Generalized Kubo formula for spin transport: A theory of linear response to non-Abelian fields

The traditional Kubo formula is generalized to describe the linear response with respect to non-Abelian fields. To fulfil the demand for studying spin transport, the SU(2) Kubo formulae are derived by two conventional approaches with different gauge fixings. Those two approaches are shown to be equivalent where the non-conservation of the SU(2) current plays an essential role in guaranteeing the consistency. Some concrete examples relating Spin Hall Effect are considered. The dc spin conductivity vanishes in the system with parabolic unperturbed dispersion relation. By applying a time-dependent Rashba field, the spin conductivity can be measured directly. Our formula is also applied to the high-dimensional representation for the interests of some important models, such as Luttinger model and bilayer spin Hall system.Comment: Revtex, 10 pages, 2 tables, typos corrected and ref adde

Two-point boundary value problems and exact controllability for several kinds of linear and nonlinear wave equations

In this paper we introduce some new concepts for second-order hyperbolic equations: two-point boundary value problem, global exact controllability and exact controllability. For several kinds of important linear and nonlinear wave equations arising from physics and geometry, we prove the existence of smooth solutions of the two-point boundary value problems and show the global exact controllability of these wave equations. In particular, we investigate the two-point boundary value problem for one-dimensional wave equation defined on a closed curve and prove the existence of smooth solution which implies the exact controllability of this kind of wave equation. Furthermore, based on this, we study the two-point boundary value problems for the wave equation defined on a strip with Dirichlet or Neumann boundary conditions and show that the equation still possesses the exact controllability in these cases. Finally, as an application, we introduce the hyperbolic curvature flow and obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves.Comment: 36 pages, 1 figur

Convergence of generalized Collatz problem in k-adic field

In this article, we define a new k-adic series transformation called Z-transformation and probe into its fixed point and periodicity. We extend the number field of the transform period problem to a wider k-adic field. Different constraints are imposed on k, then different periodic columns are formed after finite Z-transformations. We obtain that their periodic sequences are M1= {1,2} and M2={1,2}\cup {n_0}\cup {n'} respectively after derivation. As an application, it can provide a reference for C problems in more complex algebraic ystems.Comment: 10 pages, 3 figure

First-order and continuous quantum phase transitions in the anisotropic quantum Rabi-Stark model

Various quantum phase transitions in the anisotropic Rabi-Stark model with both the nonlinear Stark coupling and the linear dipole coupling between a two-level system and a single-mode cavity are studied in this work. The first-order quantum phase transitions are detected by the level crossing of the ground-state and the first-excited state with the help of the pole structure of the transcendental functions derived by the Bogoliubov operators approach. As the nonlinear Stark coupling is the same as the cavity frequency, this model can be solved by mapping to an effective quantum oscillator. All energy levels close at the critical coupling in this case, indicating continuous quantum phase transitions. The critical gap exponent is independent of the anisotropy as long as the counter-rotating wave coupling is present, but essentially changed if the counter-rotating wave coupling disappears completely. It is suggested that the gapless Goldstone mode excitations could appear above a critical coupling in the present model in the rotating-wave approximation.Comment: 9 pages, 4 figure

Quantum Rabi-Stark model: Solutions and exotic energy spectra

The quantum Rabi-Stark model, where the linear dipole coupling and the nonlinear Stark-like coupling are present on an equal footing, are studied within the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived in a compact way, much simpler than previous ones obtained in the Bargmann representation. The zeros of transcendental functions reproduce completely the regular spectra. In terms of the explicit pole structure of these functions, two kinds of exceptional eigenvalues are obtained and distinguished in a transparent manner. Very interestingly, a first-order quantum phase transition indicated by level crossing of the ground state and the first excited state is induced by the positive nonlinear Stark-like coupling, which is however absent in any previous isotropic quantum Rabi models. When the absolute value of the nonlinear coupling strength is equal to twice the cavity frequency, this model can be reduced to an effective quantum harmonic oscillator, and solutions are then obtained analytically. The spectra collapse phenomenon is observed at a critical coupling, while below this critical coupling, infinite discrete spectra accumulate into a finite energy from below.Comment: 16 pages, 4 figure

Super-resolution MRI through Deep Learning

Magnetic resonance imaging (MRI) is extensively used for diagnosis and image-guided therapeutics. Due to hardware, physical and physiological limitations, acquisition of high-resolution MRI data takes long scan time at high system cost, and could be limited to low spatial coverage and also subject to motion artifacts. Super-resolution MRI can be achieved with deep learning, which is a promising approach and has a great potential for preclinical and clinical imaging. Compared with polynomial interpolation or sparse-coding algorithms, deep learning extracts prior knowledge from big data and produces superior MRI images from a low-resolution counterpart. In this paper, we adapt two state-of-the-art neural network models for CT denoising and deblurring, transfer them for super-resolution MRI, and demonstrate encouraging super-resolution MRI results toward two-fold resolution enhancement

The mixed quantum Rabi model

The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of $4 \times 4$ determinants responsible for the exact solutions are derived. These so-called $G$-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of $G$-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.Comment: 11 pages, 8 figure

Quantum Entanglement transfer between spin-pairs

We investigate the transfer of entanglement from source particles (SP) to target particles (TP) in the Heisenberg interaction $H=\vec s_{1} \cdot \vec s_{2}$. In our research, TP are two qubits and SP are two qubits or qutrits. When TP are two qubits, we find that no matter what state the TP is initially prepared in, at the specific time $t=\pi$, the entanglement of TP can attain to 1 after interaction with SP which stay on the maximally entangled state. For the TP are two qutrits, we find that the maximal entanglement of TP after interaction is relative to the initial state of TP and always cannot attain to 1 to almost all of initial states of TP. But we discuss an iterated operation which can make the TP to the maximal entangled state.Comment: 6 pages; 4 figs. Accepted for publication in International Journal of Quantum Informatio

Invariant and hyperinvariant subspaces for amenable operators

There has been a long-standing conjecture in Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to every non-scalar amenable operator has a non-trivial hyperinvariant subspace and equivalent to every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two decompositions for amenable operators, which supporting the conjecture.Comment: 11 page

Quantum phase transitions of a generalized compass chain with staggered Dzyaloshinskii-Moriya interaction

We consider a class of one-dimensional compass models with staggered Dzyaloshinskii-Moriya exchange interactions in an external transverse magnetic field. Based on the exact solution derived from Jordan-Wigner approach, we study the excitation gap, energy spectra, spin correlations and critical properties at phase transitions. We explore mutual effects of the staggered Dzyaloshinskii-Moriya interaction and the magnetic field on the energy spectra and the ground-state phase diagram. Thermodynamic quantities including the entropy and the specific heat are discussed, and their universal scalings at low temperature are demonstrated.Comment: 8 page, 10 figure