arXiv:1810.11533 · Journal of the Physical Society of Japan

Third Boundary of the Shastry–Sutherland Model by Numerical Diagonalization

Hiroki Nakano, Tôru Sakai

Published Nov 15, 2018• 30 citations

Abstract

The Shastry-Sutherland model --- the $S=1/2$ Heisenberg antiferromagnet on the square lattice accompanied by orthogonal dimerized interactions --- is studied by the numerical-diagonalization method. Large-scale calculations provide results for larger clusters that have not been reported yet. The present study successfully captures the phase boundary between the dimer and plaquette-singlet phases and clarifies that the spin gap increases once when the interaction forming the square lattice is increased from the boundary. Our calculations strongly suggest that in addition to the edge of the dimer phase given by $J_{2}/J_{1}\sim 0.675$ and the edge of the N$\acute{\rm e}$el-ordered phase given by $J_{2}/J_{1}\sim 0.76$, there exists a third boundary ratio $J_{2}/J_{1}\sim 0.70$ that divides the intermediate region into two parts, where $J_{1}$ and $J_{2}$ denote dimer and square-lattice interactions, respectively.

DOI: 10.7566/jpsj.87.123702

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arXiv:1810.11533 · Journal of the Physical Society of Japan

Third Boundary of the Shastry–Sutherland Model by Numerical Diagonalization

Hiroki Nakano, Tôru Sakai

Published Nov 15, 2018• 30 citations

arXiv HTML PDF

Abstract

The Shastry-Sutherland model --- the

S=1/2

Heisenberg antiferromagnet on the square lattice accompanied by orthogonal dimerized interactions --- is studied by the numerical-diagonalization method. Large-scale calculations provide results for larger clusters that have not been reported yet. The present study successfully captures the phase boundary between the dimer and plaquette-singlet phases and clarifies that the spin gap increases once when the interaction forming the square lattice is increased from the boundary. Our calculations strongly suggest that in addition to the edge of the dimer phase given by

J_{2}/J_{1}\sim 0.675

and the edge of the N

\acute{\rm e}

el-ordered phase given by

J_{2}/J_{1}\sim 0.76

, there exists a third boundary ratio

J_{2}/J_{1}\sim 0.70

that divides the intermediate region into two parts, where

J_{1}

and

J_{2}

denote dimer and square-lattice interactions, respectively.

DOI: 10.7566/jpsj.87.123702

Related papers

Large-Scale Numerical-Diagonalization Study of the Shastry–Sutherland Model

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Spiral and Mixed Plaquette-Dimer Phases in the $S=1$ and $3/2$ Shastry-Sutherland Heisenberg ModelSpiral and Mixed Plaquette-Dimer Phases in the $S=1$ and $3/2$ Shastry-Sutherland Heisenberg Model

Han-Qing Wu, Muwei Wu, Shou-Shu Gong

We investigate the ground-state phase diagram of the $S=1$ and $S=3/2$ Heisenberg model on the two-dimensional Shastry-Sutherland lattice (SSL) using density matrix renormalization group (DMRG) and cluster mean-field theory (CMFT). Between the dimer phase and Néel antiferromagnetic phases, we identify two intermediate phases: a mixed plaquette-dimer (MPD) phase and a spiral phase. These phases are characterized via bond energies and spin-spin correlation functions; phase boundaries are located from the ground-state energy derivative and entanglement entropy. The MPD phase exhibits strong intradimer correlations and weak tetramerization on the empty plaquettes, and its transitions to the dimer and spiral phases are first order. Combining our results with the known boundaries for $S=1/2$ and the classical limit $S\to\infty$ , we construct a global $S$ - $g$ phase diagram. This diagram reveals the progressive suppression of quantum effects with increasing $S$ and offers a theoretical framework for larger- $S$ SSL materials.We investigate the ground-state phase diagram of the $S=1$ and $S=3/2$ Heisenberg model on the two-dimensional Shastry-Sutherland lattice (SSL) using density matrix renormalization group (DMRG) and cluster mean-field theory (CMFT). Between the dimer phase and Néel antiferromagnetic phases, we identify two intermediate phases: a mixed plaquette-dimer (MPD) phase and a spiral phase. These phases are characterized via bond energies and spin-spin correlation functions; phase boundaries are located from the ground-state energy derivative and entanglement entropy. The MPD phase exhibits strong intradimer correlations and weak tetramerization on the empty plaquettes, and its transitions to the dimer and spiral phases are first order. Combining our results with the known boundaries for $S=1/2$ and the classical limit $S\to\infty$, we construct a global $S$-$g$ phase diagram. This diagram reveals the progressive suppression of quantum effects with increasing $S$ and offers a theoretical framework for larger-$S$ SSL materials.

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Identifying the ground state phases by spin-patterns in the Shastry-Sutherland model

Yun-Tong Yang, Fuzhou Chen, Hong‐Gang Luo

Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond ( $J_1$ ) and the next-nearest-neighbor one ( $J_2$ ). In the two limits of $α=J_2/J_1$ , i.e. $α\ll 1$ and $α\gg 1$ , the corresponding phases are the N{é}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{é}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size $L=4 \times4$ with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter $κ=α/ (1+α)$ . Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{é}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around $α_c = 1.5$ is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around $α_c = 1.277$ , where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond ($J_1$) and the next-nearest-neighbor one ($J_2$). In the two limits of $α=J_2/J_1$, i.e. $α\ll 1$ and $α\gg 1$, the corresponding phases are the N{é}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{é}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size $L=4 \times4$ with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter $κ=α/ (1+α)$. Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{é}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around $α_c = 1.5$ is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around $α_c = 1.277$, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.

arXiv:2404.16330Apr 25, 20240 citations

Numerical Diagonalization Study of the Phase Boundaries of the S = 2 Heisenberg Antiferromagnet on the Orthogonal Dimer Lattice

Hiroki Nakano, Tôru Sakai, Yuko Hosokoshi

arXiv:2601.21328Journal of the Physical Society of JapanFeb 16, 20261 citation

The Shastry-Sutherland model --- the

S=1/2

J_{2}/J_{1}\sim 0.675

and the edge of the N

\acute{\rm e}

el-ordered phase given by

J_{2}/J_{1}\sim 0.76

, there exists a third boundary ratio

J_{2}/J_{1}\sim 0.70

that divides the intermediate region into two parts, where

J_{1}

and

J_{2}

denote dimer and square-lattice interactions, respectively.

J

perpendicular to the boundary and ${J}^{\ensuremath{'}}$ parallel to the boundary. We find that in the latter model, as ${J}^{\ensuremath{'}}/J$ is reduced from unity, the dimer correlations near the edge become longer ranged. In contrast, in the frustrated model, with increasing

{J}_{2}

, dimer correlations are strengthened near the boundary but they decrease rapidly with distance. These results imply that deep inside the N\'eel phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, dimer correlations remain short ranged. Hence, if there is a direct transition between the two, it is either first order or there is a very narrow critical region.

Spiral and Mixed Plaquette-Dimer Phases in the

S=1

and

3/2

Shastry-Sutherland Heisenberg Model

We investigate the ground-state phase diagram of the

S=1

and

S=3/2

Heisenberg model on the two-dimensional Shastry-Sutherland lattice (SSL) using density matrix renormalization group (DMRG) and cluster mean-field theory (CMFT). Between the dimer phase and Néel antiferromagnetic phases, we identify two intermediate phases: a mixed plaquette-dimer (MPD) phase and a spiral phase. These phases are characterized via bond energies and spin-spin correlation functions; phase boundaries are located from the ground-state energy derivative and entanglement entropy. The MPD phase exhibits strong intradimer correlations and weak tetramerization on the empty plaquettes, and its transitions to the dimer and spiral phases are first order. Combining our results with the known boundaries for

S=1/2

and the classical limit

S\to\infty

, we construct a global

S

g

phase diagram. This diagram reveals the progressive suppression of quantum effects with increasing

S

and offers a theoretical framework for larger-

S

SSL materials.

J_1

) and the next-nearest-neighbor one (

J_2

). In the two limits of

α=J_2/J_1

, i.e.

α\ll 1

and

α\gg 1

, the corresponding phases are the N{é}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{é}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size

L=4 \times4

with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter

κ=α/ (1+α)

. Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{é}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around

α_c = 1.5

is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around

α_c = 1.277

, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.