Effective Quantum Dimer Model for the Kagome Heisenberg Antiferromagnet: Nearby Quantum Critical Point and Hidden Degeneracy
EEffective Quantum Dimer Model for the Kagome Heisenberg Antiferromagnet: Nearby QuantumCritical Point and Hidden Degeneracy
Didier Poilblanc, Matthieu Mambrini, and David Schwandt Laboratoire de Physique Th´eorique, CNRS and Universit´e de Toulouse, F-31062 Toulouse, France (Dated: October 31, 2018)The low-energy singlet dynamics of the Quantum Heisenberg Antiferromagnet on the Kagome lattice isdescribed by a quantitative Quantum Dimer Model. Using advanced numerical tools, the latter is shown toexhibit Valence Bond Crystal order with a large 36-site unit cell and hidden degeneracy between even and oddparities. Evidences are given that this groundstate lies in the vicinity of a Z dimer liquid region separated by aQuantum Critical Point. Implications regarding numerical analysis and experiments are discussed. PACS numbers: 74.20.Mn, 67.80.kb, 75.10.Jm, 74.75.Dw, 74.20.Rp
The Kagome lattice, a two-dimensional corner-sharing ar-ray of triangles shown on Fig. 1(a), is believed to be one of themost frustrated lattices leading to finite entropy in the ground-state (GS) of the classical Heisenberg model [1]. Hence, theQuantum S=1/2 Heisenberg Antiferromagnet (QHAF) on theKagome lattice is often considered as the paradigm of quan-tum frustrated magnetism [2] where, in contrast to conven-tional broken symmetry phases of spin systems (as e.g. mag-netic phases), exotic quantum liquids or crystals could be re-alized. Among the latter, the algebraic (gapless) spin liquid isone of the most intriguing candidate [3].The herbertsmithite [4] compound is one of the few ex-perimental realizations of the S=1/2 Kagome QHAF. The ab-sence of magnetic ordering [5] down to temperatures muchsmaller than the typical energy scale of the exchange coupling J suggests that, indeed, intrinsic properties of the KagomeQHAF can be observed, even-though Nuclear Magnetic Res-onance and Electron Spin Resonance reveal a small fraction ofnon-magnetic impurities [6] and small Dzyaloshinsky-Moriyaanisotropy [7]. So far, the nature of the non-magnetic phase isunknown and confrontation to new theoretical ideas have be-come necessary. Alternatively, ultra-cold atoms loaded on anoptical lattice with tunable interactions might enable to alsoexplore the physics of extended Kagome QHAF [8].The QHAF on the Kagome lattice has been addressed the-oretically by Lanczos Exact Diagonalization (LED) of smallclusters [9, 10]. Despite the fact that the accessible clustersizes remain very small, these data are consistent with a fi-nite spin gap and an exponential number of singlets within thegap, in agreement with a recent Density Matrix Renormaliza-tion Group study [11]. In addition, an analysis of the four-spin correlations pointed towards a short-range dimer liquidphase [12]. Alternatively, a large-N approach [13] and var-ious mappings to low-energy effective Hamiltonians withinthe singlet subspace [14–16] have suggested the formation oftranslation-symmetry breaking Valence Bond Crystals (VBC).Recently, recent series expansions around the dimer limit [17]showed that a 36-site VBC unit cell is preferred (see Fig 1(a)).In this context, the interpretation of the LED low-energy sin-glet spectrum remains problematic [18].In this Letter, we use a quantitative dimer-projected effec- EC BAD
108 sites48 sites (b)(a)
108 sites36 sites
FIG. 1: (color online) (a) Sketch of the 36-site VBC on the Kagomelattice showing “perfect hexagons” and (yellow) “star” resonances.Its unit cell is delimited by dashed (red) lines. The extension ofthe 108-site cluster (which fits exactly three 36-site cells) is (lightly)shaded and delimited by long-dashed (blue) lines. Hardcore dimers(formed by two neighboring spin- ) are located on the bonds (onlyshown on the shaded hexagons). (b) First Brillouin zone and avail-able momenta (labeling consistent with Ref. [18]). tive model to describe the low-energy singlet subspace of theKagome QHAF. This is based on a controlled loop expan-sion [19] along the lines initiated by Rohksar and Kivelson(RK) [20] in the context of High-Temperature Superconduc-tivity and by Zeng and Elser [21] (ZE) for the Kagome lat-tice. LED of the effective model enable to reach unprece-dented sizes of periodic clusters which can accommodate afinite number of candidate VBC unit cells. The major resultsare (i) the numerical evidence of VBC order with a large 36- a r X i v : . [ c ond - m a t . s t r- e l ] F e b site unit cell and a hidden degeneracy, (ii) the vicinity of a Z dimer liquid region separated by a Quantum Critical Point.Such results transposed to the original QHAF could explainsome of its former puzzling numerical findings. Model – The method consists in projecting the QHAF, H = J (cid:80) (cid:104) i,j (cid:105) S i · S j , i and j being nearest neighbor (NN)lattice sites, into the manifold formed by NN Valence Bond(VB) coverings, an approximation shown to be excellent forthe Kagome lattice [21–23]. A transformation is then per-formed that turns the non-orthogonal VB basis into an orthog-onal Quantum Dimer basis. Overlaps between NN VB statescan be written [24] (up to a sign) as α N − n l where N is thesystem size, n l the number of loops obtained by superimpos-ing the two configurations, and α = 1 / √ . This enables asystematic expansion in powers of α involving at order α p ( p ≥ ) loops of sizes up to L = 2 p + 2 [19, 25] leading to ageneralized Quantum Dimer Model (QDM) which, restrictingto loops encircling only single hexagons, readswhere a sum over all the hexagons of the lattice of Fig. 1(a)is implicit. Kinetic terms (1) promote cyclic permutations ofthe dimers around the loops and diagonal terms (2) count thenumbers of “flippable” loops. Here we use the approximatevalues (14-th order) , V = 0 . , V = 0 . , V = V = 0 , J = − . , J = 0 . , J = − . , J = 0 (in unitsof J ). Note that the kinetic processes J L are quite close toZE [21] initial lowest-order estimates, J = − / , J = 1 / and J = − / , a good sign of convergence of the expan-sion. Very importantly, we also include here the potential V L terms which appear in our expansion scheme only at order L − but play a major role. Note that an exact resumma-tion of the weights of (1), (2) up to all orders can be carriedout (leading to tiny deviations) and that the “star” amplitudes J and V vanish at all orders [19]. Symmetry analysis – Although the generalized QDM ()bears a sign problem, it can be addressed by LED of peri-odic × L × L clusters which possess all the infinite latticesymmetries and L unit cells. We shall consider L = 4 and L = 6 corresponding to the 48-site and 108-site clusters (seeFig. 1(a)), a tremendous improvement in terms of system sizecompared to the original QHAF, and use all available latticesymmetries (see available momenta in Fig. 1(b)) to block-diagonalize the Hamiltonian in its irreducible representations(IR) [26]. We further make use of a topological symmetrywhich splits the Hilbert space into 4 topological sectors (TS) { n , n , n } , where n α = 0 ( ) for an even (odd) number of dimers cut by each crystal axis α enclosing the torus [27]. VBC Γ A B C D E N = 12 (+ , + , ± ) ( × , + , ± ) N = 36 (+ , + , ± ) ( × , + , ± ) (+ , × , ± ) ( × , × , ± ) N = 48 (+ , + , ± ) ( × , + , ± ) ( × , × , ± ) ( × , × , ± ) Z (+ , + , +) TABLE I: Quantum numbers of the GS multiplet for VBC with N -site cells ( N = 3 × m × m or N = 3 × n √ × n √ ) and degen-eracy N / . The letters refer to the momenta of Fig. 1. Invarianceunder π/ rotations ( r = + ) and parity under inversion ( r = ± )and/or reflection about the momentum direction ( σ ) are denoted as ( r , r , σ ) consistently with Ref. [18]. σ = ± corresponds to evenor odd GS and “ × ” means “symmetry not relevant”. The Z dimerliquid (last line) GS in TS ∗ is degenerate with the other TS GS [27]. An analysis of the low-energy spectrum and a careful in-spection of its quantum numbers provide invaluable informa-tions on the nature of the GS. For a VBC breaking discrete lat-tice symmetries, a multiplet structure is expected giving risein the thermodynamic limit to a degenerate GS separated by agap from the rest of the spectrum. We list in Table I the quan-tum numbers of the GS multiplet for the most popular VBCin the literature. λ . . . . ( u n i t s o f J ) E ( n ) N − E G S N (b) N =108 QCP × (c) δλ =10 − N =48 N =108 χ λ λ . . . . ( u n i t s o f J ) E ( n ) N − E G S N Z dimer liquid (a) N =48 TS* (+,+,+) GS (+, × ,+)B (+, × ,–)B ( × ,+,–)A 2-visonTS* (All IR)Other TS (+,+,–) 4-vison ( × ,+,+)A 2-vison FIG. 2: (color online) (a) Complete low-energy excitation spectrumof the 48-site cluster as a function of the parameter λ (see text). Thelowest 25 levels are displayed in each TS (see [27]) using differentsymbols/colors. The 2-vison (blue lines) and 4-vison (dashed blackline) gaps vanish around λ = 1 . (b) Close-up around λ = 1 showingthe excitation-energy of the lowest level of the most relevant IR (seeTable I) of the -site cluster: a QCP is identified from (i) a collapseof these excitations (b) and (ii) a very sharp peak in χ λ (c) (data for N = 108 divided by 200 to fit the scale). Note the transition is farmore abrupt for N = 108 than for N = 48 (c). Quantum Critical Point – The QDM with only kinetic pro-cesses of equal amplitudes, J = J = J = J (= 1 / provides an exactly solvable model [29] H RK with a gapped Z dimer liquid GS with short-range dimer correlations, sim-ilarly to the RK-point of the triangular QDM [30]. Interest-ingly, the first excitations (of energy J) correspond to pairsof (localized) topological vortices (visons). It is tempting toconstruct a simple interpolation H ( λ ) = λ H eff + (1 − λ ) H RK between this known limit and the effective model (1), (2). Itslow-energy spectrum on the -site and -site clusters areshown as a function of λ in Fig. 2(a) and Fig. 2(b), respec-tively. In the 48-site cluster, a high density of low-energy lev-els accumulate just above the GS at λ ∼ . This stronglysuggests the vicinity of a Quantum Critical Point (QCP) char-acterized by vortex (vison) condensation as in the triangularQDM [31]. A closer look around λ = 1 on the larger clus-ter reveals a sudden collapse of the low-energy excitations,clearly before λ = 1 . Remarkably, as shown in Fig. 2(c), thiscollapse coincides exactly with a very sharp peak of the sec-ond derivative of GS energy [28] χ λ = − ∂ ( E GS N /N ) /∂λ ,enabling to locate the QCP at λ QCP ∼ . . Note that theeven-parity ( σ = + ) multi-vison excitations at momenta K A and K B merge with the GS precisely at λ QCP (see below). -0.05 -0.025 0 0.025 0.0500.010.020.030.040.050.060.070.080.090.1 J ( u n i t s o f J ) E ( n ) N − E G S N (+,+,+) (+,+,–) ( × ,+,+) ( × ,+,–) ( × , × ,+) ( × , × ,–) ( × , × ,+) ( × , × ,–)AADDEETS*Other TS (VBC N = 48 even) VBC N = 48 odd VBC N = 12 even FIG. 3: (color online) Complete low-energy excitation spectrum ofthe 48-site cluster as a function of J . The lowest levels aredisplayed in each TS (same symbols as Fig. 2(a)). The quantumnumbers of the lowest energy states for J < and J > areprovided using different line types. VBC and hidden degeneracy – Next, to identify the orderedphase for λ > λ
QCP we add a finite star resonance amplitude J to H eff . As seen in Figs. 3 and 4, this term has a crucialrole. In fact, J = 0 is highly singular, with a (almost exact)degeneracy between odd ( σ = − ) and even ( σ = + ) states.Physically, we believe it is related to the hidden Ising variables(introduced in Ref. 17) associated to the resonance parities ofstars (see Fig. 1(a) for the 36-site VBC). Incidentally, it is re-markable that our effective model picks up such a feature via avanishing effective J . For finite but still very small J , this degeneracy is lifted (favoring a “ferromagnetic” Ising config-uration) allowing to characterize any candidate VBC (whichshould exist with both parity) from its lowest energy states.For the 48-site cluster, a close inspection of the associatedquantum numbers and a comparison with Table I suggest aVBC with a 48-site (12-site) unit cell for J < ( J > ).For J > slow fluctuations towards the pattern of a N = 48 VBC are signaled by the presence of intermediate mid-gapstates. Note that the lowest-energy excitations in the other TS(red lozenges) which barely depend on J set roughly the(very small) VBC gap scale. Also, the high-density of levelswithin a small energy window of ∼ . J above the GS isreminiscent of the singlet sector of the QHAF [18]. -0.03 -0.015 0 0.015 0.03 J . . . ( u n i t s o f J ) E ( n ) N − E G S N (b) VBC N = 36 evenVBC N = 36 odd N =48 N =48 N =108 N =108 -0.05 -0.025 0 0.025 0.05 J - . - . - . - . E G S N / N = (cid:30) S i . S j (cid:29) (a) l e v e l c r o ss i n g N =48 N =108 Other TS TS* (+,+,+) (+,+,–) ( × ,+,+) ( × ,+,–) (+, × ,+) (+, × ,–)AABB FIG. 4: (color online) (a) GS energies per site versus J for K = K Γ and σ = ± . The minimum defines the absolute GS energy E GSN /N . Data for sites (lines) and sites (circles) are shown.An energy shift and scale transformation E N → ( E N − N/ al-low a direct comparison with twice the bond-energy of the QHAF. (b)Same as Fig. 2(b) but as a function of J and compared to the N=48excitation at the Γ -point (full lines). Note the splittings between Γ ,A and B levels of a few − J are invisible at this scale. A similar analysis on the larger 108-site cluster [26] pro-vides definite evidence in favor of the 36-site (degenerate)VBC schematically depicted in Fig. 1(a), for λ > λ
QCP .First, Fig. 4(a) shows a kink at J = 0 of the GS energy per site , coinciding with the crossing of two GS of oppo-site parity and leading to almost the same slope discontinu-ity as for N = 48 (size effects are small). As for N = 48 ,this, in fact, corresponds to the level crossings of two groupsof quasi-degenerate GS with opposite parities as shown inFig. 4(b). We note an extremely fast decrease with N of thegaps between the A and Γ quasi-degenerate GS common toboth clusters (a few − J for N = 48 to less than − J for N = 108 ). Furthermore, the fact that a new state withmomentum K B (not allowed for N = 48 ) belongs to thesetwo groups of quasi-degenerate GS definitely points toward a N = 36 unit cell [32] (which does not fit N = 48 ). This isfurther supported by the values of the average number N L offlippable length- L loops compared in Table II to their valuesin VBC proposed in the literature. Groundstates N /N H N /N H N /N H N /N H N = 108 ; J = 0 . .
154 0 .
274 0 .
491 0 . N = 108 ; J = − . .
153 0 .
275 0 .
492 0 . × × [14] .
75 0 0 . × √ × √ [13, 15, 17] .
167 0 .
25 0 . . Z dimer liquid [29] /
32 15 /
32 15 /
32 1 / TABLE II: Average number N L of flippable length- L loops normal-ized to the total number N H of hexagons. The last 3 lines correspondto the ”frozen” limit of VBC and to the Z dimer-liquid. In summary, we introduced a generalized QDM to describethe low-energy physics of the QHAF on the Kagome lattice.In contrast to the latter, its groundstate properties can be ad-dressed by numerical simulations with unprecedented accu-racy for a frustrated quantum magnet. In particular, we pro-vide evidence of (i) a 36-site VBC order (with hidden degen-eracy), in agreement with recent series expansion [17], and(ii) the vicinity of a QCP towards a topological Z dimer-liquid (cf. schematic phase diagram on Fig. 3 of Ref. 19).Interestingly, a double Chern-Simons field-theory [33] alsodescribes such a Quantum Critical Point. The above re-markable features of the generalized QDM transposed to theQHAF would resolve mysteries of the (small cluster) QHAFspectrum such as low-energy singlets carrying unexpectedquantum numbers [18] and exceptional sensitivity to smallperturbations [34]. Experimentally, Kagome spin-1/2 sys-tems generically contain small amounts of lattice and/or spinanisotropies [7] or even longer-range exchange interactionsand the proximity to a QCP should render the experimentalsystems very sensitive to them. However, if under some con-ditions (pressure, chemical substitution,...) low-temperaturespin-induced VBC order establishes, it could be revealed viasmall lattice modulations mediated by some magneto-elasticcoupling. Lastly, we note that the above-mentioned perturba-tions as well as magnetic excitations [35] can be included inour scheme.We acknowledge support from the French National Re-search Agency (ANR) and IDRIS (Orsay, France). D.P.thanks G. Misguich and A. Ralko for discussions. [1] J.T. Chalker, P.C.W. Holdsworth, E.F. Shender, Phys. Rev. Lett. , 855 (1992).[2] For review see e.g. “Two-dimensional quantum antiferromag-nets” , G. Misguich & C. Lhuillier, in ”Frustrated spin systems”,Ed. H. T. Diep, World-Scientific (2005).[3] M. Hermele, Y. Ran, P. A. Lee, and X.-G. Wen, Phys. Rev. B77, 224413 (2008).[4] M. Shores, E. Nytko, B. Bartlett, and D. Nocera, J. Am. Chem.Soc. , 13462 (2005). [5] P. Mendels et al., Phys. Rev. Lett. , 077204 (2007).[6] A. Olariu et al., Phys. Rev. Lett. , 087202 (2008).[7] A. Zorko et al., Phys. Rev. Lett. , 026405 (2008).[8] J. Ruostekoski, Phys. Rev. Lett. , 080406 (2009).[9] P. Lecheminant et al., Phys. Rev. B , 2521 (1997).[10] P. Sindzingre et al., Phys. Rev. Lett. , 2953 (1999).[11] H. C. Jiang, Z. Y. Weng, and D. N. Sheng, Phys. Rev. Lett. ,117203 (2009).[12] P.W. Leung and Veit Elser, Phys. Rev. B , 5459 (1993).[13] J. B. Marston and C. Zeng, J. Appl. Phys. , 5962 (1991);SU(N) VBC have been introduced by N. Read and S. Sachdev,Phys. Rev. B , 4568 (1990).[14] A. V. Syromyatnikov and S. V. Maleyev, Phys. Rev. B ,132408 (2002).[15] P. Nikolic and T. Senthil, Phys. Rev. B , 214415 (2003).[16] Ran Budnik and Assa Auerbach, Phys. Rev. Lett. , 187205(2004).[17] R.R.P. Singh and D.A. Huse, Phys. Rev. B , 180407 (2007).[18] G. Misguich and P. Sindzingre, J. Phys. Cond. Matt. , 145202(2007).[19] D. Schwandt, M. Mambrini and D. Poilblanc, arXiv:1002.0774.The weights of higher-order resonances (involving 2 or morehexagons) decay rapidly with L .[20] D.S. Rohksar and S.A. Kivelson, Phys. Rev. Lett. , 2376(1988).[21] Chen Zeng and Veit Elser, Phys. Rev. B , 8318 (1995).[22] M. Mambrini and F. Mila, Eur. Phys. J. B , 651 (2000).[23] For the frustrated QHAF on the square lattice see M. Mambrini,A. L¨auchli, D. Poilblanc, and F. Mila, Phys. Rev. B 74, 144422(2006).[24] B. Sutherland, Phys. Rev. B , 3786 (1988); N. Read andB. Chakraborty, Phys. Rev. B , 7133 (1989).[25] For the same procedure on the frustrated QHAF on the squarelattice see A. Ralko, M. Mambrini and D. Poilblanc, Phys. Rev.B , 184427 (2009).[26] E.g. the IR of momenta K Γ ( r = r = + ), K B ( r =+ ) and K A ( r = + ) in TS ∗ [27] of the 108-site cluster( σ = ± ) contain
79 548 096 ,
159 073 536 and
238 642 176 fully symmetrized states, respectively. Other momenta (withlarger Hilbert spaces) have not been considered. However, apartfrom K C , they should not be involved in the GS manifold (seeTable I).[27] Studied dimer liquids and VBC belong to the { , , } [ { , , } ] TS of the 48-site [108-site] cluster, named as TS ∗ .“Other TS” refers to the 3 other (degenerate) TS.[28] As the fidelity, this quantity is a useful tool to identify quantumphase transitions. See S. Chen, L. Wang, Y. Hao and Y. Wang,Phys. Rev. A , 032111 (2008).[29] G. Misguich, D. Serban, and V. Pasquier, Phys. Rev. Lett. ,137202 (2002).[30] R. Moessner and S. L. Sondhi, Phys. Rev. Lett. , 1881 (2001).[31] A. Ralko et al., Phys. Rev. B 76, 140404 (2007).[32] Reversely, the N = 48 cell does not fit in the 108-site cluster.The × √ × √ cluster ( sites) which can accommodate both N = 36 and N = 48 cells is out of reach.[33] C. Xu and S. Sachdev, Phys. Rev. B , 064405 (2009).[34] P. Sindzingre and C. Lhuillier, Eur. Phys. Lett. , 27009(2009).[35] A. Ralko, F. Becca and D. Poilblanc, Phys. Rev. Lett.101