text: · T [K] η [Pa s] λ [s] p 0 × − 377 0.10 2.9 10 8 0.29 × − 452 0.022 2.7 10 8 0.21 × − 528 0.0055 2.6 10 8 0.13 × − 603 0.0016 6.7 10 9 0.09 TABLE I. Values of parameters η , λ, and p from the Carreau-Yasuda formula, for 0 diﬀerent temperatures T . · · T [K] η [Pa s] K [s] η∞ [Pa s] m 0 × − 377 0.13 1.6 10 8 0.0013 0.79 × − 452 0.040 3.4 10 8 0.00064 0.55 × − 528 0.0061 4.2 10 9 0.00085 0.70 × − 603 0.0018 6.2 10 10 0.00040 0.56 TABLE II. Values of parameters η , K, η∞, and m from the Cross formula, for diﬀerent 0 temperatures T . perimentally, the glass transition of bitumen depends on its exact composition and is typically around 250 K [38], which is well below the temperature range used in this work. The Arrhenius equation has been shown to ﬁt the variation of the exper- imental zero shear rate viscosity with temperature quite well for some bitumen in a temperature range going from 273 to 363 K [8]. The variation with temperature of the zero shear rate viscosity obtained with the Carreau-Yasuda and Cross ﬁts is compared to an Arrhenius behavior in the inset of Fig. 3. Both sets of values follow an Arrhenius behavior. The parameters of the Arrhenius ﬁt are found to be ∞ × − · E/k = 4073 K and η = 2.27 10 6 Pa s for the Carreau-Yasuda set of data and B ∞ × − · E/k = 4245 K and η = 2.00 10 6 Pa s. Taking into account the data from B the Carreau-Yasuda ﬁt, the Arrhenius equation predicts a value for the viscosity · of 2.5 Pa s at 293 K. Again, it is at the lowest end of the broad range of experimen- tal values for the zero shear rate viscosity at this temperature. These values can 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T

text: [K]

text: η

text: [Pa

text: s]

text: λ

text: [s]

text: p

text: TABLE

text: I.

text: Values

text: of

text: parameters

text: η

text: λ,

text: and

text: p

text: from

text: the

text: Carreau-Yasuda

text: formula,

text: for

text: diﬀerent

text: temperatures

text: T

text: T

text: [K]

text: η

text: [Pa

text: s]

text: K

text: [s]

text: η∞

text: [Pa

text: s]

text: m

text: TABLE

text: II.

text: Values

text: of

text: parameters

text: η

text: K,

text: η∞,

text: and

text: m

text: from

text: the

text: Cross

text: formula,

text: for

text: diﬀerent

text: temperatures

text: T

text: perimentally,

text: the

text: glass

text: transition

text: of

text: bitumen

text: depends

text: on

text: its

text: exact

text: composition

text: and

text: is

text: typically

text: around

text: K

text: which

text: is

text: well

text: below

text: the

text: temperature

text: range

text: used

text: in

text: this

text: work.

text: The

text: Arrhenius

text: equation

text: has

text: been

text: shown

text: to

text: ﬁt

text: the

text: variation

text: of

text: the

text: exper-

text: imental

text: zero

text: shear

text: rate

text: viscosity

text: with

text: temperature

text: quite

text: well

text: for

text: some

text: bitumen

text: in

text: a

text: temperature

text: range

text: going

text: from

text: to

text: K

text: The

text: variation

text: with

text: temperature

text: of

text: the

text: zero

text: shear

text: rate

text: viscosity

text: obtained

text: with

text: the

text: Carreau-Yasuda

text: and

text: Cross

text: ﬁts

text: is

text: compared

text: to

text: an

text: Arrhenius

text: behavior

text: in

text: the

text: inset

text: of

text: Fig.

text: Both

text: sets

text: of

text: values

text: follow

text: an

text: Arrhenius

text: behavior.

text: The

text: parameters

text: of

text: the

text: Arrhenius

text: ﬁt

text: are

text: found

text: to

text: be

text: E/k

text: K

text: and

text: η

text: Pa

text: s

text: for

text: the

text: Carreau-Yasuda

text: set

text: of

text: data

text: and

text: B

text: E/k

text: K

text: and

text: η

text: Pa

text: s.

text: Taking

text: into

text: account

text: the

text: data

text: from

text: B

text: the

text: Carreau-Yasuda

text: ﬁt,

text: the

text: Arrhenius

text: equation

text: predicts

text: a

text: value

text: for

text: the

text: viscosity

text: of

text: Pa

text: s

text: at

text: K.

text: Again,

text: it

text: is

text: at

text: the

text: lowest

text: end

text: of

text: the

text: broad

text: range

text: of

text: experimen-

text: tal

text: values

text: for

text: the

text: zero

text: shear

text: rate

text: viscosity

text: at

text: this

text: temperature.

text: These

text: values

text: can

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: · − · range from 1 Pa s [4] up to 104 106 Pa s [5, 8, 9, and 16]. The fact that the viscosity of the Cooee bitumen is quite low compared to exper- imental viscosities of bitumen at ambient pressure and temperature can be related to the observation that the density of the Cooee bitumen is itself quite diﬀerent from experimental densities. At a temperature of 377 K, the density of the Cooee · − bitumen is 1007 kg m 3. It is to be compared with the thorough experiments carried out on Athabasca bitumen [39] which ﬁnd a linear dependency of bitumen density on temperature at ambient pressure. Extrapolating their results to 377 K leads to a · − bitumen density around 955 kg m 3, which is signiﬁcantly lower than the density of the Cooee bitumen at the same pressure and temperature. This diﬀerence between the equation of state of our model and the equation of state obtained experimen- tally could be due to the speciﬁc type of asphaltene molecules that we chose. It has very short alkyl chains aside from the aromatic plane compared to the resin molecule. We know from a previous work [18], that adding resin molecules while removing asphaltene molecules from the mixture has a tendency to lower the den- sity at constant pressure. Thus, considering asphaltene molecules with longer alkyl chains could lower the density of the model and could also change the viscosity. In the future, we plan to investigate this possible eﬀect of the length of the alkyl chains on the density and viscosity of the mixture by comparing two mixtures with alkyl chains of diﬀerent lengths. In an attempt to extrapolate not only the zero shear rate viscosity but also the shear rate dependent viscosity to ambient temperature, we tested the time- temperature superposition principle. This principle has been shown to be generally valid experimentally [40] and is also discussed in another numerical model of bitu- men [41]. The reduced viscosity and reduced shear rate are deﬁned as η/a and γ˙ a , η T respectively, where η (T, ρ) 0 a = , (7) η η (T , ρ ) 0 ref ref T ρ ref ref a = a . (8) T η T ρ 10 ##LTLine## ##LTLine##

label: paragraph

text: range

text: from

text: Pa

text: s

text: up

text: to

text: Pa

text: s

text: and

text: The

text: fact

text: that

text: the

text: viscosity

text: of

text: the

text: Cooee

text: bitumen

text: is

text: quite

text: low

text: compared

text: to

text: exper-

text: imental

text: viscosities

text: of

text: bitumen

text: at

text: ambient

text: pressure

text: and

text: temperature

text: can

text: be

text: related

text: to

text: the

text: observation

text: that

text: the

text: density

text: of

text: the

text: Cooee

text: bitumen

text: is

text: itself

text: quite

text: diﬀerent

text: from

text: experimental

text: densities.

text: At

text: a

text: temperature

text: of

text: K,

text: the

text: density

text: of

text: the

text: Cooee

text: bitumen

text: is

text: kg

text: m

text: It

text: is

text: to

text: be

text: compared

text: with

text: the

text: thorough

text: experiments

text: carried

text: out

text: on

text: Athabasca

text: bitumen

text: which

text: ﬁnd

text: a

text: linear

text: dependency

text: of

text: bitumen

text: density

text: on

text: temperature

text: at

text: ambient

text: pressure.

text: Extrapolating

text: their

text: results

text: to

text: K

text: leads

text: to

text: a

text: bitumen

text: density

text: around

text: kg

text: m

text: which

text: is

text: signiﬁcantly

text: lower

text: than

text: the

text: density

text: of

text: the

text: Cooee

text: bitumen

text: at

text: the

text: same

text: pressure

text: and

text: temperature.

text: This

text: diﬀerence

text: between

text: the

text: equation

text: of

text: state

text: of

text: our

text: model

text: and

text: the

text: equation

text: of

text: state

text: obtained

text: experimen-

text: tally

text: could

text: be

text: due

text: to

text: the

text: speciﬁc

text: type

text: of

text: asphaltene

text: molecules

text: that

text: we

text: chose.

text: It

text: has

text: very

text: short

text: alkyl

text: chains

text: aside

text: from

text: the

text: aromatic

text: plane

text: compared

text: to

text: the

text: resin

text: molecule.

text: We

text: know

text: from

text: a

text: previous

text: work

text: that

text: adding

text: resin

text: molecules

text: while

text: removing

text: asphaltene

text: molecules

text: from

text: the

text: mixture

text: has

text: a

text: tendency

text: to

text: lower

text: the

text: den-

text: sity

text: at

text: constant

text: pressure.

text: Thus,

text: considering

text: asphaltene

text: molecules

text: with

text: longer

text: alkyl

text: chains

text: could

text: lower

text: the

text: density

text: of

text: the

text: model

text: and

text: could

text: also

text: change

text: the

text: viscosity.

text: In

text: the

text: future,

text: we

text: plan

text: to

text: investigate

text: this

text: possible

text: eﬀect

text: of

text: the

text: length

text: of

text: the

text: alkyl

text: chains

text: on

text: the

text: density

text: and

text: viscosity

text: of

text: the

text: mixture

text: by

text: comparing

text: two

text: mixtures

text: with

text: alkyl

text: chains

text: of

text: diﬀerent

text: lengths.

text: In

text: an

text: attempt

text: to

text: extrapolate

text: not

text: only

text: the

text: zero

text: shear

text: rate

text: viscosity

text: but

text: also

text: the

text: shear

text: rate

text: dependent

text: viscosity

text: to

text: ambient

text: temperature,

text: we

text: tested

text: the

text: time-

text: temperature

text: superposition

text: principle.

text: This

text: principle

text: has

text: been

text: shown

text: to

text: be

text: generally

text: valid

text: experimentally

text: and

text: is

text: also

text: discussed

text: in

text: another

text: numerical

text: model

text: of

text: bitu-

text: men

text: The

text: reduced

text: viscosity

text: and

text: reduced

text: shear

text: rate

text: are

text: deﬁned

text: as

text: η/a

text: and

text: γ˙

text: a

text: η

text: T

text: respectively,

text: where

text: η

text: (T,

text: ρ)

text: a

text: η

text: η

text: (T

text: ρ

text: ref

text: ref

text: T

text: ρ

text: ref

text: ref

text: a

text: a

text: T

text: η

text: T

text: ρ

text: ##LTLine##

text: ##LTLine##


text: 10 [2] R. Buyya, “Introduction to the ieee transactions on cloud computing,” [24] S. Pandey, W. Linlin, S. Guru, and R. Buyya, “A particle swarm Cloud Computing, IEEE Transactions on, vol. 1, no. 1, pp. 3–21, Jan optimization-based heuristic for scheduling workﬂow applications in 2013. cloud computing environments,” in Advanced Information Networking [3] C. Delimitrou and C. Kozyrakis, “Qos-aware scheduling in and Applications (AINA), 2010 24th IEEE International Conference on, heterogeneous datacenters with paragon,” ACM Trans. Comput. April 2010, pp. 400–407. Syst., vol. 31, no. 4, pp. 12:1–12:34, Dec. 2013. [Online]. Available: [25] E. Gelenbe and J. Fourneau, “Random neural networks http://doi.acm.org/10.1145/2556583 with multiple classes of signals,” Neural Computation, [4] P. Pradeep, K. G. Shin, X. Zhu, M. Uysal, Z. Wang, S. Singhal, vol. 11, no. 4, pp. 953–963, 1999. [Online]. Available: A. Merchant, and K. Salem, “Adaptive control of virtualized resources http://dx.doi.org/10.1162/089976699300016520 in utility computing environments,” in Proceedings of the 2Nd ACM [26] E. Gelenbe, S. Timotheou, and D. Nicholson, “Fast distributed SIGOPS/EuroSys European Conference on Computer Systems 2007, near-optimum assignment of assets to tasks,” The Computer Journal, ser. EuroSys ’07. New York, NY, USA: ACM, 2007, pp. 289–302. vol. 53, no. 9, pp. 1360–1369, Nov. 2010. [Online]. Available: [Online]. Available: http://doi.acm.org/10.1145/1272996.1273026 http://dx.doi.org/10.1093/comjnl/bxq010 [5] A. Berl, E. Gelenbe, M. Di Girolamo, G. Giuliani, H. De Meer, M. Q. [27] S. Zaman and D. Grosu, “A combinatorial auction-based dynamic vm Dang, and K. Pentikousis, “Energy-efﬁcient cloud computing,” The provisioning and allocation in clouds,” in Cloud Computing Technology Computer Journal, vol. 53, no. 7, pp. 1045–1051, 2010. and Science (CloudCom), 2011 IEEE Third International Conference [6] E. Gelenbe and R. Lent, “Energy-qos trade-offs in mobile service on, Nov 2011, pp. 107–114. selection,” Future Internet, vol. 5, no. 2, pp. 128–139, 2013. [Online]. [28] C. Lin and S. Lu, “Scheduling scientiﬁc workﬂows elastically for cloud Available: http://dx.doi.org/10.3390/ﬁ5020128 computing,” in Cloud Computing (CLOUD), 2011 IEEE International [7] ——, “Optimising server energy consumption and response time,” Conference on, July 2011, pp. 746–747. Theoretical and Applied Informatics, no. 4, pp. 257–270, Jan 2013. [29] I. S. Moreno, P. Garraghan, P. Townend, and J. Xu, “Analysis, modeling [8] J. Zhan, L. Wang, X. Li, W. Shi, C. Weng, W. Zhang, and X. Zang, and simulation of workload patterns in a large-scale utility cloud,” Cloud “Cost-aware cooperative resource provisioning for heterogeneous work- Computing, IEEE Transactions on, vol. PP, no. 99, pp. 1–1, 2014. loads in data centers,” Computers, IEEE Transactions on, vol. 62, no. 11, [30] B. Palanisamy, A. Singh, and L. Liu, “Cost-effective resource provision- pp. 2155–2168, Nov 2013. ing for mapreduce in a cloud,” Parallel and Distributed Systems, IEEE [9] A. Iosup, S. Ostermann, M. Yigitbasi, R. Prodan, T. Fahringer, and Transactions on, vol. PP, no. 99, pp. 1–1, 2014. D. H. J. Epema, “Performance analysis of cloud computing services [31] S. Zhuravlev, S. Blagodurov, and A. Fedorova, “Addressing shared for many-tasks scientiﬁc computing,” Parallel and Distributed Systems, resource contention in multicore processors via scheduling,” SIGPLAN IEEE Transactions on, vol. 22, no. 6, pp. 931–945, June 2011. Not., vol. 45, no. 3, pp. 129–142, Mar. 2010. [Online]. Available: http://doi.acm.org/10.1145/1735971.1736036 [10] S. Zhuravlev, S. Blagodurov, and A. Fedorova, “Addressing shared [32] N. Bhatti and R. Friedrich, “Web server support for tiered services,” resource contention in multicore processors via scheduling,” SIGPLAN Network, IEEE, vol. 13, no. 5, pp. 64–71, Sep 1999. Not., vol. 45, no. 3, pp. 129–142, Mar. 2010. [Online]. Available: [33] Y. Song, Y. Sun, and W. Shi, “A two-tiered on-demand resource http://doi.acm.org/10.1145/1735971.1736036 allocation mechanism for vm-based data centers,” Services Computing, [11] A. N. Tantawi and D. Towsley, “Optimal static load balancing in IEEE Transactions on, vol. 6, no. 1, pp. 116–129, First 2013. distributed computer systems,” Journal of the ACM (JACM), vol. 32, [34] E. Gelenbe, R. Lent, and M. Douratsos, “Choosing a local or remote no. 2, pp. 445–465, 1985. cloud,” in Second Symposium on Network Cloud Computing and [12] C. Kim and H. Kameda, “An algorithm for optimal static load balancing Applications, NCCA 2012, London, United Kingdom, December 3-4, in distributed computer systems,” IEEE Transactions on Computers, 2012. IEEE Computer Society, 2012, pp. 25–30. [Online]. Available: vol. 41, no. 3, pp. 381–384, 1992. http://doi.ieeecomputersociety.org/10.1109/NCCA.2012.16 [13] H. Kameda, J. Li, C. Kim, and Y. Zhang, Optimal Load Balancing in [35] E. Gelenbe and R. Lent, “Trade-offs between energy and quality of Distributed Computer Systems. Springer, 2011. service,” in Sustainable Internet and ICT for Sustainability (SustainIT), [14] J. J. Wolff, “Dynamic load balancing of a network of client and server 2012. IEEE, 2012, pp. 1–5. computers,” Feb. 6, 2001, u. S. Patent 6,185,601. [36] Q. Zhang, M. Zhani, R. Boutaba, and J. Hellerstein, “Dynamic [15] B. P. Rimal, E. Choi, and I. Lumb, “A taxonomy and survey of heterogeneity-aware resource provisioning in the cloud,” Cloud Com- cloud computing systems,” in INC, IMS and IDC, 2009. NCM’09. Fifth puting, IEEE Transactions on, vol. 2, no. 1, pp. 14–28, March 2014. International Joint Conference on. Ieee, 2009, pp. 44–51. [37] T. Dillon, C. Wu, and E. Chang, “Cloud computing: issues and chal- [16] Z. Zhang and X. Zhang, “A load balancing mechanism based on lenges,” in Advanced Information Networking and Applications (AINA), ant colony and complex network theory in open cloud computing 2010 24th IEEE International Conference on. Ieee, 2010, pp. 27–33. federation,” in Industrial Mechatronics and Automation (ICIMA), 2010 [38] E. Gelenbe, R. Lent, and A. Nunez, “Self-aware networks and QoS,” 2nd International Conference on, vol. 2. IEEE, 2010, pp. 240–243. Proceedings of the IEEE, vol. 92, no. 9, pp. 1478–1489, Sep. 2004. [17] W. Tian, Y. Zhao, Y. Zhong, M. Xu, and C. Jing, “A dynamic and [39] E. Gelenbe and S. Timotheou, “Random neural networks with synchro- integrated load-balancing scheduling algorithm for cloud datacenters,” nized interactions,” Neural Computation, vol. 20, no. 9, pp. 2308–2324, in Cloud Computing and Intelligence Systems (CCIS), 2011 IEEE 2008. [Online]. Available: http://dx.doi.org/10.1162/neco.2008.04-07- International Conference on. IEEE, 2011, pp. 311–315. 509 [18] X. Zhu, X. Qin, and M. Qiu, “Qos-aware fault-tolerant scheduling for [40] E. Gelenbe, “Sensible decisions based on qos,” Computational manage- real-time tasks on heterogeneous clusters,” Computers, IEEE Transac- ment science, vol. 1, no. 1, pp. 1–14, 2003. tions on, vol. 60, no. 6, pp. 800–812, June 2011. [41] E. Gelenbe, Z. Xu, and E. Seref, “Cognitive packet networks,” in [19] H. Topcuouglu, S. Hariri, and M. you Wu, “Performance-effective and Proceedings of the 11th International Conference on Tools with Artiﬁcial low-complexity task scheduling for heterogeneous computing,” IEEE Intelligence, Nov. 1999, pp. 47–54. Trans. Parallel Distrib. Syst., vol. 13, no. 3, pp. 260–274, Mar. 2002. [42] E. Gelenbe, “Steps toward self-aware networks,” Communications ACM, [Online]. Available: http://dx.doi.org/10.1109/71.993206 vol. 52, no. 7, pp. 66–75, July 2009. [20] G. Sih and E. Lee, “A compile-time scheduling heuristic for [43] R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction. interconnection-constrained heterogeneous processor architectures,” Par- MIT Press, 1998. allel and Distributed Systems, IEEE Transactions on, vol. 4, no. 2, pp. [44] E. Gelenbe and I. Mitrani, Analysis and Synthesis of Computer Systems. 175–187, Feb 1993. World Scientiﬁc, Imperial College Press, 2010. [21] Y.-K. Kwok and I. Ahmad, “Dynamic critical-path scheduling: an ef- [45] E. Gelenbe, “Probabilistic models of computer systems,” Acta fective technique for allocating task graphs to multiprocessors,” Parallel Informatica, vol. 12, pp. 285–303, 1979. [Online]. Available: and Distributed Systems, IEEE Transactions on, vol. 7, no. 5, pp. 506– http://dx.doi.org/10.1007/BF00268317 521, May 1996. [46] E. Gelenbe and J. Fourneau, “Random neural networks with multiple [22] E. Hou, N. Ansari, and H. Ren, “A genetic algorithm for multiprocessor classes of signals,” Neural Computation, vol. 11, no. 4, pp. 953–963, scheduling,” Parallel and Distributed Systems, IEEE Transactions on, 1999. vol. 5, no. 2, pp. 113–120, Feb 1994. [47] E. Gelenbe, “Stability of the random neural network model,” Neural [23] W. Chen and J. Zhang, “An ant colony optimization approach to a Computation, vol. 2, no. 2, pp. 239–247, 1990. grid workﬂow scheduling problem with various qos requirements,” [48] E. Gelenbe and S. Timotheou, “Synchronized interactions in spiked Systems, Man, and Cybernetics, Part C: Applications and Reviews, IEEE neuronal networks,” The Computer Journal, vol. 51, no. 6, pp. 723– Transactions on, vol. 39, no. 1, pp. 29–43, Jan 2009. 730, 2008.

label: reference

text: R.

text: Buyya,

text: “Introduction

text: to

text: the

text: ieee

text: transactions

text: on

text: cloud

text: computing,”

text: S.

text: Pandey,

text: W.

text: Linlin,

text: S.

text: Guru,

text: and

text: R.

text: Buyya,

text: “A

text: particle

text: swarm

text: Cloud

text: Computing,

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: Jan

text: optimization-based

text: heuristic

text: for

text: scheduling

text: workﬂow

text: applications

text: in

text: cloud

text: computing

text: environments,”

text: in

text: Advanced

text: Information

text: Networking

text: C.

text: Delimitrou

text: and

text: C.

text: Kozyrakis,

text: “Qos-aware

text: scheduling

text: in

text: and

text: Applications

text: (AINA),

text: IEEE

text: International

text: Conference

text: on,

text: heterogeneous

text: datacenters

text: with

text: paragon,”

text: ACM

text: Trans.

text: Comput.

text: April

text: pp.

text: Syst.,

text: vol.

text: no.

text: pp.

text: Dec.

text: [Online].

text: Available:

text: E.

text: Gelenbe

text: and

text: J.

text: Fourneau,

text: “Random

text: neural

text: networks

text: with

text: multiple

text: classes

text: of

text: signals,”

text: Neural

text: Computation,

text: P.

text: Pradeep,

text: K.

text: G.

text: Shin,

text: X.

text: Zhu,

text: M.

text: Uysal,

text: Z.

text: Wang,

text: S.

text: Singhal,

text: vol.

text: no.

text: pp.

text: [Online].

text: Available:

text: A.

text: Merchant,

text: and

text: K.

text: Salem,

text: “Adaptive

text: control

text: of

text: virtualized

text: resources

text: in

text: utility

text: computing

text: environments,”

text: in

text: Proceedings

text: of

text: the

text: ACM

text: E.

text: Gelenbe,

text: S.

text: Timotheou,

text: and

text: D.

text: Nicholson,

text: “Fast

text: distributed

text: SIGOPS/EuroSys

text: European

text: Conference

text: on

text: Computer

text: Systems

text: near-optimum

text: assignment

text: of

text: assets

text: to

text: tasks,”

text: The

text: Computer

text: Journal,

text: ser.

text: EuroSys

text: New

text: York,

text: NY,

text: USA:

text: ACM,

text: pp.

text: vol.

text: no.

text: pp.

text: Nov.

text: [Online].

text: Available:

text: [Online].

text: Available:

text: A.

text: Berl,

text: E.

text: Gelenbe,

text: M.

text: Di

text: Girolamo,

text: G.

text: Giuliani,

text: H.

text: De

text: Meer,

text: M.

text: Q.

text: S.

text: Zaman

text: and

text: D.

text: Grosu,

text: “A

text: combinatorial

text: auction-based

text: dynamic

text: vm

text: Dang,

text: and

text: K.

text: Pentikousis,

text: “Energy-efﬁcient

text: cloud

text: computing,”

text: The

text: provisioning

text: and

text: allocation

text: in

text: clouds,”

text: in

text: Cloud

text: Computing

text: Technology

text: Computer

text: Journal,

text: vol.

text: no.

text: pp.

text: and

text: Science

text: (CloudCom),

text: IEEE

text: Third

text: International

text: Conference

text: E.

text: Gelenbe

text: and

text: R.

text: Lent,

text: “Energy-qos

text: trade-offs

text: in

text: mobile

text: service

text: on,

text: Nov

text: pp.

text: selection,”

text: Future

text: Internet,

text: vol.

text: no.

text: pp.

text: [Online].

text: C.

text: Lin

text: and

text: S.

text: Lu,

text: “Scheduling

text: scientiﬁc

text: workﬂows

text: elastically

text: for

text: cloud

text: Available:

text: computing,”

text: in

text: Cloud

text: Computing

text: (CLOUD),

text: IEEE

text: International

text: “Optimising

text: server

text: energy

text: consumption

text: and

text: response

text: time,”

text: Conference

text: on,

text: July

text: pp.

text: Theoretical

text: and

text: Applied

text: Informatics,

text: no.

text: pp.

text: Jan

text: I.

text: S.

text: Moreno,

text: P.

text: Garraghan,

text: P.

text: Townend,

text: and

text: J.

text: Xu,

text: “Analysis,

text: modeling

text: J.

text: Zhan,

text: L.

text: Wang,

text: X.

text: Li,

text: W.

text: Shi,

text: C.

text: Weng,

text: W.

text: Zhang,

text: and

text: X.

text: Zang,

text: and

text: simulation

text: of

text: workload

text: patterns

text: in

text: a

text: large-scale

text: utility

text: cloud,”

text: Cloud

text: “Cost-aware

text: cooperative

text: resource

text: provisioning

text: for

text: heterogeneous

text: work-

text: Computing,

text: IEEE

text: Transactions

text: on,

text: vol.

text: PP,

text: no.

text: pp.

text: loads

text: in

text: data

text: centers,”

text: Computers,

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: B.

text: Palanisamy,

text: A.

text: Singh,

text: and

text: L.

text: Liu,

text: “Cost-effective

text: resource

text: provision-

text: pp.

text: Nov

text: ing

text: for

text: mapreduce

text: in

text: a

text: cloud,”

text: Parallel

text: and

text: Distributed

text: Systems,

text: IEEE

text: A.

text: Iosup,

text: S.

text: Ostermann,

text: M.

text: Yigitbasi,

text: R.

text: Prodan,

text: T.

text: Fahringer,

text: and

text: Transactions

text: on,

text: vol.

text: PP,

text: no.

text: pp.

text: D.

text: H.

text: J.

text: Epema,

text: “Performance

text: analysis

text: of

text: cloud

text: computing

text: services

text: S.

text: Zhuravlev,

text: S.

text: Blagodurov,

text: and

text: A.

text: Fedorova,

text: “Addressing

text: shared

text: for

text: many-tasks

text: scientiﬁc

text: computing,”

text: Parallel

text: and

text: Distributed

text: Systems,

text: resource

text: contention

text: in

text: multicore

text: processors

text: via

text: scheduling,”

text: SIGPLAN

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: June

text: Not.,

text: vol.

text: no.

text: pp.

text: Mar.

text: [Online].

text: Available:

text: S.

text: Zhuravlev,

text: S.

text: Blagodurov,

text: and

text: A.

text: Fedorova,

text: “Addressing

text: shared

text: N.

text: Bhatti

text: and

text: R.

text: Friedrich,

text: “Web

text: server

text: support

text: for

text: tiered

text: services,”

text: resource

text: contention

text: in

text: multicore

text: processors

text: via

text: scheduling,”

text: SIGPLAN

text: Network,

text: IEEE,

text: vol.

text: no.

text: pp.

text: Sep

text: Not.,

text: vol.

text: no.

text: pp.

text: Mar.

text: [Online].

text: Available:

text: Y.

text: Song,

text: Y.

text: Sun,

text: and

text: W.

text: Shi,

text: “A

text: two-tiered

text: on-demand

text: resource

text: allocation

text: mechanism

text: for

text: vm-based

text: data

text: centers,”

text: Services

text: Computing,

text: A.

text: N.

text: Tantawi

text: and

text: D.

text: Towsley,

text: “Optimal

text: static

text: load

text: balancing

text: in

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: First

text: distributed

text: computer

text: systems,”

text: Journal

text: of

text: the

text: ACM

text: (JACM),

text: vol.

text: E.

text: Gelenbe,

text: R.

text: Lent,

text: and

text: M.

text: Douratsos,

text: “Choosing

text: a

text: local

text: or

text: remote

text: no.

text: pp.

text: cloud,”

text: in

text: Second

text: Symposium

text: on

text: Network

text: Cloud

text: Computing

text: and

text: C.

text: Kim

text: and

text: H.

text: Kameda,

text: “An

text: algorithm

text: for

text: optimal

text: static

text: load

text: balancing

text: Applications,

text: NCCA

text: London,

text: United

text: Kingdom,

text: December

text: in

text: distributed

text: computer

text: systems,”

text: IEEE

text: Transactions

text: on

text: Computers,

text: IEEE

text: Computer

text: Society,

text: pp.

text: [Online].

text: Available:

text: vol.

text: no.

text: pp.

text: H.

text: Kameda,

text: J.

text: Li,

text: C.

text: Kim,

text: and

text: Y.

text: Zhang,

text: Optimal

text: Load

text: Balancing

text: in

text: E.

text: Gelenbe

text: and

text: R.

text: Lent,

text: “Trade-offs

text: between

text: energy

text: and

text: quality

text: of

text: Distributed

text: Computer

text: Systems.

text: Springer,

text: service,”

text: in

text: Sustainable

text: Internet

text: and

text: ICT

text: for

text: Sustainability

text: (SustainIT),

text: J.

text: J.

text: Wolff,

text: “Dynamic

text: load

text: balancing

text: of

text: a

text: network

text: of

text: client

text: and

text: server

text: IEEE,

text: pp.

text: computers,”

text: Feb.

text: u.

text: S.

text: Patent

text: Q.

text: Zhang,

text: M.

text: Zhani,

text: R.

text: Boutaba,

text: and

text: J.

text: Hellerstein,

text: “Dynamic

text: B.

text: P.

text: Rimal,

text: E.

text: Choi,

text: and

text: I.

text: Lumb,

text: “A

text: taxonomy

text: and

text: survey

text: of

text: heterogeneity-aware

text: resource

text: provisioning

text: in

text: the

text: cloud,”

text: Cloud

text: Com-

text: cloud

text: computing

text: systems,”

text: in

text: INC,

text: IMS

text: and

text: IDC,

text: Fifth

text: puting,

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: March

text: International

text: Joint

text: Conference

text: on.

text: Ieee,

text: pp.

text: T.

text: Dillon,

text: C.

text: Wu,

text: and

text: E.

text: Chang,

text: “Cloud

text: computing:

text: issues

text: and

text: chal-

text: Z.

text: Zhang

text: and

text: X.

text: Zhang,

text: “A

text: load

text: balancing

text: mechanism

text: based

text: on

text: lenges,”

text: in

text: Advanced

text: Information

text: Networking

text: and

text: Applications

text: (AINA),

text: ant

text: colony

text: and

text: complex

text: network

text: theory

text: in

text: open

text: cloud

text: computing

text: IEEE

text: International

text: Conference

text: on.

text: Ieee,

text: pp.

text: federation,”

text: in

text: Industrial

text: Mechatronics

text: and

text: Automation

text: (ICIMA),

text: E.

text: Gelenbe,

text: R.

text: Lent,

text: and

text: A.

text: Nunez,

text: “Self-aware

text: networks

text: and

text: QoS,”

text: International

text: Conference

text: on,

text: vol.

text: IEEE,

text: pp.

text: Proceedings

text: of

text: the

text: IEEE,

text: vol.

text: no.

text: pp.

text: Sep.

text: W.

text: Tian,

text: Y.

text: Zhao,

text: Y.

text: Zhong,

text: M.

text: Xu,

text: and

text: C.

text: Jing,

text: “A

text: dynamic

text: and

text: E.

text: Gelenbe

text: and

text: S.

text: Timotheou,

text: “Random

text: neural

text: networks

text: with

text: synchro-

text: integrated

text: load-balancing

text: scheduling

text: algorithm

text: for

text: cloud

text: datacenters,”

text: nized

text: interactions,”

text: Neural

text: Computation,

text: vol.

text: no.

text: pp.

text: in

text: Cloud

text: Computing

text: and

text: Intelligence

text: Systems

text: (CCIS),

text: IEEE

text: [Online].

text: Available:

text: International

text: Conference

text: on.

text: IEEE,

text: pp.

text: X.

text: Zhu,

text: X.

text: Qin,

text: and

text: M.

text: Qiu,

text: “Qos-aware

text: fault-tolerant

text: scheduling

text: for

text: E.

text: Gelenbe,

text: “Sensible

text: decisions

text: based

text: on

text: qos,”

text: Computational

text: manage-

text: real-time

text: tasks

text: on

text: heterogeneous

text: clusters,”

text: Computers,

text: IEEE

text: Transac-

text: ment

text: science,

text: vol.

text: no.

text: pp.

text: tions

text: on,

text: vol.

text: no.

text: pp.

text: June

text: E.

text: Gelenbe,

text: Z.

text: Xu,

text: and

text: E.

text: Seref,

text: “Cognitive

text: packet

text: networks,”

text: in

text: H.

text: Topcuouglu,

text: S.

text: Hariri,

text: and

text: M.

text: you

text: Wu,

text: “Performance-effective

text: and

text: Proceedings

text: of

text: the

text: International

text: Conference

text: on

text: Tools

text: with

text: Artiﬁcial

text: low-complexity

text: task

text: scheduling

text: for

text: heterogeneous

text: computing,”

text: IEEE

text: Intelligence,

text: Nov.

text: pp.

text: Trans.

text: Parallel

text: Distrib.

text: Syst.,

text: vol.

text: no.

text: pp.

text: Mar.

text: E.

text: Gelenbe,

text: “Steps

text: toward

text: self-aware

text: networks,”

text: Communications

text: ACM,

text: [Online].

text: Available:

text: vol.

text: no.

text: pp.

text: July

text: G.

text: Sih

text: and

text: E.

text: Lee,

text: “A

text: compile-time

text: scheduling

text: heuristic

text: for

text: R.

text: S.

text: Sutton

text: and

text: A.

text: G.

text: Barto,

text: Reinforcement

text: Learning:

text: An

text: Introduction.

text: interconnection-constrained

text: heterogeneous

text: processor

text: architectures,”

text: Par-

text: MIT

text: Press,

text: allel

text: and

text: Distributed

text: Systems,

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: E.

text: Gelenbe

text: and

text: I.

text: Mitrani,

text: Analysis

text: and

text: Synthesis

text: of

text: Computer

text: Systems.

text: Feb

text: World

text: Scientiﬁc,

text: Imperial

text: College

text: Press,

text: Y.-K.

text: Kwok

text: and

text: I.

text: Ahmad,

text: “Dynamic

text: critical-path

text: scheduling:

text: an

text: ef-

text: E.

text: Gelenbe,

text: “Probabilistic

text: models

text: of

text: computer

text: systems,”

text: Acta

text: fective

text: technique

text: for

text: allocating

text: task

text: graphs

text: to

text: multiprocessors,”

text: Parallel

text: Informatica,

text: vol.

text: pp.

text: [Online].

text: Available:

text: and

text: Distributed

text: Systems,

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: May

text: E.

text: Gelenbe

text: and

text: J.

text: Fourneau,

text: “Random

text: neural

text: networks

text: with

text: multiple

text: E.

text: Hou,

text: N.

text: Ansari,

text: and

text: H.

text: Ren,

text: “A

text: genetic

text: algorithm

text: for

text: multiprocessor

text: classes

text: of

text: signals,”

text: Neural

text: Computation,

text: vol.

text: no.

text: pp.

text: scheduling,”

text: Parallel

text: and

text: Distributed

text: Systems,

text: IEEE

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: Feb

text: E.

text: Gelenbe,

text: “Stability

text: of

text: the

text: random

text: neural

text: network

text: model,”

text: Neural

text: W.

text: Chen

text: and

text: J.

text: Zhang,

text: “An

text: ant

text: colony

text: optimization

text: approach

text: to

text: a

text: Computation,

text: vol.

text: no.

text: pp.

text: grid

text: workﬂow

text: scheduling

text: problem

text: with

text: various

text: qos

text: requirements,”

text: E.

text: Gelenbe

text: and

text: S.

text: Timotheou,

text: “Synchronized

text: interactions

text: in

text: spiked

text: Systems,

text: Man,

text: and

text: Cybernetics,

text: Part

text: C:

text: Applications

text: and

text: Reviews,

text: IEEE

text: neuronal

text: networks,”

text: The

text: Computer

text: Journal,

text: vol.

text: no.

text: pp.

text: Transactions

text: on,

text: vol.

text: no.

text: pp.

text: Jan


text: 11 space is identical to Eq. (52). But the frequency ω has a k differe(cid:114)nt form, where √1 − 4K πk · · · − φ (k) = e 2πikn/L, (PBC), ω = ω2 + sin( )2, k = 1, , L 1, (DBC). n k 0 M 2L L (55) √2 φ (k) = sin(πkn/L), (DBC). (59) n L The covariance matrix is constructed from the two-point correlators (cid:18) (cid:19) Therefore, the time evolution of the covariance matrix is (cid:104) | | (cid:105) (cid:104) | | (cid:105) 0 q q 0 0 q p 0 γ(t) = S(t)γ(0)S(t)T, (60) γ = Re (cid:104) | n m| (cid:105) (cid:104) | n m| (cid:105) . (56) n,m 0 p q 0 0 p p 0 n m n m where the ev(cid:88)olution matrix is For PBC, the correlators are (cid:20) (cid:21) (cid:88) (cid:32) ∗ (cid:33) − S (t) = φ (n)φ (m) (cid:104) | | (cid:105) 1 L 1 2πk(n m) n,m k k 0 qnqm 0 = 2L k(cid:88)=1 Mωk cos(cid:20) L− (cid:21) , ×k −√√1M cos ωkt √1√M ωk−1 sin ωkt . (cid:104) | | (cid:105) 1 L 2πk(n m) Mωk sin ωkt M cos ωkt 0 p p 0 = Mω cos , n m 2L k L (61) k=1 (cid:104) | | (cid:105) 0 qnpm 0 = iδn,m/2. (57) The entanglement properties are encoded in the reduced density matrix, which can be extracted from the the covariance For DBC, the correlators are (cid:18) (cid:19) (cid:18) (cid:19) matrix γA associated with the the subsystem A. The logarith- (cid:88) mic negativity is deﬁned by the partial transposition of the ∪ reduced density matrix ρ with the subsystem A = A A − E | | A 1 2 (cid:104) | | (cid:105) 1 L 1 1 πkn πkm as = lnTr ρT2 . We ﬁrst consider the partial transposition 0 qnqm 0 = L (cid:88) Mω sin(cid:18) L (cid:19) sin(cid:18) L (cid:19) , of γA, which cAan be constructed by inverting the signs of the k=1 k momenta corres(cid:18)ponding to A(cid:19) [9]. (cid:18) (cid:19) − 2 (cid:104) | | (cid:105) 1 L 1 πkn πkm 0 pnpm 0 = L Mωk sin L sin L , I 0 · · I 0 (cid:104) | | (cid:105) k=1 γAT2 = 0lA RlA γA 0lA RlA , (62) 0 q p 0 = iδ /2. (58) lA A2 lA A2 n m n,m where (cid:18) (cid:19) B. Evolution matrix and the logarithmic negativity (cid:101) (cid:101) (cid:101) (cid:101) R 1 0 ∈ From the H−eisenberg equation of motion, q˙k(t) = M1 pk(t) [ A2]n,m = 0 −1 δn,m, n, m A2, (63) and p˙ (t) = Mω2q (t), we have k(cid:101) k k (cid:101) (cid:101) and l is the number of sites in region A. The symplectic A q(cid:101)k(t) = √√1M (cos ωktqk(0(cid:101)) + ωk−1 sin ωkt(cid:101)pk(0)), smpuelcttirpulmyinogf γwATit2hcaansybme opble(cid:18)tcatiincemd baytriexxaΣ(cid:19)ct diagonalization after − pk(t) = M( ωk sin ωktqk(0) + cos ωktpk(0)). 0 I Σ = −I lA . (64) 0 The time-dependent canonical variables in the real space lA are  (cid:88) · q (t) =√1(cid:0) φ∗(n)φ (m) (cid:1) eoifgAtehnfetvearplaucreotsima{lp±utrtλainn1g,sp±tohλsei2t,iso·pn·e·co±trfuλtmhleAo;rfλedµiΣuc>edγ0A}d,ewnthsieitthytrtamhceaetsrneioxtromisf n M k k [9, 22] (cid:18) (cid:19)  √× c(cid:88)oks,mωk∗tqm(0) + ωk−1 sin ωktpm(0) | | (cid:89)lA 1 (cid:89)lA 1 pn(t) = M φk(n)φk(m) Tr ρTA2 = |λ + 1| − |λ − 1| = max 1, 2λ . k,m ν=1 ν 2 ν 2 ν=1 µ × − (65) ( ω sin ω tq (0) + cos ω tp (0)) , k k m k m ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: space

text: is

text: identical

text: to

text: Eq.

text: But

text: the

text: frequency

text: ω

text: has

text: a

text: k

label: equation

text: form,

text: where

text: √1

text: πk

text: φ

text: (k)

text: e

text: (PBC),

text: ω

text: ω2

text: sin(

text: k

text: L

text: (DBC).

text: n

text: k

text: M

text: L

text: √2

text: φ

text: (k)

text: sin(πkn/L),

text: (DBC).

text: n

text: L

text: The

text: covariance

text: matrix

text: is

text: constructed

text: from

text: the

text: two-point

text: correlators

text: Therefore,

text: the

text: time

text: evolution

text: of

text: the

text: covariance

text: matrix

text: is

text: q

text: q

text: q

text: p

text: γ(t)

text: γ

text: Re

text: n

text: m|

text: n

text: m|

text: n,m

text: p

text: q

text: p

text: p

text: n

text: m

text: n

text: m

text: where

text: the

text: matrix

text: is

text: For

text: PBC,

text: the

text: correlators

text: are

text: S

text: (t)

text: φ

text: (n)φ

text: (m)

text: L

text: m)

text: n,m

text: k

text: k

text: qnqm

text: Mωk

text: L−

text: ×k

text: −√√1M

text: cos

text: ωkt

text: √1√M

text: ωk−1

text: sin

text: ωkt

text: L

text: m)

text: Mωk

text: sin

text: ωkt

text: M

text: cos

text: ωkt

text: p

text: p

text: Mω

text: cos

text: n

text: m

text: k

text: L

text: qnpm

text: The

text: entanglement

text: properties

text: are

text: encoded

text: in

text: the

text: reduced

text: density

text: matrix,

text: which

text: can

text: be

text: extracted

text: from

text: the

text: the

text: covariance

text: For

text: DBC,

text: the

text: correlators

text: are

text: matrix

text: γA

text: associated

text: with

text: the

text: the

text: subsystem

text: A.

text: The

text: logarith-

text: mic

text: negativity

text: is

text: deﬁned

text: by

text: the

text: partial

text: transposition

text: of

text: the

text: reduced

text: density

text: matrix

text: ρ

text: with

text: the

text: subsystem

text: A

text: A

text: A

text: E

text: A

text: L

text: πkn

text: πkm

text: as

text: lnTr

text: We

text: ﬁrst

text: consider

text: the

text: partial

text: transposition

text: qnqm

text: L

text: Mω

text: L

text: L

text: of

text: γA,

text: which

text: cAan

text: be

text: constructed

text: by

text: inverting

text: the

text: signs

text: of

text: the

text: k

text: momenta

text: to

text: L

text: πkn

text: πkm

text: pnpm

text: L

text: Mωk

text: sin

text: L

text: sin

text: L

text: I

text: I

text: RlA

text: γA

text: RlA

text: q

text: p

text: iδ

text: lA

text: lA

text: n

text: m

text: n,m

text: where

text: B.

text: Evolution

text: matrix

text: and

text: the

text: logarithmic

text: negativity

text: R

text: From

text: the

text: H−eisenberg

text: equation

text: of

text: motion,

text: q˙k(t)

text: pk(t)

text: −1

text: δn,m,

text: n,

text: m

text: and

text: p˙

text: (t)

text: (t),

text: we

text: have

text: k

text: k

text: and

text: l

text: is

text: the

text: number

text: of

text: sites

text: in

text: region

text: A.

text: The

text: symplectic

text: A

text: √√1M

text: (cos

text: ωk−1

text: sin

text: smpuelcttirpulmyinogf

text: diagonalization

text: after

text: pk(t)

text: M(

text: ωk

text: sin

text: cos

text: I

text: Σ

text: −I

text: lA

text: The

text: time-dependent

text: canonical

text: variables

text: in

text: the

text: real

text: space

text: lA

text: are

text: q

text: (t)

text: φ∗(n)φ

text: (m)

text: n

text: M

text: k

text: k

text: ωk−1

text: sin

text: pn(t)

text: M

text: φk(n)φk(m)

text: Tr

text: |λ

text: |λ

text: max

text: 2λ

text: k,m

text: ν

text: ν

text: µ

text: ω

text: sin

text: ω

text: tq

text: cos

text: ω

text: tp

text: k

text: k

text: m

text: k

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16  (cid:112) The real time evolution can be obtained by replacing τ with  it in the last step. By expanding to the second order in (cid:15), ρ cos θ and ρ sin θ are expressed as follows. For l > 0, one 2(cid:112)l(l + t), t < l has (cid:32) (cid:33) w = (cid:15) (88)  (cid:2) (cid:3) 2˜3 2(cid:15) t(l + t). t > l → (cid:32) (cid:15)2 (cid:33) ρ cos θ max[l, t] 1 + 2 (cid:2)(max[l, t])2 − (min[l, t])2 (cid:3) Based on the expressions of wij, we can obtain the cross- → (cid:15)2 ratio ηij as follows. ρ sin θ i min[l, t] 1 + 2 (min[l, t])2 − (max[l, t])2 .  − l t  For l < 0, one has (cid:32) (cid:33) l + t, t < l  (cid:2) (cid:3) η12 = l2(cid:15)2 · 1− , t > l (89) → | | (cid:32) (cid:15)2 (cid:33) 4 t2(t2 l2) ρ cos θ max[ l , t] 1 + 2 (m(cid:2)ax[|l|, t])2 − (min[|l|, t])2 (cid:3)   → − | | (cid:15)2 ρ sin θ i min[ l , t] 1 + 2 (min[|l|, t])2 − (max[|l|, t])2 . 1, t < l η13 = l2(cid:15)2 (90) A. Appendix I: w for symmetric adjacent intervals − . t > l ij (t2 l2)2 (cid:114)    − (cid:112)l +l t t < l ll + tt, t < l w12 = l (83) η23 = l2(cid:15)2 · 1 (91) , t > l − . t > l t(t + l) 4 t2(t2 l2) (cid:114) (cid:28) −  It is found that for the cases t l, t = l + 0 and t > l, one always has η = 1 or 0. ij  (cid:112)−l t < l B. Appendix II: wij for asymmetric adjacent intervals w = l t (84) 1˜2 2 − For asymmetric adjacent intervals, w and w have the t(t l), t > l 12 1˜2 (cid:15) same form as those in the symmetric case, and we only need  to change l with l . Similarly, for w and w , we just change (cid:112) 1 23 2˜3  l with l2. Therefore, what we need to recalculate here are w13 w13 = 2(cid:15) (cid:114)l2 −1t2, t < l (85) and w1˜3 as follo2w(cid:113)(cid:112)s. lI2n−thte2c,ase of l1 < l2, onte<hals 2l − , t > l (cid:15) 2 1  (cid:112) t2 l2 w13 = (cid:115)2(cid:15) (l1 + l2)(l2 − t), l1 < t < l2 (92) (l + l )2 2(cid:112)l2 − t2, t < l (t −1l )(t2+ l ), t > l2 w = (cid:15) (86) 2 1 1˜3 2 −  (cid:113)  (cid:15)(cid:112)t2 l2, t > l 2(cid:112) − (cid:115)2(cid:15) l(l − t), t < l w1˜3 = 2(cid:15)(cid:15) (cid:112)l(22t + lt22),(l2 − l1), l1 < tt << ll12 (93) w = (87) 2 − 23 l2 (t + l )(t l ). t > l − , t > l (cid:15) 2 1 2 t(t l) In the case of l > l , one has 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: The

text: real

text: time

text: evolution

text: can

text: be

text: obtained

text: by

text: replacing

text: τ

text: with

text: it

text: in

text: the

text: last

text: step.

text: By

text: expanding

text: to

text: the

text: second

text: order

text: in

text: ρ

text: cos

text: θ

text: and

text: ρ

text: sin

text: θ

text: are

text: expressed

text: as

text: follows.

text: For

text: l

text: one

text: t),

text: t

text: l

text: has

text: w

text: t(l

text: t).

text: t

text: l

text: ρ

text: cos

text: θ

text: max[l,

text: t]

text: (min[l,

text: Based

text: on

text: the

text: expressions

text: of

text: wij,

text: we

text: can

text: obtain

text: the

text: cross-

text: ratio

text: ηij

text: as

text: follows.

text: ρ

text: sin

text: θ

text: i

text: min[l,

text: t]

text: (min[l,

text: (max[l,

text: l

text: t

text: For

text: l

text: one

text: has

text: l

text: t,

text: t

text: l

text: 1−

text: t

text: l

text: ρ

text: cos

text: θ

text: max[

text: l

text: t]

text: (min[|l|,

text: ρ

text: sin

text: θ

text: i

text: min[

text: l

text: t]

text: (min[|l|,

text: (max[|l|,

text: t

text: l

text: A.

text: Appendix

text: I:

text: w

text: for

text: symmetric

text: adjacent

text: intervals

text: t

text: l

text: ij

text: +l

text: t

text: t

text: l

text: ll

text: tt,

text: t

text: l

text: l

text: t

text: l

text: t

text: l

text: t(t

text: l)

text: It

text: is

text: found

text: that

text: for

text: the

text: cases

text: t

text: l,

text: t

text: l

text: and

text: t

text: l,

text: one

text: always

text: has

text: η

text: or

text: ij

text: t

text: l

text: B.

text: Appendix

text: II:

text: wij

text: for

text: asymmetric

text: adjacent

text: intervals

text: w

text: l

text: t

text: For

text: asymmetric

text: adjacent

text: intervals,

text: w

text: and

text: w

text: have

text: the

text: t(t

text: l),

text: t

text: l

text: same

text: form

text: as

text: those

text: in

text: the

text: symmetric

text: case,

text: and

text: we

text: only

text: need

text: to

text: change

text: l

text: with

text: l

text: Similarly,

text: for

text: w

text: and

text: w

text: we

text: just

text: change

text: l

text: with

text: Therefore,

text: what

text: we

text: need

text: to

text: recalculate

text: here

text: are

text: t

text: l

text: and

text: as

text: of

text: onte<hals

text: t

text: l

text: t),

text: t

text: (l

text: l

text: t

text: l

text: (t

text: l

text: t

text: w

text: t

text: l

text: l(l

text: t),

text: t

text: l

text: tt

text: w

text: (t

text: l

text: )(t

text: l

text: t

text: l

text: t

text: l

text: t(t

text: l)

text: In

text: the

text: case

text: of

text: l

text: l

text: one

text: has

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 17  w13 = (cid:115)(cid:115)2(cid:15) (cid:113)((ltl122−+−ll2t2)2)(,(ll12++tt)), l2 < tt << ll21 (94) w14 = 22(cid:15)(cid:15) (cid:112)(cid:112)((dd ++(cid:115)ll ++ tt))((dd ++ ll1−− tt)),, d < t <td<+dl 2(d + l) − − , t > d + l (l + l )2 (t + d + l)(t d l) 1 2− , t > l (t + l )(t l ) 1 (99) 1 2  (cid:112) w1˜3 = (cid:115)2(cid:15) (cid:113)(cid:112)((lll122−−−ltt2)2)(,(tt−+ll2)), l2 < tt << ll21 (95) w1˜3 = (cid:115)22(cid:15) (cid:112)(t(−d +dl()tt()d+(d+d−−)l −t)−,t), d < t <td<+dl (100) 1 2 (t + d)(t d l), t > d + l 2 − (cid:15) (cid:15) (t + l2)(t l1). t > l1  (cid:112) Without loss of generality, here we choose l1 < l2, as what 2 (cid:114)d2 − t2, t < d we did in the main text. Then one has (cid:15) l − t w23 = 2d(cid:114)t2 −1 d2 , d < t < d + l (101) 1 , t < l 1 l + t 1 2d − , t > d + l η12 = ll1112244(cid:15)(cid:15)22 ·· tt22((tt2211−− ll1122)),, l1 < tt >< ll22 (96) w2˜4 = 22(cid:15)(cid:15) (cid:112)(cid:112)(cid:112)(l(dd++ltl2++tt)d)(,2d + l − t), d < t <td<+dl l − t 2(cid:15) (d + l + t)(t − d), t > d + l 2 , t < l (102) l + t 1 η23 = ll222 −+ tt, l1 < t < l2 (97) 2(cid:112)(cid:112) − l2(cid:15)2 · 1 (d + l + t)(d + l t), t < d 24 t2(t2 − l22). t > l2 w1˜4 = 2(cid:15)(cid:15) (cid:112)(d + l + t)(d + l − t), d < t < d + l   2(cid:15) (t + d + l)(t − d − l), t > d +(1l03) 1, t < l 1 η13 = ((ll22(cid:15)−+2(ttl))((+ll12 l+−)ll221)), l1 < t < l2 (98) (cid:115)2(cid:112)(d + t(cid:18))(d − t), (cid:19) t < d − 1 2− , t > l (cid:15) (t2 l2)(t2 l2) 2 It is found that for the c2ases t (cid:28)1l , t = l +0− and t > l , w13 = (cid:115)(t + d) (cid:18)t −1 d − t + 1d + l (cid:19), d < t < d + l 1 2 2 one always has ηij = 1 or 0. 1 − 1 C. Appendix III: w for symmetric disjoint intervals (2d + l) − , t > d + l ij t d t + d + l (104) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: tt

text: tt))((dd

text: tt)),,

text: d

text: t

text: <td<+dl

text: l)

text: t

text: d

text: l

text: (l

text: l

text: (t

text: d

text: l)(t

text: d

text: l)

text: 2−

text: t

text: l

text: (t

text: l

text: )(t

text: l

text: tt

text: +dl()tt()d+(d+d−−)l

text: −t)−,t),

text: d

text: t

text: <td<+dl

text: (t

text: d)(t

text: d

text: l),

text: t

text: d

text: l

text: (t

text: t

text: Without

text: loss

text: of

text: generality,

text: here

text: we

text: choose

text: as

text: what

text: 2

text: t

text: d

text: we

text: did

text: in

text: the

text: main

text: text.

text: Then

text: one

text: has

text: l

text: t

text: −1

text: d

text: t

text: d

text: l

text: t

text: l

text: l

text: t

text: t

text: d

text: l

text: tt

text: l

text: t),

text: d

text: t

text: <td<+dl

text: l

text: t

text: (d

text: l

text: t)(t

text: d),

text: t

text: d

text: l

text: t

text: l

text: l

text: t

text: ll222

text: tt,

text: t

text: (d

text: l

text: t)(d

text: l

text: t),

text: t

text: d

text: t

text: l

text: t)(d

text: l

text: t),

text: d

text: t

text: d

text: l

text: (t

text: d

text: l)(t

text: d

text: l),

text: t

text: d

text: t

text: l

text: t

text: t),

text: t

text: d

text: 2−

text: t

text: l

text: It

text: is

text: found

text: that

text: for

text: the

text: t

text: t

text: l

text: and

text: t

text: l

text: d)

text: −1

text: d

text: t

text: l

text: d

text: t

text: d

text: l

text: one

text: always

text: has

text: ηij

text: or

text: C.

text: Appendix

text: III:

text: w

text: for

text: symmetric

text: disjoint

text: intervals

text: l)

text: t

text: d

text: l

text: ij

text: t

text: d

text: t

text: d

text: l

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ytivitageN 5 3.0 3.0 (a) l=25 (b) l=25 l=50 l=50 2.5 l=75 2.5 l=75 l=100 l=100 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 0 20 40 60 80 100 0 20 40 60 80 100 Time Time E FIG. 5: Entanglement negativity for two symmetric adjacent intervals as a function of time. Here we choose the central charge c = 1, (cid:15) = 0.1, l = 25, 50, 75 and 100, respectively. Shown in (a) is the CFT result, and (b) is the numerical calculation based on a critical harmonic chain. (cid:10) (cid:11) (cid:48)   Now it is possible to eliminate a and c˜ in Eq. (19) in terms T T T of (cid:15), and then one can get 1 n(w1) ¯n2(w2) n(w3) RHP  (cid:81) c˜ (cid:101) η∆(n2)−2∆n 1/2 F { } E = −4c ln (cid:15)2 (cid:15)+2 t2 . (22) = 3i=1 |(wi −nwi)/a|∆(i) η11∆,,23(n2)η2∆,3(n2) ( ηj,k ), (25) (cid:29) In the limit t (cid:15), one ends with (cid:101) where η are cross ratios which can be constructed from i,j the endpoints w (and their images w ) of the intervals in the i i E c t RHP as follows = ln , (23) (cid:101) (cid:101) 2 (cid:15) − (cid:101) (cid:101) − (w w )(w w ) which was observed in the numerical calculations based on η = i − j i − j (26) a critical harmonic chain [27]. (cid:101) i,j (wi wj)(wi wj) 2. Symmetric ﬁnite intervals − with w = w¯ being the image of w (see Fig.3). The i i F { } i In this part, we consider the case of symmetric ﬁnite inter- nonuniversal functioFn {( ηj},k ) depends on the full operator vals with l = l = l, i.e., A ∈ [−l, 0] and A ∈ (0, l], as content of the CFT. ( ηj,k ) is usually difﬁcult to calculate 1 2 1 2 and only known for several speciﬁc CFTs and BCFTs. But shown in Fig. 4. In this case, ρ ∪ represents a mixed state, → ∞ and there is no correspondencAe 1beAtw2 een the logarithmic neg- iFt is{ foun}d that in the limits ηi,j 0, 1, or , the function ativity and the Renyi entropies. By using Eq. (10) and doing ( ηj,k ) is just a constant, which follows from the long- and short-distance expansions of the correlation functions of twist a c(cid:16)onformal(cid:17)mapp(cid:10)ing onto the RHP, one(cid:11)has operators [3, 32, 36–38]. For symmetric intervals in this part, we calculate the cross ratios η explicitly in the appendix. It Tr ρTA21∪A2 n = (cid:89)Tn(cid:12)(cid:12)(cid:12)(cid:12)(z1)T(cid:12)(cid:12)(cid:12)(cid:12)¯n2(z2(cid:10))Tn(z3) (cid:11) its=folu+nd0t−haatnodnte >alwl.aIynsohtahseηriiw,jjo=rd1s,ooru0r rfeosrutlhtseacraesuesnitve(cid:28)rsall, 3 dw ∆(i) T T T for the above three cases. = dz n(w1) ¯n2(w2) n(w3) RHP , By using Eqs. (5) and (10), and neglecting various nonuni- i=1 zi versal terms, we have (24) where the scaling dimensions ∆(1) = ∆(3) = ∆n and E = − c ln (cid:15)2 − c ln η1,3 , (27) ∆ = ∆(2). The three-point correlation function on the 4 (cid:15)2 + t2 8 η1,2η2,3 (2) n RHP can be expressed as ##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##

text: ytivitageN

font: default

text: (a)

text: (b)

text: Time

text: Time

text: E

label: caption

text: FIG.

text: Entanglement

text: negativity

text: for

text: two

text: symmetric

text: adjacent

text: intervals

text: as

text: a

text: function

text: of

text: time.

text: Here

text: we

text: choose

text: the

text: central

text: charge

text: c

text: l

text: and

text: respectively.

text: Shown

text: in

text: (a)

text: is

text: the

text: CFT

text: result,

text: and

text: (b)

text: is

text: the

text: numerical

text: calculation

text: based

text: on

text: a

text: critical

text: harmonic

text: chain.

text: Now

text: it

text: is

text: possible

text: to

text: eliminate

text: a

text: and

text: c˜

text: in

text: Eq.

text: in

text: terms

text: T

text: T

text: T

text: of

text: and

text: then

text: one

text: can

text: get

text: RHP

text: c˜

text: F

text: E

text: ln

text: |(wi

text: −nwi)/a|∆(i)

text: ηj,k

text: In

text: the

text: limit

text: t

text: one

text: ends

text: with

text: where

text: η

text: are

text: cross

text: ratios

text: which

text: can

text: be

text: constructed

text: from

text: i,j

text: the

text: endpoints

text: w

text: (and

text: their

text: images

text: w

text: of

text: the

text: intervals

text: in

text: the

text: i

text: i

text: E

text: c

text: t

text: RHP

text: as

text: follows

text: ln

text: (w

text: w

text: )(w

text: w

text: which

text: was

text: observed

text: in

text: the

text: numerical

text: calculations

text: based

text: on

text: η

text: i

text: j

text: i

text: j

text: a

text: critical

text: harmonic

text: chain

text: i,j

text: (wi

text: wj)(wi

text: wj)

text: Symmetric

text: ﬁnite

text: intervals

text: with

text: w

text: w¯

text: being

text: the

text: image

text: of

text: w

text: (see

text: The

text: i

text: i

text: F

text: i

text: In

text: this

text: part,

text: we

text: consider

text: the

text: case

text: of

text: symmetric

text: ﬁnite

text: inter-

text: nonuniversal

text: functioFn

text: ηj},k

text: depends

text: on

text: the

text: full

text: operator

text: vals

text: with

text: l

text: l

text: l,

text: i.e.,

text: A

text: [−l,

text: and

text: A

text: l],

text: as

text: content

text: of

text: the

text: CFT.

text: ηj,k

text: is

text: usually

text: difﬁcult

text: to

text: calculate

text: and

text: only

text: known

text: for

text: several

text: speciﬁc

text: CFTs

text: and

text: BCFTs.

text: But

text: shown

text: in

text: Fig.

text: In

text: this

text: case,

text: ρ

text: represents

text: a

text: mixed

text: state,

text: and

text: there

text: is

text: no

text: correspondencAe

text: een

text: the

text: logarithmic

text: neg-

text: iFt

text: is{

text: foun}d

text: that

text: in

text: the

text: limits

text: ηi,j

text: or

text: the

text: function

text: ativity

text: and

text: the

text: Renyi

text: entropies.

text: By

text: using

text: Eq.

text: and

text: doing

text: ηj,k

text: is

text: just

text: a

text: constant,

text: which

text: follows

text: from

text: the

text: long-

text: and

text: short-distance

text: expansions

text: of

text: the

text: correlation

text: functions

text: of

text: twist

text: a

text: onto

text: the

text: RHP,

text: operators

text: For

text: symmetric

text: intervals

text: in

text: this

text: part,

text: we

text: calculate

text: the

text: cross

text: ratios

text: η

text: explicitly

text: in

text: the

text: appendix.

text: It

text: Tr

text: n

text: dw

text: ∆(i)

text: T

text: T

text: T

text: for

text: the

text: above

text: three

text: cases.

text: dz

text: RHP

text: By

text: using

text: Eqs.

text: and

text: and

text: neglecting

text: various

text: nonuni-

text: zi

text: versal

text: terms,

text: we

text: have

text: where

text: the

text: scaling

text: dimensions

text: ∆n

text: and

text: E

text: c

text: ln

text: c

text: ln

text: The

text: three-point

text: correlation

text: function

text: on

text: the

text: n

text: RHP

text: can

text: be

text: expressed

text: as

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 which may be further expressed as − E ∼ tα (l t)β E = − c ln (cid:15)2 − c ln w13w1˜2w2˜3 , (28) ln l−ρ (l + t)−γ . (33) 4 (cid:15)2 + t2 4 w w w 1˜3 12 23 (cid:101) (cid:39) (cid:101) (cid:101) | − | | − | By ﬁtt(cid:39)ing the numerical−results, the(cid:39)y f−ound α = 1/2, β apnednwWdwhiitex˜ihr˜j,et=ohwneee|wehxciaap−vnreewsodsbjeiot|ﬁa,nnirsneesdotpfhweewcitjieijvn=ettalhynaw.gtilaermeewcnajtlc,nuwelgai˜jtaet=diviitnwytihEe aAwspja- 0roγe.n)s1eu5=l,htsaγs0in.αEOq0=..n1(313th0/ae)2n,,obdβtyhρse=er=tthi0na.ng2(d5cβ,,=E+iγn1γ=tha)en−dl0itma.2ki05tin.t2ag8nt(cid:28).dheFρloli,rm=boitou−ttrh((cid:29)CβEF+q(cid:15)T., function oftime as follows (30) and Eq. (33) collapse to = 12 ln t. We attribute th≤e  al,bboevceaduisseagwreeenmeegnlet/catgtrheeemnoenn-tutnoivtheersfaolllfouwncintigonfascFt. (F{oηr t }) j,k E  − 4c ln (cid:15)2 (cid:15)+2 t2 − 4c ln ll +− tt, t < l wthheircehfomreamyanyont obteocbotanisntatnhtes,coorurrecretssuclatlsinagrebenhoatvaicocr.urFaoter tan(cid:28)d = − c (cid:15)2 − c 4t2 (29) l, however, our CFT results are universal and independent≤of ln ln , t > l. the speciﬁc CFT. To reproduce the numerical results for t l 4 (cid:15)2 + t2 4 (cid:15)l in Ref. [27], we have to consider the nonuniversal functions F { } (cid:29) E ( η ), which is a difﬁcult task, and out of the scope of In the limit l, t  (cid:15), can be simpliﬁed as our wjo,krk.  Nevertheless, by comparing the values of the plateau be- − tween CFT results and numerical results in Fig. 5, it is found  c ln t2 + (cid:15)2 (l t), t < l they are very close to each other. To be concrete, let’s take E 4 (cid:15)2 (l + t) E = (30) (t = l ) for example. Based on Eq. (30), one can get c l 2 ln + const, t > l. 4 2(cid:15) E l 1 l E CFT(t = ) = ln + const. (34) Shown in Fig. 5(a) is the plot of with different l. At the 2 2 2(cid:15) (cid:28) very beginning of the local quench t l, based on Eq. (30), one has On the other hand, from the scaling behavior in Ref. [27], one can ﬁnd E c t = ln , (31) 2 (cid:15) E l 1 l − numerical(t = ) = ln + 0.13 ln 3 0.28 ln 2. (35) 2 2 2 which agrees with the result of semi-inﬁnite intervals as (cid:29) shown in Eq. (23). This is reasonable because in the limit In the large l limit, i.e., l (cid:15), one always has (cid:28) t l, the quasiparticles essentially propagate without notic- ing the ﬁnite size effect. For t < l, the entanglement saturates for a cerEtain time. Then for t > l, we get the ground-state ECFT(t = l ) (cid:39) Enumerical(t = l ) (cid:39) 1 ln l . (36) value of , i.e., 2 2 2 2 Before we end this part, we mention that it is interesting to EG = 4c ln 2l(cid:15), (32) cEh(etc<k hlo)wdeEp(etn)dbseohnavtehseinnoont-huenrilvaetrtiscael fmuondcteilosn. sCFon(s{iηderi}n)g, j,k which varies for different CFTs, we expect that for other crit- which is also observed in the numerical calculations in Ref. ical lattice models such as the critical Ising model one may E E [27]. Note that in the numerical calculations, tends to the observe different scaling behaviors in (t < l). E ground-state value gradually. In our CFT results, drops 3. Asymmetric ﬁnite intervals to the ground-state value immediately after t = l. This is because all quasiparticles propagate at the same velocity in In this part, we consider the case of asymmetric ﬁnite inter- ∈ − ∈ CFTs. In lattice models, however, the dispersion relation is vals with A [ l , 0] and A (0, l ], as shown in Fig. 4. 1 1 2 2 not linear for all momentum vectors, and therefore not all Without loss of generality, we suppose l < l . The calcula- 1 2 quasiparticles propagate at the same velocity, as discussed in tions are similar to the symmetric case, and we need to eval- detail in section IV. uate the three point correlation functions in Eq. (24). First, as E In addition, the scaling behavior of for a harmonic chain shown in the appendix, we calculate the cross ratio η explic- ij was numerically studied in Ref. [27]. For t < l, they proposed itly. It is found that one always has η = 1 or 0 for the cases (cid:28) − ij the ansatz t l , t = l + 0 and t > l . That is to say, our results 1 2 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: which

text: may

text: be

text: further

text: expressed

text: as

text: E

text: tα

text: (l

text: t)β

text: E

text: c

text: ln

text: c

text: ln

text: ln

text: l−ρ

text: (l

text: t)−γ

text: w

text: w

text: w

text: By

text: the

text: numerical−results,

text: f−ound

text: α

text: β

text: aAwspja-

text: function

text: oftime

text: as

text: follows

text: and

text: Eq.

text: collapse

text: to

text: ln

text: t.

text: We

text: attribute

text: th≤e

text: (F{oηr

text: t

text: j,k

text: E

text: ln

text: ln

text: ll

text: tt,

text: t

text: l

text: wthheircehfomreamyanyont

text: c

text: c

text: l,

text: however,

text: our

text: CFT

text: results

text: are

text: universal

text: and

text: independent≤of

text: ln

text: ln

text: t

text: l.

text: the

text: speciﬁc

text: CFT.

text: To

text: reproduce

text: the

text: numerical

text: results

text: for

text: t

text: l

text: in

text: Ref.

text: we

text: have

text: to

text: consider

text: the

text: nonuniversal

text: functions

text: F

text: E

text: η

text: which

text: is

text: a

text: difﬁcult

text: task,

text: and

text: out

text: of

text: the

text: scope

text: of

text: In

text: the

text: limit

text: l,

text: t

text: can

text: be

text: simpliﬁed

text: as

text: our

text: wjo,krk.

text: Nevertheless,

text: by

text: comparing

text: the

text: values

text: of

text: the

text: plateau

text: be-

text: tween

text: CFT

text: results

text: and

text: numerical

text: results

text: in

text: Fig.

text: it

text: is

text: found

text: c

text: ln

text: (l

text: t),

text: t

text: l

text: they

text: are

text: very

text: close

text: to

text: each

text: other.

text: To

text: be

text: concrete,

text: let’s

text: take

text: E

text: (l

text: t)

text: E

text: (t

text: l

text: for

text: example.

text: Based

text: on

text: Eq.

text: one

text: can

text: get

text: c

text: l

text: ln

text: const,

text: t

text: l.

text: E

text: l

text: l

text: E

text: CFT(t

text: ln

text: const.

text: Shown

text: in

text: Fig.

text: is

text: the

text: plot

text: of

text: with

text: different

text: l.

text: At

text: the

text: very

text: beginning

text: of

text: the

text: local

text: quench

text: t

text: l,

text: based

text: on

text: Eq.

text: one

text: has

text: On

text: the

text: other

text: hand,

text: from

text: the

text: scaling

text: behavior

text: in

text: Ref.

text: one

text: can

text: ﬁnd

text: E

text: c

text: t

text: ln

text: E

text: l

text: l

text: numerical(t

text: ln

text: ln

text: ln

text: which

text: agrees

text: with

text: the

text: result

text: of

text: semi-inﬁnite

text: intervals

text: as

text: shown

text: in

text: Eq.

text: This

text: is

text: reasonable

text: because

text: in

text: the

text: limit

text: In

text: the

text: large

text: l

text: limit,

text: i.e.,

text: l

text: one

text: always

text: has

text: t

text: l,

text: the

text: quasiparticles

text: essentially

text: propagate

text: without

text: notic-

text: ing

text: the

text: ﬁnite

text: size

text: effect.

text: For

text: t

text: l,

text: the

text: entanglement

text: saturates

text: for

text: a

text: cerEtain

text: time.

text: Then

text: for

text: t

text: l,

text: we

text: get

text: the

text: ground-state

text: ECFT(t

text: l

text: Enumerical(t

text: l

text: ln

text: l

text: value

text: of

text: i.e.,

text: Before

text: we

text: end

text: this

text: part,

text: we

text: mention

text: that

text: it

text: is

text: interesting

text: to

text: EG

text: ln

text: cEh(etc<k

text: fmuondcteilosn.

text: sCFon(s{iηderi}n)g,

text: j,k

text: which

text: varies

text: for

text: different

text: CFTs,

text: we

text: expect

text: that

text: for

text: other

text: crit-

text: which

text: is

text: also

text: observed

text: in

text: the

text: numerical

text: calculations

text: in

text: Ref.

text: ical

text: lattice

text: models

text: such

text: as

text: the

text: critical

text: Ising

text: model

text: one

text: may

text: E

text: E

text: Note

text: that

text: in

text: the

text: numerical

text: calculations,

text: tends

text: to

text: the

text: observe

text: different

text: scaling

text: behaviors

text: in

text: (t

text: l).

text: E

text: ground-state

text: value

text: gradually.

text: In

text: our

text: CFT

text: results,

text: drops

text: Asymmetric

text: ﬁnite

text: intervals

text: to

text: the

text: ground-state

text: value

text: immediately

text: after

text: t

text: l.

text: This

text: is

text: because

text: all

text: quasiparticles

text: propagate

text: at

text: the

text: same

text: velocity

text: in

text: In

text: this

text: part,

text: we

text: consider

text: the

text: case

text: of

text: asymmetric

text: ﬁnite

text: inter-

text: CFTs.

text: In

text: lattice

text: models,

text: however,

text: the

text: dispersion

text: relation

text: is

text: vals

text: with

text: A

text: l

text: and

text: A

text: l

text: as

text: shown

text: in

text: Fig.

text: not

text: linear

text: for

text: all

text: momentum

text: vectors,

text: and

text: therefore

text: not

text: all

text: Without

text: loss

text: of

text: generality,

text: we

text: suppose

text: l

text: l

text: The

text: calcula-

text: quasiparticles

text: propagate

text: at

text: the

text: same

text: velocity,

text: as

text: discussed

text: in

text: tions

text: are

text: similar

text: to

text: the

text: symmetric

text: case,

text: and

text: we

text: need

text: to

text: eval-

text: detail

text: in

text: section

text: IV.

text: uate

text: the

text: three

text: point

text: correlation

text: functions

text: in

text: Eq.

text: First,

text: as

text: E

text: In

text: addition,

text: the

text: scaling

text: behavior

text: of

text: for

text: a

text: harmonic

text: chain

text: shown

text: in

text: the

text: appendix,

text: we

text: calculate

text: the

text: cross

text: ratio

text: η

text: explic-

text: ij

text: was

text: numerically

text: studied

text: in

text: Ref.

text: For

text: t

text: l,

text: they

text: proposed

text: itly.

text: It

text: is

text: found

text: that

text: one

text: always

text: has

text: η

text: or

text: for

text: the

text: cases

text: ij

text: the

text: ansatz

text: t

text: l

text: t

text: l

text: and

text: t

text: l

text: That

text: is

text: to

text: say,

text: our

text: results

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10  0− 8c ln ((dd1 ++ dd2)+(dl2)(−d(cid:115)t)+(dl2−+tl)(+dt)+(dt2)(−dd1−+dl)) d1 < tt << dd12 1 2 2 2 2 1 E =  −− 48cc llnn 2[t(2(cid:15)d(−1d+1(d+d12d+(+2d)l)l)2−](dd(1d)+1(d+d(d2+l1)2−+(dddl22+−+)(ltd)l)23(−(d+t2d2l1+−)−(lddd−211)+)t()d(dt122++−l2d+l21))t()t(2d−2 +d22l)+ t) d1 +d2l << tt << dd12 ++ ll (47) 2 1 1 2 1 − c (d + d + 2l)(d + d ) ln 1 2 1 2 t > d + l 4 (d + d + l)2 2 1 2 One can check that when d = d = d, the result in Eq. (45) and position operators with canonical commutation relations 1 2 is reproduced. [p , p ] = [q , q ] = 0 and [q , p ] = iδ . E n m n m n m n,m According to Eq. (47), we plot (t) with different (d , d ) For periodic boundary condition (PBC), the Fourier trans- 1 2 in Fig. 8(a). Compared to the symmetric case, the ‘light-cone’ form of the canonical variables are eEf(fte)ctinicsrsetaisllesobqsueicrvkelyd.noTwhehadpifpfeernesnacte is that the time when  (cid:88) (cid:101) L √1 t = max[d1, d2], (48)  q(cid:101)n = k(cid:88)=1 qk Le2πikn/L, (51) E L √1 − and the time when (t) decreases quickly happens at qk = qn e 2πikn/L, L n=1 t = min[d + l, d + l], (49) 1 2 · · · where n = 1, , L. For p , the Fourier transform is iden- n tical to q . The Hamiltonian is diagonalized in the momentum n which is also in agreement with the quasiparticle picture. space (cid:18) (cid:19) III. NUMERICAL EVALUATION OF THE NEGATIVITY (cid:88) FOR A HARMONIC CHAIN AFTER A LOCAL QUENCH (cid:101) (cid:101) L 1 Mω2 In this section, to conﬁrm our CFT results, we study the H = 2M p2k + 2 k qk2 , (52) time evolution of the logarithmic negativity after a local quan- k=1 tum quench in a lattice model, a critical harmonic chain. The entanglement negativity for a harmonic chain has been nu- where(cid:114) merically studied in several works [9, 22, 26, 27, 39]. Here we follow the method developed in these works, and apply it to the local quench problem. We will ﬁrst introduce the lat- 4K πk · · · ω = ω2 + sin( )2, k = 1, , L, (PBC). tice model and the covariance matrix in part A. In part B, we k 0 M L introduce the evolution matrix and show how to calculate the (53) entanglement negativity. In part C, we apply the method to the cases studied with CFT approach, and compare the results For the Dirichlet boundary condition (DBC), the Fourier accordingly. transform is not valid due to the breaking of translational sym- A. Harmonic chain and the covariance matrix metry. However, the Fourier(cid:114)sine tran(cid:18)sform c(cid:19)an be deﬁned as The Hamiltonian of the harmonic chain is  (cid:88)− (cid:101) (cid:88) L 1 2 πkn H = N [2pM2n + M2ω02 qn2 + K2 (qn+1 − qn)2], (50) q(cid:101)n = Lk(cid:88)=−11 qk(cid:114)L2 sin (cid:18)πLkn(cid:19), (54) n=1 qk = qn L sin L , n=1 where N is the number of sites of the chain, M is the · · · mass scale, ω is the characteristic frequency, and K is the where n = 1, , L. For p , the Fourier sine transforma- 0 n nearest-neighbor coupling. p and q denote the momentum tion is deﬁned similarly. The Hamiltonian in the momentum n n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: 0−

text: ln

text: tt

text: E

text: llnn

text: t)

text: tt

text: ll

text: c

text: (d

text: d

text: d

text: ln

text: t

text: d

text: l

text: (d

text: d

text: One

text: can

text: check

text: that

text: when

text: d

text: d

text: d,

text: the

text: result

text: in

text: Eq.

text: and

text: position

text: operators

text: with

text: canonical

text: commutation

text: relations

text: is

text: reproduced.

text: [p

text: p

text: [q

text: q

text: and

text: [q

text: p

text: iδ

text: E

text: n

text: m

text: n

text: m

text: n

text: m

text: n,m

text: According

text: to

text: Eq.

text: we

text: plot

text: (t)

text: with

text: different

text: (d

text: d

text: For

text: periodic

text: boundary

text: condition

text: (PBC),

text: the

text: Fourier

text: trans-

text: in

text: Fig.

text: Compared

text: to

text: the

text: symmetric

text: case,

text: the

text: ‘light-cone’

text: form

text: of

text: the

text: canonical

text: variables

text: are

text: is

text: that

text: the

text: time

text: when

text: L

text: √1

text: t

text: qk

text: E

text: L

text: √1

text: and

text: the

text: time

text: when

text: (t)

text: decreases

text: quickly

text: happens

text: at

text: qk

text: qn

text: e

text: L

text: t

text: min[d

text: l,

text: d

text: l],

text: where

text: n

text: L.

text: For

text: p

text: the

text: Fourier

text: transform

text: is

text: iden-

text: n

text: tical

text: to

text: q

text: The

text: Hamiltonian

text: is

text: diagonalized

text: in

text: the

text: momentum

text: n

text: which

text: is

text: also

text: in

text: agreement

text: with

text: the

text: quasiparticle

text: picture.

text: space

text: III.

text: NUMERICAL

text: EVALUATION

text: OF

text: THE

text: NEGATIVITY

text: FOR

text: A

text: HARMONIC

text: CHAIN

text: AFTER

text: A

text: LOCAL

text: QUENCH

text: L

text: In

text: this

text: section,

text: to

text: conﬁrm

text: our

text: CFT

text: results,

text: we

text: study

text: the

text: H

text: k

text: time

text: evolution

text: of

text: the

text: logarithmic

text: negativity

text: after

text: a

text: local

text: quan-

text: tum

text: quench

text: in

text: a

text: lattice

text: model,

text: a

text: critical

text: harmonic

text: chain.

text: The

text: entanglement

text: negativity

text: for

text: a

text: harmonic

text: chain

text: has

text: been

text: nu-

text: merically

text: studied

text: in

text: several

text: works

text: Here

text: we

text: follow

text: the

text: method

text: developed

text: in

text: these

text: works,

text: and

text: apply

text: it

text: to

text: the

text: local

text: quench

text: problem.

text: We

text: will

text: ﬁrst

text: introduce

text: the

text: lat-

text: πk

text: ω

text: ω2

text: sin(

text: k

text: L,

text: (PBC).

text: tice

text: model

text: and

text: the

text: covariance

text: matrix

text: in

text: part

text: A.

text: In

text: part

text: B,

text: we

text: k

text: M

text: L

text: introduce

text: the

text: evolution

text: matrix

text: and

text: show

text: how

text: to

text: calculate

text: the

text: entanglement

text: negativity.

text: In

text: part

text: C,

text: we

text: apply

text: the

text: method

text: to

text: the

text: cases

text: studied

text: with

text: CFT

text: approach,

text: and

text: compare

text: the

text: results

text: For

text: the

text: Dirichlet

text: boundary

text: condition

text: (DBC),

text: the

text: Fourier

text: accordingly.

text: transform

text: is

text: not

text: valid

text: due

text: to

text: the

text: breaking

text: of

text: translational

text: sym-

text: A.

text: Harmonic

text: chain

text: and

text: the

text: covariance

text: matrix

text: metry.

text: However,

text: the

text: be

text: deﬁned

text: as

text: The

text: Hamiltonian

text: of

text: the

text: harmonic

text: chain

text: is

text: L

text: πkn

text: H

text: N

text: sin

text: qk

text: qn

text: L

text: sin

text: L

text: where

text: N

text: is

text: the

text: number

text: of

text: sites

text: of

text: the

text: chain,

text: M

text: is

text: the

text: mass

text: scale,

text: ω

text: is

text: the

text: characteristic

text: frequency,

text: and

text: K

text: is

text: the

text: where

text: n

text: L.

text: For

text: p

text: the

text: Fourier

text: sine

text: transforma-

text: n

text: nearest-neighbor

text: coupling.

text: p

text: and

text: q

text: denote

text: the

text: momentum

text: tion

text: is

text: deﬁned

text: similarly.

text: The

text: Hamiltonian

text: in

text: the

text: momentum

text: n

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 Rev. B 71, 085320 (2005). 026805 (2006). [33] L. N. Bulaevskii and G. Ortiz, Phys. Rev. Lett. 90, [55] J. Y. Luo, X.-Q. Li, and Y. J. Yan, Phys. Rev. B 76, 040401 (2003). 085325 (2007). [34] Z. Nussinov, M. F. Crommie, and A. V. Balatsky, Phys. [56] S. A. Gurvitz, D. Mozyrsky, and G. P. Berman, Phys. Rev. B 68, 085402 (2003). Rev. B 72, 205341 (2005). [35] R. Schleser, E. Ruh, T. Ihn, K. Ensslin, D. C. Driscoll, [57] D. K. C. MacDonald, Noise and Fluctuations: An Intro- and A. C. Gossard, Appl. Phys. Lett. 85, 2005 (2004). duction (Wiley, New York, 1962), ch. 2.2.1. [36] L. M. K. Vandersypen, J. M. Elzerman, R. N. Schouten, [58] C. Flindt, T. Novotny´, and A.-P. Jauho, Physica E 29, L. H. W. van Beveren, R. Hanson, and L. P. Kouwen- 411 (2005). hoven, Appl. Phys. Lett. 85, 4394 (2004). [59] D. Urban and J. Ko¨nig, Phys. Rev. B 79, 165319 (2009). [37] S. Gustavsson, R. Leturcq, B. Simovic, R. Schleser, [60] H.-F. Lu, J.-R. Zhang, T. Wu, X.-T. Zu, and H.-W. T. Ihn, P. Studerus, K. Ensslin, D. C. Driscoll, and A. C. Zhang, J. Appl. Phys. 107, 034314 (2010). Gossard, Phys. Rev. Lett. 96, 076605 (2006). [61] J. Y. Luo, H. J. Jiao, Y. Shen, G. Cen, X.-L. He, and [38] T. Fujisawa, T. Hayashi, R. Tomita, and Y. Hirayama, C. Wang, J. Phys.: Condens. Matter 23, 145301 (2011). Science 312, 1634 (2006). [62] R.-Q. Wang, L. Sheng, L.-B. Hu, B. G. Wang, and D. Y. [39] H. Jiao, X.-Q. Li, and J. Y. Luo, Phys. Rev. B 75, 155333 Xing, Phys. Rev. B 84, 115304 (2011). (2007). [63] B. H. Wu and J. C. Cao, Phys. Rev. B 81, 125326 (2010). [40] T. Gilad and S. A. Gurvitz, Phys. Rev. Lett. 97, 116806 [64] J. Y. Luo, J. Hu, X. L. Lang, Y. Shen, X.-L. He, and (2006). H. J. Jiao, Phys. Lett. A 378, 892 (2014). [41] J. Y. Luo, S.-K. Wang, X.-L. He, X.-Q. Li, and Y. J. [65] N. Ubbelohde, C. Fricke, F. Hohls, and R. J. Haug, Phys. Yan, J. Appl. Phys. 108, 083720 (2010). Rev. B 88, 041304 (2013). [42] J. Y. Luo, Y. Shen, X.-L. He, X.-Q. Li, and Y. J. Yan, [66] M. Misiorny, I. Weymann, and J. Barna´s, Phys. Rev. B Phys. Lett. A 376, 59 (2011). 79, 224420 (2009). [43] J. Y. Luo, H. J. Jiao, B. T. Xiong, X.-L. He, and [67] I. Weymann, B. R. Bul(cid:32)ka, and J. Barna´s, Phys. Rev. B C. Wang, J. Phys.: Condens. Matter 25, 155304 (2013). 83, 195302 (2011). [44] H.-B. Xue, Ann. Phys. 339, 208 (2013). [68] Y. Okazaki, S. Sasaki, and K. Muraki, Phys. Rev. B 87, [45] A. N. Jordan and M. Bu¨ttiker, Phys. Rev. Lett. 95, 041302 (2013). 220401 (2005). [69] H. S. Goan, G. J. Milburn, H. M. Wiseman, and H. B. [46] Y. M. Blanter and M. Bu¨ttiker, Phys. Rep. 336, 1 (2000). Sun, Phys. Rev. B 63, 125326 (2001). [47] Y. V. Nazarov(Ed.), Quantum Noise in Mesoscopic [70] D. V. Averin, Phys. Rev. Lett. 88, 207901 (2002). Physics (Kluwer Academic Publishers, Dordrecht, 2003). [71] A. N. Jordan and M. Bu¨ttiker, Phys. Rev. B 71, 125333 [48] S. A. Gurvitz and Y. S. Prager, Phys. Rev. B 53, 15932 (2005). (1996). [72] J. Y. Luo, H. J. Jiao, B. T. Xiong, X.-L. He, and C. R. [49] R. Aguado and T. Brandes, Phys. Rev. Lett. 92, 206601 Wang, J. Appl. Phys. 114, 173703 (2013). (2004). [73] S.-K. Wang, J. S. Jin, and X.-Q. Li, Phys. Rev. B 75, [50] J. Y. Luo, X.-Q. Li, and Y. J. Yan, J. Phys.: Cond. Matt. 155304 (2007). 20, 345215 (2008). [74] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. [51] F. Li, H. J. Jiao, J. Y. Luo, X.-Q. Li, and S. A. Gurvitz, Murch, R. Naik, A. N. Korotkov, and I. Siddiqi, Nature Physica E 41, 1707 (2009). 490, 77 (2012). [52] C. Emary and R. Aguado, Phys. Rev. B 84, 085425 [75] B. Ku¨ng, O. Pfa¨ﬄi, S. Gustavsson, T. Ihn, K. Ensslin, (2011). M. Reinwald, and W. Wegscheider, Phys. Rev. B 79, [53] C. Flindt, T. Novotny´, A. Braggio, M. Sassetti, and A.-P. 035314 (2009). Jauho, Phys. Rev. Lett. 100, 150601 (2008). [54] A. Braggio, J. Ko¨nig, and R. Fazio, Phys. Rev. Lett. 96,

text: Rev.

text: B

text: L.

text: N.

text: Bulaevskii

text: and

text: G.

text: Ortiz,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: Y.

text: Luo,

text: X.-Q.

text: Li,

text: and

text: Y.

text: J.

text: Yan,

text: Phys.

text: Rev.

text: B

text: Z.

text: Nussinov,

text: M.

text: F.

text: Crommie,

text: and

text: A.

text: V.

text: Balatsky,

text: Phys.

text: S.

text: A.

text: Gurvitz,

text: D.

text: Mozyrsky,

text: and

text: G.

text: P.

text: Berman,

text: Phys.

text: Rev.

text: B

text: Rev.

text: B

text: R.

text: Schleser,

text: E.

text: Ruh,

text: T.

text: Ihn,

text: K.

text: Ensslin,

text: D.

text: C.

text: Driscoll,

text: D.

text: K.

text: C.

text: MacDonald,

text: Noise

text: and

text: Fluctuations:

text: An

text: Intro-

text: and

text: A.

text: C.

text: Gossard,

text: Appl.

text: Phys.

text: Lett.

text: duction

text: (Wiley,

text: New

text: York,

text: ch.

text: L.

text: M.

text: K.

text: Vandersypen,

text: J.

text: M.

text: Elzerman,

text: R.

text: N.

text: Schouten,

text: C.

text: Flindt,

text: T.

text: Novotny´,

text: and

text: A.-P.

text: Jauho,

text: Physica

text: E

text: L.

text: H.

text: W.

text: van

text: Beveren,

text: R.

text: Hanson,

text: and

text: L.

text: P.

text: Kouwen-

text: hoven,

text: Appl.

text: Phys.

text: Lett.

text: D.

text: Urban

text: and

text: J.

text: Ko¨nig,

text: Phys.

text: Rev.

text: B

text: S.

text: Gustavsson,

text: R.

text: Leturcq,

text: B.

text: Simovic,

text: R.

text: Schleser,

text: H.-F.

text: Lu,

text: J.-R.

text: Zhang,

text: T.

text: Wu,

text: X.-T.

text: Zu,

text: and

text: H.-W.

text: T.

text: Ihn,

text: P.

text: Studerus,

text: K.

text: Ensslin,

text: D.

text: C.

text: Driscoll,

text: and

text: A.

text: C.

text: Zhang,

text: J.

text: Appl.

text: Phys.

text: Gossard,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: Y.

text: Luo,

text: H.

text: J.

text: Jiao,

text: Y.

text: Shen,

text: G.

text: Cen,

text: X.-L.

text: He,

text: and

text: T.

text: Fujisawa,

text: T.

text: Hayashi,

text: R.

text: Tomita,

text: and

text: Y.

text: Hirayama,

text: C.

text: Wang,

text: J.

text: Phys.:

text: Condens.

text: Matter

text: Science

text: R.-Q.

text: Wang,

text: L.

text: Sheng,

text: L.-B.

text: Hu,

text: B.

text: G.

text: Wang,

text: and

text: D.

text: Y.

text: H.

text: Jiao,

text: X.-Q.

text: Li,

text: and

text: J.

text: Y.

text: Luo,

text: Phys.

text: Rev.

text: B

text: Xing,

text: Phys.

text: Rev.

text: B

text: B.

text: H.

text: Wu

text: and

text: J.

text: C.

text: Cao,

text: Phys.

text: Rev.

text: B

text: T.

text: Gilad

text: and

text: S.

text: A.

text: Gurvitz,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: Y.

text: Luo,

text: J.

text: Hu,

text: X.

text: L.

text: Lang,

text: Y.

text: Shen,

text: X.-L.

text: He,

text: and

text: H.

text: J.

text: Jiao,

text: Phys.

text: Lett.

text: A

text: J.

text: Y.

text: Luo,

text: S.-K.

text: Wang,

text: X.-L.

text: He,

text: X.-Q.

text: Li,

text: and

text: Y.

text: J.

text: N.

text: Ubbelohde,

text: C.

text: Fricke,

text: F.

text: Hohls,

text: and

text: R.

text: J.

text: Haug,

text: Phys.

text: Yan,

text: J.

text: Appl.

text: Phys.

text: Rev.

text: B

text: J.

text: Y.

text: Luo,

text: Y.

text: Shen,

text: X.-L.

text: He,

text: X.-Q.

text: Li,

text: and

text: Y.

text: J.

text: Yan,

text: M.

text: Misiorny,

text: I.

text: Weymann,

text: and

text: J.

text: Barna´s,

text: Phys.

text: Rev.

text: B

text: Phys.

text: Lett.

text: A

text: J.

text: Y.

text: Luo,

text: H.

text: J.

text: Jiao,

text: B.

text: T.

text: Xiong,

text: X.-L.

text: He,

text: and

text: I.

text: Weymann,

text: B.

text: R.

text: and

text: J.

text: Barna´s,

text: Phys.

text: Rev.

text: B

text: C.

text: Wang,

text: J.

text: Phys.:

text: Condens.

text: Matter

text: H.-B.

text: Xue,

text: Ann.

text: Phys.

text: Y.

text: Okazaki,

text: S.

text: Sasaki,

text: and

text: K.

text: Muraki,

text: Phys.

text: Rev.

text: B

text: A.

text: N.

text: Jordan

text: and

text: M.

text: Bu¨ttiker,

text: Phys.

text: Rev.

text: Lett.

text: H.

text: S.

text: Goan,

text: G.

text: J.

text: Milburn,

text: H.

text: M.

text: Wiseman,

text: and

text: H.

text: B.

text: Y.

text: M.

text: Blanter

text: and

text: M.

text: Bu¨ttiker,

text: Phys.

text: Rep.

text: Sun,

text: Phys.

text: Rev.

text: B

text: Y.

text: V.

text: Nazarov(Ed.),

text: Quantum

text: Noise

text: in

text: Mesoscopic

text: D.

text: V.

text: Averin,

text: Phys.

text: Rev.

text: Lett.

text: Physics

text: (Kluwer

text: Academic

text: Publishers,

text: Dordrecht,

text: A.

text: N.

text: Jordan

text: and

text: M.

text: Bu¨ttiker,

text: Phys.

text: Rev.

text: B

text: S.

text: A.

text: Gurvitz

text: and

text: Y.

text: S.

text: Prager,

text: Phys.

text: Rev.

text: B

text: J.

text: Y.

text: Luo,

text: H.

text: J.

text: Jiao,

text: B.

text: T.

text: Xiong,

text: X.-L.

text: He,

text: and

text: C.

text: R.

text: R.

text: Aguado

text: and

text: T.

text: Brandes,

text: Phys.

text: Rev.

text: Lett.

text: Wang,

text: J.

text: Appl.

text: Phys.

text: S.-K.

text: Wang,

text: J.

text: S.

text: Jin,

text: and

text: X.-Q.

text: Li,

text: Phys.

text: Rev.

text: B

text: J.

text: Y.

text: Luo,

text: X.-Q.

text: Li,

text: and

text: Y.

text: J.

text: Yan,

text: J.

text: Phys.:

text: Cond.

text: Matt.

text: R.

text: Vijay,

text: C.

text: Macklin,

text: D.

text: H.

text: Slichter,

text: S.

text: J.

text: Weber,

text: K.

text: W.

text: F.

text: Li,

text: H.

text: J.

text: Jiao,

text: J.

text: Y.

text: Luo,

text: X.-Q.

text: Li,

text: and

text: S.

text: A.

text: Gurvitz,

text: Murch,

text: R.

text: Naik,

text: A.

text: N.

text: Korotkov,

text: and

text: I.

text: Siddiqi,

text: Nature

text: Physica

text: E

text: C.

text: Emary

text: and

text: R.

text: Aguado,

text: Phys.

text: Rev.

text: B

text: B.

text: Ku¨ng,

text: O.

text: Pfa¨ﬄi,

text: S.

text: Gustavsson,

text: T.

text: Ihn,

text: K.

text: Ensslin,

text: M.

text: Reinwald,

text: and

text: W.

text: Wegscheider,

text: Phys.

text: Rev.

text: B

text: C.

text: Flindt,

text: T.

text: Novotny´,

text: A.

text: Braggio,

text: M.

text: Sassetti,

text: and

text: A.-P.

text: Jauho,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Braggio,

text: J.

text: Ko¨nig,

text: and

text: R.

text: Fazio,

text: Phys.

text: Rev.

text: Lett.