Lectures on insulating and conducting quantum spin liquids

TL;DR

The notes present a unified framework for insulating and conducting quantum spin liquids, focusing on fractionalization and emergent gauge fields to address cuprate phenomenology. They compare bosonic and fermionic spinon descriptions, derive U(1) and Z2 gauge theories, and develop the Ancilla Layer Model (ALM) to realize the FL* phase, holon metals, and doped superconducting states. A central thread is how confinement and topological excitations (visons, monopoles) shape the phase diagram, enabling transitions from spin liquids to d-wave superconductors with anisotropic nodal velocities and to pseudogap metals with pocket Fermi surfaces. The work connects SDW reconstruction, deconfined criticality, and doped spin liquids to observable signatures such as ADMR pockets and nodal structure, offering a coherent route from microscopic spin models to cuprate phenomenology.

Abstract

Two of the iconic phases of the hole-doped cuprate materials are the intermediate temperature pseudogap metal and the lower temperature $d$-wave superconductor. Following the prescient suggestion of P.W. Anderson, there were numerous early theories of these phases as doped quantum spin liquids. However, these theories have had difficulties with two prominent observations: (i) angle-dependent magnetoresistance measurements (ADMR), including observation of the Yamaji effect, present convincing evidence of small hole pockets which can tunnel coherently between square lattice layers, and (ii) the velocities of the nodal Bogoliubov quasiparticles in the $d$-wave superconductor are highly anisotropic, with $v_F \gg v_Δ$. These lecture notes review how the fractionalized Fermi Liquid (FL*) state, which dopes quantum spin liquids with gauge-neutral electron-like quasiparticles, resolves both difficulties. Theories of insulating quantum spin liquids employing fractionalization of the electron spin into bosonic or fermionic partons are discussed. Doping the bosonic parton theory leads to a holon metal theory: while not appropriate for the cuprate pseudogap, this theory is argued to apply to the Lieb lattice. Doping the fermionic parton theory leads to a $d$-wave superconductor with nearly isotropic quasiparticle velocities. The construction of the FL* state is described using a quantum dimer model, followed by a more realistic description using the Ancilla Layer Model (ALM), which is then used to obtain the theory of the pseudogap and the $d$-wave superconductor.

Figures (32)

• Figure 1: Feynman diagram leading to (\ref{['spara3']}).
• Figure 2: Antiferromagnetic (Néel) order $\mathcal{N}$ of the spin density wave state. This can be an insulator at $p=0$, provided $\Delta = \mathcal{N}$ is large enough, as shown in Fig. \ref{['fig:sdw0']}. Otherwise, it is a metal with electron and/or hole pocket Fermi surfaces, as shown in Fig. \ref{['fig:sdw0']}-\ref{['fig:sdw2']}.
• Figure 3: Fermi surface of the original dispersion $\varepsilon_{\bm k}$, and those obtained by scattering off the antiferromagnetic order by a momentum ${\bm K}$. The intersections of the Fermi surfaces are the hotspots, where the antiferromagnetism opens a gap. In this figure only, the center is the momentum ${\bm K}$, around which the large hole-like Fermi surface is centered in the cuprates.
• Figure 4: Fermi surfaces of the Néel state at half-filling, i.e. doping $p=0$. The pockets intersecting the diagonals of the Brillouin zone have both bands in (\ref{['bcs52']}) empty and so form hole pockets, while the remaining pockets have both bands occupied and form electron pockets. The dashed line in the insulator shows the boundary of the Brillouin zone of the Néel state.
• Figure 5: Fermi surfaces of the Néel state at $p>0$. The pockets are as in Fig. \ref{['fig:sdw0']}. From (\ref{['eq:sdwLuttinger']}), the area of each hole pocket on the left (when $\Delta$ is large and there are no electron pockets) is $p/4$, in units with the square lattice Brillouin zone having unit area. This follows from the existence of 2 independent hole pockets in the magnetic Brillouin zone, each with a spin degeneracy of 2.