Gapped spin liquid is a topological phase of matter that hosts deconfined fractionalized quasiparticles. Quasiparticle statistics are an intrinsic property of spin liquid. However, it is difficult to identify explicitly in a microscopic model. Here we perform an exact diagonalization study of the Z2 quasiparticle statistics in the 'easy-axis' S = 1/2 Heisenberg J1 − J2 − J3 model on the kagome lattice, which hosts a robust Z2 spin liquid. Using the entanglement entropy measurement, we identify multiple independent minimal entangled states (MESs) on the torus geometry, which represents the eigenstates of the Wilson loop operators encircling the torus that measure the Z2 flux threading it. Based on the MESs, we obtain that: (1) The scaling of the finite size entropy leads to a topological entanglement entropy γ ≈ 0.637, indicating a Z2 topological spin liquid. (2) The modular transformation of MESs gives rise to the modular matrices and , which unambiguously fixes the Z2 quasiparitcle statistics including quasiparticle quantum dimensions, fusion rules and topological spins. (3) Through adiabatical threading a 2π flux in the hole of the torus, spinon state ψs (vacuum state ψ1) evolves into ψ1 (ψs), as predicted by Z2 gauge theory. Away from the 'easy-axis' limit with intermediate spin-z coupling, we identify some signature evidence of the chiral spin liquid based on the entanglement spectrum of the state using density matrix renormalization group simulation.
Gapped spin liquid is a topological phase of matter that hosts deconfined fractionalized quasiparticles. Quasiparticle statistics are an intrinsic property of spin liquid. However, it is difficult to identify explicitly in a microscopic model. Here we perform an exact diagonalization study of the Z2 quasiparticle statistics in the 'easy-axis' S = 1/2 Heisenberg J1 − J2 − J3 model on the kagome lattice, which hosts a robust Z2 spin liquid. Using the entanglement entropy measurement, we identify multiple independent minimal entangled states (MESs) on the torus geometry, which represents the eigenstates of the Wilson loop operators encircling the torus that measure the Z2 flux threading it. Based on the MESs, we obtain that: (1) The scaling of the finite size entropy leads to a topological entanglement entropy γ ≈ 0.637, indicating a Z2 topological spin liquid. (2) The modular transformation of MESs gives rise to the modular matrices and , which unambiguously fixes the Z2 quasiparitcle statistics including quasiparticle quantum dimensions, fusion rules and topological spins. (3) Through adiabatical threading a 2π flux in the hole of the torus, spinon state ψs (vacuum state ψ1) evolves into ψ1 (ψs), as predicted by Z2 gauge theory. Away from the 'easy-axis' limit with intermediate spin-z coupling, we identify some signature evidence of the chiral spin liquid based on the entanglement spectrum of the state using density matrix renormalization group simulation.
We perform a density-matrix renormalization group (DMRG) study of the S=1/2 Heisenberg antiferromagnet on the kagome lattice to identify the conjectured spin liquid ground state. Exploiting SU(2) spin symmetry, which allows us to keep up to 16,000 DMRG states, we consider cylinders with circumferences up to 17 lattice spacings and find a spin liquid ground state with an estimated per site energy of -0.4386(5), a spin gap of 0.13(1), very short-range decay in spin, dimer and chiral correlation functions and finite topological entanglement gamma consistent with gamma=ld 2, ruling out gapless, chiral or non-topological spin liquids. At the same time, DMRG results provide strong evidence for a gapped topological Z2 spin liquid.
Abstract We use the topological entanglement entropy (TEE) as an efficient tool to fully characterize the Abelian phase of a Z 2 × Z 2 spin liquid emerging as the ground state of topological color code (TCC), which is a class of stabilizer states on the honeycomb lattice. We provide the fusion rules of the quasiparticle (QP) excitations of the model by introducing single- or two-body operators on physical spins for each fusion process which justify the corresponding fusion outcome. Besides this, we extract the TEE from Renyi entanglement entropy (EE) of the TCC, analytically and numerically by finite size exact diagonalization on the disk shape regions with contractible boundaries. We obtain that the EE has a local contribution, which scales linearly with the boundary length in addition to a topological term, i.e. the TEE, arising from the condensation of closed strings in the ground state. We further investigate the ground state dependence of the TEE on regions with non-contractible boundaries, i.e. by cutting the torus to half cylinders, from which we further identify multiple independent minimum entropy states (MES) of the TCC and then extract the U and S modular matrices of the system, which contain the self and mutual statistics of the anyonic QPs and fully characterize the topological phase of the TCC. Eventually, we show that, in spite of the lack of a local order parameter, TEE and other physical quantities obtained from the ground state wave function such as the entanglement spectrum (ES) and ground state fidelity are sensitive probes to study the robustness of a topological phase. We find that the topological order in the presence of a magnetic field persists until the vicinity of the transition point, where the TEE and fidelity drops to zero and the ES splits severely, signaling breakdown of the topological phase of the TCC.
We perform a density-matrix renormalization group (DMRG) study of the S=1/2 Heisenberg antiferromagnet on the kagome lattice to identify the conjectured spin liquid ground state. Exploiting SU(2) spin symmetry, which allows us to keep up to 16,000 DMRG states, we consider cylinders with circumferences up to 17 lattice spacings and find a spin liquid ground state with an estimated per site energy of -0.4386(5), a spin gap of 0.13(1), very short-range decay in spin, dimer and chiral correlation functions, and finite topological entanglement γ consistent with γ=log(2)2, ruling out gapless, chiral, or nontopological spin liquids in favor of a topological spin liquid of quantum dimension 2, with strong evidence for a gapped topological Z(2) spin liquid.
We perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground-state properties of the spin-1/2 antiferromagnetic square lattice Heisenberg J1-J2 model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region $0.41\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.62$, separating conventional N\'eel and striped antiferromagnetic states for smaller and larger J2/J1, respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and nonzero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a nonzero topological entanglement entropy $\ensuremath{\gamma}=0.70\ifmmode\pm\else\textpm\fi{}0.02$, extremely close to $\ensuremath{\gamma}=\mathrm{ln}(2)\ensuremath{\approx}0.69$ (expected for a Z2 quantum spin liquid) and a nontrivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of opposite sign to the topological entanglement entropy.
The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a spin-½ isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.
We perform an exact-diagonalization study of the spin-21 XXZ Heisenberg antiferromagnet on the kagome lattice at finite magnetization m=32 with an emphasis on the XY point (Jz=0) and in the presence of a small chiral term. Recent analytic work by Kumar et al. [K. Kumar, K. Sun, and E. Fradkin, Phys. Rev. B 90, 174409 (2014)] on the same model, using a newly developed flux attachment transformation, predicts a plateau at this value of the magnetization described by a chiral spin liquid (CSL) with a spin Hall conductance of ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$. Such a state is topological in nature, has a ground-state degeneracy, and exhibits fractional excitations. We analyze the degeneracy structure in the low-energy manifold, identify the candidate topological states, and use them to compute the modular matrices and Chern numbers, all of which strongly agree with expected theoretical behavior for the ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ CSL. In the limit of zero chirality, we find on most (not all) clusters that the topological invariants are still those of a CSL.