97 research outputs found
Time Evolution within a Comoving Window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains
We present a modification of Matrix Product State time evolution to simulate
the propagation of signal fronts on infinite one-dimensional systems. We
restrict the calculation to a window moving along with a signal, which by the
Lieb-Robinson bound is contained within a light cone. Signal fronts can be
studied unperturbed and with high precision for much longer times than on
finite systems. Entanglement inside the window is naturally small, greatly
lowering computational effort. We investigate the time evolution of the
transverse field Ising (TFI) model and of the S=1/2 XXZ antiferromagnet in
their symmetry broken phases after several different local quantum quenches.
In both models, we observe distinct magnetization plateaus at the signal
front for very large times, resembling those previously observed for the
particle density of tight binding (TB) fermions. We show that the normalized
difference to the magnetization of the ground state exhibits similar scaling
behaviour as the density of TB fermions. In the XXZ model there is an
additional internal structure of the signal front due to pairing, and wider
plateaus with tight binding scaling exponents for the normalized excess
magnetization. We also observe parameter dependent interaction effects between
individual plateaus, resulting in a slight spatial compression of the plateau
widths.
In the TFI model, we additionally find that for an initial Jordan-Wigner
domain wall state, the complete time evolution of the normalized excess
longitudinal magnetization agrees exactly with the particle density of TB
fermions.Comment: 10 pages with 5 figures. Appendix with 23 pages, 13 figures and 4
tables. Largely extended and improved versio
Complete devil's staircase and crystal--superfluid transitions in a dipolar XXZ spin chain: A trapped ion quantum simulation
Systems with long-range interactions show a variety of intriguing properties:
they typically accommodate many meta-stable states, they can give rise to
spontaneous formation of supersolids, and they can lead to counterintuitive
thermodynamic behavior. However, the increased complexity that comes with
long-range interactions strongly hinders theoretical studies. This makes a
quantum simulator for long-range models highly desirable. Here, we show that a
chain of trapped ions can be used to quantum simulate a one-dimensional model
of hard-core bosons with dipolar off-site interaction and tunneling, equivalent
to a dipolar XXZ spin-1/2 chain. We explore the rich phase diagram of this
model in detail, employing perturbative mean-field theory, exact
diagonalization, and quasiexact numerical techniques (density-matrix
renormalization group and infinite time evolving block decimation). We find
that the complete devil's staircase -- an infinite sequence of crystal states
existing at vanishing tunneling -- spreads to a succession of lobes similar to
the Mott-lobes found in Bose--Hubbard models. Investigating the melting of
these crystal states at increased tunneling, we do not find (contrary to
similar two-dimensional models) clear indications of supersolid behavior in the
region around the melting transition. However, we find that inside the
insulating lobes there are quasi-long range (algebraic) correlations, opposed
to models with nearest-neighbor tunneling which show exponential decay of
correlations
An orthogonal C-H borylation - cross-coupling strategy for the preparation of tetrasubstituted "A(2)B(2)''-chrysene derivatives with tuneable photophysical properties
Cl-substituents serve as a functionalisable regiocontrol element for the orthogonal functionalisation of chrysene.</p
Optimal Matrix Product States for the Heisenberg Spin Chain
We present some exact results for the optimal Matrix Product State (MPS)
approximation to the ground state of the infinite isotropic Heisenberg spin-1/2
chain. Our approach is based on the systematic use of Schmidt decompositions to
reduce the problem of approximating for the ground state of a spin chain to an
analytical minimization. This allows to show that results of standard
simulations, e.g. density matrix renormalization group and infinite time
evolving block decimation, do correspond to the result obtained by this
minimization strategy and, thus, both methods deliver optimal MPS with the same
energy but, otherwise, different properties. We also find that translational
and rotational symmetries cannot be maintained simultaneously by the MPS ansatz
of minimum energy and present explicit constructions for each case.
Furthermore, we analyze symmetry restoration and quantify it to uncover new
scaling relations. The method we propose can be extended to any translational
invariant Hamiltonian.Comment: 10 pages, 3 figures; typos adde
Optimal Investment-Consumption Problem with Constraint
In this paper, we consider an optimal investment-consumption problem subject to a closed convex constraint. In the problem, a constraint is imposed on both the investment and the consumption strategy, rather than just on the investment. The existence of solution is established by using the Martingale technique and convex duality. In addition to investment, our technique embeds also the consumption into a family of fictitious markets. However, with the addition of consumption, it leads to nonreflexive dual spaces. This difficulty is overcome by employing the so-called technique of \relaxation-projection" to establish the existence of solution to the problem. Furthermore, if the solution to the dual problem is obtained, then the solution to the primal problem can be found by using the characterization of the solution. An illustrative example is given with a dynamic risk constraint to demonstrate the method
Multifaceted implications of the competition between native and invasive crayfish: a glimmer of hope for the native's long-term survival
Out of equilibrium dynamics with Matrix Product States
Theoretical understanding of strongly correlated systems in one spatial
dimension (1D) has been greatly advanced by the density-matrix renormalization
group (DMRG) algorithm, which is a variational approach using a class of
entanglement-restricted states called Matrix Product States (MPSs). However,
DRMG suffers from inherent accuracy restrictions when multiple states are
involved due to multi-state targeting and also the approximate representation
of the Hamiltonian and other operators. By formulating the variational approach
of DMRG explicitly for MPSs one can avoid errors inherent in the multi-state
targeting approach. Furthermore, by using the Matrix Product Operator (MPO)
formalism, one can exactly represent the Hamiltonian and other operators. The
MPO approach allows 1D Hamiltonians to be templated using a small set of finite
state automaton rules without reference to the particular microscopic degrees
of freedom. We present two algorithms which take advantage of these properties:
eMPS to find excited states of 1D Hamiltonians and tMPS for the time evolution
of a generic time-dependent 1D Hamiltonian. We properly account for
time-ordering of the propagator such that the error does not depend on the rate
of change of the Hamiltonian. Our algorithms use only the MPO form of the
Hamiltonian, and so are applicable to microscopic degrees of freedom of any
variety, and do not require Hamiltonian-specialized implementation. We
benchmark our algorithms with a case study of the Ising model, where the
critical point is located using entanglement measures. We then study the
dynamics of this model under a time-dependent quench of the transverse field
through the critical point. Finally, we present studies of a dipolar, or
long-range Ising model, again using entanglement measures to find the critical
point and study the dynamics of a time-dependent quench through the critical
point.Comment: 35 pages, 10 figures, submitted to NJP for the focus issue "Out of
equilibrium dynamics in strongly interacting 1D systems
Concatenated tensor network states
We introduce the concept of concatenated tensor networks to efficiently
describe quantum states. We show that the corresponding concatenated tensor
network states can efficiently describe time evolution and possess arbitrary
block-wise entanglement and long-ranged correlations. We illustrate the
approach for the enhancement of matrix product states, i.e. 1D tensor networks,
where we replace each of the matrices of the original matrix product state with
another 1D tensor network. This procedure yields a 2D tensor network, which
includes -- already for tensor dimension two -- all states that can be prepared
by circuits of polynomially many (possibly non-unitary) two-qubit quantum
operations, as well as states resulting from time evolution with respect to
Hamiltonians with short-ranged interactions. We investigate the possibility to
efficiently extract information from these states, which serves as the basic
step in a variational optimization procedure. To this aim we utilize known
exact and approximate methods for 2D tensor networks and demonstrate some
improvements thereof, which are also applicable e.g. in the context of 2D
projected entangled pair states. We generalize the approach to higher
dimensional- and tree tensor networks.Comment: 16 pages, 4 figure
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