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 b
roken 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 $langle 111 rangle$ (${hat z}$) 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.
We carry out an analytical study of quantum spin ice, a U$(1)$ quantum spin liquid close to the classical spin ice solution for an effective spin $1/2$ model with anisotropic exchange couplings $J_{zz}$, $J_{pm}$ and $J_{zpm}$ on the pyrochlore latti
ce. Starting from the quantum rotor model introduced by Savary and Balents in Phys. Rev. Lett. 108, 037202 (2012), we retain the dynamics of both the spinons and gauge field sectors in our treatment. The spinons are described by a bosonic representation of quantum XY rotors while the dynamics of the gauge field is captured by a phenomenological Hamiltonian. By calculating the one-loop spinon self-energy, which is proportional to $J_{zpm}^2$, we determine the stability region of the U$(1)$ quantum spin liquid phase in the $J_{pm}/J_{zz}$ vs $J_{zpm}/J_{zz}$ zero temperature phase diagram. From these results, we estimate the location of the boundaries between the spin liquid phase and classical long-range ordered phases.
Motivated by recent experimental and theoretical progress on the Er2Ti2O7 pyrochlore XY antiferromagnet, we study the problem of quantum order-by-disorder in pyrochlore XY systems. We consider the most general nearest-neighbor pseudo spin-1/2 Hamilto
nian for such a system characterized by anisotropic spin-spin couplings J_e = [J_pm, J_{pmpm}, J_{zpm}, J_{zz}] and construct zero-temperature phase diagrams. Combining symmetry arguments and spin-wave calculations, we show that the ground state phase boundaries between the two candidate ground states of the Gamma_5 irreducible representation, the psi_2 and psi_3 (basis) states, are rather accurately determined by a cubic equation in J_{pm}J_{pmpm})/J_{zpm}^2. Depending on the value of J_{zz}, there can be one or three phase boundaries that separate alternating regions of psi_2 and psi_3 states. In particular, we find for sufficiently small J_{zz}/J_{pm} a narrow psi_2 sliver sandwiched between two psi_3 regions in the J_{pmpm}/J_pm vs J_{zpm}/J_pm phase diagram. Our results further illustrate the very large potential sensitivity of the ground state of XY pyrochlore systems to minute changes in their spin Hamiltonian. Using the experimentally determined J_3 and g-tensor values for Er2Ti2O7, we show that the heretofore neglected long-range 1/r^3 magnetostatic dipole-dipole interactions do not change the conclusion that Er2Ti2O7 has a psi_2 ground state induced via a quantum order-by-disorder mechanism. We propose that the CdDy2Se4 chalcogenide spinel, in which the Dy^{3+} ions form a pyrochlore lattice and may be XY-like, could prove interesting to investigate.
We study the problem of partially ordered phases with periodically arranged disordered (paramagnetic) sites on the pyrochlore lattice, a network of corner-sharing tetrahedra. The periodicity of these phases is characterized by one or more wave vector
s k=(1/2 1/2 1/2). Starting from a general microscopic Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar interactions and second- and third-nearest neighbor exchange, we identify using standard mean-field theory (s-MFT) an extended range of interaction parameters that support partially ordered phases. We demonstrate that thermal fluctuations ignored in s-MFT are responsible for the selection of one particular partially ordered phase, e.g. the 4-k phase over the 1-k phase. We suggest that the transition into the 4-k phase is continuous with its critical properties controlled by the cubic fixed point of a Ginzburg-Landau theory with a 4-component vector order-parameter. By combining an extension of the Thouless-Anderson-Palmer method originally used to study fluctuations in spin glasses with parallel-tempering Monte-Carlo simulations, we establish the phase diagram for different types of partially ordered phases. Our results elucidate the long-standing puzzle concerning the origin of the 4-k partially ordered phase observed in the Gd2Ti2O7 dipolar pyrochlore antiferromagnet below its paramagnetic phase transition temperature.
