19 research outputs found
Entanglement renormalization
In the context of real-space renormalization group methods, we propose a
novel scheme for quantum systems defined on a D-dimensional lattice. It is
based on a coarse-graining transformation that attempts to reduce the amount of
entanglement of a block of lattice sites before truncating its Hilbert space.
Numerical simulations involving the ground state of a 1D system at criticality
show that the resulting coarse-grained site requires a Hilbert space dimension
that does not grow with successive rescaling transformations. As a result we
can address, in a quasi-exact way, tens of thousands of quantum spins with a
computational effort that scales logarithmically in the system's size. The
calculations unveil that ground state entanglement in extended quantum systems
is organized in layers corresponding to different length scales. At a quantum
critical point, each rellevant length scale makes an equivalent contribution to
the entanglement of a block with the rest of the system.Comment: 4 pages, 4 figures, updated versio
Entanglement renormalization in fermionic systems
We demonstrate, in the context of quadratic fermion lattice models in one and
two spatial dimensions, the potential of entanglement renormalization (ER) to
define a proper real-space renormalization group transformation. Our results
show, for the first time, the validity of the multi-scale entanglement
renormalization ansatz (MERA) to describe ground states in two dimensions, even
at a quantum critical point. They also unveil a connection between the
performance of ER and the logarithmic violations of the boundary law for
entanglement in systems with a one-dimensional Fermi surface. ER is recast in
the language of creation/annihilation operators and correlation matrices.Comment: 5 pages, 4 figures Second appendix adde
Statistical Models with a Line of Defect
The factorization condition for the scattering amplitudes of an integrable
model with a line of defect gives rise to a set of Reflection-Transmission
equations. The solutions of these equations in the case of diagonal -matrix
in the bulk are only those with . The choice corresponds to
the Ising model. We compute the transmission and reflection amplitudes relative
to the interaction of the Majorana fermion with the defect and we discuss their
relevant features.Comment: 14 pages, LATEX file, ISAS/EP/94/30 (Figures added, originally missed
for E-mail transmission problem.
Connected Green function approach to ground state symmetry breaking in -theory
Using the cluster expansions for n-point Green functions we derive a closed
set of dynamical equations of motion for connected equal-time Green functions
by neglecting all connected functions higher than order for the
-theory in dimensions. We apply the equations to the
investigation of spontaneous ground state symmetry breaking, i.e. to the
evaluation of the effective potential at temperature . Within our momentum
space discretization we obtain a second order phase transition (in agreement
with the Simon-Griffith theorem) and a critical coupling of
as compared to a first order phase transition and
from the Gaussian effective potential approach.Comment: 25 Revtex pages, 5 figures available via fpt from the directory
ugi-94-11 of [email protected] as one postscript file (there
was a bug in our calculations, all numerical results and figures have changed
significantly), ugi-94-1
Scattering Theory and Correlation Functions in Statistical Models with a Line of Defect
The scattering theory of the integrable statistical models can be generalized
to the case of systems with extended lines of defect. This is done by adding
the reflection and transmission amplitudes for the interactions with the line
of inhomegeneity to the scattering amplitudes in the bulk. The factorization
condition for the new amplitudes gives rise to a set of Reflection-Transmission
equations. The solutions of these equations in the case of diagonal -matrix
in the bulk are only those with . The choice corresponds to
the Ising model. We compute the exact expressions of the transmission and
reflection amplitudes relative to the interaction of the Majorana fermion of
the Ising model with the defect. These amplitudes present a weak-strong duality
in the coupling constant, the self-dual points being the special values where
the defect line acts as a reflecting surface. We also discuss the bosonic case
which presents instability properties and resonance states. Multi-defect
systems which may give rise to a band structure are also considered. The exact
expressions of correlation functions is obtained in terms of Form Factors of
the bulk theory and matrix elements of the defect operator.Comment: 50 pages, LATEX file, ISAS/EP/94-12