Ground state fidelity in bond-alternative Ising chains with Dzyaloshinskii-Moriya interactions

A systematic analysis is performed for quantum phase transitions in a bond-alternative one-dimensional Ising model with a Dzyaloshinskii-Moriya (DM) interaction by using the fidelity of ground state wave functions based on the infinite matrix product states algorithm. For an antiferromagnetic phase, the fidelity per lattice site exhibits a bifurcation, which shows spontaneous symmetry breaking in the system. A critical DM interaction is inversely proportional to an alternating exchange coupling strength for a quantum phase transition. Further, a finite-entanglement scaling of von Neumann entropy with respect to truncation dimensions gives a central charge c = 0.5 at the critical point.Comment: 6 pages, 4 figure

Higher-order principal component analysis for the approximation of tensors in tree-based low-rank formats

This paper is concerned with the approximation of tensors using tree-based tensor formats, which are tensor networks whose graphs are dimension partition trees. We consider Hilbert tensor spaces of multivariate functions defined on a product set equipped with a probability measure. This includes the case of multidimensional arrays corresponding to finite product sets. We propose and analyse an algorithm for the construction of an approximation using only point evaluations of a multivariate function, or evaluations of some entries of a multidimensional array. The algorithm is a variant of higher-order singular value decomposition which constructs a hierarchy of subspaces associated with the different nodes of the tree and a corresponding hierarchy of interpolation operators. Optimal subspaces are estimated using empirical principal component analysis of interpolations of partial random evaluations of the function. The algorithm is able to provide an approximation in any tree-based format with either a prescribed rank or a prescribed relative error, with a number of evaluations of the order of the storage complexity of the approximation format. Under some assumptions on the estimation of principal components, we prove that the algorithm provides either a quasi-optimal approximation with a given rank, or an approximation satisfying the prescribed relative error, up to constants depending on the tree and the properties of interpolation operators. The analysis takes into account the discretization errors for the approximation of infinite-dimensional tensors. Several numerical examples illustrate the main results and the behavior of the algorithm for the approximation of high-dimensional functions using hierarchical Tucker or tensor train tensor formats, and the approximation of univariate functions using tensorization

Re-entrance and entanglement in the one-dimensional Bose-Hubbard model

Re-entrance is a novel feature where the phase boundaries of a system exhibit a succession of transitions between two phases A and B, like A-B-A-B, when just one parameter is varied monotonically. This type of re-entrance is displayed by the 1D Bose Hubbard model between its Mott insulator (MI) and superfluid phase as the hopping amplitude is increased from zero. Here we analyse this counter-intuitive phenomenon directly in the thermodynamic limit by utilizing the infinite time-evolving block decimation algorithm to variationally minimize an infinite matrix product state (MPS) parameterized by a matrix size chi. Exploiting the direct restriction on the half-chain entanglement imposed by fixing chi, we determined that re-entrance in the MI lobes only emerges in this approximate when chi >= 8. This entanglement threshold is found to be coincident with the ability an infinite MPS to be simultaneously particle-number symmetric and capture the kinetic energy carried by particle-hole excitations above the MI. Focussing on the tip of the MI lobe we then applied, for the first time, a general finite-entanglement scaling analysis of the infinite order Kosterlitz-Thouless critical point located there. By analysing chi's up to a very moderate chi = 70 we obtained an estimate of the KT transition as t_KT = 0.30 +/- 0.01, demonstrating the how a finite-entanglement approach can provide not only qualitative insight but also quantitatively accurate predictions.Comment: 12 pages, 8 figure

Anomalous diffusion on a fractal mesh

An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets of both backbones and fingers are constructed on the entire (infinite) $y$ and $x$ axises. To this end we suggested a special algorithm of this special construction. The transport properties of the fractal mesh is studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $\langle x^2(t)\rangle\sim t^{\beta}$, where the transport exponent $\beta<1$ is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with $\beta>1$ has been observed as well when the environment is controlled by means of a memory kernel

Balanced Pod for Model Reduction of Linear PDE Systems: Convergence Theory

We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3), 997-1013, 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For exponentially stable systems with bounded finite rank input and output operators, we prove that the balanced realization can be expressed in terms of balancing modes, which are related to the Hankel operator. The balancing modes are required to be smooth, and this can cause computational difficulties for PDE systems. We show how this smoothness requirement can be lessened for parabolic systems, and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems. We prove entrywise convergence of the matrices in the approximate reduced order models in both cases, and present numerical results for two example PDE systems

On a game-theoretic semantics for the Dialectica interpretation of analysis

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2017-2018, Tutor: Joost J. JoostenGödel's Dialectica interpretation is a tool of practical interest within proof theory. Although it was initially conceived in the realm of Hilbert's program, after Kreisel's fundamental work in the 1950's it has become clear that Dialectica, as well as other popular interpretations, can be used to extract explicit bounds and approximations from classical proofs in analysis. The program that was then started, consisting of using methods of proof theory to analyse and extract new information from classical proofs, is called proof mining. The first extension of the Dialectica interpretation to analysis was achieved by Spector by means of a principle called bar recursion. Recently, Escardó and Oliva presented a new extension using a principle called "product of selection functions", which provides a game-theoretic semantics to the interpreted theorems of analysis. This eases the task of understanding the constructive content and meaning of classical proofs, instead of only extracting quantitative information from them. In this thesis we present the Dialectica interpretation and its extensions to analysis, both using bar recursion and the product of selection functions. A whole chapter is thus devoted to exposing the theory of sequential games by Escardó and Oliva. In their paper "A Constructive Interpretation of Ramsey's Theorem via the Product of Selection Functions", Oliva and Powell gave a constructive proof of the Dialectica interpretation of the Infinite Ramsey Theorem for pairs and two colours using the product of selection functions. This yields an algorithm, which can be understood in game-theoretic terms, computing arbitrarily good approximations to the infinite monochromatic set. In this thesis we revisit this paper, extending all the results for the case of any finite number of colours