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text: dΞuvpq = dξundχvndηpndζqndξ¯un−ˆ1dχ¯vn−ˆ1dη¯pn−ˆ2dζ¯qn−ˆ2, (A.2) n n n n n n n n n and the tensor elements are not zeros only when (cid:0) (cid:1) u + v + p + q + u − + v − + p − + q − mod 2 = 0 (A.3) n n n n n ˆ1 n ˆ1 n ˆ2 n ˆ2 T holds. The tensor is made of the fermionic one T (φ ) in F n unvnpnqnun−ˆ1vn−ˆ1pn−ˆ2qn−ˆ2 eq. (3.11) and the boson one T (K) in eq. (3.31) as in eq. (3.39). The indices u , v , p , q take two values 0Bor 1wwnshniwlen−wˆ1sn,−sˆ2 run from 1 to D as seen in section . n n n n n n init The total indices X and T are given by X = (u , v , w ) and T = (p , q , s ), and they × n× n n n n n n n n n run from 1 to 2 2 D . As mentioned in section , we set D = D /2 for the actual init init cut computations. The coarse-graining of a tensor network mainly consists of three steps: the SVD of tensors, a decomposition of Grassmann measures, and a contraction of the indices and taking the integrals of Grassmann variables deﬁned on Γ. The SVD and the decomposition for Grassmann measures are performed in a diﬀerent manner for even and odd sites. We will see that the coarse-grained tensors take the same form as (A.1) with v = q = 0 and are deﬁned on the coarse-grained lattice (cid:26) (cid:12) (cid:27) (cid:0) (cid:1) (cid:12) (cid:12) (cid:12) 1 ∈ Γ(cid:63) = n + ˆ1 + ˆ2 n = (n , n ) Γ, where n + n is an even integer. . (A.4) 1 2 1 2 2 This means that Γ(cid:63) is a set of the center of the plaquette (n, n+ˆ1, n+ˆ1+ˆ2, n+ˆ2) with − even sites n. The unit vectors of Γ(cid:63) are ˆ1(cid:63) = ˆ1 + ˆ2 and ˆ2(cid:63) = ˆ1 ˆ2. The correspondence between n and n(cid:63) is shown in ﬁgure 10. ∈ T First, on even sites n Γ, we just take the truncated SVD of like eq. (4.2): (cid:88) T ≈ Dcut † U1 σ13 V 3 , (A.5) XnTnXn−ˆ1Tn−ˆ2 (XnTn)wn(cid:63)−ˆ1(cid:63) wn(cid:63)−ˆ1(cid:63) wn(cid:63)−1ˆ(cid:63)(Xn−ˆ1Tn−ˆ2) wn(cid:63)−ˆ1(cid:63)=1 where (cid:0) (cid:1) 1 ∈ n(cid:63) = n + ˆ1 + ˆ2 Γ(cid:63). (A.6) 2 The Grassmann measures are divided into two pieces as – 28 – ##LTLine## ##LTLine##

text: dΞuvpq

label: equation

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: and

label: paragraph

text: the

text: tensor

text: elements

text: are

text: not

text: zeros

text: only

text: when

text: u

text: v

text: p

text: q

text: u

text: v

text: p

text: q

text: mod

text: n

text: n

text: n

text: n

text: n

text: ˆ1

text: n

text: ˆ1

text: n

text: ˆ2

text: n

text: ˆ2

text: T

text: holds.

text: The

text: tensor

text: is

text: made

text: of

text: the

text: fermionic

text: one

text: T

text: (φ

text: in

text: F

text: n

text: eq.

text: and

text: the

text: boson

text: one

text: T

text: (K)

text: in

text: eq.

text: as

text: in

text: eq.

text: The

text: indices

text: u

text: v

text: p

text: q

text: take

text: two

text: values

text: run

text: from

text: to

text: D

text: as

text: seen

text: in

text: section

text: n

text: n

text: n

text: n

text: n

text: n

text: init

text: The

text: total

text: indices

text: X

text: and

text: T

text: are

text: given

text: by

text: X

text: (u

text: v

text: w

text: and

text: T

text: (p

text: q

text: s

text: and

text: they

text: n×

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: run

text: from

text: to

text: D

text: As

text: mentioned

text: in

text: section

text: we

text: set

text: D

text: D

text: for

text: the

text: actual

text: init

text: init

text: cut

text: computations.

text: The

text: coarse-graining

text: of

text: a

text: tensor

text: network

text: mainly

text: consists

text: of

text: three

text: steps:

text: the

text: SVD

text: of

text: tensors,

text: a

text: decomposition

text: of

text: Grassmann

text: measures,

text: and

text: a

text: contraction

text: of

text: the

text: indices

text: and

text: taking

text: the

text: integrals

text: of

text: Grassmann

text: variables

text: deﬁned

text: on

text: Γ.

text: The

text: SVD

text: and

text: the

text: decomposition

text: for

text: Grassmann

text: measures

text: are

text: performed

text: in

text: a

text: diﬀerent

text: manner

text: for

text: even

text: and

text: odd

text: sites.

text: We

text: will

text: see

text: that

text: the

text: coarse-grained

text: tensors

text: take

text: the

text: same

text: form

text: as

text: with

text: v

text: q

text: and

text: are

text: deﬁned

text: on

text: the

text: coarse-grained

text: lattice

text: n

text: ˆ1

text: ˆ2

text: n

text: (n

text: n

text: Γ,

text: where

text: n

text: n

text: is

text: an

text: even

text: integer.

text: This

text: means

text: that

text: is

text: a

text: set

text: of

text: the

text: center

text: of

text: the

text: plaquette

text: (n,

text: with

text: even

text: sites

text: n.

text: The

text: unit

text: vectors

text: of

text: are

text: ˆ1

text: ˆ2

text: and

text: ˆ1

text: The

text: correspondence

text: between

text: n

text: and

text: is

text: shown

text: in

text: ﬁgure

text: T

text: First,

text: on

text: even

text: sites

text: n

text: Γ,

text: we

text: just

text: take

text: the

text: truncated

text: SVD

text: of

text: like

text: eq.

text: T

text: Dcut

text: V

text: where

text: n

text: ˆ1

text: ˆ2

text: The

text: Grassmann

text: measures

text: are

text: divided

text: into

text: two

text: pieces

text: as

text: ##LTLine##

text: ##LTLine##


text: [1] N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and conﬁnement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B426 (1994) 19–52, [hep-th/9407087]. [2] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B431 (1994) 484–550, [hep-th/9408099]. [3] N. Seiberg, Electric - magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B435 (1995) 129–146, [hep-th/9411149]. [4] J. M. Maldacena, The Large N limit of superconformal ﬁeld theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133, [hep-th/9711200]. [5] E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B202 (1982) 253. [6] J. Wess and B. Zumino, Supergauge Transformations in Four-Dimensions, Nucl. Phys. B70 (1974) 39–50. [7] J. Bartels and J. B. Bronzan, SUPERSYMMETRY ON A LATTICE, Phys. Rev. D28 (1983) 818. [8] F. Synatschke, H. Gies and A. Wipf, Phase Diagram and Fixed-Point Structure of two dimensional N=1 Wess-Zumino Models, Phys. Rev. D80 (2009) 085007, [0907.4229]. [9] S. Cecotti and L. Girardello, Stochastic Processes in Lattice (Extended) Supersymmetry, Nucl. Phys. B226 (1983) 417–428. [10] N. Sakai and M. Sakamoto, Lattice Supersymmetry and the Nicolai Mapping, Nucl. Phys. B229 (1983) 173–188. [11] Y. Kikukawa and Y. Nakayama, Nicolai mapping versus exact chiral symmetry on the lattice, Phys. Rev. D66 (2002) 094508, [hep-lat/0207013]. [12] J. Giedt and E. Poppitz, Lattice supersymmetry, superﬁelds and renormalization, J. High Energy Phys. 09 (2004) 029, [hep-th/0407135]. N [13] D. Kadoh and H. Suzuki, Supersymmetry restoration in lattice formulations of 2D = (2, 2) WZ model based on the Nicolai map, Phys. Lett. B696 (2011) 163–166, [1011.0788]. [14] P. H. Dondi and H. Nicolai, Lattice Supersymmetry, Nuovo Cim. A41 (1977) 1. [15] D. Kadoh and H. Suzuki, Supersymmetric nonperturbative formulation of the WZ model in lower dimensions, Phys. Lett. B684 (2010) 167–172, [0909.3686]. [16] K. Asaka, A. D’Adda, N. Kawamoto and Y. Kondo, Exact lattice supersymmetry at the quantum level for N = 2 Wess?Zumino models in 1- and 2-dimensions, Int. J. Mod. Phys. A31 (2016) 1650125, [1607.04371]. [17] A. D’Adda, N. Kawamoto and J. Saito, An Alternative Lattice Field Theory Formulation Inspired by Lattice Supersymmetry, J. High Energy Phys. 12 (2017) 089, [1706.02615]. [18] M. F. L. Golterman and D. N. Petcher, A Local Interactive Lattice Model With Supersymmetry, Nucl. Phys. B319 (1989) 307–341. [19] M. Beccaria, G. Curci and E. D’Ambrosio, Simulation of supersymmetric models with a local Nicolai map, Phys. Rev. D58 (1998) 065009, [hep-lat/9804010]. [20] S. Catterall and S. Karamov, Exact lattice supersymmetry: The Two-dimensional N=2 Wess-Zumino model, Phys. Rev. D65 (2002) 094501, [hep-lat/0108024]. – 32 –

label: reference

text: N.

text: Seiberg

text: and

text: E.

text: Witten,

text: Electric

text: magnetic

text: duality,

text: monopole

text: condensation,

text: and

text: conﬁnement

text: in

text: supersymmetric

text: Yang-Mills

text: theory,

text: Nucl.

text: Phys.

text: N.

text: Seiberg

text: and

text: E.

text: Witten,

text: Monopoles,

text: duality

text: and

text: chiral

text: symmetry

text: breaking

text: in

text: supersymmetric

text: QCD,

text: Nucl.

text: Phys.

text: N.

text: Seiberg,

text: Electric

text: magnetic

text: duality

text: in

text: supersymmetric

text: nonAbelian

text: gauge

text: theories,

text: Nucl.

text: Phys.

text: J.

text: M.

text: Maldacena,

text: The

text: Large

text: N

text: limit

text: of

text: superconformal

text: ﬁeld

text: theories

text: and

text: supergravity,

text: Int.

text: J.

text: Theor.

text: Phys.

text: E.

text: Witten,

text: Constraints

text: on

text: Supersymmetry

text: Breaking,

text: Nucl.

text: Phys.

text: J.

text: Wess

text: and

text: B.

text: Zumino,

text: Supergauge

text: Transformations

text: in

text: Four-Dimensions,

text: Nucl.

text: Phys.

text: J.

text: Bartels

text: and

text: J.

text: B.

text: Bronzan,

text: SUPERSYMMETRY

text: ON

text: A

text: LATTICE,

text: Phys.

text: Rev.

text: F.

text: Synatschke,

text: H.

text: Gies

text: and

text: A.

text: Wipf,

text: Phase

text: Diagram

text: and

text: Fixed-Point

text: Structure

text: of

text: two

text: dimensional

text: Wess-Zumino

text: Models,

text: Phys.

text: Rev.

text: S.

text: Cecotti

text: and

text: L.

text: Girardello,

text: Stochastic

text: Processes

text: in

text: Lattice

text: (Extended)

text: Supersymmetry,

text: Nucl.

text: Phys.

text: N.

text: Sakai

text: and

text: M.

text: Sakamoto,

text: Lattice

text: Supersymmetry

text: and

text: the

text: Nicolai

text: Mapping,

text: Nucl.

text: Phys.

text: Y.

text: Kikukawa

text: and

text: Y.

text: Nakayama,

text: Nicolai

text: mapping

text: versus

text: exact

text: chiral

text: symmetry

text: on

text: the

text: lattice,

text: Phys.

text: Rev.

text: J.

text: Giedt

text: and

text: E.

text: Poppitz,

text: Lattice

text: supersymmetry,

text: superﬁelds

text: and

text: renormalization,

text: J.

text: High

text: Energy

text: Phys.

text: N

text: D.

text: Kadoh

text: and

text: H.

text: Suzuki,

text: Supersymmetry

text: restoration

text: in

text: lattice

text: formulations

text: of

text: WZ

text: model

text: based

text: on

text: the

text: Nicolai

text: map,

text: Phys.

text: Lett.

text: P.

text: H.

text: Dondi

text: and

text: H.

text: Nicolai,

text: Lattice

text: Supersymmetry,

text: Nuovo

text: Cim.

text: D.

text: Kadoh

text: and

text: H.

text: Suzuki,

text: Supersymmetric

text: nonperturbative

text: formulation

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text: the

text: WZ

text: model

text: in

text: lower

text: dimensions,

text: Phys.

text: Lett.

text: K.

text: Asaka,

text: A.

text: D’Adda,

text: N.

text: Kawamoto

text: and

text: Y.

text: Kondo,

text: Exact

text: lattice

text: supersymmetry

text: at

text: the

text: quantum

text: level

text: for

text: N

text: Wess?Zumino

text: models

text: in

text: and

text: Int.

text: J.

text: Mod.

text: Phys.

text: A.

text: D’Adda,

text: N.

text: Kawamoto

text: and

text: J.

text: Saito,

text: An

text: Alternative

text: Lattice

text: Field

text: Theory

text: Formulation

text: Inspired

text: by

text: Lattice

text: Supersymmetry,

text: J.

text: High

text: Energy

text: Phys.

text: M.

text: F.

text: L.

text: Golterman

text: and

text: D.

text: N.

text: Petcher,

text: A

text: Local

text: Interactive

text: Lattice

text: Model

text: With

text: Supersymmetry,

text: Nucl.

text: Phys.

text: M.

text: Beccaria,

text: G.

text: Curci

text: and

text: E.

text: D’Ambrosio,

text: Simulation

text: of

text: supersymmetric

text: models

text: with

text: a

text: local

text: Nicolai

text: map,

text: Phys.

text: Rev.

text: S.

text: Catterall

text: and

text: S.

text: Karamov,

text: Exact

text: lattice

text: supersymmetry:

text: The

text: Two-dimensional

text: Wess-Zumino

text: model,

text: Phys.

text: Rev.


text: this sign problem we employ the TRG method, whose ﬁrst step is to represent eq. (2.17) as a network of uniform tensors, which is explained in the next section. 3 Tensor network representation of partition function 3.1 Fermion Pfaﬃan We construct a tensor network representation for the fermion part of eq. (2.17) (cid:90) (cid:80) D − ZF = ψe 12 n∈Γ ψ¯nDψn, (3.1) which yields the Pfaﬃan after integrating the fermion ﬁeld as found in eq. (2.25). The basic idea follows from refs. [32, 34] which deal with the Dirac fermions. We describe the procedure for the Majorana fermions with any value of the Wilson parameter r. Now we use the following representations for γ and C that satisfy eqs. (2.2) and (2.4): µ − γ = σ , γ = σ , C = iσ , (3.2) 1 1 2 3 2 where σ is the standard Pauli matrix. The method presented in this section is appli- i cable to any possible choice of γ and C, and they just lead to diﬀerent tensors. Then the µ Majorana spinor takes the form (cid:32) (cid:33) (cid:16) (cid:17) ψ − ψ = n,1 , ψ¯ = ψ , ψ , (3.3) n n n,2 n,1 ψ n,2 and we obtain (cid:26) (cid:18) (cid:19) (cid:88) (cid:88) (cid:16) (cid:17) −1 (cid:18) 1 +(cid:19) r (cid:27) ψ¯ Dψ = (cid:16) ψ˜ ψ˜ + ψ ψ(cid:17) (cid:0) (cid:1) 2 ∈ n n ∈ 2 n+ˆ1,2 n,1 n+ˆ2,2 n,1 n Γ n Γ − 1 r (cid:48)(cid:48) + ψ˜ ψ˜ + ψ ψ + W (φ ) + 2r ψ ψ , 2 n+ˆ1,1 n,2 n+ˆ2,1 n,2 n n,1 n,2 (3.4) where √1 ψ˜ = (ψ + ψ ) , (3.5) n,1 n,2 n,1 2 √1 − ψ˜ = (ψ ψ ) , (3.6) n,2 n,2 n,1 2 – 7 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: this

text: sign

text: problem

text: we

text: employ

text: the

text: TRG

text: method,

text: whose

text: ﬁrst

text: step

text: is

text: to

text: represent

text: eq.

text: as

text: a

text: network

text: of

text: uniform

text: tensors,

text: which

text: is

text: explained

text: in

text: the

text: next

text: section.

label: section

text: Tensor

text: network

text: representation

text: of

text: partition

text: function

text: Fermion

text: Pfaﬃan

text: We

text: construct

text: a

text: tensor

text: network

text: representation

text: for

text: the

text: fermion

text: part

text: of

text: eq.

text: D

text: ZF

text: ψe

text: n∈Γ

text: ψ¯nDψn,

text: which

text: yields

text: the

text: Pfaﬃan

text: after

text: integrating

text: the

text: fermion

text: ﬁeld

text: as

text: found

text: in

text: eq.

text: The

text: basic

text: idea

text: follows

text: from

text: refs.

text: which

text: deal

text: with

text: the

text: Dirac

text: fermions.

text: We

text: describe

text: the

text: procedure

text: for

text: the

text: Majorana

text: fermions

text: with

text: any

text: value

text: of

text: the

text: Wilson

text: parameter

text: r.

text: Now

text: we

text: use

text: the

text: following

text: representations

text: for

text: γ

text: and

text: C

text: that

text: satisfy

text: eqs.

text: and

text: µ

text: γ

text: σ

text: γ

text: σ

text: C

text: iσ

text: where

text: σ

text: is

text: the

text: standard

text: Pauli

text: matrix.

text: The

text: method

text: presented

text: in

text: this

text: section

text: is

text: appli-

text: i

text: cable

text: to

text: any

text: possible

text: choice

text: of

text: γ

text: and

text: C,

text: and

text: they

text: just

text: lead

text: to

text: diﬀerent

text: tensors.

text: Then

text: the

text: µ

text: Majorana

text: spinor

text: takes

text: the

text: form

text: ψ

text: ψ

text: ψ¯

text: ψ

text: ψ

text: n

text: n

text: ψ

text: and

text: we

text: obtain

text: −1

text: r

text: ψ¯

text: Dψ

text: ψ˜

text: ψ˜

text: ψ

text: n

text: n

text: n

text: Γ

text: n

text: Γ

text: r

text: ψ˜

text: ψ˜

text: ψ

text: ψ

text: W

text: (φ

text: ψ

text: ψ

text: n

text: where

text: √1

text: ψ˜

text: (ψ

text: ψ

text: √1

text: ψ˜

text: (ψ

text: ψ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ Racetrack Fixed Field Alternating Gradient accelerator for variable energy extraction M. Haj Tahar, D.J. Brenner and G. Randers-Pehrson † ‡ In this paper, a novel concept of ﬁxed ﬁeld alternating gradient (FFAG) accelerator is presented which solves the major problems of conventional FFAG and cyclotron accelerators. This concept combines stable racetrack ﬁxed ﬁeld conﬁguration with dispersion free straight sections to enable variable energy extraction. A time-varying synchrotron-like alternating gradient focusing structure makes it possible for the ﬁrst time to correct the beam optics of an FFAG accelerator in order to heuristically maximize the beam transmission by minimizing the tune excursion and avoiding the crossing of harmful betatron resonances. This concept makes it possible to design a compact variable energy extraction machine which is particularly useful for research and medical applications such as heavy ion therapy. I. INTRODUCTION the change of the average horizontal restoring force as well as to the magnetic ﬂutter and induces a variation A Fixed Field Alternating Gradient (FFAG) accelera- of the betatron wave number leading to the crossing of tor is a concept that was invented in the 1950s almost transverse resonances during the acceleration cycle. For independently in the US, Japan and Russia [1][2]. Sev- instance, a gradient ﬁeld error of the order of 5% yields eral electron machines were built in the US. However, it a variation of the betatron wave number of the order of was not until the 1990s that the interest for FFAGs was 7% leading to the losses of the majority of the beam [13]. revived in Japan. One main feature of FFAG and cy- This is mainly due to the large fringing ﬁeld caused by clotrons is that they both use ﬁxed ﬁeld combined func- the yoke free magnets [14] and to the slow acceleration, tion magnets in order to achieve a stable particle motion i.e. to the low energy gain per turn. The main chal- in the accelerator. Therefore, for heavy ion therapy accel- lenge with optics correction in ﬁxed ﬁeld accelerators is erators requiring energies up to several 100 MeV/nucleon, that the beam moves outward radially during the accel- they are the most viable option for a compact design that eration. Therefore, a correction scheme should be imple- ﬁts best into a hospital environment. In addition, with mented along the radius of the magnet to produce the the recent advances in the superconducting magnet tech- desired ﬁeld proﬁle and is practically expensive and dif- nology, the latter has become canonical for many appli- ﬁcult to implement since it modiﬁes simultaneously the cations. bending as well as the focusing of the beam in both the An FFAG accelerator has more design ﬂexibility com- horizontal and vertical plane. In addition, other errors pared to other types of ﬁxed ﬁeld accelerators. Unlike may contribute as well to the beam losses such as mis- cyclotrons where the main target is to achieve isochro- alignment errors. nism for all energies, the magnetic ﬁeld of an FFAG can This letter shows that a racetrack FFAG conﬁgura- obey any radial or azimuthal dependence. Thus, with tion can achieve a ﬁxed tune even in presence of ﬁeld the worldwide interest many FFAG designs emerged tai- and alignment errors and thus overcome resonance cross- lored to areas ranging from industrial processes to med- ings problem. The method is based on creating disper- ical [3][4][5][6] and fundamental research [7] [8] [9]. In sion free straight sections where a time varying alternat- the early days, almost all designs were based on highly ing gradient focusing structure is placed. The merit of symmetric rings to reduce the operating restrictions that this approach is that the time varying ﬁeld structure can may arise with systematic ﬁeld errors. However, recently, be programmed in a way to maximize the overall beam several racetrack concepts were proposed [10][11]. This transmission of the built machine. is generally considered as a solution to practical issues The idea of creating a dispersion-free long insertion such as the construction cost, the injection and extrac- in an FFAG was ﬁrst discussed by Meads in 1983 [15]. tion problem, as well as to accommodate and simplify However, the design was not optimized and focused on the design of the RF cavity. using scaling FFAG type magnets to achieve a ﬁxed tune Although recently built FFAGs demonstrated the con- machine. In addition, the sensitivity to misalignments as cept, several problems remain. In [12] it is shown that well as the correction of the optics were not discussed. small gradient ﬁeld errors in a ring FFAG accelerator lead The design discussed in this paper is greatly simpliﬁed to a non-scaling of the orbits. The latter contributes to with the use of both scaling and non-scaling type of mag- nets to create a racetrack and compact microtron-like conﬁguration. ∗ As illustrated in Fig. 1, a conﬁguration with 4 iden- † tical radial sector magnets (m1) is privileged since it ‡ provides enough space between the magnets to place in- ##LTLine##

text: Racetrack

label: title

text: Fixed

text: Field

text: Alternating

text: Gradient

text: accelerator

text: for

text: variable

text: energy

text: extraction

text: M.

label: author

text: Haj

text: Tahar,

text: D.J.

text: Brenner

text: and

text: G.

text: Randers-Pehrson

text: In

label: abstract

text: this

text: paper,

text: a

text: novel

text: concept

text: of

text: ﬁxed

text: ﬁeld

text: alternating

text: gradient

text: (FFAG)

text: accelerator

text: is

text: presented

text: which

text: solves

text: the

text: major

text: problems

text: of

text: conventional

text: FFAG

text: and

text: cyclotron

text: accelerators.

text: This

text: concept

text: combines

text: stable

text: racetrack

text: ﬁxed

text: ﬁeld

text: conﬁguration

text: with

text: dispersion

text: free

text: straight

text: sections

text: to

text: enable

text: variable

text: energy

text: extraction.

text: A

text: time-varying

text: synchrotron-like

text: alternating

text: gradient

text: focusing

text: structure

text: makes

text: it

text: possible

text: for

text: the

text: ﬁrst

text: time

text: to

text: correct

text: the

text: beam

text: optics

text: of

text: an

text: FFAG

text: accelerator

text: in

text: order

text: to

text: heuristically

text: maximize

text: the

text: beam

text: transmission

text: by

text: minimizing

text: the

text: tune

text: excursion

text: and

text: avoiding

text: the

text: crossing

text: of

text: harmful

text: betatron

text: resonances.

text: This

text: concept

text: makes

text: it

text: possible

text: to

text: design

text: a

text: compact

text: variable

text: energy

text: extraction

text: machine

text: which

text: is

text: particularly

text: useful

text: for

text: research

text: and

text: medical

text: applications

text: such

text: as

text: heavy

text: ion

text: therapy.

text: I.

text: INTRODUCTION

text: the

text: change

text: of

text: the

text: average

text: horizontal

text: restoring

text: force

text: as

text: well

text: as

text: to

text: the

text: magnetic

text: ﬂutter

text: and

text: induces

text: a

text: variation

text: A

text: Fixed

text: Field

text: Alternating

text: Gradient

text: (FFAG)

text: accelera-

text: of

text: the

text: betatron

text: wave

text: number

text: leading

text: to

text: the

text: crossing

text: of

text: tor

text: is

text: a

text: concept

text: that

text: was

text: invented

text: in

text: the

text: almost

text: transverse

text: resonances

text: during

text: the

text: acceleration

text: cycle.

text: For

text: independently

text: in

text: the

text: US,

text: Japan

text: and

text: Russia

text: Sev-

text: instance,

text: a

text: gradient

text: ﬁeld

text: error

text: of

text: the

text: order

text: of

text: yields

text: eral

text: electron

text: machines

text: were

text: built

text: in

text: the

text: US.

text: However,

text: it

text: a

text: variation

text: of

text: the

text: betatron

text: wave

text: number

text: of

text: the

text: order

text: of

text: was

text: not

text: until

text: the

text: that

text: the

text: interest

text: for

text: FFAGs

text: was

text: leading

text: to

text: the

text: losses

text: of

text: the

text: majority

text: of

text: the

text: beam

text: revived

text: in

text: Japan.

text: One

text: main

text: feature

text: of

text: FFAG

text: and

text: cy-

text: This

text: is

text: mainly

text: due

text: to

text: the

text: large

text: fringing

text: ﬁeld

text: caused

text: by

text: clotrons

text: is

text: that

text: they

text: both

text: use

text: ﬁxed

text: ﬁeld

text: combined

text: func-

text: the

text: yoke

text: free

text: magnets

text: and

text: to

text: the

text: slow

text: acceleration,

text: tion

text: magnets

text: in

text: order

text: to

text: achieve

text: a

text: stable

text: particle

text: motion

text: i.e.

text: to

text: the

text: low

text: energy

text: gain

text: per

text: turn.

text: The

text: main

text: chal-

text: in

text: the

text: accelerator.

text: Therefore,

text: for

text: heavy

text: ion

text: therapy

text: accel-

text: lenge

text: with

text: optics

text: correction

text: in

text: ﬁxed

text: ﬁeld

text: accelerators

text: is

text: erators

text: requiring

text: energies

text: up

text: to

text: several

text: MeV/nucleon,

text: that

text: the

text: beam

text: moves

text: outward

text: radially

text: during

text: the

text: accel-

text: they

text: are

text: the

text: most

text: viable

text: option

text: for

text: a

text: compact

text: design

text: that

text: eration.

text: Therefore,

text: a

text: correction

text: scheme

text: should

text: be

text: imple-

text: ﬁts

text: best

text: into

text: a

text: hospital

text: environment.

text: In

text: addition,

text: with

text: mented

text: along

text: the

text: radius

text: of

text: the

text: magnet

text: to

text: produce

text: the

text: the

text: recent

text: advances

text: in

text: the

text: superconducting

text: magnet

text: tech-

text: desired

text: ﬁeld

text: proﬁle

text: and

text: is

text: practically

text: expensive

text: and

text: dif-

text: nology,

text: the

text: latter

text: has

text: become

text: canonical

text: for

text: many

text: appli-

text: ﬁcult

text: to

text: implement

text: since

text: it

text: modiﬁes

text: simultaneously

text: the

text: cations.

text: bending

text: as

text: well

text: as

text: the

text: focusing

text: of

text: the

text: beam

text: in

text: both

text: the

text: An

text: FFAG

text: accelerator

text: has

text: more

text: design

text: ﬂexibility

text: com-

text: horizontal

text: and

text: vertical

text: plane.

text: In

text: addition,

text: other

text: errors

text: pared

text: to

text: other

text: types

text: of

text: ﬁxed

text: ﬁeld

text: accelerators.

text: Unlike

text: may

text: contribute

text: as

text: well

text: to

text: the

text: beam

text: losses

text: such

text: as

text: mis-

text: cyclotrons

text: where

text: the

text: main

text: target

text: is

text: to

text: achieve

text: isochro-

text: alignment

text: errors.

text: nism

text: for

text: all

text: energies,

text: the

text: magnetic

text: ﬁeld

text: of

text: an

text: FFAG

text: can

text: This

text: letter

text: shows

text: that

text: a

text: racetrack

text: FFAG

text: conﬁgura-

text: obey

text: any

text: radial

text: or

text: azimuthal

text: dependence.

text: Thus,

text: with

text: tion

text: can

text: achieve

text: a

text: ﬁxed

text: tune

text: even

text: in

text: presence

text: of

text: ﬁeld

text: the

text: worldwide

text: interest

text: many

text: FFAG

text: designs

text: emerged

text: tai-

text: and

text: alignment

text: errors

text: and

text: thus

text: overcome

text: resonance

text: cross-

text: lored

text: to

text: areas

text: ranging

text: from

text: industrial

text: processes

text: to

text: med-

text: ings

text: problem.

text: The

text: method

text: is

text: based

text: on

text: creating

text: disper-

text: ical

text: and

text: fundamental

text: research

text: In

text: sion

text: free

text: straight

text: sections

text: where

text: a

text: time

text: varying

text: alternat-

text: the

text: early

text: days,

text: almost

text: all

text: designs

text: were

text: based

text: on

text: highly

text: ing

text: gradient

text: focusing

text: structure

text: is

text: placed.

text: The

text: merit

text: of

text: symmetric

text: rings

text: to

text: reduce

text: the

text: operating

text: restrictions

text: that

text: this

text: approach

text: is

text: that

text: the

text: time

text: varying

text: ﬁeld

text: structure

text: can

text: may

text: arise

text: with

text: systematic

text: ﬁeld

text: errors.

text: However,

text: recently,

text: be

text: programmed

text: in

text: a

text: way

text: to

text: maximize

text: the

text: overall

text: beam

text: several

text: racetrack

text: concepts

text: were

text: proposed

text: This

text: transmission

text: of

text: the

text: built

text: machine.

text: is

text: generally

text: considered

text: as

text: a

text: solution

text: to

text: practical

text: issues

text: The

text: idea

text: of

text: creating

text: a

text: dispersion-free

text: long

text: insertion

text: such

text: as

text: the

text: construction

text: cost,

text: the

text: injection

text: and

text: extrac-

text: in

text: an

text: FFAG

text: was

text: ﬁrst

text: discussed

text: by

text: Meads

text: in

text: tion

text: problem,

text: as

text: well

text: as

text: to

text: accommodate

text: and

text: simplify

text: However,

text: the

text: design

text: was

text: not

text: optimized

text: and

text: focused

text: on

text: the

text: design

text: of

text: the

text: RF

text: cavity.

text: using

text: scaling

text: FFAG

text: type

text: magnets

text: to

text: achieve

text: a

text: ﬁxed

text: tune

text: Although

text: recently

text: built

text: FFAGs

text: demonstrated

text: the

text: con-

text: machine.

text: In

text: addition,

text: the

text: sensitivity

text: to

text: misalignments

text: as

text: cept,

text: several

text: problems

text: remain.

text: In

text: it

text: is

text: shown

text: that

text: well

text: as

text: the

text: correction

text: of

text: the

text: optics

text: were

text: not

text: discussed.

text: small

text: gradient

text: ﬁeld

text: errors

text: in

text: a

text: ring

text: FFAG

text: accelerator

text: lead

text: The

text: design

text: discussed

text: in

text: this

text: paper

text: is

text: greatly

text: simpliﬁed

text: to

text: a

text: non-scaling

text: of

text: the

text: orbits.

text: The

text: latter

text: contributes

text: to

text: with

text: the

text: use

text: of

text: both

text: scaling

text: and

text: non-scaling

text: type

text: of

text: mag-

text: nets

text: to

text: create

text: a

text: racetrack

text: and

text: compact

text: microtron-like

text: conﬁguration.

text: As

text: illustrated

text: in

text: Fig.

text: a

text: conﬁguration

text: with

text: iden-

text: tical

text: radial

text: sector

text: magnets

text: is

text: privileged

text: since

text: it

text: provides

text: enough

text: space

text: between

text: the

text: magnets

text: to

text: place

text: in-

text: ##LTLine##


text: F. Proof of Theorem 4 The main idea of the proof is the following: we ﬁrst approximate the feedback capacity characterization (4), C , fb by invoking a FIR ﬁlter Q of order N and sampling the ﬁlter in frequency domain with equal spacing π . We denote m this two-degree approximation by CNfb(m) and prove that CNfb(m) converges to Cfb as N and m increase. Then we show the constructed code is essentially an optimal solution to CN (m), implying that the proposed feedback code fb is C -achieving as N and m are sufﬁciently large. We next prove Theorem 4 by providing detailed derivations in fb each aforementioned step. First of all, we present a ﬁnite dimensional optimization and we show that the dual of the optimization is equivalent to (23). This is a necessary result in order to show that the proposed code construction is essentially an optimal solution to CN (m) fb We start from (4), rewritten as follows (cid:90) (cid:16) (cid:17) Cfb =mU,aQx(cid:90)21π(cid:16)−ππ 12 log U(ejθ)+(cid:17)|1+Q(ejθ)|2 dθ, s.t. 21π −ππ U(ejθ)+|Q(ejθ)|2 Sw(ejθ)dθ ≤ P, (67) U(ejθ) ≥ 0, Q(ejθ) ∈ RH 2, strictly causal, where U(ejθ) = SSwv((eejjθθ)) since by assumption Sw(ejθ) > 0 for all θ ∈ [−π,π]. Letting c2(θ) = SSwv((eejjθθ)) and following the same steps as in (14), we obtain its generalization as (cid:90) Cfb =mΓax(cid:90)41π(cid:0) −ππ log(c2(θ)+(1+a((cid:1)θ))2 +b(θ)2)dθ s.t. (cid:90)21π −ππ a2(θ)+b2(θ)(cid:90)+c2(θ) Sw(θ)dθ ≤ P, (68) π π ≥ a(θ)cos(nθ)dθ + b(θ)sin(nθ)dθ = 0, n 0 − − π π Γ ={a(θ),b(θ),c(θ) : [−π,π] → R|a(θ),b(θ),c(θ) ∈ L2}. Note that we recover (14) if c(θ) = 0. We are now ready to consider a ﬁnite-dimensional approximation of the above problem. Deﬁne ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F.

text: Proof

text: of

text: Theorem

text: The

text: main

text: idea

text: of

text: the

text: proof

text: is

text: the

text: following:

text: we

text: ﬁrst

text: approximate

text: the

text: feedback

text: capacity

text: characterization

text: C

text: fb

text: by

text: invoking

text: a

text: FIR

text: ﬁlter

text: Q

text: of

text: order

text: N

text: and

text: sampling

text: the

text: ﬁlter

text: in

text: frequency

text: domain

text: with

text: equal

text: spacing

text: π

text: We

text: denote

text: m

text: this

text: two-degree

text: approximation

text: by

text: CNfb(m)

text: and

text: prove

text: that

text: CNfb(m)

text: converges

text: to

text: Cfb

text: as

text: N

text: and

text: m

text: increase.

text: Then

text: we

text: show

text: the

text: constructed

text: code

text: is

text: essentially

text: an

text: optimal

text: solution

text: to

text: CN

text: (m),

text: implying

text: that

text: the

text: proposed

text: feedback

text: code

text: fb

text: is

text: C

text: -achieving

text: as

text: N

text: and

text: m

text: are

text: sufﬁciently

text: large.

text: We

text: next

text: prove

text: Theorem

text: by

text: providing

text: detailed

text: derivations

text: in

text: fb

text: each

text: aforementioned

text: step.

text: First

text: of

text: all,

text: we

text: present

text: a

text: ﬁnite

text: dimensional

text: optimization

text: and

text: we

text: show

text: that

text: the

text: dual

text: of

text: the

text: optimization

text: is

text: equivalent

text: to

text: This

text: is

text: a

text: necessary

text: result

text: in

text: order

text: to

text: show

text: that

text: the

text: proposed

text: code

text: construction

text: is

text: essentially

text: an

text: optimal

text: solution

text: to

text: CN

text: (m)

text: fb

text: We

text: start

text: from

text: rewritten

text: as

text: follows

text: Cfb

text: log

text: dθ,

text: s.t.

text: −ππ

text: Sw(ejθ)dθ

text: P,

text: U(ejθ)

text: Q(ejθ)

text: RH

text: strictly

text: causal,

text: where

text: U(ejθ)

text: SSwv((eejjθθ))

text: since

text: by

text: assumption

text: Sw(ejθ)

text: for

text: all

text: θ

text: [−π,π].

text: Letting

text: SSwv((eejjθθ))

text: and

text: following

text: the

text: same

text: steps

text: as

text: in

text: we

text: obtain

text: its

text: generalization

text: as

text: Cfb

text: −ππ

text: s.t.

text: −ππ

text: Sw(θ)dθ

text: P,

text: π

text: π

text: a(θ)cos(nθ)dθ

text: b(θ)sin(nθ)dθ

text: n

text: π

text: π

text: Γ

text: ={a(θ),b(θ),c(θ)

text: [−π,π]

text: R|a(θ),b(θ),c(θ)

text: Note

text: that

text: we

text: recover

text: if

text: c(θ)

text: We

text: are

text: now

text: ready

text: to

text: consider

text: a

text: ﬁnite-dimensional

text: approximation

text: of

text: the

text: above

text: problem.

text: Deﬁne

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:113) (cid:113) We≥can rescale QmN(ejθ) by αN,m = pPmN = 1− P+δmδNm so that αN,mQmN(ejθ) uses power P. When δNm ≤ 0, αN,m 1 and we improve the lower bound, CˆNfb(m), by the rescaling. When δNm > 0, CˆNfb(m) is not a lower bound on the rate, however, αN,m < 1, and we obtain a lower bound using αN,mQmN(ejθ) in place of QmN(ejθ) . (cid:90) Based on Theorem 2, the last step of feedback code construction leads to a rate RN(m) = 21π −ππ log|1+αN,mQmN(ejθ)|dθ. We next note that − − − − δm = pm P = pm P+Pm Pm = ηm Pm . N N N fN fN N fN Furthermore, − − δNm = ηNm PfmN +PfN PfN, | | ≤ | | | | | − | (cid:82) δNm ηNm + PfN + PfmN PfN , where PfN = −ππ UN(ejθ)Sw(ejθ)dθ. As m → ∞, |ηNm| → 0 and |PfmN −PfN| → 0. Based on Remark 1, as N → ∞, | | → P 0. Thus fN Nli→m∞mli→m∞δNm = 0, and Nli→m∞mli→m∞αN,m = 1. Therefore, for large enough N and m, | − | CˆNfb(m) RN(m) < ε, and − Cfb RN(m) < 5ε. The proof is complete. G. Proof of Lemma 8 We consider the dual of (70). Since the Slater’s condition holds and we can always have a feasible primal solution, e.g.,Wi = xi = 1 and bi = 0, the dual problem has no gap. Moreover, the optimal dual solution exists. From standard derivations of dual problems involving LMI constraints [48], the dual problem is given by the following optimization, − Cfb(m,h) = λ≥0,η0,η∈mRha,xν˜i,ξ4i,ξ6i,ξ5i gD(λ,η0,η,νi,ξ4i,ξ6i) ≥ s.t. Ξi 0, i = 1,...,2m, r1i = 2λSw(θi)+η(cid:48)A(θi)+η0, (81) (cid:48) r2i = η B(θi), − π − θi = π + m(i 1), where ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We≥can

text: rescale

text: QmN(ejθ)

text: by

text: αN,m

text: pPmN

text: 1−

text: P+δmδNm

text: so

text: that

text: αN,mQmN(ejθ)

text: uses

text: power

text: P.

text: When

text: δNm

text: αN,m

text: and

text: we

text: improve

text: the

text: lower

text: bound,

text: CˆNfb(m),

text: by

text: the

text: rescaling.

text: When

text: δNm

text: CˆNfb(m)

text: is

text: not

text: a

text: lower

text: bound

text: on

text: the

text: rate,

text: however,

text: αN,m

text: and

text: we

text: obtain

text: a

text: lower

text: bound

text: using

text: αN,mQmN(ejθ)

text: in

text: place

text: of

text: QmN(ejθ)

text: Based

text: on

text: Theorem

text: the

text: last

text: step

text: of

text: feedback

text: code

text: construction

text: leads

text: to

text: a

text: rate

text: RN(m)

text: −ππ

text: We

text: next

text: note

text: that

text: δm

text: pm

text: P

text: pm

text: P+Pm

text: Pm

text: ηm

text: Pm

text: N

text: N

text: N

text: fN

text: fN

text: N

text: fN

text: Furthermore,

text: δNm

text: ηNm

text: PfmN

text: +PfN

text: PfN,

text: δNm

text: ηNm

text: PfN

text: PfmN

text: PfN

text: where

text: PfN

text: −ππ

text: UN(ejθ)Sw(ejθ)dθ.

text: As

text: m

text: |ηNm|

text: and

text: |PfmN

text: −PfN|

text: Based

text: on

text: Remark

text: as

text: N

text: P

text: Thus

text: fN

text: Nli→m∞mli→m∞δNm

text: and

text: Nli→m∞mli→m∞αN,m

text: Therefore,

text: for

text: large

text: enough

text: N

text: and

text: m,

text: CˆNfb(m)

text: RN(m)

text: ε,

text: and

text: Cfb

text: RN(m)

text: The

text: proof

text: is

text: complete.

text: G.

text: Proof

text: of

text: Lemma

text: We

text: consider

text: the

text: dual

text: of

text: Since

text: the

text: Slater’s

text: condition

text: holds

text: and

text: we

text: can

text: always

text: have

text: a

text: feasible

text: primal

text: solution,

text: e.g.,Wi

text: xi

text: and

text: bi

text: the

text: dual

text: problem

text: has

text: no

text: gap.

text: Moreover,

text: the

text: optimal

text: dual

text: solution

text: exists.

text: From

text: standard

text: derivations

text: of

text: dual

text: problems

text: involving

text: LMI

text: constraints

text: the

text: dual

text: problem

text: is

text: given

text: by

text: the

text: following

text: optimization,

text: Cfb(m,h)

text: s.t.

text: Ξi

text: i

text: η

text: B(θi),

text: π

text: θi

text: π

text: m(i

text: where

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text:   When the optimal dual solution(cid:20)is such that ν˜i > 0, then fro(cid:21)m(83) this condition reduces to     √  Wi xi bi  ν˜i −2√r1iν˜ −2√r2iν˜ xi 1 0 = 0. i i b 0 1 i which when solved leads to Wi = xi2 + b2i , ai = rν1i − 1 and bi = rν2i , where νi = 2ν˜i. Since ci = Wi − xi2 − b2i in i i Corollary 2, Wi = xi2 +b2i indicates ci = 0. Namely, when the optimal ν˜i > 0, then ci = 0. In other words, the optimal primal solution is exclusively obtained by feedback and the feedforward contribution must be zero. Moreover, in Q this case, the primal optimal solution (25), the components of , can be computed from the dual solution. Namely (cid:48) ai = 2λSw(θi)+νη A(θi)+η0 −1, (cid:48) i (84) bi = η Bν(θi), i where νi = 2ν˜i. REFERENCES [1] C. Li and N. Elia, “Control approach to computing the feedback capacity for stationary ﬁnite dimensional Gaussian channels,” 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1038–1045, 2015. [2] Y. H. Kim, “Feedback capacity of stationary Gaussian channels,” IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 57–85, 2010. [3] B. Francis, “Linear systems and control,” [online] Available: http://individual.utoronto.ca/brucefrancis1/second level/ECE1638 Francis. pdf, 2006. [4] E. J. Hannan and M. Deistler, The Statistical Theory of Linear Systems (Classics in Applied Mathematics). SIAM-Society for Industrial and Applied Mathematics, 2012. [5] S.Mitter, “Towards a uniﬁed theory of communication and control,” Seminar, LIDS-MIT, 2001. [6] N. Elia, “When Bode meets Shannon: control-oriented feedback communication schemes,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1477–1488, 2004. [7] J. L. Liu and N. Elia, “Convergence of fundamental limitations in feedback communication, estimation, and feedback control over Gaussian channels,” Communications in Information and Systems (CIS), vol. 14, no. 3, pp. 161–211, 2014. [8] O. Shayevitz and M. Feder, “Optimal feedback communication via posterior matching,” IEEE Transactions on Information Theory, vol. 57, no. 3, pp. 1186 – 1222, 2011. [9] E. Ardestanizadeh and M. Franceschetti, “Control-theoretic approach to communication with feedback,” IEEE Transactions on Automatic Control, vol. 57, no. 10, pp. 2576–2587, 2012. [10] E. Ardestanizadeh, P. Minero, and M. Franceschetti, “Lqg control approach to Gaussian broadcast channels with feedback,” IEEE Transactions on Information Theory, vol. 58, no. 8, pp. 5265–5278, 2012. [11] P. Elias, “Channel capacity without coding,” MIT Research Lab. of Electronics, Cambridge, MA, Quarterly Progr. Rep., 1956. [12] ——, “Networks of Gaussian channels with applications to feedback systems,” IEEE Transactions on Information Theory, vol. 13, pp. 493 – 501, 1967. [13] J. P. M. Schalkwijk and T. Kailath, “A coding scheme for additive noise channels with feedback, part i: no bandwidth constraint,” IEEE Transactions on Information Theory, vol. IT-12, no. 2, pp. 172–182, 1966. [14] J. P. M. Schalkwijk, “A coding scheme for additive noise channels with feedback, part ii: band-limited signals,” IEEE Transactions on Information Theory, vol. IT-12, no. 2, pp. 183–189, 1966. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: When

text: the

text: optimal

text: dual

text: such

text: that

text: ν˜i

text: then

text: this

text: condition

text: reduces

text: to

text: Wi

text: xi

text: bi

text: ν˜i

text: −2√r1iν˜

text: −2√r2iν˜

text: xi

text: i

text: i

text: b

text: i

text: which

text: when

text: solved

text: leads

text: to

text: Wi

text: ai

text: and

text: bi

text: where

text: νi

text: Since

text: ci

text: Wi

text: in

text: i

text: i

text: Corollary

text: Wi

text: indicates

text: ci

text: Namely,

text: when

text: the

text: optimal

text: ν˜i

text: then

text: ci

text: In

text: other

text: words,

text: the

text: optimal

text: primal

text: solution

text: is

text: exclusively

text: obtained

text: by

text: feedback

text: and

text: the

text: feedforward

text: contribution

text: must

text: be

text: zero.

text: Moreover,

text: in

text: Q

text: this

text: case,

text: the

text: primal

text: optimal

text: solution

text: the

text: components

text: of

text: can

text: be

text: computed

text: from

text: the

text: dual

text: solution.

text: Namely

text: ai

text: i

text: bi

text: η

text: Bν(θi),

text: i

text: where

text: νi

text: REFERENCES

text: C.

text: Li

text: and

text: N.

text: Elia,

text: “Control

text: approach

text: to

text: computing

text: the

text: feedback

text: capacity

text: for

text: stationary

text: ﬁnite

text: dimensional

text: Gaussian

text: channels,”

text: Annual

text: Allerton

text: Conference

text: on

text: Communication,

text: Control,

text: and

text: Computing

text: (Allerton),

text: pp.

text: Y.

text: H.

text: Kim,

text: “Feedback

text: capacity

text: of

text: stationary

text: Gaussian

text: channels,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: B.

text: Francis,

text: “Linear

text: systems

text: and

text: control,”

text: [online]

text: Available:

text: Francis.

text: pdf,

text: E.

text: J.

text: Hannan

text: and

text: M.

text: Deistler,

text: The

text: Statistical

text: Theory

text: of

text: Linear

text: Systems

text: (Classics

text: in

text: Applied

text: Mathematics).

text: SIAM-Society

text: for

text: Industrial

text: and

text: Applied

text: Mathematics,

text: S.Mitter,

text: “Towards

text: a

text: uniﬁed

text: theory

text: of

text: communication

text: and

text: control,”

text: Seminar,

text: LIDS-MIT,

text: N.

text: Elia,

text: “When

text: Bode

text: meets

text: Shannon:

text: control-oriented

text: feedback

text: communication

text: schemes,”

text: IEEE

text: Transactions

text: on

text: Automatic

text: Control,

text: vol.

text: no.

text: pp.

text: J.

text: L.

text: Liu

text: and

text: N.

text: Elia,

text: “Convergence

text: of

text: fundamental

text: limitations

text: in

text: feedback

text: communication,

text: estimation,

text: and

text: feedback

text: control

text: over

text: Gaussian

text: channels,”

text: Communications

text: in

text: Information

text: and

text: Systems

text: (CIS),

text: vol.

text: no.

text: pp.

text: O.

text: Shayevitz

text: and

text: M.

text: Feder,

text: “Optimal

text: feedback

text: communication

text: via

text: posterior

text: matching,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: E.

text: Ardestanizadeh

text: and

text: M.

text: Franceschetti,

text: “Control-theoretic

text: approach

text: to

text: communication

text: with

text: feedback,”

text: IEEE

text: Transactions

text: on

text: Automatic

text: Control,

text: vol.

text: no.

text: pp.

text: E.

text: Ardestanizadeh,

text: P.

text: Minero,

text: and

text: M.

text: Franceschetti,

text: “Lqg

text: control

text: approach

text: to

text: Gaussian

text: broadcast

text: channels

text: with

text: feedback,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: P.

text: Elias,

text: “Channel

text: capacity

text: without

text: coding,”

text: MIT

text: Research

text: Lab.

text: of

text: Electronics,

text: Cambridge,

text: MA,

text: Quarterly

text: Progr.

text: Rep.,

text: “Networks

text: of

text: Gaussian

text: channels

text: with

text: applications

text: to

text: feedback

text: systems,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: pp.

text: J.

text: P.

text: M.

text: Schalkwijk

text: and

text: T.

text: Kailath,

text: “A

text: coding

text: scheme

text: for

text: additive

text: noise

text: channels

text: with

text: feedback,

text: part

text: i:

text: no

text: bandwidth

text: constraint,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: J.

text: P.

text: M.

text: Schalkwijk,

text: “A

text: coding

text: scheme

text: for

text: additive

text: noise

text: channels

text: with

text: feedback,

text: part

text: ii:

text: band-limited

text: signals,”

text: IEEE

text: Transactions

text: on

text: Information

text: Theory,

text: vol.

text: no.

text: pp.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: K Starting with the decoder, we decompose as follows. We present the simplest solution here, others are possible. This coding scheme is illustrated in Fig.2. Linear Decoder K The decoder runs driven by Y. Xs(k +1) = AsXs(k)+BsY(k), Xs(0) = 0. Xˆu(k +1) = AuXˆu(k)+BuY(k), Xˆu(0) = 0. It produces two signals: an estimate of the negative initial condition of the encoder − − Xˆu0(k) = Au k 1Xˆu(k +1). (cid:104) (cid:105) and a feedback signal   Uˆ (k) = C C Xs(k) . s u Xˆu(k) Linear Encoder The encoder runs the following dynamics X˜u(k +1) = AuX˜u(k), X˜u(0) = Xu,0, U˜u(k) = CuX˜u(k), where X represents the message index. It receives Uˆ and produces the channel input u,0 U(k) = U˜u(k)+Uˆ (k) = Cu(X˜u(k)+Xˆu(k))+CsXs(k). Since the closed loop is stable, U(k) goes to zero with time if the noise is not present. Given the system is → − − observable, this implies that Xˆu(k) X˜u(k). Thus, Xˆu0(k) is an estimate at time k of X˜u(0) = Xu,0. In the presence of noise, the above coding scheme leads to Xˆu0(k) (cid:118) N (−Xu,0,A−u kE[Xˆu(k)Xˆu(k)(cid:48)](A−u k)(cid:48)) for large k, where E[Xˆu(k)Xˆu(k)(cid:48)] represents the state covariance matrix. Note that, since the system is observable and controllable, the matrix E[Xˆu(k)Xˆu(k)(cid:48)] is positive deﬁnite and so is A−u kE[Xˆu(k)Xˆu(k)(cid:48)](A−u k)(cid:48) (Theorem 4.3 in [6]). Remark 4: The above proposed coding scheme can be viewed as a generalized Schalkwijk-Kailath scheme. Based ≥ on the Schalkwijk’s scheme in [14], the channel input (encoder) and the message estimate (decoder) for k 2 can be summarized as follows by using the notati(cid:113)ons in this paper. (cid:113) − − − U(k) = A2u 1Aku 1(Xˆu0(k 1)+Xu0), (11) (cid:114) − − − − − Xˆu0(k) =Xˆu0(k 1) Au k 1 A2u 1Y(k), where Au = P+σσ2w2 and σw is the(cid:115)variance of the addi(cid:112)tive white Gaussia(cid:113)n noise in the forward channel 5. Now, w if in our scheme we let − P+σ2 − A2 1 − Au = σ2 w , Bu = Au , Cu = A2u 1, w u 5We clarify that in our scheme Xˆu0(k) is the estimate of −Xu0, or −θ in [14], at time instance k. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K

text: Starting

text: with

text: the

text: decoder,

text: we

text: decompose

text: as

text: follows.

text: We

text: present

text: the

text: simplest

text: solution

text: here,

text: others

text: are

text: possible.

text: This

text: coding

text: scheme

text: is

text: illustrated

text: in

text: Linear

text: Decoder

text: K

text: The

text: decoder

text: runs

text: driven

text: by

text: Y.

text: Xs(k

text: AsXs(k)+BsY(k),

text: Xˆu(k

text: AuXˆu(k)+BuY(k),

text: It

text: produces

text: two

text: signals:

text: an

text: estimate

text: of

text: the

text: negative

text: initial

text: condition

text: of

text: the

text: encoder

text: Au

text: k

text: and

text: a

text: feedback

text: signal

text: Uˆ

text: (k)

text: C

text: C

text: Xs(k)

text: s

text: u

text: Xˆu(k)

text: Linear

text: Encoder

text: The

text: encoder

text: runs

text: the

text: following

text: dynamics

text: X˜u(k

text: AuX˜u(k),

text: U˜u(k)

text: CuX˜u(k),

text: where

text: X

text: represents

text: the

text: message

text: index.

text: It

text: receives

text: Uˆ

text: and

text: produces

text: the

text: channel

text: input

text: U(k)

text: U˜u(k)+Uˆ

text: (k)

text: Cu(X˜u(k)+Xˆu(k))+CsXs(k).

text: Since

text: the

text: closed

text: loop

text: is

text: stable,

text: U(k)

text: goes

text: to

text: zero

text: with

text: time

text: if

text: the

text: noise

text: is

text: not

text: present.

text: Given

text: the

text: system

text: is

text: observable,

text: this

text: implies

text: that

text: Xˆu(k)

text: X˜u(k).

text: Thus,

text: is

text: an

text: estimate

text: at

text: time

text: k

text: of

text: In

text: the

text: presence

text: of

text: noise,

text: the

text: above

text: coding

text: scheme

text: leads

text: to

text: N

text: for

text: large

text: k,

text: where

text: represents

text: the

text: state

text: covariance

text: matrix.

text: Note

text: that,

text: since

text: the

text: system

text: is

text: observable

text: and

text: controllable,

text: the

text: matrix

text: is

text: positive

text: deﬁnite

text: and

text: so

text: is

text: A−u

text: (Theorem

text: in

text: Remark

text: The

text: above

text: proposed

text: coding

text: scheme

text: can

text: be

text: viewed

text: as

text: a

text: generalized

text: Schalkwijk-Kailath

text: scheme.

text: Based

text: on

text: the

text: Schalkwijk’s

text: scheme

text: in

text: the

text: channel

text: input

text: (encoder)

text: and

text: the

text: message

text: estimate

text: (decoder)

text: for

text: k

text: can

text: be

text: summarized

text: as

text: follows

text: by

text: using

text: the

text: in

text: this

text: paper.

text: U(k)

text: Au

text: k

text: where

text: Au

text: and

text: σw

text: is

text: of

text: the

text: white

text: noise

text: in

text: the

text: forward

text: channel

text: Now,

text: w

text: if

text: in

text: our

text: scheme

text: we

text: let

text: Au

text: σ2

text: w

text: Bu

text: Au

text: Cu

text: w

text: u

text: clarify

text: that

text: in

text: our

text: scheme

text: is

text: the

text: estimate

text: of

text: or

text: −θ

text: in

text: at

text: time

text: instance

text: k.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 low energy approximation under which we identiﬁed the pseudoparticles with L´evy-Leblond fermions. In the window between these two values of the radius of curvature, we expect that deforming the few–layer graphene sheets provides an eﬃcient way to increase or reduce an energy band gap between conduction and valence bands, since few layers graphene has several degrees of freedom with respect to which deformations can be introduced: both along the sheets and perpendicular to it. CONCLUSIONS In this work we have extended the geometrical approach and the results of our previous work on curvatronics with bilayer graphene of Ref.[16] to the case of three and four layers of AB stacked graphene. We found that L´evy-Leblond fermions are still present at the bottom of the parabolic bands, which we call Galilei points of the Brillouin zone, and we gave a geometrical interpretation of the solutions in terms of the p orbitals of graphene associated to them, on z the basis of their non-trivial spatial overlap. We have discussed in detail the eﬀective electronic masses present for three and four layers, comparing them with the bilayer graphene case, as well as the explicit form of the solutions. We analysed the L´evy-Leblond equation and showed how energy eigenstates are given by the superpositions of states with deﬁnite pseudospin. Using the relationship between the 2D L´evy-Leblond equation and the 4D massless Dirac equation, we were able to relate 2D pseudospin with chirality in 4D. Lastly, we discussed the fact that curving few layers graphene provides a tool to tune the energy band gap, which has implications for tunable electronics based on curvature, or curvatronics with few-layer graphene systems. For instance, a graphene–based device with alternating regions of positive or negative curvature, inducing insulating or conductive states, could be exploited to produce non–volatile memories. It would be interesting for future research to generalize our results to a higher number of layers, and to study the fermionic excitations away from the Galilei points. Another avenue of possible future research is studying the physics of pseudospin in settings with non-trivial curvature or topology, which might lead to novel eﬀects. Acknowledgments – M. Cariglia is funded by CNPq under project 303923/2015-6, and by a Pesquisador Mineiro project n. PPM-00630-17. The authors acknowledge the collaboration within the MultiSuper International Network (http://www.multisuper.org) for exchange of ideas and suggestions. [1] Geim, A. K.; Grigorieva, I. V. Van der Waals heterostructures, Nature 499, 419 (2013). [2] Perali, A.; Neilson, D.; Hamilton, A.R. High-Temperature Superﬂuidity in Double-Bilayer Graphene, Phys. Rev. Lett. 110, 146803 (2013). [3] Zarenia, M.; Perali, A.; Neilson, D; Peeters, F.M. Enhancement of electron-hole superﬂuidity in double few-layer graphene, Scientiﬁc Reports 4, 7319 (2014). [4] Ferrari, A. C. et al. Raman Spectrum of Graphene and Graphene Layers, Phys. Rev. Lett. 97, 187401 (2006). [5] Zhang, Y.; Small, J. P.; Pontius, W. V.; Kim, Ph. Fabrication and electric-ﬁeld-dependent transport measurements of mesoscopic graphite devices, Appl. Phys. Lett. 86, 073104 (2005). [6] Berger, C. et al. Ultrathin Epitaxial Graphite: 2D Electron Gas Properties and a Route toward Graphene-based Nano- electronics, J. Phys. Chem. B 108, 19912 (2004). [7] Shih, C. J. et al. Bi- and trilayer graphene solutions, Nat. Nanotechnol. 6, 439–445 (2011). [8] Mahanandia, P.; Simon, F.; Heinrich; G. and Nanda; K. K. An electrochemical method for the synthesis of few layer graphene sheets for high temperature applications, Chem. Commun. 50, 4613 (2014). [9] Craciun, M. F. et al. Trilayer graphene is a semimetal with a gate-tunable band overlap, Nature Nanotech 4, 383 (2009). [10] Bao, W. et al. Stacking-dependent band gap and quantum transport in trilayer graphene, Nature Phys. 7, 948 (2011). [11] Mak, K. F.; Shan, J.; Heinz, T. F. Electronic Structure of Few-Layer Graphene: Experimental Demonstration of Strong Dependence on Stacking Sequence, Phys. Rev. Lett. 104, 176404 (2010). [12] Zhang, Y. et al. Direct observation of a widely tunable bandgap in bilayer graphene, Nature 459, 820 (2009). [13] Lui, Ch. H. et al. Observation of an electrically tunable band gap in trilayer graphene, Nature Phys. 7, 944 (2011). [14] Zou, K.; Zhang, F.; Clapp, C.; MacDonald, A. H.; Zhu, J. Transport Studies of Dual-Gated ABC and ABA Trilayer Graphene: Band Gap Opening and Band Structure Tuning in Very Large Perpendicular Electric Fields, Nano Lett. 13, 369 (2013). [15] Avetisyan, A. A.; Partoens, B.; Peeters, F. M. Stacking order dependent electric ﬁeld tuning of the band gap in graphene multilayers, Phys. Rev. B 81, 115432 (2010). [16] Cariglia, M; Giamb`o, R.; Perali, A. Curvature-tuned electronic properties of bilayer graphene in an eﬀective four- dimensional spacetime. Phys. Rev. B 2017, 95, 245426, DOI. [17] Sepehri, A.; Pincak, R.; Bamba, K.; Capozziello, S.; Saridakis, E.N. Current density and conductivity through modiﬁed gravity in the graphene with defects, Int. J. Mod. Phys. D 2017 26, 1750094. [18] Weyl, H. Elektron und gravitation. I., Z. Phys. 56, 330-352 (1929). ##LTLine## ##LTLine## ##LTLine##

text: low

text: energy

text: approximation

text: under

text: which

text: we

text: identiﬁed

text: the

text: pseudoparticles

text: with

text: L´evy-Leblond

text: fermions.

text: In

text: the

text: window

text: between

text: these

text: two

text: values

text: of

text: the

text: radius

text: of

text: curvature,

text: we

text: expect

text: that

text: deforming

text: the

text: few–layer

text: graphene

text: sheets

text: provides

text: an

text: eﬃcient

text: way

text: to

text: increase

text: or

text: reduce

text: an

text: energy

text: band

text: gap

text: between

text: conduction

text: and

text: valence

text: bands,

text: since

text: few

text: layers

text: graphene

text: has

text: several

text: degrees

text: of

text: freedom

text: with

text: respect

text: to

text: which

text: deformations

text: can

text: be

text: introduced:

text: both

text: along

text: the

text: sheets

text: and

text: perpendicular

text: to

text: it.

text: CONCLUSIONS

text: In

text: this

text: work

text: we

text: have

text: extended

text: the

text: geometrical

text: approach

text: and

text: the

text: results

text: of

text: our

text: previous

text: work

text: on

text: curvatronics

text: with

text: bilayer

text: graphene

text: of

text: to

text: the

text: case

text: of

text: three

text: and

text: four

text: layers

text: of

text: AB

text: stacked

text: graphene.

text: We

text: found

text: that

text: L´evy-Leblond

text: fermions

text: are

text: still

text: present

text: at

text: the

text: bottom

text: of

text: the

text: parabolic

text: bands,

text: which

text: we

text: call

text: Galilei

text: points

text: of

text: the

text: Brillouin

text: zone,

text: and

text: we

text: gave

text: a

text: geometrical

text: interpretation

text: of

text: the

text: solutions

text: in

text: terms

text: of

text: the

text: p

text: orbitals

text: of

text: graphene

text: associated

text: to

text: them,

text: on

text: z

text: the

text: basis

text: of

text: their

text: non-trivial

text: spatial

text: overlap.

text: We

text: have

text: discussed

text: in

text: detail

text: the

text: eﬀective

text: electronic

text: masses

text: present

text: for

text: three

text: and

text: four

text: layers,

text: comparing

text: them

text: with

text: the

text: bilayer

text: graphene

text: case,

text: as

text: well

text: as

text: the

text: explicit

text: form

text: of

text: the

text: solutions.

text: We

text: analysed

text: the

text: L´evy-Leblond

text: equation

text: and

text: showed

text: how

text: energy

text: eigenstates

text: are

text: given

text: by

text: the

text: superpositions

text: of

text: states

text: with

text: deﬁnite

text: pseudospin.

text: Using

text: the

text: relationship

text: between

text: the

text: L´evy-Leblond

text: equation

text: and

text: the

text: massless

text: Dirac

text: equation,

text: we

text: were

text: able

text: to

text: relate

text: pseudospin

text: with

text: chirality

text: in

text: Lastly,

text: we

text: discussed

text: the

text: fact

text: that

text: curving

text: few

text: layers

text: graphene

text: provides

text: a

text: tool

text: to

text: tune

text: the

text: energy

text: band

text: gap,

text: which

text: has

text: implications

text: for

text: tunable

text: electronics

text: based

text: on

text: curvature,

text: or

text: curvatronics

text: with

text: few-layer

text: graphene

text: systems.

text: For

text: instance,

text: a

text: graphene–based

text: device

text: with

text: alternating

text: regions

text: of

text: positive

text: or

text: negative

text: curvature,

text: inducing

text: insulating

text: or

text: conductive

text: states,

text: could

text: be

text: exploited

text: to

text: produce

text: non–volatile

text: memories.

text: It

text: would

text: be

text: interesting

text: for

text: future

text: research

text: to

text: generalize

text: our

text: results

text: to

text: a

text: higher

text: number

text: of

text: layers,

text: and

text: to

text: study

text: the

text: fermionic

text: excitations

text: away

text: from

text: the

text: Galilei

text: points.

text: Another

text: avenue

text: of

text: possible

text: future

text: research

text: is

text: studying

text: the

text: physics

text: of

text: pseudospin

text: in

text: settings

text: with

text: non-trivial

text: curvature

text: or

text: topology,

text: which

text: might

text: lead

text: to

text: novel

text: eﬀects.

text: Acknowledgments

text: M.

text: Cariglia

text: is

text: funded

text: by

text: CNPq

text: under

text: project

text: and

text: by

text: a

text: Pesquisador

text: Mineiro

text: project

text: n.

text: The

text: authors

text: acknowledge

text: the

text: collaboration

text: within

text: the

text: MultiSuper

text: International

text: Network

text: for

text: exchange

text: of

text: ideas

text: and

text: suggestions.

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text: A.

text: K.;

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##