Symmetries and boundary conditions with a twist

Abstract

Interest in finite-size systems has risen in the last decades, due to the focus on nanotechnological applications and because they are convenient for numerical treatment that can subsequently be extrapolated to infinite lattices. Independently of the envisioned application, special attention must be given to boundary condition, which may or may not preserve the symmetry of the infinite lattice. Here we present a detailed study of the compatibility between boundary conditions and conservation laws. The conflict between open boundary conditions and momentum conservation is well understood, but we examine other symmetries, as well: we discuss gauge invariance, inversion, spin, and particle-hole symmetry and their compatibility with open, periodic, and twisted boundary conditions. We develop the reasoning in the framework of the one-dimensional half-filled Hubbard model, whose Hamiltonian displays a variety of symmetries. Our discussion includes analytical and numerical results. Our analytical survey shows that boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian. The exception is twisted boundary condition with the special torsion $\Theta =\pi L/2$, where $L$ is the lattice size. Our numerical results for the ground-state energy at half-filling and the energy gap for $L=2$--$7$ show how the breaking of symmetry affects the convergence to the $L\to \infty$ limit. We compare the computed energies and gaps with the exact results for the infinite lattice drawn from the Bethe-Ansatz solution. The deviations are boundary-condition dependent. The special torsion yields more rapid convergence than open or periodic boundary conditions. For sizes as small as $L=7$, the numerical results for twisted condition are very close to the $L\to \infty$ limit. We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.

arXiv commentAccepted for publication by the Brazilian Journal of Physics. The final publication is available at Springer via http://dx.doi.org/10.1007/s13538-017-0517-9

References in corpus (24)

• Spin and Charge Resolved Quantum Gas Microscopy of Antiferromagnetic Order in Hubbard Chains
• Twist-averaged Boundary Conditions in Continuum Quantum Monte Carlo
• Observation of Spatial Charge and Spin Correlations in the 2D Fermi-Hubbard Model
• Site-resolved measurement of the spin-correlation function in the Hubbard model
• Two Fermions in a double well: Exploring a fundamental building block of the Hubbard model
• Twisted Boundary Conditions in Lattice Simulations
• The one-dimensional Hubbard model: A reminiscence
• On the discretization of physical momenta in lattice QCD
• Quantum Simulation of the Hubbard Model with Dopant Atoms in Silicon
• Twisted valence quarks and hadron interactions on the lattice
• Integrable Boundary Conditions for the One-Dimensional Hubbard Model
• A Numerical Study of Partially Twisted Boundary Conditions
• Disorder-Induced Stabilization of the Pseudogap in Strongly Correlated Systems
• Control of the finite size corrections in exact diagonalization studies
• Flavor Twisted Boundary Conditions, Pion Momentum, and the Pion Electromagnetic Form Factor
• Time-dependent density functional theory with twist-averaged boundary conditions
• Temperature correlators in the one-dimensional Hubbard model in the strong coupling limit
• Fractional instantons and bions in the principal chiral model on ${\mathbb{R}}^2\times S^1$ with twisted boundary conditions
• Hubbard model for atomic impurities bound by the vortex lattice of a rotating BEC
• Bound states on the lattice with partially twisted boundary conditions
• Mesoscopic Bose-Einstein condensates as quantum simulators
• Quantum Magnetism with Mesoscopic Bose-Einstein Condensates
• Size and shape of Mott regions for fermionic atoms in a two-dimensional optical lattice
• Pseudoscalar mesons in a finite cubic volume with twisted boundary conditions

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