Kagomé Lattice Antiferromagnet Stripped to Its Basics

Kagomé Lattice Antiferromagnet Stripped to Its Basics P. Azaria 1 , C. Hooley 2 , P. Lecheminant 3 , C. Lhuillier 1 , and A. M. Tsvelik 2 arXiv:cond-mat/9807228v1 [cond-mat.str-el] 15 Jul 1998 1 Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France 2 Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK 3 Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, Site de Saint Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France (Received: ) We study a model of a spin S = 1/2 Heisenberg antiferromagnet on a one dimensional lattice with the local symmetry of the two dimensional kagomé lattice. Using three complementary approaches, it is shown that the low energy spectrum can be described by two critical Ising models with different velocities. One of these velocities is small, leading to a strongly localized Majorana fermion. These excitations are singlet ones whereas the triplet sector has a spectral gap. PACS No: 75.10.Jm, 75.40.Gb The famous kagomé lattice antiferromagnet still remains largely a mystery after a decade of extensive studies. This system exhibits both frustration and low coordinance and classically it has infinite continuous degeneracies. Local distortions allow it to explore its many ground states with no cost in energy and lead to a very specific linear spin wave spectrum with a whole branch of zero energy excitations [1]. In the quantum case (S=1/2), the system is likely to be a spin liquid with a gap for the magnetic excitations [2]. For finite samples, the system has a huge number of singlet states below the first triplet [3] which is a rather unexpected feature for a twodimensional quantum antiferromagnet. Moreover, the analysis of the specific heat shows the existence of unusual low-lying excitations [4]. The presence of these low-lying singlet excitations below the spin gap gives a picture of an intriguing spin liquid that deserves understanding from a general point of view. In this paper, we shall study a one-dimensional model of spin S = 1/2 Heisenberg antiferromagnet on a lattice presented on Fig. 1 which retains the local symmetry of the kagomé net. Insight into the behavior of the twodimensional kagomé antiferromagnet might be gained by investigating this simplified system. In particular, the one dimensional model may help us to identify the slow degrees of freedom of the problem. According to the conventional wisdom the model we study may be viewed as a version of a three-leg spin ladder which in the low-energy limit one should expect to fall into the universality class of the S = 1/2 Heisenberg spin chain. The latter means that the low-lying excitations (often called spinons) are represented by one gapless bosonic mode (in the language of the theory of critical phenomena this means that the central charge of the model is equal to C = 1 [5]). However, the frustration may play its tricks and, as we shall see, a somewhat different scenario is realized. Namely, the bosonic mode decouples into two modes of real (Majorana) fermions (C= 1/2 each) having different spectra. Jk ✉ ✉ ❏ ✡ ✡ ❏ ✡ ❏ J⊥ ❏ ✉✡ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ❏✉ ✉ ✡ ✉ ❏ ❏ ❏ ✉ ✡ ✡ ✡ ❏ ✉✡ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏✉ ✉ ✲ 2j 2j + 1 2(j + 1) 2(j + 1) + 1 FIG. 1. One dimensional version of the kagomé lattice. The Hamiltonian for the model shown in Fig. 1 may be written X X [Jk (Sa,2j · Sa,2j+1 + Sa,2j+1 · Sa,2j+2 ) H= a=1,2 j +J⊥ s2j+ 21 · (Sa,2j + Sa,2j+1 )] (1) where s2j+ 21 are the “middle spins”, and Sa,j are the “rail spins”, the index a taking two values (a = 1, 2), one for each rail. All interactions are antiferromagnetic and we shall also consider the case where interaction between spins belonging to the rails of the lattice is much stronger than the interactions with the middle spins: Jk >> J⊥ . Under this condition, one can describe the chains in the continuous approximation, representing the spins on each chain as a sum of ferromagnetic (Ma ) and staggered (na ) parts: Sa,j → Ma (x) + (−1)j na (x). It is crucial for our analysis that the middle spins interact only with the ferromagnetic part of the magnetization. The latter has the following remarkable property - it can be written as a sum of “currents” Ma = Ja + J̄a , where the currents satisfy the same commutation relations as bilinear combinations of left- and right-moving Dirac fermions [6]: 1 + J̄a = Ra,α ~σαβ Ra,β /2, Ja = L+ σαβ La,β /2 a,α ~ (2) 0 [e2 (xa + 1/g)e2(xa − 1/g)]N eN 1 (xa ) = where ~σ are the Pauli matrices. The corresponding algebra (SU(2)1 ) is called the level k = 1 SU(2) Kac-Moody (KM) algebra where k refers to the number of species of spin-1/2 fermions (one in the given case). Now we notice that the interaction includes the sum of two SU(2)1 spin currents J = J1 + J2 which, by definition, is a level k = 2 current. These currents have the same commutation relations as currents of three Majorana fermions (the explicit expression is given later) [7,8]. Since each Dirac fermion can be represented as a linear combination of two Majorana (real) fermions, it is clear that only 3/2 of the low-energy degrees of freedom of rails are involved in the interaction. Obviously the other decoupled degrees of freedom remain critical. Using the results of Ref. [8] we can represent the continuous limit of the Hamiltonian (1) as follows: H = Hs + Ht with Z iv Hs = − dx (r0 ∂x r0 − l0 ∂x l0 ) (3) 2 Z πv dx [J2 (x) + J̄2 (x)] Ht = 2 X Z g dx δ(x − aj) sj · [J(x) + J̄(x)]. (4) + M Y b=1 e2 (xa − xb ) (6) where 2N and N0 are the numbers of spins on the rails and in the middle, respectively, M is the number of up spins (M = 2N +N0 /2−S z ) and en (x) = (x−in/2)/(x+ in/2). The total energy takes the form E= 1 X ln[e2 (xa + 1/g)/e2(xa − 1/g)]. 2i a (7) Following the standard procedures of the Bethe ansatz, we derive the following system of integral equations for the free energy: Z i h F = −N T dx [s(x + 1/g) + s(x − 1/g)] ln 1 + eǫ2 (x)/T Z h i − N0 T dx s(x) ln 1 + eǫ1 (x)/T , (8) with i ih h ǫj (x) = T s ∗ ln 1 + eǫj−1 (x)/T 1 + eǫj+1 (x)/T j −δj,2 ∆ cosh πx Here v ∼ Jk is the spin velocity which we shall later put to unity and the coupling constant is g ∼ J⊥ . The Majorana fermion (r0 , l0 ) describes non-magnetic, gapless excitation. It represents Ising degrees of freedom that do not interact with the central spins and are associated with a discrete Z2 interchange symmetry between the surface chains. We shall study the Hamiltonian (4) using several alternative approaches. First, we shall study an integrable deformation of model (4) . To make sure that the exact solution reproduces the qualitative features of the spectrum, we shall consider an anisotropic version of (4) bearing in mind that it may simplify in the limit of strong anisotropy (the so-called Emery-Kivelson limit [9]). Finally, a direct mean field treatment of the lattice in the isotropic limit will be performed. Exact solution of a related model. Since the currents satisfying the level k = 2 SU(2) KM algebra can be represented as fermionic bilinears, the effective theory for the central spins corresponds to a special version of the two-channel (k = 2!) Kondo-lattice model where “electrons” do not experience backscattering. A similar model (with k = 1) was considered in Ref. [10]; the result is that integrability is achieved if one adds an additional interaction: Z ′ Hex = Ht + g dx J(x) · J̄(x) (5) (9) and limn→∞ ǫn (x)/n = H, H being the magnetic field. In (9), one has ∆ ∼ exp(−π/g), s(x) = [2 cosh πx]−1 , and the convolution product is denoted by ∗. At low temperatures T << ∆, one has ǫ2 (x) ≈ −∆ cosh πx, ǫ1 (x) = O(e−∆/T ). In this case the first term in the free energy (8) gives an exponentially small contribution corresponding to excitations with √ a spectral gap ∆ and the last term gives F ≈ −T√N0 ln 2 showing that each central spin contributes ln 2 to the ground state entropy. A lesson we learn from the exact solution is that there are degrees of freedom presumably localized on central spins (and their number corresponds exactly to a single Majorana mode) which remain decoupled. One may conjecture that since the magnetic modes are separated from the ground state by a gap, the soft modes will remain soft even when one departs from the integrable point. Below we shall give more rigorous arguments to support this conjecture. The Emery-Kivelson limit. We shall now study a U(1) version of the Hamiltonian (4), characterized by anisotropic interactions with g → gk , g⊥ where in the limit of strong anisotropy, the low energy degrees of freedom can be identified. This approach has been fruitfully applied in quantum impurity problems and gives a simple description of the two-channel Kondo model [9] and also of the Kondo lattice [11]. In particular, for the twochannel Kondo problem, Emery-Kivelson identified the residual zero point entropy stemming from a unique Majorana zero mode. To reformulate the Hamiltonian (4) as a fermionic theory similar to Kondo problems, we first introduce three right- and left-moving Majorana fermions where the ratio g ′ /g is fixed. The Bethe ansatz equations for model (5) are derived in the same way as in Ref. [10] with the only difference that “conduction electrons” now belong to the spin-1 representation of the SU(2) group. The result is 2 rb , lb (b = 1, 2, 3) to use the fermionic representation of the SU(2)2 spin current: J¯a = −iǫabc rb rc /2 with a similar relation for the left-moving current [7]. The next step in our solution is to combine the two Majorana fermions (r1 , l1 ) and (r2 , l2 ) to form a single Dirac spinor (R, L) which in turn can be bosonized. Introducing a massless bosonic field Φ and its dual field Θ, one can write: SF = XZ j 1 1 dτ [ ξ1,j ∂τ ξ1,j + ξ2,j ∂τ ξ2,j 2 2 1 (14) + χj ∂τ χj − iχj+1 χj − i∆ξ2,j χj ] 2 where χ is a Majorana fermion express in terms of the right- and left-moving √ Majorana fields (r3 , l3 ): χj = (l3,j + (−1)j r3,j )/ 2. The mean-field parame√ √ ǫ r1 + ir2 ters are√defined√ by: λ = −ig⊥ hξ2 χi/ πa0 and ∆ = exp[i π(Φ − Θ)] = √ R= √ 2πa0 g⊥ hcos( πΦ)i/ πa0 . Solving the self-consistency equa2 √ tions, we find that the bosonic field Φ becomes massive ǫ l1 + il2 exp[−i π(Φ + Θ)] (10) =√ L= √ 4/3 2 with a gap of the order ∆ ∼ g⊥ ln1/6 (1/g⊥ a0 ). In 2πa0 2 the fermionic sector, the Majorana fermion ξ2 hybridizes where a0 is a short distance cut-off. The anticommutawith the Majorana field χ which in terms of the original tion relation between R and L is taken into account by spins stems from the rails spins. The resulting excitation the commutator: [Θ(y), Φ(x)] = −iθ(y − x). The real spectrum is reminiscent of the one found in Kondo latfermionic zero mode (ǫ) is necessary to establish the cortices [12] with a small gap ∆g ∼ ∆2 /Jk ≪ ∆. Finally, rect anticommutation relations with the third Majorana there is still a singlet localized Majorana fermion ξ1 which fermion (r3 , l3 ). The interacting part of the Hamiltonian decouples from the conduction sea at the Toulouse point. (4) is then given by: This mode gives a zero point entropy of magnitude 12 ln 2 per central spin as found above in the integrable model. X  gk √ g⊥ i πΘj Away from the Toulouse point (δgk = gk − π 6= 0), this − √ szj (∂x Φ)j + √ Hint = ǫ j s+ e π 4πa0 j mode will acquire a small dispersion and will contribute j    √ to the specific heat. Coming back to the original model, √ ei πΦj e−iπ/4 l3,j + e−i πΦj eiπ/4 r3,j + h.c . (11) adding the contribution of the singlet Majorana fermion (r0 , l0 ) which has decoupled from the interaction, the total central charge in the long-distance limit is C=1. Let Following Emery and Kivelson, we absorb the phase fac√ us stress that the two Majorana modes, in the singlet tor e±i πΘj into the spin operators by a unitary transsector, contributing to the central charge are of different formation U : nature. The Majorana field (r0 , l0 ) describes a critical √ P z −i π Θj sj j Ising model whereas ξ1 is a strongly localized Majorana U =e . (12) fermion. In contrast, the triplet sector has a small specAs a result the interaction becomes tral gap. As a result of this spectrum, the middle spins  are disordered and have short-ranged spin correlations. X (gk − π) z The experience gained from the study of Kondo models − √ Hint → U Hint U † = sj (∂x Φ)j π leads us to expect that the preceeding results obtained j  in the Emery-Kivelson limit of the model will extend to  √  √ g⊥ i πΦj j −i πΦj e a+ + h.c. (13) the isotropic point. To show this, we shall now develop a l + (−1) e r +√ 3,j 3,j j 4πa0 mean-field theory directly at the isotropic limit keeping track of the lattice structure more accurately than in the where we have replaced the combinations of spin-1/2 opprevious approach. erators and fermionic zero mode by local fermions (aj ) Direct mean-field approach. The Hamiltonian (4) using the Jordan-Wigner transformation. We have also is reformulated in terms of a fermionic model on the lat+ j absorbed a phase factor (−i) in the definition of aj . tice: At a special point gk = π (called the Toulouse point), X ig X part of the interaction vanishes and the low energy χaj+1 χaj − Ht = iJk s 1 . ([~ χ2j × χ ~ 2j ] 2 j 2j+ 2 physics can be studied by a simple mean-field theory. j Since the scaling dimension d of the bosonic exponents + [~ χ2j+1 × χ ~ 2j+1 ]) . (15) in (13) is pretty small (d = 1/4), we shall replace them by their averages and try to solve the problem selfIn the continuum limit, the rail of χa fermions will consistently. Introducing two Majorana fermions (ξ√ ) contain left- and right- moving Majorana fermions 1,2 (ra , la ; a = 1, 2, 3) of the SU(2)2 spin current: χaj = associated with the complex fermion a: a = (ξ1 +iξ2 )/ 2, j we find that the effective action decouples into bosonic −la (x)−(−1) ra (x) ; with this identification, the Hamiland fermionic parts in the mean field limit: SMF = tonian (4) is then reproduced. The model (15) correSB + SF with sponds to local moments (s2j+1/2 ) interacting with a sea Z of three Majorana fermions (χaj ). To describe the mid√
1 2 dle spins we use the Majorana representation for spins SB = dxdτ [ (∂µ Φ) + λ cos πΦ ] 2 S=1/2 (see Ref. [12] and references therein): 3 i 1 sn = − [~γn × ~γn ], n = 2k + , k ∈ Z (16) 2 2 where γna are local Majorana fermions satisfying the anb ticommutation relations {γna , γm } = δnm δ ab . This representation (16) reproduces the spin commutation relations and gives the correct value of the Casimir operator: s2n = 3/4. A model very similar to (15) was analyzed in Ref. [13]. Following this analysis we substitute (16) into (15) and decouple the interaction with an auxiliary field V living on the links connecting rails with middle spins. In the mean-field approximation the variables V are considered as static with their values determined self-consistently by minimizing the free energy. We find that the minimum is achieved when a unit cell contains two middle spins (see Fig. 2) with either U+ = V− = 0, U− = ±V+ ≡ ∆ or U− = V+ = 0, U+ = ±V− ≡ ∆. U− ✉ χ1 γ1 ✉ ❅ U ❅ + ❅✉ χ2 V− ✉ χ3 In conclusion, the version of the kagomé lattice studied in this paper can be reformulated as a fermionic theory similar to models of Kondo lattices. Its low-energy excitations are two spin-singlet Majorana modes with different spectra: a critical Ising mode and a strongly localized Majorana fermion, whereas the triplet sector has a small spectral gap. The physics of the localized low-energy mode is similar to physics of the two-channel Kondo model. This picture retains some properties of the kagomé antiferromagnet with very soft singlet modes and a gap for magnetic excitations. It is tempting to conjecture that the singlet degrees of freedom and the gapful magnetic excitation identified here might be responsible for the additional structure seen in the specific heat of the kagomé magnet at low temperature. A. M. Tsvelik acknowledges the kind hospitality of Ecole Normale Supérieure during his stay in Paris. The authors thank P. Chandra, P. Coleman, B. Douçot, H.-U. Everts, and Ch. Waldtmann for important discussions. γ2 ✉ ❅ V ❅ + ❅✉ χ4 [1] J. T. Chalker, P. C. W. Holdsworth, and E. S. Shender, Phys. Rev. Lett. 68, 855 (1992); A. B. Harris, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 45, 2899 (1992); A. Chubukov, Phys. Rev. Lett. 69, 832 (1992); P. Chandra, P. Coleman, and I. Ritchey, J. Phys. I France 3, 591 (1993). [2] C. Zeng and V. Elser, Phys. Rev. B 42, 8436 (1990); J. T. Chalker and J. F. G. Eastmond, Phys. Rev. B 46, 14201 (1992); R. R. P. Singh and D. A. Huse, Phys. Rev. Lett. 68, 1766 (1992); S. Sachdev, Phys. Rev. B 45, 12 377 (1992); P. W. Leung and V. Elser, Phys. Rev. B 47, 5459 (1993). [3] P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and P. Sindzingre, Phys. Rev. B 56, 2521 (1997). [4] N. Eltsner and A.P. Young, Phys. Rev. B 50, 6871 (1994). [5] G. Sierra, J. Phys. A 29, 3299 (1996). [6] I. Affleck, Nucl. Phys. B265, 409 (1986). [7] A. B. Zamolodchikov and. V. A. Fateev, Sov. J. Nucl. Phys. 43, 657 (1986). [8] D. G. Shelton, A. A. Nersesyan, and A. M. Tsvelik, Phys. Rev. B 53, 8521 (1996); D. Allen and D. Sénéchal, Phys. Rev. B 55, 299 (1997). [9] V. J. Emery and S. A. Kivelson, Phys. Rev. B 46, 10 812 (1992). [10] E. H. Rezayi and J. Sak, Phys. Lett. 89A, 451 (1982). [11] O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. Lett. 77, 1342 (1996); P. Coleman, A. Georges, and A. M. Tsvelik, J. Phys. Cond. Matt. 9, 345 (1997). [12] A. M. Tsvelik, Phys. Rev. Lett. 69, 2142 (1992); P. Coleman, E. Miranda, and A. M. Tsvelik, Phys. Rev. B 49, 8955 (1994). [13] P. Coleman, L. B. Ioffe, and A. M. Tsvelik, Phys. Rev. B 52, 6611 (1995). FIG. 2. The unit cell in the mean-field approximation. All branches of the spectrum are found to be gapful. The low energy band is rather flat towards the edge of the Brillouin zone in agreement with the Emery-Kivelson limit of the model in the triplet sector. However, as was demonstrated in Ref. [13], the local Z2 -degeneracy of the ground state generates a local real fermionic zero mode γ0 which is coupled to the three Majorana band fermions with the Lagrangian: X1 [ γj0 ∂τ γj0 + g ′ γj0 χ1j χ2j χ3j ]. L= (17) 2 j This singlet zero mode acquires a small dispersion with the bandwidth ∆g ∼ ∆2 /Jk ≪ ∆. The total amount of entropy accumulated in this band is 21 ln 2 per central spin and this degeneracy will be slightly lifted and results in the coherent dispersion of this mode. This mode corresponds to the strongly localized Majorana fermion found in the Emery-Kivelson limit. This field will manifest itself in the spin-spin correlation functions of the central spins since at low energies, they behave like s ∼ γ0 χ ~ . Due to the small dispersion of the fermionic mode γ0 , correlation functions of middle spins are strongly localized in space, but not in time where the characteristic scale is ∼ ∆g−1 : hhs(τ, j)s(0, j)ii ∼ K1 (∆g |τ |) (18) which for small times is proportional to 1/|τ |, like for the two-channel Kondo problem. 4