Dynamical Properties of a Haldane Gap Antiferromagnet

a r X i v :c o n d -m a t /9212014v 1 9 D e c 1992DYNAMICAL PROPERTIES OF A

HALDANE GAP ANTIFERROMAGNET

O.Golinelli,Th.Jolic?ur,*and https://www.doczj.com/doc/0a1781078.html,caze ?

Service de Physique Th′e orique**

C.E.Saclay

F-91191Gif-sur-Yvette CEDEX,France

ABSTRACT We study the dynamic spin correlation function of a spin one antiferromagnetic chain with easy-plane single-ion anisotropy.We use exact diagonalization by the Lancz?o s method for chains of lengths up to N=16spins.We show that a single-mode approximation is an excellent description of the dynamical properties.A variational calculation allows us to clarify the nature of the excitations.The existence of a two-particle continuum near zero wavevector is clearly seen both in ?nite-size e?ects and in the dynamical structure factor.The recent neutron scattering experiments on the quasi-one-dimensional antiferromagnet NENP are fully explained by our results.

Submitted to:J.Phys.C:Condensed Matter

November 1992

PACS No:75.10J,74.50M,75.50E SPhT/92-135

I.INTRODUCTION

It was?rst argued by Haldane[1,2]that generic spin-S one-dimensional Heisen-berg antiferromagnets have an excitation gap for integer S.This picture is quite dif-

ferent from the usual,higher-dimensional,picture of antiferromagnets with massless

Goldstone magnons.On the theoretical side,there is now convincing evidence from numerical studies[3-10]of the S=1Heisenberg chain that it has a nonzero gap in

the thermodynamic limit.Exact diagonalizations by the Lancz?o s method have been

able to reach18spins and Monte-Carlo simulations extend the range to32spins.

All these data are suggestive of a Haldane gap.This leads to a ground state spin correlation length which is?nite and about six lattice spacings.Such a picture is in

striking contrast with the S=1/2solvable chain which is gapless and whose ground

state has algebraically decaying correlations.A spin1Heisenberg model with bi-quadratic exchange has also been discovered with an exactly solvable ground state

and a nonzero gap[11],thus reinforcing the belief in the Haldane conjecture.

On the experimental side there are several candidates to exhibit this quantum gap.The?rst experimental evidence came from neutron-scattering on CsNiCl3[12-

15].In this compound there are chains of spin one Ni2+ions with superexchange

through Cl?anions.There is however moderately small interchain couplings that

complicate the picture:this is revealed in recent experiments on the related com-pound RbNiCl3[16].The best candidate so far seems to be Ni(C2H8N2)2NO2ClO4

(NENP)[17-20],as shown by inelastic neutron-scattering(INS)and magnetization

measurements.The ratio of the interchain to intrachain magnetic coupling is es-timated to be J′/J≈4×10?4.No transition to N′e el order is found down to

1.2K which is consistent with the hypothesis of a?nite zero-temperature correla-

tion length.Nickel ions have spin1and are described by the following anisotropic

Heisenberg Hamiltonian[21]:

H=J i S i· S i+1+D i(S z i)2.(1.1) Best?t of INS gap values leads to J/k B=43.5K and easy-plane D/J=0.18[10].

There is also evidence for a smaller in-plane anisotropy that can be described by

adding to H a term E i[(S x i)2?(S y i)2].This perturbation is small and will be mostly ignored in the remainder of this paper.Its qualitative role will be discussed

in section V.

The?rst INS measurements[17-19]have concentrated on the neighborhood of Q=π(where Q is the wavevector along the chain)and showed the existence of two gaps:one for the in-plane(IP)magnetic?uctuations and a higher one for out-of-plane(OP)?uctuations.This splitting is due to the presence of a sizable easy-plane single-ion anisotropy D.The dispersion of the magnetic excitations was then studied in the range Q/π=0.9?1.0.Recent experiments have extended our knowledge throughout the whole Brillouin zone[22].Detailed theoretical work is required to test the hypothesis that the simple model Hamiltonian(1.1)is able to reproduce the dynamical properties of NENP.Two approaches have been followed up to now:small cluster numerical studies[3-10,23]and the use of e?ective?eld

theories[24,25].The previous numerical studies were limited to spectral calculations or have ignored the presence of in-plane anisotropy.

In this paper we present the results of a study of the dynamical structure factor S(Q,ω)of chains of length up to16spins by means of an exact diagonalization method.We also apply a variational technique proposed in ref.[26]to the anisotropic chain.This physically motivated method reproduces extremely well the numerical results.The results presented are valid in the absence of an external magnetic?eld and in the zero temperature limit,i.e.for temperatures well below the gap.In sect. II we present some general properties of the spin-1chain with easy-plane single-ion anisotropy.In sect.III we explain the methods used to obtain dynamical quantities. Sect.IV contains our results.They are extremely well reproduced in terms of a physically appealing single-mode approximation.The results of our variational calculation are presented there and we use them to understand the nature of the elementary excitations near Q=0and Q=π.We show in sect.V that they are in very good agreement with existing neutron data and we suggest an additional test of the theory.Sect.VI contains our conclusions.

II.GENERAL PROPERTIES

We brie?y review some known results about the isotropic Heisenberg spin1 antiferromagnetic(AF)chain:

H0=J

N

i=1 S i· S i+1.(2.1)

The vectors S i are quantum spin operators satisfying the SU(2)rotation algebra with length S2i=2.They are located along a one dimensional lattice of N sites with periodic boundary conditions.The exchange integral J is positive in the AF case.For any?nite lattice the ground state is a singlet and the higher-lying levels have energy increasing with increasing spin as is known rigorously[27].Above the singlet ground state one?nds a triplet state presumably for all values of N.In an AF quantum magnet,in the case of broken symmetry one expects that in the thermodynamic limit the triplet becomes degenerate with the ground state as do other states with spin S=2,3,...,in order to form the degenerate ground state of the in?nite volume limit.This can occur only for dimension greater than2(or equal).Haldane has argued that the spin-1chain is quantum disordered and that the singlet-triplet gap remains nonzero in the thermodynamic limit.This should be true for any integer value of the spin.This argument is based on a mapping of the in?nite-spin,semiclassical limit of the quantum chain onto a O(3)nonlinearσmodel [1].In the spin-1case of interest,numerical studies clearly points towards a gap close to≈0.41J[3-10,23].The physics of the O(3)nonlinearσmodel is that of a triplet of massive bosons with nontrivial scattering properties.This suggests a very simple e?ective?eld theory:a free theory of three massive bosons[25].Another route starting from an integrable model leads to an e?ective?eld theory of three massive fermions[24].

In such approaches,one has to adjust the gap values which are no longer deducible from the microscopic model but one can obtain simple and de?nite pre-diction on dynamical quantities.However both are approximate theories and their respective domain of validity is di?cult to assess.On the contrary,?nite chains calculations o?er unbiased theoretical predictions provided one is able to carefully

control?nite-size e?ects.Concerning the gap values this can be achieved by the use of the so-called Shanks transformation,suited to the removal of exponential tran-sients in sequences of?nite-chain data[28].It has been found that the convergence towards in?nite-volume is very good[6,10,23]as expected since we are dealing with a massive theory.

As shown with the Perron-Frobenius theorem[29],the lowest-lying triplet has

wavevector Q=π,while the singlet ground state has Q=0.Most theoretical and experimental studies have concentrated on this Q=πpart of the spectrum.In the nonlinearσmodel,there are no bound states and,if we believe that it is the e?ective theory of the spin1chain,this implies that the Q=0gap is due to two massive Q=πparticles and thus twice the Q=πgap[2].This has been checked

by a quantum Monte-Carlo simulation in the isotropic case[8]and Lancz?o s studies [23]have shown that this property persists in the presence of anisotropy(as long as it is not too large).As a consequence,the states near Q=0should be members of a two-particle continuum contrary to the states near Q=πthat should appear as long-lived well-de?ned excitations.INS measurements for Q=0[30]reveal a

vanishing structure factor in this regime.The curve of the lowest excited triplet at wavevector Q is thus bell-shaped but asymmetrical with respect to Q=π/2.

Let us now discuss the in?uence of easy-plane single-ion anisotropy,i.e.adding to H0a term of the following form:

H=H0+D i(S z i)2.(2.2) The full rotational symmetry of Hamiltonian(2.1)is broken to rotational symmetry around the z axis.Only the z component S z= i S z i of the total spin is conserved. There is also a discrete symmetry S z→?S z that is preserved in the Hamiltonian

(2.2).As a consequence of the anisotropy,the?rst excited triplet state is split into

a higher-energy singlet S z=0and a lower-lying doublet S z=±1.These three states retain their wavevectors unchanged(Q=π)with respect to the D=0case (by continuity)since the Hamiltonian(2.2)still possesses translational symmetry. The Haldane gap is split into two components:one gap G(?)between the ground state(with S z=0)and the doublet S z=±1and one gap G(+)inside the S z=0 subspace between the?rst two levels.G(?)decreases while G(+)increases with increasing D as shown quantitatively in ref.[10].

We expect this simple picture to be true for all wavevectors0≤Q≤πsince the triplet states with arbitrary Q will be split also by the anisotropy.There is thus two distinct magnetic modes throughout the Brillouin zone:one IP(in-plane) mode(doubly degenerate)and one OP(out-of-plane)mode.The IP mode is seen in the S z=±1sector while the OP mode appears in the S z=0sector.The dispersion of the IP mode has been obtained for various anisotropies including that

of NENP in[23].INS experiments[17-19,22]have resolved these two modes in the neighborhood of Q=π:the OP mode is found at2.5meV and the IP mode is even further resolved into two components at1.05meV and1.25meV.This splitting is due to a small in-plane anisotropy of the type E i[(S x i)2?(S y i)2]that lifts the degeneracy of the doublet states S z=±1.These values of the gaps(ignoring the in-plane anisotropy)leads to J=44K and D/J=0.18[10].Detailed interpretation of INS requires a calculation of the structure factor S(Q,ω).

III.EV ALUATING DYNAMICAL QUANTITIES

We describe in this section the methods used to compute the dynamical struc-ture factor:

Sαα(Q,ω)= dt e iωt<0|Sα?Q(t)SαQ(0)|0>,α=x,y,z.(3.1)

Here we denote the ground state by|0>and the components of the Fourier trans-form of the spin vectors are SαQ(t)in the Heisenberg representation.We?rst write the structure factor as:

Sαα(Q,ω)= nδ(ω?(?n??0))||2,(3.2)

where the sum over|n>means over all excited eigenstates of the system and?n denotes the energy of|n>.In a?nite system calculation,one obtains a?nite set of delta functions with weights given by the matrix elements appearing in Eq.(3.2). The Lancz?o s method which is commonly used to obtain the ground state wavefunc-tion and the?rst few excited levels is suited to the study of the dynamical properties [31].

We proceed as follows:

i)One needs to know?rst the vector|0>.We use the Lancz?o s algorithm applied to the Hamiltonian H.In fact,any kind of algorithm can be used at this level.

ii)One constructs the state:

|Φ0>=SαQ|0>.(3.3) This state|Φ0>is then used to build a new Lancz?o s sequence of states|Φn>by the standard procedure of applying the Hamiltonian to|Φk>and orthonormal-ization with respect to the last two vectors|Φk>and|Φk?1>.The Hamiltonian H is then tridiagonal in the basis{|Φn>}.The nonzero elements of H in this basis are directly provided by the orthonormalization procedure.We exhaust the corresponding subspace and store the tridiagonal form of the Hamiltonian.

iii)One diagonalizes the tridiagonal Hamiltonian by a standard routine.This leads to a set of eigenenergies?n as well as the eigenvectors whose?rst coordinate in the Lancz?o s basis are precisely the overlap matrix elementsthat appear in the de?nition of the structure factor Eq.(3.2).We thus obtain the weight of each delta function peak.This has to be done separately for each Q as well as forα=x,z.

In the isotropic D=0case the ground state is a singlet:SαQ=0|0>=0and thus Sαα(Q=0,ω)=0.The vanishing with Q will occur quadratically.In the pres-ence of easy-plane anisotropy the ground state is invariant only under z rotations: S z Q=0|0>=0and thus S zz(Q=0,ω)=0,while S xx(=S yy)will be nonzero at Q=0and of order O(D2).With in-plane anisotropy E i[(S x i)2?(S y i)2]even S zz(Q=0)will be nonzero and O(E2).

IV.RESULTS FOR S(Q,ω)

We have computed the structure factors S xx(Q,ω)and S zz(Q,ω)for chains of lengths N=4,6,8,10,12,14,16.Some Q=0and Q=πparts of the spectrum have been computed also for N=18.We have concentrated on the case D/J=0.18 quantitatively relevant to the study of NENP.Our results extend smoothly for not too large anisotropy D

According to the de?nition(3.2),we will discuss?rst the peak positions and then the corresponding weights.What we observe is that for all?xed wavevector Q,as a function ofωthe lowest lying peak concentrates almost all of the spectral weight.This remark will be made quantitatively precise at the end of this section. This lowest-lying peak occurs when the frequencyωmatches the energy of the?rst excited level with the right quantum numbers.

In the correlation S xx we?nd the excitation spectrum from S z=0to S z=±1that we obtained previously[23].This is plotted in?g.1as a function of the momenta by crosses of various sizes.For Q=0and Q=πthere is enough data to allow an extrapolation to the thermodynamic limit by use of the Shanks tranformation.We have also computed the these gaps(Q=0and Q=π)for a N=18chain.The extrapolation leads to0.301J at Q=πand0.986J at Q=0.In the interval[0,π]there is only the middle of the Brillouin zone Q=π/2where we obtain several data points from N=4,8,12,16.Shanks extrapolation leads then to 2.75J for Q=π/2.

In the correlation S zz we?nd the excitation spectrum inside the S z=0sub-space.The lowest lying excitation is plotted in?g.1with empty octogons.At Q=πthe?nite-lattice data extrapolate to0.655J.The convergence in this case Q=πis very good as expected from a massive theory yielding well-separated eigenvalues in the?nite systems[23].On the contrary at Q=0the convergence is extremely bad and one can only suggest a value of0.60±0.05J for this gap.This is clearly due to the presence of a continuum of states starting immediately above this gap.Such a phenomenon is expected if one describes the spectrum near Q=0as due to two particles with momenta near Q=π[8].

We note that the excitations in this sector(zz)are above the in-plane excita-tions in the neighborhood of Q=πbut they cross for Q/π≈0.75and on the top of the dispersion curve it is the in-plane mode which has highest energy:2.75J against 2.65J(extrapolated value)for the out-of-plane mode.This behaviour persists in the region0≤Q≤π/2of the Brillouin zone.

To gain understanding of the nature of the excitations we have performed a variational calculation along the lines proposed by G′o mez-Santos[26].One discards the states with parallel spins,either if they are nearest-neighbors or separated by any number of zeros.The typical states in this subspace are of the form:

|...↑↓↑0↓00↑↓...>(4.1) There is strict N′e el ordering of the S z=±1sites but there can be any number of intermediate zeros.In this subspace the true degrees of freedom are then the”spin-zero defects”i.e.sites with S z i=0.These domain walls are then represented as

fermions(which is a natural way of enforcing no double occupancy).The fermionic

Hamiltonian can then be treated by approximate methods.It has been shown that

a simple Hartree-Fock decoupling leads to a good approximation of the spectrum

for D=0.The best results are obtained by using the variational improvement

introduced by G′o mez-Santos where one allows a small admixture of states with

parallel spins.The Haldane gap,as computed by this method,is found to be0.45J

for D=0.It is straightforward to include anisotropy of the form D i(S z i)2.We have obtained a spectrum of massive fermions E(Q)that is plotted as a continuous

curve in?g.1.This curve should be the excitation spectrum in the S z=0subspace

due to the restriction to the subspace(4.1).The gap at Q=πis found to be≈0.70J

when D/J=0.18.The single-fermion curve E(Q)reproduces extremely well the OP

dispersion in the regionπ/2

Figure1

Dispersion of magnetic excitations S z=0and S z=±1at D=0.18J for momenta

ranging from0toπ.The continuous line is the result of a variational calculation.

The dashed line is the corresponding edge of two-particle excitations.

We now turn to the discussion of the spectral weight associated to the peak positions.For all chain lengths studied for all wavevectors in the interval[≈0.3?1.0]×πwe?nd that both S xx and S zz are dominated by a single peak as a function of the frequency.This peak concentrates at least98per cent of the spectral weight. Multimagnon contributions are thus extremely small in the whole range[≈0.3?1.0]×π.As typical examples we plot in?g.2S xx(Q=π,ω)+S zz(Q=π,ω) from the N=16chain.The peaks have been broadened for clarity.In?g.3we plot the Q=π/2case showing the interchange of the two modes.A single-mode approximation will be an extremely good description of the physics in this part of the Brillouin zone.The simple picture described above breaks down for small values of Q.In the N=16chain,for Q=3π/8there is still the one-peak structure while for Q=π/4there are several peaks as a function ofω:see?g.4.In the N=14chain we ?nd that the continuum appears already for Q=2π/7but is not there for3π/7.It is di?cult to give a precise value of the wavevector at which this phenomenon appears since the discretization of the momenta imposed by the chain length is coarse.As discussed below,this can be interpreted as evidence for a two-particle continuum. Our present data suggest that the continuum sets in at≈0.3πfor both IP and OP sectors.We see no reason why the continuum boundary should the same for both modes although we are unable to see any quantitative di?erence in our present data between IP and OP modes in this respect.

Let us discuss the nature of the two-particle excitations.If a single-particle ex-citation has a dispersion E(Q),the lower edge of the continuum of two-particle exci-tations is given by E(2)(Q)=min K(E(K)+E(Q?K))in the non-interacting case. Such a free picture seems to apply to the spin1chain[1,2,24].In the anisotropic system it is possible to construct several continua since there are two distinct modes (IP and OP).

In the S z=±1subspace two-particle states are obtained from one S z=±1 excitation and one S z=0excitation.The gap of this continuum at Q=0will be the sum of the IP and the OP gap at Q=πi.e.≈0.301J+0.655J in agreement with our extrapolation of0.986J.The continuum is also seen in the poor convergence of ?nite-size data.Its progressive build-up is seen in?g.4.

In the S z=0subspace there are two ways of building two-particle states: either with two S z=0states or with one excitation S z=+1and one excitation S z=?1.In the?rst case we can obtain an approximation for the continuum boundary by using the fermionic method cited above:the curve E(2)(Q)for two fermionic excitations belonging to the S z=0sector is plotted as a dash-dotted line in?g.1.The two-fermion continuum is clearly above the OP(and IP)mode for 0

Figure2,3,4

The structure factor S(xx)(Q,ω)+S(zz)(Q,ω),for Q=π,π/2,π/4respectively,as a function of the frequencyω(in units of J)for chain length N=16.In?g.2each of the two functions S(αα)has a single-peak structure:a single-mode approximation works very well.The IP mode is doubly degenerate and appears only in S(xx)while the OP appears in S(zz).In?g.3the IP and OP modes are interchanged with respect to?g.2.In?g.4a single-mode approximation no longer holds and the two-particle continuum is seen:spectral weight is transferred to multimagnon excitations.

The agreement between the ab initio data and the variational calculation con-?rms the relevance of domain walls in the whole Haldane phase,as is also pointed out in[32].The disordered-?at phases obtained in the solid-on-solid picture of the spin-1chain[33]also correspond to the subspace of spin-zero defects which is so successfull in the description of the elementary excitations.

Finally we quote results for the static structure factors S(αα)(Q)obtained by integration over frequency.In?g.5these quantities are plotted in a logarithmic scale as a function of momentum.Due to our limited chain length,we cannot make any?rm statement about the prediction[1]of square-root Lorentzian behaviour near Q=π.Our data are compatible with previous studies[34,35]concluding to a correlation length≈6lattice spacings in the D=0case:the correlation length does not change much when D/J=0.18.The behaviour of S(zz)(Q)≈Q2sets in only for very small wavevectors:for the largest part of the Brillouin zone excepting the neighborhood of Q=πit is rather close to Q3/2.

Figure5

The static structure factors S(xx)and S(zz)for all chain lengths as a function of momentum.The symbol sizes are the same as in?gs.1and6.Crosses stand for S(xx)and octogons for S(zz).

https://www.doczj.com/doc/0a1781078.html,PARISON WITH NEUTRON SCATTERING EXPERIMENTS The two magnetic modes in the neighborhood of Q=πhave been observed by INS in ref.[17-20].The values of the gaps lead to J≈3.8meV and D/J=0.18. These values are very close to those extracted from magnetization measurements [36].The dispersion has been studied throughout the whole Brillouin zone[22].A good?t is obtained through the following formula:

ω±(Q)= 2).(5.1)

The parameters are?+=2.5meV,??=1.2meV,v=9.6meV and A=6.1meV. Scaling by J=4meV we plotω±(Q)as continuous lines in?g.6.For clarity we have also plotted our Lancz?o s points.The edges of two-particle continuum extracted from the?ts(5.1)are also plotted in?g.6.The lower dashed line corresponds to the(S z=+1)+(S z=?1)continuum while the upper dashed line corresponds to (S z=+1)+(S z=0).

Figure6

Lancz?o s results compared to experimental?ts.The two continuous curves are taken from Eq.(5.1).They reproduce the?ndings of ref.22.The two dash-dotted lines are the two-magnon continuum boundaries obtained also from(5.1).The size of the symbols is chosen as in?g.1.

There is very good agreement with the theoretical results in the neighborhood of Q=πwhere the IP and OP modes are separated by the experimental resolution [17-20,22].In the regionπ/2

With respect to what has already been done,it would be very interesting to separate the two modes nearπ/2where intensity is not dramatically weak.The observation of the two-particle continuum on the contrary should be quite di?cult since it appears only for Q<0.25πwhere magnetic scattering is very weak.The observation of very long-lived modes[22]is thus in very good agreement with the physics of an anisotropic spin1chain.

Finally we comment on the uncertainties in the numerical data.For the gap values there is very good convergence for Q=πand there one can use all the chain lengths at our disposal.For other values of Q it is only at the top of the spectrum that an extrapolation can be performed.Concerning the values of Q at which some-thing interesting happens(i.e.crossing of the two modes or entering the continuum) without being able to make re?ned extrapolations we can only observe that they do not depend much on the chain length.The corresponding numbers should be regarded as tentative(i.e.0.75πfor the crossing and0.3πfor the continuum).

High resolution INS experiments[30]have revealed that the IP mode is split by in-plane anisotropy E i[(S x i)2?(S y i)2].Our results neglecting this further splitting thus will apply to the description of actual experiments as long as the resolution is not very high.This e?ect is not expected to change much the numerical?gures.

VI.CONCLUSION

We have studied the dynamical properties of a spin1chain with single-ion easy-plane anisotropy thought to be relevant to the magnetic behaviour of the compound NENP.The dynamical structure factor S(αα)(Q,ω)has been computed by a Lancz?o s method on chains of lengths up to16spins while some parts of the spectrum have been obtained on a18spins chain.We have performed a variational calculation to obtain the spectrum of elementary excitations in the S z=0sector.Both ab initio and approximate methods agree very well,elucidating the role of the spin-zero defects in the Haldane phase.

We?nd that the magnetic excitations are described by two distinct long-lived modes throughout most of the Brillouin zone0.3?1.0πwhere the lowest excited state in each sector(S z=0and S z=±1)carries almost all of the spectral weight: a single-mode approximation is thus adequate.We?nd that these modes merge into a two-particle continuum for Q<0.3πas seen in the frequency dependence of the structure factor.We have shown that these results reproduce the INS experiments on NENP.The dispersion of the magnetic modes as well as their intensity is close to that found theoretically.

Our results show that in principle it should be possible to observe the crossing of the two IP and OP magnetic modes.At the top of the dispersion curve their separation and intensity should allow observation.On the contrary the edge of the two-particle continuum appear only for very small Q<0.3πwhere magnetic intensity is very weak as found in recent experiments[22].We mention?nally that it would be very interesting to measure the static form factors of an anisotropic chain by a method that allows to reach the neighborhood of Q=πsince here we are limited by the coarse-graining due to our small chains.Quantum Monte-Carlo calculation that works e?ciently for static quantities can reach this goal.

ACKNOWLEDGEMENTS We thank S.Ma et al.for sending us a copy of ref.22prior to publication.Thanks are also due to L.P.Regnault for discussions about neutron scattering experiements on NENP.We also thank Th.Garel and J. Miller for reading our manuscript.

Note added:while preparing this manuscript,we received a preprint by M.Taka-hashi which addresses similar questions.

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