Dimensional crossover of weakly coupled Hubbard chains: charge-gap closing, spinon-confinement dynamics, and the ordering threshold beyond the quantum Monte Carlo window

In the half-filled Hubbard model on an anisotropic square lattice — chains coupled by a weak interchain hopping $t_\perp$ with a frustrating diagonal hopping $t'=-t_\perp/4$ — determinant quantum Monte Carlo (DQMC) combined with stochastic analytic continuation showed in 2013 that spinons confine into magnons as the chains are coupled, while the charge sector remained gapped throughout the numerically accessible window ($U/t=3$, $t_\perp/t\le 0.35$, $T\ge t/20$); beyond that window the sign problem blocked everything. The open problem asks what happens outside that window: whether the charge gap closes at a finite critical interchain hopping and where/how the dimensional-crossover-driven insulator-to-metal transition occurs in the $(U,t_\perp)$ plane; how the spinon confinement/deconfinement crossover and the magnon decay channels evolve when $U/t$ is varied substantially away from 3; the fate and spectral weight of the longitudinal spin excitation near the disorder transition under weak frustration; and whether antiferromagnetic order appears at $T\to0$ for arbitrarily weak interchain coupling at stronger frustration. Later cluster mean-field and narrow-cylinder studies settle the qualitative existence of a $t_\perp$-driven gap closing at weak-to-moderate coupling, but the quantitative phase boundary for this model and all spin-dynamics facets remain open.

Background

In a strictly one-dimensional (1D) electronic system at one electron per site and a half-filled band, onsite Coulomb repulsion $U$ produces a Mott insulator: umklapp scattering — processes that transfer electrons between the two Fermi points, allowed at half filling — is relevant for any repulsive $U$ and opens a gap $\Delta$ in the charge sector. This is the 1D half-filled Hubbard model. Its spin sector stays gapless and critical (a Tomonaga-Luttinger liquid, in which correlations decay as power laws), and a spin flip fractionalizes into two spinons, independent spin-1/2 collective excitations; the dynamical spin structure factor $S(\mathbf{q},\omega)$ is therefore a broad two-spinon continuum rather than sharp magnon modes.

Coupling many such chains by a weak interchain hopping $t_\perp$ drives a dimensional crossover from 1D to quasi-two-dimensional (quasi-2D) behavior. This crossover is the theoretical core of quasi-1D organic conductors such as the Bechgaard-Fabre salts, which undergo an insulator-to-metal transition as pressure effectively increases $t_\perp$. On a square lattice with hopping $t$ along the chains and $t_\perp$ between them, the half-filled Fermi surface is perfectly nested — parallel Fermi-surface segments are connected by a single wavevector, which maximizes the antiferromagnetic instability — so any nonzero $t_\perp$ is expected to produce antiferromagnetic (AF) long-range order at zero temperature; adding a diagonal next-nearest-neighbor hopping $t'=-t_\perp/4$ removes part of the nesting and introduces geometric frustration, making the crossover nontrivial and closer to the real materials.

Raczkowski and Assaad (2013) studied this model with determinant quantum Monte Carlo (DQMC — a finite-temperature auxiliary-field method that is numerically exact up to statistical and time-discretization errors) combined with stochastic analytic continuation (SAC — a probabilistic inversion of imaginary-time data to real frequencies). Because the diagonal hopping $t'$ triggers the fermion sign problem (exponential growth of statistical noise with system size and inverse temperature), they could reach only $U/t=3$, $t_\perp/t\le0.35$, and $T\ge t/20$ on lattices up to $20\times20$. Within this window they found: (a) the charge sector stays gapped (the system remains insulating); (b) the static spin structure factor supports the direct onset of AF long-range order at the smallest $t_\perp$ studied; (c) in $S(\mathbf{q},\omega)$ a low-energy transverse magnon — a bound pair of confined spinons, well described by linear spin-wave theory (the standard $1/S$ expansion about the ordered state) — coexists with a higher-energy two-spinon continuum whose weight fades as $t_\perp$ grows; and (d) the magnon decays into pairs of spinons, while at $t_\perp/t=0.3$, where the magnon bandwidth approaches the charge gap, decay into charge excitations was suggested as an additional channel.

Two further elements set the open questions. First, quasi-1D antiferromagnets near a magnetic disorder transition are predicted, from the quantum sine-Gordon effective theory of weakly coupled spin chains — the low-energy field theory in which weak interchain couplings lock the staggered magnetization and turn its amplitude fluctuations into a massive excitation (Schulz 1996; Essler, Tsvelik, and Delfino 1997) — to host a longitudinal spin excitation — an anomalous mode corresponding to fluctuations of the staggered-magnetization amplitude, observed by neutron scattering in KCuF$_3$ (Lake, Tennant, and Nagler 2000); the 2013 study computed only transverse modes. Second, cluster mean-field studies of the same model family (Raczkowski and Assaad 2012) had proposed a dimensional-crossover-driven Mott transition at larger $t_\perp$, so the persistence of the charge gap seen by DQMC could be a finite-window artifact. The paper's conclusions thus leave a precisely bounded set of unfinished questions: everything outside the sign-problem-accessible window.

Problem Statement

For the half-filled Hubbard model on an anisotropic square lattice with intrachain hopping $t$, interchain hopping $t_\perp$, and frustrating diagonal next-nearest-neighbor hopping $t'=-t_\perp/4$ (together with its stronger-frustration generalizations, in which the diagonal hopping is set independently of the interchain coupling, e.g. at fixed $t'/t=-0.25,-0.5,-0.75,\dots$ while $t_\perp/t$ is varied), determine what happens beyond the sign-problem-accessible window ($t_\perp/t\le0.35$, $T\ge t/20$) of the 2013 DQMC study at $U/t=3$, namely:

1. Charge sector: does the charge gap close at a finite critical interchain hopping $t_{\perp,c}$ at $U/t=3$, and what are the location and the nature (continuous versus first-order; Mott-Heisenberg — interaction-driven gap opening with simultaneous antiferromagnetic ordering — versus magnetically, i.e. Slater-like, driven, where the gap is a consequence of the ordering itself) of the dimensional-crossover-driven insulator-to-metal transition in the $(U/t,t_\perp/t)$ plane in the thermodynamic quasi-2D limit?
2. Spin dynamics versus interaction: how do the spinon confinement/deconfinement crossover in the dynamical spin structure factor $S(\mathbf{q},\omega)$ and the magnon decay channels (decay into two-spinon continua, and into charge excitations once the magnon bandwidth reaches the charge gap) evolve when $U/t$ is varied substantially above and below the studied value of 3?
3. Longitudinal mode: what is the fate and spectral weight of the longitudinal spin excitation predicted near the magnetic disorder transition, in the presence of weak frustration (the 2013 results cover transverse modes only)?
4. Ordering threshold: in these stronger-frustration regimes, where frustration is fixed independently of the interchain coupling, what is the $T\to0$ behavior — does antiferromagnetic long-range order appear for arbitrarily small interchain coupling, or only above a nonzero critical $t_\perp$, and which ordered or incommensurate states intervene?

These four facets jointly express one objective: a controlled determination of the ground-state phase diagram and of the spin and charge excitation spectra of weakly coupled, weakly frustrated half-filled Hubbard chains across the full one-dimensional-to-two-dimensional dimensional crossover. Answers may come from any controlled or systematically extrapolable method (sign-problem-free or sign-problem-tolerant quantum Monte Carlo, density-matrix renormalization group on cylinders with width extrapolation, tensor networks, or controlled weak-coupling interchain treatments), provided they are benchmarked against the 1D limit and the already-published data inside the accessible window.

References

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