Many-Body Flatband Localization

We generate translationally invariant systems exhibiting many-body localization from All-Bands- Flat single particle lattice Hamiltonians dressed with suitable short-range many-body interactions. This phenomenon - dubbed Many-Body Flatband Localization (MBFBL) - is based on symmetries of both single particle and interaction terms in the Hamiltonian, and it holds for any interaction strength. We propose a generator of MBFBL Hamiltonians which covers both interacting bosons and fermions for arbitrary lattice dimensions, and we provide explicit examples of MBFBL models in one and two lattice dimensions. We also explicitly construct an extensive set of local integrals of motion for MBFBL models. Our results can be further generalized to long-range interactions as well as to systems lacking translational invariance.

Introduction -Understanding the lack of thermalization in quantum interacting systems has been an active topic since Anderson predicted in 1958 the absence of transport in single particle lattices due to spatial disorder [1]. This localization phenomenon has been extensively studied theoretically and experimentally [2], with the impact of interaction between localized particles as one of the main open questions. Weak interactions were predicted to preserve the absence of transport of interacting particles [3,4] about fifty years after Anderson original work, leading to the phenomenon of Many-Body Localization (MBL). The study of MBL systems and their properties is nowadays a very active topic of research with several open issues and active fronts -for a survey of the state of the art, see [5,6].
MBL was initially predicted for interacting disordered systems emerging as an interplay of disorder and weak interactions. However it was later realized that the presence of disorder is not essential, launching the search for disorder-free MBL systems. Several possible scenarios emerged as a result: from non-ergodic behavior in networks of Josephson junctions [7] to 1D fermionic lattices involving different species of particles [8] or the presence of d.c. field [9], local constraints due to gauge invariance [10], presence of a large number of conserved quantities [11,12], quasi-periodic long-range interactions [13], among others. Some proposals also explored the connection to glasses, predicting MBL in glassy systems [14][15][16][17], e.g. kinetically constrained models [18] and geometrically frustrated models [19]. However, the validity of some of the proposals were later doubted, as it was shown that several disorder-free MBL systems rely on vastly different energy scales and finite-size constraints [20]. In other cases instead (e.g. [7]), disorder-free MBL requires high temperatures or specific strong interaction regimes, likewise the original MBL requests weak interaction regimes.
In this letter, we propose a generator of disorder-free MBL systems which is free of the above-mentioned re-quirements (specific interaction or temperature regimes, finite-size constraints, type of many-body statistics, among others) and applies for arbitrary spatial dimensions. This generator relies on geometrical frustration of the translationally invariant single particle Hamiltonians which yields no single particle dispersion -i.e. all Bloch bands are dispersionless (or flat) -and suitably chosen many-body interactions. The resulting models exhibit non-ergodic behavior with lack of transport of particles for any interaction strength, and this phenomenon is dubbed Many-Body Flatband Localization (MBFBL). The study of networks with one or several flatbands (FB) is an active topic of research on its own. They were first discussed in the context of groundstate ferromagnetism [21], but were later identified in various other systems [22,23] and they have been experimentally realized in several setups, using e.g. ultra cold atoms [24] and photonic lattices [25][26][27]. An important property of FB systems is the presence of compact localized states (CLS) -eigenstates with strictly finite support. These were used to systematically construct FB models [28][29][30] along with other methods [31][32][33][34][35]. Their fine-tuned character makes FB systems an ideal platform to study diverse localization phenomena in the presence of onsite disorder [36][37][38][39], DC fields [40], and nonlinearities [41,42], among many others.
We introduce MBFBL networks formed by single particle All-Bands-Flat lattice Hamiltonians dressed with suitable short-range many-body interactions, and provide explicit examples in one and two spatial dimensions. We also discuss distinct interaction terms (including long-range interactions) in order to cover different types of particle statistics. We construct an extensive set of local integrals of motion present in MBFBL networks, and explicitly derive these integrals for some of the examples presented. We extend our generator scheme by removing the assumption of translation invariance of the lattice.
with both single particle partĤ sp and interactionĤ int written as sums of local operatorsf k andĝ κ . The integers k and κ label unit cells of the lattice in a direct space for two different unit cell choices A and B. We assume that the sites from one unit cell of e.g. choice A belong to different unit cells of choice B. Regardless of the choice, each unit cell contains ν lattices sites or single particle levels. The operators are expressed through creation and annihilation operatorsĉ † k,a ,ĉ k,a which create or annihilate a single particle on a given lattice site k, a with 1 ≤ a ≤ ν. Then the local operators read We assume the interaction HamiltonianĤ int to be twobody, so that the local operators arê By the above definitions both single particle and interaction Hamiltonians are semi-detangled (SD) as [f k ,f k ] = [ĝ κ ,ĝ κ ] = 0 for any k, k , κ, κ . The spectrum of the single particle eigenvalue problem withĤ sp yields ν flatbands with each being an eigenenergy of any of the local operators f k . It follows thatĤ sp enforces full localization and absence of transport. The same is true forĤ int . However, because of the different unit cell choices A, B, in general it follows that [f k ,ĝ κ ] = 0 for any given k and at least a pair of different values of κ (and vice versa). Consequently, the combination of botĥ H sp andĤ int into H in general yields transporting manybody eigenstates [43][44][45][46].
If t ab = t aa δ a,b (with the Kronecker symbol δ a,b ), thê H sp is coined fully detangled (FD) [31] since it depends on the particle number operatorsn =ĉ †ĉ only, and does not move any particles from any lattice site to any other one. Together withĤ int being SD, the full Hamiltonian H preserves full localization of particles, which is an example of many-body flatband localization (MBFBL). Likewise, if we assume that J αβγδ = J αβαβ δ α,γ δ β,δ it follows that H int is FD and does not move any particles from site to site. Together withĤ sp being SD, we again arrive at the result that the full Hamiltonian H lacks transporting eigenstates and is MBFBL. The relation between the FD/SD character of the Hamiltonians and the presence of MBFBL is summarized in Table I. We refer to all the other types of Hamiltonians as non-detangled (ND).Ĥ  We generate MBFBL Hamiltonians by choosing any of the FD/SD MBFBL models from Table (I). We then perform a unitary transformation (rotation) on each unit cell in either of the two unit cell choices A, B. This results in general in some complicated Hamiltonian witĥ H sp being ND andĤ int being FD/SD, or vice versa -H sp being FD/SD andĤ int being ND -depending on which unit cell type the transformation was applied to. Furthermore these transformations can be chosen unit cell dependent resulting in non-translationally invariant Hamiltonians.
Conventional disordered MBL systems are known to possess an extensive set of local integrals of motion [5,47,48], though explicit derivations are complicated. These integrals are used to explain relevant properties of these systems. Local integrals of motion can be explicitly derived for MBFBL networks. With our proposed scheme and considering a SD single particle HamiltonianĤ sp in H (1), it follows that the expectation values of the oper-atorsÎ k = ν a=1n k,a measure the number of particles in each local unitf k ofĤ sp . These numbers are conserved in the presence of a FD interactionĤ int (sinceĤ int does not move particles from one to another site). It follows that eachÎ k commutes with the full Hamiltonian.
The unitary transformations used to recastĤ sp as ND yield N local integrals of motionÎ k expressed in the new basis for the generated MBFBL lattice. The very same follows if a pair of FD single particleĤ sp and a SD inter-actionĤ int is picked from Table I. In this case, the opera-torsÎ κ = ν α=1n κ,α defined in each local unitĝ κ ofĤ int as well lead to N local integrals of motion of the MBFBL lattice after the unitary transformations have been applied. In the case of FD-FD HamiltoniansĤ sp ,Ĥ int , the extensive set of local integrals of motion contains ν × N elements, since each particle number operatorn k,a commutes with the full Hamiltonian H.
Most of the generated MBFBL models, while being appealing from a mathematical point of view, could be hard to implement in experiments due to the complicated structure of the interactionĤ int spanning several unit cells. Experimental feasibility instead favors fully detan-gledĤ int , which result e.g. from Coulomb interactions between density operators in real space [49]. Therefore we refine our generator scheme by choosing SD single particleĤ sp and FD interactionĤ int , and recastĤ sp to a ND Hamiltonian via unitary transformations that keep H int fully detangled. This algorithm works for any number of bands ν ofĤ sp , in any dimension, and any type of many-body statistics.
Results -We will now discuss concrete examples in one and two spatial dimensions. We consider the SD HamiltonianĤ sp and conveniently restate it in the unit cell representation B ofĤ int . We then apply the subsequent unitary transformations. This change of unit cell introduces hopping terms between neighboring unit cells in each local Hamiltonianf κ . Without loss of generality, we assume nonzero hoppings between nearestneighboring unit cells only, and we adopt the conventions used in Refs. [28,29] for flatband networks generators. Then a possible D = 1 HamiltonianĤ sp readŝ where we grouped the annihilation (creation) operatorŝ c κ,a (ĉ † κ,a ) in ν-dimensional vectorsĈ κ (Ĉ † κ ). The matrices H 0 , H 1 describe intra-and intercell hopping respectively, and are chosen so as to enforce the SD condition [f κ ,f κ ] = 0 for all κ, κ . We remark that thisĤ sp is only one of the infinitely many realizations of a SD single particle Hamiltonian.
1D networks -We now present two concrete examples of MBFBL networks. We first start with the simplest MBFBL network with ν = 2 bands. It is based on the HamiltonianĤ sp in Eq. (4) with (b) (a) and a free complex parameter t. It is straightforward to check that this Hamiltonian is SD and has all bands flat. Next we pick the extended Hubbard interactionĤ int (6). The structure ofĤ sp andĤ int is shown in Fig. 1(a) with solid lines and red shaded rods respectively. The rotation U ab (5) recasts H 0 , H 1 (7) as and makes HamiltonianĤ sp ND, whileĤ int remains FD. The resulting MBFBL network is shown in Fig. 1(b). The local integrals of motion read (after the rotation) For three bands ν = 3 with operators a κ , b κ , c κ corresponding to the three sites of the unit cell, the SD HamiltonianĤ sp (6) has the following hopping matrices with two free complex (t 1 , t 2 ) and one free real (µ) parameters. This network is shown in Fig. 2(a) with gray solid lines. The interactionĤ int consists of the extended Hubbard interaction (6) between the top and the bottom sites (a κ , b κ ) of each plaquette (red shaded rods in Fig. 2(a)) and an additional optional onsite Hubbard interaction for the central site c κ . Then the rotation U ab (5) is applied to the pair (a κ , b κ ) only while leaving the sites c κ untouched. This recasts H 0 , H 1 (10) into defining a ND HamiltonianĤ sp while the interactionĤ int remains FD. The resulting diamond-shaped MBFBL network is shown in Fig. 2(b). That diamond-shape profile has been realized in diverse experimental setups for flatband and compact localized state studies [50][51][52][53][54]. Experimentally, the selective extended Hubbard interaction involving only the top and bottom sitesâ κ ,b κ of the diamond plaquette might be achieved by reducing the distance between these sites as compared to the distance to the middle siteĉ κ . The parameter t 1 could be used to adjust the hoppings. The local integrals of motionÎ κ for this model are given by Eq. (9) plus the additional particle number operatorn c,κ for the central siteĉ κ of the lattice, since it is unaffected by the rotation. 2D networks -Construction of higher dimensional MBFBL networks follows a procedure similar to that of 1D systems. In the simplest setting, the single particle HamiltonianĤ sp can be taken as a straightforward extension of Eq. describing the intercell hopping along different spatial directions. The matrices are chosen to ensure thatĤ sp is SD. Now taking a suitable FD interactionĤ int , Eq. (6) or its generalizations, and picking a unitary transformation that leaves thisĤ int FD, we obtain a ND HamiltonianĤ sp . The full Hamiltonian H exhibits MBFBL [55].
A notable two dimensional lattice exhibiting MBFBL obtained by applying these rules is the decorated Lieb lattice [56]. This is a five-band ν = 5 network, whose SD HamiltonianĤ sp is shown in Fig. 3(a), with matrices H 0 , H 1 , H 1 . In each unit cell, we use the extended Hubbard HamiltoniansĤ int (6) for the two site pairs indicated by red shaded rods in Fig. 3(a), and an onsite Hubbard interaction for the central site. The two rotations U ab (5) applied to the highlighted pairs (leaving the central site untouched) yield a NDĤ sp shown in Fig. 3(b), and the resulting full Hamiltonian H is MBFBL. The local integrals of motion for the decorated Lieb lattice can be easily derived and have similar but more involved expressions to those of the previous models (9).
Perspectives -The proposed scheme relies on the twobody HamiltonianĤ int with onsite terms in the interaction, restricting the interacting particles to bosons or spinful fermions. However, the same construction can be implemented for spinless fermions by e.g. choosing local κ+σ,βĉ κ,γĉκ+σ,δ + h.c. with exclusively inter-site interaction terms between unit cell κ and unit cell κ + σ. In particular,Ĥ int is FD for J σ αβγδ = Jδ α,γ δ β,δ and it is preserved as FD by the same transformation (5). This yields a generator of D-dimensional ν-band MFBFL lattices for spinless fermions, with the recent work of Kuno et al. [57] being a particular D = 1 ν = 2 band example. The construction can be further extended to long-range allto-all interaction HamiltoniansĤ int by settingĝ κ = σ v σĝ σ κ and even infinite-range interactionsĤ int = J/N κ =κ ,an κ,anκ ,a . The latter example is valid because the interaction is a function of the total densitŷ ρ = κ,an κ,a only and is therefore invariant under the transformation (5). We note that it is possible to extend the generator by abandoning the translational invariance of the Hamiltonian H. We can choose the hopping parameters t ab and the interaction matrix elements J αβγδ in the starting HamiltoniansĤ sp andĤ int respectively to be unit cell dependent. To stick with the proposed scheme whereĤ int is FD and is preserved by unitary transformations (5), the unit cell dependent terms are restrained to the SD H sp only (e.g. onsite or hopping disorder). The unitary transformations (5) used to recastĤ sp as ND induce correlations between the onsite energies of the pairs of sites involved. In the models presented - Figs.1(b), 2(b), 3(b)these correlations are between the sites within the same red-shaded area. These correlations depend of the parameters z, w defining U a,b in Eq. (5). These parameters may also be chosen to vary upon changing κ if the unitary transformations considered differ from unit cell to unit cell. Let us additionally observe that the breaking of translation invariance does not destroy the existence of the extensive set of local integrals of motion -they are given by the same operators as in the translationally invariant case.
Conclusions -We have introduced a generator of Many-Body Localized disorder-free Hamiltonians by applying unitary transformations to suitably detangled Hamiltonians -a feature that assumes all-band-flat single particle Hamiltonians. This new phenomenon -coined Many-Body Flatband Localization -implies strict localization of any number of particles irrespective of dimensionality or interaction strength, and it does not require vastly different energy scales similar some models supposed to exhibit disorder-free MBL. Our work substantially extends previous studies of localization phenomena of interacting quantum many-body platforms with All-Band-Flat lattice single particle Hamiltonians [44,46,[57][58][59][60][61][62][63]. In particular, we propose a flexible and general set of many-body localized systems which may be experimentally feasible. A novel and unique feature of these systems is the existence of unitary mappings that recast them into a detangled form. This very property can be employed to study the impact of additional perturbations of the proposed networks which lift MBFBL and modify the proposed local integrals of motion in a systematic and analytical form. Hence, these systems offer innovative and powerful tools to potentially perform systematic analytical studies of conventional properties of MBL networks which typically relay on heavy numerical studies.