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AIP conference proceedings

Ground state degeneracy and dependence of quasi local correlator for frustrated system

Sushamana Sharma

Published Jan 1, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0224815

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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.
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.
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.
Abstract In quantum many-body systems with local interactions, the effects of boundary conditions are considered to be negligible, at least for sufficiently large systems. Here we show an example of the opposite. We consider a spin chain with two competing interactions, set on a ring with an odd number of sites. When only the dominant interaction is antiferromagnetic, and thus induces topological frustration, the standard antiferromagnetic order (expressed by the magnetization) is destroyed. When also the second interaction turns from ferro to antiferro, an antiferromagnetic order characterized by a site-dependent magnetization which varies in space with an incommensurate pattern, emerges. This modulation results from a ground state degeneracy, which allows to break the translational invariance. The transition between the two cases is signaled by a discontinuity in the first derivative of the ground state energy and represents a quantum phase transition induced by a special choice of boundary conditions.
In selected lattice systems, it is impossible to satisfy all the antiferromagnetic interactions between the spins simultaneously. Due to this frustration the degeneracy of the ground state becomes very large, often scaling with the size of the system. An antiferromagnetic model of Ising spins with on the three-dimensional Swedenborgite lattice will similarly be frustrated and will behave like a spin liquid in all parameter regimes. \n\nFluctuations, which may be either thermal or quantum-mechanical in nature, between the states in the large (disordered) ground state manifold may be biased towards one of them. This causes the system to select a unique (ordered) ground state, in an effect called order-by-disorder. Our objective is now to investigate the effects of quantum fluctuations on the ground states of the Swedenborgite Ising model and to examine it for potential order-by-disorder effects. This will ultimately result in the construction of a tentative phase diagram of this system.
We study a spin- 1/2 Heisenberg model H=J 1 Sigma i=2 N S 1 S i +J 2 Sigma i=2 N S i S i+1 ; J 1 , J 2 >or=0 (Heisenberg star) which may be considered either as an essential structure element of a lattice with frustration or, alternatively, as an antiferromagnetic linear chain with a perturbation. We discuss general relations for the energy eigenvalues, the eigenstates as well as the spin correlation of the system in dependence on J 2 /J 1 . The correlation function (S 1 S i ) scales with 1/Z, where Z =N-1 is the number of nearest neighbours i of the central spin 1. In the mean-field limit (Z to infinity ) (S 1 S i ) goes to -0.25 for small J 2 /J 1 , but vanishes for strong frustration J 2 /J 1 . Adding exact numerical results for N=5, 7, ..., 23 we discuss the ground-state phase diagram, in particular, the ground-state spin correlations versus J 2 /J 1 . Analysing the spin correlation of frustrated and non-frustrated stars we suggest an upper bound -0.25 for the ground-state correlator (S i S j )0 of two antiferromagnetically interacting spins i and j in a non-frustrated Heisenberg spin- 1/2 antiferromagnet. We argue that any measured nearest-neighbour spin correlation (S i S j )0 larger than this bound indicates of frustration in quantum spin Heisenberg antiferromagnets.
We study the ground state in a quasi-one-dimensional triangular Hubbard model using the exact numerical diagonalization, the constrained-path Monte Carlo, and the cluster perturbation theory. As the frustration hopping ttt is small the system exhibits short-range antiferromagnetic correlation. As ttt is greater than a critical point tc{t}_{c}tc​, there is a transition from an antiferromagnetic state to a frustrated state, which is accompanied by a jump of local entanglement and an abrupt change of spin correlation. As ttt is greater than another critical point ${t}_{{c}^{\ensuremath{'}}}$, the insulator-metal transition takes place. The successive order of two transitions depends on the on-site Coulomb interaction. The calculated spectra indicates that the spin-charge separation is destroyed by the frustration hopping. The frustrated state corresponds to the appearance of an additional dispersion band near the Fermi surface.