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arXiv:1312.3922 · Physical Review B

Topology and criticality in the resonating Affleck-Kennedy-Lieb-Tasaki loop spin liquid states

Wei Li, Shuo Yang, Meng Cheng, Zheng-Xin Liu, Hong-Hao Tu

Published May 9, 2014• 39 citations
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Abstract

We exploit a natural projected entangled-pair state (PEPS) representation for the resonating Affleck-Kennedy-Lieb-Tasaki loop (RAL) state. By taking advantage of PEPS-based analytical and numerical methods, we characterize the RAL states on various two-dimensional lattices. On square and honeycomb lattices, these states are critical since the dimer-dimer correlations decay as a power law. On the kagome lattice, the RAL state has exponentially decaying correlation functions, supporting the scenario of a gapped spin liquid. We provide further evidence that the RAL state on the kagome lattice is a ${\mathbb{Z}}_{2}$ spin liquid, by identifying the four topological sectors and computing the topological entropy. Furthermore, we construct a one-parameter family of PEPS states interpolating between the RAL state and a short-range resonating valence bond state and find a critical point, consistent with the fact that the two states belong to two different phases. We also perform a variational study of the spin-1 kagome Heisenberg model using this one-parameter PEPS.

DOI
10.1103/physrevb.89.174411

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We exploit a natural projected entangled-pair state (PEPS) representation for the resonating Affleck-Kennedy-Lieb-Tasaki loop (RAL) state. By taking advantage of PEPS-based analytical and numerical methods, we characterize the RAL states on various two-dimensional lattices. On square and honeycomb lattices, these states are critical since the dimer-dimer correlations decay as a power law. On the kagome lattice, the RAL state has exponentially decaying correlation functions, supporting the scenario of a gapped spin liquid. We provide further evidence that the RAL state on the kagome lattice is a Z2{\mathbb{Z}}_{2}Z2​ spin liquid, by identifying the four topological sectors and computing the topological entropy. Furthermore, we construct a one-parameter family of PEPS states interpolating between the RAL state and a short-range resonating valence bond state and find a critical point, consistent with the fact that the two states belong to two different phases. We also perform a variational study of the spin-1 kagome Heisenberg model using this one-parameter PEPS.
We construct a class of projected entangled pair states which is exactly the resonating valence bond wave functions endowed with both short range and long range valence bonds. With an energetically preferred resonating valence bond pattern, the wave function is simplified to live in a one-parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin-1/2 J(1)-J(2) antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with an even or odd number of fluxes penetrating the cylinder and an even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a lnL correction to the entanglement entropy, indicating a gapless spin-liquid phase at the optimum parameter.
We construct a class of projected entangled pair states which is exactly the resonating valence bond wave functions endowed with both short range and long range valence bonds. With an energetically preferred resonating valence bond pattern, the wave function is simplified to live in a one-parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin-1/2 J[subscript 1]-J[subscript 2] antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with an even or odd number of fluxes penetrating the cylinder and an even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a ln⁡L correction to the entanglement entropy, indicating a gapless spin-liquid phase at the optimum parameter.
Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of SU(3)\text{SU}(3)SU(3), relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating SU(3)\text{SU}(3)SU(3) chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its SU(2)\text{SU}(2)SU(2) analog. We discuss the SPT feature of the state.
We examine in details the connections between topological and entanglement properties of short-range resonating valence bond (RVB) wave functions using projected entangled pair states (PEPS) on kagome and square lattices on (quasi)infinite cylinders with generalized boundary conditions (and perimeters with up to 20 lattice spacings). By making use of disconnected topological sectors in the space of dimer lattice coverings, we explicitly derive (orthogonal) ``minimally entangled'' PEPS RVB states. For the kagome lattice, using the quantum Heisenberg antiferromagnet as a reference model, we obtain the finite-size scaling with increasing cylinder perimeter of the vanishing energy separations between these states. In particular, we extract two separate (vanishing) energy scales corresponding (i) to insert a vison line between the two ends of the cylinder and (ii) to pull out and freeze a spin at either end. We also investigate the relations between bulk and boundary properties and show that, for a bipartition of the cylinder, the boundary Hamiltonian defined on the edge can be written as a product of a highly nonlocal projector, which fundamentally depends upon boundary conditions, with an emergent (local) SU(2)(2)(2)-invariant one-dimensional (superfluid) ttt-JJJ Hamiltonian, which arises due to the symmetry properties of the auxiliary spins at the edge. This multiplicative structure, a consequence of the disconnected topological sectors in the space of dimer lattice coverings, is characteristic of the topological nature of the states. For minimally entangled RVB states, it is shown that the entanglement spectrum, which reflects the properties of the (gapless or gapped) edge modes, is a subset of the spectrum of the local Hamiltonian, e.g., half of it for the kagome RVB state, providing a simple argument on the origin of the topological entanglement entropy ${S}_{0}=\ensuremath{-}\mathrm{ln}2$ of the Z2{\mathbb{Z}}_{2}Z2​ spin liquid. We propose to use these features to probe topological phases in microscopic Hamiltonians, and some results are compared to existing density matrix renormalization group data.
A one-parameter family of long-range resonating-valence-bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-1/2 frustrated Heisenberg model. We provide evidence that this RVB state in fact also realizes a topological (long-range entangled) Z2{\mathbb{Z}}_{2}Z2​ SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry of the local tensor. This Rapid Communication shows that, on one hand, spin-1/2 topological SL with C4v{C}_{4v}C4v​ point-group symmetry and SU(2)SU(2)SU(2) spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state. Forms of parent Hamiltonians hosting the Z2{\mathbb{Z}}_{2}Z2​ SL are suggested.
We perform a systematic investigation on the hexagon-singlet solid (HSS) states, which are a class of spin liquid candidates for the spin-1 kagome antiferromagnet. With the Schwinger boson representation, we show that all HSS states have exponentially decaying correlations and can be interpreted as a (special) subset of the resonating Affleck-Kennedy-Lieb-Tasaki (AKLT) loop states. We provide a compact tensor network representation of the HSS states, with which we are able to calculate physical observables efficiently. We find that the HSS states have vanishing topological entanglement entropy, suggesting the absence of intrinsic topological order. We also employ the HSS states to perform a variational study of the spin-1 kagome Heisenberg antiferromagnetic model. When we use a restricted HSS ansatz preserving lattice symmetry, the best variational energy per site is found to be ${e}_{0}=\ensuremath{-}1.3600$. In contrast, when allowing lattice symmetry breaking, we find a trimerized simplex valence-bond crystal with a lower energy, ${e}_{0}=\ensuremath{-}1.3871$.