When things stop falling, chaos is suppressed

This note is devoted to the investigation of Susskind's proposal(arXiv:1802.01198) concerning the correspondence between the operator growth in chaotic theories and the radial momenta of the particle falling in the AdS black hole. We study this proposal and consider the simple example of an operator with the global charge described by the charged particle falling to the Reissner-Nordstrom-AdS black hole. Different charges of the particle lead to qualitatively different behavior of the particle momenta and consequently change of the operator size behavior. This holographic result is supported by different examples of chaotic models at a finite chemical potential where the suppression of chaos has been observed.


Introduction
In the recent years the quantum chaotic systems and their connection to the gravity has attained a lot of attention. The systematic investigation of this correspondence revealed new concepts like scrambling, quantum complexity [2]- [5]. The notion of operator size, roughly speaking, describes how many operators taken as "basis operators" are involved in the description of evolution after perturbation. It was shown, that in the chaotic systems this growth is exponential and its power is called the Lyapunov exponent. In the AdS/CFT correspondence, the operators in the quantum theory can be related to the particle in the bulk and in the context of "GR=QM" proposal it was found that this exponential operator size growth on the gravity side corresponds to the (exponential) growth of the radial momentum of the particle falling under the black hole horizon [1,2].
In this paper we investigate the above mentioned GR=QM proposal [1,2] and the correspondence between the operator growth in chaotic holographic theories and the radial particle momenta by a simple example of an operator with a global charge described by a charged particle falling in the Reissner-Nordstrom-AdS black hole. We find that the different charges of the particle, lead to qualitatively different behaviour of the particle. We interpret this difference as a supression of a chaotic behaviour in a systems with finite chemical potential. Also we discuss the quantum mechanical model exhibiting the similar behaviour.
The paper is organized as follows. First we introduce the main ingredients of the correspondence. In the third section we compute the evolution of momentum growth for charged particles numerically solving equations of motion. In the fourth section we discuss the SYK model at finite chemical potential and future directions of investigation. We end with the discussion.

Charge falling in the Reissner-Nordstrom-AdS black hole
Let us consider the planar d-dimensional Reissner-Nordstrom-AdS black hole with the outer horizon at z h . This geometry is given by the metric where the gauge field supporting this solution is and parameters M , Q are defined to be Here T is the temperature of the black hole and µ is the chemical potential in the dual quantum field theory living on the boundary z = 0. The proposal of Susskind [1] states the correspondence between radial component of momentum p z of the neutral particle falling under the black hole horizon and the growth of the operator size. This correspondence is formulated as where the operator size can be defined as follows. If we define the perturbation of a boundary theory by the time-evolving operator W (t) and expand it in the basis of some elementary operators ψ as then s is called the operator size (see for example [4,5]). In [1,2] it was shown, that in a theory dual to the d-dimensional black hole, the neutral operator W corresponds to the massive particle in the bulk. At a late times this momenta (i.e. when the particle is in the near horizon zone) exhibits exponential growth and consequently the operator size also growth exponentially, where β is the inverse temperature.
In this framework it is natural to assume, that charged particle falling in the charged black hole background corresponds to evolution of the charged operator after a perturbation of the system at finite chemical potential. The action of the charged particle has the form where the momentum is defined as In the following we take m = 1 without loss of generality. If we take charge q of the same sign as Q (i.e. positive) then after some critical value q crit the qualitative behaviour of the particle changes. If q < q crit particle falls under the horizon, while for q > q crit it oscillates between the boundary and horizon.

Charge stops the fall
Let us take a closer look on the motion of the particle with a charge (2.11) in the RN-AdS black hole given by (2.1). Taking the particle worldline parametrization z = z(t) we get the action The energy E is We consider the case of the positive particle charge q > 0. When the energy is negative, the particle oscillates between two turning points z * ,± . In this case the charge of the particle is larger than the critical one given by where A t is given by (2.3). We take d = 3 for definiteness and solve the equations of motion corresponding to (2.11) numerically and plot the particle worldline z(t) in Fig.1 for different values of the charge.
The radial momentum of the charged particle corresponding to the metric (2.1) is given by In Fig.2 and Fig.3 we plot the time evolution of the particle momentum for the charge above and below the critical value.  Note the following details concerning the momentum and, consequently, operator size growth following from the Fig.3 and Fig.2: • The late time growth of the particle momentum for all q < q crit is independent on q. The onset of this regime occurs at a later times for a smaller values of 1 − q/q crit . This implies the scrambling time growth. This effect is present for example in the random circuit models (see discussion for more details).
• For the critical charge the operator size stops evolving after some time.
• Above the critical value of the charge, the initial period of rapid linear growth is replaced by slow nonlinear growth, and the particle reaches its maximum velocity. Then a period of nonlinear deceleration occurs, which goes into fast linear deceleration and after a short nonlinear deceleration the particle reaches its minimum velocity. After slow growth, the particle reaches again a rapid linear growth, and so on. This oscillating behavour can be seen as the localization of the operator on some RG scale.
• Note, that for the values of charge above the critical the momentum value could be negative. It is natural to assume that the operator size corresponds to the absolute value of the momentum.
For the positive charge the repulsion between the black hole and particle is absent. The late time exponential asymptotic for the operator size evolution is the same as for the case of the neutral particle. However similarly to the [2] the size value is large in comparison to the neutral particle case.

Discussion and connection with SYK-like models
In this paper we have obtained that the chemical potential lead to the operator size growth suppression for the charged operators. Our results rely on the holographic proposal of [1]. Now let us briefly list a few results observed in melonic theories at a finite chemical potential that support this observation. In [6] it was shown, that in the SYK model with the complex fermions introduction of the chemical potential lead to the effect of chaos damping. There is a little difference with our results. In our case the full suppression occurs only at the infinite µ in contrast to the full suppression in the complex fermions SYK model appeared at finite value of µ. This may be related with the large q approximation used in [6].
It is known [7] that a special large D limit of matrix quantum mechanics is dominated by the melonic diagrams. This fermionic model with a mass term playing the role of the chemical potential exhibits sharp phase transition that may be related to a chaos damping.
Another interesting model exhibiting some of these features is the random circuit model studied in [8,9]. These works show, that the operator scrambling time becomes exponentially large in chemical potential if we consider the correlators with some conservation law. Also there could be some relation between the localization in random circuits constrained by U (1) conservation considered in [10] on one side and localization of operator size described by oscillating bulk particle.
All these are in the accordance with the holographic example of GR = QM discussed in this paper.
To conclude, let us briefly summarize the results of the paper and future directions to investigate. In this paper we have considered the holographic model of the operator growth at a finite chemical potential. This model consists of the charged particle moving in the background of the charged black hole. This holographic model shows that the chemical potential leads to the chaos suppression above the critical charge. This is consistent with the chaos suppression observed in different melonic and random circuit theories mentioned above. Also note a few future possible research directions. The first one is to elaborate analytical estimate for the charged operators and the quantitative tests with some quantum theory at finite temperature. For example this includes a generalization of results from [5]. Another direction is to explore holographic calculations of the size growth to general backgrounds including hyperscaling-violating geometries and the dS case.