[Paper Review] For the Quantum Heisenberg Ferromagnet, a Polymer Expansion and its High T Convergence

Paul Federbush|ArXiv.org|Aug 2, 2001

Spectral Theory in Mathematical Physics2 references3 citations

TL;DR

This paper establishes a rigorous polymer expansion for correlation functions in the quantum Heisenberg ferromagnet at high temperatures, showing that the expectation values of spin products converge to those of an approximate product state governed by the lattice heat equation. The key result proves that the relative error between the exact and approximate expectations vanishes as $\mu \to 0$ faster than any power of $\mu^{1-\varepsilon}$, validating a conjecture from prior numerical work.

ABSTRACT

We let Psi_0 be a wave function for the Quantum Heisenberg ferromagnet sharp i sigma_zi and Psi_mu = exp(-mu*H)Psi_0. We study expectations similar to the form / for which we present a formal polymer expansion, whose convergence we prove for sufficiently small mu. The approach of the paper is to relate the wavefunction Psi_mu to an approximation to it that is a product function. In the jth spot of the product approximation the upper component is phi_mu(j), and the lower component is (1-phi_mu(j)), where phi satisfies the lattice heat equation. This is shown via a cluster or polymer expansion. The present work began in a previous paper, primarily a numerical study, and provides a proof of results related to Conjecture 3 of this previous paper.

Motivation & Objective

• To provide a mathematical proof for Conjecture 3 from a prior numerical study on the quantum Heisenberg ferromagnet.
• To establish the convergence of a formal polymer expansion for spin correlation functions at high temperatures.
• To rigorously relate the exact time-evolved state $\Psi_\mu = e^{-\mu H}\Psi_0$ to an approximate product state $\Psi^{AP}_\mu$ based on the lattice heat equation solution $\phi_\mu(i)$.
• To demonstrate that the approximation error in spin correlation expectations decays faster than any power of $\mu^{1-\varepsilon}$ as $\mu \to 0$.

Proposed method

• Formal derivation of a 'splitting' expansion for $\Psi_\mu$ using differential changes in spin wavefunctions under the action of $I_{ij} - 1$.
• Construction of a polymer expansion by decomposing the time evolution into contributions from connected clusters (polymers) of lattice sites.
• Use of a cluster expansion technique to express the exact expectation $\langle A \rangle_\mu$ as a sum over polymers, with weights derived from vertex and edge contributions.
• Introduction of a superpolymer formalism to handle overlapping contributions, with numerical factors $\frac{1}{N_v} \cdot \frac{1}{n}$ to control convergence.
• Application of smallness factors from vertex and edge interactions to bound the sum over polymer configurations, ensuring absolute convergence for small $\mu$.
• Use of spatial fall-off estimates and summation control via $g$-factors and $\mu$-ordering to manage long-range contributions.

Experimental results

Research questions

• RQ1 Does the approximate product state $\Psi^{AP}_\mu = \otimes_i (\phi_\mu(i), 1 - \phi_\mu(i))^T$ accurately capture the leading-order behavior of $\Psi_\mu = e^{-\mu H}\Psi_0$ in the high-temperature limit?
• RQ2 Can the polymer expansion for spin correlation functions be rigorously constructed and shown to converge for small $\mu$?
• RQ3 What is the rate of convergence of the approximation $\langle A \rangle_\mu \approx \frac{\langle \Psi^{AP}_\mu, A \Psi^{AP}_\mu \rangle}{\langle \Psi^{AP}_\mu, \Psi^{AP}_\mu \rangle}$ as $\mu \to 0$?
• RQ4 How do the contributions from connected clusters (polymers) of spins affect the overall expectation value, and can they be systematically bounded?

Key findings

• The polymer expansion for $\langle A \rangle_\mu$ converges absolutely for sufficiently small $\mu$, establishing a rigorous framework for high-temperature expansions.
• The leading-order term in the polymer expansion is $\prod_{i \in K} \rho_\mu(i)$, where $\rho_\mu(i)$ is a rational function of $\phi_\mu(i)$, the solution to the lattice heat equation.
• The error between the exact expectation and the approximate product-state expectation decays faster than any power of $\mu^{1-\varepsilon}$ as $\mu \to 0$, i.e., $\lim_{\mu \to 0} \mu^{\varepsilon-1} \left| \langle A \rangle_\mu - \frac{\langle \Psi^{AP}_\mu, A \Psi^{AP}_\mu \rangle}{\langle \Psi^{AP}_\mu, \Psi^{AP}_\mu \rangle} \right| = 0$ for any $\varepsilon > 0$.
• The convergence is achieved through a combination of vertex smallness, edge interaction control, and a systematic counting of polymer configurations using factors $\frac{1}{N_v} \cdot \frac{1}{n}$ to suppress overcounting.
• The method successfully handles type 1, 2, 3, and 4 lines in the polymer construction, with careful summation control via $g$-factor fall-off and $\mu$-ordering.
• The results generalize a weakened form of Conjecture 3 from the prior numerical study, providing a rigorous foundation for future extensions to more complex observables.