Spectral Gap in the Ferromagnetic Heisenberg Spin-1⁄2 Quantum Spin Chain

Colby Austin Brown

Published Dec 1, 2019• 0 citations

Abstract

The ferromagnetic Heisenberg spin-1⁄2 quantum spin chain describes a quantum spin system of spin-1⁄2 particles on a one-dimensional lattice. In the thermodynamic limit, the spectral gap of this system closes at rate of O(n−2). We demonstrate this by proving an upper bound on the spectral gap using the variational principle and a lower bound using the martingale method.

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OpenAlex

Abstract

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z 1

under summability hypotheses, and highlight quote ``the problem of determining the optimal values of

z

'' quote as a function of the interaction decay, as well as the broader open problem of developing direct gap methods for long-range frustration-free systems. We resolve a precise special case for a non-abelian, genuinely quantum model. For every

N 2

and

>0

define the long-range ferromagnetic Heisenberg Hamiltonian on

N

qubits equation* H_N^() = _1 j3$ (pure-power

z=2

). All assumptions are listed explicitly. The identity

(H_N^())=_2(M_N^())

is machine-checked against full many-body diagonalisation for all

N 8

and a grid of $$ values (scripts/verify\_spectrum.py). Supplementary materials: LaTeX source, verification scripts, and unit tests for the long-range ferromagnetic Heisenberg spectral gap results.

The mean-field Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin-1/2 interactions. This non-commuting Hamiltonian is permutation and

\mathsf{SU}(2)

invariant, and its Gibbs states undergo an

\mathsf{SU}(2)

symmetry breaking phase transition at inverse temperature

β=2

. We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight asymptotic estimates for its spectral gap at all noncritical temperatures. For fixed

β<2

, the gap as a function of number of qubits

n

Θ(1)

, while for fixed

β>2

the gap is

Θ(n^{-1})

. The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature (

β>2

) slowdown is associated with broken continuous symmetry. Two key ingredients in our approach are a comparison argument, which introduces auxiliary generators to bound dissipation on nontrivial representations of the symmetry groups

\mathsf{SU}(2)

and

\mathsf{S}_n

, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.

We prove rigorous lower and upper bounds for the mass gap of the ferromagnetic spin 1/2 XXZ chain. The two bounds coincide asymptotically in the Ising limit

Δ\to\infty

. Near the isotropic point,

Δ=1

, the estimates are good enough to determine the critical behaviour of the mass gap unambiguously. The derivation does not rely on exact solutions.

\mathrm{XY}

chain, for which an exact solution is found.