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Order-by-disorder in magnets with frustrated spin interactions -- classical and large-$S$ limits via the spin functional integral

Peng Rao, Johannes Knolle

TL;DR

This paper addresses order-by-disorder in magnets with frustrated spin interactions by developing a spin functional-integral framework that captures both quantum and thermal fluctuations. It reveals that ObD arises from gradient-dependent anisotropic interactions among pseudo-Goldstone modes, which generate a gradient-independent effective potential and open a gap $oldsymbol{Δ}$ in the pseudo-Goldstone spectrum. Explicit one-loop calculations for the type-II cubic compass model and the type-I square compass model reproduce the established large-$S$ and classical low-$T$ scalings, $oldsymbol{Δ \\sim O(S)}$ or $\\sim T$ (type-II) and $oldsymbol{Δ \\sim \sqrt{S}}$ or $\\\\sim \sqrt{T}$ (type-I). The approach provides a unified route to ObD in higher-spin and higher-$m$-spin interaction systems, extending beyond the Holstein-Primakoff regime and suggesting directions for AFM ObD and non-collinear magnets.

Abstract

We investigate spin systems with extensive degeneracies in the classical ground states due to anisotropic frustrated spin interactions, where the degeneracy is not protected by symmetry. Using spin functional integration, we study the lifting of the degeneracies by fluctuations called order-by-disorder (ObD), and the associated gap in the spin-wave spectrum. It is shown that ObD corresponds to gradient-dependent anisotropic interactions of the pseudo-Goldstone modes, which vanish for a classical uniform spin configuration. Fluctuations generate a gradient-independent effective potential which determines the ground state and the pseudo-Goldstone gap. Furthermore, we recover previous predictions for the pseudo-Goldstone gap in type-I and II ObD with two-spin interactions in the large spin-$S$ limit or the classical small temperature limit, by computing the gap explicitly for the type-II cubic compass model and the type-I square compass model. We show that these two limits correspond to the one-loop approximation for the effective potential. We also discuss other types of order by disorder due to $m$-spin interactions where $m>2$.

Order-by-disorder in magnets with frustrated spin interactions -- classical and large-$S$ limits via the spin functional integral

TL;DR

This paper addresses order-by-disorder in magnets with frustrated spin interactions by developing a spin functional-integral framework that captures both quantum and thermal fluctuations. It reveals that ObD arises from gradient-dependent anisotropic interactions among pseudo-Goldstone modes, which generate a gradient-independent effective potential and open a gap in the pseudo-Goldstone spectrum. Explicit one-loop calculations for the type-II cubic compass model and the type-I square compass model reproduce the established large- and classical low- scalings, or (type-II) and or (type-I). The approach provides a unified route to ObD in higher-spin and higher--spin interaction systems, extending beyond the Holstein-Primakoff regime and suggesting directions for AFM ObD and non-collinear magnets.

Abstract

We investigate spin systems with extensive degeneracies in the classical ground states due to anisotropic frustrated spin interactions, where the degeneracy is not protected by symmetry. Using spin functional integration, we study the lifting of the degeneracies by fluctuations called order-by-disorder (ObD), and the associated gap in the spin-wave spectrum. It is shown that ObD corresponds to gradient-dependent anisotropic interactions of the pseudo-Goldstone modes, which vanish for a classical uniform spin configuration. Fluctuations generate a gradient-independent effective potential which determines the ground state and the pseudo-Goldstone gap. Furthermore, we recover previous predictions for the pseudo-Goldstone gap in type-I and II ObD with two-spin interactions in the large spin- limit or the classical small temperature limit, by computing the gap explicitly for the type-II cubic compass model and the type-I square compass model. We show that these two limits correspond to the one-loop approximation for the effective potential. We also discuss other types of order by disorder due to -spin interactions where .

Paper Structure

This paper contains 8 sections, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Functional integration for spins
  3. Cubic Compass model
  4. Square Compass model
  5. Scaling arguments for ObD
  6. Conclusion
  7. Funding information
  8. Effective Lagrangian for the square compass model

Figures (3)

  • Figure 1: (a) the cubic lattice for the cubic compass model \ref{['eq:Hamiltonian-cubic-compass']}. (b) the square lattice for the square compass model \ref{['eq:Hamiltonian-square-compass']}. The nearest neighbour $x,y,z$ bonds are shown in orange, green and red respectively.
  • Figure 2: Diagrammatic representation of the cubic compass model Eq. \ref{['eq:Lagrangian-cubic-compass-1']}. (a) the interaction vertex corresponding to Eq. \ref{['eq:vertex-expansion']} for $n=0, 1$. The dashed lines represent the gradient-independent term given by the slow modes $\bm{\psi}_+$. (b) The second order effective potential where only the $n=0$ vertices enter. (c) The second order effective potential with $n=1$ vertices.
  • Figure 3: One-loop expansion of the gradient-dependent vertices Eq. \ref{['eq:vertex-general']}.