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Probing frustrated spin systems with impurities

Maksymilian Kliczkowski, Jakub Grabowski, Maciej M. Maśka

Abstract

We investigate the effective interaction between two localized spin impurities embedded in a frustrated spin-1/2 $J_1\!-\!J_2$ Heisenberg chain. Treating the impurity spins as classical moments coupled locally to the host, we combine second--order perturbation theory with large--scale density matrix renormalization group (DMRG) calculations to determine the impurity--impurity interaction as a function of separation, coupling strength, and magnetic frustration. In the weak--coupling regime, we show that the interaction is governed by the the static spin susceptibility of the host and exhibits oscillatory power--law decay in the gapless phase, modified by universal logarithmic corrections at the SU(2)--symmetric critical point. In the gapped dimerized phase, the interaction decays exponentially with distance. For intermediate and strong impurity--host coupling, we observe a crossover to a boundary--dominated regime characterized by pronounced parity effects associated with the length of the chain segment between impurities, signaling a breakdown of the simple RKKY--like description. Our results establish impurity--impurity interactions as a sensitive probe of frustrated quantum spin liquids and provide a controlled framework for distinguishing gapless and gapped phases through local perturbations.

Probing frustrated spin systems with impurities

Abstract

We investigate the effective interaction between two localized spin impurities embedded in a frustrated spin-1/2 Heisenberg chain. Treating the impurity spins as classical moments coupled locally to the host, we combine second--order perturbation theory with large--scale density matrix renormalization group (DMRG) calculations to determine the impurity--impurity interaction as a function of separation, coupling strength, and magnetic frustration. In the weak--coupling regime, we show that the interaction is governed by the the static spin susceptibility of the host and exhibits oscillatory power--law decay in the gapless phase, modified by universal logarithmic corrections at the SU(2)--symmetric critical point. In the gapped dimerized phase, the interaction decays exponentially with distance. For intermediate and strong impurity--host coupling, we observe a crossover to a boundary--dominated regime characterized by pronounced parity effects associated with the length of the chain segment between impurities, signaling a breakdown of the simple RKKY--like description. Our results establish impurity--impurity interactions as a sensitive probe of frustrated quantum spin liquids and provide a controlled framework for distinguishing gapless and gapped phases through local perturbations.
Paper Structure (8 sections, 1 theorem, 21 equations, 3 figures)

This paper contains 8 sections, 1 theorem, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Weak-coupling regime
  4. Strong-coupling regime
  5. General case
  6. Numerical results
  7. Summary and Outlook
  8. Symmetry constraint on the impurity--impurity interaction

Key Result

Lemma 1

Consider a spin system described by the Hamiltonian where: Then the spectrum of $H$, and in particular its ground-state energy, is invariant under a simultaneous rotation of the impurity spins, As a consequence, for fixed impurity spin magnitudes the ground-state energy can depend on the impurity orientations only through rotationally invariant combinations. In particular, for two impurities,

Figures (3)

  • Figure 1: Illustration how the chain is divided into three segments for $J_c\to\infty$.
  • Figure 2: Illustration how the components of the interaction energy \ref{['eq:int_en']} is calculated.
  • Figure 3: The effective interaction, $V(r,\theta=0)$ (cf. Eq. \ref{['eq:int_en']}), for different values of $J_2$ and $J_c$. The first and second columns show the results for $J_2=0.1$ and $J_2=J_2^c$, respectively. The third column shows the results for $J_2=0.3$ and $J_2=0.4$. The first, second, third, and fourth rows show the results for $J_c$ equal to $0.01, 0.05, 0.1$, and $0.5$, respectively.

Theorems & Definitions (2)

  • Lemma 1: Rotational invariance of the impurity energy
  • proof