Twisting the Hubbard model into the Momentum-Mixing Hatsugai-Kohmoto Model

Abstract

The Hubbard model is a standard theoretical tool for studying materials with strong electron-electron interactions, such as the cuprate superconductors. Unfortunately, interaction-driven phenomena such as the transition into the strongly correlated Mott insulator phase are difficult to treat with established theoretical techniques. However, the exactly solvable Hatsugai-Kohmoto model displays similar Mott physics. Here we show how the Hatsugai-Kohmoto model can be deformed continuously into the Hubbard model. The trick is to systematically re-introduce all the momentum mixing the original Hatsugai-Kohmoto model omits. This can be accomplished by grouping $n$-momenta into a cell and hybridizing them resulting in the momentum-mixing Hatsugai-Kohmoto (MMHK) model. We recover the Bethe ansatz ground state energy of the one-dimensional Hubbard model to within 1$\%$ from only ten mixed momenta. Overall the convergence scales as $1/n^2$ as opposed to the inverse linear behaviour of standard finite-cluster techniques. Our results for a square lattice reproduce all known features from state-of-the-art simulations also with only a few mixed momenta. Consequently, we believe the MMHK model offers an alternative tool for strongly correlated quantum matter.

Figures (6)

• Figure 1: Mott transition in band HK model and the scheme for the $n$-MMHK model.a, The DOS under different $U$ displaying the Mott transition in the half-filled band HK model at inverse temperature $\beta=200/t$. b, Evolution of the reduced Brillouin zone as the number of mixed momenta $n$ increases, leading to a purely local (in real space) model when $n= N$.
• Figure 2: Ground-state energy deviation of $n$-MMHK from Bethe ansatz. Comparison of the difference between the ground-state energy with $n$-MMHK model, $E_{\rm HK}^n$, and the infinite-system size Hubbard Bethe ansatz energy, $E_{\rm HB}^\infty$, both in one dimension. a, MMHK is solved by exact diagonalization. b, $n$-MMHK is solved by DMRG including as many as $n=40$ mixed momenta at various $U$. The dashed lines are polynomial regression fitting with extrapolation to $1/n\rightarrow 0$ ($n\rightarrow\infty$). The fitting curves can be well represented as $f(U=4)=0.45(1/n^{1.83})+a$, $f(U=6)=0.51(1/n^{2.01})+a$ and $f(U=8)=0.45(1/n^{2.07})+a$. The asymtotic values $a$ at $1/n=0$ are $-5.4\times10^{-5}~(U=4)$, $-9\times10^{-5}~(U=6)$, $-4\times10^{-5}~(U=8)$.
• Figure 3: Mott transition, antiferromagnetic susceptibility and spectral functions for the half-filled $n$-MMHK model.a, DOS representing the Mott transition for the half-filled $4$-MMHK model at $\beta=200/t$. b, Antiferromagnetic spin susceptibility of the half-filled $4$-MMHK model plotted as a function $\beta$ for various $U$ (same legend as in a). Its zero-temperature limit is shown in panel c at $U=8$ for $4$-, $8$- and $16$-MMHK. d,e show Spectral function $A({\rm k},\omega)$ at half-filling of the $16$-MMHK model at $t'=0$ and $t'/t=-0.25$ respectively, with $U=8$, zero temperature, and a broadening factor of $0.2$.
• Figure 4: Double occupancy and dynamical spectral weight transfer for $n$-MMHK.a, $D_n$ at half-filling ($x=0$) as a function of $U$ for the $n$-MMHK model with various $n$. The green and gray dots are from auxiliary-field QMC (AFQMC)shiweizhang and finite-temperature QMC (FTQMC) calculationsWhite of the standard double occupancy of the 2D Hubbard model. The inset of a compares the $4$-MMHK and Hubbard-AFQMC resultsshiweizhang at $1/8$-hole-doping ($x=0.125$). LESW and DSWT are the exact solution of the $4$-MMHK model ($U=10$) for the hopping parameters shown in b and c respectively at $\beta=30$. The solid line shown with slope $2x$ is the band HK or atomic Hubbard result. The dashed and dotted lines depict the semiconductor and Fermi liquid results respectively. Note, there is no qualitative difference with the ED results for the Hubbard modelsawatzkySawatzkyprl. As $t/U$ increases, so does the DSWT.
• Figure 5: Observation of pseudogap in $4$-MMHK. DOS at varying hole-doped densities with ( a) and without ( b) the next-nearest-neighbor hopping $t'$ for the $4$-MMHK model ($U/t=10, \beta=30/t$). a and b share the same legend. Only for $t'\ne 0$ is $\rho(\omega)$ suppressed at zero frequency in the under-doped region, thereby indicating a pseudogap. c and d show $\rho(\omega=0)$ with $t'=-0.25$ and $0$ respectively, as a function of hole-doped density under various $U$.