Weyl corrections to diffusion and chaos in holography

Using holographic methods in the Einstein-Maxwell-dilaton-axion (EMDA) theory, it was conjectured that the thermal diffusion in a strongly coupled metal without quasi-particles saturates an universal lower bound that is associated with the chaotic property of the system at infrared (IR) fixed points~\cite{blake:1705}. In this paper, we investigate the thermal transport and quantum chaos in the EMDA theory with a small Weyl coupling term. It is found that the Weyl coupling correct the thermal diffusion constant $D_Q$ and butterfly velocity $v_B$ in different ways, hence resulting a modified relation between the two at IR fixed points. Unlike that in the EMDA case, our results show that the ratio $D_Q/ (v_B^2\tau_L)$ always contains a {\it non-universal} Weyl correction which depends on the matter fields as long as the $U(1)$ current is relevant in the IR.


I. INTRODUCTION
Investigation of the thermoelectric transport in metallic systems is one of core topics in modern condensed matter physics. In contrast to the weakly coupled metals whose dynamics are governed by long-lived quasi-particles, the transport properties of the strongly correlated metals with no single particle excitations are described by the emergent hydrodynamic like degrees of freedom. Moreover, a wide class of such systems exhibit an universal Planckian relaxation timescale, τ p ∼ /(k B T ) (set = k B = 1) [2,3].
A well-known category of the strongly correlated metals are the so-called "bad metals" or "incoherent metals". In these systems, the resistivity increases linearly with temperature and violates the Mott-Ioffe-Regel (MIR) bound, and there is no sharp Drude peak in the AC conductivities at high temperatures due to rapid momentum relaxation. Because of the breakdown of the single particle approximation and other perturbative methods, these features still lack a deep understanding within the conventional QFT. Motivated by the observation that in incoherent metals the momentum dissipation depends heavily on the microscopic details of materials, which should not be the underlying reason of the universal strange metal, S. Hartnoll proposed that strange metals could be explained by the saturation of diffusion bounds D c,Q v 2 F /T where v F is the Fermi velocity [4] 1 . However, the Fermi velocity is in general not sharply defined in the systems without quasi-particles.
The holographic duality provides us an tractable approach to the physics with no quasiparticles. It has been widely applied to studying the transport properties of strongly correlated systems. In holography, the DC conductivities can be captured by fluid like dynamics near the black hole horizon via the membrane paradigm [6,7]. Based on the Einstein-Maxwell-dilaton-axion (EMDA) theories, M. Blake proposed a connection between the thermoelectric transport and quantum chaos in strongly coupled systems that [8,9] where C c,Q are constants only depending on the scaling properties of the IR fixed points, v B is the butterfly velocity characterizing the speed of information spreading, τ L is the Lyapunov timescale characterizing the growth of the chaos which reaches its maximum 1 2πT ∼ τ p in holographic systems and the Sachdev-Ye-Kitaev(SYK) models [10][11][12] but is much longer in quasi-particle systems [13,14]. Then this bound seems valid for arbitrary chaotic systems with or without Fermi velocity. Whereas, it has been found that the bound on the charge diffusion can be violated in striped systems [15] or theories with higher derivative terms [16].
Recently, it was pointed out that v 2 B τ L may bound only the thermal diffusion instead of the charge diffusion with: at generic non-relativistic fixed points, where z is the dynamical exponent [1]. Then the ratio of D Q to v 2 B τ L is quite universal, as it only depends on the scaling property of the IR theory, regardless of the UV parameters of the matter fields, say, the chemical potenial/charge density, magnitude of the lattice, etc.
However, it is still unclear wether (2) universally holds or not in holography. The bottomup approach allows us to touch this question in any (generalized) gravity theories with selfconsistency. The bound (2) has been checked in many cases, and seems to work well in holography so far [15][16][17][18][19][20] 2 . Nevertheless, the condensed matter models studied in [22] and [23] have already revealed two counter-examples. Then, it is worth exploring to what extent (2) holds in holography. Suppose the proposed universal C Q is somehow changed, it should be the two following situations: a. C Q is still geometry-dependent only, but the relation (2) is modified due to certain pure gravity corrections.
b. C Q may also depend on the details of matter fields due to other kinds of corrections, which makes its expression totally non-universal.
Either case provides a necessary condition for the complete violation of the bound.
In this paper, we focus on the second one. A practicable way of modifying the holographic theory is to add the Weyl coupling terms, which couples the gauge field with the Weyl tensor. Previously, the effects of this kind of terms have been studied in a variety of holographic models [3,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. Here, we consider an EMDA action coupled with a small Weyl coupling term and investigate the Weyl corrections on the thermal diffusivity, the butterfly velocity and the ratio C Q . The content of the paper is as follows: In section II, we introduce the holographic action and the black hole solutions. In section III, we analyze the thermal diffusion, butterfly velocity and their relation at low temperatures. In section IV, we conclude. And the technical details are shown in the appendix.
Note added: As this work was being completed, [43] appeared which has some overlap with our discussions.
2 In a recent paper [21], it was reported that the diffusion bound can be violated in a higher derivative gravity theory. However, in these kinds of theories, there are two distinct butterfly velocities even in isotropic systems which seems quite odd from the angle of condensed matter physics.

II. HOLOGRAPHIC ACTION AND BLACK HOLES
We consider the four dimensional Einstein-Maxwell-dilaton theory coupled to two axionic scalars χ I associated with the translational symmetry breaking and a Weyl coupling term.
with the indexes I = x, y and the Weyl coupling γ. In the action above we have taken 16πG = L = 1 and Einstein's convention for convenience. By definition, the Weyl tensor in four dimensions is given by Adding Weyl couplings will, in general, bring about higher order differential equations which makes the problem mathematically difficult. So we will only consider the charged case with a small γ coupling and expand the results up to the linear power in γ. The generic ansatz for isotropic solutions should be ds 2 = −f (r)dt 2 + h(r)dr 2 + g(r)(dx 2 + dy 2 ) , whose IR geometry can be classified into several distinct cases, depending on the couplings U , V , W and Z.

Lifshitz/Hyperscaling violating geometries
In the EMDA theory without Weyl corrections, the background solution can be Lifshitz/Hyperscaling violating geometry in the IR at low temperatures. This have been analyzed and classified into several different cases depending on the scaling properties in the IR [44] . So here we just review this briefly. These solutions can be achieved by setting the following exponential potentials which gives the near extremal IR solution where z and θ are dynamical and hyperscaling violating exponents respectively, ϕ 0 depends only on the scaling exponents z, θ and ζ, while L,L, a 0 depend not only on the scaling exponents but also V 0 and the magnitude of the axionic lattice, k. In the extremal limit, the black hole solution flows towards different IR fixed points with the following features: These extremal solutions with a non-vanishing horizon can be supported either by the relevant current or the axion, and realized when the IR behavior of the dilaton is a constant It is allowed to choose a proper value of ϕ 0 and take δ = λ = η = u = 0, such that The full analytic solution of black hole has been found in [41], it is where f 0 and g 0 are the metric without the Weyl correction, µ is the chemical potential, q is the charge density, G(r) = q 2 9r 2 , Y (r) and H(r) are complicated functions of q, k and r whose forms are not important in our discussions. In the extremal limit, we have f (r h ) = 0 with r h = 0 as long as the current or/and the axion is/are non-vanishing. Then, the IR geometry should be AdS 2 × R 2 .

III. THERMAL DIFFUSION AND BUTTERFLY VELOCITY
For convenience, we introduce a new radial coordinate as in [1,20] Then, the background metric can be rewritten as where Performing the Donos-Gauntlett strategy [7], we can express the DC conductivities just in terms of the metric components and A t at the horizon (See the details in Appendix C.). Our result implies that the time-reversal symmetry is violated at O(γ) when A t = 0 according to the Onsager relation [46]. Moreover, it has been revealed in [15,16,47,48] that the conjectured bounds on the electric conductivity σ = 1 as well as that on the charge diffusion D c ∼ v 2 B τ L can both be violated in general holographic models. Therefore, in this work, we focus only on the thermal transport.
The open-circuit thermal conductivity at low temperatures is given by Now the prime refers to the derivative with respect tor. In contrast to that in Einstein gravity, it can never be expressed merely in terms of the near horizon geometry. 5 Then the thermal diffusivity can be calculated via the following Einstein relation: where c q is the heat capacity with fixed charge density which is defined as Following [1], we will compute D Q and compare it with the results of the butterfly velocity at the IR fixed points that we have discussed in the previous section.

Generic fixed points
The entropy density can be calculated by the Wald formula [49][50][51], which gives Obviously, the factor U (r h )A t (r h ) 2 , plays a crucial role of modifying the thermal diffusion in (14) and the entropy density in (16), hence the heat capacity as well. In the small γ expansions, we can just take the value of U (r h )A t (r h ) 2 in the EMDA theory.
When the current is marginally relevant, i.e, ζ = θ − 2 and which is temperature-independent. Here, we should restrict that z = θ so that the correction term is not divergent. Then (13) and (15) can be rewritten as κ ≡ Aκ 0 , c q ≡ Ac q0 , 5 If one try to eliminate A t by using the Einstein equation, the final result will also depend on k 2 W , Z and U .
where κ 0 and c q0 represent the thermal conductivity and heat capacity obtained in the EMDA theory. Applying (11) and (14), the thermal diffusion is obtained as which is not modified by the Weyl coupling. On the other hand, the butterfly velocity can be obtained by performing the shockwave calculations. The details have been shown in appendix D. It can also expressed in terms of the horizon data This further requires that θ = 2. Finally, we obtain that the ratio of (19) to (20) is at the generic fixed points when the current is marginally relevant in the IR. The interesting thing is that there is always a non-universal correction that comes from the Weyl corrections, as one can see from (12), (B7) and (B8) that the constant A 0 highly depends on the details of the matter fields in the IR region.
While if the current is irrelevant and the axion is marginally relevant in the IR, the Weyl correction is vanishing. Then, If the current and axion are both irrelevant, z = 1. In this case D Q is controlled by an irrelevant deformation and C Q 1, which is not universal[1].
AdS 2 × R 2 fixed points In the extremal limit, the black hole solution (9) flows towards the AdS 2 ×R 2 fixed points in the IR. For simplicity, we denote f (r) as with the location of the extremal horizon at r = r e and R is a dimensionless constant that depends on γ, the gauge field and axion at the horizon. For this class of geometries, in contrast to the Lifshitz/hyperscaling violating cases, c q and v B should determined by the leading irrelevant deformation of the fixed point solution. Turning on a small temperature, the black hole solution is slightly deformed as where r is a small deviation from the extremal horizon and the external horizon is r = r h = r e + r . Then r = 2πT R . Expanding g(r h ) around the extremal value, we obtain g(r h ) = g(r e ) + g (r e )r + ..., In addition, the thermal conductivity and the entropy density reduce to 6 Then the entropy density is where s e is the extremal entropy. Then, the thermal diffusion is given by where R 0 = 6 − 6k 2 2k 2 +µ 2 ∈ [3, 6] and g 0 (r e ) = r 2 e are the quantities without Weyl corrections. Finally, we achieve that at low temperatures and meanwhile T √ 2k 2 +µ 2 µ 2 γ. Then we find that, for charged cases there is again a non-universal correction, and for γ > 0 the ratio C Q can be greater than 1. 6 In this case the thermal conductivity can also be expressed just in terms of the metric as κ = 4πf (r h ) 16 3 πγf (r h ). 7 Through out the paper, we do the small γ expansion before taking the low temperature limit.
In this paper, we have studied the thermal transport and butterfly effects by performing holographic calculations in the EMDA theory coupled with a small Weyl coupling term.
It is found that the ratio of thermal diffusion to the butterfly velocity multiplied by the Lyapunov timescale, C Q , contains a non-universal Weyl correction when the Weyl corrections are marginally relevant in the IR.
When extremal IR geometry is Lifshitz or hyperscaling violating type, the Weyl correction in C Q also depends explicitly on the parameter of the gauge field in the IR, A 0 , and the Weyl coupling γ. When the IR geometry is AdS 2 × R 2 , the non-universal part can be simply expressed in terms of the γ, µ and k. In both cases, the conjectured universal bound on C Q can be "slightly violated" by the Weyl corrections. While, in the "incoherent limit" [52,53] which implies that T is finite and the value of k is far bigger than T and any other parameters of the matter fields in the IR. Then we can just simply neglect the effect of A 0 or µ in the IR, and the Weyl corrections in C Q is vanshing. This suggests that the proposed diffusion bound in [1] could be valid only in the incoherent limit.

Appendix A: Covariant form of the equations of motion
The equations of motion from the holographic action (3) are given by with the Weyl corrections: where the Laplacian is defined by = ∇ µ ∇ µ .
Appendix B: Analysis of the IR geometries In the extremal limit, the IR solution (7) reduces to Plugging this into the Einstein equation, dilaton equation as well as Maxwell equation, one obtains that From now on, we assume that the Weyl corrections are at the same order in powers of the radial coordinate as the original Maxwell term. With (B2)-(B6), following the analysis in [44], we conclude that To obtain above expressions, we have used the small γ expansion.
In order to calculate the DC conductivities, we introduce the following perturbations around the background where f , g, A t are the background fields in (11) and we have omitted the tilde symbol for the radial coordinater for simplicity. All the linearized eoms can then be obtained by applying the ansatz (C1) to (A1)-(A5).
From the equation of motion of a x , we can define a conserved current along the radial direction in the bulk: which one can check that it agrees with the U (1) current in the dual field theory. 8 To obtain the heat current, we need to find another radially conserved current. For general gravity theories, this current has already been constructed in [54] which is similar as Wald's procedure: where ξ = ∂ t is the time-like Killing vector. On the other hand, the Weyl correction can also be re-expressed as the DC conductivities can be expressed in terms of the horizon data as follows If we use the horizon relation: to eliminate f and set γ = 0, they reduce to the results in the EMDA theory [7]. The thermal conductivity in the open circuit condition is defined by Eliminating k 2 W by using (C17), it is finally obtained as At low temperatures, the last term with f 2 | r=r h can be neglected. Then it agrees with (13) in the main text.
Furthermore, from (C14) and (C15), we find that α −ᾱ = 16πγU A t = 0, which implies that the time-reversal symmetry is broken according to the Onsager relation. This feature should be attributed to the introduction of the Weyl term since the difference between α andᾱ is O(γ). We leave the detailed analysis for future investigation.

Appendix D: Butterfly velocity with Weyl corrections
The butterfly velocity characterizes the propagation of information in a chaotic quantum system and can be measured through the out-of-time correlator(OTOC): where W and V are two generic local Hermitian operators, λ L is the Lyapunov exponent, t * is the scrambling time and v B is the butterfly velocity. In holography, the OTOC has been widely calculated in many gravity theories by solving a shockwave solution in a two-sided black hole [1, 8-11, 15-21, 43, 55-70].
For simplicity, we rewrite the Einstein equation (A4) into where T µν is the stress tensor. In Kruskal coordinates, the black hole solution (11) can be re-expressed as The horizon location r = r h in the original coordinates now is uv = 0. And the Kruskal coordinates are defined by: where dr * = dr f (r) . Moreover the functions appearing in the metric are related by the following relations: where f (r), g(r) and A t (r) are the metric components and gauge field in the original coordionates. We perturb the spacetime with an operator at x i = 0 9 and t L = t W , i.e. a localized shock-wave; the butterfly velocity corresponds to the rate of growth of this perturbation.
The localized stress tensor of such a perturbation is given by: Then for large distance |x| 1, one can replace a(x) with a delta function approximately.
The shockwave solution corresponds to the geometry where there is a shift v → v + h(x, t W ) once one crosses the horizon u = 0. The backreaction produces a perturbation in the spacetime metric of the form: and the stress tensor should get modified as [56,71]: where the second term is the leading contribution from the deformed geometry. Then the first order Einstein equation becomes 10 where the effective mass reads: Solving the equation, we find that at large distances the solution takes the form: where t * ∼ 1 λ L Log 1 G is the scrambling time. As is pointed in [72], the profile of the shockwave, h(x, t W ), corresponds to the OTOC of two generic local operators inserted at different locations and times with the spatial interval x and temporal interval t W . Then, the Lyapunov exponent and the butterfly velocity can be extracted as The final step is to re-express A(0), B(0), C(0) and their derivatives in the original (t, r) coordinates. Near the horizon we expand the quantities as follows uv = −κ 0 (r − r h ) + . . . , g(r) = g(r h ) + g (r h )(r − r h ) + . . . , where κ is a positive constant whose value is not important. On top of this, we have Then (D10) can be re-expressed as This is the result for general values of γ. In this work, we focus on the physics at both of small γ and low temperature limits. If we take the small γ limit first, we obtain that At low temperatures, v B ∼ √ T plus some small Weyl corrections. However, an interesting thing happens in the near AdS 2 × R 2 case when we perform T ∼ f (r h ) → 0 before taking the small γ limit. If so, the last term in (D20) dominates, which just gives m 2 = 2γf (r h )g(r h )A t (r h ) 2 6 + 8γA t (r h ) 2 . (D22) In this case, we have which implies that v B is slower than that in the EMA theory. Therefore, the result of v B highly depends on the order of manipulating the two limits.