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Low energy spectrum of the XXZ model coupled to a magnetic field

Simone Del Vecchio, Jürg Fröhlich, Alessandro Pizzo, Alessio Ranallo

TL;DR

The paper proves that a broad class of short-range perturbations of the Ising chain, including the XXZ model in a magnetic field, maintains a strictly positive spectral gap uniformly in system size when the transverse coupling is sufficiently small. The authors develop a refined, locally supported Lie-Schwinger block-diagonalization flow that enlarges intervals to manage boundary effects and hooked terms arising from non-ultralocal unperturbed parts, enabling precise control of the low-energy spectrum. Key contributions include a robust treatment of boundary-induced level splittings in the antiferromagnetic case, a priori bounds on the norms of emergent potentials, and uniform gap estimates for both ferromagnetic and antiferromagnetic regimes under open boundary conditions. The approach advances rigorous understanding of stability of spectral gaps in 1D spin chains with symmetry-breaking ground states and has implications for connections to LTQO-like behavior and potential fermionic mappings via Jordan-Wigner transformations.

Abstract

For a class of Hamiltonians of $XXZ$ spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length.

Low energy spectrum of the XXZ model coupled to a magnetic field

TL;DR

The paper proves that a broad class of short-range perturbations of the Ising chain, including the XXZ model in a magnetic field, maintains a strictly positive spectral gap uniformly in system size when the transverse coupling is sufficiently small. The authors develop a refined, locally supported Lie-Schwinger block-diagonalization flow that enlarges intervals to manage boundary effects and hooked terms arising from non-ultralocal unperturbed parts, enabling precise control of the low-energy spectrum. Key contributions include a robust treatment of boundary-induced level splittings in the antiferromagnetic case, a priori bounds on the norms of emergent potentials, and uniform gap estimates for both ferromagnetic and antiferromagnetic regimes under open boundary conditions. The approach advances rigorous understanding of stability of spectral gaps in 1D spin chains with symmetry-breaking ground states and has implications for connections to LTQO-like behavior and potential fermionic mappings via Jordan-Wigner transformations.

Abstract

For a class of Hamiltonians of spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length.
Paper Structure (20 sections, 13 theorems, 140 equations, 5 figures, 1 table)

This paper contains 20 sections, 13 theorems, 140 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definition of the model
  3. Local Hamiltonians
  4. Statement of the main result and organization of the paper
  5. Proof strategy
  6. Local interaction terms and projections
  7. Outline of the block-diagonalization flow
  8. "Hooked" unperturbed terms.
  9. Degeneracy of the "bulk" ground-state energy level in the antiferromagnetic case
  10. The block-diagonalization algorithm
  11. Conjugation formulae
  12. The algorithm: definition and consistency
  13. Operator norms and control of the flow
  14. Gap estimates
  15. Spectral Gap of $G_{\mathcal{I}^*_{+1}}$ in the ferromagnetic case
  16. ...and 5 more sections

Key Result

Proposition 1.1

Under the assumption that $h$ and $J$ are positive, the Hamiltonians $H^0_{\mathcal{I}}$ and $H^C_{\mathcal{I}}$ have only one ground-state, denoted $\Psi_{\mathcal{I}}$, corresponding to the vector Moreover, under the condition that $\frac{J}{h}+\frac{3}{2}< \text{card}(\mathcal{I})$, the spectral gaps above the ground-state energies of the Hamiltonians $H^0_{\mathcal{I}}$ and $H^C_{\mathcal{I}}

Figures (5)

  • Figure 1: The picture illustrates the interval $\mathcal{I}$, with $Q(\mathcal{I})=J$ and $\ell(\mathcal{I})=1$, and the subsequent one, $\mathcal{I}_{+1}$ .
  • Figure 2: The picture displays the relation between $\mathcal{I}$, with $Q(\mathcal{I})=J$ and $\ell(\mathcal{I})=1$, and $\mathcal{I}^\ast$.
  • Figure 3: The picture displays the relation between $\mathcal{I}^\ast$ and $\overline{\mathcal{I}^*}$.
  • Figure 4: The picture shows how $\mathcal{I}$, with $Q(\mathcal{I})=J$ and $\ell(\mathcal{I})=1$, relates to $\widetilde{\mathcal{I}^\ast}$.
  • Figure 5: On the left, we display the relation between Theorem \ref{['th-norms']} and Lemma \ref{['control-LS']} in the inductive step of the block-diagonalization procedure. On the right, we show how Lemma \ref{['control-LS']} implies S2) in Theorem \ref{['th-norms']} by means of Lemmata \ref{['gap-bound-ferr']} and \ref{['gap-bound']}.

Theorems & Definitions (37)

  • Proposition 1.1
  • Proposition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 27 more