# Longitudinal and transverse spectral functions in the three-dimensional O(4) model

## TL;DR

This study uses high-statistics simulations and maximum entropy analysis to explore the spectral functions of the 3D O(4) model, revealing sharp peaks and continuum states across different temperature regimes, and confirming theoretical relations.

## Contribution

It provides detailed spectral function analysis of the 3D O(4) model near criticality using maximum entropy, and confirms the Patashinskii-Pokrovskii relation.

## Key findings

- Transverse spectral function shows a sharp peak at all T and H.
- Longitudinal spectral function exhibits a peak at T>T_c that broadens near T_c.
- Below T_c, a continuum of states appears at higher energies.

## Abstract

We have performed a high statistics simulation of the O(4) model on a three-dimensional lattice of linear extension L=120 for small external fields H. Using the maximum entropy method we analyze the longitudinal and transverse plane spin correlation functions for T<T_c and T>=T_c. In the transverse case we find for all T and H a single sharp peak in the spectral function, whose position defines the transverse mass m_T, the correlator is that of a free particle with mass m_T. In the longitudinal case we find in the very high temperature region also a single sharp peak in the spectrum. On approaching the critical point from above the peak broadens somewhat and at T_c its position m_L is at 2m_T for all our H-values. Below T_c we find still a significant peak at omega=2m_T and at higher omega-values a continuum of states with several smaller peaks with decreasing heights. This finding is in accord with a relation of Patashinskii and Pokrovskii between the longitudinal and the transverse correlation functions. We test this relation in the following. As a by-product we calculate critical exponents and amplitudes and confirm our former results.