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Symmetries and boundary conditions with a twist

D'Amico, Irene, de Oliveira, L. N., Zawadzki, K.

Published Jul 17, 2017• 12 citations

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.In the interest of clarity, 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, as a rule, boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian.The exception is twisted boundary condition with the special torsion Θ = πL/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 → ∞ 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 → ∞ limit.We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.

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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.In the interest of clarity, 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, as a rule, boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian.The exception is twisted boundary condition with the special torsion Θ = πL/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 → ∞ 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 → ∞ limit.We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.

Systems with conserved dipole moment have drawn considerable interest in light of their realization in recent experiments on tilted optical lattices. An important issue regarding such systems is delineating the conditions under which they admit a unique gapped ground state that is consistent with all symmetries. Here, we study one-dimensional translation-invariant lattices that conserve

\mathrm{U}(1)

charge and

{\mathbb{Z}}_{L}

dipole moment, where discreteness of the dipole symmetry is enforced by periodic boundary conditions, with

L

the system size. We show that in these systems a symmetric, gapped, and nondegenerate ground state requires not only integer charge filling, but also a fixed value of the dipole filling, while other fractional dipole fillings enforce either a gapless or symmetry-breaking ground state. In contrast with prior results in the literature, we find that the dipole filling constraint depends both on the charge filling as well as the system size, emphasizing the subtle interplay of dipole symmetry with boundary conditions. We support our results with numerical simulations and exact results.

The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary conditions, which we call duality-twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along noncontractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of a transfer matrix for the critical ferromagnetic Ising model on the M×N square lattice wrapped on the torus with a specific defect line. As a result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality-twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of a perturbating conformal approach by calculating the universal structure constant C_{n1n} for descendent states generated by the operator product expansion of the primary fields. For such states the calculations of an universal structure constants is a difficult task, since it involves knowledge of the four-point correlation function, which in general is not fixed by conformal invariance except for some particular cases, including the Ising model.

We have discussed a consistency condition of Berry phases defined by a local gauge twist and spatial symmetries of the many-body system. It imposes a nontrivial gap closing condition under the gauge twist in both finite- and infinite-size systems. It also implies a necessary condition for the gapped and unique ground state. As for the simplest case, it predicts an inevitable gap closing in the Heisenberg chain of half-integer spins. Its relation to the Lieb-Schultz-Mattis theorem is discussed based on the symmetries of the twisted Hamiltonian. The discussion is also extended to the (approximately) degenerate multiplet and fermion cases. It restricts the number of the states in the low energy cluster of the spectrum by the filling of the fermions. Constraints by the reflection symmetry are also discussed.

We study the effect of varying the boundary condition on: the spectral function of a finite one-dimensional Hubbard chain, which we compute using direct (Lanczos) diagonalization of the Hamiltonian. By direct comparison with the two-body response functions and with the exact solution of the Bethe ansatz equations, we can identify both spinon and holon features in the spectra. At half-filling the spectra have the well-known structure of a low-energy holon band and its shadow-which spans the whole Brillouin zone-and a spinon band present for momenta less than the Fermi momentum. Features related to the twisted boundary condition are cusps in the spinon band. We show that the spectral building principle, adapted to account for both the finite system size and the twisted boundary condition, describes the spectra well in terms of single spinon and holon excitations. We argue that these finite-size effects are a signature of spin-charge separation and that their study should help establish the existence and nature of spin-charge separation in finite-size systems.

We study the effect of varying the boundary condition on the spectral function of a finite one-dimensional Hubbard chain, which we compute using direct (Lanczos) diagonalization of the Hamiltonian. By direct comparison with the two-body response functions and with the exact solution of the Bethe ansatz equations, we can identify both spinon and holon features in the spectra. At half-filling the spectra have the well-known structure of a low-energy holon band and its shadow---which spans the whole Brillouin zone---and a spinon band present for momenta less than the Fermi momentum. Features related to the twisted boundary condition are cusps in the spinon band. We show that the spectral building principle, adapted to account for both the finite system size and the twisted boundary condition, describes the spectra well in terms of single spinon and holon excitations. We argue that these finite-size effects are a signature of spin-charge separation and that their study should help establish the existence and nature of spin-charge separation in finite-size systems.

We discuss the integrable boundary conditions for the one-dimensional (1D) Hubbard Model in the framework of the Quantum Inverse Scattering Method (QISM). We use the fermionic R-matrix proposed by Olmedilla et al. to treat the twisted periodic boundary condition and the open boundary condition. We determine the most general form of the integrable twisted periodic boundary condition by considering the symmetry matrix of the fermionic R-matrix. To find the integrable open boundary condition, we shall solve the graded reflection equation, and find there are two diagonal solutions, which correspond to a) the boundary chemical potential and b) the boundary magnetic field. Non-diagonal solutions are obtained using the symmetry matrix of the fermionic R-matrix and the covariance property of the graded reflection equation. They can be interpreted as the SO(4) rotations of the diagonal solutions.