Solving the Bose-Hubbard model in new ways

Artur Sowa; Jonas Fransson

doi:10.22331/q-2022-06-02-728

Solving the Bose-Hubbard model in new ways

Artur Sowa¹ and Jonas Fransson²

¹Department of Mathematics and Statistics, University of Saskatchewan, Canada
²Department of Physics and Astronomy, University of Uppsala, Sweden

Published:	2022-06-02, volume 6, page 728
Eprint:	arXiv:2106.09324v3
Doi:	https://doi.org/10.22331/q-2022-06-02-728
Citation:	Quantum 6, 728 (2022).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We introduce a new method for analysing the Bose-Hubbard model for an array of bosons with nearest neighbor interactions. It is based on a number-theoretic implementation of the creation and annihilation operators that constitute the model. One of the advantages of this approach is that it facilitates accurate computations involving multi-particle states. In particular, we provide a rigorous computer assisted proof of quantum phase transitions in finite systems of this type.
Furthermore, we investigate properties of the infinite array via harmonic analysis on the multiplicative group of positive rationals. This furnishes an isomorphism that recasts the underlying Fock space as an infinite tensor product of Hecke spaces, i.e., spaces of square-integrable periodic functions that are a superposition of non-negative frequency harmonics. Under this isomorphism, the number-theoretic creation and annihilation operators are mapped into the Kastrup model of the harmonic oscillator on the circle. It also enables us to highlight a kinship of the model at hand with an array of spin moments with a local anisotropy field. This identifies an interesting physical system that can be mapped into the model at hand.

► BibTeX data

@article{Sowa2022solvingbosehubbard,
  doi = {10.22331/q-2022-06-02-728},
  url = {https://doi.org/10.22331/q-2022-06-02-728},
  title = {Solving the {B}ose-{H}ubbard model in new ways},
  author = {Sowa, Artur and Fransson, Jonas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {728},
  month = jun,
  year = {2022}
}

► References

[1] S. R White, Density Matrix Formulation for Quantum Renormalization Groups, Phys. Rev. Lett. 69 (1992), 2863–2866.
https://doi.org/10.1103/physrevlett.69.2863

[2] T. D. Kühner and H. Monien, Phases of the one-dimensional Bose-Hubbard model, Phys. Rev. B, 58 (1998), 14741(R).
https://doi.org/10.1103/physrevb.58.r14741

[3] T. D. Kühner, S. R. White, and H. Monien, One-dimensional Bose-Hubbard model with nearest-neighbor interaction, Phys. Rev. B, 61 (2000), 12474.
https://doi.org/10.1103/physrevb.61.12474

[4] M. Rizzi, D. Rossini, G. De Chiara, S. Montangero, and R. Fazio, Phase Diagram of Spin-1 Bosons on One-Dimensional Lattices, Phys. Rev. Lett. 95 (2005), 240404.
https://doi.org/10.1103/physrevlett.95.240404

[5] A. Roy, J. Hauschild, and F. Pollmann, Quantum phases of a one-dimensional Majorana-Bose-Hubbard model, Phys. Rev. B, 101 (2020), 075419.
https://doi.org/10.1103/physrevb.101.075419

[6] W. J. Hu and N. -H. Tong, Dynamical mean-field theory for the Bose-Hubbard model, Phys. Rev. B, 80 (2009), 245110.
https://doi.org/10.1103/physrevb.80.245110

[7] B. Hetényi, L. M. Martelo, and B. Tanatar, Superfluid weight and polarization amplitude in the one-dimensional bosonic Hubbard model, Phys. Rev. B, 100 (2019), 174517.
https://doi.org/10.1103/physrevb.100.174517

[8] J. Links, A. Foerster, A.P. Tonel, G. Santos, The two-site Bose–Hubbard model, Ann. Henri Poincaré 7 (2006), 1591.
https://doi.org/10.1007/s00023-006-0295-3

[9] A.P. Tonel, J. Links, A. Foerster, Quantum dynamics of a model for two Josephson-coupled Bose–Einstein condensates, J. Phys. A: Math. Gen. 38 (2005), 1235.
https://doi.org/10.1088/0305-4470/38/6/004

[10] M.T. Batchelor, A. Foerster, Yang-Baxter integrable models in experiments: from condensed matter to ultracold atoms, J. Phys. A: Math. Theor. 49 (2016), 173001.
https://doi.org/10.1088/1751-8113/49/17/173001

[11] J. I. Cirac, M. Lewenstein, K. Mølmer, and P. Zoller, Quantum superposition states of Bose-Einstein condensates, Phys. Rev. A 57 (1998), 1208.
https://doi.org/10.1103/physreva.57.1208

[12] F. Pan, and J. P. Draayer, Quantum critical behavior of two coupled Bose-Einstein condensates, Phys. Lett. A: General, Atomic and Solid State Physics, 339 (2005), 403–407.
https://doi.org/10.1016/j.physleta.2005.03.027

[13] N. Oelkers and J. Links, Ground-state properties of the attractive one-dimensional Bose–Hubbard model, Phys. Rev. B 75 (2007), 115119.
https://doi.org/10.1103/physrevb.75.115119

[14] J. Javanainen and U. Shrestha, Nonlinear Phenomenology from Quantum Mechanics: Soliton in a Lattice, Phys. Rev. Lett. 101 (2008), 170405.
https://doi.org/10.1103/physrevlett.101.170405

[15] P. Buonsante, V. Penna, and A. Vezzani, Quantum signatures of the self-trapping transition in attractive lattice bosons, Phys. Rev. A 82 (2010), 043615.
https://doi.org/10.1103/physreva.82.043615

[16] J.-B. Bost and A. Connes, Hecke Algebras, Type III Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory, Selecta Mathematica New Series 1 (1995) 411–457.
https://doi.org/10.1007/bf01589495

[17] J. Dereziński, C. Gérard, Mathematics of Quantization and Quantum Fields, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2013.
https://doi.org/10.1017/cbo9780511894541

[18] Peter D.T.A. Elliott, Duality in Analytic Number Theory, Cambridge University Press 1997.
https://doi.org/10.1017/cbo9780511983405

[19] T. Holstein and H. Primakoff, Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet, Phys. Rev. 58 (1940), 1098–1113.
https://doi.org/10.1103/physrev.58.1098

[20] H.A. Kastrup, Quantization of the Optical Phase Space $S^2 = \{\phi \mbox{ mod } 2\pi, I >0 \}$ in Terms of the Group $\mbox{SO}^{\uparrow}(1,2)$, Fortschr. Phys. 51 (2003), 975–1134.
https://doi.org/10.1002/prop.200310115

[21] M. W. Jack and M. Yamashita, Bose-Hubbard model with attractive interactions Phys. Rev. A 71 (2005), 023610.
https://doi.org/10.1103/physreva.71.023610

[22] Linda E. Reichl, A Modern Course in Statistical Physics, Fourth, revised Edition, Wiley-vch 2016.
https://doi.org/10.1002/9783527690497

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2025-04-05 00:34:22). On SAO/NASA ADS no data on citing works was found (last attempt 2025-04-05 00:34:23).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.