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Damping of magnons in a two-dimensional S = 12 Heisenberg antiferromagnet
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13 November 1995
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PHYSICS LETTERS
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Physics letters A 207 (1995) 390-396
Damping of magnons in a two-dimensional S = l/2 Heisenberg antiferromagnet A.F. Barabanov a, L.A. Maksimov b a Institutefor High Pressure Physics. 142092 Troitsk, Moscow Region, Russian Federation h Russion Research Center, Kurchatov Institute, Kurchatov Square 46, 1231832 Moscow, Russian Federation Received 15 July 1995; accepted for publication 6 September 1995 Communicated by V.M. Agrauovich
Abstract The spin excitation damping is examined for a singlet state of a 2D antiferromagnet on a square lattice using the rotationally invariant approach for two-time spin Green functions. It is shown that the lowest boundary of weakly damped
spin waves has a scale k of the order (T/T,) 3/2. This leads to an inverse correlation 15-l which is much larger than L; ’ - exp(T,/T) g’tven by the Dyson-Maleev formalism.
1. It is well known that many unusual properties of high-temperature superconductors are determined by the Cu-0, planes where Cu-ions form an antiferromagnetic S = l/2 square spin lattice. At any non-zero temperature there is no long-range (LR) order for such a two-dimensional (2D) system; the average site spin is equal to zero, (S,) = 0, and the spin correlation function (S&Y;> has a correlation length
At T<< To - J the correlation length is very large. Then the existence of weakly damped spin waves with wave vector k larger than the inverse correlation length (k > L;‘) may be supposed. As it was shown by TyE and Halperin [I], for asymptotically low temperatures and long wavelengths the spin-wave damping r, is small at T-K T,,, and r, has an essential dependence on T and k. This result was obtained in the framework of the Dyson-Maleev formalism [2] by replacing the spin Hamiltonian by a boson one. This formalism explicitly supposes the presence of two sublattices for a spin system. The main approximation involved in Ref. [l] was to boson states which correspond to a population by more ignore the requirement to project out the “unphysical” than 2S bosons in a lattice site. The application of the mentioned approximation is not obvious if we want to treat the problem for the ultraquantum case S = l/2. Moreover, it is impossible to use the Dyson-Maleev formalism if we treat the ARM state as a spherically symmetric singlet state. That is why it seems important to investigate the problem of S = l/2 spin excitation damping using the initial spin operators. 0375-9601/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00693-l
A.F. Barabanov. L.A. Maksimov / Physics Letters A 207 (1995) 390-396
391
In the present paper we examine the spin excitation damping for a singlet state of a 2D antiferromagnet on a square lattice. For such a state the average site spin is zero, (S,) = 0, and the spin correlation functions are rotationally invariant, (spsop)
= 6@( - l)“V+“Vc,,
a, p=x,
r=n,g,+n,g,,
y, Z.
( 1.2)
g.,, gY are the basic vectors of a square lattice. This state may be treated as a superposition of terms, each of which being a product of singlet site pairs (a state of resonance-valence-bond type [3]). It is obvious that under the action of any site operator S,X, SJ or S; the singlet pair with the site r will be transformed into one of the excited triplet states of the pair and the singlet state of the total system will be also excited to a triplet state. Such excitations correspond to three Green functions,
These three Green functions must have the same dependence on q and w. So in contrast to the two-sublattice analysis [l] the system is described by three degenerated branches of spin excitations as in the I-D case [4]. We shall examine the spin excitation damping in the framework of the Plakida-Tserkovnicov method [5] for two-time spin Green functions. It will be shown that the lowest boundary of weakly damped spin waves has a scale kmin of the order (T/To>‘/* which is much larger than the inverse correlation length L; ‘. As the temperature decreases, this means that in contrast to (1.1) the spin-wave damping rq leads to an increase of the correlation length according to a power law, L _ T-j/z. Let us mention
( ’ .4) that the consideration
based on fulfilling
the sum rule at finite T leads to the estimate
kmin N T
[61 2. The antiferromagnetic neighbors (NN) is governed H = +J~S,. i,g
(AFM) S = l/2 spin system on a square lattice with interaction by the Hamiltonian
Si+g,
between
lel=l,
nearest
(2.1)
where g are NN vectors (g = fg,, fg,), the interaction constant J and lattice constant g are put to unity. As mentioned above we shall suppose that at T 2: 0 the equilibrium system state is described by the rotationally invariant spin correlation functions (1.2) and that the average site spin is equal to zero, (S,) = 0. Taking into account the equivalence of three branches of spin excitations we shall discuss one branch corresponding to operators Si. Let us use the notations b;l)=N-V2C
,-iq*rs;,
b’2’ = &I’ 4
4
’
b(3) (I
=
&p) Q
(2.2)
’
r
(2.3) where the matrix element K~= ([b;‘, For spin operators
of the K matrix is
(br’)+])=8C,(l-y,), we consider
yq=2-‘(~0s
the two-time
I B) = -i6(t)([B(t),
B+]).
q,+cos
q,),
C,=
-(S;S,$.
(2.4)
retarded matrix Green function
(2.5)
392
A.F. Barabanov, LA. Maksimov / Physics Letters A 207 (1995) 390-396
The equation of motion for its Fourier transformation has the form c.o(BIB)=K+(AIB)=K+(BtA),
dAIB)=([A,
B+])+(AIA).
t 2.6)
Following Ref. [Sl, one can obtain from (2.6) (w-n)(BIB)=K,
(2.7)
where 0 is expressed through the irreducible Green function ( A I A)i”, .n=(([A,
B+])+(AIA)‘“)K-‘,
(2.8)
(AIA)‘“=(AIA)-(AIB)((BIB))-‘(BIA).
(2.9)
The spin wave spectrum is determined by the poles of the Green function (B 1B) or by the matrix 0. As for the matrix L?, it is expressed through a cumbersome Green function ( A I A). We do not need the equation of motion for ( A I A) if we restrict ourselves by the usual mean-field approximation [7,8] which leads to
6’“’ = fi$?b;’ , 4
(2.10)
L?$)= (1 - y,)[2-‘A2+32C,(1 2-‘A2 = 2 + 8FF=
c
+ y,)],
(2.11)
24C,,
(2.12) (2.13)
(S#).
8’ (g’+g) In this approximation ( A 1A)irr is equal to zero and ( B I B) takes the form (2.14)
(BIB)“‘=
Expressions (2.1 l&(2.14) give the spectrum cq of undamped magnons, sq=@@.
(2.15)
Near the point q = (0.0) the spectrum cq is linear for small it has a gap A,
q
and at the antiferromagnetic vector qA = (r, ?r>
(2.16)
cE = (32C,)“2.
Cq=/_*
Using the Green function (2.14) we find the spin correlation functions (S,ZSG). The sum rule ((Sf)2) determines the value of the gap A - exp( - To/T). For q 5* L;’ we may neglect the gap and take
= l/4
(2.17)
Eq = EE( 1 - y42)“2.
The numerator K of (2.14) is proportional to 1 - -y, and tends to zero at q = (0, 0). As a result at low temperature the spin correlation functions are determined by the magnons with q close to qA = (T, T> where the cq is small, Ek = c( k - f”#), c = 4c;/2 t
q=q,,,+k,
v = &s/L+k= c(k/k)(l L-’
-eke
- “#),
(Yt= l/3,
1.
The wave vector k is counted off qA. Hereafter the energy is taken in units c = 4Ci12.
(2.18)
A.F. Barabanov,
393
L.A. Maksimov / Physics Letters A 207 (1995) 390-396
3. The magnon damping comes from magnon-magnon scattering which is described by the Green function ( A I A)‘“, or more precisely, by its component ( 6bt3’ I 6bf’) where 6bq(3) = by’ - if). If we neglect the renormalization of all other functions except (bf’ I bf)), then the expression for (B 1B) has the same form as (2.14) where we must put L&t instead of L@, a,,
= fl;;) +
So the damping r, =
K-
4
of a magnon
(3.1)
with energy .sk is given by
Im (6bf’l
-E;‘K~
The operators
Sbc3’ Q 1S bc3’) 4 *
’ (
6br)),=.,.
Sbq(‘) have the following
ab’“’ = b”’ _ @,b;t’ 4 4
= N-t/2
(3.2)
expression c r.a,g.g’
e -q
-(S,“S;+L)irr,s;+g,
(g'+g.a+z)
+(s;+gs;+Jrr,
where for the irreducible ( S;S;)irr
= s;s,s
After performing the form
-
part of the operators
(sps;+g_J3;+g
+(sps~+g)'~s:+,-,~],
we take notations
- (S,aS,4).
in (3.3) the transformation
Qw,.4z.43 = 16N-‘%.9,+42+rll[( -4-Y-Yq2+q3
from local spin operators to spin-wave
operators the operators take
- %,%, + x7,%* - %2+43%2 + %2+43%) (3.5)
+ -Yq,+q2- %, + y4*)].
As mentioned above we are interested in wave vectors q close to the antiferromagnetic close to zero (q = qA + k). For small wave vectors k we may write r,=
-Yk=
Q k.k,.k2.k,
-I+&, =
0 < & -=K
Yq+q = 3,(2t,";:.
16N-"k,k,+k2+k
-
Im
-(EkKk)-’
c k,,k2,k3.a
a,
p=x,
y.
,=
k,,k;.k;.P
Q
vector q,, = (TT, 7) or k
1.
'tk,) +4-i(-tk2+kC)
Let us note that the last expression iS prOpOrtiOIId to tk - k2. The substitution of (3.4) in (3.2) leads to the general expression r, =
(3.3)
(3.6) +
tk I +k 2 +
for damping
tk,
-
(3.7)
tk2)].
of magnons,
k.k,.k,.k,~;.k;,k;.k;((~~,~~~)irrS;,~(S~S~~)irr~~~)~~~
L
3
(3.8)
Expression (3.8) recalls “Fermi’s golden rule” where the functions Q and Q* represent the scattering amplitudes, and the imaginary part of the Green function plays the role of density of states. That is why in the we shall treat magnons as an ideal gas in the calculations of the irreducible Green “Born approximation” function in (3.8). In this approximation we can use the following decoupling ignoring the damping,
(3.9)
394
A. F. Barabanov, L.A. Maksimov / Physics Letters A 207 (1995) 390-396
Using the spectral representation for Green functions we obtain the following result from (3.91, Im((~~,~~~)i’T~~~I(~~,~~*)irr~~;)~_=~
X
exp
X
[exp(PC j 6J,j -
SGj= SG(kj,
uj+iO)
I]SCW2~G3)
= -2 Im
ok,)]
= 2%-Kkj sgn( Wj) S( 0; - Cl,),
no= [exp( pw) - l]-‘,
j= 1,2,3,
(3.11)
/3=T-‘.
It is known that the requirements for conservation of energy and momentum forbid the spontaneous decay of one magnon into three magnons [9]. This means that the scattering of two magnons plays the main role. As wlr > 0, for the processes in (3.10) we must take one of the Wj negative and two positive. For example, w, = -&Q ‘+ = .!?k,, ‘+ = ‘?k,. Then -ee-pcb)n,*I(I
(1 -e @B%E**%*Z%I1 = -(l
+n,J(l
+n,J.
(3.12)
As a result (3.8) takes the form
rk=3(&kKk)-1(1-e-B”‘)C I&,p,r,s12(~K,/~r)(~Kr/&r)(~K~/&~)np(l +n&l + ‘5) PJ
XS(Ek+Ep-E,-Es),
k+p=r+s.
(3.13)
4. Let us restrict ourselves when estimating the integral (3.13) by the order of magnitude. Then we can take that all kj = 1, 1Q I 2 = cM = max(Ekj) and write down l-k/
Ek
=
-*(I -e-PQ)/p
Ek
XS(Ek+&p-E,-&E,).
dp d’pp / s ds dqs ~~(.s~e~~,)-‘n,(l
+n,)(l
+n,)
(4.1)
A.F. Barabanov, LA. Maksimov / Physics Lxtters A 207 (I 995) 390-396
395
It is obvious that scattering by thermal magnons plays the main role. That is why in (4.1) we can take cP, E, < T and n&l + n,> = T2/ep~,. This approximation is confirmed by straightforward calculations. As it was shown in Ref. [l], the region of integration, where the momentum of one scattered magnon is large and that of the other small, gives the main contribution to the integral of such type. Let us at first consider the damping of soft magnons (A e ck a T), such as E,, Ed -=x .F~, 8,. Then &r=&p+k-s=&p+vp(kE*.-sv), Let us calculate QS=/
first the angle interval of soft magnon
(4.2)
‘p, .
scattering,
6(~~+~p-~-[~p+v~(k/.~-~v)])=[(~v~)~-(~-b)~]-”~,
dv (1 - v~)-“~
b=k(l
v = cos
Er.= 03s UP9
-v,p).
Then we integrate
(4.3)
over the soft magnon
energy,
@, = / ds 6qS( 1 + n,) = T/b’.
(4.4)
The integral over ‘pP by order of magnitude ds? = 1 d’Pp CD,= _/- drp, rT/b
is equal to
= Tk-‘(1
-v;)-“*
= Tk-‘p-‘o-“2.
(4.5)
Note that the damping increases with decreasing nonlinearity (Y of the magnon spectrum. As the last step, we integrate over the large energy of the p-magnon, rk = T-’ /p
dp E&,( 1 + n,)@,
We see that the damping T312 << Ek << T
of the soft magnon
is small in a large range of frequencies, (4.7)
&, = Ek+p-s = Ek + uk( p,-‘The integration over the angle Integration over s leads to
of a rigid magnon
~~
<<
1). Let us discuss
the frequency
range
(4.8)
‘p, gives
a form analogous
-II&-’
to (4.3) with the substitution
k +p,
p + k.
+ rrk-‘a-“2.
over q,,,
d’Pp CD,=k- ‘a-‘/*[TO(T-p)p-’
Q2=/
(T-K
sv).
rTB(T-p)p-‘(1
ds (1 +n,)&=
Then we integrate
(4.6)
.
Now we shall estimate the damping ES, EP << Sk, E,, for which
CD’=/
= T3cr-“*~;‘.
+ 11.
(4.9)
The last integral has the form rk =
E; /
d p n,@, = a-
‘/*ok
T
dp Tp-‘(Tp-’
The first integral diverges at the lower boundary. damping, e.g. by p = T3/*. As a result we find r, = CX-‘/~E, [ T’12 + T ln( T-l/*) Thus, magnons T-== 1.
+ 1) + La dp (e
P/T._
,)-I).
(4.10)
(1
with energy
It must be cut there at the energy of magnons
+ T] .
T < Ek (< 1 (unity
with weak (4.11)
is the scale of exchange
energy
J) have small damping
at
396
A.F. Barabanoo, L.A. Mak.simov/
Physics Letters A 207 (I995) 390-396
5. We have presented a detailed analysis of magnon damping in order to show the difference between our calculations and the analogous calculations in Ref. [l]. The main physical reason for the discrepancy between our results and the results of Ref. [ 11 is the following: our approach is based on the rotationally invariant state of the 2D antiferromagnet at T = 0; in contrast, the approach of Ref. [1] originates from the scenario according to which the state of the system may be considered as the two-sublattice state with spontaneously broken rotational symmetry at the scale of the correlation length. That is why our general expression for the vertex (3.8) differs from the corresponding formula (3.8) in Ref. [l]. We found that the damping of magnons with energy tzP
<
Ed,,,,
=
T3j2
(5.1)
is large and, consequently, the spectrum with the gap (2.12) is physically meaningless. On the other hand, it is known that at low T the values of the spin correlation functions are essentially determined by the spectrum region (5.1). Our calculations of r, are based on the pole approximation for Green functions and cannot describe this region. We can only assert that the correlation length L - k$ is given by formula (1.4).
This work was supported by the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union (Grant No. INTAS-93-2851, by the Russian Scientific Foundation for Fundamental Researches (Grant No. 95-02-04239-a) and by the International Scientific Fund (Grant No. MAA300).
References [I] S. TyE and B.I. Halperin, Phys. Rev. B 42 (1990) 2096. [2] F.J. Dyson, Phys. Rev. 102 (1956) 1217, 1230; S.V. Maleev, Zh. Eksp. Teor. Fiz. 33 (1957) 1010 [Sov. Phys. JETP 6 (1958) 7761. [3] S. Liang, B. Doucot and P.W. Anderson, Phys. Rev. Lett. 61 (1988) 365. [4] M. Gaudin, La fonction d’onde de Bethe (Masson, Paris, 1983). [5] N.N. Plakida, Phys. Lett. A 43 (1973) 481; YuA. Tserkovnicov, Teor. Mat. Fiz. 7 (1971) 250; 49 (1981) 219. [6] H. Capellman, J. Liitfering and 0. Scharpf, Z. Phys. B 80 (1992) 181. [7] H. Shimahara and S. Takada, J. Phys. Sot. Japan 60 (1991) 2394. [8] A. Barabanov and 0. Starykh, J. Phys. Sot. Japan 61 (1992) 704. [9] A.B. Harris. D. Kumar, B.1. Halperin and C. Hohenberg, Phys. Rev. B 3 (1971) 961.
13 November 1995
__ B
PHYSICS LETTERS
ELSEVIER
A
Physics letters A 207 (1995) 390-396
Damping of magnons in a two-dimensional S = l/2 Heisenberg antiferromagnet A.F. Barabanov a, L.A. Maksimov b a Institutefor High Pressure Physics. 142092 Troitsk, Moscow Region, Russian Federation h Russion Research Center, Kurchatov Institute, Kurchatov Square 46, 1231832 Moscow, Russian Federation Received 15 July 1995; accepted for publication 6 September 1995 Communicated by V.M. Agrauovich
Abstract The spin excitation damping is examined for a singlet state of a 2D antiferromagnet on a square lattice using the rotationally invariant approach for two-time spin Green functions. It is shown that the lowest boundary of weakly damped
spin waves has a scale k of the order (T/T,) 3/2. This leads to an inverse correlation 15-l which is much larger than L; ’ - exp(T,/T) g’tven by the Dyson-Maleev formalism.
1. It is well known that many unusual properties of high-temperature superconductors are determined by the Cu-0, planes where Cu-ions form an antiferromagnetic S = l/2 square spin lattice. At any non-zero temperature there is no long-range (LR) order for such a two-dimensional (2D) system; the average site spin is equal to zero, (S,) = 0, and the spin correlation function (S&Y;> has a correlation length
At T<< To - J the correlation length is very large. Then the existence of weakly damped spin waves with wave vector k larger than the inverse correlation length (k > L;‘) may be supposed. As it was shown by TyE and Halperin [I], for asymptotically low temperatures and long wavelengths the spin-wave damping r, is small at T-K T,,, and r, has an essential dependence on T and k. This result was obtained in the framework of the Dyson-Maleev formalism [2] by replacing the spin Hamiltonian by a boson one. This formalism explicitly supposes the presence of two sublattices for a spin system. The main approximation involved in Ref. [l] was to boson states which correspond to a population by more ignore the requirement to project out the “unphysical” than 2S bosons in a lattice site. The application of the mentioned approximation is not obvious if we want to treat the problem for the ultraquantum case S = l/2. Moreover, it is impossible to use the Dyson-Maleev formalism if we treat the ARM state as a spherically symmetric singlet state. That is why it seems important to investigate the problem of S = l/2 spin excitation damping using the initial spin operators. 0375-9601/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00693-l
A.F. Barabanov. L.A. Maksimov / Physics Letters A 207 (1995) 390-396
391
In the present paper we examine the spin excitation damping for a singlet state of a 2D antiferromagnet on a square lattice. For such a state the average site spin is zero, (S,) = 0, and the spin correlation functions are rotationally invariant, (spsop)
= 6@( - l)“V+“Vc,,
a, p=x,
r=n,g,+n,g,,
y, Z.
( 1.2)
g.,, gY are the basic vectors of a square lattice. This state may be treated as a superposition of terms, each of which being a product of singlet site pairs (a state of resonance-valence-bond type [3]). It is obvious that under the action of any site operator S,X, SJ or S; the singlet pair with the site r will be transformed into one of the excited triplet states of the pair and the singlet state of the total system will be also excited to a triplet state. Such excitations correspond to three Green functions,
These three Green functions must have the same dependence on q and w. So in contrast to the two-sublattice analysis [l] the system is described by three degenerated branches of spin excitations as in the I-D case [4]. We shall examine the spin excitation damping in the framework of the Plakida-Tserkovnicov method [5] for two-time spin Green functions. It will be shown that the lowest boundary of weakly damped spin waves has a scale kmin of the order (T/To>‘/* which is much larger than the inverse correlation length L; ‘. As the temperature decreases, this means that in contrast to (1.1) the spin-wave damping rq leads to an increase of the correlation length according to a power law, L _ T-j/z. Let us mention
( ’ .4) that the consideration
based on fulfilling
the sum rule at finite T leads to the estimate
kmin N T
[61 2. The antiferromagnetic neighbors (NN) is governed H = +J~S,. i,g
(AFM) S = l/2 spin system on a square lattice with interaction by the Hamiltonian
Si+g,
between
lel=l,
nearest
(2.1)
where g are NN vectors (g = fg,, fg,), the interaction constant J and lattice constant g are put to unity. As mentioned above we shall suppose that at T 2: 0 the equilibrium system state is described by the rotationally invariant spin correlation functions (1.2) and that the average site spin is equal to zero, (S,) = 0. Taking into account the equivalence of three branches of spin excitations we shall discuss one branch corresponding to operators Si. Let us use the notations b;l)=N-V2C
,-iq*rs;,
b’2’ = &I’ 4
4
’
b(3) (I
=
&p) Q
(2.2)
’
r
(2.3) where the matrix element K~= ([b;‘, For spin operators
of the K matrix is
(br’)+])=8C,(l-y,), we consider
yq=2-‘(~0s
the two-time
I B) = -i6(t)([B(t),
B+]).
q,+cos
q,),
C,=
-(S;S,$.
(2.4)
retarded matrix Green function
(2.5)
392
A.F. Barabanov, LA. Maksimov / Physics Letters A 207 (1995) 390-396
The equation of motion for its Fourier transformation has the form c.o(BIB)=K+(AIB)=K+(BtA),
dAIB)=([A,
B+])+(AIA).
t 2.6)
Following Ref. [Sl, one can obtain from (2.6) (w-n)(BIB)=K,
(2.7)
where 0 is expressed through the irreducible Green function ( A I A)i”, .n=(([A,
B+])+(AIA)‘“)K-‘,
(2.8)
(AIA)‘“=(AIA)-(AIB)((BIB))-‘(BIA).
(2.9)
The spin wave spectrum is determined by the poles of the Green function (B 1B) or by the matrix 0. As for the matrix L?, it is expressed through a cumbersome Green function ( A I A). We do not need the equation of motion for ( A I A) if we restrict ourselves by the usual mean-field approximation [7,8] which leads to
6’“’ = fi$?b;’ , 4
(2.10)
L?$)= (1 - y,)[2-‘A2+32C,(1 2-‘A2 = 2 + 8FF=
c
+ y,)],
(2.11)
24C,,
(2.12) (2.13)
(S#).
8’ (g’+g) In this approximation ( A 1A)irr is equal to zero and ( B I B) takes the form (2.14)
(BIB)“‘=
Expressions (2.1 l&(2.14) give the spectrum cq of undamped magnons, sq=@@.
(2.15)
Near the point q = (0.0) the spectrum cq is linear for small it has a gap A,
q
and at the antiferromagnetic vector qA = (r, ?r>
(2.16)
cE = (32C,)“2.
Cq=/_*
Using the Green function (2.14) we find the spin correlation functions (S,ZSG). The sum rule ((Sf)2) determines the value of the gap A - exp( - To/T). For q 5* L;’ we may neglect the gap and take
= l/4
(2.17)
Eq = EE( 1 - y42)“2.
The numerator K of (2.14) is proportional to 1 - -y, and tends to zero at q = (0, 0). As a result at low temperature the spin correlation functions are determined by the magnons with q close to qA = (T, T> where the cq is small, Ek = c( k - f”#), c = 4c;/2 t
q=q,,,+k,
v = &s/L+k= c(k/k)(l L-’
-eke
- “#),
(Yt= l/3,
1.
The wave vector k is counted off qA. Hereafter the energy is taken in units c = 4Ci12.
(2.18)
A.F. Barabanov,
393
L.A. Maksimov / Physics Letters A 207 (1995) 390-396
3. The magnon damping comes from magnon-magnon scattering which is described by the Green function ( A I A)‘“, or more precisely, by its component ( 6bt3’ I 6bf’) where 6bq(3) = by’ - if). If we neglect the renormalization of all other functions except (bf’ I bf)), then the expression for (B 1B) has the same form as (2.14) where we must put L&t instead of L@, a,,
= fl;;) +
So the damping r, =
K-
4
of a magnon
(3.1)
with energy .sk is given by
Im (6bf’l
-E;‘K~
The operators
Sbc3’ Q 1S bc3’) 4 *
’ (
6br)),=.,.
Sbq(‘) have the following
ab’“’ = b”’ _ @,b;t’ 4 4
= N-t/2
(3.2)
expression c r.a,g.g’
e -q
-(S,“S;+L)irr,s;+g,
(g'+g.a+z)
+(s;+gs;+Jrr,
where for the irreducible ( S;S;)irr
= s;s,s
After performing the form
-
part of the operators
(sps;+g_J3;+g
+(sps~+g)'~s:+,-,~],
we take notations
- (S,aS,4).
in (3.3) the transformation
Qw,.4z.43 = 16N-‘%.9,+42+rll[( -4-Y-Yq2+q3
from local spin operators to spin-wave
operators the operators take
- %,%, + x7,%* - %2+43%2 + %2+43%) (3.5)
+ -Yq,+q2- %, + y4*)].
As mentioned above we are interested in wave vectors q close to the antiferromagnetic close to zero (q = qA + k). For small wave vectors k we may write r,=
-Yk=
Q k.k,.k2.k,
-I+&, =
0 < & -=K
Yq+q = 3,(2t,";:.
16N-"k,k,+k2+k
-
Im
-(EkKk)-’
c k,,k2,k3.a
a,
p=x,
y.
,=
k,,k;.k;.P
Q
vector q,, = (TT, 7) or k
1.
'tk,) +4-i(-tk2+kC)
Let us note that the last expression iS prOpOrtiOIId to tk - k2. The substitution of (3.4) in (3.2) leads to the general expression r, =
(3.3)
(3.6) +
tk I +k 2 +
for damping
tk,
-
(3.7)
tk2)].
of magnons,
k.k,.k,.k,~;.k;,k;.k;((~~,~~~)irrS;,~(S~S~~)irr~~~)~~~
L
3
(3.8)
Expression (3.8) recalls “Fermi’s golden rule” where the functions Q and Q* represent the scattering amplitudes, and the imaginary part of the Green function plays the role of density of states. That is why in the we shall treat magnons as an ideal gas in the calculations of the irreducible Green “Born approximation” function in (3.8). In this approximation we can use the following decoupling ignoring the damping,
(3.9)
394
A. F. Barabanov, L.A. Maksimov / Physics Letters A 207 (1995) 390-396
Using the spectral representation for Green functions we obtain the following result from (3.91, Im((~~,~~~)i’T~~~I(~~,~~*)irr~~;)~_=~
X
exp
X
[exp(PC j 6J,j -
SGj= SG(kj,
uj+iO)
I]SCW2~G3)
= -2 Im
ok,)]
= 2%-Kkj sgn( Wj) S( 0; - Cl,),
no= [exp( pw) - l]-‘,
j= 1,2,3,
(3.11)
/3=T-‘.
It is known that the requirements for conservation of energy and momentum forbid the spontaneous decay of one magnon into three magnons [9]. This means that the scattering of two magnons plays the main role. As wlr > 0, for the processes in (3.10) we must take one of the Wj negative and two positive. For example, w, = -&Q ‘+ = .!?k,, ‘+ = ‘?k,. Then -ee-pcb)n,*I(I
(1 -e @B%E**%*Z%I1 = -(l
+n,J(l
+n,J.
(3.12)
As a result (3.8) takes the form
rk=3(&kKk)-1(1-e-B”‘)C I&,p,r,s12(~K,/~r)(~Kr/&r)(~K~/&~)np(l +n&l + ‘5) PJ
XS(Ek+Ep-E,-Es),
k+p=r+s.
(3.13)
4. Let us restrict ourselves when estimating the integral (3.13) by the order of magnitude. Then we can take that all kj = 1, 1Q I 2 = cM = max(Ekj) and write down l-k/
Ek
=
-*(I -e-PQ)/p
Ek
XS(Ek+&p-E,-&E,).
dp d’pp / s ds dqs ~~(.s~e~~,)-‘n,(l
+n,)(l
+n,)
(4.1)
A.F. Barabanov, LA. Maksimov / Physics Lxtters A 207 (I 995) 390-396
395
It is obvious that scattering by thermal magnons plays the main role. That is why in (4.1) we can take cP, E, < T and n&l + n,> = T2/ep~,. This approximation is confirmed by straightforward calculations. As it was shown in Ref. [l], the region of integration, where the momentum of one scattered magnon is large and that of the other small, gives the main contribution to the integral of such type. Let us at first consider the damping of soft magnons (A e ck a T), such as E,, Ed -=x .F~, 8,. Then &r=&p+k-s=&p+vp(kE*.-sv), Let us calculate QS=/
first the angle interval of soft magnon
(4.2)
‘p, .
scattering,
6(~~+~p-~-[~p+v~(k/.~-~v)])=[(~v~)~-(~-b)~]-”~,
dv (1 - v~)-“~
b=k(l
v = cos
Er.= 03s UP9
-v,p).
Then we integrate
(4.3)
over the soft magnon
energy,
@, = / ds 6qS( 1 + n,) = T/b’.
(4.4)
The integral over ‘pP by order of magnitude ds? = 1 d’Pp CD,= _/- drp, rT/b
is equal to
= Tk-‘(1
-v;)-“*
= Tk-‘p-‘o-“2.
(4.5)
Note that the damping increases with decreasing nonlinearity (Y of the magnon spectrum. As the last step, we integrate over the large energy of the p-magnon, rk = T-’ /p
dp E&,( 1 + n,)@,
We see that the damping T312 << Ek << T
of the soft magnon
is small in a large range of frequencies, (4.7)
&, = Ek+p-s = Ek + uk( p,-‘The integration over the angle Integration over s leads to
of a rigid magnon
~~
<<
1). Let us discuss
the frequency
range
(4.8)
‘p, gives
a form analogous
-II&-’
to (4.3) with the substitution
k +p,
p + k.
+ rrk-‘a-“2.
over q,,,
d’Pp CD,=k- ‘a-‘/*[TO(T-p)p-’
Q2=/
(T-K
sv).
rTB(T-p)p-‘(1
ds (1 +n,)&=
Then we integrate
(4.6)
.
Now we shall estimate the damping ES, EP << Sk, E,, for which
CD’=/
= T3cr-“*~;‘.
+ 11.
(4.9)
The last integral has the form rk =
E; /
d p n,@, = a-
‘/*ok
T
dp Tp-‘(Tp-’
The first integral diverges at the lower boundary. damping, e.g. by p = T3/*. As a result we find r, = CX-‘/~E, [ T’12 + T ln( T-l/*) Thus, magnons T-== 1.
+ 1) + La dp (e
P/T._
,)-I).
(4.10)
(1
with energy
It must be cut there at the energy of magnons
+ T] .
T < Ek (< 1 (unity
with weak (4.11)
is the scale of exchange
energy
J) have small damping
at
396
A.F. Barabanoo, L.A. Mak.simov/
Physics Letters A 207 (I995) 390-396
5. We have presented a detailed analysis of magnon damping in order to show the difference between our calculations and the analogous calculations in Ref. [l]. The main physical reason for the discrepancy between our results and the results of Ref. [ 11 is the following: our approach is based on the rotationally invariant state of the 2D antiferromagnet at T = 0; in contrast, the approach of Ref. [1] originates from the scenario according to which the state of the system may be considered as the two-sublattice state with spontaneously broken rotational symmetry at the scale of the correlation length. That is why our general expression for the vertex (3.8) differs from the corresponding formula (3.8) in Ref. [l]. We found that the damping of magnons with energy tzP
<
Ed,,,,
=
T3j2
(5.1)
is large and, consequently, the spectrum with the gap (2.12) is physically meaningless. On the other hand, it is known that at low T the values of the spin correlation functions are essentially determined by the spectrum region (5.1). Our calculations of r, are based on the pole approximation for Green functions and cannot describe this region. We can only assert that the correlation length L - k$ is given by formula (1.4).
This work was supported by the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union (Grant No. INTAS-93-2851, by the Russian Scientific Foundation for Fundamental Researches (Grant No. 95-02-04239-a) and by the International Scientific Fund (Grant No. MAA300).
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