Matrix Product States Algorithms and Continuous Systems

A generic method to investigate many-body continuous-variable systems is pedagogically presented. It is based on the notion of matrix product states (so-called MPS) and the algorithms thereof. The method is quite versatile and can be applied to a wide variety of situations. As a first test, we show how it provides reliable results in the computation of fundamental properties of a chain of quantum harmonic oscillators achieving off-critical and critical relative errors of the order of 10^(-8) and 10^(-4) respectively. Next, we use it to study the ground state properties of the quantum rotor model in one spatial dimension, a model that can be mapped to the Mott insulator limit of the 1-dimensional Bose-Hubbard model. At the quantum critical point, the central charge associated to the underlying conformal field theory can be computed with good accuracy by measuring the finite-size corrections of the ground state energy. Examples of MPS-computations both in the finite-size regime and in the thermodynamic limit are given. The precision of our results are found to be comparable to those previously encountered in the MPS studies of, for instance, quantum spin chains. Finally, we present a spin-off application: an iterative technique to efficiently get numerical solutions of partial differential equations of many variables. We illustrate this technique by solving Poisson-like equations with precisions of the order of 10^(-7).Comment: 22 pages, 14 figures, final versio

Finite-size scaling exponents and entanglement in the two-level BCS model

We analyze the finite-size properties of the two-level BCS model. Using the continuous unitary transformation technique, we show that nontrivial scaling exponents arise at the quantum critical point for various observables such as the magnetization or the spin-spin correlation functions. We also discuss the entanglement properties of the ground state through the concurrence which appears to be singular at the transition.Comment: 4 pages, 3 figures, published versio

Scaling of entanglement between separated blocks in spin chains at criticality

We compute the entanglement between separated blocks in certain spin models showing that at criticality this entanglement is a function of the ratio of the separation to the length of the blocks and can be written as a product of a power law and an exponential decay. It thereby interpolates between the entanglement of individual spins and blocks of spins. It captures features of correlation functions at criticality as well as the monogamous nature of entanglement. We exemplify invariant features of this entanglement to microscopic changes within the same universality class. We find this entanglement to be invariant with respect to simultaneous scale transformations of the separation and the length of the blocks. As a corollary, this study estimates the entanglement between separated regions of those quantum fields to which the considered spin models map at criticality.Comment: 4 pages, 3 figures; comments welcom

Entanglement and Quantum Phase Transition Revisited

We show that, for an exactly solvable quantum spin model, a discontinuity in the first derivative of the ground state concurrence appears in the absence of quantum phase transition. It is opposed to the popular belief that the non-analyticity property of entanglement (ground state concurrence) can be used to determine quantum phase transitions. We further point out that the analyticity property of the ground state concurrence in general can be more intricate than that of the ground state energy. Thus there is no one-to-one correspondence between quantum phase transitions and the non-analyticity property of the concurrence. Moreover, we show that the von Neumann entropy, as another measure of entanglement, can not reveal quantum phase transition in the present model. Therefore, in order to link with quantum phase transitions, some other measures of entanglement are needed.Comment: RevTeX 4, 4 pages, 1 EPS figures. some modifications in the text. Submitted to Phys. Rev.

El modelo de Ausubel en la didáctica de la física : una aproximanción experimental al proceso de E/A de contenidos que presentan constructos poco elaborados por los aprendices

This paper shows the results of a research on physics education about «Magnetism» in secondary school. We investigate a constructivist approach to instructional design based upon Ausubel's model in the treatment of subjects on which pupils have poor conceptual representations and alternative frameworks. The first results with respect to inmediate and lasting learning, are shown

Superballistic Diffusion of Entanglement in Disordered Spin Chains

We study the dynamics of a single excitation in an infinite XXZ spin chain, which is launched from the origin. We study the time evolution of the spread of entanglement in the spin chain and obtain an expression for the second order spatial moment of concurrence, about the origin, for both ordered and disordered chains. In this way, we show that a finite central disordered region can lead to sustained superballistic growth in the second order spatial moment of entanglement within the chain.Comment: 5 pages, 1 figur

Boundary and impurity effects on entanglement of Heisenberg chains

We study entanglement of a pair of qubits and the bipartite entanglement between the pair and the rest within open-ended Heisenberg $XXX$ and XY models. The open boundary condition leads to strong oscillations of entanglements with a two-site period, and the two kinds of entanglements are 180 degree out of phase with each other. The mean pairwise entanglement and ground-state energy per site in the $XXX$ model are found to be proportional to each other. We study the effects of a single bulk impurity on entanglement, and find that there exists threshold values of the relative coupling strength between the impurity and its nearest neighbours, after which the impurity becomes pairwise entangled with its nearest neighbours.Comment: 6 pages and 6 figure

Quantum data compression, quantum information generation, and the density-matrix renormalization group method

We have studied quantum data compression for finite quantum systems where the site density matrices are not independent, i.e., the density matrix cannot be given as direct product of site density matrices and the von Neumann entropy is not equal to the sum of site entropies. Using the density-matrix renormalization group (DMRG) method for the 1-d Hubbard model, we have shown that a simple relationship exists between the entropy of the left or right block and dimension of the Hilbert space of that block as well as of the superblock for any fixed accuracy. The information loss during the RG procedure has been investigated and a more rigorous control of the relative error has been proposed based on Kholevo's theory. Our results are also supported by the quantum chemistry version of DMRG applied to various molecules with system lengths up to 60 lattice sites. A sum rule which relates site entropies and the total information generated by the renormalization procedure has also been given which serves as an alternative test of convergence of the DMRG method.Comment: 8 pages, 7 figure