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Thermodynamics of the pyrochlore-lattice quantum Heisenberg antiferromagnet

Patrick Müller, Andre Lohmann, Johannes Richter, and Oleg Derzhko
Phys. Rev. B 100, 024424 – Published 23 July 2019
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  • INTRODUCTION
  • MODEL
  • METHODS
  • ZERO-TEMPERATURE PROPERTIES
  • FINITE-TEMPERATURE PROPERTIES
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We use the rotation-invariant Green's function method (RGM) and the high-temperature expansion to study the thermodynamic properties of the Heisenberg antiferromagnet on the pyrochlore lattice. We discuss the excitation spectra as well as various thermodynamic quantities, such as spin correlations, uniform susceptibility, specific heat, and static and dynamical structure factors. For the ground state we present RGM data for arbitrary spin quantum numbers S. At finite temperatures we focus on the extreme quantum cases S=12 and 1. We do not find indications for magnetic long-range order for any value of S. We discuss the width of the pinch point in the static structure factor in dependence on temperature and spin quantum number. We compare our data with experimental results for the pyrochlore compound NaCaNi2F7 (S=1). Thus, our results for the dynamical structure factor agree well with the experimentally observed features at 3 ...8 meV for NaCaNi2F7. We analyze the static structure factor Sq to find regions of maximal Sq. The high-temperature series of the Sq provide a fingerprint of weak order by disorder selection of a collinear spin structure, where (classically) the total spin vanishes on each tetrahedron and neighboring tetrahedra are dephased by π.

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    • Received 26 January 2019
    • Revised 22 June 2019

    DOI:https://doi.org/10.1103/PhysRevB.100.024424

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Frustrated magnetism
    1. Physical Systems
    AntiferromagnetsPyrochlores
    1. Techniques
    Green's function methodsHeisenberg model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Patrick Müller1, Andre Lohmann1, Johannes Richter1,2, and Oleg Derzhko3,2,4

    • 1Institut für Physik, Otto-von-Guericke-Universität Magdeburg, P.O. Box 4120, 39016 Magdeburg, Germany
    • 2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
    • 3Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitskii Street 1, 79011 L'viv, Ukraine
    • 4Department for Theoretical Physics, Ivan Franko National University of L'viv, Drahomanov Street 12, 79005 L'viv, Ukraine

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    Vol. 100, Iss. 2 — 1 July 2019

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    • Figure 1
      Figure 1

      (Top) The pyrochlore lattice visualized here as a three-dimensional structure which consists of alternating kagome (green) and triangular (blue) planar layers. The four-site unit cell is marked with the red bonds. A red bond also indicates the nearest-neighbor correlation function c100. The path which connects the two sites entering the correlation function c110 (c200) is colored in cyan (violet) (see the main text). (Bottom) The first Brillouin zone of a face-centered-cubic Bravais lattice. The points Γ, X, W, K, U, and L in the q space are given by Γ=(0,0,0), X=(0,2π,0), W=(π,2π,0), K=(3π/2,3π/2,0), U=(π/2,2π,π/2), and L=(π,π,π) (see, e.g., Refs. [18, 31]).

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    • Figure 2
      Figure 2

      RGM results (blue symbols) for the ground-state energy E0/(NS2) (top) and the ground-state uniform susceptibility χ0 (bottom) of the PHAF (J=1) as a function of the inverse spin-quantum number 1/S. The black symbols in the upper panel correspond to the results of Ref. [18] (filled circles), Refs. [6, 10] (open circles), Ref. [71] (pentagons), Ref. [7] (up triangles), Ref. [9] (crosses), Ref. [17] (down triangles); squares and diamonds correspond to exact-diagonalization data for N=28 and 36, respectively [23]. The inset contains some data for S=12 using an enlarged scale. The black square in the lower panel corresponds to the result of classical Monte Carlo simulations [72].

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    • Figure 3
      Figure 3

      (Top) Dispersion of the excitation energies ωγq/S [Eq. (3.3), J=1] at zero temperature T=0 for S=12 (thick), S=1 (thin), and S=3 (very thin). The points Γ, X, W, and K in the first Brillouin zone of a face-centered-cubic Bravais lattice are given by Γ=(0,0,0), X=(0,2π,0), W=(π,2π,0), and K=(3π/2,3π/2,0) (see Fig. 1, bottom). (Bottom, main panel) The ground-state excitation energies ωγq/S in dependence on the inverse spin-quantum number 1/S at q=(3π/2,3π/2,0) (K point). (Bottom, inset) Normalized RGM ground-state excitation velocity v/S in dependence on 1/S.

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    • Figure 4
      Figure 4

      Ground-state correlation functions 〈Ŝ0·ŜR〉/[S(S+1)] within a range of separation |R|/d≤3, where d=2/4≈0.35 is the nearest-neighbor separation, c100<0 (R=d=2/4≈0.35), c110>0 (R=6/4≈0.61), c200>0 (R=2/2≈0.71), c2−20<0 (R=2/2≈0.71), c21−1<0 (R=10/4≈0.79), c210<0 (R=14/4≈0.94), c22−2>0 (R=1), c300<0 (R=32/4≈1.06), and c3−30>0 (R=32/4≈1.06), for the quantum PHAF obtained within the minimal-version RGM for S=12 (crosses), S=1 (down triangles), S=32 (up triangles), and S=3 (circles). Note that for several separations R inequivalent correlators exist.

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    • Figure 5
      Figure 5

      (Top) The absolute value of the ground-state correlation functions |〈Ŝ0·ŜR〉| as a function of the scaled distance |R|/u along the direction (0,12,12) for the S=12 PHAF (u=1/2, black) and for the S=12 simple-cubic HAFM along the direction (1,0,0) (u=1, blue). (Bottom) The absolute value of the ground-state correlation functions |〈Ŝ0·ŜR〉|/[S(S+1)] as a function of the separation |R| (scaled by the nearest-neighbor separation d) for the pyrochlore lattice (d=2/4≈0.35, filled symbols) and for the kagome lattice [66] (d=1, open symbols) for S=12 (red) and S=1 (black).

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    • Figure 6
      Figure 6

      Correlation length ξ as a function of the normalized temperature T/[S(S+1)] (J=1) for S=12 (red) and S=1 (black).

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    • Figure 7
      Figure 7

      Normalized static structure factor Sq/[S(S+1)] of the PHAF obtained within the RGM at T=0 for S=12 (top) and S=1 (bottom). We consider the two planes qy=qx (left) and qz=0 (right) within the (extended) Brillouin zone. The black squares in the right panels show the q points which yield the (same) maximal value of S(qx,qy,0).

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    • Figure 8
      Figure 8

      Horizontal cut (q,q,4π) (top) and vertical cut (0,0,q) (bottom) through the pinch point at (0,0,4π) for S=12 (red) and S=1 (black). RGM results at T=0 are shown by thick lines. We also show the results at T=1.7S(S+1) by thin lines (solid lines correspond to RGM results and dashed lines correspond to HTE results) (see also Sec. 5). Note that all thin lines for T=1.7S(S+1) almost coincide. The inset (bottom) shows RGM results for the width-at-half-maximum of the pinch point Δq* as a function of 1/S at T=0.

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    • Figure 9
      Figure 9

      Dynamical structure factor Sqzz(ω) of the S=12 PHAF (J=1) at qx=qy=3.6π,4π,4.4π along the line −4π≤qz≤4π for T=0. We set ε=0.1. The white lines correspond to the excitation energies ωγq (3.3).

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    • Figure 10
      Figure 10

      Dynamical structure factor Sqzz(ω) of the S=1 PHAF (J=1) at qx=qy=3.6π,4π,4.4π along the line −4π≤qz≤4π for T=0. We set ε=0.1. The white lines correspond to the excitation energies ωγq [Eq. (3.3)]. These theoretical plots may be compared to experimental data reported in the left part of Fig. 3(a) of Ref. [75].

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    • Figure 11
      Figure 11

      Dynamical structure factor Sqzz(ω) of the S=12 PHAF (J=1) at qz=3.6π,4π,4.4π along the line −4π≤qx=qy≤4π for T=0. We set ε=0.1. The white lines correspond to the excitation energies ωγq [Eq. (3.3)].

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    • Figure 12
      Figure 12

      Dynamical structure factor Sqzz(ω) of the S=1 PHAF (J=1) at qz=3.6π,4π,4.4π along the line −4π≤qx=qy≤4π for T=0. We set ε=0.1. The white lines correspond to the excitation energies ωγq [Eq. (3.3)]. Our theoretical plots may be compared to experimental data reported in the right part of Fig. 3(a) of Ref. [75].

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    • Figure 13
      Figure 13

      (Top) Dispersion of the excitation energies ωγq/S(S+1) [Eq. (3.3), J=1] at temperature T=1.5 for S=12 (thick) and S=1 (thin) and in the infinite-temperature limit T→∞ (very thin dashed). Note that ωγq/S(S+1) is independent of S at T→∞. The points Γ, X, W, and K in the first Brillouin zone of a face-centered-cubic Bravais lattice are given by Γ=(0,0,0), X=(0,2π,0), W=(π,2π,0), and K=(3π/2,3π/2,0) (see Fig. 1, bottom). (Bottom) Temperature dependence of the excitation energies ωγq/S(S+1) at the X point for S=12 (lines) and S=1 (symbols). Note that at the X point ω3q=ω4q is valid for all temperatures.

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    • Figure 14
      Figure 14

      The absolute value of correlation functions |〈Ŝ0·ŜR〉|/[S(S+1)] between nearest neighbors (thick), next-nearest neighbors (normal), and between third-nearest neighbors straight along two bonds (thin) as a function of the normalized temperature T/[S(S+1)] (J=1) for S=12 (red) and S=1 (black). Note that for S=12 the next-nearest- and third-nearest-neighbor correlators almost coincide.

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    • Figure 15
      Figure 15

      Specific heat of the PHAF obtained by RGM (solid) and HTE (dashed, Padé [6, 7] for S=12 and Padé [5, 6] for S=1) as a function of the normalized temperature T/[S(S+1)] (J=1) for S=12 (red) and S=1 (black).

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    • Figure 16
      Figure 16

      Uniform susceptibility of the PHAF within the RGM (solid) and the HTE (dashed, Padé [6,7] for S=12 and Padé [5,6] for S=1) as a function of the normalized temperature T/[S(S+1)] (J=1) for S=12 (red) and S=1 (black).

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    • Figure 17
      Figure 17

      Normalized static structure factor Sq/[S(S+1)] along two paths in q space for S=12 (red) and S=1 (black) obtained within RGM (solid) at T=0 (thick) and T=1.7S(S+1) (thin) compared with HTE data for T=1.7S(S+1) (thin dashed, 12th order for S=12 and 10th order for S=1). Here, Γ=(0,0,0), X=(0,2π,0), W=(π,2π,0), and K=(3π/2,3π/2,0) (see Fig. 1, bottom). The points 2X=(0,4π,0), Q1=(2π,4π,0), and Q0=(4π,4π,0) are located along the upper line of the black square in the right part of Fig. 7.

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    • Figure 18
      Figure 18

      Difference of the static structure factor ΔSQ=SQ1−SQ0 [scaled by S(S+1)] with Q1=(2π,4π,0) and Q0=(4π,4π,0) within the HTE approach for S=12 (12th order) and S=1,32,2,3 (10th order) as a function of the normalized temperature T/[S(S+1)].

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