Spin-background-mediated pairing of charge carriers in a two-dimensional doped Mott insulator: microscopic mechanism and finite-density fate

Unbiased simulations show that two charge carriers in a two-dimensional Mott insulator described by the t-J model attract each other at experimentally accessible temperatures, yet neither the microscopic mechanism of this spin-background-mediated attraction nor its fate at finite carrier density is known. The problem is to determine, for the unmodified two-dimensional $t$ - $J$ model, which candidate mechanism produces the attraction and whether the attraction survives doping and, if so, whether it yields a phase-coherent superconducting (RVB-type) state or only locally preformed pairs.

Background

A Mott insulator is a lattice system in which strong on-site repulsion $U$ suppresses charge motion even when band theory predicts a metal; doping it with charge carriers (holes) is the setting of cuprate high-temperature superconductivity. The minimal model is the $t$ - $J$ Hamiltonian on a square lattice, $\hat H=-t\sum_{\langle ij\rangle,\sigma}\hat c^\dagger_{i\sigma}(1-\hat n_{i\bar\sigma})\hat c_{j\sigma}(1-\hat n_{j\bar\sigma})+J\sum_{\langle ij\rangle}(\hat{\mathbf S}_i\cdot\hat{\mathbf S}_j-\hat n_i\hat n_j/4)$ , which describes holes (empty sites) hopping with amplitude $t$ through an antiferromagnetic (AFM) spin background coupled by superexchange $J=4t^2/U$ . Because a moving hole frustrates the surrounding antiferromagnetic correlations, the carriers interact through the spin background rather than through any bare force.

Several pictures compete for the resulting effective attraction. In the broken-bond picture an isolated hole interrupts superexchange on its four bonds, and two holes sharing a link save magnetic energy; this is quantitatively inadequate at realistic $t/J$ , where kinetic delocalization competes with magnetic energy and 'kinetic-magnetic' frustration near the carriers has been proposed as the attraction source. In Anderson's resonating valence bond (RVB) picture the doped system gains energy by letting the spins form a fluctuating superposition of singlets that the carriers move through, which at finite doping would develop into a phase-coherent superconducting state. The preformed-pair picture instead holds that carriers bind into pairs well above any superconducting transition temperature $T_c$ , since $J$ in the cuprates corresponds to roughly 1500 K. Modern work adds further candidates: magnetic polarons (a charge carrier dressed by a cloud of distorted spin correlations, also called a meson in this context) whose pairs differ from tightly bound bipolarons, with an emergent Feshbach-resonance-like hybridization between the two channels; and the phase-string mechanism, in which the irreducible sign structure picked up by holes moving in an antiferromagnet is identified as the root of pairing.

Unbiased progress has been possible only at the edges of the problem. Worm-algorithm Monte Carlo (a sign-problem-free sampling method for the two-hole sector in which the number of carriers is kept fixed while the spin sector is grand canonical) gave the first completely unbiased finite-temperature results for exactly two holes: attraction sets in at $J\beta\approx 2.1$ when $U=8t$ , with detailed predictions for spin correlations near the carriers. However, as the number of fermionic carriers grows the fermion sign problem — the exponential signal-to-noise degradation of quantum Monte Carlo caused by negative sampling weights from exchanged indistinguishable fermions — makes unbiased sampling exponentially costly: for Gutzwiller-projected theories the sign problem is extensive in the number of charge carriers rather than in the system size, so addressing a small number of dopants in a sufficiently large system is feasible in principle, but unbiased finite-temperature results have so far been obtained only for two carriers. Finite-density evidence therefore comes mainly from restricted settings: density matrix renormalization group (DMRG, a variational method exact in the limit of large kept states, applied to long but narrow cylinders) finds quasi-long-range superconducting correlations at finite doping in four- to six-leg cylinders; infinite projected entangled-pair states (iPEPS, a variational tensor-network ansatz working directly in the thermodynamic limit) at finite temperature find only short-ranged $d$ -wave pairing correlations down to $T/t=0.1$ up to doping $x\approx 0.25$ ; and ultracold-atom experiments with fermionic ladders have directly observed magnetically mediated hole pairing, but only in quasi-one-dimensional two-leg geometry. Which mechanism is realized, and whether the two-hole attraction extrapolates to a superconducting phase at finite density in two dimensions, remain unsettled.

Problem Statement

For the square-lattice SU(2)-symmetric $t$ - $J$ model with nearest-neighbor hopping $t$ and superexchange $J=4t^2/U$ — the strong-coupling effective model of a doped Mott insulator — determine, by theory and/or experiment (unbiased or thermodynamic-limit evidence is especially valuable here, because the phase diagrams of strongly correlated models have proven highly sensitive to uncontrolled approximations):

The microscopic mechanism by which the antiferromagnetic spin background mediates attraction between doped charge carriers, in the regime where unbiased two-hole quantum Monte Carlo has established that the attraction exists at finite temperature. A complete answer must identify the microscopic mechanism (or mechanisms, if their relative role is regime-dependent) actually realized — leading candidates discussed in the literature include broken-bond magnetic energy, kinetic-magnetic frustration, phase-string sign structure, and bipolaron–magnetic-polaron two-channel/Feshbach-resonance physics — through discriminating observables such as the pair binding energy as a function of $t/J$ and temperature, the internal symmetry and spatial structure of the pair, dopant-conditioned spin-correlation profiles, and the two-particle spectrum.
The fate of this attraction at finite carrier density $x>0$ : does the attraction persist as the number of carriers grows beyond two and, if so, does it develop phase coherence — quasi-long-range (algebraic) $d$ -wave pairing order at finite temperature in two dimensions, or true long-range order (off-diagonal long-range order) at zero temperature — into a $d$ -wave superconducting (RVB-type) state, including the temperature scale, if any, at which coherence sets in; does it instead remain localized as incoherent preformed pairs without phase coherence; or is pairing lost altogether at finite density?

These two questions are linked components of one objective — the origin of pairing in a doped Mott insulator: the RVB-versus-preformed-pairs contention posed at the two-carrier level concerns precisely whether the spin-background-mediated attraction leads to superconducting phase coherence in the doped system. The target is the same, unmodified two-dimensional $t$ - $J$ Hamiltonian in the two-dimensional thermodynamic limit (or a directly corresponding ultracold-atom realization: a two-dimensional cold-atom system realizing the same unmodified $t$ - $J$ model and resolving the same question). Results restricted to quasi-one-dimensional ladder or cylinder geometry, and models with the sign structure or spin symmetry modified, address different settings and constitute partial progress on the present problem rather than an answer to it.

References

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