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arXiv:1608.00705 · Physical Review Letters

Fractionalized Z 2 Classical Heisenberg Spin Liquids

J. Rehn, Arnab Sen, Roderich Moessner

Published Jan 27, 2017• 36 citations
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Abstract

Quantum spin systems are by now known to exhibit a large number of different classes of spin liquid phases. By contrast, for classical Heisenberg models, only one kind of fractionalized spin liquid phase, the so-called Coulomb or U(1) spin liquid, has until recently been identified: This exhibits algebraic spin correlations and impurity moments, "orphan spins," whose size is a fraction of that of the underlying microscopic degrees of freedom. Here, we present two Heisenberg models exhibiting fractionalization in combination with exponentially decaying correlations. These can be thought of as a classical continuous spin version of a Z_{2} spin liquid. Our work suggests a systematic search and classification of classical spin liquids as a worthwhile endeavor.

DOI
10.1103/physrevlett.118.047201

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Quantum spin systems are by now known to exhibit a large number of different classes of spin liquid phases. By contrast, for classical Heisenberg models, only one kind of fractionalized spin liquid phase, the so-called Coulomb or U(1) spin liquid, has until recently been identified: This exhibits algebraic spin correlations and impurity moments, "orphan spins," whose size is a fraction of that of the underlying microscopic degrees of freedom. Here, we present two Heisenberg models exhibiting fractionalization in combination with exponentially decaying correlations. These can be thought of as a classical continuous spin version of a Z_{2} spin liquid. Our work suggests a systematic search and classification of classical spin liquids as a worthwhile endeavor.
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Motivated by the recent appearance of the trillium lattice in the search for materials hosting spin liquids, we study the ground state of the classical Heisenberg model on its linegraph, the trilline lattice. We find that this network realises the recently proposed notion of a fragile spin liquid in three dimensions. Additionally, we analyze the Ising case and argue for a possible Z2\mathbb{Z}_2Z2​ quantum spin liquid phase in the corresponding quantum dimer model. Like the well-known U(1)U(1)U(1) spin liquids, the classical phase hosts moment fractionalisation evidenced in the diluted lattice, but unlike these, it exhibits exponential decay both in spin correlations and interactions between fractionalised moments. This provides the first instance of a purely short-range correlated classical Heisenberg spin liquid in three dimensions.
We study the spin-12\frac{1}{2}21​ Heisenberg model on the triangular lattice with antiferromagnetic first- (J1)({J}_{1})(J1​) and second- (J2)({J}_{2})(J2​) nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped Z2{Z}_{2}Z2​ spin liquids.
Topological Spin Liquid Phase in the Spin-1/21/21/2 Triangular Heisenberg Model
We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.
We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin SSS, we find three different symmetric Z2{Z}_{2}Z2​ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large SSS. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.
Algebraic spin liquids are quantum disordered phases of insulating magnets that exhibit fractionalized gapless excitations and power-law correlations. Quantum spin liquids in this category include the experimentally established one-dimensional (1D) Luttinger liquid, as well as the U(1) Dirac spin liquid (DSL) that has been a focus of recent candidate materials searches. Most notably, several exchange-frustrated Heisenberg materials on the triangular lattice have shown evidence of the U(1) DSL. In this work, we measure the algebraic correlations of spin-singlet excitations in the J 1 − J 2 antiferromagnetic Heisenberg model on the triangular lattice, prompting a detailed investigation of this model’s stability under spin-lattice coupling with dynamical phonons using variational Monte Carlo. As seen before in 1D spin chains, we observe a low-temperature transition from a U(1) DSL to valence bond order and predict the parameter regime where the model realizes a stable DSL ground state. To achieve this, we employ a series of finite-size scaling Ansätze inspired by the low-energy DSL’s conformal description in terms of quantum electrodynamics and show that emergent monopole operators drive the instability. We compare the physics of this transition to the 1D Luttinger liquid throughout our analysis. We derive the regime of stability against spin-Peierls ordering and argue that the DSL ground state might still be achievable in candidate materials, despite its tendency to valence bond solid ordering.