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Journal of Physics A Mathematical and Theoretical

Spin Hamiltonian with large fourth order terms: triple well potentials and Bloch sphere visualization

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

Published Dec 19, 2024• 2 citations
Publisher

Abstract

Abstract We present a study of a general spin Hamiltonian with terms up to fourth order. The semiclassical potential is obtained as the expectation value of the parameter dependent Hamiltonian in the coherent state basis, and using catastrophe theory its parameter space is constructed. Considering the components of the magnetic field as the free parameters, we found regions where the semiclassical potential has three stability wells when the parameters of the fourth order terms are large. Using the coherent states we are able to visualize the localization of the ground state eigenfunction on the Bloch sphere as the magnetic field is varied.

DOI
10.1088/1751-8121/ada1b3

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Abstract We present a study of a general spin Hamiltonian with terms up to fourth order. The semiclassical potential is obtained as the expectation value of the parameter dependent Hamiltonian in the coherent state basis, and using catastrophe theory its parameter space is constructed. Considering the components of the magnetic field as the free parameters, we found regions where the semiclassical potential has three stability wells when the parameters of the fourth order terms are large. Using the coherent states we are able to visualize the localization of the ground state eigenfunction on the Bloch sphere as the magnetic field is varied.
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.
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 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.
Based on the correspondence between the conventional Poincaré and Bloch spheres, we proposed a higher-order Bloch spheres, which is an extension of the conventional Bloch spheres. By calculating the expectation value of the spin components using the Pauli spin matrices, we found that the new spin state is formed in a ring shape and the orientation of the spin changes depending on the azimuthal angle and topological charge of orbital angular momentum. We also realize the coherent transfer of the azimuth-dependent polarization state of photons to the electron spin state in a GaAs quantum well with a V-shaped three-level system.
Using the similarity between spin states on the Bloch sphere (BS) and polarization states on the Poincare sphere (PS), we construct higher-order spin states on the higher-order BS corresponding to higher-order polarization states of photons on the higher-order PS. We investigate the time evolution of higher-order spin states in a magnetic field and establish an extended form of the conventional Larmor precession. The results provide insights on coherent transfer from extended photons to extended spin qubit systems with spin and orbital angular momenta and the operation of extended spin qubits.