Matrix Product Form Research Articles

Exact Matrix Product States at the Quantum Lifshitz Tricritical Point in a Spin-1/2 Zigzag-Chain Antiferromagnet with Anisotropic Γ Term.

Quantum anisotropic exchange interactions in magnets can induce competitions between phases in a different manner from those typically driven by geometrically frustrated interactions. We study a one-dimensional spin-1/2 zigzag chain with such an interaction, Γ term, in conjunction with the Heisenberg interactions. We find a ground state phase diagram featuring a multicritical point where five phases converge: a uniform ferromagnet, two antiferromagnets, Tomonaga-Luttinger liquid, and a dimer-singlet coexisting with nematic order. This multicritical point is simultaneously quantum tricritical and Lifshitz, and most remarkably, it hosts multidegenerate ground state wave functions with the degeneracy increasing in squares of system size. The exact ground states are obtained in the matrix product form opening wide applications to frustration-free models.

On two reversible cellular automata with two particle species

We introduce a pair of time-reversible models defined on the discrete space–time lattice with three states per site, specifically, a vacancy and a particle of two flavours (species). The local update rules reproduce the rule 54 reversible cellular automaton when only a single species of particles is present, and satisfy the requirements of flavour exchange (C), space-reversal (P), and time-reversal (T) symmetries. We find closed-form expressions for three local conserved charges and provide an explicit matrix product form of the grand canonical Gibbs states, which are identical for both models. For one of the models this family of Gibbs states seems to be a complete characterisation of equilibrium (i.e. space and time translation invariant) states, while for the other model we empirically find a sequence of local conserved charges, one for each support size larger than 2, hinting to its algebraic integrability. Finally, we numerically investigate the behaviour of spatio-temporal correlation functions of charge densities, and test the hydrodynamic prediction for the model with exactly three local charges. Surprisingly, the numerically observed ‘sound velocity’ does not match the hydrodynamic value. The deviations are either significant, or they decay extremely slowly with the simulation time, which leaves us with an open question for the mechanism of such a glassy behaviour in a deterministic locally interacting system.

Multi-Modal Image Fusion Based on Matrix Product State of Tensor.

Multi-modal image fusion integrates different images of the same scene collected by different sensors into one image, making the fused image recognizable by the computer and perceived by human vision easily. The traditional tensor decomposition is an approximate decomposition method and has been applied to image fusion. In this way, the image details may be lost in the process of fusion image reconstruction. To preserve the fine information of the images, an image fusion method based on tensor matrix product decomposition is proposed to fuse multi-modal images in this article. First, each source image is initialized into a separate third-order tensor. Then, the tensor is decomposed into a matrix product form by using singular value decomposition (SVD), and the Sigmoid function is used to fuse the features extracted in the decomposition process. Finally, the fused image is reconstructed by multiplying all the fused tensor components. Since the algorithm is based on a series of singular value decomposition, a stable closed solution can be obtained and the calculation is also simple. The experimental results show that the fusion image quality obtained by this algorithm is superior to other algorithms in both objective evaluation metrics and subjective evaluation.

Tilt angle correction based forming algorithm for high steepness aspheric surfaces polishing

According to the varying characteristics of removal function in steep aspheric surface polishing, the calculation algorithm of removal function at arbitrary polishing position was given in the best fitting plane of steep aspheric based on the experiment of removal function with several inclined angles and the feature of polishing surface. The forming model was then discretized into the form of large sparse matrix product, and the dwell time was solved efficiently by SBB algorithm. The discrete matrix model describing nonlinear polishing process of steep aspheric surface was established, and finally the process flow was verified by experiment. The experiment was conducted on a 200 mm aperture off-axis aspheric surface, finally the root mean square error of the surface was 0.022 λ and the processing convergence rate was 3.23 after two polishing runs. The research results show that the forming model with plane projection and tilt-angle-correction removal function is reasonable, and the dwell time solving algorithm based on SBB principle is a fast and efficient surface shape control technology, which could fulfill deterministic and accurate shaping for steep aspheric surface.

Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

Numerical analysis of fractional partial differential equations applied to polymeric visco-elastic Euler-Bernoulli beam under quasi-static loads

• The governing equation of visco-elastic beam based on fractional derivativeis established. • The numerical solutions of deflection, stress and strain of the polymeric beam are obtained directlyin the time domain. • Error correction is proposed to improve the accuracy of the numerical algorithm. • The bending resistance of the beam with two viscoelastic polymers is discussed. In this paper, an effective numerical algorithm based on shifted Chebyshev polynomials is proposed to solve the fractional partial differential equations applied to polymeric visco-elastic problems in the time-space domain under quasi-static loads. The governing equations using local fractional rheological models based on visco-elastic properties with fractional derivatives are established. The integer and fractional differential operator matrices of polynomials are derived according to the properties of shifted Chebyshev polynomials. The fractional order governing equation is rewritten into the form of matrix product by using the polynomial to approximate the unknown function. The collocation method is used to discretize the variables and transform the original problem into an algebraic equation system. The numerical solutions of the governing equations are obtained directly in the time-domain. In addition, an error analysis including the correction method is performed. The numerical examples have been performed to identify the sensibility of the proposed governing equations and to evaluate the efficiency and accuracy of the proposed algorithm.

Matrix product state of multi-time correlations

For an interacting spatio-temporal lattice system we introduce a formal way of expressing multi-time correlation functions of local observables located at the same spatial point with a time state, i.e. a statistical distribution of configurations observed along a time lattice. Such a time state is defined with respect to a particular equilibrium state that is invariant under space and time translations. The concept is developed within the rule 54 reversible cellular automaton, for which we explicitly construct a matrix product form of the time state, with matrices that act on the three-dimensional auxiliary space. We use the matrix-product state to express equal-space time-dependent density-density correlation function, which, for special maximum-entropy values of equilibrium parameters, agrees with the previous results. Additionally, we obtain an explicit expression for the probabilities of observing all multi-time configurations, which enables us to study distributions of times between consecutive excitations and prove the absence of decoupling of timescales in the rule 54 model.

Pericellular hyaluronan modulates IL-2 responsiveness through CD44

Abstract CD44 is ubiquitously expressed on human T cells. Stimulation through TCR results in upregulation of CD44. Through interactions with an extracellular matrix ligand, hyaluronan, CD44 classically mediates cell-cell contact for promoting adherence and migration. Here we report a distinct function of CD44 in regulating cellular production and responsiveness to the T cell growth factor IL-2. Overexpression of CD44 in IL-2 responsive cell lines results in a decreased responsiveness to IL-2. Upon CD44 cross-linking, we observe an increased sensitivity to IL-2. The enhanced response to IL-2 is accompanied by diminished Socs1 expression. These data suggest interactions between CD44 and pericellular hyaluronan function to alter cellular responses to IL-2. We propose that this is an underappreciated way inflammation, in the form of an extracellular matrix product, contributes to clonal T cell expansion.

Sum of characteristic polynomial coefficients of cycloparaphenylene graphs as topological index

ABSTRACT Algorithm for obtaining characteristic polynomial (CP) coefficients of an alternant edge-weighted cycle is used to arrive at the algorithm for that of the cycloparaphenylene (CPP) graphs in matrix product form. The algorithm gives a recursive relation in expressing the sum of the CP coefficients of a CPP in terms of that of its two immediately preceding analogues which ultimately ends up with the use of transfer matrix in an analytical form. The sum of CP coefficients, being combinatorial in nature, is found to be used as a topological index showing much similarity with Hosoya index (sum all matching polynomial coefficients), cardinality and number of Kekulé valence structures of CPP graphs compared to the Wiener index which is the distance sum of all pairs of vertices in the graph. The sum of CP coefficients has been found to model the physical properties like strain energy and diameter of CPPs that are verified by the respective excellent correlations.

Matrix product representation of the stationary state of the open zero range process

Many one-dimensional lattice particle models with open boundaries, like the paradigmatic asymmetric simple exclusion process (ASEP), have their stationary states represented in the form of a matrix product, with matrices that do not explicitly depend on the lattice site. In contrast, the stationary state of the open 1D zero-range process (ZRP) takes an inhomogeneous factorized form, with site-dependent probability weights. We show that in spite of the absence of correlations, the stationary state of the open ZRP can also be represented in a matrix product form, where the matrices are site-independent, non-commuting and determined from algebraic relations resulting from the master equation. We recover the known distribution of the open ZRP in two different ways: first, using an explicit representation of the matrices and boundary vectors; second, from the sole knowledge of the algebraic relations satisfied by these matrices and vectors. Finally, an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden Markov chains.

Matrix product solution to multi-species ASEP with open boundaries

We study a class of multi-species ASEP with open boundaries. The boundaries are chosen in such a way that all species of particles interact non-trivially with the boundaries, and are present in the stationary state. We give the exact expression of the stationary state in a matrix product form, and compute its normalisation. Densities and currents for the different species are then computed in terms of this normalisation.

Integrable Floquet dynamics, generalized exclusion processes and “fused” matrix ansatz

We present a general method for constructing integrable stochastic processes, with two-step discrete time Floquet dynamics, from the transfer matrix formalism. The models can be interpreted as a discrete time parallel update. The method can be applied for both periodic and open boundary conditions. We also show how the stationary distribution can be built as a matrix product state. As an illustration we construct parallel discrete time dynamics associated with the R-matrix of the SSEP and of the ASEP, and provide the associated stationary distributions in a matrix product form. We use this general framework to introduce new integrable generalized exclusion processes, where a fixed number of particles is allowed on each lattice site in opposition to the (single particle) exclusion process models. They are constructed using the fusion procedure of R-matrices (and K-matrices for open boundary conditions) for the SSEP and ASEP. We develop a new method, that we named “fused” matrix ansatz, to build explicitly the stationary distribution in a matrix product form. We use this algebraic structure to compute physical observables such as the correlation functions and the mean particle current.

Determination of Parameters Influencing Thermal Comfort in a Building

This paper presents a theoretical and an experimental study on the determination of thermal phase shift time and damping factors of building walls. The work consisted in evaluating the thermal comfort in a building. This study is a contribution to put in place a thermal regulation in Burkina Faso, a country with a dry tropical climate. This regulation will, in its application, reduce the energy consumption related to air conditioning in buildings. To do this, we theoretically determine the parameters influencing this comfort using the admittance method. This method makes it possible to analyze the thermal response of a building to a periodic energy demand. It is based on the modeling of the building component by the quadrupole method resulting from the writing of the thermal problem in the form of transfer matrix products. The principle is to link the flows and temperatures of both sides of a homogeneous wall through the equation of heat and the Fourier transform. The calculation method is validated by experimental results. For simple sheet walls based on their compositions, damping factors range from about 37.9% to 79.7% and thermal phase shift time from about 3.9 hours to 7.4 hours. However, the values of these parameters are better when the walls are double-sheeted, the maximum value of the damping factor is 11.5% and the minimum value of the thermal phase shift time is 9.7 h. The values of Mean Bias Difference (MBD) and Root Mean Square Difference (RMSD) show that the theoretical and experimental results are quite concordant. The admittance method can be used to evaluate the performance of building walls.

Exact Solution to Integrable Open Multi-species SSEP and Macroscopic Fluctuation Theory

We introduce a multi-species generalization of the symmetric simple exclusion process with open boundaries. This model possesses the property of being integrable and appears as physically relevant because the boundary conditions can be interpreted as the interaction with particles reservoirs with fixed densities of each species. The system is driven out-of-equilibrium by these reservoirs. The steady state is analytically computed in a matrix product form. This algebraic structure allows us to obtain exact expressions for the mean particle currents and for the one and two-point correlation functions. An additivity principle is also derived from the matrix ansatz and permits the computation of the large deviation functional of the density profile. We also propose a description of the model in the context of the macroscopic fluctuation theory and we check the consistency with the exact computations from the finite size lattice.

Detecting topological order with ribbon operators

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with abelian topological order by finding ribbon operators for $\mathbb{Z}_d$ quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting nonabelian topological order.

Integrable dissipative exclusion process: Correlation functions and physical properties.

We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory.

Inhomogeneous Generalization of a Multispecies Totally Asymmetric Zero Range Process

The $n$-species totally asymmetric zero range process ($n$-TAZRP) on one-dimensional periodic chain studied recently by the authors is a continuous time Markov process where arbitrary number of particles can occupy the same sites and hop to the adjacent sites only in one direction with a priority constraint according to their species. In this paper we introduce an $n$-parameter generalization of the $n$-TAZRP having inhomogeneous transition rate. The steady state probability is obtained in a matrix product form and also by an algorithm related to combinatorial $R$.

An efficient matrix product operator representation of the quantum chemical Hamiltonian.

We describe how to efficiently construct the quantum chemical Hamiltonian operator in matrix product form. We present its implementation as a density matrix renormalization group (DMRG) algorithm for quantum chemical applications. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - abelian and non-abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program.

Open two-species exclusion processes with integrable boundaries

We give a complete classification of integrable Markovian boundary conditions for the asymmetric simple exclusion process with two species (or classes) of particles. Some of these boundary conditions lead to non-vanishing particle currents for each species. We explain how the stationary state of all these models can be expressed in a matrix product form, starting from two key components, the Zamolodchikov–Faddeev and Ghoshal–Zamolodchikov relations. This statement is illustrated by studying in detail a specific example, for which the matrix ansatz (involving nine generators) is explicitly constructed and physical observables (such as currents, densities) calculated.

Z3symmetry-protected topological phases in the SU(3) AKLT model

We study $\mathbb{Z}_3$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with $Z_3 \times Z_3$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial $\mathbb{Z}_3$ phases and their parent Hamiltonian from a cocycle of the group cohomology $H^2(Z_3\times Z_3,U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU($N$) case, the SU($N$) AKLT Hamiltonian, is also presented which realizes nontrivial $\mathbb{Z}_N$ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\theta$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial $\mathbb{Z}_3$ SPT phase is found for a range of the parameter $\theta$ including the point of vanishing biquadratic term ($\theta=0$) as well as the SU(3) AKLT point [$\theta=\arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\theta \approx -0.027\pi$, with a central charge $c\approx3.2$. For SU(3) symmetric cases we define string order parameters for the $\mathbb{Z}_3$ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.