Topological Phases of One-Dimensional Fermions: An Entanglement Point of View

The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by Fidkowski \emph{et. al.} (Phys. Rev. B 81, 134509 (2010)), we find that in the presence of interactions there are only eight distinct phases, which obey a $\mathbb{Z}_8$ group structure. This is in contrast to the $\mathbb{Z}$ classification in the non-interacting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.Comment: 12 pages, 1 figure; journal ref. adde

Topological phases in gapped edges of fractionalized systems

Recently, it has been proposed that exotic one-dimensional phases can be realized by gapping out the edge states of a fractional topological insulator. The low-energy edge degrees of freedom are described by a chain of coupled parafermions. We introduce a classification scheme for the phases that can occur in parafermionic chains. We find that the parafermions support both topological symmetry fractionalized phases as well as phases in which the parafermions condense. In the presence of additional symmetries, the phases form a non-Abelian group. As a concrete example of the classification, we consider the effective edge model for a $\nu= 1/3$ fractional topological insulator for which we calculate the entanglement spectra numerically and show that all possible predicted phases can be realized.Comment: 11 pages, 7 figures, final versio

Berry phase induced dimerization in one-dimensional quadrupolar systems

We investigate the effect of the Berry phase on quadrupoles that occur for example in the low-energy description of spin models. Specifically we study here the one-dimensional bilinear-biquadratic spin-one model. An open question for many years about this model is whether it has a non-dimerized fluctuating nematic phase. The dimerization has recently been proposed to be related to Berry phases of the quantum fluctuations. We use an effective low-energy description to calculate the scaling of the dimerization according to this theory, and then verify the predictions using large scale density-matrix renormalization group (DMRG) simulations, giving good evidence that the state is dimerized all the way up to its transition into the ferromagnetic phase. We furthermore discuss the multiplet structure found in the entanglement spectrum of the ground state wave functions.Comment: 4.5 pages + 4 pages supplementary material, 4 figure

Symmetry protection of topological order in one-dimensional quantum spin systems

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-$S$ Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\pi$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{x,y,z} \rightarrow - S^{x,y,z}$; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B \textbf{81}, 064439 (2010)]. On the other hand, an even-$S$ Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.Comment: 9 pages, 6 figures, extended version: several new examples and numerical results added. Journal reference adde

Detection of Symmetry Protected Topological Phases in 1D

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped phases in 1D systems can be completely characterized using tools related to projective representations of the symmetry groups. We show how to determine the matrices of these representations in a simple way in order to distinguish between different phases directly. From these matrices we also point out how to derive several different types of non-local order parameters for time reversal, inversion symmetry and $Z_2 \times Z_2$ symmetry, as well as some more general cases (some of which have been obtained before by other methods). Using these concepts, the ordinary string order for the Haldane phase can be related to a selection rule that changes at the critical point. We furthermore point out an example of a more complicated internal symmetry for which the ordinary string order cannot be applied.Comment: 12 pages, 9 Figure

Axion Cosmology and the Energy Scale of Inflation

We survey observational constraints on the parameter space of inflation and axions and map out two allowed windows: the classic window and the inflationary anthropic window. The cosmology of the latter is particularly interesting; inflationary axion cosmology predicts the existence of isocurvature fluctuations in the CMB, with an amplitude that grows with both the energy scale of inflation and the fraction of dark matter in axions. Statistical arguments favor a substantial value for the latter, and so current bounds on isocurvature fluctuations imply tight constraints on inflation. For example, an axion Peccei-Quinn scale of 10^16 GeV excludes any inflation model with energy scale > 3.8*10^14 GeV (r > 2*10^(-9)) at 95% confidence, and so implies negligible gravitational waves from inflation, but suggests appreciable isocurvature fluctuations.Comment: 10 PRD pages, 4 figs, V3: updated to match published versio

Drying High Moisture Alfalfa Hay

We all recognize the value of alfalfa in horse, dairy and beef rations. That\u27s why we harvest over 17 million acres of this crop in the United States every year. Hay is a good way to harvest alfalfa because it stores well, provides long fiber in rations and we can market the surplus as a cash crop. Higher yields and higher quality mean more profit so we push to be sure we use the best management practices. All too often though, we lose part of all of a crop to rain damage. Some yield and quality is lost due to leaf shatter and respiration no matter how ideal the conditions. The goal of our research program is to develop hay harvesting and storage systems that minimize these losses at the lowest possible cost

Theory of finite-entanglement scaling at one-dimensional quantum critical points

Studies of entanglement in many-particle systems suggest that most quantum critical ground states have infinitely more entanglement than non-critical states. Standard algorithms for one-dimensional many-particle systems construct model states with limited entanglement, which are a worse approximation to quantum critical states than to others. We give a quantitative theory of previously observed scaling behavior resulting from finite entanglement at quantum criticality: the scaling theory of finite entanglement is only superficially similar to finite-size scaling, and has a different physical origin. We find that finite-entanglement scaling is governed not by the scaling dimension of an operator but by the "central charge" of the critical point, which counts its universal degrees of freedom. An important ingredient is the recently obtained universal distribution of density-matrix eigenvalues at a critical point\cite{calabrese1}. The parameter-free theory is checked against numerical scaling at several quantum critical points.Comment: 4 pages + 2 pages supplementary informatio

Elective affinities of the Protestant ethic : Weber and the chemistry of capitalism

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