Fundamentals of the Physics of Solids / 15-Elementary Excitations in Magnetic Systems

		15.5 Low-Dimensional Magnetic Systems		565
	A312	= ei(φ(k3 ,k1)+φ(k3,k2 )) .	(15.5.89)
	A123

This result may be interpreted as follows. The scattering of three magnons by each other can be understood in terms of a series of two-particle scattering events. The phase shifts in each two-particle process are of the previously seen form, irrespective of the state of the third particle. In the language of scattering theory: the S-matrix of three-particle scattering is factorizable.

Furthermore the boundary condition leads to the generalization of (15.5.76),

eik1N a = ei(φ(k1,k2)+φ(k1 ,k3)) ,
eik2N a = ei(φ(k2,k1)+φ(k2 ,k3)) ,	(15.5.90)
eik3N a = ei(φ(k3,k1)+φ(k3 ,k2)) ,
or equivalently, to the generalization of (15.5.77),
k1N a = 2πI1 + φ(k1, k2) + φ(k1, k3) ,
k2N a = 2πI2 + φ(k2, k1) + φ(k2, k3) ,	(15.5.91)
k3N a = 2πI3 + φ(k3, k1) + φ(k3, k2) ,

where I1, I2 and I3 are integers such that 0 ≤ Ii < N .

The spectrum of the allowed energies ω(k) with k = k1 + k2 + k3 has a broad continuum corresponding to the scattering states of three essentially independent magnons, but in addition two-magnon bound states and also three-magnon bound states appear in it. The dispersion relation for the latter

is	−	2	4	+2 − 1	.	(15.5.92)
ω = J
				cos ka

In the isotropic ferromagnetic point the energy expression of this bound state simplifies to

ω = 31 J [1 − cos ka] .

(15.5.93)

The energy of the three-magnon bound state is therefore lower than that of its two-magnon counterpart.

In m-magnon states, that is when m spins are reversed, the wavefunction may be written as

|Ψm =	c(xl1	, . . . xlm ) ψ(xl1 , . . . , xlm )	,	(15.5.94)

xl1 <xl2 <···<xlm

where

ψ(xl1 , . . . , xlm )

−	−	|F .	(15.5.95)
= σl1	. . . σlm

According to the Bethe ansatz, this state may be characterized by m di erent wave numbers k1, . . . , km,

566 15 Elementary Excitations in Magnetic Systems

	m
c(xl1 , . . . xlm ) = A(P ) exp i	kPj xlj	,	(15.5.96)
P	j=1

where P stands for all possible permutations of the wave numbers. The excitation energy and total wave number of the state appear again as the sum for m apparently independent magnons,

m	m
E = J ( − cos kj a) and	k = kj .	(15.5.97)
j=1	j=1

In fact, the wave numbers kj cannot be chosen arbitrarily. Once more, the phase factors φ(ki, kj ) are involved in the coe cients A(P ) of the wavefunction, therefore, when periodic boundary conditions are used to determine the wave numbers, we get

kiN a = 2πIi + φ(ki, kj ) ,	(15.5.98)

j

where Ii is again an integer, and the phase shift can be calculated from the generalization of (15.5.75),

	cot 21 φ(ki , kj ) =	sin[(ki − kj )a/2]	.	(15.5.99)
		cos[(ki + kj )a/2] − cos[(ki − kj )a/2]

The solutions of these equations provide a wide spectrum of excitations anew, and bound states appear as well. The excitation energy of the lowest-lying bound state is

E =	J	(1	− cos ka) .	(15.5.100)
	m

15.5.4 The Ground State of the Antiferromagnetic Chain

It follows from our earlier considerations on the anisotropic Heisenberg model that the fully aligned ferromagnetic state can be the true ground state for ≥ 1 only. Magnons, the continuum of multi-magnon excitations, and their

bound states appear as low-energy excitations above the energy E0 = −14 N J of the ground state in this case. When < −1, the ground state exhibits antiferromagnetic order, with the sublattice magnetization pointing in the direction of the z-axis. Between the two, in the planar regime −1 < < 1, the spin components in the (x, y) plane are more strongly coupled than the component along the quantization axis, hence in the classical limit the spins would lie in the (x, y) plane, exhibiting ferromagnetic order, and the mean value of the z component would vanish. However, in the ground state of one-dimensional models no continuous symmetry of the Hamiltonian can be broken unless the order parameter is conserved. This means that in the anisotropic Heisenberg

15.5 Low-Dimensional Magnetic Systems

567

model the continuous rotational symmetry around the z-axis cannot be broken in the region −1 < < 1. In addition to the z component, the mean values of the x and y components must also vanish. In the ground state half of the spins must point upward and the other half downward, without any spatial or temporal regularity in the spin fluctuation pattern. This ground state can be found using the Bethe ansatz in the subspace where m = N/2 spins are reversed with respect to the fully aligned state. The situation is the same in the ground state of the isotropic antiferromagnetic chain. We shall first study this case.

As it was mentioned in connection with (15.5.57), the isotropic antiferromagnetic model can be defined in two ways. In the customary approach the exchange interaction is assumed to be negative, while in the alternative one adopted there J > 0 and = −1 are assumed. The results obtained for the ferromagnetic system can be more easily utilized in the first approach: the equations derived from the Bethe ansatz for the wave numbers and phase shifts are identical to those of the isotropic ferromagnetic model. Therefore we shall use the Hamiltonian

H = J	21	σl+σl−+1 + σl−σl++1	+ 41	σlz σlz+1 − 1	+	41 N J (15.5.101)

l

with J > 0 to study the isotropic antiferromagnetic chain.

Since N/2 spins are reversed, the same number of di erent wave numbers have to appear in the Bethe ansatz of the wavefunction, and they have to

satisfy the following system of equations:

N aki = 2πIi + 2

arccot

21 (cot(kia/2) − cot(kj a/2)) .

(15.5.102)

j=i

The energy and total wave number of the corresponding state are

E = 41 N J − J

(1 − cos kia)

and

k =

ki .

(15.5.103)

i

i

In terms of the rapidities λi = cot(kia/2) instead of the wave number ki,

i

i

j=i

2

(15.5.104)

2N arccot λ

= 2πI + 2

arccot

λi − λj

,

which can be transformed into the algebraic form

λi + i

N

λi − λj + 2i

,

=

?

λi − i

j=i

λi − λj − 2i

(15.5.105)

while the energy takes the form

2

E = 41 N J − J

.

(15.5.106)

1 + λ2

i

i

568 15 Elementary Excitations in Magnetic Systems

The solution of the Bethe equations would lead us too far afield. We shall content ourselves with noting that when half of the spins are reversed (N↓ = N/2), there is only one state in which all wave numbers and rapidities are real and finite. In this state, which turns out to be the ground state, the quantum numbers Ii take all odd integer values in the interval 1 ≤ Ii < N . Provided that N is an integral multiple of four, the wave number of the ground state is k = 0 and the state is a singlet, i.e., its total spin is zero.

The numerical solution of the Bethe equations reveals that the wave numbers fill the whole interval (0, 2π/a), albeit not uniformly. Instead of ki = 2πIi/N a they tend to be located more densely around the middle of the interval on account of the phase shifts arising from scattering processes, as shown in Fig. 15.7.

2 a

ki

a

0					1 xi Ii N
0	0.2	0.4	0.6	0.8

Fig. 15.7. The wave numbers ki obtained from the numerical solution of the Bethe ansatz equations for the ground state of the isotropic antiferromagnetic Heisenberg model. The dashed line shows the wave numbers for the case when the phase shift is neglected

The ground-state energy can be calculated exactly in the large-N limit:

E0 = 41 N J − N J ln 2 = −0.443 N J .

(15.5.107)

This energy is lower than that of the Néel state, E0 = −14 N J, which would be obtained if there were no phase shifts.

Before proceeding any further it is useful to give the ground state in

another parametrization. If instead of the previously used						formula λi =
cot(kia/2) rapidity is defined by
	λi = − cot(kia/2) ,					(15.5.108)
then the following equations are obtained instead of (15.5.104):
i	i	j=i		2		(15.5.109)
2N arctan λ	= 2πJ + 2		arctan	λi − λj .

15.5 Low-Dimensional Magnetic Systems

569

Provided the chain consists of an even number of atoms, Ji is an integer if the number of reversed spins is odd, and a half-integer if it is even. The energy and wave number of the state are given by

								2
E = 41 N J − J									(15.5.110)
							1 + λ2
				i				i
and						N↓
		π		2π
k = N			+					J .	(15.5.111)
	↓ a			N a
								i

i=1

Using this parametrization all rapidities and wave numbers are real if each Ji is in the interval

−21 (N − N↓) < Ji < 21 (N − N↓) .

(15.5.112)

When N↓ = N/2, there are exactly N/2 consecutive integers or half-integers in the above interval:

−41 N + 21 , −41 N + 21 + 1, . . . , 41 N − 21 .

(15.5.113)

When this set is chosen for the quantum numbers Ji, N/2 real rapidities are obtained. In this parametrization this particular choice of the Bethe quantum numbers and the corresponding rapidities yield the ground state.

15.5.5 Spinon Excitations in the Antiferromagnetic Chain

The excited states above the singlet ground state can be characterized on the one hand their wave number, and on the other hand by their total spin and its z component. The spin of low-energy excitations is expected to di er from the ground-state spin by at most one unit, therefore excited states will be sought in the singlet and triplet subspaces. The distribution of the Bethe quantum numbers Ii , or Ji, will be di erent, too. For low-energy excitations the set of these quantum numbers is expected to di er little from the corresponding set of the ground state. In what follows, the quantum numbers Ji will be used.

We shall first consider triplet excitations with a spin projection Stotz = 1. Since in these states the number of reversed spins is one less than in the ground state, N↓ = N/2 − 1, specifying this state requires one less quantum number ki or Ji, too. On the other hand, the Bethe equations allow N −N↓ = N/2 + 1 di erent values for Ji. Since the parity of the number of reversed spins is changed with respect to the ground state, the quantum numbers Ji have to be chosen from the sequence

−41 N , −41 N + 1 , . . . , 41 N − 1, 41 N .

(15.5.114)

Real wave numbers are obtained if N/2−1 di erent numbers are chosen out of the N/2 + 1 possible values for Ji. In other words, out of the N/2 + 1 allowed

570 15 Elementary Excitations in Magnetic Systems

Bethe quantum numbers N/2−1 are occupied and two are empty in this state. Thus a set of states characterized by two parameters (the position of the two holes) is obtained. The energies of these excited states do not determine a sharp dispersion curve but a continuum whose lower and upper bounds are given by

ωmin = 21 πJ| sin ka| ,

ωmax = πJ| sin(ka/2)| .

(15.5.115)

Similar results are obtained for the Stotz = 0 component of the triplet. This continuum of excitations is shown in Fig. 15.8.

!

J

k

a

a

Fig. 15.8. The continuum of triplet excitations in an isotropic antiferromagnetic spin-1/2 chain

Excitation energies can be determined analytically in the N → ∞ limit. After some rather lengthy calculation the energy and wave number of these excitations can be written in the form

E − E0 = 21 πJ sin k1a + 21 πJ sin k2a , k = k1 + k2 ,

(15.5.116)

where k1 and k2 lie in the interval (0, π/a). Each true (physically realisable) triplet excitation appears to be composed of a pair of “elementary” excitations with the dispersion relation

εs(k) = 21 πJ sin ka .

(15.5.117)

This picture becomes even more pronounced when singlet excited states are considered. States are then characterized by N/2 quantum numbers, but some of the Ji are identical, and therefore there are complex conjugate pairs among the wave numbers and rapidities. The low-energy part of the excitation spectrum can be determined analytically in the large-N limit, and exactly the same result is obtained as for triplet excitations – i.e., singlet excitations can also be interpreted as pairs of “elementary” excitations whose dispersion relation is given by (15.5.117). We may therefore say that the elementary excitations of the antiferromagnetic Heisenberg chain are spin-1/2 spinons, defined over only half of the Brillouin zone, in the region (0, π/a), but in any physical process they are created in pairs. The physical meaning of this

15.5 Low-Dimensional Magnetic Systems

571

s(k)

J/2

				k
0			/a

Fig. 15.9. Dispersion relation for spinons in an isotropic antiferromagnetic chain

statement will become clear soon. The dispersion relation of the spinons is shown in Fig. 15.9.

Freely moving spinon excitations can be best illustrated by considering a chain in which the spins are arranged in a Néel-type antiferromagnetic order, except at one site, where the spin is reversed. Compared to the antiferromagnetic sequence, two bonds are not satisfied. Since this state is not an eigenstate of the Heisenberg Hamiltonian, two oppositely directed spins can flip each other. When one of the spins is in a “bad” bond, the antiferromagnetic order will be reestablished in this bond after the spin flip but, as shown in Fig. 15.10, a next-nearest bond will become unsatisfied, with two parallel spins. As the spin flip processes continue, the two “bad” bonds move independently. A Néel-type antiferromagnetic order exists between them, but in the opposite phase. “Bad” bonds can therefore be considered as “domain walls”. In this picture spinons are these moving domain walls. Their motion makes the Néel-type order unstable, and gives rise to a spin-liquid state.

Fig. 15.10. Freely propagating spinons (domain walls) in an antiferromagnetic Heisenberg chain with one reversed spin. Dotted lines denote the „bad” bonds, and the arrows point to the spins that will be flipped in the next step

When two spins next to a domain wall are flipped by the exchange interaction, the domain wall jumps by two lattice constants. Constructing the spinon wavefunction as the linear combination of spin configurations containing such domain walls, states of wave number k and k + π/a are equivalent since the spinon propagates along the chain as if the lattice constant were doubled. This

572 15 Elementary Excitations in Magnetic Systems

explains why the dispersion relation is defined over only half of the Brillouin zone, in the range (0, π/a).

When the spectrum of excited states and the corresponding wavefunctions are known, the correlation function between two spins a distance r apart could be calculated, in principle. However, the calculation cannot be performed analytically along these lines. We shall return to this problem in the third volume. Here we just mention that the correlation functions shows a power-law decay, as expected for a gapless critical model.

15.5.6 The One-Dimensional XY Model

Since the Bethe ansatz is satisfied even in the anisotropic spin-1/2 Heisenberg model, as has been discussed for the anisotropic ferromagnetic case, the behavior of the system can be studied in the planar regime, −1 < < 1, too. The behavior is qualitatively similar to that observed in the isotropic antiferromagnetic point. The ground state is a singlet, low-lying singlet and triplet excitations form a gapless continuum, and the energy and wave number of the excitations inside the continuum can be constructed as if they consisted of pairs of spin-1/2 spinons with dispersion relation

εs(k) = 21 πJ	sin Θ	sin ka ,	(15.5.118)
	Θ
where Θ is related to the anisotropy parameter			via cos Θ = ,11 and the

spinons are defined in the interval (0, π/a). The continuum of excitations is similar to that shown in Fig. 15.8 for the isotropic antiferromagnetic point, only the scale is di erent. The boundaries of the continuum are given by

ωmin = 21 πJ	sin Θ	| sin ka| ,	ωmax = πJ	sin Θ	| sin(ka/2)| .	(15.5.119)
	Θ			Θ

To get an even better picture of why low-energy excitations appear as pairs of fictitious particles, it is worthwile to study the special case = 0. This is the XY model. The Jordan–Wigner transformation (15.2.62) allows us to express the Hamiltonian of this model in terms of spinless fermions in a particularly simple form:

H = −21 J	j	cj†cj+1 + cj†+1cj .	(15.5.120)

The negative sign is chosen for later convenience, and J > 0 is assumed. Using the Fourier transforms of the creation and annihilation operators, the Hamiltonian is rewritten as

11Here, too, the choice J > 0, = 1 corresponds to the isotropic antiferromagnetic point.

15.5 Low-Dimensional Magnetic Systems

573

H = εkck† ck ,	(15.5.121)
k

where εk = −J cos ka. The magnetic system is thus equivalent to a gas of free spinless fermions with a simple cosine dispersion relation. To specify the allowed values of the wave number it should be noted that (15.5.99) implies that for = 0 the phase shifts φ(ki, kj ) assume the values ±π. Depending on the number of reversed spins relative to the ferromagnetic ground state with all spins pointing upward, the net phase shift is either 0 or π. Denoting the number of downward spins by N↓,

	ki =	2π		(Ii	+ 1 ) ,	(15.5.122)
		N a			2
	if N↓ is even, and
	ki =		2π		Ii ,	(15.5.123)
			N a

if N↓ is odd. Figure 15.11 shows the spectrum of a finite chain with the possible values of ki.

	k		k
/a	a k	/a	a k

Fig. 15.11. The spectrum of free spinless fermions in the XY model, and the allowed values of the wave number for N↓ even and odd, respectively. N = 16, N↓ = 8 on the left side, and N = 18, N↓ = 9 on the right side. Full (empty) circles indicate states that are occupied (unoccupied) in the ground state

When J > 0, wave numbers in the range −π/2a ≤ k ≤ π/2a appear in the ground-state wavefunction, since the associated one-particle states have negative energy. One may say that states characterized by such wave numbers are occupied. As k has as many possible values in the Brillouin zone −π/a ≤ k ≤ π/a as there are lattice sites, and half of the corresponding states are occupied, one may speak of a half-filled band. The Jordan–Wigner transformation also implies that

1
Sjz = cj†cj − 21 = N ck† ck − 21 = 0		(15.5.124)

k

in this half-filled case, that is the expectation values of the spin and the magnetic moment vanish at each lattice site in the ground state. Likewise,

574 15 Elementary Excitations in Magnetic Systems

the expectation values for the x and y components vanish, too. Furthermore, if the number of lattice sites is even, then the magnitude of the total spin also vanishes – in other words, the system has a singlet ground state. The spins are completely disordered, therefore we may call the ground state a spin liquid.

There are two ways to create low-energy excitations above this ground state. The first option is leaving the number of spinless fermions unaltered, and choosing the wave numbers in the wave function di erently – in other words, changing the distribution of occupied states. In the simplest case instead of a state of wave number ki filled in the ground state, another state of wave number kj will be occupied. Wave numbers associated with such low-energy excited states are shown in Fig. 15.12.

	k		k
/a	a k	/a	a k

Fig. 15.12. Two possible wave-number configurations for N = 16 and N↓ = 8 in low-energy excited states in the XY model. Occupied states are indicated by full circles

In terms of spinless fermions, these excitations of the spin model may be interpreted as particle–hole excitations. Taking away a particle with wave number k1 and adding another with wave number k2, the energies of excitations with wave numbers k = k2 − k1 form a continuum, as depicted in Fig. 15.13. The continuum is bounded by

ωmin = J| sin ka|

and

ωmax = 2J| sin(ka/2)| .

(15.5.125)

This result was derived in light of the fact that the creation of holes is possible only in the interval −π/2a < k1 < π/2a, while that of particles only in the complementary range of the Brillouin zone.

Another type of excitation is obtained when a spinless fermion is added to or removed from the system – corresponding to raising or lowering the z component of the total spin by unity. In contrast to the two-particle excitations discussed above, they might seem to be one-particle excitations. In reality, they are particle–hole excitations, too, but with respect to a modified configuration. Namely, it has to be taken into account that spin reversal changes the parity of N↓ with respect to the ground state, and therefore all wave numbers are shifted. In the lowest-energy state of the subspace Stotz = 1 the wave numbers are located symmetrically anew. To obtain excited states above it,