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The flux phase problem on the ring

Fumihiko Nakano

Published Jan 1, 2000• 7 citations
OpenAlex

Abstract

We give a simple proof to derive the optimal flux which minimizes the ground state energy in one dimensional Hubbard model, provided the number of particles is even. KEY WORDS: Hubbard model, flux phase problem. 1

Related papers

The flux phase problem on the ring

Fumihiko Nakano

We give a simple proof to derive the optimal flux which minimizes the ground state energy in the one-dimensional Hubbard model, provided the number of particles is even.

arXiv:math-ph/0006022•Journal of Physics A Mathematical and General•Jul 19, 2000•8 citations

Spin of the ground state and the flux phase problem on the ring

Fumihiko Nakano

As a continuation of our previous work, we derive the optimal flux phase which minimizes the ground state energy in the one-dimensional many-particle systems, when the number of particles is odd in the absence of on-site interaction and external potential. Moreover, we study the relationship between the flux on the ring and the spin of the ground state through which we derive some information on the sum of the lowest eigenvalues of one-particle Hamiltonians.

arXiv:math-ph/0402044•Journal of Physics A Mathematical and General•Mar 17, 2004•0 citations

Exact solutions for two interacting particles in a one-dimensional ring with a magnetic flux

Qiming Li, J. Callaway

We solve the one-dimensional extended Hubbard model with a magnetic flux analytically for the case of two particles. We find that bound states are ground states only for attractive interactions. We have studied the variation of the ground-state energy as a function of the flux. For the bound state, the period of the oscillation is half of a flux quantum. The dependence of the magnitude of the energy barrier on the size of the ring is discussed in connection with the possible existence of the flux quantization in the thermodynamic limit.

Physical review. B, Condensed matter•Feb 1, 1991•6 citations

Fermi–Hubbard model on nonbipartite lattices: flux problem and emergent chirality

Wayne Zheng

Abstract On several one-dimensional (1D) and 2D nonbipartite lattices, we study both free and Hubbard interacting lattice fermions when some magnetic fluxes are threaded or gauge fields coupled. First, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-1/2 fermions at half-filling on a ring lattice consisting of odd-numbered sites, the optimal flux turns out to be ± π /2. We prove this conclusion for Hubbard interacting fermions utilizing a generalized reflection positivity technique, which can lead to further applications on 2D nonbipartite lattices such as triangular and Kagome. At half-filling the optimal flux patterns on the triangular and Kagome lattice are ascertained to be ±[ π /2, π /2], ±[ π /2, π /2, 0], respectively (see the meaning of these notations in the main text). We also find that chirality emerges in these optimal flux states. Then, we verify these exact conclusions and further study some other fillings with the numerical exact diagonalization method. It is found that when it deviates from half-filling, Hubbard interactions can alter the optimal flux patterns on these lattices. Moreover, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states— both in 1D and 2D—are discussed and interpreted as some kind of non-Fermi liquid feature.

arXiv:1910.06354•Journal of Statistical Mechanics Theory and Experiment•Jul 1, 2020•1 citation

Fermion Hubbard model on non-bipartite lattices: flux problem, gauge fields and emergent chirality

Wayne Zheng

On some one dimensional (1D) and 2D non-bipartite lattices, from scratch we study both free and Hubbard interacting lattice fermions when there are some magnetic fluxes threaded or appropriate gauge fields coupled. On one hand, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-$1/2$ fermions at half-filling on a ring lattice with an odd number of sites, the optimal flux is determined to be $\pm{\pi}/2$. We prove this conclusion for Hubbard interacting fermions by means of a generalized reflection positivity technique. It can further lead to some applications towards 2D non-bipartite lattices such as triangle and Kagome. At half-filling the optimal flux pattern on the triangular lattice can be ascertained to be $[{\pi}/2, {\pi}/2]$. We find that chirality emerges in these optimal flux states. On the other hand, we verify these conclusions and further study other fillings away from half with the numerical exact diagonalization (ED) method and find that, when it deviates from half-filling, Hubbard interactions can even alter the optimal flux patterns on these lattices. Both in 1D and 2D, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states are addressed and interpreted as some non-Fermi liquid (NFL) features.

Bulletin of the American Physical Society•Oct 14, 2019•0 citations

Exact solution of the one-dimensional Hubbard model in a magnetic field

Gang Su, Bao‐Heng Zhao, Mo‐Lin Ge

By means of the Bethe-ansatz method, the one-dimensional Hubbard model with repulsive on-site coupling in an external magnetic field is solved exactly. It is found that the energy of the ground state depends on the magnetic flux \ensuremath{\varphi} through the ring formed by a periodic lattice chain, and the lowest-energy state corresponds to \ensuremath{\varphi}=n(hc/2e), with n an integer and -e the electron charge. The calculations illustrate that the strongly correlated Hubbard ring with periodic boundary conditions in its ground state is a superconducting ring. It can also be shown that the system possesses an internal SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) symmetry.

Physical review. B, Condensed matter•Dec 1, 1992•4 citations
We solve the one-dimensional extended Hubbard model with a magnetic flux analytically for the case of two particles. We find that bound states are ground states only for attractive interactions. We have studied the variation of the ground-state energy as a function of the flux. For the bound state, the period of the oscillation is half of a flux quantum. The dependence of the magnitude of the energy barrier on the size of the ring is discussed in connection with the possible existence of the flux quantization in the thermodynamic limit.
Abstract On several one-dimensional (1D) and 2D nonbipartite lattices, we study both free and Hubbard interacting lattice fermions when some magnetic fluxes are threaded or gauge fields coupled. First, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-1/2 fermions at half-filling on a ring lattice consisting of odd-numbered sites, the optimal flux turns out to be ± π /2. We prove this conclusion for Hubbard interacting fermions utilizing a generalized reflection positivity technique, which can lead to further applications on 2D nonbipartite lattices such as triangular and Kagome. At half-filling the optimal flux patterns on the triangular and Kagome lattice are ascertained to be ±[ π /2, π /2], ±[ π /2, π /2, 0], respectively (see the meaning of these notations in the main text). We also find that chirality emerges in these optimal flux states. Then, we verify these exact conclusions and further study some other fillings with the numerical exact diagonalization method. It is found that when it deviates from half-filling, Hubbard interactions can alter the optimal flux patterns on these lattices. Moreover, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states— both in 1D and 2D—are discussed and interpreted as some kind of non-Fermi liquid feature.
On some one dimensional (1D) and 2D non-bipartite lattices, from scratch we study both free and Hubbard interacting lattice fermions when there are some magnetic fluxes threaded or appropriate gauge fields coupled. On one hand, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-1/21/21/2 fermions at half-filling on a ring lattice with an odd number of sites, the optimal flux is determined to be ±π/2\pm{\pi}/2±π/2. We prove this conclusion for Hubbard interacting fermions by means of a generalized reflection positivity technique. It can further lead to some applications towards 2D non-bipartite lattices such as triangle and Kagome. At half-filling the optimal flux pattern on the triangular lattice can be ascertained to be [π/2,π/2][{\pi}/2, {\pi}/2][π/2,π/2]. We find that chirality emerges in these optimal flux states. On the other hand, we verify these conclusions and further study other fillings away from half with the numerical exact diagonalization (ED) method and find that, when it deviates from half-filling, Hubbard interactions can even alter the optimal flux patterns on these lattices. Both in 1D and 2D, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states are addressed and interpreted as some non-Fermi liquid (NFL) features.
By means of the Bethe-ansatz method, the one-dimensional Hubbard model with repulsive on-site coupling in an external magnetic field is solved exactly. It is found that the energy of the ground state depends on the magnetic flux \ensuremath{\varphi} through the ring formed by a periodic lattice chain, and the lowest-energy state corresponds to \ensuremath{\varphi}=n(hc/2e), with n an integer and -e the electron charge. The calculations illustrate that the strongly correlated Hubbard ring with periodic boundary conditions in its ground state is a superconducting ring. It can also be shown that the system possesses an internal SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) symmetry.