Strictly localised triplet dimers on one- and two-dimensional lattices
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Strictly localised triplet dimers on one- andtwo-dimensional lattices
S. Jackson and J.H.Samson
Department of Physics, Loughborough University, Loughborough, LE11 3TU, UKE-mail: [email protected], [email protected]
Abstract.
Electrons may form inter-site pairs (dimers) by a number of mechanisms. Forexample, long-range (Fr¨ohlich) electron-phonon interactions and strong on-site Hubbard U allowformation of small light bipolarons in some lattices. We identify circumstances under whichtriplet dimers are strictly localised by interference in certain one- and two-dimensional lattices.We assume a U-V Hamiltonian with nearest- and next-nearest-neighbour hopping integrals t and t ′ , large positive U and attractive nearest- and next-nearest-neighbour interactions V and V ′ . Inthe square ladder and some two-dimensional bilayers, if the dimer Hilbert space is restricted tonearest- and next-nearest-neighbour dimers, triplet dimers become strictly localised for certainvalues of these parameters. For example, in a square ladder with t ′ = t and V ′ = V , alltriplet bands become flat due to exact cancellation of hopping paths. We identify the localisedeigenstates for all flat bands in each lattice. We show that many of the flat bands persist forarbitrary t/t ′ so long as other restrictions still apply. Electrons may form inter-site pairs (dimers) by a number of mechanisms. For example, long-range (Fr¨ohlich) electron-phonon interactions and strong on-site Hubbard U allow formation ofsmall light bipolarons in some lattices. We identify circumstances under which triplet dimers arestrictly localised by interference in certain one- and two-dimensional lattices. Our calculationsare not specific to the pairing mechanism but our model can be obtained from the Coulomb-Fr¨ohlich model [1, 2, 3, 4, 5, 6]. In this model electrons interact with lattice distortions onneighbouring sites. The electron together with the distortion forms a new quasiparticle – thepolaron. An effective attraction between polarons can exist because two polarons can deformthe lattice more effectively together than separately.Applying the Lang-Firsov transformation [7] we obtain the
U V model, valid for strongcoupling [8]: H = − X ijσ t ij c † iσ c jσ + U X i n i ↑ n i ↓ + 12 X i X j = i X σσ ′ V ij c † iσ c iσ c † jσ ′ c jσ ′ (1)where c † iσ and c jσ are electron creation and annihilation operators respectively, t ij is arenormalized hopping, U is the onsite Coulomb repulsion and V ij is the polaron-polaron potentialcontaining the Coulomb repulsion between electrons on neighbouring sites i and j and the (non-retarded) effective attraction due to the Fr¨ohlich electron-phonon interaction (EPI). We takethe on-site repulsion U to be infinite so that basis states with two electrons on a single site aresuppressed.he Hilbert space of two particles on on a d -dimensional tight-binding lattice can berepresented as a particle on a 2 d -dimensional lattice, the tensor product of the two Hilbertspaces. However, if the attraction between the particles is strong enough to bind them into adimer over the whole Brillouin zone, the low-lying states will have large amplitude only near a d -dimensional subspace where the particles are in close proximity. Accordingly a truncation ofthe Hilbert space of dimers to a small number of bond lengths will capture the essential physics.We call the truncated Hilbert space of the dimers the dimer lattice . Let D i = { j : 0 < r ij ≤ L max } (2)be the set of sites j whose distance r ij from a site i is no more than L max . We shall call D i theneighbours of i . A dimer will have (spin-independent) diagonal potential V ij . We write V forthe nearest-neighbour potential and V ′ for the next-nearest-neighbour potential.If a lattice Λ has N sites and the mean number of neighbours | D i | is ν then the single-electronHilbert space is 2 N -dimensional, the two-electron Hilbert space is N (2 N − ν N -dimensional. We can further reduce to one singlet and threetriplet spaces, each of dimensionality νN/
2. The dimer spaces for S z = 0 are S = span (cid:26) √ | i ↑ j ↓i + | j ↑ i ↓i ) : i ∈ Λ , j ∈ D i (cid:27) (3) T = span (cid:26) √ | i ↑ j ↓i − | j ↑ i ↓i ) : i ∈ Λ , j ∈ D i (cid:27) (4)for singlets and triplets respectively. In each case basis states are double-counted in the span.We note that triplets are spatially antisymmetric and singlets are symmetric; if we representthe above dimer basis states as arrows pointing from i to j , then | i −→ j i = ±| j −→ i i with+ for singlets. The above formalism enables us to write the Hamiltonian of the dimers in eachsector as a tight-binding Hamiltonian on a dimer lattice constructed by placing a node on theline joining each site i to each point j ∈ D i . If j ∈ D i and k ∈ D i , and t jk = 0, then the dimercan hop from ij to ik . A dimer hopping vector is then drawn between the two nodes on thedimer lattice with hopping integral t jk .We consider a square ladder with nearest- and next-nearest-neighbour dimers, nearest andnext-nearest-neighbour hopping t and t ′ respectively and nearest- and next-nearest-neighbourinteractions V and V ′ respectively. Each sector of the truncated Hilbert space may be reducedto the five basis states indicated in figure 1(a), where n is the index of the unit cell. The dimerlattice comprises a chain of corner-sharing octahedra as indicated in figure 1(b). We find thatsome or all of the bands are flat depending on the ratios V ′ /V and t ′ /t . For V ′ = V we obtainthe triplet band structure: E ( k ) − V = ± t (5) E ( k ) − V = 0 (6) E ( k ) − V = ± s t cos (cid:18) ka (cid:19) + 8 t ′ sin (cid:18) ka (cid:19) (7)For V ′ = V and t ′ = t we obtain the triplet band structure E ( k ) − V = 0, E ( k ) − V = ± t , E ( k ) − V = ± √ t . Since all bands are flat we infer that triplets are strictly localised for V ′ = V and t ′ = t . This is because the points | A, n i in the dimer lattice (see figure 1(b)) are bottlenecksmeaning that a dimer propagating along the ladder must repeatedly pass through these points.Any path from | A, n i to a neighbouring bottleneck | A, n ± i can be replaced by a path ofopposite sign (and equal magnitude if t ′ = t ) by interchanging | B, n i with | C, n i and | D, n i with E, n i . For example | A, n i → |
B, n i → |
A, n + 1 i is replaced with | A, n i → |
C, n i → − |
A, n + 1 i .Since each path can be replaced by another path of opposite sign, and equal magnitude if t ′ = t ,the paths cancel each other for t ′ = t and triplet dimers become strictly localised. We haveidentified five linearly-independent localised eigenstates of the Hamiltonian.We also consider square and honeycomb bilayers with restricted dimer-lengths and find thatif some additional restrictions apply all triplet bands are flat for V ′ = V and t ′ = t . As inthe square ladder this is due to the existence of bottlenecks and the exact cancellation of pathsconnecting neighbouring bottlenecks. The band structure and localised eigenstates for thesebilayers are essentially the same as for the square ladder.We conclude that triplets are less mobile than singlets on a square ladder and that for t ′ = t and V ′ = V , if dimer-lengths are restricted, triplets become strictly localised. This is also truefor square and honeycomb bilayers. Figure 1. (a) Triplet basis states in the square ladder for nearest- and next-nearest-neighbourdimers. (b) Unit cell of the triplet dimer lattice. Adjacent cells are connected by the points | A, n i which form bottlenecks. [1] A. S. Alexandrov 1996 Phys. Rev. B J.Phys: Condensed matter Phys. Rev. Lett. , 807[4] P. E. Spencer, J. H. Samson, P. E. Kornilovitch and A. S. Alexandrov 2005 Phys. Rev. B J.Phys: Condensed matter Physical Review Letters Zh. Eksp. Teor. Fiz. Sov. Phys.-JETP Rep. Prog. Phys.57