text: 2 In the ﬁrst application, we show that with a sophisticated cost function choice, buffer sizes of particular queues can be steered towards beneﬁcial operating points. The choice of the underlying cost function is crucial for the performance of the resulting control policy. Therefore, to numerically compare different cost functions, we deﬁne the notion of queue outage, which both incorporates buffer overﬂows to capture effects of high packet holding costs or even dropped packets as well as buffer underﬂows to capture service interruptions. In the second application, we use our framework to design a control policy for an energy constrained wireless network, where trafﬁc is routed via battery limited intermediate nodes. We assume that when a node has no trafﬁc in each of its queues, it can switch to an energy-saving mode. Therefore, the goal is to maximize the “down time” of as many nodes as possible, while guaranteeing stability of the network. Eventually, in the third application, we consider cross-layer optimization of wireless networks, and we combine network- layer control with physical layer interference mitigation. Thereby, we demonstrate how our framework can be combined with physical layer power control based on successive convex approximation (SCA). B. Related Work After the seminal work in [1], signiﬁcant research activities followed. Many focussed on delay reduction of backpressure based policies, such as [3]. A general class of throughput optimal policies with improved delay performance has been recently presented in [5], and a general survey on recent policy synthesis techniques can be found, for example, in [8]. Considering queueing networks, previous works consider underﬂow constraints mainly in broadcast or simpler networks [9]. For example, in [9], the problem is investigated in a network of multiple transmitter-receiver pairs with cooperating transmitters. However, since it requires that buffer levels for each user progress independently of other users, it cannot be easily generalized to the case of arbitrary (multihop) networks (not even to a simple tandem network). Recently, there has been some interest in queueing network control with an arbitrary underlying cost-metric [10], allowing to incorporate application-dependent constraints in the control policy. However, stability and also cost performance crucially depend on the parameter choice, and this parameter choice has to be found in simulations. The MaxWeight policy is also frequently applied in wireless cross-layer control problems, in particular, in joint routing and power control problems (see, e.g., [11]). A common assumption on the physical layer is a fully coupled wireless network, where nodes treat interference as noise. In this case, power control results in a weighted sum-rate maximization problem which is known to be NP-hard [12]. In [13], an SCA method was introduced as a means to treat the inherent non-convexity. Compared to traditional power control in wireless networks [14], it eliminates the need to use the common unrealistic high- SINR approximation together with a log-transform of variables (see, e.g., [15]). In [16], a cross-layer control formulation was used based on the MaxWeight technique. C. Organization The rest of the paper is organized as follows. The system model is described in Section , and important preliminaries are summarized in Section . The µ-MaxWeight policy, together with important stability results, is introduced in Section . In Section , we treat the important issue of cost-function choice, and we show, how an appropriate choice can be used to steer buffers towards certain target values. In Section , we apply our framework in different scenarios, such as multimedia trafﬁc, energy constrained nodes, and cross-layer network control. In Section , we draw some relevant conclusions. D. Notation We use boldface letters to denote vectors as well as matrices, and common letters with subscript are the elements, such that A is the i-th element of vector A and B is the element in row i and column j of matrix B. Moreover, AT refers to i ij E{ } the transpose of A. X denotes the expected value of random variable X. Let I denote the identity matrix of appropriate dimension. Furthermore, we denote 1 the vector of all ones. diag(a , a , ...) refers to a diagonal matrix built from the elements 1 2 (cid:107) · (cid:107) (cid:107) (cid:107) A a , a , ..., denotes the l vector norm, and x is an arbitrary norm. Furthermore, we use c to denote the complement 1 2 i i A A {A} (cid:104)· ·(cid:105) I{·} of a set . The probability of is denoted as Pr . , denotes the scalar product of two vectors. The indicator takes the value 1 if the argument is true, and it takes the value 0 otherwise.

text: In

text: the

text: ﬁrst

text: application,

text: we

text: show

text: that

text: with

text: a

text: sophisticated

text: cost

text: function

text: choice,

text: buffer

text: sizes

text: of

text: particular

text: queues

text: can

text: be

text: steered

text: towards

text: beneﬁcial

text: operating

text: points.

text: The

text: choice

text: of

text: the

text: underlying

text: cost

text: function

text: is

text: crucial

text: for

text: the

text: performance

text: of

text: the

text: resulting

text: control

text: policy.

text: Therefore,

text: to

text: numerically

text: compare

text: different

text: cost

text: functions,

text: we

text: deﬁne

text: the

text: notion

text: of

text: queue

text: outage,

text: which

text: both

text: incorporates

text: buffer

text: overﬂows

text: to

text: capture

text: effects

text: of

text: high

text: packet

text: holding

text: costs

text: or

text: even

text: dropped

text: packets

text: as

text: well

text: as

text: buffer

text: underﬂows

text: to

text: capture

text: service

text: interruptions.

text: In

text: the

text: second

text: application,

text: we

text: use

text: our

text: framework

text: to

text: design

text: a

text: control

text: policy

text: for

text: an

text: energy

text: constrained

text: wireless

text: network,

text: where

text: trafﬁc

text: is

text: routed

text: via

text: battery

text: limited

text: intermediate

text: nodes.

text: We

text: assume

text: that

text: when

text: a

text: node

text: has

text: no

text: trafﬁc

text: in

text: each

text: of

text: its

text: queues,

text: it

text: can

text: switch

text: to

text: an

text: energy-saving

text: mode.

text: Therefore,

text: the

text: goal

text: is

text: to

text: maximize

text: the

text: “down

text: time”

text: of

text: as

text: many

text: nodes

text: as

text: possible,

text: while

text: guaranteeing

text: stability

text: of

text: the

text: network.

text: Eventually,

text: in

text: the

text: third

text: application,

text: we

text: consider

text: cross-layer

text: optimization

text: of

text: wireless

text: networks,

text: and

text: we

text: combine

text: network-

text: layer

text: control

text: with

text: physical

text: layer

text: interference

text: mitigation.

text: Thereby,

text: we

text: demonstrate

text: how

text: our

text: framework

text: can

text: be

text: combined

text: with

text: physical

text: layer

text: power

text: control

text: based

text: on

text: successive

text: convex

text: approximation

text: (SCA).

text: B.

text: Related

text: Work

text: After

text: the

text: seminal

text: work

text: in

text: signiﬁcant

text: research

text: activities

text: followed.

text: Many

text: focussed

text: on

text: delay

text: reduction

text: of

text: backpressure

text: based

text: policies,

text: such

text: as

text: A

text: general

text: class

text: of

text: throughput

text: optimal

text: policies

text: with

text: improved

text: delay

text: performance

text: has

text: been

text: recently

text: presented

text: in

text: and

text: a

text: general

text: survey

text: on

text: recent

text: policy

text: synthesis

text: techniques

text: can

text: be

text: found,

text: for

text: example,

text: in

text: Considering

text: queueing

text: networks,

text: previous

text: works

text: consider

text: underﬂow

text: constraints

text: mainly

text: in

text: broadcast

text: or

text: simpler

text: networks

text: For

text: example,

text: in

text: the

text: problem

text: is

text: investigated

text: in

text: a

text: network

text: of

text: multiple

text: transmitter-receiver

text: pairs

text: with

text: cooperating

text: transmitters.

text: However,

text: since

text: it

text: requires

text: that

text: buffer

text: levels

text: for

text: each

text: user

text: progress

text: independently

text: of

text: other

text: users,

text: it

text: cannot

text: be

text: easily

text: generalized

text: to

text: the

text: case

text: of

text: arbitrary

text: (multihop)

text: networks

text: (not

text: even

text: to

text: a

text: simple

text: tandem

text: network).

text: Recently,

text: there

text: has

text: been

text: some

text: interest

text: in

text: queueing

text: network

text: control

text: with

text: an

text: arbitrary

text: underlying

text: cost-metric

text: allowing

text: to

text: incorporate

text: application-dependent

text: constraints

text: in

text: the

text: control

text: policy.

text: However,

text: stability

text: and

text: also

text: cost

text: performance

text: crucially

text: depend

text: on

text: the

text: parameter

text: choice,

text: and

text: this

text: parameter

text: choice

text: has

text: to

text: be

text: found

text: in

text: simulations.

text: The

text: MaxWeight

text: policy

text: is

text: also

text: frequently

text: applied

text: in

text: wireless

text: cross-layer

text: control

text: problems,

text: in

text: particular,

text: in

text: joint

text: routing

text: and

text: power

text: control

text: problems

text: (see,

text: e.g.,

text: A

text: common

text: assumption

text: on

text: the

text: physical

text: layer

text: is

text: a

text: fully

text: coupled

text: wireless

text: network,

text: where

text: nodes

text: treat

text: interference

text: as

text: noise.

text: In

text: this

text: case,

text: power

text: control

text: results

text: in

text: a

text: weighted

text: sum-rate

text: maximization

text: problem

text: which

text: is

text: known

text: to

text: be

text: NP-hard

text: In

text: an

text: SCA

text: method

text: was

text: introduced

text: as

text: a

text: means

text: to

text: treat

text: the

text: inherent

text: non-convexity.

text: Compared

text: to

text: traditional

text: power

text: control

text: in

text: wireless

text: networks

text: it

text: eliminates

text: the

text: need

text: to

text: use

text: the

text: common

text: unrealistic

text: high-

text: SINR

text: approximation

text: together

text: with

text: a

text: log-transform

text: of

text: variables

text: (see,

text: e.g.,

text: In

text: a

text: cross-layer

text: control

text: formulation

text: was

text: used

text: based

text: on

text: the

text: MaxWeight

text: technique.

text: C.

text: Organization

text: The

text: rest

text: of

text: the

text: paper

text: is

text: organized

text: as

text: follows.

text: The

text: system

text: model

text: is

text: described

text: in

text: Section

text: and

text: important

text: preliminaries

text: are

text: summarized

text: in

text: Section

text: The

text: µ-MaxWeight

text: policy,

text: together

text: with

text: important

text: stability

text: results,

text: is

text: introduced

text: in

text: Section

text: In

text: Section

text: we

text: treat

text: the

text: important

text: issue

text: of

text: cost-function

text: choice,

text: and

text: we

text: show,

text: how

text: an

text: appropriate

text: choice

text: can

text: be

text: used

text: to

text: steer

text: buffers

text: towards

text: certain

text: target

text: values.

text: In

text: Section

text: we

text: apply

text: our

text: framework

text: in

text: different

text: scenarios,

text: such

text: as

text: multimedia

text: trafﬁc,

text: energy

text: constrained

text: nodes,

text: and

text: cross-layer

text: network

text: control.

text: In

text: Section

text: we

text: draw

text: some

text: relevant

text: conclusions.

text: D.

text: Notation

text: We

text: use

text: boldface

text: letters

text: to

text: denote

text: vectors

text: as

text: well

text: as

text: matrices,

text: and

text: common

text: letters

text: with

text: subscript

text: are

text: the

text: elements,

text: such

text: that

text: A

text: is

text: the

text: i-th

text: element

text: of

text: vector

text: A

text: and

text: B

text: is

text: the

text: element

text: in

text: row

text: i

text: and

text: column

text: j

text: of

text: matrix

text: B.

text: Moreover,

text: AT

text: refers

text: to

text: i

text: ij

text: E{

text: the

text: transpose

text: of

text: A.

text: X

text: denotes

text: the

text: expected

text: value

text: of

text: random

text: variable

text: X.

text: Let

text: I

text: denote

text: the

text: identity

text: matrix

text: of

text: appropriate

text: dimension.

text: Furthermore,

text: we

text: denote

text: the

text: vector

text: of

text: all

text: ones.

text: diag(a

text: a

text: refers

text: to

text: a

text: diagonal

text: matrix

text: built

text: from

text: the

text: elements

text: A

text: a

text: a

text: denotes

text: the

text: l

text: vector

text: norm,

text: and

text: x

text: is

text: an

text: arbitrary

text: norm.

text: Furthermore,

text: we

text: use

text: c

text: to

text: denote

text: the

text: complement

text: i

text: i

text: A

text: A

text: {A}

text: I{·}

text: of

text: a

text: set

text: The

text: probability

text: of

text: is

text: denoted

text: as

text: Pr

text: denotes

text: the

text: scalar

text: product

text: of

text: two

text: vectors.

text: The

text: indicator

text: takes

text: the

text: value

text: if

text: the

text: argument

text: is

text: true,

text: and

text: it

text: takes

text: the

text: value

text: otherwise.


text: 11 6000 3000 µ−MaxWeight µ−MaxWeight 5000 MaxWeight 2500 MaxWeight 4000 2000 3000 1500 2000 1000 1000 500 0 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 Mean Arrival Rate (α) Mean Arrival Rate (α) (a) Equal Link Capacities (b) Unequal Link Capacities Fig. 5. Sum idle time (aggregated over all relevant queues) as a measure of energy efﬁciency, plotted over the average arrival rate α. The performance of the µ-MaxWeight policy with a linear cost function is compared to the MaxWeight baseline algorithm. This required behaviour is certainly not achieved when the MaxWeight policy is used, which, by contrast, uses all routes in an attempt to balance buffer states. Therefore, we apply the µ-MaxWeight framework with a suitable cost function in order to obtain the desired behaviour at low trafﬁc load. A straight forward cost function choice in this setup is the linear cost function (11). Let c be the cost factor associated with the buffer of node i. In order to force the trafﬁc towards the central route, we i ∈ { } ∈ { } should choose our cost weights as c >> c , for j 2, 4, 5, 7 , i 3, 6 . j i We simulate the system of Figure 4 with the following parameters. The arrival rate α to the queue associated with node 1 ∈ is varied in the interval of α [0.1, 0.5]. All other queues have no external arrivals. As a benchmark, we use the MaxWeight policy again. The results are depicted in Figure 5. T(cid:88)he ﬁ(cid:88)gure shows the total aggregated idle time, deﬁned as T I{ } q (t) = 0 (18) m t=1 m ∈ over the load α. Thereby, we consider only the energy-limited intermediate nodes, that is, m [2, 7] in (18). In Figure 5(a), we set the capacities of all links to 0.5, while in Figure 5(b) we set the capacities of the central route higher than those of the upper and lower route. Clearly, the proposed control framework outperforms MaxWeight in terms of energy efﬁciency. Even in case of equal routes, where the relative gains are smaller, the performance is still signiﬁcantly better. Note that as the trafﬁc load increases, the behaviour of µ-MaxWeight gets closer to that of MaxWeight, which is necessary to ensure stability. C. Application III: Cost-Based Cross-Layer Control As a last example, we show how the proposed network control framwork can be combined with physical layer resource allocation in a common cross-layer control problem. While in network models with ﬁxed link capacities the coupling of link rates due to interference is usually not considered, we here assume a fully coupled wireless network, where nodes treat interference as noise. To cope with the non-convex power control problem, we apply the SCA method from [13]. In [16], cross-layer control based on MaxWeight was already successfully combined with the SCA technique. N Consider a wireless multihop network comprising a set of nodes . Trafﬁc can potentially be generated at all nodes in the network and is categorized by its destination. That is, trafﬁc for node c is called commodity-c trafﬁc. Nodes are connected ∈ L ∈ N O by links l , which are deﬁned as ordered node tuples (i, j), with i, j . Furthermore, we denote (j) the set of all I outgoing links of node j. The set of all incoming links is accordingly deﬁned as (l). We deﬁne o(l) as the transmitter of link l and d(l) as the receiver of link l. Let g be the path gain from node o(l) to node d(m). Thus, the gain of link l is denoted lm g . Accordingly, let d be the distance between node i and node j. A frequently used simpliﬁcation for static gains is that, ll ij ∈ R depending on their distance, the link between two nodes can be modeled as: g = 1 , with ρ being the path-loss ij (dij)ρ emiT eldI muS emiT eldI muS ##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##

label: paragraph

font: default

text: µ−MaxWeight

text: µ−MaxWeight

text: MaxWeight

text: MaxWeight

text: Mean

text: Arrival

text: Rate

text: (α)

text: Mean

text: Arrival

text: Rate

text: (α)

text: (a)

text: Equal

text: Link

text: Capacities

text: (b)

text: Unequal

text: Link

text: Capacities

text: Fig.

text: Sum

label: caption

text: idle

text: time

text: (aggregated

text: over

text: all

text: relevant

text: queues)

text: as

text: a

text: measure

text: of

text: energy

text: efﬁciency,

text: plotted

text: over

text: the

text: average

text: arrival

text: rate

text: α.

text: The

text: performance

text: of

text: the

text: µ-MaxWeight

text: policy

text: with

text: a

text: linear

text: cost

text: function

text: is

text: compared

text: to

text: the

text: MaxWeight

text: baseline

text: algorithm.

text: This

text: required

text: behaviour

text: is

text: certainly

text: not

text: achieved

text: when

text: the

text: MaxWeight

text: policy

text: is

text: used,

text: which,

text: by

text: contrast,

text: uses

text: all

text: routes

text: in

text: an

text: attempt

text: to

text: balance

text: buffer

text: states.

text: Therefore,

text: we

text: apply

text: the

text: µ-MaxWeight

text: framework

text: with

text: a

text: suitable

text: cost

text: function

text: in

text: order

text: to

text: obtain

text: the

text: desired

text: behaviour

text: at

text: low

text: trafﬁc

text: load.

text: A

text: straight

text: forward

text: cost

text: function

text: choice

text: in

text: this

text: setup

text: is

text: the

text: linear

text: cost

text: function

text: Let

text: c

text: be

text: the

text: cost

text: factor

text: associated

text: with

text: the

text: buffer

text: of

text: node

text: i.

text: In

text: order

text: to

text: force

text: the

text: trafﬁc

text: towards

text: the

text: central

text: route,

text: we

text: i

text: should

text: choose

text: our

text: cost

text: weights

text: as

text: c

text: c

text: for

text: j

text: i

text: j

text: i

text: We

text: simulate

text: the

text: system

text: of

text: Figure

text: with

text: the

text: following

text: parameters.

text: The

text: arrival

text: rate

text: α

text: to

text: the

text: queue

text: associated

text: with

text: node

text: is

text: varied

text: in

text: the

text: interval

text: of

text: α

text: All

text: other

text: queues

text: have

text: no

text: external

text: arrivals.

text: As

text: a

text: benchmark,

text: we

text: use

text: the

text: MaxWeight

text: policy

text: again.

text: The

text: results

text: are

text: depicted

text: in

text: Figure

text: shows

text: the

text: total

text: aggregated

text: idle

text: time,

text: deﬁned

text: as

text: T

text: I{

text: q

text: (t)

text: m

text: m

text: over

text: the

text: load

text: α.

text: Thereby,

text: we

text: consider

text: only

text: the

text: energy-limited

text: intermediate

text: nodes,

text: that

text: is,

text: m

text: in

text: In

text: Figure

text: we

text: set

text: the

text: capacities

text: of

text: all

text: links

text: to

text: while

text: in

text: Figure

text: we

text: set

text: the

text: capacities

text: of

text: the

text: central

text: route

text: higher

text: than

text: those

text: of

text: the

text: upper

text: and

text: lower

text: route.

text: Clearly,

text: the

text: proposed

text: control

text: framework

text: outperforms

text: MaxWeight

text: in

text: terms

text: of

text: energy

text: efﬁciency.

text: Even

text: in

text: case

text: of

text: equal

text: routes,

text: where

text: the

text: relative

text: gains

text: are

text: smaller,

text: the

text: performance

text: is

text: still

text: signiﬁcantly

text: better.

text: Note

text: that

text: as

text: the

text: trafﬁc

text: load

text: increases,

text: the

text: behaviour

text: of

text: µ-MaxWeight

text: gets

text: closer

text: to

text: that

text: of

text: MaxWeight,

text: which

text: is

text: necessary

text: to

text: ensure

text: stability.

text: C.

text: Application

text: III:

text: Cost-Based

text: Cross-Layer

text: Control

text: As

text: a

text: last

text: example,

text: we

text: show

text: how

text: the

text: proposed

text: network

text: control

text: framwork

text: can

text: be

text: combined

text: with

text: physical

text: layer

text: resource

text: allocation

text: in

text: a

text: common

text: cross-layer

text: control

text: problem.

text: While

text: in

text: network

text: models

text: with

text: ﬁxed

text: link

text: capacities

text: the

text: coupling

text: of

text: link

text: rates

text: due

text: to

text: interference

text: is

text: usually

text: not

text: considered,

text: we

text: here

text: assume

text: a

text: fully

text: coupled

text: wireless

text: network,

text: where

text: nodes

text: treat

text: interference

text: as

text: noise.

text: To

text: cope

text: with

text: the

text: non-convex

text: power

text: control

text: problem,

text: we

text: apply

text: the

text: SCA

text: method

text: from

text: In

text: cross-layer

text: control

text: based

text: on

text: MaxWeight

text: was

text: already

text: successfully

text: combined

text: with

text: the

text: SCA

text: technique.

text: N

text: Consider

text: a

text: wireless

text: multihop

text: network

text: comprising

text: a

text: set

text: of

text: nodes

text: Trafﬁc

text: can

text: potentially

text: be

text: generated

text: at

text: all

text: nodes

text: in

text: the

text: network

text: and

text: is

text: categorized

text: by

text: its

text: destination.

text: That

text: is,

text: trafﬁc

text: for

text: node

text: c

text: is

text: called

text: commodity-c

text: trafﬁc.

text: Nodes

text: are

text: connected

text: L

text: N

text: O

text: by

text: links

text: l

text: which

text: are

text: deﬁned

text: as

text: ordered

text: node

text: tuples

text: (i,

text: j),

text: with

text: i,

text: j

text: Furthermore,

text: we

text: denote

text: (j)

text: the

text: set

text: of

text: all

text: I

text: outgoing

text: links

text: of

text: node

text: j.

text: The

text: set

text: of

text: all

text: incoming

text: links

text: is

text: accordingly

text: deﬁned

text: as

text: (l).

text: We

text: deﬁne

text: o(l)

text: as

text: the

text: transmitter

text: of

text: link

text: l

text: and

text: d(l)

text: as

text: the

text: receiver

text: of

text: link

text: l.

text: Let

text: g

text: be

text: the

text: path

text: gain

text: from

text: node

text: o(l)

text: to

text: node

text: d(m).

text: Thus,

text: the

text: gain

text: of

text: link

text: l

text: is

text: denoted

text: lm

text: g

text: Accordingly,

text: let

text: d

text: be

text: the

text: distance

text: between

text: node

text: i

text: and

text: node

text: j.

text: A

text: frequently

text: used

text: simpliﬁcation

text: for

text: static

text: gains

text: is

text: that,

text: ll

text: ij

text: R

text: depending

text: on

text: their

text: distance,

text: the

text: link

text: between

text: two

text: nodes

text: can

text: be

text: modeled

text: as:

text: g

text: with

text: ρ

text: being

text: the

text: path-loss

text: ij

text: (dij)ρ

text: emiT

text: eldI

text: muS

text: emiT

text: eldI

text: muS

text: ##LTFigure##

text: ##LTFigure##

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text: LIE ALGEBRAS AND GEOMETRIC COMPLEXITY OF AN ISOCHRONOUS CENTER CONDITION 11 the following conjecture: Conjecture: The number glin(d) is equal to (d, 1) corresponding to holomorphic isochronous center. References [1] J. Chavarriga, M. Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst. 1 (1999), 1-70. [2] R.Conti, Centers of planar polynomial systems. A review. LE MATEMATCHE Vol. LIII(1998)- Fasc.II, pp.207-240. [3] Ecalle J., Singularités non abordables par la géométrie, Ann. Inst. Fourier, 42 (1-2), 1992, 73-164. [4] Ecalle J., Schlomiuk D., The nilpotent and distinguished form of resonant vector ﬁelds or diﬀeo- morphisms, Ann. Inst. Fourier 43 5 (1993) 1407-1483. [5] J. Ecalle, B. Vallet, Correction and linearization of resonant vector ﬁelds and diﬀeomorphisms, Math. Z. 229 (1998) 249-318. [6] J-P. Francoise, R. Pons, Les conditions du centre pour un champ de vecteurs quadratiques, CRAS 318 (1994), p. 909-912. [7] J-P. Francoise, Géométrie analytique et systèmes dynamiques, PUF, 1995. [8] W.S. Loud, Behaviour of the period of solutions of certain plane autonomous systems near centers, Contrib. Diﬀerential Equations 3 (1964) 21-36. [9] C. Reutenauer, Free Lie algebras, London Math. Soc. Monographs, new series 7 (1993). [10] B. Schuman, Correction et linéarisation des champs de vecteurs polynomiaux,unpublished, 1999. [11] J-P. Serre, Lie algebras and Lie groups, W.C. Benjamin Inc (1965). [12] H.Zoladek, Quadratic systems with center and their perturbations, Journal of diﬀerential equations 109,p.223-273, 1994. Jacky Cresson Jordy Palafox ##LTLine##

text: LIE

text: ALGEBRAS

text: AND

text: GEOMETRIC

text: COMPLEXITY

text: OF

text: AN

text: ISOCHRONOUS

text: CENTER

text: CONDITION

text: the

text: following

text: conjecture:

text: Conjecture:

text: The

text: number

text: glin(d)

text: is

text: equal

text: to

text: (d,

text: corresponding

text: to

text: holomorphic

text: isochronous

text: center.

text: References

label: reference

text: J.

text: Chavarriga,

text: M.

text: Sabatini,

text: A

text: survey

text: of

text: isochronous

text: centers,

text: Qual.

text: Theory

text: Dyn.

text: Syst.

text: R.Conti,

text: Centers

text: of

text: planar

text: polynomial

text: systems.

text: A

text: review.

text: LE

text: MATEMATCHE

text: Vol.

text: Fasc.II,

text: Ecalle

text: J.,

text: Singularités

text: non

text: abordables

text: par

text: la

text: géométrie,

text: Ann.

text: Inst.

text: Fourier,

text: Ecalle

text: J.,

text: Schlomiuk

text: D.,

text: The

text: nilpotent

text: and

text: distinguished

text: form

text: of

text: resonant

text: vector

text: ﬁelds

text: or

text: diﬀeo-

text: morphisms,

text: Ann.

text: Inst.

text: Fourier

text: J.

text: Ecalle,

text: B.

text: Vallet,

text: Correction

text: and

text: linearization

text: of

text: resonant

text: vector

text: ﬁelds

text: and

text: diﬀeomorphisms,

text: Math.

text: Z.

text: J-P.

text: Francoise,

text: R.

text: Pons,

text: Les

text: conditions

text: du

text: centre

text: pour

text: un

text: champ

text: de

text: vecteurs

text: quadratiques,

text: CRAS

text: p.

text: J-P.

text: Francoise,

text: Géométrie

text: analytique

text: et

text: systèmes

text: dynamiques,

text: PUF,

text: W.S.

text: Loud,

text: Behaviour

text: of

text: the

text: period

text: of

text: solutions

text: of

text: certain

text: plane

text: autonomous

text: systems

text: near

text: centers,

text: Contrib.

text: Diﬀerential

text: Equations

text: C.

text: Reutenauer,

text: Free

text: Lie

text: algebras,

text: London

text: Math.

text: Soc.

text: Monographs,

text: new

text: series

text: B.

text: Schuman,

text: Correction

text: et

text: linéarisation

text: des

text: champs

text: de

text: vecteurs

text: polynomiaux,unpublished,

text: J-P.

text: Serre,

text: Lie

text: algebras

text: and

text: Lie

text: groups,

text: W.C.

text: Benjamin

text: Inc

text: H.Zoladek,

text: Quadratic

text: systems

text: with

text: center

text: and

text: their

text: perturbations,

text: Journal

text: of

text: diﬀerential

text: equations

text: Jacky

text: Cresson

text: Jordy

text: Palafox

text: ##LTLine##


text: 4 JACKY CRESSON & JORDY PALAFOX (cid:26) x˙ = ξx + p x2 + p xy + p y2, (5) − 2,0 1,1 0,2 y˙ = ξy + p¯ x2 + p¯ xy + p¯ y2, 0,2 1,1 2,0 where y = x¯, p ∈ C, i + j = 2, i = 0, 1, 2. i,j The following results can be founded in [6, 12]: Theorem 2. — A real quadratic planar vector ﬁeld X is an isochronous center if and only if at least one of the following conditions is satisﬁed: i) p = p = 0 (Holomorphic isochronous center), 1,1 0,2 ii) p = p¯ , p = 0 (Uniform isochronous center), 2,0 1,1 0,2 5 | | 4| | iii) p = p¯ , p 2 = p 2, 2,0 1,1 1,1 0,2 2 9 7 | | | | iv) p = p¯ , p 2 = 4 p 2. 2,0 1,1 1,1 0,2 6 Our approach suggests to look for these conditions by studying ﬁrst their consequences on ∈ the Lie algebra generated by the B , n A(X). n Lemma 1. — A quadratic vector ﬁeld satisfying (6) p = p¯ , p = 0, 2,0 1,1 0,2 or (7) p = p = 0, 1,1 0,2 { } is such that the Lie algebra b generated by B(X) = B ∈ is nilpotent of order 1, i.e. ∈ n n A(X) for all n, m A(X) we have (8) [B , B ] = 0. n m Using this Lemma and the structure of the series, we recover easily that these conditions correspond to an isochronous center. Indeed, we have the following general observation : Lemma 2. — If the Lie algebra b is nilpotent of order 1 then any prenormal form associated • to a mould P ran is reduced to (cid:88) (9) P ran(X) = X + P rannB , lin n ∈ n A(X), ω(n)=0 and the correction is given by ##LTLine## ##LTLine## ##LTLine##

text: JACKY

text: CRESSON

text: JORDY

text: PALAFOX

label: equation

text: x˙

text: ξx

text: p

text: p

text: xy

text: p

text: y˙

text: ξy

text: p¯

text: p¯

text: xy

text: p¯

text: where

text: y

text: x¯,

text: p

text: C,

text: i

text: j

text: i

text: i,j

text: The

text: following

text: results

text: can

text: be

text: founded

text: in

text: Theorem

text: A

text: real

text: quadratic

text: planar

text: vector

text: ﬁeld

text: X

text: is

text: an

text: isochronous

text: center

text: if

text: and

text: only

text: if

text: at

text: least

text: one

text: of

text: the

text: following

text: conditions

text: is

text: satisﬁed:

text: i)

text: p

text: p

text: (Holomorphic

text: isochronous

text: center),

text: ii)

text: p

text: p¯

text: p

text: (Uniform

text: isochronous

text: center),

text: iii)

text: p

text: p¯

text: p

text: p

text: iv)

text: p

text: p¯

text: p

text: p

text: Our

text: approach

text: suggests

text: to

text: look

text: for

text: these

text: conditions

text: by

text: studying

text: ﬁrst

text: their

text: consequences

text: on

text: the

text: Lie

text: algebra

text: generated

text: by

text: the

text: B

text: n

text: A(X).

text: n

text: Lemma

text: A

text: quadratic

text: vector

text: ﬁeld

text: satisfying

text: p

text: p¯

text: p

text: or

text: p

text: p

text: is

text: such

text: that

text: the

text: Lie

text: algebra

text: b

text: generated

text: by

text: B(X)

text: B

text: is

text: nilpotent

text: of

text: order

text: i.e.

text: n

text: n

text: A(X)

text: for

text: all

text: n,

text: m

text: A(X)

text: we

text: have

text: [B

text: B

text: n

text: m

text: Using

text: this

text: Lemma

text: and

text: the

text: structure

text: of

text: the

text: series,

text: we

text: recover

text: easily

text: that

text: these

text: conditions

text: correspond

text: to

text: an

text: isochronous

text: center.

text: Indeed,

text: we

text: have

text: the

text: following

text: general

text: observation

text: Lemma

text: If

text: the

text: Lie

text: algebra

text: b

text: is

text: nilpotent

text: of

text: order

text: then

text: any

text: prenormal

text: form

text: associated

text: to

text: a

text: mould

text: P

text: ran

text: is

text: reduced

text: to

text: P

text: ran(X)

text: X

text: P

text: rannB

text: lin

text: n

text: n

text: A(X),

text: and

text: the

text: correction

text: is

text: given

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 JACKY CRESSON & JORDY PALAFOX [B − − ] replacing y by x. (m1 1,0),...,(mr 1,0) A resonant word must mix letters with positive weight and negative weight. Assume that a given word begins with some (0, n − 1), . . . , (0, n − 1) for a given k ∈ N∗. One must 1 k have at least one operator of the form B − in order to have a resonant sequence. As (m1 1,0) [B − − ] depends only on y∂ and B − of x∂ , the corresponding Lie bracket is z(0e,rno1. 1T),h...e,(n0,nakll [1B) ] such that ω(n) =y0 is zero(m. 1Th1,i0s)concluxdes the proof. n 4. Uniform isochronism Following R. Conti [2], an isochronous center of a vector ﬁeld X is said to be uniform if § all the periodic orbits have the same period. This condition is equivalent to (see [2], .19, Deﬁnition 19.1 p.28) the following relation on the polynomial P : (UI) yP (x, y) = xP (x, y). This conditions induces speciﬁc relations on the coeﬃcients of the polynomial: Lemma 6. — A polynomial P of degree d satisﬁes the uniform isochronicity condition (UI) if and only if the coeﬃcients p , i + j = n, n = 2, . . . , d satisfy i,j (21) p = 0 and p − = p − − , n = 2, . . . , d, i = 1, . . . , n. 0,n i,n i n i+1,i 1 Using this Lemma and the structure Lemma 4 for the Lie algebra generated by the comoulds in B(X), we deduce the following Lemma: Lemma 7. — Let X be a polynomial vector ﬁeld of the form (1) where P is homogeneous of degree d. Assume that the coeﬃcients of P satisfy (22) p = 0, p − = p − − , i = 1, . . . , d. 0,d i,d i d i+1,i 1 Moreover, if d is odd with d = 2m + 1, we suppose that p = 0. Then, the vector ﬁeld m+1,m X is isochronous. Proof. — By Lemma 4, the ﬁrst conditions imply that the Lie algebra b is nilpotent of order 1. We deduce from Lemma 2 that any prenormal(cid:88)form is reduced to (23) P ran(X) = X + P rannB . lin n ∈ n A(X), ω(n)=0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JACKY

text: CRESSON

text: JORDY

text: PALAFOX

text: [B

text: replacing

text: y

text: by

text: x.

text: A

text: resonant

text: word

text: must

text: mix

text: letters

text: with

text: positive

text: weight

text: and

text: negative

text: weight.

text: Assume

text: that

text: a

text: given

text: word

text: begins

text: with

text: some

text: n

text: n

text: for

text: a

text: given

text: k

text: N∗.

text: One

text: must

text: k

text: have

text: at

text: least

text: one

text: operator

text: of

text: the

text: form

text: B

text: in

text: order

text: to

text: have

text: a

text: resonant

text: sequence.

text: As

text: [B

text: depends

text: only

text: on

text: y∂

text: and

text: B

text: of

text: x∂

text: the

text: corresponding

text: Lie

text: bracket

text: is

text: such

text: that

text: ω(n)

text: is

text: zero(m.

text: the

text: proof.

text: n

text: Uniform

text: isochronism

text: Following

text: R.

text: Conti

text: an

text: isochronous

text: center

text: of

text: a

text: vector

text: ﬁeld

text: X

text: is

text: said

text: to

text: be

text: uniform

text: if

text: all

text: the

text: periodic

text: orbits

text: have

text: the

text: same

text: period.

text: This

text: condition

text: is

text: equivalent

text: to

text: (see

text: Deﬁnition

text: the

text: following

text: relation

text: on

text: the

text: polynomial

text: P

text: (UI)

text: yP

text: (x,

text: y)

text: xP

text: (x,

text: y).

text: This

text: conditions

text: induces

text: speciﬁc

text: relations

text: on

text: the

text: coeﬃcients

text: of

text: the

text: polynomial:

text: Lemma

text: A

text: polynomial

text: P

text: of

text: degree

text: d

text: satisﬁes

text: the

text: uniform

text: isochronicity

text: condition

text: (UI)

text: if

text: and

text: only

text: if

text: the

text: coeﬃcients

text: p

text: i

text: j

text: n,

text: n

text: d

text: satisfy

text: i,j

text: p

text: and

text: p

text: p

text: n

text: d,

text: i

text: n.

text: i,n

text: i

text: n

text: Using

text: this

text: Lemma

text: and

text: the

text: structure

text: Lemma

text: for

text: the

text: Lie

text: algebra

text: generated

text: by

text: the

text: comoulds

text: in

text: B(X),

text: we

text: deduce

text: the

text: following

text: Lemma:

text: Lemma

text: Let

text: X

text: be

text: a

text: polynomial

text: vector

text: ﬁeld

text: of

text: the

text: form

text: where

text: P

text: is

text: homogeneous

text: of

text: degree

text: d.

text: Assume

text: that

text: the

text: coeﬃcients

text: of

text: P

text: satisfy

text: p

text: p

text: p

text: i

text: d.

text: i,d

text: i

text: d

text: Moreover,

text: if

text: d

text: is

text: odd

text: with

text: d

text: we

text: suppose

text: that

text: p

text: Then,

text: the

text: vector

text: ﬁeld

text: X

text: is

text: isochronous.

text: Proof.

text: By

text: Lemma

text: the

text: ﬁrst

text: conditions

text: imply

text: that

text: the

text: Lie

text: algebra

text: b

text: is

text: nilpotent

text: of

text: order

text: We

text: deduce

text: from

text: Lemma

text: that

text: any

text: is

text: reduced

text: to

text: P

text: ran(X)

text: X

text: P

text: rannB

text: lin

text: n

text: n

text: A(X),

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the ice in the sample holder can be considered a parallel plate capacitor and the charging time t calculates to c U εε A 0 t = (1) c i d × − − In the above equation, i is the ion current, ε = 8.85 10 12 F m 1 is 0 the vacuum permittivity, ε is the relative permittivity of cold porous ice, d = 0.009 m is the thickness of the ice layer, and A = πr2 = 0.0013 m2 is the ice surface area inside the sample holder. Upon irradiation with ions, the ice surface will charge up to an average surface potential U that will deﬂect any additional ion onto the copper ring around the ice. If we again use the parallel plate capacitor as a model, this surface potential U is related to the acceleration voltage of the ions U via i U sin(α) cos(α)U i = (2) d r ◦ with α = 30 the incidence angle between the ion beam and the ice plane. The other unknown in Equation 1 is ε. In this study, we will only consider the real part of ε, since its imaginary part is orders of magnitude lower (M¨atzler et al., 2006). Shi et al. (2012) adopted the value of ε = 3.0 from Wang et al. (2008) for compact water ice. A similar value of ε = 3.1 is recommended by Ma¨tzler et al. (2006) for ordinary water ice (Ih) below T = 240 K. The permittivity of water ice hardly varies with temperature for the range of 90 to 140 K considered in our study (M¨atzler et al., 2006). The density of our ice samples closely resembles dry snow. We will therefore rely on Sugiyama et al. (2010) who measured ε = 1.5 at microwave frequencies for dry snow with − a density of 300 kg m 3. This agrees with the value given by Brouet et al. − (2016) for pure porous water ice with 300 kg m 3 bulk density. Equations 1 ≈ ≈ and 2 then yield t 4 s and a surface potential of U 2 kV for an ion c − beam with i = 1 nA and an energy U e = 10 keV. i Shi et al. (2010, 2012) directly measured the surface potential of amor- phous water ice ﬁlms with a Kelvin probe. They vapour deposited ice ﬁlms with hundreds to thousands of nm thickness onto the tip of a microbalance cooled to temperatures between 20 K and 160 K. They found that the sur- face potential increased asymptotically to an equilibrium during irradiation with an Ar+ beam. The timescale to charge the ice surface was hundreds 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: ice

text: in

text: the

text: sample

text: holder

text: can

text: be

text: considered

text: a

text: parallel

text: plate

text: capacitor

text: and

text: the

text: charging

text: time

text: t

text: calculates

text: to

text: c

text: U

text: εε

text: A

text: t

text: c

text: i

text: d

text: In

text: the

text: above

text: equation,

text: i

text: is

text: the

text: ion

text: current,

text: ε

text: F

text: m

text: is

text: the

text: vacuum

text: permittivity,

text: ε

text: is

text: the

text: relative

text: permittivity

text: of

text: cold

text: porous

text: ice,

text: d

text: m

text: is

text: the

text: thickness

text: of

text: the

text: ice

text: layer,

text: and

text: A

text: is

text: the

text: ice

text: surface

text: area

text: inside

text: the

text: sample

text: holder.

text: Upon

text: irradiation

text: with

text: ions,

text: the

text: ice

text: surface

text: will

text: charge

text: up

text: to

text: an

text: average

text: surface

text: potential

text: U

text: that

text: will

text: deﬂect

text: any

text: additional

text: ion

text: onto

text: the

text: copper

text: ring

text: around

text: the

text: ice.

text: If

text: we

text: again

text: use

text: the

text: parallel

text: plate

text: capacitor

text: as

text: a

text: model,

text: this

text: surface

text: potential

text: U

text: is

text: related

text: to

text: the

text: acceleration

text: voltage

text: of

text: the

text: ions

text: U

text: via

text: i

text: U

text: sin(α)

text: cos(α)U

text: i

text: d

text: r

text: with

text: α

text: the

text: incidence

text: angle

text: between

text: the

text: ion

text: beam

text: and

text: the

text: ice

text: plane.

text: The

text: other

text: unknown

text: in

text: Equation

text: is

text: ε.

text: In

text: this

text: study,

text: we

text: will

text: only

text: consider

text: the

text: real

text: part

text: of

text: ε,

text: since

text: its

text: imaginary

text: part

text: is

text: orders

text: of

text: magnitude

text: lower

text: (M¨atzler

text: et

text: al.,

text: Shi

text: et

text: al.

text: adopted

text: the

text: value

text: of

text: ε

text: from

text: Wang

text: et

text: al.

text: for

text: compact

text: water

text: ice.

text: A

text: similar

text: value

text: of

text: ε

text: is

text: recommended

text: by

text: Ma¨tzler

text: et

text: al.

text: for

text: ordinary

text: water

text: ice

text: (Ih)

text: below

text: T

text: K.

text: The

text: permittivity

text: of

text: water

text: ice

text: hardly

text: varies

text: with

text: temperature

text: for

text: the

text: range

text: of

text: to

text: K

text: considered

text: in

text: our

text: study

text: (M¨atzler

text: et

text: al.,

text: The

text: density

text: of

text: our

text: ice

text: samples

text: closely

text: resembles

text: dry

text: snow.

text: We

text: will

text: therefore

text: rely

text: on

text: Sugiyama

text: et

text: al.

text: who

text: measured

text: ε

text: at

text: microwave

text: frequencies

text: for

text: dry

text: snow

text: with

text: a

text: density

text: of

text: kg

text: m

text: This

text: agrees

text: with

text: the

text: value

text: given

text: by

text: Brouet

text: et

text: al.

text: for

text: pure

text: porous

text: water

text: ice

text: with

text: kg

text: m

text: bulk

text: density.

text: Equations

text: and

text: then

text: yield

text: t

text: s

text: and

text: a

text: surface

text: potential

text: of

text: U

text: kV

text: for

text: an

text: ion

text: c

text: beam

text: with

text: i

text: and

text: an

text: energy

text: U

text: e

text: keV.

text: i

text: Shi

text: et

text: al.

text: directly

text: measured

text: the

text: surface

text: potential

text: of

text: amor-

text: phous

text: water

text: ice

text: ﬁlms

text: with

text: a

text: Kelvin

text: probe.

text: They

text: vapour

text: deposited

text: ice

text: ﬁlms

text: with

text: hundreds

text: to

text: thousands

text: of

text: nm

text: thickness

text: onto

text: the

text: tip

text: of

text: a

text: microbalance

text: cooled

text: to

text: temperatures

text: between

text: K

text: and

text: K.

text: They

text: found

text: that

text: the

text: sur-

text: face

text: potential

text: increased

text: asymptotically

text: to

text: an

text: equilibrium

text: during

text: irradiation

text: with

text: an

text: Ar+

text: beam.

text: The

text: timescale

text: to

text: charge

text: the

text: ice

text: surface

text: was

text: hundreds

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 both the case of a simpliﬁed contact interaction and the ulation inversion in 2D DMs can be realized by optical screened Coulomb potential. pumping, as shown schematically in Fig. 1(a). We show that the critical temperature and the size of Under realistic conditions, the transient state has a the excitonic gap is controlled by the interplay between ﬁnite lifetime, limited to hundreds of femtoseconds (fs) the enhanced density of states at the non-equilibrium in the case of graphene12–16 but extending to at least chemical potentials and metallic screening which be- picoseconds (ps) in 3DTIs17. Moreover, a recent time- comes stronger with increasing the chemical potentials, resolved ARPES study by Neupane et al.18 revealed long- the value of the coupling constant and the Dirac cone lived transient 3DTI surface states with lifetimes of a few degeneracy. Based on this we derive a set of criteria to microseconds (µs). Although the nature of such gigantic identify the best material candidates for observing the lifetimes is not entirely understood, this is a strong in- transient collective states. dication that long-lived (quasiequilibrium) excited Dirac Among the existing DMs, we predict the largest eﬀect, states can be achieved in these systems. with the size of the gap of the order 10meV, in undoped We begin our analysis by assuming the lifetime of the suspended graphene in which optical pumping is realized transient populations of electrons and holes in pumped selectively on a single valley, e.g. using circularly po- 2D DMs is suﬃciently long that the system can be consid- larized light28,29. Such gap sizes are large enough to be ered to be in quasiequilibrium. In this case, the system is detected by angle-resolved photoemmision spectroscopy described by the(cid:88)Hamiltonian H = He +Hh +Hint where (ARPES). We also ﬁnd that DMs with a single non- degenerate Dirac cone and large coupling constants such H = (cid:88)ξ(e/h)ψ† ψ as large-gap 3DTIs with small Dirac velocities and small e/h k e/h,k e/h,k dielectric constants, if realized, are the most promising k † † (1) candidates for observing the transient excitonic conden- Hint = Vqψe,k+qψh,k(cid:48)−qψh,k(cid:48)ψe,k. sate. Our theoretical estimates indicate that gaps of the q,k,k(cid:48) order of 100 meV could be achieved in such materials. − − − For all examples considered, we ﬁnd critical temperatures Here ξ(e) = vk µ (ξ(h) = vk µ ) is the elec- k e k h that are several orders of magnitude larger than the esti- tron (hole) dispersion measured from the electron (hole) mated maximum critical temperature for excitonic con- chemical potential, µ (µ ), where v is the velocity of the densate in double layer graphene30. Dirac states (in this we orkh we set (cid:126) ≡ 1); ψ† (ψ ) cre- τ,k { τ,k} The rest of the paper is organized as follows. In Sec- ates (annihilates) a Dirac state in band τ = e, h with tion , we describe the details of our theoretical model. In momentum k. Note that we consider a spinless model Section , we present the results of our calculations. In of a DM. V = 2πvα/(q + κ) is the screened Coulomb q particular, we provide approximate expressions for the potential modeled using the Thomas-Fermi theory (see critical temperature of the transient excitonic conden- derivation below), where α = e2/εv is the dimensionless sate within a particular regime and we show the phase coupling constant in the DM, ε is the eﬀective dielectric (cid:80) diagram for the excitonic condensate computed numeri- constant and κ is the screening wave vector7,34. cally. In Section , we discuss our results and present the The order parameter, or gap, for an excitonic conden- (cid:104) † (cid:105) efosrtimtwaoteismopfotrhtaenetxcciatsoensicofga2pDaDndMcsristtiucdalietdeminpeerxapteurrie- sate in this system is given by ∆k = k(cid:48) Vk−k(cid:48) ψe,−k(cid:48)ψh,k(cid:48) . Combining this deﬁnition of the gap with the Hamilto- ments, i.e. graphene and 3DTI surface states, as well nian in Eq. (1), one can show that the gap equation is as for a hypothetical 2D DM with parameters tuned in given by (cid:90) (cid:0) (cid:1) (cid:0) (cid:1) such a way as to reduce the screening eﬀects. We also (cid:113) propose several schemes for experimentally detecting the (cid:48) (cid:48) − (cid:48) transient excitonic instabilities in pumped DMs. Finally, ∆ = 1 dk V − (cid:48)∆ (cid:48) nF ω+(k ) nF ω−(k ) , in Section we oﬀer concluding remarks. k 2 (2π)2 k k k | (cid:48)| − (v k µ−)2 + ∆2(cid:48) II. THEORETICAL MODEL (cid:113) k (2) As a ﬁrst step, we consider a simple model of a pumped − ± | | − ≡ where ω±(k) = µ (v k µ−)2 + 1∆2, µ± 2D DM in which the electronic states take on a tran- ± + 4 k sient distribution governed by two chemical potentials, (µ µ )/2, n (ω) = 1/(eω/T + 1) is the Fermi-Dirac e h F one for the electrons µ and one for the holes µ as shown distribution, and T is the temperature (assumed to be e h in Fig. 1(a). While the nature of photoexcited carriers identical for both photoexcited electrons and holes). The in DMs at short time delays is an open issue, multiple order parameter deﬁned in Eq. (2) represents the pairing optical pump-probe experiments have shown that such between electrons and holes in a single non-degenerate a transient population inversion could be achieved in Dirac cone and is assumed to be unaﬀected by the degen- graphene12–14 and 3DTIs17,18,31,32. While there are other eracy of the Dirac states, which can be diﬀerent from 1, observations reporting a single hot Fermi-Dirac distribu- for instance, in graphene. However, the degeneracy will tion in graphene tens of fs after the photoexcitation33, strongly aﬀect the screening and will be taken into ac- for the purposes of this paper, we assume that the pop- count30. By solving for the gap self-consistently we can ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: both

text: the

text: case

text: of

text: a

text: simpliﬁed

text: contact

text: interaction

text: and

text: the

text: ulation

text: inversion

text: in

text: DMs

text: can

text: be

text: realized

text: by

text: optical

text: screened

text: Coulomb

text: potential.

text: pumping,

text: as

text: shown

text: schematically

text: in

text: Fig.

text: We

text: show

text: that

text: the

text: critical

text: temperature

text: and

text: the

text: size

text: of

text: Under

text: realistic

text: conditions,

text: the

text: transient

text: state

text: has

text: a

text: the

text: excitonic

text: gap

text: is

text: controlled

text: by

text: the

text: interplay

text: between

text: ﬁnite

text: lifetime,

text: limited

text: to

text: hundreds

text: of

text: femtoseconds

text: (fs)

text: the

text: enhanced

text: density

text: of

text: states

text: at

text: the

text: non-equilibrium

text: in

text: the

text: case

text: of

text: but

text: extending

text: to

text: at

text: least

text: chemical

text: potentials

text: and

text: metallic

text: screening

text: which

text: be-

text: picoseconds

text: (ps)

text: in

text: Moreover,

text: a

text: recent

text: time-

text: comes

text: stronger

text: with

text: increasing

text: the

text: chemical

text: potentials,

text: resolved

text: ARPES

text: study

text: by

text: Neupane

text: et

text: revealed

text: long-

text: the

text: value

text: of

text: the

text: coupling

text: constant

text: and

text: the

text: Dirac

text: cone

text: lived

text: transient

text: surface

text: states

text: with

text: lifetimes

text: of

text: a

text: few

text: degeneracy.

text: Based

text: on

text: this

text: we

text: derive

text: a

text: set

text: of

text: criteria

text: to

text: microseconds

text: (µs).

text: Although

text: the

text: nature

text: of

text: such

text: gigantic

text: identify

text: the

text: best

text: material

text: candidates

text: for

text: observing

text: the

text: lifetimes

text: is

text: not

text: entirely

text: understood,

text: this

text: is

text: a

text: strong

text: in-

text: transient

text: collective

text: states.

text: dication

text: that

text: long-lived

text: (quasiequilibrium)

text: excited

text: Dirac

text: Among

text: the

text: existing

text: DMs,

text: we

text: predict

text: the

text: largest

text: eﬀect,

text: states

text: can

text: be

text: achieved

text: in

text: these

text: systems.

text: with

text: the

text: size

text: of

text: the

text: gap

text: of

text: the

text: order

text: in

text: undoped

text: We

text: begin

text: our

text: analysis

text: by

text: assuming

text: the

text: lifetime

text: of

text: the

text: suspended

text: graphene

text: in

text: which

text: optical

text: pumping

text: is

text: realized

text: transient

text: populations

text: of

text: electrons

text: and

text: holes

text: in

text: pumped

text: selectively

text: on

text: a

text: single

text: valley,

text: e.g.

text: using

text: circularly

text: po-

text: DMs

text: is

text: suﬃciently

text: long

text: that

text: the

text: system

text: can

text: be

text: consid-

text: larized

text: Such

text: gap

text: sizes

text: are

text: large

text: enough

text: to

text: be

text: ered

text: to

text: be

text: in

text: quasiequilibrium.

text: In

text: this

text: case,

text: the

text: system

text: is

text: detected

text: by

text: angle-resolved

text: photoemmision

text: spectroscopy

text: described

text: by

text: H

text: He

text: +Hh

text: +Hint

text: where

text: (ARPES).

text: We

text: also

text: ﬁnd

text: that

text: DMs

text: with

text: a

text: single

text: non-

text: degenerate

text: Dirac

text: cone

text: and

text: large

text: coupling

text: constants

text: such

text: H

text: ψ

text: as

text: large-gap

text: with

text: small

text: Dirac

text: velocities

text: and

text: small

text: e/h

text: k

text: e/h,k

text: e/h,k

text: dielectric

text: constants,

text: if

text: realized,

text: are

text: the

text: most

text: promising

text: k

text: candidates

text: for

text: observing

text: the

text: transient

text: excitonic

text: conden-

text: Hint

text: sate.

text: Our

text: theoretical

text: estimates

text: indicate

text: that

text: gaps

text: of

text: the

text: order

text: of

text: meV

text: could

text: be

text: achieved

text: in

text: such

text: materials.

text: For

text: all

text: examples

text: considered,

text: we

text: ﬁnd

text: critical

text: temperatures

text: Here

text: ξ(e)

text: vk

text: µ

text: (ξ(h)

text: vk

text: µ

text: is

text: the

text: elec-

text: k

text: e

text: k

text: h

text: that

text: are

text: several

text: orders

text: of

text: magnitude

text: larger

text: than

text: the

text: esti-

text: tron

text: (hole)

text: dispersion

text: measured

text: from

text: the

text: electron

text: (hole)

text: mated

text: maximum

text: critical

text: temperature

text: for

text: excitonic

text: con-

text: chemical

text: potential,

text: µ

text: (µ

text: where

text: v

text: is

text: the

text: velocity

text: of

text: the

text: densate

text: in

text: double

text: layer

text: Dirac

text: states

text: (in

text: this

text: we

text: orkh

text: we

text: set

text: ψ†

text: (ψ

text: cre-

text: τ,k

text: τ,k}

text: The

text: rest

text: of

text: the

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec-

text: ates

text: (annihilates)

text: a

text: Dirac

text: state

text: in

text: band

text: τ

text: e,

text: h

text: with

text: tion

text: we

text: describe

text: the

text: details

text: of

text: our

text: theoretical

text: model.

text: In

text: momentum

text: k.

text: Note

text: that

text: we

text: consider

text: a

text: spinless

text: model

text: Section

text: we

text: present

text: the

text: results

text: of

text: our

text: calculations.

text: In

text: of

text: a

text: DM.

text: V

text: κ)

text: is

text: the

text: screened

text: Coulomb

text: q

text: particular,

text: we

text: provide

text: approximate

text: expressions

text: for

text: the

text: potential

text: modeled

text: using

text: the

text: Thomas-Fermi

text: theory

text: (see

text: critical

text: temperature

text: of

text: the

text: transient

text: excitonic

text: conden-

text: derivation

text: below),

text: where

text: α

text: is

text: the

text: dimensionless

text: sate

text: within

text: a

text: particular

text: regime

text: and

text: we

text: show

text: the

text: phase

text: coupling

text: constant

text: in

text: the

text: DM,

text: ε

text: is

text: the

text: eﬀective

text: dielectric

text: diagram

text: for

text: the

text: excitonic

text: condensate

text: computed

text: numeri-

text: constant

text: and

text: κ

text: is

text: the

text: screening

text: wave

text: cally.

text: In

text: Section

text: we

text: discuss

text: our

text: results

text: and

text: present

text: the

text: The

text: order

text: parameter,

text: or

text: gap,

text: for

text: an

text: excitonic

text: conden-

text: sate

text: in

text: this

text: system

text: is

text: given

text: by

text: ∆k

text: Combining

text: this

text: deﬁnition

text: of

text: the

text: gap

text: with

text: the

text: Hamilto-

text: ments,

text: i.e.

text: graphene

text: and

text: surface

text: states,

text: as

text: well

text: nian

text: in

text: Eq.

text: one

text: can

text: show

text: that

text: the

text: gap

text: equation

text: is

text: as

text: for

text: a

text: hypothetical

text: DM

text: with

text: parameters

text: tuned

text: in

text: given

text: by

text: such

text: a

text: way

text: as

text: to

text: reduce

text: the

text: screening

text: eﬀects.

text: We

text: also

text: propose

text: several

text: schemes

text: for

text: experimentally

text: detecting

text: the

text: transient

text: excitonic

text: instabilities

text: in

text: pumped

text: DMs.

text: Finally,

text: dk

text: V

text: nF

text: ω+(k

text: nF

text: ω−(k

text: in

text: Section

text: we

text: oﬀer

text: concluding

text: remarks.

text: k

text: k

text: k

text: k

text: (v

text: k

text: µ−)2

text: II.

text: THEORETICAL

text: MODEL

text: k

text: As

text: a

text: ﬁrst

text: step,

text: we

text: consider

text: a

text: simple

text: model

text: of

text: a

text: pumped

text: where

text: ω±(k)

text: µ

text: (v

text: k

text: µ−)2

text: µ±

text: DM

text: in

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text: 12 FIG. 6. Dynamics of (a) the order parameter and (b) electron occupation obtained from numerical integration of the SBE, (cid:48) (cid:48) Eq. (C2), for T >> T with diﬀerent T . We use parameters for graphene with α = 0.2, T = 0 and µ = µ = 500 meV. The 1 1 1 e h inset in panel (b) shows distribution of ne at t = 0. k 31 M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Tal4e4b-TIb. rWahinimzei,r,FE..NMavaalir´ci,n,aMnd. MA.oKntneovrerr,dPe,hPys..ARuebva.nB-S8en7z,i1e6r,5413 (2013). C. Pasquier, N. Moisan, et al., Nature communications 5 (201445). T. Stroucken, J. H. Gro¨nqvist, and S. W. Koch, Phys. Rev. B 84, 205445 (2011). 32 R. V. Aguilar, J. Qi, M. Brahlek, N. Bansal, A. Azad, J. Bow46laMn,. SN.eOupha,nAe,. ST.a-Yyl.oXr,uR, .RP. rSaasnaknakru,mNa.rA, liadnodusDt,.GY.aBroiatsnk,i,C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin, A. Bansil, Applied Physics Letters 106, 011901 (2015). F. Chou, and M. Z. Hasan, Nature Communications 5, 3786 (2014). 33 J. C. Johannsen, S. Ulstrup, F. Cilento, A. Crepaldi, M. Zac4c7hiSg.n-aM,.CH.uCanacgh,oS,.-IY. .CX. uE,.I.TBurecloup, oEls.kSi,pCri.n-gCa.tLe,eeF,.GF.roCmhamn,g, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, and P. HofmannA, P. Bhyasn.sRil,evH..LLeitnt,. 1a1n1d,M02.7Z4.03Ha(2sa0n13, )N. ature Communications 6, 7373 (2015). 34 S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mo4d8. SP.h-Yys..X8u3,, I4.0B7e(l2o0p1o1ls)k. i, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. C. Chang, Z. Yuan, C.-C. Lee, S.-M. 35 H. Haug and S. W. Koch, Quantum theory of the optical and electrHonuiacnpgr,oHpe.rZtiheesnogf,sJe.mMicao,nDdu.cSto.rSsa,nVcohle.z5, (BW. oWrladnSgc,iAen.tBiﬁacn, sil, F. Chou, P. P. Shibayev, H. Lin, S. J. Jia, and M. Z. Hasan, 2004). Science 349, 613 (2015). 36 C. Klingshirn, Semiconductor Optics (Springer, 2005) p. 522. 49 H. P. Dahal, Y. N. Joglekar, K. S. Bedell, and A. V. Balatsky, Phys. Rev. B 74, 233405 (2006). 37 For numerical calculations with the screened Coulomb potential5i0nDeq.uKilhibvreisuhmch, ewnekuo,seJaousrmnaalllobfuPt hﬁynsiitcesv:aCluoenodfenthseedscMreaetntienrg21, 075303 (2009). wavevector; variations of this value do not change the value of5α10Sa.pRpyruec,iCab.lMy. udry, C.-Y. Hou, and C. Chamon, Phys. Rev. B 80, 205319 (2009). c 38 D. Brida, A. Tomadin, C. Manzoni, Y. Kim, A. Lombardo, S.52MEila.nVa., GRo.rNbaarir,,VK..PN. oGvuossyelnoivn,, AV.. FAe.rrMariri,anGs.kyC,eraunlldo,I. aAn.dShovkovy, Phys. Rev. B 66, 045108 (2002). M. Polini, Nature Communications 4, 1987 (2013). 53 E. J. G. Santos and E. Kaxiras, Nano Letters 13, 898 (2013). 39 D. Kim, S. Cho, N. P. Butch, P. Syers, K. Kirshenbaum, S. A5d4amW,.JR.iPchatgelrioanned, Ca.ndR.MB.eSck.eFru, hPrheyr,siNcaatSutraetuPshSyosilcidsi8(,B4)5894, 619 (1977). (2012). 55 H. Beidenkopf, P. Roushan, J. Seo, L. Gorman, I. Drozdov, Y. Hor, R. Cava, and A. Yazdani, Nature Physics 7, 939–943 40 C. Triola, J.-X. Zhu, A. Migliori, and A. V. Balatsky, Phys. Rev.(2B01912)., 045401 (2015). 41 A. Pertsova, C. M. Canali, and A. H. MacDonald, Phys. Rev.56BM94.,L1in21d4b0e9rg(2a0n1d6)S.. W. Koch, Phys. Rev. B 38, 3342 (1988). 42 M. Hirata, K. Ishikawa, K. Miyagawa, M. Tamura, C. Berth57ierE,. AM.aBlica,skTo.,WDiennziesr,BEa.nBdoKbkoibna,yaasnhdi,AG..KMnoartrs,uPnoh,ys.aRndev. B 84, 205406 (2011). K. Kanoda, Nature Comm. 7, 12666 (2016). 58 G. Stefanucci and R. Van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction 43 M. Hirata, K. Ishikawa, G. Matsuno, A. Kobayashi, K. Mi(yCaagmawbari,dgMe .UnTiavmerusirtay, PCre.ssB, e2r0t1h3ie)r., and K. Kanoda, arXiv:1702.00097[cond-mat.str-el]. ∗ 11 D. Elias, R. Gorbachev, A. Mayorov, S. Morozov, A. Zhukov, P. Blake, L. Ponomarenko, I. Grigorieva, K. Novoselov, 1 T. Wehling, A. M. Black-Schaﬀer, and A. V. Balatsky, AdvancesFi.nGPuhiynseiac,se6t3a,l.1, (N2a0t1u4r)e. Physics 7, 701 (2011). 2 H. P. Dahal, T. O. Wehling, K. S. Bedell, J.-X. Zhu, and A. B12alaPt.sAky.,GPehoyrsgiec,aJB. :StCroanitd,eJn.sDedawMlaattyt,erS.4S0h5i,va2r2a4m1 a(n20, 1M0)..Chandrashekhar, F. Rana, and M. G. Spencer, Nano Letters 8, 4248 3 L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 ((22000078)).. 4 L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007). 13 S. Gilbertson, G. L. Dakovski, T. Durakiewicz, J.-X. Zhu, K. M. Dani, A. D. Mohite, A. Dattelbaum, and G. Rodriguez, 5 H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. ZhangT, hNeatJuoruernPahlyosficPsh5y,s4ic3a8l (C2h0e0m9)i.stry Letters 3, 64 (2011). 6 J. Zhang, C. Triola, and E. Rossi, Phys. Rev. Lett. 112, 09681042 T(2.0L1i4,)L. . Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, Phys. Rev. Lett. 108, 167401 (2012). 7 V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. Cas1t5roI.NGeiteor,z,RJe.vC. .MPoedte.rPsehny,s.M8.4M, i1t0r6a7no(,2C01.2C).acho, I. C. E. Turcu, E. Springate, A. Sto¨hr, A. Ko¨hler, U. Starke, and A. Cavalleri, 8 J. E. Drut and T. A. La¨hde, Phys. Rev. Lett. 102, 026802 (2009)N. ature Materials 12, 1119 (2013). 9 O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin, Phys. Rev1.6BI.8G1i,e0rz7,54F2.9C(a2l0e1g0a)r.i, S. Aeschlimann, M. Ch´avez Cervantes, C. Cacho, R. T. Chapman, E. Springate, S. Link, U. Starke, 10 O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin, Phys. Rev. BC8. 0R,.1A65st4,29an(2d0A09.)C. avalleri, Phys. Rev. Lett. 115, 086803 (2015). ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Dynamics

text: of

text: (a)

text: the

text: order

text: parameter

text: and

text: (b)

text: electron

text: occupation

text: obtained

text: from

text: numerical

text: integration

text: of

text: the

text: SBE,

text: Eq.

text: for

text: T

text: T

text: with

text: diﬀerent

text: T

text: We

text: use

text: parameters

text: for

text: graphene

text: with

text: α

text: T

text: and

text: µ

text: µ

text: meV.

text: The

text: e

text: h

text: inset

text: in

text: panel

text: (b)

text: shows

text: distribution

text: of

text: ne

text: at

text: t

text: k

text: M.

text: Hajlaoui,

text: E.

text: Papalazarou,

text: J.

text: Mauchain,

text: L.

text: Perfetti,

text: A.

text: rWahinimzei,r,FE..NMavaalir´ci,n,aMnd.

text: C.

text: Pasquier,

text: N.

text: Moisan,

text: et

text: al.,

text: Nature

text: communications

text: T.

text: Stroucken,

text: J.

text: H.

text: Gro¨nqvist,

text: and

text: S.

text: W.

text: Koch,

text: Phys.

text: Rev.

text: B

text: R.

text: V.

text: Aguilar,

text: J.

text: Qi,

text: M.

text: Brahlek,

text: N.

text: Bansal,

text: A.

text: Azad,

text: J.

text: SN.eOupha,nAe,.

text: ST.a-Yyl.oXr,uR,

text: .RP.

text: rSaasnaknakru,mNa.rA,

text: liadnodusDt,.GY.aBroiatsnk,i,C.

text: Liu,

text: I.

text: Belopolski,

text: T.-R.

text: Chang,

text: H.-T.

text: Jeng,

text: H.

text: Lin,

text: A.

text: Bansil,

text: Applied

text: Physics

text: Letters

text: F.

text: Chou,

text: and

text: M.

text: Z.

text: Hasan,

text: Nature

text: Communications

text: J.

text: C.

text: Johannsen,

text: S.

text: Ulstrup,

text: F.

text: Cilento,

text: A.

text: Crepaldi,

text: M.

text: .CX.

text: uE,.I.TBurecloup,

text: B.

text: Wang,

text: N.

text: Alidoust,

text: G.

text: Bian,

text: M.

text: Neupane,

text: C.

text: Zhang,

text: S.

text: Jia,

text: C.

text: Raidel,

text: T.

text: Seyller,

text: F.

text: Parmigiani,

text: M.

text: Grioni,

text: and

text: P.

text: HofmannA,

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text: Bhyasn.sRil,evH..LLeitnt,.

text: )N.

text: ature

text: Communications

text: S.

text: Das

text: Sarma,

text: S.

text: Adam,

text: E.

text: H.

text: Hwang,

text: and

text: E.

text: Rossi,

text: Rev.

text: i,

text: N.

text: Alidoust,

text: M.

text: Neupane,

text: G.

text: Bian,

text: C.

text: Zhang,

text: R.

text: Sankar,

text: G.

text: C.

text: Chang,

text: Z.

text: Yuan,

text: C.-C.

text: Lee,

text: S.-M.

text: H.

text: Haug

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text: S.

text: W.

text: Koch,

text: Quantum

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text: oWrladnSgc,iAen.tBiﬁacn,

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text: F.

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text: J.

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text: and

text: M.

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text: Hasan,

text: Science

text: C.

text: Klingshirn,

text: Semiconductor

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text: (Springer,

text: p.

text: H.

text: P.

text: Dahal,

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text: N.

text: Joglekar,

text: K.

text: S.

text: Bedell,

text: and

text: A.

text: V.

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text: Phys.

text: Rev.

text: B

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text: D.

text: Brida,

text: A.

text: Tomadin,

text: C.

text: Manzoni,

text: Y.

text: Kim,

text: A.

text: Lombardo,

text: GRo.rNbaarir,,VK..PN.

text: oGvuossyelnoivn,,

text: AV..

text: FAe.rrMariri,anGs.kyC,eraunlldo,I.

text: aAn.dShovkovy,

text: Phys.

text: Rev.

text: B

text: M.

text: Polini,

text: Nature

text: Communications

text: E.

text: J.

text: G.

text: Santos

text: and

text: E.

text: Kaxiras,

text: Nano

text: Letters

text: D.

text: Kim,

text: S.

text: Cho,

text: N.

text: P.

text: Butch,

text: P.

text: Syers,

text: K.

text: Kirshenbaum,

text: S.

text: Ca.ndR.MB.eSck.eFru,

text: H.

text: Beidenkopf,

text: P.

text: Roushan,

text: J.

text: Seo,

text: L.

text: Gorman,

text: I.

text: Drozdov,

text: Y.

text: Hor,

text: R.

text: Cava,

text: and

text: A.

text: Yazdani,

text: Nature

text: Physics

text: C.

text: Triola,

text: J.-X.

text: Zhu,

text: A.

text: Migliori,

text: and

text: A.

text: V.

text: Balatsky,

text: Phys.

text: A.

text: Pertsova,

text: C.

text: M.

text: Canali,

text: and

text: A.

text: H.

text: MacDonald,

text: Phys.

text: W.

text: Koch,

text: Phys.

text: Rev.

text: B

text: M.

text: Hirata,

text: K.

text: Ishikawa,

text: K.

text: Miyagawa,

text: M.

text: Tamura,

text: C.

text: B

text: K.

text: Kanoda,

text: Nature

text: Comm.

text: G.

text: Stefanucci

text: and

text: R.

text: Van

text: Leeuwen,

text: Nonequilibrium

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text: A

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text: Introduction

text: M.

text: Hirata,

text: K.

text: Ishikawa,

text: G.

text: Matsuno,

text: A.

text: Kobayashi,

text: K.

text: Mi(yCaagmawbari,dgMe

text: .UnTiavmerusirtay,

text: PCre.ssB,

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text: Kanoda,

text: D.

text: Elias,

text: R.

text: Gorbachev,

text: A.

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text: S.

text: Morozov,

text: A.

text: Zhukov,

text: P.

text: Blake,

text: L.

text: Ponomarenko,

text: I.

text: Grigorieva,

text: K.

text: Novoselov,

text: T.

text: Wehling,

text: A.

text: M.

text: Black-Schaﬀer,

text: and

text: A.

text: V.

text: Balatsky,

text: Physics

text: H.

text: P.

text: Dahal,

text: T.

text: O.

text: Wehling,

text: K.

text: S.

text: Bedell,

text: J.-X.

text: Zhu,

text: and

text: A.

text: F.

text: Rana,

text: and

text: M.

text: G.

text: Spencer,

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text: Letters

text: L.

text: Fu,

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text: L.

text: Kane,

text: and

text: E.

text: J.

text: Mele,

text: Phys.

text: Rev.

text: Lett.

text: L.

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text: L.

text: Kane,

text: Phys.

text: Rev.

text: B

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text: Gilbertson,

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text: L.

text: Dakovski,

text: T.

text: Durakiewicz,

text: J.-X.

text: Zhu,

text: K.

text: M.

text: Dani,

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text: D.

text: Mohite,

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text: Dattelbaum,

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text: Fang,

text: and

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text: ZhangT,

text: Letters

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text: Zhang,

text: C.

text: Triola,

text: and

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text: Rossi,

text: Phys.

text: Rev.

text: Lett.

text: Luo,

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text: Hupalo,

text: J.

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text: Tringides,

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text: Wang,

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text: Lett.

text: V.

text: N.

text: Kotov,

text: B.

text: Uchoa,

text: V.

text: M.

text: Pereira,

text: F.

text: Guinea,

text: and

text: A.

text: I.

text: C.

text: E.

text: Turcu,

text: E.

text: Springate,

text: A.

text: Sto¨hr,

text: A.

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text: U.

text: Starke,

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text: J.

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text: Gorbar,

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text: Phys.

text: S.

text: Aeschlimann,

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text: C.

text: Cacho,

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text: T.

text: Chapman,

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text: ##LTLine##

text: ##LTLine##


text: 6 a c α=0.7 g=2 α=0.7 g=1 b d α=1.0 g=1 α=1.0 g=2 FIG. 3. Phase diagram of the transient excitonic condensate. The maximum value of the gap evaluated at k = k as a function F of µ− and T for matched chemical potentials (µ = 0) computed numerically by solving Eq. (2) self-consistently using the + Thomas-Fermi potential shown in Eq. (7) for (a, c) α = 0.7 and (b, d) α = 1.0. Left column is for g = 1, right column is for g = 2. Other parameters are the same as in Fig. 1. The color shade represents the maximum value of the gap in meV’s. Note that we exclude the region around µ− = 0 since it is not properly described in the Thomas-Fermi model. toluminescence due to the recombination of electron-hole order of magnitude smaller than that found in graphene pairs27. due to the large dielectric constants of 3DTIs39. Using ≈ ≈ × ≈ values of ε 100 and v 5 105 m/s we ﬁnd that α 0.1 It should be noted that the peak temperature of the in- and that the predicted excitonic gap is < 1 meV, which verted carrier distributions inferred from pump-probe ex- periments can be as large as a few thousand Kelvin14,15. is below the current resolution of typical ARPES exper- iments. This drawback could be overcome in a 3DTI While this is above the predicted T for the typical DMs c with a larger eﬀective coupling constant [see Table I], in Table I, an increased lifetime of the transient state which would correspond to either a smaller dielectric con- should lead to lower local electronic temperatures as the stant or Fermi velocity40. Smaller Fermi velocities can be carriers have more time to cool down. Importantly, the found in anisotropic Dirac cones on various crystal facets predicted critical temperatures for pumped graphene are of topological insulators41. Recently, tilted Dirac cones several orders of magnitude larger than the estimated with Fermi velocities of the order of 104 m/s leading to maximum critical temperature of excitonic condensation in double layer graphene in the static regime (Tmax (cid:46) large α have been found in some quasi-two-dimensional 1mK)30. Our calculations predict that even largerc T of organic conductors42,43. c hundreds of K could be in principle achieved in future In the above analysis of the transient inverted popula- DMs with carefully designed material properties. tion quasiequilibrium was assumed; however, due to mul- In Table I, we see that α in 3DTIs appears to be an tiple scattering processes, the non-equilibrium inverted ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: a

text: c

text: b

text: d

text: FIG.

text: Phase

text: diagram

text: of

text: the

text: transient

text: excitonic

text: condensate.

text: The

text: maximum

text: value

text: of

text: the

text: gap

text: evaluated

text: at

text: k

text: k

text: as

text: a

text: function

text: F

text: of

text: µ−

text: and

text: T

text: for

text: matched

text: chemical

text: potentials

text: (µ

text: computed

text: numerically

text: by

text: solving

text: Eq.

text: self-consistently

text: using

text: the

text: Thomas-Fermi

text: potential

text: shown

text: in

text: Eq.

text: for

text: (a,

text: c)

text: α

text: and

text: (b,

text: d)

text: α

text: Left

text: column

text: is

text: for

text: g

text: right

text: column

text: is

text: for

text: g

text: Other

text: parameters

text: are

text: the

text: same

text: as

text: in

text: Fig.

text: The

text: color

text: shade

text: represents

text: the

text: maximum

text: value

text: of

text: the

text: gap

text: in

text: meV’s.

text: Note

text: that

text: we

text: exclude

text: the

text: region

text: around

text: µ−

text: since

text: it

text: is

text: not

text: properly

text: described

text: in

text: the

text: Thomas-Fermi

text: model.

text: toluminescence

text: due

text: to

text: the

text: recombination

text: of

text: electron-hole

text: order

text: of

text: magnitude

text: smaller

text: than

text: that

text: found

text: in

text: graphene

text: due

text: to

text: the

text: large

text: dielectric

text: constants

text: of

text: Using

text: values

text: of

text: ε

text: and

text: v

text: m/s

text: we

text: ﬁnd

text: that

text: α

text: It

text: should

text: be

text: noted

text: that

text: the

text: peak

text: temperature

text: of

text: the

text: in-

text: and

text: that

text: the

text: predicted

text: excitonic

text: gap

text: is

text: meV,

text: which

text: verted

text: carrier

text: distributions

text: inferred

text: from

text: pump-probe

text: ex-

text: periments

text: can

text: be

text: as

text: large

text: as

text: a

text: few

text: thousand

text: is

text: below

text: the

text: current

text: resolution

text: of

text: typical

text: ARPES

text: exper-

text: iments.

text: This

text: drawback

text: could

text: be

text: overcome

text: in

text: a

text: While

text: this

text: is

text: above

text: the

text: predicted

text: T

text: for

text: the

text: typical

text: DMs

text: c

text: with

text: a

text: larger

text: eﬀective

text: coupling

text: constant

text: [see

text: Table

text: I],

text: in

text: Table

text: I,

text: an

text: increased

text: lifetime

text: of

text: the

text: transient

text: state

text: which

text: would

text: correspond

text: to

text: either

text: a

text: smaller

text: dielectric

text: con-

text: should

text: lead

text: to

text: lower

text: local

text: electronic

text: temperatures

text: as

text: the

text: stant

text: or

text: Fermi

text: Smaller

text: Fermi

text: velocities

text: can

text: be

text: carriers

text: have

text: more

text: time

text: to

text: cool

text: down.

text: Importantly,

text: the

text: found

text: in

text: anisotropic

text: Dirac

text: cones

text: on

text: various

text: crystal

text: facets

text: predicted

text: critical

text: temperatures

text: for

text: pumped

text: graphene

text: are

text: of

text: topological

text: Recently,

text: tilted

text: Dirac

text: cones

text: several

text: orders

text: of

text: magnitude

text: larger

text: than

text: the

text: estimated

text: with

text: Fermi

text: velocities

text: of

text: the

text: order

text: of

text: m/s

text: leading

text: to

text: maximum

text: critical

text: temperature

text: of

text: excitonic

text: condensation

text: in

text: double

text: layer

text: graphene

text: in

text: the

text: static

text: regime

text: (Tmax

text: large

text: α

text: have

text: been

text: found

text: in

text: some

text: quasi-two-dimensional

text: Our

text: calculations

text: predict

text: that

text: even

text: largerc

text: T

text: of

text: organic

text: c

text: hundreds

text: of

text: K

text: could

text: be

text: in

text: principle

text: achieved

text: in

text: future

text: In

text: the

text: above

text: analysis

text: of

text: the

text: transient

text: inverted

text: popula-

text: DMs

text: with

text: carefully

text: designed

text: material

text: properties.

text: tion

text: quasiequilibrium

text: was

text: assumed;

text: however,

text: due

text: to

text: mul-

text: In

text: Table

text: I,

text: we

text: see

text: that

text: α

text: in

text: appears

text: to

text: be

text: an

text: tiple

text: scattering

text: processes,

text: the

text: non-equilibrium

text: inverted

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##