Quantum-classical transition of the escape rate of a biaxialferromagnetic spin with an external magnetic field

S.A. Owerre, M.B. ParanjapeGroupe de physique des particules, Département de physique, Université de Montréal, C.P. 6128, succ. centre-ville, Montréal, Québec, Canada H3C 3J7

a r t i c l e i n f o

Article history:Received 6 January 2014Received in revised form14 January 2014Available online 31 January 2014

Keywords:Phase transitionEffective potentialInstanton trajectorySingle model magnets

a b s t r a c t

We study the model of a biaxial single ferromagnetic spin Hamiltonian with an external magnetic fieldapplied along the medium axis. The phase transition of the escape rate is investigated. Two different butequivalent methods are implemented. Firstly, we derive the semi-classical description of the modelwhich yields a potential and a coordinate dependent mass. Secondly, we employ the method of spin-particle mapping which yields a similar potential to that of semi-classical description but with a constantmass. The exact instanton trajectory and its corresponding action, which have not been reported in anyliterature is being derived. Also, the analytical expressions for the first- and second-order crossovertemperatures at the phase boundary are derived. We show that the boundary between the first-and thesecond-order phase transitions is greatly influenced by the magnetic field.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, the study of single ferromagnetic spin systemshas been of considerable interest to condensed matter physicists.These systems have been pointed out [1,3] to be a good candidatefor investigating first- and second-order phase transitions of thequantum-classical escape rate. The quantum-classical escape ratetransition takes place in the presence of a potential barrier. At verylow temperature (close to zero), transitions occur by quantumtunnelling through the barrier and the rate is governed byΓ � e�B, where B is the instanton (imaginary time solution ofthe classical equation of motion) action. At high temperatures, theparticle has the possibility of hopping over the barrier (classicalthermal activation), in this case transition is governed byΓ � e�ΔV=T , where ΔV is the energy barrier. At the critical pointwhen these two transition rates are equal, there exists a crossover

temperature (first-order transition) T ð1Þ0 from a quantum to a

thermal regime, it is estimated as T ð1Þ0 ¼ΔV=B. In principle these

transitions are greatly influenced by the anisotropy constants andthe external magnetic fields. The second-order phase transitionoccurs for particles in a cubic or quartic parabolic potential, it takes

place at the temperature T ð2Þ0 , below T ð2Þ

0 one has the phenomenon

of thermally assisted tunnelling and above T ð2Þ0 transition occurs

due to thermal activation to the top of the potential barrier [1,3].The order of these transitions can also be determined from the

period of oscillation τðEÞ near the bottom of the inverted potential.Monotonically increasing τðEÞ with the amplitude of oscillationgives a second-order transition while nonmonotonic behaviour ofτðEÞ (that is a minimum in the τðEÞ vs E curve, with E being theenergy of the particle) gives a first-order transition [1].

The model of a uniaxial single ferromagnetic spin with atransverse magnetic field, which is believed to describe themolecular magnet MnAc12 was considered by Garanin and Chud-

novsky [1], the Hamiltonian is of the form H ¼ �DS2z �hxSx, using

the spin-particle mapping version of this Hamiltonian [5–7], theyshowed that the transition from the thermal to the quantumregime is of the first-order in the regime hxosD=2 and of thesecond-order in the regime sD=2ohxo2sD. For other single-molecule magnets such as Fe8, a biaxial ferromagnetic spin modelis a good approximation. In this case, Lee et al. [13] considered the

model H ¼ KðS2z þλS2

y Þ�2μBhySy, using spin coherent state pathintegral, they obtained a potential and a coordinate dependentmass fromwhich they showed that the boundary between the firstand the second-order transition set in at λ¼0.5 for hy¼0 while theorder of the transitions is greatly influenced by the magnetic fieldand the anisotropy constants for hya0. Zhang et al. [14] studied

the model H ¼ K1S2z þK2S

2y using spin-particle mapping and

periodic instanton method. The phase boundary between the first-and the second-order transitions was shown to occur at K2 ¼ 0:5K1.The model with z-easy axis in an applied field has also been studiedby numerical and perturbative methods [2]. In this paper, we studya biaxial spin system with an external magnetic field applied alongthe medium axis using spin-coherent state path integral and the

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Journal of Magnetism and Magnetic Materials

0304-8853/$ - see front matter & 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jmmm.2014.01.065

E-mail addresses: solomon.akaraka.owerre@umontreal.ca (S.A. Owerre),paranj@lps.umontreal.ca (M.B. Paranjape).

Journal of Magnetism and Magnetic Materials 358-359 (2014) 93–97

formalism of spin-particle mapping. Unlike other models with anexternal magnetic field [4,12,14], the spin-particle mapping yields asimplified potential and a constant mass which allows us to solvefor the exact instanton trajectory and its corresponding action in thepresence of a magnetic field. We also present the analytical resultsof the crossover temperatures for the first- and the second-ordertransitions at the phase boundary.

2. Spin model and spin coherent state path integral

Consider the Hamiltonian of a biaxial ferromagnetic spin(single-molecule magnet) in an external magnetic field

H ¼DS2z þES2

x �hxSx ð1Þ

where D≫E40, and Si; i¼ x; y; z, is the components of the spin.This model possesses an easy XOY plane with an easy-axis alongthe y-direction and an external magnetic field along the x-axis.At zero magnetic field, there are two classical degenerate groundstates corresponding to the minima of the energy located at 7y,these ground states remain degenerate for hxa0 in the easyXY plane. The semi-classical form of the quantum Hamiltoniancan be derived using spin coherent state path integral. In thecoordinate dependent form, spin-coherent-state is defined by[15,16]

jn⟩¼ cos12θ

� �2s

exp tan12θ

� �eiϕS �

� �Js; s⟩ ð2Þ

where n ¼ sð sin θ cos ϕ; sin θ sinϕ; cos θÞ is the unit vectorparametrizing the spin on a two-sphere S2. The overlap betweentwo coherent states is found to be

⟨n 0jn⟩¼ cos12θ cos

12θ0 þ sin

12θ sin

12θ0e� iΔϕ

� �2sð3Þ

where Δϕ¼ϕ0 �ϕ. The expectation value of the spin operator inthe large s limit is approximated as ⟨n 0jS jn⟩� s½nþOð ffiffi

sp Þ�⟨n 0jn⟩.

For an infinitesimal separated angle, Δθ¼ θ0 �θ, Eq. (3) reduces to

⟨n 0jn⟩� 1� isΔϕð1� cos θÞ: ð4ÞThese states satisfy the overcompleteness relation (resolution

of identity)

NZ

dϕ dð cos θÞjn⟩⟨nj ¼ I : ð5Þ

Using these equations, the transition amplitude is easily obtainedas

⟨nf je�βH jn i⟩¼Z

DϕDð cos θÞe� S ð6Þ

The Euclidean action ðt-� iτÞ is given by S¼ R β=2�β=2 dτ L, with

L¼ is _ϕð1� cos θÞþVðθ;ϕÞ ð7Þ

Vðθ;ϕÞ ¼Ds2 cos 2 θþEs2 sin 2 θ cos 2 ϕ�shx sin θ cos ϕ ð8ÞThese two equations (7) and (8) describe the semi-classical

dynamics of the spin on S2. Two degenerate minima exist forhxohc ¼ 2Es, which are located at θ¼ π=2 : ϕ¼ 2πn7arccos αx,where αx ¼ hx=hc , nAZ, and the maximum is at θ¼ π=2: ϕ¼ nπwith the height of the barrier (n¼0) given by

ΔV ¼ Es2ð1�αxÞ2 ð9ÞTaking into consideration the fact that D≫E, the deviation awayfrom the easy plane is very small, thus one can expand θ¼ π=2�η,where η≪1. Integration over the fluctuation η in Eq. (6) yields an

effective theory described by

Leff ¼ is _ϕþ12mðϕÞ _ϕ2þVðϕÞ ð10Þ

where

VðϕÞ ¼ Es2ð cos ϕ�αxÞ2 ð11Þand

mðϕÞ ¼ 12Dð1�κ cos 2 ϕþ2αxκ cos ϕÞ ð12Þ

with κ ¼ E=D. An additional constant of the form Es2α2x has been

added to the potential for convenience. The first term in theeffective Lagrangian is a total derivative which does not contributeto the classical equation of motion, however, it has a significanteffect in the quantum transition amplitude, producing a quantumphase interference in spin systems [10,11]. The two classicaldegenerate minima which correspond to ϕ¼ 2πn7arccos αx areseparated by a small barrier at ϕ¼0 and a large barrier at ϕ¼ π.The phase transition of the escape rate of this model can beinvestigated using the potential Eq. (11) and the mass Eq. (12) [13],in this paper, however, we will study this transition via themethod of mapping a spin system onto a quantum mechanicalparticle in a potential field. A classical trajectory (instanton) existsfor zero magnetic field, in this case the classical equation ofmotion

mðϕÞ €ϕþ12mðϕÞ0 _ϕ ¼ dV

dϕð13Þ

integrates to

sin ϕ ¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�κÞ

ptanhðωτÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�κ tanh2ðωτÞq ð14Þ

where ω¼ 2sffiffiffiffiffiffiffiEDp

and the upper and lower signs are for instantonand anti-instanton respectively. The corresponding action for thistrajectory yields [10,17] S0 ¼ B7 isπ

B¼ s ln1þ ffiffiffi

κp

1� ffiffiffiκ

p� �

ð15Þ

For small anisotropy parameters, κ≪1, the coordinate dependentmass can be approximated as m� 1=2D, the approximate instan-ton trajectory in this limit yields

sin ϕ ¼ 72

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�αx

1þαx

rtanhðωτÞ

1þ1�αx

1þαxtanh2ðωτÞ

� � ð16Þ

where ω¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEDð1�α2

x Þp

and the corresponding action is

B¼ 2sffiffiffiκ

p ½ffiffiffiffiffiffiffiffiffiffiffiffiffi1�α2

x

q7αx arcsinð

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�α2

x

qÞ� ð17Þ

The upper and the lower sign in the action correspond to the largeand small barriers respectively while that in the trajectory is forinstanton and anti-instanton. At zero magnetic field, the instantoninterpolates between the classical degenerate minima ϕ ¼ 7π=2at τ¼ 71. For coordinate dependent mass the classical trajectorycan be integrated in terms of the Jacobi elliptic functions. Thissolution will be presented in the next section using a simplermethod.

3. Particle mapping

In this section, we will consider the formalism of mapping aspin system to a quantum-mechanical particle in a potential field[5]. In this formalism one introduces a non-normalized spincoherent state, the action of the spin operators on this state yields

S.A. Owerre, M.B. Paranjape / Journal of Magnetism and Magnetic Materials 358-359 (2014) 93–9794

the following expressions [6,7]:

Sx ¼ s cos ϕ� sin ϕddϕ

; Sy ¼ s sin ϕþ cos ϕddϕ

S z ¼ � iddϕ

ð18Þ

The Shrödinger equation can be written as

HΦðϕÞ ¼ EΦðϕÞ ð19Þwhere the generating function is defined as

ΦðϕÞ ¼ ∑s

m ¼ � s

Cmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs�mÞ!ðsþmÞ!

p eimϕ ð20Þ

with periodic boundary condition Φðϕþ2πÞ ¼ e2iπsΦðϕÞ. UsingEqs. (1), (18) and (19), the differential equation for ΦðϕÞ yields

�Dð1þκ sin 2 ϕÞd2Φdϕ

� E s�12

� �sin 2 ϕ�hx sin ϕ

� �dΦdϕ

þðEs2 cos 2 ϕþEs sin 2 ϕ�hxs cos ϕÞΦ¼ EΦ ð21ÞNow let us introduce the incomplete elliptic integral of the firstkind

x¼ Fðϕ;λÞ ¼Z ϕ

0dφ

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�λ2 sin 2 φ

q ð22Þ

with amplitude ϕ and modulus λ2 ¼ κ. The trigonometric func-tions are related to the Jacobi elliptic functions by snðx; λÞ ¼ sin ϕ,

cnðx; λÞ ¼ cos ϕ and dnðx; λÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�λ2 sn2ðx; λÞ

q. In this new vari-

able, Eq. (21) transforms into the Schrödinger equation HΨ ðxÞ ¼EΨ ðxÞ with

H ¼ � 12m

d2

dx2þV ðxÞ; m¼ 1

2D ð23Þ

The effective potential is given by

VðxÞ ¼ E ~s2½cnðx; λÞ�αx�2dn2ðx; λÞ ð24Þ

Ψ ðxÞ ¼ ΦðϕðxÞÞ½dnðx; λÞ�s exp � ~sαx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiκ

ð1�κÞ

r�arccot

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiκ

ð1�κÞ

rcnðx; λÞ

� ��ð25Þ

where ~s ¼ ðsþ12Þ and αx ¼ hx=2E ~s. In order to arrive at this potential

we have used the large s limit sðsþ1Þ � ~s2 and shifted theminimum energy to zero by adding a constant of the form

E ~s2α2x . Unlike the spin coherent state version, the mass of the

particle is constant in this case which appears to be the approx-imate form of Eq. (12) in the limit of small anisotropy parameters,but the potentials Eq. (11) and (24) are of similar form, in fact theyare equal when λ-0 except for the quantum renormalization ~s. Atzero magnetic field the potential Eq. (24) reduces to a well-knownpotential studied by the periodic instanton method [14]. In manymodels with an external magnetic field [4,12,14], the resultingeffective potential from spin-particle mapping is always toocomplicated for one to solve for the instanton trajectory, howeverin this case the effective potential is in a compact form, allowing usto find the exact classical trajectory (see the next section).

4. Phase transition and instanton solution

We will now study the phase transition of the escape rate ofthis model and the instanton solution in the presence of amagnetic field. The potential Eq. (24) has minima atx0 ¼ 4nKðλÞ7cn�1ðαxÞ and maxima at xsb ¼ 74nKðλÞ for a small

barrier and at xlb ¼ 72ð2nþ1ÞKðλÞ for a large barrier, where KðλÞis the complete elliptic function of first kind i.e. Fðπ=2; λÞ.The heights of the potential for small and large barriers are given by

ΔVsb ¼ E ~s2ð1�αxÞ2

ΔVlb ¼ E ~s2ð1þαxÞ2 ð26Þ

The Euclidean Lagrangian corresponding to the particle Hamil-tonian is

L¼ 12m _x2þVðxÞ ð27Þ

It follows that the classical equation of motion is

m €x ¼ dVdx

ð28Þ

which corresponds to the motion of the particle in the invertedpotential �VðxÞ. Upon integration, Eq. (28) gives the instantonsolution

snðx; λÞ ¼ 72

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�αx

1þαx

rtanhðωτÞ

1þ1�αx

1þαxtanh2ðωτÞ

� � ð29Þ

where ω¼ ~sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEDð1�α2

x Þp

. This trajectory has not been reported inthe literature. It is the exact classical trajectory in the presence ofan external magnetic field. The instanton (upper sign) interpolatesfrom the left minimum xðτÞ ¼ �sn�1ð

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�α2

x

pÞ at τ¼ �1 to the

center of the barrier xðτÞ ¼ 0 at τ¼ 0 and reaches the rightminimum xðτÞ ¼ sn�1ð

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�α2

x

pÞ at τ¼1. At zero magnetic field,

Eq. (29) is equivalent to the well-known instanton solution [8],which is equivalent to Eq. (14). It is noted that this trajectory is thesame as Eq. (16) except that the trigonometric sine function isbeing replaced by the Jacobi elliptic sine function and s-~s,however, in the limit λ-0, both solutions are the same, sincethe potential equations. (11) and (24) and the mass equations (12)and (23) are the same in this limit (the Jacobi elliptic functionsbecome the trigonometric functions). The action for the trajectory,Eq. (29), yields

B¼ ~s ln1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiκð1�α2

x Þp

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiκð1�α2

x Þp

!"72αx

ffiffiffiffiffiffiffiffiffiffiffiκ

1�κ

rarctan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�κÞð1�α2

x Þp

αx

!#

ð30Þ

When αx ¼ 71, there are no large and small barriers, thetrajectory and its action reduce to xðτÞ ¼ 0¼ B, hence there is notunnelling. It is noted that this action reduces to Eq. (17) in thelimit κ≪1 and to Eq. (15) when αx ¼ 0 except that s is beingreplaced by ~s. At nonzero energy (finite temperature), the particlehas the possibility of hopping over the potential barrier (thermalactivation), the escape rate (transition amplitude) of the particlecan be either the first- or the second-order depending on theshape of the potential. In order to investigate the analogy of thistransition to Landau's theory of phase transition, consider theescape rate of a particle at finite temperature through a potentialbarrier in the quasiclassical approximation [3,10]

Γ �Z

dE WðEÞe�ðE�EminÞ=T ð31Þ

where WðEÞ is the tunnelling probability of a particle at an energyE, and Emin is the energy at the bottom of the potential. Thetunnelling probability in imaginary time is given as WðEÞ � e� SðEÞ,therefore we have

Γ � e�Fmin=T ð32Þ

S.A. Owerre, M.B. Paranjape / Journal of Magnetism and Magnetic Materials 358-359 (2014) 93–97 95

where Fmin is the minimum of the free energy F � EþTSðEÞ�Emin

with respect to E. The imaginary time action is expressed as

SðEÞ ¼ 2ffiffiffiffiffiffiffi2m

p Z xðEÞ

�xðEÞdx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVðxÞ�E

pð33Þ

where 7xðEÞ are the turning points for the particle with energy�E in an inverted potential. Introducing a dimensionless quantityQ ¼ ðVmax�EÞ=ðVmax�VminÞ where VmaxðVminÞ corresponds to thetop (bottom) of the potential, the expansion of the imaginary timeaction around xb gives [4]

SðEÞ ¼ 2πΔVω0

½Q2þbQ2þOðQ3Þ� ð34Þ

where

b¼ ΔV

48ðV″ðxÞÞ3½5ðV‴ðxÞÞ2�3V⁗ðxÞV″ðxÞ�x ¼ xb

and ω20 ¼ �V″ðxbÞ=m40 is the frequency of oscillation at the

bottom of the inverted potential, xb corresponds to the maximumof the potential.

By the analogy with the Landau theory of phase transition, thephase boundary between the first- and second-order transitions(see Fig. 1) is obtained by setting the coefficient of Q2 to zero i.e.b¼0. Using the maximum of the small and large barriers of thepotential Eq. (24) at xsb and xlb we obtain

bsb ¼ ðκ�κþsb ðαxÞÞðκ�κ�

sb ðαxÞÞ ð35Þ

blb ¼ ðκ�κþlb ðαxÞÞðκ�κ�

lb ðαxÞÞ ð36Þwhere

κ7sb ðαxÞ ¼

3�4αxþα2x 7ð1�αxÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�4αxþα2

x

p4ð1�2αxþα2

x Þð37Þ

κ7lb ðαxÞ ¼

3þ4αxþα2x 7ð1þαxÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ4αxþα2

x

p4ð1þ2αxþα2

x Þð38Þ

Thus by setting b¼0 we obtain the four solution in Eqs. (37)and (38). At αx ¼ 0, the critical values at the phase boundary areκc ¼ 1 or 1

2 for the plus or the minus signs respectively [4,9,14].Expanding for a small field αx≪1, we obtain κþ

sb=lb � 17αx=4 and

κ�sb=lb � 1

2ð1732αxÞ, where the plus and minus signs correspond to

the small and large barriers respectively. The phase diagrams ofEqs. (37) and (38) are shown in Fig. 2, with the value κ� increasing

x

0.0

0.2

0.4

0.6

0.8

1.0

1.2

x

v(x)

= 0.1,α κ = 0.2

0 2 4−2−4

Fig. 1. The plot of the effective potential, Eq. (24) for αx ¼ 0:1, κ¼ 0:2, where vðxÞ ¼VðxÞ=E ~s2.

First-order transition

Second-order transition

0.00 0.05 0.10 0.15 0.20 0.250.4

0.5

0.6

0.7

0.8

0.9

Second-order transition

First-order transition

0.2

0.3

0.4

0.5

0.6

x

xα

0.0 0.2 0.4 0.6 0.8 1.0α

lb()−

()−

sb

Fig. 2. The phase diagram κ� vs αx at the phase boundary for a small barrier (a) anda large barrier (b).

Small barrier

Large barrier

0.00 0.05 0.10 0.15 0.20 0.25

0.20

0.25

0.30

0.35

0.40

0.45

Small barrier

Large barrier

0.00 0.05 0.10 0.15 0.20 0.25

0.4

0.5

0.6

0.7

0.8

xα

xα

a

b

T 0(c

)T 0

(c)

Fig. 3. Dependence of the crossover temperatures on the magnetic field at thephase boundary: (a) second-order (solid line) and its maximum (dashed line) forthe small and large barriers and (b) first-order for the small and large barriers.These graphs are plotted with T ðcÞ

0 ¼ T ðcÞ0 =E ~s .

S.A. Owerre, M.B. Paranjape / Journal of Magnetism and Magnetic Materials 358-359 (2014) 93–9796

with increasing magnetic field for the small barrier while itdecreases with increasing magnetic field for the large barrier, thefirst-order phase transition occurs in the regime κ�

sb=lb41=2 in

both cases. The crossover temperature for the first-order transition

is estimated as T ð1Þ0 ¼ΔV=B which is easily obtained from Eqs. (26)

and (30). Expanding for αx≪1 at the phase boundary (with theexpressions for κ�

sb=lbðαxÞ), we obtain the crossover temperatures

as T ðcÞ0 � E ~s=ðln½ð3þ2

ffiffiffi2

pÞe73αx=

ffiffi2

p�Þ, where the upper and lower

signs correspond to the small and large barriers respectively.

Both temperatures coincide at αx ¼ 0 ) κ�sb=lb ¼ 1=2 with T ðcÞ

0 ¼E ~s=lnð3þ2

ffiffiffi2

pÞ as shown in Fig. 3(a). In the case of second-order

transition the crossover temperature is estimated as T ð2Þ0 ¼ω0=2π.

This is easily obtained as

T ð2Þ0 ¼ E ~s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið17αxÞ

pπ

1�ð17αxÞκκ

� �1=2

ð39Þ

The maximum of this function occurs at αx ¼ 7 ð1�2κÞ=2κ,with

T ðmaxÞ0 ¼ E ~s

2πκð40Þ

where the upper and lower signs correspond to the large andsmall barriers respectively. Substituting the expressions forκ�sb=lbðαxÞ into Eqs. (39) and (40) we obtain the temperatures at

the phase boundary as shown in Fig. 3(a). The critical temperatureat the phase boundary decreases with increasing magnetic fieldfor the small barrier while for the large barrier it increases withincreasing magnetic field. In the regime of a small field αx≪1, itbehaves linearly as T ðcÞ

0 � E ~sð1732αxÞ=π. Both barriers coincide at

αx ¼ 0 ) κ�sb=lb ¼ 1=2, with T ðcÞ

0 ¼ E ~s=π which is smaller than that ofthe first-order.

5. Conclusions

In conclusion, we have investigated an effective particleHamiltonian which corresponds exactly to a biaxial spin model.

Using this Hamiltonian we studied the phase transition of theescape rate of a particle at zero and nonzero temperatures. Theanalytical expressions for the instanton trajectories and the cross-over temperatures were obtained. We showed that the boundarybetween the first- and second-order phase transitions is greatlyinfluenced by the magnetic field.

Acknowledgments

The authors would like to thank NSERC of Canada for financialsupport.

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