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arXiv:1108.5268 · Physical Review B

Order by disorder and phase transitions in a highly frustrated spin model on the triangular lattice

A. Honecker, D. C. Cabra, H. U. Everts, Pierre Pujol, F. Stauffer

Published Dec 14, 2011• 12 citations
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Abstract

Frustration has proven to give rise to an extremely rich phenomenology in both quantum and classical systems. The leading behavior of the system can often be described by an effective model in which only the lowest-energy degrees of freedom are considered. In this paper, we study a system corresponding to the strong trimerization limit of the spin-$1/2$ kagome antiferromagnet in a magnetic field. It has been suggested that this system can be realized experimentally by a gas of spinless fermions in an optical kagome lattice at $2/3$ filling. We investigate the low-energy behavior of both the spin-$1/2$ quantum version and the classical limit of this system by applying various techniques. We study in parallel both signs of the coupling constant $J$ since the two cases display qualitative differences. One of the main peculiarities of the $J>0$ case is that, at the classical level, there is an exponentially large manifold of lowest-energy configurations. This renders the thermodynamics of the system quite exotic and interesting in this case. For both cases, $J>0$ and $J<0$, a finite-temperature phase transition with a breaking of the discrete dihedral symmetry group ${D}_{6}$ of the model is present. For $J<0$, we find a transition temperature ${T}_{c}^{<}/|J|=1.566\ifmmode\pm\else\textpm\fi{}0.005$, i.e., of order unity, as expected. We then analyze the nature of the transition in this case. While we find no evidence for a discontinuous transition, the interpretation as a continuous phase transition yields very unusual critical exponents violating the hyperscaling relation. By contrast, in the case $J>0$, the transition occurs at an extremely low temperature, ${T}_{c}^{>}\ensuremath{\approx}0.0125\phantom{\rule{0.16em}{0ex}}J$. Presumably this low transition temperature is connected with the fact that the low-temperature ordered state of the system is established by an order-by-disorder mechanism in this case.

DOI
10.1103/physrevb.84.224410

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Frustration has proven to give rise to an extremely rich phenomenology in both quantum and classical systems. The leading behavior of the system can often be described by an effective model in which only the lowest-energy degrees of freedom are considered. In this paper, we study a system corresponding to the strong trimerization limit of the spin-1/21/21/2 kagome antiferromagnet in a magnetic field. It has been suggested that this system can be realized experimentally by a gas of spinless fermions in an optical kagome lattice at 2/32/32/3 filling. We investigate the low-energy behavior of both the spin-1/21/21/2 quantum version and the classical limit of this system by applying various techniques. We study in parallel both signs of the coupling constant JJJ since the two cases display qualitative differences. One of the main peculiarities of the J>0J>0J>0 case is that, at the classical level, there is an exponentially large manifold of lowest-energy configurations. This renders the thermodynamics of the system quite exotic and interesting in this case. For both cases, J>0J>0J>0 and J<0J<0J<0, a finite-temperature phase transition with a breaking of the discrete dihedral symmetry group D6{D}_{6}D6​ of the model is present. For J<0J<0J<0, we find a transition temperature Tc</∣J∣=1.566\ifmmode±\else\textpm\fi0.005{T}_{c}^{<}/|J|=1.566\ifmmode\pm\else\textpm\fi{}0.005Tc<​/∣J∣=1.566\ifmmode±\else\textpm\fi0.005, i.e., of order unity, as expected. We then analyze the nature of the transition in this case. While we find no evidence for a discontinuous transition, the interpretation as a continuous phase transition yields very unusual critical exponents violating the hyperscaling relation. By contrast, in the case J>0J>0J>0, the transition occurs at an extremely low temperature, Tc>\ensuremath≈0.0125J{T}_{c}^{>}\ensuremath{\approx}0.0125\phantom{\rule{0.16em}{0ex}}JTc>​\ensuremath≈0.0125J. Presumably this low transition temperature is connected with the fact that the low-temperature ordered state of the system is established by an order-by-disorder mechanism in this case.
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A spin-1 Heisenberg model on trimerized kagome lattice is studied by doing a low-energy bosonic theory in terms of plaquette triplons defined on its triangular unit cells. The model considered has an intratriangle antiferromagnetic exchange interaction JJJ (set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector q=(0,0)\mathbf{q}=(0,0)q=(0,0). Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this q=(0,0)\mathbf{q}=(0,0)q=(0,0) phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.
We consider the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional square-kagome lattice with almost dispersionless lowest magnon band. For a general exchange coupling geometry we elaborate low-energy effective Hamiltonians which emerge at high magnetic fields. The effective model to describe the low-energy degrees of freedom of the initial frustrated quantum spin model is the (unfrustrated) square-lattice spin-1/2 XXZ model in a z-aligned magnetic field. For the effective model we perform quantum Monte Carlo simulations to discuss the low-temperature properties of the square-kagome quantum Heisenberg antiferromagnet at high magnetic fields. We pay special attention to a magnetic-field driven Berezinskii–Kosterlitz– Thouless phase transition which occurs at low temperatures.