206 research outputs found
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
Random transverse-field Ising chain with long-range interactions
We study the low-energy properties of the long-range random transverse-field
Ising chain with ferromagnetic interactions decaying as a power alpha of the
distance. Using variants of the strong-disorder renormalization group method,
the critical behavior is found to be controlled by a strong-disorder fixed
point with a finite dynamical exponent z_c=alpha. Approaching the critical
point, the correlation length diverges exponentially. In the critical point,
the magnetization shows an alpha-independent logarithmic finite-size scaling
and the entanglement entropy satisfies the area law. These observations are
argued to hold for other systems with long-range interactions, even in higher
dimensions.Comment: 6 pages, 4 figure
Entanglement between random and clean quantum spin chains
The entanglement entropy in clean, as well as in random quantum spin chains
has a logarithmic size-dependence at the critical point. Here, we study the
entanglement of composite systems that consist of a clean and a random part,
both being critical. In the composite, antiferromagnetic XX-chain with a sharp
interface, the entropy is found to grow in a double-logarithmic fashion , where is the length of the chain. We have also
considered an extended defect at the interface, where the disorder penetrates
into the homogeneous region in such a way that the strength of disorder decays
with the distance from the contact point as . For
, the entropy scales as , while for , when the extended interface defect
is an irrelevant perturbation, we recover the double-logarithmic scaling. These
results are explained through strong-disorder RG arguments.Comment: 12 pages, 7 figures, Invited contribution to the Festschrift of John
Cardy's 70th birthda
Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit
We consider the random-bond Potts model in the large- limit and calculate
the excess entropy, , of a contour, , which is given by the
mean number of Fortuin-Kasteleyn clusters which are crossed by . In two
dimensions is proportional to the length of , to which -
at the critical point - there are universal logarithmic corrections due to
corners. These are calculated by applying techniques of conformal field theory
and compared with the results of large scale numerical calculations. The
central charge of the model is obtained from the corner contributions to the
excess entropy and independently from the finite-size correction of the
free-energy as: , close to previous
estimates calculated at finite values of .Comment: 6 pages, 7 figure
Complex quantum network models from spin clusters
In the emerging quantum internet, complex network topology could lead to
efficient quantum communication and enhanced robustness against failures.
However, there are some concerns about complexity in quantum communication
networks, such as potentially limited end-to-end transmission capacity. These
challenges call for model systems in which the feasibility and impact of
complex network topology on quantum communication protocols can be explored.
Here, we present a theoretical model for complex quantum communication networks
on a lattice of spins, wherein entangled spin clusters in interacting quantum
spin systems serve as communication links between appropriately selected
regions of spins. Specifically, we show that ground state
Greenberger-Horne-Zeilinger clusters of the two-dimensional random transverse
Ising model can be used as communication links between regions of spins, and we
show that the resulting quantum networks can have complexity comparable to that
of the classical internet. Our work provides an accessible generative model for
further studies towards determining the network characteristics of the emerging
quantum internet.Comment: 16 pages, 3 figure
Universal logarithmic terms in the entanglement entropy of 2d, 3d and 4d random transverse-field Ising models
The entanglement entropy of the random transverse-field Ising model is
calculated by a numerical implementation of the asymptotically exact strong
disorder renormalization group method in 2d, 3d and 4d hypercubic lattices for
different shapes of the subregion. We find that the area law is always
satisfied, but there are analytic corrections due to E-dimensional edges
(1<=E<=d-2). More interesting is the contribution arising from corners, which
is logarithmically divergent at the critical point and its prefactor in a given
dimension is universal, i.e. independent of the form of disorder.Comment: 6 pages, 5 figure
Renormalization group study of random quantum magnets
We have developed a very efficient numerical algorithm of the strong disorder
renormalization group method to study the critical behaviour of the random
transverse-field Ising model, which is a prototype of random quantum magnets.
With this algorithm we can renormalize an N-site cluster within a time
N*log(N), independently of the topology of the graph and we went up to
N~4*10^6. We have studied regular lattices with dimension D<=4 as well as
Erdos-Renyi random graphs, which are infinite dimensional objects. In all cases
the quantum critical behaviour is found to be controlled by an infinite
disorder fixed point, in which disorder plays a dominant role over quantum
fluctuations. As a consequence the renormalization procedure as well as the
obtained critical properties are asymptotically exact for large systems. We
have also studied Griffiths singularities in the paramagnetic and the
ferromagnetic phases and generalized the numerical algorithm for another random
quantum systems.Comment: 12 pages, 12 figure
Unveiling universal aspects of the cellular anatomy of the brain
Recent cellular-level volumetric brain reconstructions have revealed high
levels of anatomic complexity. Determining which structural aspects of the
brain to focus on, especially when comparing with computational models and
other organisms, remains a major challenge. Here we quantify aspects of this
complexity and show evidence that brain anatomy satisfies universal scaling
laws, establishing the notion of structural criticality in the cellular
structure of the brain. Our framework builds upon understanding of critical
systems to provide clear guidance in selecting informative structural
properties of brain anatomy. As an illustration, we obtain estimates for
critical exponents in the human, mouse and fruit fly brains and show that they
are consistent between organisms, to the extent that data limitations allow.
Such universal quantities are robust to many of the microscopic details of
individual brains, providing a key step towards generative computational brain
models, and also clarifying in which sense one animal may be a suitable
anatomic model for another.Comment: 17 pages, 10 figure
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