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arXiv:1912.08284 · Physical review. B./Physical review. B

Gapped Z2 spin liquid in the breathing kagome Heisenberg antiferromagnet

Mohsin Iqbal, Didier Poilblanc, Norbert Schuch

Published Apr 28, 2020• 12 citations
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Abstract

We investigate the spin-1/2 Heisenberg antiferromagnet on a kagome lattice with breathing anisotropy (i.e., with weak and strong triangular units), constructing an improved simplex resonating valence bond (RVB) ansatz by successive applications (up to three times) of local quantum gates, which implement a filtering operation on the bare nearest-neighbor RVB state. The resulting projected entangled pair state involves a small number of variational parameters (only one at each level of application) and preserves full lattice and spin-rotation symmetries. Despite its simple analytic form, the simplex RVB provides very good variational energies at strong and even intermediate breathing anisotropy. We show that it carries ${\mathbb{Z}}_{2}$ topological order which does not fade away under the first few applications of the quantum gates, suggesting that the RVB topological spin liquid becomes a competing ground state candidate for the kagome antiferromagnet at large breathing anisotropy.

DOI
10.1103/physrevb.101.155141

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Gapped ${\mathbb{Z}}_{2}$ spin liquids have been proposed as candidates for the ground state of the $S=1/2$ quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order $2%--3%$ in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological ${\mathbb{Z}}_{2}$ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.

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Variational study of the ground state and spin dynamics of the spin- 1 2 kagome antiferromagnetic Heisenberg model and its implication for herbertsmithite ZnCu 3 ( OH ) 6 Cl 2

Chun Zhang, Tao Li

We find that the best resonating valence bond (RVB) state of the spin-$\frac{1}{2}$ kagome antiferromagnetic Heisenberg model (spin-$\frac{1}{2}$ KAFH) is described by a ${Z}_{2}$ mean field ansatz belonging to the ${Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ class, which hosts a mean field spinon dispersion very different from that of the widely studied $U(1)$ Dirac spin liquid state. However, we find that the physical spin fluctuation spectrum calculated from the Gutzwiller projected random phase approximation (RPA) (GRPA) theory above such an RVB state is actually gapless and is almost identical to that above the $U(1)$ Dirac spin liquid state. We find that such a peculiar behavior can be attributed to the unique flat band physics on the kagome lattice, which makes the mapping between the mean field ansatz and the RVB state noninjective. We find that the spin fluctuation spectrum of the spin-$\frac{1}{2}$ KAFH is not at all featureless, but is characterized by a prominent spectral peak at about $0.25J$ around the $\mathbf{M}$ point, which is immersed in a broad continuum extending to $2.7J$. Based on these results, we argue that the spectral peak below 2 meV in the inelastic neutron scattering spectrum of hebertsmithite ${\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}$, which has been attributed to the contribution of ${\mathrm{Cu}}^{2+}$ impurity spins occupying the ${\mathrm{Zn}}^{2+}$ site, should rather be understood as the intrinsic contribution from the kagome layer. We propose to verify such a picture by measuring the Knight shift on the Cu site, rather than the O site, which is almost blind to the spin fluctuation at the $\mathbf{M}$ point as a result of the strong antiferromagnetic correlation between nearest neighboring spins on the kagome lattice.

arXiv:2008.07829•Physical review. B./Physical review. B•Nov 4, 2020•27 citations
We investigate the spin-1/2 Heisenberg antiferromagnet on a kagome lattice with breathing anisotropy (i.e., with weak and strong triangular units), constructing an improved simplex resonating valence bond (RVB) ansatz by successive applications (up to three times) of local quantum gates, which implement a filtering operation on the bare nearest-neighbor RVB state. The resulting projected entangled pair state involves a small number of variational parameters (only one at each level of application) and preserves full lattice and spin-rotation symmetries. Despite its simple analytic form, the simplex RVB provides very good variational energies at strong and even intermediate breathing anisotropy. We show that it carries Z2{\mathbb{Z}}_{2}Z2​ topological order which does not fade away under the first few applications of the quantum gates, suggesting that the RVB topological spin liquid becomes a competing ground state candidate for the kagome antiferromagnet at large breathing anisotropy.
Gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids have been proposed as candidates for the ground state of the S=1/2S=1/2S=1/2 quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order 22%--3%2 in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z2{\mathbb{Z}}_{2}Z2​ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.
The nature of the ground state of the spin S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e., with different superexchange couplings J▵{J}_{▵}J▵​ and J▿{J}_{▿}J▿​ within elementary up- and down-pointing triangles) is investigated within the framework of Gutzwiller projected fermionic wave functions and Monte Carlo methods. We analyze the stability of the U(1)\text{U}(1)U(1) Dirac spin liquid with respect to the presence of fermionic pairing that leads to a gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid. For several values of the ratio J▿/J▵{J}_{▿}/{J}_{▵}J▿​/J▵​, the size scaling of the energy gain due to the pairing fields and the variational parameters are reported. Our results show that the energy gain of the gapped spin liquid with respect to the gapless state either vanishes for large enough system size or scales to zero in the thermodynamic limit. Similarly, the optimized pairing amplitudes (responsible for opening the spin gap) are shown to vanish in the thermodynamic limit. Our outcome is corroborated by the application of one and two Lanczos steps to the gapless and gapped wave functions, for which no energy gain of the gapped state is detected when improving the quality of the variational states. Finally, we discuss the competition with the ``simplex'' Z2{\mathbb{Z}}_{2}Z2​ resonating-valence-bond spin liquid, valence-bond crystal, and nematic states in the strongly anisotropic regime, i.e., ${J}_{▿}\ensuremath{\ll}{J}_{▵}$.
Spin-12\frac{1}{2}21​ kagome Heisenberg antiferromagnet with strong breathing anisotropy
We study the zero-temperature phase diagram of the spin-1/2 Heisenberg model with breathing anisotropy (i.e., with different coupling strength on the upward and downward triangles) on the kagome lattice. Our study relies on large scale tensor network simulations based on infinite projected entangled-pair state and infinite projected entangled-simplex state methods adapted to the kagome lattice. Our energy analysis suggests that the U(1) algebraic quantum spin-liquid (QSL) ground-state of the isotropic Heisenberg model is stable up to very large breathing anisotropy until it breaks down to a critical lattice-nematic phase that breaks rotational symmetry in real space through a first-order quantum phase transition. Our results also provide further insight into the recent experiment on vanadium oxyfluoride compounds which has been shown to be relevant platforms for realizing QSL in the presence of breathing anisotropy.
The nature of the kagome Heisenberg antiferromagnet (HAFM) is under ongoing debate. While recent evidence points towards a Z2{\mathbb{Z}}_{2}Z2​ topological spin liquid, the exact nature of the topological phase is still unclear. In this paper, we introduce semionic resonating valence-bond (RVB) states, this is, resonating valence-bond states which are in the Z2{\mathbb{Z}}_{2}Z2​-ordered double-semion phase, and study them using projected entangled pair states. We investigate their physics and study their suitability as an ansatz for the HAFM, as compared to a conventional RVB state which is in the toric-code Z2{\mathbb{Z}}_{2}Z2​ topological phase. In particular, we find that a suitably optimized ``semionic simplex RVB'' outperforms the equally optimized conventional ``simplex RVB'' state, and that the entanglement spectrum (ES) of the semionic RVB behaves very differently from the ES of the conventional RVB, which suggests using the ES to discriminate the two phases. Finally, we also discuss the possible relevance of space-group symmetry breaking in valence-bond wave functions with double-semion topological order.
We construct a short-range resonating-valence-bond (RVB) state on the ruby lattice, using projected entangled-pair states with bond dimension D=3D=3D=3. By introducing nonlocal moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining a nonzero topological contribution, $\ensuremath{\gamma}=\ensuremath{-}\mathrm{ln}\phantom{\rule{4pt}{0ex}}2$, consistent with that of a Z2{\mathbb{Z}}_{2}Z2​ topological quantum spin liquid. We also calculate the ground-state energy of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.
We find that the best resonating valence bond (RVB) state of the spin-12\frac{1}{2}21​ kagome antiferromagnetic Heisenberg model (spin-12\frac{1}{2}21​ KAFH) is described by a Z2{Z}_{2}Z2​ mean field ansatz belonging to the ${Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ class, which hosts a mean field spinon dispersion very different from that of the widely studied U(1)U(1)U(1) Dirac spin liquid state. However, we find that the physical spin fluctuation spectrum calculated from the Gutzwiller projected random phase approximation (RPA) (GRPA) theory above such an RVB state is actually gapless and is almost identical to that above the U(1)U(1)U(1) Dirac spin liquid state. We find that such a peculiar behavior can be attributed to the unique flat band physics on the kagome lattice, which makes the mapping between the mean field ansatz and the RVB state noninjective. We find that the spin fluctuation spectrum of the spin-12\frac{1}{2}21​ KAFH is not at all featureless, but is characterized by a prominent spectral peak at about 0.25J0.25J0.25J around the M\mathbf{M}M point, which is immersed in a broad continuum extending to 2.7J2.7J2.7J. Based on these results, we argue that the spectral peak below 2 meV in the inelastic neutron scattering spectrum of hebertsmithite ZnCu3(OH)6Cl2{\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}ZnCu3​(OH)6​Cl2​, which has been attributed to the contribution of Cu2+{\mathrm{Cu}}^{2+}Cu2+ impurity spins occupying the Zn2+{\mathrm{Zn}}^{2+}Zn2+ site, should rather be understood as the intrinsic contribution from the kagome layer. We propose to verify such a picture by measuring the Knight shift on the Cu site, rather than the O site, which is almost blind to the spin fluctuation at the M\mathbf{M}M point as a result of the strong antiferromagnetic correlation between nearest neighboring spins on the kagome lattice.