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International Journal of Modern Physics B

ORDER PARAMETERS OF FRUSTRATED SPIN SYSTEMS

Antimo Angelucci

Published Feb 20, 1991• 4 citations
Publisher

Abstract

We present a study of the ground states and of the low energy fluctuations of the frustrated Heisenberg spin chain, both classical and quantum. Analyzing this simple model we draw some general conclusions about the antiferro-helix “transition” of general classical Heisenberg models. We derive the low energy action describing the fluctuations about the helical phases and we infer the possible existence of nontrivial topological terms in the quantum counterparts.

DOI
10.1142/s0217979291000365

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We present a study of the ground states and of the low energy fluctuations of the frustrated Heisenberg spin chain, both classical and quantum. Analyzing this simple model we draw some general conclusions about the antiferro-helix “transition” of general classical Heisenberg models. We derive the low energy action describing the fluctuations about the helical phases and we infer the possible existence of nontrivial topological terms in the quantum counterparts.
Magnetic frustrations are the fundamental cause of many interesting phenomena in materials that have been intensely studied since the middle of the 20th century. Magnetic systems with competing interactions, ranging from insulating antiferromagnets to arrays of superconducting qubits, exhibit nontrivial quantum e ects that arise from the need to satisfy a jagged free energy landscape. The search for and characterization of quantum spin liquids, nontrivial topological phases, and quantum phase transitions is conducted through investigations of such frustrated models. Within the large umbrella of frustrated magnetism I have studied some speci c models with the purpose of answering the following questions: (1) What are the interesting phases and less-interesting phases, and how do they fall in the phase diagram relative to one another? (2) What properties do the phases have, and how can we characterize what happens when one phase transitions to another? Armed with the answers to these questions, researchers can further explore these interesting phenomena and apply them towards new materials and devices. In this thesis, I rst develop a computational methodology for studying both classical and quantum spin models using various forms of Monte Carlo simulation. I use classical Monte Carlo to study highly-frustrated models of Ising spins both at nite temperatures and in the T = 0 limit, to map the phase diagram and determine the universality class of phase transitions. Using these methods, I show that in a dissipative chain of qubits, bath-induced qubit-qubit interactions lead to collective decoherence above a critical dissipation strength. This causes the qubits to lose all stored information and become fully correlated with one another, an e ect that one would need to mitigate in order to perform quantum computation. In another study, I show that in antiferromagnetic insulators with anisotropic interactions, frustration can cause an intermediate paramagnetic phase with oblong \droplets" of correlated spins. This phenomenon is quite general and the presence of these droplets can help explain experimental signatures that occur at unexpected temperatures in such materials. In addition to the classical simulations, I develop a variational Monte Carlo (VMC) technique to search for ground states in Heisenberg-type models, speci cally for spins S = 1. The case of S = 1 is more complicated than the case for S = 1=2 due to an additional local degree of freedom, which has interesting implications for the possible states that can be realized. Using semiclassical energy minimization as well as VMC, I show that spin S = 1 diamond-lattice antiferromagnets experience higher-order spinspin interactions that destroy previously theorized liquid-like spiral spin orderings. However, the presence of these higher-order interactions can also potentially allow other phases such as quantum spin liquids and topological paramagnets. Such results can provide an avenue for explaining anomalous phenomena like low-temperature paramagnetism in future experimental materials.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
The Heisenberg antiferromagnetic spin chain with both dimerization and frustration is studied. The classical ground state has three phases (a Neel phase, a spiral phase and a colinear phase), around which a planar spin-wave analysis is performed. In each phase, we discuss a non-linear sigma model field theory describing the low energy excitations. A renormalization group analysis of the SO(3) matrix-valued field theory of the spiral phase leads to the conclusion that the theory becomes SO(3)×SO(3)SO(3) \times SO(3)SO(3)×SO(3) and Lorentz invariant at long distances. This theory is analytically known to have a massive spin-1/2 excitation. We also show that Z2 Z_2 ~Z2​  solitons in the field theory lead to a double degeneracy in the spectrum for half-integer spins.
We investigate quantum order by disorder in a frustrated spin nanotube formed by wrapping a J1J_1J1​-J2J_2J2​ Heisenberg model at 45∘^\circ∘ around a cylinder. Using Schwinger boson theory and Density Matrix Renormalization Group (DMRG), we have computed the ground-state phase diagram to reveal a Z2\mathbb{Z}_2Z2​ phase in which collinear spin stripes form a right or left handed helix around the nanotube. We have derived an analytic estimate for the critical ηc=J1/2J2η_c=J_1/2J_2ηc​=J1​/2J2​ of the Z2\mathbb{Z}_2Z2​ helical phase transition, which is in agreement with the DMRG results. By evaluating the entanglement spectrum and nonlocal string order parameters we discuss the topology of the Z2\mathbb{Z}_2Z2​-helical phase.
Quantum corrections of the biquadratic interaction in the 1D spin-1/2 frustrated ferromagnetic Heisenberg model are studied. The biquadratic interaction for spin-1/2 chains is eliminated and transformed to the quadratic interaction. Doing a numerical experiment, new insight as to how the classical phases get modified on the inclusion of quantum fluctuations is provided. Observed results suggest the existence of an intermediate region in the ground state phase diagram of the frustrated ferromagnetic spin-1/2 chains with combination of dimer and chiral orders. In addition, from the quantum entanglement view point, differences between quantum phases are also obtained. The nearest neighbor spins never be entangled in the frustrated ferromagnetic chains but are entangled up to the Majumdar-Ghosh point in the frustrated antiferromagnetic chains. On the other hand, the next nearest neighbor spins in the mentioned intermediate region are entangled.