Introducing disorder to interacting quantum systems breaks ergodicity and thermalization, a phenomenon known as Many-Body Localization (MBL) and the focus of this project. Systems exhibiting MBL have gathered interest in the last decade because the eigenstates of the underlying Hamiltonian violate the Eigenstate Thermalization Hypothesis (ETH), which marks a departure from ergodic systems where quantum statistical mechanics applies. In MBL, the ETH violation is due to the full system failing to act as a thermal bath for its subsystems . This has several implications, such as the lack of bulk-transport of local observables, but more importantly that MBL eigenstates have low entanglement which scales with the subsystem's boundaries, so-called area law scaling. In contrast, ergodic eigenstates have high non-local entanglement, scaling with the subsystem's volume. The low entanglement of MBL eigenstates can be leveraged to make efficient numerical algorithms. In this case we use the Density Matrix Renormalization Group (DMRG), a numerical technique that uses tensor network formalism to describe many-body quantum states efficiently by keeping only the relevant entanglement information. Naturally, this is most accurate when describing low-entanglement states and therefore well suited to study groundstates of Hamiltonians as well as MBL eigenstates.
In this project we study the self-dual quantum Ising model, describing a one-dimensional spin-lattice system. It has been observed that, depending on disorder strength, the highly excited eigenstates of this model can exhibit either an ergodic phase, a "paramagnetic" MBL phase or a "spin-glass" MBL phase. Our aim is to clarify the nature of the phase transition between the two MBL regimes. This transition has eluded a full understanding due to the rapid growth of entanglement near the critical region, which poses a formidable computational challenge.