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Physical Review B

Charge degrees of freedom in frustrated lattice structures

Erich Runge, Peter Fulde

Published Dec 15, 2004• 40 citations
Publisher

Abstract

We study numerically spinless fermions with strong nearest-neighbor repulsion $V$ on frustrated lattice structures that show macroscopically many ground states in the absence of a kinetic energy term (hopping term). A finite hopping amplitude $t$ lifts the macroscopic degeneracy and leads to a small number of degenerate ground states. These can be characterized by topological quantum numbers and transformation properties under symmetry operations such as particle-hole interchange. Results for the criss-crossed checkerboard lattice (two-dimensional pyrochlore lattice) with up to 32 fermions support the scenario that translational invariance is restored in the thermodynamic limit, making half-charged quasiparticles possible. A finite entropy of approximately $(3∕4)\mathrm{ln}(4∕3)$ per site is released at low temperatures ${k}_{B}T\ensuremath{\approx}{t}^{3}∕{V}^{2}$, much below the bare energies $t,V$. This is familiar from the widely studied models involving spins on frustrated lattices, but in a spinless-fermion model these low-energy excitations are necessarily related to charge degrees of freedom.

DOI
10.1103/physrevb.70.245113

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Systems of strongly correlated fermions on certain geometrically frustrated lattices at particular filling factors support excitations with fractional charges ±e/2. We calculate quantum mechanical ground states, low–lying excitations and spectral functions of finite lattices by means of numerical diagonalization. The ground state of the most thorough-fully studied case, the criss-crossed checkerboard lattice, is degenerate and shows long–range order. Static fractional charges are confined by a weak linear force, most probably leading to bound states of large spatial extent. Consequently, the quasi-particle weight is reduced, which reflects the internal dynamics of the fractionally charged excitations. By using an additional parameter, we fine–tune the system to a special point at which fractional charges are manifestly deconfined—the so–called Rokhsar–Kivelson point. For a deeper understanding of the low–energy physics of these models and for numerical advantages, several conserved quantum numbers are identified.
We apply the contractor renormalization (CORE) method to the spin half Heisenberg antiferromagnet on the frustrated checkerboard and pyrochlore lattices. Their ground states are spin-gapped singlets which break lattice symmetry. Their effective Hamiltonians describe fluctuations of orthogonal singlet pairs on tetrahedral blocks, at an emergent low energy scale. We discuss low temperature thermodynamics and new interpretations of finite size numerical data. We argue that our results are common to many models of quantum frustration.
We show that a wide class of spin chains with topological frustration cannot develop any local order. In particular, we consider translational-invariant one-dimensional chains with frustrated boundary conditions, i.e. periodic boundary conditions and an odd number of sites, which possess a global SU(2) symmetry. This condition implies, even at a finite sizes, an exact degeneracy of the ground state and is quite general in absence of external fields. We directly evaluate the expectation value of operators with support over a finite range of lattice sites and show that, except for some precise conditions, they all decay algebraically, or faster, with the chain length and vanish in the thermodynamic limit. The exceptions that admit a finite order are cases with a higher ground state degeneracy in which the translational symmetry is broken by the ground state choice.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
Coulomb repulsion between electrons moving on a frustrated lattice can give rise, at simple commensurate electronic densities, to exotic insulating phases of matter. Such a phenomenon is illustrated using an extended $t\text{\ensuremath{-}}J$ model on a planar pyrochlore lattice for which the work on the quarter-filled case [D. Poilblanc et al., Phys. Rev. B 75, 220503 (2007)] is complemented and extended to 1∕81∕81∕8 and 3∕83∕83∕8 fillings. The location of the metal-insulator transition as a function of the Coulomb repulsion is shown to depend strongly on the sign of the hopping. Quite generally, the metal-insulator transition is characterized by lattice symmetry breaking but the nature of the insulating Mott state is more complex than a simple charge density wave. Indeed, in the limit of large Coulomb repulsion, the physics can be described in the framework of (extended) quantum fully packed loop or dimer models carrying extra spin degrees of freedom. Various diagonal and off-diagonal plaquette correlation functions are computed and the low-energy spectra are analyzed in detail in order to characterize the nature of the insulating phases. We provide evidence that, as for an electronic density of n=1∕2n=1∕2n=1∕2 (quarter filling), the system at n=1∕4n=1∕4n=1∕4 or n=3∕4n=3∕4n=3∕4 exhibits also plaquette order by forming a (lattice rotationally invariant) resonant singlet pair crystal, although with a quadrupling of the lattice unit cell (instead of a doubling for n=1∕2n=1∕2n=1∕2) and a fourfold degenerate ground state. Interestingly, qualitative differences with the bosonic analog (e.g., known to exhibit columnar order at n=1∕4n=1∕4n=1∕4) emphasize the important role of the spin degrees of freedom in, e.g., stabilizing plaquette phases with respect to rotational symmetry-breaking phases.
Spinless fermions on a lattice with nearest-neighbor repulsion serve as a toy-version Hubbard model and have a symmetry-broken even/odd superlattice at half filling. At infinite repulsion, doped holes form charged stripes that are antiphase walls ( as noted by Mila [Phys. Rev. B 49,49,49, 14 047 (1994)]). Exact-diagonalization data for systems up to 36 sites around 1/4 filling, and also for one or two holes added to a stripe of length up to 12, indicate stability of the stripe-array state against phase separation. In the boson version of the model, the same behavior can be stabilized by addition of a term with four creation/annihilation operators.