21 research outputs found
Nonuniversal and anomalous critical behavior of the contact process near an extended defect
We consider the contact process near an extended surface defect, where the
local control parameter deviates from the bulk one by an amount of
, being the distance from the
surface. We concentrate on the marginal situation, , where
is the critical exponent of the spatial correlation length, and
study the local critical properties of the one-dimensional model by Monte Carlo
simulations. The system exhibits a rich surface critical behavior. For weaker
local activation rates, , the phase transition is continuous, having an
order-parameter critical exponent, which varies continuously with . For
stronger local activation rates, , the phase transition is of mixed
order: the surface order parameter is discontinuous, at the same time the
temporal correlation length diverges algebraically as the critical point is
approached, but with different exponents on the two sides of the transition.
The mixed-order transition regime is analogous to that observed recently at a
multiple junction and can be explained by the same type of scaling theory.Comment: 8 pages, 8 figure
Nonequilibrium dynamics of the Ising chain in a fluctuating transverse field
We study nonequilibrium dynamics of the quantum Ising chain at zero
temperature when the transverse field is varied stochastically. In the
equivalent fermion representation, the equation of motion of Majorana operators
is derived in the form of a one-dimensional, continuous-time quantum random
walk with stochastic, time-dependent transition amplitudes. This type of
external noise gives rise to decoherence in the associated quantum walk and the
semiclassical wave-packet generally has a diffusive behavior. As a consequence,
in the quantum Ising chain, the average entanglement entropy grows in time as
and the logarithmic average magnetization decays in the same form. In
the case of a dichotomous noise, when the transverse-field is changed in
discrete time-steps, , there can be excitation modes, for which coherence
is maintained, provided their energy satisfies
with a positive integer . If the dispersion of is quadratic,
the long-time behavior of the entanglement entropy and the logarithmic
magnetization is dominated by these ballistically traveling coherent modes and
both will have a time-dependence.Comment: 12 pages, 10 figure
Long-range epidemic spreading in a random environment
Modeling long-range epidemic spreading in a random environment, we consider a
quenched disordered, -dimensional contact process with infection rates
decaying with the distance as . We study the dynamical behavior
of the model at and below the epidemic threshold by a variant of the
strong-disorder renormalization group method and by Monte Carlo simulations in
one and two spatial dimensions. Starting from a single infected site, the
average survival probability is found to decay as up to
multiplicative logarithmic corrections. Below the epidemic threshold, a
Griffiths phase emerges, where the dynamical exponent varies continuously
with the control parameter and tends to as the threshold is
approached. At the threshold, the spatial extension of the infected cluster (in
surviving trials) is found to grow as with a
multiplicative logarithmic correction, and the average number of infected sites
in surviving trials is found to increase as with
in one dimension.Comment: 12 pages, 6 figure
Exact relationship between the entanglement entropies of XY and quantum Ising chains
We consider two prototypical quantum models, the spin-1/2 XY chain and the
quantum Ising chain and study their entanglement entropy, S(l,L), of blocks of
l spins in homogeneous or inhomogeneous systems of length L. By using two
different approaches, free-fermion techniques and perturbational expansion, an
exact relationship between the entropies is revealed. Using this relation we
translate known results between the two models and obtain, among others, the
additive constant of the entropy of the critical homogeneous quantum Ising
chain and the effective central charge of the random XY chain.Comment: 6 page
Griffiths-McCoy Singularities in the Random Transverse-Field Ising Spin Chain
We consider the paramagnetic phase of the random transverse-field Ising spin
chain and study the dynamical properties by numerical methods and scaling
considerations. We extend our previous work [Phys. Rev. B 57, 11404 (1998)] to
new quantities, such as the non-linear susceptibility, higher excitations and
the energy-density autocorrelation function. We show that in the Griffiths
phase all the above quantities exhibit power-law singularities and the
corresponding critical exponents, which vary with the distance from the
critical point, can be related to the dynamical exponent z, the latter being
the positive root of [(J/h)^{1/z}]_av=1. Particularly, whereas the average spin
autocorrelation function in imaginary time decays as [G]_av(t)~t^{-1/z}, the
average energy-density autocorrelations decay with another exponent as
[G^e]_av(t)~t^{-2-1/z}.Comment: 8 pages RevTeX, 8 eps-figures include
Finite-size scaling of the entanglement entropy of the quantum Ising chain with homogeneous, periodically modulated and random couplings
Using free-fermionic techniques we study the entanglement entropy of a block
of contiguous spins in a large finite quantum Ising chain in a transverse
field, with couplings of different types: homogeneous, periodically modulated
and random. We carry out a systematic study of finite-size effects at the
quantum critical point, and evaluate subleading corrections both for open and
for periodic boundary conditions. For a block corresponding to a half of a
finite chain, the position of the maximum of the entropy as a function of the
control parameter (e.g. the transverse field) can define the effective critical
point in the finite sample. On the basis of homogeneous chains, we demonstrate
that the scaling behavior of the entropy near the quantum phase transition is
in agreement with the universality hypothesis, and calculate the shift of the
effective critical point, which has different scaling behaviors for open and
for periodic boundary conditions.Comment: 17 pages, 5 figures, v2: references added+update
Transverse-field Ising spin chain with inhomogeneous disorder
We consider the critical and off-critical properties at the boundary of the
random transverse-field Ising spin chain when the distribution of the couplings
and/or transverse fields, at a distance from the surface, deviates from its
uniform bulk value by terms of order with an amplitude . Exact
results are obtained using a correspondence between the surface magnetization
of the model and the surviving probability of a random walk with time-dependent
absorbing boundary conditions. For slow enough decay, , the
inhomogeneity is relevant: Either the surface stays ordered at the bulk
critical point or the average surface magnetization displays an essential
singularity, depending on the sign of . In the marginal situation,
, the average surface magnetization decays as a power law with a
continuously varying, -dependent, critical exponent which is obtained
analytically. The behavior of the critical and off-critical autocorrelation
functions as well as the scaling form of the probability distributions for the
surface magnetization and the first gaps are determined through a
phenomenological scaling theory. In the Griffiths phase, the properties of the
Griffiths-McCoy singularities are not affected by the inhomogeneity. The
various results are checked using numerical methods based on a mapping to free
fermions.Comment: 11 pages (Revtex), 11 figure
Entanglement entropy of aperiodic quantum spin chains
We study the entanglement entropy of blocks of contiguous spins in
non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg,
XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and
relevant aperiodic modulations, the entanglement entropy is found to be a
logarithmic function of the block size with log-periodic oscillations. The
effective central charge, c_eff, defined through the constant in front of the
logarithm may depend on the ratio of couplings and can even exceed the
corresponding value in the homogeneous system. In the strong modulation limit,
the ground state is constructed by a renormalization group method and the
limiting value of c_eff is exactly calculated. Keeping the ratio of the block
size and the system size constant, the entanglement entropy exhibits a scaling
property, however, the corresponding scaling function may be nonanalytic.Comment: 6 pages, 2 figure
Griffiths-McCoy singularities in random quantum spin chains: Exact results through renormalization
The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study
singular quantities in the Griffiths phase of random quantum spin chains. For
the random transverse-field Ising spin chain we have extended Fisher's
analytical solution to the off-critical region and calculated the dynamical
exponent exactly. Concerning other random chains we argue by scaling
considerations that the RG method generally becomes asymptotically exact for
large times, both at the critical point and in the whole Griffiths phase. This
statement is checked via numerical calculations on the random Heisenberg and
quantum Potts models by the density matrix renormalization group method.Comment: 4 pages RevTeX, 2 figures include
Percolation in random environment
We consider bond percolation on the square lattice with perfectly correlated
random probabilities. According to scaling considerations, mapping to a random
walk problem and the results of Monte Carlo simulations the critical behavior
of the system with varying degree of disorder is governed by new, random fixed
points with anisotropic scaling properties. For weaker disorder both the
magnetization and the anisotropy exponents are non-universal, whereas for
strong enough disorder the system scales into an {\it infinite randomness fixed
point} in which the critical exponents are exactly known.Comment: 8 pages, 7 figure