Competition between supersolid phases and magnetization plateaus in the frustrated easy-axis antiferromagnet on a triangular lattice

Luis Seabra and Nic Shannon

Phys. Rev. B 83, 134412 – Published 12 April 2011

Abstract

The majority of magnetic materials possess some degree of magnetic anisotropy, either at the level of a single ion, or in the exchange interactions between different magnetic ions. Where these exchange interactions are also frustrated, the competition between them and anisotropy can stabilize a wide variety of new phases in applied magnetic field. Motivated by the hexagonal delafossite 2 $H$ -AgNiO $_{2}$ , we study the Heisenberg antiferromagnet on a layered triangular lattice with competing first- and second-neighbor interactions and single-ion easy-axis anisotropy. Using a combination of classical Monte Carlo simulation, mean-field analysis, and Landau theory, we establish the magnetic phase diagram of this model as a function of temperature and magnetic field for a fixed ratio of exchange interactions, but with values of easy-axis anisotropy $D$ extending from the Heisenberg ( $D = 0$ ) to the Ising ( $D = \infty$ ) limits. We uncover a rich variety of different magnetic phases. These include several phases which are magnetic supersolids (in the sense of Matsuda and Tsuneto or Liu and Fisher), one of which may already have been observed in AgNiO $_{2}$ . We explore how this particular supersolid arises through the closing of a gap in the spin-wave spectrum, and how it competes with rival collinear phases as the easy-axis anisotropy is increased. The finite temperature properties of this phase are found to be different from those of any previously studied magnetic supersolid.

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Received 10 November 2010

DOI:https://doi.org/10.1103/PhysRevB.83.134412

Authors & Affiliations

Luis Seabra and Nic Shannon

H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK

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Vol. 83, Iss. 13 — 1 April 2011

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Figure 1
(a) Layered triangular lattice considered in this paper, with first- and second- neighbor interactions $J_{1}$ and $J_{2}$ in each plane, coupled by interactions between perpendicular neighbors $J_{⊥}$ . (b) Second-neighbor interactions transform the natural unit cell within the plane into a diamond plaquette.Reuse & Permissions
Figure 2
Cartoons of different three- and four-sublattice ground states found at mean-field level as a function of field. (a) Collinear stripe phase. (b) “Spin flopped” version of the stripe phase. (c) Novel magnetic supersolid state, the main focus of this paper. (d) $m = 1 / 3$ magnetization plateau. (e) 2:1:1 canted phase. (f) $m = 1 / 2$ magnetization plateau. (g) 3:1 canted phase. The staggered moment in the $S^{x}$ - $S^{y}$ plane is shown as a vector for states (b), (c), (e), and (g).Reuse & Permissions
Figure 3
Zero field $J_{2}$ - $D$ phase diagram for $J_{1} = 1$ and $J_{⊥} = - 0.15$ from zero temperature mean-field theory. The model supports three different ground states: a three-sublattice coplanar state, a two-sublattice collinear “stripe” antiferromagnet, and an incommensurate helicoidal phase.Reuse & Permissions
Figure 4
(a) Collinear stripe phase favored by second-neighbor interactions, showing the three possible stripe orientations. (b) Three-sublattice $m = 1 / 3$ plateau state with two spins up and one down. (c) First Brillouin zone in $q_{x}$ - $q_{y}$ plane showing momentum vectors $M$ and $M^{'}$ associated with the stripe phases. The blue rectangle shows the first magnetic Brillouin zone for the stripe phase with ordering vector $M$ (top cartoon). (d) The red hexagon shows the first magnetic Brillouin zone for a generic four-sublattice state centered on $M^{'}$ . (e) Ordering vectors $K$ of the $m = 1 / 3$ plateau state and related first magnetic Brillouin zone, shaded blue.Reuse & Permissions
Figure 5
Finite-field $J_{2}$ - $D$ phase diagram for $J_{1} = 1$ and $J_{⊥} = - 0.15$ , showing the first mean-field instability with applied field. Three different states are found: $m = 1 / 3$ and $m = 1 / 2$ magnetization plateaux and magnetic supersolid state. Red dots along dashed vertical line show studied anisotropy values.Reuse & Permissions
Figure 6
Magnetic phases of a layered triangular-lattice antiferromagnet with exchange interactions $J_{1} = 1$ , $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ and easy-axis anisotropy $D = 0$ . (a) Phase diagram as a function of temperature and magnetic field. Open symbols on the $h$ axis show transitions obtained in mean-field theory. Phase boundaries at finite temperature are obtained from Monte Carlo simulation for a cluster of $24 \times 24 \times 8$ spins and determined by peaks in the relevant order parameter susceptibilities. All phase transitions are first order, except where shown with a dashed line. All phases share a four-sublattice unit cell. (b) Phase diagram as a function of temperature and magnetization. Solid lines running left-right show cuts at constant magnetic field $h$ taken from simulations. The coexistence regions associated with first-order phase transitions are colored white.Reuse & Permissions
Figure 7
Structure of the ground state of the $J_{1}$ - $J_{2}$ Heisenberg model on a triangular lattice for $J_{2} / J_{1} > 1 / 8$ . (a) Degenerate manifold of states for $T = 0$ , $h = 0$ , built of two independent Néel sublattices. (b) Collinear antiferromagnetic stripe phase selected by thermal fluctuations ( $h = 0$ ). (c) “Spin flop” phase formed by direct canting of antiferromagnetic stripes in applied magnetic field. (d) Supersolid phase formed from the same two canted sublattices.Reuse & Permissions
Figure 8
Phase transition at $D = 0, h = 0$ from the paramagnet into the isotropic collinear antiferromagnetic stripe state. (a) Stripe order parameter from broken $Z_{3}$ lattice symmetry, Eq. (3), and (b) heat capacity. The first-order behavior is clear in the bimodal distribution of the energy histograms close to the critical temperature, inset to (a).Reuse & Permissions
Figure 9
Selection of simulation results for $D = 0$ , $h = 4.0$ , spanning the canted “spin flop” phase, collinear $m = 1 / 2$ plateau, 3:1 canted state, and high-temperature paramagnet. (a) $Z_{3}$ order parameter $O_{Z_{3}}$ associated with broken lattice rotation symmetry in the spin-flop state [Eq. (3)], and spin structure factor $S^{zz} (q_{M})$ associated with broken translational symmetry in the 3:1 canted and collinear $m = 1 / 2$ plateau states [Eq. (6)]; (b) in-plane staggered magnetization $O_{U (1)}$ [Eq. (5)]; (c) spin stiffness $ρ_{S}$ ; (d) heat capacity $C_{h}$ ; (e) magnetic susceptibility $χ$ . The inset to (a) shows the energy distribution at the transition from paramagnet to 3:1 canted phase. The inset to (b) shows the crossing of the Binder cumulants associated with $O_{U (1)}$ at the transition from the 3:1 canted phase to the collinear $m = 1 / 2$ plateau state.Reuse & Permissions
Figure 10
Magnetic phases of a layered triangular-lattice antiferromagnet with exchange interactions $J_{1} = 1$ , $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ and easy-axis anisotropy $D = 0$ .02. (a) Phase diagram as a function of temperature and magnetic field. Open symbols on the $h$ axis show transitions obtained in mean-field theory. Phase boundaries at finite temperature are obtained from Monte Carlo simulation for a cluster of $24 \times 24 \times 8$ spins and determined by peaks in the relevant order parameter susceptibilities. All phase transitions are first order, except where shown with a dashed line. Thick purple dashed line is obtained through a Landau expansion for the supersolid transition. (b) Phase diagram as a function of temperature and magnetization. Solid lines running left-right show cuts at constant magnetic field $h$ taken from simulations. Thick purple dashed lines show phase boundaries obtained through a low- $T$ expansion. Finite anisotropy lifts $T = 0$ degeneracy in favor of the stripe and supersolid phases, relegating the spin flop to finite-temperature only.Reuse & Permissions
Figure 11
Comparison of zero-temperature mean-field energies for the canted spin flop (blue, dashed line) and magnetic supersolid (red solid line) states for parameters $J_{2} = 0.15$ and $J_{⊥} = - 0.15$ as a function of magnetic field $h$ . A continuous phase transition into the supersolid state occurs at $h_{SSD} = 2 D$ , anticipating the first-order spin-flop transition which would otherwise have occurred at $h_{SPF}$ , Eq. (11).Reuse & Permissions
Figure 12
Classical spin-wave dispersion ( $S^{x}$ modes only) for $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ , and $D = 0.02$ , at a magnetic field $h = 2 D$ associated with continuous transition from collinear stripe phase into supersolid through closing of the spin gap at $M^{'}$ . Inset shows low-energy detail.Reuse & Permissions
Figure 13
Magnetic phases of a layered triangular-lattice antiferromagnet with exchange interactions $J_{1} = 1$ , $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ and easy-axis anisotropy $D = 0.25$ . (a) Phase diagram as a function of temperature and magnetic field. Open symbols on the $h$ axis show transitions obtained in mean-field theory. Phase boundaries at finite temperature are obtained from Monte Carlo simulation for a cluster of $24 \times 24 \times 8$ spins and determined by peaks in the relevant order parameter susceptibilities. All phase transitions are first order, except where shown with a dashed line. Thick purple dashed line is obtained through a Landau expansion for the supersolid transition. (b) Phase diagram as a function of temperature and magnetization. Solid lines running left-right show cuts at constant magnetic field $h$ taken from simulations. The coexistence regions associated with first-order phase transitions are colored white. Thick purple dashed lines show phase boundaries obtained through a low- $T$ expansion. Easy-axis anisotropy stabilizes the new phases for a wide range of values.Reuse & Permissions
Figure 14
Double transition from the paramagnet into the stripe collinear antiferromagnet and then into the supersolid phase for $D = 0.25, h = 1.0$ . Translational $S^{zz} (q_{M})$ and rotational $Z_{3}$ [Eq. (18)] symmetries are broken at the same temperature $T \approx 0.37$ (a), where system transitions into the collinear phase. The first-order character of the outer transition is confirmed by the double peak in energy histograms near the critical temperature in the inset to (a). A further continuous symmetry is broken at $T \approx 0.16$ at the onset of the supersolid phase, as observed in the $U (1)$ order parameter (b) and spin stiffness (c). The different nature of these transitions is clear in heat capacity (d) and magnetic susceptibility (e).Reuse & Permissions
Figure 15
Continuous behavior of the collinear-supersolid transition for $D = 0.25, h = 1.0$ at $T \approx 0.17$ . (a) Data collapse for the supersolid $U (1)$ order parameter susceptibility using 3D $XY$ universality class critical exponents. (b) The Binder cumulant for the same order parameter calculated for different lattice sizes all cross at a single temperature $T = 0.168 (1)$ .Reuse & Permissions
Figure 16
Classical spin-wave dispersion ( $S^{x}$ modes only) at $D = 0.25$ for values of magnetic field associated with continuous transitions at high field. (a) Closing of the spin gap at $M$ for $h = 1.95$ , associated with transition from the supersolid phase into the 2:1:1 phase. (b) Closing of the spin gap at $Γ$ for $h = 5.48$ , associated with transition from the $m = 1 / 2$ plateau into the 3:1 canted phase. The points $M$ , $M^{'}$ , and $Γ$ are defined in Fig. 4b.Reuse & Permissions
Figure 17
Double phase transition at $D = 0.25, h = 2.5$ from the paramagnet into $m = 1 / 2$ plateau and then into 2:1:1 canted state at lower temperature. (a) Translational symmetry measured by $S^{zz} (q_{M})$ is broken at $T \approx 0.4$ in the first-order transition—cf. bimodal energy distribution (inset)—into the plateau. (b) $U (1)$ continuous symmetry and (c) spin stiffness are broken in the continuous transition—cf. crossing of $U (1)$ order parameter Binder cumulants in inset to (b)—from the plateau into the canted phase at $T \approx 0.21$ . (d) The 2:1:1 canted state is also characterized by in-plane magnetization $m_{⊥}$ (d). The different nature of these transitions is resolved in heat capacity (e) and magnetic susceptibility (f).Reuse & Permissions
Figure 18
Double phase transition at $D = 0.25, h = 5.3$ from the paramagnet to 3:1 canted state and then into $m = 1 / 2$ plateau. The 3:1 canted state is heralded by its discrete $S^{zz} (q_{M})$ (a) and continuous $U (1)$ order parameters (b), and also by finite spin stiffness (c). This state also has a finite in-plane magnetization $m_{⊥}$ (d). The upper transition first-order character is clear in the double-peaked energy energy distribution close to the critical temperature in inset to (a), while the continuous character of the lower transition is shown by the Binder cumulant for the $U (1)$ order parameter, in inset to (b). The different nature of these transitions is resolved in heat capacity (e) and magnetic susceptibility (f).Reuse & Permissions
Figure 19
Temperature dependence of order parameters associated with 3:1 canted phase close to saturation. The continuous nature of the phase transition is confirmed by the single-valued energy histogram close to the critical temperature in inset to (a), and crossing of Binder cumulants associated with $U (1)$ order parameter in inset to (b).Reuse & Permissions
Figure 20
Vanishing discontinuities in magnetization for $D = 0.25$ , extracted from temperature cuts in Fig. 13, indicate a crossover from first- to second-order phase transition at the paramagnet–3:1 canted state transition, where the critical endpoint is found at $T \approx 0.14$ , $h \approx 6.5$ . Line is a guide to the eye.Reuse & Permissions
Figure 21
Region where four phases compete at intermediate field for $D = 0.25$ (a) and $D = 0.325$ (b) for system size $L = 8$ . Phase transitions are labeled according to the symmetries broken. Three distinct critical points are identified in a narrow field range; however, these never merge into a single tetracritical point. This structure is insensitive to changes in the easy-axis anisotropy as long as the $T = 0$ mean-field analysis yields the same phases. All phase transitions are first order, except where shown with a dashed line. Dashed horizontal lines in (a) show cuts at fixed field studied in Fig. 22.Reuse & Permissions
Figure 22
Successive phase transitions at $D = 0.25$ for system size $L = 8$ as a function of field as resolved in the susceptibilities associated with breakdown of translational symmetry $S^{zz} (q_{M})$ , spin-rotation symmetry $U (1)$ , lattice rotation symmetry $Z_{3}$ , and in-plane magnetization $m_{⊥}$ . (a) At high field, $h = 1.85$ , the $m = 1 / 2$ plateau transforms into the 2:1:1 canted state, followed by the supersolid phase. (b) For intermediate field, $h = 1.70$ , there is a direct first-order phase transition from the $m = 1 / 2$ plateau into the supersolid state, as evidenced by the coincidence the susceptibility peaks at $T \approx 0.28$ . (c) At lower field, $h = 1.55$ , the system transitions from paramagnet into $m = 1 / 2$ plateau, then into the stripe phase and finally into into supersolid state. Phase transitions are first order except where indicated with dashed line.Reuse & Permissions
Figure 23
Discontinuities in magnetization $Δ m^{z}$ on entering the $m = 1 / 2$ plateau from the supersolid and 2:1:1 canted phases for $D = 0.25$ and intermediate field, extracted from temperature cuts in Fig. 13. The collapse in $Δ m^{z}$ for $h = 1.81 (2)$ indicates that the transition from the 2:1:1 canted phase to the $m = 1 / 2$ plateau becomes first order before merging with the transition from the supersolid to the $m = 1 / 2$ plateau. This precludes a multicritical point. The line is a guide to the eye.Reuse & Permissions
Figure 24
Magnetic phases of a layered triangular-lattice antiferromagnet with $J_{1} = 1$ , $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ and easy-axis anisotropy $D = 0.5$ . (a) Phase diagram as a function of temperature and magnetic field. Open symbols on the $h$ axis show transitions obtained in mean-field theory. Phase boundaries at finite temperature are obtained from Monte Carlo simulation for a cluster of $24 \times 24 \times 8$ spins and determined by peaks in the relevant order parameter susceptibilities. All phase transitions are first order, except where shown with a dashed line. Thick purple dashed line is obtained through a Landau expansion for the supersolid transition. (b) Phase diagram as a function of temperature and magnetization. Solid lines running left-right show cuts at constant magnetic field $h$ taken from simulations. The coexistence regions associated with first-order phase transitions are colored white. Thick purple dashed lines show phase boundaries obtained through a low- $T$ expansion. Increasing anisotropy gives rise to an $m = 1 / 3$ plateau with a different unit cell from the other phases.Reuse & Permissions
Figure 25
Double phase transition from the paramagnet to $m = 1 / 2$ plateau and then into the $m = 1 / 3$ plateau at $D = 0.5, h = 1.89$ . Temperature dependence of the structure factor measured at momenta corresponding to four-sublattice ordering $S^{zz} (q_{M})$ (a), and momenta corresponding to three-sublattice ordering $S^{zz} (q_{K})$ (b). Both paramagnet– $m = 1 / 2$ plateau and plateau-plateau phase transitions are first order, as seen in the energy histograms in insets to (a) and (b); however, the magnetization (c) does not exhibit the characteristic jump across the inner transition.Reuse & Permissions
Figure 26
Magnetic phases of a layered triangular-lattice antiferromagnet with $J_{1} = 1$ , $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ and easy-axis anisotropy $D = 0.675$ . (a) Phase diagram as a function of temperature and magnetic field. Open symbols on the $h$ axis show transitions obtained in mean-field theory. Phase boundaries at finite temperature are obtained from Monte Carlo simulation for a cluster of $24 \times 24 \times 8$ spins and determined by peaks in the relevant order parameter susceptibilities. All phase transitions are first order, except where shown with a dashed line. Thick purple dashed line is obtained through a Landau expansion for the supersolid transition. (b) Phase diagram as a function of temperature and magnetization. Solid lines running left-right show cuts at constant magnetic field $h$ taken from simulations. The coexistence regions associated with first-order phase transitions are colored white. Thick purple dashed lines show phase boundaries obtained through a low- $T$ expansion. Further increasing anisotropy leads to suppression of supersolid phases.Reuse & Permissions
Figure 27
Double first-order transition at $D = 0.675, h = 6.4$ from the paramagnet into 3:1 canted state, and from 3:1 canted state into the $m =$ 1/2 plateau. Temperature dependence of the structure factor $S^{zz} (q_{M})$ (a), $U (1)$ order parameter (b), and magnetic susceptibility (c). The first-order character of both transitions is clear in the order parameter jumps and double-peaked energy distributions close to the critical temperatures in insets to (a) and (b). The lower transition is continuous for smaller $D$ , e.g., $D = 0.25$ (Fig. 18).Reuse & Permissions
Figure 28
(a) Magnetic phases of a layered triangular-lattice antiferromagnet with $J_{1} = 1$ , $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ and strong easy-axis anisotropy $D = 1.5$ . (a) Phase diagram as a function of temperature and magnetic field. Open symbols on the $h$ axis show transitions obtained in mean-field theory. Phase boundaries at finite temperature are obtained from Monte Carlo simulation for a cluster of $24 \times 24 \times 8$ spins and determined by peaks in the relevant order parameter susceptibilities. All phase transitions are first order. (b) Phase diagram as a function of temperature and magnetization. Solid lines running left-right show cuts at constant magnetic field $h$ taken from simulations. The coexistence regions associated with first-order phase transitions are colored white. Thick purple dashed lines show magnetization curves obtained through a low- $T$ expansion. At high anisotropy noncollinear phases have disappeared and Ising-like physics is recovered.Reuse & Permissions
Figure 29
(a) Magnetic phases of a layered triangular-lattice Ising antiferromagnet with $J_{1} = 1$ , $J_{2} = 0.15$ , $J_{⊥} = - 0.15$ . (a) Phase diagram as a function of temperature and magnetic field. Open symbols on the $h$ axis show transitions obtained in mean-field theory. Phase boundaries at finite temperature are obtained from Monte Carlo simulation for a cluster of $24 \times 24 \times 8$ spins and determined by peaks in the relevant order parameter susceptibilities. All phase transitions are first order. (b) Phase diagram as a function of temperature and magnetization. Solid lines running left-right show cuts at constant magnetic field $h$ taken from simulations. The coexistence regions associated with first-order phase transitions are colored white.Reuse & Permissions

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