• area laws
• chebyshev kicker
• density matrix renormalization group
• DMRG
• ground state search
• hamiltonian kicker
• imaginary time evolution kicker
• kicker strategy
• many body
• matrix product states
• MPS
• strongly correlated

We focus on many-body systems whose constituents are finite dimen-
sional (spins) and reside on the sites of a lattice. The hamiltonians that
govern these systems contain local and interaction terms, such as nearest-
neighbor interactions, second-nearest-neighbor interactions and so on. They
can be seen as effective hamiltonians that abstractly describe the relevant
physics of a real system. Anyway, recent advances in the field of quantum
optics allow the creation of quite arbitrary and tunable hamiltonians of this
kind, introducing ultracold and dilute atom gases in optical lattices. This is
one of the reasons why theoretical interest in the field has increased in the
past years.
From a historical point of view, early attempts to solve analytically these
kinds of problems can be traced back to the introduction of the Heisenberg
model and its solution, to the quantum version of the Ising model and to the
Bethe ansatz. The latter is one of the very few tools that, in one dimension,
allows to find some answers. Perturbation theory fails here because these
systems present very strong interactions. The first good numerical method
was the Numerical Renormalization Group (NRG), proposed by Wilson in
the 1970s, that successfully solved the Kondo problem. Unfortunately, this
approach was working only for small quantum impurities in a classical lattice
and not for full quantum spins.
Addressing some of the weaknesses of NRG, White introduced the Den-
sity Matrix Renormalization Group (DMRG) in 1992 and it still is the best
instrument to search ground states of strongly correlated systems that we
have. DMRG came in two flavours, one for infinite systems, where the size
of the system grows continuously adding new sites and minimizing the en-
ergy every step, the other for finite systems, where the system is swept back
and forth minimizing the energy at every site of the lattice. Very soon, for
both of these versions, it was recognised that the states produced in the
process were Matrix Product States (MPSs). MPSs are a particularly at-
tractive form to write a many-body quantum state as the product of many
matrices, but for some time their connection with DMRG was disregarded.
The reasons why this connection reemerged are very important.
A first reason is linked with the fact that the success of DMRG is not
universal: its best performance is only in one dimension. For some time
it was unclear why, but the explanation came from Quantum Information
and Entanglement Theory in the early 2000s. The so-called “area laws”,
that we will discuss in details, offer an elegant theoretical justification and
a remarkable simplification for DMRG in one dimension. They state that
ground states of local, gapped hamiltonians possess relatively small entan-
glement and we will see how this means relatively few parameters to process.
The importance of MPSs enters here because they are objects with a built-in
area law and thus they constitute the optimal class of states where to look
for the ground state.
A second reason is that, in the same years, Cirac, Verstraete, Vidal et al.
began to reformulate DMRG in MPS language and they found many new
algorithms, hard to notice in a pure DMRG approach. Among them we cite
real-time evolution and finite-temperature algorithms. Today, the theory of
MPSs in one dimension is well developed and we will use it extensively.
The original contribution, in our work, is a new algorithm for ground
state search that, like the procedures cited above, exploits the MPS language
as an economical way to express very large vectors. The idea is to start from
a random vector and apply to it an operator K, called the kicker. Now we
possess two vectors that span a two-dimensional subspace; we minimize the
energy of the reduced hamiltonian for that subspace and we generate the
vector that points in that direction. This strategy always finds lower or equal
energy values, because the initial vector is not excluded from the possible
choices. We apply the same process to this new vector and we repeat so
until the energy converges.
A lot of work has been dedicated to the research of the optimal kicker
K: we compare and list advantages and disadvantages of at least three
different classes of kickers such as the hamiltonian kicker, the imaginary time
evolution kickers and the Chebyshev kickers. No optimal solution was found,
though, because while the hamiltonian always results in a good kicker, the
performance of the other two types could be better, but it is very problem-
dependent. We test our kickers on the Ising model and we compare them
with DMRG.
Since we do not minimize the energy site by site as in DMRG, but we
change completely the state vector every step, we hope to find novel be-
haviours that will improve the performance of ground state search. Indeed,
we will see that this method has a very fast initial energy descent that, espe-
cially with very large systems, is faster than DMRG, but it also has a very
long asymptotic convergence at the end. Finally, we propose to create an
hybrid algorithm, where we begin with the new algorithm, initially faster,
and at some point we switch back to DMRG, that is optimal at the end.
A major effort in this work has been the writing of the program that
implements the algorithm suggested here. This program includes a small
library of functions to work with an arbitrary number of MPSs (and their
generalization to operators, MPOs). This effort repaid itself, since the pro-
gram is now a very versatile tool with which many simulations are possible.
The chosen programming language is Fortran 90/95 for the performance with
linear algebra routines and it has been compiled with ifort, the Intel For-
tran compiler freely available for students, and its Math Kernel Library that
provided the optimized LAPACK routines. SlocCOUNT, a famous program
to count the physical lines of code, computes more than 4900 lines of code
and this is why it was not possible to add the code in appendix, but it is
freely available under GNU/GPL licence for anyone who wants it.