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Quantum tunneling at excited states and macroscopic quantum coherence in ferromagnetic particles
31 March 1997 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 228 (1997) 97-102
Quantum tunneling at excited states and macroscopic quantum coherence in ferromagnetic particles J . - Q . L i a n g a,b,c, H . J . W . M t i l l e r - K i r s t e n a, J i a n - G e Z h o u a, F.-C. P u c,a a Department of Physics, University of Kaiserslautern, D-67653 Kaiserslautern, Germany b Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China c Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China d Department of Physics, Guangzhou Teachers College, Guangzhou 510400, China
Received 2 October 1996; acceptedfor publication 17 January 1997 Communicated by J. Flouquet
Abstract
The consideration of the level splitting of the ground state of a single ferromagnetic particle due to tunneling through energy barriers is extended to low-lying excited states with the help of the LSZ method in field theory, and is calculated up to the one loop correction. Contrary to the leading order WKB approximation, the prefactor of the exponential increases with the spin s of the particle, and therefore decreases the slope of the resonant frequency as a function of the volume of the particle. The temperature dependence of the tunneling probability is also calculated by means of a Boltzmann average. The results should be useful in analysing results of experimental tests of macroscopic quantum coherence. @ 1997 Elsevier Science B.V.
Quantum tunneling between macroscopically distinct states is of great interest both theoretically and experimentally [ 1 ]. Macroscopic magnetisation tunneling has therefore been extensively investigated in the last decade owing to the possible experimental observation of this phenomenon. In the context of these investigations the usual terminology is that macroscopic quantum tunneling (MQT) refers to the decay of a metastable state, while macroscopic quantum coherence (MQC) refers to the resonance between neighboring degenerate states [2]. A theoretical analysis of tunneling for both MQT and MQC in small ferromagnetic particles using the instanton method was begun some time ago [3], but was carried out only in the leading order WKB approximation. It was pointed out recently [4] that in the theoretical analysis of MQC
the periodic potential results formally in an energy "band" structure which is quenched to a single level for half-integer spin s. The one loop correction to the level splitting was also given for ferromagnetic particles. This provides an explicit formula for a quantitative evaluation of the resonance frequency toc = A E / h . A few years ago it was reported that MQC was observed for antiferromagnetic particles [5] in resonance experiments. However, a controversy exists [2,5,6]. The physical explanation of the result of an earlier resonance experiment on ferromagnetic particles is not clear since fundamental discrepancies remain between the experimental data and theoretical expectations on the basis of magnetic quantum tunneling [7,8]. Apart from other reasons [7,8], which are obstacles for the acceptance of the observation as a
0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PHS0375-9601(97)00071-6
98
J.-Q. Liang et aL/Physics LettersA 228 (1997) 97-102
crucial proof of MQC, there are two essential difficulties related to the existing theory of quantum tunneling itself in the absence of an external magnetic field. One difficulty was pointed out in Ref. [2]. The argument of the WKB exponential of the tunneling for a ferromagnetic particle is 2v/As with h = K2/KI, Kl and K2 being the hard and medium axis energies. Since s ,-~ 500 to 5000 unless K2/KI << 10 -4, we is expected to be unobservably small. Even though the phenomenon does not seem totally unobservable, it is not known that such a small A is realistic. The other difficulty is that according to the WKB leading order approximation the theoretical relation between the volume of the particle and the observed resonant frequency is quite different from the best fit of experimental data. The question is whether or not the above problem is due to a shortcoming of the leading order approximation itself. It is important therefore to perform a complete investigation of tunneling including the correction of quantum fluctuations and tunneling at lowlying levels. In the spirit of the method of Lehmann, Symanzik and Zimmermann (LSZ) in field theory [9], which provides a formula for the reduction of S-matrix elements to the Green's function expressed as a timeordered product of field operators, we construct here Euclidean field operators. These operators allow the calculation of the instanton induced transition amplitude between excited states of neighboring potential wells as demonstrated recently [ 10]. An explicit formula for the level splitting of low-lying levels is obtained which includes the contribution of quantum fluctuations. It is remarkable that the prefactor of the leading order exponential increases with s contrary to the decreasing behavior of the exponential. This result may be of importance in theoretical analysis of the relation between the volume of the ferromagnetic particle and the observed resonant frequency obtained from experiments. The probability enhances significantly if tunneling at an excited state is considered. This enhancement opens the possibility to observe MQC for much less anisotropy A ,,~ 10 -4 of the ferromagnetic particle. We begin with the Hamilton operator of the ferromagnetic particle
/~/= Kl.~z2 + K 2 g 2,
(1)
which describes [3] XOY easy plane anisotropy and an easy axis along the x direction with K1 > K2 > 0. In Eq. (1) gi, i = x, y, z, are spin operators obeying the usual commutation relation [~i, gi] = ieijk~k (using natural units throughout). Starting from the coherent state representation [ 11 ] of the time evolution operator with Hamiltonian given by Eq. ( 1 ) and with the help of the coherent state path integral we obtain
(nf]e-2iI:Ir[ni)
=
e-i(qlf-~bi)sK~(t~f,tf; ~bi,/i),
(2)
where If
IC(qbf, t f ; f b i , t i ) = f D c k D p e x p ( i / £ ( g b ,
p) dt)
ti
(3) is the path integral in phase space with canonical variables ~b and p -- s cos 0. Also
£. = (bp - H( dp,p)
(4)
is the phase space (or first order) Lagrangian. The Hamiltonian p2 H = - 4- V(q~) 2m(ff)
(5)
has position dependent mass m(~b) and potential m(~b) =
1
2K1 ( 1 - a sin 2 4,) '
V(dp) = K2s 2 sin 2 ~b,
(6)
respectively, where A -----K2/KI. The kets [ni) and Inf) denote the initial and final spin-coherent states and tf - ti --- 2T denotes the difference of final and initial times. Here s = s(sin 0 cos ~b, sin 0 sin ~b, cos 0) is visualized as a classical spin vector with spin number s, polar angle 0 and azimuthal angle ~b. In the above derivation, the large spin limit s >> 1 has been used since giant spins with spin quantum number s >> 1 are believed to describe correctly ferromagnetic grains. A novel feature of the transition amplitude given by Eq. (2) is the phase factor e -i(~f-4'Os which can be put into the Lagrangian, i.e. the expression q~f - t~i = fti' q~dt, and identified as a Wess-Zumino term [ 12]. The effect of the phase factor which indicates anomalous behavior of the quantum spin system has been
99
J.-Q. Liang et al./Physics Letters A 228 (1997) 97-102
studied extensively [4,13]. Integrating out the momentum in the path integral Eq. (3), we obtain the usual Feynman propagator in configuration space, i.e. tf
lC(4af,tf;cbi,ti)=fg)4~exp(if£(4~,4~)dt), ti
(7) where 73~b is the measure-modified functional differential resulting from the ~b-dependent mass, i.e.
which can be evaluated with the help of the instanton method of path integrals. Here Um(~b, q~K) are the usual eigenfunctions of ~0. Passing to imaginary time by Wick rotation r = it, the instanton configuration is a classical solution which minimizes the Euclidean action SE, i.e. t~SE = t~ f / ~ E d r = 0, where •E = ½m(~b)t~2 + V(~b) is the Euclidean Lagrangian and = d~b/dr denotes from now on the imaginary time derivative. The instanton configuration is then found to be ~bc = arcsin[cosh 2 too(7- - To)
•t-1 m/-~-7) d~bk, / ) & = H V 2,n-iE k
- A sinh 2 too(7- - 7-0) ] -1/2,
( 11 )
which is more convenient for the instanton method used in the following. Since the potential V(~b) is periodic, the low-lying discrete levels possess formally a "band structure" and are obtained in terms of"Bloch waves", i.e.
which can be visualized as the classical trajectory of a pseudoparticle in the barrier. At time 7- --~ -cx~ it starts from one potential well, for example that at q'r = K~r, and reaches the neighboring well at qSX+l = (K + 1)Tr at 7- --* c~. In Eq. (11) 7-o is the position of the instanton. We begin with the instanton induced amplitude for transitions between degenerate eigenstates in any two neighboring wells (say those for K = 0, 1 ), i.e.
E(s + ~:) = e,, -- 2Aem cos'n'(s + ~:),
An/.i = (m, ~, le-ZVf/lm, qO0) = e -2Te" sinh2TAem.
with the second-order Lagrangian = ½m(q~) ~2 _ V(q~),
(8)
(9)
where sc denotes the "Bloch wave vector" which is integral here (determined by the requirement of 2~- periodicity of the "Bloch states"). The quantity Em is the eigenvalue of Hamiltonian //0 such that /~/01m, 4,n) = emlm, q~n); here /~0 is the oscillatorapproximated Hamiltonian in the nth well near the local minimum qbn = n~- and is seen to be p2
/4o = ~
+ ½m0too2~2
in the large s limit, where we have expanded the potential V(~b) and mass m(~b) around the local minima, and it is seen that mo = 1/2K1 and coo2 = 4K1K2s2; in this approximation we have Em = ( m + ½)too. Each "energy band" has only two levels corresponding to even and odd integral values of s + ( and shrinks to a single level for half-integer spin s in agreement with Kramer's degeneracy. The quantity 2Aem is the level splitting defined by Aem = - f
u~n(~b, ~/<+1 )/~/Um(q~, ~K) d~b
(10)
(12) In the following the left hand side of Eq. (12) is evaluated with the help of the Euclidean path-integral method. After evaluation the result is compared with the right hand side of Eq. (12) to obtain Aem. The vacuum instanton, Eq. (11 ), which satisfies the vacuum boundary condition is only suitable for the calculation of the tunneling effect between degenerate vacuum states (m = 0). A method for calculating the tunneling effect at excited states was developed recently with the help of the nonvacuum instanton (or periodic instanton) which is a saddle point configuration [ 14]. In this note we adopt an alternative but equivalent approach [ 10,15] to evaluate the transition amplitude using the (vacuum) instanton solution of Eq. ( 11 ). In field theory the creation operator of a particle (or pseudoparticle) is related to the field operator q~ (position in quantum mechanics) in the interaction picture by a t = -i,~ 2 ~ e -i~°°' --~ ~ ( t ) . v too o~t
(13)
100
J.-Q. Liang et al.IPhysics Letters A 228 (1997) 97-102
With this relation the S-matrix can be related to the Green's function through the procedure known as the LSZ reduction technique [9]. In the spirit of the LSZ method we now calculate the amplitude for a transition through the barrier by constructing interacting Euclidean fields ~b4- in the classically forbidden region with the help of the instanton configuration of Eq. ( 11 ). Thus we define ~b+ :=Tr-~bc,
~b_ :=~bc
lim
Sm
rf~,~
1 f i ( - ~ 2 ~ e ' ° ° r ~ 00~tf) /=1 \ v too
\ v too = --
_ __
m! l=1 \
(14)
x
o
~
etOO(~t-r~)
too/ + too
07-1cgrI
+ tooG Or~
(19) '
such that the interaction fields vanish in their respective asymptotic regions, i.e.
where the 2m-point Green's function is defined as usual, i.e.
lim ~b+ = 0,
G = <0l~+(Zfm)...l~+
lim
7-~OO
7"'-*--OO
~b_ = 0.
(15)
(20)
The subscripts "-" and "+" here denote the wells with minimum at #0 = 0 and ~1 = 7r, respectively: The Euclidean creation and annihilation operators &4--Tand h+ which create and annihilate an effective boson in wells "+" and "-", respectively, are related to the interaction field operators 64- as in Eq. (13) by continuation to imaginary time. Thus
2]r~0
- o J 7- ~
&t, : V T 0 " e
° ~-~r(~+(~'), 4"-*
54- = - V ~ o ~ e °
(T~)(~--(Til)...(~-- (Tira)10).
~ (~4-(I").
(16)
In the limit of long times Irl the interaction fields approach their asymptotic values. The Green's function is then seen to be G=
lim ~b+(~m)...~b+(~-fl) ~-f~
X ~ _ ( T i l ) . . . (~_ (7"im)A0,i
(21)
which vanishes. The only contribution to the S-matrix Eq. (19) is that from the second derivative of the Green's function. The last factor in Eq. (21) is identiffed as the transition amplitude between degenerate ground states Af0,i = ./(01(~+ (7"f))((~-- ('/'i)[0)
The transition amplitude of Eq. (12) can now be written as A m = Sme-2Tnuoo
f,i
(17)
f,i
with S-matrix element S[i ,, = # ~lim -~
1
a + ( r f1 )
Tf~
× at_ (,J,)... at_
10).
(18)
The S-matrix element can be evaluated in terms of the Green's function G which arises in its evaluation. Thus
x ICE(~b+, rf; ~b_, ~'i) d~b+ d~b_,
(22)
where /CE(~b+, ~'f; ~b_, ~'i) denotes the Euclidean Feynman propagator and is evaluated around the instanton configurations of Eq. (1 I). The Euclidean propagator ]~E was calculated following the same procedure as that in Ref. [ 16]. However, the position dependent mass of the spin system requires particular care in the path integral calculation. The result is found to be ICE = 2T4V~(1 - A ) - l / 2 s 2 K 2 e - 2 r ( ° ~ ° / 2 ) e - S ¢ , (23) ¢r where the classical action is evaluated along the instanton trajectory [ 3,4], i.e.
J.-Q. Liang et al./Physics Letters A 228 (1997) 97-102
I+,/X
Sc = sin - 1-V~"
(24)
n
m=O~
Substituting Eq. (23) into Eq. (22) and performing the end point integration with the help of the end point wave functions as in Ref. [ 14] we obtain A°
f,i ----
KIK2 ,~1/2/~l/4s3/2e-Sc . (25)
2T x 8 x
101
~
4
( f --~ 7 ) 7"g/
3
The S-matrix is thus reduced to
m=2
m=4
2
sm
1 fi(
f,i---- ~. I I=l
2mo'~(d~b_~f'r~) d~b_('rl)) Ao -- OA'---O-/ k
dr,
dri
]
f,i"
(26)
0
2
3
S (10 3)
4
5
Using the instanton solution ofEq. ( 11 ) and taking the large time limit ~ ---* - c ~ , r~ ~ c~z after performing the imaginary time derivative we finally obtain the transition amplitude by observing that each pair of vertices in Eq. (26) contributes a factor - 4 w 2. Then
Fig. 1. Theoretical relation between the spin s of the ferromagnetic particle and the resonant frequency Wc. The dashed curve is that of the leading order approximation of tunneling at the ground state (toc = tOMe-2sv'~ with to M -- l011 HE). The solid curves are those of Eq. (29) with m = 0, 2, 4, 8 respectively.
A mf, i = ~1 (8mOtoo)me_2Tm~ooAOi.
expand Aem in the region of small values of A. The level splitting is then seen to be
(27)
Comparing this result with Eq. (12), the level splitting for the mth level (i.e. twice the level shift) is seen to be
2Aem = 2-'~. (8A1/2s)rnAeo,
× exp
-sln
23m+43.~;(m+l/2)sm+3/2
× (~--~-~--) 1/ 2e- 2sv/~.
(29)
(28)
with 16( KIK2 2Ae0 = \ ( 1 7_
1
2Aem = m.l
,~1/2,~.1/4S3/2
-1
and reduces to the ground state level splitting 2Aeo for m = 0. In the above comparison of Eq. (27) and Eq. (12) we can use either the one instanton sector of the amplitude given by Eq. (27) and expand sinh2TAem in Eq. (12) as a power series in 2TAem up to the first order, or we can calculate the contributions of the infinite number of instanton-antiinstanton pairs to achieve the exponentiation of the transition amplitude following the procedure in Ref. [ 14]. It has been stated [2] that the level splitting may be observable only if A is extremely small. We may
This formula has been checked by comparison with the level splitting in the limit of negligible field dependence of the effective mass of Eq. (6) in which case the level splitting is that of the Mathieu equation [ 17 ]. The resonance frequency of MQC for the ferromagnetic particle is Wc = Aem/h. The prefactor of the exponential depends on A and s.Thus the leading order approximation, namely the exponential alone, does not suffice to describe the volume-dependence of the resonance frequency. The exponential, of course, dominates the behavior of ae,, for large s and not too small values of a. Fig. 1 shows the plots of the resonance frequency toe versus the spin s which is proportional to the volume of the particle. The slope decreases if tunneling at excited states is considered. In Fig. 1 the curve of the leading order approximation is plotted in terms of the expression Wc OgMe-2sv/~ with the preconstant of the exponential given by WM = 10 al Hz as in Ref. [7], which describes a magnetic susceptibility =
102
J.-Q. Liang et al./Physics Letters A 228 (1997) 97-102
experiment and considers the effect as that of a single ferromagnetic particle but not a tunneling effect in view of the discrepancies in the comparison with the leading order of tunneling theory [7,8]. Thus a = 8 x 10 -6 so that lntoe vanishes at s ~-, 4.5 × 103. The curves of the formula of the level splitting Eq. (29) are plotted with the same value of h, and the factor (KiK2/~r)1/2 is fixed by setting the frequency tOc equal to that of the leading order approximation at s = 3.8 × 103 for the ground state (m = 0). Tunneling from the ground state requires only a small value of ,~ of the order of 10 -6. However, if tunneling at excited states is included, ,~ increases significantly. For example, to obtain the observed frequencies of Ref. [7], ,~ need only be of the order of 10 -4 for s = 1.4 x 103 and tunneling at the fourth excited state. The tunneling splitting of Eq. (29) provides a possibility of separate resonant frequencies in the magnetic susceptibility experiment for a given size of the magnetic particle. The experiments therefore presumably observe a thermal average of the frequency. It is interesting to calculate the Boltzmann average of the splitting of Eq. (29) in terms of the complete set of eigenstates of (say) H0 (for our convenience), i.e. 1
~e = ~ ~
Aeme - # ' ,
(30)
m
where fl = 1/kBO, where ka denotes the Boltzmann constant and O the temperature. Here
Zo = ~"~ e -#~" m
is the partition function. Substituting Eq. (29) into Eq. (30) we obtain ,~e = exp (2Al/2se -#'°°) (1 -- e-/~'°°)Ae0,
(31)
which may be useful in the analysis of further experiments of MQC. We conclude: A complete theory of tunneling including the tunneling at low-lying excited states and the correction of quantum fluctuations suggests a better understanding of MQC in ferromagnetic particles. The tunneling probability enhances at excited states
and the dependence of the probability on the volume of particles changes significantly with respect to that of the leading order approximation. The formula Eq. (31 ) predicts the relation between the frequency and particle size and also the temperature dependence of the frequency for a fixed size of the particle. J.-Q. Liang acknowledges support of the Deutsche Forschungsgemeinschaft and J.-G. Zhou support of the A. von Humboldt Foundation. J.-Q.L. and E-C.P. also acknowledge support of the National Natural Science Foundation of China. References [ 1] A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.EA. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59 (1987) 1. 121 A. Garg, Phys. Rev. Lett. 71 (1993) 4249. [3] E.M. Chudnovsky and L. Gunther, Phys. Rev. Lett. 60 (1988) 661. [4] J.-Q. Liang, H.J.W. Miiller-Kirsten and J.-G. Zhou, University of Kaiserslautern report no. KL-TH 96/6, to be published in Z. Phys. B (1997). [5] D.D. Awschalom, J.F. Smyth, G. Grinstein, D.E DiVincenzo and D. Loss, Phys. Rev. Lett. 68 (1992) 3092. [61 D.D. Awsehalom, D.P. DiVincenzo, G. Grinstein and D. Loss, Phys. Rev. Lett. 71 (1993) 4276. [7] D.D. Awschalom, M.A. McCord and G. Grinstein, Phys. Rev. Lett. 65 (1990) 783. [8] L. Gunther, Phys. World 3 (1990) 28. 19] S.S. Schweber, An introduction to relativistic quantum field theory (Harper and Row, New York, 1961), p. 691. [10] J.-G. Zhou, J.-Q. Liang, J. Burzlaff and H.J.W. MiillerKirsten, Phys. Lett. A 224 (1996) 142. [11] A. Perelomov, Generalized coherent states and their applications (Springer, Berlin, 1986). [12] E. Fradkin, Field theories of condensed matter systems (Addison-Wesley, New York, 1991). [ 13] D. Loss, D.P. DiVincenzo and G. Grinstein, Phys. Rev. Lett. 69 (1992) 3232. [14] J.-Q. Liang and H.J.W. Mtiller-Kirsten, Phys. Rev. D 46 (1992) 4685; D 51 (1995) 718. [151 J.-Q. Liang and H.J.W. Miiller-Kirsten, Phys. Lett. B 332 (1994) 129. [ 16] J.-G. Zhou, F. Zimmerschied, J.-Q. Liang and H.J.W. MiillerKirsten, Phys. Lett. B 365 (1996) 163. [ 17] R.B. Dingle and H.J.W. MGller, J. Reine Angew. Math. 211 (1962) 11; P. Achuthan, H.J.W. Miiller-Kirsten and A. Wiedemann, Fortschr. Phys. 38 (1990) 142.
ELSEVIER
Physics Letters A 228 (1997) 97-102
Quantum tunneling at excited states and macroscopic quantum coherence in ferromagnetic particles J . - Q . L i a n g a,b,c, H . J . W . M t i l l e r - K i r s t e n a, J i a n - G e Z h o u a, F.-C. P u c,a a Department of Physics, University of Kaiserslautern, D-67653 Kaiserslautern, Germany b Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China c Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China d Department of Physics, Guangzhou Teachers College, Guangzhou 510400, China
Received 2 October 1996; acceptedfor publication 17 January 1997 Communicated by J. Flouquet
Abstract
The consideration of the level splitting of the ground state of a single ferromagnetic particle due to tunneling through energy barriers is extended to low-lying excited states with the help of the LSZ method in field theory, and is calculated up to the one loop correction. Contrary to the leading order WKB approximation, the prefactor of the exponential increases with the spin s of the particle, and therefore decreases the slope of the resonant frequency as a function of the volume of the particle. The temperature dependence of the tunneling probability is also calculated by means of a Boltzmann average. The results should be useful in analysing results of experimental tests of macroscopic quantum coherence. @ 1997 Elsevier Science B.V.
Quantum tunneling between macroscopically distinct states is of great interest both theoretically and experimentally [ 1 ]. Macroscopic magnetisation tunneling has therefore been extensively investigated in the last decade owing to the possible experimental observation of this phenomenon. In the context of these investigations the usual terminology is that macroscopic quantum tunneling (MQT) refers to the decay of a metastable state, while macroscopic quantum coherence (MQC) refers to the resonance between neighboring degenerate states [2]. A theoretical analysis of tunneling for both MQT and MQC in small ferromagnetic particles using the instanton method was begun some time ago [3], but was carried out only in the leading order WKB approximation. It was pointed out recently [4] that in the theoretical analysis of MQC
the periodic potential results formally in an energy "band" structure which is quenched to a single level for half-integer spin s. The one loop correction to the level splitting was also given for ferromagnetic particles. This provides an explicit formula for a quantitative evaluation of the resonance frequency toc = A E / h . A few years ago it was reported that MQC was observed for antiferromagnetic particles [5] in resonance experiments. However, a controversy exists [2,5,6]. The physical explanation of the result of an earlier resonance experiment on ferromagnetic particles is not clear since fundamental discrepancies remain between the experimental data and theoretical expectations on the basis of magnetic quantum tunneling [7,8]. Apart from other reasons [7,8], which are obstacles for the acceptance of the observation as a
0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PHS0375-9601(97)00071-6
98
J.-Q. Liang et aL/Physics LettersA 228 (1997) 97-102
crucial proof of MQC, there are two essential difficulties related to the existing theory of quantum tunneling itself in the absence of an external magnetic field. One difficulty was pointed out in Ref. [2]. The argument of the WKB exponential of the tunneling for a ferromagnetic particle is 2v/As with h = K2/KI, Kl and K2 being the hard and medium axis energies. Since s ,-~ 500 to 5000 unless K2/KI << 10 -4, we is expected to be unobservably small. Even though the phenomenon does not seem totally unobservable, it is not known that such a small A is realistic. The other difficulty is that according to the WKB leading order approximation the theoretical relation between the volume of the particle and the observed resonant frequency is quite different from the best fit of experimental data. The question is whether or not the above problem is due to a shortcoming of the leading order approximation itself. It is important therefore to perform a complete investigation of tunneling including the correction of quantum fluctuations and tunneling at lowlying levels. In the spirit of the method of Lehmann, Symanzik and Zimmermann (LSZ) in field theory [9], which provides a formula for the reduction of S-matrix elements to the Green's function expressed as a timeordered product of field operators, we construct here Euclidean field operators. These operators allow the calculation of the instanton induced transition amplitude between excited states of neighboring potential wells as demonstrated recently [ 10]. An explicit formula for the level splitting of low-lying levels is obtained which includes the contribution of quantum fluctuations. It is remarkable that the prefactor of the leading order exponential increases with s contrary to the decreasing behavior of the exponential. This result may be of importance in theoretical analysis of the relation between the volume of the ferromagnetic particle and the observed resonant frequency obtained from experiments. The probability enhances significantly if tunneling at an excited state is considered. This enhancement opens the possibility to observe MQC for much less anisotropy A ,,~ 10 -4 of the ferromagnetic particle. We begin with the Hamilton operator of the ferromagnetic particle
/~/= Kl.~z2 + K 2 g 2,
(1)
which describes [3] XOY easy plane anisotropy and an easy axis along the x direction with K1 > K2 > 0. In Eq. (1) gi, i = x, y, z, are spin operators obeying the usual commutation relation [~i, gi] = ieijk~k (using natural units throughout). Starting from the coherent state representation [ 11 ] of the time evolution operator with Hamiltonian given by Eq. ( 1 ) and with the help of the coherent state path integral we obtain
(nf]e-2iI:Ir[ni)
=
e-i(qlf-~bi)sK~(t~f,tf; ~bi,/i),
(2)
where If
IC(qbf, t f ; f b i , t i ) = f D c k D p e x p ( i / £ ( g b ,
p) dt)
ti
(3) is the path integral in phase space with canonical variables ~b and p -- s cos 0. Also
£. = (bp - H( dp,p)
(4)
is the phase space (or first order) Lagrangian. The Hamiltonian p2 H = - 4- V(q~) 2m(ff)
(5)
has position dependent mass m(~b) and potential m(~b) =
1
2K1 ( 1 - a sin 2 4,) '
V(dp) = K2s 2 sin 2 ~b,
(6)
respectively, where A -----K2/KI. The kets [ni) and Inf) denote the initial and final spin-coherent states and tf - ti --- 2T denotes the difference of final and initial times. Here s = s(sin 0 cos ~b, sin 0 sin ~b, cos 0) is visualized as a classical spin vector with spin number s, polar angle 0 and azimuthal angle ~b. In the above derivation, the large spin limit s >> 1 has been used since giant spins with spin quantum number s >> 1 are believed to describe correctly ferromagnetic grains. A novel feature of the transition amplitude given by Eq. (2) is the phase factor e -i(~f-4'Os which can be put into the Lagrangian, i.e. the expression q~f - t~i = fti' q~dt, and identified as a Wess-Zumino term [ 12]. The effect of the phase factor which indicates anomalous behavior of the quantum spin system has been
99
J.-Q. Liang et al./Physics Letters A 228 (1997) 97-102
studied extensively [4,13]. Integrating out the momentum in the path integral Eq. (3), we obtain the usual Feynman propagator in configuration space, i.e. tf
lC(4af,tf;cbi,ti)=fg)4~exp(if£(4~,4~)dt), ti
(7) where 73~b is the measure-modified functional differential resulting from the ~b-dependent mass, i.e.
which can be evaluated with the help of the instanton method of path integrals. Here Um(~b, q~K) are the usual eigenfunctions of ~0. Passing to imaginary time by Wick rotation r = it, the instanton configuration is a classical solution which minimizes the Euclidean action SE, i.e. t~SE = t~ f / ~ E d r = 0, where •E = ½m(~b)t~2 + V(~b) is the Euclidean Lagrangian and = d~b/dr denotes from now on the imaginary time derivative. The instanton configuration is then found to be ~bc = arcsin[cosh 2 too(7- - To)
•t-1 m/-~-7) d~bk, / ) & = H V 2,n-iE k
- A sinh 2 too(7- - 7-0) ] -1/2,
( 11 )
which is more convenient for the instanton method used in the following. Since the potential V(~b) is periodic, the low-lying discrete levels possess formally a "band structure" and are obtained in terms of"Bloch waves", i.e.
which can be visualized as the classical trajectory of a pseudoparticle in the barrier. At time 7- --~ -cx~ it starts from one potential well, for example that at q'r = K~r, and reaches the neighboring well at qSX+l = (K + 1)Tr at 7- --* c~. In Eq. (11) 7-o is the position of the instanton. We begin with the instanton induced amplitude for transitions between degenerate eigenstates in any two neighboring wells (say those for K = 0, 1 ), i.e.
E(s + ~:) = e,, -- 2Aem cos'n'(s + ~:),
An/.i = (m, ~, le-ZVf/lm, qO0) = e -2Te" sinh2TAem.
with the second-order Lagrangian = ½m(q~) ~2 _ V(q~),
(8)
(9)
where sc denotes the "Bloch wave vector" which is integral here (determined by the requirement of 2~- periodicity of the "Bloch states"). The quantity Em is the eigenvalue of Hamiltonian //0 such that /~/01m, 4,n) = emlm, q~n); here /~0 is the oscillatorapproximated Hamiltonian in the nth well near the local minimum qbn = n~- and is seen to be p2
/4o = ~
+ ½m0too2~2
in the large s limit, where we have expanded the potential V(~b) and mass m(~b) around the local minima, and it is seen that mo = 1/2K1 and coo2 = 4K1K2s2; in this approximation we have Em = ( m + ½)too. Each "energy band" has only two levels corresponding to even and odd integral values of s + ( and shrinks to a single level for half-integer spin s in agreement with Kramer's degeneracy. The quantity 2Aem is the level splitting defined by Aem = - f
u~n(~b, ~/<+1 )/~/Um(q~, ~K) d~b
(10)
(12) In the following the left hand side of Eq. (12) is evaluated with the help of the Euclidean path-integral method. After evaluation the result is compared with the right hand side of Eq. (12) to obtain Aem. The vacuum instanton, Eq. (11 ), which satisfies the vacuum boundary condition is only suitable for the calculation of the tunneling effect between degenerate vacuum states (m = 0). A method for calculating the tunneling effect at excited states was developed recently with the help of the nonvacuum instanton (or periodic instanton) which is a saddle point configuration [ 14]. In this note we adopt an alternative but equivalent approach [ 10,15] to evaluate the transition amplitude using the (vacuum) instanton solution of Eq. ( 11 ). In field theory the creation operator of a particle (or pseudoparticle) is related to the field operator q~ (position in quantum mechanics) in the interaction picture by a t = -i,~ 2 ~ e -i~°°' --~ ~ ( t ) . v too o~t
(13)
100
J.-Q. Liang et al.IPhysics Letters A 228 (1997) 97-102
With this relation the S-matrix can be related to the Green's function through the procedure known as the LSZ reduction technique [9]. In the spirit of the LSZ method we now calculate the amplitude for a transition through the barrier by constructing interacting Euclidean fields ~b4- in the classically forbidden region with the help of the instanton configuration of Eq. ( 11 ). Thus we define ~b+ :=Tr-~bc,
~b_ :=~bc
lim
Sm
rf~,~
1 f i ( - ~ 2 ~ e ' ° ° r ~ 00~tf) /=1 \ v too
\ v too = --
_ __
m! l=1 \
(14)
x
o
~
etOO(~t-r~)
too/ + too
07-1cgrI
+ tooG Or~
(19) '
such that the interaction fields vanish in their respective asymptotic regions, i.e.
where the 2m-point Green's function is defined as usual, i.e.
lim ~b+ = 0,
G = <0l~+(Zfm)...l~+
lim
7-~OO
7"'-*--OO
~b_ = 0.
(15)
(20)
The subscripts "-" and "+" here denote the wells with minimum at #0 = 0 and ~1 = 7r, respectively: The Euclidean creation and annihilation operators &4--Tand h+ which create and annihilate an effective boson in wells "+" and "-", respectively, are related to the interaction field operators 64- as in Eq. (13) by continuation to imaginary time. Thus
2]r~0
- o J 7- ~
&t, : V T 0 " e
° ~-~r(~+(~'), 4"-*
54- = - V ~ o ~ e °
(T~)(~--(Til)...(~-- (Tira)10).
~ (~4-(I").
(16)
In the limit of long times Irl the interaction fields approach their asymptotic values. The Green's function is then seen to be G=
lim ~b+(~m)...~b+(~-fl) ~-f~
X ~ _ ( T i l ) . . . (~_ (7"im)A0,i
(21)
which vanishes. The only contribution to the S-matrix Eq. (19) is that from the second derivative of the Green's function. The last factor in Eq. (21) is identiffed as the transition amplitude between degenerate ground states Af0,i = ./(01(~+ (7"f))((~-- ('/'i)[0)
The transition amplitude of Eq. (12) can now be written as A m = Sme-2Tnuoo
f,i
(17)
f,i
with S-matrix element S[i ,, = # ~lim -~
1
a + ( r f1 )
Tf~
× at_ (,J,)... at_
10).
(18)
The S-matrix element can be evaluated in terms of the Green's function G which arises in its evaluation. Thus
x ICE(~b+, rf; ~b_, ~'i) d~b+ d~b_,
(22)
where /CE(~b+, ~'f; ~b_, ~'i) denotes the Euclidean Feynman propagator and is evaluated around the instanton configurations of Eq. (1 I). The Euclidean propagator ]~E was calculated following the same procedure as that in Ref. [ 16]. However, the position dependent mass of the spin system requires particular care in the path integral calculation. The result is found to be ICE = 2T4V~(1 - A ) - l / 2 s 2 K 2 e - 2 r ( ° ~ ° / 2 ) e - S ¢ , (23) ¢r where the classical action is evaluated along the instanton trajectory [ 3,4], i.e.
J.-Q. Liang et al./Physics Letters A 228 (1997) 97-102
I+,/X
Sc = sin - 1-V~"
(24)
n
m=O~
Substituting Eq. (23) into Eq. (22) and performing the end point integration with the help of the end point wave functions as in Ref. [ 14] we obtain A°
f,i ----
KIK2 ,~1/2/~l/4s3/2e-Sc . (25)
2T x 8 x
101
~
4
( f --~ 7 ) 7"g/
3
The S-matrix is thus reduced to
m=2
m=4
2
sm
1 fi(
f,i---- ~. I I=l
2mo'~(d~b_~f'r~) d~b_('rl)) Ao -- OA'---O-/ k
dr,
dri
]
f,i"
(26)
0
2
3
S (10 3)
4
5
Using the instanton solution ofEq. ( 11 ) and taking the large time limit ~ ---* - c ~ , r~ ~ c~z after performing the imaginary time derivative we finally obtain the transition amplitude by observing that each pair of vertices in Eq. (26) contributes a factor - 4 w 2. Then
Fig. 1. Theoretical relation between the spin s of the ferromagnetic particle and the resonant frequency Wc. The dashed curve is that of the leading order approximation of tunneling at the ground state (toc = tOMe-2sv'~ with to M -- l011 HE). The solid curves are those of Eq. (29) with m = 0, 2, 4, 8 respectively.
A mf, i = ~1 (8mOtoo)me_2Tm~ooAOi.
expand Aem in the region of small values of A. The level splitting is then seen to be
(27)
Comparing this result with Eq. (12), the level splitting for the mth level (i.e. twice the level shift) is seen to be
2Aem = 2-'~. (8A1/2s)rnAeo,
× exp
-sln
23m+43.~;(m+l/2)sm+3/2
× (~--~-~--) 1/ 2e- 2sv/~.
(29)
(28)
with 16( KIK2 2Ae0 = \ ( 1 7_
1
2Aem = m.l
,~1/2,~.1/4S3/2
-1
and reduces to the ground state level splitting 2Aeo for m = 0. In the above comparison of Eq. (27) and Eq. (12) we can use either the one instanton sector of the amplitude given by Eq. (27) and expand sinh2TAem in Eq. (12) as a power series in 2TAem up to the first order, or we can calculate the contributions of the infinite number of instanton-antiinstanton pairs to achieve the exponentiation of the transition amplitude following the procedure in Ref. [ 14]. It has been stated [2] that the level splitting may be observable only if A is extremely small. We may
This formula has been checked by comparison with the level splitting in the limit of negligible field dependence of the effective mass of Eq. (6) in which case the level splitting is that of the Mathieu equation [ 17 ]. The resonance frequency of MQC for the ferromagnetic particle is Wc = Aem/h. The prefactor of the exponential depends on A and s.Thus the leading order approximation, namely the exponential alone, does not suffice to describe the volume-dependence of the resonance frequency. The exponential, of course, dominates the behavior of ae,, for large s and not too small values of a. Fig. 1 shows the plots of the resonance frequency toe versus the spin s which is proportional to the volume of the particle. The slope decreases if tunneling at excited states is considered. In Fig. 1 the curve of the leading order approximation is plotted in terms of the expression Wc OgMe-2sv/~ with the preconstant of the exponential given by WM = 10 al Hz as in Ref. [7], which describes a magnetic susceptibility =
102
J.-Q. Liang et al./Physics Letters A 228 (1997) 97-102
experiment and considers the effect as that of a single ferromagnetic particle but not a tunneling effect in view of the discrepancies in the comparison with the leading order of tunneling theory [7,8]. Thus a = 8 x 10 -6 so that lntoe vanishes at s ~-, 4.5 × 103. The curves of the formula of the level splitting Eq. (29) are plotted with the same value of h, and the factor (KiK2/~r)1/2 is fixed by setting the frequency tOc equal to that of the leading order approximation at s = 3.8 × 103 for the ground state (m = 0). Tunneling from the ground state requires only a small value of ,~ of the order of 10 -6. However, if tunneling at excited states is included, ,~ increases significantly. For example, to obtain the observed frequencies of Ref. [7], ,~ need only be of the order of 10 -4 for s = 1.4 x 103 and tunneling at the fourth excited state. The tunneling splitting of Eq. (29) provides a possibility of separate resonant frequencies in the magnetic susceptibility experiment for a given size of the magnetic particle. The experiments therefore presumably observe a thermal average of the frequency. It is interesting to calculate the Boltzmann average of the splitting of Eq. (29) in terms of the complete set of eigenstates of (say) H0 (for our convenience), i.e. 1
~e = ~ ~
Aeme - # ' ,
(30)
m
where fl = 1/kBO, where ka denotes the Boltzmann constant and O the temperature. Here
Zo = ~"~ e -#~" m
is the partition function. Substituting Eq. (29) into Eq. (30) we obtain ,~e = exp (2Al/2se -#'°°) (1 -- e-/~'°°)Ae0,
(31)
which may be useful in the analysis of further experiments of MQC. We conclude: A complete theory of tunneling including the tunneling at low-lying excited states and the correction of quantum fluctuations suggests a better understanding of MQC in ferromagnetic particles. The tunneling probability enhances at excited states
and the dependence of the probability on the volume of particles changes significantly with respect to that of the leading order approximation. The formula Eq. (31 ) predicts the relation between the frequency and particle size and also the temperature dependence of the frequency for a fixed size of the particle. J.-Q. Liang acknowledges support of the Deutsche Forschungsgemeinschaft and J.-G. Zhou support of the A. von Humboldt Foundation. J.-Q.L. and E-C.P. also acknowledge support of the National Natural Science Foundation of China. References [ 1] A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.EA. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59 (1987) 1. 121 A. Garg, Phys. Rev. Lett. 71 (1993) 4249. [3] E.M. Chudnovsky and L. Gunther, Phys. Rev. Lett. 60 (1988) 661. [4] J.-Q. Liang, H.J.W. Miiller-Kirsten and J.-G. Zhou, University of Kaiserslautern report no. KL-TH 96/6, to be published in Z. Phys. B (1997). [5] D.D. Awschalom, J.F. Smyth, G. Grinstein, D.E DiVincenzo and D. Loss, Phys. Rev. Lett. 68 (1992) 3092. [61 D.D. Awsehalom, D.P. DiVincenzo, G. Grinstein and D. Loss, Phys. Rev. Lett. 71 (1993) 4276. [7] D.D. Awschalom, M.A. McCord and G. Grinstein, Phys. Rev. Lett. 65 (1990) 783. [8] L. Gunther, Phys. World 3 (1990) 28. 19] S.S. Schweber, An introduction to relativistic quantum field theory (Harper and Row, New York, 1961), p. 691. [10] J.-G. Zhou, J.-Q. Liang, J. Burzlaff and H.J.W. MiillerKirsten, Phys. Lett. A 224 (1996) 142. [11] A. Perelomov, Generalized coherent states and their applications (Springer, Berlin, 1986). [12] E. Fradkin, Field theories of condensed matter systems (Addison-Wesley, New York, 1991). [ 13] D. Loss, D.P. DiVincenzo and G. Grinstein, Phys. Rev. Lett. 69 (1992) 3232. [14] J.-Q. Liang and H.J.W. Mtiller-Kirsten, Phys. Rev. D 46 (1992) 4685; D 51 (1995) 718. [151 J.-Q. Liang and H.J.W. Miiller-Kirsten, Phys. Lett. B 332 (1994) 129. [ 16] J.-G. Zhou, F. Zimmerschied, J.-Q. Liang and H.J.W. MiillerKirsten, Phys. Lett. B 365 (1996) 163. [ 17] R.B. Dingle and H.J.W. MGller, J. Reine Angew. Math. 211 (1962) 11; P. Achuthan, H.J.W. Miiller-Kirsten and A. Wiedemann, Fortschr. Phys. 38 (1990) 142.