continuous but not diﬀerentiable solution B U of = . Except for B = 1 and B = 2, there is no evidence that ϕ is converging to a BPS skyrmion with axial symmetry. In the case B = 2, ϕ appears to B converge to a BPS skyrmion with axial symmetry, but not one for which is spherically symmetric. This casts doubt on the validity of phenomenological ◦ studies of BPS and near BPS skyrmions based on ﬁelds of the form ϕ ψ H B (see equation (2.13)) [5, 8, 9, 10, 11]. Such studies perform a rigid body quantization, and depend(cid:90)strongly on the isospin and spin inertia tensors of the BPS skyrmions used, (cid:90) ∧ ∧ ∗ ∧ I (ϕ) = dϕ dϕ (dϕ dϕ ) + O(ε), ij 0 i 0 j R 3 B − J (ϕ) = (x)2(r2δ x x )d3x + O(ε), (2.16) ij ij i j R 3 ◦ respectively. For ϕ ψ , J in particular is isotropic to leading order, H B I J = const + O(ε), a very strong constraint which is not supported by 3 the numerical data: see ﬁgure 4. Since all the minimizers are axially symmetric for B = 1 and B = 2, the 8 ##LTFigure##

text: Figure

text: Level

text: sets

text: of

text: baryon

text: density

text: for

text: skyrmions

text: in

text: the

text: sextic

text: model

text: with

text: ε

text: (left)

text: and

text: ε

text: (right),

text: degrees

text: B

text: to

text: B

text: On

text: each

text: surface

text: B

text: {B

text: max

text: (x)

text: U

text: Also

text: plotted

text: is

text: the

text: (square

text: root)

text: potential

text: density

text: for

text: ε

text: suggesting

text: that

text: ϕ

text: is

text: converging,

text: as

text: ε

text: to

text: a

text: continuous

text: but

text: not

text: diﬀerentiable

text: solution

text: B

text: U

text: of

text: Except

text: for

text: B

text: and

text: B

text: there

text: is

text: no

text: evidence

text: that

text: ϕ

text: is

text: converging

text: to

text: a

text: BPS

text: skyrmion

text: with

text: axial

text: symmetry.

text: In

text: the

text: case

text: B

text: ϕ

text: appears

text: to

text: B

text: converge

text: to

text: a

text: BPS

text: skyrmion

text: with

text: axial

text: symmetry,

text: but

text: not

text: one

text: for

text: which

text: is

text: spherically

text: symmetric.

text: This

text: casts

text: doubt

text: on

text: the

text: validity

text: of

text: phenomenological

text: studies

text: of

text: BPS

text: and

text: near

text: BPS

text: skyrmions

text: based

text: on

text: ﬁelds

text: of

text: the

text: form

text: ϕ

text: ψ

text: H

text: B

text: (see

text: equation

text: Such

text: studies

text: perform

text: a

text: rigid

text: body

text: quantization,

text: and

text: on

text: the

text: isospin

text: and

text: spin

text: inertia

text: tensors

text: of

text: the

text: BPS

text: skyrmions

text: used,

text: I

text: (ϕ)

text: dϕ

text: dϕ

text: (dϕ

text: dϕ

text: O(ε),

text: ij

text: i

text: j

text: R

text: B

text: J

text: (ϕ)

text: x

text: x

text: O(ε),

text: ij

text: ij

text: i

text: j

text: R

text: respectively.

text: For

text: ϕ

text: ψ

text: J

text: in

text: particular

text: is

text: isotropic

text: to

text: leading

text: order,

text: H

text: B

text: I

text: J

text: const

text: O(ε),

text: a

text: very

text: strong

text: constraint

text: which

text: is

text: not

text: supported

text: by

text: the

text: numerical

text: data:

text: see

text: ﬁgure

text: Since

text: all

text: the

text: minimizers

text: are

text: axially

text: symmetric

text: for

text: B

text: and

text: B

text: the

text: ##LTFigure##


text: SUBMITTED PAPER 5 and the other analytical results in this paper. The achievable rate Theorem 1: Consider an RZF precoded downlink transmission computed based on this model is an upper-bound for the case from a BS having M antennas to K single-antenna UEs employ- where the SINR varies symbol-by-symbol. ing TDD in the presence of oscillator phase noise. Let α > 0, ≥ ∈ Based on the effective SINR in (17), an upper bound for the β 1, q [0, 1], and SINR denote the effective SINR at the 0 k achievable rate of the kth UE for the CO and the DO setups for kth UE. Then, a given block of symbols (i.e., given τ) is [25] − −→→∞ SINR SINR M,K 0 (23) C(SINR ) ≤ log (1 + SINR ). (18) k rzfk k 2 k almost surely, and the effective SINR associated with the kth UE This upper bound, which corresponds to the AWGN channel for the GO setup is(cid:16)given as (cid:17) capacity, is generally tight for low-to-medium SINR values. E | | p t2q T 2 ADbynOoLtsahepetirudpuospt,hpweerht ibacolh.un[i3sd0gf]eo,nraentrhdaelliysacgthiiigvehevntababltyehri(cid:0)agthe SfoIrNtRh,e wCaOs adnedrivthe(cid:1)de SINRrzfk = Mt2 (cid:12)1 − tq0Eφk|TP(cid:12)N0|2φ− t(Pq1N0+Emφ|(T−PαN)|)2 + σξw22 (24) (cid:12) (cid:12) with (cid:12) (cid:12) C(SINRk) ≤ 12 log2(2πSINRk) − 12 log2 2πeτ(σϕ2 + δpnσφ2()19,) Eφ|TP(cid:88)N|2 (cid:44) Eφ (cid:12)M1 tr {∆Φτ}(cid:12)2 = 1 −Me−τσφ2 + e−τσφ2 (25) osc (cid:48) − K where δpn = 1, when Mosc = 1, and δpn = 0, otherwise. t = p m ( −α) , (26) In (19), the second term represents the differential entropy of 2 k1 (1 + m( α))2 trheseulpthianse(1n9o) isheoldpsroucnedsse,r ϕth(τek)as−suϕm(0pkt)io+n tδhpant(φthτe−phφas0e)-.noTihsee kk11=(cid:54)=−1k, (cid:16) (cid:17) where t, m( α), ξ, and T are as given in (10)-(13). Specif- process is stationary, and has a ﬁnite differential-entropy rate. E | | PN− ically, T 2 = exp τσ2 for the DO setup, and CthoemDbOingse(t1u8p)s acnadn (b1e9)t,igthhtely aucphpieevr-abboleunrdaeted afosrCth(SeINCRO )an≤d Eφ|TPN|2φ= P1Nfor the CO setup. φ min{Rate in (18), Rate in (19)} [25]. k Proof: Refer to Appendices B and C. Remark 2: Theorem 1 captures the effect of phase noise on There are no results available for the achievable rates for ≤ ∞ the SINR as an additional penalty to the quality of the channel the MISO phase noise channel in (8), when 2 M < . osc estimate—when phase noise is present, the quality of the channel Hfaodwinegvecrh,atnhneerlsa.ndAosmsunmesisngoferTgPo(cid:32)Ndicinitythfiosrctahsee eisfferec(cid:33)mtivineisccheanntnoefl estimate, q0, degrades to q0Eφ|TPN|2 in the GO setup (25). in (16), the achievable rate is written as [26, Lemma 1] The quality of the effective chan(cid:16)nel est(cid:17)imate decreases as Mosc increases (25). Also, the effective quality is reduced when τ or | | C(SINRk) = Eφ lo(cid:32)g2 1 + (cid:107)I I(cid:107)si2g+2 σ2(cid:33) (20) σdiφ2miinncisrehaesse.bIyn athefaDctOorseotfupe,xtphe −quτaσli2ty .ofHtohwe ecvhearn,nienl etshteimCatOe int w φ E | | setup, there is no reduction in the quality of the channel estimate ≈ I 2 | | log2 1 + Eφ(cid:107)Iφint(cid:107)si2g+ σw2 , (21) dueClteoarnloy,istyheosecfiflelcattoorsf caht atnhneeBl SagoinrgthoenUthEes SasINTRPiNnc2re=as1e.s as where E denotes the expectation operation with respect to Mosc increases—the SINR in (24) decreases as Mosc increases. φ This results from the degradation of the desired signal power the phase noise at the BS. The accuracy of this approximation E | | ≥ increases with increasing M . Also, I and I are not by a factor φ TPN 2 < 1 when Mosc 2, implying that the required to be independent. Tohsics motivatessigthe deﬁninittion of an desired signal power decreases as Mosc increases. But, the MU effective SINR for 2 ≤ M < ∞ as interference power increases with Mosc, as can be seen in (24). osc This is because the CSI quality deteriorates due to phase noise E | | I 2 (25), thereby reducing the interference suppression capability of SINRk = E (cid:107)Iφ (cid:107)si2g+ σ2 . (22) the RZF precoder. Speciﬁcally, to sum for the CO setup, there φ int w is no effect of phase noise on the SINR since the desired signal The effective SINR in (22) reduces to (17) for the CO and the power and the MU interference power is the same as when only DO setups. Also, note that the achievable rate computed in (21) AWGN is present. does not account for the differential entropy rates of the phase In the sequel, we will use the SINR result in Theorem 1 in noise processes at the BS and the UEs. order to derive the effective SINR for the ZF and MF precoders IV. SINR ANALYSIS AND OPTIMAL α (Corollaries 1 and 2, respectively). In this section, we ﬁrst present the analytical results for the SINR achievable at a given UE for the considered precoders. B. SINR of the ZF Precoder We introduce Theorem 1, which provides the effective SINR Corollary 1: Consider a ZF precoded transmission from a BS for the GO setup. Then, the theorem is used to obtain the effective SINR when the ZF and the MF precoders are used at having M antennas to K single-antenna UEs e→mploying TDD i∈n the BS. Furthermore, we analytically determine the optimal α, the presence of oscillator phase noise. Let α 0, β > 1, q0 which maximizes the effective SINR at a given UE for the RZF [0, 1], and assume that the minimum eigenvalue of 1 Hˆ Hˆ H is M bounded away from zero. Then, precoder. A. SINR of the RZF Precoder − −→→∞ SINR SINR M,K 0 (27) k zfk ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SUBMITTED

text: PAPER

text: and

text: the

text: other

text: analytical

text: results

text: in

text: this

text: paper.

text: The

text: achievable

text: rate

text: Theorem

text: Consider

text: an

text: RZF

text: precoded

text: downlink

text: transmission

text: computed

text: based

text: on

text: this

text: model

text: is

text: an

text: upper-bound

text: for

text: the

text: case

text: from

text: a

text: BS

text: having

text: M

text: antennas

text: to

text: K

text: single-antenna

text: UEs

text: employ-

text: where

text: the

text: SINR

text: varies

text: symbol-by-symbol.

text: ing

text: TDD

text: in

text: the

text: presence

text: of

text: oscillator

text: phase

text: noise.

text: Let

text: α

text: Based

text: on

text: the

text: effective

text: SINR

text: in

text: an

text: upper

text: bound

text: for

text: the

text: β

text: q

text: and

text: SINR

text: denote

text: the

text: effective

text: SINR

text: at

text: the

text: k

text: achievable

text: rate

text: of

text: the

text: kth

text: UE

text: for

text: the

text: CO

text: and

text: the

text: DO

text: setups

text: for

text: kth

text: UE.

text: Then,

text: a

text: given

text: block

text: of

text: symbols

text: (i.e.,

text: given

text: τ)

text: is

text: SINR

text: SINR

text: M,K

text: C(SINR

text: log

text: SINR

text: k

text: rzfk

text: k

text: k

text: almost

text: surely,

text: and

text: the

text: effective

text: SINR

text: associated

text: with

text: the

text: kth

text: UE

text: This

text: upper

text: bound,

text: which

text: corresponds

text: to

text: the

text: AWGN

text: channel

text: for

text: the

text: GO

text: setup

text: as

text: capacity,

text: is

text: generally

text: tight

text: for

text: low-to-medium

text: SINR

text: values.

text: E

text: p

text: T

text: ADbynOoLtsahepetirudpuospt,hpweerht

text: SfoIrNtRh,e

text: wCaOs

text: SINRrzfk

text: with

text: C(SINRk)

text: 2πeτ(σϕ2

text: Eφ

text: tr

text: −Me−τσφ2

text: e−τσφ2

text: osc

text: K

text: where

text: δpn

text: when

text: Mosc

text: and

text: δpn

text: otherwise.

text: t

text: p

text: m

text: −α)

text: In

text: the

text: second

text: term

text: represents

text: the

text: differential

text: entropy

text: of

text: m(

text: isheoldpsroucnedsse,r

text: where

text: t,

text: m(

text: α),

text: ξ,

text: and

text: T

text: are

text: as

text: given

text: in

text: Specif-

text: process

text: is

text: stationary,

text: and

text: has

text: a

text: ﬁnite

text: differential-entropy

text: rate.

text: E

text: PN−

text: ically,

text: T

text: exp

text: τσ2

text: for

text: the

text: DO

text: setup,

text: and

text: acnadn

text: aucphpieevr-abboleunrdaeted

text: afosrCth(SeINCRO

text: )an≤d

text: the

text: CO

text: setup.

text: φ

text: min{Rate

text: in

text: Rate

text: in

text: k

text: Proof:

text: Refer

text: to

text: Appendices

text: B

text: and

text: C.

text: Remark

text: Theorem

text: captures

text: the

text: effect

text: of

text: phase

text: noise

text: on

text: There

text: are

text: no

text: results

text: available

text: for

text: the

text: achievable

text: rates

text: for

text: the

text: SINR

text: as

text: an

text: additional

text: penalty

text: to

text: the

text: quality

text: of

text: the

text: channel

text: the

text: MISO

text: phase

text: noise

text: channel

text: in

text: when

text: M

text: osc

text: estimate—when

text: phase

text: noise

text: is

text: present,

text: the

text: quality

text: of

text: the

text: channel

text: estimate,

text: degrades

text: to

text: in

text: the

text: GO

text: setup

text: in

text: the

text: achievable

text: rate

text: is

text: written

text: as

text: Lemma

text: The

text: quality

text: of

text: the

text: effective

text: decreases

text: as

text: Mosc

text: increases

text: Also,

text: the

text: effective

text: quality

text: is

text: reduced

text: when

text: τ

text: or

text: C(SINRk)

text: Eφ

text: athefaDctOorseotfupe,xtphe

text: .ofHtohwe

text: ecvhearn,nienl

text: etshteimCatOe

text: int

text: w

text: φ

text: E

text: setup,

text: there

text: is

text: no

text: reduction

text: in

text: the

text: quality

text: of

text: the

text: channel

text: estimate

text: I

text: caht

text: atnhneeBl

text: SagoinrgthoenUthEes

text: as

text: where

text: E

text: denotes

text: the

text: expectation

text: operation

text: with

text: respect

text: to

text: Mosc

text: increases—the

text: SINR

text: in

text: decreases

text: as

text: Mosc

text: increases.

text: φ

text: This

text: results

text: from

text: the

text: degradation

text: of

text: the

text: desired

text: signal

text: power

text: the

text: phase

text: noise

text: at

text: the

text: BS.

text: The

text: accuracy

text: of

text: this

text: approximation

text: E

text: increases

text: with

text: increasing

text: M

text: Also,

text: I

text: and

text: I

text: are

text: not

text: by

text: a

text: factor

text: φ

text: TPN

text: when

text: Mosc

text: implying

text: that

text: the

text: required

text: to

text: be

text: independent.

text: Tohsics

text: motivatessigthe

text: deﬁninittion

text: of

text: an

text: desired

text: signal

text: power

text: decreases

text: as

text: Mosc

text: increases.

text: But,

text: the

text: MU

text: effective

text: SINR

text: for

text: M

text: as

text: interference

text: power

text: increases

text: with

text: Mosc,

text: as

text: can

text: be

text: seen

text: in

text: osc

text: This

text: is

text: because

text: the

text: CSI

text: quality

text: deteriorates

text: due

text: to

text: phase

text: noise

text: E

text: I

text: thereby

text: reducing

text: the

text: interference

text: suppression

text: capability

text: of

text: SINRk

text: E

text: σ2

text: the

text: RZF

text: precoder.

text: Speciﬁcally,

text: to

text: sum

text: for

text: the

text: CO

text: setup,

text: there

text: φ

text: int

text: w

text: is

text: no

text: effect

text: of

text: phase

text: noise

text: on

text: the

text: SINR

text: since

text: the

text: desired

text: signal

text: The

text: effective

text: SINR

text: in

text: reduces

text: to

text: for

text: the

text: CO

text: and

text: the

text: power

text: and

text: the

text: MU

text: interference

text: power

text: is

text: the

text: same

text: as

text: when

text: only

text: DO

text: setups.

text: Also,

text: note

text: that

text: the

text: achievable

text: rate

text: computed

text: in

text: AWGN

text: is

text: present.

text: does

text: not

text: account

text: for

text: the

text: differential

text: entropy

text: rates

text: of

text: the

text: phase

text: In

text: the

text: sequel,

text: we

text: will

text: use

text: the

text: SINR

text: result

text: in

text: Theorem

text: in

text: noise

text: processes

text: at

text: the

text: BS

text: and

text: the

text: UEs.

text: order

text: to

text: derive

text: the

text: effective

text: SINR

text: for

text: the

text: ZF

text: and

text: MF

text: precoders

text: IV.

text: SINR

text: ANALYSIS

text: AND

text: OPTIMAL

text: α

text: (Corollaries

text: and

text: respectively).

text: In

text: this

text: section,

text: we

text: ﬁrst

text: present

text: the

text: analytical

text: results

text: for

text: the

text: SINR

text: achievable

text: at

text: a

text: given

text: UE

text: for

text: the

text: considered

text: precoders.

text: B.

text: SINR

text: of

text: the

text: ZF

text: Precoder

text: We

text: introduce

text: Theorem

text: which

text: provides

text: the

text: effective

text: SINR

text: Corollary

text: Consider

text: a

text: ZF

text: precoded

text: transmission

text: from

text: a

text: BS

text: for

text: the

text: GO

text: setup.

text: Then,

text: the

text: theorem

text: is

text: used

text: to

text: obtain

text: the

text: effective

text: SINR

text: when

text: the

text: ZF

text: and

text: the

text: MF

text: precoders

text: are

text: used

text: at

text: having

text: M

text: antennas

text: to

text: K

text: single-antenna

text: UEs

text: e→mploying

text: TDD

text: i∈n

text: the

text: BS.

text: Furthermore,

text: we

text: analytically

text: determine

text: the

text: optimal

text: α,

text: the

text: presence

text: of

text: oscillator

text: phase

text: noise.

text: Let

text: α

text: β

text: which

text: maximizes

text: the

text: effective

text: SINR

text: at

text: a

text: given

text: UE

text: for

text: the

text: RZF

text: and

text: assume

text: that

text: the

text: minimum

text: eigenvalue

text: of

text: Hˆ

text: Hˆ

text: H

text: is

text: M

text: bounded

text: away

text: from

text: zero.

text: Then,

text: precoder.

text: A.

text: SINR

text: of

text: the

text: RZF

text: Precoder

text: SINR

text: SINR

text: M,K

text: k

text: zfk

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: DIAGRAMMATIC SELF-CONSISTENT THEORY OF ANDERSON LOCALIZATION FOR THE TIGHT-BINDING MODEL J. Kroha Institut fu¨r Theorie der Kondensierten Materie, Universita¨t Karlsruhe, P.O.B. 6980, D-7500 Karlsruhe, Fed. Rep. Germany E-mail: kroha@physik.uni-bonn.de A self-consistent theory of the frequency dependent diﬀusion coeﬃcient for the Anderson localization problem is presented within the tight-binding model of non-interacting electrons on a lattice with randomly distributed on-site energy levels. The theory uses a diagrammatic expansion in terms of (extended) Bloch states and is found to be equivalent to the expansion in terms of (localized) Wannier states which was derived earlier by Kroha, Kopp and Wo¨lﬂe. No adjustable parameters enter the theory. The localization length is calculated in 1, 2 and 3 dimensions as well as the frequency dependent conductivity and the phase diagram of localization in 3 dimensions for various types of disorder distributions. The validity of a universal scaling function of the length dependent conductance derived from this theory is discussed in the strong coupling region. Quantitative agreement with results from numerical diagonalization of ﬁnite systems demonstrates that the self-consistent treatment of cooperon contributions is suﬃcient to explain the phase diagram of localization and suggests that the system may be well described by a one-parameter scaling theory in certain regions of the phase diagram, if one is not too close to the transition point. 1. Introduction The concept of localization of a quantum particle in a random potential was introduced by P.W. Anderson [1] in his pioneering work in 1958. Although considerable progress has been made since then, the problem still resists a complete theoretical understanding. In further developing ideas by Thouless [2], Abrahams et al. [3] formulated a real space scaling theory of localization in 1979, which is based on the assumption that the dimensionless conductance g(L) of a sample with ﬁnite length L is the only relevant scaling parameter. The assumptions of the one- parameter scaling theory were supported by the work of Wegner [4] and others [5-7], who mapped Anderson’s original Hamiltonian onto non-linear σ models of interacting matrices and applied perturbative renormalization group calculations in d = 2 + ε dimensions. Self- consistent theories [8, 9], developed at the same time, also yielded scaling [10] in agreement with these ﬁeld theoretical treatments. In this way it was established that the Anderson transition is a continuous phase transition characterized by a diverging length, critical exponents and an order parameter.

text: DIAGRAMMATIC

label: title

text: SELF-CONSISTENT

text: THEORY

text: OF

text: ANDERSON

text: LOCALIZATION

text: FOR

text: THE

text: TIGHT-BINDING

text: MODEL

text: J.

label: author

text: Kroha

text: Institut

text: fu¨r

text: Theorie

text: der

text: Kondensierten

text: Materie,

text: Universita¨t

text: Karlsruhe,

text: P.O.B.

text: Karlsruhe,

text: Fed.

text: Rep.

text: Germany

text: E-mail:

text: kroha@physik.uni-bonn.de

label: footer

text: A

label: abstract

text: self-consistent

text: theory

text: of

text: the

text: frequency

text: dependent

text: diﬀusion

text: coeﬃcient

text: for

text: the

text: Anderson

text: localization

text: problem

text: is

text: presented

text: within

text: the

text: tight-binding

text: model

text: of

text: non-interacting

text: electrons

text: on

text: a

text: lattice

text: with

text: randomly

text: distributed

text: on-site

text: energy

text: levels.

text: The

text: theory

text: uses

text: a

text: diagrammatic

text: expansion

text: in

text: terms

text: of

text: (extended)

text: Bloch

text: states

text: and

text: is

text: found

text: to

text: be

text: equivalent

text: to

text: the

text: expansion

text: in

text: terms

text: of

text: (localized)

text: Wannier

text: states

text: which

text: was

text: derived

text: earlier

text: by

text: Kroha,

text: Kopp

text: and

text: Wo¨lﬂe.

text: No

text: adjustable

text: parameters

text: enter

text: the

text: theory.

text: The

text: localization

text: length

text: is

text: calculated

text: in

text: and

text: dimensions

text: as

text: well

text: as

text: the

text: frequency

text: dependent

text: conductivity

text: and

text: the

text: phase

text: diagram

text: of

text: localization

text: in

text: dimensions

text: for

text: various

text: types

text: of

text: disorder

text: distributions.

text: The

text: validity

text: of

text: a

text: universal

text: scaling

text: function

text: of

text: the

text: length

text: dependent

text: conductance

text: derived

text: from

text: this

text: theory

text: is

text: discussed

text: in

text: the

text: strong

text: coupling

text: region.

text: Quantitative

text: agreement

text: with

text: results

text: from

text: numerical

text: diagonalization

text: of

text: ﬁnite

text: systems

text: demonstrates

text: that

text: the

text: self-consistent

text: treatment

text: of

text: cooperon

text: contributions

text: is

text: suﬃcient

text: to

text: explain

text: the

text: phase

text: diagram

text: of

text: localization

text: and

text: suggests

text: that

text: the

text: system

text: may

text: be

text: well

text: described

text: by

text: a

text: one-parameter

text: scaling

text: theory

text: in

text: certain

text: regions

text: of

text: the

text: phase

text: diagram,

text: if

text: one

text: is

text: not

text: too

text: close

text: to

text: the

text: transition

text: point.

text: Introduction

text: The

text: concept

text: of

text: localization

text: of

text: a

text: quantum

text: particle

text: in

text: a

text: random

text: potential

text: was

text: introduced

text: by

text: P.W.

text: Anderson

text: in

text: his

text: pioneering

text: work

text: in

text: Although

text: considerable

text: progress

text: has

text: been

text: made

text: since

text: then,

text: the

text: problem

text: still

text: resists

text: a

text: complete

text: theoretical

text: understanding.

text: In

text: further

text: developing

text: ideas

text: by

text: Thouless

text: Abrahams

text: et

text: al.

text: formulated

text: a

text: real

text: space

text: scaling

text: theory

text: of

text: localization

text: in

text: which

text: is

text: based

text: on

text: the

text: assumption

text: that

text: the

text: dimensionless

text: conductance

text: g(L)

text: of

text: a

text: sample

text: with

text: ﬁnite

text: length

text: L

text: is

text: the

text: only

text: relevant

text: scaling

text: parameter.

text: The

text: assumptions

text: of

text: the

text: one-

text: parameter

text: scaling

text: theory

text: were

text: supported

text: by

text: the

text: work

text: of

text: Wegner

text: and

text: others

text: who

text: mapped

text: Anderson’s

text: original

text: Hamiltonian

text: onto

text: non-linear

text: σ

text: models

text: of

text: interacting

text: matrices

text: and

text: applied

text: perturbative

text: renormalization

text: group

text: calculations

text: in

text: d

text: ε

text: dimensions.

text: Self-

text: consistent

text: theories

text: developed

text: at

text: the

text: same

text: time,

text: also

text: yielded

text: scaling

text: in

text: agreement

text: with

text: these

text: ﬁeld

text: theoretical

text: treatments.

text: In

text: this

text: way

text: it

text: was

text: established

text: that

text: the

text: Anderson

text: transition

text: is

text: a

text: continuous

text: phase

text: transition

text: characterized

text: by

text: a

text: diverging

text: length,

text: critical

text: exponents

text: and

text: an

text: order

text: parameter.


text: → D(ω 0) vanish at the transition point. Also because of this divergence, the term ∼ → q2R(E) in eq. (27) may be dropped for ω, q 0. 4. Infrared divergence of the irreducible vertex and self-consistent theory It was shown [8] that the infrared divergence of the diﬀusion ladder Γ as an L • internal part of the total irreducible vertex URA(cid:48), is cancelled by vertex corrections pp There is, however, another class of diagrams that introduces such a singularity into URA(cid:48), namely the class of maximally crossed diagrams known from ”weak pp localization” [32, 33] (ﬁg. 5a). Unlike the diﬀusion pole, this singularity has no classical analog. It is only present in systems with time reversal symmetry. Due to − (cid:48) this symmetry, the p = p singularity (”2k singularity”, ”cooperon”) present in F the particle-particle channel is carried over to the p-h channel. In this way, the sum of maximally crossed diagrams is obtained as 2πiN(E)(URA)2 Λ (cid:48) = Γ (Q) = (cid:48)0 . (29) pp L ω + i(p + p )2D 0 (cid:48) Here we have introduced Q = p + p . − (cid:48) In any p-h vertex diagram containing a Λ block the singularity at p = p will be strongest if Λ is crossing the total diagram diagonally, since otherwise the singularity is integrated over and thus weakened. Therefore – provided that there are no other singular contributions besides cooperons – the infrared divergent behavior of each diagram contained in the irreducible vertex URA(cid:48) can be classiﬁed by the pp Figure 5. (a) Sum of maximally crossed diagrams Λ (cid:48). (b) Exact classiﬁcation of all pp diagrams of the irreducible vertex in terms of cooperons (wavy lines). ##LTFigure## ##LTFigure## ##LTLine##

text: D(ω

text: vanish

text: at

text: the

text: transition

text: point.

text: Also

text: because

text: of

text: this

text: divergence,

text: the

text: term

text: in

text: eq.

text: may

text: be

text: dropped

text: for

text: ω,

text: q

text: Infrared

text: divergence

text: of

text: the

text: irreducible

text: vertex

text: and

text: self-consistent

text: theory

text: It

text: was

text: shown

text: that

text: the

text: infrared

text: divergence

text: of

text: the

text: diﬀusion

text: ladder

text: Γ

text: as

text: an

text: L

text: internal

text: part

text: of

text: the

text: total

text: irreducible

text: vertex

text: is

text: cancelled

text: by

text: vertex

text: corrections

text: pp

text: There

text: is,

text: however,

text: another

text: class

text: of

text: diagrams

text: that

text: introduces

text: such

text: a

text: singularity

text: into

text: namely

text: the

text: class

text: of

text: maximally

text: crossed

text: diagrams

text: known

text: from

text: ”weak

text: pp

text: localization”

text: (ﬁg.

text: Unlike

text: the

text: diﬀusion

text: pole,

text: this

text: singularity

text: has

text: no

text: classical

text: analog.

text: It

text: is

text: only

text: present

text: in

text: systems

text: with

text: time

text: reversal

text: symmetry.

text: Due

text: to

text: this

text: symmetry,

text: the

text: p

text: p

text: singularity

text: singularity”,

text: ”cooperon”)

text: present

text: in

text: F

text: the

text: particle-particle

text: channel

text: is

text: carried

text: over

text: to

text: the

text: p-h

text: channel.

text: In

text: this

text: way,

text: the

text: sum

text: of

text: maximally

text: crossed

text: diagrams

text: is

text: obtained

text: as

text: Λ

text: Γ

text: (Q)

text: pp

text: L

text: ω

text: i(p

text: p

text: Here

text: we

text: have

text: introduced

text: Q

text: p

text: p

text: In

text: any

text: p-h

text: vertex

text: diagram

text: containing

text: a

text: Λ

text: block

text: the

text: singularity

text: at

text: p

text: p

text: will

text: be

text: strongest

text: if

text: Λ

text: is

text: crossing

text: the

text: total

text: diagram

text: diagonally,

text: since

text: otherwise

text: the

text: singularity

text: is

text: integrated

text: over

text: and

text: thus

text: weakened.

text: Therefore

text: provided

text: that

text: there

text: are

text: no

text: other

text: singular

text: contributions

text: besides

text: cooperons

text: the

text: infrared

text: divergent

text: behavior

text: of

text: each

text: diagram

text: contained

text: in

text: the

text: irreducible

text: vertex

text: can

text: be

text: classiﬁed

text: by

text: the

text: pp

text: Figure

text: (a)

text: Sum

text: of

text: maximally

text: crossed

text: diagrams

text: Λ

text: (b)

text: Exact

text: classiﬁcation

text: of

text: all

text: pp

text: diagrams

text: of

text: the

text: irreducible

text: vertex

text: in

text: terms

text: of

text: cooperons

text: (wavy

text: lines).

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: (cid:88) ΦRA(cid:48)(E, ω, q) = GR (E + ω)GA (E)δ (cid:48) (13) pp p+ p− pp 1 + GR (E + ω)GA (E) URA(cid:48)(cid:48)(E, ω, q)ΦR(cid:48)A(cid:48) (cid:48)(E, ω, q) , p+ p− N (cid:48)(cid:48) pp p p p where URA(cid:48), denotes the p-h irreducible vertex function. The B-S equation is pp supplemented by an exact Ward identity, valid for all frequencies [8]: (cid:88) − 1 − ΣR (E + ω) ΣA (E) = URA(cid:48)[GR (E + ω) GA (E)] (14) p+ p− N (cid:48) pp p+ p− p (cid:80) Before we will attempt to solve the B-S equation, it is useful to consider the density correlation function Φ = (1/N2) (cid:48) ΦRA(cid:48)(q) in the simplest, i.e. ladder ρρ pp pp approximation. That is, the irreducible vertex U is calculated as the sum of all irreducible p-h diagrams without crossings (single-site approxmation, ﬁg. 3). Note that this class of diagrams is generated from the CPA diagrams of section 2 via the Ward identity. The result may be obtained by direct summation (in analogy to the Figure 2. Disordered density of states in CPA for Gaussian distribution. Data points taken from ref. [24] for comparison. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

label: equation

text: ω,

text: q)

text: GR

text: (E

text: ω)GA

text: (E)δ

text: pp

text: p+

text: p−

text: pp

text: GR

text: (E

text: ω)GA

text: (E)

text: ω,

text: ω,

text: q)

text: p+

text: p−

text: N

text: pp

text: p

text: p

text: p

text: where

text: denotes

text: the

text: p-h

text: irreducible

text: vertex

text: function.

text: The

text: B-S

text: equation

text: is

text: pp

text: supplemented

text: by

text: an

text: exact

text: Ward

text: identity,

text: valid

text: for

text: all

text: frequencies

text: ΣR

text: (E

text: ω)

text: ΣA

text: (E)

text: (E

text: ω)

text: GA

text: (E)]

text: p+

text: p−

text: N

text: pp

text: p+

text: p−

text: p

text: Before

text: we

text: will

text: attempt

text: to

text: solve

text: the

text: B-S

text: equation,

text: it

text: is

text: useful

text: to

text: consider

text: the

text: density

text: correlation

text: function

text: Φ

text: in

text: the

text: simplest,

text: i.e.

text: ladder

text: ρρ

text: pp

text: pp

text: approximation.

text: That

text: is,

text: the

text: irreducible

text: vertex

text: U

text: is

text: calculated

text: as

text: the

text: sum

text: of

text: all

text: irreducible

text: p-h

text: diagrams

text: without

text: crossings

text: (single-site

text: approxmation,

text: ﬁg.

text: Note

text: that

text: this

text: class

text: of

text: diagrams

text: is

text: generated

text: from

text: the

text: CPA

text: diagrams

text: of

text: section

text: via

text: the

text: Ward

text: identity.

text: The

text: result

text: may

text: be

text: obtained

text: by

text: direct

text: summation

text: (in

text: analogy

text: to

text: the

text: Figure

text: Disordered

text: density

text: of

text: states

text: in

text: CPA

text: for

text: Gaussian

text: distribution.

text: Data

text: points

text: taken

text: from

text: ref.

text: for

text: comparison.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##