Spin-charge bound states and emerging fermions in a quantum spin liquid

TL;DR

The work addresses hole dynamics in a $ ext{Z}_2$ quantum spin liquid described by a single-band extended $t$-$J$ model, showing that spinon-holon interactions can bind to form electron-like fermions. Through a field-theoretic, parton-based approach and a reduction to an effective Schrödinger problem, it identifies bound-state energies $E^{n}_{\mathbf p}$ and relative wavefunctions $\phi^{n,\mathbf p}_{\bf k}$, yielding long-lived bound states embedded in a spin-liquid background. These emergent fermions give rise to a fractionalized Fermi liquid (FL*) at finite hole density, with hole pockets and Fermi-arc-like ARPES signatures arising purely from the internal structure of the bound states and the symmetries of the RVB background. The results connect microscopic Hubbard-model physics to pseudogap phenomenology in cuprates and point to observable probes via ARPES and quantum-gas microscopy, offering a concrete mechanism for FL* and a path to exploring pseudogap physics in strongly correlated systems.

Abstract

The complex interplay between charge and spin dynamics lies at the heart of strongly correlated quantum materials, and it is a fundamental topic in basic research with far reaching technological perspectives. We explore in this paper the dynamics of holes in a single band extended $t-J$ model where the background spins form a $\mathbb{Z}_{2}$ quantum spin liquid (QSL). Using a field theory approach based on a parton construction, we show that while the electrons for most momenta fractionalize into uncorrelated charge carrying holons and spin carrying spinons as generally expected for a QSL, the spinon-holon scattering cross-section diverges for certain momenta signalling strong correlations. By deriving an effective low-energy Hamiltonian describing this dynamics, we demonstrate that these divergencies are due to the formation of long lived spinon-holon bound states. We then show that quantum gas microscopy with atoms in optical lattices provides an excellent platform for verifying and probing the internal spatial structure of these emergent fermions. The fermions will furthermore show up as clear quasiparticle peaks in angle-resolved photoemission spectroscopy with an intensity determined by their internal structure. For a non-zero hole concentration, the fermions form hole pockets with qualitatively the same location, shape, and intensity variation in the Brillouin zone as the so-called Fermi arcs observed in the pseudogap phase. Such agreement is remarkable since the Fermi arcs arise from the delicate interplay between the symmetry of the QSL and the internal structure of the emerging fermions in a minimal single band model with no extra degrees of freedom added. Our results, therefore, provide a microscopic mechanism for the conjectured fractionalized Fermi liquid and open up new pathways for exploring the pseudogap phase and high temperature superconductivity as arising from a QSL.

Figures (13)

• Figure 1: (a) A physical spin $\uparrow$ hole fractionalizes into an unbound spinon (arrow) and holon (green circle) pair in a spin liquid formed by spin singlets (blue ovals). (b) For some momenta, however, the holon and spinon bind to form a new fermionic quasiparticle with well-defined spin and charge, and a spatial structure determined by the relative spinon-holon wave function $|\Psi_{\mathbf p}\rangle$ extending over a few lattice sites as indicated by the red coloring. (c) Mimicking the spatial structure of the bound state, the hole-spin correlation function $M^{n}_{{\bf p}}({\bf d})$ effectively describes the likelihood of finding the unpaired spin at a given lattice with the holon (green circle) at the center. (d) The spectral response in the Brillouin zone of these fermions at their Fermi energy as measured by angle-resolved photoemission spectroscopy for a nonzero hole doping $x=0.01$, where they form a fractionalized Fermi liquid. This clearly shows a hole pocket around ${\bf p}=(3\pi/4,3\pi/4)$ with a dimly lit back side due to the interplay between the internal symmetry of the fermions and the quantum spin liquid singlets. There are identical hole pockets in the other corners of the Brillouin zone. The shape and spectral weight of these pockets are similar to those of the Fermi arcs observed in the pseudogap phase of the cuprates.
• Figure 2: (a) The Green's function $G({\bf p},\omega)$ of a physical hole is a two-body Green's function of a spinon and a holon. Solid lines are the spinon propagator and double wavy lines are the dressed holon propagator Eq. \ref{['eq.holonprop']}. (b) Holon self-energy $\Sigma_h({\bf p},\omega)$ giving the dressing of holons by spinons. (c) Spinon-holon scattering matrix $\Gamma({\bf p},{\bf q}_1,{\bf q}_2\omega)$, with red and blue balls the scattering $h$ and $g$ vertices, respectively, in Eq. \ref{['Eq.vertex']}.
• Figure 3: (a) Spinon-holon energy spectrum obtained from Eqs. \ref{['Schrodinger']}-\ref{['eq.Eig']} for a $64\times64$ lattice along straight paths in the Brillouin zone. The red, yellow, and green bands are bound states. (b) Bound state wave functions for ${\bf p}=(\pi,\pi)$ in momentum space. The two leftmost panels illustrate the real and imaginary parts of one of the degenerate $p$-wave states. For the other $p$-wave state, the imaginary part has changed sign, while the real part is identical. (c) Plots of the gauge-averaged relative wave function in real space, with the left panel showing the $p$-wave states. This shows that the spin and charge of the quasiparticles are located within a few lattice sites.
• Figure 4: Spin-hole correlation function $M^{n}_{{\bf p}}({\bf d})$ given by Eq. \ref{['eq.M']} for ${\bf p} =(\pi,\pi)$, $J_{1}/t_{1}=0.6$, and a $64\times 64$ lattice. The left panel shows the $p$-wave bound state, and the right panel shows a delocalized state in the scattering continuum. Note that the scale in the right panel is multiplied by $N$.
• Figure 5: Hole spectral function given by Eq. \ref{['eq.spec']} for two different momenta. The vertical dashed lines show the energy of the bound states with the same color code as in Fig. \ref{['Fig.EnergySpectrum']}. The inset shows an enlarged section around the peak.