Convergence of the Bethe-root series solution of the Heisenberg and XXZ spin chains in a magnetic field

The ground-state free energy and magnetization of the spin-1/2 Heisenberg and XXZ chains in a magnetic field can be written as series — in the deviation from saturation $m=\frac{1}{2}-\langle\sigma^z\rangle$ and in $\sqrt{h_c-h}$ respectively — all of whose coefficients are computable in closed form from a recurrence obtained by expanding each Bethe root as a double series in its Bethe number and $m$ (Granet, Lykke Jacobsen and Saleur, 2019). The construction is, however, established numerically rather than rigorously, and three foundational questions remain open. First, no general criterion is known for when the Bethe-root series converges: the derivation assumes the roots and their differences lie inside the disks of convergence of the functions $s$ and $r$, an assumption that is empirically violated while the free-energy series still converges, including for roots lying on a curve in the complex plane. Second, the radius of convergence of the magnetization series is not rigorously established for anisotropy $\Delta$ near 1; finite-order numerics (more than 50 terms) suggests it fails already at zero field ($h_r>0$) for the isotropic chain. Third, the identity $g^*=1/g$ between the two coefficient series of the $N^{-2}$ finite-size correction — the condition that the finite-size spectrum is a free compactified boson — is verified only order by order up to $m^{28}$.

Background

The spin-1/2 XXZ chain is the one-dimensional quantum spin Hamiltonian $H=\sum_i(\sigma_i^x\sigma_{i+1}^x+\sigma_i^y\sigma_{i+1}^y+\Delta\,\sigma_i^z\sigma_{i+1}^z)/4-h\sum_i\sigma_i^z$, with $\Delta$ the anisotropy; $\Delta=1$ gives the isotropic Heisenberg (XXX) chain. For $-1<\Delta\le 1$ the antiferromagnetic chain is gapless, and a magnetic field $h$ polarizes the ground state continuously up to the saturation field $h_c$, above which the ground state is fully polarized. The eigenstates are parametrized by rapidities $\lambda_i$ (one complex variable per spin flipped from the fully polarized reference state) solving the Bethe equations, and in the thermodynamic limit these roots condense on a density determined by a singular integral equation — the thermodynamic Bethe ansatz (TBA). The magnetization curve was traditionally accessible only by numerically solving the TBA equation or through Wiener-Hopf analysis, which is intractable over the full range of fields.

Granet, Lykke Jacobsen and Saleur (2019) found a different exact route. Writing the Bethe equations as $s(\lambda_p)=\frac{k}{N}+\frac{1}{N}\sum_j r(\lambda_p-\lambda_j)$, where $k$ is the Bethe number of the root $\lambda_p$ (the integer or half-integer quantum number labeling that root in the chosen eigenstate), $N$ is the chain length, and $s$ and $r$ are model-dependent functions analytic at the origin (for the isotropic chain $s(\lambda)=\frac{1}{\pi}\arctan 2\lambda$, with radius of convergence $1/2$, and $r(\lambda)=\frac{1}{\pi}\arctan\lambda$, with radius $1$), they expanded each root as a double series $\lambda_i=\sum_{a,b}c_{ab}\alpha_i^a m^b$ in its Bethe number and in $m=\frac{1}{2}-\langle\sigma^z\rangle$, the deviation from saturation. The Bethe equations then reduce to a nonlinear recurrence on the coefficients $c_{ab}$ involving only algebraic operations, from which the free energy at fixed magnetization, $F(m)$, is obtained as a series in $m$ with closed-form coefficients. Numerically this series converges over the whole physical range $0\le m\le\frac{1}{2}$ (observed radius $1/2$), and the acquired magnetization follows by Legendre transform as a series $\sigma^*$ in $\sqrt{h_c-h}$, converging for $h_r<h\le h_c$. At $\Delta=0$ one finds $h_r=-h_c$, and $h_r$ stays negative at least for $\Delta\lesssim 0.6$; but for $\Delta$ close to $1$ the truncated series keeps oscillating at small fields, and finite-order data (more than 50 terms) suggest $h_r>0$, i.e. failure of the magnetization series already at $h=0$.

The same recurrence yields the $N^{-2}$ finite-size corrections to the free energy. Under conformal invariance — the gapless chain being described by the $c=1$ Gaussian conformal field theory, a free compactified boson whose compactification radius encodes the Luttinger parameter (the dimensionless stiffness of the gapless one-dimensional mode), known exactly for the XXZ chain from Lukyanov's low-energy effective Hamiltonian — the correction for a state with integer quantum numbers $n$ and $n^*$ (labeling how the state differs from the ground state: $n$ roots removed and the sign of $n^*/2$ roots changed) takes the form $2\pi v_F(-\frac{c}{12}+\frac{n^2}{4}g+\frac{(n^*)^2}{4}g^*)$, with $v_F$ the velocity of the gapless excitations and $c$ the central charge, and where $g$ and $g^*$ are two further series in $m$; the spectrum is that of a free compactified boson precisely if $g^*=1/g$. This identity was checked order by order up to $m^{28}$ but not proven. Rigorous groundwork exists nearby: Kozlowski (2018) proved that the Bethe roots of the XXZ chain condense on an interval and that the counting function admits an all-order asymptotic expansion for any $\Delta\ge -1$, but this concerns the large-$N$ expansion and not the convergence of the series in $m$.

Problem Statement

Give the Bethe-root series solution of the Heisenberg and XXZ chains in a magnetic field a rigorous foundation by settling three linked questions about the construction of Granet-Lykke Jacobsen-Saleur (2019):

1. General convergence criterion. The derivation assumes that every root $\lambda_i$ and every difference $\lambda_i-\lambda_j$ lies inside the disk of convergence of $s$ and of $r$ respectively, yet the resulting free-energy series $F(m)$ is empirically convergent at values of $m$ for which this assumption fails, and the method also works when the roots lie on a curve in the complex plane (the chain with imaginary extensive twist $i\varphi$: an additive shift $i\varphi/\pi$ in the Bethe equations, realized by an inhomogeneous chain whose site-dependent couplings accumulate a total twist proportional to the chain length, which pushes the roots off the real axis). Establish a general criterion — in terms of the functions $s$ and $r$ and the root configuration, real or complex — under which the series for the roots and for the free energy converge, and determine the actual radius of convergence of $F(m)$.

2. Radius of the magnetization series near $\Delta=1$. Determine rigorously the radius of convergence of the magnetization series $\sigma^*(\sqrt{h_c-h})$ as a function of the anisotropy for $\Delta$ close to 1; in particular, decide whether $h_r>0$ for the isotropic Heisenberg chain — whether the series genuinely fails to converge at $h=0$, as finite-order numerics suggest — or whether the observed oscillations of the truncated series are compatible with convergence extending to zero field.

3. All-orders identity. Prove, or disprove, that $g^*=1/g$ at all orders in $m$: show that the two coefficient series of the $N^{-2}$ finite-size correction produced by the recurrence satisfy the exact relation required for the finite-size spectrum to be that of a free compactified boson, beyond the existing verification up to order $m^{28}$.

These three questions are components of a single objective, a convergence and structure theory for the Bethe-root series: (1) concerns the validity of the thermodynamic-limit construction itself, (2) the analyticity domain of its Legendre transform in the field, and (3) the conformal-field-theory structure of its finite-size sector. A complete answer must address all three.

The same source additionally proposes extending the method to nested Bethe-ansatz models (for example the Hubbard model) and to quantum transfer matrices, and exploiting its tree representation of the recurrence to obtain closed-form results. These are genuine open research directions, but they are deliberately not part of the completion boundary of the problem stated here.

References

1. E. Granet, J. Lykke Jacobsen, H. Saleur, "Analytical results on the Heisenberg spin chain in a magnetic field", J. Phys. A: Math. Theor. 52 (2019) 255302, DOI 10.1088/1751-8121/ab1f97, arXiv:1901.05878, https://arxiv.org/abs/1901.05878
2. E. Granet, J. Lykke Jacobsen, H. Saleur, "Analytic continuation of Bethe energies and application to the thermodynamic limit of the SL(2,C) non-compact spin chains", J. High Energ. Phys. 2020 (2020) 069, DOI 10.1007/JHEP08(2020)069, arXiv:2005.06030, https://arxiv.org/abs/2005.06030
3. K. K. Kozlowski, "On condensation properties of Bethe roots associated with the XXZ chain", Commun. Math. Phys. 357 (2018) 1009-1069, DOI 10.1007/s00220-017-3066-8, arXiv:1508.05741, https://arxiv.org/abs/1508.05741
4. S. Lukyanov, "Low energy effective Hamiltonian for the XXZ spin chain", Nucl. Phys. B 522 (1998) 533-549, DOI 10.1016/S0550-3213(98)00249-1, arXiv:cond-mat/9712314, https://arxiv.org/abs/cond-mat/9712314
5. Y. Liu, "Analytic asymptotics of the integrable XXX critical spin chains in the small external-field limit from TBA", arXiv:2310.15134, https://arxiv.org/abs/2310.15134