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arXiv:1907.10241 · Physical review. B./Physical review. B

Chiral spin liquids with crystalline Z2 gauge order in a three-dimensional Kitaev model

Petr A. Mishchenko, Yasuyuki Kato, Kevin M. O'Brien, Troels Arnfred Bojesen, Tim Eschmann, Maria Hermanns, Simon Trebst, Yukitoshi Motome

Published Jan 15, 2020• 10 citations
arXivHTMLPDF

Abstract

Chiral spin liquids (CSLs) are exceptional magnetic states that break time-reversal symmetry without long-range magnetic ordering. Here, the authors showcase a family of CSLs in a three-dimensional extension of the Kitaev model on the hypernonagon lattice, using quantum Monte Carlo simulations supplemented by variational calculations. Notably, all the CSLs are found to not only break time-reversal symmetry but concurrently also a number of lattice symmetries by forming a crystalline ordering of the underlying \ensuremath{\mathbb{Z}}${}_{2}$ gauge structure. It is further shown that some of the CSLs exhibit topological features with gapless Majorana modes.

DOI
10.1103/physrevb.101.045118

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Chiral spin liquids (CSLs) are exceptional magnetic states that break time-reversal symmetry without long-range magnetic ordering. Here, the authors showcase a family of CSLs in a three-dimensional extension of the Kitaev model on the hypernonagon lattice, using quantum Monte Carlo simulations supplemented by variational calculations. Notably, all the CSLs are found to not only break time-reversal symmetry but concurrently also a number of lattice symmetries by forming a crystalline ordering of the underlying \ensuremath{\mathbb{Z}}2{}_{2}2​ gauge structure. It is further shown that some of the CSLs exhibit topological features with gapless Majorana modes.
Chiral spin liquids (CSLs) in three dimensions and thermal phase transitions to paramagnet are studied by unbiased Monte Carlo simulations. For an extension of the Kitaev model to a three-dimensional tricoordinate network dubbed the hypernonagon lattice, we derive low-energy effective models in two different anisotropic limits. We show that the effective interactions between the emergent Z2{Z}_{2}Z2​ degrees of freedom called fluxes are unfrustrated in one limit, while highly frustrated in the other. In both cases, we find a first-order phase transition to the CSL, where both time-reversal and parity symmetries are spontaneously broken. In the frustrated case, however, the CSL state is highly exotic---the flux configuration is subextensively degenerate while showing a directional order with broken C3{C}_{3}C3​ rotational symmetry. Our results provide two contrasting archetypes of CSLs in three dimensions, both of which allow approximation-free simulation for investigating the thermodynamics.
We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. This CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSL's separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number +/-1) CSL obey non-Abelian statistics.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
We study the interplay of competing interactions in spin-12\frac{1}{2}21​ triangular Heisenberg model through tuning the first- (J1{J}_{1}J1​), second- (J2{J}_{2}J2​), and third-neighbor (J3{J}_{3}J3​) couplings. Based on a large-scale density-matrix renormalization group calculation, we identify a quantum phase diagram of the system and discover a gapless chiral spin-liquid (CSL) phase in the intermediate J2{J}_{2}J2​ and J3{J}_{3}J3​ regime. This CSL state spontaneously breaks time-reversal symmetry with finite scalar chiral order, and it has gapless excitations implied by a vanishing spin triplet gap and a finite central charge on the cylinder. Moreover, the central charge grows rapidly with the cylinder circumference, indicating emergent spinon Fermi surfaces. To understand the numerical results we propose a parton mean-field spin-liquid state, the U(1)U(1)U(1) staggered flux state, which breaks time-reversal symmetry with chiral edge modes by adding a Chern insulator mass to Dirac spinons in the U(1)U(1)U(1) Dirac spin liquid. This state also breaks lattice rotational symmetries and possesses two spinon Fermi surfaces driven by nonzero J2{J}_{2}J2​ and J3{J}_{3}J3​, which naturally explains the numerical results. This realizes an example of a gapless CSL state with coexisting spinon Fermi surfaces and chiral edge states, demonstrating the rich family of interesting quantum phases emergent from competing interactions in triangular-lattice magnets.
The breaking of chiral and time-reversal symmetries provides a pathway to exotic quantum phenomena and topological phases. Recent work has extensively explored the resulting emergence of chiral charge orders and chiral spin liquids (CSLs) on the kagome lattice. Such CSLs are closely tied to bosonic fractional quantum Hall states with anyonic quasiparticles; however, their connection to nearby ordered states has remained a mystery. Here, we use spin-wave theory, parton Gutzwiller wave functions, and exact diagonalization to show that two distinct magnetic orders with uniform scalar chirality---the XYZ umbrella state and the octahedral spin crystal--emerge as competing orders in close proximity to the CSL. In this letter, we highlight the intimate link between a topologically ordered liquid and broken symmetry states with nontrivial real-space topology.
In the field of frustrated magnetism, Kitaev models provide a unique framework to study the phenomena of spin fractionalization and emergent lattice gauge theories in two and three spatial dimensions. Their ground states are quantum spin liquids, which can typically be described in terms of a Majorana band structure and an ordering of the underlying Z2{\mathbb{Z}}_{2}Z2​ gauge structure. Here we provide a comprehensive classification of the ``gauge physics'' of a family of elementary three-dimensional Kitaev models, discussing how their thermodynamics and ground state order depends on the underlying lattice geometry. Using large-scale, sign-free quantum Monte Carlo simulations we show that the ground-state gauge order can generally be understood in terms of the length of elementary plaquettes---a result which extends the applicability of Lieb's theorem to lattice geometries beyond its original scope. At finite temperatures, the proliferation of (gapped) vison excitations destroys the gauge order at a critical temperature scale, which we show to correlate with the size of vison gap for the family of three-dimensional Kitaev models. We also discuss two notable exceptions where the lattice structure gives rise to ``gauge frustration'' or intertwines the gauge ordering with time-reversal symmetry breaking. In a more general context, the thermodynamic gauge transitions in such 3D Kitaev models are one of the most natural settings for phase transitions beyond the standard Landau-Ginzburg-Wilson paradigm.