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arXiv:cond-mat/9802212 · Journal of Physics A Mathematical and General

A semiclassical analysis of order from disorder

Benoît Douçot, Pascal Simon

Published Jul 17, 1998• 11 citations
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Abstract

In this paper we present several examples of order due to disorder phenomena and mainly focus on the Heisenberg antiferromagnet with nearest neighbour interactions on the Husimi cactus, a system which locally has the same topology as the Kagomé lattice. This system has a huge classical degeneracy corresponding to an extensive number of degrees of freedom. We show that unlike thermal fluctuations, quantum fluctuations partially lift this degeneracy and favour a discrete subset of classical ground states. In order to clarify the origin of these effects, we have set up a general semiclassical analysis of the order from disorder phenomenon and clearly identified the differences between classical and quantum fluctuations. This semiclassical approach also enables us to classify various situations where a selection mechanism still occurs. Moreover, once a discrete set of ground states has been preselected, our analysis suggests that tunelling processes within this set should be the dominant effect underlying the strange low-energy spectrum of Kagomé-like lattices.

DOI
10.1088/0305-4470/31/28/005

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In this paper we present several examples of order due to disorder phenomena and mainly focus on the Heisenberg antiferromagnet with nearest neighbour interactions on the Husimi cactus, a system which locally has the same topology as the Kagomé lattice. This system has a huge classical degeneracy corresponding to an extensive number of degrees of freedom. We show that unlike thermal fluctuations, quantum fluctuations partially lift this degeneracy and favour a discrete subset of classical ground states. In order to clarify the origin of these effects, we have set up a general semiclassical analysis of the order from disorder phenomenon and clearly identified the differences between classical and quantum fluctuations. This semiclassical approach also enables us to classify various situations where a selection mechanism still occurs. Moreover, once a discrete set of ground states has been preselected, our analysis suggests that tunelling processes within this set should be the dominant effect underlying the strange low-energy spectrum of Kagomé-like lattices.
We study in this paper the Heisenberg antiferromagnet with nearest neighbours interactions on the Husimi cactus, a system which has locally the same topology as the Kagomé lattice. This system has a huge classical degeneracy corresponding to an extensive number of degrees of freedom. We show that unlike thermal fluctuations, quantum fluctuations lift partially this degeneracy and favour a discrete subset of classical ground states. In order to clarify the origin of these effects, we have set up a general semi-classical analysis of the order from disorder phenomenon and clearly identified the differences between classical and quantum fluctuations. This semi-classical approach also enables us to classify various situations where a selection mechanism still occurs. Moreover, once a discrete set of ground states has been preselected, our analysis suggests that tunelling processes within this set should be the dominant effect underlying the strange low energy spectrum of Kagomé-like lattices.
We investigate the general structure of orbital exchange physics in Mott-insulating states of p-orbital systems in optical lattices. Orbital orders occur in both the triangular and kagome lattices. In contrast, orbital exchange in the honeycomb lattice is frustrated as described by a novel quantum 120 degrees model. Its classical ground states are mapped into configurations of the fully packed loop model with an extra U(1) rotation degree of freedom. Quantum orbital fluctuations select a six-site plaquette ground state ordering pattern in the semiclassical limit from the "order from disorder" mechanism. This effect arises from the appearance of a zero energy flat band of orbital excitations.
Selection of the ground state of the kagome-lattice XXZXXZXXZ antiferromagnet by quantum fluctuations is investigated by combining nonlinear spin-wave and real-space perturbation theories. The two methods unanimously favor q=0\mathbf{q}=0q=0 over $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ magnetic order in a wide range of the anisotropy parameter $0\ensuremath{\le}\mathrm{\ensuremath{\Delta}}\ensuremath{\lesssim}0.72$. Both approaches are also in accord on the magnitude of the quantum order-by-disorder effect generated by topologically nontrivial, looplike spin-flip processes. A tentative $S\text{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}$ phase diagram of the model is proposed.
Accidental ground state degeneracies-those not a consequence of global symmetries of the Hamiltonian-are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as "order-by-disorder" (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative signatures that distinguish ObD from conventional energetic selection. We show that for order by thermal disorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap, Δ, and show that at low temperatures it scales as the square root of temperature, sqrt[T]. We establish that a power-law temperature dependence of the gap is a general consequence of ObTD, showing that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.
We identify and discuss the ground state of a quantum magnet on a triangular lattice with bond-dependent Ising-type spin couplings, that is, a triangular analog of the Kitaev honeycomb model. The classical ground-state manifold of the model is spanned by decoupled Ising-type chains, and its accidental degeneracy is due to the frustrated nature of the anisotropic spin couplings. We show how this subextensive degeneracy is lifted by a quantum order-by-disorder mechanism and study the quantum selection of the ground state by treating short-wavelength fluctuations within the linked cluster expansion and by using the complementary spin-wave theory. We find that quantum fluctuations couple next-nearest-neighbor chains through an emergent four-spin interaction, while nearest-neighbor chains remain decoupled. The remaining discrete degeneracy of the ground state is shown to be protected by a hidden symmetry of the model.
This thesis presents novel theoretical results on two cold atomic systems with particular emphasis on the order-by-disorder mechanism that they exhibit. The latter selects one or a number of classically degenerate states of lowest energy as the true ground states on the basis of Helmholtz free energy contributions of fluctuations about the classical solutions. It has traditionally played an important role in solid-state systems but has so far not been conclusively observed. The first system considered is that of cold bosons hopping in the two-dimensional dice lattice in the presence of an artificial gauge field, tuned to provide half an elementary flux per plaquette. The single-particle band structure consists entirely of flat bands. Many-particle behaviour is captured by a Bose-Hubbard model with contact interactions. Mean-field analysis yields a large degeneracy of classical ground states which is lifted through Order by Disorder. A closer analysis yields a picture of free-energy-mediated interactions between domain walls separating distinct, classically degenerate regions. The second system is that of spin-2 species in a tightly confining spatial potential, with a generalised quadratic-Zeeman coupling. It is analysed by means of an exact mapping of the many-body Hamiltonian onto that of a five-dimensional rotor. The new Hamiltonian is in general non-Hermitian, but Hermitianising transforms may be found in a number of regimes. Analytical results for the ordinary quadratic Zeeman potential are presented and shown to yield qualitatively different behaviour from the mean-field analysis. In particular, there are no signs of a fluctuation-induced phase transition, predicted by mean-field theory. Motivated by this, an alternative potential, breaking less symmetry, is considered within the rotor framework and shown to display microscopic parallels of Order by Disorder.