Microscopic origin of the two-component internal structure of a single doped hole in two-leg Hubbard and $t$-$J$ ladders

A single hole doped into a two-leg Hubbard or $t$-$J$ ladder at isotropic couplings was shown by large-scale density-matrix renormalization group calculations to behave as a quasiparticle at long wavelengths, while its spin and charge components remain only loosely bound with a nontrivial mutual statistics (phase-string) structure inside the quasiparticle. The ground state was phenomenologically decomposed into a long-wavelength quasiparticle component and a short-range component attributed to this mutual statistics, which was argued to remove the nodes that a conventional quasiparticle with the same incommensurate dispersion would exhibit in its hole-density profile. What remains unresolved is the microscopic status of this two-component picture: a model-level derivation of the short-range component from the ladder Hamiltonian, a quantitative link between the phase-string sign structure and the absence of nodes, experimental identification of the predicted local wavefunction modification, and whether the same mechanism persists beyond two-leg ladders. Later work has produced variational and tensor-network constructions incorporating the phase string, sign-structure-switching numerics demonstrating its necessity, and dynamical confirmations in the optical response, but no controlled derivation, no experimental observation, and no independent settlement of the generality question; the problem remains open.

Background

The Hubbard model — electrons on a lattice with nearest-neighbor hopping and on-site repulsion $U$ — and its strong-coupling limit, the $t$-$J$ model (no double occupancy, with antiferromagnetic exchange $J$ between neighboring spins), are the minimal models of doped Mott insulators underlying the theory of cuprate superconductivity. A two-leg ladder is a strip of two coupled chains; its rungs and legs can carry different hoppings and exchanges, parameterized by an anisotropy $\alpha$, and the undoped ladder has a finite spin gap with a rung-singlet ground state (the two spins on each rung paired into a spin singlet). Doping a single hole into such a gapped antiferromagnetic ladder is among the simplest controlled settings in which to ask how a charge moves through a quantum spin background.

For a conventional Bloch-Landau quasiparticle, the doped hole would propagate coherently, with a finite quasiparticle weight $Z$ (the squared overlap of the doped ground state with a hole acting on the undoped state) and a smooth dispersion. Before 2016 two conflicting pictures existed. White, Scalapino, and Kivelson argued from density-matrix renormalization group (DMRG) calculations — a variational technique yielding essentially exact ground states of quasi-one-dimensional systems — that the doped two-leg $t$-$J$ ladder is adiabatically connected to a noninteracting band insulator, so the hole is a conventional quasiparticle. Zhu and Weng argued instead that above a critical anisotropy $\alpha_c$ the quasiparticle collapses: the charge self-localizes while a spin-$1/2$ excitation stays deconfined. Liu, Jiang, and Devereaux (2016) found an intermediate result: at isotropic coupling the hole is a quasiparticle at lengths $L$ much larger than the spin-charge correlation length $\xi$ (its energy gap scales as $1/L^2$), but at scales of order $\xi$ its spin and charge are only loosely bound, the density shows a charge modulation, and a 'residual effect' appears: the oscillating ground-state hole-density profile $\langle n_x^h\rangle$ never vanishes (it has no nodes), unlike the hole-density profile of a free-fermion band with the same incommensurate dispersion minimum $k_0$, which vanishes repeatedly — in the source's language, the nodes at $k\neq\pm k_0$ are absent.

The proposed explanation invokes the phase string, an exact sign structure of the $t$-$J$ model identified by Sheng, Chen, and Weng (1996): each hole hop that exchanges positions with spins accumulates a many-body phase, equivalent to a mutual Chern-Simons gauge field tying the charge sector to the spin sector — a 'mutual statistics' between the doped charge and the spins. In this language the ground state is decomposed phenomenologically as $|\psi_0\rangle = |\psi_0^L\rangle + |\psi_0^\xi\rangle$: a long-wavelength quasiparticle part plus a short-range part generated by the mutual statistics, the latter held responsible for destroying the nodes (the authors note the nodes cannot be lifted by any simple finite-ranged Wannier function). A useful computational probe is the $\sigma t$-$J$ model, in which the phase-string sign structure is switched off while the magnitude structure of the Hamiltonian is preserved, so its ground state serves as a reference isolating the effect of the sign structure. The 2016 study left open whether the two-component picture can be derived microscopically from the Hamiltonian and how the residual effect could be observed experimentally; its conclusion calls for future experiments 'to identify this nontrivial effect in other systems'.

Problem Statement

For a single hole doped into a two-leg Hubbard ladder or its strong-coupling $t$-$J$ counterpart (with or without the three-site correlated hopping terms) in the isotropic regime where the undoped ladder is spin-gapped, establish the microscopic origin of the quasiparticle's internal structure:

1. Derive from the Hamiltonian the two-component decomposition of the single-hole ground state, $|\psi_0\rangle = |\psi_0^L\rangle + |\psi_0^\xi\rangle$: give the split a precise meaning — the long-wavelength component $|\psi_0^L\rangle$ governing scales $L\gg\xi$ and the short-range component $|\psi_0^\xi\rangle$ at scales of order $\xi$, proposed only phenomenologically in the source — rather than assuming it; characterize $|\psi_0^\xi\rangle$ at the model level, including whether its structure is the proposed mutual-statistics (phase-string) one; and demonstrate quantitatively which microscopic mechanism generates this component and removes the nodes at $k\neq\pm k_0$ (the zero crossings exhibited by the free-fermion reference profile with the same $k_0$) from the oscillating ground-state hole-density profile, discriminating the phase-string sign structure from conventional alternatives such as a spread finite-ranged Wannier function.

2. Identify the observable consequences of the short-range component (the 'residual effect' on the local wavefunction) and specify how they can be detected experimentally — in materials or cold-atom/synthetic-lattice realizations — in a way that distinguishes the mutual-statistics structure from a conventional quasiparticle.

These two components express one common objective — replacing the phenomenological two-component description of the doped hole with a microscopic, testable account of the quasiparticle's internal structure. Whether the same mechanism extends to other ladder geometries, other interaction strengths, or higher dimensions is an active related direction, but it is not part of the problem posed here.

References

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