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Journal of Physics C Solid State Physics

New insight into the transition to the disordered state in the 2D frustrated Ising model

J. Poulter, John A. Blackman

Published Feb 10, 1986• 4 citations
Publisher

Abstract

The authors consider the 2D frustrated Ising model with concentrations (1-p) and p of +J and -J bonds. A transition occurs in the ground-state properties for p approximately=12%. It is shown that this can be understood in terms of a transition from localised to extended states within a fermion representation.
DOI
10.1088/0022-3719/19/4/017

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We study the ground state of the two-dimensional (2D) disordered Hubbard model by means of the projector quantum Monte Carlo (PQMC) method. This approach allows us to investigate the ground state properties of this model for lattice sizes up to 10×1010 \times 1010×10, at quarter filling, for a broad range of interaction and disorder strengths. Our results show that the ground state of this system of spin-1/2 fermions remains localised in the presence of the short-ranged Hubbard interaction.
This chapter has been co-authored with T. Senthil, and adapted from the Ph.D. thesis of T. Senthil, submitted to Yale University (1997, unpublished). The last two chapters of this book move beyond the study of regular Hamiltonians that have the full translational symmetry of an underlying crystalline lattice and consider the physically important case of disordered systems described by Hamiltonians with couplings that vary from point to point in space. By the standards of the regular systems we have already discussed, the quantum phase transitions of disordered systems are very poorly understood, and only a few well-established results are available. A large amount of theoretical effort has been expended toward unraveling the complicated phenomena that occur, and they remain active topics of current research. The aims of our discussion here are therefore rather limited: we highlight some important features that are qualitatively different from those of nondisordered systems, make general remarks about insights that can be drawn from our understanding of the finite- T crossovers in Part II, and discuss the properties of some simple solvable models. In keeping with the general strategy of this book, we introduce some basic concepts by studying the effects of disorder on the magnetic ordering transitions of quantum Ising/rotor models studied in Part II; we also make some remarks in Section 21.4 on the effects of disorder on the ordering transitions of Fermi liquids considered in Chapter 18. Models with much stronger disorder and frustrating interactions that have new phases not found in ordered systems are considered in Chapter 22.
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 present exact results on the low-temperature behavior of the square lattice ±J Ising model, with a concentration c of ferromagnetic bonds and a concentration 1−c of antiferromagnetic bonds. We find that for T below a characteristic temperature T0, the system is “frozen” in the sense that its ground state properties coincide with measured quantities for systems of macroscopic size. We find that T0 is remarkably large, being roughly room temperature if J/kB ∼ 1000 K. We also study the effect on the free energy of various spatial arrangements of the ferromagnetic bonds.
We consider the 2D J 1 − J 2 classical XY model on a square lattice. In the frustrated phase corresponding to J 2 > J 1 /2, an Ising order parameter emerges by an "order due to disorder" effect. This leads to a discrete symmetry plus the O (2) global one. We formulate the problem in a Coulomb gas language and show by a renormalization group analysis that only two phases are still possible: a locked phase at low temperature and a disordered one at high temperature. The transition is characterized by the loss of Ising and XY order at the same point. This analysis suggests that the 2D J 1 − J 2 XY model is in the same universality class as XY-Ising models.