The Kagome Antiferromagnet: A Schwinger-Boson Mean-Field Theory Study
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov The Kagom´e Antiferromagnet: A Schwinger-Boson Mean-Field Theory Study
Peng Li, Haibin Su
Division of Materials Science, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798
Shun-Qing Shen
Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong, China (Dated: October 29, 2018)The Heisenberg antiferromagnet on the Kagom´e lattice is studied in the framework of Schwinger-boson mean-field theory. Two solutions with different symmetries are presented. One solution givesa conventional quantum state with q = 0 order for all spin values. Another gives a gapped spinliquid state for spin S = 1 / q = 0 and √ × √ S > /
2. We emphasize that the mixed state exhibits two sets of peaks in the static spin structurefactor. And for the case of spin S = 1 /
2, the gap value we obtained is consistent with the previousnumerical calculations by other means. We also discuss the thermodynamic quantities such as thespecific heat and magnetic susceptibility at low temperatures and show that our result is in a goodagreement with the Mermin-Wagner theorem.
PACS numbers: 75.10.Jm, 75.30.Ds, 75.40.Cx
I. INTRODUCTION
Two-dimensional (2D) geometrically frustrated Heisenberg antiferromagnets (AFMs) are potential candiadates inthe search of spin liquids (spin-disordered states) from both theoretical and experimental considerations. The firstsuggested candidate for spin liquid is the triangular lattice with nearest-neighbor (NN) couplings, but unfortunatelyit was finally revealed to exhibit 120 ◦ spin long-range order by extensive studies . People began to resort tointeractions beyond the NN coupling to realize the spin-disordered state. Another potential candidate is the Kagom´eAFM . It has been already known that there are two possible ordered states in this system, the so-called q = 0state and √ × √ (Fig. 1), while numerical studies do not support any long range orders for the spin-1/2system . This debate is still going on since numerical studies are usually limited to finite lattices. It was alsoinvestigated extensively whether the excitation spectra are gapless or not even if the system is disordered. Numericalstudy with up to 36 spins gives an estimation of the energy gap smaller than 1 /
20 of the exchange interaction . Ina scenario of valence bond crystal with translational symmetry breaking, the gap of the system is found to be verysmall . Experimentally, several Kagom´e-like systems have been found . One of the most intriguing experimentalresults is the large T coefficient of the specific heat of spin- SrCr Ga O , which suggests that a large linearterm exists in the density of states (DOS), D ( E ) ∼ ηE . A numerical study of the spin- system suggests thatthe T law of the specific heat can be inferred from the Heisenberg Hamiltonian with the NN couplings at verylow temperatures . A contractor renormalization calculation finds a columnar dimer order and re-produces the T specific heat for the spin- system, but it still cannot tell whether it is gapless or not . A very recent projectedwavefunction study suggests that the gapless mode may be missed due to the limitation of finite lattice sites, andthe gapless U(1)-Dirac state produces the T specific heat . So far many aspects of the ground state remain to bemysteries. FIG. 1: (Color online) The q = 0 and √ × √ FIG. 2: (Color online) The primitive cell and the first Brillouin zone (with an area of A BZ = 8 π / √
3) of the Kagom´elattice. The primitive translation vectors of the direct and reciprocal lattices are ` a = (1 , , a = (1 / , √ / ´ and ` b = ` π, − π/ √ ´ , b = ` , π/ √ ´´ , respectively. In the theoretical aspect, it was known that the Schwinger-boson mean-field theory (SBMFT) may provide a reliabledescription for both quantum ordered and disordered antiferromagnets based on the picture of the resonating valencebond (RVB) state . As a merit, it does not prescribe any long-range order for the ground state in advance,which should emerge naturally if the Schwinger bosons condense at low temperatures. It was supposed that such amean-field theory should be reliable for large spins where quantum fluctuation is believed to be weak. The theoryhas successfully captured the long-range order (LRO) of the Heisenberg antiferromagnets on the square andtriangular lattices at zero temperature, and is in excellent agreement with the Mermin-Wagner theorem even forsmall spins. Of course, it also has shortcomings, such as it predicts wrongly an energy gap for a one-dimensionalhalf-integer spin chains . In previous works, SBMFT had been already applied to the Kagom´e system. Manuel et. al. gave a Schwinger-boson approach to the q = 0 state and √ × √ . In a recent work by Wang, a new quantum disordered state is proposed for the systems with physicalspin values if ring exchange interactions are introduced . In this paper, we employ SBMFT to study the Heisenbergantiferromagnet with physical spin values on the Kagom´e lattice in a different approach and discover some featuresof the system quantities. The gauge freedom due to the geometry gives two solutions corresponding to two differenttypes of the states . The first solution gives the q = 0 ordered state while the second solution gives a mixed statewith q = 0 order and √ × √ S > , and a state with a very small gap for S = . It was shown that thestrong quantum fluctuation of quantum spin 1/2 may destroy the order states of higher spin and drive the system tobe disordered. The coexistence of two orders in a quantum state is one of the key results in this paper. This resultis revealed by the detailed analysis of the static spin structure factors. For the ordered states in both solutions, weshow that the low-energy spectra for the quasi-particles are linear in the momentum | k − k ∗ | at the gapless point k ∗ .As a result, the density of state is linear in energy, the specific heat obeys the T law, and the uniform magneticsusceptibility is finite at zero temperature.The rest of the paper is arranged as follows. The general formalism of the Schwinger-boson mean field theory ispresented in Sec. II. We introduce two types of mean field parameters, and expect to capture the key features ofquantum spin state on the Kagome lattice. A set of mean field equation is established by means of the MatsubaraGreen function techniques. In Sec. III, the numerical solutions of the mean field equations are given. We focus onthe ground state properties for the system and show that the ground state of spin 1/2 has a finite energy gap to theexcited states and is spin disordered while for larger spin the ground state is spin ordered. Finally, a brief discussionis presented in Sec. IV. II. SCHWINGER-BOSON MEAN-FIELD THEORY
We start with the Heisenberg Hamiltonian on the Kagom´e lattice, H = J X h i,m ; j,m ′ i S i,m · S j,m ′ , (1)where i and j are indices of the periodic Bravis lattice, m and m ′ are sublattice indices, A, B, or C as indicated inFig. 2, and the notation h i, m ; j, m ′ i means all possible NN pairs of lattice sites. The exchange interaction will be setas the unit of energy, J = 1. Note that the lattice constant is double of the triangle parameter l and we set 2 l = 1for simplicity. We choose the primitive cell and the first Brillouin zone as in Fig. 2. In the framework of Schwingerboson theory , a pair of hard-core bosons is introduced to represent one quantum spin S at each site, S + i,m = b † i,m, ↑ b i,m, ↓ , S − i,m = b † i,m, ↓ b i,m, ↑ , S zi,m = 12 (cid:16) b † i,m, ↑ b i,m, ↑ − b † i,m, ↓ b i,m, ↓ (cid:17) , (2)with the constraint, b † i,m, ↑ b i,m, ↑ + b † i,m, ↓ b i,m, ↓ ≡ S. In this representation, the Hamiltonian can be expressed as H = − X h i,m ; j,m ′ i (cid:16) † i,m ; j,m ′ · ∆ i,m ; j,m ′ + S (cid:17) + X i,m λ i,m ( b † i,m, ↑ b i,m, ↑ + b † i,m, ↓ b i,m, ↓ − S ) , (3)where ∆ i,m ; j,m ′ ≡ ( b i,m, ↑ b j,m ′ , ↓ − b i,m, ↓ b j,m ′ , ↑ ) and N Λ is the number of primitive cells (i.e. the total numberof lattice sites is 3 N Λ ). The Lagrange multipliers λ i,m is introduced to realize the constraints of the number ofSchwinger bosons at each site. Due to translational symmetry, we will set λ i,m = λ to simplify the problem. In themean field approach, one can introduce the mean-field parameter h ∆ i,m ; j,m ′ i , and decompose ∆ † i,m ; j,m ′ · ∆ i,m ; j,m ′ into D ∆ † i,m ; j,m ′ E ∆ i,m ; j,m ′ + ∆ † i,m ; j,m ′ h ∆ i, m ; j, m ′ i − | ∆ i,m ; j,m ′ | where h· · · i represents the thermodynamic average of thephysical quantity. This procedure can also be formulated equivalently as the Hubbard-Stratanovich transformation .In a suitable gauge, the bond parameter h ∆ i,m ; j,m ′ i can be taken to be real . Notice there are two different solutionsthat can be obtained by the relation of the mean fields on the two adjacent triangles (labeled as g and h in Fig. 2).Following the spirit of Sachdev’s discussion on gauge freedom of the mean-field parameter , we present two schemes:(i) ∆ = D ∆ gi,m ; j,m ′ E = (cid:10) ∆ hi,m ; j,m ′ (cid:11) ; (ii) ∆ = D ∆ gi,m ; j,m ′ E = − (cid:10) ∆ hi,m ; j,m ′ (cid:11) . They produce two physically distinct statesthat cannot be transformed into each other by gauge transformations.In both schemes the effective Hamiltonian is decomposed into the quadratic form of b i,m, ↑ and b i,m, ↓ , and themean-field theories for the two schemes have almost the same formalism. In the following deduction, we shall pointout their differences where it is appropriate. We can introduce three pairs of b † k ,m,µ and b k ,m,µ and µ = ↑ or ↓ in theFourier transform such that the effective Hamiltonian can be written in a compact form with the help of Kroneckerproduct, H eff = X k Ψ † k H mf Ψ k + ε , (4a) H mf = λI + ∆ σ x ⊗ [ γ Ω z − γ Ω y + γ Ω x ] ⊗ σ y , (4b) ε = 12 N Λ ∆ − λN Λ (2 S + 1) + 6 N Λ S , (4c)where the Nambu spinor is introducedΨ † k = ( ψ † k , ψ − k ) , ψ † k = ( b † k ,A, ↑ b † k ,A, ↓ b † k ,B, ↑ b † k ,B, ↓ b † k ,C, ↑ b † k ,C, ↓ ) , (5) I is a 12 ×
12 unit matrix, σ α ( α = x, y, z ) are 2 × α ( α = x, y, z ) are 3 × x = − i i , Ω y = i − i , Ω z = − i i , (6)and for the two schemes we have (i) γ = cos k x , γ = cos k x + √ k y , γ = cos k x −√ k y ; and (ii) γ = sin k x , γ =sin k x + √ k y , γ = sin k x −√ k y , respectively.Let us define the Matsubara Green’s function (a 12 ×
12 matrix) by the outer product of the Nambu spinor Eq.(5), G ( k , τ ) = − D T τ Ψ k ( τ ) Ψ † k E , (7)where τ is the imaginary time and Ψ k ( τ ) = e τH eff Ψ k e − τH eff . Then physical quantities concerning the average of twooperators can be readily expressed by the matrix elements, e.g. D b k , , ↑ b † k , , ↑ E = − G , ( k , τ = 0 + ). And the physcialquantities concerning the average of four operators, such as the correlation functions, can be decomposed into theaverages of two operators through the Wick theorem. We shall use these facts to establish the mean-field equationsand the static spin structure factors later.It is easy to prove that the Matsubara Green’s function in Matsubara frequency ω n = 2 nπ/β ( n is an integer forbosons) can be worked out by (also a 12 ×
12 matrix) G ( k , iω n ) = [ iω n σ z ⊗ Ω ⊗ σ − H mf ] − , (8)where Ω and σ are 3 × × ω ,µ = λ and other four-fold degenerate bands are ω ,µ ( k ) = ω ,µ ( k ) = ω ( k )with ω ( k ) = p λ − ∆ γ ( k ) with γ ( k ) = γ + γ + γ . The mean-field parameter ∆ and the Lagrangian multiplier λ should be determined self-consistently. The mean field can be evaluated by reading the elements of the MatsubaraGreen function matrix after the Fourier transformation, e.g.∆ = 12 ( h b i,A, ↑ b i,B, ↓ i − h b i,A, ↓ b i,B, ↑ i )= − βN Λ X k ,iω n e − ik x / − iω n + [ G , ( k , iω n ) + G , ( k , iω n )] . (9)Another constraint is that we should use the average value in the thermodynamic limit D b † i,m, ↑ b i,m, ↑ + b † i,m, ↓ b i,m, ↓ E =2 S to replace the original constraint. These two facts lead to a set of the mean-field equations for ∆ and λ ,3 S + 1 = 2 n B ( λ ) + 1 N Λ X k λω ( k ) [1 + 2 n B ( ω ( k ))] , (10a)∆ = 16 1 N Λ X k ∆ γ ( k ) ω ( k ) [1 + 2 n B ( ω ( k ))] , (10b)where n B ( ω ( k )) = (cid:2) e ω ( k ) /T − (cid:3) − is the Bose-Einstein distribution function with temperature T . In the thermo-dynamical limit N Λ → ∞ , the momentum sum is replaced by the integral over the first Brillouin zone (Fig. 2), N Λ P k → R d kA BZ , where A BZ = π √ is the area of the first Brillouin zone. When the Schwinger bosons condensationoccurs, i.e. the solution gives a gapless spectrum ω ( k ∗ ) = 0, we can extract a condensation term in the momentumsummation of the first equation, Eq. (10a), ρ ( T ) = λN Λ ω ( k ∗ ) [1 + 2 n B ( ω ( k ∗ ))] . (11)With the help of the mean-field equations at zero temperature, we can obtain the simplified form of ground energyper bond E / N Λ = − + S . (12) III. SOLUTIONS
To solve the mean-field equations, let us introduce dimensionless quantities, e ∆ = ∆ λ and e T = Tλ . Then the mean-fieldequations become, 3 S + 1 = coth (cid:18) e T (cid:19) − ρ ( e T ) + I ( e T ) , (13a)∆ = e ∆ γ ( k ∗ )6 ρ ( e T ) + I ( e T ) , (13b)with the definitions of two integrals I ( e T ) = Z d kA BZ q − e ∆ γ ( k ) coth q − e ∆ γ ( k )2 e T , (14a) I ( e T ) = 16 Z d kA BZ e ∆ γ ( k ) q − e ∆ γ ( k ) coth q − e ∆ γ ( k )2 e T . (14b)
FIG. 3: (Color online) The asymptotic behavior of the gap near zero temperature for the case of S = 1 / c ≈ − .
36 and c ≈ .
34, which gives ∆ gap ≈ . e − . /T as T → S = 3 /
2. The upperspectrum is for scheme (i). The lower one is for scheme (ii). The interior area of the blue hexagon is the first Brillouin zone.
First, we solve the equations at zero temperature. For the scheme (i), we found the first integral is bounded fromabove by the value I = 1 . e ∆ = √ . We notice that a nonzero condensation density persists as long as S ≥ S c ≈ . ρ = 3 S − . , ∆ = √ S + 0 . , and λ = S + 0 . k ∗ = (0 , I = 2 . e ∆ = and the critical spin value is S c ≈ . S = , we obtain a spinliquid because the spectrum is gapped with numerical solution ρ = 0 , ∆ = 0 . , and λ = 0 . gap = 0 .
120 11 .For larger spins
S > , numerical solutions are ρ = 3 S − . , ∆ = S + 0 . , and λ = S + 0 . K points, e.g. k ∗ = (cid:0) π , (cid:1) .Now, we discuss the asymptotic behavior of the solution near zero temperature. Our numerical results showthat the condensation only occurs at zero temperature, which is in agreement with the Mermin-Wagner theorem fortwo-dimensional Heisenberg systems. Numerical calculations tell us that the gapless solution can only exist at zerotemperature and an energy gap opens at finite temperature, which behaves as ∆ gap ∝ e − c/T as T →
0. We give anexample for the case of S = in the scheme (i). The curve for ln ∆ gap ∼ T is plotted in Fig. 3. The lowest finitetemperature we approach is T ≈ . J ), where we get a small gap ∆ gap ≈ . × − . FIG. 5: (Color online) The spin structure factors χ S z ( Q ) for the two mean-field schemes in the text. The upper one is forscheme (i), and the lower one is for scheme (ii). Here we set S = 3 /
2. The divergent peaks signal the existence of LRO. Otherspin values have similar results, except for the case of S = 1 / In the following, we turn to the relevant thermodynamical quantities and the patterns of LRO at zero temperature.The gapless spectra for both ordered states are exemplified in Fig. 4. It is clear that the cone-shaped spectra arelinear in the momentum ω ( k ) ≈ α | k − k ∗ | , (15)where α = λ √ for the scheme (i) and α = λ √ for the scheme (ii). Correspondingly, the density of state (DOS) ofthe quasiparticles linear in the low energy, D ( E ) ≈ ηE + O ( E ) , (16)where η = √ πλ for the scheme (i) and η = πλ for the scheme (ii). As a result, the T law of specific heat is anticipatedfor both ordered states C V /N Λ ∼ ζ (3) η (cid:18) TJ (cid:19) (17)at very low temperatures.The non-zero value of ρ means the condensation of the quasi-particles and the existence of LRO at zero-temperature.To disclose the ordered patterns of the ground states we need to calculate the static spin structure factor χ S z ( Q ) = lim τ → (cid:10) T τ S z Q ( τ ) S z − Q (cid:11) , (18)where τ is the imaginary time and S z Q = √ N Λ P i,m S zi,m e i Q · r i,m . A detailed calculation shows that the static spinstructure factor is isotropic, (cid:10) S x Q S x − Q (cid:11) = D S y Q S y − Q E = (cid:10) S z Q S z − Q (cid:11) , which indicates that the ground state is invariantunder the spin rotation. Analytically, we have that the total spin (cid:10) S tot (cid:11) = 3 N Λ χ S z (0) = 0 for both ordered states,which is consistent with the consequence of exact diagonalization techniques for . The divergent peaks of χ S z ( Q )signal the existence of LRO as shown in Fig. 5. (Notice that for the case of S = 1 / χ S z ( Q ∗ ) ∝ λ [1 + 2 n B ( ω ( k ∗ ))] [1 + 2 n B ( ω ( k ∗ + Q ∗ ))] N Λ ω ( k ∗ ) ω ( k ∗ + Q ∗ ) = N Λ ρ . (19) ∝ Notice the primitive cell contains more than one site, the replicative area of χ S z ( Q ) is 4 times of the area of thefirst Brillouin zone A BZ (Fig. 5). This can be easily verified by the definition, Eq. (18).For the scheme (i), a characteristic feature of the static structure factor χ S z ( Q ) is the six divergent peaks, whichare located at the wave vectors, Q ∗ ∈ {± Q , ± Q , ± ( Q + Q ) } with Q = b and Q = b . At the divergentpeaks, say e.g. at Q = ± Q , one gets S z ± Q = √ N Λ P i (cid:0) S zi,A + e ± iπ S zi,B + S zi,C (cid:1) . Combining it with the facts that, χ S z (0) = 0, we draw a conclusion that the configuration of LRO has the q = 0 order. The q = 0 order is marked bythe neutron scattering peak at the distance | Q | = 4 π √ . (20)should mark the neutron scattering peak position.For the scheme (ii), a similar analysis leads to another type of order pattern. In this case the divergent peaks arelocated at the wave vectors, Q ∗ ∈ {± Q , ± Q , ± ( Q + Q ) , ± Q , ± Q , ± ( Q + Q ) } with Q = (2 b + b ) and Q = ( − b + b ). The patterns corresponding Q and Q peaks are the usual √ × √ q = 0 order and √ ×√ √ × √ | Q | = 8 π . (21)Our calculation found the peak at | Q | is a little stronger than that at | Q | , since the ratio of the two sets of divergentpeaks is χ S z ( Q ) χ S z ( Q ) = 32 . (22)The uniform magnetic susceptibility can be obtained by the analytic continuation χ M = lim Q → lim iω n → χ S z ( Q , iω n )= Z d kA BZ ∆ (cid:2) γ γ + γ γ + γ γ (cid:3) ω ( k ) [ λ − ω ( k )] [ λ + ω ( k )] . (23) χ M has a finite value at zero temperature since the divergent denominator is annihilated by the linear DOS, χ M ∼ R D ( E ) E dE ∼ f inite . IV. DISCUSSION
There are several materials which possess the structures of the spin Kagom´e lattice. Experimental data by MuonSpin Relaxation show that the compound SrCr − x Ga x O lacks a long-range order until 0 . . The entropymeasurement of the same compound gives a T law for the specific heat at low temperatures, which was regarded asan evidence to support the absence of long-range order . However, spontaneous breaking of a continuous symmetryproduces massless Goldstone modes , and LRO can only exist at zero temperature for 2D systems according toMermin-Wagner theorem . From the present calculation of SBMFT, the special structure of the Kagom´e latticeleads to the cone structure of the spectra for the quasi-particles in the momentum space. The existence of long-rangecorrelation is consistent with the picture of the gapless spectrum of quasi-particles and the ordered ground state canproduce the T law of the specific heat. Of course, the possible existence of the additional next-nearest-neighborcoupling could further weaken the long-range correlation.We thank C. Broholm for helpful discussions. This work was supported by the COE-SUG Grant (No. M58070001)of NTU and the Research Grant Council of Hong Kong under Grant No.: HKU 703804. APPENDIX A: STATIC SPIN STRUCTURE FACTORS
The spin structure factor at zero temperature (see Fig. 5) contains both intra-sublattice and inter-sublatticecontributions, χ S ( Q ) = (cid:10) S z Q S z − Q (cid:11) = 1 N Λ X m,n ∈ A,B,C h S zm S zn i e i Q · ( r m − r n ) = χ AA ( Q ) + χ BB ( Q ) + χ CC ( Q )+ χ AB ( Q ) + χ BA ( Q ) + χ AC ( Q )+ χ CA ( Q ) + χ BC ( Q ) + χ CB ( Q ) (A1)where we have defined the spin density wave operator S z Q = X m ∈ A,B,C S zm e i Q · r m . (A2)The intra-sublattice and inter-sublattice contributions are χ AA ( Q ) = Z d k A BZ [ O a ( k + Q ) + P a ( k + Q )] Q a ( k ) ,χ BB ( Q ) = Z d k A BZ [ O a ( k + Q ) + P a ( k + Q )] Q a ( k ) ,χ CC ( Q ) = Z d k A BZ [ O a ( k + Q ) + P a ( k + Q )] Q a ( k ) ,χ AB ( Q ) = χ BA ( Q )= Z d k A BZ { [ − O b ( k + Q ) + P b ( k + Q )] Q b ( k ) − R ( k ) R ( k + Q ) } ,χ BC ( Q ) = χ CB ( Q )= Z d k A BZ { [ − O b ( k + Q ) + P b ( k + Q )] Q b ( k ) − R ( k ) R ( k + Q ) } ,χ CA ( Q ) = χ AC ( Q )= Z d k A BZ { [ − O b ( k + Q ) + P b ( k + Q )] Q b ( k ) − R ( k ) R ( k + Q ) } , where O a ( k ) = ∆ γ ( k )1 − ω ( k ) , O a ( k ) = ∆ γ ( k )1 − ω ( k ) , O a ( k ) = ∆ γ ( k )1 − ω ( k ) ,P a ( k ) = ∆ (cid:2) γ ( k ) + γ ( k ) (cid:3) ω ( k ) [1 − ω ( k )] , P a ( k ) = ∆ (cid:2) γ ( k ) + γ ( k ) (cid:3) ω ( k ) [1 − ω ( k )] , P a ( k ) = ∆ (cid:2) γ ( k ) + γ ( k ) (cid:3) ω ( k ) [1 − ω ( k )] ,Q a ( k ) = ∆ (cid:2) γ ( k ) + γ ( k ) (cid:3) ω ( k ) [1 + ω ( k )] , Q a ( k ) = ∆ (cid:2) γ ( k ) + γ ( k ) (cid:3) ω ( k ) [1 + ω ( k )] , Q a ( k ) = ∆ (cid:2) γ ( k ) + γ ( k ) (cid:3) ω ( k ) [1 + ω ( k )] ,O b ( k ) = ∆ γ ( k ) γ ( k )1 − ω ( k ) , O b ( k ) = ∆ γ ( k ) γ ( k )1 − ω ( k ) , O b ( k ) = ∆ γ ( k ) γ ( k )1 − ω ( k ) , P b ( k ) = ∆ γ ( k ) γ ( k )2 ω ( k ) [1 − ω ( k )] , P b ( k ) = ∆ γ ( k ) γ ( k )2 ω ( k ) [1 − ω ( k )] , P b ( k ) = ∆ γ ( k ) γ ( k )2 ω ( k ) [1 − ω ( k )] ,Q b ( k ) = ∆ γ ( k ) γ ( k )2 ω ( k ) [1 + ω ( k )] , Q b ( k ) = ∆ γ ( k ) γ ( k )2 ω ( k ) [1 + ω ( k )] , Q b ( k ) = ∆ γ ( k ) γ ( k )2 ω ( k ) [1 + ω ( k )] ,R ( k ) = ∆ γ ( k )2 ω ( k ) , R ( k ) = ∆ γ ( k )2 ω ( k ) , R ( k ) = ∆ γ ( k )2 ω ( k ) . P. W. Anderson, Mater. Res. Bull. , 153 (1973); P. Fazekas and P. W. Anderson, Philos. Mag. , 423 (1974). D. A. Huse and V. Elser, Phys. Rev. Lett. , 2531 (1988). Th. Jolicoeur and J. C. Le Guillou, Phys. Rev. B , 2727 (1989). B. Bernu, C. Lhuillier, and L. Pierre, Phys. Rev. Lett. , 2590 (1992); B. Bernu, P. Lecheminant, C. Lhuillier, and L.Pierre, Phys. Rev. B , 10048 (1994). L. Capriotti, A. E. Trumper, and S. Sorella, Phys. Rev. Lett. , 3899 (1999). S. Sachdev, Phys. Rev. B , 12377 (1992). See a review in G. Misguich and C. Lhuillier,
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