marianna maltseva and piers coleman- failure of geometric frustration to preserve a...

8/3/2019 Marianna Maltseva and Piers Coleman- Failure of geometric frustration to preserve a quasi-two-dimensional spin fluid

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Failure of geometric frustration to preserve a quasi-two-dimensional spin fluid

Marianna Maltseva* and Piers ColemanCenter for Materials Theory, Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road,

Piscataway, New Jersey 08854, USA

Received 14 February 2005; revised manuscript received 26 September 2005; published 11 November 2005

Using spin-wave theory, we show that geometric frustration fails to preserve a two-dimensional spin fluid.Even though frustration can remove the interlayer coupling in the ground state of a classical antiferromagnet,

spin layers inevitably develop a quantum-mechanical coupling via the mechanism of order from disorder. We

show how the order from disorder coupling mechanism can be viewed as a result of magnon pair tunneling, a

process closely analogous to pair tunneling in the Josephson effect. In the spin system, the Josephson coupling

manifests itself as a biquadratic spin coupling between layers, and for quantum spins, these coupling terms

become comparable with the in-plane coupling terms. An alternative mechanism for decoupling spin layers

occurs in classical XY models in which decoupled sliding phases of spin fluid can form in certain finely

tuned conditions. Unfortunately, these finely tuned situations appear equally susceptible to the strong-coupling

effects of quantum tunneling, forcing us to conclude that, in general, geometric frustration cannot preserve a

two-dimensional spin fluid.

DOI: 10.1103/PhysRevB.72.174415 PACS numbers: 75.30.Ds, 71.27.a, 71.10.w

I. INTRODUCTION

This study is motivated by recent theories of heavy elec-tron systems1,2 which propose that the formation of magneti-cally decoupled layers of spins plays a central role in thedepartures from Fermi liquid behavior observed near a mag-netic quantum critical point. A wide variety of heavy electronmaterials develop logarithmically divergent specific heat co-efficients and quasilinear resistivities in the vicinity of quan-tum critical points.116 Several theories explaining these un-usual properties have been proposed.13,1114,1820 Thestandard model for these quantum phase transitions, pro-posed by Hertz and Moriya, involves a soft, antiferromag-

netic mode coupled to a Fermi surface. The Hertz-Moriyaspin-density wave SDW theory can account for the loga-rithmically divergent specific heat coefficients and quasilin-ear resistivities,1,17 but only if the spin fluctuations are quasi-two-dimensional. An alternative local quantum criticaldescription, based on the extended dynamical mean fieldtheory, also requires a quasi-two-dimensional spin fluid.2

Each of these theories can only account for the anomalies ofquantum critical heavy electron materials if the spin fluctua-tions of these systems are quasi-two-dimensional.1,3,14,1820

The hypothesis that heavy electrons involve decoupledlayers of spins motivates a search for a mechanism that cangenerate a quasi-two-dimensional environment for the spinfluctuations out of a metal that is manifestly three dimen-sional. One such frequently cited mechanism is geometricfrustration.1,3 Here, the idea is that the frustration leads to acancellation of the interlayer Weiss fields, so that spin layersdecouple in the classical ground state see Fig. 1.1,3

In this paper, we use the Heisenberg antiferromagnet as asimple example to explore this line of reasoning. The pri-mary objective of our paper is not to examine whether inter-layer coupling is relevant or irrelevant at the quantum criticalpointbut rather to examine whether frustration can set upan environment in which the interlayer coupling is suffi-ciently weak for us to consider the system to be quasi-two-

dimensional. A key issue in this discussion is the effect offluctuations and their potential to generate interlayer cou-pling via the mechanism of order from disorder.22,23 Usingthe spin-wave theory we show that, in general, zero-pointfluctuations of the spin act as an extremely powerful forcefor coupling two-dimensional spin layers, making geometricfrustration an unlikely candidate for the development of aquasi-two-dimensional spin fluid.

To illustrate the main points of our argument, considertwo separate layers of Heisenberg spins. The Hamiltonian is

H0 = HB + HT , 1

where HT and HB are the Hamiltonians for the top andbottom layers. Namely,

H0 = J

i,

SiBSi+

B + Si+T Si++

T . 2

Here Si

B Si

T is the spin variable defined at the site i in thebottom top layer. The vector denotes a displacement tothe nearest neighbor sites within the plane, =a , 0 or0 , a. = a/2, a/ 2 defines a shift between layers.

FIG. 1. Color online A lattice structure.

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At T=0 each layer is antiferromagnetically ordered andspin waves run along the layers. Now consider the effect of a

small frustrated interlayer coupling. The Hamiltonian is thenH= H0 + V, 3

with the interlayer coupling

V= Ji,

SiBSi

T + SiBSi+

T . 4

Figure 1 illustrates the nature of the magnetic couplingsbetween layers. Any given spin on a layer is coupled to fourspins in the neighboring layer, two pointing in one direction,the other two pointing in the opposite direction, so that theWeiss fields of one layer on spins in the neighboring layerexactly cancel one another. The interlayer coupling is thus

frustrated, and there is no preferred orientation of the spinson one layer with respect to the neighboring layer. Classi-cally, the spin layers are thus decoupled, and there is nounique classical ground state. Let us now see how this pic-ture changes when we take account of quantum zero-pointfluctuations.

In the quantum-mechanical picture, even a small inter-layer coupling enables magnons to virtually tunnel betweenlayers. An antiferromagnet can be regarded as a long rangeresonating valence bond RVB state,21 so individual magnontransfer is energetically unfavorable, and the transfer of mag-nons between the layers tends to occur in pairs, as in Joseph-son tunneling see Fig. 2. Interlayer magnon pair tunnelingis ubiquitous in three-dimensional spin systems, frustratedand unfrustrated alike. So unless the interlayer coupling con-stant is set exactly to zero, magnons travel between the lay-ers, producing a coupling closely analogous to Josephsoncoupling of superconducting layers. Such a coupling is analternative way of viewing the phenomenon of order fromdisorder,22,23 whereby the free energy of zero-point or ther-mal fluctuations depends on the relative orientation of theclassical magnetization.

If we use the analogy between superconductors and anti-ferromagnets, then spin rotations of an antiferromagnet maponto gauge transformations of the electron phase in a super-

conductor. In a superconducting tunnel junction, the Joseph-son energy is determined by the product of the order param-eters in the two layers, i.e.,

EJ t

2

Re2

2 11 cos2 1,

where t is the tunneling matrix element, the supercon-

ducting gap energy, and l l =1,2 represents an electronfield in leads one and two. By analogy, in a correspondingspin junction, the coupling energy is determined by theproduct of the spin-pair amplitudes. Suppose for simplicitythat the system is an easy-plane XY magnet, then

ES J

2

JReS2

+iS2+jS1

iS1j

J

2

JS cos2 ,

where Sli represents the spin raising, or lowering operator

at site i in plane l, parallel to the local magnetization. The

factor 2arises because the spin pair carries a phase whichis twice the angular displacement of the magnetization S+

Sx + iSy Sei. In other words, the effective Hamiltonian

is

Hef f = HB + HT

J

2

JS3 ,i,

SiSi+

2, 5

where = T, B is a layer index, i is a site index, is adisplacement to the nearest neighbor sites within a layer, and is a numerical constant. Obviously, the geometric frustra-tion leads to a new ground state with

ES

2J

2

JS

cos

2

+ const,

so the interlayer coupling induced by spin tunneling is ex-pected to be biquadratic in the relative angle between thespins. Clearly, this is a much oversimplified argument. Weneed to take account of the O3, rather than the U1 sym-metry of a Heisenberg system. Nevertheless, this simple ar-gument captures the spirit of the coupling between spin lay-ers, as we shall now see in a more detailed calculation.

II. SPIN-WAVE SPECTRUM FOR DECOUPLED

LAYERS

Consider a Heisenberg model with nearest-neighbor anti-ferromagnetic interaction in its ground state defined on thebody-centered tetragonal lattice. This choice of model is mo-tivated by the structure of CePd2Si2, one of the compoundsfor which the idea of quasi-two-dimensionality was origi-nally proposed.3 In this lattice structure Fig. 1, square lat-tices stack with a shift of a/2, a/ 2 between adjacent layersa is the lattice constant within the layer. For simplicity, thedistance between the layers is also a. The spins of the nearestneighbors in each layer are antiparallel. In the classicalground state the spins in different layers are decoupled andmay assume any relative alignment.

FIG. 2. The contrasting a Josephson tunneling between pairedsuperconductors and b magnon tunneling between antiferromag-nets, viewed within a resonating valence bond RVB picture.

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For simplicity, let us consider just two adjacent layers, theargument being easily generalized to an infinite number oflayers. The Hamiltonian is then given by Eqs. 24. Sincethe coupling between layers is small JJ, we may treatthis model using perturbation theory where the ratio of cou-pling constants J/J is taken as a small parameter.

For our purposes, it is sufficient to consider a simple casewith the spins lying in the planes of the two-dimensional

lattice. At sites i = la , ma and i += la + 12 a , ma + 12 a thespins are

SiXB = S 1l+m, Si

YB = 0; 6

Si+XT = S 1l+m+1 cos , Si+

YT = S 1l+m+1 sin ; 7

where X and Y are mutually perpendicular directions in theplane and l and m are integers.

Following a standard procedure,24,25 we use the Holstein-Primakoff approximation for the spin operators to determinethe spin-wave spectrum:

Si+ 2Sai, Si 2Sai+, Siz = S ai+ai ,

8

where

ai,aj

+ = ij, = B,T. 9

The Fourier transforms of ai

B, a

i

+B, a

i

T, and a

i

+Tare

aqB =

1

Niai

Beiqi, aq

+B =1

Niai

+Beiqi , 10

aqT =

1

Niai+

Teiqi+, aq

+T =1

Niai+

+Teiqi+,

11where N is the number of spin sites in the layer.

aq

+and a

q

are the spin-wave creation and annihilation

operators.

aq,aq

+ = qq. 12

The single-layer Hamiltonian H becomes

H = 4NS2J

+ q

8SJaq

+aq

+ SJqaq

aq

+ H.c.,

13

and on diagonalization the Hamiltonian H0 for the decoupled

layers can be written as

H0 = E0 + =T,B

q

q

bq+bq

. 14

The ground state energy of the decoupled two-layer systemis then

E0 = 8NSS + 1J

+ q

q

. 15

q

defines the spectrum of spin waves propagating in each ofthe layers

q

= 4SJ4 cos qxa + cos qya2 . 16

III. MAGNON PAIR TUNNELING BETWEEN THE

LAYERS

Now we express the perturbation V in 4 in terms ofa

q

+, a

q

as

V= Sq

Aqaq

+TaqB + H.c. + S

q

Bqaq

TaqB + H.c. ,

17

where Aq and Bq

are defined as

Aq = 2Jcosqxa

2cosqya

2 + sinqxa

2sinqya

2cos ,

18

Bq = 2J

cos

qxa

2 cos

qya

2 sin

qxa

2 sin

qya

2 cos

.

19

In terms of bq

+, b

q

V= Sq

qbqTbq

B + bq+Tbq

+B + Vph , 20

where q = Aq sinh 2uq +Bq

cosh 2uq describes the ampli-tude for magnon pair tunneling, and Vph = SqAq

cosh 2uq+Bq

sinh 2uqbq+T

bq

B+ b

q

+Bb

q

T describes single magnontunneling between layers.

It is straightforward to see that the particle-hole terms do

not affect the ground-state energy, for VphGS =0, whereGS denotes the ground state wave function of the system.The second order correction to the ground state energy E0 isthen

E02 =

VGS2

E EGS, 21

where denotes a state with two magnons being trans-ferred between layers. Thus,

E02 =

q

S2q2/2q

. 22

To understand the nature of coupling between the layersdipolar or quadrupolar, let us retrieve the dependence ofE0

2on the angle .

E02 =

S

2

J2

J

C0 + C2 cos2

= S

2

J2

J C0 + C22 +

C2

2cos 2 . 23

The particular form of the coefficients is

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C0 =

dx dy

221 14 cos x + cos y2

cos2x

2cos2

y

21 1

2cos x + cos y2 , 24

C2

=

dx dy

221 1

4

cos x + cos y2

cos2 sin2x

2sin2

y

21 + 1

2cos x + cos y2. 25

The interlayer coupling is indeed quadrupolar in nature, asforeseen earlier.

IV. DISCUSSION

In the above calculation, we considered an orderedHeisenberg antiferromagnet at zero temperature. In practice,provided the spin-spin correlation length is large comparedwith the lattice constant a, a, a biquadratic interlayer

coupling will still develop. Moreover, at finite temperatures,thermal fluctuations will produce further interlayer coupling.Both thermal and quantum interlayer coupling processes aremanifestations of order from disorder. The main differencebetween the thermal and quantum coupling processes lies inthe replacement of the magnon occupation numbers with aBose-Einstein distribution function, and, in general, both thesign and the angular dependences of the two couplings areexpected to be the same.22 In general, geometrical frustrationis an extremely fragile mechanism for decoupling spin layersand will always be overcome by quantum and thermal fluc-tuations. Our work was motivated by heavy electron sys-tems. These are much more complex systems than insulating

antiferromagnets, but if our mechanism for the formation oftwo-dimensional spin fluid is to be frustration, it is difficultto see how similar interlayer coupling effects might beavoided. We are led to conclude that for the hypothesis of thereduced dimensionality of the spin fluid in heavy fermionmaterials to hold, a completely different decoupling mecha-nism must be at work.

In the special case of XY magnetism there is, in fact, onesuch alternative mechanism, related to sliding phases. Herethe idea is to create an environment in which interlayer cou-pling, though present, becomes irrelevant, ultimately scalingto zero at long distances. Some heavy fermion systems, suchas YbRh2Si2, are XY-like; most others, such as CeCu6, areIsing-like. It is, therefore, instructive to consider whether thesliding phase mechanism might be generalized to Heisenbergor Ising spin systems to provide an escape from the fluctua-tion coupling that we have discussed.

The existence of a sliding phase in weakly coupledstacks of two-dimensional 2D XY models was predicted byOHern, Lubensky, and Toner.26 Sliding phases are of par-ticular current interest in the context of Josephson junctionarrays.27 In addition to Josephson interlayer couplings,OHern et al. included higher-order gradient couplings be-tween the layers. In the absence of Josephson couplings,these gradient couplings preserve the decoupled nature of the

spin layers, only modifying the power-law exponents of the2D correlation functions, SiSj r

. As the temperature is

raised, Josephson interlayer couplings become irrelevantabove a particular decoupling temperature Td. One can al-ways select interlayer gradient couplings to satisfy TdTKTand produce a stable sliding phase in the temperature win-dow TdTTKT.

To see this in a bit more detail, consider the continuous

version of the Hamiltonian of two layers of XY models, H=H0 + V, where H0 is a sum of independent layer Hamilto-nians and V is the usual Josephson-type interlayer coupling

H0 =J

2 d2rTr2 + J

2 d2rBr2, 26

V= J d2rcosTr Br . 27At low temperature, when the interlayer coupling J is

zero, the average of the intralayer spin-spin correlation func-tion with respect to H0 is

2r0 = lnL/b, 28

and

cosr 00 L/b, 29

where = T/ 2J, L is the sample width, and b is a short-distance cutoff in the XY plane.

The average of Josephson interlayer coupling V scales asV0 L

2, so Josephson couplings become irrelevant at

Td= 4J. At temperatures above the Kosterlitz-Thouless

transition temperature TKT=J/2, thermally excited vortices

destroy the quasi-long-range order and drive the system todisorder. In this simple example, it happens that TdTKT,

which does not permit a sliding phase. However, higher-order gradient interlayer couplings between the layers, whenadded to this model, suppress Td below TKT, producing astable sliding phase for TdTTKT.

So can the sliding phase concept be generalized toHeisenberg spin systems? A sliding phase develops in the XYmodel because power-law spin correlations introduce ananomalous scaling dimension, but unfortunately, a finite tem-perature Heisenberg model has no phases with power-lawcorrelations.28 In general, biquadratic interlayer couplingswill always remain relevant in Heisenberg models. In thequantum-mechanical picture, as soon as a frustrated inter-layer coupling is introduced, the order-from-disorder phe-

nomenon22,23

generates a coupling

SJ2

/J

between thelayers.

H= 2

i

ni2 +

2

i

ni ni+12, 30

where = S2a2J. This coupling gives us a length scale l0determined from l0

2 /or l0 aSJ/J. Once the spincorrelation length a exp2JS2/T within a layer growsto become larger than l0, i.e., l0a exp2J

S2/T, a 3D-ordering phase transition occurs. An estimate of the 3D-ordering transition temperature is then Tc 2J

S2/

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lnSJ/J. The answer is essentially identical in the clas-sical picture, for here, thermal fluctuations generate an en-tropic interlayer coupling maxSJ2/J , TS2, so at highenough temperatures, for large S, TJ2/J2, l0 SaJ 3/2/JT. A classical estimate of the 3D-orderingtemperature is Tc 2J

S2/lnJ/J.Another interesting question is whether XY models permit

sliding phases at T=0. The decoupling temperature, as foundby OHern, Lubensky, and Toner,26 is

Tdp =4

fo1 fp

1 . 31

One sees no obvious mechanism of suppressing Td to zero. A2D sliding phase is equivalent to a 3D finite temperaturesliding phase, so the existence of a sliding phase in the XYmodel at zero temperature would mean a power-law phase inthe 3D XY model. Since no power-law phase exists in 3DXY-like systems, sliding phases at T=0 are extremely un-likely. In conclusion, the sliding phase scenario also fails toprovide a valid general mechanism for decoupling layers in

Ising-like and Heisenberg-like systems.

Let us return momentarily to consider the implications of

these conclusions for the more complex case of heavy elec-

tron materials. It is clear from our discussion that simple

models of frustration do not provide a viable mechanism for

decoupling spin layers. One of the obvious distinctions be-

tween an insulating and a metallic antiferromagnet is the

presence of dissipation which acts on the spin fluctuations.

The interlayer coupling we considered here relies on short-

wavelength spin fluctuations, and these are the ones that are

most heavily damped in a metal. Our exclusion of such ef-

fects does hold open a small possibility that order-from-

disorder effects might be substantially weaker in a metallic

antiferromagnet. However, if we are to take this route, then

we can certainly no longer appeal to the analogy of the in-

sulating antiferromagnet while discussing a possible mecha-

nism for decoupling spin layers.

This research is supported by the National Science Foun-

dation Grant No. NSF DMR 0312495. We would particularly

like to thank Tom Lubensky for a discussion relating to the

sliding phases of XY antiferromagnets.

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London 394, 39 1998.4

D. Sokolow, M. C. Aronson, Z. Fisk, J. Chan, and J. Millicanunpublished.5 H. v. Lohneysen, T. Pietrus, G. Portisch, H. G. Schlager, A.

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