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Fluctuations and phase transitions in quantum Ising systems

R. H. McKenzie

Published Jan 1, 2016• 1 citation
Publisher

Abstract

The quantum Ising model is perhaps the simplest possible model of a quantum magnetic material. Despite its simplicity, its versatility and wide range of applications, from quantum computation, to combinatorial optimization, to biophysics, make it one of the most important models of modern physics. In this thesis, we develop a general framework for studying quantum Ising systems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantum fluctuations, and the quantum phase transition between paramagnetic and ferromagnetic states that occurs when a magnetic field is applied transverse to the easy axis of the system. The magnetic insulating crystal LiHoF₄ is a physical realization of the quantum Ising model, with the additional features that the dominant coupling between spins is the long range dipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom. These nuclear degrees of freedom constitute a spin bath environment acting on the system. In this thesis, we present an effective low temperature Hamiltonian for LiHoF₄ that incorporates both these features, and we analyze the effects of the nuclear spin bath on the system. We find the lowest energy crystal field excitation in the system is gapped at the quantum critical point by the presence of the nuclear spins, with spectral weight being transferred down to a lower energy electronuclear mode that fully softens to zero at the quantum critical point. Furthermore, we present a toy model, the spin half spin half model, that illustrates the effects of an anisotropic hyperfine interaction on a quantum Ising system. We find the critical transverse field is increased when the longitudinal hyperfine coupling is dominant, as well as an enhancement of both the longitudinal electronic susceptibility and an applied longitudinal field. In addition, we present a field theoretic formalism for incorporating the effects of fluctuations beyond the random phase approximation in general quantum Ising systems. We find that any regular on site interaction, such as a nuclear spin bath, does not fundamentally alter the critical properties of a quantum Ising system. This formalism is used to calculate corrections to the magnetization of LiHoF₄.

DOI
10.14288/1.0314166

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The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic \nfield B_x perpendicular to the Ising spin direction has long been used \nto study tunable quantum phase transitions in pure and random disordered systems. \nWe first present analytical and numerical evidences for the validity of an effective spin-1/2 \napproach to the description of a general dipolar spin \nglass model with strong uniaxial Ising anisotropy and subject to weak \nB_x. \n We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. \n We show that an effective spin-1/2 \n model is able to capture both the qualitative and quantitative aspects of the \nphysics at small B_x. \nAfter confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the \nx direction, \ncombined with the local random dilution-induced \ndestruction of crystalline mirror symmetries \ngenerates, via the predominant dipolar interactions between Ho^{3+} ions, \n random fields along the Ising z direction. \nThis identifies LiHo_xY_{1-x}F_4 in B_x as a new \nrandom field Ising system. \nWe show that the random fields explain the smearing \nof the nonlinear susceptibility at the spin glass transition \nwith increasing B_x. \nIn this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing \nMonte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical \npoint, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on \nweak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant \nexpansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective \nperturbative classical Hamiltonian, on which MC simulations are performed. With this method we \ninvestigate different proposed sources of uncertainty which can affect the numerical results. \n We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that \n none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small \nsemi-classical B_x regime.

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The concept of T = 0 quantum phase transi-tions has emerged as an overarching theme in strongly correlated electron physics [1–9]. The nature of quantum fluctuations near the quantum critical point, however, remains enigmatic [10]. How well does the quantum criticality account for finite temperature properties? How high in tem-perature does the effect of the quantum critical point persist?[10, 11] The dearth of appropriate model materials has not permitted experimental-ists to address these fundamental questions even for the transverse field Ising chain (TFIC), a cel-ebrated textbook example of quantum criticality [6]. Here we use NMR (Nuclear Magnetic Res-onance) to map the evolution of quantum spin fluctuations in CoNb2O6 [12–14], which has re-cently been proposed as an ideal model material for the TFIC [15, 16]. We will demonstrate that the effect of the quantum critical point persists to a remarkably high temperature, T ∼ 0.4J, where the exchange coupling J is the only energy scale in the problem.

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The quantum Ising model is perhaps the simplest possible model of a quantum magnetic material. Despite its simplicity, its versatility and wide range of applications, from quantum computation, to combinatorial optimization, to biophysics, make it one of the most important models of modern physics. In this thesis, we develop a general framework for studying quantum Ising systems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantum fluctuations, and the quantum phase transition between paramagnetic and ferromagnetic states that occurs when a magnetic field is applied transverse to the easy axis of the system. The magnetic insulating crystal LiHoF₄ is a physical realization of the quantum Ising model, with the additional features that the dominant coupling between spins is the long range dipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom. These nuclear degrees of freedom constitute a spin bath environment acting on the system. In this thesis, we present an effective low temperature Hamiltonian for LiHoF₄ that incorporates both these features, and we analyze the effects of the nuclear spin bath on the system. We find the lowest energy crystal field excitation in the system is gapped at the quantum critical point by the presence of the nuclear spins, with spectral weight being transferred down to a lower energy electronuclear mode that fully softens to zero at the quantum critical point. Furthermore, we present a toy model, the spin half spin half model, that illustrates the effects of an anisotropic hyperfine interaction on a quantum Ising system. We find the critical transverse field is increased when the longitudinal hyperfine coupling is dominant, as well as an enhancement of both the longitudinal electronic susceptibility and an applied longitudinal field. In addition, we present a field theoretic formalism for incorporating the effects of fluctuations beyond the random phase approximation in general quantum Ising systems. We find that any regular on site interaction, such as a nuclear spin bath, does not fundamentally alter the critical properties of a quantum Ising system. This formalism is used to calculate corrections to the magnetization of LiHoF₄.
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 LiHoF4{\text{LiHoF}}_{4}LiHoF4​ system, in which spin-8 electronic spins couple to a spin-7/27/27/2 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.
LiHoF _{4} 4 is a magnetic material known for its Ising-type anisotropy, making it a model system for studying quantum magnetism. However, the theoretical description of LiHoF _{4} 4 using the quantum Ising model has shown discrepancies in its phase diagram, particularly in the regime dominated by thermal fluctuations. In this study, we investigate the role of off-diagonal dipolar terms in LiHoF _{4} 4 , previously neglected, in determining its properties. We analytically derive the low-energy effective Hamiltonian of LiHoF _{4} 4 , including the off-diagonal dipolar terms perturbatively, both in the absence and presence of a transverse field. Our results encompass the full phase diagram, confirming the significance of the off-diagonal dipolar terms in reducing the zero-field critical temperature and determining the critical temperature’s dependence on the transverse field. We also highlight the sensitivity of this mechanism to the crystal structure by comparing our calculations with the Fe _{8} 8 system.
Two objects are entangled when their quantum mechanical wavefunctions cannot be written in a separable product form. Entangling dissimilar quantum objects, or hybridization, has been suggested as a promising route to efficient quantum information processors, but mostly realized on a limited scale. Hybrid nuclear-electronic many-body systems remain a largely unexplored challenge to both experiments and theories. The prototypical transverse-field Ising ferromagnet LiHoF4 is an ideal platform to address this issue. The Ising model is considered as an archetype both for the investigation of quantum criticality and for the evaluation of quantum simulators. The hyperfine coupling strength of a Ho ion is exceptionally large, promoting a strong hybridization or entanglement between the nuclear and electronic moments. The magnetic coupling between the Ho ions that leads to ferromagnetic ordering is predominantly through long-range dipole interactions, while nearest-neighbor exchange interaction is negligibly weak. Applying a transverse field induces a zero temperature quantum phase transition driven by quantum fluctuations. Altogether LiHoF4 represents a unique nuclear-electronic quantum magnet, whose wavefunctions can be readily obtained by diagonalizing the Hamiltonian using the mean-field approximation. In this thesis we develop an experimental setup to probe the entangled nuclear-electronic states in a model transverse-field Ising system LiHoF4. Using magnetic resonance the field and temperature evolution of the nuclear-electronic states are successfully traced across the whole phase diagram. We develop a theoretical framework based on mean-field calculations which provides close agreement with the experimental observations. Having established experimentally that the mean-field wavefunctions are an excellent approximation of the actual wavefunction, we used them to calculate the ground-state entanglement entropy between the electronic and nuclear magnetic moments. We find that the entanglement entropy between the nuclear and electronic moments exhibits a peak at the quantum phase transition. This suggests that the electronic entanglement is encoded onto each nuclear-electronic state. Our results pave the way for new theoretical and experimental investigations of quantum entanglement.
The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic \nfield B_x perpendicular to the Ising spin direction has long been used \nto study tunable quantum phase transitions in pure and random disordered systems. \nWe first present analytical and numerical evidences for the validity of an effective spin-1/2 \napproach to the description of a general dipolar spin \nglass model with strong uniaxial Ising anisotropy and subject to weak \nB_x. \n We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. \n We show that an effective spin-1/2 \n model is able to capture both the qualitative and quantitative aspects of the \nphysics at small B_x. \nAfter confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the \nx direction, \ncombined with the local random dilution-induced \ndestruction of crystalline mirror symmetries \ngenerates, via the predominant dipolar interactions between Ho^{3+} ions, \n random fields along the Ising z direction. \nThis identifies LiHo_xY_{1-x}F_4 in B_x as a new \nrandom field Ising system. \nWe show that the random fields explain the smearing \nof the nonlinear susceptibility at the spin glass transition \nwith increasing B_x. \nIn this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing \nMonte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical \npoint, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on \nweak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant \nexpansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective \nperturbative classical Hamiltonian, on which MC simulations are performed. With this method we \ninvestigate different proposed sources of uncertainty which can affect the numerical results. \n We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that \n none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small \nsemi-classical B_x regime.
For a system at absolute zero temperature, all thermal fluctuations are frozen out, while quantum fluctuations prevail.These microscopic quantum fluctuations can induce a quantum phase transition (QPT) in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value.This paper presents a detailed and quantitative study of the magnetic properties of Ising two-spin system at zero temperature in the presence of longitudinal, transverse and mixed fields, and also at finite temperatures.At the critical longitudinal magnetic field, the system is found to undergo a QPT to a ferromagnetic state.For this system, an infinitesimal departure from zero transverse field is found to trigger off QPT and also remove the degeneracies that persisted even in the presence of longitudinal field.These quantum fluctuations, leading to QPT are more evident in the mixed field than the separate effect of longitudinal and transverse fields.The observed quantum effects at zero temperature for this system are found to be completely wiped out at finite temperatures due to thermal fluctuations.A temperature increase favours antiferromagnetic alignment of spins, while a decrease in temperature favours ferromagnetic alignment.
The concept of T = 0 quantum phase transi-tions has emerged as an overarching theme in strongly correlated electron physics [1–9]. The nature of quantum fluctuations near the quantum critical point, however, remains enigmatic [10]. How well does the quantum criticality account for finite temperature properties? How high in tem-perature does the effect of the quantum critical point persist?[10, 11] The dearth of appropriate model materials has not permitted experimental-ists to address these fundamental questions even for the transverse field Ising chain (TFIC), a cel-ebrated textbook example of quantum criticality [6]. Here we use NMR (Nuclear Magnetic Res-onance) to map the evolution of quantum spin fluctuations in CoNb2O6 [12–14], which has re-cently been proposed as an ideal model material for the TFIC [15, 16]. We will demonstrate that the effect of the quantum critical point persists to a remarkably high temperature, T ∼ 0.4J, where the exchange coupling J is the only energy scale in the problem.