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Three-dimensional unfrustrated and frustrated quantum Heisenberg magnets. Specific heat study

T. Krokhmalskii, T. Hutak, O. Derzhko

TL;DR

The paper investigates the finite-temperature thermodynamics of the $S=1/2$ Heisenberg model on four three-dimensional lattices—simple cubic, diamond, pyrochlore, and hyperkagome—with both ferromagnetic and antiferromagnetic exchanges to isolate lattice-geometry effects on the specific heat $c(T)$. It combines quantum Monte Carlo (for ferromagnets and bipartite antiferromagnets) with high-temperature expansion data augmented by the entropy method (for frustrated antiferromagnets) to obtain $c(T)$ over broad temperature ranges and to estimate ground-state energies and low-energy scales. The results reveal conventional spin-wave behavior for bipartite magnets ($c(T) o T^{3}$ at low $T$) and expansive, broad maxima with no finite-$T$ order for frustrated lattices, along with hints of a possible hidden low-energy scale. Together, the study provides a cohesive framework for interpreting thermodynamic signatures in 3D frustrated magnets and tests the proposed two-scale scenario for low-energy excitations in these systems.

Abstract

We examine the $S=1/2$ Heisenberg magnet on four three-dimensional lattices - simple-cubic, diamond, pyrochlore, and hyperkagome ones - for ferromagnetic and antiferromagnetic signs of the exchange interaction in order to illustrate the effect of lattice geometry on the finite-temperature thermodynamic properties with a focus on the specific heat $c(T)$. To this end, we use quantum Monte Carlo simulations or high-temperature expansion series complemented with the entropy method. We also discuss a recent proposal about hidden energy scale in geometrically frustrated magnets.

Three-dimensional unfrustrated and frustrated quantum Heisenberg magnets. Specific heat study

TL;DR

The paper investigates the finite-temperature thermodynamics of the Heisenberg model on four three-dimensional lattices—simple cubic, diamond, pyrochlore, and hyperkagome—with both ferromagnetic and antiferromagnetic exchanges to isolate lattice-geometry effects on the specific heat . It combines quantum Monte Carlo (for ferromagnets and bipartite antiferromagnets) with high-temperature expansion data augmented by the entropy method (for frustrated antiferromagnets) to obtain over broad temperature ranges and to estimate ground-state energies and low-energy scales. The results reveal conventional spin-wave behavior for bipartite magnets ( at low ) and expansive, broad maxima with no finite- order for frustrated lattices, along with hints of a possible hidden low-energy scale. Together, the study provides a cohesive framework for interpreting thermodynamic signatures in 3D frustrated magnets and tests the proposed two-scale scenario for low-energy excitations in these systems.

Abstract

We examine the Heisenberg magnet on four three-dimensional lattices - simple-cubic, diamond, pyrochlore, and hyperkagome ones - for ferromagnetic and antiferromagnetic signs of the exchange interaction in order to illustrate the effect of lattice geometry on the finite-temperature thermodynamic properties with a focus on the specific heat . To this end, we use quantum Monte Carlo simulations or high-temperature expansion series complemented with the entropy method. We also discuss a recent proposal about hidden energy scale in geometrically frustrated magnets.

Paper Structure

This paper contains 15 sections, 10 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting up the problem
  3. Lattices
  4. Methods
  5. Quantum Monte Carlo
  6. High-temperature expansion series augmented by the entropy method
  7. Specific heat $c(T)$
  8. Ferromagnets
  9. Antiferromagnets
  10. Bipartite lattices
  11. Frustrated lattices
  12. Low-temperature peak and a hidden energy scale
  13. Discussion and summary
  14. Pyrochlore-lattice antiferromagnet
  15. Hyperkagome-lattice antiferromagnet

Figures (8)

  • Figure 1: (Colour online) Simple-cubic (upper left), diamond (upper right), pyrochlore (lower left), and hyper-kagome (lower right) lattices which are considered in this paper. Different colors of discs denote different sublattices, i.e., two sublattices for the diamond lattice and four sublattices for the pyrochlore lattice. The unit cell contains 1 (simple cubic), 2 (diamond), 4 (pyrochlore), or 12 (hyperkagome) sites (magnetic atoms). For a detailed description see the main text, section \ref{['s2a']}.
  • Figure 2: (Colour online) Specific heat $c(T)$ for the $S=1/2$ Heisenberg ferromagnet on four three-dimensional lattices. Quantum Monte Carlo simulations for the number of unit cells ${\cal L}=40$ (simple cubic) and ${\cal L}=30$ (pyrochlore, diamond, hyperkagome) and a small symmetry breaking field $h=10^{-4}$. Inset presents $c$ as a function of $T/T_{\rm C}$.
  • Figure 3: (Colour online) Towards a sloped shoulder in the paramagnetic phase for the hyperkagome-lattice $S = 1/2$ Heisenberg ferromagnet: high-order Padé approximants $[u,d](T)$, $u + d \leqslant 20$, to the specific heat series $c(\beta)$ versus quantum Monte Carlo simulations (which were reported in figure \ref{['fig02']}).
  • Figure 4: (Colour online) Specific heat $c(T)$ for the $S=1/2$ Heisenberg antiferromagnet on two bipartite three-dimensional lattices. Quantum Monte Carlo simulations for the number of unit cells ${\cal L}=40$ (simple cubic) and ${\cal L}=30$ (diamond) and a small symmetry breaking field $h=10^{-4}$. Inset presents $c$ as a function of $T/T_{\rm N}$.
  • Figure 5: (Colour online) Specific heat $c(T)$ for the $S=1/2$ Heisenberg antiferromagnet on two frustrated three-dimensional lattices. Entropy method results. Top: pyrochlore lattice. Solid curves correspond to $e_0=-0.49$Hagymasi2021 whereas dashed curves correspond to $e_0=-0.477$Astrakhantsev2021 with $[8,9](e)$ approximant; dash-dotted curve is taken from reference Derzhko2020. In addition, we present numerical-linked-cluster-expansion data taken from Schaefer2020 (circles) as well as simple Padé approximants $[8,9](T)$ and $[8,8](T)$ which constitute the boundary of the gray area. Bottom: hyperkagome lattice. The ground-state energy is determined within the entropy method and we took $[10,10](e)$ approximant to obtain all these entropy-method curves; dash-dotted curves are taken from the previous study Hutak2024. In addition, we present simple Padé approximants $[8,9](T)$ and $[10,10](T)$ which constitute the boundary of the gray area.
  • ...and 3 more figures