Lieb–Schultz–Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems

We formulate and prove the local twist version of the Yamanaka–Oshikawa–Affleck theorem, an extension of the Lieb–Schultz–Mattis theorem, for one-dimensional systems of quantum particles or spins. We can treat almost any translationally invariant system with global U(1) symmetry. Time-reversal or inversion symmetry is not assumed. It is proved that, when the “filling factor” is not an integer, a ground state without any long-range order must be accompanied by low-lying excitations whose number grows indefinitely as the system size is increased. The result is closely related to the absence of topological order in one-dimension. The present paper is written in a self-contained manner, and does not require any knowledge of the Lieb–Schultz–Mattis and related theorems.     

Vortex rings and solitary waves in trapped Bose–Einstein condensates

We discuss nonlinear excitations in an atomic Bose–Einstein condensate which is trapped in a harmonic potential. We focus on axially symmetric solitary waves propagating along a cylindrical condensate. A quasi one-dimensional dark soliton is the only nonlinear mode for a condensate with weak interactions. For sufficiently strong interactions of experimental interest solitary waves are hybrids of one-dimensional dark solitons and three-dimensional vortex rings. The energy-momentum dispersion of these solitary waves exhibits characteristics similar to a mode proposed sometime ago by Lieb in a strictly 1D model, as well as some rotonlike features. We subsequently discuss interactions between solitary waves. Head-on collisions between dark solitons are elastic. Slow vortex rings collide elastically but faster ones form intermediate structures during collisions before they lose energy to the background fluid. Solitary waves and their interactions have been observed in experiments. However, some of their intriguing features still remain to be experimentally identified.     

Soliton dynamics in a spin nematic

One-dimensional localized waves, which can be considered as soliton elementary excitations, exist in a magnet with a unit spin and comparable bilinear and biquadratic spin-spin interactions, with which the state of spin nematic is realized. These excitations are characterized by a certain momentum P and a certain energy E. The structure of these solitons has been found, and the E = E(P) dependence, which plays the role of the dispersion law of these soliton elementary excitations, has been constructed. The energy of a soliton with a certain momentum is shown to be lower than that of the quasiparticles of a linear theory. At small momenta, these E = E(P) dependences of the soliton and quasiparticles coincide asymptotically. The dependence of the soliton energy on the soliton momentum is a periodic function with a period P ₀ = π?/a, whose value does not depend on exchange integrals and depends only on a single crystal parameter, namely, the interatomic distance a. These soliton excitations have common features with the so-called Lieb states, which are well known in many condensed matter models.     

Lieb's correlation inequality for plane rotors

We prove a conjecture by E. Lieb, which leads to the Lieb inequality for plane rotors. As in the Ising model case, this inequality implies the existence of an algorithm to compute the transition temperature of this model.     

Number of Bound States of Schrödinger Operators with Matrix-Valued Potentials

We give a Cwikel–Lieb–Rozenblum type bound on the number of bound states of Schrödinger operators with matrix-valued potentials using the functional integral method of Lieb. This significantly improves the constant in this inequality obtained earlier by Hundertmark.     

Finite size effects on the helical edge states on the Lieb lattice

For a two-dimensional Lieb lattice,that is,a line-centered square lattice,the inclusion of the intrinsic spin–orbit(ISO)coupling opens a topologically nontrivial gap,and gives rise to the quantum spin Hall(QSH) effect characterized by two pairs of gapless helical edge states within the bulk gap.Generally,due to the finite size effect in QSH systems,the edge states on the two sides of a strip of finite width can couple together to open a gap in the spectrum.In this paper,we investigate the finite size effect of helical edge states on the Lieb lattice with ISO coupling under three different kinds of boundary conditions,i.e.,the straight,bearded and asymmetry edges.The spectrum and wave function of edge modes are derived analytically for a tight-binding model on the Lieb lattice.For a strip Lieb lattice with two straight edges,the ISO coupling induces the Dirac-like bulk states to localize at the edges to become the helical edge states with the same Dirac-like spectrum.Moreover,it is found that in the case with two straight edges the gapless Dirac-like spectrum remains unchanged with decreasing the width of the strip Lieb lattice,and no gap is opened in the edge band.It is concluded that the finite size effect of QSH states is absent in the case with the straight edges.However,in the other two cases with the bearded and asymmetry edges,the energy gap induced by the finite size effect is still opened with decreasing the width of the strip.It is also proposed that the edge band dispersion can be controlled by applying an on-site potential energy on the outermost atoms.     

Lieb's Spin-Reflection-Positivity Method and Its Applications to Strongly Correlated Electron Systems

In this paper, we discuss the spin-reflection-positivity method introduced by Lieb [E. H. Lieb, Phys. Rev. Lett. 62:1201 (1989)] and its applications to strongly correlated electron systems in a pedagogical manner. We emphasize the important role played by the sign rule of the ground-state wave function in studying a many-body system. To make our explanation more readable, we shall first review some well-known one-dimensional examples and recall the Lieb–Mattis theorem on the Heisenberg localized spin models. Then, after introducing the general theory of spin-reflection positivity, we show in detail how to use it to overcome the sign problem caused by the fermion characteristics of itinerant electrons in strongly correlated models. Finally, we establish several rigorous results on the Hubbard model, the periodic Anderson model and the Kondo lattice model.     

基于NiosII的红外图像灰度变换系统设计

提出了一种基于NiosII的红外图像灰度变换设计方案,并详细介绍了红外图像灰度变换算法以及系统硬、软件设计以及实现方法。针对红外图像的特点,提出了一种基于平台直方图均衡的红外图像灰度变换算法,并在CycloneII系列FPGA上实现了该算法,其处理单元采用嵌入式处理器软核NiosII。实验结果表明,该系统能很好地实现红外图形灰度变换,并在一定程度上抑制了背景,提升了目标。     

Data processing for the sandwiched Rényi divergence: a condition for equality

The \(\alpha \)-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for \(\alpha \ge 1/2\). In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the \(\alpha \)-sandwiched Rényi divergence for all \(\alpha \ge 1/2\). For the range \(\alpha \in [1/2,1)\), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.     

Optical properties of highly excited direct gap semiconductors

The electronic excitations in direct gap semiconductors interact strongly with the photon field. We discuss both the experimental and the theoretical aspects of the optical properties of these materials under strong optical excitation. We distinguish between intermediate excitation levels at which the electronic excitations form a dense system of excitons and excitonic molecules and very high excitation levels at which a degenerate electron-hole plasma occurs. The optical spectra of dense excitonic systems, which are mainly observed in copper halides and II–VI compounds, are shown to be determined mainly by the interaction processes between excitonic molecules, polaritons and free carriers. The optical properties of the electron-hole plasma, which has been observed in II–VI and especially in III–V compounds, can be understood only by taking into account many-body effects, such as dynamical screening of the Coulomb interactions, plasmon-assisted transitions and excitonic enhancement.     

缀饰格子中时间反演对称破缺的量子自旋霍尔效应

<正>研究了缀饰格子中的量子自旋霍尔效应,模型中同时考虑了Rashba自旋轨道耦合和交换场的作用.缀饰格子具有简立方对称性,以零能平带和单狄拉克锥结构为主要特点.在缀饰格子中,不论是实现量子自旋霍尔效应还是量子反常霍尔效应,都需要一个不为零的内禀自旋轨道耦合作用来打开一个完全的体能隙,这与石墨烯等六角格子模型有着很大的不同.在交换场破坏了时间反演对称性的情况下,以自旋陈数为标志的量子自旋霍尔效应仍然能够存在,边缘态和极化率的相关结果也证明了这一结论.结果表明自旋陈数比z2拓扑数在表征量子自旋霍尔效应方面有着更广泛的适用范围,相应的结论为利用磁场控制量子自旋霍尔效应提出了一个理论模型和依据.     

An analogy between the theory of superfluidity and the theory of dielectrics

It is shown that there is a fruitful correspondence between the properties of dielectrics and the properties of excitations in helium II.     

Quantum shock waves and domain walls in the real-time dynamics of a superfluid unitary Fermi gas

We show that in the collision of two superfluid fermionic atomic clouds one observes the formation of quantum shock waves as discontinuities in the number density and collective flow velocity. Domain walls, which are topological excitations of the superfluid order parameter, are also generated and exhibit abrupt phase changes by π and slower motion than the shock waves. The domain walls are distinct from the gray soliton train or number density ripples formed in the wake of the shock waves and observed in the collisions of superfluid bosonic atomic clouds. Domain walls with opposite phase jumps appear to collide elastically.     

Exact Perturbative Results for the Lieb–Liniger and Gaudin–Yang Models

Journal of Statistical Physics - We present a systematic procedure to extract the perturbative series for the ground state energy density in the Lieb–Liniger and Gaudin–Yang models,...     

Green's function approach to the impurity state in the Lieb lattice

Wavefunction of an impurity attached to the Lieb lattice is considered by directly computing the Green's function (GF) for the nearest-neighbor tight-binding model. By replacing the lattice GF with a GF array including all the nine GFs defining on the three-atom unit cell of the Lieb lattice, an accurate and efficient numerical technique is developed. Agreement of both the real and imaginary components of the GF between numerical simulation of the lattice GF and continuum-space Fourier transformation is achieved in the whole resonant-energy range. Both results demonstrate that the wavefunction amplitude decays in a power-law pattern while the resonant energy is small and it decays in a pattern stronger than the power law and weaker than exponentially while the resonant energy is large. The decaying exponents depend on the adatom type and location in the unit cell, which directly modifies the temperature-dependent conductance under the variable-range hopping theory.     

The topological quantum phase transitions in Lieb lattice driven by the Rashba SOC and exchange field

The quantum spin Hall (QSH) effect and the quantum anomalous Hall (QAH) effect in Lieblattice are investigated in the presence of both Rashba spin-orbit coupling (SOC) anduniform exchange field. The Lieb lattice has a simple cubic symmetry, which ischaracterized by the single Dirac-cone per Brillouin zone and the middle flat band in theband structure. The intrinsic SOC is essentially needed to open the full energy gap in thebulk. The QSH effect could survive even in the presence of the exchange field. In terms ofthe first Chern number and the spin Chern number, we study the topological nature and thetopological phase transition from the time-reversal symmetry broken QSH effect to the QAHeffect. For Lieb lattice ribbons, the energy spectrum and the wave-function distributionsare obtained numerically, where the helical edge states and the chiral edge states revealthe non-trivial topological QSH and QAH properties, respectively.     

Excitonic metal–insulator phase transition of the Mott type in compressed calcium

It has been experimentally found that, under the static compression of a calcium crystal at room temperature, it undergoes a series of structural phase transitions: face-centered cubic lattice → body-centered cubic lattice → simple cubic lattice. It has been decided to investigate precisely the simple cubic lattice (because it is an alternative lattice) with the aim of elucidating the possibility of the existence of other (nonstructural) phase transitions in it by using for this purpose the Hubbard model for electrons with half-filled ns-bands and preliminarily transforming the initial electronic system into an electron–hole system by means of the known Shiba operators (applicable only to alternative lattices). This transformation leads to the fact that, in the new system of fermions, instead of the former repulsion, there is an attraction between electrons and holes. Elementary excitations of this new system are bound boson pairs—excitons. This system of fermions has been quantitatively analyzed by jointly using the equation-of-motion method and the direct algebraic method. The numerical integration of the analytically exact transcendental equations derived from the first principles for alternative (one-, two-, and three-dimensional) lattices has demonstrated that, in systems of two-species (electrons + hole) fermions, temperature-induced metal–insulator phase transitions of the Mott type are actually possible. Moreover, all these crystals are in fact excitonic insulators. This conclusion is in complete agreement with the analytically exact calculations of the ground state of a one-dimensional crystal (with half-filled bands), which were performed by Lieb and Wu with the aim to find out the Mott insulator–metal transition of another type.     

An Entropic Uncertainty Principle for Positive Operator Valued Measures

Extending a recent result by Frank and Lieb, we show an entropic uncertainty principle for mixed states in a Hilbert space, relatively to pairs of positive operator valued measures that are independent in some sense. This yields spatial-spectral uncertainty principles and log-Sobolev inequalities for invariant operators on homogeneous spaces, which are sharp in the compact case.