Finite-temperature thermodynamics and finite-size structure of the one-dimensional Hubbard model with nonparallel boundary magnetic fields

The one-dimensional half-filled Hubbard chain with integrable nonparallel boundary magnetic fields - boundary fields at the two chain ends that are not collinear and therefore break the U(1) symmetry associated with conserved spin projection - was solved exactly by the off-diagonal Bethe ansatz, and its zero-temperature thermodynamic limit was analyzed by Sun, Qiao, Yang, Cao and Yang (Phys. Rev. B 110, 035153 (2024)): the ground-state Bethe-root patterns fall into five regions of the boundary field magnitudes, the ground-state energy density and surface energy were computed exactly, and stable boundary bound states (boundary strings) were identified. That work explicitly left open the extension beyond this zero-temperature, leading-order result: the finite-temperature thermodynamics (free energy and its boundary contribution) of the model, for which the standard thermodynamic Bethe ansatz is blocked by the inhomogeneous source term in the Bethe ansatz equations; the subleading finite-size corrections, including the $O(N^{-1})$ boundary terms in which nonparallel fields first differ from equivalent parallel fields; and the characterization of the finite-N solution structure of the full inhomogeneous Bethe ansatz equations. This problem asks for these three linked extensions, which together would give complete thermodynamic control of the model.

Background

The Hubbard model describes interacting electrons on a lattice: electrons hop between neighboring sites with amplitude $t$, and two electrons occupying the same site repel with energy $U$. In one spatial dimension with periodic boundary conditions the model is exactly solvable by the Bethe ansatz, which represents each eigenstate as a superposition of plane waves whose quasi-momenta $k_j$ (and spin rapidities $\lambda_j$) are fixed by a coupled system of algebraic equations, the Bethe ansatz equations (BAEs), as shown by Lieb and Wu in 1968. Its finite-temperature thermodynamics in the thermodynamic limit is likewise exactly known: in the thermodynamic Bethe ansatz (TBA) the rapidities form continuous densities together with strings (bound-state complexes of rapidities sharing a real center), and the free energy follows from nonlinear integral equations supplemented by an entropy term, as systematized for the Hubbard and related chains by Takahashi.

The integrable open chain is obtained by adding magnetic fields $\mathbf{h}_1$ and $\mathbf{h}_N$ acting on the electron spin at the first and last sites. If the two fields are parallel, the number of electrons with a fixed spin projection remains conserved (a $U(1)$ symmetry, the phase-rotation group associated with that conserved number), the boundary reflection matrices are diagonal, and the boundary energies and boundary bound states were worked out by standard methods in the 1990s. If the fields are nonparallel (not collinear), the boundary reflection mixes spin projections, $U(1)$ is broken, and the conventional Bethe ansatz fails because no reference state (a trivial vacuum about whose occupation pattern the wavefunction is organized) exists. The off-diagonal Bethe ansatz (ODBA) of Cao, Yang, Shi and Wang (2013) bypasses the reference state by using functional relations of the transfer matrix (the generating function of the model's conserved quantities); it yielded the exact spectrum of the Hubbard chain with arbitrary boundary fields (Li, Cao, Yang, Shi and Wang, 2014) in terms of inhomogeneous BAEs: compared with the parallel case, they contain an extra additive term proportional to $c=2(\varepsilon|\mathbf{h}_1||\mathbf{h}_N|-\mathbf{h}_1\cdot\mathbf{h}_N)$, where $\varepsilon$ is the sign of $\mathbf{h}_1\cdot\mathbf{h}_N$. This term measures the non-collinearity of the two fields and vanishes only when they are parallel; it cannot be treated by taking logarithms, which is the step on which the TBA rests, so computing thermodynamic quantities remained an open task.

Sun, Qiao, Yang, Cao and Yang (2024) analyzed the thermodynamic limit at half filling (one electron per site). By finite-size scaling with density matrix renormalization group (DMRG, the standard numerical method for one-dimensional quantum systems) cross-checks, they showed that the total-energy difference $\delta_e=|E-E_{hom}|$ between the full inhomogeneous BAEs and their homogeneous part decays as a power law $\gamma N^{\tau}$ with $\tau<0$, so that the reduced (homogeneous) BAEs determine all $O(N^0)$ energies. The ground-state Bethe-root patterns fall into five regions of the boundary field magnitudes $\alpha=|\mathbf{h}_1|$ and $\beta=|\mathbf{h}_N|$ (up to exchanging the two ends), delimited by two critical values: $|\mathbf{h}|=1$, at which the corresponding boundary factors cancel from the spin-sector BAEs, and $|\mathbf{h}|=h_0$, where $h_0$ solves $(1-h_0^2)/(2h_0)=U/4$, so that $h_0\to1$ as $U\to0$ and $h_0\to0$ as $U\to\infty$ ($h_0=0.726$ at $U=1.3$). In several regions the spin rapidities include purely imaginary roots - boundary strings, exponentially localized boundary bound states such as $\lambda_M=i[(1-\beta^2)/(2\beta)-U/4]$ in one of the regions - and these boundary bound states are stable. From the root densities the authors computed the ground-state energy density and the surface energy $e_b=\lim_{N\to\infty}(E_g-Ne_\infty)$, with $e_\infty$ the ground-state energy density of the periodic chain, finding agreement with DMRG. A notable outcome is that to order $O(N^0)$ the energies coincide with those of parallel fields of the same magnitudes: nonparallelism first manifests at order $O(N^{-1})$, which the paper identifies as calculable but leaves uncomputed, and the eigenstates are helical (the local spin-polarization direction rotates along the chain) rather than uniformly spin-polarized. The analogous surface-energy problem for the XXZ spin chain with nondiagonal boundary fields was solved by Qiao, Cao, Yang, Shi and Wang (2021), where helical spinons appear; the present model is the interacting-fermion counterpart.

The conclusion of Sun et al. states that further exploration of these results at finite temperatures, together with the finite-size effect and the patterns of the inhomogeneous BAEs, are open future research topics. These three directions are the subject of this problem: no finite-temperature TBA or equivalent construction exists for this model, the finite-N corrections beyond the power-law decay of $\delta_e$ are unknown, and the finite-N solution structure of the inhomogeneous BAEs - how the inhomogeneous term modifies root patterns, excited states, and finite-size spectra - is uncharacterized.

Problem Statement

For the one-dimensional half-filled Hubbard chain with integrable nonparallel boundary magnetic fields $\mathbf{h}_1$ and $\mathbf{h}_N$ - the model and conventions of Sun et al. (Phys. Rev. B 110, 035153 (2024)), with repulsive $U$, hopping set to $t=1$, and the off-diagonal Bethe ansatz solution in terms of charge rapidities $\{k_j\}$ and spin rapidities $\{\lambda_j\}$ satisfying inhomogeneous BAEs whose source term is proportional to $c=2(\varepsilon|\mathbf{h}_1||\mathbf{h}_N|-\mathbf{h}_1\cdot\mathbf{h}_N)$ - extend the exact thermodynamic analysis beyond the zero-temperature leading-order surface energy by resolving the three linked open directions identified in that work:

(1) Finite-temperature thermodynamics. Derive the free energy per site and its boundary (surface) contribution at $T>0$ for the five boundary-parameter regions, by any exact or controlled method - for example a thermodynamic Bethe ansatz or nonlinear integral equation approach adapted to the reduced BAEs and their boundary strings, a quantum transfer matrix (an auxiliary transfer matrix whose leading eigenvalue gives the thermal free energy) or a $t$-$W$-relation construction (functional relations of the transfer matrix from which the free energy follows without a reference state) extended to non-diagonal boundary reflections, or an equivalent scheme (a chemical potential may be introduced in the standard way if the treatment leaves half filling). The result must recover the known $T\to0$ surface energy in each region, the parallel-field limit in which $c=0$, and the vanishing-field free open-chain limit.

(2) Finite-size structure. Compute the subleading finite-$N$ corrections to the ground-state energy beyond the fitted power-law decay $\delta_e=\gamma N^{\tau}$ of the inhomogeneous-term contribution - in particular the $O(N^{-1})$ boundary terms in which nonparallel fields first differ from equivalent parallel fields of the same magnitudes - as functions of $U$ and the boundary parameters, including their dependence on the region.

(3) Inhomogeneous-BAE solution structure. Characterize the solutions of the full inhomogeneous BAEs at finite $N$: the root configurations (including how the inhomogeneous term deforms the five ground-state patterns), and how it modifies excited states and finite-size spectra, beyond its established vanishing contribution to the leading ground-state energy.

These deliverables jointly realize one scientific objective - thermodynamic control of the model beyond the leading zero-temperature order - because the inhomogeneous source term is the common obstruction: it is what blocks the standard TBA in (1), it generates the first finite-$N$ terms where nonparallel and parallel fields differ in (2), and it is precisely the object to be characterized in (3).

All three numbered directions are required, with partial results in any one of them counting as partial progress rather than as a resolution; in particular, (3) is resolved only by an analytic, string-hypothesis-level characterization of the deformed finite-$N$ root patterns and of the modified excited states and finite-size spectra, not merely by numerical samples of root configurations at selected $N$.

References

1. Pei Sun, Yi Qiao, Tao Yang, Junpeng Cao, Wen-Li Yang, "Exact surface energy of the Hubbard model with nonparallel boundary magnetic fields", Physical Review B 110, 035153 (2024). DOI: 10.1103/PhysRevB.110.035153 (arXiv:2401.14356). https://doi.org/10.1103/PhysRevB.110.035153
2. Yuan-Yuan Li, Junpeng Cao, Wen-Li Yang, Kangjie Shi, Yupeng Wang, "Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic fields", Nuclear Physics B 879, 98-109 (2014). DOI: 10.1016/j.nuclphysb.2013.12.004 (arXiv:1311.0432). https://doi.org/10.1016/j.nuclphysb.2013.12.004
3. Elliott H. Lieb, F. Y. Wu, "Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension", Physical Review Letters 20, 1445-1448 (1968). DOI: 10.1103/PhysRevLett.20.1445. https://doi.org/10.1103/PhysRevLett.20.1445
4. Junpeng Cao, Wen-Li Yang, Kangjie Shi, Yupeng Wang, "Off-Diagonal Bethe Ansatz and Exact Solution of a Topological Spin Ring", Physical Review Letters 111, 137201 (2013). DOI: 10.1103/PhysRevLett.111.137201. https://doi.org/10.1103/PhysRevLett.111.137201
5. Yi Qiao, Junpeng Cao, Wen-Li Yang, Kangjie Shi, Yupeng Wang, "Exact surface energy and helical spinons in the XXZ spin chain with arbitrary nondiagonal boundary fields", Physical Review B 103, L220401 (2021). DOI: 10.1103/PhysRevB.103.L220401. https://doi.org/10.1103/PhysRevB.103.L220401
6. Minoru Takahashi, "Thermodynamics of One-Dimensional Solvable Models", Cambridge University Press (1999). DOI: 10.1017/cbo9780511524332. https://doi.org/10.1017/cbo9780511524332
7. Pengcheng Lu, Yi Qiao, Junpeng Cao, Wen-Li Yang, Kangjie Shi, Yupeng Wang, "T-W relation and free energy of the Heisenberg chain at a finite temperature", Journal of High Energy Physics 2021, 133 (2021). DOI: 10.1007/JHEP07(2021)133 (arXiv:2105.01352). https://doi.org/10.1007/JHEP07(2021)133
8. Pengcheng Lu, Junpeng Cao, Wen-Li Yang, Ian Marquette, Yao-Zhong Zhang, "T-W relation and free energy of the antiperiodic XXZ chain with η=iπ/3 at a finite temperature", Journal of Physics A: Mathematical and Theoretical 57, 445205 (2024). DOI: 10.1088/1751-8121/ad85b3 (arXiv:2402.04849). https://doi.org/10.1088/1751-8121/ad85b3
9. Zhouzheng Ji, Pei Sun, Xiaotian Xu, Yi Qiao, Junpeng Cao, Wen-Li Yang, "Tensor-Network Analysis of Root Patterns in the XXX Model with Open Boundaries", arXiv:2512.24182 (2025). https://arxiv.org/abs/2512.24182
10. Renjie Song, Mingchen Zheng, Junpeng Cao, Yupeng Wang, "An integrable Anderson-impurity problem embedded in the one-dimensional Hubbard model", arXiv:2509.07799 (2025). https://arxiv.org/abs/2509.07799