text: Draft version May 27, 2020 Preprint typeset using LATEX style emulateapj v. 12/16/11 ELECTRON HEATING IN A RELATIVISTIC, WEIBEL-UNSTABLE PLASMA Rahul Kumar, David Eichler, and Michael Gedalin 0 Physics Department, Ben-Gurion University, Be’er-Sheba 84105, Israel Draft version May 27, 2020 ABSTRACT The dynamics of two initially unmagnetized relativistic counter-streaming homogeneous ion-electron plasma beams are simulated in two dimensions using the particle-in-cell (PIC) method. It is shown that current ﬁlaments, which form due to the Weibel instability, develop a large scale longitudinal electric ﬁeld in the direction opposite to the current carried by the ﬁlaments as predicted by theory. Fast moving ions in the current ﬁlaments decelerate due to this longitudinal electric ﬁeld. The same longitudinal electric ﬁeld, which is partially inductive and partially electrostatic, is identiﬁed as the main source of acceleration of electrons in the current ﬁlaments. The transverse electric ﬁeld, though larger than the longitudinal one, is shown to play a smaller role in heating electrons, contrary to previous claims. It is found that, in 1D, the electrons become strongly magnetized and are not accelerated beyond their initial kinetic energy. Rather, the heating of the electrons is enhanced by the bending and break-up of the ﬁlaments, which releases electrons that would otherwise be trapped within a single ﬁlament and hence slow the development of the Weibel instability (i.e. the magnetic ﬁeld growth) via induction as per Lenz’s law. In 2D simulations electrons are heated to about one quarter of the initial kinetic energy of ions. The magnetic energy at maximum is about 4 percent, decaying to less than 1 percent by the end of the simulation. Most of the heating of electrons takes place while the longitudinal electric ﬁeld is still growing while only a small portion of the heating is a result of subsequent magnetic ﬁeld decay. The ions are found to gradually decelerate until the end of the simulation by which time they retain a residual anisotropy less than 10 percent. Keywords: 1. INTRODUCTION relatively less energetic upstream electrons are signiﬁ- cantly heated to energy comparable to the energy of ions Collisionless shocks forming in astrophysical environ- as they cross the foreshock, which appears to be in agree- ments are believed to be mediated by electromagnetic ment with the electromagnetic observations of supernova instabilities (Kennel and Sagdeev 1967; Parker 1961; remnants and gamma-ray bursts afterglows, where the Eichler 1979; Blandford and Eichler 1987) in which ions high energy radiation is believed to be synchrotron radi- are scattered by the resulting magnetic ﬁeld ﬂuctua- ation originating from gyration of high energy electrons tions. In particlular, the Weibel instability (Weibel in a magnetic ﬁelds signiﬁcantly higher than the inter- 1959), which causes fast growth of strong magnetic ﬁeld stellar magnetic ﬁeld (Panaitescu and Kumar 2002; Pi- at small-scale in anisotropic plasma ﬂow, has received ran 2005; Gehrels and M´esz´aros 2012). It however re- much attention as the main isotropization mechanism mains unclear how the upstream electrons are energized that leads to shock transition in free-streaming ejecta in the foreshock region, both in nature and in the numer- from violent astrophysical events (Medvedev and Loeb ical simulations of collisionless shocks, and whether the 1999; Wiersma and Achterberg 2004; Lyubarsky and Weibel instability-induced magnetic ﬁeld should persist Eichler 2006; Achterberg and Wiersma 2007; Bret 2009; over astrophysically signiﬁcant length scales behind the Yalinewich and Gedalin 2010; Shaisultanov et al. 2012). shock. As the dominant modes of the Weibel instability are less Numerical simulations suggest presence of large scale than the ion gyroradius, which renders them ineﬃcient electric ﬁeld in and around the current ﬁlaments forming scatterers of ions, the longstanding question about its just ahead of the shock and ions are found to decelerate role in collisions shocks has been how well it competes due to this electric ﬁeld. The same electric ﬁeld that de- with other mechanisms (e.g. Galeev et al. 1964; Bland- celerates ions should also accelerate electrons (Blandford ford and Eichler 1987; Lyubarsky and Eichler 2006). and Eichler 1987; Lyubarsky and Eichler 2006; Gedalin Clearly mechanisms that require a preexisting magnetic et al. 2008, 2012). However, the details of the elec- ﬁeld are questionable when the magnetic ﬁeld is weak. tric ﬁeld and the acceleration mechanism have not been On the other hand, a slow shock mechanism may ac- worked out. tually suppress a faster mechanism because it creates a In order to understand the heating of electrons in fore- broader shock transition, and thus has greater ”reach” shock region of relativistic shocks we simulate, using the upstream of the shock. kinetic PIC method, development of the Weibel instabil- The role of Weibel instability in forming the shock ity in two relativistic counter-streaming plasma beams, transition in weakly magnetized plasmas has recently which resembles the precursor of a relativistic shock, but been established through several numerical experiments simpler and more idealized and therefore more suitable (Nishikawa et al. 2003; Frederiksen et al. 2004; Nishikawa for resolving certain fundamental questions. The beams et al. 2005; Spitkovsky 2008; Keshet et al. 2009). Numeri- are taken to be homogeneous and fully interpenetrating cal simulations of relativistic shocks have shown that the at the beginning, so the time development of the instabil-

text: Draft

text: version

text: May

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: ELECTRON

text: HEATING

text: IN

text: A

text: RELATIVISTIC,

text: WEIBEL-UNSTABLE

text: PLASMA

text: Rahul

text: Kumar,

text: David

text: Eichler,

text: and

text: Michael

text: Gedalin

text: Physics

text: Department,

text: Ben-Gurion

text: University,

text: Be’er-Sheba

text: Israel

text: Draft

text: version

text: May

text: ABSTRACT

text: The

text: dynamics

text: of

text: two

text: initially

text: unmagnetized

text: relativistic

text: counter-streaming

text: homogeneous

text: ion-electron

text: plasma

text: beams

text: are

text: simulated

text: in

text: two

text: dimensions

text: using

text: the

text: particle-in-cell

text: (PIC)

text: method.

text: It

text: is

text: shown

text: that

text: current

text: ﬁlaments,

text: which

text: form

text: due

text: to

text: the

text: Weibel

text: instability,

text: develop

text: a

text: large

text: scale

text: longitudinal

text: electric

text: ﬁeld

text: in

text: the

text: direction

text: opposite

text: to

text: the

text: current

text: carried

text: by

text: the

text: ﬁlaments

text: as

text: predicted

text: by

text: theory.

text: Fast

text: moving

text: ions

text: in

text: the

text: current

text: ﬁlaments

text: decelerate

text: due

text: to

text: this

text: longitudinal

text: electric

text: ﬁeld.

text: The

text: same

text: longitudinal

text: electric

text: ﬁeld,

text: which

text: is

text: partially

text: inductive

text: and

text: partially

text: electrostatic,

text: is

text: identiﬁed

text: as

text: the

text: main

text: source

text: of

text: acceleration

text: of

text: electrons

text: in

text: the

text: current

text: ﬁlaments.

text: The

text: transverse

text: electric

text: ﬁeld,

text: though

text: larger

text: than

text: the

text: longitudinal

text: one,

text: is

text: shown

text: to

text: play

text: a

text: smaller

text: role

text: in

text: heating

text: electrons,

text: contrary

text: to

text: previous

text: claims.

text: It

text: is

text: found

text: that,

text: in

text: the

text: electrons

text: become

text: strongly

text: magnetized

text: and

text: are

text: not

text: accelerated

text: beyond

text: their

text: initial

text: kinetic

text: energy.

text: Rather,

text: the

text: heating

text: of

text: the

text: electrons

text: is

text: enhanced

text: by

text: the

text: bending

text: and

text: break-up

text: of

text: the

text: ﬁlaments,

text: which

text: releases

text: electrons

text: that

text: would

text: otherwise

text: be

text: trapped

text: within

text: a

text: single

text: ﬁlament

text: and

text: hence

text: slow

text: the

text: development

text: of

text: the

text: Weibel

text: instability

text: (i.e.

text: the

text: magnetic

text: ﬁeld

text: growth)

text: via

text: induction

text: as

text: per

text: Lenz’s

text: law.

text: In

text: simulations

text: electrons

text: are

text: heated

text: to

text: about

text: one

text: quarter

text: of

text: the

text: initial

text: kinetic

text: energy

text: of

text: ions.

text: The

text: magnetic

text: energy

text: at

text: maximum

text: is

text: about

text: percent,

text: decaying

text: to

text: less

text: than

text: percent

text: by

text: the

text: end

text: of

text: the

text: simulation.

text: Most

text: of

text: the

text: heating

text: of

text: electrons

text: takes

text: place

text: while

text: the

text: longitudinal

text: electric

text: ﬁeld

text: is

text: still

text: growing

text: while

text: only

text: a

text: small

text: portion

text: of

text: the

text: heating

text: is

text: a

text: result

text: of

text: subsequent

text: magnetic

text: ﬁeld

text: decay.

text: The

text: ions

text: are

text: found

text: to

text: gradually

text: decelerate

text: until

text: the

text: end

text: of

text: the

text: simulation

text: by

text: which

text: time

text: they

text: retain

text: a

text: residual

text: anisotropy

text: less

text: than

text: percent.

text: Keywords:

text: INTRODUCTION

text: relatively

text: less

text: energetic

text: upstream

text: electrons

text: are

text: signiﬁ-

text: cantly

text: heated

text: to

text: energy

text: comparable

text: to

text: the

text: energy

text: of

text: ions

text: Collisionless

text: shocks

text: forming

text: in

text: astrophysical

text: environ-

text: as

text: they

text: cross

text: the

text: foreshock,

text: which

text: appears

text: to

text: be

text: in

text: agree-

text: ments

text: are

text: believed

text: to

text: be

text: mediated

text: by

text: electromagnetic

text: ment

text: with

text: the

text: electromagnetic

text: observations

text: of

text: supernova

text: instabilities

text: (Kennel

text: and

text: Sagdeev

text: Parker

text: remnants

text: and

text: gamma-ray

text: bursts

text: afterglows,

text: where

text: the

text: Eichler

text: Blandford

text: and

text: Eichler

text: in

text: which

text: ions

text: high

text: energy

text: radiation

text: is

text: believed

text: to

text: be

text: synchrotron

text: radi-

text: are

text: scattered

text: by

text: the

text: resulting

text: magnetic

text: ﬁeld

text: ﬂuctua-

text: ation

text: originating

text: from

text: gyration

text: of

text: high

text: energy

text: electrons

text: tions.

text: In

text: particlular,

text: the

text: Weibel

text: instability

text: (Weibel

text: in

text: a

text: magnetic

text: ﬁelds

text: signiﬁcantly

text: higher

text: than

text: the

text: inter-

text: which

text: causes

text: fast

text: growth

text: of

text: strong

text: magnetic

text: ﬁeld

text: stellar

text: magnetic

text: ﬁeld

text: (Panaitescu

text: and

text: Kumar

text: Pi-

text: at

text: small-scale

text: in

text: anisotropic

text: plasma

text: ﬂow,

text: has

text: received

text: ran

text: Gehrels

text: and

text: M´esz´aros

text: It

text: however

text: re-

text: much

text: attention

text: as

text: the

text: main

text: isotropization

text: mechanism

text: mains

text: unclear

text: how

text: the

text: upstream

text: electrons

text: are

text: energized

text: that

text: leads

text: to

text: shock

text: transition

text: in

text: free-streaming

text: ejecta

text: in

text: the

text: foreshock

text: region,

text: both

text: in

text: nature

text: and

text: in

text: the

text: numer-

text: from

text: violent

text: astrophysical

text: events

text: (Medvedev

text: and

text: Loeb

text: ical

text: simulations

text: of

text: collisionless

text: shocks,

text: and

text: whether

text: the

text: Wiersma

text: and

text: Achterberg

text: Lyubarsky

text: and

text: Weibel

text: instability-induced

text: magnetic

text: ﬁeld

text: should

text: persist

text: Eichler

text: Achterberg

text: and

text: Wiersma

text: Bret

text: over

text: astrophysically

text: signiﬁcant

text: length

text: scales

text: behind

text: the

text: Yalinewich

text: and

text: Gedalin

text: Shaisultanov

text: et

text: al.

text: shock.

text: As

text: the

text: dominant

text: modes

text: of

text: the

text: Weibel

text: instability

text: are

text: less

text: Numerical

text: simulations

text: suggest

text: presence

text: of

text: large

text: scale

text: than

text: the

text: ion

text: gyroradius,

text: which

text: renders

text: them

text: ineﬃcient

text: electric

text: ﬁeld

text: in

text: and

text: around

text: the

text: current

text: ﬁlaments

text: forming

text: scatterers

text: of

text: ions,

text: the

text: longstanding

text: question

text: about

text: its

text: just

text: ahead

text: of

text: the

text: shock

text: and

text: ions

text: are

text: found

text: to

text: decelerate

text: role

text: in

text: collisions

text: shocks

text: has

text: been

text: how

text: well

text: it

text: competes

text: due

text: to

text: this

text: electric

text: ﬁeld.

text: The

text: same

text: electric

text: ﬁeld

text: that

text: de-

text: with

text: other

text: mechanisms

text: (e.g.

text: Galeev

text: et

text: al.

text: Bland-

text: celerates

text: ions

text: should

text: also

text: accelerate

text: electrons

text: (Blandford

text: ford

text: and

text: Eichler

text: Lyubarsky

text: and

text: Eichler

text: and

text: Eichler

text: Lyubarsky

text: and

text: Eichler

text: Gedalin

text: Clearly

text: mechanisms

text: that

text: require

text: a

text: preexisting

text: magnetic

text: et

text: al.

text: However,

text: the

text: details

text: of

text: the

text: elec-

text: ﬁeld

text: are

text: questionable

text: when

text: the

text: magnetic

text: ﬁeld

text: is

text: weak.

text: tric

text: ﬁeld

text: and

text: the

text: acceleration

text: mechanism

text: have

text: not

text: been

text: On

text: the

text: other

text: hand,

text: a

text: slow

text: shock

text: mechanism

text: may

text: ac-

text: worked

text: out.

text: tually

text: suppress

text: a

text: faster

text: mechanism

text: because

text: it

text: creates

text: a

text: In

text: order

text: to

text: understand

text: the

text: heating

text: of

text: electrons

text: in

text: fore-

text: broader

text: shock

text: transition,

text: and

text: thus

text: has

text: greater

text: ”reach”

text: shock

text: region

text: of

text: relativistic

text: shocks

text: we

text: simulate,

text: using

text: the

text: upstream

text: of

text: the

text: shock.

text: kinetic

text: PIC

text: method,

text: development

text: of

text: the

text: Weibel

text: instabil-

text: The

text: role

text: of

text: Weibel

text: instability

text: in

text: forming

text: the

text: shock

text: ity

text: in

text: two

text: relativistic

text: counter-streaming

text: plasma

text: beams,

text: transition

text: in

text: weakly

text: magnetized

text: plasmas

text: has

text: recently

text: which

text: resembles

text: the

text: precursor

text: of

text: a

text: relativistic

text: shock,

text: but

text: been

text: established

text: through

text: several

text: numerical

text: experiments

text: simpler

text: and

text: more

text: idealized

text: and

text: therefore

text: more

text: suitable

text: (Nishikawa

text: et

text: al.

text: Frederiksen

text: et

text: al.

text: Nishikawa

text: for

text: resolving

text: certain

text: fundamental

text: questions.

text: The

text: beams

text: et

text: al.

text: Spitkovsky

text: Keshet

text: et

text: al.

text: Numeri-

text: are

text: taken

text: to

text: be

text: homogeneous

text: and

text: fully

text: interpenetrating

text: cal

text: simulations

text: of

text: relativistic

text: shocks

text: have

text: shown

text: that

text: the

text: at

text: the

text: beginning,

text: so

text: the

text: time

text: development

text: of

text: the

text: instabil-


text: 2ryE ,2xcE ,2ryE ,2xcE yvr,cyE ,xvr,xcE 2/1)2c γ m n π4( / kE 0 i 0 x 2/1)2c γ m n π4( / kB 0 i 0 z 7 −2 10 Eyc2 Er2 y Er2 −3 x 10 Ec2 x 0 100 200 300 400 time × ω pe Fﬁgiguruere3)9.of tTheemeploercatrlicvaﬁreialdtiocnomopf otnheentms eEanr, sEquca,reEr(,asanind tEhce Figure 11. Power spectrum for Bz, Ex, and Ey at 100 ωp−e1 ∇ · x∇ ×x y y is shown in the left, middle, and right panel panel, respectively. (E(cid:126) = E(cid:126)r + E(cid:126)c, such that E(cid:126)r = 0, and E(cid:126)c = 0. Sub- The power spectrum shows broadband nature of the Weibel insta- script x and y indicate components along longitudinal (streaming bility. The spectrum of Bz shows that the transverse waves are direction) and transverse direction, respectively) are shown by the mostly magnetic in nature. The existence of parallel and oblique solid red, solid blue, dashed red, and dashed blue curves, respec- modes, which are rather electrostatic in nature, is apparant from tively. During the ﬁlamentation stage the transverse electric ﬁeld the spectrum of Ex. Ey is mostly electrostatic and the longitudinal electric ﬁeld Ex is partially inductive and partially electrostatic. x 10−4 x 10−4 1 8 Electron Ion 0.8 0.04 0.04 6 0.6 0.02 vy 0.02 4 Eycvy c,ry Eycvy 0.4 E 0 0 Eyrvy v,x Eyrvy 0.2 2 −0.02 Excvx c,rEx −0.02 Excvx 0 0 −0.04 Exrvx −0.04 Exrvx 0 100 t2im00e × 3ω0p0e 400 500 0 100 t2im00e × 3ω0p0e 400 500 0 200 400 0 200 400 Figure 12. The left (right) panel shows temporal evolution of time × ω time × ω the average amplitude of sinusoidal variation in the longitudinal pe pe electric (magnetic) ﬁeld Ek (Bk) with the wave vector being along Figure 10. The left (right) panel shows the instantaneous rate the streaming direction (xx-axisz). The red, green, blue, and cyan of energy gained (lost) by the electrons (protons) due to the ro- colored curve show the ﬁeld amplitude of modes with 2π/k= 410, tational and compressive components of the longitudinal electric 90, 16, and 8 c/ωpe, respectively. The average amplitude shown ﬁeld in solid red and solid blue, respectively. The dashed red and here are obtained after taking average of amplitude of any given blue curves shows the same due to the rotational and compressive mode over an ensemble of several slices parallel to the y-axis at components of the transverse electric ﬁeld Ey, respectively. Nor- equally spaced locations along the x-axis. malization and computation of the mean shown here are same as in the ﬁgure 8. tric ﬁeld in purely transverse, purely longitudinal, and an oblique mode. The general trend is that the purely ble with the inductive heating due to transverse Weibel transverse modes grow the fastest and the rate of growth modes. decreases with the angle between the wave vector and 3.3. Parallel and oblique modes y-axis. The oblique modes, like the purely transverse We perform two dimensional spectral analysis of the mode, are suppressed in the linear regime by the induc- electromagnetic ﬁeld components for the simulation M tive response of the electrons which are accelerated by, 64 to show the simultaneous growth of several other waves and therefore move along the inductive electric ﬁelds. in addition to the purely transverse Weibel waves. In Note, however, that the purely transverse mode stops ﬁgure 12, we show the average amplitude of sinusoidal growing and at time of about 250 ω (ﬁgure) while the pe variation in electromagnetic ﬁelds along the streaming oblique mode continues to grow and in fact catches up direction. The amplitude of variation in the magnetic with the purely transverse mode. This can be inter- ﬁeld serves as a proxy for bending of the current ﬁlaments preted as follows: The transverse mode, being the fastest since the parallel modes are mostly electrostatic and do growing, is the ﬁrst to trap the electrons into the reso- not contribute to the growth of the magnetic ﬁeld. As nance, so that the number of resonant electrons actually evident from the ﬁgure 12, the growth of electric ﬁeld increases. This includes those electrons that resonated in the same length scale is faster than the growth of the with the oblique modes until trapped by the purely trans- magnetic ﬁeld, hence by comparison a rather stronger verse mode. We can say that these electrons are in a electric ﬁeld in waves with their wave vector along the non-linear resonance with the purely trapped mode. At longitudinal direction can be attributed to the growth this point, the growth of the purely transverse mode is of electrostatic modes along the streaming direction in impeded by the even larger number of non-linearly reso- addition to the contribution from the bending of current nant electrons, so it stops growing.The oblique modes, in ﬁlaments. contrast, suﬀer less from the inductive eﬀects of the elec- In ﬁgure 14 we show the growth of magnetic and elec- trons so they keep growing. until they have spoiled the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: yvr,cyE

text: ,xvr,xcE

text: γ

text: m

text: n

text: kE

text: i

text: x

text: γ

text: m

text: n

text: kB

text: i

text: z

text: −2

text: y

text: −3

text: x

text: x

text: time

text: ω

text: pe

label: caption

text: tTheemeploercatrlicvaﬁreialdtiocnomopf

text: otnheentms

text: eEanr,

text: sEquca,reEr(,asanind

text: tEhce

text: Figure

text: Power

text: spectrum

text: for

text: Bz,

text: Ex,

text: and

text: Ey

text: at

text: x∇

text: ×x

text: y

text: y

text: is

text: shown

text: in

text: the

text: left,

text: middle,

text: and

text: right

text: panel

text: panel,

text: respectively.

text: such

text: that

text: and

text: Sub-

text: The

text: power

text: spectrum

text: shows

text: broadband

text: nature

text: of

text: the

text: Weibel

text: insta-

text: script

text: x

text: and

text: y

text: indicate

text: components

text: along

text: longitudinal

text: (streaming

text: bility.

text: The

text: spectrum

text: of

text: Bz

text: shows

text: that

text: the

text: transverse

text: waves

text: are

text: direction)

text: and

text: transverse

text: direction,

text: respectively)

text: are

text: shown

text: by

text: the

text: mostly

text: magnetic

text: in

text: nature.

text: The

text: existence

text: of

text: parallel

text: and

text: oblique

text: solid

text: red,

text: solid

text: blue,

text: dashed

text: red,

text: and

text: dashed

text: blue

text: curves,

text: respec-

text: modes,

text: which

text: are

text: rather

text: electrostatic

text: in

text: nature,

text: is

text: apparant

text: from

text: tively.

text: During

text: the

text: ﬁlamentation

text: stage

text: the

text: transverse

text: electric

text: ﬁeld

text: the

text: spectrum

text: of

text: Ex.

text: Ey

text: is

text: mostly

text: electrostatic

text: and

text: the

text: longitudinal

text: electric

text: ﬁeld

text: Ex

text: is

text: partially

text: inductive

text: and

text: partially

text: electrostatic.

text: x

text: x

text: Electron

text: Ion

text: vy

text: Eycvy

text: c,ry

text: Eycvy

text: E

text: Eyrvy

text: v,x

text: Eyrvy

text: Excvx

text: c,rEx

text: Excvx

text: Exrvx

text: Exrvx

text: Figure

text: The

text: left

text: (right)

text: panel

text: shows

text: temporal

text: evolution

text: of

text: time

text: ω

text: time

text: ω

text: the

text: average

text: amplitude

text: of

text: sinusoidal

text: variation

text: in

text: the

text: longitudinal

text: pe

text: pe

text: electric

text: (magnetic)

text: ﬁeld

text: Ek

text: (Bk)

text: with

text: the

text: wave

text: vector

text: being

text: along

text: Figure

text: The

text: left

text: (right)

text: panel

text: shows

text: the

text: instantaneous

text: rate

text: the

text: streaming

text: direction

text: (xx-axisz).

text: The

text: red,

text: green,

text: blue,

text: and

text: cyan

text: of

text: energy

text: gained

text: (lost)

text: by

text: the

text: electrons

text: (protons)

text: due

text: to

text: the

text: ro-

text: colored

text: curve

text: show

text: the

text: ﬁeld

text: amplitude

text: of

text: modes

text: with

text: tational

text: and

text: compressive

text: components

text: of

text: the

text: longitudinal

text: electric

text: and

text: c/ωpe,

text: respectively.

text: The

text: average

text: amplitude

text: shown

text: ﬁeld

text: in

text: solid

text: red

text: and

text: solid

text: blue,

text: respectively.

text: The

text: dashed

text: red

text: and

text: here

text: are

text: obtained

text: after

text: taking

text: average

text: of

text: amplitude

text: of

text: any

text: given

text: blue

text: curves

text: shows

text: the

text: same

text: due

text: to

text: the

text: rotational

text: and

text: compressive

text: mode

text: over

text: an

text: ensemble

text: of

text: several

text: slices

text: parallel

text: to

text: the

text: y-axis

text: at

text: components

text: of

text: the

text: transverse

text: electric

text: ﬁeld

text: Ey,

text: respectively.

text: Nor-

text: equally

text: spaced

text: locations

text: along

text: the

text: x-axis.

text: malization

text: and

text: computation

text: of

text: the

text: mean

text: shown

text: here

text: are

text: same

text: as

text: in

text: the

text: ﬁgure

text: tric

text: ﬁeld

text: in

text: purely

text: transverse,

text: purely

text: longitudinal,

text: and

text: an

text: oblique

text: mode.

text: The

text: general

text: trend

text: is

text: that

text: the

text: purely

text: ble

text: with

text: the

text: inductive

text: heating

text: due

text: to

text: transverse

text: Weibel

text: transverse

text: modes

text: grow

text: the

text: fastest

text: and

text: the

text: rate

text: of

text: growth

text: modes.

text: decreases

text: with

text: the

text: angle

text: between

text: the

text: wave

text: vector

text: and

text: Parallel

text: and

text: oblique

text: modes

text: y-axis.

text: The

text: oblique

text: modes,

text: like

text: the

text: purely

text: transverse

text: We

text: perform

text: two

text: dimensional

text: spectral

text: analysis

text: of

text: the

text: mode,

text: are

text: suppressed

text: in

text: the

text: linear

text: regime

text: by

text: the

text: induc-

text: electromagnetic

text: ﬁeld

text: components

text: for

text: the

text: simulation

text: M

text: tive

text: response

text: of

text: the

text: electrons

text: which

text: are

text: accelerated

text: by,

text: to

text: show

text: the

text: simultaneous

text: growth

text: of

text: several

text: other

text: waves

text: and

text: therefore

text: move

text: along

text: the

text: inductive

text: electric

text: ﬁelds.

text: in

text: addition

text: to

text: the

text: purely

text: transverse

text: Weibel

text: waves.

text: In

text: Note,

text: however,

text: that

text: the

text: purely

text: transverse

text: mode

text: stops

text: ﬁgure

text: we

text: show

text: the

text: average

text: amplitude

text: of

text: sinusoidal

text: growing

text: and

text: at

text: time

text: of

text: about

text: ω

text: (ﬁgure)

text: while

text: the

text: pe

text: variation

text: in

text: electromagnetic

text: ﬁelds

text: along

text: the

text: streaming

text: oblique

text: mode

text: continues

text: to

text: grow

text: and

text: in

text: fact

text: catches

text: up

text: direction.

text: The

text: amplitude

text: of

text: variation

text: in

text: the

text: magnetic

text: with

text: the

text: purely

text: transverse

text: mode.

text: This

text: can

text: be

text: inter-

text: ﬁeld

text: serves

text: as

text: a

text: proxy

text: for

text: bending

text: of

text: the

text: current

text: ﬁlaments

text: preted

text: as

text: follows:

text: The

text: transverse

text: mode,

text: being

text: the

text: fastest

text: since

text: the

text: parallel

text: modes

text: are

text: mostly

text: electrostatic

text: and

text: do

text: growing,

text: is

text: the

text: ﬁrst

text: to

text: trap

text: the

text: electrons

text: into

text: the

text: reso-

text: not

text: contribute

text: to

text: the

text: growth

text: of

text: the

text: magnetic

text: ﬁeld.

text: As

text: nance,

text: so

text: that

text: the

text: number

text: of

text: resonant

text: electrons

text: actually

text: evident

text: from

text: the

text: ﬁgure

text: the

text: growth

text: of

text: electric

text: ﬁeld

text: increases.

text: This

text: includes

text: those

text: electrons

text: that

text: resonated

text: in

text: the

text: same

text: length

text: scale

text: is

text: faster

text: than

text: the

text: growth

text: of

text: the

text: with

text: the

text: oblique

text: modes

text: until

text: trapped

text: by

text: the

text: purely

text: trans-

text: magnetic

text: ﬁeld,

text: hence

text: by

text: comparison

text: a

text: rather

text: stronger

text: verse

text: mode.

text: We

text: can

text: say

text: that

text: these

text: electrons

text: are

text: in

text: a

text: electric

text: ﬁeld

text: in

text: waves

text: with

text: their

text: wave

text: vector

text: along

text: the

text: non-linear

text: resonance

text: with

text: the

text: purely

text: trapped

text: mode.

text: At

text: longitudinal

text: direction

text: can

text: be

text: attributed

text: to

text: the

text: growth

text: this

text: point,

text: the

text: growth

text: of

text: the

text: purely

text: transverse

text: mode

text: is

text: of

text: electrostatic

text: modes

text: along

text: the

text: streaming

text: direction

text: in

text: impeded

text: by

text: the

text: even

text: larger

text: number

text: of

text: non-linearly

text: reso-

text: addition

text: to

text: the

text: contribution

text: from

text: the

text: bending

text: of

text: current

text: nant

text: electrons,

text: so

text: it

text: stops

text: growing.The

text: oblique

text: modes,

text: in

text: ﬁlaments.

text: contrast,

text: suﬀer

text: less

text: from

text: the

text: inductive

text: eﬀects

text: of

text: the

text: elec-

text: In

text: ﬁgure

text: we

text: show

text: the

text: growth

text: of

text: magnetic

text: and

text: elec-

text: trons

text: so

text: they

text: keep

text: growing.

text: until

text: they

text: have

text: spoiled

text: the

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text: 12 KARL K. BRUSTAD remembering the assumption (5.3). This function is automatically up- per semi-continuous but as the deﬁnition of p-superharmonicity re- quires lower semi-continuity, continuity has to be shown. R We claim that W δ is p-superharmonic in n provided the series (cid:80) converges at least at some point. Again we may assume that the convergence is at the origin. That is ∞ | | ∞ a y α < . Since 0 < α < 1, we get i=1 i i | − | ≤ | | | | x y α ( x + y )α i i ≤ | | | | x α + y α (cid:88) (cid:88) (cid:88)i so since ∞ ∞ ∞ | − | ≤ | | | | ∞ a x y α x α a + a y α < i i i i i i=1 i=1 i=1 → R we see that W δ W δ locally uniformly in n. We infer that W δ is (cid:12) RN (cid:12) continu(cid:12)ous in n(cid:12). (cid:12) (cid:12) ⊂⊂ R ∈ For(cid:12) part iii)(cid:12), assume D n and h C(D) is p-harmonic in D ≤ ≤ with h W δ . Since W(cid:12) δ W δ(cid:12) and W δ is p-superharmonic we ∂≤D ∂D (cid:12) N(cid:12) N get h W δ for all N. So given any (cid:15) > 0 D N D ≤ h W δ + (cid:15) D D by uniformity on the bounded set D. This proves the claim. → (cid:88) Next, let δ 0. Then ∞ − | − | − W (x) := ai x yi pp−n1 + K(x) i=1 R is p-superharmonic in n by the same argument as when 2 < p < n. This settles the case p > n. 6. Epilogue: Evolutionary superposition. The superposition of fundamental solutions has been extended to p- Laplace equations in the Heisenberg group, see [GT10]. When it comes to further extensions, a natural question is whether such a superposi- tion is valid for the evolutionary p-Laplace equation (6.1) u = ∆ u, t p or for the homogeneous equation ∂ | | − (6.2) ( u p 2u) = ∆ u. p ∂t The following shows it does not. ##LTLine## ##LTLine## ##LTLine##

text: KARL

text: K.

text: BRUSTAD

text: remembering

text: the

text: assumption

text: This

text: function

text: is

text: automatically

text: up-

text: per

text: semi-continuous

text: but

text: as

text: the

text: deﬁnition

text: of

text: p-superharmonicity

text: re-

text: quires

text: lower

text: semi-continuity,

text: continuity

text: has

text: to

text: be

text: shown.

text: R

text: We

text: claim

text: that

text: W

text: δ

text: is

text: p-superharmonic

text: in

text: n

text: provided

text: the

text: series

text: converges

text: at

text: least

text: at

text: some

text: point.

text: Again

text: we

text: may

text: assume

text: that

text: the

text: convergence

text: is

text: at

text: the

text: origin.

text: That

text: is

text: a

text: y

text: α

text: Since

text: α

text: we

text: get

text: i

text: i

text: x

text: y

text: α

text: x

text: y

text: )α

text: i

text: i

text: x

text: α

text: y

text: α

text: so

text: since

text: a

text: x

text: y

text: α

text: x

text: α

text: a

text: a

text: y

text: α

text: i

text: i

text: i

text: i

text: i

text: R

text: we

text: see

text: that

text: W

text: δ

text: W

text: δ

text: locally

text: uniformly

text: in

text: n.

text: We

text: infer

text: that

text: W

text: δ

text: is

text: RN

text: in

text: R

text: part

text: assume

text: D

text: n

text: and

text: h

text: C(D)

text: is

text: p-harmonic

text: in

text: D

text: with

text: h

text: W

text: δ

text: Since

text: δ

text: W

text: and

text: W

text: δ

text: is

text: p-superharmonic

text: we

text: ∂≤D

text: ∂D

text: N

text: get

text: h

text: W

text: δ

text: for

text: all

text: N.

text: So

text: given

text: any

text: D

text: N

text: D

text: h

text: W

text: δ

text: D

text: D

text: by

text: uniformity

text: on

text: the

text: bounded

text: set

text: D.

text: This

text: proves

text: the

text: claim.

text: Next,

text: let

text: δ

text: Then

text: W

text: (x)

text: ai

text: x

text: yi

text: K(x)

text: R

text: is

text: p-superharmonic

text: in

text: n

text: by

text: the

text: same

text: argument

text: as

text: when

text: p

text: n.

text: This

text: settles

text: the

text: case

text: p

text: n.

text: Epilogue:

text: Evolutionary

text: superposition.

text: The

text: superposition

text: of

text: fundamental

text: solutions

text: has

text: been

text: extended

text: to

text: p-

text: Laplace

text: equations

text: in

text: the

text: Heisenberg

text: group,

text: see

text: When

text: it

text: comes

text: to

text: further

text: extensions,

text: a

text: natural

text: question

text: is

text: whether

text: such

text: a

text: superposi-

text: tion

text: is

text: valid

text: for

text: the

text: evolutionary

text: p-Laplace

text: equation

text: u

text: u,

text: t

text: p

text: or

text: for

text: the

text: homogeneous

text: equation

text: u

text: p

text: u.

text: p

text: ∂t

text: The

text: following

text: shows

text: it

text: does

text: not.

text: ##LTLine##

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text: 5 light amplitude, we should have an ordinary insulator It is interesting that the low-energy Hamiltonians for phase in the regime of high frequency. In this strong the Weyl point pairs astride both k and k have the light regime, there can be a pair of Weyl points astride 0 1 same form. If we let k be equal to either 0 or √2 π, k when the frequency becomes enough low. The com- yc 1 3a the v term will disappear and the quadric term will plete phase diagram is presented in Fig. 4. We have 23 return, resuming a semi-Dirac point again. six phases: semi-metal (SM), Weyl semi-metal with one pair of Weyl fermions astride k (WSM1-A), Weyl semi- 0 metal with one pair of Weyl fermions astride k (WSM1- 1 B), Weyl semi-metal with two pair of the Weyl fermions (WSM2), ordinary insulator (I), and Weyl semi-metal (WSM). There are Dirac points in the former four phases, but there exists no Dirac points in the I and WSM phases. 4 3.5 SM I 3 2.5 WSM1-A 2 WSM1-B 1.5 WSM2 WSM 1 0.5 ± FIG. 3: The band structures of the m = 0 and m = 1 00 1 2 3 4 5 sub-bands in the k = 0 plane, with parameters A0a z η = γ/7, A0a = 1.5, w = 2γ, d = 2a, and θ = 4π/9. FIG. 4: The phase diagram with experimentally Two pairs of Weyl points are symmetrical astride k0 achievable31 parameters: d = 2a, η = γ/7, and and k1, respectively. η = 4π/9. There are six phases: semi-metal (SM), Weyl semi-metal with one pair of Weyl fermions astride k 0 We can calculate the Weyl number using the deﬁnition (WSM1-A), Weyl semi-metal with one pair of Weyl fermions astride k (WSM1-B), Weyl semi-metal with 1 two pair of the Weyl fermions (WSM2), ordinary C =sgn[det[v]] √ insulator (I), and Weyl semi-metal (WSM). The six √ =sgn[4 3a2γJ2(12A0√a) sin (kyc 23a)∗ (16) triangles shsotwrutchtuer(eωs/sγh,oAw0na)invaFluige.s2f.or the band 1 3 − (J ( A a) cos (k a) J (A a))] 2 2 0 yc 2 2 0 It can be proved that the two Weyl points possess op- D. Surface states and ﬁnite multilayers posite Weyl number or chirality. The Weyl points created ± by the band crossing of the m = 1 sub-bands, with their One of the most striking phenomena in a WSM phase Weyl numbers labeled, are shown in Fig. 3. is its Fermi arc. We can address it in a simple way. In our case, HW(k , k , k ) with a given k can be considered as eﬀ x y z y a 2D (k ,k ) Hamiltonian and one can calculate its k - x z y C. Phase diagram dependent Chern number C . It is well known that for ky a 2D Hamiltonian one can change its Chern number by When the frequency decreases, the distance between changing its parameters. In our case, the Chern number the two Weyl points in a pair become more separated will change by and k can approach k but cannot exceed it. Thus, in yc y0 this frequency regime, the system will stay in a WSM − phase, with one pair of Weyl points or two if one or ∆C = C C (cid:48) (17) ky ky both of the semi-Dirac points (responding to k and k ) 0 1 were created. On the other hand, for the very strong γ/w ##LTFigure## ##LTFigure## ##LTFigure## ##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##

text: light

text: amplitude,

text: we

text: should

text: have

text: an

text: ordinary

text: insulator

text: It

text: is

text: interesting

text: that

text: the

text: low-energy

text: Hamiltonians

text: for

text: phase

text: in

text: the

text: regime

text: of

text: high

text: frequency.

text: In

text: this

text: strong

text: the

text: Weyl

text: point

text: pairs

text: astride

text: both

text: k

text: and

text: k

text: have

text: the

text: light

text: regime,

text: there

text: can

text: be

text: a

text: pair

text: of

text: Weyl

text: points

text: astride

text: same

text: form.

text: If

text: we

text: let

text: k

text: be

text: equal

text: to

text: either

text: or

text: √2

text: π,

text: k

text: when

text: the

text: frequency

text: becomes

text: enough

text: low.

text: The

text: com-

text: yc

text: the

text: v

text: term

text: will

text: disappear

text: and

text: the

text: quadric

text: term

text: will

text: plete

text: phase

text: diagram

text: is

text: presented

text: in

text: Fig.

text: We

text: have

text: return,

text: resuming

text: a

text: semi-Dirac

text: point

text: again.

text: six

text: phases:

text: semi-metal

text: (SM),

text: Weyl

text: semi-metal

text: with

text: one

text: pair

text: of

text: Weyl

text: fermions

text: astride

text: k

text: Weyl

text: semi-

text: metal

text: with

text: one

text: pair

text: of

text: Weyl

text: fermions

text: astride

text: k

text: B),

text: Weyl

text: semi-metal

text: with

text: two

text: pair

text: of

text: the

text: Weyl

text: fermions

text: ordinary

text: insulator

text: (I),

text: and

text: Weyl

text: semi-metal

text: (WSM).

text: There

text: are

text: Dirac

text: points

text: in

text: the

text: former

text: four

text: phases,

text: but

text: there

text: exists

text: no

text: Dirac

text: points

text: in

text: the

text: I

text: and

text: WSM

text: phases.

text: SM

text: I

text: WSM

text: FIG.

text: The

text: band

text: structures

text: of

text: the

text: m

text: and

text: m

text: sub-bands

text: in

text: the

text: k

text: plane,

text: with

text: parameters

text: z

text: η

text: w

text: d

text: and

text: θ

text: FIG.

text: The

text: phase

text: diagram

text: with

text: experimentally

text: Two

text: pairs

text: of

text: Weyl

text: points

text: are

text: symmetrical

text: astride

text: parameters:

text: d

text: η

text: and

text: and

text: respectively.

text: η

text: There

text: are

text: six

text: phases:

text: semi-metal

text: (SM),

text: Weyl

text: semi-metal

text: with

text: one

text: pair

text: of

text: Weyl

text: fermions

text: astride

text: k

text: We

text: can

text: calculate

text: the

text: Weyl

text: number

text: using

text: the

text: deﬁnition

text: Weyl

text: semi-metal

text: with

text: one

text: pair

text: of

text: Weyl

text: fermions

text: astride

text: k

text: Weyl

text: semi-metal

text: with

text: two

text: pair

text: of

text: the

text: Weyl

text: fermions

text: ordinary

text: C

text: =sgn[det[v]]

text: insulator

text: (I),

text: and

text: Weyl

text: semi-metal

text: (WSM).

text: The

text: six

text: sin

text: (kyc

text: triangles

text: the

text: band

text: (J

text: A

text: a)

text: cos

text: (k

text: a)

text: J

text: (A

text: a))]

text: yc

text: It

text: can

text: be

text: proved

text: that

text: the

text: two

text: Weyl

text: points

text: possess

text: op-

text: D.

text: Surface

text: states

text: and

text: ﬁnite

text: multilayers

text: posite

text: Weyl

text: number

text: or

text: chirality.

text: The

text: Weyl

text: points

text: created

text: by

text: the

text: band

text: crossing

text: of

text: the

text: m

text: sub-bands,

text: with

text: their

text: One

text: of

text: the

text: most

text: striking

text: phenomena

text: in

text: a

text: WSM

text: phase

text: Weyl

text: numbers

text: labeled,

text: are

text: shown

text: in

text: Fig.

text: is

text: its

text: Fermi

text: arc.

text: We

text: can

text: address

text: it

text: in

text: a

text: simple

text: way.

text: In

text: our

text: case,

text: HW(k

text: k

text: k

text: with

text: a

text: given

text: k

text: can

text: be

text: considered

text: as

text: eﬀ

text: x

text: y

text: z

text: y

text: a

text: (k

text: ,k

text: Hamiltonian

text: and

text: one

text: can

text: calculate

text: its

text: k

text: x

text: z

text: y

text: C.

text: Phase

text: diagram

text: dependent

text: Chern

text: number

text: C

text: It

text: is

text: well

text: known

text: that

text: for

text: ky

text: a

text: Hamiltonian

text: one

text: can

text: change

text: its

text: Chern

text: number

text: by

text: When

text: the

text: frequency

text: decreases,

text: the

text: distance

text: between

text: changing

text: its

text: parameters.

text: In

text: our

text: case,

text: the

text: Chern

text: number

text: the

text: two

text: Weyl

text: points

text: in

text: a

text: pair

text: become

text: more

text: separated

text: will

text: change

text: by

text: and

text: k

text: can

text: approach

text: k

text: but

text: cannot

text: exceed

text: it.

text: Thus,

text: in

text: yc

text: this

text: frequency

text: regime,

text: the

text: system

text: will

text: stay

text: in

text: a

text: WSM

text: phase,

text: with

text: one

text: pair

text: of

text: Weyl

text: points

text: or

text: two

text: if

text: one

text: or

text: ∆C

text: C

text: C

text: ky

text: ky

text: both

text: of

text: the

text: semi-Dirac

text: points

text: (responding

text: to

text: k

text: and

text: k

text: were

text: created.

text: On

text: the

text: other

text: hand,

text: for

text: the

text: very

text: strong

text: γ/w

text: ##LTFigure##

text: ##LTFigure##

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text: 6 when k crosses a Weyl point where the mass term frequency levels. y changes sign. It is easy to get C = 1 when k is between In addition, some self-assembled graphene-like lattices ky y the two Weyl points in each pair, and C = 0 otherwise. of CdSe nanostructures28,36,37 can be used to realize such ky Semi-Dirac points do not change the Chern number be- Floquet Weyl fermions. Because their hopping parame- cause the HSD near it has no mass term. We already ters are approximately two order of magnitude smaller eﬀ illustrate the Chern numbers in Fig. 2. Each nontrivial compared to graphene and their lattice constant is one 2D Hamiltonian have edge states located at its boundary order of magnitude larger, much lower frequency and and these edge states make the Fermi arc which connects weaker amplitude are needed to realize Floquet Weyl the projections of each pair of Weyl points in the surface fermions in good multilayer structures of such CdSe BZ. Therefore, as long as the surface is parallel to the nanostructures31. k axis, a Fermi arc will emerge in the surface Brillouin In conclusion, we have proposed an interesting method y zone. to create Floquet Weyl fermions with a two-component semi-Dirac parent in a circularly-polarised-light-irradiated It is clear that to experimentally realize the Flo- 3D stacked graphene system instead of a four-component quet Weyl fermions, some appropriate multilayer struc- Dirac parent. One or two semi-Dirac points can ap- tures are better than the inﬁnitely-stacked graphene pear at k = (0, 0, 0) or/and k = (0, √2 π, 0) in model. On the basis of the above surface construction, 0 1 3a we can construct ﬁnite stacked graphene models un- momentum space when the frequency w of light is on der circularly-polarized light. Such circularly-polarised- the order of magnitude of hopping parameters γ and light-irradiated ﬁnitely-stacked models can be realized η. Upon decreasing the light frequency, each semi- experimentally. They can host Floquet Weyl fermions Dirac point will split into two symmetrical Weyl points and Fermi arcs. with opposite chirality. Further decreasing the fre- quency will make the Weyl points move in the mo- IV. FURTHER DISCUSSION AND mentum space, and the Weyl points can approach to CONCLUDION the Dirac points when the frequency becomes very small. The frequency-amplitude phase diagram has For experimentally realizing Floquet Weyl fermions been worked out, indicating that the Floquet Weyl in such graphene-based models, one has to prepare the femions always appear in pair. Furthermore, it has been circularly-polarized light of suitable frequency and am- shown that there exist Fermi arcs in the surface Bril- ≈ plitude. Because the NN hopping constants (γ 2.8eV, louin zones in circularly-polarised-light-irradiated semi- ≈ a 1.4˚A) are very large, we usually need a high fre- inﬁnitely-stacked and ﬁnitely-multilayered graphene sys- ∼ quency (order of magnitude of 1016Hz) and a strong tems. These theoretical results can lead to a new plat- ∼ amplitude (order of magnitude of 108V/m). As form to create Weyl fermions in graphene-based and shown in the phase diagram, however, the Floquet Weyl similar systems. fermions can also be realized when the frequency is low ACKNOWLEDGMENTS or the amplitude is weak. It should be pointed out that in the regime of low frequency, other Floquet sub-bands, This work is supported by Nature Science Foundation such as those with ∆m = 4, 6, 8, will play roles in creating of China (Grant Nos. 11174359 and 11574366), by Chi- Floquet Weyl points at (cid:15) = 0, and therefore some com- nese Department of Science and Technology (Grant No. plex situations may appear. Fortunately, in the regime 2012CB932302), and by the Strategic Priority Research of weak ﬁeld amplitude, one can realize one or two pairs Program of the Chinese Academy of Sciences (Grant No. of Floquet Weyl fermions at experimentally achievable XDB07000000). ∗ Phys. Rev. Lett. 107, 186806 (2011). † bgliu@iphy.ac.cn 8 K.-Y. Yang, Y.-M. Lu, and Y. Ran, Quantum Hall eﬀects in a Weyl semimetal: Possible application in pyrochlore iridates, ‡ Phys. Rev. B 84, 075129 (2011). refercolor 9 Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Dirac semimetal and topological phase 1 H. Weyl, Electron and gravitation, Z. Phys. 56, 330 (1929). transitions in A Bi (A=Na,K,Rb), Phys. Rev. B 85, 195320 (2012). 3 2 H. B. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw anom1a0lyDa.nTd.WSoenylafnerdmBio.nZs.inSpaivcarky,stCahl,irPalhyasn.oLmeatlty. Ban1d3c0l,as3s8ic9al negative magnetoresistance of Weyl metals, Phys. Rev. B 88, (1983). 104412 (2013). 3 S. Murakami, Phase transition between the quantum spin Hall a11ndZi.nWsualantgo,rHph. aWseesnign, 3QD.:Wemue,rXge.nDceaio,faantdopZo.loFgaincga,l gTahprleees-sdimensional Dirac semimetal and quantum transport in Cd As , 3 2 phase, New J. Phys. 9, 356 (2007). Phys. Rev. B 88, 125427 (2013). 4 A. A. Burkov and L. Balents, Weyl semimetal in a topological1i2nsOu.laVtoarfemkualntidlaAye.r,VPishhywsa.nRaetvh., LDeitrta.c1f0e7rm, 1io2n7s20in5 (s2o0li1d1s):. from high Tc cuprates and graphene to topological insulators and 5 X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, ToWpeoylolgsiecmalimseemtailms,eAtanlnaun.dRFever.mCio-anrdcenssu.rfMacaettsetra.tPeshyins. t5h,e83 (2014). electronic structure of pyrochlore iridates, Phys. Rev. B 83, 2015310Z1. K(2.0L1i1u),. B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, 6 L. Balents, Viewpoint: Weyl electrons kiss, Physics 4, 36 (2011). and Y. L. Chen, Discovery of a three-dimensional topological Dirac semimetal Na Bi, Science 343, 864 (2014). 3 7 G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chern semim14etZa.l Kan.dLituh,eJq.uJainantigz,edB.anZohmoua,loZu.sJH. aWllaenﬀge,cYt.inZhHagnCg,r HS.eM, . Weng, D. Prabhakaran, S. K. Mo, H. Peng, P. Dudin, T. Kim, 2 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: when

text: k

text: crosses

text: a

text: Weyl

text: point

text: where

text: the

text: mass

text: term

text: frequency

text: levels.

text: y

text: changes

text: sign.

text: It

text: is

text: easy

text: to

text: get

text: C

text: when

text: k

text: is

text: between

text: In

text: addition,

text: some

text: self-assembled

text: graphene-like

text: lattices

text: ky

text: y

text: the

text: two

text: Weyl

text: points

text: in

text: each

text: pair,

text: and

text: C

text: otherwise.

text: of

text: CdSe

text: can

text: be

text: used

text: to

text: realize

text: such

text: ky

text: Semi-Dirac

text: points

text: do

text: not

text: change

text: the

text: Chern

text: number

text: be-

text: Floquet

text: Weyl

text: fermions.

text: Because

text: their

text: hopping

text: parame-

text: cause

text: the

text: HSD

text: near

text: it

text: has

text: no

text: mass

text: term.

text: We

text: already

text: ters

text: are

text: approximately

text: two

text: order

text: of

text: magnitude

text: smaller

text: eﬀ

text: illustrate

text: the

text: Chern

text: numbers

text: in

text: Fig.

text: Each

text: nontrivial

text: compared

text: to

text: graphene

text: and

text: their

text: lattice

text: constant

text: is

text: one

text: Hamiltonian

text: have

text: edge

text: states

text: located

text: at

text: its

text: boundary

text: order

text: of

text: magnitude

text: larger,

text: much

text: lower

text: frequency

text: and

text: and

text: these

text: edge

text: states

text: make

text: the

text: Fermi

text: arc

text: which

text: connects

text: weaker

text: amplitude

text: are

text: needed

text: to

text: realize

text: Floquet

text: Weyl

text: the

text: projections

text: of

text: each

text: pair

text: of

text: Weyl

text: points

text: in

text: the

text: surface

text: fermions

text: in

text: good

text: multilayer

text: structures

text: of

text: such

text: CdSe

text: BZ.

text: Therefore,

text: as

text: long

text: as

text: the

text: surface

text: is

text: parallel

text: to

text: the

text: k

text: axis,

text: a

text: Fermi

text: arc

text: will

text: emerge

text: in

text: the

text: surface

text: Brillouin

text: In

text: conclusion,

text: we

text: have

text: proposed

text: an

text: interesting

text: method

text: y

text: zone.

text: to

text: create

text: Floquet

text: Weyl

text: fermions

text: with

text: a

text: two-component

text: semi-Dirac

text: parent

text: in

text: a

text: circularly-polarised-light-irradiated

text: It

text: is

text: clear

text: that

text: to

text: experimentally

text: realize

text: the

text: Flo-

text: stacked

text: graphene

text: system

text: instead

text: of

text: a

text: four-component

text: quet

text: Weyl

text: fermions,

text: some

text: appropriate

text: multilayer

text: struc-

text: Dirac

text: parent.

text: One

text: or

text: two

text: semi-Dirac

text: points

text: can

text: ap-

text: tures

text: are

text: better

text: than

text: the

text: inﬁnitely-stacked

text: graphene

text: pear

text: at

text: k

text: or/and

text: k

text: √2

text: π,

text: in

text: model.

text: On

text: the

text: basis

text: of

text: the

text: above

text: surface

text: construction,

text: we

text: can

text: construct

text: ﬁnite

text: stacked

text: graphene

text: models

text: un-

text: momentum

text: space

text: when

text: the

text: frequency

text: w

text: of

text: light

text: is

text: on

text: der

text: circularly-polarized

text: light.

text: Such

text: circularly-polarised-

text: the

text: order

text: of

text: magnitude

text: of

text: hopping

text: parameters

text: γ

text: and

text: light-irradiated

text: ﬁnitely-stacked

text: models

text: can

text: be

text: realized

text: η.

text: Upon

text: decreasing

text: the

text: light

text: frequency,

text: each

text: semi-

text: experimentally.

text: They

text: can

text: host

text: Floquet

text: Weyl

text: fermions

text: Dirac

text: point

text: will

text: split

text: into

text: two

text: symmetrical

text: Weyl

text: points

text: and

text: Fermi

text: arcs.

text: with

text: opposite

text: chirality.

text: Further

text: decreasing

text: the

text: fre-

text: quency

text: will

text: make

text: the

text: Weyl

text: points

text: move

text: in

text: the

text: mo-

text: IV.

text: FURTHER

text: DISCUSSION

text: AND

text: mentum

text: space,

text: and

text: the

text: Weyl

text: points

text: can

text: approach

text: to

text: CONCLUDION

text: the

text: Dirac

text: points

text: when

text: the

text: frequency

text: becomes

text: very

text: small.

text: The

text: frequency-amplitude

text: phase

text: diagram

text: has

text: For

text: experimentally

text: realizing

text: Floquet

text: Weyl

text: fermions

text: been

text: worked

text: out,

text: indicating

text: that

text: the

text: Floquet

text: Weyl

text: in

text: such

text: graphene-based

text: models,

text: one

text: has

text: to

text: prepare

text: the

text: femions

text: always

text: appear

text: in

text: pair.

text: Furthermore,

text: it

text: has

text: been

text: circularly-polarized

text: light

text: of

text: suitable

text: frequency

text: and

text: am-

text: shown

text: that

text: there

text: exist

text: Fermi

text: arcs

text: in

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text: ACKNOWLEDGMENTS

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text: (Grant

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text: plex

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text: Fortunately,

text: in

text: the

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text: by

text: the

text: Strategic

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text: Research

text: of

text: weak

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text: amplitude,

text: one

text: can

text: realize

text: one

text: or

text: two

text: pairs

text: Program

text: of

text: the

text: Chinese

text: Academy

text: of

text: Sciences

text: (Grant

text: No.

text: of

text: Floquet

text: Weyl

text: fermions

text: at

text: experimentally

text: achievable

text: Phys.

text: Rev.

text: Lett.

text: bgliu@iphy.ac.cn

text: K.-Y.

text: Yang,

text: Y.-M.

text: Lu,

text: and

text: Y.

text: Ran,

text: Quantum

text: Hall

text: eﬀects

text: in

text: a

text: Weyl

text: semimetal:

text: Possible

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text: in

text: pyrochlore

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text: Phys.

text: Rev.

text: B

text: refercolor

text: Z.

text: Wang,

text: Y.

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text: Weng,

text: X.

text: Dai,

text: and

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text: Fang,

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text: Weyl,

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text: negative

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text: Phys.

text: Rev.

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text: Hall

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text: A.

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text: Turner,

text: A.

text: Vishwanath,

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text: Y.

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text: ejections

text: (CMEs)


text: THE FIBER DIMENSION OF A GRAPH TOBIAS WINDISCH Abstract. Graphs on integer points of polytopes whose edges come from a set of allowed diﬀerences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the ﬁber dimension of the given graph. The ﬁber dimension is determined for various classes of graphs and an upper bound in terms of the chromatic number is stated. 1. Introduction The study of geometric properties of graphs is a key ingredient in understanding their algorithmic behaviour and combinatorial structure [17, 15]. In [11], the dimension of a graph ∈ N R was introduced, which is the smallest n such that the graph can be embedded in n with every edge having unit length. Recently, isometrically embeddings of graphs into discrete objects like hypercubes or lattices were paid a lot of attention in the literature and lead to a lot of variations like the isometric dimension [12], the lattice dimension [10, 14], or the Fibonacci dimension [4] of a graph. In this paper, a new notion is added to the list of graph dimensions (Remark 2.6). Our concept has its origin in algebraic statistics, an emerging ﬁeld which explores statistical questions with algebraic tools [8, 9, 13, 1]. A main task there is to construct connected graphs on integer points of polytopes in order to draw samples by performing a random walk [16, ⊂ Q M ⊂ Z ∩ Z Chapter 5]. For a given polytope P d and a symmetric set d, a graph on P d − ∈ M is given by connecting two nodes u and v by an edge if u v . Graphs which can be obtained in that way are often referred to as ﬁber graphs in the literature and can be understood as a discrete analogue of unit distance graphs [3]. At ﬁrst glance, it seems that ﬁber graphs are distinguished graphs with their own rich structure. However, as it turns out, every graph can be represented as a ﬁber graph (Propo- sition 2.3). This motivates the question for the smallest dimension in which a graph G can be represented as a ﬁber graph, the ﬁber dimension of G (Deﬁnition 2.5). We explore general properties of this dimension and state upper bounds in terms of the chromatic number (Theorem 3.5) in the spirit of [11]. We then determine the ﬁber dimension for a variety of graphs. The ﬁber dimension of a cycle of length n depends on Euler’s totient function and ∈ N \ { } we show that fdim(C ) = 1 if and only if n 3, 4, 6 . Cycles whose length is one of the ∈n { } exceptional cases n 3, 4, 6 have ﬁber dimension 2 (Proposition 4.4). Its proof uses the ∈ N ∈ { } well-known fact that Euler’s totient function of n is 2 if and only if n 3, 4, 6 . We also determine the ﬁber dimension of complete graphs and show that it is logarithmic in the Date: May 28, 2020. 2010 Mathematics Subject Classiﬁcation. Primary: 05C62, 52B20 Secondary: 05C70. Key words and phrases. Graph embedding, graph dimension, ﬁber graphs, lattice polytopes. 1 ##LTLine##

text: THE

text: FIBER

text: DIMENSION

text: OF

text: A

text: GRAPH

text: TOBIAS

text: WINDISCH

text: Abstract.

text: Graphs

text: on

text: integer

text: points

text: of

text: polytopes

text: whose

text: edges

text: come

text: from

text: a

text: set

text: of

text: allowed

text: diﬀerences

text: are

text: studied.

text: It

text: is

text: shown

text: that

text: any

text: simple

text: graph

text: can

text: be

text: embedded

text: in

text: that

text: way.

text: The

text: minimal

text: dimension

text: of

text: such

text: a

text: representation

text: is

text: the

text: ﬁber

text: dimension

text: of

text: the

text: given

text: graph.

text: The

text: ﬁber

text: dimension

text: is

text: determined

text: for

text: various

text: classes

text: of

text: graphs

text: and

text: an

text: upper

text: bound

text: in

text: terms

text: of

text: the

text: chromatic

text: number

text: is

text: stated.

text: Introduction

text: The

text: study

text: of

text: geometric

text: properties

text: of

text: graphs

text: is

text: a

text: key

text: ingredient

text: in

text: understanding

text: their

text: algorithmic

text: behaviour

text: and

text: combinatorial

text: structure

text: In

text: the

text: dimension

text: of

text: a

text: graph

text: N

text: R

text: was

text: introduced,

text: which

text: is

text: the

text: smallest

text: n

text: such

text: that

text: the

text: graph

text: can

text: be

text: embedded

text: in

text: n

text: with

text: every

text: edge

text: having

text: unit

text: length.

text: Recently,

text: isometrically

text: embeddings

text: of

text: graphs

text: into

text: discrete

text: objects

text: like

text: hypercubes

text: or

text: lattices

text: were

text: paid

text: a

text: lot

text: of

text: attention

text: in

text: the

text: literature

text: and

text: lead

text: to

text: a

text: lot

text: of

text: variations

text: like

text: the

text: isometric

text: dimension

text: the

text: lattice

text: dimension

text: or

text: the

text: Fibonacci

text: dimension

text: of

text: a

text: graph.

text: In

text: this

text: paper,

text: a

text: new

text: notion

text: is

text: added

text: to

text: the

text: list

text: of

text: graph

text: dimensions

text: (Remark

text: Our

text: concept

text: has

text: its

text: origin

text: in

text: algebraic

text: statistics,

text: an

text: emerging

text: ﬁeld

text: which

text: explores

text: statistical

text: questions

text: with

text: algebraic

text: tools

text: A

text: main

text: task

text: there

text: is

text: to

text: construct

text: connected

text: graphs

text: on

text: integer

text: points

text: of

text: polytopes

text: in

text: order

text: to

text: draw

text: samples

text: by

text: performing

text: a

text: random

text: walk

text: Q

text: M

text: Z

text: Z

text: Chapter

text: For

text: a

text: given

text: polytope

text: P

text: d

text: and

text: a

text: symmetric

text: set

text: d,

text: a

text: graph

text: on

text: P

text: d

text: M

text: is

text: given

text: by

text: connecting

text: two

text: nodes

text: u

text: and

text: v

text: by

text: an

text: edge

text: if

text: u

text: v

text: Graphs

text: which

text: can

text: be

text: obtained

text: in

text: that

text: way

text: are

text: often

text: referred

text: to

text: as

text: ﬁber

text: graphs

text: in

text: the

text: literature

text: and

text: can

text: be

text: understood

text: as

text: a

text: discrete

text: analogue

text: of

text: unit

text: distance

text: graphs

text: At

text: ﬁrst

text: glance,

text: it

text: seems

text: that

text: ﬁber

text: graphs

text: are

text: distinguished

text: graphs

text: with

text: their

text: own

text: rich

text: structure.

text: However,

text: as

text: it

text: turns

text: out,

text: every

text: graph

text: can

text: be

text: represented

text: as

text: a

text: ﬁber

text: graph

text: (Propo-

text: sition

text: This

text: motivates

text: the

text: question

text: for

text: the

text: smallest

text: dimension

text: in

text: which

text: a

text: graph

text: G

text: can

text: be

text: represented

text: as

text: a

text: ﬁber

text: graph,

text: the

text: ﬁber

text: dimension

text: of

text: G

text: (Deﬁnition

text: We

text: explore

text: general

text: properties

text: of

text: this

text: dimension

text: and

text: state

text: upper

text: bounds

text: in

text: terms

text: of

text: the

text: chromatic

text: number

text: (Theorem

text: in

text: the

text: spirit

text: of

text: We

text: then

text: determine

text: the

text: ﬁber

text: dimension

text: for

text: a

text: variety

text: of

text: graphs.

text: The

text: ﬁber

text: dimension

text: of

text: a

text: cycle

text: of

text: length

text: n

text: depends

text: on

text: Euler’s

text: totient

text: function

text: and

text: N

text: we

text: show

text: that

text: fdim(C

text: if

text: and

text: only

text: if

text: n

text: Cycles

text: whose

text: length

text: is

text: one

text: of

text: the

text: ∈n

text: exceptional

text: cases

text: n

text: have

text: ﬁber

text: dimension

text: (Proposition

text: Its

text: proof

text: uses

text: the

text: N

text: well-known

text: fact

text: that

text: Euler’s

text: totient

text: function

text: of

text: n

text: is

text: if

text: and

text: only

text: if

text: n

text: We

text: also

text: determine

text: the

text: ﬁber

text: dimension

text: of

text: complete

text: graphs

text: and

text: show

text: that

text: it

text: is

text: logarithmic

text: in

text: the

text: Date:

text: May

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Primary:

text: Secondary:

text: Key

text: words

text: and

text: phrases.

text: Graph

text: embedding,

text: graph

text: dimension,

text: ﬁber

text: graphs,

text: lattice

text: polytopes.

text: ##LTLine##


text: (cid:80) THE FIBER DIMENSION OF A GRAPH 3 { } { ∈ Q } Proof. Let G = ( v , . . . , v , E) be graph and let P := x n≥ : n x = 1 be the − 1 n ∩Z { } M 0 { −i=1 i{ } ∈ } (n 1)-dimensional simplex, then P n = e , . . . , e . Consider := e e : v , v E , M 1 n i j i j (cid:3) then P ( ) is isomorphic to G. The restriction on graphs without loops is necessary, since, in a ﬁber graph, either every node has a loop, or none. The next lemma states that every ﬁber graph can be written as a ﬁber graph in a full dimensional polytope. ⊂ Q M ⊂ Z Lemma 2.4. Let P m be a d-dimensional polytope and m a set of m∼oves. There (cid:48) ⊂ Q M(cid:48) ⊂ Z M (cid:48) M(cid:48) exists a d-dimensional polytope P d and moves d such that P ( ) = P ( ). Proof. We can assume that d < m. Translation of P does not change the graph structure M ⊂ Q of P ( ) and thus we can assume that P m≥ . Since P is a rational polytope and since ∈ Z × 0 ≥ ∈ Z d < m, there exists a matrix A n m with dim k(cid:18)erZ(A) =(cid:19)0 and n m and b n such { ∈ Q ≤ } that P = x m≥ : Ax b . Consider the injective and aﬃne map 0 Q → Q (cid:55)→ x φ : m m+n, x − . b Ax { ∈ Q } Then φ(P ) = (x, y)T m≥+n : Ax + y = b and dim(φ(P )) = dim(P ) = d. The set N { ∈ M}0 N M := (m, Am)T : m is a set of moves and the graphs φ(P )( ) and P ( ) are isomorphic. Hence, it suﬃces to show the statement for d-dimensional polytopes of the { ∈ Q } ∈ Z ∈ Z × form P = x k≥ : Bx = b for some b n, a matrix B n k, and a set of moves M ⊂ 0 ≥ − ≥ kerZ(B) with the property that k n, rank(B) = n, and dim(kerZ B) = k n d. − We can add rows to B without changing P such that dim(kerZ B) = k n = d. First, we · transform B into its Hermite normal form, that is, we write B = (H, 0) C for an unimodular ∈ Z × ∈ Z × − ∈ Q × − ∈ Q × matrix C k k and a matrix H n n of full rank. Let H 1 n n and C 1 k k be − ∈ Z × the inverse matrices of H and C respectively. Since (cid:18)C is un(cid:19)imodular, C 1 k k and thus ∈ Z × ∈ Z × − let C k n and C k d such that C 1 = (C , C ) and consider the aﬃne map 1 2 1 2 − Z → Z (cid:55)→ − H 1b ψ : d k, x C 1 . x (cid:48) { ∈ Q · ≤ − } ⊂ Q Clearly, ψ is injective and the image of the polytope P := v d : C v C H 1b d ∩Z (cid:54) ∅ − ∈ Z 2 1 is P . Since P k = , H 1b n (see [7, Theorem 2.3.6]) and since C is unimodular, integer (cid:48) (cid:48) (cid:48) points of P get mapped to integer points of P . Thus dim(P ) = dim(P ) = d. That is, P is Q full dimensional in d. Consider M(cid:48) { − − − ∈ ∩ Z − ∈ M} := ψ 1(v) ψ 1(u) : v, u P k, v u , M(cid:48) −M(cid:48) M(cid:48) (cid:48) (cid:48) then = and cannot contain multiples. Let ψ(v ) = v and ψ(u ) = u for (cid:48) (cid:48) ∈ (cid:48) ∩ Z (cid:48) − (cid:48) ∈ M(cid:48) − ∈ M (cid:48) M(cid:48) v , u P d, then v u if and only if v u . Thus, all edges in P ( ) are M (cid:3) mapped bijective to edges in P ( ) under ψ, which proves that these graphs are isomorphic.

text: THE

text: FIBER

text: DIMENSION

text: OF

text: A

text: GRAPH

text: Q

text: Proof.

text: Let

text: G

text: v

text: v

text: E)

text: be

text: graph

text: and

text: let

text: P

text: x

text: n≥

text: n

text: x

text: be

text: the

text: n

text: ∩Z

text: M

text: i{

text: (n

text: simplex,

text: then

text: P

text: n

text: e

text: e

text: Consider

text: e

text: e

text: v

text: v

text: E

text: M

text: n

text: i

text: j

text: i

text: j

text: then

text: P

text: is

text: isomorphic

text: to

text: G.

text: The

text: restriction

text: on

text: graphs

text: without

text: loops

text: is

text: necessary,

text: since,

text: in

text: a

text: ﬁber

text: graph,

text: either

text: every

text: node

text: has

text: a

text: loop,

text: or

text: none.

text: The

text: next

text: lemma

text: states

text: that

text: every

text: ﬁber

text: graph

text: can

text: be

text: written

text: as

text: a

text: ﬁber

text: graph

text: in

text: a

text: full

text: dimensional

text: polytope.

text: Q

text: M

text: Z

text: Lemma

text: Let

text: P

text: m

text: be

text: a

text: d-dimensional

text: polytope

text: and

text: m

text: a

text: set

text: of

text: m∼oves.

text: There

text: Q

text: Z

text: M

text: exists

text: a

text: d-dimensional

text: polytope

text: P

text: d

text: and

text: moves

text: d

text: such

text: that

text: P

text: P

text: Proof.

text: We

text: can

text: assume

text: that

text: d

text: m.

text: Translation

text: of

text: P

text: does

text: not

text: change

text: the

text: graph

text: structure

text: M

text: Q

text: of

text: P

text: and

text: thus

text: we

text: can

text: assume

text: that

text: P

text: m≥

text: Since

text: P

text: is

text: a

text: rational

text: polytope

text: and

text: since

text: Z

text: Z

text: d

text: m,

text: there

text: exists

text: a

text: matrix

text: A

text: n

text: m

text: with

text: dim

text: and

text: n

text: m

text: and

text: b

text: n

text: such

text: Q

text: that

text: P

text: x

text: m≥

text: Ax

text: b

text: Consider

text: the

text: injective

text: and

text: aﬃne

text: map

text: Q

text: Q

text: x

text: φ

text: m

text: m+n,

text: x

text: b

text: Ax

text: Q

text: Then

text: φ(P

text: (x,

text: y)T

text: m≥+n

text: Ax

text: y

text: b

text: and

text: dim(φ(P

text: dim(P

text: d.

text: The

text: set

text: N

text: N

text: M

text: (m,

text: Am)T

text: m

text: is

text: a

text: set

text: of

text: moves

text: and

text: the

text: graphs

text: φ(P

text: and

text: P

text: are

text: isomorphic.

text: Hence,

text: it

text: suﬃces

text: to

text: show

text: the

text: statement

text: for

text: d-dimensional

text: polytopes

text: of

text: the

text: Q

text: Z

text: Z

text: form

text: P

text: x

text: k≥

text: Bx

text: b

text: for

text: some

text: b

text: n,

text: a

text: matrix

text: B

text: n

text: k,

text: and

text: a

text: set

text: of

text: moves

text: M

text: kerZ(B)

text: with

text: the

text: property

text: that

text: k

text: n,

text: rank(B)

text: n,

text: and

text: dim(kerZ

text: B)

text: k

text: n

text: d.

text: We

text: can

text: add

text: rows

text: to

text: B

text: without

text: changing

text: P

text: such

text: that

text: dim(kerZ

text: B)

text: k

text: n

text: d.

text: First,

text: we

text: transform

text: B

text: into

text: its

text: Hermite

text: normal

text: form,

text: that

text: is,

text: we

text: write

text: B

text: (H,

text: C

text: for

text: an

text: unimodular

text: Z

text: Z

text: Q

text: Q

text: matrix

text: C

text: k

text: k

text: and

text: a

text: matrix

text: H

text: n

text: n

text: of

text: full

text: rank.

text: Let

text: H

text: n

text: n

text: and

text: C

text: k

text: k

text: be

text: Z

text: the

text: inverse

text: matrices

text: of

text: H

text: and

text: C

text: respectively.

text: Since

text: is

text: C

text: k

text: k

text: and

text: thus

text: Z

text: Z

text: let

text: C

text: k

text: n

text: and

text: C

text: k

text: d

text: such

text: that

text: C

text: (C

text: C

text: and

text: consider

text: the

text: aﬃne

text: map

text: Z

text: Z

text: H

text: ψ

text: d

text: k,

text: x

text: C

text: x

text: Q

text: Q

text: Clearly,

text: ψ

text: is

text: injective

text: and

text: the

text: image

text: of

text: the

text: polytope

text: P

text: v

text: d

text: C

text: v

text: C

text: H

text: d

text: ∩Z

text: Z

text: is

text: P

text: Since

text: P

text: k

text: H

text: n

text: (see

text: Theorem

text: and

text: since

text: C

text: is

text: unimodular,

text: integer

text: points

text: of

text: P

text: get

text: mapped

text: to

text: integer

text: points

text: of

text: P

text: Thus

text: dim(P

text: dim(P

text: d.

text: That

text: is,

text: P

text: is

text: Q

text: full

text: dimensional

text: in

text: d.

text: Consider

text: Z

text: M}

text: ψ

text: ψ

text: v,

text: u

text: P

text: k,

text: v

text: u

text: then

text: and

text: cannot

text: contain

text: multiples.

text: Let

text: ψ(v

text: v

text: and

text: ψ(u

text: u

text: for

text: Z

text: M

text: v

text: u

text: P

text: d,

text: then

text: v

text: u

text: if

text: and

text: only

text: if

text: v

text: u

text: Thus,

text: all

text: edges

text: in

text: P

text: are

text: M

text: mapped

text: bijective

text: to

text: edges

text: in

text: P

text: under

text: ψ,

text: which

text: proves

text: that

text: these

text: graphs

text: are

text: isomorphic.