Diffusion and butterfly velocity at finite density

Journal of High Energy Physics, Jun 2017

We study diffusion and butterfly velocity (v B ) in two holographic models, linear axion and axion-dilaton model, with a momentum relaxation parameter (β) at finite density or chemical potential (μ). Axion-dilaton model is particularly interesting since it shows linear-T -resistivity, which may have something to do with the universal bound of diffusion. At finite density, there are two diffusion constants D ± describing the coupled diffusion of charge and energy. By computing D ± exactly, we find that in the incoherent regime (β/T ≫ 1, β/μ ≫ 1) D + is identified with the charge diffusion constant (D c ) and D − is identified with the energy diffusion constant (D e ). In the coherent regime, at very small density, D ± are ‘maximally’ mixed in the sense that D +(D −) is identified with D e (D c ), which is opposite to the case in the incoherent regime. In the incoherent regime D e ∼ C − ℏv B 2 /k B T where C − = 1/2 or 1 so it is universal independently of β and μ. However, \( {D}_c\sim {C}_{+}\hslash {v}_{{}^B}^2/{k}_BT \) where C + = 1 or β 2 /16π 2 T 2 so, in general, C + may not saturate to the lower bound in the incoherent regime, which suggests that the characteristic velocity for charge diffusion may not be the butterfly velocity. We find that the finite density does not affect the diffusion property at zero density in the incoherent regime.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29030.pdf

Diffusion and butterfly velocity at finite density

Received: April Di usion and butter y velocity at nite density Chao Niu 0 1 2 3 4 Keun-Young Kim 0 1 2 3 4 di usion. At 0 1 2 3 4 regime ( =T 0 1 2 3 4 0 describing the coupled 1 Gwangju 61005 , Korea 2 School of Physics and Chemistry, Gwangju Institute of Science and Technology 3 Open Access , c The Authors 4 exactly, we nd that in the incoherent We study di usion and butter y velocity (vB) in two holographic models, linear axion and axion-dilaton model, with a momentum relaxation parameter ( ) at nite density or chemical potential ( ). Axion-dilaton model is particularly interesting since it shows linear-T -resistivity, which may have something to do with the universal bound of nite density, there are two di usion constants D di usion of charge and energy. By computing D ArXiv ePrint: 1704.00947 nite; density; Holography and condensed matter physics (AdS/CMT); Gauge-gravity cor- - 1) D+ is identi ed with the charge di usion constant (Dc) and D at very small density, D are `maximally' mixed in the sense that D+(D ) is identi ed regime De and . However, Dc C ~vB2=kBT where C 2=16 2T 2 so, in general, C+ respondence Contents 1 Introduction Methods 2 3 Di usion constants Butter y velocity Linear axion model Axion-dilaton model Zero density Finite density Conclusion Thermodynamics and transport coe cients Introduction heavy fermions etc.), resistivity ( ) is linear in temperature (T ) so called Homes' law1 have been observed [2, 3]. While such interesting phenomena in holographic methods. s(T = 0) = C DC(Tc)Tc ; where C is a material independent universal number. so called `Planckian' time scale [3, 10] If Dc saturates to the bound and v2 is temperature independent, 1=T hence linear-T At nite density, there are two di usion constants D describing the coupled di usion D+ + D relaxation time ( P ) [11]: = Dc , where is the charge susceptibility. The Einstein relation with (1.4) is the thermoelectric susceptibility. If the charge density is zero, since = 0, D are decoupled and D+ and D constant (De) respectively. In [11], it was proposed the di usion constants are bounded as 2The conductivities may be diagonalized as in [12]. di usivity [14]. inhomogeneous SYK model [26]. perature limit. Unlike the previous studies, we rst study D at nite density without between D on the Gubser-Rocha model. We nd that there are two branches of classical solutions. both at zero and nite density. In section 5, we conclude. models, it has been also observed in condensed matter systems [14, 16{18]. a dynamical transition to an many-body localization phase. be written as where C B T 2 T D v 2 T D which will be computed and displayed in section 3 and 4. 1 and = is chemical is governed by di usion of charge and energy [11]. Di usion constants From (1.6) and (1.7) two di usion constants are computed as + M ; ( ; ; c ) and conductivities ( ; ; ). First, thermodynamic susceptibilities are de ned as @T 2 = T G = thermodynamics. S = 1 X2 ( I ( )@ I ) with the ansatz ds2 = U dt2 + + V1dx2 + V2dy2 ; A = adt ; I = I Iixi ; Z( )s 4 V1 12 1( ) r=rh 12 1( ) r=rh 12 1( )s r=rh h[W (t; ~x); V (0; 0)]2i ; i denotes thermal average. The function general, C(t; ~x) takes the following form conductivity, is Butter y velocity or the following average of the commutator squared: C(t; ~x) = e L(t t vj~xBj ) + \butter y e ect cone". Outside of the cone C(t; ~x) 1, even if the operators V and W chaos spatially propagates through the system. 2 kBT =~ = 2 = P ; terms of the characteristic parameters in quantum chaos & vB2 P & 2 vB2= L : with an infrared geometry ds2d+2 = U (r)dt2 + + V (r)d~x2d : L = B2 = V 0(rh) the bound (2.13), the butter y velocity is not. Linear axion model S = 1 F 2 The second term is nothing but with the negative cosmological constant, a system at last term, two massless scalar elds of the form I = Iixi = Iixi are introduced to relaxation e ect so makes conductivity nite [27]. An advantage of this ansatz for massless I , a classical solution of the action (3.1) is ds2 = f (r)dt2 + f (r) = A = I = we set L = 1. is interpreted as the chemical potential of the boundary eld theory, T = f 0(rh) so rh is expressed in terms of T; and : The entropy density is rh = s = 4 rh2 = = rh = B2 = 4 T + p6 2 + 16 2T 2 + 3 2 : 4 T + p6 2 + 16 2T 2 + 3 2 2 4 T + p6 2 + 16 2T 2 + 3 2 : 6 T 4 T + p6 2 + 16 2T 2 + 3 2 The butter y velocity (2.16) is from the metric (3.2) and (3.4). 4 T + p6 2 + 16 2T 2 + 3 2 6 2 + 16 2T 2 + 6 2 ! 1 + p6 2 + 16 2T 2 + 3 2 c = T c = c 9p6 2 + 16 2T 2 + 3 2 9(6 2 + 6 2 + 4 T (4 T + p6 2 + 16 2T 2 + 3 2)) so the electrical, thermal, thermoelectric conductivities are = 1 + 2 = 4 T + p6 2 + 16 2T 2 + 3 2 2 4 T + p6 2 + 16 2T 2 + 3 2 ; here for convenience, D+ = B2 = p 4 T 3p6 2 + 16 2p6T 2 4 T + p6 2 + 16 2T 2 + 3 2 2 c1 = c2 = + M ; M = The analytic formulas of D can be obtained by plugging (3.8), (3.9), (3.11), and (3.12) into (3.14). Because the nal expressions are complicated and not very illuminating we 2 D . The green curve means vB2 (3.7). As =T increases both 2 T D and vB2 go to zero, but all of them behave as 1=( =T ). Thus, 2 T D =vB2 saturate the nite lower bound as shown in gure 1(b). 1 and = 1) can be read also from the analytic expression of D and vB2 at large : = 0; I = 1; Z = 1; V = I = (a) Di usion constants (2 T D ) and the butter y velocity (vB2) 1 and = =T = 0:1; 1; 5 from which yield C+ = 2 T D+ = 2 + 2 T D = 1 + 2 We nd that C Indeed, in this regime, D+ and D can be identi ed with Dc and De respectively because bounds C+ = 2 and C identify Dc = and Dc gure 2, where the term is important and for e ect is `maximal' in the sense D is identi ed with Dc and D+ is identi ed with De, T M = p (a) Di usion constants: 2 T Dc and 2 T De 2 ▲ ▲ ▲ ▲▼▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 2 ▲ ▲▼▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ mixing term, the solid curves in gure 1. which implies that the mixing term may be small if =T is large even for small =T . In and energy di usions are negligible so D+ and D may be identi ed with Dc and De respectively. (Because =T 6 5 in gure 2, =4 T 5 is the incoherent regime.) 1). To see it more clearly let us consider a di erent case 1 and which yield the following expansions: 2 T 2 T B2 = D+ = C+ = 2 T D+ = 2 T D = 1 + p 2 T T Thus C+ is not universal while C in a class of holographic models that ow to AdS2 xed points in the infrared. Our ; (3.21) D+ = B2 = M = energy di usion is expanded as of our solution is as follows. and D = De and C+ = 2 T Dc = 2 T De = 1 + p and , which and a function of 1) it becomes universal. i.e. C+ = 2. Axion-dilaton model linear axion model in the previous section. The action is S = 1 X2 (@ I ) in low frequency approximation. { 11 { which belongs to (2.7) and is reduced to (3.1) if h(r)dt2 + dx2 + dy2 + (Q + rh)3 (Q + r)3 I = 2(Q + rh)2 r2g(r)h(r) 2(Q + rh)2 Q + rh Q + r g(r) = ds2 = h(r) = 1 A = r2g(r) 2(Q + r)2 3Q(Q + rh) log(g(r)) ; where rh > 0 and rh > here we set L = 1. Thermodynamics and transport coe cients chemical potential and charge density are T = s = rh2g(rh) rh2g(rh)h(rh) 0 prh(6(Q + rh)2 8 (Q + rh)3=2 = 4 prh(Q + rh)3=2 ; 3Q(Q + rh) 1 = (Q + rh) 3Q(Q + rh) 1 2(Q + rh)2 2(Q + rh)2 t~ = t rh ; x~ = x rh ; y~ = y rh ; ~ = Q~ = Consequently, we may de ne the eld theory variables as: T~ = ~ = ~ = { 12 { To nd a possible range of Q~, we impose physical condition T Without loss of generality, we can take 0. All these inequalities imply 4 T = 0 >><0 < 4 T 1 < Q~ Q~. They correspond to the green region in gure 3. The boundary of the green region is nothing but the condition for gure 3: the red curve temperature [29] G = (Q + rh)3 G(Q~ < 0). At zero density, there is no positive Q~ To compute the di usion constants D we rst need to compute thermodynamic sus(4.6) and entropy and . In principle, rh and Q can be expressed in terms of T and from (4.3) and (4.5) but their analytic expressions are very complicated except for ities read and from (2.16) the butter y velocity is = e (rh) + B2 = Q + 4rh Q + rh { 13 { (a) =T = 0:1 (b) =T = 5 G(Q~ < 0) is shown. The positive Q~ solution is always Zero density Let us rst consider a neutral case, i.e. For Q~ = 0, the dilaton eld vanishes and the solution (4.2) is reduced to (3.2) with and susceptibilities are given as 4 T + p6 2 + 16 2T 2 ; c = 4 T + p6 2 + 16 2T 2 2 9p6 2 + 16 2T 2 = 0 and 7The energy di usion constant was also computed in [46]. Dc = De = 4 T + p6 2 + 16 2T 2 p6 2 + 16 2T 2 De. The green curve displays vB2. Q~ = The butter y velocity is B2 = 6 T 4 T + p6 2 + 16 2T 2 2 T Dc = 2 ; 2 T De = 1 + 2 T (a) Q~ = 0 (b) Q~ = 1 + p~ 4 T + p6 2 + 16 2T 2 : { 15 { For Q~ = 1 + p , there is a nontrivial coe cients and susceptibilities are8 = 0 and = 2 ; Dc = De = 2 T 4 T ; B2 = for =T 2 T Dc = 2 T De = 16 2T 2 for =T for =T c = 4 2 2 The butter y velocity is Q~ = 0, Dc; De For Q~ = 1 + ~=p2, Dc Finite density 1 + ~=p2 1 + ~=p2, we nd 1. For T = and v2 1=T and De T = 2 while vB2 The horizon position rh can be expressed in terms of T and by eliminating Q in (4.3) and (4.5). However, unlike the previous cases, if { 16 { 2 T ; v 2 T De { 17 { (a) Di usion constants (2 T D ) and the butter y velocity (vB2) 1 and = limit, which we positive Q branch, rh= ! 0 and Q= are interested in for the universal bound, can be read o analytically. For large in the = 2 ; Therefore, the charge and energy di usion constants are 4 T Together with the butter y velocity at large we have axion term. 2 T D+ 8The electric conductivity was also computed in [47]. (a) Di usion constants: 2 T Dc and 2 T De =T = results with the mixing term, the solid curves in gure 5. Thus we nd that the bounds for D at zero density (4.22) still hold at nite density. The correction by may be identi ed for Dc and De in term, where we simply identify Dc and Dc is identi ed with Dc and D+ is identi ed with De. di usion (1.5) is not realized. Even though realized because is temperature-independent in this model. Conclusion cases, the axion eld is of the form I = Iixi and plays a role of momentum relaxation, where large { 18 { Linear axion Axion-dilaton p3=2 1 and = a ground state by comparing their grand potentials. There are two di usion constants D describing the coupled di usion of charge and We have showed the exact relation between D and (Dc; De) in gure 2 and suppressed so D+ and D can be identi ed with Dc and De respectively. However, in the decoupled and D is `maximal' in the sense that D is identi ed with Dc and D+ is identi ed with De, which ten as 2 T where C agree to the values at zero density and C is universal independently of density. In [20], 6 1 at low temperature limit. Because the IR geometry of the axion model is AdS2 = 1 can be anticipated Rd so the Rd) studied in [20]. However, our result C Rd, which can be allows i.e. 1=2 < C The charge di usion constant is written as 2 T { 19 { 2 T : Thus, if there is more relevant velocity scale than the butter y velocity and if it is temperature-independent, the relation between the universality of charge di usion and di usion by the following observations. susceptibilities, and , the full di usion constant in the incoherent regime is the ratio of to c , it can be written only butter y velocity is robust [49]. Acknowledgments the GIST in 2017. Open Access. { 20 { [INSPIRE]. 430 (2004) 539 [cond-mat/0404216] [INSPIRE]. Relaxation, JHEP 04 (2015) 152 [arXiv:1501.00446] [INSPIRE]. [INSPIRE]. (2016) 144 [arXiv:1608.04653] [INSPIRE]. [arXiv:1102.4628] [INSPIRE]. [arXiv:1405.3651] [INSPIRE]. [arXiv:1505.05092] [INSPIRE]. arXiv:1610.05845 [INSPIRE]. Rev. Lett. 117 (2016) 091601 [arXiv:1603.08510] [INSPIRE]. [arXiv:1609.01251] [INSPIRE]. 95 (2017) 060201 [arXiv:1608.03280] [INSPIRE]. 114 (2017) 1844 [arXiv:1611.00003] [INSPIRE]. [arXiv:1604.01754] [INSPIRE]. [arXiv:1611.09380] [INSPIRE]. Sachdev-Ye-Kitaev models, arXiv:1609.07832 [INSPIRE]. localization transition, arXiv:1703.02051 [INSPIRE]. matter, JHEP 10 (2016) 143 [arXiv:1608.03286] [INSPIRE]. [INSPIRE]. 181 [arXiv:1401.5436] [INSPIRE]. JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE]. Phys. Rev. Lett. 112 (2014) 071602 [arXiv:1310.3832] [INSPIRE]. [INSPIRE]. [arXiv:1306.0622] [INSPIRE]. arXiv:1610.02669 [INSPIRE]. arXiv:1702.08803 [INSPIRE]. [1] S.A. Hartnoll , A. Lucas and S. Sachdev , Holographic quantum matter, arXiv:1612. 07324 [2] C.C. Homes et al., Universal scaling relation in high-temperature superconductors , Nature [3] J. Zaanen , Superconductivity: Why the temperature is high , Nature 430 ( 2004 ) 512. [4] J. Zaanen , Y.-W. Sun , Y. Liu and K. Schalm , Holographic Duality in Condensed Matter Physics, Cambridge University Press, Cambridge U.K. ( 2015 ). [5] M. Ammon and J. Erdmenger , Gauge/gravity duality, Cambridge University Press, Cambridge U.K. ( 2015 ). [6] J. Erdmenger , B. Herwerth , S. Klug , R. Meyer and K. Schalm , S-Wave Superconductivity in Anisotropic Holographic Insulators, JHEP 05 ( 2015 ) 094 [arXiv:1501.07615] [INSPIRE]. [7] K.-Y. Kim , K.K. Kim and M. Park , A Simple Holographic Superconductor with Momentum [8] K.K. Kim , M. Park and K.-Y. Kim , Ward identity and Homes' law in a holographic [9] K.-Y. Kim and C. Niu , Homes' law in Holographic Superconductor with Q-lattices , JHEP 10 [10] S. Sachdev and B. Keimer , Quantum Criticality , Phys. Today 64N2 (2011) 29 [11] S.A. Hartnoll , Theory of universal incoherent metallic transport , Nature Phys . 11 ( 2015 ) 54 [12] R.A. Davison and B. Gouteraux , Dissecting holographic conductivities , JHEP 09 ( 2015 ) 090 [13] J.A.N. Bruin , H. Sakai , R.S. Perry and A.P. Mackenzie , Similarity of scattering rates in metals showing t -linear resistivity, Science 339 ( 2013 ) 804. [14] J.C. Zhang et al., Anomalous Thermal Di usivity in Underdoped YBa2Cu3O6 +x, [17] B. Swingle and D. Chowdhury , Slow scrambling in disordered quantum systems , Phys. Rev . B [18] A.A. Patel and S. Sachdev , Quantum chaos on a critical Fermi surface , Proc. Nat. Acad. Sci. [19] M. Blake , Universal Di usion in Incoherent Black Holes , Phys. Rev . D 94 ( 2016 ) 086014 [21] Y. Gu , X.-L. Qi and D. Stanford , Local criticality, di usion and chaos in generalized [22] R. A. Davison , W. Fu , A. Georges , Y. Gu , K. Jensen and S. Sachdev , Thermoelectric [23] S.-K. Jian and H. Yao , Solvable SYK models in higher dimensions: a new type of many-body [24] A. Lucas and J. Steinberg , Charge di usion and the butter y e ect in striped holographic [25] M. Baggioli , B. Gouteraux , E. Kiritsis and W.-J. Li , Higher derivative corrections to incoherent metallic transport in holography , JHEP 03 ( 2017 ) 170 [arXiv:1612.05500] [26] Y. Gu , A. Lucas and X.-L. Qi , Energy di usion and the butter y e ect in inhomogeneous Sachdev-Ye-Kitaev chains , SciPost Phys . 2 ( 2017 ) 018 [arXiv:1702.08462] [INSPIRE]. [27] T. Andrade and B. Withers , A simple holographic model of momentum relaxation , JHEP 05 [28] B. Gouteraux , Charge transport in holography with momentum dissipation , JHEP 04 ( 2014 ) [29] M.M. Caldarelli , A. Christodoulou , I. Papadimitriou and K. Skenderis , Phases of planar AdS black holes with axionic charge , JHEP 04 ( 2017 ) 001 [arXiv:1612.07214] [INSPIRE]. [30] S.S. Gubser and F.D. Rocha , Peculiar properties of a charged dilatonic black hole in AdS5 , Phys. Rev . D 81 ( 2010 ) 046001 [arXiv:0911.2898] [INSPIRE]. [31] A. Donos and J.P. Gauntlett , Thermoelectric DC conductivities from black hole horizons , [32] K.-Y. Kim , K.K. Kim , Y. Seo and S.-J. Sin , Coherent/incoherent metal transition in a [33] K.-Y. Kim , K.K. Kim , Y. Seo and S.-J. Sin , Gauge Invariance and Holographic [34] K.-Y. Kim , K.K. Kim , Y. Seo and S.-J. Sin , Thermoelectric Conductivities at Finite [35] M. Blake and D. Tong , Universal Resistivity from Holographic Massive Gravity, Phys. Rev. [37] Y. Sekino and L. Susskind , Fast Scramblers , JHEP 10 ( 2008 ) 065 [arXiv:0808. 2096 ] [38] S.H. Shenker and D. Stanford , Black holes and the butter y e ect , JHEP 03 ( 2014 ) 067 [39] D.A. Roberts , D. Stanford and L. Susskind , Localized shocks, JHEP 03 ( 2015 ) 051 [40] J. Maldacena , S.H. Shenker and D. Stanford , A bound on chaos, JHEP 08 ( 2016 ) 106 [41] D.A. Roberts and B. Swingle , Lieb-Robinson Bound and the Butter y E ect in Quantum Field Theories , Phys. Rev. Lett . 117 ( 2016 ) 091602 [arXiv:1603.09298] [INSPIRE]. [42] Y. Ling , P. Liu and J.-P. Wu , Holographic Butter y E ect at Quantum Critical Points, [43] M. Alishahiha , A. Davody , A. Naseh and S.F. Taghavi , On Butter y e ect in Higher Derivative Gravities , JHEP 11 ( 2016 ) 032 [arXiv:1610.02890] [INSPIRE]. [44] R.A. Davison , K. Schalm and J. Zaanen , Holographic duality and the resistivity of strange metals , Phys. Rev . B 89 ( 2014 ) 245116 [arXiv:1311.2451] [INSPIRE]. [45] Z. Zhou , Y. Ling and J.-P. Wu , Holographic incoherent transport in Einstein-Maxwell-dilaton Gravity , Phys. Rev . D 94 ( 2016 ) 106015 [arXiv:1512.01434] [INSPIRE]. [46] R.A. Davison and B. Gouteraux , Momentum dissipation and e ective theories of coherent and incoherent transport , JHEP 01 ( 2015 ) 039 [arXiv:1411.1062] [INSPIRE]. [47] S.-F. Wu , B. Wang , X.-H. Ge and Y. Tian , Universal di usion in quantum critical metals , [48] A. Amoretti , A. Braggio , N. Magnoli and D. Musso , Bounds on charge and heat di usivities in momentum dissipating holography , JHEP 07 ( 2015 ) 102 [arXiv:1411.6631] [INSPIRE]. [49] D.-J Ahn, Y.-J Ahn, K.-Y . Kim , W.-J. Li and C. Niu , work in progress.


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29030.pdf

Chao Niu, Keun-Young Kim. Diffusion and butterfly velocity at finite density, Journal of High Energy Physics, 2017, 1-24, DOI: 10.1007/JHEP06(2017)030