Spin-glass instability of short-range spherical ferromagnet
aa r X i v : . [ c ond - m a t . d i s - nn ] J un Spin-glass instability of short-range spherical ferromagnet
P. N. Timonin ∗ Physics Research Institute at Southern Federal University, 344090 Rostov-on-Don, Russia (Dated: November 19, 2018)In structurally disordered ferromagnets the weak random dipole-dipole exchange may transformthe polydomain state into a spin-glass one. To some extent the properties of such phase in disorderedisotropic ferromagnet can be qualitatively described by the spherical model with the short-rangeferromagnetic interaction and weak frustrated infinite-range random-bond exchange. This modelis shown to predict that spin-glass phase substitute the ferromagnetic one at the arbitrary smalldisorder strength and that its thermodynamics has some similarity to that of polydomain statealong with some significant distinctions. In particular, the longitudinal susceptibility at small fieldsbecomes frozen below transition point at a constant value depending on the disorder strength, whilethe third order nonlinear magnetic susceptibilitiy exhibits the temperature oscillations in small fieldnear the transition point. The relation of these predictions to the experimental data for somedisordered isotropic ferromagnets is discussed.
PACS numbers: 64.60.Cn, 05.70.Jk, 64.60.Fr
The spherical model with short-range exchange shares the basic qualitative features with real isotropic ferromagnets.It has phase transition only in space dimensions d > n -component model in the limit n → ∞ [1]. So the scalar magnetization of short-rangespherical model corresponds to the magnetization module of isotropic ferromagnets and this makes this model veryuseful for the studies of qualitative features of their thermodynamics. Yet real ferromagnets have also the long-range dipole-dipole interaction. Being a weak relativistic effect it nevertheless determines crucially the nature offerromagnetic transition which usually results in the appearance of inhomogeneous polydomain state. It shows up inthe freezing of longitudinal magnetic susceptibility at the value χ = (4 πκ ) − below T c at fields H < πκM s , κ is thedepolarizing coefficient along the field direction, M s is the spontaneous magnetization [2, 3]. It is rather natural tosuppose that when some non-magnetic disorder such as structural defects or non-magnetic impurities is present in acrystal the polydomain state may transform into the spin-glass one [4].To describe the qualitative features of such spin-glass state in random isotropic ferromagnets we may turn tothe spherical model with weak long-range frustrated disorder imitating the random dipole-dipole exchange in thestructurally disordered media. The influence of such (infinite-range) disorder on the thermodynamics of the mean-field spherical ferromagnet was studied in Ref. 5. In this model the spin-glass phase instead of ferromagnetic onedo appear when disorder becomes sufficiently strong. Here we consider more realistic short-range spherical model offerromagnet with the same infinite-range frustrated random exchange. We find that contrary to the mean-field modelin the short-range one the spin-glass substitutes the ferromagnetic phase at arbitrary weak random exchange. Wealso show that the magnetic properties of this spin-glass phase in small magnetic fields have some similarity to thoseof polydomain ferromagnetic state along with some significant distinctions.The Hamiltonian of the spherical model has the form H = − S ˆ J S − HS . Here S is N -component vector subjected to the constraint S = N , H is the external field and J i,j is the matrix ofexchange integrals.Partition sum of the model can be represented as Z = a + i ∞ Z a − i ∞ dλ πi exp [ − N βF ( λ )] , (1) − βF ( λ ) = λ − N − T r ln ˆ G − ( λ ) + N − β H ˆ G ( λ ) H , (2)ˆ G ( λ ) = (cid:16) λ ˆ I − β ˆ J (cid:17) − . (3) ∗ Electronic address: [email protected] β = 1 /T is the inverse temperature. The parameter a in the integral over λ can be arbitrary provided it obeys thecondition a > βJ max , J max being the largest eigenvalue of ˆ J . Thus Eqs. (1-3) are valid for any ˆ J with the spectrumlimited from above. For the equilibrium thermodynamic potential F we have from Eq.(1) at N → ∞ F = min λ F ( λ ) = F h λ (cid:16) ˆ J (cid:17)i . (4)Here λ (cid:16) ˆ J (cid:17) is the value which provides the minimum of F ( λ ). It obeys the equation of state ∂F ( λ ) ∂λ = 0 (5)Solving Eq. (5) and substituting the λ (cid:16) ˆ J (cid:17) found into Eq. (3) we get the equilibrium potential F and can then findall thermodynamic variables of the system. In particular, we get for the average local spins h S i T = − N ∂F∂ H = β ˆ G ( λ ) H (6)When ˆ J is a random matrix we should average F over it. It can be easily done if we assume λ (cid:16) ˆ J (cid:17) to be theself-averaging quantity. Then while averaging of Eqs. (4, 5) we can just substitute λ (cid:16) ˆ J (cid:17) by its average value¯ λ = D λ (cid:16) ˆ J (cid:17)E J . Thus we get from Eqs. (2, 4) − β ¯ F ≡ − β h F ( λ ) i J = ¯ λ − Z dερ ( ε ) ln(¯ λ − ε ) + N − β H D ˆ G (cid:0) ¯ λ (cid:1)E J H , (7)where ρ ( ε ) is the average spectral density of the matrix β ˆ J , ρ ( ε ) = 1 πN lim δ → Im T r D ˆ G ( ε − iδ ) E J . (8)From Eqs. (2, 3, 5) we get the equation for ¯ λ , D (cid:0) ¯ λ (cid:1) + Q (cid:0) ¯ λ (cid:1) = 1 , (9) D (cid:0) ¯ λ (cid:1) ≡ N − T r D ˆ G (cid:0) ¯ λ (cid:1)E J , (10) Q (cid:0) ¯ λ (cid:1) ≡ N − β H D ˆ G (cid:0) ¯ λ (cid:1)E J H = N − D h S i T E J (11)The last equality in Eq. (11) follows from Eq. (6). It shows that Q (cid:0) ¯ λ (cid:1) is the Edwards-Anderson spin-glass orderparameter.Here we consider the Gaussian disorder for the exchange integrals with the mean h J i,j i = ¯ J ( r i − r j )and the deviation D(cid:0) J ′ i,j (cid:1) E = ∆ N , J ′ i,j ≡ J i,j − ¯ J ( r i − r j )We assume ¯ J ( r i − r j ) to describe the short range ferromagnetic interactions so its Fourier transform ¯ J ( k ) have amaximum at k = 0 and near it ¯ J ( k ) ≈ ¯ J − Ak . Then on a three-dimensional lattice the spectral density of β ¯ J ( k ), ρ ( ε ) = Z d k (2 π ) δ (cid:2) ε − β ¯ J ( k ) (cid:3) , would have the square-root behavior at the upper edge of the spectrum which describes the most relevant long-rangeferromagnetic fluctuations, ρ ( ε ) ∼ q β ¯ J − ε. So we choose ρ ( ε ) = 2 π (cid:0) β ¯ J (cid:1) ϑ h(cid:0) β ¯ J (cid:1) − ε i q(cid:0) β ¯ J (cid:1) − ε . (12)Here θ is the Heaviside step function. This ρ ( ε ) correctly behaves at the upper edge and makes further calculationsquite easy. The explicit form of ¯ J ( k ) appears to be irrelevant for the homogeneous external field we consider belowand all thermodynamics is determined solely by ρ ( ε ).Now we can find D ˆ G (cid:0) ¯ λ (cid:1)E J for such random ensemble where the weak infinite-range random exchange fluctuationsof arbitrary sign coexist with non-random short-range ferromagnetic interactions. Expanding ˆ G (cid:0) ¯ λ (cid:1) in the powerseries of J ′ i,j and averaging this expansion with the Gaussian distribution we find in the large N limit the followingexpression for the Fourier transform of D ˆ G (cid:0) ¯ λ (cid:1)E J ,¯ G − (cid:0) ¯ λ, k (cid:1) = ¯ λ − β ∆ D (cid:0) ¯ λ (cid:1) − β ¯ J ( k ) . (13)Then for D (cid:0) ¯ λ (cid:1) (10) we have the equation D (cid:0) ¯ λ (cid:1) = Z dε ρ ( ε )¯ λ − β ∆ D (cid:0) ¯ λ (cid:1) − ε = 2 (cid:0) β ¯ J (cid:1) (cid:20) ¯ λ − β ∆ D (cid:0) ¯ λ (cid:1) − q(cid:2) ¯ λ − β ∆ D (cid:0) ¯ λ (cid:1)(cid:3) − (cid:0) β ¯ J (cid:1) (cid:21) The solution to this equation is D (cid:0) ¯ λ (cid:1) = 2 c (cid:0) β ¯ J (cid:1) (cid:20) ¯ λ − q ¯ λ − c − (cid:0) β ¯ J (cid:1) (cid:21) , c ≡ (cid:18) ¯ J (cid:19) − (14)Eqs. (13, 14) define ¯ G (cid:0) ¯ λ, k (cid:1) . From these equations we can also find the Fourier transform of D ˆ G (cid:0) ¯ λ (cid:1)E J ,¯ G (cid:0) ¯ λ, k (cid:1) = − ∂∂ ¯ λ ¯ G (cid:0) ¯ λ, k (cid:1) = ¯ G (cid:0) ¯ λ, k (cid:1) (cid:2) − β ∆ D ′ (cid:0) ¯ λ (cid:1)(cid:3) . (15)From (8, 10, 14) we also get ρ ( ε ) = 1 π lim δ → Im D ( ε − iδ ) = 2 c π (cid:0) β ¯ J (cid:1) ϑ h c − (cid:0) β ¯ J (cid:1) − ε i q c − (cid:0) β ¯ J (cid:1) − ε (16)Thus we have all that is needed to obtain the explicit expressions for the average thermodynamic potential (7) andthe equation of state (9). Further we consider the homogeneous external field, H i = H, i = 1 , , N . It is convenient tointroduce the new variable z, < z <
1, instead of ¯ λ ,¯ λ = β ¯ J c (cid:0) z − + z (cid:1) (17)Then we have from Eqs. (9-11, 13-15, 17) the equation which defines z , h z (1 + cz ) = (1 − tz ) (cid:0) − z (cid:1) (1 − cz ) , (18) h ≡ H/T g , T g ≡ q ¯ J / , t ≡ T /T g . From Eqs. (7, 14, 16, 17) we get the average potential, − F /T g = t ln t + z + z − + zh (1 − cz ) + t (cid:18) ln z − z (cid:19) (19)It can be easily checked that Eq. (18) is equivalent to the equation ∂ ¯ F∂z = 0 and that the solution of it provides theminimum of potential in the interval 0 < z <
1. Other thermodynamic parameters can be also expressed via z . Thusaveraging Eq. (6) over random exchange we get the average magnetization M = βH ¯ G (cid:0) ¯ λ, k = 0 (cid:1) = zh (1 − cz ) , (20)while from (9, 10, 17) we get for the equilibrium value of the Edwards-Anderson order parameter Q = 1 − tz. (21)Also from Eq. (19) we obtain the entropy S = 12 (cid:18) tz − z (cid:19) , (22)and the heat capacity C = 12 (cid:20) (cid:0) − z (cid:1) d ln zd ln t (cid:21) . (23)Thus Eqs. (20-23) supplied with the solution to Eq. (18) for z = z ( t, h, c ) give full description of the thermodynamicsof the model. Here we should note that parameter c defined in Eq. (14) determine the relative strength of theshort-range ferromagnetic bonds. It varies in the interval 0 ≤ c ≤ c = 1 corresponds to the pure short-rangeferromagnet while at c = 0 only random infinite-range glassy exchange is present in the system. So at c = 1 wehave ordinary ferromagnetic transition at t = 1 , h = 0 with anomalies usual to the pure spherical model. In this case Q = M . Yet at all c < t = 1 , h = 0.Indeed, when h → z → /t for t > z → t <
1. So at all c < h = 0 M is zero, butspontaneous Q appears at t < Q = 1 − t . This is in sharp contrast with the model where instead of short-range¯ J ( k ) the infinite-range mean-field ferromagnetic interaction of the form ¯ J ( k ) = δ k , ¯ J/N is introduced [5]. Thenspin-glass transition substitutes the ferromagnetic one only at c < / T g (18) on the relative strength (∆ / ¯ J ) of frustrated disorderis shown in Fig. 1. FIG. 1: (color online) The dependence of T g on the relative disorder strength ∆ / ¯ J . At h = 0 we have for all tS = (cid:26) (cid:0) − t (cid:1) , t > (cid:0) + ln t (cid:1) , t < C = (cid:26) t , t > , t < χ ≡ ∂M∂h = ( t ( t − c ) , t > − c ) , t < FIG. 2: (color online) Field dependence of magnetization for c = 0 . t = 3 (solid line), t = 1 (dashedline), t = 0 . So the zero-field thermal properties of the model do not depend on c , while the magnetic susceptibility χ is essentiallydefined by it. Fig. 2 shows the field dependence of magnetization at various temperatures. Note the steep rise of M at low fields below T g . Here the slope of M ( h ) in small fields is limited by the value of zero-field susceptibility,(1 − c ) − , while in the polydomain ferromagnet it is limited by χ = (4 πκ ) − . We can easily find M ( t, h ) from Eqs.(18, 20) for small fields h ≪ t (1 − c ) , M = h h (1 − c ) + (cid:0) − c (cid:1) (cid:16) τ + p τ + bh (cid:17)i − τ ≡ t − t , b ≡ c t (1 − c ) . (24)Fig. 3 presents this M ( t ) and χ ( t ) in small fields for c = 0 .
9. They are rather similar to those of pure ferromagnetundergoing the transition into polydomain state albeit with the disorder-dependent saturation values.
FIG. 3: (color online) Temperature dependencies of M (a) and χ (b) for c = 0 . h = 5 × − (dotted lines),3 × − (dashed lines), 10 − (solid lines). Yet more spectacular anomalies are exhibited by the nonlinear magnetic susceptibilities of the model. They areknown to diverge at spin-glass transition in zero field in various mean-field spin-glass models [6] including the sphericalone [7]. In the last case these divergences result from the specific non-analyticity of M ( t, h, c ) at t = 1 , h = 0 in Eq.(24) which also give rise to temperature and field oscillations of nonlinear susceptibilities near the transition. Neartransition point at c = 1 and for | τ | ≪ − c we get from Eq. (24) two first nonlinear magnetic susceptibilities, χ ≡ − ∂ M∂h = 2 b h (cid:0) τ + 2 bh (cid:1) ( τ + bh ) / χ ≡ − ∂ M∂h = 6 b τ ( τ + bh ) / (25)They exhibit highly anisotropic behavior near the singular point τ = 0, h = 0. In the polar coordinates defined as r = p τ + bh ϕ = tan − √ bhτ ! we have χ = 2 b / sin ϕ (cid:0) ϕ (cid:1) , χ = 6 b r cos ϕ (26)Thus at ϕ = 0( h = 0) χ = 0 , χ = 6 b | τ | − , while at ϕ = π/ τ = 0) χ = 2 b / sign ( h ), χ = 0. The behavior of χ and χ near the singular point τ = 0 , h = 0 is shown in Fig. 4. FIG. 4: (color online) Nonlinear susceptibilities χ /b / (a) and χ /b (b) near the singular point τ = 0 , h = 0.FIG. 5: (color online)(a) - field dependencies of χ /b / at τ = 0.01 (solid line), 0.03 (dashed line), 0.05 (dotted line); (b) -temperature dependencies of χ /b at small fields h = 0 (solid line), 0.001 (dashed line), 0.002 (dotted line). These complex anomalies result in specific field dependence of χ and temperature oscillations of χ as seen in Fig.5. The behavior of nonlinear susceptibilities similar to that of Fig. 5 is observed in isotropic ferromagnet N d . Ba . M nO [8] and in the polycrystalline samples of RuSr GdCu O [9]. In the toroidal polycrystallinesamples of La . Ba . M nO with demagnetization factor κ ≈ χ ( T ) same as in Fig. 3(b) is foundmanifesting the transition into the glass state [10]. There are many other examples of such step-like behavior of χ ( T )in disordered isotropic magnets, see, for example, Refs. [11], [12]. But it is often impossible to check the relation χ = (4 πκ ) − below T c to distinguish between the polydomain and the spin-glass states as some experimental paperslacks the values of κ calculated from the sample shape. It is quite possible that such check will show that manyallegedly polydomain ferromagnets are actually the spin-glasses.Yet now it is not clear if the present result on the spin-glass instability of spherical magnet does apply to thereal dipolar Heisenberg magnets which may have some threshold disorder strength to become the spin-glasses. Toresolve this issue further theoretical studies of the role of random dipole-dipole interaction in isotropic ferromagnetsare needed.I gratefully acknowledge the useful discussions with V.B. Shirokov and E.D. Gutlianskii. [1] H. E. Stanley, Phys. Rev. , 718 (1968).[2] V. G. Bar’yakhtar, A. N. Bogdanov and D. A. Yablonskii, Usp. Fiz. Nauk. , 47 (1988).[3] P. J. Wojtovich and M. Rayl, Phys. Rev. Lett. , 1489 (1968).[4] A. Aharony and M. J. Stephen, J. Phys. C , 1665 (1981).[5] J. M. Kosterlitz, D. J. Thouless and R. C. Jones, Phys. Rev. Lett. , 1217 (1976).[6] K. Binder and A. P. Young, Rev. Mod. Phys. , 801 (1986).[7] L. F. Cugliandolo, D. S. Dean and H. Yoshino, cond-mat/0612086.[8] V. A. Ryzhov, A. V. Lazuta, V. P. Khavronin, I. I. Larionov, I. O. Troaynchuk and D. D. Khalyavin, Sol. St. Comm. ,804 (2004).[9] M. R. Cimberle, R. Masini, F. Canepa, G. Costa, A. Vecchione , M. Polichetti and R. Ciancio, Phys. Rev. B , 214424(2006).[10] A. B. Beznosov, V. V. Eremenko, E. L. Fertman, V. A. Desnenko and D. D. Khalyavin, Low Temp. Phys. , 762 (2002).[11] I. Abu-Aljarayesh and M. R. Said, JMMM , 73 (2000).[12] S. Singh, G. Sheet, P. Raychaudhuri and S. K. Dhar, Appl. Phys. Lett.88