Pair formation of hard core bosons in flat band systems. (English) Zbl 1395.82052
Summary: Hard core bosons in a large class of one or two dimensional flat band systems have an upper critical density, below which the ground states can be described completely. At the critical density, the ground states are Wigner crystals. If one adds a particle to the system at the critical density, the ground state and the low lying multi particle states of the system can be described as a Wigner crystal with an additional pair of particles. The energy band for the pair is separated from the rest of the multi-particle spectrum. The proofs use a Gerschgorin type of argument for block diagonally dominant matrices. In certain one-dimensional or tree-like structures one can show that the pair is localised, for example in the chequerboard chain. For this one-dimensional system with periodic boundary condition the energy band for the pair is flat, the pair is localised.
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B10 | Quantum equilibrium statistical mechanics (general) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
References:
| [1] | Bloch, I, Ultracold quantum gases in optical lattices, Nat. Phys., 1, 23-30, (2005) · doi:10.1038/nphys138 |
| [2] | Bloch, I; Dalibard, J; Zwerger, W, Many-body physics with ultracold gases, Rev. Mod. Phys., 80, 885, (2008) · doi:10.1103/RevModPhys.80.885 |
| [3] | Bollobás, B.: Graph Theory. Springer, Berlin (1979) · Zbl 0411.05032 · doi:10.1007/978-1-4612-9967-7 |
| [4] | Bunde, A., Havlin, S.: Fractals in Science. Springer, Berlin (1994) · Zbl 0796.00013 · doi:10.1007/978-3-662-11777-4 |
| [5] | Derzhko, O; Richter, J; Honecker, A; Schmidt, HJ, Universal properties of highly frustrated quantum magnets in strong magnetic fields, Low. Temp. Phys., 33, 745, (2007) · doi:10.1063/1.2780166 |
| [6] | Drescher, M., Mielke, A.: Hard-core bosons in flat band systems above the critical density. Eur. Phys. J. B90, 217 (2017) |
| [7] | Feingold, DG; Varga, RS, Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem, Pac. J. Math., 12, 1241-1250, (1962) · Zbl 0109.24802 · doi:10.2140/pjm.1962.12.1241 |
| [8] | Fisher, M; Weichman, P; Grinstein, G; Fisher, D, Boson localization and the superfluid-insulator transition, Phys. Rev. B, 40, 546, (1989) · doi:10.1103/PhysRevB.40.546 |
| [9] | Greiner, M; Mandel, O; Esslinger, T; Hänsch, TW; Bloch, I, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature, 415, 39-44, (2002) · doi:10.1038/415039a |
| [10] | Grémaud, B; Batrouni, GG, Haldane phase in the sawtooth lattice: edge states, entanglement and the flat band, Phys. Rev. B, 95, 165,131, (2017) · doi:10.1103/PhysRevB.95.165131 |
| [11] | Gutzwiller, MC, Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett., 10, 159, (1963) · doi:10.1103/PhysRevLett.10.159 |
| [12] | Hubbard, J, Electron correlations in narrow energy bands, Proc. R. Soc. A, 276, 238, (1963) · doi:10.1098/rspa.1963.0204 |
| [13] | Huber, SD; Altman, E, Bose condensation in flat bands, Phys. Rev. B, 82, 184-502, (2010) · doi:10.1103/PhysRevB.82.184502 |
| [14] | Jo, GB; Guzman, J; Thomas, CK; Hosur, P; Vishwanath, A; Stamper-Kurn, DM, Ultracold atoms in a tunable optical Kagome lattice, Phys. Rev. Lett., 108, 45,305, (2012) · doi:10.1103/PhysRevLett.108.045305 |
| [15] | Kanamori, J, Electron correlation and ferromagnetism of transition metals, Prog. Theor. Phys., 30, 275, (1963) · Zbl 0151.46703 · doi:10.1143/PTP.30.275 |
| [16] | Lieb, EH, Two theorems on the Hubbard model, Phys. Rev. Lett., 62, 1201, (1989) · doi:10.1103/PhysRevLett.62.1201 |
| [17] | Lieb, EH; Baeriswyl, D (ed.); Campbell, DK (ed.); Carmelo, JMP (ed.); Guinea, F (ed.); Louis, E (ed.), The Hubbard model: some rigorous results and open problems, 1-19, (1995), New York · Zbl 0886.60098 |
| [18] | Masumoto, N; Kim, NY; Byrnes, T; Kusudo, K; Löffler, A; Höfling, S; Forchel, A; Yamamoto, Y, Exciton-polariton condensates with flat bands in a two-dimensional Kagome lattice, New J. Phys., 14, 065,002, (2012) · doi:10.1088/1367-2630/14/6/065002 |
| [19] | Mielke, A, Ferromagnetism in the Hubbard model on line graphs and further considerations, J. Phys. A, 24, 3311-3321, (1991) · doi:10.1088/0305-4470/24/14/018 |
| [20] | Mielke, A, Exact ground states for the Hubbard model on the Kagome lattice, J. Phys. A, 25, 4335-4345, (1992) · doi:10.1088/0305-4470/25/16/011 |
| [21] | Mielke, A; Pavarini, E (ed.); Koch, E (ed.); Coleman, P (ed.), The Hubbard model and its properties, No. 5, (2015), Jülich |
| [22] | Motruk, J; Mielke, A, Bose-Hubbard model on two-dimensional line graphs, J. Phys. A, 45, 225,206, (2012) · Zbl 1243.82022 · doi:10.1088/1751-8113/45/22/225206 |
| [23] | Pariser, R; Parr, RG, A semi-empirical theory of the electronic spectra and electronic structure of complex unsaturated molecules. I, J. Chem. Phys., 21, 466-471, (1953) · doi:10.1063/1.1698929 |
| [24] | Petrosyan, D; Schmidt, B; Anglin, JR; Fleischhauer, M, Quantum liquid of repulsively bound pairs of particles in a lattice, Phys. Rev. A, 76, 033,606, (2007) · doi:10.1103/PhysRevA.76.033606 |
| [25] | Phillips, LG; Chiara, G; Öhberg, P; Valiente, M, Low-energy behaviour of strongly-interacting bosons on a flat-banded lattice above the critical filling factor, Phys. Rev. B, 91, 54,103, (2015) · doi:10.1103/PhysRevB.91.054103 |
| [26] | Pople, JA, Electron interaction in unsaturated hydrocarbons, Trans. Faraday Soc., 49, 1375-1385, (1953) · doi:10.1039/tf9534901375 |
| [27] | Pudleiner, P; Mielke, A, Interacting bosons in two-dimensional flat band systems, Eur. Phys. J. B, 88, 207, (2015) · doi:10.1140/epjb/e2015-60371-3 |
| [28] | Schulenburg, J; Honecker, A; Schnack, J; Richter, J; Schmidt, HJ, Macroscopic magnetization jumps due to independent magnons in frustrated quantum spin lattices, Phys. Rev. Lett., 88, 167,207, (2002) · doi:10.1103/PhysRevLett.88.167207 |
| [29] | Takayoshi, S; Katsura, H; Watanabe, N; Aoki, H, Phase diagram and pair Tomonaga-Luttinger liquid in a Bose-Hubbard model with flat bands, Phys. Rev. A, 88, 063,613, (2013) · doi:10.1103/PhysRevA.88.063613 |
| [30] | Tasaki, H, Ferromagnetism in the Hubbard models with degenerate single-electron ground states, Phys. Rev. Lett., 69, 1608, (1992) · Zbl 0968.82530 · doi:10.1103/PhysRevLett.69.1608 |
| [31] | Tasaki, H.: From Nagaoka’s ferromagnetism to flat-band ferromagnetism and beyond: an introduction to ferromagnetism in the Hubbard model. arXiv:cond-mat/9712219 (1997) |
| [32] | Tovmasyan, M; Nieuwenburg, E; Huber, S, Geometry induced pair condensation, Phys. Rev. B, 88, 220510(r), (2013) · doi:10.1103/PhysRevB.88.220510 |
| [33] | Voss, H.J.: Cycles and Bridges in Graphs. DVW, Berlin (1991) · Zbl 0731.05031 |
| [34] | Winkler, K; Thalhammer, G; Lang, F; Grimm, R; Denschlag, HJ; Daley, AJ; Kantian, A; Büchler, HP; Zoller, P, Repulsively bound atom pairs in an optical lattice, Nature, 441, 853, (2006) · doi:10.1038/nature04918 |
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