Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0002099 · Physical review. B, Condensed matter

Evaluation of low-energy effective Hamiltonian techniques for coupled spin triangles

C. Raghu, Indranil Rudra, S. Ramasesha, Diptiman Sen

Published Oct 1, 2000• 18 citations
arXivHTMLPDF

Abstract

Motivated by recent work on Heisenberg antiferromagnetic spin systems on various lattices made up of triangles, we examine the low-energy properties of a chain of antiferromagnetically coupled triangles of half-odd-integer spins. We derive the low-energy effective Hamiltonian to second order in the ratio of the coupling ${J}_{2}$ between triangles to the coupling ${J}_{1}$ within each triangle. The effective Hamiltonian contains four states for each triangle which are given by the products of spin-1/2 states with the states of a pseudospin 1/2. We compare the results obtained by exact diagonalization of the effective Hamiltonian with those obtained for the full Hamiltonian using exact diagonalization and the density-matrix renormalization group method. It is found that the effective Hamiltonian gives an accurate value for the ground-state energy only if the ratio ${J}_{2}{/J}_{1}$ is less than about 0.2 and that too for the spin-1/2 case with linear topology. The chain of spin-1/2 triangles shows interesting properties like spontaneous dimerization and several singlet and triplet low-energy (possibly gapless) states which lie close to the ground state. We have also studied the spin-3/2 case and find the low-energy effective Hamiltonians (LEH's) to be less accurate there than in the spin-1/2 case. Finally, we have studied nonlinear topologies where the LEH results deviate further from the exact results.

DOI
10.1103/physrevb.62.9484

Related papers

Antiferromagnetic sawtooth chain with spin- 12 and spin- 1 sites

V. Ravi Chandra, Diptiman Sen, Nedko B. Ivanov +1 more

We study the low-energy properties of a sawtooth chain with spin-1's at the bases of the triangles and spin-$\frac{1}{2}\mathrm{'}\mathrm{s}$at the vertices of the triangles. The spins have Heisenberg antiferromagnetic interactions between nearest neighbors, with a coupling ${J}_{2}$ between a spin-1 and a spin-$\frac{1}{2},$ and a coupling ${J}_{1}=1$ between two spin-1's. Analysis of the exact diagonalization data for periodic chains containing up to $N=12$ unit cells shows that the ground state is a singlet for exchange couplings up to approximately ${J}_{2}=3.8,$ whereas for larger ${J}_{2}$ the system exhibits a ferrimagnetic ground state characterized by a net ferromagnetic moment per unit cell of 1/2. In the region of small interactions ${J}_{2},$ the mixed spin sawtooth chain maps on to an effective isotropic spin model representing two weakly interacting and frustrated spin-$\frac{1}{2}$ Heisenberg chains composed of spin-$\frac{1}{2}$ sites at odd and even vertices, respectively. Finally, we study the phenomenon of a macroscopic magnetization jump which occurs if a magnetic field is applied with a value close to the saturation field for ${J}_{2}=2.$

arXiv:cond-mat/0307492•Physical Review B•Jun 3, 2004•22 citations

Study of Low-energy States of Clusters of Spin-1/2 and Spin-1 Triangles with Kagome-like Geometries

Indranil Rudra, S. Ramasesha, Diptiman Sen

We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.

arXiv:cond-mat/0210122•Oct 6, 2002•0 citations

Effective Hamiltonian and low-lying eigenenergy clustering patterns of four-sublattice antiferromagnets

N.-G. Zhang, Christopher L. Henley, Christian Rischel +1 more

We study the low-lying eigenenergy clustering patterns of quantum antiferromagnets with p sublattices (in particular $p=4).$ We treat each sublattice as a large spin, and using second-order degenerate perturbation theory, we derive the effective (biquadratic) Hamiltonian coupling the p large spins. In order to compare with exact diagonalizations, the Hamiltonian is explicitly written for a finite-size lattice, and it contains information on energies of excited states as well as the ground state. The result is applied to the face-centered-cubic Type-I antiferromagnet of spin 1/2, including second-neighbor interactions. A 32-site system is exactly diagonalized, and the energy spectrum of the low-lying singlets follows the analytically predicted clustering pattern.

arXiv:cond-mat/0106384•Physical review. B, Condensed matter•Jan 22, 2002•6 citations

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

Sahinur Reja, Satoshi Nishimoto

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.

arXiv:2406.04030•Physical review. B./Physical review. B•Aug 23, 2024•4 citations

Size-dependent low-energy excitations in an alternating spin-1/spin- 12 antiferromagnetic chain: Spin-wave theory and density-matrix renormalization-group studies

Sasmita Mohakud, Swapan Kumar Pati, Seiji Miyashita

We study the properties of the ground state and the low-lying excited states of an alternating spin-1 and spin-$\frac{1}{2}$ antiferromagnetic chain with ferromagnetic next-nearest-neighbor coupling with a variety of methods, namely, spin-wave analysis, density-matrix renormalization-group (DMRG) method, and exact-diagonalization method. The ground state of this model with $2N$ spins is ferrimagnetic with the total spin ${S}_{G}=N∕2$ for all parameter values. In the long chain limit, a gapless excitation is found with characteristics of goldstone mode of the ferromagnetic order with total spin $S={S}_{G}\ensuremath{-}1$. The lowest gapped excitation, however, exists in the total spin $S={S}_{G}+1$ sector. Interestingly, we find that with the increase in ferromagnetic coupling, spin-wave gapped excitation becomes lower than the massless mode for the large wave numbers. Correspondingly, from DMRG analysis we find that the excitation in the total spin $S={S}_{G}+1$ is lower than that in the total spin $S={S}_{G}\ensuremath{-}1$ when the size of the chain is short and the ferromagnetic interaction is large. Characteristics of these excitations are also reflected in the low temperature thermodynamic quantities, where the low-energy properties are strongly affected by the system size and the coupling strength.

Physical Review B•Jul 27, 2007•15 citations

Topological effects in short antiferromagnetic Heisenberg chains

Jizhong Lou, Shaojin Qin, Tai Kai Ng +1 more

The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to $5/2.$ For open chains with spin magnitudes $S=5/2$ to $S=5,$ we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes $S>~1.$

arXiv:cond-mat/0006409•Physical review. B, Condensed matter•Jan 28, 2002•23 citations
Motivated by recent work on Heisenberg antiferromagnetic spin systems on various lattices made up of triangles, we examine the low-energy properties of a chain of antiferromagnetically coupled triangles of half-odd-integer spins. We derive the low-energy effective Hamiltonian to second order in the ratio of the coupling J2{J}_{2}J2​ between triangles to the coupling J1{J}_{1}J1​ within each triangle. The effective Hamiltonian contains four states for each triangle which are given by the products of spin-1/2 states with the states of a pseudospin 1/2. We compare the results obtained by exact diagonalization of the effective Hamiltonian with those obtained for the full Hamiltonian using exact diagonalization and the density-matrix renormalization group method. It is found that the effective Hamiltonian gives an accurate value for the ground-state energy only if the ratio J2/J1{J}_{2}{/J}_{1}J2​/J1​ is less than about 0.2 and that too for the spin-1/2 case with linear topology. The chain of spin-1/2 triangles shows interesting properties like spontaneous dimerization and several singlet and triplet low-energy (possibly gapless) states which lie close to the ground state. We have also studied the spin-3/2 case and find the low-energy effective Hamiltonians (LEH's) to be less accurate there than in the spin-1/2 case. Finally, we have studied nonlinear topologies where the LEH results deviate further from the exact results.
We study the low-energy properties of a sawtooth chain with spin-1's at the bases of the triangles and spin-12′s\frac{1}{2}\mathrm{'}\mathrm{s}21​′sat the vertices of the triangles. The spins have Heisenberg antiferromagnetic interactions between nearest neighbors, with a coupling J2{J}_{2}J2​ between a spin-1 and a spin-12,\frac{1}{2},21​, and a coupling J1=1{J}_{1}=1J1​=1 between two spin-1's. Analysis of the exact diagonalization data for periodic chains containing up to N=12N=12N=12 unit cells shows that the ground state is a singlet for exchange couplings up to approximately J2=3.8,{J}_{2}=3.8,J2​=3.8, whereas for larger J2{J}_{2}J2​ the system exhibits a ferrimagnetic ground state characterized by a net ferromagnetic moment per unit cell of 1/2. In the region of small interactions J2,{J}_{2},J2​, the mixed spin sawtooth chain maps on to an effective isotropic spin model representing two weakly interacting and frustrated spin-12\frac{1}{2}21​ Heisenberg chains composed of spin-12\frac{1}{2}21​ sites at odd and even vertices, respectively. Finally, we study the phenomenon of a macroscopic magnetization jump which occurs if a magnetic field is applied with a value close to the saturation field for J2=2.{J}_{2}=2.J2​=2.
We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.
We study the low-lying eigenenergy clustering patterns of quantum antiferromagnets with p sublattices (in particular p=4).p=4).p=4). We treat each sublattice as a large spin, and using second-order degenerate perturbation theory, we derive the effective (biquadratic) Hamiltonian coupling the p large spins. In order to compare with exact diagonalizations, the Hamiltonian is explicitly written for a finite-size lattice, and it contains information on energies of excited states as well as the ground state. The result is applied to the face-centered-cubic Type-I antiferromagnet of spin 1/2, including second-neighbor interactions. A 32-site system is exactly diagonalized, and the energy spectrum of the low-lying singlets follows the analytically predicted clustering pattern.
We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from S=32S=\frac{3}{2}S=23​ to S=112S=\frac{11}{2}S=211​ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and 2S+12\frac{2S+1}{2}22S+1​ 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 SSS and converge to a single point at J2J1=14\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}J1​J2​​=41​ 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=12S=\frac{1}{2}S=21​, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ values.
We study the properties of the ground state and the low-lying excited states of an alternating spin-1 and spin-12\frac{1}{2}21​ antiferromagnetic chain with ferromagnetic next-nearest-neighbor coupling with a variety of methods, namely, spin-wave analysis, density-matrix renormalization-group (DMRG) method, and exact-diagonalization method. The ground state of this model with 2N2N2N spins is ferrimagnetic with the total spin SG=N∕2{S}_{G}=N∕2SG​=N∕2 for all parameter values. In the long chain limit, a gapless excitation is found with characteristics of goldstone mode of the ferromagnetic order with total spin $S={S}_{G}\ensuremath{-}1$. The lowest gapped excitation, however, exists in the total spin S=SG+1S={S}_{G}+1S=SG​+1 sector. Interestingly, we find that with the increase in ferromagnetic coupling, spin-wave gapped excitation becomes lower than the massless mode for the large wave numbers. Correspondingly, from DMRG analysis we find that the excitation in the total spin S=SG+1S={S}_{G}+1S=SG​+1 is lower than that in the total spin $S={S}_{G}\ensuremath{-}1$ when the size of the chain is short and the ferromagnetic interaction is large. Characteristics of these excitations are also reflected in the low temperature thermodynamic quantities, where the low-energy properties are strongly affected by the system size and the coupling strength.
The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to 5/2.5/2.5/2. For open chains with spin magnitudes S=5/2S=5/2S=5/2 to S=5,S=5,S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes S> 1.S>~1.S> 1.