A short proof of stability of topological order under local perturbations

Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological quantum order for which the unperturbed Hamiltonian $H_0$ can be written as a sum of local pairwise commuting projectors on a $D$-dimensional lattice. We consider a perturbed Hamiltonian $H=H_0+V$ involving a generic perturbation $V$ that can be written as a sum of short-range bounded-norm interactions. We prove that if the strength of $V$ is below a constant threshold value then $H$ has well-defined spectral bands originating from the low-lying eigenvalues of $H_0$. These bands are separated from the rest of the spectrum and from each other by a constant gap. The width of the band originating from the smallest eigenvalue of $H_0$ decays faster than any power of the lattice size.Comment: 15 page

Making Almost Commuting Matrices Commute

Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B] \Vert \leq \delta$). Are they close to a commuting pair of Hermitian matrices, $A',B'$, with $\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon$? A theorem of H. Lin shows that this is uniformly true, in that for every $\epsilon>0$ there exists a $\delta>0$, independent of the size $N$ of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how $\delta$ depends on $\epsilon$. We give uniform bounds relating $\delta$ and $\epsilon$. We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices and a fully constructive method in that case. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a {\it projective} measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements.Comment: 22 pages; tighter bounds; Note: fixed mistake in proof pointed out by Filonov and Kachkovski

Hastings-Levitov aggregation in the small-particle limit

We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web

Exponential Decay of Correlations Implies Area Law

We prove that a finite correlation length, i.e. exponential decay of correlations, implies an area law for the entanglement entropy of quantum states defined on a line. The entropy bound is exponential in the correlation length of the state, thus reproducing as a particular case Hastings proof of an area law for groundstates of 1D gapped Hamiltonians. As a consequence, we show that 1D quantum states with exponential decay of correlations have an efficient classical approximate description as a matrix product state of polynomial bond dimension, thus giving an equivalence between injective matrix product states and states with a finite correlation length. The result can be seen as a rigorous justification, in one dimension, of the intuition that states with exponential decay of correlations, usually associated with non-critical phases of matter, are simple to describe. It also has implications for quantum computing: It shows that unless a pure state quantum computation involves states with long-range correlations, decaying at most algebraically with the distance, it can be efficiently simulated classically. The proof relies on several previous tools from quantum information theory - including entanglement distillation protocols achieving the hashing bound, properties of single-shot smooth entropies, and the quantum substate theorem - and also on some newly developed ones. In particular we derive a new bound on correlations established by local random measurements, and we give a generalization to the max-entropy of a result of Hastings concerning the saturation of mutual information in multiparticle systems. The proof can also be interpreted as providing a limitation on the phenomenon of data hiding in quantum states.Comment: 35 pages, 6 figures; v2 minor corrections; v3 published versio

Random quantum channels I: graphical calculus and the Bell state phenomenon

This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel. As an application, we study variations of random matrix models introduced by Hayden \cite{hayden}, and show that their eigenvalues converge almost surely. In particular we obtain for some models sharp improvements on the value of the largest eigenvalue, and this is shown in a further work to have new applications to minimal output entropy inequalities.Comment: Several typos were correcte

Diffusion Limited Aggregation with Power-Law Pinning

Using stochastic conformal mapping techniques we study the patterns emerging from Laplacian growth with a power-law decaying threshold for growth $R_N^{-\gamma}$ (where $R_N$ is the radius of the $N-$ particle cluster). For $\gamma > 1$ the growth pattern is in the same universality class as diffusion limited aggregation (DLA) growth, while for $\gamma < 1$ the resulting patterns have a lower fractal dimension $D(\gamma)$ than a DLA cluster due to the enhancement of growth at the hot tips of the developing pattern. Our results indicate that a pinning transition occurs at $\gamma = 1/2$, significantly smaller than might be expected from the lower bound $\alpha_{min} \simeq 0.67$ of multifractal spectrum of DLA. This limiting case shows that the most singular tips in the pruned cluster now correspond to those expected for a purely one-dimensional line. Using multifractal analysis, analytic expressions are established for $D(\gamma)$ both close to the breakdown of DLA universality class, i.e., $\gamma \lesssim 1$, and close to the pinning transition, i.e., $\gamma \gtrsim 1/2$.Comment: 5 pages, e figures, submitted to Phys. Rev.

Diffusion limited aggregation as a Markovian process. Part I: bond-sticking conditions

Cylindrical lattice Diffusion Limited Aggregation (DLA), with a narrow width N, is solved using a Markovian matrix method. This matrix contains the probabilities that the front moves from one configuration to another at each growth step, calculated exactly by solving the Laplace equation and using the proper normalization. The method is applied for a series of approximations, which include only a finite number of rows near the front. The matrix is then used to find the weights of the steady state growing configurations and the rate of approaching this steady state stage. The former are then used to find the average upward growth probability, the average steady-state density and the fractal dimensionality of the aggregate, which is extrapolated to a value near 1.64.Comment: 24 pages, 20 figure

A Multi-Dimensional Lieb-Schultz-Mattis Theorem

For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C\log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent result by Hastings.Comment: final versio

Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.Comment: 18 pages, 4 figures. Some sections rewritten (including title and abstract), numerical results added, references added. Accepted for publication in J. Stat. Phy

Laplacian growth as one-dimensional turbulence

A new model of Laplacian stochastic growth is formulated using conformal mappings. The model describes two growth regimes, stable and turbulent, separated by a sharp phase transition. The first few Fourier components of the mapping define the web, an envelope of the cluster. The web is used to study the transition and the dynamics of large-scale features of the cluster characterized by evolution from macro- to micro-scales. Also, we derive scaling laws for the cluster size.Comment: 4 pages, RevTex, 4 figure