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arXiv:1106.0761 · Physical Review B

Quantum spin liquid in a spin- 12 XY model with four-site exchange on the kagome lattice

Long Dang, Stephen Inglis, Roger G. Melko

Published Oct 14, 2011• 23 citations
arXivHTMLPDF

Abstract

We study the ground-state phase diagram of a two-dimensional kagome-lattice spin-1/2 $XY$ model ($J$) with a four-site ring-exchange interaction ($K$) using quantum Monte Carlo simulations. We find that the superfluid phase, existing in the regime of small ring exchange, undergoes a direct transition to a ${\mathbb{Z}}_{2}$ quantum spin liquid phase at ${(K/J)}_{c}\ensuremath{\approx}22$, which is related to the phase proposed by L. Balents, M. P. A. Fisher, and S. M. Girvin [Phys. Rev. B 65, 224412 (2002)]. The quantum phase transition between the superfluid and the spin liquid phase has exponents $z$ and $\ensuremath{\nu}$ falling in the three-dimensional $XY$ universality class, making it a candidate for an exotic $XY$* quantum critical point, mediated by the condensation of bosonic spinons.

DOI
10.1103/physrevb.84.132409

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We study the ground-state phase diagram of a two-dimensional kagome-lattice spin-1/2 XYXYXY model (JJJ) with a four-site ring-exchange interaction (KKK) using quantum Monte Carlo simulations. We find that the superfluid phase, existing in the regime of small ring exchange, undergoes a direct transition to a Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid phase at (K/J)c\ensuremath≈22{(K/J)}_{c}\ensuremath{\approx}22(K/J)c​\ensuremath≈22, which is related to the phase proposed by L. Balents, M. P. A. Fisher, and S. M. Girvin [Phys. Rev. B 65, 224412 (2002)]. The quantum phase transition between the superfluid and the spin liquid phase has exponents zzz and \ensuremathν\ensuremath{\nu}\ensuremathν falling in the three-dimensional XYXYXY universality class, making it a candidate for an exotic XYXYXY* quantum critical point, mediated by the condensation of bosonic spinons.
The quantum spin liquid (QSL) state, proposed more than three decades ago by Fazekas and Anderson remains surprisingly elusive. Although recent experiments provide a strong evidence of their existence in the frustrated spin systems, the microscopic model for this state is still rare. The extensive theoretical framework, developed over decades, continues to extend further motivated by these and other discoveries from large-scale computer simulations of a relatively small number of models. In this work, we discuss the physics of the ground-state phase diagram of a two-dimensional Kagome lattice spin-1/2 XY model with a four-site ring-exchange interaction using quantum Monte Carlo simulation. We found the second order phase transition from superfluid state to a Z2 quantum spin liquid phase driven by the four-site ring exchange interaction. We have characterized the QSL by its vanishing order parameters such as the spin-spin structure factor, the plaquette-plaquette structure factor. Moreover, we have found the large anomalous exponent ηXY* ≈ 1.325 which belongs to a different universality class other than 3D XY universality class. There is no signal of supersolid phase intervening between the superfluid state and QSL state.
By means of quantum Monte Carlo simulations, combined with a stochastic analytic continuation, we examine the spin dynamics of the spin-1/2 planar (XYXYXY) ferromagnet on the kagome lattice with additional four-site ring exchange terms. Such exchange processes were previously considered to lead into an extended Z2{Z}_{2}Z2​ quantum spin liquid phase beyond a quantum critical point from the XYXYXY ferromagnet. We examine the dynamical spin structure factor in the nonmagnetic regime and probe for signatures of spin fractionalization. Furthermore, we contrast our findings and the corresponding energy scales of the excitation gaps in the ring exchange model to those emerging in a related Balents-Fisher-Girvin model with a Z2{Z}_{2}Z2​ quantum spin liquid phase, and monitor the softening of the magnon mode upon approaching the quantum critical point from the XYXYXY-ferromagnetic regime.
We study the impact of trimerization (breathing) of the nearest-neighbor (NN) exchange on the planar XY spin-1/21/21/2 ferromagnet on the kagome lattice, including additional four-site ring exchange. For uniform NN exchange, this model has previously been shown to transit from a long-range ordered ferromagnet into a Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid by virtue of the ring exchange. Using quantum Monte-Carlo calculations, based on the stochastic series expansion, we present results for the spin stiffness, the quantum phase diagram, the longitudinal static, as well as the transverse dynamic structure factor. Our results corroborate 3D XY universality also at finite trimerization and suggest a simple continuation of the quantum critical point of the uniform case into a line in terms of rescaled exchange parameters. Moreover, at any trimerization and in the ordered phase the elementary excitations can be understood very well in terms of linear spin wave theory, while beyond the critical line, in the spin liquid phase we find signatures of spinon continua.
Employing large-scale quantum Monte Carlo simulations, we study the extended XXZ model on the kagome lattice. A Z_{2} quantum spin liquid phase with effective even Ising gauge field structure emerges from the delicate balance among three symmetry-breaking phases including stripe solid, staggered solid, and ferromagnet. This Z_{2} spin liquid is stabilized by an extended interaction related to the Rokhsar-Kivelson potential in the quantum dimer model limit. The phase transitions from the staggered solid to a spin liquid or ferromagnet are found to be first order and so is the transition between the stripe solid and ferromagnet. However, the transition between a spin liquid and ferromagnet is found to be continuous and belongs to the 3D XY^{*} universality class associated with the condensation of spinons. The transition between a spin liquid and stripe solid appears to be continuous and associated with the condensation of visons.
We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU (2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al . We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.
We present the linear spin wave theory calculation of the superfluid phase of a hard-core boson J-K model with nearest neighbour exchange J and four-particle ring-exchange K at half filling on the triangular lattice, as well as the phase diagrams of the system at zero and finite temperatures. A similar analysis has been done on a square lattice (Schaffer et al. Phys. Rev. B, 80, 014503 (2009)). We find similar behaviour to that of a square lattice but with different spin wave values of the thermodynamic quantities. We also find that the pure J model (XY model), which has a well-known uniform superfluid phase with an ordered parameter [Formula: see text] at zero temperature is quickly destroyed by the inclusion of negative-K ring-exchange interactions, favouring a state with a (4π/3, 0) ordering wavevector. We further study the behaviour of the finite-temperature Kosterlitz–Thouless phase transition (TKT) in the uniform superfluid phase, by forcing the universal quantum jump condition on the finite-temperature spin wave superfluid density. We find that for K < 0, the phase boundary monotonically decreases to T = 0 at K/J = −4/3, where a phase transition is expected and TKT decreases rapidly, while for positive K, TKT reaches a maximum at some K ≠ 0. It has been shown on a square lattice using quantum Monte Carlo (QMC) simulations that for small K > 0 away from the XY point, the zero-temperature spin stiffness value of the XY model is decreased (Melko and Sandvik. Ann. Phys. 321, 1651 (2006)). Our result seems to agree with this trend found in QMC simulations for two-dimensional systems.