Real-space analysis of Hatsugai-Kohmoto interaction

The Hatsugai-Kohmoto interaction model has gained a lot of attention in recent years, due to the fact it is exactly solvable in momentum space in any dimension while capturing some key features of the Mott phase. Here a one-dimensional lattice model with this interaction is approached from the real-space perspective, to explore how breaking the translation invariance of a lattice affects the intuition built by studying the exact solution in $k$ space. The ground state properties of chains with periodic and open boundary conditions are calculated and compared with both the exact solution in momentum space as well as with analogous solutions of the Hubbard model. The results show that introducing hard edges enhances the ferromagnetic correlations and the system undergoes a magnetic transition before reaching the strong coupling limit. Understanding the impact of hard edges is a crucial step toward answering the looming question of the existence of edge states and other topological phenomena in systems with this type of interaction.

Figure 1

Dispersion relation for a one-dimensional single band Hatsugai-Kohmoto model for three interaction strengths $U$ representing weakly interacting metal (left), strongly correlated metal (middle), and insulating (right) states. The colors of the bands represent their spectral weight, which has only three values: zero, one-half, and one.

Figure 2

Comparison of the LDOS of the H-K dimer with OBC (color) and with PBC (grey) for various interaction strengths across the transition from the weak ($U=0$) to strong coupling limit ($U=5t$). Both models undergo a change in the ground state as a function of $U$. For the periodic dimer $U_c=4t$ and the dimer with OBC has the transition at $U\approx 2.83t$. In both cases, the transition is accompanied by a rapid change in the spectrum.

Figure 3

Comparison of the LDOS of the H-K dimer with OBC (color) and the corresponding Hubbard dimer (grey) for various interaction strengths $U$. The former shows a rapid change around $U\approx 2.83t$, where the ground state moves from the spin-singlet to the spin-triplet subspace.

Figure 4

Finite size scaling of the two-point spin-spin correlator between the nearest neighboring sites as a function of system size $N$ for the H-K chain with OBC and PBC. All results are for $U=5$ which is large enough to secures that the system is in the high-$U$ solution.

Figure 5

Two-point spin-spin correlator for a chain with OBC of $N=8$ sites with H-K (solid line) and Hubbard (dashed line) interactions, for various interactions strengths $U$. The area between the two curves is colored to highlight the differences between them. The $U$ values show the transition of the spin-spin correlations in the H-K model from antiferromagnetic (top) to ferromagnetic (bottom).

Figure 6

Two-point correlator for a chain with OBC, $N=8$ sites with H-K (solid line) and Hubbard (dashed line) interactions, for various interaction strengths $U$. The region between the two curves is colored to highlight the differences between them.

Figure 7

Comparison of local density of states at the middle of $N=8$ site chain ($\gamma =4$), referred to in the text as bulk, for various interaction strength $U$. The grey peaks represent the poles of the Greens function of the chain with H-K interaction and PBC. The colored peaks are the excisions of a system with OBC. The interaction strength spans from $U=0$ (noninteracting) to $U=5t$ (strong coupling limit).

Figure 8

The bulk LDOS (in the middle) of a chain with OBC of length $N=8$ with H-K interaction(color) and Hubbard interaction (grey) for various strengths $U$ from weakly to strongly interacting limit.