text: − A nucleon resonance with spin-parity JP = 3/2 and mass about 2.1 GeV is essential to reproduce the photoproduction cross sections for Λ(1520) released by the LEPS and CLAS Collaborations. It can be explained − as the third nucleon resonance state [3/2 ]3 in the constitu−ent quark model so that there is no position to settle the N(1875) which is listed in the PDG as the third N3/2 nucleon resonance. An interpretation is proposed that the N(1875) is from the interaction of a decuplet baryon Σ(1385) and a octet meson K, which is favored by a calculation of binding energy and decay pattern in a Bethe-Salpeter approach. PACS numbers: 14.20.Gk,11.10.St, 14.20.Pt − I. INTRODUCTION be assigned to the third state [N3/2 ]3 in the constituent quark model in line with the theoretical framework in Ref. [15]. − In new versions of the Review of Particle physics (PDG) There are ﬁve N3/2 states in N = 3 band, of which the − radiative and Λ(1520)K decay amplitudes were predicted in after the year 2012 [1], there are four N3/2 states, N(1520), Refs. [4, 5] as listed in Table. I. N(1700), N(1875) and N(2120). The two-star state N(2080) in previous versions has been split into a three-star N(1875) and a two-star N(2120) based on the evidence from BnGa − TABLE I: The N3/2 nucleon resonances and their decay amplitudes analysis [2]. predicted in the relativistic quark model [4, 5]. The mass m , helicity R wiUthsuoarblliytatlhaenNgu(1la5r20m)oamndenthtuemN(L17=001) ainreqausasrigknmedodtoels,taatneds aumniptloitfuMdeesVA, 11/02,−3/32/a√nGdepVaratinadl w√aMveedVe,craeyspaemcptilviteulyd.es G((cid:96)) are in the mofixthinegseeﬀsteactetsis[3v]e.ryTihmepsoirtutaantitotno feoxrptlhaieninthteerndaelcasytrupcattutererns State− mR A1p/2 A3p/2 G−((cid:96)1) G−((cid:96)2) of two N3/2− states with higher mass, the N(1875) and the [N3/2−]3 1960 36 -43 −2.6 0.2 N(2120), is much less unclear. In quark model, the N(1875) [N3/2 ]4 2055 16 0 0.5 0.0 − and the N(2120) are in the mass region of N = 3 band states [N3/2−]5 2095 −-−9 -−1−4 0.4 0.0 of which the masses and decay patterns were predicted [4, 5]. [N3/2−]6 2165 −− −− 0.4 0.0 However, the explicit correspondence between predicted and [N3/2 ]7 2180 1.1 0.1 observed states is unclear. In Large N QCD, the third and − ± c ± fourth N3/2 states have masses 2101 14 and 2170 42 MeV, The predicted radiative and strong decay amplitudes sug- respectively [6]. Klempt a−nd others claimed that the N−(1875) gest the importance of the nucleon resonance [N3/2−]3 in the is the missing third N(3/2 ) state in mass region 1800 1900 Λ(1520) photoproduction, which is the ﬁrst state in N = 3 MeV with orbit angular momentum L = 1 and radial exci- band states and the third state in all nucleon resonances with tation number N = 1 [7], which is also supported by the JP = 3/2− predicted in the constituent quark model. Ads/QCD [8]. Their conclusion is only based on a compari- In Ref. [15], based on the high precision experimental data son between predicted and observed masses. As enlightened released by the CLAS and LEPS Collaborations recently, the by Isgur, “in a complex system like the baryon resonances, interaction mechanism of the photoproduction of Λ(1520) oﬀ predicting the spectrum of states is not a very stringent test of a proton target is investigated within a Regge-plus-resonance a model” [9]. Decay pattern provides more information about − approach. The inclusion of the N(2120) as state [N3/2 ]3 in hadron internal structure. − the constituent quark model reduced the χ2 obviously. In that Many analyses suggested that a N3/2 state with mass work, mass and width are ﬁxed at 2.12 GeV and 0.33 GeV, about 2.1 GeV is essential to explain experimental results [10– respectively. Here, a mass scan is made for the N(2120) by − 13]. Before the year 2012, it is related to the only N3/2 ﬁtting the data from the CLAS and LEPS Collaborations. Ex- state listed in the PDG with mass higher than −1.8 GeV, the cept mass, the width ΓR, which was ﬁxed at 0.33 GeV in pre- N(2080), and explained as the third state [N3/2 ]3 predicted vious work [15], is also set as a free parameter. The behavior in the constituent quark model. For example, the N(2080) is of χ2 is presented in Fig. 1. found to play the most important role in the photoproduction Here the results with assuming the N(2120) as state oCfolΛla(b1o5r2a0ti)oonﬀatprJoetﬀoenrstoarngeNta[t1io2n,a1l3A].ccReelecreantotlry,FtahceiliCtyLAreS- [vNid3e/d2.−I]f3thanedNw(2it1h2a0s)siusmasinsugmtheedNto(2b1e2t0h)eatshi[rNd3s/ta2t−e][4Na3re/2p−ro]3- leased their exclusive photoproduction cross section for the in the constituent quark model, the change of χ2 will decrease Λ(1520) for energies from near threshold up to a center of and reach minimum at 2.13 GeV with the increase of mass. If mass energy W of 2.85 GeV with large range of the K pro- assumed to be the fourth state [N3/2−]4, the change of χ2 keep duction angle [14]. The reanalyses about the new data in stable around 150, which means that the experimental data Refs. [15, 19] conﬁrmed the previous conclusion that a nu- can not be well reproduced. Obviously, the N(2120) should cleon resonance near 2.1 GeV, N(2120), is essential to repro- − − be assigned as state [N3/2 ]3 instead of state [N3/2 ]4 in the duce the experimental data [12, 13]. constituent quark model. Since the N(1875) is much lower II. ROLE OF THE N(2120) IN THE Λ(1520) than the N(2120), it is unnatural to assign it to the fourth or PHOTOPRODUCTION higher states. The ﬁrst and second states in the constituent quark model have been assigned to the four-star N(1520) and In the following it will be shown why the N(2120) should the three-star N(1700) in the PDG, which has been conﬁrmed ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: VVAYZG+txsy

text: A

text: nucleon

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text: and

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text: photoproduction

text: cross

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text: the

text: LEPS

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text: It

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text: An

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text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: be

text: assigned

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text: the

text: third

text: state

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text: the

text: constituent

text: quark

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text: line

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text: the

text: theoretical

text: framework

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text: mass

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text: helicity

text: R

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text: w√aMveedVe,craeyspaemcptilviteulyd.es

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text: correspondence

text: between

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text: is

text: unclear.

text: In

text: Large

text: N

text: QCD,

text: the

text: third

text: and

text: c

text: fourth

text: states

text: have

text: masses

text: and

text: MeV,

text: The

text: predicted

text: radiative

text: and

text: strong

text: decay

text: amplitudes

text: sug-

text: respectively

text: Klempt

text: a−nd

text: others

text: claimed

text: that

text: the

text: gest

text: the

text: importance

text: of

text: the

text: nucleon

text: resonance

text: in

text: the

text: is

text: the

text: missing

text: third

text: state

text: in

text: mass

text: region

text: photoproduction,

text: which

text: is

text: the

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text: in

text: N

text: MeV

text: with

text: orbit

text: angular

text: momentum

text: L

text: and

text: radial

text: exci-

text: band

text: states

text: and

text: the

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text: state

text: in

text: all

text: nucleon

text: resonances

text: with

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text: number

text: N

text: which

text: is

text: also

text: supported

text: by

text: the

text: JP

text: predicted

text: in

text: the

text: constituent

text: quark

text: model.

text: Ads/QCD

text: Their

text: conclusion

text: is

text: only

text: based

text: on

text: a

text: compari-

text: In

text: Ref.

text: based

text: on

text: the

text: high

text: precision

text: experimental

text: data

text: son

text: between

text: predicted

text: and

text: observed

text: masses.

text: As

text: enlightened

text: released

text: by

text: the

text: CLAS

text: and

text: LEPS

text: Collaborations

text: recently,

text: the

text: by

text: Isgur,

text: “in

text: a

text: complex

text: system

text: like

text: the

text: baryon

text: resonances,

text: interaction

text: mechanism

text: of

text: the

text: photoproduction

text: of

text: oﬀ

text: predicting

text: the

text: spectrum

text: of

text: states

text: is

text: not

text: a

text: very

text: stringent

text: test

text: of

text: a

text: proton

text: target

text: is

text: investigated

text: within

text: a

text: Regge-plus-resonance

text: a

text: model”

text: Decay

text: pattern

text: provides

text: more

text: information

text: about

text: approach.

text: The

text: inclusion

text: of

text: the

text: as

text: state

text: in

text: hadron

text: internal

text: structure.

text: the

text: constituent

text: quark

text: model

text: reduced

text: the

text: χ2

text: obviously.

text: In

text: that

text: Many

text: analyses

text: suggested

text: that

text: a

text: state

text: with

text: mass

text: work,

text: mass

text: and

text: width

text: are

text: ﬁxed

text: at

text: GeV

text: and

text: GeV,

text: about

text: GeV

text: is

text: essential

text: to

text: explain

text: experimental

text: results

text: respectively.

text: Here,

text: a

text: mass

text: scan

text: is

text: made

text: for

text: the

text: by

text: Before

text: the

text: year

text: it

text: is

text: related

text: to

text: the

text: only

text: ﬁtting

text: the

text: data

text: from

text: the

text: CLAS

text: and

text: LEPS

text: Collaborations.

text: Ex-

text: state

text: listed

text: in

text: the

text: PDG

text: with

text: mass

text: higher

text: than

text: GeV,

text: the

text: cept

text: mass,

text: the

text: width

text: ΓR,

text: which

text: was

text: ﬁxed

text: at

text: GeV

text: in

text: pre-

text: and

text: explained

text: as

text: the

text: third

text: state

text: predicted

text: vious

text: work

text: is

text: also

text: set

text: as

text: a

text: free

text: parameter.

text: The

text: behavior

text: in

text: the

text: constituent

text: quark

text: model.

text: For

text: example,

text: the

text: is

text: of

text: χ2

text: is

text: presented

text: in

text: Fig.

text: found

text: to

text: play

text: the

text: most

text: important

text: role

text: in

text: the

text: photoproduction

text: Here

text: the

text: results

text: with

text: assuming

text: the

text: as

text: state

text: leased

text: their

text: exclusive

text: photoproduction

text: cross

text: section

text: for

text: the

text: in

text: the

text: constituent

text: quark

text: model,

text: the

text: change

text: of

text: χ2

text: will

text: decrease

text: for

text: energies

text: from

text: near

text: threshold

text: up

text: to

text: a

text: center

text: of

text: and

text: reach

text: minimum

text: at

text: GeV

text: with

text: the

text: increase

text: of

text: mass.

text: If

text: mass

text: energy

text: W

text: of

text: GeV

text: with

text: large

text: range

text: of

text: the

text: K

text: pro-

text: assumed

text: to

text: be

text: the

text: fourth

text: state

text: the

text: change

text: of

text: χ2

text: keep

text: duction

text: angle

text: The

text: reanalyses

text: about

text: the

text: new

text: data

text: in

text: stable

text: around

text: which

text: means

text: that

text: the

text: experimental

text: data

text: Refs.

text: conﬁrmed

text: the

text: previous

text: conclusion

text: that

text: a

text: nu-

text: can

text: not

text: be

text: well

text: reproduced.

text: Obviously,

text: the

text: should

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text: resonance

text: near

text: GeV,

text: is

text: essential

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text: repro-

text: be

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text: experimental

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text: the

text: is

text: much

text: lower

text: II.

text: ROLE

text: OF

text: THE

text: IN

text: THE

text: than

text: the

text: it

text: is

text: unnatural

text: to

text: assign

text: it

text: to

text: the

text: fourth

text: or

text: PHOTOPRODUCTION

text: higher

text: states.

text: The

text: ﬁrst

text: and

text: second

text: states

text: in

text: the

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text: ##LTLine##

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text: 6 AMEYA PITALE, ABHISHEK SAHA, AND RALF SCHMIDT ◦ Remark 3.3. It follows from Proposition 3.2 that A(Γ) is an admissible (g, K)- n-ﬁn module. Knowing Proposition 3.2, it is now easy to derive the following Structure Theorem for cuspidal nearly holomorphic modular forms. Theorem 3.4 (Structure theorem for cusp forms). Fix non-negative integers k, p and (cid:77) Q a congruence subgroup Γ of SL ( ). There is an orthogonal direct sum decomposition 2 ◦ − Np(Γ) = R(k (cid:96))/2 (S (Γ)) . (9) k (cid:96) ≥ ≡ (cid:96) 1 (cid:96)−k m≤od≤2 k 2p (cid:96) k ◦ ◦ In particular, Np(Γ) = 0 and Np(Γ) = S (Γ). 0 1 1 Proof. Let f ∈ Np(Γ). Deﬁne a function Φ on SL (R) by Φ(g) = (f| g)(i). Then ∈ ◦ k 2 k Φ A (Γ) by Lemma 3.1. If f has weight 0, then f = 0, since the weight 0 does k n-ﬁn(cid:80) ◦ ≥ not occur in A (Γ) by Proposition 3.2. Assume in the following that k 1 and k n-ﬁn (cid:80) that f is non-zero. ∈ ◦ Write Φ = Φ , where each Φ A (Γ) generates an irreducible (g, K)-module ∼ i ≥ i k n-ﬁn V = D − with (cid:96) 1; this is possible by Proposition 3.2. Evidently, f = Φ˜ , wihere Φ˜(cid:96)i is1,+the functiion on H corresponding to Φ via (7). i i ∈ ◦ 1 i Since f Np(Γ) , it follows from (2) of Lemma 1.1 that Lp+1f = 0, thus Lp+1Φ = 0 k − − by Lemma 2.1, and then also Lp+1Φ = 0 for all i. The weight of Lp+1Φ being k 2p 2, i − i ≥ − it follows that V only contains weights greater or equal to k 2p. Hence (cid:96) k 2p i i for all i. ∈ ≤ ≡ Let Φ V be a lowest weight vector; thus Φ has weight (cid:96) k, and (cid:96) k mod i,0 i i,0 i i 2. The corresponding function Φ˜ on H is an element of S (Γ). Since every weight i,0− 1 (cid:96)i occurs only once in Vi, we have R(k (cid:96)i)/2Φi,0 = ciΦi for some non-zero constant ci. By Lemma 2.1, it follows that − (cid:88) (cid:88) R(k (cid:96)i)/2Φ˜ = c Φ˜(cid:88), i,0 i i and hence − − ∈ − f = Φ˜ = c 1R(k (cid:96)i)/2Φ˜ R(k (cid:96))/2 (S (Γ)) . i i i,0 (cid:96) ≥ ≡ (cid:96) 1 (cid:96)−k m≤od≤2 k 2p (cid:96) k This proves that the left hand side of (9) is contained in the right hand side. The orthogonality of the right hand side follows from the above construction and the fact that the isotypical components in Proposition 3.2 are orthogonal; observe that the map (cid:55)→ Φ Φ˜ is isometric with respect to the L2-scalar product on the left hand side and the (cid:3) Petersson inner product on the right hand side. Remark 3.5. It is well known, or follows from an easy calculation, that Ω acts on D − − − (cid:96) 1,+ by the scalar 1(cid:96)(1(cid:96) 1). Hence, by Lemma 2.1, Ω acts on the subspace R(k (cid:96))/2 (S (Γ)) ◦ 2 2 − ◦ (cid:96) of Np(Γ) by 1(cid:96)(1(cid:96) 1)). In particular, Ω acts diagonalizably on Np(Γ) , and the k 2 2 k pieces in the decomposition (9) can be intrinsically characterized as the eigenspaces with respect to Ω. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: AMEYA

text: PITALE,

text: ABHISHEK

text: SAHA,

text: AND

text: RALF

text: SCHMIDT

text: Remark

text: It

text: follows

text: from

text: Proposition

text: that

text: A(Γ)

text: is

text: an

text: admissible

text: (g,

text: K)-

text: n-ﬁn

text: module.

text: Knowing

text: Proposition

text: it

text: is

text: now

text: easy

text: to

text: derive

text: the

text: following

text: Structure

text: Theorem

text: for

text: cuspidal

text: nearly

text: holomorphic

text: modular

text: forms.

text: Theorem

text: (Structure

text: theorem

text: for

text: cusp

text: forms).

text: Fix

text: non-negative

text: integers

text: k,

text: p

text: and

label: equation

text: Q

text: a

text: congruence

text: subgroup

text: Γ

text: of

text: SL

text: There

text: is

text: an

text: orthogonal

text: direct

text: sum

text: decomposition

text: Np(Γ)

text: R(k

text: (S

text: (Γ))

text: k

text: k

text: k

text: In

text: particular,

text: Np(Γ)

text: and

text: Np(Γ)

text: S

text: (Γ).

text: Proof.

text: Let

text: f

text: Np(Γ).

text: Deﬁne

text: a

text: function

text: Φ

text: on

text: SL

text: (R)

text: by

text: Φ(g)

text: (f|

text: g)(i).

text: Then

text: k

text: k

text: Φ

text: A

text: (Γ)

text: by

text: Lemma

text: If

text: f

text: has

text: weight

text: then

text: f

text: since

text: the

text: weight

text: does

text: k

text: not

text: occur

text: in

text: A

text: (Γ)

text: by

text: Proposition

text: Assume

text: in

text: the

text: following

text: that

text: k

text: and

text: k

text: n-ﬁn

text: that

text: f

text: is

text: non-zero.

text: Write

text: Φ

text: Φ

text: where

text: each

text: Φ

text: A

text: (Γ)

text: generates

text: an

text: irreducible

text: (g,

text: K)-module

text: i

text: i

text: k

text: n-ﬁn

text: V

text: D

text: with

text: this

text: is

text: possible

text: by

text: Proposition

text: Evidently,

text: f

text: Φ˜

text: wihere

text: functiion

text: on

text: H

text: corresponding

text: to

text: Φ

text: via

text: i

text: i

text: i

text: Since

text: f

text: Np(Γ)

text: it

text: follows

text: from

text: of

text: Lemma

text: that

text: thus

text: k

text: by

text: Lemma

text: and

text: then

text: also

text: for

text: all

text: i.

text: The

text: weight

text: of

text: being

text: k

text: i

text: i

text: it

text: follows

text: that

text: V

text: only

text: contains

text: weights

text: greater

text: or

text: equal

text: to

text: k

text: Hence

text: k

text: i

text: i

text: for

text: all

text: i.

text: Let

text: Φ

text: V

text: be

text: a

text: lowest

text: weight

text: vector;

text: thus

text: Φ

text: has

text: weight

text: k,

text: and

text: k

text: mod

text: i

text: i

text: i

text: The

text: corresponding

text: function

text: Φ˜

text: on

text: H

text: is

text: an

text: element

text: of

text: S

text: (Γ).

text: Since

text: every

text: weight

text: occurs

text: only

text: once

text: in

text: Vi,

text: we

text: have

text: R(k

text: ciΦi

text: for

text: some

text: non-zero

text: constant

text: ci.

text: By

text: Lemma

text: it

text: follows

text: that

text: R(k

text: c

text: i

text: i

text: and

text: hence

text: f

text: Φ˜

text: c

text: R(k

text: (S

text: (Γ))

text: i

text: i

text: k

text: k

text: This

text: proves

text: that

text: the

text: left

text: hand

text: side

text: of

text: is

text: contained

text: in

text: the

text: right

text: hand

text: side.

text: The

text: orthogonality

text: of

text: the

text: right

text: hand

text: side

text: follows

text: from

text: the

text: above

text: construction

text: and

text: the

text: fact

text: that

text: the

text: isotypical

text: components

text: in

text: Proposition

text: are

text: orthogonal;

text: observe

text: that

text: the

text: map

text: Φ

text: Φ˜

text: is

text: isometric

text: with

text: respect

text: to

text: the

text: product

text: on

text: the

text: left

text: hand

text: side

text: and

text: the

text: Petersson

text: inner

text: product

text: on

text: the

text: right

text: hand

text: side.

text: Remark

text: It

text: is

text: well

text: known,

text: or

text: follows

text: from

text: an

text: easy

text: calculation,

text: that

text: Ω

text: acts

text: on

text: D

text: by

text: the

text: scalar

text: Hence,

text: by

text: Lemma

text: Ω

text: acts

text: on

text: the

text: subspace

text: R(k

text: (S

text: (Γ))

text: of

text: Np(Γ)

text: by

text: In

text: particular,

text: Ω

text: acts

text: diagonalizably

text: on

text: Np(Γ)

text: and

text: the

text: k

text: k

text: pieces

text: in

text: the

text: decomposition

text: can

text: be

text: intrinsically

text: characterized

text: as

text: the

text: eigenspaces

text: with

text: respect

text: to

text: Ω.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: REPRESENTATIONS AND NEARLY HOLOMORPHIC FORMS 7 ∈ ◦ For f, g N (Γ) with at least one of them(cid:90)in N (Γ) , we deﬁne the Petersson inner (cid:104) (cid:105) k k product f, g by the equation (cid:104) (cid:105) \H − dxdy f, g = vol(Γ ) 1 f(τ)g(τ) . 1 y2 \H Γ 1 It can be easily checked that (cid:104) (cid:105) (cid:104) (cid:105) f, g = Φ , Φ , (10) f g | where Φ (h) = (f h)(i) (and Φ is deﬁned sim(cid:90)ilarly) and the inner product of Φ f k g f and Φ is deﬁned by g (cid:104) (cid:105) 1 Φf, Φg = vol(SL (Z)\SL (R)) Φf(h)Φg(h) dh. 2 2 Z \ R SL2( ) SL2( ) ◦ We deﬁne the subspace E (Γ) to be the orthogonal complement of N (Γ) in N (Γ). ∩ k k k Let Ep(Γ) = E (Γ) Np(Γ). We write E (Γ) to mean E0(Γ). In Corollary 4.4 below k k k k k ◦ we will prove that Np(Γ) is the orthogonal direct sum of Np(Γ) and Ep(Γ). k k k ∈ ∈ Lemma 3.6. Let k be a non-negative integer. Let f E (Γ), and let Φ A(Γ) R k ◦f n-ﬁn be the corresponding function on SL ( ). Then Φ is orthogonal to A(Γ) . 2 f n-ﬁn ∈ ◦ (cid:104) (cid:105) Proof. Let Ψ A(Γ) ; we have to show that Φ , Ψ = 0. We may assume that Ψ n-ﬁn ≥f generates an irreducible module D − for some (cid:96) 1. Since Φ has weight k, we may (cid:96) 1,+ f ◦ assume that Ψ does as well. But then Ψ corresponds to an element g of N (Γ) . By (cid:104) (cid:105) (cid:104) (cid:105) k (cid:3) hypothesis f, g = 0. Hence Φ , Ψ = 0 by (10). f ≥ ≥ ◦ Lemma 3.7. Let k 1 and v 0 be integers. Then Rv takes S (Γ) to Nv (Γ) and k k+2v E (Γ) to Ev (Γ). k k+2v ◦ Proof. The fact that Rv takes S (Γ) to Nv (Γ) is an immediate consequence of k k+2v | the fact that the diﬀerential operator R commutes with the operator and does not k increase the support of the Fourier coeﬃcients. ∈ Let us show that Rv takes E (Γ) to Ev (Γ). Let f E (Γ). In view of (10), it k k+2v ∈ k ◦ suﬃces to show that Rv(Φ ) and Φ are orthogonal for all g Nv (Γ) . But note that f g k+2v U(gC)Φ and U(gC)Φ are orthogonal submodules of A(Γ) (as U(gC)Φ is completely f ◦ g g ◦ contained in A(Γ) and U(gC)Φ is contained in the orthogonal complement of A(Γ) f (cid:3) by Lemma 3.6). Hence Rv(Φ ) and Φ are orthogonal. f g ∈ ≥ Lemma 3.8. Let f S (Γ). Then for all v 0, there exists a constant c (depending k k,v only on k, v) such that (cid:104) (cid:105) (cid:104) (cid:105) Rv(f), Rv(f) = c f, f . k,v ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: REPRESENTATIONS

text: AND

text: NEARLY

text: HOLOMORPHIC

text: FORMS

text: For

text: f,

text: g

text: N

text: (Γ)

text: with

text: at

text: least

text: one

text: of

text: N

text: (Γ)

text: we

text: deﬁne

text: the

text: Petersson

text: inner

text: k

text: k

text: product

text: f,

text: g

text: by

text: the

text: equation

text: \H

text: dxdy

text: f,

text: g

text: vol(Γ

text: f(τ)g(τ)

text: \H

text: Γ

text: It

text: can

text: be

text: easily

text: checked

text: that

text: f,

text: g

text: Φ

text: Φ

text: f

text: g

text: where

text: Φ

text: (h)

text: (f

text: h)(i)

text: (and

text: Φ

text: is

text: deﬁned

text: and

text: the

text: inner

text: product

text: of

text: Φ

text: f

text: k

text: g

text: f

text: and

text: Φ

text: is

text: deﬁned

text: by

text: g

text: Φf,

text: Φg

text: vol(SL

text: (Z)\SL

text: (R))

text: Φf(h)Φg(h)

text: dh.

text: Z

text: R

text: We

text: deﬁne

text: the

text: subspace

text: E

text: (Γ)

text: to

text: be

text: the

text: orthogonal

text: complement

text: of

text: N

text: (Γ)

text: in

text: N

text: (Γ).

text: k

text: k

text: k

text: Let

text: Ep(Γ)

text: E

text: (Γ)

text: Np(Γ).

text: We

text: write

text: E

text: (Γ)

text: to

text: mean

text: In

text: Corollary

text: below

text: k

text: k

text: k

text: k

text: k

text: we

text: will

text: prove

text: that

text: Np(Γ)

text: is

text: the

text: orthogonal

text: direct

text: sum

text: of

text: Np(Γ)

text: and

text: Ep(Γ).

text: k

text: k

text: k

text: Lemma

text: Let

text: k

text: be

text: a

text: non-negative

text: integer.

text: Let

text: f

text: E

text: (Γ),

text: and

text: let

text: Φ

text: A(Γ)

text: R

text: k

text: ◦f

text: n-ﬁn

text: be

text: the

text: corresponding

text: function

text: on

text: SL

text: Then

text: Φ

text: is

text: orthogonal

text: to

text: A(Γ)

text: f

text: n-ﬁn

text: Proof.

text: Let

text: Ψ

text: A(Γ)

text: we

text: have

text: to

text: show

text: that

text: Φ

text: Ψ

text: We

text: may

text: assume

text: that

text: Ψ

text: n-ﬁn

text: ≥f

text: generates

text: an

text: irreducible

text: module

text: D

text: for

text: some

text: Since

text: Φ

text: has

text: weight

text: k,

text: we

text: may

text: f

text: assume

text: that

text: Ψ

text: does

text: as

text: well.

text: But

text: then

text: Ψ

text: corresponds

text: to

text: an

text: element

text: g

text: of

text: N

text: (Γ)

text: By

text: k

text: hypothesis

text: f,

text: g

text: Hence

text: Φ

text: Ψ

text: by

text: f

text: Lemma

text: Let

text: k

text: and

text: v

text: be

text: integers.

text: Then

text: Rv

text: takes

text: S

text: (Γ)

text: to

text: Nv

text: (Γ)

text: and

text: k

text: E

text: (Γ)

text: to

text: Ev

text: (Γ).

text: k

text: Proof.

text: The

text: fact

text: that

text: Rv

text: takes

text: S

text: (Γ)

text: to

text: Nv

text: (Γ)

text: is

text: an

text: immediate

text: consequence

text: of

text: k

text: the

text: fact

text: that

text: the

text: diﬀerential

text: operator

text: R

text: commutes

text: with

text: the

text: operator

text: and

text: does

text: not

text: k

text: increase

text: the

text: support

text: of

text: the

text: Fourier

text: coeﬃcients.

text: Let

text: us

text: show

text: that

text: Rv

text: takes

text: E

text: (Γ)

text: to

text: Ev

text: (Γ).

text: Let

text: f

text: E

text: (Γ).

text: In

text: view

text: of

text: it

text: k

text: k

text: suﬃces

text: to

text: show

text: that

text: Rv(Φ

text: and

text: Φ

text: are

text: orthogonal

text: for

text: all

text: g

text: Nv

text: (Γ)

text: But

text: note

text: that

text: f

text: g

text: U(gC)Φ

text: and

text: U(gC)Φ

text: are

text: orthogonal

text: submodules

text: of

text: A(Γ)

text: (as

text: U(gC)Φ

text: is

text: completely

text: f

text: g

text: g

text: contained

text: in

text: A(Γ)

text: and

text: U(gC)Φ

text: is

text: contained

text: in

text: the

text: orthogonal

text: complement

text: of

text: A(Γ)

text: f

text: by

text: Lemma

text: Hence

text: Rv(Φ

text: and

text: Φ

text: are

text: orthogonal.

text: f

text: g

text: Lemma

text: Let

text: f

text: S

text: (Γ).

text: Then

text: for

text: all

text: v

text: there

text: exists

text: a

text: constant

text: c

text: (depending

text: k

text: k,v

text: only

text: on

text: k,

text: v)

text: such

text: that

text: Rv(f),

text: Rv(f)

text: c

text: f,

text: f

text: k,v

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 INTRODUCTION Free nonrelativistic electrons in a magnetic ﬁeld behave as a system of harmonic oscillators, with selection rules allowing only the transitions between neighboring states with equal probabilities. Therefore they cannot be used as an active medium for lasers and masers. One way to get around this limitation is to accelerate electrons to high enough speeds that the relativistic eﬀects become important. This leads to an anharmonicity in the electron spectrum and possibility of the maser action by accelerated electron beams, which has been so impressively implemented in vacuum electronic devices such as gyrotrons [1]. Free carriers in semiconductors seem to oﬀer a similar opportunity as the electron dispersion can show signiﬁcant nonparabolicity above the bottom of the conduction band. Moreover, semiconductors oﬀer a ﬂexibility to grow heterostructures with diﬀerent cyclotron transition energies which could be used for carrier injection into a given LL; see the proposal for a LL laser in the quantum Hall regime [2]. In practice, however, an ultrafast energy and momentum relaxation in semiconductors would quickly destroy population inversion between the Landau levels (LLs). As a result, there are no viable ”solid state gyrotrons”, although Landau level quantization does help with reducing scattering rate and improving performance of quantum cascade lasers [3] that operate through population inversion between quantum well subbands. Graphene seems to be an ideal material for the realization of LL lasers. Low energy excitations near the Dirac points in graphene have a linear conical spectrum which is obviously extremely nonparabolic. In a transverse magnetic ﬁeld the 2D conical spectrum splits into a series of non-equidistant LLs with energies scaling as a square root of the magnetic ﬁeld and the principal quantum number. It was suggested in [4] that the optical pumping to an arbitrary ≥ excited state n 1 will lead to electrons cascading down the LLs preferentially emitting photons, which would potentially lead to the EM ﬁeld ampliﬁcation on any of these downward transitions. Unfortunately, the proposal [4] assumed that the radiative transitions are the fastest ones in graphene. It did not include most important nonradiative relaxation channels and did not attempt to calculate actual LL populations. In particular, it turns out that the Auger relaxation is a very powerful relaxation mechanism for Dirac electrons in a magnetized graphene that proceeds much faster than radiative transitions and washes out any population inversion over the time scale of few ps; see below and also recent theoretical calculations of the Auger relaxation rate [5] and experimental measurements in [6]. A recently proposed, more sophisticated pumping scheme [7] takes into account Auger relaxation processes and still leads to only a transient population inversion existing over a ps timescale. Here we propose what we believe is a viable inter-LL laser scheme for graphene that takes into account all relevant relaxation processes and in fact utilizes them to reach a steady-state population inversion, vital for any viable laser. (cid:29) ≥ Our scheme is transferable to thin (λ ∆z 5 nm) ﬁlms of 3D topological insulators such as Bi Se where the 2 3 Landau levels associated with massless metallic surface states [8, 9] should demonstrate similar coupling to the EM ﬁeld despite diﬀerent chirality [10]. Not that our scheme provides the population inversion in a steady state, i.e. under a continuous-wave pumping, in contrast to previous proposals, with or without the magnetic ﬁeld, that could provide only a transient gain during a picosecond time interval [4, 7, 11]. We solve kinetic and density matrix equations coupled with Maxwell’s equations to calculate populations, gain and laser threshold conditions as a function of the optical pumping power. The calculation details are in the sections below. Here we present a general idea of the laser scheme. It is illustrated in Fig. 1. It shows one speciﬁc implementation of − the scheme with an optical pumping originated from level n = 2 to obtain maximum population inversion between − − − − levels -1 and -2. However, the scheme can me implemented for any pair of LLs ( n, n 1) as long as level n stays deep enough below the Fermi level. The lasing wavelength can be from sub-THz to the mid-infrared range, depending on the value of n, the magnetic ﬁeld, and the substrate used. (cid:112) − − The general idea is as follows. In the equilibrium (without pumping) labels n = 1 and 2 are fully occupied to degeneracy surface density N = g g /2πl 2, where spin and valley degeneracy factors are g = 2 and g = 2 s s v c s v for graphene, and l = c¯h/eB is the magnetic length. The Fermi level is placed at the Dirac point in the ﬁgure, c − assuming intrinsic graphene. However, this can be changed, as long as level n = 1 is fully occupied in equilibrium. − → − An optical pumping resonant to the transition 2 1 moves part of the carriers up from level n = 2 creating a − − population inversion between a fully occupied level n = 1 and level 2. In order for this population inversion to exist in a steady state, i.e. under a continuous-wave optical pumping, the relaxation of carriers back to lower laser − − state 2 should be slower than the relaxation rate to the upper laser state 1. Unfortunately, the Auger mechanism does not satisfy this criterion. Our simulations show that there is no population inversion in the steady state, no − matter how strong the optical pumping power is. This is because an increase in the depopulation rate of level 2 by an optical pumping is compensated by an increase in the Auger scattering rate to level 2, primarily through the − − scattering of electrons from states in levels 1 and 1 to states in levels 2 and 2, respectively. In order to overcome → → − this obstacle, the magnetic ﬁeld needs to be tuned in order to bring the transitions 1 0 and 0 1 (of the same ##LTLine##

text: INTRODUCTION

label: section

text: Free

text: nonrelativistic

text: electrons

text: in

text: a

text: magnetic

text: ﬁeld

text: behave

text: as

text: a

text: system

text: of

text: harmonic

text: oscillators,

text: with

text: selection

text: rules

text: allowing

text: only

text: the

text: transitions

text: between

text: neighboring

text: states

text: with

text: equal

text: probabilities.

text: Therefore

text: they

text: cannot

text: be

text: used

text: as

text: an

text: active

text: medium

text: for

text: lasers

text: and

text: masers.

text: One

text: way

text: to

text: get

text: around

text: this

text: limitation

text: is

text: to

text: accelerate

text: electrons

text: to

text: high

text: enough

text: speeds

text: that

text: the

text: relativistic

text: eﬀects

text: become

text: important.

text: This

text: leads

text: to

text: an

text: anharmonicity

text: in

text: the

text: electron

text: spectrum

text: and

text: possibility

text: of

text: the

text: maser

text: action

text: by

text: accelerated

text: electron

text: beams,

text: which

text: has

text: been

text: so

text: impressively

text: implemented

text: in

text: vacuum

text: electronic

text: devices

text: such

text: as

text: gyrotrons

text: Free

text: carriers

text: in

text: semiconductors

text: seem

text: to

text: oﬀer

text: a

text: similar

text: opportunity

text: as

text: the

text: electron

text: dispersion

text: can

text: show

text: signiﬁcant

text: nonparabolicity

text: above

text: the

text: bottom

text: of

text: the

text: conduction

text: band.

text: Moreover,

text: semiconductors

text: oﬀer

text: a

text: ﬂexibility

text: to

text: grow

text: heterostructures

text: with

text: diﬀerent

text: cyclotron

text: transition

text: energies

text: which

text: could

text: be

text: used

text: for

text: carrier

text: injection

text: into

text: a

text: given

text: LL;

text: see

text: the

text: proposal

text: for

text: a

text: LL

text: laser

text: in

text: the

text: quantum

text: Hall

text: regime

text: In

text: practice,

text: however,

text: an

text: ultrafast

text: energy

text: and

text: momentum

text: relaxation

text: in

text: semiconductors

text: would

text: quickly

text: destroy

text: population

text: inversion

text: between

text: the

text: Landau

text: levels

text: (LLs).

text: As

text: a

text: result,

text: there

text: are

text: no

text: viable

text: ”solid

text: state

text: gyrotrons”,

text: although

text: Landau

text: level

text: quantization

text: does

text: help

text: with

text: reducing

text: scattering

text: rate

text: and

text: improving

text: performance

text: of

text: quantum

text: cascade

text: lasers

text: that

text: operate

text: through

text: population

text: inversion

text: between

text: quantum

text: well

text: subbands.

text: Graphene

text: seems

text: to

text: be

text: an

text: ideal

text: material

text: for

text: the

text: realization

text: of

text: LL

text: lasers.

text: Low

text: energy

text: excitations

text: near

text: the

text: Dirac

text: points

text: in

text: graphene

text: have

text: a

text: linear

text: conical

text: spectrum

text: which

text: is

text: obviously

text: extremely

text: nonparabolic.

text: In

text: a

text: transverse

text: magnetic

text: ﬁeld

text: the

text: conical

text: spectrum

text: splits

text: into

text: a

text: series

text: of

text: non-equidistant

text: LLs

text: with

text: energies

text: scaling

text: as

text: a

text: square

text: root

text: of

text: the

text: magnetic

text: ﬁeld

text: and

text: the

text: principal

text: quantum

text: number.

text: It

text: was

text: suggested

text: in

text: that

text: the

text: optical

text: pumping

text: to

text: an

text: arbitrary

text: excited

text: state

text: n

text: will

text: lead

text: to

text: electrons

text: cascading

text: down

text: the

text: LLs

text: preferentially

text: emitting

text: photons,

text: which

text: would

text: potentially

text: lead

text: to

text: the

text: EM

text: ﬁeld

text: ampliﬁcation

text: on

text: any

text: of

text: these

text: downward

text: transitions.

text: Unfortunately,

text: the

text: proposal

text: assumed

text: that

text: the

text: radiative

text: transitions

text: are

text: the

text: fastest

text: ones

text: in

text: graphene.

text: It

text: did

text: not

text: include

text: most

text: important

text: nonradiative

text: relaxation

text: channels

text: and

text: did

text: not

text: attempt

text: to

text: calculate

text: actual

text: LL

text: populations.

text: In

text: particular,

text: it

text: turns

text: out

text: that

text: the

text: Auger

text: relaxation

text: is

text: a

text: very

text: powerful

text: relaxation

text: mechanism

text: for

text: Dirac

text: electrons

text: in

text: a

text: magnetized

text: graphene

text: that

text: proceeds

text: much

text: faster

text: than

text: radiative

text: transitions

text: and

text: washes

text: out

text: any

text: population

text: inversion

text: over

text: the

text: time

text: scale

text: of

text: few

text: ps;

text: see

text: below

text: and

text: also

text: recent

text: theoretical

text: calculations

text: of

text: the

text: Auger

text: relaxation

text: rate

text: and

text: experimental

text: measurements

text: in

text: A

text: recently

text: proposed,

text: more

text: sophisticated

text: pumping

text: scheme

text: takes

text: into

text: account

text: Auger

text: relaxation

text: processes

text: and

text: still

text: leads

text: to

text: only

text: a

text: transient

text: population

text: inversion

text: existing

text: over

text: a

text: ps

text: timescale.

text: Here

text: we

text: propose

text: what

text: we

text: believe

text: is

text: a

text: viable

text: inter-LL

text: laser

text: scheme

text: for

text: graphene

text: that

text: takes

text: into

text: account

text: all

text: relevant

text: relaxation

text: processes

text: and

text: in

text: fact

text: utilizes

text: them

text: to

text: reach

text: a

text: steady-state

text: population

text: inversion,

text: vital

text: for

text: any

text: viable

text: laser.

text: Our

text: scheme

text: is

text: transferable

text: to

text: thin

text: (λ

text: ∆z

text: nm)

text: ﬁlms

text: of

text: topological

text: insulators

text: such

text: as

text: Bi

text: Se

text: where

text: the

text: Landau

text: levels

text: associated

text: with

text: massless

text: metallic

text: surface

text: states

text: should

text: demonstrate

text: similar

text: coupling

text: to

text: the

text: EM

text: ﬁeld

text: despite

text: diﬀerent

text: chirality

text: Not

text: that

text: our

text: scheme

text: provides

text: the

text: population

text: inversion

text: in

text: a

text: steady

text: state,

text: i.e.

text: under

text: a

text: continuous-wave

text: pumping,

text: in

text: contrast

text: to

text: previous

text: proposals,

text: with

text: or

text: without

text: the

text: magnetic

text: ﬁeld,

text: that

text: could

text: provide

text: only

text: a

text: transient

text: gain

text: during

text: a

text: picosecond

text: time

text: interval

text: We

text: solve

text: kinetic

text: and

text: density

text: matrix

text: equations

text: coupled

text: with

text: Maxwell’s

text: equations

text: to

text: calculate

text: populations,

text: gain

text: and

text: laser

text: threshold

text: conditions

text: as

text: a

text: function

text: of

text: the

text: optical

text: pumping

text: power.

text: The

text: calculation

text: details

text: are

text: in

text: the

text: sections

text: below.

text: Here

text: we

text: present

text: a

text: general

text: idea

text: of

text: the

text: laser

text: scheme.

text: It

text: is

text: illustrated

text: in

text: Fig.

text: It

text: shows

text: one

text: speciﬁc

text: implementation

text: of

text: the

text: scheme

text: with

text: an

text: optical

text: pumping

text: originated

text: from

text: level

text: n

text: to

text: obtain

text: maximum

text: population

text: inversion

text: between

text: levels

text: and

text: However,

text: the

text: scheme

text: can

text: me

text: implemented

text: for

text: any

text: pair

text: of

text: LLs

text: n,

text: n

text: as

text: long

text: as

text: level

text: n

text: stays

text: deep

text: enough

text: below

text: the

text: Fermi

text: level.

text: The

text: lasing

text: wavelength

text: can

text: be

text: from

text: sub-THz

text: to

text: the

text: mid-infrared

text: range,

text: depending

text: on

text: the

text: value

text: of

text: n,

text: the

text: magnetic

text: ﬁeld,

text: and

text: the

text: substrate

text: used.

text: The

text: general

text: idea

text: is

text: as

text: follows.

text: In

text: the

text: equilibrium

text: (without

text: pumping)

text: labels

text: n

text: and

text: are

text: fully

text: occupied

text: to

text: degeneracy

text: surface

text: density

text: N

text: g

text: g

text: where

text: spin

text: and

text: valley

text: degeneracy

text: factors

text: are

text: g

text: and

text: g

text: s

text: s

text: v

text: c

text: s

text: v

text: for

text: graphene,

text: and

text: l

text: c¯h/eB

text: is

text: the

text: magnetic

text: length.

text: The

text: Fermi

text: level

text: is

text: placed

text: at

text: the

text: Dirac

text: point

text: in

text: the

text: ﬁgure,

text: c

text: assuming

text: intrinsic

text: graphene.

text: However,

text: this

text: can

text: be

text: changed,

text: as

text: long

text: as

text: level

text: n

text: is

text: fully

text: occupied

text: in

text: equilibrium.

text: An

text: optical

text: pumping

text: resonant

text: to

text: the

text: transition

text: moves

text: part

text: of

text: the

text: carriers

text: up

text: from

text: level

text: n

text: creating

text: a

text: population

text: inversion

text: between

text: a

text: fully

text: occupied

text: level

text: n

text: and

text: level

text: In

text: order

text: for

text: this

text: population

text: inversion

text: to

text: exist

text: in

text: a

text: steady

text: state,

text: i.e.

text: under

text: a

text: continuous-wave

text: optical

text: pumping,

text: the

text: relaxation

text: of

text: carriers

text: back

text: to

text: lower

text: laser

text: state

text: should

text: be

text: slower

text: than

text: the

text: relaxation

text: rate

text: to

text: the

text: upper

text: laser

text: state

text: Unfortunately,

text: the

text: Auger

text: mechanism

text: does

text: not

text: satisfy

text: this

text: criterion.

text: Our

text: simulations

text: show

text: that

text: there

text: is

text: no

text: population

text: inversion

text: in

text: the

text: steady

text: state,

text: no

text: matter

text: how

text: strong

text: the

text: optical

text: pumping

text: power

text: is.

text: This

text: is

text: because

text: an

text: increase

text: in

text: the

text: depopulation

text: rate

text: of

text: level

text: by

text: an

text: optical

text: pumping

text: is

text: compensated

text: by

text: an

text: increase

text: in

text: the

text: Auger

text: scattering

text: rate

text: to

text: level

text: primarily

text: through

text: the

text: scattering

text: of

text: electrons

text: from

text: states

text: in

text: levels

text: and

text: to

text: states

text: in

text: levels

text: and

text: respectively.

text: In

text: order

text: to

text: overcome

text: this

text: obstacle,

text: the

text: magnetic

text: ﬁeld

text: needs

text: to

text: be

text: tuned

text: in

text: order

text: to

text: bring

text: the

text: transitions

text: and

text: (of

text: the

text: same

text: ##LTLine##


text: 2− dna 1− = n neewteb niag 12 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 1 2 3 4 5 pumping intensity (W/cm2) x 105 FIG. 5. Dependence of the gain between n = -1 and -2 LLs per graphene monolayer on the pumping intensity. One can also see in Fig. 4 that the population inversion exists also between states n = 2 and 1, albeit at a two times lower level. This seems unexpected, given that the optical pumping brings carriers only to state 1. However, a closer look at the rate equations shows that the population inversion between levels 1 and 2 is a consequence of a strongly non-equilibrium carrier distribution below the Fermi level created by the optical pumping, namely the population − − − inversion between states 2 and 1. Indeed, when f− > f− , the Auger scattering rate from states (1, 1) to states − 1 2 (2, 2) is greater than the scattering rate in the opposite direction. This creates the population inversion f > f and 2 1 the gain for the ˆl⊕ polarization, which is about two times smaller than the ˆl(cid:9) gain. In conclusion, we show the feasibility of the Landau level THz laser in a magnetized graphene. Despite ultrafast Auger relaxation, steady-state operation of the laser under continuous wave optical pumping is possible by utilizing surface or interface phonon relaxation. The scheme is scalable to thin ﬁlms of 3D topological insulators such as Bi Se 2 3 or Bi Te . 2 3 ACKNOWLEDGMENTS This work has been supported by NSF Grants OISE-0968405 and EEC-0540832, and by the AFOSR grant FA9550- 14-1-0376. M. D. Tokman acknowledges support by the Russian Foundation for Basic Research Grants No. 13-02- 00376 and No. 14-22-02034. ∗ † belyanin@tamu.edu [1] G. Nusinovich, Introduction to the Physics of Gyrotrons (Johns Hopkins University Press, 2004). [2] H. Aoki, Applied Physics Letters 48, 559 (1986). [3] A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. Williams, Q. Hu, and J. Reno, Nature Photonics 3, 41 (2008). [4] T. Morimoto, Y. Hatsugai, and H. Aoki, Phys. Rev. B 78, 073406 (2008). [5] F. Wendler, A. Knorr, and E. Malic, Nature communications 5, 3703 (2014). [6] M. Mittendorﬀ, F. Wendler, E. Malic, A. Knorr, M. Orlita, M. Potemski, C. Berger, W. A. de Heer, H. Schneider, M. Helm, et al., Nature Physics 11, 75 (2014). [7] F. Wendler and E. Malic, ArXiv e-prints (2014), arXiv:1410.2080 [cond-mat.mes-hall]. ##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##

text: 2−

font: default

text: dna

text: 1−

text: n

text: neewteb

text: niag

text: pumping

text: intensity

text: x

text: FIG.

text: Dependence

label: caption

text: of

text: the

text: gain

text: between

text: n

text: and

text: LLs

text: per

text: graphene

text: monolayer

text: on

text: the

text: pumping

text: intensity.

text: One

text: can

text: also

text: see

text: in

text: Fig.

text: that

text: the

text: population

text: inversion

text: exists

text: also

text: between

text: states

text: n

text: and

text: albeit

text: at

text: a

text: two

text: times

text: lower

text: level.

text: This

text: seems

text: unexpected,

text: given

text: that

text: the

text: optical

text: pumping

text: brings

text: carriers

text: only

text: to

text: state

text: However,

text: a

text: closer

text: look

text: at

text: the

text: rate

text: equations

text: shows

text: that

text: the

text: population

text: inversion

text: between

text: levels

text: and

text: is

text: a

text: consequence

text: of

text: a

text: strongly

text: non-equilibrium

text: carrier

text: distribution

text: below

text: the

text: Fermi

text: level

text: created

text: by

text: the

text: optical

text: pumping,

text: namely

text: the

text: population

text: inversion

text: between

text: states

text: and

text: Indeed,

text: when

text: f−

text: f−

text: the

text: Auger

text: scattering

text: rate

text: from

text: states

text: to

text: states

text: is

text: greater

text: than

text: the

text: scattering

text: rate

text: in

text: the

text: opposite

text: direction.

text: This

text: creates

text: the

text: population

text: inversion

text: f

text: f

text: and

text: the

text: gain

text: for

text: the

text: ˆl⊕

text: polarization,

text: which

text: is

text: about

text: two

text: times

text: smaller

text: than

text: the

text: gain.

text: In

text: conclusion,

text: we

text: show

text: the

text: feasibility

text: of

text: the

text: Landau

text: level

text: THz

text: laser

text: in

text: a

text: magnetized

text: graphene.

text: Despite

text: ultrafast

text: Auger

text: relaxation,

text: steady-state

text: operation

text: of

text: the

text: laser

text: under

text: continuous

text: wave

text: optical

text: pumping

text: is

text: possible

text: by

text: utilizing

text: surface

text: or

text: interface

text: phonon

text: relaxation.

text: The

text: scheme

text: is

text: scalable

text: to

text: thin

text: ﬁlms

text: of

text: topological

text: insulators

text: such

text: as

text: Bi

text: Se

text: or

text: Bi

text: Te

text: ACKNOWLEDGMENTS

text: This

text: work

text: has

text: been

text: supported

text: by

text: NSF

text: Grants

text: and

text: and

text: by

text: the

text: AFOSR

text: grant

text: M.

text: D.

text: Tokman

text: acknowledges

text: support

text: by

text: the

text: Russian

text: Foundation

text: for

text: Basic

text: Research

text: Grants

text: No.

text: and

text: No.

text: belyanin@tamu.edu

text: G.

text: Nusinovich,

text: Introduction

text: to

text: the

text: Physics

text: of

text: Gyrotrons

text: (Johns

text: Hopkins

text: University

text: Press,

text: H.

text: Aoki,

text: Applied

text: Physics

text: Letters

text: A.

text: Wade,

text: G.

text: Fedorov,

text: D.

text: Smirnov,

text: S.

text: Kumar,

text: B.

text: Williams,

text: Q.

text: Hu,

text: and

text: J.

text: Reno,

text: Nature

text: Photonics

text: T.

text: Morimoto,

text: Y.

text: Hatsugai,

text: and

text: H.

text: Aoki,

text: Phys.

text: Rev.

text: B

text: F.

text: Wendler,

text: A.

text: Knorr,

text: and

text: E.

text: Malic,

text: Nature

text: communications

text: M.

text: Mittendorﬀ,

text: F.

text: Wendler,

text: E.

text: Malic,

text: A.

text: Knorr,

text: M.

text: Orlita,

text: M.

text: Potemski,

text: C.

text: Berger,

text: W.

text: A.

text: de

text: Heer,

text: H.

text: Schneider,

text: M.

text: Helm,

text: et

text: al.,

text: Nature

text: Physics

text: F.

text: Wendler

text: and

text: E.

text: Malic,

text: ArXiv

text: e-prints

text: [cond-mat.mes-hall].

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 So the polarizability becomes (cid:88) − 2A f f | | Π0((cid:126)q, ω) = − nα nβ G (q , q ) 2 . (23) πl2 (cid:15) (cid:15) + ¯hω + iδ nαnβ y x c nαnβ nα nβ | | One can check that G (q , q ) only depends on the magnitude of (cid:126)q, so Π0((cid:126)q, ω) and V ((cid:126)q, ω) are only functions | | nαnβ y x s of q = (cid:126)q . We calculated the form factor numerically and checked that it agreed with the analytical expression in [15]. Then, using Eq. (17) and Eq. (29), the Coulomb matrix element is (cid:88) V = δ δ δ δ abcd sa,sc sb,sd ξa,ξc ξb,ξd × − − − Vs((cid:126)q, ω)δqy,ka−kce iqxlc2kcGnanc(qy, qx)δ−qy,kb−kdeiqxlc2kdGnbnd( qy, qx) (cid:88)q(cid:126) (cid:12)(cid:12) (cid:12) = δ δ δ δ δ sa,sc sb,sd ξa,ξc ξb,ξd ka+kb,kc+kd × − − − − qx Vs ((cid:126)q, ω) e iqxlc2(kc kd)Gnanc(qy, qx)Gnbnd( qy, qx) qy=ka−kc . (24) − ∝ − For a ﬁxed k , this matrix element decays quickly when k k is large, since G (q , q ) exp( (ql )2/4) when a c a − nanc −y x c q is large [15]. If kc is bounded, then kb is bounded too, otherwise the term e iqxlc2(kc kd) would oscillate too fast with q , which essentially makes the summation over q to vanish. x x The Auger scattering rate between t(cid:88)wo pair(cid:88)s of La(cid:88)ndau l(cid:88)evels (na, nb)(cid:18)and (nc, nd) is (cid:19) 1 ×1 Γ ↔ = Γ ↔ if n = n or n = n , (25) nanb ncnd 2A/πl2 ab cd 2 a b c d c ξa,ka,sa ξb,kb,sb ξc,kc,sc ξd,kd,sd where the factor 1/2 in the parenthesis is because of the double counting the initial or ﬁnal states. In Eq. (25), one of the summations can be dropped immediately, since the result from the other three summations will be independent of the forth set of quantum numbers. This summation will give exactly the degeneracy 2A/πl2, so it will cancel with c the pre-factor. One summation of k can also be eliminated due to the conservation of momentum k + k = k + k . a b c d As the energy is fully quantized, we will replace the δ function in Eq. (20) with a Lorentzian of line width which can be attributed to impurity scattering [15]. PHONON SCATTERING General formulas The interaction Hamiltonian between phonons(cid:88)and electrons can generally be written as † † Hph = F(q)c c (b + b− ) , (26) int (cid:126)k+q(cid:126) (cid:126)k q(cid:126) q(cid:126) (cid:126)k, q(cid:126) † † where c and c are annihilation and creation operators for electrons, b and b are annihilation and creation operators for phonons, and F(q) is deﬁned below. Using Fermi’s golden rule, the scattering rate from an initial electronic state | (cid:105) | (cid:105) (cid:88) ϕ to a ﬁnal state ϕ is i f 2π | | | | − Γp→h = (cid:88) (n + 1) F(q) 2 M ((cid:126)q) 2δ((cid:15) + Eph (cid:15) ) i f ¯h q fi f q i q(cid:126) 2π | | | | − − + n F(q) 2 M ((cid:126)q) 2δ((cid:15) Eph (cid:15) ) , (27) ¯h q fi f q i q(cid:126) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: So

text: the

text: polarizability

text: becomes

text: f

text: f

text: ω)

text: nα

text: nβ

text: G

text: (q

text: q

text: ¯hω

text: iδ

text: nαnβ

text: y

text: x

text: c

text: nαnβ

text: nα

text: nβ

text: One

text: can

text: check

text: that

text: G

text: (q

text: q

text: only

text: depends

text: on

text: the

text: magnitude

text: of

text: so

text: ω)

text: and

text: V

text: ω)

text: are

text: only

text: functions

text: nαnβ

text: y

text: x

text: s

text: of

text: q

text: We

text: calculated

text: the

text: form

text: factor

text: numerically

text: and

text: checked

text: that

text: it

text: agreed

text: with

text: the

text: analytical

text: expression

text: in

text: Then,

text: using

text: Eq.

text: and

text: Eq.

text: the

text: Coulomb

text: matrix

text: element

text: is

text: V

text: δ

text: δ

text: δ

text: δ

text: abcd

text: sa,sc

text: sb,sd

text: ξa,ξc

text: ξb,ξd

text: ω)δqy,ka−kce

text: qy,

text: qx)

text: δ

text: δ

text: δ

text: δ

text: δ

text: sa,sc

text: sb,sd

text: ξa,ξc

text: ξb,ξd

text: ka+kb,kc+kd

text: qx

text: Vs

text: ω)

text: e

text: kd)Gnanc(qy,

text: qx)Gnbnd(

text: qy,

text: qx)

text: qy=ka−kc

text: For

text: a

text: ﬁxed

text: k

text: this

text: matrix

text: element

text: decays

text: quickly

text: when

text: k

text: k

text: is

text: large,

text: since

text: G

text: (q

text: q

text: exp(

text: (ql

text: when

text: a

text: c

text: a

text: nanc

text: −y

text: x

text: c

text: q

text: is

text: large

text: If

text: kc

text: is

text: bounded,

text: then

text: kb

text: is

text: bounded

text: too,

text: otherwise

text: the

text: term

text: e

text: kd)

text: would

text: oscillate

text: too

text: fast

text: with

text: q

text: which

text: essentially

text: makes

text: the

text: summation

text: over

text: q

text: to

text: vanish.

text: x

text: x

text: The

text: Auger

text: scattering

text: rate

text: between

text: of

text: (na,

text: (nc,

text: nd)

text: is

text: ×1

text: Γ

text: Γ

text: if

text: n

text: n

text: or

text: n

text: n

text: nanb

text: ncnd

text: ab

text: cd

text: a

text: b

text: c

text: d

text: c

text: ξa,ka,sa

text: ξb,kb,sb

text: ξc,kc,sc

text: ξd,kd,sd

text: where

text: the

text: factor

text: in

text: the

text: parenthesis

text: is

text: because

text: of

text: the

text: double

text: counting

text: the

text: initial

text: or

text: ﬁnal

text: states.

text: In

text: Eq.

text: one

text: of

text: the

text: summations

text: can

text: be

text: dropped

text: immediately,

text: since

text: the

text: result

text: from

text: the

text: other

text: three

text: summations

text: will

text: be

text: independent

text: of

text: the

text: forth

text: set

text: of

text: quantum

text: numbers.

text: This

text: summation

text: will

text: give

text: exactly

text: the

text: degeneracy

text: so

text: it

text: will

text: cancel

text: with

text: c

text: the

text: pre-factor.

text: One

text: summation

text: of

text: k

text: can

text: also

text: be

text: eliminated

text: due

text: to

text: the

text: conservation

text: of

text: momentum

text: k

text: k

text: k

text: k

text: a

text: b

text: c

text: d

text: As

text: the

text: energy

text: is

text: fully

text: quantized,

text: we

text: will

text: replace

text: the

text: δ

text: function

text: in

text: Eq.

text: with

text: a

text: Lorentzian

text: of

text: line

text: width

text: which

text: can

text: be

text: attributed

text: to

text: impurity

text: scattering

text: PHONON

text: SCATTERING

text: General

text: formulas

text: The

text: interaction

text: Hamiltonian

text: between

text: electrons

text: can

text: generally

text: be

text: written

text: as

text: Hph

text: F(q)c

text: c

text: (b

text: b−

text: int

text: where

text: c

text: and

text: c

text: are

text: annihilation

text: and

text: creation

text: operators

text: for

text: electrons,

text: b

text: and

text: b

text: are

text: annihilation

text: and

text: creation

text: operators

text: for

text: phonons,

text: and

text: F(q)

text: is

text: deﬁned

text: below.

text: Using

text: Fermi’s

text: golden

text: rule,

text: the

text: scattering

text: rate

text: from

text: an

text: initial

text: electronic

text: state

text: ϕ

text: to

text: a

text: ﬁnal

text: state

text: ϕ

text: is

text: i

text: f

text: 2π

text: Γp→h

text: (n

text: F(q)

text: M

text: Eph

text: i

text: f

text: ¯h

text: q

text: fi

text: f

text: q

text: i

text: 2π

text: n

text: F(q)

text: M

text: Eph

text: ¯h

text: q

text: fi

text: f

text: q

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 (cid:88) over all its bases with equal weights, d orbitals conserved in the r -th row and the c -th col- yz i i (cid:48) umn, respectively. We can order electrons in each row | (cid:105) |{ } { }(cid:105) ΨSz = h , σ . (11) from the left most particle to the right most one, fol- ref lowed by the ordering in each column from the top to | (cid:105) Since ΨSz is symmetric under exchanging spins of any bottom. The corresponding many-body basis can be set ref (cid:89) (cid:89) two orbitals, it is a fully spin-polarized state with the to- up as tal spin S = N /2 and its z component S . Apparently, (cid:104)inΨSGHzS|Ψz,Sretzfh(cid:105)es(cid:54)=e t0tow.toSinnocne-o|ΨrtSGhzo(cid:105)goisnathl estuanteiqsumezugsrtoushnadresttahtee |R, S(cid:105)NX,NY = jL=x1 d†yz,cj jL=y1 d†zx,rj|0(cid:105) same good quantum numbers of S and S . Q.E.D. † · · · † † † · · · † † | (cid:105) z = d d d d d d 0 , Because of the spin SU(2) symmetry, all ground states yz,cLx yz,c2 yz,c1 zx,rLy zx,r2 zx,r1 | (cid:105) H − ≤ ≤ ΨSGz in diﬀerent Sz with Ntot/2 Sz Ntot/2 are (12) degenerate, and form a set of spin multiplets with the R where j denotes the index of columns and rows; = maximal total spin S = Ntot/2. {rrj; rcj| all i’s and j’s} represents the coordinates of Remark: Theorem 1 is true for both the periodic and occi upiied sites; S = {αrj; βcj| all i’s and j’s} represents open boundary conditions. i i † † their the spin conﬁgurations. The operator d (d ) Based on Theorem 1, we have the following two corol- (cid:81) zx,rj yz,cj creates a whole line of N (N ) d (d ) electrons in the laries with their proofs presented in Appendix . rj cj zx yz row r (column c ) ordered from left to right (from top to j † j † † · · · † bottom), d = ∈ d (r )d (r ) d (r ), Corollary 1 Under condition I and a modiﬁed condition † zx,rj ri rowrj zx 1 zx 2 zx Nrj II: There is one and only one doubly occupied orbital for and d can be similarly deﬁned. yz,cj each orbital band in each layer; we have that the Hamil- Now, let us consider the additional quasi-2D d band xy tonian of Eqs. (2) and (3) also possesses the fully spin- with one and only one hole. The basis for this layer of | { }(cid:105) polarized FM ground state which is unique up to the triv- d orbital hxy, σ is deﬁned following Eq. (5) but xy ial spin degeneracy. without the layer index. Then, the basis for the Hilbert H space of this 2D system can be constructed by the 2D Corollary 2 If there is one and only one particle in each direct product of the basis for the 1D bands and that for orbital band in each layer, we also have that the ground the 2D band, state is fully spin-polarized and unique up to the trivial |R S(cid:105) ⊗ | { }(cid:105) spin degeneracy for any values of J > 0 and V . , NX,NY hxy, σ . (13) (cid:76) Again, because of the conservation of the z component of the total spin, this Hilbert space can be decomposed H H IV. FERROMAGNETISM IN THE 2D t2g as 2D = 2SDz . Following the same steps in Ref. [33] ORBITAL LAYER and in Sec. , it is straightforward to show that for the basis deﬁned in Eq. (13), and under condition (III) for In this section, we will consider the same multiorbital 1D bands, the oﬀ-diagonal matrix elements of the many- Hubbard Hamiltonian of Eqs. (2) and (3) but in a sin- body Hamiltonian are non-positive. gle layer along the xy-plane. The d orbital band re- Below, we further show the transitivity of the Hamil- xy H mains 2D; while, the dzx- and dyz orbitals form crossed tonian matrix in the sub-Hilbert space 2SDz under con- 1D bands with dispersion perpendicular to each other. dition (IV) for d - and d bands. Since the loca- zx yz The FM ground state of this system will be discussed tions of electrons can be easily adjusted by applying when both 1D and 2D bands present. hopping terms, it suﬃces to show the transitivity be- | (cid:105) When only the two quasi-1D bands are considered, the tween two bases only diﬀer by spin conﬁgurations, u = |R S(cid:105) ⊗ | { }(cid:105) | (cid:105) |R S(cid:48)(cid:105) ⊗ FM ground state has been proved in Ref. [33] under con- | , {NX(cid:48)},(cid:105)NY hxy, σ and v = , NX,NY dition (I) and the following two conditions: hxy, σ . Again, we only need to show that for the | (cid:105) (III): Open boundary condition or periodic (anti-periodic) state of u , we can exchange any two diﬀerent spins by boundary condition with odd (even) number of particles applying hopping and Hund’s interaction terms. If these in each row or column; two electrons are both in quasi-1D bands d and d , zx yz (IV): Arbitrary ﬁlling with at least one hole and one par- this situation has been proved in Ref. [33]. If these two ticle in each row and each column. electrons are both in the d band, it is reduced to the xy To describe the part of d and d bands with general usual case of the Nagaoka system. zx yz ﬁllings, let us ﬁrst recapture the many-body bases con- Now let us consider the case of one electron in the structed for the quasi-1D system in Ref. [33] and rewrite quasi-1D bands, without loss of generality, in the d ↑ zx them in terms of d and d bands. By Lemma 1, for any orbital band with spin- ; and another electron in the d zx yz ↓ xy generic ﬁlling, we can always specify a partition of parti- band with spin- . We denote their locations as r and X { · · · } zx cle numbers into rows = r = 1, , L and columns r , respectively. Let us identify the site r which is in the Y { · · · } Ni { }yN { } xy c = c = 1, , L as, X = N , Y = N , same row of the d electron and in the same column of i x ri ci zx where N and N are the particle numbers of d - and the d electron, and consider the d orbital at this site. ri ci zx xy yz

text: over

text: all

text: its

text: bases

text: with

text: equal

text: weights,

text: d

text: orbitals

text: conserved

text: in

text: the

text: r

text: -th

text: row

text: and

text: the

text: c

text: -th

text: col-

text: yz

text: i

text: i

text: umn,

text: respectively.

text: We

text: can

text: order

text: electrons

text: in

text: each

text: row

text: ΨSz

text: h

text: σ

text: from

text: the

text: left

text: most

text: particle

text: to

text: the

text: right

text: most

text: one,

text: fol-

text: ref

text: lowed

text: by

text: the

text: ordering

text: in

text: each

text: column

text: from

text: the

text: top

text: to

text: Since

text: ΨSz

text: is

text: symmetric

text: under

text: exchanging

text: spins

text: of

text: any

text: bottom.

text: The

text: corresponding

text: many-body

text: basis

text: can

text: be

text: set

text: ref

text: two

text: orbitals,

text: it

text: is

text: a

text: fully

text: spin-polarized

text: state

text: with

text: the

text: to-

text: up

text: as

text: tal

text: spin

text: S

text: N

text: and

text: its

text: z

text: component

text: S

text: Apparently,

text: estuanteiqsumezugsrtoushnadresttahtee

text: |R,

text: d†yz,cj

text: same

text: good

text: quantum

text: numbers

text: of

text: S

text: and

text: S

text: Q.E.D.

text: z

text: d

text: d

text: d

text: d

text: d

text: d

text: Because

text: of

text: the

text: spin

text: symmetry,

text: all

text: ground

text: states

text: yz,cLx

text: zx,rLy

text: H

text: ΨSGz

text: in

text: diﬀerent

text: Sz

text: with

text: Sz

text: are

text: degenerate,

text: and

text: form

text: a

text: set

text: of

text: spin

text: multiplets

text: with

text: the

text: R

text: where

text: j

text: denotes

text: the

text: index

text: of

text: columns

text: and

text: rows;

text: maximal

text: total

text: spin

text: S

text: {rrj;

text: rcj|

text: all

text: i’s

text: and

text: j’s}

text: represents

text: the

text: coordinates

text: of

text: Remark:

text: Theorem

text: is

text: true

text: for

text: both

text: the

text: periodic

text: and

text: occi

text: upiied

text: sites;

text: S

text: {αrj;

text: βcj|

text: all

text: i’s

text: and

text: j’s}

text: represents

text: open

text: boundary

text: conditions.

text: i

text: i

text: their

text: the

text: spin

text: conﬁgurations.

text: The

text: operator

text: d

text: (d

text: Based

text: on

text: Theorem

text: we

text: have

text: the

text: following

text: two

text: corol-

text: zx,rj

text: yz,cj

text: creates

text: a

text: whole

text: line

text: of

text: N

text: (N

text: d

text: (d

text: electrons

text: in

text: the

text: laries

text: with

text: their

text: proofs

text: presented

text: in

text: Appendix

text: rj

text: cj

text: zx

text: yz

text: row

text: r

text: (column

text: c

text: ordered

text: from

text: left

text: to

text: right

text: (from

text: top

text: to

text: j

text: j

text: bottom),

text: d

text: d

text: (r

text: )d

text: (r

text: d

text: (r

text: Corollary

text: Under

text: condition

text: I

text: and

text: a

text: modiﬁed

text: condition

text: zx,rj

text: ri

text: rowrj

text: zx

text: zx

text: zx

text: Nrj

text: II:

text: There

text: is

text: one

text: and

text: only

text: one

text: doubly

text: occupied

text: orbital

text: for

text: and

text: d

text: can

text: be

text: similarly

text: deﬁned.

text: yz,cj

text: each

text: orbital

text: band

text: in

text: each

text: layer;

text: we

text: have

text: that

text: the

text: Hamil-

text: Now,

text: let

text: us

text: consider

text: the

text: additional

text: d

text: band

text: xy

text: tonian

text: of

text: Eqs.

text: and

text: also

text: possesses

text: the

text: fully

text: spin-

text: with

text: one

text: and

text: only

text: one

text: hole.

text: The

text: basis

text: for

text: this

text: layer

text: of

text: polarized

text: FM

text: ground

text: state

text: which

text: is

text: unique

text: up

text: to

text: the

text: triv-

text: d

text: orbital

text: hxy,

text: σ

text: is

text: deﬁned

text: following

text: Eq.

text: but

text: xy

text: ial

text: spin

text: degeneracy.

text: without

text: the

text: layer

text: index.

text: Then,

text: the

text: basis

text: for

text: the

text: Hilbert

text: H

text: space

text: of

text: this

text: system

text: can

text: be

text: constructed

text: by

text: the

text: Corollary

text: If

text: there

text: is

text: one

text: and

text: only

text: one

text: particle

text: in

text: each

text: direct

text: product

text: of

text: the

text: basis

text: for

text: the

text: bands

text: and

text: that

text: for

text: orbital

text: band

text: in

text: each

text: layer,

text: we

text: also

text: have

text: that

text: the

text: ground

text: the

text: band,

text: state

text: is

text: fully

text: spin-polarized

text: and

text: unique

text: up

text: to

text: the

text: trivial

text: |R

text: spin

text: degeneracy

text: for

text: any

text: values

text: of

text: J

text: and

text: V

text: NX,NY

text: hxy,

text: σ

text: Again,

text: because

text: of

text: the

text: conservation

text: of

text: the

text: z

text: component

text: of

text: the

text: total

text: spin,

text: this

text: Hilbert

text: space

text: can

text: be

text: decomposed

text: H

text: H

text: IV.

text: FERROMAGNETISM

text: IN

text: THE

text: as

text: Following

text: the

text: same

text: steps

text: in

text: Ref.

text: ORBITAL

text: LAYER

text: and

text: in

text: Sec.

text: it

text: is

text: straightforward

text: to

text: show

text: that

text: for

text: the

text: basis

text: deﬁned

text: in

text: Eq.

text: and

text: under

text: condition

text: (III)

text: for

text: In

text: this

text: section,

text: we

text: will

text: consider

text: the

text: same

text: multiorbital

text: bands,

text: the

text: oﬀ-diagonal

text: matrix

text: elements

text: of

text: the

text: many-

text: Hubbard

text: Hamiltonian

text: of

text: Eqs.

text: and

text: but

text: in

text: a

text: sin-

text: body

text: Hamiltonian

text: are

text: non-positive.

text: gle

text: layer

text: along

text: the

text: xy-plane.

text: The

text: d

text: orbital

text: band

text: re-

text: Below,

text: we

text: further

text: show

text: the

text: transitivity

text: of

text: the

text: Hamil-

text: xy

text: H

text: mains

text: while,

text: the

text: dzx-

text: and

text: dyz

text: orbitals

text: form

text: crossed

text: tonian

text: matrix

text: in

text: the

text: sub-Hilbert

text: space

text: under

text: con-

text: bands

text: with

text: dispersion

text: perpendicular

text: to

text: each

text: other.

text: dition

text: (IV)

text: for

text: d

text: and

text: d

text: bands.

text: Since

text: the

text: loca-

text: zx

text: yz

text: The

text: FM

text: ground

text: state

text: of

text: this

text: system

text: will

text: be

text: discussed

text: tions

text: of

text: electrons

text: can

text: be

text: easily

text: adjusted

text: by

text: applying

text: when

text: both

text: and

text: bands

text: present.

text: hopping

text: terms,

text: it

text: suﬃces

text: to

text: show

text: the

text: transitivity

text: be-

text: When

text: only

text: the

text: two

text: bands

text: are

text: considered,

text: the

text: tween

text: two

text: bases

text: only

text: diﬀer

text: by

text: spin

text: conﬁgurations,

text: u

text: |R

text: |R

text: FM

text: ground

text: state

text: has

text: been

text: proved

text: in

text: Ref.

text: under

text: con-

text: hxy,

text: σ

text: and

text: v

text: NX,NY

text: dition

text: (I)

text: and

text: the

text: following

text: two

text: conditions:

text: hxy,

text: σ

text: Again,

text: we

text: only

text: need

text: to

text: show

text: that

text: for

text: the

text: (III):

text: Open

text: boundary

text: condition

text: or

text: periodic

text: (anti-periodic)

text: state

text: of

text: u

text: we

text: can

text: exchange

text: any

text: two

text: diﬀerent

text: spins

text: by

text: boundary

text: condition

text: with

text: odd

text: (even)

text: number

text: of

text: particles

text: applying

text: hopping

text: and

text: Hund’s

text: interaction

text: terms.

text: If

text: these

text: in

text: each

text: row

text: or

text: column;

text: two

text: electrons

text: are

text: both

text: in

text: bands

text: d

text: and

text: d

text: zx

text: yz

text: (IV):

text: Arbitrary

text: ﬁlling

text: with

text: at

text: least

text: one

text: hole

text: and

text: one

text: par-

text: this

text: situation

text: has

text: been

text: proved

text: in

text: Ref.

text: If

text: these

text: two

text: ticle

text: in

text: each

text: row

text: and

text: each

text: column.

text: electrons

text: are

text: both

text: in

text: the

text: d

text: band,

text: it

text: is

text: reduced

text: to

text: the

text: xy

text: To

text: describe

text: the

text: part

text: of

text: d

text: and

text: d

text: bands

text: with

text: general

text: usual

text: case

text: of

text: the

text: Nagaoka

text: system.

text: zx

text: yz

text: ﬁllings,

text: let

text: us

text: ﬁrst

text: recapture

text: the

text: many-body

text: bases

text: con-

text: Now

text: let

text: us

text: consider

text: the

text: case

text: of

text: one

text: electron

text: in

text: the

text: structed

text: for

text: the

text: system

text: in

text: Ref.

text: and

text: rewrite

text: bands,

text: without

text: loss

text: of

text: generality,

text: in

text: the

text: d

text: zx

text: them

text: in

text: terms

text: of

text: d

text: and

text: d

text: bands.

text: By

text: Lemma

text: for

text: any

text: orbital

text: band

text: with

text: spin-

text: and

text: another

text: electron

text: in

text: the

text: d

text: zx

text: yz

text: xy

text: generic

text: ﬁlling,

text: we

text: can

text: always

text: specify

text: a

text: partition

text: of

text: parti-

text: band

text: with

text: spin-

text: We

text: denote

text: their

text: locations

text: as

text: r

text: and

text: X

text: zx

text: cle

text: numbers

text: into

text: rows

text: r

text: L

text: and

text: columns

text: r

text: respectively.

text: Let

text: us

text: identify

text: the

text: site

text: r

text: which

text: is

text: in

text: the

text: Y

text: Ni

text: }yN

text: xy

text: c

text: c

text: L

text: as,

text: X

text: N

text: Y

text: N

text: same

text: row

text: of

text: the

text: d

text: electron

text: and

text: in

text: the

text: same

text: column

text: of

text: i

text: x

text: ri

text: ci

text: zx

text: where

text: N

text: and

text: N

text: are

text: the

text: particle

text: numbers

text: of

text: d

text: and

text: the

text: d

text: electron,

text: and

text: consider

text: the

text: d

text: orbital

text: at

text: this

text: site.

text: ri

text: ci

text: zx

text: xy

text: yz


text: 6 We assume that there is an electron of the d orbital The proof is very similar to that of Theorem 2, Nev- yz at r . If not, because of condition (IV), we can always ertheless, a new situation needs to be addressed: both c ↑ ↓ move a d electron of that column to r by hopping. electrons are in the d band with spin- and - , respec- yz c xy (cid:48) And the conﬁguration in this column can be restored by tively. Their locations are denoted as r and r , respec- reversing the hopping afterward. If the electron of the d tively. Then we can choose an electron in the d band, ↑ yz z↑x orbital at r has spin- , it can ﬁrst be moved to r by and, without loss of generality, assume its spin- . Then c xy hoppings. Then, it can exchange the spins with the d according to the proof of Theorem 2, we can ﬁrst ﬂip xy (cid:48) electron at r by Hund’s interaction. After reversing the the pair of electrons d ↑ and d ↓(r ), then their spins xy zx, (cid:48) xy, hopping, this d electron can be moved back to r but become d ↓ and d ↑(r ). Next, we consider the pair ↓ yz c zx, xy, with spin- . Further, it can be moved to r to exchange of d ↓ and d ↑(r) and exchange their spins. The net zx zx, xy, the spins with the d electron and be moved back to r result is the exchange of the spins of two d electrons. zx ↑ c xy again with its original spin- recovered. The net eﬀect is With both results of non-positivity and transitivity, it the exchange of spin conﬁgurations between the d and is also straightforward to arrive at Corollary 3 by similar xy d electrons without aﬀecting other conﬁgurations. The proof of Theorem 1. Q.E.D. zx ↓ case of the d electron at r with spin- can be similarly yz c proved. So far, we have shown both the non-positivity of oﬀ- V. DISCUSSION ON EXPERIMENTS diagonal matrix elements and the transitivity of the H Hamiltonian matrix in the sub-Hilbert space 2SDz . Then, Although Theorems 1 and 2 are under ideal conditions following the same reasoning in the proof of Theorem 1, and limits, they do have close connections to realistic it is straightforward to have the following theorem systems of transition-metal oxides. For the multiorbital Theorem 2 (2D FM Ground State) Consider the Hubbard Hamiltonian of Eqs. (2) and (3), they are ac- case in which there is one and only one hole in the dxy tually a good approximation of the t2g orbital systems band. Under conditions (I), (III), and (IV ), for any of transition metal oxides in 3D. For example, the itin- values of V and J > 0, the ground states are fully spin- erant FM SrRuO belongs to this class of materials37–39, 3 polarized which is unique apart from the trivial spin de- which is a t -active material of 4d electrons in a cubic 2g generacy. lattice. Even though, typical interaction strength in the Next, we consider the situation in which the dxy band 4d electron systems are intermediately strong, it already is half-ﬁlled, i.e., there is no hole. In this case, the dxy exhibits the FM phase with Tc = 165K. Furthermore, band by itself is not ferromagnetic. Because of the cou- the magnetic moment of this system is observed as 1.6µ B pling to the quasi-1D band, we have the following theo- per site with the electron ﬁlling in SrRuO as four elec- 3 rem trons per site. Therefore, the FM ground state stated in Theorem 1 would possibly persist to the intermediate in- Corollary 3 If the d band is half-ﬁlled, under the xy teraction regime and with ﬁnite electron or hole density same conditions in Theorem 2, for any values of V and away from half-ﬁlling. Nevertheless, the magnetization J > 0, the ground states are fully spin-polarized which is would be no longer fully polarized but partially polar- unique apart from the trivial spin degeneracy. ized to save the kinetic energy cost. Proof: We ﬁrst deﬁne the basis for the local moments Another important system is the LaAlO /SrTiO in- 3 3 for the half-ﬁlled dxy ban(cid:89)d, which can be ordered in an terface between two component insulators. This interface arbitrary way as is experimentally found metallic and ferromagnetic with |{ }(cid:105) † | (cid:105) large magnetization40,41. This is a t orbital active ma- 2g σ = d (i) 0 , (14) xy,σ terial with 3d electrons in 2D layered systems, whose d zx i and d are quasi-1D orbital bands while its d orbital yz xy (cid:104) (cid:105) forms the quasi-2D band. For 3d electrons, the interac- where σ is an arbitrary spin distribution. Then for the tion strength is stronger than that of 4d materials. The combined system, the basis is deﬁned as RKKY, itinerant, and double-exchange mechanisms were |R S(cid:105) ⊗ |{ }(cid:105) , NX,NY σ . (15) proposed to explain the FM in this system42–44.Here, we have shown that the ground state itinerant FM is fully Again becHa(cid:48)use of spin conservation, the Hilbert space in spin polarized and robust for general densities in the dzx this case can be further decomposed into the direct and d bands under strong intra orbital interactions. 2D H(cid:48) ⊕H (cid:48) yz sum of diﬀerent sectors of S ’s as = Sz . VI. CONCLUSIONS z 2D 2D Similarly to Theorem 2, the oﬀ-diagonal elements of the Hamiltonian matrix is non-positive. We next show In summary, we have investigated the Nagaoka type the transitivity of theHHam(cid:48) iltonian matrix in each physi- itinerant FM in t2g orbital systems in a 3D cubic lattice. cal sub-Hilbert space Sz . Again, we only need to show The hole motion in each orbital band is constrained in H 2D(cid:48) that for any state in Sz , opposite spins of any two elec- the plane parallel to the orbital orientation. Eﬀectively, 2D trons can be exchanged by applying hopping and Hund’s this system behaves as crossing planes of 2D Nagaoka FM interaction without aﬀecting other parts of the system. coupled by on-site inter orbital Hund’s coupling. Conse-

text: We

text: assume

text: that

text: there

text: is

text: an

text: electron

text: of

text: the

text: d

text: orbital

text: The

text: proof

text: is

text: very

text: similar

text: to

text: that

text: of

text: Theorem

text: Nev-

text: yz

text: at

text: r

text: If

text: not,

text: because

text: of

text: condition

text: (IV),

text: we

text: can

text: always

text: ertheless,

text: a

text: new

text: situation

text: needs

text: to

text: be

text: addressed:

text: both

text: c

text: move

text: a

text: d

text: electron

text: of

text: that

text: column

text: to

text: r

text: by

text: hopping.

text: electrons

text: are

text: in

text: the

text: d

text: band

text: with

text: spin-

text: and

text: respec-

text: yz

text: c

text: xy

text: And

text: the

text: conﬁguration

text: in

text: this

text: column

text: can

text: be

text: restored

text: by

text: tively.

text: Their

text: locations

text: are

text: denoted

text: as

text: r

text: and

text: r

text: respec-

text: reversing

text: the

text: hopping

text: afterward.

text: If

text: the

text: electron

text: of

text: the

text: d

text: tively.

text: Then

text: we

text: can

text: choose

text: an

text: electron

text: in

text: the

text: d

text: band,

text: yz

text: z↑x

text: orbital

text: at

text: r

text: has

text: spin-

text: it

text: can

text: ﬁrst

text: be

text: moved

text: to

text: r

text: by

text: and,

text: without

text: loss

text: of

text: generality,

text: assume

text: its

text: spin-

text: Then

text: c

text: xy

text: hoppings.

text: Then,

text: it

text: can

text: exchange

text: the

text: spins

text: with

text: the

text: d

text: according

text: to

text: the

text: proof

text: of

text: Theorem

text: we

text: can

text: ﬁrst

text: ﬂip

text: xy

text: electron

text: at

text: r

text: by

text: Hund’s

text: interaction.

text: After

text: reversing

text: the

text: the

text: pair

text: of

text: electrons

text: d

text: and

text: d

text: ↓(r

text: then

text: their

text: spins

text: xy

text: zx,

text: xy,

text: hopping,

text: this

text: d

text: electron

text: can

text: be

text: moved

text: back

text: to

text: r

text: but

text: become

text: d

text: and

text: d

text: ↑(r

text: Next,

text: we

text: consider

text: the

text: pair

text: yz

text: c

text: zx,

text: xy,

text: with

text: spin-

text: Further,

text: it

text: can

text: be

text: moved

text: to

text: r

text: to

text: exchange

text: of

text: d

text: and

text: d

text: ↑(r)

text: and

text: exchange

text: their

text: spins.

text: The

text: net

text: zx

text: zx,

text: xy,

text: the

text: spins

text: with

text: the

text: d

text: electron

text: and

text: be

text: moved

text: back

text: to

text: r

text: result

text: is

text: the

text: exchange

text: of

text: the

text: spins

text: of

text: two

text: d

text: electrons.

text: zx

text: c

text: xy

text: again

text: with

text: its

text: original

text: spin-

text: recovered.

text: The

text: net

text: eﬀect

text: is

text: With

text: both

text: results

text: of

text: non-positivity

text: and

text: transitivity,

text: it

text: the

text: exchange

text: of

text: spin

text: conﬁgurations

text: between

text: the

text: d

text: and

text: is

text: also

text: straightforward

text: to

text: arrive

text: at

text: Corollary

text: by

text: similar

text: xy

text: d

text: electrons

text: without

text: aﬀecting

text: other

text: conﬁgurations.

text: The

text: proof

text: of

text: Theorem

text: Q.E.D.

text: zx

text: case

text: of

text: the

text: d

text: electron

text: at

text: r

text: with

text: spin-

text: can

text: be

text: similarly

text: yz

text: c

text: proved.

text: So

text: far,

text: we

text: have

text: shown

text: both

text: the

text: non-positivity

text: of

text: oﬀ-

text: V.

text: DISCUSSION

text: ON

text: EXPERIMENTS

text: diagonal

text: matrix

text: elements

text: and

text: the

text: transitivity

text: of

text: the

text: H

text: Hamiltonian

text: matrix

text: in

text: the

text: sub-Hilbert

text: space

text: Then,

text: Although

text: Theorems

text: and

text: are

text: under

text: ideal

text: conditions

text: following

text: the

text: same

text: reasoning

text: in

text: the

text: proof

text: of

text: Theorem

text: and

text: limits,

text: they

text: do

text: have

text: close

text: connections

text: to

text: realistic

text: it

text: is

text: straightforward

text: to

text: have

text: the

text: following

text: theorem

text: systems

text: of

text: transition-metal

text: oxides.

text: For

text: the

text: multiorbital

text: Theorem

text: FM

text: Ground

text: State)

text: Consider

text: the

text: Hubbard

text: Hamiltonian

text: of

text: Eqs.

text: and

text: they

text: are

text: ac-

text: case

text: in

text: which

text: there

text: is

text: one

text: and

text: only

text: one

text: hole

text: in

text: the

text: dxy

text: tually

text: a

text: good

text: approximation

text: of

text: the

text: orbital

text: systems

text: band.

text: Under

text: conditions

text: (I),

text: (III),

text: and

text: (IV

text: for

text: any

text: of

text: transition

text: metal

text: oxides

text: in

text: For

text: example,

text: the

text: itin-

text: values

text: of

text: V

text: and

text: J

text: the

text: ground

text: states

text: are

text: fully

text: spin-

text: erant

text: FM

text: SrRuO

text: belongs

text: to

text: this

text: class

text: of

text: polarized

text: which

text: is

text: unique

text: apart

text: from

text: the

text: trivial

text: spin

text: de-

text: which

text: is

text: a

text: t

text: -active

text: material

text: of

text: electrons

text: in

text: a

text: cubic

text: generacy.

text: lattice.

text: Even

text: though,

text: typical

text: interaction

text: strength

text: in

text: the

text: Next,

text: we

text: consider

text: the

text: situation

text: in

text: which

text: the

text: dxy

text: band

text: electron

text: systems

text: are

text: intermediately

text: strong,

text: it

text: already

text: is

text: half-ﬁlled,

text: i.e.,

text: there

text: is

text: no

text: hole.

text: In

text: this

text: case,

text: the

text: dxy

text: exhibits

text: the

text: FM

text: phase

text: with

text: Tc

text: Furthermore,

text: band

text: by

text: itself

text: is

text: not

text: ferromagnetic.

text: Because

text: of

text: the

text: cou-

text: the

text: magnetic

text: moment

text: of

text: this

text: system

text: is

text: observed

text: as

text: B

text: pling

text: to

text: the

text: band,

text: we

text: have

text: the

text: following

text: theo-

text: per

text: site

text: with

text: the

text: electron

text: ﬁlling

text: in

text: SrRuO

text: as

text: four

text: elec-

text: rem

text: trons

text: per

text: site.

text: Therefore,

text: the

text: FM

text: ground

text: state

text: stated

text: in

text: Theorem

text: would

text: possibly

text: persist

text: to

text: the

text: intermediate

text: in-

text: Corollary

text: If

text: the

text: d

text: band

text: is

text: half-ﬁlled,

text: under

text: the

text: xy

text: teraction

text: regime

text: and

text: with

text: ﬁnite

text: electron

text: or

text: hole

text: density

text: same

text: conditions

text: in

text: Theorem

text: for

text: any

text: values

text: of

text: V

text: and

text: away

text: from

text: half-ﬁlling.

text: Nevertheless,

text: the

text: magnetization

text: J

text: the

text: ground

text: states

text: are

text: fully

text: spin-polarized

text: which

text: is

text: would

text: be

text: no

text: longer

text: fully

text: polarized

text: but

text: partially

text: polar-

text: unique

text: apart

text: from

text: the

text: trivial

text: spin

text: degeneracy.

text: ized

text: to

text: save

text: the

text: kinetic

text: energy

text: cost.

text: Proof:

text: We

text: ﬁrst

text: deﬁne

text: the

text: basis

text: for

text: the

text: local

text: moments

text: Another

text: important

text: system

text: is

text: the

text: LaAlO

text: /SrTiO

text: in-

text: for

text: the

text: half-ﬁlled

text: dxy

text: which

text: can

text: be

text: ordered

text: in

text: an

text: terface

text: between

text: two

text: component

text: insulators.

text: This

text: interface

text: arbitrary

text: way

text: as

text: is

text: experimentally

text: found

text: metallic

text: and

text: ferromagnetic

text: with

text: large

text: This

text: is

text: a

text: t

text: orbital

text: active

text: ma-

text: σ

text: d

text: (i)

text: xy,σ

text: terial

text: with

text: electrons

text: in

text: layered

text: systems,

text: whose

text: d

text: zx

text: i

text: and

text: d

text: are

text: orbital

text: bands

text: while

text: its

text: d

text: orbital

text: yz

text: xy

text: forms

text: the

text: band.

text: For

text: electrons,

text: the

text: interac-

text: where

text: σ

text: is

text: an

text: arbitrary

text: spin

text: distribution.

text: Then

text: for

text: the

text: tion

text: strength

text: is

text: stronger

text: than

text: that

text: of

text: materials.

text: The

text: combined

text: system,

text: the

text: basis

text: is

text: deﬁned

text: as

text: RKKY,

text: itinerant,

text: and

text: double-exchange

text: mechanisms

text: were

text: |R

text: NX,NY

text: σ

text: proposed

text: to

text: explain

text: the

text: FM

text: in

text: this

text: we

text: have

text: shown

text: that

text: the

text: ground

text: state

text: itinerant

text: FM

text: is

text: fully

text: Again

text: of

text: spin

text: conservation,

text: the

text: Hilbert

text: space

text: in

text: spin

text: polarized

text: and

text: robust

text: for

text: general

text: densities

text: in

text: the

text: dzx

text: this

text: case

text: can

text: be

text: further

text: decomposed

text: into

text: the

text: direct

text: and

text: d

text: bands

text: under

text: strong

text: intra

text: orbital

text: interactions.

text: ⊕H

text: yz

text: sum

text: of

text: diﬀerent

text: sectors

text: of

text: S

text: ’s

text: as

text: Sz

text: VI.

text: CONCLUSIONS

text: z

text: Similarly

text: to

text: Theorem

text: the

text: oﬀ-diagonal

text: elements

text: of

text: the

text: Hamiltonian

text: matrix

text: is

text: non-positive.

text: We

text: next

text: show

text: In

text: summary,

text: we

text: have

text: investigated

text: the

text: Nagaoka

text: type

text: the

text: transitivity

text: of

text: iltonian

text: matrix

text: in

text: each

text: physi-

text: itinerant

text: FM

text: in

text: orbital

text: systems

text: in

text: a

text: cubic

text: lattice.

text: cal

text: sub-Hilbert

text: space

text: Sz

text: Again,

text: we

text: only

text: need

text: to

text: show

text: The

text: hole

text: motion

text: in

text: each

text: orbital

text: band

text: is

text: constrained

text: in

text: H

text: that

text: for

text: any

text: state

text: in

text: Sz

text: opposite

text: spins

text: of

text: any

text: two

text: elec-

text: the

text: plane

text: parallel

text: to

text: the

text: orbital

text: orientation.

text: Eﬀectively,

text: trons

text: can

text: be

text: exchanged

text: by

text: applying

text: hopping

text: and

text: Hund’s

text: this

text: system

text: behaves

text: as

text: crossing

text: planes

text: of

text: Nagaoka

text: FM

text: interaction

text: without

text: aﬀecting

text: other

text: parts

text: of

text: the

text: system.

text: coupled

text: by

text: on-site

text: inter

text: orbital

text: Hund’s

text: coupling.

text: Conse-