The exact critical field for ring exchange Hamiltonians in 3He

NE 5

Physica 108B (1981) 855-856 North.Holland PublishingCompany

THE EXACT CRITICAL FIELD FOR RING EXCHANGE HAMILTONIANS IN 3HE

J. H. Hetherington

Department of Physics Michigan State University East Lansing, Michigan 48824 Mean Field Theory gives the exact critical field (H c) at zero temperature for the transition from the spin-flop to the completely polarized (ferromagnetic) state for the nearest neighbor antiferromagnetic Hamiltonian. This result is extended to Hamiltonians of interest in the theory of magnetism in solid 3He. The lower bound for H c from Mean Field Theory and the upper bound for H c from the Anderson cluster method are equal for many axially symmetric Hamiltonians. This equality is proved for: i) the Heisenberg spin ½ antiferromagnet on sq and bcc lattices with first and second neighbor interaction; ii) the bcc spin ½ Hamiltonian with cyclic exchange of 2, 3 and 4 spins. This observation allows measurement of a different linear combination of the exchange constants than the Curie Weiss constant, Q • W

The fact that the Mean Field Approximation (MFA) leads to an incorrect result for T c when it is applied to the Heisenberg Hamiltonian is well known. The fact that MFA gives the exact result for the critical field (H c) at T = 0 for the antiferromagnetic Heisenberg Hamiltonian is not widely known, but the point is made in the literature.l, 2 It is shown below that the result can be extended to spin Hamiltonians of interest in solid 3He due to 4-atom ring exchange. The procedure of proof must be applied to each Hamiltonian in question. This suggests that it is possible to find Hamiltonians for which MFA gives the incorrect H c. The method of proof for each case so far investigated is very simple, involving only the determination of the eigenstates of a small cluster of spins. It is also easy to find systems for which H c (MFA) is probably exact but for which the proof fails because a large cluster would be required. An important consequence of an exactly calculable H c is that a known linear combination of the exchange constants can be measured experimentally. For cases which contain more than one parameter the linear combination differs from the linear combination in the Curie-Weiss constant. The method of proof is as follows: The MFA solution (really just a Ritz Variational problem if T = 0) gives an upper bound on the energy as a function of field. The value of H c must be greater than or equal to the value from MFA: MFA requires that the energy drop below the ferromagnetic state's energy when H < H M F K Therefore at some field H ~ HMF A we must have a transition away from the ferromagnetic state. On the other hand, the value of H c must be lower than a value determined from the Anderson cluster 3 lower limit on the energy: The lower limit on E coincides with the ferromagnetic state's energy above some value, HAn d , so that the ferromagnetic state is the ground state of the system for fields above HAn d . Therefore 0378-4363/81/0000~00/$02.50

© North-HollandPublishingCompany

H c is below HAn d and above HMF A. When they coincide then HMF A must be exact. The Anderson cluster method relies on the fact that the lowest eigenvalue %o of an operator H which is the sum of operators H i (H = ~ H i) is greater than or equal to the sum of the lowest eigenvalues ~ of the operators H i (~o ~ ~ ) " By breaking the Hamiltonian H into parts corresponding to clusters, one may find various lower limits to the ground state energy of H. The ferromagnetic state can be shown to be the ground state when H is large enough as follows: If+the magnetic term is of the usual form (-H'~ s i) and if the remainder of H is axially symmetric about H then every state is an eigenstate of S z (the z component of the total spin of the cluster) and the states all have energies of the form E o - M H S z. (Where E o and M are constants). The ferromagnetic state has the maximum possible value for S z since all spins are aligned. Therefore when H is large enough the ferromagnetic state will have lowest energy. A fairly systematic method is to: a) Find a cluster large enough that the antlferromagnetic (spin flopped) state can be expressed on the cluster so that MFA applied to the cluster will be the same as the MFA applied to the whole system and b) examine the elgenstates at H = HMFA to prove that no states have lower energy than the S = S z = S m a x (ferromagnetic state at that field). A) The first and second nei~hbor~ spin ~ Helsenberg antiferromagnet on the sq or on the bcc lattice. nn ÷ ~ nnn ~ + n

855

856

where Oi= 2s i , and where the sums over the pairs are taken over nearest neighbors (nn) and over next nearest neighbors (nnn). It is assumed that Jl < 0 and J2 < 0. The mean field result (that uH c = - q l J l

when (-qlJl) > 2(-q2J2)

and that uH c =½(-qlJl ) + (-q2J2) when (-qlJl) < 2(-q2J2) ) is exact.

The cluster used

is a four particle cluster in a square with the edges first neighbor interactions and the diagonals second neighbor interactions. B) Hamiltonians of interest in the theory of solid 3He magnetism 4 which result from ring exchange of 2, 3 and 4 particles. For the bcc lattice there are two possible 4 particle rings: folded and planar. Proof has been obtained when either kind of 4-ring exchange is possible. H = (-J

) [ p.._ nn 13

(_j_) ~ (p.. -i t 13k+Pijk)

-i - HU [ O z i + (-K) (Pij k£+Pij k£ ) i

where the P's are the cyclic spin permutation operators and where the sums are over all nearest neighbor spins, triangles with two nearest neighbors and one second neighbor sides, and either planar or folded quadralaterals with nearest neighbor sides. In these cases the cluster is either a planar or folded quadralateral with appropriate interaction. Assuming that Jnn <0' Jt <0' K < 0 and that

(-Jnn)+6(-K)

> 6 ( - J t ) one f i n d s t h a t t h e NFA

critical field uH c = - 8 J n n + 4 8 ( J t - K ) is exact. As an example, the cluster Hamiltonian for the planar 4-ring is i {~i~iPi,i+ I- B =

[

(p_+p-l) +y(p4+P41)

i=1,3

i

i

i=l where c ~ = - J

; 8 = - 6 J t ; y = - 6 K , and where P nn i represents a triangle on the three atoms not including i, P4 is the 4-ring exchange on all 4 atoms. The cluster Hamiltonian above has eigenvalues i I 2 2 l e(~-S13-$24+2)

2 2 2 - 8(S +S13-$24-2)

2 3 2 ~)]3 + y[½($2-3)2-2 - 2(S13-~)'($24~

S zI )

where S13 and $24 are the spin of the spin pairs 13 and 24, and S and S

are the total spin Z

and its z component. For H > H c the state of lowest energy is the, S 1 3 z S 2 4 = i, S = S z = 2

state (ferromagnetic), it is degenerate with the S 1 3 = $ 2 4 = I, S = S Z = 1 state when H = H C . For the Hamiltonian proposed in ref. 4 the critical field (at T = 0 and at melting pressure) is 15.72 Tesla. Since the exchange parameters are very sensitive to pressure this field should be reduced to an experimentally accessible value for specific volumes below about 23 cc/mole. Measurement of this quantity, Ow, and the specific heat coefficient at the same volume would determine three independent combinations of the exchange coefficients. REFERENCES: [i] Harold Falk, Phys. Rev. 133A, 1382 (1964). [2] Fredrick Keffer, Handbuch der Physik XVIII/2, i (1966). [3] P. W. Anderson, Phys. Rev. 83, 1260 (1951). [4] M. Roger, J. M. Delrieu, J. H. Hetherlngton, Phys. Rev. Lett. 45, 137 (1980).