5 research outputs found
Particle number conservation in quantum many-body simulations with matrix product operators
Incorporating conservation laws explicitly into matrix product states (MPS)
has proven to make numerical simulations of quantum many-body systems much less
resources consuming. We will discuss here, to what extent this concept can be
used in simulation where the dynamically evolving entities are matrix product
operators (MPO). Quite counter-intuitively the expectation of gaining in speed
by sacrificing information about all but a single symmetry sector is not in all
cases fulfilled. It turns out that in this case often the entanglement imposed
by the global constraint of fixed particle number is the limiting factor.Comment: minor changes, 18 pages, 5 figure
Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas
The problem of how complex quantum systems eventually come to rest lies at
the heart of statistical mechanics. The maximum entropy principle put forward
in 1957 by E. T. Jaynes suggests what quantum states one should expect in
equilibrium but does not hint as to how closed quantum many-body systems
dynamically equilibrate. A number of theoretical and numerical studies
accumulate evidence that under specific conditions quantum many-body models can
relax to a situation that locally or with respect to certain observables
appears as if the entire system had relaxed to a maximum entropy state. In this
work, we report the experimental observation of the non-equilibrium dynamics of
a density wave of ultracold bosonic atoms in an optical lattice in the regime
of strong correlations. Using an optical superlattice, we are able to prepare
the system in a well-known initial state with high fidelity. We then follow the
dynamical evolution of the system in terms of quasi-local densities, currents,
and coherences. Numerical studies based on the time-dependent density-matrix
renormalization group method are in an excellent quantitative agreement with
the experimental data. For very long times, all three local observables show a
fast relaxation to equilibrium values compatible with those expected for a
global maximum entropy state. We find this relaxation of the quasi-local
densities and currents to initially follow a power-law with an exponent being
significantly larger than for free or hardcore bosons. For intermediate times
the system fulfills the promise of being a dynamical quantum simulator, in that
the controlled dynamics runs for longer times than present classical algorithms
based on matrix product states can efficiently keep track of.Comment: 8 pages, 6 figure
Rigorous mean-field dynamics of lattice bosons: Quenches from the Mott insulator
We provide a rigorous derivation of Gutzwiller mean-field dynamics for
lattice bosons, showing that it is exact on fully connected lattices. We apply
this formalism to quenches in the interaction parameter from the Mott insulator
to the superfluid state. Although within mean-field the Mott insulator is a
steady state, we show that a dynamical critical interaction exists, such
that for final interaction parameter the Mott insulator is
exponentially unstable towards emerging long-range superfluid order, whereas
for the Mott insulating state is stable. We discuss the implications
of this prediction for finite-dimensional systems.Comment: 6 pages, 3 figures, published versio
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
Local correlations in the 1D Bose gas from a scaling limit of the XXZ chain
We consider the K-body local correlations in the (repulsive) 1D Bose gas for
general K, both at finite size and in the thermodynamic limit. Concerning the
latter we develop a multiple integral formula which applies for arbitrary
states of the system with a smooth distribution of Bethe roots, including the
ground state and finite temperature Gibbs-states. In the cases K<=4 we perform
the explicit factorization of the multiple integral. In the case of K=3 we
obtain the recent result of Kormos et.al., whereas our formula for K=4 is new.
Numerical results are presented as well.Comment: 23 pages, 2 figures, v2: minor modifications and references adde