This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals.Three(alternating direction)minimization sch...This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals.Three(alternating direction)minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model:a standard gradient descent method,the“heavy-ball”method,and Nesterov’s method.The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state,as well as the total computational time(wall time).The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly,despite the hard uniaxial constraint that is not“smooth”when defects are present.Moreover,our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.展开更多
Theories and experiments on dirty superconductors are complex but important in terms of both theoretical fundamentals and practical applications.These activities are even more challenging when magnetic fields are pres...Theories and experiments on dirty superconductors are complex but important in terms of both theoretical fundamentals and practical applications.These activities are even more challenging when magnetic fields are present because the field distribution,electron density of states,and superconducting pairing potentials become nonuniform.Here,we present tunneling microspectroscopic experiments on NbC single crystals and demonstrate that NbC is a homogeneous dirty superconductor.When applying magnetic fields to the samples,we found that the zero-energy local density of states and the pairing energy gap followed the explicit scaling relation proposed by de Gennes for homogeneous dirty superconductors in high magnetic fields.More significantly,our experimental findings indicate that the validity of the scaling relation extends to magnetic field strengths far below the upper critical field,calling for a new nonperturbative understanding of this fundamental property in dirty superconductors.On the practical side,we used the observed scaling relation to derive a simple and straightforward experimental scheme for estimating the superconducting coherence length of a dirty superconductor in magnetic fields.展开更多
Based on Landau–de Gennes theory and two-dimensional finite-difference iterative method, the spontaneous distortion in hybrid alignment nematic cells with M = ±1/2 disclination lines is investigated by establish...Based on Landau–de Gennes theory and two-dimensional finite-difference iterative method, the spontaneous distortion in hybrid alignment nematic cells with M = ±1/2 disclination lines is investigated by establishing two models. The fine structures of defect cores are described in the order space S^2/Z2. The joint action of elastic anisotropy（L2/L1） and biaxiality of defects induces the spontaneous twist distortion, accompanied by the movement of the defect center to the upper or lower plate. For each model, four mixed defect structures appear with the same energy, which are defined as energetically degenerated quadruple states.展开更多
The defect structures of s = ±1/2 twist disclinations in twisted nematic and twisted chiral liquid crystals have been investigated within the Landau-de Gennes theory numerically. Our results show that there exist...The defect structures of s = ±1/2 twist disclinations in twisted nematic and twisted chiral liquid crystals have been investigated within the Landau-de Gennes theory numerically. Our results show that there exists eigenvalue exchange across the defect core of both the two models. The defect core is essentially biaxial and never isotropic. The defect centre is uniaxial and is surrounded by a strong biaxial region.展开更多
By using a model Hamiltonian with competing antiferromagnetic （AFM） spin density wave （SDW） and d-wave superconductivity orders, the effect of next-nearest-neighbour （nnn） hopping on spin and charge structures i...By using a model Hamiltonian with competing antiferromagnetic （AFM） spin density wave （SDW） and d-wave superconductivity orders, the effect of next-nearest-neighbour （nnn） hopping on spin and charge structures in high-temperature superconductors is investigated at finite temperatures. For an optimally doped sample, we find that the AFM order magnitude in the vortex core is firstly enhanced and then suppressed, accompanied with a positively → negatively → positively＂ charged vortex structure transition with increasing nnn hopping strength, which implies that the AFM order is unnecessarily bounded to an electron-rich vortex core. In addition, a charge ordering pattern with four negatively charged peaks localized in a small region is also found around the vortex core centre without net charge. Recent scanning-tunneling-microscopy experimental observations of the checkerboard structure are hopefully understood.展开更多
基金supported in part by NSF grant DMS-1555222(CAREER).
文摘This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals.Three(alternating direction)minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model:a standard gradient descent method,the“heavy-ball”method,and Nesterov’s method.The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state,as well as the total computational time(wall time).The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly,despite the hard uniaxial constraint that is not“smooth”when defects are present.Moreover,our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.
基金supported by National Key R&D Program of China(Grant No.2022YFA1403201)National Natural Science Foundation of China(Grant No.12274205 and No.11874205).
基金supported by the National Key Research and Development Program of the Ministry of Science and Technology of China(Grant Nos.2017YFA0302903,2016YFA0300602,2016YFA0300604,and 2016YFJC010282)the National Natural Science Foundation of China(Grant Nos.11227903,12004416,and U2032204)+3 种基金the Beijing Municipal Science and Technology Commission,China(Grant Nos.Z181100004218007 and Z191100007219011)the National Basic Research Program of China(Grant No.2015CB921304)the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant Nos.XDB07000000,XDB28000000 and XDB33000000)supported by the US Department of Energy(Basic Energy Sciences Grant No.DE-FG02-99ER45747).
文摘Theories and experiments on dirty superconductors are complex but important in terms of both theoretical fundamentals and practical applications.These activities are even more challenging when magnetic fields are present because the field distribution,electron density of states,and superconducting pairing potentials become nonuniform.Here,we present tunneling microspectroscopic experiments on NbC single crystals and demonstrate that NbC is a homogeneous dirty superconductor.When applying magnetic fields to the samples,we found that the zero-energy local density of states and the pairing energy gap followed the explicit scaling relation proposed by de Gennes for homogeneous dirty superconductors in high magnetic fields.More significantly,our experimental findings indicate that the validity of the scaling relation extends to magnetic field strengths far below the upper critical field,calling for a new nonperturbative understanding of this fundamental property in dirty superconductors.On the practical side,we used the observed scaling relation to derive a simple and straightforward experimental scheme for estimating the superconducting coherence length of a dirty superconductor in magnetic fields.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11374087 and 11447179)the Key Subject Construction Project of Hebei Province University,China
文摘Based on Landau–de Gennes theory and two-dimensional finite-difference iterative method, the spontaneous distortion in hybrid alignment nematic cells with M = ±1/2 disclination lines is investigated by establishing two models. The fine structures of defect cores are described in the order space S^2/Z2. The joint action of elastic anisotropy（L2/L1） and biaxiality of defects induces the spontaneous twist distortion, accompanied by the movement of the defect center to the upper or lower plate. For each model, four mixed defect structures appear with the same energy, which are defined as energetically degenerated quadruple states.
文摘The defect structures of s = ±1/2 twist disclinations in twisted nematic and twisted chiral liquid crystals have been investigated within the Landau-de Gennes theory numerically. Our results show that there exists eigenvalue exchange across the defect core of both the two models. The defect core is essentially biaxial and never isotropic. The defect centre is uniaxial and is surrounded by a strong biaxial region.
基金supported by the National Natural Science Foundation of China (Grant No. 60671042)by Innovation Funds of Shanghai University,China
文摘By using a model Hamiltonian with competing antiferromagnetic （AFM） spin density wave （SDW） and d-wave superconductivity orders, the effect of next-nearest-neighbour （nnn） hopping on spin and charge structures in high-temperature superconductors is investigated at finite temperatures. For an optimally doped sample, we find that the AFM order magnitude in the vortex core is firstly enhanced and then suppressed, accompanied with a positively → negatively → positively＂ charged vortex structure transition with increasing nnn hopping strength, which implies that the AFM order is unnecessarily bounded to an electron-rich vortex core. In addition, a charge ordering pattern with four negatively charged peaks localized in a small region is also found around the vortex core centre without net charge. Recent scanning-tunneling-microscopy experimental observations of the checkerboard structure are hopefully understood.
