5 research outputs found
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
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
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