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Frustrated Antiferromagnets at High Fields: Bose-Einstein Condensationin Degenerate Spectra

G. Jackeli1 and M. E. Zhitomirsky2

1Institute Laue Langevin, B.P. 156, 38042 Grenoble, France2Commissariat a l’Energie Atomique, DSM/DRFMC/SPSMS, 38054 Grenoble, France

(Received 2 December 2003; published 2 July 2004)

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Quantum phase transition at the saturation field is studied for a class of frustrated quantumantiferromagnets. The considered models include (i) the J1-J2 frustrated square-lattice antiferromagnetwith J2 �

12 J1 and (ii) the nearest-neighbor Heisenberg antiferromagnet on a face centered cubic lattice.

In the fully saturated phase the magnon spectra for the two models have lines of degenerate minima.Transition into a partially magnetized state is treated via a mapping to a dilute gas of hard-core bosonsand by complementary spin-wave calculations. Momentum dependence of the exact four-point bosonvertex removes the degeneracy of the single-particle excitation spectra and selects the ordering wavevectors at ��;�� and ��; 0; 0� for the two models. We predict a unique form for the magnetization curve�M � S�M ’ ��d�1�=2�log���d�1�, where � is a distance from the quantum critical point.

DOI: 10.1103/PhysRevLett.93.017201 PACS numbers: 75.10.Jm, 03.75.Nt, 05.30.Jp, 75.30.Kz

its critical point J2 �1 J1, which has lines of minima at dilute in the limit �! 0�. In this case, the leading order

Heisenberg antiferromagnets exhibit a quantum phasetransition between a fully polarized state and a state withpartial magnetization at the saturation field Hc. A par-tially magnetized state typically breaks spin-rotationalsymmetry about the field direction and has a long-rangeorder of transverse spin components. Such a zero tempera-ture transition belongs to the XY universality class.Various properties of quantum antiferromagnets in the vi-cinity of this quantum critical point are well understoodin terms of the Bose condensation of magnons below Hc[1,2]. The antiferromagnetic wave vector corresponds tothe minimum in the magnon spectrum, whereas an arbi-trary phase of the condensate describes sublattice orien-tation in the plane perpendicular to the field.

The above simple picture fails, however, for so-calledfrustrated antiferromagnets, which have degenerate clas-sical ground states between zero and saturation fields[3,4]. The excitation spectra of frustrated antiferromag-nets above Hc are quite unusual. The magnon disper-sion q has a continuous set of energy minima givenby a d0-dimensional hypersurface embedded in thed-dimensional reciprocal space. The problem of the Bosecondensation of particles with such degenerate spectraremains unexploited to a large extent. Numerical inves-tigations [5,6] of a specific model (model A below) show astrong singularity of the magnetization near the satura-tion field, which has not been interpreted so far. In thepresent work we investigate two problems: (i) how thequantum fluctuations remove degeneracy of the excitationspectra and select a certain ordering wave vector and(ii) how the asymptotic behavior of the magnetizationcurve close to the quantum critical point is modified bythe presence of large phase space of soft-mode fluctua-tions. Specifically, we consider two models with d0 � 1 ind � 2 and d � 3 spatial dimensions: (model A) the J1-J2antiferromagnetic Heisenberg model on a square lattice at

2

0031-9007=04=93(1)=017201(4)$22.50

��; q� and �q;�� and (model B) the nearest-neighborHeisenberg model on a face centered cubic lattice (fcc)with minima at ��; q; 0� and equivalent lines.

A convenient approach to deal with quantum antifer-romagnets near the saturation field is to employ a hard-core boson representation of spin-1=2 operators: Szi �

12�

byi bi, S�i � bi, and S�i � byi with byi bi � 0 or 1 [7]. The

hard-core constraint is imposed by an on-site repulsionU ! 1. The Heisenberg spin Hamiltonian HH �P

hi;jiJijSiSj �HPiSzi is, then, transformed to

HH �Xq

q ���byqbq �1

2N

Xq;k;k0

Vqbykb

yk0bk0�qbk�q;

(1)

where q � 12 ��q � �min�, �q �

PjJije

iqrij is the Fouriertransform of the exchange interaction Jij, � � Hc �H isa boson chemical potential, and Hc �

12 ��0 � �min� is the

saturation field. A bare four-point boson vertex is given byVq � U� �q. The exact ground state of the Hamiltonian(1) is the boson vacuum for �< 0 (H > Hc); it corre-sponds to a ferromagnetic alignment of spins. In d � 2the ground state with finite boson density hbyi bii � 0 at� > 0 should be, generally, a superfluid state: hbqi � 0for a particular wave vector q � Q, such that Q ��vanishes at the saturation field.

In the above two models, the bare spectrum q haslines of degenerate minima (see below) and the conden-sate wave vector remains undetermined. The degeneracycannot be lifted by the bare interaction Vq as it dependson momentum transfer only. This is a manifestation of aninfinite classical degeneracy of the ground state of theoriginal spin problem below the saturation field. Such an‘‘accidental’’ degeneracy can be removed by quantumfluctuations [8]. The number of bosons vanishes at thequantum critical point and hence the system becomes

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corrections to the scattering vertex are generated by theprocesses in the particle-particle channel. According tothe standard procedure, the bare interaction is, then,replaced by a solution of the integral equation for thescattering amplitude [9].We show that does depend onthe incoming momenta and thus lifts the degeneracy ofthe original problem. The ordering wave vector corre-sponds to the smallest ; i.e., condensate occurs at themomentum at which bosons interact less with each otherin order to minimize their repulsion.

The Bethe-Salpeter equation for the scattering func-tion with zero total frequency reads as

q�k;k0� � Vq �1

N

Xp

Vq�p p�k;k0�

k�p � k0�p(2)

and is graphically represented in Fig. 1. In the limitU ! 1, Eq. (2) is reduced to a system

q�k;k0� � �q � h i �1

N

Xp

�q�p p�k;k0�

k�p � k0�p;

1

N

Xp

p�k;k0�

k�p � k0�p� 1;

(3)

where h i � �1=N�P

q q�k;k0� and the identity h�i �Jii � 0 has been used. By expanding q�k;k0� in latticeharmonics of the wave vector q, the integral equations (3)are transformed to a system of algebraic equations. Sinceonly a few harmonics appear in such an expansion of q�k;k0�, essentially those which are present in �q, theresulting algebraic system can be easily solved analyti-cally. We now describe further details for the two modelsseparately.

Model A.—A frustrated antiferromagnet on a squarelattice with the nearest-neighbor exchange constantJ1 � 1 and the diagonal coupling of strength J2 �

12 J1

has infinitely many classical ground states for 0<H <Hc [10,11]. The magnon spectrum in the ferromagneticphase at H � Hc is q � �1� cosqx��1� cosqy�. Themagnon energy has lines of minima spanned by wavevectors fq�g: ��; q� and �q;��. Besides the well-knownsingularity related to vanishing of a scattering amplitudefor two quantum particles in two dimensions (2D) [12],the present 1D type degeneracy leads to an extra infrareddivergence in the kernel of the integral equation when theexternal momenta are fixed to k;k0 � q�. Away from thecritical point � > 0, the singularity can be cured byintroducing an infrared cutoff for single-particle energiesdefined by k;k0 � � [12]. The physics behind this regu-larization procedure is as follows: at the energy scale of

k’k k+q

k’-q

FIG. 1. Graphical representation of the integral equation forthe four-point vertex.

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the order of � the interaction effects become importantand modify the form of the bare spectrum. As it follows,an infrared behavior of the excitation spectrum is sound-like and, hence, removes in a self-consistent way thedivergence of the kernel below Hc.

The condensate wave vector is chosen by consideringthe scattering amplitude at zero momentum transfer andexternal momenta set to q�. The two main candidatestates are the Neel state with Q � ��;��, which is stablefor a frustrated square-lattice antiferromagnet withweaker diagonal bonds J2 <

12 J1, and the columnar state

ordered on ��; 0� [�0; ��], which is stable for strongerdiagonal bonds J2 >

12 J1 [10]. Instead of presenting a

rather lengthy solution of the algebraic system, we obtainan approximate but physically transparent solution byneglecting the momentum-transfer dependence of p�q�;q�� � p�q�� near p � 0 under the integral inEq. (3). The result is 0�q�� � 1=��q�� with ��q�� �1=N

Pp p= q��p � q��p�: It is evident that 0�q�� is

smallest for a q� at which the kernel ��q�� has the stron-gest divergence. This singles out the Neel wave vectorQ � ��;��, which has the softest excitations "Q�k ’k2xk

2y. Hence, for � > 0 magnons condense at q� � Q

and a transverse antiferromagnetic order should beformed below the saturation field. Direct estimation ofthe integral in the kernel ��q�� yields 0�Q� ��1=2=j log�j, whereas away from ordering wave vector 0�q�� ��1=2. The square root behavior of the four-pointvertex is intrinsic to 1D systems. This is a consequence ofthe quasi 1D low-energy part of the bare spectrum of thefrustrated model. In addition, there is an extra logarith-mic correction to the 1D behavior at the ordering wavevector Q related to vanishing stiffness in both directionsleading to a Van Hove-type singularity in the kernel.

In the ordered state below Hc, the condensate densityn0 � hbQi2 is found from minimization of the groundstate energy density eg:s: � ��n0 �

12 0�Q�n20: n0 �

�= 0�Q� ’ �1=2j log�j, where we have neglected thenoncondensate contribution. The excitation spectrumcan be obtained by following the standard Bogoliubovscheme and replacing the bare vertex Vq with the fullscattering amplitude:

!2q � � q ��� �11q �2 � ��12q �2;

�11q � n0 0�q;Q� � q�Q�q;Q��;

�12q � n0 q�Q�Q�:

(4)

As is seen from Eq. (4), the self-energy acquires anadditional momentum dependence thanks to the de-pendence of on the incoming momenta. The magnonspectrum (4) is no longer degenerate and has the uniquezero-energy mode at q � Q. For q ! Q the magnon en-ergy becomes !q �

�������������������������������������������4� n0 0�q;Q� ���

p. Expanding

the vertex in small jkj � jq�Qj we obtain an acousticmode !k ’ sk with the velocity s2 ’ �=j log�j, which issmaller than the velocity s2 ’ � in nonfrustrated 2D and

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3D antiferromagnets near the saturation. Away from theordering wave vector, along the degeneracy lines, mag-nons acquire a dynamically generated gap ���j log�j.

The magnetization of the spin system is related to thetotal density of particles M�H� � 1

2� �n0 � n0�, whichincludes both the condensate n0 and noncondensate n0

parts. The noncondensate density of bosons is given byn0 � 1=N

Pq q ��� �11q �!q�= 2!q�. The largest

contribution to n0 is determined by wave vectors awayfrom Q, along the degeneracy lines q 2 fq�g. The ratio ofthe two densities is estimated as n0=n0 ’ 1=j log�j1=2 andis logarithmically small, although it exceeds the conden-sate depletion found for a nondegenerate 2D Bose gas(n0=n0 � 1=j log�j) [12]. Near Hc, the magnetizationcurve exhibits a strong singularity

M�H� ��1=2j log�j ������������������Hc �H

pj log�Hc �H�j; (5)

which fits well to the available numerical data [5,6].The above results obtained in the hard-core boson

picture are valid in the vicinity of Hc and should becompared with the standard linear spin-wave theory(LSWT), which applies for all fields 0<H <Hc. In theLSWT approach, one selects a few classical ground statesand calculates energies of zero-point oscillations E0 �12

Pk!k, which are different for each state. In this way we

have found that the Neel state has lower energy than acollinear (stripe) state in the whole range of fields.Another candidate state for 1

2Hc < H <Hc is a four-sublattice state, which has identical polarization in threesublattices with the fourth sublattice compensating thenet transverse magnetization. Such a partially collinearstate is a natural favorite if quantum effects are taken intoaccount via effective biquadratic exchange interactionderived in a second-order real-space perturbation theory[13]. LSWT shows instead that the Neel state has again alower energy for a frustrated square-lattice antiferromag-net. This discrepancy is explained by a nonanalytic de-pendence of the ground state energy on applied magneticfield determined by a large number of soft (zero) modes,whereas the real-space perturbation approach capturesonly analytic contributions. Furthermore, the Neel anti-ferromagnetic order is stable even at H � 1

2Hc, where ithas a lower energy of zero-point oscillations E0 � 0:694Sthan a fully collinear up-up-up-down (uuud) state withE0 � 0:703S. This result corrects the previous spin-wavecalculation [11] and is in good agreement with the exactdiagonalization studies [6,11], which show that the1=2-magnetization plateau appears in the present modelfor 0:5< J2=J1 & 0:65. In the vicinity of Hc, the LSWTmagnetization curve for the Neel state shows the sametype of singularity (5) with M � Shf1�

������������������2�1� h�

p�

ln 128�1� h�=�4e2�=�2Sg, where h � H=Hc.Model B.—A nearest-neighbor antiferromagnet (J�1)

on an fcc lattice is another frustrated model with degen-erate ground states at 0<H <Hc. The spectrum of mag-non excitation at the saturation field is q � 2 1�

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cosqx cosqy � cosqx cosqz � cosqy cosqz� and has linesof zeros at ��; q; 0� and the cubic symmetry related lines.We then follow the same scheme as for the previousmodel and examine the kernel for the scattering ampli-tude with the new spectrum. The lowest value of therenormalized four-point boson vertex corresponds to thethree wave vectors Qi � ��; 0; 0�, �0; �; 0�, and �0; 0; ��.The corresponding vertex is estimated as � 0�Qi;Qi� � 1=j log�j2. It has a 2D logarithmic behaviordue to a quasi-2D form of the spectrum in the vicinity ofthe ��; q; 0� line with an extra logarithmic singularityrelated again to a vanishing stiffness for k � 0: Q1�k ’k2x �

14 k

2yk2z . Because of the presence of three equally

singular wave vectors, one must check now whether theground state of the systems is characterized by a Bosecondensation at a single Qi wave vector or at all threewave vectors simultaneously [14]. For this we write theground state energy in the Landau form:

E � ��Xi

j ij2 �

2

Xi

j ij4

�Xi�j

�� j ij2j jj2 �

1

2~ � �2

i 2j � �2

j 2i �

�; (6)

where i � hbQii is a complex order parameter, � �

0�Qi;Qj� � Qi�Qj�Qi;Qj�, and ~ � Qi�Qj

�Qi;Qi�. Asingle component phase is stabilized for a sufficientlystrong repulsion between components < � � j~ j,whereas in the opposite case all three components ofthe Bose condensate appear with equal weights. Directcalculations show that � � 1=j log�j and ~ � 1=j log�j2.Hence, in the vicinity of the saturation field dominated bythe logarithmic behavior � � , ~ and the single-k stateare energetically favorable. The boson density and asymp-totic behavior of the magnetization curve are given byn � 1=2�M�H� ��j log�j2. We again find a d� 1-likeform for the magnetization with a logarithmic correction.

Corrections beyond the leading logarithms can, how-ever, change the energy balance. From an explicit esti-mate of the prefactors, one finds that � � only in anextremely narrow interval near the saturation field for� & �2e�20. Beyond this interval, � < and a multi-kstate with a real (~ < 0) superposition of all three modeswith equal amplitudes are stabilized as the ground state.Such a spin structure corresponds to a partially collinearspin configuration described above in our discussion ofmodel A. The LSWT calculations confirm the above con-clusion, showing that for magnetic fields as close to thesaturation field as �H=Hc � 10�3 the partially collinearstate is energetically more favorable than a single-k state.In contrast to the behavior of a frustrated square-latticeantiferromagnet, we have also found an uuud configura-tion for the ground state at H � 1

2Hc. This state has thezero-point oscillation energy of E0 � 1:66S, while asingle-k state has E0�1:74S. Thus, a 1=2-magnetization

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plateau should appear on the magnetization curve of anfcc antiferromagnet.

Finally, we make a few remarks on finite temperaturebehavior. Models A and B are magnetic analogs of theweak-crystallization model [15]. In this model the pho-non spectrum softens for wave vectors lying on a sphere.The phase transition to a crystal state appears to be of thefirst order even if a mean-field theory would predict acontinuous transition. Such a fluctuation-driven first-order transition is explained by a singular Hartree cor-rection generated by thermal fluctuations, which have alarge phase space [15].

For a 3D quantum fcc antiferromagnet, a finite tem-perature transition to the ordered state is expected for allH <Hc. In zero magnetic field neglecting quantum fluc-tuations the Hartree correction is estimated as �H �P

qT= q � �� � CTj log�j2, where � � T � TMF�=TMFis a distance from a mean-field critical point TMF. Theself-consistent equation for the excitation gap becomes� � �� CTj log�j2. The point of absolute instability� � 0 cannot be reached at any finite T indicating afirst-order transition. For classical Heisenberg and XYmodels on an fcc lattice, the first-order transition atH � 0 has been confirmed by the Monte Carlo simula-tions [16]. The transition between paramagnetic and anti-ferromagnetic states is, therefore, of the first order at lowH and at high T. The zero temperature phase transition atHc is, instead, of the second order. This opens two pos-sible scenarios for the H–T phase diagram: (a) a finitetemperature tricritical point separates a low-field (high-T)first-order transition line from the XY transition at highfields, or (b) a fluctuation-induced first-order transitionsurvives down to zero temperature and terminates at thequantum critical end point. For the two-dimensionalmodel A, similar questions can be raised on the interplaybetween a Berezinski-Kosterlitz-Thoulees (BKT) transi-tion and a fluctuation-induced first-order transition. Athigh temperatures (low fields), a high degree of degener-acy of the spectra may induce the first-order transitionand make the BKT instability inaccessible, while at lowtemperatures (high fields) a usual BKT transition maytake place. Such a scenario is, for example, realized in thetheory of weak crystallization of films [17]. These inter-esting issues require further investigations.

In summary, we have studied a quantum phase tran-sition at high magnetic fields for a class of frustratedantiferromagnets in d � 2 and d � 3, which have degen-erate excitation spectra with lines of minima. Momentumdependence of the exact four-point boson vertex removesthe degeneracy and selects the ordering wave vector Q.The new spectra are soundlike near Q and acquire dy-namically generated gaps away from it. The asymptoticbehavior of the magnetization curve shows the samesingularity as a nonfrustrated model in the d� 1 dimen-

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sion with an additional logarithmic correction. The de-veloped scheme can be applied to other frustratedantiferromagnets near the saturation and to singletground state systems with degenerate gapped triplet ex-citations, such as, e.g., SrCu2�BO3�2 [18] or Cs3Cr2Br9[19], near the triplet condensation field Hc1. In addition,recent experimental progress on the Bose condensationof alkali atoms on optical lattices [20] opens a newpossibility for experimental study of condensate phenom-ena on frustrated lattices, the systems to which our resultsalso apply.

We thank A. Honecker and E. I. Kats for interestingdiscussions. M. E. Z. acknowledges financial support fromthe 21st Century COE program of Kyoto University dur-ing his stay at the Yukawa Institute for TheoreticalPhysics.

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