[Paper Review] Strong pairing from small Fermi surface beyond weak coupling: Application to La$_3$Ni$_2$O$_7$

Hui Yang, Hanbit Oh+1 more|arXiv (Cornell University)|Sep 26, 2023

Physics of Superconductivity and MagnetismPhysics and Astronomy3 citations

TL;DR

This paper proposes a controlled theoretical framework—via a generalized slave boson theory on an ESD t-J model in a bilayer square lattice with strong inter-layer spin coupling but no hopping—that realizes a small Fermi surface phase violating the Luttinger theorem yet consistent with Oshikawa's non-perturbative proof. It demonstrates a doping-tuned BCS-BEC crossover and a superconducting dome centered at x=0.5, with distinct normal states: a conventional Fermi liquid (x>0.5) and an unconventional second Fermi liquid (x<0.5), providing a plausible mechanism for superconductivity in La₃Ni₂O₇.

ABSTRACT

The studies of high-temperature superconductors raise a fundamental question: Can a small Fermi surface phase, which violates the Luttinger theorem, exist and give rise to superconductivity? Our work provides a positive answer through a controlled theory based on a bilayer model with strong inter-layer spin-spin coupling ($J_\perp$) but no inter-layer hopping ($t_\perp$). Then small hole doping of the rung-singlet insulator with two electrons per rung naturally leads to small hole pockets with Fermi surface volume per flavor smaller than the free fermion result by $1/2$ of the Brillouin zone(BZ). We construct a new t-J model on a bilayer square lattice, so called ESD t-J model and employ a generalized slave boson theory, which captures this small Fermi surface phase at small hole doping $x$. This metallic state is an intrinsically strongly correlated Fermi liquid beyond weak coupling theory, violating the perturbative Luttinger theorem but consistent with the Oshikawa's non-perturbative proof. We further show that it transitions into an inter-layer paired $s'$-wave superconductor at lower temperature through Feshbach resonance with a virtual Cooper pair, with a surprising doping-induced crossover from Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) at higher hole doping levels. This leads to a superconducting dome centered around $x=0.5$, with the normal state changing from the conventional Fermi liquid in the $x&gt;0.5$ to the unusual small Fermi surface state in the $x&lt;0.5$ side. Our theoretical findings including phase diagrams are also confirmed by density matrix renormalization group (DMRG) simulation in quasi one dimension. Applying our theoretical framework, we provide a plausible scenario for the recently found nickelate La$_3$Ni$_2$O$_7$ materials.

Motivation & Objective

• To resolve the fundamental question of whether a small Fermi surface phase—violating the Luttinger theorem—can exist and support superconductivity in strongly correlated systems.
• To construct a controlled, non-perturbative theory beyond weak coupling that captures a small Fermi surface without symmetry breaking.
• To explain the emergence of a superconducting dome centered at x=0.5 in nickelate materials like La₃Ni₂O₇.
• To identify a doping-induced BCS to BEC crossover in a strongly correlated metallic state with unconventional Fermi surface volume.
• To provide a theoretical framework consistent with DMRG simulations and applicable to bilayer nickelates with tunable inter-layer coupling.

Proposed method

• Introduces the ESD t-J model on a bilayer square lattice with strong inter-layer spin-spin coupling (J⊥) but no inter-layer hopping (t⊥), simplifying the Hilbert space to six states per rung: one empty, four singlons, and one doublon.
• Employs a generalized slave boson theory to describe the strongly correlated metallic state, capturing the small Fermi surface phase at low hole doping (x<0.5).
• Uses the Feshbach resonance mechanism to describe inter-layer s′-wave pairing mediated by virtual Cooper pairs, leading to superconductivity at low temperatures.
• Analyzes the Fermi surface volume via the 2kF singularity in the spin-spin static structure factor ⟨S(q)·S(−q)⟩, confirmed by DMRG simulations in quasi-1D.
• Controls the layer pseudospin polarization via a small layer-pseudospin interaction (Hp) to stabilize the Mott insulating rung-singlet phase at large J⊥.
• Performs DMRG simulations with large on-site repulsion (V=100) to suppress pairing and isolate Fermi surface size, validating the small Fermi surface at x<0.5.

Figure 1: (a-b) Illustrated phase diagram of standard one-orbital $t-J$ model from doping a single-layer spin 1/2 Mott insulator and the ESD t-J model. The black and red lines indicate the pairing scale and the coherence scale. FL (sF) stands for Fermi liquid (second Fermi liquid) respectively. SC (

Experimental results

Research questions

• RQ1 Can a small Fermi surface phase exist in a strongly correlated system without symmetry breaking, and does it support superconductivity?
• RQ2 How does the Fermi surface volume deviate from the Luttinger theorem in a non-perturbative, strong coupling regime?
• RQ3 What is the nature of the pairing mechanism in a system with a small Fermi surface and no conventional spin-fluctuation pairing?
• RQ4 How does the BCS-BEC crossover manifest in a strongly correlated metallic state with unconventional Fermi surface volume?
• RQ5 Can the ESD t-J model explain the superconducting dome and normal state evolution observed in La₃Ni₂O₇?

Key findings

• The ESD t-J model with strong J⊥ and no t⊥ realizes a small Fermi surface phase at x<0.5, with Fermi surface volume per flavor AFS = -x/2, violating the conventional Luttinger theorem but consistent with Oshikawa’s non-perturbative proof.
• For x>0.5, the normal state is a conventional Fermi liquid with AFS = (1−x)/2, matching the free fermion result.
• A superconducting dome centered at x=0.5 emerges via Feshbach resonance with a virtual Cooper pair, leading to inter-layer s′-wave pairing.
• A doping-induced BCS to BEC crossover is identified as x approaches 0.5, with the normal state evolving from sFL (small Fermi surface) to FL (large Fermi surface).
• DMRG simulations confirm the existence of a small Fermi surface at x=0.3 for large J⊥ (J⊥=4), with a 2kF peak at q=x/2, while the Fermi surface size remains fixed at x=0.7.
• The model predicts a small-to-large Fermi surface evolution at x=0.3 when decreasing J⊥, suggesting a novel quantum critical point without symmetry breaking.

Figure 2: Illustrated six states in the ESD-t-J model. Each state is defined by combining the top and bottom layer, as boxed together. The red empty (solid) circles represent the holes (filled electrons) within the one-orbital. The arrow is depicting the spin-1/2 moment of the electron. (a-b) The $b