Quantum Criticality in the infinite-range Transverse Field Ising Model

TL;DR

This paper investigates quantum criticality in the infinite-range Transverse-Field Ising Model, revealing differences from mean-field theory in gap behavior, entanglement, and quantum fluctuations, using numerical diagonalization techniques.

Contribution

It provides new insights into quantum critical phenomena in the infinite-range model, highlighting the role of symmetries and entanglement near the critical point.

Findings

Excitation gap closes at the quantum critical point from both sides.

Quantum Fisher Information becomes large at the critical point, indicating long-range entanglement.

Low energy quantum fluctuations are confined near the critical field and persist at finite temperatures.

Abstract

We study quantum criticality in the infinite range Transverse-Field Ising Model. We find subtle differences with respect to the well-known single-site mean-field theory, especially in terms of gap, entanglement and quantum criticality. The calculations are based on numerical diagonalization of Hamiltonians with up to a few thousand spins. This is made possible by the enhanced symmetries of the model, which divide the Hamiltonian into many block-diagonal sectors. The finite temperature phase diagram and the characteristic jump in heat capacity closely resemble the behavior in mean-field theory. However, unlike mean-field theory where excitations are always gapped, the excitation gap in the infinite range model goes to zero from both the paramagnetic side and from the ferromagnetic side on approach to the quantum critical point. Also, contrary to mean-field theory, at the quantum critical point the Quantum Fisher Information becomes large, implying long-range multi-partite entanglement. We find that the main role of temperature is to shift statistical weights from one conserved sector to another. However, low energy excitations in each sector arise only near the quantum critical point implying that low energy quantum fluctuations can arise only in the vicinity of the quantum critical field where they can persist up to temperatures of order the exchange constant.