Surface states in crystals with low-index surfaces

For most of the conventional crystals with low-index surfaces, the hopping between the nearest neighbor （1NN） crystal planes （CPs） is dominant and the ones from the nNN （2 〈 n 〈 ∞） CPs are relatively weak, considered as small perturbations. The recent theoretical analysisIll has demonstrated the absence of surface states at the level of the hopping approximation between the INN CPs when the original infinite crystal has the geometric reflection symmetry （GRS） for each CP. Meanwhile, based on the perturbation theory, it has also been shown that small perturbations from the hopping between the nNN （2 〈 n 〈 ∞） CPs and surface relaxation have no impact on the above conclusion. However, for the crystals with strong intrinsic spin-orbit coupling （SOC）, the dominant terms of intrinsic SOC associate with two INN bond hoppings. Thus SOC will significantly contribute the hoppings from the INN and/or 2NN CPs except the ones within each CP. Here, we will study the effect of the hopping between the 2NN CPs on the surface states in model crystals with three different type structures （Type I： “……P-P-P-P……”, Type II： “……-P-Q-P-Q……” and Type III：“……P=Q-P=Q……” where P and Q indicate CPs and the signs “-” and “=” mark the distance between the INN CPs）. In terms of analytical and numerical calculations, we study the behavior of surface states in three types after the symmetric/asymmetric hopping from the 2NN CPs is added. We analytically prove that the symmetric hopping from the 2NN CPs cannot induce surface states in Type I when each CP has only one electron mode. The numerical calculations also provide strong support for the conclusion, even up to 5NN. However, in general, the coupling from the 2NN CPs （symmetric and asymmetric） is favorable to generate surface states except Type I with single electron mode only.

Gauss quadrature based finite temperature Lanczos method

The finite temperature Lanczos method(FTLM),which is an exact diagonalization method intensively used in quantum many-body calculations,is formulated in the framework of orthogonal polynomials and Gauss quadrature.The main idea is to reduce finite temperature static and dynamic quantities into weighted summations related to one-and twodimensional Gauss quadratures.Then lower order Gauss quadrature,which is generated from Lanczos iteration,can be applied to approximate the initial weighted summation.This framework fills the conceptual gap between FTLM and kernel polynomial method,and makes it easy to apply orthogonal polynomial techniques in the FTLM calculation.

Dynamical properties of the Haldane chain with bond disorder

By using Lanczos exact diagonalization and quantum Monte Carlo combined with stochastic analytic continuation,we study the dynamical properties of the S=1 antiferromagnetic Heisenberg chain with different strengths of bond disorder.In the weak disorder region,we find weakly coupled bonds which can induce additional low-energy excitation below the one-magnon mode.As the disorder increases,the average Haldane gap closes at δ_(∆)~0.5 with more and more low-energy excitations coming out.After the critical disorder strength δ_(C)~1,the system reaches a random-singlet phase with prominent sharp peak atω=0 and broad continuum atω>0 of the dynamic spin structure factor.In addition,we analyze the distribution of random spin domains and numerically find three kinds of domains hosting effective spin-1/2 quanta or spin-1 sites in between.These“spins”can form the weakly coupled longrange singlets due to quantum fluctuation which contribute to the sharp peak atω=0.

General interaction quenches in a Luttinger liquid

We discuss a general interaction quench in a Luttinger liquid described by a paired bosonic Hamiltonian.By employing su(1,1)Lie algebra,the post-quench time-evolved wavefunctions are obtained analytically,from which the time evolution of the entanglement in momentum space can be investigated.We note that depending on the choice of Bogoliubov quasiparticles,the expressions of wavefunctions,which describe time-evolved paired states,can take different forms.The correspondence between the largest entanglement eigenvalue in momentum space and the wavefunction overlap in quench dynamics is discussed,which generalizes the results of Dora et al(2016,Phys.Rev.Lett.117,010603).A numerical demonstration on an XXZ lattice model is presented via the exact diagonalization method.

Beam combination setup for dual-frequency laser with orthogonal linear polarization

A beam combination setup for a dual-frequency laser with orthogonal linear polarization is proposed. It consists of two polarizing beam splitters（PBSs） whose polarization axes are orthogonal to each other. A theoretical analysis demonstrates that a combined dual-frequency laser beam with this setup strictly meets orthogonal linear relation. The experimental results show that compared with the conventional setup, the ellipticity and nonorthogonality of the combined dual-frequency laser beam are significantly reduced.