- Email: [email protected]
Ground state selection in a Kagomé antiferromagnet
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1693-1694
Journal of magnetism and magnetic
A• ,~
m~erlals
ELSEVIER
Ground state selection in a Kagom6 antiferromagnet C.L. Henley *, E.P. Chan Dept. of Physics, Cornell University, Ithaca, NY 14853, USA Abstract We address selection of a spin-ordered ground state in the quantum Heisenberg nearest-neighbor antiferromagnet with large spin S on the highly frustrated Kagom6 lattice. By treating the anharmonic terms in the spin-wave expansion, we obtain an effective Hamiltonian bilinear in Ising variables which parametrize the infinity of coplanar classical ground states. This favors 'vC3 × x/3'spin order, in accordance with prior expectations.
The Kagom6 lattice consists of comer-sharing triangles (obtainable by placing a spin on every bond of a triangular lattice). We study the model with Heisenberg spins on each site antiferromagnetic nearest-neighbor exchange: ~'~exch = ~(ij) [J [ si" sj (with I J [ -= 1). We will use the spin-wave expansion of ~exch (see Eqs. (1)-(3)) about a classical ground state configuration, in which 1 / S is the small parameter. The goal of this work is to determine the spatial pattern of spin order (assuming it exists) in the ground state. Often (e.g. on the triangular lattice) this is given at zero order since the classical ground state is unique (modulo spin rotations). But the Kagom6 antiferromagnet has many degenerate ground states unrelated by symmetry. In many systems, such as fcc antiferromagnets [1], such a degeneracy is resolved by summing the O(JS) zero-point energy Y'.k½hto(k), which implicitly depends on which ground state we expanded around. In the Kagom6 case this zeropoint energy favors coplanar ground states [2,3]. There is a discrete infinity of coplanar ground states any configuration in which each triangle of sites has spins at angles 0, 27r/3 and - 2 " r r / 3 . It is convenient to label them by 'chiralities' T/,~= _ 1, where a are sites (forming a honeycomb lattice) at the centers of Kagom6 triangles. Here r/~ -= + 1( - 1) depending on whether the spins rotate clockwise (counterclockwise) as one walks around the triangle. The Q = 0 state and the vc3 × v~ state (ordering wavevector Q = (27r/3, 21r/3)) are special, symmetric coplanar ground states, corresponding, respectively, to ferromagnetic and antiferromagnetic patterns of {,/,~}. The spin-wave expansion around an arbitrary coplanar ground state is .X(sw= "~2 + "U3 + a~4, where the terms have schematic form:
* Corresponding author. Email: [email protected].
X z = JO(crx2 + cry2); ~x"3 = Y'- "q,,"~3,,,
ot
(1) (2a)
with operators independent of the {~/,~} and just coupling spin pairs within one triangle at o~:
'~3ct = (J//S)O([°'i?) + cri(xs)] °y2) , "~4 = ( J/SZ)O(cri~yS)4) •
(2b) (3)
Here 'Cry' and 'Crx' are deviations out of and within the spin plane, respectively [4]. Also, ,cr/(s), and 'Crib)' denote the projections of Crx and Cry onto 'soft modes' [ks), a special branch of spin waves with t%(k)=-0 throughout the Brillouin zone [5]. Finally, ,cri~o), and 'tri~°)' refer to the other ('ordinary') modes I ks). The only dependence of Xsw on the discrete state is via the prefactors ~,~ in ,Yl3. Being derived from ,U2, oJ(k) is identical for every coplanar ground state, and so E k l h w ( k ) fails to select one of them. In the rest of this paper, we use (2) and (3) to find an effective Ising Hamiltonian Ei({r/,,}), approximately equal to (,Usw) from anharmonic fluctuations around the given ground state. Our motivation is that the couplings {,Y~t~} (see (6)-(8) below) carry much more information than the energies of the Q = 0 and ~ × ~ states. We could use E,({7/,,}) as the Hamiltonian in a partition function to describe finite-temperature behavior. Alternatively, a quantum Potts Hamiltonian could be constructed from E~ plus additional terms for the amplitudes to tunnel among coplanar states, to study how tunneling might (partially) disorder the ground state [8]. Our approach can be understood as a variant of perturbation theory. Due to the soft modes, most terms in an expansion about "g'~2 would be divergent. Instead we expand about ~ = " ~ 2 - JaY~li< jcriyO'jy' where '57' runs over third-nearest neighbors. The J3 term controls the
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00684-9
1694
C.L. Henley, E.P. Chan /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1693-1694
mean-square soft fluctuations, while preserving the (soft) Goldstone behavior oJs(k) ot ] k - Q I [3,5]. Then:
( Gri(.x °)2 ) ~ < O'i(y°)2 ) ~ S;
(4a)
ws(k) ~ < Ori(xs)2) ~ I J3 J 1/2S, (O'/(s)2) ~ I J3 ] -1/25
(48) (4c)
(J3, to be set variationally, will vanish as S ~ o0.) So our estimate for the ground state energy is Eg = e 0 4- ((~2 -a~)) 4- <'¢~4)nt-E2({T~a}),
(5)
E2 = - ~,f~'q,,'qt3,
(6)
Hence (J(s'°) labels contributions from s and o subspaces), ff~(~n) = N - 1 EeiK.r=~,(m)(K)-llA(m)(K)] 2.
(12)
K
Eq. (12) can produce ,y~, in two ways: (i) Approximate g,(m) = const -= ~(m); then
,~a(a,~) = ~ - 1 < a~3a/~(m)c~3fl )
(13)
which is negative for nearest-neighbor ( a , /3), regardless of the sign of J3. (ii) Approximate I A(m)(K) I 2 --. const =
J~= ~ , < O J , , ~ l n ) ( E , - E o ) - l < n [ ~ l [ ) ) ,
(7) n where 10) is the ground state of ,,~, and In) are excited states with energies E,,. Then Eg = ffSg+EI({~,~}) , where E~ is E 2 excluding the a =/3 terms, and ffSg= E 0 + b l J3 ] 1/25 + (Cl -Jaot )1 J3 [-1,
(8)
exhibiting the J3 and S dependences, which could be deduced from (3) and (4), along with J,~t3 ~ J,~t3 I J3 I -1, and ~ - c a -j,~,~. (The b and c a terms in (8) arise from the second and third terms of (5).) Our rule for the choice of J3 is to minimize E__g. Anticipating j,~ts/~ ~ 1/10, we instead just minimize Eg. If ? > 0, this happens at
I J3 ] = [2c./b] 2/35-2/3, ffSg-- e 0 = ( 3 / 2 1 / 3 ) b 2 / 3 c l / 3 5 2 / 3
(9a) .
(9b)
With (4), this implies ( o-/~)2 ) ~ 54/3; and w~(k) ~ ffSg~
52/3, as found in Ref. [3]. While optimizing J3, we should also try both signs, J3 > 0 or J~ < 0, which really denote different forms of 'g(~w expanded about a Q = 0 or v ~ X I/~ ground state. The two cases give different numerical values of b and ~; this affects Eg more than E n does. Hence we should use whichever sign of J3 minimizes (8), for all patterns of {~7,~}. Now, the distribution of values {to~(k)} has (slightly) less dispersion in the J3 > 0 case. This reduces the sums of negative powers of w~(k) contributing to ~. Hence it is expected that (9b) is smallest in the J3 > 0 case. We should find the configuration of {~L} that minimizes Ei({r/a}); this depends on knowing ,Y~t3- To extract pJ~ot~ (given J3 ) we first group terms in (7) as follows. Let be the projection operator for the subspace of intermediate states - both one-magnon I n ) = ] K o ) and threemagnon I klO, k2s, k3s), with k 1 + k 2 + k 3 = K; we define /9(~) likewise but with all soft modes (] K s ) or I klS, k2s, k3s>). (Other intermediate states give higher-order contributions.) We write (a'~3aff(rn),¢~3/3 > ~ N -11 a ( m ) ( K ) 12 e ig'r"~, (10) which defines ]A(m)(K)]; here N is the number of K vectors, and (m) = (s, o). Similarly, we can write
(,¢~35 l~(m ' (a.~ -- E0 ) - 1a~3/3 ) ------~'(m)(K) -- 1(~3~ t~(m),c~3B >.
(11)
X Fourier transform of [ ~ ' ( m ) ( K ) - 1].
(14)
Presuming that ~ ( K ) - 1 is most enhanced when the contributing modes kl,2,3 are near the Goldstone point Q, (14) gives a nearest-neighbor j,~0 with the same sign as the input J3. As an approximation we replaced the excited states In) by 'triangle excitations' ]K, m) a EeiXr~Q (m) ]0) (normalized); here Q~m) is the term in "~3,, of O(~r~m)~i~)2). Then we took ~(m)(K) = ( K , m l , , ~ Eo)[ K, m) and a(m)(K) ~ (6 1'~3,~ I K, m). (This was originally justified variationally.) With this scheme, for J3 < 0, we found b = 0.2206, c I = 0.0201, and Ja,~ = 0.0952. Note that we obtained < 0 in (8), which is unphysical. Our method grossly overestimates ]c2]; hence we cannot find the absolute scale of O~s(k), of the quantum fluctuations, or of , y (the scale of energy differences among different coplanar states). However, our results for the form of EI({~,~}) are insensitive to the magnitude of I J3 ]. Thus we proceed as if ~ > 0, and get j , ~ = -0.0125 for nearest neighbors. We found that contribution type (i) dominates, whichever sign of J3 we use. Hence we conclude that the vr3 × V~ state is selected, as has been commonly believed [5-7]. Acknowledgements: We thank A. Chubukov for helpful conversations. This work was supported by NSF grant DMR-9214943. References
[1] E.F. Shender, Sov. Phys. JETP 56 (1982) 178; C.L. Henley, Phys. Rev. Lett. 62 (1989) 2056. [2] I. Ritchey, P. Coleman and P. Chandra, Phys. Rev. B 47 (1993) 15342. [3] A. Chubukov, Phys. Rev. Lett. 69 (1992) 832. [4] J. Chalker, P.C.W. Holdsworth and E.F. Shender, Phys. Rev. Lett. 68 (1992) 855. [5] A.B. Harris, C. Kallin and A.J. Berlinsky, Phys. Rev. B 45 (1992) 2899. [6] S. Sachdev, Phys. Rev. B 45 (1992) 12377. [7] A.P. Ramirez, G.P. Espinosa and A.S. Cooper, Phys. Rev. Lett. 64 (1990) 2070; C. Broholm et al., Phys. Rev. Lett. 65 (1990) 3173. [8] J. von Delft and C.L. Henley, Phys. Rev. B 48 (1993) 965.
Journal of magnetism and magnetic
A• ,~
m~erlals
ELSEVIER
Ground state selection in a Kagom6 antiferromagnet C.L. Henley *, E.P. Chan Dept. of Physics, Cornell University, Ithaca, NY 14853, USA Abstract We address selection of a spin-ordered ground state in the quantum Heisenberg nearest-neighbor antiferromagnet with large spin S on the highly frustrated Kagom6 lattice. By treating the anharmonic terms in the spin-wave expansion, we obtain an effective Hamiltonian bilinear in Ising variables which parametrize the infinity of coplanar classical ground states. This favors 'vC3 × x/3'spin order, in accordance with prior expectations.
The Kagom6 lattice consists of comer-sharing triangles (obtainable by placing a spin on every bond of a triangular lattice). We study the model with Heisenberg spins on each site antiferromagnetic nearest-neighbor exchange: ~'~exch = ~(ij) [J [ si" sj (with I J [ -= 1). We will use the spin-wave expansion of ~exch (see Eqs. (1)-(3)) about a classical ground state configuration, in which 1 / S is the small parameter. The goal of this work is to determine the spatial pattern of spin order (assuming it exists) in the ground state. Often (e.g. on the triangular lattice) this is given at zero order since the classical ground state is unique (modulo spin rotations). But the Kagom6 antiferromagnet has many degenerate ground states unrelated by symmetry. In many systems, such as fcc antiferromagnets [1], such a degeneracy is resolved by summing the O(JS) zero-point energy Y'.k½hto(k), which implicitly depends on which ground state we expanded around. In the Kagom6 case this zeropoint energy favors coplanar ground states [2,3]. There is a discrete infinity of coplanar ground states any configuration in which each triangle of sites has spins at angles 0, 27r/3 and - 2 " r r / 3 . It is convenient to label them by 'chiralities' T/,~= _ 1, where a are sites (forming a honeycomb lattice) at the centers of Kagom6 triangles. Here r/~ -= + 1( - 1) depending on whether the spins rotate clockwise (counterclockwise) as one walks around the triangle. The Q = 0 state and the vc3 × v~ state (ordering wavevector Q = (27r/3, 21r/3)) are special, symmetric coplanar ground states, corresponding, respectively, to ferromagnetic and antiferromagnetic patterns of {,/,~}. The spin-wave expansion around an arbitrary coplanar ground state is .X(sw= "~2 + "U3 + a~4, where the terms have schematic form:
* Corresponding author. Email: [email protected].
X z = JO(crx2 + cry2); ~x"3 = Y'- "q,,"~3,,,
ot
(1) (2a)
with operators independent of the {~/,~} and just coupling spin pairs within one triangle at o~:
'~3ct = (J//S)O([°'i?) + cri(xs)] °y2) , "~4 = ( J/SZ)O(cri~yS)4) •
(2b) (3)
Here 'Cry' and 'Crx' are deviations out of and within the spin plane, respectively [4]. Also, ,cr/(s), and 'Crib)' denote the projections of Crx and Cry onto 'soft modes' [ks), a special branch of spin waves with t%(k)=-0 throughout the Brillouin zone [5]. Finally, ,cri~o), and 'tri~°)' refer to the other ('ordinary') modes I ks). The only dependence of Xsw on the discrete state is via the prefactors ~,~ in ,Yl3. Being derived from ,U2, oJ(k) is identical for every coplanar ground state, and so E k l h w ( k ) fails to select one of them. In the rest of this paper, we use (2) and (3) to find an effective Ising Hamiltonian Ei({r/,,}), approximately equal to (,Usw) from anharmonic fluctuations around the given ground state. Our motivation is that the couplings {,Y~t~} (see (6)-(8) below) carry much more information than the energies of the Q = 0 and ~ × ~ states. We could use E,({7/,,}) as the Hamiltonian in a partition function to describe finite-temperature behavior. Alternatively, a quantum Potts Hamiltonian could be constructed from E~ plus additional terms for the amplitudes to tunnel among coplanar states, to study how tunneling might (partially) disorder the ground state [8]. Our approach can be understood as a variant of perturbation theory. Due to the soft modes, most terms in an expansion about "g'~2 would be divergent. Instead we expand about ~ = " ~ 2 - JaY~li< jcriyO'jy' where '57' runs over third-nearest neighbors. The J3 term controls the
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00684-9
1694
C.L. Henley, E.P. Chan /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1693-1694
mean-square soft fluctuations, while preserving the (soft) Goldstone behavior oJs(k) ot ] k - Q I [3,5]. Then:
( Gri(.x °)2 ) ~ < O'i(y°)2 ) ~ S;
(4a)
ws(k) ~ < Ori(xs)2) ~ I J3 J 1/2S, (O'/(s)2) ~ I J3 ] -1/25
(48) (4c)
(J3, to be set variationally, will vanish as S ~ o0.) So our estimate for the ground state energy is Eg = e 0 4- ((~2 -a~)) 4- <'¢~4)nt-E2({T~a}),
(5)
E2 = - ~,f~'q,,'qt3,
(6)
Hence (J(s'°) labels contributions from s and o subspaces), ff~(~n) = N - 1 EeiK.r=~,(m)(K)-llA(m)(K)] 2.
(12)
K
Eq. (12) can produce ,y~, in two ways: (i) Approximate g,(m) = const -= ~(m); then
,~a(a,~) = ~ - 1 < a~3a/~(m)c~3fl )
(13)
which is negative for nearest-neighbor ( a , /3), regardless of the sign of J3. (ii) Approximate I A(m)(K) I 2 --. const =
J~= ~ , < O J , , ~ l n ) ( E , - E o ) - l < n [ ~ l [ ) ) ,
(7) n where 10) is the ground state of ,,~, and In) are excited states with energies E,,. Then Eg = ffSg+EI({~,~}) , where E~ is E 2 excluding the a =/3 terms, and ffSg= E 0 + b l J3 ] 1/25 + (Cl -Jaot )1 J3 [-1,
(8)
exhibiting the J3 and S dependences, which could be deduced from (3) and (4), along with J,~t3 ~ J,~t3 I J3 I -1, and ~ - c a -j,~,~. (The b and c a terms in (8) arise from the second and third terms of (5).) Our rule for the choice of J3 is to minimize E__g. Anticipating j,~ts/~ ~ 1/10, we instead just minimize Eg. If ? > 0, this happens at
I J3 ] = [2c./b] 2/35-2/3, ffSg-- e 0 = ( 3 / 2 1 / 3 ) b 2 / 3 c l / 3 5 2 / 3
(9a) .
(9b)
With (4), this implies ( o-/~)2 ) ~ 54/3; and w~(k) ~ ffSg~
52/3, as found in Ref. [3]. While optimizing J3, we should also try both signs, J3 > 0 or J~ < 0, which really denote different forms of 'g(~w expanded about a Q = 0 or v ~ X I/~ ground state. The two cases give different numerical values of b and ~; this affects Eg more than E n does. Hence we should use whichever sign of J3 minimizes (8), for all patterns of {~7,~}. Now, the distribution of values {to~(k)} has (slightly) less dispersion in the J3 > 0 case. This reduces the sums of negative powers of w~(k) contributing to ~. Hence it is expected that (9b) is smallest in the J3 > 0 case. We should find the configuration of {~L} that minimizes Ei({r/a}); this depends on knowing ,Y~t3- To extract pJ~ot~ (given J3 ) we first group terms in (7) as follows. Let be the projection operator for the subspace of intermediate states - both one-magnon I n ) = ] K o ) and threemagnon I klO, k2s, k3s), with k 1 + k 2 + k 3 = K; we define /9(~) likewise but with all soft modes (] K s ) or I klS, k2s, k3s>). (Other intermediate states give higher-order contributions.) We write (a'~3aff(rn),¢~3/3 > ~ N -11 a ( m ) ( K ) 12 e ig'r"~, (10) which defines ]A(m)(K)]; here N is the number of K vectors, and (m) = (s, o). Similarly, we can write
(,¢~35 l~(m ' (a.~ -- E0 ) - 1a~3/3 ) ------~'(m)(K) -- 1(~3~ t~(m),c~3B >.
(11)
X Fourier transform of [ ~ ' ( m ) ( K ) - 1].
(14)
Presuming that ~ ( K ) - 1 is most enhanced when the contributing modes kl,2,3 are near the Goldstone point Q, (14) gives a nearest-neighbor j,~0 with the same sign as the input J3. As an approximation we replaced the excited states In) by 'triangle excitations' ]K, m) a EeiXr~Q (m) ]0) (normalized); here Q~m) is the term in "~3,, of O(~r~m)~i~)2). Then we took ~(m)(K) = ( K , m l , , ~ Eo)[ K, m) and a(m)(K) ~ (6 1'~3,~ I K, m). (This was originally justified variationally.) With this scheme, for J3 < 0, we found b = 0.2206, c I = 0.0201, and Ja,~ = 0.0952. Note that we obtained < 0 in (8), which is unphysical. Our method grossly overestimates ]c2]; hence we cannot find the absolute scale of O~s(k), of the quantum fluctuations, or of , y (the scale of energy differences among different coplanar states). However, our results for the form of EI({~,~}) are insensitive to the magnitude of I J3 ]. Thus we proceed as if ~ > 0, and get j , ~ = -0.0125 for nearest neighbors. We found that contribution type (i) dominates, whichever sign of J3 we use. Hence we conclude that the vr3 × V~ state is selected, as has been commonly believed [5-7]. Acknowledgements: We thank A. Chubukov for helpful conversations. This work was supported by NSF grant DMR-9214943. References
[1] E.F. Shender, Sov. Phys. JETP 56 (1982) 178; C.L. Henley, Phys. Rev. Lett. 62 (1989) 2056. [2] I. Ritchey, P. Coleman and P. Chandra, Phys. Rev. B 47 (1993) 15342. [3] A. Chubukov, Phys. Rev. Lett. 69 (1992) 832. [4] J. Chalker, P.C.W. Holdsworth and E.F. Shender, Phys. Rev. Lett. 68 (1992) 855. [5] A.B. Harris, C. Kallin and A.J. Berlinsky, Phys. Rev. B 45 (1992) 2899. [6] S. Sachdev, Phys. Rev. B 45 (1992) 12377. [7] A.P. Ramirez, G.P. Espinosa and A.S. Cooper, Phys. Rev. Lett. 64 (1990) 2070; C. Broholm et al., Phys. Rev. Lett. 65 (1990) 3173. [8] J. von Delft and C.L. Henley, Phys. Rev. B 48 (1993) 965.