---
abstract: 'We study the effects of random bonds on spin chains that have an excitation gap in the absence of randomness. The dimerized spin-1/2 chain is our principal example. Using an asymptotically exact real space decimation renormalization group procedure, we find that dimerization is a relevant perturbation at the random singlet fixed point. For weak dimerization, the dimerized chain is in a Griffiths phase with short range spin-spin correlations and a divergent susceptibility. The string topological order, however, is not destroyed by bond randomness and dimerization is stabilized by the confinement of topological defects. We conjecture that random integer spin chains in the Haldane phase exhibit similar thermodynamic and topological properties.'
address:
- '$^1$Physics Department, Indiana University, Bloomington, IN 47405'
- '$^2$Department of Electrical Engineering, Princeton University, Princeton, NJ 08544'
author:
- 'R. A. Hyman$^1$, Kun Yang$^2$, R. N. Bhatt$^2$ and S. M. Girvin$^1$'
title: Random Bonds and Topological Stability in Gapped Quantum Spin Chains
---

Extensive theoretical work on random quantum magnetic systems has been carried out since the late 1970’s [@mdh; @bl; @fisher0; @df; @fisher1; @fisher2]. Systems that behave critically in the absence of randomness are unstable against weak randomness and flow to the random singlet (RS) phase[@mdh; @fisher1]. In the RS phase, spins far apart in space form weakly bound singlet pairs in a more or less random manner. This phase is also referred to as the valence bond glass (VBG) phase[@bl]. The low temperature thermodynamic properties of these systems are dominated by the weakly bound pairs and are universal[@mdh; @bl; @fisher1]. For instance, the susceptibility of the undimerized random antiferromagnetic Heisenberg or $XXZ$ spin-1/2 chain diverges as $[T\log^2T]^{-1}$ at low $T$, independent of the details of the randomness[@fisher1]. Universal power law behavior has also been found in disorder averaged spin-spin correlation functions[@fisher1]. Experiments, however, seem to find power law divergent susceptibilities with nonuniversal exponents[@experiments]. It is of interest to study if there exist relevant perturbations at the RS fixed point that drive the system towards a state exhibiting the nonuniversal behavior found experimentally.

A related issue is the effect of randomness on spin chains that have an excitation gap in the absence of randomness[@sorensen]. The most prominent examples of such chains are integer spin chains in the Haldane phase[@spin1]. Other examples include dimerized spin-1/2 chains[@singh] and spin chains with spontaneous dimerization[@next-near; @nomura]. All of these systems have topolologically ordered[@rn] ground states. One might think that strong enough randomness will inevitably destroy the topological order of the ground state. However Haldane has suggested[@duncan] that there exists a class of random perturbations for which the the topological order in the ground state of integer spin chains is stable regardless of the strength of the perturbations. We will show that the analog of this prediction for random bond dimerized spin-one-half chains is, in fact, correct.

An explicit example that provides strong support to Haldane’s conjecture (sic!) can be found in the random version of the AKLT model:[@aklt] $$H=\sum_i{J_i\left[{\bf S_{i}}\cdot{\bf S_{i+1}}
+{1\over 3}({\bf S_{i}}\cdot{\bf S_{i+1}})^2\right]},$$ where $J_i>0$. The exact ground state of this random model is identical to that of the pure model,[@aklt] i.e., a valence bond solid. Its excitation spectrum and thermodynamic properties will certainly depend on the distribution of $J_i$, yet the perfectly topologically ordered ground state is completely unaffected by randomness.

In the absence of disorder, spin-one chains in the Haldane phase and dimerized spin-one-half chains exhibit similar physical properties[@hida]. They both have a nondegenerate ground state with an excitation gap, and more importantly, they both have string-topological order[@hida]. Hida has shown that they can be continuously connected to each other without closing the gap or removing the topological order; i.e., they are in the same phase.[@hida] It is natural to expect that they also behave similarly in the presence of randomness.

In this paper we study the random bond dimerized spin-1/2 chain in detail. Using the asymptotically exact real space decimation renormalization group introduced by Ma, Dasgupta and Hu[@mdh] and extended by Fisher[@fisher1], we find that enforced dimerization is a relevant operator at the RS fixed point that drives the system to a random dimer (RD) phase. The low temperature thermodynamic properties of the RD phase are [*nonuniversal*]{}. For weak dimerization, the spectrum of the RD phase is gapless and the susceptibility diverges as $\chi\sim T^{-1+\alpha}$ with $\alpha>0$ and dependent on the bond distribution, (similar to the behavior found experimentally and in qualitative agreement with the RS thermodynamics), but the averaged spin-spin correlation function remains short ranged. Thus for weak dimerization the RD phase is an example of a Griffiths phase. More importantly, we find that the string-topological order is [*not*]{} destroyed by random bonds. We conjecture that these results also apply to random bond integer spin chains in the Haldane gapped phase. Comparison will also be made with spontaneously dimerized spin-chains which behave very differently upon introducing disorder.

Consider the model Hamiltonian $$H=\sum_i{J_i\left[S_i^xS_{i+1}^x+S_i^yS_{i+1}^y+\Delta S_i^zS_{i+1}^z\right],}
\label{model}$$ where $S_i^\alpha$ are spin-one-half operators, $J_i$ are (random) positive coupling constants, and $0\le \Delta\le 1$. Here we will concentrate on the cases $\Delta=0$ ($XX$ chain[@notexx]) and $\Delta=1$ (Heisenberg chain), since it has been shown that, for the case of random bonds, the Ising coupling is irrelevant when $\Delta<1$ and that the system flows to the random $XX$ chain in the low energy limit[@fisher1]. We assume that the distribution of the couplings, $J_i$, depends on whether $i$ is even or odd. We write the distribution functions for even and odd $J$’s as $P_e(J, J_0)$ and $P_o(J, J_0)$ respectively. Here $J_0$ is the cutoff in the distribution function corresponding to the strongest bond in the system. As Fisher[@fisher1] has shown, in the absence of dimerization, i.e., when $P_e(J, J_0)=P_o(J, J_0)$, the low-energy, long-distance behavior of Eq. (\[model\]) is universal, and the $XX$ and Heisenberg chains behave in essentially the same way.

Following Fisher[@fisher1], we introduce a decimation renormalization group procedure, in which we pick the bond in the system with the largest $J$ (i.e., the strongest bond), say $J_2$ between spins 2 and 3. Since this is such a strong bond, spins 2 and 3 are likely to form a singlet pair and become unimportant at low energies (on scales much smaller than $J_2$). The major physical effect of the existence of spins 2 and 3 is to generate an induced coupling between their neighboring spins 1 and 4. For the $XX$ chain: $\tilde{H}_{1-4}=\tilde{J}_{14}(S_1^xS_4^x+S_1^yS_4^y)$ where $\tilde{J}_{14}=J_1J_3/J_2+O(1/J_2^2)$ and for the Heisenberg chain: $\tilde{H}_{1-4}=\tilde{J}_{14}\bf{S_1}\cdot\bf{S_4}$ where $\tilde{J}_{14}=J_1J_3/(2J_2)+O(1/J_2^2)$. The effect of this decimation procedure is to get rid of the strongest bond (and also its two neighbors) in the system, generate a weaker bond between the spins neighboring the decimated ones, and lower the overall energy scale. This procedure becomes asymptotically exact in the low energy limit[@fisher1]. The new energy cutoff is then lowered to $\Omega={\rm max}\{\tilde{J}\}$. Following Fisher and anticipating that the bond distribution will become broad on logarithmic scales at low energy[@fisher1], we transform to logarithmic variables and define $\Gamma=-\log(\Omega/J_0)$ and $\zeta=\log(\Omega/\tilde{J})/\Gamma$, so that both $\Gamma$ and $\zeta$ are positive and a larger $\Gamma$ and a larger $\zeta$ correspond to a lower energy scale and a weaker bond respectively. The recursion relations now become (keeping the leading term only)

$$\tilde{\zeta}_{1-4}=\zeta_1+\zeta_3-\zeta_2+\kappa=\zeta_1+\zeta_3+\kappa$$

where we used the fact that $\zeta_2=0$ since $J_2=\Omega$ is the strongest bond in the system. $\kappa=0$ for the spin-one-half $XX$ chain and $\log(2)$ for the spin-one-half Heisenberg chain. The flow equations for the bond strength distribution functions $\rho_e(\zeta, \Omega)$ and $\rho_o(\zeta, \Omega)$ in terms of $\zeta$ are then $$\begin{aligned}
&&{\partial \rho_{e,o}(\zeta, \Gamma)\over
\partial \Gamma}={\partial \rho_{e,o}\over \partial \zeta}
+[\rho_{e,o}(0, \Gamma)-\rho_{o,e}(0, \Gamma)]\rho_{e,o}\nonumber\\
&&+
\rho_{o,e}(0, \Gamma)\int\int{d\zeta_1d\zeta_2}\rho_{e,o}(\zeta_1, \Gamma)
\rho_{e,o}(\zeta_2, \Gamma)\delta(\zeta-\zeta_1-\zeta_2-\kappa).
\label{flow1}\end{aligned}$$ When $\kappa=0$ (i.e., the $XX$ chain case), these flow equations are identical to those encountered in the transverse field Ising model if we identify the even bonds as the bonds between Ising spins and odd bonds as the transverse fields[@fisher2; @notefuture]. In order to find fixed point solutions of the renormalization group (RG) flow, it is necessary to rescale variables. Following Fisher[@fisher1; @fisher2], we introduce the rescaled variable $\eta=\zeta/\Gamma$ and the new distribution function $Q_{e,o}(\eta, \Gamma)=\Gamma
P_{e,o}(J,\Omega)$. The flow equations for $Q$ are $$\begin{aligned}
&&\Gamma{\partial Q_{e,o}\over \partial \Gamma}=Q_{e,o}
+(1+\eta){\partial Q_{e,o}\over
\partial\eta}
+[Q_{e,o}(0,\Gamma)-Q_{o,e}(0,\Gamma)]Q_{e,o}\nonumber\\
&&+Q_{o,e}(0,\Gamma)\int\int
{d\eta_1 d\eta_2}Q_{e,o}(\eta_1)Q_{e,o}(\eta_2)\delta(\eta - \eta_1 - \eta_2
- {\kappa\over\Gamma}).
\label{flow2}\end{aligned}$$ As Fisher has shown[@fisher2; @fisher1], the flow equations (\[flow2\]) have only one generic fixed point[@note1] $$Q_e=Q_o=Q^*(\eta)=e^{-\eta}\Theta(\eta).
\label{fixedpoint}$$ This fixed point distribution corresponds to the random spin-1/2 chain without dimerization, a model studied extensively before[@fisher1]. Going back to the original variable $\zeta$, we find the fixed point distribution corresponds to $$\rho(\zeta)={1\over \Gamma}e^{-\zeta/\Gamma},$$ i.e., the width of the distribution on the logarithmic scale grows linearly with the log of the energy scale $\Gamma$. For small deviation away from the fixed point $Q_e=Q^*+q_e$ and $Q_o=Q^*+q_o$, there is only one [*relevant*]{} eigenperturbation[@fisher2] behaving as $q_{e,o}(\eta, \Gamma)=q_{e,o}(\eta)\Gamma^\lambda$ with eigenvalue $\lambda=1$, and the eigenvector is $q_e=(\eta-1)e^{-\eta}$ and $q_o=-(\eta-1)e^{-\eta}$. The relevant perturbation, like the fixed point distribution, is independent of $\kappa$; hence the $XX$ and Heisenberg chains will behave similarly. However, non-zero $\kappa$ does affect the irrelevant perturbations.

The relevant perturbation corresponds to the [*difference*]{} in the distributions for even and odd bonds. Therefore we find that dimerization is a [*relevant*]{} perturbation near the RS fixed point, with eigenvalue $+1$.

For weak dimerization, the system barely knows that there is a small difference between even and odd bonds in the early stages of the RG flow. Both distributions initially flow toward the RS fixed point solution with a small relevant perturbation reflecting the extent of dimerization: $$\begin{aligned}
Q_o(\Gamma)&=&Q^* +\delta\Gamma(\eta - 1)e^{-\eta},\nonumber\\
Q_e(\Gamma)&=&Q^* -\delta\Gamma(\eta - 1)e^{-\eta}\end{aligned}$$ where $\delta$ characterizes the strength of the dimerization (distance from criticality). In general, $\delta$ depends in a complicated way on the shape of the original distributions and at what energy scale it is defined. As the flow away from the RS point continues, the even (odd) bonds get much weaker than the odd (even) bonds if originally the even (odd) bonds were only slightly weaker than the odd (even) bonds. The relevant perturbation grows linearly with $\Gamma$ and becomes of order $O(1)$ as $\Gamma=\Gamma_0\sim 1/|\delta|$. The flow equation for the density of spins that have not yet formed singlets at energy scale $\Gamma$ is $${\partial n(\Gamma)\over\partial \Gamma} = -2Q(0,\Gamma)
\label{density}$$ Using the RS fixed point distribution Eq. (\[fixedpoint\]) in Eq. (\[density\]) we find that $n\sim {1/\Gamma^2}$[@fisher1; @fisher2] so that when the relevant perturbation becomes large, the density of active spins is $n\sim\delta^2$. The corresponding length scale $L$, which is the typical distance between the remaining spins, is $L_0\sim\Gamma_0^2\sim 1/\delta^2$. At this stage the existence of dimerization becomes dominant and under RG most of the bonds decimated are odd bonds.

The fact that a small difference in the bond distributions grows as one lowers the energy is physically easy to see. Assume the odd bonds are slightly stronger than the even bonds in general. Then in the decimation procedure, it is slightly more likely that an odd bond gets decimated. When that happens, typically two intermediate strength neighboring even bonds also disappear, and a [*much weaker even bond*]{} gets generated. Hence, the width of the even bond distribution grows faster than the odd bond distribution, and its over all strength also decreases faster. Thus, in the low energy limit, the system can be viewed as a trivially soluble collection of uncoupled spin pairs (isolated odd bonds). We refer to this phase as the Random Dimer (RD) phase.

After renormalization the distribution of odd bonds takes the form $$\rho_o(\zeta)\sim {1\over \Gamma_0}e^{-\zeta/\Gamma_0}\Theta(\zeta).$$ In terms of the original variables the odd bond distribution is $$P_o(J)={\alpha\over \Omega_0}({J\over \Omega_0})^{-1+\alpha}
\Theta(1-{J\over \Omega_0}),
\label{distribution}$$ where $\Omega_0=J_0\exp(-\Gamma_0)$ and $\alpha\propto\delta$.

This effective independent pair Hamiltonian with a power law bond distribution is identical to that introduced by Clark and Tippie[@ct] to explain the low temperature thermodynamics of the random spin chains. Here we have [*derived*]{} it using RG from a realistic model. The leading temperature dependence of thermodynamic properties can be determined by assuming that all spins connected by bonds with energy greater than the temperature have paired up into singlets and all spins connected by bonds with energy less than the temperature are essentially free. This is a good approximation for broad bond distributions. In this way, the specific heat and susceptibility in the low temperature limit can be easily calculated. As the temperature goes to zero, the the spin susceptibility (in any direction, with possible direction dependent prefactors) diverges like $\chi\sim T^{\alpha-1}$, and the specific heat goes to zero like $C_v\sim T^{\alpha}$. The averaged spin-spin correlation function is short ranged, with the correlation length (distance between spins) $\xi\sim |\delta|^{-\nu}\sim |\delta|^{-2}$. The existence of a divergent magnetic susceptibility away from the critical point is characteristic of a Griffiths phase. The divergent susceptibility arises from magnetically active gapless excitations.[@grifnote] For the RS phase discussed by Fisher[@fisher1], the averaged spin-spin correlation function decays as $1/R^2$ at long distance so the system is critical and one expects a divergent susceptibility. The Griffiths phase in the random dimerized spin-one-half chain is exactly analogous to the Griffiths phase that appears in the random transverse field Ising chain[@fisher2]. If the initial dimerization is large, the flow begins far from the RS phase and the bond distribution of the stronger bonds does not flow to a power law and the gap does not close up. When this is the case, thermodynamic properties will depend strongly on the initial distribution and the susceptibility will remain finite.

The dimer phase has a novel kind of topological order that measures the dimerization of the chain. The “string-topological correlation function" is $$T_{ij}=\left\langle\Psi_0\left|S^z_i
\exp\left[i\pi\sum_{i<k<j}{S^z_k}\right] S^z_j\right|\Psi_0\right\rangle,$$ where $|\Psi_0\rangle$ is the ground state. $T_{ij}$ is similar to the topological correlation function for the spin-one chain[@rn] and it maps onto the spin-spin correlation function of the transverse field Ising chain.[@others] For a completely dimerized ground state $T_{ij}=-1/4$ if $i$ is a left spin of a dimer and $j$ is the right spin of a (possibly different) dimer. This is because every spin between $i$ and $j$ in the completely dimerized model is paired up with another spin between $i$ and $j$. $T_{ij}=0$ otherwise. Therefore this special topological correlation function is long-ranged although there is only short range spin-spin correlation, a situation similar to the special kind of off-diagonal long range order (ODLRO) in the fractional quantum Hall effect (FQHE)[@gm]. In the pure case this topological order is closely related to the existence of a gap[@rn], again just like in the FQHE.

For a general spin-one-half chain with randomness, we introduce a topological order parameter $$T=\lim_{j-i\rightarrow\infty}(\overline{T_{2i,2j-1}}-\overline{T_{2i+1,2j}}),$$ where the overbar stands for average over randomness. In the absence of dimerization, $T$ vanishes. For a random system, $T$ measures the probability that the two end spins survive decimation until the dimerization becomes large, and the low energy physics becomes that of the completely dimerized chain. This probability is just the square of the density of spins at the dimerization crossover scale. Therefore, for small $\delta$, $T$ scales like $$T\sim -|\delta|^{2\beta}{\rm sgn}(\delta),$$ with $\beta=\nu=2$, and ${\rm sgn}(\delta)$ is $+$ if even bonds are stronger and $-$ if odd bonds are stronger. In the absence of randomness $\beta$ is found to be $1/12$[@hida].

The dimerization described in this paper is premeditated because its magnitude and sign is determined by the details of the bond distributions. If the even bonds are stronger than the odd bonds then it costs less energy to form singlet pairs over even bonds than over odd bonds. In the pure case, premeditated dimerization leads to a unique ground state. In contrast, spontaneously dimerized spin chains can dimerize over even or odd bonds with equal energy cost. In the pure case, spontaneous dimerization leads to degenerate ground states. The fact that there is a right way and a wrong way to dimerize a premeditated chain and no wrong way to dimerize a spontaneous chain leads to profound differences in the response of these systems to random perturbations.

For the case of premeditated spin chains, the random perturbation and quantum and statistical fluctuations will generate regions where the chain has dimerized in the wrong way. Between regions of even and odd dimerization there has to be a topological soliton which is essentially a spin unpaired to its neighbors. In the presence of dimerization, the energy cost of the wrong region is proportional to its length (at least at long enough length scales); hence the topological solitons are confined by a linear confining potential and the topological order persists[@rs]. The confinement length in the weak dimerization limit is essentially the length scale at which dimerization becomes significant under RG, which is also the spin-spin correlation length. We hence find that although the gap vanishes in the presence of strong randomness, the dimer phase is stable and the topological order persists due to the confinement of topological defects. The situation for the FQHE is similar: incompressibility is lost upon introducing randomness, while ODLRO survives[@sondhi]. Another example of this phenomena occurs in dirty superconductors. Moderate disorder can remove the energy gap without destroying the condensate.

In contrast, topological solitons are unconfined in spontaneously dimerized systems because there are no wrong regions. Therefore, unlike the case of premeditated order, weak randomness will destroy spontaneously generated dimerization, despite of the existence of a gap in the absence of randomness.[@future]

We have shown that the dimerized spin-one-half chain is stable against disorder. This can be seen as due to the confinement of topological defects. When the dimerization is weak or randomness is strong the system is in a Griffiths phase in which the spin-spin correlation function is short ranged yet the susceptibility diverges. The susceptibility and specific heat follow simple power laws with nonuniversal exponents at low temperatures. The string-topological order is not be destroyed by disorder. We conjecture that these results apply to other systems with premeditated (not spontaneously generated) topological order such as the spin-one chain in the Haldane phase. We also conjecture that spontaneously generated topological order is unstable against disorder. A detailed analysis using a real space decimation procedure that is appropriate for the spin-1 and higher spin chains will be presented elsewhere[@future]. Experimentally there has been some work on effects of hole doping in the spin-1 chains[@ditusa]. The effects of hole doping is probably very different from random bonds in both spin-1 chains and dimerized spin-1/2 chains because it introduces unconfined topological defects into the system. Further investigation is underway[@future].

We are extremely grateful to Daniel Fisher and Duncan Haldane for numerous stimulating discussions and important suggestions. This work was supported by NSF grants DMR-9224077 and DMR-9416906.

S. K. Ma, C. Dasgupta and C-K Hu, Phys. Rev. Lett. [**43**]{}, 1434 (1979); C. Dasgupta and S. K. Ma, Phys. Rev. B [**22**]{}, 1305 (1979).

R. N. Bhatt and P. A. Lee, Phys. Rev. Lett. [**48**]{}, 344 (1982); R. N. Bhatt, Phys. Scr. [**T14**]{}, 7 (1986).

D. S. Fisher, Phys. Rev. Lett. [**69**]{}, 534 (1992); R. Shankar and G. Murthy, Phys. Rev. B [**36**]{}, 536 (1987).

C. A. Doty and D. S. Fisher, Phys. Rev. B [**45**]{}, 2167 (1992); S. Sachdev and J. Ye, Phys. Rev. Lett. [**70**]{}, 3339 (1993); J. Ye, S. Sachdev and N. Read, Rev. Lett. [**70**]{}, 4011 (1993).

D. S. Fisher, Phys. Rev. B [**50**]{}, 3799 (1994).

D. S. Fisher, Phys. Rev. B [**51**]{}, 6411 (1995).

L.C. Tippie and W.G. Clark, Phys. Rev. B [**23**]{}, 5846 (1981), and references therein.

Z.-Y. Lu, Z.-B. Su and Lu Yu, preprint (1994); E. S. Sorensen and Ian Affleck, Phys. Rev. B [**51**]{}, to appear (1995).

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

For recent work, see, for example, R. R. P. Singh, M. P. Gelfand and D. A. Huse, Phys. Rev. Lett. [**61**]{}, 2484 (1988), and work cited therein.

C. K. Majumdar and D. K. Ghosh, J. Math. Phys. [**10**]{}, 1388 and 1399 (1969); J. Phys. C [**3**]{}, 911 (1970); F. D. M. Haldane, Phys. Rev. B [**25**]{} 4925 (1982).

K. Nomura and S. Takada, J. Phys. Soc. Japan [**60**]{}, 389 (1991), and work cited therein.

K. Rommelse and M. den Nijs, Phys. Rev. Lett [**59**]{}, 2578 (1987); M. den Nijs and K. Rommelse, Phys. Rev. B [**40**]{}, 4709 (1989); S. M. Girvin and D. P. Arovas, Phys. Scr. [**T27**]{}, 156 (1988).

F. D. M. Haldane, (private communication).

Ian Affleck [*et al.*]{}, Phys. Rev. Lett. [**59**]{}, 799 (1987).

K. Hida, Phys. Rev. B [**45**]{}, 2207 (1992).

Here and throughout the paper we follow the terminology of Ref. . The $XX$ chain is called the $XY$ chain in many other contexts.

Further insight can be obtained by mapping the $XXZ$ system to a pair of transverse field Ising spin chains. See Refs. .

In the absence of dimerization this is a stable fixed point that controls all the low energy physics. There are other singular fixed points that cannot be reached unless the initial distribution is singular. See Ref. .

W. G. Clark and L. C. Tippie, Phys. Rev. B [**20**]{}, 2914 (1979).

The area of the phase diagram where the Griffiths phase occurs is not really a separate phase. The gap closes up but there is no level crossing.

M. Kohmoto and H. Tasaki, Phys. Rev. B [**46**]{}, 3486 (1992).

S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett. [**58**]{}, 1252 (1987); N. Read, Phys. Rev. Lett. [**62**]{}, 86 (1989).

This is known in the case of integer spin chains as confinement of spinon excitations. See N. Read and S. Sachdev, Phys. Rev. Lett. [**62**]{}, 1694 (1989). In their discussion there is no randomness. Randomness does not change this result at long enough length scales however.

S. L. Sondhi and M. P. Gelfand, Phys. Rev. Lett. [**73**]{}, 2119 (1994).

R. A. Hyman [*et al.*]{}, (unpublished).

J. F. DiTusa [*et al.*]{}, Phys. Rev. Lett. [**73**]{}, 1857 (1994).
