In this research work, frustrations arising from the geometries of triangular lattices have been studied with the aid of Ising and Heisenberg models. The study reveals that geometrical frustrations can generate multiple degeneracies in the ground state. The quantum spin flip terms in the Heisenberg model are observed to play a vital role in the partial lifting up of these degeneracies. Hence, multiple degeneracies as consequence of frustrations are more pronounced for the Ising systems, which are devoid of quantum fluctuations. The observed six- and four-fold ground state degeneracies at zero field for three spins Ising and Heisenberg systems respectively are broken down to half at finite longitudinal fields. For this three-spin system, quantum phase transitions (QPT) are observed at critical longitudinal fields of J and 1.5J respectively for the Ising and Heisenberg models. At these critical fields, the ground states are observed to shift from quasi-antiferromagnet to ferromagnet. However, for the Heisenberg three-spin system in the presence of a transverse field, no transition is observed.
The Ising model, often seen as the paradigmatic spin model, has been heavily studied for its mathematical description of ferromagnetism in statistical mechanics. We explore a quantum version of this model, the transverse field Ising model, and investigate how the quantum property of magnetization fluctuates with geometrical frustration, a phenomenon arising from the geometry of a spin system's ground state. We introduce a general measure for frustration and then implement a computational model built in Python that translates spin lattices into adjacency matrices, solves the spectrum of the transverse field Ising model, calculates the expected magnetization of the ground state, plots the variable over randomly generated spin systems with controllable degrees of geometrical frustration, and compares the evolution of these plots over an increasing transverse field. Our finite-size studies exhibit an expected decrease in magnetization with increasing frustration and suggest the possibility of robust phases in the large N limit over finite ranges of frustration, which may be precursors to spin liquids or spin glasses.
We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at
T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.
The concept of frustration and non-classical nature of antiferromagnetism for Ising and Heisenberg lattice is examined. The confusion in spin alignment makes the lattice a possible host for quantum entanglement, a requirement for quantum technology. The effect of external field on degeneracy in ground state induces phase transition in system from ferromagnet to antiferromagnets in three spin ½ spin Ising model. The dependence of quasi local correlator on the distance between sites and on the external field is discussed for both even and odd number of spins.
We present the zero-temperature phase diagram of a Heisenberg antiferromagnet on a frustrated triangular lattice with nearest-neighbor
(J1) and next-nearest-neighbor
(J2) interactions, in a magnetic field. We show that the classical model has an accidental degeneracy for all
J2/J1 and all fields, but the degeneracy is lifted by quantum fluctuations. We show that at large spin
S, for
J2/J1<1/8, quantum fluctuations select the same sequence of three sublattice co-planar states in a field as for
J2=0, and for
1/8<J2/J1<1, they select the canted stripe state for all nonzero fields. The transition between the two states is first order in all fields, with the hysteresis width set by quantum fluctuations. We study the model with arbitrary
S, including
S=1/2, near the saturation field by exploring the fact that near saturation the density of bosons is small for all
S. We show that for
S>1, the transition remains first order, with a finite hysteresis width, but for
S=1/2 and, possibly,
S=1, there appears a new intermediate phase without a quasiclassical long-range order.
The spin-
21 Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-
21 Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins.
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.