[en] A fast method is presented for simulating the dielectric-breakdown model using iterated conformal mappings. Numerical results for the dimension and for corrections to scaling are in good agreement with the recent renormalization group prediction of an upper critical η_c=4 , at which a transition occurs between branching fractal clusters and one-dimensional nonfractal clusters

[en] It is argued that the dielectric-breakdown model has an upper critical η_c equal to 4, for which the clusters become one dimensional. A renormalization group treatment of the model is presented near the critical η

[en] We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system that has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the renormalization group flow

[en] We show how to combine the light-cone and matrix product algorithms to simulate quantum systems far from equilibrium for long times. For the case of the XXZ spin chain at Δ=0.5, we simulate to a time of ≅22.5. While part of the long simulation time is due to the use of the light-cone method, we also describe a modification of the infinite time-evolving bond decimation algorithm with improved numerical stability, and we describe how to incorporate symmetry into this algorithm. While statistical sampling error means that we are not yet able to make a definite statement, the behavior of the simulation at long times indicates the appearance of either 'revivals' in the order parameter as predicted by Hastings and Levitov (e-print arXiv:0806.4283) or of a distinct shoulder in the decay of the order parameter.

[en] We show that randomly choosing the matrices in a completely positive map from the unitary group gives a quantum expander. We consider Hermitian and non-Hermitian cases, and we provide asymptotically tight bounds in the Hermitian case on the typical value of the second largest eigenvalue. The key idea is the use of Schwinger-Dyson equations from lattice gauge theory to efficiently compute averages over the unitary group

[en] There can exist topological obstructions to continuously deforming a gapped Hamiltonian for free fermions into a trivial form without closing the gap. These topological obstructions are closely related to obstructions to the existence of exponentially localized Wannier functions. We show that by taking two copies of a gapped, free fermionic system with complex conjugate Hamiltonians, it is always possible to overcome these obstructions. This allows us to write the ground state in matrix product form using Grassmann-valued bond variables, and show insensitivity of the ground state density matrix to boundary conditions. (letter)

[en] We consider wavefunctions which are non-negative in some tensor product basis. We study what possible teleportation can occur in such wavefunctions, giving a complete answer in some cases (when one system is a qubit) and partial answers elsewhere. We use this to show that a one-dimensional wavefunction which is non-negative and has zero correlation length can be written in a “coherent Gibbs state” form, as explained later. We conjecture that such holds in higher dimensions. Additionally, some results are provided on possible teleportation in general wavefunctions, explaining how Schmidt coefficients before measurement limit the possible Schmidt coefficients after measurement, and on the absence of a “generalized area law” [D. Aharonov et al., in Proceedings of Foundations of Computer Science (FOCS) (IEEE, 2014), p. 246; e-print arXiv.org:1410.0951] even for Hamiltonians with no sign problem. One of the motivations for this work is an attempt to prove a conjecture about ground state wavefunctions which have an “intrinsic” sign problem that cannot be removed by any quantum circuit. We show a weaker version of this, showing that the sign problem is intrinsic for commuting Hamiltonians in the same phase as the double semion model under the technical assumption that TQO-2 holds [S. Bravyi et al., J. Math. Phys. 51, 093512 (2010)]

[en] We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension (1D). The area law for a 1D system states that for the ground state, the entanglement of any interval is upper bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap Δ is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small Δ. To help resolve this asymptotic behavior, we construct a family of 1D local systems for which some intervals have entanglement entropy, which is polynomial in 1/Δ, whereas previously studied systems, such as free fermion systems or systems described by conformal field theory, had the entropy of all intervals bounded by a constant time log(1/Δ).

[en] We study pseudo-optimal solutions to multi-objective optimization problems by introducing partial minima defined as follows. Point x k-dominates x' when at least k of the coordinates of x are smaller than the corresponding coordinates of x'. A point not k-dominated by any other point in the set is a k-minimum or a partial minimum, generalizing the global minimum. We study statistical properties of partial minima for a set of N points independently distributed inside the d-dimensional unit hypercube using exact probabilistic methods and heuristic scaling techniques. The average number of partial minima, A, decays algebraically with the total number of points, A ∼ N^-(d-k)/k, when 1 ≤ k < d. Interestingly, there are k - 1 distinct scaling laws characterizing the largest coordinates: the distribution P(y_j) of the jth largest coordinate, y_j, decays algebraically, P(y_j)∼(y_j)^α_j-1, with α_j=j(d-k)/(k-j) for 1 ≤ j ≤ k - 1. The average number of partial minima grows logarithmically, A≅1/(d-1)factorielle(ln N)^d-1, when k = d. The full distribution of the number of minima is obtained in closed form in two dimensions. (fast track communication)

[en] We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with κ≅2.1. An argument is given that their fractal dimension d_f is related to their interface energy exponent θ by d_f-1=3/[4(3+θ)], which is consistent with the commonly quoted values d_f≅1.27 and θ≅-0.28