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A Semi-Classical Analysis of Order from Disorder

Benoît Douçot, Pascal Simon

Published Jan 1, 1998• 5 citations
OpenAlex

Abstract

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.

Related papers

A semiclassical analysis of order from disorder

Benoît Douçot, Pascal Simon

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.

arXiv:cond-mat/9802212•Journal of Physics A Mathematical and General•Jul 17, 1998•11 citations

Tunneling-induced restoration of classical degeneracy in quantum kagome ice

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Quantum effect is expected to dictate the behavior of physical systems at low temperature. For quantum magnets with geometrical frustration, quantum fluctuation usually lifts the macroscopic classical degeneracy, and exotic quantum states emerge. However, how different types of quantum processes entangle wave functions in a constrained Hilbert space is not well understood. Here, we study the topological entanglement entropy and the thermal entropy of a quantum ice model on a geometrically frustrated kagome lattice. We find that the system does not show a ${Z}_{2}$ topological order down to extremely low temperature, yet continues to behave like a classical kagome ice with finite residual entropy. Our theoretical analysis indicates an intricate competition of off-diagonal and diagonal quantum processes leading to the quasidegeneracy of states and effectively, the classical degeneracy is restored.

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Alexander A. Donkov, Andrey V. Chubukov

We study ground-state degeneracy and spin-wave excitations in a 1D version of a Kagome antiferromagnet —a Heisenberg antiferromagnet on a Kagome stripe. We show that for nearest-neighbor interaction, the classical ground state is infinitely degenerate. For any spin configuration from the degenerate set, the classical spin-wave spectrum contains, in addition to Goldstone modes, a branch of zero-energy excitations, and a zero mode in another branch. We demonstrate that the interactions beyond nearest neighbors lift the degeneracy, eliminate a zero mode, and give a finite dispersion to formerly zero-energy branch, leaving only Goldstone modes as zero-energy excitations.

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Quantum Selection of Order in an XXZ Antiferromagnet on a Kagome Lattice

A. L. Chernyshev, Mike E. Zhitomirsky

Selection of the ground state of the kagome-lattice $XXZ$ antiferromagnet by quantum fluctuations is investigated by combining nonlinear spin-wave and real-space perturbation theories. The two methods unanimously favor $\mathbf{q}=0$ 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.

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Order and disorder in SU( N ) simplex solid antiferromagnets

Yury Kiselev, S. A. Parameswaran, Daniel P. Arovas

Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.

Journal of Statistical Mechanics Theory and Experiment•Jan 1, 2016•1 citation
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.
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.
Quantum effect is expected to dictate the behavior of physical systems at low temperature. For quantum magnets with geometrical frustration, quantum fluctuation usually lifts the macroscopic classical degeneracy, and exotic quantum states emerge. However, how different types of quantum processes entangle wave functions in a constrained Hilbert space is not well understood. Here, we study the topological entanglement entropy and the thermal entropy of a quantum ice model on a geometrically frustrated kagome lattice. We find that the system does not show a Z2{Z}_{2}Z2​ topological order down to extremely low temperature, yet continues to behave like a classical kagome ice with finite residual entropy. Our theoretical analysis indicates an intricate competition of off-diagonal and diagonal quantum processes leading to the quasidegeneracy of states and effectively, the classical degeneracy is restored.
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.
We study ground-state degeneracy and spin-wave excitations in a 1D version of a Kagome antiferromagnet —a Heisenberg antiferromagnet on a Kagome stripe. We show that for nearest-neighbor interaction, the classical ground state is infinitely degenerate. For any spin configuration from the degenerate set, the classical spin-wave spectrum contains, in addition to Goldstone modes, a branch of zero-energy excitations, and a zero mode in another branch. We demonstrate that the interactions beyond nearest neighbors lift the degeneracy, eliminate a zero mode, and give a finite dispersion to formerly zero-energy branch, leaving only Goldstone modes as zero-energy 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.
Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.