A ground state search algorithm for strongly correlated 1D quantum systems

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