PHYS 612 Quantum Field Theory II Spring 2013 Final Project Optical Conductivity with Holographic Lattices Kubra Yeter Instructor: George Siopsis Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996-1200, USA e-mail address: kyeter@tennessee.edu This paper is written for the final project of Quantum Field Theory class and it does not claim any originality.
2 1 Introduction The temperature and frequency dependence of the free carrier optical conductivity of the high-t c cuprates has been the subject of debate since the discovery of the high temperature superconductors. The interpretation of these data is relevant and important, because the departure from simple Drude behavior reflects the nature of the interactions responsible for the transport properties of the normal state and of the superconductivity occurring in these materials [5. In recent years, strongly correlated condensed matter systems are reproduced in weakly coupled gravitational theories using the gauge/gravity duality. In this project, we discuss two methods to introduce so-called holographic lattice. Introduction of the holographic lattice is important to understand the low frequency behavior of the optical conductivity of the holographic model. The content of this paper is inspired by [1 and [2 papers. The data used are also taken from these papers. 2 A Holographic Lattice In this section we will present a holographic framework to calculate the optical conductivity. We use an Einstein-Maxwell theory in AdS 4 spacetime and additionally a neutral scalar field Φ which will form the lattice structure. Consider a system consisting of a U(1) gauge field, A µ corresponding field strength F µν = µ A ν ν A µ, and a real scalar field Φ which is neutral under U(1) group. They live in a spacetime of negative cosmological constant Λ = 6/L 2. The action is 1 S = d 4 x [ g R + 6 16πG N L 1 2 2 F abf ab 2 a Φ a Φ 4V (Φ). (2.1) where we choose the potential V (Φ) = Φ2 L 2 (2.2) which corresponds to a massive scalar field with mass m 2 = 2/L 2. Then the Lagrangian density becomes We obtain the Einstein s equations from L = R + 6 L 2 1 2 F abf ab 2 a Φ a Φ + 4 Φ2 L 2. (2.3) G ab = 8πG N T ab + 3 L 2 g ab (2.4)
3 where T ab = T (Φ) ab + T (EM) ab is the stress energy tensor. T (Φ)ab = L(Φ) a Φ b Φ g ab L (Φ) = 4 a Φ b Φ g (R ab + 6 ) L 2 aφ a Φ 4V (Φ) 2 (2.5) = 2 a Φ b Φ + 4g ab V (Φ) g ab R g ab 6 L 2 G ab R ab + 3 L g 2 ab 2 [ a Φ b Φ V (Φ)g ab (F ac F cb 14 ) g abf cd F cd = 0. (2.6) We also obtain the Maxwell s, and scalar field equations by varying the Lagrangian density with respect to A a, and the scalar field respectively. Maxwell s and scalar field equations read a F a b = 0, (2.7) Φ V (Φ) = 0. (2.8) We parametrize the holographic radial direction by coordinate z and to set the boundary conditions we write the metric at the boundary of the asymptotically AdS space in terms of Poincare coordinates. ds 2 = dt 2 + dx 2 + dy 2 (2.9) The gravitational lattice background is introduced by a spatially inhomogeneous source for neutral scalar field. In the asymptotic region (2.9) the scalar field Φ takes the form Φ zφ 1 + z 2 φ 2 + O(z 3 ). (2.10) In AdS/CFT dictionary φ 1 is regarded as a source for a dimension two operator dual to φ, say O φ, while φ 2 represents the expectation value O φ. We will start with a onedimensional source φ 1 to be φ 1 (x) = A 0 cos(k 0 x). (2.11) where k 0 refers to the lattice wavenumber and A 0 refers to its amplitude. The lattice size is determined by k 0 = 2π l. 2.1 The Lattice Our background depends only on the coordinates x and z since it is both static and translationally invariant in the y direction. The most general static, electrically charged, black hole solution has the metric of the form ds 2 = L2 z 2 [ (1 z)p (z)q tt dt 2 + Q zzdz 2 P (z)(1 z) + Q xx(dx + z 2 Q xz dz) 2 + Q yy dy 2, (2.12) where Q ij, for ijɛ{t, x, y, z}. Also we define Φ(z, x) = zφ(z, x) (2.13)
4 and A(z, x) = (1 z)ψ(z, x)dt (2.14) where φ and ψ are arbitrary functions of x and z which are obtained by solving (2.7, 2.8). The reason why we introduced the factor (1 z) is that because we want to make sure that the functions Φ and A are smooth at the non-extremal horizon at z = 1, assuming Q tt (x, 1) = Q zz (x, 1) and Q xx, Q yy, Q xz, ψ and φ are smooth functions at z = 1. Finally, the factor P (z) is chosen to be P (z) = 1 + z + z 2 µ2 1z 3 then the Hawking temperature is calculated by 2 (2.15) T H = 4πL = 6 µ2 1 8πL. (2.16) Note that if Q tt = Q zz = Q xx = Q yy = 1, Q xz = φ = 0, ψ = µ = µ 1, we recover the familiar planar Reissner-Nordström black hole. Since the Einstein s equations do not have a definite character, in the PDE sense, we will use the DeTurck method. 2.1.1 DeTurck Method Einstein-DeTurck equation is G H ab G ab (a ξ b) = 0 (2.17) where ξ a = g [ cd Λ a cd (g) Λ a cd (ḡ) and Λ(ḡ) is the Levi-Civita connection associated with a reference metric ḡ. The reference metric is chosen to be such that it has the same asymptotics and horizon structure as g. For our particular case, we choose ḡ to be the line element (2.12) with Q tt = Q zz = Q xx = Q yy = 1 and Q xz = 0 which gives ds 2 = L2 z 2 [ (1 z)p (z)dt 2 dz 2 + P (z)(1 z) + dx2 + dy 2. (2.18) In our numerical method we will solve Eq.(2.17) together with Eq.s(2.7) and (2.8). 2.1.2 Boundary Conditions The asymptotic behavior at the boundary z 0 can be obtained from Eq.(2.17) with ξ = 0.
5 Q tt (x, z) = 1 z2 2 φ 1(x) 2 + O(z 3 ), Q zz (x, z) = 1 + 4z2 3 φ 2(x)φ 1 (x) + O(z 3 ), Q xz (x, z) = z 2 φ 1(x)φ 1(x) + O(z 2 ), Q xx (x, z) = 1 z2 2 φ 1(x) 2 + O(z 3 ), Q yy (x, z) = 1 z2 2 φ 1(x) 2 + O(z 3 ), φ(x, z) = φ 1 (x) + zφ 2 (x) + O(z 3 ), ψ(x, z) = µ + [µ ρ(x)z + O(z 2 ). (2.19) To form the lattice structure we choose a source φ 1 (x) with a nontrivial x dependence of the form φ 1 (x) = A 0 cos(kx). (2.20) The stress-energy tensor being quadratic in Φ makes the effective wave number to be 2k 0 on the metric, gauge field components Q ij, and ψ. It is easy to see from Eq.s(2.19) that the boundary conditions at z = 0 is of the Dirichlet as the following. Q tt (x, 0) = Q zz (x, 0) = Q xx (x, 0) = Q yy (x, 0) = 1, Q xz (x, 0) = 0, φ(x, 0) = φ 1 (x), ψ(x, 0) = µ. (2.21) To satisfy the regularity at the horizon (z = 1) we choose the following expansion Q ij (x, z) = Q (0) ij (x) + (1 z)q(1) ij (x) + O ( (1 z) 2), φ(x, z) = φ (0) (x) + (1 z)φ (1) (x) + O ( (1 z) 2), ψ(x, z) = ψ (0) (x) + (1 z)ψ (1) (x) + O ( (1 z) 2). (2.22) 2.2 Perturbing the lattice In this section we will explore the transport properties by studying the perturbations about the lattice background. We will write the Eqs. (2.6, 2.7, 2.8) after perturbing the fields g ab = ĝ ab + h ab, A a = Âa + b a, Φ = ˆΦ + η, (2.23) where h ab, b a, and η are small quantities. Let s try to find Eqs. (2.6, 2.7, 2.8) explicitly. g ab = ĝ ab + g ac g bd h cd (2.24)
6 Figure 1: Q xz (left panel ) and φ (right panel), for k 0 = 2, A 0 = 1, µ = 1.4 and T/µ = 0.1. where we can use g ab and g ab to raise or lower the indices, since the corrections would be of higher order in the perturbation. We also need to find the Christoffel symbols, which are given by Γ ρ µν = 1 2 gρλ ( µ g νλ + ν g λµ λ g µν ) = 1 2ĝρλ ( µ h νλ + ν h λµ λ h µν ) (2.25) Since these connection coefficients are in first order the only contribution to the Riemann tensor will come from the derivatives of these coefficients not from the squares. Using the definition of Riemann tensor Then we find R ρ σµν = µ Γ ρ νσ ν Γ ρ µσ + Γ ρ µλ Γλ νσ Γ ρ νλ Γλ µσ (2.26) R µνρσ = ĝ µλ ρ Γ λ νσ ĝ µλ σ Γ λ νρ = 1 2 ( ρ ν h ρσ + σ µ h νρ σ ν h µρ ρ µ h νσ ) (2.27) After contracting over µ and ρ the Ricci tensor is obtained. R µν = 1 2 ( σ ν h σ µ + σ µ h σ ν µ ν h h µν ) (2.28)
7 Let s start with the scalar equation. Φ V (Φ) = g ρσ ρ σ Φ g ρσ Γ λ ρσ λ Φ + 2 Φ L 2 = (ĝ ρσ + g ρµ g σν h µν ) ρ σ (ˆΦ + η) (ĝ ρσ + g ρµ g σν h µν ) Γ λ ρσ λ (ˆΦ + η) + 2 ˆΦ + η L 2 = ĝ ρσ ρ σ ˆΦ + g ρµ g σν h µν ρ σ ˆΦ + ĝ ρσ ρ σ η + g ρµ g σν h µν ρ σ η ĝ ρσ Γ λ ρσ λ η ĝ ρσ Γ λ ρσ λ ˆΦ h ρσ Γ λ ρσ λ ˆΦ + 2 ˆΦ + η L 2 = ˆ η + h ρσ ρ σ ˆΦ h ρσ Γ λ ρσ λ ˆΦ ĝ ρσ Γ λ ρσ λ ˆΦ V (ˆΦ)η = ˆ η + h ρσ ˆ ρ ˆ σ ˆΦ ˆ ρ hρσ ˆ σ ˆΦ V (ˆΦ)η = 0 (2.29) where we used ˆ ˆΦ V (ˆΦ) = 0 and ignored the second order perturbations. We also define h ab = h ab hĝ ab /2 which is called the trace-reversed metric perturbation. Similarly, we can find the Maxwell s equation in perturbed background. b F ab = b F ab + F cb Γ a cb + F ac Γ b ca = b ( a A b b A a ) + ( c A b b A c )Γ a cb + ( a A c c A a )Γ b ca (2.30) Substituting A a = g ab A b = (ĝ ab + g ac g bd h cd )(Âb + b b ) and Eq.(2.25)we obtain the Maxwell s equations as the following. ˆ b a ˆR ac b c ˆ a ( ˆ c b c ) ˆ c hcd ˆFda h cd ˆ c ˆFda ˆF cd ˆ c h da = 0 (2.31) Finally, from Einstein s equation we obtain 1 [ ˆ h ab 2 2 ˆR acbd h cd + 2 ˆR c(ah b)c + 2 ˆ (a ˆ c hb)c = 3 L h 2 ab + 4 ˆ (a η ˆ b) ˆΦ + 2V (ˆΦ)ηĝ ab + 2V (ˆΦ)h ab c + 2f (a ˆFb)c ˆF ac ˆFbd h cd h ab 4 ˆF cd ˆFcd ĝab 2 f cd cd ˆF + ĝab 2 hcd p ˆFcp ˆF d (2.32) where f = db. The above system of equations are invariant under the following set of linear transformations and b a b a + ˆ χ h ab h ab b a b a + β c ˆ c  a + Âd ˆ d β a h ab h ab + 2 ˆ (a β b)
8 Figure 2: The charge density, ρ(x) (left panel) and the absolute value of the non-zero coefficients of its Fourier series (right panel). where χ is an arbitrary scalar function and β a the components of an arbitrary fourdimensional vector. The first set of transformations reflects the U(1) gauge freedom associated with electromagnetism, whereas the secind reflects diffeomorphism invariance. When faced with a system that is invariant under some kind of gauge transformations, our first instinct is to fix a gauge. We will choose to fix the Lorentz and the de Donder gauge, respectively, ˆ a hab = 0, ˆ a b a = 0. (2.33) Now, we can study the changes in the transport properties of induced by our lattice by specializing the equations for the Eqs. (2.12-2.14). Since our background is invariant under a time translation Killing field t, we can Fourier decompose our perturbations, h ab (t, x, y, z) = h ab (x, y, z)e iωt, β a (t, x, y, z) = b a (x, y, z)e iωt, η(t, x, y, z) = η(x, y, z)e iωt. (2.34) Since we assume translational invariance in y direction there are 11 unknown functions, { h tt, h tz, h tx, h zz, h zx, h xx, h yy, b t, b z, b x, η}, in two variables x and z. Once we count the number of nontrivial differential equations we find that number to be 15. These equations are {tt, tz, tx, zz, zx, xx, yy} components of Eq.(2.32), the {t, z, x} components of Eq.(2.31), the scalar equation Eq.(2.29), the {t, z, x} components of the first equation in Eq.(2.33) and the last equation in Eq.(2.33). However, due to gauge invariance, the {tt, tz, yy} components of Eq.(2.32) and the {t} component of Eq.(2.31) are automatically satisfied if the remaining equations are satisfied. The remaining 11 equations need to be solved using the numerical techniques.
9 Figure 3: The real and imaginart parts of the perturbation of the scalar field are shown for µ = 1.4, T/µ = 0.115 k 0 = 2, A 0 = 1.5. The two figures on the top have ω/t = 0.006, whereas the two figures on the bottom have ω/t = 0.6.
10 2.2.1 Numerical implementation of the perturbation theory To have reasonable results we need to factor out the non-analytic behavior of the near horizon expansion. We apply the ingoing boundary conditions near horizon as the following. h [ h(0) tt (x, z) = (1 z) iω tt (x) + (1 z) h (1) tt (x) + O((1 z) 2 ) h [ h(0) tx (x, z) = (1 z) iω tx (x) + (1 z) h (1) tx (x) + O((1 z) 2 ) 1 h [ h(0) iω tz (x, z) = (1 z) tz (x) + (1 z) h (1) tz (x) + O((1 z) 2 ) h [ h(0) xx (x, z) = (1 z) iω xx (x) + (1 z) h (1) xx (x) + O((1 z) 2 ) 1 h [ h(0) iω xz (x, z) = (1 z) xz (x) + (1 z) h (1) xz (x) + O((1 z) 2 ) 2 h [ h(0) iω zz (x, z) = (1 z) zz (x) + (1 z) h (1) zz (x) + O((1 z) 2 ) (2.35) h [ h(0) yy (x, z) = (1 z) iω yy (x) + (1 z) h (1) yy (x) + O((1 z) 2 ) [ b(0) bt (x, z) = (1 z) iω t (x) + (1 z) b (1) t (x) + O((1 z) 2 ) [ b(0) bx (x, z) = (1 z) iω x (x) + (1 z) b (1) x (x) + O((1 z) 2 ) 1 [ b(0) iω bz (x, z) = (1 z) z (x) + (1 z) b (1) z (x) + O((1 z) 2 ) η(x, z) = (1 z) iω [ η (0) (x) + (1 z) η (1) (x) + O((1 z) 2 ) We will not work with the actual variables, for example, instead of using η itself, we work with q(x, z) = z(1 z 3 ) iω η(x, z) (2.36) 3 Conductivity In this section we will discuss the numerical results of the calculations but before we do that we will first review the properties of the conductivity in a translationally invariant holographic background. For boundary theories with two spatial dimensions, the conductivity is dimensionless and at the conformal point, with µ = 0, is known to be a independent of ω reflecting an underlying electron-vortex duality. In the presence of the chemical potential, µ, the optical conductivity shows more structure and it depends on ω. As one can see from the dashed lines in Fig.(4) at large frequency, ω µ, both the real and imaginary parts of the conductivity tends towards a constant, real value observed at the conformal point. At lower frequencies, ω < µ, there is drop in Reσ which reveals a depletion in the density of charged states. There is also a delta function spike at ω = 0 in Reσ which does not show up in numerical plots. This delta function can be seen from pole at ω = 0 at the plot of Imσ using the Kramers-Kronig relation Imσ(ω) K ω as ω 0 (3.1)
11 Figure 4: The real (left panel) and imaginary (right panel) parts of the optical conductivity, both without the lattice (dashed line) and with the lattice (solid line and data points) for µ = 1.4, T/µ = 0.115. The lattice has k 0 = 2, A 0 = 1.5 and it only changes the low frequency behavior. The pole in Imσ without the lattice reflects the existence of a ω = 0 delta function in Reσ. where K is a constant. The formation of the delta function can be explained by conservation of momentum in the boundary theory. In the presence of lattice structure although the high frequency, ω µ, behavior remains unchanged the dissipative part of the of the conductivity, Reσ, now rises at low ω because of the redistribution of the spectral weight of the delta function, as shown with solid lines in Fig.(4). On the other hand, at the low frequency the responsive part of the conductivity, Imσ 0 as ω 0 showing that there is no delta function in DC conductivity. 3.1 The Drude Peak The numerical results show that at low frequency both the real and imaginary parts of the conductivity can be fit by the two-parameter Drude form σ(ω) = Kτ 1 iωτ. (3.2) Despite the fact that not having well-defined quasi-particles in our holographic system, it is interesting to see that the low-frequency behavior is nicely captured by the exact Drude form, as shown in Fig.(5). 3.2 DC Resistivity By introducing the lattice structure we resolved the issue of the infinite DC conductivity and now we can calculate the DC resistivity by ρ = (Kτ) 1. In the resistivity expression the only term that depends on temperature is the scattering time, τ. K is independent of
12 Figure 5: A blow up of the low frequency optical conductivity with the lattice shown in Fig.(4). The data points are the fit by the two-parameter Drude form. Figure 6: The left panel shows the DC resistivity plotted as a function of temperature for various lattice spacings. On the right hand side we factor out the scaling (3.3, 3.4) and re-plot the same data on a log scale. Both plots arise from a background with µ = 1.4 and the lattice amplitude A 0 = k 0 /2.
13 T. When we plot the resistivity with respect to dimensionless parameter T/µ we see that the resistivity depends strongly on the lattice wavenumber, k 0, as shown in Fig.(6). The plot of resistivity can be explained using the information in Hartnoll and Hofman s study. In their paper, they studied the DC conductivity in a local critical theory. The local critical theory is described as the dual theory to an extremal Reissner-Nordström AdS black hole, AdS 2 R 2, which is invariant under rescalings of time, with no rescaling of space. Hartnoll and Hofman showed that the DC conductivity can be extracted from the two point function of the charge density, evaluated at the lattice wavenumber. They then calculated this two point function by perturbing the Reissner-Nordström AdS black hole and found ρ T 2ν 1 (3.3) where ν = 1 ( ) 2 k 5 + 2 4 1 + 2 µ ( ) 2 k. (3.4) µ The exponent can be viewed as arising from the dimension = ν 1 of the operator dual 2 to the charge density in the near horizon AdS 2 region, evaluated at the lattice wavenumber k. The plot on the right hand side of Fig.(6) shows ρ/t 2ν 1 for different k 0 values. The data in this plot is fitted to ρ 0 = T 2ν 1 (a 0 +a 1 T +a 2 T 2 +a 3 T 3 ). The fact that the curves all approach nonzero, but finite, constants at low temperature shows that our data confirms the low temperature scaling (3.3) with exponent (3.4) predicted in Hartnoll and Hofman s paper [3. Note that as the temperature goes to zero, the dissipation goes to zero and the DC resistivity vanishes. Thus the DC conductivity becomes infinite, as expected for a perfect lattice with no dissipation. 3.3 Power-law Optical Conductivity Fig.(4) can be analyzed in three frequency regimes, i.e. Drude regime (ω/t 1), midinfrared regime, and continuum regime. For ω/t 1 the optical conductivity exhibits a power-law fall-off between 2 < ωτ < 8 for all lattices examined, even those with different temperatures, lattice spacing and amplitude. The data in the mid-infrared regime is fit by σ(ω) = B + C. (3.5) ω2/3 Fig.(8) illustrates the fact that the exponent of the power law does not depend on the parameters chosen. The offset C depends on k 0 and T therefore it is subtracted from the optical conductivity on a log-log plot. Also on the right panel of Fig.(7) the phase angle of the optical conductivity is depicted. The phase angle has a small dependence on ω, but varies between 65 and 80, as k 0 varies from 1 to 3.
14 Figure 7: The modulus σ and argument argσ of the conductivity. Both plots arise from a background with µ = 1.4 and temperature T/µ = 0.115. Figure 8: The magnitude of the optical conductivity with the offset removed, on a loglog plot. On the left, the plot has T/µ =.115, and shows three different wavenumbers: diamonds denote k 0 = 3, the squares denote k 0 = 1, and the circles denote k 0 = 2. On the right, the plot has k 0 = 2 and shows three different temperatures: the diamonds have T/µ =.098, the circles have T/µ =.115 and the squares have T/µ =.13. In both plots, A 0 /k 0 = 3/4. The fact that the lines are parallel for ωτ > 2 shows that the fit to the power law (3.5) is robust.
15 3.4 Comparison to the Cuprates The infrared conductivity of the high-t c cuprates in the normal state has a characteristic deviation from the normal Drude behavior of metals, which has sometimes been described as an additional, distinct mid-infrared absorption and sometimes as an extended tail of the low-frequency peak [4. In condensed matter literature two peculiar aspects of the infrared optical conductivity in the normal state are pointed out. These are, In the far infrared range the optical conductivity of optimally doped cuprates is characterized by a universal ω/t scaling function of the form σ(ω) = T 1 g(ω/t ) (3.6) where the function g(x) is to very good approximation given by g(x) = C/(1 iax) with A 0.8 for x = ω/t < 1.5. This corresponds to a Drude response where the scattering rate has a linear temperature dependence. A consequence of this behavior is a collapse of the spectra when T σ 1 (ω, T ) is plotted as a function of ω/t for different temperatures. At ω/t 1.5 a cross-over appears to take place to a power law behavior. For ω/t > 1.5 a collapse of the spectra plotted versus ω/t required that the conductivity is multiplied no longer with the temperature T, but with either T 0.5 or ω 0.5.This change of behavior is also evident from a plot of the phase of the optical conductivity (arctan(σ 2 /σ 1 )), which displays a plateau with a phase angle of approximately 60 degrees for all temperatures and frequencies in access of k B T. In the same frequency range the absolute value of the conductivity, σ(ω), follows a power law behavior, σ(ω) ω 2/3. Both the frequency dependence of σ(ω) and of the phase angle are manifestations of the fact that the optical conductivity follows approximately a power law for frequencies in the range T < ω < 0.7eV. Experimentally, the ω/t scaling and constant phase angle are most closely obeyed for samples close to optimal doping. As such they appear to be a direct manifestation of quantum critical behavior when the doping is tuned to match exactly a quantum phase transition. However, it remained unclear whether a single scaling function could be dened covering both aspects 1) and 2) of the optical conductivity[5. The data collected for this paper show that the optical conductivity has a behavior that is similar to the behavior seen in the cuprates. The results show that at low frequencies, ω/t 1.5, there is a Drude form behavior and this turns out to be a power-law behavior in which σ ω γ with γ 0.65 and the phase is roughly constant around 60. The differences between the data collected and the experimental data for cuprates include the power-law behavior behavior of the DC resistivity with an exponent that depends on the lattice wavenumber instead of the robust linear behavior characteristic of the strange metal regime. In addition to that, there is no off-set constant C is seen in cuprates.
16 Figure 9: The optical conductivity of optimally doped Bi 2 Sr 2 Ca 0.92 Y 0.08 Cu 2 O 8+δ [6. 3.5 An Ionic Lattice Now, we will only use the Einstein-Maxwell action in four dimensions to produce the socalled ionic lattice. This lattice is introduced by a spatially varying chemical potential. The reason why this kind of lattice is called ionic is since the varying chemical potential can be viewed as representing the potential felt by the electrons in an array of ions. The action is, 1 S = d 4 x g [R + 6L 16πG 12 F abf ab. (3.7) 2 N The chemical potential is determined by the boundary value of the gauge potential A t and as z 0 we choose it to be A t µ(x) µ[1 + A 0 cos(k 0 x). (3.8) Einstein-Maxwell equations of the system can be solved and the physics of the system is determined by the dimensionless quantities A 0, k 0 / µ, and T/ µ. Here we will only give the results of these calculations. Comparing the charge density obtained from scalar lattice (Fig.(2)) and the charge density obtained from the ionic lattice (Fig.(10)) we can say that the former one is varying at 1% level on the contrary the latter one varies at order one. The other important difference between these two is that the ionic lattice has a modulation of k 0, on the other hand the scalar lattice has a modulation of 2k 0 which is coming from φ 2. The optical conductivity with the ionic lattice has similar behavior as the optical conductivity obtained from scalar lattice. Namely, in the low frequency regime we observe the simple two parameter Drude form (Fig. (12)). In the mid-infrared regime the optical conductivity is well fit by the power law behavior with the same exponent γ = 2/3 and this exponent is independent of the lattice spacing and temperature. 3.6 Conclusion We have analyzed the effect of holograhic lattice to low frequency behavior of the optical conductivity of the simplest model of holographic conductor. The holographic lattice
17 Figure 10: Charge density variations in the ionic lattice with k 0 = 2, A 0 =.5, µ = 2, T/ µ =.055. Figure 11: The optical conductivity both with the ionic lattice (solid line and data points) and without (dashed line) for the 2+1 dimensional conductor with k 0 = 2, A 0 =.5, µ = 2, T/ µ =.055. Figure 12: A blow up of the low frequency regime of Fig.(11) for the ionic lattice.
18 structure is first introduced by a scalar and then by a modulated chemical potential. Both of these lattice structures the delta function at the zero frequency is smeared out. The low frequency behavior of the optical conductivity is fit by the simple two parameter Drude form and finally, the mid-infrared frequency regime fits to a power-law behavior with exponent γ = 2/3 with an off-set constant. This power-law behavior is compatible with the power-law obtained for the cuprates with the off-set constant exception. References [1 G. T. Horowitz, J. E. Santos, D. Tong, Optical Conductivity with Holographic Lattices, [arxiv:1204.0519v2[hep-th. [2 G. T. Horowitz, J. E. Santos, D. Tong, Further Evidence for Lattice-Induced Scaling, [arxiv:1209.1098v1[hep-th. [3 S. A. Hartnoll and D. M. Hofman, Locally critical umklapp scattering and holography, arxiv:1201.3917 [hep-th. [4 P. W. Anderson, Infrared Conductivity of Cuprate Metals: Detailed Fit Using Luttinger- Liquid Theory, Phys. Rev. B 55, 11785 (1997) cond-mat/9506140. [5 D. van der Marel, F. Carbone, A. B. Kuzmenko, E. Giannini, Scaling properties of the optical conductivity of Bi-based cuprates, Ann. Phys. 321 1716 (2006) condmat/0604037. [6 D. van der Marel, et al., Power-law optical conductivity with a constant phase angle in high T c superconductors, Nature 425 (2003) 271 [arxiv:cond-mat/0309172