Analytical results on the Heisenberg spin chain in a magnetic field

Abstract

We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field $h$ as a series in $\sqrt{h_c-h}$, where $h_c$ is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. The radius of convergence of the series in the full range $0<h\le h_c$ is studied numerically. To that end we express the free energy at mean magnetization per site $-1/2\le ⟨{\sigma }_i^z⟩\le 1/2$ as a series in $1/2-⟨{\sigma }_i^z⟩$ whose coefficients can be similarly recursively computed in closed form. This series converges for all $0\le ⟨{\sigma }_i^z⟩\le 1/2$. The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in $1/2-⟨{\sigma }_i^z⟩$. It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose radius can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.

20 pages, 5 figures; minor changes and typos corrected

Cited by in corpus (9)

• Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions
• HamLib: A library of Hamiltonians for benchmarking quantum algorithms and hardware
• Kondo effect in the isotropic Heisenberg spin chain
• Discretized quantum adiabatic process for free fermions and comparison with the imaginary-time evolution
• Exact time evolution formulae in the XXZ spin chain with domain wall initial state
• A Comprehensive Cross-Model Framework for Benchmarking the Performance of Quantum Hamiltonian Simulations
• Unveiling chiral states in the XXZ chain: Finite-size scaling probing symmetry-enriched $c=1$ conformal field theories
• Analytic continuation of Bethe energies and application to the thermodynamic limit of the $SL(2,\mathbb{C})$ non-compact spin chains
• Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium