Correlated Fermions on a Checkerboard Lattice

Abstract

A model of strongly correlated spinless fermions on a checkerboard lattice is mapped onto a quantum FPL 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.The interplay between quantum and geometric frustration can result in many unusual properties. From that point of view, strongly correlated spinless fermions on a checkerboard lattice present an interesting case. At certain fillings, fractionally charged excitations have been predicted for the case of large nearest-neighbor repulsion [1]. The subject is not purely academic since the checkerboard lattice is a projection of a pyrochlore lattice onto a plane. There is experimental evidence that electrons in pyrochlore lattices can be strongly correlated [2]. We shall show that at half filling, this problem can be mapped onto a quantum fully packed loop (FPL) model -an analog of the quantum dimer model (QDM) -and discuss a number of consequences. QDMs, originally introduced in the context of quantum magnetism [3], became a focus of recent interest following the discovery of a gapped liquid phase on a triangular lattice [4]. In particular, it has been established that a gapped phase with deconfined excitations exists in 2D for QDMs on non-bipartite lattices [4,5,6,7] while on bipartite lattices, such as a square lattice, systems typically crystallize into a phase with a broken translation/rotation symmetry. The liquid phase is "shrunk" into a quantum critical point with gapless excitations [3,8,9]. In both cases, an effective gauge theory is a U(1) theory for such a critical point and a Z 2 theory for a deconfined phase [10,11]. Several related models, such as the quantum six/eight-vertex models [12,13,14,15] have been shown to conform to the same dichotomy: a model with orientable loops is critical and is described by a U(1) gauge theory.The aforementioned models share one important feature: the matrix elements connecting various states of the lowenergy Hilbert space are all non-negative. The Hilbert space separates into different sectors so that all states within a sector are connected by the quantum dynamics while states belonging to different sectors are not. Then, by Perron-Frobenius theorem, the ground state (GS) of each sector is unique and nodeless [4,16]. Much less is known about models with nonFrobenius dynamics which results in some negative matrix elements. A striking difference between Frobenius vs. nonFrobenius QDMs on the kagomé lattice was found in [7,17]: while the conventional QDM exhibits a gapped Z 2 topological phase, its counterpart with different signs has an extensive GS degeneracy. A different non-Frobenius model of lattice fermions [18] has also been found to have an extensive GS...

Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice

“…It might also be interesting to revisit two-dimensional ice-type materials, such as the proton bonded ferroelectric copper formate tetrahydrate 41 . While two-dimensional quantum ice models are known to order at low temperatures [110][111][112][113][114][115] , they are described by the same class of lattice gauge theory, and possess the same spinon excitations as their three-dimensional counterparts 40,111,115 . These excitations will be confined in the ordered state, but might be visible at finite energy, and above the ordering temperature.…”

Exotic Mott phases of the extendedt−Jmodel on the checkerboard lattice at commensurate densities

“…This condition still preserves an extensive (e.g. macroscopic) manifold of states 7,10,11,12 . Although quantum fluctuations occur between such configurations (in second order in t), the GS is an insulator since no charge can be driven between two distant locations in the system without violating the local constraints.…”

“…6) and, otherwise, are tractable by weak coupling expansion around a periodic pattern of appropriately chosen clusters 9 and by Exact Diagonalisation (ED) techniques. In the case of spinless fermions on the checkerboard lattice, the dynamics within the constrained Hilbert space is introduced via 6-sites ring processes on hexagons involving 3 particles simultaneously and gives rise to a rich phase diagram 10 and, under some conditions, to particle fractionalization 11 . The case of spinful fermions which has been recently investigated at and around quarter-filling 12 involves a second-order kinetic process on the "empty squares" (i.e.…”

Classical correlations of defects in lattices with geometrical frustration in the motion of a particle

“…7 In particular, our results can be extended to the motion of strongly correlated spinless fermions on the checkerboard lattice. 8 TABLE I. The summary of the defect-defect correlations we obtained for the four different lattices and the two distinct dimer coverings.…”

“…7 Furthermore, in our case, one can expect classical correlations to hold more generally since the low-energy excitations can be described equivalently by hard-core bosons or fermions. 8 Historically, studies of monomer and dimer models started very early. 9 At zero doping, the Pfaffian method introduced by Kasteleyn allows the derivation of the classical correlations analytically.…”