A variational framework for the inverse Henderson problem of statistical mechanics

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse of dimensionality is addressed by parametrizing the many-body state in terms of a network of partially contracted tensors. These tensors form a substantially reduced set of effective degrees of freedom. In practical algorithms, functionals like energy expectation values or overlaps are optimized over certain sets of TNS. Concerning algorithmic stability, it is important whether the considered sets are closed because, otherwise, the algorithms may approach a boundary point that is outside the TNS set and tensor elements diverge. We discuss the closedness and geometries of TNS sets, and we propose regularizations for optimization problems on non-closed TNS sets. We show that sets of matrix product states (MPS) with open boundary conditions, tree tensor network states, and the multiscale entanglement renormalization ansatz are always closed, whereas sets of translation-invariant MPS with periodic boundary conditions (PBC), heterogeneous MPS with PBC, and projected entangled pair states are generally not closed. The latter is done using explicit examples like the W state, states that we call two-domain states, and fine-grained versions thereof.

     

Computational difficulty of finding matrix product ground states

We determine the computational difficulty of finding ground states of one-dimensional (1D) Hamiltonians, which are known to be matrix product states (MPS). To this end, we construct a class of 1D frustration-free Hamiltonians with unique MPS ground states and a polynomial gap above, for which finding the ground state is at least as hard as factoring. Without the uniqueness of the ground state, the problem becomes NP complete, and thus for these Hamiltonians it cannot even be certified that the ground state has been found. This poses new bounds on convergence proofs for variational methods that use MPS.     

Enhancement of stability and accuracy of the moving particle semi-implicit method

As a Lagrangian gridless particle method, the MPS (Moving Particle Semi-implicit) method has been proven useful in a wide range of engineering applications. Up to now, most of MPS applications have been limited to problems with compressive stress states. This paper investigates the performance and stability of MPS method in simulation of more general hydrodynamic problems, including those characterized by tensile stresses or by changes in the stress states. It is shown that MPS-based simulations are prone to become destabilized in presence of attractive interparticle forces. This instability appears to be similar to the so-called tensile instability in SPH method. Two new modifications, namely, a modified Poisson Pressure Equation and a corrective matrix for pressure gradient model, are proposed to resolve this shortcoming. These two new modifications together with two previously proposed improvements are shown to stabilize and enhance the performance of MPS method.     

Scale-renormalized matrix-product states for correlated quantum systems

A generalization of matrix product states (MPS) for interacting quantum systems in two and three dimensions is introduced. These scale-renormalized matrix-product states (SR-MPS) are based on a coarse graining of the lattice in which the blocks at each level are associated with matrix products that are further transformed (scale renormalized) with other matrices before they are assembled to form blocks at the next level. Using the two-dimensional transverse-field Ising model as a test, it is shown that the SR-MPS converge much more rapidly with the matrix size than a standard MPS. It is also shown that the use of lattice symmetries speeds up the convergence very significantly.     

The density-matrix renormalization group in the age of matrix product states

The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms.     

On the properties of a periodic fluid

The Born-Green equation is used to clarify and estimate the error introduced into Monte Carlo and molecular dynamics simulations of dense fluids by the use of periodic boundary conditions. This theory is applied to the Lennard-Jones fluid and the theoretical predictions are found to be in reasonable agreement with experiment. The implications for size dependence of pressure, for anisotropy of the radial distribution function, for size dependence of the supercooling limit for crystal nucleation, and for the orientations of nucleated crystals are discussed.This research was supported under grants MPS73-04743 and MPS73-04922 of the National Science Foundation, Washington D.C.     

Electronic properties of silicon with ultrasmall germanium clusters

The electronic states of silicon with a periodic array of spherical germanium clusters are studied within the pseudopotential approach. The effects of quantum confinement in the energies and wave functions of the localized cluster states are analyzed. It is demonstrated that clusters up to 2.4 nm in size produce one localized s state whose energy monotonically shifts deep into the silicon band gap as the cluster size increases. The wave function of the cluster level corresponds to the single-valley approximation of the effective-mass method. In the approximation of an abruptly discontinuous potential at the heterointerface, the quantities calculated using the effective-mass method for clusters containing more than 200 Ge atoms are close to those obtained by the pseudopotential method. For smaller clusters, it is necessary to take into account the smooth potential at the interface.     

Calculation of electronic density of states using a self-consistent cluster-Bethe-lattice method

The electronic density of states for a periodic crystal is calculated by embedding a cluster of atoms in a Bethe-lattice medium. The structure of the Bethe-lattice Hamiltonian is determined by self-consistency requirements. The results for the density of states using this self-consistent cluster-Bethe-lattice method show excellent agreement with the exact density of states and significant improvement over previous calculations using the cluster-Bethe-lattice approach.     

Equations of motion in the null formalism

An outline is presented for an approach to a two-body problem. The method uses null coordinates attached to one body and an expansion away from flat space. Unfortunately, the method fails to exhibit the Newtonian interaction between two particles. Therefore, no detailed calculations are given.Research supported in part by National Science Foundation under grant No. MPS74-15246.Based in part on a dissertation for the Ph.D. at Syracuse University, June 1971.     

非牛顿流体二维溃坝问题的数值模拟

移动粒子半隐式法(Moving Particle Semi-implicit,MPS)是一种广泛应用于不可压缩自由表面流动的粒子方法。本文通过把非牛顿流体转化为变黏度牛顿流体的处理方法将MPS方法拓展至非牛顿自由表面流动,以Casson流体和Cross流体为例计算了非牛顿流体的二维溃坝问题。将计算结果与前人数据进行了对比,结果吻合较好。同时对比了非牛顿流体与牛顿流体的计算结果,发现非牛顿流体的溃坝前端发展速度较慢。     

2D electrons in lateral superlattice in a strong B⊥

A periodic 1D potential acting upon the surface electrons on vicinal planes smears up the Landau levels into energy bands. The density of states singularities result in a specific line shape in cyclotron absorption and inelastic light scattering. In both cases there are two maxima caused by the density of states increasing at the band edges.     

Matrix Product State Simulations of Non-Equilibrium Steady States and Transient Heat Flows in the Two-Bath Spin-Boson Model at Finite Temperatures

Simulating the non-perturbative and non-Markovian dynamics of open quantum systems is a very challenging many body problem, due to the need to evolve both the system and its environments on an equal footing. Tensor network and matrix product states (MPS) have emerged as powerful tools for open system models, but the numerical resources required to treat finite-temperature environments grow extremely rapidly and limit their applications. In this study we use time-dependent variational evolution of MPS to explore the striking theory of Tamascelli et al. (Phys. Rev. Lett. 2019, 123, 090402.) that shows how finite-temperature open dynamics can be obtained from zero temperature, i.e., pure wave function, simulations. Using this approach, we produce a benchmark dataset for the dynamics of the Ohmic spin-boson model across a wide range of coupling strengths and temperatures, and also present a detailed analysis of the numerical costs of simulating non-equilibrium steady states, such as those emerging from the non-perturbative coupling of a qubit to baths at different temperatures. Despite ever-growing resource requirements, we find that converged non-perturbative results can be obtained, and we discuss a number of recent ideas and numerical techniques that should allow wide application of MPS to complex open quantum systems.     

THE CHARACTERISTIC RECEPTANCE METHOD FOR DAMAGE DETECTION IN LARGE MONO-COUPLED PERIODIC STRUCTURES

An approach based on the concept of wave propagation to detect the structural damage in large mono-coupled periodic structures is presented in this paper. The free vibration analysis of a finite mono-coupled periodic structure with a single disorder has been conducted by the characteristic receptance method, and the sensitivity of the natural frequencies to the disorder in flexibility has been discussed. Based on the sensitivity analysis, the locations and magnitude of damage in large mono-coupled periodic systems have been estimated using measured changes in the natural frequencies. The paper also introduces a substructure-based method for improving the computational efficiency and the accuracy of damage detection in large mono-coupled periodic structures. Numerical results from two periodic mass-spring-structures show that the proposed method can provide good predictions of both the locations and magnitude of damage at one or more sites. Furthermore, the proposed method, in which a priori information about the nature such as stiffness of the undamaged structure is not needed, and only measurements of the change in a few of the structure's natural frequencies between the undamaged and damaged states are required, is particularly attractive in practice. However, some issues such as the role of noise in actual measurements, application to multi-coupled periodic structures with complex boundary conditions remain to be resolved before this approach becomes a truly variable method of structural damage assessment.     

A three-dimensional frequency selective structure based on the modes coupling of spoof surface plasmon and waveguide transmission

A three-dimensional frequency selective structure (3D FSS) was proposed by applying the spoof surface plasmon (SSP) structures into periodic waveguides, which realizes an enhanced broadband transmission in low frequencies, below the cut-off frequency of periodic waveguides. The transmission characteristics of SSP structures and periodic waveguides as well as their coupling modes are investigated by simulation and experiment method. According to our analysis, the pass-band of this 3D FSS can be customized through changing the values of structure parameters and it also has a good angular stability. In addition, this paper also provides a simple assembly plan for sample preparation. This proposal may be useful for improving filtering performances and gives an effective approach for designing 3D FSSs.     

Ground-state approximation for strongly interacting spin systems in arbitrary spatial dimension

We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g., energy) and include states with diverging correlation length and unbounded multiparticle entanglement. As a demonstration, we apply our approach to the Ising model on 1D, 2D, and 3D square lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories.     

Relaxation mode analysis of nonlinear birth and death processes

Nonlinear birth and death processes with one variable are considered. The general master equations describing these processes are analyzed in terms of their eigenmodes and eigenvalues using the method of a WKB approximation. Formulas for the density of eigenstates are obtained. The lower lying eigenmodes are calculated to investigate long-time relaxation, such as relaxations of metastable and unstable states. Anomalous accumulation of the lower lying eigenvalues is shown to exist when the system is infinitesimally close to a critical or marginal state. The general results obtained are applied to some instructive examples, such as the kinetic Weiss-Ising model and a stochastic model of nonlinear chemical reactions.The work at UCSD was supported in part by the National Science Foundation under Grants MPS72-04363A03 and CHE75-20624.Adapted from the author's Ph.D. dissertation at University of Tokyo, December 1974.     

一种抖动与孤立像素点分离相结合的灰度级增强方法

为了使由等离子体显示器(PDP)反γ校正和有限灰度级显示方法引起的灰度损失得到改善,在分析灰度损失原因及有序抖动和随机抖动特点的基础上,提出了一种抖动与孤立像素点分离相结合的灰度级增强方法。该方法首先采用有序抖动和孤立像素点分离对γ校正后的图像进行灰度增强;然后把图像数据映射到有限灰度级,计算有限灰度级与需要显示的灰度级之间的误差;最后采用随机抖动和孤立像素点分离对有限灰度级图像进行灰度增强。实验结果表明,反γ校正和有限灰度级图像的灰度均得到了显著增强,低灰度级和中高灰度级过渡平滑,没有周期性抖动噪声,孤立像素点的亮度显著降低,图像显示效果明显优于仅采用随机抖动或有序抖动的图像。     

Quantum phase transitions about parity breaking in matrix product systems

According to our scheme to construct quantum phase transitions (QPTs) in spin chain systems with matrix product ground states, we first successfully combine matrix product state (MPS) QPTs with spontaneous symmetry breaking. For a concrete model, we take into account a kind of MPS QPTs accompanied by spontaneous parity breaking, though for either side of the critical point the GS is typically unique, and show that the kind of MPS QPTs occur only in the thermodynamic limit and are accompanied by the appearance of singularities, diverging correlation length, vanishing energy gap and the entanglement entropy of a half-infinite chain not only staying finite but also whose first derivative discontinuous.     

Forcing and control of localized states in optical single feedback systems

Under feedback extended nonlinear optical systems spontaneously show a variety of periodic patterns and structures. Control gives new insight into these phenomena and it can open the way for potential application of nonlinear optical structures. We briefly review methods to control localized states in single feedback experiments. Application of a Fourier control method allows to modify interaction behavior of the localized states. As a further approach we study a forcing method, using externally created light fields as additional input to the system. Recent experiments show that the forcing method enables to favor addressing positions for localized structures. We demonstrate static addressing and favoring of addressing positions. We extend the forcing method to a dynamic forcing scheme, which allows to move and reposition localized states. Additionally forcing is used to balance experimental imperfections. PACS 05.45.Gg; 42.60.Jf; 42.65.Tg     

Bound states and band structure—A unified treatment through the quantum Hamilton-Jacobi approach

We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton-Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues are obtained without having to solve for the corresponding eigenfunctions explicitly. We also demonstrate how to find the eigenfunctions through this method.