text: − Then setting B = (ζI + A) 1 for small values of ∆t, from (53), we get the system (cid:16) (cid:17) O u + ζbT Bev + bT BA˜f(U) = bT BAp(U) + (∆t) t x x xx 1 − 1 − O (54) v + bT p(U) = bT V ˜bT f(U) + (∆t), t ε2 x ε2 which is similar to Eq. (35), except that now A and bT appear in front of p(U) terms in place of A˜ and ˜b respectively. Therefore, we have the same characteristic speeds as in the uniﬁed IMEX-RK approach presented in the previous section and the same conclusions on the hyperbolic CFL condition holds true. → Concerning the AP property, in the limit ζ 0 one has from (51) − − V = p(U) + A 1A˜f(U), x − and by the GSA property, i.e. bT A 1A˜ = eT A˜ = ˜bT , we get from (53) s − − − vn+1 = eT (p(U) A 1A˜f(U)). s x Thus, scheme (52)-(53), becomes an IMEX-RK scheme for the convection-diﬀusion equation (6) − U une + A˜f(U) = Ap(U) , x xx ∆t − (55) un+1 un + ˜bT f(U) = bT p(U) . x xx ∆t Clearly, scheme (55) is a consistent approximation of the limit equation (6) where now the diﬀusion term is evaluated implicitly, therefore the CFL condition of such scheme is uniquely ∼ determined by the hyperbolic restriction ∆t ∆x. ∈ A similar analysis in the case α [0, 1) for scheme (49)-(50) under the GSA assumption produces the explicit Runge-Kutta scheme − U = un ∆tA˜f(U) , x − (56) un+1 = un ∆t˜bT A˜f(U) . x Therefore we can summarize the results in the following Theorem 2 If the IMEX-RK scheme (49)-(50), applied to () for α = 1, satisﬁes the GSA → property then, as ε 0, it becomes the IMEX-RK method characterized by the pairs (A˜,˜b) and ∈ (A, b) for the limit convection-diﬀusion equation (6). Otherwise for α [0, 1) the IMEX-RK → scheme as ε 0 yields the explict RK method characterized by the pair (A˜,˜b) for the limit scalar conservation law (4). 4 Numerical applications In this section we present numerical results that conﬁrm the validity of the new approach presented in section . In all tests we used the last approach described in section , so that we 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Then

text: setting

text: B

text: (ζI

text: A)

text: for

text: small

text: values

text: of

text: ∆t,

text: from

text: we

text: get

text: the

text: system

text: O

text: u

text: ζbT

text: Bev

text: bT

text: BA˜f(U)

text: bT

text: BAp(U)

text: (∆t)

text: t

text: x

text: x

text: xx

text: O

text: v

text: bT

text: p(U)

text: bT

text: V

text: ˜bT

text: f(U)

text: (∆t),

text: t

text: ε2

text: x

text: ε2

text: which

text: is

text: similar

text: to

text: Eq.

text: except

text: that

text: now

text: A

text: and

text: bT

text: appear

text: in

text: front

text: of

text: p(U)

text: terms

text: in

text: place

text: of

text: A˜

text: and

text: ˜b

text: respectively.

text: Therefore,

text: we

text: have

text: the

text: same

text: characteristic

text: speeds

text: as

text: in

text: the

text: uniﬁed

text: IMEX-RK

text: approach

text: presented

text: in

text: the

text: previous

text: section

text: and

text: the

text: same

text: conclusions

text: on

text: the

text: hyperbolic

text: CFL

text: condition

text: holds

text: true.

text: Concerning

text: the

text: AP

text: property,

text: in

text: the

text: limit

text: ζ

text: one

text: has

text: from

text: V

text: p(U)

text: A

text: x

text: and

text: by

text: the

text: GSA

text: property,

text: i.e.

text: bT

text: A

text: eT

text: A˜

text: ˜bT

text: we

text: get

text: from

text: s

text: eT

text: (p(U)

text: A

text: s

text: x

text: Thus,

text: scheme

text: becomes

text: an

text: IMEX-RK

text: scheme

text: for

text: the

text: convection-diﬀusion

text: equation

text: U

text: une

text: A˜f(U)

text: Ap(U)

text: x

text: xx

text: ∆t

text: un

text: ˜bT

text: f(U)

text: bT

text: p(U)

text: x

text: xx

text: ∆t

text: Clearly,

text: scheme

text: is

text: a

text: consistent

text: approximation

text: of

text: the

text: limit

text: equation

text: where

text: now

text: the

text: diﬀusion

text: term

text: is

text: evaluated

text: implicitly,

text: therefore

text: the

text: CFL

text: condition

text: of

text: such

text: scheme

text: is

text: uniquely

text: determined

text: by

text: the

text: hyperbolic

text: restriction

text: ∆t

text: ∆x.

text: A

text: similar

text: analysis

text: in

text: the

text: case

text: α

text: for

text: scheme

text: under

text: the

text: GSA

text: assumption

text: produces

text: the

text: explicit

text: Runge-Kutta

text: scheme

text: U

text: un

text: ∆tA˜f(U)

text: x

text: un

text: ∆t˜bT

text: A˜f(U)

text: x

text: Therefore

text: we

text: can

text: summarize

text: the

text: results

text: in

text: the

text: following

text: Theorem

text: If

text: the

text: IMEX-RK

text: scheme

text: applied

text: to

text: for

text: α

text: satisﬁes

text: the

text: GSA

text: property

text: then,

text: as

text: ε

text: it

text: becomes

text: the

text: IMEX-RK

text: method

text: characterized

text: by

text: the

text: pairs

text: (A˜,˜b)

text: and

text: (A,

text: b)

text: for

text: the

text: limit

text: convection-diﬀusion

text: equation

text: Otherwise

text: for

text: α

text: the

text: IMEX-RK

text: scheme

text: as

text: ε

text: yields

text: the

text: explict

text: RK

text: method

text: characterized

text: by

text: the

text: pair

text: (A˜,˜b)

text: for

text: the

text: limit

text: scalar

text: conservation

text: law

text: Numerical

text: applications

text: In

text: this

text: section

text: we

text: present

text: numerical

text: results

text: that

text: conﬁrm

text: the

text: validity

text: of

text: the

text: new

text: approach

text: presented

text: in

text: section

text: In

text: all

text: tests

text: we

text: used

text: the

text: last

text: approach

text: described

text: in

text: section

text: so

text: that

text: we

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: remove the parabolic restriction in the limit of small ε and α = 1. All the numerical examples presented here refer to the IMEX-RK schemes reported in the Appendix. We shall use the notation NAME(ν, σ, p), where the triplet (ν, σ, p) where ν, σ and p represent respectively the number of explicit function evaluations, the number of implicit function evaluations and the order of accuracy. In order to avoid spurious numerical oscillations arising near discontinuities of the solutions, we use interpolating non-oscillatory algorithms, like WENO method, [36]. In these numerical test, as emphasized in Remark 1, we use classical hyperbolic-type schemes, like ﬁnite diﬀerence discretization with WENO reconstruction, for the space derivatives characterized the hyperbolic part while for the second order term p(u) we used the standard 2-th and 4-th order ﬁnite xx diﬀerence technique. Note that when we consider third-order IMEX R-K scheme we use 4-th order ﬁnite diﬀerence to discretize the term p(u) except at the nearby boundary points where xx a 3-rd order formula was implemented, and this guarantees to achieve a global third-order. In all our numerical results we take ∆t = λ ∆x in all regimes. The precise choice of λ CFL CFL is reported in the ﬁgure captions. As a mathematical model for our numerical experiments we consider the Ruijgrook-Wu model of the discrete kinetic theory of rareﬁed gases. The model describes a two-speed gas in one space dimension and corresponds to the system [17, 25, 35]  − 1 − − − −  M∂tf+ + ∂xf+ = Kn(af+ bf cf+f ), (57) − − − 1 − − − − M∂ f ∂ f = (af+ bf cf+f ), t x Kn − where f+ and f denote the particle density distribution at time t, position x and with velocity − +1 and 1 respectively. Here Kn is the Knudsen number, M is the Mach number of the system and a,b and c are positive constants which characterize the microscopic interactions. The local (Maxwellian) equilibrium is deﬁned by − bf f+ = − − . (58) a cf The macroscopic variables for the model are the density u and momentum v deﬁned by − − − u = f+ + f , v = (f+ f )/M. (59) The nondimensional multiscale problem is obtained taking M = εα and Kn = ε, the Reynolds − number of the system is then deﬁned as usual according to Re = M/Kn = 1/ε1 α. The model, as we will see, has the nice feature to provide nontrivial limit behaviors for several values of α including the corresponding compressible Euler (α = 0) limit and the incompressible Euler ∈ (α (0, 1)) and Navier-Stokes (α = 1) limits. Test 1. Diﬀusive scaling in the linear case − For this numerical test we consider the case α = 1 with c = 0, a = 1 + Aε and b = 1 Aε in the r.h.s. of (57). Adding and subtracting the two equations in (57) one obtains the following macroscopic  equations for u and v    ∂tu + ∂xv = 0, (60) 1 − 1 − ∂ v + ∂ u = (v Au). t ε2 x ε2 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: remove

label: paragraph

text: the

text: parabolic

text: restriction

text: in

text: the

text: limit

text: of

text: small

text: ε

text: and

text: α

text: All

text: the

text: numerical

text: examples

text: presented

text: here

text: refer

text: to

text: the

text: IMEX-RK

text: schemes

text: reported

text: in

text: the

text: Appendix.

text: We

text: shall

text: use

text: the

text: notation

text: NAME(ν,

text: σ,

text: p),

text: where

text: the

text: triplet

text: (ν,

text: σ,

text: p)

text: where

text: ν,

text: σ

text: and

text: p

text: represent

text: respectively

text: the

text: number

text: of

text: explicit

text: function

text: evaluations,

text: the

text: number

text: of

text: implicit

text: function

text: evaluations

text: and

text: the

text: order

text: of

text: accuracy.

text: In

text: order

text: to

text: avoid

text: spurious

text: numerical

text: oscillations

text: arising

text: near

text: discontinuities

text: of

text: the

text: solutions,

text: we

text: use

text: interpolating

text: non-oscillatory

text: algorithms,

text: like

text: WENO

text: method,

text: In

text: these

text: numerical

text: test,

text: as

text: emphasized

text: in

text: Remark

text: we

text: use

text: classical

text: hyperbolic-type

text: schemes,

text: like

text: ﬁnite

text: diﬀerence

text: discretization

text: with

text: WENO

text: reconstruction,

text: for

text: the

text: space

text: derivatives

text: characterized

text: the

text: hyperbolic

text: part

text: while

text: for

text: the

text: second

text: order

text: term

text: p(u)

text: we

text: used

text: the

text: standard

text: and

text: order

text: ﬁnite

text: xx

text: diﬀerence

text: technique.

text: Note

text: that

text: when

text: we

text: consider

text: third-order

text: IMEX

text: R-K

text: scheme

text: we

text: use

text: order

text: ﬁnite

text: diﬀerence

text: to

text: discretize

text: the

text: term

text: p(u)

text: except

text: at

text: the

text: nearby

text: boundary

text: points

text: where

text: xx

text: a

text: order

text: formula

text: was

text: implemented,

text: and

text: this

text: guarantees

text: to

text: achieve

text: a

text: global

text: third-order.

text: In

text: all

text: our

text: numerical

text: results

text: we

text: take

text: ∆t

text: λ

text: ∆x

text: in

text: all

text: regimes.

text: The

text: precise

text: choice

text: of

text: λ

text: CFL

text: CFL

text: is

text: reported

text: in

text: the

text: ﬁgure

text: captions.

text: As

text: a

text: mathematical

text: model

text: for

text: our

text: numerical

text: experiments

text: we

text: consider

text: the

text: Ruijgrook-Wu

text: model

text: of

text: the

text: discrete

text: kinetic

text: theory

text: of

text: rareﬁed

text: gases.

text: The

text: model

text: describes

text: a

text: two-speed

text: gas

text: in

text: one

text: space

text: dimension

text: and

text: corresponds

text: to

text: the

text: system

text: M∂tf+

text: ∂xf+

text: Kn(af+

text: bf

text: cf+f

text: M∂

text: f

text: f

text: (af+

text: bf

text: cf+f

text: t

text: x

text: Kn

text: where

text: f+

text: and

text: f

text: denote

text: the

text: particle

text: density

text: distribution

text: at

text: time

text: t,

text: position

text: x

text: and

text: with

text: velocity

text: and

text: respectively.

text: Here

text: Kn

text: is

text: the

text: Knudsen

text: number,

text: M

text: is

text: the

text: Mach

text: number

text: of

text: the

text: system

text: and

text: a,b

text: and

text: c

text: are

text: positive

text: constants

text: which

text: characterize

text: the

text: microscopic

text: interactions.

text: The

text: local

text: (Maxwellian)

text: equilibrium

text: is

text: deﬁned

text: by

text: bf

text: f+

text: a

text: cf

text: The

text: macroscopic

text: variables

text: for

text: the

text: model

text: are

text: the

text: density

text: u

text: and

text: momentum

text: v

text: deﬁned

text: by

text: u

text: f+

text: f

text: v

text: (f+

text: f

text: )/M.

text: The

text: nondimensional

text: multiscale

text: problem

text: is

text: obtained

text: taking

text: M

text: εα

text: and

text: Kn

text: ε,

text: the

text: Reynolds

text: number

text: of

text: the

text: system

text: is

text: then

text: deﬁned

text: as

text: usual

text: according

text: to

text: Re

text: M/Kn

text: α.

text: The

text: model,

text: as

text: we

text: will

text: see,

text: has

text: the

text: nice

text: feature

text: to

text: provide

text: nontrivial

text: limit

text: behaviors

text: for

text: several

text: values

text: of

text: α

text: including

text: the

text: corresponding

text: compressible

text: Euler

text: (α

text: limit

text: and

text: the

text: incompressible

text: Euler

text: (α

text: and

text: Navier-Stokes

text: (α

text: limits.

text: Test

text: Diﬀusive

text: scaling

text: in

text: the

text: linear

text: case

text: For

text: this

text: numerical

text: test

text: we

text: consider

text: the

text: case

text: α

text: with

text: c

text: a

text: Aε

text: and

text: b

text: Aε

text: in

text: the

text: r.h.s.

text: of

text: Adding

text: and

text: subtracting

text: the

text: two

text: equations

text: in

text: one

text: obtains

text: the

text: following

text: macroscopic

text: equations

text: for

text: u

text: and

text: v

text: ∂tu

text: ∂xv

text: v

text: u

text: (v

text: Au).

text: t

text: ε2

text: x

text: ε2

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − Table 2: Test 1. Converge rates for v for the scheme CK(2,2,2) with ε = 10 6. Method N L∞ v-error Order of v − −− CK(2,2,2) 40 1.3156e 02 − − CK(2,2,2) 80 1.4897e 02 0.1793 − CK(2,2,2) 160 1.1520e 03 3.6928 − − CK(2,2,2) 320 1.2584e 03 0.1274 − − CK(2,2,2) 640 1.2877e 03 0.0332 − Table 3: Test 1. Converge rates for u and v for the scheme (81) with ε = 10 6. Method N L∞ u-error Order u L∞ v-error Order v − −− − −− BPR(4,4,2) 40 1.9129e 04 2.8704e 04 − − BPR(4,4,2) 80 4.9963e 05 1.9368 8.0261e 05 1.8385 − − BPR(4,4,2) 160 1.4374e 05 1.7974 2.0603e 05 1.9618 − − BPR(4,4,2) 320 3.5895e 06 2.0016 5.2702e 06 1.9669 − − BPR(4,4,2) 640 9.0120e 07 1.9939 1.4011e 06 1.9113 (cid:26) We consider two diﬀerent initial conditions. The ﬁrst one is given by two local Maxwellian characterized by − u = 1.0, v = 0, 10 < x < 0, L L (65) u = 2.0, v = 0, 0 < x < 10, R R − with v = [(1 + u2ε2)1/2 1]/ε2. We show in Figure ?? the numerical solution for u in the case α = 1 using BPR(4,4,2) and − BPR(3,4,3) schemes in the rareﬁed (ε = 0.4) and parabolic (ε = 10 6) regimes with ∆x = 0.2 at the ﬁnal time T = 0.2. The solution of both schemes is in very good agreement with the reference solution computed with ∆x = 0.04. Finally, we present the diﬀerent IMEX-RK schemes, up to order three, used in our numerical tests. Below, these schemes are represented as usual by the double Butcher tableau. On the left we have the explicit part and on the right the implicit part of the IMEX schemes. All the schemes satisfy the GSA property, but only the new scheme BPR(4, 4, 2) satisﬁes the additional order conditions (??). • First order ARS(1, 1, 1) scheme 0 0 0 0 0 0 1 1 0 1 0 1 (78) 1 0 0 1 • Second order ARS(2, 2, 2) scheme [1] 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: caption

text: Table

text: Test

text: Converge

text: rates

text: for

text: v

text: for

text: the

text: scheme

text: with

text: ε

text: Method

label: table

text: N

text: L∞

text: v-error

text: Order

text: of

text: v

text: Table

text: Test

text: Converge

text: rates

text: for

text: u

text: and

text: v

text: for

text: the

text: scheme

text: with

text: ε

text: Method

text: N

text: L∞

text: u-error

text: Order

text: u

text: L∞

text: v-error

text: Order

text: v

text: We

text: consider

text: two

text: diﬀerent

text: initial

text: conditions.

text: The

text: ﬁrst

text: one

text: is

text: given

text: by

text: two

text: local

text: Maxwellian

text: characterized

text: by

text: u

text: v

text: x

text: L

text: L

text: u

text: v

text: x

text: R

text: R

text: with

text: v

text: We

text: show

text: in

text: Figure

text: the

text: numerical

text: solution

text: for

text: u

text: in

text: the

text: case

text: α

text: using

text: and

text: schemes

text: in

text: the

text: rareﬁed

text: (ε

text: and

text: parabolic

text: (ε

text: regimes

text: with

text: ∆x

text: at

text: the

text: ﬁnal

text: time

text: T

text: The

text: solution

text: of

text: both

text: schemes

text: is

text: in

text: very

text: good

text: agreement

text: with

text: the

text: reference

text: solution

text: computed

text: with

text: ∆x

text: Finally,

text: we

text: present

text: the

text: diﬀerent

text: IMEX-RK

text: schemes,

text: up

text: to

text: order

text: three,

text: used

text: in

text: our

text: numerical

text: tests.

text: Below,

text: these

text: schemes

text: are

text: represented

text: as

text: usual

text: by

text: the

text: double

text: Butcher

text: tableau.

text: On

text: the

text: left

text: we

text: have

text: the

text: explicit

text: part

text: and

text: on

text: the

text: right

text: the

text: implicit

text: part

text: of

text: the

text: IMEX

text: schemes.

text: All

text: the

text: schemes

text: satisfy

text: the

text: GSA

text: property,

text: but

text: only

text: the

text: new

text: scheme

text: satisﬁes

text: the

text: additional

text: order

text: conditions

text: First

text: order

text: scheme

text: Second

text: order

text: scheme

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: [3] S. Boscarino, On an accurate third order implicit-explicit Rungeutta method for stiﬀ problems, Applied Numerical Mathematics, Vol. 59, pp. 1515–1528, 2009. [4] C. Bardos, C.D. Levermore, F. Golse, Fluid dynamic limit of kinetic equations II: Convergence proofs for the Boltzmann equations, Comm. Pure Appl. Math., 46, (1993), 667-753. [5] S. Boscarino, L. Pareschi, On the asymptotic properties of IMEX Runge–Kutta schemes for hyperbolic balance laws, J. Comput. Appl. Math. (to appear). [6] S. Boscarino, L. Pareschi, G. Russo, Implicit-Explicit Runge-Kutta schemes for hy- perbolic systems and kinetic equations in the diﬀusion limit, SIAM J. Sci. Comput., Vol. 35, No. 1, pp. A2251. [7] S. Boscarino, G. Russo, On a class of uniformly accurate IMEX Runge-Kutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput. Vol. 31, No. 3, pp. 1926-1945. [8] S. Boscarino, G. Russo, Flux-Explicit IMEX Runge-Kutta schemes for hyperbolic to parabolic relaxation problems, SIAM J. NUMER. ANAL., Vol. 51, No. 1, 163–190. [9] S. Boscarino, P. Le Floch, G. Russo, High-Order asymptotic-preserving methods for fully non linear relaxation problems, SIAM J. Sci. Comput., Vol. 36, No. 2, pp. A377–A395. [10] R.E. Caflisch, S. Jin, G. Russo, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Num. Anal., 34, 1, (1997), 246-281. [11] M.H. Carpenter, C.A. Kennedy, Additive Runge-Kutta schemes for convection- diﬀusion-reaction equations Appl. Numer. Math. 44 (2003), no. 1-2, 139–181. [12] C. Cercignani, R. Illner, M. Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, (1994). [13] G.Q. Chen, C.D. Levermore, T.P. Liu, Hyperbolic conservation laws with stiﬀ relax- ation terms and entropy, Comm. Pure and App. Math., XLVII, (1994), 787-830. [14] C. Cercignani, The Boltzmann equation and its applications, Springer Varlag New York, (1988). [15] G. Dimarco, L. Pareschi, Asymptotic-preserving IMEX Runge-Kutta methods for non- linear kinetic equations, SIAM J. Num. Anal. (2013), 1064–1087. [16] F. Filbet, S. Jin, A class of asymptotic preserving schemes for kinetic equations and related problems with stiﬀ sources, J. Comp. Phys., 229 (2010), pp. 7625-7648. [17] E. Gabetta, B. Perthame, Scaling limits of the Ruijgrook-Wu model of the Boltzmann equation, Proceedings of the International Conference on Nonlinear Equations and Appli- cations, Bangalore 19-23 August 1996. Springer-Verlag. [18] E. Gabetta, L. Pareschi, G. Toscani, Relaxation schemes for nonlinear kinetic equa- tions, SIAM J. Numer. Anal. 34, (1997), no. 6, 2168–2194. [19] E. Hairer, G. Wanner, Solving Ordinary Diﬀerential Equation II: stiﬀ and Diﬀerential Algebraic Problems. Springer Series in Comput. Mathematics, Vol. 14, Springer-Verlag 1991, Second revised edition 1996. 22

label: reference

text: S.

text: Boscarino,

text: On

text: an

text: accurate

text: third

text: order

text: implicit-explicit

text: Rungeutta

text: method

text: for

text: stiﬀ

text: problems,

text: Applied

text: Numerical

text: Mathematics,

text: Vol.

text: pp.

text: C.

text: Bardos,

text: C.D.

text: Levermore,

text: F.

text: Golse,

text: Fluid

text: dynamic

text: limit

text: of

text: kinetic

text: equations

text: II:

text: Convergence

text: proofs

text: for

text: the

text: Boltzmann

text: equations,

text: Comm.

text: Pure

text: Appl.

text: Math.,

text: S.

text: Boscarino,

text: L.

text: Pareschi,

text: On

text: the

text: asymptotic

text: properties

text: of

text: IMEX

text: Runge–Kutta

text: schemes

text: for

text: hyperbolic

text: balance

text: laws,

text: J.

text: Comput.

text: Appl.

text: Math.

text: (to

text: appear).

text: S.

text: Boscarino,

text: L.

text: Pareschi,

text: G.

text: Russo,

text: Implicit-Explicit

text: Runge-Kutta

text: schemes

text: for

text: hy-

text: perbolic

text: systems

text: and

text: kinetic

text: equations

text: in

text: the

text: diﬀusion

text: limit,

text: SIAM

text: J.

text: Sci.

text: Comput.,

text: Vol.

text: No.

text: pp.

text: S.

text: Boscarino,

text: G.

text: Russo,

text: On

text: a

text: class

text: of

text: uniformly

text: accurate

text: IMEX

text: Runge-Kutta

text: schemes

text: and

text: applications

text: to

text: hyperbolic

text: systems

text: with

text: relaxation,

text: SIAM

text: J.

text: Sci.

text: Comput.

text: Vol.

text: No.

text: pp.

text: S.

text: Boscarino,

text: G.

text: Russo,

text: Flux-Explicit

text: IMEX

text: Runge-Kutta

text: schemes

text: for

text: hyperbolic

text: to

text: parabolic

text: relaxation

text: problems,

text: SIAM

text: J.

text: NUMER.

text: ANAL.,

text: Vol.

text: No.

text: S.

text: Boscarino,

text: P.

text: Le

text: Floch,

text: G.

text: Russo,

text: High-Order

text: asymptotic-preserving

text: methods

text: for

text: fully

text: non

text: linear

text: relaxation

text: problems,

text: SIAM

text: J.

text: Sci.

text: Comput.,

text: Vol.

text: No.

text: pp.

text: R.E.

text: Caflisch,

text: S.

text: Jin,

text: G.

text: Russo,

text: Uniformly

text: accurate

text: schemes

text: for

text: hyperbolic

text: systems

text: with

text: relaxation,

text: SIAM

text: J.

text: Num.

text: Anal.,

text: M.H.

text: Carpenter,

text: C.A.

text: Kennedy,

text: Additive

text: Runge-Kutta

text: schemes

text: for

text: convection-

text: diﬀusion-reaction

text: equations

text: Appl.

text: Numer.

text: Math.

text: no.

text: C.

text: Cercignani,

text: R.

text: Illner,

text: M.

text: Pulvirenti,

text: The

text: mathematical

text: theory

text: of

text: dilute

text: gases,

text: Applied

text: Mathematical

text: Sciences,

text: Springer-Verlag,

text: New

text: York,

text: G.Q.

text: Chen,

text: C.D.

text: Levermore,

text: T.P.

text: Liu,

text: Hyperbolic

text: conservation

text: laws

text: with

text: stiﬀ

text: relax-

text: ation

text: terms

text: and

text: entropy,

text: Comm.

text: Pure

text: and

text: App.

text: Math.,

text: XLVII,

text: C.

text: Cercignani,

text: The

text: Boltzmann

text: equation

text: and

text: its

text: applications,

text: Springer

text: Varlag

text: New

text: York,

text: G.

text: Dimarco,

text: L.

text: Pareschi,

text: Asymptotic-preserving

text: IMEX

text: Runge-Kutta

text: methods

text: for

text: non-

text: linear

text: kinetic

text: equations,

text: SIAM

text: J.

text: Num.

text: Anal.

text: F.

text: Filbet,

text: S.

text: Jin,

text: A

text: class

text: of

text: asymptotic

text: preserving

text: schemes

text: for

text: kinetic

text: equations

text: and

text: related

text: problems

text: with

text: stiﬀ

text: sources,

text: J.

text: Comp.

text: Phys.,

text: pp.

text: E.

text: Gabetta,

text: B.

text: Perthame,

text: Scaling

text: limits

text: of

text: the

text: Ruijgrook-Wu

text: model

text: of

text: the

text: Boltzmann

text: equation,

text: Proceedings

text: of

text: the

text: International

text: Conference

text: on

text: Nonlinear

text: Equations

text: and

text: Appli-

text: cations,

text: Bangalore

text: August

text: Springer-Verlag.

text: E.

text: Gabetta,

text: L.

text: Pareschi,

text: G.

text: Toscani,

text: Relaxation

text: schemes

text: for

text: nonlinear

text: kinetic

text: equa-

text: tions,

text: SIAM

text: J.

text: Numer.

text: Anal.

text: no.

text: E.

text: Hairer,

text: G.

text: Wanner,

text: Solving

text: Ordinary

text: Diﬀerential

text: Equation

text: II:

text: stiﬀ

text: and

text: Diﬀerential

text: Algebraic

text: Problems.

text: Springer

text: Series

text: in

text: Comput.

text: Mathematics,

text: Vol.

text: Springer-Verlag

text: Second

text: revised

text: edition


text: ≤ ≤ ≈ O ≈ (cid:28) where 0 φ(ε) 1 is such that φ(ε) 1 for ε = (1) and φ(ε) 0 when ε 1. For example { } φ(ε) = min ε2, 1 or the smoother approximation φ(ε) = tanh(ε2) were considered in [23, 25]. → In the limit ε 0 we have the asymptotic behavior of the partitioned approach, whereas for larger values of ε we have the usual additive approach. The method can be naturally extended to ∈ the general multiscale case and provides a consistent discretization also for any value of α [0, 1). We refer to [24, 29] for further details. One of the main advantages of this approach is that it results in a convex approximation at diﬀerent times of the space derivative p(u) , therefore x one can use diﬀerent space discretizations for the derivative appearing at time n and the one at time n + 1. Typically, at time n one can choose a standard hyperbolic discretization that works for large values of ε whereas at time n + 1 classical central diﬀerence schemes that works O in the parabolic limit suﬃces to avoid a CFL condition of the type ∆t = (ε). In this sense the function φ(ε) can be also interpreted as an interpolation parameter between diﬀerent ﬂuxes in the evaluation of the space derivatives in (10). The determination of the optimal expression of φ(ε) in terms of stability and accuracy is a delicate aspect which is beyond the scope of the present paper. 3 A uniﬁed IMEX Runge-Kutta approach In this section we present a diﬀerent approach which overcomes some of the drawbacks of the above mentioned methods. 3.1 Description of the method Again let us consider initially, for simplicity of notation, the case α = 1. We now consider the following discretization for system () − un+1 un − (cid:0) (cid:1) = vn+1, ∆t x − (15) vn+1 vn − − ε2 = p(un) + vn+1 f(un) . x ∆t Solving the second equation for vn+1 one obtains ε2 − ∆t − vn+1 = vn (p(un) f(un)) . (16) ε2 + ∆t ε2 + ∆t x Making use of this relation in the ﬁrst equation we get − un+1 un ε2 ∆t − + vn = (p(un) f(un) ) . ∆t ε2 + ∆t x ε2 + ∆t xx x Note that, at variance with all the previous approaches, the ﬁrst equation now uses only two time levels and the space derivatives appear all at the same time level n. Therefore we can rewrite the scheme in the equivalent fully explicit form − un+1 un ε2 ∆t ∆t + vn + f(un) = p(un) , ∆t ε2 + ∆t x ε2 + ∆t x ε2 + ∆t xx − (17) vn+1 vn 1 − 1 − + p(un) = (vn f(un)) . ∆t ε2 + ∆t x ε2 + ∆t In particular, for small values of ∆t the scheme (17) corresponds to the system ε2 ∆t ∆t O u + v + f(u) = p(u) + (∆t), t ε2 + ∆t x ε2 + ∆t x ε2 + ∆t xx (18) 1 − 1 − O v + p(u) = (v f(u)) + (∆t), t ε2 + ∆t x ε2 + ∆t 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: O

text: where

text: φ(ε)

text: is

text: such

text: that

text: φ(ε)

text: for

text: ε

text: and

text: φ(ε)

text: when

text: ε

text: For

text: example

text: φ(ε)

text: min

text: or

text: the

text: smoother

text: approximation

text: φ(ε)

text: were

text: considered

text: in

text: In

text: the

text: limit

text: ε

text: we

text: have

text: the

text: asymptotic

text: behavior

text: of

text: the

text: partitioned

text: approach,

text: whereas

text: for

text: larger

text: values

text: of

text: ε

text: we

text: have

text: the

text: usual

text: additive

text: approach.

text: The

text: method

text: can

text: be

text: naturally

text: extended

text: to

text: the

text: general

text: multiscale

text: case

text: and

text: provides

text: a

text: consistent

text: discretization

text: also

text: for

text: any

text: value

text: of

text: α

text: We

text: refer

text: to

text: for

text: further

text: details.

text: One

text: of

text: the

text: main

text: advantages

text: of

text: this

text: approach

text: is

text: that

text: it

text: results

text: in

text: a

text: convex

text: approximation

text: at

text: diﬀerent

text: times

text: of

text: the

text: space

text: derivative

text: p(u)

text: therefore

text: x

text: one

text: can

text: use

text: diﬀerent

text: space

text: discretizations

text: for

text: the

text: derivative

text: appearing

text: at

text: time

text: n

text: and

text: the

text: one

text: at

text: time

text: n

text: Typically,

text: at

text: time

text: n

text: one

text: can

text: choose

text: a

text: standard

text: hyperbolic

text: discretization

text: that

text: works

text: for

text: large

text: values

text: of

text: ε

text: whereas

text: at

text: time

text: n

text: classical

text: central

text: diﬀerence

text: schemes

text: that

text: works

text: O

text: in

text: the

text: parabolic

text: limit

text: suﬃces

text: to

text: avoid

text: a

text: CFL

text: condition

text: of

text: the

text: type

text: ∆t

text: (ε).

text: In

text: this

text: sense

text: the

text: function

text: φ(ε)

text: can

text: be

text: also

text: interpreted

text: as

text: an

text: interpolation

text: parameter

text: between

text: diﬀerent

text: ﬂuxes

text: in

text: the

text: evaluation

text: of

text: the

text: space

text: derivatives

text: in

text: The

text: determination

text: of

text: the

text: optimal

text: expression

text: of

text: φ(ε)

text: in

text: terms

text: of

text: stability

text: and

text: accuracy

text: is

text: a

text: delicate

text: aspect

text: which

text: is

text: beyond

text: the

text: scope

text: of

text: the

text: present

text: paper.

text: A

text: uniﬁed

text: IMEX

text: Runge-Kutta

text: approach

text: In

text: this

text: section

text: we

text: present

text: a

text: diﬀerent

text: approach

text: which

text: overcomes

text: some

text: of

text: the

text: drawbacks

text: of

text: the

text: above

text: mentioned

text: methods.

text: Description

text: of

text: the

text: method

text: Again

text: let

text: us

text: consider

text: initially,

text: for

text: simplicity

text: of

text: notation,

text: the

text: case

text: α

text: We

text: now

text: consider

text: the

text: following

text: discretization

text: for

text: system

text: un

text: ∆t

text: x

text: vn

text: ε2

text: p(un)

text: f(un)

text: x

text: ∆t

text: Solving

text: the

text: second

text: equation

text: for

text: one

text: obtains

text: ε2

text: ∆t

text: vn

text: (p(un)

text: f(un))

text: ε2

text: ∆t

text: ε2

text: ∆t

text: x

text: Making

text: use

text: of

text: this

text: relation

text: in

text: the

text: ﬁrst

text: equation

text: we

text: get

text: un

text: ε2

text: ∆t

text: vn

text: (p(un)

text: f(un)

text: ∆t

text: ε2

text: ∆t

text: x

text: ε2

text: ∆t

text: xx

text: x

text: Note

text: that,

text: at

text: variance

text: with

text: all

text: the

text: previous

text: approaches,

text: the

text: ﬁrst

text: equation

text: now

text: uses

text: only

text: two

text: time

text: levels

text: and

text: the

text: space

text: derivatives

text: appear

text: all

text: at

text: the

text: same

text: time

text: level

text: n.

text: Therefore

text: we

text: can

text: rewrite

text: the

text: scheme

text: in

text: the

text: equivalent

text: fully

text: explicit

text: form

text: un

text: ε2

text: ∆t

text: ∆t

text: vn

text: f(un)

text: p(un)

text: ∆t

text: ε2

text: ∆t

text: x

text: ε2

text: ∆t

text: x

text: ε2

text: ∆t

text: xx

text: vn

text: p(un)

text: (vn

text: f(un))

text: ∆t

text: ε2

text: ∆t

text: x

text: ε2

text: ∆t

text: In

text: particular,

text: for

text: small

text: values

text: of

text: ∆t

text: the

text: scheme

text: corresponds

text: to

text: the

text: system

text: ε2

text: ∆t

text: ∆t

text: O

text: u

text: v

text: f(u)

text: p(u)

text: (∆t),

text: t

text: ε2

text: ∆t

text: x

text: ε2

text: ∆t

text: x

text: ε2

text: ∆t

text: xx

text: O

text: v

text: p(u)

text: (v

text: f(u))

text: (∆t),

text: t

text: ε2

text: ∆t

text: x

text: ε2

text: ∆t

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: where we only used − − un+1 un O vn+1 vn O = u + (∆t), = v + (∆t) t t ∆t ∆t (cid:32) (cid:115) (cid:33) leaving all other terms. Note that the left part of system (18) is hyperbolic with characteristic speeds 1 ξ ± 4ε2 λ1±(∆t, ε) = 1 c c2 + , (19) 2 (∆t)2 ∈ (cid:48) (cid:48) with ξ = ∆t/(ε2 + ∆t) (0, 1), c = f (u) and for simplicity we have set p (u) = 1. 1 → Now, if ∆t 0 for a ﬁxed ε, system (18) converges to the original system () for α = 1 and by (19), the characteristics speeds converge to the usual ones, i.e. ±1 λ1±(0, ε) = . ε On the other hand, for a ﬁxed ∆t, the characteristic speeds λ1 and λ1− in (19) are respectively → + decreasing and increasing functions of ε and as ε 0 they converge to 1 ± | | λ1±(∆t, 0) = (c c ) . (20) 2 Therefore, if we denote by ∆x the space discretization parameter, we obtain the expected ≤ | | hyperbolic CFL condition ∆t ∆x/ c . Concerning the second order derivative on the right hand side of (18) this induces a stability ∼ → restriction of type ∆t (∆x)2/ξ . In particular, since the discrete system (17) as ε 0 relaxes 1 towards − un+1 un + f(un) = p(un) , (21) x xx ∆t in such a limit we have the natural stability condition which links the time step to the square ∼ of the mesh space ∆t (∆x)2. ∈ Similarly, in the case α [0, 1), for small values of ∆t the scheme applied to () corresponds to the system − ε1+α ∆t ∆t ε1 α O u + v + f(u) = p(u) + (∆t), t ε1+α + ∆t x ε1+α + ∆t x ε1+α + ∆t xx − (22) ε1 α − 1 − O v + p(u) = (v f(u)) + (∆t). t ε1+α + ∆t x ε1+α + ∆t (cid:32) (cid:115) (cid:33) The left-hand side in (22) now has characteristic speeds ξ ± 4ε2 λα±(∆t, ε) = α c c2 + , (23) 2 (∆t)2 ∈ → with ξ = ∆t/(ε1+α + ∆t) [0, 1). As before, if we ﬁx ε and send ∆t 0 we obtain the usual α characteristic speeds ± 1 λα±(0, ε) = . εα 1The subscript 1 in the next expression indicates α = 1. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: we

text: only

text: used

text: un

text: O

text: vn

text: O

text: u

text: (∆t),

text: v

text: (∆t)

text: t

text: t

text: ∆t

text: ∆t

text: leaving

text: all

text: other

text: terms.

text: Note

text: that

text: the

text: left

text: part

text: of

text: system

text: is

text: hyperbolic

text: with

text: characteristic

text: speeds

text: ξ

text: ε)

text: c

text: with

text: ξ

text: ∆t)

text: c

text: f

text: (u)

text: and

text: for

text: simplicity

text: we

text: have

text: set

text: p

text: (u)

text: Now,

text: if

text: ∆t

text: for

text: a

text: ﬁxed

text: ε,

text: system

text: converges

text: to

text: the

text: original

text: system

text: for

text: α

text: and

text: by

text: the

text: characteristics

text: speeds

text: converge

text: to

text: the

text: usual

text: ones,

text: i.e.

text: ±1

text: ε)

text: ε

text: On

text: the

text: other

text: hand,

text: for

text: a

text: ﬁxed

text: ∆t,

text: the

text: characteristic

text: speeds

text: λ1

text: and

text: λ1−

text: in

text: are

text: respectively

text: decreasing

text: and

text: increasing

text: functions

text: of

text: ε

text: and

text: as

text: ε

text: they

text: converge

text: to

text: (c

text: c

text: Therefore,

text: if

text: we

text: denote

text: by

text: ∆x

text: the

text: space

text: discretization

text: parameter,

text: we

text: obtain

text: the

text: expected

text: hyperbolic

text: CFL

text: condition

text: ∆t

text: ∆x/

text: c

text: Concerning

text: the

text: second

text: order

text: derivative

text: on

text: the

text: right

text: hand

text: side

text: of

text: this

text: induces

text: a

text: stability

text: restriction

text: of

text: type

text: ∆t

text: In

text: particular,

text: since

text: the

text: discrete

text: system

text: as

text: ε

text: relaxes

text: towards

text: un

text: f(un)

text: p(un)

text: x

text: xx

text: ∆t

text: in

text: such

text: a

text: limit

text: we

text: have

text: the

text: natural

text: stability

text: condition

text: which

text: links

text: the

text: time

text: step

text: to

text: the

text: square

text: of

text: the

text: mesh

text: space

text: ∆t

text: Similarly,

text: in

text: the

text: case

text: α

text: for

text: small

text: values

text: of

text: ∆t

text: the

text: scheme

text: applied

text: to

text: corresponds

text: to

text: the

text: system

text: ε1+α

text: ∆t

text: ∆t

text: ε1

text: α

text: O

text: u

text: v

text: f(u)

text: p(u)

text: (∆t),

text: t

text: ε1+α

text: ∆t

text: x

text: ε1+α

text: ∆t

text: x

text: ε1+α

text: ∆t

text: xx

text: ε1

text: α

text: O

text: v

text: p(u)

text: (v

text: f(u))

text: (∆t).

text: t

text: ε1+α

text: ∆t

text: x

text: ε1+α

text: ∆t

text: The

text: left-hand

text: side

text: in

text: now

text: has

text: characteristic

text: speeds

text: ξ

text: λα±(∆t,

text: ε)

text: α

text: c

text: with

text: ξ

text: ∆t)

text: As

text: before,

text: if

text: we

text: ﬁx

text: ε

text: and

text: send

text: ∆t

text: we

text: obtain

text: the

text: usual

text: α

text: characteristic

text: speeds

text: λα±(0,

text: ε)

text: εα

text: subscript

text: in

text: the

text: next

text: expression

text: indicates

text: α

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: 13 (1980). 44 V. Dotsenko, J. Stat. Phys. 122, 197 (2006); V. Dotsenko, 39 W. Wiethege and D. Sherrington, J. Phys. C 19 69832 J. Phys. A: Math Gen. 27, 3397 (1994). (1986) 45 M. Randeria, J. P. Sethna, and R.G. Palmer, Phys. Rev. 40 F. M. Zimmer, S. G. Magalhaes, Phys. Rev. B 74, 012202 Lett. 54, 1321 (1985). (2006). 46 M. Guo, R. N. Bhatt, D. A. Huse, Phys. Rev. B 54, 3336 41 D. Thihumalai, Q. Li, and T. R. Kirkpatrick, J. Phys. A (1996). 22, 3339 (1989). 47 H. Rieger, A. P. Young, Phys. Rev. B 54, 3328 (1996). 42 W. Wu, B. Ellman, T. F. Rosenbaum, G. Aeppli, D. H. 48 A. P. Young, H. Rieger, Phys. Rev. B 53, 8486 (1996). Reich, Phys. Rev. Lett 67, 2076 (1991). 49 A. Biltmo and P. Henelius, Nature Comm. 3:857 (2012). 43 R.B. Griﬃths, Phys. Rev. Lett. 23, 17 (1969).

text: V.

text: Dotsenko,

text: J.

text: Stat.

text: Phys.

text: V.

text: Dotsenko,

text: W.

text: Wiethege

text: and

text: D.

text: Sherrington,

text: J.

text: Phys.

text: C

text: J.

text: Phys.

text: A:

text: Math

text: Gen.

text: M.

text: Randeria,

text: J.

text: P.

text: Sethna,

text: and

text: R.G.

text: Palmer,

text: Phys.

text: Rev.

text: F.

text: M.

text: Zimmer,

text: S.

text: G.

text: Magalhaes,

text: Phys.

text: Rev.

text: B

text: Lett.

text: M.

text: Guo,

text: R.

text: N.

text: Bhatt,

text: D.

text: A.

text: Huse,

text: Phys.

text: Rev.

text: B

text: D.

text: Thihumalai,

text: Q.

text: Li,

text: and

text: T.

text: R.

text: Kirkpatrick,

text: J.

text: Phys.

text: A

text: H.

text: Rieger,

text: A.

text: P.

text: Young,

text: Phys.

text: Rev.

text: B

text: W.

text: Wu,

text: B.

text: Ellman,

text: T.

text: F.

text: Rosenbaum,

text: G.

text: Aeppli,

text: D.

text: H.

text: A.

text: P.

text: Young,

text: H.

text: Rieger,

text: Phys.

text: Rev.

text: B

text: Reich,

text: Phys.

text: Rev.

text: Lett

text: A.

text: Biltmo

text: and

text: P.

text: Henelius,

text: Nature

text: Comm.

text: R.B.

text: Griﬃths,

text: Phys.

text: Rev.

text: Lett.


text: 3 Γ0. account only the sites occupied by one fermion in the c In the classical case, it is well known that χ is in- partition-function trace39,40. SG versely proportional to the Almeida-Thouless eigenvalue In order to deal appropriately with the experimental λ , the so-called replicon2,31. Therefore, the diverging behavior of χ in the LiHo Y − F compound, we focus AT 3 x 1 x 4 behavior of χ at the SG transition, our calculations on the 2S model by proposing a relation- 3 ship between ∆ (the width of the distribution of random ∝ − − ﬁelds hi) and Γ, following the approach introduced by χ3 [(T Tf)/Tf] γ , (7) Tabei and collaborators22. The main characteristic ob- served experimentally in χ concerns the peak for small 3 H (classical limit) being replaced by a rounded maxi- is a direct consequence of λ = 0 at T , occurring t AT f mum which becomes increasingly rounded for large H together with the onset of RSB. Similarly, in the quan- t (quantum limit). Besides the progressive smearing of χ , tum case, one expects that the divergence of χ at Γ (T) 3 3 f the amplitude of its maximum also decreases as H in- should coincide with the onset of RSB. t creases. Therefore, the eﬀects of the RF triggered by Indeed, the presence of a RF can produce deep changes H , as suggested by Tabei and collaborators22, should in the scenario described previously. For instance, in the t provoke simultaneously both eﬀects, i.e., the smearing of classical SK model6, the RF induces the RS order param- the peak and the decrease of the maximum amplitude eter q, which becomes ﬁnite at any temperature32,33. As value of χ . For the present fermionic Ising SG model, a consequence, q versus temperature presents a smooth 3 we have tested a relationship involving ∆ and Γ, partic- behavior, being no more appropriate for identifying a SG ∝ (cid:48) ularly in the power-like form, ∆/J (Γ/J)B , where J transition in the SK model. Nevertheless, such a transi- represents the width of the Gaussian distribution for the tion may still be related with the onset of RSB, signaled { } couplings J . Considering the interval for the expo- by λ = 034. In spite of this, the derivative of q with re- ij (cid:48) AT nent, 1.8 < B < 2.5, we have been able to obtain χ spect to the temperature increases as one approaches T 3 f as a function of temperature and Γ resembling qualita- from above; such an increase is the ultimate responsible ∗ tively the experimental behavior for χ described above. for the rounded maximum in χ at the temperature T , 3 3 As already mentioned, the transverse ﬁeld Γ used in the which does not coincide with the SG transition tempera- ∗ eﬀective model to describe LiHo Y − F is expected to ture T (T > T ). In fact, the maximum value of χ at x 1 x 4 ∗ f f 3 be related to the experimental applied ﬁeld H 11,12; in T reﬂects the eﬀects of the RF inside the paramagnetic ∝ t fact, at least for low H , Γ H2 (see, e.g., Ref.42). phase, instead of the non-trivial ergodicity breaking of t t The paper is structured as follows: in Section II we the SG phase transition35. Therefore, one can raise the deﬁne the model and ﬁnd its grand-canonical poten- question of whether such scenario for χ , found in the 3 tial within the one-step RSB scheme; in Section III we classical SK model, is robust in the corresponding quan- present a detailed discussion of the order parameters, the tum model, when the transverse ﬁeld is considered, i.e., (cid:54) susceptibility χ , and some phase diagrams. Finally, the Γ = 0, and no longer independent from the RF, as pro- 3 last section is reserved to conclusions. posed by Tabei and collaborators22. II. MODEL The purpose of the present work is to study the suscep- tibility χ , using the so-called fermionic Ising SG model 3 The model is deﬁned by the Hamiltonian in the presence of a longitudinal RF hi and a transverse (cid:88) (cid:88) (cid:88) ﬁeld Γ. In this model, the spin operators are written in terms of fermionic occupation and destru{ctio}n op- − − N − N erators36,37, whereas the spin-spin couplings J and H = J SˆzSˆz h Sˆz 2Γ Sˆx, (8) { } ij ij i (cid:80)j i i i random ﬁelds hi follow Gaussian distributions. The (i,j) i=1 i=1 grand-canonical potential is obtained in the functional integral formalism, and the disorder is treated using the where the summation applies to all distinct pairs (i,j) { } replica method; moreover, the SG order parameters are of spin operators, whereas the couplings J and mag- obtained in the static approximation38,41 and investi- { } ij netic ﬁelds h are quenched random variables, following gated within the one-step RSB scheme7. It should be i independent(cid:20)Gaussia(cid:21)n distrib(cid:20)utions, (cid:21) remarked that the fermionic Ising SG model is deﬁned on the Fock space, where there are four possible states per site: one state with no fermions, two states with a single N 1/2 − N − P(J ) = exp (J J /N)2 , fermion, and one state with two fermions, leading to two ij 32πJ2 32J2 ij 0 nonmagnetic states. In particular, one can consider two (9) cases: the 4S model that allows the four possible states per site and the 2S model, which restricts the spin op- and (cid:20) (cid:21) (cid:20) (cid:21) erators to act on a space where the nonmagnetic states are forbidden. In the present work we will consider the later model, by imposing a restriction to remove the con- 1 1/2 − 1 P(h ) = exp h2 . (10) tribution of these nonmagnetic states, i.e., taking into i 32π∆2 32∆2 i ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: account

text: only

text: the

text: sites

text: occupied

text: by

text: one

text: fermion

text: in

text: the

text: c

text: In

text: the

text: classical

text: case,

text: it

text: is

text: well

text: known

text: that

text: χ

text: is

text: in-

text: partition-function

text: SG

text: versely

text: proportional

text: to

text: the

text: Almeida-Thouless

text: eigenvalue

text: In

text: order

text: to

text: deal

text: appropriately

text: with

text: the

text: experimental

text: λ

text: the

text: so-called

text: Therefore,

text: the

text: diverging

text: behavior

text: of

text: χ

text: in

text: the

text: LiHo

text: Y

text: F

text: compound,

text: we

text: focus

text: AT

text: x

text: x

text: behavior

text: of

text: χ

text: at

text: the

text: SG

text: transition,

text: our

text: calculations

text: on

text: the

text: model

text: by

text: proposing

text: a

text: relation-

text: ship

text: between

text: (the

text: width

text: of

text: the

text: distribution

text: of

text: random

label: equation

text: ﬁelds

text: hi)

text: and

text: Γ,

text: following

text: the

text: approach

text: introduced

text: by

text: χ3

text: [(T

text: Tf)/Tf]

text: γ

text: Tabei

text: and

text: The

text: main

text: characteristic

text: ob-

text: served

text: experimentally

text: in

text: χ

text: concerns

text: the

text: peak

text: for

text: small

text: H

text: (classical

text: limit)

text: being

text: replaced

text: by

text: a

text: rounded

text: maxi-

text: is

text: a

text: direct

text: consequence

text: of

text: λ

text: at

text: T

text: occurring

text: t

text: AT

text: f

text: mum

text: which

text: becomes

text: increasingly

text: rounded

text: for

text: large

text: H

text: together

text: with

text: the

text: onset

text: of

text: RSB.

text: Similarly,

text: in

text: the

text: quan-

text: t

text: (quantum

text: limit).

text: Besides

text: the

text: progressive

text: smearing

text: of

text: χ

text: tum

text: case,

text: one

text: expects

text: that

text: the

text: divergence

text: of

text: χ

text: at

text: Γ

text: (T)

text: f

text: the

text: amplitude

text: of

text: its

text: maximum

text: also

text: decreases

text: as

text: H

text: in-

text: should

text: coincide

text: with

text: the

text: onset

text: of

text: RSB.

text: t

text: creases.

text: Therefore,

text: the

text: eﬀects

text: of

text: the

text: RF

text: triggered

text: by

text: Indeed,

text: the

text: presence

text: of

text: a

text: RF

text: can

text: produce

text: deep

text: changes

text: H

text: as

text: suggested

text: by

text: Tabei

text: and

text: should

text: in

text: the

text: scenario

text: described

text: previously.

text: For

text: instance,

text: in

text: the

text: t

text: provoke

text: simultaneously

text: both

text: eﬀects,

text: i.e.,

text: the

text: smearing

text: of

text: classical

text: SK

text: the

text: RF

text: induces

text: the

text: RS

text: order

text: param-

text: the

text: peak

text: and

text: the

text: decrease

text: of

text: the

text: maximum

text: amplitude

text: eter

text: q,

text: which

text: becomes

text: ﬁnite

text: at

text: any

text: As

text: value

text: of

text: χ

text: For

text: the

text: present

text: fermionic

text: Ising

text: SG

text: model,

text: a

text: consequence,

text: q

text: versus

text: temperature

text: presents

text: a

text: smooth

text: we

text: have

text: tested

text: a

text: relationship

text: involving

text: and

text: Γ,

text: partic-

text: behavior,

text: being

text: no

text: more

text: appropriate

text: for

text: identifying

text: a

text: SG

text: ularly

text: in

text: the

text: power-like

text: form,

text: ∆/J

text: (Γ/J)B

text: where

text: J

text: transition

text: in

text: the

text: SK

text: model.

text: Nevertheless,

text: such

text: a

text: transi-

text: represents

text: the

text: width

text: of

text: the

text: Gaussian

text: distribution

text: for

text: the

text: tion

text: may

text: still

text: be

text: related

text: with

text: the

text: onset

text: of

text: RSB,

text: signaled

text: couplings

text: J

text: Considering

text: the

text: interval

text: for

text: the

text: expo-

text: by

text: λ

text: In

text: spite

text: of

text: this,

text: the

text: derivative

text: of

text: q

text: with

text: re-

text: ij

text: AT

text: nent,

text: B

text: we

text: have

text: been

text: able

text: to

text: obtain

text: χ

text: spect

text: to

text: the

text: temperature

text: increases

text: as

text: one

text: approaches

text: T

text: f

text: as

text: a

text: function

text: of

text: temperature

text: and

text: Γ

text: resembling

text: qualita-

text: from

text: above;

text: such

text: an

text: increase

text: is

text: the

text: ultimate

text: responsible

text: tively

text: the

text: experimental

text: behavior

text: for

text: χ

text: described

text: above.

text: for

text: the

text: rounded

text: maximum

text: in

text: χ

text: at

text: the

text: temperature

text: T

text: As

text: already

text: mentioned,

text: the

text: transverse

text: ﬁeld

text: Γ

text: used

text: in

text: the

text: which

text: does

text: not

text: coincide

text: with

text: the

text: SG

text: transition

text: tempera-

text: eﬀective

text: model

text: to

text: describe

text: LiHo

text: Y

text: F

text: is

text: expected

text: to

text: ture

text: T

text: (T

text: T

text: In

text: fact,

text: the

text: maximum

text: value

text: of

text: χ

text: at

text: x

text: x

text: f

text: f

text: be

text: related

text: to

text: the

text: experimental

text: applied

text: ﬁeld

text: H

text: in

text: T

text: reﬂects

text: the

text: eﬀects

text: of

text: the

text: RF

text: inside

text: the

text: paramagnetic

text: t

text: fact,

text: at

text: least

text: for

text: low

text: H

text: Γ

text: (see,

text: e.g.,

text: phase,

text: instead

text: of

text: the

text: non-trivial

text: ergodicity

text: breaking

text: of

text: t

text: t

text: The

text: paper

text: is

text: structured

text: as

text: follows:

text: in

text: Section

text: II

text: we

text: the

text: SG

text: phase

text: Therefore,

text: one

text: can

text: raise

text: the

text: deﬁne

text: the

text: model

text: and

text: ﬁnd

text: its

text: grand-canonical

text: poten-

text: question

text: of

text: whether

text: such

text: scenario

text: for

text: χ

text: found

text: in

text: the

text: tial

text: within

text: the

text: one-step

text: RSB

text: scheme;

text: in

text: Section

text: III

text: we

text: classical

text: SK

text: model,

text: is

text: robust

text: in

text: the

text: corresponding

text: quan-

text: present

text: a

text: detailed

text: discussion

text: of

text: the

text: order

text: parameters,

text: the

text: tum

text: model,

text: when

text: the

text: transverse

text: ﬁeld

text: is

text: considered,

text: i.e.,

text: susceptibility

text: χ

text: and

text: some

text: phase

text: diagrams.

text: Finally,

text: the

text: Γ

text: and

text: no

text: longer

text: independent

text: from

text: the

text: RF,

text: as

text: pro-

text: last

text: section

text: is

text: reserved

text: to

text: conclusions.

text: posed

text: by

text: Tabei

text: and

text: II.

text: MODEL

text: The

text: purpose

text: of

text: the

text: present

text: work

text: is

text: to

text: study

text: the

text: suscep-

text: tibility

text: χ

text: using

text: the

text: so-called

text: fermionic

text: Ising

text: SG

text: model

text: The

text: model

text: is

text: deﬁned

text: by

text: the

text: Hamiltonian

text: in

text: the

text: presence

text: of

text: a

text: longitudinal

text: RF

text: hi

text: and

text: a

text: transverse

text: ﬁeld

text: Γ.

text: In

text: this

text: model,

text: the

text: spin

text: operators

text: are

text: written

text: in

text: terms

text: of

text: fermionic

text: occupation

text: and

text: destru{ctio}n

text: op-

text: N

text: N

text: whereas

text: the

text: spin-spin

text: couplings

text: J

text: and

text: H

text: J

text: SˆzSˆz

text: h

text: Sˆz

text: 2Γ

text: Sˆx,

text: ij

text: ij

text: i

text: i

text: i

text: i

text: random

text: ﬁelds

text: hi

text: follow

text: Gaussian

text: distributions.

text: The

text: (i,j)

text: grand-canonical

text: potential

text: is

text: obtained

text: in

text: the

text: functional

text: integral

text: formalism,

text: and

text: the

text: disorder

text: is

text: treated

text: using

text: the

text: where

text: the

text: summation

text: applies

text: to

text: all

text: distinct

text: pairs

text: (i,j)

text: replica

text: method;

text: moreover,

text: the

text: SG

text: order

text: parameters

text: are

text: of

text: spin

text: operators,

text: whereas

text: the

text: couplings

text: J

text: and

text: mag-

text: obtained

text: in

text: the

text: static

text: and

text: investi-

text: ij

text: netic

text: ﬁelds

text: h

text: are

text: quenched

text: random

text: variables,

text: following

text: gated

text: within

text: the

text: one-step

text: RSB

text: It

text: should

text: be

text: i

text: remarked

text: that

text: the

text: fermionic

text: Ising

text: SG

text: model

text: is

text: deﬁned

text: on

text: the

text: Fock

text: space,

text: where

text: there

text: are

text: four

text: possible

text: states

text: per

text: site:

text: one

text: state

text: with

text: no

text: fermions,

text: two

text: states

text: with

text: a

text: single

text: N

text: N

text: P(J

text: exp

text: (J

text: J

text: fermion,

text: and

text: one

text: state

text: with

text: two

text: fermions,

text: leading

text: to

text: two

text: ij

text: ij

text: nonmagnetic

text: states.

text: In

text: particular,

text: one

text: can

text: consider

text: two

text: cases:

text: the

text: model

text: that

text: allows

text: the

text: four

text: possible

text: states

text: per

text: site

text: and

text: the

text: model,

text: which

text: restricts

text: the

text: spin

text: op-

text: and

text: erators

text: to

text: act

text: on

text: a

text: space

text: where

text: the

text: nonmagnetic

text: states

text: are

text: forbidden.

text: In

text: the

text: present

text: work

text: we

text: will

text: consider

text: the

text: later

text: model,

text: by

text: imposing

text: a

text: restriction

text: to

text: remove

text: the

text: con-

text: P(h

text: exp

text: tribution

text: of

text: these

text: nonmagnetic

text: states,

text: i.e.,

text: taking

text: into

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 should be mentioned that q is the only quantity appear- the one occurring in Fig. 6(a). 2 ∗ ing in χ3 [either in Eq. (A1), or in Eq. (A13)], which may It should be emphasized that the temperature T plays present a divergence at ﬁnite temperatures. Moreover, a role diﬀerent from T , in the sense that it does not cor- f one can show that for ∆ = 0, the so-called “dangerous” respond to a phase transition, but rather to a crossover eigenvalue31 in Eq−. (A10) is equal to the denominator of between two distinct regions of the paramagnetic phase. q2, i.e., λAT = 1 2(βJ)2I0(Γ), even for Γ > 0. Th∗e The previous analysis of the SK model in the presence mechanism behind the ﬂattening of the χ3 peak at T of a Gaussian random ﬁeld Ref.35, which should corre- is illustrated in Fig. 5, where we plot the quantity b of spond herein to the region of high temperatures and low Eq. (24), λAT, and χ3, versus the dimensionless tem- Γ∗(i.e., the classical regime), has also found a temperature perature, for typical choices of Γ/J and ∆/J. Results T , associated with the rounded maximum of χ , with ∗ ∗ 3 for Γ and ∆ independent are presented in Fig. 5(a); the T > T . In this case, T was interpreted as an eﬀect of f full lines [cases (∆/J) = 0.0] show that the quantities the RFs acting inside the paramagnetic phase, instead of b and λAT become zero together, being associated with some type of non-trivial ergod∗icity breaking. Herein, we the divergence of χ3 [according to Eq. (7)], signalling the claim that the temperature T , although it may be also SG phase-transition temperature Tf. However, the re- aﬀected by the transverse ﬁeld Γ, should be interpreted sults for (∆/J) = 0.1 show that such a small value for in a similar manner. Hence, along the line signaled by the width of the RFs distribution yields b > 0, which T , the growth of Γ produces an enhancement of quan- ∗ f presents a smooth minimum around a temperature T , tum ﬂuctuations, which become increasingly dominant being directly associated with the rounding behavior of as compared with thermal ﬂuctuations, driving the non- χ3; on the other h∗and, one has λAT = 0 at a temperature trivial ergodicity breaking of the SG phase transiti∗on to Tf, with Tf < T . In Fig. 5(b) we present the denomi- a QCP. For temperatures in the region Tf < T < T , the nator b and λAT, for the cases where Γ and ∆ are inde- enhancement of quantum ﬂuctuations by Γ along with pendent (dashed lines), and where these quantities are re- the spin ﬂuctuations due to the RFs inside the paramag- lated through the power law (∆/J) = 0.02(Γ/J)2. In this netic phase create two distinct scenarios, more precisely later case, since one has (∆/J) > 0 for any (Γ/J) > 0, concerning the PM1 region, as discussed next: (a) For the denominator b will always display a minimum value ﬁxed ∆ [e.g., Fig. 6(a)], one has Γ and ∆ independent, ∗ around a temperature T , higher than Tf. Particularly, so that the smearing of the nonlinear susceptibility is by means of this relation, higher values of Γ imply on caused only by the RFs, leading to the eﬀect that the higher values of ∆, increasing the values of b at the min- full and dashed lines remain essentially parallel to one ima, resulting in a decrease in the amplitude of the max- another, up to zero temperature; (b) The phase diagram ima of χ3. of Fig. 6(b), for which ∆ and Γ are related through the parabolic behavior shown in the inset, the appearance of In Fig. 6 we present phase diagrams T/J versus Γ/J ∗ T occurs for Γ > 0 (i.e., ∆ > 0). Hence, the region showing a decrease in the temperature T for increas- f PM1 starts very narrow for low Γ, and gets enlarged for ing values of Γ (full lines). These lines delimit the SG increasing values of Γ, showing that the rounding of χ phase and were identiﬁed with the onset of RSB, by set- 3 is dominated by the enhancement of the RFs, leading to ting λ = 0; throughout the whole SG phases ones has AT ∗ spins ﬂuctuations due to the RFs inside the paramagnetic λ < 0. The temperature T (dashed lines), associated AT phase. with the maximum of the nonlinear susceptibility χ , sig- 3 IV. CONCLUSIONS nals a crossover between two regions of the paramagnetic phase (PM1 and PM2), as will be discussed next. The phase diagram shown in Fig. 6(a) corresponds to a ﬁxed We have investigated a quantum spin-glass model in value of ∆ [(∆/J) = 0.25], and was obtained by consider- the presence of a uniform transverse ﬁeld Γ, as well as ing Γ and ∆ as independent quantities. In this case, one of a longitudinal random ﬁeld h , the later following a i notices that the two lines (full and dashed lines) remain Gaussian distribution characterized by a width propor- essentially parallel to one another, up to zero tempera- tional to ∆. The model was considered in the limit ture, where the full line reaches a QCP, which appears of inﬁnite-range interactions and studied through the √ to be shifted towards lower values of Γ, when compared replica formalism, within a one-step replica-symmetry- ≈ with the QCP for ∆ = 0, Γ0 = 2 2J 2.828J40. The breaking procedure. The spin-glass critical frontier, sig- c case shown in Fig. 6(b) corresponds to Γ and ∆ follow- naled by the temperature T , was identiﬁed with the on- f ing the relation (∆/J) = 0.02(Γ/J)2 (see inset), so that set of replica-symmetry breaking, calculated through the ∗ for Γ = 0, one has ∆ = 0, giving T = T . By increas- Almeida-Thouless eigenvalue (replicon) λ , i.e., by set- f AT ing values of Γ, the width of RFs also increases, leading ting λ = 0. In this approach, the whole spin-glass AT to a rounded peak in the nonlinear susceptibility, yield- phase becomes characterized by λ < 0, and conse- ∗ AT ing T > T , and consequently, the region PM1 emerges. quently, it was treated through replica-symmetry break- f Due to the joint increase of both Γ and ∆, as shown in the ing. Such analysis was motivated by experimental inves- inset, the region PM1 gets enlarged up to zero tempera- tigations on the LiHo Y − F compound. In this sys- x 1 x 4 ture, where one gets a QCP, shifted towards lower values tem, the application of a transverse magnetic ﬁeld yields of Γ as compared with the QCP for ∆ = 0, similarly to rather intriguing eﬀects, particularly related to the be- ##LTLine##

text: should

text: be

text: mentioned

text: that

text: q

text: is

text: the

text: only

text: quantity

text: appear-

text: the

text: one

text: occurring

text: in

text: Fig.

text: ing

text: in

text: χ3

text: [either

text: in

text: Eq.

text: or

text: in

text: Eq.

text: which

text: may

text: It

text: should

text: be

text: emphasized

text: that

text: the

text: temperature

text: T

text: plays

text: present

text: a

text: divergence

text: at

text: ﬁnite

text: temperatures.

text: Moreover,

text: a

text: role

text: diﬀerent

text: from

text: T

text: in

text: the

text: sense

text: that

text: it

text: does

text: not

text: cor-

text: f

text: one

text: can

text: show

text: that

text: for

text: the

text: so-called

text: “dangerous”

text: respond

text: to

text: a

text: phase

text: transition,

text: but

text: rather

text: to

text: a

text: crossover

text: in

text: Eq−.

text: is

text: equal

text: to

text: the

text: denominator

text: of

text: between

text: two

text: distinct

text: regions

text: of

text: the

text: paramagnetic

text: phase.

text: i.e.,

text: λAT

text: even

text: for

text: Γ

text: Th∗e

text: The

text: previous

text: analysis

text: of

text: the

text: SK

text: model

text: in

text: the

text: presence

text: mechanism

text: behind

text: the

text: ﬂattening

text: of

text: the

text: χ3

text: peak

text: at

text: T

text: of

text: a

text: Gaussian

text: random

text: ﬁeld

text: which

text: should

text: corre-

text: is

text: illustrated

text: in

text: Fig.

text: where

text: we

text: plot

text: the

text: quantity

text: b

text: of

text: spond

text: herein

text: to

text: the

text: region

text: of

text: high

text: temperatures

text: and

text: low

text: Eq.

text: λAT,

text: and

text: versus

text: the

text: dimensionless

text: tem-

text: Γ∗(i.e.,

text: the

text: classical

text: regime),

text: has

text: also

text: found

text: a

text: temperature

text: perature,

text: for

text: typical

text: choices

text: of

text: Γ/J

text: and

text: ∆/J.

text: Results

text: T

text: associated

text: with

text: the

text: rounded

text: maximum

text: of

text: χ

text: with

text: for

text: Γ

text: and

text: independent

text: are

text: presented

text: in

text: Fig.

text: the

text: T

text: T

text: In

text: this

text: case,

text: T

text: was

text: interpreted

text: as

text: an

text: eﬀect

text: of

text: f

text: full

text: lines

text: [cases

text: (∆/J)

text: show

text: that

text: the

text: quantities

text: the

text: RFs

text: acting

text: inside

text: the

text: paramagnetic

text: phase,

text: instead

text: of

text: b

text: and

text: λAT

text: become

text: zero

text: together,

text: being

text: associated

text: with

text: some

text: type

text: of

text: non-trivial

text: ergod∗icity

text: breaking.

text: Herein,

text: we

text: the

text: divergence

text: of

text: χ3

text: [according

text: to

text: Eq.

text: signalling

text: the

text: claim

text: that

text: the

text: temperature

text: T

text: although

text: it

text: may

text: be

text: also

text: SG

text: phase-transition

text: temperature

text: Tf.

text: However,

text: the

text: re-

text: aﬀected

text: by

text: the

text: transverse

text: ﬁeld

text: Γ,

text: should

text: be

text: interpreted

text: sults

text: for

text: (∆/J)

text: show

text: that

text: such

text: a

text: small

text: value

text: for

text: in

text: a

text: similar

text: manner.

text: Hence,

text: along

text: the

text: line

text: signaled

text: by

text: the

text: width

text: of

text: the

text: RFs

text: distribution

text: yields

text: b

text: which

text: T

text: the

text: growth

text: of

text: Γ

text: produces

text: an

text: enhancement

text: of

text: quan-

text: f

text: presents

text: a

text: smooth

text: minimum

text: around

text: a

text: temperature

text: T

text: tum

text: ﬂuctuations,

text: which

text: become

text: increasingly

text: dominant

text: being

text: directly

text: associated

text: with

text: the

text: rounding

text: behavior

text: of

text: as

text: compared

text: with

text: thermal

text: ﬂuctuations,

text: driving

text: the

text: non-

text: on

text: the

text: other

text: h∗and,

text: one

text: has

text: λAT

text: at

text: a

text: temperature

text: trivial

text: ergodicity

text: breaking

text: of

text: the

text: SG

text: phase

text: transiti∗on

text: to

text: Tf,

text: with

text: Tf

text: T

text: In

text: Fig.

text: we

text: present

text: the

text: denomi-

text: a

text: QCP.

text: For

text: temperatures

text: in

text: the

text: region

text: Tf

text: T

text: T

text: the

text: nator

text: b

text: and

text: λAT,

text: for

text: the

text: cases

text: where

text: Γ

text: and

text: are

text: inde-

text: enhancement

text: of

text: quantum

text: ﬂuctuations

text: by

text: Γ

text: along

text: with

text: pendent

text: (dashed

text: lines),

text: and

text: where

text: these

text: quantities

text: are

text: re-

text: the

text: spin

text: ﬂuctuations

text: due

text: to

text: the

text: RFs

text: inside

text: the

text: paramag-

text: lated

text: through

text: the

text: power

text: law

text: (∆/J)

text: In

text: this

text: netic

text: phase

text: create

text: two

text: distinct

text: scenarios,

text: more

text: precisely

text: later

text: case,

text: since

text: one

text: has

text: (∆/J)

text: for

text: any

text: (Γ/J)

text: concerning

text: the

text: region,

text: as

text: discussed

text: next:

text: (a)

text: For

text: the

text: denominator

text: b

text: will

text: always

text: display

text: a

text: minimum

text: value

text: ﬁxed

text: [e.g.,

text: Fig.

text: one

text: has

text: Γ

text: and

text: independent,

text: around

text: a

text: temperature

text: T

text: higher

text: than

text: Tf.

text: Particularly,

text: so

text: that

text: the

text: smearing

text: of

text: the

text: nonlinear

text: susceptibility

text: is

text: by

text: means

text: of

text: this

text: relation,

text: higher

text: values

text: of

text: Γ

text: imply

text: on

text: caused

text: only

text: by

text: the

text: RFs,

text: leading

text: to

text: the

text: eﬀect

text: that

text: the

text: higher

text: values

text: of

text: increasing

text: the

text: values

text: of

text: b

text: at

text: the

text: min-

text: full

text: and

text: dashed

text: lines

text: remain

text: essentially

text: parallel

text: to

text: one

text: ima,

text: resulting

text: in

text: a

text: decrease

text: in

text: the

text: amplitude

text: of

text: the

text: max-

text: another,

text: up

text: to

text: zero

text: temperature;

text: (b)

text: The

text: phase

text: diagram

text: ima

text: of

text: of

text: Fig.

text: for

text: which

text: and

text: Γ

text: are

text: related

text: through

text: the

text: parabolic

text: behavior

text: shown

text: in

text: the

text: inset,

text: the

text: appearance

text: of

text: In

text: Fig.

text: we

text: present

text: phase

text: diagrams

text: T/J

text: versus

text: Γ/J

text: T

text: occurs

text: for

text: Γ

text: (i.e.,

text: Hence,

text: the

text: region

text: showing

text: a

text: decrease

text: in

text: the

text: temperature

text: T

text: for

text: increas-

text: f

text: starts

text: very

text: narrow

text: for

text: low

text: Γ,

text: and

text: gets

text: enlarged

text: for

text: ing

text: values

text: of

text: Γ

text: (full

text: lines).

text: These

text: lines

text: delimit

text: the

text: SG

text: increasing

text: values

text: of

text: Γ,

text: showing

text: that

text: the

text: rounding

text: of

text: χ

text: phase

text: and

text: were

text: identiﬁed

text: with

text: the

text: onset

text: of

text: RSB,

text: by

text: set-

text: is

text: dominated

text: by

text: the

text: enhancement

text: of

text: the

text: RFs,

text: leading

text: to

text: ting

text: λ

text: throughout

text: the

text: whole

text: SG

text: phases

text: ones

text: has

text: AT

text: spins

text: ﬂuctuations

text: due

text: to

text: the

text: RFs

text: inside

text: the

text: paramagnetic

text: λ

text: The

text: temperature

text: T

text: (dashed

text: lines),

text: associated

text: AT

text: phase.

text: with

text: the

text: maximum

text: of

text: the

text: nonlinear

text: susceptibility

text: χ

text: sig-

text: IV.

text: CONCLUSIONS

text: nals

text: a

text: crossover

text: between

text: two

text: regions

text: of

text: the

text: paramagnetic

text: phase

text: and

text: as

text: will

text: be

text: discussed

text: next.

text: The

text: phase

text: diagram

text: shown

text: in

text: Fig.

text: corresponds

text: to

text: a

text: ﬁxed

text: We

text: have

text: investigated

text: a

text: quantum

text: spin-glass

text: model

text: in

text: value

text: of

text: [(∆/J)

text: and

text: was

text: obtained

text: by

text: consider-

text: the

text: presence

text: of

text: a

text: uniform

text: transverse

text: ﬁeld

text: Γ,

text: as

text: well

text: as

text: ing

text: Γ

text: and

text: as

text: independent

text: quantities.

text: In

text: this

text: case,

text: one

text: of

text: a

text: longitudinal

text: random

text: ﬁeld

text: h

text: the

text: later

text: following

text: a

text: i

text: notices

text: that

text: the

text: two

text: lines

text: (full

text: and

text: dashed

text: lines)

text: remain

text: Gaussian

text: distribution

text: characterized

text: by

text: a

text: width

text: propor-

text: essentially

text: parallel

text: to

text: one

text: another,

text: up

text: to

text: zero

text: tempera-

text: tional

text: to

text: The

text: model

text: was

text: considered

text: in

text: the

text: limit

text: ture,

text: where

text: the

text: full

text: line

text: reaches

text: a

text: QCP,

text: which

text: appears

text: of

text: inﬁnite-range

text: interactions

text: and

text: studied

text: through

text: the

text: to

text: be

text: shifted

text: towards

text: lower

text: values

text: of

text: Γ,

text: when

text: compared

text: replica

text: formalism,

text: within

text: a

text: one-step

text: replica-symmetry-

text: with

text: the

text: QCP

text: for

text: Γ0

text: The

text: breaking

text: procedure.

text: The

text: spin-glass

text: critical

text: frontier,

text: sig-

text: c

text: case

text: shown

text: in

text: Fig.

text: corresponds

text: to

text: Γ

text: and

text: follow-

text: naled

text: by

text: the

text: temperature

text: T

text: was

text: identiﬁed

text: with

text: the

text: on-

text: f

text: ing

text: the

text: relation

text: (∆/J)

text: (see

text: inset),

text: so

text: that

text: set

text: of

text: replica-symmetry

text: breaking,

text: calculated

text: through

text: the

text: for

text: Γ

text: one

text: has

text: giving

text: T

text: T

text: By

text: increas-

text: Almeida-Thouless

text: eigenvalue

text: (replicon)

text: λ

text: i.e.,

text: by

text: set-

text: f

text: AT

text: ing

text: values

text: of

text: Γ,

text: the

text: width

text: of

text: RFs

text: also

text: increases,

text: leading

text: ting

text: λ

text: In

text: this

text: approach,

text: the

text: whole

text: spin-glass

text: AT

text: to

text: a

text: rounded

text: peak

text: in

text: the

text: nonlinear

text: susceptibility,

text: yield-

text: phase

text: becomes

text: characterized

text: by

text: λ

text: and

text: conse-

text: AT

text: ing

text: T

text: T

text: and

text: consequently,

text: the

text: region

text: emerges.

text: quently,

text: it

text: was

text: treated

text: through

text: replica-symmetry

text: break-

text: f

text: Due

text: to

text: the

text: joint

text: increase

text: of

text: both

text: Γ

text: and

text: as

text: shown

text: in

text: the

text: ing.

text: Such

text: analysis

text: was

text: motivated

text: by

text: experimental

text: inves-

text: inset,

text: the

text: region

text: gets

text: enlarged

text: up

text: to

text: zero

text: tempera-

text: tigations

text: on

text: the

text: LiHo

text: Y

text: F

text: compound.

text: In

text: this

text: sys-

text: x

text: x

text: ture,

text: where

text: one

text: gets

text: a

text: QCP,

text: shifted

text: towards

text: lower

text: values

text: tem,

text: the

text: application

text: of

text: a

text: transverse

text: magnetic

text: ﬁeld

text: yields

text: of

text: Γ

text: as

text: compared

text: with

text: the

text: QCP

text: for

text: similarly

text: to

text: rather

text: intriguing

text: eﬀects,

text: particularly

text: related

text: to

text: the

text: be-

text: ##LTLine##


text: 12 MOTIEJUS VALIUNAS The result of Theorem 5 may be therefore more delicate than one might think. Example 15. Suppose Γ is a complete bipartite graph K , i.e. Γ has vertex k,k set { } V (Γ) = u , . . . , u , v , . . . , v 1 k 1 k and edge set {{ } | ≤ ≤ } E(Γ) = u , v 1 i, j k , i j ∼ Z { − } ∈ and let H(u) = with genera∼tors ×X(u) = xu, xu 1 for each u V (Γ). In this case one has G(Γ, H) = F F (direct product of two free groups k k of rank k) and so one can calculate sphere sizes in G(Γ, H) and its special subgroups easily. Note that clearly (by the deﬁnition of link) every element ≤ ≤ of G G commutes with every element of G G. Now consider the A { } {linkA } case where A = u , . . . , u and so link A = v , . . . , v . It follows that 1 k 1 k { ∈ | } ⊇ × (x, y) B(n)2 [x, y] = 1 B (n) B (n). GA GlinkA An explicit computation shows that − − k(2k 1)n 1 B (n) = B (n) = − GA GlinkA k 1 and − 2k2n(2k 1)n − B (n) = − − + e (2k 1)n + e G 1 2 (k 1)(2k 1) (cid:18) (cid:19) where e = e (k) and e = e (k) are some constants. It follows that 1 1 2 2 − ≥ B (n)B (n) ∼ 2k 1 2 d GA GlinkA n B (n)2 2kn G → ∞ ∞ as n . In particular, the sequence (d ) converges to zero only at a n n=0 polynomial rate for G = G(Γ, H). The proof of Theorem 5 is based on the fact that if (7) held for inﬁnitely ⊆ many n then there would exist a subset A V (Γ) such that the growth of both G and G would be comparable to that of G. More precisely, the A linkA outline of the proof is as follows: ⊆ (i) ﬁnding such a subset A V (Γ) and showing that G is not negligible A in G, i.e. BGA(n) (cid:57) 0 as n → ∞ (subsection ); BG(n) ∼ × (ii) ﬁnding a graph of groups G(Γˆ, Hˆ ) = G H such that H has linkA B (n) (cid:57) → ∞ polynomial growth and G(Γˆ,Hˆ) 0 as n (subsection ); ×BG(n) ⊆ (iii) using the embedding G G G and Theorem 1 to obtain a A linkA contradiction (subsection ). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MOTIEJUS

text: VALIUNAS

text: The

text: result

text: of

text: Theorem

text: may

text: be

text: therefore

text: more

text: delicate

text: than

text: one

text: might

text: think.

text: Example

text: Suppose

text: Γ

text: is

text: a

text: complete

text: bipartite

text: graph

text: K

text: i.e.

text: Γ

text: has

text: vertex

text: k,k

text: set

text: V

text: (Γ)

text: u

text: u

text: v

text: v

text: k

text: k

text: and

text: edge

text: set

text: E(Γ)

text: u

text: v

text: i,

text: j

text: k

text: i

text: j

text: Z

text: and

text: let

text: H(u)

text: with

text: genera∼tors

text: ×X(u)

text: xu,

text: xu

text: for

text: each

text: u

text: V

text: (Γ).

text: In

text: this

text: case

text: one

text: has

text: G(Γ,

text: H)

text: F

text: F

text: (direct

text: product

text: of

text: two

text: free

text: groups

text: k

text: k

text: of

text: rank

text: k)

text: and

text: so

text: one

text: can

text: calculate

text: sphere

text: sizes

text: in

text: G(Γ,

text: H)

text: and

text: its

text: special

text: subgroups

text: easily.

text: Note

text: that

text: clearly

text: (by

text: the

text: deﬁnition

text: of

text: link)

text: every

text: element

text: of

text: G

text: G

text: commutes

text: with

text: every

text: element

text: of

text: G

text: G.

text: Now

text: consider

text: the

text: A

text: {linkA

text: case

text: where

text: A

text: u

text: u

text: and

text: so

text: link

text: A

text: v

text: v

text: It

text: follows

text: that

text: k

text: k

text: (x,

text: y)

text: [x,

text: y]

text: B

text: (n)

text: B

text: (n).

text: GA

text: GlinkA

text: An

text: explicit

text: computation

text: shows

text: that

text: B

text: (n)

text: B

text: (n)

text: GA

text: GlinkA

text: k

text: and

text: B

text: (n)

text: e

text: e

text: G

text: (k

text: where

text: e

text: e

text: (k)

text: and

text: e

text: e

text: (k)

text: are

text: some

text: constants.

text: It

text: follows

text: that

text: B

text: (n)B

text: (n)

text: d

text: GA

text: GlinkA

text: n

text: B

text: G

text: as

text: n

text: In

text: particular,

text: the

text: sequence

text: (d

text: converges

text: to

text: zero

text: only

text: at

text: a

text: n

text: polynomial

text: rate

text: for

text: G

text: G(Γ,

text: H).

text: The

text: proof

text: of

text: Theorem

text: is

text: based

text: on

text: the

text: fact

text: that

text: if

text: held

text: for

text: inﬁnitely

text: many

text: n

text: then

text: there

text: would

text: exist

text: a

text: subset

text: A

text: V

text: (Γ)

text: such

text: that

text: the

text: growth

text: of

text: both

text: G

text: and

text: G

text: would

text: be

text: comparable

text: to

text: that

text: of

text: G.

text: More

text: precisely,

text: the

text: A

text: linkA

text: outline

text: of

text: the

text: proof

text: is

text: as

text: follows:

text: (i)

text: ﬁnding

text: such

text: a

text: subset

text: A

text: V

text: (Γ)

text: and

text: showing

text: that

text: G

text: is

text: not

text: negligible

text: A

text: in

text: G,

text: i.e.

text: BGA(n)

text: as

text: n

text: (subsection

text: BG(n)

text: (ii)

text: ﬁnding

text: a

text: graph

text: of

text: groups

text: G(Γˆ,

text: Hˆ

text: G

text: H

text: such

text: that

text: H

text: has

text: linkA

text: B

text: (n)

text: polynomial

text: growth

text: and

text: G(Γˆ,Hˆ)

text: as

text: n

text: (subsection

text: ×BG(n)

text: (iii)

text: using

text: the

text: embedding

text: G

text: G

text: G

text: and

text: Theorem

text: to

text: obtain

text: a

text: A

text: linkA

text: contradiction

text: (subsection

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: RATIONAL GROWTH AND DEGREE OF COMMUTATIVITY 15 − ∈ ∈ which have an expression g = p 1g˜p with p B (s) and g˜ B (n). It g g g G GA follows that |{ ∈ | | | ≤ }| ≤ g B(n) supp(g˜) = A, p s B (s)B (n) ∗ g G GA and so ( ) implies that ∗∗ ε˜ ≤ B (n) ≤ ( ) GA 1 B (s) B (n) G G for inﬁnitely many n, where the second inequality comes from the fact that ⊆ B (n) B (n). GA G † In order to use ( ), one needs to consider forms of centralisers of elements ∈ ∈ g G with supp(g˜) = A. Fix an element g G with supp(g˜) = A and note − | | ≤ that one clearly has C (g) = p 1C (g˜)p , so if p s then one has G g G g g | ∩ | ≤ | ∩ | (13) C (g) B(n) C (g˜) B(n + 2s) . G G ∈ Now deﬁne an element g G to be cyclically normal (in the sense of [2]) if ≤ ≥ · · · ∈ either (cid:96) (g) 1, or n := (cid:96) (g) 2 and for any normal form w = w w ∗ n ∈ n∗ ∈ (cid:54) 1 n X of g, where w X(v ) for some v V (Γ), one has v = v . Then one i i i 1 n has ∈ Lemma 17. For any g G with supp(g˜) = A, there exists an element ∈ − p˜ G such that gˆ := p˜ g˜p˜ 1 is cyclically normal and supp(gˆ) = A. g A g g ≤ Proof. If (cid:96) (g˜) 1 then p˜ = 1 does the job. Thus suppose that n := ≥ n g (cid:96) (g˜) 2. Let n { | · · · ∈ ∗ E(g˜) := g w = w w X is a normal form for g˜ where 0 1 n ∈ ∗ ∈ } w X(v ) for some v V (Γ), and w represents g i i i n 0 (cid:81) be a ﬁnite subset of G . By Theorem 12, any two elements in E(g) commute, ∈ A ∈ and so if g H(v ) and g H(v ) are distinct elements of E(g˜) then (cid:54) 1 1 2 2 ∈ v = v . Now deﬁne p˜ := ∈ g . Then p˜ G , and following the 1 2 g gn E(g˜) n g − A proof of [2, Lemma 23] one can see that gˆ := p˜ g˜p˜ 1 is cyclically normal. ⊆ g g ⊆ Since supp(p˜ ) A and supp(g˜) = A, it is clear that supp(gˆ) A. It also g ∪ ⊇ follows by [2, Lemma 18] that supp(gˆ) supp(p˜ ) A. Thus one only needs ⊆ g to check that supp(p˜ ) supp(gˆ). g ∈ \ Suppose for contradiction that there exists a vertex v supp(p˜ ) supp(gˆ), ∈ ∩ g and let g E(g˜) H(v) be the (unique) element. It is easy to see that ∈ v \ { } v / link(A v ): otherwise any normal form of gˆ would contain a subword ∗ ∈ in X(v) representing g and so v supp(gˆ). Then, following again the proof v − ≤ − − of [2, Lemma 23], one has n˜ := (cid:96) (g g˜g 1) n 1, with n˜ = n 1 if and only · · · n v v ∈ ∗ ∈ if g˜ has no normal form w w , where w X(v ) for some v V (Γ), 1 n i i i ##LTLine## ##LTLine##

text: RATIONAL

text: GROWTH

text: AND

text: DEGREE

text: OF

text: COMMUTATIVITY

text: which

text: have

text: an

text: expression

text: g

text: p

text: with

text: p

text: B

text: (s)

text: and

text: g˜

text: B

text: (n).

text: It

text: g

text: g

text: g

text: G

text: GA

text: follows

text: that

text: g

text: B(n)

text: supp(g˜)

text: A,

text: p

text: s

text: B

text: (s)B

text: (n)

text: g

text: G

text: GA

text: and

text: so

text: implies

text: that

text: ε˜

text: B

text: (n)

text: GA

text: B

text: (s)

text: B

text: (n)

text: G

text: G

text: for

text: inﬁnitely

text: many

text: n,

text: where

text: the

text: second

text: inequality

text: comes

text: from

text: the

text: fact

text: that

text: B

text: (n)

text: B

text: (n).

text: GA

text: G

text: In

text: order

text: to

text: use

text: one

text: needs

text: to

text: consider

text: forms

text: of

text: centralisers

text: of

text: elements

text: g

text: G

text: with

text: supp(g˜)

text: A.

text: Fix

text: an

text: element

text: g

text: G

text: with

text: supp(g˜)

text: A

text: and

text: note

text: that

text: one

text: clearly

text: has

text: C

text: (g)

text: p

text: (g˜)p

text: so

text: if

text: p

text: s

text: then

text: one

text: has

text: G

text: g

text: G

text: g

text: g

text: C

text: (g)

text: B(n)

text: C

text: (g˜)

text: B(n

text: G

text: G

text: Now

text: deﬁne

text: an

text: element

text: g

text: G

text: to

text: be

text: cyclically

text: normal

text: (in

text: the

text: sense

text: of

text: if

text: either

text: (g)

text: or

text: n

text: (g)

text: and

text: for

text: any

text: normal

text: form

text: w

text: w

text: w

text: n

text: n∗

text: n

text: X

text: of

text: g,

text: where

text: w

text: X(v

text: for

text: some

text: v

text: V

text: (Γ),

text: one

text: has

text: v

text: v

text: Then

text: one

text: i

text: i

text: i

text: n

text: has

text: Lemma

text: For

text: any

text: g

text: G

text: with

text: supp(g˜)

text: A,

text: there

text: exists

text: an

text: element

text: p˜

text: G

text: such

text: that

text: gˆ

text: p˜

text: g˜p˜

text: is

text: cyclically

text: normal

text: and

text: supp(gˆ)

text: A.

text: g

text: A

text: g

text: g

text: Proof.

text: If

text: (g˜)

text: then

text: p˜

text: does

text: the

text: job.

text: Thus

text: suppose

text: that

text: n

text: n

text: g

text: (g˜)

text: Let

text: n

text: E(g˜)

text: g

text: w

text: w

text: w

text: X

text: is

text: a

text: normal

text: form

text: for

text: g˜

text: where

text: n

text: w

text: X(v

text: for

text: some

text: v

text: V

text: (Γ),

text: and

text: w

text: represents

text: g

text: i

text: i

text: i

text: n

text: be

text: a

text: ﬁnite

text: subset

text: of

text: G

text: By

text: Theorem

text: any

text: two

text: elements

text: in

text: E(g)

text: commute,

text: A

text: and

text: so

text: if

text: g

text: H(v

text: and

text: g

text: H(v

text: are

text: distinct

text: elements

text: of

text: E(g˜)

text: then

text: v

text: v

text: Now

text: deﬁne

text: p˜

text: g

text: Then

text: p˜

text: G

text: and

text: following

text: the

text: g

text: gn

text: E(g˜)

text: n

text: g

text: A

text: proof

text: of

text: Lemma

text: one

text: can

text: see

text: that

text: gˆ

text: p˜

text: g˜p˜

text: is

text: cyclically

text: normal.

text: g

text: g

text: Since

text: supp(p˜

text: A

text: and

text: supp(g˜)

text: A,

text: it

text: is

text: clear

text: that

text: supp(gˆ)

text: A.

text: It

text: also

text: g

text: follows

text: by

text: Lemma

text: that

text: supp(gˆ)

text: supp(p˜

text: A.

text: Thus

text: one

text: only

text: needs

text: g

text: to

text: check

text: that

text: supp(p˜

text: supp(gˆ).

text: g

text: Suppose

text: for

text: contradiction

text: that

text: there

text: exists

text: a

text: vertex

text: v

text: supp(p˜

text: supp(gˆ),

text: g

text: and

text: let

text: g

text: E(g˜)

text: H(v)

text: be

text: the

text: (unique)

text: element.

text: It

text: is

text: easy

text: to

text: see

text: that

text: v

text: v

text: link(A

text: v

text: otherwise

text: any

text: normal

text: form

text: of

text: gˆ

text: would

text: contain

text: a

text: subword

text: in

text: X(v)

text: representing

text: g

text: and

text: so

text: v

text: supp(gˆ).

text: Then,

text: following

text: again

text: the

text: proof

text: v

text: of

text: Lemma

text: one

text: has

text: n˜

text: (g

text: g˜g

text: n

text: with

text: n˜

text: n

text: if

text: and

text: only

text: n

text: v

text: v

text: if

text: g˜

text: has

text: no

text: normal

text: form

text: w

text: w

text: where

text: w

text: X(v

text: for

text: some

text: v

text: V

text: (Γ),

text: n

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##