We study the bootstrap method in harmonic oscillators in one-dimensional quantum mechanics. We find that the problem reduces to the Diracs ladder operator problem and is exactly solvable. Thus, harmonic oscillators allow us to see how the bootstrap method works explicitly.
Recently, novel numerical computation on quantum mechanics by using a bootstrap was proposed by Han, Hartnoll, and Kruthoff. We consider whether this method works in systems with a $theta$-term, where the standard Monte-Carlo computation may fail due
to the sign problem. As a starting point, we study quantum mechanics of a charged particle on a circle in which a constant gauge potential is a counterpart of a $theta$-term. We find that it is hard to determine physical quantities as functions of $theta$ such as $E(theta)$, except at $theta=0$ and $pi$. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any $theta$. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a $theta$-term exists in the system.
Out-of-time-order correlator (OTOC) $langle [x(t),p]^2 rangle $ in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. The quantum Lyapunov exponent computed through the OTOC precisely agrees wit
h the classical one. Besides, it does not show any quantum fluctuations for arbitrary states. Hence, the OTOC may be regarded as ideal indicators of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the OTOCs might be seen in various situations too. In order to clarify this point, as a first step, we investigate the OTOCs in one dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growths reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using suitably localized states near the peak and the second one is taking a double scaling limit akin to the non-critical string theories.
We propose that Hawking radiation-like phenomena may be observed in systems that show butterfly effects. Suppose that a classical dynamical system has a Lyapunov exponent $lambda_L$, and is deterministic and non-thermal ($T=0$). We argue that, if we
quantize this system, the quantum fluctuations may imitate thermal fluctuations with temperature $T sim hbar lambda_L/2 pi $ in a semi-classical regime, and it may cause analogous Hawking radiation. We also discuss that our proposal may provide an intuitive explanation of the existence of the bound of chaos proposed by Maldacena, Shenker and Stanford.
We investigate critical phenomena of the Yang-Mills (YM) type one-dimensional matrix model that is a large-$N$ reduction (or dimensional reduction) of the $D+1$ dimensional $U(N)$ pure YM theory (bosonic BFSS model). This model shows a large-$N$ phas
e transition at finite temperature, which is analogous to the confinement/deconfinement transition of the original YM theory. We study the matrix model at a three-loop calculation via the principle of minimum sensitivity and find that there is a critical dimension $D=35.5$: At $D le 35$, the transition is of first order, while it is of second order at $Dge 36$. Furthermore, we evaluate several observables in our method, and they nicely reproduce the existing Monte Carlo results. Through the gauge/gravity correspondence, the transition is expected to be related to a Gregory-Laflamme transition in gravity, and we argue that the existence of the critical dimension is qualitatively consistent with it. Besides, in the first order transition case, a stable phase having negative specific heat appears in the microcanonical ensemble, which is similar to Schwarzschild black holes. We study some properties of this phase.
A new method to study nuclear physics via holographic QCD is proposed. Multiple baryons in the Sakai-Sugimoto background are described by a matrix model which is a low energy effective theory of D-branes of the baryon vertices. We study the quantum m
echanics of the matrix model and calculate the eigenstates of the Hamiltonian. The obtained states are found to coincide with known nuclear and baryonic states, and have appropriate statistics and charges. Calculated spectra of the baryon/nucleus for small baryon numbers show good agreement with experimental data. For hyperons, the Gell-Mann--Okubo formula is approximately derived. Baryon resonances up to spin $5/2$ and isospin $5/2$ and dibaryon spectra are obtained and compared with experimental data. The model partially explains even the magic numbers of light nuclei, $N=2,8$ and $20$.
Recently the bound on the Lyapunov exponent $lambda_L le 2pi T/ hbar$ in thermal quantum systems was conjectured by Maldacena, Shenker, and Stanford. If we naively apply this bound to a system with a fixed Lyapunov exponent $lambda_L$, it might predi
ct the existence of the lower bound on temperature $T ge hbar lambda_L/ 2pi $. Particularly, it might mean that chaotic systems cannot be zero temperature quantum mechanically. Even classical dynamical systems, which are deterministic, might exhibit thermal behaviors once we turn on quantum corrections. We elaborate this possibility by investigating semi-classical particle motions near the hyperbolic fixed point and show that indeed quantum corrections may induce energy emission which obeys a Boltzmann distribution. We also argue that this emission is related to acoustic Hawking radiation in quantum fluid. Besides, we discuss when the bound is saturated and show that a particle motion in an inverse harmonic potential and $c=1$ matrix model may saturate the bound although they are integrable.
By exploring a phase space hydrodynamics description of one-dimensional free Fermi gas, we discuss how systems settle down to steady states described by the generalized Gibbs ensembles through quantum quenches. We investigate time evolutions of the F
ermions which are trapped in external potentials or a circle for a variety of initial conditions and quench protocols. We analytically compute local observables such as particle density and show that they always exhibit power law relaxation at late times. We find a simple rule which determines the power law exponent. Our findings are, in principle, observable in experiments in an one dimensional free Fermi gas or Tonks gas (Bose gas with infinite repulsion).
In our previous work arXiv:1704.08675, we pointed out that various multi-cut solutions exist in the Chern-Simons (CS) matrix models at large-$N$ due to a curious structure of the saddle point equations. In the ABJM matrix model, these multi-cut solut
ions might be regarded as the condensations of the D2-brane instantons. However many of these multi-cut solutions including the ones corresponding to the condensations of the D2-brane instantons were obtained numerically only. In the current work, we propose an ansatz for the multi-cut solutions which may allow us to derive the analytic expressions for all these solutions. As a demonstration, we derive several novel analytic solutions in the pure CS matrix model and the ABJM matrix model. We also develop the argument for the connection to the instantons.
Classical particle motions in an inverse harmonic potential show the exponential sensitivity to initial conditions, where the Lyapunov exponent $lambda_L$ is uniquely fixed by the shape of the potential. Hence, if we naively apply the bound on the Ly
apunov exponent $lambda_L le 2pi T/ hbar$ to this system, it predicts the existence of the bound on temperature (the lowest temperature) $T ge hbar lambda_L/ 2pi$ and the system cannot be taken to be zero temperature when $hbar eq 0$. This seems a puzzle because particle motions in an inverse harmonic potential should be realized without introducing any temperature but this inequality does not allow it. In this article, we study this problem in $N$ non-relativistic free fermions in an inverse harmonic potential ($c=1$ matrix model). We find that thermal radiation is {em induced} when we consider the system in a semi-classical regime even though the system is not thermal at the classical level. This is analogous to the thermal radiation of black holes, which are classically non-thermal but behave as thermal baths quantum mechanically. We also show that the temperature of the radiation in our model saturates the inequality, and thus, the system saturates the bound on the Lyapunov exponent, although the system is free and integrable. Besides, this radiation is related to acoustic Hawking radiation of the fermi fluid.
