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

Journal of Physics Condensed Matter

Matrix formalism for the VBS-type models and hidden order

Kiyohito Totsuka, Masuo Suzuki

Published Feb 20, 1995• 72 citations
Publisher

Abstract

We give a simple method to reduce the VBS-type states, usually expressed by the valence bonds, to the matrix-product representation. In this representation, peculiar properties of the VBS-type states are quite manifest. Using it, we investigate hidden order in the spin-S VBS model. The possibility that the Haldane phase for higher spin S is characterized by string order parameters is discussed. A picture of an approximate excitation is also presented.

DOI
10.1088/0953-8984/7/8/012

Related papers

Hidden order and dynamics in supersymmetric valence-bond solid states: Super-matrix product state formalism

Kazuki Hasebe, Keisuke Totsuka

Supersymmetric valence-bond solid models are extensions of the VBS model, a paradigmatic model of ``solvable'' gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix-product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin or the charge excitation spectrum, the superconducting order parameter. A generalized hidden order is proposed, and the corresponding string nonlocal order parameter is also calculated. The behavior of the string-order parameter is discussed in the light of the entanglement spectrum.

arXiv:1105.3529•Physical Review B•Sep 15, 2011•12 citations

Valence bond solid states with symplectic symmetry

Dirk Schuricht, Stephan Rachel

We introduce a one-dimensional valence bond solid (VBS) state with symplectic symmetry $\text{SP}(n)$ and construct the corresponding parent Hamiltonian. We argue that there is a gap in the spectrum. We calculate exactly the static correlation functions, which fall off exponentially. Hence the model introduced here shares all properties of the Haldane scenario for integer-spin quantum antiferromagnets. We further show that the VBS state possesses string order and discuss its generalization to higher dimensions.

arXiv:0805.3918•Physical Review B•Jul 30, 2008•37 citations

Generalized String Order in the Spin-1/2 Alternating Heisenberg Chain

Kazuo Hida

The phase-dependent generalized string order introduced by Oshikawa in the VBS (valence bond solid)-type states of the integer spin chain is extended to the spin-1/2 J - J ′ alternating Heisenberg chain with antiferromagnetic J . The numerical diagonalization and the bosonization analysis show that the generalized string order remains finite in the ground state of the present model for -∞< J ′ < J , taking the maximum at J ′ =0. The characteristic feature of the phase dependence of the generalized string order also coincides with that of the spin-1 VBS state. This reconfirms the notion that the Haldane state of the spin-1 chain and the dimerized state of the alternating chain belong to a single phase characterized by the local singlet-type correlation.

Journal of the Physical Society of Japan•Feb 15, 1993•16 citations

Valence-bond-solid order in antiferromagnets with spin-lattice coupling

Chenglong Jia, Jung Hoon Han

We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when $2S$ is not an integer multiple of the coordination number $z$. $S=2$ spins on the triangular lattice with $z=6$ is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.

arXiv:cond-mat/0505573•Physical Review B•May 18, 2006•3 citations

Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry

Kazuki Hasebe, Keisuke Totsuka

Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking, but most typically manifest a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate, regardless of the parity of bulk (super)spins. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.

arXiv:1303.2287•Symmetry•Apr 26, 2013•6 citations

Hidden Z 2 *Z 2 symmetry in quantum spin chains with arbitrary integer spin

Masaki Oshikawa

The author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.

Journal of Physics Condensed Matter•Sep 7, 1992•222 citations
We give a simple method to reduce the VBS-type states, usually expressed by the valence bonds, to the matrix-product representation. In this representation, peculiar properties of the VBS-type states are quite manifest. Using it, we investigate hidden order in the spin-S VBS model. The possibility that the Haldane phase for higher spin S is characterized by string order parameters is discussed. A picture of an approximate excitation is also presented.
Supersymmetric valence-bond solid models are extensions of the VBS model, a paradigmatic model of ``solvable'' gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix-product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin or the charge excitation spectrum, the superconducting order parameter. A generalized hidden order is proposed, and the corresponding string nonlocal order parameter is also calculated. The behavior of the string-order parameter is discussed in the light of the entanglement spectrum.
We introduce a one-dimensional valence bond solid (VBS) state with symplectic symmetry SP(n)\text{SP}(n)SP(n) and construct the corresponding parent Hamiltonian. We argue that there is a gap in the spectrum. We calculate exactly the static correlation functions, which fall off exponentially. Hence the model introduced here shares all properties of the Haldane scenario for integer-spin quantum antiferromagnets. We further show that the VBS state possesses string order and discuss its generalization to higher dimensions.
The phase-dependent generalized string order introduced by Oshikawa in the VBS (valence bond solid)-type states of the integer spin chain is extended to the spin-1/2 J - J ′ alternating Heisenberg chain with antiferromagnetic J . The numerical diagonalization and the bosonization analysis show that the generalized string order remains finite in the ground state of the present model for -∞< J ′ < J , taking the maximum at J ′ =0. The characteristic feature of the phase dependence of the generalized string order also coincides with that of the spin-1 VBS state. This reconfirms the notion that the Haldane state of the spin-1 chain and the dimerized state of the alternating chain belong to a single phase characterized by the local singlet-type correlation.
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when 2S2S2S is not an integer multiple of the coordination number zzz. S=2S=2S=2 spins on the triangular lattice with z=6z=6z=6 is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking, but most typically manifest a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate, regardless of the parity of bulk (super)spins. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.
The author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.