Semiclassical theory of transport in antidot lattices

Motivated by a recent experiment by Weiss et al. [Phys. Rev. Lett. 70, 4118 (1993)], we present a detailed study of quantum transport in large antidot arrays whose classical dynamics is chaotic. We calculate the longitudinal and Hall conductivities semiclassically starting from the Kubo formula. The leading contribution reproduces the classical conductivity. In addition, we find oscillatory quantum corrections to the classical conductivity which are given in terms of the periodic orbits of the system. These periodic-orbit contributions provide a consistent explanation of the quantum oscillations in the magnetoconductivity observed by Weiss et al. We find that the phase of the oscillations with Fermi energy and magnetic field is given by the classical action of the periodic orbit. The amplitude is determined by the stability and the velocity correlations of the orbit. The amplitude also decreases exponentially with temperature on the scale of the inverse orbit traversal time/T . The Zeeman splitting leads to beating of the amplitude with magnetic field. We also present an analogous semiclassical derivation of Shubnikov-de Haas oscillations where the corresponding classical motion is integrable. We show that the quantum oscillations in antidot lattices and the Shubnikov-de Haas oscillations are closely related. Observation of both effects requires that the elastic and inelastic scattering lengths be larger than the lengths of the relevant periodic orbits. The amplitude of the quantum oscillations in antidot lattices is of a higher power in Planck's constant and hence smaller than that of Shubnikov-de Haas oscillations. In this sense, the quantum oscillations in the conductivity are a sensitive probe of chaos.This paper is dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Transport in quasi one-dimensional spin-1/2 systems

We present numerical results for the spin and thermal conductivity of one-dimensional (1D) quantum spin systems. We contrast the properties of integrable models such as the spin-1/2 XXZ chain against nonintegrable ones such as frustrated and dimerized chains. The thermal conductivity of the XXZ chain is ballistic at finite temperatures, while in the nonintegrable models, this quantity is argued to vanish. For the case of frustrated and dimerized chains, we discuss the frequency dependence of the transport coefficients. Finally, we give an overview over related theoretical work on intrinsic and extrinsic scattering mechanisms of quasi-1D spin systems.     

Darboux Transformations of Bispectral Quantum Integrable Systems

We present an approach to higher-dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct nontrivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero–Moser.     

Sensitivity of energy eigenstates to perturbation in quantum integrable and chaotic systems

We study the sensitivity of energy eigenstates to small perturbation in quantum integrable and chaotic systems.It is shown that the distribution of rescaled components of perturbed states in unperturbed basis exhibits qualitative difference in these two types of systems:being close to the Gaussian form in quantum chaotic systems,while,far from the Gaussian form in integrable systems.     

Quantum Integrable Toda-Like Systems in Deformation Quantization

We define and discuss the notion of quantum integrability of a classically integrable system within the framework of deformation quantization, i.e. the question whether the classical conserved quantities (which are already in involution with respect to the Poisson bracket) commute with respect to some star product on the phase space after possible quantum corrections. As an example of this method, we show by means of suitable 2 by 2 quantum R-matrices that a list of Toda-like classical integrable systems given by Y. B. Suris is quantum integrable with respect to the usual star product of the Weyl type in flat 2n-dimensional space.     

Quantum Toda systems and Lax pairs

We describe a general method for constructing a Lax pair representation of certain quantum mechanical systems that are integrable at the classical level. This is then used to find conserved quantities at the quantum level for the Toda systems.     

An Approach to Loop Quantum Cosmology Through Integrable Discrete Heisenberg Spin Chains

The quantum evolution equation of Loop Quantum Cosmology (LQC)—the quantum Hamiltonian constraint—is a difference equation. We relate the LQC constraint equation in vacuum Bianchi I separable (locally rotationally symmetric) models with an integrable differential-difference nonlinear Schrödinger type equation, which in turn is known to be associated with integrable, discrete Heisenberg spin chain models in condensed matter physics. We illustrate the similarity between both systems with a simple constraint in the linear regime.     

Quantum hydrodynamics, the quantum benjamin-ono equation, and the Calogero model

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.     

人造量子系统的理论研究与代数动力学

从控制与利用微观系统的量子工程的观点,讨论了人造量子系统的基本物理问题。针对人造量子系统中的一大类———非自治量子系统的求解问题,提出了代数动力学理论方法。运用代数动力学,对人造量子系统进行了理论研究;对可积的非自治系统,详细介绍了线性系统和非线性可积系统的求解问题;对不可积系统,用代数动力学观点研究了量子规则运动和无规运动的特征,它们之间的过渡,以及它们对时间有关外场的不同响应。     

Role of orbital dynamics in spin relaxation and weak antilocalization in quantum dots

We develop a semiclassical theory for spin-dependent quantum transport to describe weak (anti)localization in quantum dots with spin-orbit coupling. This allows us to distinguish different types of spin relaxation in systems with chaotic, regular, and diffusive orbital classical dynamics. We find, in particular, that for typical Rashba spin-orbit coupling strengths, integrable ballistic systems can exhibit weak localization, while corresponding chaotic systems show weak antilocalization. We further calculate the magnetoconductance and analyze how the weak antilocalization is suppressed with decreasing quantum dot size and increasing additional in-plane magnetic field.     

Effect of rare fluctuations on the thermalization of isolated quantum systems

We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular, we investigate the prerequisites for thermalization, focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the microcanonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some nonintegrable systems of finite size, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and discuss two scenarios for thermalization.     

Quantum vs. Classical Integrability in Ruijsenaars–Schneider and Calogero–Moser Systems

The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider (R-S) and Calogero-Moser systems is addressed. The classical Calogero and Sutherland systems (based on any root system) at equilibrium have many remarkable properties; for example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all integer valued. These are related to the energy eigenvalues of the quantum Calogero and Sutherland systems. Similar features and results hold for the R-S type of integrable systems based on the classical root systems.     

Nodal domains statistics: a criterion for quantum chaos

We consider the distribution of the (properly normalized) numbers of nodal domains of wave functions in 2D quantum billiards. We show that these distributions distinguish clearly between systems with integrable (separable) or chaotic underlying classical dynamics, and for each case the limiting distribution is universal (system independent). Thus, a new criterion for quantum chaos is provided by the statistics of the wave functions, which complements the well-established criterion based on spectral statistics.     

CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy

We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1:1:2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO2 molecule. We uncover quantum monodromy of CO2 as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states.     

On ground-state wavefunctions for Sutherland-Calogero systems in an external field

Conditions are considered under which the ground-state wavefunctions of quantum systems of interacting particles in an external field are factorizable and can be found explicitly. The corresponding classical systems of particles are completely integrable; in the quantum case an extra integral of motion is constructed for a two-particle system.     

Integrability of the Rabi model

The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.     

A Remark on the Notion of Independence of Quantum Integrals of Motion in the Thermodynamic Limit

Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, surprisingly enough, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by dwelling upon an important aspect of quantum integrability—the notion of independence of quantum integrals of motion (QIMs). We point out that a widely accepted definition of functional independence of QIMs is flawed, and suggest a new definition. Our study is motivated by the PXP model—a model of N spins 1/2 possessing an extensive number of binary QIMs. The number of QIMs which are independent according to the common definition turns out to be equal to the number of spins, N. A common wisdom would then suggest that the system is completely integrable, which is not the case. We discuss the origin of this conundrum and demonstrate how it is resolved when a new definition of independence of QIMs is employed.

     

Study of the “metal–insulator” transition induced by the impurity fluctuation potential using the Shubnikov–de Haas effect

Heterostructures with a GaAs/InGaAs/GaAs quantum well and aMn magnetic impurity layer separated from it, which have different conductivity types, are studied. At a Mn content not exceeding the amount corresponding to 0.5 monolayer of MnAs, a percolation cluster formed in the quantum well plane is not simply connected, but consists of metal drops separated by low-conductivity interspaces. Despite the absence of the simply connected conducting channel, Shubnikov–de Haas oscillations are observed in all studied systems, which are controlled by carrier properties in conducting drops, independent of Mn content. The estimate of drop sizes corresponds to theoretical values.     

Similar Early Growth of Out-of-time-ordered Correlators in Quantum Chaotic and Integrable Ising Chains

Previous studies show that, in quantum chaotic and integrable systems, the so-called out-of-time-ordered correlator(OTOC) generically behaves differently at long times, while, it may show similar early growth in one-body systems. In this paper, by means of numerical simulations, it is shown that OTOC has similar early growth in two quantum many-body systems, one integrable and one chaotic.     

Point-contact tunneling spectroscopy between a Nb tip and an ideal topological insulator Sn-doped Bi1.1Sb0.9Te2S

We find that the quantum-classical correspondence in integrable systems is characterized by two time scales. One is the Ehrenfest time below which the system is classical; the other is the quantum revival time beyond which the system is fully quantum. In between, the quantum system can be well approximated by classical ensemble distribution in phase space. These results can be summarized in a diagram which we call Ehrenfest diagram. We derive an analytical expression for Ehrenfest time, which is proportional to h~(-1/2). According to our formula, the Ehrenfest time for the solar-earth system is about 10~(26) times of the age of the solar system. We also find an analytical expression for the quantum revival time, which is proportional to h~(-1). Both time scales involve ω(I), the classical frequency as a function of classical action. Our results are numerically illustrated with two simple integrable models. In addition, we show that similar results exist for Bose gases, where 1/N serves as an effective Planck constant.