1 Introduction

Semi-classical bound on Lyapunov exponent

and acoustic Hawking radiation in matrix model

Department of Physics, Shizuoka University,

836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan

A classical particle motion in an inverse harmonic potential shows the exponential sensitivity of the initial condition, and the Lyapunov exponent is uniquely fixed by the shape of the potential. Hence, if we naively apply the bound on the Lyapunov exponent , it predicts the existence of the bound on temperature (the lowest temperature) in this system. We consider non-relativistic free fermions in the inverse harmonic potential ( matrix model), and show that thermal radiation with the lowest temperature is induced quantum mechanically similar to the Hawking radiation. This radiation is related to the instability of the fermi sea through the tunneling in the bosonic non-critical string theory, and it is also related to acoustic Hawking radiation by regarding the fermions as a fermi fluid. Thus the inverse harmonic potential may be regarded as a thermal bath with the lowest temperature and the temperature bound may be saturated.

## 1 Introduction

Understanding the quantum gravity is the one of the most important problem in theoretical physics. Through the developments in the gauge/gravity correspondence [1, 2], many people expect that large- gauge theories may illuminate the natures of the quantum gravity. However only special classes of the large- gauge theories which possess certain properties may describe the gravity, and understanding what are the essential properties in the gauge theories to have their gravity duals is a crucial question.

Recently the idea of the maximal Lyapunov exponent was proposed, and it might capture the one of the essence of the gauge/gravity correspondence [3]. The authors of [3] conjectured that thermal many-body quantum systems have an upper bound on the Lyapunov exponent:

 λL≤2πTℏ, (1.1)

where is temperature of the system and is the Lyapunov exponent. (We take in this article.) Particularly, if a field theory at a finite temperature has the dual black hole geometry, the gravity calculation predicts that the field theory should saturate this bound [4, 5]. (This would be related to the conjecture that the black hole may provide the fastest scrambler in nature [6].) This is called maximal Lyapunov exponent, and the properties of this bound is actively being studied. One remarkable example is the SYK model [7, 8]. We can show that this model saturates the bound from the field theory calculation, and now people are exploring what is the dual gravity of this model [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

The purpose of this article is understanding the mechanism of the appearance of the bound on the Lyapunov exponent in a semi-classical regime through a simple model. Although, in the recent studies of the chaos bound [3, 29, 30, 31, 32], the Lyapunov exponents defined through the out-of-time-ordered correlator (OTOC) [33] are mainly investigated, we focus on classical chaos and consider the Lyapunov exponent defined by

 δq(t)∼δq(0)exp(λLt) (1.2)

Here is the value of an observable at time and is the deviation of through the change of the initial condition by . Such an exponential sensitivity of the initial condition is a characteristic behavior of chaos, and it appears ubiquitously in nature. Since many studies on chaos have been developing in classical systems, it would be valuable to investigate the chaos bound in the semi-classical regime.

The simplest dynamical model in which we can observe the exponential sensitivity of the initial condition (1.2) would be the classical motion of a point particle in an inverse harmonic potential

 m¨x(t)=−V′(x),V(x)=−α2x2. (1.3)

Here the solution of the motion is given by

 x(t)=c1e√α/mt+c2e−√α/mt, (1.4)

where and are constants determined by the initial condition, and it shows the sensitivity of the initial condition (1.2) with Lyapunov exponent

 λL=√αm. (1.5)

Of course the motion (1.4) is too simple and not chaotic. However, if even such a simple model shows the chaos bound (1.1), we may gain some important insights into the bound in the actual chaotic systems. For this reason, we investigate this model.

If we apply the bound on the Lyapunov exponent (1.1) to the model (1.3), we obtain the relation

 T≥ℏ2π√αm. (1.6)

from (1.5). Since the right hand side is just a constant determined by the parameters in the model, this relation predicts the existence of the lower bound on temperature in this system,

 TL:=ℏ2π√αm. (1.7)

Although this temperature becomes zero in the strict classical limit , it might be meaningful in a semi-classical regime.

We will argue that indeed the lowest temperature plays an important role. To see it clearly, we consider non-relativistic free fermions in the inverse harmonic potential (1.3). This model is related to a one-dimensional matrix quantum mechanics called matrix model which describes a two dimensional gravity through the non-critical string theory (see, e.g. the review of [34, 35, 36]).

We consider the configuration of the fermions which describes the bosonic non-critical string theory (figure 1). (In the following arguments, the correspondence to the string theory is not important.) Although this configuration is stable in the strict classical limit, it is unstable via the quantum tunneling of the fermions. We will show that this instability causes a thermal flux with the temperature (1.7) in the asymptotic region which flows from around . This radiation is an analogous of the Hawking radiation [37, 38] in the sense that the system is not thermal classically whereas a thermal spectrum is induced quantum mechanically. Therefore the inverse harmonic potential can be regarded as a black body with the temperature for the observer in the asymptotic region, and the temperature bound (1.6) may be saturated.

We will also argue that this radiation is related to the acoustic Hawking radiation in a supersonic fluid [39]. (A related study in the context of condensed matter physics has been done in [40] and developed in [41, 42].) The collective motion of the fermions are described as a fermi fluid, and it classically obeys the hydrodynamic equations [43, 44, 45]. Hence, if the fluid velocity exceeds the speed of sound, acoustic Hawking radiation may be emitted. We will argue how the supersonic region can be realized in the matrix model, and show that the acoustic event horizon arises only at the tip of the inverse harmonic potential if the flow is stationary. Thus the radiation may arise from when we turn on the quantum corrections. We will show that the temperature of this acoustic Hawking radiation is always given by the lowest temperature , and argue the connection to the instability of the bosonic non-critical string theory.

Note that related works on the particle creations in the matrix model have been done in the context of the non-critical string theory [46, 47]. They include time dependent background and tachyon condensation [48, 49, 50, 51], and high energy scattering and black hole physics [52, 53, 54, 55, 56].

The organization of this article is as follows. In section 2, we argue the appearance of the thermal radiation with the temperature (1.7) in the matrix model in the configuration of the fermions corresponding to the bosonic non-critical string theory. In section 3, we show the connection between this radiation and acoustic Hawking radiation. A brief review on the derivation of the acoustic Hawking radiation is included. Section 4 is discussions and future problems.

## 2 The c=1 matrix model and the temperature bound

We consider the matrix model which is equivalent to non-relativistic free fermions. The fermions classically obey the equation of motion (1.3), and the Hamiltonian for the single fermion is given by

 ^H=−ℏ22m∂2∂x2−α2x2. (2.1)

This Hamiltonian is unbounded from below, and usually we introduce suitable boundaries so that the fermions are confined. These boundaries are irrelevant as far as we focus on phenomena around . In this section, we introduce infinite potential walls at .

We consider the configuration of the fermions which describes the two dimensional bosonic non-critical string theory. We put all the fermions on the left of the inverse harmonic potential, and tune the parameters and so that and the fermi energy () is close but below zero. We sketch this configuration in figure 1.

Now we show that this configuration involves thermal radiation with the temperature defined in (1.7). Obviously the above configuration cannot be stationary. The fermions will leak to the right side of the potential through the quantum tunneling. Such a tunneling effect is significant for the fermions close to the fermi surface. For a single fermion with energy , the tunneling probability can be calculated as [57],

 PT(E)=1exp(−2πℏ√mαE)+1. (2.2)

The derivation is shown in appendix A. This formula is suggestive. If we regard as energy, is identical to the Fermi-Dirac distribution with the temperature

 T=ℏ2π√αm (2.3)

which precisely agrees with the lowest temperature (1.7)111This tunneling probability is for a single particle, and the appearance of the Fermi-Dirac distribution is nothing to do with the fact that we are considering the fermions.. Thus the tunneling of the fermions might cause thermal radiation.

In order to establish this expectation, we evaluate the contributions of the tunneling to physical quantities and ask whether they are thermal or not. We will investigate the local density of an observable . First we will compute this quantity in the classical limit (). Then we will calculate the same quantity including the quantum effect and, by comparing these two results, we will evaluate the contributions of the tunneling.

Let us start from the calculation in the classical limit . In this limit, the fermion with negative energy is confined in the left region , where is the classical turning point. (See figure 1.) The WKB wave function for such a fermion is given by

 ϕE(x)=C(E)√p(E,x)1√2(eiℏ∫x−Ldyp(E,y)−e−iℏ∫x−Ldyp(E,y)) (2.4)

where is the classical momentum for the particle,

 p(E,x)=√2m(E+α2x2), (2.5)

and is the normalization factor satisfying

 |C(E)|2=(∫−x0(E)−Ldx|p(E,x)|)−1, (2.6)

so that is normalized to 1. Through the boundary condition , energy is quantized as

 ∫−x0(E)−Ldx|p(E,x)|=πℏn,(n=1,2,⋯), (2.7)

where we have ignored the fragment terms. By using this WKB wave function, we evaluate the expectation value of the observable for the fermions as

 ⟨O(^x,^p)⟩N= N∑i=1⟨i|O(^x,^p)|i⟩=∫−μEGdEρ(E)∫−x0(E)−Ldx|ϕE(x)|2O(x,p(E,x)) (2.8)

where is the ground state energy and is the energy density given by

 ρ(E)=∂n∂E=mπℏ∫−x0(E)−Ldx|p(E,x)| (2.9)

through the quantization condition (2.7). Here we have assumed that is sufficiently large and the spectrum is continuous. From (2.8), the density of the observable at position is given by

 ∫−μE0(x)dEρ(E)|ϕE(x)|2O(x,p(E,x))=mπℏ∫−μE0(x)dE|p(E,x)|O(x,p(E,x)). (2.10)

Here is the minimum energy at given . (See figure 1.)

Now we evaluate the quantum correction to the density (2.10). In the WKB wave function (2.4), can be regarded as the incoming wave toward the inverse harmonic potential, while can be regarded as the reflected wave. Then through the tunneling effect (2.2), a part of the incoming wave will penetrate to the right side of the potential, and the reflected wave will be decreased by the factor . Hence the WKB wave function (2.4) will be modified as

 ϕE(x)→C(E)√p(E,x)1√2(eiℏ∫x−Ldyp(E,y)−√1−PT(E)e−iℏ∫x−Ldyp(E,y)),(−L≪x≪−x0(E)). (2.11)

Note that this result is not reliable near the turning point , since the WKB approximation does not work. Also we have assumed that the boundary is sufficiently far, and we can ignore the further reflections there. This correction modifies the left hand side of (2.10), and the density of the observable becomes,

 mπℏ∫−μE0(x)dE|p(E,x)|(1−12PT(E))O(x,p(E,x)) (2.12)

By subtracting the classical result (2.10) from this result, we obtain the contributions of the quantum tunneling to the density of the observable as

 −m2πℏ∫−μE0(x)dE|p(E,x)|O(x,p(E,x))e−βLE+1=−m2πℏ∫∞μdE|p(−E,x)|O(x,p(−E,x))eβLE+1 (2.13)

where denotes the inverse temperature and we have approximated by assuming that we are considering the density in a far region222In this region, the tunneling of the particle with the energy is sufficiently suppressed. . We also flipped in the last step.

We can interpret the result (2.13) as follows. Due to the tunneling, the holes with positive energy appear in the fermi sea. They move to the left and contribute to the quantity . Importantly, (2.13) shows that these holes are thermally excited and obey the Fermi-Dirac distribution at the temperature . Therefore the quantum tunneling indeed induces thermal radiation.

For example, if we choose in (2.13), we obtain the energy density at in the far region,

 −m2πℏ∫∞0dE|p(−E,x)|−EeβLE+1=1(−x)TL48(1+O(TLαx2)) (2.14)

Here we have taken for simplicity. The obtained energy density is positive reflecting the fact that the holes carry the positive energies.

Our result indicates that the inverse harmonic potential plays a role of a black body at the temperature . This result reminds us the Hawking radiation [37, 38] in which the thermal excitation arises through the quantum effect even though the system is not thermal in the classical limit. Indeed we will see that the thermal excitation in the matrix model is related to the acoustic Hawking radiation in quantum fluid.

Notice that, since the thermal radiation at the temperature arises, the temperature bound (1.6) may be saturated in this system333This interpretation is subtle. Since the matrix model is free and integrable, normal thermalization may not occur. (In integrable systems, thermalization to the Generalized Gibbs ensemble (GGE) [58, 45] may occur. It may be interesting to consider the role of the temperature in terms of GGE.) Another issue is that, at late times, the system will settle down into the state in which the half of the fermions are in the left side of the potential and the other half are in the right. There the formula (2.13) would not work. Thus the argument of the temperature bound would be valid only for the initial stage of the time evolution starting from the meta-stable configuration sketched in figure 1 such that the formula (2.13) is reliable. .

## 3 Thermal radiation in the c=1 matrix model and acoustic Hawking radiation

We will show that the thermal radiation in the matrix model argued in the previous section is related to the acoustic Hawking radiation [39]. (Several arguments in this section have been done by [40] in the context of condensed matter physics.)

### 3.1 Acoustic Hawking radiation in the c=1 matrix model

As is well known, the fermions in the matrix model compose a two dimensional ideal fluid. The fermion density field and velocity field classically obey the following the continuity equation and the Euler equation with pressure [43, 44, 45],

 ∂tρ+∂x(ρv)=0,∂tv+∂x(12v2+ℏ2π22m2ρ2+1mV(x))=0. (3.1)

Here . (Many of the arguments in this section will work for general .) We ignore the infinite potential walls at in this section. Hence we can apply the story of the acoustic Hawking radiation to this system.

Here we briefly show the derivation of the acoustic Hawking radiation [39]. (See e.g. a review article [59].) We consider the following small expansion

 ρ(x,t)= ρ0(x,t)+ϵρ1(x,t)+ϵ2ρ2(x,t)+⋯, v(x,t)= v0(x,t)+ϵv1(x,t)+ϵ2v2(x,t)+⋯, (3.2)

Here we regard that and describe a background current, and and describe the phonons propagating on it, and and () are the higher corrections. We will see that when we turn on the quantum effect, and may show a thermal excitation if there is a supersonic region in the background.

#### 3.1.1 Acoustic geometry for phonons

In order to derive the acoustic Hawking radiation, we consider the equations for the phonons. By substituting the expansion (3.2) into the hydrodynamic equation (3.1) and expanding them with respect to , we will obtain the equations for and order by order. Especially, at order , we obtain the equations for and ,

 ∂tρ1+∂x(ρ0v1+v0ρ1)=0,∂tv1+∂x(v0v1+ℏ2π2ρ0m2ρ1)=0. (3.3)

Here we introduce the velocity potential such that . Then from the second equation, we obtain

 ρ1=m2ℏ2π2ρ0(∂tψ+v0∂xψ). (3.4)

By substituting this result to the first equation of (3.3), we obtain the wave equation for the velocity potential as

 0= ∂μ√−ggμν∂νψ (3.5)

where () is given by

 gμν=1γρ0(c2−v20v0v0−1), (3.6)

and denotes the speed of sound defined by

 c(x,t):=√∂p∂ρ=ℏπmρ0(x,t). (3.7)

Here is the pressure defined above (3.1). Note that in is an arbitrary constant, which cannot be fixed 444In two dimension, the wave equation (3.5) is invariant under a scale transformation, and we have the ambiguity .. This wave equation is identical to the wave equation of a scalar field on a curved manifold with the metric (3.6). For this reason, is called acoustic metric.

Importantly, if the fluid velocity is increasing along -direction and exceeds the speed of sound at a certain point, occurs there and the -component of the metric (3.6) vanishes similar to the event horizon of a black hole. Correspondingly, we can show that no phonon can propagate from the supersonic region to the subsonic region across the point . For this reason, the point is called acoustic event horizon. Then we expect that the phonon in the subsonic region () may be thermally excited similar to the Hawking radiation. We will confirm it later.

#### 3.1.2 Acoustic black holes and white holes

When does occur in our matrix model? In order to see it, it is convenient to define as

 P±(x,t):=mv(x,t)±ℏπρ(x,t). (3.8)

It means that

 ρ(x,t) =12πℏ(P+(x,t)−P−(x,t)),v(x,t)=12m(P+(x,t)+P−(x,t)). (3.9)

Here describe the boundaries of the droplet of the fermions in the phase space as shown in figure 2. (We have implicitly excluded the case in which the folds [35, 60] appear on the boundaries of the droplet.) Then from (3.7) and (3.9), we can rewrite and by using ,

 c±v0=±1mP0±(x,t), (3.10)

where we have expanded

 P±=P0±+ϵP1±+ϵP2±+⋯, (3.11)

corresponding to the expansion (3.2). Therefore occurs when or vanishes, and an acoustic event horizon may appear there.

Now we solve the fluid equation (3.1), and find the acoustic metric involving an acoustic event horizon. By substituting (3.9) to (3.1), we obtain equations for ,

 ∂tP++∂x(12mP2++V(x))=0,∂tP−+∂x(12mP2−+V(x))=0. (3.12)

By using the expansion (3.11) in these equations and considering terms, we obtain equations for which are just in (3.12). We assume that the background is stationary . Then the solution of these equations are given by

 P0±=σ√2m(E±−V(x)), (3.13)

where are constants and is or .

To realize the acoustic event horizon, or has to cross 0. We can easily show that it occurs only when

 P0+=√2mE++mαx2,P0−=σ√mαx, or P0+=σ√mαx,P0−=−√2mE−+mαx2, (3.14)

with positive and . Here the acoustic event horizon appears at in both cases. Then depending on the choice of , there are four possibilities. We call them as black hole (right), black hole (left), white hole (right) and white hole (left). See figure 3. These names refer to the ways of the classical propagation of the phonons, and (left) and (right) are related by parity . To see the phonon propagation, we consider the black hole (right) case as an example. As shown in figure 3, in this case are given by

 P0+=√2mE++mαx2,P0−=√mαx. (3.15)

From (3.10) (and the plots of and in figure 3), is the subsonic region () while is the supersonic region (). Then we expect that no phonon can propagate into from . Let us confirm it. The propagation of the phonons can be read off from the wave equation (3.5). By using new coordinates (called time-of-flight coordinates)

 z±:=t−m∫xdx′P0±(x′), (3.16)

the wave equation (3.5) becomes . Thus the solution is given by

 ψ=ψ+(z+)+ψ−(z−). (3.17)

Then become

 P1±=mv1±ℏπρ1=2m2P0±ψ′±(z±). (3.18)

This relation shows (the well-known result for experts) that the fluctuation of the boundaries of the droplet describes the excitations of the phonons and they propagate depending on .

In the black hole (right) case (3.15), always propagates toward the right direction, since is always positive in (3.16). On the other hand, we need to take care . Since becomes zero at and diverges there, covers only the left region () or the right region () as

 z−= t−√mαlogx+const.(x>0) = t−√mαlog(−x)+const.(x<0). (3.19)

Thus propagates to left in and to right in . Therefore both cannot propagate into from as we expected. This is similar to the black hole geometry in which no mode can propagate to the outside from the inside. For this reason we call this case as “black hole (right)”, by identifying the supersonic region () as the inside of the black hole555Note that this result is valid only for sufficiently small fluctuations. The ways of the propagation of the finite fluctuations of can be read off from the particle trajectories as shown in figure 4. Particularly if the fluctuations are large and they cross the separatrix , even the fluctuations in the supersonic region can propagate into the subsonic region. Besides, it is known that arbitrary non-stationary fluctuations develop to folds at late times. .

Lastly, we take a detour and discuss the white hole (right) case. In figure 3 (c), is given by

 P0+=−√mαx,P0−=−√2mE−+mαx2. (3.20)

In this case, from (3.10), is always negative and the background current flows from the right to left. Importantly the fluid velocity decelerates along the flow, and the supersonic flow in becomes subsonic at . As a result, in the subsonic region () cannot propagate into the supersonic region. This is an analogue of a white hole in fluid mechanics.

However, this conclusion is valid only for limit. is unstable against the perturbations and even tiny fluctuations in the subsonic region can propagate into the supersonic region as shown in figure 4.

Now we are ready to derive the acoustic Hawking radiation. We consider the black hole (right) case. In this case, the classical solution of is given by (3.17), and importantly the coordinate (3.19) covers only either or . This is an analogue of the tortoise coordinates in the black hole geometries, and is not suitable to define the vacuum of the quantized phonon.

A coordinate which is well defined on is a Kruskal-like coordinate , where we have defined “the surface gravity” 666 Up to an overall factor, the line element of the acoustic metric (3.6) can be written as , where . Then the surface gravity can be written as the familiar formula . Also is written as . The equations in this footnote work even for a general potential if the local maximum is at and the acoustic event horizon appears there. ,

 κ:=√αm. (3.21)

The vacuum of the phonon should be defined by using this coordinate. It implies that in the asymptotic region (large negative ) will be thermally excited at temperature

 T=ℏκ2π=ℏ2π√αm (3.22)

through the Bogoliubov transformation. Here the obtained temperature precisely agrees with the lowest temperature (1.7).

For example, we can calculate the expectation value of the energy momentum tensor of the phonon by using the two dimensional conformal field theory technique [62, 63, 64],

 ⟨Tz−z−⟩=−cψ24π{U,z−}=ℏ48πκ2,⟨Tz+z+⟩=0, (3.23)

where is the Schwarzian derivative and is the central charge which we have taken since phonon is a boson. Note that , since is well defined on .

### 3.2 Acoustic Hawking radiation from quantum mechanics

So far we have seen that the acoustic Hawking radiation occurs in the matrix model, since this model can be regarded as a two dimensional fluid. Interestingly the Hawking temperature (3.22) is given by the temperature which is also coincident with the temperature (2.3) derived from the quantum mechanics of the fermions in section 2. Although the configuration of the fermions for the bosonic non-critical string case sketched in figure 1 and the acoustic Hawking radiation case sketched in figure 3 are different777In terms of the fluid variables, the configuration of the bosonic non-critical string is given by , (). See figure 1 (RIGHT)., the same temperature is obtained. We argue why these two temperatures agree.

In the classical particle picture, both the bosonic non-critical string and the acoustic black hole are described by the flow of the fermions from the left. The difference of these two configurations are just the maximum energies. The energy of the fermions are filled up to the fermi energy in the bosonic non-critical string case while up to in the acoustic black hole case. Correspondingly all the fermions are reflected by the potential in the classical limit in the bosonic non-critical string case, and, in the acoustic black hole case, they are reflected if and go through the potential if . As we have seen in section 2, for the fermions with , the holes appear in the fermi sea through the quantum tunneling, and they cause the thermal radiation. In the acoustic black hole case, in addition to these holes, even the fermions with are reflected quantum mechanically, and they may also contribute to the thermal radiation. Indeed, the probability of the reflectance of this process for the fermion with energy can be calculated as [57, 40]

 PR(E)=1exp(2πℏ√mαE)+1 (3.24)

The derivation is shown in appendix A. Again this probability can be interpreted as the Fermi-Dirac distribution at the temperature . Thus the Hawking radiation in the acoustic black hole may be explained as the net effect of the holes and reflected fermions888Thus the acoustic Hawking radiation is related to the quantum tunneling of the fermions. This result may remind us the derivation of the Hawking radiation through the quantum tunneling proposed by Parikh and Wilczek [65]. However they are not directly connected, since the former is the tunneling of the fermions which compose the fluid and the latter is the tunneling of the phonons in the acoustic geometry.. See figure 5. Since both the holes and reflected fermions obey the same Fermi-Dirac distribution, the temperature of the bosonic non-critical string case and the acoustic black hole case are the same.

Now it is natural to interpret the thermal radiation in the bosonic non-critical string case as a variety of the acoustic Hawking radiation. Although there is no supersonic region in the bosonic non-critical string case, the mechanism of the appearance of the radiation is related to the Hawking radiation in the acoustic black hole.

Finally, in order to confirm that the Hawking radiation in the acoustic black hole can be explained by the quantum mechanics of the fermions, we evaluate the density of the observable in the two different ways: the quantum mechanics of the fermions and the second quantized phonon. In appendix B, we show that these two derivations provide the same result,

 m2πℏ∫∞0dEeβLE+1(O(x,p(E,x))|p(E,x)|−O(x,p(−E,x))|p(−E,x)|).

The first term is the contribution of the reflected fermions and the second one is that of the holes which is equivalent to (2.13). (One important point to derive this result in the phonon calculation is that we need to evaluate the correction and in the expansion (3.2), since they are the same order to which is the leading contribution of the Hawking effect.) In this way, we can show that the acoustic Hawking radiation of the phonon is equivalent to the quantum tunneling and reflection of the fermions in the quantum mechanics.

## 4 Discussions

We have studied the thermal radiation in the matrix model which is related to the acoustic Hawking radiation in the supersonic fluid. Remarkably, although the system is not chaotic, the temperature (1.7) of this radiation saturates the chaos bound (1.1) [3]. The mechanism of the appearance of the bound in our model is very simple. The classical motion of the particle in the inverse harmonic potential (1.4) receives the quantum correction (2.2) or (3.24), and it causes the thermal radiation. This mechanism would be universal since only the dynamics around the tip of the potential is relevant for the radiation and it does not depend on the details of the potential in the far region. Thus our result may be applicable to various other systems. Particularly, applications to the classical chaotic systems would be interesting. It would be valuable to investigate whether any thermal radiation associated with the classical Lyapunov exponent arises in these systems in the semi-classical regime. If the mode which classically shows the exponential sensitivity of the initial condition (1.2) effectively feels the inverse harmonic potential approximately, thermal radiation might be induced through the quantum effect.

Besides the acoustic black hole may provide a chance to explore the nature of the chaos bound in experiment. Several interesting phenomena related to chaos may occur near the event horizon of black holes (e.g. [3, 6, 66, 67]), and some of them may occur even in the acoustic event horizon. Since acoustic black holes may be realized in laboratories [68, 69, 70], these predictions of the chaotic behaviors might be tested by these experiments.

##### Acknowledgements

The author would like to thank Koji Hashimoto, Pei-Ming Ho, Shoichi Kawamoto, Manas Kulkarni, Gautam Mandal, Yoshinori Matsuo, Joseph Samuel, Tadashi Takayanagi and Asato Tsuchiya for valuable discussions and comments. The work of T. M. is supported in part by Grant-in-Aid for Scientific Research (No. 15K17643) from JSPS.

## Appendix A Quantum mechanics in the inverse harmonic potential

In this appendix, we review the derivation of the transmittance (2.2) and the reflectance (3.24) by solving the Schrödinger equation (2.1). We start from the Schrödinger equation in the inverse harmonic potential (2.1)

 Eϕ(x)=^Hϕ(x),^H=−ℏ22m∂2x−α2x2⇒ϕ′′+mαℏ2x2ϕ+2mEℏ2ϕ=0. (A.1)

By defining the variables

 w=(4mαℏ2)1/4x,a=−√mαEℏ, (A.2)

the Schrödinger equation becomes

 ∂2wϕ+(14w2−a)ϕ=0. (A.3)

The two independent solutions of this equation are given by the parabolic cylinder function and its complex conjugate , where we follow the notation of [71].

To derive the transmittance and the reflectance, we consider the connection between and the WKB wave function

 ϕE±(x)=A(E)√p(E,x)exp(±iℏ∫xdy p(E,y)),p(E,x)=√2mE+mαx2. (A.4)

where is a normalization factor which is irrelevant in the following discussion. As shown in 19.24.1 of [71], oscillates for large as

 E(a,w)=Feiχ χ≃14w2−alogw+⋯=iℏ(√mα2x2+√mαElog((4mαℏ2)1/4x)+⋯) F≃√2w(1+O(w−2))=√ℏ(mαx2)1/4+⋯ (A.5)

On the other hand, the momentum in the WKB wave function (A.4) behaves

 p(ϵ,x)≃√mαx+√mαEx+⋯,(x2≫|E|/α). (A.6)

Thus by comparing this with (A.5), we can read off the connection between and the WKB wave function (A.4) as

 E(a,w)→√ℏ√p(E,x)exp(iℏ∫xdy p(E,y)),(w≫√|a|) (A.7)

Hence and describe the right () and left () moving wave in , respectively.

The final task for obtaining the transmittance and the reflectance is finding the relation between the WKB wave function (A.4) in and in . For this purpose, the relation 19.18.3 of [71] is useful,

 √1+e2πaE(a,w)=eπaE∗(a,w)+iE∗(a,−w), ⇒ E(a,w)=i√1+e2πaE∗(a,−w)−ieπaE(a,−w) (A.8)

Then, for a large negative , we obtain

 E(a,w)→i√1+e2πa√2−we−i(w24+⋯)−ieπa√2−wei(w24+⋯) (A.9)

from (A.5). Note that the momenta carried by the first term and the second term at a large negative are positive and negative, respectively, as

 −i∂w(√2−we−i(w24+⋯))≃−w2(√2−we−i(w24+⋯)) −i∂w(√2−wei(w24+⋯))≃w2(√2−wei(w24+⋯)). (A.10)

Thus the first term represents the in-coming wave towards the inverse harmonic potential and the second term describes the reflected wave. On the other hand, at large positive can be regarded as the transmitted wave. These waves are connected to the WKB wave function (A.4) in the asymptotic region as we have seen in (A.7).

By normalizing the coefficient of the in-coming wave to 1 in the relation (A.8), we obtain

 E∗(a,−w)= 1√e−2πa+1E(a,−w)−i1√e2πa+1E(a,w) = 1√exp(2πℏ√mαE)+1E(a,−w)−i1√exp(−2πℏ√mαE)+1E(a,w). (A.11)

See figure 6. Then, from the coefficients of the reflected wave and the transmitted wave , we can read off the reflectance and transmittance

 PR(E)=1exp(2πℏ√mαE)+1,PT(E)=1exp(−2πℏ√mαE)+1, (A.12)

respectively. Note that and satisfy as , and as . They agree with the classical motion of the particles.

## Appendix B Comparison of the thermal radiation derived from the quantum mechanics and fluid dynamics

As we argued in section 3.2, the temperature derived from the quantum mechanics of the fermions (2.3) and the temperature derived from the Bogoliubov transformation of the phonon (3.22) agree999The choice of the vacuum in the Bogoliubov transformation would correspond to the choice of the state in the quantum mechanics. In the Bogoliubov transformation, we implicitly chose the Unruh vacuum to describe the acoustic Hawking radiation. In the quantum mechanics, this vacuum would correspond to the state in which the fermions flows from the left to right. This state is not stationary through the quantum effects, and the radiation appears.. However you may wonder whether these differently obtained radiation are really equivalent. In fact, the fermions obey the Fermi-Dirac distribution (2.13) while the phonons obey the Bose-Einstein distribution. Thus it is important to evaluate the radiation quantitatively and confirm whether they agree. In this appendix, we evaluate the density of the observable in the black hole (right) case (3.15) both from the fermions and the phonons, and we will see the agreement.

### b.1 Radiation through the quantum mechanics of the fermions

We evaluate the density of the operator in the acoustic black hole case (3.15) by using the quantum mechanics of the fermions. The derivation is almost same to the derivation of (2.13) in the bosonic non-critical string case studied in section 2. As we argued in section 3.2, the difference between the bosonic non-critical string and the acoustic black hole is the existence of the fermions with energy only. Thus what we should do is evaluating the contributions of these fermions to the radiation.

A tiny technical issue for these fermions is the boundary condition of the WKB wave function (A.4) at . If we impose , () which we employed for the fermions with in section 2, the fermions are reflected at and they cannot compose the stationary configuration (3.15) in the classical limit. In order to avoid this issue, we impose the periodic boundary condition only for the fermions with positive energy . Then the configuration (3.15) becomes stationary in the classical limit101010If we impose the periodic boundary condition to the fermions with negative energy, the fermions cannot compose a stationary bosonic non-critical string configuration. Thus we need to apply the two different boundary conditions depending on the sign of the fermion energy. We can avoid such a energy dependent boundary condition by employing the periodic potential instead of the inverse harmonic potential and impose the periodic boundary condition , and put the fermions appropriately so that the configuration near becomes (3.15). . (The boundary condition at should be irrelevant for the radiation, if is sufficiently large.)

Then, in the classical limit , the WKB wave function for the fermion with energy is given by

 ϕE±(x)=A(E)√p(E,x)exp(±iℏ∫xdy p(E,y)),p(E,x)=√2mE+mαx2. (B.1)

Here the energy is quantized as

 ∫L−Ldx|p(E,x)|=2πℏn,(n=1,2,⋯), (B.2)

due to the periodic boundary condition. The normalization factor is given by

 |A(E)|2=(∫L−Ldx|p(E,x)|)−1. (B.3)

By using this WKB wave function, we evaluate the density of the observable in the far region . Particularly the leading quantum corrections to this quantity would correspond to the Hawking radiation. Recall that, in the classical limit (), the fermions with energy just go through the potential from the left to right, and the left moving fermions appear only through the quantum reflection with the probability . Hence we should evaluate the leading contributions of the left moving fermions to to obtain the Hawking radiation. Through the similar calculation to the case (2.10), such contributions to the density of can be obtained as

 ∫E+0dEρ(E)PR(E)|ϕE−(x)|2O(x,p(E,x)). (B.4)

Here the energy density is given by

 ρ(E)=∂n∂E=m2πℏ∫L−Ldx|p(E,x)|,(E>0), (B.5)

through the quantization condition (B.2). Then the density (B.4) can be calculated as

 ∫E+0dEρ(E)PR(E)|ϕE−(x)|2O(x,p(E,x))=m2πℏ∫E+0dE|p(E,x)|O(x,p(E,x))eβLE+1. (B.6)

This is the contributions to the Hawking radiation from the fermions with . By assuming that is sufficiently large and can be approximated as , and adding the contributions of the fermions with negative energy (2.13) with , we obtain the density of the observable induced by the Hawking radiation as

 m2πℏ∫∞0dEeβLE+1(O(x,p(E,x))|p(E,x)|−O(x,p(−E,x))|p(−E,x)|),−L≪x≪−(ℏ2mα)1/4. (B.7)

The first term is the contribution of the reflected fermions and the second term is that of the holes, and both obey the Fermi-Dirac distribution at the temperature (1.7).

More concretely, we take as and evaluate (B.7) as111111We need to take care of the sign of .