995 research outputs found
Ground state entanglement in quantum spin chains
A microscopic calculation of ground state entanglement for the XY and
Heisenberg models shows the emergence of universal scaling behavior at quantum
phase transitions. Entanglement is thus controlled by conformal symmetry. Away
from the critical point, entanglement gets saturated by a mass scale. Results
borrowed from conformal field theory imply irreversibility of entanglement loss
along renormalization group trajectories. Entanglement does not saturate in
higher dimensions which appears to limit the success of the density matrix
renormalization group technique. A possible connection between majorization and
renormalization group irreversibility emerges from our numerical analysis.Comment: 26 pages, 16 figures, added references, minor changes. Final versio
A Generic Renormalization Method in Curved Spaces and at Finite Temperature
Based only on simple principles of renormalization in coordinate space, we
derive closed renormalized amplitudes and renormalization group constants at 1-
and 2-loop orders for scalar field theories in general backgrounds. This is
achieved through a generic renormalization procedure we develop exploiting the
central idea behind differential renormalization, which needs as only inputs
the propagator and the appropriate laplacian for the backgrounds in question.
We work out this generic coordinate space renormalization in some detail, and
subsequently back it up with specific calculations for scalar theories both on
curved backgrounds, manifestly preserving diffeomorphism invariance, and at
finite temperature.Comment: 15pp., REVTeX, UB-ECM-PF 94/1
Renormalization group transformations on quantum states
We construct a general renormalization group transformation on quantum
states, independent of any Hamiltonian dynamics of the system. We illustrate
this procedure for translational invariant matrix product states in one
dimension and show that product, GHZ, W and domain wall states are special
cases of an emerging classification of the fixed points of this
coarse--graining transformation.Comment: 5 pages, 2 figur
Fine-grained entanglement loss along renormalization group flows
We explore entanglement loss along renormalization group trajectories as a
basic quantum information property underlying their irreversibility. This
analysis is carried out for the quantum Ising chain as a transverse magnetic
field is changed. We consider the ground-state entanglement between a large
block of spins and the rest of the chain. Entanglement loss is seen to follow
from a rigid reordering, satisfying the majorization relation, of the
eigenvalues of the reduced density matrix for the spin block. More generally,
our results indicate that it may be possible to prove the irreversibility along
RG trajectories from the properties of the vacuum only, without need to study
the whole hamiltonian.Comment: 5 pages, 3 figures; minor change
The Hidden Spatial Geometry of Non-Abelian Gauge Theories
The Gauss law constraint in the Hamiltonian form of the gauge theory
of gluons is satisfied by any functional of the gauge invariant tensor variable
. Arguments are given that the tensor is a more appropriate variable. When the Hamiltonian
is expressed in terms of or , the quantity appears.
The gauge field Bianchi and Ricci identities yield a set of partial
differential equations for in terms of . One can show that
is a metric-compatible connection for with torsion, and that the curvature
tensor of is that of an Einstein space. A curious 3-dimensional
spatial geometry thus underlies the gauge-invariant configuration space of the
theory, although the Hamiltonian is not invariant under spatial coordinate
transformations. Spatial derivative terms in the energy density are singular
when . These singularities are the analogue of the centrifugal
barrier of quantum mechanics, and physical wave-functionals are forced to
vanish in a certain manner near . It is argued that such barriers are
an inevitable result of the projection on the gauge-invariant subspace of the
Hilbert space, and that the barriers are a conspicuous way in which non-abelian
gauge theories differ from scalar field theories.Comment: 19 pages, TeX, CTP #223
Quantum Phase Transitions and Bipartite Entanglement
We develop a general theory of the relation between quantum phase transitions
(QPTs) characterized by nonanalyticities in the energy and bipartite
entanglement. We derive a functional relation between the matrix elements of
two-particle reduced density matrices and the eigenvalues of general two-body
Hamiltonians of -level systems. The ground state energy eigenvalue and its
derivatives, whose non-analyticity characterizes a QPT, are directly tied to
bipartite entanglement measures. We show that first-order QPTs are signalled by
density matrix elements themselves and second-order QPTs by the first
derivative of density matrix elements. Our general conclusions are illustrated
via several quantum spin models.Comment: 5 pages, incl. 2 figures. v3: The version published in PRL, including
a few extra comments and clarifications for which there was no space in the
PR
Quantum data compression, quantum information generation, and the density-matrix renormalization group method
We have studied quantum data compression for finite quantum systems where the
site density matrices are not independent, i.e., the density matrix cannot be
given as direct product of site density matrices and the von Neumann entropy is
not equal to the sum of site entropies. Using the density-matrix
renormalization group (DMRG) method for the 1-d Hubbard model, we have shown
that a simple relationship exists between the entropy of the left or right
block and dimension of the Hilbert space of that block as well as of the
superblock for any fixed accuracy. The information loss during the RG procedure
has been investigated and a more rigorous control of the relative error has
been proposed based on Kholevo's theory. Our results are also supported by the
quantum chemistry version of DMRG applied to various molecules with system
lengths up to 60 lattice sites. A sum rule which relates site entropies and the
total information generated by the renormalization procedure has also been
given which serves as an alternative test of convergence of the DMRG method.Comment: 8 pages, 7 figure
Entropy and Exact Matrix Product Representation of the Laughlin Wave Function
An analytical expression for the von Neumann entropy of the Laughlin wave
function is obtained for any possible bipartition between the particles
described by this wave function, for filling fraction nu=1. Also, for filling
fraction nu=1/m, where m is an odd integer, an upper bound on this entropy is
exhibited. These results yield a bound on the smallest possible size of the
matrices for an exact representation of the Laughlin ansatz in terms of a
matrix product state. An analytical matrix product state representation of this
state is proposed in terms of representations of the Clifford algebra. For
nu=1, this representation is shown to be asymptotically optimal in the limit of
a large number of particles
Adiabatic quantum computation and quantum phase transitions
We analyze the ground state entanglement in a quantum adiabatic evolution
algorithm designed to solve the NP-complete Exact Cover problem. The entropy of
entanglement seems to obey linear and universal scaling at the point where the
mass gap becomes small, suggesting that the system passes near a quantum phase
transition. Such a large scaling of entanglement suggests that the effective
connectivity of the system diverges as the number of qubits goes to infinity
and that this algorithm cannot be efficiently simulated by classical means. On
the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR
Entanglement in the XX spin chain with an energy current
We consider the ground state of the XX chain that is constrained to carry a
current of energy. The von Neumann entropy of a block of neighboring spins,
describing entanglement of the block with the rest of the chain, is computed.
Recent calculations have revealed that the entropy in the XX model diverges
logarithmically with the size of the subsystem. We show that the presence of
the energy current increases the prefactor of the logarithmic growth. This
result indicates that the emergence of the energy current gives rise to an
increase of entanglement.Comment: 4 pages, 4 figure
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