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:0807.3189 · Physical Review B

Order and disorder in AKLT antiferromagnets in three dimensions

S. A. Parameswaran, Shivaji L. Sondhi, Daniel P. Arovas

Published Jan 9, 2009• 26 citations
arXivHTMLPDF

Abstract

The models constructed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)] describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin $S$ is an integer multiple $M$ of half the lattice coordination number. The equal-time quantum correlations in their ground states may be computed as finite temperature correlations of a classical $O(3)$ model on the same lattice, where the temperature is given by $T=1/M$. In dimensions $d=1$ and 2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in $d=3$ where the nature of the AKLT states is now a question of detail depending upon the choice of lattice and spin; for sufficiently large $S$ some form of N\'eel order is almost inevitable. On the unfrustrated cubic lattice, we find that all AKLT states are ordered, while for the unfrustrated diamond lattice the minimal $S=2$ state is disordered while all other states are ordered. On the frustrated pyrochlore lattice, we find (conservatively) that several states starting with the minimal $S=3$ state are disordered. The disordered AKLT models we report here are a significant addition to the catalog of magnetic Hamiltonians in $d=3$ with ground states known to lack order on account of strong quantum fluctuations.

DOI
10.1103/physrevb.79.024408

Related papers

Order and disorder in SU( N ) simplex solid antiferromagnets

Yury Kiselev, S. A. Parameswaran, Daniel P. Arovas

Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.

Journal of Statistical Mechanics Theory and Experiment•Jan 1, 2016•1 citation

Order and disorder in SU(N) simplex solid antiferromagnets

Yury Kiselev, S. A. Parameswaran, Daniel P. Arovas

We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)]. Whereas the AKLT states are created by application of bond singlet operators for SU(2) spins, the simplex solid construction is based on N-simplex singlet operators for SU(N) spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a CP^{N-1} coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of SU(4) edge- and SU(8) face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the SU(3) models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to `order by disorder'.

arXiv:1410.2288•Oct 8, 2014•0 citations

Emergence of XY-like phase in deformed spin- 32 AKLT systems

Ching-Yu Huang, Maximilian Anton Wagner, Tzu-Chieh Wei

Affleck, Kennedy, Lieb, and Taski (AKLT) constructed an exemplary spin-3/2 valence-bond solid (VBS) state on the hexagonal lattice, which is the ground state of an isotropic quantum antiferromagnet and possesses no spontaneous magnetization but finite correlation length. This is distinct from the N\'eel ordered state of the spin-3/2 Heisenberg model on the same lattice. Niggemann, Kl\"umper, and Zittartz then generalized the AKLT Hamiltonian to one family invariant under spin rotation about the $z$ axis. The ground states of this family can be parameterized by a single parameter that deforms the AKLT state, and this system exhibits a quantum phase transition between the VBS and N\'eel phases, as the parameter increases from the AKLT point to large anisotropy. We investigate the opposite regime when the parameter decreases from the AKLT point and find that there appears to be a Berezinskii-Kosterlitz-Thouless-like transition from the VBS phase to an XY phase. Such a transition also occurs in the deformation of other types of AKLT states with triplet-bond constructions on the same lattice. However, we do not find such an XY-like phase in the deformed AKLT models on other trivalent lattices, such as square-octagon, cross, and star lattices. On the star lattice, the deformed family of AKLT states remain in the same phase as the isotropic AKLT state throughout the whole region of the parameter. However, for two triplet-bond generalizations, the triplet VBS phase is sandwiched between two ferromagnetic phases (for large and small deformation parameters, respectively), which are characterized by spontaneous magnetizations along different axes. Along the way, we also discuss how various deformed AKLT states can be used for the purpose of universal quantum computation.

arXiv:1605.08417•Physical review. B./Physical review. B•Oct 14, 2016•14 citations

Simplex solid states of SU(N) quantum antiferromagnets

Daniel P. Arovas

I define a set of wave functions for $\mathrm{SU}(N)$ lattice antiferromagnets, analogous to the valence bond solid states of Affleck et al. [Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 (1988)], in which the singlets are extended over $N$-site simplixes. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing one to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, I show that the quantum correlations in each SS state are calculable as the finite temperature correlations of an associated classical model with $N$-spin interactions on the same lattice. In three and higher dimensions, the SS states can spontaneously break $\mathrm{SU}(N)$ and exhibit $N$-sublattice long-ranged order as a function of a discrete parameter which fixes the local representation of $\mathrm{SU}(N)$. I analyze this transition using a classical mean field approach. For $N>2$, the ordered state is selected via an ``order by disorder'' mechanism. As in the Affleck-Kennedy-Lieb-Tasaki case, the bulk representations fractionalize at an edge, and the ground state entropy is proportional to the volume of the boundary.

arXiv:0711.3921•Physical Review B•Mar 6, 2008•59 citations

AKLT models with quantum spin glass ground states

Chris R. Laumann, S. A. Parameswaran, Shivaji L. Sondhi +1 more

We study AKLT models on locally treelike lattices of fixed connectivity and find that they exhibit a variety of ground states depending upon the spin, coordination, and global (graph) topology. We find (a) quantum paramagnetic or valence-bond solid ground states, (b) critical and ordered N\'eel states on bipartite infinite Cayley trees, and (c) critical and ordered quantum vector spin glass states on random graphs of fixed connectivity. We argue, in consonance with a previous analysis [C. R. Laumann, S. A. Parameswaran, and S. L. Sondhi, Phys. Rev. B 80, 144415 (2009)], that all phases are characterized by gaps to local excitations. The spin glass states we report arise from random long-ranged loops which frustrate N\'eel ordering despite the lack of randomness in the coupling strengths.

arXiv:1002.4593•Physical Review B•May 17, 2010•18 citations

Spectral properties for a family of two-dimensional quantum antiferromagnets

Andrew S. Darmawan, Stephen D. Bartlett

We study the spectral properties of a family of quantum antiferromagnets on two-dimensional (2D) lattices. This family of models is obtained by a deformation of the well-studied 2D quantum antiferromagnetic model of Affleck, Kennedy, Lieb, and Tasaki (AKLT); they are described by two-body, frustration-free Hamiltonians on a three-colorable lattice of spins. Although the existence of a spectral gap in the 2D AKLT model remains an open question, we rigorously prove the existence of a gap for a subset of this family of quantum antiferromagnets. Along with providing progress for the gap problem in AKLT-type antiferromagnets in 2D, this result has implications for the theory of quantum computation as it provides a family of two-body Hamiltonians for which the ground state is a resource for universal quantum computation and for which a spectral gap is proven to exist.

arXiv:1509.04719•Physical review. B./Physical review. B•Jan 21, 2016•5 citations
The models constructed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)] describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin SSS is an integer multiple MMM of half the lattice coordination number. The equal-time quantum correlations in their ground states may be computed as finite temperature correlations of a classical O(3)O(3)O(3) model on the same lattice, where the temperature is given by T=1/MT=1/MT=1/M. In dimensions d=1d=1d=1 and 2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in d=3d=3d=3 where the nature of the AKLT states is now a question of detail depending upon the choice of lattice and spin; for sufficiently large SSS some form of N'eel order is almost inevitable. On the unfrustrated cubic lattice, we find that all AKLT states are ordered, while for the unfrustrated diamond lattice the minimal S=2S=2S=2 state is disordered while all other states are ordered. On the frustrated pyrochlore lattice, we find (conservatively) that several states starting with the minimal S=3S=3S=3 state are disordered. The disordered AKLT models we report here are a significant addition to the catalog of magnetic Hamiltonians in d=3d=3d=3 with ground states known to lack order on account of strong quantum fluctuations.
Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.
We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)]. Whereas the AKLT states are created by application of bond singlet operators for SU(2) spins, the simplex solid construction is based on N-simplex singlet operators for SU(N) spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a CP^{N-1} coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of SU(4) edge- and SU(8) face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the SU(3) models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to `order by disorder'.
Affleck, Kennedy, Lieb, and Taski (AKLT) constructed an exemplary spin-3/2 valence-bond solid (VBS) state on the hexagonal lattice, which is the ground state of an isotropic quantum antiferromagnet and possesses no spontaneous magnetization but finite correlation length. This is distinct from the N\'eel ordered state of the spin-3/2 Heisenberg model on the same lattice. Niggemann, Kl\"umper, and Zittartz then generalized the AKLT Hamiltonian to one family invariant under spin rotation about the zzz axis. The ground states of this family can be parameterized by a single parameter that deforms the AKLT state, and this system exhibits a quantum phase transition between the VBS and N\'eel phases, as the parameter increases from the AKLT point to large anisotropy. We investigate the opposite regime when the parameter decreases from the AKLT point and find that there appears to be a Berezinskii-Kosterlitz-Thouless-like transition from the VBS phase to an XY phase. Such a transition also occurs in the deformation of other types of AKLT states with triplet-bond constructions on the same lattice. However, we do not find such an XY-like phase in the deformed AKLT models on other trivalent lattices, such as square-octagon, cross, and star lattices. On the star lattice, the deformed family of AKLT states remain in the same phase as the isotropic AKLT state throughout the whole region of the parameter. However, for two triplet-bond generalizations, the triplet VBS phase is sandwiched between two ferromagnetic phases (for large and small deformation parameters, respectively), which are characterized by spontaneous magnetizations along different axes. Along the way, we also discuss how various deformed AKLT states can be used for the purpose of universal quantum computation.
I define a set of wave functions for SU(N)\mathrm{SU}(N)SU(N) lattice antiferromagnets, analogous to the valence bond solid states of Affleck et al. [Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 (1988)], in which the singlets are extended over NNN-site simplixes. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing one to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, I show that the quantum correlations in each SS state are calculable as the finite temperature correlations of an associated classical model with NNN-spin interactions on the same lattice. In three and higher dimensions, the SS states can spontaneously break SU(N)\mathrm{SU}(N)SU(N) and exhibit NNN-sublattice long-ranged order as a function of a discrete parameter which fixes the local representation of SU(N)\mathrm{SU}(N)SU(N). I analyze this transition using a classical mean field approach. For N>2N>2N>2, the ordered state is selected via an ``order by disorder'' mechanism. As in the Affleck-Kennedy-Lieb-Tasaki case, the bulk representations fractionalize at an edge, and the ground state entropy is proportional to the volume of the boundary.
We study AKLT models on locally treelike lattices of fixed connectivity and find that they exhibit a variety of ground states depending upon the spin, coordination, and global (graph) topology. We find (a) quantum paramagnetic or valence-bond solid ground states, (b) critical and ordered N\'eel states on bipartite infinite Cayley trees, and (c) critical and ordered quantum vector spin glass states on random graphs of fixed connectivity. We argue, in consonance with a previous analysis [C. R. Laumann, S. A. Parameswaran, and S. L. Sondhi, Phys. Rev. B 80, 144415 (2009)], that all phases are characterized by gaps to local excitations. The spin glass states we report arise from random long-ranged loops which frustrate N\'eel ordering despite the lack of randomness in the coupling strengths.
We study the spectral properties of a family of quantum antiferromagnets on two-dimensional (2D) lattices. This family of models is obtained by a deformation of the well-studied 2D quantum antiferromagnetic model of Affleck, Kennedy, Lieb, and Tasaki (AKLT); they are described by two-body, frustration-free Hamiltonians on a three-colorable lattice of spins. Although the existence of a spectral gap in the 2D AKLT model remains an open question, we rigorously prove the existence of a gap for a subset of this family of quantum antiferromagnets. Along with providing progress for the gap problem in AKLT-type antiferromagnets in 2D, this result has implications for the theory of quantum computation as it provides a family of two-body Hamiltonians for which the ground state is a resource for universal quantum computation and for which a spectral gap is proven to exist.