Family of Exactly Solvable Models with an Ultimate Quantum Paramagnetic Ground State

Kai Phillip Schmidt and Mukul Laad

Lehrstuhl fur Theoretische Physik I, Otto-Hahn-Straße 4, TU Dortmund, D-44221 Dortmund, Germany(Received 10 December 2009; revised manuscript received 5 May 2010; published 7 June 2010)

We present a family of two-dimensional frustrated quantum magnets solely based on pure nearest-

neighbor Heisenberg interactions which can be solved quasiexactly. All lattices are constructed in terms of

frustrated quantum cages containing a chiral degree of freedom protected by frustration. The ground states

of these models are dubbed ultimate quantum paramagnets and exhibit an extensive entropy at zero

temperature. We discuss the unusual and extensively degenerate excitations in such phases. Implications

for thermodynamic properties as well as for decoherence free quantum computation are discussed.

DOI: 10.1103/PhysRevLett.104.237201 PACS numbers: 75.10.Jm, 03.67.�a, 05.30.Pr

The search for exotic phases of strongly correlatedquantum matter possessing unusual physical properties isa fascinating and active research area. Exactly solvablemodels played an essential role toward this end. Im-portant examples like the Heisenberg chain or the frus-trated Majumdar-Ghosh point with free or gapped spinonexcitations exist in one dimension [1–3]. In two dimen-sions, the frustrated Shastry-Sutherland model constitutesthe paradigm example for a valence bond solid [4]. Itsexperimental realization in SrCu2ðBO3Þ2 has spurred in-tense activity. Finally, the exactly solvable toric code [5,6]and Kitaev’s honeycomb model [7] have become standardmodels rigorously exhibiting topologically ordered phases.These have received enormous attention in the context oftopological quantum computation.

As already indicated above, one driving knob to tunequantum fluctuations is geometrical frustration. Indeed, themost promising candidates for experimental realizations ofa quantum spin liquid (SL) are strongly frustrated systemslike the so-called herbertsmithite [8,9] or the undoped par-ent compound �-ðBEDT-TTFÞ2Cu2ðCNÞ3 of the organicsuperconductors [10]. In herbertsmithite, the geometricalfrustration in the Heisenberg model on the kagome latticeis proposed to stabilize a SL state [11], while multispininteractions are relevant for the organic compound [12].

Theoretically, different quantum SL possessing verydifferent physical properties have been proposed. GappedSL have been exactly demonstrated in the quantum dimermodel on nonbipartite lattices [13,14] and in the toric code[5,6]. Further interesting examples are the U(1) critical SLdiscussed in the framework of the kagome model [11] andBose-metal phases discussed for triangular topologies withmultispin interactions [15]. Finally, there are intriguingproposals for so-called ultimate cooperative paramagnetsin the context of the transverse field Ising model on thekagome lattice [16]. In all these cases, no rigorous ex-amples of such exotic phases in two dimensions are known.Here, we present a family of quasiexactly solvable micro-scopic models having quantum disordered, extensivelydegenerate ground states and ultrashort-ranged spin corre-

lations. We dub this distinct class of SL ultimate quantumparamagnets (UQP).Our models might well be realized in experiment and,

additionally, could be relevant for quantum information(QI). First, the models consist solely of pure nearest-neighbor spin-1=2 Heisenberg antiferromagnets; i.e.,

coupled sites i and j interact via ~Si � ~Sj. This kind of

exchange is realized in many physical systems. In the QIcontext, we are especially motivated by coupled quantum-dot systems [17–19], possible patterning studies of quan-tum cellular automata (QCA) [20], trapped ions [21,22],and by using triangular magnetic clusters like Cu3 [23,24].Suitably designed Josephson junction arrays (JJA) are alsoof interest here, since the JJA can be mapped onto S ¼ 1=2Heisenberg-like models [25]. We draw attention to the factthat the core element in QCA computation is a bistable‘‘cell’’ capable of interacting with its neighbors, which isprecisely the strategy used here. Indeed, protected chiralquantum bits as also engineered in our case in a scalablefashion have been the focus of recent works in QI [23,24].Let us first introduce the construction recipe for the

lattices we study in this work. This is crucial since allinteresting physical properties emerge from the specialfrustrated topology [26]. The elementary building blockis a finite chain of Nt coupled triangles with periodicboundary conditions, known as the sawtooth chain [seeFig. 1(a)] [27]. Clearly, the ground state of this model is

J 1 2 Nt

a) b)

b)J

J

FIG. 1 (color online). (a) Heisenberg model with exchange Jon a periodically coupled chain of Nt triangles. (b) Illustration ofthe two degenerate ground states. Thick (green) bonds representsinglet states. (c) The shortest segment of a Shastry-Sutherlandchain consists of two vertical and one horizontal dimer. Note thatthe two dimers on the edges are part of the quantum cage.

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twofold degenerate for Nt > 2. These states are singletcoverings which are either oriented to the right or the left[see Fig. 1(b)]. Thus, this two-level system represents achiral pseudospin.

Next, we put this chain segment on a ring as shown inFig. 2 for Nt 2 f3; 4; 6g. Now we add Nt dimers on theboundary to form cagelike structures. Since the boundarydimers are coupled to the inner triangles in a Shastry-Sutherland fashion, singlets are formed on the outer dimersin the ground state of the full cage and the twofold chiraldegeneracy remains protected. Hence, we dub this unit achiral quantum cage.

The final step is to couple such cages. We introduce oneadditional dimer between two cages such that the intercagecoupling involves the shortest segment of a one-dimensional Shastry-Sutherland chain [see Fig. 1(c)] asillustrated in the right-hand side of Fig. 2. The boundarydimers are coupled to the connector with strength J0 ¼ xJ.In this way, one can build any two-dimensional lattice ofcoupled cages. Bravais lattices can be built only for Nt 2f3; 4; 6g. We are only aware of two works dealing withHubbard [28] and Heisenberg [29] models on decoratedlattices in the context of unconventional order, but gappedSL ground states have not yet been considered.

We want to study the ground states and elementaryexcitations as a function of x ¼ J0=J. Our focus will bethe regime x < xc, where the ground states of these modelscan be found exactly. To see this, notice that the full latticecan be covered by singlet configurations on each cage. Theintercage coupling involves the basic Shastry-Sutherlandunit; i.e., two vertical singlets from neighboring cages arelinked by one horizontal singlet. Since, as a result, the

chiral degrees of freedom on neighboring cages are effec-tively decoupled, the exact ground states for x < xc are adirect product of the dimer coverings on each of the Nc

cages. In contrast to the dimer solid in the Shastry-Sutherland model [4], the crucial point here is that thisnumber is extensively large. The ground state degeneracyscales like 2Nc , leading to a finite entropy lnð2Þ per cage atT ¼ 0 in the thermodynamic limit. As x increases, a phasetransition at x ¼ xc will certainly occur (see below).Let us first discuss the elementary excitations in the

UQP ground state for x < xc. They can also be determinedquasiexactly. One finds three sorts of massive particleswith total spin one above the infinitely degenerate groundstate: (i) excitations on the connector, (ii) excitations on theboundary dimers, and (iii) excitations inside the cage.Excitations on the connector are localized triplets. This

is a direct consequence of the peculiar Shastry-Sutherlandgeometry in one dimension which suppresses any fluctua-tions to neighboring dimers. The connecting dimer canonly be in a singlet or a triplet state, both eigenstates ofthe full problem. The total Hilbert space separates intodifferent blocks belonging to a different set of singlet-triplet configurations. A single triplet costs the exact en-ergy J independent of x. Clearly, the true low-energyexcitations of the model belong to the sector with singletson all connecting dimers.The second sort of excitations are on dimers on the

boundary of a cage. These excitations also stay local. Buttriplets are not exact eigenstates, since they can virtuallycreate and annihilate triplets on the inner star as well as onthe attached connector, thereby reducing its energy. Theseexcitations are local triplons [30] and constitute the exci-tations with lowest energy. This mode is displayed in Fig. 3for Nt ¼ 3. One finds a gap 0.23 J at x ¼ 0. The gap islowered as x increases and the mode crosses the low-energy chiral manifold at x � 0:7. This signals a phasetransition as long as no other levels cross before. Below, weargue this to be the case here.Finally, one has the excitations inside the star of a cage.

The lowest modes correspond to very weakly dispersive,massive triplons. Let us note that these excitations do notdepend on x and can therefore be identified as the horizon-tal gaps in Fig. 3 for Nt ¼ 3. Studies on the infinite chainNt ¼ 1 found completely local excitations with an energygap 0.22 J [31]. For finite Nt the energy gap is slightlylarger. One finds 0.32 J for Nt ¼ 3. The weakly dispersingcharacter is nicely understood from the dimerized limit ofthe inner star of a cage by introducing a modulation J � �on the spikes of the triangles (see Fig. 1). In the limit � ¼ 1the ground state is the product state of singlets and excita-tions are local triplets. Turning on a finite �, a first hoppingprocesses of the triplet occurs in perturbation order

�2ðNt�1Þ. One first excites all triplets of the star one afterthe other (Nt � 1 operations) and subsequently annihilatesNt � 1 triplets such that the triplet has effectively hopped.Thus, one directly understands that the excitations becomeincreasingly local with increasing Nt.

a)

b)

c)

FIG. 2 (color online). Three representative lattice models with(a) Nt ¼ 3, (b) Nt ¼ 4, and (c) Nt ¼ 6. The left-hand sideillustrates the chiral quantum cage of each lattice which containsa triangle chain shielded by surrounding dimers. The right-handside represents 2D lattices made by adding dimers which couplecages. The dimer connecting two cages forms the shortest seg-ment of a Shastry-Sutherland chain (illustrated in black).

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A fourth exactly known class of excitations is the com-bined creation of a triplon on the outer shell plus excita-tions of the star. These modes certainly depend on x due tothe presence of the local triplon. Interestingly, a strongattraction is found for certain modes which leads to thepresence of levels at rather low energy (see Fig. 3).

In summary, we have seen that ground states and allrelevant low-energy excitations for x < xc can be foundquasiexactly. Excitations on the outer shell of the cageshave the lowest energy. We stress that these excitations arelocalized on the quantum cage. This implies an extensivedegeneracy for these states as well. Since ground statecorrelations are strictly zero for distances larger than theextension of a quantum cage, we have found a family ofexactly solvable models with UQP ground states.

This can actually be extended in two respects. First, onecan change the stars to different geometries also possessinga twofold degenerate ground state, e.g., a fully frustratedplaquette with the minimal number of four spins. If onecouples the fully frustrated plaquettes in a completelyanalogous fashion as before, one again finds exactly anUQP ground state for x < xc [see Fig. 4(a)].

A second route is to change the nature of the connector.Indeed, a direct coupling xJ of stars by vertical dimers alsoprotects the UQP ground state for x not too large. A graph-ical illustration for the case Nt ¼ 3 is shown in Fig. 4(b).Again, excitations inside the star and the local triploncentered on the connecting dimer will be exact excitationsof the full problem.

As x increases, a phase transition will certainly occur.Details depend on the nature of the connector. For the caseof a direct star coupling just discussed, this will likely bethe triplon mode on the connector. If true, a magnetically

ordered triplon condensate will be stabilized for large x.This triplon mode can be exactly captured by studying acluster of two neighboring stars coupled by a single dimer.We find a critical value xc � 0:75 for Nt ¼ 3.In contrast, for the connector composed of the 1D seg-

ment of a Shastry-Sutherland chain discussed first, a differ-ent scenario is likely. One knows for the Shastry-Sutherland model that triplons strongly attract in the sectorwith S ¼ 0 when placed on neighboring vertical dimers,meaning the two ends of a connector in our case [32]. Wefind strong evidence for this scenario for coupled fullyfrustrated plaquettes. Here, one observes on a cluster oftwo plaquettes and one connector that the UQP becomesunstable for xc � 0:85. Instead of a single triplon, it isagain a singlet state keeping the ground state degeneracy ofthe UQP phase. The two triplons on neighboring cagesbind into a singlet bound state.We stress that the above considerations do not fix the

nature of the phase transition. It might well be that thetransition for both cases turns out to be first order as it isknown for the one-dimensional Shastry-Sutherland chain[33]. If so, the phase transition point would tend slightly tolower values of x.Next, wewant to discuss a connection of the caseNt ¼ 6

with Shastry-Sutherland connector to the strongly debatedHeisenberg model on the kagome lattice. There are twopossible close packings of chiral quantum cages on thekagome topology [see Fig. 4(c)]. The first packing is suchthat honeycombs remain between the cages. Assuming aspontaneous dimerization, one recovers the valence bondsolid consistent with the 36-site unit cell proposed to belinked to the true ground state of the Heisenberg model onthe kagome lattice [34,35]. The second packing is denser

FIG. 3 (color online). Localized excitations on a chiral quan-tum cage with Nt ¼ 3 as a function of x. Thick gray line atenergy J corresponds to the exact triplet on the connectingdimer. Blue horizontal lines represent excitations inside thestar. Dashed lines involve localized triplons on the outer shellwhich are the only single modes depending on x. The lowest ofthese modes is the energy of a single triplon. Other dashed linescorrespond to a triplon plus an excitation inside the star. Thenumber in square brackets gives the degeneracy on the clustershown in the inset which has been diagonalized.

a) b)

c)

FIG. 4 (color online). (a) The fully frustrated plaquette is thesmallest inner star. (b) A direct coupling of stars (Nt ¼ 3 shown)leads to a different family of models. (c) Illustration of the twopossible close packings of Nt ¼ 6 quantum cages on the kagomelattice. Left-hand figure represents the proposed valence bondsolid ground state with the 36-site unit cell. Right-hand figurerepresents a diluted kagome lattice where sites marked by graysquares are removed.

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but leaves dangling spins, which is clearly unpreferable.But removing these spins yields a depleted kagome lattice.Quite remarkably, one can show that the depleted modelhas valence bond solid (VBS) order with a 24-site unit cell.The easiest way to see this is by identifying the chiralpseudospin states as exact eigenstates of a single cage. In2nd order intercage degenerate perturbation theory, theeffective pseudospin model turns out to be precisely anIsing ferromagnet in a transverse ‘‘magnetic’’ field (seealso Ref. [36]): Heff ¼ �j

P�;�T

z�T

z� � h

P�T

x�, with

j; h � J. The ‘‘field’’ stabilizes a VBS state with ‘‘ferro-chiral’’ order. The elementary excitations are thereforeweakly dispersive ‘‘pseudospin’’ flips involving coopera-tive flipping of the six dimers on the internal star. Thissuccinctly illustrates the crucial influence of the unusualground states found here on geometry, with disorderedvalence bond liquid, valence bond solid, and magneticallyordered phases arising for differing geometries.

Let us finally discuss the specific heat. For x < xc, thedecoupled (disordered) phase is characterized by a specificheat entirely due to the fluctuating chiral doublet, givingCðTÞ ’ T�2, reflecting the finite T ¼ 0 entropy per site. Inthe case where magnetic order arises for x > xc, quenchingof this entropy will yield CðTÞ ’ T2. For the depletedkagome magnet, two weakly dispersive bands of singlets,

defined by E�k ¼ �ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2k þ �2

k

q� hÞ with �k ¼ 2j½cosky þ

2 cosð ffiffiffi3

pkx=2Þ cosðky=2Þ� and �k ¼ ð2=3Þj sinðky=2Þ�

½cosð ffiffiffi3

pkx=2Þ � cosðky=2Þ� for h � j, give rise to phonon-

like excitations with velocity v ’ j, and, in two dimen-sions, this will yield CðTÞ ’ expð�h=tÞ for T ! 0 andCðTÞ ’ T2 for higher temperatures.

In conclusion, we have presented families of frustratedquantum magnets with hitherto undiscovered gapped SLground states. The ground state as well as elementaryexcitations can be determined exactly, and both possess amacroscopic degeneracy at T ¼ 0. One interesting ques-tion is the effect of finite temperatures and finite magneticfields because both will induce a finite density of excita-tions on the lattice leading to an effective intercage cou-pling. An extensive entropy at T ¼ 0 is also known forcertain frustrated quantum magnets in high magnetic fieldsdue to exactly localized magnon states [37]. These kinds ofextensive localized magnon states also occur in our modelsin high fields, giving rise to an enhanced magnetocaloriceffect. The effects of thermal fluctuations are expected tobe strongest close to xc where energy gaps are small,resulting in sizable intercage correlations resembling thephase for x > xc at T ¼ 0.

For experimental realizations, we only demand flexibil-ity on the lattice design since our models solely containnearest-neighbor Heisenberg couplings. We suggest thatexperimental realizations of QCA in suitable geometries,artificially fabricated JJA, or trapped ions may exhibitsome of the unusual features found in this work. Arraysof magnetic clusters, e.g., of Cu3 triangles suitably engi-

neered on a substrate, may also reveal parts of the exoticaproposed here. In particular, given the protected chirality,an exciting option for future study is to use this aspect in asetting favorable for quantum computation in ‘‘interactionfree subspaces’’ [38] to minimize the troublesome deco-herence problem in quantum information.K. P. S. acknowledges ESF and EuroHorcs for funding

through his EURYI. We thank F. Mila, J. Stolze, and G. S.Uhrig for fruitful discussions.

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