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arXiv:0903.1479 · Physical Review B

Effects of impurities in spin Bose-metal phase on a two-leg triangular strip

Hsin-Hua Lai, Olexei I. Motrunich

Published Jun 12, 2009• 5 citations
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Abstract

We study the effects of nonmagnetic impurities in a spin Bose-metal (SBM) phase discovered in a two-leg triangular strip spin-1/2 model with ring exchanges (D. N. Sheng et al., arXiv:0902.4210). This phase is a quasi-one-dimensional (quasi-1D) descendant of a two-dimensional (2D) spin liquid with spinon Fermi sea and the present study aims at interpolating between the 1D and 2D cases. Different types of defects can be treated as local-energy perturbations, which we find are always relevant. As a result, a nonmagnetic impurity generically cuts the system into two decoupled parts. We calculate bond energy and local spin susceptibility near the defect, both of which can be measured in experiments. The spin Bose metal has dominant correlations at characteristic incommensurate wave vectors that are revealed near the defect. Thus, the bond energy shows a static texture oscillating as a function of distance from the defect and decaying as a slow power law. The local spin susceptibility also oscillates and actually increases as a function of distance from the defect, similar to the effect found in the 1D chain [S. Eggert and I. Affleck, Phys. Rev. Lett. 75, 934 (1995)]. We calculate the corresponding power-law exponents for the textures as a function of one Luttinger parameter of the SBM theory.

DOI
10.1103/physrevb.79.235120

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We study the effects of nonmagnetic impurities in a spin Bose-metal (SBM) phase discovered in a two-leg triangular strip spin-1/2 model with ring exchanges (D. N. Sheng et al., arXiv:0902.4210). This phase is a quasi-one-dimensional (quasi-1D) descendant of a two-dimensional (2D) spin liquid with spinon Fermi sea and the present study aims at interpolating between the 1D and 2D cases. Different types of defects can be treated as local-energy perturbations, which we find are always relevant. As a result, a nonmagnetic impurity generically cuts the system into two decoupled parts. We calculate bond energy and local spin susceptibility near the defect, both of which can be measured in experiments. The spin Bose metal has dominant correlations at characteristic incommensurate wave vectors that are revealed near the defect. Thus, the bond energy shows a static texture oscillating as a function of distance from the defect and decaying as a slow power law. The local spin susceptibility also oscillates and actually increases as a function of distance from the defect, similar to the effect found in the 1D chain [S. Eggert and I. Affleck, Phys. Rev. Lett. 75, 934 (1995)]. We calculate the corresponding power-law exponents for the textures as a function of one Luttinger parameter of the SBM theory.
Recent experiments on triangular lattice organic Mott insulators have found evidence for a two-dimensional (2D) spin liquid in close proximity to the metal-insulator transition. A Gutzwiller wave function study of the triangular lattice Heisenberg model with a four-spin ring exchange term appropriate in this regime has found that the projected spinon Fermi sea state has a low variational energy. This wave function, together with a slave particle-gauge theory analysis, suggests that this putative spin liquid possesses spin correlations that are singular along surfaces in momentum space, i.e., ``Bose surfaces.'' Signatures of this state, which we will refer to as a ``spin Bose metal'' (SBM), are expected to manifest in quasi-one-dimensional (quasi-1D) ladder systems: the discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes. Here, we search for a quasi-1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a two-leg triangular strip (zigzag chain). Using density matrix renormalization group (DMRG) supplemented by variational wave functions and a bosonization analysis, we map out the full phase diagram. In the absence of ring exchange the model is equivalent to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, and we find the expected Bethe-chain and dimerized phases. Remarkably, moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram. Spin and dimer correlations possess singular wave vectors at particular ``Bose points'' (remnants of the 2D Bose surface) and allow us to identify this phase as the hoped for quasi-1D descendant of the triangular lattice SBM state. We use bosonization to derive a low-energy effective theory for the zigzag spin Bose metal and find three gapless modes and one Luttinger parameter controlling all power law correlations. Potential instabilities out of the zigzag SBM give rise to other interesting phases such as a period-3 valence bond solid or a period-4 chirality order, which we discover in the DMRG. Another interesting instability is into a spin Bose-metal phase with partial ferromagnetism (spin polarization of one spinon band), which we also find numerically using the DMRG.
We consider Zeeman field effects on a spin Bose-metal (SBM) phase on a two-leg triangular ladder. This phase was found in a spin-1/2 model with ring exchanges [D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Phys. Rev. B 79, 205112 (2009)] and was also proposed to appear in an interacting electronic model with longer-ranged repulsion [H.-H. Lai and O. I. Motrunich, Phys. Rev. B 81, 045105 (2010)]. Using bosonization of a spinon-gauge theory, we study the stability of the SBM phase and its properties under the field. We also explore phases arising from potential instabilities of the SBM; in all cases, we find a gap to spin-1 excitations while spin-nematic correlations are power law. We discuss two-dimensional analogues of these phases where spinons can pair with their own species.
We consider an electronic model for realizing the spin Bose-metal (SBM) phase on a two-leg triangular strip---a spin liquid phase found by Sheng et al. [Phys. Rev. B 79, 205112 (2009)] in a spin-1/2 model with ring exchanges. The SBM can be viewed as a ``C1S2'' Mott insulator of electrons where the overall charge transporting mode is gapped out. We start from a two-band ``C2S2'' metal and consider extended repulsion motivated by recent ab initio derivation of electronic model for $\ensuremath{\kappa}\text{-ET}$ spin liquid material [K. Nakamura et al., J. Phys. Soc. Jpn. 78, 083710 (2009)]. Using weak coupling renormalization group analysis, we find that the extended interactions allow much wider C2S2 metallic phase than in the Hubbard model with on-site repulsion only. An eight-fermion umklapp term plays a crucial role in producing a Mott insulator but cannot be treated in weak coupling. We use bosonization to extend the analysis to intermediate coupling and study phases obtained out of the C2S2 metal upon increasing overall repulsion strength, finding that the SBM phase is a natural outcome for extended interactions.
The recent experimental realizations of spin-1/2 gapless quantum spin liquids in two-dimensional triangular lattice organic compounds EtMe3Sb[Pd(dmit)2]2 and κ-(ET)2Cu2(CN)3 have stimulated the investigation of the gapless spin liquid theories. The models in dimensions greater than one (D>1) usually involve multispin interactions, such as ring exchange interactions, that are difficult to study, while effective gauge theory descriptions are not well-controlled to give reliable physics information. Driven by the need for a systematic and controlled analysis of such phase, such models on ladders are seriously studied. This thesis first focuses on such ladder models. We propose that the gapless spin liquid phase can be accessed from a two-band interacting electron model by metal-Mott insulator phase transition. We use Bosonization analysis and weak-coupling Renormalization Group to further study the gapless spin liquid state in the presence of Zeeman magnetic fields or orbital magnetic fields. Several new exotic gapless spin liquids with dominant spin nematic correlations are predicted. In such a ladder spin liquid, we also consider the impurity effects. We conclude that the local energy textures and oscillating spin susceptibilities around the impurities are nontrivial and can be observed in the experiments. We then shift our focus to another theoretical candidate, an SU(2)-invariant spin liquid with Majorana excitations, which can also qualitatively explain the experimental phenomenology. We construct an exactly solvable Kitaev-type model realizing the long-wavelength Majorana spin liquid state and study its properties. We find that the state has equal power-law spin and spin-nematic correlations and behaves nontrivially in the presence of Zeeman magnetic fields. Finally, we realize such Majorana spin liquid states on a two-leg ladder and further explore their stability. We conclude the states can be stable against short-range interactions and gauge field fluctuations.
Magnetic correlations are studied in a two-leg Hubbard ladder doped with nonmagnetic impurities using finite temperature quantum Monte Carlo (FTQMC) method. A number of relevant macroscopic quantities calculated as a function of temperature T and impurity concentration are found to exhibit significantly renormalized behavior, a feature noted in many experiments. The uniform magnetic susceptibility χ0(T) for the impurity doped system shows a cusplike behavior followed by a sharp increase at lower temperatures, signaling the formation of impurity induced free magnetic moments, while it diminishes rapidly for the pure system. The spin correlations are found to increase in the vicinity of an impurity. To investigate the effect of nonmagnetic impurities on the spin wave modes in the system, spin wave velocity vsw is calculated as a function of impurity concentration. The result shows that vsw is reduced by nonmagnetic impurities, indicating softening of the modes. Simulations are performed slightly away from half filling and kinetic energy of the electrons in the neighborhood of an impurity site is calculated to study the interplay between impurities and holes. However, contrary to previous experimental observations on two dimensional lattices where nonmagnetic impurities reduce hole mobility, no noticeable effect of impurity on the mobility of the neighboring electrons is found in our calculation.
We investigate the effective interaction between two localized spin impurities embedded in a frustrated spin-12\frac {1}{2}21​ J1J_1J1​–J2J_2J2​ 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