What is Minimal Model of 3He Adsorbed on Graphite? -Importance of Density Fluctuations in 4/7 Registered Solid -
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What is Minimal Model of He Adsorbed on Graphite?–Importance of Density Fluctuations in 4/7 Registered Solid–
Shinji
Watanabe and Masatoshi
Imada
Department of Applied Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-8656,Japan
We show theoretically that the second layer of He adsorbed on graphite and solidifiedat 4/7 of the first-layer density is close to the fluid-solid boundary with substantial densityfluctuations on the third layer. The solid shows a translational symmetry breaking as incharge-ordered insulators of electronic systems. We construct a minimal model beyond themultiple-exchange Heisenberg model. An unexpectedly large magnetic field required for themeasured saturation of magnetization is well explained by the density fluctuations. Theemergence of quantum spin liquid is understood from the same mechanism as in the Hubbardmodel and in κ -(ET) Cu (CN) near the Mott transitions. KEYWORDS: quantum spin liquid, He, 4/7 phase, saturation field, Mott insulator, chargeorder He layers adsorbed on graphite substrate is a unique two-dimensional correlated Fermionsystem and have continuously offered fundamental issues in condensed matter. In particular,adsorption of He to the 2nd layer under the corrugation potential of the 1st-layer solidshows a variety of phenomena ranging from a correlated Fermi liquid to a solidification at thecommensurate density of 4/7 relative to the 1st layer. The solid phase bahaves as a quantumspin liquid (QSL),
2, 3 the nature of which is a long-standing theoretical challenge. This solidified He monolayer has widely been studied by the Heisenberg model with mul-tiple spin exchange (MSE).
5, 6
However, exact diagonalization studies on realistic MSE modelssuggest an opening of spin excitation gap in contrast to the gapless nature of QSL revealedby specific heat
2, 8 and magnetic susceptibility measured down to 10 µ K.
3, 6
Furthermore, theMSE model predicts that the magnetization m saturates above the field h sat ∼
7, 9 whereas a recent experiment up to 10 Tesla indicates the saturation at much higher h sat .A gapless QSL was reported in numerical studies as the ground state of the two-dimensional Hubbard model with geometrical frustration effect near the Mott transition supplemented by a report showing numerically the absence of various symmetry breakings. Itsupports the realization of a genuine Mott insulating state without any translational symmetrybreaking as initially proposed by Anderson. This series of studies is indeed relevant and pro-vides a realistic model for a subsequently discovered gapless spin liquid in κ -(ET) Cu (CN) .Although a charge gap exists in Mott insulating states, density fluctuations allowing doubly . Phys. Soc. Jpn. Letter occupied sites in the Hubbard model near the Mott transition is crucial for the stabilizationof the QSL.However, when we consider the hard core of the interatomic interaction between Heatoms, the Hubbard model with a moderate onsite interaction U near the Mott transitionwith a crucial role of density fluctuations looks unrealistic as a model of He monolayer.In this letter, we show that the 4/7-density solid is actually located in the vicinity of thefluid-solid boundary implying essentially the same character as the QSL found in the Hub-bard model with substantial density fluctuations, contrary to the conventional picture. Moreprecisely, the density fluctuation in the solid between the 2nd and 3rd layers accompanied bya translational symmetry breaking on the 2nd layer solves the puzzles: It causes enhancementof the ratio of h sat to the exchange interaction as is revealed in the recent experiment. Fur-thermore, it naturally explains why the MSE model is insufficient to describe the 4/7 phaseof He.In the 4/7 phase, the 3/4 of He atoms on the 2nd layer occupy points just above midpoints of the edges of triangles formed by the 1st-layer atoms whereas the 1/4 occupy pointsjust above the 1st-layer atoms in a regular fashion as shown in Fig. 1(a). Here, open circlesrepresent the atoms on the 1st layer and shaded circles represent actual locations of Heatoms on the 2nd layer when solidified. If He atoms are adsorbed on the 1st layer, it forms atriangular lattice with the lattice constant a = 3 . ρ = 0 . . The location of the 2nd-layer atoms is in principle determined as stable points in continuumspace. In the present treatment, we simplify the continuum by discretizing it with as much aslarge number of lattice points kept as candidates of the stable points in the solid. To illustratethe discretization, we cut out from Fig. 1(a) a parallelogram whose corners are just above4 atoms on the 1st layer as in Fig. 1(b). Possible stable locations of He atoms on the 2ndlayer are (1) the points just above the mid points of the 1st-layer atoms, (2) the centers ofthe regular triangles and (3) the points just above the 1st-layer atoms. Therefore, we employtotally 6 points as the discretized lattice points in a parallelogram as circles in Fig. 1(b).Since a unit cell in Fig. 1(a) contains 7 parallelograms, it contains 42 lattice points in total asillustrated as circles in Fig. 1(c). Now the 4/7 solid phase is regarded as a regular alignmentof 4 atoms on 42 available lattice points in the unit cell shown in Fig. 1(c).We employ the Lennard-Jones potential V LJ ( r ) = 4 ǫ h ( σ/r ) − ( σ/r ) i , (1)for the inter-helium interactionwhere ǫ = 10 . σ = 2 .
56 ˚A. More refined Aziz potentialis expected to give similar results under this discretization. In the inset of Fig. 2, V LJ ( r ) vs. r in the unit of a is shown by the bold solid curve. The interaction term of the lattice model is . Phys. Soc. Jpn. Letter (a) (b) (c) a Fig. 1. (Color online) (a) Lattice structure of the 4/7 phase of He. Both 1st-layer atoms (opencircles) and 2nd-layer atoms (shaded circles) form triangular lattices in the solid phase. The areaenclosed by the solid line represents the unit cell for the solid of the 2nd-layer atoms (see text).The lattice constant of the 1st layer is a . (b) Possible stable location of the 2nd-layer atoms areshown by circles on top of a a × a parallelogram constructed from the 4 neighboring 1st-layeratoms. (c) Structure of discretized lattice for the 2nd-layer model. Lattice points are shown bycircles. given by H V = P ij V ij n i n j ( n i is a number operator of a Fermion on the i -th site) with V ij taken from the spatial dependence of eq. (1) on the lattice points. In the actual He system,the chemical potential of the 3rd layer is estimated to be 16 K higher than the 2nd layer.
He atoms may fluctuate into the 3rd layer over this chemical potential difference and it issignaled by an increase of the specific heat for
T >
8, 17, 18 as is reflected by the entropy persite larger than k B log 2. To take account of this density fluctuation, we here mimic the allowedoccupation on the 3rd-layer by introducing a simple finite cutoff V cutoff for V ij within the sameform of Hamiltonian: When V LJ ( r ij ) for r ij ≡ | r i − r j | exceeds V cutoff , we take V ij = V cutoff andotherwise V ij = V LJ ( r ij ). This allows taking account qualitative but essential part of possibleoccupation on the 3rd layer by the atoms overcoming V cutoff . We show the case of V cutoff = 16K as indicated by an arrow in the inset of Fig. 2. Here, the open circles show V ( r ij ) on thelattice sites in Fig. 1(c) for r/a ≤ H = H K + H V consists of the kinetic energy H K = − P h ij i ( t ij c † i c j + H . C . ) and H V . By using the unit-cell index s and the site index l inthe unit cell, we have r i = r s + r l .After the Fourier transform, c i = c s,l = P k c k ,l e i k · r s / √ N u , the mean-field (MF) approxi-mation with the diagonal order parameter h n k ,l i leads to H V ∼ H MFV = 1 N u 42 X l,m =1 X s ′ V lm ( s ′ ) X k , p (cid:20) h n k ,l i n p ,m − h n k ,l ih n p ,m i (cid:21) , where the inter-atom interaction is expressed as V ij = V lmst = V lm ( s ′ ) with r s ′ = r s − r t . Then,we have the MF Hamiltonian H MF = H K + H MFV . By diagonalizing the 42 ×
42 Hamiltonian . Phys. Soc. Jpn.
Letter V cutoff [K] D c [ K ] V cutoff D c -1001020 r / a V ( r ) [K] Fig. 2. (Color online) V cutoff dependence of the “charge gap” ∆ c . The inset shows the He-He inter-action V ( r ) vs. r (see text). matrix for each k , we obtain the energy bands H MF = P k P l =1 E l ( k ) c † k ,l c k ,l . Here we show the results by taking account of the transfers and interactions for | r ij | /a ≤ V ij and t ij for the ij pairs up to theshortest-19th r ij are retained. For the kinetic energy, several choices of t ij are examined andhere we show the result for t ij = t /r ij , assuming that it is proportional to ~ / (2 mr ij ). Wenote that the kinetic energy per atom for the 4/7 phase is estimated as 20 K by the path-integral Monte Calro (PIMC) simulation. Hence, we evaluate t by imposing the condition, P h ij i t /r ij h c † i c j + H . C . i / (4 N u ) = 20 K. We thus obtain t = 0 . ǫ and σ in eq. (1) are given by ǫ/t = 260 .
14 and σ/a = 0 . t is determined so as to reproduce the total kinetic energy of the PIMC result,the main result measured in the unit of K shown below is quite insensitive to the choice of t ij . By solving the MF equations for H MF , we have the solution of the √ × √ V cutoff ≥ t ≡ V ccutoff . The “charge gap” opens for V cutoff ≥ V ccutoff , as shown in Fig. 2. Here, the “charge gap” is defined by ∆ c = E min5 ( k )- E max4 ( k ), where E αl ( k ) denotes the minimum or maximum value of the l -th band from the lowest. The left(right) and bottom (top) axes represent ∆ c and V cutoff in the unit of K ( t ), respectively. Fromthe specific heat data, ∆ c is estimated to be ∼ V cutoff /t ∼
300 (namely, 12K), which is consistent with the chemical potential difference of the 3rd layer ∼
16 K. Since V cutoff /V ccutoff ∼ .
1, the 4/7 phase is located close to the fluid-solid boundary. The effect of3 K higher potential on top of the 1st layer He than other lattice points of the 2nd layer merely shifts the ∆ c - V cutoff line toward larger V cutoff : V ccutoff is changed from ∼
11 K to ∼
14K and hence the above conclusion does not change.To further understand the nature of the solid near the fluid-solid boundary, we next . Phys. Soc. Jpn.
Letter consider a minimal model ˜ H = ˜ H K + ˜ H U + ˜ H V with ˜ H K = − P σ P h ij i (cid:16) t ij c † iσ c jσ + H . C . (cid:17) , ˜ H U = U P i n i ↑ n i ↓ and ˜ H V = P h ij i V ij P σ,σ ′ n iσ n jσ ′ , where n iσ = c † iσ c iσ and h ij i denotes thepair of the sites. To simulate the quantum phase transition between fluid and commensuratesolid, we consider a triangular lattice with N = 12 sites with N e = 4 Fermions (see insetof Fig. 4). When the nearest neighbor repulsion V ≡ V ij is large in comparison with thetransfer, a commensurate solid is expected to be realized. To make accurate estimates ofphysical quantities we employ the exact diagonalization. Here the transfer integrals with the α th nearest-neighbor t α for α ≤ V are retained. Wetake t = t = t = 1 and U = V to express the large kinetic energy and the effect of V cutoff for He. Figure 3 shows the “charge gap”. Here, we calculate the ground-state energyby introducing the phase factor for the transfer integral: t ij = ˜ t ij exp[ i~φ · ( r i − r j )], where ~φ = φ b + φ b with b i being a reciprocal lattice vector which satisfies b i · a j = δ ij . Toreduce the finite-size effects, the “charge gap” is defined by ∆ c ≡ max { µ +min − µ − max , } , where µ +min = min φ [ E ( N e + 2) − E ( N e )] / µ − max = max φ [ E ( N e ) − E ( N e − / E beingthe ground-state energy. We take φ ξ = γπ/ ξ = x, y and the integer γ running from0 to 8, i.e., totally 81 mesh points for N = 12 and N = 18 at the filling n = N e /N = 1 / V = V c ∼
10 in the bulk limit. The inset in Fig. 3 shows the V dependence of the peakvalue of the equal-time charge and spin correlation functions with the periodic boundarycondition (b.c.), ( φ x , φ y ) = (0 , N ( q ) = P i,j exp[i q · ( r i − r j )] ( h n i n j i − h n i ih n j i ) /N and S ( q ) = P i,j exp[i q · ( r i − r j )] h S i · S j i / (3 N ). The peak of N ( q ) at ( q x , q y ) = (2 π/ , π/ √ V = V c . The peak in S ( q ) at ( q x , q y ) = ( π/ , π/ √
3) jumps at a higher V s > V c suggesting that a commensurate solid for V > V c is stabilized without a spin order for V < V s implying the QSL for V c < V < V s . The realistic choice of V /V c ∼ . h n i n j i averaged over the 81 phase factors in N = 12 for V /V c ∼ . N = 12 sites with the periodic b.c., the exchangeinteraction J is estimated from high-temperature part of χ ( T ) by the fitting of the high-temperature expansion χ ( T ) = (1 − J/T ) /T on the triangular lattice. By plotting (1 / ( χT ) − T vs. T as in Fig. 4, we estimate J from the flat part indicated by the arrows. The system-size dependence of J is quite small as known in the Hubbard chain, where χ ( T ) at high T is determined by the local process. Figure 5 (open circle) shows J for each V obtained in thisway.The magnetization is calculated by adding the Zeeman term to ˜ H : ˜ H − h P i S zi . We definethe saturation field h sat at which the total magnetization m = P i h S zi i /N reaches its saturationvalue, m sat = n/
2. Figure 5 shows h sat for the N = 18 sites (open triangle) under the periodic . Phys. Soc. Jpn. Letter V D c N V N ( q p ea k ) S ( q p ea k ) Fig. 3. (Color online) V dependence of “charge gap” on N = 12 (open circle) and N = 18 (filledtriangle) triangular lattices for t = t = t = 1 and V = U at n = 1 /
3. The inset shows V dependence of the peak value of N ( q ) (filled square) and S ( q ) (open triangle) for N = 12 underperiodic b.c. T ( / ( c T )- ) T Fig. 4. (Color online) Temperature and V dependences of susceptibility on N = 12 triangular latticeat n = 1 / t = t = t = 1 and U = V . The inset shows a triangular lattice with N = 12sites. b.c. The present saturation field at V = U = 0 for N = 18 reproduces the exact bulk limit h sat = 18 .
0, which is nothing but the width of the n -filled band at h = 0. This reproductiontogether with slightly smaller h sat for N = 12 (see the inset of Fig. 5) suggests that h sat inFig. 5 is close to the bulk-limit. This is one of our central results: h sat and hence h sat /J aswell largely increase in the commensurate solid near the solid-fluid boundary, V = V c . FromFig. 5, we see that 10 Tesla shown as thin lines is still below the saturation magnetic field fora realistic choice of V = 11 . Although the enhancement of h sat /J also appears at t = t = 0 (not shown), the en-hancement is more prominent when t is switched on. This is understood by the pertur-bation from the large V (= U ) limit. When t = t = 0, J appears first in the 4th orderas J (4) = 20 t / (3 V ), whereas the 2nd-order term J (2) = 4 t /V and the 3rd-order term . Phys. Soc. Jpn. Letter V h s a t / JJ , h s a t h / J m / m sat Fig. 5. (Color online) V dependence of exchange interaction J (open circle), saturation field h sat (open triangle) and its ratio h sat /J (filled square) for t = t = t = 1 and V = U at n = 1 / V = 11 . N = 12 (solid bold line) and N = 18(broken line) under periodic b.c. The difference between N = 12 and 18 may come from differentcommensurate structures allowed at m ∼ m sat /
2. Thin lines (in the inset as well) represent h/J corresponding to 10 Tesla in the experiment when we employ J = 0 . J (3) = − t t /V appear for t = 0. For t > J (3) becomes ferromagnetic (FM), whichpartially cancels J (4) in J = J (2) + J (3) + J (4) . This is similar to the cancellation among anti-ferromagnetic (AF) J >
0, FM − J < J > n -body exchangeinteractions ( − n J n . In short, the enhancement of h sat /J is largely driven by the densityfluctuations near the fluid-solid boundary, supplemented by the reduction of AF exchangethrough partial cancellation by FM MSE.Let us discuss the significance of the density fluctuations near the fluid-solid boundaryin terms of the observed QSL. The QSL in κ -ET Cu (CN) is found in the region of atiny charge gap, which consistently reproduces the QSL numerically found near the metal-insulator boundary in the Hubbard model on the triangular lattice.
11, 12, 14
The QSL is sup-pressed when the density fluctuations are suppressed at large U consistently with the absenceof the QSL phase reported in the spin-1/2 Heisenberg model on the triangular lattice. Theysuggest the importance of density fluctuations for the realization of the QSL in the 4/7 phaseof He.The density excitations over the energy ∆ c ∼ C ( T ) at T ∼ T = 10 − ∼ J (2) = 4 t ij /V cutoff ∼ × − ( t ij /t ) K. The double-peak structure is indeed found in C ( T ) for V ≥
10 in the N = 12 cluster study(not shown) as is observed in He.
2, 8
The fluid-solid transition occurs at a very large V cutoff /t ∼
300 as seen in Fig. 2, whichreflects the general tendency that the commensurate solid phase dramatically shrinks when . Phys. Soc. Jpn.
Letter the period of the density order becomes long. This explains why the fluid-solid boundary islocated near such a large chemical potential difference of the 3rd layer.In summary, we have shown that the minimal model for He adsorbed on the graphiteshould consider the density fluctuation to the upper layers. In particular, the properties of the4/7-solid phase on the 2nd layer are understood only by considering the density fluctuationson the 3rd layer, which makes the real system close to the fluid-solid transition beyond thedescription by the MSE model. The magnetic field required for the magnetization saturationis largely enhanced in agreement with the experiments. The density fluctuations also serveas a key for stabilizing the QSL. Our study predicts that when the lattice constant of the1st-layer solid can be changed, the 4/7 solid phase easily changes to fluid. Experimental testswould be highly desired.
Acknowledgment
We thank H. Ishimoto for supplying us with experimental data prior to publication. Thiswork is supported by Grants-in-Aid for Scientific Research on Priority Areas under the grantnumbers 17071003, 16076212 and 18740191 from MEXT, Japan. A part of our computationhas been done at the Supercomputer Center in ISSP, University of Tokyo. . Phys. Soc. Jpn.
Letter
References
1) V. Elser: Phys. Rev. Lett. (1989) 2405.2) K. Ishida, M. Morishita, K. Yawata and H. Fukuyama: Phys. Rev. Lett. (1997) 3451.3) R. Masutomi, Y. Karaki and H. Ishimoto: Phys. Rev. Lett. (2004) 025301.4) P. Fazekas and P. W. Anderson: Phil. Mag. (1974) 423.5) M. Roger, C. Bauerle, Yu. M. Munkov, A.-S. Chen and H. Godfrin: Phys. Rev. Lett. (1998)1308.6) H. Ikegami, R. Masutomi, K. Obara and H. Ishimoto: Phys. Rev. Lett. (2000) 5146.7) G. Misguich, B. Bernu, C. Lhuillier and C. Waldmann: Phys. Rev. Lett. (1998) 1098.8) Y. Matsumoto, D. Tsuji, S. Murakawa, C. B¨auerle, H. Kambara and H. Fukuyama: unpublished.9) T. Momoi, H. Sakamoto and K. Kubo: Phys. Rev. B (1999) 9491.10) H. Nema, A. Yamaguchi and H. Ishimoto: unpublished.11) T. Kashima and M. Imada: J. Phys. Soc. Jpn. (2001) 3052.12) H. Morita, S. Watanabe and M. Imada: J. Phys. Soc. Jpn. (2002) 2109.13) T. Mizusaki and M. Imada: Phys. Rev. B (2006) 014421.14) S. Watanabe: J. Phys. Soc. Jpn. (2003) 2042.15) J. de Boer and A. Michels: Physica (1938) 945.16) P. A. Whitlock, G. V. Chester and B. Krishnamachari: Phys. Rev. B (1998) 8704.17) S. W. Van Sciver and O. E. Vilches: Phys. Rev. B (1978) 285.18) D. S. Greywall: Phys. Rev. B (1990) 1842.19) F. F. Abraham, J. Q. Broughton, P. W. Leung and V. Elser: Europhys. Lett. (1990) 107.20) In case of t ij = − t for ij pairs up to the shortest-19th r ij , the result of ∆ c K vs. V cutoff K isnearly the same as Fig. 2.21) G. Boato, P. Cantini, C. Guidi, R. Tatarek and G. P. Felcher: Phys. Rev. B (1979) 3957.22) T. Koretsune, Y. Motome and A. Furusaki: J. Phys. Soc. Jpn. (2007) 074719.23) W. Opechowski: Physica (1937) 181.24) In the Hubbard chain, the high-temerature part of χ ( T ) in a few sites is quite close to the bulk-limit value; H. Shiba and P. A. Pincus: Phys. Rev B (1972) 1966. See also H. Shiba: Prog. Theor.Phys. (1972) 2171.25) E. Collin, S. Triqueneaux, R. Harakaly, M. Roger, C. Bauerle, Yu.M. Bunkov and H. Godfrin: Phys.Rev. Lett. (2001) 2447.26) Y. Shimizu, K. Miyagawa, K. Kanoda, M. Masato and G. Saito: Phys. Rev. Lett. (2003) 107001.27) I. Kezsmarki, Y. Shimizu, G. Mih´aly, Y. Tokura, K. Kanoda and G. Saito: Phys. Rev. B (2006)201101R.28) B. Bernu, P. Leceminant, C. Lhuillier and L. Pierre: Phys. Rev. B (1994) 10048.29) Y. Noda and M. Imada: Phys. Rev. Lett. (2002) 176803.(2002) 176803.