Order-by-disorder near criticality in $XY$ pyrochlore magnets

Abstract

We consider a system of spins on the sites of a three-dimensional pyrochlore lattice of corner-sharing tetrahedra interacting with a predominant effective $xy$ exchange. In particular, we investigate the selection of a long-range ordered state with broken discrete symmetry induced by thermal fluctuations near the critical region. At the standard mean-field theory (s-MFT) level, in a region of the parameter space of this Hamiltonian that we refer to as ${\Gamma }_5$ region, the ordered state possesses an accidental $U(1)$ degeneracy. In this paper, we show that fluctuations beyond s-MFT lift this degeneracy by selecting one of two states (so-called ${\psi }_2$ and ${\psi }_3$) from the degenerate manifold, thus exposing a certain form of order-by-disorder (ObD). We analytically explore this selection at the microscopic level and close to criticality by elaborating upon and using an extension of the so-called TAP method, originally developed by Thouless, Anderson and Palmer to study the effect of fluctuations in spin glasses. We also use a single-tetrahedron cluster-mean-field theory (c-MFT) to explore over what minimal length scale fluctuations can lift the degeneracy. We find the phase diagrams obtained by these two methods to be somewhat different since c-MFT only includes the shortest-range fluctuations. General symmetry arguments used to construct a Ginzburg-Landau theory to lowest order in the order parameters predict that a weak magnetic moment, $m_z$, along the local $⟨111⟩$ ($\hat{z}$) direction is generically induced for a system ordering into a ${\psi }_2$ state, but not so for ${\psi }_3$ ordering. Both E-TAP and c-MFT calculations confirm this weak fluctuation-induced $m_z$ moment. Using a Ginzburg-Landau theory, we discuss the phenomenology of multiple phase transitions below the paramagnetic phase transition and within the ${\Gamma }_5$ long-range ordered phase.

arXiv comment17 pages, 8 figures. ArXiv version essentially identical to the version published in Physical Review B

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• Spin freezing in the disordered pyrochlore magnet NaCaCo${}_2$F${}_7$: NMR studies and Monte-Carlo simulations
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• Competing Gauge Fields and Entropically-Driven Spin Liquid to Spin Liquid Transition in non-Kramers Pyrochlores
• Low-field behavior of an XY pyrochlore antiferromagnet: emergent clock anisotropies
• Field induced phase diagram of the XY pyrochlore antiferromagnet Er${}_2$Ti${}_2$O${}_7$
• Experimental evidence for field induced emergent clock anisotropies in the XY pyrochlore Er${}_2$Ti${}_2$O${}_7$
• The frustrated Ising model on the body-centered cubic lattice
• Intertwined Magnetic Dipolar and Electric Quadrupolar Correlations in the Pyrochlore Tb${}_2$Ge${}_2$O${}_7$
• The free energy of anisotropic quantum spin systems: Functional integral representation
• Programmable order by disorder effect and underlying phases through dipolar quantum simulators
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