Deconfined QuantumCritical Points

T. Senthil,1* Ashvin Vishwanath,1 Leon Balents,2 Subir Sachdev,3

Matthew P. A. Fisher4

The theory of second-order phase transitions is one of the foundations ofmodern statistical mechanics and condensed-matter theory. A central conceptis the observable order parameter, whose nonzero average value characterizesone or more phases. At large distances and long times, fluctuations of the orderparameter(s) are described by a continuumfield theory, and these dominate thephysics near such phase transitions. We show that near second-order quantumphase transitions, subtle quantum interference effects can invalidate this par-adigm, and we present a theory of quantum critical points in a variety ofexperimentally relevant two-dimensional antiferromagnets. The critical pointsseparate phases characterized by conventional “confining” order parameters.Nevertheless, the critical theory contains an emergent gauge field and “de-confined” degrees of freedom associated with fractionalization of the orderparameters. We propose that this paradigm for quantum criticality may be thekey to resolving a number of experimental puzzles in correlated electron sys-tems and offer a new perspective on the properties of complex materials.

Much recent research in condensed-matterphysics has focused on the behavior of matternear zero-temperature “quantum” phase tran-sitions that are seen in several strongly cor-related many-particle systems (1). Indeed, apopular view ascribes many properties of cor-related materials to the competition betweenqualitatively distinct ground states and theassociated phase transitions. Examples ofsuch materials include the cuprate high-temperature superconductors and the rareearth intermetallic compounds (known as theheavy fermion materials).

The traditional guiding principle behindthe modern theory of critical phenomena isthe association of the critical singularitieswith fluctuations of an order parameter thatencapsulates the difference between thetwo phases on either side of the criticalpoint (a simple example is the averagemagnetic moment, which distinguishesferromagnetic iron at room temperaturefrom its high-temperature paramagneticstate). This idea, developed by Landau andGinzburg (2), has been eminently success-ful in describing a wide variety of phase-transition phenomena. It culminated in the

sophisticated renormalization group theoryof Wilson (3), which gave a general pre-scription for understanding the critical sin-gularities. Such an approach has beenadapted to examine quantum critical phe-nomena as well and provides the generallyaccepted framework for theoretical descrip-tions of quantum transitions.

We present specific examples of quantumphase transitions that do not fit into thisLandau-Ginzburg-Wilson (LGW) paradigm(4). The natural field theoretic description oftheir critical singularities is not in terms of theorder parameter field(s) that describe the bulkphases, but in terms of new degrees of freedomspecific to the critical point. In our examples,there is an emergent gauge field that mediatesinteractions between emergent particles thatcarry fractions of the quantum numbers of theunderlying degrees of freedom. These fraction-al particles are not present (that is, are confined)at low energies on either side of the transitionbut appear naturally at the transition point.Laughlin has previously argued for fractional-ization at quantum critical points on phenome-nological grounds (5).

We present our examples using phase tran-sitions in two-dimensional (2D) quantum mag-netism, although other points of view are alsopossible (6). Consider a system of spin S � 1/2moments �Sr on the sites, r, of a 2D squarelattice with the Hamiltonian

H � J��rr��

�Sr � �Sr� � . . . (1)

where J � 0 is the antiferromagnetic ex-

change interaction, and the ellipses representother short-range interactions that may betuned to drive various zero-temperaturephase transitions.

Considerable progress has been made inelucidating the possible ground states of sucha Hamiltonian. The Neel state has long-rangemagnetic order (Fig. 1A) and has been ob-served in a variety of insulators, including theprominent parent compound of the cuprates:La2CuO4. Apart from such magnetic states, itis now recognized that models in the class ofH can exhibit a variety of quantum paramag-netic ground states. In such states, quantumfluctuations prevent the spins from develop-ing magnetic long-range order. One class ofparamagnetic states is the valence bond solids(VBS) (Fig. 1B). In such states, pairs ofnearby spins form a singlet, resulting in anordered pattern of valence bonds. Typically,such VBS states have an energy gap to spin-carrying excitations. Furthermore, for spin-1/2 systems on a square lattice, such statesalso necessarily break lattice translationalsymmetry. A second class of paramagnetshas a liquid of valence bonds and need notbreak lattice translational symmetry, but wewill not consider such states here. Our focusis on the nature of the phase transition be-tween the ordered magnet and a VBS. Wealso restrict our discussion to the simplestkinds of ordered antiferromagnets: those withcollinear order, where the order parameter isa single vector (the Neel vector).

1Department of Physics, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA. 2Depart-ment of Physics, University of California, Santa Bar-bara, CA 93106–4030, USA. 3Department of Physics,Yale University, P.O. Box 208120, New Haven, CT06520–8120, USA. 4Kavli Institute for TheoreticalPhysics, University of California, Santa Barbara, CA93106–4030, USA.

*To whom correspondence should be addressed. E-mail: senthil@mit.edu

Fig. 1. (A) The magnetic Neel ground state ofthe Hamiltonian Eq. 1 on the square lattice. Thespins, �Sr, fluctuate quantum-mechanically inthe ground state, but they have a nonzeroaverage magnetic moment, which is orientedalong the directions shown. (B) A VBS quantumparamagnet. The spins are paired in singletvalence bonds, which resonate among themany different ways the spins can be paired up.The valence bonds crystallize, so that the pat-tern of bonds shown has a larger weight in theground state wavefunction than its symmetry-related partners (obtained by 90° rotations ofthe above states about a site). This groundstate is therefore fourfold degenerate.

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Both the magnetic Neel state and the VBSare states of broken symmetry. The formerbreaks spin rotation symmetry and the latterthat of lattice translations. The order param-eters associated with these two different bro-ken symmetries are very different. A LGWdescription of the competition between thesetwo kinds of orders generically predicts eithera first-order transition or an intermediate re-gion of coexistence where both orders aresimultaneously present. A direct second-order transition between these two brokensymmetry phases requires fine-tuning to amulticritical point. Our central thesis is thatfor a variety of physically relevant quantumsystems, such canonical predictions of LGWtheory are incorrect. For H, we will show thata generic second-order transition is possiblebetween the very different kinds of brokensymmetry in the Neel and VBS phases. Ourcritical theory for this transition is, however,unusual and is not naturally described interms of the order parameter fields of eitherphase. A picture related to the one developedhere applies also to transitions between va-lence bond liquid and VBS states (7) and totransitions between different VBS states (8)in the quantum dimer model (9, 10).

Field theory and topology of quantumantiferromagnets. In the Neel phase orclose to it, the fluctuations of the Neel orderparameter are captured correctly by the well-known O(3) nonlinear sigma model field the-ory (11–13), with the following action Sn inspacetime [we have promoted the lattice co-ordinate r � (x, y) to a continuum spatialcoordinate, and � is imaginary time]:

Sn �1

2g� d� � d2r� 1

c2 ��n

���2

� r n�2�� iS�

r �1� rAr (2)

Here n � ( 1)r �Sr is a unit three-componentvector that represents the Neel order param-eter [the factor ( 1)r is �1 on one checker-board sublattice and 1 on the other]. Thesecond term is the quantum-mechanical Ber-ry phase of all the S � 1/2 spins: Ar is the areaenclosed by the path mapped by the time

evolution of nr on a unit sphere in spin space.These Berry phases play an unimportant rolein the low-energy properties of the Neelphase (12) but are crucial in correctly de-scribing the quantum paramagnetic phase(14). We show here that they also modify thequantum critical point between these phases,so that the exponents are distinct from thoseof the LGW theory without the Berry phasesstudied earlier (12, 15).

To describe the Berry phases, first notethat in two spatial dimensions, smooth con-figurations of the Neel vector admit topolog-ical textures known as skyrmions (Fig. 2).The total skyrmion number associated with aconfiguration defines an integer topologicalquantum number Q

Q �1

4��d2r n � �xn � �yn (3)

The sum over r in Eq. 2 vanishes (1, 11)for all spin time histories with smooth equal-time configurations, even if they containskyrmions. For such smooth configurations,the total skyrmion number Q is independentof time. However, the original microscopicmodel is defined on a lattice, and processeswhere Q changes by some integer amount areallowed. Specifically, such a Q changingevent corresponds to a monopole (or hedge-hog) singularity of the Neel field n(r,�) inspacetime (a hedgehog has n oriented radiallyoutward in all spacetime directions awayfrom its center). Haldane (11) showed that thesum over r in Eq. 2 is nonvanishing in thepresence of such monopole events. Precisecalculation (11) gives a total Berry phaseassociated with each such Q changingprocess, which oscillates rapidly on foursublattices of the dual lattice. This leads todestructive interference, which effectivelysuppresses all monopole events unless theyare quadrupled (11, 14) (that is, they changeQ by 4).

The sigma model field theory augmentedby these Berry phase terms is, in principle,powerful enough to correctly describe thequantum paramagnet. Summing over the var-ious monopole tunneling events shows that inthe paramagnetic phase, the presence of the

Berry phases leads to VBS order (14). Thus,Sn contains within it the ingredients describ-ing both the ordered phases of H. However, adescription of the transition between thesephases has so far proved elusive and will beprovided here.

Our analysis of this critical point is aidedby writing the Neel field n in the so-calledCP1 parametrization

n � z†��z (4)

with �� a vector of Pauli matrices. Here z �z(r, �) � (z1, z2) is a two-component complexspinor of unit magnitude, which transformsunder the spin-1/2 representation of theSU(2) group of spin rotations. The z1,2 are thefractionalized “spinon” fields. To understandthe monopoles in this representation, let usrecall that the CP1 representation has a U(1)gauge redundancy. Specifically, the localphase rotation

z 3 ei�(r,�)z (5)

leaves n invariant and hence is a gauge ofdegree of freedom. Thus, the spinon fields arecoupled to a U(1) gauge field, a � [the space-time index � � (r, �)]. As is well known, themagnetic flux of a � is the topological chargedensity of n appearing in the integrand of Eq.3. Specifically, configurations where the a �

flux is 2� correspond to a full skyrmion (inthe ordered Neel phase). Thus, the monopoleevents described above are spacetime mag-netic monopoles (instantons) of a� at which2� gauge flux can either disappear or becreated. The fact that such instanton eventsare allowed means that the a� gauge field isto be regarded as compact.

We now state our key result for thecritical theory between the Neel and VBSphases. We argue below that the Berryphase–induced quadrupling of monopoleevents renders monopoles irrelevant at thequantum critical point. So in the criticalregime (but not away from it in the para-magnetic phase), we may neglect the com-pactness of a � and write down the simplestcritical theory of the fractionalized spinonsinteracting with a noncompact U(1) gaugefield with action Sz � �d2rd�Lz and

Lz � ���1

N

� (�� ia�) z��2 � s�z�2

� u�z�2�2� �(������a�)

2 (6)

Where N � 2 is the number of z components,we have softened the length constraint on thespinons, with �z�2 � �� � 1

N �z� �2 allowed tofluctuate and the value of s is to be tuned sothat Lz is at its scale-invariant critical point.The irrelevance of monopole tunnelingevents at the critical fixed point implies thatthe total gauge flux �d2r(�x ay �yax), orequivalently the skyrmion number Q, is as-

Fig. 2. A skyrmionconfiguration of thefield n (r). Note thatn � ( 1)r �Sr, and sothe underlying spinshave a rapid sublatticeoscillation, which isnot shown. Theskyrmion above has n(r � 0) � (0,0, 1)and n ( � r� 3 �) �(0,0,1).

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ymptotically conserved. This emergent glob-al topological conservation law provides pre-cise meaning to the notion of deconfinement.It is important to note that the critical theorydescribed by Lz (16) is distinct from theLGW critical theory of the O(3) nonlinearsigma model obtained from Eq. 2 by drop-ping the Berry phases and tuning g to acritical value (17). In particular, the lattermodel has a nonzero rate of monopole tun-neling events at the transition, so that theglobal skyrmion number Q is not conserved.

Among the important physical conse-quences of the theory Lz (7, 18) are thepresence of two diverging length scales uponapproaching the critical point from the VBSside (the spin correlation length and a longerscale beyond which two spinons interact witha linear confining potential) and a largeanomalous dimension for the Neel order pa-rameter (because it is a composite of thecritical spinons).

The critical theory Lz is actually impliedby existing results in the N 3 � limit (18).The following section illustrates the origin ofLz by a physical derivation for the case of“easy-plane” anisotropy, when the spins pre-fer to lie in the xy plane. Such arguments canbe generalized to the isotropic case (7, 18).

Duality transformations with easy-plane anisotropy. For the easy-plane case,duality maps and an explicit derivation of adual form of Lz are already available in theliterature (6, 19). Here we obtain this theoryusing simple physical arguments.

The easy-plane anisotropy reduces thecontinuous SU(2) spin rotational invarianceto the U(1) subgroup of rotations about the zaxis of spin, along with a Z2 (Ising) spinreflection symmetry along the z axis. Withthese symmetries, Eq. 2 allows an additionalterm uep �d�d2r(nz)2, with uep � 0.

The classical Neel ground state of the easy-plane model n is independent of position andlies in the spin xy plane. Topological defectsabove this ground state play an important role.These are vortices in the complex fieldn� � nx � iny, and along a large loop aroundthe vortex the phase of n� winds by 2�m, withm an integer. In the core, the XY order issuppressed and the n vector will point along the�z direction. This corresponds to a nonzerostaggered magnetization of the z component ofthe spin in the core region. Thus, at the classicallevel, there are two kinds of vortices, oftencalled merons, depending on the direction ofthe n vector at the core (Fig. 3). Either kind ofvortex breaks the Ising-like nz3 nz symme-try at the core. Let us denote by �1 the quantumfield that destroys a vortex whose core points inthe up direction and by �2 the quantum fieldthat destroys a vortex whose core points in thedown direction.

Clearly, this breaking of the Ising symme-try is an artifact of the classical limit: Once

quantum effects are included, the two brokensymmetry cores will be able to tunnel intoeach other, and there will be no true brokenIsing symmetry in the core. This tunneling isoften called an “instanton” process that con-nects two classically degenerate states.

Surprisingly, such an instanton event isphysically the easy-plane avatar of the space-time monopole described above for the fullyisotropic model. This may be seen pictorially.Each classical vortex of Fig. 3 really repre-sents half of the skyrmion configuration ofFig. 2. Now imagine the �2 meron at time� 3 � and the �1 meron at time � 3 �.These two configurations cannot be smoothlyconnected, and there must be a singularity inthe n configuration, which we place at theorigin of spacetime. A glance at Fig. 3 showsthat the resulting configuration of n can besmoothly distorted into the radially symmet-ric hedgehog/monopole event. Thus, the tun-neling process between the two merons isequivalent to creating a full skyrmion. This isprecisely the monopole event. Hence, askyrmion may be regarded as a composite ofan “up” meron and a “down” antimeron, andthe skyrmion number is the difference in thenumbers of up and down merons.

The picture so far has not accounted forthe Berry phases. The interference effect dis-cussed above for isotropic antiferromagnetsapplies here too, leading to an effective can-cellation of instanton tunneling events be-tween single �1 and �2 merons. The onlyeffective tunnelings are those in which four�1 merons come together and collectivelyflip their core spins to produce four �2 mer-ons, or vice versa.

A different perspective on the �1,2 meronvortices is provided by the CP1 representa-tion. Ordering in the xy plane of spin spacerequires condensing the spinons

��z1�� � ��z2�� � 0 (7)

so that n� � z*1 z2 is ordered and there is noaverage value of nz � �z1�2 �z2�2. Now,clearly, a full 2� vortex in n� can beachieved by either having a 2� vortex in z1

and not in z2, or a 2� antivortex in z2 and no

vorticity in z1. In the first choice, the ampli-tude of the z1 condensate will be suppressedat the core, but �z2� will be unaffected. Con-sequently, nz � �z1�2 �z2�2 will be nonzeroand negative in the core, as in the �2 meron.The other choice also leads to nonzero nz,which will now be positive, as in the �1

meron. Clearly, we may identify the �2 (�1)meron vortices with 2� vortices (antivorti-ces) in the spinon fields z1 (z2). Note that interms of the spinons, paramagnetic phasescorrespond to situations in which neither spin-on field is condensed.

The above considerations and the generalprinciples of boson duality in three spacetimedimensions (20) determine the form of the dualaction Sdual � �d�d2rLdual for �1,2 (6, 19)

Ldual � ���1,2

�(�� iA�)�� �2 � rd ���2

� ud���2�2

� vd��1�2��2�2

� �d ������A�)2 �[(�2*�2)

4

� (�2*�1)

4] (8)

where

���2 � ��1�2 � ��2�2

The correctness of this form may be arguedas follows: First, from the usual boson-vortexduality transformation (20), the dual �1,2

vortex fields must be minimally coupled to adual noncompact U(1) gauge field A�. Thisdual gauge invariance is not related to Eq. 5but is a consequence of the conservation ofthe total Sz: the “magnetic” flux ε�����A� isthe conserved Sz current (20). Second, underthe Z2 reflection symmetry, the two vorticesget interchanged; that is, �17 �2. The dualaction must therefore be invariant under in-terchange of the 1 and 2 labels. Finally, ifmonopole events were to be disallowed byhand, the total skyrmion number - (the dif-ference in the number of up and down meronvortices) would be conserved. This wouldimply a global U(1) symmetry [not to beconfused with the U(1) spin symmetry about

Fig. 3. The meron vor-tices �1 (above) and�2 in the easy-planecase. The �1 meronabove has n (r � 0) �(0,0,1) and n ( �r� 3�) � (x,y,0)/�r�; the �2meron has n (r � 0) �(0,0, 1) and the samelimit as �r� 3 �. Eachmeron above is halfthe skyrmion in Fig.2: A composite of�1 and �2* makesone skyrmion.

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the z axis] under which

�13 �1expi�) ; �23 �2 exp( i�) (9)

where � is a constant. However, monopoleevents destroy the conservation of skyrmionnumber and hence this dual global U(1) sym-metry. But because the monopoles are effec-tively quadrupled by cancellations from theBerry phases, skyrmion number is still con-served modulo 4. Thus, the symmetry in Eq.9 must be broken down to the discrete cyclicgroup of four elements, Z4.

The dual Lagrangian in Eq. 8 is the sim-plest one that is consistent with all theserequirements. In particular, we note that inthe absence of the � term, the dual globalU(1) transformation in Eq. 9 leaves the La-grangian invariant. The � term breaks thisdown to Z4 as required. Thus, we may iden-tify this term with the quadrupled monopoletunneling events. Berry phases are thereforeexplicitly included in Ldual.

In this dual vortex theory, the XY orderedphase is simply characterized as a dual “para-magnet,” where ��1,2� � 0 and fluctuationsof �1,2 are gapped. On the other hand, spinparamagnetic phases such as the VBS statescorrespond to condensates of the fields �1,2,which break the dual gauge symmetry. Inparticular, if both �1 and �2 condense withequal amplitude ���1�� � ���2�� � 0, then weobtain a paramagnetic phase where the globalIsing symmetry is preserved. Note the re-markable complementarity between the de-scription of the phases in this dual theorywith that in terms of the spinon fields ofthe CP1 representation: The descriptionsmap onto one another upon interchangingboth z1,2 7 �1,2 and the role of the XYordered and paramagnetic phases. This is asymptom of an exact duality between thetwo descriptions that obtains close to thetransition (17, 18).

The combination �*1�2 � ��1�2�ei(�

1 �2)

actually serves as an order parameter for thetranslation symmetry–broken VBS ground state.This may be seen from the analysis of (6, 19).Alternatively, we may use the identification (14)of the skyrmion creation operator with the orderparameter for translation symmetry breaking.

Such a condensate of �1,2 breaks the global Z4

symmetry of the action in Eq. 8. The preferreddirection of the angle �1 �2 depends on thesign of �. The two sets of preferred directionscorrespond to columnar and plaquette patterns oftranslational symmetry breaking (Fig. 4). Also,the breaking of the dual U(1) symmetry in Eq. 9by � corresponds to a linear confinement ofspinons in the paramagnet.

Despite its importance in the paramagnet,the � term is irrelevant at the critical point(18). In critical phenomena parlance, it is adangerously irrelevant perturbation (15).Consequently the critical theory is decon-fined, and the z1,2 spinons (which are frac-tions of n�), or in the dual description the�1,2 merons (which are fractions of askyrmion), emerge as natural degrees of free-dom right at the critical point. The spinonsare confined in both adjacent phases, butthe confinement length scale diverges onapproaching the critical point. At a moresophisticated level, the critical fixed pointis characterized by the emergence of anextra global U(1) symmetry in Eq. 9 that isnot present in the microscopic Hamiltonian.This is associated with the conservation ofskyrmion number and follows from the ir-relevance of monopole tunneling eventsonly at the critical point.

The absence of monopoles at the criticalpoint, when generalized to the isotropic case(18), provides one of the justifications for theclaimed critical theory in Eq. 6.

Discussion. Our results offer a new per-spective on the phases of Mott insulators in twodimensions: Liquid resonating–valence-bond-like states, with gapless spinon excitations, canappear at isolated critical points between phasescharacterized by conventional confining orders.It appears probable that similar considerationsapply to quantum critical points in doped Mottinsulators, between phases with a variety of spin-and charge-density-wave orders and d-wave su-perconductivity. If so, the electronic properties inthe quantum critical region of such critical pointswill be strongly non–Fermi-liquid-like, raisingthe prospect of understanding the phenomenol-ogy of the cuprate superconductors.

On the theoretical side, our results also

illuminate studies of frustrated quantum anti-ferromagnets in two dimensions. A theory ofthe observed critical point between the Neeland VBS phases (21) is now available, andprecise tests of the values of critical expo-nents should now be possible. A variety ofother SU(2)-invariant antiferromagnetshave been studied (22), and many of themexhibit VBS phases. It would be interestingto explore the characteristics of the quan-tum critical points adjacent to these phasesand test our prediction of deconfinement atsuch points.

Our results also caricature interesting phe-nomena (23, 24) in the vicinity of the onset ofmagnetism in the heavy fermion metals. Re-markably, the Kondo coherence that charac-terizes the nonmagnetic heavy Fermi liquidseems to disappear at the same point at whichmagnetic long-range order sets in. Further-more, strong deviations from Fermi liquidtheory are seen in the vicinity of the quantumcritical point. All of this is in contrast to naıveexpectations based on the LGW paradigm forcritical phenomena. However, this kind ofexotic quantum criticality between two con-ventional phases is precisely the physics dis-cussed in the present paper.

References and Notes1. S. Sachdev, Quantum Phase Transitions (Cambridge

Univ. Press, Cambridge, 1999).2. L. D. Landau, E. M. Lifshitz, E. M. Pitaevskii, Statistical

Physics (Butterworth-Heinemann, New York, 1999).3. K. G. Wilson, J. Kogut, Phys. Rep. 12, 75 (1974).4. The Landau paradigm is also known to fail near 1D

quantum critical points (or 2D classical criticalpoints), such as the model considered by Haldane(25). This failure is caused by strong fluctuations in alow-dimensional system, a mechanism that does notgeneralize to higher dimensions.

5. R. B. Laughlin, Adv. Phys. 47, 943 (1998).6. C. Lannert, M. P. A. Fisher, T. Senthil, Phys. Rev. B 63,

134510 (2001).7. T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, M. P. A.

Fisher, http://xxx.lanl.gov/abs/cond-mat/0312617.8. A. Vishwanath, L. Balents, T. Senthil, http://xxx.lanl.gov/

abs/cond-mat/0311085.9. D. S. Rokhsar, S. A. Kivelson, Phys. Rev. Lett. 61, 2376

(1988).10. R. Moessner, S. L. Sondhi, E. Fradkin, Phys. Rev. B 65,

024504 (2002).11. F. D. M. Haldane, Phys. Rev. Lett. 61, 1029 (1988).12. S. Chakravarty, B. I. Halperin, D. R. Nelson, Phys. Rev. B

39, 2344 (1989). These authors correctly noted that theBerry phases could at least be neglected in the Neelphase, but perhaps not beyond it; their critical theoryapplies to square lattice models with the spin S an eveninteger (but not to S half-odd-integer) and to dimerizedor double-layer antiferromagnets with an even numberof S � 1/2 spins per unit cell [such as the model in (13)].

13. M. Troyer, M. Imada, K. Ueda, J. Phys. Soc. Jpn. 66,2957 (1997).

14. N. Read, S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).15. A. V. Chubukov, S. Sachdev, J. Ye, Phys. Rev. B 49,

11919 (1994). Speculations on the dangerous irrele-vancy of Berry phase effects appeared here.

16. Halperin et al. (26 ) studied the critical theory of Lzusing expansions in 4 D (D is the dimension ofspacetime) and in 1/N. The former yielded a first-order transition and the latter second-order transi-tion. Subsequent duality and numerical studies (17,20) have shown the transition is second-order in D �3 for N � 1,2.

17. O. Motrunich, A. Vishwanath, http://xxx.lanl.gov/abs/cond-mat/0311222.

Fig. 4. Pattern ofsymmetry-breaking inthe two possible VBSstates (A and B) predict-ed by Eq. 8. The lastterm in Eq. 8 leads to apotential, �cos[4(�1 �2)], and the sign of �chooses between thetwo states above. Thedistinct lines representdistinct values of � �Sr � �Sr’�on each link. Note thatthe state in (A) isidentical to that inFig. 1B.

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18. Scaling analyses, generalizations, and physical conse-quences appear in the supporting material on ScienceOnline.

19. S. Sachdev, K. Park, Ann. Phys. N.Y. 298, 58 (2002).20. C. Dasgupta, B. I. Halperin, Phys. Rev. Lett. 47, 1556

(1981).21. A. W. Sandvik, S. Daul, R. R. P. Singh, D. J. Scalapino,

Phys. Rev. Lett. 89, 247201 (2002).22. C. Lhuillier, G. Misguich, http://xxx.lanl.gov/abs/cond-

mat/0109146.23. P. Coleman, C. Pepin, Q. Si, R. Ramazashvili, J. Phys.

Condens. Matt. 13, 723 (2001).

24. Q. Si, S. Rabello, K. Ingersent, J. L. Smith, Nature 413,804 (2001).

25. F. D. M. Haldane, Phys. Rev. B 25, 4925 (1982).26. B. I. Halperin, T. C. Lubensky, S.-k. Ma, Phys. Rev. Lett.

32, 292 (1974).27. This research was generously supported by NSF

under grants DMR-0213282, DMR-0308945 (T.S.),DMR-9985255 (L.B.), DMR-0098226 (S.S.), andDMR-0210790 and PHY-9907949 (M.P.A.F.). Wealso acknowledges funding from the NEC Corpora-tion (T.S.), the Packard Foundation (L.B.), the Al-fred P. Sloan Foundation (T.S. and L.B.), a Pappa-

lardo Fellowship (A.V.), and an award from TheResearch Corporation (T.S.). We thank the AspenCenter for Physics for hospitality.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/303/5663/1490/DC1SOM TextReferences and Notes

23 September 2003; accepted 7 January 2004

REPORTSTerahertz Magnetic Response

from Artificial MaterialsT. J. Yen,1* W. J. Padilla,2* N. Fang,1* D. C. Vier,2 D. R. Smith,2

J. B. Pendry,3 D. N. Basov,2 X. Zhang1†

We show that magnetic response at terahertz frequencies can be achieved ina planar structure composed of nonmagnetic conductive resonant elements.The effect is realized over a large bandwidth and can be tuned throughout theterahertz frequency regime by scaling the dimensions of the structure. Wesuggest that artificial magnetic structures, or hybrid structures that combinenatural and artificial magneticmaterials, can play a key role in terahertz devices.

The range of electromagnetic material responsefound in nature represents only a small subsetof that which is theoretically possible. Thislimited range can be extended by the use ofartificially structured materials, or metamateri-als, that exhibit electromagnetic properties notavailable in naturally occurring materials. Forexample, artificial electric response has beenintroduced in metallic wire grids or cell meshes,with the spacing on the order of wavelength (1);a diversity of these meshes are now used inTHz optical systems (2). More recently, meta-materials with subwavelength scattering ele-ments have shown negative refraction atmicrowave frequencies (3), for which both theelectric permittivity and the magnetic perme-ability are simultaneously negative. The negative-index metamaterial relied on an earlier theoret-ical prediction that an array of nonmagneticconductive elements could exhibit a strong, res-onant response to the magnetic component ofan electromagnetic field (4). In the present work,we show that an inherently nonmagnetic metama-

terial can exhibit magnetic response at THz fre-quencies, thus increasing the possible range inwhich magnetic and negative-index materials canbe realized by roughly two orders of magnitude.

Conventional materials that exhibit magneticresponse are far less common in nature than ma-terials that exhibit electric response, and they areparticularly rare at THz and optical frequencies.The reason for this imbalance is fundamental inorigin: Magnetic polarization in materials followsindirectly either from the flow of orbital currentsor from unpaired electron spins. In magnetic sys-tems, resonant phenomena, analogous to the pho-nons or collective modes that lead to an enhancedelectric response at infrared or higher frequencies,tend to occur at far lower frequencies, resulting inrelatively little magnetic material response at THzand higher frequencies.

Magnetic response of materials at THz andoptical frequencies is particularly important forthe implementation of devices such as compactcavities, adaptive lenses, tunable mirrors, isola-tors, and converters. A few natural magneticmaterials that respond above microwave fre-quencies have been reported. For example,certain ferromagnetic and antiferromagneticsystems exhibit a magnetic response over afrequency range of several hundred gigahertz(5–7) and even higher (8, 9). However, themagnetic effects in these materials are typicallyweak and often exhibit narrow bands (10),which limits the scope of possible THz devices.The realization of magnetism at THz and higher

frequencies will substantially affect THz opticsand their applications (11).

From a classical perspective, we can view amagnetic moment as being generated by micro-scopic currents that flow in a circular path. Suchsolenoidal currents can be induced, for exam-ple, by a time-varying magnetic field. Althoughthis magnetic response is typically weak, theintroduction of a resonance into the effectivecircuit about which the current flows can mark-edly enhance the response. Resonant solenoidalcircuits have been proposed as the basis forartificially structured magnetic materials (12),although they are primarily envisaged for lowerradio-frequency applications. With recent ad-vances in metamaterials, it has become increas-ingly feasible to design and construct systems atmicrowave frequencies with desired magneticand/or electric properties (3, 13, 14). In partic-ular, metamaterials promise to extend magneticphenomena because they can be designed to

1Department of Mechanical and Aerospace Engineer-ing, University of California at Los Angeles, 420 West-wood Plaza, Los Angeles, CA 90095, USA. 2Depart-ment of Physics, University of California, San Diego,9500 Gilman Drive, La Jolla, CA 92093–0319, USA.3Condensed Matter Theory Group, The Blackett Lab-oratory, Imperial College, London SW7 2AZ, UK.

*These authors contributed equally to this work.†To whom correspondence should be addressed. E-mail: xiang@seas.ucla.edu

Fig. 1. Illustration depicting the orientation of the30° ellipsometry experiment. The polarizationshown is S-polarization, or transverse electric, forexcitation of the magnetic response. P-polarization,transverse magnetic, was also measured. (Inset) Asecondary ion image of sample D1, taken by fo-cused ion-beam microscopy. For each SRR, the splitgap is 2 �m. The corresponding gap between theinner and outer ring (G), thewidth of themetal lines(W), the length of the outer ring (L), and the latticeparameter were 2, 4, 26, and 36 �m for sample D1;3, 4, 32, and 44�m for sample D2, and 3, 6, 36, and50 �m for sample D3, respectively. These valueswere used in both design and simulation. K, thewave vector of incident light; H, the magnetic fieldintensity; –E, the electrical field intensity.

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