Optical conductivity and string theory

نویسنده

  • Jakub Jankowski
چکیده

In this note we summarize some facts about optical conductivity and the corresponding holographic models attempting to describe it. 1. Optical conductivity The optical properties of solids provide an important tool to study their microscopic dynamics. In certain materials with strong electron correlations a quantum critical state of matter can occur. In this situation the system looks on average the same, regardless of the timeand length scale on which it is observed. This implies, that scale invariant correlations should drive the response functions, revealing some universal patterns determined by the quantum mechanical nature of the fluctuations. Such regularities were reported for the mid-infrared frequency range of the optical conductivity of optimally doped copper-oxide superconductors [1, 2]. In more detail in ref. [1] two interesting regions of frequencies were identified from the data. The low frequency region (ω < T ), where measurement time was large compared to the time scale set by inverse temperature. In this region system follows classical relaxation dynamics with a relaxation time τ . Conductivity in this case is given by the usual Drude formula σ(ω) = Kτ 1− iωτ . (1) The second region (T < ω < Ω) directly probes the scale invariance of the quantum critical state. A non-universal cut-off Ω is called the optical gap and is required by the f-sum rule [1]. In this region the optical conductivity σ(ω) = σ1(ω) + iσ2(ω) follows the universal scaling law |σ(ω)| ∼ ω , (2) with a frequency independent phase angle arctan(σ2(ω)/σ1(ω)) ≈ 60o. The experimental data are shown in figure 1. A remarkable thing is the quite robust temperature independence of this scaling relation. 2. Holographic models Up to date we have no microscopic understanding of how quantum criticality emerges from the strong correlations of the normal state near optimal doping. An emergent scale invariance implies that the low energy effective theory takes a form of a strongly coupled conformal field theory (CFT) [3]. A widely used tool to analyze certain strongly coupled CFT systems is the anti-de Sitter conformal field theory (AdS/CFT) correspondence, a setup in which a 15th International Conference on Strangeness in Quark Matter (SQM2015) IOP Publishing Journal of Physics: Conference Series 668 (2016) 012062 doi:10.1088/1742-6596/668/1/012062 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 Figure 1. Universal power law of the optical conductivity from ref. [1]. nongravitational system is mapped to a theory with gravity in a higher dimensional spacetime. Over the past few years, it has been applied to model many interesting phenomena, including superconductivity and superfluidity, Fermi surfaces and non-Fermi liquids (see [4] for a review). The main advantage of this method is that it is based on conserved currents, rather than on a quasi-particle description, which makes it more suitable for understanding the relation between the infrared and the ultraviolet degrees of freedom. A natural question to ask is whether any form of power-like scaling can occur in the optical conductivity computed in a holographic model? The simplest bulk description of a translationally invariant system at finite temperature and charge density is given by a charged black hole solution [4]. This was then improved by the coupling to a bulk scalar field, which could provide a model for a solid state lattice and in turn account for a realistic description of the charge transport [5]. Because the relevant physics is in two dimensional layers, the CFT in question lives in (2 + 1) spacetime dimensions. Thus, the gravitational dual is a theory in four dimensional spacetime. In more detail, the action used in [5] in d = 3 + 1 dimensions reads S = 1 16πGN ∫ dx √ −g [ R− 1 2 ∇aφ∇φ+ 6 + φ ] . (3) States in a CFT are mapped to solutions of the equations of motion following from the action (3), which have some specified boundary conditions. In Poincare coordinates {x, z} = {t, x, y, z} any solution of the Klein-Gordon equation for chosen parameters has the near boundary asymptote φ(x, z) ∼ φ0(x)z + φ1(x)z . (4) In particular, in the boundary theory this corresponds to the operator Ô of conformal dimension ∆ = 2 that has a source φ0(x) and corresponding expectation value 〈Ô(x)〉 = φ1(x). The simplest model of a lattice was to impose a one dimensional source of the form of a single Fourier mode, i.e. φ0(x) = A0 cos(kx) . (5) Such a choice causes explicit translational symmetry breaking in a chosen direction. The claim of ref. [5] is that with this assumption both frequency regions described in the previous section are reproduced. This has motivated further research and explorations of more realistic and transparent models of holographic lattices [6, 7, 8]. 15th International Conference on Strangeness in Quark Matter (SQM2015) IOP Publishing Journal of Physics: Conference Series 668 (2016) 012062 doi:10.1088/1742-6596/668/1/012062

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تاریخ انتشار 2015