Exact value of the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest and next-nearest neighbour interactions

Broek, P.M. van den

doi:10.1016/0375-9601(80)90662-3

Cited by 72 publications

(14 citation statements)

References 8 publications

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“…This model and the above states were first discovered by Majumdar and Ghosh [31,32], and in [15] it was proved that these states were ground states. Some exact excited states were constructed in [19,39].…”

Section: ψL=( + )> ψ2=(-)mentioning

confidence: 80%

Valence bond ground states in isotropic quantum antiferromagnets

et al. 1988

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Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds. Table of Contents 1. Introduction 478 2. The One-Dimensional Model (1.1) 2.1. The Ground State 481 2.2. The Ground State Two-Point Correlation Function 485 2.3. The Energy Gap 487 2.4. The Infinite Chain 492 3. The Spin 3/2 Model on the Hexagonal Lattice 3.1. The Ground State 496 3.2. Properties of the Ground State 499 3.3. Proof of Exponential Decay 500 4. VBS States on an Arbitrary Lattice 4.1. The Ground States 504 4.2. The VBS State on the Cayley Tree 506 4.3. SU{n) VBS State and Random Loop Representation 510 4.4. SU(n) Quantum "Spin" System 514

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Section: ψL=( + )> ψ2=(-)mentioning

confidence: 80%

Valence bond ground states in isotropic quantum antiferromagnets

et al. 1988

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“…As mentioned above, the first rigorous proof of the properties of the Haldane phase [39,40] (in particular the existence of a spectral gap) was given by Affleck, Kennedy, Lieb and Tasaki in a particular spin-1 VBS chain [12,11], which is by now called the AKLT-model, and it is fair to say that it served as a paradigm for many of the subsequent studies on massive quantum spin chains. In that paper the authors also gave a detailed analysis of some other VBS models [41,42,43,44,45,46,47], which had been studied in the literature before, and thus introduced a new class of quantum spin Hamiltonians for which exact ground states with non-trivial properties can be constructed. Since then various other VBS models were introduced [15,37,48,49,50], including some interesting two-dimensional models [51,52,53,54,55,56,57].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

The spectral gap for some spin chains with discrete symmetry breaking

1996

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We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples.

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“…In principle, it is possible to generate Ψ 0 (M ) starting from M = 1 and using Eqs. (4) and (9). However, it is more convenient to obtain the recurrent formulae for expectation values (norm and correlators) with respect to the function Ψ 0 (M ) directly.…”

Section: Frustrated Spin Chain At Ferromagnet-antiferromagnet Tramentioning

confidence: 99%

Exact ground states for a class of one-dimensional frustrated quantum spin models

1997

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We have found the exact ground state for two frustrated quantum spin-1/2 models on a linear chain. The first model describes ferromagnet -antiferromagnet transition point. The singlet state at this point has double-spiral ordering. The second model is equivalent to special case of the spin-1/2 ladder. It has non-degenerate singlet ground state with exponentially decaying spin correlations and there is an energy gap. The exact ground state wave function of these models is presented in a special recurrent form and recurrence technics of expectation value calculations is developed.

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Exact value of the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest and next-nearest neighbour interactions

Cited by 72 publications

References 8 publications

Valence bond ground states in isotropic quantum antiferromagnets

Valence bond ground states in isotropic quantum antiferromagnets

The spectral gap for some spin chains with discrete symmetry breaking

Exact ground states for a class of one-dimensional frustrated quantum spin models

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