Motivated by recent studies of the projected dice lattice Hamiltonian [K. Swaminathan , ], we introduce the on-site/bond singlet (OBS) model, a one-dimensional model of a flat band superconductor, in order to better understand the quasiparticle localization and interesting coexistence of ergodic and nonergodic behavior present in the former model. The OBS model is the sum of terms that have direct counterparts in the projected dice lattice Hamiltonian, each of which is parameterized by a coupling constant. Exact diagonalization reveals that the energy spectrum and nonequilibrium dynamics of the OBS model are essentially the same as that of the dice lattice for some values of the coupling constants. The quasiparticle localization and breaking of ergodicity manifest in a striking manner in the level spacing distribution. Its near Poissonian form provides evidence for the existence of local integrals of motion and establishes the OBS model as a nontrivial integrable generalization of the projected Creutz ladder Hamiltonian. These results show that level spacing statistics is a promising tool to study quasiparticle excitations in flat band superconductors.
Motivated by recent studies of the projected dice lattice Hamiltonian [K. Swaminathan , ], we introduce the on-site/bond singlet (OBS) model, a one-dimensional model of a flat band superconductor, in order to better understand the quasiparticle localization and interesting coexistence of ergodic and nonergodic behavior present in the former model. The OBS model is the sum of terms that have direct counterparts in the projected dice lattice Hamiltonian, each of which is parameterized by a coupling constant. Exact diagonalization reveals that the energy spectrum and nonequilibrium dynamics of the OBS model are essentially the same as that of the dice lattice for some values of the coupling constants. The quasiparticle localization and breaking of ergodicity manifest in a striking manner in the level spacing distribution. Its near Poissonian form provides evidence for the existence of local integrals of motion and establishes the OBS model as a nontrivial integrable generalization of the projected Creutz ladder Hamiltonian. These results show that level spacing statistics is a promising tool to study quasiparticle excitations in flat band superconductors.
Motivated by recent studies of the projected dice lattice Hamiltonian [K. Swaminathan et al., Phys. Rev. Research 5, 043215 (2023) [https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.043215]], we introduce the on-site/bond singlet (OBS) model, a one-dimensional model of a flat band superconductor, in order to better understand the quasiparticle localization and interesting coexistence of ergodic and non-ergodic behavior present in the former model. The OBS model is the sum of terms that have direct counterparts in the projected dice lattice Hamiltonian, each of which is parameterized by a coupling constant. Exact diagonalization reveals that the energy spectrum and non-equilibrium dynamics of the OBS model are essentially the same as that of the dice lattice for some values of the coupling constants. The quasiparticle localization and breaking of ergodicity manifest in a striking manner in the level spacing distribution. Its near Poissonian form provides evidence for the existence of local integrals of motion and establishes the OBS model as a non-trivial integrable generalization of the projected Creutz ladder Hamiltonian. These results show that level spacing statistics is a promising tool to study quasiparticle excitations in flat band superconductors.
In lattice models, destructive interference can quench single-particle motion and produce (nearly) dispersionless Bloch bands, known as flat bands. Suppressing kinetic energy enhances the role of interactions and can lead to pairing, constrained dynamics, and localization-like behavior even in translationally invariant settings. This thesis studies attractive Hubbard-type interactions in such flat-band systems using an isolated-band (projected) description, which yields an effective Hamiltonian in a Wannier-orbital basis. The π-flux dice lattice is a convenient focus because its two lowest bands are perfectly flat and degenerate, and their subspace admits compactly localized Wannier functions. This makes the projection straightforward to implement and yields a projected Hamiltonian with a simple, short-range structure. A one-dimensional reference model is used alongside the dice lattice to separate and benchmark the effects of different processes. In Publication I, we construct a class of exact many-body eigenstates of the projected Hamiltonian of generic lattice models with flat bands in which localized unpaired quasiparticles coexist with a background of paired particles in the zero-momentum state. This general result motivates the study of a specific model, the π-flux dice-lattice. Using exact diagonalization and realtime evolution from few-particle initial states, we show that two-body bound states behave ergodically. At the same time, the density and spin distribution retain strong memory of the initial quasiparticle positions at long times. These results demonstrate localization-like dynamics in a translationally invariant flat-band model, where memory arises from projection-induced constraints rather than quenched disorder. In Publication II, we introduce a one-dimensional on-site/bond singlet model whose terms mirror those of the projected dice-lattice Hamiltonian, with tunable couplings controlling their relative strengths. Exact diagonalization shows that, for some couplings, the spectrum and dynamics of the on-site/bond-singlet model closely reproduce those of the projected dice lattice. We then use spectral statistics to probe how the different dynamical regimes manifest in energy-level correlations. In Publication III, we apply spectral statistics to the projected dice lattice using higher-order spacing and gap-ratio diagnostics. We find a robust parity dependence: even particle-number sectors show multi-block GOE behavior within fixed symmetry sectors, while odd particle-number sectors exhibit exact doublets and inter-doublet correlations that are GUE at even orders. The appearance of distinct random-matrix ensembles (GOE and GUE) within the same physical model is an unusual and interesting feature, which calls for a deeper understanding of the hidden symmetry structure of the Hamiltonians obtained by projecting onto the flat-band subspace in the dice lattice and other lattice models with flat bands realized with ultracold gases in optical lattices and in the newly discovered class of moiré materials.
Abstract We study disorder-free many-body localization in the flat-band Creutz ladder, which was recently realized in cold-atoms in an optical lattice. In a non-interacting case, the flat-band structure of the system leads to a Wannier wavefunction localized on four adjacent lattice sites. In the flat-band regime both with and without interactions, the level spacing analysis exhibits Poisson-like distribution indicating the existence of disorder-free localization. Calculations of the inverse participation ratio support this observation. Interestingly, this type of localization is robust to weak disorders, whereas for strong disorders, the system exhibits a crossover into the conventional disorder-induced many-body localizated phase. Physical picture of this crossover is investigated in detail. We also observe non-ergodic dynamics in the flat-band regime without disorder. The memory of an initial density wave pattern is preserved for long times.
We construct a class of exact eigenstates of the Hamiltonian obtained by projecting the Hubbard interaction term onto the flat band subspace of a generic lattice model. These exact eigenstates are many-body states in which an arbitrary number of localized fermionic particles coexist with a sea of mobile Cooper pairs with zero momentum. By considering the dice lattice as an example, we provide evidence that these exact eigenstates are, in fact, a manifestation of local integrals of motions of the projected Hamiltonian. In particular, the spin and particle densities retain memory of the initial state for a very long time if localized unpaired particles are present at the beginning of the time evolution. This shows that many-body localization of quasiparticles and superfluidity can coexist even in generic two-dimensional lattice models with flat bands, for which it is not known how to construct local conserved quantities. Our results open interesting perspectives on the old condensed-matter problem of the interplay between superconductivity and localization.
Abstract Flat-band systems provide an ideal platform for studying strong electron correlations, in which multiple quasi-degenerate charge orders frequently coexist. Motivated by this context, we investigate the pair-density-wave (PDW) superconducting state in a quasi-one-dimensional Creutz lattice featuring two strictly flat bands. Using a renormalized mean-field approach for the strongly correlated t - J model, we obtain self-consistent solutions that demonstrate a PDW phase—characterized by finite-center-of-mass-momentum Cooper pairing—coexisting with charge density wave (CDW) and bond density wave (BDW) orders. Physically, this coexistence phenomenon originates from the perfect band nesting inherent to the flat-band system, which allows for changes in the particle momentum distributions without energy cost. Here, a pure PDW solution without CDW and BDW orders emerges in the underdoped regime when magnetic effects are neglected, while strong Hubbard correlations favor a pure CDW phase. We conclude that a robust PDW phase, accompanied by CDW and BDW orders, occupies the majority of the parameter space. Furthermore, both the PDW order parameter and its critical temperature scale linearly with the superexchange coupling strength—a hallmark of flat-band superconductivity. Overall, our results reveal that a PDW superconducting state can be realized on the Creutz lattice with strong correlations.
We study the interplay between on-site disorder and fermion pairing on the quasi-one-dimensional flat-band Creutz lattice. Both disorder and flat bands localize particles, but an attractive interaction results in pair formation and delocalization giving rise to superconductivity. In this work, we examine the attractive Hubbard model on the Creutz lattice to study the competition between these two effects and elucidate the properties of the superconducting phase and the localization quantum phase transition as the disorder strength is increased. Our main result is that flat-band superconductivity is robust against disorder: The critical disorder strength Wc required to localize the fermion pairs and destroy superconductivity is finite at any interaction strength U and is proportional to the superconducting weight Ds of the clean system. Using large-scale density matrix renormalization group computations, we show that this transition is of the Berezinskii-Kosterlitz-Thouless form. In addition, even at very small interaction strength, the localization is not due to single-fermion localization but to pair localization. For completeness, we briefly study this disorder-induced localization with mean field theory and show that Wc can be accurately determined by using an appropriate scaling function.