Momentum-Mixing Hatsugai-Kohmoto Model

• Momentum-Mixing Hatsugai-Kohmoto (MMHK) is a cluster-based, numerically exact approach for modeling strongly correlated SU(N) lattice systems with explicit momentum mixing.
• The model employs a unitary twist transformation to map to the conventional Hubbard model, accurately reproducing key bulk observables such as Mott transitions and response functions.
• Rapid 1/n^2 convergence and a sign-problem-free framework enable direct real-frequency computation of dynamical properties and reliable phase diagram analysis.

The momentum-mixing Hatsugai-Kohmoto (MMHK) model, also referred to as the orbital Hatsugai-Kohmoto (OHK) model, is a numerically exact approach to simulating strongly correlated systems with SU($N$) symmetry in $d$-dimensional lattices. It extends the original Hatsugai-Kohmoto construction by incorporating explicit momentum mixing via cluster (supercell) formation in real space, resulting in powerful convergence properties. The MMHK model is formally unitarily equivalent to the conventional Hubbard model under a twist transformation, and reproduces all bulk charge-sector observables of the Hubbard model—including Mott transitions and response functions—in the large-cluster limit (Hackner et al., 13 Mar 2025, Bai et al., 2 Dec 2025).

1. Hamiltonian Formulation

The MMHK model is constructed by partitioning the real-space lattice into clusters (supercells) of $n$ sites, labeled $\alpha=1,\dots,n$. In momentum space, for each momentum $k$ in the reduced Brillouin zone (rBZ), the system is described by an $n \times n$ "orbital" (cluster) index and an SU($N$) flavor index $\sigma=1,\dots,N$. The Hamiltonian for the SU($N$) MMHK model is: $$\begin{aligned} H_{\rm OHK}^N = & \sum_{k\in \rm rBZ} \sum_{\alpha,\beta=1}^n \sum_{\sigma=1}^N \xi_{k}^{\alpha\beta} c^\dagger_{\alpha k\sigma} c_{\beta k\sigma} \ & + \frac{U}{2} \sum_{k\in {\rm rBZ}} \sum_{\alpha=1}^n \sum_{\substack{\sigma,\sigma'=1\\sigma\neq\sigma'}}^N n_{\alpha k\sigma} n_{\alpha k\sigma'}\,, \end{aligned}$$ where $c^\dagger_{\alpha k\sigma}$ creates a fermion of flavor $\sigma$ on cluster site $\alpha$ labeled by momentum $k$, $$n_{\alpha k\sigma} = c^\dagger_{\alpha k\sigma} c_{\alpha k\sigma}$$, $U$ is the on-site Hubbard interaction, and $\xi_k^{\alpha\beta}$ is the hopping matrix derived from Fourier transforming nearest-neighbor hopping $-t\sum_{\langle ij\rangle} c_i^\dagger c_j$ onto the $n$-site cluster. Off-diagonal ($\alpha \ne \beta$) terms encode momentum mixing. In the $n \to \infty$ limit, the model reduces to the usual SU($N$) Hubbard Hamiltonian, with $\xi_q = -2t(\cos q_x + \cos q_y) - \mu$ on the full Brillouin zone (Hackner et al., 13 Mar 2025, Bai et al., 2 Dec 2025).

2. Momentum Mixing, Brillouin Zone Folding, and Twist Equivalence

Grouping sites into clusters of $n$ introduces explicit momentum mixing, absent in the non-mixing (band) HK model. The momentum-mixing arises from off-diagonal elements in $\xi_k^{\alpha\beta}$, which mix the $n$ cluster orbitals within each $k$ in rBZ. This construction can also be described in the $c_{qR,\sigma}$ basis, where $q$ labels rBZ patches and $R$ the real-space sites within each cluster: $$c_{qR,\sigma} = \frac{1}{\sqrt{n}} \sum_{K\in B_n} e^{iK\cdot R} c_{K+q,\sigma}\,.$$ The kinetic term $T_n$ acquires a $q$-dependent phase $e^{-iq\cdot(R_1-R_2)}$: $$T_n = \sum_{q \in \mathrm{rBZ}_n} \sum_{R_1,R_2,\sigma} t_{R_1,R_2} e^{-i q\cdot(R_1-R_2)} c^\dagger_{qR_1,\sigma} c_{qR_2,\sigma}\,.$$ A unitary twist transformation $U_{\rm twist}$ can remove these phases: $$U_{\rm twist} = \exp \left[ i \sum_{q,\sigma} \sum_{R} q \cdot R N_{qR,\sigma}\right], \quad N_{qR,\sigma} = c^\dagger_{qR,\sigma} c_{qR,\sigma},$$ yielding the real-space Hubbard Hamiltonian. Thus, the MMHK Hamiltonian is unitarily equivalent to the ordinary Hubbard model; this equivalence holds for all observables invariant under the twist, most notably in the charge sector (Bai et al., 2 Dec 2025).

3. SU(N) Symmetry and Thermodynamic Observables

The MMHK model is fully SU($N$) symmetric: the flavor index only enters the kinetic term diagonally and in a symmetrized Hubbard interaction $$\frac{U}{2} \sum_{\sigma \ne \sigma'} n_\sigma n_{\sigma'}$$. Global SU($N$) rotations act as $$c_{\alpha k\sigma} \to U_{\sigma\sigma'} c_{\alpha k\sigma'}$$ and leave the Hamiltonian invariant.

Key thermodynamic observables include:

• Filling per site:

$$\langle \hat n \rangle = \frac{1}{n V_{\rm rBZ}} \sum_{k,\alpha,\sigma} \langle n_{\alpha k\sigma} \rangle$$

• Double occupancy:

$$\mathcal{D} = \frac{1}{2 n V_{\rm rBZ}} \sum_{k \in {\rm rBZ}} \sum_{\alpha=1}^n \sum_{\substack{\sigma,\sigma'=1\\sigma\neq\sigma'}}^N \langle n_{\alpha k\sigma} n_{\alpha k\sigma'} \rangle$$

• Compressibility:

$$\chi = \frac{\partial \langle \hat n \rangle}{\partial \mu} = \beta \left( \langle \hat n^2 \rangle - \langle \hat n \rangle^2 \right)$$

Low-temperature and dynamical quantities, including dynamical structure factors, are computed directly in real frequency on the cluster without analytic continuation (Hackner et al., 13 Mar 2025).

4. Convergence and Numerical Benchmarks

The MMHK model exhibits rapid $1/n^2$ convergence in all thermodynamic and dynamical observables as $n$ increases. For $n \approx 10$, quantitative agreement with determinantal quantum Monte Carlo (DQMC) is reached in double occupancy, and qualitative-to-quantitative agreement for filling and compressibility across the Mott transition.

The table below summarizes key numerical benchmarks (Hackner et al., 13 Mar 2025):

Observable	MMHK (n=4–9)	Hubbard Model/DQMC
Double occupancy vs. $\langle n\rangle$	Quantitative agreement	DQMC (except for high densities)
Critical $U_c$ for Mott gap (SU(3))	$U_c(4) \approx 7t$, $U_c(8) \approx 6t$, $U_c(9) \approx 5.4t$	$U_c \approx 6t$ (DQMC), $U_c \approx 5.5t$ (AFQMC)
SU(3) structure factor $S(q)$ peak	Peaks at $(2\pi/3,2\pi/3)$ for $U > U_c$	Matches SU(3) Hubbard results

Direct real-frequency computations at $T\approx 0$ and finite $T$ are possible with no sign problem, enabling simulation at high densities and low temperatures.

5. Charge-Sector Equivalence and Physical Implications

The MMHK model, under the twist transformation $U_{\rm twist}$, produces the same single-particle Green's function, spectral function $A(\omega, k)$, and density response $\chi_c(q,i\nu)$ as the conventional Hubbard model. Watanabe's insensitivity theorem guarantees that all bulk charge-sector observables—Mott gap, phase boundaries, lower and upper Hubbard bands, and suppressed charge fluctuations—match identically in the large-$n$ limit.

Transport coefficients computed via Kubo formulas (e.g., optical conductivity, thermopower) show equivalence up to exponentially small finite-size/twist effects. While magnetic (spin) physics may differ in detail (twisted vs. periodic spin waves), bulk susceptibilities and transition temperatures are unchanged. Thus, MMHK provides a numerically exact alternative route to obtain Hubbard model charge-sector physics with fast convergence and superior computational efficiency compared to conventional real-space cluster methods (Bai et al., 2 Dec 2025).

6. Advantages, Applications, and Outlook

The MMHK model establishes itself as a powerful, sign-problem-free, cluster-based simulator for strongly interacting SU($N$) quantum lattice systems:

• Rapid $1/n^2$ convergence enables accurate results for $n \sim 10$ clusters at all temperatures.
• Direct access to real-frequency dynamical (spectral) quantities without reliance on analytic continuation.
• No sign problem, allowing simulations at large system size, high densities, and low temperatures.
• Formal equivalence—via a unitary twist—to the Hubbard model for all charge-related bulk physics, ensuring unbiased results for phase diagrams and response functions.

The MMHK approach bridges the gap between exactly solvable models and intractable fully interacting systems, providing both deep theoretical insight and practical computational schemes for high-precision studies of Mott physics, SU($N$) magnetism, and correlated lattice models (Hackner et al., 13 Mar 2025, Bai et al., 2 Dec 2025).