Integrable Free and Interacting Fermions

Abstract

Integrability conditions on local Hamiltonians for one-dimensional quantum systems to be free and interacting fermions are introduced. The definition of free fermion is the simultaneous satisfaction of the Yang-Baxter equation and Shastry's decorated star-triangle relation by the $R$-matrix, which is more general than the previous `free-fermion algebra' by Maassarani and more special than free fermions as in the context of exactly solvable quantum models or integrable classical two-dimensional vertex models dual to quantum spin chains. Free fermionic $R$-matrices are of the difference form and have a conjugation symmetry. These free Hamiltonians may sometimes be deformed by the conjugation operator to describe an integrable interacting system with non-relativistic $R$-matrices, as are the cases of the Hubbard model and the XY model in a longitudinal field. A further criterion is obtain on precisely when such deformations remain integrable. A practical procedure is proposed to iteratively solve the free fermionic $R$-matrices from local Hamiltonians, which can be used to construct non-relativistic $R$-matrices if the conditions are met.

Figures (5)

• Figure 1: Diagrammatic representations of the YBE and DYBE. The positive time direction is upward, and a line with an arrow in the opposite direction can be interpreted as an anti-particle, with the sign of its rapidity reversed. (a) The YBE $R_{12}(\mu)R_{13}(\lambda)R_{23}(\lambda-\mu)=R_{23}(\lambda-\mu)R_{13}(\lambda)R_{12}(\mu)$ corresponding to the braided form \ref{['eq:YBE']}. (b) The DYBE $R_{12}(\mu)R_{13}(\lambda)R_{23}(-\lambda-\mu)=R_{23}(\lambda+\mu)R_{13}(-\lambda)R_{12}(-\mu)$ or \ref{['eq:DYBE']} in the braided form. The black dots in (b) denote the conjugation $C_3$ that appear in the original form of \ref{['eq:Shastry']}$R_{12}(\mu)R_{13}(\lambda)C_3R_{23}(\lambda+\mu)=R_{23}(\lambda+\mu)C_3R_{13}(\lambda)R_{12}(\mu)$, which are interpreted as time-reversals in this picture.
• Figure 2: The Hubbard model viewed as two free Fermion chains coupled by a diagonal interaction.
• Figure 3: Holons hopping on a sawtooth lattice feel a magnetic $\pi$-flux through a triangle per vacancy due to kinetic frustration, which can be chosen to reverse the sign of the hopping strength across the horizontal bond.
• Figure 4: Basis vectors in the ansatz eigenstates.
• Figure 5: Energy spectrum $E_n$ versus total momentum $\theta$ of the $n$ holes for the infinite-$U$ Hubbard model with spinless fermions. $L=\infty$ for $n=1$, $L=100$ for $n=2$, $L=30$ for $n=3$, and $L=15$ for $n=4$. The labels on the right, placed at the mean energy for each band, show the total number of positive and negative values in $\bm s$ for the different bands. To better visualize the different bands, each band is shifted by $\theta \to \theta + \sum_i s_i\pi/6L$.