Spin-Wave Theory of Quantum Antiferromagnets

Spin-wave theory of quantum antiferromagnets

Abstract

All high-temperature superconducting cuprate compounds are antiferromagnetic insulators before doping, however neither the microscopic mechanism responsible for this phenomena or the simplest Hamiltonian which describes it is not yet agreed upon. Antiferromagnetism in those insulators is described as two alternating spin alignments on a lattice, which is then divided up into two interconnecting sub-lattices: one with all spins in one direction, say zˆ and the other with all spins  zˆ . After doping, those cuprate compounds become superconducting still with characteristics of the strong spin fluctuations. Therefore, understanding quantum antiferromagnetic interactions is important to fully explain high-temperature superconductivity. In this report, the quantum mechanics of antiferromagnetism using the Heisenberg model of many interacting spins on a lattice will be investigated. This will then be extended to infinite systems on a lattice by a many-body technique, the spin wave theory. The spin-wave theory can be visualised as many configurations of wave-like structures which align the spins along the lattice. The methods of bosonization of spin operators, diagonalization of an approximate quadratic many-boson Hamiltonian after Fourier transformations will also be explored. Numerical results for the ground-state energy, excitation energy, and spin-wave velocity along with other measures will be found by theoretical methods and compared to recent experimental results.

1. Spin waves in an antiferromagnet

In this section, we follow the method used by Anderson [1]. Here, the zero point energy of the quantised vibrations of the lattice is taken into consideration, in contrast to other studies where the significance of the zero point energy has not been realised. Coincidently, the zero point energy can be ignored in ferromagnetism since classically, we can considered all spins to be aligned parallel giving the energy from each nearest 2 neighbour interaction as  JSi Sj  Sc where J is the coupling exchange integral and

Sc  SS1 is the classical angular momentum used for atoms with a spin quantum number S . If this classical approximation is taken, the zero point energy changes the value of the ground state energy, increasing it to the correct value  JS2 . The exact solution of the ground state of the antiferromagnetic linear chain of spin 1 2 atoms has been found by Bethe [2], however higher dimension geometries (e.g. the square and simple cubic lattices) necessitate computational numerical integration, found by Wolfram Mathematica [3]. In spite of various treatments of the antiferromagnetic ground state, no rigorous solution has been found and none has been fully verified by experimentation, although there is supporting evidence that ordered antiparallel arrangements do exist in some substances. Observed magnetisation of the sublattices of

1 various substances, as well as the connection to superconductivity cause the subject to still be of interest after approximately a century of reports.

1.1. The Hamiltonian for spin waves

We will start this treatment assuming that the Heisenberg exchange coupling is responsible for the phenomena and using the familiar Hamiltonian for interaction between spins:

H  J S j Sk , (1) j,k where J  0 , and j,k indicates that the sum is to be taken over nearest neighbours j and k, which can be thought of as D-dimensional vectors where D is the dimensionality of the space we are considering.

In the derivation of this “semi-classical” description of the spin waves, we will assume that the state in which antiferromagnetism exists does not greatly depart from the classical case of the ground state in which the spins in one sublattice point in precisely the same direction and the spins of the second sublattice point opposite. This assumption cannot be pre-emptively justified by external reasoning as it can in ferromagnetism, but will be used on the condition that it is internally consistent with the theory and coalesces with other physical considerations. Following this assumption, we write the z components of the spins as:

Szj  S, Szk  S, (2) where we adopt the notation of labelling the atoms on one sublattice as j and the atoms on the other sublattice as k. Now:

2 2 2 2 2 2 1 2 S S  S x  S y  S z  Sc  SS 1  S z  Sc 1 S x  S y  2SC  , which can be binomially expanded to give expressions of the z component spin about the values in (2) as:

2 2 2 2 Szj  Sc  Sxj  S yj  2Sc , Szj  Sc  Sxj  S yj  2Sc . (3)

By simple substitution of (3), the Hamiltonian (1) can be shown to be

1 1 H   ZNJS 2  ZJ S 2  S 2  S 2  S 2  J S S  S S . (4) c  j  xj yj  k  xk yk   xj xk yj yk  2 2 j,k where N is the number of atoms on the lattice. This Hamiltonian is valid to first order in 2 2 the small terms Sx ,S y and smaller terms are ignored.

Two distinct sets of spin waves are now introduced, one set for each sublattice:

S  2S N exp ir  j Q , S  2S N exp  ir  j P , (5a) xj r   r yj r   r S  2S N exp  ir k R , S   2S N exp ir  j S , (5b) xk r   r yk r   r all summed over the wavenumber r. The corresponding inverses are given by:

Q  2 NS exp  ir  j S , P  2 NS exp ir  j S , (6a) r  j   xj r  j   yj R  2 NS exp ir k S , S   2 NS exp  ir k S , (6b) r k   xk r k   yk

It should be noted that the wavenumber r ranges over N/2 values between  and , altogether giving 2N coordinates; i.e. considering the linear chain of atoms of length N:

r  2m N, m   N 2,  N 2  2 , 0 N 2  2, N 2 (7)

Verifying the commutations between the expressions (6a) and (6b):

Q , P  2 NS exp  ir  j S , 2 NS exp ir j S  r r    j   xj  j   yj     2 NS exp ir  j  r  j S xj , S yj  j j

   2 NS exp ir  j  r  j i jj S zj j j

 2 NS N rr 2iS  i rr , where we have used the standard relation exp i r  r  j  N 2, and our  j     rr assumption (2). Similarly we can write: Rr , Sr   i rr and that the commutations between Qr and Qr , Rr and S r are all zero. Recalling the Hamiltonian (4), we note that the quadratic terms still need to be evaluated:

S 2  2S N exp i r  r  j Q Q  2S N N 2 Q Q  S Q2 , (8) j xj      r r  rr  r r  r r,r, j r,r r and for the cross terms:

   S xjS xk  S yjS yk   2S N  expir  jQr exp irkRr  exp ir  jPr expirkSr  j,k j,k  r r r r 

  S expir j  kQ r Rr expir k  jPr Sr . j,k Here it should be noticed that as the summation is taken over nearest neighbours, then the summations over the differences j  k and k  j are equivalent, hence we write:

, (9) S xjS xk  S yjS yk SQr Rr  Pr S r  expir  k  j  2D r SQr Rr  Pr S r  j,k r k j r where we define:

2D r expir k  j, (10) k j

where D is defined as previous. The quantity  r has different definitions dependant on the geometry of the space we are working in, but for the linear, square and simple cubic lattices we have:

D cosr i , (11)  r   i1 D

for the D components ri of the wave-vector r . The Hamiltonian can now be expressed, in spin wave coordinates, as:

2 2 2 2 2 , (12) H  NDJSc  DJSPr  Qr  Rr  Sr  2 r Qr Rr  Pr Sr  r where we have used the fact that for the lattices we are considering, Z  2D where Z is the number of nearest neighbours. The Hamiltonian (12) is not quite in a useful set of coordinates, but can be transformed into a normal coordinate system by using the canonical transformation [1]:

1 1 Pr  p1r  p2r , Qr  q1r  q2r , (13a) 2 2

1 1 Sr  p1r  p2r , Rr  q1r  q2r , (13b) 2 2

2 2 2 2 2 . (14)  H  DNJSc  DJSp1r 1 r  q1r 1  r  q2r 1 r  p2r 1  r  r

This Hamiltonian is now of a more useful and familiar form, possessing terms of a harmonic oscillator nature. Comparing (14) to the standard harmonic oscillator Hamiltonian:

p2 H   m 2q2 , eigenvalues E  2n 1 , (15) HO m n allows us to write:

p2 1  DJSp2 1    DJS1  (16a) m 1r r m r

DJS m 2 q 2  DJS q 2 1     2   DJS2 1  2  (16b) 1r r m r

2 2 1 2 2 1 2  H  DNJSc  DJS2n1r 11  r   2n2r 11  r  . (16c) r

For the ground state of the system, with all nr  0 , the energy is given by:

2 2 1 2 Eg  DJNSc  2DJS1  r  , (17) r where it can be seen that the second term in (17) is the energy due to the spin waves of the system, with the frequencies of the spin waves given by:

2 1 2 r  DJS1 r  (18)

The dispersion relation of the spin waves is sinusoidal and the first Brillouin zone is shown in figure 1. Recalling that 2 for small r, corresponding to long  r  cosr ~1 r wavelength, then it follows that r ~ r , i.e. a linear dispersion. This dispersion relation is different to ferromagnetism with is quadratic in the wavenumber.

Figure 1. The dispersion relation of spin waves in 1-dimension. It should be noted that for small wavenumber, i.e. long wavelength, that the dispersion is linear in wavenumber r.

Noting that for a harmonic oscillator, the average kinetic and potential energies are equal and each equivalent to half of the total energy, then it can be written that:

1 1 q 2 1    p 2 1    1  2 2 1r avg r 1r avg r 2 r

1 1 1 1 1 2 2 2 2 2 1   1  2 1  2 1 1   2 1 1    q  r  r r   r  , p   r  (19) 1r avg   1r avg   21  r  21  r  2 1  r  2 1  r 

Now taking the limit of long wavelength, r  0 :

1 1 1 2 2 2 2 2 2 1 1 1 r 2 1  2  r 2  1  4  p        1 . 1r  2   2   2  avg 2 1 1 r 2 2  r 2  2  r 

Enforcing the long wavelength limit gives:

1 2 2 1  4  1 p1r     , (20) avg 2  r 2  r

i.e. the mean amplitude per quantum of p 2 diverges as the wavelength for long 1r avg wavelengths.

1.2. Quantum corrections

Now that an expression has been found for the ground state energy (17), it can now be computed by noting that the values of r are dense according to (7) for large lattices approximation a continuous wavenumber, allowing the discrete summation to be replaced by an integral, such that:

1 2 1 N 1 2 D N    D cosr   2 N 1  2 2  1  2 2  dr dr 1  i    I , r r 2 r 2  1 D  D 2 D avg    i1  

1 2    D cosr   2 which defines the integral: I  2 D dr dr 1  i   . (21) D  1 D  D    i1  

For the linear chain D 1, and the integral becomes easily solvable:

1  1 1  2 I  1 cos2 r 2 dr  sin r dr  . (22) 1     2  2  

However, for two and three dimensions, the integral becomes more complicated and can be solved by expanding the term in the square root and integrating higher terms, or by computation means. In this paper, the latter has been chosen and the integrals have been numerically solved in Wolfram Mathematica giving:

I2  0.842, I3  0.903 (23)

Recalling the ground state energy (17) and rewriting it in terms of (21):

2 2 2  1  Eg  DJNSc  DJSNID  DJNS  S  SI D  DJNS 1 1 I D  , (24)  S  and evaluating for one, two and three dimensions using 2  0.363 the Eg  JNS 1 , results (22) and (23). D1  S 

2  0.158 (25) Eg  2JNS 1 , D2  S 

2  0.098 Eg  3JNS 1  D3  S  6

These energies (25) fall between the rigorous limits that were found using the variational method [4].

In order to parameterise the state of long-range order of the antiferromagnetic lattice, the total spin of the two sublattices is considered as it is the physical quantity that it is measured in neutron diffraction experiments. In these experiments, it is the average spin per atom which is parallel to the total spin on the sublattices that is measured, and in turn is related to the total spin of the lattice. The total zˆ component of the spin of each sublattice can be found by summing the spin (3) over the sublattices:

 S 2  S 2  1 S 2  S 2 S 1  S  S  xj yj   NS  xj yj ,  z t  zj  c  c  (26) j j  2Sc  2 j 2Sc 1 where Sz t is the total zˆ component of the spin located on sublattice 1. Now from (8) and (13), it can be written: S S 2  S 2   S Q 2  P 2   q 2  p 2  q 2  p 2  q q  p p . (27)  j xj yj r r r 2 r 1r 1r 2r 2r 1r 2r 1r 2r

Thus the total spin of sublattice 1 is

1 S S 1  NS  q 2  p 2  q 2  p 2  q q  p p . (28)  z t c r  1r 1r 2r 2r 1r 2r 1r 2r  2 4Sc

The total zˆ component spin of the whole lattice is not a constant of motion, however the summation of the zˆ component of the total spin on the two sublattices is a constant which can be shown from the original Hamiltonian (1). This inconsistency restricts us to using average values of the spins represented by (28). Noting that the last two terms of (28) disappear on the average and substituting in (19):

 1 1  2 2 1 1 S 1   1    1 S 1 S  NS   r    r   NS  . (29)  z t c   c  1 avg 2 4S 1   1   2 4S c r  r   r   c r 1  2    r 

Defining the quantity J D :

1 2  2 1    D cosr   2 J   2 D dr dr 1  i   , (30) D N  1  1 D  D r 2    i1   1  r   

1 For the linear chain, J1  sin r with diverges as r  0 which suggests that the assumption that a long range order exists in the linear chain is incorrect and that all

7 components of the sublattice spin are not finite, causing the method to break down and is consistent with the results of Hulthén and Bethe [3]. Similar to how the values of (23) were computed, we find:

J2 1.39, J3 1.16. (31)

From (29):

1 1 SN N  1 1  S z t  NS c  J D  S   J D  , (32) avg 2 2Sc 2  2 2 

where Sc  S 1 2 . Find the values of (29) for two and three dimensions:

N N S z t  S  0.197, S z t  S  0.078. (33) avg D2 2 avg D3 2

While these corrections to the spins are small and are expected to become smaller for Z  6 , it may be possible to observe these corrections in neutron diffraction experiments [5]. As average values for the spins have been taken, it can be said that while the spins of the two sublattices are pointing in opposite directions while in the ground state, the spins cannot be “pinned down” in any way, and there is not a defined direction which the spins prefer to point in. It can be thought that there is an equal probability that the spins are pointed in all directions. The pair of spins are not restricted such that they can rotate freely in space and can be physically seen by introducing a form of anisotropy, which every real lattice will possess. The anisotropic terms in the Hamiltonian can be expressed by:

H  K S 2  S 2  S 2  S 2  KS Q2  R2  P2  S 2 (34) anis  j  xj yj  k  xk yk  r  r r r r  where K is the anisotropy energy per atom, the Hamiltonian makes the zˆ axis the preferred direction and has been expressed in spin-wave coordinates by (5). Adding this anisotropic term to the Hamiltonian (14), it can be shown that the quantum amplitudes (19) gain a term in K : 1 1 2 2 2 2 1  1 K DJ  r  2 2 1 1 K DJ  r  (35) q1r  p2r    , p1r  q2r    , avg avg 2 1 K DJ   avg avg 2 1 K DJ   r   r  and it can be shown that the averaged difference in the xˆ components of the spins is:

1 2  DJ  2 S  S  NS , (36)  j xj k xk    avg  2K  if K is small. This can be thought of as saying that the root mean square values of the components of the spins is small for K ~ N and is therefore negligible compared to the zˆ components of the spins, so that may be considered as a convergence factor. Therefore,

8 on the average, the total spins of the sublattices is well approximated by the values given by (33).

2. Bosonisation of the spin waves

While these corrections of Anderson’s to the energy and spins are valid, they can be improved upon by methods employed by Kubo [6] and Oguchi [7] where the boson annihilation and creation operators are introduced. Consider a different form of the Hamiltonian (1) given by:

z z H  2 J Sl Sm  h1 Sl  h2 Sm , (37) l,m l m

where h1 and h2 are the magnetic fields on the two sublattices, which do not necessarily z have to be the same, and Sk is the zˆ component of the spin at the kth atom position. The two sublattices will be considered independently as before so a set of spin operators are required for each sublattice, i.e. for the sublattice l of “up” spins and the sublattice m of “down” spins. Performing a Holstein-Primakoff transformation on the usual spin operators gives the spin raising and lowering operators in terms of the boson annihilation and creation operators a, a on sublattice l and b , b on sublattice m :

  z Sl  2S fl Sal , Sl  2Sal  fl S, Sl  S  al al , (38a)

  z Sm  2Sbm  fm S, Sm  2S fm Sbm , Sm  S bm bm  S  nm , (38b)

1 2  n  2 n 1 n f S  1 j   1 j  j  (38c) j     2  2S  2S 32 S

where the semiclassical result is obtained by taking f j S 1. The operators a, a , b and b all follow the commutation relations:

c ,c   c c c c   , c  a,b . (39)  j l  j l l j ij

Noticing that the form of (38) suggests that (37) will lead to coupled equations in the boson operators of the two sublattices, therefore it is convenient to introduce the Fourier transforms of the boson operators similar to (5):

ak  2 N l expiklal , ak   2 N l exp iklal , (40) bk  2 N m exp ikmbm bk   2 N m expikmbm ,

Also, we introduce the operators k ,k  and k ,k  via the Bogoliubov transformation:

a   cosh   sinh , a    cosh   sinh , k k k k k k k k k k (41) bk   k sinhk   k coshk , bk   k sinhk   k coshk ,

which diagonalise the Hamiltonian, and the hyperbolic coefficients were determined by

enforcing that k and k obey the bosonic commutation relation (39). Using the equations (38)-(41), the Hamiltonian (37) can be written:

   1   2  H  Eh   Ak nk  Ak nk  B n1n2  n1n2  B n1n2  (42) k k1 ,k2

with:

nk k k , nk  k k , (43)

 2  1   2   1  2  1   G   k   E  2z J S  SN1 h  h  G   2  G  1 , (44) h  1 2  k    1   2 k   2SN k  2 2    G   2       k   

  1 1 1 G   2  G   2  A  G 2   2 2  h  h  k k 1 , (45) k k 2 1 2 SN 1  1  2 2 2 k  2 2 2  G   k   G   k  

    z J  G   2 G   2  2z J  G   2 G   2  B 1   1 2 1 , B 2   1 2 1 , (46) N  1 1  N  1 1   2 2 2 2 2 2   2 2 2 2 2 2   G   1  G   2    G   1  G   2  

h h h G  1 1  2 , h  q , (47) 2 2 q 2z J S

1 ir  r  e , (48) z 

where all terms to order 1 S 0 have been included and (48) is a sum over the vectors to nearest neighbours, with z being the number of them. Employing the standard method of

calculating the grand partition function of the system Z par and then using the thermodynamics relation to find the Helmholtz free energy F [8,9]:

1 Z par  TraceexpH,   , F  kBT ln Z par , (49) kBT

  F  Eh  kBT ln1 exp Ak  ln1 exp Ak  k (50)   1 1 1 1 B1         k1 ,k2  expA1 1 expA2 1 expA1 1 expA2 1 1 1   B 2 . 10    expA1 1 expA2 1 Taking the external magnetic field to zero such that h1  h2  0 , the free energy of the system becomes:

2  1  2 2 0 4z J  1    F  E  k T ln1 exp A   k , (51) 0 0 B  k  0  k N  k expAk 1  

0   where Ak  Ak  Ak from (45). Equation (44) now gives the zero-point energy:

2 2 E0  z J NS  cS  c 4, (52)

where c  1 I D  , the values of which is given by (23). Comparing with (25), it can be seen that Oguchi’s treatment of the antiferromagnet gives quantities to a higher order (to order 1 S 0 ) and are more rigorous than Anderson’s which are to order S 1 . A higher order term is also given in the energy of the spin-waves of the system, which is given by:

c 1 0   2 2 Ak  2z J S1 1 k  , (53)  2S  where the term in c2 has been neglected. It should be noted that the spin wave dispersion remains to be linear with the wavenumber for long wavelengths. A comparison of Anderson’s and Oguchi’s results for the dispersion relation of the spin-waves is shown in figure 2.

Figure 2. A graphical representation of Anderson’s and Oguchi’s results of the spin-wave energy which are both linear with the wavenumber in the region of long wavelengths. Note that the correction in Oguchi’s result shifts the gradient of the linear part of the dispersion.

The magnetisation M can be found by differentiating (50) with respect to the chosen external magnetic field, i.e. h1 , then taking the fields and temperature to zero yields:

M N  1  (54)  S  c, g 2  2  11 where c 1 J3 , the value of which is given by (31) and  is the Bohr magneton. It should be noted that for zero temperature, (54) is equivalent to Anderson’s result (33), showing that the assumptions made by Anderson are valid, however Oguchi’s treatment is more powerful as the full derivation (which is not considered here) gives the temperature dependence of the magnetisation.

Spin-waves have been observed by many different research groups as the phenomena of antiferromagnetism that is closely linked with a microscopic mechanism for high temperature superconductivity, which is of great interest. An example of which is led by Zhao et al. [10] whom use inelastic neutron scattering to observe the magnetic excitations in iron-arsenic based compounds. This study observed a linear dispersion relationship along the [H,0,0] and [0,0,L] directions of the reciprocal lattice, where H and L are odd integers, shown in figure 3.

Figure 3. The observed spin-wave dispersion of SrFe2As2 along the [H,0,0] and [0,0,L] directions of the reciprocal lattice at 160 K. [10]

The intensities of the magnetic excitations observed also were showed to obey Bose statistics, enforcing that the treatment of bosonic annihilation and creation operators by Oguchi is valid.

3. The triangular lattice

As linear spin wave (LSW) theory has shown to be effective when considering antiferromagnetic simple square and cubic structures, many cuprate and iron-based superconductors have a triangular lattice structure with anisotropic effects [11]. It is important to have a deeper understanding of antiferromagnetic triangular lattices to fully comprehend the connection between the electron interactions and the superconductivity they exhibit. Recent studies have suggested that near the boundary of the phase change from antiferromagnetism to the so-called spiral phase, the model shows superconductive behaviour, mediated by spin fluctuations [12].

3.1. The isotropic lattice

Instead of considering 3 sublattices, each with their own set of boson operators, it is convenient to rotate the projections of the quantum spin operators along their classical directions:

x ˆ x ˆ z Si  Si cosi  Si sini , y ˆ y Si  Si , (55) z ˆ z ˆ x Si  Si cosi  Si sini ,

where i  Qri and Q is named the ordering wave-vector. For the triangular lattice at zero temperature, the spins align into a formation where there exists a 120 angle between then, and such the ordering wave-vector for this state is Q  4 3,0. Using (55), the isotropic spin-1/2 Hamiltonian (1) becomes:

ˆ y y z z x z z x x z H  JSi S j  cosi  j Si S j  Si S j  sini  j Si S j  Si S j , (56) ij where all terms have their usual meanings. After expressing (56) in terms of the Holstein-Primakoff boson operators (40) and grouping terms, it is found:

Hˆ  Hˆ  Hˆ  Hˆ  Hˆ  SW 0 2 3 4 , (57)

ˆ with H n containing terms of the power n in the boson operators. Hence with n  0, the ˆ 2 first term is the classical energy H0  3JS N 2. The term quadratic in boson operators, describes non-interaction magnons whereas the interaction cubic and quartic terms are treated as perturbations in linear spin wave theory. Transforming into Fourier space, the quadratic term reads:

 1  ˆ † † † , (58) H 2  Ak ak ak  Bk ak ak  ak ak  k  2 

9 A  3JS1  , B  JS , k k k 2 k

where  k is defined similarly to (48), which sums over nearest neighbours:

1 1   k  3k   k  3k    eik  cosk  cos x y   cos x y  k z  3  x  2   2        . (59)    1  k  3k y  cosk  2cos x cos  3  x 2  2      

This is then diagonalised via a Bogolyubov transformation with associated conditions such that the new boson operators obey the appropriate commutation relations:

† ak  ukbk  vkbk , (60)

2 2 2 2 Ak Bk uk  vk  1, uk  vk  , 2uk vk  . (61) 2 2 2 2 Ak  Bk Ak  Bk

Performing this transformation results in the linear spin-wave Hamiltonian taking the form:

3  1  ˆ ˆ † (62) H 0  H 2   JSS 1N   k bk bk  . 2 k  2 

2 2  k  Ak  Bk  3JSk , k  1 k 1 2 k  , (63) where in the harmonic approximation, the spin-waves can be described as non-interacting bosons with energy  k and frequency k . The behaviour of the frequency is shown in figure 4.

Figure 4. Plots of the behaviour of the spin-wave frequency in k-space of the triangular antiferromagnet: (a)

along the ky = 0 direction and (b) a contour plot with darker and lighter colours representing lower and higher values respectively, with 0 being the minimum.

Clearly, the dispersion relation of the spin waves is linear near the origin as expected, and for values of the wavevector about the zeros of the frequency, the spin waves can be shown to have velocities:

(0) 3 (0) 3 v0  3JS , vQ  3JS , (64) 2 2 2 about k  0 and k  Q respectively. It should be noted that the shape of the dispersion relation of the triangular antiferromagnet is more complicated than that of the square lattice due to features like the three-fold symmetry about the points where k  Q , i.e. the corners of the Brillouin zone. To compute the magnetisation, we use the original form of the spin operator in boson form (38) with the spin deviation found as follows:

† 2 1 Ak 1 1 1  k 2 1 ai ai  vk     , (65)   2 2  k 2 k 2 2 k  2 Ak  Bk k where the relations (61) have been used. Thus it is found that the linear spin wave correction to the magnetisation, for a spin-1/2 system is:

† S  S  ai ai  0.5  0.2613032  0.2386968. (66)

It is worth noting here that the correction to the magnetisation of the triangular antiferromagnetic is larger than the corresponding correction for the square lattice from (33), indicating that the larger quantum correction is due to frustration of the spins. The ground state energy can be found by setting the occupation number of the bosons to zero in (62):

3 1 3 3 ˆ ˆ 2 . (67) H0  H 2   JSS 1N   k .   JS N  JSN1k  2 2 k 2 2 k

After computing the integral in (67) over the first Brillouin zone, the ground state energy is found as:

Egs 3  0.436824   JS 2 1  , (68) N 2  2S  to first order in S, which agrees well with the literature [13].

3.2. Anisotropy

It is possible to vary the ratio of the antiferromagnetic exchanges between the diagonal directions and nearest neighbours J 2 J1  by applying a uniaxial strain to a superconducting material, so we will explore a range of these ratios.

Here, we consider the Hubbard model of an anisotropic triangular lattice with one electron per lattice site. If the Coulomb interactions between two electrons on the same lattice site are large enough, the ground state becomes an insulator and the spin degrees of freedom are well approximated by the spin 1 2 Heisenberg Hamiltonian:

H  J1 Si S j  J 2 Sl Sm , (69) ij lm where ij denotes bonds between nearest neighbours as before and lm denotes bond between the electrons on the north-east diagonal on the lattice, with J 1 and J 2 being the corresponding antiferromagnetic exchanges. It is the competition between these two exchanges which leads to magnetic frustration, as on a triangle of spins not all three spins can be antiparallel to each other (the Neél order) causing them to bend away from each other (the spiral order), as shown in figure 5:

a) b)

Figure 5. Diagrams showing the antiferromagnetic a) Neél order and the b) spiral order, the transition15

between which is dependent of the ratio of the AF exchanges. In the limit of large spin, the model reduces to that of the classical Heisenberg model, and can be computed as a function of the ratio of the exchanges if it is assumed that the possible spin configurations on the lattice can be described by a spiral form

Si  Suexpiqri , where u is an arbitrary vector expressed in terms of an orthonormal basis of the lattice and the spiral vector q describes the relative orientation of the spins on the lattice. Substituting the spiral form into the Hamiltonian (69) gives:

Eq ,q  x y  J cosq  cosq  J cosq  q  Jq , (70) NS 2 1 x y 2 x y where E is the classical energy of the system in terms of the spiral vector and N is the number of sites of the lattice. When the spiral vector has value q   ,  , the ground state has Neél order and corresponds to the region where J 2 J1  0.5 . For S  1 2 and

J 2  0 , the model reduces to the Heisenberg model on a square lattice, which we have used previously. However if both J1 , J 2 are non-zero but small, then magnetic frustration is introduced which will reduce the magnitude of the order parameter (the magnetisation) in the Neél state. For this model, there is no consensus on the ground state as many theories have been put forward, for example Anderson has suggested that the ground state consists of a “spin liquid” which has no long range magnetic order [14]. There have also been studies that there is a long range order, however quantum fluctuations lower this to an order of magnitude lower than the classical value [12]. If J 1 is small but non- zero, the lattice consists of weakly coupled chains on the diagonals, which have been extensively studied and there is a general good understanding [15]. These chains correspond to J 2  J1 and lead to an energy gap in the spectrum, but a long range order still exists.

Instead of considering 3 sublattices, each with their own set of boson operators, it is convenient to rotate the projections of the quantum spin operators along their classical directions:

x ˆ x ˆ z Si  Si cosi  Si sini  y ˆ y Si  Si (71) z ˆ z ˆ x Si  Si cosi  Si sini 

Following a similar procedure as before where we use the standard Holstein-Primakoff and Bogoliubov transformations, expand them to order 1 S , and diagonalise the Hamiltonian, we find that:

2 1 1 B  H  NS Jq NBS  S Jk  q Jk  2Jqk,B 2 k 2 S 

  k,B k  k , (72) k

where the summation runs over the first Brillouin zone, k creates magnons, and we have introduced a magnetic field B directed in the zˆ -direction. The dispersion relation of these magnons is given by:

1  B 1 B  2 k,B  2SJk  Jq Jk  q Jk  q  Jq . (73)  S 2 S 

It should be noted that the ground state is given by (72) when there are no excitations present in the system, i.e. the occupation of the magnons is zero and the last term vanishes. The magnetisation can also be computed by differentiating the ground state energy with respect to the applied field:

1 Jk  q Jk  q Jk 2Jq z 1 dE0 1 S 2 Si   lim  S   . (74) B0  N dB 2 2N k k,B  0

After being numerically integrated, the behaviour of the ground state energy (72) and the magnetisation (74) are shown in figure 6 and 7 respectively:

Figure 6. The total ground state energy of the triangular antiferromagnet by linear spin wave theory (full curve) and the classical ground state (dashed curve) as functions of the anisotropy J2/J1.

Figure 7. The magnetisation of the triangular antiferromagnet by linear spin wave theory as a function of the 17 anisotropy J2/J1. At the transition from the Néel to the spiral phase (J2/J1=0.5), there is a large correction to the magnetisation, with the possibility of the existence of a quantum disordered state near the transition. The classical ground state energy has also been included in figure 6 to indicate how the ground state is affected by quantum fluctuations. The ground state energy found by linear spin wave theory is in good agreement with that of a series expansion in the literature [16]. In addition, the magnetisation shown in figure 7 is also in agreement with the series expansion and with that of the square lattice that has J 2 J1  0 and magnetisation  S  0.30339 [17]. As J 2 J1  0.5, the correction to the magnetisation is large yet finite, reducing the magnetisation and therefore the ordering at this point. This suggests the possibility of a quantum disordered phase around this transition, however it is difficult to determine the nature of the state in this region without further work. It is worth mentioning however that a similar transition point has been found by the series expansion around 0.7  J 2 J1  0.9 [16]. The quantum fluctuations around the Néel- spiral transition can be better understood by investigating the dispersion relation about the zero bosonic modes, i.e. k  0,0 and k  , . The behaviour of the dispersion in the k  k, k direction is shown in figure 8:

Figure 8. The dispersion relation in the (k,k) direction is shown for the for the values J /J =0 (dashed), 2 1 J2/J1=0.25 (dot-dashed) and J2/J1=0.5 (full). As the system reaches the transition, the dispersion vanishes linearly with wavevector and resembles a quadractic curve.

Nearing the transition J 2 J1  0.5, the excitations vanish linearly with wavevector and we define the spin wave velocity in the direction as:

c  2SJ1 21 2J 2 J1  , (75) which disappears at the Néel-spiral transition, which can be seen in figure 8. Now considering the long-wavelength, low energy magnons in the region of the transition and near the zero mode k  kx ,k y  0,0, it can be found that the correction to the magnetisation can be approximated as:

z 1 1 1 dk dk 1 kc Si      log2kc  logkc 2 2, (76) 2 2 8 2  4 2 2 2 2 k 2  k

which is represented in the diagonal directions k  kx  k y , and kc is the cut-off in the momentum. Reassuringly, the term in k  falls away quickly as the spin wave velocity in this region vanishes by (75). This behaviour of “mode-softening” occurring only in one direction has also been found in studies of the isotropic triangular lattice where next- nearest neighbours have been considered [18]. The integral (76) is finite and therefore the lowering of the energy along the k x direction gives a large but finite modification to the quantum correction of the magnetisation. Experimental work by Coldea et al. [19] shows that the spin wave theory accurately describes the magnetic salt Cs2CuCl4, shown in figure 9:

Figure 9. The dispersion relation of magnetic excitations in Cs2CuCl4 at temperature T <0.1K. The filled symbols represent the experimental data taken via neutron scattering, with the solid line being the dispersion relation predicted by spin wave theory, I.e. a dispersion taking the form of (73). The open circles indicate estimated experimental boundaries of the scattering.

Many other studies of the realisation of spin – ½ triangular Heisenburg antiferromagnets also exist in the literature, for example, the work done by Shirata et al on the compound

Ba3CoSb2O9 [20] which provides further validity to the spin wave theory.

4. Conclusion

In this report, the spin wave theory of antiferromagnetism has been presented for the simple square lattice in multiple dimensions and also for the triangular lattice where anisotropy has been considered in both cases. Using this model, it has been possible to derive the energy, magnetisation (or order parameter), spin wave frequencies, and spin wave velocities. While these properties can in principle be found by classical means, this

19 model provides quantum corrections as an expansion in the spin S which have been truncated an appropriate places. Even though the expansion can be taken to higher order terms, these involve considering the various interactions of magnons by performing various analytical techniques on complicated and non-diagonalisable Hamiltonians. While these corrections are finite and not negligible, they only yield small corrections compared to the lower terms [21], so have been left out of the calculations for brevity. For similar reasons, the region of the anisotropy where the Néel state exists has been investigated while the spiral state remains for further work, as many compounds that exhibit antiferromagnetism have large J 2 J1 due to their structure.

Spin wave theory is powerful, but it is not perfect. Many compound deviate from the idea model due to the presence of other interactions and couplings, giving rise to other models which accurately describe their behaviour, like the spin-liquid state originally proposed by Anderson [1]. The field of spin waves is still active and its future lies in more complex geometries (e.g. semi-infinite and Kagome lattices and more work is to be done in investigating the resonance valance bond theory and its connection with superconductivity if a consensus is ever to be reached.

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