# «G. F´th and J. S´lyom a o Research Institute for Solid State Physics H-1525 Budapest, P. O. Box 49, Hungary (February 1, 2008) ...»

Solitonic excitations in the Haldane phase of a S=1 chain

G. F´th and J. S´lyom

a o

Research Institute for Solid State Physics

H-1525 Budapest, P. O. Box 49, Hungary

(February 1, 2008)

arXiv:cond-mat/9309011v1 13 Sep 1993

Abstract

We study low-lying excitations in the 1D S = 1 antiferromagnetic valencebond-solid (VBS) model. In a numerical calculation on ﬁnite systems the

lowest excitations are found to form a discrete triplet branch, separated from

the higher-lying continuum. The dispersion of these triplet excitations can be satisfactorily reproduced by assuming approximate wave functions. These wave functions are shown to correspond to moving hidden domain walls, i.e.

to one-soliton excitations.

PACS numbers: 75.10. Jm Typeset using REVTEX

## I. INTRODUCTION

It was almost a decade ago that Haldane [1] conjectured the existence of a new type of ground state for isotropic Heisenberg antiferromagnets (HAF) of integer spin S. The Haldane phase was proposed to be characterized by a unique, disordered ground state with exponential decay of the correlation functions and a ﬁnite energy gap to the excited states.Since then one-dimensional quantum spin chains with S = 1 have been studied intensively and it is claimed that the gapful behavior is a generic feature of integer-spin models [2].

The ﬁrst rigorous example of a S = 1 antiferromagnetic model with Haldane phase was given by Aﬄeck et al. [3]. They showed that the S = 1 isotropic bilinear-biquadratic model deﬁned by the Hamiltonian N N 1 β 1 Sj · Sj+1 + (Sj · Sj+1 )2 + H= hj (β) =, (1) j=1 j=1 has a short range valence-bond-solid ground state for β = 1/3 (VBS or AKLT model). At that point hj (1/3) is a special projection operator, that projects out the quintuplet state of the two neighboring spins, and it is positive semi-deﬁnite. Therefore, a state Ω for which hj Ω = 0 for any j is necessarily a ground state with ground-state energy EGS = 0. Such an Ω state could be constructed using nearest-neighbor valence bonds. They were also able to prove rigorously [3] that in the inﬁnite chain limit this state is the only ground state, it is separated by a ﬁnite gap from the excited states and the two-point correlation functions decay exponentially.

According to Haldane’s conjecture such a phase should not appear for half-integer values of the spin. This was proven rigorously by Aﬄeck and Lieb [4] and independently by Kolb [5]. It was shown for a wide class of models that in the case when the ground state is a spin singlet, the energy spectrum as a function of momentum k is symmetric under reﬂections with respect to k = mπ/2 (m integer), and therefore the ground state should be at least doubly degenerate.

A similar proof fails in the integer S case, allowing for the existence of a unique singlet ground state [4,5]. The excitation spectrum is in general symmetric with respect to k = mπ only. Of course, higher symmetry can also appear in integer S chains, as e.g. in the spontaneously dimerized phase of the general bilinear-biquadratic S = 1 model [6].

The S = 1 bilinear-biquadratic model is integrable [7] at the critical point β = 1, that separates the dimerized phase from the Haldane phase. Although the spin is integer, at this point the excitations can be described in exactly the same way as for the spin-1/2 HAF.

More generally it is known since the work by Faddeev and Takhtajan [7], that there are integrable spin models for arbitrary S in which the elementary excitations are in fact spinsolitons with a dispersion independent of the spin length S. The observable excitations are composite particles, since due to topological reasons the solitons can appear in singlet or triplet pairs only. As the energy of such a soliton pair can be described by two parameters, the excitations form a continuum in momentum space.

Away from the integrable point, where the symmetry properties of the excitation spectra are diﬀerent for integer and half-integer S, the above mentioned picture of composite excitations may not hold. In this paper we will study this problem.

We will restrict ourselves to the S = 1 case only, where the non-integrability appears in the most dramatic way in the Haldane phase. We will show that the lowest excitations are real spin-1, one-particle excitations, they cannot be decomposed into pairs of spin-1/2 solitons. These triplet excitations are, however, not usual antiferromagnons but rather some sort of hidden spin-1 solitons.

The solitonic nature of the excitations of the integer spin models was predicted by Hal

This prediction was later veriﬁed numerically by several authors [9]. The appearance of the hidden long-range order was further discussed by Kennedy and Tasaki [10]. They showed, α using a nonlocal unitary transformation, that Ostring 0 corresponds to the spontaneous breaking of a hidden Z2 × Z2 symmetry of the model. Similarly, the fact that the four lowest states of an open chain are exponentially close to each other is also a consequence of this broken symmetry. It is generally expected, that the breaking of a discrete symmetry in the ground state leads to an excitation gap, since Goldstone bosons do not appear. Excitations of the model can then be thought of as some sort of (hidden) domain walls, separating regions with diﬀerent ground states. This picture was made more explicit by Elstner and Mikeska [11], who used spin-zero defects [12] to disorder the antiferromagnetic state. The spin-zero defects are in fact solitons. One of the main goals of this paper is to further examine this problem.

We will use numerical and analytical methods to study the low-lying excitations in the β = 1/3 case. Beside the fact that the ground state of the VBS model can be constructed analytically, there is another good reason to focus on this model. In a recent study of the general bilinear-biquadratic model of Eq. (1), we observed [6] that the convergence of various ﬁnite-size estimates to their thermodynamic limit is extremely fast in the close vicinity of β = 1/3. This is certainly not true for general β. Moving away from β = 1/3 ﬁnite-size corrections become stronger, and one must consider longer and longer chains in order to see the real asymptotic behaviour. The rapid convergence at the VBS point may not be very surprising, if we remember, that at this point in the ground state ﬁrst-neighbor valence bonds are only present and the ground-state energy density becomes independent of the chain length. Although the excited states do show some dependence on N, this is found to be exponentially small for the most relevant levels. Therefore extrapolation from ﬁnitesize calculations allows to draw quite reliable conclusions on the spectrum and it can be You are reading a preview. Would you like to access the full-text?

with |j = φ1 ⊗ φ2 ⊗... φj ⊗ φj+1 ⊗ φj+2 ⊗... φN. In this form k is a variational parameter rather then a momentum, because of the open boundary condition. In the thermodynamic limit, however, the boundary condition should not matter (although it might bring a constant momentum shift q in the ﬁnal result, since we have the freedom to redeﬁne |j with e.g. an arbitrary phase factor |j → eiqj |j ), and the variational energy as a function of k is in fact the dispersion of the excitations in this simplest domain wall approach.

The computation would proceed similarly to that in the previous section. As we have ˜ seen there, the important quantities are j|j ′ and j|HVBS|j ′. However, in this case they are trivial because of the tensor product form. A straightforward calculation gives

In summary, we studied the elementary excitations in the valence-bond point of the S = 1 bilinear-biquadratic model. Numerical calculations on ﬁnite-size systems were used to predict the spectrum in the thermodynamic limit. The lowest-lying excited states above the k = 0 singlet ground state form a discrete triplet branch with a minimum at k = π.

Near this minimum this branch is separated from the higher-lying scattering continuum.

The energy needed to excite the lowest k = 0 excitation was found to be twice the gap value at k = π. Similarly, the energy of the next lowest excitation at k = π is three times the gap value. These excitations belong to the continuum and can be interpreted as being composed of two or three S = 1 elementary excitations.

Comparison with the numerical results show that the separate branch of excitations can be reasonably described with a trial wave function, where one singlet bond is replaced by a moving triplet bond. In the representation where the conﬁgurations are given in terms of the S z eigenstates of the spins, a triplet bond in the sea of singlet bonds has a solitonic character. In the dilute system of + and − spin states there is a single domain wall. While this feature is hidden in the usual valence-bond description, it becomes apparent when the nonlocal Kennedy-Tasaki transformation is used. We have shown that the approximate wave functions of the excited states transform into explicit domain walls in the transformed system.

This research was supported in part by the Hungarian Research Fund (OTKA) Grant Nos. T4473 and 2979. GF was also supported by the Hungarian Scientiﬁc Foundation.

[1] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983); Phys. Lett. 93A, 464 (1983).

[2] For an overall review see I. Aﬄeck, J. Phys. Condensed Matter 1, 3047 (1989).

[3] I. Aﬄeck, T. Kennedy, E. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 (1988).

[4] I. Aﬄeck and E. H. Lieb, Lett. Math. Phys. 12, 57 (1986).

[5] M. Kolb, Phys. Rev. B 31, 7494 (1985).

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[7] L. D. Faddeev and L. A. Takhtajan, Phys. Lett. 85A, 375 (1981);

L. A. Takhtajan, Phys. Lett. 87A, 479 (1982).

[8] M. den Nijs and K. Rommelse, Phys. Rev. B 40, 4709 (1989).

[9] S. M. Girvin and D. P. Arovas, Phys. Scr. T27, 156 (1989); Y. Hatsugai and M.

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B 46, 13914 (1992).

[10] T. Kennedy and H. Tasaki, Phys. Rev. B 45, 304 (1992); Commun. Math. Phys. 147, 431 (1992).

[11] N. Elstner and H.-J. Mikeska, Z. Phys. B 89, 321 (1992).

[12] G. Gomez-Santos, Phys. Rev. Lett. 63, 790 (1989).

[13] M. Takahashi, Phys. Rev. Lett. 62, 2313 (1989).

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N = 16 sites. Labels denote the total spin ST of the states. Dashed line shows the energy of the trial state with the moving hidden soliton.

FIG. 2. Energy gaps ∆A, ∆B /2, and ∆C /3 plotted vs 1/N. Dashed lines indicate the suggested large N behavior. 2 shows the energy of the trial wave function at k = π.