P6: Many-body localization in quasi-periodic and random potentials
Over the last decade the study of the interplay of one-particle disorder and the two-particle interaction developed into one of the most quickly growing fields of quantum many-body physics (see, e.g., ). While disorder tends to localize quantum states, the two-particle interaction counteracts this trend leading to a competition which is the origin of interesting many-body physics. In one spatial dimension it is by now well established that a system being localized by disorder might remain so, even in the presence of two-particle scattering; the system is said to be in a many-body localized (MBL) state . The physics becomes particularly rich if, for fixed disorder strength but varying amplitude of the two-particle interaction and/or varying energy of the considered state, a mobility edge exists beyond which the state turns into a delocalized one. The same type of phenomenology can also be observed in lattice models without disorder which feature an incommensurate, quasi-periodic one-particle potential [2,3]; details of the transition might, however, differ . Quasi-periodic systems are not only of interest as a deepened understanding of the MBL transition in these might lead to a more detailed understanding of “disorder MBL” but also as they can be realized in cold atomic gases and thus studied experimentally in a rather controlled setting .
Up to now MBL has primarily been explored numerically by exact diagonalization of comparably small systems as many more advanced tools of quantum many-body physics have difficulties to access excited states. Furthermore, in many methods studying transport requires additional approximations, while transport properties provide the most direct access to the question of localization versus delocalization. For this reason, the details of the MBL transition remain fuzzy and apparently established results were repeatedly challenged. Therefore, MBL provides a field in which the use of alternative methods and further developing these can make an impact.
In the present highly explorative project we will combine FRG  and DMRG  to study MBL; recent methodological progress of both renders the complementary application a promising approach. On the side of FRG the latest developments are: (i) A lowest-order truncated FRG applied to disordered spinless fermions with nearest-neighbor interaction was shown to feature the weak disorder Luttinger-liquid power laws predicted by field theory . (ii) A fully consistent (dynamic) second order scheme  was shown to capture (ground state) phase transitions  as part of the RTG project of the doctoral researcher Lisa Markhof from the first funding period. (iii) It was shown how to compute expectation values in excited states using FRG . We note in passing that studying transport through inhomogeneous, one-dimensional, and correlated system by FRG is well established  and that all one-particle potentials are treated exactly in FRG. On the DMRG side several pro-posals were made how to efficiently access generic excited eigenstates to study MBL [13,14,15,16,17].
We will start out studying the Aubry-Andre model with an incommensurate, quasi-periodic potential complemented by a nearest-neighbor interaction (see  and references therein) employing the FRG approach used in  and DMRG. As an indicator for the MBL transition we will compute the bipartite fluctuations of the particle number for which, within FRG, a flow equation has to be derived and which shows a characteristic similar to the entanglement entropy [3,17]. To get a first idea which of the properties of the MBL the second order FRG scheme can capture we first do not investigate a specific excited many-body state but rather consider finite temperatures or stop the RG flow at a fixed (energy) scale by hand. From this starting point we will move on towards (a) transport geometries (semi-infinite leads), (b) systems with disorder, and (c) the study of properties in excited states with fixed energy. We expect that, in this way new, insights into MBL physics can be gained.
 Annalen der Physik 529, Special Issue: Many-Body Localization (2017)
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