A matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions

We study a matrix product state (MPS) algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [1], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions of the quantum Ising and the antiferromagnetic Heisenberg spin-1/2 model.Comment: 13 pages, 11 figures, 5 table

Entropy growth of shift-invariant states on a quantum spin chain

We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length $N$ are typically mixed and have therefore a non-zero entropy $S_N$ which is, moreover, monotonically increasing in $N$. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterised by a measurable subset of the unit interval. As the entropy density is known to vanishes, $S_N$ is sublinear in $N$. For states corresponding to unions of finitely many intervals, $S_N$ is shown to grow slower than $(\log N)^2$. Numerical calculations suggest a $\log N$ behaviour. For the case with infinitely many intervals, we present a class of states for which the entropy $S_N$ increases as $N^\alpha$ where $\alpha$ can take any value in $(0,1)$.Comment: 18 pages, 2 figure

Rigorous free fermion entanglement renormalization from wavelet theory

We construct entanglement renormalization schemes which provably approximate the ground states of non-interacting fermion nearest-neighbor hopping Hamiltonians on the one-dimensional discrete line and the two-dimensional square lattice. These schemes give hierarchical quantum circuits which build up the states from unentangled degrees of freedom. The circuits are based on pairs of discrete wavelet transforms which are approximately related by a "half-shift": translation by half a unit cell. The presence of the Fermi surface in the two-dimensional model requires a special kind of circuit architecture to properly capture the entanglement in the ground state. We show how the error in the approximation can be controlled without ever performing a variational optimization.Comment: 15 pages, 10 figures, one theore

Thermal States as Convex Combinations of Matrix Product States

We study thermal states of strongly interacting quantum spin chains and prove that those can be represented in terms of convex combinations of matrix product states. Apart from revealing new features of the entanglement structure of Gibbs states our results provide a theoretical justification for the use of White's algorithm of minimally entangled typical thermal states. Furthermore, we shed new light on time dependent matrix product state algorithms which yield hydrodynamical descriptions of the underlying dynamics.Comment: v3: 10 pages, 2 figures, final published versio

Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

Projected entangled pair states (PEPS) provide a natural ansatz for the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We develop a framework to describe on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, in terms of virtual symmetries of the local tensors expressed as a set of matrix product operators (MPOs) labeled by distinct group elements. These MPOs describe the possibly anomalous symmetry of the edge theory, whose local degrees of freedom are concretely identified in a PEPS. A classification of SPT phases is obtained by studying the obstructions to continuously deforming one set of MPOs into another, recovering the results derived for fixed-point models [X. Chen et al., Phys. Rev. B 87, 155114 (2013)]. Our formalism accommodates perturbations away from fixed point models, opening the possibility of studying phase transitions between different SPT phases. We also demonstrate that applying the recently developed quantum state gauging procedure to a SPT PEPS yields a PEPS with topological order determined by the initial symmetry MPOs. The MPO framework thus unifies the different approaches to classifying SPT phases, via fixed-points models, boundary anomalies, or gauging the symmetry, into the single problem of classifying inequivalent sets of matrix product operator symmetries that are defined purely in terms of a PEPS.Comment: 16 + 19 pages, 13 figures; v2 substantial changes to all sections, new appendices added; v3 published versio

Extending additivity from symmetric to asymmetric channels

We prove a lemma which allows one to extend results about the additivity of the minimal output entropy from highly symmetric channels to a much larger class. A similar result holds for the maximal output $p$-norm. Examples are given showing its use in a variety of situations. In particular, we prove the additivity and the multiplicativity for the shifted depolarising channel.Comment: 8 pages. This is the latest version of the first half of the original paper. The other half will appear in another pape

The Spatial Limitations of Current Neutral Models of Biodiversity

The unified neutral theory of biodiversity and biogeography is increasingly accepted as an informative null model of community composition and dynamics. It has successfully produced macro-ecological patterns such as species-area relationships and species abundance distributions. However, the models employed make many unrealistic auxiliary assumptions. For example, the popular spatially implicit version assumes a local plot exchanging migrants with a large panmictic regional source pool. This simple structure allows rigorous testing of its fit to data. In contrast, spatially explicit models assume that offspring disperse only limited distances from their parents, but one cannot as yet test the significance of their fit to data. Here we compare the spatially explicit and the spatially implicit model, fitting the most-used implicit model (with two levels, local and regional) to data simulated by the most-used spatially explicit model (where offspring are distributed about their parent on a grid according to either a radially symmetric Gaussian or a ‘fat-tailed’ distribution). Based on these fits, we express spatially implicit parameters in terms of spatially explicit parameters. This suggests how we may obtain estimates of spatially explicit parameters from spatially implicit ones. The relationship between these parameters, however, makes no intuitive sense. Furthermore, the spatially implicit model usually fits observed species-abundance distributions better than those calculated from the spatially explicit model's simulated data. Current spatially explicit neutral models therefore have limited descriptive power. However, our results suggest that a fatter tail of the dispersal kernel seems to improve the fit, suggesting that dispersal kernels with even fatter tails should be studied in future. We conclude that more advanced spatially explicit models and tools to analyze them need to be developed