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arXiv:2306.16192 · Physical review. B./Physical review. B

Phase diagram of the chiral SU(3) antiferromagnet on the kagome lattice

Yi Xu, Sylvain Capponi, Ji-Yao Chen, Laurens Vanderstraeten, Juraj Hašík, Andriy H. Nevidomskyy, Matthieu Mambrini, Karlo Penc, Didier Poilblanc

Published Nov 30, 2023• 14 citations

Abstract

Motivated by the search for chiral spin liquids (CSL), we consider a simple model defined on the kagome lattice of interacting SU(3) spins (in the fundamental representation) including two-site and three-site permutations between nearest neighbor sites and on triangles, respectively. By combining analytical methods and various numerical techniques, namely, exact Lanczos diagonalizations and tensor network variational approaches, we find a rich phase diagram with short-range entangled (i.e., nontopological) and topological (possibly chiral) gapped spin liquids (SLs). Short-range entangled spin liquids include an Affleck-Kennedy-Lieb-Tasaki (AKLT)-like phase---argued to be a Symmetry-Protected Topological (SPT) phase---and a trimerized phase breaking the inversion center between the up and down triangles of the kagome lattice. A topological SL is stabilized in a restricted part of the phase diagram by the time-reversal symmetry breaking (complex) three-site permutation term. Analyzing the chiral edge modes of this topological SL on long cylinders or on finite disks, we have come up with two competing scenarios, either a CSL or a double Chern-Simons SL characterized by a single or by two counter-propagating Wess-Zumino-Witten $\mathrm{SU}{(3)}_{1}$ chiral mode(s), respectively. In the vicinity of the extended ferromagnetic region, we have found a magnetic phase corresponding either to a modulated canted ferromagnet or to a uniform partially magnetized ferromagnet.

DOI: 10.1103/physrevb.108.195153

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arXiv:2203.07321•npj Quantum Materials•Feb 2, 2024•16 citations

Motivated by the search for chiral spin liquids (CSL), we consider a simple model defined on the kagome lattice of interacting SU(3) spins (in the fundamental representation) including two-site and three-site permutations between nearest neighbor sites and on triangles, respectively. By combining analytical methods and various numerical techniques, namely, exact Lanczos diagonalizations and tensor network variational approaches, we find a rich phase diagram with short-range entangled (i.e., nontopological) and topological (possibly chiral) gapped spin liquids (SLs). Short-range entangled spin liquids include an Affleck-Kennedy-Lieb-Tasaki (AKLT)-like phase---argued to be a Symmetry-Protected Topological (SPT) phase---and a trimerized phase breaking the inversion center between the up and down triangles of the kagome lattice. A topological SL is stabilized in a restricted part of the phase diagram by the time-reversal symmetry breaking (complex) three-site permutation term. Analyzing the chiral edge modes of this topological SL on long cylinders or on finite disks, we have come up with two competing scenarios, either a CSL or a double Chern-Simons SL characterized by a single or by two counter-propagating Wess-Zumino-Witten

\mathrm{SU}{(3)}_{1}

chiral mode(s), respectively. In the vicinity of the extended ferromagnetic region, we have found a magnetic phase corresponding either to a modulated canted ferromagnet or to a uniform partially magnetized ferromagnet.

We propose an SU(3) symmetric Hamiltonian with short-range interactions on the kagome lattice and show that it hosts an Abelian chiral spin liquid (CSL) state. We provide numerical evidence based on exact diagonalization to show that this CSL state is stabilized in an extended region of the parameter space and can be viewed as a lattice version of the Halperin 221 fractional quantum Hall state of two-component bosons. We also construct a parton wave function for this CSL state and demonstrate that its variational energies are in good agreement with exact diagonalization results. The parton description further supports that the CSL is characterized by a chiral edge conformal field theory of the

\mathrm{SU}{(3)}_{1}

Wess-Zumino-Witten type.

The kagome spin-

\frac{1}{2}

systems have attracted intensive attention in recent years as the primary candidate for hosting different gapped spin liquids (SLs). To uncover the nature of the novel quantum phase transition between the SL states, we study a minimum

XY

model with the nearest-neighbor (NN) (

{J}_{xy}

), the second-NN, and the third-NN couplings (${J}_{2xy}={J}_{3xy}={J}_{xy}^{\ensuremath{'}}$). We identify the time-reversal-symmetry-broken chiral SL (CSL) with the turn on of a small perturbation ${J}_{xy}^{\ensuremath{'}}\ensuremath{\sim}0.06{J}_{xy}$, which is fully characterized by the fractionally quantized topological Chern number and the conformal edge spectrum as the $\ensuremath{\nu}=1/2$ fractional quantum Hall state. Interestingly, the NN

XY

model (${J}_{xy}^{\ensuremath{'}}=0$) is shown to be a critical SL state adjacent to the CSL, characterized by the gapless spin singlet and spin triplet excitations. The quantum phase transition from the CSL to the gapless critical SL is driven by the collapsing of the neutral (spin singlet) excitation gap. The effect of the NN spin-

z

coupling

{J}_{z}

is also studied, which leads to a quantum phase diagram with an extended regime for the gapless SL.

The infinite projected entangled simplex state (iPESS), a type of tensor network (TN) state, has been used successfully for simulating and characterizing {\it non-chiral} spin liquids on the kagome lattice. Here, we demonstrate that iPESS also provides a faithful representation of a {\it chiral} spin liquid (CSL) on the same lattice, namely the ground state of the spin-

1/2

kagome Heisenberg antiferromagnet with a scalar chirality. By classifying local tensors according to SU

(2)

and point group symmetries, we construct a chiral ansatz breaking reflection

{J}_{z}

and time reversal

T

symmetries while preserving

PT

. The variational TN states are shown to host, for bond dimension

D\ge 8

, a chiral gapless entanglement spectrum following SU

(2)_1

conformal field theory. The correlation function shows a small weight long-range tail complying with the prediction of the TN bulk-edge correspondence. %{\color{red}With more constraints included, the chiral ansatz is reduced to a non-chiral one which preserves full point group symmetries due to an emergent {\it tensor conservation law} and is of relevance to the ground state at the Heisenberg point. Lastly, by simulations in the complete ansatz family we discuss the transition from the non-chiral spin liquid to the CSL induced by the scalar chirality term.} We identify a non-chiral manifold spanned by only a subset of symmetric tensors where a new emergent {\it tensor conservation law} is realized. This allows us to both probe the stability of the non-chiral spin liquid and discuss its transition to CSL induced by a scalar chirality term.

We construct a topological spin-liquid (TSL) model on the kagome lattice, with

\mathrm{SU}(3)

symmetry with the fundamental representation at each lattice site, based on projected entangled pair states (PEPS). Using the PEPS framework, we can adiabatically connect the model to a fixed-point model (analogous to the dimer model for resonating valence bond states) which we prove to be locally equivalent to a

{\mathbb{Z}}_{3}

quantum double model. Numerical study of the interpolation reveals no sign of a phase transition or long-range order, characterizing the model conclusively as a gapped TSL. We further study the entanglement spectrum of the model and find that while it is gapped, it exhibits branches with vastly different velocities, with the slow branch matching the counting of a chiral

\mathrm{SU}{(3)}_{1}

Conformal Field Theory, suggesting that it can be deformed to a model with chiral

\mathrm{SU}{(3)}_{1}

entanglement spectrum.

The infinite projected entangled simplex state (iPESS), a type of tensor network (TN) state, has been used successfully for simulating and characterizing nonchiral spin liquids on the kagome lattice. Here, we demonstrate that iPESS also provides a faithful representation of a chiral spin liquid (CSL) on the same lattice, namely the ground state of the spin-

1/2

kagome Heisenberg antiferromagnet with a scalar chirality. By classifying local tensors according to

\mathrm{SU}(2)

and point group symmetries, we construct a chiral ansatz breaking reflection

P

and time reversal

T

symmetries while preserving

PT

. The variational TN states are shown to host, for bond dimension $D\ensuremath{\ge}8$, a chiral gapless entanglement spectrum following

\mathrm{SU}{(2)}_{1}

conformal field theory. The correlation function shows a small weight long-range tail complying with the prediction of the TN bulk-edge correspondence. We identify a nonchiral manifold spanned by only a subset of symmetric tensors where a new emergent tensor conservation law is realized. This allows us to both probe the stability of the nonchiral spin liquid and discuss its transition to CSL induced by a scalar chirality term.

Abstract The nature of the ground state for the S = 1/2 Kagome Heisenberg antiferromagnet (KHAF) has been elusive. We revisit this challenging problem and provide numerical evidence that its ground state is a chiral spin liquid. Combining the density matrix renormalization group method with analytical analyses, we demonstrate that the previously observed chiral spin liquid phase in the KHAF with longer-range couplings is stable in a broader region of the phase diagram. We characterize the nature of the ground state by computing energy derivatives, revealing the ground-state degeneracy arising from the spontaneous breaking of the time-reversal symmetry, and targeting the semion sector. We further investigate the phase diagram in the vicinity of the KHAF and observe a

\sqrt{3}\times \sqrt{3}

3 × 3 magnetically ordered phase and two valence-bond crystal phases.