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arXiv:2206.07237 · Physical Review A

Detection of long-range entanglement in gapped quantum spin liquids by local measurements

Shi Feng, Yanjun He, Nandini Trivedi

Published Oct 14, 2022• 9 citations
arXivHTMLPDF

Abstract

Topological order, reflected in long-range patterns of entanglement, is quantified by the topological entanglement entropy $\ensuremath{\gamma}$. We show that for gapped quantum spin liquids it is possible to extract $\ensuremath{\gamma}$ using two-spin local correlators. We demonstrate our method for the gapped ${\mathbb{Z}}_{2}$ Kitaev spin liquid on a honeycomb lattice with anisotropic interactions. We show that the $\ensuremath{\gamma}=ln2$ for ${\mathbb{Z}}_{2}$ topological order can be simply extracted from local two-spin correlators across two different bonds that involve only Majorana fermions, with an accuracy comparable to or higher than the Kitaev-Preskill construction. This implies that even though the ground state can be factorized into the product of Majorana and gauge sectors, the different superselection sectors of ${\mathbb{Z}}_{2}$ gauge theory determined by global Wilson loop operators can be reflected locally in the Majorana sector.

DOI
10.1103/physreva.106.042417

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Topological order, reflected in long-range patterns of entanglement, is quantified by the topological entanglement entropy \ensuremathγ\ensuremath{\gamma}\ensuremathγ. We show that for gapped quantum spin liquids it is possible to extract \ensuremathγ\ensuremath{\gamma}\ensuremathγ using two-spin local correlators. We demonstrate our method for the gapped Z2{\mathbb{Z}}_{2}Z2​ Kitaev spin liquid on a honeycomb lattice with anisotropic interactions. We show that the \ensuremathγ=ln2\ensuremath{\gamma}=ln2\ensuremathγ=ln2 for Z2{\mathbb{Z}}_{2}Z2​ topological order can be simply extracted from local two-spin correlators across two different bonds that involve only Majorana fermions, with an accuracy comparable to or higher than the Kitaev-Preskill construction. This implies that even though the ground state can be factorized into the product of Majorana and gauge sectors, the different superselection sectors of Z2{\mathbb{Z}}_{2}Z2​ gauge theory determined by global Wilson loop operators can be reflected locally in the Majorana sector.
While most of the features of the Kitaev honeycomb model are already clarified from the combined Z2{\mathbb Z}_2Z2​ plaquette and Majorana fermion picture, we show that retrieving the exact quantum many-body form gives a clue to fix the remaining issues on its spin liquid property. We provide an exact two-dimensional tensor network description of a full set of solutions from a random product state, by sequential quantum-number projection and Gutzwiller projection, both expressed as local matrix-product operators of dimension D=2D=2D=2. The ground and excited states share the same bond dimension yet exhibit distinct Schmidt spectra, distinguishing their area-law and volume-law entanglement. We demonstrate that the topological entanglement entropy emerges {\it for all eigenstates only after} the Gutzwiller projection onto the physical spin Hilbert space, where the gauge constraint removes exactly half of the Schmidt states. This establishes that excited states are likewise topologically ordered. We further show that there exists a projection-induced collapse of the wave function into a single gauge sector in the Parton mean-field states that is known to give a good description of the spin liquids. The demonstrations on the S=1/2S=1/2S=1/2 kagome and triangular Heisenberg models suggest that the present perspective is applied to a wider class of models hosting Z2{\mathbb Z}_2Z2​ spin liquids. Our finding visualizes a bulk-boundary correspondence with the Wilson-loop characterization of topological order.
The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge symmetry. We mapped out the phase diagram of the model and found gapped and gapless spin-liquid phases. While the low-energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in small magnetic fields, where we showed that the ground state has a finite topological entanglement entropy. We argue that the gapped phases could be possibly described by exotic excitations, and their corresponding spectrum is richer than the Ising phase of the Kitaev model.
A spin fractionalizes into matter and gauge fermions in the Kitaev spin liquid on the honeycomb lattice, which is topologically ordered for certain parameters. To what extent does only one constituent of a fractionalized topological system signal the existence of topological order? To answer this, the authors focus on the reduced density matrix of the gauge sector on a cylinder after tracing out the matter degrees of freedom. The ensuing gauge entanglement Hamiltonian contains infinitely long-range correlations along the symmetry axis of the cylinder. Rather small correctly chosen gauge partitions can still account for the topological entanglement entropy in spite of long-range correlations in the gauge entanglement Hamiltonian.
The spin-1/2 Kitaev model offers the exactly solvable example of quantum spin liquids.Possible material realizations of the spin-1/2 Kitaev systems and the prospect of using the Majorana fermion excitations for quantum computations have revolutionized quantum spin liquids research.Recently it has been suggested that higher-spin, especially spin-1, Kitaev exchange interactions can be realized in a variety of materials.Numerical computations on small clusters indicate that the ground state of the spin-1 Kitaev model may also be a quantum spin liquid.On the other hand, the nature of the ground state remains elusive since the spin-1 model is not exactly solvable in contrast to the spin-1/2 model.In this work, using the quantum-entanglement based tensor network approach, we construct an explicit ground-state wave function for the spin-1 Kitaev model, which is written only in terms of physical spin operators.We establish the existence of distinct topological sectors on a torus by constructing the minimally entangled states in the degenerate ground-state manifold and evaluating topological entanglement entropy.Our results suggest that the ground state of the spin-1 Kitaev model is a gapped quantum spin liquid with Z 2 gauge structure and Abelian quasiparticles.We explain the subtle differences between the spin-1/2 and spin-1 Kitaev quantum spin liquids.
Quantum spin liquids (QSLs) are long-range entangled phases of matter and a natural setting for exploring how quantum correlations generate quantum complexity. Motivated by the emergence of magic, or nonstabilizerness, as a diagnostic of many-body complexity beyond entanglement, we study magic of QSLs by computing the stabilizer Rényi entropy (SRE) of the Kitaev honeycomb model. We derive a correspondence between Pauli strings and products of itinerant Majorana operators in the model's free-fermion description; this enables the sampling of SRE using an optimized algorithm for Gaussian states in systems with thousands of spins. Our results show that magic is largest in the gapless phase, where subleading volume-law corrections indicate nonlocal contributions, while in the gapped phases, SRE decreases in quantitative agreement with a perturbative expansion that we develop in the anisotropic limit. At the topological phase transition, SRE exhibits universal scaling dictated by critical exponents.
Topological properties are associated with nonlocal entanglement and global properties. On the other hand, in some topological systems, it has been shown that quasilocal operators detect topological transitions. We show in this work that in the case of noninteracting long-range symmetry-protected topological models, energy bonds signal topological transitions. Interestingly, we also find that they display some signatures at the Berezinskii-Kosterlitz-Thouless transition that occurs in a spin chain with first and second neighbor interactions, if we consider the first excited state instead of the ground state of the system. Moreover, we show that the ground-state topological transition in the spin Kitaev model in a honeycomb lattice, which displays topological long-range order, is also detected by the energy bond correlator. Despite the model being interacting, in the ground state it reduces to a noninteracting model. Even for the spin-liquid phase of the two-dimensional Heisenberg model with first and second neighbor interactions, where the system has true long-range entanglement and topological order, the local bonds do signal the topological phase transition.