text: not be used to replace the gradient-based methods. On the other hand, the topol- ogy evolution in deep neural networks was not explored extensively yet. The use of traditional algorithms, like NEAT and GNARL, is limited by the time which has to be spent on training and testing a single individual. We believe that to unlock the potential of those algorithms, research has to focus on making the evaluation time shortest as possible. 2.3 Co-evolution of ﬁtness predictors Co-evolution is a kind of evolutionary algorithm where one individual within the same or a separate population, is used to determine the relative ranking between other individuals [48], [49]. In other words, whether individual A is in- ferior or superior to individual B may depend of a third individual C rather than on some external ﬁtness metric which would provide an absolute ranking. There is a number of diﬀerent forms of co-evolution: antagonistic (e.g. predator- prey), cooperative (e.g. symbiosis) or nonsymmetric systems (e.g. host-parasite or teacher-learner). Fitness in the context of co-evolution has two notions: ob- jective and subjective. The former is the well deﬁned absolute ordering metric used in classical evolutionary algorithms. The latter is deﬁned by the coevolving individuals and may be only weakly correlated with the objective ﬁtness. One of the major limiting factors in the evolutional computations is the time of single individual evaluation. One approach to tackle this problem is to use the ﬁtness modelling [50] techniques, which attempt to approximate the exact ﬁtness by using a model or coarse simulation of analyzed system. In the context of evolutionary computations the major techniques include: – ﬁtness inheritance - ﬁtness values are transferred from parents to children during crossover – ﬁtness imitation - individuals are clustered by using a distance metric. The central individual of each cluster is evaluated in full and the resulting ﬁtness is assigned to all elements of the cluster. – partial evaluation - ﬁtness for some individuals is calculated exactly, others are inherited or modeled. A chosen modelling method can be incorporated into the evolutionary process in many ways, e.g. to initialize the population, guide the crossover and mutation or replace ﬁtness evaluations. Such an approach has numerous advantages: it re- duces the evaluation cost and frequency while maintaining evolutionary progress, destabilizes local optima, helps avoiding bloated solutions and it can be applied in a situation where no explicit ﬁtness function exists. Unfortunately there are also many challenges which come with using ﬁtness approximation, like choosing the correct model, training it properly or dealing with a loss of ﬁtness accuracy. Both ideas (co-evolution and ﬁtness approximation) have been combined and applied successfully to a ﬁeld, which also suﬀers from long evaluation times on big data sets - symbolic regression [51], [18], [19]. In this approach, the ﬁtness prediction technique was used. It replaces exact ﬁtness evaluations with a light- weight approximation which adapts together with the solution population. To

text: not

text: be

text: used

text: to

text: replace

text: the

text: gradient-based

text: methods.

text: On

text: the

text: other

text: hand,

text: the

text: topol-

text: ogy

text: evolution

text: in

text: deep

text: neural

text: networks

text: was

text: not

text: explored

text: extensively

text: yet.

text: The

text: use

text: of

text: traditional

text: algorithms,

text: like

text: NEAT

text: and

text: GNARL,

text: is

text: limited

text: by

text: the

text: time

text: which

text: has

text: to

text: be

text: spent

text: on

text: training

text: and

text: testing

text: a

text: single

text: individual.

text: We

text: believe

text: that

text: to

text: unlock

text: the

text: potential

text: of

text: those

text: algorithms,

text: research

text: has

text: to

text: focus

text: on

text: making

text: the

text: evaluation

text: time

text: shortest

text: as

text: possible.

text: Co-evolution

text: of

text: ﬁtness

text: predictors

text: Co-evolution

text: is

text: a

text: kind

text: of

text: evolutionary

text: algorithm

text: where

text: one

text: individual

text: within

text: the

text: same

text: or

text: a

text: separate

text: population,

text: is

text: used

text: to

text: determine

text: the

text: relative

text: ranking

text: between

text: other

text: individuals

text: In

text: other

text: words,

text: whether

text: individual

text: A

text: is

text: in-

text: ferior

text: or

text: superior

text: to

text: individual

text: B

text: may

text: depend

text: of

text: a

text: third

text: individual

text: C

text: rather

text: than

text: on

text: some

text: external

text: ﬁtness

text: metric

text: which

text: would

text: provide

text: an

text: absolute

text: ranking.

text: There

text: is

text: a

text: number

text: of

text: diﬀerent

text: forms

text: of

text: co-evolution:

text: antagonistic

text: (e.g.

text: predator-

text: prey),

text: cooperative

text: (e.g.

text: symbiosis)

text: or

text: nonsymmetric

text: systems

text: (e.g.

text: host-parasite

text: or

text: teacher-learner).

text: Fitness

text: in

text: the

text: context

text: of

text: co-evolution

text: has

text: two

text: notions:

text: ob-

text: jective

text: and

text: subjective.

text: The

text: former

text: is

text: the

text: well

text: deﬁned

text: absolute

text: ordering

text: metric

text: used

text: in

text: classical

text: evolutionary

text: algorithms.

text: The

text: latter

text: is

text: deﬁned

text: by

text: the

text: coevolving

text: individuals

text: and

text: may

text: be

text: only

text: weakly

text: correlated

text: with

text: the

text: objective

text: ﬁtness.

text: One

text: of

text: the

text: major

text: limiting

text: factors

text: in

text: the

text: evolutional

text: computations

text: is

text: the

text: time

text: of

text: single

text: individual

text: evaluation.

text: One

text: approach

text: to

text: tackle

text: this

text: problem

text: is

text: to

text: use

text: the

text: ﬁtness

text: modelling

text: techniques,

text: which

text: attempt

text: to

text: approximate

text: the

text: exact

text: ﬁtness

text: by

text: using

text: a

text: model

text: or

text: coarse

text: simulation

text: of

text: analyzed

text: system.

text: In

text: the

text: context

text: of

text: evolutionary

text: computations

text: the

text: major

text: techniques

text: include:

text: ﬁtness

text: inheritance

text: ﬁtness

text: values

text: are

text: transferred

text: from

text: parents

text: to

text: children

text: during

text: crossover

text: ﬁtness

text: imitation

text: individuals

text: are

text: clustered

text: by

text: using

text: a

text: distance

text: metric.

text: The

text: central

text: individual

text: of

text: each

text: cluster

text: is

text: evaluated

text: in

text: full

text: and

text: the

text: resulting

text: ﬁtness

text: is

text: assigned

text: to

text: all

text: elements

text: of

text: the

text: cluster.

text: partial

text: evaluation

text: ﬁtness

text: for

text: some

text: individuals

text: is

text: calculated

text: exactly,

text: others

text: are

text: inherited

text: or

text: modeled.

text: A

text: chosen

text: modelling

text: method

text: can

text: be

text: incorporated

text: into

text: the

text: evolutionary

text: process

text: in

text: many

text: ways,

text: e.g.

text: to

text: initialize

text: the

text: population,

text: guide

text: the

text: crossover

text: and

text: mutation

text: or

text: replace

text: ﬁtness

text: evaluations.

text: Such

text: an

text: approach

text: has

text: numerous

text: advantages:

text: it

text: re-

text: duces

text: the

text: evaluation

text: cost

text: and

text: frequency

text: while

text: maintaining

text: evolutionary

text: progress,

text: destabilizes

text: local

text: optima,

text: helps

text: avoiding

text: bloated

text: solutions

text: and

text: it

text: can

text: be

text: applied

text: in

text: a

text: situation

text: where

text: no

text: explicit

text: ﬁtness

text: function

text: exists.

text: Unfortunately

text: there

text: are

text: also

text: many

text: challenges

text: which

text: come

text: with

text: using

text: ﬁtness

text: approximation,

text: like

text: choosing

text: the

text: correct

text: model,

text: training

text: it

text: properly

text: or

text: dealing

text: with

text: a

text: loss

text: of

text: ﬁtness

text: accuracy.

text: Both

text: ideas

text: (co-evolution

text: and

text: ﬁtness

text: approximation)

text: have

text: been

text: combined

text: and

text: applied

text: successfully

text: to

text: a

text: ﬁeld,

text: which

text: also

text: suﬀers

text: from

text: long

text: evaluation

text: times

text: on

text: big

text: data

text: sets

text: symbolic

text: regression

text: In

text: this

text: approach,

text: the

text: ﬁtness

text: prediction

text: technique

text: was

text: used.

text: It

text: replaces

text: exact

text: ﬁtness

text: evaluations

text: with

text: a

text: light-

text: weight

text: approximation

text: which

text: adapts

text: together

text: with

text: the

text: solution

text: population.

text: To


text: Fig. 2. Maximum ﬁtness in the population in subsequent iterations for increasing dataset size (min = 100, max = 4000). 2. The network was trained using the samples speciﬁed by the genotype of a given individual. We used the same training method as described earlier. 3. Network accuracy was evaluated using the full test dataset. The results of evolution with diﬀerent sizes can be compared reliably thanks to the use of the same validation set for each ﬁtness predictor size. The maximum accuracy across the population of the subsets in the subse- quent evolution iterations is presented in Fig. 2. It is clear that with increasing the size of the ﬁtness predictor, the accuracy of the trained neural network is higher. Regardless of the size, the evolution was able to improve the predic- tion results. The eﬀect is weaker for bigger ﬁtness predictors, however it is clear that using just a random subset might result in poor performance even if the model has an optimal architecture. The diﬀerences between the worst and best individuals are presented in Table 4. Based on these results we conclude that training with only a subset of avail- able training samples can be used to approximate the prediction accuracy of the full training dataset. However, the subset has to be chosen very carefully. It is possible to pick 250 and 1000 element datasets which will result in training a model with a similar recognition accuracy. 3.2 Time of processing Cutting down the time of evaluating a single individual (neural network) is the main objective of this research. In this section we compare the time of ##LTFigure##

text: Fig.

label: paragraph

text: Maximum

label: caption

text: ﬁtness

text: in

text: the

text: population

text: in

text: subsequent

text: iterations

text: for

text: increasing

text: dataset

text: size

text: (min

text: max

label: list

text: The

text: network

text: was

text: trained

text: using

text: the

text: samples

text: speciﬁed

text: by

text: the

text: genotype

text: of

text: a

text: given

text: individual.

text: We

text: used

text: the

text: same

text: training

text: method

text: as

text: described

text: earlier.

text: Network

text: accuracy

text: was

text: evaluated

text: using

text: the

text: full

text: test

text: dataset.

text: The

text: results

text: of

text: evolution

text: with

text: diﬀerent

text: sizes

text: can

text: be

text: compared

text: reliably

text: thanks

text: to

text: the

text: use

text: of

text: the

text: same

text: validation

text: set

text: for

text: each

text: ﬁtness

text: predictor

text: size.

text: The

text: maximum

text: accuracy

text: across

text: the

text: population

text: of

text: the

text: subsets

text: in

text: the

text: subse-

text: quent

text: evolution

text: iterations

text: is

text: presented

text: in

text: Fig.

text: It

text: is

text: clear

text: that

text: with

text: increasing

text: the

text: size

text: of

text: the

text: ﬁtness

text: predictor,

text: the

text: accuracy

text: of

text: the

text: trained

text: neural

text: network

text: is

text: higher.

text: Regardless

text: of

text: the

text: size,

text: the

text: evolution

text: was

text: able

text: to

text: improve

text: the

text: predic-

text: tion

text: results.

text: The

text: eﬀect

text: is

text: weaker

text: for

text: bigger

text: ﬁtness

text: predictors,

text: however

text: it

text: is

text: clear

text: that

text: using

text: just

text: a

text: random

text: subset

text: might

text: result

text: in

text: poor

text: performance

text: even

text: if

text: the

text: model

text: has

text: an

text: optimal

text: architecture.

text: The

text: diﬀerences

text: between

text: the

text: worst

text: and

text: best

text: individuals

text: are

text: presented

text: in

text: Table

text: Based

text: on

text: these

text: results

text: we

text: conclude

text: that

text: training

text: with

text: only

text: a

text: subset

text: of

text: avail-

text: able

text: training

text: samples

text: can

text: be

text: used

text: to

text: approximate

text: the

text: prediction

text: accuracy

text: of

text: the

text: full

text: training

text: dataset.

text: However,

text: the

text: subset

text: has

text: to

text: be

text: chosen

text: very

text: carefully.

text: It

text: is

text: possible

text: to

text: pick

text: and

text: element

text: datasets

text: which

text: will

text: result

text: in

text: training

text: a

text: model

text: with

text: a

text: similar

text: recognition

text: accuracy.

text: Time

text: of

text: processing

text: Cutting

text: down

text: the

text: time

text: of

text: evaluating

text: a

text: single

text: individual

text: (neural

text: network)

text: is

text: the

text: main

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text: of

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text: research.

text: In

text: this

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text: 7 more complicated, and it is essential to propose novel low- and PA, RA for FD MC MIMO-NOMA, low-complexity RA, complexity RA schemes to ensure the superiority of NOMA and security-aware RA, which provides the basis for further over OMA. research directions on the RA for NOMA systems. C. RA for NOMA with Soft Frequency Reuse tentials and challenges,” IEEE Commun. Surv. Tuts., vol. 19, no. 2, pp. To solve the inter-cell interference problem, soft frequency 721–742, May 2017. reuse (SFR) is an indispensable element of LTE systems. [2] L. Song, Y. Li, Z. Ding, and H. V. Poor, “Resource management in non- orthogonal multiple access networks for 5G and beyond,” IEEE Network, Under SFR, the primary band is assigned to the cell edge users, vol. 31, no. 4, pp. 8–14, Jul. 2017. and the secondary band is allocated to the cell center users. [3] S. M. R. Islam, M. Zeng, and O. A. Dobre, “NOMA in 5G systems: Ex- On the other hand, if NOMA is integrated with the SFR-based citing possibilities for enhancing spectral efﬁciency,” IEEE 5G Tech. Fo- cus, vol. 1, no. 2, May 2017. [Online]. Available: http://5g.ieee.org/tech- LTE systems, the signals of strong users (cell center users) focus. and weak users (cell edge users) may be overlapped on the [4] Z. Ding, P. Fan, and H. V. Poor, “Impact of user pairing on 5G primary band. However, such an arrangement is unfavorable nonorthogonal multiple-access downlink transmissions,” IEEE Trans. Veh. Technol., vol. 65, no. 8, pp. 6010–6023, Aug. 2016. to fairness as it increases the data rate of strong users and [5] Z. Q. Al-Abbasi and D. K. C. So, “User-pairing based non-orthogonal decreases the data rate of the weak users. Additionally, the multiple access (NOMA) system,” in Proc. IEEE Veh. Technol. Conf., use of the primary band at the edge of the cell would generate May. 2016, pp. 1–5. inter-cell interference. In this regard, the design of NOMA RA [6] M. B. Shahab, M. F. Kader, and S. Y. Shin, “A virtual user pairing scheme to optimally utilize the spectrum of unpaired users in non- for SFR-based cellular systems is an open problem. D. Low-Complexity RA orthogonal multiple access,” IEEE Signal Process. Lett., vol. 23, no. 12, pp. 1766–1770, Dec. 2016. The existing fairness models exploit multiple parameters to [7] A. Benjebbovu, A. Li, Y. Saito, Y. Kishiyama, A. Harada, and T. Naka- adjust the fairness level, which introduces high complexity mura, “System-level performance of downlink NOMA for future LTE enhancements,” in Proc. IEEE Global Commun. Conf., Dec. 2013, pp. in determining the corresponding RA in NOMA. However, 66–70. α-fairness uses a single scalar, denoted by α, to achieve dif- [8] B. Di, L. Song, and Y. Li, “Sub-channel assignment, power allocation, ferent user fairness levels and well-known efﬁciency-fairness and user scheduling for non-orthogonal multiple access networks,” IEEE Trans. Wireless Commun., vol. 15, no. 11, pp. 7686–7698, Nov. 2016. tradeoffs [20]. The RA can be investigated for sum throughput [9] Z. Chen, Z. Ding, X. Dai, and G. K. Karagiannidis, “On the application optimization of the NOMA system with fairness constraints to of quasi-degradation to MISO-NOMA downlink,” IEEE Trans. Signal obtain low-complexity algorithms. Processing, vol. 64, no. 23, pp. 6174–6189, Dec. 2016. E. Security-Aware RA [10] Z. Chen and X. Dai, “MED precoding for multiuser MIMO-NOMA downlink transmission,” IEEE Trans. Signal Processing, vol. 66, no. 6, NOMA-based communication comes with security con- pp. 5501–5505, Jun. 2017. cerns, as the user with strong channel condition needs to [11] L. Lei, D. Yuan, C. K. Ho, and S. Sun, “Power and channel allocation decode the signal of the user with weak channel condition. for non-orthogonal multiple access in 5G systems: Tractability and computation,” IEEE Trans. Wireless Commun., vol. 15, no. 12, pp. 8580– Thus, when the weak user becomes malicious or is under 8594, Dec. 2016. attack, the signal decoding operations of both strong and weak [12] M. Zeng, A. Yadav, O. A. Dobre, and H. V. Poor, “Energy-efﬁcient users are no longer reliable. For example, the attack might be power allocation for MIMO-NOMA with multiple users in a cluster,” IEEE Access, vol. PP, no. 99, pp. 1–1, 2017. in the form of an alteration of the channel quality indicator [13] S. Timotheou and I. Krikidis, “Fairness for non-orthogonal multiple during feedback. Therefore, the design of the RA schemes access in 5G systems,” IEEE Signal Process. Lett., vol. 22, no. 10, pp. becomes more critical in the presence of security concerns. To 1647–1651, Oct. 2015. overcome this hurdle, introducing appropriate physical layer [14] F. Fang, H. Zhang, J. Cheng, and V. C. M. Leung, “Energy-efﬁcient re- source allocation for downlink non-orthogonal multiple access network,” security measures is necessary, which is an interesting open IEEE Trans. Commun., vol. 64, no. 9, pp. 3722–3732, Sep. 2016. problem for the research community. [15] S. Shi, L. Yang, and H. Zhu, “Outage balancing in downlink nonorthog- VI. CONCLUDING REMARKS onal multiple access with statistical channel state information,” IEEE Trans. Wireless Commun., vol. 15, no. 7, pp. 4718–4731, Jul. 2016. This article provides an overview of the RA algorithms for [16] Z. Yang, Z. Ding, P. Fan, and N. Al-Dhahir, “A general power allocation downlink NOMA in a categorized fashion. It is clear that the scheme to guarantee quality of service in downlink and uplink NOMA systems,” IEEE Trans. Wireless Commun., vol. 15, no. 11, pp. 7244– RA algorithms play a pivotal role in achieving the maximum 7257, Nov. 2016. beneﬁts of NOMA. Additionally, it is shown that the fairness [17] Z. Ding, R. Schober, and H. V. Poor, “A general MIMO framework for among the NOMA clusters can be attained by controlling the NOMA downlink and uplink transmission based on signal alignment,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 4438–4454, Jun. sum rate gain through the D-NLUPA algorithm. Moreover, 2016. for a general MIMO-NOMA system with pre-deﬁned QoS [18] M. Zeng, A. Yadav, O. A. Dobre, G. I. Tsiropoulos, and H. V. Poor, requirement for each user, simulations are conducted, which “Capacity comparison between MIMO-NOMA and MIMO-OMA with multiple users in a cluster,” IEEE J. Select. Areas Commun., vol. 35, show that MIMO-NOMA outperforms MIMO-OMA in terms no. 10, pp. 2413–2424, Oct 2017. of outage probability and effective sum rate when optimal [19] B. Wang, L. Dai, Z. Wang, N. Ge, and S. Zhou, “Spectrum and energy ef- PA is applied for both. Finally, the article concludes with a ﬁcient beamspace MIMO-NOMA for millimeter-wave communications discussion on open issues, including joint optimization of UP using lens antenna array,” IEEE J. Select. Areas Commun., vol. 35, no. 10, pp. 2370–2382, Oct. 2017. REFERENCES [20] J. Mo and J. Walrand, “Fair end-to-end window-based congestion control,” IEEE/ACM Transactions on Networking, vol. 8, no. 5, pp. 556– [1] S. M. R. Islam, N. Avazov, O. A. Dobre, and K. S. Kwak, “Power- 567, Oct. 2000. domain non-orthogonal multiple access (NOMA) in 5G systems: Po-

text: more

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text: and

text: it

text: is

text: essential

text: to

text: propose

text: novel

text: low-

text: and

text: PA,

text: RA

text: for

text: FD

text: MC

text: MIMO-NOMA,

text: low-complexity

text: RA,

text: complexity

text: RA

text: schemes

text: to

text: ensure

text: the

text: superiority

text: of

text: NOMA

text: and

text: security-aware

text: RA,

text: which

text: provides

text: the

text: basis

text: for

text: further

text: over

text: OMA.

text: research

text: directions

text: on

text: the

text: RA

text: for

text: NOMA

text: systems.

text: C.

label: section

text: RA

text: for

text: NOMA

text: with

text: Soft

text: Frequency

text: Reuse

text: tentials

label: reference

text: and

text: challenges,”

text: IEEE

text: Commun.

text: Surv.

text: Tuts.,

text: vol.

text: no.

text: pp.

text: To

text: solve

text: the

text: inter-cell

text: interference

text: problem,

text: soft

text: frequency

text: May

text: reuse

text: (SFR)

text: is

text: an

text: indispensable

text: element

text: of

text: LTE

text: systems.

text: L.

text: Song,

text: Y.

text: Li,

text: Z.

text: Ding,

text: and

text: H.

text: V.

text: Poor,

text: “Resource

text: management

text: in

text: non-

text: orthogonal

text: multiple

text: access

text: networks

text: for

text: and

text: beyond,”

text: IEEE

text: Network,

text: Under

text: SFR,

text: the

text: primary

text: band

text: is

text: assigned

text: to

text: the

text: cell

text: edge

text: users,

text: vol.

text: no.

text: pp.

text: Jul.

text: and

text: the

text: secondary

text: band

text: is

text: allocated

text: to

text: the

text: cell

text: center

text: users.

text: S.

text: M.

text: R.

text: Islam,

text: M.

text: Zeng,

text: and

text: O.

text: A.

text: Dobre,

text: “NOMA

text: in

text: systems:

text: Ex-

text: On

text: the

text: other

text: hand,

text: if

text: NOMA

text: is

text: integrated

text: with

text: the

text: SFR-based

text: citing

text: possibilities

text: for

text: enhancing

text: spectral

text: efﬁciency,”

text: IEEE

text: Tech.

text: Fo-

text: cus,

text: vol.

text: no.

text: May

text: [Online].

text: Available:

text: LTE

text: systems,

text: the

text: signals

text: of

text: strong

text: users

text: (cell

text: center

text: users)

text: focus.

text: and

text: weak

text: users

text: (cell

text: edge

text: users)

text: may

text: be

text: overlapped

text: on

text: the

text: Z.

text: Ding,

text: P.

text: Fan,

text: and

text: H.

text: V.

text: Poor,

text: “Impact

text: of

text: user

text: pairing

text: on

text: primary

text: band.

text: However,

text: such

text: an

text: arrangement

text: is

text: unfavorable

text: nonorthogonal

text: multiple-access

text: downlink

text: transmissions,”

text: IEEE

text: Trans.

text: Veh.

text: Technol.,

text: vol.

text: no.

text: pp.

text: Aug.

text: to

text: fairness

text: as

text: it

text: increases

text: the

text: data

text: rate

text: of

text: strong

text: users

text: and

text: Z.

text: Q.

text: Al-Abbasi

text: and

text: D.

text: K.

text: C.

text: So,

text: “User-pairing

text: based

text: non-orthogonal

text: decreases

text: the

text: data

text: rate

text: of

text: the

text: weak

text: users.

text: Additionally,

text: the

text: multiple

text: access

text: (NOMA)

text: system,”

text: in

text: Proc.

text: IEEE

text: Veh.

text: Technol.

text: Conf.,

text: use

text: of

text: the

text: primary

text: band

text: at

text: the

text: edge

text: of

text: the

text: cell

text: would

text: generate

text: May.

text: pp.

text: inter-cell

text: interference.

text: In

text: this

text: regard,

text: the

text: design

text: of

text: NOMA

text: RA

text: M.

text: B.

text: Shahab,

text: M.

text: F.

text: Kader,

text: and

text: S.

text: Y.

text: Shin,

text: “A

text: virtual

text: user

text: pairing

text: scheme

text: to

text: optimally

text: utilize

text: the

text: spectrum

text: of

text: unpaired

text: users

text: in

text: non-

text: for

text: SFR-based

text: cellular

text: systems

text: is

text: an

text: open

text: problem.

text: D.

text: Low-Complexity

text: RA

text: orthogonal

text: multiple

text: access,”

text: IEEE

text: Signal

text: Process.

text: Lett.,

text: vol.

text: no.

text: pp.

text: Dec.

text: The

text: existing

text: fairness

text: models

text: exploit

text: multiple

text: parameters

text: to

text: A.

text: Benjebbovu,

text: A.

text: Li,

text: Y.

text: Saito,

text: Y.

text: Kishiyama,

text: A.

text: Harada,

text: and

text: T.

text: Naka-

text: adjust

text: the

text: fairness

text: level,

text: which

text: introduces

text: high

text: complexity

text: mura,

text: “System-level

text: performance

text: of

text: downlink

text: NOMA

text: for

text: future

text: LTE

text: enhancements,”

text: in

text: Proc.

text: IEEE

text: Global

text: Commun.

text: Conf.,

text: Dec.

text: pp.

text: in

text: determining

text: the

text: corresponding

text: RA

text: in

text: NOMA.

text: However,

text: α-fairness

text: uses

text: a

text: single

text: scalar,

text: denoted

text: by

text: α,

text: to

text: achieve

text: dif-

text: B.

text: Di,

text: L.

text: Song,

text: and

text: Y.

text: Li,

text: “Sub-channel

text: assignment,

text: power

text: allocation,

text: ferent

text: user

text: fairness

text: levels

text: and

text: well-known

text: efﬁciency-fairness

text: and

text: user

text: scheduling

text: for

text: non-orthogonal

text: multiple

text: access

text: networks,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: Nov.

text: tradeoffs

text: The

text: RA

text: can

text: be

text: investigated

text: for

text: sum

text: throughput

text: Z.

text: Chen,

text: Z.

text: Ding,

text: X.

text: Dai,

text: and

text: G.

text: K.

text: Karagiannidis,

text: “On

text: the

text: application

text: optimization

text: of

text: the

text: NOMA

text: system

text: with

text: fairness

text: constraints

text: to

text: of

text: quasi-degradation

text: to

text: MISO-NOMA

text: downlink,”

text: IEEE

text: Trans.

text: Signal

text: obtain

text: low-complexity

text: algorithms.

text: Processing,

text: vol.

text: no.

text: pp.

text: Dec.

text: E.

text: Security-Aware

text: RA

text: Z.

text: Chen

text: and

text: X.

text: Dai,

text: “MED

text: precoding

text: for

text: multiuser

text: MIMO-NOMA

text: downlink

text: transmission,”

text: IEEE

text: Trans.

text: Signal

text: Processing,

text: vol.

text: no.

text: NOMA-based

text: communication

text: comes

text: with

text: security

text: con-

text: pp.

text: Jun.

text: cerns,

text: as

text: the

text: user

text: with

text: strong

text: channel

text: condition

text: needs

text: to

text: L.

text: Lei,

text: D.

text: Yuan,

text: C.

text: K.

text: Ho,

text: and

text: S.

text: Sun,

text: “Power

text: and

text: channel

text: allocation

text: decode

text: the

text: signal

text: of

text: the

text: user

text: with

text: weak

text: channel

text: condition.

text: for

text: non-orthogonal

text: multiple

text: access

text: in

text: systems:

text: Tractability

text: and

text: computation,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: Thus,

text: when

text: the

text: weak

text: user

text: becomes

text: malicious

text: or

text: is

text: under

text: Dec.

text: attack,

text: the

text: signal

text: decoding

text: operations

text: of

text: both

text: strong

text: and

text: weak

text: M.

text: Zeng,

text: A.

text: Yadav,

text: O.

text: A.

text: Dobre,

text: and

text: H.

text: V.

text: Poor,

text: “Energy-efﬁcient

text: users

text: are

text: no

text: longer

text: reliable.

text: For

text: example,

text: the

text: attack

text: might

text: be

text: power

text: allocation

text: for

text: MIMO-NOMA

text: with

text: multiple

text: users

text: in

text: a

text: cluster,”

text: IEEE

text: Access,

text: vol.

text: PP,

text: no.

text: pp.

text: in

text: the

text: form

text: of

text: an

text: alteration

text: of

text: the

text: channel

text: quality

text: indicator

text: S.

text: Timotheou

text: and

text: I.

text: Krikidis,

text: “Fairness

text: for

text: non-orthogonal

text: multiple

text: during

text: feedback.

text: Therefore,

text: the

text: design

text: of

text: the

text: RA

text: schemes

text: access

text: in

text: systems,”

text: IEEE

text: Signal

text: Process.

text: Lett.,

text: vol.

text: no.

text: pp.

text: becomes

text: more

text: critical

text: in

text: the

text: presence

text: of

text: security

text: concerns.

text: To

text: Oct.

text: overcome

text: this

text: hurdle,

text: introducing

text: appropriate

text: physical

text: layer

text: F.

text: Fang,

text: H.

text: Zhang,

text: J.

text: Cheng,

text: and

text: V.

text: C.

text: M.

text: Leung,

text: “Energy-efﬁcient

text: re-

text: source

text: allocation

text: for

text: downlink

text: non-orthogonal

text: multiple

text: access

text: network,”

text: security

text: measures

text: is

text: necessary,

text: which

text: is

text: an

text: interesting

text: open

text: IEEE

text: Trans.

text: Commun.,

text: vol.

text: no.

text: pp.

text: Sep.

text: problem

text: for

text: the

text: research

text: community.

text: S.

text: Shi,

text: L.

text: Yang,

text: and

text: H.

text: Zhu,

text: “Outage

text: balancing

text: in

text: downlink

text: nonorthog-

text: VI.

text: CONCLUDING

text: REMARKS

text: onal

text: multiple

text: access

text: with

text: statistical

text: channel

text: state

text: information,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: Jul.

text: This

text: article

text: provides

text: an

text: overview

text: of

text: the

text: RA

text: algorithms

text: for

text: Z.

text: Yang,

text: Z.

text: Ding,

text: P.

text: Fan,

text: and

text: N.

text: Al-Dhahir,

text: “A

text: general

text: power

text: allocation

text: downlink

text: NOMA

text: in

text: a

text: categorized

text: fashion.

text: It

text: is

text: clear

text: that

text: the

text: scheme

text: to

text: guarantee

text: quality

text: of

text: service

text: in

text: downlink

text: and

text: uplink

text: NOMA

text: systems,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: RA

text: algorithms

text: play

text: a

text: pivotal

text: role

text: in

text: achieving

text: the

text: maximum

text: Nov.

text: beneﬁts

text: of

text: NOMA.

text: Additionally,

text: it

text: is

text: shown

text: that

text: the

text: fairness

text: Z.

text: Ding,

text: R.

text: Schober,

text: and

text: H.

text: V.

text: Poor,

text: “A

text: general

text: MIMO

text: framework

text: for

text: among

text: the

text: NOMA

text: clusters

text: can

text: be

text: attained

text: by

text: controlling

text: the

text: NOMA

text: downlink

text: and

text: uplink

text: transmission

text: based

text: on

text: signal

text: alignment,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: Jun.

text: sum

text: rate

text: gain

text: through

text: the

text: D-NLUPA

text: algorithm.

text: Moreover,

text: for

text: a

text: general

text: MIMO-NOMA

text: system

text: with

text: pre-deﬁned

text: QoS

text: M.

text: Zeng,

text: A.

text: Yadav,

text: O.

text: A.

text: Dobre,

text: G.

text: I.

text: Tsiropoulos,

text: and

text: H.

text: V.

text: Poor,

text: requirement

text: for

text: each

text: user,

text: simulations

text: are

text: conducted,

text: which

text: “Capacity

text: comparison

text: between

text: MIMO-NOMA

text: and

text: MIMO-OMA

text: with

text: multiple

text: users

text: in

text: a

text: cluster,”

text: IEEE

text: J.

text: Select.

text: Areas

text: Commun.,

text: vol.

text: show

text: that

text: MIMO-NOMA

text: outperforms

text: MIMO-OMA

text: in

text: terms

text: no.

text: pp.

text: Oct

text: of

text: outage

text: probability

text: and

text: effective

text: sum

text: rate

text: when

text: optimal

text: B.

text: Wang,

text: L.

text: Dai,

text: Z.

text: Wang,

text: N.

text: Ge,

text: and

text: S.

text: Zhou,

text: “Spectrum

text: and

text: energy

text: ef-

text: PA

text: is

text: applied

text: for

text: both.

text: Finally,

text: the

text: article

text: concludes

text: with

text: a

text: ﬁcient

text: beamspace

text: MIMO-NOMA

text: for

text: millimeter-wave

text: communications

text: discussion

text: on

text: open

text: issues,

text: including

text: joint

text: optimization

text: of

text: UP

text: using

text: lens

text: antenna

text: array,”

text: IEEE

text: J.

text: Select.

text: Areas

text: Commun.,

text: vol.

text: no.

text: pp.

text: Oct.

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text: 20 D. MARTÍ-PETE ∈ N ∈ Z Theorem 6.3. For every N and n , there exists a holomorphic self- C∗ (cid:62) map of with ind that is a transcendental entire function, if , f (f) = n n 0 or a transcendental meromorphic function, if , and has a cycle of hyperbolic n < 0 Baker domains of period . N Proof. Let ω := e2πi/N and deﬁne N √ ⊆ C \ D (cid:54) V := ωm N W for 0 m < N, m N where is the closed half-plane from Lemma 6.1. We denote by the union of W V (cid:54) R { ∈ C∗ all V for 0 m < N, and let R := −, if N is odd, or R := z : arg z = −m } , if is even. Then put π(1 1/N) N √ { − } d := min ( N 2 1)/3, dist(V, R)/4 , (6.2) and deﬁne the closed connected set { ∈ C (cid:62) (cid:62) } dist and dist (6.3) B := z : (z, V ) d (z, R) d , (cid:48) ⊆ which satisﬁes int (see Figure 9). B := D(1, d) B B f V 1 2π (cid:48) f p B V 0 R f V 2 f Figure 9. Sketch of the construction in the proof of Theorem 6.3 (cid:54) with . The sets and , , are shaded in grey. N =3 B V 0 m < N m ∪ Observe that the closed set satisﬁes the hypothesis of Lemma 4.5; F := B V C \ C \ ⊆ namely ˆ is connected and ˆ is locally connected at inﬁnity, and F F F W α with . We now deﬁne a function on : α = 2π gˆ F ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D.

text: MARTÍ-PETE

text: N

text: Z

text: Theorem

text: For

text: every

text: N

text: and

text: n

text: there

text: exists

text: a

text: holomorphic

text: self-

text: C∗

text: map

text: of

text: with

text: ind

text: that

text: is

text: a

text: transcendental

text: entire

text: function,

text: if

text: f

text: (f)

text: n

text: n

text: or

text: a

text: transcendental

text: meromorphic

text: function,

text: if

text: and

text: has

text: a

text: cycle

text: of

text: hyperbolic

text: n

text: Baker

text: domains

text: of

text: period

text: N

text: Proof.

text: Let

text: ω

text: and

text: deﬁne

text: N

text: C

text: D

text: V

text: ωm

text: N

text: W

text: for

text: m

text: N,

text: m

text: N

text: where

text: is

text: the

text: closed

text: half-plane

text: from

text: Lemma

text: We

text: denote

text: by

text: the

text: union

text: of

text: W

text: V

text: R

text: C∗

text: all

text: V

text: for

text: m

text: N,

text: and

text: let

text: R

text: if

text: N

text: is

text: odd,

text: or

text: R

text: z

text: arg

text: z

text: −m

text: if

text: is

text: even.

text: Then

text: put

text: N

text: d

text: min

text: N

text: dist(V,

text: and

text: deﬁne

text: the

text: closed

text: connected

text: set

text: C

text: dist

text: and

text: dist

text: B

text: z

text: (z,

text: V

text: d

text: (z,

text: R)

text: d

text: which

text: satisﬁes

text: int

text: (see

text: Figure

text: B

text: d)

text: B

text: B

text: f

text: V

text: 2π

text: f

text: p

text: B

text: V

text: R

text: f

text: V

text: f

text: Figure

text: Sketch

text: of

text: the

text: construction

text: in

text: the

text: proof

text: of

text: Theorem

text: with

text: The

text: sets

text: and

text: are

text: shaded

text: in

text: grey.

text: N

text: B

text: V

text: m

text: N

text: m

text: Observe

text: that

text: the

text: closed

text: set

text: satisﬁes

text: the

text: hypothesis

text: of

text: Lemma

text: F

text: B

text: V

text: C

text: C

text: namely

text: ˆ

text: is

text: connected

text: and

text: ˆ

text: is

text: locally

text: connected

text: at

text: inﬁnity,

text: and

text: F

text: F

text: F

text: W

text: α

text: with

text: We

text: now

text: deﬁne

text: a

text: function

text: on

text: α

text: 2π

text: gˆ

text: F

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 26 D. MARTÍ-PETE B∞ ∞ f f V 1 (cid:48) ˜ B 0 V˜ 0 V˜ 0 B∞(cid:48) 1 0 ∞ V ˜ 0 B 0 f f f f ∞ V 2 f Figure 11. Sketch of the construction of the function in the proof f ∞∞ ∞ of Theorem 1.2 with . e = 00 References [Ara64] N. U. Arakelyan, Uniform approximation on closed sets by entire functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187–1206. [Bak63] I. N. Baker, Multiply connected domains of normality in iteration theory, Math. Z. 81 (1963), 206–214. [Bak76] , An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A 22 (1976), no. 2, 173–176. [Bak84] , Wandering domains in the iteration of entire functions, Proc. London Math. Soc. (3) 49 (1984), no. 3, 563–576. [Bak87] , Wandering domains for maps of the punctured plane, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 2, 191–198. [BD98] I. N. Baker and P. Domínguez-Soto, Analytic self-maps of the punctured plane, Com- plex Variables Theory Appl. 37 (1998), no. 1-4, 67–91. [Ber93] W. Bergweiler, Iteration of meromorphic functions, Bulletin of the American Mathe- matical Society 29 (1993), no. 2, 151–188. [Ber95] , On the Julia set of analytic self-maps of the punctured plane, Analysis 15 (1995), no. 3, 251–256. [Bis15] C. J. Bishop, Constructing entire functions by quasiconformal folding, Acta Math. 214 (2015), no. 1, 1–60. [BRS13] W. Bergweiler, P. J. Rippon, and G. M. Stallard, Multiply connected wandering do- mains of entire functions, Proc. Lond. Math. Soc. (3) 107 (2013), no. 6, 1261–1301. [BZ12] W. Bergweiler and J.-H. Zheng, Some examples of Baker domains, Nonlinearity 25 (2012), no. 4, 1033–1044. [Cow81] C. C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), no. 1, 69–95. [EL92] A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: MARTÍ-PETE

text: B∞

text: f

text: f

text: V

text: B

text: V˜

text: V˜

text: V

text: B

text: f

text: f

text: f

text: f

text: V

text: f

text: Figure

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text: Bergweiler

text: and

text: J.-H.

text: Zheng,

text: Some

text: examples

text: of

text: Baker

text: domains,

text: Nonlinearity

text: no.

text: C.

text: C.

text: Cowen,

text: Iteration

text: and

text: the

text: solution

text: of

text: functional

text: equations

text: for

text: functions

text: analytic

text: in

text: the

text: unit

text: disk,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: no.

text: A.

text: E.

text: Eremenko

text: and

text: M.

text: Yu.

text: Lyubich,

text: Dynamical

text: properties

text: of

text: some

text: classes

text: of

text: entire

text: functions,

text: Ann.

text: Inst.

text: Fourier

text: (Grenoble)

text: no.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ Magnetotransport experiments on Weyl semimetals are essential for investigating the intriguing topological and low-energy properties of Weyl nodes. If the transport direction is perpendicular to the applied magnetic ﬁeld, experiments have shown a large positive magnetoresistance. In this work, we present a theoretical scattering matrix approach to transversal magnetotransport in a Weyl node. Our numerical method conﬁrms and goes beyond the existing perturbative analytical approach by treating disorder exactly. It is formulated in real space and is applicable to mesoscopic samples as well as in the bulk limit. In particular, we study the case of clean and strongly disordered samples. PACS numbers: I. INTRODUCTION transport of a single Weyl node Hamiltonian.26 He used the Kubo formula and Born approximation to include the eﬀect of weak disorder while focusing on the quan- Materials realizing a nodal dispersion, such as Dirac tum limit. Indeed, this approach results in a linear and and Weyl semimetals, are currently one of the main non-saturating magnetoresistance under the assumption research topics in condensed matter physics. Early proposals for materials featuring Weyl nodes1–3 were of screened Coulomb impurities. soon followed by spectroscopic experiments conﬁrming The recent experiments triggered a renewed interest the nodal bulk dispersions and hallmark surface states, in the problem leading to the appearance of a num- which appear in the form of Fermi arcs.4–7 Weyl ma- ber of theoretical studies reproducing and extending terials show peculiar magnetotransport properties8 that Abrikosov’s earlier results.27–34 Notably, the more re- are reﬂected in two distinct experimental observations. alistic situation with a pair of nodes was accounted First, when the electric ﬁeld is aligned parallel to the for in Refs. 27, 28, 30, and 33, while disorder was as- (cid:107) magnetic ﬁeld, i.e., E B, the observed unusual nega- sumed to be of (uncorrelated) Gaussian white noise type tive magnetoresistance9–16 is understood in terms of the in Refs. 28, 29, and 32 and to have a ﬁnite range in chiral anomaly.17,18 In a simpliﬁed picture, as a magnetic Ref. 30. However, most of the approaches in the recent ﬁeld turns the Weyl node into a highly degenerate chiral papers follow Abrikosov’s method by treating disorder state, backscattering has to occur at the inter-node level. using the Born approximation and computing the con- This is much less eﬀective than intra-node scattering rel- ductivity making use of the Kubo formula with bare evant for the B = 0 case and hence leads to an increased current vertices. Some slight but noteworthy variations conductivity. Second, and of main interest for the rest are the numerical evaluation of overlap integrals for dis- of the paper, we consider transversal magnetotransport, order scattering in Ref. 31 and the extension to the ⊥ i.e., E B, where the chiral anomaly is not operational. self-consistent Born approximation (SCBA) in Refs. 29 Experimentally, a large transversal magnetoresistivity and 32. Moreover, the role of vertex corrections was has been observed in a variety of Dirac and Weyl materi- scrutinized in Refs. 29 and 32. Staying in the quan- als at low temperatures around T = 2 K. Early studies of the Dirac semimetal Cd As 10,19–21 with multiple Dirac 3 2 cones were followed by measurements on the single-Dirac cone material TiBiSSe22 and Weyl materials NbAs23, Lead Lead TaAs24 and NbP.25 Note that a Dirac node splits into two Weyl nodes when a magnetic ﬁeld is applied. The transversal resistivity can increase up to a factor of a thousand compared to the (B = 0)-resistivity upon the application of a magnetic ﬁeld with a strength on the order of 10 T.22 The observed linear and unsaturated growth of the re- sistivity with the applied magnetic ﬁeld is in stark con- trast to Boltzmann magnetotransport theory in metals where the magnetoresistivity is much smaller, quadratic in B for small B and saturating if the cyclotron frequency FIG. 1. Sketch of modeled transport experiment. The mag- exceeds the scattering time. Thus, alternative theoreti- netic ﬁeld points in the z-direction perpendicular to the trans- cal approaches are necessary to explain the experimental port direction (x). The lower panel shows the doping distri- results. bution V (x) (green) that deﬁnes the leads, and the choice for Abrikosov studied the problem of transverse magne- the vector potential A (blue). The piecewise constant ap- y toresistance long before the conﬁrmation of Weyl and proximation of Ay, shown with a dotted dotted line, is used Dirac materials in the laboratory. He developed a per- in the scattering matrix approach for the situation including disorder U(r). turbative analytical approach for transversal magneto- ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Magnetotransport

label: abstract

text: experiments

text: on

text: Weyl

text: semimetals

text: are

text: essential

text: for

text: investigating

text: the

text: intriguing

text: topological

text: and

text: low-energy

text: properties

text: of

text: Weyl

text: nodes.

text: If

text: the

text: transport

text: direction

text: is

text: perpendicular

text: to

text: the

text: applied

text: magnetic

text: ﬁeld,

text: experiments

text: have

text: shown

text: a

text: large

text: positive

text: magnetoresistance.

text: In

text: this

text: work,

text: we

text: present

text: a

text: theoretical

text: scattering

text: matrix

text: approach

text: to

text: transversal

text: magnetotransport

text: in

text: a

text: Weyl

text: node.

text: Our

text: numerical

text: method

text: conﬁrms

text: and

text: goes

text: beyond

text: the

text: existing

text: perturbative

text: analytical

text: approach

text: by

text: treating

text: disorder

text: exactly.

text: It

text: is

text: formulated

text: in

text: real

text: space

text: and

text: is

text: applicable

text: to

text: mesoscopic

text: samples

text: as

text: well

text: as

text: in

text: the

text: bulk

text: limit.

text: In

text: particular,

text: we

text: study

text: the

text: case

text: of

text: clean

text: and

text: strongly

text: disordered

text: samples.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: transport

text: of

text: a

text: single

text: Weyl

text: node

text: He

text: used

text: the

text: Kubo

text: formula

text: and

text: Born

text: approximation

text: to

text: include

text: the

text: eﬀect

text: of

text: weak

text: disorder

text: while

text: focusing

text: on

text: the

text: quan-

text: Materials

text: realizing

text: a

text: nodal

text: dispersion,

text: such

text: as

text: Dirac

text: tum

text: limit.

text: Indeed,

text: this

text: approach

text: results

text: in

text: a

text: linear

text: and

text: and

text: Weyl

text: semimetals,

text: are

text: currently

text: one

text: of

text: the

text: main

text: non-saturating

text: magnetoresistance

text: under

text: the

text: assumption

text: research

text: topics

text: in

text: condensed

text: matter

text: physics.

text: Early

text: proposals

text: for

text: materials

text: featuring

text: Weyl

text: were

text: of

text: screened

text: Coulomb

text: impurities.

text: soon

text: followed

text: by

text: spectroscopic

text: experiments

text: conﬁrming

text: The

text: recent

text: experiments

text: triggered

text: a

text: renewed

text: interest

text: the

text: nodal

text: bulk

text: dispersions

text: and

text: hallmark

text: surface

text: states,

text: in

text: the

text: problem

text: leading

text: to

text: the

text: appearance

text: of

text: a

text: num-

text: which

text: appear

text: in

text: the

text: form

text: of

text: Fermi

text: Weyl

text: ma-

text: ber

text: of

text: theoretical

text: studies

text: reproducing

text: and

text: extending

text: terials

text: show

text: peculiar

text: magnetotransport

text: that

text: Abrikosov’s

text: earlier

text: Notably,

text: the

text: more

text: re-

text: are

text: reﬂected

text: in

text: two

text: distinct

text: experimental

text: observations.

text: alistic

text: situation

text: with

text: a

text: pair

text: of

text: nodes

text: was

text: accounted

text: First,

text: when

text: the

text: electric

text: ﬁeld

text: is

text: aligned

text: parallel

text: to

text: the

text: for

text: in

text: Refs.

text: and

text: while

text: disorder

text: was

text: as-

text: magnetic

text: ﬁeld,

text: i.e.,

text: E

text: B,

text: the

text: observed

text: unusual

text: nega-

text: sumed

text: to

text: be

text: of

text: (uncorrelated)

text: Gaussian

text: white

text: noise

text: type

text: tive

text: is

text: understood

text: in

text: terms

text: of

text: the

text: in

text: Refs.

text: and

text: and

text: to

text: have

text: a

text: ﬁnite

text: range

text: in

text: chiral

text: In

text: a

text: simpliﬁed

text: picture,

text: as

text: a

text: magnetic

text: Ref.

text: However,

text: most

text: of

text: the

text: approaches

text: in

text: the

text: recent

text: ﬁeld

text: turns

text: the

text: Weyl

text: node

text: into

text: a

text: highly

text: degenerate

text: chiral

text: papers

text: follow

text: Abrikosov’s

text: method

text: by

text: treating

text: disorder

text: state,

text: backscattering

text: has

text: to

text: occur

text: at

text: the

text: inter-node

text: level.

text: using

text: the

text: Born

text: approximation

text: and

text: computing

text: the

text: con-

text: This

text: is

text: much

text: less

text: eﬀective

text: than

text: intra-node

text: scattering

text: rel-

text: ductivity

text: making

text: use

text: of

text: the

text: Kubo

text: formula

text: with

text: bare

text: evant

text: for

text: the

text: B

text: case

text: and

text: hence

text: leads

text: to

text: an

text: increased

text: current

text: vertices.

text: Some

text: slight

text: but

text: noteworthy

text: variations

text: conductivity.

text: Second,

text: and

text: of

text: main

text: interest

text: for

text: the

text: rest

text: are

text: the

text: numerical

text: evaluation

text: of

text: overlap

text: integrals

text: for

text: dis-

text: of

text: the

text: paper,

text: we

text: consider

text: transversal

text: magnetotransport,

text: order

text: scattering

text: in

text: Ref.

text: and

text: the

text: extension

text: to

text: the

text: i.e.,

text: E

text: B,

text: where

text: the

text: chiral

text: anomaly

text: is

text: not

text: operational.

text: self-consistent

text: Born

text: approximation

text: (SCBA)

text: in

text: Refs.

text: Experimentally,

text: a

text: large

text: transversal

text: magnetoresistivity

text: and

text: Moreover,

text: the

text: role

text: of

text: vertex

text: corrections

text: was

text: has

text: been

text: observed

text: in

text: a

text: variety

text: of

text: Dirac

text: and

text: Weyl

text: materi-

text: scrutinized

text: in

text: Refs.

text: and

text: Staying

text: in

text: the

text: quan-

text: als

text: at

text: low

text: temperatures

text: around

text: T

text: K.

text: Early

text: studies

text: of

text: the

text: Dirac

text: semimetal

text: Cd

text: As

text: with

text: multiple

text: Dirac

text: cones

text: were

text: followed

text: by

text: measurements

text: on

text: the

text: single-Dirac

text: cone

text: material

text: and

text: Weyl

text: materials

text: Lead

text: Lead

text: and

text: Note

text: that

text: a

text: Dirac

text: node

text: splits

text: into

text: two

text: Weyl

text: nodes

text: when

text: a

text: magnetic

text: ﬁeld

text: is

text: applied.

text: The

text: transversal

text: resistivity

text: can

text: increase

text: up

text: to

text: a

text: factor

text: of

text: a

text: thousand

text: compared

text: to

text: the

text: (B

text: upon

text: the

text: application

text: of

text: a

text: magnetic

text: ﬁeld

text: with

text: a

text: strength

text: on

text: the

text: order

text: of

text: The

text: observed

text: linear

text: and

text: unsaturated

text: growth

text: of

text: the

text: re-

text: sistivity

text: with

text: the

text: applied

text: magnetic

text: ﬁeld

text: is

text: in

text: stark

text: con-

text: trast

text: to

text: Boltzmann

text: magnetotransport

text: theory

text: in

text: metals

text: where

text: the

text: magnetoresistivity

text: is

text: much

text: smaller,

text: quadratic

text: in

text: B

text: for

text: small

text: B

text: and

text: saturating

text: if

text: the

text: cyclotron

text: frequency

text: FIG.

text: Sketch

text: of

text: modeled

text: transport

text: experiment.

text: The

text: mag-

text: exceeds

text: the

text: scattering

text: time.

text: Thus,

text: alternative

text: theoreti-

text: netic

text: ﬁeld

text: points

text: in

text: the

text: z-direction

text: perpendicular

text: to

text: the

text: trans-

text: cal

text: approaches

text: are

text: necessary

text: to

text: explain

text: the

text: experimental

text: port

text: direction

text: (x).

text: The

text: lower

text: panel

text: shows

text: the

text: doping

text: distri-

text: results.

text: bution

text: V

text: (x)

text: (green)

text: that

text: deﬁnes

text: the

text: leads,

text: and

text: the

text: choice

text: for

text: Abrikosov

text: studied

text: the

text: problem

text: of

text: transverse

text: magne-

text: the

text: vector

text: potential

text: A

text: (blue).

text: The

text: piecewise

text: constant

text: ap-

text: y

text: toresistance

text: long

text: before

text: the

text: conﬁrmation

text: of

text: Weyl

text: and

text: proximation

text: of

text: Ay,

text: shown

text: with

text: a

text: dotted

text: dotted

text: line,

text: is

text: used

text: Dirac

text: materials

text: in

text: the

text: laboratory.

text: He

text: developed

text: a

text: per-

text: in

text: the

text: scattering

text: matrix

text: approach

text: for

text: the

text: situation

text: including

text: disorder

text: U(r).

text: turbative

text: analytical

text: approach

text: for

text: transversal

text: magneto-

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 (a) The two spinors belong to the left- and right-propagating modes in the leads, i.e., the eigenvectors of the velocity operator vσ . To obtain solutions for t, we redeﬁne ≡ x ± ψ±(x) ψ↑(x) ψ↓(x), such that the coupled system of equations reads (cid:0) (cid:1) − − ∓ ∂ ψ±(x) = k xl 2 ik ψ∓(x), (8) x y B z (b) and the bounda√ry conditions become ψ−(L/2) = 0 and − √ ψ ( L/2) = 2/ 2. The transmission coeﬃcient t = + ψ (L/2)/ 2 can be found by numerically solving Eq. (8). + In the inset of Fig. 2(a), the transmission eigenvalue | | ≡ T = t 2 is shown as a function of (k , k ) k⊥ for y z L/l = 3. We observe that the presence of the magnetic B ﬁeld causes a gauge-dependent k -k asymmetry whereas − y | z | the B = 0 result T = cosh 2 (L k⊥ )35 is rotationally (c) symmetric. Nevertheless, the k⊥ = 0 mode still shows perfect transmission. The conductance G is calculated via the Landauer formula (cid:2) (cid:3) e2 † G = tr tt , (9) h and(cid:0)is (cid:1)shown in Fig. 2(a) where the dotted line rep- → resents the known result in the limit B 0, G lB2 = W2 eh2 l2nπ2 lLB 235 with the sy(cid:29)stem’s transversal width W = W = W . For L/l 1, when the magnetic ﬁeld y z B becomes relevant, the conductance vanishes rapidly FIG. 2. Numerical results for clean transversal magneto- winitFhigL. /2l(Bc). reTvehaelspaloftunocftilonn[Gal(floBr/mWG)2WlB2h2/e∼2]e−vαs.(LL/l/Bl)B2 tltirrnaaennssinpmdoiirsctsa.iot(enas)etihCgeoennBvdau=lcut0earnTecseuavlsterGasuf∼usnsLcy−tsit2oe.nmTohlfeentgirnatshne,stvtshehresoadwlostmttheode- with α = 1/4 (dotted line). menta for L/l = 3. (b) The Fano factor versus system B Although we did not succeed in d(cid:29)eriving an analytical length. The dotted line corresponds to the B = 0 result, expression for G in the limit L/l 1, the exponential F = 0.57. (c) Conductance versus (L/l )2. For the case B (cid:29) B suppression can be understood from the form of the bulk L/l 1, the logarithm of the conductance obeys a linear B − wave functions in the vicinity of zero energy. The zeroth relation with (L/lB)2. The slope of the dotted line is 1/4. Landau level is chiral and (cid:18)reads (cid:19) ∝ −1 − |↑(cid:105) ψ0,+ (r) eikyy+ikzz exp 2 [x/lB kylB]2 (10) is a measure of shot noise in quantum t(cid:28)ransport38 and is shown in Fig. 2(b). In the limit L l , we ﬁnd B (cid:126) |↑(cid:105) F = 0.57 in agreement with the pseudo-ballistic regime with energy ε = + vk and = (1, 0)T. The x 0,+ z described in Ref. 35. With increasing ﬁeld B, the Fano position of the exponentially localized wave function is factor also increases and reaches up to F = 0.74 in the determined by the transversal momentum k . Due to y large-B regime. translational invariance (i.e., the absence of disorder), k y Upon the inclusion of disorder, i.e., K > 0, the mo- is conserved and coherent transport between the leads menta are no longer conserved and we expect a diﬀu- only proceeds via the exponential tail of the orbitals leak- sion process between localized orbitals leading to drastic ing across the sample. This qualitatively explains the ob- change of transport behavior. Indeed, we show in the served exponential suppression of the conductance with next section that in the large system limit the transport length. with K > 0 becomes diﬀusive. The Fano factor (cid:2) (cid:0) (cid:1)(cid:3) IV. NUMERICAL MAGNETOTRANSPORT IN THE PRESENCE OF DISORDER † − † tr tt 1 tt F = † (11) tr [tt ] We use a stepwise strategy to ﬁnd the scattering ma- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: (a)

font: default

text: The

text: two

text: spinors

text: belong

text: to

text: the

text: left-

text: and

text: right-propagating

text: modes

text: in

text: the

text: leads,

text: i.e.,

text: the

text: eigenvectors

text: of

text: the

text: velocity

text: operator

text: vσ

text: To

text: obtain

text: solutions

text: for

text: t,

text: we

text: redeﬁne

text: x

text: ψ±(x)

text: ψ↑(x)

text: ψ↓(x),

text: such

text: that

text: the

text: coupled

text: system

text: of

text: equations

text: reads

label: equation

text: ψ±(x)

text: k

text: xl

text: ik

text: ψ∓(x),

text: x

text: y

text: B

text: z

text: (b)

text: and

text: the

text: bounda√ry

text: conditions

text: become

text: and

text: ψ

text: The

text: transmission

text: coeﬃcient

text: t

text: ψ

text: can

text: be

text: found

text: by

text: numerically

text: solving

text: Eq.

text: In

text: the

text: inset

text: of

text: Fig.

text: the

text: transmission

text: eigenvalue

text: T

text: t

text: is

text: shown

text: as

text: a

text: function

text: of

text: (k

text: k

text: k⊥

text: for

text: y

text: z

text: L/l

text: We

text: observe

text: that

text: the

text: presence

text: of

text: the

text: magnetic

text: B

text: ﬁeld

text: causes

text: a

text: gauge-dependent

text: k

text: -k

text: asymmetry

text: whereas

text: y

text: z

text: the

text: B

text: result

text: T

text: cosh

text: (L

text: k⊥

text: is

text: rotationally

text: (c)

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text: Feng,

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text: D.

text: Wu,

text: Z.

text: Wang,

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text: J.

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