Do quantum spin liquids occur in the cuprates? The spin-liquid identity of the pseudogap metal

Soon after the discovery of high-temperature superconductivity in the cuprates, Anderson proposed a connection to quantum spin liquids, while observations since then have shown that the low-temperature cuprate phase diagram is dominated by conventional competing states — superconductivity and charge-ordered phases that break translational symmetry. The surviving modern form of the question is whether the pseudogap metal, the intermediate-temperature low-doping regime, is a doped quantum spin liquid — a fractionalized Fermi liquid (FL*) whose small Fermi pockets violate the Luttinger count and whose confining instabilities produce the observed d-wave superconductor and charge order — or whether a conventional mechanism (spin-density-wave/stripe reconstruction, phenomenological pseudogap bands, structural reconstruction, loop-current order, or preformed pairs) accounts for it. The question remains actively contested: recent magnetotransport evidence for small pockets without observed symmetry breaking is consistent with FL*, but controlled computations of the doped Hubbard model, stripe-based scenarios, structural-reconstruction proposals, and non-fractionalized phenomenologies reproduce overlapping subsets of the same data. No resolution has been established; the problem is open.

Background

The hole-doped cuprate superconductors are layered copper oxides whose undoped parent compounds are antiferromagnetic Mott insulators — insulators in which electron-electron repulsion, rather than band filling, localizes the charge carriers on the CuO$_2$ planes. Doping with a hole concentration $p$ destroys the antiferromagnetic order and produces, at low temperature, a superconducting dome with the highest transition temperatures known, flanked at low $p$ by charge-ordered 'stripe' states that break translational symmetry and compete with superconductivity (Chang et al. 2012; Fradkin, Kivelson and Tranquada 2015).

Between the antiferromagnet and the superconductor lies the pseudogap metal: above the superconducting transition temperature $T_c$ and below a doping-dependent crossover scale $T^*(p)$, the electronic excitation spectrum is partially depleted near the antinodal momenta of the Brillouin zone, as seen in angle-resolved photoemission spectroscopy (ARPES), leaving so-called Fermi arcs of remaining spectral weight (Damascelli, Hussain and Shen 2003). Quantum oscillations in high magnetic fields have long indicated small closed pockets in underdoped YBa$_2$Cu$_3$O$_{6.5}$ (Doiron-Leyraud et al. 2007), and the 2025 observation of the Yamaji effect — an extremum of interlayer resistivity versus field angle that signals a closed, weakly warped quasi-two-dimensional Fermi surface — in HgBa$_2$CuO$_{4+\delta}$ above $T_c$ found pockets occupying only about 1.3% of the Brillouin zone with no evidence for the long-range translational symmetry breaking that conventional Fermi-surface reconstruction requires (Chan et al. 2025).

A quantum spin liquid is a magnetically disordered state of a Mott insulator that nevertheless retains long-range quantum entanglement and fractionalized excitations, such as spin-$1/2$ spinons carrying the electron's spin without its charge, described by an emergent gauge field. Anderson (1987) proposed that doping such a resonating-valence-bond (RVB) state — a fluctuating superposition of singlet valence bonds — underlies high-temperature superconductivity. In the modern fractionalized Fermi liquid (FL$^*$) construction, the doped spin liquid survives as a background with topological order while gauge-neutral, electron-like quasiparticles occupy small pockets whose total area violates the Luttinger theorem, which would otherwise pin the Fermi volume to the total electron density (Sachdev, Scammell, Scheurer and Tarnopolsky 2019; Sachdev 2024). This yields a sharp fermiological discriminant: hole pockets of area $p/8$ of the Brillouin zone each for FL$^*$, versus $p/4$ for reconstruction by spin-density-wave (SDW) order. Lowering temperature confines the fractionalized degrees of freedom, and the FL$^*$ program predicts the observed descendants of the pseudogap — the $d$-wave superconductor with its anisotropic nodal quasiparticles, and charge-density-wave order — as confining instabilities of the spin liquid (Christos and Sachdev 2024; Bonetti, Christos, Nikolaenko, Patel and Sachdev 2026).

Whether this picture, or one of the conventional alternatives — SDW/stripe reconstruction, phenomenological pseudogap band structures such as the Yang-Rice-Zhang model (Rice, Yang and Zhang 2011), structural reconstruction assisted by spin-orbit coupling, loop-current order breaking time-reversal symmetry, or preformed pairs — describes the actual cuprates remains one of the central open questions of strongly correlated electron physics (Keimer et al. 2015; Sachdev 2024).

Problem Statement

Determine whether any regime of the hole-doped cuprate phase diagram — most prominently the intermediate-temperature pseudogap metal at low hole doping $p$ — is a doped quantum spin liquid, as proposed in Anderson's resonating-valence-bond scenario and elaborated in the fractionalized-Fermi-liquid ($\mathrm{FL}^*$) framework. A complete answer must jointly settle three linked questions that together identify the pseudogap state: (i) Fermiology — does the pseudogap Fermi surface consist of small closed pockets whose areas violate the Luttinger count in the fractionalized manner (hole pockets of area $p/8$ of the Brillouin zone each, versus $p/4$ for spin-density-wave reconstruction), with no symmetry-breaking order present that could conventionally account for the Fermi-surface reconstruction? (ii) Instabilities — do the observed low-temperature phases, the $d$-wave superconductor and the charge-density-wave/stripe orders, arise as confining instabilities of such a fractionalized state, with the quasiparticle anisotropies, vortex structure, and charge-order halos that the confinement scenario predicts? (iii) Exclusion — are the conventional alternatives (spin/stripe reconstruction, phenomenological pseudogap band models, structural reconstruction with spin-orbit coupling, loop-current order, preformed pairs) quantitatively excluded across the pseudogap regime? These components serve one overarching objective, the identification of the pseudogap metal and its descendants; an answer may be delivered by experiment, by theory, or by unbiased computation on the relevant microscopic models (Hubbard, $t$-$J$, ancilla-layer), but it must be specific enough that a qualified expert can decide from it which class of states — doped quantum spin liquid or conventional metal — underlies the pseudogap regime of the cuprates.

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