255,806 research outputs found
Empirical risk minimization as parameter choice rule for general linear regularization methods.
We consider the statistical inverse problem to recover f from noisy measurements Y = Tf + sigma xi where xi is Gaussian white noise and T a compact operator between Hilbert spaces. Considering general reconstruction methods of the form (f) over cap (alpha) = q(alpha) (T*T)T*Y with an ordered filter q(alpha), we investigate the choice of the regularization parameter alpha by minimizing an unbiased estiate of the predictive risk E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. The corresponding parameter alpha(pred) and its usage are well-known in the literature, but oracle inequalities and optimality results in this general setting are unknown. We prove a (generalized) oracle inequality, which relates the direct risk E[parallel to f - (f) over cap (alpha pred)parallel to(2)] with the oracle prediction risk inf(alpha>0) E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. From this oracle inequality we are then able to conclude that the investigated parameter choice rule is of optimal order in the minimax sense. Finally we also present numerical simulations, which support the order optimality of the method and the quality of the parameter choice in finite sample situations
The Coupled Cluster Method Applied to Quantum Magnets: A New LPSUB Approximation Scheme for Lattice Models
A new approximation hierarchy, called the LPSUB scheme, is described for
the coupled cluster method (CCM). It is applicable to systems defined on a
regular spatial lattice. We then apply it to two well-studied prototypical
(spin-1/2 Heisenberg antiferromagnetic) spin-lattice models, namely: the XXZ
and the XY models on the square lattice in two dimensions. Results are obtained
in each case for the ground-state energy, the ground-state sublattice
magnetization and the quantum critical point. They are all in good agreement
with those from such alternative methods as spin-wave theory, series
expansions, quantum Monte Carlo methods and the CCM using the alternative
LSUB and DSUB schemes. Each of the three CCM schemes (LSUB, DSUB
and LPSUB) for use with systems defined on a regular spatial lattice is
shown to have its own advantages in particular applications
Low-energy parameters and spin gap of a frustrated spin- Heisenberg antiferromagnet with on the honeycomb lattice
The coupled cluster method is implemented at high orders of approximation to
investigate the zero-temperature phase diagram of the frustrated
spin- ---- antiferromagnet on the honeycomb lattice.
The system has isotropic Heisenberg interactions of strength ,
and between nearest-neighbour, next-nearest-neighbour and
next-next-nearest-neighbour pairs of spins, respectively. We study it in the
case , in the window
that contains the classical tricritical point (at ) of maximal frustration, appropriate to the limiting value of the spin quantum number. We present results for the magnetic
order parameter , the triplet spin gap , the spin stiffness
and the zero-field transverse magnetic susceptibility for the
two collinear quasiclassical antiferromagnetic (AFM) phases with N\'{e}el and
striped order, respectively. Results for and are given for the
three cases , and , while those for
and are given for the two cases and . On
the basis of all these results we find that the spin- and spin-1
models both have an intermediate paramagnetic phase, with no discernible
magnetic long-range order, between the two AFM phases in their phase
diagrams, while for there is a direct transition between them. Accurate
values are found for all of the associated quantum critical points. While the
results also provide strong evidence for the intermediate phase being gapped
for the case , they are less conclusive for the case . On
balance however, at least the transition in the latter case at the striped
phase boundary seems to be to a gapped intermediate state
Transverse Magnetic Susceptibility of a Frustrated Spin- ---- Heisenberg Antiferromagnet on a Bilayer Honeycomb Lattice
We use the coupled cluster method (CCM) to study a frustrated
spin- ---- Heisenberg antiferromagnet
on a bilayer honeycomb lattice with stacking. Both nearest-neighbor (NN)
and frustrating next-nearest-neighbor antiferromagnetic (AFM) exchange
interactions are present in each layer, with respective exchange coupling
constants and . The two layers are
coupled with NN AFM exchanges with coupling strength . We calculate to high orders of approximation within the CCM
the zero-field transverse magnetic susceptibility in the N\'eel phase.
We thus obtain an accurate estimate of the full boundary of the N\'eel phase in
the plane for the zero-temperature quantum phase diagram. We
demonstrate explicitly that the phase boundary derived from is fully
consistent with that obtained from the vanishing of the N\'eel magnetic order
parameter. We thus conclude that at all points along the N\'eel phase boundary
quasiclassical magnetic order gives way to a nonclassical paramagnetic phase
with a nonzero energy gap. The N\'eel phase boundary exhibits a marked
reentrant behavior, which we discuss in detail
Collinear antiferromagnetic phases of a frustrated spin- ---- Heisenberg model on an -stacked bilayer honeycomb lattice
The zero-temperature quantum phase diagram of the spin-
---- model on an -stacked bilayer honeycomb
lattice is investigated using the coupled cluster method (CCM). The model
comprises two monolayers in each of which the spins, residing on
honeycomb-lattice sites, interact via both nearest-neighbor (NN) and
frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg
exchange iteractions, with respective strengths and . The two layers are coupled via a comparable Heisenberg
exchange interaction between NN interlayer pairs, with a strength
. The complete phase boundaries of two
quasiclassical collinear AFM phases, namely the N\'{e}el and N\'{e}el-II
phases, are calculated in the half-plane with .
Whereas on each monolayer in the N\'{e}el state all NN pairs of spins are
antiparallel, in the N\'{e}el-II state NN pairs of spins on zigzag chains along
one of the three equivalent honeycomb-lattice directions are antiparallel,
while NN interchain spins are parallel. We calculate directly in the
thermodynamic (infinite-lattice) limit both the magnetic order parameter
and the excitation energy from the ground state to the
lowest-lying excited state (where is the total
component of spin for the system as a whole, and where the collinear ordering
lies along the direction) for both quasiclassical states used (separately)
as the CCM model state, on top of which the multispin quantum correlations are
then calculated to high orders () in a systematic series of
approximations involving -spin clusters. The sole approximation made is then
to extrapolate the sequences of th-order results for and to the
exact limit,
A high-order study of the quantum critical behavior of a frustrated spin- antiferromagnet on a stacked honeycomb bilayer
We study a frustrated spin-
------ Heisenberg antiferromagnet on an
-stacked bilayer honeycomb lattice. In each layer we consider
nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor
antiferromagnetic (AFM) exchange couplings , , and ,
respectively. The two layers are coupled with an AFM NN exchange coupling
. The model is studied for arbitrary values of
along the line that includes the most
highly frustrated point at , where the classical ground
state is macroscopically degenerate. The coupled cluster method is used at high
orders of approximation to calculate the magnetic order parameter and the
triplet spin gap. We are thereby able to give an accurate description of the
quantum phase diagram of the model in the plane in the window , . This includes two AFM phases with
N\'eel and striped order, and an intermediate gapped paramagnetic phase that
exhibits various forms of valence-bond crystalline order. We obtain accurate
estimations of the two phase boundaries, , or
equivalently, , with (N\'eel) and 2
(striped). The two boundaries exhibit an "avoided crossing" behavior with both
curves being reentrant
Spin-gap study of the spin- -- model on the triangular lattice
We use the coupled cluster method implemented at high orders of approximation
to study the spin- -- model on the triangular
lattice with Heisenberg interactions between nearest-neighbour and
next-nearest-neighbour pairs of spins, with coupling strengths and
, respectively. In the window we find that the 3-sublattice 120 N\'{e}el-ordered and
2-sublattice 180 stripe-ordered antiferromagnetic states form the
stable ground-state phases in the regions
and , respectively. The spin-triplet gap is
found to vanish over essentially the entire region of the intermediate phase
Ground-state phases of the spin-1 -- Heisenberg antiferromagnet on the honeycomb lattice
We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg
antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange
coupling and frustrating next-nearest-neighbor coupling , using the coupled cluster method implemented to high orders
of approximation, and based on model states with different forms of classical
magnetic order. For each we calculate directly in the bulk thermodynamic limit
both ground-state low-energy parameters (including the energy per spin,
magnetic order parameter, spin stiffness coefficient, and zero-field uniform
transverse magnetic susceptibility) and their generalized susceptibilities to
various forms of valence-bond crystalline (VBC) order, as well as the energy
gap to the lowest-lying spin-triplet excitation. In the range
we find evidence for four distinct phases. Two of these are quasiclassical
phases with antiferromagnetic long-range order, one with 2-sublattice N\'{e}el
order for , and another with 4-sublattice
N\'{e}el-II order for . Two different
paramagnetic phases are found to exist in the intermediate region. Over the
range we find a gapless
phase with no discernible magnetic order, which is a strong candidate for being
a quantum spin liquid, while over the range we find a gapped phase, which is most likely a lattice nematic
with staggered dimer VBC order that breaks the lattice rotational symmetry
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