Exact thermodynamic limit of the open one-dimensional Hubbard chain with boundary magnetic fields away from half filling

The one-dimensional Hubbard chain with magnetic fields acting on its two end sites is integrable for arbitrary, in particular nonparallel, boundary fields: the exact spectrum follows from an inhomogeneous Bethe-ansatz solution combining the coordinate Bethe ansatz with the off-diagonal Bethe ansatz. However, every exact thermodynamic-limit result for this model known to date, namely the root densities, ground-state energy density, surface energy, and the five-region classification of boundary bound states, was derived at half filling. The open problem is the zero-temperature thermodynamic limit away from half filling: classify the ground-state Bethe-root patterns as functions of electron density, on-site repulsion, and boundary-field strengths; derive the root densities, bulk energy density, and surface energy including boundary-string contributions; and determine under what conditions stable boundary bound states survive in the doped regime, where the spin sector is gapless and competes with the boundary-localized states.

Background

The Hubbard model is the minimal lattice model of correlated electrons: spinful fermions hop between neighboring sites and pay an energy $U>0$ when two occupy the same site. In one dimension it is exactly solvable by the Bethe ansatz (Lieb and Wu, 1968), an exact wave-function construction in which each energy eigenvalue is parameterized by Bethe roots, namely charge quasi-momenta $k_j$ and spin rapidities $\lambda_l$, determined by coupled nonlinear Bethe ansatz equations (BAEs). Macroscopic quantities are extracted from the distribution of these roots as the chain length $N$ tends to infinity (the thermodynamic limit), usually by taking logarithms of the BAEs and passing to linear integral equations for the root densities.

Replacing periodic by open boundary conditions introduces two boundary terms, here magnetic fields $\vec h_1$ and $\vec h_N$ acting on the end sites: Zeeman terms $h^z(n_\uparrow-n_\downarrow)$ plus spin-flip terms $h^\pm c^\dagger_\uparrow c_\downarrow$. If the two fields are parallel, the spin projection is conserved and the conventional Bethe ansatz applies. If they are nonparallel, the $U(1)$ spin symmetry is broken and no reference state — a simple product pseudo-vacuum from which the algebraic Bethe ansatz constructs eigenstates — exists, so the standard algebraic Bethe ansatz fails; the model was solved exactly only in 2013 (Li, Cao, Yang, Shi, and Wang) by solving the charge sector with the coordinate Bethe ansatz and mapping the spin sector to an inhomogeneous XXX spin chain (the isotropic Heisenberg spin chain, with site-dependent inhomogeneities inherited from the charge-sector quasi-momenta) solved by the off-diagonal Bethe ansatz (ODBA). ODBA is a method that determines eigenvalues through functional $T$-$Q$ relations, where $T(u)$ is the transfer matrix generating the commuting conserved quantities and $Q(u)$ encodes the Bethe roots, without requiring a reference state. The price is that the resulting BAEs are inhomogeneous: they contain an extra term that prevents the logarithmic thermodynamic limit, so the usual thermodynamic Bethe ansatz machinery does not apply directly.

For an open chain the ground-state energy decomposes as $E(N)=N e_{\text{bulk}}+e_{\text{surf}}+o(1)$; the surface energy $e_{\text{surf}}$ (order $N^0$) isolates the effect of the two boundary fields. A boundary bound state is an eigenstate localized at a chain end whose wavefunction decays exponentially into the bulk; in Bethe-ansatz language it is a complex root, or boundary string, pinned at a singular point of the BAEs, here $\lambda=i(1-|h|^2)/(2|h|)-iU/4$ for a boundary field of magnitude $|h|$.

At half filling (one electron per site) the repulsive model is a Mott insulator, that is, a state insulating because interaction-driven charge localization opens a gap in the charge sector rather than because of band filling, and this simplifies the analysis. Sun, Qiao, Yang, Cao, and Yang (2024) completed the half-filling program for nonparallel boundary fields: the ground-state root patterns fall into five regions according to the field magnitudes $\alpha=|\vec h_1|$ and $\beta=|\vec h_N|$ relative to two critical values, $h_0$ (defined by $(1-h_0^2)/(2h_0)=U/4$) and $1$; they computed the densities of states, the ground-state energy density, and the surface energy in each region, found stable boundary bound states whenever a boundary field exceeds $h_0$, verified the results against the density matrix renormalization group (DMRG, a variational method for one-dimensional quantum systems), and showed that the energies of chains with nonparallel and with the equivalent parallel fields agree up to order $N^0$, so that the surface energy depends only on the field magnitudes and the analysis can be carried out with the reduced Bethe ansatz equations, the homogeneous parallel-field counterparts of the inhomogeneous equations. They also observed numerically ($\delta e=\gamma N^\tau$ with $\tau<0$) that the inhomogeneous term contributes vanishingly in the thermodynamic limit, a fact proved for the XXX spin chain by fusion arguments (fusion being the construction of higher-spin transfer matrices whose functional relations constrain the inhomogeneous terms).

Earlier work on open Hubbard chains handled integrable boundary chemical potentials (coupling to the total end-site density), for which boundary bound states and boundary-state spectra were obtained at half filling (Tsuchiya and Yamamoto; Bedürftig and Frahm), and parallel boundary magnetic fields, for which BAEs, finite-size corrections, and boundary conformal field theory analyses were established (Asakawa and Suzuki; Shiroishi and Wadati; Deguchi and Yue); boundary conformal field theory describes critical one-dimensional systems with boundaries and fixes universal finite-size corrections. None of these works computed the surface energy or the boundary-bound-state structure of boundary magnetic fields away from half filling, where the charge gap closes and both charge and spin sectors are gapless.

Problem Statement

Determine the exact zero-temperature thermodynamic-limit physics of the open one-dimensional repulsive ($U>0$) Hubbard chain with boundary magnetic fields $\vec h_1$, $\vec h_N$ (the Hamiltonian of Li et al., Nucl. Phys. B 879 (2014) 98, Eq. (1.1)) at electron densities away from half filling, that is, with the electron number per site $n$ held fixed away from $1$ as the chain length $N\to\infty$. Concretely: (1) classify the ground-state Bethe-root patterns of the Bethe ansatz equations — the reduced (parallel-field) or inhomogeneous (nonparallel-field) equations of the solved model — as functions of the density $n$, the coupling $U$, and the boundary-field magnitudes $\alpha=|\vec h_1|$, $\beta=|\vec h_N|$ — the analogue of the five-region half-filling classification — and establish the status of the inhomogeneous term at finite doping (whether its contribution still vanishes at the orders that determine the bulk and surface energies); (2) from these patterns derive the root densities and compute the ground-state energy density (order $N$) and the surface energy (order $N^0$), including all boundary-string contributions, as explicit functions of $n$, $U$, $\alpha$, $\beta$; and (3) determine the conditions under which stable boundary bound states (boundary strings) exist away from half filling, where the spin sector is gapless and boundary-localized states compete with the continuum of bulk spinons (spinons being the gapless spin-carrying elementary excitations of the doped chain). All three sub-questions are jointly required for a resolution — the pattern classification and the finite-doping status of the inhomogeneous term in (1), the root densities together with the bulk energy density and surface energy in (2), and the boundary-bound-state existence condition in (3) — and (3) is an independent requirement, not a corollary of the pattern classification in (1). Answers must reduce, in the half-filling limit, to the known five-region classification and surface energies of Sun et al. (Phys. Rev. B 110 (2024) 035153), and should be checked against finite-size Bethe-ansatz solutions and/or density-matrix-renormalization-group computations.

References

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