[Paper Review] Expansions for one quasiparticle states in spin 1/2 systems

Nilanjana Datta, Tom Kennedy|arXiv (Cornell University)|Apr 11, 2001

Physics of Superconductivity and Magnetism9 references4 citations

TL;DR

This paper develops a convergent expansion method for one-quasiparticle states in quantum spin-1/2 systems, extending the Kirkwood-Thomas approach beyond the Perron-Frobenius condition. It proves the dispersion relation for a single quasiparticle has a convergent power series in the coupling strength, with the minimum energy occurring at momentum $k = (\pi,\dots,\pi)$ for $\epsilon > 0$. The method relies on momentum-based ansatz and fixed-point equations for expansion coefficients, ensuring convergence via contraction mapping arguments.

ABSTRACT

Convergent expansions of the wavefunctions for the ground state and low-lying excited states of quantum transverse Ising systems are obtained. These expansions are employed to prove that the dispersion relation for a single quasi-particle has a convergent expansion.

Motivation & Objective

• To extend the Kirkwood-Thomas method for ground states to low-lying excited states in quantum spin-1/2 systems, removing the need for a Perron-Frobenius condition.
• To develop a convergent expansion for one-quasiparticle states using a momentum-based ansatz, enabling rigorous analysis of their spectral properties.
• To prove that the dispersion relation for a single quasiparticle has a convergent power series expansion in the coupling parameter $\epsilon$.
• To establish the existence of the infinite-volume limit for the quasiparticle wavefunctions and their associated energy dispersion.

Proposed method

• A momentum-based ansatz is used to construct quasiparticle states, assuming they transform as plane waves under spatial translations.
• The quasiparticle wavefunction is expanded as $\psi_k^\alpha(\sigma) = \psi(\sigma) \sum_X e_\alpha(X) \phi_{X,k}(\sigma)$, with coefficients $e_\alpha(X)$ satisfying a fixed-point equation.
• The fixed-point equation for $e_\alpha(X)$ is derived from the Schrödinger equation and solved using the contraction mapping theorem, ensuring convergence for small $\epsilon$.
• The method employs a norm on the space of functions on finite sets, with bounds $|e_s| \leq (|\epsilon|M)^{|s|} ||e||$, ensuring exponential decay and convergence.
• The infinite-volume limit is established by showing the difference between finite-volume and infinite-volume solutions decays exponentially with system size $L$.
• The dispersion relation $\Delta(k)$ is computed via the expansion, with the first-order term $2\epsilon \sum_{i=1}^d \cos(k_i)$ and higher-order corrections of order $\epsilon^2$.

Experimental results

Research questions

• RQ1 Does the dispersion relation for a single quasiparticle in a quantum spin-1/2 system admit a convergent power series expansion in the coupling strength $\epsilon$?
• RQ2 Can the Kirkwood-Thomas method be generalized to excited states without requiring the Perron-Frobenius condition on local interactions?
• RQ3 Where does the minimum of the dispersion relation occur for the transverse Ising model, and how does it depend on the sign of $\epsilon$?
• RQ4 Are the lowest excited states characterized by definite momentum, and can they be constructed via a convergent expansion?
• RQ5 How do the quasiparticle wavefunctions and their energy spectrum behave in the infinite-volume limit?

Key findings

• The dispersion relation $\Delta(k)$ has a convergent expansion in powers of $\epsilon$, with the first-order term $2\epsilon \sum_{i=1}^d \cos(k_i)$.
• For $\epsilon > 0$, the minimum of the dispersion relation occurs at $k = (\pi, \dots, \pi)$, and this remains true for small $\epsilon$ despite higher-order corrections.
• The coefficients $e_s$ in the expansion are of order $|\epsilon|^{|s|}$, and the full solution $e_\alpha(X)$ is locally unique via the contraction mapping theorem.
• The infinite-volume limit of the quasiparticle wavefunctions exists and is exponentially localized, with the difference from finite-volume versions decaying as $e^{-cL}$.
• The second eigenfunction $\psi'$ is shown to be a multiple of $\psi_{k_0}$, confirming that the constructed states are indeed the lowest excited states.
• The method applies to Hamiltonians without the Perron-Frobenius condition, such as $H = -\sum_i \sigma^x_i + \epsilon \sum_{\langle ij\rangle} (\sigma^z_i\sigma^z_j + J\sigma^y_i\sigma^y_j)$ with $J \neq 1$.