Integrable Model of a Superconductor with non-Fermi liquid and Mott Phases

TL;DR

We construct an exactly solvable, number-conserving fermionic model with momentum-pair entanglement between $\mathbf{k}$ and $-\mathbf{k}$ that realizes superconducting, multiple metallic, and Mott-insulating phases, including non-Fermi-liquid behavior with multiple many-body Fermi surfaces. The model belongs to the Richardson-Gaudin integrable class and exhibits a macroscopic ground-state degeneracy, with a structural connection to the Hatsugai-Kohmoto model via projection. The exact single-particle Green's function displays four poles corresponding to four many-body Fermi surfaces, signaling quadruply fractionalized quasiparticles and Luttinger-theorem violation, while the phase diagram comprises ten metallic and three Mott-insulating phases, including Lifshitz-type transitions in $d\ge2$. A projected HK-like ground-state energy emerges under a pairless projection, and Cooper-pair instabilities vary across the phase diagram, suggesting multiple competing superconducting channels; extensions to $p+ip$ and $so(5)$-type pairings are discussed. Overall, the work provides a tractable analytical framework to study strong correlations, fractionalization, and unconventional superconductivity in integrable many-body systems.

Abstract

We present and analyze an exactly solvable interacting fermionic pairing model, which features interactions that entangle states at momenta $\mathbf{k}$ and $-\mathbf{k}$. These interactions give rise to novel correlated ground states, leading to a rich phase diagram that includes superconducting, multiple metallic, and Mott-insulating phases. At finite interaction strengths, we observe the emergence of multiple many-body Fermi surfaces, which violate Luttinger's theorem and challenge the conventional Landau-Fermi liquid paradigm. A distinguishing feature of our model is that it remains quantum integrable, even with the addition of pairing interactions of various symmetries, setting it apart from the Hatsugai-Kohmoto model. Our results provide an analytically tractable framework for studying strong correlation effects that give rise to fractionalized excitations and unconventional superconductivity, offering valuable insights into a broad class of integrable many-body systems.

Figures (12)

• Figure 1: Occupation numbers for the ground state of phase “(4,3,2,1,0)” of $H_{\sf n}$ in one spatial dimension.
• Figure 2: Quantum phase diagram for our model (left) in $d=1$ contrasted with the HK model (right). (Refer to Appendix \ref{['AppendixHK']} for a detailed discussion of the quantum phase diagram of the HK model.) Band insulators (BI) are found at full filling ($\rho_F =2$) and empty filling ($\rho_F =0$). Mott insulators (MI) are found at $\rho_F = 1/2$, $1$ and $3/2$ for $U>U_c(\rho_F)$. The blue lines represent second order and the red lines represent first order transitions. The red lines coincide with the Mott-insulating phases.
• Figure 3: Quantum phase diagram for our model (left) in $d=2$ contrasted with the HK model (right). Refer to Appendix \ref{['AppendixHK']} for a detailed discussion of the quantum phase diagram of the HK model.) Band insulators (BI) are found at full filling ($\rho_F=2$) and empty filling ($\rho_F =0$). Mott insulators (MI) are found at $\rho_F = 1/2$, $1$ and $3/2$ for $U>U_c(\rho_F)$. The purple dotted lines indicate the third order transitions driven by the van-Hove singularity in $d=2$. The blue lines represent second order and the red lines represent first order transitions. The red lines coincide with the Mott-insulating phases.
• Figure 4: Quantum phase diagram for our model (left) in $d=2$ contrasted with the HK model (right). Band insulators (BI) are found at full filling ($\rho_F =2$) and empty filling ($\rho_F =0$). Mott insulators (MI) are found at $\rho_F = 1/2$, $1$ and $3/2$ for $U>U_c(\rho_F)$. The purple dotted lines indicate the third order transitions driven by the van-Hove singularity in $d=2$.
• Figure 5: Occupation numbers for the ground state of phase “(4,3,2,1,0)” of $H_{\sf n}$, as a function of density $\rho_F$, in $d=2$ and $U=0.7$.