[Paper Review] Thermodynamics of a Quantum Ising system coupled to a spin bath: Zero Temperature Results

R. D. McKenzie, P. C. E. Stamp|arXiv (Cornell University)|Nov 15, 2017

Quantum many-body systems60 references18 citations

TL;DR

This paper investigates the impact of a spin bath on a quantum Ising system at zero temperature using field-theoretic methods. It demonstrates that despite the formation of a gap in the Ising mode spectrum, a new hybridized mode between the Ising and bath spins fully softens at the quantum phase transition, confirming the persistence of the quantum Ising transition even in the presence of environmental coupling.

ABSTRACT

We study the effect of coupling a spin bath environment to a system which, at low energies, can be modeled as a quantum Ising system. A field theoretic formalism incorporating both thermal and quantum fluctuations is developed to derive results for the thermodynamic properties and response functions, both for a toy model and for the $LiHoF_4$ system, in which spin-8 electronic spins couple to a spin-$7/2$ nuclear spin bath: the phase transition then occurs in a system of electronuclear degrees of freedom, coupled by long-range dipolar interactions. The quantum Ising phase transition still exists, and one hybridized mode of the Ising and bath spins always goes soft at the transition.

Motivation & Objective

• To understand the effect of a spin bath environment on quantum Ising systems, particularly at zero temperature.
• To determine whether the quantum Ising phase transition survives coupling to a spin bath.
• To develop a field-theoretic framework that incorporates both quantum and thermal fluctuations in such systems.
• To apply the formalism to the LiHoF4 system, a real-world realization of a quantum Ising system with electronuclear degrees of freedom.
• To identify experimentally accessible signatures of the hybridized mode emerging at the quantum critical point.

Proposed method

• Develops a field-theoretic formalism to treat both quantum and thermal fluctuations in the coupled Ising-spin bath system.
• Introduces an auxiliary order parameter field to represent fluctuations, enabling a trace over all degrees of freedom except the order parameter.
• Derives an effective scalar field theory by integrating out the Ising and bath spins, capturing the combined quantum and environmental effects.
• Applies the Random Phase Approximation (RPA) to compute dynamic susceptibilities in both a toy model and the LiHoF4 system.
• Performs higher-order perturbative calculations up to fourth order in fluctuation fields to assess quantum corrections beyond RPA.
• Uses known microscopic parameters from the LiHoF4 Hamiltonian to construct a realistic model for quantitative predictions.

Experimental results

Research questions

• RQ1 Does the quantum Ising phase transition persist when coupled to a spin bath environment at zero temperature?
• RQ2 What is the nature of the collective mode that emerges at the quantum critical point in the presence of a spin bath?
• RQ3 How do quantum and thermal fluctuations combine to influence the critical behavior in such systems?
• RQ4 Can the hybridized mode between Ising and bath spins be probed experimentally, and what are its signatures?
• RQ5 How does the spin bath affect the magnetization and response functions near the quantum critical point?

Key findings

• The quantum Ising phase transition remains intact even when coupled to a spin bath, as evidenced by the full softening of a hybridized mode at the critical point.
• A new hybridized mode, formed from entanglement between Ising and bath spins, becomes gapless at the quantum critical point, signaling the transition.
• Despite the formation of a gap in the pure Ising mode spectrum, the system still exhibits a true quantum phase transition due to the softening of the hybridized mode.
• The magnetization and phase diagram are quantitatively modified by the spin bath, but the critical behavior is preserved.
• The hybridized mode is identifiable via NMR and other spectroscopic probes, as demonstrated in recent experiments on LiHoF4.
• Theoretical predictions are consistent with experimental observations of suppressed mode softening, now understood as a consequence of hybridization rather than absence of criticality.