# Griffiths Singularities in the Random Quantum Ising Antiferromagnet: A Tree Tensor Network Renormalization Group Study

## TL;DR

This study uses a tree tensor network renormalization group approach to analyze the quantum phase transitions and Griffiths singularities in a disordered antiferromagnetic quantum Ising chain, revealing an infinite-randomness critical point and the effects of longitudinal fields.

## Contribution

The paper introduces a new matrix product operator representation for high-order moments, enabling efficient detection of quantum phase transitions and Griffiths effects in disordered quantum chains.

## Key findings

- Identifies an infinite-randomness quantum critical point with large dynamical exponent z>1.
- Demonstrates pronounced Griffiths singularities due to rare regions with slow fluctuations.
- Shows that longitudinal fields suppress Griffiths effects and alter the phase transition dynamics.

## Abstract

The antiferromagnetic Ising chain in both transverse and longitudinal magnetic fields is one of the paradigmatic models of a quantum phase transition. The antiferromagnetic system exhibits a zero-temperature critical line separating an antiferromagnetic phase and a paramagnetic phase; the critical line connects an integrable quantum critical point at zero longitudinal field and a classical first-order transition point at zero transverse field. Using a strong-disorder renormalization group method formulated as a tree tensor network, we study the zero-temperature phase of the quantum Ising chain with bond randomness. We introduce a new matrix product operator representation of high-order moments, which provides an efficient and accurate tool for determining quantum phase transitions via the Binder cumulant of the order parameter. Our results demonstrate an infinite-randomness quantum critical point in zero longitudinal field accompanied by pronounced quantum Griffiths singularities, arising from rare ordered regions with anomalously slow fluctuations inside the paramagnetic phase. The strong Griffiths effects are signaled by a large dynamical exponent $z>1$, which characterizes a power-law density of low-energy states of the localized rare regions and becomes infinite at the quantum critical point. Upon application of a longitudinal field, the quantum phase transition between the paramagnetic phase and the antiferromagnetic phase is completely destroyed. Furthermore, quantum Griffiths effects are suppressed, showing $z<1$, when the dynamics of the rare regions is hampered by the longitudinal field.