Efficient MPS Algorithm for Periodic Boundary Conditions and Applications

  • M. Weyrauch Physikalisch-Technische Bundesanstalt
  • M. V. Rakov Faculty of Physics, Taras Shevchenko National University of Kyiv
Keywords: matrix product representation for quantum states (MPS) and Hamiltonians (MPO), spin-1 Heisenberg ring, density matrix renormalization group (DMRG)

Abstract

We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current.

References



  1. S.R. White and D.A. Huse, Phys. Rev. B 48, 3844 (1993).
     https://doi.org/10.1103/PhysRevB.48.3844

  2. F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004).
     https://doi.org/10.1103/PhysRevLett.93.227205

  3. P. Pippan, S.R. White, and H.G. Evertz, Phys. Rev. B 81, 081103R (2010).
     https://doi.org/10.1103/PhysRevB.81.081103

  4. D. Porras, F. Verstraete, and J.I. Cirac, Phys. Rev. B 73, 014410 (2006).
     https://doi.org/10.1103/PhysRevB.73.014410

  5. D. Rossini, V. Giovanetti, and R. Fazio, J. Stat. Mech. P05021 (2011).

  6. F.D.M. Haldane, Phys. Lett. 93A, 464 (1983).
     https://doi.org/10.1016/0375-9601(83)90631-X

  7. E.S. Sørensen and A. P. Young, Phys. Rev. B 42, 754 (1990).
     https://doi.org/10.1103/PhysRevB.42.754

  8. A. Gendiar, R. Krcmar, and M. Weyrauch, Phys. Rev. B 79, 205118 (2009).
     https://doi.org/10.1103/PhysRevB.79.205118


Published
2018-10-10
How to Cite
Weyrauch, M., & Rakov, M. (2018). Efficient MPS Algorithm for Periodic Boundary Conditions and Applications. Ukrainian Journal of Physics, 58(7), 657. https://doi.org/10.15407/ujpe58.07.0657
Section
Solid matter