Exact analysis of the interplay of charge order and unconventional pairings in the 2D Hatsugai-Kohmoto model

TL;DR

The study addresses how charge ordering and unconventional pairing tendencies compete or cooperate in the exactly solvable 2D HK model. By computing order-parameter susceptibilities for spin-singlet/triplet SC and CDW (including PDW) and mapping phase diagrams as functions of $u=U/W$, doping $x$, Zeeman field $B$, strain $\delta$, and hopping modifications $t'$, the work reveals an intertwined CDW and PDW sector at intermediate-to-strong coupling and shows how strain and magnetic fields selectively stabilize PDW or CDW relative to SC. Key contributions include demonstrating that CDW and PDW can intertwine and that unidirectional PDW can be favored by strain, while magnetic fields shift phase boundaries and can enhance CDW in certain regimes; the t' term tends to suppress CDW and bolster SC, and an orbital HK generalization suppresses ferromagnetic fluctuations, linking to Hubbard-like physics in the large-$n$ limit. Overall, the HK model provides an analytically tractable platform to study fluctuation-driven superconductivity and competing charge orders, with potential relevance to cuprates and to understanding Hubbard-like physics through OHK generalizations.

Abstract

We provide here a study of some competing ordering tendencies exhibited by the exactly solvable 2D Hatsugai-Kohmoto (HK) model on a square lattice. To this end, we investigate the interplay between superconductivity, charge-density wave (CDW) and pair-density wave (PDW) orders as a function of interaction, doping parameter, magnetic field, and uniaxial strain. As a result, we confirm the intertwined nature of CDW and PDW fluctuating orders for intermediate-to-strong couplings. We also verify that, while an applied magnetic field favors the formation of a CDW and allows the subsequent emergence of a PDW as a secondary order, strain effects favor unidirectional PDW as a primary order over the subdominant appearance of a stripe-like CDW. These results underscore the value of the HK model as an interesting platform in order to investigate (via an exactly solvable framework) the emergence of charge order and unconventional superconductivity in fermionic systems with strong interactions. Finally, we briefly discuss an orbital generalization of the HK model, which has been recently argued to be relevant to describe the properties of realistic strongly correlated systems.

Figures (5)

• Figure 1: Phase diagram of dimensionless interaction $u=U/W$ versus doping parameter $x$ of the competing ordering tendencies that emerge in the 2D HK model with $V = 0.65$, with no magnetic field ($B=0$) and no strain ($\delta=0$). The label 'CDW + PDW' means that both CDW and PDW at $\mathbf{q}=(\pi,\pi)$ with $s$-wave symmetry are found within this region. The beige color denotes a region where a CDW at $\mathbf{q}=(\pi,\pi)$ with $s$-wave symmetry (with a subdominant PDW) appears in the model. The red part denotes $s$-wave singlet SC. At intermediate-to-strong coupling, CDW orders allow the emergence of PDW as a secondary order with the same symmetry and the same wavevector. The dashed lines are only a guide to the eye.
• Figure 2: Phase diagram of dimensionless interaction $u=U/W$ versus doping parameter $x$ of the competing orders in the 2D HK model for $B = 0.05$, $V = 0.4$ and $\delta=0$. The light blue regions correspond to $d$-wave CDW with $\mathbf{q} = (\pi, 0)$, and the beige color denotes the $s$-wave CDW with $\mathbf{q} = (\pi, \pi)$. The dashed lines are only a guide to the eye.
• Figure 3: Phase diagram of dimensionless interaction $u=U/W$ versus doping parameter $x$ of the 2D HK model for the strain anistropy given by $\delta_c=W/8$ and interaction $V=0.15$. Both PDW orders represented above are unidirectional and have ordering vector near $\mathbf{q} = (\pi, 0)$. The yellow region denotes the $d$-wave CDW phase near $\textbf{q} = (\pi, \pi)$, as the beige color corresponds the $s$-wave CDW near $\mathbf{q} = (\pi, \pi)$. The dashed lines are only a guide to the eye.
• Figure 4: Phase diagram of dimensionless interaction $u=U/W$ versus doping parameter $x$ of the 2D HK model for $t'=0.45t$ and $V=0.15$ (with $\delta=0$ and $B=0$). The grey region corresponds to the NFL metallic phase. The dashed lines are only a guide to the eye.
• Figure 5: Ferromagnetic susceptibility for $x=0.1$ and $u=1.5$ for different numbers of orbitals ($n$) included in the 2D OHK model. It is clear that beyond four orbitals ($n\geq 4$), the ferromagnetic susceptibility of the model becomes suppressed at low temperatures. Here, we assume units such that $\mu_B=1$.