Orbital and spin dynamics in geometrically frustrated lattices

Pal, Santanu (2019) Orbital and spin dynamics in geometrically frustrated lattices. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

This thesis reports the studies of various aspects of orbital and spin dynamics in strongly correlated geometrically frustrated systems. Frustration leads very naturally to macroscopic degeneracy in the ground state and a concomitant lack of equilibrium spin ordering. A large number of theoretical as well as experimental studies of several systems with different types of geometrically frustrated lattices (e.g., kagome, pyrochlore etc.) have sought novel ground states such as spin liquids and spin ice, as well as states possessing topological order and fractionalized excitations. Another interesting aspect of geometrically frustrated spin systems is that they can possess non-trivial plateaus at zero and fractional magnetisation. The existence of such plateaus indicates a finite gap in the energy spectrum and the possibility of ground states with non-trivial topological features analogous to the quantum Hall effects. In our work we discuss how geometry and topology play important roles in deciding ground state and excited state properties of such systems. In doing so, we extend the use of existing theoretical techniques in some cases, while for others, we develop new analytical methods. For instance, we extend the superexchange formalism to the case of strongly correlated electrons in spinel lattices. Further, we develop spectral flow-based twist operator technique to the kagome, triangular and pyrochlore Heisenberg spin-1/2 antiferromagnets, as well as formulate a renormalisation group (RG) formalism for the case of magnetisation plateaus in the kagome XXZ spin-1/2 antiferromagnet. Besides, we occasionally employ numerical methods suited for the problem at hand in order to check the predictions obtained from our analytical calculations. The following are the precise questions whose answers we attempt in this thesis. (i) How do orbital dynamics and a spin-lattice coupling help in deciding the ground state properties of geometrically frustrated spin systems? How does the answer to this question shape the phenomenology of a material system possessing the spinel lattice? (ii) Can we formulate arguments along the lines of the Lieb-Schultz-Mattis (LSM) theorem and the Oshikawa-Yamanaka-Affleck (OYA) criterion for frustrated spin systems on the kagome and triangular lattices in predicting the ground state properties at zero and finite external magnetic fields? (iii) What are the properties of the 1/3 magnetization plateau state of the S= 1/2 XXZ antiferromagnet Hamiltonian on the kagome lattice? How many different phases can exist at 1/3 magnetization for various Ising and XY exchange interaction strengths? Can we characterise the properties of the different phases by a renormalization group study? (iv) Last but not least: can we employ symmetry-based arguments (i.e., based on twist operator and spin-parity operator calculations) to predict possible magnetization plateaus ground states of the S= 1/2 Heisenberg antiferromagnet on the pyrochlore lattice? The first chapter of the thesis is an introductory chapter, where we present a brief review of the present understanding and developments in the field of strongly correlated electrons and quantum magnetism in geometrically frustrated lattices. We also provide a discussion of the basic aspects of various analytical tools and techniques we have used in this thesis. In the second and third chapters, we present a comprehensive theoretical study of the geometrically frustrated strongly correlated magnetic insulator Mn3O4 spinel oxide based on a microscopic Hamiltonian involving lattice, spin, and orbital degrees of freedom. Possessing the physics of degenerate eg orbitals, this system shows a strong Jahn-Teller effect at high temperatures. Further, careful attention is paid to the special nature of the superexchange physics arising from the 90° Mn-O-Mn bonding angle. Motivated by recent experiments, the Jahn-Teller and superexchange-based orbital-spin Hamiltonians are then analyzed in order to track the dynamics of orbital and spin ordering. We find that a high-temperature structural transition results in orbital ordering the nature of which is mixed with respect to the two originally degenerate eg orbitals. This ordering of orbitals is shown to relieve the intrinsic geometric frustration of the spins on the spinel lattice, leading to ferrimagnetic Yafet-Kittel ordering at low temperatures. Finally, we develop a model for a magnetoelastic coupling in Mn₃O₄, enabling a systematic understanding of the experimentally observed complexity in the low-temperature structural and magnetic phenomenology of this spinel. Our analysis predicts that a quantum fluctuation-driven orbital-spin liquid phase may be stabilized at low temperatures upon the application of pressure. In chapter four, we present the formulation of a twist operator argument for the geometrically frustrated quantum spin systems on the kagome and triangular lattices. We have, thereby, extended the application of the Lieb-Schultz-Mattis (LSM) and Oshikawa-Yamanaka-Affleck (OYA) theorems to these systems. The equivalent large gauge transformation for the geometrically frustrated lattice differs from that for nonfrustrated systems due to the existence of multiple sublattices in the unit cell and nonorthogonal basis vectors. Our study for the S = 1/2 kagome Heisenberg antiferromagnet at zero external magnetic field gives a criterion for the existence of a two-fold degenerate ground state with a finite excitation gap and fractionalized excitations. At finite field, we predict various plateaus at fractional magnetisation, in analogy with integer and fractional quantum Hall states of the primary sequence. These plateaus correspond to gapped quantum liquid ground states with a fixed number of singlets and spinons in the unit cell. A similar analysis for the triangular lattice predicts a single fractional magnetization plateau at 1/3. Our results are in broad agreement with numerical and experimental studies. In chapter five, we analyse the antiferromagnetic S=1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a nonperturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g., gapped or gapless spectrum etc.) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al. (Phys. Rev. B 90, 174409 (2014)), we use a Jordan-Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbour interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetisation per site) in the XY regime, we analyse the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU(2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by recent studies. We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses. This chapter is based on the reference: New Journal of Physics 21, 023019 (2019) In chapter six, we extend the LSM and OYA-like arguments to the S = 1/2 pyrochlore Heisenberg antiferromagnets in predicting the existence of magnetization plateaus states. The predicted plateaus states are supported by results obtained from calculations based on a non-local spin-parity operator. This includes possible fractional magnetization plateaus at 0, 1/8, 1/4, 1/2, 3/8, 5/8, 7/8, 3/4 etc. This chapter is based on the reference: Physical Review B 100, 104421 (2019). We end with a brief concluding chapter where we present discussions on further directions that are opened by the investigations presented in this thesis.

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