Abstract
We study a relation between the thermal diffusivity (DT ) and two quantum chaotic properties, Lyapunov time (τL) and butterfly velocity (vB ) in strongly correlated systems by using a holographic method. Recently, it was shown that {mathbb{E}}_i:={D}_{T,i}/left({v}_{{}^{B,i}}^2{tau}_Lright)left(i=x,yright) is universal in the sense that it is determined only by some scaling exponents of the IR metric in the low temperature limit regardless of the matter fields and ultraviolet data. Inspired by this observation, by analyzing the anisotropic IR scaling geometry carefully, we find the concrete expressions for {mathbb{E}}_i in terms of the critical dynamical exponents zi in each direction, {mathbb{E}}_i={z}_i/2left({z}_i-1right) . Furthermore, we find the lower bound of {mathbb{E}}_i is always 1/2, which is not affected by anisotropy, contrary to the η/s case. However, there may be an upper bound determined by given fixed anisotropy.
Highlights
Law states that ρs(T = 0)/(σDC(Tc)Tc) is universal, which means it is independent of the components and structures of superconducting materials
We study a relation between the thermal diffusivity (DT ) and two quantum chaotic properties, Lyapunov time and butterfly velocity in strongly correlated systems by using a holographic method
By analyzing the anisotropic IR scaling geometry carefully, we find the concrete expressions for Ei in terms of the critical dynamical exponents zi in each direction, Ei = zi/2(zi − 1)
Summary
Which is the Einstein-Maxwell-Dilaton theory coupled to ‘Axion’ fields χi We introduce the axions as many as. For simplicity we introduce anisotropy minimally by two couplings W1 and W2. We will further assume the axion fields have the form χi = k1xi We introduced another anisotropy by k1 and k2. We introduced two kinds of anisotropy: i) in the action, W1 and W2 ii) in the solution k1 and k2. The action yields the following Einstein equations:. Ds2 = −D(r)dt2 + B(r)dr2 + C1(r) dx2i + C2(r)dy , i=1 φ = φ(r) , A = At(r)dt , χi = k1xi , χp−1 = k2y , we obtain the Einstein equations
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