Critical properties of the antiferromagnetic quantum phase transition of the Gossamer superconductor

The Gossamer superconductor — a d-wave BCS state partially Gutzwiller-projected so that it carries strong on-site Coulomb repulsion while remaining adiabatically continuable to a regular superconductor — is the exact ground state of the Gossamer Hamiltonian. Bernevig, Laughlin, and Santiago (Phys. Rev. Lett. 91, 147003 (2003)) showed that at half filling and strong projection this Hamiltonian is effectively a Hubbard model plus a d-wave pairing attraction, and that its spin susceptibility, computed in a random-phase-approximation (RPA) ladder resummation, diverges at $U\approx 1.43t$ (for gap $\Delta_0=0.4t$), signaling a zero-temperature transition to an antiferromagnetic insulator. The authors explicitly cautioned that this crude ladder treatment cannot give the correct critical coupling or critical exponents, asserted (only at that level) a continuous transition with antiferromagnetism developing underneath the superconductivity, and claimed the instability is pinned exactly at half filling for this Hamiltonian. Two decades of follow-up work has mapped gossamer-type physics only in neighboring models — effective Hubbard, Hubbard-Heisenberg, and t-J-U models at mean-field, variational, or dynamical-mean-field level — with mutually conflicting conclusions on the order of the transition, and no controlled calculation for the Gossamer Hamiltonian itself, whose repulsion is generated by the partial projection rather than added by hand and whose ground state is exact, unlike the fully projected and hand-modified BCS Hamiltonians of references 11–13. The open problem is to determine the magnetic phase boundary and critical properties of the Gossamer superconductor: the true critical projection strength (equivalently critical $U/t$), the order and universality class of the transition, whether superconducting and antiferromagnetic orders coexist at the boundary, and how the instability locus moves with doping and with modifications of the Hamiltonian.

Background

The parent compounds of the cuprate high-temperature superconductors are antiferromagnetic Mott insulators: at half filling (one hole per copper site) the strong on-site Coulomb repulsion $U$ localizes the charge, and the antiferromagnetic exchange generated by virtual hopping orders the spins; hole doping destroys this state and produces superconductivity with a $d_{x^2-y^2}$-symmetric energy gap. How superconductivity and antiferromagnetism connect across this phase diagram is a long-standing theoretical question. Anderson's resonating-valence-bond (RVB) idea describes the doped Mott insulator using wavefunctions in which doubly occupied sites are removed entirely — a Gutzwiller projection. Laughlin proposed the opposite emphasis: start from a Bardeen-Cooper-Schrieffer (BCS) ground state of Cooper pairs with gap $\Delta_{\vec{k}}=\Delta_0(\cos k_x a-\cos k_y a)$ on a square lattice, and apply a partial Gutzwiller projection, a product over sites $j$ of factors $z_0^{(n_{j\uparrow}+n_{j\downarrow})/2}(1-\alpha_0 n_{j\uparrow}n_{j\downarrow})$ that only suppress, rather than forbid, double occupancy. Here $0\le\alpha_0<1$ is the projection strength and $z_0$ is a normalization (a 'quantum fugacity') that keeps the particle number fixed. The resulting 'Gossamer' superconductor retains an intact d-wave tunneling gap while its quasiparticle spectral weights and superfluid density (the density of coherent, phase-stiff superconducting carriers) collapse as $\alpha_0\to 1$, so near half filling it becomes spectroscopically almost indistinguishable from a doped Mott insulator, with upper and lower Hubbard bands (the split remnants of the conduction band separated by a correlation gap) and a faint midgap band. Because the partial projector is invertible for $\alpha_0<1$, this state is adiabatically continuable to a regular BCS superconductor.

The Gossamer ground state $|\Psi\rangle=\Pi_\alpha|\Phi\rangle$ is the exact ground state of a specific parent operator, the Gossamer Hamiltonian, built from projected Bogoliubov quasiparticle operators $B_{\vec{k}\sigma}=\Pi_\alpha b_{\vec{k}\sigma}\Pi_\alpha^{-1}$ (with $b_{\vec{k}\sigma}$ the usual Bogoliubov combination of $c_{\vec{k}\sigma}$ and $c^\dagger_{-\vec{k}\sigma}$): $H=\sum_{\vec{k}\sigma}E_{\vec{k}}B^\dagger_{\vec{k}\sigma}B_{\vec{k}\sigma}\ge 0$ and $B_{\vec{k}\sigma}|\Psi\rangle=0$. Bernevig, Laughlin, and Santiago analyzed this Hamiltonian at half filling by expanding it: the on-site part generates an explicit Hubbard repulsion $\sum_j U n_{j\uparrow}n_{j\downarrow}$ with $U=(1/N)\sum_{\vec{k}}2E_{\vec{k}}\,\alpha_0(2-\alpha_0)/(1-\alpha_0)$, which diverges as $\alpha_0\to 1$, while the kinetic and d-wave pairing terms are renormalized by the same factor — so at strong projection the half-filled Gossamer Hamiltonian is effectively a Hubbard model with $U/t$ of order unity or larger, plus a d-wave pairing attraction. Computing the spin susceptibility in the random-phase approximation (RPA, a ladder resummation $\chi_{\vec{q}}(\omega)=\chi^0_{\vec{q}}(\omega)/[1+U\chi^0_{\vec{q}}(\omega)]$ of the bare bubble $\chi^0$ built from the projected Green function), they found that $\mathrm{Im}\,\chi_{(\pi,\pi)}(\omega)$ diverges as $U$ approaches $\approx 1.43t$ for $\Delta_0=0.4t$, signaling the onset of antiferromagnetic order: a zero-temperature (quantum) phase transition from the Gossamer superconductor to an antiferromagnetic insulator driven by correlation strength. In parallel, Zhang showed that Gutzwiller-variational treatments of effective Hubbard-type models also support a gossamer-superconductor-to-Mott-insulator boundary at half filling.

The authors were explicit that this RPA treatment 'will not provide the right critical value of $U$ for which the transition to AFM order occurs, nor will it provide the correct critical exponents' — it gives only a qualitative picture. Within that approximation they asserted a continuous transition with antiferromagnetism developing underneath the superconductivity (opening a gap at the d-wave node, i.e., at the diagonal momentum directions where the gap $\Delta_{\vec{k}}$ vanishes) rather than displacing it, and claimed that for the Gossamer Hamiltonian the instability is pinned exactly at half filling, whereas for modified Hamiltonians, for example with an extra Hubbard term added by hand, it can shift to finite doping or stabilize stripe order (static, unidirectional modulations of spin and charge density) instead of uniform antiferromagnetism. None of these assertions — critical coupling, order of the transition, universality class, coexistence, or the half-filling pinning of the instability — has been checked by a controlled calculation.

Problem Statement

Determine the antiferromagnetic quantum phase transition of the Gossamer Hamiltonian (the partially Gutzwiller-projected d-wave BCS Hamiltonian on the square lattice with projection strength $\alpha_0$ and exact superconducting ground state). The single overarching objective is the magnetic phase diagram and critical characterization of the Gossamer superconductor, addressed through the four jointly supporting facets listed below, none sufficient on its own:

1. Critical locus: compute the true critical projection strength $\alpha_0^c$ — equivalently the critical ratio $U/t$ (or $U/\Delta_0$) at which antiferromagnetic order onsets in the half-filled Gossamer Hamiltonian — replacing the uncontrolled RPA estimate $U_c\approx 1.43t$ obtained for $\Delta_0=0.4t$ at $\vec{q}=(\pi,\pi)$ (the antiferromagnetic ordering wavevector).

2. Order and critical behavior: establish whether the superconductor-to-antiferromagnetic-insulator transition is continuous or first order, and, if continuous, its universality class and critical exponents. (The source asserts continuity only at the level of the crude ladder approximation, and mean-field and variational treatments of neighboring gossamer-type models have found both first-order boundaries and second-order coexistence regimes.)

3. Coexistence versus displacement: determine whether antiferromagnetic order develops underneath the superconductivity — leaving an arbitrarily small but nonzero superfluid density and opening a gap at the d-wave node — or destroys superconducting order at the boundary, together with the resulting quasiparticle spectrum (nodal gap, midgap spectral weight, Hubbard-band weights).

4. Instability locus versus doping and Hamiltonian modifications: verify or refute the claim that the instability is pinned exactly at half filling for the Gossamer Hamiltonian; separately, verify or refute the claim that modifications of the model (for example, an explicit extra Hubbard $U$ added by hand) shift the instability to finite doping or stabilize stripe order instead of uniform antiferromagnetism; and map the phase boundary $\alpha_0^c(\delta)$ (equivalently $U_c(\delta)$) of the Gossamer Hamiltonian, where $\delta$ denotes the hole doping, i.e., the deviation from half filling.

The target is the Gossamer Hamiltonian itself as defined by its projected quasiparticle operators, not proxy models such as the Hubbard-Heisenberg, t-J-U, or t-J Hamiltonians studied in the follow-up literature.

References

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2. P. W. Anderson, 'The Resonating Valence Bond State in La2CuO4 and Superconductivity', Science 235, 1196-1198 (1987), DOI 10.1126/science.235.4793.1196, https://doi.org/10.1126/science.235.4793.1196
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13. Hyunwoong Kwon and Kwon Park, 'Projected BCS theory for the unification of antiferromagnetism and strongly correlated superconductivity', Phys. Rev. Research 4, 013116 (2022), DOI 10.1103/PhysRevResearch.4.013116, https://doi.org/10.1103/PhysRevResearch.4.013116 (preprint arXiv:1912.13064)
14. Hui-Ke Jin, Guangyue Ji, Zhan Wang, Jie Wang, and Fu-Chun Zhang, 'Gossamer Superconductivity in Moiré WSe2 Bilayer', arXiv:2605.03766 (2026), https://arxiv.org/abs/2605.03766
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