Glue or no glue: retarded versus unretarded d-wave pairing in the t-J-V model and the screening role of the no-double-occupancy constraint

The t-J-V model describes the CuO2 planes of the cuprates as a doped Mott insulator: electron hopping $t$, antiferromagnetic superexchange $J$, strict exclusion of double site occupancy, and a nearest-neighbor Coulomb repulsion $V$. Large-$N$ calculations (a controlled expansion in the spin degeneracy $N$ of the electron, with $N=2$ the physical case) by Zinni, Bejas, and Greco [Phys. Rev. B 103, 134504 (2021)] found that at mean-field level a repulsive $V$ of order $J$ destroys d-wave superconductivity, but that fluctuations of the no-double-occupancy constraint screen $V$, leaving an unretarded (instantaneous) $J$-driven d-wave pairing that survives even for $V \gg J$. The same approximation, however, suppresses spin fluctuations, so it cannot decide whether the physical ($N=2$) system's pairing is retarded — mediated by a bosonic glue such as spin fluctuations or flux-phase bond-charge fluctuations (the flux phase being a state of staggered circulating bond currents) — or unretarded. The open problem is to determine, by controlled analytic or unbiased numerical means, the frequency-resolved structure of the d-wave pairing interaction in the t-J-V model (equivalently the extended Hubbard model in its large-$U$ limit, retaining the nearest-neighbor repulsion $V$) with the constraint enforced beyond mean field: which channel dominates, whether the constraint-induced screening and robustness against $V$ survive at $N=2$, whether bond-charge instabilities can serve as the retarded glue, and what superconducting $T_c$ follows once retardation is treated consistently.

Background

The superconducting copper oxides (cuprates) have resisted a settled microscopic explanation since their discovery in 1986 (Keimer, Kivelson, Norman, Uchida, and Zaanen, Nature 518, 179 (2015)). Their minimal description starts from the two-dimensional Hubbard model, which in the regime of large on-site repulsion $U \gg t$ and low hole doping maps onto the t-J model: electrons hop with amplitude $t$ on a square lattice while double occupation of any site is strictly forbidden (the no-double-occupancy constraint, the defining feature of a doped Mott insulator — a Mott insulator is one made insulating by this inter-electron repulsion rather than by band filling), and neighboring spins are coupled by the antiferromagnetic superexchange $J \sim 4t^2/U$. The t-J-V model adds a nearest-neighbor Coulomb repulsion $V$, the leading remnant of the long-range Coulomb tail once the on-site part is absorbed into $U$. In conventional superconductors pairing is retarded: an effectively instantaneous bare repulsion is overcome because the attractive part of the interaction is mediated by slow bosons (phonons), and this frequency-dependent attractive part — the pairing 'glue' — controls the superconducting transition temperature $T_c$ through retardation (cutoff) effects. For the cuprates, P. W. Anderson proposed early on (Science 235, 1196 (1987)) that pairing is instead carried by the superexchange $J$ itself, an unretarded (instantaneous) attraction in the $d$-wave channel that requires no glue, while a major school attributes pairing to a retarded glue made of antiferromagnetic spin fluctuations. A numerical decomposition by Maier, Poilblanc, and Scalapino (Phys. Rev. Lett. 100, 237001 (2008)) found that in the t-J model both an instantaneous $J(\mathbf{k}-\mathbf{k}')$ contribution and a retarded spin-fluctuation contribution are present, with the retarded part dominant — sharpening the dichotomy rather than settling it. Zinni, Bejas, and Greco (Phys. Rev. B 103, 134504 (2021)) revisited the question in the t-J-V model using a large-$N$ expansion (an approximation controlled in $1/N$, i.e., in the inverse spin degeneracy) built on a path integral over Hubbard operators (electron operators with double occupancy projected out), in which the constraint is imposed by a Lagrange multiplier whose fluctuations enter a bosonic propagator mixing on-site charge and bond-charge (link) fluctuations. Their findings: (a) at mean-field level (bare boson propagator), a repulsive $V$ of order $J$ destroys d-wave superconductivity; (b) with the boson propagator dressed by constraint fluctuations, $V$ is screened in a random-phase-approximation-like (RPA-like) way — the standard summation of charge-bubble insertions — and d-wave pairing remains robust even for $V \gg J$, the surviving attractive channel being the unretarded $J(\mathbf{k}-\mathbf{k}')$ term; (c) a retarded contribution arises from the flux phase — a state of staggered circulating bond currents introduced by Affleck and Marston (Phys. Rev. B 37, 3774 (1988)) in the large-$n$ analysis of the Hubbard model, equivalent to the $d$-form-factor density wave proposed as hidden order in the pseudogap regime by Chakravarty, Laughlin, Morr, and Nayak (Phys. Rev. B 63, 094503 (2001)) — but it enhances pairing only near the flux-phase critical doping $\delta_c \approx 0.13$. The authors themselves state what their scheme cannot decide: the large-$N$ approximation suppresses spin fluctuations relative to charge fluctuations, so for the physical case $N=2$ a dominant retarded spin-mediated channel cannot be ruled out; the flux-phase interpretation of the pseudogap remains controversial (numerical studies disagree — Dong and Gull (Phys. Rev. B 101, 195115 (2020)) found only short-range charge fluctuations in the two-dimensional Hubbard model — and photoemission sees Fermi arcs rather than the pockets a flux state would produce); the extension to the large-$U$ Hubbard model with a retarded/unretarded decomposition was left as future work; and quantitative $T_c$ prediction requires treating retardation (the frequency dependence of the effective interaction and cutoff frequencies) beyond the weak-coupling Fermi-surface projection used. Unbiased tools that could address the question include the dynamical cluster approximation (DCA, a quantum-cluster extension of dynamical mean-field theory that maps the lattice onto a finite cluster embedded in a self-consistently determined bath) and density matrix renormalization group (DMRG, a variational method that optimally truncates the many-body Hilbert space to a matrix-product state) simulations on cylinders.

Problem Statement

Determine the dynamical (frequency-resolved) structure of the $d_{x^2-y^2}$-wave pairing interaction in the t-J-V model — equivalently, the extended Hubbard model in its large-$U$ limit, retaining the nearest-neighbor repulsion $V$ — with the no-double-occupancy constraint treated beyond mean field, and thereby decide whether superconductivity in this canonical model of the cuprates needs a glue at all. Concretely: (i) which contribution dominates the physical ($N=2$, unapproximated) d-wave pairing — the retarded, boson-mediated channel (spin fluctuations, or flux-phase/bond-charge fluctuations) or the unretarded exchange term $J(\mathbf{k}-\mathbf{k}')$; (ii) whether the screening of the nearest-neighbor Coulomb repulsion $V(\mathbf{q})$ by fluctuations of the constraint — which in the large-$N$ treatment renders d-wave superconductivity robust even for $V \gg J$ — survives in unbiased treatments of the physical model, including its interplay with the charge-ordered, phase-separated, and pseudogap-like states that $V$ favors; (iii) whether the flux phase or related bond-charge instabilities can supply the retarded glue, and how robust that contribution is beyond the large-$N$ scheme; and (iv) what superconducting $T_c$ follows once the frequency dependence of the effective pairing interaction and retardation cutoffs are treated consistently. These facets jointly serve one objective — fixing the with-glue versus without-glue character of the pairing — and a study that resolves the dominant-channel question and the fate of the $V$-screening for the constrained model at physical parameter values would complete it, with (iii) and (iv) supplying the identification of the glue (if retarded dominates) and its quantitative consequence.

References

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