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arXiv:0704.0521 · International Journal of Modern Physics B

FRACTIONALLY CHARGED EXCITATIONS ON FRUSTRATED LATTICES

Erich Runge, Frank Pollmann, Peter Fulde

Published May 30, 2007• 1 citation

Abstract

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.

DOI: 10.1142/s0217979207043609

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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 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.

A model of strongly correlated spinless fermions on a checkerboard lattice is mapped onto a quantum fully packed loop model. We identify a large number of fluctuationless states specific to the fermionic case. We also show that for a class of fluctuating states the fermionic sign problem can be gauged away. This claim is supported by numerical evaluation of the low-lying states. Furthermore, we analyze excitations at the Rokhsar-Kivelson point of this model using the relation to the height model and the single-mode approximation.

We consider fermionic fully packed loop and quantum dimer models which serve as effective low-energy models for strongly correlated fermions on a checkerboard lattice at half- and quarter-filling, respectively. We identify a large number of fluctuationless states specific to each case and which are due to fermionic statistics. We discuss the symmetries and conserved quantities of the system and show that, for a class of fluctuating states in the half-filling case, the fermionic sign problem can be gauged away. This claim is supported by a numerical evaluation of the low-lying states and can be understood by means of an algebraic construction. The elimination of the sign problem then allows us to analyze excitations at the Rokhsar-Kivelson point of the models using the relation to the height model and its excitations, within the single-mode approximation. We then discuss a mapping to a U(1) lattice gauge theory which relates the considered low-energy model to the compact quantum electrodynamics in

2+1

dimensions. Furthermore, we point out consequences and open questions in the light of these results.

The single-particle spectral function for an incompressible fractional quantum Hall state in the presence of a scalar short-ranged attractive impurity potential is calculated via exact diagonalization within the spherical geometry. In contrast to the noninteracting case, where only a single bound state below the lowest Landau level forms, electron–electron interactions strongly renormalize the impurity potential, effectively giving it a finite range, which can support many quasi-bound states (long-lived resonances). Averaging the spectral weights of the quasi-bound states and extrapolating to the thermodynamic limit, for filling factor ν = 1/3 we find evidence consistent with localized fractionally charged e /3 quasi-particles. For ν = 2/5, the results are slightly more ambiguous, due to finite size effects and possible bunching of Laughlin-quasi-particles.

We address the question whether features known from quantum chromodynamics (QCD) can possibly also show up in solid-state physics. It is shown that spinless fermions of charge

e

on a checkerboard lattice with nearest-neighbor repulsion provide for a simple model of confined fractional charges. After defining a proper vacuum the system supports excitations with charges

\pm e/2

attached to the ends of strings. There is a constant confining force acting between the fractional charges. It results from a reduction of vacuum fluctuations and a polarization of the vacuum in the vicinity of the connecting strings.