text: 8 ± to model this tendency, we posit that the coeﬃcients η for ﬁrm i are given by: i − η = η max(1 + ΓΦ (t), 0) i 0 − i η+ = Rη max(1 ΓΦ (t), 0) (12) i 0 i where η is a ﬁxed coeﬃcient, identical for all ﬁrms, and R is the propensity ratio discussed in the previous section. 0 The factor Γ > 0 measures how the ﬁnancial fragility of ﬁrms inﬂuences their hiring/ﬁring policy, since a larger value of Φ then leads to a faster downward adjustment of the workforce when the ﬁrm is over-producing, and a slower i ± (more cautious) upward adjustment when the ﬁrm is under-producing. The above deﬁnition however ensures that η always remain non-negative, i.e. the reaction of the ﬁrms is always in the intuitive direction. It is plausible that the ﬁnancial fragility of the ﬁrm also aﬀects its wage policy: we give in Appendix A the wage update rules of Mark-0 and their modiﬁcation to account for ﬁnancial fragility, through the very same parameter Γ. In essence, deeply indebted ﬁrms seek to reduce wages more rapidly, whereas ﬂourishing ﬁrms tend to increase wages quickly. ≡ The baseline Mark-0 model corresponds to Γ 0, and leads to the phase diagram shown in Fig. 1 above. Interest- ingly, a non-zero value of Γ = Γ (constant across ﬁrms and in time) changes substantially the nature – but not the 0 existence – of the phase transition between the full employment (FE) and full unemployment (FU) phase. The ﬁrst ∞ order (discontinuous) transition for Γ = 0, Θ = found in [1] and shown in Fig. 1, is replaced by a second order 0 (continuous) transition when the ﬁrms adapt their behavior as a function of their ﬁnancial fragility, i.e. when Γ > 0. 0 Moreover, the “bad” FU phase for R < R becomes a partial unemployment phase with u < 1 that continuously c varies with R: see Appendix B and Fig. 9 for full details. As ﬁrms become more careful, employment can be to some extent preserved – as expected. The “good” phase of the economy, on the other hand, is only mildly aﬀected by a non zero Γ – for example the 0 FE region of Fig. 1 expands downwards, which is expected since ﬁrms manage more carefully their balance sheet, reducing the occurrence of defaults. c. The inﬂuence of interest rates on the strategy of ﬁrms. We now argue that Γ should in fact depend on the diﬀerence between the interest rate and the inﬂation: high cost of credit makes ﬁrms particularly wary of going into debt and their sensitivity to their ﬁnancial fragility should be increased. Therefore, we postulate that interest rates feedback into the behavior of the ﬁrm primarily through the Γ parameter, that we model in the simplest possible way (cid:101) (cid:101) as: { − } Γ = max α (ρ(cid:96)(t) π(t)), Γ , (cid:101) (cid:101) (13) Γ 0 − where α (similarly to α above) captures the inﬂuence of the real interest rate ρ(cid:96)(t) π(t) on the hiring/ﬁring policy Γ c of the ﬁrms. Whenever the real interest rate stays below Γ /α , the response of ﬁrms to changes of interest rates 0 Γ is negligible (perhaps as reported in [11]), whereas larger rates lead to a substantial change in the ﬁrms policy. The case α = 0 but Γ > 0 corresponds to the above discussion and it is interesting in itself (see Appendix B). However, Γ 0 since we will be mostly concerned with policy issues, we will concentrate below on the other extreme case, α > 0 Γ and Γ = 0, keeping in mind that reality is probably in-between. Note that Γ as deﬁned above is now zero when real 0 interest rates are negative, and is positive otherwise. C. Summary 1. How many new assumptions? In our attempt to include interest rates and monetary policy into the Mark-0 framework, we have made new behavioral assumptions, that we tried to keep as simple and as parsimonious as possible: • For the Central Bank, we have merely adapted the standard Taylor rule to our setting, introducing three ∗ monetary policy parameters ρ (the natural interest rate) and φ , φ (the Taylor rule parameters), and two ∗ ∗ π ε targets: inﬂation π and employment ε . • For the banking sector, we have essentially used trivial accounting rules to ensure a no-proﬁt condition on loans and deposits. The only new parameter is f and determines how much of the cost of bankruptcy must be paid by loans or by deposits. However, this parameter plays very little role in the qualitative behaviour of the economy, so we set it to f = 1/2 henceforth. • For the households, we assume an Euler-like behaviour of the consumption budget as a function of the real interest rate on deposits, i.e. the consumption budget decreases when the real rate is high and increases when it is low, with a slope given by parameter α . c

text: to

text: model

text: this

text: tendency,

text: we

text: posit

text: that

text: the

text: coeﬃcients

text: η

text: for

text: ﬁrm

text: i

text: are

text: given

text: by:

text: i

text: η

text: η

text: ΓΦ

text: (t),

text: i

text: i

text: η+

text: Rη

text: ΓΦ

text: (t),

text: i

text: i

text: where

text: η

text: is

text: a

text: ﬁxed

text: coeﬃcient,

text: identical

text: for

text: all

text: ﬁrms,

text: and

text: R

text: is

text: the

text: propensity

text: ratio

text: discussed

text: in

text: the

text: previous

text: section.

text: The

text: factor

text: Γ

text: measures

text: how

text: the

text: ﬁnancial

text: fragility

text: of

text: ﬁrms

text: inﬂuences

text: their

text: hiring/ﬁring

text: policy,

text: since

text: a

text: larger

text: value

text: of

text: Φ

text: then

text: leads

text: to

text: a

text: faster

text: downward

text: adjustment

text: of

text: the

text: workforce

text: when

text: the

text: ﬁrm

text: is

text: over-producing,

text: and

text: a

text: slower

text: i

text: (more

text: cautious)

text: upward

text: adjustment

text: when

text: the

text: ﬁrm

text: is

text: under-producing.

text: The

text: above

text: deﬁnition

text: however

text: ensures

text: that

text: η

text: always

text: remain

text: non-negative,

text: i.e.

text: the

text: reaction

text: of

text: the

text: ﬁrms

text: is

text: always

text: in

text: the

text: intuitive

text: direction.

text: It

text: is

text: plausible

text: that

text: the

text: ﬁnancial

text: fragility

text: of

text: the

text: ﬁrm

text: also

text: aﬀects

text: its

text: wage

text: policy:

text: we

text: give

text: in

text: Appendix

text: A

text: the

text: wage

text: update

text: rules

text: of

text: and

text: their

text: modiﬁcation

text: to

text: account

text: for

text: ﬁnancial

text: fragility,

text: through

text: the

text: very

text: same

text: parameter

text: Γ.

text: In

text: essence,

text: deeply

text: indebted

text: ﬁrms

text: seek

text: to

text: reduce

text: wages

text: more

text: rapidly,

text: whereas

text: ﬂourishing

text: ﬁrms

text: tend

text: to

text: increase

text: wages

text: quickly.

text: The

text: baseline

text: model

text: corresponds

text: to

text: Γ

text: and

text: leads

text: to

text: the

text: phase

text: diagram

text: shown

text: in

text: Fig.

text: above.

text: Interest-

text: ingly,

text: a

text: non-zero

text: value

text: of

text: Γ

text: Γ

text: (constant

text: across

text: ﬁrms

text: and

text: in

text: time)

text: changes

text: substantially

text: the

text: nature

text: but

text: not

text: the

text: existence

text: of

text: the

text: phase

text: transition

text: between

text: the

text: full

text: employment

text: (FE)

text: and

text: full

text: unemployment

text: (FU)

text: phase.

text: The

text: ﬁrst

text: order

text: (discontinuous)

text: transition

text: for

text: Γ

text: Θ

text: found

text: in

text: and

text: shown

text: in

text: Fig.

text: is

text: replaced

text: by

text: a

text: second

text: order

text: (continuous)

text: transition

text: when

text: the

text: ﬁrms

text: adapt

text: their

text: behavior

text: as

text: a

text: function

text: of

text: their

text: ﬁnancial

text: fragility,

text: i.e.

text: when

text: Γ

text: Moreover,

text: the

text: “bad”

text: FU

text: phase

text: for

text: R

text: R

text: becomes

text: a

text: partial

text: unemployment

text: phase

text: with

text: u

text: that

text: continuously

text: c

text: varies

text: with

text: R:

text: see

text: Appendix

text: B

text: and

text: Fig.

text: for

text: full

text: details.

text: As

text: ﬁrms

text: become

text: more

text: careful,

text: employment

text: can

text: be

text: to

text: some

text: extent

text: preserved

text: as

text: expected.

text: The

text: “good”

text: phase

text: of

text: the

text: economy,

text: on

text: the

text: other

text: hand,

text: is

text: only

text: mildly

text: aﬀected

text: by

text: a

text: non

text: zero

text: Γ

text: for

text: example

text: the

text: FE

text: region

text: of

text: Fig.

text: expands

text: downwards,

text: which

text: is

text: expected

text: since

text: ﬁrms

text: manage

text: more

text: carefully

text: their

text: balance

text: sheet,

text: reducing

text: the

text: occurrence

text: of

text: defaults.

text: c.

text: The

text: inﬂuence

text: of

text: interest

text: rates

text: on

text: the

text: strategy

text: of

text: ﬁrms.

text: We

text: now

text: argue

text: that

text: Γ

text: should

text: in

text: fact

text: depend

text: on

text: the

text: diﬀerence

text: between

text: the

text: interest

text: rate

text: and

text: the

text: inﬂation:

text: high

text: cost

text: of

text: credit

text: makes

text: ﬁrms

text: particularly

text: wary

text: of

text: going

text: into

text: debt

text: and

text: their

text: sensitivity

text: to

text: their

text: ﬁnancial

text: fragility

text: should

text: be

text: increased.

text: Therefore,

text: we

text: postulate

text: that

text: interest

text: rates

text: feedback

text: into

text: the

text: behavior

text: of

text: the

text: ﬁrm

text: primarily

text: through

text: the

text: Γ

text: parameter,

text: that

text: we

text: model

text: in

text: the

text: simplest

text: possible

text: way

text: as:

text: Γ

text: max

text: α

text: π(t)),

text: Γ

text: Γ

text: where

text: α

text: (similarly

text: to

text: α

text: above)

text: captures

text: the

text: inﬂuence

text: of

text: the

text: real

text: interest

text: rate

text: π(t)

text: on

text: the

text: hiring/ﬁring

text: policy

text: Γ

text: c

text: of

text: the

text: ﬁrms.

text: Whenever

text: the

text: real

text: interest

text: rate

text: stays

text: below

text: Γ

text: /α

text: the

text: response

text: of

text: ﬁrms

text: to

text: changes

text: of

text: interest

text: rates

text: Γ

text: is

text: negligible

text: (perhaps

text: as

text: reported

text: in

text: whereas

text: larger

text: rates

text: lead

text: to

text: a

text: substantial

text: change

text: in

text: the

text: ﬁrms

text: policy.

text: The

text: case

text: α

text: but

text: Γ

text: corresponds

text: to

text: the

text: above

text: discussion

text: and

text: it

text: is

text: interesting

text: in

text: itself

text: (see

text: Appendix

text: B).

text: However,

text: Γ

text: since

text: we

text: will

text: be

text: mostly

text: concerned

text: with

text: policy

text: issues,

text: we

text: will

text: concentrate

text: below

text: on

text: the

text: other

text: extreme

text: case,

text: α

text: Γ

text: and

text: Γ

text: keeping

text: in

text: mind

text: that

text: reality

text: is

text: probably

text: in-between.

text: Note

text: that

text: Γ

text: as

text: deﬁned

text: above

text: is

text: now

text: zero

text: when

text: real

text: interest

text: rates

text: are

text: negative,

text: and

text: is

text: positive

text: otherwise.

text: C.

text: Summary

text: How

text: many

text: new

text: assumptions?

text: In

text: our

text: attempt

text: to

text: include

text: interest

text: rates

text: and

text: monetary

text: policy

text: into

text: the

text: framework,

text: we

text: have

text: made

text: new

text: behavioral

text: assumptions,

text: that

text: we

text: tried

text: to

text: keep

text: as

text: simple

text: and

text: as

text: parsimonious

text: as

text: possible:

text: For

text: the

text: Central

text: Bank,

text: we

text: have

text: merely

text: adapted

text: the

text: standard

text: Taylor

text: rule

text: to

text: our

text: setting,

text: introducing

text: three

text: monetary

text: policy

text: parameters

text: ρ

text: (the

text: natural

text: interest

text: rate)

text: and

text: φ

text: φ

text: (the

text: Taylor

text: rule

text: parameters),

text: and

text: two

text: π

text: ε

text: targets:

text: inﬂation

text: π

text: and

text: employment

text: ε

text: For

text: the

text: banking

text: sector,

text: we

text: have

text: essentially

text: used

text: trivial

text: accounting

text: rules

text: to

text: ensure

text: a

text: no-proﬁt

text: condition

text: on

text: loans

text: and

text: deposits.

text: The

text: only

text: new

text: parameter

text: is

text: f

text: and

text: determines

text: how

text: much

text: of

text: the

text: cost

text: of

text: bankruptcy

text: must

text: be

text: paid

text: by

text: loans

text: or

text: by

text: deposits.

text: However,

text: this

text: parameter

text: plays

text: very

text: little

text: role

text: in

text: the

text: qualitative

text: behaviour

text: of

text: the

text: economy,

text: so

text: we

text: set

text: it

text: to

text: f

text: henceforth.

text: For

text: the

text: households,

text: we

text: assume

text: an

text: Euler-like

text: behaviour

text: of

text: the

text: consumption

text: budget

text: as

text: a

text: function

text: of

text: the

text: real

text: interest

text: rate

text: on

text: deposits,

text: i.e.

text: the

text: consumption

text: budget

text: decreases

text: when

text: the

text: real

text: rate

text: is

text: high

text: and

text: increases

text: when

text: it

text: is

text: low,

text: with

text: a

text: slope

text: given

text: by

text: parameter

text: α

text: c


text: 10 10 100 8 80 2 - RU 6 60 4 - FE Θ αΓ 2 - RU 4 40 4 - FE 2 20 3 - EC 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 ρ∗ (%) ρ∗ (%) ∗ − FIG. 2: Left: Phase diagram of the “natural” state of the economy in the ρ Θ plane (i.e. with no monetary policy, φ = φ = 0) when both ﬁrms’ and households’ decisions are sensitive to the rate level: α = 50 and α = 4. For small enough ε π∗ ∗∗ ∗ ∗∗ Γ c rates ρ < ρ , one recovers the FE-EC-RU phases as Θ is decreased. When ρ > ρ , however, the RU phase prevails for all ∗ − values of Θ, which becomes irrelevant. Right: Phase diagram in the ρ α plane for Θ = 3, showing the dependence of the ∗∗ Γ critical value ρ on α . The other parameters of the model are set to: R = 2 (with η−0 = 0.1), c = 0.5, Γ = 0, ϕ = 0.1, Γ 0 0 γ = γ = 0.05, β = 2, δ = 0.02, f = 0.5, N = 2000 (see Appendix A for a deﬁnition of all these parameters). p w F more unemployment that spirals into a destabilizing feedback loop. This is one of the “dark corners” that ABMs can help uncovering. B. Coupling between interest rates and household behavior Perhaps unexpectedly, the coupling between interest rate and consumption (captured by parameter α ) appears to c have much smaller inﬂuence on the “natural” state of the economy, at least when α is chosen within a reasonable c range. Its main inﬂuence is to increase the output ﬂuctuations around the steady state, by amplifying price trends (cid:29) through the resulting reduction/increase in consumption. Interestingly, we ﬁnd that for Θ 1 and independently of ∗ ρ , α has a stabilising eﬀect on the economy: R shifts to lower values as α increases (in the absence of monetary c c c policy). Clearly, micro-data is needed to estimate the value of α for realistic applications; but since this parameter c plays a small role we will choose rather arbitrarily α = 4, unless explicitly stated. This corresponds to a moderate c sensitivity to inﬂation/interest rates: a rise of the interest rate of 1%/year corresponds to a decrease of 4% of the consumption propensity. IV. MONETARY POLICY EXPERIMENTS We now consider a simple policy framework where the Central Bank adjusts the base interest rate ρ (t) in order 0 to achieve its inﬂation and employment targets. Given the simplicity of our model we are mainly interested here in gaining a qualitative understanding of the possible consequence of a Taylor-rule based monetary policy in a stylized Agent-Based framework. We defer parameter calibration, detailed comparison with simple DSGE models and more quantitative insights to future studies. Our major ﬁnding is that provided its policy is not too aggressive and the economy not too close to a phase transition, the CB is successful in steering the economy towards its targets. However, the mere presence of diﬀerent equilibrium states of the economy separated by phase boundaries (i.e. “dark corners”) may deeply alter the impact of monetary policy. Indeed, we will exhibit cases where the monetary policy by itself triggers large instabilities and is counter-productive. As in the previous section, we will refer to the state obtained without any response of the CB, i.e. when φ = φ = 0, π ε as the “natural” state of the economy (for a given set of parameters). The corresponding “natural” value of a variable x will be denoted by x . In order to simplify the analysis and since most of the parameters of our model play little nat role in the qualitative behavior of the economy we set once and for all some of them to the values given in the caption ∗ ∗ of Fig. 2 and choose ρ = 2%. We only focus on the four parameters deﬁning the CB policy (i.e. φ , φ , ε and ∗ π ε π ), the parameters of the transmission channels (i.e. α and α ), and the two parameters locating the system in the c Γ phase diagram of Fig. 1 (i.e. the hiring/ﬁring ratio R and the bankruptcy threshold Θ). ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: RU

text: FE

text: Θ

text: αΓ

text: RU

text: FE

text: EC

text: ρ∗

text: ρ∗

label: caption

text: FIG.

text: Left:

text: Phase

text: diagram

text: of

text: the

text: “natural”

text: state

text: of

text: the

text: economy

text: in

text: the

text: ρ

text: Θ

text: plane

text: (i.e.

text: with

text: no

text: monetary

text: policy,

text: φ

text: φ

text: when

text: both

text: ﬁrms’

text: and

text: households’

text: decisions

text: are

text: sensitive

text: to

text: the

text: rate

text: level:

text: α

text: and

text: α

text: For

text: small

text: enough

text: ε

text: π∗

text: Γ

text: c

text: rates

text: ρ

text: ρ

text: one

text: recovers

text: the

text: FE-EC-RU

text: phases

text: as

text: Θ

text: is

text: decreased.

text: When

text: ρ

text: ρ

text: however,

text: the

text: RU

text: phase

text: prevails

text: for

text: all

text: values

text: of

text: Θ,

text: which

text: becomes

text: irrelevant.

text: Right:

text: Phase

text: diagram

text: in

text: the

text: ρ

text: α

text: plane

text: for

text: Θ

text: showing

text: the

text: dependence

text: of

text: the

text: Γ

text: critical

text: value

text: ρ

text: on

text: α

text: The

text: other

text: parameters

text: of

text: the

text: model

text: are

text: set

text: to:

text: R

text: (with

text: η−0

text: c

text: Γ

text: ϕ

text: Γ

text: γ

text: γ

text: β

text: δ

text: f

text: N

text: (see

text: Appendix

text: A

text: for

text: a

text: deﬁnition

text: of

text: all

text: these

text: parameters).

text: p

text: w

text: F

text: more

text: unemployment

text: that

text: spirals

text: into

text: a

text: destabilizing

text: feedback

text: loop.

text: This

text: is

text: one

text: of

text: the

text: “dark

text: corners”

text: that

text: ABMs

text: can

text: help

text: uncovering.

text: B.

text: Coupling

text: between

text: interest

text: rates

text: and

text: household

text: behavior

text: Perhaps

text: unexpectedly,

text: the

text: coupling

text: between

text: interest

text: rate

text: and

text: consumption

text: (captured

text: by

text: parameter

text: α

text: appears

text: to

text: c

text: have

text: much

text: smaller

text: inﬂuence

text: on

text: the

text: “natural”

text: state

text: of

text: the

text: economy,

text: at

text: least

text: when

text: α

text: is

text: chosen

text: within

text: a

text: reasonable

text: c

text: range.

text: Its

text: main

text: inﬂuence

text: is

text: to

text: increase

text: the

text: output

text: ﬂuctuations

text: around

text: the

text: steady

text: state,

text: by

text: amplifying

text: price

text: trends

text: through

text: the

text: resulting

text: reduction/increase

text: in

text: consumption.

text: Interestingly,

text: we

text: ﬁnd

text: that

text: for

text: Θ

text: and

text: independently

text: of

text: ρ

text: α

text: has

text: a

text: stabilising

text: eﬀect

text: on

text: the

text: economy:

text: R

text: shifts

text: to

text: lower

text: values

text: as

text: α

text: increases

text: (in

text: the

text: absence

text: of

text: monetary

text: c

text: c

text: c

text: policy).

text: Clearly,

text: micro-data

text: is

text: needed

text: to

text: estimate

text: the

text: value

text: of

text: α

text: for

text: realistic

text: applications;

text: but

text: since

text: this

text: parameter

text: c

text: plays

text: a

text: small

text: role

text: we

text: will

text: choose

text: rather

text: arbitrarily

text: α

text: unless

text: explicitly

text: stated.

text: This

text: corresponds

text: to

text: a

text: moderate

text: c

text: sensitivity

text: to

text: inﬂation/interest

text: rates:

text: a

text: rise

text: of

text: the

text: interest

text: rate

text: of

text: corresponds

text: to

text: a

text: decrease

text: of

text: of

text: the

text: consumption

text: propensity.

text: IV.

text: MONETARY

text: POLICY

text: EXPERIMENTS

text: We

text: now

text: consider

text: a

text: simple

text: policy

text: framework

text: where

text: the

text: Central

text: Bank

text: adjusts

text: the

text: base

text: interest

text: rate

text: ρ

text: (t)

text: in

text: order

text: to

text: achieve

text: its

text: inﬂation

text: and

text: employment

text: targets.

text: Given

text: the

text: simplicity

text: of

text: our

text: model

text: we

text: are

text: mainly

text: interested

text: here

text: in

text: gaining

text: a

text: qualitative

text: understanding

text: of

text: the

text: possible

text: consequence

text: of

text: a

text: Taylor-rule

text: based

text: monetary

text: policy

text: in

text: a

text: stylized

text: Agent-Based

text: framework.

text: We

text: defer

text: parameter

text: calibration,

text: detailed

text: comparison

text: with

text: simple

text: DSGE

text: models

text: and

text: more

text: quantitative

text: insights

text: to

text: future

text: studies.

text: Our

text: major

text: ﬁnding

text: is

text: that

text: provided

text: its

text: policy

text: is

text: not

text: too

text: aggressive

text: and

text: the

text: economy

text: not

text: too

text: close

text: to

text: a

text: phase

text: transition,

text: the

text: CB

text: is

text: successful

text: in

text: steering

text: the

text: economy

text: towards

text: its

text: targets.

text: However,

text: the

text: mere

text: presence

text: of

text: diﬀerent

text: equilibrium

text: states

text: of

text: the

text: economy

text: separated

text: by

text: phase

text: boundaries

text: (i.e.

text: “dark

text: corners”)

text: may

text: deeply

text: alter

text: the

text: impact

text: of

text: monetary

text: policy.

text: Indeed,

text: we

text: will

text: exhibit

text: cases

text: where

text: the

text: monetary

text: policy

text: by

text: itself

text: triggers

text: large

text: instabilities

text: and

text: is

text: counter-productive.

text: As

text: in

text: the

text: previous

text: section,

text: we

text: will

text: refer

text: to

text: the

text: state

text: obtained

text: without

text: any

text: response

text: of

text: the

text: CB,

text: i.e.

text: when

text: φ

text: φ

text: π

text: ε

text: as

text: the

text: “natural”

text: state

text: of

text: the

text: economy

text: (for

text: a

text: given

text: set

text: of

text: parameters).

text: The

text: corresponding

text: “natural”

text: value

text: of

text: a

text: variable

text: x

text: will

text: be

text: denoted

text: by

text: x

text: In

text: order

text: to

text: simplify

text: the

text: analysis

text: and

text: since

text: most

text: of

text: the

text: parameters

text: of

text: our

text: model

text: play

text: little

text: nat

text: role

text: in

text: the

text: qualitative

text: behavior

text: of

text: the

text: economy

text: we

text: set

text: once

text: and

text: for

text: all

text: some

text: of

text: them

text: to

text: the

text: values

text: given

text: in

text: the

text: caption

text: of

text: Fig.

text: and

text: choose

text: ρ

text: We

text: only

text: focus

text: on

text: the

text: four

text: parameters

text: deﬁning

text: the

text: CB

text: policy

text: (i.e.

text: φ

text: φ

text: ε

text: and

text: π

text: ε

text: π

text: the

text: parameters

text: of

text: the

text: transmission

text: channels

text: (i.e.

text: α

text: and

text: α

text: and

text: the

text: two

text: parameters

text: locating

text: the

text: system

text: in

text: the

text: c

text: Γ

text: phase

text: diagram

text: of

text: Fig.

text: (i.e.

text: the

text: hiring/ﬁring

text: ratio

text: R

text: and

text: the

text: bankruptcy

text: threshold

text: Θ).

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text: not clear to me if for arbitrarily large values of α there exist instances (X, d) that have a clustering C that is (1 + α)-weakly-deletion stable and yet (X, d) is not (1 + α)-weakly-deletion stable. An easier goal than coming up with a useful notion of clusterability that is efﬁciently testable, is to come up with a notion of niceness of a given clustering C, such that one can efﬁciently test if a clustering solution C for an instance (X.d) satisﬁes that requirement, and so that if it does, it is guaranteed to be an optimal clustering for the (X, d). As far as I am aware no such notion currently exits. As noted above, it seems to be an open question whether the notion of a clustering C being (1 + α)-weakly-deletion stable (for some sufﬁciently large α) implies that the domain set of such a clustering is necessarily (1 + α)-weakly-deletion stable. 5.4 Implications for common practical clustering algorithms Among all the works surveyed in this note, only one, the results of [22], ad- dress (a feasible variant of) a practical algorithm - the popular Lloyd clustering algorithm. It would be very interesting to come up with results showing that some popular clustering algorithm (or an application of a practical approximation algo- rithm) efﬁciently yield guaranteed good quality clusterings, under some other, or more relaxed, niceness of data conditions. The recent work of [3] can be viewed as a step in that direction. They ask under which separation condition do various convex relaxations exactly recover the “correct” clustering. However, that work addresses a different version of clustering problems, in which one assumes that the data is generated by some parameterized generative model (a balanced mix- ture of spherical Gaussians, in the case of that paper), and aims to recover those parameters. 6 Conclusions Several notions of clusterability have been proposed so far. Depending on the values of the parameters deﬁning those notions, each of them ranges from being very lenient to a highly constraining data requirement. For each notion there is a parameter range so that, for data conforming to the clusterability requirement in that range, an optimal clustering can be rather trivially found. Clusterability with parameter values that sufﬁce for the currently available efﬁcient clustering results turns out to be rather strong requirements, that eminently restricts the practical signiﬁcance of the currently available results. The current failure to support the CDNM thesis can stem from various sources. 20

text: not

text: clear

text: to

text: me

text: if

text: for

text: arbitrarily

text: large

text: values

text: of

text: α

text: there

text: exist

text: instances

text: (X,

text: d)

text: that

text: have

text: a

text: clustering

text: C

text: that

text: is

text: α)-weakly-deletion

text: stable

text: and

text: yet

text: (X,

text: d)

text: is

text: not

text: α)-weakly-deletion

text: stable.

text: An

text: easier

text: goal

text: than

text: coming

text: up

text: with

text: a

text: useful

text: notion

text: of

text: clusterability

text: that

text: is

text: efﬁciently

text: testable,

text: is

text: to

text: come

text: up

text: with

text: a

text: notion

text: of

text: niceness

text: of

text: a

text: given

text: clustering

text: C,

text: such

text: that

text: one

text: can

text: efﬁciently

text: test

text: if

text: a

text: clustering

text: solution

text: C

text: for

text: an

text: instance

text: (X.d)

text: satisﬁes

text: that

text: requirement,

text: and

text: so

text: that

text: if

text: it

text: does,

text: it

text: is

text: guaranteed

text: to

text: be

text: an

text: optimal

text: clustering

text: for

text: the

text: (X,

text: d).

text: As

text: far

text: as

text: I

text: am

text: aware

text: no

text: such

text: notion

text: currently

text: exits.

text: As

text: noted

text: above,

text: it

text: seems

text: to

text: be

text: an

text: open

text: question

text: whether

text: the

text: notion

text: of

text: a

text: clustering

text: C

text: being

text: α)-weakly-deletion

text: stable

text: (for

text: some

text: sufﬁciently

text: large

text: α)

text: implies

text: that

text: the

text: domain

text: set

text: of

text: such

text: a

text: clustering

text: is

text: necessarily

text: α)-weakly-deletion

text: stable.

text: Implications

text: for

text: common

text: practical

text: clustering

text: algorithms

text: Among

text: all

text: the

text: works

text: surveyed

text: in

text: this

text: note,

text: only

text: one,

text: the

text: results

text: of

text: ad-

text: dress

text: (a

text: feasible

text: variant

text: of)

text: a

text: practical

text: algorithm

text: the

text: popular

text: Lloyd

text: clustering

text: algorithm.

text: It

text: would

text: be

text: very

text: interesting

text: to

text: come

text: up

text: with

text: results

text: showing

text: that

text: some

text: popular

text: clustering

text: algorithm

text: (or

text: an

text: application

text: of

text: a

text: practical

text: approximation

text: algo-

text: rithm)

text: efﬁciently

text: yield

text: guaranteed

text: good

text: quality

text: clusterings,

text: under

text: some

text: other,

text: or

text: more

text: relaxed,

text: niceness

text: of

text: data

text: conditions.

text: The

text: recent

text: work

text: of

text: can

text: be

text: viewed

text: as

text: a

text: step

text: in

text: that

text: direction.

text: They

text: ask

text: under

text: which

text: separation

text: condition

text: do

text: various

text: convex

text: relaxations

text: exactly

text: recover

text: the

text: “correct”

text: clustering.

text: However,

text: that

text: work

text: addresses

text: a

text: different

text: version

text: of

text: clustering

text: problems,

text: in

text: which

text: one

text: assumes

text: that

text: the

text: data

text: is

text: generated

text: by

text: some

text: parameterized

text: generative

text: model

text: (a

text: balanced

text: mix-

text: ture

text: of

text: spherical

text: Gaussians,

text: in

text: the

text: case

text: of

text: that

text: paper),

text: and

text: aims

text: to

text: recover

text: those

text: parameters.

text: Conclusions

text: Several

text: notions

text: of

text: clusterability

text: have

text: been

text: proposed

text: so

text: far.

text: Depending

text: on

text: the

text: values

text: of

text: the

text: parameters

text: deﬁning

text: those

text: notions,

text: each

text: of

text: them

text: ranges

text: from

text: being

text: very

text: lenient

text: to

text: a

text: highly

text: constraining

text: data

text: requirement.

text: For

text: each

text: notion

text: there

text: is

text: a

text: parameter

text: range

text: so

text: that,

text: for

text: data

text: conforming

text: to

text: the

text: clusterability

text: requirement

text: in

text: that

text: range,

text: an

text: optimal

text: clustering

text: can

text: be

text: rather

text: trivially

text: found.

text: Clusterability

text: with

text: parameter

text: values

text: that

text: sufﬁce

text: for

text: the

text: currently

text: available

text: efﬁcient

text: clustering

text: results

text: turns

text: out

text: to

text: be

text: rather

text: strong

text: requirements,

text: that

text: eminently

text: restricts

text: the

text: practical

text: signiﬁcance

text: of

text: the

text: currently

text: available

text: results.

text: The

text: current

text: failure

text: to

text: support

text: the

text: CDNM

text: thesis

text: can

text: stem

text: from

text: various

text: sources.


text: First, of course, maybe the thesis is just false. My personal belief is that, while it may very well be the case that some practical clustering tasks are indeed com- putationally hard for some real data instances, there are many more cases where data of practical interest does yield not-too-hard-to-ﬁnd meaningful clusterings (though, of course, most of the time we have no way of knowing whether those are optimal clusterings in any formal sense of optimality). Another explanation to the shortcomings of current results is that they may just be an artifact of the algorithms and proof techniques that we currently have. Maybe one could eventually come up with efﬁcient algorithms that will cluster well under much less restrictive parameter settings of the clusterability notions listed in this note. Indeed, for most of those notions we do not have any close- to-matching computational hardness lower bounds. I doubt if that is indeed the case. As mentioned above, for the notion of α-center stability, the gap between the parameter values sufﬁcien√t for efﬁcient clustering and those that imply NP hardness is very small, (1 + 2) vs 2. Furthermore, the α-center stability is a central notions, in the sense that almost any other of the notions of clusterability discussed above implies it (or some variants of that condition), and the current results rely on those implications for proving the efﬁciency of clustering under those conditions. I believe that part of the answer is that we have not yet discovered the appro- priate notions of clusterability. In light of the results surveyed in this paper, I think that notions of clusterability that aim to substantiate the CDNM statement should not be just a way of formalizing large between-clusters separation. Apparently, as demonstrated above, such assumptions become too restrictive before they yield efﬁcient clustering results. Finally, in the last paragraph of concluding remarks below, I would like to argue that if we really wish to model clustering as it is required and used in appli- cations, the formulation of clustering tasks as computational problems should be revisited and revised. 6.1 Call for a change of perspective on the complexity of clus- tering All the papers surveyed above, as well as most of the current theoretical work on the computational complexity of clustering, focus on concrete clustering ob- jectives aiming to ﬁnd the best clustering for a given number of clusters. However, the practice of clustering is widely varied. There are applications, like clustering for detecting record duplications in data bases (say, records of patients from var- 21 ##LTLine##

text: First,

text: of

text: course,

text: maybe

text: the

text: thesis

text: is

text: just

text: false.

text: My

text: personal

text: belief

text: is

text: that,

text: while

text: it

text: may

text: very

text: well

text: be

text: the

text: case

text: that

text: some

text: practical

text: clustering

text: tasks

text: are

text: indeed

text: com-

text: putationally

text: hard

text: for

text: some

text: real

text: data

text: instances,

text: there

text: are

text: many

text: more

text: cases

text: where

text: data

text: of

text: practical

text: interest

text: does

text: yield

text: not-too-hard-to-ﬁnd

text: meaningful

text: clusterings

text: (though,

text: of

text: course,

text: most

text: of

text: the

text: time

text: we

text: have

text: no

text: way

text: of

text: knowing

text: whether

text: those

text: are

text: optimal

text: clusterings

text: in

text: any

text: formal

text: sense

text: of

text: optimality).

text: Another

text: explanation

text: to

text: the

text: shortcomings

text: of

text: current

text: results

text: is

text: that

text: they

text: may

text: just

text: be

text: an

text: artifact

text: of

text: the

text: algorithms

text: and

text: proof

text: techniques

text: that

text: we

text: currently

text: have.

text: Maybe

text: one

text: could

text: eventually

text: come

text: up

text: with

text: efﬁcient

text: algorithms

text: that

text: will

text: cluster

text: well

text: under

text: much

text: less

text: restrictive

text: parameter

text: settings

text: of

text: the

text: clusterability

text: notions

text: listed

text: in

text: this

text: note.

text: Indeed,

text: for

text: most

text: of

text: those

text: notions

text: we

text: do

text: not

text: have

text: any

text: close-

text: to-matching

text: computational

text: hardness

text: lower

text: bounds.

text: I

text: doubt

text: if

text: that

text: is

text: indeed

text: the

text: case.

text: As

text: mentioned

text: above,

text: for

text: the

text: notion

text: of

text: α-center

text: stability,

text: the

text: gap

text: between

text: the

text: parameter

text: values

text: sufﬁcien√t

text: for

text: efﬁcient

text: clustering

text: and

text: those

text: that

text: imply

text: NP

text: hardness

text: is

text: very

text: small,

text: vs

text: Furthermore,

text: the

text: α-center

text: stability

text: is

text: a

text: central

text: notions,

text: in

text: the

text: sense

text: that

text: almost

text: any

text: other

text: of

text: the

text: notions

text: of

text: clusterability

text: discussed

text: above

text: implies

text: it

text: (or

text: some

text: variants

text: of

text: that

text: condition),

text: and

text: the

text: current

text: results

text: rely

text: on

text: those

text: implications

text: for

text: proving

text: the

text: efﬁciency

text: of

text: clustering

text: under

text: those

text: conditions.

text: I

text: believe

text: that

text: part

text: of

text: the

text: answer

text: is

text: that

text: we

text: have

text: not

text: yet

text: discovered

text: the

text: appro-

text: priate

text: notions

text: of

text: clusterability.

text: In

text: light

text: of

text: the

text: results

text: surveyed

text: in

text: this

text: paper,

text: I

text: think

text: that

text: notions

text: of

text: clusterability

text: that

text: aim

text: to

text: substantiate

text: the

text: CDNM

text: statement

text: should

text: not

text: be

text: just

text: a

text: way

text: of

text: formalizing

text: large

text: between-clusters

text: separation.

text: Apparently,

text: as

text: demonstrated

text: above,

text: such

text: assumptions

text: become

text: too

text: restrictive

text: before

text: they

text: yield

text: efﬁcient

text: clustering

text: results.

text: Finally,

text: in

text: the

text: last

text: paragraph

text: of

text: concluding

text: remarks

text: below,

text: I

text: would

text: like

text: to

text: argue

text: that

text: if

text: we

text: really

text: wish

text: to

text: model

text: clustering

text: as

text: it

text: is

text: required

text: and

text: used

text: in

text: appli-

text: cations,

text: the

text: formulation

text: of

text: clustering

text: tasks

text: as

text: computational

text: problems

text: should

text: be

text: revisited

text: and

text: revised.

text: Call

text: for

text: a

text: change

text: of

text: perspective

text: on

text: the

text: complexity

text: of

text: clus-

text: tering

text: All

text: the

text: papers

text: surveyed

text: above,

text: as

text: well

text: as

text: most

text: of

text: the

text: current

text: theoretical

text: work

text: on

text: the

text: computational

text: complexity

text: of

text: clustering,

text: focus

text: on

text: concrete

text: clustering

text: ob-

text: jectives

text: aiming

text: to

text: ﬁnd

text: the

text: best

text: clustering

text: for

text: a

text: given

text: number

text: of

text: clusters.

text: However,

text: the

text: practice

text: of

text: clustering

text: is

text: widely

text: varied.

text: There

text: are

text: applications,

text: like

text: clustering

text: for

text: detecting

text: record

text: duplications

text: in

text: data

text: bases

text: (say,

text: records

text: of

text: patients

text: from

text: var-

text: ##LTLine##


text: I am grateful to Shalev Ben-David, Lev Rayzin and Ruth Urner for insightful discussions concerning this paper. References [1] Margareta Ackerman and Shai Ben-David. Clusterability: A theoretical study. In AISTATS, pages 1–8, 2009. [2] Margareta Ackerman, Shai Ben-David, and David Loker. Characterization of linkage-based clustering. In COLT 2010 - The 23rd Conference on Learn- ing Theory, Haifa, Israel, June 27-29, 2010, pages 270–281, 2010. [3] Pranjal Awasthi, Afonso Bandera, Moses Charikar, Ravishankar Krish- naswami, Soledad Voilar, and Rachel Ward. Relax, no need to round: In- tegrality of clustering formulations. CoRR, Stat.ML, 2014. [4] Pranjal Awasthi, Avrim Blum, and Or Sheffet. Stability yields a ptas for k-median and k-means clustering. In FOCS, pages 309–318, 2010. [5] Pranjal Awasthi, Avrim Blum, and Or Sheffet. Center-based clustering under perturbation stability. Inf. Process. Lett., 112(1-2):49–54, 2012. [6] Maria-Florina Balcan, Avrim Blum, and Anupam Gupta. Approximate clus- tering without the approximation. In SODA, pages 1068–1077, 2009. [7] Maria-Florina Balcan, Avrim Blum, and Anupam Gupta. Clustering under approximation stability. J. ACM, 60(2):8, 2013. [8] Maria-Florina Balcan and Yingyu Liang. Clustering under perturbation re- silience. In ICALP (1), pages 63–74, 2012. [9] Shai Ben-David. Alternative measures of computational complexity with applications to agnostic learning. In TAMC, pages 231–235, 2006. [10] Shai Ben-David. A framework for statistical clustering with constant time approximation algorithms for k-median and k-means clustering. Machine Learning, 66(2-3):243–257, 2007. [11] Shai Ben-David, Benny Chor, Oded Goldreich, and Michael Luby. On the theory of average case complexity. In STOC, pages 204–216, 1989. 25

text: I

text: am

text: grateful

text: to

text: Shalev

text: Ben-David,

text: Lev

text: Rayzin

text: and

text: Ruth

text: Urner

text: for

text: insightful

text: discussions

text: concerning

text: this

text: paper.

text: References

label: reference

text: Margareta

text: Ackerman

text: and

text: Shai

text: Ben-David.

text: Clusterability:

text: A

text: theoretical

text: study.

text: In

text: AISTATS,

text: pages

text: Margareta

text: Ackerman,

text: Shai

text: Ben-David,

text: and

text: David

text: Loker.

text: Characterization

text: of

text: linkage-based

text: clustering.

text: In

text: COLT

text: The

text: Conference

text: on

text: Learn-

text: ing

text: Theory,

text: Haifa,

text: Israel,

text: June

text: pages

text: Pranjal

text: Awasthi,

text: Afonso

text: Bandera,

text: Moses

text: Charikar,

text: Ravishankar

text: Krish-

text: naswami,

text: Soledad

text: Voilar,

text: and

text: Rachel

text: Ward.

text: Relax,

text: no

text: need

text: to

text: round:

text: In-

text: tegrality

text: of

text: clustering

text: formulations.

text: CoRR,

text: Stat.ML,

text: Pranjal

text: Awasthi,

text: Avrim

text: Blum,

text: and

text: Or

text: Sheffet.

text: Stability

text: yields

text: a

text: ptas

text: for

text: k-median

text: and

text: k-means

text: clustering.

text: In

text: FOCS,

text: pages

text: Pranjal

text: Awasthi,

text: Avrim

text: Blum,

text: and

text: Or

text: Sheffet.

text: Center-based

text: clustering

text: under

text: perturbation

text: stability.

text: Inf.

text: Process.

text: Lett.,

text: Maria-Florina

text: Balcan,

text: Avrim

text: Blum,

text: and

text: Anupam

text: Gupta.

text: Approximate

text: clus-

text: tering

text: without

text: the

text: approximation.

text: In

text: SODA,

text: pages

text: Maria-Florina

text: Balcan,

text: Avrim

text: Blum,

text: and

text: Anupam

text: Gupta.

text: Clustering

text: under

text: approximation

text: stability.

text: J.

text: ACM,

text: Maria-Florina

text: Balcan

text: and

text: Yingyu

text: Liang.

text: Clustering

text: under

text: perturbation

text: re-

text: silience.

text: In

text: ICALP

text: pages

text: Shai

text: Ben-David.

text: Alternative

text: measures

text: of

text: computational

text: complexity

text: with

text: applications

text: to

text: agnostic

text: learning.

text: In

text: TAMC,

text: pages

text: Shai

text: Ben-David.

text: A

text: framework

text: for

text: statistical

text: clustering

text: with

text: constant

text: time

text: approximation

text: algorithms

text: for

text: k-median

text: and

text: k-means

text: clustering.

text: Machine

text: Learning,

text: Shai

text: Ben-David,

text: Benny

text: Chor,

text: Oded

text: Goldreich,

text: and

text: Michael

text: Luby.

text: On

text: the

text: theory

text: of

text: average

text: case

text: complexity.

text: In

text: STOC,

text: pages


text: noissimsnarT π/σ 4 1 0.5 A + A A + A B = J(cid:107) − 0 1 , B = J(cid:107) − 0 −1 0.8 1 ω (E + iγ) 2 ω (E iγ) d d 0.6 0 and obtain the eﬀective defect equation 0.4 0.2 ωAn = JAn−1 + JAn+1 + Fd(A0 + A1)δn0 (a) (b) 0 −0.5 +F (A + A )δ , (5) −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 d 0 1 n1 ω ω in which FIG. 3: (CPolTor online) (a) Transmission coeﬃcient for non- Hermitian -symmetric model H with diﬀerent γ: the red a solid line stands for γ = 0, the green dash one for γ = 0.1, and − ω E the blue dot-dash one for γ = 0.2. (b) indicate the scattering Fd = 2J(cid:107)2 (ω − E )2d+ γ2 phase σ/π for the corresponding condition. Here J = 0.5, d J(cid:107) = 0.3 and E = 0.5. d is an entirely real function acting as an eﬀective local- ized potential. This equation reveals that this problem as long as γ is smaller than a critical value γ = J(cid:107)2/J. c can be simpliﬁed to the propagating problem along the main chain with scattering center at site 0 and 1, and the In Fig. 3, we show the scattering results of the model defect subsystem acts as the eﬀective localized potential. (a) for diﬀerent γ. When γ = 0, the condition has When the module of this eﬀective localized potential is be discussed above with only one resonance energy at zero, the perfect transmission occurs, and correspond- − ω = E 2γ . The scattering phase σ = arg(t) shows a π- d c ingly when it is inﬁnite, the perfect reﬂection appears. jump correspondingly, shown as the red solid line in Fig. For the scattering p(cid:26)roblem, the wave function can be ex- 3(b). While for γ = 0.1, which is smaller than γ = 0.18, c pressed in the form there exists two resonance energies determined by Eq. − (8), at which the transmission is completely suppressed Ieikn + re ikn, n < 0, A = and the scattering phase σ experiences an abrupt π-jump n teikn, n > 1, − or π-jump. What’s interesting is that at ω = E , t = 1 ≡ d due to F 0. It means that a perfect transmission oc- where I, r, and t stand for the amplitude of the incom- d curs at the energy E , which is caused by the balanced ing, reﬂected and transmitted wave, respectively, and k d± non-Hermitian terms iγ, shown as the green dash line in is assumed to be positive, indicating the left incidence. Fig. 3. When γ = 0.2, which is larger than γ , there ex- The momentum k is related to the ω by the dispersion c ists no solution for the perfect reﬂection, and the scatter- equation ω = 2J cos k. By substituting the expression ing phase σ experiences no abrupt jump. But the perfect A into the eﬀective defect equation and applying the n transmission is still found for arbitrary γ, shown as the transfer-matrix method, we obtain the analytical form of blue dot-dash line in Fig. 3. The formula of reﬂection the transmission amplitude | | coeﬃcient R = r 2 /I2 can be obtained by the similar method, and it’s found that the summation of transmis- i sin k[F + J] sion and reﬂection coeﬃcients equals to 1 for arbitrary γ t = I d , (6) ≡ F (1 + eik) + iJ sin k value, i.e., R + T 1. That means this non-Hermitian d system acts as a Hermitian one in the scattering process PT because of symmetry [38]. and the transmission coeﬃcient As for the model described by H , it can be related b to the model (a) by breaking the couplings between site T = |t|2 /I2. (7) d1, d2 and site 1, and introducing a coupling J⊥ between d and d . Nevertheless, it shows some special features 1 2 due to its speciﬁc geometric structure. Following similar Obviously, the condition for the perfect reﬂection T = procedures as dealing with the model (a), we can obtain 0 is F + J = 0, which leads t(cid:113)o two solutions of ω the eﬀective defect equation d − J(cid:107)2 ± J(cid:107)4 − J2γ2 ωAn = JAn−1 + JAn+1 + F˜dA0δn0 (9) ω = E , (8) d J with ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: noissimsnarT

text: π/σ

text: A

label: equation

text: A

text: A

text: A

text: B

text: B

text: −1

text: ω

text: (E

text: iγ)

text: ω

text: (E

text: iγ)

text: d

text: d

text: and

text: obtain

text: the

text: eﬀective

text: defect

text: equation

text: ωAn

text: (a)

text: (b)

text: +F

text: (A

text: A

text: )δ

text: −1

text: −1

text: d

text: ω

text: ω

text: in

text: which

text: FIG.

text: (CPolTor

text: online)

text: (a)

text: Transmission

text: coeﬃcient

text: for

text: non-

text: Hermitian

text: -symmetric

text: model

text: H

text: with

text: diﬀerent

text: γ:

text: the

text: red

text: a

text: solid

text: line

text: stands

text: for

text: γ

text: the

text: green

text: dash

text: one

text: for

text: γ

text: and

text: ω

text: E

text: the

text: blue

text: dot-dash

text: one

text: for

text: γ

text: (b)

text: indicate

text: the

text: scattering

text: Fd

text: (ω

text: E

text: γ2

text: phase

text: σ/π

text: for

text: the

text: corresponding

text: condition.

text: Here

text: J

text: d

text: and

text: E

text: d

text: is

text: an

text: entirely

text: real

text: function

text: acting

text: as

text: an

text: eﬀective

text: local-

text: ized

text: potential.

text: This

text: equation

text: reveals

text: that

text: this

text: problem

text: as

text: long

text: as

text: γ

text: is

text: smaller

text: than

text: a

text: critical

text: value

text: γ

text: c

text: can

text: be

text: simpliﬁed

text: to

text: the

text: propagating

text: problem

text: along

text: the

text: main

text: chain

text: with

text: scattering

text: center

text: at

text: site

text: and

text: and

text: the

text: In

text: Fig.

text: we

text: show

text: the

text: scattering

text: results

text: of

text: the

text: model

text: defect

text: subsystem

text: acts

text: as

text: the

text: eﬀective

text: localized

text: potential.

text: (a)

text: for

text: diﬀerent

text: γ.

text: When

text: γ

text: the

text: condition

text: has

text: When

text: the

text: module

text: of

text: this

text: eﬀective

text: localized

text: potential

text: is

text: be

text: discussed

text: above

text: with

text: only

text: one

text: resonance

text: energy

text: at

text: zero,

text: the

text: perfect

text: transmission

text: occurs,

text: and

text: correspond-

text: ω

text: E

text: 2γ

text: The

text: scattering

text: phase

text: σ

text: arg(t)

text: shows

text: a

text: π-

text: d

text: c

text: ingly

text: when

text: it

text: is

text: inﬁnite,

text: the

text: perfect

text: reﬂection

text: appears.

text: jump

text: correspondingly,

text: shown

text: as

text: the

text: red

text: solid

text: line

text: in

text: Fig.

text: For

text: the

text: scattering

text: the

text: wave

text: function

text: can

text: be

text: ex-

text: While

text: for

text: γ

text: which

text: is

text: smaller

text: than

text: γ

text: c

text: pressed

text: in

text: the

text: form

text: there

text: exists

text: two

text: resonance

text: energies

text: determined

text: by

text: Eq.

text: at

text: which

text: the

text: transmission

text: is

text: completely

text: suppressed

text: Ieikn

text: re

text: ikn,

text: n

text: A

text: and

text: the

text: scattering

text: phase

text: σ

text: experiences

text: an

text: abrupt

text: π-jump

text: n

text: teikn,

text: n

text: or

text: π-jump.

text: What’s

text: interesting

text: is

text: that

text: at

text: ω

text: E

text: t

text: d

text: due

text: to

text: F

text: It

text: means

text: that

text: a

text: perfect

text: transmission

text: oc-

text: where

text: I,

text: r,

text: and

text: t

text: stand

text: for

text: the

text: amplitude

text: of

text: the

text: incom-

text: d

text: curs

text: at

text: the

text: energy

text: E

text: which

text: is

text: caused

text: by

text: the

text: balanced

text: ing,

text: reﬂected

text: and

text: transmitted

text: wave,

text: respectively,

text: and

text: k

text: d±

text: non-Hermitian

text: terms

text: iγ,

text: shown

text: as

text: the

text: green

text: dash

text: line

text: in

text: is

text: assumed

text: to

text: be

text: positive,

text: indicating

text: the

text: left

text: incidence.

text: Fig.

text: When

text: γ

text: which

text: is

text: larger

text: than

text: γ

text: there

text: ex-

text: The

text: momentum

text: k

text: is

text: related

text: to

text: the

text: ω

text: by

text: the

text: dispersion

text: c

text: ists

text: no

text: solution

text: for

text: the

text: perfect

text: reﬂection,

text: and

text: the

text: scatter-

text: equation

text: ω

text: cos

text: k.

text: By

text: substituting

text: the

text: expression

text: ing

text: phase

text: σ

text: experiences

text: no

text: abrupt

text: jump.

text: But

text: the

text: perfect

text: A

text: into

text: the

text: eﬀective

text: defect

text: equation

text: and

text: applying

text: the

text: n

text: transmission

text: is

text: still

text: found

text: for

text: arbitrary

text: γ,

text: shown

text: as

text: the

text: transfer-matrix

text: method,

text: we

text: obtain

text: the

text: analytical

text: form

text: of

text: blue

text: dot-dash

text: line

text: in

text: Fig.

text: The

text: formula

text: of

text: reﬂection

text: the

text: transmission

text: amplitude

text: coeﬃcient

text: R

text: r

text: can

text: be

text: obtained

text: by

text: the

text: similar

text: method,

text: and

text: it’s

text: found

text: that

text: the

text: summation

text: of

text: transmis-

text: i

text: sin

text: k[F

text: J]

text: sion

text: and

text: reﬂection

text: coeﬃcients

text: equals

text: to

text: for

text: arbitrary

text: γ

text: t

text: I

text: d

text: F

text: eik)

text: iJ

text: sin

text: k

text: value,

text: i.e.,

text: R

text: T

text: That

text: means

text: this

text: non-Hermitian

text: d

text: system

text: acts

text: as

text: a

text: Hermitian

text: one

text: in

text: the

text: scattering

text: process

text: PT

text: because

text: of

text: symmetry

text: and

text: the

text: transmission

text: coeﬃcient

text: As

text: for

text: the

text: model

text: described

text: by

text: H

text: it

text: can

text: be

text: related

text: b

text: to

text: the

text: model

text: (a)

text: by

text: breaking

text: the

text: couplings

text: between

text: site

text: T

text: and

text: site

text: and

text: introducing

text: a

text: coupling

text: J⊥

text: between

text: d

text: and

text: d

text: Nevertheless,

text: it

text: shows

text: some

text: special

text: features

text: due

text: to

text: its

text: speciﬁc

text: geometric

text: structure.

text: Following

text: similar

text: Obviously,

text: the

text: condition

text: for

text: the

text: perfect

text: reﬂection

text: T

text: procedures

text: as

text: dealing

text: with

text: the

text: model

text: (a),

text: we

text: can

text: obtain

text: is

text: F

text: J

text: which

text: leads

text: two

text: solutions

text: of

text: ω

text: the

text: eﬀective

text: defect

text: equation

text: d

text: ωAn

text: ω

text: E

text: d

text: J

text: with

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text: T+R\R\T T+R\R\T 6 1.5 2.5 metry exists, it is interesting to ask whether the property (a) (b) T of R + T = 1 is a general feature of one-dimensional sys- 2 R PT 1 R+T tems with symmetry? As a general proof is lack, we 1.5 would like to indicate that this property is not general PT 1 to all the non-Hermitian systems with symmetry by 0.5 considering (cid:88)an example model described by 0.5 0 0 † † † (c) 3 (d) H = Jφˆn−1φˆn + J⊥dˆ1φˆ0 + J⊥dˆ2φˆ1 + h.c. 1.5 n † − † 2 +(E + iγ)dˆ dˆ + (E iγ)dˆ dˆ , (15) 1 d 1 1 d 2 2 0.5 1 which is schematically displayed in Fig. 6(a). This PT 0 0 model is symmetric with the vertical line between −1 0 0.4 1 −0.5 00 0.4 1 ω ω site 0 anPd site 1 as thePmirPror axis. In thisPmoPdel the eﬀect of operator is φˆ = φˆ− and dˆ = dˆ P P n T n+1 1T T 2 FIG. 5: (Color online) Transmission coeﬃcient for the non- ( dˆ = dˆ ), and the eﬀect of operator is i = PT − 2 1 -symmetric case of Hb. The red solid line represents T, i. Following previous procedures, we can derive the the blue dash line represents R and the black dot-dash line transmission and reﬂection amplitudes as represents R + T. Parameters are J = 0.5, (a) J = J = − 1 2 0.4, E = E = 0.4, but γ = 0.05, γ = 0.15. (b) J = d1 d2 − 1 2 − 1 − JE2 ==0.E4, γ1==0.4γ, γ2 ==0.−05γ, b=ut0.E05d1, b=ut0.J4, E=d20=.4, J0.=5. 0(.c6). t = Ie−ik − − 2iJ2∗s−in k − − , (16) d1 d2 1 2 1 2 (G Je ik)(G Je ik) J2 (d) All parameters do−not obey balanced ru−les, J1 = 0.4, J2 = − d − d∗ − − 0.6, Ed1 = 0.4, Ed2 = 0.5, γ1 = 0.05, γ2 = 0.15. r = I J2 −(Gd− Jeik∗)−(Gd − Je−ik) (17) (G Je ik)(G Je ik) J2 d d that R + T is no longer a unit constant. Alternatively, with one can also understand this from the eﬀective defect equation described by Eq. (9), in which the eﬀective de- fect potential F˜ is generally complex and becomes real J⊥2 Pd T G = − , only under the -symmetric condition. d ω (E + iγ) d (a) d1 d2 and the transmission coeﬃcient T and reﬂection coef- ﬁcient R are given by φ-3 φ-2 φ-1 φ0 φ1 φ2 φ3 φ4 T = |t|2 /I2, R = |r|2 /I2. (18) (b) (c) 1 T 4 R R+T While we have R + T = 1 for the Hermitian case with 3 γ = 0 as shown in Fig. 6(b), but the transmission coef- ﬁcient can be larger than 1 around the resonance energy 0.5 2 E when γ is nonzero as shown in Fig. 6(c), which indi- d (cid:54) cates that R+T = 1 in this regime. Our numerical result 1 shows that this model does not obey R+T = 1 although PT PT 0 0 it is a -symme(cid:54) tric system. Other -symmetric sys- −1 0 1 −1 0 1 tems with R + T = 1 can be found in references [40, 41]. ω ω FIG. 6: (Color online) (a) Schematic diagram of another PT IV. SUMMARY symmetric model. (b) Transmission and reﬂection co- eﬃcient for the Hermitian case with γ = 0. (c) is for the non- Hermitian case with γ = 0.1. The red solid line represents T, In summary, we study two kinds of discrete non- the blue dash line represents R and the black dot-dash line Hermitian models, consisting of an inﬁnite linear chain PT represents R+T. Parameters are J = 0.5, J⊥ = 0.3, Ed = 0.2. and two defect points with -symmetric complex on- site chemical potentials, which can be regarded as the As we have already displayed that R+T = 1 holds true generalization of the Fano-Anderson model. By solv- PT for both the model (a) and (b) as long as the sym- ing the wave propagation problems, we derive analytical ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T+R\R\T

text: T+R\R\T

text: metry

text: exists,

text: it

text: is

text: interesting

text: to

text: ask

text: whether

text: the

text: property

text: (a)

text: (b)

text: T

text: of

text: R

text: T

text: is

text: a

text: general

text: feature

text: of

text: one-dimensional

text: sys-

text: R

text: PT

text: R+T

text: tems

text: with

text: symmetry?

text: As

text: a

text: general

text: proof

text: is

text: lack,

text: we

text: would

text: like

text: to

text: indicate

text: that

text: this

text: property

text: is

text: not

text: general

text: PT

text: to

text: all

text: the

text: non-Hermitian

text: systems

text: with

text: symmetry

text: by

text: considering

text: example

text: model

text: described

text: by

text: (c)

text: (d)

text: H

text: Jφˆn−1φˆn

text: J⊥dˆ1φˆ0

text: J⊥dˆ2φˆ1

text: h.c.

text: n

text: +(E

text: iγ)dˆ

text: dˆ

text: (E

text: iγ)dˆ

text: dˆ

text: d

text: d

text: which

text: is

text: schematically

text: displayed

text: in

text: Fig.

text: This

text: PT

text: model

text: is

text: symmetric

text: with

text: the

text: vertical

text: line

text: between

text: −1

text: ω

text: ω

text: site

text: anPd

text: site

text: as

text: thePmirPror

text: axis.

text: In

text: thisPmoPdel

text: the

text: eﬀect

text: of

text: operator

text: is

text: φˆ

text: φˆ−

text: and

text: dˆ

text: dˆ

text: P

text: P

text: n

text: T

text: T

text: FIG.

text: (Color

text: online)

text: Transmission

text: coeﬃcient

text: for

text: the

text: non-

text: dˆ

text: dˆ

text: and

text: the

text: eﬀect

text: of

text: operator

text: is

text: i

text: PT

text: -symmetric

text: case

text: of

text: Hb.

text: The

text: red

text: solid

text: line

text: represents

text: T,

text: i.

text: Following

text: previous

text: procedures,

text: we

text: can

text: derive

text: the

text: the

text: blue

text: dash

text: line

text: represents

text: R

text: and

text: the

text: black

text: dot-dash

text: line

text: transmission

text: and

text: reﬂection

text: amplitudes

text: as

text: represents

text: R

text: T.

text: Parameters

text: are

text: J

text: (a)

text: J

text: J

text: E

text: E

text: but

text: γ

text: γ

text: (b)

text: J

text: γ2

text: t

text: Ie−ik

text: k

text: (G

text: Je

text: ik)(G

text: Je

text: ik)

text: (d)

text: All

text: parameters

text: do−not

text: obey

text: balanced

text: ru−les,

text: d

text: d∗

text: γ1

text: γ2

text: r

text: I

text: −(Gd−

text: Jeik∗)−(Gd

text: Je−ik)

text: (G

text: Je

text: ik)(G

text: Je

text: ik)

text: d

text: d

text: that

text: R

text: T

text: is

text: no

text: longer

text: a

text: unit

text: constant.

text: Alternatively,

text: with

text: one

text: can

text: also

text: understand

text: this

text: from

text: the

text: eﬀective

text: defect

text: equation

text: described

text: by

text: Eq.

text: in

text: which

text: the

text: eﬀective

text: de-

text: fect

text: potential

text: F˜

text: is

text: generally

text: complex

text: and

text: becomes

text: real

text: Pd

text: T

text: G

text: only

text: under

text: the

text: -symmetric

text: condition.

text: d

text: ω

text: (E

text: iγ)

text: d

text: (a)

text: and

text: the

text: transmission

text: coeﬃcient

text: T

text: and

text: reﬂection

text: coef-

text: ﬁcient

text: R

text: are

text: given

text: by

text: φ0

text: φ1

text: φ2

text: φ3

text: φ4

text: T

text: R

text: (b)

text: (c)

text: T

text: R

text: R+T

text: While

text: we

text: have

text: R

text: T

text: for

text: the

text: Hermitian

text: case

text: with

text: γ

text: as

text: shown

text: in

text: Fig.

text: but

text: the

text: transmission

text: coef-

text: ﬁcient

text: can

text: be

text: larger

text: than

text: around

text: the

text: resonance

text: energy

text: E

text: when

text: γ

text: is

text: nonzero

text: as

text: shown

text: in

text: Fig.

text: which

text: indi-

text: d

text: cates

text: that

text: R+T

text: in

text: this

text: regime.

text: Our

text: numerical

text: result

text: shows

text: that

text: this

text: model

text: does

text: not

text: obey

text: R+T

text: although

text: PT

text: PT

text: it

text: is

text: a

text: tric

text: system.

text: Other

text: -symmetric

text: sys-

text: −1

text: −1

text: tems

text: with

text: R

text: T

text: can

text: be

text: found

text: in

text: references

text: ω

text: ω

text: FIG.

text: (Color

text: online)

text: (a)

text: Schematic

text: diagram

text: of

text: another

text: PT

text: IV.

text: SUMMARY

text: symmetric

text: model.

text: (b)

text: Transmission

text: and

text: reﬂection

text: co-

text: eﬃcient

text: for

text: the

text: Hermitian

text: case

text: with

text: γ

text: (c)

text: is

text: for

text: the

text: non-

text: Hermitian

text: case

text: with

text: γ

text: The

text: red

text: solid

text: line

text: represents

text: T,

text: In

text: summary,

text: we

text: study

text: two

text: kinds

text: of

text: discrete

text: non-

text: the

text: blue

text: dash

text: line

text: represents

text: R

text: and

text: the

text: black

text: dot-dash

text: line

text: Hermitian

text: models,

text: consisting

text: of

text: an

text: inﬁnite

text: linear

text: chain

text: PT

text: represents

text: R+T.

text: Parameters

text: are

text: J

text: J⊥

text: Ed

text: and

text: two

text: defect

text: points

text: with

text: -symmetric

text: complex

text: on-

text: site

text: chemical

text: potentials,

text: which

text: can

text: be

text: regarded

text: as

text: the

text: As

text: we

text: have

text: already

text: displayed

text: that

text: R+T

text: holds

text: true

text: generalization

text: of

text: the

text: Fano-Anderson

text: model.

text: By

text: solv-

text: PT

text: for

text: both

text: the

text: model

text: (a)

text: and

text: (b)

text: as

text: long

text: as

text: the

text: sym-

text: ing

text: the

text: wave

text: propagation

text: problems,

text: we

text: derive

text: analytical

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text: 2 skyrmions in spintronic devices, Eq. (1) is however not sufﬁ- tuating, stochastic ﬁeld creating random torques on the mag- cient. A generalized version of Eq. (1) valid for weak but also netic moments to model the effects of thermal ﬂuctuations, arbitrarily time-dependent forces can be written as see below. The Hamiltonian H[M] is given by (cid:88) (cid:0) (cid:1) − G 1(ω)V(ω) = F (ω) + F (ω) + F (ω) (2) c g th∇ = (cid:82)Sc(ω)vs(ω) + Sg(ω) Bz(ω) + Fth(ω) H[M] = −(cid:88)J (cid:0) M · M + M (cid:1) r r+aex r+aey r − (cid:88) × · × · Here V (ω) = eiωtR˙ (t)dt is the Fourier transform of λ M M e + M M e r r+aex x r r+aey y the velocity of the skyrmion, vs(ω) is the (spin-) drift ve- r locity of the conduction electrons, directly proportional to − · ∇ B M (4) the current, B (ω) describes a magnetic ﬁeld gradient in r z r frequency space. The role of the random thermal forces, F (ω), is special as their dynamics is directly linked via th | | the ﬂuctuation-dissipation theorem to the left-hand side of the We use J = 1, γ = 1, M = 1, λ = 0.18J for the × − r equation, see below. The 2 2 matrix G 1(ω) describes the strength of the Dzyaloshinskii-Moriya interaction and B = dynamics of the skyrmion; its small-ω expansion deﬁnes the (0, 0, 0.0278J) for all plots giving rise to a skyrmion with a terms written on the left-hand side of Eq. (1). One can expect radius of about 15 lattice sites, see Appendix . For this pa- strongly frequency-dependent dynamics for the skyrmion be- rameter set, the ground state is ferromagnetic, thus the sin- cause the external forces in combination with the motion of gle skyrmion is a topologically protected, metastable excita- × the skyrmion can induce a precession of the spin and also ex- tion. Typically we simulate 100 100 spins for the analysis × cite spinwaves in the surrounding ferromagnet, see Fig. 1. of diffusive and current driven motion and 200 200 spins for We will, however, show that the frequency dependence of the the force-driven motion. For these parameters lattice effects right-hand side of the Eq. (2) is at least as important: not only are negligible, see appendix . Typical microscopic parameters ∼ the motion of the skyrmion but also the external forces excite used, are J = 1 meV (this yields T 10 K) which we use to c internal modes. Depending on the frequency range, there is estimate typical time scales for the skyrmion motion. an effective screening or antiscreening of the forces described Following Ref. 27, we assume that the ﬁeld bﬂ(t) is gen- by the matrices S (ω) and S (ω). Especially for the current- erated from a Gaussian stochastic process with thre following c g driven motion, there will be for all frequencies an almost exact statistical properties − cancellation of terms from G 1(ω) and S (ω). As a result (cid:10) (cid:11) c the skyrmion will follow almost instantaneously any change of the current despite its large mass. (cid:10) (cid:11) In this paper, we study the dynamics of a driven skyrmion bﬂ (t) = 0 r,i by solving numerically the stochastic Landau-Lifshitz-Gilbert k T − (LLG) equation. Our strategy will be to determine the pa- bﬂr,i(t)bﬂr(cid:48),j(s) = 2α γBM δijδrr(cid:48)δ(t s) (5) rameters of Eq. (2) such that this equation reproduces the re- sults of the LLG equation. Section introduces the model and (cid:104) (cid:105) outlines the numerical implementation. Three driving mecha- where i and j are cartesian components and . . . denotes nisms are considered: section studies the diffusive motion of an average taken over different realizations of the ﬂuctuating the skyrmion due to thermal noise, section the skyrmion mo- ﬁeld. The Gaussian property of the process stems from the in- tion due to time-dependent magnetic ﬁeld gradient and section teraction of M with a large number of microscopic degrees r the current-driven dynamics. We conclude with a summary of freedom (central limit theorem) which are also responsi- and discussion of the results in Sec. . ble for the damping described by α, reﬂecting the ﬂuctuation- II. MODEL dissipation theorem. The delta-correlation in time and space in Eq. (5) expresses that the autocorrelation time and length of Our study is based on a numerical analysis of the stochastic bﬂ(t) is much shorter than the response time and length scale r Landau-Lifshitz-Gilbert (sLLG) equations27 deﬁned by of the magnetic system. For a numerical implementation of Eq. (3) we follow Ref. 27 and use Heun’s scheme for the numerical integration dM × r = γ M [B + b (t)] which converges quadratically to the solution of the general dt r eﬀ ﬂ system of stochastic differential equations (when interpreted − α × × γ M (M [B + b (t)]) . (3) in terms of the Stratonovich calculus). M r r eﬀ ﬂ For static driving forces, one can calculate the drift veloc- ity R˙ following Thiele20. Starting from the Landau-Lifshitz Here γ is the gyromagnetic moment and α the Gilbert Gilbert equations, Eq. (3), we project onto the translational − dataemdpbiyngt;heBseuﬀrr=oundδinHδMg[Mmr ]agisnaenticefmfeocmtiveentms aangdnebtic(ﬁt)eladﬂcurec-- mspoadcee2b1–y23m. ultiplying Eq. (3) with ∂iMr and integrating over ﬂ ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: skyrmions

text: in

text: spintronic

text: devices,

text: Eq.

text: is

text: however

text: not

text: sufﬁ-

text: tuating,

text: stochastic

text: ﬁeld

text: creating

text: random

text: torques

text: on

text: the

text: mag-

text: cient.

text: A

text: generalized

text: version

text: of

text: Eq.

text: valid

text: for

text: weak

text: but

text: also

text: netic

text: moments

text: to

text: model

text: the

text: effects

text: of

text: thermal

text: ﬂuctuations,

text: arbitrarily

text: time-dependent

text: forces

text: can

text: be

text: written

text: as

text: see

text: below.

text: The

text: Hamiltonian

text: H[M]

text: is

text: given

text: by

text: G

text: F

text: (ω)

text: F

text: (ω)

text: F

text: (ω)

text: c

text: g

text: th∇

text: Sg(ω)

text: Bz(ω)

text: Fth(ω)

text: H[M]

text: M

text: M

text: M

text: r

text: r+aex

text: r+aey

text: r

text: Here

text: V

text: (ω)

text: eiωtR˙

text: (t)dt

text: is

text: the

text: Fourier

text: transform

text: of

text: λ

text: M

text: M

text: e

text: M

text: M

text: e

text: r

text: r+aex

text: x

text: r

text: r+aey

text: y

text: the

text: velocity

text: of

text: the

text: skyrmion,

text: vs(ω)

text: is

text: the

text: (spin-)

text: drift

text: ve-

text: r

text: locity

text: of

text: the

text: conduction

text: electrons,

text: directly

text: proportional

text: to

text: B

text: M

text: the

text: current,

text: B

text: (ω)

text: describes

text: a

text: magnetic

text: ﬁeld

text: gradient

text: in

text: r

text: z

text: r

text: frequency

text: space.

text: The

text: role

text: of

text: the

text: random

text: thermal

text: forces,

text: F

text: (ω),

text: is

text: special

text: as

text: their

text: dynamics

text: is

text: directly

text: linked

text: via

text: th

text: the

text: ﬂuctuation-dissipation

text: theorem

text: to

text: the

text: left-hand

text: side

text: of

text: the

text: We

text: use

text: J

text: γ

text: M

text: λ

text: for

text: the

text: r

text: equation,

text: see

text: below.

text: The

text: matrix

text: G

text: describes

text: the

text: strength

text: of

text: the

text: Dzyaloshinskii-Moriya

text: interaction

text: and

text: B

text: dynamics

text: of

text: the

text: skyrmion;

text: its

text: small-ω

text: expansion

text: deﬁnes

text: the

text: for

text: all

text: plots

text: giving

text: rise

text: to

text: a

text: skyrmion

text: with

text: a

text: terms

text: written

text: on

text: the

text: left-hand

text: side

text: of

text: Eq.

text: One

text: can

text: expect

text: radius

text: of

text: about

text: lattice

text: sites,

text: see

text: Appendix

text: For

text: this

text: pa-

text: strongly

text: frequency-dependent

text: dynamics

text: for

text: the

text: skyrmion

text: be-

text: rameter

text: set,

text: the

text: ground

text: state

text: is

text: ferromagnetic,

text: thus

text: the

text: sin-

text: cause

text: the

text: external

text: forces

text: in

text: combination

text: with

text: the

text: motion

text: of

text: gle

text: skyrmion

text: is

text: a

text: topologically

text: protected,

text: metastable

text: excita-

text: the

text: skyrmion

text: can

text: induce

text: a

text: precession

text: of

text: the

text: spin

text: and

text: also

text: ex-

text: tion.

text: Typically

text: we

text: simulate

text: spins

text: for

text: the

text: analysis

text: cite

text: spinwaves

text: in

text: the

text: surrounding

text: ferromagnet,

text: see

text: Fig.

text: of

text: diffusive

text: and

text: current

text: driven

text: motion

text: and

text: spins

text: for

text: We

text: will,

text: however,

text: show

text: that

text: the

text: frequency

text: dependence

text: of

text: the

text: the

text: force-driven

text: motion.

text: For

text: these

text: parameters

text: lattice

text: effects

text: right-hand

text: side

text: of

text: the

text: Eq.

text: is

text: at

text: least

text: as

text: important:

text: not

text: only

text: are

text: negligible,

text: see

text: appendix

text: Typical

text: microscopic

text: parameters

text: the

text: motion

text: of

text: the

text: skyrmion

text: but

text: also

text: the

text: external

text: forces

text: excite

text: used,

text: are

text: J

text: meV

text: (this

text: yields

text: T

text: K)

text: which

text: we

text: use

text: to

text: c

text: internal

text: modes.

text: Depending

text: on

text: the

text: frequency

text: range,

text: there

text: is

text: estimate

text: typical

text: time

text: scales

text: for

text: the

text: skyrmion

text: motion.

text: an

text: effective

text: screening

text: or

text: antiscreening

text: of

text: the

text: forces

text: described

text: Following

text: Ref.

text: we

text: assume

text: that

text: the

text: ﬁeld

text: bﬂ(t)

text: is

text: gen-

text: by

text: the

text: matrices

text: S

text: (ω)

text: and

text: S

text: (ω).

text: Especially

text: for

text: the

text: current-

text: erated

text: from

text: a

text: Gaussian

text: stochastic

text: process

text: with

text: thre

text: following

text: c

text: g

text: driven

text: motion,

text: there

text: will

text: be

text: for

text: all

text: frequencies

text: an

text: almost

text: exact

text: statistical

text: properties

text: cancellation

text: of

text: terms

text: from

text: G

text: and

text: S

text: (ω).

text: As

text: a

text: result

text: c

text: the

text: skyrmion

text: will

text: follow

text: almost

text: instantaneously

text: any

text: change

text: of

text: the

text: current

text: despite

text: its

text: large

text: mass.

text: In

text: this

text: paper,

text: we

text: study

text: the

text: dynamics

text: of

text: a

text: driven

text: skyrmion

text: bﬂ

text: (t)

text: r,i

text: by

text: solving

text: numerically

text: the

text: stochastic

text: Landau-Lifshitz-Gilbert

text: k

text: T

text: (LLG)

text: equation.

text: Our

text: strategy

text: will

text: be

text: to

text: determine

text: the

text: pa-

text: 2α

text: γBM

text: s)

text: rameters

text: of

text: Eq.

text: such

text: that

text: this

text: equation

text: reproduces

text: the

text: re-

text: sults

text: of

text: the

text: LLG

text: equation.

text: Section

text: introduces

text: the

text: model

text: and

text: outlines

text: the

text: numerical

text: implementation.

text: Three

text: driving

text: mecha-

text: where

text: i

text: and

text: j

text: are

text: cartesian

text: components

text: and

text: denotes

text: nisms

text: are

text: considered:

text: section

text: studies

text: the

text: diffusive

text: motion

text: of

text: an

text: average

text: taken

text: over

text: different

text: realizations

text: of

text: the

text: ﬂuctuating

text: the

text: skyrmion

text: due

text: to

text: thermal

text: noise,

text: section

text: the

text: skyrmion

text: mo-

text: ﬁeld.

text: The

text: Gaussian

text: property

text: of

text: the

text: process

text: stems

text: from

text: the

text: in-

text: tion

text: due

text: to

text: time-dependent

text: magnetic

text: ﬁeld

text: gradient

text: and

text: section

text: teraction

text: of

text: M

text: with

text: a

text: large

text: number

text: of

text: microscopic

text: degrees

text: r

text: the

text: current-driven

text: dynamics.

text: We

text: conclude

text: with

text: a

text: summary

text: of

text: freedom

text: (central

text: limit

text: theorem)

text: which

text: are

text: also

text: responsi-

text: and

text: discussion

text: of

text: the

text: results

text: in

text: Sec.

text: ble

text: for

text: the

text: damping

text: described

text: by

text: α,

text: reﬂecting

text: the

text: ﬂuctuation-

text: II.

text: MODEL

text: dissipation

text: theorem.

text: The

text: delta-correlation

text: in

text: time

text: and

text: space

text: in

text: Eq.

text: expresses

text: that

text: the

text: autocorrelation

text: time

text: and

text: length

text: of

text: Our

text: study

text: is

text: based

text: on

text: a

text: numerical

text: analysis

text: of

text: the

text: stochastic

text: bﬂ(t)

text: is

text: much

text: shorter

text: than

text: the

text: response

text: time

text: and

text: length

text: scale

text: r

text: Landau-Lifshitz-Gilbert

text: (sLLG)

text: deﬁned

text: by

text: of

text: the

text: magnetic

text: system.

text: For

text: a

text: numerical

text: implementation

text: of

text: Eq.

text: we

text: follow

text: Ref.

text: and

text: use

text: Heun’s

text: scheme

text: for

text: the

text: numerical

text: integration

text: dM

text: r

text: γ

text: M

text: [B

text: b

text: (t)]

text: which

text: converges

text: quadratically

text: to

text: the

text: solution

text: of

text: the

text: general

text: dt

text: r

text: eﬀ

text: ﬂ

text: system

text: of

text: stochastic

text: differential

text: equations

text: (when

text: interpreted

text: α

text: γ

text: M

text: (M

text: [B

text: b

text: (t)])

text: in

text: terms

text: of

text: the

text: Stratonovich

text: calculus).

text: M

text: r

text: r

text: eﬀ

text: ﬂ

text: For

text: static

text: driving

text: forces,

text: one

text: can

text: calculate

text: the

text: drift

text: veloc-

text: ity

text: R˙

text: following

text: Starting

text: from

text: the

text: Landau-Lifshitz

text: Here

text: γ

text: is

text: the

text: gyromagnetic

text: moment

text: and

text: α

text: the

text: Gilbert

text: Gilbert

text: equations,

text: Eq.

text: we

text: project

text: onto

text: the

text: translational

text: ]agisnaenticefmfeocmtiveentms

text: aangdnebtic(ﬁt)eladﬂcurec--

text: ultiplying

text: Eq.

text: with

text: ∂iMr

text: and

text: integrating

text: over

text: ﬂ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##