Conjecture on the butterfly velocity across a quantum phase transition

Journal of High Energy Physics, Jul 2018

Abstract We study an anisotropic holographic bottom-up model displaying a quantum phase transition (QPT) between a topologically trivial insulator and a non-trivial Weyl semimetal phase. We analyze the properties of quantum chaos in the quantum critical region. We do not find any universal property of the Butterfly velocity across the QPT. In particular it turns out to be either maximized or minimized at the quantum critical point depending on the direction of propagation. We observe that instead of the butterfly velocity, it is the dimensionless information screening length that is always maximized at a quantum critical point. We argue that the null-energy condition (NEC) is the underlying reason for the upper bound, which now is just a simple combination of the number of spatial dimensions and the anisotropic scaling parameter.

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Conjecture on the butterfly velocity across a quantum phase transition

Received: May Conjecture on the butter y velocity across a quantum Matteo Baggioli 0 1 2 4 Bikash Padhi 0 1 2 3 Philip W. Phillips 0 1 2 3 Chandan Setty 0 1 2 3 0 1110 W. Green Street, Urbana, IL 61801 , U.S.A 1 71003 Heraklion , Greece 2 Department of Physics, University of Crete 3 Department of Physics and Institute for Condensed Matter Theory, University of Illinois , USA 4 Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics We study an anisotropic holographic bottom-up model displaying a quantum phase transition (QPT) between a topologically trivial insulator and a non-trivial Weyl semimetal phase. We analyze the properties of quantum chaos in the quantum critical region. We do not nd any universal property of the Butter y velocity across the QPT. In particular it turns out to be either maximized or minimized at the quantum critical point depending on the direction of propagation. We observe that instead of the butter y velocity, it is the dimensionless information screening length that is always maximized at a quantum critical point. We argue that the null-energy condition (NEC) is the underlying reason for the upper bound, which now is just a simple combination of the number of spatial dimensions and the anisotropic scaling parameter. Holography and condensed matter physics (AdS/CMT); AdS-CFT Corre- 1 Introduction 2 The holographic model 3 4 2.1 2.2 2.3 Weyl semimetals Holographic Weyl semimetal Anomalous transport Quantum chaos & universality Conclusion A The holographic background B Butter y velocities in anisotropic backgrounds 1 4 h [V(~x; t) ; W(0; 0)]2 i e L (t t j~xj=vB); (1.1) where V ; W are two local Hermitian operators, L the Lyapunov exponent, t is the so called scrambling time and is just the thermal timescale. The appearance of the butter y velocity in this correlation function motivates it as the relevant velocity for de ning bounded transport [2]. The monotonic growth of the Lyapunov exponent at a quantum critical point and its subsequent decrease away from the critical point determined by some non-thermal coupling constant, g is sketched in gure 1. See also [3] for an exploration of the connection between quantum chaos and thermal phase transitions. Moreover, this behavior is believed to hold also at nite but low temperature inside the quantum critical region. Preliminary studies connected to the proposal of [ 1 ] and to the onset of quantum chaos across a QPT have been already performed within the holographic bottom-up framework in [4]. Nevertheless a complete study, beyond simple models, is still lacking. A full analysis of this problem appears to be in order in view of the recent experiments where the OTOC has been measured using Lochsmidt echo sequences [5] and NMR techniques [6]. { 1 { line) at the quantum critical point g = gc. It is not clear, and indeed the purpose of our investigation, if the butter y velocity vB display a similar behaviour or not. The aim of this paper is indeed to understand the onset of quantum chaos across a quantum phase transition in more complicated holographic models displaying a quantum phase transition. In particular, we will perform our computations in the holographic bottom-up model introduced in [7, 8] which exhibits a QPT between a trivial insulating state and a Weyl semimetal. The particular new wrinkle we bring to bear on the transition is the presence of anisotropy. In other words, the rotational group SO(3) is broken to the SO(2) subgroup by an explicit source in the theory. As a consequence of the underlying anisotropy, we can de ne two butter y velocities that we will denote as v and v , where denotes the direction(s) of the anisotropy, while ? in the direction perpendicular to it. Throughout the paper, we use this notation for all such directional quantities. ? The results of our paper show that while the perpendicular velocity v ? displays a behavior similar to that in [ 1 ], the parallel one v does not. In particular, the butter y velocity along the anisotropic direction will not display a maximum at the critical point g = gc but rather a minimum. We pinpoint as the origin of this violation, the presence of anisotropy itself.1 Interestingly, the bound on the viscosity is also violated in an anisotropic system [13, 14] and by a strong magnetic eld [15, 16]. Here the mechanism leading to the violation of the bound are very analogous, that is explicit breaking of the SO(3), leading to spatial-anisotropy. Note that [17] spontaneous breaking of rotation symmetry, despite leading to a non-universal value for =s, does not provide a violation of the KSS bound. We expect this to be case for butter y velocity as well. Here is the viscosity and s is the entropy density. 1E ects of anisotropy on the butter y velocity were previously investigated in [9{12]. { 2 { To understand if any universal statement can be made about the butter y velocity, especially in the presence of anisotropy, we identify a quantity related to the spatial-spread of information which is insensitive to the breaking of the SO(D) symmetry, where D is the number of space dimensions. We do so by computing the OTOC holographically. Given an anisotropic bulk spacetime of the form ds2 = gtt(r) dt2 + grr(r) dr2 + h?(r) d~x2? + + h (r) d~x2 ; where we denote by the D anisotropic directions and with ? the D ? remaining directions. The butter y velocities can be computed for this background as ( v = L=M ; where L = 2 = is the Lyapunov exponent and all the quantities are computed at the horizon. The parameter M(?; ) controls the screening of the information =?; ) spreading in the (?; ) directions, and it clearly depends on the warp factor h . As a consequence, the butter y velocity can not represent a good and universal quantity in the presence of anisotropy. Contrastly, we can de ne a dimensionless quantity controlling the screening of information through (t; x ) e L t M jxij; 2 M 2 h (r0) : (1.2) (1.3) (1.4) (1.5) The factor h (r0) is indeed the reason why we see dissimilar result from [ 1 ] in an anisotropic setup. The important point is that our new physical parameter has no spatial dependence and hence, it is completely insensitive to any anisotropy present in the system. Our proposal is to consider the dimensionless information screening length L, which can be de ned as L 1= . Our claim can be rephrased as the dimensionless information screening length L, which can be de ned via the OTOC, is always maximum at the quantum critical point. Moreover, for a theory passing through a Lifshitz-like critical point, given the number of spatial directions D ? which scales similarly as time and the number of the directions D which has an anisotropic scaling, 0, the conjecture regarding L can be restated as 2L 1 D ? + We will later see that such a bound can be justi ed from NEC and in our model this is saturated at the quantum critical point g = gc. In a similar spirit, [18] points out a bound on the butter y velocity for an isotropic space with di erent warp factors appearing along the r, t directions, gtt(r); grr(r). Since in our case gtt(r)grr(r) = 1, we always saturate their bound. The paper is organized as follows. In section 2, we present the holographic model and its main, and known, transport features. In section 3, we study the onset of quantum chaos in the model and in particular the butter y velocity and the related conjectured bound. Conclusions are reached in section 4. In appendices A and B we provide more technical details about our computations. { 3 { We begin by reviewing the holographic model of [7, 8] which exhibits a QPT from a topologically non-trivial Weyl semimetal to a trivial insulating phase. Although the boundary theory exhibiting this topological transition in eq. (2.1) is a free theory, the holographic bulk theory strictly describes a strongly correlated system. The hope is that they share the same set of symmetries, thereby capturing the essential properties of the phase transition, if not all the details of the transport pertaining to interacting physics. Note that this is a phase transition in a certain topological invariant (such as Chern number) and not in the symmetries; thus, one can not probe it through the free energy density as it never depends on any topological term in the action. The order parameter is represented by the anomalous Hall conductance, AHE which is zero in the trivial gapped phase and nite in HJEP07(218)49 the Weyl semimetal phase. 2.1 Weyl semimetals Weyl semimetals are a class of three dimensional topological materials characterized by (point) singularities in the Brillouin zone (BZ) at which the band gap is zero. This peculiar property gives rise to exotic transport phenomena (see [20] for a comprehensive review). Quasiparticle excitations near such band-touching points, also called Weyl nodes, can be described by (left- or right-handed) Weyl spinors. In a time-reversal symmetry broken insulator, the left- and right- Weyl nodes are separated in the BZ which can be controlled by a chiral or axial gauge potential, ~b. It is the interplay of this axial eld and the (chiral) mass of the spinor, M , that gives rise to di erent phases (see gure 1). Deep in the semimetal phase, b ( node-separation equal to [21] be = b (1 j~bj) is much larger and M simply renormalizes it causing a reduced M 2)1=2, where M M=b. On the other hand, for a larger M , renormalization by a weaker b reduces the gap to Me = b (M 2 1)1=2. Thus, the semimetal-insulator phase transition occurs at Mc capturing this physics is [22] O(1). The continuum description L = eB= 5 ~ ~b + M ) : Here the slash denotes contraction by Dirac gamma matrices, . The matrix i 0 1 2 3 allows one to project the Dirac spinors, , into the chiral sectors, 5 (1 ) . B is the electromagnetic gauge potential; without loss of generality [23], we choose the axial gauge potential to be ~b = b e^z. The axial symmetry, however, is anomalous can be seen [24] in the response of the axial current, J~5 ~ b e (dJ5 6= 0), leading to a non-conservation of the number of particles of given chirality. This dA~, that is the anomalous Hall conductance, AHE be . This clearly vanishes in the insulating phase, that is for su ciently large M . The mass term and the axial term act as relevant deformations. Thus, with increasing M , the theory moves from UV to IR thereby traversing through a xed point at Mc. (2.1) 5 = L;R = { 4 { Now we turn to the holographic model of the above phase transition. The bulk action takes the form ( xing 2G2N = L = 1, where GN is Newton's constant, and L the AdS radius): S = Z d5x p g R + 12 1 4 F 2 1 4 F52 + 3 A F 5 F 5 + 3 F F (D ) (D ) V ( ) The bulk elds are an electromagnetic vector U(1) gauge eld B with elds strength iqA , and the scalar potential is chosen to be V ( ) = m2j j2 + 2 j j4. Since the phase of the scalar eld is not a dynamical variable, with out loss of generality we assume it to be real. The mass of the eld, m = p ( d) controls the scaling dimension, , of the boundary operator corresponding to . Throughout the paper, we will use d as the space-time dimension of the boundary eld theory, occasionally denoting the boundary spatial dimension as D = d 1. From the mass deformation in eq. (2.1) and the above relation, it is clear that one needs to choose m2 = 3 (see [19] for di erent choices of m2 and [25, 26] for further studies of the model), such that the dual operator has conformal dimension = 3. Note that this imaginary mass is perfectly allowed within AdS/CFT since it is with in the Breitenlohner-Freedman (BF) bound, m2 d2=4. The UV boundary conditions for the vector and scalar eld are chosen to be (2.2) (2.3) (2.4) (2.5) Although not necessary for computing the butter y velocity, we will rst discuss the behavior of zero-temperature solutions for understanding the various low-temperature limits. For nite temperature, we assume the presence of a black hole horizon at r = r0 such that f (r0) = 0. For the zero temperature background there is a Poincare horizon at r0 = 0, { 5 { lim r r!1 = M ; r!1 lim Az = b ; where both M and b represent a source for the corresponding dual operators. The parameter b can be thought as an axial magnetic eld that explicitly breaks the rotational SO(3) symmetry of the boundary to the SO(2) subgroup. From gure 2, one can see that this controls the e ective separation between Weyl nodes. On the contrary, the source M for the scalar eld is simply introducing the mass scale required by the physics of the problem. Note the presence of two more (bulk) free parameters in the problem; the quartic coupling, , controls the location of the quantum critical point (QCP) by changing the depth of the e ective potential of , and the charge q relates to the mixing between the operators dual to and A . Following [7, 8] we x these parameters to q = 1, and = 1=10, which xes Mc to 0:744. The generic solution of the system is given by the following ansatz dr2 f (r) ds2 = the bands cross) separated by 2~be in momentum. The band structure on the (top) right is that of a topologically trivial insulator with an explicit band-gap 2Me . At the QCP, the two Dirac cones merge together, giving rise to a Lifshitz xed point (black dot in the bottom gure) with a scaling anisotropy along the same direction as ~b. In the holographic picture, away from the QCP, the theory ows to two di erent types of (deep IR) near-horizon geometries, AdS5 (Weyl semimetal phase) or domain wall-AdS5 (trivial insulator phase). The gure shares some resemblances with those in [7, 19]. and f (r) = g(r). There are tree types of solutions at zero temperature | (i) insulating background (for M > Mc), (ii) critical background (for M = Mc), and (iii) semimetal background (for M < Mc). These solutions can be obtained by solving the equations of motion, the details of which we discuss in the appendix A. We quote the results here (up to leading order near the IR). Insulating background. | Similar to a zero-temperature superconductor, the near-horizon geometry of a topologically trivial insulator is an AdS5 domain-wall f (r) = 1 + r2 ; h(r) = r2 ; Az(r) = a1r 1 ; (r) = + 1r 2 : (2.6) r 3 3 8 Here a1 is xed to 1 and 1 is treated as a shooting parameter. Exponents 1;2 can be expressed as functions of (m; ; q), and are (2:69; 0:29) for our choice of parameters. Thus, the near-horizon value of Az is always zero, and that of is p3= (for = 1=10, it is (r0) ' 5:477). parametrized by 0 , Critical background. | This solution is exact and displays an anistropic Lifshitz-like scaling f (r) = f0r2 ; h(r) = h0r2 0 ; Az(r) = r 0 ; (r) = 0 : (2.7) The scaling anisotropy is explicitely induced by the source of the axial gauge eld A , hence is along the direction of ~b. The parameters (f0; h0; 0; 0) are determined by x{ 6 { ing (m; ; q). For the parameter choice mentioned previously, we have (f0; h0; 0; 0) ' (1:468; 0:344; 0:407; 0:947). From the zero-temperature equations of motion, it can be shown that 0 = the semimetal phase, which is simply AdS5 f (r) = r2 = h(r) ; Az(r) = a1 + a dependence is hidden in higher order terms. Note in this case, the near horizon solution of Az is nite; a1, however, (r0) vanishes. Figures 8 and 9 of appendix A provides the full A(r) and (r) functions for various values of M . The apparent deviations of A(r0) and (r0) from the IR asymptotes described above owes to the fact that we obtain the solutions for a small but nite temperature up to order O(T ), where T treat M and T as the free parameters in the theory to control the phase transition. T =b. We will 2.3 Anomalous transport As mentioned before, the order parameter for the QPT is the anomalous Hall conductivity. The DC, limit of all the conductivities can be extracted from (for both zero and nite temperatures) horizon data as follows AHE = 8 Az(r0) ; ? = ph(r0) ; = g(r0) ph(r0) : (2.9) Here is just a short hand for zz and `?' refers to the conductivity matrix elements, xx; yy, and should not be mistaken for the transverse conductivity. In gures 3 and 4, we plot the above conductivities as functions of M , for various temperatures T . We discuss them individually, starting from their zero-temperature behavior. In order not to sacri ce numerical stability, we con ne our lowest temperature value to T = 0:005 and treat it as zero temperature. diag ' Note that AHE Az(r0), and from the discussion of the zero-temperature solutions, we see AHE is nite only for M < Mc. A more physical picture could be that since in the IR, the axial gauge eld is completely screened [27], there are no degrees of freedom that could be coupled to it and hence, it can not be probed any further. As the temperature is increased, the sharp phase transition slowly becomes a cross-over. At zero-temperature, the onset of the semimetal phase is well tted by AHE / Mc M 0:21. For M = 0 (or, M = 0), the near-horizon geometry is the deformed AdS5 background of eq. (2.8). With our choice of normalizations, for low temperatures, g(r0) = h(r0) = 2T 2 and hence, T , which clearly vanishes at T = 0. The subscript `diag' collectively refers to all the diagonal components of the conductivity matrix, xx; yy; zz. There are two features of diag of interest. First, for vanishing b (or, M 1) the near-horizon geometry is the domain-wall AdS5 geometry of eq. (2.6), which makes diag ' c T , where c < 1 and { 7 { xy) as a function of the dimensionless mass parameter M for temperatures T = 0:1; 0:05; 0:005 (from green to orange). Note for a very low temperature the conductivity sharply drops to zero at a critical value, Mc 0:74. This marks the semimetal-insulator topological phase transition. 0.75 0.5 0.25 0 0 0.5 1 ¯ M 1.5 2 0.5 1.5 2 1 ¯ M M for temperatures T = 0:1; 0:05; 0:005 (from green to orange). independent of temperature. This is due to the fact that it is a phase transition between a semimetal-insulator transition and some degrees of freedom are now gapped out in the trivial phase. The reason why the conductivity is still nite in the insulating phase can be understood by computing the ratio of the gapped to un-gapped degrees of freedom [19], which eventually becomes a statement about the geometry or more precisely about the holographic a-theorem [28]. This ratio can be made to vanish by controlling m2 and . Second, and the most relevant for our discussion, is the fact that at the critical point, there { 8 { ε 4πη 0.5 s 0.75 0.25 0 0 1 ¯ M 0.5 1.5 2 1 ¯ M (Left) The anisotropy parameter "0 evaluated at the horizon for various T = 0:1; 0:05; 0:005 (from green to orange). (Right) Viscosity to entropy ratio, 4 k=s along the anisotropic direction for T = 0:005; 0:05; 0:1. The viscosity is given in terms of the horizon data as k = g2(r0)=ph(r0) [29]. The violation of the KSS bound is evident. On the contrary the ratio along the isotropic direction saturates exactly the KSS bound 1=4 and it is not shown here. are strong divergences at zero temperature. This can be attributed to the anisotropy of the critical point. For convenience, we de ne the ratio "0 at the horizon (also see gure 5a), as the measure of spatial anisotropy along the z direction at the horizon. More precisely, from the expressions of the diag in eq. (2.9), one can see that the ratio of the two at zero temperature becomes "0 h(r0) g(r0) 1 ? = h(r0) g(r0) r 0 2( 0 1) ; 2 HJEP07(218)49 (2.10) (2.11) which clearly diverges at the quantum critical point Mc. Another way of achieving the same conclusion is to analyze the AC conductivities [30]. From there, or simply from eq. (2.11), we can indeed conclude that ? = later see that this ratio "0 plays a key role in the behaviour of the butter y velocity. In some sense, such a result is not surprising [11, 12] since in theories with anisotropic scalings, one also observes a violation of the KSS bound [13, 14, 31]. As shown in [29], in the model we consider, the viscosity along the anisotropic direction violates the KSS bound (see gure 5b). It is important to note that the ratio between the ? quantities and their relatives is always xed by the anisotropic parameter de ned previously, !2( 0 1), which blows up at the DC limit. We will ? = ? = 1 + "0 : (2.12) We will next see that this will still be true for the butter y velocities vB2 and will ultimately be responsible for the violation of the maximization hypothesis. We show the behavior of the anisotropy parameter "0 is a function of M in gure 5a. As already discussed, the { 9 { anisotropy parameter is peaked around the quantum critical point and it blows up at T = 0 following eq. (2.11). 3 Quantum chaos & universality In this section, we compute the butter y velocity for the above holographic model. After obtaining a general expression of vB in terms of the near-horizon data for a given background, we (numerically) solve it near the quantum phase transition. Consider an anisotropic black brane metric dr2 f (r) (not to be confused with viscosity) counts the number of di erent warp factors, h( )(r), present in the = f~x( )g sub-manifold of the above background; thus, D = P d , where d = dim( ). The growth of the commutator in eq. (1.1) can be studied in holography by perturbing a black hole with a localized operator V(~x; t) [32, 33]. After a su ciently long time, (t > tr = ) the backreaction of this perturbation grows enormously, giving rise to a shockwave pro le, (~x; t), spreading at a speed vB. Before the perturbation has been completely scrambled (t < ts + j~xj=vB), the OTOC behaves as appendix B we solve the shock-pro le for the above background and obtain the butter y (~x; t)2. In velocities for an anisotropic AdS background. Note that in an anisotropic background, the velocity of the shockwave-front will depend on the spatial sector , and the full pro le (~x; t) can be approximated as a product of the shock-pro le of each sector. Doing so, we obtain Note that 1= de nes a theory-dependent, dimensionless IR length-scale in the problem, a screen length over which the shock-pro le (exponentially) decays, see eq. (B.14). This quantity plays an important role in our discussion and below we analyze this further. An alternative way to express this is through the following near-horizon quantities | surface gravity, = 2 T , and the area density of the r-slices, which relates the horizon with the entropy density of our dual QFT. We de ne the density of an r-slice which is simply proportional to the area of the spatial surface, A2(r) h(d )(r). Thus is 2 = log A r=r0 : (3.2) (3.3) For the holographic model considered in the previous section, we have one anisotropic direction z, that is, two butter y velocities. The velocity along the z-axis is denoted v ? and that on the xy-plane is denoted v . Now we use eq. (3.2) to obtain the butter y velocities for the background in eq. (2.5). Since this a holographic theory, the Lyapunov exponent naturally saturates the Maldacena bound [ 34 ], L = 2 = . In the unit of ~ = 1 = kB, the maximal Lyapunov exponent is equal to surface gravity, L = ; however, to avoid ambiguity relating the source of the thermal factor, we continue distinguishing them and write v ? = 2 = 2 pg1 g2 + g1 ; h2 2h1 v = Here we have used the near-horizon expansion of the metric functions, g(r) = g1 +g2(r r0) and h(r) = h1 + h2(r r0) discussed in appendix A, which involves Az(r0) Az1 and 1. Also, we have set the horizon radius to r0 = 1. As discussed in the previous section, the boundary theory is described by two dimensionless parameters, (M ; T ). In turn, this xes two near-horizon quantities, ( 1; Az1). All other IR variables are functions of (M ; T ), through ( 1; Az1). In gure 6 we numerically obtain the behavior of the butter y velocities. Although, as noted in [35], there is a characteristic behavior of vBs near the critical point; however, there is a clear departure from the result of [ 1 ] since the velocity along the anisotropic direction seems to attain a local minimum around the critical point, instead of a local maximum. The apparent inability of v to attain a maximum can be traced back to the anisotropic scaling. As before, this can be seen from the ratio, 2 v v ?2 = h(r0) g(r0) = 1 + "0 : (3.4) (3.5) (3.6) ? Since we observe nite v2 at g = gc, the divergence of this ratio at the critical point causes v to vanish. In other words, it is the length scale appearing in the formula of the butter y velocity that sources the deviation from the maximization behavior. Hence, modulo this B length scale, v( ) maximizes only when is minimized. Hence, if we consider the dimensionless information screening length L 1= instead, perhaps a universal statement can be made irrespective of the anisotropic scaling of the QPT. In this regard, we conjecture that L, and not the butter y velocity vB, maximizes across a quantum phase transition. Notice that in the isotropic case, the two statements are perfectly equivalent, and therefore the previously conjectured bound holds. Before discussing this more generally, we analyze the asymptotic limits of 2 in our system, using eq. (3.5) as a guide. Firstly, at M = 0, since there is no perturbation, we have UV = p of 6 is simply twice the spatial-dimension of the boundary CFT, 2D = 2 P d , which also xes the butter y velocity of a d-dimensional Schwarzschild black hole background [37]. At zero temperature, as M is increased, until one crosses Mc, there is no condensate, causing 2 to stay unchanged. At the critical point (using eq. (2.7) for the critical background) 6 ' 2:45. The factor we have c = (4 + 2 0)1=2 observes no transition in 2 causes 2 to sharply decrease at the critical point. For an isotropic system ( 0 = 1), one ' 2:19. As discussed before, NEC forces 0 < 1. In turn this . This sharp transition at the critical point for 0 6= 1 smears out becoming a cross-over behavior at nite temperature. A nal question is whether or not 2 monotonically decreases after the transition or if it increases. The IR asymptotic 2We thank Viktor Jahnke for pointing this out. This bound was observed to be violated in [11, 12]. 0.5 1.5 2 0.5 show the QCP. As one lowers the temperature the behavior of vBs near the critical point becomes increasingly non-analytic. Note the longitudinal (w.r.t. anisotropy direction) butter y velocity behaves exactly opposite to its maximization observed in [ 1 ]. The vB values have been normalized by their asymptotic values at M = 0, that is, 2=p6. This is obtained from eq. (3.5), and is equal to the bound in [ 36 ], vB2 = (D + 1)=2D, which is clearly violated2 by v? at larger M . value of 2, using the data of eq. (2.6), is larger than c; in fact it is bound to be larger than IR = (6 + 9=4 )1=2 ' 5:34. Clearly this is UV as well since is always positive. At nite temperature this asymptotic value softens but stays larger than the critical value for low enough temperature. We plot the behavior of in gure 7 which conforms to our inference and conforms to c UV IR or, L c L UV L IR : (3.7) Now, in the spirit of [19], we attempt to understand whether this conclusion remains valid if the boundary operator assumes any other scaling dimension. This discussion is con ned just to the insulating phase since the scalar deformation operator condenses only for large M . In other words, when the second- or higher- order terms in in eq. (3.5). We focus on the behavior of at low temperature, and when M 2 are turned on Mc 1, so that we can simplify our treatment by using the scalar hair 1 as a perturbation parameter. Also, since away from the critical point, behaves analytically and monotonically so as to the QPT, we rst consider m2 only. At this order, 2 6 establish our lower-bound conjecture, it su ces to justify that starts increasing as one enters slightly into the insulating phase. The coe cient of O( 21) term is simply the e ective mass of the scalar hair, me2 = m2 + gzzq2Az2. Since at low temperature gzz = 1=h1 ! 0 at m2 21=2, and only for m2 < 0 one has increasing . Recall [38] that the mass of a bulk scalar eld is xed by the scaling dimension of the dual boundary operator as m2 = ( d). The BF instability prevents this mass from becoming smaller than mBF = conjecture, m2 < 0 is true as long as d2=3 (in this case, mBF = 4). For our < d, or the perturbation is relevant. It should 0.44 0.41 L T = 0:1; 0:05; 0:005 (from green to orange). In the IR limit it asymptotes to L IR and in the UV this is L UV. The inset zooms into the behavior around the critical point. For low temperature L maximizes around the critical temperature and reaches the maximum, L c = (4 + 2 0) 1=2. Since NEC ensures 0 < 1, thus L c is always larger than L IR. In the text we argue for this maximum to be a universal property. be noted that this is a fundamental requirement in order to generate a QPT, since by perturbing a UV with an irrelevant operator, one can never generate a non-trivial RG ow towards an IR xed point. This is indeed the case as noted in the numerical studies of [19]. Thus, irrespective of the scaling dimension of the boundary deformation operator, one can de ne a lower bound on the length scale of information scrambling, which is xed by the CFTd. For a non-relativistic CFTd with a scaling anisotropy 0, along a D -dimensional sub-space (D = D D?), the upper bound is (using eq. (3.3) for a generic background) 2L 1 D + ( 0 1) D 2Lc ; (3.8) and the equality is saturated exactly at the quantum critical point,3 g = gc as illustrated in the gure above. Note that ultimately it is the NEC that restricts 0 to be less than one, and 3Since the anisotropic geometry turns out to be the critical geometry in the above model, the saturation happens at the QCP leading to the violation of the maximization-result. However, a system exhibiting such geometries in the UV or IR might saturate this bound away from the QCP. Thus, the signi cance of the bound should not necessarily be attached to quantum criticality but rather should be seen more as a universal feature of the near-horizon IR geometry. We thank Elias Kiritsis for discussing this issue. hence, makes the critical value Lc larger as compared to any other asymptotic value. In the case of isotropy, the maximum on the information screening length L becomes translated to the maximum of the butter y velocity vB since vB L L. Nevertheless, as we showed, in the presence of anisotropy ( 0 6= 1), the statement about the butter y velocity does not hold anymore and it has to be replaced by the behavior of the dimensionless information screening length L. 4 Throughout this work, we studied the onset of quantum chaos on an anisotropic quantum phase transition in a holographic bottom-up model. In particular, we focused on the behavior of the butter y velocities in the quantum critical region and across the quantum phase transition. We observed a disagreement with the results proposed in [ 1 ]. More precisely, the butter y velocity along the anisotropic direction does not develop a maximum but rather a minimum at the quantum critical point. We reiterate the similarity of our conclusions with the violation of the Kovtun-Son-Starinets (KSS) lower bound on the viscosity to entropy density ratio [13, 14]. In either cases, the presence of the anisotropic scaling, 0 seems to play an identical role. The viscosities have indeed been computed [29] within the holographic model we considered and, as expected and already mentioned, the =s ratio along the anisotropic direction violates the KSS bound, recall gure 5b. As a remedy, we propose an improved conjecture which also holds in the presence of anisotropy, and is stated in eq. (3.8). This involves a length scale, L, from the bulk perspective which can be computed using eq. (3.3). For the boundary theory this may be indirectly extracted by measuring the ballistic growth of a local perturbation through the OTOC and combining this with the measurement of various transport properties such as viscosity or conductivity along speci c anisotropic directions. This is needed since the factors g(r0) or h(r0) can only be made relevant to the boundary theory through these quantities, such as in eq. (2.9). In an anisotropic case, we observe L c L UV L IR; however for the isotropic case we do not expect L to have a local maximum at the critical point, that is L c = L UV. It would be interesting to understand the physics behind this L more precisely, especially to see if the emergence of this length scale in a strongly correlated theory can be better understood without making any reference to AdS/CFT. Acknowledgments We thank Panagiotis Betzios, Alessio Celi, Thomas Faulkner, Karl Landsteiner, Yan Liu, Napat Poovuttikul, Valentina Giangreco Puletti, for useful discussions and comments about this work. We thank Ben Craps, Dimitrios Giataganas, Viktor Jahnke and Elias Kiritsis for valuable and constructive comments on the rst version of this paper. We are grateful to Wei-Jia Li for reading a preliminary version of the draft. We acknowledge support from Center for Emergent Superconductivity, a DOE Energy Frontier Research Center, Grant No. DE-AC0298CH1088. We also thank the NSF DMR-1461952 for partial funding of this project. MB is supported in part by the Advanced ERC grant SM-grav, No 669288. MB would like to thank Marianna Siouti for the unconditional support. MB would like to thank University of Iceland for the \warm" hospitality during the completion of this work and Enartia Headquarters for the stimulating and creative environment that accompanied the writing of this manuscript. A The holographic background We discuss some more details about the gravitational background here and some aspects of the pertaining numerics. We follow closely [8]. The equations of motions derived combining the action in eq. (2.2) with our ansatz in eq. (2.5) are (note in order to be consistent with HJEP07(218)49 the notations in Landsteiner et al. we have switched f ! u; g ! f ): Here the primes denote derivative with respect to the radial-coordinate. We want to nnumerically integrate the system of equations (A.1) from the horizon r = r0 to the boundary r = 1. In order to do so we rst try to nd the asymptotic behavior of the solutions near the IR boundary (horizon) and UV (conformal) boundary. Close to the UV boundary, the bulk elds have the following leading order asymptotic expansion: u = r2 + : : : ; f = r2 + : : : ; h = r2 + : : : ; Az = b + : : : ; = + : : : : (A.2) M r Note that we have rescaled the boundary values of the three di erent metric functions to unity, such that the boundary eld theory depends only on the following free parameters, T; b; M . The removal of the boundary values of the metric is achieved by invoking the following (three) scaling symmetries 1. (x; y) ! a(x; y); f ! a 2f ; 2. z ! az; h ! a 2h; Az ! a 1Az; 3. r ! ar; (t; x; y; z) ! (t; x; y; z)=a; (u; f; h) ! a2(u; f; h); Az ! aAz . Owing to there symmetries we only have two dimensionless scales, T and M , which control the entire of the solution space. The near-horizon expansion up to O(r r0) can be 1.0 0.8 at T = 0:05. The various colors (from blue to brown) are M = 0:66; 0:724; 0:736; 0:743; 0:757; 0:8. The phase transition can be seen from the a large shift og the near-horizon values of the bulk elds when M exceeds 0:744. HJEP07(218)49 1.2 1.0 Left: the values of (Az1; 1) for the horizon shooting. Center: the value of 1 in function of M . Right: the value of f1 in function of M . written as u ' 4 T (r r0) + u2 (r r0) ; Az ' Az1 + Az2 (r r0) ; f ' f1 + f2 (r r ' 1 + 2 (r r0) ; r0) : h ' h1 + h2 (r r0) ; (A.3) Here Az1 and 1 are the only free parameters, being controlled by the boundary data T and M . From now onward, we also set the horizon radius to r0 = 1. In summary, while the horizon data are (T; r0; f1; h1; Az1; 1), using the (three) scaling symmetries they get reduced to (T; Az1; 1). At the conformal boundary they take the form of (T; M; b). We can now use shooting to construct the numerical background on the 2D plane of (M ; T ). An example of the bulk pro les for the Az(r) and (r) elds is shown in gure 8. Butter y velocities in anisotropic backgrounds Here we set up the shock wave equation in a generic anisotropic (in the spatial eld theory directions) background with constant curvature. For this we closely follow the derivations presented in [ 2, 33, 39 ]. Consider the following d-dimensional background with a black hole counts the number of di erent warp factors, h( )(r), present in the manifold of the above background. The treatment of Sfetsos con nes to = f~x( )g sub= 1, however, here we are interested in the case when > 1. The black hole (or black brane) horizon is assumed to be located r0, such that f (r0) = 0 with non-vanishing a(r0) and b(r0). The temperature of the black hole is, 4 T = 2 gravity. The background is assumed to be sourced by a stress tensor, T (0). For further = f 0(r0)pa(r0) b(r0), here is the surface simpli cations we rst move to tortoise coordinate, In the last line we've done a near-horizon expansion of r which is justi ed since r (r0) blows up. Next we move to Kruskal coordinate by exponentiating the null coordinates of t r space, u = e2 T (r t) ; v = e2 T (r +t) =) r = ln(uv) ; t = 1 4 T 1 4 T ln v u In this coordinate the horizon is at uv = 0 and the boundary is at uv = 1. The black hole singularity is at uv = 1. The above relation can be used to express the background in Kruskal coordinates ds(20) = 2A(uv)dudv + X h( )(uv)d~x(2 ) ; 2A(uv) = a((2r)fT()r2) e 4 T r : ds(20) = a(r)f (r) dr2 dt2 + r (r) = Z r dr0 r0 f (r0)pa(r0) b(r0) ; 1 4 T ln r r0 r0 : (B.1) (B.2) (B.3) (B.4) (B.5) (B.7) We will need the following relations later, h0(0) = r0h0(r0), and using near-horizon expansion of f (r) we have, 2A(0) = (2 rT0 )2 a(r0)f 0(r0) and 2A0(0) = (2 rT0 )2 (a(r)f 0(r))0jr0. 2 One can think of the above background is being generated from stress tensor T (0) by using Einstein equation, G(0) = 8 T (0), where G(0) is the Einstein tensor corresponding to ds(20) and T (0) = Tu(0v) dudv + Tu(0u) du2 + Tv(v0) dv2 + T (0) d~x(2 ) + Tu(0) dudx : (B.6) Starting from eq. (B.5) we now obtain the butter y velocity. For that we perturb our background with a point particle that is released from ~x = 0 at time tw in the past. The particle is localized onn the u = 0 horizon but moves in the direction of v with light speed. For late time, tw > its energy density can be written as [40] Tupu = E0e 2 tw (u) (~x) v, we replace dv ! dv perturbed metric and the stress tensor is (along with T p) We want to compute the backreaction of this stress tensor on our background. This can be done perturbatively for a small energy density. One can start with an ansatz solution that v gets shifted by (~x) only for u > 0, v ! v + (u) (~x). This new geometry is the shockwave geometry and we want to solve for (~x), that is the shockwave. By relabeling (u) (~x)du. Plugging this in the above metric we obtain the Since ds(21) doesn't generate nite Einstein tensor, G(u1v) = 0, we can demand (u)Tv(v0) = 0 = (u)G(v0v). There remains only one relevant Einstein equation that gives rise to the shock wave equation (which is subject to the previous contstraint) Or, X A(0) h( )(0) ( ) dim( ) h0( )(0) ! 2h( )(0) G(u1u) = 8 Tupu (u) (~x)G(u0v) : (~x) = 8 E0e 2 tw (~x) ; =) ( ) where, M 2 = h( )(0) X dim( ) h0( )(0) 2A(0)h( )(0) (B.8) (B.9) (B.10) (B.11) (B.12) (B.13) (B.14) HJEP07(218)49 ( ) In the second last line, assuming linear order, we have divided the solution space into di erent anisotropy sectors, labeled by . Clearly, for the isotropic case, = 1 = , one recovers the shock equations of [2, 33], with dim( ) = d 2. Also if the eld theory living at a constant r; t-slice is curved then the shock front is no longer planar but depends on the curvature of the spatial slice, thus its dynamics involves curved space Laplacian, pg( ) g( ) @ , rather than the at space Laplacian used above. This a ects the spatial-pro le of the shock but not its speed, that is the butter y velocity [ 41 ]. We want to solve this equation, which is equivalent to solving the Green's function of the at space Laplacian. At very long distance (x M 1) the solution becomes (~x( ); t) e 2 (tw t) M j~x( )j j~x( )j 2 d 3 : Note that the factor 2 = is the Lyapunov exponent for Einstein gravity. Note that M 1 de nes the screening length-scale in the problem and ter y velocity, as can be seen in the above equation, is a ratio of these two scales L1 de nes the timescale. The but M(2 ) = h( )(r)b(r)f 0(r) X dim( ) h0( )(r) 4h( )(r) r0 : (B.15) Here vB( ) is the velocity corresponding to the shockwave propagating in the subspace. In de ning M( ) we have used the expression in eq. (B.13) and switched from Kruskal coordinates to usual Schwarzschild coordinates using the identities discussed previously. For simplicity, we set a(r) = b(r) = 1 and rewrite M(2 ) in terms of a dimensionless quantity , such that ( ) h( )(r) = T X dim( ) h0( )(r) h( )(r) r0 : (B.16) Open Access. 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Matteo Baggioli, Bikash Padhi, Philip W. Phillips, Chandan Setty. Conjecture on the butterfly velocity across a quantum phase transition, Journal of High Energy Physics, 2018, 49, DOI: 10.1007/JHEP07(2018)049