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Journal of Physics A Mathematical and Theoretical

Semiclassical study of single-molecule magnets and their quantum phase transitions

D.S. Lohr-Robles, Enrique López‐Moreno, Peter Oto Hess

Published Nov 16, 2023• 3 citations
Publisher

Abstract

Abstract We present a study on systems of single-molecule magnet systems using semiclassical analysis and catastrophe theory. Separatrices in the parameter space are constructed, which are useful for determining the structure of the Hamiltonian energy levels. In particular the Maxwell set separatrix determines the behaviour of the ground state of the system. We consider an external magnetic field with two components: one parallel to the easy magnetisation axis of the molecule and the other perpendicular to it. Using fidelity and heat capacity we were able to detect the signals of the quantum phase transitions as a function of the magnetic field components.

DOI
10.1088/1751-8121/ad0d1f

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Abstract We present a study on systems of single-molecule magnet systems using semiclassical analysis and catastrophe theory. Separatrices in the parameter space are constructed, which are useful for determining the structure of the Hamiltonian energy levels. In particular the Maxwell set separatrix determines the behaviour of the ground state of the system. We consider an external magnetic field with two components: one parallel to the easy magnetisation axis of the molecule and the other perpendicular to it. Using fidelity and heat capacity we were able to detect the signals of the quantum phase transitions as a function of the magnetic field components.
A general spin system with a nonaxially symmetric Hamiltonian containing Jx, Jz-linear and Jz-quadratic terms, widely used in many-body fermionic and bosonic systems and in molecular magnetism, is considered for the variations of general parameters describing intensity interaction changes of each of its terms. For this model Hamiltonian, a semiclassical energy surface (ES) is obtained by means of the coherent-state formalism. An analysis of this ES function, based on catastrophe theory, determines the separatrix in the control parameter space of the system Hamiltonian: the loci of singularities representing semiclassical phase transitions. Here we show that distinct regions of qualitatively different spectrum structures, as well as a singular behavior of quantum states, are ruled by this separatrix: here we show that the separatrix not only describes ground-state singularities, which have been associated with quantum phase transitions, but also reveals the structure of the excited spectrum, distinguishing different quantum phases within the parameter space. Finally, we consider magnetic susceptibility and heat capacity of the system at finite temperature, in order to study thermal properties and thermodynamical phase transitions in the perspective of the separatrix of this Hamiltonian system.
Abstract The dynamical properties of the Lipkin–Meshkov–Glick model with longitudinal and transverse fields are studied by considering quantum quenches. The model describes a many-body quantum system with long-range interactions. A semiclassical analysis is performed to obtain the critical points of quantum phase transitions. These critical points consist in the static separatrix, which is found by analyzing the stability structure of the semiclassical potential and is useful in the identification of ground state and excited states quantum phase transitions, and in the dynamical separatrix, which depends on initial conditions and is found by studying the classical equations of motion for the Hamiltonian. The static and dynamical separatrices divide the parameter space in regions of different dynamical behavior, and it is found that both separatrices play an important role in the determination of the occurrences of the dynamical quantum phase transitions. Graphical abstract
The characterization of the material properties of Single Molecule Magnets (SMMs) has grown in importance over the last few decades with the rise of novel applications such as high-density magnetic storage and quantum computation. Many of the applications require the probing of SMMs with spectroscopic methods that make use of electromagnetic radiation. The interaction with these timedependent fields leads to energy shifts, which can be attributed to the geometric phase acquired by the system or the Bloch-Siegert shift. We model an SMM by a giant spin Hamiltonian, and use Floquet perturbation theory to find the geometric phase shifts. The locations where the phase shift between two levels is zero is useful for performing accurate spectroscopies, whereas the regions where relative phase differences exist are useful in applications like quantum computing. Using the same giant spin Hamiltonian, we can use Floquet theory and Salwen perturbation theory to determine the Bloch-Siegert shift and derive a modified version of the Rabi formula for transition probabilities between the energy states Eα → Eα±1, Eα → Eα±3, and Eα → Eα±5, where α is the index of an arbitrary initial state. The shifted eigenvalues and modified transition probabilities can be useful in spectroscopies where accurate values for the energy-splitting between magnetic states needs to be determined.
The systems exhibiting quantum phase transitions (QPT) are investigated within the Ising model in the transverse field and Heisenberg model with easy-plane single-site anisotropy. Near QPT a correspondence between parameters of these models and of the quantum ${\ensuremath{\varphi}}^{4}$ model is established. A scaling analysis is performed for the ground-state properties. The influence of the external ``longitudinal'' magnetic field on the ground-state properties is investigated, and the corresponding magnetic susceptibility is calculated. Finite-temperature properties are considered with the use of the scaling analysis for the effective classical model proposed by Sachdev. Analytical results for the ordering temperature and temperature dependences of the magnetization and energy gap are obtained in the case of a small ground-state moment. The forms of dependences of observable quantities on the bare splitting (or magnetic field) and renormalized splitting turn out to be different. A comparison with numerical calculations and experimental data on systems demonstrating magnetic and structural transitions (e.g., into a singlet state) is performed.
Abstract Motivated by recent experimental progress on the quasi-one-dimensional quantum magnet NiNb2O6, we study the spin dynamics of an S = 1 ferromagnetic Heisenberg chain with single-ion anisotropy by using a semiclassical molecular dynamics approach. This system undergoes a quantum phase transition from a ferromagnetic to a paramagnetic state under a transverse magnetic field, and the magnetic response reflecting this transition is well described by our semiclassical method. We show that at low temperature the transverse component of the dynamical structure factor depicts clearly the magnon dispersion, and the longitudinal component exhibits two continua associated with single- and two-magnon excitations, respectively. These spin excitation spectra show interesting temperature dependence as effects of magnon interactions. Our findings shed light on the experimental detection of spin excitations in a large class of quasi-one-dimensional magnets.
We present a study of a general spin Hamiltonian with terms up to fourth order. With the coherent states the semiclassical potential is obtained and with catastrophe theory its parameter space is constructed. When the fourth order parameters are large enough the parameter space has regions where the semiclassical potential has three wells. By applying an oscillating magnetic field a trajectory in parameter space crosses the Maxwell set multiple times resulting in many ground state quantum phase transitions. Using the coherent states we are able to visualize the localization of the ground state on the Bloch sphere as the magnetic field is varied.