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arXiv:1305.5553 · Physical Review B

Quantum phase transition between the Z2 spin liquid and valence bond crystals on a triangular lattice

Kevin Slagle, Cenke Xu

Published Mar 21, 2014• 16 citations
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Abstract

We study the quantum phase transition between the ${Z}_{2}$ spin liquid and valence bond solid (VBS) orders on a triangular lattice. With a fully isotropic triangular lattice, the transition from a columnar or resonating-plaquette VBS order can either be first order or there could be two transitions with an intermediate phase. If the transition splits into two, then the ${Z}_{2}$ spin liquid will first experience a first-order $q=3$ Potts transition to a new nematic ${Z}_{2}$ spin liquid that breaks the $2\ensuremath{\pi}/3$ lattice rotation symmetry (but retain translation symmetry, unlike the VBS states). The second transition will then take this new nematic ${Z}_{2}$ spin liquid to a columnar or resonating-plaquette VBS state through a second-orderthree-dimensional (3D) $X{Y}^{*}$ transition. On a distorted triangular lattice, the degeneracy between some of the different columnar VBS orders is lifted, and the phase transition can reduce to a single 3D $X{Y}^{*}$ transition.

DOI
10.1103/physrevb.89.104418

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We study the quantum phase transition between the Z2{Z}_{2}Z2​ spin liquid and valence bond solid (VBS) orders on a triangular lattice. With a fully isotropic triangular lattice, the transition from a columnar or resonating-plaquette VBS order can either be first order or there could be two transitions with an intermediate phase. If the transition splits into two, then the Z2{Z}_{2}Z2​ spin liquid will first experience a first-order q=3q=3q=3 Potts transition to a new nematic Z2{Z}_{2}Z2​ spin liquid that breaks the 2\ensuremathπ/32\ensuremath{\pi}/32\ensuremathπ/3 lattice rotation symmetry (but retain translation symmetry, unlike the VBS states). The second transition will then take this new nematic Z2{Z}_{2}Z2​ spin liquid to a columnar or resonating-plaquette VBS state through a second-orderthree-dimensional (3D) XY∗X{Y}^{*}XY∗ transition. On a distorted triangular lattice, the degeneracy between some of the different columnar VBS orders is lifted, and the phase transition can reduce to a single 3D XY∗X{Y}^{*}XY∗ transition.
Motivated by recent numerical results, we study the quantum phase transitions between Z2{Z}_{2}Z2​ spin-liquid, N\'eel-ordered, and various valence-bond solid (VBS) states on the honeycomb and square lattices, with emphasis on the staggered VBS. In contrast to the well-understood columnar VBS order, the staggered VBS is not described by an XY\mathit{XY}XY-order parameter with ZN{Z}_{N}ZN​ anisotropy close to these quantum phase transitions. Instead, we demonstrate that on the honeycomb lattice, the staggered VBS is more appropriately described as an O(3)- or CP(2)-order parameter with cubic anisotropy, while on the square lattice it is described by an O(4)- or CP(3)-order parameter.
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).
We revisit the phase transition from the N\'eel ordered to a valence bond solid (VBS) state in the two-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model. In the first part we address the question whether or not this transition could be an example of a second-order phase transition due to a deconfinement of spinons. We give arguments based on series expansion and spin-wave theory that this is not the case and the transition is most likely first order. The method proposed here to detect first-order phase transitions seems to be very sensitive and might be useful in other models as well. In the second part we analyze possible VBS patterns in the magnetically disordered phase based on numerical data for different susceptibilities, obtained in the ordered phase, which test the breaking of lattice symmetries. We conclude that a columnar dimerization pattern is the most likely candidate.
Recent numerical studies of the J1{J}_{1}J1​-J2{J}_{2}J2​ model on a square lattice suggest a possible continuous phase transition between the N\'eel state and a gapped spin-liquid state with Z2{Z}_{2}Z2​ topological order. We show that such a phase transition can be realized through two steps: First bring the N\'eel state to the U(1) deconfined quantum critical point, which has been studied in the context of N\'eel--valence bond solid (VBS) state phase transition. Then condense the spinon pair--skyrmion/antiskyrmion bound state, which carries both gauge charge and flux of the U(1) gauge field emerging at the deconfined quantum critical point. We also propose a Schwinger boson projective wave function to realize such a Z2{Z}_{2}Z2​ spin liquid state and find that it has a relatively low variational energy ($\ensuremath{-}0.4893{J}_{1}$/site) for the J1{J}_{1}J1​-J2{J}_{2}J2​ model at J2=0.5J1{J}_{2}=0.5{J}_{1}J2​=0.5J1​. The spin liquid state we obtain breaks the fourfold rotational symmetry of the square lattice and therefore is a nematic spin liquid state. This direct continuous phase transition from the N\'eel state to a spin liquid state may be realized in the J1{J}_{1}J1​-J2{J}_{2}J2​ model, or the anisotropic J1x{J}_{1x}J1x​-J1y{J}_{1y}J1y​-J2{J}_{2}J2​ model.
We map out the ground state phase diagram of the spin-12\frac{1}{2}21​ isotropic Kekul\'e-Kitaev model on the honeycomb lattice in the presence of the Heisenberg exchange couplings. Our study relies on large-scale tensor network simulations based on a graph-based projected entangled pair state approach and simple tensor update in the thermodynamic limit. We find that, on top of the quantum spin liquid and conventional magnetically ordered phases which are typical of the Kitaev-Heisenberg model, the Kekul\'e-Heisenberg phase diagram hosts two plaquette valence bond solid (VBS) phases with vanishing magnetic order. While the VBS phases preserve the symmetries of the original Hamiltonian, they differ markedly from the Kitaev spin liquid by having decorated plaquette ordering which is distinguished by a plaquette order parameter.
In this thesis we will study recent examples of exotic, topological, and many body localized quantum phase transitions. In Chapter 2 we study the quantum phase transition between the Z_2 spin liquid and valence bond solid (VBS) orders on a triangular lattice. We find a possible nematic Z_2 spin liquid intermediate phase and predict a continuous 3d XY* transition to the neighboring columnar and resonating-plaquette VBS phases. In Chapter 3 we demonstrate that an extended Kane-Mele Hubbard model on a bilayer honeycomb lattice has two novel quantum phase transitions. The first is a quantum phase transition between the weakly interacting gapless Dirac fermion phase and a strongly interacting fully gapped and symmetric trivial phase, which cannot be described by the standard Gross-Neveu model. The second is a quantum critical point between a quantum spin Hall insulator with spin S^z conservation and the previously mentioned strongly interacting fully gapped phase. We argue that the first quantum phase transition is related to the Z_16 classification of the topological superconductor ^3He-B phase with interactions, while the second quantum phase transition is a topological phase transition described by a bosonic O(4) nonlinear sigma model field theory with a Theta-term. In Chapter 4 we propose that if the highest and lowest energy eigenstates of a Hamiltonian belong to different SPT phases, then this Hamiltonian can't be fully many body localized. In Chapter 5 we study the disordered XYZ spin chain and its marginally many body localized critical lines, which we find to be characterized by an effective central charge c'=ln2 and continuously varying critical exponents.