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arXiv:1403.0575 · Physical review. B./Physical review. B

Unification of bosonic and fermionic theories of spin liquids on the kagome lattice

Yuan-Ming Lu, Gil Young Cho, Ashvin Vishwanath

Published Nov 28, 2017• 31 citations
arXivHTMLPDF

Abstract

Recent numerical studies have provided strong evidence for a gapped ${Z}_{2}$ quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either Bose (Schwinger bosons) or Fermi (Abrikosov fermions) statistics, which are then treated within mean-field theory. A long-standing question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways. Here we show that all $8{Z}_{2}$ spin liquid phases in the Schwinger-boson mean-field (SBMF) construction can also be described in terms of Abrikosov fermions, unifying pairs of theories that seem rather distinct. The key idea is that for ${Z}_{2}$ spin liquid states that admit a SBMF description on the kagome lattice, the symmetry fractionalization of visions is uniquely fixed. Two promising candidate states for the kagome Heisenberg model, Sachdev's ${Q}_{1}={Q}_{2}$ SBMF state and the Lu-Ran-Lee ${Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ Abrikosov-fermion state, are found to describe the same symmetric spin liquid phase. We expect these results to aid in a complete specification of the numerically observed spin liquid phase. We also discuss a set of ${Z}_{2}$ spin liquid phases in the fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states that do not admit a SBMF description.

DOI
10.1103/physrevb.96.205150

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Recent numerical studies have provided strong evidence for a gapped Z2{Z}_{2}Z2​ quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either Bose (Schwinger bosons) or Fermi (Abrikosov fermions) statistics, which are then treated within mean-field theory. A long-standing question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways. Here we show that all 8Z28{Z}_{2}8Z2​ spin liquid phases in the Schwinger-boson mean-field (SBMF) construction can also be described in terms of Abrikosov fermions, unifying pairs of theories that seem rather distinct. The key idea is that for Z2{Z}_{2}Z2​ spin liquid states that admit a SBMF description on the kagome lattice, the symmetry fractionalization of visions is uniquely fixed. Two promising candidate states for the kagome Heisenberg model, Sachdev's Q1=Q2{Q}_{1}={Q}_{2}Q1​=Q2​ SBMF state and the Lu-Ran-Lee Z2[0,\ensuremathπ]\ensuremathβ{Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}Z2​[0,\ensuremathπ]\ensuremathβ Abrikosov-fermion state, are found to describe the same symmetric spin liquid phase. We expect these results to aid in a complete specification of the numerically observed spin liquid phase. We also discuss a set of Z2{Z}_{2}Z2​ spin liquid phases in the fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states that do not admit a SBMF description.
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge fields. Requiring U(1)\text{U}(1)U(1) spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-12\frac{1}{2}21​ systems on square, triangular, and kagome lattices, we show that all gapped symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.
Motivated by recent numerical discovery of a gapped spin liquid phase in spin-1/21/21/2 triangular-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, we classify symmetric Z2{Z}_{2}Z2​ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped Z2{Z}_{2}Z2​ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic Z2{Z}_{2}Z2​ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).
Quantum spin liquids exhibit fractionalized spin excitations as a consequence of strong quantum many-body effects. The kagome antiferromagnetic Heisenberg model is a promising candidate for a quantum spin-liquid ground state; however, the nature of its excitation spectrum remains controversial, particularly regarding the presence of a spin gap and the gauge structure coupled to fractional quasiparticles. To address these issues, parton approaches have been extensively employed, where spin operators are represented in terms of fermionic or bosonic quasiparticles within the Abrikosov fermion and Schwinger boson frameworks. Thus far, these approaches have been pursued independently, and it has remained unclear how the results obtained from these frameworks compare, particularly with respect to the spin dynamics and gauge structure of the kagome antiferromagnet. Here, we investigate the dynamical spin structure factor of the antiferromagnetic Heisenberg model with a Dzyaloshinskii-Moriya interaction on the kagome lattice, relevant to herbertsmithite, by employing both approaches. We find that the dynamical spin structure factor obtained from the Abrikosov fermion mean-field theory exhibits dome-shaped features, and that its continuum structure significantly depends on the gauge structure of the spin-liquid ansatz. On the other hand, the Schwinger boson mean-field theory yields a concave-down structure in the low-energy region, distinct from that obtained using the Abrikosov fermion approach. Moreover, incorporating many-body effects beyond the mean-field approximation substantially reduces the low-energy gap and enhances the low-energy spectral weight, consistent with experimental observations. Our results suggest the importance of many-body effects in the Schwinger boson theory for capturing the low-energy spin dynamics of kagome antiferromagnets.
The interplay of symmetry and topological order leads to a variety of distinct phases of matter, the symmetry enriched topological (SET) phases. Here we discuss physical observables that distinguish different SETs in the context of Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids with SU(2) spin rotation invariance. We focus on the cylinder geometry, and show that ground-state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally, these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as a 1D spin chain. In particular, we show that the kagome spin liquid SET can be determined by measurements on one ground state, by wrapping the kagome in a few different ways on the cylinder. In addition to guiding numerical studies, this approach provides a transparent way to connect bosonic and fermionic mean-field theories of spin liquids. When fusing quasiparticles, it correctly predicts nontrivial phase factors for combining their space group quantum numbers.
We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z2{Z}_{2}Z2​ gauge field excitations. We classify them by the projective symmetry group method, and find six spin-liquid states that are potentially relevant to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin-liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wave vector at the Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.
Motivated by recent experiments on the vanadium oxyfluoride material DQVOF, we examine possible spin liquid phases on a breathing kagome lattice of S=1/2S=1/2S=1/2 spins. By performing a projective symmetry group analysis, we determine the possible phases for both fermionic and bosonic Z2{\mathbb{Z}}_{2}Z2​ spin liquids on this lattice, and establish the correspondence between the two. The nature of the ground state of the Heisenberg model on the isotropic kagome lattice is a hotly debated topic, with both Z2{\mathbb{Z}}_{2}Z2​ and U(1) spin liquids argued to be plausible ground states. Using variational Monte Carlo techniques, we show that a gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid emerges as the clear ground state in the presence of this breathing anisotropy. Our results suggest that the breathing anisotropy helps to stabilize this spin liquid ground state, which may aid us in understanding the results of experiments and help to direct future numerical studies on these systems.