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Acta Physica Polonica B

Probing Frustrated Spin Systems with Impurities

Maksymilian Kliczkowski, J. Grabowski, Maciej M. Maśka

Published May 15, 2026• 0 citations
Publisher

Abstract

We investigate the effective interaction between two localized spin impurities embedded in a frustrated spin-\(\frac {1}{2}\) \(J_1\)–\(J_2\) 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 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. Abstract Published by the Jagiellonian University 2026 authors

DOI
10.5506/aphyspolb.57.5-a8

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We investigate the effective interaction between two localized spin impurities embedded in a frustrated spin-(\frac {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 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. Abstract Published by the Jagiellonian University 2026 authors
We investigate the effective interaction between two localized spin impurities embedded in a frustrated spin-1/2 J1 ⁣− ⁣J2J_1\!-\!J_2J1​−J2​ 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.
The competition between the Kondo screening and indirect magnetic exchange in systems with two magnetic impurities coupled to a conventional sss-wave superconductor gives rise to a nontrivial ground-state phase diagram. Here, we utilize the density matrix renormalization group (DMRG) method and exploit the non-Abelian spin-SU(2) symmetry to study the phase diagram for two quantum spin-12\frac{1}{2}21​ impurities locally exchange coupled to large one-dimensional chains. The nonlocal interimpurity exchange is treated as an emergent Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling. We find qualitatively different phase diagrams for impurity spins coupled to sites with odd or even distances ddd on the chain and a partial Kondo-screened spin-doublet phase that extends over the whole range of local exchange couplings JJJ in the limit of weak superconducting pairing strength $\mathrm{\ensuremath{\Delta}}$. Our numerical DMRG studies are complemented by exact diagonalization of small (quantum-box) systems and by perturbative-in-JJJ computations of the ddd- and $\mathrm{\ensuremath{\Delta}}$-dependent RKKY interaction. It is thereby demonstrated that the specific system geometry is essential for our understanding of magnetic impurity interactions in superconducting hosts, and thus for insights into the control of quantum-state properties in nanoparticle systems and topological superconductivity.
In magnets with strong quantum fluctuations, paramagnetic ground states with or without confinement of spinon excitations can be realized. Here we discuss the physics of the confined phase in the vicinity of a confinement-deconfinement transition: this regime, likely relevant to a multitude of frustrated spin systems, is characterized by multiple length scales. In addition to the magnetic correlation length, a confinement length can be defined, which can be probed, e.g., by local static measurements near nonmagnetic impurities. We illustrate the ideas by explicit calculations for dimerized spin chains, but our qualitative results remain valid in higher dimensions as well. In particular, we study crossover from weak to strong confinement visible in the antiferromagnetic polarization cloud around a nonmagnetic impurity. We also discuss the effective magnetic interaction between impurities, relevant for impurity-induced magnetic order, and consequences for nuclear magnetic resonance and neutron-scattering experiments.
The Kondo effect in a one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model in the gapless XY\mathrm{XY}XY regime $(\ensuremath{-}1<\ensuremath{\Delta}<~1)$ is studied both analytically and numerically. In our model an impurity spin (S=1/2)(S=1/2)(S=1/2) is coupled to a single spin in the XXZ\mathrm{XXZ}XXZ spin chain. Perturbative renormalization-group (RG) analysis is performed for various limiting cases to deduce low-energy fixed points. It is shown that in the ground state the impurity spin is screened by forming a singlet with a spin in the host XXZ\mathrm{XXZ}XXZ chain. In the antiferromagnetic side $(0<\ensuremath{\Delta}<~1)$ the host chain is cut into two semi-infinite chains by the singlet. In the ferromagnetic side $(\ensuremath{-}1<\ensuremath{\Delta}<0),$ on the other hand, the host XXZ\mathrm{XXZ}XXZ chain remains as a single chain through ``healing'' of a weakened bond in the low-energy (long-distance) limit. The density-matrix renormalization-group method is used to study the size scaling of finite-size energy gaps and the power-law decay of correlation functions in the ground state. The numerical results are in good agreement with the predictions of the RG analysis. Low-temperature behaviors of specific heat and susceptibility are also discussed.
We consider a magnetic S = 1/2 impurity in the antiferromagnetic spin chain as a function of two coupling parameters: the symmetric coupling of the impurity to two sites in the chain J1 and the coupling between the two sites J2. By using field theory arguments and numerical calculations we can identify all possible fixed points and classify the renormalization flow between them, which leads to a nontrivial phase diagram. Depending on the detailed choice of the two (frustrating) coupling strengths, the stable phases correspond either to a decoupled spin with Curie law behavior or to a non-Fermi-liquid fixed point with a logarithmically diverging impurity susceptibility as in the two-channel Kondo effect. Our results resolve a controversy about the renormalization flow.
Quantum impurities can host exotic many-body states that serve as sensitive probes of bath correlations. However, quantitative and non-perturbative methods for determining impurity thermodynamics in such settings remain scarce. Here, we introduce an impurity renormalization group approach that merges the tensor-network representation with the numerical renormalization group cutoff scheme. This method overcomes conventional limitations by treating bath correlations and impurity interactions on an equal footing. Applying our approach to the finite-temperature quantum critical regime of quantum spin systems, we uncover striking impurity-induced phenomena. In a coupled Heisenberg ladder, the impurity triggers a fractionalization of the local magnetic moment. Moreover, the derivative of the impurity susceptibility develops cusps that mark the crossover into the quantum critical regime. We also observe an exotic evolution of the spin correlation function driven by the interplay between bath correlations and the impurity. Our results demonstrate that this method can efficiently solve correlated systems with defects, opening new pathways to discovering novel impurity physics beyond those in non-interacting thermal baths.