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arXiv:1912.13042

Frustration -- Exactly Solved Frustrated Models

H. T. Diep, Héctor Giacomini

Published Dec 25, 2019• 0 citations
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Abstract

After a short introduction on frustrated spin systems, we study in this chapter several two-dimensional frustrated Ising spin systems which can be exactly solved by using vertex models. We show that these systems contain most of the spectacular effects due to the frustration: high ground-state degeneracy, existence of several phases in the ground-state phase diagram, multiple phase transitions with increasing temperature, reentrance, disorder lines, partial disorder at equilibrium. Evidences of such effects in non solvable models are also shown and discussed.

DOI
10.48550/arxiv.1912.13042

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After a short introduction on frustrated spin systems, we study in this chapter several two-dimensional frustrated Ising spin systems which can be exactly solved by using vertex models. We show that these systems contain most of the spectacular effects due to the frustration: high ground-state degeneracy, existence of several phases in the ground-state phase diagram, multiple phase transitions with increasing temperature, reentrance, disorder lines, partial disorder at equilibrium. Evidences of such effects in non solvable models are also shown and discussed.
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.
In this work we investigate themes related to many-body systems in which multiple ground states are accessible, a condition known as frustration. Frustration can arise in a number of contexts, and we consider the consequences of this situation with some examples from condensed-matter physics. In some magnetic materials interactions between spins are such that no single spin configuration provides a unique ground state. In the class of frustrated magnets where the number of ground states is extensive, thermal fluctuations are strong even at temperatures significantly below the interaction strength. At such temperatures spins are highly correlated, and small perturbations may have profound consequences. In this thesis we provide an example of this. By considering classical n-component spins with nearest-neigbour exchange on a frustrated octahedral lattice we show that – in the limit where exchange interactions are large – the system is in a disordered, correlated phase where correlations have the form of a dipole field. This is termed a Coulomb phase. From this phase we induce an ordering transition, lifting the degeneracy with weak, additional short-range interactions. By studying the transition in the solvable limit of n → ∞, we discover that the transition has identical thermodynamics to that of a magnetic system interacting through long-range, dipolar forces. Finally, we provide a more apposite characterisation of the transition, where the high-temperature side of the transition is described through the fluctuations of solenoidal fields, and the ordering corresponds to a condensation of these fields. In a separate part of the thesis, we investigate the influence of disorder on frustrated lattices. We study a two-dimensional tight-binding model with nearest-neighbour hopping and on-site disorder. Restricting the allowed states to being those from the low-lying manifold of ground states, the disorder feeds through to act as effective disorder in the hopping terms, which decay algebraically with distance. The quasi-long range nature of this effective hopping leads to a situation in which the resultant single-particle eigenstates are critical, and we probe their behaviour numerically with a transfer matrix calculation.
We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.
We solve exactly a generalized Kagom\'e Ising lattice with three kinds of interactions by transforming the system into a 16-vertex model that satisfies the free-fermion condition. The phase diagram shows several interesting phenomena due to the frustration generated by the competing interactions: successive transitions with reentrance, partial disorder, and disorder lines. The reentrance region in the space of interaction parameters is found to be infinite, unlike previous exactly solved models. Two disorder solutions with interesting behavior are found.
Through the direct decoration transformation approach, we obtain a general solution for the pentagonal Ising model, showing its equivalence to the isotropic free-fermion eight-vertex model. We study the ground-state phase diagram, in which one ferromagnetic (FM) state, one ferrimagnetic (FIM) state, and one frustrated state are found. Using the exact solution of the pentagonal Ising model, we discuss the finite-temperature phase diagrams and find a phase transition between the FIM state and the disordered state as well as a phase transition between the disordered state and the FM state. We also discuss some additional remarkable properties of the model, such as the magnetization, entropy, and specific heat, at finite temperature and at its low-temperature asymptotic limit. Because of the influence of the second-order phase transition between the frustrated and ferromagnetic phases, we obtain surprisingly low values of the entropy and the specific heat until the critical temperature is reached.
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.