7,999 research outputs found
Universal Probability-Free Conformal Prediction
We construct universal prediction systems in the spirit of Popper's
falsifiability and Kolmogorov complexity and randomness. These prediction
systems do not depend on any statistical assumptions (but under the IID
assumption they dominate, to within the usual accuracy, conformal prediction).
Our constructions give rise to a theory of algorithmic complexity and
randomness of time containing analogues of several notions and results of the
classical theory of Kolmogorov complexity and randomness.Comment: 27 page
Universal Boundary Entropies in Conformal Field Theory: A Quantum Monte Carlo Study
Recently, entropy corrections on nonorientable manifolds such as the Klein
bottle are proposed as a universal characterization of critical systems with an
emergent conformal field theory (CFT). We show that entropy correction on the
Klein bottle can be interpreted as a boundary effect via transforming the Klein
bottle into an orientable manifold with nonlocal boundary interactions. The
interpretation reveals the conceptual connection of the Klein bottle entropy
with the celebrated Affleck-Ludwig entropy in boundary CFT. We propose a
generic scheme to extract these universal boundary entropies from quantum Monte
Carlo calculation of partition function ratios in lattice models. Our numerical
results on the Affleck-Ludwig entropy and Klein bottle entropy for the
-state quantum Potts chains with show excellent agreement with the
CFT predictions. For the quantum Potts chain with , the Klein bottle
entropy slightly deviates from the CFT prediction, which is possibly due to
marginally irrelevant terms in the low-energy effective theory.Comment: 10 pages, 4 figures. Published versio
Corner contribution to percolation cluster numbers
We study the number of clusters in two-dimensional (2d) critical percolation,
N_Gamma, which intersect a given subset of bonds, Gamma. In the simplest case,
when Gamma is a simple closed curve, N_Gamma is related to the entanglement
entropy of the critical diluted quantum Ising model, in which Gamma represents
the boundary between the subsystem and the environment. Due to corners in Gamma
there are universal logarithmic corrections to N_Gamma, which are calculated in
the continuum limit through conformal invariance, making use of the
Cardy-Peschel formula. The exact formulas are confirmed by large scale Monte
Carlo simulations. These results are extended to anisotropic percolation where
they confirm a result of discrete holomorphicity.Comment: 7 pages, 9 figure
Entanglement spectrum of random-singlet quantum critical points
The entanglement spectrum, i.e., the full distribution of Schmidt eigenvalues
of the reduced density matrix, contains more information than the conventional
entanglement entropy and has been studied recently in several many-particle
systems. We compute the disorder-averaged entanglement spectrum, in the form of
the disorder-averaged moments of the reduced density matrix, for a contiguous
block of many spins at the random-singlet quantum critical point in one
dimension. The result compares well in the scaling limit with numerical studies
on the random XX model and is also expected to describe the (interacting)
random Heisenberg model. Our numerical studies on the XX case reveal that the
dependence of the entanglement entropy and spectrum on the geometry of the
Hilbert space partition is quite different than for conformally invariant
critical points.Comment: 11 pages, 10 figure
Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance
We conjecture an exact form for an universal ratio of four-point cluster
connectivities in the critical two-dimensional -color Potts model. We also
provide analogous results for the limit that corresponds to
percolation where the observable has a logarithmic singularity. Our conjectures
are tested against Monte Carlo simulations showing excellent agreement for
.Comment: 29 pages, 9 Figures. Published version: improved discussion,
additional numerical tests and reference
Critical Percolation in Finite Geometries
The methods of conformal field theory are used to compute the crossing
probabilities between segments of the boundary of a compact two-dimensional
region at the percolation threshold. These probabilities are shown to be
invariant not only under changes of scale, but also under mappings of the
region which are conformal in the interior and continuous on the boundary. This
is a larger invariance than that expected for generic critical systems.
Specific predictions are presented for the crossing probability between
opposite sides of a rectangle, and are compared with recent numerical work. The
agreement is excellent.Comment: 10 page
Entanglement negativity and conformal field theory: a Monte Carlo study
We investigate the behavior of the moments of the partially transposed
reduced density matrix \rho^{T_2}_A in critical quantum spin chains. Given
subsystem A as union of two blocks, this is the (matrix) transposed of \rho_A
with respect to the degrees of freedom of one of the two. This is also the main
ingredient for constructing the logarithmic negativity. We provide a new
numerical scheme for calculating efficiently all the moments of \rho_A^{T_2}
using classical Monte Carlo simulations. In particular we study several
combinations of the moments which are scale invariant at a critical point.
Their behavior is fully characterized in both the critical Ising and the
anisotropic Heisenberg XXZ chains. For two adjacent blocks we find, in both
models, full agreement with recent CFT calculations. For disjoint ones, in the
Ising chain finite size corrections are non negligible. We demonstrate that
their exponent is the same governing the unusual scaling corrections of the
mutual information between the two blocks. Monte Carlo data fully match the
theoretical CFT prediction only in the asymptotic limit of infinite intervals.
Oppositely, in the Heisenberg chain scaling corrections are smaller and,
already at finite (moderately large) block sizes, Monte Carlo data are in
excellent agreement with the asymptotic CFT result.Comment: 31 pages, 10 figures. Minor changes, published versio
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