arXiv:cond-mat/9812329 · Physical review. B, Condensed matter

Spin-liquid phase of the multiple-spin exchange Hamiltonian on the triangular lattice

Grégoire Misguich, C. Lhuillier, B. Bernu, Ch. Waldtmann

Published Jul 1, 1999• 201 citations

Abstract

We performed an exact diagonalization study of the spin-liquid phase of the multiple-spin exchange model on a triangular lattice. It is characterized by no N\'eel long-range order, short-ranged magnetic correlations, and a spin gap. We found no long-range order in any local order parameter we investigated (chiral, dimer, etc.) The probable asymptotic ground-state degeneracy is discussed. We argue that it could be of topological origin and that the system is probably not a chiral spin liquid. A possible relation to the Affleck-Kennedy-Lieb-Tasaki, or valence-bond solid phase is discussed.

DOI: 10.1103/physrevb.60.1064

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arXiv:cond-mat/9812329 · Physical review. B, Condensed matter

Spin-liquid phase of the multiple-spin exchange Hamiltonian on the triangular lattice

Grégoire Misguich, C. Lhuillier, B. Bernu, Ch. Waldtmann

Published Jul 1, 1999• 201 citations

arXiv HTML PDF

Abstract

We performed an exact diagonalization study of the spin-liquid phase of the multiple-spin exchange model on a triangular lattice. It is characterized by no N'eel long-range order, short-ranged magnetic correlations, and a spin gap. We found no long-range order in any local order parameter we investigated (chiral, dimer, etc.) The probable asymptotic ground-state degeneracy is discussed. We argue that it could be of topological origin and that the system is probably not a chiral spin liquid. A possible relation to the Affleck-Kennedy-Lieb-Tasaki, or valence-bond solid phase is discussed.

DOI: 10.1103/physrevb.60.1064

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Exact diagonalization study of the triangular Heisenberg model with four-spin ring-exchange interaction

Yu Zheng, Muwei Wu, Dao‐Xin Yao +1 more

Using Lanczos exact diagonalization (ED), we study the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model with the four-spin ring-exchange interaction ${J}_{r}$ on a triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer, and chiral correlation functions. The ground state has rich phases including several magnetic-ordered phases like the zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction and help to understand the relevant triangular materials.Using Lanczos exact diagonalization (ED), we study the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model with the four-spin ring-exchange interaction ${J}_{r}$ on a triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer, and chiral correlation functions. The ground state has rich phases including several magnetic-ordered phases like the zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction and help to understand the relevant triangular materials.

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Nature of quantum spin liquids of the S = 1 2 Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study

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We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin- $1/2$ antiferromagnetic Heisenberg model on the triangular lattice with nearest- $({J}_{1})$ , next-nearest- $({J}_{2})$ , and third-neighbor $({J}_{3})$ interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with $\mathrm{SU}(2)$ spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to ${L}_{x}=48$ and circumference ${L}_{y}=6--12$ . Our results suggest that the QSL phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is gapped, which is characterized by the absence of a gapless mode and by short-range spin-spin and dimer-dimer correlations. In the presence of ${J}_{3}$ interaction, we find that a critical QSL with a single gapless mode emerges. While both spin-spin and scalar chiral-chiral correlations are short ranged, dimer-dimer correlations are quasi long ranged and decay as a power law at long distances.We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin-$1/2$ antiferromagnetic Heisenberg model on the triangular lattice with nearest- $({J}_{1})$, next-nearest- $({J}_{2})$, and third-neighbor $({J}_{3})$ interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with $\mathrm{SU}(2)$ spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to ${L}_{x}=48$ and circumference ${L}_{y}=6--12$. Our results suggest that the QSL phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is gapped, which is characterized by the absence of a gapless mode and by short-range spin-spin and dimer-dimer correlations. In the presence of ${J}_{3}$ interaction, we find that a critical QSL with a single gapless mode emerges. While both spin-spin and scalar chiral-chiral correlations are short ranged, dimer-dimer correlations are quasi long ranged and decay as a power law at long distances.

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

Yi Xu, Sylvain Capponi, Ji-Yao Chen +6 more

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.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.

arXiv:2306.16192Physical review. B./Physical review. BNov 30, 202314 citations

Using Lanczos exact diagonalization (ED), we study the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model with the four-spin ring-exchange interaction

{J}_{r}

on a triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer, and chiral correlation functions. The ground state has rich phases including several magnetic-ordered phases like the zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction and help to understand the relevant triangular materials.

We study the

S=1/2

Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor interactions

{J}_{1}

and

{J}_{2}

with the density matrix renormalization group, on long open cylinders with widths up to nine lattice spacings. In an intermediate

{J}_{2}

region $0.06\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.17$, we find evidence for a spin liquid (SL) state with short range spin-spin, bond-bond, and chiral correlation lengths, bordered by a classical ${120}^{\ensuremath{\circ}}$ N\'eel ordered state at small

{J}_{2}

and by a two sublattice collinear magnetically ordered state at larger

{J}_{2}

. Focusing on

{J}_{2}/{J}_{1}=0.1

, we find a number of signatures of a gapped SL phase.

We investigate the ground state of the

{d}^{1}

spin-orbital model for triply degenerate

{t}_{2g}

orbitals on a triangular lattice that unifies intrinsic frustration of spin and orbital interactions with geometrical frustration. Using full or Lanczos exact diagonalization of finite clusters, we establish that the ground state of the spin-orbital model that interpolates between superexchange and direct exchange interactions on the bonds is characterized by valence-bond correlations. In the absence of Hund's exchange the model describes a competition between various possible valence-bond states. By considering the clusters with open boundary conditions we demonstrate that orbital interactions are always frustrated, but this frustration is removed by pronounced spin singlet correlations that coexist with dimer orbital correlations supporting them. Such local configurations contribute to the disordered ground states found for the clusters with periodic boundary conditions that interpolate between a highly resonating, dimer-based, entangled spin-orbital liquid phase and a valence-bond state with completely static spin-singlet states. We argue that these states are also realized for the infinite lattice and anticipate that pronounced transitions between different regimes found for particular geometries will turn out to smooth crossovers in the properties of the spin-orbital liquid in the thermodynamic limit. Finally, we provide evidence that the resonating spin-orbital liquid phase involves entangled states on the bonds. In such a phase classical considerations based on the mean-field theory cannot be used, spin exchange interactions do not determine spin bond correlations, and quantum fluctuations play a crucial role in the ground states and magnetic transitions.

59%

from its classical value and the ground state energy per site is ${e}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.5458\ifmmode\pm\else\textpm\fi{}0.0001$ in units of the exchange coupling. We have identified the ground state correlations that are important at short distances.

We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin-

1/2

antiferromagnetic Heisenberg model on the triangular lattice with nearest-

({J}_{1})

, next-nearest-

({J}_{2})

, and third-neighbor

({J}_{3})

interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with

\mathrm{SU}(2)

spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to

{L}_{x}=48

and circumference

{L}_{y}=6--12

. Our results suggest that the QSL phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is gapped, which is characterized by the absence of a gapless mode and by short-range spin-spin and dimer-dimer correlations. In the presence of

{J}_{3}

interaction, we find that a critical QSL with a single gapless mode emerges. While both spin-spin and scalar chiral-chiral correlations are short ranged, dimer-dimer correlations are quasi long ranged and decay as a power law at long distances.

\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.