Charged BTZ-like black hole solutions and the diffusivity-butterfly velocity relation

Journal of High Energy Physics, Jan 2018

Xian-Hui Ge, Sang-Jin Sin, Yu Tian, Shao-Feng Wu, Shang-Yu Wu

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Charged BTZ-like black hole solutions and the diffusivity-butterfly velocity relation

HJE Charged BTZ-like black hole solutions and the di usivity-butter y velocity relation Xian-Hui Ge 0 1 3 6 Sang-Jin Sin 0 1 3 4 Yu Tian 0 1 3 5 Shao-Feng Wu 0 1 3 6 Shang-Yu Wu 0 1 2 3 0 Hsinchu 300 , Taiwan, R.O.C 1 Beijing , 100049 , China 2 Department of Electrophysics, National Chiao Tung University 3 Shanghai 200444 , China 4 Department of Physics, Hanyang University 5 School of Physics, University of Chinese Academy of Sciences 6 Department of Physics, Shanghai University We show that there exists a class of charged BTZ-like black hole solutions in Lifshitz spacetime with a hyperscaling violating factor. The charged BTZ black hole is characterized by a charge-dependent logarithmic term in the metric function. As concrete examples, we give ve such charged BTZ-like black hole solutions and the standard charged BTZ metric can be regarded as a special instance of them. In order to check the recent proposed universal relations between di usivity and the butter y velocity, we rst compute the di usion constants of the standard charged BTZ black holes and then extend our calculation to arbitrary dimension d, exponents z and . Remarkably, the case d = z = 2 is a very special in that the charge di usion Dc is a constant and the energy di usion AdS-CFT Correspondence; Black Holes; Holography and condensed matter - and De might be ill-de ned, but vB2 diverges. We also compute the di usion constants for the case that the DC conductivity is nite but in the absence of momentum relaxation. 1 Introduction 2 3 4 5 6 1 Introduction The general formalism Dimensional reduction and Jackiw-Teitelboim theory Di usion and butter y velocity of disordered BTZ black holes Di usion of BTZ-like black holes with a hyperscaling violating factor Conclusion and discussions Black hole solutions are extremely useful in studying the holographic properties of strongly coupled quantum matters. For example, the Reissner-Nordstrom-Anti de Sitter black holes are of particular usage in the study of the electronic structure of the dual quantum systems [1]. The holographic methods also yield disordered quantum systems with no quasiparticle excitations [2, 3]. However, the precise bulk dual of the recent proposed model, the Sachdev-Ye-Kitaev (SYK) model, is still elusive [4, 5]. The SYK models are theories of Majorana fermions in zero spatial dimensions with q body in nite-range random interactions in Fock space [6{8]. As one of the simplest strongly interacting system with a gravity dual, the SYK model has many interesting features including thermodynamic properties, correlation functions and the absence of quasiparticle in a non-trivial solvable limit in the presence of disorder at low temperature [9{40]. All these properties suggest a gravity-dual interpretation of the model in the low-temperature strong-coupling limit and it is believed that the SYK models are connected holographically to black holes with AdS2 horizons with non-vanishing entropy in the T ! 0 limit. It has been conjectured that the bulk gravity dual of the SYK model is the two dimensional Jackiw-Teitelboim model of dilaton-gravity with a negative cosmological constant, while there are also some hints that it is actually Liouville theory. In [41], the authors show that the spectrum of the SYK model can be interpreted as that of a three-dimensional scalar coupled to gravity. They further conjectured that the bulk dual of the SYK model is indeed a three dimensional theory. The main purpose of this paper is to study a class of charged Banados-TeitelboimZanelli-like (BTZ-like) black hole solutions in general d + 2-dimensional spacetime with a momentum dissipating source. The BTZ black hole sultions are asymptotically AdS3 and can be dimensionally reduced to solutions of various two dimensional Jackiw-Teitelboim theories. In [42], a bulk theory with the BTZ black hole solution are utilized to study the { 1 { late time behavior of the analytically continued partition function Z( + it)Z( holographic 2d CFTS. We will also compare the transport coe cients of such BTZ-like solutions with those of the SYK models. Recently, aimed to build a connection between transport at strong coupling and quantum chaos, it has been conjectured that there is a universal relation between di usion constant and the butter y velocity D fundamental quantum thermal timescale ~=kBT [43{49]. In the holographic setup, the butter y velocity in an isotropic system is de ned as [50] vB2 = dgx0x(rH) 4 T grr pgt0t=gr0r, where the prime indicates a radial derivative and d is the dimension of spatial coordinates. This relation is somehow inspired by the shear viscosity bound proposed by Kovtun, Son and v 2 B with a Starinets [ 51 ]. But recent developments have shown that the shear viscosity bound can be strongly violated in anisotropic systems and momentum dissipated systems [52{59] (see also [60{64] for discussions on the di usion bounds). In incoherent metals without a Drude peak, transports are dominated by di usive physics in terms of charge and energy instead of momentum di usion. One naturally guesses that the charge di usion constant Dc and energy di usion constant De could play a crucial role. In the di usion-butter y e ect scenario, the electron-phonon interactions of strongly correlated materials behave as a composite, strongly correlated soup with an e ective velocity vB [45]. A natural candidate for such a velocity is provided by the butter y e ect [3, 45]. We will study this relation for BTZ-like black holes. Interestingly, we also check the di usivity-butter y velocity relation in an extremal case: in a d + 2-dimensional Lifshitz spacetime with a hyperscaling violating factor, the butter y velocity behaves as vB2 = 2 T rHz) 2 . As d ! (d , the butter y velocity becomes divergent. Now the question is that do the di usion constants also diverges in the d ! limit? We are going to study this special condition in details. Last but not least, in some of the previous literature [3, 65], the relation D studied with non-zero dynamical exponent z and hyperscaling violating factor . But only one gauge eld was considered in the action. Actually, to realize Lifshitz geometry with z > 1, one needs to introduce at least one auxiliary gauge eld in additional to the real Maxwell eld [66{68]. The auxiliary gauge eld is responsible for supporting the Lifshitzlike vacuum of the background, while the Maxwell eld makes the black hole charged. It v 2 B is interesting to check the relation D in the presence of two gauge elds. Moreover, it was found in [ 69 ] that nite DC electric conductivity can be realized simply because of the presence of the auxiliary U(1) charge even without translational symmetry breaking. Another purpose of this paper is to examine the di usion-butter y velocity relation with such nite DC conductivity but without translational symmetry breaking. v 2 B was As a byproduct, we will also verify the universal formula of dc electric conductivity proposed in [70] for translational-symmetry broken BTZ-like black holes. This formula d 2) states that the ratio of the determinant of the dc electrical conductivities along any spatial directions to the black hole area density A in zero-charge (i.e. Q i iijqi=0 = Qi ZidA limit has a universal value. The structure of this paper is organized as follows. In section 2, we present the general formalism for the black hole solutions. Five concrete examples of BTZ-like black holes are derived. We brie y address dimensional reduction of BTZ black holes and the Jackiw{ 2 { Teitelboim theory in section 3. The di usivity and butter y-velocity of charged BTZ black hole with momentum dissipation are discussed in section 4. The transport properties of BTZ-like black holes with an extra hyperscaling violating factor are presented in section 5. Discussions and conclusions are presented in section 6. In the appendix, we provide DC transport coe cients for general d + 2-dimensional black holes in the presence of two U(1) 1 2 gauge elds. 2 The general formalism hyperscaling violating factor S = 1 eral (d + 2)-dimensional action with an arbitrary Lifshitz dynamical exponent z and a where we will use the notation Zi = e i and Y ( ) = e 2 in what follows and i = is a collection of d massless linear axions introduced to break the translational symmetry and denotes the strength of momentum relaxation and disorder of the dual condensed matters. The action consists of Einstein gravity, axion elds, and U(1) gauge elds and a dilaton eld. For simplicity, we only consider two U(1) gauge Fr(t1) and Fr(t2) in which the rst gauge eld plays the role of an auxiliary eld, making the geometry asymptotic Lifshitz, and the second gauge eld is the exact Maxwell eld making the black hole charged. The Einstein equation is given as R = X ZiF(2i) V ( ) d 2 g : The equations of motion for the dilaton eld and axion elds are obtained as 0 = + V 0( ) 0 = r Y ( )r i : i 4 1 X Zi0( )F(2i) 1 2 i The Maxwell eld equation is 0 = r Zi( )F(i) : The dilaton eld can be solved from the combinations of the (r,r) and (t,t) components (i.e. Rrr Rtt) of the Einstein equation and the solution read = ln r = p2(d )(z 1 =d) ln r: Assumed the metric takes the form ds2 = r2 r2zf (r)dt2 + dr2=r2f (r) + r2d~x2d = =d, the Maxwell equation can be solved as F(i)rt = Z 1 i ( )r (2 d)+z d 1Qi: { 3 { # (2.1) Iixi (2.2) (2.3) (2.4) (2.5) and (2.6) where we have set V ( ) = V0r 2 , Zi( ) = e i and Y ( ) = 1=Z2. We normalize the rst term in the metric function to be one, so we have 2 = and V0 = (d + z + d 1)(d + z + d). By further using the equation of motion for the dilaton eld, we nally arrive at 4 d( +1) 8 V = X e i F(2i) i i We then obtain the expressions for i, Q1 and d( +1) The associated black hole solution was rst obtained by some of us in [71{73]: 1 2d i X i ; = 2 = = The metric function f (r) can be solved from the (x; x)-component of the Einstein equation rd( +1)+zf (r) 0 = r (d+2)+z+d 1 ( + 1) V0r d X r 2d( +1) i Q 2 i mr d z d The above metric function holds under the condition that Q2 and 2 are nite and Q1 does not contribute to the metric function, since F(1)rt is introduced to realizing the Lifshitz scaling. The constraints from the null energy condition of the gravity yields (d )[d(z 1) ] 0 and (z mass parameter m in terms of rH: m = rHd+z + Q2rH 2 2 d z+ r H 2 d z +2 =d : By further introducing an coordinate u = rH=r, we can recast f (r) as Q22 The corresponding Hawking temperature is given by Hd . The speci c heat of this black hole can be v B2 = detail derivation of the DC transport coe cients in this background was obtained in [ 71, 73 ] (see also [74{80] for more recent work). We now present several examples when black hole solutions become \critical" as the charge density Q2 or axion density 2 is formally zero. In other words, when one of the term in f (r) is zero, we need to work out the solution very carefully because those terms would produce a logarithmic term in the metric function. This can also be seen in the di erential equation of f (r), i.e. (2.7). Such a logarithmic term can greatly modify the solution especially the behavior of the electric eld. The result is that the metric function f in any d 1 dimension can be recast in a form similar to charged BTZ black holes in 2 + 1 dimensions. In general, there are ve conditions that the logarithmic term appears in the metric function as shown in table 1. Case I. Critical black hole solutions at d + z that in this case, we have d2 + 2 be achieved in the following form [ 71, 73 ] where m, q1 and q2 are nite physical parameters without divergence as (d + z The metric function can recover that of charged BTZ black hole solution as { 5 { (2.13) (2.14) (2.15) (2.16) (2.17) 2) ! 0. = 0. A only d + z the mass and Q2 related terms are degenerated the mass and axion related terms are degenerated the Q2 and axion related terms are degenerated (d = ; z = 2) or (d = 1; z = 1 + ) all mass; Q2 and axions related terms are degenerated careful examination of (2.17) reveals that they satisfy the corresponding Einstein equation and Maxwell equation. The Hawking temperature is given by HJEP01(28)6 The metric function and gauge elds in this case take their forms where we have used the relation = = Q22 ! r2(d+z Q22 r2(d+z r2(d+z q 2 2 2 2 1) 1) 1) 2 d (d (d (d { 6 { We can also express the metric function in terms of the event horizon radius Note that only the linear scalar term in the metric function becomes a logarithmic term. The black hole temperature yields T = H (d + z ) 2 r H d z + q22rH 2(d+z but z 6= 2, the two terms containing Q22 and 2 in the metric function (2.18) (2.19) (2.20) ln rH : r (2.21) and q22 := Q22 + 2. So in this condition, the metric function can be written as Here q2 and 0 are nite and physical parameters under the d ! 0 limit (z 6= 2), instead of diverging Q2 and . The metric function written in terms of the event horizon is given by here and obtain a very particular black hole solution The scalar potentials are given by The chemical potentials and charge density are given by 1 and 2 = q2 ln rH, respectively. Notice that 1 does not correspond to the chemical potential of the black hole and as r ! 1, A(1)t ! 1. The event horizon locates at r = rH satisfying f (rH) = 0. The Hawking temperature is given by The metric function is actually a three-dimensional BTZ-like black hole solution with a Lifshitz dynamical exponent and a hyperscaling factor. As = 0 and then q1 = 0, we recover the standard charged BTZ black hole solution obtained in (4.1) since d + z = 2 infers z = 1. The entropy density and the butter y velocity are s = r 1 H 4G ; v B2 = The relation between the butter y velocity and di usion constant will be studied in detail in what follows. becomes Case V. As to d = and z = 2, the null energy condition is satis ed. The metric HJEP01(28)6 ds2 = temperature, but constant entropy density. The associated speci c heats are also vanishing. The butter y velocity becomes divergent as d ! . The transport coe cients in this case are also very special 4rH2 q22+ 02 . The entropy density is (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) (2.38) Both dc electric conductivity 22 driven by the real Maxwell eld and the thermoelectric conductivities 1 and 2 are constants. It is worth future investigating whether this black hole solution and its boundary dual has any more physical meaning. In the following discussions, we rst discuss the relation between the charged BTZ black hole and its dimensioanl reduction to Jackiw-Teitelboim theory. Then we study di usions and butter y velocity for momentum dissipated-charged BTZ black holes. The hyperscaling violating factor modi ed BTZ black holes will also be examined. 3 Dimensional reduction and Jackiw-Teitelboim theory The gravitational duality of the original SYK model is argued to be a two-dimensional dilaton AdS2 gravity [81], i.e. the Jackiw-Teitelboim theory. A three-dimensional gravity can be reduced to Jackiw-Teitelboim theory by dimensional reduction. In the context of the AdS3=CFT2 duality, a two-dimensional CFT has a holographic bulk dual. At the high { 8 { temperature the thermal state of the theory is dual to a BTZ black hole. Consider a three-dimensional gravity in general The corresponding black hole solution is the BTZ black hole solution of the form S = 1 where i; j = 0; 1. The action (3.1) simply reduces to that of the Jackiw-Teitelboim theory The BTZ solutions written in the form as (3.4) yield a solution to the Jackiw-Teitelboim model S = Z d x 2 p h R + 2 ds2 = Assume that there is a single coordinate called ', which is independent of the gravitational eld in three dimensions and the metric takes the form ds2 = hij (xi)dxidxj + 2(xi)d'2; In [82], some of us consider a new higher dimensional SYK model with complex fermions on bipartite lattices and obtain linear in temperature resistivity, thermal conductivity and speci c heat. 4 Di usion and butter y velocity of disordered BTZ black holes A special case is d = 1, = 0, V ( ) = 2, Y ( ) = 1 and Z( ) = 1. This corresponds to the Case IV discussed in section 2. In this case, the action reduces to that of BTZ black hole with a momentum dissipation term: S = 1 d x 2 p hK + Sct; It is well known that the naive free energy of the charged BTZ black hole is logarithmically divergent. Jensen found the divergence is due to the Weyl anomaly of the boundary CFT [81]. The counter-term is given by [81] Sct = 1 Z 8 G3 r=r d x 2 p h 1 + R[h] 2 1 4 i 1 Z 2 r=r d x 2 p hFraF ra ln r : { 9 { The black hole temperature is given by T = 4 zh2(q2+ 02) . The charge density and chemical potential can be obtained from the near-boundary expansion of the gauge eld jt = q and 8 zh 12 0 ln(2 T + pq12 + 02 + 4 2T 2). The horizon radius expressed in terms of where = T , q and 0 goes as The entropy density reads The butter y velocity can be evaluated as The DC conductivity is obtained as ds2 = 1 z2 f (z) = 1 At(z) = q ln f (z)dt2 + dx2 + metric and a linear scalar eld (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) 02 + q2 + 2 T (2 T + p 02 + q2 + 4 2T 2). The speci c heat at xed charge T T q 2 0 = = The compressibility and the thermoelectric susceptibility can be expressed as = = p 02 + q2 + 4 2T 2 + 2 T a11 log 12 p 02 + q2 + 4 2T 2 + T + q2 ; ; = 4 2 T + p 8 3T 2 02 + q2 + 4 2T 2 : The temperature dependence of the entropy density and the speci c heat obtained here is similar to that of 1 + 1-dimensional SYK model with complex fermions on bipartite lattices [82]. The Seebeck coe cient and the thermal conductivity are given by = = = = The mixing term M has similar temperature dependence as D term v 2 expanded at large 0 yields Compared with D , the mixing term M can be neglected in the incoherent regime. The In general, the di usion constants D are related to the transport coe cients and thermal quantities through the Einstein relations [83] The di usion constants are solved as where c2 = + c + M; c3 = T ( c 2 ) 2 : In the absence of the mixing term M, D+ and D Dc and the energy di usion De. We then momentum dissipation limit ( 0=T 1) turn out to be simply reduce to the charge di usion nd the two di usion constants in the strong D+D D+ + D = = c + ; c D = c2 D+ = D = 2 0 2 0 + T ( ) 2 : pc2 2 2 c ; : (4.12) implies that the BTZ black hole indeed satis es the proposed relation between the di usion constant and the butter y velocity. As shown in gure 1, the di usion constant D v with = ~=kBT . This approximate to a value parallel to the line vB2 ature is xed. However, as the axion 0= is xed, D as the temperature goes up. in the incoherent regime when the tempercoincides with vB2 and D+ v 2 B The low temperature and low charge density expansion of the di usion constants and butter y velocity are as follows The mixing term between the charge- and energy- di usion is given as M = 2q2 the charge di usion. ductivity proposed in [70] which cannot be ingored at low temperature but nite charge density so D+ and D may not interpreted as charge di usion Dc and energy di usion De. As shown in gure 1, both in the incoherent regime and in the low temperature limit, v has the trend to go to zero, implying there is no propagation of the chaos in these cases. One can also check the dimension of these quantities here: [q] = [ 0] = 1; [ ] = 1; [r] = [1=z] = 1. Assuming [T ] = 1, we have [ ] = consistent. Notice that in [82], 1 + 1-dimensional SYK model with complex fermions on bipartite lattices shows similar temperature dependence of the electric conductivity and On the other hand, we are able to check the validity of the universal dc electric conQ A i ii d 2 qi=0 = Y Zd i i r=rH : Substituting equation (4.6) at zero charge density, the black hole area and gauge coupling Z2 = 1 into (4.25) and evaluating at the event horizon, we can see that (4.25) is satis ed. 5 Di usion of BTZ-like black holes with a hyperscaling violating factor In order to check the universal relations between di usivity and the butter y velocity, let us rst work with arbitrary parameters d, z and . After obtaining the general ansatz for D+ and D , we then discuss the ve black hole solutions given in section 2 and check the 12 A 2 B (4.21) (4.22) (4.23) (4.24) (4.25) and the butter y e ect vB2 as functions of the axion eld and temperature. (Top) The di usion constants D+ (red), D (green) and v (blue) as functions of 2 B 0, where we have set = 1 and T = 1. (Bottom) The di usion constants D+, D 2 and vB as functions of 0, where we have set = 1 and 0 = 1. di usion-butter y velocity relation. For simplicity of calculation, we mainly consider the incoherent limit as follows 0 1; 0 T 1; with nite: T In addition, we only focus on the charge- and energy-di usions and thus neglect the mixing term M in (4.16). The metric function and the Hawking temperature are given in (2.14) and (2.15), respectively. The speci c heat at xed charge density is given by pgrr )rHd vB τ D + 2 vBτ 15 20 (5.1) (5.2) chemical potential of Reinssner-Nordstrom black hole 2) and one can obtain the = q2=rH as d = 2, z = 1 and = 0. In general, the chemical potential depends on the details of the full bulk geometry. However, the black hole solution obtained here mainly describe the IR geometry. We assume that the infra-red region of the geometry that dominates the behavior of the charge compressibility in what follows [46]. The compressibility is found to be HJEP01(28)6 = T = (2 + d z)rHd+z 2 Since there are two U(1) gauge elds, the DC electric conductivity in this case is actually 2 matrix. The general ansatz has been obtained by some of us in [73] One may notice that even q1 is zero now, 11 = r 2 2 =d 2d has non-trivial contributions H from the pair production of the boundary dual quantum eld. That is to say, once the uctuations of the auxiliary gauge eld are turned on, there exists a discontinuity from the expressions of transport coe cients of Lifshitz spacetime to those of the AdS geometry. Assuming the charge current induced by the auxiliary gauge eld is vanishing J1 = 0, we are able to focus on the diagonal elements [ 71 ] DC = rHd+2z 4 + (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) Similarly, we have the thermal conductivity at zero electric current 16 2T rH3d+2z DC = Considering only the charge di usion Dc and the energy di usion De and in the strong momentum relaxation limit, we obtain Dc = De = cq = = d r H + z r Hz 2(dz 2 + O 2 ) d 2 d2(d 1 2 0 d ) ; dz + 2 + O 1 2 0 Hereafter, our discussion work in the strong momentum relaxation limit. Singularities can be observed as d + z = 2 and d = . We may able to remove the singularities since the metric become critical in these cases. The butter y velocity can be computed by considering a shock wave geometry and written in terms of the metric at the horizon [50, 84] (5.11) (5.12) (5.13) (5.14) (5.15) (5.16) (5.17) (5.18) (5.19) (5.20) In our case vB2 = 2 T rHz) 2 . From (5.12) and (5.14), we see that the relationship (d v B2 = pgxxm2 4 2T 2 Dc = De = 2 (d (dz rH ; (d ) + z 2 )(d2 m2 = d 2 d2 T gxdxp grrgtt rH : 2) vB2 ; dz + 2 ) vB2 : holds independently of the details of the bulk solution. However, from the expressions of the di usion constants (5.12), we cannot obtain the di usion constants of BTZ black holes evaluated in section 4. So do the ve examples listed in table 1. The reasons are that the black hole temperature and the chemical potential are greatly modi ed for those cases. As ! 0, The relations between di usion constants and the butter y velocity are given by Dc = De = d 2 (d + z 2) vB2 ; 2 z)z vB2 : (d d We discuss the di usion constants of the ve special cases listed in table 1 in details: Case I: d + z 2 = 0, (d2 dz + 2 ) 6= 0. In this case, the chemical potential becomes logarithmic = q ln rH. The Hawking temperature is reformulated in (2.18). The di usion constants behave as + : Notice that as d = 2 or d = 1, the energy di usion constant De at leading order is vanishing and in this case, one shall consider the O(1= 02) order instead. The butter y velocity multiplied by is given by v 2 B = 2 r (2 H z) Note that rH is a function 0, T and charge density. The ratios Dc=vB2 all nite in this case. Case II: d + z dz + 2 ) = 0. The di usion constants read The butter y velocity multiplied by is given by The ration between the di usion constants and v2 is given by Dc=vB2 d = 4 (d 1) and De=vB2 = 2d((zd 22)) . Case III: d = , z 6= 2. The di usion constants in this case yield Dc = De = cq = = r H z 2 r Hz 2(d + ; (d 2) ln rH ) + : The butter y velocity multiplied by is given by As d ! nite vDB2e = (d 1)[(z 22) ln rH 2] . both the energy di usion and the butter y velocity diverges, but their ratio is Case IV: d = 1, z = 1 + . The di usion constants are given by (5.21) (5.22) (5.23) (5.24) (5.25) (5.26) (5.27) (5.28) (5.29) (5.30) One can easily verify that as d = 1, z = 1 and = 0 and in the strong incoherent limit, the di usion constants Dc and De recover the ansatz given in the previous section (4.21){(4.22). This re ects that the calculations in this section is consistent with the previous section. The butter y velocity multiplied by is given by We can see that the ratio between De and vB2 is nite De=vB2 H ) : Dc = De = cq = = r ; r H 2(2 z) + : v 2 B = 2 r (2 z) ! nite: Case V: d = , z = 2. The di usion constants are obtained as We can express the charge di usion in terms of the black hole temperature Dc = ln 2 2 T = 0, where we have used rH = T + pq22 + 4 2T 2 + 02=2. In this case the speci c heat cq is zero and the energy di usion might be ill-de ned. The butter y velocity times Notice that in this case, the metric takes a very special form constant value. The universal dc electric conductivity Q for all the above ve cases. The entropy density in this case is s = 1=4G and the Hawking temperature is q2+ 02 . The speci c heat becomes vanishing since the entropy density takes a i iijqi=0 = Qi ZidAd 2 is satis ed Di usivity in the absence of disorder parameter. This is a very intriguing situation that even without translational symmetry breaking (i.e. = 0), nite DC electric conductivity can still be realized because of the presence of the auxiliary U(1) charge q1 [64]. In the strong auxiliary gauge eld limit q1 ! 1, the charge di usion constant Dc reads Dc = d r H + z 2 : The energy di usion constant De is given by De = (d+z (d )q12)z rz+ d 2 =d + q(d+z H q12(d ) 2)(2d+z 2 2 ) 2 3d z+3 2 =d r H : (5.37) In this case, the behavior of the energy di usion is di erent from the charge di usion. Notice that the non-vanishing of the auxiliary charge density q1 is a consequence of z 6= 1. We then consider a special 2 + 1-dimensional black hole solution with a non-zero hyperscaling violating factor. Note that as d = 1 and d + z 2 = 0, we have a very particular black hole solution as v 2 B = 2 r (d H ) ! 1: ds2 = r 2 H r2 q22 + 0 2 2(1 )r1+z 2(1 ; dr2 The event horizon locates at r = rH satisfying f (rH) = 0. The Hawking temperature is given by As = 0 and then q1 = 0, we recover the standard BTZ black hole solution obtained in (4.1) since = 0 and d = 1 infers z = 1. The entropy density and the butter y velocity are T = H r1+ 2 1 The transport coe cients are obtained as ! 0, the quantity q1 becomes vanished but the DC conductivity 11 = rH 3 is not vanishing. The situation is very subtle in that if we set z = 1 and = 0 in the action (2.1), then the auxiliary gauge eld, 11 and 12 do not appear at the all. This re ects that once the uctuations of the auxiliary gauge eld is turned on, there exists a discontinuity in the this limit [72]. ! 0, (d + z ) ! 2 and q1 ! 0 limit since 11 does not vanish in On the other hand, as = 1, the black hole solution shows its strange behaviors since rH2 < (q2 + 02)=8 corresponds to T < 0 and rH2 (q2 + 02)=8 corresponds to T over, the black hole has its maximal Hawking temperature Tmax = 21 as rH2 0. More(q2 + 02)=8. Moreover, the null energy condition is violated as = 1 and d = 1. It seems that this black hole solution is not very physical. 6 Conclusion and discussions In summary, we obtain a class of black hole solutions analogous to charged BTZ black holes by considering d + 2-dimensional action with non-trivial Lifshitz dynamical exponent z and hyperscaling violating factor . Those BTZ-like black hole solutions can be realized because special combinations of d, z and lead to divergence of the mass-, charges- and axionsrelated terms. Such divergences can be annihilated by renormalizing the mass parameter. As summarized in table 1, there are ve concrete cases that such charged BTZ-like black hole solutions can be realized. We then show that the action of the charged BTZ black hole can be reduced to the Jackiw-Teitelboim theory by dimensional reduction. We nd that the relation D is well obeyed by the standard charged BTZ black holes in the incoherent limit. We thus study the di usions for general d, z and and obtain general expressions for the charge and v 2 B the energy di usions. We carefully evaluate the di usions for those ve special cases. We can see that for cases I and II, the ratio Dc=vB2 is nite, while De v 2 B is valid for cases I, II, III, IV and V. In this sense, the energy di usion seems more general than the charge di usion. However, for case V, the charge di usion is nite and the energy di usion seems ill-de ned, while v2 B is divergent. Since there are two U(1) gauge elds in the theory, we calso calculate the di usion constants in the absence of momentum relaxation parameter . In this case, the charge di usion is same as that of the momentum dissipated case, but the energy di usion has O(1=q12) dependence. We also examine the universial electrical DC conductivity formula and nd that for Lifshitz spacetime with auxiliary U(1) gauge elds, this formula is satis ed. Acknowledgments The study was partially supported by NSFC, China (grant No. 11375110); NSFC (grant No. 11475179); the Ministry of Science and Technology (grant No. MOST 104-2811-M-009068) and National Center for Theoretical Sciences in Taiwan (grant No. 105-2112-M-009001-MY3); and by NSFC China (grant No. 11675097). SJS was supported by the NRF, Korea (NRF-2013R1A2A2A05004846). Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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Xian-Hui Ge, Sang-Jin Sin, Yu Tian, Shao-Feng Wu, Shang-Yu Wu. Charged BTZ-like black hole solutions and the diffusivity-butterfly velocity relation, Journal of High Energy Physics, 2018, 68, DOI: 10.1007/JHEP01(2018)068