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

15.5 Low-Dimensional Magnetic Systems					555
S = kB ln	R		2	(15.5.23)
			.
	a

Consequently, the free energy of a system with a single vortex is given by

F = E − T S = (πJq2 − 2kBT ) ln(R/a) .

(15.5.24)

This indicates that individual vortices of vorticity q may appear spontaneously above T = 12 πJq2/kB. As it was pointed out by Berezinskii8 and later by Kosterlitz and Thouless,9 the behavior of the system – i.e., the character of the spin–spin correlations – changes at the temperature where the first free vortices appear,

TBKT = 21 πJ/kB .

(15.5.25)

This is the BKT (Berezinskii–Kosterlitz–Thouless) transition.

At high temperatures – as it is usual in disordered phases –, the correlation function decays exponentially with distance due to the disordered spatial distribution of vortices. The correlation length ξ(T ) characterizing the decay increases as the temperature is lowered, and diverges at TBKT. At this temperature the decay of the correlation function is no longer exponential but power-law-like, as usual in a critical point. The particular feature of the BKT transition is that this power-law behavior survives even below TBKT, in the low-temperature regime, where thermal fluctuations prevent ordering, but with a temperature-dependent exponent:

Γ (r)	r	η(T )	(15.5.26)
		.
	1

In the simplest approximation, where only the e ects of spin-wave-like harmonic fluctuations are taken into account, η(T ) = kBT /(2πJ). To demonstrate this, consider the correlation function

S(0) · S(r) = cos(θ(0) − θ(r))

= 21		ei(θ(0)−θ(r))		+ e−i(θ(0)−θ(r))		(15.5.27)
						.

If spin waves alone are taken into account – that is, if we employ the harmonic approximation and calculate the thermal average with the weight factor from the Hamiltonian (15.5.15) – then, according to (13.3.11), the averaging procedure can be performed in the exponent,

S(0) · S(r) = exp	− 21		(θ(0) − θ(r))2	.	(15.5.28)

Writing the exponent in terms of Fourier components,

8V. L. Berezinskii, 1970.

9J. M. Kosterlitz and D. J. Thouless, 1972.

556 15 Elementary Excitations in Magnetic Systems

21	(θ(0) − θ(r))2	=	(1 − cos k · r) |θ(k)|2	,	(15.5.29)

k

and using the Boltzmann weight factor that follows from (15.5.15), we have

1		2			kBT
2	(θ(0) − θ(r))		=	(1 − cos k · r)		.	(15.5.30)
					Jk2
				k

Replacing the sum over the Brillouin zone by an integral over a circle of radius π/a, for large values of r

1	(θ(0)	− θ(r))	2	=	kBT	ln(πr/a) .	(15.5.31)
2					2πJ

Substituting this into the exponent, as required by (15.5.28), the form (15.5.26) is indeed recovered for the correlation function. Below the transition point, throughout the temperature range from T = 0 up to TBKT the system behaves as it were critical, however, the critical exponents are not universal but depend on the coupling and the temperature. This is the so-called

Berezinskii–Kosterlitz–Thouless phase (BKT phase).

Although free vortices appear only above the transition point, configurations that can be regarded as bound states of two oppositely “charged” vortices may exist at lower temperatures as well. Such configurations are shown in Fig. 15.6.

Fig. 15.6. Configurations corresponding to the bound state of vortices of opposite vorticity

Assume that the system contains only two vortices, at r1 and at r2, and that their quantum numbers are q1 and q2. The angular variable θ(r) can be chosen in the form

15.5		Low-Dimensional Magnetic Systems				557
θ(r) = q1 arctan	y − y1	+ q2 arctan	y − y2	.	(15.5.32)
	x − x1
			x − x2

The path integral of θ(r) around a closed circuit gives 2πq1, 2πq2 or 2π(q1 + q2), depending on whether the circuit goes around the first, the second or both vortices.

As will be shown later (see (15.5.54)), the energy of this configuration is

21 J	dr θ(r)		2	= E1 + E2 + 2πJq1q2 ln	r1	R r2		,	(15.5.33)
					|	−	|

where E1 and E2 are the energies of the individual vortices, as given in (15.5.22). The third term, the interaction energy diverges logarithmically as the size of the system increases. Writing the total energy in the form

2

1

2

−

1

2

|

a

|

(15.5.34)

1 J dr θ(r)

2

= πJ(q

+ q

)2 ln R

2πJq

q

ln

r1

− r2

,

the size-dependent divergent term is seen to disappear when the two vortices are of opposite vorticity, q1 + q2 = 0. The energy depends logarithmically on the distance between the two vortex cores, and it might be much smaller than the energy of two individual vortices, especially if the distance of the cores is on the order of the lattice constant. If this energy is small enough, a pair of vortices with opposite vorticities may be thermally excited. Such vortex pairs are indeed observed in the low-temperature Berezinskii–Kosterlitz–Thouless phase of the planar XY model, where the system is “neutral” in the sense that the total vorticity is zero.

Thus, below TBKT, spin-wave-like configurations and vortex pairs are present simultaneously. To determine the total energy we assume that the density of vortices is given by

ρ(r) =		qiδ(r − ri) .			(15.5.35)
		i
The generalization of (15.5.18) for this case is
0	θ(r) · dl = 2π		i	qi ,	(15.5.36)
C

where the sum is over all vortices inside the closed path C. Transforming, by means of Stokes’ theorem, the line integral into an integral over the region enclosed by C,

	× θ(r) dr = 2π		(15.5.37)
ez ·		qi .
		i
This can be satisfied for any closed region if
× θ(r) = 2πez qiδ(r − ri) = 2πez ρ(r) .			(15.5.38)

i

558 15 Elementary Excitations in Magnetic Systems

Seeking the solution in the form

θ(r) = θ0(r) + ψ(r) ,

(15.5.39)

where ψ(r) is assumed to be a continuous, single-valued function, its gradient is vortex-free, i.e., × ψ(r) = 0, and

0

ψ(r) · dl = 0 .

(15.5.40)

C

Thus ψ(r) describes the fluctuations in the orientation of the spin vectors due to spin waves, while θ0(r) comes from the vortices or vortex pairs. If θ0(r) is chosen so that

is satisfied, then	θ0(r) = − × ez χ(r)	(15.5.41)
	× θ0(r) = ez 2χ(r) ,	(15.5.42)
which implies that χ(r) satisfies the two-dimensional Poisson equation
	2χ(r) = 2π qiδ(r − ri) = 2πρ(r) .	(15.5.43)

i

The solution of this equation can be written as the sum of two-dimensional Coulomb potentials:

χ(r) = qi ln(|r − ri|) .	(15.5.44)

i

A more precise form is obtained in terms of the Green function of the Laplacian, which satisfies the equation

2g(r) = δ(r) ,		(15.5.45)
which leads to	dr g(r − r )ρ(r ) .
χ(r) = 2π		(15.5.46)

In two dimensions the asymptotic solution for the Green function for large

distances from the core of the vortex is:							2π ln (R/a) ,		(15.5.47)
g(r) = −	(2π)2		ek2·	≈	2π ln (|r|/a) −
	dk		ik	r	1		1

where R is the radius of the sample.

The energy of the configuration that contains vortices as well can be determined using (15.5.14),

E = E0 + 21 J

dr

ψ(r) + θ0(r)

2

(15.5.48)

= E0 + 21 J

ψ(r)

2

+

21 J

θ0(r)

2

dr

dr

,

15.5 Low-Dimensional Magnetic Systems		559
since the mixed term containing the integral of ψ(r) · × (ez χ(r))		van-
ishes, as the reader may verify by a simple integration	by parts. This means

that the energy contributions of spin waves and vortices are independent of each other. Below we shall only deal with the contribution of vortices. Since

	(15.5.41),			∂χ(r) ∂χ(r)
	θ0(r) = −								,		(15.5.49)
					∂y			∂x
	implies	θ0(r) 2 =				χ(r) 2 ,
											(15.5.50)
	be obtained from
	the energy of vortices can
	E = E0 + 21 J				dr		χ(r)			2 ,	(15.5.51)
	or alternatively, after an integration by parts, from
	E = E0 − 21 J			drχ(r) 2χ(r) .							(15.5.52)
	Using the expressions obtained for χ(r) and 2χ(r),
	E = E0 − 2π2J		dr dr ρ(r)g(r − r )ρ(r ) .								(15.5.53)

From the explicit form of the Green function and assumption (15.5.35) for the density of vortices

#

i$

2

0 −

i=j

i

j

ln

+ πJ

i

,

(15.5.54)

a

a

E = E

πJ

q q

ri − rj

q

ln R

in agreement the expression anticipated for the energy of a vortex pair. The

term depending on the size of the sample vanishes if i qi = 0, i.e., if the total vorticity is zero.

In the foregoing discussion the correlation function in the low-temperature phase and the transition temperature were determined under the assumption that only spin waves are excited below TBKT. A more precise treatment requires the inclusion of vortex pairs as well. A rather tedious calculation, whose details cannot be given here, shows that the e ects of the vortex pairs can be absorbed into the parameters of the vortex-free model. The behavior of the full system is then similar to a simpler one in which only spin waves are present but the parameters are modified, renormalized. Below the transition point the correlation function shows a power-law decay but the exponent η(T ) is renormalized. The transition temperature itself is also renormalized. The first corrections coming from the vortex pairs give

πJ	2
	≈ 1 + 1.3πe−π	J/2kBTBKT .	(15.5.55)
2kBTBKT

560 15 Elementary Excitations in Magnetic Systems

The most accurate Monte-Carlo simulations give a much lower critical temperature, kBTBKT = 0.893J. Nevertheless, the critical exponent η at the true transition point is η(TBKT) = 1/4 – the same as the result obtained when vortex pairs were neglected.

While for increasing temperatures more and more vortex pairs are excited thermally in the Berezinskii–Kosterlitz–Thouless phase, they are all confined to neutral pairs made up of nearby vortices. Above the transition point they become deconfined (liberated). The neutral bound pairs disintegrate into freely moving individual vortices. The correlation function changes character: from this point it decays exponentially. It can also be shown that the finite correlation length ξ is not proportional to some inverse power of t = (T − TBKT)/TBKT, as it occurs above ordinary critical points but it depends exponentially on t:

ξ		exp(bt−1/2) .	(15.5.56)

15.5.3 The Spin-1/2 Anisotropic Ferromagnetic Heisenberg Chain

It was mentioned earlier that the energy of states with two raised or lowered spins relative to the ground state could, in principle, be calculated in systems of arbitrary dimensionality and spin. The exact determination of further states, involving more flipped spins, is impossible for general S, even in one-dimensional spin chains. The spin-1/2 Heisenberg chain is particular, for an exact determination of excited states with more reversed spins is possible, and the calculation can be easily extended to the anisotropic case, too.

Using the Pauli operators instead of the spin operators, and separating the energy of the ferromagnetic state, the Hamiltonian is customarily cast in the form

H = −J	21	σl+σl−+1 + σl−σl++1 +	41	σlz σlz+1 − 1	−	41 N J , (15.5.57)

l

where again, similarly to (15.2.77) a factor of two has been dropped. When J > 0, = 1 corresponds to an isotropic ferromagnet, and = 0 to the pure planar model. J < 0 would lead to antiferromagnets, but via the rotation of every second spin through 180◦ about the z-axis, this case is found to

be equivalent to J > 0,		< 0. Consequently, the parameters J > 0,	=
−1 and J < 0,	= 1 can equally be used in the discussion of isotropic
antiferromagnets.

Taking again the state with all spins aligned upward as the ferromagnetic ground state, we shall seek eigenstates with one reversed spin in the form

|Ψ = c(xl)σl−|F .	(15.5.58)

l

When the Hamiltonian acts on this state, the following eigenvalue equation is obtained:

15.5 Low-Dimensional Magnetic Systems 561

Ec(xl) = −41 N JΔc(xl ) + 12 J [2Δc(xl) − c(xl+1) − c(xl−1)] . (15.5.59)

We shall try to find the solutions using the trial function c(xl) = Aeikxl . The periodic boundary condition implies c(xl) = c(xl + N a), hence

k =	2π		where	I = 0, ±1, ±2, . . . , N/2 .	(15.5.60)
		I,
	N a
Substitution of this formula into the eigenvalue equation leads to
			E = −41 N J	+ J ( − cos ka) ,	(15.5.61)
thus the excitation energy is
			ωk = J ( − cos ka) .		(15.5.62)

In the isotropic case, the spin-wave spectrum starting o as k2 is recovered. For > 1, the excitation spectrum features a finite gap. For < 1, on the other hand, the energy of the long-wavelength excitations turns out to be negative, indicating that the ferromagnetic state is no longer the ground state. We shall determine the true ground state later.

When two spins are reversed, the wavefunction is chosen as

|		=		|			(15.5.63)
			l l
Ψ2			c(xl, xl )σ−σ−	F		,

ll

in line with (15.2.82), and xl < xl is assumed. This restriction is necessary as in a spin-1/2 chain the spin cannot be reversed twice at the same lattice site. When the two reversed spins are not at adjacent lattice sites, the Schrödinger equation yields the following relation for the coe cients:

E c(xl , xl ) = −41 N JΔc(xl , xl ) + 2JΔc(xl, xl )

−	21 Jc(xl−1, xl ) −	21 Jc(xl+1 , xl )	(15.5.64)
− 21 Jc(xl, xl −1) − 21 Jc(xl , xl +1) .

Writing the solution in the plane-wave-like form

c(xl, xl ) = A12ei(k1xl +k2 xl ) + A21ei(k2xl +k1 xl ) ,

(15.5.65)

the equation for the energy eigenvalue becomes

E = −41 N J + J ( − cos k1a) + J ( − cos k2a) .

(15.5.66)

The excitation energy is again equal to the sum of the energies of two free magnons, therefore these excitations are expected to form a continuum. To determine the possible values of k1 and k2, we have to examine the case when spins are reversed at two adjacent lattice sites. The following equation holds for the amplitude of such states:

562	15 Elementary Excitations in Magnetic Systems
	E c(xl, xl+1) = −41 N JΔc(xl , xl+1) + JΔc(xl , xl+1)	(15.5.67)
	− 21 Jc(xl−1, xl+1) − 21 Jc(xl, xl+2) .

By demanding that this equation also hold with the previously given forms of the wavefunction and the energy – (15.5.65) and (15.5.66) –, restrictions are imposed on the amplitudes A12 and A21. To determine these, we shall assume that (15.5.64) formally holds in the l = l + 1 case, too, that is,

	E c(xl, xl+1) = − 41 N JΔc(xl, xl+1) + 2JΔc(xl, xl+1)
		− 21 Jc(xl−1, xl+1) − 21 Jc(xl+1 , xl+1)					(15.5.68)
		− 21 Jc(xl, xl) − 21 Jc(xl , xl+2) .
When combined with the previous equation, consistency requires
	2Δc(xl, xl+1) = c(xl, xl) + c(xl+1 , xl+1) .						(15.5.69)
Using the assumed form (15.5.65) for the coe cients,
2	A12eik2 a + A21eik1a				= (A12 + A21) 1 + ei(k1+k2 )a ,		(15.5.70)
whence		A12			2 eik1a − 1 − ei(k1+k2 )a
			=	−		.	(15.5.71)
		A21			2 eik2a − 1 − ei(k1+k2 )a

Expressing the amplitudes in terms of a phase di erence φ(k1, k2) defined on the interval (−π, π),

A12 = A eiφ(k1,k2 )/2 ,	A21 = Ae−iφ(k1,k2 )/2 ,	(15.5.72)
which corresponds to choosing the coe cient c(xl, xl ) as
c(xl, xl ) = A ei[k1xl +k2xl +φ(k1,k2)/2] + ei[k2xl +k1 xl −φ(k1,k2 )/2] .		(15.5.73)

The two terms can be interpreted as follows. Two magnons propagate in the system. In their interaction – scattering by one another – the total wave number is not the only conserved quantity: k1 and k2 are conserved separately. On the other hand, when the two magnons pass through each other, the wavefunction undergoes a phase shift of φ(k1, k2).

From the expression

	eiφ(k1,k2 ) =			2			eik1a − 1 − ei(k1+k2 )a	,
				−	2		eik2a − 1 − ei(k1+k2 )a
which is implied by (15.5.71) and (15.5.72),
cot 1	φ(k1 , k2) = i	eiφ(k1,k2 )		+ 1
		eiφ(k1,k2 )		− 1
2
	=				sin[(k1 − k2)a/2]
			cos[(k1	+ k2)a/2] − cos[(k1 − k2)a/2]
	=			cot(k1a/2) − cot(k2a/2)					.
			(1 + ) − (1 − ) cot(k1a/2) cot(k2a/2)

(15.5.74)

(15.5.75)

15.5 Low-Dimensional Magnetic Systems

563

For the isotropic spin-1/2 Heisenberg model this is equivalent to the result (15.2.95).

The periodic boundary condition provides another relation between the as yet undetermined wave numbers and the phase. Just like in the calculation for the isotropic Heisenberg chain with arbitrary S, we arrive at equations (15.2.87) and (15.2.88) anew,

eik1N a = eiφ(k1,k2 ),	eik2N a = e−iφ(k1,k2),	(15.5.76)
or equivalently
k1N a = 2πI1 + φ(k1, k2) ,	k2N a = 2πI2 − φ(k1, k2) ,	(15.5.77)

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

To get the excitation spectrum, the closed set of equations can be solved numerically or analytically in the large-N limit. It is found that the energies associated with real values of ki fill almost continuously the region between the curves

ω−(k) = 2J( − cos 12 ka) and ω+(k) = 2J( + cos 12 ka) . (15.5.78)

Apart from these solutions, others, associated with complex conjugate pairs (k2 = k1 ) arise, too, with an energy below the continuum,

ω(k) = J − cos 12 ka + cos 12 ka , (15.5.79)

where k = k1 + k2 = 2 Re k1.

In the isotropic case the results obtained from (15.2.93) and (15.2.96) by setting S = 1/2 are recovered. It will prove convenient to introduce the

variables

λ1 = cot(k1a/2)

and

λ2 = cot(k2a/2) ,

(15.5.80)

called rapidities. Then making use of the ensuing formulas

eik1 a =

λ1 + i

and

eik2 a =

λ2

+ i

,

(15.5.81)

λ1 − i

λ2

− i

along with

cot(φ(k1

, k2)/2)

+ i

eiφ(k1 ,k2) =

,

(15.5.82)

cot(φ(k1

, k2)/2)

− i

the boundary condition leads to the following algebraic equations for the rapidities

λ1

− i

N

− λ2

− 2i

λ2

− i

N

− λ1

− 2i

(15.5.83)

λ1

λ2

λ1

+ i

=

λ1

− λ2

+ 2i

,

λ2

+ i

=

λ2

− λ1

+ 2i

.

564 15 Elementary Excitations in Magnetic Systems

It was first pointed out by Bethe10 that in the special case of a spin-1/2 anisotropic Heisenberg chain the method above can be generalized for states with arbitrarily many reversed spins.

States with three reversed spins may be written as

|Ψ3 =	1				(15.5.84)
		c(xl1 , xl2 , xl3 ) ψ(xl1 , xl2 , xl3 )		,
xl <xl2 <xl3
where		ψ(xl1 , xl2 , xl3 )	= σl−1 σl−2 σl−3 |F .		(15.5.85)

Substituting this into the Schrödinger equation, we may follow the method used for states with two reversed spins – that is, separate equations have to be written for the cases when there are no neighbors and when there are neighbors among xl1 , xl2 and xl3 . In analogy to (15.5.65), we shall use the ansatz

c(xl1 , xl2 , xl3 ) = A123ei(k1xl1 +k2xl2 +k3 xl3 ) + A132ei(k1 xl1 +k3 xl2 +k2xl3 )

+A213ei(k2xl1 +k1 xl2 +k3 xl3 ) + A231ei(k2xl1 +k3xl2 +k1 xl3 )

+A312ei(k3xl1 +k1 xl2 +k2 xl3 ) + A321ei(k3xl1 +k2xl2 +k1 xl3 ) ,

(15.5.86)

which indicates immediately that the total wave number of the state is k = k1 + k2 + k3.

From the equations for nonadjacent lattice sites, the excitation energy

ω = J ( − cos k1a) + J ( − cos k2a) + J ( − cos k3a)

(15.5.87)

emerges, regardless of the amplitude values – as if the energies of three independent magnons were summed. The allowed values of the wave numbers and the coe cients in the wavefunctions are determined by the boundary condition and the requirement that the same energy values have to satisfy the other types of equations, in which two or all three spin reversals take place at adjacent lattice sites. It is found that if an interchange of two wave numbers makes the only di erence between two terms, then the ratio of the corresponding coe cients may be written in terms of precisely the same phase shift φ(ki, kj ) as that which appeared in the wavefunction with two reversed spins. For example,

	A213	= eiφ(k2,k1 ) .	(15.5.88)
	A123

When two wave number permutations are needed to reach one term from another, the amplitude ratio contains the sum of the two phase shifts associated with the permutations. For example,

10H. Bethe, 1931. Hans Albrecht Bethe (1906–2005) was awarded the Nobel Prize in 1967 “for his contributions to the theory of nuclear reactions, especially his discoveries concerning the energy production in stars”.