Minimal loop currents in doped Mott insulators

Abstract

For the $t$-$J$ model, variational wave functions can generally be constructed based on an accurate description of antiferromagnetism (AFM) at half-filling and an exact phase-string sign structure under doping. The single-hole-doped and two-hole-doped states, as determined by variational Monte Carlo (VMC) simulations, display sharply contrasting behaviors. The single-hole state constitutes a ``cat state'' that resonates strongly between a quasiparticle component and a local loop-current component, with approximately equal weights. In the ground state, the quasiparticle spectral weight $Z_{\mathbf{k}}$ peaks at momenta $\mathbf{k}_0 \equiv (\pm\fracπ{2},\pm\fracπ{2})$. The total-energy dispersion versus $\mathbf{k}$ agrees remarkably well with the Green function Monte Carlo results. However, Landau's one-to-one correspondence hypothesis for quasiparticles breaks down here with the incoherent component exhibiting intrinsic magnetization originating from a minimal $2\times2$ loop current that forms a $4\times4$ pattern on the square lattice--a finding in excellent agreement with density matrix renormalization group (DMRG) calculations. In the two-hole ground state, a new pairing mechanism is revealed: the two holes are automatically fused into a tightly bound object consisting of an incoherent $d_{xy}$ pairing along the diagonal direction by compensating the local loop currents. This hole pair is again a ``cat state'' that resonates strongly between the incoherent $d_{xy}$ and a coherent $d_{x^2-y^2}$ Cooper channel to gain substantial hopping energy. Its size extends over an area of about $4\times 4$ lattice spacings, much smaller than the divergent AFM correlation length, implying that it should survive as a minimal superconducting building block even in the dilute doping regime. Experimental implications and the generalization to the finite-doping case are briefly addressed.

Figures (20)

• Figure 1: Schematic illustration of doped-hole wavefunctions. (a) A bare hole of $S^z=\pm 1/2$ is transformed into a twisted hole [Eq. (\ref{['eq:twistedc']})], a composite of charge and spin that encircle one another in the transverse ($x$-$y$) plane with chirality and orbital angular momentum $L_z=\pm 1$. While the bare hole's motion is blocked by the phase‑string effect, the twisted hole is mobile on the AFM background via the hopping term PhysRevB.99.205128. (b) The single‑hole variational wave function [Eq. (\ref{['eq:singlehole']})] describes a bound state between a twisted hole and an antimeron (centered at the cross). Both exhibit vortex‑like distortions in the spin $x$–$y$ plane with opposite chiralities, thereby removing the logarithmically divergent superexchange energy associated with the twisted hole alone. The bare‑hole state $|\Psi_{\mathrm{qp}}\rangle_{\mathrm{1h}}$ can be recovered when the vortex–antivortex pair annihilates at short distance. This single‑hole state thus forms a quantum "cat state", arising from a strong resonance between the quasiparticle component $|\Psi_{\mathrm{qp}}\rangle_{\mathrm{1h}}$ and incoherent component $|\Psi_{\mathrm{inc}}\rangle_{\mathrm{1h}}$ driven by the hopping term. (c) The two‑hole ground state [Eq. (\ref{['eq:twohole']})]. Here, the hopping term again drives a strong resonance between a coherent Cooper‑pair component $|\Psi_{\mathrm{Cooper}}\rangle_{\mathrm{2h}}$ with $d_{x^2-y^2}$ pairing symmetry on the AFM background $|\phi_0\rangle$—a condensate of spin‑singlet pairs on opposite sublattices Liang1988—and an incoherent component $|\Psi_{\mathrm{inc}}\rangle_{\mathrm{2h}}$ with $d_{xy}$ pairing symmetry, accompanied by the same sublattice spin‑singlet pair (connected by the orange bond), whose amplitude is no longer sign-definite and depends on the hopping history of the holes.
• Figure 2: Single-particle spectral function $A^n(\mathbf{k},\omega)$ defined in Eq. (\ref{['eq:negbias']}) (with $\eta=0.15$). Intensity of the spectral function plotted along high-symmetry momentum directions in the Brillouin zone (scan paths indicated by yellow arrows in the inset) for energies $\omega<0$. Results are from VMC calculations on a $14\times 14$ system with $t/J = 2.5$, where the energy bottom is set to $\omega=0$. Crosses denote benchmark dispersion data from Green Function Monte Carlo simulations BONINSEGNI1994330. Inset: Spectral weight $Z_{\mathbf{k}}$ for the ground state, showing pronounced peaks at momenta $\mathbf{k}_0$. The white dashed lines indicate the magnetic Brillouin zone boundaries.
• Figure 3: Hole and spin currents for the degenerate one-hole ground state ($L_z = 1$ and $S^z =-\frac{1}{2}$). Calculated hole current $J^h$ (a, b) and total spin current $J^s_{\mathrm{tot}}$ (c, d) on an $8\times 8$ lattice. Results from DMRG (a, c) and VMC (b, d) are shown, respectively. Arrow thickness indicates relative current strength; hole and spin currents are displayed on separate scales.
• Figure 4: Magnetic moment $M_{\mathrm{loop}}$ of the loop current as a function of system size $L$. Inset: the energy changes, $\Delta E = E(\Phi_{\mathrm{p}}) - E(\Phi_{\mathrm{p}} =0)$, induced by the polarization of $M_{\mathrm{loop}}$ via magnetic flux $\Phi_{\mathrm{p}}$ on a $12\times 12$ system.
• Figure 5: (a, b) Hole–spin correlator $\langle n^h_{i} S^z_{j}\rangle$ and (c, d) neutral spin current pattern $\langle n^h_{i} J^s_{jk}\rangle$ for the degenerate ground state with quantum numbers $L_z = 1$, $S^z = -\frac{1}{2}$. The projected position of the hole $i$ is marked by a blue circle, and the results obtained with DMRG (left column) and VMC (right column) are comparatively shown. In (a, b), the direction and length of each arrow represent the sign and magnitude of the correlator. In (c, d), arrow thickness indicates the relative magnitude of the current. Panels (a, b) share a common scale, and panels (c, d) share a separate common scale. Note that the hole position is slightly off the center of the $6\times 6$ lattice, such that the $S^z = -\frac{1}{2}$ and spin current surrounding the hole look asymmetric.