Macroscopic quantum tunneling and quantum-classical phase transitions of the escape rate in large spin systems
MMacroscopic quantum tunneling and quantum-classical phase transitions ofthe escape rate in large spin systems
S. A. Owerre ∗ and M. B. Paranjape † Groupe de physique des particules, D´epartement de physique, Universit´e de Montr´eal, C.P. 6128, succ. centre-ville,Montr´eal, Qu´ebec, Canada, H3C 3J7
This article presents a review on the theoretical and the experimental developments on macro-scopic quantum tunneling and quantum-classical phase transitions of the escape rate in large spinsystems. A substantial amount of research work has been done in this area of research over theyears, so this article does not cover all the research areas that have been studied, for instance theeffect of dissipation is not discussed and can be found in other review articles. We present thebasic ideas with simplified calculations so that it is readable to both specialists and nonspecialistsin this area of research. A brief derivation of the path integral formulation of quantum mechanicsin its original form using the orthonormal position and momentum basis is reviewed. For tunnel-ing of a particle into the classically forbidden region, the imaginary time (Euclidean) formulationof path integral is useful, we review this formulation and apply it to the problem of tunnelingin a double well potential. For spin systems such as single molecule magnets, the formulationof path integral requires the use of non-orthonormal spin coherent states in (2 s + 1) dimensionalHilbert space, the coordinate independent and the coordinate dependent form of the spin coherentstate path integral are derived. These two (equivalent) forms of spin coherent state path integralare applied to the tunneling of single molecule magnets through a magnetic anisotropy barrier.Most experimental and numerical results are presented. The suppression of tunneling for half-oddinteger spin (spin-parity effect) at zero magnetic field is derived using both forms of spin coherentstate path integral, which shows that this result (spin-parity effect) is independent of the choiceof coordinate. At nonzero magnetic field we present both the experimental and the theoreticalresults of the oscillation of tunneling splitting as a function of the applied magnetic field appliedalong the spin hard anisotropy axis direction. The experimental and the theoretical results ofthe tunneling in antiferromagnetic exchange coupled dimer model are also reviewed. As the spincoherent state path integral formalism is a semi-classical method, an alternative exact mappingof a spin Hamiltonian to a particle Hamiltonian with a potential field (effective potential method)is derived. This effective potential method allows for the investigation of quantum-classical phasetransitions of the escape rate in large spin systems. We present different methods for investigatingquantum-classical phase transitions of the escape rate in large spin systems. These methods areapplied to different spin models. PACS numbers: 75.45.+j, 75.50.Tt, 75.30.Gw, 03.65.Sq,75.10.Jm, 75.60.Ej, 61.46.+w
Contents
I. Introduction II. Path integral formulation
III. Macroscopic quantum tunneling of large spin systems z -axis uniaxial spin model in a magnetic field 122. Biaxial spin model and quantum phase inteference 143. Biaxial spin model with an external magnetic field 174. Landau Zener effect 195. Antiferromagnetic exchange coupled dimer model 21B. Coordinate independent formalism 241. Equation of motion and Wess-Zumino action 24 ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ c ond - m a t . o t h e r] O c t
2. Coordinate independent uniaxial spin model in a magnetic field 253. Coordinate independent biaxial model and suppression of tunneling 26
IV. Effective potential (EP) method
V. Quantum-classical phase transitions of the escape rate in large spin systems z -axis biaxial spin model with a magnetic field 441. Introduction 442. Effective potential method 453. Phase boundary and crossover temperatures 454. An alternative model 495. Phase boundary and crossover temperatures 49E. Phase transition in easy x -axis biaxial spin model with a medium axis magnetic field 501. Effective potential of medium axis magnetic field model 502. Phase boundary and crossover temperatures 513. Free energy 52F. Phase transition in exchange-coupled dimer model 531. Model Hamiltonian 532. Effective potential 533. Periodic Instanton at zero magnetic field 544. Vacuum instanton at zero magnetic field 565. Free energy and phase transition at zero magnetic field 576. Free energy with magnetic field 587. Phase boundary and crossover temperatures 60 VI. Conclusion and discussion Acknowledgments References I. INTRODUCTION
One of the remarkable manifestations of quantum mechanics is the concept of quantum tunneling. This involves thepresence of a potential barrier, that is the region where the potential energy is greater than the energy of the particle.In classical mechanics, the tunneling of a particle through this barrier is prohibited as it requires the particle tohave a negative kinetic energy, however, in quantum mechanics we find a nonzero probability for finding the particlein the classically forbidden region. Thus, a quantum particle can tunnel through the barrier. In one dimensionalsystems, the tunneling amplitude (whose modulus squared gives the probability) is usually computed using twofundamental methods, namely, the Wentzel-Kramers-Brillouin (WKB) method (Landau and Lifshitz, 1977) and the“instanton” method(Coleman, 1977, 1985; Dashen, Hasslacher and Neveu, 1974; Gervais and Sakita, 1975; Gervais,Jevicki and Sakita, 1975; Jackiw and Rebbi, 1976; Langer, 1967; Polyakov, 1977) via the Feynman path integralformulation(Feynman and Hibbs, 1965) of quantum mechanics. The term “instanton” refers to classical solutions ofthe equations of motion when the time coordinate has been continued to Euclidean time, t → − iτ . For particle ina double well potential with two degenerate minima, the basic understanding is that in the absence of tunneling theclassical ground states of the system, which correspond to the minima of the potential, remain degenerate. Tunnelinglifts this degeneracy and the true ground state and the first excited state become the symmetric and antisymmetriclinear superposition of the classical ground states with an energy splitting between them(Coleman, 1985; Landau andLifshitz, 1977). In some cases the two minima of the potential are not degenerate. The state with lower energy is thetrue vacuum, while the state with higher energy is the false vacuum, which is then rendered unstable due to quantumtunneling. In this case one looks for the decay rate of the false vacuum(Coleman, 1977; Callan and Coleman, 1977).Such a scenario plays a vital role in cosmology, especially in the theory of early universe and inflation. Additionally,in some quantum systems, tunneling does not involve the splitting of the classical ground states or the decay of thefalse vacuum, but rather a dynamic oscillation of the (phase) difference between two macroscopic order parameters(Cooper, 1956), which are separated by a thin normal layer, through tunneling of the microscopic effective excitations,such as Cooper pairs as in Josephson effect( Esposito et al, et al. , 2007; Josephson, 1986).In the last few decades, the tunneling phenomenon has been extended to other branches of physics. Tunneling hasbeen predicted in single, molecular, large magnetic spin systems such as MnAc , Mn and Fe ( Chudnovsky andGunther, 1988; Enz and Schilling, 1986; Van Hemmen and S¨ut¨o, 1986; Wernsdorfer and Sessoli, 1999). These singlemolecule magnets (SMMs) are composed of several molecular magnetic ions, whose spins are coupled by intermolecularinteractions giving rise to an effective single giant spin, which can tunnel through its magnetic anisotropy barrier,hence the name “macroscopic quantum spin tunneling ”. Van Hemmen and S¨ut¨o (1986) first studied the tunnelingin a uniaxial ferromagnetic spin model with an applied magnetic field using the WKB method. Enz and Schilling(1986) considered a biaxial model with a magnetic field using instanton technique, subsequently, Chudnovsky andGunther (1988) studied a more general biaxial spin model by solving the instanton trajectory of the Landau Lifshitzequation. These studies were based on a semi-classical description, that is by representing the spin operator as a unitvector parameterized by spherical coordinates. In this description, the spin is represented by a particle on a two-dimensional sphere S , however, in the presence of a topological term, called the Berry’s phase term or Wess-Zuminoaction (Berry, 1984; Wess and Zumino, 1971; Witten , 1979), which effectively corresponds to the magnetic fieldof a magnetic monopole at the centre of the two sphere. Based on this semi-classical description, it was predictedthat for integer spins tunneling is allowed, while for half-odd integer spin tunneling is completely suppressed at zero(external) magnetic field (Henley and Delft, 1992; Loss, DiVincenzo and Grinstein, 1992). The vanishing of tunnelingfor half-odd integer spins is understood as a consequence of destructive interference between tunneling paths, whichis directly related to Kramers degeneracy(Kramers, 1930; Messiah, 1962) due to the time reversal invariance of theHamiltonian. In the presence of a magnetic field applied along the spin hard axis, Garg (1993) showed that thetunneling splitting does not vanish for half-odd integer spins, but rather oscillates with the field and only vanishes ata certain critical value of the field, which was later observed experimentally in Fe molecular cluster (Sessoli et al. ,2000; Wernsdorfer and Sessoli, 1999; Wernsdorfer et al. , 2000). In this case tunneling suppression is not related toKramers degeneracy due to the presence of a magnetic field.An exact mapping of spin system was considered by Scharf, Wreszinski and Hemmen (1987) and Zaslavskii (1990a);Zaslavskii and Ulyanov (1992). They studied the exact mapping of a spin system unto a particle in a potential field incontrast to the semi-classical approach. This method, which is called the effective potential method, deals with an exactcorrespondence between a spin Hamiltonian and a particle in a potential field. It gives the possibility for investigatingspin tunneling just like a particle in a one-dimensional double well potential. In recent years spin tunneling effect hasbeen observed in many small ferromagnetic spin particles such as Fe (Sangregorio, et al. , 1997), Mn Ac ( Friedman et al. , 1996; Thomas, et al. , 1996; Zhang, et al. , 1996), in ferrimagnetic nanoparticles (Wernsdorfer et al. , 1997) andalso in antiferromagnetic particles (Awshalom, et al. , 1992; Gider, et al. , 1995; Tejada, et al. , 1997),antiferromagneticexchange coupled dimer [Mn ] (Hill, et al. , 2003; Tiron, et al. , 2003a) and antiferromagnetic ring clusters with evennumber of spins (Meier and Loss, 2001; Meier, et al. , 2003; Taft, et al. , 1994). These molecular magnets also play adecisive role in quantum computing (Leuenberger and Loss, 2001; Tejada, et al. , 2001). An extensive review on theexperimental analysis of SMMs can be found in (Gatteschi and Sessoli, 2003).The possibility of quantum tunneling, which is mediated by a vacuum instanton trajectory, requires a very lowtemperature T →
0. For pure quantum tunneling, the transition amplitude in the stationary phase approximationis Γ = A e − B , where B is the vacuum instanton action and A is a pre-factor. At nonzero temperature, quantumtunneling becomes inconsequential, then the particle has the possibility of crossing over the barrier, a process calledclassical thermal activation (see Fig.(1)). The study of thermal activation dates back to the work of Kramers (1940) forthe diffusion of a particle over the barrier. A review of this subject for both particle and spin system can be found inthe existing literature(Coffey, Kalmykov and Waldron, 1996; H¨anggi, Talkner and Borkovec, 1990; Stamp, Chudnovskyand Barbara, 1992). In this case the transition is governed by the Van’t Hoff-Arrhenius Law (H¨anggi, Talkner andBorkovec, 1990) Γ = B e − β ∆ U , where ∆ U is the height of the potential barrier , β is the inverse temperature and B In most literature, macroscopic quantum tunneling refers to tunneling in a bias (metastable) potential while macroscopic quantumcoherence refers to tunneling in a potential with degenerate minima (Leggett, 1995). We will use the former to refer to both systems. (a) (b)
FIG. 1 (a): A sketch of a metastable potential showing the regions of quantum tunneling at low temperature and classicalthermal activation at high temperature. (b): The inverted potential. The coordinates x and x are the classical turningpoints. is a pre-factor.The basic understanding of quantum-classical phase transitions of the escape rate is as follows: for a particlein a metastable cubic potential or double well quartic parabolic potential U ( x ), with no environmental influence(dissipation), transition at finite temperature is dominated by thermon (periodic) instanton trajectory , whose actionis given by S p ( E ) (Chudnovsky, 1992), where E is the energy of the particle in the inverted potential − U ( x ). Theescape rate is defined by taking the Boltzmann average over tunneling probabilities at finite energy(Affleck, 1981). Atthe bottom of the barrier we have S p ( E ) → S ( U min ), where S ( U min ) is the action at the bottom of the barrier, whileat the top of the barrier S p ( E ) → S = β ∆ U , which is the action of a constant trajectory at the top of the barrier.Now, if we compare the plot of the thermon action S p and that of the thermodynamic action S against tem-perature(Chudnovsky, 1992), there exist a critical temperature T c at which the thermodynamic action crosses thethermon action. If this intersection is sharp , the critical temperature T c can be thought of as a first-order “phasetransition” (crossover) temperature from classical (thermal) to quantum regimes. At this temperature T c = T (1)0 ,there is a discontinuity in the first-derivative of the action S p (Gorokhov and Blatter, 1997). The approximate form ofthis crossover temperature can be estimated by comparing the quantum action S ( U min ) at the bottom of the barrierand that of the classical action at the top of the barrier S (Stamp, Chudnovsky and Barbara, 1992) : T (1)0 = 1 β (1)0 = ∆ US ( U min ) = ∆ U B . (1)For a particle with a constant mass, the physical understanding for a sharp first-order phase transition to occur isthat the top of the barrier should be flat( Chudnovsky, et al. , 1998). This condition is not widely accepted. It hasbeen argued that the necessary condition for a sharp first-order phase transition to occur is that the top of the barriershould be wider so that tunneling through the barrier from the ground state is more auspicious than that from theexcited states(Zhang, et al. , 1999). For a particle with a position dependent mass, the necessary condition for a sharpfirst-order phase transition to occur requires the mass of the particle at the top of the barrier to be heavier thanthat at the bottom of the barrier. In this case tunneling from higher excited states is inauspicious. Thus, thermalactivation competes with ground state tunneling leading to first-order phase transition. Thermally assisted tunneling(TAT), that is tunneling from excited states which reduces to ground state tunneling at T = 0 occurs for temperaturesbelow T (1)0 . In this case the particle tunnels through the barrier at the most favourable energy E ( T ), which goes fromthe top of the barrier to the bottom of the barrier as the temperature decreases ( Chudnovsky, et al. , 1998). This is simply the solution of the imaginary time classical equation of motion with an energy E . Actually, the thermon action is defined over the whole period of oscillation of a particle in the inverted potential. In other words, theparticle crosses the barrier twice. Thus, B = S ( U min ) / However, if the intersection of the thermon action S p and that of the thermodynamic action S is smooth , thecritical temperature is said to be of second-order T c = T (2)0 . The second derivative of the thermon action in this casehas a jump at T (2)0 . This crossover temperature is defined as (Goldanskii, 1959a,b) T (2)0 = 1 β (2)0 = ω b π , (2)where ω b is the frequency of oscillation at the bottom of the inverted potential − U ( x ), that is ω b = − U (cid:48)(cid:48) ( x s ) m . Thisformula follows from equating the Van’t Hoff-Arrhenius exponential factor β ∆ U at finite nonzero temperature andthe approximate form of the WKB exponential factor 2 π ∆ U/ω b at zero temperature.Using functional integral approach, Affleck (1981) and Larkin and Ovchinnikov (1983, 1984) demonstrated that,in the regime T < T (2)0 , there is a competing effect between thermal activation and quantum tunneling leading toTAT. For T (cid:29) T (2)0 , quantum tunneling is suppressed and assisted thermal activation becomes the dominant factorin the escape rate. For T ≈ T (2)0 , the two regimes smoothly join with a jump of the second derivative of the escaperate. Thus, T (2)0 corresponds to the crossover temperature from thermal regime to TAT. In term of the potential, fora constant mass particle a smooth second-order crossover is favourable with a potential with a parabolic barrier top.An alternative criterion for the first- and the second-order quantum-classical phase transitions was demonstrated byChudnovsky (1992) based on the shape of the potential. He showed that for a first-order phase transition, the periodof oscillation β ( E ) is nonmonotonic function of E , in other words, β ( E ) has a minimum at some point E < ∆ U andthen rises again, while for second-order phase transition β ( E ) is monotonically increasing with decreasing E . M¨uller,Park and Rana (1999) derived a general criterion formula for investigating first- and second-order phase transitions,which is similar to the criterion formula derived by Kim (1999).In this report, we will review the theoretical and the experimental developments on macroscopic quantum tunnelingand quantum-classical phase transitions of the escape rate in large spin systems. The article is organized as follows.In section(II.A), we will introduce the basic idea of path integral for a one-dimensional particle from Feynman pointof view and review its application to the tunneling of a particle in a double well potential. In section(II.B) we willapply this idea to spin systems using spin coherent states. The path integral for spin systems will be derived in thethe coordinate independent form. We will show the steps on how to move from coordinate independent to coordinatedependent form. In section (III) we will then apply this coordinate dependent formalism to tunneling problem ofSMMs. The quantum phase interference (quenching of tunneling splitting) will be derived and some experimentalresults will be presented. Due to lack of solution of these models in coordinate independent form in most of theliterature, we will show that both the instanton trajectory and the quantum phase interference can be recovered usingthe coordinate independent formalism. We will further extend our consideration to tunneling in an exchange coupleddimer model and to an antiferromagnetic spin model in general. Section(IV) deals with the effective potential method,we will review the mapping of a large spin model onto a particle Hamiltonian that consists of a potential energy anda mass. In section(V) we will present different methods for studying the quantum-classical phase transitions of theescape rate. We will also apply these methods to both SMMs and exchange coupled dimer model. Theoretical,numerical and experimental results will be presented. In section(VI) we will summarize our analysis and comment ontheir significance. II. PATH INTEGRAL FORMULATIONA. Position state path integral
In this section we start with a brief review of path integral formulation of quantum mechanics. This formulation is anelegant alternative method of quantum mechanics. It reproduces the Schr¨odinger formulation of quantum mechanicsand the principle of least action in classical mechanics. In this method the classical action enters into the calculationof a quantum object, the transition amplitude, thereby allowing for a quantum interpretation of a solution of theclassical equations of motion. The basic idea of the path integral is that unlike a classical particle with a uniquetrajectory or path, a quantum particle follows an infinite set of possible trajectories to go from an initial state say | x (cid:105) at t = 0 to a final state say | x (cid:48) (cid:105) at time t = t (cid:48) . The sum over all the possible paths (histories of the particle)appropriately weighted, determines the quantum amplitude of the transition. The weight for each path is exactlythe phase corresponding to the exponential of the classical action of the path, multiplied by the imaginary number i .Consider a particle moving in one dimension, the Hamiltonian of this system is of usual form:ˆ H = ˆ p m + U (ˆ x ) . (3)Let us introduce the complete, orthonormal eigenstates of the position ˆ x and the momentum ˆ p operators:ˆ x | x (cid:105) = x | x (cid:105) , ˆ p | p (cid:105) = p | p (cid:105) , (4) (cid:104) x (cid:48) | x (cid:105) = δ ( x (cid:48) − x ) , (cid:104) p (cid:48) | p (cid:105) = δ ( p (cid:48) − p ) , (5)with (cid:104) x | p (cid:105) = e ipx/ (cid:126) . (6)The resolution of identities are (cid:90) dx | x (cid:105) (cid:104) x | = ˆI = (cid:90) dp π (cid:126) | p (cid:105) (cid:104) p | . (7)Expressing the unitary operator e − i ˆ Ht as [ e − i ˆ Ht/N ] N and using Eqs.(3)–(7), the transition amplitude in the limit N → ∞ is given by (Feynman and Hibbs, 1965; Feynman, 1948) A ( x (cid:48) , t (cid:48) ; x,
0) = (cid:104) x (cid:48) | e − i ˆ Ht (cid:48) / (cid:126) | x (cid:105) = (cid:90) D x ( t ) e iS [ x ( t )] / (cid:126) , (8)where D x ( t ) is the measure for integration over all possible classical paths x ( t ) that satisfy the boundary conditions x (0) = x and x ( t (cid:48) ) = x (cid:48) , where S [ x ( t )] = (cid:90) t (cid:48) dtL, L = 12 m (cid:18) dxdt (cid:19) − U ( x ) , (9)is the classical action and the Lagrangian of the system. We have written down the path integral for a one-dimensionalparticle, generalization to higher dimensions is straightforward.The well-known classical equation of motion can be derived in a very simple way. In the semiclassical limit, i.e., (cid:126) →
0, the phase e iS [ x ( t )] / (cid:126) oscillates very rapidly in such a way that nearly all paths cancel each other. The maincontribution to the path integral comes from the paths for which the action is stationary, i.e., δS [ x ( t )] = 0, whichyields the classical equation of motion.
1. Imaginary time path integral formalism
The main motivation of imaginary time propagator comes from the partition function in statistical mechanics,which is given by Z = T r ( e − β ˆ H ) , (10)where β = 1 /T is the inverse temperature of the system. Inserting the position resolution of identity in Eqn.(7) intothe RHS of Eqn.(10) gives Z = (cid:90) dx A ( x, β ; x, , (11)where A ( x, β ; x,
0) = (cid:104) x | e − β ˆ H | x (cid:105) . (12)Suppose we consider the time in Eqn.(8) to be purely imaginary, which can be written as t (cid:48) = − iβ , where β is real.Then, substituting into Eqn.(8) we obtain the propagator evaluated at imaginary time ( MacKenzie, 2000; Polyakov,1977; Weiss and Walter, 1983): A E = (cid:104) x (cid:48) | e − β ˆ H/ (cid:126) | x (cid:105) = (cid:90) D x ( τ ) e − S E [ x ] / (cid:126) , (13)where the action is now given by the appropriate analytical continuation of the action, nominally defined as S E [ x ] = (cid:90) − iβ dt (cid:20) m (cid:18) dxdt (cid:19) − U ( x ) (cid:21) . (14)Then setting x (cid:48) = x in Eqn.(13) yields the partition function Eqn.(12). Thus, the propagator continued to imaginarytime gives the partition function. This method is very useful in finding the ground state of a physical system instatistical physics and condensed matter physics. The analytical continuation is obtained by defining a real variable τ = it , which is called the “imaginary or Euclidean time”, we see that τ and t are related as follows: t : 0 → − iβ , τ : 0 → β . Thus, S E [ x ( τ )] = − iS [ x ( t → − iτ )]. Typically, if S [ x ( t )] = (cid:82) dt ( T − V ), the Euclidean action is given by S E [ x ( τ )] = (cid:82) dτ ( T + V ), as the kinetic energy changes sign with the continuation to imaginary time. The Euclideanaction and the Lagrangian are S E [ x ( τ )] = (cid:90) β/ − β/ dτ L E ; L E = 12 m (cid:18) dxdτ (cid:19) + U ( x ) , (15)using time translation invariance. The boundary conditions for the imaginary time propagator are x ( − β/
2) = x and x ( β/
2) = x (cid:48) . This analysis of imaginary time propagator plays a decisive role in tunneling problems, such as that ofa particle in a one dimensional double well potential, since the period of oscillation or the momentum of the particleis imaginary in the tunneling region E < ∆ U (Landau and Lifshitz, 1977; Weiss and Walter, 1983), which is neatlycompensated by the imaginary time. Thus, it is almost always convenient to use imaginary time corresponding to thereplacement t → − iτ (Polyakov, 1977; Weiss and Walter, 1983) when considering tunnelling problems.
2. Instantons in the double well potential
In many textbooks of quantum mechanics, tunneling (barrier penetration) is usually studied using the WKB method.In the tunneling region, the WKB exponent is imaginary, the wave function in the becomes ψ ( x ) ∝ (cid:112) | p | exp (cid:20) − (cid:90) x − x | p | (cid:126) dx (cid:21) , (16)where p = (cid:112) m ( U ( x ) − E ) is the momentum of the particle, and ± x are the classical tunneling points U ( ± x ) = E .At the ground state, the energy splitting is given by (Landau and Lifshitz, 1977; Weiss and Walter, 1983)∆ = (cid:126) ω √ eπ exp (cid:20) − (cid:90) a − a | p | (cid:126) dx (cid:21) , (17)where ± a are such that U ( ± a ) = E . The instanton approach, however, uses the imaginary time formulation of pathintegral to find this ground state energy splitting. If we consider the classical equation of motion in imaginary time δS E = 0 we get: m ¨ x = dU ( x ) dx , where ¨ x ≡ d xdτ , (18)which is the equation of motion with − U ( x ). In other words, it describes the motion of a particle in an invertedpotential as shown in Fig.(2). Upon integration, one finds that the analog of the total “energy” is conserved: E = 12 m (cid:18) dxdτ (cid:19) − U ( x ) . (19)There are at least three possible solutions of this equation of motion. The first solution corresponds to a particlesitting on the top of the left hill x = − a in Fig.(2) , and the second solution corresponds to a particle sitting on thetop of the right hill x = a . These are constant solutions which do not give any tunneling. However, there is a thirdsolution in which the particle starts at the left hill at τ → −∞ rolls over through the dashed line, and finally arrivesat the right hill at τ → ∞ . This solution corresponds exactly to the barrier penetration in the WKB method. Suchtrajectory mediates tunneling and it is called an instanton. Quantum mechanically, the propagator for this instantontrajectory is given by A (cid:18) − a, − β a, β (cid:19) = (cid:104) a | e − β ˆ H/ (cid:126) | − a (cid:105) . (20)For instance, the potential could be taken to be U ( x ) = ω x − a ) , (21) FIG. 2 A sketch of an inverted double well potential with two minima at ± a . There are two trivial solutions corresponding toa fixed motion of the particle at the top of the left or right hill of the potential. Tunneling is achieved by a nontrivial solutionin which the particle starts at the top of the left hill at τ → −∞ and roll through the dashed line and emerges at the top ofthe right hill at τ → + ∞ . Such a solution is called an instanton. but it is actually not necessary to make a specific choice, just the general form pictured in Fig.2 needs to be satisfied.Tunneling between the two minima of U ( x ) requires the computation of the transition amplitudes: (cid:104)± a | e − β ˆ H/ (cid:126) | − a (cid:105) . (22)In order to calculate this amplitude one has to know the solution of the classical equation of motion that obeys theboundary condition of Eqn.(13) as β → ∞ . There are two trivial solutions corresponding to no motion with theparticle fixed at the top of the left or right hill of the potential. Tunneling is achieved by a nontrivial solution inwhich the particle starts at the top of the left hill at τ → −∞ , roll through the dashed line in Fig.(2), and emerges atthe top of the right hill at τ → + ∞ . This nontrivial solution has zero “energy” E = 0 since initially it starts at thetop of the hill at − a where the potential is zero and its kinetic energy is zero. The solution of Eqn.(19) correspondingto the explicit potential Eqn. (21), is given by ( MacKenzie, 2000; Polyakov, 1977) x ( τ ) = a tanh[ ω τ − τ )] , ω = γa /m, (23)where τ is an integration constant which corresponds to the time at which the solution crosses x = 0.The action for the solution is given by B = (cid:90) β/ − β/ dτ (cid:34) m (cid:18) dxdτ (cid:19) + U ( x ) (cid:35) , (24)= (cid:90) β/ − β/ dτ (cid:112) mU ( x ) dxdτ , (25)= (cid:90) a − a dx (cid:112) mU ( x ) , (26)= 2 √ m ω a , (27)where E = 0 from Eqn.(19) is used in the second line, and only in the last equation is the specific potential Eqn.(21)used. This action is exactly the WKB exponent in Eqn.(17). In the approximation of the method of steepest descent,the path integral, Eqn.(13) is dominated by the path which passes through the configuration for which the action isstationary, i.e., Eqn.(23), and the integral is given by the Gaussian approximation about the stationary point. Then,the one instanton contribution to the transition amplitude is (Coleman, 1977, 1985) (cid:104) a | e − β ˆ H/ (cid:126) | − a (cid:105) ∝ e − B/ (cid:126) [1 + O ( (cid:126) )] . (28)In fact, one must consider other critical points which correspond to a dilute instanton gas. The justification of thedilute instanton gas approximation is beyond the purview of this review, we refer the reader to dedicated expositionsof the subject, (Coleman, 1977, 1985). The upshot is that one must sum over all sequences of one instanton followedby any number of anti-instanton/instanton pairs, the total number of instantons and anti-instantons is odd for thetransition − a ↔ a but even for the transition − a → − a ( a → a ). The result of this summation yields (Coleman,1985) (cid:104)± a | e − β ˆ H/ (cid:126) | − a (cid:105) = N
12 [exp( D βe − B/ (cid:126) ) ∓ exp( − D βe − B/ (cid:126) )] , (29)where N is the overall normalization including the square root of the free determinant which is given by N e − β E where E = (cid:126) ω is the unperturbed ground state energy and N is a constant from the ground state wave function. D is the ratio of the square root of the determinant of the operator governing the second order fluctuations aboutthe instanton excluding the time translation zero mode, and that of the free determinant. It can in principle becalculated. A zero mode, occurring because of time translation invariance, is not integrated over, and is taken intoaccount by integrating over the Euclidean time position of the occurrence of the instanton giving rise to the factor of β . The left hand side of Eqn.(29) can also be written as (cid:104)± a | e − β ˆ H/ (cid:126) | − a (cid:105) = (cid:88) n (cid:104)± a | n (cid:105) (cid:104) n | − a (cid:105) e − β E n , (30)where ˆ H | n (cid:105) = E n | n (cid:105) . Taking the upper sign on both sides of Eqs.(29) and (30) and comparing the terms, one findsthat the non-perturbative energy splitting between the ground and the first excited states is given by∆ = E − E = 2 (cid:126) D e − B/ (cid:126) . (31)In a similar manner, by comparing the coefficients one obtains symmetric ground state |E (cid:105) = 1 √ | a (cid:105) + |− a (cid:105) ) , (32)and an antisymmetric first excited state |E (cid:105) = 1 √ | a (cid:105) − |− a (cid:105) ) . (33)The analysis in the first part of this review will be based on computing the instanton trajectory, its action, and thecorresponding energy splitting for any given model that possesses tunneling. B. Spin coherent state path integral
For a spin system, the basic idea of path integral formulation is retained, however, instead of the orthogonal position | x (cid:105) and momentum | p (cid:105) basis, a basis of spin coherent states is used (Klauder, 1979; Lieb, 1973; Perelomov, 1986;Radcliffe, 1971). This basis is defined through the following construction. Let | s, s (cid:105) be the highest weight vector in aparticular representation of the rotation group, taken as its simply connected covering group SU (2). This state is aneigenstate of the operators ˆ S z and ˆS :ˆ S z | s, s (cid:105) = s | s, s (cid:105) ; ˆS | s, s (cid:105) = s ( s + 1) | s, s (cid:105) . (34)The spin operators ˆ S i , i = x, y, z form an irreducible representation of the Lie algebra of SU (2),[ ˆ S i , ˆ S j ] = i(cid:15) ijk ˆ S k , (35)where (cid:15) ijk is the totally antisymmetric tensor symbol and summation over repeated indices is implied in Eqn.(35).The coherent state, | ˆn (cid:105) , an element of the 2 s + 1 dimensional Hilbert (representation) space for the spin states, isdefined as (Eduardo and Stone, 1988; Eduardo, 1991; Klauder, 1979; Lieb, 1973; Perelomov, 1986; Zhang, Feng andGilmore, 1990) | ˆn (cid:105) = e iθ ˆm · ˆS | s, s (cid:105) = s (cid:88) m = − s M s ( ˆn ) ms | s, m (cid:105) , (36)0 FIG. 3 The directions of the unit vectors ˆz and ˆn on a two-sphere . where ˆn = (cos φ sin θ, sin φ sin θ, cos θ ) is a unit vector ie. ˆn = 1 and ˆm = ( ˆn × ˆz ) / | ˆn × ˆz | is a unit vector orthogonalto ˆn and where ˆz is the quantization axis pointing from the origin to the north pole of a unit sphere and ˆn · ˆz = cos θ as shown in Fig.(3). Rotating the unit vector ˆz about the ˆm direction by the angle θ brings it exactly to the unitvector ˆn . | ˆn (cid:105) corresponds to a rotation of an eigenstate of ˆ S z , i.e | s, s (cid:105) , to an eigenstate with a quantization axis along ˆn on a two-dimensional sphere S = SU (2) /U (1). The matrices M s ( ˆn ) satisfy the relation M s ( ˆn ) M s ( ˆn ) = M s ( ˆn ) e i G ( ˆn , ˆn , ˆn ) ˆ S z , (37)where G ( ˆn , ˆn , ˆn ) is the area of a spherical triangle with vertices ˆn , ˆn , ˆn . Note that Eqn.(37) is not a groupmultiplication, thus the matrices M s ( ˆn ) do not form a group representation. Unlike the position and momentumeigenstates in Eqn.(5), the inner product of two coherent states is not orthogonal: (cid:104) ˆn | ˆn (cid:48) (cid:105) = e is G ( ˆn , ˆn (cid:48) , ˆz ) [ 12 (1 + ˆn · ˆn (cid:48) )] s . (38)It has the following property: ˆn · ˆS | ˆn (cid:105) = s | ˆn (cid:105) ⇒ (cid:104) ˆn | ˆS | ˆn (cid:105) = s ˆn . (39)The resolution of identity is given by ˆI = 2 s + 14 π (cid:90) d ˆn δ ( ˆn − | ˆn (cid:105) (cid:104) ˆn | , (40)where ˆI is a (2 s + 1) × (2 s + 1) identity matrix, and the delta function ensures that ˆn = 1. The derivation ofspin coherent state path integral now follows a similar fashion with Sec.(II.A). Using the expression in Eqn.(38) andEqn.(40) one can express the imaginary time transition amplitude between | ˆn i (cid:105) and | ˆn f (cid:105) as a path integral. Theanalogous form of Eqn.(13) for spin system is given by (Eduardo, 1991; Zhang, Feng and Gilmore, 1990) (cid:104) ˆn f | e − β ˆ H ( ˆS ) | ˆn i (cid:105) = (cid:90) D ˆn e − S E [ ˆn ] , (41)where S E [ ˆn ] = isS W Z + (cid:90) dτ U ( ˆn ( τ )) , U ( ˆn ( τ )) = (cid:104) ˆn | ˆ H | ˆn (cid:105) , (42)and S W Z arises because of the additional phase e is G ( ˆn , ˆn (cid:48) , ˆz ) in Eqn.(38). We have set (cid:126) = 1 in the path integral. TheWess-Zumino (WZ) action, S W Z is given by (Eduardo and Stone, 1988; Eduardo, 1991; Novikov, 1982; Wess and An alternative way of deriving this equation can be found in (Blasone and Jizba, 2012). S W Z = (cid:90) S dτ dξ ˆn ( τ, ξ ) · [ ∂ τ ˆn ( τ, ξ ) × ∂ ξ ˆn ( τ, ξ )] , (43)where ˆn ( τ ) has been extended over a topological half-sphere S in the variables τ, ξ . In the topological half-spherewe define ˆn with the boundary conditions ˆn ( τ,
0) = ˆn ( τ ) , ˆn ( τ,
1) = ˆ z , (44)so that the original configuration lies at the equator and the point ξ = 1 is topologically compactified by the boundarycondition. This can be easily obtained by imagining that the original closed loop ˆn ( τ ) at ξ = 0 is simply pushedup to along the meridians to ˆn ( τ ) = ˆ z at ξ = 1. The Wess-Zumino term originates from the non-orthogonality ofspin coherent states in Eqn.(38). Geometrically, it defines the area of the closed loop on the spin space, defined bythe nominally periodic, original configuration ˆn ( τ ). It crucial to note that there is an ambiguity of modulo 4 π , sincedifferent ways of pushing the original configuration up can give different values for the area enclosed by the closedloop as one can imagine that the closed loop englobes the whole two sphere any integer number of times, but thisambiguity has no physical significance since e i Nπs = 1 for integer and half-odd integer s . The action, Eqn.(42) is validfor a semiclassical spin system whose phase space is S . It is the starting point for studying macroscopic quantumspin tunneling between the minima of the energy U ( ˆn ). III. MACROSCOPIC QUANTUM TUNNELING OF LARGE SPIN SYSTEMSA. Coordinate dependent formalism
Most often a coordinate dependent version of Eqn.(43) is used in the condensed matter literature. It seems thatmost people find it difficult to study macroscopic quantum spin tunneling in the coordinate independent form. Inthis section, we will show how one can use any coordinate system of interest. In section (III.B), we will show that thecoordinate independent form can reproduce all the known results in quantum spin tunneling. Since the spin particlelives on a two-sphere, the most convenient choice of coordinate are spherical polar coordinates. Parametrizing theunit vector as ˆn ( τ, ξ ) = (cos φ ( τ ) sin θ ξ ( τ ) , sin φ ( τ ) sin θ ξ ( τ ) , cos θ ξ ( τ )), with θ ξ ( τ ) = (1 − ξ ) θ ( τ ), which satisfies theboundary conditions, Eqn.(44) at ξ = 0 and ξ = 1. Then ∂ τ ˆn = ˆ θ ˙ θ ξ ( τ ) + ˆ φ sin θ ξ ( τ ) ˙ φ ( τ ) , (45)and ∂ ξ ˆn = ˆ θ ( − θ ( τ )) , (46)where ˆ θ and ˆ φ are the usual polar and azimuthal unit vectors which form an orthogonal triad with ˆn such that ˆ θ × ˆ φ = ˆn (and cyclic permutations). Thus we find the triple product becomes ˆn ( τ, ξ ) · ( ∂ τ ˆn ( τ, ξ ) × ∂ ξ ˆn ( τ, ξ )) = ˙ φ ( τ ) θ ( τ ) sin θ ξ ( τ ) . (47)Thus, the WZ term, Eqn.(43) simplifies to (Khare and Paranjape, 2011; Owerre and Paranjape, 2013) S W Z = (cid:90) dτ (cid:90) dξ ˙ φ ( τ ) θ ( τ ) sin θ ξ ( τ ) = (cid:90) dτ ˙ φ ( τ )(1 − cos θ ( τ )) . (48)This is the coordinate dependent form of WZ term or Berry phase (Berry, 1984), which is the expression found inmost condensed matter literature. It corresponds to the area of the unit two-sphere swept out by ˆn ( τ ) as it formsa closed path on S . To understand this explicitly, one can think of the integral in Eqn.(48) as a line integral of agauge field, which only has a φ component, integrated over a closed path on the two sphere, parametrized by τ . Wedenote the closed path as C and it is the boundary of a region S , with evidently C = ∂ S , then (cid:90) dτ ˙ φ ( τ )(1 − cos θ ( τ )) = (cid:73) C A φ dφ. (49)Then using Stokes theorem, we have (cid:73) C A φ dφ = (cid:90) S d ( A φ dφ ) , (50)2written in the notation of differential forms. However, the gauge field (cid:126)A = A φ ˆ φ = (1 − cos θ ) ˆ φ corresponds exactlyto the gauge field of a magnetic monopole located at the centre of the sphere. Such a gauge field was first describedby Dirac (Dirac , 1931), and gives rise to a constant radial magnetic field, apart from a string singularity located atthe south pole, which is an unobservable gauge artefact if the magnetic charge is appropriately quantized. The nonobservability of this string singularity in quantum mechanics was the seminal observation by Dirac if s , in Eqn.(42),is quantized to be a half integer. Explicitly, the corresponding magnetic field is simply d ( A φ dφ ) = ∂ θ A φ dθ ∧ dφ =sin θdθ ∧ dφ which is the area element in spherical polar coordinates on the unit two sphere. Thus (cid:72) C A φ dφ = (cid:82) S d ( A φ dφ ) = (cid:82) S sin θdθ ∧ dφ = area ( S ).The general form of the Euclidean action in coordinate dependent formalism is then S E = is (cid:90) dτ ˙ φ ( τ ) + S , (51)where S = (cid:90) dτ [ − is ˙ φ ( τ ) cos θ ( τ ) + U ( θ ( τ ) , φ ( τ ))] . (52)The first term in Eqn.(51) is a boundary term, which does not affect the classical equation of motion. It can beintegrated out as is (cid:90) β − β dτ ˙ φ ( τ ) = is [ φ ( β/ − φ ( − β/
2) + 2 πN ] , (53)where N is a winding number, that is the number of times φ ( τ ) winds around the north pole of S as τ progressesfrom − β/ β/
2. This term is insensitive to any continuous deformation of the field on S , thus it is topological.Its effect on the transition amplitude will be studied later.
1. Easy z -axis uniaxial spin model in a magnetic field Having derived the coordinate dependent action for a spin system, we will now turn to specific models where thisformula can be implemented. Consider a uniaxial system with an easy ˆz axis (direction of minimum energy) and amagnetic field along the ˆx axis, the corresponding Hamiltonian is given by ( Chudnovsky and Gunther, 1988; VanHemmen and S¨ut¨o, 1986) ˆ H = − D ˆ S z − H x ˆ S x , (54)where D > H x = gµ B h , h is the magnitude of the field, g is the spin g -factor and µ B is the Bohr magneton. This model is a special case of the Lipkin-Meshkov-Glick model introduced in nuclearphysics (Lipkin, Meshkov and Glick, 1965), which has been recently exactly solved (Ribeiro, Vidal and Mosseri, 2007,2008). This Hamiltonian is a good approximation for Mn acetate molecular magnet with a ground state of s = 10( Chudnovsky, et al. , 1998; Chudnovsky and Garanin, 1997; Friedman et al. , 1996; Novak and Sessoli, 1994; Paulsenand Park, 1994). An experimental review of this molecular magnet can be found in (Gatteschi and Sessoli, 2003). Thedescription of the tunneling of spin in the quantum spin terminology is as follows. For H x = 0, the Hamiltonian hasa two fold degenerate ground state corresponding to the two ground states in the ˆ S z representation, i.e, |↑(cid:105) and |↓(cid:105) ,where |↑(cid:105) ≡ | s (cid:105) and |↓(cid:105) ≡ |− s (cid:105) . For H x (cid:54) = 0, these two states are no longer degenerate since ˆ S x = ( ˆ S + + ˆ S − ) / S + |− s (cid:105) ∝ |− s + 1 (cid:105) and ˆ S − | s (cid:105) ∝ | s − (cid:105) . In the limit of small magnetic field, perturbation theory on the magneticfield term shows that the two degenerate ground states are split with an energy difference which is given by (Garanin, 1991; Zaslavskii and Ulyanov, 1999)∆ = 4 Ds / π / (cid:18) eh x (cid:19) s , h x = H x / Ds. (55)The factor h sx signifies that the splitting arises from 2 s th order in degenerate perturbation theory. This implies thatthe two quantum states |↑(cid:105) and |↓(cid:105) can tunnel to each other through a magnetic energy barrier, a process called3quantum spin tunneling . Thus, the ground and the first excited states become the symmetric and antisymmetriclinear superposition of the degenerate states: | g (cid:105) = 1 √ |↑(cid:105) + |↓(cid:105) ); | e (cid:105) = 1 √ |↑(cid:105) − |↓(cid:105) ) . (56)In the absence of the perturbative or splitting term, the energy splitting in Eqn.(55) vanishes, which directly impliesthat tunneling is only allowed when the Hamiltonian does not commute with the quantization axis, in this case ˆ S z .In the semi-classical analysis, the spin operator becomes a vector parametrized by spherical coordinate of length: S x + S y + S z = s . (57)The corresponding classical energy of Eqn.(54) is given by U ( θ, φ ) = Ds sin θ − H x s sin θ cos φ + H x / D, (58)where an additional constants have been added to normalize the minimum of the potential to zero. The minimumenergy requires ∂U∂θ (cid:12)(cid:12)(cid:12)(cid:12) φ =0 = 0 and ∂ U∂θ (cid:12)(cid:12)(cid:12)(cid:12) φ =0 > , (59)which yields two classical degenerate minima at ( φ, θ ) = (0 , θ ) and ( φ, θ ) = (0 , π − θ ) with sin θ = h x = H x /H c ,provided H x < H c = 2 Ds . The maximum energy corresponds to ( φ, θ ) = (0 , π/ FIG. 4 The description of a classical spin (thick arrows) on a two-sphere with two classical ground states at φ = 0 . Themagnetic field is applied parallel to the x -axis. The x -axis has been rotated on the right hand side for proper view. These two classical minima correspond to the spin pointing in ± zx plane (see Fig.(4)), which are analogous to thetwo quantum states |↑(cid:105) and |↓(cid:105) . The barrier height is∆ U = U max − U min = Ds (1 − h x ) . (60)Due to tunneling the degeneracy of these ground states will be lifted and one finds that the true ground state isthe linear superposition of the two unperturbed ground states. This tunneling is mediated by an instanton which is In the semi-classical description, tunneling means the rotation of the two equivalent directions of the spin on a two-sphere as shown inFig.(4)
4a solution of the classical equations of motion: is ˙ θ sin θ = ∂U∂φ , (61) is ˙ φ sin θ = − ∂U∂θ . (62)These equations are obtained from the least-action principle, whose solution gives the classical path for which theaction, Eqn.(51) is stationary δS E = 0. Although one is usually interested in a real, physical trajectory, these equationsare in fact, incompatible, unless one variable (either θ or φ ) becomes imaginary. The energy along the trajectory hasto vanish, since it is conserved by the dynamics, and normalized to zero at the starting point. This can be seen bymultiplying Eqn.(61) by ˙ φ and Eqn.(62) by ˙ θ and subtracting the resulting equations which yields ∂U∂φ ˙ φ + ∂U∂θ ˙ θ = 0 ⇒ U ( θ, φ ) = const. = 0 . (63)The transition amplitude, Eqn.(41), in the coordinate dependent form can be written as (cid:104) θ f , φ f | e − β ˆ H | θ i , φ i (cid:105) = (cid:90) D φ D (cos θ ) e − S E , (64)which defines the transition from an initial state | θ i , φ i (cid:105) at τ = − β/ | θ f , φ f (cid:105) at τ = β/
2, subjectto the boundary conditions ( φ ( − β/ , θ ( − β/ φ i , θ i ) and ( φ ( β/ , θ ( β/ φ f , θ f ). In most cases of physicalinterest, either φ i = φ f or θ i = θ f . In the present problem φ i = φ f = 0 while θ i = θ and θ f = π − θ . Similar to thedouble well problem in Fig.(2), the boundary conditions require that the real tunneling trajectory (either θ or φ notboth) approaches the two minima of U at τ = ±∞ . Using Eqn.(58) one obtains from Eqn.(63)sin( φ/
2) = ± i (sin θ − sin θ ) / (cid:112) sin θ sin θ . (65)From Eqs.(58), (61) and (65), the classical trajectory (instanton) is found to be ( Chudnovsky and Gunther, 1988;Garg and Kim, 1992) cos θ ( τ ) = − cos θ tanh( ω h τ ) , ω h = Ds cos θ , (66)which interpolates from θ ( τ ) = θ at τ = −∞ to θ ( τ ) = π − θ at τ = ∞ . Since the energy remains constant (which isnormalized to zero) along the instanton trajectory, the action for this trajectory is determined only by the WZ termin Eqn.(51). It is found to be (Garg and Kim, 1992) B = 2 s (cid:20)
12 ln (cid:18) θ − cos θ (cid:19) − cos θ (cid:21) . (67)Absence of tunneling when h x = 0 corresponds to B = ∞ . The energy splitting in the dilute instanton gas approxi-mation is given by (Garg and Kim, 1992; Garg, 2000)∆ = 8 Ds / cos / θ π / sin θ (cid:18) − cos θ θ (cid:19) s + e s cos θ . (68)In the perturbative limit, that is for a very small magnetic field, θ →
0, the splitting, Eqn.(68) reduces to∆ = 8 Ds / (1 − h x / / e s (1 − h x / π / (4 − h x ) s + h sx . (69)The factor h sx reproduces the correct order of perturbation theory result as given in Eqn.(55).
2. Biaxial spin model and quantum phase inteference
Let us consider the biaxial spin model in the absence of an external magnetic field ( Chudnovsky and Gunther,1988; Enz and Schilling, 1986; Loss, DiVincenzo and Grinstein, 1992)ˆ H = D ˆ S z + D ˆ S x ; D > D > . (70)5In the classical terminology, this model possesses an XOY -easy-plane anisotropy with an easy-axis along the y -direction, hard-axis along the z -direction and medium axis along the x -direction. Quantum mechanically, the easyaxis corresponds to the quantization axis, since the Casimir operator ˆS = ˆ S x + ˆ S y + ˆ S z = s ( s + 1), can be used torewrite Eqn.(70) as ˆ H = − D ˆ S y + ( D − D ) ˆ S z + const. (71)The first term is the unperturbed term while the second term is the transverse or splitting term which does notcommute with the unperturbed term. Thus, the minimum energy of this Hamiltonian requires a representation inwhich ˆ S y is diagonal. This means that different representations of a biaxial spin Hamiltonian in the absence of anexternal magnetic field can be related to each other by redefining the anisotropy constants. For instance Eqn.(70)is related to ˆ H = − A ˆ S x + B ˆ S z (Enz and Schilling, 1986) by D = A , D = A + B . Thus, it suffices to consider justEqn.(70). Semiclassically, the corresponding classical energy is U ( θ, φ ) = D s cos θ + D s sin θ cos φ. (72)The minimum energy corresponds to ( φ, θ ) = ( ± π/ , π/ ± ˆy as shown in Fig.(6), and themaximum is located at ( φ, θ ) = (0 , π/ θ = ± i √ λ cos φ (cid:112) − λ cos φ , λ = D /D . (73)Taking into account that the deviation of the spin away from the easy plane is very small, an alternative method toeliminate θ from the equation of motion is to integrate out cos θ in Eqn.(64)(Chudnovsky and Martinez, 2000; Enzand Schilling, 1986; Zhang, et al. , 1998). In this case the resulting action has a quadratic first order derivative term,a coordinate ( φ ) dependent mass and a potential . Integration of the classical equation of motion Eqn.(62) yields (Chudnovsky and Gunther, 1988; Enz and Schilling, 1986; Zhang, et al. , 1998)sin φ ( τ ) = √ − λ tanh( ωτ ) (cid:113) − λ tanh ( ωτ ) , ω = 2 s (cid:112) D D , (74)which corresponds to the tunneling of the spin from φ = π/ τ = ∞ to φ = − π/ τ = −∞ . The instantonaction for this trajectory is S c = is (cid:90) π − π dφ + B, (75)where B is given by B = s √ λ (cid:90) π − π dφ cos φ (cid:112) − λ cos φ = ln (cid:32) √ λ − √ λ (cid:33) s . (76)Now, consider for example the path ( φ ( τ ) , θ ( τ )) connecting the two anisotropy minima at ( φ, θ ) = ( ± π/ , π/ S , Eqn.(52) (that is excluding the total derivative term), the path( − φ ( τ ) , π − θ ( τ )) will also solve the classical equations of motion and B will be the same for both paths but the totalderivative term will be reversed: is (cid:82) ± π ∓ π dφ = ± isπ . Since the path integral in Eqn.(64) contains all paths, in thesemiclassical (small (cid:126) ) approximation (Coleman, 1977, 1985; Weiss and Walter, 1983), the contributions of these twopaths can be combined to give e iπs e − B + e − iπs e − B = 2 cos( πs ) e − B . (77)More appropriately, to obtain the tunneling rate one has to use the dilute-instanton gas approximation that is bysumming over a sequences of one instanton followed by any number of anti-instanton/instanton pairs, with an odd In the presence of a magnetic field, different representation of a biaxial spin models can also be related by the anisotropy constants orrotation of axes (cid:104) π | e − β ˆ H | − π (cid:105) = N sinh (cid:2) D β cos( πs ) e − B (cid:3) , (78)where D is the fluctuation determinant (Coleman, 1977; Callan and Coleman, 1977; Coleman, 1985). The computationof D can be done explicitly. N is a normalization constant and B is the action for the instanton. The tunneling rate(energy splitting) from Eqn.(78) gives (Loss, DiVincenzo and Grinstein, 1992)∆ = 4 D | cos( πs ) | e − B , (79)The factor cos( πs ) is responsible for interference effect and it has markedly different consequences for integer and half-odd integer spins. For integer spins (bosons), the interference is constructive cos( πs ) = ( − s , and the tunneling rate isnon-zero, however, for half-odd-integer spins (fermions), the interference is destructive cos( πs ) = 0 and the tunnelingrate vanishes. This suppression of tunneling for half-odd-integer spins in this model can be related to Kramersdegeneracy (Kramers, 1930; Messiah, 1962) due to the time reversal invariance of Eqn.(70). This directly implies thatthe ground state is at least two-fold degeneracy in the semi-classical picture. This semi-classical degeneracy sometimesimplies that the two degenerate quantum ground states of the unperturbed term, |↑(cid:105) and |↓(cid:105) are exact ground statesof the quantum Hamiltonian for half-odd integer spin (Henley and Delft, 1992). -0.02 0 0.02 0.04 0.06 0.08 0.1012345678 (cid:54)(cid:54)(cid:54)(cid:54) / (cid:54)(cid:54)(cid:54)(cid:54) H trans / H a Mn S = 9/2 Mn S = 8 F e S = 10
FIG. 5 Measured tunnel splittings obtained by the Landau-Zener method as a function of transverse field for all three SMMs.The tunnel splitting increases gradually for an integer spin, whereas it increases rapidly for a half-integer spin. Adapted withpermission from Wernsdorfer et al. , 2002
In this biaxial model we have just reviewed, the quantum phase interference appeared naturally from the topologicalterm in the action, Eqn.(53) since the instanton trajectory is in the φ variable. If we had considered the z -easy axismodel such as ˆ H = − k z ˆ S z + k y ˆ S y , k z , k y > , (80)then the situation would have been different. This Hamiltonian is related to Eqn.(70) by k z = D , k y = D − D orby rotation of axis ˆ S z ↔ ˆ S y . Suppose we wish to solve Eqn.(80) as it is, then the corresponding classical energy is U ( θ, φ ) = ( k z + k y sin φ ) s sin θ, (81)One finds from the conservation of energy that φ ( τ ) is an imaginary constant and θ ( τ ) is the real tunneling trajectorywhich is given by (Owerre and Paranjape, 2014a) θ ( τ ) = 2 arctan[exp( ω ( τ − τ ))] , (82)where ω = 2 s (cid:112) k z ( k y + k z ), and θ ( τ ) → , π as τ → ∓∞ . The fact that φ ( τ ), although imaginary, is just a constantsimply implies that the topological term in Eqn.(53) which is responsible for the phase interference vanishes. The7transition amplitude arises from the necessity to translate φ from some fiducial value, taken without loss of generalityto be zero, to the complex constant value before the instanton trajectory in θ and then followed by the translation of φ back to its fiducial value after the instanton trajectory. It was explicitly shown, that translation of φ in the complexplane yields the transition amplitude and the corresponding energy splitting is of the form: (Owerre and Paranjape,2014a, 2013) ∆ = 2 D (1 + cos(2 πs )) e − B , (83)where B = (cid:40) s ln (cid:16) k z k y (cid:17) if k y (cid:28) k z , s ( k z /k y ) / if k y (cid:29) k z . (84)The fluctuation determinant is calculated to be D = 8 √ k z s / /π / for k y (cid:28) k z and D = 8( sk z k y ) / /π / for k y (cid:28) k z (Garg and Kim, 1992). Thus, we recover that tunneling is restricted for half-odd integer spins. For integerspin and the semiclassical limit s (cid:29)
1, simple operatorial quantum mechanical perturbation theory in the splittingterm for k y (cid:28) k z gives (Garanin , 1991) ∆ = 8 k z s / π / (cid:18) k y k z (cid:19) s , (85)which is consistent with Eqn.(83) for integer spin s . The experimental confirmation of this spin-parity effect (i.esuppression of tunneling for half-odd integer spin) in spin systems was reported by Wernsdorfer et al. (2002). Theystudied three SMMs in the presence of a transverse field using Landau-Zener method to measure the tunnel splittingas a function of transverse field. They established the spin-parity effect by comparing the dependence of the tunnelingsplitting on the transverse field for integer and half-odd integer spin systems. Observation showed that an integerspin system is insensitive to small transverse fields whereas a half-odd integer spin system is much more sensitive asshown in Fig.(5). This observation is analogous to the fact that half-odd integer spin does not tunnel.
3. Biaxial spin model with an external magnetic field
The quantum phase interference (quenching of tunneling splitting) we saw in the previous section is a zero magneticfield effect. In the presence of a magnetic field complete destructive interference for half-odd integer spins does notoccur instead oscillation occurs. Consider the biaxial spin model with an external magnetic field applied along thehard-axis (Garg, 1993, 1999, 2001) ˆ H = D ˆ S z + D ˆ S x − h z ˆ S z , (86)where h z = gµ B h , h is the magnitude of applied field and g is the spin g -factor and µ B is the Bohr magneton. ThisHamiltonian can also be written asˆ H = − D ˆ S y + ( D − D ) ˆ S z − h z ˆ S z + const. (87)Thus, we see explicitly that the easy (quantization) axis is along the y -direction. Unlike the previous model thisHamiltonian is no longer time reversal invariant due the presence of the magnetic field, so Kramers theorem is nolonger applicable. This Hamiltonian has been studied experimentally for Fe molecular cluster (Sangregorio, et al. ,1997; Sessoli et al. , 2000; Wernsdorfer and Sessoli, 1999). There are 2 s + 1 energy level spectra where s = 10 and aquantum number m = − , − , · · · ,
10. At very low temperature (
T < . K ) only the lowest states m = ±
10 areoccupied which can tunnel macroscopically. In the semi-classical analysis, the classical energy up to an additionalconstant is U ( θ, φ ) = D s (cos θ − α ) + D s sin θ cos φ, (88)with α = h z /h c , h c = 2 D s being the coercive field.There are two classical degenerate minima located at cos θ = α, φ = − π/ θ = α, φ = π/ h z < h c . These ground states lie in the xz and yz planes at an angle θ = ± arccos α as shown in Fig.(6). From energyconservation , Eqn.(63) the expression for cos θ in terms of φ yieldscos θ = α + iλ / cos φ (1 − α − λ cos φ ) / − λ cos φ , (89)8 FIG. 6 The description of a classical spin (thick arrows) on a two-sphere with two classical ground states. For h z = 0, θ = ± π/ ± y directions which are joined by two tunneling paths in the equator. For h z > θ = ± arccos α , the two classical ground states lie in the yz plane. Reproduced from Schilling (1995) We have chosen the positive solution in Eqn.(89) for convenience. Using this equation and Eqn.(62), one obtains theinstanton solution: sin φ ( τ ) = √ − λ H tanh( ω H τ ) (cid:113) − λ H tanh ( ω H τ ) , (90) α ∆ / D s =10 s =19 / FIG. 7 Color online: Oscillation of the tunneling splitting as a function of the magnetic field parameter α . Solid line is forinterger spins while dotted line is for half-odd integer spins. where ω H = 2 s (cid:112) D D (1 − α ) and λ H = λ/ (1 − α ). The classical action for this instanton path is S c = iπ Θ + B, (91)where Θ = s π ( S + − S − ) , (92)9and S + − S − is the area enclosed by the two tunneling paths on a 2-sphere as shown in Fig.(6), which is given by S ± = (cid:90) ± π ∓ π dφ (cid:18) − α − λ cos φ (cid:19) = ± π (cid:18) − α √ − λ (cid:19) . (93)The instanton action is given by B = s ln (cid:32) √ − α + √ λ √ − α − √ λ (cid:33) − sα √ − λ ln (cid:32) (cid:112) (1 − α )(1 − λ ) + α √ λ (cid:112) (1 − α )(1 − λ ) − α √ λ (cid:33) . (94)In this problem the imaginary path of the instanton action, Eqn.(91) has acquired an additional term due to thepresence of the magnetic field. In the dilute instanton gas approximation, one obtains that the tunneling rate is thengiven by ∆ = ∆ | cos( π Θ) | , ∆ = 4 D e − B , (95)which clearly reduces to Eqn.(79) in the limit of zero magnetic field. Now, the tunneling splitting is no longersuppressed for half-odd integer spin but rather oscillates with the magnetic field (see Fig.(7)) with a period ofoscillation of ∆ h = 2 D √ − λgµ B , (96)only vanishes at Θ = ( n + 1 /
2) or α = √ − λ ( s − n − / /s, (97)where n is an integer. It is crucial to note that the quenching of tunneling at a critical field only occurs for biaxialspin system with a magnetic applied along the hard anisotropy axis.
4. Landau Zener effect
The uniaxial and the biaxial models we have studied so far can be mapped to a two-level pseudospin particlesystem (Chudnovsky and Garanin , 2010; Chudnovsky, 2014; Owerre, 2014). Let us consider a two-level systemwhich is described by an unperturbed Hamiltonian ˆ H ( η ) that depends explicitly on a parameter η . Suppose that theeigenstates of this Hamiltonian are | m (cid:105) and | m (cid:48) (cid:105) , then the eigenvalue equation yieldsˆ H ( η ) | m (cid:105) = ζ ( η ) | m (cid:105) , (98)ˆ H ( η ) | m (cid:48) (cid:105) = ζ ( η ) | m (cid:48) (cid:105) , (99)where ζ , ( η ) are the corresponding eigenenergies. It is assumed that the eigenstates | m (cid:105) and | m (cid:48) (cid:105) are independent ofthe parameter η , and that at some value of η , ˆ H ( η ) possesses a symmetry which allows level crossing (degeneracy)of the two eigenvalues ζ , ( η ). The parameter η could be an applied magnetic field(Wernsdorfer, et al. , 2005). In thepresence of a perturbative term ˆ V , the total Hamiltonian can be written asˆ H = ˆ H + ˆ V . (100)The Hamiltonian can be diagonalized in the basis (cid:8) | m (cid:105) , | m (cid:48) (cid:105) (cid:9) , the corresponding matrix is given byˆ H ( η ) = (cid:18) ε ( η ) ∆∆ ∗ ε ( η ) (cid:19) , (101)where ε = ζ ( η ) + (cid:104) m | ˆ V | m (cid:105) , (102) ε = ζ ( η ) + (cid:104) m (cid:48) | ˆ V | m (cid:48) (cid:105) , (103)∆ = 2 | (cid:104) m | ˆ V | m (cid:48) (cid:105) | ⇒ (cid:104) m | ˆ V | m (cid:48) (cid:105) = 12 ∆ e − iφ . (104)0Diagonalizing Eqn.(101), one obtains the eigenvalues: ε + = 12 (cid:2) ( ε + ε ) + (cid:0) ε ( η ) + 4 | ∆ | (cid:1) / (cid:3) , (105) ε − = 12 (cid:2) ( ε + ε ) − (cid:0) ε ( η ) + 4 | ∆ | (cid:1) / (cid:3) , (106)where ε ( η ) = ε − ε . If both the unperturbed energies are degenerate at some critical value η c where ε ( η c ) = 0,we see that the two levels ε ± never cross each other unless the avoided crossing term ∆ vanishes. Let us consider thetime-dependent Schr¨odinger equation: ˆ H | ψ ( t ) (cid:105) = i ∂ | ψ ( t ) (cid:105) ∂t . (107)The wave function can be taken as a linear combination of the unperturbed states: | ψ ( t ) (cid:105) = C ( t ) e − i (cid:82) ε dt | m (cid:105) + C ( t ) e − i (cid:82) ε dt | m (cid:48) (cid:105) . (108)Using Eqn.(102)–(104), the time-dependent Schr¨odinger equation can be written as i ˙ C = ∆ e − i (cid:82) t ε ( t (cid:48) ) dt (cid:48) C , (109) i ˙ C = ∆ ∗ e i (cid:82) t ε ( t (cid:48) ) dt (cid:48) C . (110)These two differential equations must be solved with the boundary conditions: C ( −∞ ) = 0 , | C ( −∞ ) | = 1 . (111)Using the fact that ∆ is time-independent, differentiating Eqn.(109) and substituting Eqn.(110) into the resultingequation yields ¨ C − iε ( t ) ˙ C + | ∆ | C = 0 . (112)Writing ε ( t ) = αt , f = ∆ e iφ and C = ye i (cid:82) t ε ( t (cid:48) ) dt (cid:48) . (113)Eqn.(112) transforms into the form: ¨ y + (cid:18) f + i α α t (cid:19) y = 0 , (114)which transforms into the Weber equation:( Whittaker, 1902) by setting n = − if /α and z = √ αe iπ/ t : d ydz + (cid:18) n + 12 − z (cid:19) y. (115)The solutions of this differential equation are parabolic cylinder functions. The general solution of Eqn.(112) hasthe form(Zener, 1932) C ( t ) = (cid:20) aD − ν − ( − i √ αe iπ/ t ) + bD ν ( √ αe iπ/ t ) (cid:21) e iε ( t ) / , (116)where a and b are constants determined by the initial conditions. In the limit t → ∞ , the asymptotic form of theexcitation probability is found to be(Jan, et al. , 1981; Landau, 1932; Landau and Lifshitz, 1977; Zener, 1932) P = 1 − | C ( ∞ ) | = 1 − exp (cid:20) − π | ∆ | dεdt (cid:21) , (117)which is the famous Landau-Zener formula. The theoretical prediction of the oscillation of tunneling splitting ofthe model in Sec.(III.A.3) has been observed experimentally in Fe molecular cluster and Mn SMMs using this1
Tunn e l s p i tt i ng (cid:54) ( - K ) Magnetic tranverse field (T) 0 ° (cid:160) = 90 ° ° ° ° ° ° A Tunn e l s p li tt i ng (cid:54) ( - K ) Magnetic transverse field (T)n = 0 n = 1n = 2 B FIG. 8 Calculated tunneling splitting as a function of the applied field using Landau-Zener method for the Hamiltonianˆ H = − AS z + B ( S x − S y ) + C ( S + S − ) − gµ B hS x . For C = 0, it is related to that of Eqn.(86) by D = A + B and D = A − B .( A ) is the quantum transition between m = ±
10 for several values of the azimuth angles φ . ( B ) is the quantum transitionbetween m = −
10 and m = 10 − n at φ = 0, where n = 0 , , , · · · , m = − s, · · · , s , and s = 10 A = 0 . K , B = 0 . K and C = − . × − K for Fe molecular cluster. Adapted with permission from Wernsdorfer and Sessoli, 1999 Landau Zener technique (Wernsdorfer and Sessoli, 1999; Wernsdorfer et al. , 2000; Wernsdorfer, Chakov and Christou,2005). In Fig.(8) we have shown the experimental confirmation of this theoretical prediction. It explicitly shows theoscillations of the tunnel splittings as a function of the magnetic field applied along the hard anisotropy axis. Thisfield is responsible for the periodic change in the avoided level crossing ∆, which we found from the semiclassicalanalysis as a destructive or constructive quantum interference, with the period of oscillation given in Eqn.(96). Thetunneling probability from the Landau Zener formula is given by (Wernsdorfer and Sessoli, 1999) P = 1 − exp (cid:20) − π | ∆ | s (cid:126) gµ B dHdt (cid:21) , (118)where dHdt is the constant field sweeping rate and g ≈ molecular cluster in Fig.(8)with D = A + B and D = A − B is ∆ h = 0 . T . The value is very small compare to its experimental measuredvalue 0 . T . In order to fix this discrepancy an additional fourth order anisotropy of the form C ( S + S − ) is requiredin Eqn.(86) (Wernsdorfer and Sessoli, 1999; Wernsdorfer et al. , 2000). The inclusion of this term involves a tedioustheoretical analysis. There is no exact instanton solution but some approximate schemes have been developed totackle this problem (Chang and Garg, 2002; Foss and Friedman, 2009; Kim, 2002).
5. Antiferromagnetic exchange coupled dimer model
We have considered only the tunneling phenomenon of single molecule magnets (SMMs) . In many cases of physicalinterest, interactions between two large spins are taken into account. These interactions can be either ferromagnetic,which aligns the neighbouring spins or antiferromagnetic, which anti-aligns the neighbouring spins. One physicalsystem in which these interactions occur is the dimerized molecular magnet [Mn ] . It comprises two Mn SMMs ofequal spins s = s = 9 /
2, which are coupled antiferromagnetically. The phenomenon of quantum tunneling of spinsin this system has been be studied both numerically and experimentally (Hill, et al. , 2003; Tiron, et al. , 2003a). Forthis system, the simplest form of the Hamiltonian in the absence of an external magnetic field can be written asˆ H = − D ( ˆ S ,z + ˆ S ,z ) + J ˆ S · ˆ S , (119)where J > D (cid:29) J > S i,z , i = 1 , z easy-axis. In this model the exchange termacts as a field bias on its neighbour. We will report here on the analysis of this model by (Owerre and Paranjape,2013), however the nature of the ground states was first proposed by (Barbara and Chudnovsky, 1990) and theenergy splitting was obtained by (Kim, 2003) and the quantum operator perturbation theoretical analysis is given in2(Chudnovsky and Tejada, 2006; Chudnovsky et al , 2007). Park, et al. (2003) demonstrated using density-functionaltheory that this simple model can reproduce experimental results in [Mn ] dimer with D = 0 . K and J = 0 . K .It also plays a crucial role in quantum CNOT gates and SWAP gates for spin 1 / z -component of the spins ˆ S z = ˆ S ,z + ˆ S ,z is a conserved quantity. However, the individual z -componentspins ˆ S ,z , ˆ S ,z and the staggered configuration ˆ S ,z − ˆ S ,z are not conserved. The Hilbert space of this system isthe tensor product of the two spaces H = H ⊗ H with dim( H )= (2 s + 1) ⊗ (2 s + 1). The basis of S zj in thisproduct space is given by | s , σ (cid:105) ⊗ | s , σ (cid:105) ≡ | σ , σ (cid:105) . We immediately specialize to the case s = s = s . In theabsence of the exchange interaction, the ground state of the Hamiltonian is four-fold degenerate corresponding to thestates where the individual spins are in their highest weight or lowest weight states, |↑ , ↑(cid:105) , |↓ , ↓(cid:105) , |↑ , ↓(cid:105) , |↓ , ↑(cid:105) , where |↑ , ↓(cid:105) = |↑(cid:105) ⊗ |↓(cid:105) ≡ | s, − s (cid:105) etc, with the exchange interaction term J , the two ferromagnetic states |↑ , ↑(cid:105) and |↓ , ↓(cid:105) arestill degenerate, exact eigenstate of the Hamiltonian, but the antiferromagnetic states |↑ , ↓(cid:105) and |↓ , ↑(cid:105) are not. Thesetwo antiferromagnetic states link with each other at 2 s th order in degenerate perturbation theory in the exchangetransverse term, that is at order J s (Kim, 2003; Owerre and Paranjape, 2013). Thus, the exchange interaction playsthe same role as the splitting terms in the uniaxial and biaxial models considered previously. This is completelyunderstandable since tunneling requires a term that does not commute with the quantization axis. However, in thismodel we will see that both integer and half-odd integer spins can tunnel but their ground and first excited statesare different. Up to an additional constant, the classical energy corresponds to U = Js (sin θ sin θ cos( φ − φ ) + cos θ cos θ + 1) + Ds (sin θ + sin θ ) . (120)The minimum energy corresponds to φ − φ = π : θ = 0, θ = π , φ − φ = π : θ = π , θ = 0 and the maximumat φ − φ = π : θ = π/ θ = π/
2. There are four classical equations of motion but we already have the constraintthat the total z -component spins is conserved, that is cos θ + cos θ = 0 ⇒ θ = π − θ = π − θ . Introducing thevariables φ = φ − φ and Φ = φ + φ (which is cyclic), one finds that the two spin problem reduces to an effectivesingle spin problem which is described by the Lagrangian: L E = is ˙ φ (1 − cos θ ) + U ( θ, φ ) , (121)where the effective energy is U ( θ, φ ) = 2 Ds sin θ (cid:18) λ φ ) (cid:19) , (122)and λ = J/D (cid:28) θ (cid:54) = 0 as θ varies as the tunneling progresses, energy conservation requirescos φ = − (cid:18) λ + 1 (cid:19) . (123)Thus, | cos φ | > λ (cid:28)
1. Therefore there is no real solution for φ as expected. It was shown that the proper choiceof φ for antiferromagnetic coupling is φ = π + iφ I , where φ I is real (Owerre and Paranjape, 2013). Plugging this intoEqn.(123) we obtain φ I ≈ ln(4 /λ ).From the classical equation of motion Eqn.(61) one finds that the classical trajectory has the form θ ( τ ) = 2 arctan (cid:16) e ω ( τ − τ ) (cid:17) , (124)where ω = Js sinh φ I = 2 Ds √ κ , κ = J/D and at τ = τ we have θ ( τ ) = π/
2. Thus θ ( τ ) interpolates from 0 to π as τ = −∞ → ∞ for the instanton and from π to 0 for an anti-instanton. The action for this trajectory is found to be S = − i sπ + 2 sφ I . (125)The energy splitting between the ground and the first excited states is given by (Kim, 2003; Owerre and Paranjape,2013) ∆ = 2 D (cid:18) J D (cid:19) s cos(2 πs ) . (126) It is crucial to note that Kramers degeneracy only applies to a system with an odd total number of half-odd integer spin. < | g (cid:105) and the first excited | e (cid:105) states are | g (cid:105) = 1 √ |↓ , ↑(cid:105) − |↑ , ↓(cid:105) ); | e (cid:105) = 1 √ |↓ , ↑(cid:105) + |↑ , ↓(cid:105) ); (127)while for integer spins ∆ > | g (cid:105) = 1 √ |↓ , ↑(cid:105) + |↑ , ↓(cid:105) ); | e (cid:105) = 1 √ |↓ , ↑(cid:105) − |↑ , ↓(cid:105) ) . (128)In this case there is no suppression of tunneling even at zero field, the phase term that arises from the imaginaryterm in Eqn.(125) switches the ground state from odd to even for half-odd integer and integer spins respectively. Thisshows that for half-odd integer spins, the ground state is the state with s = 0. This result has been experimentallyshown that [Mn ] represents an unequivocal and unprecedented example of quantum tunneling in a monodisperseantiferromagnet with no uncompensated spin ( s = 0) in the ground state (Wernsdorfer, et al. , 2004). In the presenceof an external magnetic field applied along the easy axis, there are (2 s + 1) × (2 s + 1) = 100 ×
100 matrices whichare sparsely populated giving rise to an exact numerical diagonalization of 100 non-zero energy states as shown inFig.(9) , (Hu, Chen and Shen, 2003; Hill, et al. , 2003; Tiron, et al. , 2003a,b; Wernsdorfer, et al. , 2004). The values ofthe anisotropy parameters that were used to fit experimental data for this dimer are J = 0 . K , D = 0 . K (Hill, etal. , 2003; Tiron, et al. , 2003a). An analogous two spin problem is that of a biaxial antiferromagnetic particle of two -40-35-30-25-20-15-1.5 - 1 -0.5 0 0.5 1 1.5 E n e r g y ( K ) µ H z (T) ( 1 ) (-9/2,-9/2)(-9/2,9/2)(-9/2,7/2)(-7/2,9/2)(9/2,9/2) (9/2,7/2)(-9/2,5/2) (9/2,5/2) (2)(3) ( 5 )( 4 ) ( 6 ) FIG. 9 Color online: The so-called exact numerical diagonalization of the dimer model plotted as a function of appliedmagnetic field with the parameters D = 0 . K , J = 0 . K . Each state is labeled by | m , m (cid:105) . Dotted lines, labeled 1 to 5,indicate the strongest tunnel resonances: 1: (-9/2,9/2) to (-9/2 ,9/2); 2: (-9/2,9/2) to (-9/2 ,7/2), followed by relaxation to(-9/2,9/2); 3: (-9/2,9/2) to (9/2 ,9/2); 4: (-9/2,-9/2) to (-9/2 ,5/2), followed by re- laxation to (-9/2,9/2); 5: (-9/2,9/2) to(7/2 ,9/2), followed by relaxation to (9/2,9/2). In order to get most of these transitions theoretical one needs to add term like J ( S +1 S +2 + S − S − ) in Eqn.(119). Adapted with permission from Tiron, et al. , 2003a collinear ferromagnetic sublattices with a small non-compensation s = s − s (cid:54) = 0. The corresponding Hamiltonian(Chudnovsky , 1995; Garg and Duan, 1994; Liang, et al. , 2000) isˆ H = (cid:88) a =1 , ( k ˆ S z a + k ˆ S y a − h ˆ S za ) + J ˆ S · ˆ S , (129)where k (cid:29) k > x -axis and xy easy plane, and the magneticfield h is applied along the hard z -axis. The two spins are unequal unlike the dimer model considered above so oneis interested in the sublattice rotation of the N´eel vector (Barbara and Chudnovsky, 1990). The classical energy is ofthe form: U = Js s (sin θ sin θ cos( φ − φ ) + cos θ cos θ ) + (cid:88) a =1 , ( k s a cos θ a + k s a sin θ a sin φ a − hs a cos θ a ) (130)The full action contains two WZ terms thus, there are four equations of motion in general. There is no operator thatcommutes with this Hamiltonian therefore there is no constraint. In order to get an effective single spin problem,4several approximations have to be made. Firstly, we have to assume that the two spins s and s are almost antiparallel.Therefore, one can replace θ and φ by θ = π − θ − (cid:15) θ and φ = π + φ + (cid:15) φ where (cid:15) θ , (cid:15) φ (cid:28) θ and φ in the action and setting s = s = s except for the terms containing s − s = s , and integratingout the fluctuations (cid:15) θ , (cid:15) φ from the path integral one obtains an effective single spin model, which can then be solvedusing the procedures outline above. However, unlike the dimer model, one finds in this case that in the absence ofthe magnetic field, tunneling of hampered when s is half-odd integer (Chudnovsky , 1995) while in the presence ofthe magnetic field, tunneling splitting oscillates with the field only vanishes at a certain critical value (Liang, et al. ,2000) B. Coordinate independent formalism
1. Equation of motion and Wess-Zumino action
The coordinate dependent formalism we have just reviewed in the previous section is widely used in most condensedmatter literature, but not much seems to be written about the solutions of these models in a coordinate independentform. The solution of a physical problem should be independent of the coordinate system. Having solutions only ina coordinate dependent form leaves a slight but persistent, irritating doubt that somehow the results may have somecoordinate dependent artefacts, which of course should not be there. In section (II.A) we derived the classical actionfor the spin system without the use of coordinates. In this section we will show that one can solve the spin models wehave considered so far in totally coordinate independent way and also recover the quantum phase interference exactlyas before. First of all, we need to know the classical path that minimizes the coordinate independent action Eqn.(42): S E [ ˆn ] = isS W Z + (cid:90) dτ U ( ˆn ( τ )) , U ( ˆn ( τ )) = (cid:104) ˆn | ˆ H | ˆn (cid:105) . (131)The variation of coordinate independent WZ term, Eqn.(43) due to small variation of ˆn gives δS W Z = (cid:90) dτ (cid:90) dξ ∂ τ [ ˆn · ( δ ˆn × ∂ ξ ˆn )] + (cid:90) dτ (cid:90) dξ ∂ ξ [ ˆn · ( ∂ τ ˆn × δ ˆn )] . (132)To obtain this variation we must remember that 0 = δ ( ˆn · ˆn ) = 2 ˆn · δ ˆn , and 0 = ∂ τ,ξ ( ˆn · ˆn ) = 2 ˆn · ∂ τ,ξ ˆn , since ˆn isa unit vector. Consequently, the volume defined by the parallelepiped traced out by the three vectors, the variationand the two derivatives, must vanish, δ ˆn · ( ∂ τ ˆn × ∂ ξ ˆn ) = 0 since any three vectors orthogonal to a given vector ˆn ,lie in the same plane. The first term in Eqn.(132) vanishes by virtue of the boundary conditions Eqn.(44) and thesecond term yields δS W Z = − (cid:90) dτ δ ˆn ( τ ) · [ ˆn ( τ ) × ∂ τ ˆn ( τ )] . (133)As δ ˆn ( τ ) is still a constrained variation, necessarily orthogonal to ˆn , therefore δS W Z δ ˆn ( τ ) (cid:54) = [ ˆn ( τ ) × ∂ τ ˆn ( τ )] . (134)What we may conclude is that the part of [ ˆn ( τ ) × ∂ τ ˆn ( τ )] which is orthogonal to ˆn will contribute to the equationof motion. The way to implement this, is to take the vector product with ˆn , which implements the projection to theappropriate orthogonal directions. Then, using the fact that ˆn ( τ ) × [ ˆn ( τ ) × ∂ τ ˆn ( τ )] = − ∂ τ ˆn ( τ ), the variation of thetotal action gives the equation of motion: is∂ τ ˆn ( τ ) = − ˆn ( τ ) × ∂U ( ˆn ( τ )) ∂ ˆn ( τ ) . (135)This is the imaginary-time equivalent for the equation for Larmor precession in the effective magnetic field δU ( ˆn ( τ )) /δ ˆn ( τ ), often called the Landau-Lifshitz equation (Landau and Lifshitz, 1935, 1991). Taking the crossproduct of Eqn.(135) with ∂ τ ˆn ( τ ), and subsequently the dot product with ˆn ( τ ), one finds immediately the equationof energy conservation: U ( ˆn ( τ )) = const. (136)Having obtained the equation of motion as a function of the trajectory ˆn ( τ ), we wish need to write the WZ action,Eqn.(43) as a function of τ alone, as done in the coordinate dependent formulation as in Eqn.(48), in order to compute5the instanton action for the trajectory ˆn ( τ ). This can only be achieved if the integration over ξ can be done leavingus with the integration over τ in terms of the unit vector ˆn ( τ ). This integration can indeed be done. Let us expressthe unit vector ˆn ( τ, ξ ) as ˆn ( τ, ξ ) = f ( τ, ξ ) n z ( τ ) ˆz + g ( τ, ξ )[ n x ( τ ) ˆx + n y ( τ ) ˆy ] , (137)with the boundary conditions given in Eqn.(44). From Eqn.(137) and ˆn · ˆn = 1 one obtains immediately g = 1 − f n z − n z . (138)Owing to the boundary conditions in Eqn.(44), these functions must obey f ( τ, ξ = 0) = 1; f ( τ, ξ = 1) = 1 n z ( τ ) ; g ( τ, ξ = 0) = 1; g ( τ, ξ = 1) = 0 . (139)A long but straightforward calculation (Owerre and Paranjape, 2014a) shows that ˆn ( τ, ξ ) · ( ∂ τ ˆn ( τ, ξ ) × ∂ ξ ˆn ( τ, ξ )) = n z ∂ ξ f − n z ( n x ˙ n y − n y ˙ n x ) . (140)The WZ term becomes (Owerre and Paranjape, 2014a) S W Z = is (cid:90) dτ ( n x ˙ n y − n y ˙ n x )1 + n z . (141)This expression defines the WZ term in the coordinate independent form as a function of time alone. By sphericalparameterization one can easily recover the coordinate dependent form given by Eqn. (48). Further simplification ofEqn. (141) yields S W Z = is (cid:90) d ( n y /n x )1 + ( n y /n x ) (1 − n z ) = is (cid:90) d [arctan( n y /n x )](1 − n z ) . (142)
2. Coordinate independent uniaxial spin model in a magnetic field
Now let us consider the uniaxial model in section(III.A.1). The corresponding classical energy in coordinate inde-pendent form is U ( ˆn ) = − Ds ( ˆn · ˆz ) − H x s ˆn · ˆx . (143)From Eqn.(135) we obtain the equation of motion is∂ τ ˆn − Ds ( ˆn · ˆz )( ˆn × ˆz ) − H x s ( ˆn × ˆx ) = 0 . (144)Taking the cross product of this equation with ∂ τ ˆn and using the fact that ˆn ( τ ) · ∂ τ ˆn ( τ ) = 0 we obtain the conservationof energy Ds (( ˆn · ˆz ) −
1) + H x s ˆn · ˆx − H x / D = 0 , (145)where an additional constants have been added for convenience. Using this expression together with the constraint ˆn · ˆn = 1 we find the relations ˆn · ˆx = 12 h x (1 + h x − ( ˆn · ˆz ) ); ˆn · ˆy = ± i h x (1 − h x − ( ˆn · ˆz ) ) . (146)The ratio of these two expressions give ˆn · ˆyˆn · ˆx = ± i − h x − ( ˆn · ˆz ) h x − ( ˆn · ˆz ) = tan χ, (147) A similar expression is given in (Blasone and Jizba, 2012; Garg , et al. , 2003; Klauder, 1979; Stone, Park and Garg, 2000) ˆz and using Eqn.(146) we obtain is∂ τ ( ˆn · ˆz ) ± iDs (1 − h x − ( ˆn · ˆz ) ) = 0 . (148)The above equation integrates as ˆn · ˆz = ± (cid:112) − h x tanh( ω h τ ) , ω h = Ds (cid:112) − h x , (149)which is the same as Eqn.(66). To determine the action for this trajectory we use Eqn.(142), that is B = is (cid:90) d ( n y /n x )1 + ( n y /n x ) (1 − n z ) . (150)From Eqn.(147) we find: B = ± s (cid:90) ± √ − h x ∓ √ − h x n z dn z − n z (1 − n z ) = 2 s (cid:20)
12 ln (cid:32) (cid:112) − h x − (cid:112) − h x (cid:33) − (cid:112) − h x (cid:21) , (151)which is exactly the coordinate dependent result in Eqn.(67).
3. Coordinate independent biaxial model and suppression of tunneling
In section(III.A.2), we reviewed the suppression of tunneling for half-odd integer spin for a biaxial single moleculemagnet a particular choice of coordinate. In this section we will show that these results can be recovered in termsof the unit vector ˆn ( τ ). Thus, the suppression of tunneling for half-odd integer spin is independent of the choice ofcoordinate. In the coordinate independent form, the classical energy of the Hamiltonian, Eqn.(70) can be written as U = D s ( ˆn · ˆz ) + D s ( ˆn · ˆx ) , (152)The classical equation of motion, Eqn.(135) yields is∂ τ ˆn + 2 D s ( ˆn · ˆz )( ˆn × ˆz ) + 2 D s ( ˆn · ˆx )( ˆn × ˆx ) = 0 . (153)From the conservation of energy and the fact that ˆn · ˆn = 1, it follows that ˆn · ˆz = ± i (cid:114) D D ˆn · ˆx = ± i (cid:114) D D − D (1 − ( ˆn · ˆy ) ); ˆn · ˆx = ± (cid:114) D D − D (1 − ( ˆn · ˆy ) ) . (154)Then ˆn · ˆyˆn · ˆx = ± ˆn · ˆy (cid:113) D D − D (1 − ( ˆn · ˆy ) ) = tan χ. (155)Taking the scalar product of Eqn. (153) with ˆx and using Eqn.(154) yields is∂ τ ( ˆn · ˆy ) − i s (cid:112) D D (1 − ( ˆn · ˆy ) ) = 0 . (156)Upon integration we obtain the instanton: ˆn · ˆy = n y = tanh ( ω ( τ − τ )) , (157)where ω = 2 s √ D D . The instanton interpolates from n y = 1 to n y = − τ → ±∞ . Thus, arctan( ˆn · ˆy / ˆn · ˆy ) →± π/ τ → ±∞ . Since the energy remains constant along the instanton trajectory, the action is determined onlyfrom the WZ term: S c = is (cid:90) π − π d [arctan( n y /n x )](1 − n z ) . (158)7From Eqn.(154) and Eqn.(155) we find ˆn · ˆz = n z = ± i √ λ (cid:114) − λ + (cid:16) n y n x (cid:17) , λ = D /D . (159)Thus, we recover the action in Eqn.(75) S c = isπ + ln (cid:32) √ λ − √ λ (cid:33) s . (160)The calculation of the energy splitting follows directly from section(III.A.2). Thus, one recovers the spin-parity effectin a coordinate independent manner. This simply means that the spin-parity effect is independent of the choice ofcoordinate. IV. EFFECTIVE POTENTIAL (EP) METHOD
As we mentioned earlier, the spin coherent state path integral formalism is valid in the large s limit, in otherwords if one imposes the commutator relation [ φ, p ] = i (cid:126) , where p = s cos θ , then the spin commutator relation[ ˆ S i , ˆ S j ] = i(cid:15) ijk ˆ S k is only recovered in the large s limit . On the other hand, the effective potential method usesan exact mapping (Scharf, Wreszinski and Hemmen , 1987; Zaslavskii, 1990a; Zaslavskii and Ulyanov, 1992). Inthis method, one introduces the spin wave function using the ˆ S z eigenstates, and the resulting eigenvalue equationˆ H | ψ (cid:105) = E | ψ (cid:105) is then transformed to a differential equation, which is further reduced to a Schr¨odinger equation withan effective potential and a constant or coordinate dependent mass. The energy spectrum of the spin system nowcoincides with the 2 s + 1 energy levels for the particle moving in a potential field. The limitations of the method areas follow:1). In the effective potential method, the WZ term (Berry phase) does not appear in the corresponding particle action,the quantum phase interference effect seems to disappear, however, in some special cases with a magnetic fieldone can recover the quenching of tunneling at the critical field from the periodicity of the particle wave function.2). The effective potential method of higher order anisotropy spin models such as ˆ H = − D ˆ S z − B ˆ S z + C ( ˆ S + ˆ S − ) − H x ˆ S x and ˆ H = D ˆ S z + D ˆ S x + C ( ˆ S + ˆ S − ) are very cumbersome to map onto a particle problem. In fact there isno effective potential method for such systems. Therefore the effective potential method is only efficient for largespin systems that are quadratic in the spin operators. A. Effective method for a uniaxial spin model with a transverse magnetic field
In this section we will consider the effective potential method of the uniaxial model we studied in section(III.A.1).The Hamiltonian of this system is given by ˆ H = − D ˆ S z − H x ˆ S x . (161)Consider the the problem of finding the exact eigenstates of this Hamiltonian. The eigenvalue equation isˆ H | ψ (cid:105) = E | ψ (cid:105) , (162)where the spin wave function in the ˆ S z representation is given by (Scharf, Wreszinski and Hemmen , 1987) | ψ (cid:105) = s (cid:88) σ = − s (cid:18) ss + σ (cid:19) − / c σ | s, σ (cid:105) . (163) The proof of this is given in (M¨uller, et al. , 2000), Appendix A S x = (cid:16) ˆ S + + ˆ S − (cid:17) andˆ S ± | s, σ (cid:105) = (cid:112) ( s ∓ σ )( s ± σ + 1) | s, σ ± (cid:105) . (164)A straightforward calculation using Eqns.(161),(164), and (163) in Eqn.(162) gives: − Dσ c σ − H x [( s − σ + 1) c σ − + ( s + σ + 1) c σ +1 ] = E c σ , (165)where σ = − s, − s + 1 , · · · , s , and c σ = 0 for | σ | > s . Introducing a generating function of the form: G ( x ) = s (cid:88) σ = − s c σ e σx , (166)the eigenvalue equation, that is Eqn.(165) transforms to a second-order differential equation of the form: b d G d x + b d G dx − b G = EG , (167)where b = − D ; b = H x sinh x ; b = H x s cosh x. (168)The spin-particle correspondence follows from a special transformation of the form Ψ( x ) = e − y ( x ) G ( x ) , (169)where y ( x ) = ˜ sh x cosh( x ), h x = H x / D ˜ s <
1, and ˜ s = ( s + ) is a quantum renormalization. This transformationin Eqn.(169) is regarded as the coordinate or particle wave function since Ψ( x ) → x → ±∞ . Plugging thistransformation into Eqn.(167) removes the first derivative term yielding the Schr¨odinger equation(Scharf, Wreszinskiand Hemmen , 1987; Zaslavskii, 1990a; Zaslavskii and Ulyanov, 1992):ˆ H Ψ( x ) = E Ψ( x ); ˆ H = − m d dx + U ( x ) , (170)where U ( x ) = D ˜ s ( h x cosh x − ; m = 12 D . (171)As before we have added a constant to normalize the potential to zero at the minimum cosh x = 1 /h x . In Eqn.(169),the generating function contains a real exponential function. This choice is usually a matter of convenience. In mostcases it is convenient to use an imaginary exponential function to avoid some technical issues, as we will see in thenext section. The minimum of the potential is now at x mim = ± arccosh(1 /h x ) and the maximum is at x max = 0 withthe height of the barrier given by ∆ U = D ˜ s (1 − h x ) . (172)It is possible to analytically solve the Schr¨odinger equation and find the energy levels of the particle in the potentialEqn.(171), such solution has been reported (Razavy, 1980). This potential is of the form of a double well we saw inSec.(II.A) with ± a = ± arccosh(1 /h x ). The instanton solution of such a problem follows the same approach (Coleman,1985). The Euclidean Lagrangian corresponds to Eqn.(15) with the mass and the potential given by Eqn.(171). Thesolution of the Euclidean classical equation of motion, Eqn.(19) yields the instanton trajectory (Zaslavskii, 1990a;Zaslavskii and Ulyanov, 1992) x ( τ ) = ± (cid:20)(cid:114) − h x h x tanh( ωτ ) (cid:21) , (173) Substituting Eqn.(169) into Eqn.(167) gives b Ψ (cid:48)(cid:48) +(2 b y (cid:48) + b ) Ψ (cid:48) +[ b y (cid:48) + b + b (cid:0) y (cid:48)(cid:48) + y (cid:48) (cid:1) ]Ψ = E Ψ. The function y ( x ) is determinedby demanding the coefficient of Ψ (cid:48) vanishes. ω = D ˜ s (cid:112) − h x . This nontrivial solution corresponds to the motion of the spin particle at the top of the lefthill at τ → −∞ , x ( τ ) → − a and roll through the dashed line in Fig.(2) and emerges at the top of the right hill at τ → ∞ , x ( τ ) → a . The corresponding action for this trajectory is B = 2˜ s (cid:20)
12 ln (cid:32) (cid:112) − h x h x (cid:33) − (cid:112) − h x (cid:21) . (174)The computation of the ground state energy splitting yields ( Chudnovsky, et al. , 1998; Zaslavskii, 1990a)∆ = 8 D ˜ s / (1 − h x ) / π / (cid:32) e √ − h x (cid:112) − h x (cid:33) s h sx , (175)which recovers the factor h sx we saw previously in the spin coherent state path integral formalism. In the presenceof a longitudinal magnetic field i.e along z -axis, the two degenerate minima of the potential become biased, one withlower energy and the other with higher energy. The problem becomes that of a quantum decay of a metastable state(Zaslavskii, 1990b). B. Effective method for biaxial spin models
1. Biaxial ferromagnetic spin with hard axis magnetic field
The biaxial spin model also possesses a particle mapping via the EP method. Consider the biaxial system studiedin sec.(III.A.3) ˆ H = D ˆ S z + D ˆ S x − h z ˆ S z . (176)A convenient way to map this system to particle Hamiltonian is by introducing a non-normalized spin coherent state(Ersin and Garg, 2003; Garg , et al. , 2003; Perelomov, 1986; Radcliffe, 1971): | z (cid:105) = e zS − | s, s (cid:105) = s (cid:88) σ = − s (cid:18) ss + σ (cid:19) / z s − σ | s, σ (cid:105) = e isφ s (cid:88) σ = − s (cid:18) ss + σ (cid:19) / e − iσφ | s, σ (cid:105) . (177)The last equality sign follows by restricting the complex variable on a unit circle, i.e z = e iφ . Acting from the left by e − isφ (cid:104) ψ | and subsequently taking the complex conjugate we obtain (cid:104) z | ψ (cid:105) = e isφ s (cid:88) σ = − s (cid:18) ss + σ (cid:19) / c σ e iσφ ≡ e isφ Φ( φ ) , (178)where c σ = (cid:104) s, σ | ψ (cid:105) and Φ( φ ) is the generating function , with periodic boundary condition Φ( φ + 2 π )= e iπs Φ( φ ).From Eqn.(177) we have (cid:104) z | ˆ S z | ψ (cid:105) = e isφ s (cid:88) σ = − s (cid:18) ss + σ (cid:19) / σc σ e iσφ = − ie isφ d Φ( φ ) dφ . (179)Similar expressions can be derived for (cid:104) z | ˆ S x | ψ (cid:105) and (cid:104) z | ˆ S y | ψ (cid:105) . Thus, the action of the spin operators on this functionyields the following expressions (Zaslavskii, 1990a; Zaslavskii and Ulyanov, 1992):ˆ S z = − i ddφ ; ˆ S x = s cos φ − sin φ ddφ ; ˆ S y = s sin φ + cos φ ddφ . (180) It is convenient to use the generating function for x or y easy axis models while Eqn.(166) is convenient for z easy axis model. In thatway one avoids the problem of a negative mass particle. H Φ( φ ) = E Φ( φ ) . (181)From Eqn.(176) and Eqn.(180) one obtains the differential (M¨uller, et al. , 2000; Zaslavskii, 1990a): − D (1 − λ sin φ ) d Φ dφ − D ( s −
12 ) sin 2 φ d Φ dφ + ih z d Φ dφ + ( D s cos φ + D s sin φ )Φ = E Φ . (182)A convenient way to obtain a Schr¨odinger equation with a constant is by introducing an incomplete elliptic integralof first kind (Abramowitz and Stegun, 1972; Byrd and Friedman, 1979) and the particle wave function: x = F ( φ, κ ) = (cid:90) φ dϕ (cid:112) − κ sin ϕ ; Ψ( x ) = e − iu ( x ) [dn( x )] − s Φ( φ ( x )) , (183)with amplitude φ and modulus κ = λ . The trigonometric functions are related to the Jacobi elliptic functions bysn( x ) = sin φ , cn( x ) = cos φ and dn( x ) = (cid:112) − κ sn ( x ). The function u ( x ) is defined by dudx = αs dn( x ) , α = h z / D s. (184)The imaginary phase is a topological shift in the wave function which is related to Aharonov Bohm effect (Aharonovand Bohm, 1959). In this new variable, Eqn.(182) transforms into a Schr¨odinger equation with H = 12 m (cid:20) − i ddx + A ( x ) (cid:21) + U ( x ); m = 12 D . (185)The effective potential and the gauge field are given by U ( x ) = η cd( x ) ; cd( x ) = cn( x )dn( x ) ; (186) A ( x ) = − (2 s + 1) α dn( x ) ; (187)where η = D s ( s + 1) + λα − λ ) . The potential has a period of 2 K ( κ ), where K ( κ ) is the complete elliptic function offirst kind that is φ = π/ et al. , 1999; Sahng, et al. , 2000)Ψ( x + 4 K ( κ )) = e i πs (1 − α/ √ − λ ) Ψ( x ) . (188)The corresponding Euclidean Lagrangian of this particle Hamiltonian is L E = 12 m ˙ x + iA ( x ) ˙ x + U ( x ) . (189)The second term of this equation drops out from the classical equation of motion, however, it is responsible forthe suppression of tunneling splitting just like the WZ term (Berry phase) in the spin coherent state path integralformalism. Thus one finds that the exact instanton solution issn[ x ( τ )] = tanh( ωτ ) , ω = 4 s ( s + 1) D D , (190)which interpolates from x i = −K ( κ ) ( φ = − π/
2) at τ = −∞ to x f = K ( κ ) ( φ = π/
2) at τ = ∞ . The action for thistrajectory is found to be S c = − i (2 s + 1) b + B, (191)where b = πα/ √ − λ and B is given by B = (cid:114) ηD ln (cid:32) √ λ − √ λ (cid:33) . (192)1By summing over instantons and anti-instantons configurations, it was shown that the energy splitting is givenby(M¨uller, et al. , 2000) ∆ = ∆ | cos( sπ + b ) | , ∆ = 4 D e − B . (193)Thus one recovers the suppression of tunneling as before.As an alternative approach of recovering the quenching of tunneling splitting, consider the transition from x = 0to x = 2 K ( κ ) and x = 0 to x = − K ( κ ). The former is counterclockwise transition while the latter is clockwisetransition, thus the total transition amplitude vanishes: A (2 K ( κ ) , t ; 0 ,
0) + A ( − K ( κ ) , t ; 0 ,
0) = 0 , (194)where A represent the Feynman propagator given in Eqn.(8). In terms of the wave function the propagator can bewritten as(Coleman, 1977; Callan and Coleman, 1977; Coleman, 1985; Feynman and Hibbs, 1965) A ( x f , t ; x i ,
0) = (cid:88) l Ψ l ( x f )Ψ ∗ l ( x i ) e − i E l t . (195)Then from Eqs.(194) and (195) one obtains the relationΨ l (2 K ( κ )) = − Ψ l ( − K ( κ )) , (196)which yields from Eqn.(188) e i πs (1 − α/ √ − λ ) = − , (197)for any quantum number l . From this equation one obtains the condition for suppression of tunneling (Sahng, et al. ,2000) α = √ − λ ( s − n − / /s, (198)just as Eqn.(97).
2. Biaxial ferromagnetic spin with medium axis magnetic field
Suppose we apply a magnetic field in the medium x -axis corresponding to the Hamiltonian:ˆ H = D ˆ S z + D ˆ S x − H x ˆ S x . (199)As we pointed out in sec.(III.A.3), the quenching of tunneling at the critical field is only seen with biaxial spin modelswith magnetic field along the hard-axis, thus this model does not possess such effect. At zero magnetic field, thereare two classical degenerate ground states corresponding to the minima of the energy located at ± ˆy , these groundstates remain degenerate for h x (cid:54) = 0 in the easy XY plane. The particle Hamiltonian is H = − m d dx + U ( x ) , m = 12 D , (200)with the effective potential and the wave function given by (Owerre and Paranjape, 2014b) U ( x ) = D ˜ s [cn( x ) − α x ] dn ( x ) ; Ψ( x ) = Φ( φ ( x ))[dn( x )] s exp (cid:20) − arccot (cid:32)(cid:115) λ (1 − λ ) cn( x ) (cid:33) (cid:21) , (201)where ˜ s = ( s + ) and α x = H x / D ˜ s . In order to arrive at this potential we have used the approximation s ( s +1) ∼ ˜ s and shifted the minimum energy to zero by adding a constant of the form D ˜ s α x . The potential, Eqn.(201) hasminima at x = 4 n K ( κ ) ± cn − ( α x ) and maxima at x sb = ± n K ( κ ) for small barrier and at x lb = ± n + 1) K ( κ )for large barrier. The heights of the potential for small and large barriers are given by (M¨uller, et al. , 1998; Owerreand Paranjape, 2014b) ∆ U sb = D ˜ s (1 − α x ) ; ∆ U lb = D ˜ s (1 + α x ) , (202)2 x U ( x ) / D ˜ s FIG. 10 Color online: The plot of the effective potential in Eqn.(201) for α x = 0 . κ = 0 . The classical trajectory yields sn[ x ( τ )] = ± (cid:113) − α x α x tanh( ωτ )[1 + − α x α x tanh ( ωτ )] , (203)and the corresponding action is (Owerre and Paranjape, 2014b; Zaslavskii, 1990a) B = ˜ s (cid:20) ln (cid:32) (cid:112) λ (1 − α x )1 − (cid:112) λ (1 − α x ) (cid:33) ± α x (cid:114) λ − λ arctan (cid:32) (cid:112) (1 − λ )(1 − α x ) α x (cid:33) (cid:21) , (204)where the upper and lower signs are for tunneling in large and small barriers respectively. The tunneling splittingcan be found in the usual way by summing over instanton and anti-instanton configurations. During our discussionof phase transition in the next section, we will return to this concept of large and small barriers in detail. In thissection we have specifically chosen biaxial spin models that possess an exact instanton solution. The transformationsin Eqns.(180)—(178) are derived by restricting the analysis on a unit circle parameterize by the angle φ . In thesetwo models, the variable φ and then x correspond exactly to the azimuthal angle φ in the spin coherent state pathintegral. In other representations of a biaxial spin system, this is not true and the EP method gives a very complicatedeffective potential, one can neither find the exact instanton solution nor the suppression of tunneling. However, withoutcomputing the explicit instanton trajectory, the action at the bottom of the potential well can be found in some casesby another elegant approach as we will see in the next section. V. QUANTUM-CLASSICAL PHASE TRANSITIONS OF THE ESCAPE RATE IN LARGE SPIN SYSTEMSA. Methods for studying quantum-classical phase transitions of the escape rate
In the preceding sections, we have reviewed quantum tunneling in spin systems which is dominated by instantontrajectory at zero temperature. As we mentioned in Section(I), transitions at finite temperature can be eitherfirst or second-order. In this section we will now discuss the phase transition of the escape rate from thermal toquantum regime at nonzero temperature. The escape rate of a particle through a potential barrier in the semiclassicalapproximation is obtained by taking the Boltzmann average over tunneling probabilities (Affleck, 1981; Chudnovsky, etal. , 1998): Γ = (cid:90) U max U min d E P ( E ) e − β ( E− U min ) , (205)where β − = T is the temperature of the system, which is much less than the height of the potential barrier. Thisdefines the temperature assisted tunneling rate, and P ( E ) is an imaginary time transition amplitude from excited3states at an energy E . The integration limits U max and U min are the top and bottom of the potential energy respectively.The transition amplitude is defined as P ( E ) = A e − S ( E ) , (206)where A is a prefactor independent of E . The Euclidean action is of the form: S ( E ) = 2 (cid:90) x ( E ) x ( E ) dx (cid:112) m ( x )( U ( x ) − E ) , (207)where x , ( E ) are the roots of the integrand in Eqn.(207), which are the classical turning points ( U ( x , ) = E ) ofa particle with energy −E in the inverted potential − U ( x ) as depicted in Fig.(1). The mass m ( x ) is coordinatedependent in general. The factor of 2 in Eqn.(207) corresponds to the back and forth oscillatory motion of theparticle in the inverted potential (see Fig.1). In other words, the particle crosses the barrier twice.
1. Phase transition with thermon action
The escape rate can as well be written as Γ = A (cid:90) U max U min d E e −S p , (208)where S p = S ( E ) + β ( E − U min ) , (209)is the thermon action (Chudnovsky, 1992). In the method of steepest decent (for small temperatures T < (cid:126) ω , ω isthe frequency at the bottom of the potential), one can introduce fluctuations around the classical path that minimizesthis thermon action, i.e d S p d E = 0. The escape rate, Eqn.(208) in this method is thus written as (Chudnovsky andGaranin, 1997) Γ ∼ e −S min ( E ) , (210)and S min ( E ) is the minimum of the thermon action in Eqn.(209) with respect to energy.In many cases of physical interest, when the energy is in the range U min < E < U max , the Euclidean action S ( E )can be computed exactly or numerically in the whole range of energy for any given potential in terms of completeelliptic integrals and hence the thermon action S p . This corresponds to the action of the periodic instanton(Liang, etal. , 2000) or thermon. At the bottom of the potential E = U min , the minimum thermon action becomes the vacuuminstanton action, that is S min ( U min ) = S ( U min ) . (211)Thus, the vacuum instanton action of the previous sections becomes B = S ( U min ) /
2, since it corresponds to half of theperiod of oscillation. Eqn.(210) becomes the transition amplitude formula for a pure quantum tunneling. However,at the top of the barrier E = U max , the Euclidean action vanishes, S ( U max ) = 0, the minimum thermon action(thermodynamic action) becomes S min ( U max ) = S = β ∆ U . (212)This corresponds to the action of a constant trajectory x ( τ ) = x s at the bottom of the inverted potential (Chudnovsky,1992). The escape rate Eqn.(210) becomes the Boltzmann formula for a pure thermal activation. As we showed insection(I), the crossover temperature from thermal to quantum regimes (“first-order phase transition”) occurs whenthe escape rate Eqn.(210) with S min ( U min ) is equal to that with S min ( U max ), which yields Eqn.(1). At this temperaturethe thermon action S p sharply intersects with the thermodynamic action S leading to a discontinuity in the first-derivative of the action S p at β (1)0 . For second-order phase transition the thermon action S p smoothly joins thethermodynamic action S at β = β (2)0 .4
2. Phase transition with thermon period of oscillation
The dominant term in Eqn.(208) comes from the minimum of the thermon action Eqn.(209), which is given by β ( E ) = − dS ( E ) d E = (cid:90) x ( E ) x ( E ) dx (cid:115) m ( x ) U ( x ) − E ≡ τ ( E ) . (213)This is the period of oscillation of a particle with energy −E in the inverted potential − U ( x ). At the bottom of thepotential E = U min , the period β ( E ) = ∞ i.e T = 0 which corresponds to the vacuum instanton of section(IV), whileat the top of the barrier E = U max , β ( E ) → β (2)0 = 2 π/ω b (Affleck, 1981). The first and second-order transitions canbe studied from the behaviour of β ( E ) as a function of E .1). If β ( E ) has a minimum at some point E < U max , β min = β ( E ) and then rises again, i.e non-monotonic, thenfirst-order phase transition occurs (Chudnovsky, 1992). At a certain energy within the range U min < E < E ,the thermon action sharply intersects with the thermodynamic action, yielding the actual crossover temperature β (1)0 = β ( E ).2). A monotonic decrease of β ( E ) with increasing E from the bottom to the top of the barrier indicates the presenceof second-order phase transition( Chudnovsky, et al. , 1998; Chudnovsky and Garanin, 1997; Chudnovsky, 1992).In this case the thermon action S p smoothly intersects with the thermodynamic action S , yielding the crossovertemperature β (2)0 (Chudnovsky, 1992; Gorokhov and Blatter, 1997), which is exactly Eqn.(2).
3. Phase transition with free energy
The semiclassical escape rate Eqn.(210) can be written in a slightly different form:Γ ∼ e − βF min , (214)where F min = β − S min ( E ) is the minimum of the effective free energy F = β − S p = E + β − S ( E ) − U min , (215)with respect to E . The crossover from thermal to quantum regimes (first-order phase transition) occurs when twominima in the F vs. E curve have the same free energy. All the interesting physics of phase transition in spin systemscan also be captured when the energy is very close (but not equal) to the top of the potential barrier, E → U max . Inthis case the free energy can then be used to characterize first- and second-order phase transitions in analogy withLandau’s theory of phase transition if one knows the exact expression of the action S ( E ) for any given mass andpotential. In most models with a magnetic field the action S ( E ) cannot be obtained exactly, one has to study the freeenergy numerically.
4. Phase transition with criterion formula
An alternative method for determining the phase transition of the escape rate, as well as the phase boundary wasconsidered by M¨uller, Park and Rana (1999). They studied the Euclidean action near the top of the potential barrier,which had been considered earlier by Gorokhov and Blatter (1997). For the general case of a particle that possessesa coordinate dependent mass, they found that near the top of the potential barrier the expression that depends onthe potential, which determines the type of phase transition is given by (M¨uller, Park and Rana, 1999) C = (cid:20) U (cid:48)(cid:48)(cid:48) ( x s ) (cid:16) g + g (cid:17) + 18 U (cid:48)(cid:48)(cid:48)(cid:48) ( x s ) + ω m (cid:48) ( x s ) g + ω m (cid:48) ( x s ) (cid:16) g + g (cid:17) + 14 ω m (cid:48)(cid:48) ( x s ) (cid:21) ω = ω b , (216)where5 g = − ω m (cid:48) ( x s ) + U (cid:48)(cid:48)(cid:48) ( x s )4 U (cid:48)(cid:48) ( x s ) , (217) g = − m (cid:48) ( x s ) ω + U (cid:48)(cid:48)(cid:48) ( x s )4 [4 m ( x s ) ω + U (cid:48)(cid:48) ( x s )] , (218) ω b = − U (cid:48)(cid:48) ( x s ) m ( x s ) ; m (cid:48) ≡ dm ( x ) dx , etc. (219)The coordinate x s represents the position of the sphaleron at the bottom of the inverted potential as shown inFig.(1). The criterion for first-order phase transition requires C <
0, while C > C = 0. The criterionformula in Eq.(216) is quite general. It can be simplified in two special cases:1). If the mass of the particle is a constant and the potential energy is an even function, Eq.(216) reduces to C = 18 U (cid:48)(cid:48)(cid:48)(cid:48) ( x s ) . (220)Thus, expanding the potential around x s , the coefficient of the fourth-order term quickly determines the first- andthe second-order phase transitions, as well as the phase boundary (Chudnovsky and Garanin, 1997).2). If mass is still a constant but the potential is an odd function, Eq.(216) reduces to C = −
524 [ U (cid:48)(cid:48)(cid:48) ( x s )] U (cid:48)(cid:48) ( x s ) + 18 U (cid:48)(cid:48)(cid:48)(cid:48) ( x s ) . (221) B. Phase transition in uniaxial spin model in a magnetic field
1. Spin model Hamiltonian
We have written down all the necessary formulae for studying the phase transition of the escape rate for a uniaxialspin model in an applied field. An extensive analysis of this model can be found in ( Chudnovsky, et al. , 1998;Chudnovsky and Tejada, 1998). In this section we will briefly review the theoretical analysis and recent experimentaldevelopment. For this system we saw that the spin Hamiltonian is given byˆ H = − D ˆ S z − H x ˆ S x . (222)As we mentioned before this system is a good approximation for Mn Ac, with a ground state of s = 10 and 21 energylevels. Transition between these states can occur either by quantum tunneling (QT) or thermally assisted tunneling(TAT) as depicted in Fig.(16).
2. Particle Hamiltonian
As we explicitly showed in Sec.(IV.A), the spin Hamiltonian in Eqn.(222) corresponds to the particle potential andthe mass: U ( x ) = D ˜ s ( h x cosh x − , m = 12 D . (223)Since the potential is an even function and the mass is constant, the quickest way to determine the regime where thefirst-order transition sets in, is by considering where the coefficient of the fourth order term changes sign near x s = 0.Expanding the potential around x s we have U ( x ) ≈ U (0) + D ˜ s [ − h x (1 − h x ) x + h x (cid:18) h x − (cid:19) x + O ( x )] . (224) Sphalerons are static, unstable, finite-energy solutions of the classical equations of motion. x in Eqn.(224) is negative for h x <
1, which corresponds to nonvanishing of the potential barrier,Eqn.(172). The coefficient of x is similar to C in Eq.(220), it is given by(M¨uller, Park and Rana, 1999) C = D ˜ s h x (cid:18) h x − (cid:19) . (225)Clearly, C changes sign for h x < , which corresponds to the regime of the first-order transition from thermal x U ( x ) / D ˜ s h x =0 . h x =0 . h x =0 . FIG. 11 Color online: The plot of the potential in Eq.(223) for several values of h x . activation to quantum tunneling. It is positive for h x > , which is the regime of second-order phase transition, andof course vanishes at the phase boundary h cx = .
3. Thermon or periodic instanton action
An alternative approach to investigate quantum-classical phase transitions of the escape rate is by computing thethermon action: S p = 2 √ m (cid:90) x − x dx (cid:112) U ( x ) − E + β ( E − U min ) , (226)where ± x are the roots of the integrand which are the classical turning points. This action corresponds to the actionof the periodic instanton trajectory of Eq.(223). That is the solution of the classical equation of motion:12 m ˙ x − U ( x ) = −E . (227)Integrating this equation using Eq.(223) one finds that the periodic instanton trajectory is given by (Zhang, et al. ,1999) x p = ± − (cid:2) ξ p sn ( ω p τ, k ) (cid:3) , (228)where k = 1 − (cid:16)(cid:112) ˜ E + h x (cid:17) − (cid:16)(cid:112) ˜ E − h x (cid:17) , ξ p = 1 − h x ± (cid:112) ˜ E h x ± (cid:112) ˜ E , (229) ω p = ( D ˜ s ) (cid:2) − ( (cid:112) ˜ E − h x ) (cid:3) , ˜ E = E /D ˜ s . (230) The thermon action in Eq.(226) only differs from the periodic instanton action by a factor of 2 in Eq.(226). So we will use the twonames interchangeably. τ → ± β , this trajectory must tend to the classical turning points x p → ± x as depicted inFig(12). Let us define dimensionless energy quantity( Chudnovsky, et al. , 1998; Chudnovsky and Garanin, 1997): FIG. 12 The plot of the periodic trajectory in Eq.(228). P = U max − E U max − U min . (231)Clearly P → E → U max , and P → E → U min . By making achange of variable y = cosh x , Eq.(226) can be reduced to complete elliptic integrals (Abramowitz and Stegun, 1972;Byrd and Friedman, 1979) in the whole range of energy. It is found to be of the form ( Chudnovsky, et al. , 1998): S T T (1)0 T (2)0 S p → B S p S T S T (2)0 S p S FIG. 13 Color online: The plot of the thermon action Eqn.(209) and thermodynamic action Eqn.(212) against temperature.Left: s = 10, D = 1 h x = 0 . B is the vacuum instanton action. Right: s = 10, D = 1, h x = 0 . S p = 4˜ s (cid:112) (1 − h x ) g + I ( α , k ) + β ∆ U (1 − P ) , (232)where g ± = P + h x (cid:16) ± √ − P (cid:17) , (233) I ( α , k ) = (1 + α ) K ( k ) − E ( k ) + ( α − k /α )(Π( α , k ) − K ( k )); α = (1 − h x ) P/g + ; k = g − /g + ; (234)8where K ( k ), E ( k ) and Π( α , k ) are the complete elliptic integral of the first, the second and the third kinds respectively.In Fig.(13) we have shown the plot of the actions Eqn.(232) and Eqn.(212) as a function of temperature. Indeedone observes the sharp and smooth intersections corresponding to the first- and the second-order phase transitionstemperatures respectively.
4. Free energy function
The free energy can also be used to study the quantum-classical phase transitions in this systems. It can be writtendown exactly from Eqn.(232). It is given by F ∆ U = 1 − P + 4 θ (cid:112) h x g + π (1 − h x ) / I ( α , k ) , (235)where θ = T /T (2)0 , and T (2)0 is given by Eq.(238). Near the top of the barrier P (cid:28)
1, the free energy of this spinmodel yields ( Chudnovsky, et al. , 1998; Chudnovsky and Garanin, 1997) F ( P ) / ∆ U = 1 + ( θ − P + θ (cid:18) − h x (cid:19) P + 3 θ (cid:18) − h x + 164 h x (cid:19) P + O ( P ) . (236)This free energy should be compared with the Landau’s free energy function: F = F + aψ + bψ + cψ . (237)The analogy between these two free energies comes from identifying the coefficient of P as the Landau coefficient b , h x π T / s D First-orderSecond-order
FIG. 14 Color online: The plot of the of the first-and second-order cross over temperatures against h x . Reproduced withpermission from Chudnovsky, et al. , 1998. which determines the regime of first-order phase transition b <
0, and that of second-order phase transition b >
0. Theboundary between the first- and the second-order phase transition corresponds b = 0. We see that these conditionsrecover the results in Sec.(V.B.2). The plot of the free energy in the whole range of energy is shown in Fig.(15). Theactual crossover temperature from thermal to quantum regimes is determined when two minima of a curve have thefree energy. For h x = 0 .
1, it is found to be at T (1)0 = 1 . T (2)0 . This crossover temperature is approximately given by T (1)0 = ∆ U/S ( E → x s = x max = 0 T (2)0 = 1 β (2)0 = D ˜ sπ (cid:112) h x (1 − h x ) . (238)In the limit h x → T (1)0 /T (2)0 ≈ . P F / ∆ U h x =0 . Bottom barrierTop barrier θ =1 . θ =1 θ =0 . θ =0 . θ =0 . P F / ∆ U h x =0 . Bottom barrierTop barrier θ =1 . θ =1 . θ =1 . θ =1 . θ =1 FIG. 15 Color online: The plot of the of the free energy as a function of P for h x = 0 . h x = 0 . Ac molecular magnet. to note that the magnetic field plays a decisive role on the crossover temperatures for the first-and the second-orderphase transitions. It is the main parameter that drives tunneling in this system. Physically, the sharp first-orderphase transition in this model occurs due to the flatness or wideness of the barrier top in the small magnetic field limitas shown in Fig.(11). In the strong magnetic field limit, the top of the barrier is of the parabolic form (see Fig.(11))which leads to suppression of first-order phase transition. In the limit of zero magnetic field the Hamiltonian Eq.(222)commutes with the z -component of the spin, thus ˆ S z is a constant of motion and the potential in Eq.(223) becomesa constant. Hence, there is no dynamics (tunneling) and no quantum-classical phase transitions .
5. Experimental results
Recently, experiments have been conducted to measure these crossover temperatures. Experimental result forMn Ac molecular magnet with the model in Eqn.(239) has confirmed the existence of an abrupt and gradual crossovertemperature ( T ∼ . K ) between thermally assisted and pure quantum tunneling ( Bokacheva, Kent and Marc, 2000;Garanin and Chudnovsky, 2000; Kent, et al. , 2000; Leuenberger and Loss, 2000) as shown in Fig.(17). Below thecrossover temperature the magnetization relaxation becomes temperature independent, which indicates that transitionoccur by QT between the m esc = ± s states. Above the crossover temperature transition favours the excited stateswith m esc < s (TAT). Quite recently, a similar result was observed in Mn -tBuAc molecular nanomagnet with a spinground state of s = 10. This molecular nanomagnet has the same magnetic anisotropy as Mn Ac but the moleculesare very isolated and the crystals have less disorder and a higher symmetry(Wernsdorfer, Murugesu, and Christou, The basic understanding is that in the zero magnetic field limit, the barrier becomes infinitely thick and tunneling cannot occur. B z / B T (K) n=5n=4n=3n=6n=710988910891010981098m esc = QT TATCrossover
FIG. 17 Peak positions of the first derivative of the magnetization plotted against the temperature in Mn Ac molecularmagnet. At the ground state M = s = 10, the peak is independent of temperature (QT) while for excited states M < s transition occurs by TAT. Adapted with permission from Bokacheva, Kent and Marc, 2000FIG. 18 Color online: Temperature dependence of hysteresis loops of (b) Mn -tBuAc and (c) Mn Ac SMMs at differenttemperatures and a constant field sweep rate as indicated in the figure. With decreasing temperature , the hysteresis increasesdue to a decrease in the transition rate of thermal assisted tunneling. Adapted with permission from Wernsdorfer, Murugesu,and Christou, 2006 H = − D ˆ S z − B ˆ S z − h z ˆ S z + ˆ H ⊥ , (239)where h z = g z µ µ B H z and ˆ H ⊥ is the splitting term which is comprised of ˆ S x and ˆ S y . In the absence of ˆ H ⊥ , the21 energy levels of Eqn.(239) can be found by the so-called exact numerical diagonalization in the ˆ S z representation.The inclusion of a small perturbation ˆ H ⊥ leads to an avoided level crossings in the degenerate energy subspace. Thecrossover temperature for the compound occurs at T ∼ . K . The hysteresis loops in Fig.(18) show a temperatureindependent quantum tunneling at the lowest energy levels below 0 . K , while the temperature dependent thermalassisted tunneling at the excited states occurs above 0 . K . C. Phase transition in biaxial spin model
1. Model Hamiltonian and spin coherent state path integral
The phase transition in biaxial spin systems follow a similar trend to that of uniaxial spin model in a magneticfield. The first work on this system was begun by Liang, et al. (1998). They studied the model:ˆ H = K ˆ S z + K ˆ S y , K /K = λ < , (240)by spin coherent state path integral and periodic instanton method. This Hamiltonian is related to that of Eqn.(70)and Eqn.(72) by K = D , K = D and φ → π/ − φ . It can also be related to any biaxial spin model in the absenceof a magnetic field. The effective Lagrangian of this system can be obtained by integrating out cos θ from the spincoherent state path integral, Eqn.(64), one finds that the effective classical Euclidean Lagrangian is L E = is ˙ φ + 12 m ( φ ) ˙ φ + U ( φ ) , (241)where m ( φ ) = 12 K (1 − λ sin φ ) , U ( φ ) = K s sin φ. (242)The potential barrier height is located at φ s = π , and the minimum energy is located at φ = 0.
2. Periodic instanton
The periodic instanton trajectory of this model can be computed from the classical equation of motion: m ( φ p ) ¨ φ p + 12 dm ( φ p ) dφ p ˙ φ p − dUdφ p = 0 . (243)Integrating once we obtain: 12 m ( φ p ) ˙ φ p − U ( φ p ) = −E , (244)where E is the integration constant. The corresponding periodic instanton solution of this equation yields φ p = arcsin (cid:20) − k sn ( ω p τ, k )1 − λ k sn ( ω p τ, k ) (cid:21) , (245)where k = n − n − λ , n = K s E ; ω p = ω (1 − λn ); ω = 4 K K s . (246)The classical action for this trajectory is found to be S p = 4 ωλK I ( k λ, k ) + β ( E − U min ) , (247) I ( k λ, k ) = [ K ( k ) − (1 − k λ )Π( k λ, k )] , (248)2 T S T (1)0 T (2)0 S p → B S p S T S T (1)0 T (2)0 S p S FIG. 19 Color online: The plot of the thermon action Eqn.(247) and thermodynamic action Eqn.(212) against temperaturefor the biaxial model. Left: s = 10, K = 1, λ = 0 . B is the vacuum instanton action. Right: s = 10, K = 1, λ = 0 . et al. , 1998). where K and Π are the complete elliptic integrals of the first and the third kinds respectively, and the period ofoscillation is given by β ( E ) = 2 √ K √ K s − E λ K ( k ) . (249)Near the top of the barrier E → U max = K s and β ( E ) → β (2)0 = 2 π/ω b , with ω b defined by Eqn.(219) and near thebottom of the barrier E → U min = 0 and β ( E ) → ∞ , then S p reduces to the usual vacuum ( T = 0) instanton solution,Eqn.(76). Fig.(19) shows the thermon and the thermodynamic actions with the crossover temperatures indicated,which is of similar trend to that of Fig.(13) .
3. Free energy and crossover temperatures
The ground state crossover temperature is determined from T (1)0 = ∆ U/ S p ( E → U min ) = K s / S p ( E → U min ). Forthe second order phase transition we have T (2)0 = ω b / π . In the limit λ →
0, one finds that T (1)0 /T (2)0 = π/ ≈ . F ∆ U = 1 − P + 4 θ (cid:112) (1 − λ )[1 − λ (1 − P )] πλ I ( k λ, k ) , (250)where θ = T /T (2)0 , and T (2)0 = sK (cid:112) λ (1 − λ ) /π . Near the top of the barrier P (cid:28)
1, the free energy reduces to(Zhang, et al. , 1999) F/ ∆ U ∼ = 1 + ( θ − P + θ − λ ) (cid:18) − λ (cid:19) P + θ − λ ) (cid:18) λ − λ + 38 (cid:19) P + O ( P ) . (251)The coefficient of the P changes sign when λ > , which corresponds to the regime of first-order phase transi-tion. In this analysis, the mass is coordinate dependent, therefore the coefficient of φ in the series expansion near φ s = π cannot determine the condition for any type of quantum-classical phase transitions, thus Eqn.(216) becomesindispensable. Using Eqn.(216) with x s = φ max = π one obtains (M¨uller, Park and Rana, 1999) C = K s − λ − λ , (252)where C is equivalent to the coefficient of the P in Eq.(251). It is evident that C < λ > , correspondingto the regime of first-order phase transition. At the phase boundary C = 0, which yields the critical value λ c = .3 P F / ∆ U λ =0 . Bottom barrierTop barrier θ =1 . θ =1 θ =0 . θ =0 . θ =0 . P F / ∆ U λ =0 . Bottom barrierTop barrier θ =1 θ =1 . θ =1 . θ =1 . θ =1 . FIG. 20 Color online. The effective free energy of the escape rate vs P. Left: for λ = 0 .
1, second-order transition. Right: for λ = 0 .
8, first-order transition. x U ( x ) / K s ( s + ) λ =0 . λ =0 . λ =0 . FIG. 21 Color online: The plot of the potential in Eq.(253) for several values of λ . The plot of Eqn.(250) in the whole range of energy is shown in Fig.(20). For λ = 0 .
8, that is first-order transition,the actual crossover from thermal to quantum regime is estimated as T (1)0 = 1 . T (2)0 corresponding to the pointwhere two minima have the same free energy. At λ = 0 .
1, there is only one minimum of T for all T > T (2)0 , i.e θ > θ <
1, the minimum continuously shifts to the bottom of the barrier with loweringtemperatures. This corresponds to a second-order phase transition. The ratio of the anisotropy constants λ in thismodel plays a similar role as h x in the uniaxial model. In general, the splitting term in the Hamiltonian is responsiblefor the dynamics of the system, and leads to the phase transition in large spin systems. However, the sharp first-orderphase transition in this model is not as result of the flatness of the barrier top as in the case of the uniaxial model. Itshould be noted that the mass of the particle at the top of the barrier m (cid:0) π (cid:1) = [2 K (1 − λ )] − is heavier than that ofthe bottom of the barrier m (0) = [2 K ] − . The former is responsible for the sharp first-order crossover. A constantmass in this model can be achieved through the effective potential method. The spin Hamiltonian corresponds to theeffective potential and the mass (M¨uller, et al. , 1999): U ( x ) = K s ( s + 1) cn ( x, λ )dn ( x, λ ) ; m = 12 K ; (253)where x s = x max = 0. It is now trivial to check that Eq.(253) yields the same result in the large spin limit s ( s +1) ∼ s .4Since the mass is now a constant, the flatness of the barrier as λ → D. Phase transition in easy z -axis biaxial spin model with a magnetic field
1. Introduction
In the presence of a magnetic field, other interesting features arise. In this case one can study how the crossovertemperatures vary with the magnetic field at the phase boundary. Many biaxial spin models in the presence of = 100 = 1000 = 100 = 1000 − ! ! ! !" " ! $ = 100 = 1000 = 100 = 1000 (cid:60) ! ! " FIG. 22 Left: The crossover temperature T ( c )0 at the phase boundary between first- and second-order transitions for the modelˆ H = − D ˆ S z + B ˆ S x − H z ˆ S z . Right: The phase boundary between the first- and the second-order phase transitions for the samemodel using perturbation theory, where h z = H z / Ds . Adapted with permission from Chudnovsky and Garanin, 1999 an external magnetic with different easy axes directions have been studied by different approaches, although thesesystems are related by their anisotropy constants. The early work on these models began with M¨uller, et al. (1998).They studied a Hamiltonian of the form:ˆ H = K ( ˆ S z + λ ˆ S y ) − µ B h y ˆ S y , λ < , (254)by spin coherent state path integral formalism. This Hamiltonian possesses an easy x -axis, y -medium axis and z hardaxis. They explicitly demonstrated numerically the influence of the magnetic field on the crossover temperatures,and the period of oscillation. This analysis is however valid only in the regime λ <
1. In the case of a field parallelto the z -axis, their approach will break down, since such magnetic field pushes the spin away from the x - y plane.Chudnovsky and Garanin (1999, 2000) have studied two biaxial spin models of the formˆ H = − D ˆ S z + B ˆ S x − H x ˆ S x , (255)by direct numerical method and ˆ H = − D ˆ S z + B ˆ S x − H z ˆ S z , (256)by perturbation theory with respect to B . However, the perturbative approach is less justified in the large B limit.These two models have an easy z axis, x hard axis and y medium axis with the magnetic field applied along the hardand easy axes respectively. The first model, i.e Eqn.(255) is related to the model in Eqn.(86) sec(III.A.3) by rescalingthe anisotropy constants . It is realized in Fe molecular cluster with s = 10, D = 0 . K and B = 0 . K . Thesecond model, Eqn.(256) has a magnetic field along the easy axis which creates a bias potential minima. For this modelthe effect of the external magnetic field on the crossover temperatures was explicitly demonstrated by perturbationtheory. In Fig.(22) we show the phase boundary and its crossover temperature T ( c )0 obtained via perturbation theoryfor the model in Eqn.(256).5
2. Effective potential method
Based on the results obtained from perturbation theory and numerical methods, Kim (1999) considered the effectivepotential method of the model: ˆ H = − K (cid:107) ˆ S z + K ⊥ ˆ S y − H x ˆ S x − H z ˆ S z . (257)For H x = 0, this model is exactly the model in Eqn.(256), and for H z = 0 the magnetic field H x is along the mediumaxis. It is related to Eqn.(255) only by the rotation of axis ˆ S y → ˆ S x . If one introduces the spin wave function inEqn.(163), then using the generating functionΦ( x ) = s (cid:88) m = − s c m e mx (cid:112) ( s − m )!( s + m )! , (258)and the particle wave function function Ψ( x ) = e − y ( x ) Φ( x ) , (259)where y ( x ) is determined in the usual way . The corresponding effective potential and the coordinate dependentmass are given by u ( x ) = 11 + k cosh 2 x [(1 − k )( h x sinh x − h z ) − h x (1 + k ) cosh x − h z k sinh 2 x + k (1 − cosh 2 x )] , (260) m ( x ) = 1 K (cid:107) (2 + k t )(1 + k cosh 2 x ) , (261)where U ( x ) = ˜ s K (cid:107) u ( x ), k = k t / (2 + k t ) and k t = K ⊥ /K (cid:107) , h x,z = H x,z / K (cid:107) ˜ s . The large s limit, i.e., s (cid:29) s drop out in the potential. x u ( x ) h x =0 . , h z =0 , k =0 . x u ( x ) h x =0 . , h z =0 . , k =0 . FIG. 23 Color online: The plot of the effective potential in Eqn.(260) with different parameters.
3. Phase boundary and crossover temperatures
We consider two cases h z = 0, h x (cid:54) = 0, and h x = 0, h z (cid:54) = 0. Let us consider the first case h z = 0 and h x (cid:54) = 0, in thiscase the potential reduces to u ( x ) = (1 + k )( h x cosh x − k cosh 2 x , (262)6where a constant of the form (1 + h x ) has been added to normalize the potential to zero at the minimum x min = ± arccosh (cid:16) h x (cid:17) . This potential is now an even function with a maximum at x s = x max = 0 . The barrier height is∆ U = K (cid:107) ˜ s (1 − h x ) (see Fig.(23)). In the limit k t →
0, Eqn.(261) and (262) reduce to that of the uniaxial modelstudied in Sec.(V.B). The thermon action is given by T S T (1)0 T (2)0 S p → B S p S T S T (1)0 T (2)0 S p → B S p S FIG. 24 Color online: The plot of the thermon action Eqn.(263) and thermodynamic action Eqn.(212) against temperaturefor the biaxial model with magnetic field h x . Left: s = 10, K || = 1, k t = 0 . h x = 0 . B is thevacuum instanton action. Right: s = 10, K || = 1, k t = 1 . h x = 0 . S p = 2˜ s (cid:114)
22 + k t (cid:90) x − x dx (cid:112) c cosh x − c cosh x + c k − k cosh x + β ∆ U (1 − P ) , (263)where c = (1 + k ) h x − k (1 − h x )(1 − P ); c = 2 h x (1 + k ); c = 1 + k − (1 − k )(1 − h x ) (1 − P ) . (264)The turning points ± x are found by setting the term in the square root to zero. Integrating the classical equationof motion one finds that the denominator in Eqn.(262) cancels the mass in Eqn.(261). Thus, this action Eqn.(263)corresponds to the action of the periodic instanton trajectory in Eqn.(228) but with ˜ E = E /K || ˜ s (1 + k ) and ω =( K || ˜ s ) (1 + k )(1 + k t )(1 − ( h x − (cid:112) ˜ E )) /
2. Mathematically, the integral in Eqn.(263) can be evaluated exactly in termsof complete elliptic integrals(Abramowitz and Stegun, 1972; Byrd and Friedman, 1979). At the bottom of the barrier P →
1, the exact vacuum instanton action is given by (Kim, 1999) S ( U min ) = 2 B = 2˜ s (cid:20) ln (cid:32) √ k t + (cid:112) (1 − h x ) √ k t − (cid:112) (1 − h x ) (cid:33) − h x √ k t arctan (cid:32) (cid:112) k t (1 − h x )(1 + k t ) h x (cid:33) (cid:21) . (265)This expression is consistent with the small barrier action in Eqn.(204), if one uses the relation of the anisotropyconstants. In Fig.(24) we have shown the plot of this action Eqn.(263) and the thermodynamic action Eqn.(212)against temperature. We notice that the plot of these two actions has a similar trend to every other model, whichindicates the presence of the quantum-classical phase transitions in each of the models. The free energy can also beobtained from Eqn.(263). Kim (1999) expanded this integral in Eqn.(263) around x s = x max in terms of P for ageneral potential and coordinate dependent mass: S p = π (cid:115) m ( x s ) U (cid:48)(cid:48) ( x s ) ∆ U [ P + bP + O ( P )] + β ∆ U (1 − P ) , (266)where b = ∆ U U (cid:48)(cid:48) (cid:20) U (cid:48)(cid:48)(cid:48)(cid:48) U (cid:48)(cid:48) + 15( U (cid:48)(cid:48)(cid:48) ) U (cid:48)(cid:48) ) + 3 (cid:18) m (cid:48) m (cid:19) (cid:18) U (cid:48)(cid:48)(cid:48) U (cid:48)(cid:48) (cid:19) + (cid:18) m (cid:48)(cid:48) m (cid:19) − (cid:18) m (cid:48) m (cid:19) (cid:21) x = x s , (267)7 U (cid:48)(cid:48) ( x s ) = − K (cid:107) ˜ s u (cid:48)(cid:48) ( x s ) / U (cid:48)(cid:48)(cid:48) ( x s ) = K (cid:107) ˜ s u (cid:48)(cid:48)(cid:48) ( x s ) / U (cid:48)(cid:48)(cid:48)(cid:48) ( x s ) = K (cid:107) ˜ s u (cid:48)(cid:48)(cid:48)(cid:48) ( x s ) / b corresponds to the coefficient of ψ in the Landau freeenergy expression Eqn.(237), and b < b = 0 determinesthe boundary between the first- and the second-order transition. One can check that this condition, Eqn.(267) yieldsexactly the same result as Eqn.(216). Applying the criterion, Eqn.(267) one obtains the phase boundary betweenfirst- and second-order: h ± x = 1 − k t + k t ± (1 + k t ) (cid:112) k t + k t − k t ) . (268)At k t = 0, corresponding to the uniaxial limit, the phase boundary reduces to h cx = h + x = 1 /
4, which is exactlythe result obtained before. At h x = 0, one obtains k t = 1 which also corresponds to the result of the biaxial modelwithout magnetic field, since the anisotropy constants are related by K (cid:107) = D and K ⊥ = D − D . In the smallanisotropy limit k t (cid:28)
1, the phase boundary behaves linearly as h + x ≈ (1 + 3 k t ) /
4. The approximate form for the h z T T ( c )0 T max FIG. 25 Color online: The second-order crossover temperature at the phase boundary and its maximum at h z = 0. T =2 πT / ˜ sK (cid:107) . Reproduced with permission from Kim, 1999 first-order crossover temperature is estimated as T (1)0 = ∆ U/S ( U min ). Notice that this model does not have a largebarrier so the concept of large and small barriers does not apply here. In Sec.(V.E) we will present a complete phasediagram for small and large barriers for the model in Eqn.(199). For the case of second-order transition the crossovertemperature and its maximum are given by T (2)0 = ˜ sK (cid:107) π (cid:112) ( k t + h x )(1 − h x ); T max0 = ˜ sK (cid:107) π (1 + k t ) . (269)Using the value of h x = h + x at the phase boundary, Eqn.(268), one obtains the crossover temperature at the phaseboundary T ( c )0 . As shown in Fig.(25) the difference between these temperatures vanishes at k t = 1 which is the criticalvalue at the phase boundary for h x = 0.For the second case h x = 0 and h z (cid:54) = 0, the potential Eqn.(260) is an odd function with a bias minima, and thebarrier height is ∆ U = K (cid:107) ˜ s (1 − h z ) . The maxima is located at x s = x max = ln (cid:16) h z − h z (cid:17) . Direct application ofEqn.(267) yields b = K || ˜ s (2 + k t )16 k t (1 + h z ) (cid:0) h z (1 + 2 k t ) − (1 − k t ) (cid:1) , (270)At the phase boundary b = 0 which yields k ct = 1 − h cz h cz . (271)8 h z k c t First-order phase transitionSecond-order phase transition h z T ( c ) / K || ˜ s FIG. 26 Color online: Left: The phase boundary between the first- and second-order transitions at h x = 0. Right: Thecrossover temperature T ( c )0 at the boundary between first- and second-order transitions. Reproduced with permission fromKim, 1999 k t h z ¯ b FIG. 27 Color online: Three dimensional plot of the Landau coefficient ¯ b = b/K || ˜ s . In this figure ¯ b < b > b = 0 is placed on the lower two dimensional plane forproper view of the upper plane. The plot of Eq.(270) is shown in Fig.(27), indicating the regions of the phase transitions based on the sign of b .For small field parameter h z (cid:28)
1, the critical value decreases as k ct ≈ − h cz . One obtains that the second-ordercrossover at the phase boundary is T ( c )0 = K (cid:107) ˜ sπ − h cz (cid:112) h cz , (272)which has the form T ( c )0 /K (cid:107) ˜ s ≈ (1 − h z ) /π for h z (cid:28)
1. The plot of Eqn.(271) and (272) is shown in Fig.(26).It clearly shows the consistency of the result with that of perturbation theory in Fig.(22). At k t = 0, there is notunneling due to the following: Quantum mechanically, in this limit the Hamiltonian commutes with ˆ S z , thus thereis no splitting term since ˆ S z is conserved quantity. In the effective potential method, this implies that the barrierbecomes infinitely thick and the spin cannot tunnel.9
4. An alternative model
It is sometimes difficult to deal with a particle Hamiltonian with a position dependent mass. It is possible to get aparticle with a constant mass from the model in Eqn.(257) using another approach. Let us consider the model ˆ H = − A ˆ S z − C ˆ S x − H x ˆ S x − H z ˆ S z . (273)This model is exactly the same as Eqn.(257) if we set A = K (cid:107) + K ⊥ and C = K ⊥ . Introducing an unconventionalgenerating function of the form G ( x ) = s (cid:88) m = − s c m (cid:112) ( s − m )!( s + m )! (cid:18) sn x + 1cn x (cid:19) s , (274)and the particle wave function function Ψ( x ) = e − y ( x ) G ( x ) . (275)As in the previous analysis, y ( x ) is determined by the usual procedure . One finds that the corresponding effectivepotential and the mass are given by ( Chang, et al. , 2000, 1999), U ( x ) = ˜ s Au ( x ) u ( x ) = 14 dn x [( α x sn x − α z cn x ) − b − bα z sn x + α x cn x )] , m = 12 A , (276)where the large s limit s ∼ s + 1 ∼ ( s + ) = ˜ s has been used. b = C/A and α x,z = H x,z /sA , the modulus of theelliptic functions is k = 1 − b . The maximum of the potential is at x s = x max = 0 for α z = 0 .
5. Phase boundary and crossover temperatures
Since the mass is now a constant and the potential is even for α z = 0, the criterion for the first-order transition,Eqn.(216) is determined only by the fourth derivative of the potential at x s or by considering where the coefficient ofthe fourth order expansion changes sign near x s . For α z = 0 we find U ( x ) ≈ U (0) + A ˜ s [ −
14 (2 − b − α x )(2 b + α x ) x + 124 ( α x − α + x )( α x − α − x ) x + O ( x )] . (277)The vanishing of the coefficient of x determines the phase boundary α ± cx = 1 − b c (1 − b c ) ± (cid:112) b c (1 − b c )4(1 − b c ) , (278)which is exactly the result obtained by Chang, et al. (2000). Eqn.(278) is consistent with Eqn.(268) by noticing that b = k t / (1 + k t ) and α x = 2(1 − b ) h x . The second-order crossover temperature is given by T (2)0 = A ˜ s π (cid:112) (2 b + α x )(2(1 − b ) − α x ) . (279)Plugging Eqn.(278) into Eqn.(279) one obtains the crossover temperature at the phase boundary. For α x = 0 inEqn.(276), the maximum occur at x s = sn − [ − α z / − b )], the potential is no longer an even function, therefore thecoefficient of the fourth order expansion near x s cannot determine the regime of first-order transition. With the helpof Eqn.(221) one obtains the phase boundary and the crossover temperature at the phase boundary ( Chang, et al. ,1999) α cz = 2(1 − b c ) (cid:114) − b c b c ; T c = 2˜ sA √ b c π (cid:114) − b c b c . (280) An alternative choice is ˆ H = K ˆ S z + K ˆ S y − H x ˆ S x . Setting H x = 0 in Eqn.(273), these models are related by K = A and K = A − B (M¨uller, et al. , 1999). FIG. 28 Color online: 3D numerical plot of the crossover temperature T c ( α cz , α cx ) / ˜ sA at the boundary as a function of fieldparameters α cx and α cz with b c = 0 .
29. From Chang, et al. (2000)
These expressions are consistent with Eqn.(271) and (272). At α cz = 0 = α + cx , one finds that b c = 1 / A = K and C = K − K , implyingthat λ = 1 − b . In the limit of small anisotropy b c (cid:28)
1, one finds α cz ≈ − b c /
2) and T c ≈ sA √ b c /π . Thephase diagram of these expressions are related to Fig.(25) and Fig.(26). For iron cluster Fe , s = 10, A = 0 . K ,and C = 0 . K (Barra, et al. , 1996; Sangregorio, et al. , 1997) one finds that T c = 0 . K . In Fig.(28) we have showna 3-dimensional plot of T c ( α cx , α cz ). It is evident that T c decreases as α z increases, while it increases with increasing α x . E. Phase transition in easy x -axis biaxial spin model with a medium axis magnetic field
1. Effective potential of medium axis magnetic field model
For the model we considered in Sec.(IV.B.2), that isˆ H = D ˆ S z + D ˆ S x − H x ˆ S x . (281)The effective potential and the mass were obtained as U ( x ) = D ˜ s [cn( x ) − α x ] dn ( x ) , m = 12 D . (282)This Hamiltonian is related to Eqn.(257) for H z = 0 if one sets D = K (cid:107) + K ⊥ and D = K (cid:107) , it is also relatedto Eqn.(273) for H z = 0 if one sets D = A and D = A − B , but unlike these models, we saw that the potential,Eqn.(282) has large and small barriers (see Fig.(10)) located at x lb = ± n +1) K ( κ ) and x sb = ± n K ( κ ) respectively,with the barrier heights given by Eqn.(202). The phase transition of the escape rate was studied by M¨uller, et al. (1998) using spin coherent state path integral. In this review we will consider it in the effective potential method.1 α x ¯ T ( c ) Small barrier(max)Large barrier(max)Small barrierLarge barrier α x ¯ T ( c ) Small barrierLarge barrier
FIG. 29 Colour online: Dependence of the crossover temperatures on the magnetic field at the phase boundary. Left: Second-order (solid line) and its maximum (dashed line) for the small and large barrier. Right: First-order for the small and the largebarrier. These graphs are plotted with T ( c )0 = T ( c )0 /D ˜ s . Adapted with permission from Owerre and Paranjape, 2014b
2. Phase boundary and crossover temperatures
Using Eqn.(216) and the maximum points x lb , x sb , the boundary between the first and second-order transition forsmall and large barriers are found to be (Owerre and Paranjape, 2014b) λ ± sb ( α x ) = 3 − α x + α x ± (1 − α x ) (cid:112) − α x + α x − α x + α x ) , (283) λ ± lb ( α x ) = 3 + 4 α x + α x ± (1 + α x ) (cid:112) α x + α x α x + α x ) . (284)For small barrier one can check that Eqn.(283) is consistent with Eqn.(278) and Eqn.(268). The crossover temperaturefor the first-order transition is estimated as T (1)0 = ∆ U/ B , which can be obtained from Eqn.(202) and (204). At thephase boundary we find that T ( c )0 ≈ D ˜ s/ (ln[(3+2 √ e ± αx √ ]) for α x (cid:28)
1, where the upper and lower signs correspondto small and large barrier respectively. Both temperatures coincide at α x = 0 , λ = with T ( c )0 = D ˜ s/ ln(3 + 2 √ T (2)0 = D ˜ s (cid:112) (1 ± α x ) π (cid:18) − (1 ± α x ) λλ (cid:19) / , (285) T (max)0 = D ˜ s πλ . (286)where the upper and lower signs correspond to the large and small barriers respectively. At the phase boundary onefinds that Eqn.(285) behaves as T ( c )0 ≈ D ˜ s (1 ± α x ) /π for α x (cid:28)
1, which coincides at α x = 0 , λ = as shown inFig.(29). The evidence of this crossover temperatures has predicted in Fe molecular cluster with s = 10. There are21 ×
21 matrices with 2 s + 1 states which can be found by the so-called exact numerical diagonalization. The energybarrier of this system is much smaller than that of Mn Ac. In the low-temperature limit, specifically for
T < . K ,only the two lowest energy level with M = ± s are occupied and tunneling is possible between these two states. Forthis system experimentally measured relaxation rate showed a temperature independent rate below 400mK whichsuggests the evidence of spin tunneling across its anisotropy energy barrier (Sangregorio, et al. , 1997) (see Fig.(30)).2 V OLUME
78, N
UMBER
24 P H Y S I C A L R E V I E W L E T T E R S 16 J
UNE
FIG. 2. ln s t d vs y T : High temperature data follow ther-mal activation law with t ≠ s and E ≠ K.Points below 400 mK show temperature independent QTM. were satisfactorily simulated with D ≠ K and E ≠ K [21].We have now extended the ac susceptibility measure-ments to lower temperatures and frequencies. The ac ex-periments were made on a polycrystalline powder samplefrom 6 kHz down to 0.005 Hz using a low field low tem-perature SQUID magnetometer equipped with a miniaturedilution refrigerator. The low frequency data give infor-mation on the relaxation down to 1 K. The relaxationtimes, t , have been extracted from the imaginary part ofthe susceptibility by assuming that the temperature of themaximum corresponds to the blocking temperature, i.e.,the temperature at which the relaxation time is equal tothe time scale of the experiment t ≠ y v . This is justi-fied because the curves are well fit by considering a singlerelaxation time. The t values are plotted as ln s t d vs y T in Fig. 2 which shows that the relaxation follows Eq. (1)with an energy barrier of 24.5 K and t ≠ s.These results when combined with previously publisheddata [21] indicate that thermal activation is more or lessobeyed over a time scale of 10 decades. The unusuallylarge value of t is similar to that which is observed inMn and has been explained as arising from a multi-step Orbach process [22].In order to investigate the magnetic relaxation at lowertemperature, dc measurements were performed using ahigh field low temperature SQUID magnetometer. Fig-ure 3 shows magnetization versus field curves measuredat 1.3 K and 80 mK after first saturating the sample in8 T, where we find M ≠ m B This is in agreement with S ≠ and g ≠ as seen in HF-EPR [21]. At zerofield M is one half its saturation value as expected forour nonoriented powder sample, and indicates that aftersaturation, all clusters within each (randomly oriented)crystal are still perfectly aligned along the easy axes ofthe crystal. The low temperature curves were taken withtwo different ramp rates and show hysteresis and “steps”at well defined field values similar to those observed for FIG. 3. M vs B : curve taken at 1.3 K shows no hysteresis.Below 400 mK curves are temperature independent, but dependon field ramp rate dB y dt as shown. dM y dB is shown for0.04 T y h; peaks correspond to faster relaxation, verified by dcmeasurements. Mn . For T , mK the steps become temperatureindependent (at fixed ramping rate dB y dt ). The hystere-sis is due to the fact that the characteristic relaxation timesare longer than the experimental ones; the steps occur atfield values where the relaxation is enhanced.dc relaxation experiments were made by first saturatingthe sample in a high field ( . T), and then reducingthe field to a predetermined value and measuring thedecay of the magnetization over a period of hours andfor up to one week. Some curves obtained at differenttemperatures are shown in Fig. 4. Above 1.1 K therelaxation is too fast to be measured by this technique,however, between 1.1 and 1 K there is an overlap withour very low frequency ac data. Below 400 mK andfor a given measuring field, the relaxation curves takenat different temperatures completely superimpose, clearlyshowing that the relaxation is temperature independent asseen in Fig. 4.A crossover from single exponential behavior to a morecomplex relaxation process is observed below 800 mK,where the magnetization curves can be well fit (over therelatively large time scales of our investigations) by astretched exponential M s t d ≠ M s d exp f s t y t d b g . (2)The stretched exponential describes a relaxation rate thatis faster at earlier times and which becomes progressivelyslower as the system evolves. It has often been usedto describe the dynamics of disordered systems suchas spin glasses [23]. However, it is also manifest inordered systems, when the drive field responsible for therelaxation changes (reduces) as the system relaxes [24].The average relaxation time is given by t , and b is related4646 FIG. 30 Temperature dependence on the relaxation time τ . Points below 400 mK show temperature independent quantumtunneling. Adapted with permission from Sangregorio, et al. , 1997
3. Free energy
In the presence of a magnetic field, the Euclidean action cannot be obtained exactly or analytically (M¨uller, et al. ,1998), thus it is studied numerically. The periodic instanton action or the thermon action is given by S p ( E ) = 2 √ m (cid:90) x x dx (cid:112) U ( x ) − E + β ( E − U min ) . (287)Setting y = sn( x, λ ) and using Eqn.(201) we have P F / ∆ U s b λ =0 . , α x =0 . θ =1 . θ =1 . θ =1 . θ =1 . θ =1 FIG. 31 Color online: The numerical plot of the free energy for κ = 0 . α = 0 . S p ( P ) = 2˜ s √ κ ˜ S ( P ) + β ∆ U (1 − P ) , (288)˜ S ( P ) = (cid:90) y y dy (cid:20) (cid:16)(cid:112) − y − α x (cid:17) − Q (1 − κy )(1 − κy ) (1 − y ) (cid:21) , (289)3where Q = (1 − α x ) (1 − P ), and the turning points y and y are determined by setting the numerator in the squarebracket to zero. The free energy can then be written as F ( P )∆ U sb = 1 − P + 2 θ (cid:112) − κ (1 − α ) π (1 − α ) / ˜ S ( P ) , (290)where θ = T /T (2)0 , T (2)0 is given in Eqn.(285), and ∆ U sb is given in Eqn.(202).In Fig.(31) we have shown the numerical plot of the free energy with some of the temperature parameters in (Zhang, et al. , 1999), and the same dimensionless anisotropy constant λ = 0 .
8, but in the presence of a small magnetic field α x = 0 .
05. We notice that the phase transition from classical to quantum regime (where two minima of a curve havethe same free energy) has been shifted to θ = 1 .
25 or T (1)0 = 1 . T (2)0 , which is larger than the zero magnetic field value T (1)0 = 1 . T (2)0 in Fig.(20)(b) (Zhang, et al. , 1999). Thus, the magnetic field increases the crossover temperaturefor this model as we found in the previous model in Sec.(V.D.4). However, for large barrier we expect the crossovertemperature to decrease. Thus, the large barrier plays a similar role as the longitudinal field H z in Sec.(V.D.4). F. Phase transition in exchange-coupled dimer model
1. Model Hamiltonian
In Sec.(III.A.5), we reviewed the problem of an antiferromagnetic exchange-coupled dimer model via spin coherentstate path integral formalism. In this section we will study the effective potential method of the model. In thepresence of a staggered magnetic field applied along easy z -axis, the Hamiltonian is given byˆ H = J ˆ S A · ˆ S B − D ( ˆ S A,z + ˆ S B,z ) + gµ B h ( ˆ S A,z − ˆ S B,z ) , (291)where J >
D > J > z -axis anisotropy, and h isthe external magnetic field, µ B is the Bohr magneton and g = 2 is the spin g -factor.
2. Effective potential
The spin wave function in this case can be written in a more general form as ψ = ψ A ⊗ ψ B = s A ,s B (cid:88) σ A = − s A σ B = − s B C σ A , − σ B F σ A , − σ B , (292)where F σ A , − σ B = (cid:18) s A s A + σ A (cid:19) − / (cid:18) s B s B − σ B (cid:19) − / | σ A , − σ B (cid:105) . (293)Following the same procedures outlined above, one finds that the effective potential U ( r ) and the coordinate dependentreduced mass µ ( r ) are given by (Owerre and Paranjape, 2014c) U ( r ) = 2 D ˜ s [2 α + κ (1 − cosh r ) + 2 ακ sinh r ](2 + κ + κ cosh r ) , (294) µ ( r ) = 12 D (2 + κ + κ cosh r ) . (295)In order to arrive at these equations we have used the fact that the two giant spins are equal s A = s B = s and theapproximation s ( s + 1) ∼ ˜ s = ( s + ) , where κ = J/D and α = gµ B h/ D ˜ s . The variable r denotes the relativecoordinate of the particles, the center of mass coordinate does not contain any information about the system.4
3. Periodic Instanton at zero magnetic field
In the absence of a magnetic field, the effective potential is now of the form U ( r ) = 2 Ds ( s + 1) u ( r ) , u ( r ) = κ (1 − cosh r )(2 + κ + κ cosh r ) . (296)Since we are considering large spin systems, the coefficient s ( s + 1) will be approximated as s . The potential is nowsymmetric with degenerate minima, and hence the turning points are ± r ( E ) with the maximum of the barrier heightlocated at r b = 0 as shown in Fig.(32). The action associated with the thermon action is given by
10 5 0 5 10 r u ( r ) E − r ( E ) r ( E ) FIG. 32 Color online: The plot of the potential for κ = 0 .
6. The minimum energy is u min = −
1, and the maximum is u max = 0.Thus, ∆ U = U max − U min = 2 Ds . S ( E ) = 2 (cid:90) r ( E ) − r ( E ) dr (cid:112) µ ( r ) ( U ( r ) − E ) . (297)This action can be integrated exactly for all possible values of the energy without computing the periodic instantontrajectory explicitly (Owerre and Paranjape, 2014c). In this paper, we will obtain this action by first calculating theperiodic instanton trajectory corresponding to the action. The Euclidean Lagrangian is given by L E = 12 µ ( r ) ˙ r + U ( r ) . (298)The Euler-Lagrange equation of motion gives µ (¯ r p )¨¯ r p + 12 dµ (¯ r p ) d ¯ r p ˙¯ r p − dUd ¯ r p = 0 . (299)Integrating once we obtain 12 µ (¯ r p ) ˙¯ r p − U (¯ r p ) = −E , (300)where E is the integration constant. Thus, the periodic instanton trajectory can be found from the solution of thisequation: τ = (cid:90) ¯ r p dr (cid:115) µ ( r )2( U ( r ) − E ) = 1 √ ω b (cid:90) ¯ r p dr (cid:113) a + b − b cosh (cid:0) r (cid:1) , (301)5where ω b = 2 Ds √ κ is the frequency of oscillation at the well of the inverted potential of Fig.(32), a = 1 − (2 + κ ) E (cid:48) , b = 1 + κ E (cid:48) , and E (cid:48) = E / Ds κ . In terms of a new variable y = cosh (cid:0) r (cid:1) , we have ω b τ = 1 √ b (cid:90) ¯ y p dy (cid:113) ( y − a + b b − y ) , (302)where ¯ y p = cosh (cid:0) ¯ r p (cid:1) . Introducing another change of variable: x = y − λ y , λ = a − ba + b . (303)The integral in Eqn.(302) becomes ω b τ = (cid:114) a + b (cid:90) ¯ x p dx (cid:112) (1 − x )(1 − λ x ) = (cid:114) a + b F (¯ θ p , λ ) = (cid:114) a + b sn − (sin ¯ θ p , λ ) , (304)where ¯ x p = sin ¯ θ p = (cid:115) ¯ y p − λ ¯ y p = 1 λ tanh (cid:16) ¯ r p (cid:17) , (305)and F (¯ θ p , λ ) is an incomplete elliptic integral of first kind with modulus λ and ¯ θ p Substituting Eqn.(305) intoEqn.(304), and solving for ¯ r b we obtain the periodic instanton:¯ r p ( τ ) = 2 arctanh[ λ sn( ω p τ, λ )] , ω p = ω b (cid:114) a + b . (306)It is required that as τ → ± β , the periodic instanton trajectory must tend to the classical turning points definedin Eqn.(297). In other words, ¯ r p → ± r ( E ) = ± arccosh (cid:0) ab (cid:1) as τ → ± β as depicted in Fig.(33). This demands thatsn( ω p τ, λ ) → ± τ → ± β . Using the fact that µ (¯ r p ) and ˙¯ r p are given by ω p τ ¯ r p r ( E ) β − r ( E ) − β FIG. 33 Color online: The periodic instanton trajectory with λ = 0 .
2. The turning points ± r ( E ) are shown in Fig.(32). µ (¯ r p ) = dn ( ω p τ, λ )4 D [1 + κ − λ sn ( ω p τ, λ )] ; ˙¯ r p = (2 λω p ) cn ( ω p τ, λ )dn ( ω p τ, λ ) , (307)and making the transformation x = sn( ω p τ, λ ), the action for the periodic instanton path can be computed as S p = (cid:90) β − β dτ µ (¯ r p ) ˙¯ r p + β ( E − U min ) = 2 s (cid:112) a + b ) κ [ K ( λ ) − (1 − γ )Π( γ , λ )] + β ( E − U min ) , (308)where γ = λ (1 + κ ) − . The functions K ( λ ) and Π( γ , λ ) are known as the complete elliptic integral of first andthird kinds respectively.6
4. Vacuum instanton at zero magnetic field
Since vacuum instanton occurs at zero temperature T →
0, which implies that β → ∞ , the energy of the particlemust be close to the minima of potential yielding tunneling between degenerate ground states. Near the bottom ofthe barrier E → U min = − Ds , a → κ ) /κ and b →
0, thus λ →
1, we get sn( v, → tanh v . The periodicinstanton trajectory, Eqn.(306) reduces to a vacuum instanton:¯ r p ( τ ) → ¯ r = 2 ω τ, ω p → ω = 2 Ds √ κ. (309)As τ → ±∞ , ¯ r → ±∞ , which corresponds to the minima of the zero magnetic field potential. A particle sitting atthe minimum of this potential is massless, µ (¯ r → ∞ ) = 0, but the vacuum instanton mass is not zero. It is given by µ (¯ r ) = [2 D (2 + κ + κ cosh(2 ω τ ))] − . (310)In Fig.(34), we have shown the dependence of the dimensionless anisotropy constant on the vacuum instanton mass.Near the top of the barrier E → U max = 0, a → b →
1, thus λ →
0, the periodic instanton reduces to a sphaleron
10 5 0 5 10 ω τ µ ( ¯ r ) =0 . . . . FIG. 34 Color online: Dependence of the dimensionless anisotropy constant on the vacuum instanton mass, with D = 1. ( static, unstable, finite-energy solutions of the classical equations of motion) at the top of the barrier:¯ r p ( τ ) → r b = 0 , ω p → ω b = 2 Ds √ κ. (311)The mass of the sphaleron is given by µ ( r b ) = [4 D (1 + κ )] − . (312)In Fig.(35) we have shown the plot of the ratio of the frequencies ω p /ω and ω p /ω b against energy for several valuesof κ . The action associated with the vacuum instanton trajectory can be obtained by expanding the elliptic integralsin Eqn.(308) near the bottom of the potential λ →
1, or simply by computing the action associated with the vacuuminstanton path in Eqn.(309). Using Eqn.(310) and the fact that ˙¯ r is given by˙¯ r = (2 ω ) . (313)One can easily confirm by direct integration that the vacuum instanton action is given by B = (cid:90) ∞−∞ dτ µ (¯ r ) ˙¯ r = 4 s arctanh (cid:18) √ κ (cid:19) = 2 s ln (cid:18) √ κ + 1 √ κ − (cid:19) . (314)This is the exact vacuum instanton action. In the perturbative limit J (cid:28) D , which implies that κ (cid:28)
1, Eqn.(314)reduces to B ≈ s ln (cid:18) κ (cid:19) = 2 s ln (cid:18) DJ (cid:19) . (315)7 P ω p /ω P ω p /ω b FIG. 35 Color online: The 3-d plot of the ratio of the periodic instanton frequency in Eqn.(306) to that of vacuum instantonin (309), and to that of sphaleron in Eqn.(311). At the bottom of the barrier ω p → ω , and at the top of the barrier ω p → ω b . This is the same action that was obtained by spin coherent state path integral in Eqn.(125), but the imaginary termin the spin coherent state path integral result, which is responsible for different ground state behaviour of integerand half-odd integer spins has disappeared. This is one of the disadvantages of mapping a spin system to a particlesystem.
5. Free energy and phase transition at zero magnetic field
We will now investigate the phase transition of the escape rate using the free energy method . Having obtained theperiodic instanton action for all possible values of the energy, that is Eqn.(308), the free energy associated with theescape rate can then be written as F ∆ U = 1 − P + 4 π θ (cid:112) κ ( κ + P )[ K ( λ ) − (1 − γ )Π( γ , λ )] , (316)where θ = T /T (2)0 is a dimensionless temperature quantity, and T (2)0 = Ds √ κ/π . The modulus of the completeelliptic integrals λ and the elliptic characteristic γ are related to P by λ = (1 + κ ) Pκ + P , γ = Pκ + P . (317)In Fig.(36) we have shown the plot of the free energy against P for κ = 0 . P = 0. For θ = 1 .
054 or T (1)0 = 1 . T (2)0 , twominima have the same free energy. This corresponds to the crossover temperature from classical to quantum regimes.As the temperature decreases from this crossover temperature, a new minimum of the free energy is formed, this newminimum becomes lower than the one at P = 0. We have pointed out that phase transition occurs near the top of thepotential barrier, so it is required that we expand this free energy close to the barrier top. Thus, near the top of thebarrier P →
0, the complete elliptic integrals can be expanded up to order P . The full simplification of Eqn.(316)yields F ∆ U = 1 + ( θ − P + θ ( κ − κ P + θ (3 κ − κ + 3)64 κ P . (318)Similar to the case of uniaxial spin model in a transverse magnetic field ( Chudnovsky, et al. , 1998; Chudnovsky andGaranin, 1997), this free energy looks more like the Landau’s free energy, which suggests that we should compare thetwo free energies. The Landau’s free energy has the form: F = F + aψ + bψ + cψ . (319)8 P F ( P ) / ∆ U Bottom barrierTop barrier θ =1 . θ =1 . θ =1 . θ =1 . θ =1 . FIG. 36 Color online: The effective free energy of the escape rate vs. P for κ = 0 . θ = T /T (2)0 , first-ordertransition.
The coefficient of P in Eqn.(318) is equivalent to the coefficient a in Landau’s free energy. It changes sign at thephase temperature T = T (2)0 . The phase boundary between the first- and the second-order phase transitions dependson the coefficient of P , it is equivalent to the coefficient b in Eqn.(319). It changes sign at κ = 1. Thus κ < β ( E ) is found to be β ( E ) = 2 √ Ds (cid:112) ( a + b ) κ K ( λ ) . (320) - 1-0.500.51 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 M / M s µ H (T)
FIG. 37 Hysteresis loops for the [Mn ] dimer at several field sweep rates and 40 mK. The tunnel transitions are labeled from1 to 5 corresponding to the plateaus. Adapted with permission from Tiron, et al. , 2003a
6. Free energy with magnetic field
In the previous section we considered the phase transition of the interacting dimer model at zero magnetic field.In this section we will study the influence of the staggered magnetic field on the phase boundary, the crossovertemperatures and the free energy. These analyses will be based on the potential and the position dependent massin Eqn.(294) and Eqn.(295). In Fig.(38) we have shown the plot of this potential for some values of the parameters.9
10 5 0 5 10 r u ( r ) E r r b r FIG. 38 Color online: The plot of the effective potential and its inverse as a function of r for κ = 0 . α = 0 . P F / ∆ U θ =1 θ =1 . θ =1 . θ =1 . θ =1 . FIG. 39 Color online: The numerical plot of the free energy with κ = 0 . α = 0 .
15. The phase transition from thermal toquantum regimes occurs at θ = 1 . θ = 1 . The potential has a maximum at r b = ln (cid:18) α − α (cid:19) , (321)and the height of the potential barrier is given by∆ U = U max − U min = 2 D ˜ s (1 − α ) . (322)In the presence of a magnetic field, the periodic instanton action or thermon action is given by S p = 2˜ s √ S ( P ) + β ( E − U min ); ˜ S ( P ) = (cid:90) r r dr √ a − a cosh r + a sinh r κ (1 + cosh r ) ; (323)where a = 2 α + κ − (2 + κ )( α − P (1 − α ) ); a = κ (1 + α − P (1 − α ) ); a = 2 κα. (324)The turning points are determined from the solution of the equation: a − a cosh r + a sinh r = 0 . (325)0 -2 -1 t ( s ) τ DC = 3.8x10 -6 s * exp(10.7/kT) FIG. 40 Color online: Arrhenius plot of the relaxations times τ vs. the inverse temperature for [Mn ] dimer with the modelHamiltonian ˆ H = J ˆ S A · ˆ S B − D ( ˆ S A,z + ˆ S B,z ) + gµ B µ h ( ˆ S A,z + ˆ S B,z ). Adapted with permission from Wernsdorfer, et al. , 2004
At zero magnetic field a = 0, the potential becomes symmetric hence r = − r . This action, however cannot beintegrated exactly either by periodic instanton method or otherwise. Thus, we have to resort to numerical analysis.The exact free energy can then be written as F ∆ U = 1 − P + θπ (1 − α ) (cid:112) κ (1 − α ) ˜ S ( P ) , (326)where the barrier height ∆ U is given in Eqn.(322), θ = T /T (2)0 , and T (2)0 = D ˜ sπ (cid:112) κ (1 − α ) . In Fig.(39) we haveshown the numerical plot of this free energy with κ = 0 . α = 0 .
15. In this case, the minimum of the free energyremains at ∆ U for the top three curves, however, the quantum-classical phase transition (where two minima of acurve have the same free energy) has been shifted down to T (1)0 = 1 . T (2)0 due the the presence of a small magneticfield. Thus, the presence of a longitudinal staggered magnetic field in this model decreases the crossover temperaturesas in the case of biaxial ferromagnetic spin models.
7. Phase boundary and crossover temperatures
The phase boundary with the help of Eqn.(216) yields α c = ± (cid:18) − κ c κ c (cid:19) . (327)One finds that the second-order transition crossover temperature T (2)0 at the phase boundary yields T ( c )0 = D ˜ sκ c π (cid:18)
31 + 2 κ c (cid:19) . (328)For [Mn ] dimer, the parameters are: s = 9 / D = 0 . K , and J = 0 . K (Hill, et al. , 2003; Tiron, et al. , 2003a),one finds that the value of the crossover temperature at the phase boundary is T ( c )0 = 0 . K , which is much smallerthan that of Fe molecular cluster. In Fig.(40), we show the experimental result of the Arrhenius plot of [Mn ] dimer. The plot shows that the relaxation rate is temperature-dependent above ca. 0 . K with τ = 3 . × − s and∆ U = 10 . K and below ca. 0 . K , the relaxation rate is temperature-independent with a relaxation rate of 8 × s indicating the quantum tunneling of the spins between the ground states (Wernsdorfer, et al. , 2004). The hysteresisloops in Fig.(37) show the tunneling transitions through plateaus as obtained from experimental measurement. Thestep heights are temperature independent below 400mK, which indicates quantum tunneling between the groundenergy states.1 VI. CONCLUSION AND DISCUSSION
In this review we discussed recent theoretical and experimental developments on macroscopic quantum tunnelingand phase transitions in spin systems. We reviewed different theoretical approaches to the problem of spin tunneling insingle molecule magnets and exchange coupled dimer models. It is now understood that the suppression of tunneling atzero magnetic field for half-odd integer spin system is independent of the coordinate representation but only dependson the WZ or Berry phase term. This is related to Kramers degeneracy, and its experimental confirmation has beenreported(Wernsdorfer et al. , 2002). Theoretically, it is still an open problem to determine the necessary conditions inwhich classical degenerate ground state for half-odd integer spin implies degenerate ground states in the pure quantumcase. In the presence of a magnetic field along the spin hard anisotropy axis, tunneling is not suppressed for half-oddinteger spins but rather oscillates with the field in accordance with the experimental observations.Experimental and theoretical research on single-molecule magnets have focused on the search for other molecularmagnets that exhibit tunneling and crossover temperatures. This research is expanding rapidly, and with the advancein technology, these molecular magnets have been used in the implementation of Grover’s algorithm and magneticqubits in quantum computing (Leuenberger and Loss, 2001; Tejada, et al. , 2001). Other interesting areas includetunneling of Ne´el vector in antiferromagnetic ring clusters with even number of spins (Meier and Loss, 2001; Meier, etal. , 2003; Taft, et al. , 1994). As far as we know the odd number of antiferromagnetic spin chain has not been reported.The present authors have suggested that this might give rise to solitons due to the spin frustration (Owerre andParanjape, 2014d). Most experimental research has focused on organizing the SMMs into layers with the possibilityof singling out the individual molecules(Leon, et al. , 1998, 2001).
Acknowledgments
The authors would like thank NSERC of Canada for financial support. We thank Ian Affleck, Sung Sik Lee andJoachim Nsofini for useful discussions.
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