On Nonlinear Higher Spin Curvature

We present the first nonlinear term of the higher spin curvature which is covariant with respect to deformed gauge transformations that are linear in the field. We consider in detail the case of spin 3 after presenting spin 2 as an example, and then construct the general spin s quadratic term of the deWit-Freedman curvature.Comment: 12 page

Strongly Correlated Phases in Flatband Lattices

In this thesis we study strongly correlated phases of bosons and fermions in lattice models where the lowest Bloch band is flat, i.e. the kinetic energy is constant and independent of the lattice momentum. The states forming the flat band are localized with zero group velocity, facilitating the construction of exact solutions of the many-body problem. Using these solutions and numerical simulations, we characterize the ground state and transport properties of these systems. Weak interactions generally stabilize phases which can be described in terms of non-interacting quasiparticles, and which are qualitatively similar to the phases of free particles. A prime example is the Fermi liquid which is smoothly connected to the free electron gas. An important avenue towards new exotic phases of matter, which are profoundly different than their non-interacting counterparts, is to consider systems where strong interactions dominate over the kinetic energy. For instance, strong enough interactions between electrons can turn a metal into an insulator, the so-called Mott insulator. In a system with a flat band in the single-particle spectrum, the characteristic energy scale of the kinetic energy vanishes. In a situation like this, the behavior of the system is fully governed by interactions, and even weak interactions between particles stabilize strongly correlated phases which cannot be described by a straightforward perturbative approach. In the first part of this thesis, we consider weakly interacting bosons in a one-dimensional lattice with two flat bands and study its ground state phase diagram. We show that instead of straightforward condensation, repulsive interactions of any strength stabilize exotic phases such as a pair-liquid, a supersolid, and a Wigner crystal at small densities. We derive an effective theory by utilizing the general framework of projecting interactions onto the lowest flat band subspace. We also introduce a variational approach used to construct exact eigenstates of the effective Hamiltonian. Our analytical results are in excellent agreement with numerical calculations. The second part of this thesis deals with interacting fermions in a lattice with an isolated flat band. We extend the framework of projecting interactions and construct an effective theory by eliminating other bands via the perturbative Schrieffer-Wolff transformation. This allows us to not only recover the theory obtained via the projection but also incorporate interband transitions. Using this general approach and numerical calculations, we show that attractive interactions give rise to a superconducting state and study the transport properties of the system. In the last part, we consider lattice models with a band structure consisting of only flat bands. In such systems, all single-particle states are localized, and therefore, single particles cannot propagate at any energy. Whether this remains true in the presence of interactions is a nontrivial question because of the highly correlated nature of the ground states. Using exact analytical arguments, we show that this property remains true also across the entire many-body spectrum, meaning that even with the addition of interactions single particles remain localized and only pairs of particles can propagate through the system

Effective theory and emergent SU(2) symmetry in the flat bands of attractive Hubbard models

In a partially filled flat Bloch band electrons do not have a well defined Fermi surface and hence the low-energy theory is not a Fermi liquid. Neverethless, under the influence of an attractive interaction, a superconductor well described by the Bardeen-Cooper-Schrieffer (BCS) wave function can arise. Here we study the low-energy effective Hamiltonian of a generic Hubbard model with a flat band. We obtain an effective Hamiltonian for the flat band physics by eliminating higher lying bands via perturbative Schrieffer-Wolff transformation. At first order in the interaction energy we recover the usual procedure of projecting the interaction term onto the flat band Wannier functions. We show that the BCS wave function is the exact ground state of the projected interaction Hamiltonian and that the compressibility is diverging as a consequence of an emergent SU(2) symmetry. This symmetry is broken by second order interband transitions resulting in a finite compressibility, which we illustrate for a one-dimensionalladder with two perfectly flat bands. These results motivate a further approximation leading to an effective ferromagnetic Heisenberg model. The gauge-invariant result for the superfluid weight of a flat band can be obtained from the ferromagnetic Heisenberg model only if the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used. Finally, we prove an important inequality D≥W2 between the Drude weight D and the winding number W, which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.Peer reviewe

Preformed pairs in flat Bloch bands

| openaire: EC/H2020/702281/EU//FLATOPSIn a flat Bloch band, the kinetic energy is quenched and single particles cannot propagate since they are localized due to destructive interference. Whether this remains true in the presence of interactions is a challenging question because a flat dispersion usually leads to highly correlated ground states. Here we compute numerically the ground-state energy of lattice models with completely flat band structure in a ring geometry in the presence of an attractive Hubbard interaction. We find that the energy as a function of the magnetic flux threading the ring has a half-flux quantum Φ0/2=hc/(2e) period, indicating that only bound pairs of particles with charge 2e are propagating, while single quasiparticles with charge e remain localized. For some one-dimensional lattice models, we show analytically that in fact the whole many-body spectrum has the same periodicity. Our analytical arguments are valid for both bosons and fermions, for generic interactions respecting some symmetries of the lattice and at arbitrary temperatures. Moreover, for the same one-dimensional lattice models, we construct an extensive number of exact conserved quantities. These conserved quantities are associated to the occupation of localized single quasiparticle states and force the single-particle propagator to vanish beyond a finite range. Our results suggest that in lattice models with flat bands preformed pairs dominate transport even above the critical temperature of the transition to a superfluid state.Peer reviewe