a/._
9 February
__ BB
1998
PHYSICS
ELSEVIER
Physics Letters A 238 (1998)
LETTERS
A
258264
Unitary transformations in weakly nonideal Bose gases Q. Shi, V.L. Safonov ‘, M. Mino, H. Yamazaki 2 Department of‘ Physics. Faculty Received
of Science, Okayama University, Okayama 700. Japan
1 April 1997; revised manuscript received I October 1997; accepted for publication 21 November Communicated
1997
by A.R. Bishop
Abstract We developed a method for constructing nonlinear unitary transformations in manybody systems. It is demonstrated how to eliminate the threeboson interaction terms which describe forbidden processes in one and twocomponent Bose gases.
The corresponding effective fourboson interaction amplitudes are calculated. 0 PACS: 05.30.5; 75.30.D Keywords: Unitary transformation; Boson; Interaction;
1998 Published by Elsevier Science
B.V.
Wave
1. Introduction Unitary transformations play an important role in solid state physics [ 11. They make it possible to simplify the problem of interacting quasiparticles by eliminating “inconvenient” interaction terms from a Hamiltonian and constructing the corresponding effective interaction terms. Typical inconvenient terms in the Hamiltonians of the theory of spin waves and magnetoelastic waves in magnetoordered crystals describe threeboson processes which are forbidden by the laws of conservation of energy and momentum. Consider, for example, the Hamiltonian
where A(k) is the Kronecker delta function (A(k) = 0 if k law of conservation of momentum. Such threeboson terms with exchange and dipoledipole interactions. For the positive interaction terms in (1) play no role in the magnon dynamics &k, f&k>
+&kj
f
0
# 0 and A(0) = I) which (7fj3’) appear in the theory energy spectrum &k > 0 it in the first approximation as
corresponds to the of a ferromagnet is obvious that the far as
for any kl f k2 t k3 = 0.
’ On leave from Russian Research Center “Kurchatov Institute”. Institute. 212I Hisakata, Tenpaku, Nagoya 468, Japan. ’ Corresponding author. Email:
[email protected] 03759601/98/$19.00 @ 1998 Published PII SO3759601(97)009493
123182 Moscow, Russia. Present address:
by Elsevier Science B.V. All rights reserved.
ISML, Toyota Technological
Q. Shi et al. /Phyic.c
Letters
A 238 (1998)
25X264
This means that the initial Hamiltonian tij3’ has an inconvenient form wave dynamics. Applying the corresponding unitary transformation to ( 1 ), threeboson terms and construct the effective fourboson interactions. The develop a regular method of constructing the unitary transformations which the Hamiltonian of weakly interacting Bose gases. 1.1. Tlw prirzcipal idea
qf the
25’)
for the theoretical study of spinwe can eliminate the “ineffective” purpose of the present paper is to eliminate “forbidden” terms from
method
A general form of unitary transformation
can be written as
A( 0) = ,“R’He“7“,
(2)
where R is an antiHermitian of the equation
operator
(‘72’ = R)
and 0 a formal parameter.
This expression
is the solution
$7&H) =[R,fi(fu] with the initial condition R(O) = 31. We can write the most general form of R( 0) as an expansion in terms of operator combinations with unknown Hdependent coefficients, and the most general form of R as antiHermitian operator combinations. After substituting these expressions into (3), one obtains a set of linear differential equations by comparing coefficients in analogous operator compositions. Solving these equations with the initial conditions, we get the transformed Hamiltonian R(0). In order to eliminate inconvenient terms one needs to put their coefficients (for example, for 0 = 1) equal to zero. This condition defines the choice of 72. The above procedure has first been proposed for the spin Hamiltonian diagonalization 13.3] and it was successfully used in physics of magnetic excitations [ 41 and in the theory of superconductivity [ 5 1. One can easily check that the quadratic Hamiltonian of Bose system in the kspace. 7L”’ = x[Akblbk
+ ;(Bkb:bt,
+ Bib&k)
13)
1,
k
is transformed
to a diagonal
I/‘FI’“lJ’
= xEpb;bk,
form by &k =
k
Sigtl(&)
J
A; 
(5)
lBk12.
Here U = exp
c( 4
R,b:btp
(6)
 Rib&,)
and tanh(41Rkl)
2. Onecomponent
= /Bkl
Ak ’
arg( Rk ) = arg( Bk >
(7)
Bose gas
In order to eliminate Hamiltonian as follows,
the forbidden
threeboson
terms from
( 1) we shall write the general
form of the
Q. Shi et aLlPhysics
260 “;i(@
[email protected]‘k+; k
A 238 (1998) 258264
[%(k~,k~,k~;8)b~,b~7b:1+h.c.]A(k,+k2+k3)
c kl ,kz,k3
~(k,,k2,k3,k4;8)b~,b~~bklbl,d(k,
c
+ ;
Letten
+ kz  kx  k4).
(8)
ki,ks,k>.k4
Here and below we omit all terms containing five and more Bose operators. Before proceeding let us make some general remarks. (a) As long as we consider a weakly interacting Bose gas, the amplitudes of interactions have to decrease with increasing the number of Bose operators in the Hamiltonian. For example, for (8) &k >> qi/3 > &,/2 In this case the order of the creation and annihilation operators in the interaction terms is not so important. If we change the order of operators, say b!s,bk,b;zbk,
=bl,b;2bk,bk,
+b~,bk,A(k2
 k3),
then the role of additional terms will be negligibly small relative to the “bare” quadratic terms. (b) It is easy to show that the result of commutation of combinations of Bose operators, [ bt v
bt
. . b , b+ b+ b . b] , w ml 112 m2
b .
4
leads to a sum of compositions containing ni + ml + n2 + m2  2 Bose operators: ni + 122 1 creation and ml +rnz  1 annihilation operators. For example, for n2 = m2 = 1 the result of commutation contains compositions with the same numbers of creation (ni) and annihilation (mz) operators. This means that if we want to remove a term
x
et.,b+,b ’
. b +h.c.
.
ml
nt
from the initial Hamiltonian, R, ,b+.,. b;u
we should use the antiHermitian
h.c. nr1
PiI 2.1. Threeboson
annihilation
Thus, in order to remove the ineffective with the antiHermitian operator RI = Calculating
c IR,(q,,q2,q3)b~,b~,bb, 414?4
threeboson
h.c.lAtq,
term from (8)) we shall use the unitary transformation
+qz +e).
(9)
the commutators
Mq,,q2,q3)b; [RI(q,,q2,q3)bfi from (3)
operator of the form
 h.c., ekb;bk]. / b;_bll q b;?bil h.c.,
9,(k,,k2,k3;B)b~,b~*b~~
obtain the
$%(k,,kz,kG’)
=3(&k,
+&k3)Rl(kl,k2>k3),
+h.c.l,
Q. Shi et al. /Physics
~m(k,.k?,k?.kd;H)=6R,(k,,kl,k,
Letten
A 238
(1998)
258264
261
k?_)9;(k~,kq,k7k4;8) c IO)
6RT(k3,kj,k~kj)9,(k,,k*,k,k?;8). Solving these equations 91(0= I) =O, we get
Rl(q,,qz,q,) =
with the initial
conditions
[email protected](0 = 0) = Pr.
$( 0 = 0) = (), and requiring
~,(q,,q,7q,) 3(Eq,
+E
II)
)
+s
42
that
Ql
and &k,,kz,ki,k4;1)
= P,(k,,kz,k,
 k4)
1
1
X
C12)
+
(
Sk,
+
&k> +
&k,k>
1
=; k,
&k3
( 1 ) contains
If the initial Hamiltonian x(4’ ,
 kz)W;(k3,k4,k3
+
Ekq
+
&kqk4
>
a “bare” fourboson
~(k,,k?.kj,kq)b:,b:lbk,bt,A(k,
interaction +kz
ki
(13)
 kj),
.k:.ki.kJ
then the effective amplitude
of bosonboson
scattering
in the transformed
ei”’ (@ = 1) is t 14)
2.2. Tile con$uence and decay processes Let us now consider ‘FI’“’ = 1
c
the threeboson
interactions
[~~(k,,k2,k3)btlb:lbk;
energy of the form
+h.c.lA(k,
i IS)
+ k2  kj).
kl,kn.kq
It is obvious
that this Hamiltonian
Sk, + ek> 
Ekl
=
0,
However, the Hamiltonian A&k,,
kz, kj) =
Ek,
contains
(15) contains +
Ek>

Ek>
#
0
%(4,~c72~43)
Eq, + %I,  Eq,
for kl + k2  k? = 0. processes
 h.c.lA(q,
We introduce a general form of Hamiltonian analogous to the above one gets
=
process
also terms for which
threeboson
c [R?(q,,q2.q3)b~,b~2bq, 41.4241
&(q,,qz.q3)
the threeboson
k, + kz  kj = 0.
In order to remove these forbidden antiHermitian operator Rz =
terms which describe
we shall use the unitary
+ 42  4).
containing
transformation
with the
c 16)
‘Ffk”’ with @z( kl , kz, k3; 0). and after calculations
( 17)
Q. Shi et al. /Physics
262
The corresponding
additional
Letters A 238 (1998) 258264
part to the effective fourboson
amplitude
( 14) has the form
1 p2(kl>
kZ, kl
+
k2)p;(k3,
k4.
k3 +
k4)
1 +
( &k,
Ek> 
Eki + &ka 
Ek,+kj
I
ly2(kl,k3_k,,k3)~y;(kj,k?kq,k2)
(
+ &k,k,
Ek,
~~(kl,k4kl,k4)~~(k3,k2k3,k2)


1V?_(kz,k4k2,k4)~‘2+(k3,kl k3,kl) This expression

Ekz >

&k? >

&k,

Ek,
1
+
&k, + &klkq
1
+ Ek?
+Ekyk _Ek 42
&k4 + Ekzka
&kq
+ Sk,kd 1
+ 4
&k, + Qk3
.
(18)
Bose gas
NOW we turn our attention
[email protected]
to a twocomponent
+ ek&k>
k +
?
&kq
I
holds &k > lfnf*/Aa.
3. Twocomponent
‘Ft =
Ek
Ekl
+
‘=kl+kd >
+
Bose gas with the Hamiltonian
[Wj(kl, k2, kdbl,bL2cii
c
+ h.c.lA(k,
+ k2 + k3)
kl ,kz.k?
c
[lyq(kl,kZ.k~)b:,b~*ck,
+
h.c.lA(kl
+ k2  k3)
kl.kz.ky +2
c
+h.c.lA(k,
[~~(k,,k?,ki)b:,bk,c:~
 kz+k?),
(19)
ki.kz,k3
where ek is the energy and ~1, Ck are the corresponding Bose operators which commute with bl, b,. We are interested in eliminating the forbidden threeboson processes from (19) and in obtaining the additional parts to the fourboson amplitudes in the Hamiltonian (13) and
c
7l?’=
?Yk,, k2, k3, kdb$$,ck,A(k,
+ kz  kg  k4).
(20)
kl.k?.kx.k4
Such a problem is very important, for example, for constructing the effective Hamiltonian in the theory of parametric resonance of magnons [ 61. It is convenient to remove the ineffective threeboson terms from (19) by means of several unitary transformations. The antiHermitian operators can be taken in the forms R3=
c
[R3(q1,q2,q3)b~,b~*c~~ h.c.lA(q,
+e+e)t
QI4243 R4
=
c
[R4(q,,q2,q3)b~,b~,cq7

h.c.lA(q, + 42  4x)+
qI424 Rs = c QI4243
[Rg(q,,q2,q3)b~,b42c~1 h.c.lA(q,
 q2 + 43).
(21)
Q. Shi et al. /Physics
After calculations
The corresponding
21y,(q,,q,,q3)
%,  %, + eP7
additional
kz,
 ps(kl,
parts to @( ki , k2, k3, k4) in ( 13 ) can be written as
 kdP;(k3,
k2, kl
+ pJ(ki.
kl
+ kz)P;(h,
k.l, k3  kl)P;(kq.
p5(kz,k3,k3

pstkz,h,k4 
kl
k2, k2  k4)
k2)P;(k3,kl,kl
 k3)
 kl,k4)ly,(k3,kz
Ek,
+ Ek2 
Ek
Concluding
ek,k:
&ks +
&kA +
&kl + &kd 
1
I
Ek4 
ekik,
?
Ek,

Ek? +
+ Sk4

Ek,
Ek,

.?kl
ekzk,
>
+
ekl_k4
1
+
&kq +
>
1
ek,k>
1 
EkZ + ek:_k4
+
&k? +
kd >
1
&k4 + ak,
1
ekii
1
+
&k3 f
ek,k4
1
+
ek,tk:
1
(23)
ekJk2

k2,
k3,
kdW;(kl
1 Ek,
+
ek,_ki
 k4r k,, kq)
&kl
(
+
Sk,k,
+ ekz
I
Eklfkz

ekA
2 
&k,
Ek3 + ek2
+
ekJ !
+Ekzk>

ek2
1 Ek,
ek:
&k<k4
1
+ +
1
Ek _k
Ekq f
+
1
Ek
I
+
+ Ek,k?
1
 ks.k2)
1
4.
&k> +
Ek
 k4,kzt)
+k?,k’)~~tk3,k3+k4,k4)
+2Ws(kl,kl
2p5Ck3
 kq)
+
Fk ( I(
k’)F;(kq,kl,kl
&k,
1
kc+,k3 + k4)
 kz,k2)P;(kvk3
_ z’J’d(k,,h
_
(
1
+
parts to Y( kl , k2, k3, k4) in (20) we get
For the additional 2%(kl,
1
kq, k3  kq)
~s(kl.k4,k4kl)ly;(k3,k2,k2k.?)

A 238 (1998) 258264
similar to the above we get
&(4,*42343) =
Pj(ki.
Lettm

Fk3tk4 + ek4 )
1
+ Ek,k.,
Ek,
(24) +
Pkl
remarks
WC have demonstrated how to eliminate the threeboson forbidden processes from a Hamiltonian. The procedure was based on the solution of a system of linear differential equations. This made it possible to take into account all terms in the expansion of the unitary operator. Recently Krasitskii [7,8] has developed a method of canonical transformation for the classical weakly nonlinear wave system. In order to remove the forbidden threewave interaction terms from the Hamiltonian he considered the expansion for complex wave amplitudes ak = bk + I’*‘(k)
+ I”‘(k),
(25)
where /(I’) (k) contains terms with different compositions of n new complex amplitudes. The transformation canonical if certain conditions for the coefficients of these compositions in the expansion (25) are satisfied.
is
264
Q. Shi et al. /Physics Lerters A 238 (1998) 258264
In our opinion, there is a simpler way to construct a nonlinear canonical transformation for classical According to Bohm’s general theory of collective coordinates [ 91, we can express the old coordinates system in terms of the new variables qold = e
system. of the
R4neweR=(4+[9r~l+~[[4r~l,~l+...)new.
Here we can use Poisson brackets multiplied by ifi instead of the commutator. Our results ( 12), ( 18) for the additional part to @(kl, kg, k3, k4) coincide with those in Ref. [7] if we use the additional conditions pl = q’;, ‘
[email protected] =
[email protected]; and Ek, + &kz= &kz+ E/q for k, + kz = kj + kq. In conclusion, we have developed a method of constructing the unitary transformations to eliminate forbidden threeboson processes from a Hamiltonian and calculated the corresponding effective fourboson interactions. The results are written in a general form and can be applied for any weakly interacting Bose gas (such as a gas of magnons, phonons and so on).
Acknowledgements This work was supported by a GrantinAid for Scientific Research (B) Science, Sport and Culture of Japan and, in part, by RFFI (970217586).
from the Ministry
of Education,
References I I I M. Wagner,
Unitary Transformations in Solid State Physics, in: Modem Problems m Condensed Matter Sciences, Vol. 15, V.M. Agranovich and A.A. Maradudin, eds. (NorthHolland, Amsterdam, 1986). 12] V.L. Safonov, preprint KIAE3691/ I. Moscow, 1982 (in Russian), 13I V.L. Safonov. Phys. Lett. A 97 ( 1983) 164. 141 V.L. Safonov, R.M. Farzetdinova. J. Magn. Magn. Mater. 98 (1991) L235. 15 I V.L. Safonov, Phys. Stat. Solidi (b) 174 ( 1992) 223. 161 V.L. Safonov, H. Yamazaki, J. Magn. Magn. Mat. 161 (1996) 275. 171 V.P. Krasitskii. Sov. Phys. JETP 71 ( 1990) 921. [S I V.P. Krasitskii, J. Fluid Mech. 272 (1994) 1. 19 1 D. Bohm, General theory of collective coordinates, in: The Many Body Problem, Les Houches  session 1958 (Wiley, New York. 1959) p. 401.