# Criticality of the Higgs mass for the long-range quantum $XY$ chain: Amplitude ratio between the Higgs and paramagnetic gaps

## TL;DR

This study investigates the Higgs mass and its ratio to the paramagnetic gap in a long-range quantum XY spin chain with power-law interactions, revealing how this ratio varies with interaction decay and connecting it to renormalization-group predictions.

## Contribution

It provides the first detailed analysis of the Higgs mass and amplitude ratio in a long-range quantum XY chain with biquadratic interactions, linking numerical results to RG theory.

## Key findings

- The Higgs-to-paramagnetic gap ratio increases with the decay exponent σ.
- Results agree with epsilon-expansion renormalization-group predictions.
- The study demonstrates the impact of long-range interactions on quantum critical properties.

## Abstract

The quantum $XY$ spin chain with the interactions decaying as a power law $1/r^{1+ \sigma}$ of the distance between spins $r$ was investigated with the exact diagonalization method. Here, the constituent spin is set to $S=1$, which enables us to incorporate the biquadratic interactions so as to realize the order-disorder transition with the O$(2)$ symmetry maintained. Thereby, in the ordered phase, we resolved the Higgs mass $m_H$ out of the Goldstone-excitation continuum by specifying Higgs-particle's quantum numbers to adequate indices. We then turn to the analysis of the critical amplitude ratio $m_H / \Delta$ ($\Delta$: paramagnetic gap in the disordered phase). As the power of the algebraic decay $\sigma$ increases, the amplitude ratio $m_H/\Delta$ gets enhanced gradually in agreement with the $\epsilon(=4-D)$-expansion-renormalization-group result; here, we resort to the $\sigma \leftrightarrow D$ relation advocated recently in order to establish a relationship between the renormalization-group result and ours.