effect of topological excitations in the two-dimensional quantum heisenberg antiferromagnet

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PHYSICAL REVIEW B VOLUME 41, NUMBER 10 1 APRIL 1990 Eff'ect of topological excitations in the two-dimensional quantum Heisenberg antiferromagnet J. P. Rodriguez* Institut Laue-Langevin, Borte Postale No. 156X, 38042 Grenoble CEDEX FI'ance (Received 20 April 1989; revised manuscript received 27 December 1989) Using a continuum large-N U(1) gauge theory (CP" ') for the low-temperature antiferromag- net in 2+1 dimensions, we recover previously obtained results for the static correlation length at the mean-field level. Phase Auctuations about mean field result in soundlike collective modes among a topological domain-wall loop liquid. To order N, the sound modes are gapless in the quantum disordered phase. In the Neel phase they have small gaps, which results in an anoma- lous contribution to the spin-wave damping near the antiferromagnetic Bragg point. The discovery of two-dimensional (2D) antiferromag- netic correlations in the Cu-0 planes of pure La2Cu04, ' which when doped is a high-T, superconductor, has sparked much theoretical effort towards understanding the ground state and the low-lying excitations of the 2D quantum Heisenberg antiferromagnet. To date it has still not been rigorously demonstrated whether or not long- range Neel order exists at zero temperature, although Chakravarty et al. have shown via a continuum rotor model for the 2D antiferromagnet the nonlinear cr model that the experimental results for the magnetic correlation length as a function of temperature can be well described by a zero-temperature ground state with Neel order. 2 This very same continuum model also supports to- pological point defects that are basically domain-wall loops in the 2D antiferromagnet. Classically, such topo- logical solitons, which are called Skyrmions, have an ener- gy cost on the order of the exchange coupling constant J, and they are thus not energetically low-lying excitations of the 2D antiferromagnet. However, the author has re- cently shown that quantum fluctuations (a) reduce the classical energy cost of one Skyrmion and (b) induce a repulsive interaction between two Skyrmions at long range. 3 The second result suggests that soundlike modes (Skyrme phonons) among the Skyrmion-anti-Skyrmion liquid created by quantum and thermal fluctuations in the antiferromagnet could exist. Below we will show explicitly that such Skyrme pho- nons do indeed exist via a continuum U(1) gauge theory for the 2D antiferromagnet. This model will be solved in the semiclassical limit. At the mean-field level we recover the one-loop static correlation length results of Chakra- varty et al. 2 for both the Neel and the quantum disor- dered phases. Beyond mean field, the phase fluctuations become dynamical and correspond precisely to the above- mentioned Skyrme phonon modes. In particular, a gap- less longitudinal Skyrme phonon exists in the quantum disordered phase of the antiferromagnet, which is the only low-lying excitation in this phase. In the Neel phase, both longitudinal and transverse Skyrme phonons exist with a small gap at long wavelength. In this phase, we find an anomalous contribution to the spin-wave damping at a point near the antiferromagnetic Bragg point, where the spin waves resonantly absorb optical (q 0) Skyrme pho- nons. Interestingly, the scale of the resonant spin-wave zz l, (2) where the vector potential A„ is constrained to the z fields y (3) A„-i (r)„i)z . Above, J is the effective exchange coupling constant and 8n c 't)„where c is the velocity of the antiferromagnet- ic spin waves. The z fields are related to n, the sense (I n I 1) of the sublattice magnetization of the Neel configuration, by n zaz. (4) It can be shown that the CP ' Lagrangian (1) is equivalent to the 0(3) nonlinear o model Lagrangian in terms of n. ' Notice that z and i are "charged" fields whereas physical- ly relevant Neel vector field n is not. The above continuum model should describe the long- wavelength low-energy physics of an isolated antiferro- magnetic plane; i. e. , spin-wave excitations. As mentioned before, however, in 2+1 dimensions this continuum model also supports a topologically nontrivial point defect called the Skyrmion, which is basically a domain-wall loop. There also exists the anti-Skyrmion, which is the mirror reflection of the Skyrmion. When an anti-Skyrmion is added to a Skyrmion the resulting configuration is topo- logically equivalent to the homogeneous Neel state. The current density of these topological solitons is given by J„-(2x) 'c„, „t)„A„. (5) From Eq. (5) it is evident that J„ is a conserved current, B„J„O. Thus the total number of Skyrmions (domain- damping rate agrees with the scale for the quasielastic linewidth obtained via large-5 calculations. Inelastic neutron scattering experiments performed on La2Cu04 al- ready show evidence for such an anomaly. To begin our discussion, we present the CP' model for quantum antiferromagnet in 2+1 dimensions, first intro- duced by Dzyaloshinskii, Polyakov, and Wiegmann. 7 The CP ' Lagrangian is given by I, - -2J „d'x I (a„-i~,)z I', where z is a complex valued doublet field of unit modulus; i.e. , z (z1, z2) with z; C C and 7326 @1990The American Physical Society

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Page 1: Effect of topological excitations in the two-dimensional quantum Heisenberg antiferromagnet

PHYSICAL REVIEW B VOLUME 41, NUMBER 10 1 APRIL 1990

Eff'ect of topological excitations in the two-dimensional quantum Heisenberg antiferromagnet

J. P. Rodriguez*Institut Laue-Langevin, Borte Postale No. 156X, 38042 Grenoble CEDEX FI'ance

(Received 20 April 1989; revised manuscript received 27 December 1989)

Using a continuum large-N U(1) gauge theory (CP" ') for the low-temperature antiferromag-net in 2+1 dimensions, we recover previously obtained results for the static correlation length atthe mean-field level. Phase Auctuations about mean field result in soundlike collective modes

among a topological domain-wall loop liquid. To order N, the sound modes are gapless in thequantum disordered phase. In the Neel phase they have small gaps, which results in an anoma-lous contribution to the spin-wave damping near the antiferromagnetic Bragg point.

The discovery of two-dimensional (2D) antiferromag-netic correlations in the Cu-0 planes of pure La2Cu04, '

which when doped is a high-T, superconductor, hassparked much theoretical effort towards understandingthe ground state and the low-lying excitations of the 2Dquantum Heisenberg antiferromagnet. To date it has stillnot been rigorously demonstrated whether or not long-range Neel order exists at zero temperature, althoughChakravarty et al. have shown via a continuum rotormodel for the 2D antiferromagnet —the nonlinear cr

model —that the experimental results for the magneticcorrelation length as a function of temperature can be welldescribed by a zero-temperature ground state with Neelorder. 2 This very same continuum model also supports to-pological point defects that are basically domain-wallloops in the 2D antiferromagnet. Classically, such topo-logical solitons, which are called Skyrmions, have an ener-

gy cost on the order of the exchange coupling constant J,and they are thus not energetically low-lying excitationsof the 2D antiferromagnet. However, the author has re-cently shown that quantum fluctuations (a) reduce theclassical energy cost of one Skyrmion and (b) induce arepulsive interaction between two Skyrmions at longrange. 3 The second result suggests that soundlike modes(Skyrme phonons) among the Skyrmion-anti-Skyrmionliquid created by quantum and thermal fluctuations in theantiferromagnet could exist.

Below we will show explicitly that such Skyrme pho-nons do indeed exist via a continuum U(1) gauge theoryfor the 2D antiferromagnet. This model will be solved inthe semiclassical limit. At the mean-field level we recoverthe one-loop static correlation length results of Chakra-varty et al. 2 for both the Neel and the quantum disor-dered phases. Beyond mean field, the phase fluctuationsbecome dynamical and correspond precisely to the above-mentioned Skyrme phonon modes. In particular, a gap-less longitudinal Skyrme phonon exists in the quantumdisordered phase of the antiferromagnet, which is the onlylow-lying excitation in this phase. In the Neel phase, bothlongitudinal and transverse Skyrme phonons exist with asmall gap at long wavelength. In this phase, we find ananomalous contribution to the spin-wave damping at apoint near the antiferromagnetic Bragg point, where thespin waves resonantly absorb optical (q 0) Skyrme pho-nons. Interestingly, the scale of the resonant spin-wave

zz l, (2)

where the vector potential A„ is constrained to the z fields

y

(3)A„-i (r)„i)z .

Above, J is the effective exchange coupling constant and8n c 't)„where c is the velocity of the antiferromagnet-ic spin waves. The z fields are related to n, the sense(I n I 1) of the sublattice magnetization of the Neelconfiguration, by

n zaz. (4)

It can be shown that the CP ' Lagrangian (1) is equivalentto the 0(3) nonlinear o model Lagrangian in terms of n. 'Notice that z and i are "charged" fields whereas physical-ly relevant Neel vector field n is not.

The above continuum model should describe the long-wavelength low-energy physics of an isolated antiferro-magnetic plane; i.e., spin-wave excitations. As mentionedbefore, however, in 2+1 dimensions this continuum modelalso supports a topologically nontrivial point defect calledthe Skyrmion, which is basically a domain-wall loop.There also exists the anti-Skyrmion, which is the mirrorreflection of the Skyrmion. When an anti-Skyrmion isadded to a Skyrmion the resulting configuration is topo-logically equivalent to the homogeneous Neel state. Thecurrent density of these topological solitons is given by

J„-(2x) 'c„,„t)„A„. (5)

From Eq. (5) it is evident that J„ is a conserved current,B„J„O.Thus the total number of Skyrmions (domain-

damping rate agrees with the scale for the quasielasticlinewidth obtained via large-5 calculations. Inelasticneutron scattering experiments performed on La2Cu04 al-ready show evidence for such an anomaly.

To begin our discussion, we present the CP' model forquantum antiferromagnet in 2+1 dimensions, first intro-duced by Dzyaloshinskii, Polyakov, and Wiegmann. 7 TheCP ' Lagrangian is given by

I, - -2J„d'x I (a„-i~,)z I',

where z is a complex valued doublet field of unit modulus;i.e., z (z1,z2) with z; C C and

7326 @1990The American Physical Society

Page 2: Effect of topological excitations in the two-dimensional quantum Heisenberg antiferromagnet

EFFECT OF TOPOLOGICAL EXCITATIONS IN THE TWO-. . . 7327

wall loops),

Q=„' d xJo, (6)

is a conserved quantity in the (2+1)-dimensional antifer-romagnet.

To proceed further we work with the CP ' generali-zation of the CP' model. ' " The z field in this case issimply given by z -(zi, . . . , zN ) and the partition func-tion by

with

Z(0) 2)z2)z2)A„2)kexp( —S), (7)

ta Ijd „d'X[I(a„—i~, ) I'+&(I I' —1)].

(8)Above, we have gone directly to imaginary time z, with

P h(kttT) '. In (7) integration over A, recovers con-straint (2) while integration over A„ in the Lorentz gauge(8„A„O)recovers constraint (3).

Mean field -theory. In the limit that N ~, mean-field theory is exact. Following D'Adda et al. 'n and Wit-ten, "we integrate out the z and i fields and minimize theresulting effective action with respect to X and A„. Thisgives (A„) 0 and

d'PJ~ g(p +ni +m ) (9)(2tr)' n

where m (A, ) and ni„2trnT are the bosonic Matsubarafrequencies. (From this point on the physical constants fi

and ke, plus the spin-wave velocity c, are set to unity).After introducing a momentum cutoff ko-a ', where ais the lattice constant, expression (9) can be evaluated ex-actly, and we find that in the low-temperature long-wavelength limit T, m «ko, the dynamical mass is given

by

m Texp( —2trPJtt), Jtt )0 (Neel), (ioa)

m 4tr I Jtt I, Jtt &0 (quantum disordered), With

+II(q)A, (q)A, ( —q)], (13)

where

Jtt J—(4ir) 'ko

is the renormalized exchange coupling constant. Similarresults have been obtained via a saddle-point approxima-tion of the analogous lattice rotor model. ' Notice thatquantum fluctuations renormalize down the effective ex-change coupling constant in the Neel phase. We remarkhere that the energy of a topological domain-wall loop(Skyrmion) in the presence of zero-point spin-wave fluc-tuations is given by 4trJg in the semiclassical limit (seeRef. 3).

In this semiclassical limit the z propagator is given by

G.p(p) -(z, (p)zti( —p) & -b.p(p '+ m ') ', (12)

after the rescaling z (NJ) 't z. In the case of CP'(N 2) it can be shown via (4) that (n(p) n( —p))

4(z, (p)z, ( —p)), and thus G„ is essentially the spin-spin correlation function for the antiferromagnet. Equa-tion (12) then indicates that the mass m is simply the in-verse static correlation length g '. The results in (10)imply, therefore, that there exists a phase transition atT 0 between a Neel phase with long-range antiferro-magnetic correlations g Do, for Jtt )0, and a quantumdisordered phase with finite correlation lengthg- I Jtt I

', for Jg & 0. We have thus recovered the one-loop nonlinear o model results of Chakravarty et al. 2 forthe correlation length at the mean-field level in CP

Collective modes. Let us now calculate the effect offluctuations in the A„and )I, fields from their mean-fieldvalues. Following D'Adda et al. ' and Witten, " thequadratic fluctuation corrections to the saddle point of theeffective action obtained from (7) after integrating out thez and i fields are given by

Ss„„-—,' +[II„„(q)a„(q)a„(—q)

q

le 2

II(q) -NT, g[k'+m'] '[(k+q)'+m'](2x)' n

(i4)

d'kII„„(q)-NT

4 2g n k +m(2k+q)„(2k+q),

(k'+m') [(k+q)'+m'](is)

The Feynman diagrams associated with (15) are shown inFig. 1(a). Above, k„denotes a Wick rotated three-vector(kn, k) with frequency components given by the discretebosonic Matsubara values ko 2xTni. For qo 2xTn'i,II„„(q)satisfies the gauge invariance property

q"II„,(q) 0 .

Relation (16) is thus true for any qn on the upper half ofthe complex plane. ' Below, we calculate such fluctuationcorrections in both the Neel phase and the quantum disor-dered phase.

(i) Neel phase. Consider the case when Jtt )0. In thiscase we have long-range antiferromagnetic order at T 0

FIG. l. (a) Feynman diagrams contributing to the polariza-bility of the medium. Solid lines represent z-quanta propagator(zz). (b) Feynman diagrams contributing to the self-energycorrection of the z-quanta propagator, which essentially mea-sures the spin-spin correlation function of the antiferromagnet.Dashed line represents "electromagnetic" propagator (A„A,).

Page 3: Effect of topological excitations in the two-dimensional quantum Heisenberg antiferromagnet

7328 J. P. RODRIGUEZ

and by (10a), m « T in the low-temperature limit. In thislimit the "polarizability" II„„in (15) can be directly com-puted. Inverting II„„in the Coulomb gauge, we find thatthe vector-potential propagator D„„(A„A„)is given by

D;; (q) -e' b;, — ',' (-,' q' —tv'e, ) ', (17a)q

2

Dao(q) -—e '(q'e/) (171 )

D, (q) -D.(q) -0.Above, e is the "electric charge" given by

2 32$mNT

(17c)

and e, and e/ are respectively the transverse and longitudi-nal "dielectric" constants given by

2' z'I/2

~, -1+ Npi (19a)Q CO

2 lit'22Ctl I

1 — 1—2 2

(19b)

m.

where tov~ is the "plasma" frequencyl]2

cop) 42xJg

(20)

From (17) we see that we have two physical excitations,the "photon" and the "plasmon, " which are respectivelypoles of D,/ and D00. Both excitations have a long-wavelength gap of toi,~. At short wavelen ths the photonspectrum is acoustic with a velocity I/ 3 times smallerthan the spin-wave velocity, and thus it will be weaklyLandau damped. The plasmon spectrum is also acousticat short wavelengths, yet its velocity is equal to that of thespin waves. Hence, it will not be Landau damped.

Above, we have seen that the vector potential excita-tions in our CP ' theory are those of a neutral 2N-component ultrarelativistic collisionless plasma in 2+ ldimensions 'The char. ged z quanta are, in this case, un-bound. But what do such excitations correspond to physi-cally in the (2+1)dimensional quantum antiferromag-net? From (5), we see that the photon and the plasmonare respectively the longitudinal Skyrme phonon and thetransverse Skyrme phonon in the liquid of domain-wallloops plus antidomain-wall loops created by quantum andthermal fluctuations in the antiferromagnet. Notice thatthe mere existence of a transverse Skyrme phonon indi-cates that this Skyrmion-anti-Skyrmion liquid is tendingto solidify near T 0. Also, the fact that these Skyrmephonons have a small yet nonzero gap reflects an out-of-phase oscillatory motion between Skyrmion-anti-Skyrmion pairs. In fact, such Skyrme phonons may beanalogous to the breather soliton of the sine-Gordon mod-el in 1+1 dimensions. 's Since these collective modes havea nonzero gap, there will be no homogeneous topologicalfluctuations in the Neel phase; i.e., (Qz) 0 [see Eqs. (5)and (6)].s Finally, it can be shown that the propagatorfor the l, field (X(q)k( —q)) II '(q) has no poles in thecomplex frequency plane in the limit of long wavelength.

We claim without proof that this behavior persists for allwavelengths, which would imply that the X, field does notcorrespond to a physical excitation. ' "

(ii) Quantum disordered phase C. onsider now the caseJ/t & 0 at low temperature. In the low-temperature limitwe have T«m by (10b), and the polarizability II„, canagain be directly calculated. If the vector-potential corre-lation function D„„(A„A,) is calculated in the Feynmangauge, one finds that

D„,(q)- ', b„„,q

(21)

where the charge e is given by

24@m

N

The only physical pole of D„„is the photon, which as men-tioned before by (5) corresponds to a longitudinal phononamong the liquid of topological domain-wall loop excita-tions created in the disordered antiferromagnet by quan-tum fluctuations. By (21) we see that, to lowest order inN ', the photon has an acoustic spectrum cv

~ q ~, withno gap. The gaplessness, however, does not persist beyondthis order after instanton eff'ects are included. s'6 We re-mind the reader that the spin excitations in the quantumdisordered phase have a gap of m. They will thus not beexcited near zero temperature.

In the quantum disordered phase we see that the gaugedegrees of freedom of the CP ' model correspond tothose of a dielectric vacuum. The exponentially smallnumber of zz pairs that exist are bound by the "Coulomb"attraction and the photon is the only low-ener'gy excitationat T 0. Topological domain-wall loop excitation are im-portant in this phase and we find, via (5), (6), and (21),that fluctuations in the Skyrmion number are given byV '(Q ) (2n) e T, where V is the total volume of thesystem. Finally, we mention that the k field propagator(X(q)k( —q)) II '(q) can be directly calculated in thislimit and can be shown to have no poles in the complexfrequency plane. Hence, as in the previous case, the A,

field does not correspond to a physical excitation of thequantum disordered antiferromagnet. ' "

Spin-wave damping Let us no. w calculate the effect ofthe preceding collective modes on the spin-wave propaga-tion. The spin-wave interaction with the Skyrme phononsresults in a self-energy correction to the z field propagatorof

G.p(p) -b.p[p'+ m ' —Z(p) ] (23)

To first order in N, the self-energy Z is given by the di-agrams in Fig. 1(b):

Z"'(p) -T q, Q[(2p —q)"(2p —q)"(2n)' n

xD„„(q)G(p—q) —2D„"(q)] .(24)

Above, the frequency components of q„are given by thebosonic Matsubara values qo 2nTni, and G is the diago-nal component of G,s in (12).

The second term above adds a positive N ' correctionto the mass, thus decreasing the static correlation length.

Page 4: Effect of topological excitations in the two-dimensional quantum Heisenberg antiferromagnet

EFFECT OF TOPOLOGICAL EXCITATIONS IN THE TWO-. . . 7329

64m T21% 2mJR

' 1/2

(25)

For spin-wave propagation p„(e,p) with p +rn =0,the first term in (24) gives an order N ' imaginary con-tribution to the self-energy of the form ImX(p) =el (p),where I (p) is twice the spin-wave damping rate. Thus, in

the large-N limit, all spin-wave modes are underdamped.The damping originates from three physical decay mecha-nisms; (a) spin-wave pair annihilation into a Skyrme pho-non, (b) emission of a Skyrme phonon by the spin wave,and (c) absorption of a Skyrme phonon by the spin wave.In the quantum disordered phase none of these decay pro-cesses are kinematically possible. Hence, to order Nspin excitations in the quantum disordered phase are notdamped. In the case of the Neel phase, however, theseprocesses do contribute to the spin-wave decay. In partic-ular, near the antiferromagnetic Bragg point at

~ p ~e —,

'tot, ~, resonant absorption of the

Skyrmion-anti-Skyrmion out-of-phase motion occurs. Atthis point spin-wave decay is dominated by spin-wavepair-annihilation into homogeneous "plasma" oscillations,which results in a contribution to the spin-wave decay rate0

This resonant absorption is a direct consequence of the in-teraction of the spin waves with both the longitudinal andtransverse Skyrmionic collective modes of the antifer-romagnet. It is interesting to note that the scale set bythis damping rate agrees with that predicted by mode-coupling theory and the dynamic scaling hypothesis forthe quasielastic width. For La2Cu04 (2trJtt —1200 K,c-0.7 eVA, ( tn '-200 A), ' this result predictsthat resonant spin-wave absorption should occur at an en-ergy of —,

' to~~-14 meV with a width of 2 I „,-4 meV, atTtv-300 K. More importantly, the above results indicatethat the q-integrated intensity for inelastic neutronscattering should show a peak at to~~, which increases withincreasing temperature. This has recently been observedin LazCu04.

We mention here that related large-N U(l) gaugetheories for the (2+1)-dimensional antiferromagnet havebeen formulated on the lattice. 's"

Finally, it is a pleasure to thank G. Baym, D. Grempel,H. Capellmann, S. M. Hayden, and N. Bonesteel for il-luminating discussions.

'Present address: Department of Physics, Temple University,Philadelphia, PA 19122.

'G. Shirane, Y. Endoh, R. J. Birgeneau, M. A. Kastner, Y. Hi-daka, M. Oda, M. Suzuki, and T. Murakami, Phys. Rev. Lett.59, 1613 (1987).

S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev.Lett. 60, 1057 (1988).

3J. P. Rodriguez, Phys. Rev. B 39, 2906 (1989).4D. R. Grempel, Phys. Rev. Lett. 61, 1041 (1988); S. Chakra-

varty, B. I. Halperin, and D. R. Nelson, Phys. Rev. B 39, 2344(1989); S. Tyc, B. I. Halperin, and S. Chakravarty, Phys.Rev. Lett. 62, 835 (1989).

5K. Yamada, K. Kakurai, Y. Endoh, T. R. Thurston, M. A.Kastner, R. J. Birgeneau, G. Shirane, Y. Hidaka, and T. Mu-rakami, Phys. Rev. B 40, 4557 (1989).

For a preliminary exposition of this work see J. P. Rodriguez,BulL Am. Phys. Soc. 34, 1009 (1989).

7I. Dzyaloshinskii, A. Polyakov, and P. Wiegmann, Phys. Lett.

A 127, 112 (1988).sA. M. Polyakov, Gauge Fields and Strings (Harwood Academ-

ic, New York, 1987).9F. Wilczek and A. Zee, Phys. Rev. Lett. 51, 2250 (1983).' A. D'Adda, M. Luscher, and P. DiVecchia, Nucl. Phys. B146,

63 (1978)."E.Witten, Nucl. Phys. B149, 285 (1979).' E. Manousakis and R. Salvador, Phys. Rev. B 40, 2205

(1989).'3G. Baym and N, D. Mermin, J. Math Phys. 2, 232 (1961).'sV. P. Silin, Zh. Eksp. Teor. Fiz. 38, 1577 (1960) [Sov. Phys.

JETP 11, 1136 (1960)].'sR. Rajaraman, Solitons and Instantons (North-Holland, Am-

sterdam, 1984).'sN. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).'7D. P. Arovas and A. Auerhach, Phys. Rev. B 38, 316 (1988);

J. B. Marston, Phys. Rev. Lett. 61, 1914 (1988); L. B. Ioffeand A. I. Larkin, Phys. Rev. B 39, 8988 (1989).