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under the assumption <strong>of</strong> the symmetry along it. gtt, gxx and gzz are the induced world-volume metric, and they are equal to the background metric (1) except for gzz = 1/z 2 + θ ′ (z) 2 . NONLINEAR CONDUCTIVITY It was found [6] that the on-shell D7-brane action becomes complex unless we choose a specific combination <strong>of</strong> J and E; the relationship between J and E is determined by the reality condition <strong>of</strong> the on-shell action, hence, J is obtained as a nonlinear function <strong>of</strong> E. The on-shell action is given by [6] ¯ SD7 = −N ∫ dzdtd 3 x √ ¯gzz|gtt| −1√ F1F2 with F1 = |gtt|gxx − E 2 and F2 = |gtt|g 2 xx cos 6 ¯ θ − gxxJ 2 /N 2 , where ¯gzz is the induced metric given by ¯ θ, which is the on-shell configuration <strong>of</strong> θ(z). Since both F1 and F2 cross zero somewhere between the boundary and the horizon, the only way to make ¯ SD7 real is to choose J and E so that F1 and F2 cross zero at the same point z = z∗. The hypersurface given by z = z∗ is <strong>of</strong>ten called the “singular shell”. Then, the reality condition F1(z∗) = F2(z∗) = 0 gives us the relationship between J and E in the form <strong>of</strong> J = σ0E [6], where σ0 = N T (e 2 + 1) 1/4 cos 3 ¯ θ(z∗). (3) Our task is to solve the equation <strong>of</strong> motion (EOM) for θ to obtain the explicit representation. ¯ θ(z) can be expanded as ¯ θ(z) = mqz + O(z 3 ), where mq is the current quark mass [13], which is a parameter <strong>of</strong> the microscopic theory. mq is related to the gap in condensed matters. Let us choose the temperature to be T = √ 2/π so that zH = 1, e = E/2 and z∗ = √ E 2 /4 + 1 − E/2. 4 We further fix NcNf = 40. NcNf governs the pair-creation rate <strong>of</strong> the charge carriers as we shall explain later. We need to solve the EOM for θ numerically. The boundary condition we employ is θ(z)/z|z=0 = mq and we request the absence <strong>of</strong> singularity in the D7-brane configuration. For earlier studies on the nonlinear conductivity by using this method, see for example, Refs. [7, 8]. RESULTS Examples <strong>of</strong> J-mq curves at several values <strong>of</strong> E are shown in Fig. 2. Of course, mq has a unique value at a given model and we need to choose some particular value <strong>of</strong> mq. We find that there are two different possible values <strong>of</strong> J at given mq and given E in some parameter region, which indicate the multi-valued nature <strong>of</strong> J(E). Furthermore, if we increase E along the given mq, the smaller J decreases while the larger J increases; the smaller-J branch shows NDR, whereas the larger-J branch has a positive differential resistivity. Note that the smaller-J branch 4 In this article, we have employed the natural units c = ¯h = kB = 1. If our scale unit is meV (mili eV), T ∼ 5 K. If we identify the unit quark charge with the unit charge <strong>of</strong> electrons, the effective fine-structure constant read from the Coulomb interaction in the inter-quark potential is ∼ 1. mq 1.34 1.33 1.32 1.31 1.30 1.29 E0.12 E0.20 E0.15 1.28 0.000 0.005 0.010 0.015 0.020 0.025 Figure 2: J-mq curves at E = 0.12, 0.15, and 0.20. mq is maximum at a nonzero but small value <strong>of</strong> J. E 0.25 0.20 0.15 0.10 0.05 0.00 0.000 0.005 0.010 0.015 0.020 0.025 Figure 3: J-E curve at mq = 1.315. Ec = 0.11 in this case. NDR appears in J ≤ 0.0031 and is absent for E ≥ 0.19. is a very narrow window in the full part <strong>of</strong> the J-mq curve. For example, the J-mq curve at E = 0.2 extends until J = 0.288, and the width <strong>of</strong> the smaller-J branch along the J axes is less than 2% <strong>of</strong> the full part. The detailed analysis shows that the highest value <strong>of</strong> mq approaches around 1.310 at the E → +0 limit, suggesting that Ec = 0 if mq < 1.310. This is consistent with the fact that the system is a conductor at sufficiently small mq in comparison with T (or sufficiently high T in comparison with mq). An example <strong>of</strong> J-E relation at mq = 1.315 is given in Fig. 3. 5 The system is an insulator for E < Ec = 0.11. If E ≥ Ec, the insulation is broken and we observe a current. NDR is realized in the smaller-J region. We always have the J = 0 branch on top <strong>of</strong> the vertical axis. Therefore, our interpretation is that Fig. 3 shows the B-C-D region <strong>of</strong> Fig. 1 with the axes swapped; our NDR falls within the Sshaped NDR. There may be a small tunneling current that almost overlaps with the vertical axis, but we could not detect it within our numerical precision. We leave the detailed analysis on the tunneling current in a future work. It is important to clarify what is the physically essential process in our NDR. Let us consider the doped cases. We can also “dope” the system by introducing finite quarkcharge density [14, 15]. In this case, the system is always a 5 If we choose our scale unit to be meV, the critical electric field Ec in Fig. 3 is Ec ∼ 5 × 10 −1 V/m, and the current density realized at E = Ec is J ∼ 1 × 10 −4 mA/mm 2 . J J
conductor. The current is given by [6] √ J = σ2 0 + d2 /(e2 + 1) E, (4) where d is related to the quark-charge density ρ through d = ρ/( π √ 2 2 λT ). Owing to the doped charges, any small E causes a current and we observe Ohm’s law in the small- J region. If we raise J, we may again observe NDR owing to the nontrivial behavior <strong>of</strong> σ0. It is indeed the case if d is small enough not to smear the contribution <strong>of</strong> σ0. In this case, the curve in Fig. 3 will be “N-shaped”, (S-shaped in the sense <strong>of</strong> Fig. 1) starting at the origin. The point is that the d-dependent term in the square root in (4) does not have any structure to produce NDR. Therefore, the σ0-part in (4) is crucial for NDR. It is understood that the current due to the σ0-part is caused by the pair creation <strong>of</strong> the charge carriers. The reasons are as follows: it contributes the current with the total system being kept neutral, and it vanishes if the mass <strong>of</strong> the charge carriers mq is infinite. [6] 6 As a conclusion, the pair-creation process is essential for our NDR. DISCUSSION We can suggest a phenomenological model <strong>of</strong> NDR. The phenomenological origins <strong>of</strong> NDR are classified into three types (except for the tunnel effect for some semiconductor junctions) in Ref. [2]: 1) nonlinearity <strong>of</strong> mobility, 2) nonlinearity <strong>of</strong> carrier density, and 3) nonlinearity <strong>of</strong> the electron temperature. 7 We have found that our NDR originates in the pair creation <strong>of</strong> the charge carriers but not in the normal current <strong>of</strong> the doped charges. This means that the above feature 2) is crucial in our NDR. Although further study is necessary to reach the final conclusion, it is natural to assume that both the normal current and the paircreated current contribute 1) and 3) regardless <strong>of</strong> the origin <strong>of</strong> the charge carriers. If this assumption is right, 1) and 3) do not seem to be important in our NDR. The behavior <strong>of</strong> our NDR is in the category <strong>of</strong> the “SNDC” in Ref. [2] and it may be attributed to the impact ionization explained in Ref. [2]. The proposal <strong>of</strong> the many-body avalanche model <strong>of</strong> NDR[17, 18] also matches our picture. It is important to study further the connection between our results and the phenomenological models <strong>of</strong> NDR. We can also see our results from the viewpoint <strong>of</strong> the quark-hadron physics and that <strong>of</strong> the excitonic insulators. Let us consider the sQGP state [11] where the quarkantiquark bound state exists in the deconfinement phase <strong>of</strong> gluons. Our results suggest that the quarks are liberated at the critical value <strong>of</strong> the electric field and their current may show NDR. We may also have a chance to observe a qualitatively similar NDR in excitonic insulators after the insulation breaking. It is important to study how general this NDR is, in quark/meson systems and in the systems <strong>of</strong> 6 We also point out that σ0 is proportional to NcNf . This suggests that it may be a one-loop contribution <strong>of</strong> the quarks, as in the perturbative computation <strong>of</strong> the pair-creation rate. Note that the quark loops have been taken into account to the 1-loop order in the probe approximation. 7 See also Ref. [16]. charge-anticharge bound states, in the presence <strong>of</strong> strong external fields. We expect that the present system is a good theoretical playground for studies on nonlinear charge transport and nonequilibrium steady states. The AdS/CFT correspondence can be a new tool for studying nonequilibrium physics as we have demonstrated here. The author thanks the organizers <strong>of</strong> PIF2010 conference. Discussions with the participants <strong>of</strong> various research areas such as plasma physics, strong-field dynamics and laser physics are quite fruitful in planning further studies along the direction <strong>of</strong> the present work. REFERENCES [1] S. Nakamura, Prog. Theor. Phys. 124 (2010), 1105. [2] E. Schöll, Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors (Cambridge University Press, Cambridge 2001). [3] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1988), 231 [Int. J. Theor. Phys. 38 (1999), 1113]. [4] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428 (1998), 105. [5] E. Witten, Adv. Theor. Math. Phys. 2 (1998), 253. [6] A. Karch and A. O’Bannon, J. High Energy Phys. 0709 (2007), 024. [7] J. Erdmenger, R. Meyer and J. P. Shock, J. High Energy Phys. 0712 (2007), 091. [8] T. Albash, V. G. Filev, C. V. Johnson and A. Kundu, J. High Energy Phys. 0808 (2008), 092. [9] N. F. Mott, Philos. Mag. 6 (1961), 287. [10] See also for a review, S. A. Moskalenko and D. W. Snoke, Bose-Einstein Condensation <strong>of</strong> Excitons and Biexcitons (Cambridge Univ. Press, Cambridge 2000). [11] M. Gyulassy and L. McLerran, Nucl. Phys. A 750 (2005), 30. [12] A. Karch, A. O’Bannon and E. Thompson, J. High Energy Phys. 0904 (2009), 021. [13] A. Karch and E. Katz, J. High Energy Phys. 0206 (2002), 043. [14] S. Nakamura, Y. Seo, S. J. Sin and K. P. Yogendran, J. Korean Phys. Soc. 52 (2008), 1734. [15] S. Kobayashi, D. Mateos, S. Matsuura, R. C. Myers and R. M. Thomson, J. High Energy Phys. 0702 (2007), 016. [16] H. C. Law and K. C. Kao, IEEE Trans. Electron Devices 17 (1970), 562. [17] Y. Tokura, H. Okamoto, T. Takao, T. Tadaoki and G. Saito, Phys. Rev. B 38 (1988), 2215. [18] T. Oka, H. Kishida and H. Aoki, talk given at JPS 2010 Annual Meeting, March 20th (2010).
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Proceedings <stron
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Conference poster
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Preface The international conferenc
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Recent progress of
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Figure 1: The Pulse Intensity-Durat
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Figure 3: The attosecond metrology
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THE QED EFFECTIVE ACTION In quantum
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Figure 2: Turning points in the com
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[5] A. Ringwald, “Pair production
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QED IN ULTRA-INTENSE LASER FIELDS
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form e + nγL → e ′ + γ where
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ection(s) associated with the exter
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This is listed in Tab.1. The effect
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the spectrum of ph
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pf f f pi p x2 x1 k ki p pi Figure
Page 36 and 37: STRONG FIELD DYNAMICS IN HEAVY-ION
Page 38 and 39: Possible effects of</strong
Page 40 and 41: Figure 3: Flux tube structure <stro
Page 42 and 43: Abstract Yoctosecond photon pulse g
Page 44 and 45: emsstrahlung or inelastic pair anni
Page 46 and 47: Abstract Fields, Instantons, and Cu
Page 48 and 49: the dispersion relation p0 = Ep −
Page 50 and 51: CRITICAL BEHAVIOR OF CHARMONIUM: QC
Page 52 and 53: different masses m 0 i ̸= mi only
Page 54 and 55: ON THE UNRUH EFFECT ∗ Ralf Schüt
Page 56 and 57: Abstract Can we detect ”Unruh rad
Page 58 and 59: Equipartition Theorem The stochasti
Page 60 and 61: Abstract Quantum fields and static
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Page 65 and 66: = γ WISP ; ; ALP HP(mγ ′ > 0) .
Page 67 and 68: Figure 4: Exclusion limit (95% C.L.
Page 69 and 70: Probing extremely light fields via
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Page 73 and 74: Abstract Dynamical view of<
Page 75 and 76: (a) longitudinal momentum distribut
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Page 79 and 80: Pair production in heat bath Finall
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Page 84 and 85: Abstract Non-linear charge transpor
Page 88 and 89: Brilliant hardγ-production ande +
Page 90 and 91: each other. Thus the E field rotate
Page 92 and 93: ACKNOWLEDGMENTS This work is based
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Page 96 and 97: of the sampling eq
Page 98 and 99: Abstract SCHWINGER LIMIT ATTAINABIL
Page 100 and 101: To find it we use the integral calc
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Page 104 and 105: A way to quantify the irreversible
Page 106 and 107: Figure 4: Pair production vs time f
Page 108 and 109: of the quantum flu
Page 110 and 111: Phenomena H. Schwarz and H. Hora Ed
Page 112 and 113: In USA, LLNL has a user’s facilit
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Page 116 and 117: QGP is studied by using the particl
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Page 120 and 121: REFERENCES [1] P.A. Franken, A.E. H
Page 122 and 123: of electron. When
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Page 126 and 127: Abstract WIDE-BAND X-RAY OBSERVATIO
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egime. As analogous to a convention
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With extremely high-peak power lase
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Abstract Flying Mirror as a tool to
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Reflected light intensity (μJ/sr/n
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4-mirror laser stacking cavity for
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CURRENT STATUS OF LFEX LASER AND EX
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which were comparable to those befo
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Abstract X-ray Emission from Magnet
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sometimes exceed the Eddington lumi
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1 0.8 0.6 0.4 0.2 B 2 z E 2 T 0.5 1
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First order quantum correction to t
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dE (1) dt = ¯he4 m 2 12πɛ0p 5 i
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Next, let us consider the origin <s
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Abstract NONCANONICAL LIE PERTURBAT
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with C (2) µ = Γ (2) µ − ˆ L
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log N (PIC particle) When the condi
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Abstract X-RAY GENERATION VIA LASER
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The electron beam is bent to the du
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INVESTIGATING THE ONE-PHOTON ANNIHI
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As a representative characteristics
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the interference terms ⟨ϕIϕh⟩
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P r o g r a m of P
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11:55 lunch & group photo [85] [Rec
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List of Participan
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