# Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields

## TL;DR

This paper investigates how quantum fluctuations and disorder affect a spin-glass model with transverse and random longitudinal fields, revealing phase transitions, susceptibility behavior, and the impact of random fields on experimental observations in LiHo$_x$Y$_{1-x}$F$_4$.

## Contribution

It introduces a detailed analysis of nonlinear susceptibility in a quantum spin glass with both uniform transverse and random longitudinal fields, aligning theoretical results with experimental data.

## Key findings

- Spin-glass phase transition occurs at a finite temperature $T_f$.
- Increasing transverse field $\Gamma$ lowers $T_f$, approaching a quantum critical point.
- Random fields cause smearing of nonlinear susceptibility and define two distinct temperature regimes.

## Abstract

The interplay between quantum fluctuations and disorder is investigated in a spin-glass model, in the presence of a uniform transverse field $\Gamma$, and a longitudinal random field following a Gaussian distribution with width $\Delta$. The model is studied through the replica formalism. This study is motivated by experimental investigations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility $\chi_3$, which have led to a considerable experimental and theoretical debate. We analyzed two situations, namely, $\Delta$ and $\Gamma$ considered as independent, as well as these two quantities related as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature $T_f$; moreover, $T_f$ decreases by increasing $\Gamma$ towards a quantum critical point at zero temperature. The situation where $\Delta$ and $\Gamma$ are related appears to reproduce better the experimental observations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility. In this later case, by increasing $\Gamma$, $\chi_3$ becomes progressively rounded, presenting a maximum at a temperature $T^*$ ($T^*>T_f$). Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature $T^*$, one for $T_f<T<T^*$, and another one for $T>T^*$. It is argued that the conventional paramagnetic state corresponds to $T>T^*$, whereas the temperature region $T_f<T<T^*$ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.