Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling

We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the (N,D) parameter plane, where N is the size of the spin chain and D is the dimension of the MPS matrices. In the first regime MPS can be used to perform finite size scaling (FSS). In the complementary regime the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large N limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS, as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendix, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.Comment: 18 pages, 13 figure

Complete devil's staircase and crystal--superfluid transitions in a dipolar XXZ spin chain: A trapped ion quantum simulation

Systems with long-range interactions show a variety of intriguing properties: they typically accommodate many meta-stable states, they can give rise to spontaneous formation of supersolids, and they can lead to counterintuitive thermodynamic behavior. However, the increased complexity that comes with long-range interactions strongly hinders theoretical studies. This makes a quantum simulator for long-range models highly desirable. Here, we show that a chain of trapped ions can be used to quantum simulate a one-dimensional model of hard-core bosons with dipolar off-site interaction and tunneling, equivalent to a dipolar XXZ spin-1/2 chain. We explore the rich phase diagram of this model in detail, employing perturbative mean-field theory, exact diagonalization, and quasiexact numerical techniques (density-matrix renormalization group and infinite time evolving block decimation). We find that the complete devil's staircase -- an infinite sequence of crystal states existing at vanishing tunneling -- spreads to a succession of lobes similar to the Mott-lobes found in Bose--Hubbard models. Investigating the melting of these crystal states at increased tunneling, we do not find (contrary to similar two-dimensional models) clear indications of supersolid behavior in the region around the melting transition. However, we find that inside the insulating lobes there are quasi-long range (algebraic) correlations, opposed to models with nearest-neighbor tunneling which show exponential decay of correlations

Time Evolution within a Comoving Window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains

We present a modification of Matrix Product State time evolution to simulate the propagation of signal fronts on infinite one-dimensional systems. We restrict the calculation to a window moving along with a signal, which by the Lieb-Robinson bound is contained within a light cone. Signal fronts can be studied unperturbed and with high precision for much longer times than on finite systems. Entanglement inside the window is naturally small, greatly lowering computational effort. We investigate the time evolution of the transverse field Ising (TFI) model and of the S=1/2 XXZ antiferromagnet in their symmetry broken phases after several different local quantum quenches. In both models, we observe distinct magnetization plateaus at the signal front for very large times, resembling those previously observed for the particle density of tight binding (TB) fermions. We show that the normalized difference to the magnetization of the ground state exhibits similar scaling behaviour as the density of TB fermions. In the XXZ model there is an additional internal structure of the signal front due to pairing, and wider plateaus with tight binding scaling exponents for the normalized excess magnetization. We also observe parameter dependent interaction effects between individual plateaus, resulting in a slight spatial compression of the plateau widths. In the TFI model, we additionally find that for an initial Jordan-Wigner domain wall state, the complete time evolution of the normalized excess longitudinal magnetization agrees exactly with the particle density of TB fermions.Comment: 10 pages with 5 figures. Appendix with 23 pages, 13 figures and 4 tables. Largely extended and improved versio

Out of equilibrium dynamics with Matrix Product States

Theoretical understanding of strongly correlated systems in one spatial dimension (1D) has been greatly advanced by the density-matrix renormalization group (DMRG) algorithm, which is a variational approach using a class of entanglement-restricted states called Matrix Product States (MPSs). However, DRMG suffers from inherent accuracy restrictions when multiple states are involved due to multi-state targeting and also the approximate representation of the Hamiltonian and other operators. By formulating the variational approach of DMRG explicitly for MPSs one can avoid errors inherent in the multi-state targeting approach. Furthermore, by using the Matrix Product Operator (MPO) formalism, one can exactly represent the Hamiltonian and other operators. The MPO approach allows 1D Hamiltonians to be templated using a small set of finite state automaton rules without reference to the particular microscopic degrees of freedom. We present two algorithms which take advantage of these properties: eMPS to find excited states of 1D Hamiltonians and tMPS for the time evolution of a generic time-dependent 1D Hamiltonian. We properly account for time-ordering of the propagator such that the error does not depend on the rate of change of the Hamiltonian. Our algorithms use only the MPO form of the Hamiltonian, and so are applicable to microscopic degrees of freedom of any variety, and do not require Hamiltonian-specialized implementation. We benchmark our algorithms with a case study of the Ising model, where the critical point is located using entanglement measures. We then study the dynamics of this model under a time-dependent quench of the transverse field through the critical point. Finally, we present studies of a dipolar, or long-range Ising model, again using entanglement measures to find the critical point and study the dynamics of a time-dependent quench through the critical point.Comment: 35 pages, 10 figures, submitted to NJP for the focus issue "Out of equilibrium dynamics in strongly interacting 1D systems

Properties and controlled release of chitosan microencapsulated limonene oil

Chitosan microcapsules containing limonene essential oil as active ingredient were prepared by coacervation using three different concentrations of NaOH (0.50, 1.00, 1.45 wt%) and fixed concentrations of chitosan and surfactant of 0.50 wt%. The produced microcapsules were fully characterized in their morphology and chemical composition, and the kinetic release analysis of the active ingredient was evaluated after deposition in a non-woven cellulose fabric. The concentration of 1.00 and 1.45 wt% clearly show the best results in terms of dimension and shape of the microcapsules as well as in the volatility results. However, at the concentration of 1 wt% a higher number of microcapsules were produced as confirmed by FTIR and EDS analysis. Free microcapsules are spherical in size with disperse diameters between 2 and 12 μm. Immobilized microcapsules showed sizes from 4 to 7 μm, a rough surface and loss of spherical shape with pore formation in the chitosan walls. SEM analysis confirms that at higher NaOH concentrations, the larger the size of the microcapsules. This technique shows that by tuning NaOH concentration it is possible to efficiently control the release rate of encapsulated active agents demonstrating great potential as insect repellent for textiles.JMS and ALC acknowledge CAPES Foundation, the Ministry of Education of Brazil, Proc. no 8976/13-9 e Proc. No 1071/13-0, respectively, and the Department of Textile Engineering of the University of Minho, Portugal. J. Molina is grateful to the Conselleria d'Educacio, Formacio i Ocupacio (Generalitat Valenciana) for the Programa VALi+D Postdoctoral Fellowship. AZ (C2011-UMINHO-2C2T-01) acknowledges funding from Programa Compromisso para a Ciencia 2008, Portugal. Shafagh Dinparast Tohidi would like to thank the Portuguese Foundation of Science and Technology for providing the PhD grant SFRH/BD/94759/2013

Optimal Investment-Consumption Problem with Constraint

In this paper, we consider an optimal investment-consumption problem subject to a closed convex constraint. In the problem, a constraint is imposed on both the investment and the consumption strategy, rather than just on the investment. The existence of solution is established by using the Martingale technique and convex duality. In addition to investment, our technique embeds also the consumption into a family of fictitious markets. However, with the addition of consumption, it leads to nonreflexive dual spaces. This difficulty is overcome by employing the so-called technique of \relaxation-projection" to establish the existence of solution to the problem. Furthermore, if the solution to the dual problem is obtained, then the solution to the primal problem can be found by using the characterization of the solution. An illustrative example is given with a dynamic risk constraint to demonstrate the method

Concatenated tensor network states

We introduce the concept of concatenated tensor networks to efficiently describe quantum states. We show that the corresponding concatenated tensor network states can efficiently describe time evolution and possess arbitrary block-wise entanglement and long-ranged correlations. We illustrate the approach for the enhancement of matrix product states, i.e. 1D tensor networks, where we replace each of the matrices of the original matrix product state with another 1D tensor network. This procedure yields a 2D tensor network, which includes -- already for tensor dimension two -- all states that can be prepared by circuits of polynomially many (possibly non-unitary) two-qubit quantum operations, as well as states resulting from time evolution with respect to Hamiltonians with short-ranged interactions. We investigate the possibility to efficiently extract information from these states, which serves as the basic step in a variational optimization procedure. To this aim we utilize known exact and approximate methods for 2D tensor networks and demonstrate some improvements thereof, which are also applicable e.g. in the context of 2D projected entangled pair states. We generalize the approach to higher dimensional- and tree tensor networks.Comment: 16 pages, 4 figure

European economic integration and migration in Romania

Considering the recent debates on the benefits of European economic integration, the purpose of this paper is to assess the impact of EU membership on the migration process in the case of Romania. The paper focussed on two directions of research: comparisons with neighbouring countries that are not member or candidates for EU and the explanation of the remittances based on the economic situation in the destination countries. The approach based on comparisons used difference-in-difference estimator as quantitative method, while the approach based on economic factors in destination countries employed mixed-effects models. The results based on these two approaches indicated that Romania did not send more migrants abroad in the period 2002–2017 compared to Ukraine and Republic of Moldova due its EU membership. On the other hand, Romania gained around 2.5 percentage points more remittances due its EU membership compared to Republic of Moldova and Ukraine. However, the unemployment and the GDP per capita in the destination countries are more important determinants of remittances rather than EU membership in the period 2010–2017. The results reveal that the remittances of Romanian migrants are conditioned by labour market issues in the destination countries, the unemployment in host country having a greater impact on remittances compared to GDP per capita and EU membership. It is expected that a future economic crisis will reduce remittances gained by Romania from other EU countries