Green’s Function Theory in Magnetism

The Green’s function method is a very general method in quantum field theoiy. It can be used for various problems in many areas such as condensed matter, nuclear physics and elementary-particle physics [115]. The general formulation is rather complicated, abstract, not suitable for a quick application. For our purpose, we present in this chapter a simplified version which, from the beginning, aims at an application of the method to systems of interacting spins. This formulation does not need a high level of knowledge in quantum field theoiy. A basic level in quantum mechanics is enough to understand and to apply the method.

As seen below, the Green's function method is an alternative technique to study spin waves, in addition to the theory of magnons presented in Chapter 3. We have seen that the theory of free magnons is exact at veiy low temperatures. However, at higher temperatures we have to take into account magnon-magnon interactions. For example, we have to treat terms of higher orders in the Holstein-Primakoff expansion [see Eqs. (3.32)-(3.34)] to modify the harmonic magnon spectrum given by Eqs. (3.39) and (3.40). This is a laborious task [91]. An alternative and, by far, simpler way to take into account some correlations is the Green’s function method which can treat the whole temperature range going from

Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com low-temperature phase up to the transition temperature T_c. This is possible because Green's function includes even at the lowest level some spin-spin correlations. We can therefore follow the evolution of the magnon spectrum with increasing temperature. However, since all correlations are not hierarchically included, its validity is still an open question. Note that there is at present no other better method to calculate the spin wave spectrum at finite temperatures.

Numerous applications of the Green's function method in surface magnetism are shown in Part II of this book.

Green’s Function Method

Definition

Let Л(Г) and B(t') be two operators in the Heisenberg representation at times t and t', respectively. We define the retarded Green's function by

and the advanced Green’s function by where

[Л(С), being the commutation relation and < • • ■ > denoting the thermal average. In spite of the complicated notation in its definition given above, Green's function is just a thermal average of a commutation relation between two operators. Green's function is connected, as will be seen below, to the physical properties of the system. So, depending on what we want to study, we choose the operators Л(Г) and B[f'). For instance, to study an electron gas, we can choose Л(Г) = o,⁺ (t) and B(f') = a,(f') where a,⁺(f) and are creation and annihilation operators of electron state /, to study phonon excitations we can choose Л (t) and B(t') as phonon creation and annihilation operators, and for a spin system we can choose Л (t) = and B(t) = S“(t') where S,⁺(t) and are spin

operators (cf. Chapter 3).

Green's function contains information on the system, in particular on the elementaiy-excitation spectrum. In what follows, we present the formulation of the method as it is applied to a spin system so that the reader can associate mathematical tools presented here to physical meanings of a phenomenon that has been seen in Chapter 3.

Formulation

We consider the following functions:

The time Fourier transformation of Fba (t' - t) is written as

We show below that where p = [квТ)~^г.

Demonstration: We use essentially the circular permutation properties of operators in Tr[- • • ] and Л(Г) = e' ^HtAe~^,Ht:

We have thus

The Fourier transformation of F_AB[t — t⁷) is

Using this formula for Green’s function we obtain

where we have used the following expression:

Integrating on (t — t') and using the formula we arrive at

In the same manner, we obtain

The difference between the retarded and advanced Green's functions is the sign in front of ie in the denominator. We write from now on these functions without superscripts r and a, but we distinguish them by the sign in their arguments. Combining these functions we write

We shall use this relation to calculate the spin-spin correlation in the following sections.

Ferromagnetism by the Green’s Function Method

We consider the Heisenberg spins with a ferromagnetic interaction between nearest neighbors. The Hamiltonian is given by

where the spin operators obey the commutation relations

For a ferromagnet, we define the following Green's function:

We set t' = 0 hereafter to simplify the writing.

Equation of Motion

The equation of motion for is written as

where the sum on p is performed on the vectors connecting spin S/ to its nearest neighbors.

Remark: We have used the identity [AB, С] = [А, С] В + A [£?, C] then (4.19) and (4.20) to calculate ['И, S,⁺] (see Problem 2 in Section 4.6).

We see that the right-hand side of Eq. (4.22) contains Green's functions of higher order with three operators. Writing the equation of motion for these functions will generate functions of five operators. In a first approximation, we reduce higher- order Green’s functions by using the so-called Tyablikov decoupling scheme [346]

We obtain on the right-hand side of Eq. (4.22) Green's functions of the same order as the one initially defined in (4.21). This decoupling bears the same spirit as the mean-field theory (Chapter 2): replacing an operator in a product by its average value, i.e., by a ''c-number.” This approximation is called sometimes "random-phase approximation” (RPA). The RPA is hierarchically higher than the mean-field theory in the sense that one operator in a three-operator product is replaced by its average value in the RPA while in the mean-field theory one operator in a two-operator product is replaced.

We notice that Sf is a constant of motion. Therefore, in a homogeneous system, we can suppose that

We obtain then

Dispersion Relation

The time Fourier transformation of Eq. (4.26) gives

We use next the following spatial Fourier transformation:

where the sum on the wave vector к is performed in the first Brillouin zone of N states. We obtain

where Z is the coordination number. We get where

As seen in Eq. (4.30), the singularity of the Green’s function is at hw = e_k. This determines the eigen-energy hw_k of the magnon of wave vector k:

This is the ferromagnetic magnon dispersion relation which is to be compared to Eq. (3.40) obtained by the theory of magnons:

We see that apart from the factor 2 due to the model defined by (3.30) the only difference is that < S^z > appears in (4.33) instead of S. As < S² > depends on the temperature, the magnon spectrum varies with T. We calculate < S² > in the following.

Magnetization and Critical Temperature

We write

where we have used (4.19). For S = 1/2, we have (Sf)² = 1/4 (see Pauli's matrices in Chapter 1). We get then

from which we have

We now calculate < Sj~S+ >. We have

from which, for t = 0,

Using (4.17) we write

Replacing G_k(ш + /e) and G_k(a>— ie) by (4.30) and using (4.14), we obtain

Equation (4.36) becomes

This is an implicit equation for < Sf >: The right-hand side contains also < Sf > in w_k. Therefore, we have to solve this equation by iteration to get < Sf >.

At low temperatures, we follow the same method as that used for (3.46). We obtain the following result:

where o, (/ = 1, 2, 3, • • •) are constants. We notice that the T³ term does not exist in the low-temperature expansion (3.49) of the theory of magnons. It may be due to the Tyablikov decoupling (4.23) and (4.24), which is the only approximation used here. To improve the decoupling, the reader is referred to the references [222, 340].

At high temperatures, < Sf >-*■ 0. An expansion of e^^/kUk yields

At T = T_c, < Sf >= 0. Equation (4.43) gives then

In the same manner, for spins of amplitude S Ф 1/2, using (Sf)² = (S/ • S/)/3 = S(S + 1)/3 in Eq. (4.34) one obtains

The transition temperature is given by

Remark: When transforming the sum on к in (4.44) into an integral, we clearly see that the integral diverges at small к for dimensions d = 1 and 2 (see the discussion in the conclusion of Chapter 3). As a consequence, T_c = 0 for these low dimensions, in agreement with exact results [231] (see Chapter 5). For d = 3, the values of T_c numerically calculated by (4.44) are

• • Simple cubic lattice: квТ_с/) ~ 0.994
• • Body-centered cubic lattice: keT_c/J ~ 1.436
• • Face-centered cubic lattice: ksTJ) ~ 2.231

These values are much lower than those given by the mean-field theory: 3, 4 and 6, respectively. The present results are better than those of the mean-field theory because in the Green's function method, fluctuations due to spin-spin correlations are partially taken into account.

Antiferromagnetism by the Green’s Function Method

In a spin system with an antiferromagnetic interaction, we have to define two Green's functions, the first one concerns the correlation between spins of the same sublattice and the second one expresses the correlation between spins belonging to the two sublattices.

We consider a system of Heisenberg spins with an antiferromagnetic interaction J between nearest neighbors

where the spin operators satisfy the commutation relations (4.19) and (4.20). We define the following Green's functions:

where /, /' belong to sublattice of f spins and m belongs to sublattice of i spins. For writing simplicity, we take t = 0.

The equations of motion for G//-(t) and are written as, after the Tyablikov decoupling,

where we have used the fact that nearest neighbors of a / site are on m sites and vice versa, namely 1 + p is a site m of the 4- sublattice and that m + p is a site / of the t sublattice. As in the ferromagnetic case we have supposed < Sf > = < S^zm+- > = - < S^zn > = — < S^z+~ > = < S^z >, independent of the lattice site. We use the Fourier transformations

where the factor 2/N comes from the fact that each sublattice has N/2 sites. The equations (4.50) and (4.51) become

where A_k = ZJ < S^z > and B_k = ZJ < S^z > yu. The solution of (4.54) and (4.55) is

where

The singularities of these functions are ±e_k. The magnon mode к has thus two opposite precessions. This degeneracy comes from the reversal symmetry of the two sublattices. The antiferromagnetic dispersion relation is

It is noted that

• • for small k, one has e_k a к instead of k² of the ferromagnetic case (cf. previous paragraph and Chapter 3)
• • as in the ferromagnetic case, the magnon spectrum depends on the temperature via < S^z > in hw_k.

To calculate the magnetization we follow the same method used in the ferromagnetic case. We have

Replacing /_k(a>and f(_k(a>each with its corresponding Green's function by using (4.17), we obtain

Using these relations in (4.60) and (4.61), we have

For 5 = 1/2, we replace < Sf S₍⁺ > by 1/2— < S^z > in (4.64). We finally arrive at

This implicit equation for < S^z > should be solved by numerical iteration.

At low temperatures, expanding the right-hand side of (4.66) when T -*■ 0 [p -»■ oc), one has

where AS is the zero-point spin contraction (at 7 = 0) (see Chapter 3). As seen, the temperature dependence of < S^z > is not the same as in the ferromagnetic case. This is a consequence of the linear dependence on к of e_k at small k.

The Neel temperature T_N is calculated by letting < S^z >-> 0 in

(4.66). One has

Green’s Function Method for Non-Collinear Magnets

In frustrated spin systems, the competition between different kinds of interaction can give rise to ground-state spin configurations which are not collinear. The incompatibility of some lattice geometries with antiferromagnetic interaction yields also non-collinear spin configurations as those in the antiferromagnetic triangular lattice, in the antiferromagnetic face-centered cubic lattice or hexagonal- close-packed lattice. Such frustrated systems show many striking properties. The reader is referred to Ref. [85] for recent reviews on many aspects of the frustration. For non-collinear magnets, the Hamiltonian can be expressed in the local coordinates. This has been done in Eq. (3.124). We define the following Green's functions:

With the use of the Tyablikov decoupling [346], the equations of motion of these functions are given by

Note that the sinus terms in Eq. (3.124) are canceled out upon summing over symmetric neighbors at each lattice site (inversion symmetiy) in the above equations. The next steps are similar to those used in the antiferromagnetic case in the previous section: Using the Fourier transformations, the solution of the resulting coupled equations gives the dispersion relation which allows us to calculate the magnetization and the transition temperature. Some applications are given as problems in Section 4.6.

Conclusion

We have presented in this chapter the Green’s function method, which is used to investigate spin waves in ferromagnetic, antiferromagnetic and non-collinear magnets. The method gives the temperature-dependence of the magnon spectrum and a compact expression which allows us to calculate the magnetization up to the transition temperature. To keep the method simple and tractable, we have used the Tyablikov decoupling scheme to reduce higher-order Green's functions. There is room for improving it but we will lose the simplicity of the method. The dispersion relation can be explicitly obtained in simple cases. However, for complicated systems where we have to define several Green's functions, we obtain a system of coupled linear equations which can be numerically solved to obtain the dispersion relation which is used to compute the temperature dependence of various physical quantities. Such applications are treated in part II of this book.

We note that the method, though efficient, is not accurate enough to allow us to calculate critical exponents of the phase transition. However, it can detect first-order transitions as well as multiple phase transitions of the system at different temperatures as seen in Ref. [285] and in part II.

Problems

Problem 1. Give proofs of the formula (4.13).

Problem 2. Give the demonstration of Eq. (4.22).

Problem 3. Helimagnet by Green’s function method:

Consider a crystal of simple cubic lattice with Heisenberg spins of amplitude 1/2. The interaction Ji between nearest neighbors is ferromagnetic. Suppose that along the у axis there exists an antiferromagnetic interaction ]₂ between next nearest neighbors, in addition to/j.

(a) Follow the method in Section 3.4, show that the ground state is helimagnetic in they direction if I/2I//1 is larger than a critical value Determine a_c.

• (b) Let в be the helical angle between two nearest neighboring spins in the у direction. Express the Hamiltonian in terms off?.
• (c) Define two Green’s functions which allow us to calculate the spin wave spectrum using the RPA decoupling scheme. Calculate that spectrum.
• (d) Show that this spectrum is reduced to the ferromagnetic and antiferromagnetic spectra when /2 = 0 and /1 ^ 0.

Problem 4. Apply the Green’s function method to a system of Ising spins S = ±1 in one dimension, supposing a ferromagnetic interaction between nearest neighbors under an applied magnetic field.

Problem 5. Apply the Green’s function method to a system of Heisenberg spins on a simple cubic lattice, supposing ferromagnetic interactions between nearest neighbors and between next-nearest neighbors.

Problem 6. Magnon spectrum in Heisenberg triangular antiferro- magnet: Green’s function method

Calculate the magnon spectrum of a triangular lattice with spins 1/2 interacting with each other via an antiferromagnetic interaction between nearest neighbors. Estimate numerically the zero-point spin contraction. Guide: Use the spin configuration obtained in Problem 6 of Section 3.6.