arXiv subjects

Subir Sachdev

Publications and source records attributed to Subir Sachdev.

At least 19 recordsLinked to original sources

arXiv2026

Deconfined criticality between an antiferromagnetic insulator and a nodal d-wave superconductor: a quantum Monte Carlo study

We present a quantum Monte Carlo study of the transition between the insulating Néel state and the nodal $d$-wave superconductor on the square lattice at half-filling. We access a regime of frustrated magnetic order without a sign problem using a parton representation of the electron in terms of fermionic spinons and bosonic chargons. Both partons move in a background $π$-flux (so the electron experiences no net flux) and are coupled to a quantum fluctuating SU(2) lattice gauge field. In contrast to earlier studies directly on the electronic degrees of freedom, we find evidence for a second-order deconfined quantum phase transition at which both the Néel and $d$-wave superconductivity orders vanish continuously. We compute correlators of the spinon-chargon composite with the same quantum numbers as the electron: we find a gapless Dirac dispersion inside the $d$-wave superconductor, turning into a gapped dispersion in the antiferromagnet.

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arXiv2026

Crossover to Sachdev-Ye-Kitaev criticality in an infinite-range quantum Heisenberg spin glass

A central question in frustrated quantum magnets is how quantum fluctuations destabilize spin-glass order and promote quantum spin-liquid behavior. We study the equilibrium dynamics of an infinite-range quantum Heisenberg model with random couplings, in which local magnetic moments arise from $\mathcal{N}_f$ flavors of spinful fermions. We employ an expansion in $\mathcal{N}_f$, which controls the strength of quantum fluctuations, and self-consistently include $1/\mathcal{N}_f$ corrections to the Luttinger-Ward functional. In the large-$\mathcal{N}_f$ limit, where quantum fluctuations are weak, the high- and low-temperature phases are respectively paramagnetic and spin glass ordered, with a transition temperature independent of $\mathcal{N}_f$. For small numbers of fermionic flavors, however, quantum fluctuations substantially suppress the ordering temperature. We show that this behavior reflects the proximity of the system to a Sachdev-Ye-Kitaev (SYK) regime, realizing quantum spin-liquid dynamics over a broad range of finite frequencies, where both fermionic and spin spectral densities display critical behavior over a broad range of finite frequencies, with the latter exhibiting the scale-invariant form $χ''(ω)\sim\operatorname{sgn}(ω)$. At the lowest energies and temperatures, spin glass dynamics ultimately take over, producing a universal sub-Ohmic dynamical spin susceptibility $χ''(ω)\sim\operatorname{sgn}(ω)\sqrt{|ω|}$. Our results highlight a direct connection between the suppression of spin-glass order and the emergence of SYK criticality in a frustrated quantum magnet.

cond-mat.str-el↗

arXiv2026

Transformer Neural-Network Quantum States for lattice models of spins and fermions: Application to the Ancilla Layer Model

We introduce a variational wave function based on Neural-Network Quantum States (NQS) to study lattice systems whose local Hilbert space contains both spin and fermionic degrees of freedom. Our approach is based on the use of the Transformer architecture, which can naturally handle composite local Hilbert spaces through a tokenization procedure closely inspired by techniques from natural language processing. The neural network predicts a set of fermionic orbitals that depend on the spin configuration in a backflow-inspired manner. We apply the method to the one-dimensional Ancilla Layer Model, consisting of a chain of mobile spin-$1/2$ fermions coupled to a two-leg spin-$1/2$ ladder. For open boundary conditions, we achieve excellent quantitative agreement with Density Matrix Renormalization Group (DMRG) results across the full range of parameters considered. We find a phase in which the chain forms an effectively decoupled Luttinger liquid (LL), and a LL* phase with a distinct Fermi wavevector in which the mobile fermions are Kondo screened by one leg of the ladder, while the other leg forms the critical Bethe spin liquid. The LL* is the analog of the phase describing the pseudogap in two dimensions. We also find a Luther-Emery (LE) phase, where the LL* state becomes unstable toward the formation of a spin gap. The Transformer Ansatz maintains comparable accuracy for periodic boundary conditions, where tensor-network methods are computationally more demanding. Together, these findings establish Transformer-based NQS as an accurate and scalable variational framework for correlated lattice systems with composite local Hilbert spaces and highlight their potential for studying higher-dimensional models where boundary effects and heterogeneous local structures pose significant challenges.

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arXiv2026

Symmetry-resolved monopoles and stability criteria for an $N_f=12$ Dirac spin liquid on the maple-leaf lattice

The $\mathrm{U}(1)$ Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED$_3$ with $N_f=12$ two-component Dirac fermions. The charge-one monopoles span a $924$-dimensional complex representation. In the spin-singlet, zero-momentum, zero-$C_6$-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations, $C_6$, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-$N_f$ and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED$_3$ in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the $N_f=12$ Dirac spin liquid.

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arXiv2026

Floquet engineering of competing antiferromagnetism and $d$-wave superconductivity on the square lattice

We propose a driven Lieb--Hubbard quantum simulator whose prethermal dynamics realize a square-lattice spin-1/2 fermion model with independently tunable repulsive on-site and attractive bond interactions. Periodically modulating the charge-transfer offset between the site ($d$) and bond ($p$) orbitals of the Lieb lattice brings a $p$-orbital doublon energetically close to a pair in the $d$ manifold while keeping all $p$-orbital singlons off resonance, thereby creating a synthetic ``negative-$U$'' center on a Lieb-lattice bond site. Two controlled eliminations then generate a compact bond attraction in the reduced $d$-only model on the square lattice, despite the microscopic repulsion in the $p$ orbital. The resulting interaction $J$ is tunable independently of the Hubbard repulsion $U_d$ on the $d$ orbitals, while interference between photon-assisted paths provides access to an intermediate-coupling regime in which $J$ and $U_d$ are both comparable to the effective hopping. At half filling, a mean-field calculation in this regime finds adjacent antiferromagnetic and $d$-wave superconducting phases, as well as narrow coexistence regions, suggesting close competition between these orders. We discuss the branch-preparation, prethermal, and higher-band conditions required to translate the formal construction into an optical-lattice protocol. More broadly, our work identifies a structural similarity between Floquet systems and electron-phonon problems that may guide the design of novel quantum-simulation protocols.

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arXiv2026

Emergence of nodal Bogoliubov quasiparticles across the transition from the pseudogap metal to the d-wave superconductor

We model the pseudogap state of the hole- and electron-doped cuprates as a metal with hole and/or electron pocket Fermi surfaces. In the absence of long-range antiferromagnetism, such Fermi surfaces violate the Luttinger requirement of enclosing the same area as free electrons at the same density. Using the Ancilla theory of such a pseudogap state, we describe the onset of conventional $d$-wave superconductivity by the condensation of a charge e Higgs boson transforming as a fundamental under the emergent SU(2) gauge symmetry of a background $π$-flux spin liquid. In all cases, we find that the $d$-wave superconductor has gapless Bogoliubov quasiparticles at 4 nodal points on the Brillouin zone diagonals with significant velocity anisotropy, just as in the BCS state. This includes the case of the electron-doped pseudogap metal with only electron pockets centered at wavevectors $(π, 0)$, $(0, π)$, and an electronic gap along the zone diagonals. Remarkably, in this case too, gapless nodal Bogoliubov quasiparticles emerge within the gap at 4 points along the zone diagonals upon the onset of superconductivity.

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arXiv2026

Higgs criticality of Dirac spin liquids on depleted triangular lattices

We investigate Higgs criticality in candidate U(1) Dirac spin liquids across a family of depleted triangular lattices: the triangular, kagome, and maple-leaf geometries. For each, we identify the symmetry-allowed spinon-pairing channel connecting the U(1) state to a proximate $\mathbb{Z}_2$ spin liquid, deriving the corresponding quantum electrodynamics (QED$_3$)-Higgs theory. While the triangular and kagome lattices share a low-energy description with $N_f=4$ Dirac fermions, the maple-leaf lattice yields an analogous theory with $N_f=12$ and a distinct nodal structure where the Dirac cones can move along high-symmetry lines in momentum space instead of gapping out. Using a large-$N_{f,b}$ expansion, we compute critical exponents and the scaling dimensions of the symmetry-allowed Yukawa couplings. Although Higgs-field fluctuations and a large fermion flavor number strongly suppress the Yukawa coupling---pushing the maple-leaf lattice closer to stability than its counterparts---we find that the coupling remains relevant at leading order in all three cases, substantially reducing its scaling dimension on the maple-leaf lattice and asymptotically breaking Lorentz and conformal invariance. Ultimately, our results provide a unified framework demonstrating how the interplay between fermion flavor count and nodal geometry dictates the fate of the QED$_3$-Higgs transition.

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arXiv2026

Meissner Effect and Josephson Radiation in Driven Dissipative Superconductors

Photo-induced superconducting signatures have been observed in several materials at temperatures far above their equilibrium critical temperatures, including magnetic field expulsion as in the Meissner effect. We propose driven-dissipative condensation of the superconducting order parameter as a mechanism for this phenomenon. In such a theory, optical pumping generates an effective gain (possibly via a parametric resonance) that overcomes the intrinsic damping of the pairing field, and stabilizes a non-equilibrium condensate whose phase rotates at an intrinsic frequency that is generally lower than, and incommensurate with, the drive frequency. We develop a phenomenological continuum theory that couples this slowly rotating order parameter to the conserved charge density and the electromagnetic gauge field. Despite its finite-frequency dynamics, the resulting state exhibits the conventional long-wavelength electromagnetic signatures of superconductivity. In three dimensions, it displays a static Meissner effect and a gapped plasmon generated by the Anderson--Higgs mechanism. In a two-dimensional sheet, it exhibits Pearl screening and the characteristic square-root plasmon dispersion. We further propose a direct experimental test based on a Josephson junction between the driven-dissipative state and an equilibrium superconductor. The junction supports an AC Josephson current at zero applied voltage and emits radiation at the intrinsic rotation frequency of the condensate. Its low, pump-dependent, and generally incommensurate frequency provides a clear signature distinguishing driven-dissipative superconductivity from equilibrium and drive-locked pairing states.

cond-mat.supr-con↗

arXiv2026

Superconducting and charge-ordered phases from Dirac quantum spin liquids on the triangular lattice

Triangular-lattice quantum spin-liquid insulators are observed to undergo transitions to superconductivity under pressure or doping, and exhibit enhanced terahertz conductivity when driven by mid-infrared light. We present a general theoretical framework for the emergence of superconducting and charge-ordered phases from a U(1) Dirac spin liquid with fermionic spinons, as well as from its gapped $\mathbb{Z}_2$ and chiral descendants. Numerical studies have provided substantial evidence for these spin-liquid states. The spin-liquid phase hosts fractionalized Dirac spinons coupled to an emergent gauge field, whereas the superconducting and charge-ordered phases are conventional, with neither fractionalized excitations nor emergent gauge dynamics. The transition between these phases is driven by the Higgs condensation of spinless charge-$e$ bosonic chargons (``doublons'' and ``holons''). We show that the projective symmetry group of the Dirac spinons uniquely determines the symmetry and dispersion of the chargons, allowing us to construct an effective low-energy theory near the chargon band minima. Gauge-invariant composites of the chargon Higgs fields provide the order parameters characterizing the phases. The resulting phase diagram contains a rich variety of ordered states, including $d+id$ superconductivity, charge-density waves, bond-density waves, and pair-density waves.

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arXiv2026

Entanglement entropy of fermions in a strange metal

The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-$N$ 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.

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arXiv2026

Lectures on insulating and conducting quantum spin liquids

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 suggestion of P. W. Anderson, there were 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) in the pseudogap metal, including observation of the Yamaji effect, present convincing evidence of small hole pockets which can tunnel coherently between square lattice layers (Fang et al., Nature Physics 18, 558 (2022); Chan et al., Nature Physics 21, 1753 (2025)) and (ii) the velocities of the nodal Bogoliubov quasiparticles in the $d$-wave superconductor are highly anisotropic, with $v_F \gg v_Δ$ (Chiao et al., Phys. Rev. B 62, 3554 (2000)). These 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 ultracold atom experiments on 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 in a single band Hubbard-type model 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. The ALM also leads to a wavefunction for the FL* state of the Hubbard model, which agrees with ultracold atom observations.

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arXiv2026

Influence of Harris disorder on quantum-critical superconductivity

In the Hertz theory, a quantum critical metal is described by the coupling of a Fermi surface to fluctuations of a Landau-damped bosonic field $ϕ$, which may represent either an order parameter or a Higgs field for a transition without symmetry breaking. Scattering from $ϕ$ produces non-Fermi-liquid behavior in the normal state, while the same fluctuations mediate enhanced Cooper pairing. By the Harris criterion, symmetry-preserving disorder couples most strongly to the coefficient of the $ϕ^2$ term, locally tuning the system toward or away from criticality. This random mass (``Harris disorder'') leads to a finite density of localized low-energy $ϕ$ modes even when the fermionic states remain extended and continue to provide Landau damping. We study the onset of pairing mediated by these localized overdamped bosonic modes. Starting from a real-space linearized Usadel equation for the two-particle propagator, we show that the localized bosonic wave functions generate both a spatially random pairing vertex and an effective random potential for Cooper pairs. Numerical solution, a self-consistent Born approximation, and Lifshitz-tail analysis reveal two regimes of the pairing instability. At high temperatures, pairing nucleates compact superconducting puddles on the most localized bosonic modes. At lower temperatures, an extended pairing eigenstate appears, but its transition scale and spatial structure remain strongly affected by mesoscopic correlation effects and enhanced probability of returns to favorable regions of the localized bosonic glue. The resulting distribution of local pairing scales has a power-law tail, in contrast to the stretched-exponential tails of disordered BCS superconductors. This mechanism provides a route to broad gap inhomogeneity and superconducting puddles in quantum-critical metals, and offers an interpretation of STM measurements of the cuprates.

cond-mat.supr-con↗

arXiv2026

Fermi surface change and $d$-wave superconductivity in the square lattice Kondo-Heisenberg model

We study the two-dimensional Kondo-Heisenberg model on a square lattice, with the conduction electrons away from half-filling, using neural network quantum states. Mapping the ground-state phase diagram as a function of the Kondo and Heisenberg couplings, we identify (i) at weak Kondo coupling, antiferromagnetic Néel order with a Fermi surface whose enclosed area counts only the conduction electrons and is insensitive to the Néel order, and (ii) at strong coupling, a heavy Fermi liquid with a Fermi surface whose enclosed area counts both the conduction electrons and the spins. In the crossover between these regimes, we find $d_{x^2-y^2}$ superconductivity, evidenced by off-diagonal long-range order in the pair-pair correlations and a pairing-amplitude dome that coexists with the underlying magnetic phase. Our results establish Fermi volume change and unconventional superconductivity as intrinsic features of the two-dimensional Kondo-Heisenberg model.

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arXiv2026

Yamaji effect in models of underdoped cuprates

Recent angle-dependent magnetoresistance measurements in underdoped cuprates have revealed compelling evidence for small hole pockets in the pseudogap regime, including observation of the Yamaji effect in HgBa$_2$CuO$_{4+δ}$ (Chan et al., Nature Physics 10.1038/s41567-025-03032-2 (2025)). A key distinction between theories is their predicted Fermi volumes, measured as fractions of the square lattice Brillouin zone: $p/4$ per pocket for spin density wave (SDW) versus $p/8$ for fractionalized Fermi liquid (FL*), where $p$ is the hole doping. We calculate the $c$-axis magnetoresistance $ρ_{zz}(θ, ϕ)$ within the semiclassical Boltzmann formalism for both states, and using the ancilla layer model (ALM) for FL* in a single-band Hamiltonian. The results from the $\text{FL}^*$ phase show good consistency with current experimental data. Conversely, the results for the SDW phase are highly sensitive to the ordering momentum along the $z$-direction. An ordering vector of $Q = (π, π, π)$ yields predictions that starkly disagree with the experiment. The only possibility for agreement within the SDW scenario is to assume an ordering momentum of $Q = (π, π, 0)$. However, even in this specific case, the SDW scenario predicts a marginally smaller Yamaji angle at $ϕ=0$ than the FL* theory, and a second Yamaji peak near in-plane angle $ϕ= 45^\circ$, which was not observed in the experiment. In reality, the Néel ordering vector is likely uncorrelated between adjacent layers, so that there is no coherent interlayer transport of hole-pocket quasiparticles in the SDW scenario, and consequently no Yamaji effect. Our results support the FL* interpretation of Fermi arcs in the pseudogap phase, and establish Yamaji angle measurements as a discriminatory tool between theoretical models.

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arXiv2026

Fractionalized Fermi liquids and the cuprate phase diagram

We review a theoretical framework for the cuprate superconductors, rooted in a fractionalized Fermi liquid (FL*) description of the intermediate-temperature pseudogap phase at low doping. The FL* theory predicted hole pockets each of fractional area $p/8$ at hole doping $p$, in contrast to the area $p/4$ in spin density wave theory. Magnetotransport measurements, including observation of the Yamaji angle, show clear evidence of hole pocket quasiparticles which can tunnel coherently between square lattice layers, and are consistent with the FL* description. The FL* phase of a single-band model is described using a layer construction with a pair of ancilla qubits on each site: the Ancilla Layer Model (ALM). Fluctuations are described by the SU(2) gauge theory of a background spin liquid with critical Dirac spinons. A Monte Carlo study of the thermal SU(2) gauge theory transforms the hole pockets into Fermi arcs in photoemission. One route to confinement of FL* upon lowering temperature yields a $d$-wave superconductor via a Kosterlitz-Thouless transition of $h/(2e)$ vortices, with nodal Bogoliubov quasiparticles featuring anisotropic velocities and vortices surrounded by charge order halos. Increasing doping from the FL* phase in the ALM drives a transition to a conventional Fermi liquid (FL) at large doping, passing through an intermediate strange metal regime. We formulate a theory of the FL*-FL metal-metal transition without a symmetry-breaking order parameter, using a critical quantum `charge' liquid of mobile electrons in the presence of disorder, developed via an extension of the Sachdev-Ye-Kitaev model to two spatial dimensions. At low temperatures, and across optimal and over doping, we address the regimes of extended non-Fermi liquid behavior by Griffiths effects near quantum phase transitions in disordered metals.

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arXiv2026

Thermal SU(2) lattice gauge theory for intertwined orders and hole pockets in the cuprates

The cuprate pseudogap phase displays Fermi arc spectral weight in photoemission and scanning tunneling microscopy (STM), while recent magnetotransport observations yield evidence for the existence of hole pockets of fractional area $p/8$, where $p$ is the doping density. We present a Monte Carlo study of a thermal SU(2) lattice gauge theory which can reconcile these observations. Our simulation includes the SU(2) gauge field $U$ of a $π$-flux spin liquid, and a SU(2) fundamental charge $e$ Higgs boson $B$. There is a Yukawa coupling between $B$, the fermionic spinons of the spin liquid, and the hole pockets of a fractionalized Fermi liquid. At the higher temperatures of the pseudogap, the finite-doping sign problem is evaded by including only thermal fluctuations of $B$ and $U$, while the fermions are diagonalized exactly for each boson background. Our study also yields a fractionalized description of intertwined orders at lower temperatures, including the onset of $d$-wave superconductivity by the expulsion of vortices with flux $h/(2e)$, each with charge-order halos. We discuss conditions under which quantum oscillations in the density of states from hole pockets of area $p/8$ could be observable in clean under-hole-doped cuprates.

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arXiv2026

From Ferrimagnetic Insulator to superconducting Luther-Emery Liquid: A DMRG Study of the Two-Leg Lieb Lattice

Motivated by recent experiments on ultracold fermionic spin-1/2 $^6$Li atoms in a Lieb lattice at various Hubbard repulsion $U$ and filling fractions $n$ (Lebrat et al., arXiv:2404.17555), we conduct a density matrix renormalization group (DMRG) analysis of the Hubbard model on a two-leg Lieb ladder. At half-filling, we find a ferrimagnetic Mott insulating ground state, consistent with Lieb's theorem. Away from half-filling, a state with finite total spin $\vec{S}^2 \neq 0$ and vanishing charge gap persists down to filling $n_c \approx 2/3$. For lower, incommensurate fillings, the system is described by a Luttinger liquid with one charge and one spin mode. Intriguingly, in a narrow window near $n_c = 2/3$, close to the onset of ferromagnetic order, we identify a superconducting Luther-Emery phase with dominant $s_{xy}$-wave pairing.

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arXiv2026

Superconductivity and fractionalized magnetic excitations in CeCoIn5

Recent experiments on CeCoIn5 -- a prototypical d-wave superconductor -- indicate that its normal state lies near an unconventional quantum critical point (QCP). One intriguing hypothesis is that quantum-critical fluctuations promote fractionalization of localized 4f moments into fermionic spinons. This fractionalized Fermi liquid (FL*) scenario provides a comprehensive framework for the unconventional QCP and superconductivity, and can reconcile a "missing" Fermi-surface volume relative to the Luttinger count in the normal state of CeCoIn5. To test this possibility, we performed inelastic neutron scattering (INS) measurements on CeCoIn5 across the superconducting transition and corresponding theoretical analysis. Our high-precision spectra reveal detailed momentum and temperature dependence of the spin resonance and a structured spin excitation continuum persisting even in the normal state, placing stringent constraints on the physical picture of pairing in a d-wave superconductor. We show that a Kondo-lattice framework incorporating proximity to FL* physics and d-wave pairing reproduces key features of the data. The model suggests that both the quasi-localized nature of the f-moments above Tc and the resonance below Tc arise from common underlying gauge dynamics, implying a unifying organizing principle linking spin fractionalization and unconventional superconductivity in strongly correlated metals.

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