arXiv:2406.04030 · Physical review. B./Physical review. B

Topological phases in half-integer higher spin J 1 − J 2 Heisenberg chains

Sahinur Reja, Satoshi Nishimoto

Published Aug 23, 2024• 4 citations

Abstract

We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from $S=\frac{3}{2}$ to $S=\frac{11}{2}$ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of $\frac{{J}_{2}}{{J}_{1}}$ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and $\frac{2S+1}{2}$ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with $S$ and converge to a single point at $\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}$ in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for $S=\frac{1}{2}$, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of $\frac{{J}_{2}}{{J}_{1}}$ values.

DOI: 10.1103/physrevb.110.054436

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Ling Wang

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arXiv:2406.04030 · Physical review. B./Physical review. B

Topological phases in half-integer higher spin J 1 − J 2 Heisenberg chains

Sahinur Reja, Satoshi Nishimoto

Published Aug 23, 2024• 4 citations

arXiv HTML PDF

Abstract

We study the ground-state properties of antiferromagnetic

{J}_{1}\text{\ensuremath{-}}{J}_{2}

chains with half-integer spins ranging from

S=\frac{3}{2}

S=\frac{11}{2}

using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of

\frac{{J}_{2}}{{J}_{1}}

containing topological phases with alternating

\frac{2S\ensuremath{-}1}{2}

and

\frac{2S+1}{2}

valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those

\mathrm{high}\text{\ensuremath{-}}S

systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with

S

and converge to a single point at

\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}

in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for

S=\frac{1}{2}

, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of

\frac{{J}_{2}}{{J}_{1}}

values.

DOI: 10.1103/physrevb.110.054436

Related papers

Spin liquid ground state of the spin- 1 2 square J 1 - J 2 Heisenberg model

Hong‐Chen Jiang, Hong Yao, Leon Balents

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Ling Wang

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Apr 25, 20127 citations

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Shou-Shu Gong, D. N. Sheng, Olexei I. Motrunich +1 more

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Topological Spin Liquid Phase in the Spin- $1/2$ Triangular Heisenberg ModelTopological Spin Liquid Phase in the Spin-$1/2$ Triangular Heisenberg Model

Wen-Jun Hu, Shou-Shu Gong, Wei Kang Zhu +1 more

We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.

Apr 2, 20151 citation

We study the ground-state properties of antiferromagnetic

{J}_{1}\text{\ensuremath{-}}{J}_{2}

chains with half-integer spins ranging from

S=\frac{3}{2}

S=\frac{11}{2}

using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of

\frac{{J}_{2}}{{J}_{1}}

containing topological phases with alternating

\frac{2S\ensuremath{-}1}{2}

and

\frac{2S+1}{2}

valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those

\mathrm{high}\text{\ensuremath{-}}S

systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with

S

and converge to a single point at

\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}

in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for

S=\frac{1}{2}

, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of

\frac{{J}_{2}}{{J}_{1}}

values.

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

{J}_{1}

{J}_{2}

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

{J}_{2}/{J}_{1}

, 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

{Z}_{2}

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.

5/2

case confirms the universality of these phenomena across half-integer higher-spin systems. Our results provide detailed insight into how fractionalization and incommensurate condensation govern the spectral properties of frustrated spin chains, offering a unified picture across different spin magnitudes.

N

. A fully gapped

\mathrm{SU}(N)

valence bond solid made of

N

sites is formed above a critical value of

{J}_{2}/{J}_{1}

for all

N

. We find that the extension of this

N

-merized phase for larger values of

{J}_{2}

strongly depends on the parity of

N

. For even

N

, the phase smoothly interpolates to the large

{J}_{2}

regime where the model can be viewed as a zigzag

\mathrm{SU}(N)

two-leg spin ladder. The phase exhibits both a

N

-merized ground state and incommensurate spin-spin correlations. In stark contrast to the even case, we show that the

N

-merized phase with odd

N

has only a finite extent with no incommensuration. A gapless phase in the

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

universality class is stabilized for larger

{J}_{2}

that stems from the existence of a massless renormalization group flow from

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

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

conformal field theories when

N

is odd.

We use the density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin-

\frac{1}{2}

Heisenberg model on a honeycomb lattice with first (

{J}_{1}

) and second (

{J}_{2}

) neighbor antiferromagnetic interactions, where a

{Z}_{2}

spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with a width slightly over 15 lattice spacings and a torus up to the size $N=2\ifmmode\times\else\texttimes\fi{}6\ifmmode\times\else\texttimes\fi{}6$. With increasing

{J}_{2}

, we find a N\'eel phase with a vanishing spin gap and a plaquette valence-bond (PVB) phase with a nonzero spin gap. By extrapolating the square of the staggered magnetic moment

{m}_{s}^{2}

on finite-size cylinders to the thermodynamic limit, we find the N\'eel order vanishing at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.22$. For $0.25<{J}_{2}/{J}_{1}\ensuremath{\le}0.35$, we find a possible PVB order, which shows a fast growing PVB decay length with increasing system width. For

0.22<{J}_{2}/{J}_{1}<0.25

, both spin and dimer orders vanish in the thermodynamic limit, which is consistent with a possible spin liquid phase. We present calculations of the topological entanglement entropy, compare the DMRG results with the variational Monte Carlo, and discuss possible scenarios in the thermodynamic limit for this region.

Topological Spin Liquid Phase in the Spin-

1/2

Triangular Heisenberg Model